Page 139 - Mechatronic Systems Modelling and Simulation with HDLs

128 6 MECHANICS IN HARDWARE DESCRIPTION LANGUAGES

true for very simple structures. Thus Chau and Wise [67] and Bota et al. [41], for
example, use analytical equations for the modelling of the square membrane of a
pressure sensor. In addition to bending mechanics, torsional mechanics can also be
considered analytically, as G´ omez-Cama et al. [122], for example, demonstrate for
a capacitive acceleration sensor and Wetsel and Strozewski [428] demonstrate for a
micromirror.
To illustrate analytical modelling, the example of a capacitive pressure sensor,
see Figure 6.12, will be considered again in what follows. The bending of the upper
plate can be described by the following equation, see Timoshenko and Woinowski-
Krieger [401] or Voßk¨ amper et al. [417]:

1 E t 3 p
u = (p + p el ), where D = (6.50)
2
D 1 − ν 12
where represents the Laplace operator, u the vertical deflection, D the bending
resistance, p the external pressure and p el an electrostatic pressure caused by the
read-out voltage applied through the plates. The bending resistance is again defined
as shown via the modulus of elasticity E, Poisson’s ratio ν and the thickness of the
plate t p . The electrostatic pressure can be described as follows using the radius:

1 U 2
p el (r) = ε 0 ε r,eff (r) (6.51)
2 t c + t i − u(r)
with the dielectric constants ε 0 and ε r,eff , the radius r, the read-out voltage U, the
thickness of the hollow space t c and the insulator thickness t i . A direct execution
of the four-fold integration of (6.50) for the solution with respect to deflection is
not possible because the electrostatic pressure in (6.51) is itself dependent upon
the deflection. A polynomial approximation of p el solves the problem, see [417]:

n
i
p el (r) ≈ a i r (6.52)
i=0

The general solution of equation (6.50) is then calculated as:
1 p 1 r 1 r
2
4
u = r + C 1 r 2 ln − 1 + C 2 r + C 3 ln
64 D 4 R 0 4 R 0
n
a i i+4

+ C 4 + r (6.53)
2
(i + 2) (i + 4) 2
i=0
with the radius of the plate R 0 and four constants C 1 to C 4 that have been yielded
by the integration, the values of which are to be determined from the boundary
conditions. With the aid of the resulting equation, further effects can be built in,
such as the restriction of the plate movement through the insulator, the influence
of plate suspension, or the dynamics of the movement.