Page 284 - Mechanics of Microelectromechanical Systems

5. Static response of MEMS 271

where the actuation torque is twice the value determined in Eq. (4.47) of
Chapter 4 (there are two parallel actuators in Fig. 5.5). The elastic torque can
be expressed as:

where is the rotation angle of the mobile hub and is the rotation
compliance of the spiral spring, which has been defined in Eq. (3.137) of
Chapter 3 for thin spiral springs. Equation (5.18) uses the simplifying
assumption that the rotation compliance is simply the inverse of the
corresponding stiffness.
By combining Eqs. (5.17), (5.18), (4.47) and (3.137) together with the
numerical data of this example, the predicted value of the rotation angle is
found to be The capacitance change, as provided by the two
sensing units, relates to the actual rotation angle according to Eq. (4.49) of
Chapter 4 (the total capacitance variation is twice the value given by Eq.
(4.49) because there are two sensing units in parallel) and a value of
results from the measurement. The relative (percentage) error between
the model and actual rotation angles is therefore equal to

3. TWO-SPRING MEMS

Two springs can be coupled either in series or in parallel and the
resulting stiffness is found as a combination of the individual springs’
stiffnesses. Figure 5.7 illustrates the models that give the equivalent stiffness
for spring parallel/serial connection.
The equivalent parallel and series stiffnesses, as well-known from
elementary mechanics, are calculated as:

Equations (5.19) are specified for linear springs, but they are also valid for
rotary springs, which can similarly be coupled either in series or parallel. The
same equations can be extrapolated in design cases where more than two
microsprings are connected in parallel/series.