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; 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

(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

(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 , 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:

(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

,

;

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. 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).

**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,

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