Page 150 - Mechatronic Systems Modelling and Simulation with HDLs

7.2 DEMONSTRATOR 1: SEMI-ACTIVE WHEEL SUSPENSION 139

will be implemented here. More complex algorithms and further input data can be
added in a straightforward manner.

7.2.3 Modelling of mechanics

The approach to the modelling of the mechanical components depends to a large
degree upon the desired application of the models. Thus, we can provide complex
multibody models for the development of the mechanics which can be used, for
example, to determine the tilt of the wheels or changes to their toe-in in relation
to the spring deflection of the car body, see for example Schmidt and Wolz [365].
However, these details are of less importance to the development of the electronics.
In this context the overall behaviour of the system is much more interesting, and
simpler models are sufficient for the consideration of this.
The model of the wheel suspension is based upon the fundamental model pre-
sented in Chapter 6. However, in this context some further boundary conditions
have to be taken into account. Firstly, we should note the fact that real shock
absorbers exhibit different characteristics for the compression and tension modes.
Typically the damping is significantly higher in tension mode. This is because the
dampers should transmit road unevenness to the passengers as little as possible.
This means that comparatively low forces should arise in the compression mode.
Consequently most of the consumption of the motion energy occurs in the tension
mode. This situation generates a first nonlinearity. A second naturally arises as a
result of the switching of the damper characteristics, which is precisely the main
purpose of the system.
It has already been demonstrated in Chapter 6 how such a model is put together
from the basic models for masses, springs, dampers, etc. for a quarter of a car. For
this reason, a system-oriented approach to modelling will be followed here. The
Lagrange equations shall form the basis for this. As shown in Chapter 6 these take
the following form:

d ∂T ∂T
− = Q i (i = 1, 2,..., n) (7.1)
dt ∂ ˙q i ∂q i
The degrees of freedom are again the y-positions. The kinetic energy of the system
T is found from the kinetic energy of the two masses:

1
1
2
q 1 = y a , q 2 = y b , T = m a ˙y + m b ˙y 2 (7.2)
2 a 2 b
The generalised forces are the spring, damper and weight forces:
Q a1 = k w (y s − y a + l 0a ),

Q a2 =−k s (y a − y b + l 0b ),
Q a3 =−b(˙y a − ˙y b ),