Page 169 - Mechanics of Microelectromechanical Systems

156 Chapter 3
It is therefore sufficient to define the compliances or stiffnesses of a unit,
because the compound serpentine springs are formed by series connection of
several base units. Figure 3.27 is a three-dimensional model of a serpentine
unit with the local reference frame. The in-plane compliances can be found
based on Fig. 3.24, which shows the applied loads and By using
Castigliano’s displacement theorem (in the case where only bending is
accounted for and the 5 segments composing the base unit are assumed to
have identical constant cross-sections), the corresponding displacements
and are determined. The compliances that form the 3 by 3 symmetric
compliance matrix of Eq. (3.19) are given below.

Figure 3.27 Three-dimensional view of a serpentine base unit with boundary conditions and
loads for stiffness calculations

As mentioned previously, the definition stiffnesses can be found by simply
taking the algebraic inverses of the compliances formulated in Eqs. (3.75)
through (3.80). If the loads need to be calculated when the
corresponding displacements at point 1 – Fig. 3.24 – are known, then a
stiffness matrix has to be determined by inverting the compliance matrix of
Eq. (3.19) and which contains the terms of Eqs. (3.75) through (3.80).