Page 311 - Mechanics of Microelectromechanical Systems

298 Chapter 5
The approach followed here is the one based on the stiffness approach of
Chapter 1 for a straight flexible member. As shown in Fig. 5.34, two sets of
reference frames are utilized here: one is the global frame XY, and the other
is the local reference frame with its x-axis aligned with the straight flexure
hinge. The actuation force F decomposes locally into the and and
therefore, the following matrix equation can be written, according to
Castigliano’s first theorem:

The supplemental part of the subscript which has been used in Chapter 1 to
denote the extremity of the flexible member which is assumed free (point 3
here) was eliminated from the notation, because the 2-3 flexure is symmetric.
As Eq. (5.90) indicates, axial and bending effects are both taken into account.
In order to determine the input stiffness, a relationship between the force
F and the corresponding displacement (taken about the global direction
X) is needed. This displacement results from adding up the two local
deformations, and namely:

The local deformations and can be expressed from the first two rows
of the matrix Eq. (5.90) as:

where it has been taken into consideration that and are the projections
of F onto the local x-and y-axes. The rotation (slope) at point 2 is zero,
because the flexure hinge is rigidly attached to the link 1-2 at that particular
point. By combining now Eqs. (5.91) and (5.92), results in the following
equation giving the input stiffness:

It should be mentioned that Eq. (5.93) is generic in the sense that it can
accommodate any shape of a flexure hinge whose required stiffnesses are
known.