Nonlinear multi-wave coupling and resonance in elastic structures
Solutions to the evolution equations describing the phase and amplitude modulation of nonlinear waves are physically interpreted basing on the law of energy conservation. An algorithm reducing the governing nonlinear partial differential equations to their normal form is considered. The occurrence of resonance at the expense of nonlinear multi-wave coupling is discussed.
The principles of nonlinear multi-mode coupling were first recognized almost two century ago for various mechanical systems due to experimental and theoretical works of Faraday (1831), Melde (1859) and Lord Rayleigh (1883, 1887). Before First World War similar ideas developed in radio-telephone devices. After Second World War many novel technical applications appeared, including high-frequency electronic devices, nonlinear optics, acoustics, oceanology and plasma physics, etc. For instance, see  and also references therein. A nice historical sketch to this topic can be found in the review . In this paper we try to trace relationships between the resonance and the dynamical stability of elastic structures.
Consider a natural quasi-linear mechanical system with distributed parameters. Let motion be described by the following partial differential equations
where denotes the complex -dimensional vector of a solution; and are the linear differential operator matrices characterizing the inertia and the stuffiness, respectively; is the -dimensional vector of a weak nonlinearity, since a parameter is small; stands for the spatial differential operator. Any time the sought variables of this system are referred to the spatial Lagrangian coordinates .
Assume that the motion is defined by the Lagrangian . Suppose that at the degenerated Lagrangian produces the linearized equations of motion. So, any linear field solution is represented as a superposition of normal harmonics:
Here denotes a complex vector of wave amplitudes; are the fast rotating wave phases; stands for the complex conjugate of the preceding terms. The natural frequencies and the corresponding wave vectors are coupled by the dispersion relation . At small values of , a solution to the nonlinear equations would be formally defined as above, unless spatial and temporal variations of wave amplitudes . Physically, the spectral description in terms of new coordinates , instead of the field variables , is emphasized by the appearance of new spatio-temporal scales associated both with fast motions and slowly evolving dynamical processes.
This paper deals with the evolution dynamical processes in nonlinear mechanical Lagrangian systems. To understand clearly the nature of the governing evolution equations, we introduce the Hamiltonian function , where . Analogously, the degenerated Hamiltonian yields the linearized equations. The amplitudes of the linear field solution (interpreted as integration constants at ) should thus satisfy the following relation , where stands for the Lie-Poisson brackets with appropriate definition of the functional derivatives. In turn, at , the complex amplitudes are slowly varying functions such that . This means that
(1) and ,
where the difference can be interpreted as the free energy of the system. So that, if the scalar , then the nonlinear dynamical structure can be spontaneous one, otherwise the system requires some portion of energy to create a structure at , while represents some indifferent case.
Note that the set (1) can be formally rewritten as
where is a vector function. Using the polar coordinates , eqs. (2) read the following standard form
(3) ; ,
where . In most practical problems the vector function appears as a power series in . This allows one to apply procedures of the normal transformations and the asymptotic methods of investigations.
As an illustrative example we consider the so-called Bernoulli-Euler model governing the motion of a thin bar, according the following equations :
with the boundary conditions
By scaling the sought variables: and , eqs. (4) are reduced to a standard form (0).
Notice that the validity range of the model is associated with the wave velocities that should not exceed at least the characteristic speed . In the case of infinitesimal oscillations this set represents two uncoupled linear differential equations. Let , then the linearized equation for longitudinal displacements possesses a simple wave solution
where the frequencies are coupled with the wave numbers through the dispersion relation . Notice that . In turn, the linearized equation for bending oscillations reads
As one can see the right-hand term in eq. (5) contains a spatio-temporal parameter in the form of a standing wave. Allowances for the this wave-like parametric excitation become principal, if the typical velocity of longitudinal waves is comparable with the group velocities of bending waves, otherwise one can restrict consideration, formally assuming that or , to the following simplest model:
which takes into account the temporal parametric excitation only.
We can look for solutions to eq. (5), using the Bubnov-Galerkin procedure:
where denote the wave numbers of bending waves; are the wave amplitudes defined by the ordinary differential equations
stands for a coefficient containing parameters of the wave-number detuning: , which, in turn, cannot be zeroes; are the cyclic frequencies of bending oscillations at ; denote the critical values of Euler forces.
Equations (7) describe the early evolution of waves at the expense of multi-mode parametric interaction. There is a key question on the correlation between phase orbits of the system (7) and the corresponding linearized subset
which results from eqs. (7) at . In other words, how effective is the dynamical response of the system (7) to the small parametric excitation?
First, we rewrite the set (7) in the equivalent matrix form: , where is the vector of solution, denotes the matrix of eigenvalues, is the matrix with quasi-periodic components at the basic frequencies . Following a standard method of the theory of ordinary differential equations, we look for a solution to eqs. (7) in the same form as to eqs. (8), where the integration constants should to be interpreted as new sought variables, for instance , where is the vector of the nontrivial oscillatory solution to the uniform equations (8), characterized by the set of basic exponents . By substituting the ansatz into eqs. (7), we obtain the first-order approximation equations in order :
where the right-hand terms are a superposition of quasi-periodic functions at the combinational frequencies . Thus the first-order approximation solution to eqs. (7) should be a finite quasi-periodic function , when the combinations ; otherwise, the problem of small divisors (resonances) appears.
So, one can continue the asymptotic procedure in the non-resonant case, i. e. , to define the higher-order correction to solution. In other words, the dynamical perturbations of the system are of the same order as 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 effective even at the small parametric excitation.
In a particular case of the external force , eqs. (7) can be highly simplified:
provided a couple of bending waves, having the wave numbers and , produces both a small wave-number detuning (i. e. ) and a small frequency detuning (i. e. ). Here the symbols denote the higher-order terms of order , since the values of and are also supposed to be small. Thus, the expressions
can be interpreted as the phase matching conditions creating a triad of waves consisting of the primary high-frequency longitudinal wave, directly excited by the external force , and the two secondary low-frequency bending waves parametrically excited by the standing longitudinal wave.
Notice that in the limiting model (6) the corresponding set of amplitude equations is reduced just to the single pendulum-type equation frequently used in many applications:
It is known that this equation can possess unstable solutions at small values of and .
Solutions to eqs. (7) can be found using iterative methods of slowly varying phases and amplitudes:
(10) ; ,
where and are new unknown coordinates.
By substituting this into eqs. (9), we obtain the first-order approximation equations
(11) ; ,