Mathematical Analysis of Vehicle Suspension System

Running Head: Mass-Spring-Damper System 1
Vehicle Suspension System
Institutional Affiliation
Assignment
Date of Submission

Mass-Spring-Damper System 2
SECTION 1: MATHEMATICAL ANALYSIS OF SYSTEM (25 marks)
The vehicle suspension system is modeled as illustrated in the figure below,
The suspension system comprises of two mass-spring-damper systems stacked on each other. A
simple mass spring damper is analyzed for each section,
X= input body or surface position, x=0
M=mass
Y= mass height with reference to the ground surface, y=0
(1) Free body diagram
(2) Motion of the system
ms ̈y ( t ) +Cs ̇y ( t ) +k s y ( t )=Cs ̇x ( t )+ ks x ( t )
̈y ( t ) + Cs
ms
̇y ( t ) + k s
ms
y ( t ) = Cs
ms
̇x ( t ) + ks
ms
x ( t )
(3) Laplace transform and resultant transfer function
s2 Y ( s )+ Cs
ms
sY ( s ) + ks
ms
Y ( s )= Cs
ms
sX ( s ) + ks
ms
X (s )

Mass-Spring-Damper System 3
Forming a transfer function,
H ( s )=
( Cs
ms
s+ ks
ms )
(s2+ Cs
ms
s+ k s
ms )
The denominator forms the characteristic equation as,
s2 + Cs
ms
s+ ks
ms
=0
(4) Damping ratio and the natural frequency
2 ζ ωn=Cs
ms
ωn
2= ks
ms
→ ωn = √ ks
ms
ζ =
1
2 ωn
∗Cs
ms
= Cs
2 ms ωn
(5) Characteristic equation and Eigen values
s2 + Cs
ms
s+ ks
ms
... .∈the S−domain
(6) Natural response based on different cases of the damping ratio
Cs=2 m √ k
m =2 mωn
ξ= c
cs
= c
2 mωn
s1,2= −c
2 m ± √ ( c
2m )2
− k
m
s1,2= (−ξ ± √ξ2−1 ) ωn

Mass-Spring-Damper System 4
x ( t ) = A e
( −ζ + √ ζ 2−1 ) ωn t +B e
( −ζ − √ ζ 2−1 ) ωn t
Damping cases:
C> Cc ... . Heavy damping
The roots are real.
x (t )= A es1 t + B es2 t
A , B ..... constants of integration
α >0 , β> 0 , β =α2 − k
m <α 2
s1=−α + β
s2=−α −β
α = c
2 m β= 1
2 m √ c2−4 mk
C=Cc ... . critical damping
β= 1
2 m √ c2−4 mk
s1=s2=−α−¿ Cc
2m =−ωn
x ( t ) = ( A +Bt ) e−ωnt
x (t )= [ x0 + ( ̇x0 + ωn x0 ) t ] e−ωn t
The motion is not vibratory and it decays to the desired equilibrium position.
0<C <Cc ... . light damping
c2 <4 mk
β=i ω¿ whereω¿= 1
2 m √ 4 mk−c2= √ ( k
m )− c2
4 m2
s1=−α +i ω¿
s2=−α −iω¿
α = c
2 m
x=e−αt ( a cos ω¿ t+ B sin ω¿ t )