Page 141 - Mechatronic Systems Modelling and Simulation with HDLs

130 6 MECHANICS IN HARDWARE DESCRIPTION LANGUAGES

The h i interpolation functions are selected such that they each fulfil the boundary
conditions. Then the residuum R is calculated as follows:

n

R = r − L a i h i (6.59)
i=1
For the exact solution the residuum is zero and, for the approximation, should at
least be sufficiently low at all points of the solution range. Then the weighting
factors a i can be determined during the approximation of the partial differential
equation. For the Galerkin method the following equations are used as the basis:

h i RdD = 0 i = 1, 2,..., n (6.60)
D

Where D is again the solution range.
Hung et al. [156] use the Galerkin method to investigate a pressure sensor,
which consists primarily of a bending beam that is fixed at both ends. A voltage and
consequently an electrostatic force is applied to this. The time that passes before the
beam ‘snaps into place’ as a result of the positive feedback of the electrostatic force,
i.e. forcefully rests upon the insulator, is strongly dependent upon the prevailing air
pressure. The modelling uses the Euler equation for bending beams and Reynolds’
equation for air damping. The authors use the Galerkin method with up to four
interpolation functions, which are determined with the aid of a FE simulation. They
thereby achieve an acceleration of the simulation by a factor of between 4 and 105
in comparison to FE simulators, with deviations from the FE simulation in the
range of 1%–14%.
This method permits the formulation of lower-order models. However, it requires
that the system can be considered as a comparatively simple structure, because
the starting point, the partial differential equations and boundary equations, either
cannot be set or can be set only with great difficulty.

6.3.4 Experimental modelling

Introduction

Experimental modelling dedicates itself to the creation of models on the basis of
measured data or FE simulations. The internal physics of the components is dis-
regarded and only the terminal behaviour considered. In this manner we obtain
so-called macromodels that can be simply formulated in a hardware description
language. We thus obtain efficient and numerically unproblematic models. This
method has its advantages if it is difficult or even impossible to derive the phys-
ical background of a component. However, its main problem is that the resulting
models are only valid for precisely one geometric form of the structure and set of
technology parameters. Every change means that a new model must be drawn up.