Triple-wave ensembles in a thin cylindrical shell — страница 4

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-triads High-frequency bending waves in the shell can be unstable with respect to small perturbations of low-frequency bending and shear waves. Figure (4) displays an example of projection of the -type resonant manifold of the shell with the same sizes as in the previous sections. The spectral parameters of the primary high-frequency bending mode are and . The phase detuning also does not exceed one percent. The triple-wave resonant coupling can be observed only in the case when the group velocity of the primary high-frequency bending mode exceeds the typical velocity of shear waves. NBEssentially, the spectral parameters of -type triads fall near the boundary of the validity domain predicted by the Kirhhoff-Love theory. This means that the real physical properties of -type

triads can be different than theoretical ones. NB-type triads are essentially two-dimensional dynamical objects, since the nonlinearity parameter goes to zero, as all the waves propagate in the same direction. -triads High-frequency bending waves in the shell can be unstable with respect to small perturbations of low-frequency bending waves. Figure (5) displays an example of the projection of the -type resonant manifold of the shell with the same sizes as in the previous sections. The wave parameters of the primary high-frequency bending mode are and . The phase detuning does not exceed one percent. The triple-wave resonant coupling cannot also be observed only in the case of long-wave processes, since in such cases the parameter cannot be small. NBThe resonant interactions of

-type are inherent in cylindrical shells only. Manly-Rawe relations By multiplying each equation of the set (11) with the corresponding complex conjugate amplitude and then summing the result, one can reduce eqs.(11) to the following divergent laws (12) Notice that the equations of the set (12) are always linearly dependent. Moreover, these do not depend upon the nonlinearity potential . In the case of spatially uniform wave processes () eqs.(12) are reduced to the well-known Manly-Rawe algebraic relations (13) where are the portion of energy stored by the quasi-harmonic mode number ; are the integration constants dependent only upon the initial conditions. The Manly-Rawe relations (13) describe the laws of energy partition between the modes of the triad. Equations (13), being

linearly dependent, can be always reduced to the law of energy conservation (14). Equation (14) predicts that the total energy of the resonant triad is always a constant value , while the triad components can exchange by the portions of energy , accordingly to the laws (13). In turn, eqs.(13)-(14) represent the two independent first integrals to the evolution equations (11) with spatially uniform initial conditions. These first integrals describe an unstable hyperbolic orbit behavior of triads near the stationary point , or a stable motion near the two stationary elliptic points , and . In the case of spatially uniform dynamical processes eqs.(11), with the help of the first integrals, are integrated in terms of Jacobian elliptic functions [1,2]. In the particular case, as or ,

the general analytic solutions to eqs.(11), within an appropriate Cauchy problem, can be obtained using a technique of the inverse scattering transform [3]. In the general case eqs.(11) cannot be integrated analytically. Break-up instability of axisymmetric waves Stability prediction of axisymmetric waves in cylindrical shells subject to small perturbations is of primary interest, since such waves are inherent in axisymmetric elastic structures. In the linear approximation the axisymmetric waves are of three types, namely bending, shear and longitudinal ones. These are the axisymmetric shear waves propagating without dispersion along the symmetry axis of the shell, i.e. modes polarized in the circumferential direction, and linearly coupled longitudinal and bending waves. It was

established experimentally and theoretically that axisymmetric waves lose the symmetry when propagating along the axis of the shell. From the theoretical viewpoint this phenomenon can be treated within several independent scenarios. The simplest scenario of the dynamical instability is associated with the triple-wave resonant coupling, when the high-frequency mode breaks up into some pairs of secondary waves. For instance, let us suppose that an axisymmetric quasi-harmonic longitudinal wave ( and ) travels along the shell. Figure (6) represents a projection of the triple-wave resonant manifold of the shell, with the geometrical sizes m; m; m, on the plane of wave numbers. One can see the appearance of six secondary wave pairs nonlinearly coupled with the primary wave. Moreover,