Triple-wave ensembles in a thin cylindrical shell (стр. 4 из 5)

-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

, 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, on the plane of wave numbers. One can see the appearance of six secondary wave pairs nonlinearly coupled with the primary wave. Moreover, in the particular case the triple-wave phase matching is reduced to the so-called resonance 2:1. This one can be proposed as the main instability mechanism explaining some experimentally observed patterns in shells subject to periodic cinematic excitations [4].

It was pointed out in the paper [5] that the resonance 2:1 is a rarely observed in shells. The so-called resonance 1:1 was proposed instead as the instability mechanism. This means that the primary axisymmetric mode (with

) can be unstable one with respect to small perturbations of the asymmetric mode (with

) possessing a natural frequency closed to that of the primary one. From the viewpoint of theory of waves this situation is treated as the degenerated four-wave resonant interaction.

In turn, one more mechanism explaining the loss of stability of axisymmetric waves in shells based on a paradigm of the so-called nonresonant interactions can be proposed [6,7,8]. By the way, it was underlined in the paper [6] that theoretical prognoses relevant to the modulation instability are extremely sensible upon the model explored. This means that the Karman-type equations and Donnell-type equations lead to different predictions related the stability properties of axisymmetric waves.

Self-action

The propagation of any intense bending waves in a long cylindrical shell is accompanied by the excitation of long-wave displacements related to the in-plane tensions and rotations. In turn, these long-wave fields can influence on the theoretically predicted dependence between the amplitude and frequency of the intense bending wave.

Moreover, quasi-harmonic bending waves, whose group velocities do not exceed the typical propagation velocity of shear waves, are stable against small perturbations within the lowest-order nonlinear approximation analysis. However amplitude envelopes of these waves can be unstable with respect to small long-wave perturbations in the next approximation.

Amplitude-frequency curve

Let us consider a stationary wave

traveling along the single direction characterized by the ''companion'' coordinate

. By substituting this expression into the first and second equations of the set (1)-(2), one obtains the following differential relations

(15)

Here

while

where

and

Using (15) one can get the following nonlinear ordinary differential equation of the fourth order:

(16)

which describes simple stationary waves in the cylindrical shell (primes denote differentiation). Here

where

and

are the integration constants.

If the small parameter

, and

satisfies the dispersion relation (4), then a periodic solution to the linearized equation (16) reads

where

are arbitrary constants, since

Let the parameter

be small enough, then a solution to eq.(16) can be represented in the following form

(17)

where the amplitude

depends upon the slow variables

, while

are small nonresonant corrections. After the substitution (17) into eq.( 16) one obtains the expression of the first-order nonresonant correction