Nonlinear multi-wave coupling and resonance in elastic structures
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
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
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
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 ,
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
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
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
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
the critical values of Euler forces.
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 :
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
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
(10) ; ,
where and are
new unknown coordinates.
this into eqs. (9), we obtain the first-order approximation equations
(11) ; ,
where is the coefficient of the parametric excitation; is the generalized phase governed by the following differential
Equations (10) and
(11), being of a Hamiltonian structure, possess the two evident first integrals
which allows one to
integrate the system analytically. At ,
there exist quasi-harmonic stationary solutions to eqs. (10), (11), as
which forms the
boundaries in the space of system parameters within the first zone of the
From the physical
viewpoint, one can see that the parametric excitation of bending waves appears
as a degenerated case of nonlinear wave interactions. It means that the study
of resonant properties in nonlinear elastic systems is of primary importance to
understand the nature of dynamical instability, even considering free nonlinear
The linear subset of eqs. (0) describes a superposition of
harmonic waves characterized by the dispersion relation
where refer the branches
of the natural frequencies depending upon wave vectors . The spectrum of the wave vectors and the
eigenfrequencies can be both continuous and discrete one that finally depends
upon the boundary and initial conditions of the problem. The normalization of
the first order, through a special invertible linear transform
leads to the
following linearly uncoupled equations
is the diagonal
matrix of differential operators with eigenvalues ; and
are reverse matrices.
uncoupled equations can be rewritten in an equivalent matrix form 
(12) and ,
using the complex
variables . Here is
the unity matrix. Here is the -dimensional Vector of nonlinear terms analytical at
the origin . So, this can be presented as a series in , i. e.
where are the vectors of homogeneous polynomials of degree , e. g.
Here and are
some given differential operators. Together with the
system (12), we consider the corresponding linearized subset
(13) and ,
solutions can be written immediately as a superposition of harmonic waves
where are constant complex amplitudes; is the number of normal waves of the -th type, so that (for
instance, if the operator is a polynomial, then , where is a scalar, is
a constant Vector, is some differentiable function. For more detail see
A question is
following. What is the difference between these two systems, or in other words,
how the small nonlinearity is effective?
According to a
method of normal forms (see for example [7,8]), we look for a solution to eqs.
(12) in the form of a quasi-automorphism, i. e.
where denotes an unknown -dimensional
Vector function, whose components can
be represented as formal power series in ,
i. e. a quasi-bilinear form:
where and are
unknown coefficients which have to be determined.
the transform (14) into eqs. (12), we obtain the following partial differential
equations to define :
It is obvious that
the eigenvalues of the operator acting
on the polynomial components of (i.
e. ) are the linear integer-valued combinational values
of the operator given at various arguments of the wave Vector .
lowest-order approximation in eqs.
components of are associated with their eigenvalues , i. e. ,
while in the lower-order approximation in .
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 .
the second-order approximation in :
the eigenvalues of
can be written in the same manner, i. e. , where ,
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
resonant processes, when and ;
generation of the
second harmonic, as and .
The most important
4-order resonant cases are the following:
processes, when ; (interaction
of two wave couples); or when and
(break-up of the high-frequency mode into tree
triple-wave resonant processes at and
generation of the
third harmonic, as and .
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
of a wave pair at and ;
self-action of a
single wave as and .
coupling is characterized as a rule by a phase modulation.
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) ; ,
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.
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 .
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).
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. It is obvious that we
confirm the main result of the theory of normal forms. The resonance takes
place provided the phase matching conditions
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
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),
, 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).
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
are satisfied. At
the same time the first-order resonance, experienced by the longitudinal wave
at the frequency , cannot be automatically predicted.
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