Magneto-Rheological Damper hysteresis characterization for

SAS systems

D. Ghorbany, H.R. Karimi, Y. Iskandarani

Department of Engineering, Faculty of Engineering and Science, University of Agder,

N-4878 Grimstad, Norway

E-mails: dariug05@student.uia.no, {hamidrk, yousef.iskandarani}@uia.no

Abstract

A semi-active control of vehicle suspension system with

magneto-rheological (MR) damper is studied for the vibration

suppression in this paper. Subsequently several mathematical

models are used to simulate and analyze hysteresis behavior

of MR brake. The second part of this work is devoted to

derivation of the dynamic model equation of the experimental

setup. The last part presents the evaluation of the dynamic

simulation modeling results with the full-scale experimental

data. Simulation results show effects of the semi-active

control on the vibration of suspension system.

1. INTRODUCTION

In recent years, the manufacturing of magneto-rheological

(MR) dampers is an important subject for engineers. Some

has used magneto-rheological dampers as control devices for

engineering applications to dissipate the energy in civil

structures (e.g. during an earthquake). Those instruments

have many advantages such as small power requirement,

reliability, and low price to manufacture [1]-[4]. Recently the

military has shown interest to using MR dampers to control

gun recoil on naval gun turrets and field artillery.

Vibration control of vehicle suspension systems has been a

very active subject of research, since it is of great importance

for drivers and passengers. There are various mechanical

systems in the world which provide isolation of a structure

from the effect of disturbances (e.g. vibrations). One example

of such a device is a rotational magneto-rheological brake

(MR brake) which creates braking torque by changing the

viscosity of the MR fluid inside the brake. The magneto-

rheological brakes are used in many applications including

prosthetics, automotive, vibration stabilization.

In this paper, there is presented examination of the

mechanical system equipped with MR brake. The main aim

of this work is to obtain the hysteresis plot by finding the

torque in MR brake dependent on different input current.

2. MR DAMPER AND EXPERIMENTAL SETUP

2.1 MR ROTARY DAMPER (brake)

In order to reduce vibration of the mechanical system as a

kinetic and potential energy it has done built MR brake with

the waveform boundary of rotary disk. The kinetic and

potential energy which is generated in the mechanical system,

are been saving in the MR brake, and then by re-cycling,

some of the energy lost in braking into electrical energy.

Rotary brakes based on controllable fluids are passive devices

with an excellent torque to weight ratio and excellent control

capability [2]. As such, they offer a strong potential to be

used in semi active suspension system like a quarter of car as

well as to decrease the vibration of the car body. Higher

torque to volume ratio, lower voltage and less sensitivity to

impurity are the some advantages of this device. MR dampers

consist of two categories:

Linear damper.

Rotary damper.

The proposed MR brake mechanism utilizes a hybrid concept

of magnetic circuit in using both axial and radial magnetic

flux to generate braking force. MR actuators provide

controlled torque through control of an applied magnetic

field. Therefore knowledge of the relationship between the

applied current and output torque is required. The measured

torque shows hysteresis effects as the current increases and

decreases [4]-[5]

3. DESIGN OF EXPERIMENTAL SETUP

The prototype of experimental setup (e.g. of a vehicle

suspension) works as shown in Fig. 1. Vibration control of

vehicle suspensions systems has been an active subject of

research, since it can provide a good performance for drivers

and passengers. Recently, many researchers have investigated

the application of MR fluids in the controllable dampers for

semi-active suspensions; it means control with the current.

Fig 1 Design of experimental setup.

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ISBN: 978-1-61804-126-5 485

This work has the purpose of characterizing, identifying the

mathematical model and simulating the behavior of a MR

fluid in car suspension systems. The MR device designed in

this research is a rotational damper through which an electro-

magnet rotates MR fluid. The semi-active suspension system

consists of a rocking lever that emulates the quarter of car

body, whereas a spring and an MR damper simulate the semi

active vibration control. A DC motor coupled to an eccentric

wheel due to the same shaft is used to simulate the vibrations

induced to the vehicle.

3.1 DYNAMIC MODEL

Based on Newton’s second law the following two dynamic

equations describe the vehicle suspension system - see Fig 2.

𝐽2𝑑2𝛼2

𝑑𝑡2− 𝑘2

𝑑𝛼2

𝑑𝑡+ 𝑚2 𝑔 𝑟2 𝑐𝑜𝑠𝛼2

− 𝑟2𝑘𝑠 𝑙0𝑠

− 𝑟2𝑐𝑜𝑠𝛼2 − 𝑟1𝑐𝑜𝑠𝛼1 2 + 𝑟2𝑠𝑖𝑛𝛼2 − 𝑟1𝑠𝑖𝑛𝛼1

2

= 𝑑𝛼1

𝑑𝑡−𝑑𝛼2

𝑑𝑡 𝑀𝑀𝑅 𝑖

(1)

Fig 2 Geometrical diagram of experimental setup.

J1d2α1

dt2 − k1dα1

dt+ m1 g1 r1 cos β − α1 +

r1ks(l0s − r2cosα2 − r1cosα1

2 + r2sinα2 − r1sinα1 2

)

−kg R cos β − α1 l0g + R sin β − α1 + r − Dx + ukin

−fg d Dx − ukin

dt−

dα1

dtR cos β − α1

= dα1

dt−

dα2

dt MMR i

(2)

3.2 METHODOLOGY OF PARAMETERS

ESTIMATION

In the following, a set of equations describe the torque of the

experimental setup.

TGravity − Tspring − Tviscous − TMR = J ω (3)

where

𝑇𝑏𝑜𝑑𝑦 ∶ 𝐽2𝑑2𝛼2

𝑑𝑡2

𝑇𝑣𝑖𝑠𝑐𝑜𝑢𝑠 ∶ 𝑘2𝑑𝛼2

𝑑𝑡

𝑇𝐺𝑟𝑎𝑣𝑖𝑡𝑦 ∶ 𝑚2 𝑔 𝑟2 𝑐𝑜𝑠𝛼2

𝑇𝑠𝑝𝑟𝑖𝑛𝑔 ∶

𝑟2𝑘𝑠 𝑙0𝑠 −

𝑟2𝑐𝑜𝑠𝛼2−𝑟1𝑐𝑜𝑠𝛼12+𝑟2𝑠𝑖𝑛𝛼2−𝑟1𝑠𝑖𝑛𝛼12

𝑇𝑀𝑅 ∶ 𝑐 𝜔 + 𝛼 𝑧

The equations (3) describe the torque from the upper beam

due to the magneto-rheological damper. In this way the 𝑇𝑀𝑅

is the function of input to the simulink block diagram, to

estimating the mathematical model such as Bouc-Wen, Dahl

and Bingham models.

Fig 3 Scheme of the upper part of the experimental setup.

By comparing those three different mathematical models by

using the experimental graph (blue) in the last three figures, it

is concluded that the Bouc-Wen model gives the best

estimation for the hysteretic characteristic. The different

control current input from 0A to 1A shows that the vibration

of the body is reduced. For the 0A the bodies vibration goes

to the stable in around 11 seconds, and by changing the

current from 0A to the 0.2A as a input the system is going to

stable in 6 seconds. In another case by increasing the control

input of current, the damper friction in the magneto-

rheological (MR) rotary brake fluid is increasing. And also

the vibration of the body is going to be stabilized in the

shorter time.

Fig 4 Damping simulation of MR brake.

TMRα1

α2X1

r2

r1

ks

ᵝ

Dx

Ukin

r

R

log

Kg

+

T body

T spring

TsT

TDT

fg

T unsrung mass

m1 g1

X

m2 g2

Spring

α2

α1 = 0

MR brake

Upper beam

Lower beam

Advances in Computer Science

ISBN: 978-1-61804-126-5 486

3.3 EQUATION OF MOTION

Fig 5 Quarter of car design.

𝐹0sin(𝜔𝑡) is the external force that is the function of the road

oscillation. The general equation of motion of the eccentricity

is described of the model in figure above like a quarter a car.

𝑀 𝑋 + 𝐶 𝑋 + 𝐾 𝑋 = 𝐹(𝑡) (4)

where the M is the mass matrix, the C is the damping matrix,

and the K is the stiffness matrix for the system.

Based on equation of motion, the system is described in

below: [7]

𝑚𝑢𝑥 1 + 𝑓𝑔𝑥 1 + 𝑘𝑔𝑥1 + 𝑐𝑠 𝑥 2 − 𝑥 1 + 𝑘𝑠 𝑥2 − 𝑥1 =

𝐹0 sin(𝜔𝑡)

(5)

𝑚𝑠𝑥 2 + 𝑐𝑠 𝑥 2 − 𝑥 1 + 𝑘𝑠 𝑥2 − 𝑥1 = 0 (6)

Fig 6 Torque (Nm) – Time(s)

4. PARAMETERS ESTIMATION AND

IDENTIFICATION

4.1 Bouc-Wen model and parameters identification

For application of MR brake in vibration control, the model

of the MR damper should be continuous in all the ranges and

be numerically tractable, and the Bouc–Wen model is

adopted here. The schematic model is shown in Fig. 7. The

torque in this system is given by [3]-[6]

Fig 7. Bouc-Wen model of MR damper

TMR = α i z + C i θ (7a)

z = −γ θ z z n − βθ z n + δθ (7b)

with

𝐶 𝑖 = 𝐶0 + 𝐶1(𝑖)

𝛼 𝑖 = 𝛼0 + 𝛼1(𝑖)

where 𝛾,𝛽, 𝛿 and n are parameters related to the shape of

hysteresis loop. By adjusting the parameters γ, β, δ and n, of

the model it is possible to control the nonlinearity the size of

hysteresis graph. The simulation result is shown as figures

below.

Fig 8 Torque – Angle (Deg)

Fig 9 Torque – Angular velocity (Deg/sec)

mu

Cs Ks

Kg

ms

fg

x1 mu

ms

Tire

x2

xin

x1

x2

F0 sin(ωt)

Road oscillation T

ө

Bouc-WenC

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ISBN: 978-1-61804-126-5 487

Table 1 Parameters identification of Bouc-Wen model

Parameters Value Unit

𝛽 737 𝛾 1 𝛿 843 N 0,9 Coa 0,0015 Ns/m

Cob 0,07 Ns/m

αa 1 N/m

αb 0,07 N/m

Simulation results confirm that the higher current is applied,

the higher torque is generated, resulting in increasing

system’s damping.

4.2 Dahl model

Dahl model of MR damper is described as follows

TMR = kx i θ t + kw i z t (8a)

z = ρ (θ − θ z) (8b)

with

kx = kxa+ kxb

(i)

kw = kwa+ kwb

(i)

The Dahl model is the Coulomb friction element in the

change of friction of the force, when the direction of motion

is changed. The equations (8) explain that the spring stiffness

(kx and kw ) are current dependent, and also

(kwa, kwb

, kxa, kxb

) and ρ are parameters that control the

shape of the hysteresis loop, see simulation result of Dahl model

in Fig. 10. It means that the spring stiffness is a function of

current, and the parameters are estimated by the output of

experimental data response. [3]

Fig 10 simulation result of Dahl model

Fig 11 Torque – Angle (Deg)

Fig 12 Torque- Angular velocity (Deg/sec)

The simulation results of the Dahl parameters show that the

identification of parameters is correct. The hysteresis loop of

Torque- Angular velocity is increasing from 0A up to the 6A.

By increasing the current, the Magneto-rheological (MR)

brake generates more torque. The hysteresis loop of the

parameters explains that the estimation of parameters is

acceptable.

4.3 Bingham model

Bingham mechanical model is described as follows:

Tmr = fc ∗ sgn θ + c0θ + f0 (9)

The Bingham model has been used to characterize MR and

ER fluids. The model consists of a Coulomb friction element

placed in parallel with a viscous damper, which is shown in

Table 2 Parameters identification of Dahl model

Parameters Value Unit

kxa 0,001 Ns/m

kxb 0,001 Ns/m

ρ 5

kwa 5 N/m

kwb 1,5 N/m

Advances in Computer Science

ISBN: 978-1-61804-126-5 488

Figs. 11 and 12. In this model the torque generated by the

device given by (eq. above) where 𝑐0 is the damping

coefficient and 𝑓𝑐 is the friction force, 𝑓0 is non zero. [3]

The Bingham parameters are shown as in below. The goal of

the Bingham model is that to reproduce a hysteretic torque

with high velocity. The simulation results are plotted in the

different parameters for 𝑓𝑐 as it is shown as in Figs. 13-15.

Fig 13 Torque – Angle (Deg)

Fig 14 Torque – Angle (Deg)

Bingham model parameters:

𝐶0 ∶ 0.002 𝑁𝑠

𝑚, 𝑓𝑐 ∶ 0.1 𝑁 , 0.2 𝑁, 𝑓0 ∶ 1

To identify the Bingham parameters, it has been used the

torque equation of the Magneto-rheological (MR) brake.

Torque-Angle hysteresis loop describes the displacement of

the body angle between the (-3.5, 3.5) deg. Torque-Angular

velocity hysteresis loop describes the rotational speed of the

body angular velocity in deg/sec.

5. Conclusions From the experiment investigation of the magneto-rheological

damper, it has been shown that the MR brake has a very

broad changeable damping force range under magnetic field

and the damping coefficient increases with the electric

current.

Fig 15 Torque - Angular velocity (Deg/sec)

The application of the physical device for the vibration

control is studied. The dynamic equation of the mechanical

SAS system is derived together with the geometrical and

graphical description of the system. Equations of motion are

established by using Newton’s 2nd Law.

The flexibility of shaping the closed-loop road excitation

frequency responses of a quarter-car model by feedback

control was investigated. Physical torque equation of the MR

brake has been investigated, and then parameters of the

different mathematical models have been estimated. Also, the

parameters identification has been done for the real

experimental data. The estimated hysteresis loops were

compared in passive and semi active suspension with the real

experimental response of the damper.

References [1] Tran Hai Nam and Kyoung Kwan Ahn, A New Structure of MR

Brake with the Waveform Boundary of Rotary Disk, In Procc.

ICCAS-SICE, 2009, pp. 2997 – 3002.

[2] Jinung An and Dong-Soo Kwon, Modeling of a Magneto-

rheological Actuator Including Magnetic Hysteresis, Journal of

Intelligent Material Systems and Structures September 2003 vol.

14 no. 9 541-550

[3] Andre N. Vavreck and Ching – Han Ho, Characterization of a

commercial magnetorheological brake/damperin oscillatory

motion. Proc. SPIE 5760, 256 (2005); doi:10.1117/12.599156

[4] Jorge A. Cortes R., Leopoldo S. Villareal- Gonzales and Manuell

Martinez M., Characterization, modeling and simulation of

magneto rheological damper behavior under triangular

excitation. AZojomo (ISSN 1833-122X) Volume 1 November

2005, DOI: 10.2240/azojomo00164.

[5] Jastrzebski, Lukasz, Simulation of Magnetic Fields in a

Generator for an MR Rotary Damper Using COMSOL

Multiphysics. XXXIV. Seminar ASR '2009 “Instruments and

Control”, Babiuch, Smutný & Škutová (eds) 2009, VŠB-TUO,

Ostrava, ISBN 978-80-248-1953-2

[6] Mauricio Fabian Zapateiro De La Hoz, Semi active control

strategies for vibration mitigation in adaptronic structures

equipped with magneto rheological dampers. Universitat de

Girona, Spain, 2009, PhD Thesis, ISBN: 9788469255223

[7] Toni Liedes, Improving the performance of the semi active

Tuned mass damper, OULU UNIVERSITY PRESS, OULU

2009, ISBN 978-951-42-9124-1

Advances in Computer Science

ISBN: 978-1-61804-126-5 489