Page 182 - Mechatronic Systems Modelling and Simulation with HDLs

8.2 DEMONSTRATOR 5: CAPACITIVE PRESSURE SENSOR 171

where R is the radius relating to the rotation. The hoop strain is calculated as:
∂w u
= (8.8)
∂z R
The element displacements u and v are, however, defined in the natural coordi-
nate system (s,t), so that the derivatives with respect to x and y must be linked
to the derivatives with respect to s and t. According to the chain rule, the follow-
ing applies:

∂/∂s ∂/∂x ∂x/∂s ∂y/∂s
= J where J = (8.9)
∂/∂t ∂/∂y ∂x/∂t ∂y/∂t
It is a prerequisite that a clear relationship between the coordinate systems exists.
In this case the following applies:

∂/∂x −1 ∂/∂s
= J (8.10)
∂/∂y ∂/∂t
By the use of the inverted Jacobi’s operator J −1 we can now set up the strain
vector for arbitrary points on the rotationally symmetrical plane elements.
We then move from the strains ε to the stresses τ, by using the material
matrix C:

T
τ = C · ε where τ = [τ xx τ yy τ xy τ zz ] (8.11)
For the material matrix of a rotationally symmetrical element the following applies
according to [19]:
ν ν
 
1 0
1 − ν
ν ν
 1 − ν 
 
1 0
 
E(1 − ν)  1 − ν 1 − ν 
C =  1 − 2ν  (8.12)
(1 + ν)(1 − 2ν)  0 0 0 
 
2(1 − ν)
 
 ν ν 
0 1
1 − ν 1 − ν
In the material matrix, E represents the modulus of elasticity and ν represents
Poisson’s ratio. To determine the stiffness matrix we first apply the elastic potential
of a plane element:
1
T
= ε τ dA (8.13)
2 A
Alternatively we can also formulate
1
T
= ˆ u Kˆu (8.14)
2