Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 1, pp. 119-131, Warsaw 2007

CHAOTIC VIBRATION OF AN AUTOPARAMETRICAL

SYSTEM WITH A NON IDEAL SOURCE OF POWER

Danuta Sado

Maciej Kot

Institute of Machine Design Fundamentals, Warsaw University of Technology

e-mail: dsa@simr.pw.edu.pl

This paper studies the dynamical coupling between energy sources and
the response of a two degrees of freedomautoparametrical system,when
the excitation comes fromanelectricmotor (with unbalancedmass m0),
whichworkswith limited power supply.The investigated systemconsists
of a pendulum of the length l and mass m, and a body of mass M
suspended on a flexible element. In this case, the excitation has to be
expressed by an equation describing how the energy source supplies the
energy to the system. The non-ideal source of power adds one degree of
freedom, which makes the system have three degrees of freedom. The
system has been searched for known characteristics of the energy source
(DC motor). The equations of motion have been solved numerically.
The influence of motor speed on the phenomenon of energy transfer has
been studied.Near the internal and external resonance region, except for
different kinds ofperiodic vibration, chaoticvibrationhasbeenobserved.
For characterizing an irregular chaotic response, bifurcation diagrams
and time histories, power spectral densities, Poincarémaps andmaximal
exponents of Lyapunov have been developed.

Key words: nonlinear dynamics, non ideal system, energy transfer, chaos

1. Introduction

Depending on whether excitation is influenced or not by the response of a
system, the vibrating system may be called ideal or non-ideal. When the for-
cing is independent of the system it acts on, then it is called ideal. The ideal
problems are the traditional ones. Formally, the excitation is expressed as a
pure function of time, for example by a sinusoidal excitation. In this case, the
excitation is independent of the system response. On the other hand, when
the forcing function depends on the response of the system, it is said to be



120 D. Sado, M. Kot

non-ideal.Whenweuse anon-ideal source of power instead of an ideal one, the
excitation should be presented as a functionwhich depends on the response of
the system. In this case, the non-ideal source of power can not be expressed as
a pure function of time but as an equation that relates the source of energy to
the system. Then the effect of energy supply is described by another different
equation, increasing the number of degrees of freedom. A non-ideal source of
power is for example a DCmotor with an unbalanced mass.

The first detailed study on non-ideal vibrating systems is amonograph by
Kononienko (1969). He obtained satisfactory results through the comparison
of numerical analysis and approximated methods. According to Kononienko
(1969), characteristics of an oscillatory system become dependent on the pro-
perties of the energy source. After that publication, the problem of non-ideal
vibrating systemshasbeen investigated byanumberof authors. Simulations of
similarmodelswere describedbyGiergiel (1990). Thenon-ideal problemswere
presented by Evan-Ivanowski (1976) or Nayfeh and Mook (1979). These au-
thors showed that, sometimes, the dynamical coupling between energy sources
and the structural responsemust not be ignored in real engineering problems.
A complete review on different theories on non-ideal vibrating systems were
discussed and presented by Balthazar et al. (2003). Non-ideal models were
researched by Krasnopolskaja and Shvets (1987). In Belato et al. (1999), the
authors studied a non-ideal similar system consisting of a pendulum whose
support point vibrated along a horizontal guide by a two-bar linkage driven
from a limited power DCmotor. Vibrations of ideal and non-ideal parametri-
cal and self-excited models were described by Pu̇st (1995) and Warmiński et
al. (2001). Calvalca et al. (1999) studied a non-linearmodel for theLaval rotor
with an unlimited power source. A model for flexible slewing structures with
DC motors was investigated by Fenili et al. (2003). Possibilities of the exi-
stence of regular and irregular motion in non-ideal parametrical models were
presented byWarmiński (2001), Belato et al. (2001) or Tsuchida et al. (2003).
Sado andKot (2002, 2003) investigated a non-ideal autoparametrical system,
where the influence of linear damping on energy transfer between modes of
vibration was studied.

This paper illustrates results of numerical simulation of a non–ideal au-
toparametrical system with non-linear damping put on the main mass M
and on the pendulum. The present work shows that in this type of a non-
ideal system, one mode of vibrations may excite or damp another mode, and
near the resonance regions, except for multiperiodic and quasiperiodic vibra-
tions, chaotic motion may appear as well. To prove the chaotic character of
this vibration, bifurcation diagrams for different damping parameters, time
histories, power spectral densities (using FFT), Poincaré maps and Lyapunov
exponents are developed. These descriptors are devoted to observe chaos, and
to better understand it (Moon, 1987; Baker and Gollub, 1996). Due to non-



Chaotic vibration of an autoparametrical system... 121

linearities andacouplednature of the equations ofmotion, numerical solutions
are used.

2. A model of an autoparametrical system with a non ideal

source of power

The investigatedmodel of an autoparametrical two-degrees-of-freedom system
with a non-ideal source of power is shown in Fig.1. The system consists of a
pendulumandabodyofmass M suspendedonaflexible element characterized
by a linear elasticity k and a non-linear viscous damping. The pendulumwith
a weightless rod of the length l and a lumped mass m is mounted to the
body M. It is assumed that the non-linear viscous damping force applied to
the hinge opposesmotion of the pendulum.The body ofmass M is subjected
to an excitation by an electric motor with an unbalanced mass m0. In this
case, this DCmotor is the non-ideal source of power.

Fig. 1. A non-ideal model of an autoparametrical system

The non-ideal source of power adds one degree of freedom, thus the sys-
temhas threedegrees of freedom.Thegeneralized co-ordinates are: thevertical
displacement y of the mainmass M, the angular displacement α of the pen-
dulummeasured from the vertical line, and the co-ordinate ϕwhich describes
the angular displacement of the unbalanced mass m0 measured from the ho-
rizontal line.

It is assumed that the elasticity force is S(y)= k(y+yst), where yst is the
static vertical displacement which can be found from the rel;ation: (M+m+
m0)g = kyst, where g is the acceleration of gravity and k is the coefficient
of elasticity. Also is assumed that the damping force acting on the body M
is Q1(ẏ) =C1ẏ+C3ẏ

3, while the resistant moment acting on the pendulum



122 D. Sado, M. Kot

is Md(α̇) = C2α̇+C4α̇
3, where C1, C2, C3, C4 are constant coefficients of

damping.

Vibrations have been researched around the static point of balance and
the equations ofmotion have been derived fromLagranges formula. The kine-
tic (T) and potential energy (V ) of the system are

T =
1

2
(I0+m0r

2)ϕ̇2+
1

2
(M+m+m0)ẏ

2+
1

2
ml2α̇2+

−m0rẏϕ̇cosϕ−mlẏα̇sinα
(2.1)

V =
1

2
k(y+yst)

2+m0g(r sinϕ−y)+mg(l− lcosα−y)−Mgy

The equations of motion of the system take the following form

(M+m+m0)ÿ−mlα̇
2cosα−mlα̈sinϕ+m0rϕ̇

2 sinϕ+

−m0rϕ̈cosϕ+ky+C1ẏ+C3ẏ
3 =0

ml2α̈+C2α̇+C4α̇
3
−mÿl sinα+mglsinα=0 (2.2)

(I+m0r
2)ϕ̈−m0ÿrcosϕ+m0grcosϕ=L(ϕ̇)−H(ϕ̇)

where L(ϕ̇) is the driving torque of theDCmotor and H(ϕ̇) is the resistance
torque.

The following dimensionless definitions have been introduced in Eqs (2.2)

τ =ω1t y1 =
y

yst
ω21 =

k

M+m+m0

ω22 =
g

l
β=
ω1
ω2

a1 =
m0r

(M+m+m0)yst

a2 =
m

M+m+m0
γ2 =

C2
ml2ω1

γ1 =
C1

(M+m+m0)ω1

γ3 =
C3l
2ω1

M+m+m0
γ4 =

C4ω1
ml2

q=
m0ryst
I0+m0r

2

G(ϕ̇)=L(ϕ̇)−H(ϕ̇) G1(ϕ̇)=
G(ϕ̇)

(I0+m0r
2)ω21

(2.3)
The characteristic curves G1(ϕ̇) of the energy source (DCmotor) are assumed
to be straight lines: G1(ϕ̇) = u1−u2ϕ̇, where the parameter u1 is related to
the voltage, and u2 is a constant parameter for each model of the motor
considered. The voltage is the control parameter of the problem.



Chaotic vibration of an autoparametrical system... 123

After transformations, the equations of motion can be written in the form

α̈= [(a1β
2ϕ̇2 sinϕ−a2α̇

2cosα+β2y1+β
2γ1ẏ1+β

6γ3ẏ
3
1)sinα+

−(qa1cos
2ϕ−1)(β2 sinα+γ2α̇+γ4α̇

3)+

−a1β
2cosϕcosα(u1−u2ϕ̇−qcosϕ)]

1

a2 sin
2α+a1qcos2ϕ−1

ϕ̈=
[(

a1ϕ̇
2 sinϕ−a2

1

β2
α̇2cosα+y1+γ1ẏ1+β

4γ3ẏ
3
1

)

qcosϕ+

+a2qcosϕsinα
(

sinα+
1

β2
γ2α̇+

1

β2
γ4α̇
3
)

+ (2.4)

+(u1−u2ϕ̇−qcosϕ)(a2 sin
2α−1)

] 1

a2 sin
2α+a1qcos2ϕ−1

ÿ1 =
[

a1ϕ̇
2 sinϕ−a2

1

β2
α̇2 cosα+y1+γ1ẏ1+β

4γ3ẏ
3
1 +

+
(

sinα+
1

β2
γ2α̇+

1

β2
γ4α̇
3
)

a2 sinα+

−(u1−u2ϕ̇−qcosϕ)a1 cosϕ
] 1

a2 sin
2α+a1qcos2ϕ−1

3. Numerical simulation results

The equations of motion have been solved numerically by the Runge-Kutta
method with a variable step length. Calculations have been done for different
values of the systemparameters and for the followingparameters of the engine:
u1 =0.2 tou1 =4,u2 =1.5 (forDCmotor),where u1 is the control parameter
which depends on voltage and the parameter u2 which depends on the type of
energy source. The calculations incorporated the following initial conditions:
ϕ̇(0) = 1, α(0) = 0.005◦, ϕ(0) = y(0) = ẏ(0) = α̇(0) = 0 and the parameters
β=0.5, q=0.2.

The resonant curves for the body of mass M and for the pendulumwith
damping put on the mass M for the conditions of the autoparametric main
internal resonance are shown in Fig.2. There are three peak amplitudes, re-
sulting from strong interactions between themass M and the pendulum.The
firstpeak is for the control parameter u1 =0.78, the secondone for u1 =1.465
and the third for u1 =1.565.

Near the internal andexternal resonances, dependingon the selection ofpa-
rameters of a physical system, the amplitudes of vibrations of the coupled sys-
temmaybe related differently. The systempresents some interesting nonlinear
phenomena.Motions y1 and α are periodic, multi-periodic or quasi-periodic,



124 D. Sado, M. Kot

Fig. 2. Amplitudes α
max
(a) and y1max (b) versus the control parameter u1 for:

u2 =1.5, a1 =0.001, a2 =0.1, q=0.2, β=0.5, γ1 =0.01, γ2 = γ3 = γ4 =0

but sometimesmotions of themass M and the pendulumare chaotic. For cha-
racterizing irregular chaotic response forms ang transition zones between one
and another type of regular steady resonant motion, bifurcation diagrams are
developed. These phenomena can be more easily observed in terms of displa-
cement, sometimes velocities, so diagrams are presented for both. Exemplary
results for small dampingputon themass M near the internal resonance (near
the principal autoparametric resonance for β = 0.5) versus the control para-
meter u1 for displacements y1 and α are shown in Fig.3, and for velocities
dϕ/dτ in Fig.4.

Fig. 3. Bifurcation diagrams for y1 and α for: u2 =1.5, q=0.2, β=0.5, a1 =0.01,
a2 =0.3, γ1 =0.001, γ2 = γ3 = γ4 =0

As we can see in diagrams presented in Fig.3, motion of the mass M
and the pendulum have different characters: may be periodic, multi-periodic,
quasi-periodic or irregular. Next, segments of the bifurcation diagrams in a
tensioned scale are presented. In Fig.4 diagrams corresponding to y1, α and
dϕ/dτ for u1 ∈ (1.49,1.51) are given. As can be seen in Fig.4, velocity of DC
motor has different values.

As it can be seen from these bifurcation diagrams, several phenomena
can be observed: the existence of simple or chaotic atractors, and various
bifurcations. All these phenomena have to be verified in the phase space. So
next the time histories, power spectral densities (their fast Fourier transforms
– FFT), Poincaré maps and the largest Lyapunov exponents corresponding to



Chaotic vibration of an autoparametrical system... 125

Fig. 4. Bifurcation diagrams for y1, α and dϕ/dτ in the region u1 ∈ (1.49,1.51) for:
u2 =1.5, q=0.2, β=0.5, a1 =0.01, a2 =0.3, γ1 =0.001, γ2 = γ3 = γ4 =0

the coordinates y1 and α have been determined. Exemplary results for the
control parameter u1 =1.499 are presented in Fig.5, for u1 =1.504 in Fig.6
and for u1 =1.72 in Fig.7.

As can be seen from these diagrams, the responses for presented values
of the control parameter u1 are chaotic (the motions look like irregular, the
frequency spectra are continuous, the Poincaré’s maps trace strange atrac-
tors and the largest Lyapunov exponents corresponding to the coordinates y1
and α are positive. As can be seen in Fig.7, in this non-ideal system after
a long time a jump in the amplitudes is sometimes possible, so we should
investigate these problems in a larger period of time.

4. Conclusions

This work is concerned with the problem of nonlinear dynamical motion of
a non-ideal vibrating system with autoparametric coupling. Several intere-
sting phenomena have been presented. The influence of linear and non-linear
damping parameters on the energy transfer cycle has been observed. The be-
haviour of the system near the internal and external resonance frequencies is
very important. Depending on the selection of physical system parameters,
the amplitudes of vibrations of coupled bodies may be related differently. It
has been shown that the examined system exhibits very rich non-linear dyna-



126 D. Sado, M. Kot

Fig. 5. Time histories, power spectral densities, Poincaré’smaps andmaximum
Lyapunov exponents corresponding to coordinates y1 and α for the control
parameter u1 =1.499 and for: u2 =1.5, q=0.2, β=0.5, a1 =0.01, a2 =0.3,

γ1 =0.001, γ2 = γ3 = γ4 =0



Chaotic vibration of an autoparametrical system... 127

Fig. 6. Time histories, Power spectral densities, Poincaré’smaps andmaximum
Lyapunov exponents corresponding to coordinates y1 and α for the control
parameter u1 =1.504 and for: u2 =1.5, q=0.2, β=0.5, a1 =0.01, a2 =0.3,

γ1 =0.001, γ2 = γ3 = γ4 =0



128 D. Sado, M. Kot

Fig. 7. Time histories, Power spectral densities, Poincaré’smaps andmaximum
Lyapunov exponents corresponding to coordinate α for the control parameter
u1 =1.72 and for: u2 =1.5, q=0.2, β=0.5, a1 =0.01, a2 =0.3, γ1 =0.001,

γ2 = γ3 = γ4 =0



Chaotic vibration of an autoparametrical system... 129

mics. Except for different kinds of periodic vibrations, also different kinds of
irregular vibrations have been found.The bifurcation diagrams for the control
parameter u1, which is related to the voltage of the DC motor, showed for
weaker dampingmany sudden qualitative changes, that is, many bifurcations
in the chaotic atractors as well as in the periodic orbits. For every value of
the control parameter u1, these phenomena were verified in the phase space.
The time histories, power spectral densities, Poincaré’s maps and maximum
Lyapunov exponents corresponding to coordinates of the system indicated a
possibility of the onset of chaos. It has been shown that after a long time
jumps in the amplitudes are possible. So this kind of vibrating systems sho-
uld be investigated in an adequately long time to be sure that the results are
correct.
In the future, we are going to continue the research using a non-linear

characteristic of the source of power.

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130 D. Sado, M. Kot

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parametrycznego układu z nieidealnym źródłem energii, X Francusko-Polskie
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Drgania chaotyczne autoparametrycznego układu z nieidealnym źródłem

energii

Streszczenie

W pracy uwzględniono wzajemne oddziaływania autoparametrycznego układu
drgającego o dwóch stopniach swobody i układu wymuszającego, którym jest silnik
elektryczny z niewyważonąmasą o znanej charakterystyce.Układpodstawowyskłada
się z wahadła o długości l i masie m podwieszonego do ciała o masie M zawieszo-
nego na elemencie sprężystym.Uwzględniając nieidealne źródło energii dodaje się do
badanego układu dodatkowy stopień swobody, bada sięwięc układ o trzech stopniach
swobody, ale czas nie występuje w równaniach w postaci jawnej. Równania ruchu



Chaotic vibration of an autoparametrical system... 131

rozwiązywanonumerycznie i badano drganiaw pobliżu rezonansuwewnętrznego i re-
zonansu zewnętrznego. W tym zakresie parametrów oprócz różnego rodzaju drgań
regularnychmogąwystąpić również drgania chaotyczne. Charakter drgań nieregular-
nychweryfikowanoanalizując diagramybifurkacyjne, przebiegi czasowe, transformaty
Fouriera, mapy Poincaré orazmaksymalne wykładniki Lapunowa.

Manuscript received August 3, 2006; accepted for print August 18, 2006