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Modelling and analysis of suspension systems 137
heights (A 1z , B 1z , C 1z and D 1z ), and the corner stiffness (k A , k B , k C and k D ),
and wheel locations are the inputs to the calculations. The unknowns that
are found from (4.16) are the body heights at three corners (A 2z , B 2z and
C 2z ). These are used to find D 2z from (4.15) leading to the force solution in
each corner from equations (4.11–4.14). Thus, the presence of an elastic
suspension for the roadwheels allows a solution to the ‘wobbly table’ problem
for vehicles with four or more wheels even when traversing terrain that
is not smooth.
4.1.2 Body isolation
The interaction of a single wheel with terrain of varying height is
frequently idealized as shown in Figure 4.5. This is a so-called ‘quarter
vehicle’ model and is widely used to illustrate the behaviour of suspension
systems. It may be thought of as a stationary system under which a ground
profile passes to give a time-varying ground input, z g . Note that at this point
we are assuming the tyre to be rigid. Whether classical models or ‘literal’
multibody system models are used, the methods used to comprehend body
isolation are the same.
Classically, the system may be formulated as a single second order differential
equation:
mz ˙˙ c( z˙z ˙
g) ks( z zg) 0
(4.20)
In order to more fully understand the isolation behaviour of the suspension,
a fully developed (steady state) harmonic solution of the equation above
may be presumed to apply. ‘Harmonic’ simply means that both input and
output may be described with sine functions and that the phase relationships
are fixed. This may be conveniently expressed in the form
z A e (jt) (4.21)
where j is the imaginary square root of 1. Thus for the two derivatives of
z it may be written:
Vehicle body
or
sprung mass
m
z
Body response
Suspension
spring and
damper
k
c
Z
Z g
Ground input
X
Time (s)
Fig. 4.5
A classical quarter vehicle ride model