Nonlinear multi-wave coupling and resonance in elastic structures (стр. 3 из 5)

which forms the boundaries in the space of system parameters within the first zone of the parametric instability.

From the physical viewpoint, one can see that the parametric excitation of bending waves appears as a degenerated case of nonlinear wave interactions. It means that the study of resonant properties in nonlinear elastic systems is of primary importance to understand the nature of dynamical instability, even considering free nonlinear oscillations.

Normal forms

The linear subset of eqs. (0) describes a superposition of harmonic waves characterized by the dispersion relation

where

refer the

branches of the natural frequencies depending upon wave vectors

. The spectrum of the wave vectors and the eigenfrequencies can be both continuous and discrete one that finally depends upon the boundary and initial conditions of the problem. The normalization of the first order, through a special invertible linear transform

leads to the following linearly uncoupled equations

where the

matrix

is composed by

-dimensional polarization eigenvectors

defined by the characteristic equation

;

is the

diagonal matrix of differential operators with eigenvalues

;

and

are reverse matrices.

The linearly uncoupled equations can be rewritten in an equivalent matrix form [5]

(12)

and

using the complex variables

. Here

is the

unity matrix. Here

is the

-dimensional vector of nonlinear terms analytical at the origin

. So, this can be presented as a series in

, i. e.

where

are the vectors of homogeneous polynomials of degree

, e. g.

Here

and

are some given differential operators. Together with the system (12), we consider the corresponding linearized subset

(13)

and

whose analytical solutions can be written immediately as a superposition of harmonic waves

where

are constant complex amplitudes;

is the number of normal waves of the

-th type, so that

(for instance, if the operator

is a polynomial, then

, where

is a scalar,

is a constant vector,

is some differentiable function. For more detail see [6]).

A question is following. What is the difference between these two systems, or in other words, how the small nonlinearity is effective?

According to a method of normal forms (see for example [7,8]), we look for a solution to eqs. (12) in the form of a quasi-automorphism, i. e.

(14)

where

denotes an unknown

-dimensional vector function, whose components

can be represented as formal power series in

, i. e. a quasi-bilinear form:

(15)

for example

where

and

are unknown coefficients which have to be determined.

By substituting the transform (14) into eqs. (12), we obtain the following partial differential equations to define

(16)

It is obvious that the eigenvalues of the operator

acting on the polynomial components of

(i. e.

) are the linear integer-valued combinational values of the operator

given at various arguments of the wave vector

In the lowest-order approximation in

eqs. (16) read

The polynomial components of

are associated with their eigenvalues

, i. e.
, where

while

in the lower-order approximation in