Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

52, 3, pp. 595-604, Warsaw 2014

DYNAMICAL JUMP ATTENUATION IN A NON-IDEAL SYSTEM

THROUGH A MAGNETORHEOLOGICAL DAMPER

Vinicius Piccirillo, Ângelo Marcelo Tusset

UTFPR – Federal Technological University of Paraná, Department of Mathematics, Ponta Grossa, Brazil

e-mail: piccirillo@utfpr.edu.br; a.m.tusset@utfpr.edu.br

José Manoel Balthazar

UNESP – São Paulo State University, Department of Statistics, Applied Mathematical and Computation, Rio Claro, Brazil

e-mail: jmbaltha@gmail.com

This paper is concerned with the Sommerfeld effect (Jump phenomena) attenuation in an
non-ideal mechanical oscillator connected with an unbalanced motor excitation with a li-
mited power supply (non-ideal system) using a magnetorheological damper (MRD). The
dynamical response of systems with MRD presents different behavior due to their nonline-
ar characteristic. MRD nonlinear response is associated with adaptive dissipation related
to their hysteretic behavior. The Bouc-Wen mathematical model is used to represent the
MRD behavior. Numerical simulations show different aspects about the Sommerfeld effect,
illustrating the influence of the different electric current applied in theMRD to control the
force developed by this damper.

Keywords: magnetorheological damper, Sommerfeld effect, nonlinear dynamics

1. Introduction

In the studyof problems that involve coupling of several systems, changes of structural characte-
ristics of machines and their component have been investigated in the recent years. In this way,
some phenomena are observed in composed dynamic systems suporting structures and rotating
machines, in which it has been verified that the unbalancing of rotating parts is themajor cause
of vibrations. In the study of such systems, for a more realistic formulation one must consider
the action of an energy source with a limited power (non-ideal), that is consider the influence
of the oscillatory system on the driving force and vice versa. Recently, a number of works have
been done in order to investigate resonant conditions of non-ideal vibrating oscillator systems
(Balthazar et al., 2003) and a number of several non-ideal vibrating systems has been studied,
for some examples (Frolov andKrasnpolskaya, 1987;Krasnopolskaya andShvets, 1994; Piccirillo
et al., 2009, 2011; Tusset and Balthazar, 2013) andmany others.
The Sommerfeld effect is a kind of problem that occurs in non-ideal systems near resonance

frequencies.This effectwas described in the classical bookbyKononenko (1969), entirely devoted
to this subject. The jump phenomena in the vibration amplitude and the increase of the power
required by the source to operate next to the system resonance are both manifestations of
this non-ideal problem. This phenomenon suggests that the vibratory response of the non-ideal
system emulates an “energy sink” in the regions next to the system resonance, by transferring
the power from the source to vibrations of the supporting structure instead of the speeding up
the driving machine (Castão et al., 2010). In other words, one of the problems confronted by
mechanical engineers is how to drive a system through the system resonance and to avoid this
“disappearance of energy” as originally described by Sommerfeld (Nayfeh andMook, 1979).
Palacios et al. (2009) presented a research which contained analysis of the Lugre friction

in elimination of the Sommerfeld effect for a non-ideal structural system (NIS). The authors



596 V. Piccirillo et al.

observed significant reductions in the resonance capture phenomenon when this friction law is
considered in NIS and consequently, the Sommerfeld effect is then eliminated. The analysis of
the Sommerfeld effect of a Duffing-Rayleigh oscillator under a non-ideal excitation (unbalanced
motor with a limited power supply) using the method of averaging and numerical computation
was investigated by Felix et al. (2009a). Furthermore, for the reduction of the Sommerfeld
effect, the jump phenomenon and resonance capture there was used a shape memory alloy
spring. According to Belato (1998), the jump phenomenon related to the Sommerfeld effect is
associated with a cyclic saddle-node bifurcation with the system losing its stability in the point
where the jump occurs.
In this work, we use a semi-active approach to reduce resonance vibrations of a non-ideal

structure (Sommerfeld effect) in non-ideal system by applying a nonlinear damping with ma-
gnetorheological fluids. The mechanism of the damper (MR) is similar to the mechanism of
hydraulic dampers inwhich the force is obtained by passage of the fluid through an orifice. This
variable resistance to fluid flow allows using a viscous fluid in MR dampers and other devices
electrically controllable. Thus, the magnetic properties of the fluid allow its use as a damper
controlled by electric voltage (V) or an electric current (A) (Tusset et al., 2009).
The use of MR damper control in the suppression of unwanted oscillations is done by the

electrical current or voltage which changes the viscosity of the internal damper fluid. The dam-
ping force will depend on the velocity of the piston of the damper and density of the internal
fluid.

2. Magnetorheological damper model with hysteresis

The MRD is a semi-active device inwhich the viscosity of the fluid can be controllable by a
change in the input voltage (Spencer et al., 1997). A large number of analytical models based
on different descriptions have been done with the objective of describing nonlinear properties of
MR dampers. The force-velocity characteristics of theMRDmeasured after various excitations
and electric currents, indicates nonlinear behavior such as hysteresis (Ma et al., 2003).
The velocity of the piston also has an important influence on the dynamic properties of the

MRDfluid. If the velocity is high, the duration inwhich the particles are in themagnetic field is
short.This results in saturation of the damping force to the upper velocity ±0.4m/s. Saturation
can also occur in relation to the applied electric current in the coil with electric currents between
0 and 1.5A. (McManus et al., 2002).

Bouc-Wen model of the MR damper

A large number of models ofMRDs have been proposed for describing their hysteretic beha-
vior (Wang and Liao, 2011), such as bi-viscous model, Bouc-Wen model, neural-network-based
model, etc. A model that can be solved numerically and used to represent the MRD dynamics
with hysteresis is the Bouc-Wen model. The Bouc-Wen model is considered to be extremely
versatile and can display a wide variety of hysteretic behavior. Figure 1 shows the layout of the
Bouc-Wen model (Dominguez et al., 2006).
The force F of the system is determined by

F = c0ẋ+k0x+αz (2.1)

and z is obtained from the equation

ż=−ς|ẋ|z|z|n−1− ξẋ|z|n+Λẋ (2.2)

This Bouc-Wen model incorporates the MR damper force f0 as an initial displacement x0
and a stiffness coefficient k0. It can be seen in Eqs, (2.1) and (2.2) that the control variable (i)



Dynamical jump attenuation in a non-ideal system... 597

Fig. 1. Bouc-Wenmodel of theMR damper

does not appear explicitly. Since the goal is to control the damper strength using the electric
current, an approximation ofEq. (2.1) can beused that depends explicitly on the electric current
(Tusset et al., 2013)

F =
3.2

3e−3.4i+1
ẋ+k0x+

8.5

1.28e−3.9i +1
z (2.3)

The electrical current to be applied can be determined by solving numerically the following
function (Tusset et al., 2013)

C(i)=
3.2

3e−3.4i+1
ẋ+k0x+

8.5

1.28e−3.9i+1
z−F (2.4)

In order to examine the response of the MRD, a comparison with Eq. (2.1) proposed by
Dominguez et al. (2006) with Eqs. (2.3) and (2.4) proposed by Tusset et al. (2013) has been
made, and this comparison is very close, as can be seen inFig. 2. The characteristics of variation
of the damping force dependingon the velocity of the piston of the damper and the applied
electric current in the coil estimated for an excitation of 1Hz and an amplitude of 0.04m, with
the parameters Λ=180, k0 =0, ξ=0, n=2, ς =0.1 and Table 1 (Yao et al., 2002).

Fig. 2. Characteristics of theMR damper as function of the electric current considering the model by
Dominguez et al. (2006) and the model by Tusset et al. (2013)

Table 1. System parameters used in Fig. 2 (Yao et al., 2002)

Electric current i Parameter c0 Parameter α0

0 1.00 5.630

0.25 1.65 4.102

0.50 2.20 6.877

0.75 2.50 7.950

1.00 2.90 8.300



598 V. Piccirillo et al.

3. Non-ideal excitation

Let us consider a vibrating system (NIS) which includes a direct current (DC) motor with a
limited power supply operating on a structure (Fig.3). The excitation of the system is limited
by a characteristic of the energy source (non-ideal energy source). Then, the coupling of the
vibrating oscillator and the DC motor takes place. As the vibration of the mechanical system
depends on theDCmotor, alsomotion of the energy source depends on vibrations of the system.
Hence, it is important to analyze what happens to the motor as the response of the system
changes.

Fig. 3. (a) Non-ideal mechanical system (NIS) and (b) electrical schematic representation
of the DC motor

Theconsideredvibrating systemconsists of amass m1, linear dampingwithviscous damping
coefficient b. The non-ideal DC motor has the driving rotor of moment of inertia J, and r is
the eccentricity of the unbalanced mass.
The electrical scheme of the DCmotor is presented in Fig. 3b. The equations governing the

motion of the DC motor are typically written in the form by Warminski and Balthazar (2003)

J
d2ϕ

dt2
=Mm(t)−Mz(t)−H(t) U(t)=RtI(t)+Lt

dI(t)

dt
+E(t) (3.1)

where time functions U(t) and I(t) are thevoltage andcurrent in the armature, Rt and Lt is the
resistance and inductance of the armature, E(t) is the internally generated voltage, Mz(t) is the
external torque applied to themotor driving shaft, H(ϕ) is the frictional torque and Mm(t) de-
notes the torque generated by themotor.The torque Mm(t) and internal generated voltage E(t)
can be expressed as

Mm(t)= cMΦI(t) E(t)= cEΦω(t) (3.2)

where cM, cE aremechanical and electrical constants, and Φ is themagnetic flux.Let us assume
that the external exciting current Im andvoltage Um are constant, and then themagnetic flux Φ
is also constant in the consideredmodel. Taking into account Eqs. (3.1) and (3.2), we can write
differential equations of the complete electro-mechanical system presented in Fig. 3 as follows

Mx′′+ bx′+F +klx+knlx
3−m0r(ϕ

′2 sinϕ+ϕ′′cosϕ) = 0

(JM +m0r
2)ϕ′′ = cMΦĨ(t)− H̃(ϕ

′)+m0rx
′′cosϕ

dĨ(t)

dt
=−
Rt

Lt
Ĩ(t)−

cEΦ

Lt
ϕ′+
Ũ(t)

Lt

(3.3)

where the prime denotes the derivative with respect to dimensional time and M = m1 +m0,
and F is theMRD restoring force. It is convenient to work with the dimensionless position and
time, in such a way that Eqs. (3.3) are rewritten in the following form

ü+ ζu̇+FMRD +u+γu
3−w1(ϕ̇

2 sinϕ+ ϕ̈cosϕ)= 0

ϕ̈= p3I(τ)+w2ücosϕ−H(ϕ) İ =U(τ)−p1I(τ)−p2ϕ̇
(3.4)



Dynamical jump attenuation in a non-ideal system... 599

where

ω20 =
kl

M
ζ =

b

Mω0
γ =
knl

kl
x2
st

I =
Ĩ

Ir

w1 =
m0r

Mxst
w2 =

m0rxst

J+m0r2
p1 =

Rt

Ltω0
p2 =

cEΦ

LtIr

p3 =
cMΦIr

(J+m0r2)ω
2
0

U(τ)=
Ũ(τ)

LtIrω0
H(ϕ) =

H̃(ϕ′)

(J+m0r2)ω0

τ =ω0t u=
x

xst
FMRD =

F

Mω20xst

and xst means the static displacement of the system, τ is the dimensionless time, Ir is rated
current in the armature anddots indicate differentiationswith respect to the dimensionless time,
the function H(ϕ) is the resistive torque applied to the motor and, in this work, H(ϕ)will be
neglected (H(ϕ) = 0).

4. Results of numerical simulation

This section considers numerical simulations that are meant to illustrate the MRD influence
on jump phenomena in a complete electro-mechanical system. The DC motor and mechanical
parameters used in numerical simulations are given in Table 2. All simulations consider the
following oscillator parameters: Rt = 1Ω, Lt = 3.7 · 10

−2H, Ir = 3.93A, cEΦ= 0.437Vs/rad,
cMΦ=0.437Nm/A.

Table 2. System parameters used in simulation (Warminski and Balthazar, 2003)

w1 w2 p1 p2 p3 ζ γ

0.2 0.3 0.3 3 0.15 0.1 0.2

Innon-idealmechanical systems, theoscillator cannotbedrivenby systems,whoseamplitude
and frequency are arbitrarily chosen, once the forcing source has a limited available energy
supply. For this kind of oscillator, the driven system cannot be considered as given a priori,
but it must be taken as a consequence of the dynamics of the whole system (oscillator and
motor). Therefore, a non-ideal oscillator is, in fact, a combined dynamical system resulting from
the coupling of passive and active oscillators which serve as the driving source for the first
ones. The resulting motion will be thus the outcome of dynamics of the combined systems.
The dimensionless voltage applied across the armature U is the control parameter in the non-
ideal system. For each value of U, the non-ideal system presents one frequency and amplitude
behavior.

It is known that the dynamics of a system close to the fundamental resonance region may
be analyzed through a frequency-response diagram,which is obtained by plotting the amplitude
of the oscillating system versus the frequency of the excitation term. For the complete electro-
mechanical system, this graph is estimatedbynumerical simulationdefining theamplitudeas the
maximum value of the amplitude of mechanical oscillation (denoted by A), and the frequency
as the mean value of the rotational speed ϕ̇ (denoted by ω).

Figure 4 represents the resonance curvewithout theMRDdevicewhen themean frequency ω
is slowly increased. The curve was calculated using an increment ∆U =0.01 as the variation of
the control parameter U. The transient response is also considered in the computation because
its evolution inside the state space determines the occurrence of the jump phenomenon during
passage through the resonance region (ω ≈ 1). In Fig. 4a, it can be seen that when the value



600 V. Piccirillo et al.

of the control parameter is U ≈ 3.1, the resonance region was reached, so large amplitude
vibrations of the system are generated, as shown in Fig. 4b, in other words the system reaches
the maximum amplitude of displacement. Note that close to the resonance (ω ≈ 1), the power
it is supplied to the DC motor to initiate the jump increase (see Figs. 4c and 4d). Then the
operating frequency increases and thus the system amplitude decreases resulting in lower power
consumption by the DCmotor.

Fig. 4. Jump phenomenon observed when the mean frequency ω is slowly increased:
(a) frequency-response diagramwithout theMRD, (b) time response, (c) energy source and

(d) zoom of Figure (c)

The jump phenomenon is characterized by sudden amplitude transition, as indicated in
Fig. 4a. This happens because there is not enough damping in the system to stop theDCmotor
from transmitting large amounts of energy to the nonlinear oscillator. Now, the same non-ideal
system is investigated, however, the MRD is introduced in this system, as observed in Fig. 3.
The Sommerfeld effect and the MRD parameter in the non-ideal system will be verified, too.
These results are comparedwith thenon-ideal systemwithout theMRD.Due toMRD,nonlinear
characteristic and dissipative behavior of the jump response tends to exhibit a small vibration
amplitude.

When the MRD is introduced to the non-ideal system, it can be observed that for i=0A,
that is the electric current applied in the MRD is zero, the reduction of amplitude in the non-
ideal system is observed, as shown in Figs. 5a and 5b. Nevertheless, the Sommerfeld effect still
happens because the damping introduced by theMRD, in this case, is not efficient to suppress
or inhibit the energy transfer from the DC motor to the nonlinear oscillator. Moreover, when
compared to the situation with the non-ideal system without an MRD, the MRD non-ideal
oscillator presents a smaller amplitude response.

Figure 6 shows the behavior of the non-ideal system with i = 0.5A. With this increase
in the applied electric current, the characteristics of variation of the damping force change
and the Sommerfeld effect starts to be controlled, for this reason the MRD damper dissipates
more vibrational energy now than previously, therefore, the amplitude of motion decreases, too.



Dynamical jump attenuation in a non-ideal system... 601

Fig. 5. Jump phenomenon observed when the mean frequency ω is slowly increased:
(a) frequency-response diagramwith theMRD (i=0A), (b) time response, (c) energy source and

(d) zoom of Figure (c)

Fig. 6. Jump phenomenon observed when the mean frequency ω is slowly increased:
(a) frequency-response diagramwith theMRD (i=0.5A), (b) time response, (c) energy source and

(d) zoom of Figure (c)



602 V. Piccirillo et al.

Fig. 7. Jump phenomenon observed when the mean frequency ω is slowly increased:
(a) frequency-response diagramwith theMRD (i=1A), (b) time response, (c) energy source and

(d) zoom of Figure (c)

Fig. 8. Jump phenomenon observed when the mean frequency ω is slowly increased:
(a) frequency-response diagramwith theMRD (i=1.5A), (b) time response, (c) energy source and

(d) zoom of Figure (c)



Dynamical jump attenuation in a non-ideal system... 603

Figures 6c and 6d show that the energy transferred from the DC motor to the structure began
to be controlled, and that meant that the MRD energy dissipation improved (or decrease) the
jump.
Figures 7 and 8 show the system, considering that the current is increased to a level of 1A

and 1.5A, respectively. In both cases, the Sommerfeld effect is completely put down because
the damping force is greater than in the previous cases, and the effect of the MRD results in
suppression of the jump phenomena. Note that the increase in the electric current causes a
decrease in the amplitude of motion in the resonance region, eliminating possible jumps. In this
case, the energy transferred from theDCmotor to the oscillator was controlled bymeans of the
MRD.
Figure 9 shows the Sommerfeld effect and its suppression for different values of the electric

current. The equivalent non-ideal response is compared with the MRD non-ideal system which
allowed one to verify the MRD effect in this dynamical system. It might be observed that the
amplitude reduction can be achieved for different electric currents.

Fig. 9. Jump phenomenon for different values of the electric current

5. Conclusion

In this paper, the attenuation and suppression of the Sommerfeld effect in a non-ideal vibrating
system, using an MRD was presented. This happens as a consequence of using an additional
electrical current introduced to theMRdamper,which increased energy supportof the combined
system. For this reason, if values of the electric current in the MRD is increased, the damping
force grows aswell and, therefore, the Sommerfeld effect and the amplitude in resonance regions
are avoided. The analysis of this system shows that the introduction of an MR damper, in this
case, is efficient to suppress or inhibit the energy transfer from the DC motor to the nonlinear
oscillator in resonance cases.

Acknowledgements

The authors acknowledge the financial support by FAPESP and CNPq (grant No. 484729/2013-6).

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Manuscript received September 10, 2013; accepted for print December 30, 2013