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Vibration mode localization in coupled microelectromechanical resonators Manav 2014

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VIBRATION MODE LOCALIZATION IN COUPLEDMICROELECTROMECHANICAL RESONATORSbyManavB. Tech., Indian Institute of Technology Kanpur, 2009A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2014c©Manav, 2014AbstractState-of-the-art resonant sensors rely on shift in resonant frequency due to a changein its mass or stiffness caused by a physical quantity to be measured. However,they require low damping operating environment. As a result, applications suchas biomolecular detection in aqueous environment pose formidable challenges. Apromising, alternative sensing paradigm, minimally affected by damping, is basedon normal mode localization in a weakly coupled, symmetric resonator systemdue to parametric changes. The higher sensitivity of mode shape compared toresonant frequency in a weakly coupled, symmetric resonator system results fromthe phenomena of eigenvalue veering and associated mode localization induced bysymmetry breaking parametric changes in the system. The method offers addedbenefit of common mode rejection.This thesis critically examines the mode localization based resonant sensingparadigm using a combination of energy based analytical theory, Simulink mod-els, and experimental studies on planar MEMS devices. Built-in asymmetry infabricated devices and its influence on achievable sensitivity are highlighted. In-creasing the number of degrees of freedom (DOF) is shown to enhance sensitivity,but a trade-off exists with the size and complexity of the device. Similarly, decreas-ing coupling enhances sensitivity at the expense of measurable range of parametricchanges. Two and three DOF coupled resonator MEMS devices with tuneable lin-ear coupling were designed, fabricated and tested to verify the above conclusions.In summary, this thesis demonstrates that mode localization based sensing isorders of magnitude more sensitive compared to resonant frequency shifts. Thesensitivity can be further increased by decreasing coupling between resonators, orincreasing number of DOF in a resonant MEMS device, or both.iiPrefaceParts of this thesis have appeared in a conference proceedings and a peer reviewedjournal. Parts of chapters 3 and 4 have been published as M Manav, G Reynen, MSharma, E Cretu and A S Phani, “Ultra sensitive resonant MEMS transducers withtuneable coupling,” The 17th International Conference on Solid-State Sensors, Ac-tuators and Microsystems, Barcelona, Spain, 2013 [1]. Some parts of chapters 3and 4 have been published as M Manav, G Reynen, M Sharma, E Cretu and A SPhani, “Ultra sensitive resonant MEMS transducers with tuneable coupling,” Jour-nal of Micromechanics and Microengineering, 2014 [2]. In the work presentedin the papers, I developed the energy-based framework to analyze mode localiza-tion as well as analytical and Simulink model of the MEMS device with guidancefrom Dr. Srikanth A. Phani. Also, I carried out experiments with help from Mri-gank Sharma on devices designed by Greg P. Reynen, under the guidance of Dr.Edmond Cretu.The contributions of the thesis are:• Chapter 2 develops an energy based framework to analyze mode localization.• In chapter 3, an analytical model to quantify electrostatic stiffness pertur-bation in a MEMS coupled resonator device will be developed using Cas-tigliano’s theorem. Moreover, a new design of a MEMS coupled resonatordevice with shaped combs to achieve linearity of electrostatic coupling springwill be presented.• In Chapter 4, asymmetry built in a MEMS device will be quantified andhigher sensitivity of a three degree of freedom (DOF) coupled resonatorMEMS device as compared to a two-DOF device will be demonstrated.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Resonant sensing . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Eigenvalue veering and mode localization . . . . . . . . . . . . . 31.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Analysis of normal mode localization in coupled resonators . . . . . 72.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Normal modes of N-DOF coupled resonators . . . . . . . . . . . 82.3 Normal modes of two-DOF coupled resonators . . . . . . . . . . 112.3.1 Sensitivity of natural frequency and normal mode . . . . . 132.4 Normal modes of three-DOF coupled resonators . . . . . . . . . . 152.4.1 Perturbation in the end resonator . . . . . . . . . . . . . . 182.4.2 Differential perturbation . . . . . . . . . . . . . . . . . . 20iv2.5 Implications for device design . . . . . . . . . . . . . . . . . . . 222.6 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . 243 MEMS device design and modeling . . . . . . . . . . . . . . . . . . 253.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2 Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.3 Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Device-1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.4.1 Device description . . . . . . . . . . . . . . . . . . . . . 293.4.2 Device models . . . . . . . . . . . . . . . . . . . . . . . 293.5 Device-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.5.1 Device description . . . . . . . . . . . . . . . . . . . . . 393.5.2 Design and fabrication of the device . . . . . . . . . . . . 403.5.3 Device models . . . . . . . . . . . . . . . . . . . . . . . 433.6 Three-DOF MEMS resonator device . . . . . . . . . . . . . . . . 513.6.1 Device description . . . . . . . . . . . . . . . . . . . . . 513.6.2 Design and fabrication of the device . . . . . . . . . . . . 533.6.3 Device models . . . . . . . . . . . . . . . . . . . . . . . 533.7 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . 564 Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.2 Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . . . 584.3 Experimental procedure . . . . . . . . . . . . . . . . . . . . . . . 584.4 Characterization of Device-1 . . . . . . . . . . . . . . . . . . . . 594.4.1 Results and discussion . . . . . . . . . . . . . . . . . . . 604.5 Characterization of Device-2 . . . . . . . . . . . . . . . . . . . . 634.5.1 Results and discussion . . . . . . . . . . . . . . . . . . . 704.6 Characterization of Device-3 . . . . . . . . . . . . . . . . . . . . 744.6.1 Results and discussion . . . . . . . . . . . . . . . . . . . 774.7 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . 80v5 Conclusions and future work . . . . . . . . . . . . . . . . . . . . . . 815.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.2.1 Differential perturbation . . . . . . . . . . . . . . . . . . 825.2.2 Dynamic perturbation . . . . . . . . . . . . . . . . . . . 825.2.3 Higher DOF system . . . . . . . . . . . . . . . . . . . . 835.2.4 Perturbation mechanisms . . . . . . . . . . . . . . . . . . 835.2.5 Energy domains . . . . . . . . . . . . . . . . . . . . . . 83Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84A Roots of a polynomial with perturbed coefficients . . . . . . . . . . . 88viList of TablesTable 3.1 Device dimensions . . . . . . . . . . . . . . . . . . . . . . . . 30Table 4.1 Voltages used in the experiment . . . . . . . . . . . . . . . . . 61Table 4.2 Voltages used in the experiment . . . . . . . . . . . . . . . . . 70viiList of FiguresFigure 2.1 An n-DOF coupled, symmetric spring-mass system with nonuni-form stiffness perturbation . . . . . . . . . . . . . . . . . . . 8Figure 2.2 A two-DOF coupled, symmetric spring-mass system with stiff-ness perturbation . . . . . . . . . . . . . . . . . . . . . . . . 11Figure 2.3 Veering diagram for two-DOF coupled, symmetric spring-masssystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Figure 2.4 Comparison of sensitivity norms for two coupling values . . . 14Figure 2.5 Three-DOF coupled, symmetric spring-mass system with stiff-ness perturbation . . . . . . . . . . . . . . . . . . . . . . . . 15Figure 2.6 Comparison of sensitivity norms for a three-DOF coupled res-onator system with stiffness perturbation in the end resonatorfor two coupling values . . . . . . . . . . . . . . . . . . . . . 21Figure 2.7 Comparison of sensitivity norms for a three-DOF coupled res-onator system with differential stiffness perturbation for twocoupling values . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 2.8 Schematic showing decrease in measurable range of perturba-tion as sensitivity increases . . . . . . . . . . . . . . . . . . . 24Figure 3.1 Two-DOF coupled resonator MEMS device-1 . . . . . . . . . 28Figure 3.2 (a) Schematic of a fixed-guided beam (b)Schematic of the up-per half of the 2nd resonator . . . . . . . . . . . . . . . . . . 31Figure 3.3 Variation of stiffness perturbation with perturbation voltage . . 35Figure 3.4 Simulink block diagram for the device shown in Figure 3.1 . . 38Figure 3.5 Comparison of sensitivity norms for 18 V coupling voltage . . 39viiiFigure 3.6 An image of the second two-DOF device . . . . . . . . . . . 41Figure 3.7 ARCHITECT model of one resonator unit of the device . . . 42Figure 3.8 Frequency response of one resonator unit with unshaped combs 43Figure 3.9 Two-DOF device as obtained from CoventorWare DESIGN . 44Figure 3.10 Shaped comb profile . . . . . . . . . . . . . . . . . . . . . . 45Figure 3.11 Simulink block diagram for the device shown in Figure 3.6 . . 49Figure 3.12 Comparison of sensitivity norms for 40 V coupling voltage . . 51Figure 3.13 An image of the three-DOF device . . . . . . . . . . . . . . . 52Figure 3.14 Three-DOF device as obtained from CoventorWare DESIGN . 54Figure 3.15 Comparison of sensitivity norms for 40 V coupling voltage forthe three-DOF device . . . . . . . . . . . . . . . . . . . . . . 55Figure 4.1 A view of the experimental set-up . . . . . . . . . . . . . . . 58Figure 4.2 Electrical power connections for 18 V coupling in device-1 . . 60Figure 4.3 Typical frequency response data and associated curve fit . . . 61Figure 4.4 Comparison of sensitivity norms for device-1 . . . . . . . . . 62Figure 4.5 Shift in origin of sensitivity plot due to built-in asymmetry . . 64Figure 4.6 Frequency response of individual resonators of Device-2 . . . 65Figure 4.7 Variation of the natural frequency with excitation force amplitude 66Figure 4.8 Natural frequency vs square of perturbation voltage . . . . . . 66Figure 4.9 Natural frequency vs square of coupling voltage . . . . . . . . 67Figure 4.10 Electrical power connections for 40 V coupling in Device-2 . 68Figure 4.11 Typical amplitude-frequency response and associated curve fit 69Figure 4.12 Veering diagram for the two-DOF device with 40 V coupling . 71Figure 4.13 Veering diagram for the two-DOF device with 35 V coupling . 71Figure 4.14 Comparison of sensitivity norms for 40 V coupling in device-2 72Figure 4.15 Comparison of sensitivity norms for 35 V coupling in device-2 73Figure 4.16 Electrical power connections for 40 V coupling in device-3 . . 75Figure 4.17 Typical amplitude-frequency response and associated curve fit 76Figure 4.18 Veering diagram for the three-DOF device with 40 V coupling 77Figure 4.19 Comparison of sensitivity norms for 40 V coupling in device-3 78ixAcknowledgmentsFirst and foremost, I would like to thank my supervisor for believing in me andgiving me the opportunity to work with him, and investing his time and effort inguiding me to produce this work.My sincere thanks to my co-supervisor for his guidance specifically in theMEMS side of the work and for bringing in interesting ideas to pursue, some ofthem yet to be worked upon.I wish to also thank Natural Sciences and Engineering Research Council ofCanada (NSERC), Canada Foundation for Innovation (CFI), Canada Research Chairs,CMC Microsystems and ICICS, UBC for supporting this research.I would like to extend my gratitude to Mrigank Sharma for helping me acquaintwith the experimental set-up as well as MEMS device design software tools and forextending a helping hand whenever my foray into experimentation hit a roadblock.I would also like to thank Greg P. Reynen, whose devices I used to get my first setof experimental data.My thanks is also due to my lab colleagues for their help in my work, inter-esting discussions and for making the lab a congenial place. My thanks to myfriends for being inseparable part of my existence here and for creating a familiar,dependable world so far away from the settings I used to recognize as my world.Last but not the least, I would like to acknowledge my family members, mygrandparents, my parents and my siblings, for being the hidden stream that keepsme alive, afloat and inspired.xChapter 1Introduction1.1 Resonant sensingA resonant sensing device works by measuring the shift in resonant frequency ofits resonator induced by a physical quantity to be measured. The physical quan-tity brings about the shift by changing effective mass or effective stiffness of theresonator. Efforts have been invested to develop resonant sensors to measure mass[3–5], pressure [6], acceleration [7], rate of rotation [8] etc. High quality fac-tors (low damping) are essential for a measurable resonant frequency shift. Thisrequirement makes sensing very challenging in applications such as biomolecu-lar detection in aqueous environment. An alternative sensing paradigm, relativelyless affected by damping, is based on measurement of the change in mode shapeof a coupled resonator system instead of the change in resonant frequency of thesystem. Change in mode shape is produced by the phenomenon of mode localiza-tion through eigenvalue veering in a weakly coupled, symmetric resonator systemdue to symmetry breaking change in a system parameter. Mode localization basedsensing has much higher sensitivity as compared to resonant frequency shift basedsensing. It also possesses intrinsic common mode rejection property (paramet-ric changes common to all the resonators of the coupled system have no effect onmode shape). This property makes it a better choice for sensing under ambient con-ditions. The pioneering work in mode localized sensing utilized mode localizationin weakly coupled MEMS resonators to measure mass [9]. The coupled, symmetric1system in the sensor consisted of two identical cantilevers coupled by an overhangbetween them. The sensor was able to measure the mass of a microsphere (with amass of ∼154 pg) by quantifying the change in mode shape after the microsphereis placed at the tip of one of the cantilevers even though shift in resonant frequencywas within natural drift of resonance frequency for the cantilevers used. They reg-istered more than two orders of magnitude better sensitivity of mode shapes com-pared to resonant frequency under ambient conditions. Extending on this work, thesame group developed an array of fifteen cantilevers coupled through overhangsbetween them to measure mass of an analyte attached to one of the cantilever tips[10]. A further increase in sensitivity of mode shapes (three orders of magnitudebetter than resonant frequency) in a low pressure environment was demonstrated.Moreover, attaching a mass at different cantilevers of the array produces a uniquepattern of shift in mode shapes, opening the possibility of simultaneous detectionof mulitple analytes [10]. However, one of the drawbacks of such an scheme is thatit requires simultaneous measurement and processing of motion of all of the cou-pled cantilevers, leaving the optimal choice of number of resonators in a coupledresonator device as an open question.Coupling between resonators plays an important role in veering: smaller thecoupling, the more sensitive the mode shapes are. So, it is imperative to havesmaller coupling. Exploiting electrostatic means to couple mechanical resonatorsprovided the answer to the need of weak coupling [11]. Electrostatic coupling hasits origin in electrostatic attraction force between two parallel plates separated by asmall gap and with a constant potential difference between the plates. The electro-static attraction force between the plates for small displacement of the plates abouttheir equilibrium position can be approximated to be proportional to the relativedisplacement of the plates, just like the force exerted by a mechanical spring al-though with negative stiffness. Moreover, the stiffness is proportional to the squareof the potential difference between the plates. Hence it can be controlled by con-trolling the potential difference between the plates, making coupling stiffness tun-able too. Utilizing electrostatic coupling, three orders of magnitude higher sensitiv-ity of mode shapes compared to resonant frequency sensitivity in electrostaticallycoupled double ended tuning fork (DETF) resonators was shown [11]. The samegroup also developed a mode localized mass sensor and demonstrated three orders2better sensitivity of mode shapes to a mass perturbation compared to that of res-onant frequency, using electrostatically coupled ring resonators [12]. They alsodemonstrated 400% tuning of eigenmode sensitivity by controlling electrostaticcoupling.The same group also devised a mode localized displacement sensor [13]. Thesensor consisted of two electrostatically coupled DETF resonators. Displacementof a large suspended proof mass is measured by transducing the displacement asa stiffness perturbation in one of the resonators. The stiffness perturbation is in-troduced by a change in gap between the proof mass and a plate at a differentelectrostatic potential attached to the resonator to be perturbed.Most of the studies on mode localized devices described above were carried outin vacuum even though relatively weaker dependence of mode shape sensitivity ondamping is one of the merits of mode localized sensing. Plate resonators withelectrostatic coupling and stiffness perturbation using electrostatic means were de-veloped to study localization in ambient conditions [14]. Use of three-DOF deviceswith differential stiffness perturbation scheme to obtain high mode shape sensitiv-ity with insensitive resonant frequency was also proposed [14]. In the differentialstiffness perturbation scheme, perturbations of equal magnitude but opposite signsare applied at the two end resonators of a three-DOF device.Though mode localized sensing is a recent development, eigenvalue veeringand associated mode localization phenomena are known in structural dynamics fora long time and have been studied extensively. A brief review of the literature onthe subject is presented in the following section.1.2 Eigenvalue veering and mode localizationOn plotting the eigenvalues of a periodic system versus symmetry breaking per-turbation in a system parameter, it is found that close eigenvalues of the systemapproach each other but veer away without crossing and their paths are exchanged.This phenomena is known as avoided crossing or eigenvalue loci veering. A re-cent study of veering is presented in [15] in the context of wave propagation in astructure. However, this phenomena is known in plate theory literature for a longtime [16]. Leissa posited that the curve veering abberation results from inaccurate3representation of physical reality [17]. Using an example of a rectangular platewith varying aspect ratio, it was shown that the inaccuracy is introduced by ap-proximations such as truncation of higher order terms in solution or discretization[17]. Validity of many of the approximate methods were rendered dubious. Lat-ter, Perkins and Mote showed, using example of a simple continuous system withexact solution, that veering indeed occurs and is not always the result of discretiza-tion [18]. They also developed criteria to distinguish veering from crossing [18].Veering requires coupling of modes and presence or absence of coupling betweenmodes in the veering region differentiates between veering and crossing [18]. Otherperspectives to describe veering were also developed. One among them is a uni-fied geometric perspective which describes veering and crossing as singularities inmapping from the complex system parameter plane to the complex frequency plane[19].While probing the behaviour of a periodic system with disorder1, a study foundthat close eigenvalues of the system veer away, and mode shapes of the system lo-calize simultaneously as a result of introduction of disorder [20]. The phenomenaof localization of vibration modes (spatial confinement of vibrational energy as-sociated with a mode) in a periodic system due to introduction of disorder in thesystem is known as mode localization [21, 22]. The study concluded that “modelocalization and eigenvalue veering are two manifestations of the same drastic phe-nomenon occurring when some type of disorder is introduced into nearly periodicstructures with close eigenvalues” [20]. Consequently, two phenomena, eigenvalueveering and mode localization, which were being studied independently till then,were brought together.The phenomena of mode localization was first observed in solid state physics,wherein it was shown that disorder may localize electron eigenstates [23]. In struc-tural dynamics, confinement of vibration induced by disorder in a periodic struc-ture was first discussed in [21]. It was shown that a small departure from symmetrymay change the transmission of vibration in a structure qualitatively and not justquantitatively. An experiment on a beaded string was carried out to demonstratemode localization [24]. While transmitting energy from one end of the string to1Disorder is perturbation distributed throughout the system and not necessarily localized to spe-cific sites of the system.4the other, satisfactory agreement between measurement and theoretical predictionof energy attenuation in the disordered case was demonstrated [24]. Furthermore,a perturbation method to study localization was developed and closed form expres-sions for strongly localized modes were obtained [22]. The frequency response of adissipative structure subject to veering was also studied to conclude that small per-turbations in such systems do not alter the frequency response considerably eventhough mode shapes undergoing localization can vary significantly [25].In the veering region, a pair of interacting eigenvalues also exchange theireigenvectors along with changing their paths [20]. The coupling between peri-odic units determines the sharpness of veering: smaller the coupling, sharper isveering and smaller the veering region. In strong localization with small veeringregion, this results in rapid rotation of eigenvectors. In fact, eigenvectors are ordersof magnitude more sensitive to small perturbation in system parameters comparedto eigenvalues in this region. This effect can be utilized in resonant sensing and ithas motivated research on mode localization based MEMS sensing.1.3 ObjectivesThis thesis aims to further the understanding of veering in MEMS devices and toimprove methods to analyze veering. With this aim, the following objectives havebeen pursued:• Development of an energy-based framework to analyze veering. This facili-tates convenient analysis of systems involving multi-energy domains.• Design and fabrication of MEMS devices utilizing shaped combs to achievelinear electrostatic coupling. This eliminates any nonlinear effect presentin parallel plate type electrostatic coupling. Moreover, it makes tuning ofcoupling straightforward as coupling stiffness varies proportionally with thesquare of the coupling voltage.• Characterization of two and three DOF coupled resonator MEMS devices tostudy issues involved in the fabrication of real devices.The devices are transducers as the perturbation in them is introduced electrostat-ically and not by any specific physical quantity to be measured. This is to study5veering with greater control on perturbation. A mechanism can be devised to intro-duce similar perturbation by a physical quantity to be measured. However, it hasnot been investigated in this thesis.1.4 OutlineIn Chapter 2, an energy-based framework to analyze mode localization in a multidegree-of-freedom (DOF) coupled resonator system due to stiffness perturbationwill be developed. The general results will be utilized for analytical study of modesensitivity in two and three DOF systems. Chapter 3 describes design, fabricationand modeling of two and three DOF coupled resonator MEMS devices designedto study veering. Analytical as well as SIMULINK models of the devices will bedeveloped. In Chapter 4, experimental results from characterization of the MEMSdevices are presented and compared with analytical and SIMULINK results. Fi-nally, conclusions and future directions will be presented in Chapter 5.6Chapter 2Analysis of normal modelocalization in coupled resonators2.1 IntroductionAnalysis of normal mode localization in a coupled, symmetric resonator systemusing eigenvalue perturbation method has been presented previously [9, 11]. Nor-mal mode localization in a two-DOF coupled, symmetric system has also beeninvestigated using state-space method [14]. Here, it will be examined employ-ing an energy based approach. This approach is adopted because it lends itself toeasier integration of multi energy domains in analysis. We start by analyzing ageneral multi-DOF spring-mass system. The results obtained are utilized to studythe specific cases of two and three DOF systems. Measures to quantify sensitivityof mode shapes and natural frequencies are described and it is shown that sensi-tivity of mode shapes is orders of magnitude higher than the sensitivity of naturalfrequencies for the same perturbation in a system parameter, a result that has beenshown in earlier works also [9, 10]. An analysis of differential perturbation schemein three-DOF systems, wherein a positive stiffness perturbation at the first resonatorand a negative stiffness perturbation of the same magnitude at the last resonator isapplied, will be presented. It has been shown in [14] that such an scheme has theadded advantage of constant resonant frequency even after a small perturbation isapplied to the system. Here the same result will be reproduced using an energy7m !" m m  !"#!1 kc "!"#!j "!"#!n  !"#!2 kc kc Figure 2.1: An n-DOF coupled, symmetric spring-mass system with nonuni-form stiffness perturbationmethod.2.2 Normal modes of N-DOF coupled resonatorsWe start by deriving normal modes of an N-DOF coupled, symmetric resonatorsystem with nonuniform stiffness perturbation. Results pertaining to mass pertur-bation will not be derived. However, it can be obtained emulating the same logicthat will be followed in the analysis below. The results obtained in this section willbe used to obtain normal modes of two-DOF and three-DOF coupled, symmetricresonator systems with specific perturbation schemes and to study mode localiza-tion in those systems.Consider an N-DOF coupled, symmetric resonator system with nonuniformstiffness perturbation scheme as shown in Figure 2.1. Symmetry here means that allthe resonators have equal mass (m) as well as equal stiffness (k) and all the couplingsprings have the same stiffness (kc). Stiffness perturbation in the jth resonator is∆k j. The general form of stiffness perturbation developed here extends the validityof this analysis to any scheme of stiffness perturbation. For example, the resultsfor the case of perturbation in only one spring can be obtained by making all otherstiffness perturbations equal to zero.We confine our study to synchronous motion of the system. In synchronousmotion, all the masses pass through their respective equilibrium positions at thesame time [26, 27]. In a conservative system, the governing equations of motionof the system are given by:mx¨ j =−∂V∂x j, (2.1)where x j denotes displacement of the jth mass of the system. V is the potential8energy of the system given asV = 12n∑i=1(k+∆ki)x2i +12kcn−1∑i=1(xi+1− xi)2. (2.2)where k is stiffness of the resonators, ∆ki is perturbation in stiffness of the ith res-onator and kc is coupling stiffness. Now, for generalized synchronous motion,displacements are algebraically related and displacement of one mass is sufficientto characterize the motion of the system:x j = x j(x1), (2.3)where the first mass is the reference mass (assuming x1 6= 0 at all times), thoughany mass with nonzero displacement can be taken as reference. Interest is in elimi-nating the time dependence in Equation 2.1 and establishing a relation between themodal amplitudes (amplitudes of vibration of the masses when they are vibrating inone of the normal modes) for synchronous motion of the system. The synchronic-ity relation in Equation 2.3 can be used to obtain the relation. Applying the chainrule, we getx˙ j =dx jdx1x˙1, (2.4)x¨ j =d2x jdx21x˙21 +dx jdx1x¨1. (2.5)Replacing the acceleration terms in Equation 2.5 using relation in Equation 2.1, weget−1m∂V∂x j=d2x jdx21x˙21−1mdx jdx1∂V∂x1. (2.6)We assume synchronous motion of the following form:x j = a jx1(t) (2.7)9where a j are constant modal parameters. Under this assumptiond2x jdx21= 0 (2.8)and Equation 2.6 transforms to time-independent modal equation∂V∂x j=dx jdx1∂V∂x1. (2.9)In the next step, we substitute the expression for potential energy Equation 2.2 inthe modal equation Equation 2.9 to get− kc(1−δ j1)x j−1 +(k+∆k j +(2−δ j1−δ jn)kc)x j− kc(1−δ jn)x j+1=dx jdx1((k+∆k1)x1 + kc(x1− x2)). (2.10)The above relation can be converted to an algebraic equation in terms of constantmodal parameters a j by applying the modal relation Equation 2.7 in the above,dividing the equation by kc and simplifying:a j−1(1−δ j1)−(∆k j−∆k1kc+(1−δ j1−δ jn)+a2)a j+a j+1(1−δ jn) = 0. (2.11)This recursive relation can be transformed into an nth order polynomial equationin a2. The solutions of the polynomial equation can be denoted as a2 j, wherej= 1, ...,n. a2 j is the modal amplitude of the second mass when the system vibratesin its jth mode. Using the relation among modal constants, modal amplitudes of allthe other masses for the jth mode vibration of the system can be calculated and thejth normal mode of the system can be written as [1 a2 j a3 j ... an j]T (a1 = 1from Equation 2.7).Natural frequency corresponding to a mode can be obtained using the govern-ing equation Equation 2.1 for the displacement of the reference mass. The equation10! m k kc k !" Figure 2.2: A two-DOF coupled, symmetric spring-mass system. ∆k is per-turbation in stiffness.of motion for the reference mass is:mx¨1 =−∂V∂x1=−((k+∆k1)x1 + kc(x1− x2)). (2.12)For jth mode vibration, x2 = a2 jx1 (from Equation 2.7). Substituting this relationin the above yields:mx¨1 =−(k+∆k1 + kc(1−a2 j))x1. (2.13)Considering harmonic motion, the jth natural frequency, ω j, of the system is ob-tained as:ω j =√(k+∆k1 + kc(1−a2 j))m. (2.14)In what follows, we use the above results for an N-DOF coupled, symmetricresonator system with stiffness perturbation to analyze two-DOF and three-DOFsystems and study vibration mode localization in them.2.3 Normal modes of two-DOF coupled resonatorsNow we consider a two-DOF coupled, symmetric spring-mass system as shownin Figure 2.2. Notice that in this resonator system, stiffness of only one of theresonators is perturbed (∆k1 = 0). We use the results from Section 2.2 to obtainnormal modes of this system. Translating Equation 2.11 for a two-DOF system bysubstituting n = 2 and j = 2 in the equation yields:a1−(∆k2kc+a2)a2 = 0. (2.15)11We recognize from Equation 2.7 that a1 = 1. Substituting this value in Equa-tion 2.15 along with ∆k2 = ∆k for simplicity, gives:a22 +∆kkca2−1 = 0. (2.16)The roots of the above modal equation define the two linear normal modes of thesystem. The roots are:a21 =−∆k2kc+√1+(∆k2kc)2; a22 =−∆k2kc−√1+(∆k2kc)2. (2.17)The two normal modes of the system are [1 a21]T (in-phase) and [1 a22]T (out-of-phase). Using the modal relation x2 = a2x1 in Equation 2.1 and considering aharmonically undamped motion givesω1 =√k+ kc(1−a21)m; ω2 =√k+ kc(1−a22)m. (2.18)Notice that for a symmetric system (∆k = 0), the two normal modes of the systemare [1 1]T (in-phase) and [1 −1]T (out-of-phase) and corresponding natural fre-quencies are√k/m and√(k+2kc)/m. So, the in-phase mode frequency of a two-DOF coupled, symmetric system is independent of coupling stiffness. Figure 2.3shows the variation of the natural frequency and the mode shape as the stiffnessperturbation is changed. If we plot the modal amplitude of the first mass (a1) onthe horizontal axis and and the modal amplitude of the second mass (a2) on thevertical axis, then the normal modes can be visualized as vectors. The angle madeby a normal mode vector with the horizontal axis is unique to the mode and it hasbeen used to characterize the mode.To normalize the relations above, we introduce ω20 = k/m and the non-dimensionalquantities stiffness perturbation δ = ∆k/k and coupling κ = kc/k in Equation 2.17and Equation 2.18 respectively, which gives,a21 =−δ2κ +√1+(δ2κ)2; a22 =−δ2κ −√1+(δ2κ)2. (2.19)122! 2" 2# $ # " !$%&'((%$')*+,,-.//01.2*3245*+6-70→8980:;<=62>5?+@.A0=5*325?0B2.C3.-DE0:−9− $<→→!!→→→!"#→→!"# →→!""→→−!""→→!!a2a2a2a2a2a2 a2 a2a2a2a2a2Figure 2.3: Veering diagram for two-DOF coupled, symmetric spring-masssystem shown in Figure 2.2 showing the change in natural frequencyand mode shape (characterized by eigenvector angle θ ) with stiffnessperturbation for negative coupling stiffness. In inset, the eigenvectorsare drawn, where modal amplitude of the first mass (a1) is plotted onthe horizontal axis and the modal amplitude of the second mass (a2)is plotted on the vertical axis. The arrow after θ denotes that θ is ap-proaching the specified value. At zero perturbation θ1 = θ10 = 45◦ andθ2 = θ20 =−45◦andω1 = ω0√(1+κ(1−a21)); ω2 = ω0√(1+κ(1−a22)). (2.20)For small values of stiffness perturbation δ , the above can be approximated toexpressions similar to those deduced by using the more standard matrix formalismfor eigenmodes in linear systems [11].Here, we have obtained normal modes and corresponding natural frequenciesof a two-DOF system. Now, we move to quantify changes in them induced bystiffness perturbation.2.3.1 Sensitivity of natural frequency and normal modeWe saw above that stiffness perturbation induces a change in natural frequency aswell as mode shape of the system. We quantify changes by calculating the differ-ence in natural frequency and mode shape of a perturbed system and corresponding13−6 −5 −4 −3 −2 −1 0020406080Stiffness Perturbation, ∆k/k (%))Sensitivity Norms (%)  Nθ: κ=−0.01Nω: κ=−0.01Nθ: κ=−0.1Nω: κ=−0.1Figure 2.4: Comparison of sensitivity norms for two coupling values. κ isnon-dimensional coupling stiffness.unperturbed (symmetric) system as follows:Nωi =∣∣∣∣ωi−ωi0ωi0∣∣∣∣ ,Nθi =∣∣∣∣θi−θi0θi0∣∣∣∣ , tanθ = a2i, (2.21)where Nωi and Nθi are sensitivity norms corresponding to natural frequency andmode shape. Subscript i denotes the mode for which we want to find sensitivitynorms and 0 in the subscript denotes parameter values in the unperturbed state ofthe system.For a two-DOF system, the norms can be obtained in terms of non-dimensionalquantities δ and κ by substituting expressions in Equation 2.19 and Equation 2.20in Equation 2.21. For in-phase mode (i = 1), the sensitivity norms are as follows:Nω =∣∣∣∣∣∣∣√√√√√1+κ1+δ2κ −√1+(δ2κ)2−1∣∣∣∣∣∣∣,Nθ =∣∣∣∣∣∣4pi tan−1−δ2κ +√1+(δ2κ)2−1∣∣∣∣∣∣. (2.22)Approximate expressions for the sensitivity norms around the unperturbed state ofthe system (sensitivity norms near origin in Figure 2.4) can be obtained by assum-14m  !"#!1 kc !!"#!3  !"#!2 kc m m Figure 2.5: Three-DOF coupled, symmetric spring-mass system with stiff-ness perturbationing ∆k << kc in Equation 2.22:Nω ≈∣∣∣∣δ4∣∣∣∣ , Nθ ≈∣∣∣∣1piδκ∣∣∣∣ . (2.23)As seen from the above and from Figure 2.4, the mode shape sensitivity, Nθ , ismuch higher (and tunable through the κ term) than the resonant frequency sensitiv-ity, Nω , for small κ values (weak coupling). The exploitation of this phenomenoncan lead to orders of magnitude improvement in sensitivity, compared to conven-tional resonant frequency monitoring systems.2.4 Normal modes of three-DOF coupled resonatorsIn this section, a three-DOF coupled, symmetric spring-mass system as shown inFigure 2.5 is considered. We use the results from Section 2.2 to obtain normalmodes of this system. For this system, n equals three in Equation 2.11. Substi-tuting j = 1 in Equation 2.11 yields the trivial relation (a2− a2 = 0) as a1 = 1.Substituting j = 2,3 yield the following relations:a1−(∆k2−∆k1kc+1+a2)a2 +a3 = 0, (2.24)a2−(∆k3−∆k1kc+a2)a3 = 0. (2.25)15Now, we convert this recursive relation to a polynomial equation in a2. Substitutinga1 = 1 in Equation 2.24 and expressing a3 in terms of a2 yields:a3 =(∆k2−∆k1kc+1+a2)a2−1. (2.26)Substituting the above relation in Equation 2.25 and expanding yields a polynomialequation in a2.a32 +(1+∆k2−∆k1kc+∆k3−∆k1kc)a22+(∆k3−∆k1kc(1+∆k2−∆k1kc)−2)a2−∆k3−∆k1kc= 0. (2.27)The above cubic equation in a2 can be solved numerically for different stiffnessperturbation schemes to obtain the modal amplitude of the second mass for thethree normal modes of the system. Using the relation above, modal amplitude ofthe third mass is calculated. Afterwards, natural frequencies are determined usingEquation 2.14. For no perturbation (i.e. ∆k1 = ∆k2 = ∆k3 = 0), the above equationconverts to:a32 +a22−2a2 = 0. (2.28)The three roots of this equation are a2 = 1,0,−2 (modal amplitudes of the secondmass for the three normal modes). a3 is evaluated by substituting these values ofa2 in Equation 2.26 and is found to be a3 = 1,−1,1 (modal amplitudes of the thirdmass for the three normal modes). Hence, the mode shapes of the unperturbedsystem are [1 1 1]T , [1 0 −1]T and [1 −2 1]T .For small stiffness perturbations, the theory of sensitivity of roots of a poly-nomial to perturbation in its coefficients [28], described in Appendix A, can beutilized to obtain an approximate analytical expression for a2. In our analysis, weget the following polynomial in a2 for the unperturbed case:g(a2) = a32 +a22−2a2. (2.29)We denote its roots by a2 j, where j = 1,2,3 (a21 = 1, a22 = 0 and a23 =−2). Theexpression on the left hand side of Equation 2.27 is a perturbed polynomial in a2.16We denote it by g¯(a2).g¯(a2) = a32 +(1+∆k2−∆k1kc+∆k3−∆k1kc)a22 (2.30)+(∆k3−∆k1kc(1+∆k2−∆k1kc)−2)a2−∆k3−∆k1kc.After rearranging the above, we can write it as:g¯(a2) = g(a2)+(∆k2−∆k1kc+∆k3−∆k1kc)a22 (2.31)+(∆k3−∆k1kc(1+∆k2−∆k1kc))a2−∆k3−∆k1kc.Evaluation of the change in roots of the polynomial requires us to find the slopeof the unperturbed polynomial at its roots. The first derivative of the unperturbedpolynomial at its roots is given by:g′(a2 j) = 3a22 j +2a2 j−2. (2.32)Substituting values of a2 j for the three modes in the above, we obtain g′(a21) = 3,g′(a22) =−2 and g′(a21) = 6. The shift in a2 j is given by:∆a2 j ≈−[(∆k2−∆k1kc+∆k3−∆k1kc)a22 j+ (2.33)(∆k3−∆k1kc(1+∆k2−∆k1kc))a2 j−∆k3−∆k1kc]/g′(a2 j).The change in a3 j (modal amplitude of the third mass in jth mode vibration) canbe obtained using Equation 2.26.∆a3 j ≈(∆k2−∆k1kc)a2 j +(1+2a2 j)∆a2 j. (2.34)Shifted mode shapes of the system are [1 1+∆a21 1+∆a31]T , [1 ∆a22 −1+∆a32]T and [1 −2+∆a23 1+∆a33]T . The shift in the jth natural frequency17is obtained using Equation 2.14 and is given by:∆ω j =ω202ω j0(∆k1k−kck∆a2 j), (2.35)where ω0 = k/m and the subscript 0 denotes the parameter value in the unperturbedstate.2.4.1 Perturbation in the end resonatorThe results obtained above are valid for general stiffness perturbation scheme in acoupled, symmetric three DOF system. In this subsection, the case with stiffnessperturbation only in the end resonator (i.e. ∆k1 = ∆k2 = 0) is considered. In thiscase, the shift in a2 j is given by:∆a2 j =−[∆k3kca22 j +∆k3kca2 j−∆k3kc]/g′(a2 j). (2.36)Shift in a3 j can be obtained using Equation 2.34. After normalizing the relationabove and Equation 2.35, by introducing the non-dimensional quantities, the stiff-ness perturbation δ = ∆k3/k and the coupling κ = kc/k, the shifted natural fre-quencies and the mode shapes of the system can be written as:ω1 = ω0(1+δ6), φ1 =[1 1−δ3κ 1−δκ]T,ω2 = ω0(√1+κ+ δ4), φ2 =[1 −δ2κ −1−δ2κ]T,ω3 = ω0(√1+3κ+ δ12), φ3 =[1 −2−δ6κ 1+δ2κ]T. (2.37)To define sensitivity, we plot modal vectors in three-dimensional cartesian spacewith standard basis and take their projection on the plane defined by the first andthe third axes (akin to X-Z plane in XYZ coordinate system). The angle θ j made bythe projection of jth modal vector with the first axis can be used to define sensitivity18of the mode.Nω j =∣∣∣∣ω j−ω j0ω j0∣∣∣∣ , Nθ j =∣∣∣∣θ j−θ j0θ j0∣∣∣∣ , tanθ j = a3 j, (2.38)where j = 1,2,3 for the three modes of the system respectively and the subscript 0denotes the parameter value in the unperturbed case.The sensitivities of the natural frequencies are given by:Nω1 =∣∣∣∣δ6∣∣∣∣ , Nω2 =∣∣∣∣δ4∣∣∣∣ , Nω3 =∣∣∣∣δ12∣∣∣∣ . (2.39)The sensitivities of the mode shapes are given by:Nθ1 =∣∣∣∣2piδκ∣∣∣∣ , Nθ2 =∣∣∣∣1piδκ∣∣∣∣ , Nθ3 =∣∣∣∣1piδκ∣∣∣∣ . (2.40)From the expressions above, it can be observed that the mode shape sensitivitiesdepend on coupling, unlike the natural frequency sensitivities, and they are verylarge compared to natural frequency sensitivities for small coupling. It should benoted that for negative coupling, ω3 is the smallest natural frequency. Hence, ω3and φ3 are the natural frequency and the mode shape corresponding to the firstmode vibration of the system. ω1 and φ1 are the natural frequency and the modeshape corresponding to the third mode vibration of the system. Figure 2.6 showsthe comparison of the sensitivities for the three modes obtained numerically for twocoupling values. The slope of a sensitivity curve near zero perturbation equals theapproximate analytical sensitivity curve slope. Comparing mode shape sensitivitiesfor a three-DOF system with perturbation in the end resonator (see Equation 2.40)with those of a two-DOF system (see Equation 2.23), it can be observed that thesensitivity of the third mode shape of a three-DOF system is twice the sensitivityof a mode shape of a two-DOF system. However, for the two other modes of athree-DOF system, the mode shape sensitivities are the same as that of a mode ofa two-DOF system. Hence, for resonant sensing the third mode is the best choice.In Figure 2.6, the sensitivity of the second mode increases near zero pertur-bation but soon starts to decrease and goes to zero when stiffness perturbation istwice the coupling. Apart from having no advantage in sensitivity of this mode as19compared to the sensitivity of a two-DOF device, this bend in the sensitivity curvealso decreases the measurable range of perturbation (perturbation range for whichsensitivity curve is linear). The Sensitivity curve of the first mode also is curvedupwards near zero perturbation and it curves down after an inflection point on thecurve, though it is not readily noticeable in the plot.2.4.2 Differential perturbationIn a differential perturbation scheme, a positive stiffness perturbation is appliedon the first resonator and a negative stiffness perturbation of the same magnitude isapplied at the last (third) resonator while keeping the middle resonator unperturbed(∆k3 = −∆k1, ∆k2 = 0). In this subsection, the shift in modal amplitudes effectedby differential perturbation is studied.The shift in a2 j is given by:∆a2 j =−[−3∆k1kca22 j−2∆k1kc(1−∆k1kc)a2 j +2∆k1kc]/g′(a2 j). (2.41)The shift in a3 j can be obtained using Equation 2.34. After normalizing the re-lations above and Equation 2.35, by introducing the non-dimensional quantities,the stiffness perturbation δ = ∆k1/k and the coupling κ = kc/k, the shifted naturalfrequencies and mode shapes of the system can be expressed as:ω1 = ω0, φ1 =[1 1+δ3κ 1+δκ]T,ω2 = ω0√1+κ, φ2 =[1δ2κ −1+δ2κ]T,ω3 = ω0√1+3κ, φ3 =[1 −2+δ6κ 1−δ2κ]T. (2.42)Notice that the three natural frequencies do not change up to first order approxi-mation, even after application of a small differential perturbation, i.e. the naturalfrequencies are insensitive to small differential perturbation. To calculate the sensi-tivity of natural frequency and mode shape, the definition of sensitivity delineated202! 2"#$ 2" 2%#$ 2% 2&#$ 2& 2'#$ ''%'!'(')'&''*+,--./0012/3+4356+,7.81!9:91;<=*/.0,+,>,+?1@73A01;<=!"#$!%  &"'#(!)*)+&$'#(!)*)+&"'#(!)*+&$'#(!)*+1 2! 2"#$ 2" 2%#$ 2% 2&#$ 2& 2'#$ ''$&'&$%'%$"'"$!'()*++,-../0-1)2134)*5,6/!787/9:;(-,.*)*<*)=/>51?./9:;!"#$!%  &"'#(!)*)+&$'#(!)*)+&"'#(!)*+&$'#(!)*+2! 2"#$ 2" 2%#$ 2% 2&#$ 2& 2'#$ ''%'!'(')'&''*+,--./0012/3+4356+,7.81!9:91;<=*/.0,+,>,+?1@73A01;<=!"#$!%  &"'#(!)*)%&$'#(!)*)%&"'#(!)*%&$'#(!)*%1 Figure 2.6: Comparison of numerically obtained sensitivity norms for athree-DOF coupled resonator system with stiffness perturbation in theend resonator for two coupling values21in Section 2.4.1 has been used. Sensitivities of the natural frequencies are:Nω1 = Nω2 = Nω3 = 0. (2.43)Sensitivities of the three normal modes are:Nθ1 =∣∣∣∣4piδκ∣∣∣∣ , Nθ2 =∣∣∣∣2piδκ∣∣∣∣ , Nθ3 =∣∣∣∣2piδκ∣∣∣∣ . (2.44)From the expressions above, it can be observed that the mode shape sensitivityis twice as compared to mode shape sensitivity in the case with perturbation onlyin the end resonator (see Equation 2.40). However, the natural frequencies areinsensitive to perturbation up to first order of approximation. This arrangementprovides the ability to measure a perturbation without any need to track the naturalfrequency to be able to measure mode shape. It should be noted that for negativecoupling, ω3 is the smallest natural frequency. Hence, ω3 and φ3 are the naturalfrequency and the mode shape corresponding to the first mode vibration of thesystem. ω1 and φ1 are the natural frequency and the mode shape correspondingto the third mode vibration of the system. Figure 2.7 shows the comparison ofsensitivities for the three modes obtained numerically. The slope of a sensitivitycurve near zero perturbation equals the approximate analytical sensitivity curveslope. From the plot, it can be seen that the third mode is the most sensitive modeas the analytical approximation suggests. The other modes are not of interest forresonant sensing.2.5 Implications for device designFrom the analysis above, we can conclude that weaker coupling enhances the sen-sitivity of the mode shape. Hence, having electrostatic coupling in a device ispreferable as it provides weak coupling along with the possibility to tune it. Thesensitivity of the mode shape can be further enhanced by increasing the numberof DOF in a device. However, increasing the mode sensitivity leads to a smallermeasurable range as shown in Figure 2.8. Another important point to note is thatthe analysis assumes that the system is symmetric to begin with. However, in fab-ricated devices built-in asymmetry is present due to fabrication tolerances. As a re-222! 2"#$ 2" 2%#$ 2% 2&#$ 2& 2'#$ ''%'!'(')'&''*+,--./0012/3+4356+,7.81!9:91;<=*/.0,+,>,+?1@73A01;<=!"#$!%  &"'#(!)*)+&$'#(!)*)+&"'#(!)*+&$'#(!)*+1 2! 2"#$ 2" 2%#$ 2% 2&#$ 2& 2'#$ ''&'%'"'!'$'()*++,-../0-1)2134)*5,6/!787/9:;(-,.*)*<*)=/>51?./9:;!"#$!%  &"'#(!)*)+&$'#(!)*)+&"'#(!)*+&$'#(!)*+2! 2"#$ 2" 2%#$ 2% 2&#$ 2& 2'#$ ''%'!'(')'&''*+,--./0012/3+4356+,7.81!9:91;<=*/.0,+,>,+?1@73A01;<=!"#$!%  &"'#(!)*)%&$'#(!)*)%&"'#(!)*%&$'#(!)*%1 Figure 2.7: Comparison of numerically obtained sensitivity norms for athree-DOF coupled resonator system with differential stiffness pertur-bation for two coupling values23Stiffness Perturbation, !k/k (%) Sensitivity Norms (%) Sensitivity  Measurable range of perturbation (shown by red     dotted line) Figure 2.8: Schematic showing decrease in measurable range of perturbationas sensitivity increasessult, the sensitivity relations obtained above work only as a guide to design. Actualsensitivity and effect of built-in asymmetry on sensitivity needs to be investigatedby testing fabricated coupled resonator devices.2.6 Summary and discussionAn analytical treatment of veering and associated mode localization phenomenahas been presented in this chapter. The approach to the analysis is energy-based.To start with, analytical results for an N-DOF coupled, symmetric spring mass sys-tem with perturbation has been obtained which in turn has been used in the studyof two-DOF and three-DOF spring-mass systems. We obtained a complete analyt-ical solution for two-DOF systems. However, for three-DOF systems, a completeanalytical solution was difficult to obtain. Therefore, we have used the theoryof sensitivity of roots of a polynomial to perturbation in its coefficients to obtainmode shapes and natural frequencies of a three-DOF system with small stiffnessperturbation. It has been shown that the mode shapes of a system are orders ofmagnitude more sensitive to a stiffness perturbation than its natural frequencies.Moreover, we show that increasing the number of DOF from two to three alsohelps improve sensitivity. Furthermore, analytical results show that the differentialperturbation scheme has the added benefit of a constant natural frequency but asharply changing mode shape as a small differential perturbation is applied to it.24Chapter 3MEMS device design andmodeling3.1 IntroductionTo verify the analytical results about sensitivity of mode shapes, obtained in Chap-ter 2, experimentally, two and three DOF coupled resonator MEMS devices weredesigned and fabricated. In this chapter, the design of the MEMS devices and theirmodels will be discussed. An overview of the design tools used and the fabrica-tion method employed will be presented. Three MEMS devices were designed.Device-1 is a two-DOF device. It utilizes the electrostatic spring effect betweentwo parallel plates with nonzero potential difference between them to couple tworesonators as well as to create a stiffness perturbation. It was designed by Greg P.Raynen, a former member of the group, but the device model was not developedand experimental results obtained from the device were also inaccurate. Here, ananalytical and a Simulink model of the device is developed and comparison be-tween sensitivities predicted by them is presented. The coupling method employedin Device-1 has inherent nonlinearity due to the fact that coupling electrostaticforce varies inversely with gap between parallel plates. To remove the nonlinearityin coupling, Device-2 was developed. Device-2 is a two-DOF device with linearelectrostatic coupling which exploits a shaped comb design to achieve linearity incoupling. To improve the sensitivity further, a three-DOF device was also devel-25oped. Device-3 is a three-DOF device which has linearized electrostatic couplingbetween its resonators. The design of these devices will be discussed and analyt-ical and Simulink models for them will be presented. These models will be usedto study mode localization and predict the effect of veering on the response of thedevices.3.2 DesignTo design the devices, software tools from CoventorWare 2010 suite were used. Weexploited CoventorWare ARCHITECT’s system-level approach to get a prelimi-nary design. CoventorWare’s DESIGNER module was employed to incorporatedesign features not obtainable through ARCHITECT [29]. The design flow startswith prescribing information about fabrication process using Process Editor and theMaterial Properties Database. Process file corresponding to Silicon-on-InsulatorMulti-User MEMS Process (SOIMUMPs) was selected for our design and the sil-icon layer thickness was chosen to be 25 µm (SOIMUMPs process is available fortwo silicon layer thicknesses: 10 µm and 25 µm). Then, using schematic editorof ARCHITECT called Saber Sketch, a schematic of a MEMS device is assem-bled by connecting different components from a library of parametrized, MEMS-specific behavioral models. The library contains electromechanical componentssuch as rigid plates with electrodes, flexible beams, electrostatic comb drives etc.Schematic of the device can be visualized using the Scene3D sub-module in AR-CHITECT. Once the schematic looks satisfactory, the physical behavior of the de-vice can be simulated and the results can be viewed using a plotting tool in ARCHI-TECT called Cosmos Scope. The design at this stage is completely parametrizedand the system-level modeling approach ensures very fast simulations. Hence, it-erations in design, to obtain a device with desired physical behavior, to optimize adesign or to study sensitivity of the design to variations in the fabrication process,can be performed very quickly. Once a device with desired characteristics has beenobtained, the design can be exported to the DESIGNER [30] module.Now, the two-dimensional layout of the design is opened in the Layout Editorof DESIGNER. Here, design features not available in ARCHITECT such as filletedcorners, shaped fingers etc. are incorporated in the design. This layout is used by26Solid Modeler along with layer stack information in the Process Editor to generatea three-dimensional solid model. This model can be preprocessed and desired fieldsolvers from ANALYZER [31] can be employed to obtain exact device behavior.Two-dimensional layout of the design from DESIGNER is saved as a GDSIIfile. CleWin 4 layout editor software was used to design layout for fabrication. Thelayout was saved as a GDSII file.3.3 FabricationTo fabricate the device, we submitted the GDSII file with layout design to CMCMicrosystems, which forwarded it to MEMSCAP for fabrication. MEMSCAP fab-ricated the device using SOIMUMPs. A brief description of the process flow [32]is presented below.SOIMUMPs is a four-mask level patterning and etching process which startswith a silicon-on-insulator (SOI) wafer. The SOI wafer used for our device has a25±1 µm thick silicon layer, a 2±0.1 µm thick oxide layer and a 400±5 µm thicksubstrate. The free surface of the silicon layer is doped by depositing a phospho-silicate glass (PSG) layer and annealing it to start the process flow. Afterwards,the PSG layer is removed via wet chemical etching. Then, a Pad Metal layer is de-posited through a liftoff process. This process is used to provide features like bondpads and electrical routing. Thereafter, the silicon layer is patterned and etcheddown to the oxide layer using deep reactive ion etch (DRIE) to obtain desired me-chanical structure. The wafer is then reversed and the substrate layer is patternedand etched from bottom side down to the silicon layer to create trench beneath thedevice layer and free it. Afterwards, remaining exposed oxide layer is removed.The blanket metal layer is then deposited and patterned using shadow maskingtechnique. This process produces a smooth metal layer and can be used to createoptical quality mirrored surfaces. Fabrication process flow ends with this process.After fabrication, CMC Microsystems arranged for packaging of the devicesand devices packaged in 68-pin ceramic pin-grid array (CPGA) packages werereturned to us along with some unpackaged devices.27!"#$%#&'$()*+,"'-+!"#$%#&'$()*+!.'$"+/%--0+!"#$%#&'$()*+1$#%2$%#"+342($'$()*+5)-&6#(7"+ 3."2$#)8$'$(2+5)%9.(*:+1%89"*8()*+,"'-+;<+9"#+9.'$"=+-'88++>+ -'88++?+>@@+A-+Figure 3.1: Two-DOF coupled resonator MEMS device-1283.4 Device-13.4.1 Device descriptionDevice-1 is a two-DOF coupled, symmetric MEMS resonator system with electro-static coupling as well as electrostatic stiffness perturbation. The device (shownin Figure 3.1) consists of two identical proof masses with in-plane dimensions of210 µm× 210 µm, suspended by beams of 500 µm in length and 6 µm width.The thickness of the device layer is 25 µm. The two masses are electrostaticallycoupled by capacitive plates of 300 µm length, with a 2 µm gap. The electrostaticcoupling stiffness is varied by changing the voltage across these plates, given bydifference in electric potentials of the two proof masses. A perturbation plate of100 µm length is attached at the mid-point of each suspension beam by a horizontalbeam (see Figure 3.1). A stiffness perturbation is induced by applying a constantvoltage (called perturbation voltage now onwards) between the perturbation beam(top right in Figure 3.1) and the perturbation plates attached to the suspension beamof the right mass in Figure 3.1. The electrostatic attraction between the perturba-tion beam and the neighboring perturbation plates introduces the stiffness change(∆k). The perturbed stiffness can be exploited for instance in displacement sensingin future sensing applications by allowing displacement of the perturbation beamin vertical direction. The device dimensions are summarized in Table 3.1.Identical perturbation plates and comb-drive actuators are present in both res-onators in order to maintain structural symmetry. However, the perturbation platesand their associated perturbation beams on the left resonator are electrically shortedto ensure no stiffness change in the left structure. The device is excited (with a DC-AC voltage combination) on the left comb-drive, while the right comb-drive has thesame DC level, in order to maintain the static symmetry. Only the left resonatorwas therefore excited externally in our experiments, while the stiffness perturbationwas induced on the right resonator only.3.4.2 Device modelsThe device was modeled as a two-DOF spring-mass system with damping. Thesystem was assumed to be proportionally damped. An analytical model as well as29Table 3.1: Device dimensionsDevice layer thickness 25 µmIn-plane dimensions of proof mass 210 µm × 210 µmSuspension beam length 500 µmSuspension beam width 6 µmDimensions of coupling capacitive plate 300 µm × 25 µmAir-gap between coupling capacitive plates 2 µmDimensions of perturbation capacitive plate 100 µm × 25 µmAir-gap between perturbation beam and a perturbation plate 2 µmNumber of comb fingers on each resonator 10a Simulink model of the device was developed. The models are described in thefollowing sections.Analytical modelTo model the device in Figure 3.1 as a spring-mass system (Figure 2.2), the equiv-alent mass and the equivalent stiffness of the resonators along with the equivalentcoupling stiffness need to be calculated. As proportional damping does not alterthe mode shape or the natural frequency of a system, we do not need to considerdamping in this model. The equivalent mass is the sum of mass of the proof massand the equivalent mass of the suspension beams at the point of attachment to theproof mass. The equivalent mass of the suspension beams is obtained using kineticenergy equivalence assuming a quasi static deformation shape of a fixed-guidedEuler-Bernoulli beam. The stiffness of the left resonator is obtained using standardbeam deflection formula for a fixed-guided Euler-Bernoulli beam [33]. However,for the right resonator, the stiffness is perturbed by application of a perturbationvoltage. In order to obtain a closed form relation between the applied pertur-bation voltage and the consequent perturbation in stiffness, Castigliano’s secondtheorem [34] along with a linear approximation for electrostatic force is used asdescribed below.The suspension beams of the right resonator mass are modeled as a fixed guidedbeam with a force at the mid point of the beam as shown in Figure 3.2(a). Theboundary condition and the displacement compatibility condition at the guided end30!!"!"#$! "%$!&'(!&'(!"#!$#!%#!%& !'#)*!"& !Guided ends Axis of Symmetry "&( !)#!%&( ! %&* !"&* !Figure 3.2: (a) Schematic of a fixed-guided beam (b)Schematic of the upperhalf of the 2nd resonator (coupling plate not shown)in Figure 3.2 are:θ2(y = L) = 0 and w2(y = L) = x2. (3.1)The reaction force P2 and moment M2 applied by the guide onto the guided endof the beam are unknown. We consider the deflection wm at the mid point of thebeam to be unknown whereas the electrostatic force Pm at the point to be knownin this part of the analysis. To determine the unknown quantities wm, P2 and M2,Castigliano’s second theorem can be used. To proceed, we need to evaluate thestrain energy, which in turn requires us to know the moment distribution in thebeam. With the aid of Figure 3.2(a), the moment distribution in the beam can beevaluated.M =−M2−P2(L− y)−Pm(L/2− y) for 0≤ y≤ L/2−M2−P2(L− y) for L/2≤ y≤ L.(3.2)31The strain energy in the beam (U) is given byU =∫ L0M22EIdy =L48EI(8L2P22 +5L2P2Pm+L2P2m+24LM2P2 +6LM2Pm+24M22), (3.3)where EI and L are in-plane bending rigidity and length of a suspension beamrespectively. Applying Castigliano’s second theorem, we get:w2 =∂U∂P2=L248EI(24M2 +16LP2 +5LPm), (3.4)θ2 =∂U∂M2=L248EI(8M2 +4LP2 +LPm), (3.5)wm =∂U∂Pm=L248EI(5LP2 +2LPm+6M2). (3.6)To evaluate unknowns P2, M2 and wm, we use Equation 3.1 in the above equationsand solve the resulting linear simultaneous equations to obtain:P2 =12EIx2L3−Pm2, (3.7)M2 =−6EIx2L2+LPm8, (3.8)wm =PmL3192EI+x22. (3.9)In our design, Pm is the electrostatic force on the perturbation plate (consequentlyon the beam) applied by the perturbation beam. Extending this analysis to thetwo-beam suspension shown in Figure 3.2(b) and using expression for electrostaticattraction, we can find the force on the mid point of the left and the right beams,Pml and Pmr (subscripts l and r are for the left and the right beams in Figure 3.2(b)),in terms of deflections, wml and wmr at those points respectively, as follows:Pml =εApV 2p2(gp−wml)2, (3.10)Pmr =εApV 2p2(gp+wmr)2, (3.11)32where ε , Ap, Vp, gp are permittivity of air, area of the perturbation plate, pertur-bation voltage and initial gap between the perturbation plate and the perturbationbeam, respectively. From Figure 3.2(b) and using Equation 3.9, we get:wml =PmlL3192EI+x22, (3.12)wmr =(−Pmr)L3192EI+x22, (3.13)where x2 is deflection of the right proof mass. Substituting Pml and Pmr in theabove with the expressions in Equation 3.10 and Equation 3.11, and simplifyinggives cubic equations in wml and wmr respectively, which can be solved numeri-cally. They are in turn used to calculate Pml (using Equation 3.10) and Pmr (usingEquation 3.11). Using Equation 3.7, the reaction force on the left beam, P2l , andon the right beam, P2r, by the right mass is evaluated. The spring force, Fk, on theright mass by the suspension beams is given by:Fk =−2(P2l +P2r), (3.14)where the factor of two accounts for the beams on the lower side of the proofmass (symmetric to the part shown in Figure 3.2(b)). The Simulink model of thedevice uses this approach to evaluate the spring force numerically. But for theanalytical model, the approximation of very small wml and wmr is introduced, asthe exact analytical solution for wml and wmr is difficult. This approximation willbe validated using the Simulink model. In the small displacement regime, Pml andPmr can be approximated as:Pml ≈εApV 2p2g2p(1+2wmlgp), (3.15)Pmr ≈εApV 2p2g2p(1−2wmrgp). (3.16)In this case, expression for wml and wmr in Equation 3.12 and Equation 3.13 convert33towml ≈(x22+αg2p)(1−2αg3p)−1, (3.17)wmr ≈(x22−αg2p)(1−2αg3p)−1, (3.18)(3.19)where α = (L3/192EI)(εApV 2p /2). These values of wml and wmr are substitutedin Equation 3.15 and Equation 3.16 to get Pml and Pmr which are then used toevaluate P2l and P2r using Equation 3.7. Now, the spring force is calculated usingEquation 3.14 and the following expression is obtained:Fk ≈−4×12EIx2L3+ x2(g3pεApV 2p−L3192EI)−1. (3.20)The equivalent stiffness of the beams iske f f ≈−Fkx2≈ 4×12EIL3−(g3pεApV 2p−L3192EI)−1. (3.21)The first term in the above expression is the combined stiffness of the suspensionbeams without any perturbation. The second term is stiffness perturbation.∆k ≈−(g3pεApV 2p−L3192EI)−1. (3.22)The nondimensional stiffness perturbation (δ ) is obtained by using the relationδ = ∆k/k, where k is the unperturbed combined stiffness of the beams:δ ≈−(kg3pεApV 2p−14)−1, k =48EIL3. (3.23)The variation of nondimensional stiffness perturbation with perturbation voltageis nonlinear and it is plotted in Figure 3.3. It should also be notes that this rela-340 5 10 15 20 25−20−15−10−50Perturbation Voltage (V)Stiffness Perturbation ∆k/k (%)Figure 3.3: Variation of nondimensional stiffness perturbation (δ = ∆k/k)with perturbation voltagetion is valid only for perturbation voltages smaller than the pull-in voltage of theperturbation spring arrangement.Now that mass and stiffness of the two resonators are known, we need to calcu-late the coupling stiffness kc. In the device, resonators are coupled electrostatically.Stiffness of such a spring can be computed as follows. For an electrostatic (capac-itive) coupling with a constant coupling voltage Vc between the parallel plates, thecontribution of coupling to the total potential energy is of the form:Uc =−12εAcg0− (x1− x2)V 2c , (3.24)where Ac, g0, x1 and x2 are the area of the coupling capacitor plate, the initial zero-voltage gap between the capacitor plates, the displacement of the left mass andthe displacement of the right mass respectively. Note that these displacements aremeasured from the initial positions (positions before any voltage is applied) of themasses. The aplied Vc changes the equilibrium gap between the coupling platesto gv and hence the equilibrium position of the masses. The new potential energyexpression is given as:Uc =−12εAcgv− (x1− x2)V 2c , (3.25)where gv is the equilibrium gap between the capacitor plates, while x1 and x2 arenow the displacements of the masses relative to the shifted equilibrium configura-35tion. The Taylor series expansion up to second order givesUc ≈−12εAcgvV 2c(1+(x1− x2gv)+(x1− x2gv)2). (3.26)The Taylor series approximations of the total potential energy, to which Uc alsocontributes, in the shifted frame can be written as:U =U0 +[∂U∂x1]eqx1 +[∂U∂x2]eqx2 + ... (3.27)with derivatives evaluated at the new equilibrium position. U0 can be taken as zeroif we consider the equilibrium position as the reference energy level, while theequilibrium state implies a zero value for the linear term. In physical terms, therestoring spring force balances the electrostatic force between the plates at equi-librium. The equations of motion depend only on the derivatives of the potentialenergy, so the position chosen for the reference zero-level energy will not affectthe results. The effective potential energy contribution of the coupling element isgiven as:Uc ≈−12εAcg3vV 2c (x1− x2)2 (3.28)Comparing the above potential energy with the potential energy expression for aspring gives the correspondencekc ≈−εAcg3vV 2c . (3.29)The nondimensional coupling strength is given byκ = kck. (3.30)Now we have a complete analytical model of the device.By substituting the δ and κ values, obtained in Equation 3.23 and Equation 3.30respectively, in Equation 2.22, the analytical sensitivity norms, introduced in Equa-tion 2.21, are obtained. The following approximate closed-form expressions can36be used to compute the sensitivity norms analytically.Nω =∣∣∣∣∣∣∣√√√√√1+κ1+δ2κ −√1+(δ2κ)2−1∣∣∣∣∣∣∣, (3.31)Nθ =∣∣∣∣∣∣4pi tan−1−δ2κ +√1+(δ2κ)2−1∣∣∣∣∣∣,δ ≈−(kg3pεApV 2p−14)−1, κ = kck,k =48EIL3, kc =−εAcV 2cg3c.In the analytical model, the material and geometric properties required to calculatek and kc are chosen based on nominal design values of the device.Simulink modelThe analytical model is based on small displacements approximation introducedin Equation 3.15 and Equation 3.16. A more general Simulink model (Figure 3.4)of the device, without the small displacement approximation, and incorporatingnonlinearities in the electrostatic forces, has been developed for a more accuratecomputation of the sensitivity norms in Equation 2.21. Values of parameters forthis model are obtained from the device geometry and experimental measurements.The mass of the model is known from the device geometry and density of silicon(2330 kg/m3). Using this and the experimentally observed values of the natural fre-quencies at zero perturbation voltage, the stiffness parameter is identified. Damp-ing is incorporated into the Simulink model based on the experimentally measuredQ factors (≈ 100). Given the inherent nonlinearities in the coupling and perturba-tion electrostatic springs, the sine sweep method with 10 Hz frequency step wasfound sufficient to obtain the frequency response of the model, including withinthe half power band width region. The voltage amplitudes were chosen based onthe corresponding experimental values (see Table 4.1). Within the range studied,a linear frequency response has been observed with no jumps. Hence, a compari-37Figure 3.4: Simulink block diagram for the coupled resonator system shownin Figure 3.1son of the sensitivities obtained from the Simulink model with that obtained fromthe analytical model would reveal the accuracy of the approximate closed-formexpressions for the sensitivity norms in Equation 3.25.Comparison of the two modelsFigure 3.5 compares sensitivity norms obtained from analytical model and Simulinkmodel for a coupling voltage of 18 V. As expected from Figure 2.4, the mode38−5 −4 −3 −2 −1 0051015Stiffness Perturbation, ∆k/k (%)Sensitivity Norms (%)  Nω−SimulationNθ−SimulationNω−AnalyticalNθ−AnalyticalFigure 3.5: Comparison of sensitivity norms obtained analytically and fromSimulink model for 18 V coupling voltageshapes are more sensitive to small changes in the stiffness than the natural fre-quencies. The difference between the sensitivities predicted by the analytical andthe Simulink model is negligible. The higher mode shape sensitivity at the highervalues of stiffness perturbation suggests energy localization to one resonator. Athigher degrees of localization, or, lower coupling strength between the resonators,nonlinearities may be important. But the analytical model does not consider it.3.5 Device-23.5.1 Device descriptionA two-DOF MEMS resonator was designed keeping electrostatic coupling springlinear. The device (shown in Figure 3.6) consists of two identical resonators placedside by side. Thickness of the device layer is 25 µm. Each resonator is suspendedby four folded suspensions. Each suspension spring is made of three beams oflengths 565 µm, 540 µm and 565 µm connected in series. In-plane thickness ofthe beams is 6 µm. The central structure is 254 µm in length and 25 µm in width.On the two sides of a resonator, there are combs (60 combs on the left side and61 on the right), half of which are shaped. Maximum thickness of a shaped combis 6 µm and its minimum thickness is 2 µm. Thicker section of a comb is 25 µmlong; thinner section is 10 µm in length and the shaped part has a length of 10 µm.39The shapes are designed such that when two resonators lie side by side with endof a rectangular comb in one resonator lying near the middle of the shaped partof another resonator as in coupling arrangement in Figure 3.6, the force betweenthe two resonators varies linearly with engagement length. Hence the couplingarrangement acts as a linear electrostatic spring, stiffness of which can be var-ied by changing potential difference between the two resonators. Where constantforce between two sets of combs is desired, overlap length between combdrivesis increased so that whole of the shaped part is within overlap region (see overlapbetween static combdrive and combs of a resonator in Figure 3.6). The total lengthof a combdrive is 966 µm. Four thick perturbation beams lie beside a resonatorat four locations (see Figure 3.6) to introduce electrostatic stiffness perturbation.There is a gap of 2 µm between a resonator and a perturbation beam and overlaplength between them is 50 µm. By controlling the voltage difference between aresonator and the associated perturbation beams (called perturbation voltage nowonwards), the stiffness perturbation is varied.Identical perturbation beams and combdrive actuators are present in both res-onators. The device is excited (with a DC-AC voltage combination) on the leftcomb-drive, while the right comb-drive has the same DC level across it as the DCpart of the excitaion voltage, in order to maintain static symmetry. Only the left res-onator was excited externally in our experiments, while the stiffness perturbationwas induced on the right resonator only.3.5.2 Design and fabrication of the deviceThe device was designed in CoventorWare ARCHITECT. The ARCHITECT modelof the device is shown in Figure 3.7. In ARCHITECT, essentially one resonatorunit was designed with all its combs rectangular in shape having in-plane thick-ness the same as the maximum in-plane thickness of a shaped comb and length thesame as the total length of a shaped comb. The dynamics of the resonator unit wasaccessed in ARCHITECT to ascertain that the desired vibration mode (with small-est natural frequency) was far from other vibration modes of the resonator. Thisensures that the undesired vibration modes of the device do not affect the deviceresponse at the desired resonant frequency. Figure 3.8 shows frequency response40!Perturbationbeam Resonator-1 Resonator-2 Coupling arrangement Suspension Excitation Combdrive Figure 3.6: An image of the second two-DOF device41Suspension Proof mass Figure 3.7: ARCHITECT model of one resonator unit of the device (combsnot shaped)of the resonator when different vibration modes are excited separately. The designobtained from ARCHITECT was modified using CoventorWare DESIGN to obtainshaped combs. DESIGN’s capability to draw a curve from a given equation wasutilized for this purpose. Afterwards, two identical resonators were placed side byside in DESIGN to obtain design of the device shown in Figure 3.9. Finally thedesign was exported as a GDSII file and CleWin was used for layout design. Usingthis layout, the devices were fabricated by MEMSCAP using SOIMUMPs and itwas returned to us packaged in a 68 pin ceramic pin-grid array (CPGA) package.42Figure 3.8: Frequency response of six modes of vibration of one resonatorunit with unshaped combs3.5.3 Device modelsThe device was modeled as a two-DOF spring-mass system with damping. Ananalytical model as well as a Simulink model of the device was developed. Themodels have been described in the following sections.Analytical modelTo model the device in Figure 3.6 as a two-DOF spring mass system, equivalentmass and equivalent stiffness of the resonators as well as equivalent coupling stiff-ness needs to be calculated. Each resonator in the coupled system is identical. So,we need to calculate equivalent mass and stiffness for only one resonator. Equiv-alent mass of a resonator is sum of mass of the proof-mass, which includes massof combdrives, and equivalent mass of the suspension beams at the point of attach-ment to the proof-mass. The equivalent mass of suspension beams was obtainedusing kinetic energy equivalence. For kinetic energy calculation, deformed shape43Figure 3.9: Design of the two-DOF device as obtained from CoventorWareDESIGN44g!"g1 l x Figure 3.10: Shaped comb profileof each beam in a suspension is assumed to follow quasi static deformation shape ofa fixed-guided Euler-Bernoulli beam. Equivalent mass of suspension for Device-1was also calculated similarly.In this device, resonators are coupled using shaped combs to ensure linearity ofcoupling. The shape is obtained by following [35]. Electrostatic potential energyUc for the arrangement of fingers as shown in Figure 3.10 with potential differenceVc is given by:Uc =−12CV 2c . (3.32)Force between two sets of fingers is given by:fc =∂U∂x =−12∂C∂x V2c . (3.33)Linearity of coupling force requires ∂C∂x to be linear in x. First we determine changein capacitance as overlap between two sets of fingers is changed from x to x+dx.Assuming slope of the curved finger to be very small such that parallel plate theoryfor capacitance is valid, change in capacitance can be written as:dC =εtdxg. (3.34)From the above, rate of change of capacitance with change in x can be determined.∂C∂x =εtg. (3.35)We can make ∂C∂x linear in x if 1/g varies linearly with x. We assume the followingform for 1/g.1g= ax+b. (3.36)45Now, constants a and b need to be determined. We use limitation on gap width.Assuming maximum gap width to be g1 at x = 0 and minimum gap width to be g2at x = l, the constants are evaluated to be:a =1l(1g2−1g1), b =1g1. (3.37)The gap profile is given by:g =[1l(1g2−1g1)x+1g1]−1. (3.38)By substituting the above in Equation 3.35, ∂C∂x is obtained.∂C∂x = εt(1l(1g2−1g1)x+1g1). (3.39)Force is calculated by substituting the above in Equation 3.33.fc =−12(1l(1g2−1g1)x+1g1)εtV 2c . (3.40)Force in a combdrive with N movable combs is given by:Fc = 2N fc =−N(1l(1g2−1g1)x+1g1)εtV 2c . (3.41)Notice that in force calculation in a combdrive the factor of two appears becauseshaped combs are designed on both sides of a rectangular comb. The force hastwo parts: the constant part is akin to force in a combdrive with rectangular combsand it acts to change equilibrium position of the resonators. The part of the forcevarying with overlap length x is the source of linear coupling spring with stiffnessgiven below.kc =−12l(1g2−1g1)εtV 2c . (3.42)Unlike mechanical springs, the stiffness is negative. A big advantage of this springis that stiffness can be controlled by controlling potential difference between thetwo sets of combs in a combdrive. We also notice in the above expression that a46smaller minimum gap (g2) gives higher magnitude of coupling stiffness, howeverfabrication process sets a minimum limit of 2 µm on gap width. In our design, wechose maximum gap to be 4 µm and length of the curved part of a comb (l) to be10 µm. Total number of combs in a combdrive (N) is 60. The choice was made toensure that total length of a combdrive is smaller than 1000 µm so that constrainton suspension beam length imposed by fabrication process is met.The analysis above ignores contribution of fringe fields. However, the experi-ments described in the Section 4.5 confirm that coupling spring is indeed linear.Perturbation is also applied through electrostatic means. There are four per-turbation beams placed beside a resonator with a gap of 2 µm between them (seeFigure 3.6). To determine perturbation in stiffness, we start with electrostatic po-tential energy.Up = 2×(−12εApgp− xV 2p −12εApgp+ xV 2p). (3.43)where Ap, gp, x and Vp are the overlap area between a perturbation beam and theresonator, the gap between a perturbation beam and the resonator, the displacementof the resonator mass and the potential difference between a perturbation beam andthe resonator respectively. Adding the two terms in the expression above, we get:Up =−2gpg2p− x2εApV 2p . (3.44)For small displacements of the mass, the potential energy Up can be approximatedby its Taylor series expansion up to first order.Up ≈−2gp(1+x2g2p)εApV 2p ≈−2gpεApV 2p −124εApV 2pg3px2. (3.45)The first term in the expression above is a constant and contributes to reference po-tential at equilibrium configuration. The second term is similar to potential energyexpression for a spring with stiffness given by:kp =−4εApV 2pg3p. (3.46)47Notice that the resulting spring has negative stiffness. This is the stiffness pertur-bation introduced by perturbation beams (∆k = kp) and it is always negative. Itsmagnitude can be controlled by changing the perturbation voltage Vp.Parameter values for analytical model of the device are based on design values,except perturbation gap and stiffness perturbation due to built-in asymmetry in thedevice. Perturbation gap was obtained by direct measurement on a device under amicroscope. Initial stiffness perturbation was obtained from dynamic response ofthe device when no external perturbation was applied.We have a complete analytical model of the device. Now, a Simulink model ofthe device is developed.Simulink modelThe analytical model is based on two assumptions: 1) gaps between a resonator andall of its perturbation beams are equal at equilibrium and 2) displacements are small(introduced in Equation 3.45). A more general Simulink model Figure 3.11 of thedevice, without these assumptions, and incorporating nonlinearities in electrostaticforces in the stiffness perturbation arrangement, is developed for the more accuratecomputation of the sensitivity norms in Equation 2.21. Unlike the analytical model,the parameter values for the Simulink model are obtained from the device geometryand experimental measurements. The mass of a resonator in the device is knownfrom the geometry and density of silicon (2330 kg/m3). Using this and the exper-imentally obtained natural frequency and mode shape, the stiffness parameters areidentified. Damping for the Simulink model is calculated from the experimentallymeasured Q factors (≈ 30 for in-phase, and≈ 24 for out-of-phase mode) assumingproportional damping. The frequency response of the Simulink model is obtainedby performing a sine-sweep test on the model numerically. For the test, the fre-quency was increased in steps of 25 Hz covering a range of 6 kHz–7.2 kHz. Thevoltage amplitudes were chosen based on the corresponding experimental values(see Table 4.2).48Figure 3.11: Simulink block diagram for the coupled resonator system shown in Figure 3.649Within the range studied, a linear frequency response is observed. Normalmodes and corresponding natural frequencies of the system are obtained from thefrequency response of the model and their sensitivities are calculated using Equa-tion 2.21. A comparison between sensitivities obtained from this model and theanalytical model would reveal the accuracy of the analytical model.Comparison of the two modelsFigure 3.12 compares the sensitivity norms obtained from the analytical modeland the Simulink model for a coupling voltage of 40 V. Note that experimentalresults suggest that the device has residual asymmetry due to fabrication tolerances,causing an initial perturbation (δ0) of 6.85% in the device. Hence, in the plot, theright most points correspond to no external perturbation. The external electrostaticperturbation is bringing the device towards symmetry.The difference between the slopes of the sensitivity norm curves obtained fromthe analytical and the Simulink models is large. This difference results from thefact that the actual coupling stiffness in the tested device (also the coupling stiff-ness value used in the Simulink model) has smaller magnitude as compared to theanalytical coupling stiffness. This difference can be attributed to fabrication toler-ances. Fabricated comb shapes are not the same as the designed shapes, resultingin weaker than predicted coupling stiffness.Another important point in the plot to note is that for the same perturbationvoltages (0, 4, 6, 8, 10, 12, 14, 15 and 16 V), the analytical perturbation is smallerthan the perturbation values obtained from the Simulink results. When the excita-tion and coupling voltages are applied to the coupled resonator system, the systemvibrates about its equilibrium configuration, which is different from the initial con-figuration. In the equilibrium configuration, the perturbation gap near the four per-turbation beams of a resonator do not remain the same. In this case, on applicationof a perturbation voltage, perturbation introduced in the system is higher than thecase when perturbation gaps are uniform. The analytical model assumes uniformperturbation gap resulting in underestimation of stiffness perturbation. But theSimulink model accurately incorporates the nonuniformity of perturbation gaps.As expected from Figure 2.4, the mode shapes are more sensitive to small changes50−1 0 1 2 3 4 5 6 701020304050Stiffness Perturbation, ∆k/k (%)Sensitivity Norms (%)in−phase  Nω−SimulationNθ−SimulationNω−AnalyticalNθ−Analytical−1 0 1 2 3 4 5 6 701020304050Stiffness Perturbation, ∆k/k (%)Sensitivity Norms (%)Out−of−phase  Nω−SimulationNθ−SimulationNω−AnalyticalNθ−AnalyticalFigure 3.12: Comparison of sensitivity norms obtained analytically and fromSimulink model for 40 V coupling voltagein the stiffness than the natural frequencies.3.6 Three-DOF MEMS resonator device3.6.1 Device descriptionA three-DOF MEMS resonator was also designed keeping the electrostatic cou-pling spring linear. The device (shown in Figure 3.13) consists of three identical51Resonator-1 Resonator-2 Resonator-3 Excitation Combdrive Suspension Coupling arrangement Perturbationbeam Figure 3.13: An image of the three-DOF device52resonators placed side by side. These resonators are the same as the ones describedin Section 3.5.1 for Device-2. Refer to Section 3.5.1 for details of the resonators.3.6.2 Design and fabrication of the deviceA resonator unit designed for Device-2 (see Section 3.5.2) was also utilized to de-sign Device-3. After a resonator unit was designed, three of them were placed sideby side in CoventorWare DESIGNER’s layout editor to obtain design of the deviceand it is shown in Figure 3.14. Finally the design was exported as a GDSII fileand CleWin was used for layout design. Designs of both of the devices , Device-2and Device-3, were placed on the same chip. Using this layout, the devices werefabricated by MEMSCAP using SOIMUMPs and it was returned to us packaged ina 68 pin ceramic pin-grid array (CPGA) package. One packaged unit had two setseach of Device-2 and Device-3.3.6.3 Device modelsThe device was modeled as a three-DOF spring-mass system with damping. Equiv-alent mass and spring stiffness of each resonator and coupling stiffness betweenresonators in the device is obtained in the same way as described for Device-2.Refer to Section 3.5.3 for analytical model. Simulink model is also similar to theabove two-DOF device (Section 3.5.3) except that three resonators are coupled inthis case instead of two in two-DOF device model.Comparison of the two modelsFigure 3.15 compares sensitivity norms obtained from analytical model and Simulinkmodel for a coupling voltage of 40 V. Note that the experimental results suggestthat the device has a small residual asymmetry due to fabrication tolerances (rightmost point in the plot corresponds to no external perturbation). It is overcome onapplication of a small perturbation voltage and the device starts to move away fromsymmetry on application of higher perturbation voltages.Analytical and Simulink based sensitivity curves have similar slopes near zerostiffness perturbation point, suggesting that non-dimensional coupling stiffness (κ)in the device is close to analytically predicted value in this device. However, the53Figure 3.14: Design of the three-DOF device as obtained from CoventorWareDESIGNcurves diverge very quickly showing the range of perturbation for which approxi-mate analytical expression for sensitivity is valid.Another important point in the plot to note is that for the same perturbationvoltages, analytical perturbation is smaller than perturbation values obtained fromSimulink results. When the excitation and coupling voltages are applied to thecoupled resonator system, the system vibrates about its equilibrium configuration,which is different from the initial configuration. In the equilibrium configuration,the perturbation gap near the four perturbation beams of a resonator do not remainthe same. In this case, on application of a perturbation voltage, perturbation intro-542!" 2# 2$ 2% 2& " &"'!"!'&"&'("('%"%''")*+,,-.//01.2*3245*+6-70!8980:;<).-/+*+=+*>0?62@/0:;<!"#$!%  &"!'()*+,-(".&#!'()*+,-(".&"!/.,+0-(1,+&#!/.,+0-(1,+1 2!" 2# 2$ 2% 2& " &"&%$#!"!&!%!$!#'()**+,--./,0(1023()4+5.!676.89:',+-)();)(<.=40>-.89:!"#$!%  &"!'()*+,-(".&#!'()*+,-(".&"!/.,+0-(1,+&#!/.,+0-(1,+2!" 2# 2$ 2% 2& " &"!"&"'"%"("$")*+,,-.//01.2*3245*+6-70!8980:;<).-/+*+=+*>0?62@/0:;<!"#$!%  &"!'()*+,-(".&#!'()*+,-(".&"!/.,+0-(1,+&#!/.,+0-(1,+1 Figure 3.15: Comparison of sensitivity norms obtained analytically and fromSimulink model for 40 V coupling voltage55duced in the system is higher than the case when perturbation gaps are uniform.The analytical model assumes uniform perturbation gap resulting in underestima-tion of stiffness perturbation. But the Simulink model accurately incorporates thenonuniformity of perturbation gaps. This is observed in Device-2 as well, how-ever relatively weaker suspension springs in Device-3 as compared to Device-2exacerbate the effect, resulting in the large difference observed in this case.3.7 Summary and discussionDesign and fabrication of devices developed to study veering has been discussedin this chapter. Fabrication process employed has been briefly described. Threecoupled resonator MEMS devices have been described. Device-1 has its two res-onators coupled electrostatically using parallel plate capacitors which has inherentnonlinearity. To remove nonlinearity in coupling, Device-2 was developed. Toimprove sensitivity, Device-3 (three-DOF device) was developed. Analytical aswell as Simulink models of the devices have been developed to predict the de-vice response and to gauge effect of different parameters on mode localization inthe coupled resonator devices. Energy based principles have been employed to de-velop analytical model of the devices. Analytically obtained sensitivity norms havebeen compared with sensitivity norms obtained from Simulink model to verify thevalidity of the analytical model. For Device-1, there is a good match between sen-sitivity norms obtained from the two models indicating accuracy of the analyticalmodel. However, for Device-2 and Device-3, match is not as good, suggestingsimple analytical models for these devices are not accurate.56Chapter 4Experiments4.1 IntroductionThe MEMS devices described in Chapter 3 were characterized and the experimen-tal results are presented in this chapter. A description of the experimental set-upwill be given and the characterization procedure will be explained. For Device-1 (Figure 3.1), only the in-phase mode of vibration was characterized for threecoupling voltages. For Device-2 (Figure 3.6), both, the in-phase as well as the out-of-phase mode, were characterized for two coupling voltages. For Device-3 (Fig-ure 3.13), all the modes with perturbation in the end resonator were characterizedusing one of the devices for one coupling voltage. However, experimental resultsfor differential perturbation could not be obtained as all the other three-DOF de-vices had large asymmetry introduced by the fabrication process. The asymmetryhindered the propagation of vibration in the system resulting in a small vibrationamplitude of the second and the third masses even when the first mass was beingexcited at its resonance frequency and its amplitude of vibration was large.From the experimental results, the sensitivity of the devices will be calculatedand it will be compared with the analytical and Simulink based sensitivity resultsobtained in Chapter 3.57!!!!!!!!!!!!!!!"#$%&'(#)*++,-#.'/0#1230*('#4+0256,#7'68#"%9+*0'(#:;0'(<65'#Figure 4.1: A view of the experimental set-up used to characterize the MEMSdevices4.2 Experimental set-upThe device was characterized using the Planar Motion Analysis (PMA) tool inPoytec Microsystem Analyzer (MSA-500) equipment [36]. Figure 4.1 shows theexperimental set-up used to characterize the MEMS devices. The set-up consistsof the Polytec MSA equipment with computer interface, a signal amplifier, anda DC power source. The Polytec MSA has an integrated signal generator. ThePMA tool of the Polytec equipment, which measures in-plane vibration, was usedin the experiment. The PMA tool uses stroboscopic video microscopy to measurevibration. It can measure the vibration of two separate objects in the field of viewof the microscope at the same time. This allows for simultaneous measurement ofvibration of two masses in a coupled resonator system.4.3 Experimental procedureTo start the measurement, the required electrical connections are made to supplyvoltages to the device to be characterized. Constant electrostatic potentials aremaintained by the DC power sources and the excitation signal is supplied by the58integrated signal generator of the MSA equipment and controlled through the PMAinterface. Thereafter, two rectangular measurement regions on two resonators (onelocation on each resonator) are selected. The PMA measures the in-plane vibra-tion of these regions. After ensuring that the required voltages are being suppliedto the device, a sine-sweep test in a frequency region surrounding the natural fre-quencies of the coupled resonator system is conducted and frequency response isobtained. For the sine-sweep test, the excitation frequency is increased in equalsteps and the step-size is chosen such that the frequency response within the half-power bandwidth, the region with very high curvature, is a smooth curve. Also, inthe sine-sweep test when the excitation frequency is increased from one value tothe next, sufficient settling time is allowed before the steady state vibration ampli-tudes of each resonator mass at the new excitation frequency is recorded. Theseparameters for the sine-sweep test are specified through the PMA interface. Atypical frequency response is shown in Figure 4.3.4.4 Characterization of Device-1To start with, the required electrical connections are made as shown in Figure 4.2.The left resonator is grounded and is excited sinusoidally by a combdrive which isconnected to the built-in signal generator of the PMA system. The coupling volt-age is maintained by controlling potential of the right resonator. The perturbationis changed by changing the potential of the perturbation beam. Once a couplingvoltage and a perturbation voltage is fixed, a sine-sweep test in a frequency regionsurrounding the in-phase normal mode is conducted and the frequency response ofthe coupled resonator system is obtained. The choice of the in-phase mode ensuredminimization of the effects of both the squeeze film damping and the nonlinear de-pendence of the coupling on the gap width, as it remains practically constant inthis vibration mode. In the sine-sweep test, the excitation frequency was increasedin steps of 10 Hz in the range of 15.5 kHz-17 kHz. The amplitude-frequency de-pendence for the given perturbation voltages were measured by allowing a settlingtime of 1 s. A least-square curve fit method is used to extract the modal parame-ters of the system. A typical frequency response and the corresponding curve-fitis shown in Figure 4.3. The modal parameters are then used to calculate the mode59!"#$#%&'%()*$+,$-./*&$01--)2$!"#$+,3$!4#$-5-$0'6*$/%7*$89'0$.:$;2<<*=&2$Figure 4.2: Schematic showing electrical power connections for 18 V cou-pling case (only symmetrical upper half is shown)sensitivity of the coupled resonator system using Equation 2.21.The experiment was carried out for three coupling voltages: 18 V, 10 V and8 V. A maximum of 18 V coupling was chosen to obtain the response of the deviceat stronger coupling levels, without being affected by pull-in instability. Couplingvoltages lower than 8 V were not applied, due to the fact that the limited qual-ity factor of the device would then limit the distinction between the two resonantmodes [37]. An intermediate coupling voltage of 10 V was also used to obtain acomparative device response. For each coupling voltage, the perturbation in thesystem was varied and the frequency response of the system was obtained. Thecoupling and perturbation voltages applied in the experiment are summarized inTable 4.1.4.4.1 Results and discussionFigure 4.4 shows the variation of the sensitivity norms with (normalized) stiffness-perturbations induced by different perturbation voltages, for the simulated andthe experimental results, respectively. A comparison of the sensitivity norms ob-6015 15.5 16 16.5 17 17.500.10.20.3Frequency (kHz)Magnitude (µm)First Resonator  MeasuredFitted15 15.5 16 16.5 17 17.500.10.20.3Frequency (kHz)Magnitude (µm) Second Resonator  MeasuredFittedFigure 4.3: Typical frequency response data and associated fourth ordercurve fitTable 4.1: Voltages used in the experimentCoupling PerturbationVoltage (V) Voltage (V)18 0, 5, 7.5, 10, 14, 16, 18, 20, 21, 22, 2310 0, 5, 10, 12, 14, 16, 18, 20, 21, 22, 23, 248 0, 2.5, 5, 7.5, 10, 12, 14, 16, 18, 20, 21, 22, 23, 24tained experimentally and predicted by the analytical expression Equation 2.22and Simulink model are shown in Figure 4.4 (a)-(c). The magnitude of the cou-pling stiffness decreases with decreasing coupling voltage, based on Equation 2.23,leading to an increase in sensitivity, as in Figure 2.4. The average slopes of the lin-ear part of the normal mode sensitivity curves for 18 V, 10 V and 8 V couplingare -2.6, -9.8 and -16.4 (slopes obtained from analytical and Simulink model arewithin 5% of the experimental values) respectively; as expected, an increase in61(a)−5 −4 −3 −2 −1 0051015Stiffness Perturbation, ∆k/k (%)Sensitivity Norms (%)  Nω−ExptNθ−ExptNω−SimulationNθ−SimulationNω−AnalyticalNθ−Analytical(b)−6 −5 −4 −3 −2 −1 0 101020304050Stiffness Perturbation, ∆k/k (%)Sensitivity Norms (%)  Nω−ExptNθ−ExptNω−SimulationNθ−SimulationNω−AnalyticalNθ−Analytical(c)−6 −5 −4 −3 −2 −1 0 1020406080Stiffness Perturbation, ∆k/k (%)Sensitivity Norms (%)  Nω−ExptNθ−ExptNω−SimulationNθ−SimulationNω−AnalyticalNθ−AnalyticalFigure 4.4: Comparison of sensitivity norms for coupling voltages a) 18 V b)10 V and c) 8 V62the slope magnitude is noted with a decrease in coupling voltage. These slopesare inversely proportional to the nondimensional coupling stiffness κ (see Equa-tion 2.23). Based on Equation 3.29, one would expect them to be inversely pro-portional to the square of coupling voltage. However, that is not the case becauseapplication of the coupling voltage changes the equilibrium gap between the cou-pling plates gv also, destroying the expected proportionality relation between thecoupling stiffness and the square of coupling voltage. Moreover, the comparisonof slopes of the normal mode sensitivity curve with that of the frequency sensitiv-ity curve near origin shows that the normal mode sensitivity is approximately 10,40 and 70 times the frequency sensitivity for the three coupling voltage cases of18 V, 10 V and 8 V respectively. However, the degree of agreement among the-ory, simulation and experiments diminishes with decreased coupling voltages. Thenormal mode sensitivity curve for smaller coupling saturates at higher stiffness-perturbation levels. This can be reconciled with a residual built-in asymmetry inthe device. The initial, externally unperturbed configuration of the system is notsymmetric and hence the axes in the plot in Figure 4.4 are actually shifted axes asshown in Figure 4.5. We can see from Figure 4.5 the linear part of the mode shapesensitivity curve is small and the sensitivity reaches saturation part of the curvevery soon. The effect of residual asymmetry is insignificant for higher couplingvoltages, as the sensitivity curve has large linear region and total stiffness pertur-bations the system experiences lie in the linear part of the mode shape sensitivitycurve.4.5 Characterization of Device-2Measurement on this device was started by characterizing each resonator to obtaintheir individual frequency response. This required disabling the coupling betweenthe resonators which was readily achieved by keeping the coupling voltage (differ-ence between potentials of the two resonators) equal to zero. Both, the resonatorsand their perturbation beams, were grounded. A rectangular measurement regionon one of the resonators was selected after this. PMA measures the vibration ofthis region. Thereafter, an excitation signal (40 V dc + 400 mV p-p sinusoidal),obtained by amplifying the signal from the built-in signal generator of the PMA632! 2" 2# 2$ 2% 2& ''%'#'!'(')*+,,-.//01.2*3245*+6-70!8980:;<<).-/+*+=+*>0?62@/0:;<  !""##$!%&%'!$"##$!%&%'!""##$!%&'!$"##$!%&'Pertubraionm  Rusmi -1r2uRCipRsueslgurSai-1r2uRCinRanrurErumixcdcviFigure 4.5: Shift in origin of sensitivity- stiffness perturbation plot caused bybuilt-in asymmetry in a coupled resonator device is shown.system, was supplied to the static combs beside the resonator to be characterized.A sine-sweep test with a 10 Hz frequency step was carried out in the range aroundthe natural frequency of the resonator and the amplitude-frequency dependencewas measured. In the test, a settling time of 1 s was allowed before the steady statedisplacement amplitude of the resonator was recorded. Figure 4.6 (a)-(b) shows thefrequency response of the two resonators. Their natural frequencies are 7.14 kHzand 6.95 kHz respectively. The difference in the natural frequencies of the two res-onators of a coupled resonator system leads us to conclude that the device has aninitial asymmetry built into it. Further experiments on the coupled system confirmthis observation and help quantify the initial asymmetry.To check the linearity of the suspension spring, a test similar to the test de-scribed above to obtain the frequency response of a resonator was carried out butwith larger amplitudes of vibration of the resonator. To increase the amplitude ofvibration, the force magnitude was increased by increasing the AC part of the exci-tation signal (DC part was fixed to be 40V). For the different force amplitudes, thefrequency response of the resonator was obtained which was in turn used to findthe natural frequency of the resonator. Figure 4.7 shows that the natural frequencydoes not change suggesting linearity of the suspension spring.To characterize the perturbation spring, the frequency response of a resonator64(a)5500 6000 6500 7000 7500 8000 850000.10.2Frequency (Hz)Amplitude (µm)5500 6000 6500 7000 7500 8000 8500−200−1000100Frequency (Hz)Phase (degree)(b)5500 6000 6500 7000 7500 8000 850000.10.2Frequency (Hz)Amplitude (µm)5500 6000 6500 7000 7500 8000 8500−200−1000Frequency (Hz)Phase (degree)Figure 4.6: Frequency response of a) resonator-1 and b) resonator-2 of a cou-pled resonator device650.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.259365938594059425944594659485950Force amplitude (µN)Natural frequency (Hz)Figure 4.7: Variation of the natural frequency with amplitude of the excita-tion force−50 0 50 100 150 200 250 3005650570057505800585059005950Square of Perturbation voltage (V2)Natural frequency (Hz)  ExperimentLinear fitFigure 4.8: Variation of the natural frequency with the square of the pertur-bation voltage66−200 0 200 400 600 800 1000 1200 1400 1600 180058505860587058805890590059105920593059405950Square of Coupling voltage (V2)Natural frequency (Hz)  ExperimentLinear fitFigure 4.9: Variation of the natural frequency of a resonator with the squareof the coupling voltage when the other resonator is constrainedwas measured after applying a perturbation voltage to the resonator. For differentperturbation voltages, the frequency response of the resonator was obtained, whichwas in turn used to find the natural frequency of the resonator. Figure 4.8 showsthe variation of the natural frequency with the square of the perturbation voltage.The linearity of the plot suggests linearity of the perturbation spring.To characterize the coupling spring, the frequency response of a resonator wasmeasured after applying a coupling voltage and constraining the motion of the otherresonator in the coupled resonator device using a probe. For different couplingvoltages, the frequency response of the resonator was obtained, which was thenused to find the natural frequency of the resonator. Figure 4.9 shows the variationof the natural frequency with the square of the coupling voltage. The linearity ofthe plot points to the linearity of the coupling spring.After characterizing each resonator, suspension spring, electrostatic perturba-tion spring and electrostatic coupling spring, we move to characterize the coupledresonator system. To start with, electrical connections, as shown in Figure 4.10, aremade. Resonator-1 (left resonator) is grounded and the excitation voltage is appliedto the static combs beside it. The coupling voltage is maintained by controlling67!"#$#!"#$#%&'#(""#)$#*+*#,-./#012/#$13-145/#%&#*60/3#,7**58#Figure 4.10: Schematic showing electrical power connections for 40 V cou-pling case (only symmetrical upper half is shown)the potential of resonator-2 (right resonator). The combdive beside resonator-2 ismaintained at a potential such that the voltage difference across it is the same as theDC part of the excitation voltage. The perturbation is changed by changing the po-tential of the perturbation beam. Thereafter, two rectangular measurement regionson the two resonators (one location on each resonator) are selected. PMA measuresthe vibration of these two regions simultaneously. Once a coupling voltage and aperturbation voltage is established, a sine-sweep test in a frequency region sur-rounding the normal modes of the system is conducted and the frequency responseof the coupled resonator system is obtained. For the sine-sweep test, the excitationfrequency was increased in steps of 10 Hz from 6 kHz to 7.2 kHz. The amplitude-frequency dependence for a given perturbation voltage was measured by allowinga settling time of 1 s before the steady state displacement amplitudes of each res-onator were recorded. A system identification tool was employed to perform leastsquare curve fit on the frequency response and obtain the modal parameters as well68(a)6000 6200 6400 6600 6800 7000 720000.20.4Frequency (Hz)Amplitude (µm)6000 6200 6400 6600 6800 7000 7200−200−1000Frequency (Hz)Phase (degree)(b)6000 6200 6400 6600 6800 7000 720000.10.2Frequency (Hz)Amplitude (µm)6000 6200 6400 6600 6800 7000 7200−2000200Frequency (Hz)Phase (degree)Figure 4.11: Typical amplitude-frequency response measured at a) resonator-1 b) resonator-2 and associated fourth order curve fit69Table 4.2: Voltages used in the experimentCoupling PerturbationVoltage (V) Voltage (V)40 0, 4, 6, 8, 10, 12, 14, 15, 1635 0, 4, 6, 8, 10, 12, 14, 15, 16as the stiffness parameters of the system. In the system identification process, weassume that the resonators have the same mass and it equals the value obtained an-alytically from the device geometry. Also, the eigenvectors obtained from the sys-tem identification are not orthogonal. Gram-Schmidt orthogonalization has beenutilized to obtain orthogonal eigenvectors assuming eigenvector corresponding tothe in-phase mode to be accurate. A typical frequency response and the associatedcurve-fit is shown in Figure 4.11. The modal parameters are used to calculate themode sensitivity of the coupled resonator system using Equation 2.21.The experiment was carried out for two coupling voltages: 40 V and 35 V. The40 V coupling voltage was chosen to obtain the response of the device at a strongercoupling level. Coupling voltages lower than 35 V were not applied, due to thefact that the limited quality factor of the device would then limit the distinctionbetween the two resonant modes [37]. For each coupling voltage, the perturbationin the system was varied and the frequency response of the system was obtained.The coupling and perturbation voltages applied in the experiment are summarizedin Table 4.2.4.5.1 Results and discussionFigure 4.12 and Figure 4.13 show veering of eigenfrequencies caused by stiffnessperturbation in a coupled two-DOF resonator device for the cases of 40 V and 35 Vcoupling respectively. By comparing the two plots, it can be observed that veeringis sharper for smaller coupling. Figure 4.14 and Figure 4.15 show the variation ofthe sensitivity norms with stiffness perturbations for the coupling voltages of 40 Vand 35 V respectively, and compares between analytical, simulation and the exper-imental results. The agreement between the experimental and the Simulink modelbased sensitivities is very good. However, the analytical results do not match well70−2 0 2 4 6 86.46.56.66.76.86.9Stiffness Perturbation, ∆k/k (%)Natural frequency (kHz)Figure 4.12: Veering diagram for the two-DOF device with 40 V coupling−4 −2 0 2 4 66.46.56.66.76.86.9Stiffness Perturbation, ∆k/k (%)Natural frequency (kHz)Figure 4.13: Veering diagram for the two-DOF device with 35 V couplingwith them. The difference between the analytical and the Simulink model basedsensitivities has been discussed in Section 3.5.3. For this device, the magnitudeof coupling stiffness decreases proportionally with the square of the coupling volt-age, leading to an increase in sensitivity. The average slopes of the linear part ofthe normal mode sensitivity curves (near origin in Figure 4.14 and Figure 4.15)for 40 V and 35 V coupling are 6.4 and 8.7 respectively; as expected, an increasein the slope magnitude is noted with a decrease in the coupling voltage. More-over, The slopes are inversely proportional to the square of the coupling voltageswith constant of proportionality showing a variation of less than 5% between thetwo cases. The comparison of the slopes of the normal mode sensitivity curvewith that of the frequency sensitivity curve near origin shows that the normal mode71−1 0 1 2 3 4 5 6 701020304050Stiffness Perturbation, ∆k/k (%)Sensitivity Norms (%)In−phase  Nω−ExptNθ−ExptNω−SimulationNθ−SimulationNω−AnalyticalNθ−Analytical−1 0 1 2 3 4 5 6 7−1001020304050Stiffness Perturbation, ∆k/k (%)Sensitivity Norms (%)Out−of−phase  Nω−ExptNθ−ExptNω−SimulationNθ−SimulationNω−AnalyticalNθ−AnalyticalFigure 4.14: Comparison of in-phase and out-of-phase mode sensitivitynorms for 40 V coupling voltagesensitivity is approximately 25 and 30 times the frequency sensitivity for the twocoupling voltage cases of 40 V and 35 V, respectively. Furthermore, it should benoted here that on the x-axis of the sensitivity plots Figure 4.14 and Figure 4.15,the total stiffness perturbation in the devices (the sum of perturbation due to built-in asymmetry and the external perturbation) is shown. The rightmost point on thetwo plots correspond to no external perturbation i.e. the devices have a built-inasymmetry resulting in an initial stiffness perturbation and the external perturba-tion brings them towards symmetry. Use of such devices in measurement requires72−3 −2 −1 0 1 2 3 4 5 605101520253035404550Stiffness Perturbation, ∆k/k (%)Sensitivity Norms (%)In−phase  Nω−ExptNθ−ExptNω−SimulationNθ−SimulationNω−AnalyticalNθ−Analytical−3 −2 −1 0 1 2 3 4 5 605101520253035404550Stiffness Perturbation, ∆ k/k (%)Sensitivity Norms (%)Out−of−phase  Nω−ExptNθ−ExptNω−SimulationNθ−SimulationNω−AnalyticalNθ−AnalyticalFigure 4.15: Comparison of in-phase and out-of-phase mode sensitivitynorms for 35 V coupling voltage73that sensitivity at the initial perturbation should lie on the linear part of the sensitiv-ity curve. This is achievable by making coupling in the devices strong. However,strong coupling makes a device less sensitive.4.6 Characterization of Device-3This device consists of the same resonator units as Device-2 (Figure 3.6). Hence,the linearity of the suspension spring, the coupling spring and the perturbationspring of this device follows from their linearity shown experimentally for Device-2 (see Section 4.5). Therefore, in this section we start from the characterization ofthe coupled resonator system. To start the measurement, the electrical connections,as shown in Figure 4.16, are made. Resonator-1 (leftmost resonator) is groundedand the excitation voltage is applied to the combdrive beside it. Resonator-3 (right-most resonator) is also grounded and the combdrive beside it is maintained at apotential such that the voltage difference across it is the same as the DC part of theexcitation voltage. The coupling voltage is maintained by controlling the poten-tial of resonator-2. This ensures an equal coupling stiffness between resonator-1and resoantor-2, and resonator-2 and resonator-3. Stiffness perturbation is changedelectrostatically by changing the potential of the perturbation beam beside resonator-3. Thereafter, two rectangular measurement regions on resonator-1 and resonator-2(one location on each resonator) are selected. PMA measures the vibration of thesetwo regions simultaneously. Once a coupling voltage and a perturbation voltage isestablished, a sine-sweep test in a frequency region surrounding the normal modesof the system is conducted and frequency response of the coupled resonator sys-tem is obtained. For the sine-sweep test, the excitation frequency was increased insteps of 10 Hz in the range from 4.2 kHz to 5.4 kHz. A settling time of 1 s wasallowed before the steady state displacement amplitudes of each resonator wererecorded. After obtaining the frequency response of the first two resonators of thecoupled resonator system, the device is moved to bring resonator-2 and resonator-3 in the field of view of the microscope. Now, two rectangular regions on thesetwo resonators are selected and a sine-sweep test with the same parameters as de-scribed for the test with the first two resonators is carried out. For resonator-2, theresponse from the two measurements is averaged. A system identification tool was74!"#$#!"#$#%&'#!""#($#)*)#+,-.#/01.#!"#$##$02,034.#%&#)5/.2#+6))47#Figure 4.16: Schematic showing electrical power connections for 40 V cou-pling case (only symmetrical upper half is shown)employed to perform least square curve fitting on the frequency response and ob-tain the modal parameters as well as the stiffness parameters of the system. In thesystem identification process, we assume that the resonators have the same massand it equals the value obtained analytically from the device geometry. Also, theeigenvectors obtained from the system identification are not orthogonal. Gram-Schmidt orthogonalization has been utilized to obtain orthogonal eigenvectors as-suming eigenvector corresponding to the highest natural frequency (third mode) tobe accurate. A typical frequency response and the associated curve-fit is shown inFigure 4.11. The modal parameters are used to calculate the mode sensitivity ofthe coupled resonator system using Equation 2.21. A typical frequency responseand associated curve-fit is shown in Figure 4.17. The modal parameters thus ob-tained are used to calculate mode sensitivity of the coupled resonator system usingEquation 2.38.The experiment was carried out for 40V coupling voltage. Coupling voltageslower than 40 V were not applied, due to the fact that propagation of vibration tothe mass farthest from the excited mass was very small introducing large noise inmeasurement. Experiments for other coupling voltages could not be carried out asthe device got pulled-in and other three-DOF devices were rendered useless dueto high degree of built-in asymmetry in them. For the given coupling voltage, the75(a)4200 4400 4600 4800 5000 5200 540000.20.4frequency (Hz)Amplitude (µm)4200 4400 4600 4800 5000 5200 5400−200−1000frequency (Hz)Phase (degree)(b)4200 4400 4600 4800 5000 5200 540000.10.2frequency (Hz)Amplitude (µm)4200 4400 4600 4800 5000 5200 5400−2000200frequency (Hz)Phase (degree)(c)4200 4400 4600 4800 5000 5200 540000.10.2frequency (Hz)Amplitude (µm)4200 4400 4600 4800 5000 5200 5400−600−400−2000frequency (Hz)Phase (degree)Figure 4.17: Typical amplitude-frequency response measured at a) resonator-1 b) resonator-2 c) resonator-3 and associated sixth order curve fit76−10 −8 −6 −4 −2 0 24.44.64.855.2Stiffness Perturbation, ∆k/k (%)Natural frequency (kHz)Figure 4.18: Veering diagram for the three-DOF device with 40 V couplingand perturbation in the end resonatorperturbation in the system was varied and the frequency response of the systemwas obtained. Experiments were carried out for perturbation voltages of 0 V, 4 V,6 V, 8 V, 10 V, 12 V and 14 V.4.6.1 Results and discussionFigure 4.18 shows the veering of the eigenfrequencies caused by the stiffness per-turbation in the third resonator of a coupled three-DOF resonator device. Fig-ure 4.19 shows the variation of the sensitivity norms with stiffness perturbation for40 V coupling voltage and compares between analytical, simulation and the ex-perimental results. There is good agreement between the experimentally measuredand the Simulink model based sensitivities for the third mode. The experimentallymeasured sensitivities follow the pattern of the Simulink model based sensitivitiesfor the other two modes also. However, there is a mismatch in the magnitudes. Thedifferences between analytical and Simulink model based sensitivity has been dis-cussed in Section 3.6.3. The average slopes of the linear part of the normal modesensitivity curves near zero perturbation are -4.6, -3.2 and -7.8 for the three modesrespectively. Theory suggests that for a symmetric coupled three-DOF resonatorsystem, the slope for the first and the second mode should be equal and half of theslope for the third mode (see Equation 2.40). However our results do not follow therelation, because of two reasons: a) All three resonators in the device has slightly772!" 2# 2$ 2% 2& " &2!""!"&"'"%"(")*+,,-.//01.2*3245*+6-70!8980:;<).-/+*+=+*>0?62@/0:;<!"#$!%  &"!'()*&#!'()*&"!+,-./0*,"1&#!+,-./0*,"1&"!210/3*,40/&#!210/3*,40/1 2!" 2# 2$ 2% 2& " &2'"'!"!'&"&'(")*+,,-.//01.2*3245*+6-70!8980:;<).-/+*+=+*>0?62@/0:;<!"#$!%  &"!'()*&#!'()*&"!+,-./0*,"1&#!+,-./0*,"1&"!210/3*,40/&#!210/3*,40/2!" 2# 2$ 2% 2& " &"!"&"'"%"("$")*+,,-.//01.2*3245*+6-70!8980:;<).-/+*+=+*>0?62@/0:;<!"#$!%  &"!'()*&#!'()*&"!+,-./0*,"1&#!+,-./0*,"1&"!210/3*,40/&#!210/3*,40/1 Figure 4.19: Comparison of sensitivity norms for coupling voltage of 40 Vfor three modes of the three-DOF device78varying stiffness. So at no external perturbation the device reaches actual symmetryas the perturbation is applied only in the third resonator. b) Lack of larger numberof points near zero perturbation in experimental as well as simulated sensitivitycurves, which results in averaging out of the slope. Furthermore, the comparisonof slopes of the normal mode sensitivity curve with that of the frequency sensitivitycurve near origin shows that normal mode sensitivity is approximately 40, 10 and60 times the frequency sensitivity for the three modes respectively.Also, it should be noted here that on the x-axis of the sensitivity plot Fig-ure 4.19, the total stiffness perturbation in the device (the sum of perturbation dueto built-in asymmetry and the external perturbation) is shown. The rightmost pointon the plots correspond to no external perturbation i.e. the device has a built-inasymmetry resulting in an initial stiffness perturbation and the external perturba-tion brings it towards symmetry.Another important comparison to be made here is the comparison between thesensitivity curve slopes for the two and the three DOF systems. We see that for40 V coupling, the slope of the sensitivity curve for the third mode of the three-DOF system (-7.8) is higher in magnitude as compared to that for the two-DOFsystem (6.4). However, it is not as large as predicted by theory, which predicts100% improvement in sensitivity (see Equation 2.23 and Equation 2.40). We foundthat the reason for this discrepancy lies in difference in non-dimensional couplingκ in the two systems. Even though coupling stiffnesses for the two devices aresimilar for 40 V coupling, non-dimensional coupling for three-DOF device is 3%larger in magnitude as compared to that for the two-DOF device. This resultsfrom the fact that stiffness of the resonators for three-DOF device is smaller thanthat for the two-DOF device, a fact evident from comparison of highest naturalfrequencies of the two system (see Figure 4.11 and Figure 4.17). It should alsobe noted here that, even with this difference in coupling, the measurable range ofstiffness perturbation (linear part of the sensitivity plots) for the three-DOF systemis smaller than that for the two-DOF system.794.7 Summary and discussionThe experimental characterization of the three MEMS devices described in Chap-ter 3 has been discussed in this chapter. The experimental results from the threedevices confirm that the sensitivity of the mode shape is much higher than thesensitivity of the natural frequency to a stiffness perturbation in a resonator. Theresults from the two-DOF devices show an increase in sensitivity with decreasingcoupling, which is controlled by varying the coupling voltage. Also, the sensitiv-ity of Device-2, which has a linear electrostatic coupling, varies inversely with thesquare of the coupling voltage. This allows easy tuning of the coupling strength.An increase in sensitivity is also noted by increasing the DOF of a device from twoto three. Apart from these, it is observed that the initial asymmetry in a device,caused by fabrication tolerances, limits the minimum allowable coupling strength.It also limits the measurable perturbation range of a device.80Chapter 5Conclusions and future work5.1 ConclusionsTheoretical analysis shows that the mode shapes of a symmetric coupled resonatorsystem are orders of magnitude more sensitive to small stiffness perturbations com-pared to their natural frequencies for a small coupling. Moreover, decreasing thecoupling strength increases the sensitivity though at the cost of decreasing the mea-surable range of perturbation. Experiments on two-DOF and three-DOF systemsconfirm these theoretical findings. Furthermore, experiments show that increasingthe number of degrees of freedom from two to three improves the sensitivity aspredicted by theory.It is observed that built-in asymmetry in a device due to fabrication tolerancesplays an important role- from limiting the range of measurable perturbations tocases where a perturbation actually brings the device towards symmetry. Moreover,Increasing the DOF increases the avenues of asymmetry thereby increasing thedeviation from predicted sensitivity.Shaped combs in the devices behave linearly. Moreover, electrostatic couplingstiffness produced by them varies proportionally with the square of the couplingvoltage, allowing convenient tuning of the coupling strength. However, the cou-pling stiffness as well as a constant force produced by them is smaller in magni-tude than predicted. This leads to a mismatch between the analytical prediction ofsensitivity and the experimentally measured sensitivity. Inaccuracy in device fab-81rication causes weaker than predicted coupling stiffness as the shape of the combsin the fabricated devices are slightly different from the designed comb shapes.5.2 Future workThis work contributes to the development of a better understanding of mode local-ization in two and three DOF systems by developing a more general energy basedframework to study veering and by examining the effect of fabrication induceddeviation of systems from the desired symmetric configuration on veering. How-ever, many of the directions emerging from the study could not be pursued in thiswork and can be explored further. They have been described briefly in the sectionsbelow.5.2.1 Differential perturbationTheoretical analysis shows that a differential perturbation scheme offers the ad-vantage of an insensitive resonant frequency even though the mode shape is highlysensitive (see Section 2.4.2). We could not show it experimentally owing to highinitial asymmetry in the available three-DOF devices. Better device design anduse of more accurate fabrication methods can yield devices with less asymmetry toallow differential perturbation characterization.5.2.2 Dynamic perturbationAll the work in mode localized sensing pertains to quasi-static sensing. Dynamicperturbation remains an unexplored area, baring use of such sensors to measure dy-namic quantities such as acceleration, rotation rate etc. Theoretical analysis as wellas experimental characterization of devices with dynamic perturbation needs to beperformed to understand their benefits as well as limitations. One key constraintin such usage is maintaining the system vibration at its resonance at all times. Avelocity-based feedback loop suggested in [38] can be exploited to achieve it dy-namically.825.2.3 Higher DOF systemIncreasing the DOF of a device enhances the sensitivity of modes. In this work, wesee an improvement in sensitivity as the DOF of a device is increased from two tothree. A further increase will make it even more sensitive. However, increasing theDOF presents a challenge in signal processing as the vibration responses of all thecoupled resonators need to be processed simultaneously.5.2.4 Perturbation mechanismsMost of the work in mode localized sensing relies on mass perturbation or elec-trostatic perturbation of stiffness. However, other perturbation mechanisms likethermal perturbation may be more apt for certain sensing applications. Studiesin this direction also need to be performed to increase applicability of the modelocalized sensing.5.2.5 Energy domainsSystems in various energy domains undergoing oscillations can be treated as aresonator and coupled to each other to create a coupled resonator system. However,most of the studies in mode localized sensing are restricted to mechanical andelectrostatic energy domains. Oscillators from other energy domains can also bestudied and employed in mode localized sensing.83Bibliography[1] M. Manav, G. Reynen, M. Sharma, E. Cretu, and A. Phani, “Ultrasensitiveresonant mems transducers with tunable coupling,” in Solid-State Sensors,Actuators and Microsystems (TRANSDUCERS EUROSENSORS XXVII),2013 Transducers Eurosensors XXVII: The 17th International Conferenceon, 2013, pp. 996–999. → pages iii[2] M. Manav, G. Reynen, M. Sharma, E. Cretu, and A. S. Phani,“Ultrasensitive resonant mems transducers with tuneable coupling,” Journalof Micromechanics and Microengineering, vol. 24, no. 5, p. 055005, 2014.→ pages iii[3] T. Thundat, E. Wachter, S. Sharp, and R. Warmack, “Detection of mercuryvapor using resonating microcantilevers,” Applied Physics Letters, vol. 66,no. 13, pp. 1695–1697, 1995. → pages 1[4] T. P. Burg, A. R. Mirza, N. Milovic, C. H. Tsau, G. A. Popescu, J. S. Foster,and S. R. Manalis, “Vacuum-packaged suspended microchannel resonantmass sensor for biomolecular detection,” J. MEMS, vol. 15, no. 6, pp.1466–1476, 2006. → pages[5] M. Godin, A. K. Bryan, T. P. Burg, K. Babcock, and S. R. Manalis,“Measuring the mass, density, and size of particles and cells using asuspended microchannel resonator,” Applied physics letters, vol. 91, no. 12,p. 123121, 2007. → pages 1[6] D. Burns, J. Zook, R. Horning, W. Herb, and H. Guckel, “Sealed-cavityresonant microbeam pressure sensor,” Sensors and Actuators A: Physical,vol. 48, no. 3, pp. 179–186, 1995. → pages 1[7] D. Burns, R. Horning, W. Herb, J. Zook, and H. Guckel, “Sealed-cavityresonant microbeam accelerometer,” Sensors and Actuators A: Physical,vol. 53, no. 1, pp. 249–255, 1996. → pages 184[8] A. A. Seshia, R. T. Howe, and S. Montague, “An integratedmicroelectromechanical resonant output gyroscope,” in Proc. 15th IEEE Int.Conf. Micro Electro Mech. Syst. IEEE, 2002, pp. 722–726. → pages 1[9] M. Spletzer, A. Raman, A. Q. 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Kuroda, “Self-excited coupled cantilevers formass sensing in viscous measurement environments,” Applied PhysicsLetters, vol. 103, no. 6, p. 063104, 2013. → pages 8287Appendix ARoots of a polynomial withperturbed coefficientsLet us consider a polynomial in x of degree n. It can be expressed as:f (x) =n∑i=0αixi. (A.1)Let us assume that its roots are real and denote them by x j, where j= 1, ...,n. Now,coefficients of the polynomial are perturbed to obtain a new polynomial given by:f¯ (x) = f (x)+n∑i=0∆αixi. (A.2)Roots of the perturbed polynomial can be expressed as x j+∆x j, where j = 1, ...,n.By substituting these roots in the perturbed polynomial, we get:f¯ (x j +∆x j) = f (x j +∆x j)+n∑i=0∆αi(x j +∆x j)i = 0. (A.3)For small ∆x j, f (x j +∆x j) can be approximated by its Taylor series expansion upto first order:f (x j +∆x j)≈ f′(x j)∆x j. (A.4)88where f ′(x j) is value of the first derivative of the polynomial f (x) at x = x j. Itshould be noted that this is valid only if f ′(x j) 6= 0. Otherwise higher order termsshould be added in the expansion in Equation A.4. Now, substituting the above inEquation A.3 and neglecting higher order terms yields:f ′(x j)∆x j +n∑i=0∆αixij ≈ 0. (A.5)The above equation is solved to obtain ∆x j, change in roots due to perturbation incoefficients.∆x j ≈−n∑i=0∆αixijf ′(x j). (A.6)89

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