Nonlinear multi-wave coupling and resonance in elastic structures (стр. 4 из 5)
So, if at least the one eigenvalue of
approaches zero, then the corresponding coefficient of the transform (15) tends to infinity. Otherwise, if 
, then 
represents the lowest term of a formal expansion in 
.Analogously, in the second-order approximation in
: 
the eigenvalues of
can be written in the same manner, i. e. 
, where 
, 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
can be a divergent function. If is an analytical function, then these systems are analytically equivalent. Otherwise, if the eigenvalue 
in the 
-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
and 
;generation of the second harmonic, as
and 
.The most important 4-order resonant cases are the following:
four-wave resonant processes, when
; 
(interaction of two wave couples); or when 
and 
(break-up of the high-frequency mode into tree waves);degenerated triple-wave resonant processes at
and 
;generation of the third harmonic, as
and 
.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
and 
;self-action of a single wave as
and 
.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
up to the order 
, 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 
.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) 
;
,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