Page 127 - Mechatronic Systems Modelling and Simulation with HDLs

116 6 MECHANICS IN HARDWARE DESCRIPTION LANGUAGES

cannot avoid the consideration of the continuum in the modelling. The associated
mechanics, and in particular its representation in hardware description languages,
are the subject of this section.
We can initially differentiate between whether the consideration is to be per-
formed statically or dynamically. For the static case each mechanical position may
be assigned an electrical quantity. Here only the steady state is considered. In the
dynamic case velocities and accelerations of mechanical quantities also play a role,
so that phenomena such as mechanical resonance are also considered. A further
distinction is supplied by the selection of a desired level of abstraction. It is a funda-
mental truth of continuum mechanics that elasticity and mass spread continuously,
thus giving rise to an infinite number of degrees of freedom. As is described in
more detail in what follows, we can perform the modelling of mechanical continua
on the basis of (geometric) structure, physical equations, and experimental data.
In this context the reader is referred to a corresponding classification of modelling
approaches in Section 2.4.

6.3.2 Structural modelling

Introduction

Structural modelling traces the generation of a model back to the composition of
basic models. In the case of continuum mechanics these basic models may be
finite elements, for example. Due to the generality and high degree of adoption
of finite elements in the framework of structural modelling, we will deal exclu-
sively with this approach. Bathe [19], Gasch and Knothe [113] and Knothe and
Wessels [202] supply a good overview of the methods of finite elements in their
works. For the modelling of finite elements, as in the Ritz procedure (see within
Section 6.3.3), we work on the basis of interpolation functions. However, these are
not formulated globally for the whole structure here, but locally for the finite ele-
ment. Thus the main difficulties of the Ritz procedure are removed. If the models
of the finite elements are available, modelling is a purely geometric task, which
primarily represents a breakdown of the continuum. In this context we also speak
of a meshing, in which finer resolutions buy more precision at the expense of
greater simulation time.
Up until now, finite elements have typically only been used to investigate the
component level, disregarding the system context. The following sections will show
that finite elements can be drawn into a circuit simulation on the basis of hardware
description languages. As we will show in what follows, the differential equation
solver of the circuit simulator in question is thus entrusted with the calculation
of the equations of the finite elements. The dynamic coupling of electronics and
mechanics then takes place automatically. Overall, this opens up a simpler, faster
and more secure way of modelling mechanical continua that is compatible with
hardware description languages and thus also with circuit simulation.