Page 143 - Mechatronic Systems Modelling and Simulation with HDLs

132 6 MECHANICS IN HARDWARE DESCRIPTION LANGUAGES

known in advance, so that we have to start from the assumption of the existence
of strong couplings and nonlinearities. Furthermore, it is required that inputs and
outputs of the FE model can be formulated in an integral manner, i.e. they are not
position dependent.
We now start with a basic model, the parameters of which should be identi-
fied with the aid of various optimisation procedures. For oscillating systems, for
example, equation (6.61) would be a good starting point, whereby the parameters a 1
and b 1 would have to be determined. For the general case these can be determined
from the criterion that the resulting model should behave as closely as possible to
the FE model. The target function of optimisation is thus the minimisation of the
behaviour difference between the predetermined and sought-after model, i.e. [150]:
#outputs #timesteps
2
(t j )) (6.62)
(f FEM i (t j ) − f MACRO i
j i
For optimisation, gradient procedures, simulated annealing, or genetic algorithms
can be used and it is also possible to switch between these. The resulting parameters
initially apply only for the selected input function. In [150] it is thus proposed to
initially define a set of input functions, which represent reality as well as possible.
Then optimisation takes place primarily for the input function, the macro model of
which exhibits the greatest differences in relation to the FE model. This procedure
corresponds with a parameter identification for nonlinear systems.
Now, if it is difficult to arrive at an acceptable solution using parameter opti-
misation, this raises the question of whether the assumptions with regard to the
structure of the solution equations were correct. Once again, the problem lies in the
nonlinearities that rule out an analytical solution of the problem. The solution pro-
posed by Hofmann et al. consists of setting operators that evaluate the differences
between the FE and macromodel, such as for example ‘rate of rise too low’, ‘over-
shoot too low’ and so on. On the basis of this information a fuzzy controller base
decides on possible structural changes. So we now go from parameter identification
to system identification.
Overall, the procedure supplies efficient and numerically unproblematic models,
that can be easily formulated in hardware description languages, e.g. HDL-A [149].
However, a significant computing time has to be expended for model generation.
Furthermore, the validation of the generated models remains difficult, since firstly
the quality and coverage of the selected input functions is sometimes questionable
and secondly the inner physical structure is not available for an investigation into
the plausibility. Finally, this type of modelling has to be performed afresh for virtu-
ally every variation of the micromechanical geometry or the underlying technology.

6.4 Summary

In this chapter, methods for the modelling of multibody mechanics and contin-
uum mechanics have been highlighted and the representation of the resulting