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

(16) ; ,

where the nonlinear terms . Here are the uniform -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 are usually truncated up to first - or second-order terms in .

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

Formally, the eigenvalues of operator 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 , 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) , as ,

where are the complex constants related to the linearized solution of the evolution equations (13); denotes the whole spatial volume occupied by the system. If the function has a jump at some given eigen values of and , 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

and .

are satisfied. Here is a number of resonantly interacting quasi-harmonic waves; are some integer numbers ; and 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

,

where , , , , , è are some appropriate constants, . This equation can be rewritten in a standard form

,

where , , . At , a solution this equation reads , where the natural frequency satisfies the dispersion relation . If , then slow variations of amplitude satisfy the following equation

where , denotes the group velocity of the amplitude envelope. By averaging the right-hand part of this equation according to (17), we obtain

, at ;

, at and ;

in any other case.

Notice, if the eigen value of 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 .

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

è .

are satisfied. At the same time the first-order resonance, experienced by the longitudinal wave at the frequency , 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.