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1 +3|q>,| V " ' +9, V " V
(3.1.8)
^ T (
922 T 2 ' 0 1 + 2cp, |cp2|2 + 9 , cp22 + 9 l cp22 ^ 2 '"' +
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| 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 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 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
105
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
106
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)
Non-autonomous Two-Degree-of-Freedom Cubic Systems
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;
108
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 6120. 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)-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
Non-autonomous Two-Degree-of-Freedom Cubic Systems
109
a 2 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)):
120
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 with various magnitudes of 8 ° - a °. As a first example of systems with different signs of bn and b22 (ot2 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 one infinite NM path and one finite EM path exist (Fig. 2.5, (a)), but at CT 0 does not differ from that for the case a °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) 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|=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, 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 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 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 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 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, 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
/[ K 0 7 i • (4.2.27)
Assuming that a*>0 and, respectively, ^ 3 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#, ' =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)
154
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
„,«-„;. 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 4b) and an imaginary ellipsis, if A>0 {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 =,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).
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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
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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):
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Localized Normal Modes in a Chain of Nonlinear Oscillators
T — (q>;, o + £ 9 y , t + £ 2 9 A 2 +•••)+ s — (7C0
g2
(..+ E 2(P;,2 +••) +
CT,
T — (
