analytical and numerical modal analysis of an automobile rear torsion beam suspension

7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension

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15thInternational Conference on Experimental Mechanics

ICEM15 1

PAPER REF: 2747

ANALYTICAL AND NUMERICAL MODAL ANALYSIS OF AN

AUTOMOBILE REAR TORSION BEAM SUSPENSIONMarco Dourado1(*), Jos Meireles11Mechanical Engineering Dept., University of Minho, Guimares, Portugal(*)

Email:[email protected]

ABSTRACT

The aim of this paper is to present the vibrational analysis of a multi-DOF system,

representative of the torsion beam type suspension that equips some vehicles. The new

approach is to get the solution of the system, incorporating the effect of the torsion beam. The

natural frequencies iand mode shapes ui, are calculated analytically and numerically, freely

in space. The analytic results are compared with the numerical results obtained in the finite

element model developed. It is presented the mode shapes and natural frequencies of the

suspension system with and without torsion beam effect.

Keywords:torsion beam, torsion bar suspension system, multi-dof system

INTRODUCTION

The main functions of suspension system are to keep the vehicle tires in contact with the road,

support the weight of the vehicle, allow the vehicle to drive with stability and absorb the

forces generated by your movement, and provide comfort to passengers (Reza, 2008). When

the vehicle is moving, especially on uneven roads, forces are transmitted to the wheels with

vertical direction and which magnitude depends heavily of uneven pavement. The wheel thus

suffers a vertical acceleration.

In a torsion beam type suspension, the vertical displacement is transferred to the trailing arm,

which is rigidly connected to the torsion beam. The torsion beam is responsible for absorbing

energy that results from differences in effort between the two ends of the system. So this can

not only, guide the local system at each wheel, but also balance the effort involved on two

wheels when required with different loads.

The number of researches that study the interaction of torsion beam with the road vehicle is

very small. However, Jia et al (2006), make the behavior study of road vehicle in contact with

the deck surface, but dont applied the torsion bar in the system. Some studies of dynamicsanalysis involving torsion bar suspension type, are applied to tracked vehicles, to evaluate the

ride performance, steerability, and stability on rough terrain (Yamakawa and Watanabe,

2004). Hohl (1986) study the influence of torsion beam in performance of tracked vehicle.

Murakami et al (1992), developed mathematical model which predicts spatial motion of

tracked vehicles on non-level terrain.

The our system is constituted by half body of a engine vehicle, assumed as a rigid massm1, by

lateral mass moment inertia Iyand front mass moment inertia Ixas shown in Fig. 1. In this

new approach, a torsion beam is included in the system with stiffness constant kt, which,

together with the right and left wheel, constitutes the mass ( )( )232 + mm . As shown in laterFig. 2, two springs with stiffness constant k1and two shock absorbers with damping constant

c1, applied in n3-n6and n4-n8nodes, support the damped mass of the suspension system in thelinkage point one to the body vehicle, together with two other springs with stiffness constant


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Porto/Portugal, 22-27 July 2012

Editors: J.F. Silva Gomes and Mrio A.P. Vaz2

k3applied in n1-n5and n2-n7nodes, located in the other linkage point to the body vehicle. The

undamped mass is supported by two springs with stiffness constant k2 and two shock

absorbers with damping constant c2 applied in n5-n9, n7-n11, n6-n10 and n8-n12 nodes, that

represents the wheels.

To simulate the effect torsion beam, we applied ktvalues of same magnitude, in n3-n6,n4-n8,

n1-n5 and n2-n7nodes, but with opposite signs. So, 3 6 tn n k , 4 8 tn n k , 51 tn n k ,

2 7 tn n k . We attributed three different ktvalues, to get three different dynamic behaviors

in ours models.

We calculated natural frequencies i and mode shapes uifor the three models. These mode

shapes represent: the vertical displacementx, the body pitch and the body roll ; the vertical

displacement x1 and x3 of m2+m3 mass applied in n5 and n6 nodes, and the vertical

displacementx2andx4of m2+m3mass applied in n7and n8nodes.

ANALYTICAL MODEL

As described in the previous point, for this system, shown in figure 1, it is assumed that the

system is undamped and natural frequencies are free in space.Then, the equation of motion

for the system under study is given by(Zienkiewicz, 1977):

0x][x][..

=+ km (1)

a) b) c)

Fig. 1 Half car analytical model: a) right view; b) front view; c) left view

The equations that govern the movement of system shown in figure 1 are:

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 0424413

3223114243

4133322131111

=++

++++++

+++++++

ddxxkddxxk

ddxxkddxxkddxxk

ddxxkddxxkddxxkxm

tt

tt

..

(2)

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 042424131

3222311142432

413313221231111

=++

++++++

+++++

ddxxkdddxxkd

ddxxkdddxxkdddxxkd

ddxxkdddxxkdddxxkdI

tt

tt

X

..

(3)

( ) ( ) ( )

( ) ( ) ( )

( ) ( ) 042444134

3223311342434

322134133431113

=+

++++++

+++++

ddxxkdddxxkd

ddxxkdddxxkdddxxkd

ddxxkdddxxkdddxxkdI

tt

tt

..

y

(4)


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ICEM15 3

( ) ( ) ( ) 0112311311112 =+++ yxkddxxkddxxkxm t ..

(5)

( ) ( ) ( ) 0222322322122 =++++++ yxkddxxkddxxkxm t ..

(6)

( ) ( ) ( ) 0132413412333 =++ yxkddxxkddxxkxm t ..

(7)

( ) ( ) ( ) 0242424424343 =+++ yxkddxxkddxxkxm t ..

(8)

So, the equation (1) is express in matricial form by:

0

000

000

000

000

000000

000000

000000

000000

000000

000000

000000

4

3

2

1

77737271

66636261

55535251

44434241

37363534333231

27262524232221

17161514131211

..

4

..

3

..

2

..

1

..

..

..

3

3

2

2

1

=

+

x

x

x

x

x

kkkk

kkkk

kkkk

kkkk

kkkkkkk

kkkkkkk

kkkkkkk

x

x

x

x

x

m

m

m

m

I

I

m

Y

x

(9)

where,3111 22 kkk += (10)

323112112112 kdkdkdkdkk ++== (11)

34133113 22 kdkdkk == (12)

tkkkk == 14114 (13)

tkkkk +== 15115 (14)

tkkkk +== 36116 (15)

tkkkk = 37117 (16)

3

2

23

2

11

2

21

2

122 kdkdkdkdk +++= (17)

tttt kddkddkddkddkddkddkddkddkk 423241313421323411313223 ++== (18)

tkdkdkk 1114224 +== (19)

tkdkdkk 2125225 +== (20)

tkdkdkk 1316226 == (21)

tkdkdkk 2327227 == (22)


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3

2

41

2

333 22 kdkdk += (23)

tkdkdkk 3134334 == (24)

tkdkdkk 3135335 +==

(25)

tkdkdkk 4346336 == (26)

tkdkdkk 4347337 +== (27)

tkkkk ++= 2144 (28)

tkkkk += 2155 (29)

tkkkk += 2366 (30)

t

kkkk ++=2377

(31)

The natural frequencies i are obtained by determination of the eigenvalues square root i ,

of matrix [ ]A , that is calculating the determinative

[ ][ ] 0det == IA (32)

em que

[ ] [ ] [ ]km 1A = (33)

e2

ii = (34)

The modes shapes iu , are calculated determining the eigenvectors of matrix [ ]A

[ ][ ] 0= iiA uI (35)

The table 1 presents the values associated with the variables envolved in the system. The

stifness constant ktassumes three diferente values. So we get three diferente models: modelwithout torsion beam system, with stifness constant kt = 0 N/m; model with torsion beam

system, with stifness constant kt= 25000 N/m; model with torsion beam system. with stiffness

constant kt= 75000 N/m.


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ICEM15 5

Table 1 Variables of system

Variable Value Units Variable Value Unitsd1 0.75 m m1 840 kg

d2 0.7 m m2 33.8 kgd3 1.47 m m3 32.2 kg

d4 1.4 m Ix 147.2 kg.m2

k1 13000 N/m Iy 576.63 kg.m

k2 200000 N/mk3 10000 N/m

kt

0

N/m25000

75000

Substituting the values of table 1, and with resource of MATLAB we solve the equations (33)

and (35) to the matrix system (9). The results are presented in the section Results.

NUMERICAL MODEL

The numerical model, as shown in figure 2, was constructed in ANSYS.

Fig. 2 Half car numerical model

The finite element model use three types of elements existents at the library of ANSYS: shell

elements (SHELL63), formed by n1, n2, n3 and n4nodes, to represent the mass m1 of body;

mass elements (MASS 21), applied in n5, n6, n7and n8nodes, that represent the torsion beam

and wheels mass distributed in the system; combination elements (COMBIN 40) in n3-n6,n4-


6/13



n8,n1-n5and n2-n7nodes, to represent the stiffness constant of springs and torsion beam,and

n5-n9, n7-n11, n6-n10and n8-n12nodes, to represent the stiffness constant of wheels.

ANALYTICAL RESULTSPresents in this chapter the natural frequencies values analytically obtaind for the three

models with different ktvalues, and the respective modes shapes.

The eingenvalues of matrix [ ]A and natural frequencies for kt= 0are, respectively:

36526

36525

86309

56307

1158

8154

50

7

6

5

4

3

2

1

.

.

.

.

.

.

=

=

=

=

=

=

=

Hz8612rad/s798036526

Hz8612rad/s788036525

Hz6512rad/s437986309

Hz6512rad/s427956307

Hz2rad/s57121158

Hz981rad/s44128154

Hz131rad/s07750

7

6

5

4

3

2

1

...

...

...

...

..

...

..

==

==

==

==

==

==

==

The eingenvalues of matrix [ ]A and natural frequencies for kt = 25000 are, respectively:

1207

87346

57092

45755

15568

8436

139

7

6

5

4

3

2

1

.

.

.

.

.

.

.

=

=

=

=

=

=

=

Hz6513rad/s718587346

Hz4113rad/s228457092

Hz0812rad/s867545755

Hz8811rad/s627415568

Hz333rad/s90208436

Hz990rad/s256139

6

5

4

3

2

1

...

...

...

...

...

...

==

==

==

==

==

==

The eingnevalues of matrix [ ]A and natural frequencies for kt = 75000 are, respectively:

41220

975

69165

88693

4446554191

7812

7

6

5

4

3

2

1

.

.

.

.

.

.

.

=

=

=

=

=

=

=

Hz2415rad/s749569165

Hz8514rad/s249388693

Hz6410rad/s902044465

Hz3110rad/s746454191

Hz544rad/s50287812

5

4

3

2

1

...

...

...

...

...

==

==

==

==

==

Although to be extracted seven eigenvalues for the trhee systems, nor all correspond to

natural frequencies, because these eigenvalues are negatives. For the case that kt= 0 N/m, we

have seven natural frequencies and seven mode shapes. For the case that kt = 25000 N/m only

six natural frequencies are considered, because the rotation aroun xaxle and yaxle, that is,body roll e body pitch modes shapes, originate only one mode, the torsion mode


7/13


ICEM15 7

shape. For the case that kt = 75000 N/m only five natural frequencies are considered, because

the displacementxis anulled, and the body roll e body pitch modes shapes originate

only one mode, the torsion mode shape.

The eigenvectors of matrix [ ]A to kt= 0 N/mare:

+

+

+

+

+

+

=

0.0023

0.0021

0.0020

0.0018

0.1026-

0.0118

0.9946

1u

+

+=

0.0033-

0.0085

0.0113-

0.0046

0.6482-

0.7599-

0.0458-

2u

+

+

+

+

=

0.0050-

0.0017-

0.0035

0.0079

0.9643

0.2114-

0.1590

3u

+

+

+

+

+

=

0.0131

0.0157-

0.7651-

0.6404

0.0113

0.0625

0.0078

4u

+

+

+

+

+

=

0.0114

0.0078

0.6342

0.7576

0.1269-

0.0077

0.0869-

5u

+

+

+

=

0.7915

0.6085-

0.0127

0.0163-

0.0128

0.0506-0.0085-

6u

+

+

+

=

0.6053-

0.7872-

0.0121

0.0073

0.0968-

0.0083-0.0654

6u

The figures 3 to 9, presents the modes shapes analytically obtained, for the model without

torsion beam, kt= 0 N/m. In the abcisses axle are represented the seven DOFs, where:Point 1 is the first mode shape associated to the vertical displacementx;

Point 2 is the second mode shape associated to the body roll ;

Point 3 is the third mode shape associated to the body pitch ;

Point 4 is the fifth mode shape associated to the vertical displacementx1;Point 5 is the fourth mode shape associated to the vertical displacementx2;Point 6 is the seventh mode shape associated to the vertical displacementx3;Point 7 is the sixth mode shape associated to the vertical displacementx4.

Fig. 1 1 mode shape Fig. 4 2 mode shape


8/13





Fig. 9 7 mode shape

The eigenvctors of matrix [ ]A to kt= 25000 N/mare:


9/13


ICEM15 9

+

+

=

00600

00280

00280

00610

00960

01820

99970

1

.

.

.

.

.

.

.

u

+

+

+

+

=

02170

00050

00040

02650

89200

44950

03350

2

.

.

.

.

.

.

.

u

+

+

+

+

+

+

+

=

00130

02890

99280

00230

08680

04110

06590

3

.

.

.

.

.

.

.

u

+

+

+

+

=

00180

98860

02930

00330

11090

05930

07770

4

.

.

.

.

.

.

.

u

+

+

+

=

13620

00270

00180

94730

20710

10580

17330

5

.

.

.

.

.

.

.

u

+

+

+

+

+

=

94480

00100

00070

13720

24370

12230

11910

6

.

.

.

.

.

.

.

u

The figures 10 to 15, presents the modes shapes analytically obtained, for the model wityh

torsion beam, kt= 25000 N/m. In the abcisses axle are represented the seven DOFs where the

second and third correspond to one DOF only. So:

Point 1 is the first mode shape associated to the vertical displacementx;

Point 2 e 3 is the second mode shape associated to the body pitch and body roll , that

is, torsion;

Point 4 is the fifth mode shape associated to the vertical displacementx1;Point 5 is the third mode shape associated to the vertical displacementx2;Point 6 is the fourth mode shape associated to the vertical displacementx3;Point 7 is the sixth mode shape associated to the vertical displacementx4.


Fig. 12 3 mode shape



10/13




The eigenvectors for the matrix [ ]A to kt= 75000 N/mare:

+

+

+

=

04690

00060

00240

05220

88990

45060

00520

1

.

.

.

.

.

.

.

u

+

+

+

+

+

+

+

=

0.0105

0.4377

0.7189

0.0136

0.1410

0.0597

0.5176

2u

+

+

+

+

=

0.0029

0.6574

0.4124-

0.0019

0.5576-

0.2670-

0.1247

3u

+

+

+

+

=

0.3780-

0.0059

0.0064

0.8120-

0.2295

0.1135-

0.3635

4u

+

+

=

0.7338

0.0016-

0.0012-

0.3436-

0.5101

0.2653-

0.1138-

5u

The figures 16 to 20, presents the modes shapes analytically obtained, for the model wityh

torsion beam, kt= 75000 N/m. In the abcisses axle are represented the seven DOFs where the

first is annulled, and the second and third correspond to one DOF only. So:

Point 2 e 3 is the first mode shape associated to the body pitch and body roll , that is,

torsion;

Point 4 is the fourth mode shape associated to the vertical displacementx1;Point 5 is the second mode shape associated to the vertical displacementx2;Point 6 is the third mode shape associated to the vertical displacementx3;Point 7 is the fifth mode shape associated to the vertical displacementx4.



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ICEM15 11



NUMERICAL RESULTS

The natural frequencies and mode shapes numerically obtained are presented in table 2.

Table 2 Numerical natural frequencies and mode shapes for different kt valueskt

(N/m)1

(Hz)

2(Hz)

3(Hz)

4(Hz)

5(Hz)

6(Hz)

7(Hz)

WithoutTorsionBeam

- 1.12 1.97 1.98 12.63 12.64 12.85 12.86

Mode

Shape

disp.

x

body

roll

body

pitch

disp.

x1 and

x2

disp.

x1 and

x2

disp.

x3and

x4

disp.

x3and

x4

WithTorsionBeam

+25000

-250000.92 3.29 11.87 12.06 13.32 13.6

Mode

Shape

disp.

x

body roll

andbody pitch(torsion)

disp.

x2 andx3

disp.

x2andx3

disp.

x1 andx4

disp.

x1 andx4

WithTorsionBeam

+75000

-75000- 4.50 9.98 10.58 14.30 15.07

ModeShape

-

body roll

andbody pitch

(torsion)

disp.

x2 and

x3

disp.

x2and

x3

disp.

x1 and

x4

disp.

x1 and

x4

The table 3 and table 4, present respectively, the comparison between the natural frequencies

and mode shapes analytically and numerically calculated. The values obtained using the two

methods, numerical and analytical, are very concordant, and the numerical models confirm

the analytical models.


12/13



Table 3 Comparison between the natural frequencies obtained using the two methods

WithoutTorsionBeam

WithTorsionBeam

WithTorsionBeam

kt(N/m) - kt(N/m)

+25000

-25000 kt(N/m)

+75000

-75000Analytical

Model

Numerical

Model

Analytical

Model

Numerical

Model

Analytical

Model

Numerical

Model1(Hz) 1.13 1.12 0.99 0.92 4.54 4.50

2(Hz) 1.98 1.97 3.33 3.29 10.31 9.98

3(Hz) 2 1.98 11.88 11.87 10.64 10.58

4(Hz) 12.65 12.63 12.08 12.06 14.85 14.30

5(Hz) 12.65 12.64 13.41 13.32 15.24 15.07

6(Hz) 12.86 12.85 13.65 13.6

7(Hz) 12.86 12.86

Table 4 Comparison between the mode shapes obtained using the two methodsWithoutTorsionBeam

WithTorsionBeam

WithTorsionBeam

kt(N/m) - kt(N/m)+25000

-25000kt(N/m)

+75000

-75000

Mode

Shapes

Analytical

Model

Numerical

Model

Analytical

Model

Numerical

Model

Analytical

Model

Numerical

Model1st disp.x disp.x disp.x disp.x - -

2nd body roll body roll body roll

andbody pitch

(torsion)

body roll

andbody pitch

(torsion)

body roll

andbody pitch

(torsion)

body roll

andbody pitch

(torsion)3rd body pitch body pitch

4th disp.x1 disp.x1 andx2 disp.x2 disp.x2 andx3 disp.x2 disp.x2 andx3

5th disp.x2 disp.x1 andx2 disp.x3 disp.x2 andx3 disp.x3 disp.x2 andx36th disp.x3 disp.x3andx4 disp.x1 disp.x1 andx4 disp.x1 disp.x1 andx4

7th disp.x4 disp.x3andx4 disp.x4 disp.x1 andx4 disp.x4 disp.x1 andx4

CONCLUSION

Depending on the increase of stiffness constant value ktof torsion beam, for constant values

k1, k2and k3,the natural frequencies that influence the vertical displacementxtend to decrease

or disappear. The body roll and the body pitch become a torsion mode shape when the

torsion beam is incorporated in the system. The fourth and fifth natural frequency value, both

associated with the vertical movementx2andx3of the undamped mass, decrease by increasing

ktvalue.The natural frequencies and mode shapes values analytically calculated, confirm the values

numerically obtained, so the new method to represent the torsion beam effect is reliable.

ACKNOWLEDGMENTS

The authors gratefully acknowledge the Centre for Mechanical and Materials Technologies

(Centro de Tecnologias Mecnicas e de Materiais CT2M) and QREN.


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ICEM15 13

REFERENCES

Hohl G.H. Torsion-Bar Spring and Damping Systems of Tracked Vehicles. Journal of

Terramechanics, Vol. 22, N 4, pp. 195-203, 1986.

Jia J., Ulfvarson A. Dynamic Analysis of Vehicle-Deck Interactions. Ocean Engineering, Vol.33, pp. 1765-1795, 2006.

Murakami H., Watanabe K., Kitano M. A Mathematical Model for Spatial Motion of Tracked

Vehicles on Soft Ground. Journal of Terramechanics, Vol. 29, N 1, pp 71-81, 1992.

Reza NJ. Vehicle Dynamics: Theory and Application. Springer, New York, 2008, p. 827-881.

Yamakawa J., Watanabe K. A Spatial Motion Analysis Model of Tracked Vehicles with

Torsion Bar Type Suspension. Journal of Terramechanics, Vol. 41, pp. 113-126, 2004.

Zienkiewicz OC. The Finite Element Method in Engineering Science, McGraw-Hill

Publishing Company, London, United Kingdom, 1977.

analytical and numerical modal analysis of an automobile rear torsion beam suspension

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