Page 10 - Mechanical design of microresonators _ modeling and applications

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Design at Resonance of Mechanical Microsystems

Design at Resonance of Mechanical Microsystems 9

X/X st

1/(2ξ)
2
0.707/( ξ)
ω 1 ω 2
ω
ω r
2ξω r
Figure 1.9 Sharpness of resonance with sidebands.

1 Ȧ r Ȧ r
Q = = = (1.30)
r
Ȧ íȦ
2ȟ
2 1 Ȧ b
Large values of the Q factor microresonators require, as Eq. (1.30) in-
dicates, high resonant frequencies and small bandwidths, which means
small damping coefficients. It should also be mentioned that the band-
width describes the ability of a resonant system to follow a sinusoidal
driving signal which is close to the resonant frequency, and the band-
4
width is proportional to the speed of response (see Ogata ).
Equation (1.27) also indicates that the quality factor can be defined
for resonance as
X r
Q = (1.31)
r
X
st
Figure 1.10 plots the quality factor of Eq. (1.26) in terms of the fre-
quency ratio for various damping ratios.
An alternative to qualifying the damping losses in mechanical
2
resonators is the loss coefficient Ș (see Thomson, for instance), which
is the inverse of the quality factor:
1 cȦ
Ș = = =2ȕȟ (1.32)
Q k
The resonance loss coefficient is obviously
1
Ș = =2ȟ (1.33)
r Q r

It is interesting to point out that the damping energy ratio can be ex-
pressed as

U d
= ȕ (1.34)
U d,r

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