Page 9 - Mechanical design of microresonators _ modeling and applications
P. 9
0-07-145538-8_CH01_8_08/30/05
Design at Resonance of Mechanical Microsystems
8 Chapter One
d
U = F dx (1.22)
d
which, for viscous damping where the damping force is proportional to
the oscillator’s velocity:
F = cx ˙ = ȦXcos(Ȧt í ́) (1.23)
d
becomes
2ʌ ˶
/
d . 2 2 (1.24)
U = c x dt = ʌcȦX
0
At resonance, the energy lost through damping is
U =2ʌȟkX 2 (1.25)
d,r
By combining Eqs. (1.21) and (1.24), the quality factor of Eq. (1.20)
becomes
k 1
Q = = (1.26)
cȦ 2ȕȟ
At resonance, the quality factor is expressed as
km 1
Q = c = 2ȟ (1.27)
r
The resonance quality factor is also called sharpness at resonance
3
2
(mostly in the mechanical vibration language, see Thomson or Rao ),
which is defined as the ratio (Ȧ 2 íȦ 1 )/ Ȧ r ,where the frequency difference
in the numerator is
Ȧ íȦ =2ȟȦ (1.28)
2 1 r
These frequency values are also called sidebands or half-power points.
It can be shown that this particular situation leads to an amplitude
ratio of
X = 1 싉 0.707
X 2 2ȟ 2ȟ (1.29)
st
and the case is pictured in the plot of Fig. 1.9.
The frequency difference Ȧ 2 íȦ 1 is called bandwidth being denoted
by Ȧ b , and by combining Eqs. (1.25), and (1.29) and Fig. 1.9, the resonant
Q factor can be expressed as
Downloaded from Digital Engineering Library @ McGraw-Hill (www.digitalengineeringlibrary.com)
Copyright © 2004 The McGraw-Hill Companies. All rights reserved.
Any use is subject to the Terms of Use as given at the website.
