Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 3, pp. 779-786, Warsaw 2017
DOI: 10.15632/jtam-pl.55.3.779

CHAOTIC VIBRATION OF AN AUTOPARAMETRICAL SYSTEM WITH

THE SPHERICAL PENDULUM1

Danuta Sado, Jan Freundlich, Anna Bobrowska

Warsaw University of Technology, Institute of Machine Design Fundamentals, Warsaw, Poland

e-mail: dsado@poczta.onet.pl

In the paper, the dynamics of a three degree of freedom vibratory system with a sphe-
rical pendulum in the neighbourhood of internal and external resonance is considered. It
has been assumed that the spherical pendulum is suspended to the main body which is
then suspended to the element characterized by some elasticity and damping. The system
is excited harmonically in the vertical direction. The equation of motion has been solved
numerically.The influence of initial conditions on the behaviour of the spherical pendulum is
investigated. In this type of the system, onemode of vibrationmay excite or damp another
one, and for different kinds of periodic vibrations there may also appear chaotic vibrations.
For characterization of an irregular chaotic response, time histories, bifurcation diagrams,
power spectral densities, Poincaré maps and the maximum Lyapunov exponents have been
calculated.

Keywords: spherical pendulum, energy transfer, coupled oscillators, chaos

1. Introduction

The subject of this work is investigation of the effect of initial conditions on the dynamics
of a three degree of freedom system with a spherical pendulum. Dynamical systems having
an element in the form of a mathematical or physical pendulum have important applications.
Different kinds of coupled autoparametric oscillators with simple pendulums are presented in
the book by Sado (2010). Furthermore, the numerical and analytical methods of researching the
dynamics of two degree and three degree of freedom vibratory systems with the pendulums are
presented in this book. There are shown: the influence of initial conditions on the behaviour of
the vibratory system, the characteristic of vibrations, and how to manipulate the vibrations.

The real pendulum is of spherical type. The spherical pendulumwas investigated by many
researches. The spherical pendulumsubjected to parametric excitationwas studied byMiles and
Zou (1993). They investigated the non-linear response of a slightly detuned spherical pendulum
with natural frequencies. In that model, no stable harmonic motion was possible, therefore the
motion of the system was either periodically modulated by a sinusoid or becomes chaotic. The
numerical integration was supported by analytical predictions, which also uncovered that the
limit cycles and chaotic motions overlaped those of stable harmonic motion.

In paper by Naprstek and Fischer (2009), the pendulum vibration damper modelled as a
two degree of freedom strongly non-linear auto-parametric system was investigated. There was
a kinematic external excitation applied to a point. In that model the excitation was considered
to be horizontal and harmonically variable in time. The solution and the stabilitywere analysed.
Therefore, the biggest attention was paid to the resonance domain. In certain domains of the
pendulum and excitation parameters, the semi-trivial solution did not exist in that domain

1
The paper was presented at the 27th Symposium on Vibrations in Physical Systems, May 9-13, Poznań-

-Będlewo, Poland.



780 D. Sado et al.

and various post-critical three-dimensional regimes occurred. Some of themwere non-stationary
despite theharmonic excitation.Threedifferent types of the resonancedomainwere investigated.
Their main properties depended on dynamic parameters of the pendulum and the external
excitation amplitude. An analytical and numerical study brought some recommendations for
designers of such devices. Their aim was to avoid any post-critical response rules risking the
pendulum functionality.

The bifurcation behaviour of a spherical pendulum where the suspension point is harmo-
nically excited in both vertical and horizontal directions was presented by Leung and Kung
(2006). The equations of motion for a lightly damped spherical pendulumwere considered. The
point of the suspensionwas harmonically excited in both vertical and horizontal directions. The
equations were solved with approximation in the neighbourhood of resonance by including the
third order terms in the amplitude.

A mathematical model of the spherical pendulum with a moving pivot was suggested by
Mitrev and Grigorov (2009) and later developed. Such a model allowed studying the influence
of different kinematic excitations applied to the pivot point. That required the kinematical
and dynamical parameters of the pendulum which also determined the force in the rope. A
numerical simulation for spatial curvilinear and planar with straight line motions trajectories
was performed.The stochastic analysis of a spring spherical pendulumwas done byViet (2015).
The vibration was reduced by a spring and damper installed in the radial direction between the
point mass and the cable. Under the sway motion, the centrifugal force resulted in the radial
motion, which in its turn produced the Coriolis force to reduce the sway motion.

The dynamics of coupled spherical pendulums (where two lower pendulums were mounted
at the end of the upper pendulum) was considered by Witkowski et al. (2014). The analysis
showed that three rotatingmodes existed. The linearmodes helped to understand the nonlinear
normal modes, which were later visualized in plots. When there was an increase of energy in
one mode, we could see a symmetry pitchfork bifurcation. In the second part of the paper, the
energy transfer between pendulumswas investigated. The results for co-rotating (all pendulums
rotated in the same direction) and counter-rotating motion (one of lower pendulum rotated in
the opposite direction) were presented.

In the present paper, it is assumed that the spherical pendulum is suspended on a flexible
element, thus in this system theremay occur an autoparametric excitation as a result of inertial
coupling.

2. A model of an autoparametric system with a spherical pendulum

The investigated system is shown inFig. 1. The system consists of a bodyofmassm1 suspended
to the flexible element. This flexible element is characterised by the stiffness k and damping c.
This system also consists of the spherical pendulum of the length l and mass m2 which is
suspended to the body of massm1. The body of massm1 is subjected to the harmonic vertical
excitation F1(t) and the spherical pendulum is subjected to the horizontal excitation F2(t).

The spherical pendulum is similar to the simple pendulum but it moves in 3D space, so
we need to introduce a new variable ϕ in order to describe rotation of the pendulum in the
planeXY . The position of the body ofmassm1 is described only by the coordinate z. However,
the position of the pendulum is described by the coordinate z and two angles: θ and ϕ. The
angle θ is the deflection of the pendulummeasured from the vertical line. This system has three
degrees of freedom. The equations of motion are derived as Lagrange’s equations.

The kinetic energy T is the sum of the energy of the two bodies

T =
m1v

2
1

2
+
m2v

2
2

2
=
m1ż

2
1

2
+
m2(ẋ

2
2+ ẏ

2
2 + ż

2
2)

2
(2.1)



Chaotic vibration of an autoparametrical system... 781

Fig. 1. Amodel of an autoparametric systemwith a spherical pendulum

where

x2 = lsinθcosϕ y2 = lsinθsinϕ z2 = z1+ lcosθ

z1 = z+zst zst =
(m1+m2)g

k

(2.2)

The kinetic energy T is given by the expression

T =
1

2
(m1+m2)ż

2+
1

2
m2(lθ̇

2+ l2ϕ̇2 sin2θ−2lżθ̇sinθ) (2.3)

The potential energy V is given by the expression

V =−(m1+m2)g(z+zst)+m2g(l− lcosθ)+
k(z+zst)

2

2
(2.4)

Assuming that the exciting forces are in the form: F1(t) =P1cos(ν1t), F2(t) =P2cos(ν2t), the
dissipation functionD= cż2/2, the equations of motion of the system are as follows

(m1+m2)z̈−m2lθ̈ sinθ−m2lθ̇
2cosθ+kz+ cż=P1cos(ν1t)

m2l
2θ̈−m2lz̈ sinθ−m2l

2ϕ̇2 sinθcosθ+m2glsinθ= lcosθsinϕP2 cos(ν2t)

m2l
2ϕ̈sin2θ+2m2l

2ϕ̇θ̇sinθcosθ= lsinθcosθP2cos(ν2t)

(2.5)

By introducing the dimensionless time and parameters

τ =ω1t ω
2
1 =

k

m1+m2
ω22 =

g

l
β=
ω2
ω1

γ =
c

(m1+m2)ω1
z̄=
z

l
a=

m2
m1+m2

A1 =
P1

(m1+m2)ω
2
1

A2 =
P2
m2lω

2
1

µ1 =
ν1
ω1

µ2 =
ν2
ω1

(2.6)

after transformation (2.5) into the dimensionless formwe obtain

z̈−aθ̈sinθ− θ̇2cosθ+z+γż=A1cos(µ1τ)

θ̈− z̈ sinθ− ϕ̇2 sinθcosθ+β2 sinθ=A2cosθ sinϕcos(µ2τ)

ϕ̈sinθ+2ϕ̇θ̇cosθ=A2cosθcos(µ2τ)

(2.7)

where the overbars denoting non-dimensionalisation are omitted for convenience.



782 D. Sado et al.

After transformation, the equations ofmotion can bewritten in a simpler form for numerical
calculations:

z̈=
1

1−asin2θ
[(A1cos(µ1τ)+ θ̇

2cosθ−z−γż)+asinθ(ϕ̇2 sinθcosθ−β2 sinθ)]

θ̈=
1

1−asin2θ

{

ϕ̇2 sinθcosθ−β2 sinθ

+[A1cos(µ1τ)+aθ̇
2cosθ−z−γż+asinθ(ϕ̇2 sinθcosθ−β2 sinθ)]sinθ

}

ϕ̈=
1

sin2θ
(A2 sinθcosϕcos(µ2τ)−2ϕ̇θ̇sinθcosθ)

(2.8)

3. Numerical simulation results

The equations of motion of the given model are solved numerically using Runge-Kutta method
with a variable step length. The calculations are carried out for different values of parameters
of the system and for different initial conditions. Exemplary time histories of the displacements
z and θ are obtained for parameters of the system a=0.8, β=0.5, γ=0,A1 =A2 =0 and for
the initial conditions: z(0)= 0.1, θ(0)= 0.005◦, ϕ(0)= 0, ż(0)= θ̇(0)= ϕ̇(0)= 0 are presented
in Fig. 2, where we can observe the energy transfer between the modes of vibration in a closed
cycle. In this case, the spherical pendulumbehaves in the sameway as a simple pendulum, and
motion of the pendulum is in the vertical plane (angleϕ is constant and it depends on the initial
conditions, so it is equal to 0 for ϕ(0)= 0).

Fig. 2. Time history for (a) oscillator, (b) pendulum for a=0.8, β=0.5, γ=0,A1 =A2 =0,
z(0)= 0.1, θ(0)=0.005◦,ϕ(0)= 0

Fig. 3. Internal resonance for: a=0.8, γ=0,A1 =A2 =0, z(0)=0.1, θ(0)= 0.005
◦, ϕ(0)=0

The diagram of internal resonance for the same initial conditions put on the displacements
is presented in Fig. 3, and it is similar to a simple pendulumpresented in thework (Sado, 2010).



Chaotic vibration of an autoparametrical system... 783

We observe the resonance excitation for the frequency ratio β = 0.5. In this case, there is an
assumption that the simple pendulum results are good.
On the other hand, when the initial conditions are put on the displacements and also on the

velocities (in this example: z(0)= 0, ż(0)= 0, θ(0)= 5◦, θ̇(0)=−0.04,ϕ(0)= 0, ϕ̇(0)=−0.96),
we can observe the influence of angle ϕ (Fig. 4). For these initial conditions, we observe the
oscillation displacements z and θ with the same frequency.

Fig. 4. Time history for: a=0.5, β=0.51, γ=0,A1 =A2 =0, z(0)= 0, ż(0)= 0, θ(0)=5
◦,

θ̇(0)=−0.04,ϕ(0)= 0, ϕ̇(0)=−0.96

With the same initial conditions for displacements and velocities, the internal resonance is
observed for the frequency ratio β=0.51 (shown in Fig. 5a).

Fig. 5. Internal resonance for: (a) a=0.5, γ=0,A1 =A2 =0, z(0)= 0, ż(0)= 0, θ(0)=5
◦,

θ̇(0)=−0.04,ϕ(0)= 0, ϕ̇(0)=−0.96; (b) a=0.2, γ=0,A1 =A2 =0, z(0)= 0, ż(0)=0.65,
θ(0)=50◦, θ̇(0)=−0.04,ϕ(0)=0, ϕ̇(0)=−0.296

For different parameters of the system: a=0.2, γ=0,A1 =A2 =0, β=0.75 and for initial
conditions which are put on the displacements and also on the velocities (z(0)= 0, ż(0)= 0.65,
θ(0)= 50◦, θ̇(0) =−0.04, ϕ(0) = 0, ϕ̇(0)=−0.296) there is visible an influence of the angle ϕ,
and the internal resonance area is observed for the frequency ratio near β = 0.75 (Fig. 5b). In
this example, the angle ϕ describes rotation of the pendulum around the axis z.



784 D. Sado et al.

Assuming themodel of thependulumasa spherical pendulum,weobtain resultsmore similar
to the real system.

Next we prepare diagrams of the external resonance. An example is considered when the
initial conditions are imposed on the displacements: a = 0.8, β = 0.5, γ = 0, A1 = 0.001,
A2 = 0, z(0) = 0, θ(0) = 0.005

◦, ϕ(0) = 0. In that case, the diagram shows the external
resonance for both the coordinate z and the angle θ with the vertical excitation force F1(t).

Fig. 6. External resonance for the coordinates z and θ for: a=0.2, β=0.5, γ=0,A1 =0.0001,
z(0)=0, γ=0

Near the internal and external resonances depending on the selection of physical parameters,
the amplitudes of vibrations of the coupled system may be related differently. The motion in
terms of z and θ are periodic or quasi-periodic vibrations, but sometimes the motion of the
body of mass m1 and the pendulum are chaotic. To characterize an irregular chaotic response,
bifurcation diagrams are constructed.

Exemplary bifurcation diagrams for the displacements z and θ versus bifurcation parame-
terA1 with parameters of the system: a=0.2, β=0.5, γ =0.00081443, µ=0.99 are presented
in Fig. 7 (it is assumed that the bifurcation parameter is the amplitude of excitation A1).

Fig. 7. Bifurcation diagrams for the coordinates z and θ for: a=0.2, β=0.5, γ=0.00081443,µ1 =0.99

As it can be seen from the bifurcation diagrams, motion of the oscillator and the spherical
pendulumdepends of the amplitude of excitationA1 and can be of different character: itmaybe
periodic, quasi-periodic or chaotic. Several phenomena canbe observed: the existence of a simple
or a chaotic atractor, and various bifurcations. All these phenomena have to be verified in
the phase space. Therefore, the Poincaré maps and Lyapunov exponents are then constructed.
Exemplary results of thePoincaré maps (Fig. 8) and themaximumLyapunov exponents (Fig. 9)
are presented for the amplitude of excitation A1 =0.0029.

It is visible that the Poincaré maps trace strange atractors and the maximum Lyapunov
exponents are positive, so the motions with respect both coordinates are chaotic.



Chaotic vibration of an autoparametrical system... 785

Fig. 8. Poincaré maps for the coordinates z (a) and θ (b) for a=0.2, β=0.5, γ=0.00081443,
µ1 =0.99,A1 =0.0029

Fig. 9. MaximumLyapunov exponent for the coordinates z (a) and θ (b) for a=0.2, β=0.5,
γ=0.00081443,µ1 =0.99,A1 =0.0029

4. Conclusions

The influence of initial conditions on the behaviour of an autoparametric systemwith a spherical
pendulum is very interesting, because sometimes when the initial conditions are imposed on the
displacements of the spherical pendulum it behaves similarly to a simple pendulum (angle ϕ
is constant), but when the initial conditions are put also on the velocities, we observe some
influence of the angle ϕ. It is important, because near the internal and external resonance
area there may appear different motion – regular or chaotic. Autoparametric systems are very
sensitive to nonlinearities and the energy is transferred betweenmodes of vibrations in a closed
cycle. The time of this cycle depends on the values of parameters, and numerical calculations
should be carried for a satisfactorily long time. Assuming the model of the pendulum as the
spherical pendulum, the obtained results aremore similar to those obtained for the systemwith
the simple pendulum.

References

1. Leung A.Y.T., Kuang J.L., 2006, On the chaotic dynamics of a spherical pendulum with a
harmonically vibrating suspension,Nonlinear Dynamics, 43, 213-238

2. Miles J.W., ZouQ.P., 1993, Parametric excitation of a detuned spherical pendulum, Journal of
Sound and Vibration, 164, 2, 237-250



786 D. Sado et al.

3. Mitrov R., Grigorov B., 2009, Dynamic behaviour of a spherical pendulum with spatially
moving pivot, Zeszyty Naukowe Politechniki Poznańskiej, 9, 81-91

4. Naprstek J., Fischer C., 2009, Auto-parametric semi-trival and post-critical response of a
spherical pendulum damper,Computers and Structures, 87, 1204-1215

5. Sado D., 2010,Regular and Chaotic Vibration in Selected Pendulum Systems (in Polish), WNT,
Warsaw

6. Viet L.D., 2015, Partial stochastic linearization of a spherical pendulum with Coriolis dam-
ping produced by radial spring and damper, Journal Vibration and Acoustics, 137, 5, 1-9, DOI:
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7. Witkowski B., Perlikowski P., Prasad A., Kapitaniak T., 2014, The dynamics of co- and
counter rotating coupled spherical pendula, The European Physical Journal Special Topics, 223,
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Manuscript received September 20, 2016; accepted for print January 16, 2017