Introduction to Non-Linear Mechanics

INTRODUCTION TO NON-LINEAR MECHANICS TOPOLOGICAL METHODS NON-LINEAR RESONANCE -

ANALYTICAL METHODS RELAXATION OSCILLATIONS

BY

N. MINORSKY, PH.D. '"

J. W. EDWARDS

1947

ANN ARBOR

Copyright 1947 by J . W. Edwa rds Origina lly Published, 1944-1 946, a s Restricted Reports by The Dallid W. Toylor Model Basi n, U. S. Navy Repub lished by Arrangement with th o Office a f Tedmical SeNlces Depor1ment af Commerce

OPERATIONS RESEARCH. INC. LIBRARY 0,/ 90

Lithoprinl ed In U.S.A. EDWARDS B RO THE RS, Ann Arbor, Michigan

I NC.

FOREWORD

This report aims to bring to the attention of technical personnel, both of the Navy and of other agencies engaged in the war effort, certain new developments in applied mathematical methods. These new methods have come to be called "Non-Linear Mechanics. II This new branch of theoretical mechanics does not introduce any new postulates but relies upon the same Newtonian principles commonly used in engineering applications. The only difference between these new methods and the old ones lies in the mathematical methods themselves. The old methods, such as that of small motions, simplify the problem in advance so as to bring it within the scope of linear different1a1 equations. The general theory of these 1s known, so that approximate solutions are obtained by standard methods. It is the purpose of the new methods to obtain more accurate solutions without recourse to artificial simplification. A few definitions and examples w1ll be given so that the subject may appear somewhat better defined. The general form of a linear differential equation of the nth order is d"~

d·-i~

d"-2~

dt" + pt(t) dt,,-i + P2(t) dt,,-2

where ~, the dependent variable, constitutes the unknown function to be determined by integrat10n, and t is the independent variable, for example, time. In a great majority of dynamical problems ~ is a spacial coordinate. The coeffic1ents Pi(t) ..• p,,(t) are known functions of t. Q(t), deSignated sometimes as a forcing function, 1s also a known function of t. According to whether Q(t) = 0 or Q(t) ~ 0 the linear equation is called homogeneous or non-homogeneous. In many engineering applications all the functions Pi(t) ... p,,(t) are constant; the linear differential equation is then said to have constant coefficients. This is the simplest form of a linear differential equation and its solut10n, as is shown 1n elementary texts, is reduced to that of solving an algebraic equation. On the other hand, 1n appl1cat10ns linear d1fferentia1 equations with variable coefficients are often encountered. For 1nstance, the well-known Mathieu equation is % + (a

2

+

b2co8pt)~ = 0

Th1s equation has a great var1ety of appl1cat10ns, such as oscillations of locomotive mechanisms, theory of modulation in radio CirCuits, OSCillations of membranes with ellipt1c boundaries, and diffraction of l1ght. Although

vi

the theory of the Mathieu equation is much more difficult than that of equations with constant coefficients, it is still a linear differential equation. The basic feature of such equations is that they obey the so-called principle 0/ superposition. Thus, if a system is governed by a linear differential equation and is capable of oscillating in certain modes, the motion will take place as if each component oscillation with a certain frequency existed alone, not influenced by the presence of oscillations with other frequencies. This constitutes a considerable simplification wh1ch permits a physicist to be guided by intuition without having to rely continuously on the mathematical methods. In order to be able to proceed in this manner, however, one must be certain that the differential equations are linear. In this connection there arise frequently serious d1fficulties as to the ltm1ts of validity of the differential equation describing a certain physical phenomenon. Clearly the only criterion of validity of a certain mathematical law is its agreement with the observed facts. If this agreement is good, within a certain range of the phenomena under consideration, it is natural to conclude that the differential equation correctly describes the phenomena.and that the hypotheses which were made in obta1ning the equation are correct. If, however, for some other range the same differential equation ceases to give a satisfactory agreement with observation this generally indicates that certain assumptions or idealizations which were sufficiently correct for a certain range cease to be correct when the range of observation is extended. Concepts of constant electric resistance of a conductor, of a constant spring constant, of a rectilinear characteristic of an electron tube, and so on, appear very often as idealizations valid only i~ a certain restricted range. If the problem is investigated within that range the idealized linear differential equations give a reasonably good agreement with observation. If, however, the same phenomenon is studied in a somewhat larger range, in which the assumed idealizations cease to be true, the departures between the theory and the observed facts become more and more eVident, as the range of the investigation is extended. The reason for such discrepancies is clearly that the problem has been over-idealized, and a more correct form of differential equation must be employed. In such cases the correct differential equations are usually non-linear. In some cases, as will appear later, non-linear equations appear at the very outset of the problem. In such cases it is impossible to describe the phenomenon mathematically by means of linear differential equations for any range, however small. A few examples which follow illustrate this general situation.

vii

THE PENDULUM The exact differential equation of the simple pendulum is

6 + ..R..sinO =0 L

[1 ]

This is a homogeneous non-linear d1fferential equation of the form 6+4>(0)0=0

with a variable "spring constant"

resulting from the expansion of sin 0 in a power series. If we assume that all terms beginning with 0 2 /6 are negl1gible we obtain the "zero approximation ll 4>(0)

= L

L

which gives [2]

This is the well-mown elementary equation of a "linearized" pendulum having a fixed period T= 2"YL/g. For the next, or first, approximation we assume 4>(0) = [(1 This gives a non-linear differential equation of the form

f-).

9+

[f (1 -

2

0 6

)]

0 =

0

[3]

The spring constant in this case does not remain constant but dim1nishes with increasing O. In this manner we obtain a more nearly correct representation of the motion. By retaining a greater number of terms in the power expansion. of sin 8 we obtain still greater accuracy but the problem becomes more complicated. In practice even the first approximation [3] gives an error of less than 1 per cent for angles of the order of 30 degrees.

FROUDEIS DIFFERENTIAL EQUATION OF ROLLING The oscillations of a ship in still water, according to Froude, are given by the equation

[4] in which I is the moment of inertia of the ship about a longitudinal axis through the center of gravity, B t , B2 are the so-called Froude1s coefficients of resistance to rolling, D is the displacement, and h(O) is the variable lever arm of the righting couple.

viii

Tnis equation is non-linear due to the presence of the term B2 6 2 and the term h(O) which departs slightly from a linear function of O. For small angles 0, h(O) is approximately of the form h(O) = hoO, where ho is the initial metacentric height. The linearization of this term is therefore permissible in the range of small angles for which ko remains substantially constant. As regards the other non-linear term B2B2, it is generally uncertain whether we can "linearize" [4] by dropping this term altogether. In case this can be done for a particular ship, the problem is reduced to a simple linear equation

+ BlB + DhoO =

19

[5]

0

If, however, the experimental evidence is such that the term B2 62 is of the same order as BlB, the differential equation is ••

•

•2

+ Dh o9

10 +B 1 6 +B 2 6

[6]

=0

and the difficulty of dealing with a non-linear d1fferential Equat10n [6) cannot be avo1ded. THE WILLIAMS BLAST GAGE The d1fferential equation of the Williams gage used for the invest1gat1on of explosive blast 1n air is it

+~P m

0

[(_L_)Y L-x

1] =

1.. pet) m

[7]

where m, A, Po' L, and " are constants and P( t) is a }mown function of t1me. Introducing the variable y = x/L, this equation acquires the form

y+

k

~ [(1 !

yV -

1] y =

[7a]

Q(t)

This is a non-homogeneous non-linear equation of the form

ii + q,(y)y where q,(y)

= ~ [(, _'y)Y - ,];

[7b]

= Q(t)

k is a constant and 0

< y < ,.

OSCILLATIONS OF A SPHERICAL GAS-FILLED CAVITY IN A FLUID The differential equation of the phenomenon is

R + : ~ R2 +

aR

3

-

R3(~-t) =

0

[8]

where R is the radius of oscillating cavity and a, b, " are certain constants. This equation can be written as

..

R

+

3R·

2R R

+

q,(R)R = 0

[9]

Ix

where the non-linear spring constant ~(R) = aR2 _ b/R Sy - 2 and the coefficient of damping b = 3R/2R. Equation [9] is also a homogeneous non-linear differential equation, where the non-linearity is found in both the spring-constant term and in the damping coefficient.

DIRECTIONAL STABILITY OF SHIPS The differential equation of the initial azimuthal motion of a ship from a straight course with the rudder amidships is Jii

+ C(a) -

M(a) = 0

[101

where J is the moment of inertia of the ship about the vertical axis through its center of gravity, C(a) is the resistance to turning, M(a) is the moment of the leeway force, and o is the angle of the initial deviation. The function C(a) is not known at present; it is probably proportional to some power of a; from physical considerations the linearization of this term by an expression of the form C(a) = Coa is not objectionable for small ranges of a. As regards the term M(o), its linearization by a term of the form MoO leads certainly to incorrect results; in fact this would mean that for an increasing 0 the term Moo, where Mo is a certain constant, increases indefinitely. Observation shows that for an increasing 0 the coupleM(a) passes through a maximum for a relatively small value of 0 and becomes zero thereafter, when the line of action of the resultant of the leeway forces recedes toward the center of graVity. This situation can be described more correctly by an approximate expression of the form M(o)

= MoO -

MI a

8

Hence a more correct d1fferential equation for the directional stabil1ty is of the form [11 )

This is a non-linear differential equation in which the non-linearity 1s localized in the spring constant (Mo - Mlol) which decreases with the angle o. In this case the spring constant is negative inasmuch as a ship proceed1ng in1tially on a straight course is in unstable equilibrium.

x

PARASITIC OSCILLATIONS OR "HUNTING" IN CONTROL SYSTEMS Somewhat different non-linear phenomena occur when the damping term of a dynamical system is initially negative and increases with increas1ng amplitudes so as to become ult1mately positive for sufficiently large values of the dependent variable. Phenomena of this kind are amenable to the Van der Pol equat10n Z - Jl (1 - z 2):i:

+z

= 0

[12]

The equat10n of Rayleigh, discovered 1n connection w1th acoustic phenomena, [13 ]

1s known to reduce to Van der Pol's equation by a change of var1ables, so that the two equations have analogous features. These equat10ns have a very extensive field of applicat10n in connection with self-excited oscillations in electron-tube circuits. A few years ago, in connection with a ship stabilization research program, it was discovered that under certain conditions the blade angle on the pump controlling the transfer of ballast between the tanks begins to flutter, interfering seriously w1th the efficiency of operation. A detailed analysis of this effect is given in Section 31 of this report. It is sufficient to mention here that the hon-linear different1al equation of this parasit1c effect is of the form [14 ]

where J 1s the aland as couple b is the c is the

moment of inertia of the ballast, including tanks and ducts, are the coefficients in the expression giving the hydrodynamic exerted by blades on the ballast, coefficient of the natural damping (friction), and coefficient of static stability of the water ballast.

Equation [14] is seen to be Rayleigh's equation mentioned previously. These few examples show the importance of the methods of non-linear mechanics for the solution of naval problems. As to the question where and how to use these methods, answers can be given only for specific cases. Each problem must be formulated explicitly and then examined to determine whether simplifying assumptions can be introduced to linearize the problem without the loss of any essential features. In cases where this is not possible one must face the situation arising from the essential non-linearity of the phenomenon and apply the more laborious methods of non-linear mechanics. The present report introduces the student who is familiar with standard methods of attack on linear problems to these more advanced methods.

CONTENTS Introduotion • . •.

1

PART I - TOPOlDGICAL METHODS OF NON-UNBAR MECHANICS Chapter I - Phase Trajectories at Linear Systems

7

1. Phase Plane and Phase Trajectories; Linear Oscillator .•

7

2. Phase Velocity . . • . . • • . . . . . . • • • • • • . .

9 9 14

,. 4. 5. 6. 7. 8. 9.

Theorem at Cauchy; Singular Points; Trajectories and Motions • Phase Trajectory of an Unstable Motlon; Saddle Polnt • • • • Phase Trajectory at an Osolllatory Damped Motlon; Focal Polnt Phase Trajectory ot an Aperiodl0 Damped Motlon; Nodal Po1nt Method ot Isocllnes • • • • • • • • • • • • • • • Systems wlth Negatlve Damplng; Froude's Pendulum Remarks Conoernlng Linear Systems • • • • • • •

Chapter iI - Phase Trajectorles of Non-Linear Conservatlve Systems • 10. General Properties ot Non-L1near Conservatlve Systems 11. Topology ot Phase Trajectorles 1n the Nelghborhood ot Singular Polnts 12.

1'. 14. 15.

A. Graphioal Method • • • • • • • • • • • • • • • • • • • • B. Analytical Method • • • • • • • • • • • • • • • • • • • • • Topology of Trajeotories ln the Phase Plane. Separatrix Behavlor of a Conservatlve System as a Funotion ot a Parameter. Cr1tioal Values of a Parameter • • • • • • • • • • • • • • : • • • • • • • • • • • Polncar~'s Theorem (14) (15) • • • • • • • • • • • • • • • • • • Motlon ot an Elastioally Constra1ned Current-Carry1ng Conduotor Relat1ve Mot10n of a Rotatlng Pendulum

Chapter III - Questlons ot Stabll1ty • • • 16. Introductory Remarks • • • • • • • • 17. Stab1llty ln the Sense ot Liapounott 18. Canonlcal Forms ot Linear Equat10ns; Charaoterlstio Equat10n; Class1fioation of Slngularltles (Po1ncare); Branch P01nts of Equ1l1br1a 19. Stabllity of Equll1br1um According to Liapounoff • . • • • • • • • • • • • • 20. Liapounorr I s Theorem , • • . • • • . . . . . . . • . • • • • • . • . . . . 21. Equ1libr1um ot a Clrou1t Conta1n1ng a Non-Linear Conductor (Electrlc Aro) Chapter IV - Limit Cyoles of Poincar~ 22. Lim1t Cycles; Det1n1tion; Analyt1cal Examples 2'. Phya1cal Examples

• • • • • • • • • • • • • •

24. Topology ot Trajector1es in the Presenoe of S1ngularities and Limit Cyoles 25. Further Propertles of Limit Cycles; Ind1ces ot Po1ncar~; Theorems of Bendixson A. Indloes ot Polncar~.. • • • • • • • • • • • • • • • B. First Theorem of Bendlxson (The Negat1ve Cr1ter10n) C. Second Theorem of Bendixson (5) • • • • • • • • • • 26. Parallel Operatlon of Series Generators • • • • • 27. Stabll1ty ot Perlodl0 Motion . . . . . Chapter V - Blfurcatlon Theory (Po1ncar6) 28. Introductory Remarks • • • • • • • • 29. Transltion of S1ngularitles. Branch Po1nts at Lim1t Cycles ,0. Selt-Excltation ot Thermionl0 Generators • • • • • • • • • ,1. Selt-Excitat1on ot Mechanical and Eleotromechan1cal Systems A. Selt-Exc1ted Oscillations of a Mechanical Control System Case 1. Weak Ampllticat10n • • • • • • Case 2. Strong Amp11flcatlon • • • • • • • • • • B. Self-Exclted Osc1llations in an Eleotromeohanioal System Chapter VI - Geometrlcal Analysls of Exlstence ot Perlodlc Solutions '2. ". '4. '5.

Introduotory Remarks • • • • • Lidnard's Method • • • • • • • Exlstenoe of Closed Trajeotories in the Lienarc1 Plane First Asymptotic Case I 11 «1. •

16 18 20

21 22

24 24 25 25 26 29

,2 ",4

'7 40 40

41 42 48 51 55 62 62 66 68 75 76 77 78 79 82 87 87 88 95

99 99 101

101 102 105

105 106

• • 108 110

xii

}6. Second Asymptot1c Case: The Parameter ~ 1s large • }7. L1mit Cycles in the Van der Pol and Lienard Planes Chapter VII .- Cy11ndr1cal Phase Space • • }S. }9. 40. 41. 42. 4}.

General Remarks. Lim1t Cycles of the P1rat and Second K1nd D1fterent1al Bquat10n ot an Blectro:echan1cal System Cylindr1cal Phase Trajector1es of a Conservat1ve System • • Cylindrical Phase Trajectories of a Non-Conservat1ve System Closed Trajectoriss of the Second Kind in Non-Conservative Systems OscIllatIons of a Synchronous Motor • • • • • • • • • • • • • • • •

PART II - ANALYTICAL METHODS OP NOIf-LINBAR MBCHA1fICS 44. Introductory Remarks "

..

11.1

11} 116 116 117 l1S 119 121 124

l}5

Chapter VIII - Method of PoIncare 45. Cond1tion of PeriodicIty • • . • • . • • . • 46. Expansions ot PoIncare; Generating SolutIons; Secular Terms 47. Sy'lltems With Tvo Degreell of Preedom • • • 4S. Stabil1ty of a PeriodIc Solution •• • • • • 49. L1m1t Cycle and Frequency of a ThermionIc Generator 50. BifurcatIon Theory For Quall1-Linear SYlltems • • 51. "Soft" and "Bard" Selt-Excitation of Thermionic Generators; Oscillation Hysteresis A. CondItion For a Soft Self-Excitation B. Condition Por a Hard Selt-Excitation

l}S l}S 141 14S 15}

Chapter IX - Method of Van der Pol

167 167 169 172 17S lSl

52. 5}. 54. 55. 56.

Rotating System of Axes; Equations of the Pirst Approximation Topology of the Plane of the Variables of Van der Pol • • • • • • • Example: "50ft" and "Bard" Self-Excitation of ThermionIc CIrcuits Example: Bquation of Lord Rayleigh; Froude's Pendulum More General Forms of Non-L1near Bquations • • • • • • • • • • • •

Chapter X - Theory of the Pirst ApproximatIon ot Kryloft and Bogoliuboff 57. Introductory Remarks • • • • • • • • • • • • • • • • 58. Eftect ot Secular Terms in Solutions by Expansions in Series 59. Bqua tions of the Pirs t Approxima tion 60. Hon-Linear Conservati~e Systems • • • • • • • . 61. Examples ot N.on-Linear ConservatIve Systems A. Pendulum • • • • • • • • • • • • • • • • B. Torsional Oscillat10ns of a Shaft C. Blectrical OSCillations of a Circuit Containing an Iron Core 62. Systems With Hon-Linear Damping ot a Dissipative Type A. Linear Damping: a Ai •• B. Quadratic Damping: r(t) .. b:f2 C. Coulomb Damping: ,(i) .. A (i) D. Mued Case81 f(i) .. cd + pi ••• 6}. Systems With Hon-Linear Variable Damping A. Van dar Pol's Equation • • • • • • • B. Rayleigh's Equation • • • • • • • • • 64. Existenoe of L1m1t Cycles; Syatems With Several Limit Cycles 65. Stability ot L1m1t Cycles; CrItical Values of a Parameter; Systems With Several

rei)

lor

lJ.r:D1 t Cycles

158 160 161 164 165

18}

lS} 18}

186 189 190 190 191 192 192 19} 194 195 195 197 197 199 200

.....•.••...•...•...

66. L1m1t Cycles in the Case ot Polynomial Charaoteristios Chapter XI - Approximations of Higher Orders 67. Introduotory Remark8• • • • •

68. Improved Pirst Approximation 69. Appl1ca tions ot the Theory of the Improved Pirs t Approxlma tion

70. 71. 72. 7}.

A. Variable "Spring Constant" •• B. Variable Damping . . . . . . . . C. Correotion Por Frequenoy ••• Approximations of Higher Orders • Motion of a Conservative Hon~L1near System With a Cubio Term H1gher Approximations Por Ron-Linear, Ron-Conllervative Systems General Porm ot Bquations of H1gher Approximations • • • • • •

209

209 210 214 215 216 217 219 222

22} 229

xiii

Chapter XII - Method of EquIvalent Linearization of Kryloff and Bogoliuboft •

74. 75. 76. 77. 78.

233 233 234 236 238 239 239

Introductory Remarks • • • • • • • • • • Method of Equivalent Linearization Prinoiple of Equivalent Balanoe of Energy Prinoiple of Harmonic Balanoe • • • • Examples of Application of the Method of Equivalent Linearization A. Non-Linear Restoring Force • • • • • • • • • • • • • • • • • • B. Non-Linear Dissipative DampIng • • • • • • • • • • • • • • • • C. Non-Linear Restoring Force and Non-Linear Dissipative Damping D. ElectrIcal Oscillations in a Cirouit Containing a Saturated Core Reaotor E. Non-Linear Conductors F. ThermIonI0 Generators

2110

241 2112

242 243 • • • 246

Referenoes PART III - NON-LINEAR RESONANCE

79. Introductory Remarks Chapter XIII - Systems With Several Degrees of Freedom 80. Method of Complex Amplitudes 81. Blements of the Theory of Linear CIrcuits • • • • 82. Analogies Between Eleotrioal and Meohanical Systems 83. ApplIcatIon of the Method of EquIvalent LinearIzatIon to the Steady State of a QuasI-Linear System • • • • • • • • • • • • • • • • • • • • • • • • 811. ApplIcatIon of the Method of EquIvalent LinearIzatIon to the TransIent State of a QuaSi-Linear System • • • • • • • • • • • • • • • • • 85. Non-Resonant Self-Excitation ot a QuaSi-Linear System 86. Resonant Selt-Excitation of a QuaSi-Linear System •

251 251 252 255

Chapter XIV - Subharmonios and Frequenoy Demultiplication

272

87. CombInation Tones; Subharmonics • • • • • • • • • • 88. Equivalent Linearization for Multiperiodio Systems 89. Internal Subharmonio Resonance

• 272 274 278 283

90. Synchroniza tion • • • • • • • • • • • • • • • • • • 91. Internal Resonance of the Order One • • • • • • • • 92. Method of Equivalent Linearization in QuaSi-Linear Systems With Several Frequenoies

Chapter XV - External Periodic Excitation of QuaSi-Linear

Syste~s

••••••••

93. Equations of the First Approximation for a Periodic Non-Resonant Excitation 94. First-Order Solution of Van der Pol's Equation With Forcing Term • • • • • 95. Improved First Approximation For a Non-Resonant External Excitation of a QuasiLinear SY8 tam • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

258 262 267 269

284

• 285 289 • 289 • 293 294

96. Heteroperiodic and Autoperiodic States of Non-Linear Systems; Asynohronous Excitation and Quenching Chapter XVI - Non-Linear External Resonanoe

97. Equations of the First Approximation For an Externally Excited Resonant System 98. Fractional-Order Resonance •• '" • • • • 99. Parametric Excitation • • • • • • • • • • • 100. Stability of Hon-Linear External Resonance; Jumps •

Chapter XVII - Subharmonic Resonance on the Basis of the Theory of Poincard 101. 102. 103. 104.

105. 106. 107.

108. 109.

Method ot Mandelstam and Papalexi • • • • • • • • • • • • • • • • • • Resonanoe of the Ordern; Differential EquatIons in Dimensionless Form Periodio Solutions of a QuaSi-Linear Bquation With a Foroing Term • • • • Stability of PeriodiC Solutions • • • • • • • • • • • • • • • • • • • • Subharmonic ReBonance of the Order One-Half for a Soft Self-Excitation Nature of Subharmonic Resonance of the Order One-Half for an Underexoited System Subharmonio Resonanoe of the Order One-Half for a Hard Self-Exoitation Subharmonic Resonanoe of the Order One-Third Experimental Results • • • • • • •

Chapter XVIII - Entrainment of Frequency

110. Introduotory Remarks

•• • • • • • 111. Differential Equations of Van der Pol 112. Representation of the Phenomenon in the Phase Plane •

295 301

301 305

308 313

• 323 323 323 • 326 328 331 335 • 336 339

340 341 341 342 344

xiv

113. 114. 115. 116.

Nature and Dlstrlbutlon of Singularitles; Trans lent State of Entralnment • Steady State of Entralnment Acoustlc Entrainment of Frequency Other Forms of Entralnment • •

Chapter XIX - Parametrlc Excitation 117. Heteroparametrio and Autoparametrlc Excltatlon • 118. Differentlal Equatlons With Periodic Coeffiolents 119. Stable and Unstable Regions of the Mathleu Equation 120. Physlcal Nature of Solutions • • • • • • • • • • • • 121. Topology of the Hill-Meissner Bquation • • • • • • • 122. Dependence of Heteroparametrio Exoitation on Frequenoy and Phase of the Parameter Varlation ••• • • • • • • • • • • • • • • • • • • l2}. Heteroparametric Excltation of a Dissipative System •• • • • • 124. Heteroparametrio l4aohine of Mandelstsm and Papalexi •• • • • • 125. Subharmonl0 Resonanoe on the Basis of the Mathleu-Hill Equation 126. Autoparametric Excltat10n

355 356 359 ,61 ,62 365 }69 }12

}1' 314

References • • • • •

PART IV - RBlAXATION OSCILLATIONS 121. Introductory Remarks. Chapter XX - Fundamentals of the Discontinuous Theory of Relaxation Osoillations

381 388

128. Solutions of a Differential Bquation in the Neighborhood of a Point of 129. 130. 131. 132.

Degenera tion • • • • • • • . . . • . . . • . . . . . . .

388

Idealizations in Physical Problems • • • • • • • • • • • Critical Points of D1fferential Bquatlons; BaSic Assumptlon Conditlon!' of ~delstsm • • • • • • • • • • • • • • • • • • Remarks Concerning Systems of Degenerate Dlfferentlal Equations

391 392 395 }96

Chapter XXI - Degenerate Systems With One Degree of Fresdom 13}. Per10dlc Solutions of Degenerate Systems of the F1rst Order 134. Relaxation OsolllationB in a Circuit Contaln1ng a Gaseous Conduotor 135; RC-Multlvibrator • • • • • • • • • • • • • • • • • • • • • • • • • • • 136. System With One Degree of Freedom Describable By Tvo Different1al Bquations of the First Order • •

399

Chapter XXII - Multiply Degenerate Systems •

410

137. Multivibrator of Abraham-Blooh • • . • • 138. Heegner's Cirouit; Analyt1c Trajectories 1}9. Transitlon Between Continuous and D1soontlnuous Solutions of Degenerate Systems Chapter XXIII - Mechanical Relaxation Osc11lations • • • • • • • 140. Introductory Remarks • • • • • • • • .'. • • • • • • • • • 141. Qualitatlve Aspeots of a Mschanical Relaxatlon Oscillatlon • 142. Mechanical Relaxation Osoillations Caused By Non-Linear Frlotion Chapter XXIV - Osoillations Maintained By Periodic Impluses 143. 144. 145. 146. 147.

Introductory Remarks • • • • • • • • • • • • • • • • • Elementary Theory of the Clock • • • • • • • • • • • • • Phase TrajeotorleB in the Presence of Coulomb Friction Blectron-Tube Osoillator,Wlth Quasl-Disoontinuous Grid Control. Phase Trajeotories of an Impulse-Exolted Osc11lator • • • • •

Chapter XXV - Effect of Parasitic Parameters on Stationary States of Dynamioal Systems 148. Parasitic Parameters • • • • • • • • • • • • • • • • • • • • 149. Influence of Parasitio Parameters on the State of Bquilibrium of a Dynamical System. • • • . • . . . . . • • • • • • • . . • . • • • • . . • • • •

150. Bffect of Parasitic Parameters on Stability of an Bleotrio Arc • • • • 151. Bffect of Parasitic Parameters on Stability of Relaxation Osoillations References • Index

399 400 402 405

410 414 415 411 411 418 420 423 423 424 421 4}0 433 436 436

1&,8 440 443 131,246 218,446 449

INTRODUCTION

Praotioally all differential equations of Meohanios and Physios are non-linear; in the applications linea~ approximations are frequently used. The Method of Small Motions is a well-known emmple of the "linearization" of problems whioh are essentially non-linear. With ~e disoovery of numerous phenomena of self-excitation of oirouits oontaining non-linear oonduotors of eleotrioity, such as eleotron tubes, aros, gaseous disoharges, eto., and in many cases of non-linear meohanica1 vibrations of special types the Method of Small Motions becomes inadequate for their ana1ytioal treatment. In faot the very existenoe of these osoillations in a steady state indioates that there is an element of non-linearity preventing the oscillations from building u~ indefinitely, whioh by energy oonsiderations is obviously impossible. In addition to this there is another important differenoe between these phenomena and those governed by linea~ equations with oonstant ooeffioients, e.g., osoillations of a pendulum with small amplitudes, in that the amplitude of the ultimate stable osoillation seems to be entirely independent of the initial oonditions, whereas in oscillations governed by linear differential equations* it depends upon the initial oonditions. Van der Pol (1)** was first to invite attention to these new osoillations and to indioate that their existenoe is inherent in the non-linearity of the differential equations Charaoterizing the process. This non-linearity appears, thus, as the very essenoe of these phenomena and, by linearizing the differential equation in the sense of the method of small motions, one simply eliminates the possibility of investigating suoh problems. Thus, it became neoessary to attaok the non-linear problems direotly instead of evading them by dropping the non-linear terms. Although the early disooveries of Van der Pol inTited the attention of mathematical physioists to these new problems, very little was done theoretioally in the follOwing years in the way of further generalizations of Van der Pol's original theory. On the other hand, the aooumulation of experimental data oontinued at a rapid rate and eaoh additional problem had to be treated mainly on its own merits and by methods whioh were not unified into one oentral dootrine. The situation was similar in some respeots to that whioh existed in the early

*

unless otherwise spec1tled the term -Unear d1tterentlal equations- will be used lD the tollowlDg lD reterrlDg to those wi th constant coettlclents.

** N\lDIbers

1n parentheses 1ndlcate references on page 131 of th1s report.

2

stage of development of the theory of differential equations when, prior to the advent of Cauchy's general theorems of existence, a few isolated methode of direct integration were the ~nly ones available. later, Cauchy's theorems nade it possible to obtain a general theory of linear differential equations. About fifteen years ago certain Russian scientists directed their attention to a further development of methode of Non-Linear Mechanics with a view toward obtaining a more general basis for a nathenatical treatment of numerous experimental facts to which modern . electronic oirouits and apparatus oontributed a large part. A line of approach was soon found in the olassical researches of Henri Poincar~ who nay be considered as a real forerunner of modern NonLinear Mechanios. In his two treatises "Sur les courbes d~f1n1es par une 9quation d1.1'f~rentielletl (2) and "Les nuSthodes nouvelles de la meoanique c~leste" (3) the great analyst opened two naolor avenues of approach. to the solution of problems of Non-Linear Mechanios, i.e., 1. The topologioal methode of qualitative integration, 2. The quantitative methode of approximations by expansions in terms of suitable parameters. These two trends persist in modern developments of NonLinear MechaIiics, frequently· supplementing eaoh other in some respects. The adaptation of these general theories, developed originally for the purposes of Celestial M;,ohanics, to the problems of applied science in general is the work of Mandelstam, Papalexi, Andronow, Kryloff, Bogoliuboff, and a number of other Russian soientists working jointly in this field during the past fifteen years or so. The two najor trende referred to, the topological method and the quantitative method of successive approximatiOns, have their respective merits as well ~s lim1tatio~. The topological methods are based on the study of the representation of solutions of differential equations in phase spacej the latter is napped by means of point-singularities and certain singular lines so as to obtain certain topologioal domains in whioh the form of the integral curves - the phase trajectories - can be investigated by relatively simple gecmietr1cal methods. The main advantage of the topological methods lies in the fact that insofar as they deal with trajectortes, and not with lams Of motton, they make it possible to obtain, so to speak, a bird's -eye view of the total tty of all possible motions which nay arise in a given system under all possible conditions. Just as a topographic nap of a locality gives an idea as to its three-dimensional form - its peaks, valleys, diVides, and other features - a topological

3

picture of a domain of integral curves permits ascertaining at once in which regions of the domain the motions are periodic and in which they are either aperiodic or asymptotic. Likewise the critical thresholds or "divides" which separate the regions of stability and instability can be easily ascertained by these methods. The principal l1m1tati"on of topological methods, as of' other qualitative methods, is that they do not lend themselves readily to numerical calculations, insofar as they deal with geometrical curves J the trajectories, and not with the laws of motion which are of interest for numerical calculations. The quantitative methods, on the other hand, possess the advantage of leading directly to numerical solutions Which are of importance in astronomical as well as in engineering applications. These quantitative methods, however, inevitably narrow the field of vision to a relatively limited region of the domain. This frequently limits the grasp of the situation as a whole, particularly if the system possesses critical thresholds, the separatrices, the branch points of equilibrium, etc., at which the qualitative features of the phenomena undergo radical changes; such critical conditions are of common occurrence in Non-Linear Mechanics. Both methods, however, very frequently supplement each other. The topological method permits a rapid exploration of the whole field of integration, and the quantitative method leads to numerical results, once a particular range of the problem has been selected for study. Another point of great importance in Non-Linear Mechanics is the question of stability. The fundamental theorem in this connection is due to Liapounoff (4). In Non-Linear Mechanics this theorem plays a role similar to the Routh-Hurwitz theorem for linear systems. Its formulation is closely related to the question of singularities of differential equations. An interesting feature of this theorem is the fact that, in a great majority of cases, it permits establishing criteria of stability for a non-linear system from the equatiOns of the fi~t approximation, which are linear; this fact simplifies the problem &i>preciably. This report reviews the progress accomplished in Non-Linear Mechanics approximately up to 1940; its preparation was greatlyfacilitated by the availability of the following two works in the Russian language. 1. 2.

"Theory of OSCillations," by A. Andronowand S. Chaikin, Moscow, 1937. "Introduction to Non-Linear Mechanics," by' N. Kryloff and N. Bogoliuboff, Kief, 1937.

4

a number of questions, particularly those treated in Part III, the original publications had to be consulted. In view of the fact that the literature, in Russian alone, comprises more than 2000 pages, to say nothing of the earlier publications of Poincare, Liapounoff, Bendixson (5), Birkhoff (6), Van der Pol, and others, no attempt was made to present a complete account of Y.hat has been accomplished in this field. For that reason the report is l1m1ted only to a few selected topics Which seem to offer more 1mmedjBte applications on the one hand, and which do not require too abstract mathenatical generalizations on the other. It is believed that in its present form the report is wi thin the grasp of an average reader having a general knowledge of the theory of ordinary differential equations in the real donain. The report as a whole talls into four major subdivisions: Part I is concerned with the topological methods; its presentation substantially follows the "'lheory of Oscillations" (19). The material is slightly rearranged, the text condensed, and a number of figures in this report were taken from the book. Chapter V, concerning Menard's analysis, was added since it constitutes an important generalization and estab~shes a connection between the topological and the analytical methods, which otherwise might appear as somewhat unrelated. Part II gives an outline of the three prinCipal analytical methods, those of Poinca~, Van der Pol, and Kryloff -Bogoliuboff •* Part III deale with the complicated phenomena of non-linear resonance with its numerous ramifications such as internal and external subharmon1c resonance, entrainment of frequency, para.metric exCitation, etc. This subject is still in a state ot development, and the classification of the numerous eJEperimental phenomena is far tram being detinitely established. Much credit for the experimental discoveries and theoretical studies of these phenomena is due to Mandels~ and Papalexi, and the school of physicists under their leadership. ~e first tour chapters of Part III represent the application of the quasi-linear theory of Xrylotf and Bogoliuboff to these problems and the last three concern the developments of Mandeletam, Papalexi, Andronow, Witt, and others, following the classical theory of Poincare. Finally, Part IV reviews the interesting developments of L. Mandelstam, S. Chaikin, and Lochakow in the theory of relaxation oscillations for large values of the parameter IJ.. This theory is based on the existence of quasi-discontinuous solutiOns of differential equations at the point of their II degeneration, " that is, when one of the coefficients approaches zero so that the differential On

* Dlr1ns the preparation of this report there appearec1

& free translation of extracts ot tile !mort and Bo80Uuboft text 1:17 profdssor 8. Lefscbetz, Princeton University Press, 1943.

5

equation "degenerates fl into one of lower order. A considerable number of experimental facts are explained on the basis of this theoretical idealization. In going over this report the reader will notice that the electrical examples are more numerous than the mechanical ones. The reason for this situation is twofold. First, electrical non-linear oscillations constitute generally useful phenomena that are utilized in radio technique, electrical engineering, television, and allied fields, whereas most of the known mechanical non-linear phenomena are of a rather undesirable, parasitic nature. Second, the determination of the parameters and characteristics is generally much easier in electrical than in mechanical problems, particularly when a mechanical system is relatively complicated. This does not mean, of course, that this state of things will always persist; in fact, mechanical engineers are becoming more and more concerned with non-linear problems (7), (8), (9), and the lack of any app,reciable progress at present is not due to a lack of interest on their part but rather to the absence of a theory sufficiently broad to cover the various cases encountered in practice. It is difficult to write a relatively short report on a subject of this scope in a form that will be satisfactory to all readers. A matheDBtician will undoubtedly find a series of drawbacks in the presentation of purely DBtheDBtical DBtter; the proofs of a number of theorems are omitted. Frequently more rigorous theorems and criteria are replaced by rather intuitive definitions and statements, and so on. It is probable that the mere idea of applying the relatively complicated and laborious methods of Poincar~ and Lindstedt for the purpose of explaining the well-known performance of a thermionic generator or of a simple mechanical system with non-linear damping or a non-linear spring constant may appear to an engineer pedantic and hence superfluous. On the other hand, it cannot be denied that in the past theoretical generalizations have been always most fruitful in the long run, although during the 1n1tial stages of development, they appeared to contemporaries somewhat laborious and conf'uaing. It is sufficient to mention as an example the modern theory of electronics with its numerous ramifications in the fields of radio technique, televiSion, controls, etc. In this case, between the spectacular experimental discoveries of Hertz, Marconi, Fleming, Lee de Forest, and others, there intervened the less spectacular but not less important theoretical work of Richardson, langmuir, and Shottky on thermionic emission, whic~, in turn, was based on the earlier statistical theory of Maxwell and Boltzmann. This permitted later a more rigorous quantitative treatment of these phenomena. Very likely the present trend toward a codification of theoretical knowledge in the field of non-linear mechanics will bring about a further progress resulting from a more general viewpoint on the whole

6

subjeot. It must be borne in. mind, however, that the period of oodification of non-linear meohanics has existed only for about the last fifteen years. Henoe, we are witnessing at present only the important initial stases of these studies rather than their final formulation. It is hoped that this review will attract the aotive attention of applied lIBthellBtioiana, physioists, and enginee~s, for the new methods seem to affeot all branches of applied scienoe either by offering more aoourate solutions of old problema or by making possible an attaok upon new problema beyond the reach of the older mathellBtical methods. The preparation of this report ws fao1litated by valuable disoussions and suggestions on the part of a number of lIBthematioiana and mathellBtioal physioists who showed an interest in this work. The writer is indebted to A. Franok and M. Levenaon who were most helpful in oonneot1on with the or1t1oal editorial work and to Professors E. B. Kennard, W. Hurevioz, S. Lefsohetz, and J. A. Shohat. The oooperat1on of the Applied MathellBtics Panel of the National Defense Research Committee, Dr. W. W. Weaver, Cha1rman, and of the Applied MathellBtios group at Brown University, Dean R. G. D. Riohardson, Chairman 1s especially aoknowledged.

PART I TOPOLOGICAL METHODS OF NON-LINEAR MECHANICS

CHAPTER I PHASE TRAJECTORIES OF LINEAR SYSTEMS

1.

PHASE PLANE AND PHASE TRAJECTORIES; LINEAR OSCILLATOR Consider the d1fferential equation of a harmonic oscillator mil:

where m and c are constants.

+

e z = 0, (e > 0)

[1 .1 ]

Its integral is

z = A

+ tP)

COB (w o t

[ 1.2]

where A and tP are constants of integration determined by the initial conditions at t = 0; and Wo = Yc/m is the angular' frequency. Since the system is conservative, the integral of energy exists. In fact, mult1plying [1.1] by ~ and integrating we get 1

1

.2

2

-mz + -ez = h 2 2

[ 1.3]

Equation [1.3] expresses the law of conservation of energy: The sum of the 2, energies of the system remains kinetic, tmz2, and of the potent1al, constant throughout the motion. Putting ~ = y, [1.3] can be written in the form

tc:a:

[1.4 ]

where a = Y2h/C, {J = V2h/m. Equation [1.4] represents an ellipse having a and {J as semi-axes; see Figure 1.1. The plane of the variables x, y is called the phase plane. The same result is obtained by starting with the integral [1.2] of [1.1]. By differentiating [1.2] we have ~ = -Awosin(wot + tP) and noting that for one single particle the phase angle tP can be made equal to zero by a suitable choice of the origin of t1me, two equations result z

= Acoswot;

~ =

y

= - AWosinwot

[ 1.5]

The el1mination of t between Equations [1.5] gives [1 .6]

8

which is identical with [1.4), provided A

=

Vc'i = a 2

and Awo

= V~ = {J.

follows that Wo :: Yelm. Although the result is the same, the derivation of [1.4) and [1.6) has been different. Equation [1.6) was obtained from the solution of the dynamical Equation [1.1), whereas [1.4) results from the law of conservation of energy, which is the first integral of the dynamical equation. In both [1.4] and [1.6] time does not appear explicitly; theyexpress, therefore, a trajectory in the phase plane, the phase traJectory, or simply, tr4jeetory, where % and y = ~ are considered as coordinates in that plane. The uni-dimensional real motion of a particle oscillating along the z-axis according to the sinusoidal law % = A coswot is thus represented in the two-dimensional phase space, the ph48e pl4ne, by elliptic trajectories, described by the representative pOint R situated at the extremity of the radius vector r. The projection of R on the %-axis gives the actual position % of the OSCillator, and the projection on the y-axiS gives its velocity y :: ~ at the instant t. In general, if some other initial condition is chosen,· another ellipse El is obtained, concentric with the first one and having the y =ic same ratio of semi-axes /J/a :: Yelm = wo, inasmuch as this ratio depends only on the constants of the system and not on the initial conditions. Thus, in this case, the phase trajectories representing the motion are a family of concentric ellipses having a constant ratio of semiaxes. These are shown in Figure 1.1. Figure 1.1 This representation frequently offers advantages as compared to the usual plot of coordinates against time in that it gives an idea of the topology of trajectories in the phase plane. As we shall see later, it enables one to visualize at once all possible motions which may arise in a system with zones separating motions of one type from those of an entirely different type.··

It

*

Such initial conditions which are represented by pointa Bituated on the s8llle trajectory are to be excluded 1DaslllUch eB th87 represent the s8llle motion with a different phaseJ 8ee Section 3.

** The representation of Illation by trajectories in the (.,i) pha8e plane will be 1II08t frequently

used in what tollowa. In 80De cases, however, different variables lIIaT be preferable. 11Dle8B Bpecitied to tha contrary, by the phaaa plane we Bhall alwaY8 mean tha plane ot variableB a and ••

9

2.

PHASE VELOCITY The veloc1ty of the point R is called the pha8e velocity. If tis the radius vector from V to R, clearly the phase velocity t = dt/d t = + ~ ~ where t and J are the unit vectors along the x, y axes of Figure 1.1. With the use of [1.5] we get

ix Jy

t=

i[-Awosinwot]

+

i[- Aw:coswotJ

[2.1]

*

from which it follows that the phase velocity tnever becomes zero if A 0, inasmuch as sin wot and cos Wo t cannot be zero simultaneously, the case Wo = o being excluded. Thus the phase velocity v may only vanish when A= 0; this is possible only when h = O. In other words, this happens only when the particle is placed initially at x = 0 with zero velocity (z = y = 0). This condition corresponds to the center V of the ellipse. On the other hand, for this point the force kx also vanishes. We are, thus, led to identify the point V with the position of equilibrium of the system. One may surmise that this position of equilibrium is stable although this question needs further consideration.

..

3. THEOREM OF CAUCHY; SINGULAR POINTS; TRAJECTORIES AND MOTIONS Cauchy's Theorem of Existence for the solution of a differential equation will play an important role in the following. The proof of this theorem can be found in any textbook of Analysis; for this reason it is sufficient to give here only its formulation. Consider a differential equation of the nth order

The theorem of Cauchy states: There exists a unique analytic* solution of [3.1] in the neighborhood of t = to such that the function x and its (n - 1) derivatives acquire for t = to the set of prescribed values x o ' zo' .. " XOCfl -1) provided the function f( t, x, x, .. '. X(fl -1») is analytic in the neighborhood of this set of values. It is obvious that [3.1] is equivalent to a system of n equations of the first order dYR-l dt

* Analytic

=

!(t

,x'Yl""'Y,,-l

at a given point means admitting a Taylor series about the point.

)

[3.2]

10

which is a special case of the system d1/' Tt

( = f it,

1/1,1/2,···,1/,,), i

=

1,2,"', n

In applications, t usually represents time, the other variables representing coordinates of a point in the phase space of a dynamical system. Of particular importance is the case of [3.1] and system [3.2] in which the function f does not depend on time t, and is a function of x, i,' .. , X("-l>, or 1/1' 1/2' .. " Y", alone. From a physical standpoint, this means that the system in question is lIautonomous,1I that is, one in which neither the forces nor the constraints vary in the course of time. This will be assumed in what follows. Henceforth we shall be concerned mainly with systems with one degree of freedom which are represented in the phase plane by phase trajectories as previously explained. The differential equations are.of the form

~:

= P(x,1/);

.£L

= Q(x,y)

[3.4 ]

A differential equation of the second order

= Q(x,z)

i

can be reduced to the form [3.4] by putting dx _ y'

dt -

1/

= x,

so that one obtains

!!1. _ Q('""', y)

• dt -

[3.4a]

It is to be noted that one will frequently encounter dynamical equations of the more general form [3.4]. Upon el1minating the independent variable t between the Equations [3.4] one obtains one equation dx = P(:z,y)

dy Q(x,y)

[3.5]

....,...,.-"-~

which gives the slope dy/d:z of the tangent to the phase trajectory passing through the point (x,y). We shall often speak of the point R(x,y), the repre8entative p01nt, as mov1ng along the trajectory. By this we mean th~t the mot10n of this pOint is governed by the law 'expressed by [3.4] or, more particularly, by [3.4a]. Since t1me does not enter explicitly at the right in [3.4], the general solution of [3.4] can be written as x = ¢» (t - to, xO' Yo) := x(t) y

= I/I(t

- to,

XOI

Yo)

==

D.6]

yet)

so that the trajectory passing through the point (xo,Yo) at t be obtained from [3.6] by el1minating (t - to).

= to may also

11

The topological methods studied in Part I will deal with the trajectories [3.5], whereas the analytical methods outlined in Part II will be concerned with the differential equations of motion [3.4]. In some cases it is useful to apply both methods, so as to obtain a more detailed picture of the behavior of a given dynamical system; in such cases [3.5] gives a qualitative idea of the nature of the trajectories in various regions of the phase plane, whereas Equations (3.4] yield the actual motion starting from given in1tial condit10ns. Such a procedure is particularly useful when a dynamical system possesses certain critical thresholds or "divides,1I in which case the application of [3.5] permits a preliminary qualitative study of the nature of the trajectories in various domains of the phase plane before the actual integration of the dynamical Equations [3.4] is undertaken. In the following, we shall make use of the concept of singularities of differential equations, and therefore we shall find it expedient to make the following definitions. A. A point (%0.110) for which P(%o, Yo) simultaneously is called a singulAr point.

= Q(%o' 1'0)

I:

0

B. Any other point of the phase plane to which Definition A does not apply is called an ordinary point. From these definitions it follows that an ordinary pOint is characterized by a definite value of the slope of the tangent to a phase trajectory passing through that pOint. For a singular point, on the contrary, the direction of the tangent is indeterminate and the trajectory corresponding to [3.4] or [3.4a] degenerates into a Single point, the si~ular point itself. Using these definitions we can formulate the theorem of Cauchy in the following form: Through every ordlnary polnt of the phase plane there passes one and only one phase trajectory.

One can also conSider Equations [3.4] as defining a vector field with components d%/dt, dy/dt which determine a non-vanishing vector for any ordinary p01nt (%,y) of the field. For a singular point (%o,Yo) we have P(%o' Yo) = Q(%o' Yo) 0, hence, both components d%/dt, dy/dt vanish. More specifically, if the vector fleld is a velOCity field, it is clear that a Singular point is a point at wh1ch the representative polnt 1s at rest. In a number of dynamical problems for which [3.4a] is applicable, an additional interpretation can be given. A singular point ln this case occurs when y = 0, dy/dt = 0, that is, when both the velOCity and acceleration of the system are zero; this clearly defines an equilibrium condition. We are thus led to identify the singular points 1n such a case with the pOints of equ1librium of a dynam1cal system. I:

12

It follows from the preceding definition that if (~o,Yo) is a singular pOint, a trajectory passing through an ordinary pOint \~,y) at a certa1n instant will never reach (~o,Yo) for any finite value of the time parameter t; for the only trajectory passing through the singular point (~o' Yo) is the degenerate trajectory consisting of this point alone. It may happen, however, that a proper trajectory, that is, one which consists of more than one point, may approach a singular point either for t = + ~ or t = -~, which means that lim ~(t) = ~o and lim yet) = Yo for t = + ~ or t = -~. It is to be understood that, when we say that a trajectory "approaches" a point (~o' 110), we mean that a representative pOint following this trajectory in accordance with [3.4] approaches the point (~o,Yo)' Our main concern will be the study of the behavior of trajectories in the neighborhood of singularities. This question is studied in more detail in the following sections. It may be useful, however, to give at the outset a brief geometrical description of the behavior of trajectories in the neighborhood of the four singularities with which we shall be concerned, leaving a more complete analysis of this subject to a later section.

1.

A vorte~ point V is a singularity which is not approached by any

trajectory. Point V in Figure 1.1 gives an example of a vortex point. Section 1 illustrates that a vortex point is surrounded by a continuum of closed trajectories such that none approaches it. In this example, the vortex point appears in connection with the elliptic trajectories of the differential equation, + wi~ = O. We shall see later that vortex pOints may occur in.connection with equations of a more general form, in which case the trajectories, while being closed, are not necessarily ellipses. In all cases, however, a vortex point V is characterized by the following two conditions: a. the trajectories are closed, enclosing the Singularity, b. there is a continuum of these trajectories.

z

2.

Figure 3.1

A saddle point S is a singularity which is

approached by four trajectories forming two distinct analytic curves, Al A2 and AaA4' as shown in Figure 3.1. Two of these trajectories, A1S and A2 S, approach the saddle pOint S for t = +~; two others, AsS and A4S, approach S for t = -~, which is equivalent to saying that the representative p01nts on these trajector1es, As and A4 , move away from S. Between these

13 four isolated trajectories, there exist four regions containing continua of hyperbolically-shaped trajectories which do not approach S. The repreF sentative point moves along these trajectories in the direction of the arrows in conformity with the directions indicated on the asymptotic trajectories A l , A 2 , As, A., which appear, thus, as "divides" for the four regions. Figure 3.2 In Section 4 we shall see that this situation arises, for instance, when one considers the equationm~ - cx = 0 of unstable motion. In Section 18 a general criterion for the existence of a saddle point will be given for systems of the form [3.41.

3.

A focal point F is a singularity which trajectories approach, without any definite direction, in the manner of the spirals Kl and K2 shown in Figure 3.2. The radius vector r of the spirals decreases as R approaches F, but the direction of approach is indeterminate since a spiral trajectory winds around a focal point F an infinite number of times as the point R approaches F. There exists an infinity of spiral trajectories one and only one of which passes through every ordinary point of the phase plane. If the spirals approach F for t = +~, the point F is called a stable focal point. If the focal pOint 1s approached for t = - ~ the direction of motion on the trajectories, as shown by the arrows in Figure 3.2, is reversed so that the trajectories "leave" the focal point. The point F is then called an unstable focal point. \ In Section 5 an example of motion in the neighborhood of a focal point will be given in connection with the equation ~ + 2bx + W 02 x = 0 of a damped oscillatory motion (b 2 - w o2 < O)i a more general criterion for the existence of a focal point in the case of the system l3.4] will be given in Section 18.

Figure 3.3

4. A nodal point N is a singularity which is approached by trajectories in the manner of the curves Kl and K2 of Figure 3.3, so that when r .. 0 the direction of the tangents to the trajectories approaches definite limits, e.g., the straight line DD. According as the point N is approached for t = + ~ or t = -~, the nodal point is called stable or unstable; in the latter case the direction of the arrows in Figure 3.3 should be reversed.

14 An example of trajectories of this kind is given in Section 6 for the case of an aperiodic damped motion corresponding to the equation i + 26z + W 02 % = 0 when 62 - W 02 > 0; a more general criterion is indicated in Section 18. It may be useful to add a remark concerning the physical interpretation of some of the preceding definitions. When we speak of a trajectory passing through an ordinary point (%o,yo) of the phase plane, we frequently understand by this that we wish to consider the motion as starting from given initial conditions %0' Yo = ZOo Furthermore the statement of the theorem of Cauchy, that through every ordinary point (%o,yo) of the phase plane there passes one and only one trajectory, is equivalent to saying that if the initial conditions are prescribed, the subsequent motion is uniquely determined. On the other hand, to a given trajectory [3.51, i.e., a geometr1cal curve in the phase plane, there corresponds an infinity of possible motions [3.51 corresponding to a different selection of the time origin. In fact, if Equations [3.41 are satisfied by solutions [3.6] for a given value of to, they are obviously satisfied by [3.6] for any other value of to' Geometrically, all solutions [3.6] for the same values of %0' Yo are represented by the same trajectory, say C, passing through the point (%o,yo), although there exists an infinity of possible motions (C) on this trajectory when the arb1trary constant to is varied.* It can also be stated that the solutions of [3.4] form a twoparameter family, while the trajectories [3.5] form a one-parameter family. 4.

PHASE TRAJECTORY OF AN UNSTABLE MOTION; SADDLE POINT The differential equation

mx - C%

=

0, (c

> 0)

[ 4.1]

haa an exponential solution of the form %

= A e rC

+ B e -rt

[4.2]

where r = + Ye/m. The solution %, thus, tends to infinity for t +~ on account of a positive root r of the characteristic equation. Equation [4.11 represents, for instance, the motion of an undamped pendulum in the neighborhood of its unstable equilibrium pOint. Replace [4.1] by the system

z = y, * In

iJ

what follows we shall. use consistenU;y this notation, Le., C will designate a geometrical curve, the traJector:y and (C) a motion of the representative point R on the trajecto17 C.

15 whence

y =i

dy

c x

dx

m y

y=-~J.

[4.3]

For x = 0, Y ~ 0, the tangent to the trajectory is horizontalj for x * 0, Y = 0, it is vertical. For x = 0, y = 0, the direction of the tangent is indeterminatej this point is a singular point. The phase trajectories are obtained by integrating [4.3] which gives C 2 Y2 - mx

=C

where C is the constant of integration. The trajectories are thus hyperbolas, shown in Figure 4.1j the asymptotes are obtained by putting C = 0 which gives y =

±~

Figure 4.1

x

The representative point R arriving from infinity will approach the origin x = y = 0 and then will depart again into infinity following the direction of the arrows. The interpretation of such motion is obtained by considering the motion of a pendulum in the neighborhood of the point of unstable equilibrium. Consider the motion of a pendulum 'having a unit moment of inertia. Let Xl be the angle measured from the position of stable equilibrium and Yl c ~1 the angular velocity of the pendulum. The law of conservation of energy gives [4.4]

l

where y 12 and V( Xl) are the kinetic and the potential energies respectively; the constant h is the total energy, communicated, for example, by means of an impulse at the beginning of the motion. It is known from Theoretical MechaniCS (10) that there are three typical motions according to h - V( 1T) ~ O. We proceed to consider each case separately. h - V( 1T) > O.

In this case the pendulum goes "over the top" and continues rotating in the same direction, i.e., Yl keeps the same sign but 1.

16

varies in magnitude passing through a maximum at at ~l = 'Ir.

2. close to

3.

~l c

0 and through a minimum

h - V(") < o. ~l c "

In this case the pendulum reaches a certain angle and turns back away from the unstable point ~l = ".

h - V(")

~o

= O.

It is shown in Theoretical MechaniCS (10) that the pendulum approaches the unstable point ~l c " in infinite time with a gradually decreasing velocity Yl approaching zero as a limit. We call motion of this kind aBY1nptotic. In this discussion the angle Zl was measured from the stable equilibrium position. If we introduce now the angle ~ = 'Ir - ~l' measured from the point of unstable equilibrium 8, the results obtained from the equation of energy acquire a simple graphical representation shown in Figure 4.1. The conditions specified in Case 1 are represented by the hyperbolic trajectories situated in the upper and lower quadrants of Figure 4.1; those corresponding to Case 2 are represented by hyperbolic trajectories in the right and the left quadrants of that figure; and the asymptotic Case 3, in which the total energy h is Just equal to the potential energy of the system at the point 8 of unstable equilibrium, is represented by the asymptotes AlB and A2 80f Figure 4.1 along which the motion of the representative point R is asymptotic. The inverse process, i.e., the motion of a pendulum placed initially in the unstable position of equilibrium, is also asymptotic; it is represented in the phase plane by asymptotes 8B l and 8B 2 • The singular point 8(0,0) of this type is a Baddle point. It is seen that four asymptotic trajectories approach the saddle pOint, namely, Al 8 and AlB for t = +~, and Bl 8 and B2B for t = -~, inasmuch as the asymptotes are also trajectories. Once agaln we observe that the polnt 8 which is a singular point, is a point of equilibrium which is, however, unstable in this case and a point at which the phase velocity vanishes.

5. PHASE TRAJECTORY OF AN OSCILLATORY DAMPED MOTION; FOCAL POINT The differentlal equation of a damped oscillator of unit mass is .. Z

provided b 2

- W0

2

<

o.

+

2b'Z

2 + woz

= 0

The solution of [5.11 is Z

= zoe -bl cos ( WIt

+ 01 )

17 where

and a are two constants of integration, b is the decrement,* and 2 is the damped angular frequency. The system is not conservaW1 = YWo2 - b tive in this case. We omit the well-known properties of motion defined by [5.1) and [5.2) and consider the representation of the motion in the phase plane. The differential Equation [5.1) of second order can be replaced by a system of two differential equations of the first order Zo

dz

dt

= Y;

d,tJ

dt

2

[5.3)

= - 2by - woz

whence

!bL _ _

2 by

dz -

+ Wo2 z y

Equat10n [5.4) is a particular form of [3.5). It 1s seen that the origin, z = y = 0, is a singular pOint since the direction of the tangent dy/dz becomes indeterminate at this pOint. Equation [5.4) is a homogeneous differential equation of the first order and can be integrated by introducing a new variable u defined by the equation y = uz. We finally get y

2

+

2bzy

+

2

woz

2

= Ce

2"!"ta -I~ n

WI

[5.5)

WI %

+-'d t = +-. The phase velocity +v = dr, iz + +-. jy is obtained by substituting for z and y their expressions obtained from (5.2). The absolute value of the phase velocity is 4 2 Iv I = 0woz + 4bzywo2 + ( 1 + 4b 2) y 2

[5.6)

The phase velOCity is zero only at the origin, z = y = 0, of the phase plane, which point coincides with the singular point of the differential equation as seen from [5.4). Equat10n [5.51 can be transformed into polar coordinates. The left-hand term can be wr1tten as {y + bZ)2 + {W 0 2 - b 2 )Z2 == {y + bZ)2 + W 12 Z2 • Introducing new coordinates u = w 1 z, V = Y + bz, [5.51 becomes

If we now introduce polar coord1nates def1ned by u the last equation becomes

=p

cos

~,

v

=p

Sin

~,

[5.7] * It

sbould be noted that dimensionally, the decr8lllent is 6/m, and not 6; bere however, m = 1.

18 where C1 depends on the initial conditions. The phase trajectory is, thus, a logarithmic spiral in the (u,v)-plane. In view of the transformation of coordinates (from ~, y to u, v) the spiral given by [5.7] undergoes a distortion when referred to the original coordinates z, y as shown in Figure 5.1. F For a positive damping. b > 0, corresponding to disSipation of energy, the exponent in [5.7] must be negative which can be taken care of by a Figure 5.1 suitable definition of the positive direction of ~. For small values of blwl the curve in the (u.v)-plane approaches the circle v 2 + 1.1. 2 = Cwhich in the (~.y)-plane corresponds to an ellipse given by the equation y2 + 2bzy + wo 2 Z Z = C. One may surmise that the point F towards which the spiral converges is an equilibrium position. We recognize in this Singular point the focal point defined previously. From physical considerations it is apparent that any trajectory of this kind approaches the focal point for t = +~; furthermore. the focal point is approached by an infinity of phase trajectories Since through every ordinary pOint of the plane there passes a spiral trajectory. y:i

6.

PHASE TRAJECTORY OF AN APERIODIC DAMPED MOTION; NODAL POINT If in [5.1] b l - wo2 > 0, the corresponding characteristic equation. has two distinct real roots, - Tl and - T I , of the same sign and the motion is aperiodiC of the form Z = Ae -rl' + Be -ra', where A and B are two constants of integration determined by the initial conditions. Differentiating we have Z. = - A Tl e - rl ' - B TI e - rl' • Eli m1 na t ing t 1me t b e t ween Z and z. * we obtain the equation of the phase trajectory in the form

[6.1] Taking as new variables v

= ZTI + y, 1.1

1.1.

=

= ZTI + y,

we get

Cu o

[6.2]

where a = TalTI > 1, T z being the absolute value of the larger root. Equation [6.2] represents parabolic curves tangent to the u-axis at the origin. for dvl du = Cau 0 - 1 is zero when u .. 0 since a - , > 0 and 11 = U = 0 is a point on the curve.

* Tbis

-r ,

-r ,

el1m1Dat10D 1s best obtained by solving the system or linear equations w1th e 1 aDd e a uaknown. By taldJIg logar1tbDa or the solutions thus obtained, the time' 1s eaaU,. e11lll1nated.

as

19 Por C = 0, the curve degenerates Into the u-axls, 1.e., v = 0; for C +~ it degenerates into the v-axis, i.e., u = 0. Purthermore, the curves [6.2) are convex towards the u-axiB since v"/v = a( a - 1 )/u 2 > 0. These curves are shown in Pigure 6.1a. If we transform the (u,v)-plane back to the (%,y)-plane, the phase trajectories have the configuration shown in Figure 6.1b. The u-axis, v = 0, is represented in the (%,y)-plane by the line y = - %r 1 ; the v-axis, u = 0, Is represented by y = - %r2 • The trajectories are tangent to the u-axls at the orlgln in the (u,v)-plane; hence, they wlll be tangent to the 11ne 11 = - %rl at the origin in the (%,y)-plane as shown in Plgure 6.1b. Furthermore, as u increases, the curves in the (u,v)-plane become parallel to the v-axls, hence the asymptotlc branches of phase trajectories approach parallellsm wlth the 11ne y = - %r 2 In the (%,y)-plane. It is seen that the trajectorles cross the %-axls, 1.e., the line y = 0, at right angles. We also find that the locus of the polnts In which the trajectories %, y have horizontal tangents 11e on the line 11 = - rlr~r2r2 %. y: i

v

Figure 6.1 These conditions determine the trajectories shown in Figure 6.1b. The origin N, Pigure 6.1 a is a Singular point, a nodal point. It is seen that an infinity of trajectories, or integral curves, approaches a nodal pOint.

20

1. METHOD OF ISOCLINES Let

:: = P(%.y); ~ = Q(%. y) be the differential equations of a dynamical system and

that of the trajectories.

Suppose that

d1l

Q(%.1/)

dz = P(z.1/)

==

()

F Z.1/

=

a. = constant

[7.1]

This equation clearly defines a curve in the (%,y)-plane along which the slope, dy/dz, of the trajectories remalns constant. Such curves are called i80cline8. It is apparent that all polnts of isoclines are necessarlly ordinary pOints since at Singular pOints the slope of trajectories is indeterminate, so that [7.1] has no meaning at such pOints. The method of isoclines makes it possible to explore the field of trajectories gra.phica."y without solving the differential equation. Once the curve F(z,1I) = a. is traced, one draws along its length small llneal elements having the prescribed slope dy/dz = a. One repeats the procedure for other values of a so that flnally one obtains a serles of curves F(%,y) = a, aI' 0. 2 , • • • , with the corresponding slopes drawn along these curves. These slopes then determine the field of directions of tangents to the trajectories in a certain region of the (z,y)-plane. Starting from a point (zo,Yo) a continuous curve can be traced following always the direction of lineal elements of the field. The curve so obtained is clearly the trajectory paseing through the point (%o'Yo)' The method is very valuable in cases where the explicit form of the solution of the differentlal equation is not known. It has been applied, for instance, by Van der Pol in his early studies of the equation ~ - JJ (1 - z 2):i:

+z

=

0

[ 7.2]

This equation can be reduced to the system :i:

=

y;

iI

= JJ (1 - Z 2) 1/ - z

whose trajectories are given by ~ = JJ (1 - Z2)1/ dz y

%

[7.3]

21

The only singularity of [7.2] is clearly the or1gin, % = Y = 0; hence the method 1s applicable everywhere except at that p01nt. Equat10n [7.1] of the 1soc11nes 1n th1s case 1s 11 = (Jl - a) - JlZ2

For a fixed Jl and for a number of d1fferent values of the slope a, a series of curves [7.4] are obtained along which the slope of the trajector1es is constant. F1gure 7.1 shows this construction of Figure 7.1 Van der Pol (1) for Jl = 1 in [7.2] wh1ch is selfexplanatory. This graphical construct10n makes it possible to establish the existence of a closed trajectory C to which the non-stationary spiral trajectories C' and C"approach both from the outside and from the inside of C. The reader can easily check, by this method, the principal types of trajectories established previously. The principal advantage of the method of isoclines lies 1n the fact that it always leads to the desired result although its application is tedious and subject to errors inherent in any graphical construction, particularly when the slope of trajectories changes rapidly in certain regions of the phase plane as shown in Figure 7.2. This figure,also taken from Van der POllS paper, represents the trajectories of [7.2] for Jl = 10.0.

8. c " : 10

-4!~~1I

Figure 7.2

SYSTEMS WITH NEGATIVE DAMPING; FROUDE'S PENDULUM In many branches of applied science one is confronted with problems in which negative da?nping occurs. From a physical standpoint the term is rather unfortunate; its just1fication lies in the sign of the damp1ng coeffic1ent b. In cases previously considered it was assumed that b > 0 corresponds to a dissipation of energy from the system. For b < 0, on the contrary, there is an addition of energy to the system from an external source according to the same law, that is, in proportion to velocity. In electr1cal problems negative damp1ng plays an important role; it 15 generally associated with circuits containing non-ohmic conductors. which exh1bit a decrease in voltage for an increase of current. In many control problems a similar result is observed if the phase of a normally damping

22

control action is reversed (11). Many years before the discovery of electrical circuits possessing negative resistance, or negative damping features, W. Froude (12) (13) discovered a sim11ar effect in the case of a pendulum mounted on a rotating shaft with a certain amount of friction. It was observed that a pendulum of this kind begins to oscillate with gradually increasing amplitudes. The following theory explains this effect: In addition to its own parameters 1, h, 1ngl = C governing the free motion as determined by the equation l~ + h~+ C~= 0, the pendulum is also acted upon by an external momentAl(w - ~), due to dry friction, which depends on the difference of angular velocities w of the shaft and ~ of the pendulum. Expanding the functionM(w - ~) in a Taylor series and keeping only the first two terms we have the following differential equatlon lfi,

+

b~

+

C~ = M(w) - ¢,AI'(w)

that Is,

[8.11 The constant termAl(w) on the right merely displaces the posltion of equilibrium and is of no further interest. Thus, interest centers on the coefficient, [h + M'(w}), of~. If, in a certain range of w, the friction is such that M'(w} < 0, 1.e., the friction decreases with w, the negative term M'(w) may outweigh the positive one, b. Thus, the motion of the pendulum occurs as if the coefficient of ~ were negative. From a formal standpoint the introduction of negative damping does not alter the discussion in Sections 5 and 6 appreciably; the only difference in the representation of motion in the phase plane conSists in the reversal of the positive direction of the phase trajectories shown in Figures 5.1 and 6.1. Instead of approaching a stable focal or nodal pOint, the trajectories "approachll these pOints for t = - 00, that is actually depart from them.

9. REMARKS CONCERNING LINEAR SYSTEMS The above examples are intended primarily to familiarize the reader with the representation of motion by phase trajectories; nothing new has been learned so far, but the known facts were merely presented in a different way. The connection between the singular points and the state of equilibrium of a system will play an important role in what follows and it is useful to mention at this stage that a linear approximation will have considerable importance in establishing the criteria of stability as will be shown in Chapter III. The topology of phase trajectories in the neighborhood of point singularities in non-linear problems remains qualitatively the same as in the simple linear

23 problems studied so far, although quantitatively the relations may be somewhat different. Thus, in non-linear conservative systems the trajectories around a vortex point are still closed curves but not necessarily ellipses. Likewise the unstable equilibrium in non-linear systems is still characterized by a saddle point but the asymptotes of the latter are not straight lines as they are in linear cases. We shall investigate these questions in detail in the following chapter but it is worth-while mentioning now that insofar as their local properties are concerned, non-linear systems behave not very differently from the linear ones. The important difference lies in the behavior in the large. It will be shown later that non-linear systems possess under certain conditions closed trajectories of a special type, the limit cycles. These new closed trajectories are entirely different from those of linear systems. The most important feature of this new type of trajectory is that it can occur only in a non-conservative system whereas a linear system can possess closed trajectories only when it is conservative. This can be stated as follows: In both linear and non-linear mechanics a similar formulation of conditions for stability is used but the corresponding stationary states of motion are quite different. A typical pattern of stationary motion in linear mechanics is the motion of a non-dissipative harmonic OSCillator, whereas a corresponding pattern in a non-linear, non-conservative system is that represented by this special closed trajectory of a new type, the limit cycle, as defined in Chapter IV. The theory of limit cycles and associated phenomena is, thus, a domain of non-linear mechanics proper for which there exists no counterpart in linear mechanics.

CHAPTER D PHASE TRAJECTORIES OF NON-LINEAR CONSERVATIVE SYSTEMS

We shall now consider non-linear systems of conservative type, that is, systems in which the dissipation of energy is negligible. The law of conservation of energy is applicable here and leads to a relatively simple topological representation without the necessity for solving the non-linear differential equation. The question of stability of motion in the vicinity of equilibrium points, left open so far, will also be clarified following this line of argument. 10.

GENERAL PROPERTIES OF NON-LINEAR CONSERVATIVE SYSTEMS

Consider first a simple motion defined by the differential equation {10.1]

in which the restoring force is a certain function of the distance x. It will be assumed that !(x) is analytic for the whole interval (- =, + =). Equation [10.1] of the second order is equivalent to the system x=y; y=!(x)

{10.2]

Eliminating time we have !(x) y

[10.3]

which specifies the field of trajectories in the phase plane. The velocity of motion is given by the ordinate y of the trajectory. and the phase velocity is [10.4]

From [10.3] it follows that the trajectories cross the x-axis (y = 0) at right angles and have horizontal tangents at points Xi which are roots of !(x), provided the trajectory does not cross the axis of abscissas (y = 0) at these pOints. We see thus that Cauchy1s theorem of uniqueness holds in all cases except at pOints (Xi'O) for which !(x i ) = O. Furthermore, from the fact that at a singular point y = 0, !(x) = 0, and, in view of relations [10.2], it follows that at singular points dx/dt = o and y = d 2 x/dt 2 = O. The latter condition is equivalent to the vanishing of the forces at th1s point. The last two cond1tions clearly def1ne a position of equi11brium. Moreover. from [10.4] 1t follows that at the singular points the phase velocity vanishes.

25 11.

TOPOLOGY OF PHASE TRAJECTORIES IN THE NEIGHBORHOOD OF SINGULAR POINTS A. GRAPHICAL METHOD For conservative systems the problem is simplified owing to the existence of the integral of energy. For a system of unit mass we have [11.1]

which expresses that the sum of the kinetic energy, ~y~ and the potential energy, V(x), remains constant. By definition V(x) = - J~f(x)dx, where f(x) is o the restoring force. For a given value of h, [11.1] represents a curve of constant energy in the phase plane. The motion is impossible if h - V(x) < 0 since the value of y is then imaginary. This is ruled out in a physical problem. Equilibrium points are characterized by the relation f(x) = 0, that is V'(x) = O. In other words, the potential energy has an extremum value at the point of equilibrium. From [11.'] and the preceding discussion the following conclusions can be formulated: 1.

The phase trajectories are symmetrical with respect to the x-axis.

2. The trajectories cross the x-axis at right angles, that is dy/dx ~; they have a horizontal tangent at the paints where f(x) = 0, provided y does not vanish at these points.

3.

The singular points are situated on the x-axis at points Xi' for which f(x) = O.

=

z

The topological picture of the phase trajectories can be best shown in two steps: a. by drawing an auxiliary diagram giving the representation of the difference h - V (x), and b. by drawing the phase trajectory y/Y2 itself, both curves being plotted against x. Figures '1.1a and b show examples of this representation. Figure 11 .1a shows the diagram of the balance of energy h - V(x) for a

0.....--------.........- ..........- ·

(0)

M'2 O~--;r----~~---x

M'I (b)

Figure 11.1

26 given shape of vex) and assumed value h = hi. The difference h - Vex) is positive to the left of All. Figure 11.lb gives the phase diagram (coordinates %, y/P2) corresponding to Figure 11.1a. The phase traje9tories are real to the left of All. This case corresponds, for instance, to the trajectories in the right (or left) quadrants of Figure 4.1 representing the motion in the vicinity of a saddle point when the pendulum does not have enough energy to overcome the "potential barrier" and is thus unable to go over the upper unstable equilibrium point.. In this case, the question of linearity assumed in connection with Figure 4.1 is waived and the discussion, therefore, can be made general. If the value of h is"increased (h = ha), the point All is shifted to Ala' and the phase traJectory is shifted accordingly (point Al~). Figure 11.2 represents an analogous construction when Vex) has a minimum for % = %1. The only essential difference between this and the preceding case is in that for % = %1; V( %tl z has now an extremum, that is, fez) = o. Hence the pOint % = %1 is a singular point, z " V(x) the vortex pOint, as previously defined. The trajectories in the phase space, Figure 11.2b, around the vortex point are obtained in the same manner as before, that is, for a given value of h ~ V(%l) we plot the oro '----f---+-t-"----L._x (0) dinates ± Yh - Vex) in Figure 11.2a; then for the next curve we change h and plot 11,+--... another curve, etc. The oval curves which thus appear are enclosed within each other and the motion is periodic; moreover, in view of non-linearity, these curves depart (b) somewhat from the elliptic form. Figure 11.2 B. ANALYTICAL METHOD The topological picture of the trajectories in the vicinity of singular pOints can also be obtained analytically. Assume first that Vex) has a minimum for z = Xl' and expand f(x) in a Taylor series about % = Xl

f(x)

= alex - Xl) a

+

1a. 22 (X - Zl)2

+

as

1·2·3

(X - Xl )3

+ . .. [11.2]

Since vex) = - 1 f(x)d%, by integrating (11.2], changing Sign, and adding the o constant of integration hOI we get

27

[11.3] where [1'.4]

Transferrlng the orlgln to the slngular polnt, i.e., puttlng % a %1 + (, y = o + ~, th1s glves, upon the substltutlon of the value of V(%1 + () as given by [11.3] lnto the integral ot energy y2/2 + V(%) = h, the equation

-~2 + ho 2

[a ~2, ~

1•2

+ . . . + 1· 2ak(k+1 + . .. ] . . . (k + U

=

h

['1.5]

Two principal cases can be considered: a 1 ~ O. In this case the curve V(%) and straight line z = ho have at % = %1 contact of the first order. Since V(%1) is minimum, V'(%1) = 0 and V"(%1) > O. Therefore a 1 co - V"(%1) < O. For h = ho ['1.5J degenerates into an isolated pOint co ~ = 0, which is clearly the vortex point of the phase trajector1es. For h - ho co a > 0, [11.5], to the first order of f(%), is 1.

e

~2

-2 +

la-lk2

!.:.l.!..l-.

2

== a

['1.6]

which is of the form

i: m2

(2

+ -;t = 1

where m 2 = 2aj n 2 = 2a/la 1 1. The trajectory in the phase plane is thus an ellipse with sem1-axes m, n. To the first approximation, for small motions, it is legit1mate therefore to cons1der the motion as a sinusoidal function of time. For a greater general1ty assume a 2 = as = ••• a k _ 1 = 0 and at ~ O. Then k is necessarily an odd 1nteger and at < 0, since z =V(z) lies above its tangent in the neighborhood % = %1' For h = ho the curve ['1.5] reduces again to an 1solated point e = ~ = 0, the vortex point of trajectories. For h - ho = a> 0, [1'.6] to the (k + ,) order of k becomes 2.

a1

= O.

~2

2'" +

la l~k+1

(k k+ 1) t = a

[1'.6]

which represents a closed curve differing from an ell1pse even for very small oscillations. It would be erroneous, therefore, to "linearize" such a motion even for small osc11lations. The topological picture of the trajectories in

28 the phase plane will have the appearance of closed curves surrounding each other around the vortex point V, Figure 11.2b, as previously found by the graphical method. If the potential energy V(x) has a maximum for % = %1 the procedure is Similar. with the exception that a l (or a k in the more general case when al = a2 = ... ak _ 1 = 0) is now positive. so that the trajectory becomes TJ2/2 .. a l ~2/2 = a to the first order (case when at :I: 0), which is of the form [11.7] with the previous notations. Equation [11.7] represents hyperbolas with TJ = + ~~ and TJ = .. ~~ as asymptotes. The point ~ = TJ = 0 is, clearly a saddle pOint. Thus to the first order. the small motions in the vicinity of the maximum of potential energy are unstable and are represented in the phase plane by hyperbolic trajectories as previously found from a study of linear systems. In the second case (a l = a 2 = .•• = ak -1 = 0; ak "" 0), we find by a Similar argument that for a k > 0 the trajectories in the phase plane are obtained from [11.5] by neglecting the terms with a k + t ' a k + 2 ' •• ' ; in this case [11.8] z

The phase trajectories, while still possessing the same general features, such as asymptotic motions, etc., depart from ordinary hyperbolas even for small motions. Furthermore the asymptotes are curvilinear.

XI

x"I

(0)

x

I

I I I I

(VS)

x" x

(b)

Figure 11.3

3. The potential energy V(x) has an inflection point at % C Xl with a horizontal tangent, Figure 11.3a. In this case at =- V"(x t ) = O. Furthermore, in the general case, the first term ak' which is not zero, must be necessarily of even order (k = 2. 4, ... ) as follows from [11.3] because the potential energy variation changes its sign for h = hi ± tJh in this case. In the phase plane, Figure ~1 .3b, the trajectory corresponding·to the ordinate h = h t of the inflection point has a Singular point VS in the sense of differential geometry, e.g., a

29 cusp, which represents the coalescence of a vortex pOint V with a saddle point S. The motion along this singular trajectory is asymptotically approaching the point VS, and reverses the direction of its motion also in an asymptotic manner. The motion is ultimately unstable; this instability is more definite for other trajectories. Everything happens as if the coalescence of a stable vortex point with an unstable saddle point were contaminating the process with an ultimate instability. In reality this process of coalescence of singularities means that the singularity at this point is no longer simple in character. The complete study of the conditions of equilibrium upon the coalescence of singularities is too complicated, and will not be attempted here. Summing up the results of this analysis we are led to the following two theorems of which the first is due to Lagrange and the second to Liapounoff. THEOREM 1.

II the potenUal energy

vb;) is minimum at the point x

= xl'

the equUibrium

1s stable. THEOREM 2.

II the potential energy Vex) has an extremum at the point Xv without being

a minimum, the equilibrium 1s unstable.

12.

TOPOLOGY OF TRAJECTORIES IN THE PHASE PLANE. SEPARATRIX The preceding method was applied to the motion occurring in the vicinity of the equilibrium pOints; it can be extended, however, to the whole phase plane. In this manner we obtain the complete picture, the topology of the phase trajectories with the critical boundaries separating motions of different types. The starting point for this representation is again the energy equation y2/2 = - V(~) + h. The various cases to be discussed are illustrated in Figure 12.1. 1. If h< Vex) for all values of ~, clearly no motion is possible. Figure 12.1 exhibits this condition between the points 0 and 1, between point 5 and 7 and to the right of point 13.

2. h > V(~) for all values of~. This case is shown in Figure 12.2. The velocity y never changes sign but varies only in magnitude. At the pOint ~l at which V(~) is maximum, the velocity decreases somewhat. This corresponds to the motion of a pendulum having an initial energy h greater than the amount required to carry it to the upper unstable equilibrium point.

>

> V min

assuming that no point of tangency exists between the straight line z = h and the curve V(~) as is shown in Figure 12.1a for z = h 2 • Between the points a and b, h - V(~) > 0; the motion is 3.

V max (x)

h

(~),

z

Figure 12.1 z h

h -V(I)

(0)

(b)

Figure 12.2

thus possible. Point 2 in the interval (a,b) corresponds to the minimum ot potential energy, hence, it is a vortex pOint 2 in the phase diagram. The trajectories surrounding the point 2 are closed curves without pOints ot intersection; by changing ha we obtain a tamily ot these trajectories around the point 2 torming an "island" in the phase plane. Similarly, between the pOints c and d (same value ha) there appears another "island" ot trajectories around the other vortex point 4.

4.

It in a certain region, e.g., to the lett ot pOint 0, in Figure 12.18, the potential energy decreases monotonically, the motion becomes possible again. The trajectory arrives trom z = -~, reaches the point 0 and then turns back. Within a limited range, e.g., between 0, 0' the motion resembles that ot a pendulum proJected initially with an energy insufficient to carry it over the upper position ot unstable equilibrium.

5.

=V(ZI)

at a certain z a zl' The line % c hs is tangent to the curve of energy V(z) at the point, ot its maximum. Between the points 2 and " the motion approaching point 3 becomes asymptotic and point 3 is a saddle point. We note that as the energy constant ha approaches the value h a , the h

31 lIisland ll of trajectories, around the vortex point 2, gradually increases in size, the motion remaining periodic. When the value z = hs is reached the motion loses its periodiC character because the time of approach to the saddle point is infinite. We infer, therefore, that this singular trajectory on which the motion is asymptotic, the Beparatrix, limiting the size of the lIislands,1I is limited by the singular trajectories issuing from the saddle point 3. The same conclusion is also valid for the separatrices issuing from the other two asymptotes of the saddle point 3 and encloSing the vortex point 4 corresponding to another position of stable equilibrium. If the value of the energy constant hs is increased slightly, the saddle point disappears and the phase trajectory encircles the saddle point 3 and the vortex points 2 and 4 as shown by the dotted lines. If, however, the energy constant is decreased slightly, there appear two closed trajectories around the vortex points 2 and 4. We thus find two kinds of trajectories, typified by those in the upper and lower quadrants, Figure 4.1, on the one hand, and those in the left and right quadrants, on the other hand. Furthermore, we are now able to correlate radical changes occurring 1n the topology of phase trajectories with a critical value of a certain quantity h s . When the line z = hs is tangent to the energy curve at two consecutive maximum points, e.g., points 9 and 11 1n Figure 12.1a, the separatrix has the appearance of a chain enclOSing three vortex pOints 8, 10, and 12, and issuing from the asymptotes of two saddle pOints 9 and 11. For h = hs + ~h (~h > 0) the phase trajectory has the appearance shown in dotted lines, since the saddle points disappear in this case. For h = ha - ~ h three lIislands ll of periodiC trajectories will appear around each of the three vortex points 8, 10, and 12. We thus meet a situation which will be of great importance in what follows. We shall frequently use the expression lithe topological structure of the phase trajectories undergoes a qualitative change II for a critical value of a certain parameter whenever we encounter two entirely different aspects of the trajectories on both sides of this critical value. The value hs of the constant of energy h in the above examples illustrates this situation. In these examples the critical or IIbifurcation ll points at which these changes occur are aSSOCiated with certain values of the constant of energy appearing in the first integral of a dynamical system. In the following section we shall consider a still more important case when a certain parameter appears in the differential equation itself. From the elementary considerations developed in this section the following theorem is apparent. Inside a closed phase curve there is always an odd number of singular pOints, the number of vortex points being greater than the number of saddle pOints by one unit (2).

32 Before closing this qualitative study of the topology of trajectories in the phase plane it may be useful to mention the analogy existing between the separatrix and the divide of common topography.* In both cases a small change in the parameter, e.g., small displacement of the source across a theoretical divide, results in an entirely different topographic picture of the ultimate process. 13.

BEHAVIOR OF A CONSERVATIVE SYSTEM AS A FUNCTION OF A PARAMETER. CRITICAL VALUES OF A PARAMETER We propose now to review briefly the theory of Polncare (14) (15) regarding the dependence of the topologlcal structure of trajectories in the phase plane on a parameter in the dlfferential equation. The qualitative analysis of the trajectorles contained in the preceding section has already prepared the ground for a more rigorous quantitative analysls of this question. ConSider a dynamical system described by a differential equation depending on a parameter A. If the parameter A is varied, the solutions of the differential equation represented by the trajectories undergo variations. By definition we shall deSignate as ordinary values of A those values which correspond to trajectories belonging to the same family, as, for instance, was the case of closed trajectories belonglng to the same "island" consldered in the previous section. Therefore, in an interval of ordinary values of A a continuous variation of A corresponds to a continuous variation of the trajectories in the phase plane without any radical changes in their topological features. The values Ai of the parameter A, at which the topologlcal structure of the trajectorles changes abruptly are called critical values.** In dynamical problems it ls convenient to lntroduce a parameter A in the expresslon of potential energy V(X,A) and, hence, of the force f(x A) = _ oV(x, A) , ox

[13.1]

As shown in Section 11, the points of equilibrium are given by the extremal values of the potential energy, that ls, by equating the expression [13.1] to zero. This equation represents a curve M, in the (x,A)-plane, Figure 13.1, which, in general, may have points of self-intersection such as F. For a given value AO of the parameter the pOSitions of equilibrium Xl' x 2 • Xs are

*

The introduction of the separalriz and of a parameter in the study of the topology of the phase trajectories is due to Poincar6.

** The

term "bifurcation value" is also used.

33 x obtained as points A, B, C of intersection of M with the straight line A = AO' XI IfI'.hW''''''''' If the value of A is changed the posiXo tions of equilibrium change. It was x2~~~~-~~~~ shown in Section 11 that the stability of equilibrium depends on the sign of the derivative !z(Z,A) = - Vzz(Z,A).* ~--=------A >AI--.--....j For a minimum of potential energy Figure 13.1 (Vzz > OJ !z < 0), the equilibrium is stable; hence the point of equilibrium is a vortex point. Furthermore for !z > 0 the equilibrium is unstable. Thus pOints of equilibrium are given by the equation

!(X,A) =

0

and the criteria of stability by

!Z(X,A) ~ 0 Differentiating the equation !(X,A)

=0

[13.21

with respect to A we get [13.3]

If by increasing A we get a value AI' at which two points A and B coalesce~ the two positions of equilibrium XI and x 2 coalesce into the equilibrium point x O ' Point D on Curve M. For A > Al there remains only one point of equilibrium zs' Point E on Curve M. When A = AI' [13.1] has a simple root Zs and a double root zO' hence !z(ZO,A) = O. From [13.3) it follows that dz/dA =~, i.e., the tangent at D is vertical. The value of A = Al at which both equilibrium points Zl and Z2 coalesce and then disappear, is the critical value of the parameter. For the point F at which the curve intersects itself, the expression dz/dA is indeterminate. This implies that both !z and !A vanish. Poincare gives a Simple rule for ascertaining the stability of motion in the vicinity of a critical value of the parameter.

POINCARE'S THEOREM (14) (15) If the region in which !(Z,A) > 0 is below the curve !(x,A) = 0 for the positive values for X and A shown in Figure 13.1 the equilibrium is stable,

* In

the following the symbol I z designates tl~ partial derivative of I with respect to

%.

i.e., a vortex point; if it is above the curve the equilibrium is unstable, hence, a saddle pOint. In fact, assume that the shaded area in Figure 13.1 represents the region in which fCZ,A) > 0 and consider a pOint in the region below the curve and proceed in the direction of increasing z, with A fixed. In such a case f(Z,A) decreases and hence fz(z,A) < 0, that is, Vzz(x,A) > 0, which indicates the existence of a minimum of potential energy and, hence, stability. The oppOSite condition occurs when the region of f(x,A) > 0 is above the curve, i.e., instability. The theorem indicates that the branches FAD and GeE correspond to stability, while the branches GF and FBD correspond to instability. It is seen thus that the lI exchange of stabil1ties" Cllechange des stabiliti~s,1I according to the expression of Poincare) occurs at the critical values of the parameter. The points of equ1l1brlum in conservative systems always appear and disappear in pairs; the disappearance of equilibrium pOints in conservative systems always results in the coalescence of a vortex point with a saddle point. As has already been mentioned, after such coalescence no stable equilibrium exists. A few simple examples illustrating the topological method outlined in this and in the preceding sections are given in the following sections. MOTION OF AN ELASTICALLY CONSTRAINED CURRENT-CARRTING CONDUCTOR Consider an elastically constrained* conductor of length I and carrying a current i, attracted by a rigidly constrained conductor of infinite length carrying a current 1; let a be the distance between the wall ~ constraining i and the fixed current 1, x being the variable distance between ~and i. It is assumed that the electrodynamic action exerted on i is limited to the length, i.e., the current i is conveyed through flexible wires pp at right angles to i so that the electrodynamiC forces act along the length l only. Furthermore, since the conductors have a finite diameter, z < a,the total force acting on the conductor i is 14.

[14.1]

where A = 2Iil/k is a parameter; the first term kx is the force due to the elastic constraint, and the second kA/(a - x) is the electrodynamiC force according to the Biot-Savart law.

* As,

for example, bT the springs ss, Figure 14.1a.

35 For A = a 2 j4 one has a critical value of the parameter because both f(X,A) and fz(X,A) vanish. The differential equation of motion in this case is

m:': + k (x - a-x A ) == 0

[14.2]

It is equivalent to the system x == y; Y =

k m

ax a- x

x2 -

+

A

[14.3]

From [14.3] it follows that

dy dx ==

[14.4]

Equation [14.4] establishes the topology of the phase trajectories. There are two singular pOints, both located on the x-axis (y = 0), given by the roots Xl and x 2 of the equation X2 - ax + A = O. The coordinates of these points are (x It 0) and (x 2 , 0) with Xl = h and x 2 = + h, where h = A. The motion has a different character according as A ~ a 2 j4. The equality A = a 2 /4 corresponds to the critical value of the parameter.

t-

V1r -

2

1. A < a 2 /4. Both roots Xl and x 2 are real and positive. Substituting the values of Xl and x 2 into the expression of fz(X,A) = - Vzz(X,A), we have for Xl = ~ - b, fz(XI,A) < 0, Vzz(XI,A) > 0, and hence the potential energy is minimum; the equilibrium is, therefore, stable and the singular point is a vortex pOint. Similarly, for %2 = ~ + h, fz(X 2 ,A) > 0, V zz (X 2 ,A) < 0, and therefore the equilibrium is unstable; hence, this singular point is a saddle point S. From l14.4] it follows that as x approaches a, the slope dy/dx of the phase trajectories approaches infinity; hence the trajectories have the vertical line X = a as asymptote. In order to complete the topological picture we have to determine the separatrix. In this case the integral of energy exists. We obtain its value by integrating the dynamical equation

Multiplying this equation by

mx -f(X,A) =0 x = y and integrating,

2"1 my 2 + 2"1 k X 2 + k A log ( a -) x

[14.5]

we have == h

[14.6]

The equation of the separatrix is obtained if h is such that the separatrix passes through the saddle point y = 0, X2 = ~ + h. Substituting

these coordinates into [14.6), the constant of energy becomes

~ k (~ + b) + k A log (~ 2

h =

[14.7]

- b)

Hence, the equation of the separatrix is 1 2 k [2 2" my + 2" 2:

(a

2" +

-

)2] + k A log ~a -_

b

2:

b

=

0

[14.8]

2

Figure 14.1b gives the topological picture of the trajectories. It is seen that, if the initial conditions are such that the representative point

'Joe--- 0

y = i.

---I'"

IfI i

I I

t :I 1

I

t

o

I I

I

~..,..,...,..:~~U

.:t

(0) (b)

Figure 14.1 for t = to is within the area limited by the separatrix D, the motion is periodic around the vortex po1nt V. On the separatrix itself the motion is asymptotic. Outside the separatrix the motion is aperiodic; the phase traJectories exhibit a IIdipll above the saddle point and approach the line 2: = a asymptotically. 2. A > a 2/4. In this case there are no singular pOints, and hence no positions of equilibrium, since the equation 2: 2 - a2: + A = 0 has no real roots. The electrodynamic force exceeds the elastic force for all points of the phase plane. The phase trajectories are shown qualitatively on the right side of Figure 14.1b between the separatrix and the line 2: = a.

-

A = a 2 /4. As A approaches this value, b =V~ A approaches zerOj both singular points, assuming that the approach takes place from the region where A < a 2/4, approach each other and coalesce for A = a 2/4. The separatrix

3.

37 still exists but exhibits a cusp at the point VS of coalescence, Figure 14.2. The motion remains unstable. This case is similar to that considered in Section 11, Case 3, when the potential energy has an extremum value without being either maximum or minimum. 15.

y;x

o

RELATIVE MOTION OF A ROTATING PENDULUM

This problem may be considered as the generalized study of a centrifugal governor in the absence of friction. Let g be the angular velocity of the plane of the pendulum of length a; m the mass of the pendulum, and 6 the coordinate angle determining its position on the circle of its relative oscillaFigure 14.2 tion, Figure 15.1. The centrifugal force acting on the particle is mg 2 a sin 6, and its moment about the axis of the pendulum is mg 2 a 2 sin 6 cos 6. The moment due to gravity is mgasin 6. The resultant total moment is

where A = g/g2 a is a parameter. The Signs occurring in [15.1] are apparent from Figure 15.1. For greater generality we may consider negative values of A; they correspond to a purely theoretical case when 9 < O. The differential equation of relative motion is [15.2] where I = ma 2 is the moment of inertia of the pendulum. The equivalent system is mo Figure 15.1 whence dw

2

mg a 2 {cos 8 - A) sin8

d8 =

[15.4]

Iw

Equation [15.4] gives the phase trajectories in the (8,w)-plane. lar points are Wi

=

0, 8

=

0; w 2 = 0, 9 2

=

7r; wa = 0, 8 3

=

COS

-1

A

The singu[15·5]

The third singular point exists only lf A = g/D 2 a < 1, that ls, for a sufficiently large value of D. The force is given by equation !(8,A) = mD 2 a 2 (cos8 - A)sin8

[15.6]

The pOints of equllibrium are given by 1(6,A) = 0; that is, 6 = 0, 6 = ", 6 = - ", and cos 6 = A. Figure 15.2 represents the (6,A)-diagram for these values. The region in which 1(6,A) > 0 is shaded; Figure 15.2 the non-shaded regions correspond to 1(8,A) < O. The branches of the curves 6 = COS-lA, 8 = 0, 6 =", 6 = - ", situated above the regions for which 1(6,A) > 0, represent the loci of saddle pOints shown by circles.* The points (6 = 0, A = 1), (6 = ", A = - 1), (6 = - ", A = - 1) are the critical pOints at which there occurs the "exchange of stabilities" of Poincare. As regards the curve cos 8 = A, it represents loci of vortex points; this region exists only between the limits A = ± 1. The integral of energy exlsts in this case, because the system is conservative by our assumption. Its expression is l

'

2

• 2

6 - mD 2a 2(8m 6 -2-2-

+ Acos 6)

= h

[15·1]

Equation [15.1] expresses the law of conservation of energy. The equation of the separatrix is obtained from the condition that the phase trajectory passes through the saddle point. This condition is satisfied by the pOints 6 = ~ ", 6 = 0 which gives the equation of the separatrix, Curve A, Figure 15.3, 2

(oJ

2

= mD1 a

2

[. 2 810 8

+

2A ( cos 6

+

1)]

There exists a second separatrix (B, Figure 15.3) passing through the second saddle point 8 = 0, iJ = 0, for which h = - mAD 2a 2 j its equation is

[15.9] Figure 15.3 shows the topology of the phase trajectories for o < A < 1. It is observed that the vortex point 6 = 0 for D = 0 becomes a

* The

following convention w1ll be used: circles the unstable singularities.

black points designate the stable singularities and the

39

saddle point for Q* 0; at the same time there appear two vortex points VI and V2 symmetrically situated with respect to the pOint 8 = 9 = 0 around which periodic motions are possible. 6 The oscillations about these vortex pOints are asymmetrical with respect to the point 8 = 9 = 0; the coordinates of the vortex pOints are 8 = COS-lX, W = O. As soon as the constant of energy reaches the critical Figure 15.3 value and the separatrix B is crossed the motion becomes again periodic and symmetrical with respect to 8 = 9 = 0, but not uniform; the velocity w = 9 decreases for 8 = O. This range corresponds to the region of the phase plane situated between the separatrices A and B. When the energy constant reaches a second critical value at which the separatrix A is crossed, the w=6 motion becomes rotary, shown by the broken line in Figure 15.3, and non-uniform; the phase trajec9 tories pass outside the separatrix

A.

Figure 15.4

To sum up, in the case o < X < 1 three kinds of motions are possible:

a., b. Two periodic oscillatory motions, one around each vortex point (V I ,V 2 ) and one around the combination of one saddle pOint S and two adjoining vortex pOints VlJ V2 • c. One periodic motion around the external separatrix A, see Chapter VII. For X = 0, i.e., Q +~, both separatrices A and B coalesce. and the vortex points move to 8 = ~ "/2 positions. The topological picture of the trajectories is shown in Figure 15.4. For Ixi > 1 the topological picture also changes; there appear vortex points 8 = 0, ", - ", but the intermediate range disappears entirely. The values X = - 1, X = O. X = + 1 are thus the critical values of the parameter X.

CHAPTER

m

QUESTIONS OF STABILITY

16.

INTRODUCTORY REMARKS In the preceding chapter the question of the equilibrium of conservative systems was investigated, and certain defin1tions were formulated. The general problem of stability was studied by Liapounoff (4). In this chapter we shall give a brief outline of this study insofar as it is related to the problem of the stability of equilibrium, postponing the question of stationary motion to a later chapter. The study made so far is still incomplete since it concerns only periodic trajectories of a very special type, namely, those occurring in conservative systems. In Chapter IV we shall extend this study in connection with an important class of special periodic trajectories, which arise 1n non-conservative and non-linear systems. From the preceding analysis of a few simple motions, it appears that the trajectories of dynamical systems exhibit the following principal properties: a. They may approach singular pOints either for t = + ~, or t = -~, or for both. These pOints, as was shown, generally correspond to the pOints of equilibrium of a system. b. The trajectories may be closed and hence correspond to periodic motions. c. The trajectories may either go to infinity or arrive from infinity. We shall be interested primarily in the stationary states of dynamical systems, that is, in the above specified Cases a and b. As regards to Case c, the non-stationary motions, it is of relatively little interest 1n applications. In fact, when we say that a trajectory "goes to infinity" or "arrives from infinity," we mean that a physical phenomenon is encountered which cannot be studied entirely within the range of observation. The trajectories of Type a characterize motions in the neighborhood of equ1librium, and the problem of stab1lity 1n this case 1s that of equilibrium. The motions of Type b possess also certain other features of stability which will be defined later. Insofar as the stability of equilibrium is concerned, the preceding definition of stability for motions occurring in the neighborhood of focal, nodal, and saddle points does not present any difficulty. In fact, the representative point R approaches the focal and nodal pOints if they are stable

41 and leaves these points (or "approaches" them for t = - 00) if they are unstable. A saddle point is always unstable* in this sense. The usual definition of stability as an asymptotic approach to the equilibrium position ceases to be convenient, however, if we consider it in connection with motions around a vortex pOint. In fact, trajectories in this case do not approach this singularity although the potential energy is minimum for that point as required for a stable equilibrium. These considerations led Liapounoff to formulate a definition of stability sufficiently broad so as to be applied to both equilibrium and stationary motions. STABILITY IN THE SENSE OF LIAPOUNOFF We shall give first an intuitive geometrical definition of stability and supplement it by an analytic definition. Consider a closed trajectory C. The motion of the representative point R on this trajectory clearly represents a periodic motion. The geometrical formulation of stability of the motion in the sense of L1apounoff can be visualized as follows. Assume that during the motion of R on C the system has received an impulse translating R ., abruptly into R', a point which lies on a neighboring closed trajectory, C', as shown in Figure 17.1. Let us consider the "perturbed motion" of R' on C' in relation to the unperturbed one of R on C. If the initial distance Po = RR', originFigure 17.1 ally small, remains small throughout the subsequent motion, the motion is stable in the sense of L1apounoffj if, however, after a certain finite time, this condition ceases to be fulfilled, the motion is unstable. It is to be noted that this statement requires that the curves C and C' be close to each other. The proximity of curves C and C' is not sufficient to insure the stability of motion occurring on these curves unless the condition that the distance RR' between the representative points following the trajectories C and C' and considered at the same instant remains small for all value of t, be fulfilled. 17.

* Only

in a purely theoretical case when the motion of R takes place along the stable separatrix or asymptote, A1S in Figure 3.1, of a saddle point, the latter may be considered as a .table singularity. In Part IV we shall see that wtder special conditions R may follow a trajectory situated in the neighborhood of a stable separatrix and may give the impression for a limited time that a saddle point 1s a point of stable equilibrium. Hence one could better describe the stability of a saddle point by introducing the term almut "".tabl ... i"f1"larit".

42 This can be seen easily from the following example. Assume that the curves C and C' are two concentric circles whose radii differ by a small quanti ty .Ii r. If, for t = 0, the pOints R and R' are situated on the same radius, the distance RR' will be small. It follows that, if the motion is to be stable in the above sense, the pOints R and R' must remain on the same radius in order that the distance between these pOints remains small for all values of t. This implies that the two motions must be strictly isochronous. In this particular instance the motion is clearly that of a conservative system with two slightly different amplitudes. We conclude therefore, that in general the motions of conservative systems may be stable in the sense of L1apounoff only when they are strictly isochronous, that is, when the period of motion of R on C is exactly equal to that of R' on C'. In general the condition of isochronism is only approximately fulfilled in conservative systems so that such motions are generally unstable in the sense of Liapounoff for all values of t, although they may be considered as temporarily stable if, for a limited time, the motions are almost isochronous in some region. We shall .see later that there exist special periodic motions observed in non-linear systems which, on the contrary, exhibit stability in the sense of L1apounoff for all values of t. After these preliminary remarks we may now formulate the precise analytic definition of stability. Let ~ = z(t, ~o' Yo) and y = yet, ~o' Yo) be the coordinates in the (~, y)-plane determining a periodic motion on a trajectory C, with a period T. By this we mean that ~(O) = z(T) and y(O) = y(T). We shall call the motion stable in the sense of LiapounoU if, to any given number E

> 0,

another number 71 can be found such that 1%0 - ~ol

- ~(t, %0'

Yo>1 < J7ily(t, zo' Yo> -

y(t,

%0'

Yo>1 <

<

E,

IYo - Yol <

J7 for all t,

E

impUes I~(t. ~o,

Yo>

where 'io' Yo are the

slightly modified initial conditions ~o' Yo' When the trajectory C reduces to a pOint, L1apounoff's definition reduces to that of stability of equilibrium. This chapter will only deal with the stability of equilibrium.

18.

CANONICAL FORMS OF LINEAR EQUATIONS; CHARACTERISTIC EQUATION; CLASSIFICATION OF SINGULARITIES (POINCARE); BRANCH POINTS OF EQUILIBRIUM We shall now investigate the problem of stability of equilibrium more systematically by showing its relation to the nature of the singularities of dynamical equations. Let us consider the system of differential equations 2:

=

a2:

+ by;

11

=

C2:

+

dy

[18.11

where a, b, e, d are constants. In the phase plane this leads to the differential equation of the trajectories dy dx =

+

ex

dy

[1a.2]

a:c + by

A general non-linear system can be represented by equations

x=

ax

+ by + P'(x, y); iI =

ex

+

dy

+

[1a.3]

Q'(x, y)

with

!!1!. _

e:c + dy + Q'(:c,y) d:c - a:c + by + P'(:c,y)

[1a.4]

where P'(x,y), Q'(:c,y) are polynomials which have neither constant terms nor linear terms in x and y. Under relatively broad assumptions, it can be shown (16) that in the neighborhood of x = y = 0 the investigation of [18.4] can be reduced to that of [18.2]. By means of a linear transformation of variables ~

=

ell:!:

+ Py;

=

77

')':C

+

[18.5]

6y

with I~ ~I:I: 0* where a, p, ,)" 6 are suitable constants, we can reduce the system [18.1] to the canonical form ~

where SI and S2 are constants.

= SI~;

~

= S2 77

[18.6]

In fact, from Equations [18.5] one has [18.71

Substituting x and iI from Equations [18.1] and ~ and ~ from Equations [18.6] into [18.7], we have identically in x and y SI(a:c

+

py)

= a(a:c + by) +

S2(')'X

+

6 y) = ')' (ax

P(cx

+

dy)

6 (ex

+

dy)

[18.8]

+

by)

+

Identifying the coefficients of :c and y one obtains a(a - SI) + pc = 0; { a b + p (d - S 1) = 0;

{')'(a - S2)

')' b

+

+

6c

6 (d - S2)

=0 =

[18.9] 0

The first system contains a and p, the second,), and 6 considered as the unknowns. Non-trivial solutions exist only when SI and S2 are the roots of the

* This

implies a one-to-one correspondence of the transformation.

44 quadratic equation b la-8

C

d- 8

/=0

[18.10)

or written explicitly 82

-

S (a

+

d)

+

(ad -

[18.11)

c b) = 0

Equation [18.11) is called the characteristic equation of system [18.1). The two pairs of equations, [18.9), reduce now to two equations one of which determines the ratio alp and the other the ratio y/', assuming 8 1 ~ 8 2 and alp ~ Y/', that is ~I ~ O. Conversely, if ~I 0, then the roots 8 1 and 8. of [18.11) are unequal as seen from [18.9) unless 8 1 = 8 2 = a = d, b = c = 0, in which case the original system [18.11 is already in canonical form. Since we have assumed that ~ ~ :1= 0, [18.5) can be solved for 2: and , yielding the set of equations

I;

I; '"

I I

[18.12) where

~ = I~ ~I.

- (4 - 82)~'

If we choose a = - c~, p = (a - 8.)~, " = c~, 6 = Equations [18.91 are satisfied and therefore [18.12] can be

written as z = {S2 - aH

+

(SI -

a)11;

y

= -

c~ -

CTJ

[18.13)

Equations [18.13] will transform the system [18.1) into the canonical form

[18.6}. Thus far the system [18.1} was considered. By a similar procedure it can be shown that the more general system [18.3) can be reduced to a canonical form given by

[18.141

~

= 8 2 11

+

C{SI

~

S2)

[CP'(~'11) + (8 2

-

a)Q'{~'11)]

where 2: and y have been replaced by their expressions [18.13}. If the roots 8 1 and 8 2 of the characteristic equation are conjugate complex, then it can be shown, see Theorem 3 on page 45, that ~ and 11 are also conjugate complex. Letting ~ = u + iv, 11 = u - iv, 8 1 = a l + ib 1 , 8 2 = a l - ib l , and equating

45

real and imaginary parts of [18.14] one gets

u

= a u - b 1

1

11 -

...!.... Q' (u 11) 2c ' [18.15]

v

= btU

+

a 1"

+

1 2b C [CP'(U,lI) t

+

(at -

a)Q'(u,lI)]

These relations, which hold when the roots 8 1 and 8 2 of [18.11] are conjugate complex, can be obtained directly from [18.3] by an application of the transformation

[18.16] The latter result is obtained by substituting the values of ~ and ~ in terms of u and 11 1nto [18.13]. Once the possibility of the canonical transformation has been established we can proceed with the analysis of the various cases arising from the nature of the roots of the characteristic Equation [18.11]. We assume hereafter that 8 1 ~ 8 2 , We further assume 8 1 8 2 :1= 0, that is, I~ O. When I~ = 0, [18.2] reduces, if a '" 0, to the Simple equation

:1:1=

:1

dy acz + ady dz == a(az + by) = THEOREM 1. has a nodal point at

c

a

When the roots Stand S 2 are real and of the same Sign, the system [18.1] %

= Y = O.

From the canonical Equations [18.6] we get

[18·17] Separating the variables and integrating we have ~ = C~

Q

[18.18]

where a = 8 2 /8 1 > 0 and C is a constant. The phase trajectories are parabolic curves. For a > " the curves are tangent to the ~-axis at the origin of coordinates (~ = ~ = 0) except for the singular curve ~ = 0 corresponding to C.~. For a < 1 the curves are tangent to the ~-axis, except for the singular curve ~ = 0, corresponding to C = O. If 8 1 and 8 2 are both negative, it follows from the canonical Equations [18.6] that the representative point in the phase plane approaches asymptotically the point ~ = ~ = 0 for t • + ~. We have then a stable nodal point. If 8 1 and 8 2 are both positive the nodal point is unstable. Passing from the (~,~) to the (z,v)-coordinates one finds

46 the topological picture of trajectories shown in Figure 6.1, already established by an elementary procedure. There is an infinity of trajectories either approaching or leaving a nodal point. THEOREM 2.

When the roots 8 1 and 8 2 are real but of opposite signs, the system

[18.1] has a saddle point at

:I: ::

y =

o.

In this case dl1 = _ a!1.

where a

I I> O.

= ~

~

~

[18.19]

Integrating, we have 11 =

C~-a

(18.20 ]

The phase trajectories are hyperbolic curves referred to the axes as asymptotes. The representative point (assuming SI > 0 and 8 2 < 0) being placed close to the ~-axis, will move away from the origin as follows from the first Equation [18.6]. However, if it is placed near the l1-axis it will first approach the origin, as follows from the second Equation (18.6], and then move away from the origin along the branch approaching the ~-axis asymptotically as shown in Figure 4.1. The picture is thus typical for a saddle point as previously defined. There are only two singular trajectories, the asymptotes, which pass through a saddle point. In the (x,v)-plane the trajectories in the neighborhood of a saddle point are deformed and the asymptotes are inclined to each other by an angle different from "/2 in general. When the roots 8 1 and 8 2 are conjugate complex the system [18.1] has a focal point at :t ::;: y = O. THEOREM 3.

For real x and V, we shall show that, when 8 1 and 8 2 are conjugate complex, ~ and 11 must be also conjugate complex. Putting ~ = u + iv and 11 = u - iv, [18.6] become

dut d

+ '1. Cit dv =

8 l'iI: = ( a 1

.) + t• b1)( U + tV [18.21 ]

That is [ 18.22]

Thus, the velocities u and v are real in the new coordinates u and v. phase plane (u,v) the differential equation of trajectories is

dv du

b1u =

+

alV

a1u - b1v

In the

(18.23)

47 Transforming into polar coordinates, u

c

r cos rp, v

=

r sin rp, we find [ 18.24]

~9

whence r = Ce 61 Thus, 1n the phase plane (u,v) the trajectories are logarithmic spirals and the point, u = v = 0, is an asymptotic point, the focal point of trajectories. Multiplying the first Equation [18.22] by u, the second by v, adding and letting u 2 + v 2 = p, we find 1 dp

"2 Cit

[18.25 ]

= alP

For a l < 0 the representative point approaches asymptotically the focal point which is thus a stable focal point. For a 1 > 0 one has an unstable focal point. If one now passes from the (u,v)plane to the original (x.y)-plane. the general nature of motion remains the same but the spirals are distorted in the (x,y)plane as shown in Figure 5.1. A Summing up these results, the following criteria can be given: If SI and S2 are real and negative one has a stable nodal pOint, 1.

Focal

2.

If SI and S2 are real and positive one has an unstable nodal polnt. If SI and S2 are real and of

Figure 18.1

opposite slgn one has a saddle point, .4. If SI and S2 are conjugate complex with R*[SI,2J < 0, one has a stable focal polnt,

5.

If SI and S2 are conjugate complex with R[SI,2J > 0, one has an unstable focal point. These various cases can be represented graphically as shown in Figure 18.1 by puttlng P = - (a + d). q = ~I; the characteristic Equation [18.11] becomes then

I:

S2 + pS + q

it

R

represents the real part of 8 1,2'

= 0

[ 18.26]

48 If one takes p and q as rectangular coordinates, the zones of distribution of the roots in Cases 1 to 5 are shown in Figure 18.1. For q < 0, i.e., ad - cb < 0, the roots SI and S2 are real and of opposite signs; hence the region to the left of the axis of ordinates, q = 0, represents the zone of saddle points. The parabola pI - 4q = 0 separates the half plane, q > 0, into four regions: 1. P > 0, pI < 4q; the roots in this case are conjugate complex with R[SI,a] < 0, hence this region (vertical shading) corresponds to the stable focal pOints. p > 0, p2 > 4q; the roots are real and negative; this zone (oblique 2. shading) corresponds to the stable nodal points.

3. p < 0, p2 < 4q; the roots are conjugate complex with R[SI,Z] > 0, hence, the unstable focal pOints (double shading). p < 0, p2 > 4q; both roots are real and positive; hence this region 4. (horizontal shading) corresponds to the unstable nodal points.

If one traces a rectangular circuit ABeD surroUnding the origin with a positive direction indicated by the arrow and proceeds from A to B, one passes from the domain of saddle points to that of stable nodal pOints and from there to stable focal points; proceed1ng along BC the stable focal points become unstable; the branch CD is inverse to AB. The origin 0 appears, thus, as a branch point of equ11ibrium, insofar as several kinds of equilibrium exist in its neighborhood. It is seen that the nodal points are the singular1ties intermediate between the saddle points (always unstable) and the focal pOints. If the latter are stable, the intermediate nodal region is also stable; if the focal points are unstable, the intermediate nodal region is unstable. The threshold p2 - 4q = 0 corresponds to the transition between nodal and focal pOints, hence between the aperiodic motions and oscillatory damped motions.

19. STABILITY OF EQUILIBRIUM ACCORDING TO LIAPOUNOFF Consider a system with one degree of freedom ~ in the neighborhood ot equilibrium ~ = ~o corresponding to t = to' According to Liapounott, the equilibrium is called stable it, tor any arbitrary small number E,one can determine another number' = '(E) such that

I~(t) - ~ol <

E

tor to ~ t < + 00 provided I~(to) - ~ol

<,

which means that it the original departure I~(to) - ~ol ot the system from the position of equilibrium ~o is small, the subsequent departures I~(t) - ~ol

49 from that position will also remain small in the course of time. We shall apply Liapounoff1s criterion to a simple case which will serve to illustrate its physical significance. Since in this particular case we are interested primarily in 8mall deviations from the position of equilibrium, x = xo' let x = Xo +~, I~I being small. Let the differential equation of motion be dx = dt

-

f(x)

where f(x) is assumed to be analytic at the point x = xo' Substituting for x its value and developing f(x o + ~) in a Taylor series we have [19.2}

By virtue of the assumed equilibrium for x = x o' Xo is a singular point, and therefore f(x o } = O. Equation [19.2] thus reduces to

d;

-=a~+a~

dt

l'i

2 'i

2

+ ...

[19·3]

where

The method of Liapounoff consists of the following procedure: the non-linear terms in ~ in [19.3] one has d~

(it = a 1 ;

Upon dropping

[19. 4 1

Liapounoff1s theorem states that under certain conditions which will be specified later and which are frequently encountered in physical problems, the information obtained from the linear equations of the first approximation is sufficient to give a correct answer to the question of stability of the non-linear system. The solution of [19.4] is ~ = CeQ,c where a 1 = f·(xo). Hence, according to Liapounoff, if a 1 < 0 the equilibrium is stable; if a 1 > 0 it is unstable; finally, if a 1 = 0 the equation of the first approximation is not applicable and a special investigation is needed. Liapounoff also indicates additional criteria required for the investigation of stability when al = O. For the present we shall limit ourselves to the more general case when a 1 ¢ 0 which occurs frequently in physical problems. The proof of the Liapounoff theorem in the more general case of Equations [3.4] is given in Section 20. In the case considered here, the

50 proof is very simple and will enable us to exhibit Liapounoff's line of argument. If one multiplies both sides of [19.2) by ~, one has

d (~2) "21 ~

= al~

2

+

a2~

3

+ ... ==

One notes that FCO) = F'CO) = 0 and F"CO) = 2al'

F

( )

s

[19.5)

This gives

dp = ...i:....F .. (9~) dt 1· 2

[19.6)

where p = ~ t 2 and 0 < 9 < 1. In view of the assumed continuity of F"U), it is apparent that ifF"{O) < 0, that is, if al < 0, F"(9t) < 0 for smalll~l, and therefore dp/dt < O. Hence, if p = !t2decreases initially, it will continue to decrease, since the right-hand term of [19.6) is negative in the neighborhood of z = zo' and, therefore the equilibrium is stable. Likewise, i f F"(O) > 0 by a similar argument, one finds that the equilibrium is unstable. It is seen from this particular example that the equation of the first approximation gives a correct answer to the question of stability of the non-linear Equation [19.5], the only limitation of the procedure being the condition that a l rI: O. One can also illustrate the preceding considerations graphically. Consider the diagrams shown in Figure 19.1 in which % = fCz) is plotted against z. Figure 19.1a shows a monz otonically decreasing fCx); furtherM more, f(xo) = O. In this case f(x) > t to the left of Zo and fCx) < 0 to the right of xo. Liapounoff's method consists in replacing the curve MXoN in (0) the neighborhood of x o' by the tangent z at xo, that is by the function N a l (x - x o ), provided a l O. The direction of velocity dp/dt is indicated on the lower part of Figure 19.1a. It (b) is seen that this velocity is positive, i.e., directed to the right, for x < Zl and negative, i.e., to the left, for x > Zo which clearly indicates a stablE equilibrium at xo' I Likewise, when f(x) is inI P'o (e) creasing, Figure 19.1b, (from negative values for x < Xo to positive ones for Figure 19.1

*

_e_

...

51

x > x o·), one concludes, by similar reasoning, that the equilibrium is unstable. When f(x) has a minimum for x = Xo with f(x o) = 0, the question of equilibrium becomes more complicated. In fact, the system approaches x = Xo for values x < Xo as a point of stable equilibrium; for x > Xo it moves away from the point x = Xo as from a point of unstable equilibrium. One can designate this case, shown in Figure 19.1c, as a half-stable equilibrium, i.e., stable for x < Xo and unstable for x > xo' This half-stable case corresponds to at = 0, in which case the equation of the first approximation ceases to be applicable, and the consideration of higher order terms becomes necessary. 20.

LIAPOUNOFF 's THEOREM

In physical problems the Liapounoff theorem is generally encountered in connection with the dynamical systems :: =

P(X,lI);

~

=

[20.1 ]

Q(X,lI)

which are represented in the phase plane by trajectories, and the formulation of the Liapounoff theory depends then on the analytical form of Equations [20.1). The differential equation of phase trajectories is of the form

dll Q(X,lI) dx = P(x,y)

(20.2)

The positions of equilibrium as was shown are identified with the singular points. The preceding analysis was made under the assumption of a linear approximation in the neighborhood of x = y = 0 when P and Q are of the form: P = ax + by; Q = ex + dYe We have to consider now a more general case, that is, Equations [18.3) and [18.4). A point (x o' Yo) of equilibrium is clearly a point of intersection of the curves P(x,y) = 0, Q(x,y) = O. In order to be able to analyze the stability of equilibrium it is necessary to give a small departure (~,~) from the equilibrium point; the new coordinates are now x = Xo + ~, y = Yo +~. To simplify the procedure, the origin of coordinates can be transferred to the point (xo,Yo); furthermore, P(x,y) and Q(x,y) can be expanded in a Taylor series. The differential Equations [18.3) become

~;

:i

= a~ =

+ b~ +

[p

II

~2 +

2 P 12

~~ +

e ~ + d ~ + [q II ~ 2 + 2 q 12 ~

~ +

P 22 ~ 2

+ . . .]

q 22 ~ 2 + ..

OPERATIONS RESEARCH, INC. LIBRARY

·1

[20.3)

52 where the non-written terms of Taylor's expansions are at least of third degree in~, 17 and a = P%(Xod/o); b = P~(xo,Yo); c = Q%(xo,yo); d = Q~(xo'yo)' The theorem of Liapouno1'f states: If the real parts of the roots of the characteristic equation corresponding to the equations of the first approximation are different from zero, the equations of the first approximation always give a correct answer to the question of stability of a non-linear system.

More specifically, if the real parts of these roots are negative, the eQuilibrium is stable; if at least one root has a positive real part, the equilibrium is unstable. The starting point for this proof is to reduce the non-linear system to a canonical form by making use of [18.13] and [18.14]. Thus we have du

dt d11

dT

= S1 U

+

(

,

2

(q;1 U

2

Pn U

+

P:2 U11

'

2)

+

P22 11

+

' V2) Q22

+ [20.4]

= S2 11

+

+

q;2 UV

+

where P;1' ••• , q ~2 are the new constants. Multiplying the first Equation [20.4] by 11. and the second by v and adding, one obtains [20.5] where P = 11. 2 + 112. Let us now investigate the behavior of the curve ~(u,v) = o in the neighborhood of the point u = v = 0 for different forms of the roots of Equation [18.10] and thus establish the above stated theorem. 1. Consider first the case when 8 1 and 8 2 are both real and negative. In this case the surface z = ~(u,v) has a maximum z = 0 at the origin and the curve ~(u,v) = 0 reduces to one paint, u = v = O. Write ~(U,11)

= S1U2 + S2V2 +

~(U,11)

whence dp "21 Cit =

S1 U

2

+

S2 11

2

+

(

~ u,v

)

[20.6]

We can find a region S, Figure 20.1, around the point u = v = 0, where 1~(u,v)1 < ~181U2 + 8 2 v 2 1, so that ~(u,v) < 0 in S with the exception of the point 11. = v = 0, where ~(O,O) = O. Moreover, for all points (u,v) in S

"2IldPI>ll( CIT = 2" SI U 2 +

) S2 V 21

[20.71

Let 6 be a circular region situated inside S. If the point R is initially placed inside 6 it is easy to show it will never cross the boundary of 6.

53 v

u

Figure 20.1

Figure 20.2

In fact, for all points of 6, with the exception of the origin, dp/dt < 0 so that lim p = Po ~ 0 necessarily exists and is reached decreasingly. If I .... Po > 0, then for t ~ to sufficiently large, R remains inside S but outside the circle with center at 0 and radius Po/2. It follows, if IS ll ~ IS 21, that ISlu2 + s2v21 ~ ISll (u 2 + v 2 ) ~ ISll Po2/4 for t ~ to' From [20.7), we have

.!.I2 ddp I => !.2 d.ls I>0 4 t

and lim pet) = ~wh1ch is impossible. I.'" Hence the equilibrium is stable.

1

I

t -L

to

We have thus shown that lim pet) I ....

= o.

2. When SI and S2 are both real and positive, the surface z = ~(u,v) has a minimum z = 0 at 0, and the curve ~(u,v) = 0 reduces to the pOint, u = v = O. Hence, there exists a region S around 0 in which ~(u,v) > 0 with the exception of point 0 for which ~(O,O) = O. As the region € we can take now a circle lying inside S, Figure 20.2. It can be shown that it is impossible to determine a region 6 such that if a point R is placed initially in 6 it would not reach the boundary of € after a finite time. Let us assume first that such a region 6 exists. Further, let R be placed in any point of 6 except at the origin. Since ~(u,v) = dp/dt > 0 inside S except at the origin, the distance OR increases monotonically as long as R remains in S. Let Po = U 0 2 + V 0 2 c OR for t = 0 and P, = 2 2 u , + V f • It is apparent that in the region between p = Po and p = P"~ the function ~(u,v) and hence dp/dt has a positive lower bound. It follows that R will move in the annular region between Po and P, with a non-zero velocity

54 and will reach the boundary of E after a finite time which is contrary to our assumption. Hence, it is impossible to specify such a region 6, and the equilibrium is therefore unstable.

I

3.

If 8 1 and 8 2 are real and of different signs, the surface % = ~(u,v) has a saddle-shaped extremum, and the curve ~(u,v) = 0 has a node with two distinct tangents at the origin. The region S has in this case two zones, say I and n II, in which ~(u,v) > 0 and two others, III and Figure 20.3 IV, in which ~(u,v) < 0 as shown in Figure 20.3. We propose to show that the equilibrium in this case is unstable. Before proceeding with the proof, it is first necessary to obtain the sign of d 2p/dt 2 • DIfferentiating Equation [20.5] and substituting du/dt and dv/dt from [20.4], one obtains III

[20.8] The surface % = ~l(U,v) has a minimum z = 0 at the origin; hence, there exists a region S' around 0 for which ~l(U,V) > 0, with the exception of the origin for which ~1(0,0) g o. Thus in this region d 2p/dt 2 > O. We now can give the proof of the instability of equilibrium in this case. For the region E take a circle situated inside of both regions, S' and S, and drawn from the origin as center. It will be shown that it is impossible to determine a region 6 containing the origin and such that if a point R is placed initially at any point of this region except the or1g1n, it would never reach the boundary of E. Assume that such a region 6 exists. Since 6 surrounds the origin, there exist pOints inside 6 for which ~(u,v) > 0, and hence dp/dt > o. If R is placed initially at such a point, it is clear that for this point d 2p/dt 2 > 0 for the region E is situated inside S', by assumption. R will start moving with an accelerated velocity and will reach the boundary of E in a finite time, as dp/dt is monotonically increasing. This is contrary to our assumption concerning the existence of such a region. Hence, it is impossible to determine a region 6 satisfying the above requirement, and the equilibrium is unstable. The same conclusions reached in Cases 1, 2, and 3 hold when we transform the (u,v)-plane into the (~,~)-plane. 4. Finally, when the roots of the characteristic equation are conjugate complex, that 1s 8 1 = a 1 + i6 1 and 8 2 = a 1 - i6 1 , then using a procedure

55 analogous to that given in Theorem 3 of Section 18, the non-linear system l20.4] can be reduced to

[20.9] where the non-written terms are at least of the second order in u l or VI' MUltip~ing the first Equation [20.9] by u I ' the second by VI' and adding, we have

[20.10] where p = UI 2 + V 1 2 , and the non-written terms are at least of the third degree. One can show as above that the existence of either maximum or minimum depends on the sign of a l . If al < 0, the equilibrium is stable; if a l > 0, it is unstable. This completes the proof of the theorem. The advantage of the Liapounoff theorem lies in the fact that it enables one to apply equivalent linear criteria of stabllity to essentially non-linear systems and establishes the conditions under which this equivalence is valid. If these conditions are fulfilled, the theorem gives a correct answer at once, if not, one is confronted generally with a more difficult problem. This occurs, for instance, at the point E in Figure 18.1 at which the circuit ABeD intersects the q-axis (p = 0). At this point the roots SI and S2 of the characteristic Equation [18.11] become purely imaginary, and the equations of the first approximation of Liapounoff cease to be applicable.* Omitting this exceptional case which generally corresponds to a branch point of equilibrium (Section 27) in a great majority of practical problems, the Liapounoff theorem yields the conditions of stability in a relatively Simple manner. An example is given in the following section.

21.

EQUILIBRIUM OF A CIRCUIT CONTAINING A NON-LINEAR CONDUCTOR (ELECTRIC ARC) Consider a circuit as shown in Figure 21.1 where A is an electric arc whose characteristic Va = ¢(i) is indicated in Figure 21.2. By Kirchhoff's laws, we have

v

= L!!i

dt

+

¢ (i); E = RI

+ V; 1 =

i

+c

dV dt

[21 .1 ]

It must be noted that, in the general theory of equilibrium, Liapounoff also considere in detail a series of particular cases which 11e outside the range of validity of equations of the first approximation. We shall encounter one such case later. It is impossiblQ, however, to give a full account of the Liapounoff theory here.

it

56

v R

L

c

Figure 21.1

Figure 21.2

Eliminating I between Equations [21.1), one has two equations dV

dT= For equilibrium (dV/dt

E - V - Ri di RC • dt

= di/dt = 0)

v -

I/I(i) L

[21 .2]

one has

[21 .3] From Figure 21.2 it follows that for the assumed form Va = I/I(i) of the characteristic there exists either three or one pOSition of equilibrium depending on the intersection of the line V = E - Ri with the curve Va' Let Vo and io be the coordinates of an equilibrium point and consider the small departures 11 and j from this point, so that the "disturbed values" of V and i are now V = Vo

+ v; i

= io

+j

[21 .4]

Developing the function I/I(i) 1n a Taylor series in the neighborhood of io we find [21 .5] Following Liapounoff1s method, the equations of the first approximation are obtained by substituting the values [21.4] and l21.5] into [21.2]. Upon canceling the steady-state terms, we find dv v dj 11 • P =-- -j. =--)-

dt

RC

C' dt

L

L

[21 .6]

where p = I/I'(i o ) is the tangent to the characteristic of the arc at the point io which has the dimension of resistance. The quantity I/I'(i o ) = p varies with io on account of non-linearity of the characteristic. On the branch Be, Figure 21.2, where the curve I/I(i) is falling, p < OJ this branch 1s commonly deSignated as the range of the negative resistance of the arc.

57

The characteristic equation of the system [21.6] is

[21.71 and its roots are S1,2 = -

L

+ RCp 2RCL

+

1

2RCL

YL 2

+ (RCp)2

- 2RLCp -

4LCR

2

[21.8]

The nature of the roots depends on four parameters R, L, C, and p. Since we are interested in the non-linear problem, involving the parameter p, three two-dimensional IIcross sectionsl! p (R,p), (L,p), and (C,p) of the fourdimensional manifold (R, L, C, p) of Stable parameters will be sufficient for Nodal Points this analysis. The parameters R, L, and C can have only positive values, whereas p may have both positive and negative values determined respectively by the branches AB and Be of the characteristic. 1. Diagram in the (R,p)-plane, Figure 21 .). The quantity under the square root in l21.8] can be written as

Figure 21.3

The condition for complex roots is therefore

[21 .9] This leads one to consider the following two equations L - RCp

+

2RYLC

=

0;

L - RCp - 2RVLC

=

0

[21.10]

These equations represent hyperbolas in the (R,p)-plane, shown by Curves 1 and 2 in Figure 21.3, corresponding respectively to the first and second Equations {21.10]. The p-axis is an asymptote for both hyperbolas. The lines p = 2 YilC and p = - 2li/c are asymptotic to Curves 1 and 2 respec ti vely • Thus, we see that the area between Hyperbolas 1 and 2 is the region of distribution of the complex roots of the characteristic equation, that is, the focal points. To distinguish between the stable and unstable focal points

58 we equate R[S] to zero and obtain L

+ Rep

= 0

[21.11 ]

Since R, L, and Care positive,Equation [21.11] implies that p < O. It is also to be noted thatCRp + L = 0 is a hyperbola having the axes Rand p as asymptotes; see Curve }, Figure 21.}. The region between Hyperbolas 2 and} to the right of the POintM, their intersection, is thus the zone of unstable focal pOints. Prom the definition of the latter it follows that, within this range, self-excitation of oscillation is possible because the oscillations increase on the divergent spiral issuing from an wlstable focal point. It is further observed that no unstable focal points, and hence no self-excitation of oscillations, are possible for p > O. This is, however, only a necessary condition. In fact, not for every negative value of p is self-excitation possible. For sufficiently large values of p < 0 which correspond to pOints in the region below Curve } both necessary and sufficient conditions of selfexcitation are satisfied. It is further seen that the limit of the range of saddle paints is determined by the straight line R + p = 0, Curve 4. For when R + P < 0, the roots of the characteristic equation are real and of opposite sign. It is seen thus that p < 0 as R is positive. Thus, the region below Curve 4 is the region of saddle paints and, hence, of instability. Physically this means that energy is supplied to the system from an external source at a rate greater than its rate of dissipation. Thus the phenomenon "runs away," that is, either the circuit is destroyed or the fuses are blown out. It is to be noted that Point M of intersection of Curves 2 and } has coordinates + YL/C and - YL/C, that is the Straight Line 4 passes through this point. This completes the picture; for as we have shown previously, the region of focal pOints is generally separated from that of saddle pOints by an intermediate region of nodal points. We infer, therefore, that the area between Curve 2 and Straight Line 4 is the zone of nodal pOints. The region of nodal points to the left of and above M corresponds to stable nodal paints; that to the right of and below Mcontains the unstable nodal paints. Point M is thus a point of hifurcation for the various types of equilibrium. Hence, in following a closed circuit ABeD surrounding the point M in the direction indicated, the transition of singularities in the order indicated in Figure 21.3, i.e., saddle paints + stable nodal points + stable focal pOints + unstable focal pOints + unstable nodal points + saddle pOints, can be found.

59 Diagram in the (L,p)-plane, Figure 21.4. The limits of the zone of distribution of complex roots, that is, of the focal pOints, is given by the equation 2.

L2

+ (RCp) 2

-

2RCLp - 4LCR 2

=

0

[21. 12]

that is [21.13] In the (L,p)-plane Curve 1 represents the two branches of Equation [21.13]. It passes through the origin and has a vertical tangent at this point. The line p = LICR is an asympp tote of this curve. Furthermore, the curve crosses the L-axis at the point L1 = 4R 2C and has a horizontal tangent passing through the point (- R, R 2C) •

The zone of distribution of .--Stable stable focal points is separated from Nodal Points that of the unstable focal pOints by Straight Line 2, given by the equation L + RCp = 0, since this equation determines the condition for which the real part of the complex roots will vanish. Figure 21.4 Furthermore since L, R, and C are positive, this condition is fulfilled for p < as def1ned by Ipl = LIRC; the slope of Line 2 1s thus tan p = - l/RC. Within the zone of the complex roots Straight Line 2 separates the region of stable focal pOints from that of the unstable ones; the self-excitation of oscillations is possible only in the latter region. One concludes, therefore, that self-excited oscillations are possible only:

°

a.

for negative p, i.e., falling characteristic of the arc, and

b.

for not too great values of the inductance L.

These facts are well known experimentally. The zone of saddle pOints is obtained, as in the first case (R,p) for - p > R. The straight line p = - R, shown as Line 3 in Figure 21 .2, is thus the threshold separating the region of saddle points from the other singularities. Finally, between the p-axis (L = 0), Straight Line 3 on the one hand and Curve 1 on the other hand, lies the region of nodal points. Following the diagram of Figure 21.4 one

60 discriminates easily between the stable and the unstable nodal points. Points O(p = L = 0) and M(p = - R, L = R 2C) are the branch pOints of equilibrium.

3.

Diagram in the (C,p)-plane, Figure 21.5. The procedure remains the same.

a. One finds the zone of the distribution of complex roots, i.e., of focal pOints. b. One finds the line separating the stable focal points from the unstable ones. This line is obtained again by equating to zero the real part of the complex roots. c. The line separating saddle points from other singularities is the same as in the other cases considered, viz: R + p = O. d. The wedges between the zones of distribution of saddle points on the one hand, and the focal points on the other, represent the zones of distribution of nodal paints, stable or unstable. The two-dimensional "cross sections" (R,p), (L,p), and (C,p) of the four-dimensional manifold (R, L, C, p) of parameters thus permit us to establish general criteria of stability of p equilibrium in a circuit of this kind. From elementary considerations of static equilibrium of a circuit conStable Nodal taining an arc it is known that there Points is, in general, either three or one Unsta ble equilibrium points as follows from the Focal Poi nts (V,i)-diagram of Figure 21.2 in which the series resistance R appears as the tangent of the angle a of the straight line intersecting the characteristic Va of the arc (tan a = R). We shan analyze the conditions of stability of equilibrium, in the case when three Figure 21.5 equilibrium points exist, at the Points 1, 2, and 3, respectively, in the light of the preceding study of the "cross sections" of the phenomenon. First, for Point 1, p > 0 as follows from all three diagrams, the singularities in this half-plane (p > 0) are stable. Point 1 is thus a position of stable equilibrium. Consider now Point 3 of equilibrium, Figure 21.2, p < O. Since in the (V, i)-diagram the resistances are represented by the tangents to the curves, it is seen that the condition of equilibrium at

61 this point is expressed by the fact that \p\ < R; that is, the absolute value of the tangent to the falling characteristic of the arc is less than the positive inclination of the ohmic drop line. There exists a rule, known as Kaufmann's criterion of stability (17) which states that, if at the point of equilibrium \ p \ < R, the equilibrium is stable; i f \ p \ > R, i t is unstable. This criterion was found useful for a majority of applications. Later we shall see that certain ambiguous cases were discovered in which this criterion gave results at variance with observations. The preceding study permits us, however, to avoid these ambiguities and gives a complete account of what happens in the various cases. In fact, from the diagram of Figure 21.3, for instance, it follows that within a certain range of p < 0, there is a zone of absolute stability (stable focal points). There is also a range of a conditional stability (unstable focal points); in this latter region the phenomenon is stable in the sense that it does not "run away," e.g., blowing fuses, but rather approaches a stationary periodic motion.* If we consider now Point 2 of equilibrium, Figure 21.2, the Kaufmann criterion states that the equilibrium is unstable. The above study indicates that absolute instability, i.e., the zone of saddle pOints, begins already for \p\ = R and sometimes a little earlier if there is a zone of unstable nodal points wedging in between the zone of saddle points and other singularities. Furthermore, from diagrams of Figures 21.3,21.4, and 21.5, it is apparent that for a given value of p < 0 the question of stability or instability at a given static point (R,p) is also influenced by the other two parameters Land C which appear dynamically in the process, that is, when a transient phenomenon occurs in the circuit. From this study it appears that the application of Liapounoff's theorem gives a rather broad approach to the investigation of problems of stability by taking into account both static and dynamic factors of equilibrium. In each particular problem one has to ascertain first which particular "cross section" of the manifold (R, p, L, C) is of importance and which can be neglected as not having an appreciable influence on the phenomenon.

* See

Chapter IV.

CHAPTER IV LIMIT CYCLES OF POINCARE

LIMIT CYCLES; DEFINITION; ANALYTICAL EXAMPLES It was shown that in conservative systems per10dic motions generally occur around the vortex pOints; the phase trajectories form a continuum of closed curves enclosing either one single vortex point or more generally, an odd number of vortex and saddle pOints, the number of the former exceeding the latter by one. In a more complicated topological picture shown 1n Figure 12.1 wh1ch is composed of several groups of vortex and saddle poInts, the continua of closed trajectories form lIislands" in the phase plane limIted by the separatrices connecting the adjoining saddle pOints. If there is one per10dic trajectory, there exists also an infinity of others depending on the initial conditionsj by varying these continuously one obtains the continuum of trajectories as long as they remain within the limits of the domain of per10dIc1ty determined by separatrices. We shall now consider motions of an entirely different type observed in autonomous non-linear and non-conservative systems. Let 22.

~:

= P(x,1/);

~

= Q(x,1/)

be the differential equations of the system, and assume that there exists a closed trajectory C In the (x,1/)-plane. Assume further that there exists also a non-closed trajectory C' represented by equations x = x(t), y c y(t) such that either for t = + ~ or for t = - ~ the representative p01nt R' movIng on C' approaches C. By this we mean that for any gIven number E > 0 one can find a value to with the property that any point [x(t), y(t») on C' is at a distance ~ E from some point on C eIther for t > to or for t < to' IntuItively this means that C' winds around C either from the inside or from the outside like the spirals of FIgure 24.1. If the closed curve C is approached in this manner by a trajectory C' we call C a limit cycle. A limit cycle C is called stable if it is approached by trajectorIes C' both from the inside and from the outside for t = +~; it is called unstable if it Is approached by the trajectories C' both from the inside and outside for t = -~, and half stable, or semi-stable, if the trajectories C' approach it from the outside for t = + ~ and from the inside for t = - ~ or v1ce versa. Limit cycles met with in practice have the property that they are approached not by merely one open trajectory C' but by every such trajectory C' originat1ng in a certain domain of the phase plane. This means also that, whatever the initial conditions provided they are represented by pOints of

the phase plane situated within a certain region, the ultimate motion approaches a definite periodic stationary motion represented by C either for t = ~ or for t = -~, independent of the initial conditions. With this property in mind we can also say that the ultimate periodic motion on a limit cycle does not depend on the initial ~onditions. It is thus seen that a pe~iodic motion of this kind differs radically from periodic motions of conservative systems around a vortex pOint in that the closed trajectories occur always in continuous families in the latter case, whereas in the former they are isolated. By this we mean that if C is one such closed trajectory of a limit-cycle type there 1s no other limit cycle distinct from C and differing very little from it. The question of the analytical existence of periodic motions for non-linear and non-conservative systems will be treated in Part II. In this chapter we shall give a qualitative geometrical theory of limit cycles which will provide a relatively simple illustration for the various phenomena of the limit-cycle type observed in physical problems. Andronow (18) was first to suggest that periodiC phenomena in non-linear and non-conservative systems can be described mathematically in terms of limit cycles which thus made it possible to establish a connection between these phenomena and the theory of Poincare developed for entirely different purposes. We shall now give a few examples illustrating the preceding definitions. 1.

Consider the following system of non-linear differential equations :: = y

+

Yx 2 :

y2

[1 -

(x

2

+

y2)]

[22.1 ]

In polar coordinates x = r cos 8, y = r sin 6, these equations become dx dt

=

y

+

-=- (1 r

_ r 2) ;

~ = _ x + Jl. (1 dt

r

_ r 2)

[22.2)

Multiplying the first Equation [22.2) by x, the second by y, adding and noting that xx + yy = rr we obtain

[22.3) Multiplying the first Equation [22.2) by y, the second by x, subtracting and noting that yx - xy = r 2 0 we have

iJ = 1

[22.4]

64 dr Furthermore, 1 _ r2

= 2(1 dr+

dr r) + 2(1 _ r) and on integrating one gets log 1

+r

1 - r

= 2t

+

10gA

where A= ~ ~ ~: is an integration constant, ro being the initial value of the radius vector. Thus 21

r=

Ae 21 Ae

-

1 1

[22.51

-~--

+

For t .. 00, r .. 1 both from the inside (ro < 1) and from the outside (ro > 1 ) of the circle r = 1. In view of the uniform rotation 8 = 1, the trajectories are spirals approaching the circle r = 1 both from the inside and the outside as shown in Figure 24.~. Equation [22.51 cannot be used when ro = O. From the inspection of the Equation [22.1] it is seen that the origin x = y = 0 is a singular point. From the preceding definitions it is apparent that the system [22.11 admits a stable limit cycle x 2 + y2 = 1 as a stationary solution. 2. As another example consider the following system of differential equations

[22.61

In polar coordinates this system becomes

T

dr2 - 1); 6 = - 1

=

[22.71

From the first equation we have

[22.8) 2

where A= ro ; 1 is a constant of integration. ro [22.81 can be written as 1

r = ---r:=====::= VI + IAI e 21

If ro < 1, then A < 0 and

[22.9]

It is seen that for t .. -00, r = 1. In other words the spiral trajectories unwind from the circle r = 1 inwards. For t .. +00, r = 0, that is the spiral trajectories approach the origin which is a singular point of Equations [22.61. In this particular case the singular point is stable as can be verified from

65 the equations of the first approximation.

For ro > 1, we have A > 0 so that [22.10]

For t + -~ this equation gives r = 1, that is, the spiral trajectories unwind from the circle r = 1 outwards. From the preceding definitions it follows that the circle r = 1 in this case is an unstable limit cycle.

3.

Consider now the following system·

[22." 1

In polar coordinates we obtain

= r(r2

dr dt

Setting r2

=u

we have du/dt

=

_ 1)2. dB • dt

1

[22.12)

= 2u(u - 1 )2. But

du du u(u - 1)2 = 7

-

du

u - 1

+

du

(u - 1)2

=

2dt

and therefore we obtain upon integrating 1 = logC u - 1

u

log u - 1

+ 2t

That is U

--:- e u - 1

Putting u - 1

=v

__1_ u-l

= Ce

2t

[22.13)

we obtain 1 (V

+

) -.! 2t 1 e v = Ce

(22.14]

We shall Investigate now the behavior of trajectories in the neighborhood r = 1 since for r = 1, dr/dt = O. If r = 1 - E, E being a small positive number, then v < 0, hence from [22.14] C < O. Thus as v + 0, i.e., r+ 1 from the inside of the circle r = 1, t + +~, which means that the circle r = 1 is a stable limit cycle for the spiral trajectories inside the circle. If, however, r ~ 1 + E, then v > 0, and hence C > O. For v + 0 (r + 1 from

* Communicated b,y Professor

G.D. Blrkhoff.

66

the outside of the circle) e2t +- 0, that is t +- - 00, which means that the circle r = 1 is an unstable limit cycle for the outside trajectories. Hence the system [22.11] admits a half-stable limit cycle as a stationary solution. 4. As an example showing the existence of a region of accumulation of limit cycles (16) consider the following system of differential equations dx dt

[22.15 ]

:~

= -

x

+

Jl(x

2

+ y2

- 1)y sin x2

+ ~2

_

1

In polar coordinates the equations of the phase trajectories are dr 1 dB = Jl r( r2 - 1) sin r2 _ 1 • r :1= 1

[22.16 ]

dr dB

=

O. r

=

1

Thus there exists an infinity of circles in the neighborhood of r = 1 which represent the solutions of Equations [22.16] corresponding to the zeros of sin 1/(r 2 - 1). Between two consecutive zeros, and, hence between the two corresponding circles, the trajectories are spirals connecting two adjoining circles; see Section 24. The value r = 1 is thus a region of accumulation of the periodic solutions. It is very easy to construct examples of limit cycles having all sorts of peculiarities by the use of polar coordinates. The above examples were obtained in this manner and the resulting equations were then transformed into rectangular coordinates; hence the apparent complications. It must be borne in mind that the unearthing of limit cycles known, or suspected, to exist in a given system is more difficult. 23.

PHYSICAL EXAMPLES Practically all self-excited oscillatory phenomena of Mechanics and Physics are governed by non-linear differential equations and illustrate limit cycles. In the past, the mathematical formulation of oscillatory phenomena has made use of some method of simplification. A typical simplification is the Method of Small Motions of Dynamics. The application of this method, as its name implies, is limited to small motions, generally studied in the vicinity of an equilibrium point. In the numerous phenomena of selfexcitation of electronic circuits, circuits containing non-ohmic conductors such as arcs, gaseous discharges, and the like, the ultimate stationary oscillation is generally limited by amplitudes which cannot be considered as small. Likewise, in a great majority of mechanical self-excited vibrations

or oscillations the phenomenon generally stabilizes itself in a range in which the non-linearity of the differential equations cannot be neglected. Such "linearized" differential equations generally leave open the question of the ultimate amplitude at which the self-excited phenomenon stabilizes itself. Thus, for example, in the investigation of a simple linearized* problem of the Froude1s pendulum, Section 8, we reach a conclusion that under certain assumed conditions the damping term is of the form - b~(b > 0). The conclusion, therefore, is that the general solution of Equation [8.1] is of the form tP c tPo e 6t sin (wt + a), which indicates that in the early stages of the motion the amplitude is gradually increasing. Actually the amplitudes of the subsequent oscillations do not increase indefinitely however, but reach a limit when the oscillation becomes stationary. This stationary oscillation is represented by a limit cycle. Likewise, in a thermionic generator in which a similar condition exists initially, the amplitude of oscillation eventually reaches a limit cycle. A linearization of this kind, while giving an indication as to what happens initially, does not give any information as to the final state of the stable oscillation. In other words, it does not permit the determination of the ultimate limit cycle towards which the initial process approaches asymptotically. Since limit cycles are not present in the linear systems, it is necessary to study problems which are essentially non-linear in character. From this preliminary survey, it appears that modern electronic circuits involving electron tubes, gaseous dischargeg, and similar non-linear conductors, offer numerous examples of the existence of limit cycles. Although there are many examples of mechanical OSCillations they have been given less attention inasmuch as they generally appear as undesirable parasitic phenomena. They occur whenever there is a so-called "closed-cycle effect," by which a certain "cause" produces an "effect" which tends to reinforce the original "cause," etc. The initial process is thus cumulative. However, 1n view of the non-linearity of the system for larger amplitudes of oscillat1ons, the stable amplitude of the stationary state generally approaches that of a limit cycle. Again, the initial conditions do not playa leading role. The final ampl1tude depends only on the parameters of the system, but not on the initial cond1tions. A commonly encountered mechanism in which limit cycles exist is an ordinary clock (19). In fact in a clock there is an OSCillatory damped system, eXCited by shocks twice per period from a source of external energy,

* The expression

IIl1Dearization" used here simp17 III80DS droppiDg the non-linear terIIIB froll the differential equation. A somewbat different meaning iB attached to thiB word b,y Kryloff ODd Bogoliubottj Bee Chapter XII, Part II.

68

e.g., weight or main spring, released by the escapement; the 1mpulses so released replenish the energy lost per half cycle of the oscillatory system and thus allow for the "closing" of the phase trajectories which then become limit cycles; see Part IV. In fact, if a clock is wound from rest, it is immaterial whether a small or a large impulse, e.g., shaking, is employed to start it; the ultimate operation of the clock, once it is started, is entirely independent of the init1al condition which has produced the starting. Perhaps it is not too great an exaggeration to say that the principal line of endeavor of non-linear mechanics at present is a search for limit cycles. These modern tendencies to consider the problem of self-excited oscillations as the problem of determining limit cycles seem to transcend even the domain of mathematical physics proper; in fact, attempts have been made to extend these new mathematical methods to the description of biological and statistical phenomena as well. Thus, for instance, Van der Pol and Van der Mark (20) gave a theory of the performance of the heart considered as a relaxation oscillation mechanism possessing a limit cycle. Likewise, V. Volterra (21) in his mathematical theory on the "Struggle for Life" gave examples in which l1mit cycles may exist. TOPOLOGY OF TRAJECTORIES IN THE PRESENCE OF SINGULARITIES AND LIMIT CYCLES Recalling the definitions of the various types of limit cycles as given in Section 22, it is seen that the trajectories in the neighborhood of stable, unstable, and half-stable l1mit cycles have the form shown in Figures 24.1, 24.2, and 24.3, respectively. Trajectories winding on (unwinding from) a limit cycle may either arrive from 1nf1n1ty (go to infinity) or may orig1nate (terminate) at singular pOints or other 11mit cycles. It will be shown that a necessary condition that a closed curve C 1n the phase plane be a trajectory 1s the existence of at least one point s1ngularity of a definite type inside it. The fact that 24.

c·

c Figure 24.1

Figure 24.2

69 the trajectories either depart from or approach singularities of the focal or nodal type makes it possible to use a rather pictorial language in describing certain topological relationships in the phase plane by considering these singularities either as "sources" or "sinks" for trajectories in their neighborhood. A stable singularity or a stable limit cycle considered from this viewpoint, is a Figure 24.3 "Sink" for trajectories which it approaches asymptotically; likewise, an unstable singularity or an unstable limit cycle appears as a "source" with respect to the trajectories from which it departs. In this manner a number of propOSitions concerning these relationships become almost self-evident. Very frequently a trajectory may approach a limit cycle for t +~ and approach a singular point for t + -~. Physically this means that the motion develops from a state of rest and passes ultimately to a stationary periodic state. This can be expressed in light of the above descriptive language by stating that for t + - ~ the trajectory starts f~om a point source and for t + + ~ it approaches a line sink. In the following we shall consider focal points. There 1s no essential difference in considering nodal pOints when the nature of the trajec~ories in the neighborhood of the singularities is taken into account. A point A on a trajectory C divides it into two h4Ij-tT4jectorie8. If the time origin is selected when the representative point is at the point A, these half-trajectories for t ~ 0 and for t ~ 0 describe the future and the past history of the system. The present condition is represented by the pOint A. If we are only interested in a physical phenomenon beginning at a certain instant and disregard its past history, the state of the system is then determined by the positive half-trajectory (t ~ 0) "originating" at the pOint A. Very frequently this viewpoint is useful when we deal with the impulsive eXCitation of a dynamical system in which case the representative pOint R is transferred discontinuously from one point of the phase plane to another, say A. Disregarding what occurred during the period of discontinuity, or prior to it, we can consider only the half-trajectory "originating" at A and study the behavior of the system from that moment. With this remark in mind we can say, for example, that to a set of initial conditions represented by the point A in Figure 24.1 there corresponds a half-trajectory winding onto the stable limit cycle C from the outside and to the initial conditions represented by the point B there corresponds a

70

" .... ---. .... "

half-trajectory winding on C from the inside • There may be, of course, more complicated / I \ situations when a number of other singulariI \ I \ ties are present but for our immediate pur\ pose it will be sufficient to consider the I I I I Simple case of one singularity surrounded by I I Fu \ one, or several, limit cycles. I \ C, Having in mind later applications , let us consider the following special case • .......... ....... --_ .... Co A point Singularity F~, which we shall assume Figure 24.4 to be an unstable focal pOint, is surrounded by several limit cycles, represented by closed curves shown as circles Co' Co', CI , etc., in Figure 24.4; the circles in full lines represent the stable limit cycles and those in broken lines, the unstable ones. The fact that we assume that the limit cycles are circles is not essential since we are primarily interested in the topology of traJectories in the various domains which we are now gOing to specify. In order to include the half-stable limit cycles, the following terminology is convenient. A limit cycle is inwardly or outwardly stable according to the side on which stability exists; Similarly, a limit cycle may be inwardly or outwardly unstable. A stable limit cycle in this terminology is one which is both inwardly and outwardly stable and an unstable limit cycle is both inwardly and outwardly unstable. We can now formulate the following theorem* given here without proof. In a succession of concentric limit cycles considered from the center outward, an outwardly stable limit cycle is followed by one that is inwardly unstable and a cycle which is outwardly unstable is followed by one that is inwardly stable. The point singularity at the center is to be considered as a degenerate limit cycle possessing only the outward stability (or instability). We shall consider first a few typical examples which stress the significance of this theorem. This will enable us to derive certain conclusions concerning more complicated cases which will be important later. One such case arises when a point singularity being originally unstable becomes stable ~r vice versa. Figure 24.4 exhibits an unstable singularity Fu surrounded by a stable limit cycle Co and a few other cycles (Co', CI ' unstable, CI stable, etc.). It is apparent that, since the state of rest Fu is unstable, a spiral /.

/' 0

"

,

,

C·>,,

* This

'

theorem is a particular case of a more general theorem formulated by I. Bendixson (5). One can also find the proof of the proposition tl~t two adjoining closed integral curves cannot both be stable in a recent paper by N. Levinson and O.K. Smith (22).

71 trajectory will originate at Fv and will approach Co which represents the state of the ultimate stable stationary oscillation on the limit cycle. It is readily seen, following this line of reasoning, that this state of stationary motion will be reached not only when the system starts from rest but also when it starts from any arbitrary initial conditions which are represented by a point of the phase plane inside the first unstable limit cycle Co' such as A. In this sense we can state that the ultimate motion on the limit cycle Co is independent of the initial conditions. It is apparent that the following stable cycle C l cannot be reached spontaneously by the system starting from rest since the oscillation cannot develop beyond Co on which it becomes stationary. However, in spite of this, the trajectories may approach the limit cycle Cl if they originate from the initial conditions represented by the points situated in the annular region between Co' and C l ' such as points Band D in Figure 24.4. It is thus seen that the unstable limit cycles Co', C l ' • • • constitute a kind of divide or IIbarrier ll for the initial conditions from which various stable limit cycles such as C]I C2 • • • can be reached by traj ectories • We are now in a position to formulate two definitions which will be important for the sequel. 1. A self-excitation of a system on a limit cycle Co is called soft if it can originate spontaneously from rest. 2. A self-excitation on a limit cycle C is called hard if it requires a certain finite disturbance, e.g., shock excitation, to transfer the initial conditions into the annular region between two consecutive unstable limit cycles in which C is situated. These definitions may also be formulated in the following manner. A stable limit cycle

Co

is said to induce a soft self-excitation, whenever it contains no other limit cycle and

Just one singular point, an unstable singularity. other cases

Co

In all

is said to induce a hard self-excitation.

This condition of IIhard ll selfexcitation is illustrated in Figure 24.5, in which the system is in stable equilibrium when at rest. If there exists a stable limit cycle Co, it follows from the theorem stated above that there must necessarily be an unstable limit cycle Co' between F. and Co. Therefore the system cannot become selfexcited and reach the stable limit cycle either from rest or from any initial conditions

c~

Figure 24.5

72

(a)

(b)

(c)

(d)

Figure 24.6 represented by a point A inside Co'. In fact, in the latter case, Figure 24.5, the trajectory starting from A will approach the state of rest at F •. The stable limit cycle Co can be approached however, if a trajectory originates either at B or at D, which represent points of the annular region between the divides Co' and CI ', that is, the system must be given initial conditions represented by any point in this annular region. In the following we shall be concerned with those cases in which the topology of the phase plane undergoes changes as a result of changes of a parameter in the differential equation. As we shall see these changes may be of different kinds. Thus, for example, the singularities may undergo a transition from stability to instability or vice versa, limit cycles may vary in size, a stable limit cycle may coalesce with an unstable one with the disappearance of both, limit cycles may shrink and coalesce with point singularities modifying the nature of the latter, etc. These various changes will be studied in later sections. In all cases, however complicated, we shall find that the theorem on limit cycles provides most useful information. As an example consider the following case which we shall encounter later. Assume that there is an unstable singularity Fw surrounded by a stable limit cycle Co as shown in Figure 24.6a. A trajectory will unwind from Fw and will approach Co; this corresponds to the case of a soft .selfexcitation. Assume now that, as the result of a variation of a parameter A in the differential equation, the singularity undergoes a transition from instability to stability and that the trajectory Co in the neighborhood of this limit cycle varies continuously, remaining a limit cycle. Then, by the foregoing theorem, an unstable limit cycle Co' must necessarily originate between Co and F. as shown in Figure 24.6b. This, however, in no way affects the stationary motion on Co, except, pOSSibly, that there may be a slight change in the "radius" of Co. A further variation of the parameter A may decrease the "radius ll of Co and increase that of Co', so that the two cycles approach each other,

73

as shown in Figure 24.6c, although the actual trajectory of the system continues to be Co. This is often the case in considering the non-linear characteristics encountered in practice. After the coalescence of these limit cycles a number of alternatives is possible with a continuous variation of A in the same sense. One possibility of frequent occurrence in physical problems is the case in which there exist no limit cycles beyond this stage. Everything then takes place as if the stable and the unstable limit cycles upon coalescing destroy each other. When this situation arises, the representative point approaches the stable focal point as shown in Figure 24.6d, so that the selfexcited process disappears gradually. The coalescence of limit cycles with point singularities will be investigated tn Section 29. It should be mentioned at this point that the question of coalescence of limit cycles remains relatively unexplored theoretically. Interesting illustrations of the above definitions and theorems can be obtained experimentally by means of a cathode-ray oscillograph arranged to record the phase trajectories of a non-linear process. It is recalled that the pattern traced by the luminous spot of a cathode-ray oscillograph is due to the alternating potentials impressed on two pairs of deflecting plates at right angles to each other. If, therefore, one pair of plates is subjected to a voltage proportional to a dynamical variable x, which may be the displacement in a mechanical oscillation or a current in a circuit, and the other pair of plates is acted on by a voltage proportional to the derivative x of that variable, it is clear that the luminous path traced by the electronic beam on the screen will give directly the phase trajectory of the process. There exist numerous, so-called differentiating circuits which give an electrical differentiation of this kind. We shall not go into a survey of these various schemes but will indicate as an example the essential points of a scheme due to Bowshewerow (23) who was one of the first (1935) to develop the experimental technique for the investigation of phase trajectories. Consider the c scheme shown in Figure 24.7 representing a thermionic generator with an adjustable inductive coupling LL'. The OSCillating circuit LC of the generator contains a relatively small resistor Figure 24.7

74 'R 2 in series and a rather large resistor Rl in parallel with L and C.

It is apparent first that the potential difference across R2 represents the oscillating current i and that across Rl the oscillating potential V, and since these quantities are approximately in quadrature with each other, it is apparent that the oscillograph M, controlled by these potential d1fferences,1s capable of record1ng the phase trajector1es of the process. By means of ad, ditiona1 deta11s not shown 1n F1gure 24.7, 1t was possible to start the phenomenon from a given point (i,V) of the phase plane and also to vary the parameter A of the process by chang1ng the coeff1cient of mutual inductance between the coils Land L'. A variable bias permits f1x1ng the equilibrium point at different pOints of the non-11near characteristic of the tube. Osci1lograms a, b, c, and d shown in Figure 24.8 represent the var10us cond1tions of a hard se1f-excitat10n recorded in this manner. Oscillogram a shows the disappearance of self-excitation by removing the coupling of the oscillator with the incident approach of the phase trajectory to a stable focal point. Oscillogram b shows a similar process but with a small amount of regeneration through a relatively weak coupling LL' below the critical value at which the energy 1nput 1nto the oscillating circuit becomes greater than its d1ssipation of energy. Oscillogram c represents osc111ation on the limit cycle. With the non-linear characterist1c employed it is seen that there are two limit cycles C2 and CI , the former being stable and the

(e)

Figure 24.8

75 latter unstable. The black pOint inside C 1 is a stable focal point. By transferring the initial conditions to the different pOints of the phase plane the stable limit cycles C1 and C2 can be reached in the manner just explained either from the inside of these cycles or from the outside. A phase trajectory approaching C2 from the outside has been recorded on the oscillogram. Oscillogram d shows the approach of phase trajectories to a stable limit cycle both from the outside (Curve C') and the inSide (Curve C"); the black spot in the middle is an unstable focal pOint. 25.

FURTHER PROPERTIES OF LIMIT CYCLES; INDICES OF POINCARE; THEOREMS OF BENDIXSON In the examples given in Section 22 the establishment of the existence of limit cycles was particularly Simple. Unfortunately, given a nonlinear differential equation, the problem of establishing the existence of a limit cycle, or cycles, is, generally, very difficult. There exist criteria which rule out limit cycles in certain cases. The method most frequently used for the establishment of the existence of limit cycles within a certain domain is based on the theorem of Bendixson formulated below. It is important to note, however, that Bendixson's theorem cannot be applied to systems with more than one degree of freedom; furthermore, even in such systems its application 1s frequently handigapped by the difficulty of determining the domain to w~ich it can be applied. Poincare has given a series of necessary criteria for the existence of limit CYCles, based on the theory of indices associated with closed curves. The whole situation when considered from the point of view of determining limit cycles can be best described in the words taken from the "Theory of Oscillations" by Andronow and Chaikin: liThe present status of the theory establ1shing the existence of limit cycles can be best compared to the game of. chess. There exists no theory by means of which a game can be won. There do exist, however, alternatives which enable a skilled partner to win a game starting from a given configuration on the chess board." We shall review now the three prinCipal methods available. A. Indices of POincare, B. Negative criterion of non-existence of closed trajectories of Bendlxson, C. A second theorem of Bendixson. There exists also a fourth method of the curve of contacts due to Poincare. We shall not go into this subject here but shall mention it in a later chapter where the use of the curve of contacts will be helpful.

76

A.

INDICES OF POINCARE Consider a closed curve C as shown in Figure 25.' in a vector field F.* Assume an arbitrary positive sense of rotation, say clockwise, and consider the moM tion of the point R on C in this positive ~~o::::::::aJr,x' direction. Let (x',y') be a frame of reference attached to R, the directions of which remain parallel to the axes x, y of a fixed coordinate system during the motion Figure 25.' of R on C. If the vector R~ with R as origin is drawn, so as to represent the vector of the field at R, this vector ~M will turn around R in the (x',y')system as the point R moves on C. When the circuit C is completed, the vector R~ will resume its original position in the (x',y')-system. Poincar~ calls the "index j" of a closed curve with respect to a vector field the algebraic number of complete revolutions of RM when R completes one circuit C. The index may be equal to zero, which means that RM executes only an oscillation in the (x",y')-system but not a complete rotation. By the index of a singularity is meant the index of a closed curve surrounding the singularity and lying in a ve6tor field determined by the phase trajectories. From this definition it follows that the index of a closed trajectory enclosing a vortex, a nodal, or a focal point is + " Figure 25.2a, b, and c, whereas that of a trajectory surrounding a saddle point is - " Figure 25.2d. Poincar~ has established a series of theorems which are given here without proof (2). Some of these theorems are obvious from geometrical consideraUons. y'

.. ..

c

(0)

(b)

Figure 25.2

* For

1,-- over a closed curve is zero. such a field the line integral ]Fda

77 1.

The index of a closed curve not containing singularities is zero.

2. The index of a closed curve containing several singularities is equal to the algebraic sum of their indices.

3.

The index of a closed curve which is a phase trajectory is always + 1. 4. The index of a closed curve, with respect to which the field vectors are directed either inwards or outwards at all pOints, is always + 1. These theorems lead to the following conclusions with regard to closed phase trajectories. 1. A closed trajectory contains in its interior at least one singularity the index of which is + 1. 2. A closed trajectory may contain several singularities in which case the algebraic sum of their indices is + 1. By virtue of the first conclusion, limit cycles may contain in their interior nodal or focal points, but not saddle pOints. By the second conclusion the number of enclosed singularities must always be odd, the number of singularities with index + 1 must exceed the number of saddle points by one unit. This theorem has already been established directly, see Section 12, from topological considerations. B.

FIRST THEOREM OF BENDIXSON (THE NEGATIVE CRITERION)

The first theorem of Bendixson establishes a condition for the non-6~i8t6nce of closed trajectories and, hence, for the impossibility of periodic motions. Let the motion in the phase plane be given by the equations

i =

P(~,y);

The theorem of Bendixson states:

~

=

Q(~,y)

[25.1 ]

If the expression fJP/fJz + 8Q/fJy does not change

Us sign within a domain D of the phase plane, no periodic motions can exist in that domain.

Consider a closed circuit in D and apply Gauss's Theorem

f (Pdy -

Q d:d =

ff(:=

+ :~) d~ dy

[25.2)

with respect to this circuit. If one assumes that the circuit is a phase trajectory satisfying [25.11. the line integral can be written I(:iydt - y:idt) which is equal to zero. As to the double integral, it can be zero only if, within the area limited by the closed circuit, the integrand (8P/8~+ fJQ/8y) changes its sign. This is contrary to the assumption; hence, the closed

78

curve cannot be a phase trajectory which implies that no periodic motion can exist in D. One can also state a series of additional criteria based on the negation of the theorems resulting from the theory of indices. No periodic motions and, hence, no limit cycles can exist in a system not having singularities. 1.

If a system has Just one singularity and if its index is not + 1, periodic motions are impossible. 2.

3.

In a system with several singularities the algebraiC sum of whose indices is different trom + 1, periodic motions on closed curves enclosing all these Singularities are impossible.

4.

In a system with Just one singularity having the index + 1, which is approached by trajectories gOing to infinity, periodic motions are impossible. C• SECOND THEOREM OF BENDIXSON (5) Let z(t), y(t) be the parametriC equations of a halt-trajectory C which remains for t + + ~ inside a finite domain D without approaching any singularity. The second theorem of Bendixson asserts that only two cases are then possible. 1.

Either C is itself a closed trajectory, or

2.

C approaches asymptotically a closed trajectory Co.

If, assuming for instance that there is only one singularity inside a closed curve CI , one succeeds in determining a domain D limited by two closed curves CI and C2 , as shown in Figure 25.3, within which the halttrajectory is confined and has no singularities either in D or on its boundaries, then the theorem asserts that there exists at least one stable limit cycle in D. An intuitive interpretation of this theorem on the basis of singularities and limit cycles considered as "sources" and "SinkS" is frequently useful. As an example consider Equations [22.3] and \ [22.4]. For r < 1, dr/dB> OJ tor 'r >\1,d~/dB < O. Choosing as the domain D an annular region l1m1ted by two concentriC circles r l = 1/2 and 'r 2 = 3/2, for ", example, drawn from the origin as center, one readily sees that D satisfies the condition of Bendixson Figure 25.3

79

since no singularities exist either in D or on its boundaries. Hence, there exists at least one limit cycle. In this particular case a direct proof of existence of a limit cycle is very simple, as appears from [22.5]. In applying the second theorem of Bendixson a difficulty often encountered consists in finding the appropriate domain D which would contain an entire half-trajectory and exclude the singularities. As an example of such a case, consider the Van der Pol equation

x-

Ii (1 -

:l: 2):i:

+

:l:

=

0

which, as is known, has a periodic solution. More specifically, for p « 1, this equation admits a closed trajectory differing very little from a circle of radius r = 2. If one attempts to apply the Bendixson theorem to a domain D limited by r 1 = 1 and r2 = 3, one easily finds that the theorem fails to indicate that the trajectories cannot leave the domain D by crossing the boundary of the circle. If, however, one succeeds in establishing a domain D for which the conditions specified by Bendixson's theorem hold, then one is certain that at least one periodic solution exists. Thus, for example, in a later chapter we shall encounter the system

[25.4]

~~

= A

+

a2:

+ yO -

r2)

where A is a constant and r2 = :I: 2 + y2. It is apparent that, for a suffiCiently large r 2 , the trajectories are directed inwards. Furthermore, by transferring the origin to the singular point by a change of variables :I: = :1:0 + f, Y = Yo + ~, the system [25.4] is reduced to that of the form

df dt

=

~ _ a~ + ...

[25.5] dJ7 = dt

a~

+

71

+ ...

It is thus seen that the origin is an unstable focal point and since no other singularities exist, the Bendixson theorem shows that there exists a stable periodiC solution contained in a finite domain.

26.

PARALLEL OPERATION OF SERIES GENERATORS

As an example of the application of the Bendixson criterion of nonexistence of closed trajectories consider two series generators connected in

80

parallel with respect to the external circuitR, as shown in Figure 26.1. Selecting as the positive direction for the currents i l and i2 the direction in which they contribute to the external load and designating the characteristics of series generators by e = ~(i), one obtains, by Kirchhoff's laws, the following two differential equations applied to the circuits (il,i) and (i 2 ,i).

R

Figure 26.1

~(il)-(r+R)il-Ri2-L~il

=0

[26. , ]

~ (ill)

- (r

+ R)i2 - Ril - L ~it

= 0

Divid1ng these equations, one f1nds di2 ~(i2) - (r + R)i2 - Ril di 1 = ~ (i l ) - (r + R)il - Ri2

[26.2]

The variables il and i2 determine the trajectories 1n the phase plane. appl1cation of the first theorem of Bendixson gives

The

[26.3] If the cond1tion of self-excitation is fulfilled, one has always ~i(i) (r + R) > OJ* hence, no period1c motions are possible. The condit10n of equilibrium is obtained by setting the numerator and denominator in Equation [26.2] equal to zero and by finding the pOints of intersection of the resulting curves in the (i l ,i 2 )-plane. One can also investigate the system of two linear equations

[26.4]

.. In fact, this inequality means that the initial supply of energy must be greater than its dissipation, see Section 21.

81 by forming the equations of the first approximation, that is by developing ¢(i) in Taylor's series and neglecting the terms of higher orders. Writing d¢( i) ( di ),=0 = (¢,),=o = p, one has (p- r -

R)i l

Ldi 1 -Ri2 dt

-

=

0

[26.5] - Ril

+

(p -

R}i2 - L di2 = 0 dt

r -

Putting 6 = d/dt. this can be written as (p - r - R - L6) i

l

-

Ri2 = 0

[26.6] - Ril

+

(p -

r -

R - L6) i2 = 0

The system [26.6] admits solutions other than the trivial ones i1 if the determinant is zero.

jp

- r - R - L6

That is, p - r - R - L6 tion are

- R

= ±R;

1-

- R 0 p - r - R - L6 -

= i2 = O. [26.7]

whence the roots of the characteristic equa-

[26.8] In practice generally p - r > 0; thus '2 > O. As regards 61 , it may be either positive or negative. In the first case there is an unstable nodal point and in the second, a saddle point. Thus, for the assumed conditions. that is, when i1 and i z add up with respect to the external circuit, R the process is unstable and therefore cannot exist in a steady state. In i : i. + is! fact. it is well known that a paralFigure 26.2 lel connection of two series machines leads to an unstable parasitic performance in which one machine generates energy and the other absorbs it so that no energy flows into the external circuit. If the field connections are crossed as shown in Figure 26.2, instead

---

82 of Equation [26.1] one has

[26.9]

I/JCi 2) -

(r

+

R)i 1

-

Ri2 - L

~il

= 0

The roots of the characteristic equation are [26.'0] If p< r + 2R, both roots are real and negative, which corresponds to a stable nodal point and, hence, to equilibrium. The machines have a stable performance, delivering energy into the external circuit. If p > r + 2R, there is a saddle point and, hence, a loss of stability. The threshold condition p = r + 2R is, thus, a branch point of eq~ilibrium corresponding to the transition from a stable nodal point to a saddle point, which is in accordance with the sequence of the zones of singularities shown in Figure

,a.'.

21. STABILITY OF PERIODIC MOTION We shall now investigate the problem of stability of periodic motion, a question which was left open in Chapter III. For this purpose a few additional theorems will be useful. Let dx

dt

= P(x,y) and

~

dt

_ - Q(x, y)

[27. , ]

be the differential equations of a dynamical system. Having in mind the representation of motion of a system with one degree of freedom in the phase plane we shall limit the number of these equations to two. The argument applies to any number of such equations. Assume that we know a non-constant periodic solution of [27.1]

-

[21.2]

We shall be interested in the properties of a neighboring perturbed solution [27.3) where ~(t) and ~(t) are the functions determining the perturbation. It will be assumed that the quantities I~I and I~I are sufficiently small so that we may neglect ~2, ~2, ~3,. " . Substituting the expressions [27.3] into the differential equations and expanding the functions in a Taylor series in the

neighborhood of Xl and Yl' one obtains the variational equations (3)

d17 Cit

+

= Qz(xl' Yl) ~

Q,(x p YI) 17

This is a system of linear equations of the type

~i

[27.5] = c(tH

+ d(t)17

where a( t), ••• , d( t) are periodic functions of t with a common period T. It is known (24) that a system of the type [27.5] admits a fundamental system of solutions of the form ~ 1 -_

'i

e 11 1'111 (t ).• 111 = e "2 'I 12 (t )

where 1111 . . . , 122 are periodic functions of time with the period T, and hi and h2 are certain constants, real or complex, which are determined only to integral multiples of 2~iT. These constants are called the characteristic exponent8 of the system. The characteristic exponents satisfy the relation

[27.71 that is, the sum of the characteristic exponents is the average of the sum of the diagonal coefficients of [27.5] taken over the period T. This can be proved as follows. Substituting the expressions [27.6] into the equations [27.5] and solving for a and d we obtain a = el112 - e2111 ; ~1112 -

~2171

d = ~1i]2 -'~2i]1 ~1712

~2711

-

The denominator of these expressions, upon substitution of the values for ~ I' ••• , 112 obtained from [27.6], becomes ~1112-~217I=e

[11 1 +11 2

]t[

]

111/22-/12/21 =e

where 6 is a periodic function with period T.

[" 1 +"2)'

6

[27.9]

Addlng the expressions [27.8]

84 one obtains after simple

calcul~tions

Averaging this expression over the period T one has T

hl+h2=

T

~J(hl+h2)dt= ;J(a+d)dt o

0

T .

'!'J'!dt To

6

Since the last term on the right side of this expression has period Tand hence vanishes, one obtains the expression [27.7). It should be noted further that if the system [27.5] admits a periodic solution with period Tat least one of the characteristic exponents is zero (3). For suppose t(t) = Cl~l(t) + C2~2(t); ~(t) = CI~I(t) + C2~2(t) is periodic and not identically equal to zero, that is, t(t + T) = ~(t); ~(t + T) = ~(t). From [27.6] one obtains easily

[27.10] Since Cl and C2 cannot vanish simultaneously and since the solutions (~l' ~I) and (~2' ~2) are assumed to be linearly independent, the relations [27.10] 1mply that either e h2' = 1 or e hI' = 1. ThUS, in the first case, we have h2 = 0; in the second, hI = O. We now assert that the system [27.4] has a non-trivial periodic solution, and that one of its characteristic exponents is zero. By differentiating [27.1] we obtain

d 2x

tfiY

=

P:r (x, y)

dx

dt +

dy P" (x, y) dT [27.11]

Comparing with [27.4] it is seen that the periodic functions dX1/dt and dYI/dt satisfy the latter system. Moreover, the solution d~l/dt, dYI/dt is a non-trivial solution of [27.4] since XI(t) and Yl(t) are not constant. Knowing that one of the characteristic exponents, say hI' is zero, we can determine the second one from [27.7] [27.12]

85 Following a procedure similar to that used in the discussion of the stability of equilibrium it can be shown that the motion defined by the functions x 1 (t), Yl(t) is stable if h < 0 and unstable if h > O. The proof is more complicated than in the case of stability of equilibr1um, owing to the fact that one of the characteristic exponents is bound to vanish. We shall omit the proof. The reader will recall that in the case of stability of equilibrium, Chapter III, the real parts of both roots of the characteristic equation must be negative in order to ensure the stability. Here, however, the stability depends on the sign of the real part of the non-vanishing characteristic exponent. A few examples below give an illustration of the application of Equation [27.12).

z-

The Van der Pol equation p(1 - x 2 )i + x = 0 is known to possess a period~c solution in the neighborhood of functions Xo = 2 cos t, Yo = Xo = - 2 sin t, when the parameter p is very small. Thus, 1.

x 1 = 2 cos II t +

WI (p

, t); y 1 = - 2 sin II t

+

W 2 (p

, t)

where II = 2-rr/T :::: 1, T:::: 2-rr, and W 1 (~, t ), W2 (p, t) are functions approaching zero uniformly in t when p + O. This gives P~(Xl'Yl) = 0; Qw(X1'Yl) = p(l - X12) = p[l - 4 cos 2 lit + wa(p,t») and by [27.12] T

h= ;J[1-4cos

2

I1t

+

wa(p,t>]dt = -

p[l +w (p)] 4

o

Since w.(p) + 0, h < 0 which proves that the periodic motion in the neighborhood of (xo,Yo) is stable, a well-known fact.

2. As further examples we shall consider the three systems (22.1), [22.6], and (22.11] of non-linear differential equations investigated in Section 22. It was shown that these systems admit limit cycles r = 1, stable for the system (22.1], unstable for [22.6), and half stable for [22.11]. We shall establish here the condition of stability,by means of Equation [27.12). a.

In case of the system [22.1) we have p

~

y2

= (X2

Qw =

+

S

+

(X2

+

y2)] _

y2)"f

X2 (X2

[1 -

a

y2)2

2X2 tx2

[1 -

(X2

+

y 2)] _

+

2y Y'X2

y2

2

+

y2

86

Since this system admits a periodic solution have

Xl =

cos t, Yl = - sin t, we

and 1 2 2 '!r [2 (sin t

h = -

which shows that the limit cycle r b.

=1

+

COS 2 t)

dt = - 2

is stable.

For the system [22.6] we obtain P z = 3x

2

+

Y

2

- 1; QII

=

3y

2

the generating solutions being the same as above. h =

and the limit cycle r

h

* O.

=1

l... [(4 2'!r

- 2)dt

2

-+

x - 1

Whence,

= 2 >0

is unstable.

It is important to note that one is able to draw conclusions as to stability only when

Should h vanish the preceding argument based on the use of [27.6] is not applicable. c. As an illustration, we shall consider two such cases: the system [22.11], second the harmonic case. For the system [22.11] one finds Pz(Xl' Yl)

so that h

+

QII(Xl'

first,

Yl) = 0

= O.

It was noted previously that the limit cycle is half stable. As regards the harmonic oscillator, whose non-dimensional equation

is

x+

x = the corresponding first-order system is dx

dt

= Y;

dy

It

0

= - :t

It is observed that P z = Q II == O. The motion of a harmonic oscillator is neither stable nor unstable but rather indifferent in the sense that if a perturbation results in a new amplitude of motion this amplitude will be maintained without any tendency either to approach to, or to depart from, the motion with the old amplitude.

CHAPTER V BIFURCATION THEORY (POINCARE)

28.

INTRODUCTORY REMARKS It has been shown in Section 13, 14, and 15 that, for certain critical or bifurcation values of a parameter in a differential equation, radical changes occur in the qualitative aspect of its trajectories. In this chapter we propose to go further into this subject by considering the important particular case in which a singularity undergoes a transition from stability to instability, or vice versa. There exists a number of more complicated cases in the theory of bifurcation such as bifurcation of limit cycles from separatrices, diSintegration of a limit cycle into a number of cycles, and so on. We shall not go into the investigation of these complicated cases but will confine our attention to the above specified case. A preliminary qualitative investigation of the subject of this chapter has been made already at the end of Section 24 where the intuitive concepts of IIsources ll and IIsinksll in the phase plane were applied; here we shall follow the analytical method of Poincar~.

In practice, the problems are very frequently simplified or IIlinll earized from the start so that some limited conclusions regarding stability may be reached. Unfortunately. such line~rized equations do not give a full account of the observed phenomena. Thus under certain conditions the normal behavior of a dynamical system, e.g., an airplane, which is predictable on the basis of a "linearized" theory suddenly gives way to self-excited parasitic oscillations of large amplitudes, the "flutter II phenomenon, frequently causing destructive effects. One recognizes in these effects a typical case of the so-called "hard ll self-excitation which has now been completely explained by means of the theory of bifurcation as applied to non-linear electronic Circuits. We are now entering the domain of non-linear mechanics proper, and the equations of the first approximations which we have been using in Chapter III in connection with the problem of the stability of equilibrium cease to be applicable. In other words, it is impossible to obtain the results of this chapter by linearizing the differential equations. The present topiC leads to results of great practical interest in connection with the problem of self-excitation of dynamical systems in general. More specifically, all problems such as self-excitation of thermionic circuits or of electrodynamical systems, "flutter" of aircraft wings, and similar phenomena fall within the scope of this theory.

88

29.

TRANSITION OF SINGULARITIES. BRANCH POINTS OF LIMIT CYCLES Consider the gener al form of the dynamical equat ions and assume that the coeff icien ts a, b, c, and d of the linea r terms as well as the nonlinea r terms P2(X, y) and Q2(X, y) are now funct ions of a parameter A. We assume furth er that the origin (0,0) is a focal point for all value s of A under consi derati on. It was seen in Sectio n 18 that, for a focal pOint, the roots of the chara cteris tic equat ion are conjugate complex 8 1 = a t (>")

+

ib 1(A)

[29. 1 ]

8 2 = at (>") - ib t (>..)

By means of the transf ormat ions [18.16] and [18.15] the gener al dynamical system can be writte n as

x-

+

P2(X. y.A)

+ al(>")y +

Q2(X. y.A)

at(A)z - b1(A)y

11 = bt(>..)x

[29.2]

where P2 and Q2 are power series in x and y beginning at least with the terms of the second degree in x and Yi compare with [20.9 ]. The focal point is stable or unsta ble according as at < 0 or at > 0, as was explained in Sectio n 19. Multi plying the first Equation [29.2] by x, the second by y, adding the two equat ions, and transf ormin g the resul tant equations into polar coord inates , one obtain s the equat ion 1 d (1' 2)

"2 ---crt

= a 1(>..)r

2

+ P2 1'cos8 +

•

Q21'sm 8

[29.3]

Multiplying the first Equation [29.2] by y, the second by x, and subtr acting the former from the latte r, one finds , simil arly d8 1 [b ( ) 2 di"= ;Z 1 >",r

• ] +Q2r cos8 -P2rs tn8

[29.4 ]

Dividing Equation [29.3] by [29.4] one obtain s the equat ion of the traJec tories [29.5] It is to be noted that for a suffiC iently small 1', and hence for small 'Ixl and Iyl, the sign of d6/dt is determined by that of b (>..). Equat ion [29.5] t

89 reduces to the form

+(pgSinO-QgcosO)g+ ••• ] bl (A)r

[29.6]

This expression can be written as a power series in r

[29.7] where

R = ~ (PgSinO - Qg COSO) 2

P1

R = ~(PgsinO 3

b(

+

+ Qg sinO b I r2

Pg cosO

r2

rll

QgCOSO)

+ a.

b!

(P

2

sinO - Q2 COSO)2

+

[29.8]

r2

Since it is deSirable to investigate the transition from stable focal pOints to the unstable ones, the branch BC of the circuit shown in Figure 18.1 must be followed; in this case q O. In fact, when passing from stable focal pOints to unstable ones the real part of the complex roots changes its sign but the imaginary part does not. In other words hl(A) in Equations [29.2] does not change its sign for Al < A < A2 corresponding to the branch BCin Figure 18.1. The series [29.7] converges, therefore, for all values of A in the interval (A 1 ,A 2) provided r < p, where p is a small fixed number not depending on either A or O. From [29.8] it is observed that RI(O,A) = al(A)/bl(A) does not depend on 0 whereas all other coefficients Rk(O,A) are periodic functions of O. The function r = f(O,ro,A) can be developed in a power series in terms of To converging for all values of 0, for Al < A < A2' and for ro < p. Thus

*

[29.9] substituting this solution into [29.7] one obtains the following set of

90 recurrent d1fferential equations determining the functions uk(8,A)

[29.10]

Since by assumption TO

= !(O,TO,A) Ul(O,A)

one finds from Equation [29.9]

=

1; 1'/c{O,A)

=0

[29.11]

for k = 2, 3, . ". These initial conditions, in conjunction with Equations [29.10] determine the functions 1' k(8,A). The first equation gives d1'l = 1'1

41

(A) d (J

b1 (>..)

and, on integrating,

[29.12] Let To be a small pos1tive number. Since the sign of d8/dt does not change for small values of T, it is apparent that the trajectory originating at the pOint (r = To, 8 = 0) is a limit cycle if, and only if

[29.13} where

In this expression TO designates the radius vector of the l1m1t cycle in the neighborhood of the T-axis. We may use the general expression ~(T,A) to designate the same function in which r is not necessarily TO. We have

[29.14] where

91 We now consider a fixed value AO of the parameter A and distinguish two cases:

1. al(A o) ~ O. This means that the stability conditions of the focal point do not change when A passes through the value AO' By the first Equation [29.15] we have al(A o) ~ 0 so that the coefficient of To in the expansion [29.14] does not vanish for A = AO' Hence, aside from the "trivial" solution To = 0, Equation [29.14] has no real solution in To and A when To and IA - A0 I are sufficiently small. This means that a sufficiently small neighborhood of the singular point remains free of llmit cycles when A varies in a small interval around Ao'

2. al(A o) ~ O. By [29.15] this implies that al(A o) = O. Furthermore it can be shown that a2(Ao) is also equal to zero. The proof of this statement is omitted here. Let us now assume that a•

a I' ( A0) :I: 0;

b.

a 3 (Ao)

¢

0

where al'(Ao) deSignates the value of the derivative da1(A)/dAat the point A = AO. Thus the singularity changes from a stable focal point to an unstable one or vice versa. We will show now that, at the point A = AO at which the stability of the singular point changes, there appears a limit cycle or cycles. Everything occurs as if the phenomenon were developing according to the following scheme: Stable singularity

~

Unstable singularity

............ stable limit cycle

from which Poincare's term "bifurcation" appears Justified in usage. In order to show this we can best proceed geometrically conSidering the (To,A)-plane. It is clear that in Case 1 the variation of the parameter A does not have any effect on TO since it remains identically equal to zero. In Case 2, however, the situation is different inasmuch as the curve ~(TO,A) = o in the neighborhood of the point TO = 0; A = A0 consists now of two branches: a straight line To = 0 and the curve

Expanding the functions al(A) and a 2(A) in Taylor's series in the neighborhood of A = AO we obtain:

92 where O;(AO) and O~(AO) des1gnate the values of d0 1 (A)/dA and d0 2 (A)/dA for A = AO' Since atfAo) ~ 0 1t follows that a;(Ao) ¢ O. If one now assumes that the first and the th1rd terms on the r1ght s1de of [29.17] are of the f1rst order, the second term 1s of the order }/2 and can be neglected. Equat10n [29.17] then becomes

whence [29.18] where

Equat10n [29.18) shows that (A - AO) must have a s1gn oppos1te to that of Clas s1nce ro is to be real. We can now d1st1ngu1sh four typical cases: 1. c > 0, os> 0; 2. c > 0, a8 < 0; }. c < 0, a 8 > 0; and 4. c < 0, Os < O. It is to be noted first that c = O;(AO) > 0 means that for the increaSing values of the parameter A the real parts of the characteristic roots change from negat1ve to positive for A >AO' Hence, the focal point (A ~ AO) orig1nally stable becomes unstable for A~ AO' For c < 0 the focal pOint (A ~ AO) originally unstable becomes stable for A ~AO' We proceed now to examine the above four cases. 1. From [29.18) it is seen that ro is real, i.e., limit cycles may exist, only for A - AO < O. The region A > AO is free of limit cycles; see Figure 29.1a. In the region A - AO < 0, in which l1mit cycles exist, the focal points are stable. Hence, the limit cycles in that region are necessarily unstable as follows from the topological conSiderations of Section 24. 2. Equation [29.18] shows that the limit cycles exist only for A ~ AO and the region A - AO ~ 0 is free of limit cycles; see Figure 29.1b. This represents the commonly encountered case of a soft self-excitation when, for a gradually increasing value of the parameter A, the self-excitation sets in for A~ AO' The limit cycles in this case are manifestly stable since the singularity is unstable.

3.

By a Similar argument one finds that the limit cycles exist for A ~ AO; see Figure 29.1c. They are unstable since the singularity is stable in that region. 4. In this case the limit cycles exist only for A ~ AO and are stable; see Figure 29.1d.

93 '0

'0

Limit Cycle

o

Limit Cycle

o

A

(a)

Ib)

'0

o

'0

A

o

(c)

A

(d)

Figure 29.1 The question of stability of limit cycles can be ascertained also . by means of the theorem of Poincar6 given in Section 13. Although the application of that theorem was made in Section 13 in connection with the question of the stability of equilibrium it can be also applied in this case as will be shown. It 1s apparent that the curves of Figure 29.1 represent Equat10n [29.17). In the ne1ghborhood ot (A C AO' r c ro) we can write t,6(r,A) = ~A - Ao)a;(Ao)

+

(A - Ao)a;(Ao)r

+

a3(Ao)r2

+.

Subtract1ng t,6(ro,A) as given by [29.17] from t,6(r,A) we have t,6(r,A) - t,6(r o ,A) = t,6(r,A)

or

[29.19 ] Recal11ng that ro2 and (A - AO) were follows that~ in the neighborhood of as A - AO and, hence, the first term order 3/2. Neglecting this term and

assumed to be of the f1rst order, 1t ro' r2 - ro2 will be ot the same order in the above expansion w1ll be of the also terms of higher order we obtain:

[29.20]

94 We propose now to ext9nd the application of the theorem of Poincar6 in connection with the question of stability of limit cycles. As an example take the second case: c > 0; a 8 < O. It will be shown later that this case is of particular interest in applications. From [29.20] it follows that ~(T,A) > 0 for r < To. Hence, the region in which ~(T, A) > 0 lies below the,curve ~(TO,A) = O. Since ~(To,A) = To~(To,A) and To is Figure 29.2 . a non-vanishing positive quantity in the region in which the limit cycles exist, it is apparent that ~(T,A) > 0 for T < To and ~(T, A) < 0 for T > To. The theorem of POincar6 states that the curve ~(To,A) is then a locus of stable points. In order to see the application of the theorem in this case, it 1s to be noted from [29.13] that, in general, for T ~ To, ~(T,A) = 1(2",T,A) - T, where T is the initial value ~f the radius vector and 1(2",~A) is the value of the radius vector after one complete rotation, 8 = 2". of the representative point on the spiral trajectory. Figure 29.2 illustrates this situation for two values of T, i.e., Tl < To and T2 > TO' For T = To the trajectory is closed, that ls, a limit cycle. Since for Tl < To, ~(TI,A) > 0, it follows that 1(2",~,A) > T1, that is, the spiral issuing from the point Tl after one turn approaches the limit cycle C as shown. For T2 > To, ~(T2,A) < 0, that is, 1(2". T2,A) < T2 , which means that the spiral trajectory issuing from a point T 2 , external with respect to the limit cycle, approaches it after one turn. This characterizes a stable limit cycle. It is worth mentioning once more that the existence of limit cycles ascertained by the more detailed study made in this section was possible only by considerlng the non-linear terms. The equations of the first approximations which were sufficient for analyzing the conditions ot the equilibrium of the system are incapable of giving any information concerning the existence ot limit cycles, whose Qetermination depends on the curvature of the characteristics inasmuch as the coefficient 08(A) is related to the curvature as can be easily ascertained.

95

30.

SELF-EXCITATION OF THERMIONIC GENERATORS Although this subject has been considered to some extent at the end of the preceding section, we propose to investigate it in more detail by introducing certain analytical approximations for the characteristics of the non-linear conductor, the electron tube. Consider the circuit shown in Figure 30.1 representing a commonly used type of thermionic generator with an inductive grid coupling. c The differential equation of the circuit is •

L ~ dt

R

I

+

R't

+

'!"J' dt C t

= ,..\ d10 dt

[30.1 ]

o

in the usual notation; the coefficient).. of muFigure 30.1 tual inductance between the anode and grid circuits will be the parameter of the preceding theory. 10 is the anode .current. Let us assume the following expression for 10 considered as a function of the grid voltage Vg [30.2]

where 8 .. 8 2 , 8 3 , ' •• are certain numerical coefficients determined so as to fit the function 10 = f(V g } to the experimental curve. For the alternating performance of the circuit the term 10 is clearly of no interest and can be dropped since it amounts to a shift of the origin of coordinates to this point. If the characteristic were perfectly symmetrical with respect to the origin, only the odd powers would be present. In order to take into account a slight asymmetry we shall retain the term 8 2 Vg 2 and shall limit the power series to the cubic term. The latter, as will be shown, is essential for what follows. Under these assumptions [30.3]

This expression for the anode current is inconvenient, however, since the· coefficients S1' 8 2 , ' • • have different physical dimensions. In order to avoid this it is convenient to introduce a dimensionless variable v = Vg/V. where V. is the grid voltage beyond which the anode current 10 does not change appreciably so that in an idealized case we can assume that it does , 2, , not change. Putting 8. = P.. 8 2 V8 = " .. and 8 3 V. = 6. where PI' " .. 6. have

now the dimension of the "transconductance," [30.3] becomes [30.4 ]

From the experimental curves of electron tubes it is observed that, for small values of Vg , and hence of v, the approximation 10 = V6 (P l v + Y1V2) is sufficiently accurate. During the process of self-excitation the oscillations of the grid potential Vg may be considerable so that the third term 6; va is Justified. It is also noted that the coefficient 61' is generally negative since the characteristic exhibits an inflection point. Putting 6 1 = - 6; we obtain the following expression for the characteristic

[30.51 in which Pi' Yl' and 61 are positive. Introducing the "dimensionless voltage" v = Vg/~. in the other terms one gets 1 v = ev.

f' dit 'V = eVa i i 1

.. v

=

1 di eVil dt

[30.6]

Differentiating Equation [30.51 one obtains dIn = dIn dv = V (p dt dv d t 8 1

+

2 .... v _ 36 v 2 ) V r t

t

[30.71

Substituting this value into [30.1] we get [30.8]

A further Simplification is obtained by introducing a "dimensionless" or ll II cyc11c time.,. = wot where "'0 = Y1/Le. This gives . dv diJ d.,. dv v = dt = d.,. dt = d.,. Wo

and Similarly .. V

d 2v

= d"t2 =

d 2v d.,.2

2 Wo

The dif·ferential Equation .l30.11 reduces then to the following dimenSionless form

where

[30.10]

97

Equation (30.9] is of the Van der Pol type. We know that the self-excited oscillations are possible. The equivalent system of the first order is dv =w d.,.

-

(30.11 ] The equations of the first approximat1on are dv

=w d.,. dw -d.,.

The point v = w equation are

=0

[30.12 ]

= - v + fJw

is a singular point.

8 1 ,2 = :

The roots of the characteristic

± V~2

-

1

Experimental evidence shows that in a great majority of cases the self-excited oscillations start in an oscillatory manner like the trajectories departing from an unstable focal pOint. There are cases of the so-called relaxation oscillation which will be studied in Part IV, in which the starting of oscillations occurs in the manner of trajectories departing from an unstable nodal point but we shall not treat this case here as the theory of these oscillations has not yet been established for the steady state. For the oscillations developing from an unstable focal point (Section 8) the roots 2

8 1,2 = f!... 2 + - y.fJ4

-

1

of the characteristic equation corresponding to the equations of the first approximation (30.12] must be conjugate complex with a positive real part. This implies that 0 < fJ < 2. Substituting for fJ its expression [30.10] we obtain the condition

The critical value of the parameter is given by the equation

98 that is \

_ RC

"'0 -

[30.1}]

P1

In order to be able to ascertain the appearance of a stable limit cycle for A~ AO we have to investigate the non-linear Equations [}O.ll]. Making use of the transformations [18.14] and [18.16] we introduce the new variables % and y by the equations with a 1 (A)

= P(A)/2 and b1 (A) = Y1 -

p 2 j4 and obtain the system

[}o.14 ] 11 = bl % + aly - ~ [4,,(a l % + b1y) - 12,J(a l % + b1y)2]Z

It is noted that the derivative with respect to A of the real part of the roots 8 1 ,2 for A = AO is positive; thus a;(Ao) > O. Hence if we show that 0S(AO) < 0 we shall be dealing with Case 2 of Section 29. For A = AO' P(AO) o and hence a 1 (A o) = 0; b1 (A O) = 1. Comparing [30.14] with [29.2], making use of ' polar coordinates and relations [29.8] we find that R I (6, )..,0)

=0

[30.15] R s (6,A o)

=

3

16"(Ao)sin 6 cos 3 6 -126(Ao)8in 2 6cos 2 6

The recurrent system [29.10] of differential equations for A = AO has the following form dU 1 d8

=

0

[}o.16]

~~s =

2U2R2(6, AO)

+

Rs(6, AO)

Hence, upon integrating Equations [30.16], we obtain u1(6, AO) = 1 u2(6,)..,0) =

3

: Y(AO)(l - (08 6)

=

99 Making use of [30.10] we get Cls(AO) = - 37TRCWo(;:) < O. Thus. by Case 2 ·of the preceding section, it follows that there is a stable limit cycle. This, in conjunction with the instabllity of the focal point for A ~ AO' creates the favorable conditions for self-excitation. Summing up the conclusions of this and of the pre~eding sections, it can be stated that the existence as well as the nature of limit cycles depends on the curvature of the non-linear characteristic of the system, whereas the condition of stability depends on its slope at the point of equilibrium. For this reason it was necessary to retain a cubic term in the serles expansion of the plate current characteristic, Equation [30.5], in investigating limit cycles, whereas for problems of equilibrium the equations of the first approximation, containing only the first powers of the dynamical variables, were sufficient. We have considered in this section only the case of a 80ft selfexcitation of the circuit which corresponds to a non-linear characteristic capable of being approximated by a polynomial of the third degree, Equation [30.5]. An entirely different kind of self-excitation occurs when the nonlinear characteristic is expressible by polynomials of a still higher degree. Self-excitation in such cases is called hard. We shall not enter into this subject here but will reserve it to a later chapter after we get acquainted with the analytical method of Poincare (Part II). 31.

SELF-EXCITATION OF MECHANICAL AND ELECTROMECHANICAL SYSTEMS Mechanical systems, also, offer numerous examples of self-eXCited non-linear OSCillations, but their study is less advanced at present than that of electrical systems. Two main reasons account for this situation. First, the self-excited mechanical oscillations in practice are always undesirable or parasitic phenomena of a "closed cycle" type (Section 23) and the main endeavor so far has been to eliminate them by breaking the closed cycle somehow, rather than to attempt to study them. Secondly, the determinatlon of the parameters of a mechanical system is generally a more difficult problem than that of electric Circuits. It is possible only in a few· particularly Simple cases in which the chain of "causes" and "effects" can be followed completely. The following two examples of self-excited mechanical oscillations may be mentioned. A.

SELF-EXCITED OSCILLATIONS OF A MECHANICAL CONTROL SYSTEM Consider the following arrangement used for the study of antirolling stabilization of ships by the method of actlvated tanks (25). Two tanks are mounted on a pendulum and located symmetrlcally with respect to

100

the axis of oscillation; the tanks are connected by a U-tube and filled with water. An 1mpeller pump having a variable blade angle a is capable of displacing the water in the system so formed; the blade angle a is controlled in response to the angular motion of the pendulum as will be specified. The system has thus two degrees of freedom, the angle 6 of the pendulum and the relative angle ~ of the water level in the tanks. In the phenomenon of non-linear oscillations analyzed below, the motion of the pendulum is exceedingly small, a fraction of one degree, and its direct action, that is, by direct mechanical couplings, on the motion of water in the tanks is negligible. There exists, however, an important action which is exerted through the blades actuated by the control system. If there is no control and the pendulum is fixed, the motion of water in the system for small OSCillations can be approximated by a linear equation J if,

+

64>

+

c~ = 0

where J, b, and c are constants the physical significance of which is obvious. Assume now that the pump is made to act on the liquid column in the U-tube and let the moment of the force exerted on the water by the pump be M(a), a function of the blade angle a. Experiment shows that this couple increases in1tially more or less in proportion to a and exhibits for larger angles a a "saturation" feature due to complicated hydrodynamical effects. One can apprOXimate, therefore M(a) = Mla - Msa s• The arrangement of control used in this case is such that the blade angle is continuously adjusted to be pr0p.0rtional to the rate of flow, i.e., to ~.* The expression for the external moment is then of the form al~ - as~s. The theory of this control is based on the linear approximation. It is found that with a control of this kind, at least within a certain range, the free oscillation of the pendulum is damped and the forced OSCillation is reduced in accordance with the linear theory; these features, are, however, of no interest here. Aside from this useful effect, predictable on the basis of a linearized equation, the following parasitic effect is observed. Under certain conditions the pendulum, the water in the tanks, and the blade angle begin to oscUlate or "nutter" spontaneously; the oscUlation of the pendulum is, however, very small and will be neglected. The oscillation of the

* The

blade angle in this particular case is made proportional to the angular acceleration of the pendulum. The latter, however, is in phase with the rate of flow through the tube. A complete study of the problem requires a consideration of the system with two degrees of freedcm 9, _ comprising the pendulUlll and the water ballast. Insofar as this stud7 is limited only to the motion of the latter, this complete st.udy 1s omitted here. AccO\Dlt. 1s taken only of 1ts final conclusion, D8IIIely, the blade angle 1s proportional to the rate of flow in the U-tube.

101

blades, and also of the water, designated usually as "hunting" perSists indef1nitely and may acquire cons~derable amplitudes. The explanation of this phenomenon follows from the preceding theory. The differential equation with the blades under control, as specitied, is

that is [31.1]

We shall consider two cases according as b - a l ~ O. It is to be noted that the coefficients al and as of the hydrodynamical couple depend on tHe amplification X used in the thermioniC circuit, so that we can consider them as functions al(X) and aa(X) 1ncreasing monotonically with X. Hence, for a small amplification X, we have b - aleX) < 0 and for a larger amplification b - al(X) > O. Case 1. Weak Amplification Dividing by J and putting [b - ad X)]/J 2 c/J = w we have

= 11.'

> 0, aa /J

= P'.

and

[,1 .2]

By a change of the independent variable r = wt we obtain the dimensionless form ['1 .,]

where 11. = n'/w and p = p'w. Forming the equations of the first approximation, the characteristic equation is S2 + nS + 1 0 and its roots are I:

11. S 1,2 =--+ 2 -

~2 --1 4

['1 .4]

Hence for 11. 2 /4 > 1 the origin, q, = dq,/dr = 0, is a stable nodal point and for 11. 2/4 < 1 the origin, q, c dq,/dr = 0, is a stable focal point. Case 2. strong Amplification In this case b - al(X) < O. Proceeding as previously and putting tal (#l) - b ]/J = 11.' > 0, aa/J c p', and c/J = w 2 we obtain [31.5]

102

where n = n'/w and p case are

= p'w.

The roots of the characteristic equation in this

S 1,2 =

n

./n2

"2 ± V"4 -

1

[31. 6]

Hence for n 2/4 > 1 the origin is an unstable nodal point, and for n 2 j4 < 1 the origin is an unstable focal pOint. In the first case the control equipment functions without any parasitic oscillations. In the second case, since the singularities are unstable the trajectories approach a stable limit cycle and this characterizes a steady state of parasitic OSCillations or IIhunting" which is observed in such systems if the amplification is too high. Moreover, this condition generally sets in abruptly at a certain critical value Al of the amplification factor for which b - al(AI) = O. The condition n 2/4 ~ 1 is equivalent to (b - a l )2/4J 2w2 ~ 1. It is the same both for stable and unstable operations of the control system. The stability or instability of the system is governed by the sign of b - al(A) as previously set forth. The condition a l > b also has Simple meaning; in fact, a l is the measure of the energy input on the part of the pump and b characterizes the dissipation of energy in the system. This inequality means that initially the input of energy is greater than its dissipation so that self-excitation can occur with a gradually increasing amplitude of OSCillation. The critical value of the parameter is given by the equation al(A I ) =,b. In this particular case there are two parameters: WI' the speed of the pump, and v, the amplification factor of the thermionic control system. By increasing either one, the steepness of the characteristic is increased, and hence also aI' so that for a certain value A = AO the critical point is reached and the oscillation stabilizes itself on a limit cycle. Beyond this point, for A > AO' the amplitude of the limit cycles increases monqtonically with A. We have, thus, a typical case of "soft" self-excitation; see Figure 24.6. If the characteristic has an inflection point, in addition to one for a = 0, it can be approximated by a polynomial M(a) c MIa + Msa s - M5a5 where M I, Ms' M5 > O. In such a case the self-excitation would appear in an entirely different manner, as will be explained in Part II. Experiments corroborate these theoretical conclusions, at least qualitatively.

.

B.

SELF-EXCITED OSCILLATIONS IN AN ELECTROMECHANICAL SYSTEM Another example of the same kind is the well-known experiment (26) in which a series generator is connected to a separately excited motor.

103

Approximating the voltage of the series generator by a polynomial of the form E= ali - asi s , where i is the current, and expressing the condition of dynamical equilibrium of electromotive forces in the circuit, we find all. - a 3 l.3 = K w

di + R'1 + L dt

[31. 7)

By differentiating this equation we find [31.8]

The quantity dw/dt can be eliminated from the equation by expressing that the electrical power Kwiabsorbed by the motor serves to accelerate the rotor

!!:...(.!. Jw 2 ) dt 2

Kwi =

dw dt

= Jw

[31.9)

where J is the moment of inertia of the rotor; whence dw/dt = Ki/J. tuting this value of dw/dt into [31.8] one obtains the equation 2 •

d t -:i"72 -

dt

(

d' m - nt.2) - t

+

dt

pt.

__

0

Substi-

[31.10]

where m = (a 1 - R )/L, n = 3a s/L, p = K2/LJ > O. This equation is of Van der POllS type. Equation [31.10] is equivalent to the system

~;

= y;

iI -

2

(m - ni )y

+

pi = 0

[31.11)

The equations of the first approximations are di dt -

y;

dy -dt

= -

.

pt

+ my

[31 .12]

The characteristic equation is S2 - mS + p = 0 and its roots are S 1,2 =

m

.lm2

"2 ± V4 -

p

The singularity is a nodal point if m 2/4 - p > 0 and is a focal point if 2 m /4 - p < O. This singularity is unstable if m = (al - R)/L > 0, that is al > R, and stable for at < R. For a sustained self-eXCited oscillation the singularity must be an unstable focal point, whence the conditions [31.13 ]

The first condition is a static criterion which has been analyzed in the preceding example. The second criterion 1s dynamical; it is generally fulfilled

104

for not too great a moment of inertia J of the motor. The critical value of the parameter occurs for a1 = R. In practice one can select either R or a 1 as the parameter A of the general theory. In the latter case it is convenient to introduce an auxiliary saturation winding on the series generator field which mod~fies the state of saturation in the machine and, hence, the coefficients a l and as of the non-linear element of the system. One can go a step further and determine the amplitude of the selfexcited OSCillation but for th1s purpose we shall need the analytical method of Po1ncar6 outlined in Part II. The remaining conclusions are the same as in the preceding example.

CHAPTER VI GEOMETRICAL ANALYSIS OF EXISTENCE OF PERIODIC SOLUTIONS'

32.

INTRODUCTORY REMARKS

In the preceding chapter we have investigated the principal properties of limit cycles which characterized periodic motion in non-linear and non-conservative systems. We shall now be concerned with the question of the existence of such motions, a rather difficult question for which the relatively Simple criteria of Poincare and Bendixson give only limited information. In Part II we shall enter into this question in more detail in connection with the analytical methods of Poincare and Liapounoff. In this chapter we shall investigate the results obtained by Lienard (27) by studying the trajectories in a special phase plane. Further progress in this direction was made recently by N. Levinson and O.K. Smith (22). These methods occupy an intermediate pOSition between the topological and analytical methods and thus present particular interest as a connecting link between them. The prinCipal aim of these geometrical methods is to formulate conditions under which trajectories become closed, i.e., when they represent periodic solutions. The starting point is the original equation of Van der Pol

x-

.u (I

-

x 2) X +

X

= 0

[32.1 ]

generalized by Lienard and the Cartans (28) to

x + I(x) it

+

x = 0

+

g(x)

[32.2]

and by N. Levinson and O.K. Smith to

x + I(x,x)x

=

0

[32.3]

The functions I and g entering into these equations are subject to certain restrictions which will be specified later. It may be worth mentioning that the original proof of the existence of periodic solutions for [32.1] by Van der Pol rested upon the graphical method of isoclines, see Section 7. Before proceeding with the geometrical analysis of the more general Equations [32.2] and [32.3] it is useful to consider the original Van der Pol Equation [32.1] from the physical standpoint. It is apparent that [32.1] may be considered dynamically as an autonomous oscillatory system with one degree of freedom possessing a variable damping - .u(1 - x 2 )x. For small deviations x the system has a negative damping, for larger x the damping becomes positive. In the light of what has been said regarding "negative damping" one concludes that for small values of

106

the system absorbs energy from an outside source so that in the early stages the motion develops with gradually increasing amplitudes; for large ~, on the contrary, the system is dissipative, hence, the amplitudes decrease. Ult~te1y a steady state is reached when absorption and diss1pation of energy balance one another throughout the cycle. This is, in fact, 1n entire agreement with observation. Unfortunately these physical considerat10ns are 1nsufficient. Since the system is non-conservat1ve, one cannot utilize the energy integral, as was the case for a conservative system, see Chapter II. It becomes thus necessary to establish conditions for closed trajectories and to infer from them that a periodic process is possible. ~

33. LIENARD'S METHOD Cons1der the following differential equation

x + w/(~) x +

W2~ = 0

where I{~) is a continuous, differentiable, even function of ~; additional properties of I( ~) will be specified later. Taking w t as a new independent variable this equation can be written as

x + lex) x + x

[33.2]

= 0

Since no further.confusion is to be feared we shall use the same symbols % and x as 1n [33.1] although occasionally we shall write [33.2] as

x + wo/(x)x +

wo2~ = 0

with Wo = 1, in order to remind one of the dimensional homogene1ty of the original Equation [33.1]. It is apparent that [33.1] is a particular case of [32.3]. If f(~) 0, the motion 1s harmonic, corresponding to the equation x ~ ~ = O. If f(~} = e, e being a constant, we have the well-known damped motion, either oscillatory (lei < 2) or aperiodic (lei> 2); see Chapter I. Setting x = v, Equation [33.2] is replaced by the system

=

:i: = v; v

~: + lex) v + ~ =

0

The second Equat10n [33.3] can be written

~ + I(~) + ~ v

d~

IntrodUCing a new variable y = v + f(~} is even, we replace [33.4] by dy dx

+

F(~},

x

[33.4]

= 0 z

where F{x)

y - F(x) =

0

= J f(x}dx o

1s odd since

107

This equation can also be written as

xdx + [y - F(x>]dy = O.

[33.6]

The system [33.3) can also be written in the form

dy

- -x =

y -

dx dx F( x > = -v

=

dt

= Wo

dt

[33.7]

Since the independent variable t does not appear in (33.6], the latter represents the phase trajectories in the (x,y)-plane of the dynamical system [33.2]. It must be noted, however, that the (x,y)-p1ane of Lienard is different from the customary (x, y= x)-p1ane considered previously so that the geometrical form of the trajectories in both cases is necessarily different, as will be shown in Section 37. The use of the Lienard plane (x, y = v + F( x)] makes it possible to obtain a relatively simple geometrical construction of the trajectories. In fact, the equation of the normal at the point (x,y) on the trajectory is (x - X)dx + (y - Y)dy = 0, and from [33.6] we see that it is satisfied for X = 0, Y = F( x) • Therefore, the normals to the phase trajectories of Lienard's Equation [33.6], for x = Xl' all pass through the same pOint N whose coordinates c are 0, F(x l ). Hence, one obtains the elements of trajectories along the line x = Xl by taking on the y-axis a point Nl whose ordinate is Yl = F(x l ) and by describing with Nl as center a series of small arcs as shown in Figure 33.1. By taking other points X = x 2 , xa, ••• and by repeating the procedure, additional Figure 33.1 elements of trajectories are obtained. Having a field of line elements and starting from a certain initial point (xo'Yo) a continuous curve can be traced following the line elements so trace~; this curve will be clearly a trajectory of Lienard's Equation [33.5] or [33.6]. Since F(O) = 0, it follows that the only singular point of Equation [33.5] is the origin x = y = O. Furthermore trajectories are symmetrical with respect to the origin, for the substitution of - x for x and - y for y does not change [33.5], since F(x) is odd. Likewise, upon replacing + y by - y, + dt by - dt, f(x} by - f(x), Equation [33.5) is not changed; hence,

108

this substitution transforms a trajectory into its symmetrical image relative to the x-axis. The curve y = F(x) and the y-axis determine four regions 1, 2, 3, 4 shown in Figure 33.1; the limits of the first and the second regions are shown by a different shading in Figure 33.1. From Equations [33.71 it follows that the differential elements of the trajectories in these regions for dt > 0 are such that dy and x are of opposite signs and that dx and y - F(x) are of,like signs. In terms of these regions the following table is apparent. x

Region 1 2

y - F(x)

+

+ +

--

+

-

4

+

-

-

3

dx

+

dy

-

+ +

From [33.6] it follows that dy/dx = ~ when y c F(x). Hence, the tangent to the trajectories is vertical at pOints C at which the trajectories intersect the curve y =F(x); the abscissa of point C thus determines the amplitude of oscillation in the (x,y)-plane. Instead of the above construction one can apply the method of isoclines, see Section 7. From [33.5] we have

*

dy dx = -

x y -

F(x)

=

a

that is, y = F(x) x. The locus of isoclines dy/dx = a = 0 is thus on the y-axis; that of dy/dx = a = ~ is on the line y = F(x); and that of dy/dx = 1 is on the line y =F(x) - x and so on. EXISTENCE OF CLOSED TRAJECTORIES IN THE LIENARD PLANE We shall now consider the most important point of the L1enard theory concerning the existence of closed trajectories in the (x,y)-plane previously defined. Instead of Lienard's original proof, we shall follow the one given by N. Levinson and O.K. Smith (22). They consider an equation 34.

:c +

f(x):i

+

9 (x) = 0

[34-.1 ]

more general than that of Van der Pol (1), but possessing essentially the same characteristics which are given explicitly below. Introduce the functions F(x) = f(x)dx and G(x) = g(x)dx. The conditions in question are as follows. 0 0

jZ

jZ

109

1. All functions are continuous. f(~) is an even function of ~, hence, F(~) is odd. g(~) is an odd function of ~, hence, G(~) is even. The sign of g(~) is that of ~.

2.

F(~)

3.

F(~)'"

has a single positive zero ~o. It is negative for 0 < ~ < ~o. For ~ > ~o it increases monotonically and hence is positive. 00

with

y

F (x)

..

o~----------~~------

~.

Figure 34.1

It is to be noted that F(~) need not be monotonic for 0 < ~ < ~o' as shown in Figure 34.1. Under these assumptions, as N. Levinson and O.K. Smith show, Equation [34.1] possesses a unique periodic solution.

Following the same procedure as in Section 33, one obtains the generalized Li'nard equation dy dx

+

g(x) == 0 y - F(x)

[34.21

We shall show that there is one, and only one, closed trajectory of [34.21. Considerations of symmetry clearly remain the same as in Section 33. This means that a closed trajectory passing through a point (O'Yo) must necessarily pass through the point (0,- Yo). Conversely, a trajectory passing through pOints (O'Yo) and (0,- Yo) must necessarily be closed, since on leaving (O'Yo) it is symmetric with respect to the origin along the arc of the curve lying between (O'Yo) and (0,- Yo). The existence and uniqueness of a periodic solution will be proved if one shows that among all trajectories there is one, and only one, with equal positive and negative intercepts OA and OB. In outline this is done as follows. Introduce the function

[34.31 2 Thus A( 0, y) co 2'1 y. In order to show that OA = OB, i t is only necessary to show that AA = AB sometime. Now Equation [34.21 may be written as

y dy

+

g(x) dx

==

F(x) dy == dA

[34.41

Thus B

fdA = A8 - AA A

8

==

JF(x) dy

[34.51

A

where, here and throughout the rest of the section, the integrals are curvilinear integrals taken along the trajectories.

110

The examination of Figure 34.2 gives rapid and convincing information. In A'r-_ _ ~~Ct, if the point C at which the trajectory intersects the curve y = F(z) is to the left F(x) of the point H, the curvilinear integral A B dA > 0, since both F(z) and dy are negative :.t in that region. As the pOints A and B move away from the origin the cont~ibution of the curvilinear integral in this region is deB B' " - - creasing monotonically while being positive. B" Since F(z) remains negative, it follows that Iy - F(z)1 increases indefinitely. FurtherFigure 34.2 more, since g(z) is bounded and positive, we see from [34.2] that dy decreases. Since F(z) increases monotonically in the region to the right of H, the contribution to the curvilinear integral along the arc G'C'K' is negative. B Hence, the curvilinear integral dA decreases monotonically as A and B move :.t away from 0 since its positive part, due to the elements to the left of H, is bounded while its negative part due to the elements situated to the right of H, is negative and increases monotonically. Hence, there exists one, and B only one, position of the points A and B on the y-axis for which :.t dA = 0 and for which there corresponds a unique closed trajectory. This fact is expressed by the condition that A " t - - -__

l

i

1

B

jF(z)dy = 0

[34.61

A

35. FIRST ASYMPTOTIC CASE: J,l« 1 We shall now consider Equation [32.2] in which g(z) == z. Lienard's form [33.6] it is

zdz + 7/dy - F(z)dy == 0

In [35.1]

In the asymptotic case, when F(z) is very small, we may replace F(z) by J,lF(z) where J,l is a small number. If J,l = 0, the trajectories are circles Z2 + 7/ 2 = constant. If J,l 0, and a periodic solution exists, clearly a closed trajectory will differ b~t little from a neighboring circle. We propose to determine the Circle in whose neighborhood there exists a closed trajectory. The criterion [34.6] which has been obtained geometrically will serve this purpose.

*

111

= R sin~;

Making use of the polar coordinates z can be written as dR d~

y

= R cos~,

[35.1]

F(Rsin~)Rsin~

= -

J,l

R - IIF(R sin~)cos~

or dR

d~ ;::. - J,l F(R sin~)sin~

[35.2]

Therefore, [34.6] becomes

. f F(R sin~) R sin~ d~ = 0 o Thus, for Van der Pol's equation, f(z) 1 -

~

= z2

f7l'

3

x - 1; F(x) =3x, [35.3] is

R3

F(Rsin~) sin~ d~ = -

o

[35.3]

8

-

R

-2 = 0

[35.4 ]

Hence when II is small, R = 2 is the radius of the circle in the vicinity of which there exists a closed trajectory. Furthermore, a simple approximation based on the smallness of J,l shows that along the closed trajectory the angular velocity Wo ;::. ~ is constant. It is to be observed that we have obtained a first order solution of Van der Pol's equation by a mixed method, viz., the criterion [34.6] has been established by means of a geometric method and from that point the argument has been analytic. It will be shown in Part II that the same result can be obtained by purely analytical methoas.

36. SECOND ASYMPTOTIC CASE: THE PARAMETER

J,l IS LARGE The case in which J,l is large is of great importance in the applications, e.g., the so-called RC oscillations in modern thermionic circuits. The oscillations are now strongly distorted, showing the presence of numerous harmonics. For this reason, the application of the analytical methods of approximation given in Part II, become too laborious owing to the poor convergence of the Fourier expansion representing the osoillation. Lienard did, however, develop an approximation method yielding considerable information regarding the wave forms of the oscillations for large values of J,l. We now propose to show this method. Setting y :: J,lZ in Equation (33.6] and replacing F(z) by IIF(z), as in Section 35, we obtain

[36.1]

112

For p very large this equation can be written approximately

[36.2] The integral curves in the (%,I)-plane consist of the principal branches I = F(%) and I = constant, joined by short arcs. Referring now to Equations [33.7], we shall examine the relative order of magnitude of the terms under the assumption that p is large; thus, terms like ~C, where C is finite, are treated as small quantities of the first order. Equations [33.7] in the new variables (%,1) are dt = wodt

= -

p

dz

%

[36.3] dt = w dt = o

d%

p[z _ F(%)]

From the first Equation [36.3] it follows that dl/dt - 1/p, where the symbol - means "of the order of," that is, of the first order, since % is finite. On the other hand, Since I ~F(%) by [36.2] dz ~ dF dx dx

= fez)

which is finite. But dl = dl dt. hence dx _l Since the velocity remains dz dt dz' , dt p. small in this finite interval, we conclude that it takes a relatively long time to traverse it. Thus, we can conclude that on the branch I = F(z) the representative point R in the (z,I)-plane moves slowly; its velocity is small, of the order of l/p. Clearly, the acceleration d 2z/dt 2 is then of the second order of smallness, 1/p2, and can be neglected in the differential equation

z + Jiwof(z) z + w;x which, moreover, is Equation [32.2] since to the first order only, [36.4] becomes

Wo

2

= 0

= 1.

IJwof(%)x + wo x = 0

[36.4] Hence, limiting the terms

[36.5]

ThUS, under the assumed approximation, the second order Equation [36.4] can be replaced by the degenerate Equation [36.5] of the first order, with an error of the order of 1/p2. This subject will be considered more fully in Part IV. One must ascertain the order of the approximation I - F(z) ~ 0, arising from the use of (36.2] instead of [36.1]. First, from the comparison ot [36.1] and [36.2] it tollows that the approximation is ot the order of

113

1/~2. One can also see this from the second Equation

[36.31, since dx/dt - l/p.

We thus infer that on the branches z ~ F(x) of integral curves, the representative point R moves slowly with a velocity of the order of 1/~ and an acceleration of the order of 1/~2. It takes, therefore, a relatively long time to traverse these arcs in view of the fact that x is finite. We shall now investigate the motion on the other typical arcs for which dz ~ 0, that is z ~ constant; z - F(x) is finite and stays away from zero. From the second Equation [36.31 we observe that dx/dt - p, that is, d 2 x/dt 2 - p2, so that we can now neglect the term W 02 x in [36.41 and write

[36.61 From [36.1] it follows that dz/dx - 1/p2; hence, in the (x,z)-plane the slope of the curve is very small for this second characteristic branch, dz ~ O. This almost horizontal branch in the (x,z)-plane is traversed very rapidly, since dx/dt - ~ and p is very large in this case.

37. LIMIT CYCLES IN THE VAN DER POL AND LIENARD PLANES We may now summarize L16nard's principal results and compare them with the earlier results obtained by Van der Pol. The experimental results in connection with Van der Pol's equation in the (x, t)-plane are shown in Figure 37.1 for three values of the parameter ~ (1). For the representation of the trajectories of [37.11, Van der Pol uses the (x,z)-plane with which we were concerned throughout the first four chapters of this report. Van der Pol originally obtained the phase trajectories for (37.1] by the graphical method of isoclines; these results are shown in Figure 37.2. ,,=0.1 (0)

" =I

" = 10 (el

Figure 37.1

114

c p

=0.1 palO

(b)

Ie)

Figure 37.2 In fact, Liapounoff's equations of the first approximation are x - x + py; the characteristic equation is

·s 2

-

#

S

+1

= 0

= y, y = [37.2)

For 0 < # < 2 the roots are conjugate complex with a positive real part, and the origin is, thus, an unstable focal point. For # = 0.1, the limit cycle C differs very little from a circle of radius 2 and the trajectories approach it both from the inside and the outside as shown in Figure 37.2a. For p = 1 .0, the origin is still an unstable focal point but the limit cycle C differs considerably from a circle as shown in Figure 37.2b corresponding to the expertmental curve in Figure 37.1b. For p = 10, the roots of [37.2) are real and positive; the origin is an unstable nodal point. The trajectories in the (x, i)-plane leave the origin along definite directions and approach the limit cycle without spiralling as shown in Figure 37.2c. It is observed that the limit cycle is now strongly distorted and acquires an elongated narrow form; the corresponding oscillation in the (x,t)-plane, Figure 37.1c, exhibits the presence of numerous harmonics. If we consider now the shape of trajectories in the Lienard plane [x, v + F(x)], where v = dx/dt, the situation is somewhat different, as shown in Figure 37.3 taken from a paper by Ph. LeCorbeiller (29). For # = 0.1, Figure 37.3a, the limit cycle C is again nearly a circle as in the (x,x)-plane. For larger values of p, Figures 37.3b and 37.3c, the limit cycle undergoes a

115

z

A

B

z

c

It

II:

0.1

(0)

o It

z

x

Vo

G II :

1.0

(b)

F II:

10

E (e)

Figure 37.3 different kind of deformation as compared to that in the (x,x)-plane and acquires an almost rectangular form for p = 10 as shown in Figure 37.3c. One recognizes in this figure the second asymptotic case, p » 1, of Lienard discussed in the preceding section. The peculiar motion of the representative point on a distorted limit cycle as shown in Figure 37.3c represents a typical case of relaxation oscillations for large values of the parameter p appearing in a great number of appl1ca tions.

CHAPTER

vn

CYLINDRICAL PHASE SPACE

38.

GENERAL REMARKS. LIMIT CYCLES OF THE FIRST AND SECOND KIND In problems dealing with the motion of a rigid body about a fixed axis, e.g., pendulum rotor of an electric machine, etc., the position and velocity are determined uniquely in terms of a certain angle ~, to within a multiple of 2", and its derivative~. One may thus assert that the dynamical state of the system is represented by a point on a cylinder with the cylindrical coordinates ~, the curvilinear coordinate measured along the arc of a right section of the cylinder, and % =~, the coordinate measured along the generating line. This mode of representation will be discussed here. The general properties of this phase space are about the same as in the case of a plane. Thus, we may have singular points, limit cycles, etc., on the Burface of the cylinder. There will appear, however, a special feature due to the fact that we are representing the trajectories on a cylinder and not on a plane. Consider the simplest case of the motion with constant velocity z = ~o. In the phase plane this motion is represented by a straight line parallel to the ~-axis. Clearly no periodicity is involved in this case. If, however, the phase plane is wrapped on a cylinder the trajectory in this case becomes a right circular section and is thus closed. We thus conceive of a periodicity "around the cylinder" although we lose the grasp of this periodicity if we unwrap the cylinder on a plane. We shall call this particular form of periodicity, inherent in the form of this particular cylindrical phase space, as periodicity of the second kind. It is apparent that this periodicity corresponds S to a closed trajectory such as S in Figure 38.1. By a slight extension of the definition of the limit cycle we can say that such a closed trajectory is a limit cycle of the 8econd kind if it is approached either for t = + ~ or for t = - ~ by non-closed trajectories going around the cylinder in the manner of the curve S'. The limit cycles in the phase plane with c which we have been concerned previously appear on Singular the surface of the cylinder as closed curves C Point bounding a region of the cylindrical surface; we shall call them limit cycle8 of the firBt kind. It 1s apparent that these cycles represent the Figure 38.1

117

same situation which we have already studied in the phase plane; the only difference lies in the fact that the plane is wrapped on the cylinder. Furthermore, the theory of Poincare remains applicable to these limit cycles. Thus, for example, we can assert that inside a closed trajectory of the first kind there must exist a number of singularities with the algebraic sum of their indices being equal to + , and so on. We cannot, however, assert this with respect to the closed trajectories of the second kind since these do not bound off any region. The important property of these closed trajectories lies in the fact that they have a period 2~ relative to ~ which does not depend on time. This means that such a limit cycle is fully described when the coordinate ~ varies by 2~. It may be observed that through a suitable transformation of coordinates the cylindrical phase space may be replaced by a plane with the origin left out. In fact, if we set p = e', where % = ;P, then the point M(~,z) of the cylinder goes into a point M'(~,p) of the plane and the correspondence between M and M' is one to one and continuous in both directions, that is, topological. Since % varies between plus and mlnus~, p varles between 0 and +~, zero being excluded. Thus the cylinder is transformed lnto the whole plane with the origin left out. Through this transformation the circle % = 0 of the cylinder goes into the circle p = , in the plane and the generating lines of the cylinder are represented in the (~,p)­ plane by the half-ray, OR, passing through the origin. The closed trajectories of the flrst kind are those which do not go around the origin 0, e.g., curve C in Figure 38.2, whereas those of the second kind are those which go around it, such as curve S. Intuitively speaking, the transformation M + M' is equivalent to a flattening out of the cylinder into a plane. Flgure 38.2 This plane representation makes it clear that the usual features of the planar systern should be expected here. In fact, in the planar representation, the closed trajectory of the second kind acquires a familiar feature, that is, the orlgin behaves now in all respects as a polnt singularity.

39.

DIFFERENTIAL EQUATION OF AN ELECTROMECHANICAL SYSTEM Consider the differential equation A9

+

B9

+

f«(J)

=

M

[39·1 )

118

where A, B, and M are constants and I( 6) is a periodic function of 9. following, we take 1(6) = C sin 6. In this case A9

+

+

Be

In the

[39.2]

C sin 6 = M

Equations [39.1] and [39.2] are non-linear on account of the presence of the terms 1(9) and C sin 9. As an analogue of [39.2] one may consider a physical pendulum with constants A, B, and C, acted upon by a constant moment M. The general character of the motion can be, in this case, either oscillatory or rotary ~ccord­ ing to the relative magnitude of M, "dead beat" or with dying-out oscillations, etc. Our purpose will be to consider the solutions of this differential equation in a cylindrical phase space in which the existence of periodic traJectories of the second kind will be established. Introducing the "dimensionless time" .,. = <.lot = t and putting = a > 0, A! 2 = ~ = {J ~ 0, [39.2] becomes . 0

JA

Vi

2

d 9 d9 W + °Cft +

. sm9 - {J

=

[39.3]

O·

We shall make use of this form in what follows.

40. CYLINDRICAL PHASE TRAJECTORIES OF A CONSERVATIVE SYSTEM Equation [39.3] is equivalent to the system d6

-

dt

dz = crt

= z·

•

az -

. 6 + {J

[40.1 ]

sm

Hence dz d6

=

For a conservative system a = O. we have

~ where

Z2

{J - 0 Z -

= {J9

= ±

[40.2]

In this case [40.2] can be integrated and

+

l k is a constant of integration. z

sin 9

Z

cos 9

+ ~

[40.3]

k

Hence

V2(P9

+

[40.4]

cos6)+k

Several cases are pOSSible, according to the value of {J. 1.

k

*

{J = 0, hence, M= O.

For k

= - 2, z = d9/dt = 0,

and 9

= 0;

for

> - 2 these are closed periodic trajectories around the singular point

Since 110 cOlltusioll can arise, the usual Ilotatiollu d 28/d' a WId d8/d'. inutead will be resumed.

or d l 8/dT a

IlIld d8/dT,

119

YI

Separatrix I I I I I I I I I zI I I I Separatrix I .--.r~....

Saddle Point

II

(0)

Saddle Point

(b)

Figure 40.1

Saddle Point Vortex Point

9

Figure 40.2

6 = 0, Z = O. The latter is, thus, a vortex point. The periodic trajectories form a continuum of closed curves for the interval - 2 < k < + 2. For k = + 2, we obtain separatrices limiting "the island" of closed trajectories in the phase plane. The separatrices issue from saddle points for 6 = ± ". Thus far the discussion has been analogous to that of Section 12. Let us wrap the plane figure on a cylinder of unit radiUS, see Figure 40.2; the points 6 = - " and 6 = + " will coincide on the cylinder. The trajectories of the second kind are those which go outside the separatrices. On the cylinder they appear as surrounding the cylinder but not the singularities.

p * O.

Let z = ± Y2(P6 + cos 6) + k =± Y2Yl + k. The curve Yl in Figure 40.1a has maxima and minima rising on the average due to the presence of the term pO. If the curve z = z(O) in Figure 40.1b is wrapped on the cylinder, the trajectories have the appearance shown in Figure 40.2. The periodic trajectories do not go around the cylinder, hence, they are not of the second kind. If P= 1, the curve Yl = z(P6 + cos 6) has only an inflection point but no maxima or minima; for p > " the curve Y1 is monotonic without inflection points. One can construct these curves in a plane and, by wrapping them on the cylinder, form an idea of their appearance. No periodic trajectories exist, and the use of a cylindrical phase space offers no particular advantages over the phase plane. 2.

41.

CYLINDRICAL PHASE TRAJECTORIES OF A NON-CONSERVATIVE SYSTEM For a non-conservative system, a ~ 0 and we must, therefore, consider [40.2). This equation cannot be integrated directly. We assume in

120

th1s and 1n the next sect10n that p • 1. the equat10ns %

= 0;

P-

The singular pOints are given by [41.1]

sin 8 = 0

They ex1st only for p < 1 in which case the second Equat10n [41.1] admits two groups of roots [41.2]

where 80 = s1n- 1p and 80 assumes the princ1pal value, that is, 0 < 80 < "/2. Accord1ngly, there are two groups of s1ngular po1nts: Ai with coordinates (8 = 8" % = 0) and Aj with coordinates (8 = 8i , I = 0). It 1s suffic1ent to consider two s1ngular p01nts, one 1n each group, say, A iO (8 = 80 , % = 0); A jo (8="-8 0 , %=0).

= 2k"+

Group Ai. For th1s group 8, departure E from 80 • We have 8 = 80 + 1.

d% d8

= dl = P -

d8

E,

80 ,

= dE,

a% - sin(8 0

= O.

I

Consider a sUght

and, by [40.2]

+

E)

[41.3]

dE I Develop1ng sin (8 0 + E) 1n this equat10n, one has sin (8 0 + E) = sin 8 0 + E cos 80 • Since, for the s1ngular point, p = sin 8 0 , we have from the equation of the f1rst approximation d%

= - al - E cos 8 0

[41.4]

%

corresponding to the system d% dt

= - al - E cos 6 0;

dE

-

dt

= I

[41.5)

The characteristic equat10n is S2 +

as + cos 60 =

0; a

>

0

[41.6)

Using the criterion given in Section 18, we find that, if 0 < 4 cos 80 , the singularity is a stable focal point; if 0 2 > 4 cos 80 , it is a stable nodal pOint. 2

2. 6J

,

=

Group A j • For this group of s1ngular1t1es, the coordinates are (2k + 1)" - 60 , z = O. Proceeding as before, one finds d%

-=

-01

+

Ecos8 0

[41.7)

cos 8 0 = 0

( 41.8]

I

and the characteristic equation is S2

+ as -

121

Its roots are 2

SI,2 = -

a ± .V'/a "2 4

+

[41.9]

COS(JO

The singularities of this second group Aj are thus saddle pOints. 42.

CLOSED TRAJECTORIES OF THE SECOND KIND IN NON-CONSERVATIVE SYSTEMS We shall now investigate periodic solutions of the second kind of a non-conservative system, a > O. Two cases are to be distinguished. 1. p > 1. In this case there exists exactly one periodic solution with the period 2~. In order to establish the existence of such a periodic solution it is sufficient to prove the existence of two particular solutions ZI((J) and Z2((J) such that for an arbitrary (J, Zl((J + 2~) ~ ZI((J), Z2((J + 2~) ~ z2(9). The existence of a solution satisfying z((J + 2~) = z((J) follows then by continuity reasons since there are no singular points if p >1. From [40.2] it is noted that the equation of the isocline dz/d(J = 0 is z = P - sin (J a

[ 42 .1 ]

Curve [42.1] in the (z,8)-plane is, therefore, a locus of pOints at which dz/d(J = O. This curve crosses the axis of (J, i.e., z = O. only if P < 1. If P > 1 the curve is above that axis. These two cases are shown in Figures 42.1a and 42.1b. From [40.2], it follows that for z = O. dz/d(J =~. Moreover, the shaded areas in Figures 42.1a and 42.1b correspond to regions in which dz/d(J > 0; the non-shaded areas correspond to dz/d(J < O. In order to find one of the two particular solutions. it is necessary to express the fact that this solution goes above the sinusoid of Figure 42.1. Since in this region dz/d(J < 0, and hence, ZI((JO + 2~) < ZI((JO)' One solution of this type can be found if we take Zl((JO) > (1 + p)/a, because this initial point lies certainly above the curve z = (P - sin (J)/a, that is, In the region in which dz/d(J < O. This solution satisfies the above condition required for ZI'

1

9

(0)

Figure 42.1

(b)

122

In order to find the second particular solution %2 satisfying the condition %2(8 0 + 2") > %2(8 0 ), consider' the minimum paint A on the curve of Figure 42.2. For this point 8 = "/2 and % = (p - 1 )/a. A curve %2(8) issuing from A is in the region where d%/d8 > 0; hence, the curve %2(8) is rising and must intersect the sinusoid at a certain point M. Since the sine curve is the locus of d%/d8 = 0, clearly, M is a maximum point for %2(8). Beyond the point M, the curve %2(8) decreases. Furtherz more, the next intersection M1, cannot be lower than the point B, since at the point of intersection M, the curve %2(8) has a horizontal tangent. It is thus seen that the condition %2 (8 0 + 2") ~ '2(8 0 ) is fulfilled, and hence, by virtue of the 9 B continuity of the sequence %(8) in the interval Figure 42.2 (%1'%2) there is a periodic solution %0 in the interval (%1'%2). It now will be shown that this periodic solution %0 is unique. In fact, integrating (40.2] between 8 1 and 81 + 2" one has 9.

~

~z2(81)

z2(8 1 + 2") -

= -

+ 271"

aJzd8 + 2"p

(42.2]

9.

If the solution is periodic then z(8 1 + 2") = %(8 1), whence 9.

J

+ 271" %

d9 =

2:

fJ

[42.3]

9.

Equation [42.3] expresses the condition of periodicity. Assume now that there are two periodic solutions '01 (8) and '02(8). Since there are no singularities, these solutions cannot intersect each other, so that one of them is always greater than the other, e.g., %01(8) > %02(9) for all 8, whence 9.

+ 271"

J

'01 (9)

9.

d9

+ 271"

> JZ02 (9) d8

This, however, is impossible by virtue of [42.3]; hence, the periodic solution is unique. 2. p < 1. Consider again the diagram of Figure 42.2. Proceeding as before, one can establish first that there exists a solution '1 (8 0 + 2,,) ~ %1(8 0 ). In order to see that there exists another solution %2(9 0 ), such that %2(8 0 + 2,,) ~ '2(8 0 ), it is oonvenient to consider two integral curves r 1 and

123

r 2 on the surface of the cylinder passing through two adjoining saddle points, singular points of group A j , separated by 2". The curve r l has the same slope at the point Ajl as the asymptote of the positive slope and r 2 has the same slope at the next saddle point Aj2 as the asymptote of the negat1ve slope. One can show that, for sufficiently small values of 0, the cond1tion z2(6 0 + 2") ~ %2(6 0 ) is fulfilled. Hence, for sufficiently small values of o and of the constant B, Equation ('9.2], a periodic solution of the second kind exists. Physically this condition is obvious; in fact, since the system is acted upon by a constant moment of forcelf, one can readily see that, if the damping is not too great, the rotary motion of the pendulum may become periodic when the energy communicated by the moment lfper cycle is just equal on the average to the energy dissipated by damping. If the damping is Just slightly below or above this critical value the rotary motion will become either damped, i.e., the trajectory approaches a focal point, or will continue with an increasing angular velocity. In the latter case a state of periodicity of the second kind will eventually be reached when the energy communicated to the system by the constant moment M will be Just equal to the energy dissipated by damping. As a result there will appear a periodic trajectory of the 8econd kind closed around the cylinder and not enclosing any point singularities on its surface. The topological picture on the surface of the cylinder will thus have the various aspects shown in Figure 42.3. The separatrices issuing from a saddle point may either approach a limit cycle of the second kind extending around the cylinder or approach a stable focal I I 9 point. In the former case the originally Limit Cycle unstable motion will have a tendency to ap"'--1-..........r- of the Second Kind proach a periodic rotary motion; in the latter case the motion will approach a definite Seporatrices angle around which the oscillations will Focal Point .r.;;;r--t-_ Sad die gradually die out. There exist also sepaPoint ratrices of the second kind, i.e., turning around the cylinder, which approach a saddle , ... point; in this case the trajectory will ap,--proach the saddle point asymptotically. ---Since the latter is unstable, it will depart from it following one of the other two asymptotes either to a periodic trajectory of Figure 42.' the second kind or to a stable focal pOint.

--

----'-'

----

--_.

124

From this study it follows that the advantage of analyzing a phenomenon of this kind in a cylindrical phase space is particularly marked when the process extends over a number of periods of the system.

43. OSCILLATIONS OF A SYNCHRONOUS MOTOR A noteworthy Situation, calling also for a cylindrical phase space, is the investigation by Vlasov (30) of the oscillations of a synchronous motor around its average angular velocity. As is known, a synChronous motor ls a mechanical system, a rotor, with one degree of freedom ~ about its axis of rotatlon, dr1ven normally at a constant angular velocity Wo by the electromagnetic driving torque T produced by a rotatlng magnetic field. This field is excited by a polyphase stator winding and "locked ll in synchronism with the corresponding fle1d of the salient poles on the rotor excited by a direct current. The manner in which both flelds on the stator and rotor are locked is represented graphically in Figure 43.1 by the lines of magnetic inductlon. It is observed that these lines cross the alr gap ob11quely as ~o""I---- ~

= II' --~

~

T

\\\\\\ ~\

It

Stotor"\

.,

IIj~L

Rotor"!

Figure 43.1 shown; this fact, on the basis of the Faraday-Maxwell theory concerning the pondermotlve forces actlng along the lines of magnetic induction, accounts for the mechanical torque applied to the rotor. It is seen, thus, that for a g1ven value of the res1sting torque Tr , there corresponds a glven lag ~o of the rotor poles behind the corresponding stator poles. If the resist1ng torque Tr suddenly increases, for instance, the relat1ve angle ~ increases which accounts for a corresponding increase of the tangential component of the lines of magnetlc induction in the air gap owing to a greater obliquity of these lines in the air gap. This occurs until a new equilibrium between T and Tr 1s reached. This transient state is, in all respects, similar to that which would exist if one had two rotating mechanical systems connected by springs. It 1s apparent, however, that 1f the reSisting torque Tr becomes

125

so great that the relative angle ~ reaches the value ~ = ", this similarity with the above mentioned mechanical picture ceases because an S-pole on the stator comes in front of an S-pole on the rotor and instability results in the interval (",2~) inasmuch as the like poles repel each other. Furthermore, the rotor has a tendency lito drop out Figure 43.2 of step" and slow down still further. It may, however, "get into step" again when Nl comes in alignment with S2 since a stable configuration reappears exactly similar to that which existed originally. It follows that, if the disturbance causing a change d~ of the angle ~ is such that the new equilibrium pOint A is within the region of stability, there appears a relative traJectory approaching the point A in the manner of a spiral approaching a focal point; see Figure 43.2. If, however, the disturbance is so great as to carry the representative point to a point A' on the other side of the saddle point SI limiting the region of stability, the spiral trajectory originating at A' will approach another focal point B situated at an angular distance ~ = 2" from the former point of equilibrium A. It may happen under special conditions, which were produced electrically by Vlasov but not commonly encountered in connection with the industrial synchronous motors, that the rotor runs at a certain speed below the synchronous speed. In such a case, the rotor poles "slip" continuously behind the stator poles and the performance in this case is called asynchronous. It is apparent that this mode of operation is possible only when a certain asymmetry exists in the driving torque so that a continuous slipping of rotor poles behind the stator poles is accompanied by a certain average value of the driving torque for the angular period ~= 2". Thus it is clear that a steady state of asynchronous performance corresponds to a limit cycle of the second kind. It is to be noted also that, under certain conditions usually eliminated in industrial motors, a steady sustained oscillation of the rotor speed about its normal synchronous speed Wo may arise. It is generally impossible to eliminate an oscillation of this kind by any damping devices and, therefore, it is particularly objectionable. These self-sustained oscillations are characterized by the existence of limit cycles of the first kind. With this physical picture of the phenomena involved we can now proceed with a brief outline of Vlasov's theory.

126

The differential equation of motion of a synchronous motor according to Dreyfuss (31) is 2

+

d " -I -:i7f p dt

mpEoVocosp {[. ( +p ) - sm" 2wx

En. Vo

~smp

En sin4p Vo 4wcosp

--><.

d" dt

+

[43.1) where I is the m is the p is the Eo is the steady

moment of inertia of the rotor, number of phases, number of pairs of poles, electromotive force induced per phase of the stator in a state, y = dEo/di e is the tangent to the saturation curve Eo = /(i e ) where ie is the d-c eXCitation, Vo is the voltage applied to one phase of the stator, 7r - " is the angle between vectors E and V, p = tan-l~ where r is the ohmic resistance and x is the resistance per phase of the stator, M is the driving moment, Re is the resistance of exciting winding, and w = 21(/ is the angular frequency where / is the frequency.

Introducing the constant factors mp 2 kV.

E

2

o -- K I 0 cosp = a -V 2pw'" 0 ' "

and the new variables dimensionless form 2

dd T ¢J2

+

sin ¢J

~

+ [;

= ,,+

2 I

p

a my 2a = b 4w ' 2 W ..."'R e --

-

at

t

= T,

c

[43.2)

,

one transforms [43.1] into the

cos p sin ¢J (sin ~ - 2 K sin p) -

_ Kb sin 4 P ] d¢J cosp dt

=

(J

Kcosp

+ Ksinp

[43.3)

This equation corresponds to the system d¢J dT

-=z

[43.4) dz

(J

dT = Kcosp

+ Ksinp

- sin¢J

+

[Kb sin4p - KC cosp sin¢J(sin¢J - 2KSin p)] z cosp

127

Consider now a cylindrical phase space with the axis of the cylinder parallel to the z-axis. This is an appropriate representation since the form of [43.4] does not change when ~ is replaced by ~ + 2Kn; the ~-axis is curvilinear along the circular cross section of the cylinder. Equations [43.4] can be written as

dz

d",

- = z · d.,. d.,. •

v - sin~

+ f("')z

+

>

=

[43.5]

where v =

fJ Kcosp

K sin p

0

and

f(~)

= Kb sin4p -

cosp

KC cospsinp(sin~ - 2Ksinp)

[43.6]

Equating the right-hand terms of [43.5] to zero, one obtains the singular pOints. If v > 1, singular pOints are absent. If v = 1, there is one singular point of a higher order representing the coalescence of two simple singular points. If v < 1, there are two singular points. Consider the last case. The coordinates of the singular pOints are clearly ;1 = sin-Iv, z = 0 and ~2 = "- ;1' z = O. In forming the equations of the first approximation, Section la, it is assumed that f(~I) ~ 0, f(~2) ~ O. Setting ~ = ~1 + E, d; = dE and observing that sin; = sin (~1

+ E)

::::s

sin "'1

+

E cos "'1

[43.5] then becomes [43.71 The characteristic equation is S2 - f(~I)S + cos ~l

= 0, and its roots are [43.8]

According as the sign of f 2("'I) - 4 cos ~1 ~ 0 one has either a nodal point or a focal point. In practice one usually encounters focal points. In such cases the equilibrium is stable for f(~I) < 0 and unstable for f(~l) > o. Following a similar procedure one finds that the second singular ~oint (~2' z = 0) is always a saddle pOint. Thus there exist two types of singularities situated on the ",-axis which have been already investigated in Section 41. The only additional feature is the existence of a threshold f(~) = 0, separating stability [f(~) < 0] from instability [f(~) > 0].

128

It is to be observed that in practical problems the quantities K= Eo/Vo, y = dEo/di o, and b may be considered as small quantities of the first order. Furthermore, from expressions [43.2] one ascertains that c/Kis also small. Thus, generally, the value of !(~) in practice is small and can be put in a form !(~) = p!o(~) where p is a small parameter. There is a certain degree of arbitrariness in the choice of this parameter, depending on the relative order of magnitude of the small quantities K, y, b, . • ., charac terizing each particular case. Under these conditions, [43.5] can be put in the form

:~ =

Z;

:~

= II -

sin~ + p!o(~)z

[43.9]

The system [43.9] is conservative for p = 0 although non-linear and can be analyzed by the method indicated in Section 40. The phase trajectories are given b J the equation dz II - sin~ d~ = z

[43.10]

which is readily integrated, giving z = ± V2(cos~ + J)~ + k); h = constant

[43.11]

This equation has the same form as [40.4]. The conclusions thus remain the same, viz: there are no trajectories of the second kind, although there may be closed periodic trajectories surrounding a vortex point and forming a continuum, "the island, II limited by separatrices issuing from saddle points. PhYSically this mean& that there will be oscillations of the rotor about its uniform speed of rotation. These oscillations, in general, are not of the IIrelaxation type ll but are ordinary oscillations of a quasi-linear type, see Section 11, depending on the initial conditions such as may arise from accidental disturbances. Oscillations of this kind are not important, however, because they are rapidly damped out by the squirrel-cage damping arrangement, the effect of which is not considered here. We shall now investigate the system [43.9] assuming it to approximate a conservative system, that is, taking p ~ 0 but very small. Physically this means that the performance, instead of being synChronous, becomes asynchronous, that is, the motor begins to "slip" and settles on a sub synchronous speed. The equation of phase trajectories becomes dz dt/J

=

J)

-

8in~

z

+

p!o(t/J)

[43.12J

129

It is seen that when 1%1 is small the trajectories differ but little from those of [43.10], that is, the separatrices still exist and the area of the "islands" of periodicity changes slightly. No trajectories of the second kind exist in the neighborhood of the ~-axis. Hence, in order to ascertain the existence of closed trajectories of the second kind, we have to investigate the regions of the cylindrical phase space when 1:1 is large. Equation [43.12) in such a case becomes

d%

d,p

Figure 43.3

\ ~

,.,Io(,p)

and the condition of periodicity of the second kind %(2n) -

%(0) =

IJrlo(~)d,p

= [Kb

:84:

+

2CC08P

8in2p] 27T

[43.13]

Two cases will be considered according to the sign of the right-hand term of [43.13]. sin p4p + 2 C cos p sin2 p < O. As a consequence we a. Kb cos have Z(27T) - %(0) < O. Thus, when 1%1 is large, % > 0, the trajectories approach the ~-axis. On the other hand, when % is small and greater than zero, they depart from the ~-axis as do the trajectories of the conservative system [43.10] for any % > O. Since all point singularities are situated on the ~-axis, there exists at least one limit cycle of the second kind and it is stable. The same reasoning applied to % < 0 shows that there is no limit cycle. b. Kb ~~~ ~p + 2 C cos p sin2 p > O. Then %(27T) - :(0) > O. In this case the trajectories depart from the ~-axis when 1%\ is large, % > 0, and also when % is small, as was shown previously. Hence, no closed trajectories of the second kind exist for % > o. If, however, % < 0, the trajectories approach the ~-axis when I~I is large and depart from it when Izi is small. Hence, for z < 0, there exists at least one limit cycle of the second kind. This situat10n 1s 11lustrated in Figure 43.3 wh1ch is self-explanatory.

By a more elaborate analysis not discussed here, involving the theory ot critical points, Vlasov has shown that, in addition to the point singularities and limit cycles ot the second kind, the operation ot a synchronous motor may a180 exhibit limit cycles of the first kind. In ordinary industrial motors these steady oscillations ot a "relaxation" type are generally eliminated by a suitable design. Vlasov succeeded in producing them by means ot a special experimental arrangement.

131

REFERENCES liOn Relaxation Oscillations, II by B. Van der Pol, Philosophical Magazine, Vol. 2, November 1926. (1)

(2) IISur les courbes d~finies par une ~quation differentielle" (On Curves Defined by a Differential Equation), by H. Poincar~, Ouevres, GauthierVillars, Paris, Vol. 1, 1928.

(3)

ilLes m~thodes nouvelles de la mecanique celeste II (New Methods in Celest1al Mechanics), by H. Poincare, Gauthier-Villars, Paris, Vol. 1, 1892. (4) "Probl~me general de la stabi11U du mouvement" (General Problem of Stab1lity of Motion), by M.A. L1apounoff, Annales de Toulouse, paris, Vol. 9, 1907. This is a translation from the Russian by E. Davaux.

(5)

IISur les courbes definies par des equations differentielles ll (On Curves Defined by Different1al Equations), by I. Bendixson, Acta Mathematica, Vol. 24,1901.

(6) IIQuelques thl§or~mes generales sur Ie mouvement des systemes dynamiques ll (Some General Theorems on the Movement of Dynamical Systems), by G.D. B1rkhoff, Bulletin Societe Mathematique, France, Vol. 40, 1912. (]) IIVibration Problems in Engineering, II by S. Timoshenko, D. Van Nostrand Company, New York, N.Y., Second Edition, 1937, Chapter 2.

(8) IIMechanical Vibrations," by J.P. Den Hartog, McGraw-Hill Book Company, New York, N.Y., Second Edition, 1940, Chapters 7 and 8. (9) liThe Engineer Grapples with Non-Linear Problems," by Th. Von Karman, Bulletin of the American Mathematical Society, Vol. 46, August 1940. (10) IITraite de la mecanique rationelle" (Treatise on Theoretical Mechan1cs), by P. Appell, Vol. 1, Paris. (1') IIProblems of Anti-Rolling Stab111zat1on of Ships," by N. Minorsky, American Society of Naval Engineers, Vol. 47, February 1935. (12) "Froude1s Pendulum," by S. Strelkoff, Journal of Technical Phys1cs, USSR, (Russian), Vol. 3, 1933. (13)

"Theory of Sound," by Lord Rayleigh, London, Vol. 1, 1894 ,

p. 212.

(14) "Sur llequil1bre d1une masse fluide animee d1un mouvement de rotation ll (On Equ1librium of a Rotating Fluid Mass), by H. Poincare, Acta Mathematica, Vol. 7.

(15)

IIFigures d1equilibre d1une masse fluide ll (Figures of Equilibrium of a Fluid Mass), by H. Poincare, Paris, 1903. IIDHferentialgleichungen ll (Differential Equations), by L. Bieberbach, J. Sp~inger, Berlin, Third Edition, 1930, Part 1, Chapter 4. (16)

(17) Article by W. Kaufmann, Annalen der Physik, Vol. 2, 1900. (18)

ilLes cycles l1mites de Poincare et la theorie des oscillations auto-entretenues (Limit Cycles of Poincare and the Theory of Self-Excited Oscillations), by A. Andronow, Comptes Rendus, PariS, Vol. 189, 1929. ll

(19)

IITheory of Oscillations, II by A. Andronow and S. Chaikin, Moscow, (Russian), 1937.

(20)

IlLe battement du coeur considere comme oscillation de relaxation" (Heart Beat Considered as a Relaxation Oscillation), by B. Van der Pol and M. Van der Mark, Onde Electrique, 1928, p. 365.

(21)

IlLe~ons

sur la theorie mathematique de la lutte pour la vie ll (Lectures on the Mathematical Theory of the Struggle for Life), by V. Volterra, Gauthier-Villars, Paris, 1931.

(22)

IIA General Equation for Relaxation Oscillations,1I by N. Levinson and O.K. Smith, Duke Mathematical Journal, Vol. 9, June 1942.

(23)

IIExperimentelle Untersuchung des Phasenraumes Autoschwingender Systeme (Experimental Investigation on the Phase Space of Self-Excited Oscillatory Systems), by V. Bowshewerow, Technical Physics, USSR, Vol. 2, 1935. ll

(24)

IIGew!:Shnl1che DHferentialgleichungen bel1ebiger Ordnung ll (Ordinary Differential Equations of Arbitrary Order), by J. Horn, Leipzig,

1905. (25)

liOn Mechanical Self-Excited OscUlations,1I by N. Minorsky, Proceedings of the National Academy of SCiences, October 1944.

(26)

"Note sur une ancienne experience dlelectriciU" (Memoir on an Old Electrical Experiment), by M. Janet, Annales des Postes Telegraphes et Telephones, December 1925.

(27)

"Etude des oscillations entretenues" (Study of Self-Excited Oscillations), by A. Lienard, Revue Generale dlElectricite, Vol. 23. 1928.

(28)

IINote sur la generation des oscillations entretenues ll (Memoir on the Generation of Self-Excited Oscillations), by E. and H. Cartan, Annales des Postes Telegraphes et Telephones, December 1925.

(29)

ilLes sysdmes auto-entretenues ll (Self-Excited Systems), by Ph. LeCorbeiller, Librairie Scientifique, Hermann et Cie., Parls, 1931.

1;; (30) 1I0scHlations of a Synchronous Motor, II by N. Vlasov, Journal of Technical Physics, USSR, (Russian), Vol. 9, '939. IIElectrotechnik und Maschinenbau ll (Electr1cal and frlechan1cal Engineering), by L. Dreyfuss, '9". (31)

INTRODUCTION TO NON-LINEAR MECHANICS PART D ANALYTICAL METBOOS OF NON-LINEAR MECHANICS·

44 . DlTRODUCTORY REMARKS It is apparent that for practical applications the general qualitative methods reviewed in Part I are not sufficient, and that quantitative methods capable ot yielding numerical solutions of differential equations are necessary. Thus, for example, a physicist or an engineer may wish to determine the amplitude and phase of an oscillatory process with a certain prescribed accuracy once the general qualitative aspects of the phenomenon have been ascertained. In general, there exist no methods capable of yielding exact solutions of non-linear differential equations, and the only methods available are those of approximations. A typical and very general class of non-linear differential equations encountered in applications is represented by the equation [44.1] The solutions of this equation are periodic with frequency w if p m O. Poincare (1)** has shown that very near some of these solutions, when p = 0, periodic solutions of Equation (44.1] may exist for very small values of the parameter p. The search for these solutions is the object of the method 0/ .."",,11 paramet,rl of Poincare. This method, with its various ramifications, constitutes the principal subject of Part II. The scope of the quantitative methods available at present is rather limited. It is restricted, in fact, to the class of non-linear differential equations of the type (44.1], and for this class it is further restricted by the condition that the parameter p should be very small, that is, p« 1. In spite of these limitations, the usefulness of the method is very great and its applications in various branches of applied science are extensive. Only in special cases when the parameter p is large does the theory

*

The text of Part II follows the presentation contained in the two treatises on Non-Linear Mechanics: "Theor,r of Oscillations," by A. Andrano. and S. Chaikin, Moscow, (Russian), 1937, Chapters VII and VIII. "Introduction to Non-Linear Mechanics," by N. Kryloff and N. Bogolluboff, Kieff, (Russian), 19)7, Chapters X, XI, and XIII.

** Numbers

in parentheses indicate references on page 112 of this report.

of Poincare cease to be applicable, and for such cases analytical methods are practically unexplored.* We shall frequently refer to Equation [44.1], with p« 1, as quasilinear, which means that the solutions of this equation do not differ appreciably from solutions of the corresponding linear equation when p = O. The important point to be noted is that, although the solutions of (44.1] do not differ much from the solutions of the corresponding linear equation if p is very small, these periodic solutions do not exist in the neighborhood of any periodic solution of the corresponding linear equation but only in the neighborhood of certain special solutions of that equation. The establishment of conditions for the existence of periodic solutions of [44.1] when p« 1 thus constitutes the crux of the theory of Poincare. Another important point is the effect of the so-called secular terms in the approximate solutions obtained by these quantitative methods. Poincare's method consists in substituting certain power series in Equation [44.1] and in determining the coefficients of these series by a recurrence procedure. As a result ot this, there may appear terms such as tft sin wt and t n cos wt, the secular terms,in which the time t appears explicitly in the expansions. It is apparent that the existence ot these terms, which grow beyond any bound as t+~,destroys the periodicity one is seeking. This can be illustrated by the following argument. Let w = a + fJ. Then sinwt

= sinat = [at -

cosfJt (at)

+ cosat

sinfJt

a t] ). 3 ] cosfJt + [1 - ( 2 ---'2"! + ... smfJt

3 ! + . .. {at)3

[44.2 ]

The terms a t cos fJ t, ~ cos fJ t, and so on, in this expression are secular terms. It is obvious that if the series expansion of Sin at and cos at is limited to a finite number of terms one cannot speak of the "periodicity" of the expression on the right side of Equation [44.2] since the polynominals in the parentheses ot this expression will increase indefinitely as t .~. It, however, n+~, the expressions in parentheses approach sin at and cos at, and the whole expression [44.2] then becomes a periodic function of time with period T = 27r/w. In practice it is necessary to stop the series expansions at a certain finite number ot terms; hence one is generally confronted with secular terms. In the original work of Poincare this difficulty was obviated by limiting the expansions to a certain finite time interval sufficient for

*

See, for example, the paper by J .A. Shohat (2).

137 astronomical purposes. Gylden and Lindstedt (3) have avoided this difficulty by eliminating the secular terms in each step of the recurrence procedure by which the coefficients of the expansion are determined. These methods will be mentioned briefly in Chapter XI. For this reason the transfer of the methods of celestial mechanics to the problems of non-linear oscillations presented certain difficulties. In fact, if one attempts to apply Poincare's method, for example, to a thermionic generator oscillating with a frequency of several megacycles per second, it is apparent that in a few seconds such a generator will pass through stages corresponding to those through which an astronomical system passes in many millions of years. The effect of secular terms in such a case should be felt within a few seconds. Nothing of the kind, however, is observed. In view of this it becomes necessary, in adapting these methods to the theory of non-linear oscillations, to follow the method of Lindstedt, which eliminates the secular terms in each step of the recurrence procedure. This was accomplished by Kryloff and Bogoliuboff. We shall return to this question in Section 58. In spite of the difficulty of using the original theory of Poincare for studies of non-linear oscillation, this theory is still capable of yielding a considerable amount of information. Chapter VIII outlines the salient points of Poincare's method as outlined in "Theory of OscUlat10ns," by A. Andronow and S. Cha1k1n, (4). Chapter IX 1s devoted to the Van der Pol method; 1ts presentat10n follows closely the text of Andronow and Cha1k1n. Chapter X concerns the theory of the first approximat10n of Kryloff and Bogo11uboff (5). These two methods of Van der Pol and of Kryloff and Bogo11uboff are analogous 1n some respects and follow a method s1milar to the method 0/ variation 0/ constants of Lagrange. Chapter XI deals w1th Lindstedt's method, as app11ed by Kryloff and Bogoliuboff to apprOximations of orders h1gher than the first. Chapter XII 1s devoted to the method 0/ equ.ivalent linearization of Kryloff and Bogo11uboff, wh1ch 1s an attempt to simplify the problem by reducing the given non-linear d1fferential equat10n to an equ1valent l1near one.

CHAPTER

vm

METHOD OF POINCARE

45 • CONDITION OF PERIODICITY Consider a system of d1fferential equations Z = (u:

+ by + JIll (z, Y)i iJ = ez +

dy

+ JlI2 (z,y)

(45.1]

where 11 and 12 are the non-linear elements of the system and J.l 1s a parameter. It w1ll be assumed that 11 and 12 are analytic funct10ns of the1r var1ables in certain intervals under cons1derat1on. For J.l = 0 the system becomes l1near. In general, we shall be 1nterested in per1od1c solut1ons of the non-l1near system. Let us cons1der, f1rst, the per10d1c solut1ons Gf the system (45.1] when J.l = O. Forming the character1st1c equat10n, we obtain

+ d)5 + (ad - be) = 0 (45.2 ] The period1c solut1ons of (45.1] for J.l = 0 correspond clearly to purely 1mag152 - (a

nary roots of Equat10n [45.2]. We thus obta1n the follow1ng cond1tions of per1od1c1ty a

+d

= 0i

ad -

be

>0

(45.3]

Under these cond1t1ons the 11near system adm1ts an inf1n1ty of per10d1c solut10ns of the form z

=

Kcos(wt

+

~)i

Y = gKsin(wt

+

~

+ X)

[45.4 ]

where w = Yad - be and g 1s a determined constant. Obv1ously, the phase angle ~ 1s arb1trary and. can be made equal to zero by a suitable cho1ce of the or1g1n of t1me. K and the relat1ve phase angle X appear as the integrat10n constants determined by the 1n1t1al cond1t1ons. The general form of per1od1c solut1ons of [45.1] for J.l = 0 1s then z = zo(t,K) = Kcoswti

Y = Yo(t,K) = gKsin(wt

+ X)

(45.5]

so that zo(t,R1 and Yo(e,R1 are per10d1c funct10ns w1th per10d 2"/w. Let us assume now that per10d1c solut1ons ex1st for small values of J.l 0 and let z = z ( t , J.l,K) and Y = Y ( t ,J.l,K) be these solut1ons. For t = 0 the solut1ons are z(O,J.l,K) and y(O,J.l,R1 and we can wr1te

*

z(O,Jl,K) = zo(O,K)

+ 111

and

y(O,Jl,K) = yo(O,K)

+ 112

(45.6]

wh1ch defines the functions 11 1(Jl) and 11 2(J.l). It 1s obv10us that 11 1(0) = 11 2 (0) = O. The method of Poincare consists in develop1ng the solut10ns z(t,Jl,K) and y( t, J.l,K) as power series in J.l, 111, and 11 2 , Poincare shows (6) that the

139

expansions converge if the values of Ipl, IpII, and Ip 2 1 are sufficiently small. Moreover, this convergence is un1form for any finite time interval 0< t < t l • The coefficients of the expansions so obtained are functions of time. By substituting these expansions in the differential equations [45.1J it is possible to determine these coefficients by equating like powers of p, PI' and P2 • One obtains in this manner a system of differential equations subject to certain initial conditions. Let us write the solutions of [45.1J in the form x :.: x(t,p,pl' P 2 ,K);

y = y(t,p,Pl'p 2 ,K)

Since we are looking for periodic solutions of [45.1] in the neighborhood of a known periodic solution with period T, when p = 0, it is logical to assume that in this neighborhood the period of the solution [45.1] will be T+ T, where T(p) is a small correction which approaches zero as p approaches zero. Our chief objective is to show that under certain conditions periodic solutions 1na~ exist provided p is small. The condition for the periodicity of [45.1] is clearly x(T

+

T, p,Pl'P 2 .K) - x(O.jl'Pl'P 2 ,K)

y(T

+

T,p,Pl'p 2 ,K) -

=

y(O,jl.P I ,P 2 ,K) =

° °

[45.7J

For given values of p and Kwe must select functions Pl(P), P 2 (p), and T(p) so as to satisfy Equations [45.7]. Furthermore, the non-linear equation becomes linear when p = 0 so that

[45.8 ] The phase is arbitrary, however, so that it is possible to assume that one of the PiS, say PI' equals zero. Putting P2 = P, the conditions [45.7] can be written as x(T

+

T,p,O,P,K) -

x(O,p,O,P,K) = rP(T,jl,p,K)

yeT

+

T,p,O,P,K) -

y(o,p,O,P,K)

°

=

[45.9 J I/J(T.jl,P;K)

apparent that, when # = and hence T(O) = p(O) = 0, the system [45.1], with the conditions expressed by [45.3], has an'infinity of periodic solutions corresponding to the arbitrary values of the integration constants K and X in Equations [45.4J. In such a case Equations [45.9] become ident1cally satisfied for any value of K. One can express th1s by wr1ting It

~s

rP(T,jl,p,K) = #rPl (T,#.P,K) =

°

[45.10]

140

The right-hand side of these equations represents a straight line p = 0 in the (~,B1-plane and a point represented by the intersection of the curves 'il (p,K) 0 and ~1 (p,K) 0, where 'il and ~1 are the functions ¢Jl and "'1 in which T(~) and P(Jl) have been expressed in terms of Jl. We can, for instance, represent T(p) and P(Jl) by power series I:

I:

[45.11 ] Expanding the functions ¢J 1(T, p, P,K) and ¢J 1 "'1

=

¢J OI

= 1/101

"'1 (T, Jl, P,K)

+ ap + bT + cp + ... = + alP +

bl T

+ clP + ...

we get

0

=

[45.12 ] 0

Substituting in these expressions the values of T(Jl) and P(p) from [45.11] and conSidering Jl as a small quantity of the first order, one obtains ¢J 1 =

"'I

=

,pOl "'01

+ p(a +

bd

+ cd l )

+ p(a 1 + bId +

cld l )

= 0

=

0

These equations hold only when p is very small so that the terms containing powers of p greater than the first are negligible. Since these two equations must be satisfied for a sufficiently small p, two conditions must be fulfilled: ¢JOI =
"'01

= "'ol(K) = 0

[45.14 ]

[45.15] + bd + cd l = 0; a l + bId + cld 1 = 0 The condition [45.14] states that the terms independent of Jl must be equal to zero and [45.15] that the system of the two equations must yield the values a

of d and d 1 which determine the quantities T(Jl) and P(Jl) to the first order. It is apparent that this is possible whenever the determinant I~ c~1 O. With ExpreSSions [45.12] taken into account, this is equivalent to the condition

*

J=

Hence, whenever the Jacobian [45.16] is d1fferent from zero, period1c solutions of the non-linear problem exist since it is possible then to determine the functions T(Jl) and p(Jl) provided Jl is suff1c1ently small and provided the cond1tions [45.14] are fulfilled. If Equat10ns [45.14] can be solved, one obtains one or several values of K; hence the problem is solved. We can recap1tulate the problem somewhat differently using the terminology of the phase plane. For Jl = 0 there is a continuum of closed

141

trajectories corresponding to different values of the integration constants, as was shown in Section 1. For ~ 0 but very small, closed trajectories may ex1st in certa1n restricted regions of the phase plane in which the condit10n [45.16] is fulf1lled; the value of the integration constant KI , the amplitude, is determined by solving Equations [45.14]. The solution of [45.1], when ~ = 0 and K = KI as Just explained, is called the generating solution. We shall see numerous examples of this procedure in what follows.

*

46. EXPANSIONS OF POINCARE; GENERATING SOLUTIONS; SECULAR TERMS Instead of Equations [45.1], we shall consider now a non-linear differential equation of the form

[46.1] Let ~ = z(t,~,PI,P2,R1 be its periodic solution in the neighborhood of ~ = O. Expanding this solution into a power series of ~, PI' and P2 , we know by the theorem of Poincare that this expansion converges in any arbitrary but finite time interval provided these quantities are sufficiently small in absolute value. We obtain

[46.2 ] where ,po' A, B, •.. are functions of t. Our purpose will be to identify the expansion [46.2] with a periodic solution of [46.1] provided I~ I, IpI(~)I, and Ip2(~)1 are small. Differentiating [46.2] with respect to t, we obtain the following equations

Z = ~o(t) + API + EP 2 + .. '; % = ~o(t> + API + EP2 + . . . [46.3] Expanding f(z,x) in a Taylor series around the values ~o' Xo we get f(z,x)

=

f(zo'x o) +

(z - zo)(:~t

+

°

1 (. . )2(02/) ( + '2 z - Zo Ii; 2 0 + z -

(x - xO)(:{)o + )(.. Zo z - ~o

! b: - Zo)2(:~)o +

>( OZ02foli: )0 + . . .

[116 II] Of'

.Of'

Substituting z - Zo = z - tPo(t); Ii; - li: o = Ii; - ~o(t) as given by [46.2] and [46.3] into [46.4] and replacing z, z, and f(z,li;) by their values [46.2], [46.3], and [46.4] in the differential equation [46.1], one obtains a series arranged in terms of P, PI' P2 , P1~' PaP, ~2, . . . which by the theorem of Poincar6 (6) converges. By equating the coefficients of ~, PI' ... , one obtains a set of differential equations. If the expansion is limited to the second order, one obtains nine differential equations, of which three are identically satisfied and the remaining six are as follows:

A+

A

=

0;

B+

B

=

0

[46.5 ]

142

E+E

=

(81) B + (a~) iJ; if' + F 8z

az

0

(8/) C + (8~) C

=

az

0

8z

0

[46.5]

0

Here the symbols (~)o and (~)o designate the partial derivatives of /with respect to the variables z and ~ in which the generating solutions Zo = ~o(t) and ~o = ~o(t) have been substituted after differentiation. Writing [46.2] and its derived equation for t = 0 1n the form z - Zo = /1 1 = A/1 1 + B/12 + Cp + D/1IP + .. . Z - Zo = /1 2 = A/1 1 + B/12 + Cp + D/1IP + .. . one obtains the following initial conditions A(O) = 1;

[46.6]

B(O) = 1

[46.7] = E(O) = teo) = 0 With these 1n1tial conditions the first two equations [46.5] have the soluB(O)

=

C(O) = D(O) = E(O)

=

F(O)

=

A(O)

=

C(O)

=

D(O)

tions A = cost;

B = sint

with period 21T. The remaining four equations [46.5] are of the form 1i + 11 = V( t ) , having the initial conditions 11(0) = v(O) = O. The solution of this equation is

,

11 =

1

[46.8]

V(u) sin(t - u) du

Replacing V(u) in Equation [46.8] by the right-hand terms of the last four equations [46.5], one obtains the following expressions: A = cost;

,

B = sin t;

C = f/[tf>o(u)'~o(u)]sin(t -u)du;

f[:~o cosu -

F

,

C =

1

/[tf>o(u), ~o(U)] cos(t - u) du

.

'[81

E =

.

f'r.81

81 ~ + 8~o cosuJ cos(t -

F=

I[:~ C +

:£ C]cos(t -

u) dUi

D = [ 8xo cosu -

:{o cosu] sin(t -

u) du;

I[:~ c + :{o C]sin(t -

u)du;

1[:~ =

= cos t

!{o sinu] sin(t -

,

E =

iJ

[46.9]

,

D =

.A.=-sint

sinu +

L8z sin u o

8/ sinu" cos(t - u) du a~o J 11.) du

u)du

where Zo = K cos t and %0 = - K sin t are the generating solutions in which the phase is taken equal to zero. Inasmuch as only periodic solutions are of interest here, it is important to know the values of A, B, .•. F after one period. Replacing t by 2" in Expressions [46.9], one obtains A(2")

=

1;

.4(2")

=

0

B(2")

=

0;

iH2")

=

1

C(2") == [f(Xo,%O)COSUdU

C(2") == - [f(Zo,%o)SinUdU;

of 2 of. D. (2") == [[-cos u - -1 . sln2u] du ozo 2 oxo

[46.10 ]

.( ) [[1'2 OZo of . of 2] sm2u + oio cos u du

E 2" =

. F(2") = [[of OXo C(u) cosu

+

of C(u) . oio cosu] du

The expressions for D and E can be further simplified by expressing the values of f cos 11.) and~ f sin u) differently. We have

tv(

ktv, (

-d du

(I cosu) =

df . -cosu - f SlnU' du •

Since the generating solutions Zo and Zo are Xo

= Kcosu;

OXo = - K smu; . OU

%0 = - Ksinu

oi = -Kcosu OU

~

hence df of . of = - K-slnu - K-. cosu du oXo oXo

-

and thus

f .

f ) == - -SlnU of . COSU - of. cos 2 u -1 - d ( cosu K du oXo oZo

K SlnU

or 1 d ( ) 1 of • of 2 - fcosu = - - -sln2u - - . cos U K du 2 oXo oXo

-

f. -SlnU K

[46.11 ]

144 and similarly

1. ..!:.... (fsinu> = - of sin 2u _1. a! sin2u + K du a:c o 2 a:co

f cosu K

[46.12]

Expressions [46.10] for D, D, E, and E, taking into account the expression for C, are simplified by means of Equations [46.11] and [46.12] and assume the following symmetrical form:

[46.13 ] 1 . K C(21T);

E(21T) = -

If C(21T)

= 0 and

e(21T)

* 0,

D(21T) =

. 1 E(21T) = K C(21T)

Equations (46.13] become

[a! du; a:c o

D(21T) = [Of du _ C(21T) a:co K

1 . E(21T) = - K C(21T);

If C(21T)

= C(21T) = 0,

[46.14 ]

. E(21T) = 0

one has D(21T)

=

[:£

du;

DC21T)

=

r:~o du [46.15 ]

Prom the latter form of the expressions forD(21T) and D(21T), it is a~parent that they represent the constant terms in the Pourier expansion of .f.land (J8f, multiplied by 2",. On the other hand, - e(21T) and t(21T), as given "':to :to by Equation [46.10], are the coefficients of sin t, cos t in the expansion of f(:to,so), multiplied by",. Hence, if f(:t o, so) is given, these coefficients can be calculated directly trom Equations [46.10]. We are now in a position to write Equations [45.10] in a new form, expressing the existence of periodic solutions. It is apparent that by the choice of generating solutions in the form :to ( t) = K cos t and So ( t) = - K sin t, the amplitude Kis already contained in the expressions for PI and P2 so that Equations [45.10] can be written as [46.16 ]

145 These equations express sufficient conditions for the existence of periodic solutions. There are thus two equations with three unknowns, r, Pl' and P2 • One of the P's, however, is arbitrary and can be taken equal to zero, as previously mentioned. If, therefore, Equations [46.16] can be solved giving rand P1 as functions of II in such a manner that for II + 0, r(Il) + 0, and P1 (Il) + 0, the problem is solved. If this is 1mpossible, there is still another alternative. We may put P1 = 0 and try to solve for rand P2 as unknown functions of II. The left-hand terms of Equations [46.16] represent the differences ~(21T + r) - ~(O) and %(211' + r) - %(0). Expanding ~(21T + r) and %(211' + r) in a Taylor series in which r is considered small, we have ~(21T

+ r)

%(211'

+ r) =

= ~(21T)

+

rz(21T)

+ ...

+

rx(21T)

+ ...

%(211')

[46.17]

Here we substitute the series expansions [46.2] and [46.3]. The coefficients A(21T) , B(21T) , ••• have already been calculated in Equations [46.10]. Considering r and II as small quantities of the first order and carrying out the expansions to the second order, one has ~(21T

+ r)

+

= ~0(21T)

•

F(21T)1l

+

A(21T)P1

rz o(21T)

+

+ B(21T)P2 + C(21T)1l +

..

rA(21T)P 1

..

D(21T)P 1 J.l

+

..

r 2 2x~(21T)

+ rB(21T)P 2 + rC(211')1l +

E(21T)P2 1l

+

[46.19]

and %0(211') = %0(0). Furthermore, A(21T) = 1, ..4(211') = 0, B(211') = 0, andB(21T) = 1. With these values of the coefficients, Equations [46.18] and [46.19] become But

~0(21T)

=

2

+

~0(0)

~(211'

+ r)

- ~(O) = - K

+

r2

.

T + rP 2 + C(21T)1l + C(21T)rll + +

D(21T)P t ll

+

Kr - rP t

+ 6(211')11 +

C(211')rll

D(21T)Plll

+

+

E(21T)P 2 J.l

F(211')11

2

0

[46.20]

= 0

[46.21 ]

=

and X(21T

+

r) - :i:(0)

=+

E(21T)P21l

+

P(211')11

2

146

One of the parameters P can be fixed as we please. Thus for a given value of one P these equations determine the other P and the correction T for the period. Since p, PI' P2 , and T are small quantities of the first order, one can obtain different conditions according to the order of the approximation. The simplest case is that in which one considers the first-order solution, dropping terms of the second order. The only term of the first order in (46.20] is C(2")p, and in [46.21] there are two terms of the first order, - KT and C(2")p. By eqqating these terms to zero, we obtain the following two equations C(2")

T

= - [!(KCOSU' - Ksinu}sinudu == t/>(K) =

= C(271'}# K

p = K

r

.

0

!(Keosu, - Ksmu}eosudu == #I/JCK)

[46.22) [46.23)

Equation [46.22] determines the amplitude K'of the generating solution in the neighborhood of which exist periodic solutions of [46.1], and Equation [46.23] gives the correction T for the period, provided C(2") ¢ O. If C(2") = 0, from (46.15], E(2") =E(2"' = O. Hence, Equation [46.201 reduces to [46.24 ] Since T in this case is zero to ~he first order, we may proceed to the second order and put T = (1#2. From [46.21], in which we can put P2 = 0, and where C(271') = 0, C(271') = 0, and E(2") = 0, we obtain 2'

•

2

- Kop + DC271'}P I # + F(2")p = 0

Dividing by p and substituting the value of PI derived from [46.24] into this expression, we obtain (1=

P(2") D(2") - F(271') b(271') K· D(271')

(46.25]

The correction T = (1#2 must be introduced each time the motion is isochronous to the first order. Thus, if K ¢ 0 and D( 271') ¢ 0, Equations (46.24] and [46.25] determine PI and T = (1#2, the amplitude K having been determined from Equation [46.22]. Substituting the values of A, B, C, and PI in Equation [46.2], we obtain z = Kcost

+ #[/f(KCOSU' -

Ksinu)sin(t - u)du -

~~~:~ cost]

[46.26]

147

where Khas been calculated from Equation [46.22].* 2 fT+T . It is to be noted that the term T + T 0 f( x O ' x o ) sin( t - u) du represents the first term of the Fourier expansion of the function appearing on the right side of Equation [46.1]. Moreover, the period has been changed because of the presence of the term T, the correction for the period. The function x(t) given by Equation [46.26] remains periodic. It should be mentioned here again that the presence of secular terms does not destroy the periodicity but merely accounts for a modification of the period, as was explained in connection with Equation [44.2]. The effect of the appearance of secular terms can also be ascertained from the following example. Assume that we have a periodic function x(t) =

.i'

[aK(p) cosKw(p)t

+ b~ll)sinKw(ll) t]

[46.271

K=O

in which both the amplitudes and frequencies are functions of a parameter p. Expansion of this function in a power series of p gives :dt)

=

I

GO

[aK(O)cosKw(O) t

+

+

bK(O) sinKw(O) t]

K=O

+ pI[a~(O)coSKw(O)t + b~(O)sinKw(O)t

-

K=O

-

aK(O)w'(O)Kt sinKw(O)t

+

bK(O)w'(O)Kt cosKw(O)t]

+ p:i; •

••

[46.281

K=O

where a~, b~, and w' deSignate the derivatives of the functions aK(Il), bK(p), and w(ll) with respect to II in which the value II = 0 has been substituted after differentiation. It is observed that, since the function x(t) is periodic, the appearance of secular terms does not destroy the periodicity in view of the summation of these terms from 0 to~. This might not be the case if only a few secular terms were considered in the expansion. With reference to [46.26], it is to be noted that the secular terms do not appear in the expansion for x(t) if the correction T can be calculated first, which requires that D(2") O. It is sufficient then to use as the period over which the functions A, B, ... of Poincare are determined, the corrected period T+ T, which amounts to the choice of generating solutions Xo and Xo in the form K cos [1 - ~'ir2] t and - K sin [1 - ~;2] t, instead of K cos t and - K sin t. The question of secular terms will be discussed in more detail in Section 58.

*

*

In order to re-establish the arbitrariness of the phase, an arbitrary phase.

I

should be replaced by

I

+

~,

where. i ..

148 47.

SYSTEMS WITH '!WO DEGREES OF FREEDOM In the preceding sections we have been concerned with the establishment of conditions for periodicity of solutions of a single non-linear differential equation [46.1) of the second order, which generally represents in applications a dynamical system with one degree of freedom. The only stationary solutions in this case are periodic ones, and the topological representation of such mot~ons in the phase plane does not present any particular difficulty, as was shown in Part I. Although theoretically the extension of Poincar6 1 s method to systems-with several degrees of freedom follows the same argument, the practical difficulties rapidly increase and the benefit derived from topological considerations disappears.* Moreover, the stationary motions in systems of more than one degree of freedom are not necessarily periodic. It thus becomes necessary to restrict the analysis somewhat by endeavoring to formulate only the conditions of stability, which in applications is equivalent to the physical possibility of a particular motion. Systems with two degrees of freedom play an important role in applications, and for that reason it may be of interest to give a brief outline of the method of Poincare in connection with such systems, omitting the details which have already been explained. The calculations of Andronow and Witt (7) (8) are given in this and the following sections. Consider a quasi-linear system of two differential equations of the second order [47.1) where ~ is a small positive number and the functions f and g are analytic functions of the indicated variables. Since there are two degrees of freedom, one has a greater variety of limit conditions for p = 0 than in the previously discussed case of a single equation [46.1). Thus we can write the limit conditions either as

*

In fact, a dynamical system with two degrees of freedom is generally reducible to a system of four differential equations of the first order, and its representation in a phase plane becomes generally impossible. Only in very special cases of the so-called "degeneration D defined in Part IV is a planar representation possible, but such "degenerate" systems possess entirely new features which are not investigated here.

149

x = Rcoswlt = lPo(t);

= 0 with period T

y

= 27r Wl

or

[47.2 ] x

=

0;

y

=

R COSW 2 t

=

with period

¢'o(t)

T

=

27r w2

The question as to which of these two generating solutions the dynamical system will "select" will form an important object of a later study. The procedure initially follows the pattern outlined in connection with a single equation [46.1]. Let us assume that we select the first alternat1ve of [47.2] and apply the perturbat10n method by putt1ng x

= lPo (t) +

In terms of the perturbations

~

and

71,

~;

y

=

0

+

71

[47.3]

Equations [47.1] become

e

~ + Wl2~ = Jl [/
[47.4 ]

2 ij + W2 71 = Jl[g(lPo,~o'O,O;O) + g%~ + gi~ + 9y71 + g;,~ + gllJl + 02(~'~'71,~,Jl)]

where the quant1ty O2 conta1ns terms of a degree higher than the first in ~, ••• Jl. According to Poincare, the solutions of these equations can be taken as power series

~ = /3 l A + /32B + /3a G + /3 4 D + Jl[E + /3 l F + /32 G + /3 3H + /3 4 K + JlL +

+ 02(/3l,/32,/33,/34,Jl)] + 02(/3l,/32,I1S,114 ) 71 =

[47·5) /3 l A + /32B + /33 G + /3 4 D + Jl[E + /3 lF + /3i; + /3 sil + /3 4 K + JlL +

+ 02(/31,/32,113,/34,Jl)] + 02(/3l,/32,/33,/34) where A, •.• L and A, ... L are functions of time and /3 l

=

HO);

/3 2 = ~(O); /3 s

=

71(0);

/3 4 = i}(O)

(47.6)

One of the /3 1 s, as will be seen, can be assumed to be equal to zero; for example, /3 2 = O. If one substitutes the expressions [47.5) into Equations [47.4) and equates the coefficients of like powers of 11 1 , ••• Jl, a system of differential equations results from Which the functions A, ... L and A, ... L can be determined. One obtains the following expressions:

150

t

F = -If
t

H = .!..f(c osw2 u'l" - w 2 sinw 2 u • Iii) sinwl( t - u)du WI

[47.7]

t

F=

-l f (COSWIU' gz -

wlsinw 1u· g;i)sin W2(t

-

u) du

W20

t

jj =

..!... f(cosw 2 u'g"

- w2 sinw 2 u'gil)sinw 2 (t - u)du

W20

The condi tions of perio dicity for the soluti ons [47.3] are [x] =

[y]

x(~ + or) -

= Y(!: + or) -

x(O) = 0;

[x] = x(~~ + or) -

x(O) = 0

[47.8 ] y(O)

=

0;

bi] = ti(~~ + or)

-

y(O)

=

0

by [47.8], These four condi tions, with the use of the symbols [ ] defined become

151

R[ - w1sinw 1t]

+ ,81[A] + 1l([E] + ,81[F] + ,8a[.H] + ,8.[K] + liLt.] + [02(,8",8a,,84tIl)])

= 0

[47.9] ,8a[C]

+ ,8.[D] + 1l([E] + ,81[F] + ,8aLH] + ,8.[i(] + 11[£] + [02(,81,,8a,,8.,p)])=

,83[C]

+ ,8.[15] + 1l([E] + ,81[F] + ,83[H] + ,84[K] +

0

Il[t] + [02(,81,,8a,,8.,p)))= 0

Developing the expressions indicated by the symbols [ ] in power series of the correction for the period, one obtains

+ T)

[A]

=

[A. ]

= A. (271 W';" + T)

A(!:

ao + a l T

+ a 2 T 2 + ...

- A. (0) = a. o + a. 1 T

+ a. 2 T 2 +...

=

- A(O)

where a o ' ••• and ao , can be calculated by Equations [47.7]. from [47.10] the following equations

T,

[47.10]

One obtains

2

- RWi T

+

,81 (a o + alT)

+

Il(eo

+ elT + I1do +

l1aho + P.ko + Illo)

+ Os (11 •• l1 a.I1•• P. T) =

0

[47.11] I1s(Co + CIT)

+ p.(iIo + ~T) + Il(eo + ~T + Plio + Paho + P."o + Pro> + 0a(l1l' 113 .11•• P.T) =

0

In these equations certain coefficients do not depend on the choice

of the functions f and g.

Thus, one always has [47.12]

Co = where Y

cosy -1i

do =

..!..siny·

w2

•

Co =

- w2 sinYi

do =

cosy - 1

W2 = 271 WI •

The second equation [47.11], taking into consideration [47.12] and 08(,81',88,,84'P), gives T, viz., [47.13]

152

where

• a-~' e 0-

a

Rw 12'

k

-~. Rw 12'

4 -

a5

=

Introducing the value of T given in Equation [47.13] into the remaining equations [47.11], one obtains the following three equations: eo + fJdo + fJ 3 hO + fJ4ko+ p(lo + aOel -

~ w12a 02 ) + 02(fJl'fJ3 ,fJ4 ,P) =

fJ3(COS" - 1) + fJ4(~2 Bin,,) + pFo + O'I'
+ fJ 4(COB"

-1)

+ Ileo + 02(fJl'/1S,fJ4 ,p) =

° °

° [47.14]

Since we are looking for a periodic solution of the system [47.1], which reduces for P = 0 to the first generating solution [47.2], it is necessary that the functions /11(p), /1s (p), and fJ. (11) approach zero as 11 approaches O. Hence, in view of [47.12], the necessary condition for the existence of a periodic solution is l

2-

eo

=

t"I/(RCOSW1U, -w 1Rsinw 1u,O,O;O) sinw 1u du

=

°

[47.15]

From this equation one obtains the amplitude of the periodic solution in the neighborhood of 11 = O. The sufficient condition for the existence of a periodic solution is ho

10

ko 1

w sin" = 2/00 -

°

[47.16]

° - w2sin" COS" ° In this condition the value of R, and hence of Jo , is the one wh1ch sat1sfies COS" - 1

2

cos,,)

=1=

1

the amp11tude equat10n [47.15]. 11' w2 =1= nw 1 , that is, if " =1= 2"n, where n is an integer, the condition for the existence of a periodic solution can be written as 2.

10 = t'I(COSW 1U' Iz

-

w1sinw 1u • 1%) sinwl'U d'U

=1=

°

11' the determinant [47.16] is zero, it is sufficient for the existence of a periodic solution that one of the determinants, obtained by equating to zero a fJ other than fJ 2 as assumed here, be different from zero. If all determinants are zero, a special study is required.

153 The periodic solution thus obtained is of the form x = t,6(t) = Rcosw 1 t + P1A+ ~[E + P1F + PaH + P4K + ~L + 02(Pl'Pa,.84'~)] [47.18] y = ~(t) = PaC! + pJj + ~[E + .81F + .8aH + .84K + pL + 02(.81.Pa,.8•• p)]

where R is determined by [47.15]; A, E. F, •.. by [47.7); and .81' .8 a • and .8. by [47.14). Replacing P1' .83 ' and .8. by their values in [47.18] and arranging the terms of the series according to the powers of P. one obtains

[47.19]

It is apparent that similar results can be obtained if.one starts with the second generating solution [47.2). 48.

STABILITY OP A PERIODIC SOLUTION

The stability of the periodic solutions [47.18] can be investigated by the perturbat10n method. Consider the perturbed solution x

= t,6(t)

+ Ui

Y = ~(t)

+v

[48.1)

The variational equations obtained from [47.1] are

[48.2 ]

In these equations the non-linear terms in u and v are left out and the functions f~ • ... g; are the derivatives of f and g with respect to the indicated variables in which x, i. y, and yare replaced by t,6(t), ~(t), ~(t) and ~(t)

respectively. These equations have periodic coefficients, and we can expect solutions of the form U

= API

+ BP2 + CPa + DP4 [48.3]

v

=

APt + BP2 + CPa + DP4

where A, ... D are unlmown functions of time and the P's are in1t1al values [48.4 ]

154

Since the funct10ns fe' ••• 0, appearing in (48.2] can be developed in power series in terms of p, the functions A, ... Dcan be assumed to be also of the form [48.5] -

-

A = Ao

-

+ pAl +

2-

P A2

+ ...

Subst1tuting the expressions (48.3] into (48.2], taking into account their form (48.5], and comparing the terms with the same powers of p, one obtains a number of differential equations from which the coefficients A o' AI' ..• 1 0 , Al can be determined. One obtains in this manner the following expressions:

= 0;

C1 = H; etc.

Do:O: 0;

DI = K; etc.

10 =

Al

Co

0;

=

F;

[48.6 ]

etc.

Introducing the notations of Poincar6, viz •• u(T) - u(O)

=

[u] = ~1;

u(T) - u(O) = [it] = ~2

(48.7] 11 (T)

- 11(0) = [11] = ~s;

veT) - v(O) = [v] = ~4

where T = ~fr1 + T, T being the correction for the period given by (47.13]. one obtains the equation for the determination of the characteristic exponents in the form 8~1

+1_

eaT

0/11

.1 =

O~2

O~2

8P t

8P2

8~1

O~l

O~l

8P2

8Ps

8P.

+1_

8~a

8~s

OP 1

8P 2

81/14 8Pt

8~.

8P2

eaT

8~s B/1s

O~2

8~2

8P3

8P.

+ 1-

=0 (48.8 )

O~S

aT

e

0/1.

8~4

8~.

8Ps

8/1.

+1_

aT

e

155 This reduces to the form [A)

+ 1 - eaT

[B] a

+1- e

[il]

L1 =

;

[B]

with [A] .. A(~ t r) - A(O), etc. written in the form

[C]

[C]

[D]

[C]

[D]

+ 1-

e

a7

Putting 1 - eaT

= p, this equation can be

an + P

a l2

als

a l4

a 21

a 22 + p

a 23

a 24

a SI

a 32

a 33 + p

a 34

a 41

au

a 43

a 44 + p

L1(p)

[48.9 ]

=0

(15)

'

[48.10]

=0

which reduces to the quartic equation

[48." ] with b =

n A22

+

II

Ass

+

II

A44

+

22

Ass

+

22 A44

3S

+ Au

[48.12 ] ss

11

where A II' ... A 44 and A 22 , ... A 44 are the diagonal minor s of L1. Since one of the characteristic exponents is always zero (9) because the equations are autonomous, d = 0, and the quart1c equation thus reduces to a cubic one ps

+

ap2

+ bp +

c

=

0

[48.13J

If the motion is stable in the sense of Liapounoff, the remaining three characteristic exponents must have negative real parts, which means that the modu11 of the quantities eaT must be less than one. This means that the complex number p = 1 - eaT must be represented in the complex plane p by pOints situated inside a circle of radius 1 whose center is on the real axis at a unit distance from the origin, see Figure 48.1. By means of the funct10n p = 1 : z' the interior of the circle in the (PI,P2)-plane is mapped into a half plane (%1'%2) so that the circles, see broken line in Figure 48.1, transform into straight lines, see broken line in Figure 48.2, parallel to the %2-axis on the axis of the negative %1'

156

--f-~~+::-----Zl

Pigure 48.1

Figure 48.2

By this transtormation the problem ot tinding the roots or [48.13] with moduli situated inside the circle ot the (Pl,P2)-plane is reduced to that ot determining the roots ot the transtormed equation having negative real parts. This brings the problem within the scope ot the Routh-Hurwitz criteria. It this transtormation is carried out, Equation [48.13] becomes

+ (8 + 4a + 26 + The Routh-Hurwitz criteria ot stability (10) (11) are Z3(-C)

26

+

z2(26

+ 3e > O.

-e

b2

•

+ 3e) + +

z(- 4a - 46 - 3e)

e 2 + 2 be

+ a6 +

ae _ e

< O.

,

8

+ 4a +

-e

26

[48.14]

e) = 0

+e >

0

[48.15]

These are the necessary and sufticient conditions tor the roots ot Equation [48.14] to have negative real parts, or, which is the same, tor the roots or [48.13] to have moduli less than two, which assures the stability ot the periodic motion. The conditions [48.15] can be written also in the torm 26

+ 3e > 0;

e

< 0;

(6

+ e)2 + a(6 + e)

- e

< 0;

8

+ 4a +

26

+ e > 0 [48.16]

In order to apply these conditions ot stability, it is necessary to

calculate the determinant [48.9]. The quantities [Al, [B], .•. can be developed in power series in terms or p, tor example

The value ot the determinant ~(O), in which are written only terms containing linear terms in p and terms independent ot p, is

157

11(0)

+ .. . 624 # + .. . + b # + ..

+ ... 62S Jl + ...

+ ... b22 Jl + ... b32 Jl + ... b42 # + ...

+ ... + ... = 631 # + ... b,n# + ...

b13 #

612 #

bu # b21 Jl

b14 #

sin), W2 34 (COS)' -1) + 644 #

(cos)' - 1) + bsa #

- W2 sin)'

+. + b. 3Jl + ...

The values of bll , b 12 , ••• are determined from Equations (48.6]. to account (48.12], one obtains the following relations

+ ...

(48.17 ] Taking in-

c = Jl2 (I - cos)')( 6u + b22 ) + 02 (#) (b

+ C)2 +

a(b

+

c) -

C

B

=

+

#2(1 - cos)')[COS)'(b 33

4a

+

2b

2b

+

3c =

+

C

=

4(1

+

+

cosy)

4(1 - coSY)

*

b44 )

+

+ sin)'(w 2 b34

+

-

~:)] + 02(#) [48.18)

01(#)

01(#)

*

1, )' 2"#, and)' 2(n + 1)",.n being an integer, the stability conditions reduce to the following

It we consider Jl«

(48.19 ]

+ b44 ) +

COS,,(6 33

The values of 611 , 622

,

•••

sin,,(w 2 b34

-

~!) < 0

are given by the expressions

2.

b ll =

~1["'I(coswlU'f%

622 =

f"'I(~1

2-

sinwlu·f%

-

wlsinwlu'fi)Sinwl(~~

- u)du

+ cOSWIU'fi)COSW1(~: - u)du

[48.20 ] 2-

b44 =

"'I( w1

J

2

. smw 2 u· gil

+

cosw 2 u· gil ) cos (27T ~ - u ) dU

-

. aOw2 sm"

2-

b34 =

;2 !"'1(;2 sinw2u' gIl + cosw2u· giJ) Sinw2(~~ 2.

b. 3 = fi:i;(cosw 2U' 9" -

w2sinw2u'9il)COSW2(~~

u) du

- u)du -

+ aocos)' 2 aOw 2 COSY

Introducing these values into the inequalities [48.19], one obtains the following conditions of stability [48.21

J

In applying these criteria one has to take :J; = R cos WI1£; :i: = - RWI sin "'11£; = 0; Y = 0; and 11 = O. R is determined by [47.15]. We shall return to this matter in Part III in connection with the question of stability of coupled electronic oscillators. This problem is expressible in terms of two non-linear differential equations of the second order. y

49.

LIMIT CYCLE AND FREQUENCY OP A THERMIONIC GENERATOR

We propose to apply the preceding theory to Equation [30.9) of a thermionic generator. The quadratic term 11V 2 in Expression [30.5] of the characteristic will be dropped, inasmuch as this term accounts for only a slight asymmetry of the characteristic and has no effect on the calculation of the stationary motion.* The simplified equation [30.9) can be written as

v+

2

v = Jl(/1 - 36v )V

[49.1)

This equation is dimensionless; p > 0 and 6 > O. The small parameter Jl is introduced so that we may consider the oscillation in the quasi-linear range and therefore be able to apply the preceding theory. Por Jl = 0 the generating solutions are of the form Vo

In this case !(v,ti)

=

~o(t)

= pv -

=

Keost;

Vo

= t.6 o(t) = -

Ksint

[49.2 )

36v 2 v and hence

[49.3 ) Making use of the condition of Poincare, Equation (46.22), upon integrating Equa tion [49.3) we obtain PK - 6K 8 = 0 and hence

-¢

[49.4]

Thus the amplitude of the generating solution to the first approximation depends on the ratio~. In other words, the amplitude of oscillation reached The reader w11l note that by tion [30.9] instead of Equation vanishes, which proves that the remark applies also to all ""en

it

retaining the quadratic term "1112 in Equation [30.5] one would have Equa[49.1]. The integral from 0 to 2" of the term 2"lIti in Equation (46.22) term "1,,2 has no effect on the calculation or the stationary motion. This terms in the polynomial [30.4). See also Section 54.

159

by the self-excited process will be greater as the value of 6 is smaller. This is physically obvious, for as 6 • 0 the self-excitation would build up indefinitely since the factor that eventually limits it is precisely the nonlinearity of the characteristic expressed by the term - 61 v S in Equation [30.5]. Returning to the non-linear term f(v,v), we find that ~ = - 66vv and ~ = fJ - 3 6v 2, so tha t v

~f =

vVo

2

2

66K cost sint = 36K sin2t

and

of

~ vVo

=

2

2

fJ - 36K cos t

From [46.14] we have, upon taking account of [49.4], D(211')

=

1

• (fJ - 36K 2 cos 2 t)dt = 211'(fJ -

One finds also that D(211')

= O.

2

36K -2-)

[49.5J

= - 211'fJ

From Equation [46.26J the correction for the

7'

[49.6 ]

=

Calculating F(211') from the last equation [46.10J, one obtains

[49.7] The coefficient C(t) given by Equation [46.9] after a calculation is C(t) = -

36K.

"""323 sm3t +

15 K36 . 32

smt

= - 4' ~ 36 fJ

fJ. sm3t

5/J /J +T ~ 36

. smt

so that the periodic solution without secular terms is then

[49.8] where ¢ is an arbitrary phase angle. It is clear that the periodic solution occurs in the neighborhood of the amplitude 2~. The correction for the period is of the second order and hence can be neglected for small values of p. The secular terms do not appear here in view of the fact that the correction 7' for the period has been calculated first.

160

50. BIFURCATION THEORY FOR QUASI-LINEAR SYSTEMS The results obtained in Section 29 can be somewhat extended by means of the quantitative method of Poincare, see Section 46. In order to obtain this extension, we shall consider instead of Equation (46.1] the equation % + :x: = #1(:x:,3:,>..)

[50.1]

in which there appears an additional parameter 'A which we have encountered in the bifurcation theory. The procedure remains substantially the same as before; that is, there appear certain generating solutions in the neighborhood of which periodic solutions exist when # is small. We propose to investigate now what happens to these generating solutions when the parameter 'A determining the state of the system varies and reaches a critical value. It was shown in Section 46 that the amplitude K of the generating solutions is given by Equation [46.22], which can be written here in the form

C~~1T) = _ 2~ [I(K cosu, Putting K2

=p

C(21T) 1T

K sinu, 'A) sinu du

=

0

[50.2 J

and multiplying Equation [50.2] by 2Y,O, we have

YP = t,6(p,'A)

= -

.!. [I(YP cosu, - yp sinu, 'A) yP sinu du 1T

[50.3]

Differentiating this equation with respect to p, we have

+

2~ [Ii du - 21T~ [I sinu du

[50.4 ]

The.first term on the right side of Equation [50.4] is equal to 1 2.. 1 21TVp[lcosu]o + 21TyP

r

Isinudu

as is easily verified by integrating the term Equation [50.4] reduces then to a simple form t,6p(p,'A) =

2~

[Ii

~

du

[ I sin 1£ du by parts.

[50.5]

161

In Section 13 it was shown that, in a conservative system containing a parameter A, the conditions for stable equilibrium are I(x,x) = 0; 1%(x,X) < 0

[50.6]

2r

Suppose we now impose the condition that L fz du < 0 so that ~p(p,X} < O. o Associating the function ~(p,A) with l(x,A}, we may infer that the limit cycles are stable if

[50.7] This is by no means a proof, but merely a plausible deduction. A proof for the conditions under which stationary motion is stable may be found in Chapter III of Liapounoff1s treatise (12). The remainder of the bifurcation theory applies directly to limit cycles, the coordinate x of equilibrium being replaced by p = K2, the square of the amplitude of the limit cycle. In a number of problems of non-linear mechanics, the function l(x,z,A} is of the form

11(x,X):i:

l(x,:i:,X) =

where

[50.8]

11 (x,X) is a polynomial of the form 11(x,X)

= ao(A)

+

+

a 1(A)x

a 2 (X)x

2

+ ...

[50.9]

To this class belong equations of the generalized Van der Pol type. Substituting I = l1:i: in Equation [50.3], with the generating solutions x = yp cos u and :i: = - yp sinu, and carrying out the integrations, one has 1[

~(p,A) = -

2

a p + 0

a p2

_2_

4

a p3 + ~ + 8 [50.10]

'" p

( X) -_.!.[ 2 ao + p,

In these equations the coefficients

51.

a2P

2

a o'

+

3a4p2

8

a 2 , a.,

+ ... J

•••

are functions of X.

"SOFT" AND II HARD II SELF-EXCITATION OF THERMIONIC GENERATORS; OSCILLATION HYSTERESIS It is well known that there are two kinds of self-excitation of thermionic circuits designated as "soft" and "hard." It is observed that by increasing the coefficient of mutual induction X between the anode and the grid circuits, self-excitation starts smoothly as soon as a critical value X = AO of this parameter is reached; for A > AO the amplitude of oscillations

162

steadily increases with increasing A, as shown in Figure 51.1 representing this "soft" case of excitation; upon decreasing A the phenomenon takes place in the opposite direction, as shown by the arrows. The theory of soft selfexcitation has been studied in Section 29. In some cases, however, a different type of self-excitation occurs, as shown in Figure 51.2. With increasing A it is observed that the selfexcitation starts abruptlr with a finite amplitude for A = Al and increases p

p

O~----~,~-------.).. "0

o Figure 51.2

Figure 51.1

smoothly for A ~ AI' For decreasing A it is observed, however, that the phenomenon is different; namely, for A = Al the seif-excitation does not disappear; it disappears at A = AO < AI' There exists a kind of "hysteresis cycle" shown by the shading in Figure 51.2. This type of self-excitation is called "hard." These phenomena are due to the non-linearity of the system, and the hysteresis cycle referred to above is sometimes called "oscillation hysteresis" (13). We have already analyzed this situation qual1tatively in Section 24. In this section we propose to investigate this effect utilizing the theory of Poincar6. Consider the circuit shown in Figure 51.3 with positive directions indicated. The differential equation of the oscillating cirCuit is L

~;

+ Ri +

~ /(i o

- 10> dt =

0

[51.1]

where 10 = f(V) is the non-linear function expressing the anode current 10 as a function of grid voltage V. Let us approximate this function by a power series in V l1m1ted to terms through VII for reasons which will appear later. We have [51 .2]

r------1IIIIt--------,

Figure 51.3 It is convenient to introduce a dimensionless variable z = V/V. where Veis the "saturation voltage," that is, a sufficiently high grid voltage beyond which the current 14 does not change appreciably. Since V = A{~; we can write M di V" dt

x = ---

[51.3 ]

From this equation we obtain di dt

-

V" M'

=-X·

Substituting these values in [51.1] and differentiating we get, after a few Simplifications,

Introducing the new independent variable ceding equation becomes 2

= woM [(a o - RC) M + 2PoV,x +

ddT2 x + x

T

= wot,

where Wo

= 1/VLC,

the pre-

3yoVs2 x 2 + 46 0 VB3 X 3 + 5EoV"4 X 4] dx dT

The condition of quasi-linearity is fulfilled if we assume that the coefficients a o ' ••. Eo of [51.2) are small. One can take one of these coefficients, for example Po, as a factor and write 2

d ~ + d.,.2 x

=

M[aoM - RC + 2 + 3yo~ 2 + 460 V,2 3 ,Wo MPo'Va x Po x Po x

PV 0

3

5 V +~

Po

x

4

]

d ~ [51.5] d.,.

164

By introducing the notations aoM - RC = (M). .0 V. a.

2M = b(M);

"'0 •

3"0 Y,M

Po

= c(M);

2

46 0 Y.t M

Po

=

3

d(M);

5EoY.t M = e(M)

Po

Equation [51.5] can be written as

:~ + % =

p[a(M)

+ b(M)% +

C(M)z2

+ d(M)%3 +

e(M)z4] : :

=

p/l(%;M)

:~ [51.6]

that 1s, the function 1(%,:i:,M) = 11 (%;M)~~ has the form [50.8]. The function ~(p~ given by [50.10] can be written, after certain transformations, as

[51.7] where

Differentiating Equation [51.7] with respect to p, one has

[51 .8] The discussion of Equations [51.7] and [51.8] yields the qualitative features of the phenomena. A.

CONDITION FOR A SOFT SELF-EXCITATION If a 1 > 0, Yo < 0, and fo = 0, Equation ,,(p,M) = (a 1

-

[51.7] becomes

a~I"oIMp)p = (aoM - RC - a 2 1"0IMp)p = 0

[51.9 ]

where a2 absorbs the constant factor V. which is of no further interest. In the (p,M)-plane this equation represents a straight line p = 0 and a hyperbola

[51.10] The point of intersection of p = 0 and the curve [51.10] is given by the equa tion Ml a o - Re = O. The value

[51.1']

165 is a critical value of the parameter AI. Following the method of Poincar~, Section 13, one obtains the d1agram of Figure 29.1b. By increasing the parameter AI from small values, one has a locus (p = 0) of stable focal points. The point AI = Alo is a branch point of equilibrium; here the focal point undergoes a transition from stability (M < Alo) to instability (AI > M o), and a stable limit cycle appears; the square of the amplitude of the latter increases with AI, following the hyperbolic branch. The asymptotic value of p for M + GO is clearlyao/a2yo, which represents the square of the amplitude for infinitely strong coupling M. The curves representing this case are shown in Figure 29.1a and b. The former represents the condition for AI> Mo, the latter, the condition for AI < Mo. The phenomenon is reversible, as shown in Figure 51.1. B.

CONDITION FOR A HARD SELF-EXCITATION If ao > 0, Yo > 0 and EO < 0, and if we designate by Eo the absolute value of EO' and put for abbreviation a~yo = m and as Eo = n, Equation [51.6] becomes [51.12] In the curve

(p~-plane,

this equation represents a straight line p (Mao - RC) + mMp - nMp2 = 0

= 0 and

a

[51.13]

°

The intersection of this curve with the line p = is clearly a 1 = 0, which gives the value Ml previously found, see Equation [51."]. The tangent to the curve [51.13] is given by the equation orp dp dM

aM

= - orp =

00

+

mp - np

2

M(m - 2np)

[51.14]

op

The curve has a vertical tangent for Po =m/2n; for this value [51.15] When Mo is compared with Ml from Equation [51.11], it is seen that Mo< MI' Furthermore, it can be shown that the curve does not go to the left of the value M = Mo and has a horizontal asymptote for p = min. This defines the curve shown in Figure 51 .4. It is easily seen that rp(p~ > 0 in the shaded area; whence, applying the criteria of Poincar~, Section 13,

166 one finds that the branch of the curve above point P is stable and below this point unstable. The Limit Cycle axis p c 0 is stable for M < MI and unstable tor M > M I ' If M is increased gradually below the value M c MI , the tocal points are stable. For M;: Ml there is a tin1te stable limit cycle and a Stobie Focol Points d1scontinuous Jump MIN, shown in Figure 51.4. If o M> Ml and M increases, there is a continuous variation of the amp11tude of the limit cycle. Figure 51.4 It, however, M> Mt and Mdecreases, when M= Ml 1s reached, the limit cycle is still stable although there is a transition of the point singularity from instability to stability. Th1s corresponds to Figures 24.6b and c. It follows, therefore, that when M decreases from Ml to Mo' the stable limit cycle is still being followed. When the point M Mo is reached, the stable and the unstable limit cycles coalesce, and no limit cycle exists tor M < Mo' which accounts tor the Jump PMo, shown in Figure 51.4. By comparing the results of this sect10n with those ot Section 29 it is seen that it the non-linear characteristic can be approximated by an expression ot the form FI(z) c a1z - a,zl, where a l and a l are positive, one has a sott type ot selt-excitation. It, however, it can be approximated by an expression F2 (z) = alz + aaz' - alz l , where ai' aa, and a l are positive, the selt-excitation is ot a hard type. In the first case the characteristic has no inflection point (except the point z = 0, wh1ch is of no interest); in the second case, there is an add1tional inflection point tor Zl = )l3a~a. Since an electon tube exhibits characteristics of both types of selt-exc1tat1on, depending on the point at wh1ch it is biased, each of these cases can be obtained in pract1ce by a suitable adjustment of the circuit. p

Q

CHAPTER IX METHOD OF VAN DER POL

52.

ROTATING SYSTEM OF AXES; EQUATIONS OF THE FIRST APPROXIMATION

Consider again the quasi-linear equation

x+

x = pf(x,x)

Its equivalent system is

z = y; y = -

x

+

[52.1 ]

J.lf(x,y)

the notations being the same as in the preceding chapter. For p the linear equation ~ + x = 0, having a harmonic solution

=

x

a cos t

+ 6 sin t,

wi th

x= -

a sin t

+ 6 cos t

= 0, one has {52.2 ]

where a and 6 are constants of integration. The phase trajectories in this case are circles of radii K = 2 2 Va + 6 • If, instead of considering a coordinate system (x,x) in a fixed phase plane, one introduces a system rotating with angular velocity w = 1 about the common origin of both systems, in this rotating phase plane Equation [52.2] will represent a fixed point A at a distance OA = K= Va 2 + 62 from origin O. The inclination of OA to a reference line in the rotating plane is given by the angle 6 = tan- 1 It is to be noted that this transformation is nothing but the usual method of representing sinusoidal quantities by vectors, used in the theory of alternating currents. We shall call the rotating plane of the variables (a,b) the Van der Pol plane. Consider now Equations [52.2] as a transformation defining x and x in terms of the new variables a and 6. This implies that

!.

[52.3 )

da.

db

- CIt sm t + dt cos t

= pf ( a cos t

+ b sin t , - a sin t + b cos t )

whence

:~

= - pi(a cos t

+

b sin t , - a sin t

+ 6 cos t) sin t [52.4 )

:~ =

pf( a cos t

+ b sin t , -

a sin t

+ b cos t) cos t

Since p is small, by assumption, da/dt and db/dt are also small because f(x,y) is bounded. In other words, a and b are slowly varying quantities in comparison with the rapidly varying trigonometric terms of frequency w;:: 1.

168

For the tirst approximat10n 1t 1s suft1c1ent therefore to cons1der a and b as constants on the right side of Equations [52.4). However. 1t ~ and yare replaced by the1r expressions [52.2], the right sides of [52.4] are periodic and can be expanded 1n a Fourier series so that [52.4] becomes da CiT

= P

[tI>o(a, 2

6) +

:r.

tl>l(a, b) C08 t + 'Pl(a, b) am t + tl>z(a, 6) co82t + ...

]

[52.5]

~:

=

p[I/Io(~,6)

+ I/Il(a,6)co8t + ifi'J(a,b)sint+ 1/12(a,6)co8 2t + ...

J

It must be noted that the system [52.5] now contains t explicitly. whereas the original system [52.1] does not. Van der Pol cons1ders the tol10w1ng "abbreviated" equations as equations ot the tirst approximat10n

[52.6] They are obtained trom [52.5] by dropping the terms conta1ning the trigonometric tunctions. On the other hand,

[52.7]

Mult1plying the first equat10n [52.6] by a. the second by b. adding and putting K2 = a 2 + 62 , one obtains 2

dK '21 dt

=

dK KdI

=

f;

r

f(aco8~ + 68in~, - asin~ + 6cost)(-asin~ + bC08~)ct~

Putting aC08~

+

bsin~

= Kcos(f - 6)

and -asin~+ 6cos~

where 9

= tan- J

:f =

= -Ksin(f-9)

!. and introducing the variable u = t - 9.

I.l tI> (K)

and, by a similar transformat1on, ::

one has

= p !f1(K}

[52.8]

where

~

t/> (K) = - 2

r

f (K cos u, - K sin u ) sin u d u

[52.9] '" (K)

= + 21TK 1 J"f(K cos u , - Ksin u) cosu du

One notes that the definite integrals in these equations coincide with the functions C(21T) and 6(21T) of Poincare, Equations (46.10J. 53.

TOPOLOGY OF THE PLANE OF THE VARIABLES OF VAN DER POL In the (z,y) phase plane the limit cycle is reached when the phase trajectory is a circle; in the Van der Pol plane (a,b) the condition for the existence of a stable limit cycle is satisfied when the end of the vector K is a point of stable equllibrium, that is, at this point ~f = IJ t/>(K) = O. Hence, limit cycles exist for radii K corresponding to the roots of 9S(K)

=

2~ liCK cosu,

-

Ksinu) sinu du = 0

It is to be noted again that this equation coincides with Equation (46.22J of Poincar6 1 s theory. A root Ki of 9S(K) = 0 will correspond to a stable limit cycle if t/I'(K.) < O. By a method similar to that applied in connection with Equation (50.5J, one finds that the condition t/I'(K.) < 0, written explicitly, gives (53.1 J The limit cycle is unstable if t/>'(K.) > O. Consider now the second equation (52.8J. interest.

Here two cases are of

Case 1.

~

'" (K) = 2 K

r

f (K cos u, - K sin u ) cos u d u =

0

[53.2 J

In this case 8 = 60 = constant. The topological picture of trajectories in the plane of variables (a,b) of Van der Pol, in this case, is shown in Figure 53.1. The equilibrium on a limit cycle at a point K. for 9 = 90 is stable if

110

b

'I

Stable Limit Cycle

•

Stable Focal Point

Figure 53.1

Unstable Limit Cycle

Figure 53.2

the representative point being displaced along the radius returns to 1(;. If this point does not return to Ki' as for example Point K;, such a l1m1t cycle is unstable. The loci of limit cycles in this case are concentric circles corresponding to all possible values of the constant 60 , If one returns to the original variables (~,y) of the phase plane, one must apply the equations of transformation [52.2], where the variables a and 6 are respectively Ki cos 90 and 1(; sin 6 0 , This gives x = a cos t

+

b sin t = Ki cos 80 cos t

+

Ki sin 80 sin t = K j cos (t - ( 0)

[53.3] y = -asint

+

bcost= -Kjcos60sint

+ Kjsin80cost

= -Kj sin(t-80)

where 8 0 is arbitrary. This arbitrariness of 6 0 for the (~,y) phase plane is due to the fact that a point of equilibrium in the plane of variables (a,6) of Van der Pol corresponds to a circle in the phase plane of the variables (z,y). The general form of trajectories in the (z,y)-p1ane is shown in Figure 53.2. Case 2. '" (K) = 27rK 1

1"f (Kcos u, -1(sinu) cos udu '*

0

Let KI , K2 • be the roots of I/I(K) = 0, and assume that these roots are different from the roots 1(11 K 2 , ••• of ~(K) = O.

171

Consider now Equations [52.8). The motion on a limit cycle is represented in the plane of variables (a,b) by points of equilibrium given by equations a

=

KiCOS[~if/{Ki) t

+ 80 ];

The stability (or instability) of a limit cycle is determined again by the sign of t/> (Ki ) and the direction of rotation by the sign of if/(K i ). The topological picture of trajectories in the (a,b)-plane is shown in Figure 53.3. The traJectories "turn back" at pOints corresponding to the roots Xli X2 , ••• of if/(K i ) , approaching the stable limit cycles which are again the pOints of equilibrium in the Van der Pol plane. The topological picture of trajectories in the fixed (x,y)plane has the same appearance as that shown in Figure 53.2. The onl7 difference between the two cases is that, in the second case, Equations [53.3J become

b = Kisin[Jlif/(K,)t +80 ]

I

x = a cos t

+

[53.4 )

b Stable Limit Cycle

a

LOCI of

Unstable Limit Cycle

Radial Tongents

Figure 53.3

b sin t = Ki cos ([1- 1.1 if/ (Ki >] t - 80 )

[53.5] y = - a sin t

+ b cos t = -

K, sin([l - J.lif/ (Ki>]t - 80 )

It is clear that in this case a correction for frequency exists expressed by Jlif/(K,). In other words, the velocity along the spiral trajectories is not uniform. By a further analysis it can be shown that when if/(K,) = 0 the correction for frequency (and hence for the period) is of the order of 1.1 2 and consequently can be neglected in the theory of the first approximation. If, however, W(K,) 0, this frequency correction appears to the first order of 1.1. To sum up the results of this section, it can be said that the use of the variables (a,b) of Van der Pol enables us, if the system is isochronous, to represent a limit cycle by a point in the plane (a,b), that is, by a constant vector. Such representation of a limit cycle is similar to the

*

172

mode of representing alternating currents by vectors. For transient conditions the representative point moves toward, or away trom, the limit cycle point along the length ot the rad1us vector, since the phase angle 80 remains constant. The phase angle in this case has no particular physical significance it one single oscillatory phenomenon is considered. It, however, ~(.1C) :/: 0, that is, i t the motion is not isochronous, in the plane ot variables (a,b) of Van der Pol the vector Jri undergoes oscillations depending on the roots ot ~(Jr) = 0. This peculiarity resembles the representation ot phase-modulated vectors in radio technique. Fixed points in the (a,b)-plane correspond to circles in the (~,y)-plane, and a radial motion in the (a,b)plane corresponds to a spiral motion in the (~,y)-plane. 54. EXAMPLE: II SOFT" AND II HARD II SELF-EXCITATION OF THERMIONIC CIRCUITS We now propose to apply the Van der Pol method to the problem previously treated by the method ot Poincare in Section 49. Consider again Equation [30.9]. Here we shall let P = PP1 , )' = P)'l' and 6 = p6 l , where P is a small parameter. [54.1]

where P I ')'1I and 61 are positive. In this case I(v,v) = (p + 2"v - 36v 2 )v. [52.9], we have

Using the first equation

[54.2)

= Jr cos

in which the generating solutions v We obtain t/>(K) =-

u, 'Ii

2~[PJr[sin2UdU +2)'Jr2rCOSUSin2udU -

r

.2

=

'If',

r

a

2

sin ucosudu

sin

U

are substituted.

S 2 2 S6Jr rCOS uSin udu]

.

Since sm udu

= - Jr

2.
smu = [-s-l =-

0

and

r Jcos

2

d=

• 2 U8lDU U

JW810 . 2U

dr. U -

4

d

J81n u u

==

'If'-

'43 'If'

'If'

== 4

[54.3)

173 we have [54.4 ]

By the first equation [52.8], the condition for the existence of a limit cyc~Ms tP(K) = O. From Equation [54.4] this takes place for K} ::r 0 and K2 = V~. For K} = 0, the limit cycle reduces to one point, the singular point. The radius of a finite limit cycle is thus

In order to ascertain that there is actually a condition of se1texcitation, one has to prove that the singularity is unstable and the limit cycle is stable, see Section 29 •. For the proot ot the first point, equations ot the first approximation in the sense of Liapounoff must be formed. The system equivalent to Equation [54.1], upon dropping the nonlinear terms, is v = y and y + 11 = ppy, that is v = y;

The characteristic equation is 8 2

y

= - 11

- p{J8

+

1

+ p{Jy

=0

[54.6 ]

whose roots are

.

Since p is small, it is seen that the roots are complex, with a positive real part. Therefore the singularity is an unstable focal point from which ~~ spiral trajectories unwind themselves approaching the limit cycle K2 = V~, provided it is stable. In order to establish the stability of the limit cycle K2 = ~, it is necessary to ascertain the Sign ot tPK(K2 }. One finds

The limit cycle is thus stable. In this discussion it has been assumed that the coefficients p, ~, and 6 entering into Equation [54.2) are positive. By waiving this assumption one could analyze addit10nal cases following the same procedure. It is of interest also to investigate the second equation [52.8], which concerns the frequency of oscillation, ~~.

174

The function

~(K) =

~(K)

2!K

in this case is

[(P + 2"K coau -

2

2

36K coa u) Kainu coaudu

Developing it, we find

~(K) =

+ 2"K 2 [ain u coa 2 udu

2;K[PKjsinucosudU

:.... 36K

a

j

8

cos usinudu] [54.8]

Each of the integrals appearing in the above expression is zero so that by the second equation [52.8] ~: = o. This means that in the phase plane the radius vector ot the representative point rotates around the origin with a constant angular velocity, and the trequency corr~ction is zero to the tirst order. In order to investigate the var1at1on ot K as a tunct10n ot time, the first equat10n [52.8] must be integrated upon substituting tor ~(K) its value g1ven by [54.4]: ~(K) = mK - nK3

P and where m = "2

11.

=.3! ~...

This g1ves dK

mK - nK3

=

!

dK

-

m K (1 - pK2) - P

dt

n .3.! where p '" m = Tfji.

dK

_ dK K

K(l- pK2) -

That is, d 10gK- d[t 10g(1 - PK 2 )]

'"

+

pKdK 1- pK 2 = mpdt

mpdt, or

K d(logYI_ pK2 ) = mpdt

which, upon integration, gives log

K

Vl-

pK2

- log

Ko

Yl-

pK2

o

= mpt

or

Finally, [54.10]

175 For t = 0, K = Ko = ~ and tor t + - 00, K ... 0, which means that, tor increas:?fot, the radius vector K increases trom zero and approaches the value Ko = V on the limit cycle. Furthermore, tor t + + 00, K ... Ko, which shows that the 11m1t cycle is stable. In order to eliminate the operation with intinities inherent in the asymptotic nature ot the process, one can select instead ot K and Ko in Equation (54.10] certain initial and tinal values K' and Ko' slightly removed trom the unstable to cal point and the stable limit cycle respectively. In such a case Equation (54.10] can be used tor numerical calculations with a view to ascertaining how rapidly the selt-excited oscillatory process builds up as a tunction ot time. It is interesting to note that the terms with Y, E, " ' , corresponding to the even powers in the approximation ot the characteristic by a series expansion, disappear trom ExpreSSion (54.10] tor the radius ot the limit cycle in the tirst approximation, as was noted in connection with the 2w vanishing ot the term 2YK2fo sin 2 u cosudu in Equation [54.8]. The inverse passage trom the phase plane (a,b) ot Van der Pol to the ordinary phase plane (%,1/) yields the expressions % = K cos t and 1/ = x = - K sin t, where K is given by Equation [54.10]. The conditions ot selt-excitation considered above represent the so-called "sott" type ot selt-excitation. Topologically, it corresponds to the existence either ot an unstable singularity surrounded by a stable 11m1t cycle or ot a stable singularity within an unstable 11m1t cycle, see Section 29. The tirst case represents the building up ot oscillations asymptotically approaching a stable limit cycle; the second, an asymptotiC disappearance ot the oscillatory process. It, however, between an unstable Singularity and a stable limit cycle an unstable limit cycle exists, one is then concerned with the so-called "hard" selt-excitation ot oscillations. This subject has already been studied in Section 51 in connection with the theory ot Poincare. It was shown that the characteristic in this case has an inflection point; in its approximation by expansion in a power series, one has to retain a term EV 6 with a negative sign. Since even terms do not have any ettect on the determination ot limit cycles, one can drop them trom the equation. Under these conditions, Van der Pol's equation becomes

5f

Ii

+v =

p(1J

+ 3cJv 2 -

In this case I(v,v) = (IJ + 3cJv 2

-

511V4) V

511V·)V

and

(54.11 ]

176

Prom this, by the first equat10n [52.9], we get 2 1 cP(K) = 2 ,J[PKSin UdU

=

+

[36K3cos2USin2UdU - [51/K5cos4USin2udU]

2~ ~[sin2udU + 36K 2 rCOS2USin2UdU -

4

2

517K4rcos usin udU]

The values of these defin1te 1ntegrals are

~,

2

~smu

d U=7rj

r

2

,2

~cosuBmu

r

. 2udu = cos4usm

dr.

2

U=JSlnU

r r

d r ·SlnU 4 d 3 1T uU=1T-4"1T=i

3 - --1T 3·5 =7r cos6udu = -1T 4 4' 6 8

cos4udu -

This g1ves 3 2- -"K 5 4) cP(K) = -K (P+ -6K 2 4 8

= 0

[511-.12]

as the condit1on for limit cycles. It 1s assumed that " 1, and 1/ are pos1tive. One root is clearly K= O. Pollowing the same procedure as before and making use of L1apounoffl s equat10ns of tho first approximation, one finds that the singularity 1s an unstable focal po1nt. The 1im1t cycles proper are given by the quadrat1c equation 523 -77S - -6S -P = 0 8 4

where S = KI. Only pos1tive roots are to be considered because 8 :: KI 1s essentially pos1t1ve. Equation [54.13] can be wr1tten as 5 2 3 2 2 2 2 q2 q2 ( q)2 q2 P = -,,8 - -6S = p S - q8 = p S - q8 + - - = pS - -8

where pi

4p2

4

= ~" and

q =

4p2

2p

4p2

t"

If th1s equation is rearranged, [511-.14 ]

Equation [54.14J represents the parabola (P - Po) = p~(S - So)2, as shown in P1gure 54.1, If we change 2the (S,p)-axes to a new system of axes (8 .. P1 ) with a new origin at (Po m So = Equation [511-.14] becomes

J/ir,

P1 =

P2S12

t),

177

The second root corresponds to the limit cycle K2 a ~ and the first one to an unstable focal point which can be ascertained as explained in the beginning of this section. For ~ < 0, that is, to the left of the origin 0, the roots of the quadratic equation p 28 2 - q8 - ~ = 0 are

____

~

____

~

_ _ _ __ L_ _ _ _ _ _ _ _ _

p

o

_ q+v'q2-41~lp2 S1,2 2p2

Figure 54.1

They are real and positive only as long as q2 - 41~lp2 ~ O. They correspond to the points 8 1 and 8 2 of intersection of the parabola with the straight q line ~ = constant. The condition for a double root is I~ol = 4;2; at this value ot the negative ~, the value of the root is 8 0 = 2 ' The tangent to the parabola at this point is vertical. In the region 0'0, where two roots exist, there are two limit cycles, K1 = JIS';" and K2 = VS;. The first is unstable and the second stable. To illustrate this point, differentiate Equation [54.12) with respect to K.

2:

[54.15] It is sufficient to substitute into this equation the values of the roots 8 1 and 8 2 for ~ in the interval (0, - q2J4 p 2), since the curve does not extend to the left of ~ = - 4q~ p and has only one root to the right . of P2 = O. Consider, for example, the middle value in this interval, ~1 = -~. The corresponding roots are 8 1 ,2

=>

- q (1 + 2p2 -

V

1 _ 21 )

that is, 8 1 = O.293q/2 p 2 and 8 2 = 1. 707q/2 p 2. Substituting these values in Equation [54.15), in which ~= 2, one finds, after a reduction, that tP'(K1 ) > 0 and tP'(K2 ) < O. Hence the limit cycles on the lower branch 8 1 ot the parabola are unstable and on the upper branch 8 2 they are stable. If ~ varies now trom negative values and reaches the point 0', there is no self-excitation throughout the range 0'0 since the unstable limit cycles 8 1 , interposed between the stable tocal points situated on the ~-axis and the stable limit cycles 8 2 , act as a barrier preventing the

8:

178

self-excitation from developing, see Chapter IV. As soon as Point 0 at which the unstable limit cycle disappears is reached, the stable limit cycle K2 = ~ is reached abruptly; if fJ is still further increased, the amplitude of the limit cycle increases continuously, the representative point S following the upper branch of the parabola to the right of Point A. If, however, fJ is decreased, the amplitude of the stable limit cycle follows the upper branch 8 2 until the point So (Ko = ~ ~) is reached. Here the seUexcited oscillation disappears abruptly. As previously mentioned, see Section 51, this type of self-excitation is called hard. It is apparent that the theory of Van der Pol gives exactly the same results as the theory of Poincare. 55.

EXAMPLE: EQUATION OF LORD RAYLEIGH; FROUDE' S PENDULUM In his researches on the maintenance of vibrations, Lord Rayleigh

(14) came across the following equation

+ Kx

ma: - (a - pih i

[55.1 ]

= 0

in which there is a predominance of negative damping for small values of the velocity Xi for larger velocities the damping becomes positive. By introducing "dimensionless time," as was explained in Section 30, Rayleigh's equation can be put in the form

x + x + Jlf(x,i)

[55.2]

= 0

Assume that the damping terms are small enough to justify the introduction of the small parameter p. In Rayleigh's equationf(x,i) = f(i) = mi 3 - ni, where m and n are constants appearing instead of a and fJ as the result of the transformation of the independent variable. One has finally

x+

+ Jl (nti 3

x

-

ni) = 0

where m > 0 and n > O. By Equations [52.9], the functions

~(K)

1 ~ 1 f mKsm.2 udu - -27T 27T 6

= -

~(K)

[55.3]

and ¢(K) are

1 !WnK 3.4 sm udu = -K(m 2

3 2 - -nK ) 4

[55.4]

and I/J(K) =

2~K [nK

3

3

sin ucos udu - 2!K [ mK sin u cos udu

[55.5]

179 Prom Equation [55.4) we see that the conditions for limit cycles are K

=

0;

K=

V

4m 3n

[55.6]

The first value K= 0 is clearly a point singularity. If the nonl1near term m3: 3 1s omitted, Liapounoff1s equations are 3: == 'II and iJ- ny + x = O. Prom this we obtain the characteristic equation S2 -nS + 1 = 0, whose roots are

=n+~

S

2

1,2

Por n < 2, one has an unstable focal point, and for n > 2, an unstable nodal pOint. In both cases the point singularity K= 0 is unstable. In order to ascertain the stabil1ty of the limit cycle K = ~, one must differentiate Equation [55.4] with respect to K f/J'(K)

Substituting K2

= m -! nK2 2

[55.7]

8

= j: in this expression, one finds f/J'

(V 4m ) = m -! n 4m = ~ - ! 3n 2 8 3n 2 2

m

<

0

The limit cycle is therefore stable. The oscillating system governed by Rayleigh1s equations, [55.1] or [55.3], thus exhibits the property of being self-excited in a steady state. If the dissipative terms were of the same Sign, that is, of the form ±(mi: + ni: 3 ), there would be no limit cycle in this case, as is easily ascertained, although the oscillation would still be governed by a non-linear differential equation of a dissipative type. The example of Section 31 belongs to the case considered in this section. It can be shown that the oscillation of Proude1s pendulum, see Section 8, is also governed by Rayleigh1s equation. In the elementary theory of this phenomenon it was established that the damping is initially negative under certain specified conditions. A linear equation does not represent the actual pnenomenon because when the coefficient of ~ is negative it indicates that the amplitudes of oscillation increase 1ndefinitely, which is clearly imposs1ble from physical considerations. The reason for th1s 1nconsistency is the fact that the equat10n was overlinearized by dropping the non-linear terms. By conserving at least the first two non-linear terms, we shall be able to establish the existence of a finite limit cycle, characterizing the

180

steady state of oscillation of Froude's pendulum which is actually observed. Expanding the function M( w - ~) of Section 8 in a Taylor series, we have M(w -

~) =

'2

M(w) - tJM'(w)

+ !L M"(w) 2!

'3

-

L M"'(w) + ' , . 3!

[55.8 ]

Here we assume that the function M is analytic, that is, that it admits derivatives at least as an idealized feature of the observed phenomenon. Dividlng Equatlon [8.1] by I, introduclng dimenslonless time, and keeping only the first two non-linear terms in the expanslon, one obtains an equatlon of the form ••

··2-3

q, + q, = - cq, + kq, - nq,

[55.9]

where c b + M'(w), and k and n are su1table constant coefflclents obtalned by substltuting the expanslon [55.8] into Equat10n [8.1], It must be noted that c ls negatlve according to what has been stated ln Section 8. Puttlng c = - m, where m > 0, one has ::3

whence

and

Hence

that ls,

It

3 3]

q, (K) = - -2 m1K - -n K 4 1

The equat10n for lim1t cycles ls

K(m 1-.!n K2) = 41

0

[55.10 ]

The solution 1[= 0 corresponds clearly to a point singularity, and the amplitude of the 11m1t cycle is K = One can calculate m l and n l explicitly from the equations of transformatlon of the original equat10n to

V;::.

181

its form involving dimensionless time. It is to be noted that the quadratic term disappears from Expression [55.10) tor the limit cycle because z.. 3 sin u du = 0, a tact which has already been noted in Section 54.

f.o

56.

MORE GENERAL FORMS OF NON-LINEAR EQUATIONS The Van der Pol method is applicable also to non-linear equations involving, in addition to variable damping, a variable " spring conBtant." It must be noted, however, that the existence of selt-excited oscillations, that is, ot limit cycles, depends only on the conditions of the variable damping, as was shown by Van der Pol (15) and generalized later by E. and H. Cartan (16) and Li6nard (11). In fact, the initial damping, tor small x in the Van der Pol equation or tor small x in the Rayleigh equation, is tirst negative then positive tor larger values of the corresponding variable (x or x). Physically this means that there exists, initially, an energy input into the system Which, in later stages of motion, becomes an energy drain from the system, which then becomes dissipative. The existence ot a steady oscillation, that is, of a limit cycle, depends thus on the average equality between the input and the drain per cycle. We shall come back to this important point later in connection with the Pr.inciple of Equivalent Balance of Energy formulated by Krylotf and Bogoliuboff. Consider, for example, a more general type of equation mx"

2 + (nx

- a )'x

+ fJ x

+)lX 2

= 0

[56.1]

which has variable damping, the term nx 2 x, as well as a variable spring constant, the term (fJ + yx)x. As long as the terms (nx 2 - a)x and YX 2 are small, we can write this equation in the form

mx

+fJx =

p[(a

1

-

2 n 1x )x - YIX2] == pq,(x,i)

[56.2 ]

We can easily reduce this equation to a form previously considered. It is sufticient to divide it by m, putting fJlm = W 0 2 • and pass to a dimensionless independent variable* in order to obtain the equation in the usual form

*

The last operation, although convenient for practical calculations, is not altogether necessary. In case one does not use it, the generating solutions should be taken in the form K cos w t, - Kw sin wt instead of K cos t, - K sin t.

182

and to substitute the generating solutions K cosu and - K sinu into the first equation [52.9] giving the limit cycles. The same procedure applies to the generalized Rayleigh equation with the variable spring constant. In all cases the existence of limit cycles requires that the coefficient of x satisfy the conditions of Cartan-Lienard. If this condition is not fulfilled, no self-excited oscillations can exist in a steady state, and the system behaves as purely dissipative, while still non-linear.

CHAPTER X THEORY OF THE FmsT APPROXIMATION OF KRYLOFF AND BOGOLIUBOFF*

57 . INTRODUCTORY REMARKS The method of Kryloff and Bogoliuboff is very similar to that of Van der Pol and is related to it in the following way. While Van der Pol applies the method of variation of constants to the basic solution x = a cos wt + b sin wt of x + w 2 x = 0, Kryloff and Bogoliuboff apply the same method to the basic solution x = a cos (wt + 4» of the same equation. Thus in Kryloff and Bogoliuboff's method, the IIvaried ll constants are a and 4> (polar coordinates), while in Van der Pol's method they are a and b (rectangular coordinates). The method of Kryloff and Bogoliuboff seems more interesting from the point of view of applications, since it deals directly with the amplitude and phase of the quasi-harmonic oscillation. Before proceeding with a review of the method of Kryloff and Bogoliuboff, a few additional remarks concerning the effect of secular terms may be helpful.

58. EFFECT OF SECULAR TERMS IN SOLUTIONS BY EXPANSIONS IN SERIES The difficulty arising from the appearance of secular terms has already been mentioned in Sections 44 and 46. In the example given in Section 46 that d1fficulty was avoided by a rather delicate change trom the lIold ll periods to the II new ,1I or corrected, ones. This change requires a lmowledge of the correction for the period, which is not always obtainable as has been demonstrated. Unfortunately, in a great majority of cases in which the approximation consists in abbreviating an infinite series by a few terms, the situation is still more difficult. In order to see this point, consider again a quaSi-linear differential equation of the form (58. , ] where ~ is a small parameter, i.e., p « ,. Since the non-linear term appears with a small coefficient p, Poisson suggested as a solution an expression of the form x = Xo

+

PX 1

+

.)

Jl~x.,

~

+

(58.2]

The subject matter of tlus and of the following two chapters is taken from the treatise of Kryloff and Bogol1uboff, Reference (5). This SUbj~ct is also treated in the free translation by s. r.efschetz of the Kryloff-Bogoliuboff text, Princeton University Press, 1943.

It

1814

For p. = 0, % = %0 = a cos fAIt is the imown generating solution. For 0 < P. «', it seems logical to consider the effect of the non-linear term p./(%,z) as a small perturbation and to assume that this perturbation will also be felt in the modification of the initial generating solution represented as a power series of P., given by Equation (58.2]. The method of Poincare, Chapter VIII, is, in fact, a further generalization of Poisson's method. If one substitutes the solution (58.2] into the differential equation (58.1], limiting the expansion up to the power p.Kof the small parameter, one obtains the following series of differential equations by equating terms containing the same powers of p.:

.. +

%0

= 0

2

W %0

(58.3]

This pr~cedure has already been outlined in connection with Equations [46.5] of the theory of Poincare. It is easy to Show, however, that a direct application of this method is handicapped by the following difficulty. Consider, for example, the simplest case, that in which /(%,i) = + ~fAI. In this case Equation [58.1) is x.. +

W 2%

+ P.fAI%.

= 0

The exact solution of this equation is

[58.4) where A and ~ are the constants of integration determined by the initial conditions. If, however, one proceeds by substituting the expansion [58.2) into Equations [58.3], the f1rst equation gives %0

= Acos{fAlt

+ ~)

Substituting this solution into the second equation [58.3], one has

This equation is satisfied by %

1

Awt 2

= - --eos(fAlt

+ ~)

[58.6]

185

Subst1tuting these values for %

=

%0

and

%1

into Equat10n [58.2], one has

A(l - Il;t)cos (wt +

q,)

It 1s clear that the ampl1tude of the approximate solut1on [58.7] increases w1th time t 1ndef1n1tely, whereas, according to the exact solut1on [58.4], it .l!!!!.!.. approaches zero, owing to the presence of the exponential term e--YAs a second example, consider the differential equation

[58.8] wh1ch may be considered as the equation of motion of a mechan1cal mass attracted to the position of equ1librium by a force proport1onal to the distance, w1th a perturbation term proportional to the cube of the distance. Proceeding as before and seeking a solution of the form % = %0 + P%, one has

From the first equat10n [58.9], %0 = A sin(wt + this value for %0 into the second equation, one has

~l +

2 W %1

8 = -w 2A8sin (wt

+ q,) = -~w2A3sin(wt + q,) +

~).

Substitut1ng

! w A sin3(wt + 2

3

q,)

This equation is satisf1ed by

:2

8

Xl

= : wtAScos(wt +

q,) -

sin 3(wt

+

q,)

whence X

=

Asin(wt + q,) +

3:

3

wtA eos(wt

3

+~) - ~: sin 3(wt +

q,)

[58.10]

The second term of this expression 1s a secular term. Thus, this expression tor the displacement has no physical meaning. Unfortunately, in th1s case the exact solut1on 1s not known, as Equation [58.8] 1s not l1near. One can, however, affirm the correctness of the above statement by invoking the law of conservation of energy, which holds 1n this case Since the system 1s conservative. In fact, multiplying Equation [58.8] by X, one can write 2

. [..

X X

+ (1 +

Jj%

2) W 2] X

.2 + -w 12 2X2 + -pw4x4) = 0 = -dtd (1-x 2

186

Prom this we can obtain the law of conservation of energy 1 2

-

.2

X

4 + -21 w 2x2 + -#W2 x = 4

constant

=

[58.11]

E

Prom this equation it follows that, for # > 0, the square of the amplitude X2 has an upper bound ~/W2. This result is to be expected, as the system has no sources of energy and is conservative. Hence there is a definite contradiction of Equation [58.10]. Prom these two examples, we see that the direct application of Poissonls method to problems of dynamiCS encounters a serious difficulty because of the presence of secular terms.

59. EQUATIONS OP THE PIRST APPROXIMATION In this section we propose to establish the fundamental points of the theory of the first approximation of Kryloff and Bogoliuboff, which will play an important role in subsequent chapters. For # = 0, Equation [58.1] reduces to a simple linear equation whose solution is x = a sin(wt

+ tP);

i = aw cos(wt + t/J)

[59.1 ]

where a and t/J are constants, the amplitude and the phase respectively. For a quasi-linear equation when # 0 but is small, it appears logical to retain the form of solutions [59.1], provided that we consider the quantities a and tP not as constants but as certain functions of time to be determined. Differentiating the first equation [59.1], one obtains

*

x=

+ awcos(wt + t/J) + atj,cos(wt + t/J) Making use of the second equation [59.1], one has asin(wt + t/J)

[59.2 ]

+ t/J) + atj, cos (wt + t/J) = 0 Differentiating the second equation [59.1], one gets

[59.3 ]

a sin (wt

!i = awcos(wt

+ t/J) - aw2 sin(wt + t/J) -aw~sin(wt + t/J)

Substituting these values for x, X, and tion [58.1], one has

x into

the original quasi-linear equa-

awcos(wt+t/J)-awq)sin(wt+ t/J)+ pf[asin(wt+tP),awcos(wt+tP)]

Solving Equations [59.3] and [59.5] for it

a and

= 0

[59.51

tP, one gets

= - ~f[a sin (wt + 4», awcos(wt + q,)Jcos (wt + q,)

[59.6 ]

187

if,

=

:wf[asin(wt + q,).awcos(wt + q,)]Sin(wt + q,)

(59.7]

It can be seen that the original equation (58.1] of the second order has been reduced to a system of two equations, (59.6] and (59.7], of the first order. The interesting feature of this transformation lies in the fact that these first-order equations are now written in terms of the amplitude and phase as dependent variables. One notes a formal analogy with Equations [52.8) of Van der Pol and also with Equations [46.22] and [46.23] of Poincare. From the form of the right side of Equations [59.6J and [59.7J, it is seen that both a and ~ are periodic functions of time. From the fact that the right-hand terms of these equations contain a small parameter ~, one can conclude that both a and ~, whjle being periodic, are functions which vary slowly during one period T= 2"/wof the trigonometric functions involved. It 1s reasonable, therefore, to consider a and q, as constant during one period T. It is possible to transform Equations [59.6J and (59.7) into a more convenient form. For this purpose, consider the Fourier expansions of the func tions

[59.8)

where 1

2..

KoCa) = -2 ffCasinq,.awcosq,)cosq,dq, "0

1 Po (a) = -2 Ii(a sinq,. awcosq,) sinq,dq, "0

K,,(a)

=

1

2..

-

fl(asin q,. awcosq,)cosq,cos nq,dq,

"0

1

Ln (a)

=-

2..

f I(a sin q,. aw cos q,) cos q, sin n q,dq,

"0

2..

Q,,(a)

=

f

.! I(a sinq,.awcosq,) sinq, sin nq,dq, "0

[59.9]

188 Equations [59.6] and [59.7) can then be written in the torm dta = d

-E..Ko(a)-~ .t"rKn(a)cosn(wt + tP) + w w ,,=1 t

L,,(a)sinn(wt + t/»]

[59.10] d,pt = LPo(a) +..1!. .t[~(a)cosn(wt + tP) + Q,,(a)sinn(wt + ,p>] d aw aW,,=1

Integrating these equat10ns between the lim1ts t and t + T, and considering aCt) and ,pet) as remain1ng approx1mately constant 1n this interval, one has as the t1rst approximat1on

s1nce I+T

I+T

f cosn(wt + tP)dt f sinn(wt + t/»dt =

I

= 0

I

Furthermore, since, by assumpt1on, the var1ations da and d,p ot amplitude and phase are small dur1ng the interval (t, t+ T), one can write Equations [59.11] to the first approximation da -dt = - -wJ.l K 0 (a)·•

dtP = -P. J.l -dt (a) aw 0

[59.12 ]

If these equations are compared w1th the exact equations [59.10], it is seen that the equations ot the t1rst approximation are obtained from the exact equations by averaging the latter equations over the period, thus eliminating the rest ot the Fourier series under the summation sign. The analogy between Equations [59.12] and the Jlabbreviated Jl equations [52.6] ot Van der Pol should be noted. Letting r/J = wt + ,p, the total phase ot the motion, we have dI/J/dt = w + d,p/dt. Making use of these relations and the relation for Ko(a) and Po(a) in [59.9], we obtain for the equations ot the tirst approximation

~; :~

= -

~ 2~rf(asintP,awcostP)COStPdtP ==

CP(a)

= w+:w 21rrrf(aSint/>,awcostP)SintPdtP == D(a)

[59.13 ] [59.14 ]

The tirst approximation will then be ~ = a sin r/J, where the amplitude a and the phase r/J are obtained from Equations [59.13] and [59.14).

189 --...

60.

NON-LINEAR CONSERVATIVE SYSTEMS Consider again a quasi-linear differential equation

x + w2x

+ pf(x)

in which f(x) does not contain the velocity da = _

dt

-L 27TW

[f

= 0

x.

Equation [59.13] gives [60.1 ]

(a sin c;6) cosc;6 dc;6

and Equation [59.14] gives

:~

= W

+ 2:aw [f(a sint,6) sint,6 d4J

[60.2 ]

Noting that 1

r

f(a sint,6) cosc;6 dt,6

=

2.

a 1/1 (a sint,6)1

=

0

o

where I/I(x) =L%f(t)dt, Equation [60.1] gives a = constant. Hence, from the o first approximation, it follows that the amplitude does not change in the course of time; the system is thus conservative. This can be seen from a priori considerations, because the function f does not depend on the velocity X, whereas dissipative forces generally do depend on z. From Equation [59.14] it follows that 1/1 = .Q (a) t

+

[60.3 ]

1/10

"since D(a) does not depend on t in this case; 1/10 is a constant of integration. Thus, the oscillations will be of the form x

=

a sin[.Q(a)t

+

1/1 0]

[60.4 )

Therefore, to the first approx1mation, the effect of a non-linearity of this type will be felt only in that the frequency of oscillation will depend on the amplitude a, that is, the oscillations are not isochronous, but the decrement of oscillation is zero since the system is conservative. Squaring Equation [59.14) and neglecting the term of the second order in p, one has

[60.5]

190

where

[60.6] On the other hand, the general form of a non-linear equation, in which the non-linearity is only in the spring constant, is of the form x + F(x) = O. If the term F(x) is not far from linearity, it can be written as F(x) = w2 x + pf(x), where f(x) is the non-linear component of F(x). Comparing this expression with Equation [60.6], one finds exactly the same result provided one substitutes the generating solution a sin ~ for x. From this we obtain the following theorem: When a system is conservative but not linear J the amplitude a remains constant and the frequency Q(a) is given by Equation [60.5], in which F(z) is the term entering into the equation

x + F(z)

;: 0, without the necessity of splitting it into a linear component w 2z and a non-linear

one, pf(x).

In the following section examples are given illustrating the application of the theory of the first approximation. EXAMPLES OF NON-LINEAR CONSERVATIVE SYSTEMS PENDULUM .. g The differential equation for a pendulum is (J + L sin (J = O. In elementary theory, which we may designate as an approximation of zero order, it is assumed that for small angles sin (J ~ (J. The well-known solution for the period, T = 27r JILlg, is obtained. It is to be noted that osc11la tions are isochronous under this assumption. For the first approximation we can take sin (J ~ (J the differ•• (J8 I 2 2 ential equation then becomes (J + w «(J - 11) = 0, where w = g L. Using Equation [60.5], one obtains 61.

A.

1r;

that 1s, D(a) = w

.r:-7 VI - "8

(

~ w 1 -

2

a ) 16

[61 . , ]

or, in terms of the period T(a) = T( 1

+ ~:)

[61.2 ]

191

It is thus seen that the oscillation is not isochronous; the period increases slightly with increasing amplitudes of oscillation. Thus, for example, for amplitudes of the order of 10 degrees, T(a) = T x 1.001; for amplitudes of the order of 20 degrees, T(a) = T x 1.006; for amplitudes of the order of 30 degrees, T(a) = T x 1.014. It should be observed that, although the expression for the period can be established in this case by means of elliptic functions, the theory of the first approximation leads to this result by a more general procedure. Furthermore, the latter method easily leads to correct results in more complicated cases for which exact methods are not available. B.

TORSIONAL OSCILLATIONS OF A SHAFT Let J 1 and J 2 be the moments of inertia of rotating masses placed at the ends of a shaft S, see Figure 61.1. If 9 1 and 92 are the two angles determining the angular position of the masses J 1 and J 2 with respect to a fixed reference angle, the torsional moment M is a certain function of s the difference (9 1 - 92 ), say C( 9 1 - 92 ) =M( 9) . * The differential equations of the coupled system are

+

J 161

C(9 1

-

92 ) = 0

and

J 2 62

C(9 1

-

-

92 ) = 0

Figure 61.1

Subtracting the second equation from the first and letting 9 = 9 1 - 92 , we obtain [61 .3 J which is the non-linear differential equation of the torsional oscillation of the system. Equat10n [61.31 can be written as (j + K 2C(9) = 0, where K2 = (J 1 + J 2 )/J1 J2 • Assume that C(9) is of the form C(9) = Co9 ± C 1 9 3 , where C 1 9 3 is a small non-linear term. From this,

and the frequency 1s given by g2(a)

*

C(9 1

-

=

_l_]w(ma 'Ira 0

sin~

+

na3sin3~) sin~ d~

9 2 ) in these notations should be read:

=

C is a function of (9 1

W 2 + !K 2C a 2 0 4 I

-

9 2 ), and not C times (9 1 - 9 2 ),

192

or 2

(1 ± :

D(a) :::

Wo

K

. 2

~~2a

)

The non-linear frequency (and therefore the period) will thus depend on the amplitude. The amplitude of the vibration is constant, but the vibration is not isochronous. C.

ELECTRICAL OSCILLATIONS OF A CIRCUIT CONTAINING AN IRON CORE

Let a circuit be composed of an inductance L and a capacity C with negligible resistance. The inductance coil is wound on an iron circuit A, see Figure 61.2, subject to magnetic saturation. The condit10n of equilibrium of electromotive i --forces in the circuit g1ves dt

t

.!..1 i dt =

d~ + e

c

L

0

where.1s the totalized flux through the coil. The condit10n of saturat10n can be approximated by the equation i = A. + B~a, whence from the preceding equation one has

A

Figure 61.2

[61 .4]

Equation [61.4] is the non-linear equat10n of oscillation. Reducing it to the standard form of a quasi-linear equat1on, where A/e = w2 = constant, and assuming that the rat10 ~2« 1, one can apply Formula [60.5] and obtain

!

g2(a) =

",!e r(Aa

sint,6

+ Ba 3 sin3t,6) sint,6 dt,6 ...

~ (1

+

3:;2)

that is, D(a) =

W

v'

1

2

3Ba +- ::: 4A

W (1

+ -83 -Ba A

2 )

[61.5]

The actual frequencyD(a) 1s thus increased in comparison w1th the frequency w for small ampl1tudes, ow1ng to a decrease ot L w1th the amp11tude a of the

osc1llation. Thus the oscillat10n is non-1sochronous. 62.

SYSTEMS WITH NON-LINEAR DAMPING OF A DISSIPATIVE TYPE

Consider the d1fferential equat10n

mz

+ K~ +

f(x)

=0

193

Dlvldlng lt by m and putting K/m

= w2 ,

we get

2 I (·) x+wx+x =0 mf

••

We shall keep wlthln the 11mlts of the quasl-harmonlc theory. ~ f(x) plays the role taken by ~f(x,x) ln the general theory. ~

fo f (a sin c;6. aw cos c;6) cos c;6 dc;6

In thls case The expresslons

~

f f (a sin c;6. a w cos c;6) sin c;6 dc;6

and

0

entering into Equatlons [59.13] and [59.14] of the first approximatlon, ln thls case are [f(aw cos c;6) cos c;6 dc;6 and

]'>

(aw cos c;6) sin c;6 dc;6

respectively, since f(x,x) reduces to f(x). We note that

r

f(awcosc;6)sinc;6dc;6

1 = - -aw

r

f(awcosc;6ld(awcosc;6)

=-

1 ~= 0 -cfJ(awcosc;6)1 aw 0

so that, by Equation [59.14), dl/1 dt = w

[62.1]

Furthermore, Equation [59.13] can be wrltten as da = dt

a=

- 2 1

rf(awcosc;6) cosc;6 dc;6 1Tmwo

[62.2 ]

It ls clear that the instantaneous frequency dl/1/dt ls equal to the constant 1I11near il frequency w, and the amplitude a varles according to Equatlon [62.2). Thus, the oscillatlon is generally of the form x = a sln(wt + 1/1 0 )' where 1/10 is a constant. The frequency is not changed since the frequency correctlon is of the second order and therefore does not appear in equations of the first approximation. A few examples given below illustrate the applicatlon of Equation [62.2] to various types of non-linear damping f(i). A.

LINEAR DAMPING: In this case

f( x) = Ai

194

and Equation [62.2] gives 1 a = - -:---- . aWA1T = 21Tmw

Aa 2m

whence

[62.3] Comparing this with the exact solution At

X

= aoe - 2m sin (WI t

+ f/l o)

where

one notes that the first approximation gives the same expression for the amplitude as the exact solution; the d1fference between the expressions for frequency wand WI is of the order of ~(A/YKm)2, that is, of the second order, 1f A 1s of the first order, as prev10usly set forth. QUADRATIC DAMPING: f(:i:) = h:i: 2 Since f(:i:) is an even function of :i:, and from physical considerations it should be an odd funct10n of :i:, we should write the above expression as f(%) = 61%1%. In this case B.

L

"

[!(a.",,. d

= 6a ....[IcosI ...· d = 6a'..' [{.... d + d 2

3"

-1:"",1 d] =

·[! + ~ + !]

6a·..

= : 6a·.. •

2

Prom Equat10n [62.2] we get da -dt

= -

2

1 1Tmw

iZ.f(awcos~)cos(6 d~

that 1s, 1 da

- a2

dt

46wa

2

= - -3;"";';"";;1Tm

-

195 From this, on integrating, we obtain 1

1

4bwt

a

o7 - - --""':"

a =

or

+

1

[62.4 ]

4bwa o t 31Tm

It 1s seen that the law of variation of the ampl1tude with time in this case is entirely different from that given for Case A. C.

COULOMB DAMPING:

!(x)

= A sgn(x)*

From this equation

z.

for a =1= O. Furthermore, 10 !(awcos~) cos~d~= 0 for a = O. [62.2 J da 2A =--dt 1Tmw

if a

=1=

By Equation

0

[62.5] da = 0 dt

if a = 0

Integrating the first equation [62.5J, for a

=1=

0, we get

2A a=a - - - t o 1Tmw

[62.6]

2A >0, and will cease for tl deThe motion will continue as long as ao - ;:;:mwt fined by the equation ao = ~~ t 1 • The motion thus lasts a finite time. D.

MIXED CASES:

!(x)

= ax + /Jx2

In applications one frequently encounters differential equations 1n which both linear and quadratic damping are present. Thus, for example, Froude1s differential equation for the rolling of a ship in still water is 18 + K 19 + K 2 2 + Wh6= 0, where I, K I , K 2 , W, and h are well-mown constants. Likewise, the so-called "surge chamber" equation** in hydraulic engineering is of the form x + pZ2 + qz + yx = O. Writing equations of this kind in the form x + w 2x + p!(x) = O. one has

e

*

The symbol sgn(i) designates a discontinuous function defined as follows: sgo(i) 1 for i < 0; and sgn(i) 0 for i O.

** The

=-

=

=

sgn(i)

=1

writer is indebted to Dr. i.F. Durand for bringing this equation to his attention.

for i

>

0;

196

Iflaw

COII,6) -<6 d,6 =

r. .

aaw

'<6 d<6 +

Jla'w'[tcos',6 d<6 + [

cos'<6 d<6

2

8 Q 2 w2 -,.,a 3

From this

where S

= 4~/3".

Separating the variables da 2 da ----=

a "2a

+ Sa

a

2

a

2 d(aS

+

i)

a as +2

we get d log [

a a] = - a2 d t

as + -2

Upon integrating and putting the constant of integration in the form C= ao/(aoS + ~), we obtain a --~-

as +

a

2

or

[62.7]

For t = 0, one finds a = ao; for S = 0, one finds a = aoe-ft, as in the case of linear damping, Case A. It is seen that the presence of quadratic damping causes a somewhat more rapid decay of amplitudes owing to the presence of the ao

term S aoS + ing.

a

2 e -'2

t

in the denominator than is found with a pure linear damp-

This fact is to be expected on physical grounds.

197

SYSTEMS WITH NON-LINEAR VARIABLE DAMPING By the expression "variable" damping occurring in this section, we shall understand a damping which, for small values of the determining variable (either % or X as the case may be), is negative and becomes positive above a certain critical value of the variable. Since negative damping means supply to the system, and positive damping means withdrawal of energy from it, the system considered here is non-conservative. Furthermore, stationary states, or limit cycles, are possible when the average supply of energy per cycle becomes equal to its average dissipation. According to the mode of production of the non-linear variable damping, there exist two principal types of non-linear self-excited oscillations governed by the following equations, which are solved here by the method of the first approximation. 63.

A.

VAN DER POL'S EQUATION The Van der Pol equation is

[63.1] We have f(%,x) = p(x 2 x - x); furthermore, w = 1, hence f(a sin~, aw cos~) is f(a sin~, a cos(,6). Since % = a sin(,6 and x = a cos(,6, i t follows that

and

Whence, by Equation [59.13], we have

~:

=-

:1T[f(aSin~,aCOS~)COS~d~

=-

:1T[aarSin2(,6COS2~d(,6

2

- a[COS (,6d(,6]

which reduces to

~~

a

= P2

(1 _ ~2)

[63.2]

Forming the expression fo2.f ( a sin(,6, a cos~} sin ~ d~, one finds that it is zero. From thls, by Equation [59.14], dl/l/dt = W= 1, that ls,I/I= t + I/Io,where 1/1 0 is arbitrary. The solution of Van der POl'S equation to the first order is then of the form x = a sin (t

where a is given by Equation [63.2].

+

1/1 0 )

198

As we shall mention shortly, the interesting feature ot this solution is the variation ot the amplitude 4 as a tunction ot time. Prom the preceding study of Van der PollS equation we know that a limit cycle exists in this case. We propose now to establish its existence by use ot Equation (63.2]. Multiplying both sides of Equation [63.2] by 24, we have

that is,

Upon integration, we obtain 42

log 4

2

- a

2

= log 4

ao 2 - ao

+

2

Expressing this relation in the equivalent form ing for a2 , we get

[63.4 ]

Pt 2

4 4_ a2 = 4 ao- 402 e II'

and solv-

2 lit

a 2 = _---::-a.....o'-e_ _ _ . 1

+ .!.4 a 02(e llt

The fundamental equation is then

~

a=

- 1) ,

= a sin ~

of the theory of the first approximation

/!J..

~

2

=

---r=~a=o=e==== sin (t + ~o) llt - 1) I /1 + 1 a 2(e

Y

4

= a sin (t

+ "'0)

(63.6]

0

It is apparent that Equation [63.6] describes the general nature ot motion previously investigated in connection with limit cycles. In fact, if for t = 0, a o = 0, ~ 0. This trivial solut1on of the Van der Pol equat10n 1s, however, unstable. Por any fin1te a o ' however small, the amplitudes increase, approach1ng the value a(t) = 2 as ~ l1mit. In the phase plane as t ++~, the trajectory spirals toward the circle of radius a = 2 from the ins1de.

=

199 B.

RAYLEIGHIS EQUATION Rayleigh1s equation* is X + x + p( - a + pz 2 ) Z = 0

[63.7]

this case w .. 1 and I( X,2:) = - 02: + p2: 8 • S1nce I(x, z) == I(z) and w .. 1, the generating solution a cos~ must be Bubstituted into 1(2:) resulting in

In

[f(a cos~) cosc;6 d~ == - aa [COS2~ dc;6 + pa S[ Hence, by Equation [59.13], with pa ( a. == 2"" 0

-

COS4~ dc;6

W=

== - aa7T + : 7Tpa 3

1, one has

3 2) 4Pa

[63.8]

The system will reach the limit cycle when 0 = -ipa2, from which the amplitude of the limit cycle is a = Y4a/3P. The radius of the lim1t cycle increases as the ratio pIa decreases, which is physically obvious since for P = 0 the system becomes linear and the amplitude, at least theoretically, increases indefinitely, the damping then being negative. To find the mode of approach to the limit cycle, one must 1ntegrate Equat10n [63.8]. 'ollow1ng the procedure explained in Section 63A, one obtains** a(t)=

Th1s gives a(t),.~= Y4a/3P, which is independent of the initial amplitude ao' as required by the condition for a limit cycle.

*

It is supposed that the original Rayleigh equation,

mz + XIII

+ (-A+Bea)i

= 0,

has first been divided

b.Y m and written as

..

2

Z+(o/Z+

where lola

**

=X/fA,

(A B '2)'z= 0 -m+mZ

after which a change of the independent variable brings it to the above form, with

101

=1.

In case Equation (6).7] is not reduced to a unit frequency and tas thtt form Z2 +

(o/2 Z + ( -

a

+

{JZ2)%

=

0

the generating solution should be taken in the form aw cos fI (instead of a cos fI ). which finally results in the following equation for a(l)

(6).9a]

200

64. EXISTENCE OF LIMIT CYCLES; SYSTEMS WITH SEVERAL LIMIT CYCLES Although the scope of the method of the first approximation has been suffICiently ascertained from the previous sections of this chapter, we now propose to introduce certain additional transformations of the form of the fundamental equations [59.13] and [59.14] of this theory. The object of these transformations is to introduce functions similar to those appearing in Equations [52.8] of the Van der Pol theory. By this procedure we will prepare the groundwork for the investigation of an important subject, namely, the existence and stability of limit cycles. It is useful for this purpose to recapitulate briefly the principal results of the theory of the first approximation. The solution of a quasi-linear equation is considered in the form x = a Sin ~, where a and ~ are given by the equations

f

2w

a= Ip

= w

2=W

-

+

I(a

sint,6.

aw cost,6)

cost,6 dt,6 ==

«P(a)

-J.l2 il(a sint,6. aw cost,6) sint,6 dt,6 == D(a) 7rwa

[64.1] [64.2]

The condition for a stationary oscillation on a limit cycle is «P(a) = 0

[64.3]

This is exactly the condition obtained from Equation [46.22] of Poincarels theory and the first equation [52.9] of Van der POllS theory. Equation [64.2] can be transformed by taking account of Equation

[60.5]

and the identity

Thus

[64.4] where F(x,i)

= w2 x + J.lf(x,i)

is the non-linear force appearing in the equation x + F(x,x) 2w= O. Likewise, Equation [64.1] can be transformed by means of the identity aw 2 sint,6 cost,6 dt,6 == 0, which gives

1.o

201

a

=

f/>(a)

= -

2:WrF(aSin¢'t awcos¢') cos¢' d¢'

[64.5]

Equations [59.13] and [59.14] appear now in the form

a=

f/>(a)

[64.6]

tiJ =

.o(a)

[64.7]

where .(a) and D(a) are given by Equations [64.5] and [64.4] in terms of the total non-linear force F(%,i). Considering the question of limit cycles generated by a harmonic % = K C08 t when Il 0, we exam1ne Equation [64.6]. If f/>(a) > 0, the ampl1tude a increases indefinitely, and hence no such limit cycle axists. If .(a) < 0, the amplitude decreases, and again no such limit cycle exists. This cond1t1on character1zes dissipative systems. If .(a1) = 0, we obtain the condit1on for a limit cycle with amplitude a 1 • The quest10n of the existence of limit cycles which are not generated by a harmonic % = Kcost when Il = 0, is not considered. We now propose to investigate a practical case in which a limit cycle exists, as shown by experiment, and to show how this existence can be ascertained analytically on the basis of this theory. For this purpose consider the differential equation of an oscillating circu1t containing a nonlinear conductor characterized by a non-linear equation of the form v = G(i). Putting i = %, the differential equat10n is :II

Lx

+ G(%) + ~J % dt =

0

[64.8]

where the constant parameters Land C designate the inductance and capacity of the circuit respectively. Differentiating this equation with respect to t, dividing it by L, and putting for abbreviation 1/LC = w2 , we obtain •• 2 ). %+W%+ L1 G '( %x=O

[64.9]

Identifying the term IG1%)x w1th pf(x,x) of the general theory, which incidentally imposes a requirement that it be small in comparison with the first two terms, we obtain on the basis of this theory

202

it

= -

'" =

w

27T~L rG'(a sin~)awcos~ . cos~ d~ == ~(a)

[64.10]

rG'(aSin~) cos~· sin~ d~

[64.111

+

2!L

== D(a)

Integrating by parts, we see that the second term on the right side of Equation [64.11] is zero, hence,

rP

= D(a) = w

[64.12 )

and", :::: wt + "'0 where "'0 is arbitrary. The oscillation is thus isochronous at least to the first order. 2r If we let ~I. G'(a sin~) COS2~ d~ =R(a), Equation [64.10) can be o written as

a = - R~~a

=

~(a)

[64.13 )

From the defin1tion of R(a) it follows that R(a) > 0, if G'(x) > 0, that is, if the voltage across the non-linear conductor increases with the current. Thus, a < 0, and the final state of the system is x = 0, as is obvious from physical considerations since a "positive resistance" characterizes a dissipation of energy. Therefore, the final state of equilibrium is stable, and the point x = 0 is either a stable focal, or a stable nodal, pOint. If G'(x) < 0 (negative resistance), that is, the voltage across the non-linear conductor decreases when the current through it increases, ~(a) is positive, and from Equation [64.13) it follows that the amplitude increases. Prom phySical considerations it is apparent that the amplitude cannot increase indefinitely. Analytically this is expressed by the condition ~(at) = 0 for a certain amplitude at which is the amplitude of the limit cycle. Figures 64.1a, b, c, and d illustrate the various possible cases. Figure 64.1a represents the case of an ohmic conductor (Curve a represents an ideal, and Curve b a real, ohmic conductor). Since G'(x) ~ 0 in this case, R(a) ~ 0; hence it < O. The amplitudes always decrease since the system is dissipative. Figure 64.1b corresponds to the case when G'(x), and hence R(a), are negative for small amplitudes and become positive for larger ones. The root at of the equation R(at) = 0 corresponds to a stable amplitude a = a l • If the oscillations are started from values a < ai' they will increase until the amplitude a = a l is reached; if, however, they are started from a value a > ai' they will decrease down to the value a = a l • This condition is indicated by

20 3

R(o)

R(o)

R(o)

R(o)

b

o o

(0)

(b)

(e)

(d)

Figure 64.1 arrows in Figure 64.1b. The amplitude a = a l thus corresponds to a stable limit cycle. Figure 64.1c shows a more complicated form of non-linear characteristic. By a similar argument, one finds that the amplitudes a = a l and a = as correspond to stable limit cycles and a = a 2 to an unstable one. Finally, Figure 64.1d shows a condition sometimes encountered in applications. For small amplitudes the "resistance" of a non-linear conductor is positive, that is, the energy is dissipated. Beg1nning with a certain critical value a = ai' the reSistance becomes "negative,1I that is, energy is conveyed into the system, and this state exists unt11 an amplitude a = U2 is reached. In such a case if the initial amplitude a o < aI' the only stable amplitude is a = 0; the system cannot acquire self-excitation. If, however, a o > all the amplitudes begin to grow and eventually become stabilized at a stable limit cycle a 2 • A condition of this kind is designated as a IIhard" type of self-excitation, which we have previously invest1gated. It is thus seen that the unstable limit cycle a = a l acts as a kind of "barrier" preventing the amplitudes from building up i f the initial amplitude is below the value corresponding to this barrier. We thus find by the theory of the first approximation a situation exactly the same as that found previously by the topological methods of Poincare.

65 . STABILITY OF LIMIT CYCLES j CRITICAL VALUES OF A PARAMETER; SYSTEMS WITH SEVERAL LIMIT CYCLES

Let

al

Let us first consider the question of the stability of limit cycles. be a root of the equation ~(al) = O. For a sl1ghtly var1ed amplitude,

a l + /Ja,

to the first order.

204

From Equation [59."] it follows that d(6a)

"(it = IPa(a t ) 6a

We shall take 6a as the absolute value ot the departure. It the initial departure has a tendency to disappear, that is, it d(6a)/dt < 0, -0(a 1 ) < 0, which is the condition for a stable limit cycle. It -0(a 1 ) > 0, by a similar argument one concludes that the limit cycle is unstable. If selt-excitation starts trom rest (a = 0), the condition tor its occurrence is IP(O)

>0

which is equivalent to the existence of an unstable singularity in the presence ot a stable limit cycle, see Part I, Chapter IV. Similarly, the condition tor a critical value ot some parameter can easily be established by this method. In fact, assume that _(a), in addition to a, also depends on a parameter A, that is, it is a function _(a,A). Consider the value of _(a,A) tor a • 0 and varying A. When a value A = AO is reached for which _o(O,A) > 0, the amplitudes begin to grow from zero and the subsequent increase of amplitudes from that moment will be determined by Equation [64."], which now has the torm

a=

IP(a,X)

The limit cycle 1s reached tor a value a 1 of the amplitude tor which

Por a given value A • AI' the limit cycle will be determined from the equation -(al,Al) = 0, and for some other value A = A2' from the equation _(a 2 ,A 2 ) = O. Hence, the amplitude ot the limit cycle in general 1s a certain function of the parameter A. In some cases the function _(a,A) can be put in the form

[65.5 ] where -1{a,A) > 0 tor all values of 4 and A, and -CO) Equation [65.5] with respect to a, one has

= O.

Differentiating

[65.6]

205

Putting a

= 0,

one has

[65.7] Self-excitation will start from rest if

since

~l(O,X)

> O.

1

Consider a curve 'Yl = ~(a) and a straight line 'Y2 = a shown in Figure 65.1. ~A(O) is the slope of the tangent to the curve ~(a) at the origin, and the slope of the straight line Y2' Condition [65.8] states that self-excitation occurs starting from rest (a = 0) only if the initial slope of the tangent to the curve ~(a) is greater than the slope of the line a/X. y As an example, one may mention the selfexcitation of a shunt generator. In this case the frequency is zero but the amplitude, Equation [59.13], is still applicable. The function ~(a) is the voltage induced in the armature, a is the exciting current, and a/X is the ohmic drop across o the field winding; whence 1/X is the resistance of the field winding plus the field resistor. In Figure 65.' Figure 65.1, 1/X = tan Ct is the slope of the straight line If ~A(O) - 1/X < 0, there is no selt-excitation. For < ~A(O}, self-excitation is possible since ~A(O,X) > O. The equilibrium condition is ~(al) = aI/X, which corresponds to the intersection of curves YI = ~(a) and Y2 = ta. It is interesting to note that the amplitude, Equation [64.10], holds in this case in spite of the tact that the trequency, Equation (64.11], is absent. From the general considerations explained in connection with Figure 64.1, the following theorems result:*

*

l

ita.

THEOREM 1. If a system possesses several stable limlt cycles forming a sequence a}t

as, ali' ••• , between each pair of consecutive stable limit cycles there is always one unstable limit cycle; these unstable cycles form another sequence a2' a 4 , a6' •••• THEOREM 2. The limit cycle reached spontaneously by a system starting from rest is always the one which corresponds to the smallest root

*

It should be borne in mind

~hat

al of

the sequence.

we are considering only limit cycles that are generated by circles.

206 THEOREM 3. The stable limit cycles corresponding to larger roots

4a, 4 6 , ••• of

the

stable sequence can be reached only If the system Is given a shock excitation carrying It beyond the corresponding unstable Umlt cycles

66.

~, 44. • •• •

LIMIT CYCLES IN THE CASE OF POLYNOMIAL CHARACTERISTICS

Let the characteristic of a non-linear conductor be approximated by a polynomial I(i) = A

+ Bi +

Differentiating and setting i ::: a /'(a sin~) COS2~ = B COS2~

+

+

Di 3

+ Ei4 +

Fi 6

(66.1)

we have

sin~,

+ 2Ca sin~ COS2~ + 3Da 2 sin2~ COS2~ +

4Ea s sin3~ COS2~

Forming the expression for R(a) ::: ~ R(a) =

Ci 2

+

5Fa4sin4~ COS2~

I"o 1'( a sin ~) COS2~ d~, we have

!rBCos2~d~ + !r2casin~cos2q,d~ + !r3Da2sin2~cos2~d~

+ (66.2]

The second and the fourth terms on the right side of this equation are zero. The remaining terms are

1 2 • 2 2 -; 3Da sm ~ cos ~

J

d~

=

[I

~ d~

2 a 2.. . 2 3D -7rsm

-

r

• 4

sm

(66.3 ]

~ d~]

3Da 2

= - ; - 7r

3 (1 - 4")

=

4"3 Da 2

From this R (a )

= B + -43 Da 2 + -85 Fa 4

(66.4 ]

It must be noted that the coefficient F in this equation must be positive; otherwise, beginning with a certain value a = ai' R(a) would become negative and remain negative for increasing a. In such a case, by Equation (62.2], da/dt would be positive. This is impossible in a physical problem.

207

Setting ~F =m > 0 and 11m! t cycle is

-¢D = ",

the condition for the existence of a

[66.5] The roots of this biquadratic equation are 2

-

a 1,2 =

11.

± Yn 2 -

4mB

2m

[66.6]

Por the existence of limit cycles at least one of the roots at or ai must be positive. Since m > 0, the necessary and sufficient condition is that B< 0, which expresses the fact that the slope of the characteristic /(i) is negative, that is, "negative" resistance. When B is positive, two cases are possible: ,.

B

> 0,

11.

> 0, that is,

D

> O.

In this case both roots a 12 and a 22

are negative; hence, the amplitudes a l and a z of the limit cycles are imaginary. In other words, no limit cycles exist, the system being dissipative. The amplitudes decrease indefinitely, and the only stable solution is a c O. 2. B> 0, 11. < 0, that is, D < O. In this case limit cycles are possible as long as 11. 2 - 4Bm> 0, which expresses the condition of the reality of the roots. If we substitute for" and m their values, this gives 11.2 9 n2 B<-=--4m 40 F

[66.7]

Summing up this discussion, one can state that with a non-linear voltage, approximated by the polynomial [66.1], of the fifth degree, the following conditions exist: 1. On physical grounds the coefficient Fmust always be positive. 2.

If B < 0, there is always one stable limit cycle.

3. If B > 0 and D > 0, no limit cycles exist and the system is dissipative. 4. If B > 0, D < 0, and B < ~ ~2, limit cycles are possible with the amplitude al (positive root). ~D2

0 and D < 0, but B> 40 F' the system behaves again as a dissipative one, and no limit cycles exist.

5.

If B>

208

Since any experimental characteristic can be approximated by a polynomial, the coefficients A, ..• F in Equation [66.1] are known quantities, and the above procedure permits ascertaining from the form of the characteristic the behavior of the system into which the non-linear element with this particular characteristic is introduced.

CHAPTER XI APPROXIMATIONS OF HIGHER ORDERS

67 .

INTRODUCTORY REMARKS

In this chapter we review the extension of the method of Kryloff and Bogoliuboff as applied to approximations of orders higher than the first. Although for practical applications the theory of the first approximation gives a satisfactory degree of accuracy, it is also interesting to consider the possibility of a further refinement of approximate solutions in case greater accuracy is reqUired. The procedure of Kryloff and Bogol1uboff is derived from the classical methods ot Gylden and Lindstedt used in celestial mechanics. As explained in Section 44, the object of these methods is to eliminate the secular terms resulting trom the resonance effect of subsequent harmonics in the recurrence procedure by which the higher-order terms are determined. The method can be summarized as follows. Assume that we wish to find a periodic solution of the differential equation with a certain unknown period T. Introducing a new independent variable .,. = 21rt/T = Dt, where D = 21r/T, we shall look for a solution x( t) = : (.,.) , where z(.,.) is a periodic function with period 21r. We shall try to represent the periodic solution in the form Z(T) =

I

00

pn %n(T)

[67.2)

n =0

where %ft("') with n = 1,2, ... , are periodic functions with period 21r. We further assume that

[67.3 ) where

aft

is constant.

The transformed equation [67.1) then becomes

If one substitutes into this equation the series expansions [67.2) and [67.3], one obtains a series of recurrent differential equations resulting from equating to zero the coefficients of equal powers of p. For the subsequent approximations one thus obtains a series of differential equations

210

[67.5 ]

............. ,'ft. It is where F(,o. '1' ...• 'ft) is a certa in polynomial in '0' '1' ... first apparent that if '0' '1' ••. , 'N and 00' 01' •.• , ON satisf y the ssions (}{+ 1) equations of the system [67.5], then the expre and

2 S2 =

I

N

pft0ft

[67.6]

n=O

ered as the also satisf y [67.1] up to the order pN+la nd, hence, may be consid (N + 1) th approximation. conThe method of Linds tedt, which Krylo ff and Bogoliuboff follow, s of the resists in determining the coeff icien ts 0. in the subsequent stage period 2". currence procedure so as to el1m1nate terms with the fundamental [67.5], they In fact, if these terms were left on the right side of Equations as prewould account for the "resonance terms, II which are of secul ar form, same time vious ly define d. The determ inatio n of 0. by this procedure at the By this methleads to the expression for frequency given by Equation [67.3]. er VIII) od the diffic ulties encountered in the theory of Poincare (see Chapt sions are elimin connection with the appearance of secul ar terms in the expan inated , and soluti ons without secul ar terms can be obtain ed. Poincare has shown by an example that the approximations gener ally pract ice, the do not converge. However, nothing bette r is avail able, and, in gives entire ly second or third approximation (and usual ly, in fact, the first) satisf actor y resul ts. comThe subje ct matte r of this chapt er is considerably abbre viated , n the reade r pared with Kryloff and Bogoliuboffls text (5), and for that reaso should refer to the origin al text for addit ional detai ls. 68.

IMPROVED PIRST APPROXIMATION

Equations [59.13] and [59.14] ot the first approximation were obthe right tained by dropping the higher harmonics in the Pouri er serie s on

211

side of Equations (59.10J. In reality, owing to the presence of these t.erms, the slowly varying quantities a and ~ undergo oscillations of a relatively high frequency. In order to take this into account, it is convenienent to consider the quantities a and ~ in Equations (59.10] as practically constant in comparison with the rapidly varying trigonometric terms cos newt + ~) and sin newt + ~). Designating the left sides of Equations [59.10] by da/dt and di/dt, and noting that - ~ Ko(a) = ;~ and ~ Po(a) = ~, upon integration of Equations [59.10], one has

a=

.t;

+ c;6)

a _ !!... Kn(a) sinn(wt w n =1

- L,,(a) cos newt

+

c;6)

nw

[68. , ] ~

=

~

+ ..E....

~ PII(a) sin newt

aW~1

+

c;6) -

QII(a) cos newt

+ ~)

nw

where a designates the first approximation for the amplitude given by Equation (59.' 3] . Substituting these values into the equation z = a sin(wt + 9r), one has X

p.

~ Kn sinn(wt + ~) - Lncosn(wt n = 1 n

= [ a - -:-:T .£. W

f" Pn sinn(wt + ~) -

+ L...

aw 2 .£. n = 1

+ ~)] Sin . [W t + .J.Y' +

Qn cosn(wt + ~)]

[68.2 ]

n

If we let

the sine term of this expression can be written as sin(wt

since p. is small. x = [a -

!;r

t.

+

c;6

+ p.S)

=

sin(wt

+

~)cosp.S

~

sin(wt

+

~)

+

cos(wt

+ p.S cos(wt +

+

c;6) sinp.S

~)

Substituting this into Equation [68.2], one has Kn sinn(wt

+ ~) ~ Lncos n (wt + ~)] [sin(wt + ~) + I1S cos(wt + ~)] [68.3]

212

whence, neglecting the terms with pi, pI, ••• , one has 3:

+ 'I'-')

. ( t = a Sin w

+

Il ~ Knsinn(wt 2 "'-' W n=1

-

all ~ Pn sinn(wt aw l ",-,

+

,,= 1

If we put

w, +

~

+ cjJ)

+

- Lncos n (wt

cjJ)

n

+

cjJ) - Qn cosn(wt n

cjJ) cos(wt

+

• ( t

Sin W

+ -') + 'I'

cjJ)

[68.4 ]

=r T,

1; n

~

Ln COsnT = U(T)

=1

1; n

KnsinnT

P" sinnT

~

[68.5] Q" cosnT = VeT)

=I

and

[68.6] We now rewrite [68.4] as 3:

= a sin T -

4w

[68.7]

W (T)

Prom Equations [68.5], in view of [59.8), one has further U'(T) = f(asinT, aWCOST) COST -

Ko(a)

V'(T) = f(asinT, awcosr) sinT -

Po(a)

[68.8] On the other hand, differentiating Equation [68.6], one has u'(T)sinr - V'(T) COST

+

u(r)cosr

+

[68.9 ]

v(ThinT = tU'(T)

If the values [68.8] are substituted into Equation [68.9J, one finds W'(T) W"(T) = -

=

Po(a)sinr -

Po(a) COST - Ko(a) sinT

UCOST

+

[68.10]

v sinr

VCOST

+ u'cosr +

usinT

+

V COST

+

Po(a)sinT - Ko(a)cosT

+

U'cOST

+ v'sinT

Ko(a)cosr - usinT

+

+

v'sinr

(68.11]

ThUB

W"(T)

+

WeT)

+

usinT - VCOST

= - Po(a)sinT -

=-

Ko(a)cosT -

U'cOST

+

v'sinr

[68.12)

213

Noting that

one obtains from Equations [68.8] to"(T)

+

WeT) = l(asinT, aWCOST) - 2Ko(a)cosT - 2Po(a)sinT [68.13]

In order to determine the corrective term W(T) of the improved first approximation, it is necessary to transform the right-hand term of [68.13] into the known Pourier series. For this purpose consider the Pourier expansion 00

ICasinT, awcosT)

=

lo(a) + I n

=1

[In(a)cosnT + gn(a) sinnT]

[68.14 ]

in which the coefficients I and 9 are given by the equations 10 (a) =

2~ [I(a sinT, aw COST) dT

In(a) =

!

gn(a) =

!tl(aSinT, aWCOST) sinnT dT

[68.15]

[1(aSinT' aWCOST) cosnT dT

On the other hand, by [59.9], Ko(a)

= ill

and Po(a)

= iOl'

From

this

00

= 10 + IUncosnT + gnsinnT) n

[68.16 ]

=2

Substituting the right-hand term of Equation [68.16] into Equation [68.13], one gets 00

W"(T)

+

WeT) = 10

+

IU"cosnT n

=2

+

unsinnT)

[68.17]

Looking for a solution of Equation [68.17] of the form "" WeT) = a o + I(ancosnT + bnsinnT) n =2

one obtains by identification of the coefficients after the substitution of this expression into the differential equation [68.17]

214

- f. -

( )WT

0

I"" In COSnTn2+- gnSinnT 1

[68.18)

n=2

Substituting this expression into Equation [68.7), one finally obtains the following expression for the improved first approximation

where a and ~ satisfy the equations of the first approximation, Equations [59.12), that is, ci = - !wJi(a) and ¢> = ~gl(a). In order to see whether the solution [68.19) satisfies the original quasi-linear equation ~ + w2z + p/(z,x) = 0, substitute the solution [68.19) in that equation. This gives, on the other hand,

where 0 1 is of the order of p2. On the other hand, in view ot Equation [68.191, one has p/(x,x) = p/[ a sin(wt

+ ~),

awcos(wt

+ ~)] +

02(p2)

whence, by [68.141, ~

+ w2x +

p/(z,x) = 01(p2) -

02(p2)

Thus the expression [68.191 satisfies the original differential equation with accuracy of the order of p2. Furthermore, it can be shown, upon developing the expressions for 01(p2) and 02(p2), that the error in the approximate solution [68.191 is uniform in the interval 0 ~ t <~.

69. APPLICATIONS OF THE THEORY OF THE IMPROVED FIRST APPROXIMATION We shall consider first oscillations of a conservative system acted on by a non-linear force represented by an odd function of the dependent variable. In this case, I(x) + f(- z) == O. Since the function 1s odd, no cosine terms are present in the Fourier expansion. Hence /(asinT) =

I,,=1"" g,,(a)sinnT

215

and 2 gn(a) = 1r

I

IF

I(a sinT) sin nT dT

By Equation [68.19], the improved first approximation 1s :t =

. ( t

a 81n w

+ ¥'.I.) +

~ gn(a)sinn(wt + t/J)

p.

- 2~ w n=2

2

n-

1

The frequency is given by the equation D(a) =

w

+

-2P. gl(a) wa

that is,

to the first order. We shall now discuss three applications of the theory of the improved first approximation. A.

VARIABLE

II

SPRING CONSTANTII

Consider the differential equation

.. :t

+

2 wox

+

p.x 3 = 0

so that

whence gl

() a

and

For the next harmonic

=

2 ["(3'4 a3SInT .

1r

-

1 s Sin • 3T ). d 3 3 4'a SInT T = 4'a

[69.2 ]

216

2["(3'4 a

Y3(a) = 1T

3

1

• . ). smr - 4"a3sm3r slO3r dr =

f.

3 3"

2 = -a- smx 61T0

d x=

From this, by Equation [69.1]

Thus the improved first approximation introduces a small corrective term ~ 32 sin 3(wot + ~) in the form of a third harmonic. B.

VARIABLE DAMPING

Consider a differential equation of the form

x+

w2 x + pl(x) x = 0

The non-linear term in this case is I(x)x

= I(asinr)awcosr

1

A

Consider the function F(A) = I(t) dt. For A = a cos ~, the deo velopment of F(a cos~) in a Fourier series gives

I

F(acos~) =

...

Fn(a)cosn~

[69.4]

n=O

Differentiating this equation, we have

I

a/(acos~) sin~ =

n

Putting ~

=r

+

2f'

GO

=1

nFn(a) sin n~

one obtains

!(asinr)awcosr

= - wl'nFn(a)sinn(wt + 3; + ~o) 11=0

where ~o is an arbitrary phase angle. tion is

Hence, by Equation [69.1J, the solu-

217

assuming that w = D, since in this case the correction for frequency is of a higher order. If one introduces 80 = ~o - "/2, Equation [69.5] has the form X

= a cos (Dt

+

.!!. I

8 0> -

co

2 n 1 Fn (a> sin n(Dt

w n =2 n -

+

80 >

For example, consider the Van der Pol equation

x+

x - .uO - x 2>% =

0

s

X In this case f(x) = x 2 - ' , F(z) ="'3 - x, and w = " whence

a cos3 3

F(acos~)

=

3

tP

(a4"" 2

- acostP

=

a

2

a - , ) and Fs(a) It follows that FICa) = a (~ By Equation [69.6], the oscillation is

)

-

1 costP

as = 12'

as + 12

the other

cos3~

Fft(a)

being zero.

3

X

= acos(t

+

~~ sin3(t +

90> -

[69.7 J

9 0>

The differential equation for the amplitude is obtained from the equations of the first approximation

~: =

a P2

(1 _ ~2)

For a steady-state condition, a = 2; Equation [69.7J in this case becomes

=

x

2 cos(t + 9 0 > -

~ sin3
C.

CORRECTION FOR FREQUENCY In the preceding notations w is the linear frequency when .u = 0, and D is the frequency of the quasi-linear oscillation when .u O. For quasiisochronous motions to the first order, w ~ D. It is to be noted first that the exact solution for stationary oscillations can always be developed in a Fourier series

*

co

X

=

a cos(Dt

+

90 )

+

In=2 [An cos n(.Qt + 9 > + Bnsinn(Dt 0

+

90 )]

[69.9J

One has identically T

f [xx

o

+

w

2

X

2

+ p/(x) xx) dt = 0

[69.'0]

218

since

On the other hand, •• :U

+ x.2

d ( .) dt xx

=

and T

T

T

2

J%Xdt + Jx dt = x:i:1 = o

in view of the periodicity.

0

0

0

Hence T

T

Jo ~X dt = - J3:

2

dt

0

L1kewise T

Jf(x)x:i: dt =

0

o

for the same reason.

Hence,

If one replaces x by 1ts expression [69.9] and :i: by its expression obta1ned by differentiating Equation [69.9], one obtains finally

.o2[a2 +

t. ,,2(A;

+

B;)]

=

W2[ a2+

t. (A;

+

B;)]

since the terms of the form ApB, cos p (.ot + ( 0 ) sin q (oOt + ( 0 ) disappear when integrated over the period It follows that

a2 +

I fI

GO

=2

,,2(A;

+

Bfl2)

On the other hand, under the assumption that f(z,:i:) • I(z):i:, by Equation [69.6], A" co 0 and B" = "I ~ , F,.(a). Substituting these values for A" and B" into Equation [69."], one has

Z

219

1 It we let

5:0

K2 and [FII!a)]

+

[n

112 ~ 2 2 _ 1 FII(a)~2 -:-:T:T ~ n wall = 2 n

=6

[69.14 J

the preceding equation can be written

11 ,

Hence, to the second order of the osmall quantityK, one has

Applying this formula to the Van der Pol equation with a 2 = 12 = 3' and FII(a) = 0 for n one has as

* },

= 2,

112 D = 1-16

70.

APPROXIMATIONS OP HIGHER ORDERS

Consider, first, the quasi-linear equation %

+

W

2

% + 11/(%)

=0

and assume that it has one or several periodic solutions. Let % = %(t) be such a solution, with an unknown period T and frequency D= 2"/T. Let % = z(Dt + ~) == 2:('7"), with '7" = Dt +~. Then :('7") has the period 2ir. Changing the independent variable in the orIginal equation 2 i + W % + 11/(%) = 0, one obtains [70.1

J

Prom now on the differential notation %will designate differentiation with respect to '7". By hypotheSiS, Equation [70.1] has a periodic solution with period 2". We shall look for solutions 2:(T) and D2 in the form of a power series in II: GO

,('7") = '£11",,,('7")

,,=0

and

[70.2 ]

220

with a o = w2 , since the existence of a periodic solution of period [70.1] with p = 0 requires that D2 = w2 • Pw-thermore,

=

f( '1.0 )

+

PZl

f '() '1.0

+

P

2['1. 2 f'() '1.0 +

2

"(

%1f2

2~for

%0)] + ...

The substitution of these values into Equation [70.1] gives rise to the following recw-rent differential equations obtained by equating to zero the coefficients of pO, pI, P 2 ,

whereF(%o, '1.1' '0. ,'I.,,) is a polynomial in %1' %2' .•. , 'I.". The solution %(T) under consideration is determined up to a translation on T. We can choose the origin, that is, ~, 1n the substitution T = Dt + ~ so that 2:(0) = 0

(70.4 ]

The condition [70.4] then yields the following conditions for Zn(T), which are periodic functions of period 2" and which we take as being represented by their Fow-ier series, %n(O) = 0

for

n=l,2,3···

Let %0' %1' .•. , zn and ai' , an be the solut10ns of the first (}{+ 1) equations of the system [70.3]; it is then clear that and

will satisfy the equation x + w 2 x + pf(x) = 0 to the order of pN+l, and hence can be considered as the (}{+ 1)th approximation of x.

221

The first equation [70.3] gives, in view of [70.5], '0 = 4 cos T, where a is an arbitrary constant. There exists, however, a certain arbitrariness in the following steps, that is, in the successive determination of '1' '2' .... We can remove this arbitrariness by requiring that no fundamental harmonic shall appear on the right side of Equations [70.3], for otherwise some would contain secular terms. We wish, however, to obtain a function '(T) = z(t) representing the periodic solution of our quasi-linear equation in the whole interval 0 ~ t < ~. Consider now the second equation ot System [70.3]

'It

The function 1(4 cos T) can be developed in a Fourier series which contains only cosine terms, that is, ~

l(acosT) = I

~

f,,(a)cosnT

=

lo(a)

+

fl(a)COST

,,=0

+

II,,(a)cosnT ,,=2

~

W2(ZI

+

'I) = - Ifn(a)COSnT

+

n=2

[ala - II(a)]cosT

[70.6 ] - lo(a)

It is noted that the secular term is bound to appear 1n th1s case unless the coefficient of cos T on the right side of Equation [70.6] is zero. Hence, the condition for the el1mination of the secular term is

wh1ch determines the approximation ('I,a 1). %\

+ '\

= -

-;'[fo(a) W

+

One has

I

,,= 2

f,,(a) cosnT]

[70.8 ]

In view of [70.5], the solution of [70.8] is

[70.9 ] with In = I,,(a) where A 1s a constant. Substitut1ng '0 and %1 1nto the right side of Equation [70.3] for '2 and annull1ng the fundamental harmonic, we obtain an equation involving a2 and A, so that A remains arb1trary. In order to s1mplify our solution, we take A = 0 so that 1 /,

'I

= - w2

0

+

~ IncosnT wr1 n=2 ~ n2 - 1

[70.10]

222

and the equation linking ClIO and A, where we set A c 0, now yields the value of ClI2' By proceeding in this manner the following terms '2' 'S' ... and a2' can be determined. We require that none of I.(n ~ 3) should contain the fundamental harmonic, by analogy with '2' The condition (70.5] is then automatically satisfied. We proceed to show by induction that any,. can be determined in this manner trom Equation [70.3]. Assume that 'I' '2' "', '. and aI' ClI2' '" , aft' satisfying the first n equations of the system [70.3], have been determined. The (n+ l)th equation is

a" ...

where '0

=a

cos T, as before. Since 'I' .,. , 1ft and 'I' ... , 'ft contain only cosine terms, }1,o, 'I' -.. , 'n) also contains terms of this kind only. 00 PuttingF(,o, ' 1 ' ""'ft) -a n -a n _ l z 2 - ••• -alz. = L6",cosmT, one can write Equation [70.11] as

'l

",=0

The condition for the absence of secular terms is again aft + 1 =

_E.!. a

[70.13 ]

Equation [70.12] now becomes [70.14 J The solution of this differential equation is [70.151 the secular term having been removed again from the solution. 71.

MOTION OF A CONSERVATIVE NON-LINEAR SYSTEM WITH A CUBIC TERM As an example of an application of this method, consider the differential equation [71. 1 ]

The integration of this equation by the method of the first approximation was given in Chapter X.

223

We propose now to determine the approximations of higher orders, following the method explained in the preceding section. Taking as generating solutions '0 ( T) = a cos T, w2 = 1, and a o = 1, we have

The elimination of the secular term gives 3

a 1 = 4- a2

[71.2]

and the solution of this equation g1ves %1 =

If we substitute tor ..

%2+ %2

al

and 3

'1

6

their values, the third equation [70.3] becomes

2

= - 32 a cos TCOS

_ (

-

3 128

a2 a -

3 3 32 a COS3T

a

3

+

T

a 2 acosT

+

27 128 COS3T

5) cos T + 128 21 5 3 a cos T

-

3 128

5

a cos

5

T

[71.4]

The condition for the absence of the secular t erm gi ves a2 -- ~ 128 a4 . Thus the solution of Equation [71.4] is 21

%2

6

= - 1024 a cos 3T

+

a6

1024 COS5T

Consequently the approximate solution satisfying Equat10n [71.1] to the order of J.l2 is :t

=

acos(wt +1,6)

+

J.l

as 32

(21) 1 -J.l3'2 cos(3wt +1,6)

+ J.l2

afJ

1024 cos(5wt

+ 1,6) [71.5]

where a and 1,6 are constants of 1ntegrat1on. The frequency D 1s given by the second express10n of [70.2], wh1ch at, a2' ... have already been determined g

72.

2

= 1

3

2

+ 4" J.la +

3 128

24

J.l a

in

[71 .6]

HIGHER APPROXIMATIONS FOR NON-LINEAR, NON-CONSERVATIVE SYSTEMS

We now consider the general form of a quas1-l1near equat10n [72.1]

224

We shall attempt to write the general solution for higher approximations in the form of an improved first approximation by replacing wt + t/i by flit + • - ~ - r/I. One then obtains trom Equation [68.' 9 ] It

.1.

P

= acosY' - w2

I:' (

1'0

,a

)

+

P

w2

~ Fn (a)cosn'" + Gn(a) sin nl/J n2 - 1

~

[72.2]

n=2

with equations of the first approximation da

Cit

p = 2w G1(a);

dl/J = D (a) dt

w + -P 2 Fl(a) wa

D(a) =

£72.3 ] £72.4 ]

The F. and G. in Equation (72.2] are the 'ourier coefficients in the development of f(acosT, - awsinT) =

I

00

[Fn(a)cosnT

+ Gn(a)sinnT]

£72 .4a]

n=o

The condition for a steady state is Gl(a)

=

0;

'"

=

D(a)t

+1/10

£72.5 ]

where r/l o is an arbitrary constant. Equation (72.2] for a steady state becomes It

= a COS[D(a)t

+.1!.. 61 2

+ 1/10] - ;,

Fo(a)

f'Fn(a)cosn[D(a)t

~ n.:2

+

+ "'0] + Gn(a)sinn[D(a)t +1/10] n2 -1

£72.6 ]

It was seen that for conservative systems Gda) == O. In such a case, Equation [72.6] has two integration constants a and "'0' as is to be expected for an equation of the second order. If, however, G1 (a) = 0 has only simple roots without being equal to zero identically, the solution £72.6] has only one integration constant 1/10 since a is determined trom the equation G1(a) D O. This case corresponds, therefore, to the existence of limit cycles corresponding to the roots of G1 (a) = O. In fact, as was shown previously, the stationary oscillation in this case does not depend on the initial amplitude. We now propose to establish the existence of periodic solutions, that is, of limit cycles in non-conservative systems.

225

If % = z(Dt + differential equation

~)

is such a solution, it must clearly satisfy the

[72.7]

We can follow the same procedure as before, assuming solutions of the form [72.8]

where Zft are periodic with period 2". Here we take zft as ~epresented by their Fourier series. Forming i and % and substituting the values of z, i, z, and D into Equation [72.7], one obtains again a series of recurrent d1fferential equat10ns by equating to zero the coeffic1ents of var10us powers of p. One has n2 Z0 oUo

+ w2 z0 --

0

[72.9 ]

·z .. ·z

-2.SJo!Jz 1 II -R(z " 0 ···z11-1'·%0 .. ·% "-1'

0

,,- l·.SJ • o

···!1,,_l)

D;

= w2 ; where Fft 1s a known function of the indicated arguments. As before, we requ1re that i(O) = O. Hence ift(O) = 0, where n = 0, 1, 2, .•. , so tha t Zo c: a cos T, where a 1s a constant to be determined, and zp Z2' ••• do not contain sine terms. Subst1tuting these values for Zo and Do into the second equation [72.9], one gets

=

-I [F"(a) cos nT +G,,(a)SinnT] + "=0

2w!J 1acosT

[72.10]

226

The cond1tion tor absence ot a secular term gives Q = F,(a) 1

Prom Equations [72."], a and [12.10] then becomes 2 W (;'I+ ZI) = -

I

FoCa) -

QI

2wa

[12.' 1]

can be determined. Equat10n

CO>

"=2

[F,.(a)OO8n '7" + G,,(a)sinn'7"]

£72.12 ]

The solut10n ot Equat10ns [72.12] is ZI = alcosT -

1\i(:) +

i"r [F,.Ca)cosn'7" + Gn(a) sin n '7"]

1

W2~

w

n2 - 1

-=2

[12.1, ]

1n which the amplitude al is to be determined by the condition tor the elimination ot secular terms on the r1ght aide ot Equat10n [72.9] tor '2' Wr1ting Equat10n £72.,,] as ' I a al cos '7" + V where

and subst1tuting 1t 1nto the third equat10n [72.9], one bas

where W(T) is a periodic funct10n conta1ning the rema1n1ng terms ot the third equat10n [72.9]. S1nce w('7") 1s per10d1c, one can represent 1t by a Four1er ser1es

Furthermore, one has the 1dent1ty

- r. a l cos

T'

+ Ij alw sin '7"

== -

8 1!.1

vp

I

[Ca + pal) COS '7", - (a + pal) wsin '7"]

M=O

whence, by Equat10n [72.4a], CO>

-f,a l OO8T +l;a l wsin'7"

=-

alI [F.'(a)cosnT + G~(a)sinn'7"] [12.15]

_=0

where F~(a) and G~(a) des1gnate the der1vat1ves of F.(a) and G~(a). Substituting I. and I. 1nto the third equation [12.9], one gets

227 00

W

2

(ZZ+

Z2)

= - a1I[ ~'(a)C08nT

"=0

+ G~(a)SinnT] + 00

+ 2w(D 1a 1 +D 2 a)COST + I[p"COSnT + q"SinnT]

"=0

In view

[72.16]

ot the periodicity ot '2' the secular terms must be el1minated aga1n.

Their el1mination gives the conditions ,

alGI (a) = ql

and

n

,wla l

+

n

,w2 a

= -

...E.t 2w

+

a1F,.' 2w

[72.17]

It tollows that

-....!!.L-

[72.18 J

a l = G;(a)

S1nce we are 1n search ot the condition tor the existence ot limit cycles, G1(a) • 0 by the tirst equation [72.3], we have to add now a second condition G:(a) :#: 0, s1nce only 1n this case does Equation [72.18] give the determination ot al. The equation G;(a) :#:0 shows that the root ot the equation G(a) = 0, which gives the limit cycle, is a S1mple root. Prom the second equation [72.17] D = Z

.!. a

(...1i +Da 2w 1

1

_ all'i') 2w

[72.19 J

Hence, Equation [72.16) can now be written as

The solution ot this equation is

+

~z..t[(p,,-al~')cosnT+(q,,-alG~)sinnT]1~n2 ,,=2

[72.21J

where a2 is again an undetermined coetticient, which, together with D., is determined by the condition tor the absence ot secular terms on the right sides ot Equations [72.9] tor 'a' and so on.

228

Pollowing this recurrence procedure, it 1s apparent that subsequent equations [72.9] permit the determination of aft and D~+1 from the equations for the elimination of secular terms. [72.22 ]

in which

*

G;(a) index n.

en

and ~n are known from the preceding recurrent operations. As long as the equation G1 (a) = 0 has only simple roots, that is, 0, the process can be continued indefinitely up to any value of the

The expressions for the non-linear oscillation ~ and its frequency D, up to the order of #N inclusive, are of the following form [72.23 J D = w

+ Il Dl + ... +

IlNDN

Pollowing this procedure, one finds, as an example, for the second approximation x = (a +Ilal) cos (Dt

+

4w

+ q,) -

Il ~ F,.(a)cosn (Dt (]i.£. n= 2

Fo(a)

+ q,) + n2

-

+

G,,(a)sin nUU

1

+ q,)

[72.24 J

where D

=

W

+ Il

})(a)

2wa

If we compare this expression with the earlier formula [72.6], it is observed that the only difference between these expressions is in the amplitude of the first harmonic, which is now a + Ilal instead of a, where a is the root of Gda) = O. This difference is due to the fact that for higher approximations, as can be shown, the amplitude equation is

~~

=

2:

GI(a) + 1l 2S1(a) +

1l3S2(a)

+ ...

[72.25 ]

If the limit cycle is reached,

Hence, in view of the factor # before G)(a), it is seen that by stopping the approximation for a certain value n = Nof the index, the error in the

229

determination of the first harmonic of the limit cycle is only of the order (N - '), and not of the rfh order. Hence, the Nth approximation determines the amplitude to the order (N + ,) and the frequency to the order (N + 2). For N = 0, one has x

=

a cos (Dt

D = w

+ ¢J) ;

+

IIFJ(a) 2wa

[72.26 ]

which represents the first approximation obtained by a different method in Chapter VIII. For N = " x 18 given by Equa tion [72.23] and D --w

+

1J Fi(a)

2w

+ II 251 2

[72.27]

In order to establish the explicit expression for this approximation, one has to determine al and .0 2 entering into these formulas.

73. GENERAL FORM OF EQUATIONS OF HIGHER APPROXIMATIONS We shall now review a generalization of the preceding theory applicable to steady oscillations as well as to the tranSient conditions of a quasi-linear system

x+ wx 2

+ IJf(x,x) =

0

[73.' ]

One may attempt to find a periodic solution of the form [73.2 ]

x = z(r/J, a)

Furthermore, by analogy with equations of the first apprOXimation, one can postulate that da = A(a)

dt

dl/l =

dt

.o(a)

Por the time being, the functions A(a) and D(a) remain unlmown. fact, their determination constitutes the object of this procedure. Proceeding formally, we obtain

x=

{)z {)'"

.0

+

{)z {)a

A

Differentiating the second time, we f1nd

In

230 2 8 z 2 z = {JI/I2 D •.

2

2

+

8 % "'A

2 81/1{Ja

M

+

+

8 % A2

{Ja 2

8% 8D A {JI/J {Ja

+

8% 8A A 8a 8a

Replac1ng z, i, and ~ 1n the quas1-11near equat10n [73.1] by the1r express10ns [73.2], [73.4], and [73.5], one gets

+ -8z -8A A + w2: + III {Ja 8a

(8: - D + -(J%) A = ' {JI/I (Ja I

0

[73.6 ]

It 1s apparent that 1f one f1nds express10ns for I, A, and D sat1sfying th1s equat10n to a certa1n degree pH of accuracy, these solut1ons w111 sat1sfy the or1ginal quas1-11near equation [73.1] to the same degree of approximat1on, cond1t1ons [73.3] be1ng sat1sf1ed. In order to apply the method of successive approximations, let us represent the solut1ons in the form

[73.7]

The method then consists 1n subst1tuting the ser1es expressions [73.7] for :(I/I,a) , A(a), and D(a) into Equat10ns [73.3] and [73.2] and equat1ng to zero the coeff1c1ents of II, p2, ••.• One obta1ns 1n this manner the following ser1es of recurrent d1fferent1al equat10ns (compare w1th the analogous method of Po1ncar', Chapter VIII) . 8 2,

{J",l + 2

'1 ( 8{J¢2

(8

2

'1

81/1

'0

= 0

+ 'I ) W2 ::: - f ({J,o) 'o'W {JI/I -

)

2 +'2 W

= _

J. ( %0' W 8'0) {JI/I 'I I

2

{J 2z o 2wDI 81/12 -

-

~

2wAI 81/1{Ja

~ ( 'o'w ~) ({J'I ~. 8%0) 8", w {JI/I + DI {J", + Al {Ja -

Ji

2 {J2%, ~ 2 8 z0 {J 2z o _ A2 {J 2z o _ wgl {J",2 - 2wA, {JI/I8a - g, 81/12 - 29l A I {J",8a • {Ja2

{J,o {JD, A

81/1 {Ja

I

_ {J,o A 8A. _ 2 D {J2,o _ 2 A {J 2z o {Ja • {Ja w 2 8",2 W 2 8",8a

[73.8]

·.............. (73.8 )

where E" is a tWlction ot '1' ... , I.; AI' ..• ,A.; Ql' ••• , D.; and their partial derivatives, which can be cons1dered as known trom the solution ot the first " equations [73.8) by a recurrence procedure. As before, the first equation ot System (73.8) is solved by putt1ng 10 • a cos~. Substituting th1s solut1on 1nto the second equation [73.8], one obta1ns

<10

=

-~[F,.(a)cosnl/l +G,,(a)sinnl/l] + 2(c.1D 1 a cos 1/1 + 2(c.1A 1 sinl/l £73.9]

S1nce we w1sh to have '1 per10dic, the secular terms on the r1ght side of Equation [73.9] must be el1m1nBted. The conditions tor this are [73.10 ]

Prom these condit10ns Al and Q 1 are determined. into Equat10n [73.9], one has

(:~~

+

11) (c.I2

= -

Fa (a)

-

to [~(a)

Substituting their values

cos n 1/1 + G,,(a) sin nl/l]

£73.11].

The solution ot this equat10n is %1

= -

1 (c.I2

Fa (a) +

1 6<10 (c.I2

n=

F.: (a) cos n 1/1 + G" (a) ,,2 - 1

sin n 1/1

(73.12)

Substituting the values ot Al and Dl from Equations £73.10) and that of '1 from Equation [73.12] into the third equat10n [73.8], one has

and G(') are certain fWlct10ns ot a. The el1m1nat1on ot secular terms aga1n permits determining A2 and D2 from the equations.

where

F(')

£73.14 )

and the substitution ot these values into Equation [73.13] reduces it to 2

0 %2 ( 0",2

+

%2

)

2

=_

W

R(II( ) .1. > a cos n .1. + G(')(a). sm n f;1 n2-1 00

1:>(1) ( ) _

rO

a

n

n

'Y

The recurrence procedure is now apparent. solution so obtained is ot the torm

'Y

[73.15]

It is thus seen that the [73.16]

where a and '" are given by the equations

:~ =

,uA1(a)

:~

W

=

+

+ ,u 2 A 2 (a) + ... +

,uD1(a)

+

,u 2D 2(a)

On the other hand bv Equations [73 10] ,

da dI

:t

1#

•

G1(a) = ,u ~ -

2

A

+ ... +

+

(73.18]

,uNDN(a)

= G.(a) and D = Fl(a) 2w

' .

(3A

,u A2 a)

(73.17]

JlNAN(a)

,u J'1s(a)

+ ... +

•

2wa '

whence

N ) ,u ~N(a

[73.19] = w +,u

:~~ +

2 ,u n 2 (a)

+ ,usns(a) + ... +

,uNnN(a)

For }{= 1, Equations [73.19] give the improved first approximation, Equation [68.19]. • Furthermore, since by the method ot el1mination of secular terms the quantities AI' A 2 , ••• , A" are expressed in terms of the subsequent first harmonics which are el1m1nated from expressions %1("',a), ••• , %"(",,a), it is apparent that the first equation (73.19] relates to the amplitude of the fundamental harmonic. The second equation [73.19], viz., d",

D(a) = dt

= w

+

F, (a)

,u ~ 2wa

+

,u2!} (a) + 2

may be deSignated as the equation of the instantaneous frequency non-linear oscillation.

[73.20] Q(a)

of the

CHAPTER XII METHOD OF EQUIVALENT LINEARIZATION OF KRYLOFF AND BOGOLIUBOFF

74.

INTRODUCTORY REMARKS The method of Kryloff and Bogoliuboff outlined in Chapter X was established by assuming a sinusoidal solution x = a sin ~ for a quasi-linear equation [58.1] and by determining the functions a(t), the amplitude, and ~(t), the total phase, so as to satisfy the differential equation [58.1] with accuracy of the order of ~2. As was mentioned, from the standpoint of formal procedure the method resembles that of the variation of constants of Lagrange. The method of the first approximation stated in Chapter X gives approximate expressions for the frequency and the amplitude of a non-linear oscillation for small values of~. It is plausible to think that these same approximate relations may be obtained from a linear equation in which the coefficients have been suitably chosen. This is essentially what Kryloff and Bogoliuboff have done and which is designated by them as the method of equivalent linearization. The essence of the method is the determination of the equivalent parameters, as is indicated in Section 75. On the basis of formal procedure it is not clear why this particular determination of parameters leads to the possibility of approximating the solutions of a quasi-linear equation by those of a corresponding linear one in which equivalent parameters appear. In order to Justify the procedure, Kryloff and Bogoliuboff'observe that a non-linear oscillatory process is generally characterized by a certain Fourier spectrum of the component frequencies resulting from the non-linearity of the system. If, however, one limits oneself to the theory of the first apprOXimation, it is logical to assume that the fundamental harmonic of the spectrum should be considered. Hence it is sufficient to determine the equivalent parameters so as to obtain in the linearized problem the same oscillation which appears as the fundamental harmonic of the quasi-linear system. In fact, if one assumes this to be an apriori propOSition, the Principle of Harmonic Balance, it can be shown that the formulas giving the equivalent parameters follow directly from this principle, see Section 77. One can also justify the introduction of equivalent parameters by postulating that the work per cycle done by a non-linear force Fg and by a corresponding linear one is the same. In fact, if one assumes a PrinCiple of Equivalent Balance of Energy of this kind, one likewise obtains the same formulas for the equivalent parameters, see Section 76. Viewed from this standpOint, the Principle of Harmonic Balance enables us to determine the equivalent parameters without actually writing the

non-linear differential equation. Krylotf and BogoliUbott show that the solutions so obtained do not difter much trom those ot the corresponding linear equation. It is to be noted, however, that this argument should not be considered aB a proof. In spite ot this, the method ot equivalent linear1zat10n, as we shall Bee particularly in Part III, plays an important role in the quasilinear theory and leads to results consistent with experimental data. Thus, tor instance, the generalization ot the concept ot equivalent parameters tor several variables makes it possible to absorb the eftect ot an external periodic excitation by the equivalent parameter and thus to explain a number ot phenomena such as asynchronous quenching and eXCitation, and similar phenomena. Moreover, when an equivalent parameter is a tunction of the ampl1tude, the approach of the phenomenon to a limit cycle in this representation amounts to the approach ot the equivalent parameter to a critical value at which the linearized decrement vanishes and the oscillation becomes stationary. We shall encounter numerous applications of the method ot equivalent linearization in Part III. In this chapter we shall establish the principal detinitions ot equivalent parameters and give a tew applications ot this method.

75.

HE'l'HOD

or

EQUIVALENT LIREARIZA'l'IOH It was shown in Chapter X that the solution ot a quasi-linear equa-

tion

mx + Kz

+ p/(z,i) = 0

can be written z = 4 cos ~,. where the amplitude 4 and the total phase ~ are given by two ditferential equations ot the tirst order. Applying Equations [59.1'] and [59.14] to Equation [75.1] with this torm ot solution, one obtains

~;

=

2:wm r/(a.C08c/J, -

4wsin9S)sin9Sd9S

= ~(4)

drp = D(a)

dt

Where

[75.4 ] *

As a utter ot fact, this soluUcn iD Chapter X was taken as _ = G siD p, which lIIerely reverses the siD ~ and cos _ under the integral sign in the 8IIIpl1 tude anel phase equations. The notation in the present chapter cClllpl1es with that used in the text ot Kr;yloff aDd Bogol1uboff.

235 The second term on the right side of Equation [75.4] is the frequency correction between the linear, = ~, and the non-linear, QI(CI), frequencies. If one defines two constants I and R, the equivalent parameters, by the equations

w'

x= K=

-L[ICacos
[75.5 ]

K + L[fCacos9S, - awsin9S) costPdtP 1ra

[75.6]

it can be shown that an "equivalent" linear equation, with coefficients J: and il, approximates the solution of the quasi-linear equation [75.'] to an accuracy ot the order ot pl. In fact, with values [75.5] and [75.6], the amplitude equation [75.2] becomes

.

X

[75.7 ]

a = --a 2m

and the phase equation ¢, = DCa) =

VI

One recognizes these expressions as the usual ones for the decrement and frequency of an ordinary linear equation of the second order. In order to make sure that the solution S D CI cos ~, with a and ~ given by Equations [75.7] and [75.8], actually satisfies the equivalent linear equation, with accuracy ot the order ot pI, substitute the values s and i into the equivalent linear equation m~

+ Xi + K~

[75.9 ]

= 0

We have X = acos¢ - asin",'

z. =

-

X.

K

.1.

,p

X acos¢ - a.Qsin¢ = - -2m

X

2

aD .

1

aX X

2

am cos¢ + m aDsm¥, + 2m a aa sm{b + 2m aa 2m a cosl/!

K

= - m ~ -

X.

m~

+

X

2m

[75 .1 0]

2 aD. 1 aX X /X\2 a a aa sm'" + 2m aa 2m a~ - \mJ "2 cos{b

Substituting s, i, and ~ from these equations into the equivalent linear equation [75.9], one sees that the quasi-linear differential equation [75.1]

reduces to a residue of the form R(#2), which proves that the equivalent linear equation is satisfied with accuracy of the order ot #2. The transtormation ot the original quasi-linear equation

mx + Kx + #!(x,x) =

0

into an equivalent linear one is accomplished by replacing the term #!(x,x) ot the quasi-linear equation by K1x + Ix, where Kl ::: X - K. It is apparent that the quantity i' ::: I/2m = - a/a, as determined by Equation [75.7], is the decrement, and D::: VR/mis the trequency ot the equivalent linear equation. It one substitutes tor Xand Xtheir exp.ressions [75.5] and [75.6], one tinds equations ot the tirst approximation. In tpis manner one obtains a purely formal connection between the equations of the tirst approximation and the equivalent linear equation with parameters I and X, as defined by Equations [75.5] and [75.6].

76. PRINCIPLE OF EQUIVALENT BALANCE OF ENERGY Prom the preceding section it appears that the method ot equivalent linearization consists in replacing a quasi-linear torce, Fg ::: #!(x,x), by a linear one, FL = K1x + Ix. Furthermore, it has been shown that if the equivalent parameters Kl and X are detined by Equations [75.5] and [75.6], with Kl = X - K, the solution ot the equivalent linear equation [75.9] differs by a small quantity of the second order from the solution of the original quasilinear equation [75.1]. We propose now to show why this particular definition of equivalent parameters Kl and I has been adopted. The physical Justification for this definition lies in the Principle of Equivalent Balance of Energy, which requires that the work per cycle of Fg and FL be the same, that is T #

T

f!(x,x)xdt = If:i 2 dt o

0

The term with Kl does not enter into this expression because the work ot a conservative force per cycle is always zero. It is to be noted, in view ot the fact that the integral in Equation [75.5] is finite, that Xis of the same order as #, that is, it is small. On the other hand, to the first approximation, x = a cos 1/1 and x = - aw sin 1/1, where 1/1= wt + 1/10' and a and 1/1 0 can be considered as approximately constant during the time interval 2"/w. The lett side of Equation (76.1], upon changing the limit of integration and substituting the generating solutions x = a cos 1/1 and x aw sin 1/1, becomes CI

-

237

f

2.

- Jl !(alcos r/lt - awsin "') a sin r/I dr/l

[76.2 ]

and the right side is clearly

r"

2.rr

I

2

2

2

a w sin ¢ dt

= Ia 2w [sin 2¢d¢ = Ia 2wll'

[76.3)

Hence, by Equation [76.1],

I

= - -Jl-[!(acosq,. - awsinq,) sin q,dq,

lI'aw

[76.4 )

This is precisely the first equation [75.5], by which the parameter I was originally defined. It is thus seen that the introduction of the equivalent parameter Xis dictated by the equivalence of work per cycle in both the quasi-linear and the equivalent linear systems. We next give a suitable physical interpretation to the other equivalent factor, K l , which does not appear in the energy equation [76.1]. For this purpose i t is helpful to utilize the definition of IIwattless li or reactive power commonly used in the theory of alternating currents. In this theory the energy (or active) component of power Mra and its wattless (or reactive) counterpart Mrr are defined as T

lYa

= :

f eicos~ dt

T

and

o

w,.

=

if

e i sin q, d t

o

where e, i, and ~ are voltage, current, and phase angle respectively. Defining the active "'a and the reactive ",r components of power for a mechanical system in a similar manner, we have 1 T F(t) x(t)dt ;

~ =T

f

o

Hence, by equating the expressions for the IIreactive powers ll for a quasi-linear and for an equivalent linear system, we obtain

Since in this equation both Jl and of the same order.

X are

of the first order, Kl is

Substituting the generating solutions z .. " cos (fAIt + 6), :i -"",sin (fAIt + 6), and T= 2"/fAI, the left side of Equation [76.5] becomes Q

a2",: rfCacosq,. - afAlsinq,)cosq,dq, and its right side is ,,'lwKt /2. Kl =

It follows that

LtfCacosq,. - awsinq,)cosq,dq,

7ra ~.

[76.6 ]

which is Equation [75.6].

77 . PRINCIPLE 0' HARMONIC BALAHCE alternative auxiliary principle serving the same purpose can be described as follows. Consider again the non-linear force Fa A(z,i) and the equivalent linear one, F£ Klz + Is. The harmonic oscillation z :: " cos (wt + 9), where w is the frequency of the "zero" approximation, is taken again as a generating solution. With this solution, F£ can be written as F£ = Fu cos (we + '£), where F£o and '£ are the amplitudes and the phase respectively ot F£. The non-linear torce F is represented by a 'ourier seriel ot which the tundamental harmonic is F:I Fo cos (we + ,). It one makes F D F£, which constitutes the Prif&cipl. 01 H"rmof&ic Bfd4ftC8, i t entails two equations, F • • F". and 9 = '£' trom which again the two parameters I and Xl can be obtained. In fact, An

:I

and the fundamental harmonic ot the non-linear force is

[77.2 ] Equating the coefficients of cos (wt + ,) and sin (we + ,) in Equations [77.1] and (77.2], since 6 = 9£, one obtains the same expressions tor KI and I as before. It is seen that both principles, that of the Equivalent Balance of Energy and that ot Harmonic Balance, are equivalent, because the work ot higher harmonics per cycle of the fundamental frequency 1s zero.

239 Summing up the results of this and of the preceding sections, one can state: 1. The Principle of Equivalent Linearization consists in defining an equivalent linear system as a system with parameters X and KI expressing the equality of work per cycle for the non-linear and the equivalent linear systems. 2. The parameter Xis obtained by equating the active components of power in both cases; K I , by equating the reactive components. }. When the equivalent parameters X and KI are so determined, the equivalent linear differential equation admits a solution differing from that of the quasi-linear equation by a small quantity of the order of p2. 4. For practical purposes the formation of the equivalent parameters X and KI is the only requirement for the solution of the quasi-linear equation, in view of Statement }.

78. EXAMPLES OF APPLICATION OF THE METHOD OF EQUIVALENT LINEARIZATION A few examples given below illustrate the application of this method. A.

NON-LINEAR RESTORING FORCE Consider the differential equation

mx + F(x) =

[78.1 ]

0

where F(x) is of a quasi-linear type. For example, F(x) = ex + px 8, where c and p are constant. The condition for quasi-linearity is that ~ ex «1. Since the system is conservative, the amplitude a remains constant, but the oscillations are not isochronous. Substituting the value of F(x) into Equation (78.1], one has

mx + ex + px 3 = = /(x)

Hence f(x,z)

... px 8

[78.2J

0

and, by Equation [76.6],

pa 2 The equivalent spring constant in this case will be c + ~, whence 2 Da=,lc+:;da) =,lw2+J.ta

V'

where

a

V'

p

= 4mw2 '

2m

::

w ·4+ J

V

pa

2

2mw 2

::::W(1+

J.la

2

4mw 2

)

=wO+aa 2 )

[78.} ]

240

The frequency of non-linear oscillation is here a function of the amplitude a. The oscillat10n is thus non-isochronous, although the amplitude a 1s constant. One can easily discover this fact formally by calculating the expression for I from Equation [75.51, which in this case gives

X = O. B.

NON-LINEAR DISSIPATIVE DAMPING Consider a differential equation of the form ntx+Kx+pj(x)= 0

In order to be more specific, assume quadrat1c damping, that Is, pf(i) = b

Ixlx

Equation [75.5] for X in this case is

I

= -

_6_ ~aw

Ji( -

aw sin,p) sin,pd,p = _b_ Ji(aw cos,p) cos,pd,p ~aw

where

rf(."

cos <1»

COl d

= • '.,'

r

l ' " <1>1 . . . ' d = .'.,' [[

T...' d

-

L

-l;;""d + ""d] - ~ .'oJ 2

2

8 baw X 4 baw Hence, ).. = '3 -;r' The equlvalent decrement IS = 2m = '3 m~; it is seen that the decrement in this case varies with the ampl1tude. Applying Equation [76.6], one finds 2

Kl = p:aw rlSin,p1 sin,pcos,pd,p = 0

Hence the non-linear correction for the frequency in this case is zero to the first order. From the fact that tor quadratlc damping the decrement varies with the amplltude, one concludes that for large amplitudes quadratic damping is more eff1cient, and for small amplitudes less efficient, than is linear damping, the decrement of whlch does not depend on the amplitude. Since the decrement "6 = ~~a == Sa, where S = !~, the motion under the effect of quadratlc damping can be determlned. We have

5

-

IS = Sa = -

1 da aa = - adt

241

whence

s Hence, on integrating,

=

! --1 a

_.!..a

2

da = ~ dt

dt

(.!..) a

=St,

so that finally

=

a0 4 bw

1

[78.4 J

+3-m1r - at 0

which coincides with Equation [62.41, obtained by the theory of the first approxima tion • C.

NON-LINEAR RESTORING FORCE AND NON-LINEAR DISSIPATIVE DAMPING The differential equation is of the form mx

+f

(x)

+ c (x)

= 0

It will again be assumed that both f(z) and c(x) are quasi-linear, that is, they are of the form f(z) = Aoz + p~(z) and c(x) = cox + v~(x), where Ao and Co are constant, p and v are small parameters, and ~(z) and ~(x) are nonlinear terms. The application of the method of equivalent linearization gives and

X i - 2m -

where Xl and X are again the equivalent parameters determined by Equations [75.51 and [76.6], applied to the function /(x,z) • p~(z) + v~(x), as explained in connection with the two previous examples. Equations of this type are of frequent occurrence in practice. For example, Froude1s well-knOwn differential equation for the rolling of a ship in still water is of the form

where I, Mr, and h are respectively the moment of inertia, the displacement, and the metacentric height of the ship, and Kl and K2 are Froude1s coeffi88 cients of reSistance to rolling. If one approximates sin 9 by 8 -~, one has, upon dividing the equation by I, the following equation

ii + b19 + b2 92 + w28 which is of the type considered here.

w2 8 3 - -6-

== 0

242

D. ELECTRICAL OSCILLATIONS IN A CIRCUIT CONTAINING A SATURATED CORE REACTOR Consider an oscillating circuit containing a constant air-core inductance L o' a variable saturation iron-core inductance L 1 , and a fixed capacity C. The non-linearity in this case is due to Lt. In tact, the flux _ through the coll L t is _ = f( i ), where, i is the current. Por a sinusoidal current i = io cos (wt + 8), the fundamental harmonic of magnetic flux is

~ rf(ioCOS~)COS~d~' cos(wt +~) According to the method of equivalent linearization, the non-linear equation f(i) can be replaced by the linear one _ = L.i, where L. is the equivalent linear coefficient of self-inductance,

_=

[78.5 ] If the constant air-core inductance Lo is relatively large compared with the non-linear inductance containing iron, the current will be quasi-harmonic and the express.lon for frequency wll1 be

[78.6] E.

be

NON-LINEAR CONDUCTORS Consider a conductor and let the voltage drop

6

across its terminal

6 :: - ' ( i).

It the current is of the form i :: i 0 cos (,., t + 8), the fundamental

harmonic of the voltage drop is - ;

r'(ioCOS~)COS~ d,p· cos(wt + 9) =

61

By putting

the non-linear conductor can be replaced by an equivalent linear one having a voltage drop 61 ::R.i. If R, > 0, the non-linear conductor dlssipates energy; if R. < 0, energy is brought lnto the system owing to the non11nearlty of the process. Llkewise, if the non-linearity appears in the form i = f(6) and the voltage executes a harmonic oscll1ation 6 :: 6 0 cos (,.,t + 9), the fundamental harmonic of current wll1 be eo cos,p) cos tP d~. Here

! fo

lr

,(

·243

again one can replace the non-linear parameter i = J(e) by a linear one, i = ae, provided that we define the equivalent conductance as

Depending upon whether q is greater than or less than zero, one has either absorption or generation of energy. These considerations are useful in analyzing circuits containing electron tubes. The anode current in this case is of the form ia = J(E o + e), where Eo is the constant voltage of the liB source, II and e is the alternating grid voltage. If e is Sinusoidal, the fundamental harmonic of ia will be

If one defines

[78.8 ] as the average transconductance of the tube, instead of a non-linear relation ia = J(E o + e), one will have an equivalent linear relation, ia = S, e. F.

THERMIONIC GENERATORS Consider a thermionic circuit arranged according to the diagram shown in Figure 78.1, which is self-explanatory. The resistance R, shown to be in parallel, is supposed to be large so as to obtain only rather small damping in the oscillating c1rcuit LC. The control voltage is e = (M - DL) ~;, where the term DL ~; takes care of the anode reaction (D« 1), The alternat1ng component of the anode current is

t where S. 1s the equivalent transconductance of the linearized problem. By Kirchhoff's law, ia = iL + iR + ie, where RiR = Li~ in the LR-meshi hence iR = i~. In the CL-mesh,

iL

L

fiR R

i

!.. C

I .d lC

t

=

L diL dt

'------L--IIIIIII q1-----' F1gure 78.'

io

ic

C

244 whence ic = LCi L". Substituting these values into Kirchhoff's equation and dropping the subscript L, one obtains

Hence finally LC -d~2 dt

S (M - DL )~ -di e dt

+ [L -R -

+ 1, =

0

This equation is an equivalent l1nearized equat10n of the process, s1nce 1t contains the linear1zed parameter S,. From this we get the decrement 6 = _1 [L - S (M-DL)~ 2LC R r ~

[78.10 ]

Equation [59.13] of the first approximation is here ' d ~ dt

=

L S,(M - DL) --R

2LC

, to'

[78.11]

The stable amplitude 1s reached when L

S, (M- DL) =R -

[78.12 ]

Since, by Equation [78.8], S, contains 6 0 , the subst1tut10n for S. of its value from [78.12] determines the equilibrium amplitude 6 0 of the gr1d voltage at which the oscillat10n reaches a steady state. As a second example, cons1der a somewhat modified scheme shown in Pigure 78.2, in which the resistance R is supposed to be small so as to be within the range of the quas1-linear theory. It the current in the oscillating circuit is designated by i and the anode current by i o , the differential equation is L!i£ +R'

dt

where

6

'

+

"!"f 'dt = M dia C dt t

= MS

de

' dt

[78.13 ]

is the grid voltage given by the equation e -l.f'dt - C 1

+ D(ME dt - L a ~) dt

If the anode reaction is neglected,

D~

e ::::

~

0, so that

f idt

[78.,4 ]

245 Introducing the var1able , instead ot i in Equat10n [78.,,], we obtain 2

d e LC dt 2

+ (RC

- MS"

) de

The stationary condition is reached when

se =

RC

M

dt +

e

=

0

[78.'5 ]

....------111111--------.

[78. ,6]

In the transient state the decrement is La

6

=

RC - MS. 2LC

[78. , 7]

Since by [78.8] the equivalent transPigure 78.2 conductance S. 1s a tunction ot the amplitude eo, the decrement d varies during the trans1ent state. It the static curve ia = fee) is approximated by a polynomial, Equation [78.8] permits calculating the amplitude e~ at which the decrement 6 vanishes and the stationary condition is reached. Assume, for example, that the constant biasing voltage Eo is such that the characterist1c ia = f(,) of the electron tube can be approximated by the polynomial

[78.,8 ] where all ~1' and "1 are pos1tive constants. [78.8], one finds

Carrying out the calculation

and, by Equation [78.,6], e' = o

V:~ (a1 3" I

RC)

M

[78.'9 ]

246 REFERENCES (1) "Les methodes nouvelles de la mecanique celeste" (New Methods in Celestial Mechanics), by H. Poincare, Gauthier-VUlars, Paris, Vol. 1, 1892, Chapters III and IV. (2) "On Van der Pol's and Related Non-Linear Ditterential Equations, II by J .A. Shohat, Journal ot Applied Physics, Vol. 15, No.1, July 1944, pp. 568-514. (3) "Ditterent1algleichungen der St8rungstheorie" (Ditterential Equations ot the Theory ot Perturbation), by And. Lindstedt, Memoires de l'Academie Imperiale des Sciences de st. P6tersbourg, Vol. XXXI, No.4, 1883. (4) "Theory ot OscUlations," by A. Andronow and S. Chaikin, Moscow, (Russian), 1931, Chapters VII and VIII. (5) "Introduction to Non-Linear Mechanics," by N. Krylott and N. Bogoliuboft, Klett, (Russian), 1931, Chapters X, XI, and XII. (6)

H. Poincare, loco cit., Chapter II.

n)

"Sur la theorie mathematique des auto-oscUlations" (On the Mathematical Theory ot Selt-Excited Oscillations), by A. Andronowand A. Witt, Comptes Rendus, Faris, 1930. (8) "Sur la theorie mathematique des systemes auto-oscillatoires a deux degres de liberte" (On the Mathematical Theory ot Self-Excited Systems with Two Degrees ot Freedom), by A. Andronow and A. Witt, Journal ot Technical Physics, USSR, (Russian), 1934. (9) H. POincare, loco cit., p. 179. (10) "Dynamics ot a System ot Rigid Bodies," by E.J. Routh, London, 1905, Chapter VI. (11) Uber die Bedingungen unter welchen eine Gleichung nur Wurzeln mit negativen reellen Theilen besitzt" (On the Conditions under which an Equation Possesses Only Roots with Negative Real Parts), by A. Hurwitz, Mathematische Annalen, Vol. 46, 1895, p. 273. (12) "Probleme general de la stabUite du mouvement" (General Problemes ot Stability ot Motion), by M.A. Liapounott, Annales de la Faculte des Sciences de Toulouse, FariS, Vol. 9, 1907. (13) "On a Type ot OscUlation Hysteresis in a Simple Triode Generator," by E.V. Appleton and B. Van der Pol, Philosophical Magazine, Vol. 42, 1921.

247

(14)

"Theory of Sound," by Lord Rayleigh, London, Vol. 1, 1894.

(15) liOn Oscillation Hysteresis in a Triode Generator," by B. Van der Pol, Philosophical Magazine, Vol. 43, 1922. (16) "Note sur 18 generation des oscillations entretenues" (Memoir on the Generation of Self-Excited Oscillations), by E. and H. Cartan, Annales des Postes Telegraphes et T~lephones, December 1925. (17) "Etude des oscillations entretenues" (Study of Self-Excited Oscillations), by A. Lienard, Revue Generale de llElectricite, Vol. 23, 1928.

INTRODUCTION TO NON-LINEAR MECHANICS PART

m

NON-LINEAR RESONANCE

79. INTRODUCTORY REMARKS The object of Part III is to outline the present status of the theory of non-linear resonance. The phenomena of non-linear resonance are far more complicated and diversified than those of ordinary linear resonance, and it does not seem possible as yet to give a unified picture of the whole subJect. Some of these phenomena appear to be more adequately discussed on the basis of the quasi-linear theory of Kryloff and Bogoliuboff, while others, on the contrary, fit better into the theory developed by Mandelstam and Papalexi, which is based on the earlier work of Poincare. For these reasons it was thought preferable to present separately the expositions of these two principal schools of thought without attempting to establish further generalizations at this time. It is apparent that this procedure inevitably reflects the somewhat unsettled state of the whole subJect and leads to a certain overlapping of topics. The reader will undoubtedly observe this in connection with certain specific topics such as parametric excitation. entrainment of frequency, and others. The first four chapters of Part III are devoted to an exposition of the theory of Kryloff and Bogoliuboff; the last three give an outline of the work done by the school of Mandelstam and Papalexi. In addition to the intrinsic difference 1n the methods used by various writers, there also exists a considerable difference in the terminology they employ. An attempt was made to remedy this situation to some extent by designating as inte~nal resonance the case when the divisors in the generalized response function become small. The term ezternal resonance is reserved exclusively for the case when an external periodic excitation exists, as in the theory of ordinary linear resonance. In the quasi-linear theory of Kryloff and Bogoliuboff the study is more or less equally divided between these two principal cases, whereas the work of the school of Mandelstam and Papalexi centers mainly about the phenomena of external resonance, which are illustrated by numerous experimental researches. The reader will find it convenient to read Chapters X and XIl of Part II before reading Chapters XIII, XIV, XV, and XVI of Part III. It is also suggested that he read Chapters I, ILl, and IV of Part I,and particularly

250

Chapter VIII ot Part II,betore reading Chapters XVII and XVIII, which are independent ot the tirst tour chapters ot Part III. Chapter XIX depends very l1ttle on any ot the preceding chapters, except posslbly on the concept of representing so~ut1ons ot dltferent1al equat10ns by phase traJector1es wh1ch 1s outl1ned in Sect10n " Part I. The essence of Chapter XIX 11es ~n the theory ot ditterential equations with period1c coefficients, outlined only briefly in Section 108. It must be admitted that these attempts to establish junction pOints between the exceedingly compl1cated phenomena ot non-linear resonance and var10us ex1st1ng theor1es are probably incomplete at present, and it is hoped that th1s survey will serve as a stimulus tor further generalizat1ons.

CHAPTER XlII SYST EMS WITH SEVERAL DEGREES O F F RE EDOM·

We now pro pose to s t udy the beha vior of quas i- linear sys t ems wi th sever al degrees of freedom. For this purpose it is useful to review the 50cal l ed method of complex amplitudes used extensive l y in the th eory of electric circu i ts . Fr om this method the definitions of impedances and admittances of electri c circ ui ts can be gener a li zed so t hat they also apply to mechanica l systems. The f urther gener a l iza ti on necessary to pass from linea r problems to quas i - l i near ones i s t hen re l at ively Simple .

80 . METHOD OF COMPLEX AMPLITUDES I n dea ling wi t h os cil l a t or y phenomena , it i s ad van t ageous to use t he exponentia l funct i on e iut "" cos wt + j si n w t, where j = 0 . A f ew well - Imown pr oposit l ons , whi ch wi l l be usef ul l a t er , are gi ve n be l ow. Multiplica t ion by l' of a s i nusoidal func tion I( t ) = e jut adva nces its phase by Tr/2. Th i s f ollows f rom Eul er 1s identity l' = e J2 ; whe nce 1'e}<.l1 = ei(ut + f ) . Li kewise . mu lt ipli ca ti on by - 1' retard s the phase of e Jyt by Tr12 . 1.

"

'

2. Multi plicat i on of e jY 1 by 1''' , where n is an i n t eger , advances or r e tards t he phase of t he vector e iY ! by n7r/2 , according to whether n i s a po si tive or a negative i nteger . Thi s fo llows from the defi nition 1'1t = j . j . . . j (n facto r s ); each mu l tipli cat i on by j advances the rotation by rr/2 . 3. The der iva tive df/d t :: jwl. Hence the der i va t i ve of a sinusoi da l vector 1 '" ae iYI cons i s t s i n the oper a t ion (j w ) on t he vector ae JUI, whi ch mul ti pl ies the ampl i tude a by w a nd adva nces t he phase by " /2 . Likewi se , d" f / dt" :: (jw) "f is the operati on wh i ch multiplies the amplitude by w" and advances t he phase by n"/Z . Symboli ca lly, 6" '" d"ldt"':, (j W) " ... j"w ". Thi s hol ds only when a i s a constant . 4 . For a linear system of s inuso i dal functions I. for examp l e , I = 11 + I: . the opera ti on (j w) i s addi tive. that i s , (j w )/"" ( j w )/1 + (jw) / z '

5.

If . in stead of ( j w) or (j W)" , one has a linear f unc ti on

example . ~(j) = A + Bj , the oper a tion {rp(j ) )j oper a ti on ~(j l produc es t wo ef f ec t s :

= AI+

j 81 .

~(j ) ,

f or

I n thi s case the

.. The subject :::atte:r of tb.1s &nd of the folla.1ng th:ree chapters 1B •... keD frOi:l the treatise of rryloff And Bolol1uboff, Reference (1).

**

**

Numbers 1n PlLren t hollee 1ml1cate roferences on palO 130 of this report .

OPERATIONS RESEARC H, INC, LI BRARY

252

a. Multiplication of the amplitude vector f by A without changing ita phase. b. Multiplication of the amplitude by B with the incident rotation of the phase by "/2. The amplitude I of the new vector [~(j)]f is thus complex, if the original amplitude a of the vector f is real.

6. More generally, if one has a relation between sinusoidal functions of the form [80.1 ]

there also exists an analogous relation tPo(j)Jo

+ tP 1(j)I1 + ... +

between their complex amplitudes 10 , 1 1 , ' 81.

tP,,(j)IfI = 0

[80.2]

Ifi'

•• ,

ELEMENTS OF THE THEORY OF LINEAR CIRCUITS

In the theory of linear electric circuits one encounters differential equations of the form

where I and E are the current and voltage in a given mesh, and Ok and Pk are constant parameters. This equation, written in terms of complex amplitudes, is

I

m

k=O

0k(jw)k I =

I

fI Pk(jw)k E

k=O

From this we obtain

[81 .1 ]

and I =

Y(jw)E

[81 .2]

253 where [81 .3 J

Z(jw)

is the complex impedance and

Y(jw) =

I " PJt(jw)Jt .... I OJt(jw)k

Jt..::;:=m="O--_

k=O

is the complex admittance of the mesh.

From these equations it follows that [81 .4 ]

Problems involving systems with one degree of freedom are thus reduced to the ultimate calculation of impedances or admittances, as the case may be. For electric circuits this procedure is too well known to need emphasis here; consequently a few words about it will suffice. For an inductance L and a capacity C the values of the complex impedances are respectively jLwand 1/jCw; the resistance is a real quantity R. For a series circuit, the impedance equation is used; for a parallel one, the admittance equation. Thus, when L 1, R 1, and C 1 are in series, Z1 = R1

+

j(L1W -

C~w)

For another series circuit (L 2 ,R2 ,Ca) one has Z2 = R2

+

j(L2W -

c~w)

If these two circuits are in parallel, the admittance is

and so on. For systems with several degrees of freedom, there arises the question of coupling between these degrees of freedom. The coupling factor can be determined by analogy with electric-circuit theory. Consider, for instance, a system with two degrees of freedom. The first circuit contains L 1 , R 1 , and an external "forcing" function. The forcing

254 function is a sinusoidal electromotive force E = Eoe N '. The second circuit contains L z , R z , and Cz • In addition, the two circuits are coupled together; Alis the coefficient of mutual inductance. Kirchhoff I s law applied to the first circuit gives

c.

L di l ldt

Figure S1.1

+

R . _ M di z = E 6 jlJt ltl dt 0

[S1 .5]

For the second circuit it gives

[S1.6] Written in terms of the complex amplitudes, these equations are

+

(jLlw jMwII

jMwI2 = Eo

R1)JI -

+ (jLzw +

Rz

+ j~w)I2

= 0

whence [S1. 7]

The cotactor ot Eo in this expression is the adm1ttance YI(jw) ot the tirst circuit. It there is no coupling, that is, it Al= 0, Equation (S1.7] gives 1I --

Eo

RI

+ jL1w

As another applioation of the theory of linear oircuits, consider two simple circuits (LI,C1 ) and (Lz'C z ) coupled inductively as shown in Figure S1.1, The complex equations are* (Ld W + - Mjwil

CI~w)il

- Mjwi2 = 0

+ (L 2jW + Cz~w) i2

(S1 .S] = 0

The condition which expresses the consistency ot the system [S1.S] tor values i l and i z other than the trivial ones, i l = i z = 0, is

*

We follow the notation of Kr,ylotf and Bogol1ubofr in choosing the positive directions as shown in

Figure 81.1.

255

[81.91

that is, [81.101

Let

This gives

w(1 -l> 4

(w~

+ wi>w 2 +

w;w; = 0

[81 .11 ]

The oscillating circuits will have frequencies [81 .12 J

2 ( WI

2) W2 -

2

2 2

+

differing somewhat from the natural frequencies

WI

2 -

82.

+

_"(

Y I WI - W2) 2(1 _ g2)

D2

49

2

2 2 WI (&12

and

W2

(81

.1, 1

of each circuit.

ANALOGIES BETWEEN ELECTRICAL AND MECHANICAL SYSTEMS

Very often the establishment of a formal analogy between the differential equations expressing two different types of problems permits a formal transfer of known solutions of problems of one type to those of the other. The method of complex amplitudes developed in connection with electric circuits has a useful application in mechanical problems where generalized definitions of mechanical impedances and admittances are involved. In acoustics the notion of "acoustic impedance ll also plays an important role. The real usefulness of these generalizations occurs in connection with systems having several degrees of freedom. It is preferable, however, to establish an analogy f1rst.for a system having one degree of freedom. The differential equatIon of a Simple (L,C,R)-circu1t acted upon by a sinusoidal electromotive force is

+

L di dt

Ri

+ ~J'i dt C o

= E ej"'t = E 0

(82.1

J

Consider, on the other hand, a mechanical system with one degree of freedom excited by an external sinusoidal force. Its equation is 2

d % m dt 2

+

h d% dt

+

k% = Foe

.

J

'"

t

= F

256 When the new variable v m

= ~~ dv dt +

is introduced, this equation becomes

J t

+

hv

k v dt = ~e

iwt

[82.2)

= F

o

One observes that Equations [82.1) and [82.2] are of the same form and that the following corresponding quantities indicate the analogy between electrical and mechanical problems: (i,v);

(L,m);

(~

(R,k);

,k);

[82.3)

(E,F)

The method of complex amplitudes in the electrical problem gives 1

o

where x

= Lw - ciw

is

=

VR2

Eo

+ x2

the reactance and tP.

e- i9•

[82.4]

= tan- 1 [ LW; dw ]. The quantity

e -19.

VR2 + (Lw -

C~)2

V+ h2

where tP m

= tan- 1 mw~;

Fo. {mw -

k.

;)2

e-i~m

Y.(jw)

By the analogy [82.3), one obtains

is the complex admittance of the circuit. v =

=

=

e- u1m

Fow 2

Y'h w

2

+

[82.5]

(mw 2 _ k)2

By further analogy, the quantity

Yh 2w2 + (mw 2

-

k)2

is the complex admittance of the mechan1cal system. In both cases tP = 0 at resonance, the variables 10 and v are in phase with the exciting forces Eo and Fo respectively, and their ampl1tudes are limited by the dissipation factors Rand L. The complex impedanceB are the inverse quantities of the admittances, that is, Z(jw) = ydw)' One could proceed to establish an analogy between differential equations of the second order by differentiating EqUation [82.1). Here one would compare the differentiated equation [82.1] with that of the mechanical system. The condition of equivalence for these equations is (i,x);

(L,m);

(R,k);

(~ Ik); (~:

I

F)

[82.6 ]

It is seen that both the electrical and mechanical problems can be treated by the concept of admittances and impedances. These electro-mechanical analog1es can easily be established for systems with several degrees of freedom. Sometimes the establishment of an

257

E

j

Ibl

Figure 82.1 analogy with an electrical problem helps considerably in the solution of a mechanical problem. As an example, consider the mechanical system shown in Figure 82.1a, whose electrical analogue is indicated in Figure 82.1b. Designating the displacements of the masses m 1 , m 2 , and ms by xl' x 2 ' and Xs and the velocities by Xl = VI' X2 = V 2 , and Xs = Vs respectively, we have, using the operational notation, [82.7] The differential equations of the mechanical system are ,

m 1 Jwv 1

kl VI + -,JW

-

k2 (

-,- V 2 -

VI

JW

)

=

0

(82.8J msJ'wva

+

k

~ (11 JW S

-

11 ) 2

=

F

In these equations, if one takes the velocity V as the dependent variable, the acceleration is clearly jwv and the displacement is v/jw. On the other hand, if, in the electric circuit of Figure 82.1b, the current i is the dependent variable, by applying Kirchhoff's law to the subsequent meshes of the circuit one gets " L IJWl 1

+

1, - C ' l1 IJW

-

1 (' - C ' t2 2JW

-

')

'I

= 0

[82.9] 1 C'ta L 8J' W1'+ a -C' aJw

' ) = E

12

258

It is seen that both systems are formally identical. In this particular case a more complicated investigat10n of the motion of the mechanical system shown in Figure 82.1a can be more conveniently conducted by utiliz1ng the electrical analogy of Figure 82.1b. In general, any mechanical problem with several degrees of freedom can be represented by an electrical analogue. Since the terminology is more definitely established for electric circuits than for mechanical systems, it is always preferable to use the "electrical language. 1I In more complicated problems it is sometimes difficult to establish an analogy because the determination of mechanical parameters generally is more difficult than that of electrical ones. Where it is possible to establish an analogy, the method of complex amplitudes leads immediately to the establishment of steady-state conditions. 83.

APPLICATION OF THE METHOD OF EQUIVALENT LINEARIZATION TO THE STEADY STATE OF A QUASI-LINEAR SYSTEM With this reminder of the principal points of the theory of linear circuits, we can now proceed to establish a generalization of the KryloffBogoliuboff theory applicable to circuits containing non-linear conductors of electricity. It will be assumed that the departure from linearity is small so that the theory of the first approximation describes the phenomena with sufficient accuracy. In what follows we shall make frequent use of the Principle of Equivalent Linearization, Chapter XII, so that the definitions of impedances and admittances can be extended to quas1-linear systems. We shall consider first a linear dissipative circuit with constant parameters R, L, and C, the decrement of which is 6 = -R/2L. Let us assume that we introduce in series with this circuit a non-linear resistor having the characteristic e = !(i). The variable resistance p of such a non-linear c'onductor is defined, as usual, by the relation p = ~~ = d~(/). In accordance with the Principle of Equivalent Linearization, the non-linear conductor can be replaced by an equivalent linear one whose resistance Re is Re =

~r!( io cos cp) cos cp dcp 1no

[83.1]

where io is the quiescent value of i around which the oscillations are to be investigated. The complex equation of the circuit is Ljw

+ R + C~JW + Re

= 0

(83.2]

Equating to zero the real and imaginary components of this equation, we obtain two equations:

259

1 Cw

Lw -

=

0

R + R~

and

=

0

[83.3J

The first equation [83.3] determines the frequency and the second the amplitude of oscillations in a stationary state. From the second equation it is noted that oscillations are possible if Re < O. In the theory of linear circuits, R~ = 0, which requires that R = 0, that is, steady oscillations are possible only if the circuit has no resistance, which is obvious. It is to be noted that the method of equivalent linearization can only be applied when the resistance R is small enough to make the decrement 6 = -~2L small, in which case the oscillations are feebly damped and the system is qUasi-linear and can be described by equations of the first approximation. Let us now consider a linear circuit with admittance A(jw) , that is, with impedance Z(jw) = A(Jw), closed on a non-linear conductor with the equivalent admittance SCa), where a is the amplitude. The circuit will consist, 1 1 therefore, of two impedances A(jw) and S(a) in series so that the resultant impedance will be 1

Zr(jw) = A(jw) + For steady-state oscillations, Zr(jw)

1 S(a)

= 0,

that is,

S(a)

=

A{jw) +

[83.41

[83.51

0

Again two equations are obtained by splitting the complex quantities into real and imaginary components. One of these equations determines the frequency and the other the amplitude of the stationary oscillations. We shall now apply the method of complex amplitudes to the study of oscillations in electron-tube circuits. The non-linear conductor, the electron tube, is represented by an equation of the form ia = !(E)

where ia is the anode or plate current and E is the control voltage. equation can also be written as ia = S(a)[eg

+

Dea]

=

S(a)e

This

[83.6J

where e g and ea are the grid and the anode voltages. D = lip is the factor of the anode reaction (here p is the amplification factor of the tube), and Sea) is the average transconductance,· a function of the amplitude a. The quantity Sea) is given by the equation

S(a)

*

The term

",",,111 cOlld""'A"". is

=

~r!(Eo 1Ta

aleo employed.

+

acos~)cos~d~

[83.1]

260

A

c

-

iz

~------~--------.B

Figure 83.1

where Eo is the quiescent point ot the control voltage. Let us consider the circuit shown in Figure 83.1 representing an electron-tube oscillator with inductive coupling. Designating by ZB(jW) the impedance ot the circuit between the points A and B and by k«jw) the mutual reactance and using other notations and positive directions as indicated in Figure 83.1, we have

[83.8] From these equations and Equation [83.6J the admittance A(jw) ot the circuit is given by the expression 1 A(jw) = M(jw) - DZa(jw)

[83.9 J

In a selt-excited state the total impedance vanishes, and by Equations [83.4J and [83.5] one must have A(jw) - 8(a)

=

0

[83.10J

In other words, th~ admittance ot the external circuit must be equal to the admittance, or transconductance, 8(a) ot the electron tube. Additional equations trom which the condit1ons of the steady state can be established are obta1ned as follows. Let us introduce a complex number K defined by the relat10n K

= ~ = M(~w) - ea

Za(;w)

(83.11J

From Equations [83.9J, [83.10J, and [83.11J, we obtain

[83.12J This equation 1s establ1shed by Barkhausen (2) and can be appl1ed to the circuit shown in Figure 83.'. By a simple calculation we find first that

The voltage between the pOints A and B is eAB == ZB(jw)i a and the current i l 1n the LR-branch of the circuit is i l == eABI(R + jLw) = iB I[ (1 - CLW2) + jCRw]. Hence the gr1d voltage eg 1s

261

Mjw

[83.141

whence Mjw[(l - LC( 2 ) + jCRw] [(1 _ LC( 2 ) + jCRw] (R + jLw)

e K

= -;:; =

Mjw R + jLw

[83.151

Substituting this value of K into Equation [83.12] and using Equation [83.131. one finds. after separating the real and the imaginary components. the following two equations: SCM - DL) = CR

[83.161

+

[83.171

1 - LCw 2

SDR = 0

Since both D and R are generally very small. the quantity SDRis of the second order and can be neglected. One then finds from Equation [83.17J that the frequency of self-excited oscillations is practically that of the OSCillating circuit. provided the amplification factor ~ = liD is sufficiently large. Equation [83.161 determines the amplitude a l of the stationary self-excited oscillation. namely. Seal)

=

1

1ra

l

JWf(Eo +

acos~) coscj d~

= M CR _ DL

Let us assume. for example, that the experimental function ia is of the form ia = !(Eo

+

acos~) = io

+

klacoscj

+

k 2(acoscj)2

where io ~ /(E o ) and k l • k 2 • ••• are constants. into Equation [83.18]. one gets Sea) = _l_[io

na

J'rcos~ dcj

+

2

kla fcos cj 0

+

[83.18 J

= f( Eo

ka(acoscj)3

+ a cos

~)

+ ...

Substituting this expression

d~

+ [83.191

The first and the third terms of this equation are zero. pansion to the first four terms. Equation [83.19] gives sea) = k1

+ :

If we limit the ex-

k3a2

whence, by Equation [83.16], the amplitude of the stationary oscillation 1s a =

V~(RC 3k M 3

-

k) 1

[83.201

262

It is seen that the quadratic term of the polynomial approximating the experimental non-linear characteristic io = I(E) does not contribute anything to the expression for the amplitude a of the stationary oscillation. The amplitude is expressed only in terms of the coefficients kl and ka of the linear and cubic terms and the paramete~s of the circuit, as was found previously by other methods; see Section 54. APPLICATION OF THE METHOD OF EQUIVALENT LINEARIZATION TO THE TRANSIENT STATE OF A QUASI-LINEAR SYSTEM The method of complex amplitudes can also be generalized for the transient state of a quasi-linear system. For this purpose it is necessary to extend ~he definition of symbolic differentiation to the complex exponential functions e P' , where p = d + jw. The quantity d is the decrement if it is negative and the increment if it is positive. The rule of symbolic differentiation remains the same as given in Section 80, Proposition 3, namely,

84.

"

dl

(iI=p'l;

The vectorial interpretation of these operations is somewhat different. Thus, for example, = pI = (d + jw)1 = (dl + jw/) means that the vector consists now of two components, one jwl leading the original vector I by "/2 and the other dl in phase with I. It follows, therefore, that the vector:f leads the vector I by an angle ~ = tan- 1 ~ which is less than "/2 if 4 > 0, that is, if d is an increment. If, however, 4 < 0, that is, if 4 is a decrement, ~ = tan- 1 _I~I which is larger than "/2. The generalization for higher derivatives does not present any difficulties. If d + 0, ~ +"/2; if, however, the condition of aperiodic damping is approached, w + 0 and ~ +0, which means that all higher derivative vectors approach the in-phase condition with the real exponential function I(t) = e 6t • For the transient state we must use the condition

M

M

,d(p)

=

0

[84.1]

instead of the condition of consistency [81.9]. Consider, for example, the self-excitation of a simple (L,C,R)circuit, as shown in Figure 84.1, closed on a non-linear conductor Nwhich has the equivalent reSistance Re(a).* In order to obtain a self-excitation of this circuit, the total impedance must be zero, that is, Z(p)

+

R, = 0

[84.2 ]

where Z(p) is the linear impedance, which we wish to consider in connection

*

For convenience we will write Re in place of Re(o) in the intermediate calculation.

with the transient state p = - IS

when 6

N

+ jw

0 but is small.* The linearized impedance of the steady state is

ZI(jW)

~

+

R,

1

= Ljw + (R + R,> + Cjw =

0

L

Figure 84.1

Hence, for the transient state, we have to substitute p for jw. This gives LCp2 1 Dividing by L C and putting LC 2

+

(R

=

Wo , we have

P +

+

R

c

R,> Cp

+

1

=

[84.3]

0

2

R

+ R, L

2

P + Wo

=

0

whence (84.4] where 6 = R;LRc is the decrement and w is the frequency. Self-excitation from rest is possible if R + R~(O) < 0, that is, if R < -R~(O). We shall call Ro = -R the critical value of the negative resistance and 60 = the linear decrement. In these notations we have

-if

6= 6[1 0

Ri

a )] o

[84.5]

At the start, that is, when a = 0, one must have IR~(O)I > IRol, that is, 6 < 0, in which case it follows from [84.4] that the amplitudes increase initially. The function IR~(a)1 is a monotonically decreasing function of a so that for a certain value a = ai' R~(al) = Ro and 6 = 0, which means that the oscillation reaches a stationary state. It is thus seen that the concept of equivalent resistance R,(a) permits formulating the condition of approach to the steady state by means of the variable decrement 6. When a+ ai' for which

R,(a) + R o, 6 + O. As a second example, consider the coupled system shown in Figure 84.2 in which ~is a non-linear conductor characterized by the equivalent resistance R,(a) as in the first example. With the same assumptions and notations as in the preceding example, the linear impedance of the system is

*

In this expression, I is the decrement if it is positive and the increment if it is negative; this corresponds to the notation of Kryloff and BogoUuboff.

264

[84.6] When the non-linear conductor N is present, we have to add R,(a) to RI in the preced1ng expression and replace jD by P in order to form the express10n for the trans1ent impedance Z(p) during self-excitat1on, which we now propose to investigate. It is apparent that the quantity Z(p) is generally of the form

N

M

Z(p) =

Figure 84.2 Z(p)

=

°and since

A(p) =

[L 1C1P 2

+

B(p)

(R 1

+

*

A(p) B(p)

[84.7]

S1nce the condition for self-excitation is 0, we express this condition as

Re)CIP

+

1][L2 C2p2 + R2C2P + 1] - M2CIC2P·

= 0

[84.8]

Let 1 L C = I

where

PI

and

P2

I

2 WI;

1

L C.

2 2

2

M

= W2; -:l£T y.ul.u2

= g;

are small dimensionless factors of the first order.

Equation

[84.8] becomes [84.9 ]

Since

and P2 are small, we can 1ntroduce a small parameter p by setting PI = Jlt 1 and P2 = pt 2 • We thus obta1n the follow1ng characteristic equ.ation for the transient state: PI

This equation can be solved by substituting for p a series expansion p = Po + PPI + Jl2 P2 + .•. and developing the function .:1(p,p) in a Taylor series around (po'O). By equatIng to zero the coefficients of like powers of Jl, one obtains a system of equatIons from which the subsequent terms PI' P2' ••• can be calculated. These equations are

265

L\(po'O) =

+

L\p(PO,O)P2

L\pp(PO'O)p;

=

°

2L\pp(PO,O)Pl

+

+

L\p(PO,O)PI

+

°

L\p(Po'O)

From the first two equations one obtains proxima tion P = Po

+

[84.11] L\pp(PO'O) =

°

and PI' which give the first ap-

Po

[84 .12 J

IIPI

Introducing this value of P into Equation [84.10] and making use of Equations [81.12J and [81.13], one obtains the following equations 1 (p2 1 - g2 L\(p,O) =

1

1 - g2

L\ ( p

0) p, -

+

2p(2p

$112)(p2 W;W22 2

+

[l/

w?w g

+

$122)

+

SJg )

g

[84.13 J

It is apparent that, since we have here a coupled system possessing two distinct coupled frequencie8 $11 and [l2' self-excitation may occur with either one of these frequencies. Thus one obtains two values for P = Po + IIPI' namely, . P =

)DI -

1 D2 - W g $12 - w 2 1 1 I 1 _ g2 "2PIW1 $1\ - $1"- 1 - gg - "2 PgW2 $1 \ $12 1 2 1 g 1

[84.14] 2

2

2

2

1 1 1 WI - $12 _. _.!.2 P1W W\ - SJ2 P - )SJ2 I $1 _ $12 1 _ g2 - "2 P2w2 $1; - $1i 1 - g2 1 2

From these expressions it follows that the frequency and the decrement during the transient state are given by either pair of the following expressions: [84.15]

(84.16] It is seen that the frequencies of a non-linear system subject to a selfexcitation to the order of approximation considered here are the same as those of the corresponding linear system, Equations [81.12) and [81.13), in which the dissipative parameters, both linear and non-linear, are neglected. Moreover, since neither [ll nor $12 depends on the amplitude, the system is isochronous.

266 The expression [84.15] for the decrement becomes - W22)Rl + R,(a) (1 _ 92)6 I (a) = (D12 D12 _ D22 2L I

+

(D12 - t.}12) R2 D12 _ D22 2L 2

[84.17]

when PI and P2 are replaced by their values. If 41 (0) < 0, self-excitation with frequency 111 occurs. With the use of Equation [84.17] this condition of self-excitation with frequency 111 can be written as IR~(o)1

> IRoll

[84.18J

where ROI is the critical value of the equivalent resistance Re. cal value is R01

=-

(R 1

+

L R 2~ L

2

gl n 2 _

2 oWl

This criti-

2

WI ) 2

[84.19)

W2

With the use of [84.19J, Equation [84.17] can be written (

1

_

2)

9

wl..&..

() _ (g12 61 a - gl2 _ D22 2Ll

The decrement 4l (al)

I1l -

+

2 Wl

~)(1

D12 - D22 2L2

_ R,(a») ROI

[84.20]

= 0 when

The stationary amplitude a l of the self-excited oscillation is then obtained from the explicit form of R.(a l ), as previously shown. Introducing the notation

[84.21 ] we can write Equation [84.20J as

. () = . [1

III

a

1101

-

R,(a)]

~ 01

[84.22]

and s1m1larly

[84.23] where 601 and

402

111 and 112 and

ROl

are the decrements of the linear circuits with frequencies and R02 are the corresponding critical values of the equivalent non-linear resistance Re(a). It is to be noted that the possibility of determining PI from the second equation [84.11] depends on the condit10n ~p(Po,O) # 0, as follows from the theorem on implicit functions. On the other hand, the expression

267

becomes

or

°

for Po ::z jD10r Po = jD 2 respectively. Hence, the condition .1,,( po'O) :I: is fulfilled only when Dl D 2 , that is, when both circuits are not tuned in resonance to the same frequency. Whenever the circuits are tuned in resonance to the same frequency, the preceding method ceases to be applicable and a speCial procedure indicated in Section 86 will be necessary.

'*

85. NON-RESONANT SELF-EXCITATION OF A QUASI-LINEAR SYSTEM We shall now generalize the conclusions of the preceding section which were obtained for a particular non-linear system with two degrees of freedom. The general condition of self-excitation of a quasi-linear system is, as previously,

Z(p,,.,) + ,.,r.Ca) = 0 where Z(p) is the transient impedance of the linear system and ,.,r.(a) = R.(a). In general, the impedance can be represented by a rational function, namely,

where A(p,,.,) and B(p,,.,) are certain polynomials prime to each other; see Equation [84.71. For the system shown in Figure 84.2, the quantity A(p,,.,) is given by Equation [84.10], and one finds

[85.3 ] In the relation [85.2], the system is conservative if ,., = 0, and the impedance consists only of inductive and capacitive reactances. Moreover, the oscillations are undamped so that p = jD and Equation [85.21 becomes

A(jD,O) B(jD,O) This expression is, therefore, purely imaginary, that is, it is an odd function of jD. Hence, if A(p,O) is even, B(p,O) must be odd and vice versa. Since B(p,O) 0, the condition for self-excitation is obviously Z( 'D) = J

'*

268

[85.5]

A(p,/l) = 0

It the system is non-dissipative, /l = 0, and we know that the decrement is then zero. We conclude therefore that the equation A(p,O) = 0

has purely imaglnary roots

=jD.

Let

p

= iDo

A(p,O) = (p2

+

Di) ,p(p)

p

be one such root; ,then

[85.7]

*

where ,p(jDo) 0. The quantlty A(p,/l) can be expanded ln a Taylor series around /l = 0; when /l is very small we can wrlte A(p,/l)

=

A(p,O)

+

[85.8]

JlA,,(p,O)

The first term, A(p,O), contalns the capacity and inductance terms; the second, /lAp( p, 0), the dlsslpat1 ve components. Since this term ls proportlonal to /l, ohmlc resistances tor a serles connectlon, or ohmlc conductances tor a parallel connection, appear in lt linearly and homogeneously. It follows, therefore, thatA(p,O) is odd and Ap(p,O) 1s even and vice versa. In view of this and of Equatlon [85.7], lt is clear that A$(~:g~ and are odd, since, if A(p,O) ls even, B(p,O) is odd and ,p(p) is even. The characteristic equation of a non-llnear system ls

!l::gi

+

A(p,/l) = A(p,Jl)

/lB(p,/l)rfj = 0

[85.9 ]

In order to solve this equatlon we assume a solution of the form p = jDo

+

/lPl

+ ...

[85.10]

By the theorem of implicit functions, this is possible since from Equation [85.7] we find

Using the second equation [84.11], we find _ 'g _ Ap(jDo,O) + B(jDo,O)re p - J 0 /l Ap(jDo,O) B(jDo,O)R. 2jDorp(jDo)

In view of the fact that both tities

A"I,p

and

BI,p

are odd, we conclude that the quan-

and are real. From Equation [85.11] we obtain the frequency and the namely,

[85.12 ] dec~ement,

269

[85.1, ] If this procedure is applied to the solution of the characteristic equation of the linear circuit A(p,JI) = 0

[85.14 ]

one obtains to the first order the expression

60 [85.15] From the second expression [85.13] it follows that the condition of P = iDo -

self-excitation with frequency D

= Do

and 6(0)

< 0 reduces to the condition

t

where Ro = - 60 is the critical value of the equivalent resistance R8(~) at which the decrement 6 vanishes and the oscillation becomes stationary. In conclusion, one can state that. if Do and 60 are the frequency and decrement of a linear system and if Do is not tuned in resonance with other frequencies of the system, the frequency and the decrement of a quasilinear system. linearized by the method of equivalent linearization. are D = Do 6 = 60 [1 -

RjLa)]

As was previously mentioned. the condition of quasi-linearity permits neglecting the dissipative parameters, which are assumed to be small quantities of the first order, as far as the frequency determination is concerned. since the error is of the second order. that is, of the order of Jl2. In other words, the quasi-linear systems in a non-resonant state are also quasi-isochronous.

86. RESONANT SELF-EXCITATION OF A QUASI-LINEAR SYSTEM We shall now investigate a quasi-linear system in which the two frequencies Dl and D2 are adjusted so that they are the same. This condition shall be called the resonant self-excitation of a system with several degrees of freedom. For simpliCity, we shall consider a system with two degrees of freedom of the kind previously investigated in connection with Figure 84.2. It is to be noted that in order to have the difference Dl - D2 of the two frequencies very small, say of the first order of smallness, it is necessary that the difference of the component, non-coupled frequencies w1 and w2 of each degree of freedom also be small and of the same order. This follows from Equations [81.12] and [81.13]. Moreover, the coefficient AI of mutual inductance

270

must also be small, that is, the coupling between the two degrees of freedom must be rather II loose ." These conditions imply that

:2

= 1

+

[86. , ]

9 = PQ

pP;

1

Equation [84.9] can now be written (Z2

+

pe 1 z

+ 1)[Z2 +

p(1

+ PP)~2Z +

pP)2] - 11 2Q2 Z4 = 0

+

(1

[86.2]

where z = plw1' Substituting in this equation the power series solution :I: = 1 + PZ1 + .•• and equating the coefficients of equal powers of p, one obtains the following equation for Zl: (2jZl

+ U1)(2jZl +

jt2

+

2P) - Q2 = 0

When rearranged, it becomes 4z: + 2z1(t 1 + e2 - 2jP) + Q2 + tlt2 - 2jPt1 = 0 [86.3] Determining its roots, and using the notations tl - t2 = M and Q2 + p2 = N, one obtains to the second order the following expression for :I: = 1 + I1Z1:

Z=

-

p(f1

1t2) ± 4Vr

VM2 - 4N

+

J'16p 2M 2 + (M 2 - 4N)2 +

[86.4 ] Putting (Re

::)2] =

R 2)2C12 "'12 - 4[g2 - (1 _

+ Rl -

21]

and

[86.5] 4(R,

~y =

- R2)2CI 2W22(1 -

+ R1

d

2

one obtains to the first order the following expression p =

+

R1

R2

+

R,

+ LlWl Y1]

+

J'd

2

+ 1]2 +

4Ll

+

1[Wl

+

2(W2 - WI)

+ w Y-

1

+ Yd z +

1]

1]Z]

[86.6 ]

which gives the frequency D and the decrement , as

~ fI

D

=

=

~

.01(4)

=

( ) = Rl

fll 4

"'1

!

"'2

+ ~1

+ Rg + R, -

(-1]

+

LlWl,t1] 4L.

J'd 2 + 1]2

+ Yd 2 +

[86.71 1]2

271

or 9 = D 2 (a) =

6 = 6 (a) = RI 2

+ W2

WI

2

+

R2

_

WI

4

+ R" +

y'-

17

+ Yd 2 +

L1Wl Y17 4L)

17 2

+ Vd 2 +

17

[86.8 ]

2

The essential feature of these expressions is that both the frequency D(a) and the decrement 6(a) are non-linear junctions of Rf(a). This leads to the following important conclusion: When the osclllatory system is tuned in resonance so that system ceases to be isochronous.

WI ~ W2 ~

w, the

Developing the expressions [86.7] and [86.8] in terms of the parameters of the system, we obtain the following two sets of expressions for D(a) and 6( a): 1.

If (R" + Rl - R 2 )C1W 61() a =

Re

>

2g, then

+

Rl

+

R2 - L1w VCR. + Rl - R2)2C12W2 4L)

4g 2

-

[86.9]

and

2.

If (R" + Rl - R 2 )C1W < 2g, then

D)(a) =

W[l

+

!

V'4g 2 - (R t

+ Rl

- R2)2CI 2W2];

[86. , 0]

and .I

U2

(

a

)

= R(J

+ Rl + 4L

R2

1

It is thus seen that the condition of resonance of a quasi-linear system with two degrees of freedom introduces a radical change in the behavior of the system. In a later chapter it will be shown that resonance in a mUltiperiodic system is characterized by another feature, namely, the differential equations of the first approximation do not permit a separation of variables although in a non-resonant condition such a separation is always possible and simplifies the problem appreciably.

CHAPTER XIV SUBHARMONICS AND FREQUENCY DEMULTIPLICATION

87. COMBINATION TONES; SUBHARMONICS The non-linearity of an oscillatory system accounts for the appearance of additional frequencies which we shall now investigate. This phenomenon was first studied by Helmholtz (3) in connection with the theory of physiological acoustics. He discovered that the ear receives sounds which are not contained in the emitted acoustic radiation, and he has shown how the slightly funnel-shaped form of the tympanic membrane of the ear may account for unsymmetrical oscillations represented by a non-linear differential equation of the form

..x +

2

Wo

x

= - IJx 2 +

X() t

where x is the displacement of the membrane and Xis the exciting force produced by the periodically varying pressure of a sound wave. If the sound wave contains two frequencies, say WI and w 2 ' the function Xis of the form X = a 1 cos WI t + a 2 cos w2 t. It can be shown that the oscillation expressed by Equation [87.1] contains, in addition to the frequencies WI and w 2 ' the frequencies WI - w2 and WI + w2 which Helmholtz calls combination tone8. It is simpler to reach these conclusions from the well-known principles of modern radio technique. Consider a non-linear conductor of electricity, an electron tube, whose characteristic is

where ia is the anode current and v is the grid potential. We shall be interested only in the alternating components of these quantities. It has been shown in preceding chapters that this experimental relation can always be expressed in the form

[87.3 ) The number of terms in this series depends on the degree of the approximation desired, and it is shown (4) that in practice the coefficients decrease with sufficient rapidity to justify the retention of only a few terms. In order to simplify the argument, let us assume that the impressed grid voltage is of the form

[87.4) that is, it is composed of two alternating voltages in series having the same amplitude k but two different frequencies WI and w 2 ' Substituting [87.4) into Equation [87.3], one obtains

273

ia

=

~k(COswlt

+

COSW 2 t)

+

a 2k 2(cosw l t

+ COSW2t)2 +

aak3(coSWlt

+

coswzt)3

[87.5]

when the expansion is limited to the first three terms. The first term contains the original frequencies WI and wz; the second gives the frequencies 2w 1 , 2w z , WI + w 2 ' and WI - w 2 ; the third term yields the frequencies wI' 3wI' w 2 ' 3w 2, 2WI + w z, 2Wl - w z , 2w2 + wp and 2w2 - WI' The current ia will contain, therefore, the harmonics of the following frequencies:

Thus, for example, if WI = 27f' X 100 and w2 = 27f' X 120, where fl = 100 cyc Ie s per second and f2 = 120 cycles per second, the frequency spectrum of ia will be composed of the following frequencies: 20, 80, 1100, 120,1 140, 200, 220, 240, 300, 320, 360 It is seen that the combination frequencies spread out on both sides of the impressed frequencies, 11 = 100 and 12 = 120. In the study of this phenomenon, attention is usually centered on the frequencies lower than the impressed ones (in this case, 1= 20 and 1= 80); these lower frequencies are called 8ubharmonic8. The term "subharmonics," instead of the expression "combination harmonics," is sometimes applied, although perhaps improperly, to terms of the form The problem of producing subharmonics in electric circuits is sometimes called the problem of Irequency demultiplication. In Equation [87.6], m and n are integers which depend upon the nature of the polynomial. Thus, for instance, in the previous example these comb ina tions are (0,0); (1,0); (2,0); (3,0); (0, 1 ); (0,2); ( 0 , 3 ); (1, 1 ) ; (l,-l)i (1,2); (1,-2); (2,1); (2,-1). The terms (0,0) are constants arising from the trigonometric transformations. As follows from the foregoing analysis, the combination frequencies are due exclusively to the non-linearity of the circuit. This can be demonstrated by the following experiment mentioned by H.J. Reich (4). The voltages from two audio-frequency oscillators are filtered so as to obtain two pure sinusoidal oscillations of angular frequencies WI and W2, which are applied to the grid of an electron tube. The voltage across the anode resistance is applied to the input of a low-pass filter. The various combination frequencies can thus be heard in a telephone connected to the output of the filter. If the oscillators I frequencies are changed, the whole spectrum of the combination frequencies changes, but the relation [87.6] holds for any

274 value of WI or w 2 • If, instead of a low-pass filter, a high-pass filter is provided, the high-frequency part of the spectrum of the combination frequencies can also be recorded. The relation [87.6) is applicable throughout the whole spectrum. If an oscillating grid potential of the form v = a l cos WIt + + a2 cos W2t is applied in the region where the electron-tube characteristic is fairly rectilinear, the spectrum of the combination frequencies fades away. This shows that the combination frequencies are due to the non-linearity of the circuit. If a combination harmonic of amplitude en,m is applied to a linear network of impedance Z(jD), the current in,m due to this harmonic will be i.. m =

l =[<00. + mO,)! + 091. + m",r

e.,. Z [j(nD I

+ mD2)

[87.7 J

88. EQUIVALENT LINEARIZATION FOR MULTIPERIODIC SYSTEMS In the preceding chapter a quasi-linear system with two degrees of freedom was discussed. It was found that solutions in terms of frequency and decrement were given, not by one set of relations, but by two such sets. Thus, for example, we always have two pairs of relations

o=

0l(a);

tJ = tJI (a)

o=

02(a);

tJ = cJ 2(a)

and In still more complicated systems there may be a still greater number of pairs of solutions:

On the basis of the method of the first apprOXimation, Chapter X, this means that the most general oscillatory system is characterized by oscillations of the form

*

From now on lIe shall designate the component frequencies by capital letters Q 1 and Da, reserving the small letters Wl and Wa for the frequencies of uncoupled linear systems, as lIas done in the preceding chapter.

275 It is recalled that ~, the total pha8e, in these expressions contains both the frequency 0" and the ordinary phase ~k' as follows from the relation ~k = D"t + ~". The most general forms of the equations of the first approximation for both a" and I/J" are k = 1, 2, ... , n

[88.' ]

If the system is linear, the principle of superposition is applicable, that ls, i = i l + i2 + . . . + in = a l cos 1/11 + a 2 cos ~2 + . . . + an cos 1/1" [88.2 ] For non-linear systems this ceases to be applicable, and, if we wish to make use of the principle of superposition, we must introduce further definitions of the equivalent parameters of multiperiodic systems. In order to do this, it is sufficient to extend the Principle of Equivalent Linearization to systems where two distinct oscillations exist. This presupposes that the oscillations are not tuned in resonance with each other. We shall consider again the system shown in Figure 84.2 and note that its impedance across the terminals of the non-linear conductor N is

[88.3] where 01 and O2 are the angular frequencies, and 601 and 602 are the decrements of the linear system given by Equations [81.12], [81.13], and [84.21], respectively, to the first order of small quantities. Equation [88.3] is obtained from Equation [84.6] by making use of Equations [84.10] and [84.14]. Since for quasi-linear systems the decrements 601 and 602 are small, one can put 601 = P(1 j 602 =P(2' and we obtain

[88.4 ] where (1' (2' and e2 are small. This impedance is impressed on a non-linear conductor having the characteristic e

= -

F(i) = -

[88.5]

pf(i}

where p emphasizes again the quasi-linearity of the example considered here. The function f{i) can be apprOXimated in practice by a polynomial. If p = 0, the system is linear and the principle of superposition holds, so that i

where ai' a 2 ,

~1'

=

il

and

+ ~2

i2

=

al cos (Dlt

+

are constants.

~I)

+

a2

cos(D2 t

+

~2)

[88.6]

276 If p * 0 but very small, one can still consider Equation [88.6] as approximately correct and substitute it in. Equation [88.5]. Thus, e = - pf(i) = - Pf(il

+

i 2) = - p.t{aICos(glt

+

q,1)

+ ~COS(.o2t +

Consider now the function of the two variables (PI and f(al cos "'I

+ a2 cos "'2)

¢2)]

(P2'

[88.7] namely,

= fez)

Since f(z) is a polynomial, it can be represented as a finite sum of harmonics of the type so that

[88.81 n

m

where the index n runs from zero through the positive integers and where m has positive integral values for n = 0 and runs through the negative integers for n O. For a polynom1al of the third degree these values for m and n were given in the preceding section. From [88.71 and [88.8] it follows that

*

e = - p I I An.tncos[(nDI n

m

+

m.o2)t

+

nq,1

+

mq,2]

[88.91

Hence the corresponding combination harmonic i n • m of the current will be i n•m = -

.uAn.m Z[j(n.QI

+

cos [(ngl

+

m.Q2)t

+

n¢1

+

mq,2]

[88.10]

m.Q2)]

and the total current will be the sum of all these harmonics. It is apparent that the amplitude of each combination harmonic will be

Written explicitly, in view of [88.4], this expression becomes

[88. '1 1

From this expression it is observed that, if for some values of nand mneither of the expressions (n.QI + m.Q2)2 - .0 12 or (n.QI + m.Q2)2 - .0 22 becomes small, the amplitudes lin.ml will be small because of the small factor p. If, however, for a given pair of numbers (n and m), one of these expressions. or, which is the same. n.Q I + (m ± 1 ).0 2 or (n ± 1).01 + m.Q2, becomes small and of the first order, or zero, both the numerator and the denominator

277

in [88.11] become small and of the same order so that lin,ml may be finite. This happens either for n = 1, m = 0 or for n = 0, m = 1. It is seen from [88.9] that all harmonics of e are small except the two principal harmonics

[88.12 ] Hence, aside from the two corresponding harmonics of current, all others are very small and can be neglected. Thus we have e =

-

,uAl,o cos (.01t + q,1) - ,uAO,1 cos (.ogt + q,g)

[88.13 ]

From Equations [88.5], [88.7], and [88.13], by the Principle of Equivalent Linearization, we obtain ,uA 1,o = ,uA O,1

=

rr 2~g rr 2~2

+

F(41 cos 1/11

42 cos I/Ig)cos 1/I1dl/l 1dl/l 2

[88.14]

+

F(41 cos 1/11

4g cos 1/12) cos I/Igdl/l l dl/l 2

Defining the equivalent non-linear parameters by the expressions

R:

= 2:241

rrF(41 cos 1/11

+

4g cos I/Ig) cos 1/11 dl/l1dl/l g

[88.15]

and

one has

and, by [88 . 13 ],

[88,16] This equation shows that in the example of non-resonance considered here the non-linear characteristic can be replaced by the corresponding linear one in which the variables i l and i2 can be separated in the first approximation. This generalization of the Principle of Equivalent Linearization for a nonresonant system simplifies the problem because the variables can be separated, We obtain the following systems of equations of the first approximation for the equivalent system: d4g = - u~"4g

d41 = _ 6«1' dt

'

~= dt

dt

dl/l2

Qt.

= gil

dt

'

Since the total phases are 1/1 1 = glt + preceding equations in the form

[88.17 ]

~1

and 1/1 2 =

ggt

+ q,g, one can write the

278 dal = _

dt

cHz I '

'

da2 = -

dt

~"ag

u

[88,18] drPl dt

= SJ' - SJ 1, '

drP2 = SJ" - SJ dt 2

In view of [88.,6J, self-excitation of the equivalent linearized system can be represented by the equation

[88.19] whence Z(jSJ' - 6') + R;

=

0;

Z(jD" - 6") + R;'

=

0

[88.20 J

Because rPl and rP2 are constants, one has from [84.23J [88.21

J

where ROI and R02 are the critical values of the equivalent parameters R,' and R," corresponding to self-excitation with frequencies DI and D2 respectively, One concludes, therefore, that in the first approximation and in the absence of resonance,

[88.22 ] the principle of superposition still holds, and the stationary amplitudes a l and a2 of the quasi-linear system are given by the system of differential equations in which the variables can be separated, namely, (88.23 J This result can be generalized for systems with n frequencies; however, this generalization will not be presented here.

INTERNAL SUBHARMONIC RESONANCE In the theory of linear oscillations, an oscillating system is said to be in "resonance" when the frequency of the external exciting force coincides with the frequency of the system. In non-linear systems the situation, as was shown, is far more complicated. We saw that a non-linear system with one degree of freedom acted upon by an exciting force with two or more component frequencies has a discrete spectrum of comhination frequencies which are more numerous than the exciting frequencies. On the other hand, a linear system with several degrees of freedom possesses the so-called coupled frequencies, the number of which is equal to that of the degrees of freedom. For non-linear systems with several degrees 89,

279

of freedom, the situation becomes very complicated. In fact, if one considers one particular circuit of the system that is coupled to other circuits, the excitation of this circuit occurs through couplings and, by virtue of the nonlinearity, is resolved into combination frequencies which, in turn, react on the other circuits and cause combination frequencies in these circuits if they are non-linear. It must be noted, however, that not all frequencies may appear in the process of self-excitation but only those for which the initial decrement Ij(O) < O. Moreover, as long as the principal frequencies do not coalesce, the method of equivalent linearization introduces a further simplification by providing a set of differential equations of the first approximation in which the variables can be separated. A new complication arises, however, when two frequencies of the spectrum are nearly the same and coalesce at the limit; we shall call this internal re80nance of the system. It occurs whenever two or more component parts of the system are tuned to the same frequency. The word internal used in this definition merely emphasizes the fact that the actuation is effected through couplings without involving any externally impressed periodic force. We shall consider again the system with two degrees of freedom shown in Figure 84.2 which was the basis for our discussion of the non-resonant condition. The impedance of the system is

[89.' ] This impedance is impressed on a non-linear element }{with characteristic e = -

F(i) = - fJf(i)

We shall now assume that the ratio of the frequencies gl and g2 of the linear system is not far from being a rational number; this fact is expressed by the following relations: with

r:l= 0 8

[89.3 ]

where, without any loss of generality, rand 8 are relatively prime integers, and a is a finite number so that fJa is small and of the first order. It is apparent that for fJ = 0 the system degenerates into a linear one with coupled frequencies

In this case the principle of superposition holds; i = il + i2 where il = a 1 cos ~1 and i2 = a2 cos W2; ~1 and W2' the total pha8e8, are equal to gl t + ~1 and g2 t + ~2 respectively; and all a 2 , ~1' and ~2 are constants determined by the initiar conditions.

280

For P :#: 0 but very small we can still use the principle of superposition to the first order of approximation, as was just shown, and write

=-

e

P/[41 cos (Dlt + q,l) +

42 cos (; Dlt

+

~)]

[89.4]

The function I is periodic with period 2if:. Since we assume, on the other hand, that this function is approximated by a polynomial, it is possible to express the function by a limited number of terms of a Fourier series. Thus we obtain 1[4ICOS(Dlt +

~l)

42 cos(; Dlt

+

~2)]

+

=

I [Am cos ;

Dlt + Bm sin : Dlt]

[89.5]

m>O

where Am and Bm are certain functions of the variables The harmonic 1m of the current will then be

41' 4 2 ,

~l'

and

~2'

[89.6 ]

Since the factor P is small, all harmonics of the current are small except the two harmonics lr and I., with frequencies ;.01 and .011 Which remain f1n1te in spite of the smallness of Pi see Equation [88.11]. If, therefore, one neglects in the first approximation all harmonics 1m for which m:#: rand m:#: 8, Expression [89.5] reduces to e

=-

[89.7]

P(Ar cos ; Dlt + Brsin ; Dlt) - p(A.cos Dlt + B, sinDlt>

The expressions in parentheses in this equation can be written Ar cos ; Dlt + Br sin ; Dlt = Cr COB (; Dlt + ~2) + Dr sin (fDlt +

q,2) [89.8 ]

where the constants Cr , Dr. C"

and D, are given by the Fourier procedure:

Cr ![1(4I COS[8T" -

~(81j62 - r~l)]+ ~cosrT")cosrT"dT"

==

(8~2 - r~l)] + 42 COS rT") sin rT"dT"

Dr

=;

Ji(4: COS[8T" -

;

c,

== ;

V(4 1cos 8T"

42 cos [rT" + ! (8~2 - r~l)]) COS 8T"dT"

D.

=

+

! ]i(41COS 8T" + ~ COS [rT" + ! (8eP2 -

rePl)]) sin

8T"~T"

[89.9 ]

281

= 9 and introduce the notations

If we put S92 - r91 R:' =

1T~2 [F[al cos(s'T

~) + a2 cos r'T] cos r'Td'T

-

[89.10 ]

the expressions in Equation [89.7J become .u{ArcosfD1t

+

+

Brsin';Dlt) = R;'a2COS(:D1t

ch) -

y'''a2 sin (;Dlt

+

92)

+ .i¥.')al cos (Dlt + 91) [89.12] When these values are substituted into Equation [89.7], it becomes = (R;

e= -

[(R; +

jy")il

+

(Il;'

+ i Y.")i2]

[89·13 ]

It is seen that in systems having internal resonance the non-linear characteristic can be replaced by the equivalent linear one given by Equation [89.13]. The difference between resonant and non-resonant systems is that in resonant systems the equivalent linear impedances are complex quantities, whereas in non-resonant systems they are real ones.

This modification of the equivalent parameters for internal resonance introduces an essential difference in the form of the equations of the first approximation. For example, let .0 I, .0", 4 I, and 4" be the frequencies and decrements of the equivalent linear system. The equations of the first approximation are then dal = dt

-

.t'

val'

'

dl/ll _ '. dt - D.

where, by [88.6], 1/1 1 = glt + tions [89.14J become

tPl

da2 = _ 6"a dt 2

dl/l2 = dt

!,}"

and 1/1 2 = g2t + tP2 are the total phases. Equa-

282

dal

,

== - 6al'

-

dt

•

da2 = _ 6"a

dt

2

(89.15 ]

dr/Ji = D' - DI . dt •

For dynamical equilibrium the impedance drop and the impressed voltage must balance each other tor both oscillations ot the l1nearized systems; this gives

Hence Z(jD' - 6')

+

(R:

+ jY,')

= 0;

Z(jD" - 6")

+

(R:

+

jY;') = 0

[89.17]

It is noted that these equations are the same as Equation [88.20] except that R, is now replaced by Re + jYe • The algebraic work for Equations [89.16] and [89.17] is identical to that for Equation [88.20], provided one does not separate the real and the complex terms. Hence the solution of Equations [89.17] is the same as that of [85.17], wnich can be written as

Without repeating the procedure, we can, therefore, write d1rectly D ' = DI

+

[89. ,8]

y'" ; 61 ROl

R'

6' = 61(1 -

-'); ROI

Hence, for the first approximation in a resonant system, one obtains i == al cos (D 1t

+ epl) + ~ cos (; Dl t + ep2)

where all a 2 , epl' and ep2 are given by the equations da. = - 6 (1 - -R~) ali 1

~

dt

ROl

[89.21 ] depl dt

=>

.!L 1': '. R 01

' •

It is noted that the four equivalent parameters R,', R,", Y:, and Y," appearing in these equations are now functions of only three variables aI' a 2 , and 9, as follows from [89.10]. The last two equations [89.21] can then be written dep2 -dB == 8 dt

dt

-

depl

r - = aD2 -

dt

rDl

+

62 y." •

8 R02

-

61 1':' •

r -

ROl

For a steady-state condition, a l 9 = constant, which gives R' (1 - - '

)al = ROI

= constant, R"

0;

(1 - - ' R02

)a2

=

a2

= constant,

and

0

The last equation can be written

[89.24 ]

= D + ~ Y and

D" = D

+.!.L Y ". R02 • Equation [89.24] shows that the stationary oscillation of the system occurs with two frequencies D' and D" whose ratio is a rational number. In addition to these frequencies there exists a frequency spectrum corresponding to the harmonics of the order ~D~ as can be shown from the study of the refined first approximation, which we omit here.

where D'

I

I

ROI

•

2

90. SYNCHRONIZATION In the previous notation the total phases are ~l = Dlt + ~l and ~2 =-fDlt + ~2' whence ~2 --f~l= ~2 -f~l= 9. For a stationary state d9/dt = 0 and 9 = constant, which gives 8~2

-

r~l

=

9

= constant

[90. , ]

Thus the total phases are IIlocked in synchronism ll with each other. The question now arises whether the condition [90.1] is stable. Since the four equivalent parameters R,/ , R e", Y/, and Y." are functions of the three variables aI' a 2 , and 9, consider special values a lO ' a 20 , and 90 of these parameters corresponding to the stationary condition of the system. Following the method of variational equations, Chapter III. and considering arbitrary small increments da l , da2' and d9 of the first order of small quantities, one obtains

284 where and

[90.3J

The question of stability is reduced to the investigation of the nature of the roots of the characteristic equation of the variational equations [90.2J. If all real parts of these roots are negative, the synchronized OSCillations are stable; if anyone of these parts is positive, the synchronization is unstable. Assume that the oscillations are stable; then, for sufficiently small departures of the parameters characterizing a disturbance in the synchronized condition [90.1J, these departures satisfy the variational equations [90.2J. It is apparent that this is possible only for a certain range of ~al' ~a2' and ~8 around the values a 10 , a 20, and 80 , If one exceeds this range, the conditions of stability may no longer be fulfilled. The interval of variation of the parameters ~al' ~a2' and ~8 which results in the stable condition of synchronized oscillations [90.1J is called the zone of 8ynchronization. Non-linear systems are characterized by the presence of such a zone. It is absent in linear systems, or, in other words, one can say that linear systems are characterized by a zone of synchronization reduced to zero. It is thus seen that the phenomenon of synchronization is a characteristic property of non-linear internal resonance.

91. INTERNAL RESONANCE OP THE ORDER ONE Consider now a particularly important case when r = 8, that is, when Dl J::I D 2 , and W2/Wl = 1 + pP, and g = pQ; see Section 86. Por p = 0, WI = W2 = wand for the first approximation we assume

[91 .1 J a

sin~,

Putting a 1 cos one has

~1

+ a 2 cos

~2

=

a cos

~

and a 1 sin

a2 = a12 + al + 2a 1 a 2 cos (~1 - ¢J2)

~1

+ a2 sin

~2

=

[91 .2 J

Equation [91.1] can be written i

= a cos (wt +

cfJ)

[91.3 J

Therefore, one has a simple relation

+ Re(a) = 0 [91.4 ] It was shown in Section 86 that the frequencies D1(a) and D2(a) and the decrements 6 1 (a) and 6 2 (a) are given by Equations [86.9]. Z(iw)

285

The equations of the first approximation for the amplitudes a l and a 2 and phases Dl and D2 in Equation [91 .1l are therefore dag = _ 6g(a)a2 dt

[91 .5] drP2 = D2(a) dt

w

where a is given by Equation [91.2] and 6t (a), 6 g (a), Dl(a), and D 2 (a) are obtained from Equations [86.7] and [86.8]. It is apparent that, if 61 (0) > 0 and 62 (0) > 0, there will be no self-excitation. If, however, at least one of these quantities is negative, self-excitation from rest will take place. For a steady-state condition the resultant amplitude a is a constant, and from Equations [91.5]

rPl = rPlO + [Dt(a) where

alO, a20'

rPIO' and

rP20

w]t;

rP2

= rPgO +

[D2(a)

-

w]t

are the integration constants.

[91 .6] From Equation

[91.2] one has

[91.7J If 61(a) = 0, then a = alO = ali if 6 g (a) = 0, then a = agO = a2' In the first case the oscillation will occur with frequency Ql(a) and its harmonics; in the second, with D2 (a) and its harmonics. The oscillations in the first case will be stable if ~~ > 0 and 62 > 0, Which means that the oscillation with frequency Dz(a) ~ill die out. In the second case the oscillations will be stable if ~~ > 0 and 61 > 0, that is, the oscillation Dl(a) will die out. For a resonant system, the investigation of the four equations [91.5] is reduced to an investigation of three equations; the first two give a l and a2' and the third gives the phase difference 9 = rPl - rP2.

92. METHOD OF EQUIVALENT LINEARIZATION IN QUASI-LINEAR

SYSTEMS WITH SEVERAL FREQUENCIES The principal points of the method of equivalent linearization were outlined in Chapter XII in connection with quasi-linear systems which have only one frequency. We now propose to generalize these conclusions so that they are applicable when, because of some couplings provided in the system, several frequencies are possible.

286

Let us assume that the oscillations are of the form % = %1

+ ... +

[92. , ]

XII

where XI = al cos (WIt + ~I)' •.• , XII = all cos (w.t + ~II)' Substituting Equation [92.1] into the expression for the characteristic of a non-linear conductor 11 = F(x) [92.2] one obtains, in general, a Fourier series. Let us keep in this series only those terms which contain the original frequencies WI' ••• , WII and let III' ••• ,1111 be these terms. Since any term IIi will have the same frequency as the corresponding term Xi' we can write IIi = Biz i so that Expression [92.2] becomes '[92.3 ] It is apparent that this procedure leads to the definition of the equivalent parameters 8 1 , ••• ,8 on the basis of the Principle of Harmonic Balance, Section 77. One notes also that the non-linear expression [92.2] has been replaced by an equivalent linear one [92.3]. Moreover, it is also clear that by adopting this procedure the terms expressing combination frequencies have been omitted inasmuch as the frequencies of these terms are generally different from the original frequencies WI' ••. , wlI appearing in ExpreSSion [92.3]. After the frequencies wie and the decrements 'i" of the equivalent linear system have been determined, the equations of the first approximation so obtained can be written in the form 11

dal - _ 6 a . dt I" I'

d~l = dt

W

II

-

W • It

.... I

..

'.

•

da ll

de d~1I

dt

= =

'""a"

Wile -

WII

[92.4 ] [92.5 ]

order to improve the accuracy of the approximation, the terms with combination harmonics omitted in the first approximation are added on the right side of these equations and appear as external forcing terms. When there are regular forcing terms, they are added to the harmonics Just mentioned. Since the equation is now linear, there is no difficulty in carrying this procedure further. In setting up the original differential equations, it is essential to introduce non-linear terms with a small parameter # so that for # = 0 one has a linear system without damping. In the equations of the first approximation only the terms containing the first power of the parameter # are retained. In

287 Let us apply these considerations to a linear system characterized by the impedance Z(jw); the relation between the current and voltage is then e = Z(jw) i. If the impedance is of the form Z(jw) = GI~wj )W' the preceding relation becomes

[92.6]

Q(iw)e = P(jw)i

Since for a steady state the operator jw is d/dt, the preceding expression can also be written

(92.7] As was shown in Section 84, Equation (92.7] is also valid for a transient state. In non-linear systems, as we saw, the current and voltage have a non-linear relation e = - F(i) == - p/(i)

so that Equation [92.7] becomes

p(te, p); =

[92.8]

- PQ(:t)/(i)

= 0 one has a linear differential equation

When p

Thus, tor example, instead of Equations [88.4] and [88.5], we can write d2 (dt2

d

+ 2ptl dt +

d2 D12)(dt2

+

d 2pt2 dt

+

d d2 D22)i = - pCWl2 dt

(dtl

[92.9]

+

2ptl W2 :t

+ W22)/W

which is of the same form as [92.8]. Since we know that for linear systems. that is, for systems in which p = o. this equation has two coupled frequencies, let us try to form a nonlinear solution (p * 0) in the torm i= alcos~l

where

%1'

.a2' ~l'

%2'

and

~2

+

a2cos~2

+

P ZI(at. a 2,I/It.I/I2)

+ p2z2 (a"a2.I/Ihl/l2) + ... [92.10]

are periodic functions with period 2" and the functions a l • satisfy expressions of the form

•••

~~l

= pXU(ah a 2)

+

p2 X12 (a" a2)

:~2

= pX21 (al, az)

+

pZX (a a2)

:~1 =

22 h

+ .. . + .. .

[92." J Dl

+ P Yu (ai, a 2) +

p2Yl2 (a" a 2)

+...

288

Substituting Expression [92.10] into Equation [92.9] and taking into account Equations [92.11], we can develop the result of the substitution into a power series of the parameter p. Equating the coefficients of equal powers of p, one obtains the expressions for %11 %2' . . . ; X lll X 12 , . . . ; X 211 X 22 , ••• ; Y ll , Y 12 , ••• ; and Y 2 1' Y 22 , • • • • Carrying out these calculations for terms containing the first power of p, that is, for %11 XIII X 211 Ylll and Y 211 one observes that the equations of the first approximation so obtained are precisely those which were obtained directly by the method of equivalent linearization. If, however, one retains the first three terms in Equation [92.10], the calculation results in what have previously been called equations of the improved first approximation, Section 68, and so on for approximations of higher orders. To summarize these conclusions, it can be stated that the method of equivalent linearization, which was more or less postulated in Part II, can be Justified on the basiS of the preceding analysis and can be generalized for approximat10ns of higher orders. It can be shown by following the argument Just outlined that 1t is possible to form linearized differential equations whose solutions satisfy the original different1al equations with accuracy of the order p, p2, •••• For a more deta1led proof of this proposition, the reader is referred to the Kryloff-Bogolluboff text, Reference (1), pages 241 to 246.

CHAPTER XV EXTERNAL PERIODIC EXCITATION OF QUASI-LINEAR SYSTEMS

In preceding chapters we have been concerned with the phenomena of self-excitation of quasi-linear systems of the autonomous type, that is, quasi-linear systems in which the independent variable t, meaning time, does not appear explicitly in the differential equations. We shall now indicate how the Kryloff-Bogoliuboff theory of quasi-linear systems can be applied to systems having an external periodic excitation. We shall return to this question later in connection with the Mandelstam-Papalexi method based on the theoryof Poincare. EQUATIONS OF THE FIRST APPROXIMATION FOR A PERIODIC NON-RESONANT EXCITATION The quasi-linear differential equation for a system with an external excitation has the form mx + kx = J.l/(t, x, i) [93.1 ]

93.

in which the time t appears explicitly. We shall consider only the case where I(t,x,i) can be written in the form N

I(t,x,x)

=

lo(x,x) + I[/"I(x,x)cos)'"t + 1"2(x,x)sin)',,t]

[93.2]

,,:0

where 10 ,

I nl , and In2 are certain polynomials in

x and

X.

In the terminology of electric-circuit theory, the motion represented by Equation [93.2] may be considered a current produced by an electromotive force e = p/( t,x,x) applied to a linear impedance Z(jwo) = mjwo +~. JWo It was mentioned in Chapter XIII that in electrical problems x corresponds to the charge in the capacitor and i to the current. When p = 0, the system [93.1] becomes a linear one whose solution is

x=

awocos (wot

+

~)

*

where Wo = Vk!m and ~ are two constants of integration. When J.l 0 but suffiCiently small, the expressions [93.3] appear as generating solutions (compare with the method of Poincare in Chapter VIII) with which we start the approximation. One may consider these expressions a8 an approximation of zero order. Our purpose will be to establish an approximation of the lirst order which will characterize the quasi-linear system with a degree of accuracy of the order of p2.

290

The non-linear term becomes e = .ul[t, a sin (wot

+ 9S), awocos(wot + 9S)]

[93.4 ]

Since the functions 10 , 1,,1' and 1,,2 are polynomials, their Fourier expansions are of the form N'

fo(asin~, awocosl/l) = I[Uk(a)coskl/l

+ hk(a)sinkl/l]

k~O

N'

I"l(a sin 1/1, awo oosl/l) = I[U"l,k(a) oos kl/l + hIl1,k(ahin kl/l]

[93.5 ]

k~O

I n2 (asinl/l, awooosl/l)

N' = I[U,,2,k(a)coskl/l k~O

+ hIl2 ,k(a) sinkl/l]

Using these expressions and also the expansion [93.2J, one can write the nonlinear term [93.4] as N'

e = .u .L'[g.(a)oosk(wot

+ ~) + hk(a)sink(wot +

~)]

+

k=O

N

N'

+ ~I I[U"I,k (a) - h"2,k(a)] cos [(kwo + ",,)t + k9S] + n=Ok"O

[93.6] It is noted that the frequencies kwo + "" and kwo - "" appearing in these expressions are cO?n6ination Irequencies like those we have previously encountered. We shall limit our study in this section to systems in which none of the combination frequencies approaches or is equal to the frequency WOo In other words [kwo + ",.]

'* Wo

and

[kwo -

"tI]

* Wo

As before, we call this case the non-resonant case. From the form of Expression [93.6J one ascertains that the only harmonic having the frequency Wo 1s the harmonic [93.7]

291

Using this expression together with the second equation [93.3], one has

[93.8J Hence, to the first order, the non-linear element of the system can be re-, placed by the equivalent linear one so that the symbolic equation of the quasi-linear system becomes

[93.9 )

Z - Ze = 0

Written explicitly, this equation is mp

+ !:.. = Lawo [gl(a) p

-

[93.10 J

hl(a)j]

where p = -6 + jw, 6 being the decrement and w the frequency of the equivalent linear system. Substitut1ng this value of p into Equation [93.10], one has

[93." ] whence, to the first order, one obtains 6

= -

P

2mw oa

g (a)' I

'

w = Wo - 2

P

mwoa

hi (a)

[93.12 J

By Bubstituting these values into the equations of the first approximation da = -

dt

6a

and

d¢> = w dt

w

0

one obtains d¢> = dt

By the introduction of the total

pha8e ~

[93. , 3 J

= wot + ¢>, these equations become [93.14J

It is thus seen that the solution to the first approximation is still of the form [93. , 5 J x = a sin~ where a and ~are given by Equations [93.14]. By analogy with the definition of the linear decrement, it is logical to introduce now a quantity X, the equivalent decrement, defined by the equation -A

= -

-P g(a)

awo

I

[93. ,6]

292

Since gl(a) is the first coefficient of the Fourier expansion, its explicit value is [93. , 7]

whence

I

= - -IJ-[fo(aSin",.awocos"')COS",d'" awo7l'

[93.18]

Following the procedure explained in Section 75, one obtains

l"fo (a sin "', awo cos"') sin 1/1 dl/l

k; = - L71'a

[93.19]

,

and to the first order w = k ~ k" • The equations of the first approximation then acquire the familiar form 2

da

([i = -

X

dl/l _ ... _

2m a;

dt -..., -

Vk

+

m

k;

[93.20 ]

If one differentiates the solution ~ = a sin'" twice, takes into account Equations [93.20], and substitutes z and %into Equation [93.1], one finds that the solution z ~ a sin", satisfies the linearized equation [93.21 ]

to within a factor of the order of #2. It should be noted that in the equation of the first approximation the dependence of the "forcing function" on time does not appear explicitly. This is due to the fact that, for the formation of these equations, only the term lo(z,z) of Expansion [93.2] has been retained. This term is expressed by the equation

fo(~' x)

= lim t.ao

~

T

f I(T,

Z,

x) dT

0

In other words, the first approximation deals with the average value of the forcing function with respect to time, and the instantaneous behavior of that function is felt only in approximations of higher orders. The rest of the discussion is centered about the linear equation [93.21]. Thus, for example, the stationary state is reached when ,,(ao) = -

~

aW07T

2r

[10 (a sin "'. awo cos (b) cos 1/1 d", = 0

[93.22]

This corresponds to Equation [46.22] of the theory of Poincar6 or to the first of the "abbreviated equations ll [52.9J of the theory of Van der Pol. The stationary state is stable if (~,,) > 0 and unstable if ( 8" ) < O. a 0 = 00 _ 8a 0 =00 Self-excitation may develop from rest if ,,(0) < 0; 1f, however, X( 0) > 0, no self-excitat10n take's place.

293

94.

FIRST-ORDER SOLUTION OF VAN DER POL'S EQUATION WITH FORCING TERM

It has been shown that the operat10n of a self-excited therm10nic generator with inductive coup11ng can be expressed by an equat10n of the Van der Pol type, that is, If an alternat1ng electromot1ve force E= Eo sin at of constant frequency 1s provided in the grid c1rcu1t shown 1n Figure 94.1, the differential equation [94.1] acquires a "forcing term" and becomes

ii +

y -

p{1 -

y2)y =

Eosinat

If we introduce a new variable ~, def1ned by the equat10n y + U sin at, where U = 1 ~oa2' Equation [94.2] becomes

[94.2]

=~

+

ln this equation f(t,x,x) =

[1 -

(x + Usinat)2][x + Uacosat]

[94.4 ]

Developing the right side of th1s equation and collecting terms not depending explicitly on t, one has fo(x,x) =

(1 -

x2

-

~2) x

[94.5]

If we let m= 1 and k = 1 1n Equat10n [93.1], then we can take as generating solutions ~ = a sin~ and x = a cos ~, since ~o = 1. Substituting these solutions into [94.5] and carrying out the integration indicated by [93.18], one obtains

Figure 94.1

~2)

[94.6)

The equation of the first approximation therefore becomes da = + 1!... dt 2

(1 _

~ 4

_

U 2

2 )

a

This equation shows that for U2 < 2 there exists a triv1al solut10n a = 0 which, however, is unstable. In fact, for a very small initial departure, the quantity in parentheses is positive, which 1ndicates that the amplitude begins to increase. The stationary amplitude at is reached when

294

[94.8) If U2 > 2, there 1s no self-excltatlon, and the trlvlal solutlon a stable. The value U

2

=0

ls

E2

° 2 )2

= (1 - (

= 2

ls thus a crltlcal threshold separating two zones: ln one zone, where U2 the system becomes self-exclted; ln the other, where U2 > 2, the external perlodlc excitatlon wlth frequency a prevents self-excitation.

< 2,

95.' IMPROVED FIRST APPROXIMATION FOR A NON-RESONANT EXTERNAL EXCITATION OF A QUASI-LINEAR SYSTEM Equatlon [93.6] gives the expression for a non-Ilnear force ln terms of the Fourier coefficlents for dlfferent combinatlon harmonics. Using the terminology of electrlc-circult theory, one can say that, lf an electromotlve force e ~ eo sin (Dt + ~) is impressed on a 11near impedance Z = mjw + -JL, the steady-state current due to this forclng term will be )w

eo

+ -

mJo.o

+ ~) =

k sin(Dt

jD

Llkewise, for e

= eo

e

+

mJU

2

sin(.Qt

+ ~)

x

°

jD

.1. +

a SID Y'

Jlgo

k

:0.0

2

cos(Dt

+~)

m(wo - D )

- e0 JOD 2 2 cos(Dt +,p) = m(Q - wo )

One obtains the follow1ng equat10ns for [93.6] by the linear impedance of the system:

=

==

cos (Dt + ~), the forced oscillatlon ls

--....lO'---:'k-cos(Dt +~) = on

e:jQ

m(wo - D )

~

eD 0

m(Q2 -

2

wo )

sin(Dt +~)

by divlding Equat10n

N

+ 1!. ~ r ' glt cos kIP + hit sin kIP + (1 _ k2) ,2 m k=2

wo

[95.1 ] A similar expresslon can be written for~. From Express10n [95.1] 1t follows that resonance occurs whenever one of the d1visors becomes e1ther zero or a small quant1ty of the first order. Under such cirCUMstances the

295 smallness of the numerator due to the factor p is offset by the smallness of the denominator, so that the resulting term remains finite. It can also be seen from Expression [95.1] that the dependent variable z(t) consists of three kinds of oscillations, namely: 1.

Free or autoperiodic oscillation, a sin rp

= a sin

[w(a)t + ~].

2. Forced or heteroperiodic oscillation with frequencies Yft given by the terms in which k = o.

3. The spectrum of combination oscillations with frequencies kw(a) t Y•. One could proceed with building approximations of higher orders in which the divisors would be of the form wo2 - (kwo + hlAl + ... + hftAft)2 and apply the preceding argument. However, the formation of these approximations is complicated and adds nothing new to the qualitative aspects contained in the improved first approximation [95.1]. In the following discussion, we prefer to use the terms "autoperiodic II and IIheteroperiodic ll instead of II free II and "forced,1I for reasons which will appear later.

96. HETEROPERIODIC AND AUTO PERIODIC STATES OF NON-LINEAR SYSTEMS; ASYNCHRONOUS EXCITATION AND QUENCHING We shall now consider Equation [95.1] when the autoperiodic oscillation is absent, that is, when a = O. In such a case gk g"l,k = 0;

g,,2,k

=

== 0;

hk

=

0;

k

== I, 2, . . . k

h"l,k = 0;

= 1,2, ...

This follows from the fact that the numbers gk' ••• for k > 0, which we have Just asserted to be zero, are merely the non-constant terms in the Fourier expansions of [93.5]. Now for a = 0 all the expressions on the left side of these equations reduce to their constant terms. This same fact is true of their Fourier expansions, and hence all coefficients of the cosine and sine terms in these expansions are zero. It we put g"l,O = A" and h ft2 ,o = B", Expression [95.1] reduces to _

J.l ~ A" cos ""t + B" sin ""t ~ 2 2 m,,=o Wo - Yft

z - -

+

p

-k go

(0 0)

'

[96.1]

Hence, for a = 0, only heteroperiodic oscillations exist with externally applied frequencies "1' Y2••• , • "N' In Section 94 it was shown that for U2 = (

the

he~eroperiodic

Eo 2 a

1 -

)2 >

2

state is the only one possible.

296

Consider the following differential equation encountered in acoustics:

[96.2] wherem, AI' and k are positive constants. Moreover. Al and 61 are small. so that we can introduce a small parameter # by writing

=

AI

f.lA;

61

=

[96.3]

f.l6

In order to make Equation 196.2] of the same form as [93.1], one can introduce a new variable x defined by the equation

[96.4 ] This change of the variable gives mx

+

kx = f.l{[-

Ai -

AElalcosalt 2

AE2a2cos(a2t

m (w; - a 1 )

+

2

m (w; - a 2

p>] -

)

[96.5] Here X = f.lA > O. Hence. by Equation [93.20), the autoperiodic oscillation dies out and only the heteroperiodic state is possible. By Equation [96.1] one obtains the following expression for the heteroperiodic oscillation: f.lAE2a2cos(agt + P) m2(w~ - a 22 )2

+

f.lIJE'.!.'.!. 2k[m(wo2 - aiW

f.lIJEI2 cos 2a l t 2m[m(w; - a l 2W(wi - 4a 12)

f.l 6E I E 2 cos[(a 2

+

3

m (w; -

2 a I )(w;

-

[96.6J

-

al)t

a 22 )[w;

+ pJ

- (a 2

-

+

IlIJE.,'.!. cos 2(a.,t

f.lIJEIE2cos[(a2 al]

+

3

m (w; -

2

2

a l )(wo -

+

p)

a 22 )]2(wo2 - 4a:n

2m[m(w:2 -

+

+ al)t + p]

ai)[wo2 -

(a 2

+

al)?]

It is seen that the heteroperiodic oscillation consists of harmonics of the fundamental frequencies al and a 2 and also of the combination frequencies 2aJl 2a 2 , a l + a 2 , and a l - a 2 • The frequency zero (the third and the fourth terms) also appears in the spectrum. As another, more complicated example in which both heteroperiodic and autoper1od1c osc11lations appear, we shall consider an electron-tube oscillator acted upon by an extraneous voltage of frequency a in addition to the feed-back voltage e. The anode current 1s ia

=

f(Eo

+ F cosat +

e)

297

Let the frequency of the oscillating circuit be tion, the preceding expression becomes

WOo

In the first approxima-

ia = f[Eo + Fcosat + acos(wot + tP)]

[96.8J

where a and ~ are the amplitude and phase of the autoperiodic OSCillation of the voltage applied to the grid. In order to apply the method of equivalent linearization, the nonlinear conductor ia = f(e g ) must be replaced by a linear one ia = Seg, where the equivalent parameter S must be so chosen that the fundamental harmonic of (96.81 is equal to the harmonic Sa cos (wot + t,6). By the Principle of Harmonic Balance, Section 77,

It was shown in Chapter XIII that in the linearized scheme in which the nonlinear conductor is replaced by an equivalent linear one, [96.101

where Wo is the frequency and 40 is the decrement of the linear circuit closed on the non-linear conductor linearized by the transconductance S. The quantity SI is the critical transconductance, that is, a particular value of the transconductance S for which the decrement 4 vanishes and the oscillation becomes stationary. It is apparent that, if the external excitation F cos at is absent, the value of S is somewhat different from its value in Equation [96.91. In fact, for an autoperiodic excitation the equivalent transconductance is given by Equation [83.7J, viz., 1 S(a) = -

11'a

2,.

jf(Eo

+

a cos tP) costP dt,6

In order to use this equation, it is sufficient to replace Eo by Eo + F cos at and to average it again over the period 211'. If the autoperiodic frequency Wo is high (for example, radio frequency) and the heteroperiodic frequency a is low (for example, audio frequency), the equivalent transconductance S(a) given by Equation [83.71 is a slowly varying function with frequency a. If the value of S(a) oscillates with frequency a in the neighborhood of the critical value of self-excitation (autoperiodic frequency) and ,the amplitude Fof the heteroperiodic frequency is sufficiently large to pass out of the zone of self-excitation, it is apparent that the appearance and disappearance of the

298 autoperlodlc state wlll also be perlodlc wlth the perlod TH = ~ of the heteroperlod1c oscillat10n. The reader will recognize the character1stic behavlor of the so-called superregenerat1ve circuit. If, however, the heteroper10dic frequency is considerably h1gher than the autoperiodic one, its effect will be felt in the virtual modificat1on of the critical transconductance of the electron tube. Depending on the form of the characteristic, this effect will sometimes manifest itself In the appearance of an autoperlodlc self-excltatlon and sometlmes In the extinctlon of an existing autoperiodic oscillation. These phenomena are sometimes referred to as asynchronous e%citation or asynchronous quenching of an autoperiodlc oscillation by a heteroperiodic one (5). The conditions for such asynchronous action are easily established by following the procedure indicated by Equation [96.9] in which one integration, say dT1 , Is carrled out with respect to the heteroperiodic period and the other dT2 with respect to the autoperiodlc one. As an example (5), consider an electron-tube oscillator with a nonlinear characterlstic glven by the polynomial

[96.11 ] It has been shown that this expression approximates sufficiently well both the soft and the hard characteristlcs. For a soft characteristic, It is sufficlent to put 4 = E = 0, whereas for a hard one, the full polynomlal should be used. Assume that the grid voltage % Is of the form %

=

acosl,6

+

[96.12]

6cosI/J

where a and 1,6 are the autoperiodic variables and band I/J the heteroperiodic ones. Replacing % in Expression [96.11] by its value [96.12] and carrylng out the integrations indicated by [96.9], vlz., 1 8(a,6) = 2a '1r2

r

[fCaocosrp

+

6 cosl/J) cos 1,6 dl,6dl/J

[96.13 ]

one obtains the following expression 8(a,uI.) = a

3 2 3 62 + + 4""a +"2"

5

SEa

4

+ "4 15 Ea 262 + 15 1.4 SEu

I

If the heteroperlodic excitation Is absent, that is, if b 8(a,0) = a

3

5

+ "4 "a 2 + "8 Ea 4

[96.14]

= 0, [96.15 ]

which is the expression for the transconductance 8 of an autoperiodic state. More speCifically, if t~e characterlstlc is soft, that Is, If y < 0 and , = E = 0, the stationary condition is

[96.16 ]

299 whlch glves [96.17 ]

whlch was obtained prevlously by the method of Polncare, see Sectlon 54.* The condltion for soft self-excitation in the absence of a heteroperiodlc frequency ls

> S(a,O)

S(O,O) = a

[96.18 ]

If, however, the heteroperlodlc frequency ls present, that is, 6 lnltlal value of the transconductance is 3

+ 2" ,,6 2

S(O,6) = a

* 0,

the

[96.19 ]

Slnce " < 0, S(0,6) < S(O,O), whlch means that the presence of the heteroperiodic frequency may prevent the occurrence of self-excitation. Therefore one concludes that, for a normally soft characteristic, asynchronous quenchlng of the autoperiodic frequency by the heteroperiodic one occurs, a fact which was mentloned at the end of Section 94. Por a hard characteristic (E < 0 and a, p, ", and 6 > 0) in the absence of the heteroperiodic excitation, the transconductance is 3

) = a S( a,O

+ '4"a

2

-

5

glE 14 a

[96.20]

This expression considered as a functlon of a 2 is a maximum when 2

a

=

3"

51EI

The maximum value of S(a,O) is then Smax(a,O ) = a

One can obtain asynchronous self-excltation (6 S (0 ,0 ) - Smax(a,O )

=

3

"2,,6

2

{96 .21 ]

9 "j;f " + 40

+

15

* 0)

SEb

4

+

if 9,,2

40 -;-

>0

[96.22]

that is, i f [96.23]

This inequality can be fulfilled lf the quantity 6 2 lies in the interval (96 .24)

*

It should be observed that tbe notation here differs frOll that of Section 54.

300

The subject of asynchronous action on a self-excited system can be summarized as follows: a. If the characteristic is 80ft, the heteroperiodic frequency may result only in quenching the autoperiodic oscillation and not in causing its appearance. b. If the characteristic is hard, the heteroperiodic frequency may cause self-excitation of the autoperiodic oscillation provided the amplitude of the heteroperiodic frequency lies in the interval indicated by the inequalities [96.24]. c. In all cases, the heteroperiodic actIon manifests itself in a virtual modification of the transconductance by which the conditions of self-excitation are influenced one way or the other according to the form of the non-linear characteristic.

CHAPTER XVI NON-LINEAR EXTERNAL RESONANCE

97.

EQUATIONS OP THE PIRST APPROXIMATION POR AN EXTERNALLY EXCITED RESONANT SYSTEM

We shall now investigate the conditions under which certain harmonics in the combination-frequency spectrum become large compared to others when a quasi-linear dynamical system is excited by an external heteroperiodic frequency. We shall call this ezternal resonance in contrast to internal resonance, which characterized a similar system without an external excitation. For simplicity. we confine our attention to systems having one degree of freedom and fractional-order resonance. In such systems r

Wo

= "8 a

[97 . 1 ]

where rand s are relatively prime. Unless otherwise stated we also assume that s > 1, because s = 1 corresponds essentially to ordinary resonance. In the "neighborhood" of resonance W

=

r

"8 a + IlQ

[97.2 ]

where p is small. Let us consider an electron-tube oscillator whose non-linear element. the electron tube, has a characteristic ia = /(e). Let the autoperiodic oscillation be e = a cos (w o t

+ rp)

= a cos

(f a t + rp)

The anode current ia is given in terms of the grid voltage by the expression [97.3 J

In this expression the quantity a is the amplitude of the auto periodic oscillation. is its frequency, and F and are the corresponding quantities for the heteroperiodic oscillation introduced in the grid circuit, for instance, through an inductive coupling. We begin by linearizing the non-linear element of the system by writing [97.4 ]

;a

a

where S is the equivalent transconductance, a function of the amplitude a of the autoperiodic oscillation. For simplicity we will write S instead of S(a). It is recalled, see Section 77, that, according to the Principle of Harmonic Balance, the fundamental harmonic of the non-linear periodiC quantity

[97.3] is equal to the linearized oscillation [97. 4 ]. By a Pourier series we obtain the following expression for this harmonic:

q,) sin(; at + q,}]dT

+ sin(faT +

[97.5 ]

This expression can be written in the form S,. a cos

(f at

+

q,) -

Si a sin (; at

+

q,)

where 2r

1 S,. = S,. (a,q,) = 7r a

ff [Eo + F

cos (8T

-

;

q,) + a cos TT]COS TT dT

o

[97.7]

Equation [97.6] can be written as Sacos(; at

+ ~)= (S,. +iSi)acos(; at+

q,)

[97.8 ]

It is seen that in a resonant system the transconductance S is a complex quantity

[97.91 whereas in a non-resonant system it is real. Moreover, the formal procedure remains the same as in Section 89, the only difference being that instead of impedances we are now dealing with admittances since transconductance is an admittance. Using Equation [96.10] for the fundamental harmonic [97.8] and separating the real and the imaginary parts, one gets

[97.10] where So, as before, designates the critical value of the transconductance S. It we put =at + ~ = VI, the equations of the first approximation become da = dt

-6 (I _ So' Sr)a. 0

dq,

Sj

r

-=w--a+6dt

0

8

0

So

[97.11 J

The variables in these equations cannot be separated since both S,. and Si are now functions of a and~. On the other hand, since these equations are of the type investigated by Poincare, we can assert that in the (a,~)-plane the only stationary motions are either positions of equilibrium, that is, singular pOints of the system [97.11]. or stationary motions of the limit-cycle type.

The f1rst case is characterized by the approach of a and tain fixed values a o and ~o when t .. ~. Thus we have

+

w _2:. a o

8

~

to cer-

1 Sda o,(,6o) == w -!.a == 0 [97.12] 0 So 8

fa;

The oscillation ultimately reaches a frequency w which is exactly equal to this means that the frequencies w and a become "locked" in a certain rational relation. We may call this ,ynchronization of the autoperiodic oscillation with the heteroperiodic one. Thus there exists one single frequency, and the stationary oscillation is aocos(wt

+

aooos(; at

t,6)=

+ (,6)

In the second case both a(t) and _(t) are periodic and have the same period as the autoperiodic solution so that the ultimate oscillation consists of the two frequencies. In other words, there are "beatsn of autoperiodic and heteroperiodic frequencies. We shall consider this subject in greater detail in Chapter XVIII. It is possible to show* that, when r"~ and, .. 00, Si" 0 and S,. approaches the value given by Equation (96.9], so that for large values of r and, the resonant case degenerates into the non-resonant one, which has already been investigated. Hence, the typical features of fractional-order resonance appear when rand, are relatively small integers. Let us consider the following example. We assume that the function !(Eo + u) can be approximated by a polynomial of the th1rd degree: where 8 1 , 8 2 , and Sa are positive. Assume further that r = , and , a 2, which gives "'0 = a/2. Placing u = Fcosat + a cos (1at+~) in [97.13], we obtain S,. = 8 1

+ 2"3 saF 2 + '21 S2Fcos 2t,6

3 2 - 4Saa ;

Equations [97.11] become da dI

~( = So S1- So

-'23

Sa F

2

dt,6 a d t = Wo - 2"

1 ) 3 10 Sa 3 + iS2Foos2t,6 a - 4 So a

-

'OS2 F . 2.1.. 2 So sm."

(97.15]

The second equation (97.15] admits separation of variables and can be integrated. Putting

*

The proof or this propos1t.1on oan be round in the Krylorr-Bogol1uborr text "Introduotion to HonLiDear Meobaaios,· Reference (1), pages 270 and 271.

= /w o - ~ / and one has two cases: Case 1, .1w < A and Case 2, .1w

.1w > A. In Case 1 it is apparent that cP(t) t _00 + cPo' where cPo is given by

the expression 2 .J. ¥'o

= sm. -l( 2S o.1W) 6S F o

{97 .16]

2

One can select for 2cPo the principal determination, that is, sin 2cPo > 0; cos 290 > O. Let us consider the asymptotic behavior of the solution of the first equation [97.15] as cP + cPo' If, for brevity, we put

~(Sl

- So - : S3F2

+ ~ S2Fcos 2 CPo)

= A;

!~S 4 So

=

B

3

the first equation [97.15] is da 3 - = Aa - Ba dt

[97.17]

When A< 0, and since a is always positive, the derivative da/dt 1s always negative and hence a cannot tend to any limit other than zero. Thus a = 0 is the only stable stationary amplitude. If, however, A > 0, one finds that the point a = 0 is unstable. The stationary autoperiodic oscillation, which can be shown to be stable, see Section 65, is then

[97.181 Since 9+ 90 when t +~, we conclude that there will be a synchronous autoper10dic oscillation. In Case 2, when .1W>A, the second equation [97.15] gives _ _d.;,.:cp-:-~ .1w sin 2cp

A

=

dcp

.1w

(1 _ .1AW sin2CP)

= dcp (1 + ~sin2cp) .1w

= dt [97 .19]

.1w

Integrating this expression, one obtains cP - CPo -

2~W

cos

2cP = .1wt

that is, cP

= .1wt

where rPo is an arbitrary constant. written as

A

+ rPo + 2.1w

cos 2rP

In view of the smallness of A this can be

[97.20 ] Substituting this value of 9 into the first equation [97.15], one obtains a differential equation with periodic coefficients.

It is noted that this equation admits a trivial solution a = 0, which expresses the condition of a heteroperiodic state, The stability of this solution depends on the sign of the expression 3 2 I -SI - So - i SaF + iS2Fcos2¢1

where cos 2116 is the average of cos 2116 per period, that is, I

cos 21/1 = T

f

T

cos 2116 dt =

f

dt cos 2116 d¢l d¢l

2..

=

1

'2

T

Jcos 2116 dt o

J"L1w _cosA2116sin 2116

1 I" dO - k sin ¢I) 2L1wk20 1 - k 2 sin ¢I = 2

= -

thus cos

2~

= 0,

[97.21 ]

d(2¢1)

I"

I =0 2 2L1wk2 logO - k u) ,,= 0 = 0

and the preceding expression becomes

sI - s0 _!s 2 3 F2

= C

[97.22]

If C < 0, the first equation [97.13] shows that the heteroperiodlc state a = 0 is stable; if, however, C > 0, the heteroperlodlc state is unstable and autoperiodic excitation sets in.

98, FRACTIONAL-ORDER RESONANCE We shall consider Equations [97.11] again in the more general case of fractional-order resonance. It was shown that a trivial solution a = 0 exists which characterizes the heteroperiodic state. We will now investigate the stability of the solution a = 0; for that purpose we develop the integrands appearing in the functions Sr and Si in terms of the small quantity a, around the point Eo + F cos (BT - t¢l) in Equations [97.7]. This gives the first terms of Taylor's expansion for these functions, namely:

!

S .. =

[fa [Eo + F cos (ST

;¢I)J

-

2 COS TT

dr

(98.1 ]

With

T

=t

+~, these expressions reduce to S,. =

2~[fa[Eo

Si = -

211' 1

+

FcosBt][I

+

cos(2rt

J"fa [Eo + F cos Bt] sin (2rt

+

+

2¢1)]dt

2116) dt

;06

because of the periodicity of the integrands.

Since

[la(Eo + FcosBt)sin2rt dt = 0

these relations reduce to Sr =

2~[la[Eo + FcosBt][l

+

cos2rtcos2~] dt

2~ [fa [Eo + FcosBt]cos2rt sin2cp dt

S. = -

If we set

2~ [fa[Eo + FCOSBt] dt

= No

[98.2]

and

2~ {fa[Eo + FcosBt]cos2rt dt =

NI

Equations [98.'] become Sr = No

+

Nt cos2cp

Si = - Nt 8in2~ With these values for Sr and Si' Equations (97."] become

da = 6 (~ _ 1) a dt 0 So dcp = dt

+ 60NI a cos2cp

(w0 _ .!.B a) -

So

[98.4 ] 60 N t

So

sin 2cp

The nature of the solutions of these equations establishes the conditions for the stability or instability of the autoperiodic oscillation. If the only stable autoperiodic solution is = 0, only heteroperiodic oscillations are poss1ble. Equat10ns (98.4) w1th periodic coefficients can be reduced to a system with constant coefficients by introducing the new variables u = a cos cp and v = a sin~. With these new variables we have

a

du

dt

=

da

dt cos cp

. dcp - a SlnCP dt

and

dv da. (it = dt 8lnCP

dcp

+ a cos~ dt

Substituting in these equations ~~ and ~ from Equations [98.4] and rearranging, one gets du = 60 (No So + Nl - 1) u - (Wo (it dv = (Wo Cit

-

-

ra) u + 6 (N.0 So - NI 0

-;

8'ra) v -

= Au - Bv

1 ) v = Bu

+ Cv

307 These equations have non-trivial solutions if (- p

+ A)

- B (- p + C)

B

=

0

[98.6]

Hence the solution a = 0 is stable if the roots Pl and P2 of [98.6] have negative real parts. Written explicitly, these roots are

According to whether the quantity under the radical is positive or negative, the roots will be real or conjugate complex. Replacing Nl by its value [98.2] gives

I. ., - : "I > 2:~, 1['.(E, + Fcom) ,os2rT dTi

[98.8]

I. ., - : "I < 2:~, 1['.(E, + Fcom) oos2rT dTi In Equation [98.8], the roots are conjugate complex, and self-excitation is possible if No > So' that is, if

2~ [io(Eo +

FCOST) dT

> So

[98.10)

In Equation (98.9], the roots are real, and self-excitation is possible when at least one root is positive.

One finds that this condition is

[98.11 ] with No

> So'

If one takes a still stronger inequality by dropping the term (No - SO)2 in (98.11], one obtains a sufficient condition for self-excitation:

[98.12 ] This means that the autoperiodic state always sets in when the external frequencya lies in the interval defined by [98.12], which requires that Nl =

2~ [

fa (Eo

+

F cos aT) cos 2rT dT :I: 0

This condition is fulfilled only when 2r/a

= k,

[98.13 )

where k is an integer.

Since

systems where rls is an integer have been el1m1nated because they do not possess subharmonic solutions, one can assert that the condition for the existence of self-excitation of the autoperiodic state excited by the extraneous frequency a is 29

r

Ii where

q

=

+

1

2

[98.14]

= 0, 1, 2, •••.

If the amplitude F of the externally applied force is very small, one can further simplify the expression [98.13] and write N = F fgF(E o) I

21T

[cos ST cos 2rT dT

[98.15]

where faF(E o) designates the derivative of fa with respect to F taken at the point Eo. From this expression it follows that NI = 0 for s =1= 2r [98.,6 ] NI = FIg~(Eo) for S = 2r which means that in the zone of self-excitation defined by (98.12] only fundamental fractional resonance of the order one-half can exist, in which case a w=o 2

[98.17]

This fact was noted by Lord Rayleigh in his experiments with oscillating systems. He observed that, if one of the parameters (L,m) or (lIC,k), in the notation of Chapter XIII, oscillates with a frequency twice as large as the frequency of the system, the system will oscillate with half the frequency of the parameter.

99.

PARAMETRIC EXCITATION Parametric excitation of a system is defined as the condition of self-excitation caused by a periodic variation of a parameter of the system. Although this subject is discussed in Chapter XIX from a different point of View, it is preferable to give an outline of the phenomenon here in order not to interrupt the argument of Kryloff and Bogoliuboft which we are following. Let us consider the circuit shown in Figure 99.1. It consists of a very small reSistance R, a constant inductance L, a non-linear inductance L1(i) containing a saturated iron core, and a variable capacitor C arranged to produce a fluctuating capacity with frequency a around its average value Co according to the law

c=

CoCl

+ p sin at)

where peo is the amplitude of the fluctuating capacity and p « 1. If we assume that R is very small and that Ll (i) « L, the problem is clearly within the scope of the quasilinear theory. Let us also assume that the circuit is tuned so that

c L,(i)

L

R

Wo

=

1

a YLC ::::: 2

[] 99 . 1

o

Figure 99.1 This condition will result in fractionalorder resonance of the order one-half, as was Just shown, and the steady-state current will be of the form i = a sin (~ t

[99.2 ]

+ 9S)

In this problem we have, in addition to the constant parameters Rand L, one perIodically varying parameter C = Co(l + p sin at) and one non-linear parameter L 1 (i). We can apply the Principle of Equivalent Linearization to the non-linear parameter Ll and write

It is recalled that the coefficient of inductance L 1s defined in all cases by the relation ~ = Li, where ~ is the flux linkages of the coil carrying the current i. When the coil is wound on an iron core, the function ~(i) has the appearance shown in Figure 99.2, when Figure 99·2 we neglect the effect of hysteresis and the inflection point near the origin in order to simplify the argument. It is apparent from the above definition that the coefficient of inductance L is a monotonically decreasing function of i in the presence of magnetic saturation. Approximating the function ~(i) by the expression ~(i) = ~0(1 - e- Ai ), one has the following expression for L(i): L(i) =

2

3

2

6

f/J,(A - ~i + ~i2 o

_ ... )

[99.41

It 1s seen from this expression that dL(i)

(ii

A3

= ~o( - "2 + 3" - ... ')..2

•

)

is negative. The argument is obviously valid for any point around which the expansion is made. It is also apparent that the term ~oA = Lo represents the

310

constant coefficient of inductance of the coil without an iron core; the re>..2· >..8·2 maining terms r/JoT; r/Jo-i-; ••• are variable terms resulting from the nonlineari ty of the function r/J ( i) • The variable parameter C = Co (' + p sin at) accounts for the electromotive force 1

e = - Co(l

psinat) [Idt = -

+

a cos(it =

Co ; (1

t

t•

Co(1 + ~Sinat)[aSin(~ t + r/J)dt

+ r/J)

[99.5]

+ psinat)

By the PrinCiple of Harmonic Balance, Section 77, the fundamental harmonic of this expression must be equal to the voltage drop of the current i, given by [99.2], across the impedance Za = r~ + jz., where Z, is the equivalent impedance of the circuit. Since the Pourier coefficients of the first harmonic at cos (~t + r/J) + hI sin t + r/J) of the periodic function [99. 4 ] are

(1

one finds, after a few elementary transformations, the following values for and z,:

r~

re = _1_ rSin(O + r/J) cos(8 + r/J) dO = _ -Lcos2r/J 7r!! 2

1

~e = 7ra

Co(l

r

+ psin20)

aCo

[99.6] 2 008

Co(l

(0 + r/J) + psin20)

2 [ dO = aCo 1

1

.

]

+ '2 psID2 r/J

2

It is thus seen that a variable capacity results in the appearance of a variable impedance Z, characterized by components re and Ze given by [99.6]. The circuit acts as if it had an equivalent capacity C = t:

Co :.::. C (1 1+.e. sin2r/J 0 2

-

!!... sin2 r/J) 2

[99.7]

We have seen that the equivalent system consists of an inductance L + L t , a resistance R - a~o cos 2t/J, and a capacity Co(1 + ~ sin 2r/J). The equivalent parameters thus appear as functions of the phase angle t/J. The decrement and frequency of the equivalent system are

311

=

If

R - - P cos2,p aCo

.::::r

..!!.. 2L

pwo cos 2,p 4

[99.8 ] w=

V(L

1

+

.::::r

L 1)C

W [1 _ L. (a) 0 2L

+ .!!... sin 2,p] 4

whence the equations of the first approximation are da (~ R) (it = 4 cos2,p - 2L a;

d,p Cit

(

=

Wo -

L.(a) 2L Wo -

(1)

2" +

WoP sin 2,p 4

[99.9]

Expanding the function ~Wo, which appears in the second equation, in a Taylor1s series around the value a = 0, we get d,p

dt

(W _ L 1 (0) W - ~) 0 2L 0 2

=

If we let

wp T

= m;

+

woP sin 2,p - ~ L'(O) - .•• 4 2L 1

R

2L = n;

Equations [99.9] become

-da - n ) a' dt = (mcos2...l ~,

~f

= p

+

msin2,p -

2~

Let us examine the behavior of the system when a points occur when m sin 2,p

+ p

L;(O)

= O.

= 0

(99.10]

The singular [99.11]

Let 2,po be one of the roots, which we will assume to be in the first quadrant (sin 2,po > 0; cos 2,po > 0). Applying the Poincare-Liapounoff criteria of stability and designating the small perturbation of the amplitude by e and the perturbation in the phase angle around ,po by ~, we have the following variational equations

~!

= (mcos2,po-n)e

d~ = ( - L~L '(0») dt e + (2m cos 2,po) ~ - 2m cos 2,po . ,po

[99.12 ]

The last constant term in the second equation clearly does not have any effect on stability and amounts to a shift of the origin in the (~,~)-plane. The characteristic equation of the system [99.12] is then S2 -

(3mcos2,po - n)S

+

2mcos2,po(mcos2,po - n) = 0

[99·13 ]

For self-excitation it is necessary that the free term as well as the coefficient of S be positive. In this case the Singularity is either an unstable nodal point or an unstable focal point. Since we have assumed that 2,po is

312

in the first quadrant, these conditions require that cos 2c;6o cos 2c;6o

n

>

m

>

n 3m

(99.14 )

The first inequality is stronger than the second and should be used. other hand, for real values of the argument, one must have

..!!. m

=

~ <1 LwoP

On the

[99.15 )

This merely imposes the additional condition that the index of modulation p should be below a certain critical value given by (99.15). From the first . inequality [99.14) and the condition [99.1'] we have p2

= m 2 sin 2 2c;6o = m2 (1

- cos 2 2c;6o>

that is, whence p2

+ n 2 < m2

Substituting the values ofm, n, and p, we obtain the condition of selfexcitation which was derived by Kryloff and Bogoliuboff:

[99. ,6 J Self-excitation does not occur if the sign of this inequality is reversed. The stationary condition is obtained when da/dt = dc;6/dt = 0 in Equations [99.9]. If we deSignate the stationary amplitude by a 1 and the phase angle by c;61' we get 2R cos2c;61 = - -

LwoP

From the second equation [99.9] we have

LI(a l )

=

L(2 - ~ + P sin2c;61) 2 wo

[99.17]

This equation gives the amplitude a l of the stationary oscillation since 2c;61 is known and since the function LI(a) is given by Equation (99.3). For the stability of a stationary state one can again apply the Poincare-Liapounoff criteria, see Chapter III, to the differential equations [99.9] by expanding the functions cos 2c;6, sin 2c;6, and L1(a) in a Taylor series around fixed values c;61 and al and by introducing the perturbation variables t and 71 given by equations c;6 = c;61 + t and a = a 1 + 71. Proceeding in this manner, one obtains the following variational equations:

313

[99.18) where m

=

Wop.

4'

R - 2L ;

n _

q

~

= 2L

It is to be noted that L1'(al) is negative, since the non-linear inductance decreases when a 1 increases. The characteristic equation of the system [99.13) is )...2 _ (3m cos 2~1 - n) ... + [( m cos 2~1 - n) 2m cos 21,61 + 2mS sin 2,pd = 0 where S clearly

= a QIL 1

1

'(a 1 )1

3mcos2~1 - n

> O. Conditions for a stable stationary solution are

<0

and

m cos 2 2,pl - ncos 2~1

From the first condition cos 2~1 <~. cos 21,61 (m cos 21,61 - n)

>-

+

Ssin2,pl

> 0 [99.19]

From the second

S sin 2~1

If the angle 21,61 is in the first quadrant (sin 2~1 > 0; cos ceding inequality can be replaced by a stronger one

2~1

> 0), the pre[99.20)

whence cos 21,61

> -mn

Comparing this with the previous one, namely, cos 2~1 < ~, we see that these two conditions are not consistent. This means that a stable stationary solution cannot exist when 2~1 is in the interval 0 < 21,61 < ~. If one now considers the lnterval ~ < 2~1<"' that ls, cos 2~1 < 0, sln 21,61> 0, it is observed that both conditions, namely. 3m cos 2~1 - n < 0 and m cos 2~1 - n < 0, are fulfilled. The argument can easily be carried out for the remaining two quadrants. This means that stable statlonary solutions may exist for certain definite ranges of the phase angle.

100. STABILITY OF NON-LINEAR EXTERNAL RESONANCE; JUMPS We shall now investigate the so-called jump phenomenon observed in non-linear systems acted upon by an external periodic excitation. We will consider the usual quasi-linear equation mx + kx = J.lf(at, x,x)

[100.1]

where f(at,x,x) is a non-linear periodic function of t with period ~. the neighborhood of external resonance we have the relation

In

314

"'0

=

T -(I

8

+

JlQ

[100.2]

where "'0 =~ i8 the undamped frequency and T and 8 are relatively prime integers. The term JlQ is a small quantity of the first order. The mechanical interpretation of Equation [100.1] is obvious. We can apply the method of equivalent linearization, Chapter XII, and for this purpose we seek a solution of the form x

= a sin (; at + ~)

[100.3 ]

According to this method, the non-linear exciting force F = Jlf(at,x,x)

[100.4 ]

is to be replaced by the equivalent linear one [100.5 ]

F = - kl:t - AIX

The equivalent paTa1neteT8 kl and harmonic of the expression

Al

are obtained by equating the fundamental

F = Jlf(at,x,x) = Jlf[at, asin(; at

a;

+ (1),

acos(; at

+ ~)J [100.6]

to the linearized terms F

= - klx - Alx = - klasin(: at +~)

-

Ala ; acos(; at +~) [100.7]

The equivalent parameters kl and Al are given by the Pourier coefficients of the first harmonic, namely, k = - L[f(S'T 1

1Ta

-!!...~ T '

a sin T'T, a TS a

COST'T)

sin T'T dT

[100.8 ] Al = - J!....

(...!-) [f(S'T - !!...~,

1Ta Ta

T

a sin TT, a.!.a cos TT) cos TT dT 8

In view of Equation [100.2] these equations can be written as

[100·9 J

The parameters of the equivalent linearized system are [100.10 ]

315 and the equations of the first approximation appear in the form da -dt =-

d_

Al 2m '

r

-dt =

-a'

r

~

-a = w0 - -a + w02k 8 8

W -

(100.11]

Replacing in these equations kI and Al by their values in [100.9J, one obtains two differential equations of the first order sufficient to determine the two quantities a and _. As an example of the application of these results, we will consider a rod of length l on which is impressed an axial periodic force F = H sin at. The partial differential equation for the transverse vibrations of the rod is EI

where

y

{J4y

8x4

+ "A

{J2y

g 7it"I +

H' t {J2y = 0 sm a 8x2

(100.12 ]

is the lateral deflection,

EI is the rigidity of the rod,

" is the weight of the rod per unit volume, A is the cross-sectional area of the rod, and 9 is the force of gravity, 32.2 feet per second squared. Assuming as boundary conditions y(O)

=

y(/)

=

(100. , 3]

Yzz(O) = yz:z(l) = 0

and seeking a solution of the form [100.14 ]

y = z(t) sin7T :

one finds, upon the substitution of [100.14] into (100.12], the following differential equation [100. , 5] If we let

H

where Wo is the fundamental frequency of the transverse oscillation of the rod and Fer is the critical Eu1er 1 s load, Equation [100.15] becomes

z+

Wo (1 -

p

sin at) z = 0

[100.16J

We are looking for a Bubharmonic oscillation of the order one-half, when Wo ;::s a/2. The solution is then of the form z

=

a sin

(~t + ~)

[100.17 ]

316 where a and

~

satisfy the equations of the first approximation, viz., da Cit

=

1

'4 pawo cos 2~ (100.18 ]

d~

Cit

= Wo -

a

1

"2 - 4"

. pW osm 2~

Comparing these equations with Equations [99.9] of the circuit excited by a periodically varying capacity, we note that they appear as a particular case of the latter, namely, the case when L 1 (a) = 0 and R= O. We can therefore use the condition of self-excitation [99.16], which gives here

1-~I < .e. 2

r

a

8.

[100.19]

2

Wo

Let us now consider fundamental non-linear resonance, that is, The non-linear function appearing in Equation [100.1] is of the form IJI(at,z,z) = - I(z,x)

We have Wo

t:::s

a.

+

[100.20 J

E sin at

The quasi-l1near equation then becomes mx

+ kz + I(z,x)

= E sinat

[100.21 ]

+

[100.22 ]

We are seek1ng a solution of the form z = a sin (a t

~)

The linearized equations of the first approximation are da A (IT = - ~a 2m

[100.23 ]

d~ = .!k+k; _ a dt V~

where the equivalent parameters Al and kl are given by the equations Al = _1_ r[/(asinT, aWocosT) - Esin(T 1TaWo JI

~)] COST dT (100.24 J

If we put A, = _1_[I
[100.25 ]

317 it is noted that the coefficients ~~(a) and k~(a) correspond to the absence of the external periodic excitation Esin at. In view of Equations [100.24] we can write

~l

~. + 2 E

2m = 2m

maw o

~. + E sin cP :::: 2m 2maa

'.1..

SlnY'

= 6e

+ 2

E

maa

sincP [100.26]

where 6. =~. Similarly,

.!iC""+k; _ V~

a :::: _1 2a

= -1 2a

(k

+ kl _ m

(2 W '

(k, _ a 2 _

a2) = _1 2a m

~ COScP) ma

E ) a 2- -coscP

-

[100.27]

ma

where w. = Yk/m. The quantities 68 and W. are the equivalent parameters for the non-linear oscillations of the system in the absence of an external periodic force. Substituting these values of ~l and ~kl in Equations [100.23], one obtains the following expressions of the first approximation:

Vk

da

Cit

E

.r

= -

Uta

2ma

-

.

.I..

SInY'

[100.28 ] 2

We

a

-

2

E

- -COScP ma

The stationary amplitude a is obtained from these equations: - E sin cP = 2maa6.

[100.29] whence a =

E 2

m yI(w.

_

a 2)2

+

46.2a 2

[100.30 )

It is observed that the stationary amplitude is given to the first order of approximation by exactly the same relation which gives the forced amplitude of a linear system, except that the equivalent parameters are to be used instead of the constant linear parameters. Although these results, which were derived from the equations of the first approximation at a glance, do not seem to yield anything new, it will be shown now that the important difference between linear resonance and non-linear resonance lies in the conditions of stability. More specifically, it will be shown that, Whereas the linear oscillation is stable throughout the Whole neighborhood around the point of resonance, the non-linear oscillation is stable only in certain regions.

318

If

we set R(a,q,) = -

E sin 9'1

m

-

2aa68

[100.31 ]

¢/(a,q,) = (W,2 - ( 2 )a _

E cos 9'1

m

Equations [100.28] become 2a

da Cit

= R(a,q,)

[100.32 ] 2aa

~~

= ¢/(a,q,)

The stationary state is given by the equations R(a,q,) = 0;

[100.33]

¢/(a,q,) = 0

In order to investigate the stability ot the stationary state we must torm variational equations. It we ~esignate the perturbations in a and 9'1 by 4a and 49'1, respectively, the variational equations are 2a

~~a

= Ra 4a

+

R_4q,

[100.34 ] 2aa

d6q,

dt

= ¢/a 6a

+

¢/_ 49'1

The characteristic equation, see Chapter III, ot the system (100.34] is aS

2

-

(aRa

+

¢/_)S

+ (Ra ",_

-

R_"'a)

= 0

[100.35]

The conditions tor stability are clearly aRa

+ "'_ < 0;

>0

Ra "'_ - R_"'a

[100.36 ]

Using Equations [100.29] and [100.31]. one has a Ra

+ .f,Y'_ --

_

2

aa

o(a6,) oa -

2

1

aau, =

_

2a o(a26~) oa

[100.37 ]

On the other hand, we have [100.38 ]

Let us consider the quantity W(a) =

2~ awo

r

f(a sinT, awo COST) COST d'T

2..

= i':jW;f[aSin(wot

+ q,),awocos(wot +

q,)]awocos(wot

+

q,)dt

[100.39]

319 It is apparent from this expression that the quantity MrCa) represents the average power dissipated by the non-linear force fCz,~) during the oacillation z = a sin (wot + ~). Hence, tor the usual law of friction, W(a) 1ncreases with a so that MraCa) > O. In this case the first cond1tion [100.36] 1s always fulf1lled, as follows from Equat10n [100.38] and from the condition ~(a) > O. Hence the stabil1ty of the stat10nary state depends on the fulf1llment of the second condition [100.36]. Differentiating the functions RCa,~) and ¢Ca,~) with respect to a, we have

- "'a whence

[100.40 ]

Ra "'~ - "'aR~ = "'a R~ - Ra "'~

From Equation [100.311 we have R~

=-

E

m cosf/J;

.,. = m E sin Y'. J.. .,,~

Ra = - 2a6e ;

I/Ia = - 2aa [100.41 ]

which gives E

I/Ia R~ - Rarp~ = 2a m (a cos 116 + 6., sin f/J)

From Equations [100.291 E (a cos f/J

m

+

6"

sin~)

= a

a 2 )a - 2a6e2a

(W.,2 -

hence so that

(Rarp~ - l/JaR~) ::

= 2aw

2

[ (w;

- a

2

) -

26.,2)

The second condition [100.36] can be written in the form

~~ >

0

if

W~2 >

if

we2

a

2

+

2

26.

[100.42 ] da

dt

<

0

<

a

2

+

26,2

Since the term 6.,2 is small and of the second order, it can be neglected, so the cond1t10ns of stab1lity become da dt

>

0

if

w~• > a

[100.43] da dt

<

0

if

we

<

a

320

o

These conditions for stability and, hence, for the existence of stationary oscillations, can be represented graphically in a simple manner. Let us trace the curve a = F(a) determined by Equation [100.30], which can be written as

Furthermore, let a = Fo(a) be the curve corresponding to the exact resonance Figure 100.1 w,(a) = a. Assume that these curves have the shape shown in Figure 100.1. On the portion of the curve F(a) situated to the left of the curve Fo(a) , the condition of stability exists in intervals such as AB, CD, ••• , where the -amplitude a increases with increasing frequency a. On the parts of the curve F(a) situated to the right of Fo(a) , on the contrary, stability exists in intervals such as EF, HL, ••• , in which the amplitude a decreases with increasing frequency a. These peculiar conditions of stability of non-linear external resonance cause the appearance of jU1nPB similar to those which we have already investigated in Part II in connection with the phenomena of hard self-excitation. Thus, for example, if we excite a non-linear system from rest by a gradually increasing frequency, the stable branch ABwill be traversed. At the point B, however, this stable zone ends and the amplitude suddenly jumps up to the point B', after which for a continuously increasing frequency of the external excitation the branch B'D will be followed. At the point n the stable region on this branch ends, so that the amplitude drops from D to D'. For a further increase of the frequency of the external periodic excitation the branch n'L will be traversed. If, however, the frequency is decreased, the amplitude will not pass through the same stages it traversed during the period when the frequency was steadily increaSing. Thus, for example, if the frequency is decreased from the value corresponding to the point L, the region D'Hwill be traversed in a stable manner. This region, as was just mentioned, was missed during the period when the frequency was steadily increasing. If the frequency continues to decrease, the amplitude will jump abruptly from the value H to the value H' and a further change will occur along the branch H'E, and so on. These phenomena of reBonance hYBtereBis may be more or less complicated, depending on the form of the curvesF(a) and Fo(a); they are usually accompanied by quasi-discontinuous Jumps and hence by similar discontinuities

321

in the energy input into the oscillating system supplied by the external periodic source of energy. As a special example, consider a non-linear system whose nonlinearity is limited only to x, that is, its non-linear function 1s of the form f(x). In practice, this corresponds to a system with a non-linear spring constant. From Equations [100.25] and [100.26], it is apparent that 4. = 0, and from the second equation [100.25) we get 2 we =

![k + 7T1a!'>
[100.44 ]

so that Equation [100.30] gives a =

+

[100.45 ]

From Equations [100.29] it follows that, for the plus sign, ~ = 7T; for the minus sign, ~ = O. From [100.45] one can build the curve a = F(a). It is noted that if w.(a) varies with a, for instance according to the relation when

w.(a o) = Wo + )..a;

the amplitude cannot increase indefinitely for any a. This circumstance is another typical feature of an undamped nonlinear resonance. Interesting examples of Jump phenomena have been obtained recently by Ludeke (6) in his experimental work on non-linear mechanical systems. By varying the non-linearity of the springs, different response curves were obtained. Figure 100.2 shows the experimental and theoretical resonance curves obtained for a particular non-l1near spring of the "increasing stiffness" type. The theoretical curve was obtained by a graphical method (7) the details of which we omit here.

)..:1:

0.032

0

;r'

0.028

J

0.024

I

0.020

IJ

i

i Ii .5

.:

- -

Experimental values Theoretical valuu

0.016

~

Q.

~

0.012

0.008

0.004

o

j

1, .......

~

.l,

50

~-,-

100

r---

150

---

200

Frequency in radians per second

Figure 100.2

250

322

The results of this section and of Section 99 were obtained on the basis of linearized equations of the first approximation. In Chapter XIX the study of the effect of a periodically varying parameter will be resumed, but from a d1fferent viewpoint; there we will use different1al equations with periodic coefficients.

CHAPTER

xvn

SUBHARMONIC RESONANCE ON THE BASIS OF THE THEORY OF POINCARE

101.

METHOD OF MANDELSTAM AND PAPALEXI

The preceding four chapters of this report have been devoted to a survey of the theory developed by Kryloff and Bogoliuboff. We now propose to review an alternative theory developed by Mandelstam and Papalexi (8) and derived from the classical theory of Poincar6, which was discussed in Chapter VIII. The essence of the Mandelstam-Papalexi method lies in extending Poincar6's theory to systems having an external periodic excitation. Certain advantages arise from this argument. First, the questions ot the existence and stability of solutions are treated in a relatively simple manner. Secondly, the description ot the behavior of a system in terms of its characteristic parameters is also relatively simple. Finally, the manipulation with generalized impedances and admittances used by Kryloff and Bogoliuboff is replaced here by the analytical method, which is probably a more familiar approach to the subject. It is noteworthy that the theory given in this chapter was found to be a useful tool in connection with the numerous experimental researches conducted by the group of scientists under the leadership of Mandelstam and Papalexi. 102.

RESONANCE OF THE ORDER ft.; DIFFERENTIAL EQUATIONS IN DIMENSIONLESS FORM

The following analysis is a discussion of differential equations of the torm % + z = II/(t, x, i:)

[102. , ]

where the non-linear function /(t,x,i:) now depends explicitly on the time t. More specif1cally, 1n what follows we shall cons1der the equat10n % + x == II/(X t i:)

+

AO sin nT

[102.2 ]

in wh1ch we let the argument of the periodic "forcing" funct10n be nT instead of T in order to prepare tor the study of subharmonic resonance of the order lIn. Many circuits of electron-tube oscillators can be represented by equations having the same form as (102.2], but we shall not enter into these generalizat10ns here. We shall consider the standard circuit shown in Figure 102.1 representing an electron-tube oscillator with an inductive coupling Al. The externally applied electromotive force E= Eo sinwt is inserted either in the anode circuit between Mand N or in the gr1d circuit between P and Q. It has been shown that, if the effects of the anode reaction and the grid current are

32 4 neglected, the differential equation of the circuit is d2 ' CL dt''l M

.---....

d' CR d;

+

i = ia

+

dE C fit

[102.3 J

where ia = f(V.) is a non-linear function representing the anode current ia as a function of the grid voltage V•. It is apparent that VB =M~:, where M is the coefficient of mutual inductance between the anode and grid circuits. It is convenient to transform Equation [102.3J into dimensionless form. We introduce the following dimensionless variables:

""E N--------, - I. c

L

+

R

til)

-r=

Figure 102.1

-i

n

i 1=:0-' 1 t o

where 10 is the saturation anode current, occurring for a sufficiently high grid voltage Vo' The change of the independent variable gives di

di d-r

di II)

-=:0---=-dt dT dt dT n d 2i

d di d di dT d 2i 11)2 = Tt(dt) = dT(dT)dt = d?;r

cw:

dE dE dT dE II) -=--=-dt dT dt dT n

[102.4 ]

In the new variable, moreover, E = Eo sin II)t becomes E ~f = Eon cos nor and, by [102.4], dE CIt

= Eo

sin nor; hence,

= EolI) cosn-r

Equation [102.3] becomes w2 d 2 i

w di + C R -dTnT

CL~p

n

If the "dimensionless current" I

= 10

+

i = ia

+

CEowcosnT

is introduced, the previous equation

becomes [102.5 ) If we assume that the resistance R of the oscillating circuit is small. the autoperiodic frequency of the circuit is equal to its undamped frequency 11)0 = 1/~ to the first order. If the impressed frequency II) is in the neighborhood of the frequency nll)o of the oscillating circuit, that is, if

325 [102.6)

where

e«

1, the coefficient of dI/d-r can be written

CRw -- = n

=

Equat10n [102.5) then becomes 2 (1 + ~) d J + (1 + 'i d-r2

~) Lw Rn

'i

+

dI d-r

+ e> Rn

(1

Lw

+

J = J

a

CEloW COSn-r

°

If we divide this equation by (1 + e), noting that 1

l+e=

l+e-e e l+e =l-l+e

we obtain

2 d 1 d-r2

Since Ia, equation

+

+

Rn dI Lw d-r

J =

1

e

+e

J

+

1

+e

1

I

+

a

= ;; = f( ~: Vo) where Vo 1s the saturation 1'0 = Mlow/n, the "dimensionless voltage

CEow [102 7] IoU + e) cos nor •

voltage defined by the

II

Mdi w d-r n Va= 1'0 MI!!!.

°n

so that /a,

Setting

dI d-r

=-

=

f(~:)

=

fl(:~)

(102.8 J

f= = 29 in Equation (102.7] and rearrangIng, we have 2

d / d-r2

+ J = 1 +1 ef,(dI) 1 d-r

29 dd:

,

+

1

ie/ +

Qcosn-r

where

Letting 1 (dI) _ 29 dd:] = F(dd:) [~~d-r , ,

[102.9]

one obtains

dd-r21 + I -_ F (dI) d-r 2

+

~ 1 + eJ

+ Q cos nor

Differentiating this equation with respect to -r, and putting dI/d-r= x, we obtain

%+ x where >'0

==

-Qn.

= F'(x)x

+ 1

! e x + >'0

sin nor

Equation [102. 1 0] has the same form as ['02.2].

[102.10]

326

103.

PERIODIC SOLUTIONS OP A QUASI-LINEAR EQUATION WITH A PORCING TERM Prom the theory ot POincar6, Chapter VIII, it tollows that a quasilinear equation ~

+

[103.1]

:a: = J.l!(:a:,x)

with J.l = 0 admits an infinity ot periodic solutions represented in the phase plane by a continuum ot concentric circles with the origin as center. Por J.l ~ 0, but small, periodic solutions may still exist in the neighborhood ot ce~tain circles, the gene~4ting 8olution8. In the rest ot the phase plane no periodic solutions exist, but the phase trajectories are spirals winding onto the closed trajectories, the limit cycles, which exist in the neighborhood ot the generating solutions. Por Equation [102.2] the situation is similar, tor when J.l = 0 there exists an infinity ot such linear solutions ot the torm :a: = 4 sin T - h cos T

+ 1

AO 2 sin 11. T

-

11.

[103 .2 ]

The tundamental problem is the determination ot the tunctions 4(J.l) and 6(J.l) which will yield periodic solutions tor the non-linear case, that is, when J.l ~ 0 but is small. It, when J.l +0, these constants reduce to 40 and 60 respectively, that is,

[103.3 ] the corresponding solution ot the linearized equation is called the principal or tundamental solution. In order to establish the conditions under which the expression [103.2] is the principal solution ot Equation [102.2], it is necessary tirst to determine the limit values [103.3] ot 4 and 6 tor J.l +0 and then to ascertain that the solution so obtained is stable in the sense ot Liapounott. In this section we shall be concerned with the tirst part ot this problem. It the new variable z = :a: - 1

AO -11.

2

•

[103.4 ]

smnT

is substituted into [102.2], that equation becomes

..Z +

!( Z +

Z = J.l

1 AO. - 2 smnT , Z. -11.

+

1 nAo 2 cosnT ) -11.

[103.5]

Introducing into this equation the variables u and v detined by the equations U = ZCOST + zsinT v = %SinT - ZCOST

[103.6 ]

327 we find u = (Z

+ Z)COST

v = (z

+

z)sinT = pl/l(u,v.T)sinT

- VCOST

+

. + 1 nAQ 1 _AQ. n 2 smnT, UCOST + vsmT _ n 2 cosnT)

= pl/l(U,V,T)COST

[103.7]

where .1.( U,V,T ) =

Y'

f( usmT .

The function I/I(U,V,T) is periodic with period 2". [103.6] one obtains X

• = u sm T

-

V

cos T

+

1

From Equations [103.4] and

AQ • 2 sm n T - n

[10~.8 ]

Since this expression is of the same form as [103.2], the principal solution of Equation [102.2] will be found when u Q = a and v Q = h for p + O. We can now apply the procedure of Poincare by assuming that, when p« 1, the quantities u and v for T = 0 differ but little from a and h, that ls, [103.9 ] where a and P are small numbers.

From Equations [103.7] one obtains r

U

= ur=Q

+

pJI/I(U.V,T)COSTdT Q

[103.10] r

V = vr = Q

+

f

p I/I(u, v, T) sin T dT Q

The functions u and v, on the other hand, can be expanded in terms of the small parameters p, a, and Pi this, in view of [103.9], gives u = a

+

a

+

pC1(T)

+

paD1(T)

+

pPE1(T)

+

p 2G1(T)

+

v = b

+

P

+

pC2(T)

+

paD2(T)

+

pPE2(T)

+

p 2G2(T)

+

where the dots deSignate terms of higher orders containing p8, p4, Comparing these expansions with [103.10], one finds r

C1(T)

= J I/I(a.b,T) COST dTi Q

[103.11 ]

... .

r

C2( T) =

f I/I(a. 6, T) sin T dT

[103. , 2 ]

Q

and also E 1(T) =

E 2(T) =

[[~~]COSTdT f[:~] sinTdT Q

[103.13 ]

where the symbols [~] and [~t] designate the partial derivatives ~ and in which # = a = P = O. If u and v are periodic. it is clear that u(2") - u(O) = 0 and v(2") - v(O) = 0; in view of [103.11]. this implies that Ct (2")

+

aDt (2")

+

PEt (2")

+

pGt (2")

+ ...

= 0

C2(2")

+

aD2(2")

+

PE2(2")

+

pG2(2")

+ ...

= 0

~

1103.14 ]

The problem of determining a and h in Equations [103.2). when # is small and hence when a(#) and p(#) are small. therefore is one of finding values of a and p which will satisfy Equations [103.14] and which will reduce to zero when p = O. Since there are two equations. it is possible to determine a and P as functions of #. provided

These equations give the first-order solution for a and b. since other terms in [103.14] contain small factors a. P. and #. For solutions valid to the second order. the equations aD1(2") aD 2(2")

+ +

PEt (2")

+

#G t (2")

+ ...

= 0

PE2(2")

+

#G2 (2")

+ ...

= 0

[103.16 )

must be satisfied for any arbitrary but small p. These equations admit singlevalued solutions with a and P approaching zero as # approaches 0 if .1 =

I

D t (2") D 2 (2")

E t (2") E 2 (2")

I

:1=

0

Hence the problem of determining the limit values of the coefficients a and b in Equation [103.2] when #.0 is solved by Equations [103.16J provided the condition [103.17] is satisfied.

104. STABILITY OF PERIODIC SOLUTIONS The condition for stability of periodic solutions can be obtained by utilizing the variational equations of Poincare. If we introduce in Equations [103.7) the quantities u = U o + ~ and v = Vo + e. where U o and Vo are periodic with period 2" and ~ and e are small perturbations. and develop the function ~(U.V.T) in a Taylor series around the values Uo and vo. we obtain the variational equations

3 29

:~ = (,.,,,,11. cos T) 17 + ("''''" cos T)~ [104.1]

!Y.. dT - (,.,,,,11. sin T) 17 +

(p"'" sin T) t

since the functions U o and Vo satisfy Equations [103.7]. Equations [104.1] have periodic coefficients. Suppose that 171(T), ~1(T) and 172(T), ~2(r) are two sets of solutions forming a fundamental system. One can assume the initial conditions 171(0) = 1, ~1(0) = 0 and 172(0) = 0, ~2(0} = 1. Since 171(r + 2"), tl(r + 2"), ••. are also solutions, one can write 171(r + 2") = Whence, for

T

= 0,

a171(T)

+

tl(r + 2") =

6172(r);

at 1(r)

+

6~2(r)

[104.2]

in view of the initial conditions, [104.3]

It is possible to select a fundamental system so as to reduce [104.2] to a canonical form where 171(T + 2") = SI171(r) •••• The formation of such a system depends on the solution of the characteristic equation [104.4] This can be written as F(S) =

82

+

p8

+

q

=

[104.5]

0

with [171(2") +

t2(2")]

[171(2")t2 (2") - 172(21TH1(21T)] [104.6] The parameter,., in [104.1] is supposed to be fixed; thus, if ,., = 0, ~ = ~ = 0, so that 17 and t remain equal to their initial values, namely,r p = -

171(r) =

and

q =

~2(T) = 1;

172(r) =

tl(r)

=

0

[104.7]

*

Hence, for,., = 0, p = -2 and q = +1. For p 0 but small, we conclude, therefore, that p < 0 and q > O. The system is stable if the real parts of the characteristic exponents hi and h2 (see Section 27) are negative, which implies that Ie 21rA1 1 < 1 and I e 2 J1'A21 < 1, that is, the moduli of the roots 8 1 and 8 2 are less than unity. Equation [104.5] has roots with absolute values less than unity only if p > - 2; l+p+q>O [104.8] which follows from the equations 1

+

P

+

q = (81

-

1)(82

-

1);

[104.9]

330

On the other hand, for p c 0, p = -2 and q = +1, as was shown. Hence, the cond1t1ons of stab1l1ty [104.8] can be satisf1ed for small positive values of p only when the first non-vanishing derivatives of p and p + w1th respect to p are positive for p = O. In order to calculate these derivatives, replace ~ and t 1n Equat10ns [104.1] by ~1 and tl and integrate between 0 and 2". Th1s g1ves 71 1(2") -

~1(0)

= 71 1(2") - 1

=- P [',,"711COST dT + P

q

[rAt1 COST dT [104.10]

t 1(2") - t 1(0)

t 1(2")

=

= p

[I/I"~18inT dT

+ P[I/IlIt! sinT dT

Different1ating these equat10ns w1th respect to p, one obta1ns

d71~~")

= [(1/1,,71 1 COS T

+

1/11'/1 COS T) dT

+

P[( 1/1"

~~1 COS T

:- 1/1'1

~~ COST) dT [104.1']

dtd~")

= [CI/I,,711sinT

Pass1ng to the l1m1t p

+ I/IlIt 1sinT)dT +

= 0 and

Jl[(I/IIl~:ISinT

+

1/111

~~1 sinT}dT

taking 1nto account [104.7J, one gets

From [104.6] and [104.12] one obtains

(~=)" =0 = -

[D 1(2")

+ E2 (2"»);

[

d(P

+ dp

'I)]

= 0; II

=0

[104.13]

This leads finally to the following conditions of stability. and

*

Conci1 tiODS [1 04 .14 ] have been fOl'lllUla ted by Mandels tam and Papalexi (8). developed by Professor W. Hurevicz.

[104.14]

The proof given here was

331

105. SUBHARMONIC RESONANCE OF THE ORDER ONE-HALF

FOR A SOFT SELF-EXCITATION We shall apply the theory outlined in the two preceding sections to the important practical case when n = 2. This case has already been investigated in Chapter XVI by the quasi-linear method of Kryloff and Bogoliuboff. We employ the usual polynomial approximation for the non-linear element of the system, the electron tube, that is,

[105.1 1 In Section 51 it was shown that for a soft self-excitation the apprOXimation can be limited to the first four terms, that is, a: = a5' = 0, whereas for a hard self-excitation the full polynomial (105.11 must be used. We have also seen that the important terms are those containing the odd powers of V., but for greater generality we shall use the full expression [105.1]. USing the notation of Section 102 and designating the "dimensionless" grid voltage ~ = Z = ~~, one has

IQ

= ft(z) = 140

+ alz + a 2z 2 +

aa z

a

+

a4z

4

+

-

a 5 z"

[105.2]

Using ExpreSSion [102.81, one obtains the following expression for p/(x,x) in

[102.2] : p/(x,i:) =

l!~ +

Hal - 29(1

2a 2 x

+ 3aaz 2 +

+ ~) + 4a 4 z

3

+

5a5 z

4 ]

x + ~z)

Since 1n practice the coefficient a 2 is very small, we can put a2 1

+

~

= p;

al - 29(1 a2

+ ~)

= k;

A3 = 3a3.

a2

A4 = 4a 4

•

;

a2

A = 5a5 5 a2

Equation (105.3] then becomes /(x.x) = (k

+

+ Aaz2 +

2x

A4Xa

+

A6X4)X

+ ..Lx a

[105.4 ]

2

For n = 2, Expression [103.2] is x = asinT - bCOST -

~OSin2T

= Xsin(T - cp) -

~o

sin2T

[105.5 ]

In this section we shall consider systems having soft self-excitation, that is, systems in which A4 = A5 = O. In order to determine the l1mit values (103.3] for a and h, one must solve the equations

f

2.

r",(a.b.T) COST dT

=

0

and

I/I(a,b,T) sinT dT = 0

Since the function ",(a,b,T) is, by defin1tion, /(z,x), in which x has the value (105.5], we obtain

332

r",
[
=

+ 2x

Carrying out this subst1tution and the integrations, one finally obtains

a[k

+

b [k +

~3 (a

2

+

~ (a 2 +

62 + 2 b +

2~:)]

=

b( - ~o

2

2~o)]

= - a

(~

+

!J

[105.7]

+ ;2)

From these expressions one obtains the square of the amplitude Xof the principal solution [105.8 ]

and the phase rp

= tan-I! a

=:

tan

-1

[105.9]

a2

The principal solution corresponds to real values of X, that is, to values of 2 X such that X > O. Since the term - j>"02 in [105.8] is always negative, it is apparent that this condition is fulfilled if the second term on the right of [105.8) is posi ti ve and is greater than the value ~ >..i • Hence, i f A sand k are negative, only the plus sign can be taken before the radical. If, however, As < 0 and k > 0, the condition X2 > 0 may be fulfilled for either sign of the radical. All depends, of course, on the magnitude of As, k, and Jt, a2 that is, on the magnitude of the constants of the circuit. Fulfillment of the cond1tion X2 > 0, however, does not mean that the principal solution exists in practice. Its existence implies that the oscillation be 8table, which requires that the conditions [104.14) be satisfied. In Equa tions [103.13], which determine the func tions D. ( 1'). D2 ( 1'). E. ( 1'), and E 2(T), appear the expressions [~] and [~] previously defined, that is, [ d"'] du

=

IJj IJx ox 011.

+

IJ/ IJz , o:i: 011.'

[d"'] dv

=

IJ/ IJx ax ov

+

IJ/ oz a:i: ov

[105. 10)

where x

=

11.

sin l' -VCOST- >"0 sin 21" 3 •

x =UCOST+ vsinT -

2>" T cos

Hence

-ox = 011.

sinT' •

o:i: 011.

=

COSTi

IJx OV

=:

- COS"i

ox = sin l' OV

21'

333

This gives

[105.11 )

Moreover, from [105.4), one has

:~ =

2(1

af

ax =

+ Asx)x +

k

+ 2x + A aX 2

(105.12 ]

The criteria of stability (104.14) can be applied now both to the principal solution [105.5) and to the heteroperiodic one, in which a = b = O. Carrying out the calculations [104.14] in which Dl(2~), .•• are replaced by their expressions (105.11] and (105.12) for both the prinCipal and heteroperiodic solutions, one finds that the conditions for stability, and hence for the existence of the principal solution, are k

+ ~3 (X2 +

2

A;)

<

0;

For the heteroperiodic solution, they are

k+

A2

Aa

~ <

0;

(k

A2

~2

+ Aa l~) + (l2 2

A:.! T

[105.14 ]

>0

The last inequalities, in view of [105.8], reduce to

-+ VA9n2 ~2 Aa 2 -"- _ -" ai + -4 X < o·•

As [+

VA-f2

~2 -;;::

]

>

0

(105.15]

These inequalities are satisfied if As < 0 and the minus sign is taken before the radical. We re-emphasize here the important point which has previously been noted in Section 51, that is, for a soft self-excitation, the coefficient of the cubic term in the polynomial approximation of the non-linear characteristic must be negative. Under these conditions Expression [105.8) for the square of the amplitude X becomes X2

Since k :::

4 [k + .fA e V~ - tl!

= + IAsl

where 29

2

= Rn/Lw,

A:]

+ Aa18

it is seen that when k

(105.16 )

>0

the

energy input from the electron tube outweighs the dissipation of energy in the circuit, whereas when k < 0 the dissipation of energy exceeds the input. From Equation [105.16] it is observed that the existence of a stable amplitude when k < 0 depends on the values of other parameters. Physically this means that the self-excitation of a system which is not normally self-excited (>"0 = 0) can be produced by the effect of the externally applied electromotive force (>"0 :I: 0). If ).,0 = 0, the quantity e does not exist and one has X2." tt~, which coincides with the equation previously obtained from the theory of Poincare. Comparing the condition of stability of the principal solution with that of the heteroperiodic solution, one finds from [105.16]

t

2

).,2

::TS~-

a2

9

-

IAaIAi)2 -

[105.17]

(k _ IA al).,l)2

[105.18 ]

(

k-

18

On the other hand, from [105.15] one finds

t

2

a;

>

~

_

18

9

It is thus seen that one solution appears at the point where the other disappears and vice versa, so that there is no interval in which both exist at the same time. From the equation

1! 4: =

~

(k _ IAalAi)2 18

_

9

[105.19]

where k =

41 -

26 _ 26~ = ko _ 26t

42

42

42

one can obtain two values e' and e" determining the limits of stab1lity. These limits determine the zone of discrepancy de = ~" - t' between wand nwo within which a stable principal solution exists. One obtains the following expression for dt valid to the second order of the quantity 6:

.I

Considering dt as a function of ).,2

T

~

).,0'

A2 2

2

~t = (' - ( = 2a2Vt(1 -

koAa -

;;0)

2

-

k0

[105.20]

it is seen that for

2 jf2" [1 - koAa a

VI -

2k oA a]

[105.21 ]

de

is real, hence the zone til - e' exists. Beginning with the value of ).,0 given by the equality sign in [105.21], the real values for ~e appear and dt increases up to a maximum value [105.22 ]

when the amplitude >"0 of the externally applied excitation reaches the value

335

[105.23] Por a further increase of AO' the interval A2 T

2

=

;p[l s

koAs

+

~e

decreases and becomes zero when

VI - 2koAs]

[105.24]

It one continues to increase Ao' the interval ~e becomes imaginary, that is, it ceases to exist. Summing up, one can say that the external periodic excitation having a frequency w which differs somewhat from twice the frequency Wo to which the oscillating circuit is tuned is capable of producing oscillations in the circuit provided the coefficient ~ = W22:';ZW02 remains inside the interval corresponding to values of AO in the interval 2 A2 a

[I -

koAa -

).. 2

2 a

Y1 - 2koA3] ~ ~ ~ Jf2 [1

-

k oA 3

+

VI -

2koA a

]

106. NATURE OP SUBHARMONIC RESONANCE OP THE ORDER ONE-HALF

POR AN UNDEREXCITED SYSTEM In the preceding section the conditions for the existence and stability of the principal solution were established without specifying the sign of the quant! ty k = al - 29(1 a2

+

e)

= ko _

2ge

a2

As has been mentioned, this quantity characterizes the stability of the system in the neighborhood of equilibrium, when the effect of the terms containing the &mall quantity % (see Equation [105.3]) is negl1g1ble. Hence when k < 0 at the pOint of equ1l1br1um (% = 0) the system is stable, and when k > 0 it is unstable. These conditions correspond to the ex1stence of either a stable (k < 0) or unstable (k > 0) Singularity in a self-excited system w1thout an external force. When subharmon1c external resonance is present, the situation is different in that even when k < 0 the pr1ncipal subharmonic oscillation may arise 1f the amplitude of the external per10dic oscillat1on is contained in the zone ~e specified in Section 105. In this section we shall 1nvestigate a system where k < 0, that 1s, a system wh1ch is stable without an external excitation. We call such a system an undere%cited one. If such a system is subjected to an external exc1tat10n with frequency w differing from 2wo by a considerable amount, only a relatively small heteroperiodic OSCillation w1ll be present. If, however, w approac~es 2wo so as to be within the limits of the zone ~e, the pr1ncipal oscillation will suddenly appear and will have e%actly the frequency w/2. If the parameter e is varied, the frequency of the pr1nc1pal oscillation will

W2

2W2

always be w/2, but it is worth noting that the variation of ~ = 2- 2 0 can Wo be accomplished in two different manners, that is, by varying either w or w o' If w is varied, the frequency of the subharmonic oscillation will follow the variations of w, since it is always w/2; if Wo is varied, the frequency of the subharmonic oscillation will remain constant. It is thus seen that when the prinCipal oscillation exists its frequency is always 1/2 (or l/n for subharmonic resonance of the nth order) of the externally applied frequency w, but the range within which it exists is influenced by ~, that is, by both wand Wo simultaneously. Moreover, the range ~~ is a function of the amplitude AOl as was shown at the end of Section 105. The phenomenon of subharmonic resonance of the order one-half (or, more generally, of the order l/n) presents features radically different from those of ordinary linear resonance. It is sufficient to investigate the behavior of the function X2 given by Equation [105.16 J. Since the range within which the prinCipal solution exists as well as the amplitude Xof this solution depend on ~ and not on wand Wo individually, it is convenient to take the quantity ~ as the independent variable instead of w as is customary for linear systems. If one plots the results previously obtained concerning the ranges ,-, / ~~ depending on the amplitude Ao of the I \ I \ external excitation, one obtains the I \ I , curves shown in Figure 106.1, which were corroborated experimentally by physicists of the Mandelstam-Papalexi school. It is observed that this phenomenon of subharmonic resonance differs radically from claSSical linear resonance, which 1 + - - - M ----''tot has the appearance shown by the dotted line. The phenomenon is entirely difFigure 106.1 ferent when k > O.

,

/------,

107.

SUBHARMONIC RESONANCE OF THE ORDER ONE-HALF FOR A HARD SELF-EXCITATION The procedure remains the same as in Section 105 except that now we have to introduce Expression [105.4] with A4 :I: 0 and A6 :I: O. Carrying out the calculations, one obtains, instead of Expressions [105.7], the following ones:

337

b k + ~S(X2 [

=

A 4 -f(X + ~~

+ 2~o) +

4

2

+

a[- Ao _ 1..a _ A. ~(X2 + 3

4

2

2

A; [5X +

A02

3

6

2

+

4a 2])

]

26 2)]

Discussion of the conditions of stability in Equations [107.1] is too complicated. One notes that in practice the coefficient A. of asymmetry is very small and can be neglected. One can further simplify the problem by assuming that the square of the amplitude of the principal solution is much larger than the forcing term, that is, )[2 » A~2. With these simplifications, one obtains the expression

Applying the same procedure as that followed in Section 105, one finds the inequalities

These inequalities can be satisfied s1multaneously if As)[2+ As < 0 and the minus sign is taken before the radical. The stable solution for X2 is then given by the equation X2

=

_~ ± . fAa: _ ~(k + ./ A02 _~)

VAs

As

As

V9

Equation [105.9] for ~ is also applicable here. If the external excitation is absent (Ao X2 = _As

+

As -

./As2 V"As2

_.!!£. As

[107. 4 ]

a2

= 0 and

e = 0) [107·5 J

The condition AI\X 2 + As < 0 implies that systems where both As and As are greater than zero are to be excluded. Hence the following combinations of signs are possible: 1. All

< 0,

As

< 0;

In the first two cases

2.

As

> 0, As < 0;

X2 can be positive only when k > 0, that is, when the system is self-excited. These two cases therefore characterize soft self-excitation in that the amplitude increases with k begInning when k = O. The third 'case yields different results, however. It is noted that, if As > 0 and Aa < 0, the characteristic of the non-linear conductor, the electron tube, exhibIts an inflection point for a certain value of the amplitude (see Section 51). In this case [107.7) It is observed that in thIs case one can also have k < 0, which means that a periodic oscillation may occur in an underexcited system. In order that no self-excitation be possible one must have AS2

[107.8 )

Ikl> 8 !Alii

It can be shown, however, that in the interval A2

~ > Ikl >

A2

sBtJ

[107.9 )

the heteroperiodic oscillation is unstable. Thus in the interval [107.9) neither the principal nor the heteroperiodic oscillation exists. This implies that Ikl must be greater than As2/61Afil in order to obtain resonance of the order one-half. Proceeding in the manner indicated at the end of Section 105 and omitting the intermediate calculations, we obtain the following results. By requiring that X2 given by [107.4) be real, one finds that there exist two intervals d~~ (for a stable principal oscillation) and d~A (for a stable heteroperiodic oscillation) with the condition [107.10) which shows that in a certain region these intervals overlap. Hence there exists a zone in which both a subharmonic and a heteroperiodic OSCillation may exist at the same time. It is recalled that for a soft self-excitation these intervals do not overlap, so that the oscillation of one type appears at the point where that of the other type disappears. Moreover, from Equation [107.4) it is apparent that the positive quantity X2 is composed of two essentially posItive parts. One of these parts (-~) is constant since it depends on the characteristic of the non-linear element. Hence, if X2 is considered as a function of ~, it is noted that the curve X2(~) cannot become zero either at the beginning or at the end of the interval in which X2 exists, but has to start from, and end at, a constant

339 value (- ~:). This feature is characteristic of the phenomenon of hard selfexcitation, as has been pOinted out in Section 51. This is illustrated by Figure 107.1, where the "discrepancy" ~ C B AI 1° I is assumed to vary while the externally I I I I I 1 applied amplitude AO remains constant. I 1 I I For ~ < 0 and for I~ I sufficiently large, I J~I I· I I 1 I I I there will be no principal oscillation 1 I r-- J~). ----t I I and only a relatively weak heteroperiI I I I ~ " ~I' 0 ~~ ~~ odic oscillation until the value t = ~: is reached; at this point a powerful Figure 107.1 oscillation of subharmonic resonance of the order one-half will set in abruptly, see Point A 1n Figure 107.1. With a further increase ot ~, the amplitude X increases relatively slowly, passing through a rather flat maximum. For t = t:' the subharmonic oscillation will suddenly disappear. If, however, one starts with large positive values of t and decreases them gradually, the subharmonic oscillation will start at the point t ::: ~;' and will disappear at C tor ~ = ~ %'. In other words, tor increaSing t, the principal oscillation starts abruptly at A and ends abruptly at D; for decreasing ~, it starts at B and ends at C in the same abrupt manner. The sudden Jump during both the appearance and disappearance ot the oscillation is numerically equal to (-~).

•

• .1 I

'08.

.

SUBHARMONIC RESONANCE OF THE ORDER ONE-THIRD For n = 3 the prinCipal oscillation [103.2) becomes x

=

+

a sin T

h cos T

~

-

[108.1 )

sin 3 T

Proceeding in the manner explained in Section 105, one obtains the following expressions (compare with Equations [105.7)): ark

+ ~(X2 + 4

b[k

!:I)] 32

+ ~(X2 + 4

+ -..t b = _ 42

(a

2_ b2) AoAa 32

[108.2)

A02)] _ .La 32 a

=

2

+ 2ab

AoAa 32

Squaring, adding, and rearranging, one obtains

X2{[k

+

Aa(x2 4

2 X. Cil } = AgAl 32

+ ~)]2 + t 32

Leaving out the trivial solution X= 0, we have

[k

+

~3(X2

+

~;)r

+

!2: = (AS:a)2x2

[108.3)

340 If we let k

+ A3 (X2 + ~) = 4 32

y

Equation [108.3] becomes 2

Y

= >"oAa ± 32 ·16

[108.4 ]

It is seen that the quantity under the radical can be positive only if k > O. In other words, a periodic oscillation with a frequency of one-third of the externally applied frequency can exist only if the system is self-excited.

'09. EXPERIMENTAL RESULTS The preceding theoretical considerations can be verified by the following experiment. An ordinary self-excited electron-tube oscillator is inductively coupled to a circuit containing an electromotive force with frequency w. The oscillating circuit is tuned so that it has a frequency of approximately w/2. The OSCillator has first been adjusted to a condit10n of soft self-exc1tation, that is, self-excitat10n starts from zero at a critical value of. the coupling k 1f one gradually changes the feed-back coup11ng. When the critical condition is thus establlshed, the coupling is decreased below that critical value (k < 0). Under such circumstances, the oscillator remains underexcited. If the external excitation of frequency w is now introduced, the previously discussed phenomena of resonance of the order one-half make their appearance. If the discrepancy t remains outside the interval ~t (Equat10n [105.20]), the circuit exhibits a vanishingly small heteroperiodic osclllation with a frequency the same as that of the external excitation. As soon as t enters the interval Atl = t t" - t t' , an intense subharmonic oscillation w1th frequency w/2 sets 1n at t = t;; this osc1llat10n passes through a max1mum tor a value of t 1n the 1nterval ~tl and disappears at t = tt, as shown by Curve 1 1n Figure 109.1. It one now reproduces the phenomenon for a somewhat smaller value >"02 of the ampliXl tude of the external excitation, one obtains Curve 2, which has a smaller maximum than Curve 1. For a sufficiently large value of >"0 the subharmonic resonance disappears entirely, wh1ch is in accordance with Equat10n [105.24]. It should be noted that in these experiments the oscillator remains below the point of selfexcitation 1f the external electroFigure 109.1 motlve force is wlthdrawn.

CHAPTER

xvm

ENTRAINMENT OF FREQUENCY

INTRODUCTORY REMARKS If a periodic electromotive force of frequency W is applied to an oscillator tuned to a frequency wO' one observes the well-known effect of beats. or heterodyning. which can be heard through a headphone in a circuit inductively coupled to the oscillator. As the difference between the two frequencies decreases. the pitch of the sound decreases, and from linear theory one may expect that the beat frequency should decrease indefinitely as Iw - wol • O. In reality. the sound in the headphone disappears suddenly at a certain value of the difference (w - wo). and it is found that the oscillator frequency Wo falls in synchronism with, or is entrained by. the external frequency w within a certain band of frequencies. This phenomenon is called entrainment of frequency. and the band of frequency in which the entrainment occurs is called the band or the zone of entrainment. Figure 110.' represents the difference Iw - wol plotted against the external frequency w; the interval ~w is the zone of entrainment in which both frequencies coalesce and there exists only one frequency w. On the basis of linear theory. the difference Iw - wol should be zero for only one value of w = wO' as shown by the broken 11nes. The phenomenon of entrainment of frequency is a manifestation of the non-linearity of the system and cannot be accounted for by linear theory. This effect was apparently recognized long ago. but its theory was not developed until recently. Thus. for example. Van der Pol. who developed the theory of the phenomenon (9). observes that "the synchronous t1mekeeping of two clocks hung on the same wall was already mown to Huygens." Before Van der Pol, Lord Rayleigh (10) observed a similar effect in connection with acoustic oscillations. Vincent (1'). M611er 1101 - 10101 (12). and Appleton (13) have also investigated the phenomenon. In recent years Russian physicists have analyzed the phenomenon in the light of modern methods of non-linear mechanics. so that at present the matter seems to be well understood and offers an inter,, "" esting field of research. particularly , " o 101 in connection with the problem of synchronizing OSCillating systems. Figure 1'0.' , '0.

The entrainment phenomenon has been analyzed in Chapter XVI on the basis of the quasi-linear theory of Kry10ff and Bogo1iuboff, but it is preferable to treat this matter in more detail, starting from the discussion of Van der Pol and concluding the analysis with the topological method of Andronow and Witt. Electron-tube circuits, as usual, offer a simple way of obtaining the differential equations from which conclusions regarding the phenomenon of entrainment can be formed. It must be noted, however, that the entrainment effect is a general property of non-linear systems acted on by a periodic excitation with a frequency in the neighborhood of the autoperiodic frequency of the system. Acoustic entrainment is also sufficiently well explored at present. As far as is known, no special studies of mechanical entrainment have been made so far, but the following example is worth mentioning. If one actuates a mechanical pendulum by a periodic non-linear torque, one obtains beats if the two frequencies are sufficiently far apart; these beats can easily be observed as the envelope of oscillations recorded on a moving chart. If the frequency w of the exciting moment approaches wo' the frequency of the pendulum, the period on the envelope becomes longer. At a certain pOint, the envelope suddenly becomes a straight line parallel to the motion of the recording paper, and beyond this point no further beats are observed. The nonlinearity of the torque in this case is generally due to the kinematics of the mechanism which drives the pendulum by springs attached to a crank. One may assume, therefore, that the phenomenon of frequency entrainment arises whenever there is a non-linearity in the differential equation of a system subject to an external periodic excitation with frequency sufficiently near the autoperiodic frequency of the system. DIFFERENTIAL EQUATIONS OF VAN DEft POL The differential equation of the circuit shown in Figure 111.1, in which E is an external electromotive force with fixed frequency Wl inserted in the oscillating circuit, is 111 •

L

~!

+

Ri

+

~ fidt o

-

M~~a

= Eo sinwlt

[111.1J

Assuming that the anode current i g , considered as a function of the grid voltage eg , is approximated by a cubic parabola, we have ia

=

f(e g} = Seg(l -

3~:2)

[1" .2J

c

Figure ",., where V. is the "saturation voltage, II which was defined in Section 5', and S is the transconductance of the electron tube. As usual, we shall neglect the grid current and the anode reaction. Introducing the notations t

I idt = .:D..-.

CVa •

MS

R

a = LC - L'

B

=

E V' 8

:::J1..

2

Wo

=

1 LC

we obtain [1'1.31

It is noted that the left side of this equation is of the same general type as Equation [44.']. Van der Pol assumes as the solution of this equation the expression [1".4J

where 6. and 62 are certain slowly varying functions of time. Substituting this expression into ["'.3J, equating like coefficients, and neglecting second derivatives, we obtain

["1.5]

where ao2 = ~

["1.6]

!" 4

It is apparent that, if 6. and 62 were constant, the solution v would be

periodic with frequency W1; according to the previous definition we would call such solutions heteroperiodic solutions. Van der Pol discusses two particular cases of Equations [111.5], namely, linear oscillation and absence of external excitation. In the first case y = 0, =~, and the equations are easily integrated. In the second case the approach to a limit cycle is ascertained, as is to be expected since iri this case we have the normal Van der Pol equation. For the general case, that is, when y :I: 0 and Eosin w1 t :I: 0, we shall follow the presentation of Andronow and Witt (14) which will permit establishing a more definite connection with the representation of the phenomenon in the phase plane. The results which will be so obtained coincide with those obtained by Van der Pol by a somewhat different argument (9). It is noted that Equations [111.5] are of the form studied by Poincare, that is,

a:

if b 1 and b2 are taken as the variables of the phase plane; thus the cond1t10ns for a stationary oscillat10n, 61 = constant and b2 = constant, reduce to P(b 1 , 62 ) c Q(b 1 , b2 ) = O. Hence the s1ngular po1nts of the system [111.5] g1ve precisely the cond1tion for the heteroper10d1c state with a single frequency w 1 • In the neighborhood of the singular pOints, b 1 and b2 are slowly varying quantities. This may be expressed by saying that the solution [111.4] is an amplitude-modulated function, the period of modulation approaching infinity as the frequencies Wo and WI approach each other. It is seen that the whole procedure is now reduced to the investigation of the singular pOints of the differential equations [111.5].

112 .

REPRESENTATION OF THE PHENOMENON IN THE PHASE PLANE

If we introduce the notations x

b

=~.

Y

tet

b

=~.

T=-

2

ao • Equations [1".5] become ao •

+

dy = ax

dT

y(1 -

r2)

+

[',2.,]

[1'2.2J A

The equation of the phase trajectories is dy dx

=

A

-

+

ax + y(1 - r2) ay + z (1 - r2)

[1'2.3J

To investigate the nature of the singular point (%o'Yo) we use the standard procedure given in Chapter III. If we let

+ e; Y the differential equations [112.2] become %

:!

=

:; =

P(t,17)

= %0

=

mt

=

pe +

+

=

Yo + 17

11.17

+ terms

q17

+ terms

with

[112.41

e

2

,

17

2

17

2

, •••

[112.5] Q(t,17)

The nature of the singular point the characteristic equation S2 -

(m

(%0'

+

q)S

with t

2

,

, •••

Yo) is given by the form of the roots of

+

(mq -

np) = 0

[112.6]

For a stable singular point it 1s necessary to ascertain first that the singularity is not a saddle point, which implies that (mq - np) should be positive. If this necessary condition is fulfilled, then the condition of stability is that the term (m+ q) should be negative. This implies that the real part of the roots should be negative; thus one has either a stable nodal point when the roots 8 1 and 8 2 are real or a stable focal point when 8 1 and 8 2 are conjugate complex. If the system [112.2] has stable singular pOints, the motion approaches an oscillation with a single frequency WI since both 61 and 6. tend to become constant for t .~. If the system [112.2] possesses a limit cycle, the functions % and y, and hence also e and 17, are periodic with period 2". The quantities 61 and 6. are also periodiC, which means that the Van der Pol solution [111.4] for this system represents beats between the heteroperiodic and autoperiodic oscUlations. We know from the theorem of Bend1xson, Chapter IV, that any nonclosed trajectory which neither goes to infinity nor approaches the singular points winds itself on a limit cycle. This limit cycle is stable for t .+~ and unstable for t • -~. From a practical standpoint only the stable limit cycles are of interest; we know from Section 25 that in the interior of such cycles there exist, generally, 211. + 1 singularities whose sum of indices is always +1. This means that, if the number of saddle pOints is n, the number of singularities with index +1 is necessarily 11. + 1 so as to make the sum of the indices +1. The "coordinates" (%0' Yo) of the singular point are given by the equations Xo

where

p

= -

ap

A;

Yo = -

= r02 is determined by the equation

pel - p) A

[112.71

["2.8] For a fixed A, Equation ["2.8] represents in the (p,a)-plane a curve of the third degree which gives the "amplitudes" yp = ro of the singular pOints for any a = 2(wo - wI)/a. The quantity A is the parameter of the family of the curves ["2.8]. Any point of the (p,a)-plane represents a singular point of the differential equations ["2.2] for a given value of the parameterA. The nature of the singular points in the (p,a)-plane depends on the nature of the roots of Equation ["2.6]. , , 3 • NATURE AND DISTRIBUTION OF SINGULARITIES; TRANSIENT STATE OF ENTRAINMENT

The curves represented by Equation ["2.8] have the appearance shown in Figure For sufficiently small values of the parameter A the curve consists of two branches, Ml and M;; the figure shows these branches for A2 = 0.', For an increasing A the branch Ml increases in size, and the branch , Ml rises until both branches Join as shown by the curve M. If A is further increased, there exists only one branch M2 shown for A2 It is noted that

"3.'.

=,.

Figure

"3.'

a

Unstable Nodal Paints

the curves Mof the family exist only above the a-axis and are symmetrical with respect to the p-axis. It one substitutes Expression ["2.4] into Equations ["2.2], one obtains Equations ["2.5] with the following values ot the coefficients m, ft, p, and q:

~[(1

*

:: =

=

-

p) -

bi]

+

71[ - (a

+

22:0 110)]

+

~2,

terms in

71

2

, •••

[113.' ]

e[a - ::r;oYoL~_-::pJ~~1l02]

+ t~rms

in

2

t • 'fl . ...

where %0 and Yo are given by Equations [112.7]. tion of the system [11,.1] is S2 -

2(1 -

2p)S

+ [(1 -

Thus the characteristic equa-

pH! - 3p)

+

a 2] = 0

[1'3.2J

From this equation one can determine the zones of separation of the roots of various types as was shown in Chapter III. These zones, when drawn in the same (p,a)-plane as Figure 113.1 in which the curves [112.8] representing the loci of singular points are drawn, will indicate the nature of singularities in that plane. We note first that the region of saddle points is determined by the ine qua l1ty (1 -

pHI - 3p)

+

a2

<

[113.3]

0

for, with this condition, the roots are real and of opposite sign. (1 -

pHI - 3p)

+ a2

The curve

= 0

is an ellipse E with its center situated on the p-axis, and Condition [1'3.3) means that the region of saddle points is situated inside this ellipse. The quantity under the radical sign in the expression for the roots 8 1 and S2 of ['13.2] is p2 - a 2 • Hence the straight lines Band B11 expressed by p

+

a = 0

and

p -

a = 0

which bisect the first and the second quadrants represent the divides between the real roots and the complex ones. These lines are tangent to the ellipse at p = 1/2. Inside the angle BOB 1 formed by these lines lies the zone of nodal points and saddle pOints; outside it. the zone of focal points. The area inside the ellipse, as was shown, is the zone of distribution or·-saddle points. The condition for negative real parts and hence for stability is clearly (1 - 2p)

<

0

Hence the line pp' of the equation p = 1/2 is the divide separating the roots with negative real parts (stable singularities) from those with positive real parts (unstable singularities). The former lie above that line; the latter, below it. This completes the picture of the distribution of the various singularities in the (p,a)-plane. From [1'2.8] one has a

= ±

./A2

VP- -

(1 - p)2

For a = 0 the ordinate of the curve is given by the equation p3 _ 2p2

+

p - A2 = 0

The condition for the reality of the three roots is A2

[113.4] < 4/27. For A2 > 4/27,

348 there exists one real root, and two conjugate complex roots which are of no interest here. For 0 < at < a 12 , where al is the abscissa of the point of intersection of the curve [112.8] with the ellipse (1 - p)(1 - 3p) + a 2 = 0, Equation [112.8] has three roots of which only one is stable as can be seen from Figure 11'.1. For al 2 < a 2 < +~, there exists only one unstable root. Hence, only in the region 0 < a 2 < al2 can the Van der Pol solution ['11.4] approach a stationary periodic solution with the heteroperiodic frequency WI' since only in that interval does there exist a stable Singularity so that the coefficients 61 and 62 in the Van der Pol equations [111.5] approach fixed values as t .. ~. From Equations ['12.2] it follows that the trajectories are directed radially inward for sufficiently large values of r2. Hence, if only one unstable singularity exists, we can assert by the Bend1xson theorem that a limit cycle exists and hence an autoperiodic oscillation Wo beating with the external frequency WI' Hence, whenever a 2 > a1 2 , which corresponds to the existence of a single unstable Singularity, the solution [11'.4] of Van der Pol has slowly varying coefficients 61 and 62 characterizing the heterodyning of the two frequencies Wo and Wt. If, however, a 2 < a1 2 , one singularity is stable w1th index +1, and the other two are unstable. No limit cycle exists in this case, and the stable singularity gives rise to a stationary heteroperiodic osCillation, as previously ment1oned. The topological study of the trajectories of the Van der Pol equations [1'2.2] in the zone of entrainment can be pursued by constructing the family of curves [112.8] for different values of the parameters with superimposed reg10ns of distribution of the various singular points, as shown 1n Figure 113.1. A topological analysis of this kind was carried out by Gaponow (15) on the basiS of the general conSiderations of Chapter IV, where singularities and limit cycles are considered as either sources (if they are unstable). or sinks (if they are stable) for the "flow" of trajectories in the phase plane. Such an analysis gives some idea of the transient state of the entrainment phenomenon under various conditions. Without going into the deta1ls of this analYSiS, since they have been given in Part I, it is sufficient to indicate a few interesting results. In pure entrainment, when only a s1ngle stable singularity eXists, the trajectories approach it in the usual manner Figure 11'.2

depending on whether the singularity lies in the region of stable nodal pOints or in that of stable focal points. When one unstable singularity exists, in view of the fact that for large values of r Equations [112.2] indicate the inward flow of traJectories, the Bendixson theorem indicates that a stable limit cycle exists, as shown in Figure 11,.2. This condition, as was just mentioned, corresponds to the quasiperiodic Van der Pol solution (111.4] when heteroperiodic and autoperiodic frequencies exist and no entrainment takes place. According to the form of curves K (112.8] and the different location of the Figure 11,., regions of stability (or instability), more complicated situations may arise, as was shown by Gaponow. Thus, for instance, for certain values of the parameters resulting in a particular shape of the curve [112.8] and for a certain range of a = 2(wo - wI)/a, one may have three singular pOints, namely, a stable nodal point, a saddle point, and a stable focal point. The flow of trajectories for this situation is shown in Figure 11,.,. Since there is a saddle point S, there also exists a separatrix K formed in the neighborhood of S by the stable asymptotes of the saddle pOint S. There is one singular trajectory SN issuing from S along its unstable asymptote and approaching the nodal point N. The focal point F is approached by a singular trajectory issuing from the other unstable asymptote of S. At a large distance from the singularities the trajectories are inwardly directed as shown. Depending on the form of the separatrix, the trajectories may approach either the nodal point or the focal point. Their approach to the nodal point will be aperiodic from a definite direction; their approach to the focal point will be in the manner of a spiral, which indicates an oscillatory damped motion. Another possibility is the combination of a stable nodal point, a saddle pOint, and an unstable focal point. This configuration is shown in Figure 11,.4. The separatrix forms a closed loop with the unstable focal point in its interior. The trajectories arriving from distant points of the phase plane approach the stable nodal point. A number of other combinations are possible, particularly when two singular points coalesce so as to form a singularIty of a higher order. Here the approach to the state of entrainment may be relatively complicated. This

350 coalescence of singular points occurs whenever the line a = a. in Figure 113.1 becomes tangent to the curve of singular pOints defined by Equation [112.8]. This transient state of the entrainment process can be studied oscillographically by analyzing the form of the envelopes of the oscillations. When entrainment is reached (16), the envelope becomes a straight line. STEADY STATE OP ENTRAINMENT

114.

Unstable Focal Point

During entrainment the heteroperiodic and the autoperiodic oscillations Pigure 113.4 become "locked," and the former imposes its frequency on the latter. In the Van der Pol solution [111.4] the quantities b. and bl then become constant and one can write 11 = bl sin WIt

+ h2 coswlt

=

Vh; + h:

= hsin(wlt

[sin wIt cos 106

+

+

cosw1t sin 106] [114.1]

106)

where

~ h

= cos 106;

and where b is the amplitude and 106 the phase of the oscillation relative to the externally applied voltage. One has tantP

= ~I = rI

Yo = 1 - p %0

a

=

(1 - ri)a 2(wo - WI)

[114.2]

where a and p are the coordinates of the stable singular point in the (p,a)plane. The amplitude b of the oscillation is h

=

Vb: + hi = aoV%; + y; = V!~ iVa

*

2

+

(1 -

p)2

=

kVt ~

[114.3]

where k = ta 2 + (1 - p2) is a factor depending on the difference of frequencies Wo - WI and the ordinate p of the stable Singular point. It 1s noted that the quantity ao =

Vi ~

[114 .4 ]

is the ampl1tude of the generat1ng solution of Poincar~; compare with Equation [54.5]. It 1s thus seen that the autoperiodic amplitude a o is affected by the entrainment factor k during the steady state of the phenomenon.

351 115. ACOUSTIC ENTRAINMENT OF FREQUENCY The preceding theory of entrainment was established in connection with an electron-tube circuit where experimentation is relatively simple and results can be established in terms of known parameters of the circuit and of the electron tube. As was mentioned in Section 110, the phenomenon of acoustic entrainment was discussed by Lord Rayleigh in his investigation on sound. More specifically, he says that if two organ pipes of slightly different frequencies are placed near each other, the beats disappear and both pipes osclllate at the same frequency. Later he reproduced an analogous experiment with electrlcally driven tunlng forks of sllghtly different frequencies; the entrainment effect is evident if the tuning forks are "coupled" by an acoustic resonator. A recent study of this effect was made by K. Theodorchlk M and E. Chaikln (17) at the sugtlon of Mandelstam and Papalexl. Without going into details, lt is sufficient to mention brlefly the experimental arrangement used. Flgure 115.1 shows an electrontube oscillator; in its anode circuit a telephone T ls inserted and in lts grid circuit a microphone M. The telephone and the microphone are also coupled acoustlcFlgure 115.1 ally by two armatures Al and A2 fixed to the same rod R. The rod is centralized by a spring and provided with a damper which ls not shown. The mechanical system AIRA2 is described by a linear differential equation of the second order having a frequency w. The oscillator is a non-linear self-excited system wlth frequency Wo on the limit cycle. If the difference w - Wo is appreciable, one finds that there are beats in the system, indicating the presence of both frequencies Wo and w. If the value of this difference is decreased, one finds that both frequencies coalesce into a single frequency w which corresponds to the external frequency mentioned in Section 110. The "non-linear frequency" Wo is thus entrained by the external one w, and it is found that the ratio w ~WQ of the zone of entrainment is proportional to the ratio a/ao where a is the amplitude of the oscillations of the mechanical system driven by the acoustic pressure emitted by the telephone, and ao is the amplitude of the autoperiodic oscillation in the electron-tube oscillator.

352 This method has been applied in measuring acoustic intensity by observing the magnitude of the band of entrainment, knowing ao' and determining the proportionality factor by calibration.

, ,6 . OTHER FORMS OF ENTRAINMENT The phenomenon of entrainment, as we have already indicated, has important applications in problems in which it is desirable to obtain synchronization of frequencies. For example, the problem of maintaining the speed of an electric motor with a high degree of accuracy can be solved by synchronizing the motor's frequency with the standard frequency of a quartz oscillator, which can be maintained with great accuracy. If such synchronization can be obtained, the motor speed can be maintained with the same accuracy. In this particular example, the frequency f9 of the quartz oscillator is generally many times greater than the rotational frequency fm of the motor. This difficulty is eliminated, however, by a frequency-demultiplication network, which permits obtaining a frequency f9/n if n is the demultiplication factor. The problem then consists of 111 0ckingll the two frequencies f9/n and f", by some kind of entrainment phenomenon. In the preceding sections of this chapter we have investigated the phenomenon of entrainment starting from a particular circuit investigated by Van der Pol. The non-linearity in this circuit is due to the characteristic of the tube which was approximated by retaining the cubic term in the representation of the non-linear function iA = f(e g } by a polynomial. For practical purposes, this type of entrainment is difficult to obtain because of the small zone of entrainment and also because it is difficult to modify the characteristic of an electron tube so as to produce more favorable conditions for entrainment. In view of this, numerous schemes have recently been developed in which the zone of entrainment is artificially made large by suitable circuits. In this manner one obtains a kind of artificially produced entrainment which is more adequate for practical purposes than the simple type investigated by Van der Pol. As an illustration we shall investigate one such scheme suggested by Kaden (,8) and shown in Figure ,,6.'. We shall omit the mathematical analYSis of the circuit, since it follows the argument previously explained, and will give only an elementary explanation of its behavior. The electron tube VI operates as an oscillator with frequency WI having C1 and LI as its oscillating circuit. The coefficient of inductance Ll can be varied within certain limits because the coil Ll is wound on an iron core whose state of magnetic saturation can be varied by changing the direct current i flowing through the coil G wound on the middle leg of the magnetic circuit M as shown. There are

353

A

C

T G

Figure 116.1 two other coils: Kl • which produces a feed-back voltage to the grid of Vl which merely maintains the oscillation with frequency wl • and the coil K2 transmitting the oscillation at that frequency to the second tube V2 working as an amplifier. The output of V2 through a transformer is coupled with the branch AB of the synchronizing network K. The network K also has a second branch CD into which the external frequency W2 is inductively transferred. The synchronizing network BACDis closed on a bridge N formed by rectifying elements; the direction of rectification is shown by the arrows. The diagonal points of the bridge N are closed on the saturation coil G of the iron-core reactor. In the circuit BAeD there are two induced voltages: El with frequency wl induced in the AB-branch. and E2 with frequency W2 induced in the CD-branch. We shall consider the case when the difference w2 - wl = ~wis small. The vector diagram is shown in Figure 116.2; the vector El can be assumed to be fixed; E2 rotates with frequency ~W in one direction or the other. depending on the Sign of the difo· ference w2 - Wl' The resultant vector E~ is the voltage between Band D and. to a certain scale. it represents the rectified current i flowing through the coil G. The frequency Wl = 1/YL l ( i)Cl where LtC i) is a non-linear function of i decreasing with increasing i. It is apparent that. if WI =1= wl • one has the relation [1,6.1]

Assume, for instance, that initially ~w > 0, which means that the extraneous frequency W2 1s greater than the frequency Wl of the oscillator. In the vector

o

Figure 116.2

354 diagram, the vector E2 rotates in the direction of the arrow A, that is, towards the advance, around the end of EI as center. The resultant Er increases, and, as with the rectifier-bridge arrangement shown, the current i is proportional to Er • The current will also increase so that L1(i) will be reduced. This accounts for an increase of the oscillator's frequency WI until it becomes equal to w2 • The equilibrium point drp [116.2) Cit = w2 - WI = 0 is stable. This can be shown easily by repeating the argument for W2 < WI provided rp is contained between 0 and". Thus, depending on the adjustment of the circuits, there is always an equilibrium phase angle rpo(O < rpo < ") for which [116.2] holds; that is, entrainment of the frequency WI by the external frequency W2 is artificially produced. If the frequencies WI and W2 are far apart, the vector E2 rotates rapidly with respect to the fixed vector E I , and in view of the fin1te time constants of the circuits the magnetic saturation control may not be sufficiently rapid to adjust the frequency WI so as to "lock" i t in synchronism with the frequency W2' One would then have beats due to the existence of both frequencies WI and W2' In the example given here the entrainment phenomenon is possible because of the non-linearity of the parameter LI(i). If the parameter LI were constant, that is, in the absence ot the saturable iron core, it would be impossible to obtain the synchronization ot the two frequencies WI and wa' and the zone of entrainment would be absent.

CHAPTER XIX PARAMETRIC EXCITATION

117. HETEROPARAMETRIC AND AUTO PARAMETRIC EXCITATION In Section 99 it was shown that it is possible to obtain selfexcitation of subharmonic oscillations by varying periodically a parameter of the system. In this chapter we shall investigate this phenomenon, called parametric excitation, from a somewhat different point of view and will introduce certain generalizations. It is noteworthy that the phenomenon of parametric excitation has been known for many years. Thus, for example, Lord Rayleigh describes in Reference (10) an old experiment of Melde (19) which he reproduces and analyzes. In this experiment a stretched string is attached to a prong of a tuning fork vibrating in the direction of the string; it is observed that periodic variations of frequency I in the string's tenSion account for the appearance of transverse vibrations of the string with a frequency of 1/2. Later, M. Brillouin (20) and H. Poincare (21) investigated a similar effect in electric circuits. Quite recently certain Russian physicists under the leadership of Mandelstam and Papalexi (22) investigated these phenomena in greater detail; we propose to give a brief outline of these researches. It is useful to define two types of parametric excitation, heteroparametric and autoparametric. In heteroparametric excitation, self-excitation is caused by the variation of a parameter expressed as an explicit function of time. In autoparametric excitation the variation of the parameter depends directly on some physical quantity and thus is an implicit periodic function of time. The vibrations of the string in Lord Rayleigh's experiment are clearly heteroparametric in that the variation of the parameter is produced by a tuning fork having a definite frequency. Parametric excitation occurs here with a frequency equal to one-half the external frequency of the tuning fork. The same remark applies to the circuit described in Section 99 where the capacity is modulated as an explicit function of time, On the other hand, as has been shown on numerous occasions, selfexcitation of electron-tube circuits can be traced to the fluctuating transconductance of the tube caused by the oscillatory process itself. In all electron-tube circuits the periodic variation of the parameter, the transconductance, appears as an explicit function of the physical quantity which characterizes the process, for instance, the grid voltage, and depends only implicitly on time, Self-excitation of electron-tube circuits therefore belongs to the autoparametric type. The concept of autoparametric excitation

356 is not particularly interesting because most excitations of the autoparametric type can be treated by the standard method of Poincare. With heteroparametric excitation, however, the situation is different. Since one or several parameters of the system appear as explicit periodic functions of time, the problem is reduced to the solution of differential equations with periodic coeffiCients, that is, equations of the Mathieu-Hill type; more specifically, the condition of self-excitation of the system is equivalent to the existence of unstable solutions of such equations, which means that an initially small departure increases because of the periodic variation of a parameter. It must be noted that, although the theory of the Mathieu-Hill equation is necessary for the establishment of the conditions of heteroperiodic excitation, there is nothing in that theory which would permit determining the amplitude of the ultimate steady state. This difficulty arises from the fact that the known types of Mathieu-Hill equations are linear equations and, as such, possess unstable solutions increasing indefinitely in their unstable region. In order to establish a theory of heteroparametric excitation approaching a definite steady state, one should apply some kind of non-linear differential equation with periodic coefficients. Unfortunately, no theory involving non-linear equations with periodic coefficients exists at present. These theoretical difficulties limit a further analysis of heteroparametric excitation. It is interesting to note that Mandelstam and Papalexi, who developed a heteroparametric generator, an electric machine described in Section 124, were able to demonstrate that, in the absence of non-linearities in the circuit, the voltage builds up indefinitely until the insulation is punctured. On the contrary, by providing a non-linear element in the circuit, the voltage builds up to a finite value, and the generator functions in a stable manner. In spite of these theoretical lImitations, equations with periodic coefficients can be used to determine the conditions of heteroparametric excitation in a general manner, as will be shown. 118.

DIFFERENTIAL EQUATIONS WITH PERIODIC COEFFICIENTS Consider a differential equation with periodiC coefficients (23) :i

+ 2p(t)z +

where p(t) = p(t + 2") and q(t) ~ defined by the equation

= q(t

q(t)z

+ 2").

z == xe

= 0

[118.1]

If we introduce a new variable

-Jpdt

[118.2)

Equation [118.1] becomes

x + M(t)% where M( t)

=q

- p2 -

=

0

P is a periodic function.

[118.3]

357 In practice one frequently encounters the following expressions for M( t):

+ a 0 2 cos kt, in which case Equation (l18.}] is called the Mathieu equation, or 2 2. M(t) is a Fourier series, wo + Al cos kt + A2 cos 2kt + ... + BI sin kt + B2 sin 2kt + ... , in which case [118.3) is called the Hill equation. 1.

M( t) :::

w0 2

Since the theory of the Hill equation is similar to that of the Mathieu equation, it is sufficient to consider the latter. The Mathieu equation [118.41 by a change of the independent variable

x+

(w 2

+

a

2

t ::: kT

can be reduced to the form

cos t) x = 0

[118.5)

where wk = Wo and ak = ao. The essential feature of the Mathieu equation [118.5] is that, although the functionM(t) = w 2 + a 2 cos t is periodic, its solutions are not necessarily periodiC although under certain conditions they may be periodic. If they are periodic, the solutions are given in terms of the so-called Mathieu functions (2}). Since Equation [118.5] is linear, one can assert that, if one knows two particular solutions II and 12 forming a fundamental system, the general solution will be of the form [118.6) where Al and A2 are arbitrary constants. Moreover, since II(t + 2~) and 12 (t + 2~) are also solutions, one can express them in terms of II(t) and 12 (t) by equations of the form

Ii (t +

2~)

= all (t) + bJ2 (t)

12 (t +

2~)

= ell (0 + dl2 (t)

[118.7]

Prom [118.6] one has also F(t

+

2~)

=

+

Alf1(t

+ A 2 f 2 (t +

2~)

[118.8]

2~)

From the theorem of Floquet (24) we know that there is a solution F such that F(t

+

2~)

= Al~ (t

+

2~)

+

A 2 f 2 (t

+

2~)

=

uF(t)

[118.9]

If we select the following initial conditions,

It (0)

= 0;

f; (0)

=

1;

fiO)

= 1;

we observe that the Wronskian II (0)

12 (0)

f; (0)

1;(0)

::1=

0

f~(O)

=

0

[118.10]

and thus the system of solutions 0 [118.7] one gets for t

11

and

12

is fundamental.

From Equations

g

II (271")

==

bi

1 2(271")

==

1;(271")

d;

== a;

1;(271") = e

[118.11]

From [118.9] one obtains F(t

+ 271")

= Al (all

+ 6/2 ) +

A 2 (e!1

+

[118.12]

d/2 )

Since 11 and 12 are the solutions of [118.5], clearly

I;' + M(t)/ = 0; I;' + M(t)/ 1

whence It/II

= 12"/12

2

=

and, therefore, Itl2 - Itll

0

= 0,

that is,

1;/2 - /./11 == h == constant

[118.13]

h = 1/(0)1;(0) - 1;(0)/1(0) == 1;(271")~(271") - ~'(2"')~ (2.,,)

[118.14]

The value of h is which, by [118.10] and [118.11], becomes [118.15]

1 = ad - 6e

From Equations [118.9] and [118.12] in view of the in1tial conditions [118.11] one gets Al (a AI"

+

(7)

+ A2e

A 2 (d -

(7)

= 0

[118.16]

== 0

Thus, in view of [118.15], the condition for the non-trivial solution of the system [118.16] is (12 -

(a

+ d)(7 + (ad -

be) =

(12 -

(a

+ d)(7 + 1

== 0

['18.17]

The roots of the characteristic equation [118.17] are (71,2

== a;d

If we put a ~d

±v(a~d)2 -1

= cos

=

a;d

±

iV

1 - (a

~d)2

[118.18]

2wp, this equation be~omes (11,2

+ j27f1l == cos 27r~ + _ j sin 27r~ == e-

[118.19]

If a ~d < 1, cos 27r~ is real; hence, p is also real, and (7 is complex with modulus equal to one. This characterizes stability, both of equilibrium and of the stationary motion, from the very definition of (1, Equation [118.9J. If ~ > 1, IJ is imaginary; hence, (7 is real, and there is a· root greater than one which indicates instability. If a ~d = 1, p = 0, and hence (7 = 1; this should be considered as the beginning of the unstable range of the Mathieu equation.

359 Taking into account the values [118.11], one can write cos 21r1l =

!

(a

+

d) =

!

[1/(21r)

+

1 2 (21r)]

[118.20]

STABLE AND UNSTABLE REGIONS OF THE MATHIEU EQUATION Let f = cos yt and ~ = ~ s1n yt be a pair of fundamental solutions in the interval 0 ~ t ~ 1r with initial conditions f( 0) = 1, ~ (0) = 0, f'(O) = 0, and ~'(O) = " and let

119.

get) = C sin 6t

+

D cos 6t;

h(t) = E sin 61

+

Fcos 6t

[119.1]

be a pair of fundamental solutions in the interval 1r ~ t ~ 21r, where y = yw 2 + a 2 and 6 = yw 2 - a 2 • Fitting these functions together at t = 1r, we obtain f(1r) = g(1r) or cos y1r = C sin 61r + Dcos 61r f'(1r)

=

g '(1r)

or

- y sin y1r

,p(1r)

=

h (1r)

or

!y sin Y1r =

,p'(1r)

=

h'(1r)

or

cos Y1r

=

=

C6 cos 61r - D6 sin 61r

E sin 61r + F cos 61r

E6 eos61r -

[119.2]

F6 sin 61r

From these equations we can determine the four constants A, B, C, and D and thus determine cos 2 1r1l

=

f(21r)

+ 2

,p' (21r)

[119.3]

=

One finds that for w 2 > a 2 > 0, cos 2 1r1l

cos 21r1l

= Yal!

= cos 1rY cos 1r6 - 2"1 (y if +

r6).sm 1rY sm. 1r6

= cos 1ry cosh 1r1'/ - ~ (; - ~) sin 1ry sinh 1r1'/

[119.5)

w2 • From these equations one may plot curves in the (w 2 ,a 2 }-plane which are the boundaries between the regions of unstable motion (shown in white in Figure 119.1) and of stable motion (shown by shading). This discussion is taken from an article by Van der Pol and Strutt (25). These authors discuss the character of the stable and unstable regions for various values of the two parameters a 2 and w 2 and derive the following conclusions:

where 1'/

1.

-

The unstable regions cover a larger area than the stable ones.

2. Below the 45-degree line in the first quadrant the motion is, in general, stable. Here a 2 < w 2 , the stepwise "ripple" appearing in Figure '19.1

360

I '~~~44'¥I-l 4r-~~-+~~---r--;-~+---+--,~~~~~~~~

Figure 119.1 does not touch the zero line, and the coefficient of x in the Mathieu equation remains positive. Thus, without the ripple, one would always have stable motion. Under certain conditions the ripple renders the motion unstable.

3. Above and to the left of the 45-degree line the motion is generally unstable; the stable areas which exist are relatively small. Without the ripple the motion is unstable in this region so that the ripple under certain conditions transforms the instability into stability. The last conclusion is illustrated experimentally by a reversed pendulum whose support undergoes a periodic vertical motion. It is found that, for a certain band of frequencies and for a certain amplitude of the motion of the support in the vertical direction, the unstable pendulum exhibits stability. In what follows we shall be interested particularly in the unstable solutions of differential equations with periodic coefficients and will extend the discussion a little further to ascertain whether self-excitation will exist under various conditions of frequency and phase of the ripple relative to the fundamental oscillation of the system. Instead of following the analytical argument of Van der Pol and Strutt, we will investigate the behavior of the phase trajectories, which will enable us to gain a more intuitive understanding of the phenomenon of heteroparametric excitation.

120.

PHYSICAL NATURE OF SOLUTIONS From the preceding analysis it follows that in certain regions of 2 2 the (w ,a )-plane the solutions of the Mathieu-Hill equation are unstable. These regions of instability have not as yet been explored to any extent because the aim of previous analytical studies has been the establishment of conditions of stability which resulted in the Mathieu functions, with which we are not concerned here. On the contrary, for parametric self-excitation, in which we are interested here, the unstable regions present greater interest. Although by introducing the Mathieu-Hill equation we lose the familiar ground of the theory of Poincare, that is, it is impossible to eliminate time between the two differential equations of the first order, the procedure is more direct, as will be shown. The main limitation of this method, as was already mentioned, is the fact that since the Mathieu-Hill equation is linear, there is no indication whatever as to how the gradually increasing oscillations of the unstable region reach a steady state. To determine this it would be necessary to investigate a non-linear equation of the Mathieu-Hill type, but, as we pointed out, no theory of such equations exists at present. Since we are unable to proceed analytically with a non-linear Mathieu-Hill equation, it is still possible to form a certain physical idea as to what happens in the unstable region of solutions of this equation by the following argument of Mandelstam. Assume that we have a non-dissipative oscillating circuit with a capacity which varies periodically between the two limits Cmll" and C mln • Let the capacitor have initially, that is, when t = 0, a certain charge q; the circuit has no current. Since there is no essential difference between the solutions of the Mathieu equation and those of the Hill equation with the function M(t) = w 2

+ .!a 2 (cos t

-

'IT'

1.3 cos 3t +

.!.cos 5t - . . . ) 5

[120.1]

representing a rectangular ripple, see Section 119, we can adopt the argument of Van der Pol and Strutt and consider abrupt variations of capacity from C ma" to C miD ' and vice versa, occurring periodically. Assume, therefore, that for t = 0 the capacity is suddenly decreased by LiC = C max - C min' Since the whole energy stored in the circuit is purely electrostatic, it is apparent that the impulsive work done during this sudden decrease of capacity is LiC

2c 2 q

2

[120.2]

This amount of energy is thus added to the initial weak electrostatic energy existing in the capaCitor prior to the instant t = O. The capaCitor will then begin to discharge through the circuit and a current will appear. Assume now

that one-quarter period later, when the capacitor is totally discharged and the energy is entirely electromagnetic (Li 2/2), we restore abruptly the original value of the capacity Cmu by giving the increment +AC to the capacity. In so doing no work will be performed since the electrostatic energy is zero at this instant. However, from the fact that during the preceding operation the electrostatic and, hence, the total energy of the system has received an increment :~q2, it is apparent that it is still present in the system Which has been assumed to be conservative. If one-quarter period later we repeat the procedure made at t = 0, that is, reduce the capacity by the amount AC, another increment of the electrostatic energy will be added, and so on for subsequent abrupt changes ACof capacity occurring periodically every quarter period of the circuit. It is thus seen that energy is injected into the system periodically at the instants t = n being an integer, when the capac~ ity is suddenly changed by the amount - AC; the restoration of the capacity (+ dC) to its maximum value occurs at the instants t = (2n + without involving any work. It is observed that the period of variation of the capacity is one-half the period of the tree OSCillatory phenomenon. The argument remains the same it, instead of capacity variations, inductance variations ± AL are used. The timing of the ripple for inductance variations is exactly the same as for capacity variations, namely, the coefficient of the inductance is decreased (-AL) at the instants 0, T/2, ••• , and increased (+ AL) at the instants T/4, 3T/4, •... To the same timing, however, there will correspond a diametrically opposite effect, that is, at the instants T/4, 3T/4, ... , when L is increased, there will be an addition ot energy since the whole energy is electromagnetic at these instants; whereas at 0, T/2, ... , when L is decreased, no work will be done since the electromagnetic energy 1s zero.

2n¢,

l)f

TOPOLOGY OF THE HILL-MEISSNER EQUATION The Hill equation with the rectangular ripple expressed by Equation [120.1] was used by Meissner in his analysis of vibrations arising in driving rods of electric locomotives (26) and was found useful by other investigators (25). Very frequently this particular form of Hillis equation is designated as the Hill-Meissner equation; we will inquire further into the nature of its solutions. The usefulness of the Hill-Meissner equation 11es in the particular torm of its periodic coefficient, the ripple, which permits a Simple discussion of its trajectories 1n ths phase plane. It is apparent that the trajectories of the Hill-Meissner equation differ somewhat from those of the Mathieu equation, but it is likely that, at least qualitatively, there is not much difference between the shapes ot integral curves for both equations, as was pointed out by Strutt (27) and as follows from experimental evidence. It

121.

is to be understood, however, that no direct quantitative comparison of the unstable solutions of equations of both types has been attempted so far, and the above assumption seems to be a plausible hypothesis convenient for a qualitative analysis of the phenomenon. With these remarks in mind, we shall write the Hill-Meissner equation in the form [121.1}

which means that we consider alternately the two equations [121.2}

during each half period ~ of the ripple, with the understanding that the solutions have to be continuous on physical grounds although not necessarily analytic at the points at which the changes from (a 2 + 6 2 ) to (a 2 - 6 2 ), or vice versa, occur. We will assume that a 2 > 62 inasmuch as we will be concerned with the problem of modulation of the quantity a 2 by a rectangular ripple t 62 • We will now elaborate somewhat the example of heteroparametric excitation discussed in the preceding section and write the differential equation of the non-dissipative circuit in the form d 2q Lo dt'l.

1

+ Cq

= 0

[121.3]

where Lo is the inductance, C is the capacity, and q is the quantity of electricity stored in the capaCitor. Let us assume that the capacity C varies between Crnax ; Co + .dC and Cm1n ~ Co - .dC In a stepWise manner. The preceding equation can then be written as .. 1 0 q+ LC(l+y)q= o 0 c

[121.4}

where Ye = .dC/Co is the index of the stepwise modulation. If l/L oCo = w0 2 and if we assume that Ye « 1, without any loss of generality Equation [121.4J becomes [121.5}

This equation, as was Just explained, should be considered as an alternate sequence of the two equations [121.6J

A trivial change of the independent variable tions into the form

t

= T/W o

transforms these equa-

[121

where a1 2 = 1 + Ye and a2 2 = 1 - Ye • the "frequency" of the ripple t IJ.C.

.7}

The two equations replace each other at

Since no confusion is to be feared, we shall deSignate by q and q the derivatives dT with respect to the new variable T, the angular time. Let us transfer the problem into the phase plane of the variables q and q. The solutions q(T) will then be represented q by the integral curves, or phase trajectories, of Equations [121.7J, and the dynamical process described by these equations will be represented by the motion of the representative point on these trajectories; see Part I. Figure 121.1 For )Ie = and al2 = a2 2 = 1, the trajectories of Equations [121.7J form a continuous family r. of concentric Circles with the origin as center. If)le ¢ 0, the trajectories form continuous families r l and r2 of concentric homothetic ellipses shown in Figure 121.1. The family r l corresponding to a l2 > , has a constant ratio 6/a = al 2 = 1 + )Ie of semiaxes; the family r 2 has a ratio 6' /a = a22 = 1 - )Ie' The family r, corresponds to the reduced value Co - ~C of the capacity and r 2 to the increased value Co + ~C. The origin 0 is the singular point of Equations [121.7J. The two families r l and r 2 thus serve as a kind of reference system determining the motion in the phase plane. For example, if for t = certain initial conditions, say (qo,O),are given and the value of C is prescribed, for example, C = Co - ~C, the process is depicted by the motion starting from the point A corresponding to the initial conditions and moving along the ellipse of the family r l passing through A. If at a later instant t = t l , corresponding to the point B on the ellipse, the capacity is changed and is then C = Co + ~C, the representative point will pass onto the elliptic trajectory belonging to the family r 2 passing through B and will continue to move on that trajectory until the next change (C= Co - ~C), and so on. This representation of the solutions q(T) of Equations [121.7J by phase trajectories is a convenient way of ascertaining the various circumstances of heteroparametric self-excitation. As an example, let us consider self-excitation when a capaCity ripple, discussed in Section 120, is present. Let us start from a point A(qo'O), see Figure 121.2, after the capacity has been reduced (C= Co - ~C). The representative point will move on the arc AB of the elliptic trajectory of the family r l • At the point B (q = 0, q = max) the capacity is increased (C = Co + ~C), and the arc BC of the family r2 is followed. At the point C the capacity is reduced, and the next arc CD is of .!!!I...

°

°

the family r 1 , and so on. After a ~ dT period 2" one reaches the point E B corresponding to q2J, > qo which shows that the energy content of the system has been increased. c It must be noted that q the phenomenon is reversible; in fact, if we replace the words "capacity is decreased" by "capacity is increased,lI and vice versa, in the argument of Section 120, it is apparent that instead of adding o energy by capacity variations, energy will be withdrawn by these Figure 121.2 variations. Physically this means that, instead of injecting energy into the system by providing external impulsive work which will overcome the electrostatic forces, energy will be withdrawn because the electrostatic forces will do the impulsive work and will thus diminish the energy content of the system. This situation is shown by the trajectory AB'C' ..• in Figure 121.2. If, starting from the point A, as before, the capacity is increased (+ de), at B' decreased, at C' increased, and so on, a convergent spiral will result which represents withdrawal of energy. In this example the trajectories are spirals made up of elliptic arcs; these spirals have continuous tangents at every pOint, although there are discontinuities in the curvature at the pOints B, C, D, E, •.• , at which th~ changes of capacity occur. In other words the trajectories of the HillMeissner equation with which we are concerned here are piecewise analytic curves, possessing continuous first derivatives but discontinuous second derivatives at pOints where a loss of analyticity occurs. In a more general case analyzed in the following section the piecewise analytic trajectories may have discontinuous first derivatives at certain points. 122.

DEPENDENCE OF HETEROPARAMETRIC EXCITATION ON FREQUENCY AND PHASE OF THE PARAMETER VARIATION

In the preceding section we studied a special case in which the discontinuous changes in the rectangular ripple occurred at the instants when the representative point crossed the coordinate axes of the (z.x)-plane and the frequency of the ripple was twice that of the circuit. This case, which is the one studied by the early investigators, is also the one most frequently encountered in practice. We will now outline a more general method of approach

to thls problem by considering dlfferent relative frequencies and phase angles of the ripple with respect to the oscillatory process in the circult. For this purpose we shall extend somewhat the study of the preceding sectlon. The radlus vector r of a phase trajectory of the family r l , for example, is glven by the equatlon Figure 122.1

r=

V'/cos I

2

¢J

+ ~ sin ¢J

[122. , J

a1

where ¢J is the angle of the radius vector; a o = OA, the semiaxis on the ~­ axis; and 0 12 = 1 + 10 as before. If the change of capacity occurs at some pOintM, see Figure 122.1, whose coordinates are ~l = r l cos ¢J l and Yl = r1 sin ¢J 1 , where r 1 corresponds to the angle ~1' the arc of the family Jr2 corresponding to C = Co + ~C will begin at the point M and will continue to the point N( r 2 , tP2) at which a change of capacity from Co + ~C to Co - ~C occurs and a new arc NP of the family r l will be traversed. It we start from a given point of the phase plane and assume a particular subdivision of angles ~2 -~l' ~8 - tP21 ••• , it can be shown (28) that the subsequent major semiaxes ai' a2' of the elliptic trajectories can be calculated by an elementary recurrence procedure. Thus, for example, starting from the point A in Figure 122.1, one obtains after lVchanges of capacity the following expressions for the major semiaxis aN: [122.2J a

= a ~

= a N

211+1

0a

. 2

11/3 .. . 1 211 + 1 ~~ J2 J 4

••• ~

[122.:5 J

J2v

where [122.4 ] From the properties of the functlons

li(~i)

it is apparent that [122.5 ]

The only case of practical interest is that in whlch all intervals are equal and are fractlons of 2k", where k ls an lnteger. We shall call the intervals in this mode of subdivision the equiphase intervals inasmuch as the phase plane is divlded into equal sectors. If lV is the number of changes of capacity in one period (2") of the process, the phase angles for lV= 4 will be

· ·,

. ..

I•

I•

•

I

"

~

~:

:J -- r -+ -- i---

I'

Jr----I-~:_1

-1t----1--


m

--

--

-

F-

1

Figure 122.2 2" rP8 = rPo + 1rr rP2 = rPo + 2' 2' and rP. = rPo' We may cal 1 rPo the pha8e angle of the ripple and the number Nthe relative frequency of the ripple. Figure 122.2 shows the relative position of the ripple, with N= 4 and a certain arbitrary angle rPo' with respect to the free oscillation of the system. It is to be noted that in a more detailed investigation of this phenomenon one has to take into account the tact that the equiphase intervals are not equitime intervals, that is, intervals of equal time, because ot the non-uniformity of the motion of the representative point on the elliptic traJectories. Although this circumstance can be taken into account by defining certain functions g,(rP,) similar to the functions fi(rP i ) Just introduced, we shall not elaborate on this subject here but will investigate the principal features ot heteroparametric excitation on the basis of the equiphase intervals. There is sufficient Justification for this because in the most important practical cases, when the changes of capacity occur on the coordinate axes of the phase diagram, both types ot intervals coincide; when they do not, the introduction of equitime intervals, while complicating the calculations someWhat, does not change the qualitative aspect of the phenomenon of heteroparametric excitation. It is convenient to consider the following four groups of numbers N. 1. N = 4 'J,I; 2 . N = (2 J,I + 1) 2 ; } . N = 2 J,I + 1; 4. N = p / q, where J,I = 1, 2, }, ••• , and p and q are relatively prime. In the first three groups N is an integer, and in the last it is a rational fraction. This covers all cases of practical interest.

rPo' rPl

1.

= rPo

" +~,

First group: N

= 4, 8,

12, •.•

Let us consider the first case, N= 4, which has previously been studied by an elementary method. We shall now apply the general method involving the use of the functions fi(rP.). The intervals are clearly rPo, rPl = rPo + ~, rP2 = rPo +.?;, rP8 = rPo +"';, and rP. = rPo, By the properties [122.5]

of the functions Ii (~d it is apparent that 10

=

12

J;.;

=

11

=

la

whence, by Equation [122.2] we have a =

4

J..1a --

ao f f

112

ao /2

[6] 122.

240 It is evident that the condition for a heteroparametric excitation is a. > a o, that is, 112 ~ 1 12 -

Figure 122.'

o

If we substitute for 11 and 12 their values from Equation [122.4], it is easy to discuss the conditions for self-excitation of the heteroparametric oscillations. We will omit these elementary calculations and merely indicate the conclusions. The zones of the phase angle ~o in which self-excitation occurs are located within the shaded sectors shown in Figure 122.'. In the non-shaded sectors, self-excitation does not occur. The lines AC and BDform the critical phase angles ~o for which self-excitation appears or disappears. The maximum increase of the amplitudes per cycle occurs for ~o = 0 and ~o = n. For these values of ~o one has the relation a

4

2.

Second group: N

= 6,

1+ " = -aa~2 a0 = -a 1 _" 0

[122.7]

2

10, 114-, 18, •.•

Consider, for instance, the case when N= 6. By Equation [122.2] a6

_ -

ao

111316 ~14/s

[122.8]

and, for the intervals in question, 10 la = Is, 11 c 1" and 12 = 1 6 , which shows that as = ao for all values of ~o' Hence, for this particular frequency of the ripple, parametric excitation is impossible. 110

3 . Third group: N

= 3,

5, 7, 9,

Let us take the case N= 3. [122.3] and obtain

Since N is odd, we apply Equation

Ilia

a

a = al- - a 0 a I 2

2

From the form of the subdiVisions and from Equation [122.5J, la the condition for self-excitation will be

~ 1110 ~ 1 a2

12

= 10 ,

Hence,

[122.9 ]

,69 By introducing for f o • fl' and f2 their values from Equations [122.4] and carrying out the calculations. one finds that no self-excitation is possible in this case. 4.

Fourth group: N = p/q

One easily ascertains that as far as self-excitation is concerned most fractions p/q are either of no interest or fall within the scope of the groups previously investigated. The only cases in which heteroparametric excitation occurs are those in which N= 4/3. 4/5. 4/7. 8/3,8/5, ... , corresponding to the intervals ~~ = 3"/2, 5"/2,7"/2, ... , which fall into the first group. 123.

HETEROPARAMETRIC EXCITATION OF A DISSIPATIVE SYSTEM If a system is dissipative, it is apparent that the injection of energy communicated by the variation of a parameter must, on the average. exceed the energy dissipated by the system. This. as we shall see, will lead to an additional condition. Let us consider the equation Loq + RoQ +

1

c

=0

q

[123.1]

of a dissipative electric circuit in which we assume that the capacity C is modulated by a ripple ± ~C so that C = Co ± ~C. If we divide by Lo and put Ro/Lo = 2po and 1

LoCoU

± "'c) =

2(1 -

+,..c

Wo

)

Equation [123.1] becomes

q + 2Poq +

w; (1

=+=

,,)q

[123.2 ]

= 0

When the new variable Q defined by the equation q = Qe-!Pndt

[123.3 ]

is introduced, Equation [123.2] becomes

Q+

Changing the independent variable from t to T

::~

[123.4]

w,2(l =+= 6)Q = 0

+

(1 =+= 6)Q

= Wit.

=0

one obtains [123.5 ]

As mentioned previously, this equation is equivalent to the alternate sequence of the following two equations replacing each other at each discontinuous change of capacity:

370

[123.6 ]

where

2

a 12 = 1 + 6 = 1

+

~ W

1

2

'V •

'c'

a 22 = 1 - 6 = 1 -

W -3 W 'e 'V

[123·7]

1

The plus sign in the first equation (123.7] corresponds to C = Co - ~C. Equations (123.6] have the same form as Equations [121.7], so that the conclusions reached for those equations are applicable here except that Equations [123.6] contain the dependent variable Q whereas Equations [121.7] contain the variable q; the two variables are related by Equation [123.3]. The trajectories of Equations (123.6] are either convergent or divergent piecewise analytic spirals formed by elliptic arcs, the closed trajectories, appearing as a threshold between the two forms of spirals. Por a closed trajectory Q is bounded; hence, by [123.3] q decreases monotonically. This means that convergent spirals in the (q'~)-Plane correspond to the closed trajectories in . the (Q'~-Plane so that noTself-excitation is possible. It is obvious that in order to have parametric excitation, the amplitude q must increase monotOnically, or at least be constant, which requires that [123.8]

where PI ~ Po' This means that the traJectories in the (Q'~-Plane must be divergent spirals with the absolute value of the average negative decrement Ipil greater than, or at least equal to, the positive decrement Po = Ro/2Lo of the diss1pative circuit (Ro,Lo'Co)' Physically this means precisely the condition stated at the beginning of this section, namely, the energy inJections into the circuit by the r1pple t~c must, on the average, be greater than the energy dissipation. On the other hand, since for a dissipative circuit with constant parameters (Ro,Lo'Co) the decrement is Po = Ro/2Lo, the ratio of the amplitudes after one turn 2" in the phase plane is q2rr

qo

[123 .9 ]

= e- po2rr

In the optimum case of parametric excitation (Nr::: 4, tPo (122.7] one has

= 0),

from Equation

which def1nes the 1ncrement (123.10 ]

371

Expressing the condition for parametric excitation, namely, PI stituting for Q(, Q22, and Po their values, one gets ...

l.~

6:

2 W0 - P02 W 2 0

ello ello

-

1

+1

=

~

Po' and sub-

Y:•

where #0 == TrRo/Lowo' Since Q 12 and Q 22 are to be positive and obtains the other limit for Ye' This gives

[123.11 ] 2 Q1

>

Q2

2

,

one

[123.12J

From these expressions it follows that Ye must be in the interval [123.131

in order to obtain heteroparametric excitation. When Ro = 0, we have #0 = 0 and Po = 0 which gives the interval (0,1). This interval decreases when Ro increases and becomes zero when Wo 2 - P02 = wt 2 == 0, that is, when Ro == 2~. The last expression is the condition for critical damping. It is therefore impossible to obtain parametric excitation of a critically damped, or overdamped, circuit. As was previously mentioned, heteroparametric excitation can be obtained by a variable inductance, instead of a variable capacity. For a non-dissipative circuit the conditions of heteroparametric excitation are identical except for the phase of energy inJect10ns, as was ment10ned at the end of Section 120. This similarity in the effects resulting from the variation of capacity and inductance arises from the fact that both these factors enter symmetrically into the expression w0 2 == 1/LoCo for the frequency. For a dissipative circuit the situation is different in that the capacity enters only into the expression for the damped frequency W I 2 == W 02 2 The - P02 (through W 0 ) and does not appear in the decrement Po = Ro/2Lo. inductance Lo appears both in the expressions for the frequency and for the decrement. Hence, a priori, one may expect different results in both cases. One can develop the theory of inductance modulation in exactly the same manner in which we have outlined the effect of capacity modulation •. The only difference lies in the fact that for the inductance ripple instead of an interval in which the index o/modulation Ye must remain in order to obtain self-excitation, the condition for self-excitation is given by an inequality. It is also noteworthy that in the preceding discussions it was assumed that y« 1, which enabled us to simplify the expressions by writing 1 y ~ 1 - y. By waiving this restriction, the calculations are considerably more complicated but the qualitative picture of the phenomenon remains substantially the same.

!

372 124.

HETEROPARAMETRIC MACHINE OF MANDELSTAM AND PAPALEXI

A differential equation with periodic coefficients of the form L(t)x

+

R(t)x

+

C(t)x = 0

[124.1]

can be reduced to an equation of the Mathieu-Hill type. It follows that the effect of a parametric excitation can be obtained by periodically varying one ·of the parameters L, C, or R. The variation of the parameters Land C has been studied suffiCiently in preceding sections. As to the parameter R, it must be noted that only those parametric variations which extend alternately into the region of negative resistance are capable of producing parametric excitation. We shall not go into this matter here, as it is clear that on physical grounds negative resistance means the supply of energy from an outside source. Mandelstam and Papalexi (22) developed an interesting generator of electrical energy, which they called a "heteroparametric machine." The arrangement conSisted of a series of coils located on the periphery of a stationary disk; the inductance of these coils was varied by the periodic passage of teeth and slots on a rotating disk parallel to the stationary disk. The frequency of the inductance variation thus obtained was of the order of 2000 cycles per second. In a circuit of this kind devoid of any source of energy other than the kinetic energy of the wheel, electrical oscillations of exactly half the frequency of the parameter variation were observed. For a linear system, corresponding to a linear Mathieu equation, the oscillations rapidly reached high amplitudes of about 1500 volts which caused a puncture of the insulation; later on, by adding a non-linear element in the circuit, the authors succeeded in obtaining stable performance of the machine. The factor of modulation during these tests was of the order of 40 per cent, and the power developed was about 4 kilowatts. The non-linearity by which the oscillations were stabilized was obtained by means of a saturatedcore reactor; an auxiliary d-c winding served the purpose of displacing the stable point on the characteristic and of adjusting the stable voltage to a desired value. Similar experiments were produced with a periodically varying capacity. The variable capacitor consisted of 25 aluminum disks with peripheral holes rotating between a corresponding number of similar stationary disks. The variable capacitor in these tests was shunted by a constant oil capacitor. The non-linearity necessary for the stabilization of oscillations was obtained by neon tubes which permitted maintaining the voltage at about 600 to 700 volts. Without the neon tubes the phenomenon is governed by a linear Mathieu equation, and the voltage rises rapidly to between 2000 and ,000 volts and the insulation is punctured. If one changes the parameters of the oscillating

373 circuit so as to deviate from the condition of exact fractional-order resonance, the amplitudes of the parametrically excited oscillations decrease, all other conditions being equal, until finally the excitation suddenly disappears at a certain critical threshold as was analyzed theoretically in Section 123. 125.

SUBHARMONIC RESONANCE ON THE BASIS OF THE MATHIEU-HILL EQUATION In order to complete the study of heteroparametric excitation, we shall now show that the differential equation [102.2] of subharmonic external resonance which we have discussed in Chapter XVII on the basis of the theory of Poincare can be reduced to an equation of the Mathieu-Hill type. For that purpose, instead of following the method of Poincare by introducing small perturbations a and p, Equation [103.9], in the value of the parameters u and v, we shall now introduce a small perturbation p in the value of the periodic solution XO(T) of Equation [102.2] since we know that it possesses periodic solutIons. Putting :I: = Xo(T) + p [125.1) and substituting it into Equation [102.2], we get, after expanding !(x,x) in a Taylor series, %0

Since

Xo

+

:1:0

+ P+

p

= p.!(xo,xo) +

AO

sin nT

+ pp.f'zo(xo, xo) + PJl!zo(xo.io) [125.2]

satisfies Equation [102.2], we obtain (125.3 ]

where !zo (x o' x o) and !:0 (xo,x o ) are known functions of Xo and xo and, hence, known periodic functions of time. Equation [125.3] is therefore an equation with periodic coefficients. If we use the substitution [118.2] which in this case is

Equation [125.3] becomes

Z+

[1- p./,.

Zo

+ E....!!:....(/,.) - Jl\/. 2 dT 2 Zo

Zo

)2]: 0 =

[125.5]

Since the quantity in brackets is a periodic function of time, this equation is of the Mathieu-Hill type whose general solution is [125.6 ] where tPl and tP2 are periodic with period 21f' and kl and k2 are the characteristic exponents of the general theory. The question of the stability of the motion can be discussed by the method indicated in Sections '18 and "9 if one knows the explicit form of the function !(xo,x o )'

374 AUTO PARAMETRIC EXCITATION Throughout this chapter we have been concerned with heteroparametric excitation because 1n a great majority of practical cases self-excitation as well as the steady state of non-linear oscillations can be discussed more conveniently on the basis of the theory of Poincare than by treating it as autoparametric excitation. In some special problems, however, the concept of o autoparametric excitation of oscillations may be convenient. In this section we propose to apply this method to an interesting problem of an elastic pendulum investIgated by Gorelik and WItt (29). These authors investIgated the motion of a physical pendulum suspended on a spring and capable of oscillating in a plane, as shown in Figure 126.1. Let m be the mass of the bob, Lo the length of the pendulum in the absence of the dynamical load, r its length under load, k the spring constant, and 9 the acceleration of graVity. Obviously the system possesses two degrees of freedom, namely, the angle ~ of the pendulum and the elonFigure 126.1 gation z of the spring. In order to investigate the condition for autoparametric excitation, we write the Lagrangian equations of motion for both degrees of freedom. The kinetic energy of the pendulum is 126.

[126.1 ]

and its potential energy is l1 26.2 ) ~2

where 1 - lr ~ cos~. The first term of V corresponds to the elasticity of the suspension and the second to gravity. If we introduce a new constant 1= r

o

+

mg

k

[126.3 )

and a new variable z

=

r -

l

l

[126.4 J

Expressions [126.1] and [126.2] become T

= ml 2

2 (Z2

+ ~2 + 2Z~2)

[126.5 ] [126.6]

375

where z and ~ are assumed to be small of the first order and we retain terms up to and including those of the third order. The Lagrangian equations are :i

+ ~Z + m

(..fL ~2 2l

-

J2)

= 0

[126.1]

It is seen that, if the terms in the parentheses of these equations are zero, the first equation will represent a simple harmonic oscillation in the z degree of freedom with frequency w. = Yk/m, and the second equation will give a similar oscillation in the ~ degree of freedom with frequency w~ =YU/l. These terms represent a non-linear coupling between the two degrees of freedom; we will now investigate this condition. It is to be noted that in the general case when w. :1= : w~ the nonlinear coupling does not contribute anything of particular interest. An interesting case arises when w, = 2w~. Assume that the spring has been stretched and released at t = 0 so that ~ = 0 initially. The initial motion Will, therefore, be [126.8 ) z = Zo cos w,t Substituting this value of z into the second equation [126.1) and regrouping the terms, one has [126.9] which is a differential equation with periodic coefficients and can be reduced to the form of the Mathieu equation. Since w. = 2w~, it is observed that the periodic variation of the coefficients occurs at twice the frequency of the oscillation in the ~ degree of freedom. If the parameters of the Mathieu equation to which Equation [126.9] can be reduced are such as to correspond to the unstable region, the oscillation in the ~ degree of freedom will gradually build up. This curious phenomenon of autoparametric self-excitation was actually observed by Gorelik and Witt. However, in view of the fact that this system is conservative, it is apparent that the building-up of the oscillation in the ~ degree of freedom implies a decrease of the original oscillation [126.8] in the z degree of freedom. In this manner the occurrence of the initial oscillation in the z degree of freedom is transferred into the ~ degree of freedom through the instrumentality of the autoparametric non-linear coupling between both degrees of freedom.

376

One could, of course, start from the ~ degree of freedom by releasing the pendulum from an angle ~ = ~o for t = 0 which would give the oscillation ~

=

~Ocosw9t

(126.10]

Substituting this expression into the first equation (126.7], one has [126.11] This is the equation of a simple harmonic oscillator with frequency w, acted on by a periodic external excitation with frequency 2w. = w,. Hence, in the % degree of freedom there will be ordinary linear resonance by which the %oscillation will gradually increase while the ~-oscillation will gradually decay, since the system is conservative. It is to be noted that in both cases the phenomenon manifests itself in the fact that the energy appearing initially in one degree of freedom is eventually transferred into the other degree of freedom. There exists, however, an asymmetry in the phenomenon depending on whether one starts with the oscillation [126.8] or [126.10). If the initial oscillation is [126.8), the excitation of the oscillation in the ~ degree of freedom occurs through the instrumentality of the unstable solution of the Mathieu equation, whereas if the initial oscillation is [126.10], the autoparametric excitation manifests itself in classical linear resonance with which the z-oscillation builds up. This difference, however, is incidental to the particular scheme employed and is of no further importance insofar as in the second case the autoparametric excitation is still present in the form of the centrifugal force whose frequency is twice the frequency of the oscillation in the ~ degree of freedom. It is noteworthy that in both cases the frequency with which the parameter varies is double that with which the self-excited oscillation occurs. If one starts the oscillation in the z degree of freedom this is apparent because W~ = 2w9 • If, however, one starts the oscillation in the ~ degree of freedom, it is noted that the variation of the z-parameter takes place under the effect of the centrifugal force so that in both cases the condition of autoparametric excitation is fulfllled and the "pumping" of energy from one degree of freedom into the other is reCiprocal. Another interesting experiment similar to that of Melde was made recently by Sekerska (22), who passed an alternating current of 50-cycle frequency through a stretched metallic wire capable of oscillating laterally with a frequency of 50 cycles. It is observed that if the wire is initially at rest the passage of alternating current eventually builds up the lateral vibration of the string. The explanation of this phenomenon is that the thermal effect of a current of 50-cycle frequency occurs at a frequency of 100

377 cycles which causes a periodic variation of the parameter, the coefficient of elasticity, at that frequency, and this, through the instrumentality of the autoparametric excitation, produces self-excitation of lateral oscillations with half the frequency of the parameter variation. These phenomena occur not only for the ratio wr/w = 2/1, where Wr is the frequency at which a parameter varies and W is the frequency of selfexcited autoparametric oscillation, but also for the ratios 2/2,2/3, ...• Migulin (30) investigated, both theoretically and experimentally. these phenomena when this ratio has the value 2/3 and found that the resonance curves then resemble those obtained by Mandelstam and Papalexi, Chapter XVII, in their studies of subharmonic resonance of the nth order. As a matter of fact, the phenomena of subharmonlc resonance and those of autoparametric excitation are closely related to each other and merely represent different aspects of the same physical phenomenon.

378

REFERENCES (1) "Introduction to Non-Linear Mechanics," by N. Kryloff and N. Bogoliuboff, Kieff, (Russian), 1937. (2) "Lehrbuch der Elektronenr6hren" (Manual of Electronic Tubes), by G.H. Barkhausen, S. Hirzel, Leipzig, Germany, Fourth Edition, 1931-1937.

0)

"Sensations of Tone," by H. Helmholtz, Longmans, Green, and Co., London, England, English Translation, Third Edition, 1895. (4) "Principles of Electron Tubes," by H.J. Reich, McGraw-Hill Book Co., New York, N.Y., First Edition, 1941, Chapter 3. (5) "Asynchronous Action on Self-Excited Oscillatory Systems," by J.B. Kobsarev, Journal of Techn1cal Physics, USSR, (Russian), Vol. 3, 1933. (6) "An Experimental Investigation of Forced Vibrations in a Mechan1cal System Hav1ng a Non-Linear Restor1ng Force," by C.A. Ludeke, to appear in the Journal of Applied Physics, July 1946. "Mechanical Vibrations," by J.P. Den Hartog, McGraw-Hill Book Co., New York, N.Y., 1940, p. 403. (7)

(8) "Ober Resonanzerscheinungen bei Frequenztellung" (On Resonance Phenomena in Frequency Demultipl1cation), by L. Mandelstam and N. Papalexi, Zeitschrift fUr Physik, Vol. 73, 1932. (9) "Forced Oscillations in a Circuit with Non-Linear ReSistance," by B. Van der Pol, Philosophical Magazine, Vol. 3, Number 13, January 1927. (10) "On Maintained Vibrations," by Lord Rayleigh, Philosophical Magazine, 5th Series, Vol. 15, April 1883. (11) "On Some Experiments in Which Two Neighboring Maintained Circuits Affect a Resonating Circuit," by J.H. Vincent, Proceed1ngs of the Physical Society, London, England, Vol. 32, Part 2, February 1919. (12) "tiber st6rungsfreien Gle1chstromempfang m1t dem Schw1ngaud1on" (On Non-D1sturbed D-C Recept10n with an Autodyne), by H.G. Maller, Ze1tschr1ft fUr drahtlose Telegraphie und Telephonie, Vol. 17, Apr1l 1921. (13) "The Automatic Synchronization of Tr10de Osc1llators," by E.V. Appleton, Proceed1ngs of the Cambridge Philosophical Soc1ety, London, England, Vol. 21, November 1922. (14) "Zur Theor1e des M1tnehmens von Van der Pol" (On Van der Pol's Theory of Entrainment), by A. Andronow and A. W1tt, Archlv fUr Elektrotechnlk, Vol. 24, 1930.

379

(15) "Theory of Entrainment in the Case of a Soft Characteristic," by V.I. Gaponow, Journal of Technical Physics, USSR, (Russian), Vol. 5, 1935. (16) "Einstellungs- und Schwebungsprozesse bei der Mitnahme ll (Tuning or Beat Phenomena in Entrainment), by P. RJasin, Journal of Technical Physics, USSR, (Russian), Vol. 5, 1935. (17) "Acoustical Entrainment," by K. Theodorchik and E. Chaikin, Journal of Technical Physics, USSR, (Russian), Vol. 2, 1932. (18) "Prequenzmesser nach dem Kompensationsverfahren" (Frequency Meter Designed According to the Compensation Method), by H. Kaden, Elektrische Nachrichtentechnik, Vol. 16, 1939. (19) "{Jber die Erregung stehender Wel1en eines fadenfHrmigen KHrpers" (On the Excitation of Standing Waves in a Filiform Body), by F. Melde, Poggendorff's Annalen der Physik, Vol. 109, 1860, p. 193. (20) "Theori6 d'un alternateur auto-excitateur ll (Theory of a SelfExciting Oscillator), by M. Brillouin, ~clairage ~lectrique, Vol. XI, April 1897. (21) "Sur quelques theoremes generaux de l'6Iectrotechnlque" (On Some General Theorems of Electrical Engineering), by H. Poincare, Eclairage Electrique, Vol. 50, March 1907. (22) "Expose des recherches recentes sur les oscillations nonl1neaires" (Outline of Recent Research on Non-Linear Oscillations), by L. Mandelstam and N. Papalexi, Journal of Technical Physics, USSR, 1934. A complete bibliography on parametric excitation is appended to this reference. (23) "Modern Analysis," by E.T. Whittaker and G.N. Watson, Cambridge University Press, 1927, Chapter XIX. (24) IISur les equations differentielles l1neaires a coefficients periodiques" (On Linear Differential Equations with Periodic Coefficients), by G. Floquet, Annales de l'Ecole Normale Superieure, Vol. XII, 1883. (25) liOn the Stability of the Solutions of Mathieu's Equation," by B. Van der Pol and M.J.O. Strutt, Philosophical Magazine, 7th Series, Vol. V, January-June, 1928. (26) IIOber SchUttelerscheinungen in Systemen mit periodlsch verflnderl1cher Elastizitflt ll (On Vibrations in Systems with Periodically Varying Elasticity), by E. Meissner, Schweizerische Bauzeitung, Vol. 72, 1918. (27) "Lame'sche, Mathieu'sche und verwandte Funktionen" (On Lame, Mathieu and Related Functions), by M.J.O. Strutt, J. Springer, Berlin, Germany, 1932. I

380 (28) "On Parametric Excitation," by N. Minor sky , Journal of the Franklin Institute, Vol. 240, Number 1, July 1945. (29) "Schwingungen eines elastischen Pendels als Beispiel fUr zwei parametrisch gebundene lineare Schwingungsysteme" (Oscillations of an Elastic Pendulum as an Example for Two Parametrically United Linear Vibratory Systems), by G. Gorelik and A. Witt, Journal of Technical Physics, USSR, Vol. 3, 1933, p. 294. (30) "Resonance Effects in a Non-Linear System with Two Degrees of Freedom," by V. Migul1n, Journal of Technical Physics, USSR, (Russian), Vol. 4, Number 2, 1937.

INTRODUCTION TO NON-LINEAR MECHANICS PART IV RELAXA TION OSCILLATIONS-

INTRODUCTORY REMARKS The term relazation oscillations, introduced by Van der Pol (1)t (2) and commonly used at present, generally designates self-excited oscillations exhibiting quasi-discontinuous features. Because of the importance of such oscillations in applications in connection with the so-called "sweep circuits" in electronics, television, and allied fields, an extensive literature exists on this subject, References (3) through (9). Ph. LeCorbeiller (10) gives an interesting survey of various devices, both mechanical and electrical, by which these phenomena can be demonstrated; some of these devices have been known for centuries. The characteristic. feature of these phenomena is that a certain physical quantity (such as coordinate, velocity, etc., in mechanical problems, and charge, current, etc •• in electrical problems) exists on two levels, remaining on each level alternately for a relatively long time but paSSing from one level to the other so rapidly that in the idealized representation the passage may be considered as instantaneous. A few examples taken from the paper by LeCorbeiller will illustrate these phenomena. Figure 127.1 shows a device consisting of a container C, of the form shown, fastened to a support R capable of rotating about an axis A, perpendicular to the plane of the paper, and provided with a weight ~sufficient to hold the system against the stop S. The container 1s slowly filled with water, and at the instant when the moment due to the weight of the water becomes greater than the moment due to the weight VV, the system tumbles over against another stop S'. The container then empties, and the weight VV brings the system back aga1nst the stop S, afc w ter which the fill1ng period begins again, and so on. In this system the two levels, previously mentioned, are the angles 6 and 6' at which the system is constrained by the stops Figure 127.1 127.

---

*

The text of Part IV follows the presentation in DTheory of Oscillations, n by Andronow and Chaikin, Moscow, (Russian), 1937. A complete bibliography on the subject of relaxation oscillations appears in this volume. t

Numbers in parentheses indicate references on page 66 of this report.

g and g' respectively, and the representation of this system in the (6,t)plane appears as a periodic rectangular ripple with the length of its horizontal stretches determined by the rates of filling and of evacuation of C. Another familiar example is the charging of a capacitor shunted by a gaseous conductor such as a neon lamp. During the charging period the capacitor's voltage gradually rises. At the point of ionization of the gaseous conductor, the neon lamp flashes and the capacitor is suddenly discharged, whereupon the gaseous conduction ceases abruptly and the charging period begins anew. Here, again, there are two levels, the voltage V. immediately before the discharge strikes and the voltage VI immediately after the extinction of the discharge. The tranSition from V2 to V. is gradual, but the inverse transition from V. to V2 is quasi-discontinuous. A phenomenon of this kind is represented in the (V,t)-plane by a so-called IIsaw-tooth ll curve. It was shown in Section 37, Part I, that a Similar situation exists for large values of the parameter p in the Van der Pol equation. Figure 37.1c shows quasi-discontinuous changes in the variable ~(t) between the two regions in which it changes but little. When these periodic phenomena are represented in the phase plane of the variable undergoing rapid changes, they appear as closed curves with regions of very large curvature, such as the curves shown in Figures 37.2c and 37.3c. By idealizing these very rapid changes as discontinuous changes, closed trajectories of this kind become piecewise analytic curves IIclosed" by the discontinuous stretches. Available analytical methods are inadequate for a rigorous treatment of these phenomena. In fact, all analytical methods presuppose that the parameter p appearing in the basic quasi-linear equation ['27 .1 ] is very small. On the contrary, in some of these oscillations, which are expressible by Van der Pol's differential equation, this parameter is large. More specifically, in Figures 37.1c, 37.2c, and 37.3c, referred to above, the value of the parameter p is 10. Attempts have been made to extend the analytical methods to oscillations in which p is large. In Section 36 it was shown that Lienard succeeded in obtaining certain conclusions regarding the qualitative aspect of the phase trajectories when p was very large. N. Levinson (11) extended the proof of the existence of closed trajectories to cover oscillations in which p is not small. In a recent publication (12) J.A. Shohat has indicated a form of series expansion formally satisfying the Van der Pol equation when p is large. These various attempts, however, did not result in any complete analytical

theory comparable to the one which has been studied in Part II in connection with oscillations in which ~ is small. Moreover, as will appear below, not all known relaxation oscillations seem to belong to the group of equations [127.1] of which the Van der Pol equation is a particular example. More specifically, it will be shown in Chapter XXI that relaxation oscillations are frequently observed in systems which are amenable to representation by differential equations of the first order which do not admit any analytic periodic solutions for the simple reason that these equations do not possess singularities, without which no closed analytic trajectories can exist; see Section 25. These difficulties led the school of physicists under the leadership of L. Mandelstam and N. Papalexi to evolve a theory, called by its authors the di8continuou8 theory of relaxation 08cillation8, whose exposition and applications will form the principal topic of Part IV. The use of the concept of mathematical discontinuities for the purpose of describing a rapidly changing dynamical process, at least during certain instants of its evolution in time, is not new. It is recalled that the classical theory of mechanical impacts uses precisely the discontinuous method by assuming an infinitely small duration of the impact process during which the dynamics of the process is entirely ignored, and the "initial" and "terminal" conditions are correlated on the basis of certain additional information not contained in the differential equations themselves. This permits obtaining the correct overall effect of the impact without knowledge of its details. For an elastic impact, such additional information is supplied by the theorems of momentum and kinetic energy; to this information is added, for non-elastic impacts, the so-called coefficient of restitution, an empirical factor characterizing the loss of energy during the impact. This coefficient depends on w w 0the material of which the colliding bodies are composed. ~--------2a--------~ One can imagine a discontinuous i periodic motion generated by impacts from A B the following example given by Andronow and Chaikin (13). Let us assume that a perfectly elastic ball rolls without fric0 Ie tion on a horizontal plane and strikes elastic walls WW perpendicular to the di0 C rection of its motion, as shown in Figure Ib 127.2a. With the usual discontinuous treatment of mechanical impacts, the phase Figure 127.2

trajectory of such motion is represented by a "closed curve" ABCOA, a rectangle. On the branches AB and CO the motion is continuous with the constant velocities z = ± vO' respectively; on the branches BC and OA, on the contrary, it is discontinuous. In the systems of Figures 121.1 and 121.2 we encounter periodiC phenomena having certain quasi-discontinuous features. The nature of these discontinuities in the two systems is, however, different. For the ball striking the walls there exists a definite external actuation, the reaction of the constraint, the wall, applied to the dynamical system, the ball; this actuation is properly "timed" by the distance 2a between the walls which determines the "period" of the motion. On the contrary, no external impact excitation exists during the discontinuities in the motion of the container shown in Figure 121.1. These ~iscontinuities are due rather to a sudden loss of equilibrium between the moment Mwof the constant weight and that of the container Me' occurring at a certain critical value, Me =Mw' The change of equilibrium position from 6 to 6' is not instantaneous, of course, but in comparison with the long periods of filling and evacuation of the container it may be considered as such in the idealized picture of the phenomenon. We can improve the idealization by making the moment of inertia of the system (flT,C) relatively small; this will render the short time interval during which 6 varies still shorter, which, in turn, makes the relative time intervals of filling and of evacuation still longer. We find it expedient to define as relaxation oscillations those quasi-discontinuous oscillations in which the rapid changes between certain levels of a physical quantity occur as the result of the loss of a certain internal equilibrium in the system, and as impulse-excited oscillations those quasi-discontinuous oscillations in which these rapid changes are due to the action of certain external impulsive causes. On this baSiS, the quasi-discontinuous oscillations of the container shown in Figure 121.1 are of a relaxation type, whereas the ball rebounding between the walls is an impulse-excited phenomenon. The essential difference between the two types of oscillations is that in relaxation oscillations the energy content stored in the system remains constant during the quasi-discontinuous changes of certain variables, whereas in impulse-excited systems, on the contrary, the energy content changes abruptly. Impulse-excited oscillations do not require any particular additional information for their treatment, as will be seen in Chapter XXIV. For relaxation oscillations proper (see Chapters XXI, XXII, and XXIII) it is necessary to specify the conditions under which the discontinuities are bound to occur in a system; it will be shown that the basic assumption given in

Section 130 provides a criterion sufficiently broad to cover all known types of relaxation oscillations. Pinally, inasmuch as the representation of a rapidly changing process by a mathematical discontinuity is always an idealization, it becomes necessary to analyze the conditions under which this idealization is Justified in practice. In analyzing the behavior of the device shown in Pigure 127.1, we have noted that the change from the angle 6 to the angle 6' may be considered as quasi-discontinuous. The smaller the moment of inertia of the system, the more accurate is the approximation. In mechanical systems, such as the examples described above, it is obviously difficult to extend the hypothesis by assuming that the moment of inertia is zero, but in electrical systems neglect of one of the oscillatory parameters is a common practice. In both types of systems, instead of a "full" differential equation of the second order, the abbreviated or degenerate equation of the first order is frequently employed. By using degenerate equations, numerous problems can be treated as discontinuous in the phase plane; this appreciably simplifies their solution. A simple example will show the appl1cation of degenerate equations for this purpose. Let us consider an oscillating circuit shown in Pigure 127.3 compris1ng an inductance L, a capacity C, and reSistors Rand r as shown. The c1rcuit may be closed on a source of d-c voltage by a switch S. It is useful to spec1fy certain idealizations which appear somewhat trivial but which will be found to be of cons1derable importance in what follows. We assume first that the left branch of the circuit ALRB has no capac1tyand that the right branch ACrBhas no inductance. In other words, we neglect the effect of small, parasitiC, distr1buted capac1t1es in the inductance Land res1stance R; likewise, we neglect the effect of a small inductance accompanying the flow of current in the branch ACrB. The second assumption will be that the opening and closs ing of S is instantaneous, that is, A occurs in an 1nfin1tely small time interval (t - 0, t + 0). c Let us assume that we open and close the switch S at some fre+ -=-E quency. The process w11l then be represented by a sequence of very R long intervals, when S is either closed or opened, separated by inB fin1tely short intervals of closing and opening. During the long interP1gure 127.3 vals there will generally be a

386

certain oscillatory process in the circuit (L,C,R+ r) describable by a differential equation of the second order whose phase trajectories are spirals converging toward a focal point, as we know from Section 5. The closing or opening of S will disturb this process by introducing a certain transient. Without any loss of generality we may consider the first closing of S at to = 0 and assume that for t < to the circuit was "dead." It is noted that, at the instant of closing, the two circuits ALRB and ACrB are in parallel, and the differential equations are L ::

where E = 0 for t finds that

~

+

Ri = E;

dV dt

rC- + V = E

= constant

for t

E (1 _At) i =R - eL ;

di

to and E

V = E ( 1 - e - .Lt) rC ;

Cit

~

[127.2]

to' With these assumptions one E L

= -e

_l!t L

[127.3]

dV E dt = rC

It is seen that for t = to the solutions of Equations [127.2], i(t) and V(t), are continuous but not analytic in the sense that their first derivatives undergo discontinuous jumps (~;)o = (~r)o = Noting that ic = C ~r we can take instead of ~r t~e variable ic and state that under the assumed idealizations the functions ~; and ic undergo discon~inuities f.and ~, respectively. If one takes the plane of the variables (~~,ic), the process occurring at t = to is represented by a discontinuous Jump of the representative pOint from the origin to the point A whose coordinates are ~ and see Figure 127.4. After the initial discontinuity the subsequent motion of the representative point will follow a continuous trajectory ABwhich, as the transient dies out, will eventually approach a damped OSCillatory motion represented by a convergent spiral. If, at a later instant t = t l , di Cit the switch is opened, another jump will occur, but this jump will generally not bring the representative pOint back to the origin but to some other point A", and so on. It is thus seen that in the phase plane of these particular variables (~; , ic) the history of the system will be represented by a sequence of disconnected, spiral arcs "joined" by discontinuous stretches such as OA, BA', ••.. We thus obtain a piecewise analytic repreFigure 127.4 sentation of such a phenomenon.

fi

:c.

f;

.

.

It Is Important to emphasIze once more that such a representation of a quasI-dIscontinuous phenomenon by discontinuitIes in the phase plane of certain variables is possible only because we have Introduced certain idealizations into the problem. a. We consider the time interval during which rapid changes occur as an infinitely short Interval. b. The effect of the parasitic parameters is neglected, which enables us to deal with the degenerate equations (127.2J of the first order instead of the full equations of the second order. c. We have selected the dynamical variables ~~ and ie which are capable of undergoing dIscontinuities under Assumption b. The necessity for Assumptions a and b is obvIous. As for AssumptIon c, It Is clear that If one selected some other variables, for example, i and V Instead of ~; and ie, dIscontInuous representation in such a phase plane would be lost since these varIables are continuous. In the preceding discussIon we have tacitly assumed an impulseexcited phenomenon as previously defined. For a pure relaxation phenomenon we must answer an addItional questIon, namely, how to determine the instants (in the tIme representation) or the points (in the phase-plane representatIon) at whIch the dIscontInuIty occurs in a system of this kind. It Is Impossible to go beyond this poInt without formulatIng some kind of a priori assumption, as will be explained in SectIon 130.

CHAPTER XX FUNDAMENTAlS OF THE DISCONTINUOUS THEORY OF RELAXATION OSCILLATIONS

128.

SOLUTIONS OF A DIFFERENTIAL EQUATION IN THE NEIGHBORHOOD OF A POINT OF DEGENERATION

We now propose to investigate the nature of the solutions of a differential equation of the second order with constant coefficients, for example,

aii +

b~

+

[128.1]

k:c = 0

when one of the coefficients approaches zero. In an electrical problem, a ~ L, 6 ~ R, and k = lie; in a mechanical one, a = 1n (mass), b is the coefficient of IIvelocity damping,1I and k is the spring constant. First, if b approaches zero, one readily sees that the oscillatory damped motion approaches the oscillatory undamped motion. We saw that in the phase plane the solutions of [128.1] with b * 0 but small are spirals approaching a stable focal point; this remains true as b approaches zero. For b = 0, the origin is a vortex point and the trajectories are closed. It is thus seen that there is a definite difference between the qualitative aspect of trajectories when b is very small and that when b is equal to zero. From a practical standpOint, however, there is hardly any difference between the two cases; no discontinuities of any kind exist in the solutions. or greater practical interest are the cases when either a + 0 or k +0. We shall examine first the case when a+ O. It is apparent that when a = 0 Equation [128.1] becomes an equation of the first order and its solution is given in terms of one constant of integration, namely,

",

:c = :Coe - T

where

:Co

~y

is that constant. •

[128.2]

differentiating [128.2] we obtain k

:c = - -:c b 0e

_ A, 6

= -

k

-:c b

[128.3 ]

It is seen that the coordinate :c and the velocity ~ are not independent but are related by Equation [128.3]. In other words, in the phase plane the trajectories of Equation [128.2] are reduced to a single line 11 = - f:c, and the rest of the plane is not involved. This fact can be expressed by stating that the phase space of a differential equation of the first order is unidi1nenBional, that is, it is a pha,e line instead of a phase plane. The limit case, a = 0, never actually occurs in practIce since in any electrical system containing resistance and capacity there is always a small residual or IIparasitic" inductance. Likewise, mechanical systems without inertia are only idealizations. For these reasons it is preferable to investigate the effect of a small coefficient a in the solution of Equation

[128.1] rather than to drop this coefficient in the differential equation itself. The solution of (128.1] is %

+

= Cieri'

C2 e r2 '

[128.4]

where C1 and Cz are the constants of integration and r 1 and r z are the roots of the characteristic equation ar2

If the initial conditions

t

= 0.

+ br +

%

= %0'

k = 0

and

~

= ~o are given. one obtains [128.5 ]

from which one obtains the values of C1 and Cz :

(128.6J

and where

.Ill! - !a

b r 1,2 == - -2a + - V4a~

.::. - ~ + ~(1 2a - 2a

- ~) 6~

[128.7 J

In this expression only one term is retained in the expansion of the square root. since a is small. This gives k

rI = - b

and

b k b r=--+-.::.-2 a b a

[128.8]

If the values (128.6] of the constants and [128.8] of the approximate expressions for the roots r l and r z are substituted in Equation [128.4J. the approximate solution %I(t) of [128.1] is in the form

[128.9 ] It is to be noted that the solution %I(t) is an approximate one because the expansion of the square root has been limited to the first two terms; this is Justified by the assumed smallness of a. On the other hand. for a = 0 the solution ot Equation [128.1] of the second order becomes the same as that of the equation of the first order given by [128.2J. To emphasize the fact that the solution [128.2] is the same as that ot Equation (128.1] when complete degeneration occurs. that is. when a = 0. we will write it as i(t)

Consider now the function

==

(128.2]

390

This function represents the difference between the approximate solution Zl(t) of [128.1] in the neighborhood of the point of degeneration, where a is very small, and the solution z(t) of a completely degenerated equation [128.1] of the first order. The function ~(t) approaches zero uniformly in the interval o < t < 00 when a + O. The expression for the derivative of this funct10n is •(

~

a,t )

=

• ( Zl a,t ) - ..!.. Z ( a,t )

=

(zOb k

+

. ) e-.!., Zo CI

-

ak. -!6 t -;;rzoe

[

128 .11 ]

For very small values of t the funct10n ~ ~ZO: + ~o' and it 1s 1mposs1ble to reduce it by reduc1ng the coeff1cient a. However, for a suffic1ently large t, which is supposed to be fixed, one can always f1nd a value of a small enough so that the value of ~(t) is smaller than a given pos1tive number E. We can express th1s by saying that whereas the funct10n ~(a,t), considered as a function of a, approaches zero uniformly 1n the interval 0 < t < 00 when a + 0, the function ~(a,t) behaves 1n a 11ke manner only when the values of tare suff1ciently large. For t = 0 the convergence of the function ~l(t) to the function i(t) when a +0 1s not uniform. In other words, to a given a, however small, one can always assign a value of t = tl such that ~ ~ Zo: + zoo Only 1n a very special case, when + ~o = 0, does th1s non-un1formity of convergence disappear, but th1s case is of no practical interest. One can also state that the d1fference between the approximate solut10n Zl(t) of a quas1-degenerate system when a 1s very small and the corresponding solut10n z(t) of a completely degenerate system when a = 0 approaches zero 1n the whole interval 0 < t <00 when a + 0 except in a very small neighborhood around the p01nt t = 0; this ne1ghborhood is smaller as a 1s smaller and in it the difference %l(t) - t(t) of the slopes of the two curves Zl(t) and z(t) cannot be reduced. This means that the function Zl(t) undergoes a quas1-discont1nuous Jump 1n th1s neighborhood. When the parameter k + 0, the problem is treated 1n a similar manner. First, for a completely degenerate equat10n with k = 0, Equation [128.1] becomes all: + bz = 0 [128.12 ] Integrat1ng it, one obtains

zof

az

+

b

where M 1s the constant of 1ntegra tion. init1al conditions, namely,

= M

[128.13 ]

The value of M 1s determined by the [128.14]

391 The solution of Equation [128.13] is %

=

M

b +

where C is a constant of integration. -% (t) =

%0

Ce

_'!t

[128,15 ]

a

One obtains finally

+ %0, ba ( 1

-

e

-

.!t) a

[128,16 ]

If, however, one proceeds with the solution of Equation [128.1] in the neighborhood of its degeneration, where k is very small, the approximate solution is

[128.17 ] By forming the functions ",(k,t) = %l(k,t) -

and

x(k,t)

one ascertains by an argument s1milar to that given in connection with the functions ~(a,t) and ~(a,t) that for a sufficiently small k the function ~(k,t) approaches zero when k + 0 uniformly in the interval 0 < t <~, whereas ",(k,t) approaches zero when k + 0 for all values of t except when t +~, for which value ",(k,t) approaches the value %0'

129. IDEALIZATIONS IN PHYSICAL PROBLEMS In applications, idealizations of quaSi-degenerate systems as abso-

lutely degenerate systems are frequently made, as was mentioned in Section 127. Thus, for example, the differential equation of the so-called (R,C)circuit is usually written in the form 1, , = 0 R -di + dt

where i is the current in the idealized circuit (L quasi-degenerate equation, with L very small, is d 2i

[129.1 ]

C

d'

LPt + R-' dt

+

1 -i = 0 C

= 0).

The corresponding

[129.2 ]

With the solution of the absolutely degenerate equation [129.1] deSignated by T(t) and that of the quasi-degenerate equation [129.2J designated by i 1 (t), the functions T(t) and i 1 (t) have the appearance shown in Figure 129.1. For Equation [129,1J, with L = 0, the current starts from the point A and decays exponentially thereafter. For the quaSi-degenerate equation [129.2J, with L very small, the current starts from zero, increases very rapidly, and follows the curve i 1 (t) which becomes practically identical with the curve i(e) after a very short time; this t1me 1s shorter as L 1s smaller. According to the

392 absolutely degenerate equation [129.1], with L = 0, the current i(t) undergoes a true mathematical discontinuity OAat t = 0, whereas the current il(t) of the quasi-degenerate equation [129.2], with L~O, has a quasi-discontinuity OA' which approaches OA when L approaches zero. Although these facts are well known if considered independently of the previous history of the system, as in the above discussion, they appear in a somewhat different form in a quasidiscontinuous stationary relaxation oscillation. Thus, for example, the oscillation depicted in Figure 127.4 is governed by the full differential equation of the second order on the analytic branches AB, A'B', .•• of the trajectories; these branches are determined in terms of the two constants of integration corresponding to the initial conditions represented by the points A and o t A of the phase plane. During the Jumps BA', Figure 129.' B'A", ••• the transition is governed by two differential equations of the first order. Since the solutions of each of these equations are determined by one constant of integration, there appears a relation between the variables ~: and ie of the phase plane which does not exist on the analytic branches AB, A'B', ••• • As a result of this, the jumps must occur along a certain direction in the phase plane. Thus the assumption of certain idealizations specified in connection with Figure 127.' not only permits a discontinuous treatment of the problem but also indicates the direction of the jumps in the phase plane. It is impossible, however, to proceed beyond this point if one attempts to apply these idealizations to the problem of discontinuous stationary relaxation oscillations. What is lacking in the system of Figure 127.4 is the mechanism by which the phenomenon of "closing" or "opening" the switch is produced 8pontaneou8ly hy the internal reaction8 of the circuit itself. In order to be able to formulate this condition and thus to complete the discontinuous theory it will be necessary to introduce an a priori propoSition whose validity is Justified only by its agreement with the observed facts. This emphasizes once more the physical nature of this theory as distinguished from the contents of Parts I and II in which the argument was purely analytical. I

130.

CRITICAL POINTS OF DIFFERENTIAL EQUATIONS; BASIC ASSUMPTION It was shown that, as a result of certain idealizations, discontinuities appear in the mathematical treatment of physical phenomena which exhibit

393 rapid changes at certain points of their cycles. The use of discontinuities is convenient 1n some respects but inevitably introduces certain complications. These facts are too well known from the theory of mechanical impacts to need any emphasis here. It is obvious that similar d1fficulties are to be expected if a discontinuous treatment of relaxation oscillations is adopted. For example, in an electronic "sweep circuitll some variables change so rapidly at certain points in the cycle that it is natural to attempt to idealize these changes by mathematical discontinuities. Obviously, any attempt to explain these changes on the basis of some kind of impact is difficult because the energy delivered by the external source, a battery, remains constant, and one cannot very well correlate the apparent continuity of the energy input into the system with the quasi-discontinuous changes of some of its variables. Very frequently a slight change of a parameter causes a disappearance of the phenomenon,and vice versa. In some particularly simple circuits in which the effect is known to exist, one succeeds in lIexplaining" i t by a more or less elementary physical argument. In more complicated circuits it is impossible to give an account of what actually happens and, still less, to predict theoretically the existence, or non-existence, of such effects. There exists no analytical theory of these oscillations which would permit a treatment of these phenomena on a uniform basis as was possible for the quasi-linear oscillations with which we were concerned in Parts I, II, and III. In order to be able to find a solution and to correlate the numerous experimental phenomena on a common basis, it becomes necessary to define terms and to introduce some kind of basic assumption, the value of which is to be justified by its agreement with the observed facts. Deflnltlon:

Critical pOints are the points at which the dlfferential equation describing

a phenomenon in a certain domain ceases to describe it. Basic Assumption:

Whenever the representative point follOWing a trajectory of the

differential equations describing a phenomenon reaches a critical point, a discontinuity occurs in some variable of the system.

Since, by virtue of the basic assumption, the occurrence of discontinuities depends on the existence of critical points, it is necessary to specify certain criteria by which their existence can be ascertained. In what follows we will encounter two principal criteria. 1.

Let us consider a system of differential equations of the form dx =

dt

P(X,1I) R(x,1I) ;

d1l

Cit

=

Q(x,y) R(x,1I)

[130.' ]

Obviously these d1fferent1al equations become meaningless and, hence, cease to

394 describe a physical phenomenon for the points (z"Y,) for whichR(z"y,) = O. This equation represents a certain locus of critical points, and by virtue of the basic assumption a discontinuity occurs each time the representative point reaches the curve R(z"y,) = O. It is important to note that, as far as the trajectory is concerned, the passage through a critical point does not in any way affect its determinateness since R cancels out in the expression :: =~. It is impossible, however, to determine the ?notion on the trajectory in the neighborhood of a critical pOint. In this respect the local properties of a critical point are opposite to those of a singular point where the trajectory is indeterminate but the motion is determinate. 2. The existence of critical points or of a locus of such points can sometimes be revealed from the study of trajectories in a certain domain of the phase plane. A typical example in which this can be done is shown in Figure 130.1. The trajectories arrive at, or depart from, a certain threshold L from both sides, 8S shown. If no singular points, that is, postions of equilibriu?n. ezist in a narrow dO?nain surrounding L, one can assert that the line L is a locus of critical points. It is apparent that the trajectories situated in the regions M and N belong to two different differential equations. Let us assume that the phenomenon is represented by the motion of the representative point R on a traFigure 130.1 jectory T of the region N. Since the singular points are absent by our assumption, R will reach a point P on L in a finite time. Having reached this point, the representative point finds itself in a kind of analytical i?npasse from which there is no normal issue, that is, along the integral curves. In fact R cannot pass onto the trajectory T' passing through P nor can it turn back on T Since, in both cases, this would be inconsistent with the different1al equations prescribing a definite direction on the trajectories of the two regions M and N. Nor can the representative point remain at the point P which is not a pOSition of equilibrium. The differential equations cease to have any meaning at the point P and therefore cease to represent the physioal phenomenon. Hence the point P is a critical pOint, and the line L is a locus of such points. By our basic assumption the discontinuities necessarily occur once the representative point has reached some p01nt on L. We are going to use the basic assumption extensively in the investigation of relaxation oscillations 1n relatively complicated circuits in which it 1s impossible to predict the nature of the phenomenon on a basis of

395 elementary intuitive reasoning. It is useful to illustrate the application of the basic assumption to the simple example given previously, Figures 127.3 and 127.4. It is apparent that at the instant when the switch is closed, or opened, the differential equation of the second order ceases to describe the phenomenon since the right and the left portions of the circuit, instead of being in series, become in parallel. Hence this instant corresponds to a critical point, and a discontinuity is to be expected as we have ascertained by the elementary argument. The same argument applies to the ball striking the walls. In these two examples the oscillations are of the impulse-excited type, and the application of the basic assumption does not yield anything of interest. We shall see that in connection with relaxation oscillations proper the basic assumption will be a useful tool by which the possibility of relaxation oscillations can be ascertained. 131.

CONDITIONS OF MANDELSTAM At the end of Section 127 certain idealizations and a choice of variables were specified so as to be able to introduce a discontinuous treatment of certain problems. It was shown that the necessary condition for such a treatment is the degeneracy of differential equations from the second to the first order if the var~ables ~~ and ~r are selected. In the preceding section we have formulated a sufficient condition for the occurrence of a discontinuity on the basis of a certain basic assumption. There still remains one question to be settled, namely, the determination of the discontinuities once we have ascertained by the assumption that the discontinuity has to occur. Using the terminology of the phase plane, we can specify this last part of the problem as follows. Let us assume that the representative point R has reached a critical point A(~l'Yl)' We may question into which other point BC~2'Y2) the representative point will jump from the point A. In discussing the solutions [127.3J of Equations [127.2J we have already touched this subject and found that in the very special case considered there the jump is from ACO,O) toB(~,f). L. Mandelstam formulated the conditions of a jump on the basis of certain plausible assumptions regarding the continuity of energy during the infinitely short time interval of the discontinuity. It is to be noted that these conditions of Mandelstam are useful for relaxation oscillations and not for impulse-excited OSCillations for reasons which will appear later. The argument of Mandelstam is based on the continuity of the functions i(t), the current through an inductance L, and Vet), the voltage across the capacitor, as was previously mentioned in connection with the expressions [127.3J representing solutions of the degenerate equations of the first order. Since

396 L -i·2

stored in itt) and V(t) are continuous, clearly the electromagnetic energy an inductance and the electrostatic energy stored in a capacitor are also continuous functions of time. One obtains the conditions of Mandelstam by writing .1V

'0 + 0 1'0 - 0

= 0

[131.1]

where (to - 0, to + 0) is the infinitely small time interval during which the discontinuity occurs. The important point to be noted in connection with these conditions is that they are applicable to an infinitely small time interval and to circuits with finite dissipative parameters. The first restriction is trivial and is nothing but the expression of a continuity of functions itt) and V(t). As to the second, it requires an additional remark. One could formulate the following case in which the conditions of Mandelstam apparently do not hold. Let us assume that a charged capacitor is suddenly short-circuited so that its energy C~2 disappears instantly; this seems to contradict the second condition [131.1] of Mandelstam. The fallacy of this reasoning lies in the fact that the only way in which the energy can disappear suddenly is to be totally converted into heat. But in order that this may occur, a,finite dissipative parameter must be present. If, however, such a dissipative parameter. exists in the circuit, there exists also a finite time constant so that the disappearance of the charge, and, hence, of the energy, cannot be instantaneous, and it is sufficient to define a small time interval consistent with the time constant of the circuit to ensure the validity of the conditions of Mandelstam. The usefulness of the conditions of Mandelstam is limited only to relaxation oscillations proper. In fact, in impulse-excited oscillations the idealization employed is of an entirely different kind, and it is assumed that the energy exchanges between the system and an external source occur instantaneously. Summing up the result of this and of the preceding sections, it can be stated that the basic assumption and the conditions of Mandelstam are useful in studies of relaxation oscillatIons but are unnecessary for impulseexcited oscillations. 132. REMARKS CONCERNING SYSTEMS OF DEGENERATE DIFFERENTIAL EQUATIONS In Section 3 it was mentioned that a differential equation of the second order can always be represented by a system of two differential equations of the first order if a new variable'll = ~~ is introduced. Likewise, a differential equation of the nth order can be reduced to a system of n differential equations of the first order by introducing the variables ~: = 'Ill , d1l1 = 'II

dt

2'

•••

•

397 With degenerate differential equations the·situation is somewhat different. Thus a completely degenerate equation of the second order is, in fact, a differential equation of the first order. As a result of this, the phase space, instead of being two-dimensional, that is, a phase plane, becomes uni-dimensional, that is, a phase line. Moreover, instead of analytiC trajectories, piecewise analytiC trajectories become possible. A system of two differential equations of the second order can generally be reduced to a system of four differential equations of the first order, which means a system of the fourth order. If, however, each of the original differential equations of the second order degenerates into one equat10n of the first order, the system of the fourth order reduces to one equation of the second order, and its solutions can be represented by trajectories in a phase plane. This resultant equation of the second order, however, represents the result of degeneration of the system of the fourth order. We can express this by saying that we have a doubly degenerate system. Since each of the two differential equations of the first order admits discontinuous solutions, the doubly degenerate system of the second order will also possess certain discontinuous stretches in its phase plane so that its trajectories, in general, will be composed of certain analytic arcs joined by these stretches. Under certain conditions a doubly degenerate system of the second order may degenerate into a single differential equation of the first order; we can call such a case a triply degenerate system. We shall encounter one such system in what follows. In a triply degenerate system one differential equation of the first order represents the result of the degeneration of the system of the fourth order. Although the use of degenerate equations extends the application of topological methods to a series of important practical cases which could not otherwise be represented in a phase plane, it should be noted that piecewise analytic trajectories of this kind "closed ll by discontinuous stretches do not exhibit the features co~~on to the regular analytiC trajectories studied in Part I. We shall encounter, for example, systems in which no singularities exist inside such IIclosed ll trajectories; in some other systems such traJectories include singularities alternating in the course of time between a focal, or nodal, point and a saddle pOint. Moreover, certain difficulties arise in the formulation of conditions for stability of such degenerate systems, as will be specified in Chapter XXV. These peculiarities are, of course, to be expected. They reflect to some extent the fact that the discontinuous theory of relaxation oscillations is based on certain idealizations; this fact makes it difficult to compare it directly with classical methods making use of analytic functions. The principal usefulness of this theory at present is that

398 it permits obtaining satisfactory qualitative information in a great majority of important practical problems whose solutions are beyond the reach of existing analytical methods.

CHAPTER XXI DEGENERATE SYSTEMS WITH ONE DEGREE OF FREEDOM

133.

PERIODIC SOLUTIONS OF DEGENERATE SYSTEMS OF THE FIRST ORDER A differential equation of the first order

dx dt

=

f(x)

[133.1J

obviously does not possess continuous analytic periodic solutions. Moreover, one can assert that if the function f{x) is Single-valued, no continuous, although not necessarily analytic, periodic solutions are possible. In fact, in order that 8Mne sort of periodicity may exist, it is necessary that the system traverse the same line x = Xl with two oppositely directed velocities; this, however, is impossible if f{x) is single-valued. In the example illustrated by Figure 127.2b, we saw that the "closed" periodic trajectory ABeDA is characterized by the fact that the function f{x) = ±vo is actually doublevalued. It so happens that, in the example referred to, the function f{x) is a constant. It is easy, however, to waive this restriction by assuming that the plane on which the ball rolls, instead of being horizontal, rises toward the right wall. Then the trajectory will be a trapezoid ABeDA, as shown in Figure 133.1. One could imagine still other cases by assuming that, instead of rolling on a plane, the ball rolls on some kind of cylindrical surface whose generating lines are parallel to the x-axis. In this w w case the trajectory would be formed x by stretches AB and eD of analytic arcs "closed" by the discontinuous stretches Be and DA. The change from one branch of the function f{x) to the other one generally occurs at critical points Figure 133.1 and is discontinuous. Very frequently this is equivalent to saying that the phenomenon is governed by two distinct differential equations during its cycle. During one fraction of the cycle the phenomenon is described by one differential equation and during the other fraction by the other equation. The change from one differential equation to the other occurs at the critical pOints. In the example illustrated by Figure 133.1 the function f{x) happens to have branches which are symmetrical with respect to the x-axis. This is not always so, as will be shown later. We will now illustrate this matter by the following two well-known examples.

400

134.

RELAXATION OSCILLATIONS IN A CIRCUIT CONTAINING A GASEOUS CONDUCTOR

In view of the fact that this subject has been explored, we shall omit the familiar details and will endeavor primarily to show the application of the discontinuous theory in order to prepare the groundwork for more complicated cases beyond the reach of elementary theory. R

~

v

c

E

N

o

Figure

v

Figure

134.1

134.2

Figure 134.1 shows a circuit with the usual notations and with the positive directions as indicated; N is a gaseous conductor, such as a neon tube. Figure 134.2 represents the characteristic* ABDof the gaseous conductor N, which can be represented by a non-linear empirical relation i 1 = t/>( V) •

The differential equations of the circuit obviously are i =

dV c dt

[134.1]

These equations reduce to the following differential equation of the first order dV = RC 1 [ E - V - Rt/>(V)] (it

[134.2 ]

This equation is valid only when the discharge exists; during its extinction il = t/>(V) = 0, and we have dV 1 = - ( E - V) dt RC

-

[134.3 ]

Since we know that the phenomenon 1s characterized by alternate str1king and extinction of the d1scharge,1t 1s apparent that 1t is represented

*

In order to avoid ~ lII1lJU1lClerstandiDg, we shall usa the term o/acsrllo'.ridie 1D tba angiDeerillg sense, that is, to designate a certaiD exper1Jllental curve cODll8ct1ng the values of certaiD ~s1cal quantities, such as current 'I through the non-linear conductor N and voltage Vacross it. We will reserve tba term 'rllJ,,'orw to lIIean an ''''',rll' IItiroe of a differential aquati'On, as we have done previously.

401

alternately by two distinct differential equations, [134.2] and [134.3]. Since these differential equations are of the first order, only one constant of integration is involved in their solutions. But as there are two dynamic variables i l = ~(V) and V, it is apparent that there must exist a definite relation between these variables, as has been mentioned in connection with Equation [128.3]. During the time intervals when the discharge exists, this relation is obviously the characteristic ABD. Since the phase space is uni-dimensional for this system, it is apparent that during the intervals when the discharge exists the characteristic appears as the phase line and during the intervals of extinction the phase line is the V-axis. Aside from these phase lines, the plane is not involved. As is well known from elementary theory, the points of equilibrium are given by the points of intersection of the characteristic and the straight line i ; EiV' On the upper branch AB of the characteristic the equilibrium is stable; on the lower branch BDit is unstable, as is easy to ascertain by an elementary procedure. The case when the straight line i = E V cuts the upper branch of the characteristic is obviously of no interest from the standpoint of oscillations. We shall confine our attention, therefore, to the case when the resistance R has been adjusted to a value at which the straight line and the characteristic intersect at some point 5 situated on the lower unstable branch BD. Assume that we start the investigation of the phenomenon at the instant when the discharge has Just appeared; this instant is represented by the point A in Figure 134.2. Since ~r < 0 on the upper branch in this case, the representative point will move from A to B on the upper branch. The arrows on the characteristic, which, as was just explained, is also the phase line, indicate the positive directions consistent with the differential equation. It follows, therefore, that having reached the point B, the representative point finds itself in a situation which was specified in connection with Figure 130.', so that the point B is a critical point of the differential equation [134.2], and by our basic assumption we can assert that a Jump must occur at this pOint. In order to determine the character of the jump we have to apply the condition of Mandelstam. The only form of stored energy is the electrostatic energy C~2 stored in the capacitor. Thus the condition of the jump is

R

.1V

'0 + 0 1

'0 - 0

=

0

which means that II during II the Jump the voltage V across the capacitor remains constant, that is, the jump occurs parallel to the ii-axis along the stretch Be. The process of extinction is thus "explained" on the basis of the discontinuous theory.

402

Beginning with the point C, Equation [134.3] describes the process of charging the capacitor with no discharge. During that time interval the representative point moves along the V-axis with a finite velocity until the point D is reached. Here the discharge strikes again, and the representative point is transferred discontinuously to the point A, after which the cycle is repeated. Since at the point D Equation [134.3) ceases to represent the phenomenon, we conclude that D is also a critical point. It is thus seen that the familiar phenomenon of relaxation oscillations of a circuit containing a gaseous conductor can be treated consistently on the basis of the discontinuous theory and can be represented by a piecewise analytiC cycle ABCDA formed by two analytic branches AB and CD "closed II by the discontinuous stretches BC and DA. We note also certain differences between the discontinuous oscillations just studied and the continuous oscillations treated by the classical theory in Part I. The closed trajectoryABCDAis not of a limit-cycle type in that the phenomenon jumps, so to say, directly into its cycle without a gradual asymptotic spiraling around it. This cycle resembles in all respects a similar cycle shown in Figures 127.2b and 133.1 describing the idealized behavior of a ball undergoing reflections from constraining walls. So far, study of such oscillations does not reveal anything new, and the main purpose of this discussion is to illustrate the application of the discontinuous theory of relaxation oscillations to a familiar phenomenon generally treated by elementary methods. As we proceed further, the usefulness of this theory will become more manifest. In the example treated in the following section, for instance, it would be relatively difficult to apply the semi-intuitive physical argument. In still further examples it would be altogether impossible. 135. RC-MULTIVIBRATOR As a second example of a system which can be described by one differential equation of the first order, we shall investigate the so-called RCmultivibrator circuit shown in Figure 135.1. The electron tube Vz is a nonlinear conductor characterized by the equation I. = ~(eg). The tube VI appears here merely as a linear amplifier amplifying the potential difference ri, the feed-back voltage, between the points Band D and applying the amplified voltage eg = kri to the grid of Vz' k being the amplification factor of VI' The circuit is idealized in the customary manner, that is, the effects of the parasitic inductance, the grid current, and the anode reaction are neglected. The differential equations of the circuit with the positive directions shown are

IIII

A

VI

ti

Ve

tl

R 89

i=

lIa

C

Ii

r

Figure 135.1

(R + r) i + V = R,p(kri);

i = C

dV dt

[135.1 )

These equations reduce to the equation 1 .

t [krR,p'(kri) - (R + r)] :: = C

The critical point i

[135.2)

= i l is given by the equation !(i l )

= [krR,p'(kri l )

-

(R

+ r>] =

0

[135.3 ]

where ~'(kril) designates the slope of the characteristic at the point i = i l • One can also apply the argument given in connection with Figure 130.1. For this purpose the graphical procedure shown in Figures 135.2a and b will be useful. Figure 135.2a shows the characteristic 1m = ,p(kri) of V2 multiplied by a constant factor R. The tube V2 is supposed to be biased at a point 0 in the middle of the rectilinear part of its characteristic. From the usual form of the characteristic ,p(kri) it is apparent that its slope ~ = ~' is maximum when i = 0 and approaches zero monotonically as Iii + 00. Let ~'(O) = Sand assume that RS>R + r, without which, obviously, no self-excitation is possible. The curve veil represents the function R,p(kri) - (R + r)i. The curve shown in Figure 135.2b is the slope curve of the function B¥(kri) multiplied by a constant factor kr and referred to the axis M'N'. It is apparent that if this curve is referred to the MN-axis parallel to M'N' and at a distance (R + r) from the origin 0, the ordinates I(i) of this curve represent the left-hand term of the expression [135.3). Hence, the points P and Q are critical pOints, and by transter~ing these points on the diagram ot Figure 135.2a one obtains the critical pOints Band D situated on the Veil-curve and symmetrical with respect to the origin under the assumed idealization ot the

404 characteristic. It is apparent also that the inner interval (-i l < i < +i l ) is unstable since f(i) > 0 in that interval. On the contrary, the outer interval (i > +i l ; i < -id is stable since f(i) < 0 in that interval, as is seen at once from Equation [135.2]. The positive directions on the I --..,---- A phase line, the V(i) curve, are I shown in Figure 135.2a. The arguI I ment specified in Section 130 is I therefore applicable. Thus, for I example, when the representative I I point has reached the point B on I (0) the stretch AB, it has to jump discontinuously, and the second condiR tion of Mandelstam specifies that the jump must occur parallel to the i-axis, since in such a case the energy stored in the capacitor does not change during the jump. The discontinuous stretch BC so traversed ends at the point C. Here M·-~------:!-...I...-----......;::-N· (b) the phenomenon is again governed by the differential equation, and the Figure 135.2 analytic stretch CD is traversed with a finite velocity. At the point D another jump occurs along the stretch DA, so that a closed piecewise analytic cycle ABCDA results. The phenomenon therefore follows a pattern substantially the same as that which has previously been investigated in connection with the ball striking the walls and also with the neon-tube oscillator. The difference between the neon-tUbe circuit and the multivibrator circuit is that for the former the phenomenon is described alternately by two separate differential equations [134.2] and [134.3], whereas for the latter the phenomenon is governed by only one differential equation [135.2]. Because of the symmetry of the curve V(i) and that of the critical points, however, the closed cycle is obtained with the origin as the center of symmetry. v

405

136.

SYSTEM WITH ONE DEGREE OP PREEDOM DESCRIBABLE BY TWO DIFFERENTIAL EQUATIONS OP THE PIRST ORDER

The systems with one degree of freedom so far considered could be expressed by one differential equation of the first order. Chaikin and Lochakow (14) have investigated an interesting case when a dynamical system with one degree of freedom is describable by two differential equations of the second order possessing critical points. We shall outline briefly the principal features of the relaxation oscillations appearing in this case. The fact that the system of differential equations is now of the second order will account for the representation of the phenomenon in the phase plane instead of its uni-d1mensional representation by a phase line, as in previous cases. It will be shown that the discontinuous theory permits establiShing the principal features of the phenomenon, although a direct intuitive argument, such as is applicable in the Simple systems of Sections 133, 134, and 135, is insufficient here. The circuit investigated by Chaikin and Lochakow is shown in Figure 136.1. It is observed that the only difference between this circuit and that shown in Figure 135.1 is that here the inductance L replaces the resistance R of the circuit of Figure 135.1. It will be shown now that the behavior of thi~ circuit exhibits features entirely different from those of the circu1t of Pigure 135.1. Using the prev10us notations, we obtain, by Kirchhoff's laws, I. = cp(kri) = I

IIII VI

+

i;

L :: -

ri -

~ Ji dt =

[136.1 ]

0

A

v

t

2

i

L

fI

vI

-

to

r

Pigure

136.1

1 i

C

406 Introducing the variables z tions, one gets d:t

dt

= kri

= _'11__

1/1(%)

I

and

'II

= ~; and differentiating these equa:t

d'll =

dt

+

krLC

1 'II kL I/I(:t)

[136.2 ]

where 1 kr

The phase trajectories are given by the equation d'll = d%

%I/I(:t) krLC'II

+..!.kL

[136.4 ]

The point z = 'II = 0 is clearly a singular point. It is apparent from the form of the characteristic of electron tubes that the function I/I(z) decreases monotonically from a positive value 1/1(0) = S - k1 to a negative value 1/1(%) = when l:tl .. 00. Hence there exist two roots :t : !%l' for which both ~~ and ~ r become infinite. According to our definition, these roots :t = !Zl are the critical point8 of the system [1,6.2]. In the phase plane the values :t = ±:t1 determine two critical thre8hold8; that is, they are loci of the critical points of this system. It is to be noted that at these thresholds the tangent to the phase trajectories

t

is determinate. It is impossible, however, to determine the ~otion of the representative pOint since the differential equations become meaningless at these points. In order to show that these thresholds z = ±:t1 separate the regions of the phase plane in which the topological structure of trajectories is radically different, we shall simplify the ~(x) problem slightly without, however, inI troducing any qualitative changes. FigI ure 1,6.2 represents the characteristic IIIII Yo of the electron tube VI shown in Figure -1 1-4 • 1,6.1. We shall exclude the critical I I 1 thresholds :t = ±%l by drawing parallels I I x, x to the 'II-axis on both sides of each threshold 80 as to obtain narrow strips of width ~:t. The function 1/1(%) will then be S(s) - k~ > 0 and -k~ in the inner and outer intervals, respectively. This will cause no qualitative changes Figure 136.2 but will change slightly the form of the

407

trajectories in the neighborhood of the strips; see Section 9, Part I. Since the equations then become linear, the standard procedure, see Section 18, Part I, shows that the origin 0 appears as a saddle point for the trajectories of the inner interval (-2: 1 < 2: < + 2: 1) and as either a focal point (if r < ~) or • a nodal point (if r > ~) in the outer interval (2: < -2: 1; 2: > +2: 1), We shall assume the existence of a focal point in the outer interval since this is commonly encountered in applications. The picture of traJectories in this case is shown in Figure 136.3. A continuum of hyperbolic Figure 136.3 trajectories corresponding to the saddle point fills the inner interval, and a continuum of spirals corresponding to the focal point appears in the outer interval. By an elementary discussion of the Sign of ~ in Equation [136.4] for different quadrants of both intervals, it can be ascertained that the positive directions on the trajectories are oriented as shown by the arrows. It is apparent that these thresholds 2: = ±2:1 are loci of critical poL~ts, and discontinuities occur whenever the representative point following a trajectory reaches one of the lines 2: = !2: 1• Since the differential equations become meaningless at the critical points, the determination of discontinuities is made by the conditions [131.1] of Mandelstam, which, in this case, are fj/

to

I

+0

to - 0

=

Ito + y dt 0

= 0;

o [136.6]

to - 0

Applying these conditions to Equations [136.1), one obtains ~(2:1) - .!L = ~(2: ) - .!!.. kr 2 kr'

where 2:2 and Y2 are the coordinates of the representative point immediately after the discontinuity. Since the function ~(2:) is an empirical curve, it is preferable to plot first the function 9(2:) = ~(2:) - k2:r as was done in Figure 135.2a. One obtains for 9(2:) a curve Similar to the curve veil of Figure

408 135.2a. By expressing the condition 9(~1) = 9(~2)' one can find , .,/ ./ /' ~2 if ~l is given. We omit this ./ graphical construction and merely mention that for characteristics .,II' ....of electron tubes commonly encountered the jump occurs from + ~l to X ·X2 -x. -~2 (with I~al >I~ll) and from 0 -~l to +~a. When ~2 is mown, the second equation [136.7] permits determining Y2 for given ~I and YI· The representation of such a quasi-discontinuous oscillation in the phase plane of the variables (~,y) is shown in Figure 136.4. It is noted that the jump Figure 136.4 from (~I'YI) to (~2'Y2) is the same for all trajectories. Assume that the representative point, following a trajectory T of the outer interval, has reached a critical point A of coordinates (~I'YI). The second equation [136.7] then permits determining the "direction" of the jump AB in the phase plane. At the point B another spiral trajectory begins which encounters at C another critical point resulting in the jump CD, and so on. It can be shown, although we omit the proof, that after one turn of the radius vector the piecewise analytic spiralABCDEwill approach the origin, which means that the point E is below the point A if the latter is suffiCiently distant from the origin O. If, however, one applies the same reasoning to a point A' near the ~-axis on the critical line ~ = !~l' one finds, on the contrary, that E' is above A'. Thus the large spirals shrink and the small ones grow with each turn of the radius vector. Hence there exists one and only one piecewise analytic spiral for which the points A and E coinCide so that the trajectory becomes closed. Such a closed trajectory may be termed a piecewi8e analytic limit cycle, and i t is stable. One notes the difference between such a trajectory and the piecewise analytic cycles described in Sections 133, 134, and , 35, which are not 01' the limit-cycle type. The difference is due to the fact that the present phenomenon is expressible by two differential equations of the first order and is represented in the pha8e plane, whereas in the sections mentioned above it was determined by one differential equation of the first order and was represented by pha8e line8 without involving any other points of the phase plane. .!I.l. dt

....-

./

Another remarkable feature of E this analysis is that the unstable hyperB bolic trajectories of the inner interval do not appear at all in this representation because that interval is traversed discontinuously. Experiments made by Chaikin and Lochakow corroborate these theoretical conclusions. In their experiments a cathoderay oscillograph was adapted to record the phase trajectories of the system, as was o explained in connection with Figure 24.7, c Part I. The record has the appearance shown in Figure 136.5. There are two spiral arcs Figure 136.5 BC and DE corresponding to a relatively slow motion of the electronic beam on the analytic trajectories of the outer interval. The quasi-discontinuous jumps EB and CD remain unrecorded because of the much higher speed of the beam in the region of the inner interval. It seems logical to assume that when the phenomenon starts from rest, one hyperbolic trajectory of the inner interval is actually traversed, but once the first critical point has been reached, the phenomenon begins to "skip" the inner interval and continues to do so thereafter. Once the essential features of this phenomenon have been ascertained by this method, it is easy to give a corresponding physical interpretation. The fact that' the jumps occur obliquely in the phase plane of the variables (x,y) means that at these instants the voltage L ~! across the inductance and the current i = c~r in the capacitor undergo quasi-discontinuities, although the current I through the inductance and the voltage V across the capacitor remain continuous. In other words, the functions I(t} and V(t} are continuous but are not analytic at the jumps in the sense that they have discontinuous first derivatives. This is in agreement with the elementary theory of an impulsive excitation of the idealized (L,R)- and (R,C)-circults. It is apparent, however, that mere knowledge of these well-known facts would be entirely insufficient for the purpose of establishing a qualitative picture of a complicated phenomenon of this nature if no general method, such as that offered by the discontinuous theory, were available.

CHAPTER XXII MULTIPLY DEGENERATE SYSTEMS

137.

MULTIVIBRATOR OF ABRAHAM-BLOCH*

As an example of a system with two degrees of freedom describable by two differential equations of the first order we shall investigate the behavior of the circuit shown in Figure 137.1. Neglecting the effect of grid current and anode reaction and using the notations and positive directions shown, we obtain the following equations:

RI2 Ia2

+ ~ =

f i2 dt

+

ri 2 = E

(137.1 ]

~(e"2) = ~(ril)

where Ia = ~(e,,) is the non-linear characteristic of the electron tubes VI and V 2 • Differentiating the second group of equations (137.1] and making use of the other two groups,we obtain the following system of differential equations: (R

dill.

+ r ) dt + C 'I +

'( .) di2 Rr~ r'2 dt = 0 [137.2 ]

=E

Figure

*

This device is described in Reference (15).

137.1

411

Th1s system reduces to the following equations: di, == dt

i (R + r)~ci - Rrq,'(ri2).!1.c R 2r 2q,'(ri l ) q,'(ri 2) - (R + r)2 (R

dig ==

+

r)

[137.3]

i - Rrq,'(ril)~

dt

which are of the form

One notes the symmetry of Equations [137.3] with respect to the variables it and ip,; this symmetry is due, of course, to the symmetry of the circuit. The phase trajectories in the (ii' i 2 )-plane are given by the equation

[137.5J Prom the explicit values of the functions Pand Q, observes that the origin i l = ip, = 0 is a singular librium point of the circuit. We first inqu1re whether closed analytic in the system. Applying the negative criterion of one ascertains that dP -d' 'I

+

dQ -d' 12

==

2(R

+

C

r)

==

Equations [137.3), one pOint, that is, an equitrajectories are possible Bendixson, see Section 25,

constant

Hence no closed analytic trajectories exist here. The nature of the singular point i l = i2 = 0 is determined by the equations of the first approximation. If we let q,'(ritlil = 0 = ~'(ri2)i2 = 0 = S and i f we assume that RrS > R + r, Equations [137.3] become di

RrS.

-dtl ==

M

RrS.

12 ;

+

R

--;.{" 11

+

M

r. '2

[137.7]

0

[137.8J

where The characteristic equation of this system is )..2 _

2(R

+

M

r»)..

+

[(R

+

r)

+

RrS][(R

M2

+

r) - RrS]

==

The origin (i l = i2 = 0) is therefore a saddle pOint. One ascertains easily that, when RrS< R+ r, the or1gin is a stable nodal point, but this case is of no interest from the standpoint of relaxation oscillations. We will assume the existence of a saddle point. Since, initially, RrS >R+ r and the origin

412

is unstable, the variables i l and i2 begin to increase. On the other hand, from the form of the characteristic Ig = ~(ri), we know, see Section 135, that I~' (ri) I + 0 when i + 00. The function

[137.91 which is initially positive, decreases monotonically when it and i2 increase and is negative when i l and i2 are very large and are equal. Hence there are certainly some values of it and i2 for which U= o. This means that the system [137.}1 has critical points so that by virtue of the basic assumption, Section 1}0, discontinuities must occur at these points. The expression U(i l , i 2 ) = 0 defines a certain curve r 1 in the plane of the variables i l and i 2 • In order to simplify the notation, let us designate by ~l and Y l the values of it and i 2 • respectively, situated on the curve r l shown in Figure 137.2. The equation of this curve is then

[137.101 From the form of the functions ~'(r~l) and ~'(rYl)' approximating the slope to the characteristic Ig = ~(e,) of the electron tube, one ascertains that the curve r t 1s a symmetrical closed curve with the origin at its center and has the form shown in Figure 137.2. As soon as the representative " point following a trajectory has reached a point (~l'Yl) on the curve r l , a Jump must occur, and the point (~2'Y2) into which the representative point Jumps is determined by the condItions of Mandelstam which we now propose to apply. Since the only form Figure 137.2 of stored energy in this case 1s purely electrostatic, namely, the energy stored in capacitors, the conditions of Mandelstam require that the potential differences V' and V" across the capacitors remain constant during the discontinuity. This gives the conditions V" = E - R~(ril) - {R

If we designate now the terminal coordinates of the Jump by ditions of Mandelstam obviously are

+ r)i 2 [137."] (~2'Y2)'

the con-

413 Rrp(rYl) -

(R

+

r)x 1 = Rrp(rY2) -

Rrp(rXl) -

(R

+

r)Yl

=

Rrp(rx 2) -

(R

+

r)x 2

(R

+

r)Y2

[137.12]

these notations (X 1'Y1) represents the critical point on the curve r 1 from which the representative point is transferred discontinuously to some other poInt (X 2'Y2) determined by these equations. Since there is generally a one-to-one correspondence between the points (X 1'Y1) and (X 2'Y2)' we conclude that the locus of the points (xz,Yz) is another curve r z which is also a closed curve symmetrical with respect to both variables i1 and i2 since the circuit is entirely symmetrical. The phase-plane representation of the behavior of the Abraham-Bloch multivibrator can then be described as follows. Assume that the representative point has reached a critical point a on the curve 1;. From this point it jumps into a corresponding point A on the curve ra' At the point A begins a stretch of a continuous analytic trajectory Ab. The point b is another critical point where another jump bB occurs. At B a second continuous trajectory begins which ends at another critical point c, from which a new Jump cC takes place; at C a continuous stretch Cd begins which ends at d, and so on. By a more detailed analysis of such piecewise analytic trajectories interrupted by discontinuities, it can be shown that the transient behavior of the Abraham-Bloch multivibrator approaches a stationary symmetrical state, which is oriented along the bisector of the i 1 - and i 2 -axes and which consists of a continuous stretch apfollowed by a jump PD, then by another continuous trajectory DE followed by another Jump Ea which closes the "cycle . II In this case, the cycle 1s un1-dimensional, that is, it is the phase line Da. It is apparent that if, instead of assuming the asymmetry of the initial conditions as we did, only a steady-state condition was aimed at, a simplification could be made in the differential equations by introducing the conditions of symmetry, that is, i l = -ia and f,6'(rid = -rp'(ria ). With this simplification, one obtains a single differential equation of the first order, namely, In

di dt =

(R + r) + Rrf,6'(ri) i R2r2[rp'(ri)]2 - (R + r)2 C

which reduces to the form [Rrf,6'(ri) -

(R

+

r)]

~~ = ~

[137.14]

This is identical with Equation [135.2], the equation of an asymmetrical relaxation oscillation. We conclude, therefore, that the transient behavior of the Abraham-Bloch multivibrator is represented by a doubly degenerate system. As far as the stationary condition described by one different1al equation

414 [137.14] of the first order 1s concerned, the multivibrator 1s represented by a triply degenerate system, to use the terminology of Section 132.

13B.

HEEGNER I S CIRCUIT; ANALYTIC TRAJECTORIES As a second example of a doubly degenerate system, we shall investigate the so-called Heegner circuit (16) shown in Figure 13B.l, which is the same as the circuit of an asymmetrical RC relaxation oscillation shown in Figure 135.1, the only difference being that there is now an additional capacitor C2 shunting the resistor R. We propose to show now that the addition of the capac1tor C2 radically mod1f1es the behavior of the circu1t. It would be impossible to see this without the criteria of Section 130.

1\1\t-..A - - - . . . . . . - - - -.....

11

R

o

Figure 13B.1 With the positive directions indicated by arrows, the application of Kirchhoff1s laws gives

Putting IG = ~(krll)' as we have prev1ously, and el1minat1ng I, we obta1n the following system of the second order: d~

dt = ~ = R -

dt

1

Rrk~'(krII) I

RrCl

+

rC II I _ 1

R

+

1

rC

2

12

r - Rrk~'(krII) I RrC2 2

[1,8.2 J

It is observed that this system of differential equations has no critical points, and hence no discontinuities are to be expected. The only singular point is II = 12 = O. Noting that ~'(O) = S is a maximum and that the function I ~'(krIl)l .. 0 when III I" 00, the characteristic equation of the system

415 [138.2] is

[138.3 ] It is noted that the singularity here is not a saddle point. Hence, it is either a nodal pOint, if the roots Al and A2 are real, or a focal point, if they are conjugate complex. In both cases the singularity is unstable if R

+

r - RrkS ~

R

fA

l1 38.4]

- CI

If the singularity is unstable, self-excitation from rest is possible. By approximating the experimental function IG = ~(krll) by a polynomial and by applying the bifurcation theory, it is possible to establish the existence of a stable limit cycle. We shall omit this calculation since the fact that the Heegner circuit is capable of producing continuous self-excited oscillations is well known. Although the Heegner circuit does not produce quasi-discontinuous relaxation oscillations, it is mentioned here as an example of the difficulties which appear if an analysis is attempted on the basis of a more or less intuitive physical argument. In the meantime, this circuit provides an additional example for testing the validity of the basic assumption.

139. TRANSITION BETWEEN CONTINUOUS AND DISCONTINUOUS SOLUTIONS OF DEGENERATE SYSTEMS On the basis of the preceding argument, one might ask whether a gradual modification of a given dynamical system, for example, an electric circuit, might cause a transition from continuous performance to quasidiscontinuous performance, or vice versa. It will now be shown that such a transition is indeed possible and that its mathematical formulation can be reduced to the question of the appearance, or disappearance, of critical points as a result of the variation of a certain parameter in the differential equations. In order to show this, let us consider a slightly modified Heegner circuit such as that shown in Figure 139.1. The modification in question is that the capacitor C21 instead of being connected to the point B, is now connected by an adjustable sliding contact to some point E along the resistor r, as shown. Let r l be the resistance between the points Band E and r 2 be that between E and D, where r l + r 2 = rand rl/r = /J. Proceeding as we did in Section 138, we obtain the following system of equations: d/ l _ dt -

_

+

I

1 (1 - /J) rCI

(1 -

I

1 1 /J)rC2 2

l1 39.1 ] dl2 = [/Jr dt

+

R -

rR~'kr([1 + /JI2)]~ (1 -

,8)[R

+

-

[r

+

R -

,8r - ,8rR~'kr(I1

rR~'kr([l + /J/2)]~ +

/J/2)]

416

VI

A

IIII

IIII

tIl

f12

Vz

fl

- +11

Cz

CI

Ia

II

r

E

III

111

Figure 139.1 It is apparent that for p ~ 1, that is, r 1 = r, the circuit reduces to that shown in Figure 135.1 where a discontinuous performance occurs. For p = 0 and r 1 = 0, we have Heegner1s circuit, which produces continuous oscillations. For some intermediate value of p, the co-factor of (1 - P) in the denominator of the second equation [139.1J may vanish, which means that the otherwise continuous oscillation will undergo a discontinuous jump parallel to the I 2 -axis as shown by the broken lines in Figure 139.2. This generally occurs when the system (139.1J is characterized by a saddle point, and the transition range is at the point where an unstable focal point degenerates into a saddle point, o see Figure 18.1. We shall not enter into a further study of these complicated and relatively unexplored phenomena. It is sufficient to emphasize once more that their apparent complexity is related to the appearance, or disappearance, of critical Figure 139.2 points in the differential equations.

CHAPTER

xxm

MECHANICAL R1i:LAXA TION OSCILLATIONS

140.

INTRODUCTORY REMARKS

To date, mechanical relaxation oscillations have been studied less than electrical ones. As was mentioned in the introduction to this report, which appears in Part I, two principal reasons account for this. First, the determination of the parameters of a mechanical system is generally more complicated than the determination of the parameters of an electric circuit. Secondly, mechanical relaxation oscillations always appear as undesirable parasitic phenomena, and the endeavor of engineers is directed toward their elimination rather than their study. These phenomena generally take the form of II jarring II motions resulting from dry friction, misalignment of machinery, and similar factors. Whether any useful applications of these effects can be found is difficult to say. Very likely this state of affairs will continue until the laws of friction, which usually account for the appearance of such effects, are better understood and can be controlled with a prescribed degree of reliability. For the time being, the whole subject of mechanical relaxation oscillations is purely academic. In addition to mechanical relaxation oscillations proper, of which we will indicate examples in Sections 141 and 142, there exists a class of oscillations maintained by periodically timed impacts. These oscillations also exhibit quasi-discontinuous behavior describable by piecewise analytic trajectories. These impact-maintained oscillations can hardly be considered to be of a relazation type, although very frequently, and perhaps improperly~ they are classified as such. Typical examples of these systems are clocks and quenched spark oscillators. in which the energy of a linear dissipative dynamical system is increased periodically in a quasi-discontinuous manner by special timing. We shall leave the investigation of impulse-excited systems to a later chapter and will investigate here a simple mechanical system of a pure relaxation type. Let us consider a non-linear differential equation of the form

mz

+

kz = -

F(z)

= -

11/(Z)

[140.1 ]

where F(z) is a certain non-linear function of velocity which we shall identify with friction. If 11 is small, Equation [140.1] is of a quasi-linear type and can be discussed by the standard methods of Part II. More speCifically, for particular forms of the function fez) self-excitation may occur, and the OSCillation may approach a stable limit cycle, as was studied in Part I in connection with Froude1s pendulum; see Section 8.

418

When ~ is large, Equation [140.1] cannot be solved by existing analytical methods, and at present the only guide to its solution is the qualitative analysis of Lienard;* see Section 36. We know from his analysis that the periodic trajectory of the differential equation when ~ is large generally consists of two pairs of branches. On one pair of branches the motion of the representative point is slow and the displacements are 'large, so the system remains for a relatively large fraction of its relaxation period in that region. On the other pair of branches, however, the motion 1s very rapid. This phase of the motion is of very short duration so that, in spite of the large values of velocity and acceleration prevailing during these short fractions of the cycle, the coord1nate has no time to change appreciably. Lienard's analys1s fails, however, to determine the "cornersll connect1ng these branches 1n a closed analyt1c curve; thus a rigorous analyt1c solut10n 1s st1ll lack1ng. The mechanical relaxation OSCillations caused by friction exh1bit the familiar picture prev10usly stud1ed. In fact, there are time intervals when the slow motion becomes a state of rest, followed by an interval of exceedingly rap1d motion, followed by another per10d of rest, and 80 on. We are thus confronted w1th a special type of relaxation oscillation appearing as a kind ot II Jarring" mot1on.

141. QUALITATIVE ASPECTS OF A MECHANICAL RELAXATION OSCILLATION Since a cond1t10n of degenerat10n, as prev10usly shown, is essent1al for the appearance ot quasi-discontinuous solut10ns of a differential equation of the second order and since we are trying to approx1mate the relaxation process by a mot10n in conformity with the Lienard analysis, we must consider a mechanical system w1th a very small mass. Under this assumption, the motion of the system 1s determined ma1nly by the balance between the restor1ng force kz and the friction force ~f(z). When this balance is momentarily lost at a certain instant of the cycle, the accelerat10n and the veloc1ty of the system may suddenly reach very large values since, by our assumption, the mass 1s very small. When m= 0, Equation [140.1] degenerates into the equat10n F(z) = - kz

[141.1]

which describes the pair of branches of the Lienard cycle on which the system remains a relatively long time and the coordinate changes appreciably. The balance between the restoring force and the friction force takes place on these branches. Differentiating this equation, one has

*

Recent research (12) of J.A. Shohat gave hope of bridging this gap, but these efforts have been interrupted by his I.Dltimely death. .

419 F '( %.).. %

= -

k'%

[141.2 ] If the function F(z) is such that F'(~) < 0 in a certain region, Equation [141.2] shows that this region is unstable. If at a certain point in this region F'(z) ~ 0, the acceleration z may acquire a very large value so that a new balance of forces will appear in which the term mz will play a r~le in spite of the assumed smallness of the mass m. We thus reach the conclusion that this condition characterizes the second pair of branches of the L1enard cycle which are traversed in a very short time. During this phase of the cycle the acceleration is very large and the velocity varies in a quasi-discontinuous manner by a finite quantity, but the coordinate does not change appreciably because of the short duration of this phase. The qualitative aspect of the motion in this case is similar in all respects to that Which occurs when a ball strikes a wall. However, the underlying dynamical facts are different. For a mechanical impact, the quasidiscontinuities in velocity and acceleration described above are due to an external force, the reaction of constraint when the ball strikes the wall. In Equations [141,1] and [141.2] these discontinuities appear as a result of the disappearance of the balance of forces expressed by Equation [141,1] and are due to the assumed peculiarities of the function F(~). The preceding argument can be condensed somewhat by putting ~ = y in Equation [141.2], which gives [141.~

]

It is apparent that this equation has a critical point whenF~y) = 0 and, hence, ceases to describe the phenomenon at that point. In order to determine the discontinuity by which we idealize the quasi-discontinuity of the physical problem, we must again apply the condition of Mandelstam. In the idealized case, m = 0; hence, the kinetic energy is zero so that the total energy is the potential energy, which is a function of the coordinate %. From the condition of Mandelstam we infer, therefore, that the function %(t) is continuous although it has a discontinuous second derivative, as is to be expected. The situation is thus similar to that which we have studied in Chapter XXI in connection with electrical relaxation oscillations. It also resembles the results obtained in the theory of mechanical impacts, with the difference, however, that the quasi-discontinuity here arises from the existence of a critical point in the differential equation and is not due to external impulsive excitation as it is for a mechanical impact where the energy also changes abruptly.

420

142.

MECHANICAL RELAXATION OSCILLATIONS CAUSED BY NON-LINEAR FRICTION Since Equation [141.2] has a critical point for F'(x) ; 0, we conclude that relaxation oscillations are possible in a system of this kind whenever the friction force considered as a function of the velocity x possesses extremum values. Moreover, the general condition specified in Section 133, that is, that F(i) should be a two-valued function of x, must also be fulfilled. In practice, these theoretical conditions are frequently encountered. Thus, when a shaft rotates in a bearing, Sommerfeld (17) has indicated that the friction force goes through a minimum for a speed VI given by the equation [142.1 ] where P is A Is r Is 6 is

the the the the

pressure, coefficient of viscosity of the lubricant, radius of the shaft, and thickness of the oil film.

Chaikin and Kaidanowski (18) have investigated mechanical relaxation oscillations by the device described in the following paragraphs. A relatively small mass a forming a Prony brake engaged frictionally a rotating shaft K, as shown in Figure 142.1. The mass a was centralized in a definite position by a rather strong spring S. A definite friction force was secured by means of another spring not shown, pressing a against the shaft K. The differential equation of motion of a, neglecting its mass (which results in the degeneration of the equation to the first order), is Figure 142.1 [142.2 ] where F(v)

= F[(D -

~)r]

is the friction force, a non-linear function of the relative peripheral velocity v = (D - ~}r, D is the angular velocity of the shaft K, ~ is the angular velOCity of a for small departures ~, and c is a constant. depending on the spring's strength.

The approximate form of the functionF(v) is shown in Figure 142.2. It is noted that the tangent to this characteristic is horizontal at the point B(v = 0). When v = 0, a moves together with K, and ~ = D. When a is not moving, ~ = 0 and v = Vo = Dr. To the right of the point v = Vo the mass a and

421

F the shaft K move in the same direction; to the left of v = Vo they A move in opposite directions. Because o of the constraining spring S no continuous motion of a is possible; there may, however, be a position of c equilibrium for which ~ = O. As long as v = 0, that is, ~ = g, the spring S is gradually stretched, and the v o friction force F is a static force Figure 142.2 which may have any value whatever provided F ~ F I , where FI is the limit of static friction. When a is moved by the amount ~l' for which C~l = FIT, the limit of static friction is reached. During this phase (v = 0) of the phenomenon, the representative point moves along the axis of Funtil it reaches the point D, the limit of static friction. Differentiating Equation [142.2J, we have

[142.3] where F' deSignates~. It is seen that F' < 0 when ~ and ~ have the same signs, and F' > 0 when they have opposite signs. Moreover, for v = Vo the quantity ~ goes through zero and changes its Sign. If the speed 9 of the shaft is relatively small, the point M will be on tne part BD of the dynamical characteristic of friction which may be considered here as the phase line of the process, see Section 134. On the part BD of the phase line, F' < 0, so that ~ and ~ have the same sign; this part of the phase line is therefore unstable, as shown by the arrows. It follows, therefore, that when the representative point has reached the point D during the "static friction period" CD of the process, it cannot pass onto the dynamic characteristic DBA. We conclude that the point D is a critical point. One can illustrate this also by putting ~ = y in Equation [142.3J, which gives dy =

cy

dt

From this expression it follows that~ = ~ = ~ at the point D since F'[(9 - ~)rJ = 0 at this point. At D, the representative point undergoes a discontinuity, the direction of which is parallel to the v-axis because here the discontinuity occurs at a constant potent1al energy, as required by the condition of Mandelstam. The Jump terminates at the point A where the characteristic is met, and the subsequent motion of the representative point

422

is again continuous along the branch BC until another critical point is encountered at the point B where F' = O. Here another jump DAoccurs, after which the phase CD of static friction begins anew; during that phase of the motion the representative point again moves continuously along the branch CD of the cycle DABCD. One notes an analogy between this example of a mechanical relaxation osc1llat10n and the example of the relaxat10n osc1llation of a neon tube, Section 134. In both examples, relaxation oscillations are poss1ble if the phenomenon is contined to an unstable region of the characterist1c. In the neontube oscillator th1s 1s accompl1shed by adjusting the resistance so that the straight line E V cuts the non-linear characteristic, the phase line of the process, on its unstable branch. W1th the brake described in this section, a similar effect is obtained by running the shatt K at a relatively slow speed D so as to have the point M in Pigure 142.2 in the region where F' < O. There are no relaxation oscillations in a neon-tube circuit if the line Ei V intersects the characteristic on its upper stable branch, see Figure 134.2; likewise, no mechanical relaxation oscillations are observed, see Pigure 142.2, if the angular velocity D of the shaft is large enough so that the point M' corresponding to vo' = Dr is on the stable branch CD of the friction characteristic. The piecewise analytic cycles ABCDA in both Figures 134.2 and 142.2 are indicated by the corresponding letters. Thus, for example, the branch CD corresponds to the period when the capacitor is charged (Figure 134.2) and to the period when the static friction force F increases with no slipping between a and K (Figure 142.1). Branch AB in both examples corresponds to continuous trajectories of a non-linear differential equation. In both figures the discontinuous stretches DAand BC are determined by the condition of Mandelstam, and so on. Plotted in the (~,t)-plane, the curves of the mechanical relaxation oscillations observed by Chaikin and Kaidanowski have a typical "saw-tooth" appearance, see Figure 142.3, characterizing electrical relaxation OSCillations of a similar nature. This discussion emphasizes o t the features which all relaxation Figure 142.3 oscillations have in common.

i

CHAPTER XXIV OSCILLATIONS MAINTAINED BY PERIODIC IMPULSES

INTRODUCTORY REMARKS We have defined "relaxation oscillations" as stationary self-excited oscillations exhibiting quasi-discontinuities at some points of their cycle. One of the fundamental properties of these oscillations is that their stationary state does not depend on the initial conditions of the system. In that respect they resemble the continuous oscillations of the limit-cycle type studied in Chapter IV with the difference, however, that because of the existence of discontinuous stretches their properties are different from those of continuous oscillations. As a somewhat different type of oscillation appear the so-called impulse-excited oscillations. In some respects these oscillations resemble relaxation oscillations as defined above; in some other respects they differ from them. The feature common to both types of oscillations is that they can be represented in a phase plane by piecewise analytic trajectories. Por both types, also, the stationary state does not depend on initial conditions; moreover, the stationary trajectories of both are "closed ll by discontinuous stretches. The essential difference between these two types of oscillations lies in the physical process during the quasi-discontinuous rapid changes, idealized as mathematical discontinuities. In the pure relaxation oscillations with which we have been concerned in preceding chapters of Part IV, the energy stored in the system during the quasi-discontinuous interval does not change appreciably. This results in an idealized picture of these phenomena with an assumption that the energy does not change in the interval (to - 0, to + O). Moreover, the existence of these discontinuities, as we saw, was formally reduced to that of the critical pOints in the differential equations describing the system. Por the impulse-excited OSCillations which we are going to investigate in this chapter, a difference exists, in that the discontinuous stretches in the representation of the phenomenon by phase trajectories correspond precisely to the impulsive changes of the energy of the system. In the investigation of relaxation oscillations proper, we had to rely mostly on the theory of electric circuits. We shall begin the study of impulse-excited oscillations by investigating the behavior of a mechanical device known for centuries, the clock. '43 •

424

144.

ELEMENTARY THEORY OF THE CLOCK

A clock is a mechanical device consisting of three principal elements: 1. An oscillatory dissipative system, for example, an ordinary pendulum, a torsional pendulum with a hair spring, etc. 2. A source of energy, for example, a weight, a main spring, etc., which has to be replenished periodically ("winding" the clock).

3.

An escapement connecting periodically the first and second

elements. The purpose of the escapement is to release periodically the energy stored in the second element in the form of an impulse and to apply this impulse to the first element at an appropriate instant of the oscillation. Since the instant at which the impulse is released by the escapement is uniquely determined by the motion of the system, the system is autonoonOU8 in the sense that all three elements, 1, 2, and 3, of the mechanism are connected so as to form a single unit, Just as a thermionic generator forms a single autonomous unit comprising an oscillating circuit, a battery, and an electron tube with its grid control circuit. We shall see later that it is possible to arrange the operation of an electron-tube circuit so that it resembles the performance of a clock, but in general the mode of operation of a clock differs in other respects from that of the thermionic oscillators commonly used. The escapement communicates to the OSCillatory system periodically timed impulses which we shall first idealize as instantaneous and of a constant value a. Let VI be the velocity of the system immediately after the first impulse, from which instant we wish to study the motion. If we assume that the system is a linear dissipative one with a decrement d per cycle, the velocity v 2 ' immediately before the second impulse will be v 2 ' = VIe-d. Immediately after the second impulse, the velocity will be v 2 = v 2' + a = vle- d + a, where a is the impulsive increment of the velocity, and so on for subsequent impulses. A stationary state will be reached when, beginning with a certain number n of impulses, v" J:::I VII + 1 J:::I VII + 2 J:::I • • • J:::I vo, which gives (144.1]

The representation of the process in the phase plane is shown in Figure 144.1, where one spiral trajectory S of the system is indicated. As was shown in Section 5, these spirals form a continuous family so that through every ordinary point of the phase plane passes one and only one such spiral. Let us deSignate as u the segment AB of the z-axis. This intercept represents

425 the change of the initial radius vector xav OA after one revolution 2". From the properties of the logarithmic spiral it is apparent that the intercept u, considered as a function of To = OA, is a monotonic function U(TO)' Since the spirals S form a continuous family, it is also obvious that the function U(To) is a continuous monotonically increasing function of the radius vector T. On c the other hand, by our hypothesis, the impulsive changes of the velocity v due to the operation of the escapement are constant and hence are represented by Figure 144.1 constant jumps a along the x-axis. It follows, therefore, that a steady-state condition will be attained when a radius vector TO = OA is reached for which U(T o ) = a. This stationary condition is represented by a piecewise analytic cycle BCAB shown in Figure 144.1 by a heavy line; this cycle consists of a convergent spiral BCA (since, by our assumption, the system is dissipative) "closed" by a discontinuous jump AB representing the impulsive increase of the velocity v caused by the impact. It is also clear that such a stationary piecewise analytic cycle BCAB is of a stable limit-cycle type in the sense that, if one starts from the relatively small values of the initial radius vector To = OA, the value of a is initially greater than the corresponding value of the intercept U(To) in this region of the phase plane. Physically this means that the energy communicated by the impulse is greater than the amount of energy which the system is able to disSipate during one cycle 2", which will result in the fact that the subsequent radius vectors TO = OA will grow initially. If, on the contrary, the initial velocity of the system is sufficiently great, the dissipation of energy per cycle is greater than the impulsive increments communicated by the escapement so that the spirals will gradually shrink. The stable condition is reached when the impulsive increments a are just equal to the intercepts U(To) in a particular region. In view of the continuity and the monotonic character of the function U(TO)' it is clear that there exists one and only one piecewise analytiC limit cycle BCAB of the clock and it is stable. This elementary discussion accounts for the fact that the clock's performance is of a limitcycle type and as such does not depend on initial conditions, see Chapter IV; in other words, the ultimate performance of a clock does not depend on how it has been started. Once it is started the operation of the clock depends

426

entirely on the parameters of the system and has nothing to do with the initial conditions. The converse is true for a mathematical pendulum. In one respect the elementary theory discussed above fails to account completely for the observed facts. For, if a clock is wound, this simplified theory indicates that the clock should start by itself even if the initial disturbance is infinitely small. In other words, it predicts a 80ft self-excitation of the clock. In reality, unless a clock is given a certain minimum disturbance, for example, shaking, it will not start. This threshold is partially due to the existence of Coulomb friction not taken into consideration here. There is, however, another reason why the elementary theory is not complete, namely, our assumption that the jump a in velocity caused by the impact remains the same whatever the velocity of the system. Andronow and Chaikin (13) have shown that by a slight refinement of the preceding theory it is possible to explain better the observed facts. Let us assume that the change of kinetic energy during the impact remains constant, that is, 2

2

mV l -2In

mvo == constant -2-

the phase-plane representation this is equivalent to the condition 222

Y1 -

[144.2]

Yo == h

where h is a constant determined by the properties of the escapement and Yo and ylare the values of the radius vector immediately before and immediately after the impact. It follows, therefore, that the jump a in the phase plane, instead of being constant as was originally assumed, is now given by the equation a ==

y'y: +

h2

-

Yo

[144.3]

that is, it decreases with increasing velocity Yo according to a hyperbolic law, as shown in Figure 144.2. a This refinement of the theory, although it leads to conclusions concerning the existence of a stable piecewise analytic limit cycle which are similar to those obtained by the original simplified theory, is not yet sufficient ~ to explain the fact that a clock is a system with a hard self-excitation. In order to extend the theory still further it is necessary to investigate how Coulomb friction manifests itself in the representation of motion Figure 144.2 by phase trajectories.

,, ,, ,

427 145.

PHASE TRAJECTORIES IN THE PRESENCE OF COULOMB FRICTION The customary idealization of Coulomb, or dry, friction consists in assuming that the friction force 10 remains constant during the motion and changes its sign with a change of the direction of motion, as shown in Figure 145.1.

The motion of a system with one degree of freedom in the presence of Coulomb friction cannot be described by one differential equation but requires two equations, one for < 0 and the other for > 0, namely,

z

mx

Putting

!=

z

+ kx

2 W0

=

mx

+ 10 ;

and I/o I

= awo 2 ,

2

x + wox == + awo2 x

+

2

""OX

2

== - awo

+ kx = - 10

[145.1]

these two equations are

<0 x >0

when x when

[145.2 ]

Introducing the variables %1 = % - a for the first equation [145.2] and %2 = %+ a for the second, one obtains two equations, namely, when

z<0 [145.3 ]

when % > 0

These equations are obviously identical, with the difference, however, that the center of oscillation is displaced from +a to -a, and vice versa, each time % changes its sign. It is to be noted that % itself does not enter into the equations of motion. The "change of equations" occurs at the instant when = O. In the (%,t)-plane such a motion can be represented by the curve shown in Figure 145.2. Assume that initially the system has been deviated to the right (%01 > 0) and then released with an initial velocity %01 = O. The system will then move to the left (% < 0). As was Just mentioned, this is a sinusoidal motion with respect to the axis tl displaced a distance +a from

z

c I - - r - - - \ - - - - - - - - i - - - + - -..... t •

~L--~-~--+_----~t2

B

Pigure 145.1

Figure 145.2

428

the t-axis. At the po1nt B the veloc1ty % 1s equal to zero, and the amp11tude is 12:021 = 12:01 - 2al with respect to the t-a2:i8. At the p01nt B the change of equations takes place, and the "acting abscissa axis" is now t2 displaced a d1stance -a from the t-axis. With respect to the t 2 -axis, the motion 1s again s1nusoidal, so that the p01nt C (at which %= 0) is at the same distance from the t 2 -ax1s as the point B was or1ginally. With respect to the t-axis, however, the amplitude of the point C is 2:03 = 2:02 - 2a. It is seen that with respect to the t-axis the amplitudes decrease in an arith'metic progression with the constant difference 2a. The motion stops after a certain number of swings. The time 1nterval between two consecutive maximums (A and C in Figure 145.2) is the same as for a harmonic oscillator, as follows from the preceding discussion. It is to be noted again that the curve ABC··· is a piecewise analytic curve which loses its analyticity at the points A, B, C, "', at which there is a discontinuity in the second derivative. The representation of such a motion by phase trajectories is obtained if we observe that each of the equations [145.}] is represented by an elliptic trajectory around the vortex point, see Section 1. If we let ~: = y, Equations [145.}] become dy d:r: dy

d2:

= _

w;(:r: - a) y

+ a)

w:(:r:

= -

Y

when y

<0 [145.4)

when y

>0

Integrating these equations, one gets (:r: - a)2

R2 1 (:r:

+ R2 2

a)2

+

when y

<0

when y

>0

[145.5 J

2

R;w; Y

= 1

where Rl and R2 are the constants of integration determined at the end of each preceding interval, that is, when x = O. Equations [145.5J represent two families of ellipses, r 1 and r 2 , whose centers are displaced on both sides of the origin by a constant quantity±a, as shown in Figure 145.}. Let us consider the motion beginning with a point A representing the initial conditions on one of the trajectories r 2 which is drawn in a heavy line. At the point B, where y = 0, the change of equations takes place, and the representative point goes onto the trajectory r l passing through B. The motion on that trajectory will continue up to the point C at which the representative point will go onto a trajectory r 2 which passes through C. On that trajectory the change of equations should occur at the point D. It is apparent, however, that such a trajectory would be confined within the zone limited by the broken lines :r: = +a and :r: = -a, which characterize the zone of static friction. Hence once a trajectory is reached which does not emerge from that zone, the motion ceases.

429

It is thus seen that idealized Coulomb friction is capable of being represented by piecewise analytic trajectories formed by elliptic arcs of the two families r 1 and r 2 with centers symmetrically displaced by a constant quantity a on both sides of the origin 0. The Junction points of the arcs are situated on the x-axis. At these points the curve has a continuous first, but a discontinuous second, derivative as may be expected from the fact that the friction force changes discontinuously at these pOints. Instead of having a point of equilibrium, Figure 145.3 systems possessing Coulomb friction have a line of equilihriunt 0102' which means that any point on the segment 0102 is a point of equilibrium and that the segment 0 10 2 itself is the zone of static friction. Making use of these conclusions regarding the form of trajectories in the presence of Coulomb friction, Andronow and Chaikin have further elaboated the theory of the clock. We shall omit the details of their theory and will merely indicate their argument as well as their final conclusions. By introducing "angular time" T = wot, as we did on a number of occasions, Equations [145.3] can be reduced to the form 2

d xI dT2

+

d 2x dTI

+

XI

= 0

when

x
X2

= 0

when

x>0

The trajectories of these equations are two families CI and C2 of concentric circles whose centers are displaced on the x-axis on both sides of the origin by a fixed quantity a. By reproducing the preceding argument and by expressing the condition that the ultimate trajectory always emerges from the "dead zonel! of static friction, these authors show that, if the trajectory is to emerge ultimately from the dead zone, the impulsive change of kinetic energy h 2 must be equal to or greater than 16102 , where 10 is the Coulomb force. This means that a clock is a mechanism with a hard self-excitation. In other words, unless the condition stated above concerning the relation between h 2 and 16f0 2 is satisfied, a clock will not start. If, however. this condition is satisfied. the clock will start and will approach its piecewise analytic limit cycle irrespective of any other aspects of the initial conditions.

146.

ELECTRON-TUBE OSCILLATOR WITH QUASI-DISCONTINUOUS GRm CONTROL

We will 1nvestigate in this section the behavior of an electron-tube circuit exhibiting features resembling those of a mechanical system possessing Coulomb friction and acted on by impulses. The analogy, as we shall see, is purely formal, but it will be helpful in approaching from a somewhat new viewpoint the difficult subject of self-excited oscillations. The interesting feature of the analysis made by Andronow and Chaikin is that it permits investigating questions concerning the establishment and stability of self-excited oscillations by a method similar to that used in the preceding section, except that here one may have nega.tive Coulomb friction maintaining the oscillations. Consider the circuit shown in Figure 146.1 with positive directions and customary notations indicated. The symbol z deSignates the current in the inductance L. The differential equation of the circuit is (146.1]

Differentiating and rearranging, we obtain L z"

1 Z + R'z + C

....

1 f( ell ) C

[146.2 ]

where 14 = f(e l1 ) is the non-linear characteristic of the electron tube. We will consider a peculiar performance of this circuit, namely, the one which appears when the coefficient of mutual inductance M is made very large. As 8 result of this, the amplitude le l1 l of the grid-voltage variation will also be large; this will cause the electron tube to act more or less as a switch operating between the points A, when 14 = 0, and B, when 14 = I,. where I, is the saturation plate current. It is apparent that when the grid potential varies between the limits tell' as shown in Figure 146.2, the interval AlBl corresponding to the variations te l1 1' whi~h are instrumental in causing the plate current to vary, is small in comparison with the total interval AB, and it is possible to idealize the performance as a discont1nuous characteristic such as 10

/

-8"

-eaJ '-----~III------'

Figure 146.1

A

7

0

Is +8"

-teal

B

Figure 146.2

B

that shown in Figure 145.1. Instead of being described by one non-linear differential equation [146.2], the phenomenon will be described alternately by two linear differential equations it

f'

+

2hz

te'

+

2 W0 x

=

0;

it

+

2hz

+ w:x

= w:ls

[146.3]

where 211. = wo = and wo I, = ~f(+e,,). It is obvious that the change from one equation to the other occurs at the point where :i changes its sign. We shall assume that the connections are made so that j,( > 0, e" > 0 when :i < 0: and e" < 0 when :i > O. We shall investigate the second alternative later. In view of our idealization, the anode current 14 will undergo discontinuities in the neighborhood of :i = O. These discontinuities have to pass through the capacitor branch of the oscillating circuit since in the inductive branch no discontinuities of current are possible. However, the discontinuities L~~ in the voltage across the inductance are possible. We conclude, therefore, that the quantities which are capable of varying discontinuously are 14 in the capaCitor branch andL~: across the inductance. The quantity x remains continuous. Another continuous quantity is the voltage V across the capacitor, as was mentioned in Section 131. It is noted that the change from one differential equation to the other, mentioned above, resembles the situation which we have already encountered in connection with Equations [145.2] describing the behavior of a system possessing Coulomb friction, and may follow a similar argument, We can write the second equation [146.3] as 2

2

.

i +

2hx

+

w:x

= 0

[146.4]

where ~ = x - 16 , In this form the first equation [146.3] and Equation [146.4] are two linear differential equations possessing focal points on the t-ax1s at a distance 16 from each other. We can, therefore, repeat the argument used in connection with Figure 145.2. Let us consider the phenomenon from the instant when the amplitude is Xl and the electron-tube switch is off, which corresponds to the first equation [146.3]. During the first half-cycle the original amplitude Xl' see Figure 146.3, will be reduced by damping and will become x 2 ' It is noted that x 2 < Xl if referred to the t-axis. At the point B the electron-tube current Jumps suddenly to the value I, and, as we saw, the new abscissa axis is now t l , d1splaced above the t-axis by the quantity I,. With respect to this tl-axis. the initial amplitude is now ~2 = x 2 + I. and although during the follow1ng half-cycle the amplitude is again reduced with respect to this axis because of the dissipation of energy. the position of the point C with respect to the t-axis may be at a higher level than that of A if

x

I---\---~---h---t-.L.-t--tl

Figure 146.,

1, is large enough. Beginning with the point C, we can resume the same argument which we used at the point A, and so on. It is seen that, because of this quasi-discontinuous timing of the electronic switch, the amplitudes in the oscillating circuit may grow, thus outweighing the effect of the dissipation of energy. It is easy to express this condition mathematically. If one designates by x the amplitudes referred to the t-axis and by X those referred to the t 1 -axis, one has

If the amplitude reaches a stationary value, %3 =

= Xl = xO' which gives

1 - e

_h1' T

[146.5 ]

This shows that a stationary amplitude Xo is determined solely by the properties of the circuit and is independent of the initial conditions. The question of the stability of these oscillations can be investigated graphically by comparing their subsequent amplitudes. For instance, take the relation between Xl and x3 ' namely,

[146.6 ] In the (x 1 ,x 3)-plane, see Figure 146.4, this relation is a straight line with AT a slope tan a = e- AT , and an intercept 1.(1 + e- lr ) = A. On the other hand, if the oscillation is stationary, Xl = x 3 ' which represents the bisector of the angle between the x1- and %3-axes. The point M of intersection of these lines determines the stationary amplitude xO' If the initial amplitude Xl' is not stationary, the subsequent amplitude will be x 3 ' and the corrected amplitude will be %3' on the xl-axis to which the following amplitude x 6 ' will correspond on the x3-axis, and its corrected value on the xl-axis will be x 6 ' •

433 By the interplay of the subsequent corrections it is seen that any non-stationary amplitude tends to approach the stationary value ~o' This argument is also valid when the initial non-stationary ampliI; tude is larger than the stationary amplitude ~o' Hence the stationary I~ amplitude is stable. It should be noted that if we reverse the connection of the coils, the displacement of the ~l-axis with respect to the ~-axis by the quant1ty I. will occur in a direction opposite to that shown in Pigure 146.3. This will result in a reduction of Pigure 146.4 the subsequent amplitudes similar to that found in connection with Pigure 145.2 illustrating the action of Coulomb friction. Viewed from this standpoint, the circuit behaves as i f it had a kind of negative Coulomb damping when M > 0 and a positive Coulomb damping when M < O. The self-excitation of the circuit occurs when the circuit is characterized by negative Coulomb damping.

147.

PHASE TRAJECTORIES OF AN IMPULSE-EXCITED OSCILLATOR The property of negative Coulomb damping of the circuit investigated in the preceding section becomes still more striking if we analyze the phase trajectories of the system (146.3) which we write in the form ii + 2hz + w:~ = 0

iii + 2hzl +

2 Wo ~I

[147.1)

= 0

where ~1 = ~ - I.. Since each of these equations describes a damped oscillator, each is represented by a spiral trajectory approaching a stable focal point; see Section 5. The focal pOints of these equat10ns are separated by a constant distance I. along the ~-ax1s in the (~,z) phase plane. Figure 147.1 shows the phase diagram of the motion corresponding to Equations [1 47.1). Point 0 is the focal point of the first equation (147.'), and 0 that of the second equation. Let us start from a certain arbitrary initial condition, represented, for instance, by a point 1 on the negative abscissa axis. Through Point 1 passes the spiral trajectory bn2 having 0 as its focal pOint and representing the first equation [147.1). At the point 2 the "change of equations"

occurs, and the phenomenon is now governed by the second equation [147.1] having its focal point at O. The second stretch of the spiral trajectory 2n3 ends at the point 3, where the first equation begins to describe the phenomenon II again; this arc of the spiral trajectory 3p4 ends at the point 4, and so on. It can be shown that the piecewise analytic trajectory nn2n3p4··· ultimately approaches a closed curve ABCDA formed by two spiral arcs, ABC with 0 as Figure 147.1 its focal point and CDA with 0 as its focal point. One obtains a similar conclu~ion if, instead of starting with a point 1, one starts with a point l' exterior to the closed curve ABCDA and repeats the argument. We shall designate by r the radius vectors with respect to the focal point 0 when the representative point moves on the upper spiral arcs, and by r the radius vectors with a focal pOint at 0 when the representative point moves on the lower arcs. For the subsequent radius vectors along the z-axis, we have the following obvious relations:

One obtains easily the following general expressions: rk == a [ 1 + e _

[

- l!!. W

r. == a 1 + e

+ e- !!!! W + . . . + e _ (k _ 2) l!!.] W + r 1e- l!!. W

+ ... + e

_ " _ 2)

l!!.] W

+ r1e

Ck - 1) "II W

_(. _09

W ;

;

r.=r.- a

where k is an odd integer and 8 is an even integer. If k and 8 increase indefinitely, that is, if the number of turns of the piecewise analyt1c spiral nn2n3p4··· increases 1ndef1nitely, the spiral approaches the closed curve ABCDA, previously mentioned, in the manner specified in Section 22. One sees this immediately from the express10ns g1ven above for r k and r. when k + = and 8 + =, namely,

[147.3 ]

435 The conclusions remain exactly the same if one repeats the argument for a piecewise analytiC spiral ,'m'2'n','p'··· starting at a relatively distant pOint ,'. It is thus apparent that the closed curve ABCDA formed by the two arcs of the logarithmic spirals is a piecewise analytic limit cycle which the oscillation approaches as t .~. Moreover, this limit cycle 1s stable. It is useful to emphasize once more that this limit cycle characterizing stationary periodic motion is determined exclusively by the constant parameters a = I., h, and w of the oscillatory system and is entirely independent of initial conditions. This is an essential property of self-excited OSCillations, as was pointed out in Section 22. It is also worth mentioning that we have obtained these results of the classical theory by replacing the actual non-linear equation [146.2] by a system [,47.1] of two alternating linear differential equations. Such a procedure becomes possible only after we have idealized the relatively complicated non-linear characteristic, shown in Figure 146.2, as a discontinuous step-function like that shown in Figure 145.1. In connection with this it may be useful to recall the statement made in Section , that the really important difference between linear and non-linear systems is their behavior in the large and that local properties are of relatively minor importance. This argument is a typical example of a situation of this kind where a certain idealization of the local properties of trajectories does not change their properties in the large and merely simplifies a problem which would otherwise be extremely complex.

CHAPTER XXV EFFECT OF PARASITIC PARAMETERS ON STATIONARY STATES OF DYNAMICAL SYSTEMS

148.

PARASITIC PARAMETERS

The study of relaxation oscillations in the preceding sections was made under the assumption of certain simplifying idealizations which resulted in degenerate equations instead of complete ones containing both oscillatory parameters (L and C in electrical problems and 1n and k in mechanical on~s). Although the advantages of such simplifications, as we saw, are numerous and the results obtained in this manner are generally found to be in agreement with experimental facts, the introduction of these simplifying assumptions is not without theoretical difficulties and, in some cases, which we will analyze here, may lead to certain complications. In spite of the fact that such cases appear generally as rare exceptions, it is important to analyze this matter in greater detail now that we are acquainted with the general method, at least in its present scope. One of the principal difficulties. noted on several occasions, is the inconsistency between the number of integration constants appearing in a degenerate problem and that in the corresponding non-degenerate one. Thus, for example, if the non-degenerate problem involves a differential equation of the second order, two constants of integration are necessary to determine the solution; these constants appear as certain definite physical "initial conditions." In the corresponding degenerate problem. where the differential equation is of the first order, one constant of integration is sufficient to determine a solution. If, however, there still exist two physical factors to which some initial values can be assigned, these factors cannot have entirely arbitrary values but must readjust themselves eventually so that a definite relation exists between them, as was explained in Section 128. The discontinuous theory of relaxation OSCillations ignores this rather delicate passage from the solutions of one form to those of the other form. Just as the classical theory of mechanical impacts ignores the unknown dynamics of the collision process. Both theories follow a somewhat similar argument, namely. the rapidly changing motion is idealized during a very short time interval as a mathematical discontinuity occurring in the infinitely short time interval (t - 0, t + 0). and the loss of information resulting from intentionally overlooking t.~ dynamics of the process is supplemented by additional physical information which permits solution of the problem in the large although certain local details are inevitably lost in such a procedure. In the theory of mechanical impacts this additional physical information is provided by the theorems of

437 momentum and kinetic energy as well as by defining a certain empirical coefficient of restitution characterizing the instantaneous dissipation of energy during the impact; likewise, in the discontinuous theory of relaxation oscillations this additional information appears in the form of the conditions of Mandelstam. The use of the concepts "infinitely short time interval," "infinitely large acceleration," and so on, instead of the more correct terms "very short time interval," "very large acceleration," and so on, while convenient for the description of a phenomenon in the large, is sometimes capable of introducing serious errors in a theoretical argument and of leading to conclusions at variance with experimental facts. Thus, for instance, in dealing with an idealized (L,R)-circuit, we describe its behavior by the equation L :;'

+

Ri = E(t)

[148.1 ]

This equation describes the phenomenon with adequate accuracy in a range sufficient for practical purposes (on the scale, say, of milliseconds). If, however, the behavior of the same circuit is studied in a different range (say, on the scale of microseconds), this equation may not give a correct answer. In fact, any resistor or inductance coil inevitably has a small parasitic capacity Cp , and if we take this capacity into account a more correct equation will be [148.2 ] It was shown in Section 128 that under certain conditions the solutions i(t) and i 1 (t) of Equations [148.1] and [148.2J may have entirely different features locally, although in the large these solutions are practically indistinguishable. For that reason an electrical engineer would prefer to use the simpler equation [148.1 J rather than the more complicated one [148.2]. These facts are too well known to need further emphasis here. However, they are frequently very troublesome. Thus, for instance, a student attempting to investigate the behavior of a simple circuit by means of a cathode-ray oscillograph, instead of observing the pattern of curves in accordance with his differential equatio~, usually observes what is commonly called "hash," that is, a far more complicated pattern of numerous harmonics with cross modulation, and so on. The reason for this is that too many parasitiC parameters have been neglected in forming the differential equation, and consequently the true differential equation is far more complicated than the assumed one. In most practical cases such discrepancies are not very serious. However, in some special instances, analyzed in what follows, difficulties may arise which lead to paradoxes already noted by certain authors (19) (20).

A further remark may be useful. of an (L,R,C)-circuit, Lq

Consider the differential equation

+ Rq + ~ 9

[148.:5 ]

= 0

and assume that L is very small. If L is neglected, one has a differential equation of the first order. The expression "L is very small" is, however, rather indefinite in the sense that it is not important to mow that L is small but rather that the consideration of the problem is restricted to the range in which the term Lqis negligible compared with the other two terms. Still more contusing is the effect of the degeneration when C+ 0, which leads to a meaningless result. If, however, we make C +00, at the limit we obtain L di dt

+ Ri

= 0

[148.4 ]

which is an absolutely degenerate equation of an idealized (L,R)-circuit. Here the degeneration occurs not because a parameter is small but because it is large. This contusion disappears if, instead of the parameter C, the capacity, we use the elastance k = &as a parameter. It is convenient, for this reason, to consider parameters as parasitic if they are small or if the corresponding terms in the differential equation are small. The first condition is merely a def1n1tion, whereas the second condition specifies the range in which a small parameter can be neglected. INFLUENCE OF PARASITIC PARAMETERS ON THE STATE OF EQUILIBRIUM OP A DYNAMICAL SYSTEM Consider a system with several degrees of freedom expressible by the differential equation (11) . . 0 [1491] ao~ + al~ (11-1) + ... + a _ . % + a _ % + a % = . 149.

1l

where

%(11)

= :;~,

and so on.

z

1l

1

n

The corresponding characteristic equation is [149.2 ]

The equilibrium is stable if the real parts of the roots of [149.2] are negative, which can be ascertained by the well-known criteria of Routh-Hurwitz. As was previously stated, we never mow the exact form of the differential equations in a physical problem because of a number of parasitic parameters either entirely unlmown or known only approximately. Let us assume that Equation [149.1] i8 a degenerate one derived from some other equation describing more correctly the behavior of the system by taking into account a parasitic parameter o. There are two ways in which this parasit1c parameter may appear, depending on whether it is of the "inductance" type or of the "elastance" type.

If it is of the inductance type, the non-degenerate equation will be a:!:

In

+ 11

+

ao:!:

(n,

+

a l :!:

(n - 1)

+

+

•••

a"_lz

+

a,,:!: == 0

[149.3 )

If it is of the elastance type, it will be [149.4 )

Differentiating this equat10n, we get [149.5 )

For both types of parasitic parameter the order of the or1g1nal different1al equation has been raised by one unit. The correspond1ng characteristic equations are ,,+

aX

1

+

n

aoX

+

n-I

alA

+. . . +

an _ 1 A

+

a" = 0

[1 49.6 ]

[1 49.7)

They now have (n + 1) roots, whereas the original characteristic equation [149.2) had only n roots. Since a is small, by our assumption, the old n roots change but little ow1ng to the appearance of the new root, and we may neglect th1s change. The matter hinges, therefore, on the new root A,,+I' We can write Equation [149.2], in terms of its roots, as ao(X - A 1)(A -

A2 ) ••• (X -

Xn ) = 0

The new characteristic equation [149.6] of the "parasitic inductance II type can accordingly be written as a(X -

A1)(X -

A2)'"

(X -

Xn)(A -

An+l)

= ~[aox" + a o

[149.9 ]

Expanding the right-hand side of this equation and identifying it with Equat10n [149.6), we get A 11+

a

1

o = . a

Applying a similar argument to Equation [149.7], we obtain A,,+

1

a = -an

[149.11 )

The sign of the new root A,,+1 indicates its effect on the stability of equilibrium. The general integral of the new system 1s [149.12 ]

440 If the conditions of stability for the original degenerate equation [149.1] are fulfilled, the stability of the new equation [149.3] or [149.5] will depend solely on the new root Xft + 1 • If this root is stable, that is, if its real part is negative, the equilibrium will still be stable. If, however, the root Xft + 1 is unstable, the non-degenerate system will be unstable although the corresponding degenerate system is stable. For a complete discussion, one must apply the Routh-Hurwitz criteria, as usual. In practical problems the existence of parasitic parameters is generally unknown; still less known is the magnitude of such parameters. For these reasons in a complicated physical problem one is never certain about the actual conditions of stability. This is why one occasionally observes somewhat puzzling departures from theoretical predictions based on a simplified study of a system expressed in terms of differential equations with known parameters. Conclusions obtained from the simple discussion given above may appear to some extent paradoxioal. Thus, for instance, if the parasitic parameter is of the lIinductance" type, see Equation [149.10], the new root X" + 1 is larger as the parasitic parameter a is smaller. If, however, it is of the lIelastance li type, see Equation [149.11], the magnitude of the root X" + 1 is proportional to a. The rather erratic "floating" of a system around its theoretical position of equilibrium can sometimes be traced to the effects of parasitic parameters, as we shall see from a few typical examples in the following sections. 150. EFFECT OF PARASITIC PARAMETERS ON STABILITY OF AN ELECTRIC ARC The question of the stability of an electric arc was the subject of considerable controversy in the past (19) (20). In Section 21 this matter was investigated in detail by means of Liapounoff's equations of the first approximation on the basis of a differential equation with finite parameters L, C, and p. The behavior of the arc is far more complex if some of these parameters are small, a condition which results in a degenerate form of the differential equation describing the process. Using the notation of Section 21, we recall that the differential equations are

L~:

=

V-

tb(i);

dV E - V - Ri C dt = R

[150.1 ]

where i is the arc current, V is the potential difference across the capacitor, and tb{i) = Va is the non-linear characteristic of the arc, see Figure 150.1. As was explained in Section 21, there are either three equilibrium

441

R

L

-

I

E

j

V

i

E - Ri • V I

i~ j

rIe

/Va' \!IIi)

Va

Figure 150.1 points (Points 1,2, and 3) or-one such point (Point 3), as shown in Figure Figure 150.2 150.2, depending on the orientation of the straight line E - Ri with respect to the characteristic ~(i). It was shown that for small departures, v of voltage and i of arc current, from the corresponding equilibrium values, the equations of the first approximation are dv

d[=

dj v .P -=--;dt L L

L.

v

RC

C'

where P = ~'(io) is the slope of the tangent to the characteristic the point i = i o • The characteristic equation of the system is

[150.2 ) ~(io)

at

(150.3 J When there are three points of equilibrium, it is observed that Point 1 corresponds to P = ~'(io) > O. This point is always stable because the real parts of the roots of Equation [150.3J are negative. It is, therefore, either a stable nodal, or a stable focal, point according to whether the roots are real or conjugate complex. At Point 2 the equilibrium is unstable because p < 0 and, moreover, Ipi > R. This point, therefore, is a saddle point. Point 3 is a point of stable equilibrium because, although at this point p < 0, Ipi < R. Having recalled these conclusions of Section 21, we propose to investigate now what happens to these various cond1tions of equilibrium when the system undergoes a degeneration, that is, when one of the two oscillatory para1neters Land C approaches zero. We may distinguish two cases of degeneration, namely, C-degeneration when C + 0 and L-degenerat1on when L + O. In order to invest1gate the C-degenerat10n, it is convenient to write Equation (150.3] 8S C'A 2

+

(.!. R

+

CP)'A

L

+

_l_(R

LR

+

p) = 0

[150.4 ]

442 For the investigation of the L-degeneration we will write it as L')...2

+ (R~ +

p) ')...

+ C~ (R +

p) = 0

[150.5]

In this manner we shall be able to use the conclusions of Section 128. Putting C= 0 in Equation [150.4], we obtain R+p

')...=_

[150.6 ]

L

Putting LaO in Equation [150.5], we have ')... = -

1

CRp (R

+

p)

[150.7]

Let us consider the equilibrium at Point 2 of Figure 150.2. It is observed that in the C-degeneration this point is still unstable. In the Ldegeneration, it is stable. A considerable controversy existed on this subject in the technical literature before this question was completely understood. The following explanation was suggested by S. Chaikin (20) (21). Let us consider the characteristic equation a')...2

+ b')... +

c

=

0

[150.8 ]

with a > 0 and c < 0, which characterizes a saddle pOint; see Section 18. We can define saddle pOints as positive when b > 0 and as negative when b < O. In the case of a-degeneration, a positive saddle point gives rise to a positive root and, hence, to unstable equilibrium. If, however, the saddle point is negative, the degenerate system has stable equllibrium at this point. Thus the a-degeneration reverses the stability at Point 2 of Figure '50.2. Since in both cases Point 2 corresponds to a saddle point of the system, we conclude that the only way to obtain stability under this condition is to have the representative point follow a stable separatrix.* If, however, it follows the stable separatrlx, there exists a fixed relation between z and y = ~ in the phase plane. But this is precisely what happens in a degenerate system described by a differential equation of the first order, see Equation [128.3). The condition of degeneration thus imposes this Singular trajectory, the only one which is stable in the neighborhood of a saddle point. In this case, therefore, a threshold exists s1m1lar to that which exists for the asymptotic motion of a pendulum approaching a point of unstable equilibrium, see Section 4, Case 3; such a case is never obtained in practice since there exist no absolutely degenerate systems, that is, systems in which a = O.

*

Only in a purely tbaoretlcal cose "hen the lIIotion of the representative point tekea place along the stable separatrix or asymptote of a saddle point, IIIII¥ the latter be considered as a ddl. slngularity. If the actual 1II0tion of the representative point occurs in the neighborhood of this theoretical motion, it lIayappear that the saddle point is a stable singularity, at least tor a limited time. In view of this, 0118 could characterize the saddle point 8S lID 111.0" singularity.

"",,061.

443 For practical purposes, if precautions are made to reduce the parasitic parameter to a very low value, the representative pOint will follow a trajectory very closely to the stable separatrix so that one has the impression that the phenomenon would ultimately IIsettle ll at Point 2 as if i t were a point of stable equilibrium. In reality, sooner or later the representative point will depart from this point as a result of accidental disturbances. By a similar argument one finds that Point 3 of stable equilibrium for a non-degenerate system may become unstable in the case of L-degeneration, as follows from Equation [150.1) in which p < 0 and I pi < R. Point 1 is stable both for the non-degenerate and the degenerate systems. It is thus seen that the assumption of absolute degeneration may lead to conclusions at variance with experimental facts as tar as the question of stability is concerned. 151. EFFECT OF PARASITIC PARAMETERS ON STABILITY OF RELAXATION OSCILLATIONS As another example of the same sort, we shall investigate the behavior of the circuit shown in Figure 135.1, taking into account the presence of the small parasitic inductances Land L' shown in Figure 151.1. KIrchhoffls equations for this circuit are f

+

i = fa;

df L dt

+

RI

-

.

rl -

Moreover, fa ::: ep(kr.i) as before. these equations results in the equations di

dt

L 1 dT di

-

1 C

f· d t = 1

[151.1]

Elimination of fa and f between

Rep(kri) - (R + r)i - V = L + L' - Lkrep'(kri) ;

dV

dt

=

i

C

The conditions of equilibrium are obviously i ::: 0 and Rep(O) - V

Figure 151.1

0

[151.2 )

= O. On the

444 other hand, for small departures from equilibrium, we can write ¢,(kri) = ¢,(O)

+

+ ...

kri¢,'(O)

Limiting the expansion to the first term and noting that ¢,'(O) Liapounoff equations of the first approximation are

v

di dt = -

,,'

where 6 = r

+

dV dt -

R(l - Skr)

and

= S,

i

the

[151.3 ]

C

,,= L' + L(l -

Skr)

[151.4]

The characteristic equation of the system [151.3] is )..2

+ .!).. +

"

and its roots are

- 6 )..1,2

±

=

1

"C

V 62

-

0

-

C

[151.5]

4"

[151.6]

2"

For " < 0 the roots are real and of opposite sign; hence, the singularity is a saddle pOint. For " > 0, the Singularity is either a nodal point or a focal pOint, depending on whether the roots are real or conjugate complex. If they are conjugate complex, with 6 > 0, the singularity is stable; with 6 < 0, it is unstable. The distribution of singularities is shown in Figure 151.2, which is the same as Figure 18.1 with the exception that now we make a distinction between "positive" and "negFigure 151.2 ative" saddle points as was explained in Section 150. For an absolute degeneration, L = L' .,. " = O. Multiplying the first equation [151.3] by", one has 6i + V = 0, that is, i = The second equation gives

-f.

dV dt

= -

V 6C

[151.7]

point of equilibrium V= R~(O) is the same as in the non-degenerate system. The equilibrium is stable for 6 > 0 and unstable for 6 < O. The points of stable equilibrium are indicated by solid pOints on the positive axis of ordinates and those of unstable equilibrium by circles on the negative axis

~~e

445 of ordinates. It is seen that the region of stable equilibrium is localized in the region of "negativell saddle points and that of unstable equilibrium in the region of "positive ll ones. It was shown, however, in the preceding section that such stability is of theoretical 1nterest only, because it is impossible in practice to prescribe the initial conditions accurately enough to make the representative polnt follow the stable separatrix, Just as it is impossible to impart to a pendulum a veloclty Just sufficient to produce an asymptotic motion. Moreover, no physical system is completely degenerate, but is only quasi-degenerate. The situation is still further complicated by the fact that stability conditions are influenced not only by the existence of parasltic parameters but also by their relative intportance. In order to show this, assume, for instance, that L :1= O. L' = 0, 1 - Skr < 0, but R is sufficiently small to render ~ = r +R(1 - Skr) positlve. In such a case the equllibrium is unstable for any value of L. however small, because the real part of the roots is pOSitive, that ls, ~ > 0 and y < O. In a degenerate system, where L = L' = 0, the equilibrium is stable, however, as follows from Equation [151.7). Since any physical system is only quasi-degenerate. we conclude that the equilibrium is unstable in this case. If we consider now the second case, when L' :1= 0, the situation is different. Let us assume, as previously, that 1 - Skr < O. It is obvious that, by giving L' a suitable value, the quantity y = L' + L(1 - Skr) can be made positive and since ~ = r + R(1 - Skr) is also positive, as previously assumed, it follows from Equation [151.6) that the equilibrium is stable. Hence, the equilibrium which was unstable for L :1= 0, L' = 0 becomes stable for L = 0, L' :1= O. Thus the change from stability to instability depends here on the ratio LIL' of the parasitic parameters. Since in practice one is seldom sure of the existence of parasitic parameters, much less of their relative values, the predetermination of stability conditions for a circuit of this kind becomes altogether meaningless. It is likely that floating. erratic excitation and similar phenomena still little explored can be explained on this basis. To sum up the results of this analYSiS, it can be stated that. in general, the effect of parasitic parameters is negligible in practice unless the system happens to be in the neighborhood of a branch point of equilibrium. If the system is in such a neighborhood. even a small "cause" may exert an appreciable lIeffect" on the system. and its behavior then becomes entirely unpredictable.

446 REFERENCES liOn Relaxation OSCillations, II by B. Van der Pol, Philosophical Magazine, 7th Series, Vol. 2, November 1926. (1)

(2) IIUber Relaxationschwingungen" (On Relaxation Oscillations), by B. Van der Pol, Zeitschrift fUr Hochfrequenztechnik, Vol. 28, 1926, and Vol. 29, 1927. (3) "Uber KippschwingUngen insbesondere bei ElektronenrHhren" (On Sweep Oscillations in Electron Tubes), by E. FriedlUnder, Archiv fUr Elektrotechnik, Vol. 17, No.1, September 1926, and Vol. 18, No.2, October 1926. (4) "Reissdiagramme von SenderrHhren" (Graphical Diagram for Transmitting Tubes), by H. Rukop, Zeitschr1ft fUr Technische Physik, Vol. 5, 1924. (5) IIEine neue Schaltung zur Erzeugung von Schwingungen mit 11nearem Spannungsverlauf" (New Wiring Connection for Generating Oscillations wi th Linear Vol tage Curvature), by G. Frtnlauf, Archiv fUr Elektrotechnik, Vol. 21, No.5, February 1929. (6) "Zur Theorie des rUckgekoppelten RHhrensenders" (On the Theory of Regenerative Coupled Vacuum Tube), by F. Kirschstein, Zeitschrift fUr Hochfrequenztechnik, Vol. 33, No.6, June 1929. IIUber Schwingungszeugung mittels eines Elektronrtshrensystems bei welchem die KapazitUt von untergeordneter Bedeutung ist ll (On the Excitation of OSCillations by Means of a Vacuum-Tube System in Which Capacity Is of Subordinate Importance), by K. Heegner and I. Watanabe, Zeitschrift fUr Hochfrequenztechnik, Vol. 34, No.2, August 1929. (7)

"tiber symmetrische Kippschwingungen und ihre Synchronisierungll (On Symmetrical Relaxation Oscillations and Their Synchronization), by H.E. Hollmann, Elektrische Nachrichtentechnik, Vol. 8, No. 10, October 1931. (8)

(9) liOn the Theory of the FrUhauf Circuit,1I by L.P. Cholodenko, Journal of Technical PhYSiCS, USSR, (Russian), Vol. 5, 1935. (10) ilLes systemes autoentretenues et les oscillations de relaxation (Self-Excited Systems and Relaxation OscIllatIons), by Ph. LeCorbeiller, Hermann, PariS, 1931. ll

('1) liOn the Existence of Periodic Solutions of Second Order Differential Equations, II by N. Levinson, Journal of Mathematics and PhYSics, Vol. 22, 1943.

447 (12) "On Van der Pol's and Related Non-Linear Differential Equations," by J.A. Shohat, Journal of Applied Physics, Vol. 15, 1944. (13) "Theory of Oscillations," by A. Andronow and S. Chaikin, Moscow, (Russian), 1937. (14) "Oscillations discontinues" (Discontinuous Oscillations), by S. Chaikin and L. Lochakow, Journal of Technical Physics, USSR,. (Russian), Vol. 5. 1935. (15) "Mesure en valeur absolue des periodes des oscillations alectriques de haute frequence ll (Absolute Value of Periods of High-Frequency Electric Oscillations), by H. Abraham and E. Bloch, Annales de Physique, Vol. 12, No.9, 1919. (16) "tiber Schwlngungszeugung mlttels elnes Elektronrohrensystems welche Selbstlnduction nicht erhalten" (On the Excitation of Oscillations by Means of an Electron-Tube System without Self-Induction), by K. Heegner, Zeitschrift fur Hochfrequenztechnik, Vol. 29, 1927. (17) IIZur hydrodynamischen Theorie der Schm1erm1ttelreibungll (On

the Hydrodynamic Theory of Lubricating-Oil Friction), by A. Sommerfeld, Zeitschrift fur Mathematik und PhYSik, Vol. 50, 1904. (18) "Mechanical Relaxation Oscillations,1I by S. Chaikin and N. Kaidanowsky, Journal of Technical PhYSics, USSR, (Russian), Vol. 3, 1933. (19) IIWarwn kehren sich die fur den Lichtbogen gUltigen StabilitUtsbedingungen bei Elektronenrbbren um?1I (Why Do the Conditions of Stability Val1d for an Electric Arc Light Reverse in Electron Tubes?), by H. Barkhausen, Physikalische Zeitschrift, Vol. 21, No.2. February 1926. (20) "tiber StabilitUtsbedingungen und ihre AbhUngigkeit von Steuerorganen (On Stability Condit1ons as Functions of Control Elements), by E. FriedlUnder, Physikalische Zeitschrift, Vol. 21, 1926. ll

(21) IIEffect of Small Parameters on the Stationary State of a Dynamical System," by S. Cha1kin, Journal of Technical Physics, USSR, (Russian), Vol. 5, 1935.

IHDKlt

Abraham-Bloch, multivibrator of, 410-414 Absolute stability, zone of, 61

ASJn.chronous .excitation, 234,295,298 Asynchronous motion, 125

Analogies betveen electrical and mechanical systems, 255-258 Analytical definition of stabil1ty, 41-42 Analytical impasse, 394

Asynchronous qUenching, 2,4,295,298-299 Asynchronous self-excitation, 299

Analytical method, 11,111,1}5-245,323,382 of Kryloff and BogoliubofC, 4 of P01ncare, 4,87,99,104-105,1}8-166 of topology of phase trajectories, 26-29 of Van der Pol, 4

Autonomous equation, 155 Autonomous system, 424 deflnition of, 10 non-linear non-conservatlve, 62 quasi-linear, 289 vlth one degree of freedom, 105

Analytical periodic solutions, 383

Autoparametric excltation, 355-356,374-377 condltion for, 374

Analytical trajectories, 383,411,4l}-414

Autoparametric non-linear coupllog, 375

Andronov, A., 2-4,6',75,135,137,148,381,383,426, 429-430 Andronov and Witt presentat10n of, 344 topological method of, 342 Aperiodic damped motion, 106 e~uation ofj'14' " phase traject9r~es of, 18-19 Aperiodic damping, 262., . Aperiod1c motion, ',36,48 Appleton, 341. Approxlma tion e~uations oC

first, 3,49-52,55-56,65,81,87,94, 97,99,101,103,114,120,127,167-168,173,176, 179,186-188,193,217,224, 229,23t!, 24h259,271, 275,277,279,281,285-286,288-293,301-305,,11, 315-317 ,'22,411,440-441,444 '. i ' . ' linear, 49 " , .... firs t, 27, 189,229,265,278,280,282 -284 .. 297: improved first, 210-214,22Jh232,29,4-29.5 application of, 214-219, e~uat10ns of, 288 linear, 1,22,100 method, 135 method of first, 197-200,22~,274" ,'" method oC L1enard,' l l i . ' . , of f1rs t order, 289 ' '. ' of hlghen or.ders, ~7,209~232,288 of zero order, 289" .... , definition of, 190 order of,. 146 " " . , . q~tltative method Of" 2~;,~5-136 advanta'ge or',' 3 ... second, 210 , ' example o'f; 2 2 8 . . theory of first, 137 ,171,183~209,233,241,258 applled to non-linear conservative system, ~9-~0 " thi~, ,210 ,',' , Asymptotic motion, 3,28,~.o-3~·,36 definitlon oC, 16 of pendu1um~ ~6,442 Asynchronous act10n condltions for, 298 on selC-excited system, ,00 "

Autoparametrlc self-excltatlon, 375,377 Autoperlodic excitation, 297-298,305 Autoperiodic rrequency, 297-299,325,,42,349 Autoper10dic oscillation, 301,345.,48,350 amp1ltude of~' '50 conditions for stabl1ity of, ,06 deflo! tion ot j' 295 existence of, 295-,00, self-excltation of, ,00 stability of, 306 stability of statlonary,,04 . synchronlzedv1thheteroperiodic osc11latlon, 303 synchronous, 304 .Auto~e~lodic se1f-e~~ltatlon, 298.300 Autoperlodlc solutlon,,306

Autop~r!Odic;:8~te."ot:'

. . of non':linea.r systems, ,295· seU'-excitation of. '08'

Autoperiodlc

v~~lables~

298

Barkhausen, 260 Barrier, 177,203 Beats, '03,,41-342,'51,354 Bend1.x$on, 4 appllcation of flrst theorem of, '79-82 criterla of, 75,77-79,105,,45,411 Clrst theorem of, 75,77-78,3 45 negatlve crHerlonof,75,77-78,34 5 second theorem of, 75,78-79,3 48-3 49 Blbliography; see, Refere.~ces Blfurcatlon oC ~lt cycles, 87 Bifurcation polnt, '1,58 (.eu also Crltlca1 point) Blfurcation theory, 415 .. for ~uaBi-llnear systems, 160-161 of Polocare" 87 -: 104 Bifurcation values of par!lJlU!.t,!,r, 87 "

,.

1

'.

•

Blot-Savart.lav, 34 Birkhoff, O. D., 4,65

449

450 Bogoliubott, 2-,,67,~5,~7,181,210,2'4 (sao 0180 Krylott and Bogoliubott) lIIethod ot equIvalent 11naarization ot, ~7,209,

2"

notatIon ot, 254,26, quasi-l1near theory ot, 4,2,4,244,249,258,"1, ,42 theory ot tirst approxllllation ot, 18,,186-188, 209,258,289,'2' treatise ot, 251,288,,0, Bol tZllllUlll, theory ot, 5 Bowshewerov, U Branch point ot equil1brium, ,,42,48,55,60,82,165,445 ot 11m1t oycles, 88

Complex amp1ltudes, 254 method ot, 251-252,255-256,258-259,262 Complex 1IIIpedance, 25',256 Conditlonal stabIlIty, range ot, 61 Conservatlve systelll, 1,2,,25,,8,40,42,106,128,161, 185,189,224,261,,62,'15-'76 oy11ndrlcal phase ~jectories ot, 118-119 equillbrlum ot, 40 1II0tlon ot, 42 oscI1latlons ot,214 perIodI0 1II0tlon in, 62-6, poInts ot equilibrium 1n, ,4 Constant linear parameter, ,11

Constant parameter, 201,252,'70,4'5 linear dissIpatIve system With, 258 Continuum of Circles, '26 Canonical equatlon, 45 ot closed curves, 62,119 ot closed trajectories, 12-1',62,140-141 Canonical tom, '29 ot phase trajectories, 62,128,407 ot linear equatIons, 42-44 ot non-l1near systelll,52 Coulomb damping, 195 negative, 4" Cartan, B. and H., 105,181 podtive, 4" Cartan-Lienard, oonditions ot, 182 Coulolllb friotion, 426-427,429-4'1,4" Cauohy's theorem existence ot, 426 ot exlstence tor solutIon ot dltterentlal nesa tive, 4,0 equation, 2,9 phase trajectorIes in presence ot, 427-429 ot exietence ot I1near dltterentical equatlon, 2 Coupled eleotronio oscIllators, stability of, 158 ot phase trajeotorles, 11,14 ot uniqueness, 24 Coupled trequenciee, 265,218 Chaik1n, B., '51 Coupl1ng autoparametrio non-linear, '75 Cha1k1n, S.~ '-4,75,~5,~1,,al,'83,405,409,420, between degrees ot freedom, 25',270 422,426,429-4,0,442 oritical value of, ,40 Change ot equatlons, 421~428,4" 1nduotive, 260 laok ot, 254 Charaoteristl0 equation, 14,18,42,44-45,47,51-58, non-l1near,'15 102,114,l2O,~8,1~,119,'1l,,~,'18,'29,'4~ '41"58,'89,411,414,4'8-4'9,44l~442,444 Coupl1ng"taotor, 25' tor translent state ot quad-l1near system, 264 Criteria of I1near circuit, 269 tor existenoe of non-l1near syetem, 268 or orltlcal po1nt, '9'-'94 Charaoteristic exponents, 8,-85,154-155,,29 ot tocal point, l' ot 11m1t oyo1e, 75-77 CharaoteristI0 parameter, '2' ot nodal point, ~ Claosltication ot SingularIties, 42,41 ot saddle point, ~ ot Bendlxson, 75,77-19,105,,45,411 Clook appl1ca tion at," 19.;.82 elelllsntary "theory ot, 424-426 of Kaufmann tor stabil1ty, 61 111111 t oycles 1n, 61-68 at ~pounott tor stabIlity, 48-49,,11-,12 selt-exoltation.ot, 426,429 appllcation ot, 49-55 theory ot, 429 of POincar4, 105,165 Coales"oence ot Poincare-Liapounott tor stability, ,11-'12 of equillbrium Po1nts, " of Routh-Hurvltz tor stabIlIty, 156,4,8,440 ot frequenoles, 219,,41,'51 ot stability, 22-2"""" of l1lll1t oycles, "72~1,,166 applIed to non-linear ayatem, ',55 ot singular pOints, 127,,49-'50 ot equilIbrium, 60-61 ot singularlties, 29,,4,,6-'1,'9 ot singularitiea, 47 Coetticient ot restitution, ,8,,4'1 Critioal dampIng, '11 Combination trequencies, 2U-274, 278-219, 286, Critioal poInts, ,1,,8,102,'95,'99,401-406,408-409, 290,296"01 412-415,419-42' oriteria tor existenoe ot; '9'-'94 Combination harmonics, 2U-214,216,286,294 loous ot, '94,406-401 CombinatIon oscillatlons, 295 ot ditterential equatIons, '92-'95 theory ot, ~O Combination tones, detinltlon at, 272 Cagplex adm1ttanoe, 25',256 BrilloUin, N., '55

.,:

451 Critlcal threshold, 3,11,294,37},406 Dlvlde)

(866 0180

Crl tlcal value ot amplloatlon tactor, 102 ot amplltude, 20) ot coupling, 340 ot damping, 12} ot equlvalent parameter,. 2}4,278 ot index ot modulatlon, }12 ot negatlve reslstance, 263 ot parameter, }1-35,}9,87,97,104,160-l61,165, 203-204 of reslstance, 266,269 ot selt-excltatlon, 297 ot transoonductance, }02 of variable, 197 Curve ot contacts ot Poincare, 75 Cylindrlca1 phase space, 116-!}0 Cy1lndrlca1 phase trajectorles ot conservatlve system,118-119 ot non-oonservatlve system,119-l2l Damped oscl1lator, dltfersntla1 equatlon ot, 16 Damped oscl1latory motlon, equatlon ot, I} Damplng aperlodlc, 262 Coulomb, 195 negative, 4}} posltive, 4)3 crltical, 371 crltlcal value of, 123 linear, 19)-196 negatlve, 105,178-179,197,199 def1n1tlon of, 21 examples of, 21-22 non-linear, 5 non-linear d1ssipatlve, 192-196,240-241 non-linear varlab1e, 105,178-179,197-199 poslt1ve, 18,105,178,197 quadratic, 194-196,240 varlab1e, 105,181,216-217 deForest, Lee, 5 Degenerate differential equation, }96-397 syatems of, 396-398 Degenerate equation, 390,436,438,440 absolutely, 391·392,438 of f1rst order, l12,385,}87,395,397 Degenerate l1m1t cycle, 70 Degenerate phase trajectory, 12 Degenerate system, 148,440,442,445 absolutely, 391,442 completely, )90 condltlons for stabllity ot, 397 of flrst order, periodic solution of, 399 transltion betveen ccntinuous and dlscontlnuous solutlons of, 415-416 vlth one degree of freedom, 399-409 Degenera tion absolute, 444 complete, 389-390 condltlon of, 418 of equatlons, 4-5,}95,420 of .ystem, 441 point of, 4-5,}88,390

Degeneratlon - contlnuod solutlons of dlfferential equatlons ln nelghborhood of, 388-391 Demu1t1p1ication of frequenoy, 272-27' Differential equatlon, 1-6,10-11,14,22,24,31-}2, '5,37,42-4',46,49,51,62,64,66-67,72,80,62,87, 90 ,95-96,101,106,112,117,1,8,192,215,241,32, oritlca1 points of, 392-395 degeneratlon of, 5,)95-'97 llnear, 1 Cauchy's theorem for exletence of, 2 vith constant coefficlents, 1 linearlzed, 67,87 non-linear; Bee Non-1lnear dlfferential equatlon of damped oscl1lator, 16 of dynamical system, 20 of electromechanlcal syatem, 117-118 of first order, 17,}88-}89,}92,408,41',4,8 degenerate, }85,,87,}95,}97 deacrlbing syatem vith one degree of freedom, 405-409 dlscontinuous solutlon of, }97 of harmoniC OScillator, 7 of mechanlcal system, 257 of mot1on, 11 of synchronous motor, 126 of non-diaaipative circuit, }63 of nth order, 9 of pendulum, 190 of aecond order, 10,17,24,}86-,88,'92,395,4}6 full, }85 quasl-dlacontinuoua ao1ution of, 418 of aynchronous motor, 126 of Van der Pol, 342-,44 quasi-linear systam of, 148 slngular pOinta of, 17,,44 singular1tles of, 3,11 sotutions of Cauchy's theorem of existence for, 2,9 ln neighborhood of polnt of degeneratlon, ,88-'91 represented by phase trajectorles, 250 systems of degenerate, '96-}98 theory of, 2,4 vith periodic coefflclents, 250,304,}22,'56-,60, }72-3n,375 Discontinulties, '9}-395,412-414,421,42,,428'431 occurrence of, 393,395-}96,~07 Discont1nuoua jump, 166,'86,392,}95,401-402,404, 408,412-413,416,421-422,425-426 Discontinuous periodic motion, ,8, Discontinuous stationary relaxation oscillations, '92 Discontlnuoua solutions of degenerate system, transltion betveen continuoua and, 415-416 of differential equatlona of first order, 397 Discontinuous stretches, ,82,422-423 Dlscontinuous theory of relaxatlon oscillatlons, ,8},,88·398,400-402,405,409,4,6- 437 Disslpative parameter, 269,'96 Disslpatlve system, 201,207,425 heteroparametrlc excltation of, 369-}11 Dlvldes, 3,11,!} examples of, 12 of common topography, }2

452 Dreyfuss, 126 Dry friction, 22,427 Durand, Dr. V. P., 195 Dynamical equilibrium, 282 Dynamical system, 10-11,,2,51,69,82,87-88,107 dIfferential equation of, 20 equilibrium of, 438-440 stationary states of, 40,436-445 Electrical oscillations in circuit containing iron core, 192 in circuit containing saturated core, 242 Electrical system, 255-258 Electrodynamical system, 87 Electrcmechanical system dIfferential equat10n of, 117-118 self-excitation of, 99,102-104 self-exclted cscillations in, 99,102-104 Entrainment of frequency, 4,249,,41-'54 acoustic, 342,351-352 artificially produced, 352 band of, 352 definition of, 341 mechanical, 342 other forms of, '52-'54 pure, 348 state of, 350 steady state of, 350 transient state of, 346,'50 zone of, 341,348,351-352,354 Equations absolutely degenerate, 391-392,4'8 autonomous, 155 canonical, 45 change of, 427-428,433 characteristic; see Characteristic equation degenerate, 390,436,438,440 of first order, 112,385,387,395,397 degeneration of, 4-5,395,420 differential;sao Differential equation Lagrangian, 374-375 linear, 80,136-137,139,179,186,407,431 approximate, 100 canonical form of, 42-44 equivalent, 2}5-236 harmonic solution of, 167 Mathieu, 372 Mathieu-Hill, 356 of first approximation, 49 system of, 18,23,80,83 with constant coeff1c1ents, 1 linear1~ed, 67,87,100 equ1 valent, 244 non-degenerate, 439 non-linear, 1,118,139,181-183,242 of osc11lation, 192 non-l1near differential; 88e Han-linear differen t1al equa tion of aper10d1c damped mot10n, 14 of damped osc11latory mot10n, 13 of first approximation, 3,49-52,55-56,6;,81,87, 9 4 ,97,99,101,103,114,120,127,167-168,173, 176,179,186-188,193,217,224,229,2,6,244,259, 271,275,277,279,281,285-286,288-29,,301-305, 311,,15-317,'22,411,440-441,444 linear, 49 of first order, 9,383,396-397,400,402

Bquations - continued of Froude for rolling of sh1p, 195,241 of H111, ,61-362 def1nit10n of, 357 of Hil1-Me1ssner, 362-363,365 topology of, ,62-365 trajector1es of, 362,,65 of improved first apprOXimation, 288 of isocl1nes, 121 of L1enard, 107,109-110 of limit cycle, 180 of Mathieu, 357-3;8,360-362,372,375-376 definit10n of, 356 linear, 372 parameter of, 375 stable and unstable reg10ns of, 359-'60 of Math1eu-Hi11, 356-357,361-,62,372-'73 linear, 356 stab111ty of, 361 subharmon1c resonance on basis of, 373,377 of n th order, 396 of phase trajectory, 66,88,128 of POincare, 187 of Rayle1gh, 178-179,181-182,199 of second order, 396-397,418 of separatrlx, 35-36,38 of unstable motion, 13 of Van der Pol, 79,85,97,103,105,108,111,113, 161,175,181,187,197-198,200,217,219,293-29 4 , 348,382-383 quasi-degenerate, 391-392 quasi-linear; soe Quas1-1inear equation variational, 83,153,283-284,311-312,318,328 w1th both linear and quadrat1c damping, 195 Bqu1librium branch point of, 3,42,48,55,60,82,165,445 condit10ns for stable, 161 cond1t10ns of, 11,29,60,80,94,205,443 dynamioal, 282 half-stable, 51 Kaufmann's cr1ter10n for stability of, 61 line of, 429 loss of, 384 motion 1n neighborhood of, 40 ne1ghborhood of, 335 of cirou1t conta1n1ng non-l1near conductor, 55-61 of conservative system, 40 of dynamical system. 438-440 of electromotive forces 1n circuit, 192 of point, 32 p01nt of, 11,24-26,29,40,51,56,60,66,7 4 ,99,125, 170-171,335,354,401,411,429,440-441,443- 444 coalescence of, 33 disappearance of, 34 1n conservat1ve system, 34 motion in vicinity of, 29 stability of, 354 stability of motion in vicin1ty of, 24,26 posit10n of, 18,22,32,38,49,185,302,39 4 ,421,440 Change of, ,811 stability of, 9 stability of, 3'-'5,40-61,71,85,87,93,127,169170,202,354,'58,401,4,8- 44 5 accordin~ to Llapounoff, 48-61 cr1teria for, 60-61 theorems for, 29 state of, 22 types of, 58 Equilibrium phase angle, 354 Equiphase interval, 366-367

453 Equitime Interval, 367 Equivalent Balance of Energy, Principle of, 181, 2}3,2}6-238 Equivalent linear equation, 2}5-2}6 Equivalent linear impedance, 281 Equivalent linoar system, 2}6,28l,286 Bquivalent linearization for multiperiodic systems, 2711-278 method of, 137,233-236,269,279,288,297,314 app11ed to steady state of quasi-lInear system, 258-262 applied to transient state of quasi-linear sys tem, 262-267 examples of application of, 239-245 1n quasi-linear systems with several frequencies, 285-288 pr1nciple of, 239,258,275,277,309 Equivalent linear1zed equation, 244 Equivalent l1noarized system, 3111 selt-excitation of, 278 Equivalent para~eter, 236-239,241,281-283,297, 310,314,316-317 critical value of, 234.278 definition of, 234-235,286 determination of, 233-236 non-l1near, 277 of multiper10dic systeD, 275 Euler's ident1ty, 251 Exchange of stability, 34,38 Excitation asynChronous, 234,295,298 autoparametric, 355-356,374-377 cond1tion for, 374 autoperiodic, 297-298,305 external, 289.335-336,340 absence of. 337-344 system with, 289 external periodic, 249,289,2911,313,317,320,335. 3'+2,376 of quasi-linear systems, 289-300 system vith, 323 heteroparametrIc, 355-356,360,363,373-374 conditIons of, 356,371 dependence on frequency and phase of parameter variatIon, 365-369 of dissipatIve system, 369-371 pr1ncipal features of, 367-369 heteroperiodlc, 298,355-356 absence of, 299 non-resonant external, of quasi-linear system, 294-295 parametric, 4,249,308-313,355-377 definition of, 308 of critically damped or overdamped cirCUit, 371 periodic non-resonant, 289-292 Existence of autoperiodic oscIllatIon, 295-300 of closed trajectories, 21,108-11},129 of critical polnt, crIteria for, 393-394 of focal point, criteria for, 13 ot limit cycles, 99,181-182,198,200-203,224229,348 condition for, 169,173,176,179,199,207,227 criteria for, 75-77 proof of, 79

Existence - contInued of nodal pOint, criteria for, 13 of pericdic solution, 105-115,121,144-145,220,224 condition for, 136,152 geometrical analysIs of, 105-115 in non-conservative system, 224-229 proof of, 105 of principal solution, 333 conditions for, 335 of saddle point, criterion for, 13 of self-eXCited OSCillatIons, 181 Expansions of POincare, 141 External excitation, 289,335-336,340 absence of, 337-344 non-resonant, 294-295 system with, 289 External frequency, 307,341,348,351,353-354 External periodiC excitation, 249,289,294,313,317, 320,335,31;2,376 of quasi-11near systems, 289-300 system with, 323 . Externally applied frequency, 336,340 Paraday-Maxwel1 theory, 124 Plemlng, 5 Ploquet, theorem of, 357 Poca1 point, 16,18,22,69-70,75-77,79,101-103,114, 120,123,125,127,,47,349,386,,88,397,407, 415-416,431,433- 1'34,441,444 critericn of ex1stence of, 13 definition of, 13 motion in ne1ghborhood of, 13 stability of, 47-48,57-60,88-89,91-92,97,99, 165-166,175-177,202 . stable, 13,202 unstable, 13,311 Pract1ona1-order resonance, 301,303,305-309,373 Franck, A., 6 Frequency acoustic entrainment of, 342,351-352 autoperiodic, 297-299,325,342,3 49 coalescence of, 279,341,351 combination, 273-27 11,278-279,286,290,296,301 correction, 174,193,217-219,240 coupled, 265,278 demu1tiplication, 272-273 domultiplication netvork, 352 entrainmenJ: of, sse Entrainment of frequenoy enterna1. 307,341,348,351,353-354 externally applied, 336,340 harmonics of, 296 hoteroperiodic, 297-301,348-349 of pendulum, 342 of quasi-linear oscillation, 217 of rIpple, 363,365,367-368 of self-excited non-linear system, 265 of selC-exc1tod oscillations, 261 of subharmonic oscillation, 336 of thermion1c generator, 158-159 synchron1zation of, 352,354 Priction Coulomb, 426-427,429-431,433 existenoe of, 426 negative, 430 phase trajectories in presence of,

427-~29

Prlotlon - continued dry, 22,427 non-linear, 420 statlc, 428-429 zone of, 429 Proude'q equatlon tor ro111ng ot ship, 195,241 Proude's pendulum, 21-22,67,178-180,417 Gaponov, 348-349 Gauss's theorem, 77 Generating solution, 86,141-144,146-147,149, 152-153,158,160-161,172,181-182,184,190,199, 223,236,238,289,293,326 amplitude ot, 1~6 detln1tlon ot, 141 ot Poincare, 3SO Geometrical analysis ot exlstenoe ot perlodl0 qo1utlons, 105-115 Geometrlca1 det1n1tlon ot stability, 41 Geometrlca1 theory ot l1m1t oyo1es, 63-86 Gore 11k, 374-375 Graphical method ot topology ot trajeotor1811, 25-26 G1yden, 137 method ot, 209 Halt-lltab1e equll1brll1111, 51 Halt-stable limit oyc1e, 70 Halt-trajectory, 69-70,78-79 Hard lIe1t-exoltatlon, 74,99,166,175,178,203,320, ,,1,3}9,426,429 oondltlon tor, 165-166 detinltlon ot, 71 example ot, 87 subbarmonio rellonanoe ot order one-halt tor, ,,6-"9 Harmonio Balance ot Bner81, Pr1nolple ot, 238-239,286,297,301,310 Harmonio motion, 106 Harmonio osoillator, 86,376,428 ditterent1al squatlon ot, 7 motion ot, 86 non-dlsI11patlve, 23 Harmonl0 solutlon ot linear equation, 167 Harmonlos, 276-277,28},290,301-302,310,314 combination, 273-274,276,286,294 ot current, 280 ot frequenoles, 296 Heegner's olroult, 414~416 Helmholtz, 272 Hertz, 5

2",

's••

Betero41D1nB, '48 Ql~ B.ats) Heteroparametrio exoitatlon, '55-'56"60,,6,, '73-)74 oon41tions ot, ,56,'71 dep.ndenoe on frequenoy and phase ot parameter variation, ,65-,69 ot dissipative system, '69-'71 prinoipal teaturel ot, '67-,69

Heteroparamotrlo generator, 356 Heteroparametrl0 machlne of Mandelstam and Papa1exl, 372 Heteroparametrl0 selt-exoltatlon, 364 Heteroparametrl0 oscl1latlonl, condltlonl tor se1t-excltatlon of, 368 Heteroperlodlc excltatlon, 298,355-356 absenoe ot, 299 Heteroperlod1c frequency, 297-301,348-349 Heteroperlodio oso11lations, 301,335,338-340, 345,350 det1n1tlon ot, 295 existence ot, 295-300 posslbl11ty ot, 306 s ta tlonary, 348 synchronlzed vlth autoperlodlo 01l011latlon, '03 Heteroperlodl0 solution, 333-334,344 Heteroperlodic lltatell, 344 oondltlon ot, 305 ot non-linear sY8tem8, 305 8tabl1lty ot, 305 Heteroperlodl0 varlab1ell, 298 Hill equatlon, 361-362 detln1tlon ot, 357 Hi11-Mell18ner equation, 362-363,365 topo1081 ot, 362-,65 trajeotorles ot, ,62,365 "Hunting," 101-102 Hurevloz, Prot. w., 6,3'0 Hurvltz, lee Routh-Hurvltz Huygen8, 341 HystereSiS, 309 ot 01l011latlon, 161-162 ot resonanoe, 320 Hysterellis oyc1e, 162 Improved tirllt approximatlon, 210-214,224,232, 294-295 application ot, 214-219 equatlons ot, 288 Impull1e-exoltsd oaol1latloDu, ,84,387,'95-396,423 Impu1l1e-exolted oaol1lator, phase trajeotorleu ot, 43,-4'5 Indloell ot Poincare, 75-77 theory of, 75-78 Index ot modulation, 371 oritloa1 value ot, '12 ot s1ngularitiea, 76-77,117,,45 ot stepwlle modulation, 36, ot trajeotor1, 76 Instabil1ty, repons ot, , nIntroduotion to Ron-l1near Meoh&n1os,· ',1}5,'0, Isl4nd ot traJeotories, ,0·,2,62,119,128-129 Isoohronism, oondltion ot, 42 Isoohronous motion, 42,146

455 Isochronous osolllatlons, 189-190,202,2}9 Isoohronous system, 171,265 Iaool1nes appllcatlon ot msthod ot, lOS det1n1tlon ot, 20 equa tion ot, 121 msthod ot, 20-21,105,1l} Jaoobian, 140 Jump, }l},}20-}21,}}9 disoont1nuous, 166,}86,}92,}95,401-402,404,40S, 412-4l},416,421-422,425-426 quasl-dlscont1nuous, }20,}90,409 Kaden, }52 Kaldanovskl, 420,422 Kaut=ann's or1ter10n tor stabl1lty ot equl1lbr1um, 61 Kennal'4, Prot. B. H., 6 Kr710tt and Bogol1ubott, 2-},67,1}5,l}7,181, 209-210,2}4 analytlcal msthod ot, 4 ILl'gument ot, }OS oond1tion ot selt-eJt01tation, }12 method ot equlvalent linelLl'1zation ot, l}7, 2Q9,2}} notatlon ot, 254,26} quas1-1inear method ot, }}1 quasl-11ne1U' theory ot, 4,2}4,244,249,2S8,}}1, }42 theory ot tirst approx1llla tion ot, 18}, 186-lBS, 209,2S8,289,}2} treat1se ot, 251,288,}O} 1a8l'8ll8e

method ot variatlon ot constants of, l}7,lB}, 2}} theol'Glll ot, 29 lagrangian equatlon, }74-}75 lanpu1r,

5

LeCorbe111er, Ph., 114,,81 Letsohetll, Prot. S., 4,6,18} "Los ~thodes nouvelles de la mAchanlque cele8te." 2 Levenson, M., 6 Levinaon, H., 70,105,lOS-109,}82 L1apounott, 4,17} or1teria ot stabl11ty ot, 48-49,}11-}12 appl1catlon ot, 49-55 equatlon ot t1rst approx1lllatlon ot, 114,176, 179,440,444 method ot, 49,56,105 stabl11ty 1n sense ot, 40-42,4S-61,155,}26 theorem ot,.},29,49,51-55,61 advantase ot, 55 proot ot, 49,51-55 treatlse ot, 161 L1dnard, 105,181,}82 1Ul&1ysio ot, 4 approximation method ot, 111 oonditions ot CaJ'tan-, 182

Lienard - COli Ullllod oyole ot, 418-419 equatlon ot, 107,109-110 methcd of, 106-1OS plane ot, 107-115 limit cyo1es ln, 1l}-115 qual1tatlve analys18 ot, 4lB theory ot, 107 L1m1t oyo1e, 90-92,169,172,177,182,199,228-229, 2}4,}02,}26,}44-}45 amplltude ot, 166 ana1ytlca1 e~p1es ot, 62-66 blfurcatlon of, 87 branoh pointe ot, 88 ooa1eacenoe ot, 72-7',166 oondlt10n tor statlonary oso11latlon on, 200 detlnltlon ot, 2},62 degenerate, 70 dislntegratlon, 87 equatlon ot, 180 ex1stence of, 99,181-182,198,200-20},224-229, }48 oond1tlon tor, 169-173,176,179,199,207,227 or1terla tor, 75-77 proot ot, 79 geometr10al theory ot, 6}-86 halt-stable, 70 1n case ot polynomial charaoterletl0, 206-2OS in olook, 67-68 ln Lienard plane, 1l}-115 1n Van der Pol plane, 1l}-1l5 inwal'41y stable, 70 motion on, 171 na ture ot, 99 ot tlrst kind, 125,1}0 detinltlon of, 116-117 ot POinoare, 62-86 of eeoond kind, 12} , 125, 129-l}0 def1n1tlon ot, 116-117 ot thermionic generator, 158-159 outwardly stable, 70 phY8lca1 examp1ss ot, 66-SS pleoew18e analytlc, 408,425- 426,429,4}5 poss1b1l1ty ot, 197 propert1es ot, 75,105 representatlon ot, 171 stabl1lty ot, 62,64-66,68-7J,75,S5,9}-9 4 ,9S-99, 102,161,165-166, 169-171, 17}, 175, 177-179,200, 20'-206,}45,}49 stable, 415,417,4}5 system8 wlth several, 200-20} theorem ot, 72-75,205-206 theorem ot stabl1lty ot, 70 topology ot trajector1es 1n presenoe of, 68-75 Lindstedt, l}7 method ot, 5,210 methcd ot Oylden and, 209 LineILl' approx1lllatlon, 1,22,100 Linear clroul t charaotarlstlc equatlon ot, 269 theory ot, 252-255,259 Linear damping, 19'-196 Linear dlfterentlal equatlon, 1 Oauohy's theorem ot exlstsnoe of, 2 wlth constant coett1clents, 1 Linsar dlaelpatlve 011'Oult vlth COJ\Jlu.nt parameters, 258

456 Unear dissipative system vith constan~ parameters, 258 Linear equation, 80, lJ6-lJ7, lJ9, 179; 186,407 ~Jl:n approxlmBte, 100 canonical form of, 42-44 equivalent, 235-236 harmonic solution of, 167 Mathieu, 372 Mathieu-Hill, 356 of first approximation, 49 system of, 83 vith constant coefficients, 1 Linear parameter, 265 Linear resonance, 249,317' Linear system, 23,28,67,138,275,284 phase trajectories of, 7-23 Routh-Hurvitz theorem for, 3 ''',' vith several degrees of freedom, 278 " L1nearized differential eqUations, 67,87' Linearized equation, 67,87,100 equivalent, 2114 L1nearized motion, 27 Lochakov, 4,405,409 lJJdeke, ;21 Mande Is tam, 2,4,361 conditions of, 395-396,401,404,407,412,419,421422,437 , '

MandelBtam and Papalexl, 351,355-356,372,377 conditions of stability of, 330 discontinuous theory of relaxation oSQillations of, ;8, heteroparametric machine of, 372 . method of, 289~323 school of, 4,249,;23,336 theory of, 2119 Marconi, 5 Mathieu equation, ;57-358,360-362,372,375-;76 definition of, 356 ' linear, 372 parnmeters of, '75 stable and unstable regiOns of, ;59-360 Mathieu funotion, ;57,;61 ',.';,:, ' MathieU-Hill equation, ;56-357,;61-;62,;72-373 linear, 356 stability of, 361 subharmonic resonance on basiS of, 373,;77 Maxvell, theory of, 5 Mechanical system, 99,124,255-258,417 differential equation of, 257 parameter of, 417 self-excited OS,cillations of, 99-102 stability of, 102 Meissner, 362 (see also Hill-Meissner Melde, 376 experiment ?f, }55 Migul1n, 377 Modulo tion critical value of index of, 312 index of, }71 index of stepvise, 363

r

'

equa~~on)

MInIer, ,41 Motion aperiodiC, 3,;6,48 aperiodic damped, 106 equation of, 14 phase trajectories of, 18-19 asymptotiC, 3,28,,0-31,36 definition Of, 16 of pendulum, 16,442 asynchronous, 125 damped osoillatory, equation of, 13 differential equation of, II' discontinuous 'periodic, 383 harmonic, 106 ' in neighborho'od of equilibrium,' 40 in neighborhood of focal point, 13 in vicinity of equilibrium point, 29 in vicinity of saddle point, 26 isochronous, 42,146 lavs of, 2-3 linearized, 27 Method of Small, 1,66 non-isochronous, 172 non-stationary, 40 of conservative ayatem, 42 of elastically constrained current-carrying conductor, 34-37 . of harmonic oscillator, 86 of pendulum, 14-15,29:';0,100,118,12' of representative point, 10 of rotating pendulum, ;7-39 of synchronous motor, diff9rentlal equation of, 126 . of lIylitelD 'Vi ttl one degree of freedom, 82 on l.tm1t cycle, 171 oscllla'torydamped, 48;106 phase trajectories of, 16-18' periodiC, ;,26,'1,39,41-42,62-6',77-78,85,105 discont1nUOUS~ '8; in conservative aystem, 62-6' ,",!Stability of, 8,2-86 ·periodicatat1oiun.y:, '63 quaai-isochronous, frequency of, 217 stable, 155 stability of, 24,41-42,85,155,359-;60,37; in neighborhood of singuiar pOint, 40-41 in viainity of critical value of parameter,' 33-)4 '" , in vicinity of equilibrium pOint, 24-26 of harmoniC oaci1lator, 86' " stationarY, f 2), 40~~1, 148, 158,}02 stability of, 161,358 stationary periodic, 61,435 uni-d.tmensional reel, 8 unstable, 28-29.37 equation of, 13 phasetfaJectoriea of, 14716 ~; :. -. l ,~ "' • . Multiperiodia aystem equivalen~'l1nearlzation for, 274-278 squivalent parametsra of, 275 resonanae ln, 271 Multiply degener.. te system, 410-416 Mult1v1brator of Abraham-Bloch,410-414 RC, 1102-404 Negative ariterion of B~ndixson, 75,77-78,345 Negative damp1ng, iO~L17~-179,.l97',199

457 Negat1ve - continued def1nition of, 21 examples of, 21-22 Nodal pOint, 18-19,22,45-46,59,69,76-77,81-82, 101-103,114,127,3 45,347,349,397,407,411, 415,441,444 def1nltlon of, 13 criter10n for ex1stence of, 13 stabi11ty of, 13,45,47-48,58,60-61,202,311 theorem for occurrence of, 45 Non-conservative system, 17,23,106 olosed trajector1es of second kind in, 121 cy11ndr1cal phase trajector1es of, 119-121 existence of period1c solut10ns in, 224-229 v1th non-linear variable damping, 197-199 Non-degenerate equat10n, 439 te system, 443-444

l~on-degenera

Non-d19s1pat1ve c1rcu1t, differential equation of, 363 Non-d1ssipat1ve harmonlc osc11lator, 23 Non-d1ssipatlve system, 268

Non-llnear oBc1llatory BYBtem, 272 Non-11near parameter, 243,265,309,354 Non-1lnear resonance, 4,249-377 external, 301-322 stabl11ty of, 313,320 internal, 284 undamped, 321 Non-linear restor1ng force, 239-24} Non-llnear self-exc1ted osc11lat10ns, 197 Non-l1near selt-exc1ted system, 351 Non-l1near system, 23,42,55,284,287 autoper10d1c state ot, 295 canonlcal torm of, 52 character1stic equation for, 268 criteria of stabl1ity for, 3,55 frequency of self-excited, 265 heteroperiodic and autoperiodic states of, 295-300,305 perlodic solut10n of, 138,140 stability of, 49,52 with one degree of freedom, 278 with several degrees of freedom, 278-279

Non-1sochronous motlon, 172 lion-lsochronous osc11lation, 191-192

Non-llnear variable damping, 105,178-179,197-199 non-conBervatlve systems With, 197-199

Non-isochronous system, 240,271

Non-resonance, 271,290

Non-linear clrcuit, self-excltation of, 1 Non-11near conductors, 1,242-243

Non-resonant oscillations, 275 Non-resonant self-excitation of quasi-linear system, 267-269

Non-linear conservative system, 23,128 behav10r of, 32-33 definltion of, 24 examples of, 190-192 general propert1es of, 24 phase trajector1es of, 24-39 theory of first approximat10n app11ed to, 189-190 vlth cub1c term, 222-223 Non-l1near coupllng, 375 autoparametrlc, 375 Non-linear damp1ng, 5 Non-linear differenat1al equat10n, 1,24,63,66, 75,85,135,137,141,201,216,23 4,272,417,431 cond1tions for per10d1c1ty of solut10ns of, 138-153 exact solutions of, 135 in dimenslonless form, 323-325 of dlsslpatlve type, 179 of oscillatlng circu1t, 162 of second order, 158 of self-exc1ted oscl1latlons, 66 of tors1nal osc11lat10n of shaft, 191 Non-llnear d1ssipatlve damp1ng, 192-196,240-241 Non-llnear.d1ssipatlve system, 182 Non-linear equat10n, 1, 118,139,181-182,242 of oscillatlon, 192 stability of, 50,52 Non-linear equivalent parameter, 277 Non-11near mechan1cal v1brat10ns, 1 Non-linear non-conservat1ve system, 23,40,63,105 autonomous, 62 hlgher ap~rox1matlons tor, 223-229 Non-linear oscl1lat10n, 100,137,233 selt-exclted, 197

Non-resonant system, 277,281,303 Non-stationary motion, 40 Ord1nary pOint, 12-14,20. deflnitlon of, 11 Ordinary value of parameter, 32 Osclllatlng system, experiments with, 308 Oscillation hysteresis, 161-162 Oscllla tions autoperiodlc, 301,345,348,350 amplitude of, 350 conditions for stability of, 306 definition of, 295 existence of, 295-300 se1f-exaitation of, 300 otabl1ity of, 306 Btabl1ity of stationary, 304 synchron1zed with heteroperiodic oscillation, 303 synchronous, 304 camb1nation, 295 electr1cal in c1rcuit containing 1ron core, 192 1n c1rcuit containing saturated core, 242 heteroparametric, conditions for self-exc1tat10n of, 368 heteroperiodic, 301,335,338-340,345,350 definition of, 295 existence of, 295-300 possib11ity of, 306 stationary, 348 synchronized w1th autoperiodic oscillation, 303 impulse-excited, 384,387,395-396,423 in stationary state, 259

458 Oso11latlons - con"nuod lsoohronous, 189-190,202,2'9 malntalned bJ perlodl0 impulses, 42,-4'5 non-isoohronQus, 191-192 non-llnear, 100,~7,2" selt-exolted, 197 non-linear equatlon of, 192 non-resonant, 275 of conservatlve slstem, 214 of l1m1t-olcle trP8, 42, of pendulum, 1 of sJDohronous motor, 124-~0 quasl-dlscontinuous, 408,417 quasl-harmonic, quasl-11naar, 128,158,'9' frequenol at, 217 relazatlon; 110 Relaxatlon oscl1latlons leli-excltatlon at, 58-59,66 selt-exolted, sse Se1f-exclted oscUlations stabllitl at, 284-285,"2,'72,4'2 statlonarl, 66-67,71,~,,269,28},297,'O,,'44 on l1m1t OJole, 200 subharmonio, ,15",a-,40 frequenol at, selt-exol ta tlon at, '55 sJDchroni&ed, stable conditlon of, 284 Oso1llator ooupled e1eotronic, stabl11tl at, 158 damped, ditterentia1 equation at, 16 eleotron-tube, 296 harmonio, 86, '76,428 difterential equatlons of, 7 mot1on of, 86 non-diss1patlve, 2, impulse-exolted, phase trajeotorles ot, 4,,-4'5 0,811latorr damped motlon, 48,106 phaae tl'ajeotoriea at, 16-18 Os011latorr parameter, ,08,,85,4,6,441 Oso1llatorr alatem, 68,271 non-l1near, 272 resonanoe at, 271

18,

,,6

Papa1ex1, 2,4 (aoo olIO ICandelatam and Papa1exl) Parameter, 284,,!i4-"5,,46,'51,'59,'90,'9' biturca t10n values at, 87 oharaoter1st10, '2'· oonetant, 201, 252,'70,4'5 l1n04l' diulpative alatem With, 258 oonatant 11naar, '17 or1t1ca1 value at, ,1-'5.'9,87,97,104,160-161, 165,20'-204 diaa1patlve, 269,,96 equivalent, 2!l6-2!l9,241,281-28',297,,10,,14, '16-'17 oritical value at, 2,4,278 det1n1tion at, 2,4-2'5,286 determlnat10n of, 2"-2,6 non-l1near, 277 of mu1tiperiodio alatem, 275 tinite, 440 twd, '29 luse va1uea at, 4,lU-1~,'82 llnear, 265 mothod ot smail, ~5-166 non-l1ne&r, 24,,265,'09.'54 ot oirouit, 262,417 ot ICa th10u oqua tion, '75 ot uobaDica1 slstom, 417

Parameter - con"nuod of eletem, 271,426 ordlnal'1 value of, ,2 oso11latorl, ,08,,85,4}6,441 paraa1 tic, ,87 etfect on statiOIl&l'1 states of c1ynam1oal slstems, 436-445 exlstence of, 440 lnt1uence on equl11brlum at dJnam1cs1 sl stau , 4,8-440 perlodic variat10n of, ,08,}22,'56 ema11 values at, 85,128,172,184,187,286,288,296, 327,,82,4,8 variatlon of, 91-92,284,'10,'55,,65,369,372-'73, '76-377,415 Parametrl0 excltatlon, 4,249,308-31','55-'77 detlnitlon at, ,08 at orltlca111 damped or overdamped olroult, '71 Parametrl0 selt~ex01tatlon,. 361 Psrasitlc capaoltl,,4'7 Paras1t10 parameter, ,87 etteot on lltationarr statee of dJnamlca1 sls tems, 4'6-445 exlstenoe of, 440 lntluenoe on equ11ibrlum of dynamlca1 systems, 4'8-440 Pendulum, ,60,445 as example at non-1lnear oonservatlve sYlitem, 190-191 asymptotl0 motion at, 16,442 differential equat10n of, 190 elastio, '74 ener81 of, 26 frequenoy at, ,42 Proude's, 21-22,67,178-180,417 ma thematl cal, 426 mechanical, ,42 motion at, 29-,0,100,118,12' In ne1ghborhood of unstable equ1librlum. 14-15 of olock. 424 oscl1latlons of. 1 Periodl0 ooeffl01ents. '29,362 differential equations wlth. 250,'04.'22,'56360,'72-'73.'75 Periodio impulaea. oscillation malntalned bJ. 42'-"'5 Periodio motlon, ,.26,,1.'9,41-42,62-6',77-78, 85.105 d1scontlnuous, ,8, In conservatlve system. 62-6' stabllity of. 82-86 Periodl0 non-resonant ex01tation, 289-292 Periodio sOlut10n, 66,79.82,84,86,122-12,,~5-l}6, lAl.146, 152-15', 159-160. 229,'7} analJ1;lca1, ,S} oon41t1on for stability of, '28-,,0 exiatenoe of, 105-115,121,144-145,220,224 oondition tor, ~6,152 soometrioa1 analysis at, 105-115 In non-oonservative SJ8tOlD, 224-229 proot ot, 105 ot desenerate oystem ot t1rst order, '99 ot non-l1near prOblem, 140 of non-linear systOlD, ~8,l40 ot quasi-linear equation, 221,,26-,28 stabi11ty of, 15'-158

459 PeriodIcity, 116,136,147 condItion of, 122,129,138-15) PerturbatIon, 184,)18,)28,)7) in phase angle, )11 of amplItude, )11 PerturbatIon method, 149,15) Perturbation tel'lll. 185 PerturbatIon varIables. ,12 Phase, total, 188.2,)-2}4.275,277,279,281,28).291 Phase angle, 7,1)8,159,172,216,)10-'13,)68 critIcal, )68 equilibrIum, )54 of rIpple, )66-)67 perturbation in. )11 Phase dIagram, 26,)0,)67,4)) Phase line, )aa,)97,401,404-405,408,413,421-422 Phase plane, 7-8,10-11,14,16-17,21-22,24-,0,)2, )6,4),45- 47,51,62-6),68-69,71-72,74-75,77, 80,82,87. 105,116-117,119,140-141,148,167, 169-170,174-175,198,)26,)44,)48-)49,,62, ,64,)66,'70,)82,)85-)aa,)92,)94-)95,'97, 405- 406,408-409,413,42)-426,4}},442 defInition of, 7-8 representatIon of phenomenon in, )44-)46 Phase space, 2.8,10,26,,88,'97,401 cylindrical, 116-130 uni-dimendona1, )aa,401 Phase trajectories, 2-,,7-2,,40-41,4,~45-46,62, 64-66,68-71,73-78,87,90,94,97,102,105-107, 109-110,11'-114,117-118,125,128, ],67,169-170,

198,)26,)44,}4~,)60,)64-}66,)70,)82,)84,)86, ,aa,'92-'94,'97,)99-400,402,406-407,409,41~

422-424,426,428-429,4)5,442-44} analytIcal, )8},411,413-414 analytIcal method of topology of, 26-29 behayIor of, In neIghborhood of sIngularItIes 12-14 ' Cauchy's theorem of, 11,14 continuum of, 62,128,407 continuum of closed, 12-13,62,140-141 cylindrIcal of conservative system, 118-119 of non-conservatIve system, 119-121 defInitIon of, 8 degenerate, 12 equations of, 66,aa,128 existence of closed, 21,108-113,129 graphIcal method of topology of, 25-26 index of, 76 in presenoe of Coulomb friotlon, 427-429 in presence of singularItIes and limit oycles 6845 ' Islands of, )0-)2,62,119,128-129 of aperIodl0 clamped motion, 18-19 of Kl11-Melssner equatIon, }62,,65 of 1mpqlse-exolted osoillator, 4,}-4}5 ot I1111t-oyolo typo, 402 ot linear system, 7-2} ot non-linear ooniervative system. 24-}9 ot oloill&tory daIIJIod motlon, 16-18 ot leoond klnd. 121-12},129 of unstable motlon, 14-16 ot Van der Pol equatlon, ;48 proper, 12 sink tor, 69,79 solutIon ot dlfterential equatlons represented by. 250

Phase trajeotories - conttnued s~uroe of, 69,78 topology of, 171,)48,406 analytioal method ot, 26-29 graphIoa1 method ot, 25-26 in nelghborhood of singular points, 25-29 In phase plane, 29-)2 in presence ot singularities and limit cyc1ea, 68-75 Phaae velocIty, 9,16-17,24 definition ot, 9 Pleoewiae ana1ytI0 ourvea, ,65,}82,428 PIeoewlse analytl0 cycle, 402,404,422,425 Plecewlse analytIc limIt cyo1e, 408,425- 426,429, 4)5 Piecewlse analytIc repreaentatlon of phenomenon, )86 PiecewIse analytl0 splral, )70~408,4)4-4'5 PIecewIse analytIc trajectory, )65,397,413,417, 42),429,4)4 Potncar6, K., 4,)4,)8,135,149,249,)02,)27,,44, )55-)56 analytical method cf, 4,87,99,104-105,138-166 bifurcatlon theory ot, 87-104 olassItlcatl~n of singularltles accordlng to, 42,47 condi tion ot, 158 orlterIa ot, 105,165 crlterIa of stabllity of L1apounoft and, )11-)12 curve of oontacts ot, 75 equa tIona of, 187 expansIons ot, 141 funotions of, 147,169 generating solutIon ot, '50 indioes of, 75-77 limlt cycles ot, 62-86 method ot, 5,1'5-166,172,184,2)0,289,299,)73 applied to systems wlth several degrees of freedom, 148 notatlon of, 154 quantitative method of approx1.mation, 2 research of, 2 rule for ascertaIning stabIlIty ot motIon in vlc1n1ty of crltical value of parameter,

))-,4 theorems of indices, 76-77 theory of, }2,6},9)-94,117,136-137,141,162,169, 175, 178,200, 209-226, 247, 289, 292,}73-}14 topological methoda of, 2,20) variational equationa ot, )28 Polsson, 18) me thod of, 184 applicatlon of, 186 Poaltlve damping, 18,105,178,191 Proper trajeotory, 12 Quadratl0 dampIng, 194-196,240 Qualltative analysl1 of L1'nard, 418 Qualitatlve methods, 1}5 Quantitatlve method of approximatlons, 2-}, 1:!5-1,6 ac1vantase of, , Quad-dogenerate equation, }91-}92

460 Quasl-degenerate system, 390-391,445 Quasl-discontinulty, 419,423 Quasl-disoontinuouu jump, 320,390,409 Quasi-discontinuous oso111ation, 408,417 Quasi-diucontinuous relaxation oscl11ation, 415 Quasi-discontinuous statlonary relaxatlon osoillatlon, 392 Quasl-dlscontinuous solution of dlfferent1al equations of second order, 418 Quasl-dlscontinuous timing of electronlc svltch, 432 Quasi-harmonic oscil1atlon, 18; Quasl-harmonlc theory, 193 QU8ul-lsochronous mot1on, frequency, 217 Quasi-isochronous system, 269 Quasi-linear equation, 167,183,186,189,192,200, 214,219,223,230,233- 236, 239, 313, 316,}82,417 defin1t1on of, 136 of system vlth external excltatlon, 289 periodlc solut1on of, 221,326-328 vlth forcing term, perlodlc solutlons of, 326-328 Quasl-1inear method of Kryloff and Bogolluboff, 331 Quasl-11near oscl11atlons, 128,158.393 frequenoy of, 217 Quasi-linear system, 160-161,229,2;3,237,275, 278,301 autonomous, 289 blfurcatlon theory for, 160-161 condltlon of resonanoe of, 271 external perlod1c excltatlon of, 289-300 Kryloff-Bogolluboff theory of, 289 method of equlvalent linear1zation applled to steady state of, 258-262 applled to trans1ent statu of, 262-267 non-resonant external excitatlon of, 294-295 non-reoonant self-excltatlon of, 267-269 of dlfferentlal equations, 148 resonanoe in, 271 reoonant oolf-excltatlon of, 269-271 oelf-exoltation of, 289 vlth several degreeo of freedom, 251-271,274 vlth several frequencles, 285,-288 Quasl-1inear theory of Kryloff and Bogol1uboff, 4,234,244,249,258,331,342 Quasl-1lnearlty, oondlt1on of, 163 Rayloigh, Lord, 341.351,355 equatlon of, 178-179,181-182,199 experiments vlth oso111ating systems, ;08 RC-multivibrator, 402-404 RC ooc111atlons in therm10nlc olrcu1ts, 111 ReferenceD, 1;1-1;;,246-247,;78-;80,446-447 Relch, H. J., 273 Relaxat10n oscl11ations, 68,128,1;0,;81-445 defin1tion of, }81,;84 dlscontinuous stat1onary, ;92 disoontinuous theory of, ;83,}88-;98,400-402, 405,409,436-437 examples of, 97,115

Relaxatlon oscillat1ons - conttnuod mechanical, 417-422 quas1-discontinuous, 415 quasl-d1scontlnuous stat1onary, ;92 stabil1ty of, 44;-445 theory of, 4 Representat1ve pOlnt, 8,10-16,;6,40.41,45,47,62 , 69,7;,94,112,115.125,170 ,172,174,;64-365, 367,}86,;9;-395,401-402,404,406-408,412-41;, 418,421-422.428,434,442-44;,445 motion of, 10 Resonance external, ;01-;22 defin1tion of, 249 fractional-order, ;01,;0;,305-;09,;7; in multlperiodic system, 271 in quasl-11near system, 271 internal, 281,284,;01 definition of, 249,279 of order one, 284-285 linear, 249,317 non-linear, 4,249-;77 external, ;01-;22 1nternal, 284 undamped, ;21 of order n, ;2; of oscl11atory system, 271 of quasl-1inear system, oondit1on of, 271 subharmonic; see Subharmonic resonance Resonance hysteres1s, 320 Reoonant self-excltatlon of quasi-linear system, 269-271 Resonant system, 281,;01-305 Restltution, coefficient of, ;83,437 Richardson, Dean R. O. D., 5-6 Ripple, 359-;60,;69-;71 capaclty, ;64 frequency of, ;6;,;65,;67-;68 phase angle of, ;66-;67 rectangular, ;62-;6;,365,}82 timing of, ;62 Routh-Hurvitz criteria of, 156,438,440 theorem for I1near system, ; Saddle point, 2;,28-;1,;4-;6,;8-;9,41,46-48,58-59, 61-62,76-77,81-82,119,121,12;,127,;45,347, ;49,;97,407,411,415-416,442,444 crlterion for ex1stence of, 1; definltion of, 12 example of, 16 motion in vlcin1ty of, 26 negatiVe, 444-445 positlve, 444-445 theorem for occurrence of, 46 Saturation voltage, 16;,343 Savart; sse Biot-Savart lav Secular termo, 141,147,159,185,221 appearance of, 147 cond1tion for absence of, 222-22;,226-227 condltion for eliminatlon of, 291 definitlon of, 1;6 effect of, 18; eliminat10n of, 1;7,209-210,221-22;,226-228, 2;1-232

461 Secular terms - contlnuod in solutions by series expansion, 18}-186 presence ot, 186 Sekerska, }76 Selt-excitation, 58-59,7},159,278-279,}08,360, }74 asynchronous, 299 autoparametric, }75,}77 autoperiodic, 298,}00 condition of, 80,99, 17}, 204-205, 264, 266-267, 269,}07,}11-}12,}16,}56,}71 crltlca1 value of, 297 disappearance of, 74 existence of, 292,}68-}69 hard, 74,99,166,175,178,20},}20,}}l,}}9,426,429 condition for, 165-166 definition of, 71 example of, 87 subharmonic resonance of order one-half for, }}6-}}9 heteroparametric, }64 lack of, 285,291,294,}38 non-resonant, of quasi-linear system, 267-269 occurrence of, 417 of autoperlodic oscillation, }OO of autoperiodic state, }08 of circuit, 433 of clock, 426,429 of electromechanical system, 99,102-104 of electronic cirCUits, 66 of electron-tube circuit, 355 of equivalent lineariz&d system, 278 of heteroparametric OSCillation, }68 of non-linear circuit, 1 of OSCillation, 58-59,66 of quasi-linear systems, 289 of shunt generator, 205 of simple Circuit, 262-267 of subharmonic oscillations, }55 of system, 203,334,338 of thermionic circuits, 87,172 of thermionic generators, 95-99,161-166,293 parametric, }61 point of, 340 }~ssibi1ity of, 307,370,403,415 prevention of, 177-178,299 resonant, of quasi-linear system, 269-271 soft, 72,99,166,175,331,}}3,}38,340,426 condition for, 164-165,299-300 definition of, 71 example Of, 92,102 subharmonic resonance Of order one-half for, }31-}35 zone of, 297 Self-excited OSCillations, 66-68,91,103,175,178179,182,376,381,415,435 amplitude of, 104 existence of, 181 frequency of, 261 non-linear, 191 non-linear differential equation of, 66 of electromechanical system, 99,102-104 of mechanical system, 99-102 of non-linear circult, 1 stationary, 261-262,423 Self-excited state, 260 system, 294,}40 asynchronous action on, }OO non-linear, 351 frequency of, 265

Se~r-excited

Self-eXCited thermionic generator, 29} Separatrlx, 3,29,31-32,}5-36,38-39,62,119,123, 128-129,349,442 bifurcation of 1imlt cycles from, 87 equation of, 35-36,38 of second kind, 123 stable, 41,443,445 Series generator, 102-104 parallel operat10n of, 79-82 Shohat, J. A., 5,136,382,418 Shottky, 5 Shunt generator, se1f-excltatlon of, 205 Singular pOint, 9,11-28,}5-38,40,49,51,64,69-70, 72,91,116-117,119-120,123,127, 173,302,}44346,394,406,411,414 classif1cation of, 42,47 coalescence of, 127,349-350 coalescence vith 11m1t cycles, 7} defin1t10n of, 11 nature of, 345 number of, }1 occurrence of, 311 of differential equations, 17,344 stabi11ty of, 38,91,345,350 topology of trajectories in nelghborhood of, 25-29,68-75 Singularity, 2,41,61,78-79,121,176,179-180,311, }35,,s},397,444 almost unstable, 41,442 classification of, 42,47 coalescence of, 29,34,36-37,39 dlstrlbutlon of, 346-}50,444 index of, 76-77,117,345 nature of, 346-350 of dlfferentia1 equat10n, 3,11 Simple, 29 stabl1lty of, 38,41,69,87,91-92,103,166,17},175, 348-349,415,442,444 criteria for, 47 transition of, 58,88,91 zone of, 82 Sink, 87,348 for trajectories, 69,79 Sm1th, O. K., 70,105,108-109 Sommerfeld, 420 Source, 87,348 of trajectories, 69-78 Stabi1lty, 3,125,121 absolute, zone of, 61 ana1ytlcal definition of, 41-42 conditions of, 55,99,148,156-158,317-320,330, }33-}35, 331, 345, 347, 361,440 condit10nal, 61 criteria of, 22-2},33,3}} applied to non-1lnear system, 3,55 of equilibrium, 60-61 of singularities, 47 defined for motion in neighborhood of singularities, 40 deflnition of, 40-41 exchange of, }4,38 geometrical definition of, 41 in sense of Liapounoff, 40-42,48-61,155,}26 Kaufmann's criteria of, 61 Liapounoff's criteria of, 48-49,311-312 applica~ion of, 49-55 of autoperiod1c oscillation, 306

462 Stabll1tT - cOlUillUed oonditlons tor, ,06 ot ooupled eleotronio oaol1latora, 158 ot degenerate sJstem, oondltlons tor, ,97 ot eleotrlc aro, 440-44, ot equillbrlum, "-'5,40-61,71,85,87,9,,127, 169-170,202,'54,,58, 401, 4,s-445 aooordins to Llapounott, 48-61 Kautmannls oriteria tor, 61 theorems tor, 29 ot taoal point, 4~-48,57-60,88-89,91-92,97,99, 165-166.175-177,202 ot heteroperiodio state, ,05 at limit cJole, 62,64-66,68-73.75.85,93-94, 98~99,102,161.165-166,169-171.1~.175.177-

179,200,20,-206,,45,,49 theorem at, 70 at Mathleu-Bil1 equatlon, ,61 ot meohanioal sJstem. 102 ot motion. 24,41-42,85,155.'59-~0.'~ in ne1ghborhood ot singular point. 40-41 in vlo1n1tJ ot oritloal value ot parameter.

,,-,4

in vlo1n1tJ at equillbrlum points. 24.26 ot harmonio osoillator. 86 ot nodal point, 1,,45.47-48,58.60-61.202.,11 ot non-linear equatlon, 50.52 ot non-linear external resonanoe, ,1",20 ot non-linear SJ8tlllll. 49,52 ot oso11latlons, 284-285",2,'72,4,2 ·ot perlodl0 lIIotlon, 82-86 ot perlodio solutlon. 15~158 oondltlons tor, ,28-,,0 ot point ot equillbrlum, '54 ot poalt1on ot elluUlbrlum, 9 ot relaxatlon oso11latlona, 44,-445 ot a1ngul.ar pOint, ,s,91.}45,'SO ot s1ngularltJ, ,s,41,69.87,91-92.10,,166,1~, 175.~-,49,415.442.444

ot statlon&rJ autoperlodio oso111atlon. ,04 at statlon&r1 motlon, 161,,58 at statloQ&rJ state. 292~'18-,19 Poincar~-Llapounott orlterla of. '11~'12 reglons of, , Routh-Hurvltz crlterla tor. 156.4,s.440 threshold of. 127 Statl0 friotlon. 428-429 zone ot, 429 Statlon&r1 motlon, 2),40-41,148,158,,02 stabl1lty ot, 161,,58 Statlon&rJ perlodl0 motlon, 61,4'5 Station&r1 selt-exolted oso11latlon, 261-262,42, StatlonBrJ oso111atlon, 66-67,71,26"269,28,, 297,,0,.,44 on limit cJale, 200 Stati0n&r1 solutlon, 64,148,,1, Statlon&rJ. state, ,18, 42,-424 etteot ot parasltic parameters on, 4,6-445 or dJDlllll10al sJstem, 40.4,6-445 ot lIIotlon. 2,,67,69 oso11latlon in. 259 stabl1lty ot, 292,,18-,19 Statlon&rJ value, 4'2-4" "Struggle tor L1te," 68 Strutt, '59-,62 Subharmonl0 oscl11atlon, ,15",s-}40 frequenoy ot,

,,6

Subharmonio oso111atlon - con'lnu.d selt-exoltatlon ot. '55 Subharmonio resonanoe, '2) external, 4,,7, tor underexolted sJOt8lll, ,;,5 internal, 4,278-28, ot nth order. '77 ot order one-halt tor bard selt-exoltatlon, ,,6-"9 tor sott selt-exoltatlon, ,,1-"5 tor underexolted system, "5-'~ ot order one-third, '39-}40 on basis ot Mathleu-Hl11 equatlon, ,~,'77 on bas~s or theory ot POinoare, .,2,-}40 phenomenon ot, '77 Subharmonio solutlons, 'OS Subharlllonios, 272-27' Superregeneratlve cirouit, 298 "Sur les oourbes d4tinles par une equat10n dlrt~rentlelle,n 2 Synchronizatlon, 28,-284 ot autoperlodlc wlth heteroperlod1c oscillatlon, ,0, ot trequencles, '52,'54 zone of, 284 Synohronlzed oso111atlons, stable oondltlon ot, 284 Synchronous lIIotor dltterentlal equatlon ot, 126 oscillations or, 124.1,0 System absolute1J degenerate, '91,442 autonOlllous, 424 detinitlon or, 10 non-linear non-oonservatlve, 62 quasl-1inear, 289 vlth one degree ot freedOlll, 105 00lllp1etely degenerate, '90 oonservat~ve, 7,2"25,,s,40,42,106,128,161, 185,189,224,267,,62.'75-'76 oJl1ndrloal phase traJeotorles ot, 118-119 equilibrium ot, 40 motion ot, 42' oso11lations ot, 214' periodl0 lIIotlon in, 62-6, pOints otequilibrlum in. ,4 degenerate, 148,440,442,445 condltions tor stabl1lty ot, ,97 ot first order,'periodlo solutlons ot, '99 transition between continuous and discontinuous solutions ot, 415-416 wlth one degree of treedOlll, ,99-409 degeneration or, 441 dlssipatlve, 201,207,425 heteroparametrlc excltation of. ~9-'71 doub1J degenerate. '97,41,-414 dJDlllll1cal, ).0-11,,2,51,69.82,87-88,107 ditterential equation ot, 20 equl1lbrium ot. ·4,s-440 statlon&r1 states of, 40,4)6-445 eleotrioal, 255-258 e1eotrodJDlllll1ca1, 87 eleotromeohanioal differentla1 equation of, 117-118 self-exoltation of, 99,102-104 self-excited osoi11ations in, 99,102-104 equlva1ent linear, 2)6,281.286 equivalent 1inearlzed, ,14

463 Sylltelll - continued selt-ex01tat10n of, 278 lsoohronous, 171,265 linear, 2',28,67,1,8,275,284 phase trajeotor1es ot, 7-2' Routh-Hurvltz theorelll for, , wlth several degrees ot freedom, 278 Ilnear dlss1pat1ve with constant parameters, 258 mechanical, 99,124,255-258,417 differential equatlon of, 257 parameter of, 417 self-exclted oscillatlons ot, 99-102 stability of, 102 multlperlodlc equIvalent linearizatlon for, 274-278 equlva1ent parameters of, 275 resonance In, 271 lIIu1tlPl1 degenerate, 410-416 non-conservatlve, 17,2,,106 closed trajectorles ot second klnd in, 121 oyllndrlcal phase trajectories ot, 119-121 existence ot periodlc solutlons ln, 224-229 wlth non-linear variable damplng, 197-199 non-degenerate, 44,-444 non-dlssipative, 268 non-isochronous, 240,271 non-linear; soo Ron-linear system non-l1near oonservatlve; S08 Non-llnear conserative system non-linear dissipative, 182 non-linear non-conservative, 2',40,6',105 autonomous, 62 higher approximations for, 22'-229 non-reaonant, 277,281,28, of degenerate dlfferential equatlons, '96-'98 of equatIcns of fIrst order, 9 of first order, 97 of linear equations, 18,2',80,8, OSCillating, sxperiments with, '08 oscl1latory, 68,271 non-linear, 272 resonance ot, 271 parameter ot, 271,426 quasl-degenerate, '90-'91,445 quasl-isochronous, 269 quasi-linear; 880 Quad-linear system rescnant, 281,,01-,05 self-excitation ot, 20""4,,,8 selt-exolted, 294,,40 asynchronous actlon on, '00 ncn-linear, '51 trlply degenerate, '97,414 underexcl ted', "5 subharmonlc resonance tor, "5-,,6 w1th external exoitatlon, 289 w1th external periodic excitation, '2' w1th lnterna1 resonance, 281 with 1II0re than one degree ot treedom, 75 with non-linear variable damplng, 197-199 with cne degrse of freedom, 10,82,427 condltlons ot per1od1city tor, 1,8-148 differentlal equat10n descr1blng, 405-409 1II0tlon of, 82 wlth several degrees ot freedom, 148 Polnoar~ method applied to, 148 with several I1m1t cycles, 200-20' with tvo degrees ot freedom, 148-15,,410 wlth variable damplng, 105

Theodoroh1k, K., '51 "Theoretical Mechan1cs," 15-16 "Theory of Oscillations," ,-4,75,1'5,137,,81 Therm10nic circuits, 101 RC osc11lat10ns in, 111 Thermlon1c emiSSion, 5 Thermlonlc generator, 5,7,,1'7,243-245,424 ampl1tude ot osc11lat10n 1n, 67 frequenoy ot, 158-159 I1m1t cycle of, 158-159 self-excitation of, 95-99,161-166,29' cond1tion for hard, 165-166 condlt10n for soft, 164-165 self-exclted, 29' Threshold, 48,'70,'94,407 critical, ,,11,294,37,,406 of stability, 127 Threshold cond1tlon, 82 Topo10g1cal lIIethods, 7-1'0 of Andronow and Vltt, ,42 of P01ncare, 203 of qual1tative 1ntegratlon, 2-4 advantage of, 2 1lmltations ot, , Topolog1ca1 representation, 148 Topo10g1ca1 structure of trajeotor1es, 406 Topolog1ca1 study of trajector1es ot Van der Pol's equat10n, '48 Topology ot Hill-Meissner squat10n, ,62-365 of phase trajector1es, 171,348,406 analytlcal method of, 26-29 graphical method of, 25-26 ln nelghborhood of slngular points, 25-29 ln phase plane, 29-'2 in presence of singularltles and I1m1t cycles, 68-75 of Van der Pol plane, 169-172,175 Trajectories; 80e Phase trajectorles Transitlon of singularlties, 58 Underexcited system, '35 sUbharmonic resonance for, Unl-d~enslonal phase space, Uni-dimensional real motion, Unlqueness, Cauchy's theorem

335-336 ,88,401 8 ot, 24

Van der Mark, 68 Van der Pol, 1,4,20-21,181,,41-,42,,45,'52,'59361,,81 abbrevlated equatlon of, 188,292 ana1ytlcal method of, 4 dlfferential equation of, 342-'44 equation of, 79,85,97,10,,105,108,111,11,,161, 175,181,187,197-198,200,217,219,293-294,348, ,82-,83 phase trajectories of, '48 method ot, 167-182

464 Van del' Pol - conUnll.ed plane, 113-115,167,169-172,175 limit cycles in, 113-115 solution, 197,,48-'50 tbeory of, 1,178,200,292 tbeory of performance of heart. 68 variables of, 169,171-172 Variable damping, 105,181,216-217 Variation of constants, metbods of, 1}7,18',2" Variat10nal equat10ns, S"l5',28,·284"11.,12,,lB of Poincar~, ,28 Velocity f1eld, 11 Vincent, ,41

V1asov, 124-125,1,0 tbeory of, 125 Volterra, V., examples of 11m1t oyo1es, 68 Vortex pOint, 2,,26~'1,"-'6,'8-'9,41,62-6',76, 119,,88,428 defin1t1on of, 12 Weaver, Dr. W. w., 6 Witt, 4,148,,42,'74·'75 (see oleo Andronav and W1tt) Wronsk1an, '57

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