Page 154 - Mechanics of Microelectromechanical Systems

3. Microsuspensions 141
When the axial deformations are negligible compared to the bending
deformations, Eqs. (3.27) and (3.28) can still be used by considering that the
axial compliances of the two segments are zero (axially rigid members).
The mention was made in Chapter 2 that the stiffnesses defined by
inverting the compliance matrix are different from the stiffnesses that are
calculated as:

and that the stiffnesses of Eqs. (3.27) and (3.28) should be used when forces
need to be calculated based on known displacements. However, Eqs. (3.29)
are used as definition relationships and their values can be obtained by using
the transformation Eqs. (2.25) of Chapter 2 from the stiffnesses of Eqs.
(3.26).
The out-of-the-plane definition stiffness can be determined by applying a
force at point 1 of Fig. 3.9 about a direction perpendicular to the bent
beam’s plane and by calculating the corresponding displacement. By taking
bending and torsion into account, the z-direction stiffness is:

Example 3.4
Calculate the mam stiffnesses of a bent beam microsuspension with
identical legs and of constant rectangular cross-section. Evaluate the errors in
calculating by its definition – Eqs. (3.29) – as opposed to the compliance
derived stiffness of Eq. (3.27).

Solution:
For this particular case, the linear in-plane stiffnesses are equal, namely:

and the z-axis stiffness is:

It has been considered that w << t (very thin cross-section) and therefore: