Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

41, 2, pp. 215-240, Warsaw 2003

ANALYSIS OF PARAMETRIC MODELS OF MR LINEAR

DAMPER

Bogdan Sapiński

Jacek Filuś

Department of Process Control, University of Mining and Metallurgy

e-mail: deep@uci.agh.edu.pl

This work deals with the analysis of parametric models of a MR linear
damper, which suit various rheological structures of the MR fluid. The
MR fluid structure and properties, damper description and parametric
damper models which are connected in various ways with the actual
behaviour of the MR fluid are presented. Based on computer simula-
tions, the effectiveness of themodels under predictedMR linear damper
behaviour is shown. The values of the parametric models used in the
simulations were determined in an identification experiment.

Keywords:MRfluid,MRdamper, rheological structure,modelling, iden-
tification, simulation

1. Introduction

MR dampers draw attention not only from a phenomenological point of
view, but also from hope associated with their practical application in vibra-
tion control of a dynamic structure. From the phenomenological viewpoint,
the most important factors are, above all, high non-linear characteristics of
these dampers (Gandhi and Chopra, 1996; Kamath andWerely, 1997; Snyder
et al., 2000; Stanway et al., 2000; Wereley et al., 1998). Accordingly, a hyste-
resis and jump-type phenomenon appears, which characterises the behaviour
ofMRfluids. The explanation for these appearances depends on velocity. The
relationship between the damping force and piston velocity for a linear MR
damper of low efficiency is illustrated in Fig.1. This means that the non-
linearity of hysteresis has greater significance for lower velocities, while for
higher velocities non-linear jumps dominate.



216 B.Sapiński, J.Filuś

Fig. 1. Character of force-velocity curve at: (a) low velocities – hysteresis
phenomenon, (b) higher velocities – jump-type phenomenon

These non-linearities create difficulties in the acceptance of such pheno-
menological MR damper models, which make the behavioural prediction in
its whole operating range impossible. They can also create limitations in the
practical application of MR dampers. However, MR dampers contain certain
features such as, quick response to changes of the control signal, full reflexivi-
ty ofMR fluid transformations and low power consumption. These create the
assumption that, in the future, MR dampers will be applied successfully to
semi-active vibration control systems. Examples of the commercial use ofMR
dampers have already been presented in, among others, Carlson and Sproston
(2000).

In the modelling, MR dampers can be divided into two main methods,
parametric and non-parametric. In the parametric method, the damper is
characterised by a system of linear and non-linear elements, which define the
parameters of springs,dashpots andothermechanical elements, controlling the
mechanismsandtheir operatingarea (Dyke et al., 1996). In thenon-parametric
method, the damper is characterised by special well- suited functions (such
as polynomials, hyperbolic tangent, delay, offset), or by a polynomial with
the power of damper piston velocity (Choi et al., 2001), and the artificial
intelligence methods (fuzzy logic or neural networks) (Sapiński, 2002).

In this paper, the description of parametric models of a MR linear dam-
per closely associating the actual MR fluid behaviour (Table 1) have been
analytically formulated based on rheological laws. In these models, the dam-
per is treated as a stationary system, whose dynamics can be described by
differential equations.



Analysis of parametric models of MR linear damper 217

Table1.Phenomenological parametricmodels ofMRdampers considering
various rheological MR fluid behavioural scenarios

MR fluid rheological behaviour
No. MR dampermodel viscotic elastic plastic histeresis

1 Bingham ✓ ✓

2 Bingham body ✓ ✓ ✓

3 Gamota-Filisko ✓ ✓ ✓

4 Li ✓ ✓ ✓

5 Bouc-Wen ✓ ✓ ✓

6 Spencer ✓ ✓ ✓

The acceptance of various rheological structures for the characterisation
of MR fluid properties shows the effectiveness associated with these models
as tools for prediction of the actual damper behaviour. Necessary for this is
the knowledge of parametric values of themodel, which are designated on the
basis of experiments undertaken for different periodic kinematic excitations
of the piston and at several levels of the control current in the damper coil
(Sapiński, 2002).

2. Structure and properties of MR fluids

MR fluids belong to a group of fluids which are non-Newtonian, rheolo-
gically stable, with a shear yield strength, and are controlled by a magnetic
field. These fluids are non-coloidal suspensionswithmagnetic particles of high
concentration in a non-magnetic fluid carrier. The materials applied as the
magnetic particles are soft magnetic compounds (most common ferrous oxide
Fe2O3 and Fe3O4 with magnetic saturation of about 1.9T). Water, silicon,
mineral or synthetic oil and glycerol act as fluid carriers. Magnetic particles
have a diameter of between 0.5µmand 8µm, therefore theMR fluids are also
known as microfluids, as opposed to nanofluids and ferrofluids (Bolter and
Janocha, 1998).

The rheological properties of MR fluids depend on the concentration of
magnetic particles, their size, shape, properties of fluid carriers,magnetic field
intensity, temperature and also other factors (Jolly et al., 1999; Weiss et al.,
1994). The interrelation of these factors is complex. The most important pa-
rameters ofMRfluids are; maximum shear yield stress (50÷ 100)kPa,maxi-
mummagnetic field ∼=250kA/m, plastic viscosity (0.1 ÷ 1.0)Pa·s, operating



218 B.Sapiński, J.Filuś

temperature (−50 ÷ 150)◦C, density (3 ÷ 4)g/cm3. An essential feature of
a MR fluid is its insensitivity to contamination. When no separate external
magnetic field is applied (H = 0), the concentration of particles is diffused
in the fluid carrier (Fig.2a), their magnetic moments are randomly directed
and resultantly attain the zero value. The MR fluid shows then Newtonian
behaviour.

Fig. 2. Behaviour of MR fluid: (a) H =0, (b) H 6=0

In a magnetic field (H 6= 0), the aggregates are polarised and their ma-
gnetic moments occur along the field lines in theMR fluid. These aggregates
create chain-like structures which run parallel to the field lines (Fig.2b) and
do not show thermal movements. The energy of the mechanisms necessary to
create the chain structures increases alongside any increase in the magnetic
field. TheseMR fluid properties change as a result of a meaningful formation
of the shear stress, which in turn creates an increase in the viscosity.

In their rheological behaviour, MR fluids can be divided into two charac-
teristic areas, pre- and post yield (Li et al., 2000), to which the characteristic
shear stresses τy,d and τy,s are respectively connected (Fig.3). The values of
τy,d and τy,s respectivelymean that; dynamic shear yield stress (Ginder, 1996)
(defined as the point of the zero-rate of the linear regression curve fit) and the
static shear yield stress (defined as the shear stress necessary to initiate MR
fluid flow) depend on themagnetic field.

In the pre-yield area theMRfluid shows visco-elastic properties and a part
of the energy required is regained (elastic behaviour) and a part is dissipated
in the form of heat (viscotic behaviour). This behaviour can be explained by
the linear visco-elastic theory.

The Bingham body model is often applied in the research of MR fluid
behaviour. In this model, in the post-yield area (for a stress greater than the
shear stress, which depends on the magnetic field strain H, i.e. for τ ­ τy,d)



Analysis of parametric models of MR linear damper 219

Fig. 3. (a) Shear stress-shear strain relationship ofMR fluid; (b) observed post-yield
shear behaviour of MR fluid

theMRfluid behaves like a viscoplasticmaterial. Assuming that τy,d = τ(H),
the shear stress in theMR fluid is given by

τ = τy(H)+ηγ̇ τ ­ τy(H) (2.1)

where
η – viscosity
γ̇ – shear strain rate.

However, in the post-yield area (for stress τ < 10−3Pa (Jolly et al., 2000)),
the MR fluid behaves as a visco-elastic material and the fluid stress can be
expressed as

τ =G∗γ τ < τy(H) (2.2)

where G∗ =G′+jG′′, is the complex shearmodulus dependent on themagne-
tic field (Kormann et al., 1994), in which G′ means the storage modulus, and
G′′ – lossmodulus.The storagemodulus is proportionally related to the avera-
ge energy required during one fluid deforming cycle in volume units, however,
the loss modulus is proportional to the average energy stored during a cycle
in fluid volume units. In reality, the MR fluid behaviour shows wide depar-
tures from Bingham’s model, the most important of which is non-Newtonian
behaviour for H =0.
The research on MR fluid reactions sometimes shows that through jump-

type changes inmagnetic fields, thefluid stress changes have anasymptotically
exponential character and gain 90% of their value in the time less than 10ms.
This means that MR fluids change their viscosity rapidly. It was stated in
(Snyder et al., 2000) that the changes inMR fluid properties from viscotic to



220 B.Sapiński, J.Filuś

elastic follow a stress lower than 0.08% of the limited stress, which may limit
the application of aMR fluid to adaptive structures, in which the stability is
required in the pre-yield area.

3. Description of a MR damper

The construction of aMR linear damper is shown in Fig.4a (longitudinal
section) andFig.4b (cross section). The damper has a cylindrical shape and is
filledwith aMRfluid.The piston constructionwith an integrated coil ensures
that the magnetic field is focused within a gap, inside a volume of the active
portion of theMR fluid.

Fig. 4. (a) Longitudinal section: 1 – piston, 2 – rod, 3 – coil, 4 – gap, 5 –MR fluid,
6 – wires, 7 – housing, 8 – accumulator; (b) cross-section: 1 – piston, 3 – coil, 4 – gap

The damper operation is based on using the MR effect, which amounts
to quick changes in the viscosity of the active portion of the fluid in the
gap (Kordoński, 1993; Shulman and Kordoński, 1978). In the absence of a
magnetic field (I = 0 ⇒ H = 0), the aggregates of magnetic particles are
suspended in the carrier fluid (Fig.2a), their magnetic moments are without
any ordered structure and the resultant magnetic moment is equal to zero.
In situations when external forces act on the piston, the damper resistance
force depends, above all, on the fluid flow strain through the gap, resulting
from the pressure differences between the cells from the fluid. Then the MR
damper behaves like a viscotic damper. In the presence of an external field
(I 6=0 ⇒ H 6=0), the aggregates are polarized and their magnetic moments
are arranged along the field lines. These aggregates form chain-like structures
parallel to the field lines (perpendicular to the fluid flow direction), thereby
increasing the shear stress and the fluid viscosity and restricting its motion
(Fig.2b). Themagnetic energy required to form such structures increases due



Analysis of parametric models of MR linear damper 221

to the increase in the magnetic field. As a result, the fluid flow through the
gapbecomes limited,which draws increased hydraulic resistance for the piston
movement and creates an additional damping force component, depending on
themagnetic field (intensity of the current in the coil).

Due to the fact that the height of the gap in the damper is considerably
less than its width and length (Fig.4a and Fig.4b) it can be seen that an
adequate, accurate and comparative description of themathematical fluid flow
through the gap is amodel with parallel plates (Spencer et al., 1998). On this
assumption, the input to the damping force consists of three parts: the force
resulting from the static friction (depending on the type used in the damper
seal), the force depending on the fluid viscosity, and the force depending on
the magnetic field (more precisely, on the distribution of the magnetic flux
density in the gap).

In order to control the MR damper, the pulse width modulation me-
thod (PWM) of the current in the coil is used. The PWM controller creates
rectangular voltage signals of an amplitude Am, and changeable co-factors
αw = Ti/Tp, where Ti means the duration of the pulse, and Tp is set as the
pulse repeat period (Fig.5). If the damper coil is supplied by the voltage si-
gnal with this shape, the magnetic flux distribution is changed, which results
in changes in the field line set-up in the damper magnetic circumference and
related the damping force process.

Fig. 5. Pulse width modulation

The theoretical evaluation of the magnetic flux density set up along the
gap l and themagnetic field in thedamper, in the casewhenthevoltage oncoil
terminals is U =0V ⇒ I =0A and U =3.0V ⇒ I =0.6A, are shown in
Fig.6 andFig.7, respectively (Sapiński et al., 2001). The experimental process
of the damping force for sinusoidal kinematic excitation of the damper piston
(at the amplitude of 4 · 10−3m and frequency 2.5Hz) and the step voltage
input, U = 3.0V (I = 0.6A) and the connection time t1 = 0.488s, and
disconnections time t2 =0.992s are shown in Fig.8 (Sapiński et al., 2001).



222 B.Sapiński, J.Filuś

Fig. 6. Distribution of magnetic flux density along the gap for I =0.0A and
I =0.6A

Fig. 7. Distribution of magnetic field lines: (a) I =0.0A, (b) I =0.6A

4. Parametric models of MR linear dampers

Theparametricdampermodelsare formulatedusingrheological structures,
which, in various ways, approximate to the MR fluid behaviour, have direct
influence on the accuracy of the damper behaviour prediction. Assuming, for
example, that the MR fluid does not show hysteresis, it may be treated as
a visco-plastic or visco-elastic material, which leads in the first case to the
Bingham model, and in the second to the Bingham body, Gamota-Filisko
and Li models. However, in view of the development in the MR fluids, the
appearance of hysteresis leads to the Bouc-Wen or Spencer models. In order
to analyse the above models, identification experiments were undertaken, the
aim ofwhichwas to designate the parameter values for thesemodels necessary
for simulation (Sapiński, 2002).



Analysis of parametric models of MR linear damper 223

Fig. 8. Displacement vs. time; curent in the coil vs. time; damping force vs. time

4.1. Bingham model and Bingham body model

The idealisation of the visco-plastic MR dampermodel presented in Dyke
et al. (1996) uses similarities in the rheological behaviour of ERandMRfluids
and the similar techniques in themodelling of ER dampers (Sims et al., 2000;
Stanway et al., 2000).

Fig. 9. Rheological structure of aMR damper for the Binghammodel

In the rheological structure inFig.9, onwhich theBinghammodel is based,
there is a Coulomb friction element fc placed parallel to the dashpot c0.
According to Bingham’s MR damper model, for non-zero piston velocities ẋ,
the damping force F can be expressed as

F = fc sgnẋ+c0ẋ+f0 (4.1)

where fc is the frictional force, c0 is the viscous damping parameter, f0 is the
forcedue to thepresenceof theaccumulator.This last simplification in themo-



224 B.Sapiński, J.Filuś

del results from the assumption that the elasticity replacing the accumulator
activity has a low stiffness and linear characteristics.

Fig. 10. Rheological structure of aMR damper for the Bingham body model

The Bingham bodymodel, whose structure is presented in Fig.10, differs
from the Binghammodel (Fig.9) by the introducing of a spring k. Thus, the
Bingham body model tries merging of three elements, that is, connecting in
parallel the elements of St.Venant (plastic bodymodel),Newton (Newtonflow
model) and the element ofHooke (elastic bodymodel).Thismodel, through its
low shear stress, presents solid body behaviour, however only through a high
shear stress, the liquid body. This occurs because, to a certain value of the
applied force fc (static friction force of the St. Venant element which refers to
the shear stress τy(H) characteristic) only the springwill deform– similarly to
the elastic Hooke body. If this force is greater than fc the Bingham bodywill
elongate (the body flows). The rate of the deformation will be proportional
to the difference of the applied force and the friction force of the St. Venant
element (i.e. γ̇∼ [τ −τy(H)]).
According to the Bingham bodymodel (see Fig.10), the damping force F

can be expressed as

F =

{

fc sgnẋ1+ c0ẋ1+f0 for |F|>fc

k(x2−x1)+f0 for |F| ¬ fc
(4.2)

where the parameters fc, c0, f0 have the same meaning as in equation (4.1)
and k represents the stiffness of the elastic body (Hooke model).

4.2. Gamota-Filisko model

An extension of theBinghamMRdampermodel is the visco-elasto-plastic
model formulated by Gamota and Filisko (Dyke et al., 1996). This extension
depends on connection of the Bingham, Kelvin-Voight body and Hooke bo-
dy models (Fig.11). The Kelvin-Voight model represents a solid body, whose
maximumelongation exclusively depends on the applied force (independent of
time).



Analysis of parametric models of MR linear damper 225

Characteristic for thismodel is theappearanceof the creepingphenomenon
(elongation gradually increases as a result of delayeddamperactivity).Hooke’s
model, in turn, for which the spring elongation is proportional to the applied
force and independent of time, represents the ideal elastic body.

Fig. 11. Rheological structure of aMR damper for the Gamota-Filiskomodel

The damping force in the Gamota-Filisko model (see Fig.11) can be de-
scribed as

F =











k1(x2−x1)+ c1(ẋ2− ẋ1)+f0 =
c0ẋ1+fc sgnẋ1+f0 = k2(x3−x2)+f0

for |F|>fc

k1(x2−x1)+ c1ẋ2+f0 = k2(x3−x2)+f0 for |F| ¬ fc

(4.3)

where
c0,f0,fc – parameters representing the viscous damping, force due

to the presence of the accumulator, frictional force (Bin-
gham model)

k1,c1 – parameters representing the stiffness and damping of the
body (Kelvin-Voigt model)

k2 – parameter representing the stiffness of the elastic body
(Hooke model).

It should be noted that when |F| ¬ fc, then ẋ1 = 0, which means that
when the friction force fc related with the new stress in the fluid is greater
than the damping force F, the piston remains motionless.

4.3. Li model

Another view of visco-elasto-plastic properties of MR fluids in the model-
ling of MR damper behaviour is the model composed by Li (Li et al., 2000).
This model is divided into two areas; pre- and post-yield. In these areas the
MR fluid shows visco-elastic and visco-plastic body properties, respectively,
which conform to the rheological structures presented in Fig.12.



226 B.Sapiński, J.Filuś

Fig. 12. Rheological structure of aMR damper for the Li model

Li proposed a description of a MR fluid state in the pre-yield area of a
visco-elasticmodel, inwhichHooke’s body (spring k2)was joinedwithKelvin-
Voight’s body (spring k1 and dashpot c1). Besides the visco-elastic force fve,
a contribution to the damping force F in the pre-yield area also carries the
static friction force fs, resulting from the applied type of seal in the damper.
Accordingly, the damping force F in the pre-yield area can be written as

F = fve+fs (4.4)

where the force fve is described by the equation

ḟve+
k1+k2
c1
fve =

k1 ·k2
c1
x+k2ẋ (4.5)

where thedamping force F crosses theplastic flow force fc (when τ ­ τy(H)),
the damper operates in the post-yield area. Then damping force is also equal
to the visco-plastic force, to which, besides the friction force connected with
the fluid shear stress fc, the viscotic force and inertial force contribute, which
can be written as

F = fc sgnẋ+ c2ẋ+mẍ (4.6)

where c2 is a co-factor of viscotic friction, and m is the mass of replaced
MRfluid dependent on the amplitude and frequency of a kinematic excitation
applied to the piston. As such, the damping force in the Limodel is expressed
as: equation (4.4) in the pre-yield area (|F| ¬ fc) and equation (4.6) in the
post-yield area (|F|>fc).

4.4. Bouc-Wen model

In structures showing hysteresis, through dynamic pressure (especially
when the considered structure has a non-elastic character), the restoring for-
ce depends not only on the moment of the displacement, but also on the



Analysis of parametric models of MR linear damper 227

past events of the structure. This fact creates certain problems in the model-
ling of structures with hysteresis creating vibration situations. To solve this
problem, various approximation methods are applied (Brokate and Sprekels,
1996), which lead to variousmodels allowing the consideration of the hystere-
sis in theMR andERdamper behaviour.McClamroch andGavin (1995) and
Hsu andMeyer (1995) used trigonometric functions to describe the hysteresis.

Fig. 13. Rheological structure of a MR damper for the Bouc-Wenmodel

This approach, however, does not allow the consideration of the damping
force saturationwhichoccurs inperiodsof highpistonvelocity. In turn,Werely
et al. (1998) presented a biviscotic model with hysteresis, Sims et al. (2000)
modified Bingham’s plastic model (Dyke et al., 1996), and the Choi (Choi
et al., 2001) model, in which the dependence of the damping force on the
velocity is a 6th order polynomial with respect to the piston velocity of the
damper. A further development of the MR damper model, considering the
appearance of the hysteresis and damping force saturation, is the Bouc-Wen
model (Dyke et al., 1996), whose rheological structure is presented in Fig.13.
In this model the hysteresis approximation method proposed by Wen (1976)
is applied. Accordingly, in structures showing hysteresis, the restoring force
Q(x,ẋ) is the sum of two components, without the hysteresis g(x,ẋ) and
with h(x)

Q(x,ẋ)= g(x,ẋ)+h(x)+ ... (4.7)

The component h(x) is defined by an equation, which refers to the displa-
cement x and z (evolutional variable), through which the position of the
equation is dependent onwhether n is an even or odd number. For an odd n,
the equation is

ż=−a|ẋ|zn−βẋ|zn|+Aẋ (4.8)

however for an even number

ż=−a|ẋ|zn−1−βẋzn+Aẋ (4.9)



228 B.Sapiński, J.Filuś

Fig. 14. Influence of parameters: n, β, γ,A, α on force-velocity curves

The influence of n on the dependence of the MR damper force on the
velocity, through the chosenparameters of theBouc-Wenmodel and for n=1,
n=2, n= 3, is shown in Fig.14a. In Wen (1976) it is stated that using this
method, the prediction accuracy of behavioural structures with hysteresis is
satisfactory as compared with the experimental results gained for n=2.
The damping force in the Bouc-Wen model can be written as

F = c0ẋ+k0(x−x0)+αz (4.10)

where the evolutional variable z is described by the equation

ż=−γ|ẋ|z|z|n−1−βẋ|z|n+Aẋ (4.11)



Analysis of parametric models of MR linear damper 229

where
β,γ,A – parameters representing the control of the linearity du-

ring unloading and the smoothness of the transition from
the pre-yield to post-yield area

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

k0 – parameter representing the stiffness of the spring associa-
ted with the nominal damper due to the accumulator

c0 – parameter representing viscous damping
c1 – parameter representing the dashpot included in the mo-

del to produce the roll-off at low velocities
x0 – parameter representing the initial displacement of the

spring with the stiffness k0.

The influence of the parameters β, γ, A, α on the accuracy of hysteresis
prediction in the damping force-velocity curve is shown for the Bouc-Wen
model in Fig.14b,c,d,e.

4.5. Spencer model

The extension of the Bouc-Wen model proposed by Spencer concerns the
introduction of an additional dashpot c1 and spring k1. This suits the rheolo-
gical structure shown in Fig.15.

Fig. 15. Rheological structure of aMR damper for the Spencer model

Damping force in Spencer’s model can be expressed as

F = az+ c0(ẋ− ẏ)+k0(x−y)+k1(x−x0) (4.12)



230 B.Sapiński, J.Filuś

or can be also written as

F = c1ẏ+k1(x−x0) (4.13)

where the displacements z and y are respectively defined as

ż=−γ|ẋ− ẏ|z|z|n−1−β(ẋ− ẏ)|z|n+A(ẋ− ẏ)
(4.14)

ẏ=
1

c0+c1
[az+ c0ẋ+k0(x−y)]

The parameter c1 suits viscotic damping at higher velocities, the parameter
k1 corresponds to the stiffness representing the accumulator, the parameter k0
stands for the stiffness at higher velocities, however, the remaining parameters
in equations (4.12)-(4.14) have the samemeaning as in the Bouc-Wen model.

5. Identification experiment

In order to simulate thedamperbehaviour through theuse of thedescribed
models it is necessary todefine thevalues of their parameters.For thispurpose,
experimental tests were carried out on the MR damper of RD-1005 type,
produced by Lord Corporation. The scheme of the experimental setup for the
damper tests is shown in Fig.16.

Fig. 16. Scheme of the experimental setup for examinations of aMR linear damper

The tests were conducted at the experimental setup, see Fig.16, with a
computer-controlled INSTRON test machine, data acquisition system with
the multi I/O board of RT-DAC3 type and in the software environment of
Matlab/Simulink, and Real Time Workshop Target. The machine was pro-
grammed to move up and down in a sinusoidal wave for seven levels of the



Analysis of parametric models of MR linear damper 231

frequency f (0.5, 1, 2.5, 4, 6, 8, 10)Hz. The responsesweremeasured for eight
levels of the control current I (0.0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.6)A. The way
of conducting the tests and the parameter identification methods for the Bin-
gham and Spencer models is carefully described by Sapiński (2002). Due to
the requirements of this work, the identificationwas extended to theBingham
body, Gamota-Filisko, Li and Bouc-Wen models.

For the parameter computation the following criteria have been adopted
(Spencer et al., 1998)

εt =

T
∫

0

(Fe−Fm)
2 dt εx =

T
∫

0

(Fe−Fm)
2
∣

∣

∣

dx

dt

∣

∣

∣
dt

(5.1)

εẋ =

T
∫

0

(Fe−Fm)
2
∣

∣

∣

dẋ

dt

∣

∣

∣ dt

where Fe and Fm denote the damping force derived from the experiments
and models, respectively. The criteria define for each model the difference
between Fe and Fm as a function of time (5.1)1, displacement (5.1)2 and
velocity (5.1)3. For computation theCONSTR subroutine available inMatlab
optimisation toolbox was used. The procedure finds the local minimumof the
function of several variables under some constrains imposed on the variables.
The parametric values of themodels obtained by this procedure are presented
in Table 2-7.

Table 2.Parameters of Bingham model

Current Value of the parameter

I [A] fc [N] c0 [Ns/m] f0 [N]

0.0 43.95 735.90 195.51

0.4 262.13 3948.70 186.28

Table 3.Parameters of Bingham bodymodel

Current Value of the parameter

I [A] fc [N] c0 [Ns/m] f0 [N] k [N/m]

0.0 43.95 735.90 195.51 895.00

0.4 262.13 3948.70 186.28 3623.00



232 B.Sapiński, J.Filuś

Table 4.Parameters of Gamota-Filisko model

Current Value of the parameter

I [A] fc [N] c0 [Ns/m] c1 [Ns/m] k1 [N/m] k2 [N/m] fs [N]

0.0 43.95 735.90 2026.00 1387.00 895.00 195.51

0.4 262.13 3948.70 16487.00 8421.00 3623.00 186.28

Table 5.Parameters of Li model

Current Value of the parameter

I [A] fc [N] fs [N] c1 [Ns/m] c2 [Ns/m]

0.0 43.95 195.51 4983.00 12120.00

0.4 262.13 186.28 9987.00 6103.00

Current Value of the parameter
I [A] k1 [N/m] k2 [N/m] m [kg]
0.0 998.10 10053.00 0.07
0.4 998.10 61432.00 0.07

Table 6.Parameters of Bouc-Wen model

Current Value of the parameter

I [A] α [N/m] c0 [Ns/m] A [-] β [m
−1]

0.0 7507.50 549.86 125.18 1097222.00

0.4 38988.04 3195.43 215.20 1880301.00

Current Value of the parameter
I [A] γ [m−1] k0 [N/m] x0 [kg]
0.0 1340385.00 1610.14 0.13
0.4 1696138.00 2169.09 0.92

Table 7.Parameters of Spencer model

Current Value of the parameter

I [A] α [N/m] c0 [Ns/m] c1 [Ns/m] k0 [N/m] A [-]

0.0 24830.30 553.90 38519.90 1037.4 25.70

0.4 86665.00 1965.10 20626.50 2807.8 29.37

Current Value of the parameter
I [A] β [m−1] γ [m−1] k1 [N/m] x0 [kg]
0.0 1387320.0 3862730.00 651.50 0.30
0.4 1198420.0 52260.00 675.30 0.27



Analysis of parametric models of MR linear damper 233

6. Simulation of behaviour of a MR linear damper

Based on the resulting parameters, the theoretical relationships were esta-
blished for: damping force vs. time, damping force vs. displacement and dam-
ping force vs. velocity, using Matlab/Simulink. As an example, these curves
compared with the experimental data for f =2.5Hz; I =0A and I =0.4A
are shown inFig.17,Fig.18 andFog.19, respectively.The lines ( ) indica-
te the experimentaldata,while the theoretical curves aremarkedwith ( ).

Fig. 17. Comparison of force-time, force-displacement and force-velocity curves

betweenmeasurement ( ) and prediction ( ) for the Binghammodel: (a),

(b), (c), for the Bingham body model: (d), (e), (f)

Analysis of the results gained from the simulations and their comparison
with the experimental data allows us to estimate the accuracy with which
we can predict the actual MR damper behaviour using the consideredmodels
with regard to various MR fluid properties (visco-plastic, visco-elasto-plastic,
and visco-elastic with hysteresis).

The quality of the visco-plastic Bingham model is its simplicity, however
its shortcoming is that for zero piston velocity the damping force does not



234 B.Sapiński, J.Filuś

Fig. 18. Comparison of force-time, force-displacement and force-velocity curves

betweenmeasurement ( ) and prediction ( ) for the Gamota-Filisko

model: (a), (b), (c), for the Li model: (d), (e), (f)

equal zero, and that the force in the friction element is equal to the applied
force. The force-velocity curve, which can be predicted with the use of this
model, is one-to-one, but does not suit the experimental data (that is not one-
to-one).At thevelocity equal to zero, themeasured force andacceleration have
opposite signs: theadditional damping force–negative acceleration (additional
displacement) and vice-versa.

As a result of the spring employed in theBinghammodel, theMRdamper
parametric model representing the visco-elasto-plastic model of the Bingham
bodywas obtained. The effect of this spring is the proportional change of the
damping force with velocity.

The Gamota-Filisko model allows us to predict the force-velocity curve
which more closely relates to the experimental data than in the Bingham
body model. The characteristic feature of this model is that if the velocity
approaches zero, the decrease in the damping c1, may generate the roll-off
observed in the force-velocity curve (Dyke et al., 1996).

The advantage of the visco-elasto-plastic model proposed by Li lies in the
possibility ofmore accurate capturing ofMRdamper dynamic properties than



Analysis of parametric models of MR linear damper 235

Fig. 19. Comparison of force-time, force-displacement and force-velocity curves

betweenmeasurement ( ) and prediction ( ) for the Bouc-Wenmodel:

(a), (b), (c), for the Spencer model: (d), (e), (f)

in theGamota-Filiskomodel. That is due to distinguishingpre- andpost-yield
areas of the MR fluid operating range. In addition, this model enables us to
investigate the influence of the MR fluid inertia on the damping force (Li et
al., 2000).

The force-displacement curves predicted by the Bouc-Wen model are ac-
curate in comparison to the experimental data (more accurate than in the
visco-elasto-plastic models described above). However, at low velocities (si-
milarly to the Bingham model) there appears a nonlinear character of the
force-velocity curves, and therefore, it does not apply to the area in which the
accelaration and the velocity have different signs.

The visco-elastic model with the hysteresis formulated by Spencer enables
us to the most accurately predicted actual MR damper behaviour (from the
discussed models) in the whole operating range, also in the area of low velo-
cities in which the acceleration and the velocity have different signs. It refers
to all curves, i.e. force-time, force-displacement and force-velocity ones.



236 B.Sapiński, J.Filuś

Fig. 17. Comparison of predicted force-velocity curves for consideredMR damper
models: f =2.5Hz, I =0.4A

7. Summary

In this paper, analysis of parametric models of MR linear dampers has
been undertaken. The analysis has been concerned with various behaviour
characteristics of a MR fluid, which fills the damper, i.e. visco-plastic, visco-
elasto-plastic and visco-elastic with a histeresis. Based on the identification
experiment, the parameters for these models were designated for a sinusoidal
kinematic excitation and constant control current. The analysis showed that
basic difficulties in formulation of the models, which could effectively portray



Analysis of parametric models of MR linear damper 237

the actual behaviour of the MR damper were caused by the hysteresis and a
jump-type phenomenon resulting from the specific properties of theMRfluid.
This is illustrated in Fig.20, in which the force-velocity curves computed for
each considered model are presented.

It can be clearly seen that if the rheological structure representing the
fluid behaviour in the model is too simple, then the predicted curves (force
vs. time, force vs. displacement and force vs. velocity) becomes less accurate.
The consequence of this is limitation of the applied models, an example of
which may be Bingham’s model, which captures damper features well, but
inaccurately represents its behaviour for near-zero piston velocity. Thismeans
that Bingham’s model is not useful in control problems. The key factor is
to take into account the magnetic field saturation, which is an inherent MR
damper feature.
There are also difficulties in finding the solution to some equations of the

analysed models. This concerns, in particular, extended models taking into
account the appearance of a hysteresis (Bouc-Wen and Spencer), which are
difficult to solve analytically. Similarly, theGamota-Filisko and Limodels are
difficult to solve numerically (numerical integration reguires a time step of
10−6 s). As experience shows, the stiff character of differential equations in the
Gamota-Filisko and Li models entails that they can not be used as governing
formulas for real-time controllers.
Which of the analysed models will be useful for the control problems will

be decided by the future research aiming at the obtaining of the actual rela-
tionship between themodel parameters and the control signal.

Acknowledgement

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

Research as a part of grant No. 8T07B03520

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Analiza modeli parametrycznych liniowego tłumika

magnetoreologicznego

Streszczenie

W pracy dokonano analizy parametrycznychmodeli fenomenologicznych liniowe-
go tłumika magnetoreologicznego (MR), które odpowiadają różnym strukturom re-
ologicznymcieczyMRwypełniającej tłumik. Przedstawionowłasności cieczy, budowę



240 B.Sapiński, J.Filuś

tłumika orazmodeleparametryczne tłumikaprzybliżającewrożny sposób rzeczywiste
zachowanie cieczy MR. W oparciu o symulacje komputerowe pokazano efektywność
modeli do przewidywania rzeczywistego zachowanie liniowego tłumika MR.Wartości
parametrówmodeli wykorzystywanew symulacjachwyznaczono na podstawie ekspe-
rymentu identyfikacyjnego.

Manuscript received October 1, 2002; accepted for print January 14, 2003