Page 138 - Mechatronic Systems Modelling and Simulation with HDLs

6.3 CONTINUUM MECHANICS 127

The bending of such a plate can be described by the following partial differential
equation, see Gasch and Knothe [113]:

3
2
2
4
∂ u Et 3 p ∂ u 2 ∂ u 1 ∂ u
=− + + + w (6.48)
2
2
∂t 2 12ρ(1 − ν ) ∂r 4 r ∂r 3 r ∂r 2
Where u is the deflection, E the modulus of elasticity, t p the thickness of the plate,
ρ the density of the plate material, ν Poisson’s ratio, w the excitation, and r the
(radial) position variable. This is then discretised over the range of the plate radius
in n nodes, see Figure 6.12. The above equation is used for each of these n points,
whereby the positional derivation is replaced according to the following plan:
∂u 1
≈ (u r(i−2) − 8u r(i−1) + 8u r(i+1) − u r(i+2) )
∂r 12h
2
∂ u 1
≈ (−u r(i−2) + 16u r(i−1) − 30u r(i) + 16u r(i+1) − u r(i+2) )
∂r 2 12h 2
(6.49)
3
∂ u 1
≈ (−u r(i−2) + 2u r(i−1) − 2u r(i+1) + u r(i+2) )
∂r 3 2h 3
4
∂ u 1
≈ (u r(i−2) + 4u r(i−1) + 6u r(i) − 4u r(i+1) + u r(i+2) )
∂r 4 h 4
As a result of the form of the terms used, the necessity arises to add two further
nodes at both ends of the discretised range. These do not describe a real expansion
of the plate but can, however, be used in addition to the boundary nodes for the for-
mulation of the boundary conditions. This yields a description with 2n + 4 degrees
of freedom, some of which are dispensed with due to the boundary conditions.
Overall, the drawing up of the equation system and its description in a form
compatible with the electronics is definitely specified, but it is very cumbersome to
achieve manually. For this reason a model generator is used in [322], [323] or [324],
which automatically converts the partial differential equation into a formulation in
an analogue hardware description language or a Spice compatible equivalent circuit
on the basis of general integrators. The procedure is so general that it can also be
used on other partial differential equations such as the heat conduction equation,
see for example Bielefeld et al. [35] and [36]. However, one remaining limiting
factor is the fact that the method is only suitable for relatively simple structures
due to the nature of the underlying partial differential equations. Furthermore, in
this model only the plate is considered and not its suspension.

Analytical modelling

For some structures, such as square or circular plates, analytical solutions to the
partial differential bending equations are known. Models can be created on this basis
if the geometric form of a micromechanical structure permits. This is particularly