4569846498

446 Multibody Systems Approach to Vehicle Dynamics
0.8 1
0.6
0.4
0.2
0.2 0
0.4
0.6
0.8
1
0 1 2 3 4 5 6 7 8 9 10
Fig. 8.4 Two-pole Butterworth filter described in equation (8.5) applied to
signal in equation (8.6) (solid) – resulting signal in dashed line
arrived at using some external filter design tool. Particular caution is
needed since some software environments use descending order for the
terms in the transfer function while some use ascending order, as illustrated
in equation (8.5), which represents a 1 Hz second order Butterworth filter
in the Laplace domain as used in Matlab and MSC.ADAMS:
39.884
39.884
gs ()
2
s 8.8869 s
2
39.884 39.884 8.8869
ss
Matlab description MSC.ADAMS Description
(8.5)
More sophisticated filtering is possible using higher order transfer functions
and using the more general state–space modelling methods, which are
available in most multibody system environments. The interested reader is
referred to Blinchikoff and Zverev (2001) for detailed discussion of filtering
methods and their repercussions. The most important thing to recognize
is that almost any form of real-time/run-time filtering introduces some
form of phase delay. Using the example transfer function in equation (8.5),
an input chirp signal in Matlab defined as:
⎛
x sin⎜2
⎝
2
time ⎞
⎟
10 ⎠
(8.6)
produces an output as shown in Figure 8.4. Note that the amplitude attenuation
is very gentle but that a phase delay has been introduced even at frequencies
well below the notional cut-off frequency.
The 2-pole Butterworth filter represents a linear second order mechanical
system and so its intelligent use can be very helpful in avoiding modelling
complexity. The interested reader is encouraged to study its formulation and
implementation in different forms – pole/zero and Laplace polynomial –
as a launchpad to a deeper understanding of real-time filtering issues and
complexities.
Adaptive damping logic can be implemented in a similar fashion to active
suspension inside an MBS model, scaling a damper spline or adding a scaled
‘damper variation’ spline to a ‘minimum damping’ spline according to a controller
demand. The 1988 Lancia Thema 8.32 is believed by the authors to
be the first production implementation of an adaptive damping system, with