Реферат: Nonlinear multi-wave coupling and resonance in elastic structures

which allows one to integrate the system analytically. At , there exist quasi-harmonic stationary solutions to eqs. (10), (11), as

,

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) ,