Page 286 - Microsensors, MEMS and Smart Devices - Gardner Varadhan and Awadelkarim

266 MICROSENSORS

8.4.7 Microgyrometers

The second type of inertial sensor is the gyroscope that measures the change in orientation
of an object. Silicon-micromachined gyroscopes have been fabricated on the basis of
coupled resonators. The basic principle is that there is a transfer of energy from one
resonator to another because of the Coriolis force. Thus, a simple mass m supported by
springs in the x- and y-axes and rotated around the z-axis at an angular velocity Q has
the following equations of motion.

mx + bx + k xx — 2m£2y = F x
(8.35)
my + by + k yy + 2m£2jc = F y

where the terms 2mfii and 2mQy describe the Coriolis forces and the resonant frequen-
cies are
w 0x = ^/k x/m and aty = Jk y/m (8.36)

Now assume that the resonators are excited and behave harmonically with the amplitudes
a(t) and b(t). By fixing the amplitude of one oscillator (a 0) by feedback and then for
synchronous oscillators (w 0x = w 0y), the equations simply reduce to
db / c \
— + ( — U + Qoo = 0 (8.37)
dt V2m/
Under a constant rotation, the steady-state solution to Equation (8.37) is a constant ampli-
tude b 0 where >j \
( — JaoQ (8.38)

Therefore, the amplitude of the undriven oscillator is linearly proportional to the rotation
or precession rate £2.
The first silicon coupled resonator gyrometer was developed by Draper Laboratory
in the early 1990s and its arrangement is shown in Figure 8.33. The device is bulk-
micromachined and supported by torsional beams with micromass made from doped
++
(p ) single-crystal silicon (SCS). The outer gimbal was driven electrostatically at a
constant amplitude and the inner gimbal motion was sensed. The rate resolution was only
-1
4 deg s and bandwidth was just 1 Hz.
More advanced gyroscopes have been fabricated using surface micromachining of
poly silicon. There are a number of examples of coupled resonator gyroscopes such as the
MARS-RR1 gyroscope reported by Geiger et al. (1998). The performance of this device
is provided in Table 8.14.
There are reports of a number of other types of device to measure precision rates; the
IDT MEMS device described in Chapter 14 is one such example. Another is the ring
gyroscope that again works by the Coriolis force transferring energy from one mode into
another at 45° (Ayazi and Najifi 1998). The basic approach is attractive but does require
a deep etch to produce viable devices. Figure 8.34 shows two ring gyroscopes. The first
was made at the University of Michigan (Ayazi and Najifi 1998), whereas the second is
a prototype made by DERA Malvern, (UK).