Nonlinear multi-wave coupling and resonance inelastic structures
Kovriguine DA
Solutions to the evolution equationsdescribing the phase and amplitude modulation of nonlinear waves are physicallyinterpreted basing on the law of energy conservation. An algorithm reducing thegoverning nonlinear partial differential equations to their normal form isconsidered. The occurrence of resonance at the expense of nonlinear multi-wavecoupling is discussed.
Introduction
The principles of nonlinear multi-modecoupling were first recognized almost two century ago for various mechanicalsystems due to experimental and theoretical works of Faraday (1831), Melde (1859)and Lord Rayleigh (1883, 1887). Before First World War similar ideas developedin radio-telephone
devices. After Second World War many novel technicalapplications appeared, including high-frequency electronic devices, nonlinearoptics, acoustics, oceanology and plasma physics, etc. For instance, see [1] andalso references therein. A nice historical sketch to this topic can be found inthe review [2]. In this paperwe try to trace relationships between the resonance and the dynamical stabilityof elastic structures.
Evolution equations
Consider a natural quasi-linear mechanicalsystem with distributed parameters. Let motion be described by the followingpartial differential equations
(0),
where denotesthe complex-dimensional vector of asolution; and are the linear differentialoperator matrices characterizing the inertia and the stuffiness, respectively; is the-dimensional vector of aweak nonlinearity, since a parameter issmall [1]; stands for the spatialdifferential operator. Any time thesought variables of this system arereferred to the spatial Lagrangian coordinates.
Assume that the motion is defined by theLagrangian. Suppose that at the degenerated Lagrangianproduces the linearizedequations of motion. So, any linear field solution is represented as asuperposition of normal harmonics:
.
Here denotesa complex vector of wave amplitudes [2]; are the fast rotating wavephases; stands for the complexconjugate of the preceding terms. The natural frequencies and the corresponding wavevectors are coupled by thedispersion relation. At small valuesof, a solution to thenonlinear equations would be formally defined as above, unless spatial andtemporal variations of wave amplitudes.Physically, the spectral description in terms of new coordinates, instead of the fieldvariables, is emphasized by theappearance of new spatio-temporal scales associated both with fast motions andslowly evolving dynamical processes.
This paper deals with the evolutiondynamical processes in nonlinear mechanical Lagrangian systems. To understandclearly the nature of the governing evolution equations, we introduce theHamiltonian function, where. Analogously, thedegenerated Hamiltonian yieldsthe linearized equations. The amplitudes of the linear field solution (interpreted asintegration constants at) shouldthus satisfy the following relation, wherestands for the Lie-Poissonbrackets with appropriate definition of the functional derivatives. In turn, at, The complex amplitudesare slowly varying functions such that.This means that
(1) and,
where the difference can be interpreted as thefree energy of the system. So that, if the scalar,then the nonlinear dynamical structure can be spontaneous one, otherwise thesystem requires some portion of energy to create a structure at, while represents someindifferent case.
Note that the set (1) can be formallyrewritten as
(2),
where isa vector function. Using the polar coordinates,eqs. (2) read the following standard form
(3);,
where.In most practical problems the vector function appearsas a power series in. This allows oneto apply procedures of the normal transformations and the asymptotic methods ofinvestigations.
Parametric approach
As an illustrative example we consider theso-called Bernoulli-Euler model governing the motion of a thin bar, accordingthe following equations [3]:
(4)
with the boundary conditions
By scaling the sought variables: and, eqs. (4) are reduced to astandard form (0).
Notice that the validity range of the modelis associated with the wave velocities that should not exceed at least thecharacteristic speed. In the case ofinfinitesimal oscillations this set represents two uncoupled lineardifferential equations. Let, thenthe linearized equation for longitudinal displacements possesses a simple wavesolution
,
where the frequencies are coupled with the wavenumbers through the dispersionrelation. Notice that. In turn, the linearizedequation for bending oscillations reads [3]
(5).
As one can see the right-hand term in eq. (5)contains a spatio-temporal parameter in the form of a standing wave. Allowancesfor the this wave-like parametric excitation become principal, if the typicalvelocity of longitudinal waves is comparable with the group velocities ofbending waves, otherwise one can restrict consideration, formally assuming thator, to the following simplestmodel:
(6),
which takes into account the temporalparametric excitation only.
We can look for solutions to eq. (5), usingthe Bubnov-Galerkin procedure:
,
where denote the wave numbers of bending waves; are the wave amplitudesdefined by the ordinary differential equations
(7).
Here
stands for a coefficient containingparameters of the wave-number detuning:,which, in turn, cannot be zeroes; arethe cyclic frequencies of bending oscillations at;denote the critical valuesof Euler forces.
Equations (7) describe the early evolutionof waves at the expense of multi-mode parametric interaction. There is a keyquestion on the correlation between phase orbits of the system (7) and thecorresponding linearized subset
(8),
which results from eqs. (7) at. In other words, howeffective is the dynamical response of the system (7) to the small parametricexcitation?
First, we rewrite the set (7) in theequivalent matrix form:,where is the vector of solution,denotes the matrix of eigenvalues, is the matrix with quasi-periodiccomponents at the basic frequencies. Followinga standard method of the theory of ordinary differential equations, we look fora solution to eqs. (7) in the same form as to eqs. (8), where the integrationconstants should to be interpreted as new sought variables, for instance, where is the vector of thenontrivial oscillatory solution to the uniform equations (8), characterized bythe set of basic exponents. Bysubstituting the ansatz intoeqs. (7), we obtain the first-order approximation equations in order:
.
where the right-hand terms are asuperposition of quasi-periodic functions at the combinational frequencies. Thus the first-orderapproximation solution to eqs. (7) should be a finite quasi-periodic function [4], when the combinations; otherwise, the problem of small divisors (resonances) appears.
So, one can continue the asymptoticprocedure in the non-resonant case, i. e. ,to define the higher-order correction to solution [5].In other words, the dynamical perturbations of the system are of the same orderas the parametric excitation. In the case of resonance the solution to eqs. (7)cannot be represented as convergent series in.This means that the dynamical response of the system can be highly effectiveeven at the small parametric excitation.
In a particular case of the external force, eqs. (7) can be highlysimplified:
(9)
provided a couple of bending waves, havingthe wave numbers and, produces both a smallwave-number detuning (ie) and a small frequencydetuning (i. e.). Here the symbols denote th...
