Modeling the MEMS gyroscope

The MEMS gyroscope affects so much of our everydaylives, in particular in consumer electronics, that engineersnow design systems that include MEMS gyros asessential components. In this age of model-based design, it iscritical to know how this device works and how to model it.

A gyroscope measures the angular velocity of a body abouta specified axis of rotation. A MEMS gyro, however, is notthe bulky, spinning-disk gyroscope we see in science museums.Figure 1 shows a physical model of a typical MEMS vibratorygyroscope designed to measure the angular velocity of thebody about the z axis of the ground reference frame.

Figure 1 This physical model shows a typical MEMS vibratory gyroscope designed to measure the angular velocity of the body about the z axis of the ground reference frame. The main principle of MEMS gyros is the transfer of energy between two modes of vibration through the Coriolis acceleration.

The main principle of MEMS gyroscopes is the transfer ofenergy between two modes of vibration—the drive and the sensemodes—through the Coriolis acceleration. How does this happen,and what are the challenges? Developing a mathematicalmodel for the physical model shown in Figure 1 will answerthose questions; every engineer thus needs to know how to do so.

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It all starts with a fundamental kinematic relationshipspecifying the absolute acceleration of the mass translating inthe body, whose rotation rate we want to know. This is shownwith a matrix relationship between the ground and body-fixedreference frames, here assumed to have a common origin:

We next apply Newton’s Second Law to the mass in both thex and y directions, resulting in two coupled differential equations of motion, shown below, where F₀ sin(ω_d t) is the drivingforce. Note the Coriolis acceleration terms, 2ωx˙ and 2ωy˙:

Assume a constant angular speed ω, and since ω is usuallymuch smaller than the natural frequencies of the system, neglect terms involving ω². The sense-mode response y is muchsmaller than the drive-mode motion x, so the term 2ωy can bedropped. The resulting equations are

The first equation can be solved independently. You can thensubstitute its solution into the second equation , which can besolved for the sense-mode response y. The resulting amplitudes ofmotion X and Y are given below; Y/ω is the gyroscope sensitivity:

The sensitivity is proportional to the oscillating mass,which puts some restrictions on the level of miniaturizationthat can be achieved. To achieve maximum sensitivity,resonance in both modes is desirable; that is, ω_d =ω_x =ω_y . Theexpression for the gyroscope sensitivity then becomes

Q_x and Q_y are the quality factors of the drive and sensemodes, respectively. High quality factors are desirable toimprove sensitivity.

All engineers should count the ability to do this kind ofwork as a critical element of their skill set; it is what differentiatesmodel-based design engineers in the 21st century.