Vibration Analysis and Modeling of Automobile Suspension Systems

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1

SECTION A
QUESTION I: 1 DOF MODELLING
In the single DOF modeling, the acceleration of the automobile is a second derivative. The mass,
stiffness and damping of the SDOF system are represented using m, k, c which undergo the
relative displacement, velocity, and acceleration. The SDOF system and free body diagram is as
illustrated below,
The equation of motion,
m ̈x ( t )+ c ̇x ( t ) + kx ( t )=0
When it is normalized as a second order characteristic equation, it is expressed as,
̈x (t ) +2 ξ ω0 ̇x ( t )+ ω0
2 x ( t )=0
The damping ratio and the natural frequency of the SDOF system are given as,
ω0= √ k
m
ξ= c
2 mω0
For free vibration, the system is considered to have no external forces. Its response is given as,
x (t )= Asin ωn t+ B cos ωn t
For the damped case,
x (t )= ( A +Bt ) e−( bt
2 m )
b=bcr=2 √ km=2 m ωn
2

For the under-damped case,
b< bcr
x ( t ) =e−( bt
2 m ) ( A sin ωd t + B cos ωd t )
The damping natural frequency of the SDOF system is given as,
ωd=ω0 √ 1−ξ2
For the forced vibration, there is an external force towards the SDOF system, and the general
solution to the characteristic equation is obtained as,
̈x ( t ) + k
m x ( t ) + c
m ̇x ( t ) =Fsin ωt
x (t )= F sin ωt +θ
k
√ (1−
( ω2
ωn
2 ) )+ ( 2 ξω
ωn )2
¿ F
k
( sin ωt +θ
√ ( 1− ( ω2
ωn
2 ) ) +
( 2 ξω
ωn )
2
)
Road surface automobile suspension response,
Using the modeled uneven road surface, a vehicle can be driven at speeds ranging from 0 to
65kph to avoid too much rumbling and discomfort of the passengers,
3

MATLAB implementation of the SDOF System
(i) Obtaining the system Laplace transform and developing a transfer function
(ii) Generating a Simulink model of the system
(iii) Performing several analysis on bode diagrams of the system response, magnification
and transmissibility factor, and derivation of some parameters.
The code snippet
%% SDOF System analysis
m=45e3; % 45000 kg
k=35e3; % stiffness of the spring
c=18500; % damping coefficient, of the damper
% Using matrix form of the SDOF system to generate the
transfer function
A=tf([m c k],[1]) % The transfer function, A
B=tf([c k],[1]) % The transfer function, B
sys=B/A % The transfer function, The system
%% Using the step function to determine the system response
figure(2)
step(sys)
grid on
%% To determine the system magnitude and phase plot using bode
diagram
figure(3)
bode(sys)
grid on
%% The natural frequency and the damping factor of the
calculations in the system
[Wn, zeta]=damp(sys)
4