An advantage of the transform (7) is that the average Lagrangian depends only upon the slowly varying complex amplitudes and their derivatives on the slow spatio-temporal scales , and . In turn, the average Lagrangian does not depend upon the fast variables.
The average Lagrangian can be formally represented as power series in :
At the average Lagrangian (8) reads
where the coefficient coincides exactly with the dispersion relation (3). This means that .
The first-order approximation average Lagrangian depends upon the slowly varying complex amplitudes and their first derivatives on the slow spatio-temporal scales , and . The corresponding evolution equations have the following form
Notice that the second-order approximation evolution equations cannot be directly obtained using the formal expansion of the average Lagrangian , since some corrections of the term are necessary. These corrections are resulted from unknown additional terms of order , which should generalize the ansatz (3):
provided that the second-order approximation nonlinear effects are of interest.
Triple-wave resonant ensembles
The lowest-order nonlinear analysis predicts that eqs.(9) should describe the evolution of resonant triads in the cylindrical shell, provided the following phase matching conditions
hold true, plus the nonlinearity in eqs.(1)-(2) possesses some appropriate structure. Here is a small phase detuning of order , i.e. . The phase matching conditions (10) can be rewritten in the alternative form
where is a small frequency detuning; and are the wave numbers of three resonantly coupled quasi-harmonic nonlinear waves in the circumferential and longitudinal directions, respectively. Then the evolution equations (9) can be reduced to the form analogous to the classical Euler equations, describing the motion of a gyro:
Here is the potential of the triple-wave coupling; are the slowly varying amplitudes of three waves at the frequencies and the wave numbers and ; are the group velocities; is the differential operator; stand for the lengths of the polarization vectors ( and ); is the nonlinearity coefficient:
Solutions to eqs.(11) describe four main types of resonant triads in the cylindrical shell, namely -, -, - and -type triads. Here subscripts identify the type of modes, namely () - longitudinal, () - bending, and () - shear mode. The first subscript stands for the primary unstable high-frequency mode, the other two subscripts denote the secondary low-frequency modes.
A new type of the nonlinear resonant wave coupling appears in the cylindrical shell, namely -type triads, unlike similar processes in bars, rings and plates. From the viewpoint of mathematical modeling, it is obvious that the Karman-type equations cannot describe the triple-wave coupling of -, - and -types, but the -type triple-wave coupling only. Since -type triads are inherent in both the Karman and Donnell models, these are of interest in the present study.
High-frequency azimuthal waves in the shell can be unstable with respect to small perturbations of low-frequency bending waves. Figure (2) depicts a projection of the corresponding resonant manifold of the shell possessing the spatial dimensions: and . The primary high-frequency azimuthal mode is characterized by the spectral parameters and (the numerical values of and are given in the captions to the figures). In the example presented the phase detuning does not exceed one percent. Notice that the phase detuning almost always approaches zero at some specially chosen ratios between and , i.e. at some special values of the parameter. Almost all the exceptions correspond, as a rule, to the long-wave processes, since in such cases the parameter cannot be small, e.g. .
NB Notice that -type triads can be observed in a thin rectilinear bar, circular ring and in a flat plate.
NBThe wave modes entering -type triads can propagate in the same spatial direction.
Analogously, high-frequency shear waves in the shell can be unstable with respect to small perturbations of low-frequency bending waves. Figure (3) displays the projection of the -type resonant manifold of the shell with the same spatial sizes as in the previous subsection. The wave parameters of primary high-frequency shear mode are and . The phase detuning does not exceed one percent. The triple-wave resonant coupling cannot be observed in the case of long-wave processes only, since in such cases the parameter cannot be small.
NBThe wave modes entering -type triads cannot propagate in the same spatial direction. Otherwise, the nonlinearity parameter in eqs.(11) goes to zero, as all the waves propagate in the same direction. This means that such triads are essentially two-dimensional dynamical objects.
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.
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.
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
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
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
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 . 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