A simplified multi-degree of freedom system model of an automobile suspension system is

shown in Figure 1. The automobile traveling along a level road at a constant horizontal

speed 0 v encounters bumps in the road shown of the shapes shown in Figure 2.

J

f

l r

l

G

M

yf t mf mr

f 1 k r1 k f 1 c

f 2 k f 2 c

r1 c

r2 k

r2 c

r y t

y t

r u t f u t

Front Suspension

Rear Suspension

Figure 1. Model of an automobile suspension system

f r u t u t

t

0.25

0.25 0.50 0.75 1.00

f r u t u t

t

0.25

0.00 0.25 0.50 0.75 1.00 1.25

f r u t u t

t

0.25

0.00 0.25 0.50 0.75 1.00 1.25

f r u t u t

t

0.25

0.00 0.25 0.50 0.75 1.00

Figure 2. Road Profile

King Fahd University of Petroleum & Minerals Mechanical Engineering Department

ME 482 Mechanical Vibrations 2/4 Term PROJECT (131)

The vehicle’s suspension system (front and rear springs and shock absorbers) is modeled by

linear springs and dampers, and the compliance of the tires is modeled by front and rear

springs. The vehicle motion is limited to heave in the vertical direction and a small amount

of pitch of the vehicle’s longitudinal axis. The tires are assumed to remain in contact with

the road surface at all times.

The road profile is responsible for the system’s input

T

f r u u u , where f u and r u are

the height of the road (with respect to some reference) underneath the front and rear tires,

respectively. The system has three translational degrees of freedom, y , f y , r y which are the

vertical displacements of the vehicle and both front and rear axles from their equilibrium

positions. The lone rotational degree of freedom is the pitch angle .

1. Assuming small motions, derive the differential equations for the pitch angle and

the vertical (heave) displacement y , f y , r y of the centroid of the vehicle using

f u t and r u t as the inputs.

2. Write the equations of motion in matrix form and identify the mass matrix M , the

stiffness matrix K and the damping matrix C .

3. Define a suitable set of state space variables and write the equations of motion in state

space form.

q A q B u

q = C q + D u

4. Find the natural frequencies and mode shapes of the system.

5. For each of the road profile shown in Figure 2, plot the responses y , f y , r y and

using MATLAB.

6. Use Simulink to answer the previous questions.

King Fahd University of Petroleum & Minerals Mechanical Engineering Department

ME 482 Mechanical Vibrations 3/4 Term PROJECT (131)

Nomenclature:

Notation Description Values Units

M Body mass 1500 kg

J Body inertia 3443 Kg.m2

mf

Front wheel mass 59 kg

mr Rear wheel mass 59 kg

f 2 k Front main stiffness 35000 N/m

r2 k Rear main stiffness 38000 N/m

f 1 k Front tire stiffness 190000 N/m

r1 k Rear tire stiffness 190000 N/m

f 2 c Front main damping 1000 N.s/m

r2 c Rear main damping 1100 N.s/m

f 1 c Front tire damping 1000 N.s/m

r1 c Rear tire damping 1100 N.s/m

f

l Front length from the center of mass 1.4 m

r

l Rear length from the center of mass 1.7 m

f u t Front road profile See Fig. 2 m

f u t Rear road profile See Fig. 2 m

f y t Front wheel response (translation) m

r y t Rear wheel response (translation) m

y t Response of the chassis (translation) m

t Response of the chassis (rotation) rad

King Fahd University of Petroleum & Minerals Mechanical Engineering Department

ME 482 Mechanical Vibrations 4/4 Term PROJECT (131)

General Requirements:

The report should include the followings:

1 Introduction: Why a car suspension system is worth studying in this course

and mention how it is linked to it.

2 The Model: Write the mathematical model of the system provided to you.

3 Analysis: Provide all the required information using the data provided to you.

You may assume reasonably any missing data. You may also use any

data from the internet as long as you reference it.

4 Discussion: Discuss the obtained results.

5 Conclusions: Briefly

6 References: You should reference all used