Page 24 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

Design at Resonance of Mechanical Microsystems 23
system. Two methods are presented next that enable calculation of the
maximum and minimum resonant frequencies without requiring cal-
culation of all the resonant frequencies defining the multiple-DOF
system.

Rayleigh’s method and the upper bound on resonant frequencies by the
stiffness method. For a multiple-DOF system, the dynamic equation
defining the free undamped vibrations is

..
M {x}+ K {x}=0 (1.62)
where [M] is the mass matrix and [K] is the stiffness matrix. In a har-
monic motion, the acceleration vector can be expressed in terms of the
displacement vector and the frequency as
.. 2
{x}= íȦ {x} (1.63)

By combining Eqs. (1.62) and (1.63), the following equation can be
written:

í1 2
( M K íȦ I ){x}=0 (1.64)
The characteristic equation corresponding to Eq. (1.64) is

í1 2
det( M K íȦ I ) =0 (1.65)
and by solving it the resonant frequencies Ȧ are obtained. The method
of finding the resonant frequencies and the corresponding modes based
on the stiffness matrix is called the stiffness method. In general, the
characteristic equation can be written as

íȦ 2n + (a + a + 썳 + a )Ȧ 2(n í 1) + 썳 =0 (1.66)
11 22 nn
where a , a , . . . , a are the diagonal terms of the dynamic matrix,
nn
11
22
which is defined as
෹1
A = M K (1.67)
Related to Eq. (1.66), it is well known from algebra that

2
2
2
Ȧ + Ȧ + 썳 + Ȧ = a 11 + a 22 + 썳 + a nn (1.68)
n
1
2
where Ȧ , Ȧ , …, Ȧ are the resonant frequencies. It follows from
1
2
n
Eq. (1.68) that
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