Page 7 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

6 Chapter One
Often Eq. (1.12) is written in the alternate form:

.. . 2 F
x +2ȟȦ x + Ȧ x = m sin(Ȧt) (1.13)
r
r
The solution to Eqs. (1.12) and (1.13), as shown by Timoshenko, 1
2
3
Thomson, or Rao, is the sum of the homogeneous solution—Eq. (1.8)
—and a particular solution which is of the form:
x (t) = Xsin(Ȧtí́) (1.14)
p
where the amplitude X is

X st X st
X = = (1.15)
2 2
2
(1 í mȦ 2 / k) + (cȦ / k) 2 (1 íȕ ) + (2ȟȕ) 2
with the frequency ratio ȕ being defined as

Ȧ
ȕ = (1.16)
Ȧ
r
and the phase angle between excitation and response ij as

2ȟȕ
́ = arctan (1.17)
1 íȕ 2
The particular solution of Eq. (1.14) is of special importance as it de-
scribes the forced response of a vibratory system. In Eq. (1.15) the static
displacement is X st and is defined as F / k. Figures 1.7 and 1.8 are plots
of the amounts X/X st and ij as functions of ȕ for various values of ȟ.
As Fig. 1.7 indicates, when the driving frequency equals the resonant
frequency (ȕ = 1), the amplitude ratio reaches a maximum, which, for
very small damping ratios, goes to infinity. Even in the presence of
moderate damping, the amplitude at resonance is large, and this fea-
ture is utilized as a working principle in mechanical microresonators.
ȕ
At resonance, when = 1, the amplitude ratio of Eq. (1.15) becomes
X
r = 1 (1.18)
X st 2ȟ

which gives an amplitude of

F 0
X = (1.19)
2kȟ

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