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

Название: Nonlinear multi-wave coupling and resonance in elastic structures
Раздел: Рефераты по физике
Тип: реферат Добавлен 16:40:54 07 декабря 2010 Похожие работы
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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) Nonlinear multi-wave coupling and resonance in elastic structures,

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

Assume that the motion is defined by the Lagrangian Nonlinear multi-wave coupling and resonance in elastic structures. Suppose that at Nonlinear multi-wave coupling and resonance in elastic structures the degenerated Lagrangian Nonlinear multi-wave coupling and resonance in elastic structures produces the linearized equations of motion. So, any linear field solution is represented as a superposition of normal harmonics:

Nonlinear multi-wave coupling and resonance in elastic structures.

Here Nonlinear multi-wave coupling and resonance in elastic structures denotes a complex vector of wave amplitudes[2] ; Nonlinear multi-wave coupling and resonance in elastic structures are the fast rotating wave phases; Nonlinear multi-wave coupling and resonance in elastic structures stands for the complex conjugate of the preceding terms. The natural frequencies Nonlinear multi-wave coupling and resonance in elastic structures and the corresponding wave vectors Nonlinear multi-wave coupling and resonance in elastic structures are coupled by the dispersion relation Nonlinear multi-wave coupling and resonance in elastic structures. At small values of Nonlinear multi-wave coupling and resonance in elastic structures, a solution to the nonlinear equations would be formally defined as above, unless spatial and temporal variations of wave amplitudes Nonlinear multi-wave coupling and resonance in elastic structures. Physically, the spectral description in terms of new coordinates Nonlinear multi-wave coupling and resonance in elastic structures, instead of the field variables Nonlinear multi-wave coupling and resonance in elastic structures, 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 Nonlinear multi-wave coupling and resonance in elastic structures, where Nonlinear multi-wave coupling and resonance in elastic structures. Analogously, the degenerated Hamiltonian Nonlinear multi-wave coupling and resonance in elastic structures yields the linearized equations. The amplitudes of the linear field solution Nonlinear multi-wave coupling and resonance in elastic structures (interpreted as integration constants at Nonlinear multi-wave coupling and resonance in elastic structures) should thus satisfy the following relation Nonlinear multi-wave coupling and resonance in elastic structures, where Nonlinear multi-wave coupling and resonance in elastic structures stands for the Lie-Poisson brackets with appropriate definition of the functional derivatives. In turn, at Nonlinear multi-wave coupling and resonance in elastic structures, the complex amplitudes are slowly varying functions such that Nonlinear multi-wave coupling and resonance in elastic structures. This means that

(1) Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures,

where the difference Nonlinear multi-wave coupling and resonance in elastic structures can be interpreted as the free energy of the system. So that, if the scalar Nonlinear multi-wave coupling and resonance in elastic structures, then the nonlinear dynamical structure can be spontaneous one, otherwise the system requires some portion of energy to create a structure at Nonlinear multi-wave coupling and resonance in elastic structures, while Nonlinear multi-wave coupling and resonance in elastic structures represents some indifferent case.

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

(2) Nonlinear multi-wave coupling and resonance in elastic structures, Nonlinear multi-wave coupling and resonance in elastic structures

where Nonlinear multi-wave coupling and resonance in elastic structures is a vector function. Using the polar coordinates Nonlinear multi-wave coupling and resonance in elastic structures, eqs. (2) read the following standard form

(3) Nonlinear multi-wave coupling and resonance in elastic structures; Nonlinear multi-wave coupling and resonance in elastic structures,

where Nonlinear multi-wave coupling and resonance in elastic structures. In most practical problems the vector function Nonlinear multi-wave coupling and resonance in elastic structures appears as a power series in Nonlinear multi-wave coupling and resonance in elastic structures. 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) Nonlinear multi-wave coupling and resonance in elastic structures

with the boundary conditions

Nonlinear multi-wave coupling and resonance in elastic structures

By scaling the sought variables: Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures, 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 Nonlinear multi-wave coupling and resonance in elastic structures. In the case of infinitesimal oscillations this set represents two uncoupled linear differential equations. Let Nonlinear multi-wave coupling and resonance in elastic structures, then the linearized equation for longitudinal displacements possesses a simple wave solution


Nonlinear multi-wave coupling and resonance in elastic structures,

where the frequencies Nonlinear multi-wave coupling and resonance in elastic structures are coupled with the wave numbers Nonlinear multi-wave coupling and resonance in elastic structures through the dispersion relation Nonlinear multi-wave coupling and resonance in elastic structures. Notice that Nonlinear multi-wave coupling and resonance in elastic structures. In turn, the linearized equation for bending oscillations reads[3]

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

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 Nonlinear multi-wave coupling and resonance in elastic structures or Nonlinear multi-wave coupling and resonance in elastic structures, to the following simplest model:

(6) Nonlinear multi-wave coupling and resonance in elastic structures,

which takes into account the temporal parametric excitation only.

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

Nonlinear multi-wave coupling and resonance in elastic structures,


where Nonlinear multi-wave coupling and resonance in elastic structures denote the wave numbers of bending waves; Nonlinear multi-wave coupling and resonance in elastic structures are the wave amplitudes defined by the ordinary differential equations

(7) Nonlinear multi-wave coupling and resonance in elastic structures.

Here

Nonlinear multi-wave coupling and resonance in elastic structures

stands for a coefficient containing parameters of the wave-number detuning: Nonlinear multi-wave coupling and resonance in elastic structures, which, in turn, cannot be zeroes; Nonlinear multi-wave coupling and resonance in elastic structures are the cyclic frequencies of bending oscillations at Nonlinear multi-wave coupling and resonance in elastic structures; Nonlinear multi-wave coupling and resonance in elastic structures 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) Nonlinear multi-wave coupling and resonance in elastic structures,

which results from eqs. (7) at Nonlinear multi-wave coupling and resonance in elastic structures. 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: Nonlinear multi-wave coupling and resonance in elastic structures, whereNonlinear multi-wave coupling and resonance in elastic structures is the vector of solution, Nonlinear multi-wave coupling and resonance in elastic structures denotes the Nonlinear multi-wave coupling and resonance in elastic structures matrix of eigenvalues, Nonlinear multi-wave coupling and resonance in elastic structures is the Nonlinear multi-wave coupling and resonance in elastic structures matrix with quasi-periodic components at the basic frequencies Nonlinear multi-wave coupling and resonance in elastic structures. 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 Nonlinear multi-wave coupling and resonance in elastic structures, where Nonlinear multi-wave coupling and resonance in elastic structures is the vector of the nontrivial oscillatory solution to the uniform equations (8), characterized by the set of basic exponents Nonlinear multi-wave coupling and resonance in elastic structures. By substituting the ansatz Nonlinear multi-wave coupling and resonance in elastic structures into eqs. (7), we obtain the first-order approximation equations in order Nonlinear multi-wave coupling and resonance in elastic structures:

Nonlinear multi-wave coupling and resonance in elastic structures.

where the right-hand terms are a superposition of quasi-periodic functions at the combinational frequencies Nonlinear multi-wave coupling and resonance in elastic structures. Thus the first-order approximation solution to eqs. (7) should be a finite quasi-periodic function [4] , when the combinations Nonlinear multi-wave coupling and resonance in elastic structures; otherwise, the problem of small divisors (resonances) appears.

So, one can continue the asymptotic procedure in the non-resonant case, i. e. Nonlinear multi-wave coupling and resonance in elastic structures, 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 Nonlinear multi-wave coupling and resonance in elastic structures. 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 Nonlinear multi-wave coupling and resonance in elastic structures, eqs. (7) can be highly simplified:


(9) Nonlinear multi-wave coupling and resonance in elastic structures

provided a couple of bending waves, having the wave numbers Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures, produces both a small wave-number detuning Nonlinear multi-wave coupling and resonance in elastic structures (i. e. Nonlinear multi-wave coupling and resonance in elastic structures) and a small frequency detuning Nonlinear multi-wave coupling and resonance in elastic structures (i. e. Nonlinear multi-wave coupling and resonance in elastic structures). Here the symbols Nonlinear multi-wave coupling and resonance in elastic structures denote the higher-order terms of order Nonlinear multi-wave coupling and resonance in elastic structures, since the values of Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures are also supposed to be small. Thus, the expressions

Nonlinear multi-wave coupling and resonance in elastic structures; Nonlinear multi-wave coupling and resonance in elastic structures

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

Nonlinear multi-wave coupling and resonance in elastic structures

It is known that this equation can possess unstable solutions at small values of Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures.

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


(10) Nonlinear multi-wave coupling and resonance in elastic structures; Nonlinear multi-wave coupling and resonance in elastic structures,

where Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures are new unknown coordinates.

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

(11) Nonlinear multi-wave coupling and resonance in elastic structures; Nonlinear multi-wave coupling and resonance in elastic structures,

where Nonlinear multi-wave coupling and resonance in elastic structures is the coefficient of the parametric excitation; Nonlinear multi-wave coupling and resonance in elastic structures is the generalized phase governed by the following differential equation

Nonlinear multi-wave coupling and resonance in elastic structures.

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

Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures,

which allows one to integrate the system analytically. At Nonlinear multi-wave coupling and resonance in elastic structures, there exist quasi-harmonic stationary solutions to eqs. (10), (11), as

Nonlinear multi-wave coupling and resonance in elastic structures,

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

Nonlinear multi-wave coupling and resonance in elastic structures,

where Nonlinear multi-wave coupling and resonance in elastic structures refer the Nonlinear multi-wave coupling and resonance in elastic structures branches of the natural frequencies depending upon wave vectors Nonlinear multi-wave coupling and resonance in elastic structures. 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

Nonlinear multi-wave coupling and resonance in elastic structures

leads to the following linearly uncoupled equations

Nonlinear multi-wave coupling and resonance in elastic structures,

where the Nonlinear multi-wave coupling and resonance in elastic structures matrix Nonlinear multi-wave coupling and resonance in elastic structures is composed by Nonlinear multi-wave coupling and resonance in elastic structures-dimensional polarization eigenvectors Nonlinear multi-wave coupling and resonance in elastic structures defined by the characteristic equation

Nonlinear multi-wave coupling and resonance in elastic structures;


Nonlinear multi-wave coupling and resonance in elastic structures is the Nonlinear multi-wave coupling and resonance in elastic structures diagonal matrix of differential operators with eigenvalues Nonlinear multi-wave coupling and resonance in elastic structures; Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures are reverse matrices.

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

(12) Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures,

using the complex variables Nonlinear multi-wave coupling and resonance in elastic structures. Here Nonlinear multi-wave coupling and resonance in elastic structures is the Nonlinear multi-wave coupling and resonance in elastic structures unity matrix. Here Nonlinear multi-wave coupling and resonance in elastic structures is the Nonlinear multi-wave coupling and resonance in elastic structures-dimensional vector of nonlinear terms analytical at the origin Nonlinear multi-wave coupling and resonance in elastic structures. So, this can be presented as a series in Nonlinear multi-wave coupling and resonance in elastic structures, i. e.

Nonlinear multi-wave coupling and resonance in elastic structures ,

where Nonlinear multi-wave coupling and resonance in elastic structures are the vectors of homogeneous polynomials of degree Nonlinear multi-wave coupling and resonance in elastic structures, e. g.

Nonlinear multi-wave coupling and resonance in elastic structures

Here Nonlinear multi-wave coupling and resonance in elastic structuresand Nonlinear multi-wave coupling and resonance in elastic structures are some given differential operators. Together with the system (12), we consider the corresponding linearized subset

(13) Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures,

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


Nonlinear multi-wave coupling and resonance in elastic structures,

where Nonlinear multi-wave coupling and resonance in elastic structures are constant complex amplitudes; Nonlinear multi-wave coupling and resonance in elastic structures is the number of normal waves of the Nonlinear multi-wave coupling and resonance in elastic structures-th type, so that Nonlinear multi-wave coupling and resonance in elastic structures (for instance, if the operator Nonlinear multi-wave coupling and resonance in elastic structures is a polynomial, then Nonlinear multi-wave coupling and resonance in elastic structures, where Nonlinear multi-wave coupling and resonance in elastic structures is a scalar, Nonlinear multi-wave coupling and resonance in elastic structures is a constant vector, Nonlinear multi-wave coupling and resonance in elastic structures 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) Nonlinear multi-wave coupling and resonance in elastic structures

where Nonlinear multi-wave coupling and resonance in elastic structures denotes an unknown Nonlinear multi-wave coupling and resonance in elastic structures-dimensional vector function, whose components Nonlinear multi-wave coupling and resonance in elastic structures can be represented as formal power series in Nonlinear multi-wave coupling and resonance in elastic structures, i. e. a quasi-bilinear form:

(15) Nonlinear multi-wave coupling and resonance in elastic structures ,

for example

Nonlinear multi-wave coupling and resonance in elastic structures

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

(16) Nonlinear multi-wave coupling and resonance in elastic structures .

It is obvious that the eigenvalues of the operator Nonlinear multi-wave coupling and resonance in elastic structures acting on the polynomial components of Nonlinear multi-wave coupling and resonance in elastic structures (i. e. Nonlinear multi-wave coupling and resonance in elastic structures) are the linear integer-valued combinational values of the operator Nonlinear multi-wave coupling and resonance in elastic structures given at various arguments of the wave vector Nonlinear multi-wave coupling and resonance in elastic structures.

In the lowest-order approximation in Nonlinear multi-wave coupling and resonance in elastic structures eqs. (16) read

Nonlinear multi-wave coupling and resonance in elastic structures .

The polynomial components of Nonlinear multi-wave coupling and resonance in elastic structures are associated with their eigenvalues Nonlinear multi-wave coupling and resonance in elastic structures, i. e. Nonlinear multi-wave coupling and resonance in elastic structures , where

Nonlinear multi-wave coupling and resonance in elastic structures

or Nonlinear multi-wave coupling and resonance in elastic structures,

while Nonlinear multi-wave coupling and resonance in elastic structures in the lower-order approximation in Nonlinear multi-wave coupling and resonance in elastic structures.

So, if at least the one eigenvalue of Nonlinear multi-wave coupling and resonance in elastic structures approaches zero, then the corresponding coefficient of the transform (15) tends to infinity. Otherwise, if Nonlinear multi-wave coupling and resonance in elastic structures, then Nonlinear multi-wave coupling and resonance in elastic structures represents the lowest term of a formal expansion in Nonlinear multi-wave coupling and resonance in elastic structures.

Analogously, in the second-order approximation in Nonlinear multi-wave coupling and resonance in elastic structures:


Nonlinear multi-wave coupling and resonance in elastic structures

the eigenvalues of Nonlinear multi-wave coupling and resonance in elastic structures can be written in the same manner, i. e. Nonlinear multi-wave coupling and resonance in elastic structures, where Nonlinear multi-wave coupling and resonance in elastic structures, etc.

By continuing the similar formal iterations one can define the transform (15). Thus, the sets (12) and (13), even in the absence of eigenvalues equal to zeroes, are associated with formally equivalent dynamical systems, since the function Nonlinear multi-wave coupling and resonance in elastic structures can be a divergent function. If Nonlinear multi-wave coupling and resonance in elastic structures is an analytical function, then these systems are analytically equivalent . Otherwise, if the eigenvalue Nonlinear multi-wave coupling and resonance in elastic structures in the Nonlinear multi-wave coupling and resonance in elastic structures-order approximation, then eqs. (12) cannot be simply reduced to eqs. (13), since the system (12) experiences a resonance.

For example, the most important 3-order resonances include

triple-wave resonant processes, when Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures;

generation of the second harmonic, as Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures.

The most important 4-order resonant cases are the following:

four-wave resonant processes, when Nonlinear multi-wave coupling and resonance in elastic structures; Nonlinear multi-wave coupling and resonance in elastic structures (interaction of two wave couples); or when Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures (break-up of the high-frequency mode into tree waves);

degenerated triple-wave resonant processes at Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures;

generation of the third harmonic, as Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures.

These resonances are mainly characterized by the amplitude modulation , the depth of which increases as the phase detuning approaches to some constant (e. g. to zero, if consider 3-order resonances). The waves satisfying the phase matching conditions form the so-called resonant ensembles .

Finally, in the second-order approximation, the so-called “non-resonant" interactions always take place. The phase matching conditions read the following degenerated expressions

cross-interactions of a wave pair at Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures;

self-action of a single wave as Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures.

Non-resonant coupling is characterized as a rule by a phase modulation .

The principal proposition of this section is following. If any nonlinear system (12) does not have any resonance, beginning from the order Nonlinear multi-wave coupling and resonance in elastic structures up to the order Nonlinear multi-wave coupling and resonance in elastic structures Nonlinear multi-wave coupling and resonance in elastic structures, then the nonlinearity produces just small corrections to the linear field solutions. These corrections are of the same order that an amount of the nonlinearity up to times Nonlinear multi-wave coupling and resonance in elastic structures.

To obtain a formal transform (15) in the resonant case, one should revise a structure of the set (13) by modifying its right-hand side:

(16) Nonlinear multi-wave coupling and resonance in elastic structures; Nonlinear multi-wave coupling and resonance in elastic structures,

where the nonlinear terms Nonlinear multi-wave coupling and resonance in elastic structures. Here Nonlinear multi-wave coupling and resonance in elastic structures are the uniform Nonlinear multi-wave coupling and resonance in elastic structures-th order polynomials. These should consist of the resonant terms only. In this case the eqs. (16) are associated with the so-called normal forms .

Remarks

In practice the series Nonlinear multi-wave coupling and resonance in elastic structures are usually truncated up to first - or second-order terms in Nonlinear multi-wave coupling and resonance in elastic structures.

The theory of normal forms can be simply generalized in the case of the so-called essentially nonlinear systems, since the small parameter Nonlinear multi-wave coupling and resonance in elastic structures can be omitted in the expressions (12) - (16) without changes in the main result. The operator Nonlinear multi-wave coupling and resonance in elastic structures can depend also upon the spatial variables Nonlinear multi-wave coupling and resonance in elastic structures.

Formally, the eigenvalues of operator Nonlinear multi-wave coupling and resonance in elastic structures can be arbitrary complex numbers. This means that the resonances can be defined and classified even in appropriate nonlinear systems that should not be oscillatory one (e. g. in the case of evolution equations).

Resonance in multi-frequency systems

The resonance plays a principal role in the dynamical behavior of most physical systems. Intuitively, the resonance is associated with a particular case of a forced excitation of a linear oscillatory system. The excitation is accompanied with a more or less fast amplitude growth, as the natural frequency of the oscillatory system coincides with (or sufficiently close to) that of external harmonic force. In turn, in the case of the so-called parametric resonance one should refer to some kind of comparativeness between the natural frequency and the frequency of the parametric excitation. So that, the resonances can be simply classified, according to the above outlined scheme, by their order, beginning from the number first Nonlinear multi-wave coupling and resonance in elastic structures, if include in consideration both linear and nonlinear, oscillatory and non-oscillatory dynamical systems.

For a broad class of mechanical systems with stationary boundary conditions, a mathematical definition of the resonance follows from consideration of the average functions

(17) Nonlinear multi-wave coupling and resonance in elastic structures, as Nonlinear multi-wave coupling and resonance in elastic structures,

where Nonlinear multi-wave coupling and resonance in elastic structures are the complex constants related to the linearized solution of the evolution equations (13); Nonlinear multi-wave coupling and resonance in elastic structures denotes the whole spatial volume occupied by the system. If the function Nonlinear multi-wave coupling and resonance in elastic structures has a jump at some given eigen values of Nonlinear multi-wave coupling and resonance in elastic structuresand Nonlinear multi-wave coupling and resonance in elastic structures, then the system should be classified as resonant one[6] . It is obvious that we confirm the main result of the theory of normal forms. The resonance takes place provided the phase matching conditions

Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures.

are satisfied. Here Nonlinear multi-wave coupling and resonance in elastic structures is a number of resonantly interacting quasi-harmonic waves; Nonlinear multi-wave coupling and resonance in elastic structures are some integer numbers Nonlinear multi-wave coupling and resonance in elastic structures; Nonlinear multi-wave coupling and resonance in elastic structuresand Nonlinear multi-wave coupling and resonance in elastic structures are small detuning parameters. Example 1. Consider linear transverse oscillations of a thin beam subject to small forced and parametric excitations according to the following governing equation

Nonlinear multi-wave coupling and resonance in elastic structures,

where Nonlinear multi-wave coupling and resonance in elastic structures, Nonlinear multi-wave coupling and resonance in elastic structures, Nonlinear multi-wave coupling and resonance in elastic structures, Nonlinear multi-wave coupling and resonance in elastic structures, Nonlinear multi-wave coupling and resonance in elastic structures, Nonlinear multi-wave coupling and resonance in elastic structures è Nonlinear multi-wave coupling and resonance in elastic structures are some appropriate constants, Nonlinear multi-wave coupling and resonance in elastic structures. This equation can be rewritten in a standard form


Nonlinear multi-wave coupling and resonance in elastic structures,

where Nonlinear multi-wave coupling and resonance in elastic structures, Nonlinear multi-wave coupling and resonance in elastic structures, Nonlinear multi-wave coupling and resonance in elastic structures. At Nonlinear multi-wave coupling and resonance in elastic structures, a solution this equation reads Nonlinear multi-wave coupling and resonance in elastic structures, where the natural frequency satisfies the dispersion relation Nonlinear multi-wave coupling and resonance in elastic structures. If Nonlinear multi-wave coupling and resonance in elastic structures, then slow variations of amplitude satisfy the following equation

Nonlinear multi-wave coupling and resonance in elastic structures

where Nonlinear multi-wave coupling and resonance in elastic structures, denotes the group velocity of the amplitude envelope. By averaging the right-hand part of this equation according to (17), we obtain

Nonlinear multi-wave coupling and resonance in elastic structures, at Nonlinear multi-wave coupling and resonance in elastic structures;

Nonlinear multi-wave coupling and resonance in elastic structures, at Nonlinear multi-wave coupling and resonance in elastic structures and Nonlinear multi-wave coupling and resonance in elastic structures;

Nonlinear multi-wave coupling and resonance in elastic structures in any other case.

Notice, if the eigen value of Nonlinear multi-wave coupling and resonance in elastic structures approaches zero, then the first-order resonance always appears in the system (this corresponds to the critical Euler force).

The resonant properties in most mechanical systems with time-depending boundary conditions cannot be diagnosed by using the function Nonlinear multi-wave coupling and resonance in elastic structures.

Example 2 . Consider the equations (4) with the boundary conditions Nonlinear multi-wave coupling and resonance in elastic structures; Nonlinear multi-wave coupling and resonance in elastic structures; Nonlinear multi-wave coupling and resonance in elastic structures. By reducing this system to a standard form and then applying the formula (17), one can define a jump of the function Nonlinear multi-wave coupling and resonance in elastic structures provided the phase matching conditions

Nonlinear multi-wave coupling and resonance in elastic structures è Nonlinear multi-wave coupling and resonance in elastic structures.

are satisfied. At the same time the first-order resonance, experienced by the longitudinal wave at the frequency Nonlinear multi-wave coupling and resonance in elastic structures, cannot be automatically predicted.

References

1. Nelson DF, (1979), Electric, Optic and Acoustic Interactions in Dielectrics, Wiley-Interscience, NY.

2. Kaup P. J., Reiman A. and Bers A. Space-time evolution of nonlinear three-wave interactions. Interactions in a homogeneous medium, Rev. of Modern Phys., (1979) 51 (2), 275-309.

3. Kauderer H (1958), Nichtlineare Mechanik, Springer, Berlin.

4. Haken H. (1983), Advanced Synergetics. Instability Hierarchies of Self-Organizing Systems and devices, Berlin, Springer-Verlag.

5. Kovriguine DA, Potapov AI (1996), Nonlinear wave dynamics of 1D elastic structures, Izvestiya vuzov. Appl. Nonlinear Dynamics, 4 (2), 72-102 (in Russian).

6. Maslov VP (1973), Operator methods, Moscow, Nauka publisher (in Russian).

7. Jezequel L., Lamarque C. - H. Analysis of nonlinear dynamical systems by the normal form theory, J. of Sound and Vibrations, (1991) 149 (3), 429-459.

8. Pellicano F, Amabili M. and Vakakis AF (2000), Nonlinear vibration and multiple resonances of fluid-filled, circular shells, Part 2: Perturbation analysis, Vibration and Acoustics, 122, 355-364.

9. Zhuravlev VF and Klimov DM (1988), Applied methods in the theory of oscillations, Moscow, Nauka publisher (in Russian)


[1] The small parameter Nonlinear multi-wave coupling and resonance in elastic structures can also characterize an amount of small damped forced and/or parametric excitation, etc.

[2] The discrete part of the spectrum can be represented as a sum of delta-functions, i.e. Nonlinear multi-wave coupling and resonance in elastic structures.

[3] The resonance appears in the system as Nonlinear multi-wave coupling and resonance in elastic structures that corresponds to any integer number of quarters of wavelengths. There is no stationary solution in the form of standing waves in this case, though the resonant solution for longitudinal waves can be simply designed using the d'Alambert approach.

[4] The conservation of quasi-periodic orbits represents a forthcoming mathematical problem in mathematics, which is in progress up to now [4].

[5] Practically, the resonant properties should be directly associated with the order of the approximation procedure. For instance, if the first-order approximation is considered, then the resonances in order Nonlinear multi-wave coupling and resonance in elastic structures have to be neglected.

[6] In applied problems the definition of resonance should be directly associated with the order of the approximation procedure. For instance, if the first-order approximation is considered, then the jupms of Nonlinear multi-wave coupling and resonance in elastic structures of order Nonlinear multi-wave coupling and resonance in elastic structures have to be neglected [9].

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