Реферат: Nonlinear multi-wave coupling and resonance in elastic structures
Nonlinear multi-wave coupling and resonance in elastic structures
Kovriguine DA
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.
Introduction
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 [1] and also references therein. A nice historical sketch to this topic can be found in the review [2]. In this paper we try to trace relationships between the resonance and the dynamical stability of elastic structures.
Evolution equations
Consider a natural quasi-linear mechanical system with distributed parameters. Let motion be described by the following partial differential equations
(0) ,
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[1];
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[2];
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
(2) ,
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.
Parametric approach
As an illustrative example we consider the so-called Bernoulli-Euler model governing the motion of a thin bar, according the following equations [3]:
(4)
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[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. 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:
(6) ,
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
(7) .
Here
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
(8) ,
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 [4], 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[5].
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:
(9)
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) ;
,
where is
the coefficient of the parametric excitation;
is
the generalized phase governed by the following differential equation
.
Equations (10) and (11), being of a Hamiltonian structure, possess the two evident first integrals
and
,
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 parametric instability.
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 oscillations.
Normal forms
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
,
where the matrix
is composed by
-dimensional polarization
eigenvectors
defined by the
characteristic equation
;
is the
diagonal matrix
of differential operators with eigenvalues
;
and
are reverse matrices.
The linearly uncoupled equations can be rewritten in an equivalent matrix form [5]
(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
,
whose analytical 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 [6]).
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.
(14)
where denotes
an unknown
-dimensional vector
function, whose components
can be
represented as formal power series in
,
i. e. a quasi-bilinear form:
(15) ,
for example
where and
are unknown coefficients
which have to be determined.
By substituting the transform (14) into eqs.
(12), we obtain the following partial differential equations to define :
(16) .
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
.
In the lowest-order approximation in eqs. (16) read
.
The polynomial components of are associated with their
eigenvalues
, i. e.
, where
or ,
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
.
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
,
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
1. Nelson DF, (1979), Electric, Optic and Acoustic Interactions in Dielectrics, Wiley-Interscience, NY.
2. 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.
3. Kauderer H (1958), Nichtlineare Mechanik, Springer, Berlin.
4. Haken H. (1983), Advanced Synergetics. Instability Hierarchies of Self-Organizing Systems and devices, Berlin, Springer-Verlag.
5. Kovriguine DA, Potapov AI (1996), Nonlinear wave dynamics of 1D elastic structures, Izvestiya vuzov. Appl. Nonlinear Dynamics, 4 (2), 72-102 (in Russian).
6. Maslov VP (1973), Operator methods, Moscow, Nauka publisher (in Russian).
7. 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.
8. 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.
9. 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.
[4]The conservation of quasi-periodic orbits represents a forthcoming mathematical problem in mathematics, which is in progress up to now [4].
[5] 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.
[6] 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].