Page 142 - Mechatronic Systems Modelling and Simulation with HDLs

6.3 CONTINUUM MECHANICS 131

A whole range of approaches extract the main corner-stones of the behaviour of a
component from measurements or simulations using finite elements and use this for
simple models consisting of few equations, see for example Ansel et al. [11], Hof-
mann et al. [149], [150], Karam et al. [179] and Nagel et al. [292]. In what follows
three approaches will be considered that aim in the aforementioned direction.

Table models

The simplest case of experimental modelling is based upon a list of input and
output values, thus arriving at a table model that only considers the static case. In
this manner it is possible, for example, to draw up a table listing pressures and
the associated capacitance values for the pressure elements described above. Such
table models lead to characteristics with kinks that can considerably detract from
the convergence of the simulator. This problem can be circumvented by using the
present value pair as a support point for the characteristic, e.g. on the basis of
splines, which typically removes the numerical problems. In this manner measured
values can be very simply integrated into a simulation. More elaborate procedures
estimate the structure of the equations and move themselves to the identification
of the associated parameter.

Identification of a harmonic oscillator

In [11], Ansel et al. consider a seismic acceleration sensor as a harmonic oscillator.
For the modelling a linear differential equation is used for the force f and the
deflection x:
n
m
df d f dx d x
a 0 f + a 1 + ··· + a m m = b 0 x + b 1 + ··· + b n n (6.61)
dt dt dt dt
For a spring-mass system, for example, m is set to 0 and n to 2. Here b 0 represents
the spring constant, b 1 the viscous damping, and b 2 the seismic mass. For the system
currently under consideration the parameters a i and b i are automatically obtained
from the results of a simulation using finite elements. For this purpose the classical
methods for system identification are used. This describes the mechanical section
of the system. In addition, there is the conversion of mechanical deflection into
capacitance based upon an interlacing comb structure. A table model is used for
this, which is also determined on the basis of simulations using finite elements.

General identification

Hofmann et al. [149] and [150] propose a general procedure in order to put together
the behaviour of a component from functional modules. The modelling is based
upon a FE model, the behaviour of which is stored in a macromodel. Thus the
complexity and nature of the underlying (partial) differential equations are not