Chaotic Motions of a Duffing Oscillator Subjected to Combined
Parametric and Quasiperiodic ExcitationChin An Tan1 and Bongsu
Kang21 Department of Mechanical Engineering, Wayne State
University, Detroit, Michigan 48202, USA, Email:
[email protected] 2 Engineering Department, Indiana
University-Purdue University at Fort Wayne, Fort Wayne, Indiana
46825, USA, Email: [email protected] (Received 28 July 2000,
Revised 27 November 2000) AbstractThe forced response of a
Mathieu-Duffing oscillator subjected to a two-frequency
quasiperiodic excitation is examined in the context when the ratio
of the excitation frequencies is large. Numerical results are
obtained by the spectral balance method and compared with those
predicted by direct numerical integrations. Characteristics of the
response as a frequency parameter is tuned are investigated in
terms of the time histories, frequency spectra, Poincar sections
and Lyapunov exponents. It is observed that routes to chaotic
motions are different for frequency ranges near the natural
frequency of the linear system and near the parametric resonance
frequency. It is also shown that the contribution of the small
frequency component is important in the prediction of chaotic
motions.
Keywords: parametric resonance, quasiperiodic, chaotic motion,
Mathieu-Duffing oscillator 1. IntroductionThe Mathieu-Duffing
oscillator is the simplest model prototypical of the dynamical
behaviour of many complex structural systems that are
parametrically excited. In this paper, the dynamic response of a
Mathieu-Duffing oscillator under a two-frequency quasiperiodic
excitation is investigated numerically. In particular, we are
interested in systems where one excitation frequency is much
smaller than the other one. Our research is motivated by a recent
study on the dynamic instability of an automotive disc brake pad
system[1,2]. Consider an illustrative model of a disc brake pad
excited by a rotating and vibrating disc, as shown in Fig. 1(a).
The rotation frequency of the disc is much smaller than the
fundamental frequency of the transverse vibration of the disc (the
ratio is about 1/200 or smaller). In [2], it was shown that a
one-mode Galerkin approximation of the equation of motion of the
brake pad leads to a Duffing oscillator that is parametrically and
quasiperiodically excited. Here, the parametric excitation is due
to the nonconservative, follower-type friction force due to the
contact between the disc (rotor) and pads during braking. The disc
excitation impacted onto the pad is modeled as a travelling wave
that is vibrating at some modal frequency. Another example of a
quasiperiodically excited Mathieu-
kx M x
cx
brake pad model
cutting tool model
kx
cxky
ynonlinear contact mechanics
Mcy
xcutting force
friction force
1tsurface of spinning disc: x s = a cos 1t cos 2 t (a) brake pad
and disc model
csurface of moving workpiece: x s = b cos 1t cos 2 t
(b) machining tool and workpiece model
Figure 1. Schematics of two models under quasiperioidc and
parametric excitations.
Duffing oscillator is the cutting tool model, as shown in Fig.
1(b). The cutting force moving at a low frequency and the workpiece
vibrating at a relatively higher frequency excite the cutting tool.
Quasiperiodic systems are found in numerous engineering
applications such as nonlinear electrical circuits[3,4] and rotors
with piecewiselinear non-linearity[5]. In such systems, the ratios
of two or more excitation frequencies may be incommensurable and
the resulting steadystate motions are aperiodic. There are in
general two kinds of aperiodic systems: (1) an almost periodic
system with an infinite number of frequency spectra, and (2) a
quasiperiodic system with a finite number of frequency spectra.
Since the periods of quasiperiodic motions are usually very long,
thus making it difficult to obtain the complete characteristics of
the response, various computation approaches have been proposed for
the response and stability analyses[3-8]. These approaches are
generally based on the harmonic balance method and the fixed point
algorithm. Zounes and Rand[9] examined a quasiperiodic Mathieu
equation and compared the stability transition
curves obtained by regular perturbation and harmonic balance.
Their results show that the perturbation method fails to converge
in the neighbourhood of resonance due to small divisor terms while
the harmonic balance method does not have this deficiency.
Yagasaki[10,11] showed by simulations and experiments that the
response of a fixed-fixed beam excited by two frequencies can be
chaotic, through cascades of doubling bifurcations of the unstable
torus. It was also shown that chaotic motions might occur in both
the single- and multi-mode equations. Irregular motions are in
general undesirable; as they are difficult to predict and can
significantly increase the wear and reduce the durability and
reliability of machinery. Since the work by Lorenz[12] on
deterministic systems exhibiting aperiodic behaviour, chaotic
motions have been shown to occur in chemical reactions[13], simple
mechanical systems with piecewise-linear characteristics[8,14,15],
nonlinear continuous structures such as harmonically excited beams
with geometric non-linearities[16], buckled beams[17,18],
fluttering buckled beams[19], beam-mass structures[20], and surface
waves in a vertically forced channel of water[21], and a
quasi-periodically forced Duffing model[22]. Extensive
references on the bifurcations and chaos of physical systems are
well documented[23-25]. To date, the forced response of nonlinear
oscillators under combined parametric and quasiperiodic excitation
has not been reported in the literature. This manuscript is
organised as follows. The governing equation of motion and a
numerical solution method by the spectral balance method are first
described. Numerical results showing the basic characteristics of
the forced response in the vicinity of the primary and parametric
resonant frequency regions are presented and discussed. The
presence of chaotic motions is confirmed by calculating the
Lyapunov exponents. Routes to chaotic motions are also
examined.
3. Spectral Balance Solution MethodForced response solution of
(1) is obtained by applying the spectral balance method (SBM).
Introduce new time variables 1 = 1t and 2 = 2 t ,
(2)
where 0 i 2 (i = 1, 2) . time derivatives become
Accordingly, the (3a)
d = 1 + 2 , dt 1 2
2 2 2 d2 + 2 2 . (3b) = 1 2 + 21 2 1 2 2 1 dt 2
Re-write (1) in terms of the new variables as:122 2 x 2 x x 2 x
+ 21 2 + 2 + (1 2 2 1 2 1 1 2
2. Problem StatementThe models of Fig. 1 can be represented by
the general equation of the form&& + x + (1 + 1 cos1t cos 2
t + 2 cos1t sin 2 t ) x & x + 2 x + 3 x = f1 cos 1t cos 2t + f
2 cos1t sin 2 t2 3
x + 2 ) + 1 x + M ( x) + N ( x; 1 , 2 ) = F ( 1 , 2 ), 2
(4)
where,M ( x) 2 x 2 + 3 x 3 ,
(1)
(5a)
The above equation may be viewed as a onemode Galerkin
approximation of some structural models. The excitation is
modulated with two frequency components 1 and 2, where 1 0, the
corresponding orbit is chaotic. By ordering 1 2 L n , the
criterion 1 0 has been used to define the existence of
chaos[24]. The computational algorithm employed in this paper is
based on the work of Wolf et al.[26], and Eckmann and Ruelle[27].
Consider an initial time t0 when the trajectories of (12) originate
from an infinitesimal n-sphere with radius di(t0). As the system
evolves, these trajectories deform into an n-ellipsoid with
principal axes di(tN) at time step N. The Lyapunov exponents are
calculated asi = LimN
i =
1 p t
ln N ik ,k =1
p
(18)
d (t ) 1 ln i N . t N t0 d i (t0 )
where N ik denotes the norm of the denominator in (17) for the
i-th vector at the k-th time step. In this paper, the Fehlberg
order 4-5 RungeKutta method for non-stiff equations and the Gears
backward difference formula for stiff equations are employed to
construct the Poincar maps and to compute the Lyapunov
exponents.
(16)
Since chaotic trajectories are locally divergent, to ensure
accurate and efficient computations, the integrations of the
equations should be stopped before the values of di(t) become too
large. The numerical integrations are then resumed after a
re-definition of the initial conditions. This procedure repeats
until all Lyapunov exponents reach asymptotic values. The
afore-outlined procedure is implemented as follows. Let the
n-orthonormal initial vectors be y m (0) , (m = 1, K , n) .
Integrate the equations of motion over a small period of time t
such that none of y m (t ) becomes too large or diverges. This new
set of vectors is then orthonormalized by the Gram-Schmidt
procedure y1 = y2 = y1 (t ) , || y1 (t ) ||
5. Results and DiscussionThe numerical parameters chosen are: =
0.01, 1 = 1, 1 = 2 = 0.4, 2 = 0.3, 3 = 0.2, f1 = f2 = 0.5, 1 =
0.005. Note that the ratio 2/1 is about or more than 200 in the
neighbourhoods of the fundamental resonance ( 2 1 ) and parametric
resonance ( 2 2 ). The steady-state frequency response amplitude of
the system around 2 1 = 1 is plotted in Fig. 2. In the
computations, response amplitudes were obtained by taking the
maximum of the time history, where 20~30% of the time history
starting from t = 0 was discarded to eliminate the transient
response. From Fig. 2, three distinct regions are identified. It is
seen that the two approaches (SBM and numerical integration (NI))
give almost the same results in Region I where the motions are
either periodic or quasiperiodic. Figure 3 shows the system
response for the case 2 1 = 0.84 (Region I). The (almost) closed
orbit in the Poincar section and the dominant peaks with uniformly
spaced sidebands in the frequency spectrum, are both indicative of
the quasiperiodic nature of the response. Note that 1 ~ 1.2 10 4
data points are collected from the orbit at intervals of 2 (1 + 2 )
to construct the Poincar section. For comparison, the fixed point
attractor corresponding to the periodic motion with 1 neglected is
also shown in the Poincar section. It is clear that the negligence
of the contributions of a small frequency parameter could lead to
erroneous conclusion.
y 2 (t ) (y 2 (t ) y1 )y1 ,L, || y 2 (t ) (y 2 (t ) y1 )y1 ||y n
(t ) n 1
yn = || y n (t )
m =1 n 1 m =1
(y n (t ) y m )y m (y n (t ) y m )y m ||
,
(17)
|| || where denotes a vector norm. Subsequently, using each of y
m as new initial conditions for the equations, another round of
nintegrations over t gives a new set of y m (t ) , and a new set of
y m by applying (16). After repeating the same procedure p times,
the Lyapunov exponents are determined from
5.0 4.5 4.0 3.5 SBM NI
Amplitude, |x|
3.0 2.5 2.0 1.5 1.0 0.5 0.0 0.6 Region I Region II
Region III
0.8
1.0
1.2
1.4
1.6
1.8
2/1
Figure 2. Frequency response solutions obtained by the spectral
balance method (SBM) and numerical integration (NI).1.5 1 x 0.5 0 -
0.5 -1 - 1.5 45000 46000 47000 48000 49000 50000
development of an inflation point in the corresponding circle
map. The presence of the inflection point means that the inverse of
the map may be multi-valued, an indication of possible chaotic
motion. Note that in Region II, the steady state frequency response
amplitudes predicted by the SBM and NI become different. As the
frequency is further increased, the wrinkled torus becomes more
distorted until the torus breaks down at 2 1 = 1.065 . Here, the
Poincar map shows a phase locking on the broken torus[30]. Although
the motion appears to be complex as seen in Fig. 5, it is
quasiperiodic as shown by the (identical) detailed plots of the
time history and by the frequency spectrum where a number of weak
frequency components appear discretely. The occurrence of phase
locking on a broken torus before the emergence of chaos has been
observed in several other studies[31-33].
2 1 = 0.9Normalized power1.0 0.8 0.6 0.4 0.2 0.0 0.5 1.0 1.5
2.0
2 1 = 0.92
Frequency (radian/sec) Frequency
Figure 3. Time history, Poincar section, and frequency spectrum
at 2 1 = 0.84 . The blank dot denotes the fixed point solution of
the periodic orbit when 1 is neglected.
2 1 = 0.94
2 1 = 1.065
Figure 4. Poincar sections of orbits in Region II.
When the excitation frequency is increased, i.e., in Region II,
the orbit begins being distorted and this distortion results in the
formation of wrinkles on the orbit, see Fig. 4. Steinmetz and
studied a four-dimensional Larter[29] quasiperiodic system and
showed that the highly wrinkled torus is associated with the
Further increase of the frequency ratio leads to fully developed
chaotic motions (Region III); see Fig. 6 at 2 = 1.18 . While it is
difficult to observe chaotic behaviour from the time history, the
corresponding Poincar maps, frequency spectrum, and Lyapunov
exponents clearly show the chaotic nature of the response. The
largest
Lyapunov exponent converges to a nonzero positive value. In this
Region, the results from the SBM and NI show significant
discrepancy, with SBM producing multiple solution branches. It is
believed that the appearance of many solution branches using the
SBM is indicative of very complex quasiperiodic motions with
multiple, incommensurate frequencies or possible chaotic motions.
It should also be noted that frequency response amplitudes obtained
by the NI in Region III might not represent the classical results
since chaos is highly sensitive to initial conditions.2 1
attractor undergoes a cascade of period-doubling bifurcations,
see Fig. 9. This is a typical torusdoubling scenario with a
sequence of perioddoubling tori culminating in chaos. Note that in
the previous case when the excitation frequencies are near the
fundamental resonance, the two-period quasiperiodic motion evolves
to chaotic motion via a torus breakdown.2 1
x
0-
1 2 3 40000 42000 44000 46000 48000 50000
time
x
0-
1 2 3 45000 46000 47000 48000 49000 50000
time2 1 0-
1Normalized power
1 0.1 0.01 0.001
2 463002 1 0
46350
46400
46450
0.5
1.0
1.5
2.0
Frequency (radian/sec)
Frequency
-
1 2Lyapunov Exponents 0.01
48800
48850
48900
48950
49000
0.00
Normalized power
1.0 0.8 0.6 0.4 0.2 0.0 0.5 1.0 1.5 2.0
-0.01 0 2000 4000 Time 6000 8000 10000
Frequency (radian/sec) Frequency
Figure 6. Response time history, Poincar section, frequency
spectrum and Lyapunov exponents when 2 1 = 1.18 .
Figure 5. Time histories and frequency spectrum of the response
at 2 1 = 1.065 .
The steady-state frequency response amplitude of the system when
excitation frequencies are near the parametric resonance of 2 1 = 2
is plotted in Fig. 7. As shown in the frequency spectrum of Fig. 8,
at 2 1 = 1.878 , the response is clearly quasiperiodic. When the
frequency is further increased from 1.878 at a small increment of
0.0001, the two-torus
In Fig. 10, at 2 1 = 1.88 , a fully developed chaotic motion is
observed, with continuously distributed frequency components
emerging from the harmonics of both the excitation and natural
frequencies. One Lyapunov exponent converges to a nonzero positive
value. It is also found that this chaotic response can only be
sustained in a narrow frequency range 2 1 = 1.88 ~ 1.89 .
4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 1.8 SBM NI
Fig. 12 are observed. These irregular motions disappear when the
excitation frequency is away from the parametric resonance.
Amplitude, |x|
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2/1
Figure 7. Frequency response solutions.0.3 0.2
x
0.1 0- 0.1 - 0.2
45000
46000
47000
48000
49000
50000
timeNormalized power1.0 0.8 0.6 0.4 0.2 0.0 1.0 1.5 2.0 2.5
Frequency (radian/sec) Frequency
Figure 8. Time history and frequency spectrum of the response at
2 1 = 1.878 .
Numerical results were also obtained for the frequency range of
2 1 = 1.89 ~ 2.08 , and the sequence of Poincar sections in Fig. 11
shows a re-appearing of quasiperiodic responses. It is also
observed that the two closely located tori evolve into two separate
ones, which continue to be distorted as the frequency is increased.
A phase-locked motion at 2 1 = 2.08 indicates the evolution of the
response into a possible chaotic motion as the frequency parameter
is further increased. As shown in Fig. 12, the chaotic response is
fully developed at 2 1 = 2.09 , with the geometric structure of the
corresponding Poincar section resembling a strange attractor often
referred to as a Cantor set. The appearance of this highly
organized geometric structure is a strong indicator of chaotic
motions[24]. As the excitation frequency is further increased,
chaotic responses similar to
Figure 9. Poincar sections showing the transition from
quasiperiodic to chaotic motions via successive bifurcations at
various frequency ratios (from 1.8782 at top left to 1.8791 at
bottom right with 0.0001 increment).
Both theoretical and experimental studies have shown that
chaotic motions do not occur in any unique way[25]. However, in
general, there are three possible scenarios: period-doublings,
torus bifurcations, and intermittency mechanisms.
0.6 0.4
x
0.2 0- 0.2
40000
42000
44000
46000
48000
50000
time1
0.1 0.01 0.001
0.0
0.5
1.0
1.5
2.0
2.5
3.0
bifurcation, resulting in a stable periodic solution with a
fundamental frequency 1 . Further Hopf bifurcation produces a
second orbit with (incommensurate) frequency 2 . The resulting
two-period quasiperiodic motion is a two-torus. Each successive
Hopf bifurcation then produces a new torus around the original
torus. The process continues until the motion is chaotic. Both of
these mechanisms were observed in our numerical simulations. The
third and fourth cases, proposed by Landau[34]
Normalized power
Frequency (radian/sec)
Frequency
0.02
Lyapunov Exponents
0.01
0.00
-0.01
1.890 2000 4000 6000 8000 10000
1.96
-0.02
Time
Figure 10. A fully developed chaotic motion at 2 1 = 1.88 , as a
result of successive torus bifurcations.
1.90
1.97 2.0
In period-doublings, as a control parameter is varied, a
periodic motion with a fundamental frequency undergoes a sequence
of bifurcations or changes to another periodic motion with twice
the period of the previous oscillation. This process continues
until a critical value of the parameter is reached, beyond which
chaotic motions sustains. This cannot happen in our quasiperiodic
system. The torus bifurcations scenario consists of four different
cases. First, a two-period quasiperiodic attractor or torus (with
incommensurate
1.92
1.93 1.95
2.05 2.08
frequencies) can evolve into chaos through a torus break down.
This case is characterized by the fact that chaos occurs following
the appearance of a two-period quasiperiodic attractor, and with
post-bifurcation states such as phased-locked or mixed-mode
oscillations. The second case for a twoperiod quasiperiodic motion
evolving into chaos is via a torus doubling sequence. As the
control parameter is varied, a fixed-point solution loses its
stability through a supercritical Hopf
Figure 11. Poincar sections showing the evolution of the
quasiperiodic orbit toward chaotic motion at various frequency
ratios (as indicated in each map).
and Ruelle and Takens[35], respectively, involve successive
bifurcations from an equilibrium solution to a quasiperiodic
solution with n incommensurate frequencies and then to chaos. These
mechanisms were not observed. The intermittency mechanisms,
proposed by Pomeau and Manneville[36], refer to oscillations that
are periodic for certain time intervals and are then interrupted by
bursts of aperiodic oscillations of finite durations. After these
bursts diminish, a new periodic phase emerges, and so on. This was
also not observed in our numerical experiments.2 1
6. ConclusionsIn this paper, the forced response of a
Mathieu-Duffing equation is investigated numerically. The
excitation has two frequencies of which 1
