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922 T 2 ' 0 1 + 2cp, |cp2|2 + 9 , cp22 + 9 l cp22 ^ 2 '"' + 2o|2) • (3.M7) T 2>—)> i-e-' i n ,. ,cp *>0) + e"3'T°(p* . ocp*_,J(5.2.10)
012
2 2int
+2vlM
e
+q>1\y2\2e4in%
^l=-s[n(q, 2 -9 2 c 2 ' n ')-i-(8 -a)(q>2 4 ^ ^ ° ' ) + a/ + % ( 9 ^ - 2 ' ° ' +3|92|2 92 +3| 92 | 2 9 2 e 2/n ' +cp23^4/"')+ 8fr (3.1.9) +
773" ((()2(Pl2e ~ 2 ' n ' + 2 ( p 2 1 9 'I' +^2(p'2 +(p2 ^> 2 g2/n' + 8 £2
+ 292|q>.rc2'n'+92 9i 2 c 4 ' n ')]. This set of equations is solved by the multiple scales method. Introducing times of various scales To = t, T/= e To , ..., we seek the solution in the form q>k=qko(T0,Tl,...)
+ Z(pkl(T0,Tl,...)+...
(£=1,2)
(3.1.10)
The standard procedure of the method yields to the equations: D0(?k0=0, Acp,, = - ^ 9 , 0 + [|(e 2
*Lr
ltk
(3.1.11)
- D - fi ^ 0 - 9 , / ' ^ ° ) "
(3.1.12)
80
Mechanics of Nonlinear Systems with Internal Resonances
(3.1.13) 2 75
-^(cfcoCflV" '" +2%J«P| <| W o + ^ o ^ + It follows from (3.1.11) that(pto (/c=l,2) do not depend upon time To: (pk0 = 9 ^ ( 7 ] ) . (In the zero approximation integrals of energy have the form |(plo|2 =Q2uf + w,2 =2EX,
|(p20|2 sn 2 H 2 2 + «2
=2E2).
From the set (3.1.12), (3.1.13) we obtain conditions of absence of secular terms: d(pl0 3ibu , ,2 ibn 2 _ 2 x -7^r+^(Pio+—-r
3/Z>27
I
|2
/ 6 1 7 ._
I
7 i r ^-p M + 1 H r9 2 »|
|2
+
(ju4b)
+ 920 9 1 0 ) - ' ( 8 -CT)q>20 = ° -
Multiplying Eq. (3.1.14a) by (p"10 and adding up the obtaining equation with the conjugate one we have ^ r | 9 i o | 2 =-4(9io+9io)-2^|9io| 2 +^7T(9I 2 O 9 2 2 0 -9io 9220) • (3-1-15) This equation governs the change of energy for the first-degree-offreedom, the first term in the right hand side being the speed of energy input, the last term-speed of the energy exchange between two modes. Similarly from Eq. (3.1.14b) one obtains the equation governing the energy for the second degree of freedom:
Non-autonomous Two-Degree-of-Freedom Cubic Systems
— r |(p 2 0 | 2 =-2(a|(p 2 0 | 2 +-^-g-((p 2 2 0 (pi 0 -92o9i 2 o)-
81
(3-1-16)
Combining Eqs. (3.1.15) and (3.1.16) yields to following equation for the total energy: rf
""°j!;W)=4«P,o
+ O - 2 , . <|
Along with the complex equations (3.1.14)—(3.1.17) consider equations in real variables (in polar coordinates). Putting t p M = Q f l t e ( 9 ' ( i t = l,2),
(3.1.18)
we obtain from (3.1.14) the following set of equations of amplitudefrequency modulation: $ + ^-^^in2(e2^=-^co£,, dlx oil 111 « 1 ^ - 5 a ; + ^ ^ + -^q^[2+cos2(9 2 -e i )]=— sirfi,, 1
(3.1.19)
^+^a2+^^a2sin2(e2-ei)^O, ^ - ( 5 - a H fli]
+
oil
^ + ^^2+co£(e2-£,)]=(). Oil
Equation (3.1.17) with account of (3.1.18) takes the form d{aX + ai) 2 d T +2v{a x+al)
=-^/a,cose,
(3.1.20)
(this equation can be easily obtained also directly from the first and third equations (3.1.19)). Combining the second and fourth Eqs. (3.1.19) yields to the equation for the phase difference y -Q2 -Q\ '• 8Q^+3^2^+fr2(^-^)(2+cos2Y)-3frrf+8aQ= dTx
^ " ^ . (3.1.21) q
o2
Mechanics of Nonlinear Systems with Internal Resonances
3.2. Undamped Systems Let us first consider an undamped system u=0 (more exactly, it is the case when the damping terms in Eqs. (3.1.2) have the order of smallness higher than 1). Then Eqs. (3.1.19) can be written as follows
^-^^sin2(92-ei)=-|co^, o i ^ - 5 Q q + ^ i ^ + ^a1^[2+cos2(e2-ei)]={sirei,
(3.2.1)
^ U ^ s i ^ -e,)=o, ax o
^ _
( 6
ax
_
C T )
^
+
^
+
^^[2+co£(e2-ei)]=0,
os
where x = 7", / Q. Equation (3.1.20) takes the form
f*£t£J)-l/fl,co«,. dx
Q
(3.2.2,
3.2.1 Stationary oscillations Omitting the apparent cases of stationary (steady-state) oscillations in each degree of freedom separately (uncoupled modes ax * 0, a2 = 0 or a, = 0, a2 *• 0) let us consider coupled modes (CSM) for which a{ * 0, a 2 * 0. Putting in (3.2.1) ak = const, Qk = const (k = 1,2) we have: }\2 <\ a?, sin2 (02 -8,) = 4/cos0,, -85Qa,+3£| l ^+Z i 2 a i ^[2+cos2(e 2 -6 1 )]-4/sirfi 1 ,
^ 2 ^a2sin2(e 2 -e,) = 0, -8(5 - a ) Q oj +3^4+hi
<%
(3.2.3)
Non-autonomous Two-Degree-of-Freedom Cubic Systems
83
The third and first Eqs. (3.2.3) give sin2(9 2 - 0 , ) = 0 , cosG, = 0 , whence y =e 2 - 9 , = knl2
(£=1,2,...), 9, = ± T I / 2 .
(3.2.4)
We see that, similarly to the case of free oscillations, two types of CSMs exist in undamped systems: a) normal modes (NM), for which y = 0 or y = 71 ; b) elliptic modes (EM), for which y = n 12 or y = -n 12 (see chapter 2). For normal modes cos2y = 1 , for elliptic modes cos2y = - 1 . Then the second and fourth Eqs. (2.2.24) give the following equations with respect to the amplitudes of CSMs:
-S8Qal+3b[^+b[2chal(2±l)=4fsix&u a2[-%(b-a)Q.+?>b22c%+b[2($ (2+1)] = 0 ,
(3.2.5)
where upper sign relates to the NMs, the lower sign - to the EMs, and sinG, equals to 1 or - 1 . For the coupled modes the expression in square brackets in the second Eq. (3.2.5) equals to 0, so 8 (8- CT) n_(2±i)j 11
36 2 2
3
*„ '
(3.2.6)
Then from the first Eq. (3.2.5) we get a cubic equation with respect to a, for the NMs a, [3(6,, b22 -b2n)a[
-SQ{b22
8 -bl2(8
- a ) ) ] = 4 / 6 2 2 sinG,, (3.2.7a)
and for the EMs:
«, U[bu hi -^. 2 2 ]«, 2 - 8 ^ 2 2 5 ~6 12 (6 -a)jj = 4/6 22 sinG,. (3.2.76) Positive roots of Eqs. (3.2.7) determine coupled stationary modes, under condition that the right hand side of (3.2.6) is nonnegative:
— [8(8-a)Q-(2±l)Z) 12 a 1 2 ]>0.
(3.2.8)
b22
Each equation (3.2.7) represents a pair of cubic equations ( sin 9, = + 1 ) . Roots of the equations of each pair are equal by modulus, but differ with their signs. So the total number of real positive
84
Mechanics ofNonlinear Systems with Internal Resonances
roots of each pair is equal to three or one. Thus the system have one or three solutions for normal modes as well as for elliptic modes. (We remind the reader that each the coupled solution corresponds to two normal modes — in-phase or anti-phase oscillations — or two elliptic modes). Inequality (3.2.8) determines a domain in the plane [a\, 8) whose boundary is a parabola. It follows from (3.2.8) that coupled solutions may exist at a, —> oo only if bl2 b22 < 0. Otherwise the CSMs may exist only in a limited range of energy of oscillations. Each of curves (3.2.7a) and (3.2.7b) consists of two branches separated by "backbone curves" for couple modes which are also parabolas 3(bu b12 -b?2)a* -8fi(Z>22 8 -bl2(S -cr)) = O (forNMs), (3.2.9a) 3|Z>n b22 --tf2)af
-8Q|6 2 2 8 --b]2(b
- a ) 1 = 0 (forEMs). (3.2.9b)
Let us find bifurcation points where the CSM paths branch off the uncoupled modes path. At these points inequality (3.2.8) is fulfilled as a strong equality (a2 = 0 in (3.2.6)), so
W^nE
i (2±D*I2
(3.2,0,
where upper and lower signs relate to NMs and EMs, respectively, and index "b" stands for bifurcation. It follows from (3.2.10) that two bifurcation points exist under condition — — >0. bn
(3.2.11)
It means that in the case bn >0 ( bn <0) bifurcation points on the first uncoupled mode exist if the excitation frequency is larger (lesser) than the natural frequency of the second uncoupled mode. If this condition does not fulfilled a CSM path may appear at a bifurcation point laying on the "backbone curve" for the second uncoupled mode, i.e. at a point «j = 0, a2 ^ 0. For this point we have from (3.2.6)
85
Non-autonomous Two-Degree-of-Freedom Cubic Systems
»
a
= i ^
<3.2..2>
V 36 22 This point is the same for the NMs and EMs, in distinction from the points (3.2.10); it exists if
— — >0. (3.2.13) b22 So the bifurcation point on the "backbone curve" for the second uncoupled mode exists in the case b22 >0 ( b22 <0) if the excitation frequency is larger (lesser) than the natural frequency of the second uncoupled mode. Note that bifurcation points (3.2.10) and (3.2.12) depend on the nonlinear coefficients and the excitation frequency and do not depend on the amplitude of the excitation force. Depending on signs of nonlinear coefficients b12 , b22 and 8 - a the following four cases are possible: a) identical signs of bX2, b22 and 5 - a ; then three bifurcation points (3.2.10), (3.2.12) exist (two points for the NMs and two points for the EMs, one point being common); one may anticipate that in this case one finite path for NMs appears, as well as for EMs; b) 612 b22 < 0 and condition (3.2.11) is satisfied; then only two bifurcation points (3.2.10) exist (infinite paths branching off the driven mode); c) bX2b22 < 0 and condition (3.2.13) is satisfied; then only one bifurcation point (3.2.12) exists (infinite paths branching off the "backbone curve" for companion mode); d) signs of b]2, b22 are identical and opposite to sign of 8 - a ; then bifurcation points are absent; it is clear that in this case CSMs are absent. Consider the particular case of exact internal and external resonances a = 0 , 8 = 0 . Then Eqs. (3.2.7), (3.2.8) give for the NMs and EMs, respectively: •, 4fb77 sin0, A a{ = i 22
3[bub22-bn)
(for NMs),
(3.2.14a)
86
Mechanics of Nonlinear Systems with Internal Resonances a3=_4/^Lsin6]_
(forEMs),
b
(3.2.14b)
i ub21--tf2\ and Eq. (3.2.6) is reduced to a
/ (2±1)612 2 =ai J-^r^-rV
3
6
22
(3 2 15)
- -
(upper sign relates to the NMs, lower one - to the EMs). It follows from (3.2.11) that the NMs and EMs exist at exact internal and external Note that in formulas resonances only under condition b]2b22<0. (3.2.10) sinGj equals either to 1 or - 1 so that the right hand sides become positive. Bifurcation points (3.2.10), (3.2.12) in the case of exact resonances coincide with the zero point. It is easily seen that formulas (3.2.14), 3.2.15) give asymptotics for Eqs. (3.2.7), (3.2.6) at / —> oo, i.e., they determine asymptotic behavior at l a r g e / i n the general case of inexact resonances. At l a r g e / w e have and bub22 -bf219 = 0 determine Conditions bub22-bf2=0 degenerated cases (then frequencies of free nonlinear oscillations in the NMs and EMs, respectively, do not depend upon their energy, see Chapter 2, formulas (2.1.30) and (2.1.31)). In these cases equations (3.2.7) become linear ones with respect to a\. If one of these conditions is fulfilled simultaneously with conditions of exact internal and external resonances then corresponding equation (3.2.7) for NMs or EMs cannot be fulfilled. It means that steady-state motions — either NMs or EMs — are impossible in such an undamped system (similar to the case of exact primary resonance in linear undamped systems). 3.2.2
Stability of the coupled stationary modes
In order to check the stability of steady state modes we used the standard procedure with respect to equations of amplitude-frequency modulation (3.2.1). Expanding the right hand sides about a stationary point (ais, a2s,Qls,Q2s) one obtains a set of linear equations with the following matrix:
87
Non-autonomous Two-Degree-of-Freedom Cubic Systems
' H
=
o
o
0
0
- ^ ^
^(2+002)0
-2^+4; 2 ^
2H.V - 2 ^
-2^(2+00^)
0
0
-60^
0
0
'(3'2-16) ,
where r\, =aha2scos2ys, ^s = / 0 sin8 ls , y, =Q2s -Qh. Eigenvalues of this matrix can be easily calculated. The point is stable if the real part of each eigenvalue is not positive. The characteristic equation for matrix (3.2.16) is a biquadratic one of the form X 4 +2A,X 2 +A 2 = 0 (expressions for A,, A2 are not presented here). It is easily seen that roots of this equation have no positive real parts only in the case when both roots A. are real and negative. So the CSMs are stable under conditions A, > 0 , A 2 > 0 , A* >A 2 .
(3.2.17)
3.2.3 Numerical analysis of the coupled stationary modes Consider some results of the numerical analysis. It is seen that the behavior of the system under consideration depends on the following dimensionless parameters: the coefficients ratios cc; = bH IbX2{i=\, 2), two detuning parameters 5, =Q8 lbn, a , =Qa Ibn and excitation parameter /„ = / lbn. As a frequency parameter we took parameter
5,. As the response of a system at forced oscillation can be intimately bound up with the behavior of the system at free oscillation, one of aims of our analysis is to trace possible links between the CSMs in nonautonomous and corresponding autonomous systems. So our analysis is partially based on the results of studying the autonomous systems presented in Chapter 2.
88
Mechanics ofNonlinear Systems with Internal Resonances
We consider consequently systems with various combinations of signs of nonlinear coefficients ratios a,- (7=1,2). It was noted above that namely the sign of a 2 determines whether the CSM paths are finite or infinite in the space {au a2,fo) (at given frequency parameters,). The case of negative coefficient a 2 . As the first example consider system with parameters a, = -0.8, a 2 = -0.5 at exact internal resonance (a =0) whose CSM paths at free oscillations were presented in Fig. 2.8 (two infinite CSM paths — the stable NM- and unstable EM-path emanating from the zero point). Amplitudes of stationary modes a,, a2 as functions of the excitation parameter f0 for this system at given excitation frequency ( 5 , = -3, i.e. the excitation frequency is lower than the eigenfrequencies) are presented in Fig. 3.1. Solid lines here and below correspond to stable modes, dashed lines to unstable ones. Points of curves for ay and a2, lying on the ordinate axis, correspond to values of the free oscillations amplitudes in normal and elliptic modes (points at the "backbone curves"). The plot presented in Fig. 3.1 becomes more convenient and clear if, instead of considering two branches in each equation (3.2.7) for sin 9, = ± 1, we account for only one branch but admit both positive and negative values of excitation parameter/ Bearing in mind that the change of the/sign means the phase shift equal to 7t, we obtain the plot given in Fig. 3.2 (Fig. 3.1 is obtained from Fig. 3.2, to reflect all curves in the second quarter into the first quarter symmetrically with respect to the ordinate axis). Using Fig. 3.2 one should remember that the response at a given has to be found for both values:/„ and - / 0 . The value of fo = f/bl2 different branches of the normal (elliptic) modes in Fig. 3.1 become parts of a smooth curve in Fig. 3.2. We see here only one stable normal path and one unstable elliptic path, similar to the corresponding autonomous system.
Non-autonomous Two-Degree-of-Freedom Cubic Systems
Fig. 3.1 Amplitudes of stationary modes as functions of the excitation parameter for system with negative coefficient CC 2 ( a , = - 0 . 8 , 0C 2 = - 0 . 5 , a =0, 8 . = -3).
Fig. 3.2 Amplitudes of stationary modes as functions of the excitation parameter for system with negative coefficient 0t 2 , admitting both signs o f /
89
90
Mechanics of Nonlinear Systems with Internal Resonances
But one should remember that in the points of intersection of a coupled curve with the ordinate-axis the phase 9, undergoes a jump, so in Fig. 3.2 we have smooth projections of a discontinuous spatial curve in 4D-space ( a , , a2,9, ,9 2 ) . It is of interest to compare the amplitudes versus the excitation parameter curves obtained for different values of the excitation frequency. In Fig. 3.3, (a), (b), these functions are presented for the same system at 8 , = 0 (exact external resonance) and 8 , = 3, respectively. At 6 , = 0 all curves, for uncoupled and coupled modes, originate from the zero. At positive 5 , the both coupled modes curves branch off the uncoupled modes paths for ax (a 2 =0) at the bifurcation point (3.2.10). At negative 8 , (Fig. 3.2) the coupled modes curves have the origin at the bifurcation point ax = 0, a2 ^ 0 (3.2.12), i.e., at the "backbone curve" for the second uncoupled mode (both paths — NM and EM — at the same point). Note that the coupled stationary oscillation in the latter case can appear only at certain initial conditions (with nonzero a2). Consider now frequency response curves for this system. In Fig. 3.4, {a), (b) these curves are constructed for normal and elliptic modes, respectively, at given excitation force parameter f0 =5 and a =0. Amplitudes a,, a2 via 8 , for coupled modes, frequency response curves for uncoupled modes ax (a 2 =0), "backbone curve" for mode a2(a] =0, free oscillations) and the bound parabola according to (3.2.8) are presented. The frequency response curves for uncoupled modes ax ( a 2 = 0 ) are determined by the first Eq. (3.2.5) at <32=0, and the "backbone" curve for mode a2- by the second Eq. (3.2.5) at ax =0,
a2*0. The bound parabolas in Fig. 3.4, (a), (b) fall out one (right hand) branch of the coupled mode paths — normal and elliptic — shown with the dotted curve.
Non-autonomous Two-Degree-of-Freedom Cubic Systems
Fig. 3.3 Amplitudes of stationary modes as functions of the excitation parameter for system with negative coefficient Ot 2 ; (a) 8 , =0, (b) 5 , =3.
91
92
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 3.4 Frequency response curves for stationary modes in system with negative coefficient (X 2 ; (a) normal modes; (6) elliptic modes.
Non-autonomous Two-Degree-of-Freedom Cubic Systems
Fig. 3.5 Frequency response curves for all stationary modes in a system with two infinite CSM paths at free oscillations ( a , = -0.8, (X2 = -0.5, a =0, / „ =5).
We see that projections of the CSM path on the planes (a, , 5 , ) and (<22,5.) consist of two infinite branches 1 and 2 separated with the backbone curve for uncoupled mode ax (a 2 =0) (this curve is not shown in Fig. 3.4.). The CSM curves / (normal and elliptic) branch off the uncoupled modes curve a\ at the point of its intersection with the bound parabola (bifurcation points (3.2.10)). The CSM curves 2 for amplitude a2 at negative values of 5, asymptotically approach the backbone curve for mode a2 (uncoupled free oscillations). At positive 5, values the CSM curves for a2 asymptotically approach the backbone curve for coupled free oscillation (in the Figs. — its projection on the plane (a2,8,) which can be obtained from the set (3.2.5) by excluding a, and putting f=0). Corresponding curve for ax
93
94
Mechanics of Nonlinear Systems with Internal Resonances
asymptotically approaches the projection of the backbone curve on the plane (a, , 5 , ) at coupled free oscillation (3.2.9). Fig. 3.5 presents all the stationary modes for the system considered. The picture looks rather complicated.
Fig. 3.6. Amplitudes of stationary modes as functions of the excitation parameter for a system with opposite signs of coefficients CO, ((X, = 1.1, a 2 = -0.1, CT. =1, 5 , =3).
As a second example, consider system with opposite signs of coefficients a ; , namely CC]>O, (X2<0. The CSM paths at free oscillations for such a system were given in Chapter 2 (Fig. 2.4, (b)) for the case a . >0 (two finite CSM paths, both stable); if a . < 0 then CSM are absent. Numerical results for forced oscillations are presented in Figs. 3.6, 3.7. In Fig. 3.6 amplitudes of stationary modes a,, a 2 a s functions of the excitation parameter f0 for CT.=1 at given excitation frequency 8 , = 3 are presented. The picture is similar to that of Fig. 3.3, (b) - two infinite CSM paths (a stable normal and an unstable elliptic path) branch off the uncoupled mode path. No essential changes appear when <7. =0 or <7.<0.
Non-autonomous Two-Degree-of-Freedom Cubic Systems
If 5, <0 then the CSM paths branch off the backbone curve for mode a2, similarly to Fig. 3.2. At a . =0 and 8, =0 all stationary modes paths emanate from the point 0. In Fig. 3.7, (a), (b), frequency response curves are presented for this system at a , =0 and/ 0 =10, for normal {a) and elliptic (b) modes. Here the bound parabola falls out the whole right hand branch of the normal modes curve and intersects with another branch remaining only a bounded coupled path. For the elliptic modes only left hand parts of the curves remain. So, in distinction from the system presented in Fig. 3.5, the CSM paths projections on the planes (ai ,8.) and ( « 2 , 8 , ) turn out to be bounded from the right. The case of positive coefficient a 2 . Consider system with both positive parameters a,: a[=0.1, a 2 =0.1. In this case, according to (2.2.21), «, = 3 a , - 3 = -2.7, e, = 3 a , - l = -0.7, n2 = - 3 a 2 +3 = 2.7, e2 = - 3 a 2 +1 = 0.7, and the situation of intervals (n x , ex), (e 2 , n2 ) is similar to that of Fig. 2.2, (4), with zero point in the middle. So the system has two infinite CSM paths at free oscillations, for any sign of CT , similarly to the above case. Plot of amplitudes of stationary modes as functions of the excitation parameter f0 for this system at 8, =3, presented in Fig. 3.8, exhibits a principal difference from that of Fig. 3.2 and Fig. 3.6. Here the CSM paths at a given 5%are finite, they exist only in a limited range of excitation parameter (due to positive a 2 ) . The normal and elliptic modes paths connect each bifurcation point at the uncoupled modes path ax ^ 0 , a2=0 (3.2.10) (different for NMs and EMs) with a common bifurcation point (3.2.12) at the backbone curve a, =0, a 2 * 0 . Note that in both the cases a 2 < 0 and a 2 > 0 the CSM paths pass across the bifurcation points (3.2.10), (3.2.12), but if a 2 >0 they connect these points with a finite curve, and if a 2 < 0 then the separate CSM paths go from each the point to infinity.
95
96
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 3.7. Frequency response curves for stationary modes in the case of opposite signs of coefficients a,. ( a 2 < 0 ); (a) normal modes; (b) elliptic modes.
Non-autonomous Two-Degree-of-Freedom Cubic Systems
Fig. 3.8. Amplitudes of stationary modes as functions of the excitation parameter for system with positive signs of coefficients a ; ( a , =0.1, (X2=0.1, a =0, 8 , =3).
Fig. 3.8 also displays essential differences in stability of the CSM paths in non-autonomous and autonomous systems. Here the CSM paths include both stable and unstable portions (in autonomous systems a whole CSM path is stable or unstable). We would like to remind the reader that each coupled modes curve is a projection of a spatial curve on a certain plane, and transition through the ordinate axis correspond to a jump to another branch of a discontinuous spatial curve in 4D-space ( a i , a2,0 j ,9 2 ). Frequency response curves for this case are presented in Fig. 3.9 at/ 0 =5, for all coupled and uncoupled stationary modes. Here condition (3.2.9) falls out the left hand sides of coupled modes paths, and the CSMs exist only at positive values of 5 , ; characteristics for both normal and elliptic modes are stiff, in distinction from the curves in Fig. 3.5.
97
98
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 3.9. Frequency response curves for coupled and uncoupled stationary modes in the case of positive signs of coefficients (X,-.
The case of opposite signs of a ; with positive a 2 ( a , = -1.1, a 2 =0.1, a =0) is presented in Fig. 3.10, where amplitudes ax and a2 via the excitation parameter /„ are given for 8, =3. The CSMs at free oscillations in this system were presented in Fig. 2.6, (b) (two infinite paths, a stable NM- and unstable EM- path). Here the CSM paths for a given 5, are finite, similarly to those of Fig. 3.8. It is of interest to consider special cases ot,a 2 =1 (bub22 - bf2 = 0) and a , a 2 = l / 9 (bub22 -bf219 = 0) when equations (3.2.7a) and (3.2.7b), respectively, become linear ones with respect to a,. In these cases backbone curve for normal (or elliptic) modes coincides with the axis of ordinates. Frequency response curves for these cases are presented in Fig. 3.11.
Non-autonomous Two-Degree-of-Freedom Cubic Systems
99
Fig. 3.10. Amplitudes of stationary modes as functions of the excitation parameter for a system with opposite signs of coefficients (X,, OC 2 > 0 .
There are observed resonance peaks at 8 =0, similarly to linear oscillations, in spite of nonzero nonlinear coefficients of the systems. Note that the coupled paths in the vicinity of these resonances are stable, and we can anticipate that the presence of such a resonance strongly affects the dynamical behavior of real systems.
100
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 3.11 Frequency response curves for stationary modes in special cases.
Non-autonomous Two-Degree-of-Freedom Cubic Systems
101
3.2.4 Nonstationary oscillations To study nonstationary oscillations in undamped systems a numerical experiment was carried out based on equations (3.2.1). The response of a system depends upon initial conditions. As a rule, we chose an initial point close to the stationary one, stable or unstable.
Fig. 3.12. Amplitudes and phases of the oscillation modes when the initial point is in a vicinity of a stable stationary point.
Fig. 3.12 illustrates typical response in the case when the initial point is in a vicinity of a stable stationary one (frequency response curves for stationary oscillations of this system are given in Fig. 3.9).
102
Mechanics of Nonlinear Systems with Internal Resonances
Amplitudes and phases of both linear modes periodically oscillate about the stationary point (normal mode al =2.649, o2=3.135, y =0, 6i=7u/2). More diverse is the response of the systems when the initial point is close to an unstable stationary one. Some types of the response are presented in Fig. 3.13, 3.14. Different types of periodical modulation of amplitudes and phases are observed. But in the most cases the modulation of amplitudes and phases is chaotic.
Fig. 3.13 Amplitudes and phases of the oscillation modes (initial point is close to an unstable stationary one).
Non-autonomous Two-Degree-of-Freedom Cubic Systems
103
Fig. 3.14. Amplitudes and phases of the oscillation modes (initial point is close to an unstable stationary one).
3.3 Damped Systems
3.3.1 Coupled stationary modes Consider now coupled stationary modes in damped systems. Putting in (3.1.19) ak = const, Qk = const {k - 1,2) we obtain the following set of equations:
104
Mechanics of Nonlinear Systems with Internal Resonances
u, a, —— a, a\ sin2v = cosG,, r ^ ' 8Q ' 2 2Q ' 2 —U- a,3 + ^ - a, a22V (2 + cos 2y7) - 5 a, = -1— sin0,, ' ' 2Q ' 8 Q ' 8Q '
(3.3.1)
2
|i +-^-a, sin2Yr = 0 , 8Q '
P
^ 2 - a 2 22 + - ^ - a 1 2 (V2 + c o s 2ny , ) - ( 5 - a ); = 0. 8Q 8Q ' Equation (3.1.20) gives integral of energy 2|j,Q(a, 2 +a 2 ) + /«!cose, - 0 ,
(3.3.2)
and equation (3.1.21) yields to the equation 7 , , 4/sirfi, 3b22a22 +bl2(af -a22X2+cos2y)-3*na,2+8CTQ+ - = 0. (3.3.3) a
i
From the third equation (3.3.1) we have sin2y=-|^-. b] 2 a,
(3.3.4)
This expression leads to following conclusions. • Coupled stationary modes do not exist in damped systems at sufficiently small energy of oscillations; they can exist only at amplitudes satisfying the condition
«?*£f. \b\2\
•
(33.5)
When ax -> oo then 2y -> k% , y ->• k n/2, (k =0, 1,...). The phase difference y = 0 or n means that the CSMs are normal modes (in-phase or anti-phase oscillations), and phase difference y = 7 t / 2 o r 37i/2 means that they are elliptic modes. Thus we can conclude that // the CSMs exist at large amplitudes they
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Non-autonomous Two-Degree-of-Freedom Cubic Systems
approach normal modes or elliptic modes (but at any finite amplitudes they are not exact normal or elliptic modes, if (Li ^ 0 / From (3.3.2) and the last equation (3.3.1) one has (3.3.6)
f\a\ cos2y =
8Q(CT
i
-5)+3b 2 2 al +2b na} '- 2LJ !2_L
(33 7)
Excluding the phase difference y =9 2 —0, from Eqs. (3.3.4), (3.3.7) we obtain equation with respect to amplitudes ax ,a2 :
(8p. Q) 2 +[8Q(a - 5 )+3b22 a\ +2b]2 a] f =(b12 a] )2 .
(3.3.8)
The second equation with respect to ax,a2 is obtained by substituting (3.3.7) into the second equation (3.3.1) and excluding 0, with use of (3.3.6):
(8^D)2(^ +^+[3(^^-^)4-8^(5 -a)^-8Q5sff =
(3.3.9)
Introducing dimensionless variables and parameters
buff 8Q|a
„ bX2a\ 8Qn
0
_ G n
2 =^ T , a , = ^ , a
2 =
0
_ 5
n ^,
(3.3.10)
8(Q|i)3 6 I2 A12 we rewrite set of equations (3.3.8), (3.3.9) in dimensionless form: l + ( a ° - 8 ° + 3 a 2 j / + 2x) 2 =x 2 ,
(3.3.11)
(x + y)2 +[^y ^x2y2)+^° -o°)y-d°x} 2 =^xQ . (3.3.12) Note that expressions (3.3.4), (3.3.6) and (3.3.7) in the new variables take the form
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Mechanics of Nonlinear Systems with Internal Resonances
sin2y=-I , cos2y = ^ = ^ a X
l Z l
2x
>
^ ^
X
2(x + y) cos9, = — , — - ,
(3.3.14)
and inequality (3.3.5) is reduced to condition x>l. Variables x, y have a definite sign (as well as Q) which is determined by sign of b\2. Therefore the coupled stationary oscillations correspond to points of intersection of the second order line (3.3.11) and the fourth order line (3.3.12), for which x and y have identical signs coinciding with the sign of bn (i.e., lying in the first or third quarter, in dependence on sign of bn). From (3.3.11) we have:
y=— 3a2
(±VX2-1-CT°+5°-2X).
v
(3.3.15)
'
The coupled stationary modes exist if expression in the right hand side of (3.3.15) has the same sign as coefficient bt2. Substituting (3.3.15) into (3.3.12), we can obtain an equation of frequency response for the CSMs (with 8 as a frequency parameter), but because of complexity of this equation it is more convenient to deal with the set (3.3.11), (3.3.12) (with account of (3.3.13), (3.3.14)). Let us consider asymptotic behavior of roots of this set at Q —> oo . Assume that there exist unlimitedly increasing roots x —> oo, y —> oo . Keeping only highest power terms in (3.3.11), (3.3.12) one obtains two solutions:
Q*]_f
y = _±x
36(a,a2-l)
a2
x3=
x'-
8
' 8 a | ,. 36(9a,a2-l)2
y-^-x 3a2
(3316)
(3.3.17)
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107
Conditions x>0, y>0 axe satisfied only when a 2 < 0 so infinite CSM paths in the space (x, y, Q) exist only under condition b]2b22 < 0 , (similar to undamped systems, see p. 3.2). Solutions (3.3.16), (3.3.17) coincide (with account of notations (3.3.10)) with those obtained in section 3.2 for the case of exact internal and external resonances in undamped systems (3.2.10). Solution (3.3.16) gives the normal modes path in undamped systems (3.2.10a), solution (3.3.17) gives the elliptic modes path (3.2.10b). Thus the asymptotic behavior of the CSMs paths at Q - » ° o in damped and undamped systems is identical. There either exist, ifa 2 < 0, two infinite CSM paths in the space (x, y, Q), one of them approaching the NM path in corresponding undamped system, another - the EM path), or these paths are absent, iffX 2 > 0. Note that for infinite CSM paths x ~ \[Q , y ~ IJQ at Q - » oo . Hence at Q-±<x>
lim4±2U0, and according to (3.3.14) cosG, -> 0 ( 6 , - > ± r c / 2 ) . There are to be noted degenerated cases when a,a2=l
ora,a2=l/9.
(3.3.18)
The relationships (3.3.18) were considered for undamped systems in p. 3.2.2 as conditions determining special cases when frequencies of the CSMs do not depend on their amplitudes. For the degenerated cases
x~Q,
y ~Qd& 2 - > ° ° -
3.3.2 Qualitative analysis of equations for CSMs In the general case solutions of set of equations (3.3.11), (3.3.12), determining the CSMs, depend upon five parameters introduced in (3.3.10): • parameters ai , a 2 , characterizing ratios of nonlinear coefficients of the system;
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Mechanics of Nonlinear Systems with Internal Resonances
• parameters a ° H 8 ° that depend upon the frequencies difference and the damping coefficient; • parameter of the external force Q. Without loss of generality one may put b\2>0 and consider only roots of the set (3.3.11), (3.3.12) lying in the first quadrant (at 612<0 all considerations are similar, only for the third quadrant). For this case Q>0. Consider the shape of curves (3.3.11) and (3.3.12) in dependence on their parameters. Equation (3.3.11) determines a hyperbole with invariants (G.A.Korn, T.M. Korn, 1968) 7 = 3 + 9a22, £> = -9a 2 2 , A = -9a22 (/>0, Z)<0, A<0); its center lies on the >>-axis at the point x0 = 0, y0 = (l/3a 2 )(8 - a ). So only one branch of the hyperbole lies to the right of the y-axis. Asymptotes of the hyperbole are lines >> = - ( l / 3 a 2 ) x , >> = - ( l / a 2 ) x (3.3.19) (compare with (3.3.16), (3.3.17)). Shape of the hyperbole depends on sign of a2. If a 2 < 0, both branches of the hyperbole far from the vertex lie in the first and third quadrants; at <x2 > 0 only a finite part of the hyperbole (adjoining to the vertex) lie in these quadrants. In Fig. 3.15 the right hand branches of hyperbole (3.3.10) at positive and negative signs of a 2 are shown (it was assumed 8 ° - a ° -2).
Fig. 3.15 Right hand branches of hyperbole (3.3.11) at different signs of a 2 ( 8 " - a ° = 2) (a) a 2 >0, {b) a 2 < 0.
These plots clear up our above statement about existence of the CSMs with unlimitedly increasing amplitudes only under condition
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109
a 2 <0. Note that at positive £12, in accordance with (3.3.10), condition a 2 <0 means that 622<0, i.e. a soft characteristics on the second degree of freedom. For negative 6 ]2 condition a.2<0 means a stiff response on the second degree of freedom. (When a.2=0 the hyperbole degenerates into two straight lines parallel to jy-axis). Equation (3.3.12) determines a fourth order curve passing across the origin. As the left hand side of (3.3.12) is nonnegative, the curve lies to the right of y-axis if bn>0 (then Q>0) and to the left - if 6,2<0 (Q<0); therefore y-axis always is tangent to this curve. At £12>0 a part of the curve necessarily falls in the first quadrant, at Z>i2<0 - in the third quadrant. In particular case a.i=0, a 2 =0 (linear characteristics in each degree of freedom separately) the equation determines an ellipsis with
invariants A = -Q2[l + (5° - a 0 ) 2 ] / 64, / = 2 + (8 °) 2 +(5 ° - a °) 2 , D = (25 ° - a ° ) 2 (I > 0, D > 0, A <0). When Q increases then the center of the ellipsis moves from the origin and its axes enlarge. The shape of curves (3.3.12) in the general case depends on signs of coefficients a I ; a 2 . When signs of a : and a 2 are different then terms with higher powers have identical signs, and curve (3.3.12) is close to an ellipsis (for a given Q). At the same signs of a, and a 2 the curve far from the origin "stretches out" along straight lines
y = ± l—x Va2
(3.3.20)
(for the first and third quadrants one has to take here sign «+»). Typical views of curves (3.3.12) are shown in Fig. 3.16, where parts of the curves lying in the first quadrant are presented for several values of Q at a ° = - 1 ; 8 ° = 1 for different and identical signs of ai and a 2 . But there are also possible more complex shapes of curve (3.3.12), presented, in particular, in Fig. 3.17 (a), (b), (c) for system ai=0.5, a 2 = l , a " = - 1 , Q=20 with different values 8 ° . When 8 "increases, this curve disintegrates into two branches (in the first quadrant). It is clear that the number of roots of set (3.3.11), (3.3.12) in the first (or third) quadrant in general case does not exceed four.
HO
Mechanics ofNonlinear Systems with Internal Resonances
Fig. 3.16. Parts of the curves (3.3.12) in the first quadrant for a" =-l; 5"=1; (a) different signs of a, and a 2 (a,=l; a 2 =-l); (b) identical signs (a, = - 1 ; a 2 =-l).
Fig. 3.17. Curves (3.3.12) in the first quadrant in the case ai=0.5, a 2 = l, a ° = - l for three values of 5 °.
Two special (degenerated) cases of both negative coefficients should be noted when one of the asymptotes of hyperbole (3.3.19) coincides with straight line (3.3.20). These cases coincide with those given by expressions (3.3.18). Then one of the roots will unlimitedly (and quickly) remove from the origin when Q^>°o, 3.3.3 Bifurcations of the CSMs Consider CSM paths in the space (x, y, Q) with special attention to points of birth and disappearance of the roots, i.e., to the bifurcation points. When the energy is small the oscillations occur only in the excited mode.
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111
Let usfindpoints of branching off'this uncoupled mode, i.e., values of its amplitudes at which CSM paths origin. Points of branching are found from set (3.3.11), (3.3.12) at _y=0. From (3.3.11) we obtain quadratic equation with respect to x 3jc 2 -4(8°-CT 0 )x + l + ( 5 0 - a 0 ) 2 = 0 .
(3.3.21)
This equation has real roots only under condition |8°-G°|>V3.
(3.3.22)
If this condition is not satisfied then the CSM paths do not branch off the uncoupled modes path (or are absent). Accounting for (3.3.10) H (3.1.3), we conclude that existence of these branching points depends on the difference between the frequency of external force and the natural frequency for the second degree of freedom. If inequality (3.3.22) is satisfied then roots of (3.3.21)
x1>2 =ifz<5° - a ° ) ± V(5°-a°) 2 -3]
(3.3.23)
determine branching points, under condition that sign of the root is identical to sign of bn. It is easily seen that signs of both the roots are the same and coincide with sign of 5 ° - a °. Therefore ifb/2>0, then the branching points exist when 8 - a
> V3 , and ifbi2<0, these points
exist when 8 — a < —v3 . Note that x\<2 values (3.3.23) do not depend on the coefficients b\\ and b2i governing the nonlinear oscillation in each degree of freedom separately. Values of the external force parameter Q*, corresponding to the branching points, are to be found from Eq. (3.3.12) by substituting _y=0 and x values from (3.3.23). Dropping the trivial root x=0 we have
0*=4jt[l + (3a 1 jc-8 o ) 2 ].
(3.3.24)
Apart these branching points, there can exist another points of bifurcation — points of touching both curves (3.3.11) and (3.3.12) at which two roots coincide. These are points of birth or disappearance of
112
Mechanics of Nonlinear Systems with Internal Resonances
two CSMs (independently of the uncoupled modes). One can obtain the analytical condition of touching (an additional fourth order algebraic equation) but it is rather complex and is not written here. We will consider the "touching points" below in the numerical analysis. It is clear that the number of bifurcational values ofQ cannot exceed five (two branching points on the x-axis and up to three points of touching). In order to study the behavior of roots of set (3.3.11), (3.3.12) as force parameter Q increases, we solved this set numerically. It was assumed that bn > 0 (hence x>0, y>0, Q>0). Curves (3.3.11), (3.3.12) were constructed for various combinations of signs of coefficients <X|, ct2 and values of the force parameter Q, In Fig. 3.18 curves (3.3.11), (3.3.12) (in the first quadrant) for the case oti > 0, a2 < 0 (a, = 0.5, a 2 = - 1, o °= - 1 , 8 ° = 1) and four values of Q (in the increasing order) are presented. Note that condition (3.3.22) is satisfied, so two branching points exist. The first root appears at Q=5 (Fig. 3.18, (a)); it is the branching point (x=0, y=l), determined by formulas (3.3.23), (3.3.24). In the range 0(5, 21.67) only one root exists gradually moving along the hyperbole 1 (Fig. 3.18, (b), Q =10). At Q =21.67 (Fig. 3.18, (c)) the second root appears (curve 2 intersects the lower branch of the hyperbole 1 at a point on x-axis; it is the second branching point (3.3.23)). As Q further increases, both roots slowly rise (Fig. 3.18, (d)); they remain at £>-><». The case of two negative coefficients ai and a 2 is presented in Fig. 3.19. It was assumed a ^ - 0.5, a 2 = -1.0. The shape of curve (3.3.12) here differs from that of Fig. 3.18 (for positive ai), and this leads to peculiarities in the behavior of the roots at increasing Q. Apart two branching points, an additional bifurcation of the CSMs appears — the point of touching both curves (3.3.11) and (3.3.12), corresponding to merging two stationary points and their disappearance (or to the birth of two roots with following their divergence at decreasing amplitudes). The first point of bifurcation leading to the appearance of the CSMs is namely a touching point (at Q =20). The following two
Non-autonomous Two-Degree-of-Freedom Cubic Systems
113
Fig. 3.18 Curves (3.3.11), (3.3.12) (in the first quadrant) in the case a, > 0, a 2 < 0 for different values of Q (a, = 0.5, a 2 = - 1, a 0 = - 1 , 8" = 1); 1-curve (3.3.11), 2-curve (3.3.12) (condition (3.3.22) is satisfied).
bifurcational points (Q =29 and 2=88.3) are points of intersection of both curves lying on the x-axis (the first point corresponds to disappearance of the first CSM path, the second point - to appearance of the second CSM path). Here also two CSM paths exist at sufficiently large energy. The case of positive a2 with a, < 0 is presented in Fig. 3.20, for system a,= -0.1, a 2 = 0.1, a ° = - 1 , 8°= 1. As in this case instead of the infinite branch of the hyperbole only its limited part close to the vertex
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Mechanics of Nonlinear Systems with Internal Resonances
falls in the first quadrant (curve 1), the behavior of the roots as Q increases essentially differs from the preceding cases.
Fig. 3.19 Curves (3.3.11), (3.3.12) (in the first quadrant) in the case a,< 0, a 2 < 0 for different values of Qt (a, =- 0.5, a 2 = - 1 , a 0 = - 1 , 8 " = 1); 1-curve (3.3.11), 2- curve (3.3.12) (condition (3.3.22) is satisfied).
The first root (the first bifurcation) appears as a touching point, similarly to Fig. 3.19 (birth of two CSM paths at Q =8.93). As Q increases, one of the roots approaches x- axis and disappears at Q= 10.76 (branching off the uncoupled mode). At Q =13.5 the second point of touching appears (curve (3.3.12) touches the right branch of the hyperbole), and else two CSM paths appear. At Q=\4.6 two roots merge and disappear; then at Q =21.67 the third root reaches the x-axis and disappears (branching off the uncoupled mode). Thus, in accord with our above considerations, the roots do not exist at rather large energy of oscillations, i.e., the CSMs exist only in a limited range of the external force parameter (8.93< Q < 21.67). Note that 2, 1, or 3 stationary points exist on different parts of this interval; the
Non-autonomous Two-Degree-of-Freedom Cubic Systems
115
number of bifurcational values of Q equals to five — three points of touching and two points of intersection on the x-axis. Graphs presented in Figures 3.18-3.20 relate to systems for which condition (3.3.22) is satisfied. If this condition is violated, i.e., 8
-a
< V3 then branching points do not exist (in particular, this
statement holds for the exact internal and external resonances).
Fig. 3.20 Curves (3.3.11), (3.3.12) (in the first quadrant) in the case a,< 0, a 2 > 0 for different values of 0 (a, = - 0 . 1 , a 2 =0.1, a 0 = - 1 , 5 " = 1); 1-curve (3.3.11), 2-curve (3.3.12) (condition (3.3.22) is satisfied).
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Mechanics of Nonlinear Systems with Internal Resonances
In Fig. 3.21 curves (3.3.11), (3.3.12) are shown for the same system as in Fig. 3.19 but with negative value ofa2, and instead of a = -1 we assume a =0, remaining 8 =1. Here only one bifurcation appears — the "touching point" at 2=10, and the CSMs remain at Q —» oo . In systems with positive a 2 values when 5 ° - a ° < V3 roots are absent because in this case no part of hyperbole (3.3.11) lies in the first (third) quarter.
Fig. 3.21 Curves (3.3.11), (3.3.12) (in the first quadrant) in the case a , > 0, a 2 < 0 for four values of Q (a, = 0.5, a 2 = - 1, a 0 = 0, 8 " = 1); 1-curve (3.3.11), 2 - curve (3.3.12) (condition (3.3.22) is not satisfied).
The presented analysis enables us to outline general topological features of the CSM paths in the space (x, y, Q). Tracing the motion of points of intersection of curves (3.3.11), (3.3.12), as Q changes, we see that these points either run through the whole finite part of the hyperbole in the first (third) quarter (if a 2 >0) or move along its both infinite branches (if a 2 <0). Let two branching points exist (i.e., 8 - a > V3 ). It is clear that the CSM paths in the space (x, y, Q) are continuous curves, which appear or disappear at the branching points. As in systems with negative values of a 2 the CSM paths are infinite in the space (x, y, Q) they cannot connect these two branching points. There must exist two infinite CSM paths having origins at the branching points. In systems with positive a 2 value only finite CSM paths in the space (x, y, Q) are possible. So there should exist a single CSM path connecting these two branching points. Let now branching points are absent ( 8 - a < -\/3 ). In systems with negative values of a 2 , where two asymptotes (3.3.19) exist, a CSM
Non-autonomous Two-Degree-of-Freedom Cubic Systems
117
path should be a continuous curve consisting of two single-valued branches (which are jointed at the "touching point"). In systems with positive a.2 value CSMs do not exist, as was noted above. Thus, the picture of the CSM paths in the damped systems has much in common with that of the undamped systems but at the same time it has peculiarities. Similar to the undamped systems, the number of the CSM paths in the space (x, y, Q) does not exceed 2, as well as the number of branching points; the CSM paths are infinite, if 012 <0, and finite, if ot2 >0; in the latter case either a single CSM path exists connecting the two branching points, or CSMs are absent (if branching points are absent). As distinct from undamped systems, in damped systems with 0:2 <0 and 8 ° - a ° < v 3 a single CSM path exists that has no intersections with the uncoupled modes path (a separate curve, which consists of two infinite branches). Let us consider frequency response curves for the CSMs on planes (x, 8 ) and (y, 8 ) for a given Q value (projections of spatial curves in the space (x, v,8 ) on the coordinate planes). It is easily seen from Eq. (3.3.12) that these curves, in distinction from those of undamped systems, cannot be infinite (for sufficiently large x or y and constant Q the first term in (3.3.12) greater than the right hand side: (x + y)2 > xQI'4, so this equation does not hold). As only two branching points exist (as 8 changes) there exists one and only one finite CSM path in the space (x, y,d ) connecting these points. Of course, this single path can determine a multi-valued function and be rather complex. The influence of magnitude of 8 - a on the number of stationary points and bifurcations at a given Q value is illustrated in Fig. 3.22 for the system with positive otj (a! =0.5, a2=l), at given values a ° = - 1 , 0=50 and different values of 8 ° . Magnitude of 8°-a0 strongly influences the topology of the curves and the number of points of intersections and bifurcations. The CSMs are absent at sufficiently small and very large values of 8 - a . Thus we can conclude that the main factors determining the topological features of the CSM paths are: • sign of nonlinear coefficients ratio ot2 = b22 / b]2,
118
Mechanics of Nonlinear Systems with Internal Resonances
• magnitude b°-u°, that is determined by the difference between the frequency of external force and the natural frequency for the second degree of freedom.
Fig. 3.22 Curves (3.3.11), (3.3.12) (in the first quadrant) in the case CX| > 0, cx2> 0 for different values of 5 " (a, = 0.5, a 2 = 1, a ° = 0, 0=50); 1-curve (3.3.11), 2- curve (3.3.12).
Non-autonomous Two-Degree-of-Freedom Cubic Systems
119
3.3.4 Stability of the CSMs To check the stability of the steady-state modes, we use the standard procedure with respect to equations of amplitude-frequency modulation (3.1.19). In dimensionless parameters (3.3.10) these equations take the form 8112(0,-6,)-^cose,, 2
l^=-x+xy 2 dx x^-=8° dx
JC-3fljx2 -xy[2+ cos 2 (92 - 9 , ) ] + ^ sinG,, 2
(3.3.25)
Y£=-y-*y si^-e,), y^=(50-cy0)>'-3fly-x>;[2+orjs2(P2-«,)],
where x = \\.TX. Expanding the right hand sides about a stationary point we obtain a set of linear equations with the following (xs,ys,Bis,d2s) matrix: '-2+2ytsw2fs-vs
2x,sin2y,
-2j»;ssin2ys
-2+x i sin2y^
-3a,-^
-2-cofiy,
^-2-co^y^
-3a2
-TI,+C, t], -2^sin2yJ+v, -2*,sin2y,
^
'
-t], 2yssin2ys ' 2xs&in2f, ,
where n, =4*^,0082/,, C = ^ 2 s i n 6 l 5 , v 5 = - — sin9 u , ys =e2s - 8 U . For the CSMs this matrix can be simplified with account of expressions (3.3.13), (3.3.14)):
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Mechanics of Nonlinear Systems with Internal Resonances
'-(i+U
-2
2^
-3
. ~
?, '~~4xT
-
TI, 0
2
~ ~
^-2-cos2ys
-n,+;, TI/
COS2Y
*
-TI,
, , ^ " !
-3a2
2
, , "^
(3-3.26)
-2,
where t>s=yjx!.. The point is stable if real parts of each eigenvalue are not positive. When checking the stability of an uncoupled mode one should put ys=0 ( ^ = r^ = 0). Then the matrix H takes the form '-2-v, 0 H
;,
0
-2+x, sin2y,
0
0
-2-cos^
v.
-3a 2
-Ir^sir^^
2xssiv2is
--3a,i
4xs2
^-costy^
0
(3
-3^
Zx^sir^^
It is apparent that matrix Ho has two eigenvalues equal to (2xs sin2y s ) and (-2+xs sin2y s ), respectively; two other ones are eigenvalues of the matrix
r-2-vf 3a
1
o ^
"47
v
'
•
(3 3 28)
- -
V ™s J This matrix, as is easily seen, coincides with the matrix governing the stability of uncoupled mode ( x ^ 0 , ^ = 0) with respect to perturbations in x and d\ only. It is worthwhile to note a difficulty caused by uncertainty of the parameter 62 (and so y) for uncoupled modes since oscillations in the second degree of freedom are absent for them. Due to this circumstance we have to check stability not of a point but of a line, i.e., to check eigenvalues for any values of 02 (or y). It is clear that the first eigenvalue (2 xs sin 2y s) is always positive for y within a certain range. But this
Non-autonomous Two-Degree-of-Freedom Cubic Systems
121
does not mean instability of the uncoupled mode (the orbital stability of the stationary point can retain) as this eigenvalue corresponds namely to change of 02. But the change of 02 (and y), associated with this eigenvalue, can influence other coefficients of matrix Ho. The second eigenvalue ( - 2+xs sin 2y s), which is associated with parameter y, depends upon y (and so upon 0 2 ). If xs >2 one can easily find a y value (or 02 value), for which the second eigenvalue is also positive. However, the direct numerical integration of equations of motion (3.3.25) not always confirms conclusion about instability of such a mode. The loss of stability of a stationary mode can demand rather large time, and coefficients of this set of equations quickly vary from positive values to negative and inversely, and therefore the mode can remain stable. In the examples presented below we checked the stability of uncoupled modes also by direct numerical integration of equations of motion (3.3.25). Note that for coupled stationary modes these difficulties do not appear due to definiteness of 02 value, and analysis of eigenvalues of matrix H is sufficient. 3.3.5 Numerical analysis ofthe CSMs In order to illustrate features of the CSM paths we present results of a systematical numerical experiment that was carried out for systems with various combinations of signs a ( and a2. In Figs. 3.23-3.29 amplitudes of the stationary modes as functions of the excitation force parameter Q (for a given 8 °) and frequency-response curves (for a given value of Q, with 5 ° as a frequency parameter) are constructed. We put bn>0 (otherwise x, y and Q are negative). Solid lines correspond to stable portions of the curves, and dashed lines to unstable ones. Taking into account the results of paragraph 3.3.3, consider separately cases a 2 <0 and a2 >0 with various magnitudes of 8 ° - a °. As a first example of systems with different signs of bn and b22 (ot2 <0) we consider system with parameters cci= 0.5, a2 = - 1 . Two values of a °
122
Mechanics of Nonlinear Systems with Internal Resonances
were assumed: a =0 (exact internal resonance) and o ° = - 1 (the mode with higher eigenfrequency is excited).
Fig. 3.23 Amplitudes of the response as functions of the amplitude of excitation; a, >0, a 2 < 0 (a,=0.5, a 2 =-l, a ° = -1); a) 8 ° =3; b) 5 ° =0).
Non-autonomous Two-Degree-of-Freedom Cubic Systems
123
In Fig. 3.23, {a), (b) amplitudes of the stationary modes as functions of the excitation force parameter Q are given for two 8 ° values: (a)8 °=3 (a ° = -1); (b) 8 °= 0 ( a °=0), respectively. In the case (a) 8 - a =4, i.e., condition (3.3.22) is satisfied. In Fig. 3.23, (a) two CSM paths (one stable and one unstable) branch off the uncoupled modes path, and the portion of the latter path between the bifurcation points A\ ,A2 becomes unstable. The graph of Fig. 3.23, (a), is topologically similar to that of Fig. 3.6 for an undamped system with ai>0, 012 <0, where one stable normal modes path and one unstable elliptic modes path branch off the uncoupled path (but for the damped system considered we may speak only about CSM paths approaching NM- or EM- paths). It is of interest also to compare this graph with the CSM paths at free oscillations. The (e2,n2) for the corresponding situation of intervals (n{,ex), autonomous undamped system was shown in Fig. 2.5, (a). At a >0 one infinite NM path and one finite EM path exist (Fig. 2.5, (a)), but at CT <0 only one infinite NM path appears. Note that at forced oscillations the plot for case a °>0 does not differ from that for the case a °<0, if condition (3.3.22) is satisfied. So at both signs of a these plots for free and forced oscillations topologically are essentially different. In the case (b) 8 - a =0, i.e. condition (3.3.22) is not satisfied. Fig. 3.23, (b) shows that only one CSM path (unstable) exists at given 8 , and this path does not branch off the uncoupled path (the bifurcation point is "touching point" of two curves (3.3.11), (3.3.12)). The appearance of the CSMs does not change stability of the uncoupled modes path. The frequency response curves of this system at a = - 1 , Q=20 are presented in Fig. 3.24. We see that in the space (x, y,5°) only one coupled modes curve exists (at given Q), which boundary points are the branching points on the uncoupled modes response curve. The CSM curve includes stable and unstable parts, and the portion of the uncoupled modes curve between the boundary points of the CSM is unstable.
124
Mechanics ofNonlinear Systems with Internal Resonances
Fig. 3.24 Frequency response curves for systemai>0, a 2 < 0 (a° = - l , g=20).
Comparing plot of Fig. 3.24 with that of the undamped system, presented for the case a 2 <0 in Figs. 3.5, 3.7, (a), (b), we see that infinite or semi-infinite stationary modes curves are transformed into a single finite curve in the space (x, y,8 ) (at a given Q value). Instead of asymptotical approaching the "backbone curve" for the companion mode, curves for damped systems acquire points of bifurcation — branching points on the uncoupled modes curve. The case of two negative coefficients a, is represented in Figs. 3.25, 3.26. In Fig. 3.25 amplitudes of the stationary modes as functions of the excitation force parameter Q are given for system with cti= a 2 = -0.1, a °= 0, 8 °=2, in Fig. 3.26 frequency response curves are presented for system ai= -0.5, a2 = - 1 , a ° = - 1 , Q=50. The graph presented in Fig. 3.25, is similar to that of Fig. 3.23, (a) — two infinite CSM paths exist in the space (x, y, Q). The picture is also similar to that of the autonomous system with equal a, described in p.2.1.2.2 (Figs. 2.5, 2.6). In both the cases (free or forced oscillations) two bifurcational values of energy exist; the first CSM is stable, the second CSM is unstable, the uncoupled mode becomes unstable after point A i, but regains its stability in point A2.
Non-autonomous Two-Degree-of-Freedom Cubic Systems
Fig. 3.25. Amplitudes of the response as functions of the amplitude of excitation; system with negative a,.: a,= -0.1, a 2 = -0.1, a °=0, 8 °=3.
Fig. 3.26. Frequency response curves for system with negative OC,.
125
126
Mechanics of Nonlinear Systems with Internal Resonances
The graph of Fig. 3.26 is topologically similar to that of Fig. 3.23, (b) — in space (x, y,S°) at given Q value we see a single finite frequency response curve for the CSMs, whose boundary points lie on the uncoupled modes curve, with stable and unstable portions. But this curve has two local maxima and one local minimum. Two peaks of the CSM curve in Fig. 3.26 correspond to two infinite branches for the normal and elliptic modes curves in undamped systems (see the frequency response curve for a similar undamped system in Fig. 3.5). All discontinuous branches of the CSM curves in Fig. 3.5 are transformed into a single continuous curve in Fig. 3.26, and range of existence of the CSMs is essentially narrowed. We would like to note an interesting peculiarity of the system presented in Fig. 3.25. The stable CSM path which begins at point A\ has nearly the constant value x (constant energy of oscillation in the first degree of freedom) in the whole considered range of Q, the energy of the second degree of freedom (y) being increased. So in this case we may speak about approximate "saturation phenomenon" which is usually noted in quadratic systems (Nayfeh, Mook, 1979).
Fig. 3.27. Amplitudes of the response as functions of the amplitude of the excitation; the case of two positive a ; ( a t =0.5, a 2 =1.0, a ° = -1.0, 5 °=6).
Non-autonomous Two-Degree-of-Freedom Cubic Systems
127
As an example of systems with the same signs of bi2 and b22 (ot2>0) we considered the case of two positive coefficients a,. (0^ =0.5, a 2 =1.0, a ° = - 1.0). Results of the numerical analysis are presented for this system in Figs. 3.27, 3.28. The graph of amplitudes via Q (Fig. 3.27) shows a finite CSM path between two bifurcational points lying on the uncoupled modes path.
Fig. 3.28. Frequency response curves in the case of two positive coefficientsCl (. (g=40).
Comparing frequency response curves in Fig. 3.28 with the graph for undamped system with a ( >0 (Fig. 3.9) we also see that different CSM branches due to the damping are transformed in a single curve. The CSM branch projections do loops and include stable and unstable portions. The appearance of the CSM branch essentially influences the primary resonance. And finally consider the response in special cases (see (3.3.18)). Fig. 3.29 represents frequency-response curves for system with aia 2 = l (a,= -0.5,a 2 = -2.0,e=100,o° = - l ) .
128
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 3.29. Frequency response curves in the case CX|
In this case the additional resonance generated by the coupled modes occurs exactly at linear natural frequency of the driven degree of freedom (8 °=0), independently of the amplitudes of oscillation. This is a stable CSM path, and with increasing Q the CSMs approach a normal mode. 3.3.6 Nonstationary oscillations To study possible nonstationary responses we solved numerically set of equations of amplitude-frequency modulation in dimensionless parameters (3.3.25) for various values of parameters oti, cc2, a 0 , 5 ° , Q and different initial conditions. Some typical responses are presented in Figs. (3.30) - (3.32). In all the Figs, plots (a), (b), (c), (d) represent amplitudes parameters x,y and phases 9i, 02, respectively. Figs. 3.30 and 3.31 relate to system a, =0.5, a2 =1, a°=^l, 5 ° = 7, Q=40 with different initial conditions. In Fig. 3.30 we see transition from
Non-autonomous Two-Degree-of-Freedom Cubic Systems
129
an uncoupled oscillation (with small perturbation j>=0.001) to a stable steady-state coupled mode, in Fig. 3.31 — transition from a coupled oscillation to a stable uncoupled one. Such a behavior is in conformity with the response curve for this system presented in Fig. 3.28.
Fig. 3.30. Numerical solution of Eqs. (3.3.25). Transition to a steady-state coupled oscillation. Initial conditions: x(0)=4.2,y(0)=0.001, 6,(0)=2.4, G2(0)=2.4.
A somewhat different behavior is shown in Fig. 3.32 (oci = -0.5, a 2 —1, a°= - 1 , 5 ° = 7, <2=80). Here at an initial time range an uncoupled oscillation is established but it is unstable and gives rise to a stable CSM. The transition to a stable mode (coupled or uncoupled) is often a rather complex motion including complicated modulations and jumps, shown, e.g., in Figs. 3.32, 3.33. But, unlike undamped systems, in all considered examples after some transition ranges a stable mode is established.
130
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 3.31. Numerical solution of Eqs. (3.3.25). Transition to a stable uncoupled oscillation (y=0). Initial conditions: 40)=l .5, y(0)=0.5, 9 t (0)=l, 92(0)=l.
Non-autonomous Two-Degree-of-Freedom Cubic Systems
131
Fig. 3.32 Numerical solution of Eqs. (3.3.25). Transition to a stationary uncoupled oscillation and then to a stable coupled steady-state mode. Initial conditions: x(0)=0.5, .y(0)=0.001, 9,(0)=0, 92(0)=0.
Fig. 3.33 Numerical solution of Eqs. (3.3.25). Transition from a coupled oscillation to the uncoupled mode (y=0). Initial conditions: x(0)=1.2,_y(0)=3.0, e,(0)=2, 92(0)=2.
132
Mechanics of Nonlinear Systems with Internal Resonances
3.4 Concluding Remarks Let us summarize the main results concerning the number and configuration of coupled steady-state modes in non-autonomous cubic systems with internal 1:1 resonance. 1. Interaction of the internal resonance and the primary external resonance gives rise to additional resonances due to appearance of coupled stationary modes (two additional resonances). 2. Two types of CSMs exist in undamped systems (similarly to the case of free oscillations): a) normal modes (NM), for which the phase difference between two linear modes y = 0 or y - n ; b) elliptic modes (EM), for which y = ±n 12. In damped systems the exact normal or elliptic modes are not possible, but as the amplitude of oscillation increases (2-»°°) the CSMs approach normal or elliptic modes (if they exist at large amplitudes). 3. Up to two CSM paths in 3D space (ax,a2,f) can exist in undamped and damped systems (at given frequency of excitation), where / is the excitation force parameter. The CSM paths are finite or infinite curves depending on signs of nonlinear coefficients br. Infinite CSM paths in the space ( a x , a2, f) exist only under condition bn b22 < 0 . 4. Boundary points of the CSM paths (one or two branching points, if they exist) in damped systems lie on the uncoupled modes path for the driven mode. In undamped systems one bifurcational point can lie on the "backbone curve" for the companion mode. The CSM paths either connect these branching points or go from one of these curves to infinity, similarly to the case of autonomous systems. Existence of the branching points depends on combination of signs of nonlinear coefficients b n, b22 and difference of detuning parameters 8 - a . In damped systems the branching points exist only under condition 5 ° - a ° > v 3 (see (3.3.10). In particular, they are absent in the case of exact internal and external resonances. If the branching points are absent then a single CSM path can exist, which does not intersect the uncoupled modes path. 5.
In damped systems a single CSM path can exist in 3D space ( a 1 ; f l 2 , § ) at given amplitude of excitation, where 8 is an excitation frequency parameter (in undamped systems the CSM path undergoes
Non-autonomous Two-Degree-of-Freedom Cubic Systems
133
fission on two infinite branches). Boundary points of this continuous spatial curve lie on the uncoupled modes path. Asymptotes of these branches are the "backbone curves" for the accompanying mode and for the coupled stationary modes (at free oscillations). 6. At exact internal and external resonances the NMs and EMs exist in undamped systems only under condition b]2b22 < 0. The NM- and originate from the zero point (i.e., EM-paths in 3D space (ax,a2,f0) they exist at any small energy). In damped systems and at approximate external or internal resonances in undamped systems the CSM paths appear after exceeding a certain energy threshold. 7. Asymptotic behavior of the CSMs at large amplitudes is governed by expressions (3.2.14), so ax ~ iff , a2 ~ iff if / —> °o. and bub22 - b?2 /9 = 0 determine 8. Conditions bnb22-bf2=0 special (degenerated) cases. Then equations for amplitudes of the CSMs (3.2.7) become linear ones with respect to a\, and a, ~ / , a2 ~ f if / —> oo. In these cases one of additional resonances generated by the couples modes occurs exactly at the linear natural frequency of the driven degree of freedom (5 ° =0), independently of the amplitudes of oscillations. This is a stable CSM, and the values of amplitudes for two degrees of freedom become nearly equal, as/increases. Historical remarks. Forced oscillations of nonlinear two-degree-offreedom systems with cubic nonlinearities having close natural frequencies were studied in (Month, Rand, 1977), (SzemplinskaStupnicka, 1980), (Vakakis, 1992)). It has been established that the internal resonance significantly affects the topology of the primary resonance curves. But because of the special type of systems considered the theoretical considerations were mainly focused on studying the correspondence between bifurcations of normal modes in unforced, undamped systems and steady-state modes in forced systems, without account of elliptic modes.
Chapter 4
Nonlinear Flexural Free and Forced Oscillations of a Circular Ring
In this chapter we consider an important example of a system with internal 1:1 resonance — a circular ring (infinitely long cylindrical shell). At flexural oscillations of thin-walled bodies of revolution circular rings, cylindrical shells, disks — the directly excited modes can be accompanied by the appearance of conjugate modes (which are geometrically similar but shifted in the circumferential direction by angle cp = 7t / In where n is the number of circumferential waves). Because of existence of the companion (conjugate) mode for each driven mode a body of revolution always is a system with the internal 1:1 resonance. Many experimental observations connected with flexural oscillations of thin-walled bodies of revolution, in particular, the formation of traveling waves, can be explained only with account of the nonlinear interaction of conjugate modes, as has been shown for the first time in (Evensen and Fulton, 1967). The second factor, essentially influencing dynamics of thin-walled bodies of revolution, is «splitting» of natural frequencies of conjugate modes due to inevitable initial imperfections. In this chapter an analytical investigation of nonlinear free and forced oscillations of circular rings with account of the interaction of conjugate modes and «splitting» of their natural frequencies is presented. The complete analysis of coupled steady-state modes (running waves) in the vicinity of the primary resonance under the harmonic excitation is carried out in dependence on two detuning parameters (the difference of natural frequencies of two conjugate modes and the difference of the frequency of the external force and the natural frequency of the driven mode).
134
Nonlinear Flexural Oscillations of a Circular Ring
135
4.1 Governing Equations and Solution by the Multiple Scales Method 4.1.1 Geometrical relationships Consider flexural oscillations of a circular ring of radius R in its plane (the cross-section of the ring has an axis of symmetry lying in this plane). Denote the radial and tangential displacements by w and v (w is positive in the direction of external normal). The axial deformation e and the curvature % are given by the expressions (see, e.g., (Alfutov, 1978)):
e=(w + v^)//?+(WjV -v)2/2R2),
x =(w/f -v)^IR2,
(4.1.1)
where (p = yl R , y is the circumferential coordinate. We consider only flexural oscillations, so it is assumed that the average membrane stress is equal to zero:
jeiWcp=0. o
(4.1.2)
The radial deflection w is taken in the form: w(9 / ) =fx (Ocosncp +/ 2 (Osinwp + /„ ( 0 .
(4.1.3)
are
Here f\if\ / 2 ( 0 independent generalized displacements (amplitude of the driven mode and the companion mode respectively), fo(t) is the axisymmetrical component of the displacement, which is determined from the condition (4.1.2) with account of periodicity condition for the tangential displacement v and expressions (4.1.1), (4.1.3): 1
2K
MO = - — J K - v)2
(4.1.4)
Tangential displacement v is determined from the condition of vanishing the linear term in expression for deformation e (4.1.1): v = (1/«)(-/| sinncp+/2coswcp)
(4.1.5)
136
Mechanics of Nonlinear Systems with Internal Resonances
(note that the terms with v give a little contribution in the solution — neglecting all these terms results in error of order \ I n comparing with unity). Then (4.1.4) gives:
/o(O = - ^ r ^ [ / , 2 ( O + /22(o] •
(4.1.6)
4.1.2 Equations of motion The potential and kinetic energy of the ring are given by expressions:
K-f%'Mp-*^"',-' ) '[A 1 W^/,'(')], r = ^ V ^ ) J M p = ^\2f2{t)+^[f2{t) o
A \_
+
n
(4-1.7) (4.1.8)
f2\A
where p, A, I are the density, area and moment of inertia of the ring cross-section, dot denotes differentiation with respect to time t). With an account of (4.1.6) we obtain: (4.1.9)
We assume that the external force F acts only on the first mode:
F,(
(4.1.10)
and damping is the same for both modes, i.e., the generalized friction forces are -\ifk(t) (£=1,2). Then the Lagrange's equations result in the following set of differential equations:
I +(!•/, W / , +2K/i(^2 +fjx t
2
2
+f2
2
+fj2) = q(t) 2
f2 +[x f2 -KB / 2 + 2K/ 2 (/, + fjx + f2 + fj2) = 0 where
_n(n2-l) f El a 2 ~ K~ • \ | p 4 « 2 + l ) ' ^
\xn2 ~npAR(n2+l)
'
(4.1.11)
137
Nonlinear Flexural Oscillations of a Circular Ring
K=
(" " ^ 2
2
, , q'{t) = q(t)
4n\n +l)R
.
2
n
(n +1> pAR
. (4.1.12)
Note that equations of motion (4.1.11) represent the system with nonlinear inertia, in distinction from systems with nonlinear stiffness considered in Chapter 2 and 3. We consider the case of harmonic excitation force: q(t) = qsinQt.
(4.1.13)
As we are interested in description of the behavior of the ring near the primary resonance the excitation frequency Q is assumed to be close to the first natural frequency co : Q = co+e8,
(4.1.14)
where E is a small parameter, 8 is a detuning parameter. Let us introduce a dimensionless time T = Qt and dimensionless displacements fk = fk I h, where h is a characteristic dimension of the ring, e.g., its height. In addition, we take into account that in real rings (having inevitable initial imperfections) the linear natural frequencies of the conjugate modes do not coincide though their difference is small («splitting» of the spectrum of conjugate modes in bodies of revolution was detected experimentally in (Tobias, 1951)). So we assume that the natural frequency for the companion mode is co2 =co + e a ,
(4.1.15)
where a is the second detuning parameter (for the driven mode notation co is preserved)'. Then the set of equations (4.1.11) takes the form:
/Ux +fi/,x -Ki-ep,)7i +xMi
+MTT +71 +/2./U)=ffo Axx •+££,„ +(l-ep 2 )/ 2 +2ic/2(/jfT +/,/ liTT +fl +/ 2 / 2(TT )=0 ' 1
J 16)
Note that imperfections also lead to the appearance of a linear link between the modes of oscillations. However, under conventional experimental conditions (where the position of nodes is not fixed) those modes of vibration are excited for which no linear link exists by virtue of their orthogonality (that is, the location of the nodes is determined by the condition of extremum for the natural frequency).
138
Mechanics ofNonlinear Systems with Internal Resonances
where • ~ H* M« 2 Q 7tp^i?fi(«2+l)'
,2
P2=^_a),^)s^inT,^ = J ^ 2
n
hn
2
(«2-l)V _ 2 Pl 4«V+l)J? ' =
2
^
(n +I)KPARhn
(in view of smallness of E it is assumed that
2
25 Q' (4.U7)
o 2 = Q2 - 2 s Q 5 ,
co j = Q 2 - 2s Q(8 - a )). Note that parameters introduced by (4.1.17) are dimensionless, in distinction from parameters (4.1.12). Let us discern in (4.1.16) all small quantities. As is seen from (4.1.17), K" is small when n is not very large: n «^JR/h). Besides, the damping and the amplitude of the excitation force are assumed to be small (as the vicinity of the resonance is considered). So we assume K =ekQ,
fi=sp0,
(4.1.18)
q=eqQ
After translating all the small terms to the right hand side we obtain: [
•
1
Axx +/2 =e [-M0/2 +P/2 -2Ko/2a' +fj^ +fl +f2Aj\
(4.1.19)
4.1.3 Equations of the amplitude-frequency modulation The set of equations (4.1.19) is solved by the method of multiple scales. (Note that in this chapter we do not pass to the complex representation of equations of motion since it does not give here perceptible advantages). Introducing «fast» and «slow» times To = t, Ti= e To , ..., we seek the solution in the form of asymptotic series fj =fjO(To,Ti,..)+efjl(To,Tl,...)+...
(/=1,2)
(4.1.20)
Accounting for expansions of the differential operators d/dt = D0+sD] +..., d2 Idt2 = D2 +2eZ)0Z)1 +..., (Do = dl dT0, Dl = dl dTx ), we obtain following sets of equations in subsequent approximations:
139
Nonlinear Flexural Oscillations of a Circular Ring
D20fj0 + fj0=0 D
0=1,2),
(4.1.21)
ofn + / n = ^o sin7 i) "2I>oA/io - ^oA)/io + Pi/io ~
-2K0/10[(Z)07,0)
2
2
+7 10 £> 0 7 10 + ( D 0 / 2 0 )
2
+/ 2 0 J D 0 / 2 0 ],
^o 721 + 72, = - 2 D 0 D J 2 0 - » 0 D J 2 0 + p 2 7 20 -2KJ20[(D0/]0)2
j
2
(412
+7 I0 Z) 2 7 I0 + P O 7 2 O ) 2 +/2O^O/2O]-
Writing the solution of (4.1.21) in the form (overscribed bar denotes the complex conjugate) fjo=^UJ(Tl)txp(iT0)
0=1,2) (4.1.24)
+ AJ(T1)exp(-iT0)]
and substituting (4.1.24) into (4.1.22), (4.1.23), we obtain from the condition of vanishing secular terms 2—±+q0+ii0Al+ifilAl+iK0Al(A?+A22) dT, 2—±+ii0A2+ip2A2
= 0, (4.1.25) 2
+iK0A2(A? +A 2) = 0.
Writing then the complex amplitudes in exponential form Aj =ajeQj
0=1,2),
(4.1.26)
and separating the real and imaginary parts in (4.1.25), we obtain the following set of four equations with respect to a .• ,9 .• (/=1,2), which determines the slow changing of the amplitudes and phases of oscillations (equations of amplitude-frequency modulation):
140
Mechanics of Nonlinear Systems with Internal Resonances
^+^ax dlx
-^ l f l 2 2 sin2(B 2 -9.) = -fcose,,
2
2
2
^+^a2A2afsin2(Q2-Qx) dT, 2 2
= 0,
,e
°'~df
+
2
1+
T"'
(4.1.27)
[ f l 2 +fl 2
'
2 cos2(02-e,)] = ^sin8, ,
«2 ^ f + Y ° 2 + y« 2 [«2 +«i cos2(92 -9,)] = 0. From the first two equations (4.1.27) we obtain:
M l ^ l l + »0(a? + a22) = -^.cotf,
(4.1.28)
dTx
4.2 Free Oscillations Consider at first free oscillations without damping. For free oscillations dimensionless time is introduced as x =(£>t, and then § = 0 , P, =0, (B 2 =-2CT/CO (from (4.1.14), (4.1.17)). Putting in (4.1.27), (4.1.28) #0=0, \i =0, we come to set of equations (here notation is introduced for the phase difference y = 0 2 —8,).: dTx
2
da7
Kn
r
'
2
r
-) . _
—f+^« 2 « 1 2 srn2y=0,
9- + ^( a i 2 + f l 2 2 cos2T) = 0, aJj
2
4(21}
--
^ - - + ^ - ( - 2 2 + « 1 2 c o s 2 T ) = 0. a7^ a) 2 with the energy integral in the first approximation (,/V is a constant)
Nonlinear Flexural Oscillations of a Circular Ring
a\ + a\ = AT.
141
(4.2.2)
Subtracting third equation (4.2.1) from the fourth, we exclude 0/ and 02 from the set and obtain equation with respect to y :
iL--iK 0 («, 2 -a 2 2 Xl-cos2y) = 0. co 2
dTx
(4.2.3)
4.2.1 Stationary oscillations Begin with uncoupled free vibration a] ^ 0, a2 = 0 (or ax = 0, a2 ^ 0 ) . It follows from (4.2.1) that the nonlinear characteristics of the ring for uncoupled free vibration is soft. Indeed, from the third equation we have (for a2 = 0 ) : (4.2.4) ^r=~K0fl,2. 2 all The frequency (the first derivative of phase 9 ] with respect to time) according to (4.1.24) and (4.1.26) equals in dimensionless time x to tD
0)SBl+*L ch
=
l+ s * L > dTx
(4.2.5)
and in real time t =x /co fijOUafl + e^L).
(4.2.6)
So co(1) =co(l--H
e
i=--Koai2;ri
+e
(4.2.7)
o =--(BK o a, 2 /+0 o
(similar expression can be obtained for the companion mode, after replacing co by co2). Apart from the uncoupled oscillations, stationary modes (a, = const, y = const) are possible only under condition sin2y = 0, as is seen from
142
Mechanics of Nonlinear Systems with Internal Resonances
the first and second equations (4.2.1). These are coupled stationary modes (CSM). It follows from Eq. (4.2.3) that cos2y * 1 (in the general case a*0). So for the CSMs cos2y = - 1 , y = ± 7 t / 2 , therefore the CSMs are elliptic modes (EM). Equation (4.2.3) for y= + n 12 yields: - + K 0 ( O 2 - f l 2 ) = 0.
(4.2.8)
CO
From (4.2.8) and (4.2.2) we obtain for amplitudes of the EMs in dependence on the energy:
"t=\{N—5-), fl* =!(# + -£_). 2
COKO
2
(4.2.9)
COK0
The EMs exist under condition that the right hand sides in (4.2.9) are positive: N>N, = —
COKO
=— , COK
(4.2.10)
where Aco =co 2 -co . So the EMs are possible only when the energy exceeds the threshold (bifurcational) value determined by (4.2.10). Frequency of oscillations of the coupled stationary mode coe is determined from the third (or fourth) equation (4.2.1) with account of (4.2.9) and cos2y = - 1 . We have ^ =— , dTx 2co
(4.2.11)
and from (4.2.6) we obtain coe = c o + ^ = co + - ^ - = I ( , B + c o 2 ) .
(4.2.12)
Thus, the frequency of coupled stationary free oscillation (EM) does not depend on its amplitude and is equal to the average of linear natural frequencies for the two conjugate modes. In Fig. 4.1 frequencies of stationary modes via the energy of oscillation are presented. The descending straight lines correspond to uncoupled vibrations characteristics (4.2.7), the horizontal line - to the
Nonlinear Flexural Oscillations of a Circular Ring
143
coupled elliptic mode (4.2.12) (the solid lines correspond to stable modes, the dashed line to unstable ones, see below). It is obvious that the appearance of the stable coupled oscillations with constant frequency (beginning from a certain bifurcational energy value) is an important factor, which can determine essential features both free and forced oscillations.
Fig. 4.1 Frequencies of steady state modes via the energy of free oscillation; bifurcation of the steady state modes.
Let us elucidate the physical sense of the elliptic mode. The radial displacement w((p,t)with account of (4.1.3), (4.1.6), (4.1.20), (4.1.24), (4.1.26) and (4.2.12) is equal to (we assume, for certainty, y=7i/2,90=0): w(cp, 0=~[( f l i +a2)cos((det + n
+ h {"~2l)
[(af-al)cos2(oet-(al+af)], (4.2.13)
where amplitudes ax and a2 are determined by (4.2.9). The first (linear) terms describe two "fast" waves running in opposite directions (with frequency coe); the amplitudes of these waves are different, and their superposition gives a standing wave (with amplitude and a running wave (with amplitude hax). As energy h(a2 -a{)/2)
144
Mechanics ofNonlinear Systems with Internal Resonances
increases, amplitudes ax and a2 draw together (see (4.2.9), amplitude of the standing mode diminishes, and amplitude of the running mode increases, so in the limit N —> oo only the running wave remains. The last (quadratic) terms in (4.2.13) describe radial vibration of axial line of the ring with double frequency. Results of this paragraph, in particular, the existence of bifurcational value of energy (4.2.10), are in accord with the experimental observations for cylindrical shells (Kubenko et al, 1984). Note that the boundary condition of the free edge in these tests make it possible to apply approximately the model of a ring (or infinitely long shell). A strong modulation (beating) was observed when free vibrations of large amplitude (of the order of 5-10 thicknesses) were excited, but the beating disappears when amplitudes decrease (due to damping). When f{ (0) = 0 (only the mode with higher frequency is excited) we obtain from (4.2.10), taking account of expression for K* (4.1.17): Mafl], 2» *&. h ' ( « 2 - l ) 2 h^ co
(4.2.14)
For example, for a shell with parameters hiR = 3.125• 10~3, co =36.9*271 , co2 =37.8*271 , «=4 we have f2(0)/h > 7, which is in good agreement with the experimental observations (Kubenko et al, 1984). 4.2.2 Integral of the amplitude-frequency modulation. Stability of the stationary modes In order to describe nonstationary oscillations (that is necessary, in particular, for studying stability of steady-state modes), we are needed in solution of the set (4.2.1). Introducing the new variable £, by the relation $=^-
(0<^
(4.2.15)
Nonlinear Flexural Oscillations of a Circular Ring
145
and accounting that a2 = N(l - \) (from (4.1.30) we obtain the following set of two first order differential equations in % and y (from the first equation (4.2.1) and (4.2.3)):
^
dix
-^-=K0A^(l-^)sin2Y,
(4.2.16a)
= -
(4.2.16b)
co
+
IK07V(2^-1)(1-COS2Y).
2
Dividing the first equation (4.2.16) by the second, we obtain an equation in complete differentials which has the integral 5(l-!;)(l-cos2y)-2a*S=C,
(4.2.17)
where a* =
-
.
(4.2.18)
The integral curves (4.2.17) constitute an "amplitude-phase portrait" (APP) which topology depends on the single dimensionless parameter a *. Without loss of generality we may assume a * > 0, i.e. co 2 > co . Value a *=1 corresponds to the bifurcational value of energy (4.2.10). In Fig. 4.2, (a) ~{d) the APPs are presented in the rectangle 0 < t, < 1, 0
^Izf,
y .|.
(«.,„
146
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 4.2 Amplitude-phase portraits for nonstationary free oscillations of the ring for different values of parameter a*.
The stationary point moves to the right as a * decreases (the energy increases), £, increases from 0 and tends to 0.5 at N —> oo . It means that afj —> a2 at N —> oo. So the radial oscillation (4.2.13), corresponding to the stationary elliptic mode, asymptotically tends to a running wave.
Nonlinear Flexural Oscillations of a Circular Ring
147
It is seen directly from Fig. 4.2 that the coupled stationary point is a stable (elliptic) point of the surface C(£,y) (4.2.17), so this elliptic mode is always a stable one. Simultaneously with this point two stationary (uncoupled) points appear on the line % =0 which are given by expressions
(-\Y
ys=1——
arccos(l-2a*) + 57i , ^ =0, (5=1,2).
These points are hyperbolic points of the surface (4.2.17), and so they are unstable. Therefore the appearance of the coupled stationary point leads to the loss of stability of the uncoupled mode al = 0, a2 =£ 0 (but the first uncoupled mode a^ ^ 0, a2 = 0, with lower eigenfrequency, remains stable at any energy of oscillation). All these statements about stability, of course, could be proved by calculating hessian of (4.2.17), similar to Chapter 2. Fig. 4.2 shows that the appearance of the coupled stationary point has an essential effect upon nonstationary oscillations. The amplitudefrequency modulation is observed which becomes more pronounced as the energy of oscillation increases. The domain of attraction of the coupled stationary point, which is bounded by a separatrix (crossing the uncoupled stationary points), simultaneously enlarges. In the case of an ideal ring ( G * = 0 ) there are a stable coupled stationary point E, =0.5 and unstable "stationary lines" £, =0 and £, =1. The APP for this case is shown in Fig. 4.3. Both uncoupled modes of oscillation become unstable. For the coupled elliptic mode we have (3] = a2, and it becomes a pure running wave. As is seen from (4.2.13), this mode is a unique regime of oscillations for which no radial vibrations of the axial line occur. We may suppose that the elliptic mode is stable namely because it minimizes the axisymmetrical radial vibrations of the ring.
148
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 4.3 Amplitude-phase portrait for nonstationary free oscillations of the ideal ring.
Fig. 4.4 The range of existence of solution of Eq. (4.1.48).
4.2.3 Nonstationary oscillations. General solution In order to obtain time dependencies for the amplitudes and phases, let us return to Eqs. (4.2.16) and integral (4.2.17). Eliminating y from (4.2.16a) and (4.2.17), we obtain the equation, which determines the dependence of £, on the slow time T\ (NK T
°
\1T\
=( 2 C T *^ + c )[ 2 ^ 1 -^-( 2 a *^ + c )]- (4-2-2°)
Due to positiveness of the right-hand side of this equation, only segments of the lines y-2o *£, +C lying within the domain bounded by the parabola y = 2£>(\-£>) (Fig. 4.4) correspond to solutions. In the case of ideal ring (o * =0) the inclined lines become horizontal ones. Consider first the case a * =0. Equation (4.2.20) reduces to ^ ^ N K ^ G - W & S ) ,
(4.2.21)
149
Nonlinear Flexural Oscillations of a Circular Ring
where ^,, ^ 2 are roots of the trinomial in the square brackets of (4.2.20): S 2 = l - S , , C = 2£,(l-5,). This equation with initial condition ^ (0) = ^o n a s solution \ =0.5-(0.5-^)sin(D7; +oc0),
£> = ± 2 K 0 J V A ,
(4.2.22)
A = ^ , ( l - ^ ) , exo=arcsin2^"^~^2 . Substituting (4.2.22) into (4.2.16b) and taking account of integral (4.2.17), we obtain for the phase difference s ;
y = y o + arctg^ a, =arctg
'
(1
^
'
^-a,
,
" 2 ^ C ° S a V y 0 = y(0).
(4.2.23)
Then we find 6, and 9 2 from the last two Eqs. (4.2.1). For example, we have
e, -e10 -J^Lgft^'+^ff-Q- 2 *')-,/, a2=a,c,gt8(g'-/2>-(1-2^),el0=el(0). Calculating ax = ^N\ time / we obtain
(42.24)
, a2 = -JNQ. — %) and returning to real
/ 1 (/) = [A^(0.5 + (0.5-^1)sin(2KAA^?+a))]1/2cos(co/+91). (4.2.25) A similar expression can be also written for f2 (t). It is seen from (4.2.25) that the oscillations are amplitude-frequency modulated vibrations, which can be regarded as a superposition of slow modulation waves (traveling waves) and fast running waves. The ratio of amplitudes of the fast and slow components depends on the closeness of the integral curve to the central (stationary) point. It follows from (4.2.25) and expression for K (4.1.17) that the modulation period is equal to
150
Mechanics of Nonlinear Systems with Internal Resonances
~ 2o fe n e r V^ i Cl — ^ i)
min
~ / 2 n 4 Ay • (« - 1 ) Mo
(4-2-26)
The period depends on energy of the vibrations and the initial ratio of amplitudes ax and a2 • In the case of integral curves, which approach lines y = 0 , y = 7i , the period tends to infinity; when approaching the stationary point (£, -> 0.5, y ->n/2) it tends to the minimal value min •
Consider now the general case a * * 0 . As previously, let £,X, t,2 ^ e roots of the polynomial on the right-hand side of (4.2.20) lying in the interval (0, 1) (only two roots can fall within this interval). The third root ^ 3 lies to the right of this interval, if CT*<0, and to the left, if rj*>0, so that, when 0 < ^ < l , one has a *(£ - % 3 ) > 0 . Equation (4.2.10) reduces to the following (the initial value of t, is assumed to be equal to the maximal value %2 > f° r simplicity) ^
-1/2
/[<**(€-Si)(&2-$)(&-53)]
^ = ± V 2 > K 0 7 i • (4.2.27)
Assuming that a*>0 and, respectively, ^ 3 <0, by making the substitution §=^2-g2-^)sin2v|/,
(4.2.28)
we can reduce the integral to an elliptic integral of the first kind and write the solution in terms of Jacobi elliptic functions (4-2.29)
%=$I-<&2-Z>XW{Z>T\),
z = T[0.5MjK0(g2-^)fi7;,Ti= ^-j^\
.
.S2 ~S3 J
For the amplitudes of oscillations we have
a,=H 2 -g 2 -^)^ 2 (2,Tl))F 2 , a2={N-aW2. (4.2.30)
Nonlinear Flexural Oscillations of a Circular Ring
151
The modulation period is expressed in terms of complete elliptic integral of the first kind
T° = 2[NKCO
ACO (fc 2
-^)Yl2K{r\).
(4.2.31)
This period depends on initial conditions and, as the calculations show, is two or three orders of magnitude greater than the period of the characteristic oscillations, which agrees with experimental observations (Kubenko et al, 1984). 4.3. Forced Oscillations 4.3.1. Uncoupled forced oscillations Let us return to Eqs. (4.1.27), which describe forced oscillations of the ring. Consider at first forced oscillations only in the driven mode. Such uncoupled oscillations can appear only under specific initial conditions (zero initial values of generalized coordinate and velocity for the companion mode). Uncoupled oscillations are described by the first and third equation (4.1.27) with a2=0: -JrL=—z(%C0^\ +Mo'3i) ' JA
1
fl.-^—^sinB, dTx
(4.3.1a)
2
dTx
-(3,0,
-K0O,3).
(4.3.1b)
2
Hence ? o c o ^ i +M-o^i ^L = -a c#, ' <70sin9, - P , a , -Koa,3
(4 3 2)
At \io=O (conservative system) this equation has integral governing the amplitude-frequency modulation (at uncoupled vibration) tf0a1sinei--p,a12--HC0a14=C.
(4.3.3)
For stationary uncoupled oscillations (in the general case u * 0), putting in (4.3.1) ai=const, 9, =const, we have
152
Mechanics of Nonlinear Systems with Internal Resonances
(jocose, +|a o a, = 0 ,
(4.3.4a)
9 o sm0, - P , a , - K 0 O , 3 = 0 .
(4.3.4b)
Excluding 0/ from (4.3.4), we obtain the equation of frequency response for uncoupled vibration in the driven mode: li20af+^l+K0af)2af=q20.
(4.3.5)
Note that the maximum value of a{ for given q0, which equals to ax = q01 (J.o, coincides with the maximum in the linear oscillations, but it is reached at P, = -Koql / [i^, i.e., for relative frequencies difference 1 -co /Q= - p , /(2s) = -Kq20 /(2^i 2 ). If to introduce new parameters
x = ^ f ? , A>=A fi = # , ^0
^0
(4.3.6)
^0
(x is proportional to the energy of oscillation) then Eq. (4.3.5) is written in the simplest form
x + x(bl+x)2=Q,
(4.3.7)
where bx and Q are generalized dimensionless parameters of frequency and amplitude of the external force. Generalized frequency response curves calculated according this equation at given values of Q, and the oscillation energy parameter x as a function of Q at given values of b{ are presented in Figs. 4.5 and 4.6. Note that the nonlinear characteristic is soft; maxima on the generalized frequency response curves lie on the line bx + x = 0 and are equal to *max =Q4.3.2 Coupled stationary oscillations Consider now the coupled forced oscillations. Combining the third and fourth equations (4.1.27) we obtain equation for the phase difference 7=62-9,:
Nonlinear Flexural Oscillations of a Circular Ring
-^ r -i[p+K 0 (l-cos2y)( f l l 2 -« 2 2 )]=-^sine i , dTx
2
2a,
153
(4.3.8)
where
p = p , - p 2 =2a/Q.
Fig. 4.5 Generalized frequency response curves for uncoupled forced oscillations of ring.
Here we consider only the coupled stationary modes (CSM) ax = const, y = const. Then 9, and hence02 also are constants (see (4.1.28)). Equation (4.1.28), the second and fourth Eqs. (4.1.27) and (4.3.8) give the following set of four equations with respect to a{, a2, y, 0, : ^oO> 2 + a 2) = -9o a i c o s e i> (Xo+Koa,2sin2y = 0 , P 2 + K 0(a22 + a 2 cos2y) = 0,
p +KO(1 -cos2y)(a,2 - a\) A i n 9 , .
(4.3.9)
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Mechanics ofNonlinear Systems with Internal Resonances
Fig. 4.6 The energy parameter x as a function of the excitation force parameter at various frequencies of the excitation (uncoupled stationary forced oscillations of ring).
Hence for the phase differences between two modes y and between the external force and the first mode 6, we have: sin2y
=-Jio_ ; C O s 2 y = _P 1 ±J£ f i Koa,
Koa,
cos0i =
jOi±^) qoat
( 4 3 1Q)
It follows from the first relationship (4.3.10) that y —»±7t / 2 when ax —> oo. Therefore the CSM approaches the elliptic mode at increasing amplitude. Excluding y from the second and third equations (4.3.9) we obtain equation with respect to flj , dj '• «1 2 =— V^O +(P 2 +K0«22)2 •
(4-3.11)
Excluding 9, from the first and fourth equations (4.3.9) (with account of (4.3.11)), we get another equation with respect to a\ ,#2 :
H o V +a22f +{pa,2 +[co(«12 + a 2 ) + p2](a12 -a 2 2 )f =q\a\. (4.3.12)
Nonlinear Flexural Oscillations of a Circular Ring
155
It follows from the above expressions that the CSMs do not exist at sufficiently small amplitudes. The first relationship (4.3.10) gives the necessary condition of existence of these modes: a, 2 >-^-.
(4.3.13)
So at small energies there exists only the uncoupled driven mode (cos n cp ). At certain amplitude a, a bifurcational (branching) point appears on the frequency response curve (4.3.5) for this uncoupled mode, and a CSM path originates from this point. Note that the CSM path corresponds to a pair of steady-state modes with phase differences y and y +7i . The branching point can be easily found from the set (4.3.9) assuming a2 =0. Denoting with star all quantities related to the branching point, we obtain:
al=—Vno+P22>sin2y.=K o
p2 ; !*° • cos2y.= (4.3.14) V^+P2 VM0+P2
Depending on sign of p 2 (i-e-> o n ^S11 of difference between the natural frequency of the companion mode and the external force), the angle 2y „. lies either in the third or fourth quadrant. From (4.3.12) we obtain the amplitude of excitation for this point (accounting that
P = (3,-P2):
& =—VMo2 +P2 Mo2 +(VMo2 +P2 +Pi) 2 ] K
o
(4-3.15)
(it conforms with (4.3.5)). The frequency response curves for the coupled stationary modes are given by Eqs. (4.3.11), (4.3.12) with 2=const and P 2 as a frequency parameter. The latter equation can be written as follows:
H 0 V +al)2 + \io(a? -4) + p,«,2 - p2a22]2 = qla] From (4.3.11) we have:
(4.3.16)
156
Mechanics of Nonlinear Systems with Internal Resonances
„,«-„;. <£t§L + 2 M. K K
o Then Eq. (4.3.16) takes the form: 2
a
2
2 2
H 0 (a, +«2 ) + - ^K
o
~1^
n 2
±
o
2
2
^ + (P1a, + P2«2 )
J
=?o«, 2 - (4-3.17)
Let us introduce variables (4.3.6) and two new variables y = -s-L, ^o
b2=^-. Ho
(4.3.18)
Set of equations (4.3.11), (4.3.17) in the new variables has the form:
(x + yf + [l + b2 + (bxx + b2y)f = Qx,
(4.3.19)
x2 -(y + b2)2 = 1 . (4.3.20) Equation (4.3.20) determines a hyperbole; equation (4.3.19) determines a second order curve with invariants (A. Corn, T.M. Corn, 1968):
/ = 2 + 6,2+Z>22, D = b2 ,A = Q{l + b2Ab-Q- J; (b = bl -b2). (4.3.21) In the general case D>0 equation (4.3.19) determines a real ellipsis, if A<0 (i.e., if Q > 4b) and an imaginary ellipsis, if A>0 {Q <4b). In the case of exact internal resonance (the conjugate modes have equal natural frequencies, 3=0) we have D=0, and equation (4.3.19) determines a parabola. As x and y may be only nonnegative, only parts of curves (4.3.19), (4.3.20), lying in the first quadrant, are of interest. Note that formulas (4.3.14), (4.3.15) for the branching point in the new variables take the form:
x,=Jl+bJ
, s i n 2 y . = - - l , cos2y,= — j J L = , (4.3.22) ^\ + b22 -i\ + b22 Q, =^U^\
+ (^bJ
+ b,)2\.
(4.3.23)
Nonlinear Flexural Oscillations of a Circular Ring
157
So the CSMs can appear only if the amplitudes of external excitation satisfy the condition
Q, >max(4b,
fi+bf
[l + (-N/l+*22 + * i ) 2 ] ) -
(4.3.24)
Set of Eqs. (4.3.19), (4.3.20) can be reduced to a quadratic equation. From (4.3.20) we have y = -62±Vx2-l.
(4.3.25)
The real solution exists only for x > 1 (this condition coincides with (4.3.13)). Substitution of (4.3.25) to (4.3.19) after some algebra gives the equation Ax2 +2Bx +C = 0,
(4.3.26)
where A = (bl-b2?[(bl+b2)2+4], 2
2
B = (2b{-2b2-Q)(b2x+bl 2
+ 2), (4.3.27)
2
C = 4{b + b 2 + b b] +1) + Q - 4Q(bt -b2). 4.3.2.1 Exact internal resonance In the case of coincident natural frequencies for the conjugate modes C0 2 =<» (perfect ring) we have a=0, (3i=p2 (see (4.1.15), (4.1.17)), and bx=b2 (see (4.3.6), (4.3.18)). Then A = 0, B = -2Q(bf +1), C = 4(6,2 +1) 2 + Q2 . So Eq. (4.3.26) becomes a linear one and has single root (4.3.28)
Corresponding value of y equals to , if y = -b] + -\
Q f
2(Z>,2+1)) ^ -\.
(4.3.29)
Here we drop the modulus sign and take only the upper sign "+" in (4.3.25). An elementary analysis shows that the only root of set (4.3.19), (4.3.20) exists in the first quadrant in the case b\=b2, for given Q and bh and this root is given by (4.3.28), (4.3.29) (even if the expression in
158
Mechanics of Nonlinear Systems with Internal Resonances
brackets (4.3.29) is negative). It is interesting to note that the point (4.3.28), (4.3.29) is a point of maximum of parabola (4.3.19), as can be easily proved. In Fig. 4.7, (a),(6), location of curves (4.3.20) and (4.3.19) and their intersection at several values of Q is shown for positive (a) and negative (b) values of b\. At positive b\ roots of set (4.3.20) and (4.3.19) in the first quadrant appear for much larger values of Q than at negative ones, due to the soft characteristic of the ring at uncoupled oscillation.
Fig. 4.7 Intersection of curves (4.3.20) (curve 1) and (4.3.19) (curves 2) for various values of Q in the cases (a)fe1=fc2=land (b) bi=b2=-l.
The value of Q, for the branching point (4.3.23) in the case b\=b2 is equal to
e»=2(l+6 1 2 )[( A /l^f + 6 1 )j-
(4.3.30)
Plot of this function is presented in Fig. 4.8. The minimum value of Q,is equal to Qfn = 1.5396(for bx= -0.57735). The asymmetry in b\ shows that at positive b\ (the excitation frequency is greater than the eigenfrequency) branching off the uncoupled mode occurs at much larger values of the excitation force than at negative bt. In the case b\=0 (exact external resonance which coincide with the exact internal resonance b=0) expressions (4.3.28), (4.3.29) give
x.liS. + l\
2(2
y.L(2.-l).
Qj '
2(2
Q)
(4.3.3!)
Nonlinear Flexural Oscillations of a Circular Ring
159
Fig. 4.8 Values of Q, for the branching points in the case of exact internal resonance.
In this case the CSM path originates from the point X = 1, y - 0 and exists when Q>2. This path is shown in Fig. 4.9, along with the uncoupled modes path. In Fig. 4.10, (a), the spatial CSM path in the space (x, y, Q) is shown for this case. As Q increases, the values of x and y increase and gradually converge; so the CSMs approach the elliptic mode corresponding to a running wave.
Fig. 4.9 Stationary modes paths at exact internal and external resonances.
160
Mechanics ofNonlinear Systems with Internal Resonances
Fig. 4.10. Stationary modes paths at exact internal resonance; (a) exact and (b) inexact external resonances.
In Fig. 4.10, (b), the stationary modes paths are constructed for the inexact external resonance (the case of negative bh i.e., the excitation frequency is less than the eigenfrequencies). No principal differences are observed here in comparison with the case Z>,=0, as well as in the case of positive bj, except of displacement of the branching point to larger values of x and Q. The generalized frequency response curves for the system under exact internal resonance condition are presented, for a given value of Q, in Fig. 4.11. The CSM curve originates and ends off the uncoupled mode curve; it exists in a certain range of frequency parameter b\. Note that the bounds of interval (b*, b"), where the CSMs exist, can be easily obtained for any Q from expression (4.3.30) or Fig. 4.8. We see that always b* < 0, b" > 0, and the interval becomes wider, as Q increases. This interval can be approximately determined using asymptotics of (4.3.30) at large (by modulus) negative and positive b\.
b'x*-Q, b?*tfQT4.
Nonlinear Flexural Oscillations of a Circular Ring
161
Fig. 4.11. Frequency response curves for the case of exact internal resonance; Q-3.5.
In Fig. 4.12 the generalized frequency response curves for larger value of Q (£>=50) are presented. We would like to note two interesting peculiarities. First, at large Q the CSM curve branches exactly off the peak of the uncoupled modes curve. It is due to the identical asymptotics of point b*(b' « -Q) and of the peak point (x=Q, b\= -Q, see above). Second, a sharp peak appears on the CSM curve at b\=0 (the exact external resonance) with practically equal values x and y (i.e., equal energies of oscillations for two linear modes). Thus appearance of the CSMs results in a new resonance peak for elliptic modes (running waves) exactly at linear eigenfrequncy, instead of the nonlinear frequency dependent resonance (which becomes unstable in whole interval ( b ' , b")).
162
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 4.12 Frequency response curves for the case of exact internal resonance; Q=50.
4.3.2.2 Inexact internal resonance There has been carried out the numerical analysis for the general case of inexact internal resonance (CO2 ^ CO ). Some results of the analysis are presented in Figs. 4.13-4.16. In Fig. 4.13 intersection of curves (4.3.20) (the right hand branch of hyperbole) and (4.3.19) (portion of an ellipsis in the first quadrant) is shown for cases (a) b= - 1 , b\= -1 (the eigenfrequency of the driven mode is greater than that of the companion mode, and the latter coincides with the excitation frequency), and (b) b= 1, b\= 0 (the eigenfrequency of the driven mode is lower than that of the companion mode, and coincides with the excitation frequency), for three levels of the excitation force. In all cases the only root of set (4.3.19), (4.3.20) exists; in distinction from the exact internal resonance, this root does not coincide with the point of maximum on the curve (4.3.19).
Nonlinear Flexuml Oscillations of a Circular Ring
163
Fig. 4.13 Intersection of curves (4.3.20) (curve 1) and (4.3.19) (curves 2) for various values of Q in the cases (a) bi=-l, b2=0 and (b) bx=0, b2=-l-
In Fig. 4.14, (a)-(d) spatial stationary modes paths are shown for various combinations of eigenfrequencies and the excitation frequency (plots (a) ,{b)' (^0 relate to cases when the eigenfrequency of the driven mode is greater than that of the companion mode). In all cases the only CSM path exists. Bifurcation to CSMs occur at lower excitation force level when the eigenfrequency of the companion mode is lower then that of the driven mode, and coincides with the excitation frequency (Fig. 4.14, (a)). Frequency response curves, presented for Q=10, b=-\ and b=\ in Figs. 4.15, 4.16, are similar to those for the case of exact internal resonance (see Fig. 4.11), but the range where the CSMs exist becomes narrower (especially for positive b) and vanishes at sufficiently large values of b. Peaks on the frequency response curves are removed from the value b\=0 to values b\==b (when the excitation frequency coincides with the eigenfrequency of the companion mode).
164
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 4.14 Stationary modes paths at inexact internal resonance; (a)b=-l,b\=-l, b2=0; {b)b=-l,bi=O,b2=l; (c)b=\,bx=Q,b2=-\\ (d) b=-l, *>,= - 3 , b2=-2.
Nonlinear Flexural Oscillations of a Circular Ring
Fig. 4.15 Fequency response curves for the case of inexact internal resonance; b=-l; Q=10.
Fig. 4.16 Frequency response curves for the case of inexact internal resonance; fc=l;Q=10.
165
166
Mechanics ofNonlinear Systems with Internal Resonances
4.4 Concluding Remarks The detailed analytical description of nonlinear free and forced oscillations in circular rings is presented based on the multiple scale method. A mathematical model taking into account the interaction of conjugate modes, "splitting" of their eigenfrequencies and participation of an axisymmetrical component of oscillations, reduces the ring to a cubic symmetric 2DOF system with "nonlinear inertia". Main results concerning free oscillations of rings. 1. When the energy of oscillations exceeds a certain threshold value, which depends on the detuning parameter (the difference of natural frequencies of two conjugate modes), the bifurcation of steadystate modes occurs, and a pair of coupled stationary modes appears, with energy exchange between two conjugate modes. These modes are elliptic modes — a combination of two conjugate modes with phase difference ±7i/2, and they differ only with the sign of phase difference. 2. The elliptic modes path branches off the uncoupled mode path fl] = 0, a2 & 0, i.e., off the mode with higher natural frequency, and after bifurcation these uncoupled oscillations become unstable (but the first uncoupled mode ax & 0, a2 = 0, with lower eigenfrequency, remains stable at any energy of oscillation). 3. The elliptic mode consists of a standing wave (with amplitude proportional to a2 - ax) and a running wave (with amplitude proportional to flj). As the energy increases the amplitude of the standing mode diminishes, the amplitude of the running mode increases, so in the limit (energy N —> oo) only the running wave remains. 4. Frequency of the coupled stationary oscillation (elliptic mode) does not depend on its amplitude and is equal to the average of the linear natural frequencies for both conjugate modes. 5. The elliptic mode is always stable. 6. This mode is a regime of oscillations, which minimizes the axisymmetrical radial vibrations of the ring (no radial vibrations of the axial line appear in the limit N —> co).
Nonlinear Flexural Oscillations of a Circular Ring
167
7. Nonstationary oscillations of the ring are amplitude-frequency modulated vibrations, which can be regarded as a superposition of standing waves, fast running waves and slow modulation waves (traveling waves). The amplitude-frequency modulation becomes more pronounced when the energy of oscillation increases. 8. The ratio of amplitudes of the fast and slow components depends on the closeness of the integral curve to the stationary point. The modulation period depends on the energy and the initial ratio of amplitudes ai and a2. In the case of integral curves, close to the stationary point, the modulation period equals approximately to the minimum value 7^in (4.2.26); at removal of the integral curve from this point the modulation period tends to infinity. Main results concerning forced oscillations of rings 1. At a certain amplitude of the excitation force a branching point appears on the frequency response curve for the uncoupled (driven) mode, and a coupled steady-state modes path originates from this point. 2. There exist the only CSMs path, which corresponds to a pair of steady-state modes with phases differing by JI. 3. The coupled steady-state modes approach the elliptic mode (a running wave) at increasing amplitude. 4. The CSMs can exist at amplitudes of external excitation satisfying condition (4.3.24). 5. The minimum value of bifurcational force parameter Q, is equal to Q?m = 1.5396 and is reached at 6,=M).57735. In the case of positive b\ (the excitation frequency is greater than the eigenfrequency) branching off the uncoupled mode occurs at much larger values of the excitation force than in the case of negative b\. 6. In case of exact internal resonance (a perfect ring) the CSM path originates from the point x -1, y = 0 and exists when Q > 2 (see Eqs. (4.3.6), (4.3.18)). 7. Frequency response curves for the coupled modes are derived in a closed form. They depend upon two detuning parameters (the difference of natural frequencies of two conjugate modes and the difference of the frequency of the external force and the natural frequency of the driven mode).
168
Mechanics of Nonlinear Systems with Internal Resonances
8. The frequency interval (b', Z>"), where the CSMs exist, depends on the excitation force parameter Q. In the case of exact internal resonance this interval is determined by relationships b' «—Q, b" ~yQ/4 (approximately, at sufficiently large Q). 9. Appearance of the CSMs results in an additional resonance peak for coupled modes in frequency response curves, with practically equal energies of oscillations in two linear modes. This resonance peak does not depend on the excitation frequency and coincides with the eigenfrequency of the companion mode. 10. In the general case of inexact internal resonance the range of existence of the couple modes ( b ' , b") diminishes when the first detuning parameter increases. The intensity of the coupled mode is larger when the driven mode has higher natural frequency than the companion mode. This investigation is restricted by analysis of the interaction of two conjugate modes. More complicated models can be studied in similar manner when clear understanding is reached for simple basic models. The obtained results can be extended and applied to analysis of a finite-length cylindrical shell. Historical remarks. Theoretical models taking into account the nonlinear interaction of conjugate modes in circular rings and cylindrical shells were proposed in the 1960s-70s (Evensen and Fulton, 1967), (Chen and Babcock, 1975) and have been developed later on (Kubenko et al, 1984), (Raouf and Nayfeh, 1990) and others. An analytical investigation of nonlinear oscillations in circular rings with account of interaction of conjugate modes and splitting their natural frequencies has been given for free oscillations in (Manevich A., 1994), for forced oscillations in (Manevich A., 2002).
Chapter 5
Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators
In this chapter we present an asymptotic approach to analysis of coupled nonlinear oscillators with asymmetric nonlinearity. We begin from 2DOF system and show that asymmetry leads to renormalization of nonlinear terms in final equations corresponding to the main 1:1 resonance approximation. Then we study localized normal modes in a chain of coupled asymmetric nonlinear oscillators. Their existence is a consequence of internal 1:1 resonance, which can be manifested in the cases of weak or effectively weak coupling between oscillators (in latter case weakness of coupling is provided by long wavelength of the considered excitations). The main goal of this chapter is to demonstrate the possibility of crucial simplification of the nonlinear problem by means of the complex representation of equations of motion, which allows efficient study of the energy localization phenomena. 5.1 A System of Two Weakly Coupled Nonlinear Oscillators Let us consider a system of two identical weakly coupled nonlinear oscillators with an asymmetric cubic potential (Fig. 5.1). Their dynamics is described by the equations w ^ ^ + 2 « ^ + c 1 £/ 1 +c 2 [/ 1 2 +c 3 L/ I 3 +c 12 (£/ 1 -£/ 2 ) = 0, at dt m^± dt
(5.1.1a)
+ 2 n ^ + CiU2 +c2U22 + c,U23 + ci2(U2 - l / , ) = 0, (5.1.1b) dt
169
170
Mechanics ofNonlinear Systems with Internal Resonances
where Uj (j = 1,2) are displacements of the oscillators. Note that, in distinction from Chapter 2, equations (5.1.1) include a linear link between Uj (j = 1,2).
Fig. 5.1 Schematic representation of a chain of coupled nonlinear oscillators.
We rewrite these equations, using the complex representation, in the form -^•-R|/y+8\(|/y+v|//)-a1s (j//-y/)2+
^
(5.1.3)
+ ia2e2{yj -v)/// -/pe 2 ||/,. - \ | / / ) - | | / t -v|/4')] = 0, where
\ j / y = v y . + zwy.,
Uf = Ujr0;
V(/y. -Vj-iUj,
Vj-dUjJdx,
-t=G>ot,
r0 is the distance between particles in the undisturbed state,
coo = yjcjm
, s « 1 ; A=3-y, y = 1,2;
8 2 y = - ^ = , 4 a , s = - ^ , 8 a 2 8 2 = - ^ , 2 e 2 p = ^ . (5.1.3) Vc,w
c
c
i
i
c
i
Introducing new variables cpy- =v|/ y - e - ft
(5.1.4)
and "slow" times x, = E T 0 , X 2 = e 2 x 0 ( x 0 = x ) and presenting solution of (5.1.2) in the form of power expansion (p y .(T 0 ,T,,T 2 ,...)=(P 7 . o +S(p ; . 1 +8 2 (p 7 . 2 +...,
(5.1.5)
one obtains after substitution of the above expressions into Eq. (5.1.2):
171
Localized Normal Modes in a Chain of Nonlinear Oscillators
T — (q>;, o + £ 9 y , t + £ 2 9 A 2 +•••)+ s — (7C0
g2
(
CT,
T — (
+ e
^(
- i s 2 p ^q> y ,o -
(5.1-6)
where MPy,o.9jIo»
— — = 0. rh
Therefore
tne
principal approximation the functions 9 ;Q depend on "slow" times only;
2) 8,:
OX0
OX[
^_^+a^ylt_2e-^j+e-»vj).
The condition of the absence of secular terms leads to equation - g ^ - = ° . s 0 9y,o=
(5.1.7)
Integrating equations for cp .• 1 with respect to variable T 0 and taking into account Eq. (5.1.7), we obtain
172
Mechanics of Nonlinear Systems with Internal Resonances
j.i= - « l ' U % ; . o + 2*""1 \pj,o( -\e~r"-
+
(7C2
or,
^k°?;,
-ia 2 U 2 ' T °cpj )O -3|(p y -o| cp ;>0 +3e- 2 ' T |(p ;j0 |
+ 'p[(py,0 -
(5.1.9)
Conditions of the absence of secular terms (j, k=\,2, k=7>-j) 5cp ,• 0 | 2 / \ -^-+y
(5.1.10)
where ,
20
a = 3 a 2 - — af 2 , provide a system of equations of motion in the principal approximation. An equation which is close to (5.1.10) (at y = 0) was considered in (Kosevich, Kovalyov, 1989) as an example of a phenomenological model ("a discrete model with self-localization"). It has been shown that this system is integrable. Besides, at some energy of oscillations the localized normal modes arise in addition to in-phase and out-of-phase normal modes of the linearized system. Their appearance is a consequence of instability of these uncoupled modes (dependent on the sign of parameter a ). Let us consider the influence of damping forces in more details. Introducing in (5.1.10) new variables O . 0 by the expressions q>y,0=e ( / P " 7 ) T 2 O ; . 0 (/-=l,2),
(5.1.11)
we obtain — M - + /p
(5.1.12a)
Localized Normal Modes in a Chain of Nonlinear Oscillators
^ - + /pO I 0 -/ae~ i y T 2 O 2 0 °20
=0
-
173
(5.1.12b)
ar2 ' We note that the equations of motion (5.1.12) at y = 0 can be written in the hamiltonian form
9 f % = j ^ _ ao*i0 = dH dz2 ao*. 0 ' dt2 aoy0' where Hamiltonian is 2
4
(5.1.13) Z
7=1
and the integral of energy in the given approximation is /7=const. Exact integrability of system (5.1.12) at y = 0 follows from the existence of this energy integral and the following integral 2
2
£ K o =N(x3,x4,...).
(5.1.14)
7=1
Returning to the full system (5.1.12) (y & 0) we conclude that the relation (5.1.14) remains valid in this case also. Indeed, multiplying the first and second equations (5.1.12) respectively by Oj g and O 2 Q , making the operation of conjugation and combining all four equations we obtain ax2
;=1
whence the integral (5.1.14) yields. Therefore, as well as in the case of zero damping, the unknown functions 0 , 0 and O 2 0 can be presented as follows: 0liO = V N C O S ^ ^ C * ^ ) ; O 2 0 = V N s i n ^ ^ , ' 5 ^ . ( 5 . 1 . 1 5 ) Substituting these expressions in (5.1.12) we obtain
-ii^iiL-^L-p r g ^ ^ ' + a A ^ W ^ O , (5.1.16a)
174
Mechanics of Nonlinear Systems with Internal Resonances
a Ne-2^ sin 2 - = 0. 2 (5.1.16b) Equating the real and imaginary parts of equations (5.1.16) to zero we come to the following equations for real functions 6 (T 2 ) and \ictg\^--^-^ctg^^ 2 dz2 oz2 2 2
+
A(T2) = S,(T2)-S2(T2):
= 2PsinA,
(5.1.17a)
dx2 9T 2
= 2p ctg$ cosA+a 7Vef2yT!cos0 .
(5.1.17b)
Stationary points, in which the right hand sides of Eqs. (5.1.17) are equal to zero, are: 1) A = 0,6 = n/2; 2) A = n,0 =n/2. They correspond to the in-phase and out-of-phase cooperative normal modes, which survive in the presence of energy dissipation also. In the case y = 0 additional out-of-phase ( a > 0) or in-phase ( a < 0) normal modes appear. They correspond to values A =TI ( a > 0 ) or A = 0 ( a < 0 ) , sinB =
—cosA, (5.1.18) OLN
and appear when the energy of oscillations exceeds the bifurcational value determined by condition 2B N>Nb = -^- . (5.1.19) a In the case y = 0 set of equations (5.2.17) has the following integral (5.1.20) sinB (2a * cos A + sinB ) = C, where a'=^-. (5.1.21) aN Integral (5.1.20) governs the amplitude-frequency modulation at non-stationary oscillations. It depends on the single dimensionless
Localized Normal Modes in a Chain of Nonlinear Oscillators
175
parameterCT*, and the bifurcational value of N (5.1.19) corresponds to value a I = ±1. In Fig. 5.2 integral curves (5.1.20) are presented (an "amplitudephase portrait") for several positive values of a *. At a * =2 and cr * =1 (Fig. 5.2, (a),(b)) two stable stationary points exist corresponding to the in-phase and out-of-phase cooperative normal modes (0 -n/2, O 2 0 = ± ( l ) i 0 ' s e e (5-1-15)); when a*
^ ""2,0
ViWi-s'V 1
Vi-Vi ^
1
(5.1.22)
176
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 5.2 Amplitude-phase portraits for different positive values of (7 .
Localized Normal Modes in a Chain of Nonlinear Oscillators
177
Fig. 5.3 Comparison of amplitude-phase portraits for positive and negative values of (7 .
At a * ->0 (that, naturally, falls outside the range of applicability of the principal approximation) the full localization of excitation upon a single oscillator would be reached. The consideration of the dissipation reveals once more important aspect of the problem. When N • e"2yT 2 > 2 P / a the energy of the system can also be originally localized on a particle. But because of presence of an exponential coefficient in the second equation of motion, the localization of excitation becomes impossible in the instant x2 = ln(l/CT*)/2y , and the energy should be redistributed between the both particles. This important effect was discussed in the Chapter 2 for symmetric systems (in certain instances the domination of one mass is replaced by a redistribution of the energy between masses). Direct computation of localized normal modes becomes difficult if the number of degrees of freedom is more than two. Therefore generalization for more complicated cases requires a more universal approach. The existence of localized modes can be associated with bifurcations and instability of cooperative modes. For the treated system the bifurcation analysis becomes simpler with the use of the Eqs. (5.1.15), i.e., in terms of real functions G ( T 2 ) , 8 ( T 2 ) . However, the appropriate extension on the systems with many degrees of freedom also
178
Mechanics of Nonlinear Systems with Internal Resonances
becomes impossible. In this connection let us perform a bifurcation analysis using the complex Eqs. (5.2.12) at y = 0 . d
(5.1.23a)
2
$20 =°-
(5.1.23b)
Let us focus on the analysis of the stability of out-of phase collective mode ^ 2fi = — O l0. Denoting w, and w2 perturbations of this mode, substituting O, 0 + w, and - ,
0
+ w2 into the equations of
motion and taking into account relations (5.1.23), we obtain
-MTOH +^i,of>M +2^,oh|2 + ^ * o +h|2>M)=^
-~+i^
•zj^+®*\ -«y$o4+ii%fwi
-2^oH 2 - ^ o +hrM2)=°. (5.1.24)
where 0
1 0
f/y~ ifp+a — IT, 2} = I—e ^ '.
By combining these equations and introducing the sum W{ and the difference W2 of w, and w2, respectively, we obtain the set of two nonlinear equations with respect to the new variables. Linearization of this system leads to two independent equations of parametrical oscillations for functions W\ and W2, describing perturbation of the initial mode with respect to in-phase and out-of-phase modes, accordingly, 2/ P+a— x, W*
r)W
£!LL 5X2
r)W
+ i(p-aN)Wl-iaNe
?ZL-i(V+aN)W2-iaNe dx2
"
^
^ - = 0, 2
2/ P+a— k , W [
l]
- ^ - = 0. 2
(5 1 25^
179
Localized Normal Modes in a Chain of Nonlinear Oscillators
Exact solution of Eqs. (5.1.17) gives the same value of critical amplitude. But as it is shown below, the applied approach allows finding the conditions of energy localization not only in finite-dimensional systems of the high dimensionality, but also in infinite-dimensional models. At value N = Nb =|2p /<x| the conditions of main parametrical resonance are satisfied and the periodic solution W\ = Ae'm2 exists. Here A is a real constant and co = P +aN. This solution corresponds to the boundary between the stability and instability regions for the first Eq. (5.1.25). On this boundary and for N > Nb the intensive transfer of energy to in-phase mode becomes possible and localized normal modes exist. In the second Eq. (5.1.25), as it is easy to verify, the instability occurs at infinitesimal amplitudes, but it is realized as a phase shift of out-of-phase oscillations. The exact solution of the Eqs. (5.1.23) gives the same value of critical amplitude. But as it is shown below, the applied approach allows finding the conditions of the energy localization not only in finitedimensional systems of the high dimensionality, but also in infinitedimensional models. 5.2 Nonlinear Dynamics of Infinite Chains of Coupled Oscillators Equations of motion for an infinite system of coupled nonlinear oscillators with damping have the form (- oo < j < co) r^+lf^+cfJj
+cfJj2 +c3U/ +c(2Uj -UH -t/y+I) = 0. (5.2.1)
We introduce the change of variables T=,P-f;
«,= — ,
(5-2.2)
where r 0 is the distance between particles. Then we have d Uj
j
ditj
—^-+2s7—J-+Uj+4a\SUj 6/L
C/X
2
•>•%•>
+8ot2s Uj
I
\
+2zIfi[2uj-Uj_}-uJ+l)=0,
180
Mechanics of Nonlinear Systems with Internal Resonances
where
4a,e=^--r0, 8as2=-^--r2, ye2 = - £ = , 2s2p = - , 8 « 1 . Let us now use the complex representation of the equations of motion
—£—Wj +s2yfry- +\|/;)-/a,E^ y -y'f
+id262(|/y -vj/j)3 -
^
where v|/ y = v j +iuj-,\y .• = v .- - /My. For new variables v|/y- = e ; > y - , we come to the set of equations
^+s2y((P;+e-2
rt
9/)-/a1s^°(P2-2e-/T°^.|2+e-3
+ /a 2 8 2 (e 2 >/ -3^.| 2 (p y +3e-2h\?j\\*
-e"
^(pj2^
>*V
- / s 2 p (2cpy -
(5.2.4)
We introduce now the "slow" times T | = s x 0 , x 2 = 8 To> x 3 = s x0,..., where T 0 = T , and considercpy as functions of the "fast" time T o and the "slow" times xuz2,---By presenting the unknown variables as expansions on the parameter e , j,o+sbj,i+*2b <7CQ
y,2+-)+ e -Tr(
(5-2.5)
Localized Normal Modes in a Chain of Nonlinear Oscillators 181 +S2((p y j O +S(p y l +S 2 (p 7 ) 2 + ") + £M(9y,O' C P*,O' ( Py,l>9*,i->S ) OX 2 - / 8 2 p { 2 ( ( p y > 0 +SCpy>, + . . . ) - ((py_10 +E(p y _ (;1 + . . . ) -fc>y+i,o +£
-(PJ-I,O
+e<
P;+i.i +-)]-[ 2 ( { P*,0 +e(P,*,i +•••)-
+'
(5-2-6)
where expression for Mwas presented above. Equating to zero the coefficients at various degrees of the small parameter s we have: 0
^ P J; 0
1) s :
' - 0 , therefore (p , 0=cp • 0(x ,,x 2 , . . . ) ,
2)s':
^ L = - ^ L + a 1 ( . S j i O - 2 e ^ | q , ^ | 2 + a - % V ) . (5.2.7) The condition of the absence of secular terms leads to the equation 3
(5.2.8)
Taking into account (5.2.8) and integrating (5.2.7) with respect to variable T O we obtain
cp J>x= - i a j e""
(5.2.9)
Let us select now the terms corresponding to second order of the small parameter.
^
-
-
^
-
^
-
^
•
•
^
*
)
+
*
'
^
"
-
182
Mechanics of Nonlinear Systems with Internal Resonances
+ 2a,i[e /t °9;, 0
+ »P [(2(Py,o -
+
2 I / \ Y Py,o-' | Py,o tPy.o - z 'Pl 2( P;,o -^y+i.o -
a(
(5-2.11)
where ,
20
a=3a2-Ya!2 . We introduce now a new change of the variable 9y>0=e-TTlO7.>0.
(5.2.12)
Then the set of equations in the principal approximation takes the form
^ - a
ie~2^ |
Let us seek the functions <E>,
0
in the form
*y,O=/o(*2)e'W+9o),
(5-2-14)
where 9 0 is the phase corresponding toy = 0, fo(j2) is a real function, f0(0)>0 (here we seek only a partial solution corresponding to a certain distribution of initial displacements). Substituting (5.2.14) in (5.2.13), one obtains
^ - - / / 0 f a e - 2 ^ | / 0 2 + 4psin 2 ^Vo. 5x 2
V
2J
(5.2.15)
Since expression in the brackets is a real quantity, vector dfo/dz2 is orthogonal to f0. Solution of (5.2.15) has the form
183
Localized Normal Modes in a Chain of Nonlinear Oscillators
/0=V7Vrexp / 4psin 2 —T 2 -a—(e" 2 y T 2 -l) , (5.2.16)
2
LI
so that
[
kr
2y
J_
V
M
4(3sin2—-T2-CX—(e"2yi2 - l ) + *r0y+80
2
2Y
J_
,
(5.2.17) where -JN is the modulus of the complex amplitude. Returning to the initial complex variables we obtain ijf. 0 =VNexp i x+4s2(3rsin2 — -ys 2 T-a—(e' 2 "^ -l)+£r 0 y+9 0 .
LI
2
2y
J_
(5.2.18) Thus, the analysis of the principal approximation in slow time leads to the conclusion that in an infinite chain of weakly coupled damped nonlinear oscillators the quasi-harmonic waves with exponentially decreasing amplitude and variable frequency can spread. At y = 0 and a = 0 they turn out to be usual harmonic waves
Vy,o = VNexp[/(coT +kroj+B „)],
(5.2.19)
where co = 1 + 4e (3 sin (kr0 IT), with a spectrum of wave numbers 0 < kr Q
184
Mechanics of Nonlinear Systems with Internal Resonances
with r0 (interparticle distance). Further we accept r0 as a unit of length, then this condition will be noted as k~l. In the case k « 1 there is one more small parameter in the problem. Its appearance requires a revision of the overall procedure of the asymptotic expansion. In the case of strong coupling \c/cx — O{\)) the complex representation becomes justified (i.e., the rotations in opposite directions are separated in the principal approximation) if the coupling between oscillators is "effectively weak" because of relative smallness of the second differences in the equations of motion (5.2.3). It means that the field of applicability of complex representation in that case coincides with the range of applicability of the continuum approximation. The equations of motion take in this case the form d2U:
, du.
,
•> \
i
—j- +2s7— J -+Uj +4zuj +&L£iuf +2p(2w. -«•_, dx dx
\
-M/+1)=0,
(5.2.20)
where P = c/cj = 0(1) . Introducing the continuum approximation — function of two variables u (x,i) (so that u • (T ) = u (* ,x ) , x• = j r0) and a dimensionless coordinate c, —sx/ro(i.e., measuring the distances between particles in units of E ~ rg), we obtain for the second differences in (5.3.21) the expansion -(2Hy-lly.1-Il7+1) =e ^
+
- ^
T
- . . . .
(5-2.21)
Thus, in spite of the fact that P « 0(1), the derivatives terms contain now a small parameter and the complex representation turns out to be justified when using the continuum approximation. As a result the equations in the principal approximation take the form (if y & 0) l)/t«l
f£^_p 5x2
310^^.2^ |
f
=Q
(5222)
8C,
2)n-k«l
i^-^+
p £ l ^ o _ + a e - 2 T ^ | a o | 2 a o = 0. (5.2.23)
Localized Normal Modes in a Chain of Nonlinear Oscillators
185
Relations (5.2.22) and (5.2.23) at y = 0 become the nonlinear Schrodinger equation (NSE), which is an integrable system (Zaharov et al, 1980). Its solution in a wide class of initial conditions can be obtained by the inverse scattering method. It is known (Whitham, 1974), (Zaharov et al, 1980), that if a < 0 ("soft" nonlinearity) the periodic wave packets described by Eq. (5.2.23) are unstable. As a result, the localized solitonlike waves ("envelope solitons ") exist:
Ofc ,T 2)= / H E ^ - ' J s e c h [[Sk -vt 2)], (5.2.24) V a
where
(amplitude and velocity of the soliton are independent parameters here). If a > 0 the wave packets are stable and solitonic solutions are absent. Thus if the coupling between oscillators is not weak, the conditions of localized excitations formation are well known and are formulated in terms of the continuum approximation. It is of interest the case of the weak coupling, when it is necessary to use a discrete description. As well as in two-degrees-of-freedom systems, in the infinite chain of nonlinear oscillators, apart from the cooperative modes, the waves with some space localization can be realized. They result from the instability of the cooperative modes. Considering small perturbations of cooperative nonlinear normal modes, we substitute in Eqs. (5.2.13) expressions O , j 0 + w / , o instead of functions ^ ^ o Then, taking into account that functions O :Q satisfy the equations (5.2.13) and condition Wj
0
« O ^
0
, we obtain, supposing y = 0 ,
186
Mechanics of Nonlinear Systems with Internal Resonances dWj
0
/
\
— - — /p (2wy>0 - wJ+l0 <**
wHfi) (5.2.25)
2
-/a^o5 >0 wJ >0 +2|
where the cooperative mode is determined by expression (5.2.17). Let us assume now that instead of the infinite system of coupled oscillators we consider a finite chain with number of oscillators N = 2«, and the conditions of periodicity are satisfied: °0,0=(I>Nfl'
°-W=N-1,0'
"O.O^AT.O'
CO
•••
(5-2.26)
-l,0=a)Ar-l>0'
and/=0, l,2,...,2w-l. Then the wave number can have magnitudes k= 2nm IN, where m= 0,1,2,...,« . Let us consider solutions of set (5.2.25) satisfying to the relations w
j+l,0=e'k>wj,0 kx = ?jmx
, Wy-1,0 =e~'*' WJQ , (5.2.27)
(m]=0,l,2,...,N/2).
Equation (5.2.25) after substitution (5.2.27) becomes uncoupled one: d\V
(
k
\
— ^ = 2/1 2psin2^+a7V\wi0+iaN(?ikje
2i
~ik\
(
l
aA/+4Psin - k 2)
2
w* . (5.2.28)
y 2 J Jt Thus, the stability analysis of the cooperative mode, characterized by wave number m, with respect to the mode with wave number mi, is reduced to the equation of parametrical oscillations considered above. The criterion of instability of this cooperative mode (i.e., of reaching the boundary between regions of stability and instability) is determined by the condition dz2
4psin 2 -+oJV = 3ayV+4psin 2 ^-.
(5.2.29)
Localized Normal Modes in a Chain of Nonlinear Oscillators
187
From here for the magnitude of squared complex amplitude corresponding to instability of the collective mode with a wave number m, with respect to the mode with a wave number mi, we obtain the expression sin2--sin2M (5.2.30) 2 2 a V ) As follows from (5.2.30), at a > 0 each mode corresponding to a certain value m with increasing intensity of excitation becomes unstable sequentially with respect to the modes with m\ = m-\, m-2,..., smaller than the wave number of the considered nonlinear normal mode. The threshold excitation energy increases with decreasing m\ at fixed m. It is necessary to focus especially on the case rri\ = m, when the critical value of excitation energy is equal to zero. In this case the instability (at any small amplitudes) is the Lyapunov instability, and actually leads only to a phase shift of the analyzed mode (similar situation arose in the system with two degrees of freedom, see section 5.2). The most important is the first nontrivial instability with respect to the mode with m\ = m-\, corresponding to the minimal excitation energy (if to eliminate the case m\ = m). At superposition of modes with close wave numbers m and m\ (space beatings!) there is a tendency to localization of the excitation, which becomes more and more noticeable with increasing energy. With a decrease of /»i, when the critical energy increases, the localization becomes weaker. Let us consider, for example, stability of the out-ofphase mode of minimum length (m = JV/2). Then the bifurcational value of the excitation energy corresponds to value of squared modulus of complex amplitude N = 2$-
H-1$-\l-*?*$.-%
(5.2.31)
This instability leads to the localization of oscillations practically on one particle, just as it happens in a system of two coupled oscillators. It is clear that the formed mode is multiply degenerated, as the localization of excitation can be realized on any oscillator. The comparison with models consisting of three and four particles (Kosevich, Kovalyov, 1989) allows us to conclude that all other critical values of energy correspond to formation of unstable modes.
188
Mechanics of Nonlinear Systems with Internal Resonances
Now, returning to the initial model, we will consider the limit case N —> oo. With increased N the arguments of trigonometric functions for the nearest m and mx in (5.3.30) differ less and less, so that in the limit the energetic barrier, corresponding to the instability, tends to zero. It means that all cooperative modes in an infinite chain are unstable (except of a homogeneous mode for which there are no values mu satisfying to the condition m> m\) and spatially localized oscillations turn out to be the unique elementary excitations. If the anharmonicity is negative (a<0), the situation is quite similar. In this case all cooperative modes, except of the out-of-phase mode with the shortest wavelength, are unstable in the infinite limit with respect to the modes with wave numbers rri\ > m. As a result of this instability the localized modes arise again. 5.3 Concluding Remarks The complex representation of dynamics of coupled nonlinear oscillators turns out to be efficient in the asymptotic analysis of the systems with weak as well as strong coupling. In the first case the equations of the principal approximation remain discrete, and in the second case the asymptotic approach leads to continual description in terms of the nonlinear Schrodinger equation and its generalizations. We have shown that in the case of 1:1 internal resonance between oscillators the localization phenomena turns out to be the most pronounced consequence of nonlinearity. Such localization is manifested clearly in the existence of envelope solitons or breathers, which can propagate along the chain and have an internal degree of freedom. Historical remarks. The chain of coupled nonlinear oscillators is one of widely used and realistic models both in Physics (waves in magnetics and Josephson junctions, dislocations in crystals) and Mechanics (a weightless string with concentrated masses on discrete elastic supports, a system of coupled pendulums, see (Lonngren, Scott, 1978), (Scott, 2003)). The study of the model of coupled oscillators in the long wavelength approximation was initiated by Frenkel and Kontorova in application to dislocations theory in crystals (Frenkel, Kontorova, 1939), (Frank, Merwe, 1949), (Frank, Merwe, 1950). It had become clear in the 1970s
Localized Normal Modes in a Chain of Nonlinear Oscillators
189
that the Sine-Gordon equation, arising in this approach, is one of fully integrable systems with infinite degrees of freedom (Zaharov et al, 1980), (Lamb, 1980) . The elementary solutions of this equation preserving their form and velocity in time are soliton excitations, such as kinks and breathers. Other important long wave length approximation, similar to Sine Gordon equation by type of nonlinearity (soft one), is cp-4 theory, which is not integrable but admits also soliton-like particular solutions The case of both soft and hard symmetric nonlinearity has been considered in (Kosevich, Kovalyov, 1989). It was found that breather-like solutions could exist in attenuation zones of linearized system. Numerous applications of localized modes to the theory of oscillatory spectra in polyatomic molecules and molecular crystals are presented in (Ovchinnikov, 1970), (Skott, Lomdahl, 1985), (Kosevich, Kovalyov, 1989), (Henry, Siebrand, 1968), ( Saje, Jornter, 1981), (Jaffe, Brumer, 1980), (Sibert et al, 1982), (Benjiamin, Levine, 1983), (Rosmalen et al, 1983), (Thiele, Wilson, 1961), (Collins, 1983). A common concept of localized nonlinear waves and applications to the problems of mechanics and polymer physics are presented in (L. Manevitch, Mikhlin, Pilipchuk, 1989), (Vakakis, L. Manevitch et al, 1996), (Manevitch L., Smirnov, 1992), (Manevitch L., Savin, Smirnov andVolkov, 1994), (Manevitch L., Savin, 1995), (Gendelman, Manevitch L., 1997), (Manevitch L., Savin, 1997), (Manevitch L., 2001b). The cases of weak and strong coupling between oscillators were consistently considered in (Manevitch L., 2001a). Interesting physical effects in nonlinear dynamics are connected with the strong space localization of vibrations and waves. This problem is the subject of growing interest during last decades. For much extent it relates to essentially discrete vibrations and waves, e.g., (Aubry, 1997). It is beyond our scope to discuss in detail all interesting results in this field, which are successfully reviewed in the above reference. The natural intersection with this field is our consideration of strongly localized modes in the case of weakly coupled oscillators.
Chapter 6
Nonlinear Dynamics of Coupled Oscillatory Chains
6.1 Introduction The coupled oscillatory chain is a realistic model of many mechanical and physical systems. The simplest model of a thin-walled structure is a one-dimensional homogeneous elastic continuum. The latter can be also obtained from an atomistic model using an asymptotic procedure. Corresponding dynamical problems were studied analytically for the linear case only. The account of nonlinearity of the links between particles in the chain and anharmonicity of interchain interactions leads to very complicated mathematical problems. We present an asymptotic approach to an analysis of a planar system of coupled oscillatory chains with the asymmetric anharmonicity. The ideas of the approach are first developed for the case of an isolated chain. To overcome mathematical difficulties one can use an asymptotic approach considering three possible limiting cases: 1) long wavelengths in both directions; 2) long wavelengths in the longitudinal direction and short ones in the transversal direction; 3) short wavelengths in both directions. It seems that the absence of anchor springs leads to the impossibility of any resonance effects at all. Analysis of dispersion relations demonstrates that the internal resonance cannot be actually realized in the long wavelength approximation for the acoustic branch. 190
191
Nonlinear Dynamics of Coupled Oscillatory Chains
However, the situation becomes different in the description of optic waves when the second of the cases mentioned above could be realized. One may consider a corresponding model as consisting of weakly coupled nonlinear oscillators. In contrast to Chapter 5, the nonlinearity is caused by the intrachain interaction (perhaps, alongside with the interchain one). In the third case we deal with distinguished 1:1 resonance. For completeness of the description all three cases are considered below. Special attention is paid to the existence of localized soliton-like excitations. The comparison with the case of coupled nonlinear oscillators is performed. 6.2. Nonlinear Dynamics of an Infinite Chain of Coupled Particles We consider an infinite chain of coupled particles, the interaction between them being described by a gradient-type asymmetric potential
V=\<\ £ (UJ-UJA)2+\C2 £ (Uj-Uj+\c3
£ (Uj-UHy. (6.2.1)
Here U: are the displacements of particles in the oscillatory chain. The system of the equations of motion after transformation to dimensionless variables can be written as follows:
-£f
+ (luj -uH -u J+l ){l + ea, (uJ+l -uH)+
+ E2a2[(uj+]-uH)2+(uj-uJ+i)(uj
^ ^
-«;-_,)]}= 0,
where cr jc7 U: c2r0 io 2 Vm r0 cx c{ r0 is the distance between particles, e « l . We will measure the distance between the particles in terms ofe~V0 and introduce an appropriate space coordinate £ .
192
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 6.1 An infinite chain of coupled particles
It is well known that in the case of long wavelengths the system considered can be reduced to the Korteveg-de Vries (KdV) equation. Actually, representing the difference expressions in (6.2.2) by their Taylor expansions, we obtain one equation in partial derivatives instead of the infinite system of the ordinary differential equations: d2u
d 2u
,
„
du
1
53w
.
fdu]2
4 — r - s 22 — - l + 2e2 a,— + -s4 a,—- + 3e a2 — dT 3 * I ^ ^ ^ J 4 - 4
(6.2.3)
12 dC where u = u(^,x), and "..." corresponds to terms of higher order of smallness by the parameter s . A natural way of the use of the small parameter consists in introduction of new space and time variables:
193
Nonlinear Dynamics of Coupled Oscillatory Chains
^=C-8-x;
x,=e3-x.
(6.2.4)
Substituting (6.2.4) in (6.2.3) we obtain d2u
1
dxfil
d2udu
+ —a,—r-— + l 2 2 d£, d^
1 8*u +
r --- = °-
24 dV
The new space coordinate £, is counted from the front of linear (sound) wave and the new time x, is slow in comparison with x. Denoting d ujdt, = w we come in the principal approximation to the KdV equation 1 dw I d3w „ +-a,w + T =0 • dx, 2 ' dt, 24 5 ^ 3
e^oo (6.2.5)
dw
In this equation all terms have identical orders by the parameter S , and it ensures the simplest exposition of nonlinear dynamics for a chain of coupled particles in the case of asymmetric anharmonicity. As it is known (Zakharov et al, 1980) the KdV equation has localized solutions (solitons and multisoliton waves). In the case of short wavelengths the situation is sharply changed. The equations of motion (6.2.2) have an exact solution in the form of nonlinear standing wave with the shortest wavelength: Uj = w, (x), uj+i = -w2 (x), uj+2 = w, (x), «7.+3 = -w2 (x), (6.2.6) where — <x> < j < +oo . In view of analysis of short-wavelength modes, we use the change of variables (L.I. Manevitch, 1999a): Uj = W,,, ft ) . »j+l = -Wy.2 (C ) ,
Uj+2 = WJ+U, (X ) , M7.+3 = -WJ+i<2 (x ) .
Then the equations of motion are written as follows: d
w
i\
(
\f
(
\
«T
+ s2a2\(wja-wJ_ll)2+(wJl
(6.2.7a)
+wJt2)(wjtl +Wy-,,2)]}=0,
194
Mechanics of Nonlinear Systems with Internal Resonances
d2wj2
~^f~l
( 2 w
\(
y.2
+W
7.i
+w
y+i,iii1 +
I 8a
\
i K - + u "wy,i)+
(627b)
+ s 2 a 2 [(w,.+1>1 - wjtX ) 2 + (wJi2 + w .+1>1 )(wJt2 + yvjj)] }= 0. Let us introduce two functions of two variables w, (C, ,x ) and w2(C, +s,x ) describing in continuum limit the nonlinear dynamics of the chain of coupled particles in the supposition that the modulations of nonlinear mode with minimum wavelength have a characteristic space scale exceeding essentially the distance between particles, which in selected units is equal to 8 . Thus W;±i,i = w,fe,x ) ± e —J- + - e
2
— ^ +... ,
(6.2.8)
where je =C, .
With taking into account of (6.2.8), the following continual equations of motion are obtained: —f+2(w,+w 2 )h- E 2 2a,-f-a 2 ( W l +w 2 ) 2
^
I
L «=
JJ
+E2—f+-=0,
^
(6.2.9)
-^-2(w,+W2)|l+e2[2al^+a2(w,+W2)2|-B^+...=0. Combining these two equations we come to relatively simple system B 2W dW ± + 4W,+4a,e,Wl—2^ / \ ^ 2
d iv2
&i
2
2
dim )
' ' a;
+ 4a2e,W? +e,
d2W ^
J- + - = 0,
(6.2.10)
2
d w2
' a; 2
where Wl - w, + w2; W2 = w, - w 2 ; s, = s 2 ; "..." corresponds to terms of higher powers by parameter s . Up to this stage the equations of motion coincide with the results of (Kosevich, Kovalyov, 1989). But in these equations different terms have different order by parameter s . So the asymptotic analysis has to be
Nonlinear Dynamics of Coupled Oscillatory Chains
195
continued. Now, unlike the case of long wavelengths, the equations of motion contain non-gradient terms, and the coupling between the particles, which is described by the gradient terms, becomes "effectively weak". Therefore the use of the complex representation is justified, but only for variable Wi. The similar situation arose at strong coupling between oscillators in Chapter 5, where the complex representation was introduced in the continual model of a system of oscillators. After the change of variable T O = 2x we use the complex functions dW vfG,T 0 ) = T - L + ^ , ;
dW ~ V*K > T 0 ) = ^ - ] - ' W I ,
(6-2.11)
for which two conjugated equations are obtained. Let us consider one of them and the second equation of system (6.2.10): cto —
is,L
( * _iX \dW is, i •„ « ,T y •„ a , m-(p e ° +—-a 2 ctoe' T ° -cp e~n°) e'n°
i(^j^e_2h\
8 \&^
2
a;
2
(6212)
)
2
d w e, d ( hn * _iT \2 i aV n iT —-+—a,—kpe °-(p e ° — s , +---=0, v ; 4 ' a; 2 dil 2 2 ^
where W = W2,\\i = en°q> .
Introducing further alongside with "fast time" x0 "slow times" T , = £ ,T 0 ; x 2 = e, x 0 . . . and presenting ty , W as power expansions by parameter Sj (p=(p o +s I cp l +s, 2 (p 2 + •••, W = W0 +slW1 +£?W2 + •••, after substitution of (6.2.13) into (6.2.12) we obtain
(6.2.13)
196
Mechanics of Nonlinear Systems with Internal Resonances
dtp I \ d I \ 2 2 —— (cp0 + e , q > , + e , 9 2 + - - - J + S , ^ — \ c p 0 + e , c p , + s , 9 2 + • • • ] + OX]
OXo
+ -/s 1 a 1 [(cp 0 + 8 , 9 , +s, 2 (p 2 + • • • ) -e
-2hJ
°lcp
*
+8,cp
*
2 *
\](SW0
+8,cp
+•••)
— - + s,
dWx
\
!- + ••• +
+-is1a2[(9o+s19,+s,292+-")e'to 5
-(9o*+s19;+s129;+--^'T°]
e*°-
_i
^h.
(d\±_d%e-*o
8 '[a;
2
a;
2
+
' a;
2
(6.2.14)
#*Le-*...Lo,
' a;
J
l^(^0+effr+...)+2El_L-(^0+e1^+.-)+ + ^ 8 , ^ 1 9 0 + 8 , 9 , + - ^ - ^ ; +8,9; +e,2*+--^-/t0] -(6.2.15)
Now we equal to zero the coefficients at each power of the parameter s,. 1 ) c °.
5(
Po_ O .
^o
5
Xo_o.
5T02
therefore
^ 0 =^ 0 (C,x,,x 2 ,...)
(6.2.16)
(we take into account the condition of the absence of secular terms). 2)s,':
Nonlinear Dynamics of Coupled Oscillatory Chains
OT0
CT,
197
0 , 0
Z
(6.2.17a) 2
2To
+3|cpo| cp^" ' -%e^ J - ^ l U
-^ T °9O)=O,
^ + 2 . ^ L + I a , ^ U/ + ( P o y*» + ^ - . ) _ i ^ = 0 . (6.2.17b)
Vo 5t o 5r, 8 ' a ; 2 l ^ 0 | Vo J 4a;2 Taking into account again the conditions of the absence of secular terms in (6.2.17) we obtain the following equations
5r02
dcp0
/
dWr, 3 /
/ 9 2 cp 0
I ,2
(6.2.18)
' a; a; 2 From the second equation (6.2.18) we have ^ - = ^,h|2-
(6-2.19)
Substitution of (6.2.19) into the first equation (6.2.18) leads to a nonlinear equation in partial derivatives describing the dynamics of the chain of particles in the main short wavelength approximation: ^ 0
ia
U
2
(f)
'd2(Po_Q
where a =-(3a 2 -4a, 2 ). This equation coincides exactly with the corresponding equation for the chain of oscillators but the sign of a depends now on the relationship between coefficients a2 and a.\. We come to results, which were obtained in the case of the chain of oscillators (Chapter 5). Namely, the solitonlike solutions (breathers) can propagate along the chain if a >0. In the
198
Mechanics ofNonlinear- Systems with Internal Resonances
case a <0 the soliton-like solutions are absent and only periodic wave packets can exist. We have for a > 0:
• " ^ ' ^ ( ^ f ^ " 1 JsecA^fc -vx,)]. (6-2.20) v2
v where k =
;
co =
2P,
4p, 2
1 S,
B, = — ; amplitude and velocity
8
of the soliton are independent parameters here. 6.3 Nonlinear Dynamics of Infinite Coupled Chains The next stage of our study is devoted to nonlinear dynamics of an infinite plane system of weakly coupled chains, each of them consisting of interacting particles with intrachain gradient potential (6.2.1). Interchain interaction is described by a periodic symmetric potential (6.3.1)
Corresponding equations of motion in dimensionless variables have the form
—^- + (2Uj -uH -tt 7 . +1 ){l + s a , (uJ+l
-uH)+
+ s 2 a 2 [{uj+] -uH ) 2 + (uj - u J + l )(uj -uH)]}+ + &2
[f(uj,k -uj,k-i)+f(uj,k
where S 2 / ( A ) =
(6.3.2)
-«M+I)]=°»
. dA
6.3.1 Long wavelengths in both directions In the case of waves with long lengths in both directions the system considered can be reduced to well-known Kadomtsev-Petviashvily equation. To show this we replace the difference expressions in (6.3.2)
199
Nonlinear Dynamics of Coupled Oscillatory Chains
by their Taylor expansions (we measure here a distance between particles in terms s ~V0 and introduce the corresponding space coordinates C, ,f]):
d2u —--s
dx2
2
w[" o 2 du 1 4 d \ fdu)2 4 — - l + 2e a, — +-e a,—-+3s a, —
25
di;2
L
i2ac4
'ac 3 ' a ; 3 ^
\acj y
J
(6.3.3)
at] 2
Here w = u (^ ,ri,x), (3 = 2 — ( 0 ) . We introduce new space and time dA variables (6.2.4) to use the smallness of parameter E . Substituting (6.2.4) in (6.3.3) we obtain in the main approximation the Kadomtsev-Petviashvily equation d1u
1 d2udu 1 d4u p 82u n +- a , — - — + - + — —r-.... = 0dxxdr\ 2 'Sri2^ 24 <5>ri4 2 dr]1
(ft'XA\ (6.3.4)
This equation is an integrable system and, in particular, its solutions are solitary plane waves (Zakharov et al, 1980). 6.3.2 Long wavelengths in the longitudinal direction The wave with minimum length in transversal direction is an exact solution of the equations of motion (6.3.2). To analyze the short wave length modes in this direction we use new variables «^=(-l)*wyit(T). Then equations of motion (6.3.2) take the form , 2 at
+ 2w
\
j,k -wj^-wj.ijtjll
+ ea^wj^
(6.3.5)
-wJ_lJt)+
+ e2a2[(wJ+lk - wj_u)2+ (wJk - wy+w)(wjJc - Wj_lk)]}+ + e 2 IAWM
+ wjMi)+ AWM
(6-3.6)
+w
i,k-\ )]= 0.
We introduce the function of three variables w = vt>(£ ,ri ,x) describing in continuum limit the nonlinear dynamics of coupled chains
200
Mechanics of Nonlinear Systems with Internal Resonances
in supposition that the modulations of nonlinear mode with the shortest length (in transversal direction) as well as waves themselves (in longitudinal direction) have characteristic space scale exceeding essentially the distance between the particles (which in selected units (s ~V0) is equal toe ). As a result we have in the main approximation the following nonlinear equation in partial derivatives:
p ^ e ^ ox
oQ
+
2sV(2W) = 0.
(6.3.7)
Small parameter s can be eliminated by transformation x, -ex , and then equation of motion takes the form
i ! ^ - ^ dx ,
dC,
+
2/(2W) = 0.
(6.3.8)
In the important particular case when f(w) = P sin w we come to well-known Sine-Gordon equation. Its localized solutions are kink-like solitons or "kinks" (Zakharov et al, 1980). 6.3.3 Short waves in both directions Alongside with (6.3.6) equations of motion (6.3.2) have the solution in the form of a wave with minimum length both in longitudinal and transversal directions. Introducing the modulations of this wave we use new variables: «M = ("I)* wj,k,i (x ).
«;±u = (- 1 )*"' wj,k,i (x ),
«y±2jt = H ) * wy±2>ti, (T ), uj±u
= (-1)*"1 w y±Ui3 (T ).
We rewrite the equations of motion in the form
(6.3.9)
201
Nonlinear Dynamics of Coupled Oscillatory Chains
-^L+(2wMi
+ V
W + wMJ {1+soti H . M
+W
MM)+
+£2a2[(^,2-w;-u-,2)2+(^,*,, +wyw)(wMJ +w y _ u J]}+ (6.3.10a)
w
d
j k2 ( TT—\2WJM
+W
JM
+w
\(
I
/-+uiJi 1 + 8 a iK-+u,i
\
-wjjt,\)+
ax +s2a2[(w/.+u>1 -wm)2+(wjk>2
+wj+w)(wj^
(6.3.10b)
+wjAl)\}+
+£ 2
lf(wj,k,2 + w ^ + i,2 )+/( w y,u + W/^-i.21 = ° Following the previous section we introduce two functions of three variables w,(C, ,r\,x ) and w2(^ ,r|,T ), and obtain for them equations + 2(Wl+w2)\l-e2
^ -
\
OX
1
2ai—^-a2(w,+w2)2 OL,
L
\ JJ
(6.3.11a)
-—-^—2(w,+w 2 ) 1 + s 2 2a,—J- + a 2 (w,+w 2 ) 2 i 1
- 8
2
L
JJ
(6.3.11b)
^ + 28 2 /(2w 1 ) + - = 0,
which describe in continuum limit the nonlinear dynamics of coupled chains in supposition that the modulations change slowly enough. Combining these two equations we have ^ 2 + 4 ^ + 4 a 1 s ] J F 1 ^ ^ + 4 a22 s 1 f F 1 3 + s 1 - ^2 + 5T d; ' ' ' dC 2
2s [/(2 Wl ) + / ( 2 w 2 ) ] + - = 0,
(6.3.12a)
202
Mechanics of Nonlinear Systems with Internal Resonances
82W2 ~d^~
dfa2) l£l
a;
d2W2 ~
£
' ^
+
(63
-12b)
2 s 2 [ / ( 2 w 1 ) - / ( 2 v v 2 ) ] + - = 0, where Wl = wx + w2', W2= wx -w2; s, = s2 . In the case of linear interchain coupling / ( 2 w , ) = 2Pw,,
/ ( 2 w 2 ) = 2Pw 2 ,
f(2wi) + f(2w2) = 2$Wl,
= 2$W2. (6.3.13)
f(2wl)-f(2w2)
Using the transformations and denotations just as in the case of isolated chain we come to the system of equations with respect to functions (po(<;,Ti,To,T1,...) and Wo =W20: (6.3.14a)
a,%L^-W0=0.
(6.3.,4b)
Particular solution of the second equation determines coupling between the functions W0,q>0:
^
= a, -|cpo|2 + | 4 2 # f e -s)}f?0(s)\2ds . (6.3.15)
Substitution
of (6.3.15)
into
(6.3.14)
and transformation
(p0 lead to following integral-differential equation:
^ - / a | O 0 | 2 O 0 - i ^ + /a, j^zVpfe -S)]\%(sf ds-% =0, (6.3.16) ct,
8 dC,
0
where a =(3a 2 -4a, 2 )/8,
Nonlinear Dynamics of Coupled Oscillatory Chains
203
which differs from NSE by the last integral term. In the case of symmetric anharmonicity (a, - 0) equation (6.3.16) coincides with NSE and has, in particular, the solution in the form of an envelope soliton which can be written as (6.2.20) after substitution O0 instead of (p0. 6.4 Concluding Remarks Despite the absence of anchor springs the resonance phenomena are manifested both in isolated oscillatory chains and in coupled oscillatory chains if considering short wavelength asymptotics. It means that a similarity to the system of coupled oscillators can be observed. It is shown that in the case of long wavelengths nonlinear dynamics can be reduced in the main approximation to the well-known Kadomtsev-Petviashvily equation (KP). Its localized solutions describe the supersonic plane solitary waves. In the case of long wavelengths in longitudinal and short ones in transversal direction we obtain the SineGordon (SG) or similar equations dependent on concrete type of the interaction between the chains. Corresponding localized solutions are subsonic solitons. At last, in the case of short wavelengths in both directions the solution of the nonlinear problem turns out to be the most efficient when it is formulated partially in the complex variables. Using multiple-scales expansions, the problem can be reduced (similarly to the case of strongly coupled oscillators) to Nonlinear Shrodinger Equation (NSE), which describes, in particular, the propagation of localized wave envelope. All the above mentioned equations corresponding to main asymptotic approximations are integrable systems. So we have a possibility to describe not only the solitonic excitations in them but also more complicated regimes for a wide class of initial conditions. The main essentially nonlinear effect caused by the resonance condition is the reduction of the system to NSE and, as a consequence, existence of localized soliton-like solutions. Historical remarks. The oscillatory chain with gradient asymmetric power nonlinearity has become the subject of steady interest beginning from famous work (Fermi, Pasta and Ulam, 1955). This model was a
204
Mechanics of Nonlinear Systems with Internal Resonances
basic one when discovering the solitons. The decisive point is that the Korteweg de Vries equation can be considered as a long wavelength approximation for such oscillatory chains. In the special case of Toda potential even the discrete system is integrable and admits the soliton solutions, which may be both long and short waves (Toda, 1981). Long wavelength excitations in oscillatory chains with different types of gradient power nonlinearities were studied in many papers (see, e.g., (Collins M.A., 1981). Short wavelength approximation was considered in (Kosevich, Kovalyov, 1989) and it has been shown that breathers can exist under a certain relationship between symmetric and asymmetric nonlinearities. This result is important both for mechanical and physical problems (Manevitch L., 2001b), (Manevitch L., 2003). Direct derivation of NSE for the description of short wavelength dynamics of oscillatory chain with asymmetric gradient power nonlinearity was given in (Manevitch L., 2003). The effect of interchain interaction was mainly studied in the approach "chain on nonlinear elastic substrate" leading to Sin-Gordon or cp-4 equations (Scott, 2003).
Chapter 7
Nonlinear Dynamics of Strongly Non-Homogeneous Chains with Symmetric Characteristics
If the inertial and elastic characteristics of coupled oscillators are weakly non-homogeneous, the solution for homogeneous chain can be considered as an appropriate first approximation. However, in many cases we deal with strongly non-homogeneous chains. Such a nonhomogeneity can be caused, e.g., by significant difference of masses or elastic characteristics. In this case, alongside small parameters relating to a weak nonlinearity or coupling between oscillators, it is necessary to take into account new small parameters characterizing inertial or elastic non-homogeneity of the chain. We present a general asymptotic approach to nonlinear dynamics of periodically non-homogeneous chains based on generalization of complex representations for homogeneous chains presented in previous chapters. In this case resonance relations do manifest themselves in vicinities of characteristic point for both acoustic and optic branches. 7.1 Initial System and Main Asymptotics Let us consider an infinite system of coupled nonlinear oscillators with a strong difference between alternating inertial or elastic characteristics. The motion of this system is described by the infinite set of nonlinear equations (— °o < j < <x>): ml——^-
+ cluJj
+c,3r02My3>1 +c(2w y ., -« y ._ ) > 2 - i / y > 2 ) = 0 , (7.1.1a)
205
206
m2—^-
Mechanics of Nonlinear Systems with Internal Resonances
+ c2uj2 +c23r02u3j>2+c(2uj2
-uJA -uj+]i)=0.
(7.1.1b)
We use again the dimensionless variables ulk=UjklrQ, where r0 is the distance between particles in the longitudinal direction (z is the corresponding coordinate), UJik are displacements of corresponding oscillators. Parameter c characterizes the linear coupling between oscillators, mk are masses, ck. and c« specify elastic properties of k-th oscillator (k =1,2). One can consider different cases depending on the magnitudes of the coefficients. 7.1.1 Inertial non-homogeneity In the case of inertial non-homogeneity we mean that /w2 lm\ = e « 1 (Fig.7.1). This relation gives us a small parameter for the system under consideration.
Fig. 7.1 An infinite system of coupled nonlinear oscillators with inertial non-homogeneity.
7.1.1.1 Acoustic branch Acoustic branch corresponds to low frequency modes, and the order of the chain oscillation period (with respect to small parameter e) is the same as for low-frequency oscillator. That is why we introduce the dimensionless time variable T~ = CO0/, (7.1.2) where (0 0 = ^jcx /mx is the frequency of the linearized low-frequency oscillator. So, the initial system is reformulated as
207
Nonlinear Dynamics of Strongly Non-Homogeneous Chains
£$• + *„ + ^ « i , +-(2»;,, C C f ' ' «T
C|
C,
- W )-O,
W
(7X3)
Cj
The most interesting situation for the acoustic branch corresponds to estimations
£l =v ~, £233L 8 V 3 , £!2i = 6 2|q
c2
i.=fr,
c2
(7.1.4)
q
where v", v^ , C3 ,'n « O(l). a) Lo«g Wavelength Approximation One can consider a continuum function uk (T,C )» C = £ z / r o being a new longitudinal coordinate for distance measuring (the distance between particles is e in units of f ) . We can use series expansions for u k^S&) n e a r m e points C C+ e in the first and second Eqs. (7.1.3), respectively. These expansions lead us to the set —^- + ul+E%uf +2r({ul-u2)-E2i]—^ox ox, s
+ ... = 0,
^ 5 " + i ; r M 2 + E 2 v ' 3 » 2 + 2 r T ( " 2 - " i ) - e ? n T 7 T + --- = 0. ox cC,
(7.1.5a) (7.1.5b)
From Eq. (7.1.5b) we have M2= 2
!
"l
2ff+v- ' 2rT+v-[ar
f + 8V,M2
2
3 2
!
2 ^2
f+--- • (7.1.6)
)
K
'
Then we introduce a new change of time variable and make some changes in parameters to simplify the equations
/—~
2fTv~ + 2ff +v" 2p +v
fT y
208
Mechanics of Nonlinear Systems with Internal Resonances
v-l±2I.vJ-£L.5,-!-.
(7.1.7)
y 8y 8y After substitution of (7.1.6) and (7.1.7) into Eq. (7.1.5a) and dividing by y we come to the set
*S. 5x2
+ Ul +
2 s ^ + 8 s V , 3 + 16e 2 ^l W 2 3 v ax2 v
v 5C 2
(7L8)
d^2
2rj 1 (d2u2 3 2 M, ] 3 3 w2 = - ! - « , - s - - ^ 2- + 8 8 V 3 M 2 - S T I - - J2- + . . . = 0. v ^ ax dC, ) v
/-Tim (7.1.9)
Then we apply the approach given in previous chapters. The first step is introduction of complex representation Vk=-^-
+ iuk, Vk=-£—*uk>
*=U,
(7.1.10)
Introducing new variable yk=eh
,
(7.1.11)
and "slow times" x, = e x o , T 2 = 8 2 x 0 , . . . ( x 0 = x ),
(7.1.12)
and considering (p* as functions of times x0, Xi, x2, ..., we present the solution in form of power expansion as follows (7.1.13)
Substituting (7.1.10)-(7.1.13) into (7.1.8), (7.1.9) one obtains 3q>!n
ax0
(dyu
i^ax0
faA
ax, j
2rdq>i.2_Ldq>uJ.dq>i,oV
{ax0
ax, ax2 j
A
209
Nonlinear Dynamics of Strongly Non-Homogeneous Chains
+ 8 ^Lr + fe
2
f^i£ 2 3 +^, 3 V'"+fe 2 ! L ^-+... = o,
-iE2-^-iE,--r-iz2^-E32e2h° v v v 2
= 0,
(7.1.15)
where Ek ={qkfi+e
Jap 2 j 0
= l,2,
rap2)l a p ^ ] [ocp*0
(7.1.14)
(7.1.16)
Tap*, a p * ^
- | ( p 2 0 + S ( p 2 J +...)+(p2,o+eq>2,i +...]e~ 2 f t o ]+/6^-—J-.
(7.1.17)
After equating to zero the coefficients at each of growing powers of parameter 8 in Eqs. (7.1.8), (7.1.9) we subsequently obtain:
920 " —
v
I
v
'
=O
(7-hl8>
-
J
Therefore 91,0 =
i.
Pl,l
Scpi.O
c*r0
^1
Tl v
(
\(
a(
.v
+
P2,0
v
91,0 = ° -
*
^92,0 - 2 n 0
^o
^0
^ J
2/ To
'( P2,O 92V" ' )j=O /
'92,i
I
2n L
9
") /
«•
"'h>2,i
v ^ l , U
2r| . ") _ L 2<2 T,To
v
9U K
)
if392o
» +-\—J-
vydx0
(7.1.19) 5
92o
+ —-i-e
dx0
-22 ft n l "°
+
j
210
Mechanics ofNonlinear Systems with Internal Resonances
lfe« + ^ e -.U.
v (^ dx0 Therefore
E:
g(p 12 1 " ^ OX
Q
dx0
+
(7.L20)
)
no=e / P l T 'Fu)(T 2 ,...,0,
C7-1-21)
9U=-^-^O(^,-,CK(''2TO+PITI).
(7.1.22)
(7-1-23)
5tp U ^ " OX j
+
P.=-^T
2 d
V
3e-4rtt.,or9,:o -cp;,o3^-6^)^» + ^ H 2 , o 3 3|q>2,orq>2.oe2fto +\?2,o\\loe2iXo - ^ e ^ e
v I [ dx0
ax, J [ ax0
^ +
ax, J e
\
2[a^2
1
(7.1.24)
a^;2 J
(we do not present the second equation). Elimination of the secular terms while taking into account of (7.1.21) - (7.1.23) gives the principal approximation _M-,-a,|
(7.1.25)
211
Nonlinear Dynamics of Strongly Non-Homogeneous Chains
v
v
v
b) Short Wavelength Approximation In this case we can consider the continuum functions as envelope waves, C, being a new variable for distance measuring. When assuming that the envelope wavelength has the order of unity, the distance between particles is ps/2r 0 , p = O(l). We introduce the change of variables "y,i = "1fc&\ uj,2 = " 2 ^ + 4 «/+i,i =-«i(f,C ± 4 «y±i,2 =-«2^.?
±e
) '
(7.1.26)
and use series expansions of u^ (F,C,) by e in vicinities of the points £ C+s for first and second Eqs. (7.1.3) respectively. As a result we have the following equations: —^- + «1+s2C3w13 +2ffM, = 0 , ^
(7.1.27)
8 —^|- + v« 2 + 8 2v~3M2 + 2rfw2 = 0. After the transformation of parameters x = ^ , Y =2n + l,r, = i , v = ^ l , v
3
= ^ - , ^3 = ^ - ,
(7-1.28)
we perform a further asymptotic analysis similarly to the previous case. The result is 91,0 =
=0
(s2)'
" X ^ + toiko < P i o + ' a 2 — " ^ = 0. where a, =
, 0^ =—*-. v v
(7-1-29) (7-1-30)
212
Mechanics of Nonlinear Systems with Internal Resonances
Equation (7.1.30) is the principal approximation for high-frequency acoustic oscillations and waves. 7.1.1.2 Optical branch Optical branch corresponds to high-frequency modes, and chain oscillation period order is the same as for high-frequency oscillator. Therefore we introduce a new time variable x =COQ' > where coo = VC2/W72 i s m e frequency of the linearized high-frequency oscillator. Then the initial system is presented as follows
i
^
+ »y,2+£^L»j.2
+
^(2"y,2-'V,i-»J>i.i)=()
P- 1 - 3 ')
The most interesting situation for optical branch corresponds to estimations
£L = f, c2
£ s i =8V3,
C
c2
-^
c2
= s%,
^-=fT, (7.1-32) c2
where £", v~3 , C3 . 'H « O(l). a) Long Wavelength Approximation When using the expansions for long-length wave optical branch, we introduce a new time variable x = ^/2ff+ l f , and then make change of parameters
£
£12n;>
Results of the analysis are given below
V3 ^
=
^
(7i33)
213
Nonlinear Dynamics of Strongly Non-Homogeneous Chains
92,o=^2,o(^,-^y (P ' Tl+p2T2+P3T3) ^ 2
2
2
2
2
(7-1-34) 4
2
P 1 =2r) ,p 2 =-2T 1 (5r 1 -0,P3=2T 1 (^ +42T 1 -14^T 1 ), 9 2 = 9 2 , o + e r i 2 e " 2 ' > 2 ) o - e ^ 2 ( 6 T l 2 - ^ " " 2 " 0 9 2 , o + 4 2 ) ' (7-'-35)
|5L = o,
(7.1.36)
where a1 = - 3 v 3 , a 2 = - 2 i i 2 .
(7.1.37)
6^ Short Wavelength Approximation Using the relation (7.1.28), we introduce a new time variable T = ^/2ff + If and then make change of parameters (7.1.33). Taking into account (7.1.14) we obtain after asymptotic analysis 92,0=92,0^3.-^)
92=92,0+o(s2J, 9 l = ^ e 2 J .
(7.1.38)
The condition of absence of secular terms in the second equation gives the principal approximation equation for the short wavelength optic branch 5(P2 o
I
2
d (p2 o
2
— ^ - + a 1 zb 20 9 2 o + a i ' r~ = °> a i = - 3 v 3 ' a 2 = 2 r l -(7.1-39) d%1 dz3 7.1.2 Elastic non-homogeneities Considering the case of elastic non-homogeneity we mean that q / c 2 =e « 1 (Fig.7.2). This relation gives us small parameter for the system under consideration.
214
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 7.2 An infinite system of coupled nonlinear oscillators with elastic non-homogeneity.
7.1.2.1 A coustic branch Acoustic branch corresponds to low frequency modes, and the chain oscillation period order is the same as for the linearized low-frequency oscillator. That is why we introduce a new time variable like that in paragraph (7.1.1.1) The most interesting situation for acoustic branch corresponds to estimations
^m, = »,C^l q
= z%,C^lq
= e%A^, c x
(7.1.40)
where \i, v~3 , £3 , r\ « 6>(l). Then the initial system is presented as —TT" + «y,i + e % u)tX +T\faujj - uH<2 - ujt2 ) = 0, ax H—-^ax
+ £-luJ2
+ff(2M.i2 -ujX
+z%u)2
-uj+u)=0.
(7.1.41a) (7.1.41b)
a) Long Wavelength Approximation Using the continuum approximation (see (7.1.1.1a)) we introduce a new time variable and make change of parameters
x=V2rfTT-r, T I = 2 - , V = I , V 3 = ^ M 3 = ^ - . y
y
8y
8y
(7.1.42)
215
Nonlinear Dynamics of Strongly Non-Homogeneous Chains
We obtain as well as earlier 9l,o=^3v..,Oe'
(PlT +P2T2+P3T3)
'
,
R - _ 2 n i R _ 2n2(n2-2n + H)
131
"
v
' ^ -
V2
P 3 = - 2 n 2 ( 2 t i 4 - 4 T l 3 - 2 ^ 2 ^ 2 - 4 ^ + 4 1 1 2 ) , (7.1.43) V
9, =9,0 -s^e^- e 2 ^^|L±iie-%; 0 j-4 2 ) f2n ") 2 f 2jJ.ri-4ri2 q>2=e —9i,o +e ' cp vv ) I v
1 0
-
2ai«-»..^+ap^']4») v2
v
DC, J
K
(7.1.44)
'
and come to the equation corresponding to principal approximation —J1L +icn ^0 ^2 _
^io+/a2
a,=-3§3,
5 ^ = 0, ar
(7.1.45)
2TI 2
a2=—i-. v
ij 5/zo/"? Wavelength Approximation Using the continuum approximation (see (7.1.1.1b)) we introduce a new time variable and make a change of parameters T=V2rfTTr, T I = 1 , v = i , v 3 = ^ _ , ^ = T ~ y y 8y 8y Equation (7.1.14) and asymptotic analysis leads to 91,0 =
(7-1-46)
(7- 1 - 4 7 )
216
Mechanics of Nonlinear Systems with Internal Resonances
The condition of the secular terms absence gives a principal equation for short wavelength acoustic oscillations and waves dcPl o
where a , = - 3 ^ 3 ,
i
2
a2 =
3 (p10
(7.1.48)
—.
v
7.1.2.2 Optical branch Optical branch corresponds to high-frequency modes, and chain oscillation period order is also the same as for high-frequency oscillator. So we introduce a new time variable •T = co0/, where co0 is the frequency of the linearized high-frequency oscillator (just as in paragraph 7.1.1.2). Then the initial system is now presented as c,3r02
-l d u,\ OX
C')
3
c (.
\
C-y
(7.1.49)
2
^£+.,, ^
-
n
^ ^(2«,2 -»„ - « ) - 0.
The most interesting situation for optical branch corresponds to estimations
£2W = e V c2
£ni=eT C2
^. =ff£)
(7.1.50)
C2
where V3 , f3 ,r\ »O(l). o^ Zo«g Wavelength Approximation Using the continuum approximation (see (7.1.1.1, (a)) we introduce, as earlier, a new time variable, but now it is the same as initial one: T = x". Then we make the parameters change
217
Nonlinear Dynamics of Strongly Non-Homogeneous Chains
r|=ff,
^ = 1 + 2ff,
v3=^-,
^3=^-.
(7-1-51)
(7-1-52)
o
o
and obtain
a -n R -felZlhl R ~
Pi - T l > P2
P4-
(VS+TI-I^V
' P3 -
~
>
Tl 2 (l6^ 2 +35n 2 -80^T 1 ) 8
cp2 =cp2;0+s ^ e - 2 S i o ) - e 2 ( ^ 1 V ^ q , i o ] + 8
I 9i = - e
) 2
(2rm(P2,o)+s [2Ti(2^Ti-^ 2 ^}p 2jO / \
1
(7.1.53)
+ O (8 2 ).
The principal approximation for low-frequency optical oscillations and waves is written below 50, o 2 520,0 - ^ ° - + ,-a1
(7 L54)
"
where a, = - 3 v 3 , a 2 = -2r] 2 . b) Short Wavelength Approximation We use again the continuum approximation (see 7.1.1.1, (b)), all computations are the same as in paragraph 7.1.2.1. Their results are 9,>0=
218
Mechanics of Nonlinear Systems with Internal Resonances
Pi =n f P 2 = - ^ , P 3 = ^ - . P 4 = ~ ^ ,
(7.1.55)
cp,=-82 ^ p - ^ - + 4 2 J ,
(7.1.56)
and we obtain the principal approximation for short-wave optic oscillations and waves — ^ - + /a,blj0 0 l o + i a 2 where a , = - 3 v 3 ,
f- = 0,
(7.1.57)
a2=2r|2.
7.2 Numerical Simulation It is of interest to compare analytical results with the numerical ones. We make comparison for the case discussed in 7.1.1.1. Constants for equations 7.1.4 and 7.1.5 are listed below: \T = l,ff = l,v~3 = -8,f 3 - - 8 e=0,2 .
(7.2.1)
The fourth-order classical Runge - Kutta method was used with stepsize 0,05. The initial shape corresponds to soliton-like solution with 5=0,5 and v=0 or v=0,3. The first case v=0, when the standing wave can be excited, is presented in Fig. 7.3. The number of first type particles is shown on horizontal axis, corresponding displacements are given on vertical one. The particles of the second type are not presented. The solid line relates to the asymptotical solution, circles relate to results of the numerical simulation.
Nonlinear Dynamics of Strongly Non-Homogeneous Chains
219
Fig. 7.3 Standing wave; numerical results correspond to X = 8 0 0 , asymptotical ones - to T = 7 9 9 . 4 .
The second case is a traveling wave (v^O). It is represented in Fig. 7.4. We see that nonlinearity provides preservation of the soliton shape, contrary to linearized systems. The asymptotic solution is in good accordance with the numerical one despite we deal with the rather large value of parameter e.
220
Mechanics of Nonlinear Systems with Internal Resonances
Fig. 7.4 Traveling wave; numeric results correspond to T = 8 0 0 ,
asymptotical - to x - 799.3.
7.3 Concluding Remarks We see that in all cases considered (for both acoustic and optical branches) the final equation corresponding to main asymptotic approach is the Nonlinear Shrodinger Equation (NSE). These cases are the most interesting because for other relations between parameters of the initial system we come in final approximation to uncoupled oscillators with effective characteristics or to the linearized couple system. In our approach the signs of nonlinear and elastic coefficients in the final NSE can be different depending on signs of the anharmonic
Nonlinear Dynamics ofStrongly Non-Homogeneous Chains
221
coefficients and the branch under consideration. When these signs are similar, NSE admits localized nonlinear waves, which are envelope solitons. In all cases mentioned above nonlocalized, cooperative waves are unstable. On the contrary in all other cases where signs of oq and second derivatives a 2 are different the cooperative harmonic-like motions are stable. Historical remarks. Although the account of a strongly nonhomogeneous structure of the chain of nonlinear oscillators is rather natural extension of the homogeneous case, analytical consideration of this problem was restricted until recent time, to our knowledge, by 2DOF systems in (Manevitch L., Cherevatsky, 1969), (Manevitch L., Cherevatsky, 1972). The linear dynamics of periodically nonhomogeneous oscillatory chains was systematically considered in (Brillouin, Parodi, 1956). Corresponding nonlinear problems were discussed in series of papers devoted to diatomic Toda lattices, see (Toda, 1981), (Gendelman, Manevitch L., 1992). The solution using the complex representation of the equations of motion and multiple scale expansions was presented in paper (Gorlov, Manevitch L., 2001). Complete analysis of strongly asymmetric nonlinear 2DOF systems, including the solution of general Cauchy problem was presented in (Gorlov, Manevitch L, 2003). Recently the interest to strongly nonhomogeneous nonlinear 2DOF systems was renewed in connection with solution of the energy pumping problem (Gendelman, Manevitch L., Vakakis, 2002), (Gendelman, Manevitch L., Vakakis, 2003). The chain of strongly non-homogeneous nonlinear oscillators is a subject of paper (Gorlov, Manevitch L., 2002).
Chapter 8
Transversal Dynamics of One-Dimensional Chain on Nonlinear Asymmetric Substrate
8.1. Introduction Investigation of the longitudinal dynamics of quasi-one-dimensional oscillatory chains on a nonlinear substrate attracts a lot of interest today in connection with localized excitations (solitons and breathers) analysis. Usually a substrate with symmetric anharmonicity is considered. This is warranted in the case of longitudinal dynamics investigation, when this substrate simulates shear elastic interaction (e.g., between the neighboring chains in polymer crystals). Meanwhile, transversal rigidity has apparently not yet been considered. Moreover, even when nonlinear dynamics of a quasi-one-dimensional asymmetric chain on a nonlinear substrate dynamics of an isolated oscillatory chain was considered, its bending flexibility was not taken into account. Difficulty in transversal dynamics analysis of chains with regard to the bending properties is caused by rising the equation order (when compared with the case of longitudinal dynamics). This is the reason for disuse of conventional procedures for localized waves search. Besides, in certain cases taking into account only symmetric anharmonicity no longer remains correct. In the present work the multiple-scale procedure is used to overcome these problems. The procedure seems to be the most effective when complex presentation of equations of motion is used. In linear approximation all asymptotics were obtained in work (Manevitch L., Oshmyan, 1999), where beam inertia was also taken into consideration.
222
Transversal Dynamics of One-Dimensional Chain on Substrate
223
In this chapter we present an analytical and numerical study of short wavelength breathers in the system of asymmetric nonlinear oscillators coupled by stretched weightless beam. The study is focused on modulations of the nonlinear normal mode with shortest wavelength. A small parameter is introduced as ratio of distance between the particles in the chain and characteristic wavelength of modulation. 8.2. Model An infinite chain of particles on a substrate is considered (Fig. 8.1). Let us assume the interaction between the particles obeys a parabolic potential interaction, which depends upon an alteration of the angle between two adjacent elements of the chain. The interaction between the particles and the substrate is described by a weekly anharmonic asymmetric fourth-power potential. Hamiltonian of the system under consideration is
a=- ZWJ^Y, £ j=-oo
/.FQ j=-cn
C
L
[2Wj -wH -wj+xy+± i (wj -wjAy+
+°°
ZTQ y'=-co
+
C
-H»
3
C
+°o
4 2^ r-^ +f^S^4>
(8-2.1)
Fig. 8.1 An infinite chain of particles on a substrate.
where the first term stands for the kinetic energy of the system and other terms for the potential energy (bending energy, stretching energy, and energy of nonlinear supports). We introduce the change of variables and parameters
0
4ea2=-&-,
8«r3=-^2-,
(8.2.2)
224
Mechanics of Nonlinear Systems with Internal Resonances
and come to the equations {-<x> < j < +co) d
w:
—f(XT
i
\
i
+ {6WJ -4w y _, -4w j + x + Wj_2 + wj_2)+ 2a0[2wj + 2eaxwj +4£3/2a2wj +8e2a3wj =0.
Wj_{
\
- wJ+l)+ (8.2.3)
=-wj+l Further we use two new types of variables uj -Wj,uj+\ describing the displacement of odd and even particles, respectively. Sign "-" takes into account that for nonlinear normal mode with shortest wavelength the displacement of neighbor particles have opposite signs. Two variables are introduced because of asymmetry of potential energy. In such a case the displacements of neighbor particles for shortest nonlinear normal mode are not equal by modulus, but the next neighbor of each particles has similar magnitude of displacement. Naturally, for nonlinear excitations close to the normal mode with shortest wavelength such a similarity disappears. So, Eq. (8.2.3) can be rewritten as follows: d u;
dr
i
\
i
\
f- + \6l4j + 4Wy_, + 4Uj+i + Uj_2 + UJ+2)+ 2aQ[2Uj + Uj_; + UJ+] ) +
2saxuj + 4£3/2a2u2j + 8s2a2u] = 0 , —£-
(8.2.4)
+ (6uJ+i + 4uj + 4uj+2 + uj-i + uj+3)+ 2a0 (2£#/+1 + uj + uj+2 )+
2ea\3j - 4£3/2a2uj + U2a3uj = 0. (8.2.5) Then one can transfer to continuum approximation considering the long wavelength modulations of nonlinear normal mode with shortest wavelength. Introducing new longitudinal coordinate C, = s zl r0 we measure the distance between particles in new units, namely s~ r0 and use power excitations in vicinities of points £ and £ + £ in (8.2.4) and (8.2.5), respectively. Then the dynamics of our nonlinear system is described by two nonlinear partial differential equations -2 p)2~ p2 — j + 7saxu + 4(2 + a0 )(u + u) + 2(2 + ao)s2 — j + 4e2 — j + 2sa\u +
Transversal Dynamics of One-Dimensional Chain on Substrate
4e3/2a2u2+&£2a3u3
+ ... = 0,
225
(8.2.6)
3 u . ~ ./. \/^ \ „/„ \ 2 d u , 2 <3 w ~ ~ — j + 2ea\U — - + 2ea\u +4(2 + ao)' M + u)+2{2 + ao)£—J+ 4s3/2a2u2
+ 8£2a3u3 + ... = 0,
(8.2.7)
where "..." corresponds to terms of higher order relative to parameter e. Then we divide our chain on the pairs of particles (odd and even ones in every cell). One can introduce the variables characterizing the average cell displacement
V=^i,
(8.2.8)
KJ^.
(8.2.9)
and relative displacement
After the introduction of these variables the equations of motion are presented as follows." ^ + 2saxV + 8(2 + ao)v + 2(4 + ao)s2 ^ j + dz d^ Ss3/2a2VV + Ss2a3v(v2
^l
dv
+3V2) = 0
+ 2£aiV-2(4 + a0y^
4£3/2a2(v2
dC,
+ V2) + 8s2a3v(v2
(8.2.10)
+
+ 3V2) = 0.
(8.2.11)
The small parameter e characterizes smallness of displacements as well as distance between neighbor masses (relative to wavelength of modulation). Further we use the following change of variables: 0 = ^*1—+ {8r
r/
\i" 2
where fi = 2(2(2+ a 0 J\ .
ifiV\, )
V=v
(8.2.12)
226
Mechanics of Nonlinear Systems with Internal Resonances
Then the equations of motions look as follows
|£-ffJ-alA|-^-?»V2^)-i(4 + a o K I * 2 ^ - t ' - / e - ^ ) -4i£V2a2M~l (q>-tp'e-2"")v + isV~' •r
,
,
(
v
n
(8.2.13)
| > 2 ( p 2 (?"" -2\
aoy—jdC,
a2\»-2(
3n
£
+ 8£2aJv2
--(pei/iT
+
-q>*e-i/tTJ~\ = O .
(8.2.14)
Introducing further the «slow» times alongside with «fast» time TO=T, TX=ST, r2 - s 2T ..., presenting
V = Vo +£
V, +£V 2 +£•
3/2
2 V3 + £ V4 + . . . ,
(8-2.15)
and employing the standard procedure of the multiple scales method, we obtain the following sets of equations. \
5r0V e—[
H~xeiax
q>T,+£ q>4+... +
) 1/2
d (
^
[(^ 0 + em
1/2
2 5 { +e
^
+ £
(v0 +£ 1 / 2 V, +£V 2 + £ 3 / 2 V 3 + £ 2 V 4 + . . . ) -
/£ 2 a 3 //- 1 (^ o +... + ^ o V 2/ '
227
Transversal Dynamics of One-Dimensional Chain on Substrate
24(vo2+...)}-/(4 + a o ) / i - 1 e 2 ^ T ( ^ o - ^ e - 2 ^ ) = O , { d2 I
A
r o
(^5r0
2 d2
^
5zi
,
r ^»O
d2 5r o 9r!
T
,
^C
2
d2 ar o 5r 2
)
"T^ . . .
(8.2.16)
A
y'
2«*,(v0 +f 1 / 2 v, +iv 2 +...)- 2(4 + « 0 > 2 ^ | - -
^ 3 / 2 a 2 {^- 2 [U+^ / 2 ^ + ...) 2 ^°-2(^ 0 + f 1 / V l +...)x 4(v o +* 1/2 v,+...) 2 } + 8a 3 f 2 ( vo + ...)x {(vo+...)2-|[U + . . . y ^ - U + - ) ^ r o ] 2 } = O.(8.2.17) Equating the coefficients at each power of s to zero, we get following equations. Terms of s -order:
-jr- = °>—T" = 0 ->^o = ^ i ^ 2 . 4 v0 =v o (r 2 ,T 3v ..). (8.2.18) Terms of £
-order:
- ^ - = 0-> ^ = ^ , ( T , , T 2 , . . . ) , ^ A = 0->v1=v,(r1,r2,...). Terms of e -order:
(8.2.19) (8.2.20)
228
Mechanics ofNonlinear Systems with Internal Resonances
(8.2.21)
^A-ia^-vle-*™)^.
Condition of absence of secular terms when integrating Eq. (8.2.21) by TQ leads to equation
-p—ialM~l
(8.2.22)
Integration of Eq. (8.2.22) gives <po=®o{T2,Ti,...)eia^.
(8.2.23)
Then equation (8.2.21) takes the form
^--/a^-VoV^^O,
(8.2.24)
and its solution is
(8225)
Selection of first order terms in Eq. (8.2.17) gives the relation | ^ + 2 / ^ + 2a,v o =O,
(8.2.26)
so VO = O , V 2 = V 2 ( T , , T 2 , . . . J .
Terms of 8
(8.2.27)
-order:
^^ia^U-ne^y*.
(8.2.28)
Condition of secular terms absence when integrating Eq. (8.2.28) by r 0 gives the equation
-^- + /a,/rVi =0. Si,
229
Transversal Dynamics of One-Dimensional Chain on Substrate
From this equation (Pl=0l(r2,r3,..)eia^~^
(8.2.29)
Then 2
DT0
Selection of e
-order terms in Eq. (8.2.18) leads to the relation
^ - + 2a,v, -aScpl e2i^ -2(po\2 +cpf e'2^"V2
=0. (8.2.30)
In this case condition of secular terms absence gives algebraic equation 2a1v1+2a2|^o|2-""2=°. which reveals the relationship between functions Vj and 0Q
v, =-—K|V 2 -
(8.2.31)
(8-2-32)
Terms of s -order: ox0
oxx
ox-t
\
'
i{4 + ao)M-1 — (po - qw~2i/tT° )= 0 . 8C2
(8.2.33)
Condition of secular terms absence gives the relation -^- -4za 2 ia-'( Po v I -2ia 3 p.~ l % % -i{4 + ao\i-1 —Y% = 0. (8.2.34) OX2
OL,
After substitution of (8.2.23) and (8.2.32) into (8.2.34) we get NSE
230
Mechanics of Nonlinear Systems with Internal Resonances
where a = 2/T1fa3 - Zafaf V~2 I P = M~l(4 + a0). If a; > 0, this equation, as it was discussed above, has localized solutions, which are envelope solitons Mei(k£-a>T2)
<2>0 = — T —
;>
ch[Ja{C-vT2)\ where a = 2M2a, co = v2 - M2al2. The amplitude Mand velocity v are independent parameters here. The final expressions for wj and wj are w.=(-lVM
sin 2g
( ^ + r) cosh[VaJ2£ 7 - vf rjl
cosh|va(2£-y-v£- r | 8.3. Numerical Simulation To confirm the asymptotic solution we have performed a numerical simulation using an initial discrete system (8.2.3). This is an important issue because a number of assumptions were made on the way to the final analytic expression. A chain of 500 oscillators with periodic boundary condition was chosen for the simulation. The Runge-Kutta method of fourth order of accuracy is used with the following system parameters: • wave amplitude M= 2.5, 2.0, 1.3; • linear rigidity ax =0.5; • nonlinear factors a 3 = 0.25 and a2 = 0.05, 0.125, 0.22; • breather velocities v = 0.0, 1.0, 2.0; • small parameter s = 0.04; • integration step = 0.005.
Transversal Dynamics of One-Dimensional Chain on Substrate
231
The relative error for chain energy was of order 10"'°. The initial coordinates and velocities are set in accordance to the expression obtained for the breather. The criterion of validity of the obtained analytical solution is the form and velocity preservation during breather propagation. In addition the propagation velocity should be equal to v, which is set as one of the initial parameters. We also compare the behavior of such wave packets in the linear and nonlinear chains (in the former case we eliminate the nonlinear interaction). Here we present some of the simulation results. Fig. 8.2 corresponds to the case when the velocity equals to zero for nonlinear and linear chains.
Fig. 8.2 Breathers in nonlinear (solid line) and linear (T =2000 - dashed line, X =4000dotted line) chains. M= 2.0, CCQ = 0.22, small bending stiffness. Initial conditions were set as breathers in both cases.
The solid line shows the initial form of the breather and its form at the moment r = 2000 for nonlinear chain, dashed and dotted lines show the form of the breather for linear chain at r = 2000 and 4000, respectively. Hereafter we show only amplitudes of oscillations on figures, so these figures present instantaneous displacements of oscillators when their velocities are close to zero. As is seen from the
232
Mechanics of Nonlinear Systems with Internal Resonances
figures, in the nonlinear chain, contrary to the linear one, the wave packet can live for a long time without any changes of the envelope.
Fig. 8.3 Passage of two breathers through each other {Mi = 2.0, M2 = 1.3, v, = 2.0, v2 = -2.0, a0 = 0.22, ax = 0.5, a2 = 0.05, a 3 = 0.25, small bending stiffness).
Fig. 8.3 shows the passage of two breathers through each other. Two breathers propagate as independent packets. One can see clearly three stages of the process: propagation of two breathers before collision, their interaction and propagation after collision without any noticeable change of their form and velocity.
Transversal Dynamics of One-Dimensional Chain on Substrate
233
Figs. 8.4 and 8.5 demonstrate qualitative influence of stretching and bending stiffness on breather parameters for fixed energy. At last, one can see from Fig. 8.6 that even strongly discrete breather can be well enough described by NSE.
Fig. 8.4 Breathers in nonlinear chain M= 2.0, v = 0, ax = 0.5, «2 = 0.05, a 3 = 0.25, 6 = 0.04, small bending stiffness.
234
Mechanics of Nonlinear Systems with Internal Resonances
(a) Dashed line: bending stiffness is absent, (6) Dashed line: bending stiffness is absent, solid line: small bending stiffness; solid line: intermediate bending stiffness;
(c) Dash line: bending stiffness is absent, solid line: large bending stiffness Fig. 8.5 Breathers in nonlinear chain M= 2.0, v = 0, a0 = 0.22, a2 = 0.05, a 3 = 0.25.
Fig. 8.6 Narrow discrete breather; M= 2.5, v = 0, a0 =0.0, a2 =0.125,5 = 0.3.
Transversal Dynamics of One-Dimensional Chain on Substrate
235
The main problem of numerical simulation is to provide the appropriate accuracy. When the accuracy is not enough the nonlinear terms give no contribution. The nonlinear contribution to the energy is much less then the linear one. The energies ratio is about 350 - 2000. However this small contribution is crucial for the existence of breathers. The point is, that the equations are integrated in "fast time" while the role of nonlinearity is developed in "slow time", and this is seen from the asymptotic solution. So, the NSE is a really appropriate tool for the description of short wavelength localized nonlinear excitations in the chain. It can be added also that all terms in the equation have the same order. 8.4. Concluding Remarks We have considered firstly the short wavelength asymptotic for the homogeneous chain of oscillators coupled by linear forces. The corresponding small parameter characterizes a strong coupling between weakly nonlinear oscillators. Preliminary transition to complex variables has allowed efficient utilizing of the multiple-scale expansion. As a result we have shown that in short wavelength approximation the very complicated problem can be considerably simplified by being reduced to NSE, whose localized solutions are breathers. The results of the analytical study are confirmed by computer simulation. Historical remarks. Investigation of the longitudinal dynamics of one-dimensional oscillatory chains on a nonlinear substrate goes back to the well-known paper (Frenkel, Kontorova, 1938). From mathematical point of view this problem is similar to that of transversal dynamics of a system of nonlinear oscillators coupled by weightless stretched string. There are three important restrictions usually accepted in such problems: i) consideration of long wavelength approximation only; ii) assumption of symmetric anharmonicity of the substrate elastic characteristics; this restriction is justified in the case of longitudinal dynamics investigation when this substrate simulates shear elastic interaction (e.g., between the neighboring chains in polymer crystals); iii) neglecting bending flexibility of the chain of nonlinear oscillators.
236
Mechanics ofNonlinear Systems with Internal Resonances
All these restrictions were removed in (Manevitch L, Oshmyan, 1999). However this paper was devoted to the linearized problem. The short wavelength longitudinal dynamics of the chain of coupled nonlinear oscillators (alongside with long wavelength) was analytically considered for the first time in (Kosevich, Kovalev, 1974). They have proved the existence of short wavelength excitations modulated by long wavelength envelopes. Strongly localized excitation of this type (discrete breathers), which can not be described in the framework of continuum approximation even for modulation functions, as it was mentioned above, is a subject of growing interest (Vedenova, Manevitch L., 1984), (Vedenova, Manevitch L., Nisichenko, Lysenko, 1984), (S. Aubry, 1997), (Flach, Willis, 1998). The transversal dynamics of a beam on asymmetric nonlinear substrate was considered in (Manevitch L., Pervouchine, 2003). The main result of this paper is that highly stable and mobile localized nonlinear excitations (breathers) are present in such a chain and can be described by continuum equations for envelopes.
Concluding Remarks
There are two guidelines in the development of Nonlinear Dynamics, which were elaborated to some extent independently. The first of them has its origin in theoretical mechanics and the second one - in general theory of nonlinear waves. The first guideline deals commonly with vibrations in systems having several degrees of freedom. Corresponding problems are formulated usually in terms of real variables, and then one of asymptotic procedures (expansions by small parameter, averaging, multiple scales expansions, transition to normal forms with subsequent application of other techniques) can be applied. The second guideline deals predominantly with nonlinear waves in the infinite systems such as chains of oscillators (with anchor springs) or oscillatory chains. The decisive point for recent development of corresponding theory was the discovery of different types of solitons and soliton-like excitations in long and short wavelength approximations for infinite chains. We attempted to consider both the guidelines mentioned from a common viewpoint. Such a possibility is provided by deep similarity between resonance vibrations in the systems with several degrees of freedom and soliton-like waves (envelope solitons or breathers) in infinite chains. The similarity relates to both propagation and standing waves. Actually, an analogy between localized nonlinear normal modes in 2DOF system and standing breathers in infinite chains as well as between elliptic modes and traveling breathers can be traced. The reason for such similarity is the presence in both cases of the resonance relations between frequencies of weakly coupled oscillators. From this point of view, consideration of such long wavelength elementary excitations as solitons of KdV or kinks of Sine-Gordon approaches (which are not the vibration-type solutions) presents a specific problem that is out of our scope. 237
238
Concluding Remarks
Consideration of vibration-type solutions (envelope solitons or breathers) in the chains leads naturally to complex variables and complex equations of NSE-type. Therefore, common descriptions of different models does require introducing the complex variables from very beginning. (The only exception is Chapter 4, where a two-mode approach is used for description of 1:1 resonance in circular rings and shells. We apply here multiple scales expansions in the terms of real variables). We would like also to propose a notion with respect to conception of nonlinear normal modes. In linear theory we deal with linear normal vibrations and linear normal waves (in infinite systems). Both them submit to the linear superposition principle. We have recently known that infinite nonlinear systems as well as finite ones with many degrees of freedom can be also fully integrable for a wide class of initial conditions. Then one can find analytical solutions, which are nonlinear superpositions of solitons, kinks, breathers and periodic waves (dependent on type of the system). So, very important limiting cases show us that in nonlinear theory we can consider all nonlinear elementary excitations as an extension of linear normal modes. We would like also to emphasize the idea that initial formulation of mechanical problems cannot be frequently considered as fully adequate. The reason is that it is not written in appropriate time and space scales. When mentioning "appropriate scales" we mean those in which nontrivial nonlinear effects may be clearly manifested. Using asymptotic procedure one can find such appropriate scales in which equations of main nonlinear approximation turn out to be adequate model of considered nonlinear phenomena, e.g., localization of energy. In the theory of nonlinear vibrations it seems that derivation of such equations is simply one of the stages of solution. However, nonlinear wave theory tells us clearly that namely such approximations as KdV, Sine - Gordon and NS equations are appropriate models of nonlinear processes in complicated mechanical and physical systems. We can make the solutions more precise, but only on the basis of these models using certain perturbation techniques.
Appendix
Inertial Forces and Methodology of Mechanics1
It is shown that the present-day methodology of mechanics treating "inertia forces" as fictitious ones is inconsistent and leads to a contradiction in continuum mechanics. Newton's laws inevitably lead to the conclusion about the existence of real "Newtonian inertial forces" at all points of accelerated bodies. But the usual formulation of the second law excludes these forces from the class of "physical forces". A real sense of the d'Alembert principle consists in changing the treatment of this law, by considering the term mW as a force, which allows one to bring both laws of mechanics in concordance. The translatory and Coriolis inertial forces are components of the real Newtonian inertial force. Forces acting at all points of a body are the same in all reference frames, inertial and non-inertial ones.
1. Introduction Fundamentals of mechanics and its methodology have been well established during the seventeenth and eighteenth centuries, in the times of Newton and Lagrange [1,2]. But there is an important point in the foundation of classical mechanics that until now has not received unambiguous treatment. This is the problem of inertia forces, their reality or fiction. It is obvious that this is not a particular question, but a principal problem of the methodology of mechanics.
' Translation of paper by A. I. Manevich published in "Reports of Ukrainian National Academy of Science", 2001, N 12, pp. 52-57. 239
240
Mechanics of Systems with Internal Resonances
Generalizing and simplifying the situation, one can distinguish three views on inertial forces — the position of physics and those of theoretical mechanics and applied mechanics. In physics courses only the translatory and Coriolis forces, appearing in description of relative motion, i.e., a motion in non-inertial reference frames, are named "inertial force". These forces depend upon the choice of the coordinate system and so are considered a priori as a convenient fib, as "pseudoforces" [3], which are introduced in order to present the equations of motion in the same form as in the inertial reference frame. In courses of theoretical mechanics the "inertial forces" are usually treated in a wider sense. The D'Alembert principle introduces inertial forces as some vectors having dimension of force, which are invented in order to use the language and notions of statics in dynamics. The simplest formulation of this principle for material points follows from Newton's second law mW = F + R (F, R are the active force and the reaction force respectively) after transferring all the terms of this equation to the right hand side and denoting -mW = F4. The obtained equation F+ R + F' = 0 is usually regarded as a formal balance of forces acting on the body, if to apply (conventionally) force F* ("D'Alembertian inertial force") to the body. It is emphasized that no balancing is actually reached, as these forces are applied not to the given body but to its braces and to the bodies causing its motion ("accelerating bodies"). In addition to the translatory, Coriolis and D'Alembertian inertial forces, theoretical mechanics considers also "Newtonian inertial forces" or "counteraction forces", which are spoken about in the Newton's third law. "Newtonian inertial forces", in distinction from other inertial forces, are treated as real ones (therefore the Newton's term is sometimes considered as unsuccessful, and it is proposed to use only term "counteraction forces"). To be distracted from the terminology arguments, we may say that theoretical mechanics treats the inertial forces applied to the moving (accelerated) bodies as fictitious, imaginary ones, but recognizes at the same time the reality of inertial forces applied to braces or accelerating bodies. A quite opposite standpoint on inertial forces is agreed upon in courses of applied mechanics and engineering. They usually regard the inertial forces applied namely to moving bodies as real ones (similarly to elastic, damping and other forces), which are responsible for many real phenomena. But such a natural (and clear for engineers) treatment has no sense within the framework of the "legalized" methodology of classical
Appendix: Inertial Forces and Methodology of Mechanics
241
mechanics and is considered as a primitive one, as a display of shortage of common sense, and the known discussions on inertial forces were devoted namely to the struggle against such notions. But it is not only the different approaches that have been well established in physics, theoretical and applied mechanics that cause confusion. Within physics (as well as theoretical mechanics) one can easily bring out an inconsistency in treatment of the subject. When considering experiments and various concrete phenomena, authors of modern courses of mechanics and physics often stand on the viewpoint of applied mechanics and deal with inertial forces as with real ones (see, e.g., Feynman physics course [3]). This inconsistency is not accidental but is a consequence of serious difficulties in the methodology of mechanics. Let us consider, for example, such a simple question: "Do centrifugal forces act in rotating bodies?". Now mechanics gives the following answer: "These forces are absent in inertial reference systems, but they exist from the viewpoint of an observer in the non-inertial reference system rotating with the body". But let us compare the forces for two cases. Let a material point and a body rotate around fixed axes (Fig.l, (a), (b), (c)). In the case of a material point the centrifugal force exists both in the inertial coordinate system (as the force exerted by the point mass on its brace, according to Newton's third law), and in the rotating system, where this force is the translatory inertia force. In the case of a rotating body centrifugal forces exist in the co-rotating reference frame but they are absent, according to the traditional methodology, in the inertial coordinate system.
Fig.l
242
Mechanics of Systems with Internal Resonances
These questions arise: why the forces acting in the rotating body are different in the inertial and rotating coordinate systems, whilst the force acting from the material point on the thread is the same in all reference systems (as well as the force acting from the thread on the mass point)? And if any rotating elementary mass exerts on its surroundings by an elementary centrifugal force, why these forces are absent in the rotating body for the observer in the inertial coordinate system? (One could put more general question: how can force represent a physical quantity if it is not an invariant or a tensor?). It is only one simple example among the many questions which the traditional methodology of mechanics cannot give an answer. The traditional viewpoint of physics and theoretical mechanics on inertial forces (which denies, in particular, the existence of centrifugal forces) leads to deadlock. All attempts of consistent treatment of the foundations of mechanics with separation of inertial forces ("fictitious") from real ("physical") forces, which were made repeatedly (e.g., see monograph by Ishlinskiy [4]), were unconvincing and led only to additional confusion. Note that Newton in "Mathematical Principles of Natural Philosophy" considered the centrifugal force as real one (regardless of the choice of coordinate system [1]). In this connection we would like to draw attention to papers by L. I. Sedov [5] and G. Ju. Stepanov [6], which assert the reality of inertial forces. But these papers give no answer on the main question: how to concord this statement with the foundations of classical mechanics? How must we change its methodology? It is shown in the paper that the cause of these difficulties lies in the very foundations of mechanics, in certain discordance of the notion "force" in its principal laws. These difficulties find natural resolution in the framework of the proposed interpretation of the fundamentals of mechanics.
2. Existence of Spatial Inertial Forces We have to begin from the very beginning. Let us consider a body 1 on which a force F acts (from a body 2) imparting to this body an accelerated motion with respect to an inertial reference system (Fig. 2, (a)). The body 1 acts on the body 2 with force F* = - F, which will be called "inertia force" (following Newton [1]).
Appendix: Inertial Forces and Methodology of Mechanics
243
Let us consider the body 1 not as a material point but as a solid (for simplicity as a uniform bar). Cutting body 1 in an arbitrary crosssection on two parts and writing equations of motion for each part, we easily ascertain using Newton's third law that the longitudinal force in the bar varies along the length from 0 to F (in this case linearly, Fig. 2, (c)). Such a change of the force in statics would testify the existence of spatial forces. Does this statement hold in dynamics, i.e., do spatial inertia forces act in the body? Dividing the body on several parts and applying Newton's laws to each part, we also easily ascertain that force F equals to sum of the Newtonian inertial forces for all these parts. Continuing such division up to infinitely small volumes, we come to the conclusion that the force F is the sum (integral) of the elementary inertia forces for all elementary parts. Thus, spatial Newtonian inertial forces act in the accelerated body 1.
Fig. 2 The reasoning presented here is elementary, but as a matter of fact even it is not necessary. It is enough to ask the question: what gives rise to force F, the existence of which is asserted by the third law? The usual answer — this force appears according to the third law — explains nothing. The cause of force may be only a change in the state of the body. But the change is that the body 1 begins to move with acceleration (with respect to the inertial reference frame). Hence force F is generated by accelerated motion of body 1 with respect to the inertial coordinate system. But if a body with mass m generates a force -mW then each elementary mass dm must generate an elementary inertial force -dmW (obviously, additiveness of mass implies namely this).
244
Mechanics of Systems with Internal Resonances
The spatial Newtonian inertia forces are the same inertia forces, which are introduced in the d'Alembert principle. Hence "D'AIembertian inertia forces" are as real as the force F*. (Of course, one may name these forces "spatial counteraction forces" but it will be shown below that our terminology is preferable). The use of the solid model instead of the material point model enables us to make sure that the standpoint of theoretical mechanics —"inertia forces are real ones but they are applied not to given body but to its braces or "accelerating bodies"— has no sense. It is inconsistent to declare that the spatial inertia forces are fictitious but the force F* is real. In this sense there is no difference between inertial forces and, e.g., gravity forces. It would be strange to assert that the pressure force of a body lying on a table is real but the gravity forces at each point of the body are fictitious. However during more than two hundred years classical mechanics declares such a statement with respect to inertial forces considering spatial inertial forces as imaginary ones but their sum F* -as a real one! Let us remember known experimental observations described in popular books on mechanics. In the middle of the nineteenth century English botanist Night plants bean seeds on the felloe of a rotating wheel. It is known that plants always grow against the gravity force. How did these plants grow? Their stems were directed inward (to the axis of rotation), and roots outwards. This experiment unambiguously has shown that each cell of the plant undergoes the action of the centrifugal force, i.e., Newtonian (or D'AIembertian, that is the same) inertia force. Thus the following reciprocally connected conclusions can be drawn from the Newton's laws: • each material point at accelerated motion gives rise to an inertial force; spatial "Newtonian inertia forces" act in accelerated bodies; • these spatial forces are identical to "D'AIembertian inertia forces" which are, hence, real forces. It is important to make the following remark. Accepting the third law we thereby acknowledge that not only a cause of accelerated motion is a force but, inversely, any accelerated motion generates a force. The inertia forces are not a cause of the motion of the body; they are themselves caused by the motion. Newton wrote: "This force is manifested by body only when another force applied to it causes a
Appendix: Inertial Forces and Methodology of Mechanics
245
change in its state"[l]. Hence for each body we should discern forces causing its motion, and forces caused by the motion, inertia forces. Distinction between forces causing motion and those caused by motion is clearly seen on the example of rotating bodies. A torque causes rotation, the rotation causes accelerations of all points and therefore centrifugal forces are induced which are balanced by centripetal forces (due to deformations of the body). Of course, the division of forces on "causing motion" and "caused by motion" one has sense only with respect to a given body, and there is no actual difference between inertial forces and those forces which are acknowledged by physics as real ones. If we consider two interacting bodies, these forces commute — the force caused by the motion of body 1 (its inertial force) causes the motion of body 2, and inversely. Generally speaking, main forces acting at collisions of bodies are inertial forces; other forces are caused by the inertial forces or negligible. 3. "Paradox" of Forces Balance. Discordance of Notion "Force" in Two Main Laws of Mechanics But let us consider now the set of forces applied to body 1. This set — force F4 and the Newtonian (or D'Alembertian) spatial inertial forces— turns out to be balanced. The question arises — how can the body move with acceleration, if appliedforces are balanced? The way out of this "contradiction" is apparent, it follows naturally from the reasoning above about distinction of forces causing motion of body, and forces generated by the motion. Inertial forces enter in the left hand side of the second law equation. Term mF* is not simply a product of mass by acceleration, but also a force, namely an inertia force (with the opposite sign). Forces causing motion of the body are written at right hand side, and forces caused by the motion at the left. Hence sense of the second law is: under action of forces a body performs such an accelerated motion that inertial forces generated by this motion counterbalance the applied forces. Thus the above "paradox" is resolved simply by changing the treatment of the second law, namely by considering it as equality (at opposite directions) of two forces — the force applied to body and causing its acceleration and the inertial force caused by this acceleration. Such an idea is an essence of the D'Alembert principle (though it was not initial idea of the principle author).
246
Mechanics of Systems with Internal Resonances
We would like to repeat that the balance of forces really occurs due to accelerated motion. Forces applied to body are balanced not only in statics but also in dynamics. The distinction between dynamics and statics is only that in dynamics a part of the forces appears as a result of accelerated motion. Physics always identified two notions — rest state (uniform motion) of the body and the balance of forces applied to the body. We should separate these notions. If forces initially applied are balanced then the body is in the rest state; but if these forces are not balanced then balancing is achieved due to inertial forces. Let us compare again two principal laws of mechanics. We should note a principal difference between these laws. Newton's third law implies all forces — causing motion and caused by motion (only due to Newtonian inertial forces does this law hold). But the second law, in its usual formulation, names "force" only forces causing motion, but not all forces acting on the body in the process of motion. Only the terms entering in the right hand side are named "force", and by that inertia forces are excluded from the class of "physical forces". Hence, in certain sense we may speak about certain discordance of notion "force" in two principal laws of mechanics. Namely this difference in treatment of notion "force" by two main laws is the cause of the contradiction in methodology of classical mechanics. This contradiction remains implicit in the dynamics of material points, but manifests itself in dynamics of solids. Recognition that -mF is also a force eliminates the contradiction between these two laws. Classical mechanics could not reject the D'Alembert principle; it has become the foundation of analytical dynamics. But classical mechanics has not changed its methodology, the inherent contradiction between both laws has not been overcome, and D'Alembertian inertial forces have been introduced in mechanics as "a convenient fib", as "a fiction", not as real forces. It should be emphasized that the very technique of mechanics is noncontradictory as analytical dynamics is based on account of the term mF as a force and on balancing forces applied to body in dynamics as well as in statics. The problem is the elimination of the contradiction between the techniques of mechanics and its methodology. It goes without saying that the inertial forces manifest themselves in stresses, deformations and failures of bodies; they perform work. They have potential, and this is the body's kinetic energy [5].
Appendix: Inertial Forces and Methodology of Mechanics
247
4. Non-inertial Reference Frames. Invariance of Forces in Mechanics Let us consider the translatory and Coriolis inertia forces in non-inertial reference frames, i.e., namely those forces with which physics relates exclusively notion "inertial forces". It is evident, in the light of the above said, that such an approach, when inertia forces are connected only with the relative motion, is principally limited and does not give possibility grasping the meaning of the problem. The more general viewpoint, which recognises the reality of D'Alembertian inertia forces, enables us easily to bring to light the sense of the translatory and Coriolis inertia forces. Inertial forces are determined by acceleration with respect to the inertial reference frame, i.e., by the absolute acceleration of the material point. In non-inertial reference frames the absolute acceleration is the sum of the translatory, Coriolis and relative accelerations Wc, Wz, Wv and so the full, or absolute, inertial force is sum of three items: F = -mW3 = -m{Wc +fVc+ Wt) =F> + F' + F', where FT' = - mWr is the relative inertia force (this term was introduced for the first time, apparently, by L. I. Sedov [5]). In non-inertial coordinate systems two components of the full inertial force — translatory and Coriolis forces — are transferred to the right hand side of the equation of motion. So the distinction of non-inertial reference frames from inertial ones is only the division of the full inertial force into three components, two of which being forces caused by motion, are regarded formally as forces causing the motion. Of course such breaking up of the full inertial force on three terms depends on the choice of the reference frame. But the arbitrariness of the division of the absolute inertia force on components cannot be a reason for treating them as "pseudoforces", similarly as projections of a force on coordinate axes are not regarded as fictitious quantities, although they depend upon the choice of a coordinate system. Thus the translatory and Coriolis inertia forces are only components of the full inertia force (Newtonian, or D'Alembertian). Now we can give an answer the following principal question: do forces acting on a body depend upon the choice of coordinate system? The key moment here is the presence of the relative inertia force.
248
Mechanics of Systems with Internal Resonances
Physics until now considered only two terms — the translatory and Coriolis inertia forces — and could not regard them as real forces since these forces (or any their combination) do not constitute an invariant and depend upon the choice of the coordinate system. Only with the addition of the third component — relative force of inertia — these forces constitute an invariant. Taking into account the relative inertial force, we obtain the same force in any reference frame (though observers in different coordinate systems may name this force differently: e. g., for one observer it may be the translatory inertial force, for another the relative inertial force). The above treatment of inertial forces entirely conforms (in distinction from the present-day dominating standpoint) with the fundamental statement of general relativity theory about equivalence of inertial and gravity forces. We must recognize that in the arguments an with engineer's intuition and common sense about reality of inertial forces, applied to bodies accelerated, mechanics is found to be wrong. With rehabilitation of D'Alembertian inertial forces and disappearance of the phantom of "fictitious" forces the methodology of mechanics becomes clear and transparent.
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Ladygina Ye. V. and Manevich, A. I. (1993). Free oscillations of a non-linear cubic system with two degrees of freedom and close natural frequencies. J. Appl. Maths Mechs, v. 57, No. 2, pp. 257 - 266. Ladygina Ye. V. and Manevich, A. I. (1997). Non-linear free oscillations of a cylindrical shell with account of the interaction of conjugate modes. Mechanics of Solids (Proc. of. Russian Acad. of Sciences), No.3, pp. 169 - 175. (in Russian). Lamb G.L., Jr. (1980). Elements of soliton theory, Wiley, New York-ChichesterBrisbane-Toronto. Lonngren, K. and Scott A. (ed.) (1978). Solitons in Action, Academic Press, New York. San Francisco- London , Lomdahl, P. S. (1985). Solitons in Josephson junctions: an overview. J. Stat. Phys., 39 (5/6), pp. 551-561. Louisell, W. H. (1960). Coupled mode and parametric electronics, New York, Wiley. Louisell, W. H. (1962). Correspondence between Pierce's coupled mode amplitudes and quantum operators, Journal of Applied.Physics, 33, pp. 2435 - 2436. Manevich, A. I. (1994). Interaction of conjugate modes at non-linear free flexural oscillations of a circular ring. Appl. Maths Mechs, v. 58, No. 6, pp. 119 - 125 (in Russian). English Transl.: J. Appl. Maths Mechs, v. 58, No. 6, Pergamon Press, 1995, Elsevier Science, pp. 1061 - 1068. Manevich, A. I. (1999). Bifurcations of stationary free oscillations of a cubic two-degreeof-freedom system having close eigenfrequencies. In: Theoretical Foundations of Civil Engineering - VII. Warsaw, OW PW, pp. 133 - 40 (in Russian). Manevich, A.I. (2002). Free and forced flexural nonlinear oscillations of a circular ring with account of interaction of conjugate modes. In: Nonlinear Dynamics of Shells with Fluid-Structure Interaction. NATO CLG Grant Report. Proceedings, Praque, pp. 141-155. Manevich, A. I. and Ladygina, E. V. (1993). Traveling waves at free flexural oscillations of cylindrical shells. In: "Theoretical Foundations of Civil Engineering". Proc. of Ukrainian-Polish Seminar-1, Warsaw, pp. 51 - 57. Manevich, A. I. and Manevich, E. L. (1999a). Internal resonances at free oscillations of several-degree-of-freedom systems with quadratic non-linearities. In: "Problems of non-linear mechanics and physics of materials", Dniepropetrovsk, 1998, pp. 124- 138 (in Russian). Manevich, A. I. and Manevitch, L. I. (1999b). Free oscillations of a dissipative cubic two-degree-of-freedom system with closed natural frequencies. Proc. of 5th Conference on Dynamical Systems — Theory and Applications. Poland, Lodz, pp. 257 - 264. Manevich, A. I. and Manevitch, L. I. (2003). Free oscillations in conservative and dissipative symmetric cubic two-degree-of freedom systems with closed natural frequencies, Meccanica, Int. J. Italian Assoc. of Theor. and Appl. Mechs., v. 38, No. 3, pp. 335 - 348.
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Index
amplitude-phase portrait at primary resonance, 42 at subharmonic resonance, 22, 23 at superharmonic resonance, 33, 34 circular ring, 145, 146, 148 2DOF autonomous system, 51,68, 69, autonomous 2DOF systems, 45-76 coupled stationary modes, 52-68 elliptic modes, 52-68, 142, 154 normal modes, 52-68, 174 governing equations, 46 nonstationary oscillations, 68-74 in dissipative systems, 71-74 in undamped systems, 68-71
soliton-lyke solution, 197 chain on nonlinear asymmetric substrate, 222-235 equations of motion, 224 hamiltonian, 223 c h a n g e o f dependent variable, 10-12 circular ring amplitude-phase portraits, 145-148 conjugate modes, 134 equations of motion, 136-138 equations of amplitude-frequency modulation, 139, 140 forced oscillations, 151-165 coupled stationary modes, 152— 165 bifurcations, 155, ] 56— 159 condition of existence, 157 exact internal resonance, 157162 frequency response curves, 155, 161, 162 inexact internal resonance,
backbone curves, 84, 88 bending flexibility, 222 bifurcations of stationary modes, 19, 20, 31,40-43, 54, 56, 84, 111, 142, 145, 174 branching points, 111, 155, 156 breather, 224, 231-236 chain of coupled particles, 191-198 equations of motion, 191 gradient-type potential, 191 long wavelengths modes, 192 short wavelengths modes, 193-197
,, . ,,„
,,„ . , ,
P a t h s l n 3 D - s P a c e - 160> 1 6 4 ran e of
8 existence, 160, 163 governing equations, 153, 156 uncoupled vibration, 151-154 257
258
Mechanics of Nonlinear Systems with Internal Resonances
frequency response curves, 153, 165 free oscillations, 140-151 coupled stationary modes, 141148 bifurcations, 142, 145 condition of existence, 142 frequency, 142
governing equations, 140
nonstationary oscillations, 148general solution, 148-151 period of modulation, 149,
stability, 147 uncoupled vibration, 141 integral of energy, 141 integral of amplitude-frequency modulation, 145 running waves, 143, 146, 149, 159 splitting of natural frequencies, 134 I37 ' traveling wave, 134, 149 complex equations of motion, 1 configurational plane, 52 continuum approximation, 184, 226 coupled chains, 198 equations of motion, 198 long wavelength modes, 199, 200 potential, 198 short wavelength modes, 200-202 coupled nonlinear oscillators, 169-170 amplitude-phase portraits, 175-177 bifurcation, 174 equations of motion, 169 equations of amplitude-frequency modulation, 172, 174 integrals, 173
localized normal modes, 177 stability, 178, 179 coupled stationary modes, 52-68 autonomous 2DOF system, 51 conditions of existence, 54 non-autonomous 2DOF system, 104 circular ring, 152 , .
JSSSSSSTSSjS^
Duffing
equation
non-autonomous, 12 • 5 _, Q wkh d elastic non-homogeneity, 214-219 ^ ^ 154 bifurcations, 54, 56-68 conditions of existence, 54 frequency 53 stability, 55, 56 envelope solitons, 185, 230 exact interna] and e x t e m a , resonances, 85-86 e ,, i p t i c m o d e S j
frequency response curves a t primary resonance, 40 a t subharmonic resonance, 24 a t superharmonic resonance, 30, 31 j n circular ring, 153 j n non-autonomous 2DOF systems, 90-100, 117, 123-128 Hamiltonian, 49 Hessian, 24 55,56,120 historical remarks, 43, 44, 76, 133, 168, 188-189, 221, 235, 236 infinite chains of coupled oscillators, 179-188 equations of motion, 179, 180 equations of amplitude-frequency
Index modulation, 182, 174 localized normal modes, 187, 188 solution in principal approximation, 183 stability, 186 strong coupling, 184 integral of amplitude-frequency modulation at subharmonic resonance, 22 at superharmonic resonance, 33 in autonomous 2DOF system, 51 in circular ring 144 145, 151 Jacobi elliptic functions, 150 Kadomtsev-Petviashvili equation, 199, 203 , w • <.• ,„_ , „ , v Korteveg de Vnes equat.on, 192, 193, multiple scales method, 2-5, 11-14, 4750 non-autonomous 2DOF systems, 77-133 govern ing eq uations, 78 damped systems, 101-131 amplitude-frequency modulation equations, 81 coupled stationary modes, 103-128 asymptotic behavior, 106,107 bifurcation points, 110-118 conditions of existence, 104, 105 dimensionless parameters, 105 governing equations, 104 numerical analysis, 121-128 qualitative analysis, 107-110 response curves, 117, 122-128 stability, 118-121 nonstationary oscillations, 128-131 undamped systems, 82-103 amplitude-frequency modulation
259 equations, 82 coupled stationary modes, 82-86 asymptotic behavior, 86 bifurcation points, 84, 85 elliptic modes, 83-86, 142, 154 governing equations, 82 normal modes, 83-86, 174 response curves, 88-100 rf-uii;*,, c< QT stability, 86, 87 nonstationary oscillations, 101-103 non-homogeneous chain of coupled oscillators, 205-220 elastic non-homogeneity, 214-218 acoustic branch, 214-216 optical branch, 216-218 inertial non-homogeneity, 206-214 B acoustic branch 206/212 optical branch, 212-214 numerical results, 218-220 nonlinear Shrodinger equation, 184 nonlinear substrate, 224, 225, 237, 238 nonresonant case, 14-16 nonstationary oscillations at primary resonance, 41-43 at subharmonic resonance in damped systems, 24-26 in undamped systems, 21 -24 at superharmonic resonance in damped systems, 35-36 in undamped systems, 32-35 in autonomous 2DOF systems, 68 in circular rings, 148-151 in non-autonomous 2DOF systems damped systems, 128-131 undamped systems, 102-104 normal modes, 52, 174 bifurcations, 54, 56-68 conditions of existence, 54
260
Mechanics of Nonlinear Systems with Internal Resonances
frequency, 53 stability, 55, 56 numerical solution autonomous 2DOF system, 70, 71 subharmonic resonance, 28 superharmonic resonance, 37 optical oscillations branch, 213-214, 217-219 orthogonality conditions, 3,4, 11 oscillatory chains, 224, 225 period of modulation, 149-151 primary resonance in 1DOF systems, 3843 nonstationary oscillations, 21-29 damped systems, 43 undamped systems, 41,42 amplitude-phase portraits, 42 integral of amplitude-frequency modulation, 42 steady-state modes, 39—41 frequency response curve, 39^41 stability, 23, 24 second approximation, 12 separatrices, 23, 34 Schrodinger nonlinear equation, 185, 188,221 Sine-Gordon equation, 200, 203 solitons, 185, 219, 220 stability of stationary modes autonomous 2DOF systems, 56-58 non-autonomous 2DOF systems, 86-87, 119-121 steady-state modes at primary resonance in 1DOF systems, 39—4] at subharmonic resonance, 17-21 at superharmonic resonance, 30-32
in autonomous 2DOF systems, 51-55 in circular rings, 141 in non-autonomous 2DOF systems undamped systems, 82 damped systems, 103 paths, autonomous 2DOF systems, 2D plots 60 61 68,159 161 ,62 65 ,„ ' ., . . . . . . . . 3D P l o t s ' 6 1 ^ 6 > 160> > 64 non-autonomous 2DOF systems, 89, 91, 94, 97 subharmonic resonance, 16-29 nonstationary oscillations, 21-29 damped systems, 24-28 direction fields, 25, 26 undamped systems, 21-24 amplitude-phase portraits, 22, 23 integral of amplitude-frequency modulation, 22 steady-state modes, 17-21 conditions of existence, 19, 20 frequency response curve, 21 stability, 23, 24 superharmonic resonance, 29-38 nonstationary oscillations, 32-38 damped systems, 35-38 direction fields, 36 undamped systems, 32-35 amplitude-phase portraits, 33, 34 integral of amplitude-frequency modulation, 33 steady-state modes, 30-32 frequency response curve, 31 stability, 33, 34 Van der Pol equation, 9