oscillatory models of vibro-impact type for essentially non-linear systems

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SPECIAL ISSUE PAPER 2007

Oscillatory models of vibro-impacttype for essentially non-linear systemsL I Manevitch1,2 and O V Gendelman1,∗1Faculty of Mechanical Engineering, Technion – Israel Institute of Technology, Technion City, Haifa, Israel2Institute of Chemical Physics, Russian Academy of Sciences, Moscow, Russia

The manuscript was received on 21 February 2008 and was accepted after revision for publication on 3 July 2008.

DOI: 10.1243/09544062JMES1057

Abstract: This paper reviews recent developments related to oscillatory systems, their impact onand relationship to the cases of smooth, essentially anharmonic (non-linearizable) potentials,and vice versa. Special methods of treatment that allow the response regimes in dynamic vibro-impact systems to be computed have been discussed. Mathematical models that approximatepurely elastic as well as inelastic impact, with the help of smooth functions, are presented andillustrated by specific examples. The use of ideas based on non-smooth time transforms to treatessentially non-linear systems with smooth potentials has also been discussed. Special attentionhas been paid to uncommon applications of vibro-impact models.

Keywords: vibro-impact systems, non-linear oscillations, targeted energy transfer, non-smoothtime transform, localization, non-linear normal modes

1 INTRODUCTION

Oscillatory systems with essential non-linearitiesoccupy a very special niche in the theory of vibrations.On the one hand, it is very important to understandtheir behaviour, both from an academic perspectiveand in view of the numerous possible applications. Onthe other hand, they can be very difficult to analyse.Besides an extremely narrow class of integrable sys-tems [1, 2], they cannot be described exactly. If one ofthese integrable cases is, in a certain sense, close to thesystem under consideration, then, often, some kind ofperturbation procedure can be realized [2, 3], yield-ing at least a qualitative understanding of the globaldynamics.

In many interesting cases, however, such options arenot available. For these cases, one is forced to restrictone’s search to particular solutions of interest. In theabsence of a general picture, these particular solutionsconvey valuable information about the dynamics ofthe system and are also useful for testing numericalapproaches. Important examples of these particular

∗Corresponding author: Faculty of Mechanical Engineering, Tech-

nion – Israel Institute of Technology, Technion City, Haifa 32000,

Israel. email: [email protected]

solutions, available in many essentially non-linear sys-tems, are non-linear normal modes (NNMs) [4, 5]. Oneshould mention that these modes are just particularsolutions that can be obtained out of symmetry con-siderations or as synchronous motions in the system.For weakly non-linear systems, they can be analyticcontinuations of the regular normal modes, althoughdo not play the same basic role – the superposition isabsent.

Vibro-impact systems, which are described in detailin many studies [6–18], demonstrate the most severenon-linearity – in fact, the strongest possible. Thesenon-linear interactions are concentrated at the pointsof impact; therefore, one can substitute the compu-tation of the complete trajectory by matching theparts via impact conditions [8, 18]. Thus, the extremenon-linearity is ‘pushed’ to the boundary or match-ing conditions. This is important simplification whencompared with the solution of the complete dynamicproblem, especially if one is interested in particular(e.g. periodic) solutions. Consequently, the extremityof the non-linear interaction may itself simplify thesystem.

Simplification arising from the extreme impact non-linearity has inspired numerous attempts to extend itto systems with less extreme behaviour [6, 9, 11].Theseworks try to construct special asymptotic approaches

JMES1057 © IMechE 2008 Proc. IMechE Vol. 222 Part C: J. Mechanical Engineering Science

2008 L I Manevitch and O V Gendelman

for different systems that are in some sense ‘close’to the vibro-impact ones. Thus, one can constructconvenient approximations for the computation ofparticular periodic orbits [17] and sometimes alsoextend the approach enough to grasp the transientdynamics, at least for certain particular cases. In othercases, the situation turns out to be rather the oppo-site – it is convenient to substitute the impact potentialby some smooth approximation and then solve theappropriate smooth problem [11].

The aim of the article is to describe attempts toinvoke the vibro-impact models in various physicalcontexts. To be clear, an exhaustive review of existingliterature on the vibro-impact models is not provided –this issue has been addressed in several reviews,including a recent, detailed paper [18]. Instead, theauthors would like to concentrate on the results thatillustrate non-trivial and unexpected manifestationsof the vibro-impact (or close to it) dynamics.

In section 2, a number of works devoted to compu-tation of families of periodic and localized solutionsin the vibro-impact systems are reviewed. In section3, a relationship between the impact and the smoothpotential function is discussed both for conserva-tive and dissipative cases. Section 4 treats the use ofasymptotic methods, inspired by the vibro-impacts,to describe smooth dynamic systems, apparently farfrom anything related to the impacts.

2 SPECIAL SOLUTIONS OF VIBRO-IMPACTSYSTEMS

2.1 The particle between rigid barriers

The case of an absolutely elastic interaction betweenthe particle and rigid barriers is considered. The par-ticle’s motion in this case is described by the Newtonequation

md2udt 2

+ P(

u,dudt

)= 0 (1)

P(

u,dudt

)= 2

dudt

[δ(u + �) − δ(u − �)] (2)

There are three natural approaches to describe vibro-impact processes. The first one was proposed in refer-ence [16] and uses the evident solution of equations(1) and (2), which is a saw-tooth sine of period 2π andamplitude 1

u(t) = �τ(φ) = arcsin[sin(φ)], dφ

dt=(

12

)πv�

(3)

where φ is the phase and v the constant velocity ofthe particle. A non-smooth change of the dependent

variable transforms equation (1) to the simplest form

d2φ

dt 2= 0 (4)

Another possibility, proposed in papers [14, 19] isbased on the change of the right-hand part of expres-sion (2) by the periodic function, reflecting a sequenceof periodic pulses. The δ function describes the impactinteraction of the particle with a rigid barrier, with theintensity of interaction depending on impulse 2p. Thegeneralized function can also be presented in morecompact form as a second derivative (in the senseof the distribution theory) of a saw-tooth periodicfunction

P(

u,dudt

)= 2p

{∑δ

[t −

(t4

+ kt − �

)]

− δ

[t −

(−T

4+ 2kt − �

)]}

= pd2

τ

dt 2(5)

The period in the right-hand part of expression (2) isdetermined by impulse intensity.

It is important to note that, contrary to the ini-tial system, when using presentation (5), the systembecomes formally linear, although with a non-smoothright-hand part.

This approach is strongly connected with the non-smooth change of the independent (temporal) vari-able, which may be naturally made by introducing thenon-smooth sine-like and cosine-like basic functionsτ(t) and e(t) (Fig. 1). The latter is considered as a gen-eralized derivative of the former (these basic functionswere first introduced by Pilipchuk in reference [14] andthe corresponding procedure was further developed inreference [19]).

Looking for the solution as

u = U (τ ) (6)

It follows that

d2Udτ 2

= 0 (7)

and

U (τ ) = �τ (8)

Therefore, the solution of the problem on a non-smooth basis is even simpler than that given fora classical linear oscillator and is presented as thestraight line: in real-time, it corresponds to a peri-odic motion with periodically repeating impacts. Both

Proc. IMechE Vol. 222 Part C: J. Mechanical Engineering Science JMES1057 © IMechE 2008

Oscillatory models for non-linear systems 2009

Fig. 1 Basic functions τ(t) and e(t)

equations (4) and (6) are tractable models of the vibro-impact system, and they can be extended to the casesof many degrees of freedom and forced vibrations.

The third natural approach is based on a canon-ical transformation of displacement and velocity toaction and angle variables. If, following paper [20], oneconsiders a particle of mass 1/2 oscillating in an infi-nite square well potential, its Hamiltonian is writtenas follows

H = p2 + b[η(w − 1) + η(−w − 1)] (9)

where w is again the position of the particle, p themomentum, η the unit step function, and b the heightof the square well.

New variables can be introduced by the transforma-tion

w = −1 +(

2ϑ

π

)sign[sin(ϑ)]

p =(

πI2

)sign(sin ϑ) (10)

ϑ ∈ [−π, π], w ∈ [−1, 1], and p are periodic functionsof ϑ with period 2π. This transformation can be easilypredicted because it reproduces the uniform periodicmotion of the particle between the rigid barriers withperiod T = 2/|p| over the time period [−τ/2, τ/2], ifI = (2

√E0)/π, ϑ = π2It/2. This solution is coordinated

with the transformed Hamiltonian

H = π2I 2

4(11)

and equations of motion

dIdt

= dHdϑ

,dϑ

dt= −dH

dI(12)

with initial conditions t = 0, I = (2/π)√

E0, and ϑ = 0.

2.2 Linear oscillator with rigid barriers

If a linear oscillator vibrates between two rigid barriersafter the introduction of non-smooth basic functionsand presentation of the solution in the form

u = U(

4T

τ

), u � � (13)

then the tractable model described by equation [19] isobtained

(4T

)2 d2

dτ 2U + ω2

0U = 0,dUdτ

(τ = ±1) =(

T4

)2

p

(14)

Taking into account (alongside equation (13)) the con-dition U = ±� when τ = ±1, one can find the solutionof the boundary problem in the form [19]

u = sin[(T /4)ω0τ ]sin[(T /4)ω0] (15)

If u(0) = 0, it means that φ = 0 and total energy

E = 12

(dudt

)2

(16)



and

ω0

(T4

)= ±1

2arccos

(1 − ω2

0�2

E

)+ kπ,

k = 0, 1, 2 . . . (17)

There is a critical value of energy

E = Ecr = ω20�

2

2(18)

such that the oscillator can reach the constraints ifE � Ecr.

In the case of a unilateral barrier (e.g. if the left-hand barrier is removed), the problem can be solvedsimilarly [19].

There is a natural limit case that can be foundfrom the analysis of equation (12). It corresponds tocondition ω0 → 0 when

u → �τ = �τ

(t

T /4+ �

)(19)

corresponding to the solution for the particle betweentwo rigid barriers.

2.3 The oscillatory chain with rigid barriers

To clarify the sense of simplification provided by theuse of the vibro-impact model, the chain of non-linear oscillators with rigid barriers is considered.Corresponding equations of motion can be written asfollows

md2wj

dt 2+ c1wj + c3w3

j + c(2wj − wj=1 − wj−1)

+ P(

wj ,dwj

dt

)= 0, |w| < � (20)

If |wj| � �, the system (20) has only a smooth potentialof interaction. When |wj| = �, the impact interactionappears, which can be replaced by periodic externalimpulses.

The specificity of the vibro-impact model is that ifthe conditions

|wj| < � (21)

are satisfied, the system is described by non-linearsmooth equations because

P(

wj ,dwj

dt

)= 0 (22)

It is possible to distinguish several important particu-lar cases.

The first possibility is a ‘small’ periodic or clampedchain in which differences of the natural linear fre-quencies are comparable with the frequencies them-selves. In this case alone, the system is essentiallya discrete one. The particular case c = 0, c3 = 0 (aweightless string with discrete masses between rigidlimiters) for the string with two and three uniformlysituated particles (in former case j = 1, 2 and w0 =w3 = 0; in latter one j = 1, 2, 3 and w0 = w4 = 0) hasbeen considered in paper [21].

Spatial transformation introduced in reference [15]was used in this article. This transformation, whichreflects the form of the exact solution for uncoupledparticles interacting with the rigid barriers mentionedabove, does not remove but changes the type of dis-continuities (after transformation, the equations ofmotion do not contain functions with discontinu-ities of the second kind). In the case of an intensiveimpact interaction, the coupling is demonstrated tobe relatively small.

This means that the transformed system contains asmall parameter that allows the introduction of a slowtime and to use a procedure of averaging with respectto fast time. On the basis of such an approach, the exis-tence of (3n − 1)/2 NNMs that essentially exceeds thenumber of degrees of freedom is revealed. Parts of theNNMs, including the symmetric one (all particles havesimilar displacements) and normal modes localizedon one or two neighbouring particles, are stable withrespect to small perturbations. The rest are unstable,which is confirmed by computer simulation.

The analysis of a smooth but strongly non-linear sys-tem on the basis of a vibro-impact solution as a startingapproximation has also been shown. These results areillustrated in Figs 2 and 3.

Fig. 2 General sketch of the system



Fig. 3 Modal shapes for n = 3

The system is schematically described in Fig. 2. Theshapes of possible modes are illustrated in Fig. 3. Thein-phase mode is stable (denoted by the ‘+’ sign, solu-tion (6) in Fig. 3). The antiphase mode is unstable anddenoted by the ‘−’ sign (solution (7)). Other modesin which the displacements of two adjacent masseshave different signs (solutions (4) and (8)), and also themode in which the fixed mass separates two deformedsprings (solution (2)), are unstable.

The question arises: what will happen when thecoupling between the adjacent oscillators increases?To get an answer, it is reasonable to deal with othertractable vibro-impact models mentioned above andthe sequence of impulses. This approach reveals muchmore complicated types of periodic motions thanthose available for study in the framework of theprevious approach. The simplest case of two-degree-of-freedom system is considered. If the energy issufficiently small, the system will have two stablenormal modes: in-phase and anti-phase. The varia-tion of the coupling parameters and energy of theregime brings about generic bifurcations (saddle-node, period-doubling, etc.) that manifest in thechanges of the impact sequences of responses. In thecase of weak coupling with increased energy, ‘saddle-node’ bifurcation of the antiphase mode leads to thegeneration of the NNMs described earlier. This bifur-cation [21] will give rise to two pairs of localized modeswith two-sided collisions of one of the masses. It turnsout that in the case of the stronger (moderate) cou-pling, increase of the energy parameter gives rise, firstof all, to the formation of stable (1a) and unstable(2a) modes with one-sided impacts of the masses on‘unlike’ stopping devices (Fig. 4). The unstable branch3 and stable branch 4 correspond to oscillations withtwo-sided collisions of the first mass, whereas the sec-ond mass is not engaged in the collisions. Finally,solution (5) of type c (one-sided collisions of eachoscillator with ‘similar’ stopping devices) is unstable.In Fig. 5, some modes of oscillations are shown in theconfiguration plane (w1, w2).

Fig. 4 Branches of solution in the E−w2 plane (moder-ate coupling)

Stable modes are denoted by the solid curves andunstable modes by the dashed curves. Following acomparison of the curves in Figs 4 and 5, importantconclusions can be drawn as to the change in the orderin which the different types of periodic states are ‘born’as the energy parameter of the system increases.

In order to demonstrate the increasingly complexbehaviour of the system at large values of the cou-pled parameter, the data for periodic vibro-impactstates in the case of strong coupling are also presented.These regimes are illustrated in Fig. 6 ‘total energyE – amplitude w2(0)’ plane and in Fig. 7 – in the con-figuration plane (w1, w2). Here, the branches 1a and2a (stable, shown in Fig. 3 by the solid curves) and3a–5a (unstable, shown by the dashed curves) rep-resent oscillations of type a (collisions with ‘unlike’stopping devices). Figure 7 shows some of the stableand unstable modes of types a, b, and c that have beenfound. The numbers on the curves in Fig. 7 correspondto the numbering of the modes in Fig. 6. The analy-sis of the results presented in Figs 6 and 7 indicatesthat the order in which modes of different types are‘born’ is changed. In particular, the stable localizedmodes, which correspond to two-sided collisions ofone of the masses and are most important in the caseof small ε values, may be realized here at very largeenergies. However, modes with one-sided collisions ofthe masses with ‘like’ stopping devices arise even atsmall energies.

2.4 Vibro-impact system in weak external periodicfield

The simplification caused by using the vibro-impactmodel is clear if one considers the effect of the externalfield.



Fig. 5 Modes of oscillation in the configuration plane w1−w2 (moderate coupling)

Fig. 6 Branches of solution in the E−w2 plane (strong coupling)

It may be considered as another possible simplemodel of the free particle moving between two rigidbarriers. For the application of canonical transforma-tions, paper [20] is followed, in which a particle of mass1/2 oscillating in an infinite square well potential in thepresence of periodic external field is described by theHamiltonian

H0 = p2 + b[η(w − 1) + η(−w − 1)] + εw cos �t ,

b = +∞(23)

In this case, the application of transformation

w = −1 +(

2ϑ

π

)sign(sin ϑ)

p =(

πI2

)sign(sin ϑ)

(24)

yields

H = π2I 2

4+ εw(ϑ) cos �t (25)

The advantages of the vibro-impact model in com-parison with strongly non-linear but smooth modelshave now to be clarified. Namely, perturbation of theconservative Hamiltonian caused by the presence ofinternal force does not depend on the action variableI . In fact, this perturbation part can be expanded intothe trigonometric series by the second variable ϑ withknown coefficients

H = π2I2

4− 4ε

π2

∞∑n=−∞

(1

n2

)cos(nϑ − �t) (26)

where n is odd.



Fig. 7 Modes of oscillation in the configuration planew1−w2 (strong coupling)

The corresponding equations of motion have to bewritten as follows

dIdt

= ∂H∂ϑ

= 4ε

π

∑(1n

)sin(nϑ − �t)

dϑ

dt= −∂H

dI= −π2I

2

This system is equivalent to a second-order equation

d2ϑ

dt 2+ 2πε

∞∑n=−∞

(1n

)sin(nϑ − �t) = 0 (27)

The presence of the small parameter indicates themultiple scale nature of the processes considered.Introducing alongside the ‘fast’ time t = to, a ‘slow’time t1 = ετ0, the solution to equation (27) can be givenas a power expansion by the small parameter ε

ϑ = ϑ0 + εϑ1 + ε2ϑ2 + · · · (28)

Taking into account that

ddt

= ∂

∂t0+ ε

∂

∂t1

one can rewrite equation of motion (27) as

∂2

∂t 20

(ϑ0 + εϑ1 + · · · ) + 2ε∂2

∂t0∂t1(ϑ0 + εϑ1 + · · · )

+ ε2 ∂2

∂t 21

(ϑ0 + εϑ1 + · · · ) + ε2π

×∞∑

n=−∞

(1n

)sin[n(ϑ0 + εϑ1 + · · · ) − �t0] = 0

(29)

Selecting the terms of a similar order with respect tothe small parameter ε and equating them to zero, onecan obtain

ε0:∂2

∂t 20

ϑ0 = 0 (30)

ε1:∂2

∂t 20

ϑ1 + 2∂2

∂t1∂t0ϑ0

+ 2π

∞∑n=−∞

(1n

)sin(nϑ0 − �t0) = 0 (31)

Equation (30) is similar to the equation for the conser-vative vibro-impact system without an external field;it has the solution

ϑ0 = ω(t1)t0 (32)

Supposing that ϑ0 coincides with the solution for aconservative system in which dϑ/dt(t = 0) = π

√E0,

one can find that

ϑ0 = π√

E0t0, so ω = π√

E0 (33)

Equation (31), with equation (32), may be written asfollows

∂2

∂t 20

ϑ1 + 2∂

∂t1ω + 2π

∞∑n=−∞

(1n

)sin[(nω − �)t0] = 0

(34)

Integrating equation (33) with respect to t0, supposingthat

ϑ1(0) = 0, ϑ1 = 2π

∞∑n=−∞

(1n

)sin(nω − �)t0

(nω − �)2(35)

This result is valid if the denominator is not close toone of the resonances

nω − � = nπ√

E0 − � = aε � 1 (36)

When these relations are not fulfilled, equation (35)provides a small periodic correction caused by thepresence of the external periodic field, and

ϑ = ϑ0 + εϑ1 + · · · (37)

In the opposite case, if one of the relations in equation(36) takes place, the corresponding term in the sum,which is the right-hand part of relation (35), is large(e.g. of the order ε−1) and has to be accounted for in the



zeroth approximation in spite of the small parameter ε

∂2

∂t 20

ϑ0 + 2πε sin(nϑ0 − �t0) = 0 (38)

After introducing a new ‘slow’-dependent variable

φ = nϑ0 − �t0 (39)

one can obtain the final equation of the main asymp-totic approximation

d2

dt 20

φ + 2πε sin φ = 0 (40)

which corresponds to a pendulum model.It is clear that this model adequately describes

the strong interaction of the vibrating particle withthe periodic external field or resonance capture. Thestationary state ϕ = 0 is a resonance regime with a con-stant energy share between the particle and the field.Other phase trajectories surrounded by the separatrixdescribe weak energy redistribution near the stablestationary point and strong energy redistribution nearthe separatrix itself. In both cases, simple analyticalrepresentations are possible: in the former case byusing a quasilinear approach and in the latter, on thebasis of non-smooth temporal transformation. Theexact solution of equation (40) can be found in terms ofelliptic functions. The phase trajectories situated outof the separatix correspond to a weak interaction of theparticle with the periodic field and may be describedusing non-resonant asymptotics if they are far enoughfrom the separatrix.

It is easy to calculate the width of the resonanceusing a simple solution of equation (40) correspondingto the separatrix

ϕs = 4 tan−1(exp√

2πεto) − π (41)

dϕs

dt0= ± 2

√2πε

cosh(√

2πεt0)(42)

where dϕs/dt0 represents the width of resonance,which is the frequency of vibrations and which turnsout to be equal to 4

√2πε.

Also, one can estimate the resonance width by theaction variable. Using denotation I = I0 + �I , where I0

corresponds to free motion without external forcing,the equations of motion are presented as follows.

dϑ

dt= −

(π2

2

)(I0 + �I ) (43)

Taking into account that due to the resonance condi-tion

dϑ0

dt0= 1

n�

that leads to the relation

dϑ0

dt0=(

π2

2

)(−I0) =

(1n

)� (44)

and I0 = −(−2/nπ2)�.Then

dφ

dt0= −(nπ2�I ),

d(�I )dt0

=(

4ε

πn

)sin ϕ

Therefore

�I = −(

2nπ2

)dϕ

dt0

The resonance width by the action variable is

�I = 8√

2ε

nπ1.5(45)

One resonance approximation is valid till the inter-resonance interaction is not essential, and it is realizedif both the external frequency and energy of the par-ticle are essentially more than zero [20]. When theyare small enough, the approximation of isolated reso-nances loses its relevance: this implies a transition tochaotic behaviour.

To estimate the conditions of such transition, theoverlapping criteria of resonances had been proposedby Chirikov [22] almost half a century ago. These cri-teria turn out to be insufficient when the resonancedensity is very large as in the case of a vibro-impactsystem in a periodic external field [22].

One way of finding a solution to this problem is touse the ideas of renormalization groups (RGs). The factthat a set of transformations changing from one wayof normalization to another forms a group was firstrevealed in 1953 [23], and already in 1954, it had beenused to improve the results of the perturbation theoryin quantum electrodynamics [24]. For RG application,it is important that the property of RG invariance isthe characteristic for whole infinite series, but not forevery term of this series itself.

In the case being studied here, it means that nei-ther approximation of isolated resonances nor eventhe overlapping of two resonances need be consid-ered, but only the full series in equation (26). Changeof normalization means rebuilding of the series tak-ing into account information with respect to distinctterms of the series and the condition of RG invariance[25].

As for the vibro-impact systems, to the best of theauthor’s knowledge, no Hamiltonian is known to datewhich allows exact computation of the RG. However,in the report in reference [25], it has been shown thatthe numeric results for the standard map fit well intothe renormalization scheme, and thus the transitionto global stochasticity may be successfully predicted.



2.5 Discrete breathers in a vibro-impactone-dimensional system

Models of one-dimensional chains are used in numer-ous applications of the solid state physics as simplebenchmark models for the study of different proper-ties of crystals. In general, these models are non-linearand hardly available for exact analytic treatment.

The Toda lattice [26] is an exception. It is describedby the following Hamiltonian

H =∑ p2

n

2+ exp[−(Qn − Qn−1)]

This system has been demonstrated to be integrableand to allow soliton solutions. Technical details are notpresented here; the interested reader may consult themonograph [26].

If one considers the Toda lattice in the limit of highenergies, it is possible to demonstrate that this latticeis asymptotically reduced to a chain of particles withthe impact potential of interaction [26]. Such a sys-tem is especially simple to analyse, as two particles ofequal mass in one dimension simply exchange theirvelocities after collision. Therefore, if a particle at oneend of such a chain is given a certain initial velocity,then the pulse will travel through the system withoutany dissipation. Of course, it is the simplest availablemodel of a spatially localized solitary wave.

The use of the vibro-impact model also allows oneto get another rather unique object – an exact analyticsolution for a discrete breather (DB). Such breathershave been an object of intensive study for many years,but exact analytic results were obtained for very fewsystems [27].

A common one-dimensional linear chain with everyparticle placed between rigid on-site barriers is con-sidered. Equations of motion are

un + c(2un − un−1 − un+1) = 0, |un| < �,

n = 0, ±1, ±2, . . . (46)

Scalar un denotes the displacement of the nth par-ticle of unit mass and c is the rigidity of the linearcoupling. The distance between the barriers at eachsite is equal to 2�. An interaction of every particle withthe rigid barrier as the displacement achieves ±� isdescribed as a purely elastic impact. It means that if theimpact occurs at time t0, then the following conditionholds for all n

limt→t0−0

un = − limt→t0+0

un

∣∣∣∣un=±�

(47)

System (46) is schematically presented in Fig. 8.Systems (46 and 47) can be considered as particu-

lar cases of discrete Klein–Gordon lattices [27]. Thesesystems are known to possess a family of localized

Fig. 8 Infinite linear chain with impact barriers

dynamic solutions referred to as DBs. Normally, onlyapproximate solutions for the DB may be constructed.System (48) is obviously non-integrable; still, due toits simplicity it is possible to obtain exact solutions forthe DB.

The solution of equations (46) and (47) is foundby supposing that only one particle is engaged inperiodic impacts with the barriers.Without loss of gen-erality, it can be suggested that it is particle n = 0.Such impacts are equivalent to the action of periodicexternal δ-pulses on this particle. In other terms, forparticular solutions, system (46) is equivalent to thefollowing equations

un + c(2un − un−1 − un+1)

= 2pδn0

∞∑k=−∞

[δ

(t − T

4+ kT

)− δ

(t + T

4+ kT

)](48)

where T is the period of the impacts, 2p is the unknownchange of the particle moment in the course of theimpact, and δij is the Kronecker symbol.

At this point, the crucial advantage of the vibro-impact model reveals itself – equation (48) is linear andmay be solved exactly. Once the solution is obtained,one should check whether it satisfies the followingconditions:

(a) maximal displacement of the particle n = 0 isequal to �;

(b) maximal displacements of all particles with n �= 0are less than � (no other impacts occur).

If both these conditions are satisfied, then the solutionof forced linear equation (48) is a genuine solution ofthe initial system (46).

It is convenient to rewrite the right-hand side ofequation (48) as a Fourier series (in the sense ofdistributions)

un + c(2un − un−1 − un+1)

= δn04pω

π

∞∑j=1

(−1)j sin[(2j − 1)ωt] (49)

Here ω = 2π/T . Thus, the conditions of the impact areequivalent to local forcing of the chain with multi-ple frequencies. The dispersion relation for travelling



waves in the linear chain is

�2 = 2c(1 − cos q) (50)

where � is the wave frequency and q the wavenumber.Consequently, the frequency spectrum of any periodiclocalized solution must be situated in the attenuationzone – above the maximum frequency

�max = 2√

c (51)

The forcing terms in equation (49) have frequenciesω, 3ω, 5ω, etc. Consequently, the forced solution ofequation (49) will be localized if

ω > �max (52)

The stationary solution of equation (49) may be easilyfound with the help of an Z transform. It can be writtenin the following form

un(t) = (−1)npω

πc

∞∑j=1

(−1)j

×

(2μ(2j − 1)2 − 1

−2√

μ2(2j − 1)4 − μ(2j − 1)2

)|n|

√μ2(2j − 1)4 − μ(2j − 1)2

× sin[(2j − 1)ωt]

μ = ω2

4c(53)

Maximum displacement of the particle n = 0 shouldbe equal to the impact threshold �. This is achievedwhen t = T /4 + kT /2. In other terms

∣∣∣∣u0

(T2

)∣∣∣∣ = pω

πc

∞∑j=1

1√μ2(2j − 1)4 − μ(2j − 1)2

= �

(54)

From equation (54), one obtains the value of theunknown coefficient p. With account of equation (54),equation (53) is reduced to the following form

un(t) = (−1)n�

×

∑∞j=1(−1) j

({[2μ(2j − 1)2 − 1

−2√

μ2(2j − 1)4 − μ(2j − 1)2]|n|/

√μ2(2j − 1)4 − μ(2j − 1)2

}sin[(2j − 1)ωt]

)∑∞

j=1 1/√

μ2(2j − 1)4 − μ(2j − 1)2

(55)

Expression (55) is the exact solution for the DB. Atthe outset, it should be mentioned that the series con-verge both in the numerator and in the denominator.

In the numerator, the decay of the coefficients for theFourier series is similar to (2j − 1)−(n+2) for a large j; inthe same limit, the series in the denominator behavessimilar to �(2j − 1)−2.

Maximum displacement of the nth particle isexpressed as

∣∣∣∣un

(T2

)∣∣∣∣ = �

∑∞j=1

{[2μ(2j − 1)2 − 1

−2√

μ2(2j − 1)4 − μ(2j − 1)2

]|n|/√

μ2(2j − 1)4 − μ(2j − 1)2}

∑∞j=1

[1/√

μ2(2j − 1)4 − μ(2j − 1)2]

(56)

It is easy to demonstrate that function

F (x) = 2x − 1 − 2√

x2 − x (57)

for x > 1 obeys 1 > F (x) > 0 and decreases mono-tonously when x increases. Thus, the followinginequalities hold

∣∣∣∣un

(T2

)∣∣∣∣ = �

∑∞j=1

{[2μ(2j − 1)2 − 1

−2√

μ2(2j − 1)4 − μ(2j − 1)2]|n|/

√μ2(2j − 1)4 − μ(2j − 1)2

}∑∞

j=1

[1/√

μ2(2j − 1)4 − μ(2j − 1)2]

<(

2μ − 1 − 2√

μ2 − μ)|n|

�

=(

2μ − 1 − 2√

μ2 − μ)|n| ∣∣∣∣u0

(T2

)∣∣∣∣ (58)

If μ > 1 (i.e. the basic frequency of the impacts ω is inthe attenuation zone), then the solution (55) is expo-nentially localized, as one should expect from the DB.Besides, for any n, the maximum displacement of theparticles is less than �, i.e. they are not engaged inthe impacts. This observation concludes the proof ofconsistency for solution (55).

It does not seem possible to derive simple com-pact expressions for the series in expression (55) ina general form, but this is not very significant. As theseries converge fast enough, no special computationdifficulties are encountered.

In order to illustrate the solution (55), the breatherprofile – maximum displacement for each particle –for μ = 3 (basic frequency far from the boundary ofthe attenuation zone, Fig. 9(a)) and μ = 1.05 (basic fre-quency close to the boundary of the attenuation zone,Fig. 9(b)) is plotted. � is adopted as unity. From obvi-ous considerations of symmetry, it is adequate to plotonly the particles with n � 0.

One can see that, as expected, the breather withbasic frequency far from the boundary of the propa-gation zone is strongly localized, whereas the DB rela-tively close to this boundary is much wider. In order



Fig. 9 Breather profile for (a) high frequencies (μ = 3)and (b) low frequencies (μ = 1.05)

to assess the type of motion exhibited by differentparticles, it is instructive to plot the time dependenceof displacement for n = 0 and 1 for the same values of� and μ as in Figs 8 to 9 (Figs 10 and 11). The basicfrequency ω in both cases is scaled to unity. Approx-imately similar localized excitations were observedexperimentally in reference [9].

3 VIBRO-IMPACT SYSTEMS TREATABLE BYANALYTIC FUNCTIONS

3.1 Elastic impacts – relation to smooth functions

As just stated, the vibro-impact model can be con-sidered as a limiting case for systems with stronglynon-linear but smooth potentials of interaction, whichmay be of the power or exponential type. This situa-tion is typical when considering solids on a molecularlevel, because the atomic potentials of interaction in

Fig. 10 Time dependence (a) u0(t), μ = 3 and (b) u1(t),μ = 3

the compression region are adequately described bythe power or exponential type functions.

The method of formally changing a strongly non-linear but smooth elastic force to impact interactioncan be briefly discussed here. The former can bedescribed, e.g. by power non-linearity of a high degreesuch as w2n+1, n 1. After introducing the variableW = w/�, where � is amplitude, the latter corre-sponds to the limiting case n → ∞. As shown inthe paper [28] one can formally change a smoothnon-linearity by impact interaction using the Laplacetransformation of the power elastic force

φ(p) = p−n−1γ (n + 1, p) (59)



Fig. 11 Time dependence (a) u0(t), μ = 1.05 and (b)u1(t), μ = 1.05

where γ (n + 1, p) is the incomplete γ function. Afterits expansion by the small parameter 1/n and returnto the original, the elastic force is presented as follows

F (W ) =(

1n + 1

)[δ(W − 1) − δ(W + 1)]

−[

1(n + 1)(n + 2)

]

×[

dδ

dt(W − 1) − dδ

dt(W + 1)

]+ · · · (60)

In this manner, the strongly non-linear power forceis replaced by impact interaction, and the equation of

motion in the main asymptotic approach is presentedas follows

d2Wdt ∂2

+ �n−1

(n + 1)[δ(W − 1) − δ(W + 1)] = 0 (61)

where the multiplier before the squared brackets playsthe role of the impulse change that is the result of‘collision’.

Therefore, the system is transformed to view,becoming typical for the particle vibrating betweentwo rigid barriers. The construction of the correc-tions to the main asymptotic approximation is alsodiscussed in reference [28].

3.2 Inelastic impacts – integrable models

As mentioned in the previous section, instead ofimposing exact impact conditions, it is possible to sim-ulate the impact-like motion with the help of smoothpotential functions [11, 29, 30]. Two such commonmodels are those of two-sided impact, where the par-ticle is allowed to move between two restraints, andone-sided impact, with only one restraint. In the for-mer case, the smooth function approximation takesthe form of potential of the shape x2n with the n – pos-itive integer, and the motion is considered for x in theinterval (−∞, +∞); however, if n is large enough, themotion of the particle is localized in the vicinity of theinterval (−1, 1). Therefore, impact restraints at points±1 are simulated. In the case of a one-sided impact,the potential x−α, α > 0 is used, simulating the impactrestraint at x = 0. In this case, the motion is restrictedto the interval (0, +∞).

The smooth function models of impact interactionsdescribed here were designed for the case of purelyelastic impacts (with unity restitution coefficient). Nogeneralization of this strongly non-linear model existsfor the case of inelastic impacts, i.e. for the restitu-tion coefficients less than unity. One should mentionthat there exists a number of models that simulate theinelastic impact with the help of viscoelastic elements[29, 30]. This approach, however, is completely linear.

The current section is devoted to the developmentof such models for the cases of both two-sided andone-sided impacts. In other words, it is intended hereto generalize the existing smooth function modelsfor the case of the restitution coefficient of less thanunity. A common one-dimensional model for simulat-ing the elastic impacts by means of smooth functionsis formulated with the help of the following equation

x + (n + 1)x2n+1 = 0 (62)

where x denotes the displacement of the particle andn is the positive integer. Limit n → ∞ corresponds tothe motion of a free particle between impact restraintsat x = ±1.



The goal of this article is to generalize the modeldescribed by equation (62) in order to describe theimpacts with non-unity restitution coefficients. Thatis, the model described by the equation must be found

x + f (x, x) + (n + 1)x2n+1 = 0 (63)

which will satisfy the following conditions

As n → ∞, f (x, x) → 0, |x| < 1

f (x, x) → ∞, |x| = 1(64)

For x = 0

κ |x(0)|before impact = |x(0)|after impact (65)

where κ is the restitution coefficient. The value of κ

should not depend on the value of the initial velocityof the particle.

The condition (65) for invariance of the restitutioncoefficient with respect to velocity imposes severerestrictions on the possible shapes of the function f .Scaling with respect to time should preserve the sym-metry of the equation; therefore, the function f musthave a shape

f (x, x) = x2m+1g(x) (66)

where m is a non-negative integer. Further simplifica-tions are based on the fact that velocity independenceof the ‘restitution coefficient’ simulated by equation(63) manifests an additional internal symmetry of thesystem, i.e. its invariance with respect to certain non-trivial groups. Infinitesimal Lie generators of suchgroups for equation (63) may be written as [31]

Z = ξ(x, t)∂

∂t+ η(x, t)

∂

∂x

Z1 = [ηt + p(ηx − ξt) − p2ξx] ∂

∂p

Z2 = [ηtt + p(2ηxt − ξtt) + p2(ηxx − 2ξxt)

− p3ξxx + r(ηx − 2ξt) − 3prξx] ∂

∂r(67)

where Z1 and Z2 are the first and the second prolon-gations respectively, of the infinitesimal operator Z ,p ≡ dx/dt , r ≡ d2x/dt 2. The symmetry condition forequation (63) along with equation (66) is

(Z + Z1 + Z2)[r + p2m+1g(x) + (n + 1)x2n+1] = 0

(68)

Equation (68) may be easily solved by the standardmethods [4, 31]. The result is summarized below

η = Cx, ξ = −Ctn provided that m = 0,

g(x) = μxn

where C and μ are constants. It is convenient to rescalen = 2k, μ = λ(2k + 1). Condition (64) imposes addi-tional restrictions (the effective damping of the systemshould be positive), and finally the required system ispresented as follows

x + λ(2k + 1)xx2k + (2k + 1)x4k+1 = 0 (69)

where k is a positive integer. Equation (69) isintegrable, although non-Hamiltonian. Substitutingw(x) = x, y(x) = x−2k−1w(x), one finally gets the inte-gral of the equation (69)

(y2 + λy + 1)1/2 exp[− λ√

1 − λ2/4

× tan−1

(y√

1 − λ2/4

)]x2k+1 = constant (70)

For initial conditions x(0) = 0, x(0) = v0, one gets inthe vicinity of the initial point: y → +∞, w(x) → v0.After one ‘impact’, y → −∞, w(x) → −v1 as x → 0.Thus, from equation (70) one obtains

κ =∣∣∣∣v1

v0

∣∣∣∣ = exp(

− πλ

2√

1 − λ2/4

)(71)

Velocity ratio at x = 0, expressed by equation (71),depends neither on the initial velocity nor on k.Consequently, it represents the genuine restitutioncoefficient in the limit k → ∞. Therefore, equation(69) provides a suitable model for the descriptionof inelastic two-sided impact by means of smoothfunctions.

The standard model for the description of the one-sided impact may be written as

x − α − 12xα

= 0 (72)

α > 0 may not be an integer, because only the motionfor x > 0 is encountered. The velocity of the particleas x → ∞ should be considered as velocity before andafter the impact. Equation (72) may also be modifiedin order to describe the inelastic one-sided impacts.The consideration is similar to the one presented inthe previous section, and it is not necessary to repeatit here. The modified model may be thus written as

x − λxα − 1

2x(α+1)/2− α − 1

2xα= 0 (73)

The restitution coefficient can also be expressed byequation (71).

3.3 On the smooth models of vibro-impact systems

In certain cases, it is convenient to take the oppositeaction, namely to study the vibro-impact system usinga smooth model.



Such a case is considered in the context of theso-called ‘pumping problem’ dealing with passive irre-versible energy transfer from a linear system to astrongly non-linear defending element (or energeticsink). It is shown that a smooth approximation ofvibro-impact interaction allows one to consider themain types of energetic sinks in a unique manner,and to draw a conclusion with respect to their relativeefficiency [17].

First, however, some results presented earlier arecompared, and related to the description of thebreathers in the linear oscillatory chain with rigidbarriers, with those obtained using smooth approxi-mation of vibro-impact interaction by the exponentialtype function [32].

The phenomenon of energy pumping in the dampedstrongly non-linear autonomous system [33–37],clearly manifesting some general regularities of transi-tional dynamical processes in the vicinities of internalresonances [34, 35], raises a series of problems of greatimportance. One of them is the possibility of opti-mization of energetic sink parameters. The efficientanalytical description of the pumping process in astrongly non-homogeneous, two-degree-of-freedomsystem, proposed in reference [17], turned out to bemost appropriate for the statement and solution ofthe optimization problem as applied to the ‘cubic’-type sink problem studied in previous papers [33, 38].Meanwhile, along with this type of sink, other stronglynon-linear sinks were discussed in the framework ofthe general energy pumping problem. However, iso-lated numerical estimations and the absence of ageneral criterion of relative sink efficiency make theircomparison difficult. Such a general criterion followsfrom the viewpoint proposed in reference [38] whenconsidering the energy pumping process as a dampedbeating. The development of this viewpoint enables‘design’ of the effective phase plane of a dynamicsystem, which is the main subject of the article.

Based on this, the sinks that have several equilib-rium states are considered and compared with thoseclose to vibro-impact systems. Some general regular-ities determining the efficiency of energy sinks arediscussed in detail.

The following system of coupled oscillators is con-sidered

Md2x1

dt 2+ μ1

dx1

dt+ η

(dx1

dt− dx2

dt

)+ k1x1

+ k3 (x1 − x2)2n−1 ± D (x1 − x2) = 0

md2x2

dt 2− η

(dx1

dt− dx2

dt

)− k3 (x1 − x2)

2n−1

± D (x2 − x1) = 0 n � 2

(74)

Non-linear coupling with multiple states of equilib-rium corresponds to sign ‘−’. As the linear primarystructure is excited by an impulse, free oscillations ofstructures with initial conditions are considered

x1(t = 0) = x2(t = 0) = dx2

dt(t = 0) = 0,

dx1

dt(t = 0) = CI

System (74) can be analysed using the perturbationtheory. When the following change of variables U1 =x1, U2 = x2 − x1 is considered, system (74) looks like

(M + m)d2U1

dt 2+ m

d2U2

dt 2+ μ1

dU1

dt+ k1U1 = 0

md2U2

dt 2+ m

d2U1

dt 2+ η

dU2

dt+ k3U 2n−1

2 + DU2 = 0

n � 2

(75)

To clarify the equations, dimensionless coefficientsand displacements are used, and the previous equa-tions are rewritten as

(1 + ε)d2U1

dτ 2+ ε

d2U2

dτ 2+ εμ1

dU1

dτ+ U1 = 0

εd2U2

dτ 2+ ε

d2U1

dτ 2+ εη

dU2

dτ+ cU 2n−1

2 + εαU2 = 0

n � 2

(76)

where ω = √k1/M , U1 = (ω/CI)U1, U2 = (ω/CI )U2,

ε = m/M , τ = ωt , εμ1 = √(1/k1M )μ1, εη = √

(1/k1M )η,εα = √

(1/k1M )D and c = C 2n−2I k3/ω2n−2k1.

ε is a small parameter representing a mass ratio thathas to be very small.

The infinitesimal order of the non-linear term inthe second of equation (76) is less than the infinites-imal order of linear terms in the same equation. Thefollowing change of variables: u1 = ε−1/(2n−2)U1, u2 =ε−1/(2n−2)U2 is introduced. Then, equation (75) lookslike

(1 + ε)d2u1

dτ 2+ (1 + ε) u1 + ε

(d2u2

dτ 2+ μ1

du1

dτ− u1

)

= 0

d2u2

dτ 2+ u2 +

(−u2 + d2u1

dτ 2+ η

du2

dτ+ cu2n−1

2 ± αu2

)

= 0

(77)

Further assume that the oscillations occur near theresonance on frequency ω. It should then be supposed



that the sum of the terms in square brackets in thesecond of equation (77) is a small quantity of the orderof ε. To accomplish this, a factor δ = 1/ε is nominallytaken as being equal to 1 during the further asymptoticanalysis (actually the sum of the terms in square brack-ets is assumed to be small). The true value of factor δ innumerical calculations should be taken into account(such a procedure, as one can see, is fully justified bythe detailed numerical analysis). Then, equation (77)looks as follows

(1 + ε)d2u1

dτ 2+ (1 + ε) u1 + ε

(d2u2

dτ 2+ μ1

du1

dτ− u1

)

= 0

d2u2

dτ 2+ u2 + εδ

(−u2 + d2u1

dτ 2+ η

du2

dτ+ cu2n−1

2 ± αu2

)

= 0

(78)

Introducing the change of variables [39]

ϕ1 = e−iτ

(du1

dτ+ iu1

)ϕ∗

1 = eiτ

(du1

dτ− i u1

)

ϕ2 = e−iτ

(du2

dτ+ iu2

)ϕ∗

2 = eiτ

(du2

dτ− i u2

)(79)

and performing a multiple scale analysis

τ0 = τ , τ1 = ετ , τ2 = ε2τ , . . . (80)

ϕ1 = ϕ10 + εϕ11 + ε2ϕ12 + · · · (81)

ϕ2 = ϕ20 + εϕ21 + ε2ϕ22 + · · · (82)

leads to the equations

∂ϕ1

∂τ1+ i

2(ϕ1 + ϕ2) + μ1

2ϕ1 = 0 (83)

and

∂ϕ2

∂τ1+ δ

[i2(ϕ1 + ϕ2) + iη

2ϕ2 |ϕ2|2n−1 ± iα

2ϕϕ2

]= 0

(84)

Multiplying equations (83) and (84) by ϕ∗1 and

ϕ∗2 , respectively, and combining these equations and

complex conjugates yields

∂ |ϕ2|2

∂τ1+ δ

∂ |ϕ1|2

∂τ1+ ηδ |ϕ2|2 + δμ1 |ϕ1|2 = 0 (85)

If there is no damping in the system (74), i.e. η =μ1 = 0, then equation (85) is the conservation law

of quantity H = |ϕ2|2 + δ|ϕ1|2 relative to time τ1. Onecan consider relation (85) as an ordinary differentialequation with respect to function |ϕ2|2, the termδ(∂ |ϕ1|2/∂τ1) + δμ1 |ϕ1|2 being the right-hand mem-ber. Applying the direct Laplace transformation toequation (85), one can obtain its solution in the form

�(s) = G(s) + |ϕ2|2 (0)

s + δη

where �(s) is a Laplace representation of function|ϕ2|2(τ1), and G(s) is a Laplace representation of func-tion −δ(∂ |ϕ1|2/∂τ1) − δμ1 |ϕ1|2. After the application ofthe inverse Laplace transformation to this equation,the following representation for function H (τ1) can befound

H (τ1) = exp(−δητ1)

[H (0) + δ(δη − μ1)

×∫ τ1

0exp(δηz) |ϕ1|2 (z)dz

](86)

To find a solution, the integral in the right-handmember of equation (86) is expanded in the form ofa Taylor series in the vicinity of point τ1 = 0. It allowsthe calculation of function H (τ1), avoiding the solu-tion of equations (84) and (85) or initial equation (74).Then, equation (86) looks like

H (τ1) = exp(−δητ1)

{H (0) + δ(δη − μ1)

[τ1 |ϕ1|2 (0)

+ τ 21

2(δη |ϕ1|2 (0) + ∂

∂τ1(|ϕ1|2)(0)) + · · ·

]}(87)

The quantities H (0) and |ϕ1|2(0) are known fromthe initial conditions. The derivative ∂ |ϕ1| (0)/∂τ1 andhigher order derivatives of function |ϕ1| at the samepoint τ1 = 0 can be found from the initial conditionsand equations of motion (74).

When energy pumping occurs, the analyticalapproximation (87) is good, as shown in Fig. 12, wherethe analytical solution H of equation (87) taking intoaccount the Taylor series up to the terms of the fifthorder on τ1 and the numerical integration of system(74) have been compared (n = 2, ω = 1, μ1 = 0, ε =0.1, η = 0.2, c = 0.8, α = 0.2, dx1/dt(t = 0) = 0.3, andsign ‘−’ are considered in equation (74)).

In this case, energy pumping occurs as shown inFig. 13 where the numerical solutions of the system(74) have been plotted with and without coupling.

Analytic approximation (87) also fairly describestime dependence of variables ϕ1 and ϕ2. One canobserve it from direct comparison with numeric sim-ulation of initial system (74) – see Fig. 14.

Initial conditions are: dx1/dt(t = 0) = 0.5, dx2/dt(t = 0) = x1(t = 0) = x2(t = 0) = 0; parameters of



Fig. 12 Time dependence of function H (t). Solid linedepicts that solution (87) taking into account theTaylor series up to the terms of the fifth order onτ1, inclusive. Dashed line depicts the numericalsolution of system (74)

Fig. 13 Responses with the numerical integration ofequation (74) with and without coupling

the system: n = 2, ω = 1, μ1 = 0, ε = 0.1, η = 0.2, c =0.8, α = 0.2, and sign ‘−’ remains in equation (74).

If sign ‘+’ is considered in equation (74), thenthe analytical expression (87) is also good as shownin Fig. 15 where dx1/dt(t = 0) = 0.3, dx2/dt(t = 0) =x1(t = 0) = x2(t = 0) = 0; parameters of the system:n = 2, ω = 1, μ1 = 0, ε = 0.1, η = 0.5, c = 0.8, and α =0.2.

Therefore, it is now possible to try to design the opti-mal energy sink owing to the calculation of H . Indeed,one can see that if sign ‘+’ is considered in equation(74) then energy pumping appears to be more efficientbecause the decrease of energy H is more abrupt. Theenergy decreases faster with sign ‘+’ in equation (74)than with sign ‘−’ (if all other parameters are fixed)

as shown in Fig. 16 where dx1/dt(t = 0) = 0.3, dx2/dt(t = 0) = x1(t = 0) = x2(t = 0) = 0, and n = 2, ω =1, μ1 = 0, ε = 0.1, η = 0.5, c = 0.8, and α = 0.2.

Thus, energy pumping is more efficient when thesign ‘+’ is considered in equation (74), as shownin Fig. 17, with a numerical integration of system(74) with the same values of parameters as shownpreviously. In this figure, it clearly appears that thevibrations are almost completely attenuated at t = 20 swhen the sign ‘+’ is considered in equation (74).

Moreover, the influence of the degree n of thenon-linearity on the efficiency of the sink can alsobe considered. Indeed, for a given set of parame-ters, there exists an optimal value of n for whichthe efficiency of energy pumping is optimal. For thisstudy, the case of the sign ‘+’ in equation (74) isconsidered because its efficiency seems better. Forexample, with dx1/dt(t = 0) = 0.4, dx2/dt(t = 0) =x1(t = 0) = x2(t = 0) = 0, and n = 2/3/4, ω = 1, μ1 =0, ε = 0.1, η = 0.2, c = 0.8, α = 0.2 and the sign ‘+’ inequation (74), then the optimal value of n is 3 (thedegree of the non-linearity is 5), as shown in Fig. 18.In this figure, it is also seen that for n = 4, the analyt-ical approximation is less good after t = 30 s becauseenergy pumping does not occur, and there is no longerany resonance.

Important information about sink efficiency can beextracted from the analysis of the corresponding con-servative system, as seen in the following. If η = μ1 =0, the equations (83) and (84) look as

∂ϕ1

∂τ1+ i

2(ϕ1 + ϕ2) = 0 (88)

and

∂ϕ2

∂τ1+ δ

[i2(ϕ1 + ϕ2) − ic

2C n−1

2n−1ϕ2 |ϕ2|2n−1 ± iα2ϕ

ϕ2

]= 0 (89)

By introducing the change of variables

ϕ1 = f1, ϕ2 = √δf2 (90)

they can be written as follows

∂f1

∂τ1+ i

2( f1 + √

δf2) = 0 (91)

and

∂f2

∂τ1+ √

δ

[i2( f1 + √

δf2) − ic22n−1

δ2n−1

2 C n−12n−1f2

∣∣f2

∣∣2n−1

± iα2ϕ

√δf2

]= 0 (92)



Fig. 14 Function H (t), Imag ϕ1(t), Re ϕ1(t), and Re ϕ2(t) are compared with the results ofintegrating initial system (74)

Fig. 15 Function H (t). Solid line depicts that solution(87) taking into account the Taylor series upto the terms of the fifth order on τ1, inclusive.Dashed line depicts the numerical solution ofsystem (74)

The system is now completely integrable with twofirst integrals of motion

H = i2

(∣∣ f1

∣∣2 + √δ∣∣f2

∣∣2)− i2( f1f ∗

2 + f2 f ∗1 )

+ icδC n−1

2n−1

22n−1

∣∣f2

∣∣2n ± iα2

δ∣∣f2

∣∣2 (93)

Fig. 16 Comparison of function H (t) with the consider-ation of sign + or − in equation (74)

and

N = | f1|2 + | f2|2 (94)

The following change of variables is introduced

f1 = √N cos θeiδ1 , f2 = √

N sin θeiδ2 (95)

and

� = δ1 − δ2 (96)



Fig. 17 Comparison of responses with the numericalintegration of equation (74) with the consider-ation of sign + or − in equation (74)

Fig. 18 Comparison of H (t) for different values of n

Finally one can obtain

∂θ

∂τ1−

√δ

2sin � = 0 (97)

and

∂�

∂τ1− δ − 1

2± αδ

2

√δ cos � cot 2θ + cδN n−1

× C n−12n−1

22n−1sin2n−2 θ = 0 (98)

Then two cases appear.

Fig. 19 Non-linear beatings: time series for averagedamplitudes

1. If −(δ − 1)/2 ± (αδ)/2 > 0 then for all values of Nthere exist only two NNMs where the solutions ofequations (97) and (98) have been plotted in the(θ − �) plane for N = 0.4.

2. If −(δ − 1)/2 ± (αδ)/2 < 0 then under a certainvalue of N there exist two NNMs where the solu-tions of equations (97) and (98) have been plotted inthe (θ − �) plane for N = 0.15; and above a certainvalue of N , there exist four NNMs where the solu-tions of equations (23) and (24) have been plottedin the (θ − �) plane for N = 0.38.

When all four NNMs appear, then the energy pumpingphenomenon occurs with the beating phenomenon.

This non-linear beating can be also seen in the twodisplacements x1 and x2, as shown in Fig. 19, with thesame values as shown previously.

4 UNEXPECTED APPLICATIONS OFVIBRO-IMPACT MODELS

4.1 Vibro-impact model in the problem of heattransfer

The authors would like to show that the vibro-impactmodel gives a unique possibility of obtaining analyti-cal results in one of the ‘hot’ problems of modern solidstate physics – substantiation of the phenomenologi-cal theory of heat conductivity.

Heat conductivity in one-dimensional lattices is aclassic problem related to the microscopic foundationof Fourier’s law. The problem started from the famouswork of Fermi, Pasta, and Ulam (FPU) [40], wherean abnormal process of heat transfer was detectedfor the first time. Non-integrability of a system is anecessary condition for normal heat conductivity. Asdemonstrated recently, [41] for the FPU lattice, dis-ordered harmonic chain, diatomic one-dimensional



gas of colliding particles, and the diatomic Toda lat-tice, non-integrability is not sufficient to get normalheat conductivity. It leads to linear distribution oftemperature along the chain for small gradients, butthe value of heat flux is proportional to 1/N α, whereN is the number of particles in the chain and thenumber exponent 0 < α < 1. Thus, the coefficient ofheat conductivity diverges in the thermodynamic limitN → ∞. Analytical estimations have demonstratedthat any chain possessing an acoustic phonon branchshould have infinite heat conductivity in the limit oflow temperatures.

Probably the most interesting question related toheat conductivity of one-dimensional models, (whichactually inspired the first investigation of Fermi et al.[40]) is whether small perturbations of an integrablemodel lead to convergent heat conduction coeffi-cients. One supposes that for the one-dimensionalchains with conserved momentum, the answer is neg-ative [42]. Still, normal heat conduction has beenobserved in some special systems with conservedmomentum [43, 44], but it can be clearly demon-strated only well apart from integrable limits. Thismeans that mere non-integrability is insufficient toensure normal heat conduction if additional an inte-gral is present.

It seems that the numeric simulation of heat con-duction in the vicinity of integrable limits is ratherdifficult not only due to weak computers or ineffectiveprocedures. In systems with conserved momentum,divergent heat conduction is fixed by a power-likedecrease of heat flux autocorrelation function with apower less than unity. Still, for the systems with on-sitepotential, exponential decrease is more typical [45].For any fixed value of the exponent, the heat conduc-tion converges; if the exponent tends to zero with thevalue of the perturbation of the integrable case, thenfor any finite value of the perturbation, the character-istic correlation time and length will be finite but maybecome very large. Consequently, they will exceed anyavailable computation time or size of the system withno conclusion on the convergence of heat conductionstill being possible.

This difficulty can be overcome by constructing amodel [46], which will be, at least to some extent,tractable analytically and will allow one to predictsome characteristic features of the heat transfer pro-cess and the behaviour of the heat conduction coef-ficient. The numerical simulation may later be usedto verify the assumptions made in the analytic treat-ment. To the best of the authors’ knowledge, no modelsbesides pure harmonic chains have been treated insuch a way to date.

It will be demonstrated here that there exist mod-els that have integrable systems as their natural limitcase, with small perturbations of the integrabilityimmediately leading to convergent heat conduction.

The mechanism of energy scattering in these kindsof systems is universal for any temperature and setsof the model parameters. The simplest example ofsuch models is a one-dimensional set of equal rigidparticles with non-zero diameter (d > 0) subjected toperiodic on-site potential. This system is completelyintegrable only if d = 0. It will be demonstrated thatany d > 0 leads to effective mixing due to unequalexchange of energy between the particles in each col-lision. This mixing leads to the diffusive mechanismof the heat transport and, subsequently, to convergentheat conduction.

The one-dimensional system of hard particles withequal masses subject to periodic on-site potential isconsidered. The Hamiltonian of this system will read

H =∑

n

[12

x2n + V (xn+1 − xn) + U (xn)

](99)

where M is the mass of the particle, xn is the coordinateof the centre of the nth particle, xn is the velocity of theparticle, and U (x) is the periodic on-site potential withperiod a[U (x) ≡ U (x + a)]. Interaction of absolutelyhard particles is described by the following potential

V (r) = ∞ if r � d and V (r) = 0 if r > d (100)

where d is the diameter of the particle. This potentialcorresponds to pure elastic impact with unit restitu-tion coefficient. A sketch of the model considered ispresented in Fig. 20.

Elastic collision of two equal particles with collinearvelocity vectors leads to exchange of their velocities.The presence of an external potential does not changethis fact, as the collision takes zero time, and thusthe effect of the external force on the energy andmomentum conservation is absent.

The one-dimensional chain of equivalent hard par-ticles without external potential is a paradigm of theintegrable non-linear chain, as all interactions arereduced to exchange of velocities. In other words,the individual values of velocities are preserved and

Fig. 20 Sketch of the hard-disc chain exposed to peri-odic on-site potential U (x) (a is the period ofthe potential, U0 its height, d is the diameterof the discs). (a) Piecewise-linear potential (108)and (b) sinusoidal potential (118)(b) are plotted



merely transferred from particle to particle. It is nat-ural therefore to introduce quasiparticles associatedwith these individual values of velocities. They arecharacterized by a pair of parameters (Ek , nk), whereEk = v2

k/2 is an energy of the quasiparticle and nk isa unit vector in a direction of its motion. Every par-ticle in every moment ‘carries’ one quasiparticle. Theelastic collisions among the particles lead to simpleexchange of parameters of the associated quasiparti-cles, therefore the quasiparticles themselves should beconsidered as non-interacting.

The situation changes if the external on-site poten-tial is present. It is easy to introduce similar quasipar-ticles (Ek will be a sum of kinetic and potential energy).The unit vector n of each quasiparticle between sub-sequent interactions may be either constant (motionin one direction) or periodically changing (vibrationof the particle in a potential well). In every collision,the particles exchange their velocity vectors, but donot change their positions. Consequently, two quasi-particles interact in a way described by the followingrelationships

E ′1 = E1 + U

(xc + d

2

)− U

(xc − d

2

)

E ′2 = E2 − U

(xc + d

2

)+ U

(xc − d

2

)n′

1 = n1

n′2 = n2

(101)

The values denoted by the apostrophes correspondto the state after the collision, and xc is a point ofcontact between the particles. It should be mentionedthat in the case of a non-zero diameter, the quasipar-ticles are associated with the centres of the carryingparticles.

If the diameter of the particles is zero, then theadditives to the energies in the first two equations ofsystem (102) compensate each other, and the ener-gies of the quasiparticles are preserved in the col-lision. Therefore, effectively, the interactions amongthe quasiparticles disappear, and the chain of equalparticles with zero size subject to any on-site poten-tial turns out to be a completely integrable system.Thus, contrary to many previous statements it is pos-sible to construct an example of a strongly non-linearone-dimensional chain without momentum conser-vation, which will have clearly divergent heat con-ductivity (even linear temperature profile will not beformed).

The situation differs if the size of the particles is notzero, as the energies of the particles are not preservedin the collisions. In order to consider the effect ofsuch interaction, a simplified semi-phenomenologicalanalytical model is proposed.

After l collisions, the energy of the quasiparticlewill be

E(l) = E0 +l∑

j=1

�Ej , �Ej = U(

xj + d

2

)− U

(xj − d

2

)

(102)

where the jth collision takes place in point xj andE0 is the initial energy of the quasiparticle. Now it issupposed that the coordinates of subsequent contactpoints {. . . , xj−1, xj , xj+1, . . .}, taken by the modulus ofthe period of the on-site potential, are not correlated.Such a proposition is equivalent to fast phase mixingin a system close to integrable one, and is widely usedin various kinetic problems [46]. The consequences ofthis proposition will be verified by direct numericalsimulation.

The average energy of the quasiparticle is equal to〈E0〉 over the ensemble of the quasiparticles, as obvi-ously 〈�Ej〉 = 0. Still, the second momentum will benon-zero

〈(E(l) − E0)2〉 = l

⟨[U(

x + d2

)− U

(x − d

2

)]2⟩

x

(103)

The right-hand side of this expression will dependonly on the exact shape of the potential function

〈(E(l) − E0)2〉 = lF (d)

F (d) = 1a

∫ a

0

[U(

x + d2

)− U

(x − d

2

)]2

dx (104)

The last expression is correct only at the limit of hightemperatures; it neglects the fact that the quasiparticlespends more time near the top of the potential barrierdue to lower velocity.

The quasiparticle with initial energy E0 > U0, whereU0 is the height of the potential barrier, is consid-ered. Therefore, vector n is constant. Equations (103)and (104) describe random walks of the energy of thequasiparticle along the energy scale axis. Therefore,after a certain number of steps (collisions), the energyof the quasiparticle will enter the zone below thepotential barrier E(l) < U0. In this case, the behaviourof the quasiparticle will change, as constant vectorn will become oscillating, as described. After someadditional collisions, the energy will again exceedU0, but the direction of motion of the quasiparticlewill be arbitrary. It means that the only mechanismof energy transfer in the system under considera-tion is associated with diffusion of the quasiparticles,which are trapped by the on-site potential and are



later released in arbitrary directions. Such traps-and-releases resemble the Umklapp processes of phonon–phonon interaction, but occur in a purely classicsystem.

The diffusion of the quasiparticles in the chainis characterized by mean free path, which may beevaluated as

λ ∼ 2a[(U0 − E0)2]

ncF (d)

∼ 2a[2(kBT )2 − 2U0kBT + U 20 ]

ncF (d)(105)

where nc is a number of particles over one periodof the on-site potential (concentration). Coefficient2 appears due to the equivalent probability of posi-tive and negative energy shift in any collision, T is thetemperature of the system, and kB is the Boltzmannconstant.

Average absolute velocity of the quasiparticle maybe estimated as

〈|v|〉 ∼ aa − ncd

√πkBT

2(106)

Here, the first multiplier takes into account the non-zero value of d and absolute rigidity of the particles.The second is due to the standard Maxwell distributionfunction for the one-dimensional case.

Heat capacity of the system over one particle isunity, as the number of degrees of freedom (i.e. thenumber of quasiparticles) coincides with the num-ber of the particles and does not depend on thetemperature and other parameters of the system.Therefore, the coefficient of heat conductivity may beestimated as [46]

κ ∼ λ〈|v|〉 ∼ 2a2

nc(a − ncd)

2(kBT )2 − 2U0kBT + U 20

F (d)

×√

πkBT2

(107)

It is already possible to conclude that according toequation (107), regardless of the concrete shape ofpotential U (x) in the limit d → 0, one can haveF (d) → 0 and therefore κ → ∞, although for everynon-zero value of d, the heat conductivity will befinite. Therefore, unlike known models with conservedmomentum, the small perturbation of the integrablecase d = 0 immediately brings about convergent heatconductivity.

It is convenient, for the following numerical simu-lation, to introduce dimensionless variables. Let themass of each particle be M = 1, on-site potentialperiod a = 2, its height U0 = 1, and Boltzmann con-stant kB = 1 in all above relationships. It is supposedthat the chain contains one particle per each period of

the potential, i.e. that nc = 1, and the particle diameter0 < d < 2.

Periodic piecewise linear on-site potential is consid-ered

U (x) = x if x ∈ [0, 1]U (x) = 2 − x if x ∈ [1, 2]U (x + 2l) = U (x) for x ∈ [0, 2], l = 0, ±1, ±2, . . .

(108)

(the shape of the potential is presented in Fig. 20).Then, it follows from equation (107) that thenon-dimensional heat conduction coefficient isexpressed as

κ = 8(2T 2 − 2T + 1)

(2 − d)F (d)

√πT2

(109)

where function

F (d) = d2 − 23d3

, for 0 < d � 1

F (d) = −43

+ 4d − 3d2 + 23d3

, for 1 � d < 2

(110)

Dynamics of the system of particles with potentialof the nearest-neighbour interaction (2) and piecewiselinear on-site potential (10) may be described exactly.Between the collisions, each particle moves underconstant force with the sign dependent on the posi-tion of the particle. Therefore, the coordinate of eachparticle depends on time t as a piecewise parabolicfunction that can be easily computed analytically. Ifthe particle centres are situated at distance equal tod, then elastic collision occurs. The particles exchangetheir momenta as described, and afterwards the par-ticle motion is again described by piecewise parabolicfunctions until the next collision.

A finite chain of N particles with periodic boundaryconditions is considered. Let at the moment t = 0, oneparticle be at each potential minimum and the Boltz-mann’s distribution of the initial velocity be chosen.Solving the equations of motion, one can find a timet1 of the first collision between some pair of the adja-cent particles, next a time t2 of the second collision, ingeneral between another pair of the adjacent particles,and so on. As a result, a sequence {ti, ni}∞

i=1, where ti isthe time of the ith collision in the system, is obtained,and ni and ni + 1 are the particles participating in thiscollision. First, the energy change of the nith particleis incorporated during the ith collision as

�Eni = 12(v′2

ni− v2

ni) = 1

2(v2

ni+1 − v2ni

)

Next, a time step �t is introduced, which is sig-nificantly less than the simulation time, but satisfies



the inequality �t t0, where t0 = limi→∞(ti/i) is themean time between successive collisions. Then, foreach k = 0, 1, . . . , the local energy flow is defined asa piecewise constant (in time) function

jn(t) = a�t

∑i∈Ikn

�Eni , k�t � t < (k + 1)�t (111)

where the integer sets Ikn’s are defined by

Ikn = [i| k�t � ti < (k + 1)�t , ni = n]The set Ikn takes into account those collisions that

occur between particles n and n + 1 during the timeinterval k�t � t < (k + 1)�t . Equilibration times typ-ically occurred in the system of the order 106. Afterthese times have passed, the time-averaged localthermal flow

Jn =< jn(t) >t≡ limt→∞

1t

∫ t

0(τ )dτ (112)

and the temperature distribution Tn = 〈v2n(t)〉t , where

vn(t) is the velocity of particle n calculated at a timet , are defined. To find these averaged quantities, timesup to 107 have been used.

To find the flow–flow correlation function C(t)numerically, the time average 〈J (τ )J (τ − t)〉τ /NT 2 wascalculated, with J (t) = ∑

n jn(t) being the total heatflow through the gas/chain system consisting of N =500 particles and temperature T = ∑

n Tn/N averagedover 104 realizations of initial thermalization.

Numerical simulation of the dynamics demon-strates exponential decrease of the autocorrelationC(t) ∼ exp(−αt) for all values of the diameter 0 <

d < 2 and temperature T > 0 where the simulationtime is plausible from the technical viewpoint. For lowtemperatures, however, the exponential decrease isaccompanied by oscillations with periods correspond-ing to the frequency of the vibrations near the potentialminima (Fig. 20). The reason is that if the tempera-tures are low, the concentration of transient particlesdecreases exponentially and the majority of the parti-cles vibrates near the potential minima. It means thatthe one-dimensional gas on the on-site potential hasfinite heat conductivity (Fig. 21). The coefficient of theexponential decrease of the autocorrelation function

α = − limt→∞

ln C(t)t

(113)

and coefficient of the heat conduction

κ =∫∞

0C(t)dt (114)

are computed numerically.Dependence of α and κ on particle diameter d is

presented in Fig. 22. Maximum of α and minimum of

Fig. 21 Correlation functions of the system of particleswith d = 0.5 under temperatures T = 0.24, 0.45,and 0.75 (curves 1, 2, and 3)

Fig. 22 Dependence of the coefficient of theexponential decrease of the autocorrela-tion function α (a) and the coefficient ofthe heat conduction κ (b) on the parti-cle diameter d of one-dimensional gas atT = 1. Curves 1 and 2 correspond to piecewiselinear on-site potential (108), curve 3 representtheoretical predictions according to formula(109)



Fig. 23 Temperature dependence of exponent coeffi-cient α (a) and heat conduction coefficient κ (b)for particle diameter d = 0.5

κ is attained at d = 1.4. As the temperature increases, αdecreases (Fig. 23(a)) and heat conduction κ increases(Fig. 23(a)).

The preceding theoretical analysis of heat con-ductivity allows only approximate (although ratherreliable, see Figs 23(b) and 24) prediction of the numer-ical value of the heat conduction coefficient κ . Still,the other question of interest is the asymptotic depen-dence of the heat conduction on the parameters of themodel. Formulae (109) and (110) lead to the followingestimations

κ ∼ T 5/2, for T → ∞ (115)

κ ∼ d−2, for d → +0 (116)

κ ∼ (2 − d)−3, for d → 2 − 0 (117)

These estimations should be compared with numeri-cal results.

In order to check estimation (115), the depen-dence of the logarithm of the heat conduction ln κ

on the logarithm of the temperature ln T is consid-ered. From Fig. 24, it is clear that in accordance withequation (115) ln κ increases as 2.5 ln T as T → ∞.Figure 25(a) demonstrates that as d → +0, the log-arithm ln κ increases as −2 ln d, in accordance withequation (116). Figure 25(b) demonstrates that as

Fig. 24 Dependence of heat conduction coefficient onthe temperature. The markers correspond tonumerical results (ln κ versus ln T ), the straightline is ln κ = 2.5 ln T + 3.45, corresponding to(115). Particle diameter d = 0.5

d → 2 − 0, the logarithm ln κ increases as −3 ln(2 − d),in accordance with equation (117). Therefore, it is pos-sible to conclude that analytical estimations (115 to117) fairly correspond to the numerical simulationsdata.

These analytical estimations imply that the type ofdependence of the characteristic exponent α and heatconductivity κ on diameter d and temperature T isnot reliant on the concrete shape of on-site poten-tial U (x) – actually, only its finiteness and periodicitydo matter. Piecewise linear periodic potential (108)was chosen because it allowed essential simplifica-tion of the numerical procedure. For comparison, thesmooth sinusoidal periodic potential also have beenconsidered

U (x) = [1 − cos(πu)]2

(118)

with period 2 and amplitude U0 = 1, similarly topotential (108).

Potential (118) does not allow exact integration, andrequires standard numerical procedures. Therefore, itis convenient to change rigid wall potential also (100)by smooth Lennard–Jones potential

V (ε; r) = ε

(1

r − d− 1

2 − d

)2

(119)

Parameter ε > 0 characterizes the rigidity of thepotential, the hard-particle potential being the



Fig. 25 Dependence of the heat conduction coefficienton the particle diameter (logarithmic coordi-nates, ln κ versus ln d (a) and versus ln(2 − d)

(b), curves 1 and 3). Lines ln κ = −2 ln d (curve 2)and ln κ = −3 ln(2 − d) (curve 4) correspondto relationships (116) and (117). TemperatureT = 1

limit case

V (r) = limε→+0

V (ε; r)

Methods of computing of the autocorrelation func-tion C(t) and the heat conduction coefficient κ inone-dimensional chains with analytic potentials ofinteraction are described in reference [46]. It shouldbe mentioned that in order to reach close to the limitof the hard particles, small values of ε (at the temper-ature T = 1 value ε = 0.01 was used) should be used.It implies a rather small value of the integration step(here standard Runge–Kutta procedure of the fourthorder with constant integration step �t = 0.0001 isused). Therefore, for the case of hard (or nearly hard)particles, the simulation with smooth on-site potential(118) is far more time-consuming than the simulationwith piecewise linear potential (108).

In the case of hard particles with smooth on-sitepotential, the autocorrelation function C(t) decreasesexponentially as t → ∞ for all range 0 < d < 2, T > 0,i.e. the heat conduction converges. Figure 26 demon-strates that the type of dependence of α and κ on

Fig. 26 Dependence of the coefficient of the exponen-tial decrease of the autocorrelation functionα (a) and the coefficient of the heat con-duction κ (b) on the particle diameter d ofone-dimensional gas at T = 1. Curves 1 and 2correspond to smooth on-site potential (118)and curve 3 represents theoretical predictionsaccording to formula (115)

parameters d and T is similar for piecewise lin-ear potential (108) and sinusoidal potential (118)(although numerical values α and κ vary slightly).For this potential function, F (d) = (1/2) sin2(πd/2).This confirms that the type of heat conduction doesnot depend on concrete choice of on-site potentialfunction.

Analytical treatment predicts that for zero diam-eter of the particles, the system will be completelyintegrable regardless of the exact shape of the on-sitepotential. Therefore, the heat conductivity will be infi-nite. For any non-zero size of the particles, the heattransfer is governed by diffusion of quasiparticles, giv-ing rise to finite heat conductivity. The value of theheat conduction coefficient computed by the analyti-cal treatment is in line with numerical simulation data.This coincidence is very profound in the context of theasymptotic scaling behaviour of the heat conductioncoefficient in cases of small and large particle sizes, aswell as for the case of high temperatures. The charac-teristic behaviour of the heat conduction coefficientdoes not depend on the exact shape of the on-sitepotential function.

These results mean that there exists a new class ofuniversality of one-dimensional chain models with



respect to their heat conductivity. The limit caseof zero-size particles is integrable, but the slightestperturbation of this integrable case by introducingthe non-zero size leads to a drastic change of thebehaviour – it becomes diffusive and the heat con-duction coefficient converges. It should be stressedthat this class of universality, unlike the systems withconserved momentum, cannot be revealed by solenumerical simulation. The reason is that the correla-tion length (as well as the heat conduction coefficient)diverges as the system approaches the integrable limit;therefore, any finite capacity of the numerical instal-lation will be exceeded. That is why the analyticalapproach is also necessary.

4.2 Problem of beats

This section is devoted to vibro-impact models arisingunexpectedly in the problems having nothing in com-mon with impacts. As just another example, classicalproblem of beats have been chosen (Fig. 27).

It is well-known that Newton, when considering thediscovery of differential equations as the most signif-icant achievement in the field of natural philosophy,had a hope that at his disposal there is a universalprocedure of their solution, namely, power expan-sion by independent variables. This hope was notjustified for several reasons. One of the most impor-tant of them was manifestations of periodicity thatimpelled the use of alternative ideas, such as averag-ing. Certainly, power expansion by small parametershas become a powerful tool of analysis in mathemat-ical physics, but it is a rather different point. As forexpansions by independent variables of the other type,the phenomenon of periodicity led to the appearanceand extraordinary development of the Fourier series,including extension on non-periodic processes due tothe introduction of the Fourier integral. Discovery ofinverse scattering transform procedure in non-lineardynamics, as one of consequences of the FPU prob-lem statement [40], was a genuine triumph of Fourierideology, being its far going and unexpected exten-sion. On analysing the reasons for such success, it isseen that besides very high symmetry of fully inte-grable dynamical systems, all they describe are thesolutions generated by perturbation of the equilib-rium state (non-topological solitons with amplitude

approaching zero with increase of their width) or tran-sitions among different equilibrium states (topologicalsolitons). Meanwhile, rather different types of theproblems arise in the systems with internal resonance.To be more concrete, the simplest non-linear problemof this kind is considered: dynamics of two weakly cou-pled non-linear oscillators with cubic restoring forcesthat is described by the set of two non-linear equations(in dimensionless form)

d2U1

dτ 21

+ U1 + 2βε(U1 − U2) + 8αεU 31 = 0

d2U2

dτ 21

+ U2 + 2βε(U2 − U1) + 8αεU 32 = 0

(120)

where

Uj = uj0

l2, τ1 =

√c1

m t, 2βε = c12

c1, and

8αε = c3l20

c1.

Introducing the complex variables

ϕ1 = e−iτ1

(dU1

dτ1+ iU1

)ϕ∗

1 = eiτ1

(dU1

dτ1− i U1

)

ϕ2 = e−iτ1

(dU2

dτ1+ iU2

)ϕ∗

2 = eiτ1

(dU2

dτ1− i U2

)(121)

and slow time τ2 = ετ1 (alongside with fast time τ1 = τ ),one can use the two-scale expansions

ϕj(τ1, τ2) =∑

n

εn−1ϕj,n(τ1, τ2), j = 1, 2 (122)

After simple calculations, one obtains the followingset of equations of the principal asymptotic approxi-mation

df1

dτ2+ iβf2 − 3iα

∣∣f1

∣∣2 f1 = 0

df2

dτ2+ iβf1 − 3iα

∣∣f2

∣∣2 f2 = 0

ϕj = eiβτ fj , j = 1, 2

(123)

Fig. 27 Dynamical system considered for the beating problem



which describes, besides the presented mechanicalsystem, some others, e.g. optic couplers. It was con-sidered a few times [47–49].

Set of equations (123) is fully integrable and has twointegrals

H = β( f2f ∗1 + f1f ∗

2 ) −3α(∣∣f1

∣∣4 + ∣∣f2

∣∣4)2

(124)

N = ∣∣f1

∣∣2 + ∣∣f2

∣∣2 (125)

Therefore, the most adequate consideration isachieved in coordinates θ , �, where

f1 = √N cos θeiδ1 , f2 = √

N sin θ eiδ2 , � = δ1 − δ2

(126)

dθ

dτ2= β sin �,

d�

dτ2sin 2θ = 2β cos 2θ cos �

+ 3αN cos 2θ sin 2θ

(127)

The integral of equation (127) has a form H0 =[β cos � + (3αN /4) sin 2θ ] sin 2θ . The system (127) isstrongly non-linear even in the case of an initiallylinear problem, which is one of the key points. Itsstationary states correspond to NNMs, the numberof which increases from 2 (if N < 2β/3α) to 4 (N >

2β/3α) (Figs 28 and 29).The regimes of this kind are not specific for the

systems with internal resonances, and they are thesimplest motions that can be presented in the con-figuration space of initial variables as straight lines(the existence of ‘straight’ normal modes in stronglynon-linear systems was first shown by Rosenberg [50]).

Efficient techniques for their construction, even inthe case when they are not straight, may be developedon the basis of the Jacobi formulation of variationalprinciple and the equations of motion in configurationspace.

Fig. 28 θ−� plane with N < 2β/3α

Fig. 29 θ−� plane with N > 2β/3α

Such a technique allows finding the NNMs usingpower expansions by an independent variable (that isone of the unknowns in this case) in the framework ofa boundary problem [4, 5]. It turns out to be possibleonly due to ‘straightness’ of generating solutions.

For the discussed problem, such expansions areunnecessary because NNMs are presented here bystationary points, and this advantage is widely usedin the papers devoted to NNMs and their bifurca-tions, looking for close regimes in damped and forcedweakly coupled systems [51, 52] (Figs 30 and 31).Theseregimes close to stationary points at the θ−� planesin the conservative systems under consideration arebeatings with small energy transfer between two oscil-lators. The equations of motions can be linearized inthe vicinities of these points and the solutions presentsmall-amplitude oscillations of both θ and � aroundtheir values, corresponding to NNMs. Thus, the mostcomplex situation arises when the phase curves arefar from stationary points. They correspond to beat-ings with maximally intensive energy transfer betweenoscillators.

Fig. 30 Four straight NNMs



Fig. 31 Non-straight NNM

Certainly, due to full integrability of the ‘θ−�’ sys-tem, the exact solution can be found. It can beexpressed by the Jacobi elliptic function and for thelimiting phase curve has a view

2θ = am(2τ2, k), � = ar cn (2τ2, k), (128)

where am and cn are Jacobi elliptic functions withparameter k, 0 � k � 1.

It can be seen that in the limit k → 0, the first ofthe formulas (128) gives an unbounded solution thatis a straight line. This behaviour clearly contradictsthe existence of actual boundaries for this variable:0 � θ(τ2) � π/2.

Infinitely small non-linearity of the initial sys-tem (N > 0) leads to the behaviour shown in Figs 32to 34. Two ‘non-physical’ phenomena are thusobserved:

(a) abrupt qualitative change of behaviour in the welldefined limiting case;

(b) instantaneous jumps of θ-variable characteriz-ing a continuous energy transfer between theoscillators.

Fig. 32 Function 2 θ = am (2 τ , k) for k = 0.1



One can clarify the situation by seeking the solu-tion for the initially linearized system from limitingthe ‘θ−�’ system directly. As this takes place, one canfirst consider the motion with phase shift � = π/2and θ = β τ2 (from the first of equation (127), thesecond one is also satisfied). Then, in at the pointβ τ2 = π/2 instantaneous change of the phase shiftsign is observed, so that � = −π/2 and θ = π/2 − β τ2

(exactly as in a vibro-impact process with velocity �

and displacement θ). It is more convenient to intro-duce two non-smooth functions τ (τ1) and e(τ1). Theyare related to basic functions τ and e introduced earlier(Fig. 1) by the following relations

τ (t) = 12

+ τ

(t − 1

2

)

e(t) = 12

+ e(t)

However, as shown in the following, the most nat-ural area for the application of similar presentationsis not construction of smooth regimes but, specifi-cally, of beatings (using the variables θ and �). In thiscase, ‘Newton’s approach’ can be applied using powerexpansions by independent variables without formu-lation of a boundary problem and gradual ‘restoration’of the oscillations period.

The solution can be presented as

θ = X1(τ ) + Y1(τ )e(τ1), � = X2(τ ) + Y2(τ )e(τ1)

(129)



where the smooth functions Xi(τ ) and Yi(τ ) satisfy theequations

∂

∂τ

{X1

Y1

}= 1

2aβ[sin(X2 + Y2) ± sin(X2 − Y2)] (130)

∂

∂τ

{X2

Y2

}= aβ[cot g2(X1 + Y1) cos(X2 + Y2)

± cot g 2(X1 − Y1) cos(X2 − Y2)]+ 3a

2αN [cos2(X1 + Y1) ± cos 2(X1 − Y1)]

(131)

Then, the solution can be sought in the form ofpower expansions by the independent variable τ

Xi =∞∑

l=0

Xi,l(k)τl , Yi =

∞∑l=0

Yi,l(k)τl (132)

where k = 3αN /4β, and the generated solution is thelinear beating

X1,0 = bτ , Y1,0 = 0, X2,0 = 0, Y2,0 = π

2(133)

satisfying exactly the ‘θ−�’ equations for the case ofthe most ‘strong’ beating. It can be demonstrated thatpresentation (129) taking into account equation (133)actually restores the exact solution of the non-linearproblem for beating with the most intensive energytransfer among oscillators. As this takes place, theexpansions (132) restore the exact local representationof the corresponding elliptic function (near τ = 0),but the expressions (129) allow the prediction of exactglobal behaviour of the beating. It is important to notethat even for large enough k, the solution is very closeto linear beating (the only change is a hardly seen cur-vature of the lines, being straight for linear-beating,and small change of the period).

However, this exact solution does not coincide withthe formal analytical one mentioned earlier and pre-sented in Fig. 32. Actually, if reminding the behaviourof functions τ (τ1) and e(τ1), it becomes clear that theunbounded behaviour of θ in the limit k → 0 disap-pears now, and abruptly jumps at half-period. It is alsoclear why the formal exact solution distorts realisticbehaviour of the sought functions for the situationwith limiting ruptures. When k = 0, the periodic-ity condition formally disappears, and unboundedbehaviour is seen (a similar situation arises if non-smooth transformation of time is introduced in vibro-impact systems and the motion by inertia is arrived atformally, without any reflections caused by the pres-ence of the walls). As for abrupt jumps for k not equalto zero, they are caused by the necessity to satisfy theperiodicity condition and, simultaneously, closenessto limiting case k = 0. The only possibility of satisfyingboth conditions is the jumps observed.

Introducing the non-smooth basic functions leadsto the realistic interpretation of the exact solution – atthe point of the jump the discontinuity of the deriva-tive is obtained (because differentiation of τ by τ2/agives the multiplier e, i.e. the only change of sign ofderivative that corresponds to real situations insteadof non-realistic θ-jumps).

Contrary to previous applications of non-smoothtransformations, in the case being considered it isunnecessary to formulate the boundary problems forcompensation of singularities arising due to the sub-stitution of non-smooth functions into equations ofmotion for the derivation of the preceding equationsin terms of smooth functions X and Y . Singularitiesarising due to the substitution of non-smooth func-tions into the second of the equations of motion areexactly compensated for the limiting phase trajectory(LPT) because sin 2θ = 0 at singular points.

The most important feature of such representationsis the unification of local and global approaches. Thelocal approach uses power expansions with unusuallygood results even in zero approximation. The expan-sion of period of oscillations by parameter ‘k’ can befound separately, after construction of the analyticalform of solution (with a still unknown period). The keyto solving this problem is to have preliminary knowl-edge of amplitude values of θ and � functions. It isonce more an important distinction from previousapplications of non-smooth transformations, in whichthis problem was solved step by step. Here, it is worthdiscussing a principal question connected with thebehaviour of arising power series. It was mentionedabove that the zeroth approximation turns out to bevery efficient even for large values of the non-linearityparameter (going far from the bifurcation point, corre-sponding to a qualitative change in the phase plane).However, the convergence of this series is slow enoughand almost does not depend on the magnitude of thenon-linearity parameter that is the modulus of theelliptic integral of the first kind. The situation resem-bles the behaviour of an asymptotic series where thefirst terms can give a reliable representation but thenext ones may make the approximation worse (Largeenough magnitudes of independent variables that arenecessary to equate the values of dependent variablesto their a priori known amplitude values are beingreferred to here). The solution may be found similarlyin the case of an asymptotic series. This refers to usingthe first terms of expansion for the construction of aPade-approximant that allows to extend essentially therange of reliable representation of a function (the clas-sical example is restoration even of exact solution inthe case f (x) = 1 − x + x2 − · · · , which is a geometricseries).

To conclude this section, the results of concrete cal-culation of the first terms of power expansion for theconsidered system are presented. If by substituting



the series presented above into ‘smooth’ equationsof motion, one can obtain from the first equationthat x2 = 0 and x3 = −aβy2

1 , then, from the secondequation, y1 = αN /2. After substitution in the pre-vious relation, x3 (as well as y1 = αN /2) actuallyrestores the corresponding terms of the elliptic func-tion power expansion. This process may be natu-rally continued. Using the first terms for construc-tion of the Pade-approximation, one can determinesmall (as it is seen from Fig. 33) corrections to theperiod.

The procedure proposed may be applied to all casesin which the processes in the system are being studied,that have internal resonances far from the stationarystate and consequently are close to beatings. Beatingsin a linearized, but strongly non-linear (in terms of θ −� language) system is a very good approximation tonon-linear beating, both in conservative and damped(and probably, in forced) systems.

4.3 Response regimes of integrable dampedstrongly non-linear oscillator under impactperiodic forcing

This section investigates the response of a stronglynon-linear damped oscillator to periodic vibro-impactexcitation [10]. An important aspect of the model cho-sen is that due to the special choice of parameters,the strongly non-linear damped oscillator is integrable[12, 53, 54], i.e. the solution for its free motion maybe expressed explicitly in terms of elliptic functionsdespite strong non-linearity and damping. If there wasno damping, then the integrability of the unforcedsystem would be trivial, but the lack of the damp-ing prohibits dissipation of transients; therefore, thedamping is necessary to allow the attractors in thestate space. Integrability of the damped system allowsa complete analytic investigation of single-periodresponse regimes and an evaluation of their stabil-ity; if stable, they are expected to serve as attractorsfor certain regions in the space of initial conditions.After that, for a given set of parameters it is instructiveto investigate the whole variety of real responses inthe complete space of initial conditions by means ofdirect numeric simulation. Comparison of these tworesults allows one to decide whether the regimes pre-dicted analytically adequately represent the multitudeof possible responses of the system. Understanding ofpossible behaviour of such strongly non-linear oscil-lators under vibro-impact excitations may pave newways for their applications.

Consider the following damped non-linear oscilla-tor with odd power non-linearity m (m � 3) and linearviscous dissipation

y + 1τ

y + ω20y + Cym = 0 (134)

where y = y(t) denotes the displacement, τ−1 the vis-cous damping coefficient (τ may be characterized ascharacteristic time of damping), and ω0 the linearizednatural frequency. The coefficients τ and C are non-negative scalars, and all coefficients are considered tobe O(1) quantities; therefore, both the non-linearityand the damping may be strong. In general, thissystem is non-integrable, of interest is the particu-lar case where the integrals of motion exist and theexact solution of motion can be expressed analytically.According to references [53, 54], the exact integrabilitycondition is

ω20τ

2 = 2(m + 1)

(m + 3)2(135)

Thus, the exact analytical solution of equation (134) isgiven by

y(t) = Ae(2t /(m+3)τ )cam(m)

×(√

2Cm + 1

m + 3m − 1

τA(m−1/2)e(−(m−1/(m+3)τ )t) + ϕ

)

(136)

where A and ϕ are related to initial conditions of theproblem and

du =∫

dcam(m)√1 − cam(m+1)

(m)

(137)

In the simplest case m = 3, the cam function isreduced to elliptic functions of Jacobi

du =∫

d cam(3)√1 − cam4

(3)

⇒ cam(3) =√

22

cn

(u,

√2

2

)

(138)

Assume C = 1. This assumption does not affect thegenerality of the treatment due to the possibility ofscaling of the dependent variables. The solution ofequation (133) is presented as

y(t) =√

22

· Ae(t/3τ) · cn

(√2

23τAe(t/3τ) + ϕ,

√2

2

)

(139)

Investigating this system under periodic impactloading, such an input to the system has to be for-mulated. An elastic impact leads to a change in thebody velocity in an infinitely small period of time. If thechange of the velocity �V occurs with period T , then



Fig. 35 Formation of pair of new solutions with the change of parameters (decrease of impactperiod), τ = 5. The left figure is computed for T = 4, �V = 4 and the right figure for T = 2,�V = 4. Solid line corresponds to the numeric solution of equation (141a) and dashed lineto equation (141b)

Fig. 36 Formation of pair of new solutions with the change of parameters (increase of impactintensity), τ = 5. The left figure is computed for T = 4, �V = 4 and the right figure forT = 4, �V = 8. Solid line corresponds to the numeric solution of equation (141a) anddashed to equation (141b)

Fig. 37 Parameters of solutions for the case T = 2, �V = 4, and τ = 5. Solid line corresponds tothe numeric solution of equation (141a) and dashed to equation (141b). Parameters A andϕ are computed for each particular solution



a single-period stationary response regime of the sys-tem should obey the following conditions of continuityand smoothness

{y(t) = y(t + T )

y(t) = y(t + T ) + �V(140)

In order to compute the response according toequation (140), one can choose t to be zero (a differentchoice will lead to phase shift only).

After substituting expression (139) to the first termof equation (140), one obtains

√2

2A · e(−0/3τ)cn

(√2

23τAe(−0/3τ) + ϕ

)

=√

22

Ae(−(0+T )/3τ)cn

(√12

3τAe(−0+T /3τ) + ϕ

)

⇒ cn

(√12

3τA + ϕ

)

= e(−T /3τ)cn

(√12

3τAe(−T /3τ) + ϕ

)(141a)

The second term of equation (140) yields

1√2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

A(

− 13τ

)e(−t/3τ)cn

(√12

3τAe(−t/3τ) + ϕ

)

+Ae(−t/3τ)

[−sn

(√12


)]

×dn

(√12


)A

3τ√2

(− 1

3τ

)×e(−t/3τ)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

− �V

= 1√2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

A(

− 13τ

)e(−t+T /3τ)cn

(√12

3τAe(−t+T /3τ) + ϕ

)

+Ae(−t+T /3τ)

[−sn

(√12

3τAe(−t+T /15) + ϕ

)]

×dn

(√12

3τAe(−t+T /3τ) + ϕ

)A

3τ√2

(− 1

3τ

)

×e(−t+T /3τ)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Fig. 38 Numerical simulation of single-period solutionsfor the case T = 2, �V = 4, and τ = 5. Solution(1): A = −5.777 840 068 and ϕ = 64.310 231 65.(a) Plot of the displacement versus time; (b)Poincaré section; (c) frequency content of thesolution – fast Fourier transform



After regrouping and substituting t = 0, one obtains

(− 1

3τ

)cn

(√12

3τA + ϕ

)+ A√

2sn

(√12

3τA + ϕ

)

× dn

(√12

3τA + ϕ

)−

√2�VA

=(

− 13τ

)e(−T /3τ)cn

(√12

3τAe(−T /3τ) + ϕ

)

+ e(−2T /3τ) A√2

[sn

(√12

3τAe(−T /3τ) + ϕ

)]

× dn

(√12

3τAe(−T /3τ) + ϕ

)(141b)

Two parameters of external excitation, T (periodbetween impacts) and �V (magnitude of impact),together with a single free parameter of the systemτ (relaxation time of the system), completely deter-mine the structure of solutions of systems (141a andb). Different solutions for ϕ and A correspond to single-period response regimes of the system with differentinitial conditions. Systems (141a and b) of transcen-dent equations are solved numerically. In order todescribe the structure of its solutions, it is convenientto introduce the new variables

x = 3√

22

τA + ϕ (142a)

y ≡ 3√

22

τAe−T /3t + ϕ (142b)

Fig. 39 Numerical simulation of single-period solutions for the case T = 2, �V = 4, and τ = 5.Solution (2): A = −5.387 553 133 and ϕ = 57.753 019 71. (a) Plot of the displacement ver-sus time; (b) detailed view of the displacement versus time for the period 5 solution; (c)Poincaré section; (d) frequency content of the solution – fast Fourier transform



Fig. 40 Numerical simulation of single-period solutionsfor the case T = 2, �V = 4, τ = 5. Solution(3): A = −2.531 297 498 and ϕ = 28.869 036 29.(a) Plot of the displacement versus time; (b)Poincaré section; (c) frequency content of thesolution – fast Fourier transform

and to present equations (142a and b) as lines in thex–y plane. The intersection points of these curvescorrespond to exact solutions of system (134) with

boundary conditions (140) and thus to single-periodresponse regimes of system (134) with periodic vibro-impact loading. Values of A and ϕ are easily computedfor every intersection point with the help of equations(142a and b).

In order to illustrate the method and possible bifur-cation scenarios for single-period response regimes,computations for selected values of T and �V are pre-sented below for fixed τ (Figs 35 and 36). Only oneparameter is varied in each pair of computation.

In both cases, the increase of the impact intensity orthe decrease of its period leads to the formation of anew pair of single-period responses. Taking a deeperlook at the correlation between period, magnitude ofimpact, and number of solutions of the system: whenincreasing �V or decreasing T , the ‘neck’ on the solidcurve becomes thinner until a couple of new solutionsare born when it intersects the dashed one (Fig. 37).

One can conclude that the number of solutions ofequations (8a and b) is determined by the number ofintersections between two families of curves, whichare either closed or go to infinity. For simple geometricreasons, it is clear that the single-period solutions areborn or disappear in pairs (at least one solution alwaysexists; therefore, the overall number of the solutions isodd). The period of the external forcing generates anatural Poincaré mapping of the state space on itself,and the appearance of a pair of the ‘newborn’ solutionscorresponds to the generic saddle-node bifurcation ofthis mapping. Therefore, one of these solutions shouldbe stable and the other unstable.

However, the variety of possible response regimesis not restricted by the single-period solutions. Oncethe single-period solution is created, it may be subjectto generic period-doubling bifurcations, with sub-sequent transition to chaos. Besides, the Poincarémaps generated by multiple periods of the exter-nal forcing may generate saddle-node bifurcations oftheir own, not manifested at the single-period map –multiple-period solutions also may be created withouta previous creation of the single-period ones.

Of course, all these solutions may in principle berevealed analytically by solving the system of equa-tions equivalent to equations (141a and b). How-ever, even for the investigation of double-periodresponses, one should already solve a system of fourtranscendent equations with four unknowns; thisproblem is extremely complicated even for numericsolution. In the following section, only the single-period responses are computed exactly and theother responses (as well as stability of the single-period ones) are determined by direct numericalsimulation.

Examining now a case of �V = 4 and T = 2 (Fig. 27),there are three solutions: two are supposed to bestable, whereas the third is not. This conclusioncan be verified by direct numeric simulation of the



single-period solutions with the parameters definedin Fig. 37. The results are presented in Figs 38 to 40.

It is obvious that solutions (1) and (3) are stable,whereas solution (2) is not. Due to computation errors,the unstable solution switches from single-periodresponse to the response of period 5. The latter is notrelated to any of the stable single-period responses andhas been created by another mechanism.

Hitherto the analytical model gave some good ‘fore-casts’ for a number of solutions and their stability ascompared to numerical simulations, but there is stilla question whether this ‘forecast’ is exhaustive. Unfor-tunately, the answer is negative. In order to reveal that,the system is directly simulated with a wide set ofinitial conditions; this task is assessable because thespace of the IC is only two-dimensional. The result ispresented in Fig. 41.

The simulation is run for different initial conditionswith a step of 0.01 in each direction. The initial condi-tions that converge to the same attractor are markedwith the same colour or background. From the picture,it is clear that there are four different periodic responseregimes within the region [−2.5, 2.5; −2.5, 2.5]. Thus,it is clear that there are more stable response regimesthat can be predicted with the single-period analyticalmodel.

This region of parameters is interesting to anal-yse, but is of small engineering interest because it

is very difficult to predict the system’s behaviour.For instance, an unnoticeable error in initial con-ditions will result in a completely different system’sresponse regime. Therefore, if any application of thestrongly non-linear oscillator is planned (for instance,for mitigation of periodic impacts), it is desirable towork in the region of parameters with predictablebehaviour. This can be done by changing the sys-tem’s time of relaxation τ . Making τ small will resultin greater damping in the system. If one chooses τ = 1instead of τ = 5 with the same conditions of loadingT = 2, �V = 4, one gets a homogenous picture witha single-period response, without any unexpectedmultiple-period solutions.

It is instructive also to investigate the sensitivity ofthe system for changing �V (Fig. 42).

It is obvious that no different single-period responseregimes appear even if the change of the impactintensity is immense (ten times).

In order to assess the effect of the relaxation time, itis instructive to investigate a possible number of thesingle-period solutions as a function of time of relax-ation τ , period of impact T , and impact intensity �V .Since τ is the parameter of the system, the dimen-sionless ratio T /τ is used in order to characterize theinfluence of �V (Fig. 43).

It is evident that the system’s behaviour differs a lotfor the different values of τ/T and �V . A small T /τ

Fig. 41 Domains of attraction of different response regimes at the plane of initial conditions aremarked by different colour or background (T = 2, �V = 4, and τ = 5)



Fig. 42 A number of solutions for different parametersof the loading for T = 2 and τ = 1. (a) �V = 40and (b) �V = 4

Fig. 43 A number of single-period solution as afunction of τ/T and τV (figure in each zone)

corresponds to a multiple-solutions region, and so itcorresponds to the previously made assumption thatan oscillating system with a large time of relaxation willnot absorb the energy of the impact completely, thus

giving rise to birth of new response regimes. The globalpicture is rather complicated, but it is clear that there isa limit of T /τ ≈ 2, above which only one single-periodresponse regime will exist for all values of the impactintensity.

5 CONCLUDING REMARKS

1. Vibro-impact models often turn out to be tractablemodels of non-linear dynamics and admit applica-tion of various analytic techniques for their efficientinvestigation. Some of these results provide unex-pected insights into various physical problems andprocesses.

2. Vibro-impact models are tightly related to thesmooth, essentially non-linear, systems, allowingone to substitute and approximate one type ofsystem by another where applicable.

3. Sometimes, vibro-impact models can arise natu-rally in problems that apparently have no relationto the impacts.

ACKNOWLEDGEMENTS

The authors are very grateful to the Israel ScienceFoundation (Grant 486/05) and to the Lady DavisFellowship Trust for financial support.

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oscillatory models of vibro-impact type for essentially non-linear systems

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