Page 147 - Mechanics of Microelectromechanical Systems

134 Chapter 3

By substituting the compliances of Eq. (3.4) in Eqs. (3.1) and (3.2) results in:

which shows that the linear direct-bending stiffness of a fixed-guided beam
here) is identical to the one of a fixed-free beam – see Example
1.1, for instance. For a fixed-guided beam of constant cross-section, the
bending stiffness (see Young and Budynas [1] for instance) is given by Eq.
(3.3), which confirms the more generic Eq. (3.5)
The stiffness that is related to bending about the y-axis can be
determined similarly, and its equation is:

with:

It can be shown that this stiffness of the fixed-guided beam is equal to the
stiffness of its corresponding fixed-free beam, namely:

Similarly, the torsion stiffness of this configuration is:

where is the torsional stiffness of the same-geometry, fixed-free
beam. When subject to axial extension, the resulting stiffness at the guided
end 1 is:

where is the axial stiffness of the fixed-free beam. Defining the axial
stiffness of a beam spring makes physical sense as point 1 of Fig. 3.3 can be
attached to a rigid mass (such as the proof mass of Fig. 3.1) that might
translate about the x-direction and cause the beam to deform axially.