Page 17 - Mechanics of Microelectromechanical Systems

4 Chapter 1
subscript c), relates a cause (the force) to an effect (the slope/rotation) that is
not a direct result of the cause, in the sense discussed thus far. A similar
causal relationship is produced when applying a moment at the free end of
the cantilever, as sketched in Fig. 1.3 (b). The moment generates a
slope/rotation, as well as a deflection at the beam’s tip, and the following
equation can be formulated:

Formally, Eqs. (1.5) and (1.6) can be written in the form:

where the matrix connecting the load vector on the left hand side to the
deformation vector in the right hand side is called bending-related stiffness
matrix.
Elastic systems where load and deformation are linearly proportional are
called linear, and a feature of linear systems is exemplified in Eq. (1.5),
which shows that part of the force is spent to produce the deflection and
the other part generates the rotation (slope) Equation (1.6) illustrates the
same feature. The cross-compliance connects a moment to a deflection,
whereas (the rotary stiffness, signaled by the subscript r) relates two
causally-consistent amounts: the moment to the slope/rotation. The
stiffnesses and can be called direct stiffnesses, to indicate a force-
deflection or moment-rotation relationship. Equations that are similar to Eqs.
(1.5) and (1.6) can be written in terms of compliances, namely:

and

where the significance of compliances is highlighted by the subscripts which
have already been introduced when discussing the corresponding stiffnesses.
Equations (1.8) and (1.9) can be collected into the matrix form: