Journal of Sound and Vibration (1996) 192(1), 43–64

ANALYTICAL STUDY OF VIBRATING SYSTEMSWITH STRONG NON-LINEARITIES BY

EMPLOYING SAW-TOOTH TIMETRANSFORMATIONS

V. N. P

Department of Higher Mathematics, State Technology and Chemistry University of Ukraine,320005 Dniepropetrovsk, Ukraine

(Received 19 October 1994, and in final form 30 June 1995)

A special saw-tooth temporal transformation technique is formulated. It is simple enoughto allow analytic computation of strongly non-linear free and forced dynamic responses,but, at the same time, can be applied to the analysis of general classes of non-linearproblems.

7 1996 Academic Press Limited

1. INTRODUCTION AND PRELIMINARIES

There exist numerous quantitative techniques for computing non-linear dynamicresponses. The majority of these techniques are carried out under the assumption of weaknon-linearity. After assuming that the non-linear system ‘‘neighbors’’ a linear one, aperturbation parameter is introduced to denote the small magnitudes of the non-linearterms, and the non-linear response is constructed ‘‘close’’ to a linear generating solution.Since the generating functions are harmonic, the weakly non-linear responses areconstructed by using complete bases of trigonometric functions. An obvious disadvantageof such techniques is that they cannot be applied to strongly non-linear or non-linearizableoscillators. To circumvent this deficiency of weakly non-linear techniques, an alternativeclass of strongly non-linear ones was developed. These techniques relax the assumptionof weak non-linearity by utilizing non-linear generating solutions, thereby assuming thatthe strongly non-linear systems under consideration neighbor simplified, but otherwise,non-linear systems. These strongly non-linear techniques are highly specialized and cannotbe employed for the analysis of general classes of non-linear problems. The main reasonis that multi-dimensional non-linear systems are generically non-integrable, and, hence, thenon-linear generating solutions are seldom available in closed form [1].

From the above remarks it is concluded that a strongly non-linear analytical techniquewith a wide range of applicability must employ non-linear generating systems which (i)are sufficiently general so that they can be used in a broad range of non-linear applications,(ii) possess a simple enough structure in order to enable the correction of efficient iterativeperturbation schemes for computing the non-linear response, and (iii) possess generatingsystem properties in addition to those of linear systems.

It must be noted that the requirement (i) seems to contradict the well-known‘‘individuality’’ of non-linear systems, which generally prohibits the conceivement of

43

0022–460X/96/160043+22 $18.00/0 7 1996 Academic Press Limited

. . 44

Figure 1. The basic pair of functions.

analytical methodologies which are applicable to general classes of strongly non-linearsystems.

The harmonic oscillator (the linear generating system) is probably the most fundamentalmodel in vibrating analysis. The reason for the wide applicability of this simple mechanicalmodel is that the generated trigonometric functions {sin t, cos t} possess a number ofconvenient mathematical properties associated with the group of motions in Euclideanspace, such as the rotation-subgroup. In the same spirit, one could consider an additionalpair of (non-smooth) functions which have relatively simple forms associated withtranslation- and reflection-subgroups in the group of Euclidean motions. These functionswill be termed the saw-tooth sine, t(t), and right-angled cosine, e(t), respectively, and aredefined as t(t)= (2/p) arcsin [sin (pt/2)] and e(t)= t'(t), where the prime denotes thegeneralized derivative (cf. Figure 1). The mechanical model which generates these functionsis the vibro-impact oscillator moving with constant velocity between two rigid barriers (cf.Figure 2).

Interestingly enough, there is a remarkable relation between the harmonic oscillator andthe vibro-impact one, since both can be viewed as limiting cases of the same non-linearoscillator,

x+ xm =0, x $R, t=0, x=0, x=1, (1, 2)

where m is an arbitrary positive odd integer; the over dot denotes differentiation withrespect to time t.

The exact solution of the initial problem (1), (2) can be expressed in closed form by usingthe special Lyapunov’s functions [2] or cam-functions [3], but these expressions aremathematically too complicated to provide the aforementioned requirement (ii).

Figure 2. The simplest vibrating models and the corresponding periodic functions.

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Considering the range 1EmEa, one obtains the following limiting cases for thesolutions:

{x, x}= {sin t, cos t} if m=1, and {x, x}:{t(t), e(t)} if m:a. (4)

The last case needs an extended concept of the solution; at the same time this case canbe interpreted by means of the first integral of motion

x2/2+ (xm+1)/(m+1)=1/2,

which is satisfied by the functions (4) almost everywhere as m tends to infinity. From theviewpoint of physics the given case corresponds to the classical particle in the square-wellpotential or the aforementioned vibro-impact system with two rigid barriers. The localizedsingularities of the functions (4) for m:a occur at time instants {t: t(t)=21} (i.e., atthe instances of contact of the vibro-impact oscillator with its rigid boundaries), and arethe cause of convergence problems of conventional analysis based on trigonometricexpansions when they are applied to strongly non-linear problems.

So, it has been shown that the oscillator (1) gives two simple limiting systems generatingtwo simple pairs of periodic functions.

It should be noted that oscillator (1) has been considered by a number of authors.So, this essentially non-linear system appears in the theory of motion stability, in whichthe so-called degenerated case is analyzed [2, 4]. Equation (1) plays a special role fornon-linear normal modes definition and description in strongly non-linear cases ofhomogeneous n-DOF systems [5], and resonant ones [6]. The solutions of equation (1)with various integers m were used for determining the frequencies of strongly non-linearoscillators having polynomial characteristics [7]. Note that the case of a third-degreecharacteristic is described by Jacobi’s functions (these functions represent a special caseof Lyapunov’s or cam-functions which have been studied more). In reference [8] and someother works this case was employed as a generating system for construction of analyticalsolutions in a broad class of non-linear oscillators. The oscillator (1) can be found in thephysical literature also [9].

In the present paper the oscillator (1) will be used for testing of the constructed analyticalprocedure. The corresponding analytical solution for all m will contain no specialfunctions. What is more, the solution admits hand calculations with no tables orcomputers.

As a second example demonstrating the physical significance of the pair of functions{t(t), e(t)} consider the Duffing oscillator with negative non-linear stiffness

x+ x− x3 =0.

Denote by T=T(E) the period of oscillation, where E is the parameter of the total energy.When the energy is in the interval, 0QEQ 1/4, the system performs periodic oscillationswith amplitude A in the neighborhood of the stable fixed point (x, x)= (0, 0). For thistype of motions, the exact solution can be expressed in terms of Jacobian elliptic functions,and it can be proved to satisfy the following asymptotic relations:

T : 2p,xA

:cosp

2(t+ a) if E:+0; T:a, x:e (t+ a) if E:1/4−0. (5)

Here t=4t/T is non-dimensioned time, and a=const is an arbitrary phase. Solution (5)for E:1/4−0 is written in terms of the right-angled cosine, and corresponds to themotion of the system on a heteroclinic orbit in phase space. In the scale ofnon-dimensioned (own) time the system performs momentary ‘‘jumps’’ between thetwo unstable equilibrium points (x, x)= (21, 0). Increasing the energy above the critical

. . 46

value E=1/4, leads to strongly non-linear, non-periodic motions outside the heteroclinicloop. For values of the energy in the range 0Q 1−4E�1, standard perturbation methodsbased on trigonometric expansions encounter convergence problems and do not lead toaccurate results. It will be shown that, by using the non-smooth functions {t(t), e(t)} asgenerating solutions, one can analytically study such essentially non-linear solutionswithout encountering convergence problems.

The principal possibility of the similar constructions was discussed in previous works[10–12]. An analysis of the technique from the viewpoint of the non-linear normal modestheory will be given in the book [13].

Note that the method of non-smooth transformations for coordinates (not for time) ofimpact systems has been developed in a number of works [14, 15]. It should be emphasizedthat the corresponding technique is a technique of non-smooth spatial transformationsof variables that can be directly applied to impact or vibro-impact systems only. Thetechnique that will be constructed here contains a saw-tooth (non-smooth) timetransformation (STTT), and the corresponding procedures will be applied to systems withanalytical restoring force characteristics.

2. REPRESENTATION OF FUNCTIONS USING STTT

The time transformations (−a, a)% t:I, where I is half-limited or limited domain,are used in a number of non-linear mechanics methods as a preliminary stage. Suchtransformations are useful for the equations of motion analysis by analytical iterationprocesses [16]. The non-smooth transformations will be used in this paper. For the simplestcases it can be written as: t:=t =. Consider its periodic version

t:t(t)0 t. (6)

It should be noted that the metric of time is preserved under expression (6) because

e2 =1, e(t)= t'(t). (7)

Recalling that derivatives and equalities should be understood in the generalized senseof the theory of distributions, the possibility of using the time parameter t is connectedwith the following propositions.

Proposition 1. The general periodic function x= x(t) with the period T=4 can beexpressed as [11]

x=X(t)+Y(t)e, (8)

where

X(t)= 12[x(t)+ x(2− t)], Y(t)= 1

2[x(t)− x(2− t)]. (9)

Proof. In the interval, equal to the period the following relations occur:

t= t, e=1 if −1Q tQ 1, t=2− t, e=−1 if 1Q tQ 3.

Taking into account equations (9)

X(t)+Y(t)e=X(t)+Y(t)= x(t) if −1Q tQ 1,

X(t)+Y(t)e=X(2− t)−Y(2− t)= x(t) if 1Q tQ 3.

The points L= {t: t(t)=21} need a special examination. If function x(t) is continuousin the neighborhood of points L, then the equality

Y =t$L =0 or Y=t=21 =0 (10)

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occurs, and the equality (8) is also true for all t $L. If function x(t) has jumps inthe points L, then these jumps are described by the function e(t) contained in theright side of the equality (8). The situation can be considered locally by means of theidentity

x(t)= 12[x(=t =)+ x(−=t =)]+ 1

2[x(=t =)− x(−=t =)]=t =' (11)

in the neighborhood of the point t=0. Note that the identity (8) is a periodic version ofthe identity (11). Hence, the term X(t) is the even component of the function x(t) withrespect to a quarter of the period (namely, the point t=1), and the term Y(t)e is the oddone.

The simplest examples are

sinp

2t0 sin $p2 t(t)%, cos

p

2t0 cos $p2 t(t)%e(t).

In particular, the first identity is illustrated by Figure 3, where behavior of the ‘‘oscillatingtime’’ −1E tE 1 and the original time −aQ tQa are shown.

Proposition 2. The elements (8) are elements of hyperbolic numbers algebra because ofequations (7). Hence, for any function f(x) we have

f(X+Ye)=Rf + Ife, (12)

where

Rf = 12[ f(X+Y)+ f(X−Y)], If = 1

2[ f(X+Y)− f(X−Y)].

Proof. This equality can be easily verified because either e=1 or e=−1.The values Rf and If will be termed the ‘‘real’’ and ‘‘imaginary’’ parts of the element

respectively. The element is equal to zero if and only if its ‘‘real’’ and ‘‘imaginary’’parts are equal to zero. A simple example is given by considering the exponentialfunction:

exp (x)= exp (X+Ye)= exp (X)[cosh (Y)+ e sinh (Y)].

Proposition 3. The result of differentiation remains in the algebra.Proof. The first order generalized derivative of expression (8) is

x=Y'+X'e+Ye , (13)

where the prime denotes differentiation with respect to t. If the necessary conditionsof continuity for function x(t), that is equalities (10), exist, then the underlined

Figure 3. A physical treatment of the identity sin (pt/2)0 sin [pt(t)/2]. The oscillating time, t, reflects frompoints t=21, and the original time, t, moves without the reflections. However, the observer sees the sameperiodic processes. Situations before (b), and after (a) the first reflection of time t have been shown.

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addend in (13) should be ignored. In fact, the derivative of e is a periodic system of Diracimpulses

e=2 sa

k=−a

[d(t+1−4k)− d(t−1−4k)],

which are ‘‘localized’’ in the points L where the multiplier Y is equal to zero because ofequation (10). In a similar manner one can consider the second derivative x, and so on.For example, the second derivative is given by

x=X0+Y0e (14)

if the conditions

X'=t=21 =0 (15)

exist.Proposition 4. The result of integration remains in the algebra, that is the equality

g (X+Ye) dt=Q+Pe

exists under the condition

g1

−1

X(t) dt=0.

Here

Q=gt

0

Y dt+C, P=gt

−1

X dt,

and C is an arbitrary constant.Proof. This result may be easily verified by differentiation with respect to t.

3. TRANSFORMATIONS OF DYNAMICAL SYSTEMS

The STTT technique is now applied to the study of the non-linear dynamics ofdiscrete oscillators in strongly non-linear regimes. As a preliminary illustration considera dynamical system of the general form

x= f(x), x $Rn, (16)

where f(x) is a continuous vector function; x= x(t). Let the system be considered on amanifold of periodic solutions with a period equal to T=4a. Note that in the autonomouscase the period is an a priori unknown value. Since any periodic function can be expressedin terms of X- and Y-components, one can seek any periodic solution in the form

x=X(t)+Y(t)e, t= t(t/a), e= e(t/a), (17)

where X(t), Y(t) are unknown components of the solution.Substitution of expressions (17) into equation (16) gives

(Y'− aRf )+ (X'− aIf )e+Ye'=0,

where Rf , If are defined by equation (12). Eliminating the periodic singular term (the termhas been underlined in the last equation) by means of the condition (10) and equalizing

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separately the ‘‘real’’ and ‘‘imaginary’’ parts to zero, one obtains

Y'= aRf , X'= aIf , Y=t=21 =0. (18)

This is a non-linear boundary value problem for the standard range −1E tE 1. It isimportant to note that due to the new parameter of time t, consideration of system isrestricted to one half-period only.

Let the dynamical system be described with the set of second-order equations, whichis written as

x+ f(x, x, t)=0, x $Rn, (19)

where the vector function f is assumed to be sufficiently smooth, and either to dependperiodically on time t with period equal to T=4a, or to have no time dependence.Substituting expressions (17) into equation (19) and using the aforesaid properties ofexpression (8) one obtains

(X0+ a2Rf )+ (Y0+ a2If )e=0

under the conditions (10), (15). Here

Rf =12 $f 0X+Y,

Y'+X'a

, at1+f 0X−Y,Y'−X'

a, 2a− at1%,

If =12 $f 0X+Y,

Y'+X'a

, at1− f 0X−Y,Y'−X'

a, 2a− at1%.

So one obtains the boundary value problem in the form

X0+ a2Rf =0, X'=t=21 =0; Y0+ a2If =0, Y=t=21 =0. (20, 21)

Although the transformed equations are formally more complicated than the originalequations, they possess certain significant advantages. Indeed, since the solution’squalitative properties ‘‘are included’’ in the system due to the ‘‘oscillating’’ variable t, thesolutions of the simplified equations

X0=0, Y0=0, (22)

as generating solutions, can be employed for the perturbation method of successiveapproximations. This leads to simple perturbation solutions for strongly non-linearcases.

Since the replacement of equations (20), (21) by equations (22) is an essential stepin our constructions, we have to discuss this from both physical and mathematicalviewpoints.

Firstly, the same step being made in terms of original variables, x(t), destroys aqualitative structure of the periodic motion, and thus does not lead to eligible generatingsystems. In fact, the corresponding equation, x=0, contains too little information aboutthe original system (19) to be used for obtaining the periodic solutions, x(t), by means ofan iterative process. Using this equation one could obtain a local expansion of typex(t)= x(0)+ x(0)t+· · ·, about the initial point, t=0, but the corresponding series areunsuitable to directly obtain the global characteristics of the vibrating motion: that is, anamplitude and a period. By comparison, the new equations (22) possess solutions,X=X(0)+X'(0)t, Y=Y(0)+Y'(0)t, t= t(t/a), which are periodic with respect to theoriginal time parameter, t. So both the radically simplified system (22) and the originalone (20), (21) are included in the set of vibrating systems.

. . 50

Secondly, the replacement of equations (20), (21) by equations (22) reflects our basicphysical assumption: that is, our wishing to approximate the vibrating system with thesimplest vibro-impact system, which is included now in equations (22). Indeed theafore-presented solution is a linear form of the saw-tooth function, t, and hencein particular describes a family of the simplest vibro-impact systems. The mathematicalcorrectness of initially neglecting the terms a2Rf , a2If , and the boundary conditions inequations (20), (21) will be confirmed by the analysis of the corresponding series.The neglected terms are assumed to be small in an integral sense, and the corresponding‘‘small parameter’’ will be given by the relationship (34) (see Remark 1 of the nextsection).

And finally, note that in the same manner the harmonic oscillator is successfully usedin the harmonic balance ideology.

4. PERIODIC SOLUTIONS OF A CONSERVATIVE SYSTEM

4.1. -

Consider the n-DOF unforced system

x+ f(x)=0, x $Rn, (23)

where f(x) is an odd analytical vector function: f(−x)=−f(x). The one-parametric(except time translation) family of periodic solutions will be constructed. Taking intoaccount the symmetry of the system (23),

x=X(t), Y0 0, t= t(t/a). (24)

Then, starting from equations (20), (21), one obtains the following simplified expressions:

X0+ a2f(X)=0, X'=t=1 =0, X(−t)=−X(t). (25)

The solutions of the non-linear boundary value problem (25) can be found in a series ofsuccessive approximations:

X=X0(t)+X1(t)+X2(t)+ · · ·, a2 = h0(1+ g1 + g2 + · · ·), (26)

where it is assumed that

O(>Xi>)�O(>Xi+1>); O(gi+1)�O(gi+2), (i=0, 1, 2, · · ·)

and the norm, > >, is defined by the expression >X>=maxt>X>Rn. The series (26)generates the following sequence of equations (the corresponding technique, of using aformal ‘‘small’’ parameter, will be given in section 6):

X00=0, X10=−h0 f(X0), X20=−h0[g1 f(X0)+ f 'x (X0)X1], . . . , (27)

under the conditions

(X0'+X1')=r=1 =0, X2'=t=1 =0; . . . . (28)

The generating solution is

X0 =A0t, Rn%A0 = const.

This solution describes an n-DOF vibro-impact oscillator with two rigid barriers; in sodoing the length of arbitrary vector A0 is equal to the barrier spacing. The direction ofthe vector will be defined in the next approximation.

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The first successive approximation of the solution is

X1 =A1t− h0 gt

0

(t− j)f(A0j) dj, (29)

where A1 is an arbitrary constant vector. Note that the first term in this expression hasthe same structure as the generating solution, and thus can be taken to be equal to zero:A1 =0. Combining two members of the expansion X0, X1, and satisfying the first boundarycondition in (28), one obtains the following non-linear eigenvector problem relating to thevector A0:

g1

0

f(A0t) dt= h−10 A0, (30)

where

h0 =A0TA0>A0T g1

0

f(A0t) dt, (31)

()T denotes the transpose of a vector.In the next step, considering the third equation of sequence (27), one obtains

X2 =A2t− h0 gt

0

(t− j)[g1 f(A0j)+ f 'x (A0j)X1(j)] dj, (32)

where A2 is an arbitrary constant vector; f'x () denotes the n× n-matrix of the first partialderivatives of f() with respect to the components of the vector x.

The corresponding boundary condition of the sequence (28) gives

A2 = h0 g1

0

[g1 f(A0t)+ f 'x (A0t)X1(t)] dt,

where the value g1 is undetermined as yet, and then an additional condition for the vectorA2 can be adopted:

A0TA2 =0.

This condition, meaning that the vector A2 is orthogonal to the corresponding vector ofthe generating solution A0, leads to the following expression for g1:

g1 =−A0T g1

0

f 'x (A0t)X1 dt>A0T g1

0

f(A0t) dt. (33)

Hence, the second approximation is completely obtained. Similar calculations can beperformed to compute higher order approximations.

Remark 1. The convergence properties of the series constructed essentially depend onthe action of, corresponding to equation (25), an integral operator F:

X=F(X)0 a26t g1

t

f [X(j)] dj+gt

0

jf [X(j)] dj7;a2 = h0 0A0T g

1

0

f(A0t) dt>A0T g1

0

f(X) dt1.

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The necessary condition is:

>F'X (X0)dX>/>dX>Q 1, (34)

where dX is an arbitrary function from the neighborhood of X0: >dX>�>X0>.For the linearized system the condition (34) leads to the set of unequalities vi /vj Q 1

for all i$ j where vj is the eigenfrequency corresponding to the basic regime.Remark 2. For the 1-DOF system all boundary conditions (28) can be satisfied by the

scalar values h0, g1, g2, . . . , only, and thus the equalities Ai =0 (i=1, 2, . . .) can beassumed. In this case the values Ai are scalar and all members of the expansions can beuniquely determined by the sequence of equations of the following integral form:

X0 =At;

X1 =−h0 gt

0

(t− j)f(Aj) dj, h0 =A> g1

0

f(Aj) dj;

Xi =− si

j=1

hj−1 gt

0

(t− j)Ri− j dj; hi−1 =− si−1

j=1

ai− jhj−1;

hi−1 = h0gi−1, (i=2, 3, . . .), (35)

where A is an arbitrary constant scalar depending on the initial conditions, and

ai =g1

0

Ri dj> g1

0

R0 dj; Ri =1i !

dif(X0 + eX1 + e2X2 + · · ·)dei be=0

, (i=0, 1, 2, . . .);

e is the formal auxiliary parameter.Example 1. Realizing two steps of the constructed algorithm for the oscillator (1),

x+ xm =0, one obtains

X1A$t−tm+2

m+2+

m2(m+2) 0 t2m+3

2m+3−

tm+2

m+21%;a2 1m+1

Am−1 01+m

2(m+2)+

m2

4(m+2)2 $1+m+2

m(2m+3)%1.

Quantitative characteristics of the convergence properties of this series are illustrated inTable 1. The tabulated data are produced by the automatic system of symbolic calculations(‘‘Reduce’’). The time forms of the approximations are shown in Figure 4. Figure 5 exhibitsthe period dependence on the integer m for the fixed energy in the first approximation only.For comparison, the exact result and the first order approximation of the harmonicalbalance method are presented.

Note that this solution leads to the following asymptotic: x:t(vt), m:a, wherev= x(0).

Example 2. Consider the two-mass chain that is shown in Figure 6. Let the elastic energyof springs be defined by the following relationships:

P=F(x1)+G(x1 − x2)+F(x2);

F(s)0 s2/2+ s4/4+ s6/6, G(s)0 g1s2/2+ g3s4/4+ g5s6/6,

where gi (i=1, 3, 5) are positive parameters.The vector function of the restoring force in equation (23) is f(x)= (1P/1x1, 1P/1x2)T.

It can be shown that the non-linear eigen-vector problem (30), (31) has the qualitative

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T 1

The ‘‘experimental’’ checking of the series convergence for the oscillator with stiffnessesf(x)= x, x3, x5. The tabulated data Xi /Xi−1=t=1 have been obtained by means of the automatic

system of symbolic calculations (‘‘Reduce’’)

i x x3 x5

1 −0·33333 −0·20000 −0·142862 0·06667 0·13333 0·164843 0·23810 0·26154 0·278204 0·30667 0·32364 0·334815 0·34651 0·36023 0·368596 0·37267 0·38432 0·390997 0·39119 0·40136 0·406918 0·40500 0·41406 0·418809 0·41569 0·42388 0·40623

Figure 4. The first three approximation X0 = t, X1 =−t7/7, X2 =5(7t13 −13t7)/1274 for A=1 and thecorresponding approximate solution, x1X0 +X1 +X2, for the oscillator x+ x5 =0. The higher order terms arelocalized about the zero velocity time points. These terms improve the solution smoothness.

Figure 5. The period dependence on the degree of non-linearity, m: 1—saw-tooth single step approximation;2—exact solution; 3—harmonical approximation. The total energy is equal to E=2.

. . 54

Figure 6. Two-mass chain with non-linear springs.

dissimilar solutions in the following cases: (a) g3 q 1/4, g5 q 1/8; (b) g3 Q 1/4, g5 q 1/8;3(4g3 −1)2 −32g1(8g5 −1)q 0; (c) g3 Q 1/4, g5 Q 1/8. Figure 7 exhibits the solutions at thefollowing quantities: (a) g1 =1/10; g3 =1/2; g5 =1/4; (b) g1 =1/10; g3 =1/8; g5 =1/7;(c) g1 =1/10; g3 =1/8; g5 =1/15. The solutions corresponding to the symmetric mode arenot shown in Figure 7. Figures 7(b, c) present cases such that the non-linear localizedmodes [12, 17, 18] can be realized.

4.2. -

An additional class of problems arising from the general set of equations (20), (21) isconcerned with non-linear free oscillations close to separatrices of dynamical systems.Consider the 1-DOF, conservative non-linear oscillator, with an odd restoring force,

x+ f(x)=0, x $R,

and assume that the phase plane possesses a stable equilibrium (x, x)= (0, 0), and twounstable saddle-points (x, x)= (2K, 0). When the total energy of the system is equal to

E=Es 0gK

0

f(x) dx, (36)

then motions on the heteroclinic orbits (separatrices) connecting the two unstableequilibria occur.

It can be verified that the computation of strongly non-linear periodic vibrations insmall neighborhoods of the heteroclinic orbits, i.e., in the energy range

0Q 1−E/Es�1, (37)

is reduced to solving the following non-linear boundary value problem in terms of theI-component:

Y0=−a2f(Y); Y =t=1 =0, Y(−t)=Y(t); X0 0. (38)

Figure 7. Manifolds of initial points (x1(0), x2(0)) for the non-linear normal modes (NNMs): (a) strongcoupling; (b) ‘‘weak’’ g3; (c) ‘‘weak’’ both g3 and g5. Starting from the points of the manifolds with zero velocity,x1(0)=0, x2(0)=0, the system moves in the NNMs-regimes. As example, thin lines in (c) indicate a pair oflocalized NNMs.

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Consider the one-parametric (except time translation) family of periodic solutions,allowing the limiting case E:Es .

The solution of the boundary value problem is expressed in a series of successiveapproximations:

x=Y(t)e, Y=K+Y1(t)+Y2(t)+ · · · ; (39)

a2 = a20 /(1− l1 − l2 − · · ·), (40)

where, as before, t= t(t/a) and e= e(t/a); a is equal to a quarter-period of the vibration.The possibility of the aforementioned limiting case means that almost everywhere, exceptthe points L= {t: t(t/a)=21}, the limit,

Y1(t)+ · · ·+YN (t):0 if E:Es , (41)

takes place for any N. Hence, the solution close to the separatrix tends to the limit

x:Ke(t ), t= t/a, (42)

and the system performs sudden ‘‘jumps’’ between the two unstable equilibrium positions(see the Introduction).

Substituting equations (39) and (40) into equations (38), one obtains the followinghierarchy of problems at various orders of approximation:

Y01 − p2h0Y1 =0, Y1=t=1 =−K;

Y02 − p2h0Y2 = l1Y01 − h0I2, Y2=t=1 =0;

Y03 − p2h0Y3 = l1Y02 + l2Y01 − h0I3, Y3=t=1 =0; . . . , (43)

where

h0 = a20 ; p2 =−f '(K); Yi (−t)0Yi (t), (i=1, 2, . . .);

I2 =12!

f 0(K)Y 21 ; I3 = f 0(K)Y1Y2 +

13!

f1(K)Y 31 ; . . . .

In the first step of the procedure one obtains the following correction to the limitingsolution:

Y1 =−K(cosh (pa0t)/cosh (pa0)). (44)

Substituting this expression into the right side of the second of equations (43) andeliminating the ‘‘resonance’’ term by setting l1 =0, one obtains the following expressionfor the second order approximation:

Y2 =K2f 0(K)

6p2[1+cosh (2pa0)] $3−cosh (2pa0t)− [3−cosh (2pa0)]cosh (pa0t)cosh (pa0) %.

In the next step one computes the first non-zero correction in the series (40), as follows:

l2 =K2

8p2 cosh2 (pa0) $53 0f0(K)p 1

2

+ f1(K)%.

Example 3. For the mathematical pendulum x+sin x=0, two steps of the procedureyield

x1 p[1−cosh (a0t)/cosh (a0)]e; t= t(t/a); e= e(t/a);

a2 1 a20 /[1− (p2/8) cosh−2 (a0)],

. . 56

where a0 is an arbitrary parameter that gives asymptotic estimation of the quarter periodclose to the separatrix periodic motion.

5. PERIODICALLY FORCED SYSTEM

In this section it will be shown that the technique can be easy modified for analyzingforced systems.

Indeed, consider the equation of motion x+ f(x, t)=0, x $R, where the vector functionf satisfies the symmetry condition f(x, t)= f(x, 2a− t), for all x, t. The relationships(35) can be used after the replacement f(X):f(X, at). Although the value a is known,the expansion a2 = h0(1+ g1 + g2 + · · ·) can be used for presenting the amplitude-fre-quency dependences.

Example 4. For example consider the saw-tooth forced strongly non-linear oscillatorx+ x+ x5 = pt, where t= t(t/a); p is a constant parameter. Two steps of the algorithm(35) with the replacement f(At):At+(At)5 − pt give

h0 =6A/(A5 +3A−3p);

h1 = [3A2(10A9 +108A5 −105A4p+42A−42p)]/[14(A15 +9A11 −9A10p

+27A7 −54A6p+27A5p2 +27A3 −81A2p+81Ap2 −27p3)];

a2 = h1 h0 + h1.

The relationship derived enables one to compute the main resonance. Figure 8 exhibitsthe amplitude frequency dependences.

6. SELF-EXCITED OSCILLATOR

To demonstrate the application of the STTT-technique for computation of periodicself-excited vibrations, consider the equation of motion

x+ g(x)x+ f(x)=0, x $R, (45)

where f(x) and g(x) are analytical functions, satisfying the following conditions:(i) G(x)= fx

0 g(u) du is an odd function such that G(0)=G(2m)=0 (mq 0); (ii) G(x):aif x:a; G(x) is monotonically increasing for xq m; (iii) f(x) is an odd function such that

Figure 8. ‘‘Amplitude frequency’’ dependences of the oscillator x+ x+ x5 = pt(t/a) for the valuesp=( j−1)/14+0·0001 ( j=1, 2, . . . , 7).

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f(x)q 0 for xq 0. It is known [19] that the system (45) has a single stable limit cycle underthese conditions.

Presenting the corresponding periodic solution in the form

x=X(t)+Y(t)e; t= t(t/a), e= e(t/a), (46)

and taking into account the STTT-properties (section 2), one obtains the followingnon-linear boundary value problem:

X0=−a2Rf − a(RgY'+ IgX')0−FX , X'=t=21 =0;

Y0=−a2If − a(IgY'+RgX')0−FY , Y =t=21 =0. (47)

Analyzing the symmetries of the system under the conditions (i), (iii), one obtains:

X(−t)0−X(t), Y(−t)0Y(t). (47a)

Note that the equations (47) admit no solutions with X(t)0 0, or Y(t)0 0, and,hence, separate consideration of R-, or I-components of the system cannot beprovided.

The solution of the boundary value problem (47) and the quarter-period, a, of the limitcycle are expressed in series of successive approximations:

X=X0(t)+ eX1(t)+ e2X2(t)+ · · · ; Y=Y0(t)+ eY1(t)+ e2Y2(t)+ · · · ;

a= q0 + eq1 + e2q2 + · · ·. (48)

The formal parameter, e, is introduced only for book-keeping purposes and helps oneto obtain the various orders of approximation. At the end of the computation thisparameter is set equal to unity. In line with the idea of the technique, the right sides ofthe equations (47) should be multiplied by e.

Substituting equations (48) into equations (47) and matching the coefficients of therespective powers of e, one obtains the sequence of boundary value problems

X00 =0, Y00 =0;

X0i+1 =−FX, i , Y0i+1 =−FY, i , Yi =t=1 =0, (i=0, 1, 2, . . .);

(X0 +X1)'=t=1 =0, X'i =t=1 =0, (i=2, 3, . . .), (49)

where

FX, i =1i !

diFX

dei be=0

, FY, i =1i !

diFY

dei be=0

.

To provide the construction of a non-zero solution, the orders of the terms X'0 and X'1in the boundary conditions are equalized.

Taking into account the relationships (47a), one represents the hierarchy of boundaryvalue problems (49) in the following integral form:

the zero-th step:

X0 =At, A=const; Y0 0 0;

. . 58

the first step:

X1 =−q20 g

t

0

(t− j)f(Aj) dj, q20 g

1

0

f(Aj) dj=A;

Y1 =−Aq0 gt

0

(t− j)g(Aj) dj, g1

0

(1− j)g(Aj) dj=0;

the (i+1)th step:

Xi+1 =−gt

0

(t− j)FX, i (j) dj, Yi+1 =−gt

1

dz gz

0

FY,i (j) dj,

g1

0

FX, i (j) dj=0, (i=1, 2, . . .).

The algorithm can be easily realized by means of the automatic system of symboliccalculations (‘‘Reduce’’).

Example 5. Consider the self-excited oscillator with a power stiffness:

x+(bx2 −1)x+ xm =0; (m=1, 3, 5, . . .).

Making two steps of the procedure, one obtains the following approximate solution:

x1A0t−tm+2

m+21+A

2v0(t2 − t4)e; t= t0t

a1, e= e0ta1;

bA2 =6,1a

1v0 $1−m

4(m+2)%, v20 =

Am−1

m+1.

The automatic system of symbolic calculations for the case m=3 gives

x1z6/b {t− t5/5+ [105t9 +900t7b−21t5(70b+9)+350t3b]/3150

− [38115t13 +1456650t11b−715t9(3260b+189)+107250t7b

+429t5(2575b+336)−439725t3b]/6756750}

+ {t2 − t4 + [216t8 −196t6 −63t4 +63t2 −20]/420

− [86736t12 +880t10(342b−77)−1584t8(445b+81)

+616t6(640b+189)+11t4(160b−279)−11t2(2960b+2121)

+8(5060b+2413)]/554400}e,

a1 2Xb6 01+

320

+2960b+2121

50400+

7367360b2 +4554992b+8659035605404800 1.

Four steps of the procedure were realized here. Figure 9 gives the phase planerepresentations of the single-step and two-step approximate solutions. For comparison,

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Figure 9. The phase plane representations of the single-step (N=1) and two-step (N=2) approximatesolutions for the limit circle (solid lines). Numerical (‘‘exact’’) solutions are shown with thin lines. (a) b=1·0,N=1 (left); N=2 (right). (b) b=0·5, N=1 (left), N=2 (right).

Figure 10. The quarter period dependence on the value b−1/2 from the first step of the analytical procedureto the fifth one. Numerical solutions are shown with small circles.

the numerical solutions are also presented. Figure 10 exhibits the quarter perioddependence of the value b−1/2. This value gives the approximate estimation for the limitcircle amplitude.

. . 60

7. DAMPED OSCILLATOR

The STTT-technique can be extended for analyzing the non-periodic (damped)vibrations. To show this consider the strongly non-linear but weakly damped single-DOFoscillator

x+2mx+ f(x)=0, 0Q m�1, x $R, (50)

where f(x) is an odd function such that f(x)q 0 for xq 0. To analyze this systemthe STTT-technique will be combined with a two-scales expansion method [20]. Therole of ‘‘fast’’ time scale will be played by the oscillating saw-tooth variable t(8),where the angular velocity, 8, will be assumed to depend on the ‘‘slow’’ time scalet0 = mt,

8=v(t0), (51)

where the explicit form of the right side is yet to be determined. To combine theaforementioned techniques, the solution of equation (50) is expressed as

x=X(t, t0)+Y(t, t0)e. (52)

Substituting expression (52) into the equation of motion (50) and assuming smoothnessof the solution at the points L= {t: t=21} (see section 2), one obtains the set of partialdifferential equations

v2(12X/1t2)=−Rf − mH(1Y/1t)− m2LX,

v2(12Y/1t2)=−If − mH(1X/1t)− m2LY, (53)

under the boundary conditions

Y =t= 2 1 =0, 1X/1t =t=21 =0. (54)

The terms Rf and If are the R- and I-components, respectively, of function f(x) (cf. section2). The differential operators, H, L, are introduced as

H0 2v(1+ 1/1t0)+dv/dt0, L0 12/1t02 +2(1/1t0).

To facilitate the application of the method of successive approximations, the firsttwo terms on the right sides of equations (53) are multiplied by a formal parameter e=1,and the last terms by e2. The solution of the problem (53), (54) is then sought in the seriesform

X=X0(t, t0)+ eX1(t, t0)+ e2X2(t, t0)+ · · · ;

Y=Y0(t, t0)+ eY1(t, t0)+ e2Y2(t, t0)+ · · · ;

v=v0(t0)+ ev1(t0)+ e2v2(t0)+ · · ·

(H=H0 + eH1 + e2H2 + · · ·).

It is more convenient to determine the coefficients of the series expansion of the frequencysquared:

v2 = l0(t0)+ el1(t0)+ e2l2(t0)+ · · ·, li = sa+ b= i

vavb , (i=0, 1, 2, . . .).

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Substituting these series into the equations (53), (54), and considering terms of the firstthree orders, one obtains the following sets of equations:

12X0/1t2 =0, 12Y0/1t2 =0;

12X1

1t2 =−1l0 0R0 + mH0

1Y0

1t 1,12Y1

1t2 =−1l0 0I0 + mH0

1X0

1t 1;12X2

1t2 =−1l0 0R1 + mH0

1Y1

1t+ mH1

1Y0

1t+ m2LX0 + l1

12X1

1t2 1,

12Y2

1t2 =−1l0 0I1 + mH0

1X1

1t+ mH1

1X0

1t+ m2LY0 + l1

12Y1

1t2 1,

where

R0 =Rf =e=0, R1 = 1Rf /1e =e=0; I0 = If =e=0, I1 = 1If /1e =e=0.

Following the idea of the STTT-technique and taking into account the symmetries ofthe system one writes the generating solution (in the zero-th order of approximation) as

X0 =A0(t0)t, Y0 0 0.

Note that, in this case, the constant of integration, A0, depends on the slow time scale t0.This constant is determined by eliminating ‘‘undesirable’’ terms at the next approximation.

Taking into account the boundary conditions (54), in the next step one obtains

X1 =−1l0 g

t

0

(t− j)f(A0j) dj, l0 0v20 =

1A0 g

1

0

f(A0t) dt;

Y1 = (m/2l0)(1− t2)H0A0.

Subordinate the function A0(t0) to the differential equation

H0A0 0 2v0(A'0 +A0)+v'0A0 =0, (55)

so that the first order approximation, likewise the generating solution, contains no termsmultiplied by the discontinuous function e= t'. The integration of this equation gives

A0 =A0zv0/v0 exp(−t0), (56)

where the constants are denoted with the over lines: v0 =v0(0), A0 =A0(0). Consideringthe last set of the sequence of equations, in the second order of e, one obtains

X2 =−1l0 g

t

0

(t− j)f '(A0j)X1(j) dj−m2

l0LA0

t3

6−

l1

l0X1,

Y2 =−m

l0 $H0 gt

1

X1(j) dj+12

H1A0(t2 −1)%,

l1 0 2v0v1 =1A0 g

1

0

f '(A0j)X1(j) dj+m2

2A0LA0.

The higher order approximations can be obtained as in the previous section.

. . 62

Figure 11. The single-step analytical solutions (are shown with thick lines) and numerical (exact) ones (thinlines) for the damped oscillator with non-linear stiffness of the mth degree. (a) m=3; (b) m=7.

Remark 3. For the linear damped oscillator the above technique gives the expansion (onthe powers of e) of the following exact solution:

x=A0v exp (−t0)

ze(1− em2)sin $ze(1− em2)

vt(vt)%, (57)

v2 = e(1− em2)/(4 arcsin2 ze/2); e=1,

and presents the expansion on the powers of e.Example 6. Consider the weakly damped oscillator with the m-degree restoring force

characteristic [15]: x+2mx+ xm =0; (m=3, 5, 7, . . . , 0Q m�1). One has above thelinear case, when m=1. For mq 1 the first two steps of the procedure give

x1A(t− tm+2/(m+2)); A=A� exp(−4mt/(m+3)), t= t(8),

where the phase variable is defined as

818a $1−exp0−2mm−1m+3

t1%;8a =

12m

m+3m−1

A�(1−m)/2

2z2(m+1), A�=const.

It is interesting to note that these analytical expressions predict that the dampedoscillator performs only a finite number of oscillations as t:a. This does not hold in thelinear case (m=1). In Figures 11(a, b) the damped responses of the strongly non-linearoscillator with degrees of non-linearity, m=3 and m=7, respectively, are presented. Theapproximate analytical expressions compare relatively well to the numerical simulations.Note that the corresponding curve is more closed in the case m=7.

8. CONCLUSIONS

A strongly non-linear analytical technique, allowing the analysis of sufficiently generalclasses of vibrating systems, has been formulated. The technique can be viewed as an

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additional one with respect to the broadly expanded quasi-linear (or quasi-harmonic)analytical methods. Indeed, both simple limiting cases of the oscillator (1), x+ xm =0,namely, the left and the right boundaries of the range 1EmEa, are covered by thequasi-linear, and the STTT-technique respectively.

To compare these tools it must be remembered that the quasi-linear methods, as a rule,contain the procedure of the secular (non-periodic) terms’ elimination in approximatesolutions. This procedure finally leads to the amplitude frequency dependences. From theviewpoint of the STTT-technique the similar terms (t, t2 . . . ; t sin (pt/2), t cos (pt/2), . . .)are periodic due to the ‘‘oscillating time’’, t= t(t/a). With regard to the amplitudefrequency dependencies, they appear as a result of the singular terms’ elimination in thetransformed systems. This elimination is provided by satisfying the boundary conditions(see the relationships (16)–(18)).

The second essential distinction between the quasi-linear, and the STTT-technique isthat the STTT-technique gives the localized higher order corrections of the time shapesof vibrations (see Figure 4). So, utilizing the STTT-technique, the smoothness propertiesof the approximate solutions are improved.

ACKNOWLEDGMENT

The author would like to thank Professor A. Vakakis for many useful discussions.

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