Nonlinear multi-wave coupling and resonance in elastic structures
2
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) Mpu tt  Cpu  Np, u ,
3
where ux, t    un x, t  denotes the complex N -dimensional vector of a
solution; M and C are the N  N
linear differential operator matrices
characterizing the inertia and the stuffiness, respectively; N is the N -dimensional
vector of a weak nonlinearity, since a parameter  is small1; p   x stands for the
spatial differential operator. Any time t the sought variables of this system ux, t 
are referred to the spatial Lagrangian coordinates x .
Assume that the motion is defined by the Lagrangian L ux, t ; . Suppose
that at   0 the degenerated Lagrangian L0  ux, t   L ux, t ;0 produces the
linearized equations of motion. So, any linear field solution is represented as a
superposition of normal harmonics:
ux, t  

 Ak exp ix, t dk    .
*

Here Ak denotes a complex vector of wave amplitudes 2; x, t   kt  kx
are the fast rotating wave phases;

*
stands for the complex conjugate of the
preceding terms. The natural frequencies k and the corresponding wave vectors
k are coupled by the dispersion relation   k . At small values of   0 , a
solution to the nonlinear equations would be formally defined as above, unless
spatial and temporal variations of wave amplitudes A  Ak, x, t  . Physically, the
spectral description in terms of new coordinates A , instead of the field variables
u,
is emphasized by the appearance of new spatio-temporal scales associated both
with fast motions and slowly evolving dynamical processes.
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.


A n k    A nj k  k  k j .
j
4
This paper deals with the evolution dynamical processes in nonlinear
mechanical Lagrangian systems. To understand clearly the nature of the governing
evolution
equations,
we
H m, u;   mu t  Lu;  ,
where
introduce
the
Hamiltonian
function
m  L / u t .
Analogously, the degenerated
Hamiltonian H0 m, u  Hm, u;0 yields the linearized equations. The amplitudes of
the linear field solution Ak (interpreted as integration constants at   0 ) should
thus satisfy the following relation dA / dt  A / t  A , H0   0 , where , stands
for the Lie-Poisson brackets with appropriate definition of the functional
derivatives. In turn, at   0 , the complex amplitudes are slowly varying functions
such that dA / dt  A / t  A, H . This means that


dA *
dA
 A , F  and
(1)
 A* , F ,
dt
dt
where the difference F  H  H0 can be interpreted as the free energy of the
system. So that, if the scalar F  0 , then the nonlinear dynamical structure can be
spontaneous one, otherwise the system requires some portion of energy to create a
structure at F  0 , while F  0 represents some indifferent case.
Note that the set (1) can be formally rewritten as



dA *
dA
*
 Z p, A , A ,
(2)
 Z * p, A , A *
dt
dt

where Zp, A , A *  is a vector function. Using the polar coordinates
A  a exp i , eqs. (2) read the following standard form
(3)




da
d
  Re Z cos   Im Z sin  ; a
  Re Z sin   Im Z cos  ,
dt
dt




5

where Zp, a,   Zp, a exp i, a * expi . In most practical problems the

vector function Z 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]:

EJ
c2 3
2
wtt  F w xxxx  c  x u x w x   2 w x ;
(4) 
2
u  c 2 u  c  w 2
xx
x
x
 tt
2
 
with the boundary conditions
u0, t   0, EFux  L, t    p0  Pt ,



w0, t   w L, t , wxx 0, t   wxx  L, t .
By scaling the sought variables: u  u and w  w , 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 c . In the case of
infinitesimal oscillations this set represents two uncoupled linear differential

equations. Let Pt    pn cos  n t , then the linearized equation for longitudinal
n 1
displacements possesses a simple wave solution
6
u x , t   
p0
pn sin  n x
x  
sin  n t ,
EF
EF n cos n L
where the frequencies  n are coupled with the wave numbers  n through
the dispersion relation n  c n2 . Notice that  n  n / 2 L . In turn, the linearized
equation for bending oscillations reads 3
(5)
wtt 



p
cos n x
EJ

wxxxx  0 wxx 
 x  wx  pn
cosn t  .
F
F
F  n1 cos n L

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 c   or  n  0 , to the
following simplest model:
(6) wtt 
Pt 
EJ
wxxxx 
w ,
F
F xx
which takes into account the temporal parametric excitation only.
We can look for solutions to eq. (5), using the Bubnov-Galerkin procedure:

w x , t    Wn t  sin k n x ,
n 1
3
The resonance appears in the system as
 n  n / 2 L 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.
7
where k n  n / L denote the wave numbers of bending waves; Wn t  are the
wave amplitudes defined by the ordinary differential equations

(7) Wl   2l Wl 
kn pm  l ,n,m
Wn cos mt .
m1 n 1 cos m L



4F
Here
 sinknlm L sinknlm L 
 sinknlm L sinknlm L 
 l ,n,m  kn  km 


   k n  k m 

knlm L 
knlm L 
 knlm L
 knlm L
stands for a coefficient containing parameters of the wave-number detuning:
k nl m  k n  k l  k m ,
which, in turn, cannot be zeroes;  n  kn EJkn2  Pn  p0  / F are
the cyclic frequencies of bending oscillations at   0 ; Pn  EJkn2 denote the critical
values of Euler forces.
Equations (7) describe the early evolution of waves at the expense of multimode parametric interaction. There is a key question on the correlation between
phase orbits of the system (7) and the corresponding linearized subset
(8) Vl   2l Vl  0 ,
which results from eqs. (7) at   0 . In other words, how effective is the
dynamical response of the system (7) to the small parametric excitation?
First,
we
  2 W  f t W ,
W
rewrite
the
set
(7)
in
the
equivalent
matrix
form:
where W  Wn t  is the vector of solution,   diag n 
denotes the n  n matrix of eigenvalues, f t  is the n  n matrix with quasi-periodic
components at the basic frequencies  m  . 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
8
as new sought variables, for instance Wt   1  ct   O 2 Vt  , where Vt  is
the vector of the nontrivial oscillatory solution to the uniform equations (8),
characterized by the set of basic exponents  m  . By substituting the ansatz Wt 
into eqs. (7), we obtain the first-order approximation equations in order  :
c  2 c  f t V .
where the right-hand terms are a superposition of quasi-periodic functions at
the combinational frequencies  m   l   m  . Thus the first-order approximation
solution to eqs. (7) should be a finite quasi-periodic function 4, when the
combinations
m  l  m  0 ;
otherwise, the problem of small divisors
(resonances) appears.
So, one can continue the asymptotic procedure in the non-resonant case, i. e.


Wt   1  ct    2 c1 t ... Vt  , to define the higher-order correction to solution5.
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 Pt   p cost , eqs. (7) can be
highly simplified:
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
neglected.
2
have to be
9
1   12W1    k 1 k 2 p W2 cos t  O  2 ;
W
4F cos L
(9)
2   22W2    k 1 k 2 p W1 cos t  O  2 ,
W
4F cos L
 
 
provided a couple of bending waves, having the wave numbers k1 and k 2 ,
produces both a small wave-number detuning k    k1  k 2 (i. e. kL  1 ) and a
small frequency detuning      1   2 (i. e. L / c  1). Here the symbols
 
O 2
denote the higher-order terms of order  2 , since the values of k1 L and
1
k2 L1 are also supposed to be small. Thus, the expressions
k    k1  k 2 ;      1   2
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 Pt  , 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:
 k2p
2

W  W  
W cos t  O  2 ;
4F cosL
 
It is known that this equation can possess unstable solutions at small values
of k    2 k and     2 .
Solutions to eqs. (7) can be found using iterative methods of slowly varying
phases and amplitudes:
10
(10) Wn t   rn t  sin  n t  ; Wn t    n rn t  cos n t  ,
where rn t  and  n t  are new unknown coordinates.
By substituting this into eqs. (9), we obtain the first-order approximation
equations
(11) r1  


r2 sin  ; r2  
r1 sin  ,
4 1
4 2
where   pk1k2 / 4F cosL is the coefficient of the parametric excitation;
  t   1 t    2 t  is the generalized phase governed by the following differential
equation
   
r 
  r2
 1  cos  .

4   1r1  2 r2 
Equations (10) and (11), being of a Hamiltonian structure, possess the two
evident first integrals
r1 t r2 t  sin  t   r1 0r2 0 sin  0 and  1r12 t    2 r22 t    1r12 0   2 r22 0 ,
which allows one to integrate the system analytically. At   0 , there exist
quasi-harmonic stationary solutions to eqs. (10), (11), as
 
r 0 
  r2 0
 1

,
4   1r1 0  2 r2 0 
which forms the boundaries in the space of system parameters within the
first zone of the parametric instability.
11
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


D n , k   det  2n Mik   Cik   0 ,
where  n k refer the N branches of the natural frequencies depending
upon wave vectors k . 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
u  pU
leads to the following linearly uncoupled equations
U tt   2 pU   1 M 1 Np , U ,
where the N  N matrix  is composed by N -dimensional polarization
eigenvectors  n defined by the characteristic equation
 
2
n

Mp  Cp  n  0 ;
12
p  diag n p is the N  N diagonal matrix of differential operators
with eigenvalues  n   n k ;  1 and M 1 are reverse matrices.
The linearly uncoupled equations can be rewritten in an equivalent matrix
form [5]
(12) Q  ipQ  if p, Q, Q *  and Q *  ipQ *  if * p, Q * , Q ,
using the complex variables Q  p 1u  iE 1u t . Here E is the N  N
unity matrix. Here f  i 1M1N1 is the N -dimensional vector of nonlinear terms
analytical at the origin  f p,0,0  0 . So, this can be presented as a series in  , i. e.






 
f p , Q , Q *  f 1 p , Q , Q *  f  2 p , Q , Q *  O  2 ,
where f  m  are the vectors of homogeneous polynomials of degree m , e. g.
N

N


f n    f lm n p Ql  Ql* Qm  Qm* 
l  1 m 1
N
N
N




 
 n
    f klm
p Qk  Qk* Ql  Ql* Qm  Qm*  O  2 .
k  1 l  1 m 1
 n
Here f lm n p and f klm
p are some given differential operators. Together with
the system (12), we consider the corresponding linearized subset
(13) q  ipq  0 and q *  ipq*  0 ,
whose analytical solutions can be written immediately as a superposition of
harmonic waves
13
  
 
Mn

qn x, t    An k j exp i  n k j t  k j x ,
j 1
where An are constant complex amplitudes; M n is the number of normal
waves of the n -th type, so that  n k j    n ik j  (for instance, if the operator p
is a polynomial, then p exp it  kxx, t   expt  kxp  ikx, t  , where  is
a scalar, k is a constant vector, x, t  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) Qx, t   qx, t   hq, q* 
where h denotes an unknown N -dimensional vector function, whose
components hn can be represented as formal power series in  , i. e. a quasibilinear form:
(15) h  h 1 q, q*   h  2  q, q*   O 2  ,
for example
N

N


hn    hlm1 q l  q l* q m  q m* 
l 1 m1
N
N
N



  
 2
     hklm
q k  q k* q l  q l* q m  q m*  O  2
k 1 l 1 m1
2
where hlm1 and hklm
are unknown coefficients which have to be determined.
14
By substituting the transform (14) into eqs. (12), we obtain the following
partial differential equations to define h :
 
(16) q h q*  qhq  h  if p, Q, Q*  .
*
It is obvious that the eigenvalues of the operator K acting on the
 
polynomial components of h (i. e. Kh  q h q*  qhq  h ) are the linear
*
integer-valued combinational values of the operator  n ik j  given at various
arguments of the wave vector k .
In the lowest-order approximation in  eqs. (16) read


Kh 1  if 1 p, q, q* .
The polynomial components of h 1   h 1 are associated with their

eigenvalues 1 , i. e. Kh 1  1 h 1 , where



1  nk  k  ik    nl  l ik    m ik   ik 

or 1  nk  k k    nl  l k     m k   k   ,
while nk  nl  2 in the lower-order approximation in  .
So, if at least the one eigenvalue of K approaches zero, then the
corresponding coefficient of the transform (15) tends to infinity. Otherwise, if
 


nk  k k    nl  l k    m k   k  , then h represents the lowest term of a formal
expansion in  .
Analogously, in the second-order approximation in  :
15





Kh 2  i q* f 1 h*1  i qf 1 h1  if  2 p, q, q*

the eigenvalues of K can be written in the same manner, i. e.
 
 


2  nk  k k    nl  l k   n p  p k    m k   k   k  , where nk  nl  n p  3 , 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 h can be a divergent function. If h is an analytical function, then these
systems are analytically equivalent. Otherwise, if the eigenvalue s  0 in the s 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 1 k 1    2 k 2    3 k 3    and
k 1  k 2  k 3  k ;
generation of the second harmonic, as
2 1 k 1    2 k 2   
and
2k 1  k 2  k .
The most important 4-order resonant cases are the following:
four-wave resonant processes, when  1 k 1    2 k 2    3 k 3    4 k 4    ;
k 1  k 2  k 3  k 4  k
(interaction
 1 k 1    2 k 2    3 k 3    4 k 4   
of
two
wave
couples);
or
when
and k 1  k 2  k 3  k 4  k (break-up of the
high-frequency mode into tree waves);
degenerated triple-wave resonant processes at 2 1 k 1    2 k 2    3 k 3   
and 2k 1  k 2  k 3  k ;
generation of the third harmonic, as 3 1 k 1    2 k 2    and 3k 1  k 2  k .
These resonances are mainly characterized by the amplitude modulation, the
depth of which increases as the phase detuning approaches to some constant (e. g.
16
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  1 k 1    2 k 2    1 k 1    2 k 2  and
k1  k 2  k1  k 2 ;
self-action of a single wave as  1 k 1    1 k 1    1 k 1    1 k 1  and
k1  k1  k1  k1 .
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 s  1 up to the
order s  n
n  1 ,
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 O n t  .
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) q  ipq  R p, q, q*  ; q *  ipq*  R * p, q, q*  ,

where the nonlinear terms R p, q, q*     n R n p, q, q*  . Here R n are the
n 1
uniform n -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.
17
REMARKS
In practice the series R 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 socalled 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 x .
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 s  1 , 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
18
1
0 V0

T
1
(17) r, k   lim
T

*
Z
p
,
A
,
A
d
x
dt , as T   ,


V0


where A are the complex constants related to the linearized solution of the
evolution equations (13); V0 denotes the whole spatial volume occupied by the
system. If the function r, k  has a jump at some given eigen values of  and k ,
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
M
 mn n   and
n 1
M
m k
n
n
 k .
n 1
are satisfied. Here M is a number of resonantly interacting quasi-harmonic
waves; mn are some integer numbers
m
1
... m M  M  ;  and k 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
wtt   2 wxxxx  Twxx  wxx cos 0t  f cos  1 x cos 1t ,
where  , T ,  ,  0 , 1 ,  1 è f are some appropriate constants,   1. This
equation can be rewritten in a standard form
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
order
 2 have to be neglected [9].
r, k  of
19
z t  i pz 
2ip 2
Re z cos  0 t  if cos  1 x cos 1 t ,
 p
where z   pw  iwt ,  p   2 p 4  Tp 2 , p   / x . At   0 , a solution this
equation reads z  A expit  kx , where the natural frequency satisfies the
dispersion relation    2 k 4  Tk 2 . If   0 , then slow variations of amplitude
satisfy the following equation

dA
ik 2

2 A * cos  0 t  A exp i  0  2 t  exp i  0  2 t
dt
2
 if cos  1 x cos  1 t exp it  kx ,


 
where d / dt   / t  vg  / x , denotes the group velocity of the amplitude
envelope. By averaging the right-hand part of this equation according to (17), we
obtain
r , k   0 , at  0  2 ;
r , k   0 , at 1   and  1  k ;
r, k   0 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 r, k .
Example 2. Consider the equations (4) with the boundary conditions
w0, t   w L, t   u0, t   0 ;
wxx 0, t   wxx  L, t   0 ;
EFux  L, t    p0   cos t .
By
reducing this system to a standard form and then applying the formula (17), one
can define a jump of the function r, k provided the phase matching conditions
20
1   2   3
è k1  k 2  k 3 .
are satisfied. At the same time the first-order resonance, experienced by the
longitudinal wave at the frequency    n  cn / 2l , cannot be automatically
predicted.
21
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)