Page 151 - Mechatronic Systems Modelling and Simulation with HDLs

140 7 MECHATRONICS

Q a4 =−m a g,
Q b1 = k s (y a − y b + l 0b ),

Q b2 = b(˙y a − ˙y b ),
Q b3 =−m b g (7.3)

where k w and k s denote the constants of wheel and body springs, Q ai and Q bi the
components of the generalised forces Q i on the bodies A and B, and b the coefficient
of damping. Furthermore, the convention is used for the springs that a positive force
increases the positions in question. The constants l 0a and l 0b correspond with the
y-positions of the two related bodies in a relaxed state. At the damper, positive
forces bring about an increase in the coordinate difference. Thus the damping force
tends to resist a positive relative velocity. The weights Q a4 and Q b3 finally effect
a reduction in the positions and are thus counted negatively. Substitution into the
Lagrange formula gives the following equation system:

m a ¨y a = k r (y s − y a + l 0r ) − k f (y a − y b + l 0f ) − b(˙y a − ˙y b ) − m a g
m b ¨y b = k f (y a − y b + l 0f ) + b(˙y a − ˙y b ) − m b g (7.4)

This can be directly formulated in an analogue hardware description language,
whereby the damper constant b can be set to the value in question using an if-
then-else construct.

7.2.4 Simulation
The test case for which the simulation should be performed is, as in Chapter 6,
driving over a step of 5 cm height. As described in Chapter 6, the model of a
quarter of a car can be used again at this point. The first result of the simulation
is shown in Figure 7.2 and represents the relative movement of wheel and vehicle
body after driving over the bump. Initially the wheel starts to move and compresses
the body spring. This movement produces an overshoot. Overall, we recognise that
the natural frequency of the wheel actually lies in the range of around ten hertz.
The oscillation, however, decays very rapidly as a result of the damping, which is
increased at the change-over point. The car body follows the vertical movement
at a significantly slower rate than the wheel. The forces in question are applied by
the compressed body spring. Here too the natural oscillation is, as expected, set
to a value of around one hertz. The change-over of the damping from the ‘soft’
to the ‘hard’ characteristic is triggered by the digital signal shown at the top and
takes place around 50 ms after the step is reached.
The difference in driving behaviour resulting from the change-over is particularly
well illustrated by the consideration of the body spring compression, which is
equated with the compression of the shock absorber. This is shown in Figure 7.3
for the same case of driving over a step of 5 cm height. In the first 50 ms there