150

CHAPTER 7

DYNAMIC MODELING OF MR DAMPERS

Quasi-static models of MR dampers have been shown to be effective for design pur-

poses. Although those models are capable of describing the MR damper force-displace-

ment behavior reasonably well, they are not sufficient to describe the nonlinear damper

force-velocity behavior. Therefore, a more accurate dynamic model of MR dampers is

necessary to describe damper behavior and for structural vibration control design and sim-

ulation with MR dampers. In this chapter, following a review of several models for con-

trollable fluid dampers, a phenomenological model based on the Bouc-Wen hysteresis

model is proposed. This model accommodates the MR fluid stiction phenomenon, as well

as inertial and shear thinning effects. Compared with other types of models based on the

Bouc-Wen model, the proposed model has been shown to be more effective, especially in

describing the force roll-off in the low velocity region, force overshoots when velocity

changes in sign, and two additional clockwise hysteresis loops at the velocity extremes.

7.1 Introduction

As clearly shown in Chapter 5, quasi-static models are not sufficient to describe the

dynamic behavior of MR dampers, especially the nonlinear force-velocity behavior. As a

direct extension of the Bingham plasticity model, an idealized mechanical model was pro-

posed by Stanway et al. (1987). This model is shown in Fig. 7.1. In this model, a Coulomb

151

friction element is placed in parallel with a linear viscous damper. The force generated by

the damper is

(7.1)

where = post-yield plastic damping coefficient; and = friction force, which is related

to the fluid yield stress. The Bingham plasticity model is intrinsically the same as the qua-

si-static models developed in the previous chapter. Therefore, it is not surprising that the

MR damper force-displacement behavior is reasonably modeled; however, the nonlinear

force-velocity behavior is not captured (Spencer et al. 1997a).

Two types of dynamic models for controllable fluid dampers have been investigated

by researchers: non-parametric models and parametric models. Ehrgott and Masri (1992)

and Gavin et al. (1996b) presented a non-parametric approach employing orthogonal Che-

bychev polynomials to predict the controllable fluid damper force using the damper dis-

placement and velocity information. Chang and Roschke (1998) developed a neural

network model to emulate the dynamic behavior of MR dampers. However, the non-para-

metric damper models are quite complicated.

x

c0

f

f f

Figure 7.1: Bingham model for MR dampers (Stanway et al. 1987).

f ff x·( ) c0x·+sgn=

c0 ff

152

Gamato and Filisko (1991) proposed a parametric viscoelastic-plastic model based on

the Bingham model. The schematic is shown in Fig. 7.2. The governing equations for this

model are

(7.2)

(7.3)

where = damping coefficient associated with the Bingham model; and = pa-

rameters associated with the linear solid material. Wereley et al. (1998) developed a non-

linear hysteretic biviscous model, as shown in Fig. 7.3. This model is an extension of the

nonlinear biviscous model having an improved representation of the pre-yield hysteresis.

The equations of the nonlinear hysteretic biviscous model are

x1

c0

x3

f

c1

x2

k2

k1

ff

Figure 7.2: Viscoelastic-plastic model proposed by Gamota and Filisko (1991).

f k1 x2 x1–( ) c1 x·2 x·1–( )+=

c0x1 ff x·1( )sgn+=

k2 x2 x1–( )=

f ff>

f k1 x2 x1–( ) c2x·2+=

k2 x3 x2–( )=

f ff≤

c0 k1 k2, c1

153

(7.4)

where = decelerating and accelerating yield velocity, given by

(7.5)

The parametric models mentioned above can describe the nonlinear force-displacement

and force-velocity behavior. However, they are not readily able to capture the force roll-

off in the low velocity region that is observed in the experimental data (Spencer et al.

1997a).

To overcome this difficulty, Spencer et al. (1997a) proposed a phenomenological

model for MR dampers based on the Bouc-Wen hysteresis model. The schematic of the

model is illustrated in Fig. 7.4. In this model, the total damper resisting force is given by

Figure 7.3: Nonlinear hysteretic biviscous model (Wereley et al. 1998).

v

fv1

v2

fy

fy–

v0

v– 0

Cpr

Cpo

Cpo

Cpr

f

Cpov fy– v v1–≤ v· 0>

Cpr v v0–( ) v1 v≤– v2≤ v· 0>

Cpov fy+ v2 v≤ v· 0>

Cpov fy+ v1 v≤ v· 0<

Cpr v v0+( ) v2 v≤– v1≤ v· 0<

Cpov fy– v v2–≤ v· 0<

=

v1 v2,

v1

fy Cprv0–

Cpr Cpo–----------------------- and v2

fy Cprv0+

Cpr Cpo–------------------------= =

154

(7.6)

where and is governed by

(7.7)

(7.8)

in which = accumulator stiffness; = viscous damping at large velocities; = vis-

cous damping for force roll-off at low velocities due to bleed or blow-by of fluid between

the piston and the cylinder; = stiffness at large velocities; and = initial displacement

of the spring . This model has shown to be effective in modeling the LORD RD-1000

seat damper (Spencer et al. 1997a).

7.2 Proposed Dynamic Model of MR Dampers

Based on the damper response analysis in Chapter 5.4.2, a phenomenological model

is proposed which considers the MR fluid stiction phenomenon, as well as inertial and

Figure 7.4: Phenomenological model of MR dampers (Spencer et al. 1997a).

fc0

k0c1

k1

Bouc-Wen

xy

f c1y· k1 x x0–( )+=

z y

z· γ x· y·– z zn 1–

– β x· y·–( ) zn

– A x· y·–( )+=

y· 1c0 c1+---------------- αz c0x· k0 x y–( )+ +{ }=

k1 c0 c1

k0 x0

k1

155

shear thinning effects. The schematic of the model is shown in Fig. 7.5. The damper force

is given by

(7.9)

where the evolutionary variable is governed by

(7.10)

In this model, = equivalent mass which represents the MR fluid stiction and inertial ef-

fect; = accumulator stiffness and MR fluid compressibility; = damper friction force

due to seals and measurement bias; and = post-yield plastic damping coefficient. To

describe the MR fluid shear thinning effect which results in the force roll-off of the damp-

er resisting force in the low velocity region, the damping coefficient is defined as a

mono-decreasing function with respect to absolute velocity . In this chapter, the post-

yield damping coefficient is assumed to have the form

(7.11)

x

k

Bouc-Wen

f - f0

c(x).

m

Figure 7.5: Proposed phenomenological model considering MR fluid stiction phenomenon, as well as inertial and shear thinning effects.

f f0– αz kx c x·( )x· mx··+ + +=

z

z· γ x· z zn 1–

– βx· zn

– Ax·+=

m

k f0

c x·( )

c x·( )

x·

c x·( ) a1ea2 x·( )p–

=

156

where and = positive constants. Note that the mass element is employed to

phenomenolocially emulate the MR fluid stiction phenomenon and fluid inertial effect.

The mass element was also used by Gamota and Filisko (1991), Powell (1995), and Ka-

math and Wereley (1997) in their models.

7.3 Model Comparison

In this section, the proposed dynamic model is compared with three other types of

models, all based on the Bouc-Wen hysteresis model. These models are illustrated in Fig.

7.6. To assess their ability to predict the MR damper behavior, these four dynamic models

are fitted to the damper response under a 1 inch, 0.5 Hz sinusoidal displacement excitation

with an input current of 2 A.

The first model employed is the simple Bouc-Wen model shown in Fig. 7.6a (Spencer

et al. 1997a), where the damping coefficient is set at a constant and the mass element is

not included. Fig. 7.7 provides a comparison between the simple Bouc-Wen model and

experimental results. As can be seen, this model cannot capture the force roll-off in the

low velocity region, as well as the two clockwise loops at the velocity extremes that are

due to the inertial effect. The parameters for the simple Bouc-Wen model are chosen to be

, , , , ,

, , and .

To accommodate the inertial effect, a mass element is added to the simple Bouc-Wen

model, as shown in Fig. 7.6b. As can been seen in Fig. 7.8, the simple Bouc-Wen model

with a mass element has an improvement on modeling the two clockwise loops at the

velocity extremes. However, it still fails to portray the force roll-off at low velocities. The

a1 a2, p m

α 318470 N= n 2.3983= γ 3819.4 m1–

= β 100.1 m1–

= A 833.45 m1–

=

c 687300 N sec m⁄⋅= k 146.81 N/m= f0 1456– N=

157

parameters for this model are chosen to be , ,

, , , ,

, , and .

In order to achieve better modeling of the force roll-off, Spencer et al. (1997a) pro-

posed a phenomenological model (Fig. 7.4). To accommodate the inertial effect, an addi-

tional mass element is included in the model as shown in Fig. 7.6c. In this model, two

different damping coefficients, and , are employed. The damping coefficient is

Figure 7.6: Four types of dynamic models of MR dampers based on the Bouc-Wen hysteresis model (Spencer et al. 1997a)

c(x)

k

Bouc-Wen

.

x

f - f0m

Bouc-Wen

c

kf - f0

x

c

k

Bouc-Wen

x

f - f0m

c0

k0

c1

k1

Bouc-Wen

xy

f - f0m

Simple Bouc-Wen Model Simple Bouc-Wen Model with Mass Element

Proposed Dynamic Model ConsideringShear Thinning and Inertial Effects

Phenomenological Model

(a) (b)

(c) (d)

with Mass Element

α 472710 N= n 2.2021=

γ 5100.7 m1–

= β 150.0 m1–

= A 550.57 m1–

= m 14424 kg=

c 643230 N sec m⁄⋅= k 279.14 N/m= f0 1449.3 N–=

c0 c1 c0

158

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-250

-200

-150

-100

-50

0

50

100

150

200

250

Time (sec)

For

ce (

kN)

-4 -2 0 2 4-250

-200

-150

-100

-50

0

50

100

150

200

250

Displacement (cm)

For

ce (

kN)

Figure 7.7: Comparison between the experimentally-obtained and predicted responses using the simple Bouc-Wen model: (a) force vs. time; (b) force vs. displacement; and

(c) force vs. velocity.

-8 -4 0 4 8-250

-200

-150

-100

-50

0

50

100

150

200

250

Velocity (cm/sec)

For

ce (

kN)

Experimental Result

Simple Bouc-Wen Model

(a)

(b) (c)

159

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-250

-200

-150

-100

-50

0

50

100

150

200

250

Time (sec)

For

ce (

kN)

-4 -2 0 2 4-250

-200

-150

-100

-50

0

50

100

150

200

250

Displacement (cm)

For

ce (

kN)

Figure 7.8: Comparison between the experimentally-obtained and predicted responses using the simple Bouc-Wen model with mass element: (a) force vs. time; (b) force vs.

displacement; and (c) force vs. velocity.

-8 -4 0 4 8-250

-200

-150

-100

-50

0

50

100

150

200

250

Velocity (cm/sec)

For

ce (

kN)

Experimental Result

Simple Bouc-Wen Model with Mass

(a)

(b) (c)

160

used for large velocities; and is used for low velocities. The parameters for this phe-

nomenological model are chosen to be , , ,

, , , ,

, , , and

. Fig. 7.9 provides a comparison between the predicted and

experimentally-obtained results. As shown in the figure, this mechanical model has only a

slight improvement over the simple Bouc-Wen models in modeling the force roll-off at

low velocities.

The proposed model utilizes a variable mono-decreasing damping coefficient to

describe the force roll-off due to the MR fluid shear thinning effect and a mass element to

accommodate the inertial effect. The comparison between the theoretical result and exper-

imental data is given in Fig. 7.10. It can be readily seen that this model predicts the

damper behavior very well in all regions, including the force roll-off at low velocities,

force overshoots when velocity changes in sign, and the two clockwise loops at velocity

extremes. The parameters for the proposed model are chosen to be ,

, , , , ,

, , , ,

and .

In addition to the graphical evidence of the superiority of the proposed model, a quan-

titative study of the errors between each of the models and the experimental data is con-

ducted. For each of the models considered here, the error between the predicted force and

the measured force has been calculated as a function of time, displacement and velocity

over one complete cycle. The following expressions have been used to represent the errors

c1

α 271790 N= n 6.6862= γ 4429.6 m1–

=

β 8446.5 m1–

= A 841.41 m1–

= m 14929 kg= k0 55.9152 N/m=

f0 1251.7– N= k1 64.0617 N/m= c0 533100 N sec m⁄⋅=

c1 28566000 N sec m⁄⋅=

α 927570 N=

n 2.7755= γ 31778 m1–

= β 21.637 m1–

= A 217.27 m1–

= m 59999 kg=

k 486250 N/m= f0 1377.32– N= a1 3308891 N sec m⁄⋅= a2 24.9054 sec/m=

p 0.5403=

161

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-250

-200

-150

-100

-50

0

50

100

150

200

250

Time (sec)

For

ce (

kN)

-4 -2 0 2 4-250

-200

-150

-100

-50

0

50

100

150

200

250

Displacement (cm)

For

ce (

kN)

Figure 7.9: Comparison between the experimentally-obtained and predicted damper responses using the phenomenological model proposed by Spencer et al. (1997a) with mass element: (a) force vs. time; (b) force vs. displacement; and (c) force vs.

velocity.

-8 -4 0 4 8-250

-200

-150

-100

-50

0

50

100

150

200

250

Velocity (cm/sec)

For

ce (

kN)

Experimental Result

Mechanical Model Proposed by Spencer et al. with Mass

(a)

(b) (c)

162

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-250

-200

-150

-100

-50

0

50

100

150

200

250

Time (sec)

For

ce (

kN)

-4 -2 0 2 4-250

-200

-150

-100

-50

0

50

100

150

200

250

Displacement (cm)

For

ce (

kN)

Figure 7.10: Comparison between the experimentally-obtained and predicted responses using the proposed phenomenological model: (a) force vs. time; (b) force vs.

displacement; and (c) force vs. velocity.

-8 -4 0 4 8-250

-200

-150

-100

-50

0

50

100

150

200

250

Velocity (cm/sec)

For

ce (

kN)

Experimental Result

Proposed Mechanical Model

(a)

(b) (c)

163

(Spencer et al. 1997a):

(7.12)

where

(7.13)

(7.14)

(7.15)

(7.16)

The resulting error norms are given in Table 7.1. In all cases, the error norms calculated

for the proposed model are smaller than those calculated for the other models considered,

indicating that the proposed model is superior to the other models for the MR damper con-

sidered.

TABLE 7.1: ERROR NORMS OF MR DAMPER DYNAMIC MODELS.

Model

Simple Bouc-Wen Model 0.0352 0.00593 0.0179

Simple Bouc-Wen Model Including Inertial Effect 0.0338 0.00551 0.0170

Phenomenological Model Proposed by Spencer et al. (1997a) 0.0246 0.00407 0.0128

Proposed Phenomenological Model 0.0171 0.00308 0.0089

Et

εt

σf----- Ex,

εx

σf----- Ex·,

εx·

σf-----= = =

εt2 1

T--- fexp fpre–( )2

td0

T

∫=

εx2 1

T--- fexp fpre–( )2 dx

dt------ td

0

T

∫=

εx·2 1

T--- fexp fpre–( )2 dx·

dt------ td

0

T

∫=

σf2 1

T--- fexp µexp–( )2 td

0

T

∫=

Et Ex Ex·

164

7.4 Generalization for Fluctuating Current

All of the data examined previously has been based on the response of the MR

damper when the input current – and hence the magnetic field – was held at a constant

level. However, optimal performance of a control system using this device is expected to

be achieved when the magnetic field is continuously varied based on the measured

response of the system to which it is attached. To use the damper in this way, a model must

be developed that is capable of predicting the behavior of the MR damper for a fluctuating

magnetic field.

To determine a model that is valid under the fluctuating input current, the functional

dependence of the parameters on the input current must be determined. Since the fluid

yield stress is dependent on input current, can be assumed as a function of the input cur-

rent . Moreover, from the experimental results, , , , and are also functions of

the input current.

In order to obtain the relationship between the input current and damper parameters

, , , , and , the damper was driven by band-limited random displacement

excitations with a cutoff frequency of 2 Hz at various constant input currents. A con-

strained nonlinear least-squares optimization scheme based on the trust-region and pre-

conditioned conjugated gradients (PCG) methods is then used (Coleman et al. 1999). The

results are provided in Table 7.2 and shown graphically in Fig. 7.11. A linear piecewise

interpolation approach is then utilized to estimate these damper parameters for current lev-

els not listed in the table. The rest of the damper parameters which are not varied with

input current are chosen to be , ,

α

i a1 a2 m n f0

i

α a1 a2 m n f0

γ 25179.04 m1–

= β 27.1603 m1–

=

165

0 0.5 1 1.5 21

1.5

2

2.5x 105

Current (A)

α (N

)

0 0.5 1 1.5 20

1

2

3

4

5x 107

Current (A)

a1

(N·s

ec/m

)

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

6000

Current (A)

a2

(sec

/m)

0 0.5 1 1.5 20

0.5

1

1.5

2

2.5x 104

Current (A)

Mas

s (k

g)

0 0.5 1 1.5 20

1000

2000

3000

4000

5000

6000

7000

Current (A)

f 0 (

N)

0 0.5 1 1.5 20

1

2

3

4

5

6

7

8

Current (A)

n

Figure 7.11: Relationship between damper parameters and input current.

166

, , and .

The transfer function between the current and current command signal is given

by

(7.17)

which is also provided in Eq. (6.29). Moreover, a first order low-pass filter is utilized to

accommodate the dynamics involved in the MR fluid reaching rheological equilibrium.

(7.18)

Figs. 7.12 and 7.13 provide comparisons between the simulated force and experimen-

tal data at constant input currents of 0 A and 2 A, respectively. In this test, the damper is

driven by a band-limited random displacement excitation with a cutoff frequency of 2 Hz.

As seen here, the model accurately predicts the behavior of the damper. The error norms

TABLE 7.2: DAMPER PARAMETERS AT VARIOUS INPUT CURRENTS UNDER RANDOM DISPLACEMENT EXCITATIONS.

Current (A)

( N)

(sec/m)

( kg)

(N)

0.0237 1.3612 0.4349 862.03 3.0 1.000 1465.82

0.2588 2.2245 2.4698 3677.01 11.0 2.0679 2708.36

0.5124 2.3270 2.8500 3713.88 16.0 3.5387 4533.98

0.7625 2.1633 3.2488 3849.91 18.0 5.2533 4433.08

1.0132 2.2347 2.4172 2327.49 19.5 5.6683 2594.41

1.5198 2.2200 3.8095 4713.21 21.0 6.7673 5804.24

2.0247 2.3002 3.5030 4335.08 22.0 6.7374 5126.79

A 1377.9788 m1–

= k 20.1595 N/m= p 0.2442=

α

105

a1

(107 N sec m⁄ )⋅

a2m

103 n

f0

i v0

i s( ) 1001.45s 1016.1+

s2

503.7s 508.05+ +--------------------------------------------------v0 s( )=

H s( ) 31.4s 31.4+-------------------=

167

0 2 4 6 8 10 12 14 16-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

Dis

plac

emen

t (cm

)

0 2 4 6 8 10 12 14 16-30

-20

-10

0

10

20

30

Time (sec)

For

ce (

kN)

Figure 7.12: Comparison between predicted and experimentally-obtained damper responses under random displacement excitation at a constant input current of 0 A:

(a) displacement vs. time; (b) force vs. time; (c) force vs. displacement; and (d) force vs. velocity.

-0.4 -0.2 0 0.2 0.4-30

-20

-10

0

10

20

30

Displacement (cm)

For

ce (

kN)

-4 -2 0 2 4-30

-20

-10

0

10

20

30

Velocity (cm/sec)

For

ce (

kN)

MeasuredPredicted

(a)

(b)

(c) (d)

168

0 2 4 6 8 10 12 14 16-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

Dis

plac

emen

t (cm

)

0 2 4 6 8 10 12 14 16-250

-200

-150

-100

-50

0

50

100

150

200

250

Time (sec)

For

ce (

kN)

Figure 7.13: Comparison between predicted and experimentally-obtained damper responses under random displacement excitation at a constant input current of 2 A:

(a) displacement vs. time; (b) force vs. time; (c) force vs. displacement; and (d) force vs. velocity.

-0.4 -0.2 0 0.2 0.4-250

-200

-150

-100

-50

0

50

100

150

200

250

Displacement (cm)

For

ce (

kN)

-4 -2 0 2 4-250

-200

-150

-100

-50

0

50

100

150

200

250

Velocity (cm/sec)

For

ce (

kN)

MeasuredPredicted

(a)

(b)

(c) (d)

169

given in Eq. (7.12) are determined to be , , and

at the input current of 0 A, and , , and

at the input current of 2 A.

Furthermore, the predicted force is also compared with experimental data when the

damper is subjected to a fluctuating input current. As in the previous test, a band-limited

random displacement excitation is applied. The displacement excitation and input current

are shown in Fig. 7.14. The comparison between the measured and predicted damper

responses is provided in Fig. 7.15. Again, excellent agreement is observed. The error

norms for this test are determined to be , , and

.

Fig. 7.16 illustrates the measured damper upside force using the force-feedback

approach and compares this with the predicted result. Fig. 7.17 provides a comparison

between the measured and predicted damper forces of the force-tracking experiment in

Chapter 6. It can be seen that the measured and predicted results match very well, and the

proposed model is shown to be effective.

7.5 Summary

In this chapter, a review of the dynamic modeling of controllable fluid dampers has

been presented. Subsequently, a new phenomenological model is proposed that overcomes

a number of shortcomings of these models and can effectively describe the typical behav-

ior of the MR damper. This model is based on the Bouc-Wen hysteresis model, which

accommodates the MR fluid stiction phenomenon, as well as inertial and shear thinning

effects. The proposed model has been shown to be effective in portraying the damper

Et 0.11908= Ex 0.00876=

Ex· 0.02785= Et 0.09551= Ex 0.00851=

Ex· 0.02607=

Et 0.35517= Ex 0.03257=

Ex· 0.10118=

170

0 2 4 6 8 10 12 14 16-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

Dis

plac

emen

t (cm

)

0 2 4 6 8 10 12 14 160

0.5

1

1.5

2

2.5

Time (sec)

Cur

rent

(A

)

Figure 7.14: (a) Random displacement excitation; and (b) random input current.

(a)

(b)

171

0 2 4 6 8 10 12 14 16-250

-200

-150

-100

-50

0

50

100

150

200

250

Time (sec)

For

ce (

kN)

-0.4 -0.2 0 0.2 0.4-250

-200

-150

-100

-50

0

50

100

150

200

250

Displacement (cm)

For

ce (

kN)

-4 -2 0 2 4-250

-200

-150

-100

-50

0

50

100

150

200

250

Velocity (cm/sec)

For

ce (

kN)

Measured

Predicted

Figure 7.15: Comparison between the measured and predicted damper responses under random displacement excitation at random input current: (a) force vs. time;

(b) force vs. displacement; and (c) force vs. velocity.

(a)

(b) (c)

172

-0.2 0 0.2 0.4 0.6 0.8 1-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Time (sec)

Cur

rent

(A

)

-0.2 0 0.2 0.4 0.6 0.8 10

20

40

60

80

100

120

140

160

180

Time (sec)

For

ce (

kN)

MeasuredPredicted

Figure 7.16: Comparison between the predicted and experimentally-obtained damper responses on force upside using force-feedback control approach: (a)

current vs. time; and (b) force vs. time.

(a)

(b)

173

0 5 10 15 20 25 30 35-3

-2

-1

0

1

2

3

Time (sec)

Dis

plac

emen

t (cm

)

0 5 10 15 20 25 30 35-3

-2

-1

0

1

2

3

Time (sec)

Cur

rent

(A

)

Figure 7.17: Comparison between the measured and predicted damper responses of force-tracking experiment: (a) measured displacement excitation; (b) measured

input current; and (c) measured damper resisting force.

0 5 10 15 20 25 30 35-200

-150

-100

-50

0

50

100

150

200

Time (sec)

For

ce (

kN)

MeasuredPredicted

(a)

(b)

(c)

174

force roll-off in the low velocity region, force overshoots when the velocity changes in

sign, and two clockwise loops at the velocity extremes. To obtain a model that reproduces

the behavior of the damper with a fluctuating input current, six parameters are assumed

which vary with the input current. When compared with experimental data, the resulting

model is shown to be accurate in predicting damper behavior under a wide variety of oper-

ating conditions. These results indicate that the proposed model can be effectively used for

control algorithm development and system evaluation.