A Study for Shimmy Reduction using Theoretical Multi-body Model

and Experimental Techniques

Ho Won Lee and Jang Moo Lee

Machine Dynamics Laboratory, Seoul National University, Korea

Abstract

In this study, both theoretical linear model and numerical model are used to predict and optimize the frequencies of

shimmy related modes and the level of the vibration for a vehicle. A theoretical multi-body linear model of 32-DOF were used for quasi-static analysis and modal analysis. The theoretical model was verified by comparing the results of modal analysis with those of modal test for front suspension system using CADA-X program. The quasi-static analysis results of the theoretical model were compared with those of ADAMS model. Also, full vehicle model using ADAMS was verified with the results of the modal test and chassis dynamometer tests for shimmy reproduction. From tests and simulations, we found wheel longitudinal vibration mode to be the most dominant source of shimmy vibration. Also, based on the test results, we selected 13 design parameters of the front suspension system, including compliance and geometric factors for reducing shimmy level. We performed 27 orthogonal simulations using Taguchi methodology and constructed an optimal combination of design parameters. Through modal analysis of theoretical model with optimized design parameters, we found that the wheel longitudinal vibration mode changed into two local vibration modes. Lastly, a simulation of the numerical model verified that the suggested design parameters resulted in reducing shimmy vibration.

Keywords: shimmy, multi-body model, suspension, modal analysis, modal test, Taguchi

1. Introduction

Shimmy is rotational vibration of steering wheel under driving condition. Shimmy is generally categorized into low speed shimmy (20~60km/h) and high speed shimmy (100~130km/h.) Non-uniformity, unbalance, eccentricity, or run-out of wheel-tire assembly are known as the source of generation of rotational moment around kingpin axis which leads to lateral vibration of tie-rod and subsequent rotational vibration of steering wheel. Shimmy occurs regardless of braking or steering. Shimmy has a significant influence on ride comfort and maneuvering stability due to its resonant characteristics. Once shimmy occurs, it does not vanish until vehicle speed is significantly increased or decreased. Since it is virtually impossible to eliminate the source of shimmy mentioned above, it is necessary to study for the optimization of suspension system which is the transmitting path of vibration from wheel-tire assembly to steering wheel.

Although a number of researches in vehicle shimmy vibration have been carried out, most of them take either

mathematical model or complex numerical simulation model into consideration. In this study, theoretical vehicle model is constructed by applying linear analysis method with small displacement assumptions, and quasi-static and modal analyses are conducted using this model. Also, full vehicle numerical simulation model is constructed using multi-body dynamics software ADAMS, and the frequency and the level of shimmy vibration are predicted through driving simulation. Besides theoretical and numerical analyses, cause of shimmy and process of vibration transmission are investigated by observing dynamic behavior of each part of front suspension system through shimmy reproduction test. Moreover, we find out shimmy related modes through modal testing. Furthermore, based on Taguchi’s methodology, sensitivity analysis for principal design parameters is conducted in order to suggest optimal design strategy to reduce shimmy vibration. Lastly, a numerical simulation is carried out to verify that applying suggested design parameters results in alleviating shimmy vibration.

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2. Theory and Modeling 2.1 Linear Elasto-Kinematic Governing Equation [13]

There have been a wide variety of researches about kinematic and dynamic analyses on multi-body systems consisting of rigid bodies. Although it is relatively easy to determine the position of the purely kinematic system, it is difficult to determine the position of the rigid body system which is connected by elastic elements and constrained by kinematic constraints. The conventional method of finding static equilibrium of multibody system is to solve nonlinear constraint equations (Eqn. 2.1) and force equilibrium equations (Eqn. 2.2) [14, 15].

0)( =Φ q (2.1)

0=+Φ ATq fλ (2.2)

In Eqn. 2.1, represents general coordinate vectors of a rigid body system. In Eqn. 2.2,

qλ , qΦ , and

represent Lagrange’s multiplier, Jacobian, and constraint force respectively. is external forces applied on a rigid body system. These are represented in the form of nonlinear equations (Eqn. 2.3), which are generally solved by iteration methods, such as Newton-Rhapson method, in order to obtain and

λTqΦ

Af

q λ . However, it is difficult to estimate the initial value of λ and to obtain exact Jacobian matrix, which results in degrading the convergence of the solution.

⎭⎬⎫

⎩⎨⎧

+ΦΦ

−=⎭⎬⎫

⎩⎨⎧ΔΔ

⎥⎥⎦

⎤

⎢⎢⎣

⎡Φ+Φ

ΦAT

qTqq

ATq

q

fq

f )())((0

λλλ (2.3)

However, with small displacement assumptions,

difference of Eqn. 2.1, , can be satisfied while the system undergoes external forces. Therefore, constraint equation (Eqn. 2.4) is satisfied for small displacement . Also, taking linear elastic force and constraint force into consideration, force equilibrium equation (Eqn. 2.5) can be obtained,

0=ΔΦ

qΔ

0=ΔΦ=ΔΦ qq (2.4)

fqKTq =Δ+Φ λ (2.5)

where q represents Jacobian matrix of the kinematic constraint equation. In matrix form, Eqn. 2.4 and Eqn. 2.5 become an elasto-kinematic governing equation as

follows.

Φ

⎭⎬⎫

⎩⎨⎧

=⎭⎬⎫

⎩⎨⎧Δ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

ΦΦ

00fqK

q

Tq

λ (2.6)

2.2 Quasi-Static Equilibrium Analysis with Piece-wise Linear Assumption [13]

If we assume small displacement, Eqn. 2.6 can be satisfied in deformed positions resulted from continuous external forces. This piece-wise linear assumption makes it possible for the above mentioned elasto-kinematic governing equation to be applied. With this assumption, we can induce the following series equation, which determines the coordinates of the next step by obtaining the displacement increment from the coordinates of the previous step .

1−Δ nq1−nq

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧Φ

+⎭⎬⎫

⎩⎨⎧Δ

=⎭⎬⎫

⎩⎨⎧Δ

⎥⎥⎦

⎤

⎢⎢⎣

⎡

Φ

Φ −−

0)(

00)()()( 11 nTn

qn

nq

TnTq

n qfqq

qqK λλ

(2.7)

According to small rotation of local coordinate

systems, coordinate transformation matrix also has to be updated by Eqn. 2.8, and any vector represented by global coordinate system can be updated by Eqn. 2.9. With these series equations, quasi-static behavior of multibody system can be analyzed easily by any linear methods such as Gauss elimination method.

1

1

11

1−

−

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

ΔΔ−Δ−ΔΔΔ−

= n

n

xy

xz

yzn AA

ππππππ

(2.8)

111 ~ −−− Δ−= nnnn vvv π (2.9)

Suspension parameters have been obtained by

applying Eqn. 2.7, 2.8, and 2.9 to a Mcpherson suspension system. A theoretical linear model of 32-DOF shown in Figure 2.1 has been constructed in order to simulate SPMD(Suspension Parameter Measurement Device) test.

2

3K

4K

1K

2K

5K

6K

7K

8K

9K

10K

11K 12K

13K 14K

15K

1 ( )q DΦ

2 ( )q CΦ

3 ( )q SΦ

4 ( )q DΦ

5 ( )q CΦ

6 ( )q SΦ

7 ( )q SΦ 8 ( )q SΦ9 ( )q IΦ 11( )q IΦ

10 ( )q TΦ 12 ( )q TΦ

xy

z

①

③

②

⑨⑧

⑦

⑥

⑤

④

Number of General Coord. 66

Number of Constraints 34

Number of Indep. Coord. 32

①, ④ Wheel & Knuckle Assembly②, ⑤ Strut③, ⑥ Lower Control Arm

⑦ Rack⑧, ⑨ Stabilizer Bar

Figure 2.1 Mathematical model for quasi-static analysis

Vertical force has been applied on the center of the jack in order for the wheel to move in vertical direction. For model verification, results of SPMD analysis have been compared with those of ADAMS simulation. The result of SPMD analysis is shown in Figure 3.3~4 in the next chapter along with ADAMS simulation result.

2.3 Theoretical Modal Analysis[13]

Dynamic characteristics of vehicle suspension system can be effectively investigated through modal analysis. In order to find out dominant modes related to shimmy and changes in those modes according to design modifications, theoretical modal analysis for front suspension system has been conducted. Based on the theory suggested above, 32-DOF mathematical model has been constructed by modeling Mcpherson suspension system as 9 rigid bodies connected by elastic elements and joints, shown in Figure 2.2.

3K

4K

1K

2K

5K

6K

7K

8K

9K

10K

11K 12K

13K 14K15K

1 ( )q DΦ

2 ( )q CΦ

3 ( )q SΦ

4 ( )q DΦ

5 ( )q CΦ

6 ( )q SΦ

7 ( )q SΦ 8 ( )q SΦ

xy

z

Figure 2.2 Mathematical model for modal analysis

From linear static equations, dynamic equation can be induced. By differentiating equation 2.5 with respect to time, velocity and acceleration constraint equations for rigid body system can be obtained with small motion assumption. Taking constraint forces into consideration, the following dynamic equation can be obtained.

)(tfqKqCqM Tq =Φ+Δ+Δ+Δ λ&&& (2.10)

e vector qΔIf we divide general coordinat and Jacobian matrix qΦ into and [ ]TTT vuq ΔΔ=Δ ,

[ ]vuq ΦΦ=Φ , respectively, constraint equation 2.4 can be written as follows.

0=ΔΦ+ΔΦ vu vu (2.11)

Since submatrix uΦ of Jacobian matrix qΦ is not

singular, Eqn. 2.11 can be arranged and written as follows.

vu vu ΔΦΦ−=Δ −1 (2.12)

That is, equation of motion of general coordinate

system can be transformed into that of independent coordinate system . Transformation equation and transformation matrix are shown below.

nk

)( kn ≥

vDq Δ=Δ (2.13)

⎥⎥⎦

⎤

⎢⎢⎣

⎡ ΦΦ−=

−

ID vu

1 (2.14)

If we carry out coordinate transformation by

substituting Eqn. 2.13 into Eqn. 2.9, and then conduct forward multiplication of TD , following equation of motion of independent coordinates can be obtained. k

DDDD fvKvCvM =Δ+Δ+Δ &&& (2.15) where

MDDM TD =

CDDC TD =

KDDK TD =

fDf TD =

3

If we remove damping term and external force for eigenvalue analysis, simple dynamic equation can be obtained as follows.

0=Δ+Δ vKvM DD && (2.16)

The solution satisfying the following eigenvalue

problem exists in this system,

[ ][ ] [ ][ ][ ]2ωΨ=Ψ DD MK (2.17)

[ ] [ ]kΨΨΨ=Ψ ,,, 21 L (2.18)

[ ]⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢

⎣

⎡

=

2

22

21

2

0

0

kω

ωω

ωO

(2.19)

where [ ]2ω and are eigenvalue matix and

eigenmode matrix, respectively. In case that there are number of independent coordinates, number of eigenvalues and eigenmodes can be obtained by coordinate transformation through Eqn. 2.13.

[ ]Ψk

k

Important mode shapes from the result of theoretical modal analysis on front suspension system in low frequency band are shown in Figure 2.3.

Figure 2.3 Result of theoretical modal analysis

3. Numerical Analysis 3.1 SPMD Simulation

ADAMS model has been constructed using geometry, inertia, stiffness, and damping data of front suspension system, and SPMD simulation under the same condition as quasi-static analysis in chapter 2.2 has been carried out. The result is compared in Figure 3.2~3.3 for model validation.

Figure 3.1 ADAMS model of front suspension system

Figure 3.2 Lateral movement Figure 3.3 Caster angle

3.2 Shimmy Simulation

Shimmy simulation has been conducted with the ADAMS simulation model which was verified to have kinematic and dynamic characteristics of the subject vehicle. Simulation has been carried out under the driving condition of constant speed of 120km/h on random road profile without steering input. Road profile of ISO D level was used in order for random excitation to be applied to wheels. Figure 3.4 shows the frequency response of rotational acceleration on the steering wheel. Shimmy occurs in 15Hz, which accords with the test result discussed in the next chapter.

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Figure 3.4 ADAMS simulation result

4. Test 4.1 Shimmy Reproduction Test

Shimmy reproduction test has been conducted in order to observe dynamic behavior of each component of suspension system using chassis dynamometer while shimmy occurs. Vehicle speed has been increased from 40km/h to 140km/h with constant acceleration for the investigation of shimmy trend with respect to vehicle speed. Also, additional unbalance masses of 0g, 20,g, 40g, 60g, and 80g have been attached on the wheel in each test for looking into the effect of wheel unbalance on shimmy.

Figure 4.1 Chassis dynamometer test

Signals have been collected through 24 channels in order to analyze dynamic behavior of each part of front suspension system. Sampling frequency was 160Hz and frequency resolution was 0.0195Hz. Data acquisition points are shown in Figure 4.2. The rotational acceleration is extracted by calculating difference of two

vertical direction signals measured on the both sides of steering wheel, and . ⑧ ⑨

Figure 4.2 Signal acquisition points

Figure 4.3 shows time response of rotational vibration

on the steering wheel when vehicle speed is increased from 40km/h to 140km/h. Shimmy begins to appear when vehicle speed reaches 100km/h and salient rotational vibration is observed around 110km/h. Shimmy decreases behind the velocity of the maximum response, which shows that shimmy is due to the resonance of specific modes of suspension system.

4 0 5 0 6 0 7 0 8 0 9 0 1 0 0 1 1 0 1 2 0 1 3 0 1 4 0- 1

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

0

0 . 2

0 . 4

0 . 6

0 . 8

1

m/s

2

k m / h

Figure 4.3 Steering wheel rotational acceleration

Figure 4.4 and 4.5 represent shimmy response in time domain and frequency domain simultaneously with horizontal axis in Hz and vertical axis in km/h. The solid line shows angular velocity of wheel in corresponding vehicle speed. The fact that the response of steering wheel coincides with the angular velocity of wheel means that the vibration on the steering wheel is due to the rotational excitation of wheel with unbalance mass. That is, sine sweeping excitation of increasing amplitude is applied on the wheel as vehicle speed increases. With this excitation, steering wheel shows 15 Hz shimmy response, and this means that the suspension system has a resonant mode in this frequency.

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Figure 4.4 Waterfall chart of steering wheel rotational vibration

Figure 4.5 Contour plot of steering wheel rotational vibration

Tendency of shimmy to the amount of unbalance mass

on the wheel is represented in Figure 4.6. As unbalance mass attached on the wheel increases, shimmy appears more remarkably in lower speed, which is due to a nonlinear characteristic [17,18] of suspension system of which resonant frequency becomes lower as the magnitude of excitation force increases due to increasing unbalance mass.

2ωrmF e=

Figure 4.6 Effect of unbalance mass

In Figure 4.7, the magnitudes of response at each

measuring point are compared in order to investigate the process of vibration transmission from wheel to steering wheel. As shimmy phenomenon becomes more evident due to increasing unbalance mass, lateral vibration at ball joint on lower arm(point 12) and longitudinal vibration at G-bush point on lower arm(point 13) increase. In contrast, vertical motion of every part and resoponse at A-bush point on lower arm(point 11) remain steady. Moreover, lateral response of connecting point of tie-rod and knuckle(point 17) increases as shimmy grows, with the same tendency as lateral response of G-bush point. In sum, centrifugal force originated from wheel unbalance excites wheel-knuckle assembly in longitudinal direction, which leads to lateral vibration of G-bush point since A-bush is stiffer than G-bush. Hence, lower arm vibrates in rotational direction on the whole with A-bush being rotational center, as shown in Figure 4.8. Consequently, tie-rod moves in lateral direction resulting in subsequent rotational vibration of steering wheel, or shimmy.

Shimmy(Left)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

11 12 13 14 15 16 17

Node number

x

y

z

Figure 4.7 Comparison of the responses on suspension

system components

Figure 4.8 Dynamic behavior of suspension system components during shimmy

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4.2 Modal Test

Shimmy reproduction test indicates that shimmy is due to a resonance of specific modes of suspension system. In order to find out dominant modes related to shimmy, modal test has been conducted for front suspension system. Front suspension is excited on the wheel by electromagnetic exciter with random function up to 256Hz which is generated by CADA-X[19]. Acceleration signals have been measured on the same points(Figure 4.2) as shimmy reproduction test. Natural frequencies and natural modes of the system have been extracted from signals using modal analysis module of CADA-X .

Figure 4.9 Electromagnetic exciter and force transducer

for modal test

Considering the result of modal test, wheel longitudinal vibration mode accords with the dynamic behavior of suspension system in shimmy reproduction test. In sum, when frequency of excitation caused by rotation of unbalance mass of wheel increases as vehicle speed becomes faster, resonance of wheel longitudinal vibration mode occurs at the time excitation frequency coincides with the natural frequency of the mode, which gives rise to lateral vibration of tie-rod that is connected to steering wheel.

Natural frequencies and mode shapes obtained from modal test and theoretical modal analysis are compared in Table 4.1. Results of test and analysis coincide well in low frequency bend, which verifies the validity of the theoretical model. Especially, as shown in Figure 4.10, there is a very good agreement in the result of test and analysis for the wheel longitudinal vibration mode which is considered to be the most dominant source of shimmy vibration.

Table 4.1 Results of theoretical modal analysis and test

Mode Shape Analysis Test

In phase 13.13Hz Wheel Hop mode

Out of phase 13.45Hz 13.99Hz

In phase 22.13Hz 22.14Hz Wheel Longitudinal

vibration mode Out of phase 22.37Hz 21.38Hz

In phase 28.24Hz 27.75Hz Wheel camber motion mode

Out of phase 29.44Hz 26.58Hz

Figure 4. 10 Wheel longitudinal vibration mode

5. Design Optimization 5.1 Sensitivity Analysis and Design Optimization

using Taguchi Methodology Taguchi methodology[20] has been applied to

suspension system in order to find out design parameters sensitive to shimmy and to determine optimal values of those parameters. 13 design parameters which are related to the wheel longitudinal vibration have been selected, and numerical simulation has been conducted in order to minimize the amplitude of vibration in frequency band of 13~18Hz. Since it is smaller-the-better type problem, signal-to-noise ratio is as follows.

7

⎥⎥⎦

⎤

⎢⎢⎣

⎡−= ∑

=

n

iiy

n 1

210

1log10η

Using orthogonal array of 13 factors including geometry and compliance of suspension system and 3 levels for each factor, 27 orthogonal simulations are conducted in order to minimize interactions among design factors, and the effect of each factor is obtained through ANOVA(Analysis of Variance.) This is shown in Table 5.1.

Among 13 factors, C, J, L, M of which F-value is over 4 are set to be design parameters and the rest to be error terms. The following optimal combination of design factors has been obtained through ANOVA.

A2, B3, C3, D2, E1, F1, G2, H1, I2, J2, K2, L2, M1

Error variation of factor effects is )( 2

eσ ± 0.2590, and this satisfies 95% confidence interval.

5.2 Estimation of Shimmy Reduction Level S/N ratio of design factors C3, J2, L2, M1 and

improvement are predicted as follows.

)(2955.2

)(533.19

)()()()( 1223

dB

dB

mmmmmmmmm

initialopt

MLJCopt

=−=Δ

=

−+−+−+−+=

ηηη

η

In order to predict changes in dynamic behavior of

suspension system after design optimization, modal analysis has been conducted by applying the optimal combination of design parameters to 32-DOF mathematical model. As a result, it is observed that there are some changes in mode shapes while natural frequencies remain almost the same. Particularly, there is a significant change in wheel longitudinal vibration mode which participates most in shimmy. Wheel longitudinal vibration mode, which moved as a whole, split into two local modes with independent movement of right wheel and left wheel respectively. This shows that shimmy can be controlled by suppressing global motion of suspension system through appropriate design modification.

Table 5.1 Design factors and contributions to shimmy

Design factor Sum of square

Mean square

F-value Contribution (%)

A A bush

X,Y 0.6189 0.3094 1.3964 4.6

B A bush

Z 0.1997 0.0998 0.4505 1.5

CG bush

X,Y 2.6084 1.3042 5.8854 19.4

DG bush

Z 0.1528 0.0764 0.3446 1.1

E

Strut bush X,Y

1.4436 0.7218 3.2572 10.8

F

Strut bush

X 0.0651 0.0325 0.1469 0.5

GA bush

X 0.2207 0.1103 0.4979 1.6

HG bush

X 0.4917 0.2458 1.1093 3.7

I

Tie-rod X

1.2704 0.6352 2.8664 9.5

J

Strut Y

1.8823 0.9411 4.2470 14.0

K

Torsion bar

spring 0.1999 0.1000 0.4511 1.5

L

G bush Axis

2.2414 1.1207 5.0572 16.7

M

Tire Stiffness

2.0225 1.0113 4.5634 15.1

8

Figure 5.1 Mode shape change after design optimization

Optimal combination of design parameters obtained above is applied to ADAMS model to predict the level of shimmy reduction after design optimization. Rotational acceleration on the steering wheel is measured under 120km/h constant speed driving condition. As a result, shimmy response of 15Hz has been decreased by more than 10 % as shown in Figure 5.2.

Figure 5.2 Simulation result after design optimization

6. Conclusion

In this study, 32-DOF theoretical linear model of front suspension system was constructed and quasi-static analysis was carried out. The result was compared with that of numerical simulation conducted with multibody dynamics analysis software ADAMS. Also, modal analysis was carried out with theoretical model and the results were compared with those of modal test in order to validate the model. In shimmy reproduction test,

dynamic behavior of each component of suspension system was measured and analyzed, and wheel longitudinal vibration mode was found to be the most dominant mode in shimmy phenomenon.

In order to suppress shimmy vibration, geometry and compliance of suspension components were selected to be design factors in Taguchi methodology. After sensitivity analysis and design optimization, lateral and axial stiffness of G-bush, y-direction position of strut mount bush, and tire stiffness were found to be important design factors which have a significant influence on shimmy.

Furthermore, it is observed that wheel longitudinal vibration mode split into two local modes with independent movement of right wheel and left wheel after optimization of design parameters. This resulted in reducing shimmy response of 15Hz and its level was predicted to be decreased by more than 10% in numerical simulation.

Theoretical and numerical models for shimmy analysis were developed and they are expected be used in estimation of shimmy level in the course of developing suspension system. Moreover, optimal combination of design parameters is suggested to improve shimmy phenomenon in the subject vehicle. Acknowledgement This work is supported by Hyundai Mobis.

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