Nonlinear multi-wave coupling and resonance in elastic structures

Kovriguine DA

Solutions to the evolution equations describing the phase and amplitude modulation of nonlinear waves are physically interpreted basing on the law of energy conservation. An algorithm reducing the governing nonlinear partial differential equations to their normal form is considered. The occurrence of resonance at the expense of nonlinear multi-wave coupling is discussed.

Introduction

The principles of nonlinear multi-mode coupling were first recognized almost two century ago for various mechanical systems due to experimental and theoretical works of Faraday (1831), Melde (1859) and Lord Rayleigh (1883, 1887). Before First World War similar ideas developed in radio-telephone devices. After Second World War many novel technical applications appeared, including high-frequency electronic devices, nonlinear optics, acoustics, oceanology and plasma physics, etc. For instance, see [1] and also references therein. A nice historical sketch to this topic can be found in the review [2]. In this paper we try to trace relationships between the resonance and the dynamical stability of elastic structures.

Evolution equations

Consider a natural quasi-linear mechanical system with distributed parameters. Let motion be described by the following partial differential equations

(0) ,

where denotes the complex -dimensional vector of a solution;  and  are the  linear differential operator matrices characterizing the inertia and the stuffiness, respectively;  is the -dimensional vector of a weak nonlinearity, since a parameter  is small[1];  stands for the spatial differential operator. Any time  the sought variables of this system  are referred to the spatial Lagrangian coordinates .

Assume that the motion is defined by the Lagrangian . Suppose that at  the degenerated Lagrangian  produces the linearized equations of motion. So, any linear field solution is represented as a superposition of normal harmonics:

.

Here  denotes a complex vector of wave amplitudes[2];  are the fast rotating wave phases;  stands for the complex conjugate of the preceding terms. The natural frequencies  and the corresponding wave vectors  are coupled by the dispersion relation . At small values of , a solution to the nonlinear equations would be formally defined as above, unless spatial and temporal variations of wave amplitudes . Physically, the spectral description in terms of new coordinates , instead of the field variables , is emphasized by the appearance of new spatio-temporal scales associated both with fast motions and slowly evolving dynamical processes.

This paper deals with the evolution dynamical processes in nonlinear mechanical Lagrangian systems. To understand clearly the nature of the governing evolution equations, we introduce the Hamiltonian function , where . Analogously, the degenerated Hamiltonian  yields the linearized equations. The amplitudes of the linear field solution  (interpreted as integration constants at ) should thus satisfy the following relation , where  stands for the Lie-Poisson brackets with appropriate definition of the functional derivatives. In turn, at , the complex amplitudes are slowly varying functions such that . This means that

(1)  and ,

where the difference  can be interpreted as the free energy of the system. So that, if the scalar , then the nonlinear dynamical structure can be spontaneous one, otherwise the system requires some portion of energy to create a structure at , while  represents some indifferent case.

Note that the set (1) can be formally rewritten as

(2) ,

where  is a vector function. Using the polar coordinates , eqs. (2) read the following standard form

(3) ; ,

where . In most practical problems the vector function  appears as a power series in . This allows one to apply procedures of the normal transformations and the asymptotic methods of investigations.

Parametric approach

As an illustrative example we consider the so-called Bernoulli-Euler model governing the motion of a thin bar, according the following equations [3]:

(4)

with the boundary conditions

By scaling the sought variables:  and , eqs. (4) are reduced to a standard form (0).

Notice that the validity range of the model is associated with the wave velocities that should not exceed at least the characteristic speed . In the case of infinitesimal oscillations this set represents two uncoupled linear differential equations. Let , then the linearized equation for longitudinal displacements possesses a simple wave solution

,

where the frequencies  are coupled with the wave numbers  through the dispersion relation . Notice that . In turn, the linearized equation for bending oscillations reads[3]

(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  or , 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. ............