Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

46, 3, pp. 511-520, Warsaw 2008

BIFURCATIONS IN AN ELECTRO-VIBROIMPACT SYSTEM

WITH FRICTION

Jee-Hou Ho

Ko-Choong Woo

The University of Nottingham, Malaysia

e-mail: jeehou.ho@nottingham.edu.my; woo.ko-choong@nottingham.edu.my

Bifurcation phenomena of an electro-vibroimpact system have been in-
vestigated by means of numerical analysis. It has been shown that the
system undergoes transition from chaotic motion to periodic motion as
the control frequency of the solid state relay (one of the system para-
meters) varies. A close co-relationship with an experimental bifurcation
diagramhas been observed. Periodicmotion has been identified to yield
better system performance over chaotic motion. The foundation of im-
plementing an optimal feedback control strategy is established.

Key words: bifurcation, electro-vibroimpact system, numerical analysis

1. Introduction

Studies on vibro-impact systems have revealed very rich system dynamics
due to the presence of nonlinearities in the system characteristics (Hinrichs
et al., 1997; Pavlovskaia andWiercigroch, 2003; Peterka, 1996). Construction
of Poincaré maps, bifurcation diagrams and basins of attraction are useful to
understand the qualitative dynamics of the system.

The considered electro-vibroimpact system is a discontinuous system, both
from a mathematical and physical point of view. A detailed approach to de-
scribe and solve dynamical systems with motion dependent discontinuities
was undertaken by Wiercigroch (2000). An important result from that piece
of work was the clarification of accurate mathematical modelling of such sys-
tems and the numerical realisation of the analytical solution. While it may
be attractive to assume a solution a priori be found numerically (Woo et al.,



512 J.-H. Ho, K.-C. Woo

2000), considerable amounts of time are required to program and implement
formulti-degree-of-freedom systems.An alternativemethod to understand the
system dynamics is to perform numerical integration as per Pavlovskaia et al.
(2003). A form of the analytical solution may be obtained from consideration
of the coefficient of restitution. Lenci andRega (2003) treated this problem for
a simple inverted impacting problem. Formulation of analytical solutions to
systemsdescribingvibro-impactmachinerywas achieved byLuo et al. (2006b).
That analysis has also facilitated the description of bifurcation.

A large variety of dynamic responses is known to exist for nonlinear di-
scontinuous systems. For example, systems exhibiting dry friction are known
to behave in a chaotic manner, as demonstrated by Stefanski et al. (2003). In
relation to that, the bifurcational phenomenon for impact systems was scruti-
nised byLuo et al. (2006a,c) and Luo andXie (1998, 2004). Especially when a
very interesting phenomenon such as intermittent chaos is reported for impact
systems (Blazejczyk et al., 1994), the associated loss in stability, and possi-
bility of chaos allows for the realisation of an ”optimal-harmonic” feedback
implementation. This was demonstrated by Lenci and Rega (2000).

Bifurcation analysis involves the study of the change in system topology
under the influence of a system parameter. It was originally used by Poin-
caré (1885) to describe the ”splitting” of equilibrium solutions in a family
of differential equations. Bifurcations of equilibria usually produce changes in
the topological type of a flow (Guckenheimer and Holmes, 1983). Blazejczyk-
Okolewska and Kapitaniak (1998) identified co-existing attractors in a me-
chanical system with impacts by means of bifurcation diagrams and basins
of attraction, and concluded that basins of some attractors are so small that
random noise prevents trajectories from reaching them.

This paper presents a flavour of bifurcation phenomena from numerical
analysis of a new electro-vibroimpact system. The system involves a solenoid
driven by a RLC circuit, coupled with a solid state relay, to generate large
electro-magnetic forces acting on ametal bar which oscillates within the sole-
noid. Impacts are generated by placing a stop in the path of bar oscillations.
The system was experimentally studied by Nguyen et al. (2007). Ho et al.
(2008) performed numerical analysis of the system and revealed a variety of
dynamic responses, ranging from periodic to chaotic. The results established
a good correlation with the experimental data. To further confirm the quali-
tative responses of the system, a bifurcation diagramneeds to be constructed,
especially to compare with the experimental bifurcation diagram previously
observed by Nguyen andWoo (2007).



Bifurcations in an electro-vibroimpact system... 513

2. System description

This piece of work scrutinises the bifurcation scenarios of the analysed system.
The basis for the physical model of the electro-vibroimpact mechanism was
described in detail by Ho et al. (2008). Figure 1 shows a schematic of the
system, whereas the physical model of the system subject to friction of rails
is shown in Fig.2.

Fig. 1. Schematic diagram of the prototype of an electro-vibroimpact device
(Nguyen, 2007)

Fig. 2. Physical model of the electro-vibroimpact system (Ho et al., 2008)

The governing equations ofmotion for the system can be expressed as (Ho
et al., 2008)

u′ = v

v′ =
( 1

m1
+
1

m2

)[1

2
y2
∂L

∂u
−µ1m1g sgn(v)−k0(u−G)H(u−G)− cv+

−k1(u−X0)
]



514 J.-H. Ho, K.-C. Woo

w′ =x

x′ =
1

m2

[

−
1

2
y2
∂L

∂u
+µ1m1g sgn(v)+k0(u−G)H(u−G)+ cv+ (2.1)

+k1(u−X0)−µ2(m1+m2)g sgn(Ẋ2)
]

y′ = z

z′ =
1

L

[

ωPctrVscos(ωt)−
(

R+2
∂L

∂u
v
)

z−
(1

C
+
∂2L

∂u2
v2+

∂L

∂u
v′
)

y
]

where u is the relative displacement of the metal bar with respect to the
molingmechanism, v is the velocity of themetal bar with respect to the base
board, w is the displacement of the mole, x is the velocity of the mole, y is
the current, z is the first derivative of the current, m1 is themass of themetal
bar, m2 is themass of themole, µ1 andµ2 are frictional coefficients of the dry
friction forces Ff1 and Ff2, g is the acceleration of gravity, k0 is the stiffness
of the obstacle block, c is the damping coefficient, k1 is the spring stiffness,
G is the gap position, X0 is the initial displacement of themetal bar, ω is the
frequency of power supply, Vs is the voltage amplitude, R is the resistance,
C is the capacitance, L is the inductance function, Pctr is the factor of the
control frequency and H(·) is the Heaveside step function defined as

H(x)=

{

1 if x> 0

0 if x¬ 0
(2.2)

In this way, the discontinuousmechanical characteristics of the abrupt change
in the stiffness which reflects an impact, and the velocity-dependent friction
are described in thismathematicalmodel. Since points of bifurcation are asso-
ciated with the loss of stability, a scan of the system dynamics while varying a
system parameter can confirm a range of system parameter values for which a
operation is optimum. In particular, the switching frequency of the solid state
relay is varied. The bar displacement relative to the base board is observed to
identify the range of the frequency for which the achieved forward progression
is maximum.

3. Bifurcation diagrams

Both experimental observation and numerical integration have revealed that
bothperiodic and chaotic trajectories exist byvarying the control frequency of



Bifurcations in an electro-vibroimpact system... 515

Fig. 3. Bifurcation diagram of relativemotion of the bar for the same system
parameters as in Nguyen (2007). The variable is the control frequency which ranges
from (a) 29rad/s to 33rad/s (4.62Hz to 5.25Hz) and (b) 33rad/s to 50rad/s
(5.25Hz to 7.96Hz). System parameters are V

s
=82.02V, µ1 =0.295, µ2 =0.235,

C =32µF, c=0.155kg/s,R=27.5Ω,m1 =0.297kg,m2 =2.94kg,G=−0.002m,
k0 =1.24 ·10

5N/m, k1 =200N/m and X0 =0.022m



516 J.-H. Ho, K.-C. Woo

the solid state relay (SSR). An experimental bifurcation diagram constructed
by Nguyen andWoo (2007) reflected the dynamics of the metal bar observed
in the laboratory. Period-2 motion was observed for frequencies lower than
5Hz, beyondwhich period-1 solution exists, valid up to a frequency of 8.3Hz.
At control frequencies greater than 8.3 Hz, the amplitude of the bar displace-
ment decreased abruptly and significantly. The progression rate of themoling
rig also dropped correspondingly. Those observations were checked against a
bifurcation diagram constructed by numerical integration of the mathemati-
cal model (using the Dynamics software, see Yorke and Nusse (1998)), and
shown in Fig.3. 960 different values of the control frequency were used in the
iteration. In Fig.3a, the frequency increases from 4.62Hz to 5.25Hz in incre-
ments of 6.5625 ·10−4Hz. In Fig.3b, the frequency increases from 5.25Hz to
7.96Hz with an increment of 2.8229 ·10−3Hz. Besides that, for each control
frequency, 60 cycles were allowed to elapse to allow transients to subside and
sample points from a steady state trajectory. Data pertaining to 200 cycles of
steady statemotion were captured to identify themain features of the system
dynamics. In Fig.3a, when the frequency increases from 4.62Hz to 5.25Hz,
chaotic motion changes to period-2motion before settling to period-1motion.

Fig. 4. Phase planes of relativemotion of the bar at a control frequency of (a) 7Hz
and (c) 8Hz. Poincaré maps are plotted for (b) 7Hz and (d) 8Hz



Bifurcations in an electro-vibroimpact system... 517

This synchronous trajectory becomes more apparent in Fig.3b for higher
frequencies. The phase portraits and Poincaré maps for 7Hz and 8Hz con-
firm that the amplitude fluctuation decreases with the increasing frequency.
This is shown in Fig.4. At a frequency of 7Hz, period-1 motion is shown in
Fig.4a. Due to the fluctuation in the amplitude about amean value, Poincaré
sampling results in 7 distinct points on the map, as shown in Fig.4b. On in-
spection of the corresponding time history in Ho et al. (2008), period-1 orbit
is confirmed. Hence, the amplitude fluctuation here caused scatter in the data
points. These may be then considered as belonging to one average amplitu-
de of motion. To construct this, 35 cycles of the displacement were allowed
to elapse before 734 data points (i.e. 734 cycles) were taken, so as to ensure
that all transients had to be subsided, and a steady state had been reached.
A similar situation is observed in Fig.4c, when the control frequency is 8Hz.
Here, the amplitude fluctuation is less than the previous case, and period-1
motion is even more apparent. Due to variation between two amplitudes of
motion very close to each other, the Poincaré map of Fig.4d shows plots of
two amplitudes very similar inmagnitude. For this frequency, data pointswere
taken after 40 transient cycles.

4. Discussion

The experimental observation that periodic relative motion of the metal bar
with respect to the base board is most beneficial to the overall progression
rate achieved by the mechanism was mentioned in the previous section. This
is confirmed by the experimental bifurcation diagram shown by Nguyen and
Woo (2007).Meanwhile, numerical integration has also revealed a similar phe-
nomenon. This is shown in Fig.5.
Experimental results revealed that the progression achieved by themecha-

nism for 5 seconds peaked at a control frequency of 8.3Hz (Nguyen andWoo,
2007). Predictions of the achieved forward progression were obtained from
numerical integration and are shown in Figure 5. Close correlations with the
experimental data from Nguyen and Woo (2007) are observed. For example,
the maximum progression achieved at a control frequency of 8.1Hz compares
favourably to 8.3Hz in the experimental result. Besides that, local maxima
were found in both cases, corresponding to 4Hz in the numerical integration
and 3.3Hz in the experiment. In the experiment, there was almost no pro-
gression right after the local maxima and the forward progression increased
gradually after that until the peak value (i.e. in the region of 3.4Hz to 8.3Hz).



518 J.-H. Ho, K.-C. Woo

Fig. 5. The achievedmole progression for 5 seconds with respect to the control
frequency obtained from numerical integration. The same set of parameters was

used as in Fig.3

The numerical integration showed a similar trend but with a different ran-
ge of the control frequency (from 4Hz to 8.1Hz). The achieved progression
dropped significantly after the peak value for both cases. However, there are
some differences in the absolute magnitudes of the achieved progression for
the simulation and the experiment. For example, themaximum achieved pro-
gression was found to be approximately 0.17m at 8.3Hz in the experiment,
whereas thenumerical studypredicted themaximumvalueof 0.256mat 8.1Hz
and 0.194m at 8.3Hz. This difference might have been caused by the impact
energy loss, which is not accounted for in the mathematical model.

5. Conclusions

Qualitative responses of an electro-vibroimpact system have been revealed
through bifurcation analysis.When the control frequency of the solid state re-
lay increases from4.62Hz to 7.96Hz, ofmotion the systemvaries from chaotic
to period-2 before settling to period-1 motion. This has been observed both
numerically and experimentally. The results further confirm that periodicmo-
tions aremost beneficial to themole progression rate.On gainingmore insight
to the systembifurcation phenomena, coupledwith an approximate analytical
solution, an optimal feedback control system can be then designed to achieve
a better forward progression rate.



Bifurcations in an electro-vibroimpact system... 519

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Bifurkacje w układzie maszyny elektro-wibracyjnej z uderzeniami przy

uwzględnieniu tarcia

Streszczenie

W artykule przedstawiono zjawiska bifurkacyjne zachodzące w układzie maszy-
ny elektro-wibracyjnej z uderzeniami za pomocą analizy numerycznej. Pokazano,
że układ wykazuje przejście z ruchu chaotycznego do periodycznego przy zmianach
częstości sterującej (jednego z parametrów układu) pracą bezstykowego przekaźnika
mocy. Zaobserwowanobliską współzależność otrzymanych diagramówbifurkacyjnych
z wynikami doświadczalnymi. Stwierdzoną lepszą wydajność urządzenia dla zakresu
parametrówzapewniających ruchperiodyczny. Sformułowanopodstawydo określenia
i wdrożenia optymalnej strategii sterowania układu opartej na sprzężeniu zwrotnym.

Manuscript received December 17, 2008; accepted for print March 2, 2008