Analysis of Vibratory Gyroscopes: Drive and Sense Mode

IEEE SENSORS JOURNAL, VOL. 17, NO. 2, JANUARY 15, 2017 347

Analysis of Vibratory Gyroscopes: Drive and SenseMode Resonance Shift by Coriolis Force

Hakan Cetin, Student Member, IEEE, and Goksen G. Yaralioglu, Member, IEEE

Abstract— In this paper, we demonstrate the analysis ofcoupling between drive and sense systems of vibratory gyro-scopes. Vibratory gyroscopes have attracted a lot of interestrecently with the development of MEMS gyroscopes. Thesegyroscopes made their way through portable devices and smartphones. Novel gyroscope architectures have been proposedand analyzed in detail. However, in most of these analyses,coupling between the sense and drive systems was ignored.We analytically show that drive and sense systems are coupledtogether via Coriolis and centrifugal force. As a result, systemresonances shift as the rotation rate increase for linear andtorsional gyroscope systems. Starting from a simple gyro system,we calculated the sense and drive resonant frequency shifts invarious configurations. Then, for more complex systems whereanalytical solution is difficult to obtain, we used commerciallyavailable FEM tools to determine corresponding frequency shift.In general, we found that the shift is small and can beignored for linear vibratory gyroscopes where Q of the sensesystem is less than 2500 for mode matched gyros. But forhigher Q systems, the frequency shift may affect the linearityof these gyroscopes. This sets a fundamental limit for thelinearity of vibratory gyroscopes. Based on our calculations thenon-linearity is above 1% for linear 2-DOF mode-matchedvibratory gyroscopes where Q is above 3000 and for torsional2-DOF vibratory gyroscopes where Q is above 600. Multi-DOFand ring vibratory gyroscopes are also examined. We find thatthe effect is less pronounced for Multi-DOF gyros, whereas thereis a large effect on the linearity of ring gyroscopes.

Index Terms— Coriolis force, frequency shift, gyroscopes.

I. INTRODUCTION

MEMS gyroscopes are finding wide range of applicationsin automotive and consumer electronics. In automo-

tive they are being used for vehicle stability control, rollover detection and advanced navigation systems. Consumerelectronics applications include gesture recognition for smartphones and tablets, input devices, image stabilization for cam-eras and camcorders [1]–[4]. These markets are demandinglower cost, smaller size, low power and high performance.Especially recently emerged applications, devices such asGPS assisted personal navigation (PND) and location-basedservices (LBS) for cell phones require low noise, high stabilityand high linearity [5]. As a result, more accurate analytical andnumerical models are required.

Manuscript received September 29, 2016; accepted November 2, 2016. Dateof publication November 8, 2016; date of current version December 20, 2016.This work was supported by the Turkish Scientific and Technological ResearchCouncil under Grant 114E592. The associate editor coordinating the reviewof this paper and approving it for publication was Prof. Sang-Seok Lee.

The authors are with the Electrical and Electronics Engineering Department,Özyegin University, 34794 Istanbul, Turkey (e-mail: [email protected];[email protected]).

Digital Object Identifier 10.1109/JSEN.2016.2626518

A typical vibratory gyroscope is composed of a drive systemand a sense system. In the drive system, a periodic oscillationof a proof mass is maintained at constant amplitude whereasin the sense system orthogonal vibrations of the proof massto the drive mode are detected. As the input rate is increased,driven proof mass starts to vibrate in the sense direction dueto the Coriolis force. As a result, drive and sense systemsare coupled together through the Coriolis force. In most ofthe previous work the coupling between drive and sensesystems have been ignored [6], [7]. In this paper we performedrigorous analyses of gyro systems to include this couplingand centrifugal force. We studied both 2-DOF and multi-DOFvibratory gyroscopes that are commonly used. These gyrosusually employ multiple proof masses coupled by a networkof spring systems [4], [8], [9], [17]. Based on our analyses weobserved that, drive and sense resonant frequencies shift as arotation rate is applied to the gyro system.

In macro-scale, the frequency shift due to gyroscopic effecthas been observed and analyzed in a rotor system which iscomposed of a cylindrical rod with a disk in the middle [10].In one of the modes which is called wobbling, the rod movesin its second flexural mode and the disk moves out of plane.The frequency of this mode shifts with the rotation of the rotordue to the gyroscopic effect which changes the stiffness of thesystem. In [11], in plane and out of plane modes of rings arestudied to see the effect of Coriolis coupling with the appliedangular velocity.

In micro-scale, the frequency shift due to gyroscopic effecthas been investigated for micro rectangular beams [12]. Forcantilever beams, two flexural modes moving orthogonal toeach other are excited to form two oscillators. Each oscillatorworks at the resonant frequency of the corresponding mode.The resonant frequencies of the two modes of vibration couplewith the applied angular velocity along the cantilever lengthdue to the Coriolis Effect and this coupling causes the resonantfrequency shift. It is claimed that this shift between twomodes can be used for designing a vibratory gyroscope basedon frequency detection. In [13]–[15], frequency modulatedgyroscopes have been studied for various structures such asquadruple mass, cantilever beam and ring. In these gyroscopes,the frequency shift is utilized for detecting the rotation rateinstead of the amplitude of the sense system.

In this paper, we extent the analysis presented in [12]–[15]and demonstrate that frequency shift exists even in amplitudemodulated gyros. We further show that the coupling betweenthe drive and sense system affects the linearity of the amplitudemodulated gyroscopes for 2 DOF systems. In order to identifythe frequency shift, in this paper, analytical solutions were

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348 IEEE SENSORS JOURNAL, VOL. 17, NO. 2, JANUARY 15, 2017

Fig. 1. Mass-spring-damper system of the 2-DOF Vibratory Gyroscope.Rotation axis is going through the point mass.

performed for simple gyro systems and numerical tools wereused for complex gyro systems.

II. ANALYSIS OF VIBRATORY GYROSCOPES

In this section, we start with the analysis of 2-DOF modematched gyroscopes. In a typical vibratory gyroscope, theproof mass is vibrated at its resonant frequency through aclosed loop system. When the gyroscope rotates there aretwo forces applied to the proof mass; 1. Coriolis force,2. Centrifugal force. The effect of the centrifugal force hasbeen studied previously. We repeated the analysis of a simpleharmonic oscillator (SHO) experiencing only centrifugal forcein Appendix 1. For centrifugal force, the location of therotation axis is important. As shown in Appendix 1, as theradius of rotation increases the total DC force on the proofmass increases. However, the AC force that is due to thecentrifugal force is only dependent on the amplitude of theproof mass vibration. Therefore, the resonant frequency shiftis independent of the DC force as shown in Appendix 1. Basedon this, in our analyses we assumed that the rotation is appliedat the center of the proof mass throughout the text without lossof generality.

A. Linear 2-DOF Vibratory Gyroscopes

A generic z-axis gyroscope which is used to detect rotationalong z-axis is shown in Fig. 1. The proof mass vibrates in thedrive direction (x) and the sense system detects the vibrationof the proof mass in the sense direction (y) due to Coriolisforce as the input rate (�z) is applied. To establish the correctequation of motion for the proof mass, one needs to considerboth the Coriolis force and the centrifugal force. We assumedthe rotation axis goes through the point mass ‘m’ as explainedabove. According to [7], if the angular input rate is constantthen the equation of motion is formed in (1) for linear 2-DOFvibratory gyroscopes in a rotating reference frame. 2�zm y and2�zmx denote Coriolis acceleration and xm�2

z and ym�2z are

centripetal accelerations. kx , ky and cx , cy denote spring and

damping constants in x and y directions, respectively.

mx + cx x + kx x = +2�zm y + xm�2z

m y + cy y + ky y = −2�zmx + ym�2z (1)

The equation of motion of the gyroscope given in (1) canbe written in the state-space notation where one variable (X)can be used to represent amplitudes of drive and sense mode:

X = AX + B

Y = C X + D (2)

where

X =

⎡⎢⎢⎣

xyxy

⎤⎥⎥⎦ (3)

A =

⎡⎢⎢⎣

0 0 1 00 0 0 1

−kxm + �2

z 0 −cxm 2�z

0 −kym + �2

z −2�z−cym

⎤⎥⎥⎦ (4)

B =

⎡⎢⎢⎣

00F0m0

⎤⎥⎥⎦ , C =

[1 0 0 00 1 0 0

], D = 0 (5)

In general A matrix has three parts as shown in Equation (6).

A =[

0 MK C

](6)

M denotes the mass part for linear and the moment ofinertia part for torsional systems, respectively. K denotesthe stiffness part (linear or rotational) and C denotes thedamping part. According to (4) damping part of A matrixconsists of Coriolis accelerations and the damping coefficientsof the system and the spring part of the A matrix consists ofcentripetal acceleration and the spring constants of the system.Mass part of A matrix consists of only one proof mass whereall springs and dampers are connected for a 2 DOF system.The Eigen values of A matrix are the roots of the followingequation:

λ4 +(

cx + cy

m

)λ3 +

(kx + ky

m+ cxcy

m2 + 2�2z

)λ2

+(

cx

m

(ky

m− �2

z

)+ cy

m

(kx

m− �2

z

))λ

+(

kx

m− �2

z

) (ky

m− �2

z

)= 0 (7)

For mode matched designs, where c = cx = cy andk = kx = ky , Eigen frequencies can be calculated from (7).

λ1,2 = i�z − c

2m∓

√( c

2m

)2 − ic

m�z − k

m

λ3,4 = −i�z − c

2m∓

√( c

2m

)2 + ic

m�z − k

m(8)

The imaginary parts of the Eigen values are the resonantfrequencies of the linear system. If the system is defined as

CETIN AND YARALIOGLU: ANALYSIS OF VIBRATORY GYROSCOPES: DRIVE AND SENSE MODE RESONANCE SHIFT 349

TABLE I

THREE DESIGNS FOR LINEAR 2-DOF VIBRATORY GYROSCOPE ANDTHEIR RESONANT FREQUENCIES WHEN �z=0

a high Q system (ignoring c), then Eigen frequencies areobtained as:

λ1 =√

k

m+ �z

λ3 =√

k

m− �z (9)

However, for non-mode matched designs, a closed form equa-tion is difficult to find. Therefore, we used Matlab to calculatethe Eigen values of matrix A. In general, there are 2 sets ofEigen values corresponding to drive mode and sense modefor 4x4 A matrix. Each set consists of two complex conjugateEigen values. The imaginary parts of the Eigen values are theresonant frequencies and the real parts determine the dampingfactor for the corresponding system.

According to [7], for the drive resonator of linear 2-DOFvibratory gyroscopes with a known Q factor, the relationbetween the oscillation amplitude and force at resonance canbe found as in (10) and damping coefficients also can becalculated as in (11).

F0 = kxx0

Qd(10)

cx =√

kxm

Qd, cy =

√kym

Qs(11)

We calculated the Eigen frequencies of three designs withdifferent uncoupled resonances that are summarized in Table I.In a typical mode matched gyro, drive and sense resonantfrequencies are very close to each other. For all the designswe assumed the spring constants of drive and sense systemsare around 1 N/m and the proof mass is exactly 20x10−12 kg.These values are consistent with typical MEMS gyroscopes.In design 1, the spring constants of the drive and senseoscillators are equal to each other (1 N/m) therefore theresonant frequencies match each other. For the second andthird design spring constants are slightly different resultingin slight mismatches in the resonances. We plotted Eigenvalues to depict how resonant frequencies for a linear 2-DOFvibratory gyroscopes shift for various input rates in Fig. 2.In Fig. 2, we assumed same Qs for the drive and the sensesystems.

For design 1 (mode matched design), when the input rateis zero, both drive and sense systems oscillate at the samefrequency. As the input rate is increased, there appear tworesonances for both drive and sense modes (total of 4 reso-nances; drive system has two resonances, sense system has tworesonances). One set of the resonances shifts down with theinput rate linearly whereas the other set increases. For the other

Fig. 2. Resonant frequencies of 3 designs for linear 2-DOF VibratoryGyroscopes while �z �=0 and Qd = Qs = 1000. Each curve representstwo resonances; one for drive, the other for sense.

Fig. 3. Frequency shift at 100 rad/s for different designs where kx and kyare around 1 N/m, and mass m is 20x10−12 kg and Qd = Qs = 1000.(Frequency shift is calculated for only one curve of Fig. 2.)

designs (Design 2 and 3) there is a non-linear relation betweenthe resonance shift and the input rate as demonstrated in Fig. 2.In general, if sense and drive resonances are placed awayfrom each other, the effect of the frequency shift diminishes,as shown in Fig. 3. In Fig. 3, we calculated the frequencyshift of one of the resonances as a function of initial frequencyseparation.

In Fig. 4, it is illustrated the shift in the resonance when thedrive and sense systems have different Q values. For perfectlymatched gyro (Design 1), the resonant frequencies do not shiftuntil the rate is increased above certain value which is 56 rad/sin this example. Beyond this rate a frequency shift is againobserved. Design 2 and Design 3, on the other hand behaveas same as the designs shown in Fig. 2, in terms of resonantfrequencies.

Next, we calculated the frequency responses of drive andsense systems. Fig. 5 and 6 show drive mode amplitude (x0)and sense mode amplitude (y0) respectively, calculated using


Fig. 4. Resonant frequencies of 3 designs for linear 2-DOF VibratoryGyroscopes while �z �=0 and Qs = 500, Qd = 1000. Each curve representstwo resonances; one for drive, the other for sense.

Fig. 5. Amplitude of the drive mode of Design 1 where Qs = Qd = 1000.

Fig. 6. Amplitude of the sense mode of Design 1 where Qs = Qd = 1000.

equation (2). In Fig. 5, it is assumed that the drive systemis driven by a constant force to get 1 μ m peak vibrationamplitude at the resonance for zero input rate. As the input rate

Fig. 7. Amplitude of the drive mode of Design 1 where Qs = Qd = 1000and �z = 150 rad/s with the effect of Coriolis force, the Centrifugal forceand both of them. (The curves with Coriolis Effect and both effects are ontop of each other)

increases the amplitude drops and frequency shifting occurs inconsistent with Fig. 2. However, in a gyro system, the driveamplitude is kept constant by a feedback loop and the force isvaried to keep the amplitude constant. This will be consideredlater. For the sense system, when there is no input rate, thereis no sense mode vibration. As the input rate increases thesense system starts vibrating due to the coupling of drivesystem through Coriolis and centrifugal force. As a resultthe amplitude increases and also frequency shifting occurs asshown in Fig. 6.

We also plotted the curve corresponding to the 150 rad/secinput rate in Fig. 7 to illustrate the effect of Coriolis force andthe Centrifugal force, separately. As shown in the figure, themain source of frequency shift and amplitude variation is theCoriolis force. When we ignored Coriolis force and includedonly centrifugal force, we do not observe any frequency shiftin the spectrum (dashed curve, Centrifugal effect). This is validfor most MEMS gyroscopes where the resonances are above30 kHz and the input rates are up to 200 rad/sec. Thereforewe can neglect the centrifugal force due to the high resonantfrequencies of the sample gyroscope and low input rates thatwe used in this paper. But, in case of low resonance vibratorygyroscopes and high input rates, one cannot neglect centrifugalforce.

Fig. 8, 9 and 10, 11 show drive mode amplitudes (x0)and sense mode amplitudes (y0) respectively in case ofQd <Qs and Qd >Qs . All the parameters are assumed assame as in Fig. 5 and 6 except Q of the drive and sensesystems. For Qd <Qs , as the input rate increases, the amplitudedrops and frequency shifting occurs in consistent with Fig. 4.The frequency shift appears at slightly lower input rates for theamplitude of drive mode in Fig. 8 compared to Fig. 5. On theother hand the frequency shift effect appears at higher inputrates for the amplitude of sense mode in Fig. 9 compared toFig. 6. For Qd >Qs , the frequency shift effect appears at higherinput rates for both amplitudes of drive and sense modes inFig. 10 and 11 compared to Fig. 5 and Fig. 6.


Fig. 8. Amplitude of the drive mode of Design 1 where Qd = 500 andQs = 1000.

Fig. 9. Amplitude of the sense mode of Design 1 where Qd = 500 andQs = 1000.

Fig. 10. Amplitude of the drive mode of Design 1 where Qd = 1000 andQs = 500.

In a typical vibratory gyroscope, the drive mode is notdriven in open-loop architecture but it is oscillated in a closed

Fig. 11. Amplitude of the sense mode of Design 1 where Qd = 1000 andQs = 500.

loop system. The amplitude of the drive mode oscillationsis a kept constant by a controller. The frequency responseof the sense system is sampled by the oscillation frequencyof the drive system. Therefore any relative change betweenthe drive and sense frequencies affects the response of thegyroscope to the input rate. To find the response of thegyroscope to the input rate we assumed a simple gyrosystem. In this simple gyro, the drive system is oscillatedat the frequency where the amplitude of the drive system ismaximum (we assumed that the controller electronics providedthe necessary phase shift). In the case of a frequency split i.e.when there are multiple peaks in the frequency spectrum ofthe drive system, we assumed that the closed loop controllertracks the lower frequency peak. We also assumed the drivevibration amplitude was kept at 1 μm by the controller. Then,we calculated the amplitude response of the sense system forthe given rate input, and found the amplitude of the sensesystem at the oscillation frequency of the drive. Note that,the rate input affects both the drive frequency and the sensefrequency response but the drive amplitude does not changeand it is kept at 1 μm. More complex gyro systems canbe considered here but for demonstration purposes we choseabove described system which is commonly used. We appliedthe calculation method to the mode-matched gyro (Design 1)which was described in Table 1. The response of the sensesystem is shown in Fig. 12 for various quality factors. We usedequations (2) to (5) where we chose F0 of the system suchthat the drive mode amplitude (x0) is 1 μm at the frequencywhere the drive amplitude is maximum. Then, the amplitudeof the sense system was calculated at the drive frequency. Thesense amplitude is a measure of the input rate. Fig. 12 plotsthe sense amplitude as a function of input rate for Design 1where the modes match perfectly. As shown in Fig. 12, thesensitivity of the system increases as the sense quality factorincreases, as expected, however, at higher sense Qs valuesthe non-linearity can be observed. Based on Fig. 12, we alsoobserved that higher drive Qd results in lower non-linearity.

Based on Fig. 12, one can calculate the percentage non-linearity which is defined by the ratio of the maximum


Fig. 12. Linearity of the linear 2-DOF Vibratory Gyroscope for Design 1.

Fig. 13. Nonlinearity of different Q systems for Design 1 whereQ = Qs = Qd .

deviation from the expected output over the full-scale range tothe full-scale. For the calculations, we assumed that the full-scale is 20 rad/s. As shown in Fig. 12, for larger input ratesthe response deviates from the linear response. The ratio ofthe deviation (the difference between the extrapolated linearresponse and the actual response) to the full-scale range isthe percentage non-linearity. Fig. 13 plots the percentage non-linearity as a function of Q of the gyroscope where driveQd and sense Qs are equal to each other. Above Q of 3000,the non-linearity increases above 1%.

B. Torsional 2-DOF Vibratory GyroscopesSimilarly one can also analyze the torsional micro-machined

gyroscopes. These gyro systems use a rotational vibratorymotion which is generated by rotational drive oscillator, vibrat-ing at a constant angular momentum. The Coriolis torqueinduces a rotational motion along the sense direction whichis measured by a sense-mode angular accelerometer. One ofthe torsional 2-DOF vibratory gyroscopes is designed as agimbal system [16]. This gimbal system for a z-axis torsional

Fig. 14. Torsional z-Axis gyroscope with drive gimbal structure. Rotationaxis is going through the inner mass.

gyroscope is shown in Fig. 14. The drive and sense-modedeflection angles of gyro structure are φd , φs respectively.The drive gimbal allows the proof mass to rotate along x-axis.Coriolis torque is induced along the y-axis as a response torotation along z-axis. Assuming angular input rate is constantand the oscillation angles are small, the equation of motion isshown in (12) for torsional 2-DOF vibratory gyroscopes [7].

(I dx + I s

x )φd + (Cdx + Cs

x)φd + K dx φd = −(I s

z + I sx − I s

y )�zφs

I sy φs +Cs

yφs + K syφs = (I s

z + I sy − I s

x )�zφd

(12)

where I is the moment of inertia for the correspondingdirection of the drive or sense systems. (I s

z + I sx -Is

y)�zφs and

(I sz + I s

y -Isx)�zφd denote Coriolis torques. K d

x , K sy and Cd,

x Csx ,

Csy denote spring and damping constants for drive and sense

systems in x and y directions respectively.Equations of motion of the gyroscope given in (12) can be

written in the state-space notation as:

X =

⎡⎢⎢⎣

φd

φs

φd

φs

⎤⎥⎥⎦ , B =

⎡⎢⎢⎣

00τ0

(I dx +I s

x )

0

⎤⎥⎥⎦ (13)

A =

⎡⎢⎢⎢⎢⎣

0 0 1 00 0 0 1

−K dx

(I dx +I s

x )0 −(Cd

x +Csx )

(I dx +I s

x )

−(I sz +I s

x −I sy )�z

(I dx +I s

x )

0 − K sy

I sy

(I sz +I s

y −I sx )�z

I sy

−Csy

I sy

⎤⎥⎥⎥⎥⎦

(14)

C and D matrices are as same as in (5). According to (14)damping part of the A matrix consists of Coriolis torquesand damping coefficients of the system. The spring part andmoment of inertia part of the A matrix consists of only springconstants and moments of inertia of the system, respectively.

For a typical MEMS gyro, 3 designs were assumed with dif-ferent spring constants around 1 kg.m2/s2 and their moment ofinertias, I d

x , Isx , Is

y and I sz are 10×10−12, 10×10−12, 20×10−12

and 200x10−12kg.m2 respectively. For these three designs,


TABLE II

THREE DESIGNS FOR TORSIONAL 2-DOF VIBRATORY GYROSCOPEAND THEIR RESONANT FREQUENCIES WHEN �z=0

Fig. 15. Resonant frequencies of three designs for torsional 2-DOF VibratoryGyroscopes while �z �=0 and Qs = 1000, Qd = 1000. Each curve representstwo resonances; one for drive, the other for sense.

drive and sense resonant frequencies are shown in Table IIfor undamped case.

After applying a rotation on to the system it is clearlyseen that the resonant frequency of one of the modes with anapplied angular velocity is linearly increasing and the otherone is decreasing for mode matched design. And the effect ofthe frequency shift diminishes as the difference between reso-nant frequencies of drive and sense mode increases in Fig. 15.

We also investigated the effect of Q differences in the threedesigns. Fig. 16 shows the frequency shift when the drive andsense Q values are different. The biggest impact is for themode matched case. For the mode matched case, there is nofrequency shift until the input rate is increased to 11 rad/sec.

As we demonstrated before for linear 2-DOF vibratorygyroscopes, the resonant frequency shift should also affect thelinearity of the torsional 2-DOF vibratory gyroscopes. In tor-sional systems, the fundamental resonant operation principleis same as linear systems so amplitudes of drive mode (φd0)in Fig. 17 and sense mode (φs0) in Fig. 18 can be calculatedfrom the state-space model for this structure given in (2).

τ0 of the system is arranged to excite its drive mode by10−6rad for all Q values and it is illustrated that the linearityof Design 1 is affected by different Q. For Fig. 19, we againassumed that the feedback system tracks the resonance of driveloop and keeps the drive amplitude constant.

We also calculated the non-linearity as a function of qualityfactor as shown in Fig. 20 for Design 1. The full-scale rangeis 20 rad/s. Above Q of 600, the non-linearity increasesabove 1%.

Fig. 16. Resonant frequencies of three designs for torsional 2-DOF VibratoryGyroscopes while �z �=0 and Qs = 500, Qd = 1000. Each curve representstwo resonances; one for drive, the other for sense.

Fig. 17. Amplitude of the drive mode of Design 1 where Qs = Qd = 500.

Fig. 18. Amplitude of the sense mode of Design 1 where Qs = Qd = 500.

C. Multi-DOF Vibratory Gyroscopes

There are various multi-DOF vibratory gyroscopes in theliterature. Examples include 3-DOF systems with 2-DOF


Fig. 19. Linearity of the torsional 2-DOF Vibratory Gyroscope for Design 1where Q = Qs = Qd .

Fig. 20. Nonlinearity of different Q systems for Design 1 whereQ = Qs = Qd .

sense-mode and 1-DOF drive mode architecture [17], 4-DOFsystem with 2-DOF drive and 2-DOF sense-modes [18]. In thispaper we examined 3-DOF vibratory gyroscope with 2-DOFsense-mode architecture as shown in Fig. 21. This multi-DOFgyroscope system has a 2-DOF sense-mode oscillator and a1-DOF drive-mode oscillator. Two interconnected proofmasses are attached to the decoupling frame through m1.The proof masses, m1 and m2 oscillate together in the drive-direction and they form 1-DOF oscillator in the drive modealong with the spring k1x . The sense system is more complex.The proof masses move in the senses direction and for the2-DOF oscillator along with k1y and k2y . According to [7],the equation of motion is shown in (15).

mT x1 + c1x x1 + k1x x1 = 2m2�z y2 + 2m1�z y1 + mT �2z x1

m1 y1 + c1y y1 + k1y y1 = k2y(y2 − y1)+m1�2z y1−2m1�z x1

m2 y2 + c2y y2 + k2y y2 = k2y y1+m2�2z y2 − 2m2�z x1 (15)

Fig. 21. Mass-spring-damper system of the 3-DOF Vibratory Gyroscopewith 2-DOF sense-mode architecture.

Equations of motion of the gyroscope given in (15) can bewritten in the form of state-space model given in (2):

X =

⎡⎢⎢⎢⎢⎢⎢⎣

x1y1y2x1y1z2

⎤⎥⎥⎥⎥⎥⎥⎦

, B =

⎡⎢⎢⎢⎢⎢⎢⎣

000F0mT

00

⎤⎥⎥⎥⎥⎥⎥⎦

, C =[

1 0 0 0 0 00 0 1 0 0 0

](16)

A

=

⎡⎢⎢⎢⎢⎢⎢⎢⎣

0 0 0 1 0 00 0 0 0 1 00 0 0 0 0 1

− k1xmT

+�2z 0 0 − c1x

mT2 m1

mT�z 2 m2

mT�z

0 − (k1y+k2y )m1

+�2z

k2ym1

−2�z − c1ym1

0

0 k2ym2

− k2ym2

+�2z −2�z 0 − c2y

m2

⎤⎥⎥⎥⎥⎥⎥⎥⎦

(17)

According to (17), damping part of the A matrix consistsof Coriolis accelerations and the damping coefficients of thesystem. The spring part of the A matrix consists of centripetalacceleration and the spring constants of the system. Mass partof the A matrix consists of proof masses mT , m1 and m2where mT is the total mass of the two proof masses.

We used the same example in [7] and for this bulkmicro-machined prototype, spring constants, k1x , k1y and k2y

are 61.2, 78.4 and 3.36 N/m respectively. The proof masses,m1, m2 and mT are 2.46x10−6 kg, 1.54x10−7 and 1.19x10−7.Their drive and sense resonant frequencies are calculated bythe Eigen values of A matrix given in (17).

There are 3 complex conjugate pairs of Eigen valuesand the imaginary parts indicate the resonant frequencies.Fig. 22 shows the resonant frequencies for the example designdescribed above.

With the applied input rate, the resonant frequencies shift.In this example, we did not observe frequency splitting for thedrive mode, and the drive mode resonance increases as shownin Fig. 22. The two sense modes on the other hand move awayfrom each other with the increased input rate.


Fig. 22. Resonances of sense and drive modes of the multi-DOF VibratoryGyroscope while �z �=0 and Qs = Qd = 1000. Each curve represents threeresonances; one for drive, the others for sense.

Fig. 23. Amplitude of the drive mode where Qs = Qd = 1000.

Fig. 24. Amplitude of the sense mode where Qs = Qd = 1000.

Amplitudes of drive mode (x0) (Fig. 23) and sensemode (y02) (Fig. 24) can be calculated from the state-spacemodel given in (16) and (17). The frequency shift affects the

Fig. 25. Linearity of the 3-DOF Vibratory Gyroscope whereQs = Qd = 1000.

Fig. 26. The HARPSS Vibrating Ring Gyroscope.

linearity of the multi-DOF vibratory gyroscopes as in the caseof linear 2-DOF vibratory gyroscopes.

To assess the linearity of the gyro system, we again assumedthat F0 of the system is adjusted to excite the drive modeby 1 μm with an automatic gain control feedback system.Then we calculated the sense response as shown in Fig. 25.The calculated non-linearity is only 0.0145% for the full-scalerange of 20 rad/s. In the 2-DOF gyro systems part, it was notedthat as the sense and drive resonances are placed away fromeach other, the effect of the frequency shift diminishes. Sincethis is the case for multi-DOF gyroscopes, we observed verylittle non-linearity. For multi-DOF gyros, the non-linearity isvery small compared to that of 2-DOF gyro systems, and basedon our calculations, the Q of the drive and sense resonanceshave minimal effect of the non-linearity.

D. Ring Vibratory GyroscopesOne of the examples of microfabricated gyroscopes is the

vibrating ring gyroscope which consists of an elastic ringsymmetrically supported by the eight identical semicircularsprings [6] as shown in Fig. 26. The gyroscope was fabricated


Fig. 27. Finite Element Analysis of the HARPSS Ring Gyroscope (a) Primaryflexural mode (b) Secondary flexural mode.

using HARPSS fabrication process [19]. This process allowsfabrication of high aspect ratio structures made of poly-silicon.(Young’s modulus, E = 150 GPa, density, ρ = 2.328 g/cm).The outer ring diameter is 1.1 mm and the ring width is 4 μm.The thickness of the structure is 80-μm. The support diameteris 120 μm and it is anchored to the substrate. The supportingsprings are in the shape of arcs which diameter is 470 μm.In this design, the ring is excited at the primary flexural modeat a fixed amplitude electrostatically and the secondary flexuralmode is formed at 45°. Coriolis force causes the energytransfer from the primary mode to the secondary flexural modewhen the rotation is applied and this amplifies the latter modewhich can be detected capacitively.

For the structure described above, we performed modalanalysis using ANSYS Simulation Tool. BEAM 188 elementswere used and rotating reference frame was defined withCoriolis option on. The geometry was clamped to supportthe post at the center. The first vibrational mode is torsionalat about 10 kHz; next two modes are translational at about20 kHz. The fourth and fifth modes are the ones that are used

Fig. 28. ANSYS and state-space model results for resonances of sense anddrive modes of the HARPSS Vibrating Ring Gyroscope while �z �=0 andQs = Qd = 5000. Each curve represents two resonances; one for drive, theother for sense.

in the gyroscope. These were flexure modes at 28.42 kHz.Mode shapes for the first two flexural modes of this geometryare shown in Fig 27. The mode shape of the second mode is45° degree rotated version of the first mode.

This ring architecture is similar to 2-DOF linear system.However, sense mode of the ring gyroscope is 45°rotatedinstead of 90° from its drive mode compared to other lineargyros. Therefore one can use 2-DOF gyro equations for ringgyroscopes. The A matrix is given by:

A =

⎡⎢⎢⎣

0 0 1 00 0 0 1

−kxm + �2

z 0 −cxm −2�z sin(45°)

0 −kym + �2

z 2�z sin(45°) −cym

⎤⎥⎥⎦

(18)

To obtain spring constant, mass and the damping coefficientwe used harmonic analysis tool of ANSYS. First, we excitedthe primary flexural mode of the vibrating ring gyroscopeby applying outward forces at its top and bottom nodes.The spring constant (k = kx = ky) of the system can becalculated from the amplitude of the drive mode (x0) at lowfrequencies and the applied force (F). The modal mass (m)at 28.42 kHz can be calculated by using the spring constant(ω0 = √

k/m). Damping coefficient (c = cx = cy) is appliedby the user. We compared the eigen value calculations ofA matrix against ANSYS modal analysis results. We got a verygood match between ANSYS and Eigen value calculations asshown in Fig. 28. The frequency of the first flexural modesis 28.42 kHz. As the input rate increases, there appear tworesonances. One of the resonant frequency increases linearly,whereas the second resonant frequency decreases as the inputrate increases.

In this system, the fundamental resonant operation principleis same as linear systems so the amplitude of drive mode canbe calculated by harmonic analysis with ANSYS SimulationTool. We excited the primary flexural mode of the vibratingring gyroscope by applying outward forces, 10−8 N at its


Fig. 29. Amplitude of the drive mode for the HARPSS Vibrating RingGyroscope where Qs = Qd = 5000. (damping coefficient is 10−4N.sec/m.)

Fig. 30. Amplitude of the sense mode for the HARPSS Vibrating RingGyroscope where Qs = Qd = 5000. (damping coefficient is 10−4N.sec/m.)

top and bottom nodes to get 1 μm radial motion and thedamping ratio is 10−4. The spring constant (k = kx = ky)of the system can be calculated by the amplitude of drivemode (x0) at low frequencies and applied force (F). The modalmass (m) can be also calculated at 28.42 kHz by the calculatedspring constant (k). So one can also acquire the dampingcoefficient (c = cx = cy) of this system.

Fig. 29 and 30 show drive mode amplitude (x0) andsense mode amplitude (y0) respectively, calculated using equa-tion (2). In Fig. 29, it is assumed that the drive system is drivenby a constant force to get 1 μ m peak vibration amplitude atthe resonance when there is no input rate. As the input rateincreases the amplitude drops and frequency shifting occursin consistent with Fig. 28. For the sense system, when thereis no input rate, there is no sense mode vibration. As theinput rate increases the sense system starts vibrating due tothe coupling of drive system through Coriolis and centrifugalforce. As a result the amplitude increases and also frequencyshifting occurs as shown in the figure.

Fig. 31. Linearity of the HARPSS Vibrating Ring Gyroscope whereQs = Qd = 5000

F0 of the system is arranged to excite its drive modeby 1 μm with an automatic gain control feedback system andit is illustrated that the linearity of this gyro structure is shownin Fig. 31. According to Fig. 31 non-linearity is calculated as15.3 % since the full-scale range is taken as 20 rad/s.

III. CONCLUSION

In this paper we presented a thorough analysis of gyroscopesystems including the coupling between the drive and sensesystems due to Coriolis force and centrifugal force. Ouranalysis showed that the main coupling mechanism betweenthe drive and sense modes is through the Coriolis force and theeffect of centrifugal force can be neglected for MEMS vibra-tory gyroscopes. The Coriolis force changes the damping ofthe system affecting the resonant frequencies. The analysis wasapplied to both mode matched and multi-DOF gyro systems.The most important finding of the analysis is the fact that theresonance frequencies of the drive and sense modes shift withthe input rotation. This affects the linearity of the gyroscopesystems by setting a fundamental limit for the linearity. Forgyros with high input range lowering the Q may result in morelinear response sacrificing the sensitivity for mode matchedgyros. For non-mode matched gyros the non-linearity due tofrequency shifting is minimal. On the other hand, all gyrosshould be prone to quality factor widening at high inputrates. The frequency splitting observed in the drive modemay result in reduced stability of the drive mode oscillators.Double peaking may result in sudden frequency jumps at largeinput rates for the drive loop oscillator increasing the outputnoise. We do not have an immediate solution for this problem.Lowering Q may solve the problem but the overall stabilityof the oscillator will be sacrificed in this case. More detailedanalysis has to be performed to address this issue.

APPENDIX

SIMPLE HARMONIC OSCILLATOR

Fig. 32 shows a simple harmonic oscillator. We assumedthe rotation axis coincides with the proof mass. NeglectingCoriolis force, the equation of motion of the linear harmonic


Fig. 32. Mass-spring-damper system of the linear harmonic oscillator.

Fig. 33. Resonant frequencies of the linear harmonic oscillator while �z �=0.

oscillator is given in (A1) with a proof-mass of m, a springconstant of kx , and a damping coefficient of cx .

mx + cx x + (kx − m�2z )x = 0 (A1)

Resonant frequencies of the system is obtained from (A1)while �z �= 0.

ωS H O =√

(kx

m− �2

z ) − c2x

4m2 (A2)

For a bulk micro-machined prototype oscillator the springconstant is 40 N/m and its proof mass is 1x10−6 kg. It isclearly seen that the resonant frequency of simple harmonicoscillator decreases as shown in Fig. 33 due to centrifugalforce since rotation is applied to the system in Fig. 32.

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Hakan Cetin (S’16) received the B.Sc. and M.Sc.degrees in electrical engineering from ÖzyeginUniversity, Istanbul, Turkey, in 2014 and 2016,respectively, where he is currently pursuing thePh.D. degree in electrical engineering. His researchinterest includes MEMS sensors and inertial sensors.

Goksen G. Yaralioglu (S’92–M’99) received theB.Sc., M.Sc., and Ph.D. degrees from BilkentUniversity, Ankara, Turkey, in 1992, 1994, and 1999,respectively, all in electrical engineering. He is cur-rently an Associate Professor with the Electricaland Electronics Engineering Department, ÖzyeginUniversity, Istanbul, Turkey. His current researchinterests include design, modeling and applicationsof micromachined ultrasonic transducers, microflu-idic channels, MEMS sensors, and inertial sensors.

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Analysis of Vibratory Gyroscopes: Drive and Sense Mode