Modeling and simulation of non-linear and hysteresis behavior of magneto-rheological dampers in the example of quarter-car model

Sulaymon Eshkabilov Dynamics & Control Lab Tashkent Automotive Road Institute Amir Temur Str. 20, Tashkent -100060, Uzbekistan Email: sulaymon@d-c-lab.com Abstract. This paper presents reviews of several models and numerical simulation models of non-linear and hysteresis behaviors of magneto-rheological liquid dampers in MATLAB®/Simulink® in the example of quarter-car model of vehicle suspension simulation, such as, Bingham, Dahl, LuGre and Bouc-Wen models. In addition, it demonstrates numerical simulation models built in MATLAB®/Simulink® and discusses results from numerical simulation models for three different input excitations from terrain.

Keywords: Bingham model, Dahl model, LuGre model, Bouc-Wen model, passive and semi-active suspension, magneto-rheological damper, quarter car model, numerical simulation, Simulink model.

1. Introduction

In general, most of the natural phenomena, operational machine processes and dynamic system behaviors are of non-linear nature that is very often linearized for the sake of simplicity in formulations and analyses. In fact, nonlinear behaviors or phenomena of processes may create difficulties in studies and engineering design processes but considering some of those non-linear characteristics of processes or behaviors of dynamic systems carefully could be also very beneficial and of great importance for efficient and accurate control, and used for operational efficiency and energy preservation or dissipation depending on their application areas. For example, nonlinear parameters and characteristics of some materials and interactions of different parts made of different materials have a great potential to apply for dampers and shock absorbers [1]. One of the good examples for such processes is a hysteresis loop observed in magnetic or magnetized materials and magneto-rheological (MR) liquids. In studies [2, 3, 4, 5], the MR liquids are found to be one of the most suitable and promising in designing vibration dampers and shock absorbers, and there are some combinatorial designs [6] of MR fluid dampers. In studies [7], feasibility of MR liquid damper modeling by employing Bouc-Wen model in association with an intelligent self-tuning PID controller for semi-active suspension modeling is studied numerically via computer modeling in MATLAB/Simulink. Nevertheless, identification of the hysteresis loop parameters is rather complex and may require considerable laboratory and numerical studies in order to apply them and get a best use of MR damper properties.

In this paper, we put some emphases on different mathematical models and formulations of the MR liquids, and their hysteresis loop parameters and numerical simulation models designed for a semi-actively controlled feedback damper for a vehicle suspension systems developed in MATLAB/Simulink. In addition, we shall try to analyze and compare efficiency and accuracy of these models in the example of the quarter-car model to design a semi-active suspension system.

2. Mathematical formulation of a quarter-car model

To derive an equation of (vertical) motion of a vehicle while driving on uneven roads, we take quarter of a vehicle by assuming that terrain roughness is evenly distributed under all wheels of a vehicle and loading from the whole vehicle body is equally distributed across all of its axles. In addition, we consider that a tire has some damping effect. With these preconditions, we draw the next physical model (Figure 1) of the system for passively and semi-actively controlled systems of a quarter-car model.

a) Pa

From the pbodies whicequations of

a) Fo

b) Fo

Where ,, and and da

(disturbancecontroller th

- the coLuGre and B

3. Mathe3.1. Bingha

To simulatefollowing:

Where isconstant;

assive suspensio

assive and semich as un-sprung f motion of the s

or passive suspe

or semi-active su

and are di are displacem

amping coefficiee) displacement hat takes into acntrol force exerBouc-Wen mode

ematical formulam model

e and identify pa

a piston’s relati is offset force

on design;

i-active suspensmass (half of a

systems are

nsion system:

uspension system

isplacement, velment, velocity anents of suspensio

and velocity wcount terrain rou

rted by the contrels and design nu

lations of the M

arameters of the

ive displacement(constant force

Figure 1. V

ion design showaxle mass and o

m:

locity and accelnd acceleration oon and tire; anwith respect to lughness , anroller, we applyumerical simulat

MR dampers

MR liquids, Bin

t and is its dervalue). The sig

Vehicle suspensi

wn in Figure 1, one wheel) a

leration of the of the un-sprung nd stiffness olongitudinal spend vertical disply several differetion models in M

ngham plastic m

rivative that is vegnum function

b) Semi-act

on models.

we can derive and sprung mas

0

sprung mass (qmass (half of ax

of suspension aneed of the vehicacement and ve

ent hysteresis efMATLAB/Simul

model [8] was pro

elocity of a pisto will take

tive suspension

equations of ms (quarter car b

quarter car bodxle mass and on

nd tire; and cle; is the flocity of the veh

ffect models, suclink.

oposed in 1985.

on; is frictione care of the dir

design.

motion of the twody mass) .

dy mass), respecne wheel), respec

are terrain rouforce generated hicle. In the modch as, Bingham

It is formulated

nal force; is darection of the fri

o mass So, the

(1)

(2)

ctively; ctively; ughness

by the del, for

m, Dahl,

d by the

(3)

amping ictional

force depcorrespond

The responsBingham mequal to the

Now we buthe semi-act

pending on the rto the displacem

se of Bingham mmodel force w

linear relationsh

ild a Simulink mtively controlled

relative velocityment and veloc

model corresponwill be equal to hip between the

Figure

model – Figure 4d suspension mod

of the hystercity of the spru

nds to the followCoulomb force force ∆ and the

e 2. Bingham me

Figure 3. The re

4 using the formdel from the exp

esis (internal) vung mass.

wing graph showplus friction fore velocity ∆ dif

echanical model

esponse of Bingh

mulation from thepression of (2) as

variable . Note

wn in Figure 3 anrce ( ). The damfferences- Figure

proposed by [6]

ham model.

e equation (3) ans shown in Figur

∆∆

that in our simu

nd it can be assumping coefficiene 3.

].

nd link it with thre 1.b.

ulation model,

umed that the shnt (constant)

he model expres

and

hape of will be

ssed for

Note that thwhich are function, i.e

3.2. Dahl m

This model

Where, and are pa

Using the ex(Figure 5), tgoing to the

he Coulomb fric and make u

e., .

model

considers quasi-

is exerted forcearameters that co

xpressions (4) athere is one feede un-sprung mass

Figure 4. Bing

tional force ( )up relative velo

-static bonds in t

e from the MR dontrol the hystere

nd (5), we builddback coming fros and sprung ma

gham model emb

) is directly relatcity the origin of fric

damper, is theesis loop shape.

d a simulation mom the un-spring

ass.

bedded in semi-a

ted to the yield in order to di

ction [9]. Dahl m

| |e control voltage

model of Dahl mog mass that is ve

active suspension

stress. In Binghirect the Coulom

model of the MR

e, is the dyna

odel in Simulinkelocity an

n control.

ham model theremb frictional for

damper [8] is fo

amic hysteresis c

k as shown in Find there is one ou

e are two input srce with the s

ormulated by:

coefficient, ,igure 5. In Dahlutput signal that

signals, signum

(4)

(5) ,

l model t is

In Dahl momass that iscoefficient. for un-sprun

3.3. LuGre

In modelingsimulation oCoulomb, st

Where ,of the bristle

In the above

Where is

The simulatvelocity fromjunction of t4 and 5.

Fig

del, there are ons a car body velThe output sign

ng mass with (+)

e model

g the hysteresis of dampers. Thitick-slip and stri

, are stiffneses), is the v

e expression,

the Coulomb fr

tion model of thm sprung mass athe sprung and u

gure 5. Dahl mod

ne input signal locity and feedinal is the control) sign.

loops, the LuGrs model [10] takibeck effects that

ss, damping and velocity of the fr

is define

iction force, i

he LuGre modeland one output sun-sprung masse

del implemented

and output signg summing junl force feeding a

re model is devekes into accountt are formulated

viscous friction riction state,

ed by [12 and 13

is the sticktion fo

l, as shown in Fsignal that is cones with (-) and (+

d for semi-active

gnal . The innction of foa summing junc

eloped within stut three types of by the followin

coefficients, res is the relative v

] that has been e

orce, and is th

Figure 6, is builntrol force f+) signs respecti

control of suspe

nput signal is coforce, and summction of input for

udies [10] and africtions observ

ng:

spectively; velocity of the sp

| |expressed with th

he Stribeck veloc

lt in Simulink wfor the suspensioively alike Bingh

ension system.

oming from the ming junction of

rces for sprung

applied in worksved in dry frictio

is the friction stprung mass.

he following

city.

with one input son system conneham and Dahl m

velocity of the f dynamic hysmass with (-) si

s [11] in modelion and fluid flow

tate (average def

signal that is a rected with a summodels shown in

sprung steresis ign and

ing and ws, viz.

(6)

flection

(7)

(8)

relative mmation

Figure

In the Simuis the MR fand (8).

3.4. Bouc-W

The MR damelements. ThThe hystereequation of

Where is parameters

link model, a funforce . The tw

Wen model

mper with Bouche schematic repsis loop has an iBouc-Wen mod

the evolutionary, and .

F

Fi

nction block witwo input signals

-Wen model is cpresentation of Binternal variable

del [8] is express

y variable that ca

Figure 7. Schem

igure 6. Simulink

th three input sigs, which are

composed of stifBouc-Wen modee that represened by the follow

an vary from a s

matic representati

k model of the L

gnals, viz. ,and , are

ffness (spring) elel of an MR damnts hysteretic behwing. | | | |sinusoidal to a qu

ion of Bouc-Wen

LuGre model. , , is eme internal variab

lement, passive mper is depicted bhavior and satisf

| |

uasi-rectangular

en model of an M

mployed to comples computed fr

damper and Bouby the next sche

fies the next exp

r function of the

MR damper.

pute a control forom the expressi

uc-Wen hysteresematic view – Fipression (9). The

time depending

rce that ons (7)

sis loop igure 7. e model

(9)

g on the

The force exthe control v

In the modevalues of thinfluence ofparameters (

The best fitsystem.

The simulatexpressed insprung massand to the uNote that inand velocity

xerted by the MRvoltage , and is

el computing dahe parameters (cof the model on (coefficients)

t parameter valu

tion model of tn (9), (10) and (s ( ) of the sy

un-sprung ( ) n the MR modely of the sprung m

R damper is the s given by

amping force of oefficients) the final force

and ar

ues of these par

he system from(11). The simulstem, and two omass with (+) p, there are two i

mass and the outp

function of the

the MR damper and hav

. The force re determined fr

rameters are de

m Bouc-Wen moation model hasutput signals for

plus sign. Note thinput signals andput signal is the

Figure 8

relative displace

r, is the stiffnve a linearly relatakes into acco

om the followin,etermined by fitt

odel shown in Fs two input sourr control force hat is equd one output sigcontrol force

. Bouc-Wen mo

ement and velo

fness of the sprinationship with th

ount pre-yield stng expressions:

tting to the expe

Figure 8 is builrces, viz. di

going to theual to and

gnal. The input s generated by

del.

ocity and the p

ng element of thhe control voltagtress of the dam

erimentally mea

lt in Simulink bisplacement and

e sprung mass ( is equal to

signals are the MR damper

parameter defi

he MR damper age and determ

mper. The values

asured response

by using the equd velocity

) with (-) min in the equatiand displa

r.

ined by

(10)

and the mine the s of the

(11)

of the

uations y of the nus sign ion (2). cement

In Bouc-Wesprung mass

Also, all of other and a in the car bo

4. Simula

The above deach modelcontroller fo(1) in the exto and controller ofvalues of sumodel (Bingparameter v

en model alike s with (-) sign an

f the four simulapassively contro

ody from the roa

Fig

ation results an

depicted mathem with respect toormulated in thexample of quarte

vibration dampf the MR dampeuspension paramgham, Dahl, Lu

values of the hys

Bingham, Dahl nd for the un-spr

ation models areolled system for ad excitations.

gure 9. Passively

d discussions

matical formulato its exerted dame system equatioer car model shoping is evaluateer models are commeters (quarter c

Gre and Bouc-Wsteresis models

and LuGre morung mass with (

e summed up asfour different ex

controlled syste

tions as implemmping force, andons (2) of motionown in Figure 1.d in the sprungmpared with the

car) are taken froWen) parametersare found by tra

odels, the contro(+) sign.

s sub-systems (Fxcitation signals

em model vs. fou

ented in Simulind vibration and n against passive In all of our sim

g mass. Displacee displacement vom the data givs are chosen froails and errors. F

Semi_Active (Bouc-

ol force feeds th

Figure 9) to comfrom the terrain

ur MR models a

nk models are sshock damping

ely controlled/damulations, the coement values of

values of a passivven in Table 1 aom the data giveFor numerical s

-Wen)

he summing jun

mpare their perfn. The system re

as sub-systems.

simulated to comg efficiency as aamped vibrationontrol force in (2f the sprung mavely controlled sand all numericaen in Table 2, 3simulations three

nction of forces

formances againsponse is displa

mpare performana semi-active vin damper formul2) is set to bass with a semisuspension systeal values for hys, 4 and 5. The re different signa

for the

nst each cement

nces of ibration lated in e equal i-active

em. The steresis rational als, viz.

random white noise, Heaviside step function and sine waves with 2.1 Hz and 20.8 Hz of oscillations, and also, a combinatorial excitation signal, a sum of sine waves and random (Gaussian white) noises, are taken. Road excitation signals are set to have maximum (absolute) magnitude of 0.075 m and oscillation frequencies of sine waves are taken by considering natural frequencies of the quarter car model.

Table 1. Data for suspension system (quarter car model). Parameter name Parameter notation Parameter value Sprung Mass 2500 Un-sprung mass 320 Stiffness of suspension 80000[ / ] Stiffness of un-spring mass (tire) 500000[ / ] Damping coefficient of sprung mass 320 [ ∙ / ] Damping coefficient of un-sprung mass 15020 [ ∙ / ]

Table 2. Data for Bingham model simulation.

Parameter name Parameter notation Parameter value Damping coefficient in Bingham model 320 [ ∙ / ] Offset force 10 Frictional force 100

Table 3. Data for Dahl model simulation.

Parameter name Parameter notation Parameter value Control voltage 5 [ ] Hysteresis parameters , , , 350, 800, 250, 25

Table 4. Data for the LuGre model simulation.

Parameter name Parameter notation Parameter value Coulomb friction force 10[ ] Sticktion force 25[ ] Stribeck velocity 0.04 [ / ] Stiffness coefficient 500 [ / ] Damping coefficient 10 [ ∙ / ] Viscous friction coefficient 0.6 [ ∙ / ]

Table 5. Data for Bouc-Wen model simulation.

Parameter name Parameter notation Parameter value Parameters of the Hysteresis shape , , , n 1, 0, 1.5, 2 Stiffness of the spring element 300 [ / ] Input voltage 5 [ ] Other parameters , , , 4400, 442, 10872, 49616 Pre-yield stress 0 [ ]

From the numerical simulations of hysteresis loop models with Bingham, Dahl, LuGre and Bouc-Wen models for the semi-active suspension system it is clear that all of the semi-active system models outperform passively controlled system model for four different excitation signals from road. Figure 10 and 11 demonstrate system responses (displacement of the car body) of the passively and semi-actively controlled models from random (Gaussian white) noise with the magnitude of 0.075 m (in the range of -0.0375 m … +0.0375 m) and from the simulation results it is clearly seen that all hysteresis models outperform passively damped system model in damping undersigned excitations from the terrain. Out of these four semi-active models, Bingham and Bouc-Wen models demonstrate much higher damping than the other two models, viz. Dahl and LuGre models.

In another eexcitation inconsiderablyseconds. Wh

Figur

Figur

excitation with Sn the car body iny better than thehereas Bingham

Car

bod

y di

spla

cem

ent,

[m]

re 10. Model res

re 11. Model res

Step (Heaviside)n comparison we other two mo

m model damps th

0 2-8

-6

-4

-2

0

2

4

6

8

x 10-3

Input signal

sponses on rando

sponses on rando

function – Figuwith passively co

dels and dissipahe excitation in a

4

Passive vs. S

l (road): Gussia

om (Gaussian wh

om (Gaussian wh

ure 12 and 13, thontrolled model. ate the step exciabout 8 seconds

6time, [sec]

emi-active hyste

an noise

hite noise) excita

hite noise) excita

he hysteresis mo In this case, L

itation with the and Dahl model

8 10

eresis models

PBDLB

ation from road.

ation from road.

odels outperformLuGre and Bouc-

magnitude of 0l in about 13 sec

12

PassiveBinghamDahlLuGreBouc-Wen

.

.

m in damping un-Wen models pe0.075 m in less conds.

desired erforms

than 2

In sine wavWen modelcontrolled sthis case, Bexcitations vibrations inPerformancrange of MR damper

Fi

Fi

e excitations wil have demonstrsystem model anBouc-Wen modewith 20.8 Hz ofn car body morees of all models5 of displacr models in term

0

0

0

0

0

0

Car

bod

y di

spla

cem

ent,

[m]

igure 12. Passive

igure 13. Passive

th 2.1 Hz of freqrated slightly bend frequency of el has outperfof frequency showe than passively s after about twcement in car bos of damped exc

0 20

0.02

0.04

0.06

0.08

0.1

0.12

0.14

e vs. semi-active

e vs. semi-active

quency shown intter in damping excitation from rmed all other wn in Figure 16controlled mod

wo seconds of simody. In this casecitation.

4

Passive vs. S

Input signa

e suspension mod

e suspension mod

n Figure 14 andmagnitude of ethe road is presmodels in term

6 and 17, all hyel by preservingmulation time h, Bouc-Wen mo

6 8time, [sec]

Semi-active hyster

al (road): Heavisid

dels on Step inp

dels on Step inp

d 15, semi-activeexcitation oscillaserved clearly a

ms of damped ysteresis models g periodic oscillahave reached to odel has perform

8 10

resis models

de Step function

put excitation.

put excitation.

ely controlled mations in compa

as a periodic signoscillation maghave dissipated

ations with respvery similar ste

med slightly poor

12

PassiveBinghamDahlLuGreBouc-Wen

models except forarison with a panal with all mod

gnitudes. In sined magnitude of ect to road exciteady state valuerer than the othe

r Bouc-assively dels. In e wave excited tations.

e in the er three

F

F

Figure 14. Passiv

Figure 15. Passiv

-0.0

-0.0

-0.0

-0.0

-0.0

0.0

0.0

0.0

0.0

0.0

Car

bod

y di

spla

cem

ent,

[m]

ve vs. semi-activ

ve vs. semi-activ

0 2

05

04

03

02

01

0

01

02

03

04

05

Input si

ve suspension on

ve suspension on

4

Passive vs. S

ignal (road): 0.75sin

n sinusoidal wav

n sinusoidal wav

6 8time, [sec]

Semi-active hysteres

n(2π*f*t); f=2.1 Hz

ve: 0.075 sin 2

ve: 0.075 sin 2

8 10

sis models

PasBinDahLuGBou

, 2.1

, 2.1

12

ssiveghamhlGreuc-Wen

excitation.

excitation.

F

F

A fourth experformancewave excitahave not rea

Figure 16. Passiv

Figure 17. Passiv

xcitation signal fes of the semi-aation with 20.8 Hached to a stable

-0

-0.0

0.0

0

Car

bod

y di

spla

cem

ent,

[m]

ve vs. semi-activ

ve vs. semi-activ

from road used tactive models foHz frequency fo steady-state val

0 2

.01

005

0

005

.01

ve suspension on

ve suspension on

to simulate the mr this excitation

or some extent alue.

4

Passive vs.

n sinusoidal wave

n sinusoidal wave

models is sine wn – Figure 18 anand in this case,

6time, [sec]

Semi-active hyst

Input signal (

e: 0.075 sin 2

e: 0.075 sin 2wave with 20.8 Hnd 19 have been

the two MR mo

8 10

eresis models

(road): 0.75sin(2π*

, 20.8

, 20.8Hz of frequencysimilar to the p

odels, viz. Bingh

12

*f*t); f=20.8 Hz

PassiveBinghamDahlLuGreBouc-Wen

excitation

excitation

y plus white noisprevious case wiham and Dahl m

se. The ith sine models,

Figure 18

Figure 19

5. Summ

The developmathematicmodel havesemi-activelpassively cocar suspensterms of thesine wave. perform bet

8. The system re

9. The system re

mary

ped simulation mal formulations o

e showed adequaly controlled damontrolled damperion system mod

e damped vibratiIn case of highter than Bouc-W

-0

-0.0

0.0

0

0.0

Car

bod

y di

spla

cem

ent,

[m]

sponses of passi

sponses of passi

models of the hyof Bingham, Daacy of these MRmper models wir model in the exdel. Amongst thons and steady-s

her (>20.8 Hz) Wen model.

0 2

0.01

005

0

005

.01

015

Input sig

ive and semi-actwhite

ive and semi-actwhite

ysteresis or non-hl, LuGre and B

R dampers for dith Bingham, Daxample of four dhese semi-activestate response timfrequency (pure

4

Passive vs. Se

gnal (road): 0.75s

tive on sinusoidanoise excitation

tive on sinusoidanoise excitation

-linear system bBouc-Wen modeldesigning vibratiahl, LuGre and Bdifferent excitatiely controlled mme in three excite periodic) excit

6 8time, [sec]

emi-active hyster

sin(2π*f*t)+Gussia

al wave ( 20n.

al wave ( 20n.

behaviors of the ls in MATLAB/ion and shock dBouc-Wen modeion signals mimi

models, Bouc-Wtation signals, vitation from road

8 10

resis models

an noise

PBDLB

0.8 ): 0.075 sin

0.8 ): 0.075 sinMR liquids use

/Simulink in the dampers. The simels have demonsicking terrain rou

Wen has outperfoiz. step, white nod, Bingham, Da

12

PassiveBinghamDahlLuGreBouc-Wen

n 2 + Gaus

n 2 + Gaus

ed in dampers byexample of qua

mulation resultsstrated superioriughness for the

ormed other mooise and low freahl and LuGre m

ssian

ssian

y using arter car s of the ty over quarter

odels in quency models

Further studparameters parameters.

AcknowledCommittee

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imental validatios with respect t

with these MR da

m the State Sc

mi-active contro

hanical structure

cal Modeling o

amper with two

gneto-rheologica

y on a Nonlinearer 12-17, 2008, Bh MR damper Aigence, Modellin

ppropriate for co

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