Page 280 - Mechanical design of microresonators _ modeling and applications

0-07-145538-8_CH05_279_08/30/05

Resonant Micromechanical Systems

Resonant Micromechanical Systems 279

(a) (b)
Figure 5.52 Modes in a vibratory ring gyroscope: (a) primary (drive) mode; (b) secondary
(sense) mode.

5.6 Tuning Forks
Tuning forks have long been used as standard pitch tools for musical
instrument calibration. Another utilization of tuning forks is in the
clock and wristwatch industry as frequency standards. Resonating
tuning forks, such as the ones introduced in Chap. 1, can also be em-
ployed as microsensors in a similar manner to gyroscopes in detecting
changes in an external angular velocity, as shown by Satoh, Ohnishi,
23
24
25
and Tomikawa; Sato, Ono, and Tomikawa; Momosaki; or
26
Matsiev. The tuning forks operate similarly to gyroscopes, and the
Coriolis effects are again the underlying principle. The classical tuning
fork sensor is sketched in Fig. 5.53, where two variants are outlined. In
the first variant, the driving is out-of-the-plane, which results in Cori-
olis accelerations pulling away the two tines in their plane
(Fig. 5.53a). In the second variant (Fig. 5.53b), driving and sensing in-
terchange, with the net result that the two tines are deformed out-of-
the-plane through Coriolis effects.
The vibrational amounts involved in one tine of the tuning fork are
sketched in Fig. 5.54. The input to the tine consists of the angular
velocity Ȧ which is applied about the z axis. A sinusoidal force is driving
the tine about the x axis, and the result is a relative velocity v of the
r
tip of the tine about the same axis. The interaction between the angular
velocity and relative velocity results in a Coriolis-type acceleration
about the y direction.
By ignoring the damping, the dynamic equations of motion about the
drive (x) and sense (y) axes can be written, using a lumped-parameter
model, as follows:

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.