Page 31 - Mechatronic Systems Modelling and Simulation with HDLs

20 2 PRINCIPLES OF MODELLING AND SIMULATION

(differential-algebraic equation set). The number of equations depends upon the
circuit and is very high, typically significantly above the number of degrees of
freedom. The resulting system matrices are however only sparsely occupied.
For multibody mechanics, the equations of motion are normally derived by means
of the application of a classical principle, e.g. that of Lagrange or D’Alembert. When
drawing up the equations it is possible to choose between two extremes. In one case
the generalised coordinates, which fully describe the state of a system and which
can also be regarded as degrees of freedom, are first determined. For n generalised
coordinates (at least for holonomous systems) n equations can be drawn up. The
constraints fall away, leaving a system of ordinary differential equations. However,
these may turn out to be very complex. Alternatively, it is possible — as in electron-
ics — to permit more unknowns and thereby obtain a system of differential equations
for the motion of bodies and algebraic equations for the constraints, which may, for
example, be caused by joints. This establishes a system of DAEs, which can be solved
using similar methods to those used in the circuit simulation, see for example Orlan-
dea et al. [304]. In both cases the number of degrees of freedom is relatively small in
comparison to those in electronics. The number of objects under consideration, such
as bodies, joints, springs, shock absorbers, etc. is generally significantly below one
hundred. However, the numerical problems caused by transitions between static and
sliding friction, mechanical impacts, three-dimensional coordinate transformations
and other effects, cannot be disregarded.
In the representation of continuum mechanics by means of finite elements the
number of degrees of freedom is significantly higher than those in multibody
mechanics. The associated system matrices normally have a band shape, which
the simulation exploits by suitably customised numerical procedures. Overall, this
normally establishes a system of ordinary differential equations, the parameters of
which, i.e. the inputs into the mass, damping and stiffness matrix, may however
have to be recalculated at runtime.

2.4.4 Experimental modelling

Introduction

Experimental modelling consists of the development of mathematical models of
dynamic systems on the basis of measured data or at least providing existing
models with parameters. So neither the underlying physics nor the internal life of
the system need necessarily play a role in model generation. In contrast to physical
modelling there are procedures for experimental modelling in which the modelling
can be wholly or partially automated.

Table model
The simplest method of incorporating measured data is by the formulation of table
models that lead to a stepped or piece-wise linear characteristic. The problem with