A Model-Based Approach to Multi-Modal Mass Tuning of a Micro ...

A Model-Based Approach to Multi-Modal Mass Tuningof a Micro-Scale ResonatorDavid Schwartz ∗ , Dennis Kim † , and Robert M’Closkey ‡Mechanical and Aerospace Engineering DepartmentUniversity of California, Los AngelesAbstract— The signal-to-noise ratio of axisymmetric vibratorygyroscopes is maximized when a pair of coriolis-coupledmodes resonate at the same frequency. The manufacturingprocess of micro-scale resonators creates random minute massand stiffness asymmetries that cause the natural frequencies ofthese modes to deviate from one another, thereby degradingsensor performance. One method of “tuning” these modalfrequencies to equality involves using electrostatic forces toselectively soften the stiffness at points in the resonant structure.This generally requires large volume electronics thatare incompatible with application requirements common tothese sensors. Alternatively, modal frequency tuning by massperturbation of the resonator is a promising approach becauseit is permanent and requires no ancillary electronics. In thispaper, a novel micro-scale resonator is presented which lendsitself to mass perturbation experiments. A resonator model,based on empirical frequency response data, is used to guidethe mass perturbation process and demonstrates how multiplepairs of modes can be tuned.I. INTRODUCTIONIn recent years there has been a resurgence of interest indeveloping compact inertial sensors for use in GPS-deniedenvironments. Before this can become a reality, though, advancementsmust first be made in the size and accuracy of thevibratory gyroscopes with axisymmetric resonators that areproposed for use in such systems. In theory, the symmetricnature of the resonator design enables high performancecharacteristics due to the natural modal degeneracy intrinsicto axisymmetric designs. Unfortunately, the manufacturingprocess invariably produces random, minute asymmetries inthe resonant structure. These asymmetries cause the pairsof modes that are used by the gyroscope to sense angularmotion to resonate at slightly different frequencies which inturn reduces the signal-to-noise ration (Fig.1) [5].A pair of modes can be “tuned” to operate at the samefrequency either by using electrostatic forces or selectivemass loading. Electrostatic tuning involves applying staticvoltages to electrodes to change the effective modal stiffness.This process can be effective, but the electronics required tomaintain the necessary voltage stability are generally largerthan the overall size requirements of the sensor [6]. Thus,mass tuning, in which a pair of modes is tuned by altering the* Email: daves@ucla.edu† Email: dongj@seas.ucla.edu‡ Corresponding author. Email: rtm@seas.ucla.edu. This work is supportedby DARPA contract P-1035-112802.G ain Magnitude (dB )G ain Magnitude (dB )3020100-10-2013.4 13.41 13.42 13.43F requency (kHz)3020100-10-2013.4 13.41 13.42 13.43F requency (kHz)S 3/D 3 S 3/D 4G ain Magnitude (dB )G ain Magnitude (dB )3020100-10-2013.44 13.4 13.41 13.42 13.43F requency (kHz)S 4/D 3 S 4/D 413.443020100-10-2013.4 13.41 13.42 13.43F requency (kHz)13.4413.44Fig. 1. The 2×2 empirical frequency response of the n=2 pair of modes ofthe UCLA Resonator (URES). If the resonator were fabricated with perfectsymmetry, these modes would resonate at the same frequency and only asingle peak would be observable in each channel. In the case presentedhere, minute asymmetries have caused the modes to exhibit a significantfrequency split. A similar split occurs in the n=3 modal pair as presentedFig. 7.mass distribution of the resonator, appears to be a necessaryendeavor.On its face, mass tuning can be a straightforward process.Fig. 2 displays a potential mass tuning scenario for thefundamental Corilolis-coupled n=2 modes of simple ringgeometry. The n=2 modes have an elliptical shape and arethe ones most commonly used for rate sensing. With noasymmetry, the ring can vibrate with an n=2 mode shapeat any angular orientation with the same frequency. If amass asymmetry exists, two modes clearly define themselveswhose frequencies are no longer matched [1], [8]. The devicecan be tuned by placing additional mass on the anti-node ofthe mode with a higher frequency.Mass tuning has been successfully implemented on largerdevices. The Hemispherical Resonator Gyroscope (HRG) [7],most notably used on the Hubble Space Telescope for instrumentstabilization, has been a benchmark for performance,though its 3 cm diameter is too large for it to be considered amicro-scale device. The HRG is tuned by perturbing its mass

Lastly, one can calculate σ cos and σ sin using the frequencysplit and the location of the high frequency anti-nodes:25S 1/D 1D. Tuning Methodsσ cos = (ω n2−ω n1 )cos(2nΨ n2 )γ 2σ sin = (ω n2−ω n1 )sin(2nΨ n2 )γ 2.(13)A single modal frequency can be modified by finding acombination of perturbations of mass m p placed at locationsθ p that solveσ cosn(0) + ∑ Nmp=1 m p cos(2nθ p ) = 0σ sinn(0) + ∑ Nmp=1 m p sin(2nθ p ) = 0(14)where σ cosn(0) and σ sinn(0) are defined by the mass asymmetrystate before any perturbations are made. This set ofequations can be solved with any number of perturbations.A single mass solution is attained with m 1 =γ 1 2√σcos 2 + σsin2and θ 1 = Ψ n2 . In other words, a specific quantity of mass isplaced precisely at the location of the anti-node of the highfrequency mode.If there are constraints on where the mass perturbationscan physically take place on the resonator, as is the casewith the URES device, then two masses are required fortuning. There are multiple two mass solutions, and onesimple method for finding a solution is to pick two locations,denoted θ 1 and θ 2 , where the perturbations are to occur anduse[ ]m1=m 2[ ] −1 [ ]cos(2nθ1 ) cos(2nφ 2 ) σcosn(0)sin(2nφ 1 ) sin(2nθ 2 ) σ sinn(0)(15)to compute m 1 and m 2 . Generally, there are two constraintson the choices of θ 1 and θ 2 . First, they must be chosen sothat the inverted matrix is nonsingular, or, equivalentlymod (θ 1 − θ 2 , π 2n ) ≠ 0.Also, the solutions m 1 and m 2 must be positive (or negativein the case mass reduction). This is guaranteed ifθ 1 ≤ Ψ n2 ≤ θ 2 , 2nθ 2 − 2nθ 1 < π,which is interpreted as choosing θ 1 and θ 2 to be on eitherside of the high frequency anti-node. The closer the twoperturbation locations actually are to the higher frequencyanti-node, the smaller the amount of mass will be requiredfor tuning.It is also possible to extend this method to manipulatingthe frequencies of multiple modal pairs. In our case, we areinterested in tuning the n=2 and n=3 modes:σ cos2(0) + ∑ Nmp=1 m p cos(4θ p ) = 0σ cos2(0) + ∑ Nmp=1 m p sin(4θ p ) = 0σ cos3(0) + ∑ Nmp=1 m p cos(6θ p ) = 0σ cos3(0) + ∑ Nmp=1 m p cos(6θ p ) = 0(16)In general, this system of equations can be solved with twoor more masses [9].G ain Magnitude (dB )20151050-5-10-1523.56 23.565 23.57 23.575 23.58F requency (kHz)Fig. 6. Example of the model fitting algorithm. The red points display themeasured frequency response magnitude, while the trace is generated usingthe frequency domain model using 2×2 positive definite mass, stiffness anddamping matrices.A. Validation of ModelsIV. EXPERIMENTAL RESULTSAn example of the model fit generated by using the methoddiscussed in Section III.A is shown in Fig. 6. Only themagnitude of the transfer function and its corresponding fitare shown. Still, the model for M,C,K, and R is a close fitfor both magnitude and phase in all four channels and ismore than 95% accurate as quantified by the H 2 norm of theresidual.The validation of the modelling process as well as estimatesof γ 22 and γ 32 are done experimentally on a devicewith results shown in Table I. If a single mass perturbationis made to a measured system, the imbalance parametersσ cos(p) and σ sin(p) can be calculated usingσ cosn(p) = σ cosn(p−1) + γ n2 m p cos(2nθ p )σ sinn(p) = σ sinn(p−1) + γ n2 m p sin(2nθ p ).(17)In order to validate the model, data is taken before and aftera mass perturbation of known mass is made. The sensitivityγ n2 and an estimate of the location of the mass perturbationare then calculated using√¯γ n2 =m 1 (σcosn(p) p− σ cosn(p−1) ) 2 + (σ σsinn(p) −σ sinn(p−1) )( 2¯θ np =2n 1 σsinn(a))tan−1 −σ sinn(p−1)σ cosn(p) −σ cosn(p−1)(18)where the bar represents an estimate being made. Table Ishows that the sensitivity of the mass perturbation is actuallyextremely variable from test to test. This can be explainedby the large variability in the amount of mass being addedas a consequence of the physical process that is employedto deposit the mass. The perturbation model does accuratelyestimate the angle at which the mass was placed. This meansthat it also accurately estimates the location of the highfrequency mode and will be a useful aid in tuning.

FrequencyModelingDataset w 2 -w 1 Y 2 (degrees)(Hz)0 3.1 35.51 1.8 7.92 7.0 1.0TABLE IPERTURBATION MODEL VALIDATIONn=3 Data (Same device)Perturbation InformationPerturbation ModelEstimatesPerturbation # # of masses location g 22 q 2p(degrees)(degrees)n=2 data1 1 -30 8.0 -30.522 - 67.5 4.7-67.8g 32 q 3pTABLE IIMASS TUNING EXAMPLEn=2 Data n=3 Data DepositionData Frequency Split Angle Estimate Frequency Split Angle Estimate Locationset (Hz) (degrees) (Hz) (Degrees) (Degrees)1 14.1 36.2 8.2 23.12 12.3 39.6 7.0 34.2 22.53 8.5 48.1 4.6 39.6 37.54 5.1 39.5 2.9 28.5 37.55 1.7 39.2 1.2 47.5 -37.5, 306 0.46 5.7 5.7 38.0 -30, 22.57 0.07 84.7 6.1 53.7 fine tuning w/ink0 6.5 31.21 12.2 222 2.8 201 1 -30 6.7 -30.32 2 -67.5 3.7 -67.420After coarse tuning (1.2 Hz split)10B efore tuning (8.2 Hz split)B. Tuning ExampleDue to the lack of a consistent mass quantum that canbe placed on the resonator, tuning a modal pair to degeneracyis difficult. Despite this challenge, in this sectionwe successfully demonstrate the model driven approach totuning, starting with an unperturbed URES resonator. Thefollowing example uses several different tuning strategies atdifferent stages to reach a final, modally tuned, state. Atfirst, masses are placed with the goal of tuning both modes.As this becomes more difficult the strategy is shifted towardachieving a tuned state for one mode.The experimental results are displayed in Table II. Eachrow of table gives the modeled parameters of the n = 2 andn = 3 models, and the location(s) of the perturbation(s) madebefore the data was taken. In the first stage, masses are placedin order to reduce the split in the n = 3 mode as much aspossible without reducing the split of the n = 2 modes. Sincethe mass of each perturbation is so variable, the perturbationsare made one at a time for the first 3 perturbations. Thesplits of both modes reduce each time, again confirmingthat the model based predictions are useful. After the fourthperturbation, the split is 1.2 Hz for the n = 3 modes and 1.7Hz for the n = 2 modes. At this point (Dataset 5), the goldball mass perturbation is too large to reduce either split withonly one mass, i.e. a single gold ball will cause “overshoot”of the desired frequencies, so any subsequent perturbationswill only address reducing the n = 2 modal split and forthis case, a two-location solution is found that reduces thesplit of the n = 2 modes to 0.46 Hz (Dataset 6). AfterDataset 6, further reduction in the n = 2 modal frequencysplit is not possible with gold ball mass perturbations soa different mechanism employing silver ink deposition isused to accomplish fine tuning. The mass quantum associatedwith the ink is several orders of magnitude smaller than thegold balls. This process took several iterations which havebeen removed for brevity but the resonator’s n = 2 modeswere tuned to less than a 0.1 Hz split (see Fig. 8) after theconclusion of the ink deposition (Dataset 7).G ain Magnitude (dB )G ain Magnitude (dB )3020100-100-10-20S 1/D 1S 2/D 2-3023.03 23.035 23.04 23.045 23.05F requency (kHz)Fig. 7.Before and after tuning of the n = 3 modesAfter fine tuning (

V. CONCLUSIONA model based approach has been developed and usedto guide the mass loading of a micro-resonator with theobjective of reducing the modal frequency split between twonominally degenerate modes. Of particular interest are then = 2 and n = 3 modal pairs since these are the modesthat can be exploited for angular rotation rate sensing inaxisymmetric vibratory gyros. The algorithm only requiresknowledge of the mode shapes in order to estimate thelocation of the anti-nodes from models fit to empiricalfrequency response data. The biggest challenge in this studywas not performing the measurements or implementing thealgorithm, but rather the process for effecting the mass perturbationis not sufficiently developed to produce a consistentmass “quanta” across all experiments. This is especiallytrue of the gold ball deposition which had the advantageof creating a large change in the modal properties of theresonator. Ultimately, a hybrid approach was used in whichsilver ink deposition was employed to further reduce themodal frequency split. The impact of mass perturbation onthe modal quality factors is also of interest since higherquality factors produce higher signal-to-noise ratios in tunedvibratory gyros. Fortunately, the current data suggest that thequality factors are not changed by these mass perturbations.REFERENCES[1] Allaei, D., Soedel, W., and Yang, T.Y., “Natural frequencies of ringsthat depart from perfect axial symmetry.” Journal of Sound andVibration, Vol. 111, pp. 9-27, 1986.[2] Charnley, T., and Perrin, R., “Perturbation studies with a thin circularring.” Acustica, Vol. 28, pp. 140-146, 1973.[3] Fell, C.P., “Method for matching vibration mode frequencies on avibrating structure” US Pat. 5739410, 1996.[4] Gallacher, B. J., “Multi-modal tuning of a ring gyroscope using laserablation” Proc. Inst. Mech. Eng. C.,Vol. 217, pp. 557-76, 2000.[5] Kim, D., and M’Closkey, R. T., “Dissecting tuned MEMS vibratorygyros,” book chapter in Feedback Control of MEMS to Atoms. Gorman,Jason J.; Shapiro, Benjamin (ed.), October 2011, Springer.[6] Kim, D-J., and M’Closkey, R.T., “A systematic method for tuningthe dynamics of electrostatically actuated vibratory gyros,”IEEE Trans. Control System Technology, Vol. 14, No. 1, pp. 69–81,2006.[7] Lynch, D.D. “Coriolis Vibratory Gyros,” Symposium Gyro Technology,Stuttgart Germany, September 1998.[8] Rourke, A.K., McWilliam S., and Fox, C.H.J. , “Multi-mode trimmingof imperfect rings,” Journal of Sound and Vibration, Vol. 248, No. 4,pp. 695724, 2001[9] Schwartz, D., “Mass Perturbation Techniques for Tuning and Decouplingof a Disk Resonator Gyroscope.” PhD thesis. University ofCalifornia, Los Angeles, 2010.[10] Schwartz, D., Kim, D.J., and M’Closkey, R.T., “Frequency Tuning of aDisk Resonator Gyro Via Mass Matrix Perturbation”, ASME Journalof Dynamic Systems, Measurement, and Controls, Vol. 131, No. 6,p. 061004 , 2009.[11] Zhbanov, Yu. K., and Zhuravlev, V. F., “On the Balancing of aHemispherical Resonator Gyro,” Mech. Solids, Vol. 33, No. 4, pp. 2–13, 1998.