Page 85 - Mechanics of Microelectromechanical Systems
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72 Chapter 2
It has been shown in Chapter 1 that by inverting the bending-related
compliance matrix of a constant rectangular cross-section microcantilever, a
stiffness matrix is obtained whose components are not the direct inverses of
their corresponding compliances. It has also been mentioned that although
the three springs characterizing the lumped-parameter model of a
microcantilever are defined by the stiffnesses
(the algebraic inverses of the stiffnesses
of Eqs. (1.16) – which are the components obtained by
inverting the compliance matrix of Eq. (1.15) – should be used when
calculating the force and moment at the free tip in terms of the deflection and
slope at the same point.
All these conclusions are also valid for a variable cross-section
microcantilever as demonstrated next. The compliance equation describing
the bending about the y-axis is:
where the matrix of the right-hand side is of Eq. (2.6). Equation (2.20)
can be rewritten by expressing the loads in terms of displacements in the
form:
where the matrix of the right-hand side is the submatrix of Eq. (2.16).
It is clear that:
and the components of are related to the components of as:
