Page 186 - Mechatronic Systems Modelling and Simulation with HDLs

8.2 DEMONSTRATOR 5: CAPACITIVE PRESSURE SENSOR 175

forces and moments, depending upon whether the degree of freedom relates to a
translational or rotational deflection. In particular, the parts of the exciting forces
and moments that are assigned to the elements adjoining the node are also added to
the currents (and current changes) at a node. It is not necessary to explicitly create
the system matrix, since its solution is yielded implicitly from the interconnection
of the finite elements.

Excitation of the mechanics

In addition to the mechanical behaviour of the structure it is necessary to describe
the external mechanical excitation. For reasons of modularity this is formulated in
a further element, which is connected to the finite plane elements in question at the
node(s) where the excitation acts. The main task of this element is to determine the
exciting forces and moments at each node point, i.e. for each degree of freedom,
and to convert these into a current change, with the resulting current being fed into
the circuit nodes of the degree of freedom in question. The same procedure can in
principle also be used in order to model the electrostatic feedback using an addi-
tional element. However, this effect could mainly be disregarded in the application
considered, so that the corresponding modelling could be dispensed with.

Geometric nonlinearity

If the deflection exceeds a certain value, then the upper plate rests upon the isolator.
If the pressure is further increased the plate ‘rolls’ out over the isolator. This corre-
sponds with a one-sided holonomous and scleronomic constraint of the movement.
Numerically this means that upon the manifestation of new boundary conditions, a
corresponding number of degrees of freedom disappear. The realisation of numeri-
cal equation solvers with a variable number of unknowns or equations is a difficult
task; only very few solvers are set up for this case. For this reason it is worth-
while approximating the prescribed situation by damping the specified degrees of
freedom to a significant degree. This again leads to the fact that only minimal
movements are possible for these, which in principle corresponds with the desired
behaviour. However, this procedure may nevertheless be numerically dangerous
because the sudden setting of a high damping — in other words the striking of the
plate on the isolator — excites the most important natural frequencies.

Calculation of the capacitance

A further element type is introduced for calculating the capacitance, which just
like the plane element or the pressure element is formulated in a rotationally
symmetrical manner. For this element the plate capacitor equation is applied to the
average deflection of a corresponding annulus. The resulting capacitance, in the