Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 3, pp. 641-664, Warsaw 2008

CONTROL AND CHAOS FOR VIBRO-IMPACT AND

NON-IDEAL OSCILLATORS

Silvio L.T. de Souza
Iberê L. Caldas

University of São Paulo, Institute of Physics, São Paulo, SP, Brazil

e-mail: thomaz@if.usp.br; ibere@if.usp.br

Ricardo L. Viana

Federal University of Paraná, Departament of Physics, Curitiba, PR, Brazil

e-mail: viana@fisica.ufpr.br

José M. Balthazar

State University of São Paulo, Department of Statistics, Computational and Applied Mathematics,

Rio Claro, SP, Brazil; e-mail: jmbaltha@rc.unesp.br

In the paper, we discuss dynamics of two kinds of mechanical systems.
Initially, we consider vibro-impact systems which havemany implemen-
tations in appliedmechanics, ranging fromdrillingmachinery andmetal
cutting processes to gear boxes. Moreover, from the point of view of
dynamical systems, vibro-impact systems exhibit a rich variety of phe-
nomena, particularly chaotic motion. In this paper, we review recent
works on the dynamics of vibro-impact systems, focusing on chaoticmo-
tion and its control. The considered systems are a gear-rattling model
and a smart damper to suppress chaotic motion. Furthermore, we inve-
stigate systems with non-ideal energy source, represented by a limited
power supply. As an example of a non-ideal system, we analyse chaotic
dynamics of the dampedDuffing oscillator coupled to a rotor. Then, we
show how to use a tuned liquid damper to control the attractors of this
non-ideal oscillator.

Key words: vibro-impact, chaos, control of chaos, non-ideal oscillators

1. Introduction

There is a steadfast interest in the theoretical and experimental study of
vibro-impact systems which have oscillating parts colliding with other vibra-
ting components or rigid walls (Blazejczyk-Okolewska et al., 2004). Vibro-
impact systems are widely found in engineering applications, like vibration



642 S.L.T. de Souza et al.

hammers, drilling machinery, milling, impact print hammers, and shock ab-
sorbers (Blazejczyk-Okolewska et al., 1999). There are also undesirable effects
coming from vibro-impact systems like gearboxes, bearings, and fuel elements
in nuclear reactors: large amplitude response leading to material fatigue and
rattling (Jerrelind and Stensson, 2000).

From the point of view of dynamical systems, however, vibro-impact sys-
tems are extremely rich models, for they exhibit a wealth of phenomena like
bifurcations, chaos, crises, multi-stability, and final state sensitivity (Peterka
andVacik, 1992; Ivanov, 1996;Wiercigroch anddeKraker, 2000). Thedistinc-
tive dynamical character of vibro-impact systems is their non-smoothness due
to impacts with several types of amplitude constraints (Nordmark, 1991; Luo,
2004; Luo et al., 2006, 2007). In this case, the impacts are treated bymodify-
ing the initial conditions of motions according to an impact rule considering
the coefficient of restitution. Another interesting and useful approach is to
introduce piecewise stiffness characteristics for describing impacting systems
(Wiercigroch, 2000; Ing et al., 2006; de Souza et al., 2007c).

Another key issue related to the dynamics of vibro-impact systems is the
control of their oscillations (Kapitaniak, 2000). In fact, chaotic oscillations,
being intrinsically random and unpredictable, are generally considered as an
undesirable, even harmful phenomenon when it occurs. Hence the control of
chaos in vibro-impact systemshas immediate practical applications as, e.g. the
suppression of rattling noise in gearboxes (Karagiannis and Pfeiffer, 1991).

Several other important features of impact oscillators with practical appli-
cations have also been analysed. Thus, impacts are also employed to describe
step disturbances in multi-body mechanical systems of many industrial ma-
chines (Czolczynski et al., 2000). Moreover, instabilities and bifurcations have
been explained by considering impact systems with bounded progressive mo-
tions (drifts) (Pavlovskaia et al., 2001) or low-velocity collisions causing the
so called grazing effect (Nordmark, 1991). Another relevant application comes
from considering impact systems with dry friction generating high amplitude
forces involving dynamic fractures required to drilling brittlematerials (Wier-
cigroch et al., 2005).

In this paper an overview of the modern research on vibro-impact sys-
tems dynamics, focusing on chaos and its control is presented.We shall briefly
present each model and how to derive its governing equations from first prin-
ciples. We shall present dynamical features like phase portraits, bifurcation
diagrams, Lyapunov exponents, and basin boundaries, including quantitative
characterisations of chaotic motion and final state sensitivity (fractal basin
boundaries).



Control and chaos for vibro-impact... 643

Theabovementionedworksdealwithoscillators that aredrivenby systems
whose amplitude and frequency can be arbitrarily chosen. However, in several
mechanical experiments this cannot be achieved because the forcing source
has a limited available energy supply. This has been called non-ideal energy
sources (Kononenko, 1969). A common example appears when the driving
comes from an unbalanced rotor linked to the oscillator. As a consequence
of this mechanical coupling, the rotor dynamics may be heavily influenced by
the oscillating systembeing forced.As a consequence, the resulting oscillations
differ from those predicted for experiments with ideal energy sources. Hence,
the driven system cannot be considered as given a priori, but itmust be taken
also as a consequence of the dynamics of the whole system (oscillator plus
rotor) (Dimentberg et al., 1997; de Souza et al., 2005a, 2007b).

This paper is organized as follows: Section 2 deals with a single-stage ge-
arbox system under periodic excitation. Section 3 considers a simple model
of vibro-impact system consisting of a linear oscillator undergoing inelastic
collisions with a fixed barrier.We study the suppression of chaotic motion by
a smart damper which changes the damping coefficient according to the sign
of the relative velocity. A non-ideal oscillator is treated in Section 4, where
a damped Duffing oscillator is coupled to a non-ideal motor driving a rotor.
This procedure introducesmore degrees of freedom in the system, but it turns
out to be a more realistic model of a finite-power external source of driving
to the system. Section 5 combines such a non-linear oscillator with a U-tube
filled with liquid as the damping mechanism. Our conclusions are left to the
last Section.

2. Gear-box system

Gear units have typically backlashes or variable clearances between adjacent
movingparts, and these backlashes need to allow thermal expansion and lubri-
fication of themoving wheels. The presence of backlashes makes gear teeth to
lose contact for a short interval generating repeated collisions andahammering
effect (Pfeiffer and Kunert, 1990). In spur gears of engines driving camshafts
and injection pump shafts, for example, rattling is the source of uncomfortable
noise. Hence, many theoretical and experimental efforts have been devoted to
the understanding of such vibro-impact problems (Karagiannis and Pfeiffer,
1991).



644 S.L.T. de Souza et al.

2.1. Gear-rattling model

We focus on the gearbox rattling model proposed by Pfeiffer and collabo-
rators, and consisting of two spur gears with different diameters and a gap
between the teeth (single-stage rattling) (de Souza and Caldas, 2001), Fig.1.
Motion of one gear is supposed to be sinusoidal the with well-defined ampli-
tude and frequency, whereas motion of the other gear results from the sys-
tem dynamics. The gears have radii R and Re, and a backlash ν between
the teeth. Motion of the driving wheel is described by a harmonic function
e(t) = −Asin(ωt). Between the impacts, motion of the second wheel is go-
verned by a linear differential equation and can be analytically determined.
Impacts are treated by modifying the initial conditions of motion, according
to the Newton impact laws.

Fig. 1. Schematic view of a single-stage gearbox system

Inanabsolute coordinate system,wedenoteby ϕ theangular displacement
of the secondgear, such that the rotationdynamics is governedby the following
equation of motion

mϕ′′+νϕ′ =−T (2.1)

where primes denote differentiation with respect to time, m is themoment of
inertia, ν is the oil drag coefficient, and T is the oil splash torque.

The relative displacement between the gears due to the backlash is thus

s=
ARe
ν
sin(ωt)−R

ν
ϕ (2.2)

in such a way that −1<s< 0.
The equation ofmotion, in the relative coordinate system, is obtained from

(2.1) and (2.2), resulting in

s̈+βṡ= ë+βė+γ (2.3)



Control and chaos for vibro-impact... 645

where dots stand for derivatives with respect to the scaled time τ = ωt, and
we have introduced the following non-dimensional parameters

α≡ ARe
ν

β≡ d
mω

γ ≡ TR
mω2

(2.4)

representing the damping coefficient, excitation amplitude, and moment of
inertia, respectively.

Since equation (2.3) is linear, it can be integrated analytically between two
impacts for the initial conditions s(τ0) = s0, and ṡ(τ0) = ṡ0. The displace-
ment s and velocity ṡ between impacts is given by

s(τ)= s0+α(sinτ− sinτ0)+
γ

β
(τ − τ0)+

+
1

β
{1− exp[−β(τ −τ0)]}

(

ṡ0−αcosτ0−
γ

β

)

(2.5)

ṡ(τ)=αcosτ+
(

ṡ0−αcosτ0−
γ

β

)

exp[−β(τ − τ0)]+
γ

β

An impact occurswhenever s=−1 or 0, since they actually correspond to
the backlash boundaries. At these points, motion is no longer smooth and we
have to reset the initial conditions according to the Newton laws of inelastic
impact

τ0 = τ s0 = s ṡ0 =−rṡ (2.6)

where 0<r< 1 is the restitution coefficient.

Adopting the instant of each collision as the time unit, we can define di-
screte variables sn, ṡn, and τn representing the displacement, velocity, and
time (modulo 2π) just before the n-th impact, respectively. Thanks to the
analytical solution of the equations of motion between two impacts, we can
substitute the initial conditions τ0 = τn, s0 = sn, and ṡ0 = −rṡn into Eqs.
(2.5).With this representation we have amapping relating the dynamical va-
riables for the n+1-th impact to their corresponding values just before the
n-th impact

sn+1 = sn+α(sinτn+1− sinτn)+
γ

β
(τn+1− τn)+

−
1

β
{1− exp[−β(τn+1− τn)]}

(

rṡn+αcosτn+
γ

β

)

(2.7)

ṡn+1 =αcosτn+1+
γ

β
− exp[−β(τn+1− τn)]

(

rṡn+αcosτn+
γ

β

)



646 S.L.T. de Souza et al.

This map is used here to calculate the Lyapunov exponents (de So-
uza and Caldas, 2004). Such exponents are computed through λi =
= limn→∞(1/n)ln |Λi(n)|, (i = 1,2), where Λi(n) are the eigenvalues of the
matrix A= J1J2 · · ·Jn, where Jn is the Jacobian matrix of the map, compu-
ted at time n. For systems without analytical solutions between the impacts,
the Lyapunov exponents can be calculated by using themethods proposed by
Jin and co-workers (Jin et al., 2006).

2.2. Fractal basin boundaries and chaotic transient

In Fig.2, we show time series for the relative displacement s(t) for three
qualitatively different classes of behaviour: (i) one impact with the wall at
s = −1 for two successive impacts with the wall at s = 0 (Fig.2a); (ii) one
impact with each wall (Fig.2b); and (iii) one impact with the wall at s = 0
for two impacts with the wall at s=−1 (Fig.2c).

Fig. 2. Time histories of three co-existing attractors for amplitude excitation
α=0.5 and restitution coefficient r=0.9

The initial conditions leading to each of these behaviour scenarios are
depicted in Fig.3awith different shades of gray. Similarly as we define a basin
of a given attractor as a set of initial conditions which generates trajectories
asymptotically tending to this attractor, we may call the sets in Fig.3a as
basins of behaviour, with the same interpretation.Thebasins of eachbehaviour



Control and chaos for vibro-impact... 647

are densely mixed and, as suggested by the magnification shown in Fig.3b,
the basin boundary looks like a fractal curve.

Fig. 3. (a) Basins of behaviour for three co-existing attractors shown in Figs. 2a-c.
(b)Magnification of the previous figure

The boundary between the behaviour basins in Fig.3 is indeed fractal,
which is quantitatively confirmed by Fig.4, where we show variation of the
uncertain fraction f(ǫ) of the phase space section shown inFig.3 as a function
of the uncertainty radius ǫ. By uncertain fraction we mean the result of the
following numerical experiment: we create a fine mesh of initial conditions in
the plane (ṡn,τn) and consider the fate of a trajectory starting from each
initial condition and another, very close, initial condition far apart from the
former by a distance ǫ. If both trajectories tend asymptotically to different
final responses, the initial condition is said to be ǫ-uncertain (McDonald et
al., 1983, 1985).

Fig. 4. Uncertain fraction versus uncertainty radius ǫ for basins of behaviour showed
in Fig.3. The solid curve is a linear regression fit with slope ̟=0.154±0.001



648 S.L.T. de Souza et al.

The uncertain fraction is the relative number of ǫ-uncertain initial con-
ditions. We expect from general arguments that this fraction scales with the
uncertainty radius as a power law: f(ǫ) ∼ ǫ̟, where ̟ = 2− dB is called
the uncertainty exponent, dB being the box-counting dimension of the basin
boundary. If 0<̟< 1 then 1<dB < 2 and the basin boundary is a fractal
curve. As shown in Fig.4, this scaling is verified for this case where a least-
squares fit shows that ̟=0.154±0.001, confirming that the basin boundary
is a fractal with final-state sensitivity. As an illustration of the consequences
of this fractality, suppose one manages to diminish the uncertainty radius by
a factor of ten. Due to the small value of the uncertainty exponent ̟, the
corresponding decrease in the uncertain fraction is a factor ∼ 10−0.15 ≈ 0.708
only.

Fig. 5. (a) Bifurcation diagram of the impact moment τn as a function of the
excitation amplitude α for R=0.9. (b) Lyapunov exponents λ1,2 as a function of α

In Fig.5a we plot a bifurcation diagram showing the asymptotic values of
the variable τn versus the excitation amplitude α. The interesting feature of
this diagram is that the chaotic behaviour of τn is suppressed as the damping
increases, as would be expected fromgeneral arguments. However, this change
does not occur smoothly but rather in an abrupt way, after a crisis at αC ≈
1.25 leading to a stable period-1 orbit corresponding to four impacts (two
with each wall) (de Souza et al., 2004). The latter is followed by a period-
doubling bifurcation cascade and a wide chaotic region for higher damping, a
surprising fact. Figure 5b shows the corresponding diagram for the Lyapunov
exponents.



Control and chaos for vibro-impact... 649

Fig. 6. Basins of transient behaviour with α=1.245 for noise levels (a) σ=0, and
(b) σ=0.03

Figure 6 illustrates the effect of additive noise in the basin boundary struc-
ture of the gearbox system (de Souza et al., 2005b). The additive noise is here
represented by a pseudo-randomvariable with uniform distribution and noise
level σ. We fixed a number, say 500 iterations of the gearbox system, and
consider whether or not the evolution is chaotic. The behaviours indicated in
Fig.6 is the following: (i) periodic evolution (black pixels); (ii) chaotic tran-
sient evolution (white pixels). For zero noise (Fig.6a) the transient basins are
already mixed up but, as we switch on the external noise, most of the basins
are related to the periodic attractor (Fig.6b). This decrease of the relative
area of the chaotic transient basin can bemeasured as a function of the noise
level. Our results support a linear decay law: 0.708−8.176σ (Fig.7).

Fig. 7. Relative area A of the chaotic transient basin as a function of noise
perturbation σ. The solid curve is a linear regression fit



650 S.L.T. de Souza et al.

3. Controlling chaos in vibro-impact systems

Since chaotic motion is quite common in vibro-impact systems, leading to
sometimes undesirable effects, it is important to devise methods to control
chaotic motion (Bishop et al., 1998; Fradkov et al., 2006; Lee and Yan, 2006).
In 1990, a scheme of chaos control was put forward byOtt,Grebogi andYorke
(Ott et al., 1990). The so-called OGY method consists on stabilizing a desi-
red unstable periodic orbit embedded in a chaotic attractor by using only
a tiny perturbation on an available control parameter. Another interesting
chaos control strategy was proposed in (Pyragas, 1992) who also considered
dynamical properties of a chaotic attractor to stabilize unstable periodic or-
bits. In that case the method implementation required a delayed feedback
signal. Another kind of feedback control methodwas proposed in recent years
(Alvarez-Ramirez et al., 2003; Tereshko et al., 2004), using a small-amplitude
control signal, applied to alter the energy of a chaotic system. We used ano-
ther approach, namely to alter the damping coefficient, to suppress chaotic
motion and steer the system to some desired periodic attractor (de Souza et
al., 2007a).

3.1. Impact oscillator

We applie a control strategy in order to suppress chaotic motion in a
vibro-impact system. Figure 8 depicts a model of an impact oscillator, which
is a periodically forced and damped linear one-dimensional oscillator whose
displacement is limited by an amplitude constraint, a fixed wall at x = xc.
Between two successive impacts, smoothmotionwithout control input, is given
by

ẍ+ cẋ+x=αcos(ωt) (3.1)

where c is the damping coefficient, F is the forcing amplitude, and ω is the
forcing frequency. Both the oscillator mass and the elastic constant have been
normalized to unity for simplicity.

Considering the control method by de Souza et al. (2007a), the equation
of motion is described by

ẍ+fd(ẋ)+x=αcos(ωt) (3.2)

where

fd(ẋ)=

{

(c−k)ẋ if ẋ­ 0
cẋ if ẋ < 0



Control and chaos for vibro-impact... 651

Fig. 8. Model of an impact oscillator

and k is a constant coefficient. In this case, thedampingcoefficient isdecreased
for positive velocities and is not changed for negative velocities.

An impact occurs whenever x = xc. After each impact, we reset the ve-
locity of the oscillator using Newton’s impact law. In other words, we model
the collisions with a rigid wall (amplitude constraint) by the law of inelastic
impact: the velocity after the impact is taken to be −r times the velocity
before the impact, where r is a constant restitution coefficient (0<r< 1).

3.2. Suppressing chaotic vibrations

Numerical simulations were performed by using the fourth-order Runge-
Kutta method. We adopted a fixed step for displacements far away from the
rigidwall (amplitudeconstraint) andanadaptive step, ifweare close enoughto
the wall. The adaptive step was obtained using theNewton-Raphsonmethod.
The control parameter values were fixed at xc = 0, r = 0.8, α = 2.0, and
ω=2.8.

For the control switched off, in Fig.9a, we show a bifurcation diagram of
the velocity ẋ just before an impact with the amplitude constraint in terms
of the damping coefficient c. Hence, as can be seen, there is a wide range of
the parameter for which the system presents chaotic attractors, with one and
two bands, occasionally interrupted by narrow periodic windows.

In order to verify the performance of the control method, we present in
Fig.9b, for c=0.7, a bifurcation diagram in terms of the parameter k, that
is associated with the damping coefficient, showing the transformation of the
chaotic attractor for small k into a period-1 orbit as k is increased from zero
through a reverse period-doubling bifurcation cascade. From these results, by
varying the damping coefficient according to velocities, we can obtain the
suppression of chaotic regimes.



652 S.L.T. de Souza et al.

Fig. 9. Bifurcation diagrams of the velocity just before the impact as a function of
(a) damping parameter c for k=0; and (b) control parameter k for c=0.7

Fig. 10. Phase portraits for c=0.7 and (a) k=0.2; (b) k=0.1. The uncontrolled
chaotic attractor (k=0) is depicted in the background

In Fig.10a we show an example of the suppression of chaotic regimes,
where we plot the phase portrait, for c= 0.7, of the period-1 orbit that was
obtained for k = 0.2. In the background of this figure, a chaotic attractor
without control (k = 0) is depicted. When k is changed to a smaller value,
the resulting orbit has period 2, Fig.10b. We present in Fig.11 an example
of the control implementation for this case, where we depict the evolution of
velocity collected justbefore the impacts as a functionof the impactnumber n.
The control is switched on at the time n= 1000 for k = 0.1 and a period-2
orbit is obtained. At n=2000 the control is switched off and at n=3000 it
is switched on again, for k = 0.025, after which we obtain a period-4 orbit.



Control and chaos for vibro-impact... 653

At n= 4000 the control is switched off and, finally, at n=5000 the control
is switched on again for k = 0.2, resulting in the period-1 orbit shown in
Fig.3a.

Fig. 11. Time series of the velocity just before an impact as a function of the impact
number n for c=0.7 and the application of control at three time instants (see text

for details)

4. Oscillator with limited power supply

Most studies of drivenoscillators assume that thedriving comes froman exter-
nal sourcewhich is not appreciablyperturbedbymotion of the oscillator (ideal
systems). However, in practical situations, the dynamics of the forcing system
cannot be considered as given a priori, and it must be taken as also a con-
sequence of the dynamics of the whole system (Kononenko, 1969). In other
words, the forcing system has a limited energy source as that provided by
an electric motor for example, and thus its own dynamics is influenced by
that of the oscillating systembeing forced (Krasnopolskaya and Shvets, 1993).
This increases the number of degrees of freedom, and is called a non-ideal
problem.

In terms of the vibrating cart model, the non-ideal system is obtained by
replacing theexternal sinusoidaldrivingof the cartby the rotor attached to the
cart, and fedby themotor (Warminski et al., 2001).Theangularmomentumof
the rotor is imparted to the cart. The application of non-ideal models to some
vibro-impact problems has been considered in recent papers (Dimentberg et
al., 1997; de Souza et al., 2005a, 2007b; Xu et al., 2007).



654 S.L.T. de Souza et al.

4.1. Non-ideal oscillator

We consider one-dimensional motion of a cart of the mass M (mass con-
sidering the motor) connected to a fixed frame by a nonlinear spring and a
dash-pot (viscous coefficient c), as shown in Fig.12. The nonlinear spring stif-
fness is given by k1X−k2X3, where X denotes the cart displacement.Motion
of the cart is due to an in-board non-ideal motor driving the rotor.We denote
by ϕ the angular displacement of the rotor with themass m0 and amassless
rod of radius R.

Fig. 12. Schematic model of a non-ideal oscillator

It is convenient toworkwith dimensionless dynamical variables, according
to the following definitions

x≡ X
R

τ ≡ t
√

k1
M

(4.1)

The equations ofmotion for both the cart and the rotor are given, respectively,
by

ẍ+βẋ−x+ δx3 = ǫ1(ϕ̈sinϕ+ ϕ̇2cosϕ)
(4.2)

ϕ̈− ǫ2cosϕ= ẍsinϕ+E1−E2ϕ̇

where the dots stand for differentiation with respect to the scaled time τ, and
the following abbreviations were introduced

β≡
c√
k1M

δ≡
k2
k1
R2 ǫ1 ≡

m0
M

ǫ2 ≡
g

Rk1
(4.3)

The parameters E1 and E2 can be estimated from the characteristic curve of
the energy source (DC-motor).



Control and chaos for vibro-impact... 655

4.2. Co-existence of attractors

Numerical simulations of nonlinear oscillator systems were performed by
using the fourth-orderRunge-Kuttamethodwith a fixed step. The systempa-
rameters were fixed at β=0.02, δ=0.1, ǫ1 =0.1, ǫ2 =1.0, and E2 =1.5. As
representative examples of typical solutions obtained for the considered non-
ideal oscillatormodel,we show inFig.13a abifurcationdiagram for the displa-
cement versus the parameter E1. In this figure, we identify the co-existence
of attractors, both periodic (of various periods) and chaotic. For example,
three periodic attractors are observed for E1 = 2.0 (the point labelled as C
in Fig.13a), one periodic and one chaotic attractor for E1 = 2.5 (the points
labelled as A and B in Fig.13a), and two quasi-periodic and one periodic
attractor at E1 = 4.0 (the point labelled as D in Fig.13a). In Fig.13b, we
depict the same bifurcation diagram for the Lyapunov exponents of the at-
tractor labelled as A in Fig.13a.We used the algorithm of (Wolf et al., 1985)
to numerically obtain the Lyapunov exponents for this model.

Fig. 13. (a) Bifurcation diagram for the cart displacement versus the control
parameter E1 showing the co-existence of attractors. (b) Three largest Lyapunov

exponents of attractors A

The situation where the chaotic attractor A coexists with the periodic
attractor B (Fig.14) deserves particular attention. Depending on the initial
condition chosen, the trajectory will asymptote to one or another attractor.
Curiously, both attractors correspond to oscillations with similar ranges both
in the position and velocity of the cart. The basins of attractions of both
attractors have a quite complicated structure, as shown in Fig.15, where the
initial conditions converging to the chaotic (periodic) attractor are depicted
with white (dark gray) pixels. Using the uncertain fraction approach outlined
in Section 2, we computed the dependence of the uncertain fraction with the



656 S.L.T. de Souza et al.

Fig. 14. Phase portraits for the control parameter E1 =2.5 showing the co-existence
of periodic and chaotic attractors

Fig. 15. (a) Basins of attraction for the two co-existing attractors shown in Fig.14.
(b)Magnification of the previous figure

uncertainty radius, obtaining a power-law scaling with the exponent ̟ =
0.445±0.008, corresponding to the fractal basin boundarywith box-counting
dimension dB =1.555 (Fig.16).

5. Non-ideal oscillator with a tube liquid damper

The need to mitigate wind, ocean wave and earthquake induced vibrations in
structures like tall buildings, long span bridges and offshore platforms has led
to a steadfast interest in damping devices. Impact dampers are a very useful
way to suppress unwanted high-amplitude vibrations in small-scale systems,
but they are somewhat difficult, if not impossible, to implement in large-scale
engineering structures (Chaterjee et al., 1995). For the latter systems, tuned



Control and chaos for vibro-impact... 657

Fig. 16. Uncertain fraction in the phase space region shown in Fig.15b versus the
uncertainty radius ǫ. The solid curve is a linear regression fit with slope

̟=0.445±0.008

liquid dampers (TLDs) and tuned liquid column dampers (TLCDs) have ga-
ined special attention by virtue of their simplicity and flexibility (Yalla and
Kareem, 2001). A tuned liquid damper is basically amass-spring-dashpot sys-
tem connected to the structure and works due to the inertial secondary sys-
temprinciple bywhich the damper counteracts the forces producing vibration
(Yalla et al., 2001; Felix et al., 2005).

A tuned liquid column damper replaces the mass-spring-dashpot system
by a U-tube-like container where motion of a liquid column absorbs a part of
the vibration on the system with a valve/orifice playing the role of damping.
ATLCDhas an additional advantage of being a low-cost application. In a tall
building, for example, the container can also be used as a building water sup-
ply, whereas in a TLD, the mass-spring-dashpot is a dead-weight component
without furtheruse. In fact, vibration control throughTLCDhasbeen recently
used in other engineering applications, such as ship and satellite stabilization.
Whereas thedampingcharacteristic of amass-spring-dashpot systemof aTLD
is essentially linear, the damping in a liquid column is amplitude-dependent
(regulated by the orifice in the bottom of the U-tube) and consequently non-
linear. Hence, the dynamics of a TLCD is far from being simple, and very few
analytical results can be obtained. Numerical explorations of dynamics of a
TLCDmounted on a structural frame, using a non-ideal motor as a source of
energy, have been performed recently (Felix et al., 2005).

The liquid damper consists of a U-shaped tube attached to the top of the
cart, containing a liquid of the total mass m and density ρ, Fig.17 (de Souza
et al., 2006). The distance between the two vertical columns is denoted by b
and the distance of the liquid levels in these columnswill be denoted by l. The
vertical displacement of the left columnwith respect to the liquid level, when



658 S.L.T. de Souza et al.

the cart is at rest, is denoted by Y . There is a valve at themiddle point of the
tube bottom whose aperture can be tuned in order to vary the resistance to
the flow through this orifice.This is the source of the nonlinear andamplitude-
dependent damping experienced by the liquidmass while flowing through the
U-tube. The coefficient of head loss of the valve is ξ.

Fig. 17. Non-ideal systemwith a liquid damper

Motion of the combined cart-liquid damper system is governed by the
following equations

(1+µ)ẍ+βẋ−x+ δx3 = ǫ1(ϕ̈sinϕ+ ϕ̇2cosϕ)−α1µÿ
ϕ̈+ ǫ2cosϕ= ẍsinϕ+E1−E2ϕ̇ (5.1)
ÿ+γ|ẏ|ẏ+σy=α1ẍ

where

µ≡
m

M
γ≡
ξr

2l
σ≡

k3
k1µ

α1 ≡
b

l
(5.2)

Figure 18 shows phase portraits for the oscillator without (in gray) and
with thedamper (inblack). In theuncontrolled case,wehave coexistenceof the
periodic and chaotic attractor. As can be seen, considering the controller, the
amplitudes of periodic and chaotic vibrations are decreased, and the chaotic
attractor is suppressed. The parameters of the damper are fixed at α1 =3.0,
γ=0.4, σ=1.0, and µ=0.01. The basin structure in this situation is shown
in Fig.19, showing a quite complicated structure of interrupted striations.
However, it should be noted that the basins here are simpler than in the case
analysed in Fig.15 for a non-ideal system.



Control and chaos for vibro-impact... 659

Fig. 18. Phase portraits for the control parameter E1 =2.5 showing the co-existence
of periodic and chaotic attractors without control (in gray); and two periodic
attractors with the liquid damper (black), for α=3.0, γ=0.4, σ=1.0, and

µ=0.01

Fig. 19. (a) Basins of attraction for the two co-existing attractors (in black) shown
in Fig.18. (b)Magnification of the previous figure

6. Conclusions

In this paper, we reviewed some recent works authors have done on complex
dynamics in vibro-impact systems, focusing chiefly on chaotic motion and its
control through different schemes. Besides their evident engineering applica-
tions, vibro-impact systems enjoy also amore fundamental interest due to the
loss of smoothness in their dynamics, leading to a plethora of complex dyna-
mical phenomena, some of them being presented throughout this paper. We
have seen that inall systems considered, chaoticmotion is aubiquitous feature,
presenting some challenging questions concerning to its control or suppression.



660 S.L.T. de Souza et al.

As an example of vibro-impact systems, we considered a gear box model and
discussed the coexistence of attractors with fractal basin boundaries as well as
the existence of long chaotic transients. In addition, we used smart dampers
to suppress chaotic vibrations of an impact oscillator.

Another feature considered in this paper is taking into account the fini-
teness of oscillator energy sources. As an example of a non-ideal system, we
analysed chaotic dynamics of the dampedDuffingoscillator coupled to a rotor.
For that system, we identified the influence of the power supply on attractors.
Moreover, we showed how to use a tuned liquid damper to control the attrac-
tors of such a non-ideal oscillator.

In conclusion, we presented examples of controlling chaotic dynamics of
vibro-impact and non-ideal oscillators. The used control procedure may help
avoiding undesirable behaviour of mechanical systems with practical applica-
tions.

Acknowledgements

This work has beenmade possible by partial financial support from the following

Brazilian government agencies: FAPESP, CNPq, and Capes. The authors would like

to thankProfessorTomaszKapitaniak for his continuous encouragementandvaluable

discussions and suggestions.

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Sterowanie i chaos w układach drgających z uderzeniami oraz

nieidealnych oscylatorów

Streszczenie

Wpracy przedyskutowano zagadnienie dynamikimechanizmówdwóch rodzajów.
Najpierw rozważono układ drgający z uderzeniami, który znajduje liczne aplikacje
praktyczne wmechanice stosowanej, począwszy od urządzeń wiertniczych przez pro-
cesy cięcia metalu do skrzyń biegów włącznie. Z punktu widzenia dynamiki maszyn



664 S.L.T. de Souza et al.

układy wibro-uderzeniowewykazują bogactwo interesujących zjawisk, wliczając w to
chaos.W pracy zaprezentowano przegląd ostatnich prac dotyczących dynamiki ukła-
dów wibro-uderzeniowych, w których zajęto się problemem chaosu i możliwości jego
sterowania.Przeanalizowanoukładymechanicznenaprzykładziemodelukół zębatych
z systemem”inteligentnego” tłumika do eliminacji ruchu chaotycznego. Zajęto się, po
drugie, mechanizmami z nieidealnym źródłem energii odwzorowanym poprzez układ
ograniczonegopoborumocy.Jakoprzykładzbadanodynamikęchaotycznątłumionego
oscylatoraDuffingapołączonegozwirnikiem.Pokazanosposóbzastosowaniapłynnego
tłumika do sterowania formą atraktorów obserwowanychw nieidealnym oscylatorze.

Manuscript received December 17, 2007; accepted for print March 19, 2008