Page 55 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH02_54_08/30/05

Basic Members: Lumped- and Distributed-Parameter Modeling and Design

54 Chapter Two

C C 0 0 0 0
l,y c,y
C C 0 0 0 0
c,y r,y
0 0 C l,z C c,z 0 0
C = (2.29)
0 0 C c,z C r,z 0 0
0 0 0 0 C a 0
0 0 0 0 0 C t

such that the deformation/displacement vector and the load vector cor-
responding to the free end of the micromember shown in Fig. 2.1 are
connected as

{d} = C {L} (2.30)
It is obvious that the compliance and stiffness matrices are inverses of
each other, namely,

C = K í1 (2.31)
A few remarks are in order here. While the axial- and torsion-related
stiffnesses are the algebraic inverses (reciprocals) of their correspond-
ing compliances, namely,

1 1
k = k =
a C a t C t (2.32)
the same is not true for the bending-related stiffness-compliance pairs,
due to the coupled effects between deflections/rotations and forces/
moments. In other words, it appears, at least from a formal point of
view, that
1 1 1
k  k  k  (2.33)
l C l r C r c C c

In cases where forces/moments need to be calculated in terms of known
deformations (such as when these are available experimentally), the
stiffnesses of Eqs. (2.33) have to be calculated either by applying
the direct approach or by inverting the compliance matrix–a more de-
3
tailed explanation of this aspect is given in Lobontiu and Garcia and
4
Lobontiu et al. It can be shown, however, that the bending-related
stiffness, which corresponds to the direct linear effects and which has
to be employed in resonant frequency calculations, is the algebraic in-
verse of the corresponding compliance. Therefore, in this book, the

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