An analytical approach to optimally desi

The optimization problem is to find optimal value of significant geometric dimensions of the ER damper, such as the ER duct length, ER duct radius, ER duct gap and the piston shaft radiu

DOI 10.1007/s11012-012-9544-3

An analytical approach to optimally design

of electrorheological fluid damper for vehicle suspension

system

Q.H Nguyen · S.B Choi · Y.G Park

Received: 18 October 2010 / Accepted: 29 February 2012

© Springer Science+Business Media B.V 2012

Abstract This work develops an analytical approach

to optimally design electrorheological (ER) dampers,

especially for vehicle suspension system The optimal

design considers both stability and ride comfort of

ve-hicle application After describing the schematic

con-figuration and operating principle of the ER damper, a

quasi-static model is derived on the basis of Bingham

rheological laws of ER fluid Based on the quasi-static

model, the optimization problem for the ER damper

is built The optimization problem is to find optimal

value of significant geometric dimensions of the ER

damper, such as the ER duct length, ER duct radius,

ER duct gap and the piston shaft radius, that maximize

damping force of the ER damper The two constrained

conditions for the optimization problem are: the

damp-ing ratio of the damper in the absence of the electric

field is small enough for ride comfort and the buckling

condition of the piston shaft is satisfied From the

pro-posed optimal design, the optimal solution of the ER

damper constrained in a specific volume is obtained

In order to evaluate performance of the optimized ER

damper, simulation result of a quarter-car suspension

system installed with the optimized ER damper is

pre-sented and compared with that of the non-optimized

Q.H Nguyen · S.B Choi () · Y.G Park

Smart Structures and Systems Laboratory, Department

of Mechanical Engineering, Inha University, Incheon

402-751, Korea

e-mail: seungbok@inha.ac.kr

url: http://www.ssslab.com

ER damper suspension system Finally, the optimal re-sults of the ER damper constrained in different vol-umes are obtained and presented in order to figure out the effect of constrained volume on the optimal design

of ER damper

Keywords Electrorheological fluid· Vehicle damper· Optimal design · Vehicle suspension ride comfort· Vehicle stability · Semi-active control

Nomenclature

A p area of damper’s piston

A s area of the piston shaft

c vis damping coefficient of the ER damper

c v viscous dampers of the suspension

system

C s control gain

d gap of the annular duct

D width of the test bump

E a applied electric field

E Young’s modulus of the piston shaft

material

F d damping force

F ER damping force due to the yield stress of

the ER fluid

I inertia moment of the shaft sectional area

k stiffness of the suspension system

k s safety coefficient

k d coefficient considering dynamic load

acting on the shaft

k1, k2 wheel and suspension stiffness of the

quarter car, respectively

L s length of the piston shaft

L d length of the annular duct

L dmax maximum available ER duct length

m mass of quarter car

m v mass of vehicle

m1, m2 unsprung and sprung masses of the

quarter car, respectively

P1 pressure in the upper chamber of the

damper

P2 pressure in the lower chamber of the

damper

P a pressure in the gas chamber

(accumulator)

P0 initial pressure in the gas chamber

Q d flow rate of ER flow in the duct

R d radius of the annular duct

R dmax maximum available radius of the ER duct

R s radius of the piston shaft

T transmissibility of the suspension system

V c vehicle velocity

V0 initial volume of the gas chamber

W n white noise

u control input

x(t ) displacement of the car body

x p displacement of piston

x0(t ) road surface input

x1(t ), x2(t ) unsprung mass and the sprung mass

deflection, respectively

X0 half of the bump height

y(t ) input excitation to the suspension system

(road profile)

α, β intrinsic values of the ER fluid

 inner electrode thickness

P d pressure drop between the upper

chamber and the lower chamber

P a pressure drop between the lower

chamber and the gas chamber

γ coefficient of thermal expansion

μ post-yield viscosity of ER fluid

 excitation frequency

ω n natural frequency of the suspension

system

ρ road roughness parameter

σ2 covariance of road irregularity

τ y yield stress of the ER fluid induced by

the applied electric field

ξ damping ratio of the suspension system

ξ min minimum tunable damping ratio

1 Introduction

It is well-known that suspension systems take a vital role in automotive technology Traditionally, a passive vehicle suspension system consists of an energy dis-sipating element which is a damper, and an energy-storing element which is a spring Essentially, the sus-pension supports the weight of the upper part of a vehicle on its axles and wheels, allows the vehicle

to travel over irregular surfaces with a minimum of up-and-down body movement (stability), reduces the load from road transferred to occupants (ride com-fort), and allows the vehicle to corner with mini-mum roll or loss of traction between the tires and the road (good handling) These goals are generally at odds Good ride comfort requires a soft suspension, whereas stability of the vehicle requires a stiff sus-pension Good handling requires a suspension setting somewhere between the two Therefore, the tuning of

a passive suspension involves finding the right com-promise between the three conflicting criteria This

is an inherent limitation of a passive suspension sys-tem

In order to compensate for the limitations of a passive suspension system, active suspension systems have been developed and applied in the real field [1 5] With an additional active force introduced as

a part of a suspension unit, the suspension system is then controlled using appropriate algorithms to make

it more responsive to different types of road profile However, this type of suspension requires high power sources, active actuators, sensors and sophisticated control logics A semi-active configuration can ad-dress these limitations by effectively integrating a tun-ing control scheme with tunable passive devices For this, active force generators are replaced by modulated variable compartments such as variable rate damper and stiffness

Recently, the possible application of electrorheo-logical (ER) and magnetorheoelectrorheo-logical (MR) fluids to the development of controllable dampers has attracted considerable interest, especially in vehicle suspension application Sturk et al proposed a high voltage sup-ply unit with ER damper and proved their effective-ness via quarter car suspension system [6] Nakano

constructed a quarter car suspension model using ER

damper and proposed a proportional control algorithm

in order to isolate vibration [7] Petek et al

con-structed a semi-active full suspension system installed

with four ER dampers and evaluated suspension

per-formance through the implementation of a skyhook

control algorithm which considers heave, pitch and

roll motions of the car body [8] Choi et al proposed

a cylindrical ER damper for passenger car and its

controllability of damping force was proved by

im-plementing a skyhook controller [9] More recently,

Choi et al have designed an ER suspension system

for middle sized passenger vehicle and a field test

un-der bump and random road conditions is unun-dertaken

The control responses for the ride quality and

steer-ing stability are also evaluated in both time and

fre-quency domains [10, 11] In order to improve ER

suspension performance, modern control algorithms

have been applied with considerable success [12–15]

As is evident from previous works, most of research

works have been focused only on design

configura-tion, damping force evaluation and controller design

of ER damper However, an optimal design of the ER

dampers considering stability, ride comfort and

han-dling of ER suspension system seems to be absent

The primary purpose of the current work is to fill this

gap

Consequently, the main contribution of this work

is to develop an analytical approach to optimally

de-sign ER dampers for vehicle suspension application

considering both stability and ride comfort After

de-scribing the schematic configuration and operating

principle of the ER damper, a quasi-static model is

derived on the basis of Bingham rheological laws

of ER fluid Based on the quasi-static model, the

optimization problem for the ER damper is built

The optimal solution of the ER damper constrained

in a specific volume is then obtained based on the

proposed optimal design In order to evaluate

per-formance of the optimized ER damper, simulation

result of a quarter-car suspension system installed

with the optimized ER damper is presented and

com-pared with that of the non-optimized one designed by

Choi et al [11] Finally, the optimal solutions of the

ER damper constrained in different volumes are

ob-tained and presented in order to figure out the effect

of constrained volume on the optimal design of ER

damper

2 Quasi-static modeling of ER damper

In this study, the cylindrical ER damper for passen-ger vehicle suspension proposed by Choi et al [11] is considered The schematic diagram of the damper is shown in Fig.1 The ER damper is divided into up-per and lower chambers by the damup-per piston These chambers are fully filled with ER fluid As the piston moves, the ER fluid flows from one chamber to the other through the annular duct between inner and outer cylinders The inner cylinder is connected to the pos-itive voltage produced by a high voltage supply unit,

playing as the positive ( +) electrode The outer

cylin-der is connected to the ground playing as the negative

( −) electrode On the other hand, a gas chamber

lo-cated outside of the lower chamber acts as an accu-mulator of the ER fluid induced by the motion of the piston In the absence of electric fields, the ER damper produces a damping force only caused by the fluid vis-cous resistance However, if a certain level of the elec-tric field is supplied to the ER damper, the ER damper produces additional damping force owing to the yield stress of the ER fluid This damping force of the ER damper can be continuously tuned by controlling the intensity of the electric field By neglecting the com-pressibility of the ER fluid, frictional force and assum-ing quasi-static behavior of the damper, the dampassum-ing force can be expressed as follows:

F d = P2 A p − P1 (A p − A s ) (1)

where Ap and As are the piston and the piston-shaft

cross-sectional areas, respectively P1and P2are pres-sures in the upper and lower chamber of the damper,

respectively The relations between P1, P2 and the

pressure in the gas chamber, Pa, can be expressed as

follows:

P2= P a + P a; P1= P a − P d (2)

where Pais the pressure drop of ER fluid flow in the connecting pipe between the lower chamber and the accumulator which is small and neglected in this study,

P d is the pressure drop of ER fluid flow through the annular duct The pressure in the gas chamber can be calculated as follows:

P a = P0



V0

V − A x

γ

(3)

Fig 1 Schematic configuration of the ER damper

where P0and V0are initial pressure and volume of the

accumulator γ is the coefficient of thermal expansion

which is ranging from 1.4 to 1.7 for adiabatic

expan-sion xpis the piston displacement From Eqs (1) and

(2), the damping force of the ER damper can be

calcu-lated by

F d = P a A s + P d (A p − A s ) (4)

By neglecting minor loss and taking note that radius of

the annular duct is much larger than its gap, the

pres-sure drop Pdcan be approximately calculated as

fol-lows [16]:

P d= 6μLd

π d3R d

Q d + c L d

where Qdis the flow rate of ER flow in the duct, given

by Qd = (A p − A s ) ˙x p ; τy is the yield stress of the

ER fluid induced by the applied electric field; μ is

the post-yield viscosity of ER fluid; Ld , R d and d are

length, average radius and gap of the annular duct,

re-spectively c is an coefficient which depends on flow

velocity profile and has a value range from a

mini-mum value of 2.07 to a maximini-mum value of 3.07 The

coefficient c can be approximately estimated as

fol-lows [16]:

12Qd μ + 0.8πR d d2τ y

(6)

Plugging Pdfrom Eq (5) into Eq (4) one obtains

F d = P a A s + c vis ˙x p + F ER sgn( ˙x p ) (7) where,

c vis= 6μLd

π R d d3(A p − A s )2;

F ER = (A p − A s ) cL d

d τ y

The first term in Eq (7) represents the elastic force from the gas compliance This term causes the damp-ing force-piston velocity curve to be shifted verti-cally and does not affect damping characteristics of the damper The second term represents the damping force due to ER fluid viscosity, thus the damping force when

no electric field is applied to the damper The third one is the force due to the yield stress of the ER fluid, which can be continuously controlled by the intensity

of the electric field applied to the damper This is the dominant term which is expected to be large enough for suppressing vibration energy

The commercial ER fluid (Rheobay, TP Al 3565) is

used in this study and induced yield stress of the ER

fluid can be experimentally estimated by [11]

τ y = αE β

Here, Ea is the applied electric field whose unit is

kV/mm The α and β are intrinsic values of the ER

fluid which are experimentally determined At room

temperature, the values of α and β of the above ER

fluid are evaluated by 591 and 1.42, respectively The

post-yield viscosity of the ER fluid is assumed to be

independent on applied voltage and is estimated from

experimental results to be 30cSt

3 Optimal design of ER damper

In this study, optimal design of the proposed ER

damper is considered based on the quasi-static model

developed in Sect.2 For vehicle suspension design,

the ride comfort and the suspension travel are the two

conflicting performance indexes to be considered In

order to reduce the suspension travel (i.e., increase

sta-bility of the vehicle), high damping force is required

On the other hand, for improving ride comfort, low

damping force is expected In order to clearly

under-stand the above mentioned, let us consider a

simpli-fied one degree of freedom (1-DOF) suspension model

shown in Fig.2(a) In this simple idealized model, the

vehicle mass mv is supported by four springs k in

par-allel with four viscous dampers cv of the suspension

system The model in Fig.2(a) can be expressed in a

more simplified model, quarter car model, shown in

Fig.2(b) The motion of the sprung mass in Fig.2(b)

can mathematically be expressed as follows:

m ¨x + c v ( ˙x − ˙y) + k(x − y) = 0 (9)

or

where m is the mass of quarter car, m = m v / 4; x(t) is

the displacement of the car body; and y(t) is the

dis-placement of the wheels It is noted that y(t) is

consid-ered as an input excitation to the suspension system

By assuming a sinusoidal excitation applied to the

un-sprung mass, the transmissibility of the above 1-DOF

suspension system can be obtained as follows:

Fig 2 1-DOF model of a vehicle

Fig 3 Transmissibility of 1-DOF quarter car suspension

T =



1+ (2ξω/ω n )2 (1− ω2/ω2)2+ (2ξω/ω n )2 (11)

where ω is the excitation frequency, ωn is natural

fre-quency of the suspension system, ωn=√k/m and

ξ is the damping ratio, ξ = c s /2√

km In practice, the natural frequency of vehicle suspension systems

is commonly around 1.5 Hz (ωn = 9.42 s−1), and in this case the dependence of the transmissibility on ex-citation frequency is presented in Fig 3 As shown from the figure, at low damping the resonant trans-missibility is relatively large, while the transmissibil-ity at higher frequencies is quite low As the damping

is increased, the resonant peaks are attenuated, but

vi-bration isolation is lost at high frequency The lack of

isolation at higher frequencies will result in a harsher

vehicle ride This illustrates the inherent tradeoff

be-tween resonance control and high frequency isolation

associated with the design of passive vehicle

suspen-sion systems It is obvious that the damping constant

of the damper determines both the stability of the

ve-hicle and the comfort of occupants A high damper (a

damper with high damping characteristics) reduces the

amplification and provides good stability, keeping the

tires in contact with the road and preventing frame

os-cillations and other problems, but it increases the force

transmissibility and transfers much of the road

excita-tion to the passenger, causing an uncomfortable ride

On the other hand, a soft damper (a damper with low

damping characteristics) increases ride comfort, but it

reduces the stability of the vehicle

It is noteworthy that the damping force of an ER

damper can be controlled continuously by applied

electric field Therefore, if the applied electric field

is proportional to the sprung mass velocity, the ER

damper behaves similarly to a semi-active suspension

system with tunable damping ratio and a minimum

tunable damping ratio is obtained when no electric

field is applied to the damper An inherent challenge

in design of ER suspension is the limitation of

tun-ing range of the damptun-ing ratio If the ER suspension

is designed with large reachable damping ratio to

at-tenuate resonant peak, its ride comfort characteristics

is low because the minimum reachable damping ratio

can not be tuned to a very small value and via versa

Obviously, a wide tunable range of damping ratio can

be achieved by using a large sized ER damper

How-ever, the large ER damper results in high cost and

re-quires large space In practice, the suspension size is

limited depending on practical application Thus, there

is an inevitable trade-off between the minimum

tun-able and the maximum reachtun-able damping ratio in

de-sign of ER suspension system From Fig.3, it is

ob-served that the isolation at high excitation frequency

approaches to a saturation when the damping ratio is

smaller than 0.1 It is also seen from practical

appli-cation of vehicle suspension that the ride comfort and

handling performance are improved very little when

the damping ratio decreased to 0.1 or smaller Thus, a

smaller value of damping ratio is not necessary and

useless Taking the aforementioned into the optimal

design of ER damper, the optimization problem can

be stated as follows: Find optimum geometric

dimen-sions of the ER damper constrained in a specific vol-ume so that the minimum tunable damping ratio can be

as small as 0.1 and the damping force is maximized For the ER damper shown in Fig.1, from Eq (7) the damping force can be can be calculated by

F d = P a π R2s + c vis ˙x p + F ERsgn( ˙x p ) (12) where,

c vis=6π μLd

R d d3



(R d − d − )2− R2

s

2

;

F ER = π(R d − d − )2− R2

s

cL d

d τ y

The minimum tunable damping ratio of the damper is calculated as follows:

ξ min= c vis

2√

km

=√3π μLd

kmR d d3



(R d − d − )2− R2

s

2

(13)

In the above,  is the inner electrode thickness From

Eqs (12) & (13), it is seen that the damping force Fd

and the minimum tunable damping ratio ξmin

signifi-cantly depends on the duct length Ld, the duct width

d , the duct radius Rd and the piston shaft radius Rs

of the ER damper The larger value of Rd and Ld is the higher damping force can be obtained However,

the large value of Rd and Ld causes an increase of minimum tunable damping ratio which results in a lost

of ride comfort Furthermore, the value of Rd and Lp

are limited by a constrain in damper size A reduction

of duct width d causes an increase of damping force

but this significantly increases the minimum tunable

damping ratio, especially at small value of d In

addi-tion, the duct gap can not be designed too small that results in high cost of fabrication and potential electric short in practical application The piston shaft radius

R s affects not only the damping ratio and the damp-ing force but also the strength of the shaft Under the damping force, the shaft may reach to a buckling state, especially when it is in compression In order to avoid the buckling in the shaft, the following condition must

be satisfied [17]

F d≤ 1

k s k d

π2EI

L2 = 1

k s k d

π3ER4s

24



k s k d L2

s F d

π3E − R s≤ 0 (15)

In the above, ks is the safety coefficient which is set

by 2 in this study kd is the coefficient considering

dynamic load acting on the shaft which is chosen as

k d = 1.5 L s is the length of the shaft, I is the

iner-tia moment of the shaft sectional area and E is the

Young’s modulus of the shaft material

It is noted again that the first term of the damping

force, Eq (12), only causes the damping force-piston

velocity curve to be shifted vertically and does not

af-fect damping characteristics of the damper From the above, the optimization problem of the ER damper is mathematically expressed as follows:

– Find the values of Lp , d, R d and Rs (design vari-ables) that maximize the following objective func-tion:

OBJ=6π μLd

R d d3



(R d − d − )2− R2

s

2

˙x p + π(R d − d − )2− R2

s

cL d

d τ y (16) – Subject to:

3π μLd

√

kmR d d3



(R d − d − )2− R2

s

2

− 0.1 ≤ 0;

24



k s k d L2

s

π3E



P a π R2s +6π μLd

R d d3



(R d − d − )2− R2

s

2

˙x p + π(R d − d − )2− R2

s

cL d

d τ y



− R s≤ 0;

0≤ L d ≤ L dmax; 0≤ R p ≤ R pmax; 0≤ d; 0≤ R s

where Ld is the ER duct length, d is the ER duct

width, Rd is the ER duct radius and Rs is the

pis-ton shaft radius of the ER damper Ldmax , R dmax

are maximum available values of the ER duct length

and the duct radius of the ER damper which are

de-termined from practical application

4 Optimal results and discussion

In this study, the above constrained optimization

prob-lem is transformed to an unconstrained one via penalty

functions The transformed unconstrained

optimiza-tion problem is then numerically solved using first

or-der method with golden-section algorithm and a local

quadratic fitting technique [18] Figure4 shows

opti-mal solution of the rear ER damper for a middle sized

vehicle suspension designed by Choi el al [11] It is

noted that, from practical application, Choi et al have

determined available space for the ER damper in

re-placement of the conventional damper of the

suspen-sion The maximum available size of the duct length

L d and duct radius Rdare respectively 280.5 mm and

18 mm In the optimal solution shown in Fig.4, the

initial values of the design variables Ld , d, R d and

R s are arbitrarily selected as follows: Ld= 270 mm;

R d = 15 mm, d = 1 mm and R s = 8 mm The in-ner electrode thickness is set equal to that designed

by Choi et al.,  = 3.5 mm The damper piston is

assumed to move relatively to the damper housing at

a velocity of 0.4 m/s ( ˙x p = 0.4 m/s) and the applied

electric field is 3 KV/mm The higher applied field po-tentially causes an electric short between the damper electrodes The convergence condition of the objec-tive function is set by 0.2 % In addition, whenever

a design variable reaches to its boundary a conver-gence of that design variable is assumed and the value

of the design variable at boundary is set as the opti-mal value The optimization process is then continued with the other design variables Figure 4 shows that the solution is converged after 23 iterations At the op-timum, the damping force reaches up to 1400 N which

is around 4 times greater than that at the initial while the damping ratio is constrained to be smaller than

0.1 The optimal values of design variables Ld , d, R

Fig 4 Optimization solution of the ER damper constrained in

a volume of R dmax = 18 mm, L dmax = 280.5 mm

and Rs are 280.5 mm; Rd = 18 mm, d = 0.825 mm

and Rs = 6.52 mm, respectively It is clearly from

the result that the duct length Ld and duct radius Rd

are reach to their maximum available values in this

case A question arises here that if the duct length Ld

and duct radius Rd always reach to their maximum

available values in this optimization problem In

or-der to answer this question, optimal solution of the ER

damper constrained in many different volumes is

con-sidered The results show that the optimal duct radius

always reaches to its maximum available value

How-ever, this is not always true for the duct length

Fig-ure5shows the optimal duct length as a function of the

constrained duct radius It is noteworthy that no

con-strain is imposed for the duct length in this case The

result shows that there exists an optimal duct length

Fig 5 Dependence the optimal solution on the constrained

ra-dius of the ER duct

Fig 6 Dependence of the optimal solution on the constrained

damping ratio

for a constrained duct radius It is also seen that the larger constrained duct radius is the higher values of the optimal duct length, duct gap and shaft radius are obtained However, the optimal duct length is much larger than the maximum available value in vehicle suspension application For instance, the optimal duct length is up to 743 mm if the constrained duct radius is

14 mm This optimal length is obviously much larger than the maximum available length Therefore, in the optimal design of ER damper for vehicle suspension system, the optimal duct radius and duct length can always be selected as large as their maximum avail-able values for simplicity The design variavail-ables then

can be reduced from four variables (Lp , d, R d , R s )to

two variables (d, Rs ) Figure 6 shows the optimal solutions of the ER damper as a functions of the constrained damping ratio (the minimum tunable damping ratio) It is observed

Fig 7 Quarter-car suspension model installed with the ER

damper

from the figure that the optimal value of the duct length

L d and duct radius Rd are not affected by the

con-strained damping ratio However, the optimal values

of shaft radius and the duct gap are significantly

af-fected The higher value of the constrained damping

ratio is the smaller optimal value of the duct gap and

the higher value of the shaft radius is It is also

ob-served that the maximum damping force increases

al-most in proportion with the constrained damping ratio

5 Simulation of the optimized ER damper

In order to evaluate the effectiveness of the above

op-timization solution, performance characteristics of the

suspension installed with the optimized ER damper

are obtained through simulation and compared with

that of the ER suspension designed by Choi et al [11]

It is noteworthy that Choi et al have designed the ER

suspension based on their experiences and performed

a number of simulation results Thus, the design is

the best choice from the simulated results By this

ap-proach, the ER suspension is expected to be

consider-ably good but not an optimal design Furthermore, it

takes time to perform a large number of simulations

Figure7shows a quarter-car model installed with the

ER suspension From the figure, the following

govern-ing equations can be derived

m1¨x1 + P a A s + c vis ( ˙x1 − ˙x2 ) + F MR sgn( ˙x1 − ˙x2 )

+ k2 (x − x2 ) + k1 (x − x0 )= 0 (17)

m2¨x2 − P a A s + c vis ( ˙x2 − ˙x1 ) + F MR sgn( ˙x2 − ˙x1 )

In the above, m1, m2are unsprung and sprung masses

of the quarter car; k1, k2 are wheel and suspension

stiffness; x0(t ), x1(t ) and x2(t )are the road surface in-put, the unsprung mass deflection and the sprung mass deflection, respectively The parameters of the suspen-sion system are determined based on the parameters of conventional suspension systems For a middle-sized passenger vehicle, the suspension parameters are as

follows: m1= 35 kg, m2 = 310 kg, k1= 309 kN/m,

k2= 20 kN/m

Firstly, bump response of the passenger vehicle equipped with the ER damper is evaluated In this study, the bump profile is mathematically described by

x0(t )=

X0[1 − cos(ωr t ) ] if t ≤ 2π/ω r

where

ω r = 2πV c /D

In the above, X0(= 0.035 m) is the half of the bump height, D ( = 0.8 m) is the width of the bump and V c

is the vehicle velocity In the bump test, the vehicle

is assumed to travel the bump with constant velocity

of 3.08 km/h (Vc = 0.856 m/s) Both the simulation

results of the optimized damper and the damper de-signed by Choi et al (non-optimized damper) are pre-sented It is noted that, based on practical experiences and simulation results, Choi et al have determined geometric dimensions of the ER damper as follows:

L d = 280.5 mm; R d = 17.88 mm, d = 0.88 mm;

R s = 6.5 mm.

Figure 8 shows the bump response of the vehi-cle when no electric field is applied to the electrodes

It is noteworthy that there are three main parameters for design and vehicle suspensions evaluation: Sprung mass vibration isolation, which determines ride com-fort Suspension stroke, which indicates the limit of the vehicle body motion and tire road contact eval-uated through tire deflection, which determines sta-bility and safety It is clearly observed form Fig 8 and 8b that the vibration of sprung mass is better suppressed by using the optimized damper than the non-optimized one It is also observed from Fig 8 and8d that the suspension deflection and the tire de-flection in case of the optimized damper suspension

Fig 8 Bump responses of the ER suspension system, the applied is E= 0 kV/mm

are smaller than those in case of the non-optimized

one Thus, optimized suspension can provide a

bet-ter performance and stability than the non-optimized

one This is an important advantage of the optimized

damper when the control system in failure condition

and the ER damper works similarly to a conventional

damper

In order to evaluate vibration control

characteris-tics of the optimized suspension, a sky-hook control

algorithm is employed to control the ER damper The

sky-hook control input is mathematically expressed as

follows:

u=

C s ˙x2 if ˙x2 ( ˙x2 − ˙x1 ) >0

where Csis the control gain The unit of control input

uis kV/mm and the unit of sprung mass velocity ˙x2is m/s Figure9shows the bump response of the quarter-vehicle suspension system featuring the ER damper

and the sky-hook controlled algorithm with Cs = 10

It is noted that this value of the control gain is chosen

by trial and error It is obvious that different values of

C s results in different performance of the ER suspen-sion However, the relative comparison between the

two ER suspensions is not affected by the value of Cs

It is observed from Fig.9that vibration of the sprung mass is significantly reduced by employing the sky-hook controller for the ER damper From Fig.9(a), it

is seen that the sprung mass acceleration of the op-timized suspension and the non-opop-timized one is al-most similar However, the suspension deflection and