Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 4, pp. 805-822, Warsaw 2003

GENERALIZED MODEL OF A MAGNETORHEOLOGICAL

FLUID DAMPER FOR FLUCTUAITING MAGNETIC FIELDS

Bogdan Sapiński

Adam Piłat

Department of Process Control, University of Science and Technology, Cracow

email: deep@uci.agh.edu.pl; ap@ia.agh.edu.pl

The paper investigates a generalizedmodel of amagnetorheologicalfluid
damper (MRD). Rheological behaviour ofMRfluids is described and so
is the Spencer model taking into account the effects of magnetic field
fluctuations in the MRD gap on the magnitude of damping force. It is
demonstrated that the Spencermodel does not fully emulate the real be-
haviour of theMRD, especially at lowcurrent levels. Somemodifications
of the model are suggested, whereby a nonlinear function is introduced,
enveloping the family of damping force curves. The thus formulated ge-
neralized model is then verified experimentally. The model efficiency is
confirmed by investigation of MRD control in a semi-active vibration
isolation system for a quarter car body.

Key word: magnetorheological fluid damper, model, identification,
control

1. Introduction

Magnetorheological fluiddampers (MRDs) are semi-active devices inwhich
damping characteristics can be continuously controlled depending on thema-
gnetic field. For this reason, MRDs are promising devices for shock and vi-
bration isolation in spite of strongly nonlinear dynamic characteristics, which
appears to be their inherent feature (Choi et al., 2001). These nonlinear cha-
racteristics aremostly a result of nonlinearity introducedby the control circuit
of theMRD, consisting of ferromagnetic elements and the portion of theMR
fluid affected by the magnetic field excited by the current in the coil (Sapiń-
ski, 2003). Hence, processes inMR fluids and properties of the control circuit
determine the dynamic behavior of MRDs.



806 B.Sapiński, A.Piłat

Among different MRDs models, the only parametric model developed by
Spencer (Dyke et al., 1996) exhibits functional dependence of the parameters
on the applied control signal and, hence, is validwhendealingwith fluctuating
magnetic fields. This model captures real behaviour of the MRD over a wide
range of operating conditions, however it does not provide fully useful tool
for control system design (Sapiński, 2002). Therefore, in order to take the full
advantage of the unique features of the MRD, a dynamic model is required,
which ensures good fidelity for both system evaluation and control algorithms
development.

The aim of this paper is to develop such a dynamic model of the MRD
that does not share shortcomings of the Spencer model, which is valid for
fluctuatingmagnetic fields and is useful for control system design. To confirm
the effectiveness of the generalized model, a linear MRD of RD-1005 series,
fabricated byLordCorporation is considered.ThisMRD, shown inFig.1, was
designed to operate in the post-yield region, at low frequencies with medium
or high amplitudes.

Fig. 1. Structure of RD-1005 damper

The MRD may generate any force, with the only constraint being the
admissible level of the current in the coil. When no current flows through the
coil, the piston motion is counteracted by the force of static friction on the
piston sealing and the force due to theMR fluid flow through the gap.When
the current appears in the coil, a magnetic field is generated around it. An
increase in the magnetic field intensity brings about an increase in viscosity
of the MR fluid present in the gap. Then the piston motion is additionally
impaired by the force due to MR effect.

The MRD control uses the pulse width modulation method (PWM). It
allows for shaping the control signal (i.e. current in the coil) depending on the
signal frequency andPWMwidth. Application of PWM signals allows for the
optimal selection of operating parameters of theMRD supplied from a power
interface (Sapiński et al., 2003).



Generalized model of a magnetorheological fluid damper... 807

2. Rheological behaviour of MR fluids

MR fluids are non-colloidal suspensions with micron-size, magnetically-
soft iron particles dispersed in a non-magnetic fluid carrier with additives
that promote homogeneity and inhibit gravitational settling. They belong to
a group of non-Newtonian, rheologically stable fluidswhich display shear yield
strength and can be controlled by amagnetic field.

The rheological properties of the MR fluid depend on the concentration
of the magnetic particles, their size and shape, properties of the carrier fluid,
magnetic field intensity, etc.When no separate external magnetic field exists,
the particles are randomly dispersed, and the MR fluid exhibits Newtonian
behaviour. In the presence of a magnetic field, the particles create chainlike
structures along the field lines, which in turn create an increase of several
orders of magnitude in theMRfluid viscosity. The key feature ofMRfluids is
the ability to reversibly change from free-flowing linear viscous liquids to semi-
solid, having controllable yield strength within milliseconds, which is used in
MR devices, such as dampers, clutches, brakes, etc. (Jolly et al., 1999).

In terms of rheological behaviour, aMRfluid can be divided into pre- and
post yield region. In the pre-yield region, the MR fluid behaves like a visco-
elasticmaterialwith a complex shearmodulusdependenton themagnetic field
intensity H, however, in the post-yield region (Fig.2), it exhibits properties
of a visco-plastic material with the viscosity η (Bingham’smodel). The values
of τy,d and τy,s, denote the dynamic shear yield stress and the static shear
yield stress which depend on H. In reality, MR fluid behaviour differs from
Bingham’s model, the most important difference being the non-Newtonian
behaviour in the absence of a magnetic field.

Fig. 2. MR fluid behavior in the post-yield region



808 B.Sapiński, A.Piłat

3. Spencer model

TheMRDmodel developed by Spencer captures both visco-elastic proper-
ties of the MR fluid and the appearance of a hysteresis and damping force
saturation (see Fig.3). This model is an extension of the model proposed
by Bouc-Wen (Dyke et al., 1996), introducing an additional dashpot c1 and
spring k1.

Fig. 3. Spencer model of theMRD

3.1. Governing equations

The governing equation for the Spencer model is given as

F = az+ c0(ẋ− ẏ)+k0(x−y)+k1(x−x0) (3.1)

It can be also written in the form

F = c1ẏ+k1(x−x0) (3.2)

The displacements z and y in equations (3.1) and (3.2) are defined by equ-
ations (3.3) and (3.4), respectively

ż=−γ|ẋ− ẏ|z |z|n−1−β(ẋ− ẏ)|z|n+A(ẋ− ẏ)
(3.3)

ẏ=
1

c0+ c1
[az+ c0ẋ+k0(x−y)]



Generalized model of a magnetorheological fluid damper... 809

Designations used in equations (3.1), (3.2) and (3.3) are explained below:

F – damping force
β,γ,A – parameters whose adjustment allows for shaping the line-

arity of the control during unloading and the smoothness
of the transition from the pre-yield to post-yield area

α – parameter representing the stiffness for the damping force
component associated with the evolution variable z

k0 – parameter representing the control of the stiffness of the
spring at higher velocities

k1 – parameter representing the stiffness of the spring associa-
ted with the nominal MRD due to the accumulator

c0 – parameter representing viscous damping observed at hi-
gher velocities

c1 – parameter representing the dashpot included in themodel
to produce the roll off at low velocities

x0 – parameter representing the initial displacement of the
spring with the stiffness k0

n=2 – this value ensures satisfactory accuracy of the predicted
damping force in comparison with the measured one.

The aboveMRDmodel was previously discussed and identified (Sapiński,
2002) for a given current applied, and hence the magnetic field intensity was
held at a constant level. In practical applications for vibration and shock
isolation, theMRD is usually employed as an actuator in the control system,
and the optimal performance is expected to be achieved when the magnetic
field is varied basing on the measured response of the system to which it is
attached. For this reason, a dynamic model of the MRD is required, which
would be capable of predicting its behaviour for fluctuating magnetic fields.
To develop the model that could be useful for control system design, the
functional dependence of the parameters on the applied control signal has
to be determined.

InSpencer’s approach, theyield stress of theMRfluid is directlydependent
on themagnetic field intensity. Thus, the parameter α in equations (3.1), (3.2)
and (3.3), is assumed tobea linear functionof the applied control signalwith a
nonzero initial value at 0V (this, partly, is due to theMRfluid,which exhibits
a small yield strength at the zeromagnetic field for the save of stability against
gravitational settling, and partly due to the friction in the piston road seal).
Similarly, the viscous damping constants c0 and c1 are also assumed as linear
functions of the applied control signal. Accordingly, the following functional



810 B.Sapiński, A.Piłat

dependence for the parameters α, c0 and c1 can be written (Dyke et al.,
1996) as

α=α(u)=αa+αbu c0 = c0(u)= c0a+ c0bu
(3.4)

c1 = c1(u)= c1a+ c1bu

where u is the control signal (voltage).

To prove the ability of the Spencer model and to reflect the MRD real
behaviour for fluctuating magnetic fields, the following parameters: β, γ, A,
k0, k1, x0, αa, αb, c0a, c0b, c1a, c1b have to be determined, which was done by
experiments.

The following modifications to the Spencer model have beenmade in fur-
ther considerations:

• the control signal in equations (3.4) is the current in the coil

• the damping force is given as

F = c1ẏ+k1x+F0 (3.5)

where F0 is the force associated with pressure in the accumulator.

3.2. Data acquisition and processing

The aim of the experiments was both to determine the influence of PWM
signalparameters (frequency,PWMwidth factor) on thecurrent in thecoil and
functional dependenceof the control signal (current) on themodel parameters.

The MRD experiments were conducted in an experimental setup with a
computer-controlled vibration testing machine (Fig.4).

The input-output data were acquired by using a data acquisition system
which was based on a PC (Pentium III/1GHz) and multipurpose I/O board
RT-DAC4 (INTECO Ltd., 2002), operating in the software environment of
Windows 2000,MATLAB 6.1/Simulink andReal TimeWindows Target (The
Math Works Inc., 1999).

Tests were run for the mid-stroke piston position in the MRD. For the
purpose of the experiments a computer program was created providing for
automatic changes of the PWM signal frequency and PWM width. On the
basis of registered measurement data, the relationship between the control
current and signal frequency as well as PWMwidth was established (Fig.5).

The model parameters: β, γ, A, k0, k1, x0, αa, αb, c0a, c0b, c1a, c1b we-
re determined by parametric identification. For this purpose, the measured



Generalized model of a magnetorheological fluid damper... 811

Fig. 4. Experimental setup forMRD data acquisition

Fig. 5. Current versus PWMparameters

data of the piston displacement and current in the coil were used. Thus obta-
ined data involved certain noise from the measured system. To minimize the
optimization errors, data pre-processing was required.

The input signal, being a sine signal, can be approximated with the follo-
wing function

g(t) = a1 sin(2πa2t+a3)+a4 (3.6)

The approximation uses the least squaremethod. As the ideal sine signal was
considered, the piston velocity with no noise could be easily determined. The



812 B.Sapiński, A.Piłat

recorded piston displacement patterns and the approximating function g(t)
are given in Fig.6; the parameters of this function are

a1 =9.9372 ·10
−3m a2 =0.9996Hz

a3 =3.3474 ·10
−3 rad a4 =−3.3130 ·10

−5m

Fig. 6. Measured (- - -) and approximated (—) displacement signals

The input data in the optimization procedure were: level of the current,
displacement approximated in accordance with formula (3.6) and velocity cal-
culated on the basis of the approximated displacement.

3.3. Force prediction

In the first stage of the experiments the machine was programmed to
generate a sinusoidal wave for six frequency levels (1.0, 2.5, 3.1, 4.1, 5.0,
6.0)Hzwhich corresponded to the following amplitudes (10.0, 4.0, 3.0, 1.5, 1.2,
1.0)·10−3m. The damping force responses (Fig.7b) were measured for eight
levels of the applied current: (0.0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.8)A (Fig.7a).

Theparametric optimization of theSpencermodelwas runusing thenonli-
near constrained optimizationwith the constr procedure available in theOpti-
mization Toolbox ofMATLAB (TheMathWorks Inc., 1999). The quality cri-
terion was the integral of the squared difference between the measured and
predicted force (i.e. predicted in the course of computer simulations). When
the model parameters were available, the supervisory optimization algorithm
started up themodel simulation procedure, utilising the approximated displa-
cement signal, (Fig.8). The simulations yielded the damping force patterns.
The error was fed back to the optimization procedure.



Generalized model of a magnetorheological fluid damper... 813

Fig. 7. Current (a) and damping force (b) for various frequencies of kinematic
excitation

Fig. 8. Diagram of parametric optimization



814 B.Sapiński, A.Piłat

The parametric optimization procedure yielded parameters of the Spencer
model, summarised in Table 1.

Table 1.Parameters of the Spencer model

Parameter Value Parameter Value

γ [1/m2] 8130143.7364 c0a [N·s/m] 5932.5033

β [1/m2] 8266348.8895 c0b [N·s/(m·A)] 310.5295

A [–] 84.4652 c1a [N·s/m] 1354301.9374

k0 [N/m] 8853.5106 c1b [N·s/(m·A)] 137534.3689

k1 [N/m] 1218.3144 F0 [N] 260.9558

αa [N/m] 485277.9998 x0 = y0 = z0 [m] 0.0000

αb [N/(m·A)] 148549.6456

Accordingly, computer simulations of the Spencer model were run. The
results are provided in Fig.9. Figure 9a shows the damping force patterns for
the current range (0.0-0.8)A. Zoomed sections for the current 0.0A and 0.1A,
and the kinematic excitation frequency 1Hz and 6Hz, are presented in Fig.9b
and Fig.9c, respectively.

Fig. 9. Comparison of damping force patterns obtained in experiment (- - -) and
simulation (—), in accordance with the Spencer model



Generalized model of a magnetorheological fluid damper... 815

The results of computer simulations reveal that the Spencer model does
not emulate the real MRD behaviour throughout the tested range of control
signal variations and the frequency (amplitude) of kinematic excitations, hence
it proves to be inadequate for control purposes.

4. Development of a generalized model

In the light of considerations presented in Section 3, it appears that the
Spencermodel requires certainmodificationsmaking themodel properly emu-
late theMRDbehaviour.Major deviations of thedamping force in theSpencer
model occurred at zero and very small currents (0.0-0.4)A. Consistently, the
measurement data were analysed again. It appeared that the envelope of the
damping force patterns for small current levels was a nonlinear function (see
Fig.10). That allowed formodification of the Spencermodel, given by formula
(4.1). This modification led to the formulation of a generalized model which
might be regarded as obligatory when dealing with fluctuatingmagnetic fields
in theMRD gap

FG = f(I)F (4.1)

where

f(I)= δexp(λI)+1 (4.2)

and
I – current in the coil
F – damping force predicted by the Spencer model
FG – damping force predicted by the generalized model
δ,λ – constant.

Fig. 10. Function f(I)



816 B.Sapiński, A.Piłat

When function (4.1) is employed and the parametric optimization proce-
dure repeated, we get the parameters of the generalizedmodel, as provided in
Table 2.

Table 2.Parameters of the Spencer model in the full operating range

Parameter Value Parameter Value

γ [1/m2] 8130143.7364 c0a [N·s/m] 5932.5036

β [1/m2] 8266348.8895 c0b [N·s/(m·A)] 310.5295
hlineA [–] 84.4652 c1a [N·s/m] 1354301.9374

k0 [N/m] 8853.5106 c1b [N·s/(m·A)] 137534.3689

k1 [N/m] 1218.3144 F0 [N] 260.9558

αa [N/m] 485277.9998 δ [–] −0.9203

αb [N/(m·A)] 148549.6456 λ [1/A] −18.3110

x0 = y0 = z0 [m] 0.0000

Fig. 11. Comparison of measured (- - -) and predicted (—) force time patterns for
the generalizedmodel

These values were then used in computer simulations of the generalized
model. The results are shown in Fig.11. Figure 11a shows the damping force
patterns for the current range (0.0-0.8)A. Zoomed sections for the current



Generalized model of a magnetorheological fluid damper... 817

intensity 0.0A and 0.1A, and the kinematic excitation frequency 1Hz and
6Hz are presented in Fig.11b and Fig.11c, respectively.

It is seen that when the linear envelope of the damping force curves is
added, the Spencer model begins to correctly emulate the MRD behaviour
throughout the tested range of the frequency and current levels.

The investigated damperRD-1005 is intended for truck driver seats. Toge-
ther with other necessary equipment, thisMRD is the key element of a vibra-
tion isolation system for the driver seat, providing adequate protection from
vibrations. The results of experimental tests (Sapiński, 2003) reveal that the
current level applicable to the control of the damperRD-1005 is (0.0-0.3)A. In
order to formulate the dynamicMRDmodel, this range of the applied current
was extensively tested.

For this purpose, the vibration testing machine was programmed to ge-
nerate a sinusoidal wave for three frequency levels (1.0, 2.5, 5.0)Hz which
corresponded to the following values of the amplitude: (10, 4, 1.2)·10−3m.
The responses were measured for the following levels of the applied current:
(0.000-0.150) with the step 0.010A, and (0.200-0.400) with the step 0.050A.

The measurement results confirmed the earlier conclusions regarding the
nonlinearity of theMRD damping force. It is seen in Fig.9 that the damping
force envelope is the nonlinear function given by formula (4.2). The repeated
optimization procedureyielded the generalizedmodel parameters, summarised
in Table 3.

Table 3.Parameters of the generalized model in control operating range

Parameter Value Parameter Value

γ [1/m2] 8150381.3585 c0a [N·s/m] 4661.23886

β [1/m2] 8273128.2292 c0b [N·s/(m·A)] 8062.2340

A [–] 168.1040 c1a [N·s/m] 1461874.4425

k0 [N/m] 8853.5105 c1b [N·s/(m·A)] 183564.9655

k1 [N/m] 1218.3144 F0 [N] 151.8174

αa [N/m] 418553.3227 δ [–] −0.9436

αb [N/(m·A)] 252504.8818 λ [1/A] −2.0987

x0 = y0 = z0 [m] 0.0000

These values were then used in computer simulations of the generalized
model. The results are shown in Fig.12. Figure 12a shows the damping force
patterns for the current range (0.0-0.4)A and the kinematic excitation frequ-
ency 1Hz. Zoomed sections for the current 0.2A and 0.15A are presented in
Fig.12b and Fig.12c.



818 B.Sapiński, A.Piłat

Fig. 12. Damping force obtained from experiments (- - -) and simulations (—) of the
generalizedmodel in the control operating range

The results confirm that the real behaviour ofMRDs is fully reflected also
in the range of control currents.

5. Application of the generalised model

This generalized model was then applied in simulations of a quarter-car
suspension with the MRD (Sapiński et al., 2003). The system performance
was tested using the experimental setup shown schematically in Fig.13.

The following values of the system parameters were assumed: wheel mass
60kg, vehicle mass 356kg, tire stiffness 7500N/m, damping factor of the tire
10Ns/m, spring stiffness 2500N/m. As a MRD in the quarter car body the
RD-1005 damper controlled by a hybrid controller was employed. The hybrid
control scheme is one of the commomly used for vehicle suspensions. It allows
for adjusting thedamping force in the rangeboundedby theminimumandma-
ximumdamping.The aim of the control was to stabilize the car body position
at a desired level. Mathematically, the hybrid control is a linear combination
of the formulation of the skyhook and groundhook control (Ahmadian, 1999),
and can be expressed as



Generalized model of a magnetorheological fluid damper... 819

Fig. 13. Schematic diagram of the experimental setup

δSKY =

{

ẋ2 for ẋ2ẋ0 > 0

0 for ẋ2ẋ0 < 0

δGND =

{

ẋ1 for −ẋ1ẋ0 > 0

0 for −ẋ1ẋ0 < 0
(5.1)

v=G[θδSKY +(1−θ)δGND]

where: ẋ0 – base velocity, ẋ1 – wheel velocity, ẋ2 – body velocity, v – control
signal (PWMwidth factor, see Fig.5). The variables δSKY and δGND are the
skyhook and groundhook components of the damping force respectively, θ is
the relative ratio between the skyhook and groundhook control, and G is a
constant gain chosen experimentally such that the allowable damping force
is fully utilized. If θ = 0 the hybrid control reduces to the pure groundhook
control, and if θ=1 the hybrid controller becomes the pure skyhook control.
The constant were assumed as θ=0.6 and G=10.

The input-output data were measured with a sampling frequency of
1000Hz. The results of the measurements are shown in Fig.14 (base excita-
tion), Fig.15 (car position), and Fig.16 (control signal – PWMwidth factor).

The model time response was achieved for both real kinematic extication
and real control signals. The comparison of the predicted (simulation) andme-
asured (experiment) system response indicates that the developed generalized
MRDmodel provides sufficient accuracy in control applications.



820 B.Sapiński, A.Piłat

Fig. 14. Base position

Fig. 15. Car position – generalizedmodel

Fig. 16. Control signal – PWMwidth factor

6. Conclusions

The paper investgates a generalized model of the linear MRD for fluctu-
ating magnetic fields. The model is created as a modification of the Spencer
dynamic model. Although the Spencermodel captures both visco-elastic pro-



Generalized model of a magnetorheological fluid damper... 821

perties and hysteretic behaviour of theMRfluid, it does not follow the repro-
duce real behaviour of the MRD with satisfactory accuracy. For this reason,
it is not good enough for the development of the control algorithm.
On one hand, the shortcomings of the Spencermodel were pointed out via

the comparison of predicted (simulation) andmeasured (experiment) results.
On the other, the model improvement, consisting in introducing a nonlinear
function in the form of an envelope for the family curves of the damping force
was presented. Such amodification of the Spencermodel ensures good fidelity
inportraying theactual behaviourof theMRD.For this reason, thegeneralized
model can be successfully used in both system evaluation and control system
design. The latter application was demonstrated on an example of a semi-
active vibration isolation system for a quarter car body.

Fig. 17. Car position – Spencer model

The comparison of time responses for the closed-loop system from Fig.15
and Fig.17, confirms the effectiveness of the generalizedmodel for fluctuating
magnetic fields and the shortcomings of the Spencer model it used in such a
case. For current levels greater than 0.4A, the generalized model behaves like
the Spencer model (see Fig.10).

Acknowledgement

The research work has been supported by the State Committee for Scientific

Research as a part of the grant No. 8T07B03520.

References

1. Ahmadian M., 1999, On the isolation properties of semi-active dampers, Jo-
urnal of Vibration and Control, 217-232



822 B.Sapiński, A.Piłat

2. Choi S.B., Lee S.K., Park Y.P., 2001, A hysteresis model for the field-
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Uogólniony model tłumika magnetoreologicznego przy fluktuacjach pola

magnetycznego

Streszczenie

W pracy opisano (uogólniony) model dynamiczny liniowego tłumika magnetore-
ologicznego uwzględniający wpływ fluktuacji pola magnetycznego w szczelinie robo-
czej na siłę tłumienia.Wykazano, że model ten dokładniej odzwierciedla rzeczywiste
zachowanie tłumikawporównaniu zmodelem sformułowanymprzez Spencera. Zapro-
ponowany model zweryfikowano eksperymentalnie. Dla potwierdzenia efektywności
modelu przedstawiono przykład jego zastosowania w układzie zawieszenia pojazdu,
w którymwyniki symulacji komputerowej zweryfikowano z przeprowadzonym ekspe-
rymentem.

Manuscript received April 14, 2003; accepted for print July 15, 2003