Eur. J. Mech. ASolids 18 1999 527–538
Elsevier, Paris
On the separation of motions in systems with a large fast excitation of general form
A. Fidlin
LuK GS GmbH, Bühl, Germany Received 27 November 1997; revised and accepted 31 July 1998
Abstract – In this study dynamic systems are considered, in which motion can be described through a system of second-order ordinary differential equations with the right sides depending both on the slow time t and on the fast time τ = ωt ω ≫ 1 is a big dimensionsless parameter. It is assumed
that the right sides are large they have the magnitude order ω and depend both on generalised coordinates and on the generalised velocities of the system. A motion separating procedure is developed for the systems described in twi ways. The procedure enables separate systems to derive for fast
oscillating and slow components of the solution. Each of these separated systems is simpler than the original one. The equivalence of both procedures is shown. The first of them is based on the multiple scales method, the second one generalises the averaging method of Krylov–Bogoliubov–Mitropolskii.
Motion of a linear oscillator excited through the large fast oscillations of the damping coefficient is analysed as an example of the established method usage. It is shown, that the excitation significantly changes the effective mass and in consequence the natural frequency of the original system. The
analytic results are compared with numerical ones.
Elsevier, Paris
oscillations theory averaging multiple scales
1. Introduction
The separation of motions is one of the main ideas for asymptotic analysis of oscillating systems with small or big parameter. It is connected with the fact that solutions of many types of dynamic systems can be represented
as a superposition of fast oscillations and slow evolution of the solution. The averaging method Krylov and Bogoliubov, 1957; Bogoliubov and Mitropolskii, 1974 and the method of multiple scales Nayfeh, 1973;
Nayfeh and Mook, 1979, which differ significantly in form, are substantially very close. Together with the method of the direct separation of motions, which finds its origins in the works of Kapitsa 1951, 1954 and is
more generally formulated in works of Blekhman 1988, 1994, these methods are the most effective practical realisations of this idea. Each of these methods has its own advantages in solving the special problems in the
theory of oscillations.
In this paper we study the response of non-linear systems to a strong excitation in general form, whose carrier frequency ω is much higher than natural frequencies of the system. Such systems are one of the main objects
of interest for the method of the direct separatoin of motions, which however is proved only for special types of fast excitation. Method of multiple scales was also successfully used to analyse the response of some particular
systems to these types of strong excitation Nayfeh and Mook, 1979; Nayfeh and Nayfeh, 1995.
We are going first to establish a formal procedure for the separation of motions in non-linear systems with strong fast excitation in its most general form based on the multiple scales method technique. We are then going
to generalise the averaging method technique, which enables us to demonstrate the internal distinctions of the considered problem and to prove developed technique. A simple example of the use of this procedure can be
found at the end of this paper.
528 A. Fidlin
2. Problem statement. A formal solution through the multiple scales technique
Consider a system x
••
= F x, x
•
, t + ω8x, x
•
, t, ωt. 1
Here x is an n-dimensional vector of the generalised coordinates and x
•
is a vector of the generalised velocities. We take F and 8 to be n-dimensional vectors of forces. The components of F have to be bounded
functions, satisfying Lipschitz conditions on the first and second arguments, and the components of 8 have to satisfy these conditions together with their first partial derivatives with respect to x and x
•
. We take also 8 to be 2π -periodic vector-function with respect to τ = ωt , and ω to be a big parameter.
This system differs from that considered in the work of Blekhman 1994, because 8 does not depend only on the generalised coordinates x, but depends also on the generalised velocities x
•
, i.e., we are considering the large fast excitation in its most general form.
In order to apply the multiple scales technique to 1, we have to convert to two independent variables t and τ , i.e., from the system of ordinary differential equations 1 to the following system with partial derivatives
∂
2
ϕ ∂t
2
+ 2ω
∂
2
ϕ ∂t∂τ
+ ω
2
∂
2
ϕ ∂τ
2
= F
ϕ, ∂ϕ
∂t +
ω ∂ϕ
∂τ , t
+ ω8
ϕ, ∂ϕ
∂t +
ω ∂ϕ
∂τ , t, τ
. 2
The relationship between 1 and 2 is defined through the condition that, if ϕt, τ is a solution of 2, then x = ϕt, ωt
is a solution of 1. In other words, system 2 is more general then 1, and any solution of 2 taken at the straight line τ = ωt satisfies 1.
We require ϕt, τ to be a 2π -periodic function of τ and try to find ϕ as a formal asymptotic expansion in terms of ε = 1ω
ϕt, τ = ψ t, τ + εψ
1
t, τ + ε
2
ψ
2
t, τ + · · · . Substituting this expression in 2 and balancing the terms with equal orders of ε, we obtain
ε
− 2
: ∂
2
ψ ∂τ
2
= 0,
3
ε
− 1
: ∂
2
ψ
1
∂τ
2
+ 2
∂
2
ψ ∂t∂τ
= 8
ψ ,
∂ψ
1
∂τ +
∂ψ ∂t
+ ω
∂ψ ∂τ
, t, τ ,
4 ε
: ∂
2
ψ
2
∂τ
2
+ 2
∂
2
ψ
1
∂t∂τ +
∂
2
ψ ∂t
2
= F +
∂8 ∂x
ψ
1
+ ∂8
∂x
•
∂ψ
1
∂t +
∂ψ
2
∂τ .
5 The general solution of 3 is
ψ t, τ = Xt + Atτ.
6 According to the periodicity of ψ
At ≡ 0.
Hence ψ =
Xt depends only on the slow time t . The objective of the following analysis is to discover the
differential equations for this still unknown function, which do not contain the fast time τ .
Separation of motions in systems with a large fast excitation of general form 529
The Eq. 4 takes the form ∂
2
ψ
1
∂τ
2
= 8
X, X
•
+ ∂ψ
1
∂τ , t, τ
. 7
The main assumption of this investigation is that we take to know a general 2 π -periodic with respect to τ solution of the system of n differential equations of the first order
∂u ∂τ
= 8X, u, t, τ
8 depending on a vector of arbitrary constants C here X and t are taken to be fixed
u = U X, t, τ, C. We determine these constants to annihilate the average of
∂ψ
1
∂τ
ψ
1
has to be periodic U X, t, τ, C
= X
•
. Here and further, h i means the averaging of the expression in brackets, calculated during the time interval of
periodicity of all the considered functions. Hence
u = U X, X
•
, t, τ and
ψ
1
=
Z
τ
U − X
•
dτ + X
1
t. The unknown function X
1
t is a small magnitude order ε slow correction to the main slow part of the
solution Xt. Denoting
9t, τ =
Z
τ
U − X
•
dτ 9
9 is a periodic function of τ and depends on t both direct and indirect through X and X
•
, we obtain ψ
1
t, τ = 9t, τ + X
1
t. 10
The Eq. 5 takes now the following form ∂
2
ψ
2
∂τ
2
= ∂8
∂x
•
∂ψ
2
∂τ +
F + ∂8
∂x ψ
1
+ ∂8
∂x
•
∂ψ
1
∂t −
2 ∂
2
ψ
1
∂t∂τ −
X
••
. 11
All the functions here should be calculated at the point X, X
•
+
∂ψ
1
∂τ
, t, τ .
System 10 is a system of inhomogeneous linear differential equations in respect to τ . All the coefficients are periodic and the unknown function is
∂ψ
2
∂τ
. As is known from the general theory of linear systems with periodic coefficients, for the existence of periodic
solutions of 11 the projections of the inhomogeneous parts of the equations on the solutions of the system
530 A. Fidlin
conjugated to the homogeneous one must vanish. [A reference to the classical work of Poincaré 1957 seems here to be in order and to underscore the close relationship of Poincaré’s method to the method of multiple
scales, which is in this case a procedure to find periodic in respect to τ solutions of 2 with a not isolated undisturbed solution 6.]
In other words, we introduce a homogeneous system ∂w
∂τ =
∂8 ∂x
•
w and the conjugated one
∂w
∗
∂τ = −
∂8 ∂x
• T
w
∗
. 12
Let us assume we have got n independent periodic solutions of 12, the matrix of which we will design as W
∗
. Then for the existence of a periodic solution of 11 is necessary that W
T ∗
F + ∂8
∂x ψ
1
+ ∂8
∂x
•
∂ψ
1
∂t −
2 ∂
2
ψ
1
∂t∂τ −
X
••
= or
X
••
= W
T ∗
− 1
W
T ∗
F + ∂8
∂x ψ
1
+ ∂8
∂x
•
∂ψ
1
∂t −
2 ∂
2
ψ
1
∂t∂τ .
13 The Eq. 13 seems to contain through ψ
1
and
∂ψ
1
∂t
side by side with the already known functions W
∗
and 8 and the desired variable X also the unknown function X
1
t [see 10]. Let us show that all the terms containing
X
1
t vanish.
Really W
T ∗
∂8 ∂x
•
X
• 1
= W
T ∗
∂8 ∂x
•
X
• 1
= ∂8
∂x
• T
W
∗ T
X
• 1
= − ∂W
∗
∂τ
T
X
• 1
. But we have assumed, that W
∗
τ is periodic. Hence
∂W
∗
∂τ =
and therefore W
T ∗
∂8 ∂x
•
X
• 1
= 0.
In order to calculate hW
T ∗
∂8 ∂x
X
1
i let us consider 8 and derive it partially in respect to X
∂ ∂τ
∂u ∂X
= ∂8
∂x +
∂8 ∂x
•
∂u ∂X
. Multiplying the last equation with W
T ∗
one obtains W
T ∗
∂8 ∂x
= ∂
∂τ W
T ∗
∂u ∂X
Separation of motions in systems with a large fast excitation of general form 531
and because of the periodicity of W
∗
, u and
∂u ∂X
W
T ∗
∂8 ∂x
X
1
= ∂
∂τ W
T ∗
∂u ∂X
X
1
= 0.
14 So the function X
1
does not occur in 13 and ψ
1
can be replaced with the known function 9t, τ 9. Let us notice in addition, that
W
T ∗
∂8 ∂x
•
∂9 ∂t
− ∂
2
9 ∂t∂τ
= − ∂
∂τ W
T ∗
∂9 ∂t
= because of the periodicity of W
∗
, 9 and
∂9 ∂t
. So we get the desired equation for Xt
X
••
= W
T ∗
− 1
W
T ∗
F + ∂8
∂x 9 −
∂
2
9 ∂t∂τ
. 15
With this procedure we have formally separated the fast oscillating part of the solution 1, which is determined through 8, 9, called according to Blekhman 1994 the equations of fast notions, from the slow
‘evolutionary’ component of the solution. This was determined through the equations of slow motions 15. So, to find Xt we have first of all to solve system 8, the order of which is two times lower than the order
of the original system, then one has to solve the linear system 12 and to determine the matrix W
∗
. At last it is possible to get the system 15, which differs from 1 because it does not contain the fast time
τ = ωt .
Equation 15 is not solved in respect to X
••
. In fact [see 9] ∂
2
9 ∂t∂τ
= ∂U
∂X X
•
+ ∂U
∂t +
∂U ∂X
•
X
••
− X
••
. Substituting this relationship in 15, we obtain
MX, X
•
, tX
••
= F X, X
•
, t + V X, X
•
, t 16
with MX, X
•
, t = W
T ∗
− 1
W
T ∗
∂U ∂X
•
, V X, X
•
, t = W
T ∗
− 1
W
T ∗
∂8 ∂x
9 − ∂U
∂X X
•
− ∂U
∂t +
W
T ∗
− 1
W
T ∗
F X, U, t − F X, X
•
, t .
Function V X, X
•
, t can be called, according to Blekhman 1994, the ‘vibration force’. The matrix
coefficient MX, X
•
, t can be interpreted as a matrix of efficient mass or inertia of the averaged system in
respect to its slow motions. We should notice that this matrix depends on the solutions of the equations of fast motions, i.e., on the amplitude of fast excitation.
This described procedure together with its validation is enough for the asymptotic analysis of 1, but we are going to investigate this system once more with the help of the averaging method in order to expose the internal
essence of the transformations carried out. At the same time we are going to validate our procedure.
532 A. Fidlin
3. The generalised averaging method. Validation of the procedure
