Page 25 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

24 Chapter One
2
Ȧ < a 11 + a 22 + 썳 + a nn (1.69)
1
In other words, the upper bound on the first (natural) frequency of
an n-DOF system is the sum of the main diagonal terms; that sum is
also known as the trace of the dynamic matrix. Equation (1.69) is the
mathematical formulation of Rayleigh’s procedure.

Dunkerley’s method and the lower bound on resonant frequencies by the
compliance method. The lower bound on the resonant frequencies is
determined by applying Dunkerley’s method, which is presented next.
The inertia force acting on the undamped dynamic system formed of n
masses is

..
{F}= ෹> M {x} (1.70)
At the same time, the displacement vector can be expressed as
{x} = C {F} (1.71)

where [C] is the compliance matrix (the inverse of the stiffness matrix
[K]). By combining Eqs. (1.62), (1.63), (1.67), and (1.71), the following
equation is obtained:
( C M í I 2) {x}=0 (1.72)
Ȧ

which is also known as the modal equation formulated by means of the
flexibility or compliance method. The characteristic equation corre-
sponding to Eq. (1.72) is
( I 2)
det C M ෹ =0 (1.73)
Ȧ

The dynamic matrix according to the compliance approach is defined as

A = C M (1.74)
and it can be shown that

1 + 1 + 썳 + 1 = C M + C M + C M
Ȧ 2 Ȧ 2 Ȧ 2 11 1 22 2 nn n (1.75)
1 2 n
where the C ii are the diagonal terms of the symmetric compliance
matrix and the M i are the terms of the diagonal mass matrix. Equation
(1.75) enables us to state that

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