Page 146 - Mechanics of Microelectromechanical Systems
P. 146
3. Microsuspensions 133
rotations about the x-axis and the y-axis, respectively. Translation about
either the x- or y-axis is possible, but at least one of the four members is
subject to compression in that case, which might lead to elastic instability (or
buckling), as will be discussed in Chapter 5.
Each beam-spring will be defined by means of stiffnesses that related to
either bending or torsion. Figure 3.3 is the model of a beam suspension, and
indicates the corresponding boundary conditions and the load generated by
the central mass inertia when the motion about the z-axis is of interest.
Figure 3.3 Beam-spring model for bending-related stiffness
In order to find the linear direct-bending stiffness (or simply the
procedure that has been applied within the previous chapters is again utilized.
A force is applied and the unknown bending moment reaction is
determined followed by calculation of the deflection which enables
specification of the sought stiffness. Both steps are solved by means of
Castigliano’s displacement theorem. It can be shown that in the case where
the beam has a variable cross-section, the stiffness is:
where:
The compliances of Eqs. (3.1) and (3.2) have been given in Chapter 1 for a
constant rectangular cross-section beam. By substituting them in Eqs. (3.1)
and (3.2), one obtains:
The bending compliances of Eqs. (3.1) and (3.2) can be expressed in terms of
their corresponding stiffnesses, as shown in Chapter 1, in the matrix form:
