Page 289 - Mechanics of Microelectromechanical Systems

276 Chapter 5

whereas Eq. (5.26) remains unchanged. It can be seen that there is a non-zero
out-of-the-plane motion about the z-direction because the second Eq. (5.27)
in combination with Eq. (5.28) give a non-zero value of The following
relative error is introduced:

when is calculated by means of the second Eqs. (5.27) and (5.28). This
error reduces to:

As Eq. (5.30) shows, the error is dependent on both the device characteristics,
such as stiffnesses and geometry, and the actuation force. The errors are zero
when:

Example 5.6
Consider the case where, due to imperfections, the actuation force makes
a small angle with the horizontal direction, in the x-z plane of Fig. 5.10.
Evaluate the errors introduced by this imperfection.

Solution:
According to Fig. 5.10, the angle formed by the left segment with the
horizontal direction is in this case:

and the horizontal projections of the two springs’ deformations are given by
Eq. (5.26) under the assumptions of small-displacements. By combining Eqs.
(5.25) and (5.32), results in the following error function, defined in Eq.
(5.29):

Equation (5.33) indicates that for the linearized, small-displacement case, the
error in the z-displacement varies linearly with the inclination angle of the
actuation force, and obviously, the error is zero when the angle of the
actuation force is zero.