Nonlinear multi-wave coupling and resonance in elastic structures (стр. 2 из 5)

(5)

As one can see the right-hand term in eq. (5) contains a spatio-temporal parameter in the form of a standing wave. Allowances for the this wave-like parametric excitation become principal, if the typical velocity of longitudinal waves is comparable with the group velocities of bending waves, otherwise one can restrict consideration, formally assuming that

, to the following simplest model:

(6)

which takes into account the temporal parametric excitation only.

We can look for solutions to eq. (5), using the Bubnov-Galerkin procedure:

where

denote the wave numbers of bending waves;

are the wave amplitudes defined by the ordinary differential equations

(7)

Here

stands for a coefficient containing parameters of the wave-number detuning:

, which, in turn, cannot be zeroes;

are the cyclic frequencies of bending oscillations at

;

denote the critical values of Euler forces.

Equations (7) describe the early evolution of waves at the expense of multi-mode parametric interaction. There is a key question on the correlation between phase orbits of the system (7) and the corresponding linearized subset

(8)

which results from eqs. (7) at

. In other words, how effective is the dynamical response of the system (7) to the small parametric excitation?

First, we rewrite the set (7) in the equivalent matrix form:

, where

is the vector of solution,

denotes the

matrix of eigenvalues,

is the

matrix with quasi-periodic components at the basic frequencies

. Following a standard method of the theory of ordinary differential equations, we look for a solution to eqs. (7) in the same form as to eqs. (8), where the integration constants should to be interpreted as new sought variables, for instance

, where

is the vector of the nontrivial oscillatory solution to the uniform equations (8), characterized by the set of basic exponents

. By substituting the ansatz

into eqs. (7), we obtain the first-order approximation equations in order

where the right-hand terms are a superposition of quasi-periodic functions at the combinational frequencies

. Thus the first-order approximation solution to eqs. (7) should be a finite quasi-periodic function [4], when the combinations

; otherwise, the problem of small divisors (resonances) appears.

So, one can continue the asymptotic procedure in the non-resonant case, i. e.

, to define the higher-order correction to solution[5]. In other words, the dynamical perturbations of the system are of the same order as the parametric excitation. In the case of resonance the solution to eqs. (7) cannot be represented as convergent series in

. This means that the dynamical response of the system can be highly effective even at the small parametric excitation.

In a particular case of the external force

, eqs. (7) can be highly simplified:

(9)

provided a couple of bending waves, having the wave numbers

and

, produces both a small wave-number detuning

(i. e.

) and a small frequency detuning

(i. e.

). Here the symbols

denote the higher-order terms of order

, since the values of

and

are also supposed to be small. Thus, the expressions

;

can be interpreted as the phase matching conditions creating a triad of waves consisting of the primary high-frequency longitudinal wave, directly excited by the external force

, and the two secondary low-frequency bending waves parametrically excited by the standing longitudinal wave.

Notice that in the limiting model (6) the corresponding set of amplitude equations is reduced just to the single pendulum-type equation frequently used in many applications:

It is known that this equation can possess unstable solutions at small values of

and

Solutions to eqs. (7) can be found using iterative methods of slowly varying phases and amplitudes:

(10)

;

where

and

are new unknown coordinates.

By substituting this into eqs. (9), we obtain the first-order approximation equations

(11)

;

where

is the coefficient of the parametric excitation;

is the generalized phase governed by the following differential equation

Equations (10) and (11), being of a Hamiltonian structure, possess the two evident first integrals

and

which allows one to integrate the system analytically. At

, there exist quasi-harmonic stationary solutions to eqs. (10), (11), as