analytical and numerical modal analysis of an automobile rear torsion beam suspension
TRANSCRIPT
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
1/13
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
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
2/13
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)
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
3/13
15thInternational Conference on Experimental Mechanics
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)
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
4/13
Porto/Portugal, 22-27 July 2012
Editors: J.F. Silva Gomes and Mrio A.P. Vaz4
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.
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
5/13
15thInternational Conference on Experimental Mechanics
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-
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
6/13
Porto/Portugal, 22-27 July 2012
Editors: J.F. Silva Gomes and Mrio A.P. Vaz6
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
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
7/13
15thInternational Conference on Experimental Mechanics
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
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
8/13
Porto/Portugal, 22-27 July 2012
Editors: J.F. Silva Gomes and Mrio A.P. Vaz8
Fig. 5 3 mode shape Fig. 6 4 mode shape
Fig. 7 5 mode shape Fig. 8 6 mode shape
Fig. 9 7 mode shape
The eigenvctors of matrix [ ]A to kt= 25000 N/mare:
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
9/13
15thInternational Conference on Experimental Mechanics
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. 10 1 mode shape Fig. 11 2 mode shape
Fig. 12 3 mode shape
Fig. 13 4 mode shape
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
10/13
Porto/Portugal, 22-27 July 2012
Editors: J.F. Silva Gomes and Mrio A.P. Vaz10
Fig. 14 5 mode shape Fig. 15 6 mode shape
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.
Fig. 16 1 mode shape Fig. 17 2 mode shape
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
11/13
15thInternational Conference on Experimental Mechanics
ICEM15 11
Fig. 18 3 mode shape Fig. 19 4 mode shape
Fig. 20 5 mode shape
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.
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
12/13
Porto/Portugal, 22-27 July 2012
Editors: J.F. Silva Gomes and Mrio A.P. Vaz12
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.
-
7/23/2019 Analytical and Numerical Modal Analysis of an Automobile Rear Torsion Beam Suspension
13/13
15thInternational Conference on Experimental Mechanics
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.