Page 118 - Mechatronic Systems Modelling and Simulation with HDLs

6.2 MULTIBODY MECHANICS 107

The first subterm represents the generalised inertial forces that arise as a result
of the change of position of the system, thus, for example, the Coriolis force.
Opposing this part is the second component, which describes the rate of change of
the generalised impulses. The above-mentioned premise

I
Q =−Q i (6.26)
i
yields the Lagrange equation

d ∂T ∂T
− = Q i (i = 1, 2,..., n) (6.27)
dt ∂ ˙q i ∂q i

By drawing up a formula for kinetic energy and the conversion of applied forces
into their generalised form we can obtain the equations of motion directly by sub-
stituting into equation (6.27). This is particularly simple because kinetic energy is
a scalar that contains no higher derivatives with respect to time than the velocities.
These are significantly easier to determine than the accelerations.

Formulation in hardware description languages

The primary purpose of analogue hardware description languages is for the mod-
elling of analogue electronic components for a circuit simulator. The variables
considered in this application — voltage and current — correspond with the duality
of a potential and a flow and can be represented by other quantities in accordance
with the analogies described in Section 3.2.2. Although the text-based formulation
of the mechanical model is based upon accelerations, velocities, positions, and
forces, the underlying calculations take place in accordance with the analogies of
the potentials and flows available.
Furthermore, the preceding section has shown that the selection of the con-
sidered unknowns of multi-body mechanics is attributed decisive importance. In
electronics the unknowns are normally in the form of node voltages, which is
because of the nodal analysis that is prevalent in circuit simulation. In the system-
oriented modelling of mechanics, on the other hand, it is of decisive importance
to specify a suitable set of generalised coordinates. For holonomous systems,
which can be described using generalised coordinates, the — sometimes very com-
plex — constraint equations are dispensed with. As was shown by the relatively
simple example from Figure 6.2, it is not a question of selecting from a fund of
existing coordinates, but one of an independent engineering task.
The methods described supply sets of ordinary differential equations in symbolic
form. These can easily be formulated in analogue hardware description languages.
This is true under the prerequisite that the size of the equation set remains within
limits. In Section 7.2.3 the obtaining and formatting of the equations of motion for
an automotive wheel suspension system using the Lagrange approach is illustrated.