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138 Multibody Systems Approach to Vehicle Dynamics
and
˙ż
Ae z
(4.22)
(4.23)
The original equation (4.20) can then be written:
m 2 z jc(z z g ) k(z z g ) 0 (4.24)
Rearranging this in the form of a transfer function gives
H( )
(4.25)
This expression relates the amplitude and phase of ground movements to
the amplitude and phase of the body movements and is commonly reproduced
in many vibration theory books and courses.
Considering the transfer function, several behaviours may be observed. At
frequencies of zero and close to zero, the transfer function is unity. This
may reasonably be expected; if the ground is moved very slowly the entire
system translates with it, substantially undistorted. The behaviour of the
system may be described as ‘static’ where its position is governed only by
the preload in the spring and the mass carried by the spring; dynamic
effects are absent.
At one particular frequency where k/m, —– the real part of the denominator
becomes zero and the transfer function is given by
H( )
( j
t )
ż
j Ae j z
2 ( j
t ) 2
(4.26)
At this frequency, the behaviour of the system is called ‘resonant’.
Examination of the transfer function shows it is at its maximum value.
Since k, c and . are all positive real numbers, the transfer function shows
that ground inputs are amplified at this frequency. If c is zero, i.e. there is
no damping present in the system, then the transfer function is infinite. If c
is very large, the transfer function has an amplitude of unity. For typical
values of c, the amplitude of the transfer function is greater than unity.
At substantially higher frequencies where k/m, —– the transfer function
is dominated by the term m 2 in the denominator and tends towards
1
H( )
m
2
( k) j( c)
2
( km ) j( c)
( k) j( c)
jk
jc ( ) jc ( )
1
(4.27)
At these higher frequencies, the amplitude of the transfer function falls
away rapidly.
It may therefore be noted that if it is desired to have the body isolated
from the road inputs, the system must operate in the latter region where
—– k/m. In fact this is a general conclusion for any dynamic system;
for isolation to occur it must operate above its resonant frequency. Given
that the mass of the vehicle body is a function of things beyond the suspension
designer’s control, it is generally true that the only variables available