EDNE December 2012 - (Page 33)

AU I TO TIVE MO AERO SP A CE M N G CE Control Systems Digital Control Systems E D S IC S A L MECHATRONICS IN DESIGN By Kevin C Craig, PhD he MEMS gyroscope affects so much of our everyday lives, in particular in consumer electronics, that engineers now design systems that include MEMS gyros as essential components. In this age of model-based design, it is critical to know how this device works and how to model it. A gyroscope measures the angular velocity of a body about a specified axis of rotation. A MEMS gyro, however, is not the bulky, spinning-disk gyroscope we see in science museums. Figure 1 shows a physical model of 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 gyroscopes is the transfer of energy between two modes of vibration—the drive and the sense modes—through the Coriolis acceleration. How does this happen, and what are the challenges? Developing a mathematical model for the physical model shown in Figure 1 will answer those questions; every engineer thus needs to know how to do so. It all starts with a fundamental kinematic relationship specifying the absolute acceleration of the mass translating in the body, whose rotation rate we want to know. This is shown with a matrix relationship between the ground and body-fixed reference frames, here assumed to have a common origin: ˆ i cos θ 0 sin θ 0 0 0 1 ˆ I ˆ J ˆ K R→O R→P M AT E R IALS PR O Control Electronics RAPHY ROG XE Software mechatronics Mechanical CAD Electromechanics Electronic Systems FreSH IdeAS ON INTeGrATING mecHANIcAL SYSTemS, eLecTrONIcS, cONTrOL SYSTemS, ANd SOFTWAre IN deSIGN T E Mechanical Systems M S M CO A S CT NSU MER PRODU Modeling the MEMS gyroscope Knowing how this ubiquitous device works is essential in the age of model-based design. T 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. neglect terms involving ω². The sense-mode response y is much smaller than the drive-mode motion x, so the term 2ωy can be dropped. The resulting equations are .. . mx + kxx + cx x = F0 sin(ωdt) .. . . my + kyy + cyy = −2mωx The first equation can be solved independently. You can then substitute its solution into the second equation, which can be solved for the sense-mode response y. The resulting amplitudes of motion X and Y are given below; Y/ω is the gyroscope sensitivity: X=(F0/m)ω 2 x 1 (1−r 2) 2+(2ζxrx) 2 x 1 (1−r ) +(2ζyry) 2 2 y 2 a = a + R→B R→O R→B ω ×(ω × r →OP B→P R→B →OP ) B→P ωx= ωy= c ω kx ζ= x r= d m x 2mωx x ωx c ω ky ζ= y r= d m y 2mωy y ωy ˆ j = −sin θ cos θ ˆ k R→P + α × r a =0 R→B R→B +a +2 ω × v →OP R→B Y 2m ωd X = ky ω .. . .. . a = x−αy−2ωy−ω2x ˆ y+αx+2ωx−ω2y ˆ i+ j ˆ ω = ωk ˆ α = αk ˆ ˆ r = xi + yj . . B→P ˆ ˆ v = xi + yj We next apply Newton’s Second Law to the mass in both the x and y directions, resulting in two coupled differential equations of motion, shown below, where F0sin(ωdt) is the driving ˙ ˙ force. Note the Coriolis acceleration terms, 2ωx and 2ωy: . . .. −kxx−cx x+F0 sin(ωdt) = m x−αy−2ωy−ω2x . .. . −kyy−cy y = m y+αx+2ωx−ω2y The sensitivity is proportional to the oscillating mass, which puts some restrictions on the level of miniaturization that can be achieved. To achieve maximum sensitivity, resonance in both modes is desirable; that is, ωd=ωx=ωy. The expression for the gyroscope sensitivity then becomes Y 2m ωd F0 Qx Qy = kx ky ω Qx = 1 2ζx Qy = 1 2ζy Assume a constant angular speed ω, and since ω is usually much smaller than the natural frequencies of the system, www.edn-europe.com Qx and Qy are the quality factors of the drive and sense modes, respectively. High quality factors are desirable to improve sensitivity. All engineers should count the ability to do this kind of work as a critical element of their skill set; it is what differentiates model-based design engineers in the 21st century.EDN december 2012 | EDN EuropE 33 U N FA C TU S NS FE DE RI NG E Y S http://www.edn-europe.com

Table of Contents for the Digital Edition of EDNE December 2012

Cover
Contents
Texas Instruments Europe
Microchip
Digi-Key
Masthead
EDN comment
Pulse
Digi-Key
Baker’s Best
Embedded World 2013
Test & Measurement World
Rohde & Schwarz
Digi-Key
Squeezing the most from battery cells with a switched-mode pump
Signal integrity
Processor architectures : one to rule them all ?
Digi-Key
Mechatronics in Design
Advances in wireless speaker performance and technology
Design Ideas
Teardown; the ultimate Consumer Product ?
Product roundup
Tales from the Cube

EDNE December 2012