Archive of SID

Iranian Journal of Science & Technology, Transaction B, Engineering, Vol. 32, No. B4, pp 325-339

Printed in The Islamic Republic of Iran, 2008

© Shiraz University

KINEMATICAL AND DYNAMICAL ANALYSIS OF MACPHERSON

SUSPENSION USING DISPLACEMENT MATRIX METHOD

*

M. S. FALLAH, M. MAHZOON

**

AND M. EGHTESAD

Dept. of Mechanical Engineering, Shiraz University, Shiraz, I. R. of Iran

Email: mahzoon@shirazu.ac.ir

Abstract In this paper, kinematics and dynamics of the MacPherson suspension are studied.

Displacement Matrix Method is utilized for this purpose. Camber and toe angle alterations are

derived by means of the rotation matrix. Kinematical characteristics of the MacPherson suspension

are displayed as functions of time and the wheel vertical displacement. Relations for velocities and

accelerations of key points are also obtained. Since internal forces and external forces are

important in stress analysis and for comfortability, these forces are calculated as functions of time

as well. Subsequently, a general analysis of the me chanism under the wheel sinusoidal

displacement is presented together with relevant figures and results.

Keywords Kinematics, dynamics, MacPherson suspension mechanism, camber angle, toe angle, kingpin angle,

caster angle, track alteration

1. INTRODUCTION

The MacPherson suspension was made in the Ford Company by Earl S. MacPherson for the first time in

1949. This type of suspension is widely employed in many modern vehicles because of its light weight

and compact size [1]. It can be used for both front and rear suspensions, but is usually found at the front,

where it provides a steering pivot (kingpin) as wel l as a suspension mounting for the wheel. A

MacPherson suspension, as shown in Fig. 1, consists of: 1) a control arm, 2) a tie rod, 3) a spindle and

piston rod, and 4) a strut; the control arm is connected to the chassis with a rotational joint. A spherical

joint connects the control arm and the strut. Wheels are mounted on the struts spindle. A cylindrical joint

connects the strut to the piston rod which is connected to the chassis with a spherical joint. A spring and a

damper are put in between the strut and the chassis along the piston rod to absorb vibration and shocks

caused by a bumpy road. The strut is also connected to the tie rod with a spherical joint [2].

The main functions of suspension systems are to adequately support the vehicle weight, to provide

effective isolation for the chassis from excitations due to rough roads, to maintain the wheels in the

appropriate steer and camber attitudes with respect to the road surface and to keep tire contact with the

ground. The wheel suspension characteristics are of vital importance for the road holding ability of the

tires, riding quality of the car, minimizing the transient forces to the body and the severity of the

environmental noise.

In the design process for choosing ride characteristics a mathematical model is essential. This can

lead to a better choice for system parameters and improvement of its behavior [3]. In order to study the

behavior of a MacPherson strut wheel suspension, various mathematical approaches and models have been

presented. The models are more or less complex depending on the requirements set by the authors. For

Received by the editors January 28, 2006; Accepted April 27, 2008.

Corresponding author

www.SID.ir

Archive of SID

M. Saber Fallah et al.

Iranian Journal of Science & Technology, Volume 32, Number B4 August 2008

326

example, a spatial model of the Macpherson suspension to conduct kinematical analysis of this type of

structure was formulated by Cronin [1], and Stensson et al., [3]. Mantaras et al., [4] proposed three 2D

nonlinear models for Macpherson suspension to analyze the dynamical behavior of this mechanism.

Jonsson [5] carried out a finite element analysis for assessing deformation of the components of this

structure. In [2] and [6] two similar 3D models of MacPherson suspension were used to estimate its

dynamical parameters. Habibi et al. [7] implemented genetic algorithm method on a three dimensional

model of MacPherson suspension to optimize its design characteristics. Sohn et al. [8]-[11] proposed a 2D

model of Macpherson suspension for control applications. Lee and Han considered a simple model of

Macpherson suspension and designed a controller to control varying roll center during cornering [12]. A

comprehensive analysis of MacPherson suspension was performed by Reimpell et al. [13]. They reviewed

the main parameters of several suspensions and compared them with each other. In [14], another approach

to study variations of handling parameters of MacPherson suspension was proposed by Raghavan.

Fig. 1. MacPherson suspension mechanism [2]

In this paper, the displacement matrix method is employed to analyze the system behavior and a

comprehensive kinematical and dynamical analysis of Macpherson suspension is presented. As a main

advantage, implementation of this method is truly convenient for computer coding and also formulation of

kinematical and dynamical relations.

The main kinematical parameters which are investigated in this paper are: 1) camber angle, 2) toe

angle, 3) caster angle, 4) kingpin angle, and 5) track alterations. As a result of functional factors when the

wheels travel in bump and rebound in travel direction, the track changes. However, alteration of the track

size causes the rolling tire to slip and, on flat crossings in particular, causes lateral forces and may even

influence the steering. Caster and kingpin angles alterations affect the selfaligning torques and

consequently affect the stability and handling of the vehicle when wheels bounce or rebounce. Camber

angle alterations are due to the rubbing of tire and produce lateral force acting on the wheel. In order to

keep the vehicle stable, any change in the gradient of the toe angle characteristics as functions of wheel

travel should be avoided. In addition, to avoid increased tire wear and rolling resistance or impeding

directional stability, no toe angle changes should occur when the wheels compress or rebound [13]. For

B

0

Link 2: Tie Rod

D

E

Link 1: Control Arm

A

1

P

Link 3: Spindle

B

1

C

1

j

1

www.SID.ir

Archive of SID

Kinematical and dynamical analysis of MacPherson

August 2008 Iranian Journal of Science & Technology, Volume 32, Number B4

327

z

0

Yaw

y

0

Pitch

Roll

dynamical analysis of the mechanism, the determination of velocities and accelerations of the key points

are necessary; therefore, relations for velocities and accelerations of these points should also be derived.

Internal forces are important in stress analysis and are crucial factors in the failure of the components

and joints. Therefore, in this paper all internal forces and moments acting on the mechanism are calculated

as functions of time. The equations for external forces and moments such as: 1) spring force; 2) damper

force; 3) road force, and 4) resisting torque are derived as they affect comfortability and also magnitude of

the internal forces.

2. DISPLACEMENT MATRIX METHOD AND CONSTRAINT EQUATIONS

One of the most effective methods to analyze mechanisms is through the use of displacement matrix

method. In this method, the general three-dimensional displacement matrix is given in terms of a

translation from a point ),,(

1111

zyxp to another point ),,(

2222

zyxp, and a rotation about a fixed

coordinate frame [15].

++

++

++

=

++

++

++

=

1000

1000

1331321312

1231221212

1131121112

1331321312333231

1231221212232221

1131121112131211

12

)zayaxa(z

)zayaxa(y

)zayaxa(x

R

)zayaxa(zaaa

)zayaxa(yaaa

)zayaxa(xaaa

]D[

D

(1)

where

ij

a are components of the rotation matrix. The rotation matrix can be described as the product of

successive rotations about the principal coordinate axes x

0

, y

0

and z

0

taken in a specific order. These

rotations define the roll, pitch, and yaw angles, which are denoted as

and,,

, and are shown in Fig. 2.

The resulting rotation matrix, R

D

, is then given by

R

D

=

++

++

=

coscossincossin

cossinsinsincossinsinsincoscoscossin

cossincossinsinsinsincoscossincoscos

333231

232221

131211

aaa

aaa

aaa

(2)

Another task in the displacement matrix method is the formulation of the holonomic constraint equations

which are specifically related to the joints between adjacent bodies. These are mathematical restrictions on

the mobility of the model so as to take away degrees of freedom of the multi-body system. The constraint

equations for the spherical-spherical (SS) link, the revolute-spherical (RS) link and spherical-cylindrical

(SC) link of the mechanism are listed below.

Fig. 2. Roll, pitch and yaw angles

www.SID.ir

Archive of SID

M. Saber Fallah et al.

Iranian Journal of Science & Technology, Volume 32, Number B4 August 2008

328

a) Spherical-spherical (SS); link constraint equations

SS link is defined by a spherical joint at a point ),,(

0000

zyxp on a fixed body and another spherical

joint at a different point ),,(

1111

zyxp on a moving body. The constraint equation that specifies the

constancy of the distance between the two spherical joints is

=++

2

01

2

01

2

01

)zz()yy()xx( constant

Therefore, the displacement constraint for tie rod is:

222222

)()()()()()(

01

0

101000

BB

B

BBBB

j

BBBBB

zzyyxxzzyyxx

jj

++=++ (3)

where j subscript indicates the displaced point, i.e. B

j

, j = 2, 3, 4, .

b) Revolute-spherical (RS) link constraint equations

An RS link connects a revolute joint at a point ),,(

0000

zyxp with rotation axis ),,(

0000 zyx

uuuU on

a fixed body and a spherical joint at another point ),,(

1111

zyxp on a moving body. The constraint

equations which specify the constancy of distance between the revolute and spherical joints and the

perpendicularity between the revolute axis and the axis defined as the link are

=++

2

01

2

01

2

01

)zz()yy()xx(

constant,

0

010101

000

=

+

+

)zz(u)yy(u)xx(u

zyx

Consequently, the displacement constraints for control arm are:

222222

)()()()()()(

0

00010101

AAAAAAAAAAAA

zzyyxxzzyyxx

jjj

++=++ (4)

0)()()(

000000

=

+

+

AAzAAyAAx

zzuyyuxxu

jjj

(5)

),,(

0000 zyx

uuuU is unit vector of the DE line in the principal axis. In the subsequent analysis, the

revolute joints at D and E will be replaced by one revolute joint at point A

0

, found from the orthogonal

projection of vector DA

1

on the revolute axis.

c) Spherical-cylindrical (SC) link constraint equations

The SC link connects a spherical joint at a point ),,(

0000

zyxp on a fixed body and a cylindrical joint

at another point ),,(

1111

zyxp on a moving body with an axis of translation/rotat ion

),,(

1111 zyx

uuuU along the link axis. The constraint equations that specify that the straight line defined by

the link, or the cylindrical joint axis, remains a straight line during any displacement are

0

0101

11

=

)zz(u)xx(u

xz

, 0

0101

11

=

)zz(u)yy(u

yz

and

1

222

111

=++

zyx

uuu

So, the displacement constraints for the strut are (j

1

is an arbitrary point on strut):

For line C

1

C

0

:

0)()(

00

=

ccxccz

zzuxxu

jjjj

, 0)()(

00

=

ccyccz

zzuyyu

jjjj

and 1

222

=++

jjj

zyx

uuu (6)

For line j

1

C

0

:

0)()(

00

=

CjxCjz

zzuxxu

jjjj

, 0)()(

00

=

CjyCjz

zzuyyu

jjjj

(7)

www.SID.ir

Archive of SID

Kinematical and dynamical analysis of MacPherson

August 2008 Iranian Journal of Science & Technology, Volume 32, Number B4

329

d) Kinematical equations

The assumptions adopted in Fig. 1 are summarized as follows:

1) The chassis is fixed; 2) All bodies are rigid; 3) Control arm is modeled by a rod.

The input to the system is the road profile which is applied at point P (Fig. 1). In this paper, the road

profile is assumed be a sinusoidal curve with the characteristics ,

)tsin(z

r

5

80

=

.

If displacement of point P in the vertical direction is equal to z

r

subject to road disturbance, point P

1

)z,y,x(

ppp

111

will move to point P

2

)z,y,x(

ppp

222

and the magnitude of its z coordinate will be equal

to:

12

PrP

zzz +=.

Consequently, the displacement matrix of spindle can be written as:

[ ]

++

++

++

=

1000

)(

)(

)(

1112

1112

1112

333231333231

232221232221

131211131211

AAAA

AAAA

AAAA

Spindle

zayaxazaaa

zayaxayaaa

zayaxaxaaa

D (8)

where

2

A is the new position of

1

A (Fig. 1) under the wheels vertical displacement. This matrix is used

to determine the new positions of points B

1

, C

1

and j

1

by the following relation:

[ ]

=

111111

111

111

111

222

222

222

jCB

jCB

jCB

Spindle

jCB

jCB

jCB

zzz

yyy

xxx

D

zzz

yyy

xxx

(9)

To solve the unknown parameters, the following constraint equations should be added to Eq. (9).

+++=

=++

++=++

=++

=

=

=

=

++=++

34333231

222222

222

222222

0

1112

020020020

0

10101020202

222

022022

022022

022022

022022

01

0

10102022

9

08

7

16

05

04

03

02

1

azayaxaz:

)zz(u)yy(u)xx(u:

)zz()yy()xx()zz()yy()xx(:

uuu:

)zz(u)yy(u:

)zz(u)xx(u:

)zz(u)yy(u:

)zz(u)xx(u:

)zz()yy()xx()zz()yy()xx(:

PPPP

AAzAAyAAx

AAAAAAAAAAAA

zyx

Cjyjjz

Cjxjjz

ccyccz

ccxccz

BB

B

BBBBBBBBB

As for the numerical scheme, the Newton-Raphson method is employed [16] and all unknown parameters

of the displacement matrix of the spindle are determined. Using this matrix, the spring force, which is vital

for dynamical analysis of MacPherson suspension mechanism, can be calculated.

3. MATHEMATICAL DETERMINATION OF THE KINEMATICAL PARAMETERS

In this section, kinematical parameters of the MacPherson suspension mechanism are described. As

mentioned before, θ and ψ correspond to camber and toe angles. Caster is the angle between the projection

of the steering axis (line A

1

C

0

) on xz plane and a line perpendicular to the road. Using this definition:

www.SID.ir

Archive of SID

M. Saber Fallah et al.

Iranian Journal of Science & Technology, Volume 32, Number B4 August 2008

330

Caster angle

)(tan

0

01

j

j

AC

AC

zz

xx

=

. Kingpin is the angle between the projection of steering axis on the yz

plane and a line perpendicular to the road; therefo re, it can be stated as: Kingpin

angle

)(tan

0

0

1

j

j

AC

AC

zz

yy

=

, and track alteration is equal to: Track alteration

j

AA

yy

=

1

.

a) Velocity and acceleration matrices

The velocity and acceleration matrices are:

[ ]

++

++

++

=

0000

)(

)(

)(

1112

1112

1112

333231333231

232221232221

131211131211

AAAA

AAAA

AAAA

spindle

zayaxazaaa

zayaxayaaa

zayaxaxaaa

D

(10)

where

D

xy

xz

yz

D

R

aaa

aaa

aaa

R

=

=

0

0

0

333231

232221

131211

, and

[ ]

++

++

++

=

0000

)(

)(

)(

1112

1112

1112

333231332331

232221232221

131211131211

AAAA

AAAA

AAAA

spindle

zayaxazaaa

zayaxayaaa

zayaxaxaaa

D

(11)

where

D

zxzyyzx

xzyyzyx

yzxzyxx

D

R

aaa

aaa

aaa

R

+

+

+

=

=

22

22

22

333231

232221

131211

,

2222

zyx

++=.

Using these matrices, velocity and acceleration of any point can be determined.

[

]

[

]

[

]

TT

zyxDzyx 10

=

;

[

]

[

]

[

]

TT

zyxDzyx 10

=

(12)

Differentiating Eqs. (3) to (7), the velocity and acceleration constraints are derived.

Set of Velocity Equations:

++=

++=

++=

+=

+=

+=

++=

++=

++=

34

222

34

222

34

222

24

222

24

222

24

222

14

222

14

222

14

222

9

8

7

6

5

4

3

2

1

ayxz:

ayxz:

ayxz:

azxy:

azxy:

azxy:

azyx:

azyx:

azyx:

jxjyj

CxCyC

BxByB

jxjzj

CxCzC

BxBzB

jyjzj

CyCzC

ByBzB

www.SID.ir

Archive of SID

Kinematical and dynamical analysis of MacPherson

August 2008 Iranian Journal of Science & Technology, Volume 32, Number B4

331

++=

=++

=++

=++

=+

=+

=+

=+

=++

34

222

202020

022022022

222222

0222202

2

22

0222202222

0222202

2

22

0222202

2

22

022022022

18

017

016

015

014

013

012

011

010

ayxz:

zuyuxu:

)zz(z)yy(y)xx(x:

uuuuuu:

)yy(uyu)zz(uzu:

)xx(uxu)zz(uzu:

)yy(uyu)zz(uzu:

)xx(uxu)zz(uzu:

)zz(z)yy(y)xx(x:

PxPyP

AzAyAx

AAAAAAAAA

zzyyxx

CjzjzCj

y

jy

CjzjzCj

x

jx

CCzCzCC

y

Cy

CCzCzCC

x

Cx

BBBBBBBBB

Set of Acceleration Equations:

++++=

=++

=+++++

=+++++

=++

=++

=++

=++

=+++++

++++=

++++=

++++=

++++=

++++=

++++=

++++=

++++=

++++=

34

2

22

222

202020

2

2

2

2

2

2022022022

2

2

2

2

2

2222222

02222220222222

02222220222222

02222220222222

02222220222222

2

2

2

2

2

2022022022

34

2

22

222

34

2

22

222

34

2

22

222

24

22

22

22

24

22

22

22

24

22

22

22

14

222

22

2

14

222

22

2

14

222

22

2

18

017

016

015

2214

02213

02212

02211

010

9

8

7

6

5

4

3

2

1

az)(y)(x)(z:

zuyuxu:

zyx)zz(z)yy(y)xx(x:

uuuuuuuuu:

)yy(uyuyu)zz(uzuzu:

)xx(uxuxu)zz(uzuzu:

)yy(uyuyu)zz(uzuzu:

)xx(uxuxu)zz(uzuzu:

zyx)zz(z)yy(y)xx(x:

az)(y)(x)(z:

az)(y)(x)(z:

az)(y)(x)(z:

az)(y)(x)(y:

az)(y)(x)(y:

az)(y)(x)(y:

az)(y)(x)(x:

az)(y)(x)(x:

az)(y)(x)(x:

PzPxzyPyzxP

AzAyAx

AAAAAAAAAAAA

zyxzzyyxx

CjzjzjzCjyjyjy

CjzjzjzCjxjxjx

CCzCzCzCCyCyCy

CCzCzCzCCxCxCx

BBBBBBBBBBBB

jzjxzyjyzxj

CzCxzyCyzxC

BzBxzyByzxB

jx'zyjyjzyxj

Cx'zyCyCzyxC

Bx'zyByBzyxB

jyzxjzyxjxj

CyzxCzyxCxC

ByzxBzyxBxB

Solving these linear sets of equations, velocity and acceleration of every key point can be calculated.

4. DYNAMICAL ANALYSIS OF MACPHERSON SUSPENSION

Figure 3 represents the MacPherson suspension model used in this paper. The suspension system is

attached to the vehicle body at points B

0

, C

0

and A

0

. The strut/shock absorber is composed of a cylindrical

joint with the axis formed by the collinear points C

0

, j

1

, C

1

, a viscoelastic damping element along the strut

www.SID.ir

Archive of SID

M. Saber Fallah et al.

Iranian Journal of Science & Technology, Volume 32, Number B4 August 2008

332

axis, an offset elastic spring between points H

1

and G

1

, and finally, a resisting torque element at point C

0

.

The MacPherson suspension has three degrees of freedom for motions of the wheel, however, two of them

are passive and only one degree of freedom is active. In as much as passive angular velocities of the tie

rod and strut do not directly contribute to the motion of the mechanism as a whole, they are ignored.

Fig. 3. Representation of MacPherson suspension

By means of sets of displacement, velocity and acceleration equations, positions, velocities and

accelerations of points B

1

, j

1

, and C

1

can be determined. Also, using kinematical relations, positions,

velocities and accelerations of mass centers of all links will be found. Subsequently, the forces exerted by

the spring and damper, resisting torque and the road force at point C

0

can be found as follows:

1-Spring force:

Direction: GH, Magnitude:)(

0

SGHk

s

,

0

S: Original length of spring,

s

k: Spring constant

2- Damper force:

Direction:

C

x

u (Along strut axis), Magnitude: )(

2

C

xk

uCd ×

where

2

C is on the wheel assembly

(spindle) and

k

d is damping constant.

3- Resisting torque )(

0

C

R:

Direction:)(

z

C

x

uu , Magnitude:

[

]

).(

zxkC

uurR

co

=, :

k

r Proportionality constant

4- Road force )(

R

F:

Using the work-energy method the road force will be found. First, kinetic and potential energies of the

system are calculated:

22222

2

1

2

1

2

1

2

1

2

1

)(I)(I)(I)(I)(I)energyKinetic(E

C

z

C

z

C

y

C

y

B

z

B

z

B

y

B

y

A

y

A

y

CCCCBBBBAA

++++=

PmIII

E

E

z

E

z

E

y

E

y

E

x

E

x

EEEEEE

2

1

)(

2

1

)(

2

1

)(

2

1

222

++++

(13)

where

p is velocity of the wheel center,

j

x

j

and

j

y

j

are x and y components of angular velocity of

link j, respectively.

i

x

i

I,

iy

i

I and

i

z

i

I are moments of inertia of link i in its body frame x, y and z. In this

equation, links identified as A, B, C and E are the control arm, the tie rod, the strut and spindle, and xyz

C

0

j

1

B

0

G

1

H

1

C

1

P

A

1

B

1

A

0

www.SID.ir

Archive of SID

Kinematical and dynamical analysis of MacPherson

August 2008 Iranian Journal of Science & Technology, Volume 32, Number B4

333

represents the local coordinates of each link, while XYZ shows the global coordinate frame.

p

in Eq. (13)

can be found using matrices,

[

]

Spindle

D and

[

]

Spindle

D

, as:

[

]

[

]

TT

SpindleSpindle

ppDD

21

= (14)

Also, potential energy is

)(

2

1

2

0

2

SGHkgzmU

srE

+= (15)

where m

E

is spindle mass. On the other hand, the work done on the system is equal to

ZCZCYCYCXCXCrR

C

CrR

)()R()()R()()R(zFRzFW

oooO

+++=×+×= (16)

where F

R

and

o

C

R are vertical force from the road and internal force acting on joint C

0

, respectively. So,

r

CC

R

z

R)energyPotential(U)energyKinetic(E

F

O

×+

=

(17)

In Eq. (13) the angular velocities of the bodies are expressed in the body local coordinate systems, in

contrast to the velocity matrices which are in terms of angular velocities expressed in the global coordinate

frame. The transformation tensor can be used to make the transformation between a vector expressed in

the global coordinate system V and the same vector expressed in the local coordinate system V

as

[

]

TT

VVL

=, where [L] is the second order transformation tensor for the body, [17, 18].

For example, the transformation tensor for body A (Control arm) is[ ]

=

AAA

AAA

AAA

z

Z

z

Y

z

X

y

Z

y

Y

y

X

x

Z

x

Y

x

X

A

uuu

uuu

uuu

L

where ),,(

AAAA

x

Z

x

Y

x

X

x

uuuu is a unit vector along the direction of

A

x (in the global coordinate system). For

the displaced positions of the mechanism, first the new direction of the unit vector should be calculated

from the rotational part of the displacement matrix as

TT

D

uuR

2

1

=, where

D

R is found from Eq. (2).

5. ANALYSIS OF JOINT REACTION FORCES AND MOMENTS

Figure 4 shows the joint reaction forces and moments for the MacPherson suspension.

Unknown reaction forces and moments are:

Control Arm, Body A

At revolute joint, A

0

:

AAAAA

z

A

x

A

z

A

y

A

x

A

MMFFF

00000

,,,,; at spherical joint, A:

AAA

z

A

y

A

x

A

FFF,,

where F and M indicate force and moment, respectively. Subscripts refer to the joint location and

superscripts refer to the vector direction.

Tie-rod, Body B

At spherical joint, B

0

:

BBB

z

B

y

B

x

B

FFF

000

,,; at spherical joint, B:

BBB

z

B

y

B

x

B

FFF,,

Strut, Body C

At spherical joint, C

0

:

CCC

z

C

y

C

x

C

FFF

000

,,; at cylindrical joint, C:

CC

z

C

y

C

FF, and J:

CC

z

j

y

j

FF,

There are a total of 21 unknown joint forces and moments. Since the motion of each body is already

known, all equations are linear in terms of unknowns. The method chosen here is to formulate Newtons

equations in the global coordinate system; the inverse of transformation tensor

1

][

L may be used to

transform a vector from a local coordinate system to the global coordinate system. Eulers equations are

formulated in the principal axes of the local coordinate system of the body. The 16 unknown forces and

www.SID.ir

Archive of SID

M. Saber Fallah et al.

Iranian Journal of Science & Technology, Volume 32, Number B4 August 2008

334

moments at joints, A, B

0

, B, C

0

, and C are determined, first by formulating the Newton-Euler equations for

bodies B, C and E in terms of these 16 unknowns.

Fig. 4. Joint reaction forces and moments for

MacPherson suspension

Strut

Spindle

Control Arm

Tie Rod

B

B

0

B

z

A

F

0

B

y

B

F

0

B

z

B

F

0

B

x

B

F

B

y

B

F

B

z

B

F

A

z

A

F

0

A

0

A

A

y

A

F

0

A

x

A

F

0

A

x

A

M

0

A

z

A

M

0

A

z

A

F

A

x

A

F

A

y

A

F

C

z

C

F

0

C

y

C

F

C

x

C

F

0

S

F

S

F

C

z

j

F

1

C

y

j

F

1

C

z

C

F

1

D

F

C

y

C

F

1

D

F

Rc

0

R

F

A

y

A

F

A

x

A

F

A

z

A

F

B

x

B

F

B

z

B

F

B

y

B

F

C

z

C

F

1

C

y

j

F

1

C

z

j

F

1

www.SID.ir

Archive of SID

Kinematical and dynamical analysis of MacPherson

August 2008 Iranian Journal of Science & Technology, Volume 32, Number B4

335

Body B:

[ ]

=

+

Bm

Bm

Bm

BB

z

y

x

z

y

x

Z

Y

X

Lm

F

F

F

F

F

F

B

B

B

B

B

B

0

0

0

and

BBBBB

HFrFr

=+

00

(y, z components) (18)

where m

B

is tie rod mass,

B

j

F and

0

B

j

F are j components of internal forces acting at joints B and B

0

,

respectively. zyx ,, are acceleration components of center of mass of link BB

0

.

j

r and

i

H

are the

position vector of the corresponding point and derivative of angular momentum of the body around the

corresponding point.

Next, Newton-Euler equations for body A is used to determine the remaining five unknowns at joint

A

0

[

]

mAAAA

ALmFF

=+

0

(x, y, z components) and

AAAA

HMTT

=++

00

(x, z components) (19)

where M

A0

is the joint reaction torque for the revolute joint at point A

0

. F and T stand for acting forces and

torques, respectively.

6. SIMULATION OF THE MACPHERSON SUSPENSION MECHANISM

Simulation results of kinematical and dynamical analysis are presented under a sinusoidal vertical

displacement of the wheel and an amplitude of 80 mm (a typical road bump) is assumed for this

displacement. The initial positions for the key points of the MacPherson mechanism (all coordinates are in

mm) and system parameters are presented as follows (these characteristics are obtained from the R&D

office of Mehvar sazan -e- Iran Khodrow for Peugout 405 GLX):

A

1

= (503.46, 687.25, 191.5), C

1

= (525.478, 583.66, 403.968), D= (789.02, 379.5, 259.37),

j

1

= (527.327, 580.3961, 500), E= (501.17, 379, 243.67), p

1

= (509.23, 748.94, 279.66)

B

0

= (670.85, 312.5, 341), G

1

= (529.217, 609.579, 605.701), B

1

= (639.178, 664.471, 292.165),

H

1

= (530, 575, 850), C

0

= (534.42, 567.875, 868.4)

System parameters:

Spring constant:

mmN.k

s

615

=

, Original length of spring, mmS 245

0

=

Damping constant:

mmsNd

k

.12

=

along strut axis, Initial velocity:

smmp 0.0

1

=

o

C

R is a resisting torque acting on the strut and is proportional to the angular displacement of the strut

from the z axis with proportionality constant

radmmNr

k

.90

=

Input: profile of road surface:

)

5

sin(80 tz

r

= mm acting in the vertical Z direction

All local coordinate systems are taken along the principal axes. Also, local coordinates and dynamical

characteristics of the links are:

Control arm (RS link)

Mass, m

A

=1.25 kg, Center of mass location: )0,0,150(),,(

1111

=

AAAm

zyxA

Local coordinate system (x

A

, y

A

, z

A

):

1010

1

AA)AA(x

A

=,

=

1

A

y along vector DE,

1

11

AAA

yxz

=

Moments of inertia at A

m1

:

2

500 mmkgI xA ×=,

2

2000 mmkgI yA ×=,

2

2500 mmkgI zA ×=

Tie-rod (SS link)

Mass, m

B

=1.4 kg, Center of mass location: )0,0,180(),,(

1111

=

BBBm

zyxB

Local coordinate system (x

B

, y

B

, z

B

) :

1010

1

BB)BB(x

B

=, 6276)0956903,0.0.0957787,(

1

=

B

y,

www.SID.ir

Archive of SID

M. Saber Fallah et al.

Iranian Journal of Science & Technology, Volume 32, Number B4 August 2008

336

1

11

BBB

yxz

=

. Moments of inertia at B

m1

:

2

500 mmkgI xB ×=,

2

2000 mmkgI yB ×=,

2

2500 mmkgI zB ×=

Strut (SC link)

Mass, m

C

=1.30 kg, Center of mass location: )0,0,230(),,(

1111

=

CCCm

zyxC

Local coordinate system (x

C

, y

C

, z

C

):

1010

1

CC)CC(x

C

=,

=

1

C

y through point H,

1

11

CCC

yxz

=

Moments of inertia at C

m1

:

2

500 mmkgI xC ×=,

2

3000 mmkgI yC ×=,

2

3500 mmkgI zC ×=

Spindle

Mass, m

E

=17.00 kg; Center of mass location: )0,0,0(),,(

1111

=

EEEm

zyxE

Local coordinate system (x

E

, y

E

, z

E

):

=

1

E

x global X,

=

1

E

y global Y ,

=

1

E

z global Z

Moments of inertia at E

m1

:

2

12500 mmkgI

xE

×=,

2

21000 mmkgI

yE

×=,

2

21000 mmkgI

zE

×=

Alterations of camber angle, caster angle, toe angle and track are indicated in Figs. 5 to 8,

respectively. Damping and spring forces are calculated and their time variations and magnitudes are

shown in Figs. 9 and 10. Internal forces at joints A

1

and B

1

are shown in Figs. 11 to 16.

Fig. 5. Camber angle alteration versus

vertical displacement of wheel

Fig. 6. Caster angle alteration versus vertical

displacement of wheel

Fig. 7. Toe angle alteration versus vertical

displacement of wheel

Fig. 8. Track alteration versus vertical

displacement of wheel

www.SID.ir

Archive of SID

Kinematical and dynamical analysis of MacPherson

August 2008 Iranian Journal of Science & Technology, Volume 32, Number B4

337

Fig. 9. Damping force versus time

Fig. 10. Spring force versus time

Fig. 11. x component of joint A

1

force versus time

Fig. 12. y component of joint A

1

force versus time

Fig. 13. z component of joint A

1

force versus time

Fig. 14. x component of joint B

1

force versus time

Fig. 15. y component force of joint B

1

versus time

Fig. 16. z component force of joint B

1

versus time

www.SID.ir

Archive of SID

M. Saber Fallah et al.

Iranian Journal of Science & Technology, Volume 32, Number B4 August 2008

338

Reimpell et al. have performed a comprehensive kinematical analysis for different types of

MacPherson suspension [13]. For example, they studied the track behavior of MacPherson strut

suspension for Opel and Audi, camber angle alterations of this suspension for a 3-series BMW and a

Mercedes, and caster angle alterations for Fiat Uno, VW Polo and a Mercedes. In addition, toe angle

alterations have been investigated for three separate cases which are different in their control arm length.

The range of alterations of each parameter for bounce and rebounce of one wheel is shown in Tables 1-4.

Comparing results presented in the tables, it is seen that there is a reasonable agreement between the

results presented in this paper and those of reference [13], and the trend of alteration of each kinematical

parameter is in accordance with the corresponding figures of the reference [13]. Also, for every

kinematical parameter, the range of alterations is acceptable. On the other hand, spring force and damper

force are proportional to displacement and velocity, respectively; as shown in Figs. 9 and 10. Because tie

rod is a spherical-spherical link, it can merely withstand force along x local direction and this is shown in

Figs. 14 to 16.

Table 1. Maximum track alteration (mm) for different types of MacPherson suspension

Opel Audi Studied Case

Bounce 3 2 3

Rebounce 11 14 24

Table 2. Maximum camber angle alteration (degree) for different types of MacPherson suspension

BMW Mercedes Studied Case

Bounce 1.3 1.2 0.5

Rebounce 1.7 2 2

Table 3. Maximum caster angle alteration (degree) for different types of MacPherson suspension

Fiat Uno Mercedes Studied Case

Bounce 0.1 2.2 0.5

Rebounce 0.2 1.2 0.8

Table 4. Maximum toe angle alteration (degree) for different types of MacPherson suspension

Case 1 Case 2 Studied Case

Bounce 1 0.6 0.18

Rebounce 1 1 0.7

7. CONCLUSION

All kinematical and dynamical relations for MacPherson suspension are derived. The displacement matrix

method has been employed as a general tool for simulating MacPherson mechanism. It requires no

commercial software but simply a non-linear algebraic equation solver. Kinematical parameters such as

camber angle, caster angle, toe angle, King-pin angle and, track alteration are investigated and their

variations are obtained. External forces such as damping and spring forces are calculated and their time

variations and their magnitudes are indicated. Internal forces, which are very important in the stress

analysis of the mechanism, are derived by the Newton-Euler method. Specifically, internal forces at joints

A

1

and B

1

are found.

www.SID.ir

Archive of SID

Kinematical and dynamical analysis of MacPherson

August 2008 Iranian Journal of Science & Technology, Volume 32, Number B4

339

REFERENCES

1. Cronin, D. L. (1981). MacPherson strut kinematics. Journal of Mechanism and Machine Theory, Vol. 16, No. 6,

pp. 631-644.

2. Chen, K. & Beale, D. G. (2003). Base dynamic parame ter estimation of a MacPherson suspension

Mechanism. Journal of Vehicle System Dynamics, Vol. 39, No. 3, pp. 227-244.

3. Stensson, A., Asplund, C. & Karlsson, L. (1994). The nonlinear behavior of a Macpherson strut wheel

suspension. Journal of Vehicle System Dynamics, Vol. 23, pp. 85-106.

4. Mantaras, D. A., Luque, P. & Vera, C. (2004). Development and validation of a three dimensional kinematic

model for the Macpherson steering and suspension mechanism. Journal of Mechanism and Machine Theory,

Vol. 39, pp. 603-619.

5. Jonsson, M. (1991). Simulation of dynamical behavior of a front wheel suspension. Journal of Vehicle System

Dynamics, Vol. 20, No. 5, pp. 269-281.

6. Kim, C. & Ro, P. I. (2000). Reduced order modeling and parameter estimation for quarter car suspension

system, Journal of Automobile Engineering, IMechE, Vol. 214, Part D, pp. 851-864.

7. Habibi, H., Shirazi, K., H. & Shishesaz, M. (2007). Roll steer minimization of McPherson-strut suspension

system using genetic algorithm method. Journal of Mechanism and Machine Theory, Article in press.

8. Hong, K. S., Jeon, D. S. & Sohn, H. C. (1999). A new modeling of the Macpherson suspension system and its

optimal pole-placement control. Proceedings of the 7th Mediterranean Conference on Control and Automation

(MED99).

9. Sohn, H. C., Hong, K. S. & Hedrick, J. K. (2000). Semi-active control of the Macpherson suspension system:

hardware-in-the-loop simulations. Proceedings of the 2000 IEEE International Conference on Control

Applications, Anchorage, Alaska, USA.

10. Sohn, H. C. & Hong, K. T. (2004). An adaptive LQG control for semi-active suspension systems. International

Journal of Vehicle Design, Vol. 34, No. 4, pp. 309-325.

11. Sohn, H. C., Hong, K. S. & Hedrick, J. K. (2002). Modified skyhook control of semi-active suspensions: a new

model, gain scheduling, and hardware-in-the-Loop tuning. Journal of Dynamic Systems, Measurement, and

Control, Vol. 124, No. 1, pp. 158-167.

12. Lee, U. & Han, C. (2000). A suspension system with a variable roll center for the improvement of vehicle

handling characteristics. Proc. Instn. Mech. Engrs, 215, Part D.

13. Reimpell, J., Stoll, H. & Betzler, W. (2001). The automotive chassis engineering principles, Butterworth-

Heinmann.

14. Raghavan, M. (1996). Number and dimensional synthesis of independent suspension mechanisms. Journal of

Mechanism and Machine Theory, Vol. 31, No. 8, pp. 1141-1153.

15. Spong, M. W. & Vidyasagar, M. (1989). Robot dynamics and control. John Wiley & Sons.

16. Chapra, S. C. & Canale, R. P. (1998). Numerical methods for engineering McGraw-Hill.

17. Kane, T. R. & Levinson, D. A. (1985) Dynamics: theory and application. McGraw-Hill.

18. Meghdari, A. & Fahimi, F. (2000). First order decoupling of equations of motion for multibody systems

consisting of rigid and elastic bodies. Iranian Journal of Science and Technology, Vol. 24, No. B3, pp. 333-343.

www.SID.ir

## Comments 0

Log in to post a comment