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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
-
Nelson DF, (1979), Electric, Optic and Acoustic Interactions in Dielectrics, Wiley-Interscience, NY.
-
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.
-
Kauderer H (1958), Nichtlineare Mechanik, Springer, Berlin.
-
Haken H. (1983), Advanced Synergetics. Instability Hierarchies of Self-Organizing Systems and devices, Berlin, Springer-Verlag.
-
Kovriguine DA, Potapov AI (1996), Nonlinear wave dynamics of 1D elastic structures, Izvestiya vuzov. Appl. Nonlinear Dynamics, 4 (2), 72-102 (in Russian).
-
Maslov VP (1973), Operator methods, Moscow, Nauka publisher (in Russian).
-
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.
-
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.
-
Zhuravlev VF and Klimov DM (1988), Applied methods in the theory of oscillations, Moscow, Nauka publisher (in Russian)
1 The small parameter 
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. 
.
3 The resonance appears in the system as 
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.
The conservation of quasi-periodic orbits represents a forthcoming mathematical problem in mathematics, which is in progress up to now [4].
4 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 
have to be neglected.
5 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 of order have to be neglected [9].
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