r, 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. 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,,,,, ГЁ 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
В
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 ; p> 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
ГЁ.
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. p> 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. p> 3. Kauderer H (1958), Nichtlineare Mechanik, Springer, Berlin. p> 4. Haken H. (1983), Advanced Synergetics. Instability Hierarchies of Self-Organizing Systems and devi...
