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Sensors and Actuators A 134 (2007) 98–108

Nonlinear dynamics of vibrating MEMS

Francesco Braghin a,∗, Ferruccio Resta a, Elisabetta Leo a, Guido Spinola b

a Mechanical Engineering Department, Politecnico di Milano, 20158 Milano, Italyb MEMS Business Unit, STMicroelectronics Srl, 20010 Cornaredo, Milan, Italy

Received 1 March 2006; received in revised form 6 October 2006; accepted 16 October 2006Available online 30 November 2006

bstract

Let us consider a MEMS translational gyroscope. When significantly displacing the proof mass, the nonlinear hardening characteristic of theupporting beams becomes visible. Thus, the resonance peak of the structure bends towards the higher frequencies. This property is useful to easilyynchronise sense and drive resonances thus increasing the sensibility of the MEMS gyroscope. Through a test structure designed to access the

igh deformation range of the supporting beams, its nonlinear vibrations were investigated both experimentally and numerically. It is shown that aimple nonlinear lumped parameter model is sufficient to schematise the gyroscope and that a semi-analytical integration method allows to quicklyetermine both stable and unstable branches of the system’s dynamic response.

2006 Elsevier B.V. All rights reserved.

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eywords: MEMS gyroscope; Nonlinear dynamics; Lumped parameter model

. Introduction

MEMS-sensors are systems in which the electronic andechanical parts are strictly integrated. Whereas the elec-

ronic part has bases well consolidated, the mechanical part,t micro level, still has to be investigated. Micromachined iner-ial sensors (accelerometers and gyroscopes with sizes in theange of 1 �m to 1 mm) are the most important and sold typesf silicon-based sensors. MEMS accelerometers are used inany fields: from automotive (air-bag sensor, rollover detec-

ion sensor, etc.) to mobile phone applications. Micromachinedyroscopes, instead, are at present at developing state. Becausef the photolithographic technology, it is much easier to pro-uce plane gyroscopes rather than spatial gyroscopes as the onesncountered at macro level. In particular, translational MEMSyroscopes are the most promising solution.

The typical structure of a tuning-fork MEMS gyroscope

Fig. 1) consists of proof masses (eventually held by frames thatllows to decouple the stiffness values along different directions)upported by polysilicon beams. The proof masses are set into

∗ Corresponding author.E-mail addresses: francesco.braghin@polimi.it (F. Braghin),

erruccio.resta@polimi.it (F. Resta), elisabetta.leo@polimi.it (E. Leo),uido.spinola@st.com (G. Spinola).

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ahl

924-4247/$ – see front matter © 2006 Elsevier B.V. All rights reserved.oi:10.1016/j.sna.2006.10.041

ibration along drive direction through combdrive actuators. Thetructure is symmetric and is designed to excite the push–pullibration mode along the x-axis (drive-direction). In presencef an angular speed (ωz) perpendicular to the drive direction,oriolis acceleration determines a force along sense directionaving the same frequency as the driving force and proportionaloth to the speed of the proof mass along the drive axis and tohe angular rate.

To increase the sensitivity of the gyroscope (i.e. to obtainigher displacements along the sense direction), the eigenfre-uencies along drive and sense directions should be equal tohe driving frequency [1] and a low damping ratio (low pressureevel) should be aimed for [2]. However, low damping meansarrow resonance peaks and thus small variations in the eigen-requencies (e.g. due to small errors in the production process,3]) would lead to a very small sensitivity. Complex contrologics have therefore to be implemented. Another possible solu-ion to achieve high sensitivity even in presence of productionrrors, is based on the relaxation the constrain that the two res-nance peaks (along drive and sense directions) have the samerequencies. This relaxation can be obtain in several differentays.

The way analysed in this paper relies on the nonlinear char-

cteristics of the device. A test structure (Fig. 2) to achieveigh displacements and thus to experimentally study the non-inear behaviour of MEMS devices was ad hoc designed and

F. Braghin et al. / Sensors and Actuators A 134 (2007) 98–108 99

pac

olmaitiiTdpti

Tr

2

t

Fi

Fig. 3. 2D FEA model of the test structure developed in FEMLAB.

(Tiht

ss9

dtd0

Fig. 1. Typical structure of a tuning-fork MEMS gyroscope.

roduced with the support of STMicroelectronics. It consists ofproof mass suspended through four beams and forced by 20

omb-drives along the x-axis (drive direction).A 2D FEA model of this test structure has been developed in

rder to determine the first eigenmodes of the device. This pre-iminary analysis showed that a very simple lumped parameter

odel is sufficient to describe the dynamics of the system up tofrequency of approximately 80 kHz. To assess the nonlinear-

ty of the supporting beams, a refined FEA model of just one ofhese supporting beams was used [4–7]. The nonlinear character-stic curve was then fitted with a cubic polynomial curve and thedentified coefficients were used in the lumped parameter model.hrough semi-analytical and numerical integration methods, theynamic response of the test structure was determined and com-ared to experimental results showing a very good agreement upo an excitation force amplitude of approximately 3 �N whichs a rather high force for the device considered.

The given methodology is then used to design a gyroscope.he identification of the modal parameters is provided and the

esults are discussed.

. FEA model

As already said in Section 1, at first a 2D FEA model ofhe test structure, shown in Fig. 2, was developed in FEMLAB

ig. 2. Layout of the test structure to assess the nonlinear behaviour of support-ng beams.

tcpm

Fig. 4. First eigenmode of the test structure.

Fig. 3) and used to determine the first eigenmodes of the device.he comb drives were schematised through equivalent masses

n order to reduce the complexity of the model. Similarly, theoles in the proof mass were not modelled explicitly but wereaken into account by reducing the density of the polysilicon.

Figs. 4 and 5 show the first two eigenmodes of the testtructure. While the first eigenmode (at 1705 Hz) is rigid, theecond one is clearly deformable but it occurs at a frequency of6,385 Hz.

It can therefore be concluded that, in this case, due to theesign of the test structure, a 1dof lumped parameter modeluned on the first modal frequency can be used to analyse theynamics of the test structure and that its validity range is fromHz up to 80 kHz.

The developed FEA model has then been reused to determine,hrough a static analysis, the supporting beam characteristic

urve, i.e. the elastic reaction force versus the proof mass dis-lacement. Due to the symmetry of the test structure the proofass is bounded to translate along x-direction. Thus, it is possi-

Fig. 5. Second eigenmode of the test structure.

100 F. Braghin et al. / Sensors and Actuators A 134 (2007) 98–108

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Fig. 7. Comparison between simulated and approximated hardening character-istic curve.

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ig. 6. Hardening characteristic curve of a single supporting beam obtainedhrough FEM analysis.

le to simplify the FEA model just considering one supportingeam. In this case, its boundary condition are an encastre at oneeam end and a sliding-block constraint at the other end.

Two different FEA models were used to study the influencef element types: one “beam model” and one “shell model”.n the “beam model” 260 beam elements were used while inhe “shell model” the supporting beam is discretized through800 shell elements. For both models, the beam cross-section isupposed to be rectangular and constant over the beam length.he simulated characteristic curve (force versus displacement)f a single supporting beam is shown in Fig. 6.

As expected, the two models give almost the same results.oreover, a hardening behaviour is clearly visible.To speed-up the dynamic simulation necessary to assess the

onlinear behaviour of the test structure, an analytical descrip-ion of this characteristic curve would be very useful. Thus, theimulated characteristic curve of a single supporting beam haseen approximated through a cubic polynomial relation:

∼= k1x + k3x3 (1)

F being the applied force at the free end of the beam (or thelastic reaction force), x the displacement at the same end, and1 and k3 the linear and cubic stiffness constants, respectively.he curve approximation is carried out using a nonlinear curve-tting algorithm. Fig. 7 shows the results of the approximationrocedure: the fitted characteristic curve and the simulated onere almost overlapped.

Changing the supporting beam’s dimensions (width, heightnd length), a sensitivity analysis of k1 and k3 parameters haseen carried out thus allowing to setup a database that corre-ates the beam’s dimensions to its stiffness parameters. Thisatabase is of great interest in the design phase. Fig. 8 shows, asn example, how the linear and cubic stiffness parameters varyor increasing beam width.

In conclusion, the developed FEA model has allowed to deter-ine the eigenfrequencies of the test structure (authorizing the

doption of a simple lumped parameter model) and has permit-ed to identify both the linear and cubic stiffness parameters to be

tabf

ig. 8. Linear and cubic stiffness parameters of the supporting beam for increas-ng beam length.

sed for the simulation of the dynamic behaviour of the devicen case of high displacements.

. Lumped parameter model

Since only the first eigenfrequency of the system is of inter-st, a lumped parameter model of the test structure has beeneveloped. Moreover, due to the fact that the test structure wasesigned to assess the nonlinear behaviour of MEMS devicesnd is not a fully working MEMS gyroscope, only the motionquation along the drive direction is of interest:

x + cx + k(x) = fdrive (2)

here x is the displacement of the test structure along drive direc-

ion, m the proof mass, c the damping coefficient along the drivexis and k(x) is the nonlinear characteristic of the supportingeams and the right-hand term of the equation is the actuationorce. Using the cubic approximation of the supporting beams’

d Act

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F. Braghin et al. / Sensors an

haracteristic, the differential equation that governs the motionlong the drive axis becomes:

x + cx + Nbk1x + Nbk3x2 = fdrive (3)

here Nb is the number of beams that support the proof mass.hile the parameter m is easily determined, the value of the

amping coefficient is dependent on the working pressure of theevice. Ambient pressure level is considered.

The following simplifying assumptions were made:

the stiffness effects due to the compressibility of the fluid canbe neglected [8] with respect to the stiffness introduced bythe supporting beams;the structural damping of polysilicon can be neglected [9,10]with respect to the viscous damping due to the air surroundingthe structure (being of an order of magnitude smaller).

To model the viscous damping, Couette flow and squeezelm damping [11] are used:

= cflow + csqueeze (4)

Approximating air as a Newtonian fluid, flow and squeezelm damping are equal to:

flow = −μA

y0(5)

squeeze = μ2Nlh3

y3comb

(6)

being the viscosity constant of air at ambient temperature andressure, A being the area of the overlapped plates, y0 beinghe distance between the parallel surfaces, N being the numberf comb fingers, l being the length of the overlapped fingers, heing the thickness of the structure (and therefore of the fingers)nd ycomb being the distance between two fingers. As alreadyaid, the test structure is set into vibration through a series ofomb drives that allow to achieve a high actuation forces andherefore high displacements. To avoid having a driving forceith multiple frequencies, the pulsating component of the volt-

ge Vac applied to the two stators of the comb drive has oppositeign. Thus, the driving force is equal to

drive = 2N

(ε

h

ycomb

)VdcVac (7)

here ε is the dielectric constant of air and Vdc and Vac arehe amplitudes of the applied constant and pulsating voltages,espectively.

The equation of motion along drive direction can thereforee rewritten as:

x + cx + Nbk1x + Nbk3x3 = |fdrive|sin(ωdrivet)( )

= 2N εh

ycombVdc|Vac|sin(ωdrivet) (8)

drive being the drive frequency. This nonlinear, constant param-ter, second order differential equation is known as Duffing

Nato

uators A 134 (2007) 98–108 101

quation. Even though simple, there is not any analytical solu-ion. Thus, to solve it two ways are possible: a semi-analyticalpproach or a numerical approach. The semi-analytical approachs based on Galerkin–Urabe’s method [12,13] and allows toetermine only regime solutions (both stable and unstable solu-ions) in a very fast and efficient way. The numerical approache.g. Runge–Kutta’s method), instead, allows to determine bothhe transient response and the stable regime solution but requires

much higher computational effort due to the high drivingrequency (that requires a small integration step) and the lowamping (that leads to a long transient response before regimes reached).

.1. Semi-analytical method

The semi-analytical method used to solve the nonlinear equa-ion of motion of the test structure along drive direction is thealerkin–Urabe method. The basic idea behind this method is

hat, although nonlinear, when forced by a sinusoidal actuationorce having frequency equal to ωdrive, the system will haveperiodic regime response. Thus, the regime response can beritten as a Fourier series:

(t) = a1 cos(ω1t) + b1 sin(ω1t) + a2 cos(ω2t)

+ b2 sin(ω2t) + . . . (9)

here the frequencies ωi are multiples (super-harmonics) orractions (sub-harmonics) of the frequency ωdrive of the actu-tion force. Due to the cubic nonlinearity introduced by theupporting beams, it can be shown that the system’s responses a function of only odd frequencies:

(t) = · · ·a1/3 cos(ωdrive

3t)

+ b1/3 sin(ωdrive

3t)

+ a cos(ωdrivet) + b sin(ωdrivet) + a3 cos(3ωdrivet)

+ b3 sin(3ωdrivet) + · · ·

= ∼=p∑

i=1

(a1/(2i+1) cos

(ωdrive

2i + 1t

)

+b1/(2i+1) sin

(ωdrive

2i + 1t

))

+q∑

i=0

(a(2j+1) cos((2j + 1)ωdrivet)

+ b(2j+1) sin((2j + 1)ωdrivet)) (10)

being the number of sub-harmonics and q being the numberf super-harmonics considered. By increasing p and/or q thepproximate system’s response tends to the exact one. Substi-uting the system’s response x(t) into Duffing equation, a systemf 2(p + q + 1) nonlinear algebraic equations is obtained. Solv-ng this system of nonlinear equations through, for example,

ewton–Raphson method, the unknown coefficients ai and bi

re determined. Table 1 shows the amplitude values of the sys-em’s response for three levels of actuation force (fdrive = 2.7 �N)btained considering 0 sub-harmonics (p = 0) and 5 super-

102 F. Braghin et al. / Sensors and Actuators A 134 (2007) 98–108

Table 1Harmonics’ amplitude values of the system’s response for three levels of actuation force (fdrive = 2.7 �N) obtained considering 0 sub-harmonics (p = 0) and 5super-harmonics (q = 5) with ωdrive equal to 2 kHz

Amplitude (m)1 × fdrive

% of mainharmonic

Amplitude (m)3 × fdrive

% of mainharmonic

Amplitude (m)10 × fdrive

% of mainharmonic

Main harmonic 1.5E−6 – 2.0E−6 – 2.7E−6 –First super-harmonic 4.5E−8 3.00 1.1E−7 5.50 4.4E−7 16.30Second super-harmonic 1.2E−9 0.08 5.9E−9 0.29 5.0E−8 1.85Third super-harmonic 3.6E−11 0.00 3.1E−10 0.02 6.1E−9 0.22Fourth super-harmonic 1.0E−12 0.00 1.6E−11 0.00 7.5E−10 0.03F 8.5E−

hso

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nstest structure is excited with an harmonic actuation force

ifth super-harmonic 2.8E−14 0.00

armonics (q = 5) with ωdrive equal to 2 kHz. It can be clearlyeen that, increasing the actuation force amplitude, the weightf higher harmonics increases.

In order to determine the system’s regime response, onlyhe amplitude of the main harmonic is taken into account.ig. 9 shows the system’s regime response in the frequencyomain obtained with 0 sub-harmonics (p = 0) and 5 super-armonics (q = 5). This response is determined by applying ariving force having a frequency that changes with discreteteps of 50 Hz from 1 kHz to 6 kHz. Three different zones areisible:

for frequencies lower than 2.5 kHz, only one regime solutionis found by the semi-analytical method;for frequencies in the range of 2.5 and 4.0 kHz, three differentregime solutions are found;for frequencies higher than 4.0 kHz, again only one regimesolution is found.

While in the first and third frequency ranges only one solu-ion is found, in the intermediate frequency range three differentolutions are determined. Thus, three different branches can bedentified:

ig. 9. Nonlinear frequency response function of the test structure obtained withhe semi-analytical method (0 sub-harmonics and 5 super-harmonics).

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13 0.00 9.2E−11 0.00

one higher branch that starts at the lowest frequency consid-ered and extend to the upper limit of the central frequencyrange;one intermediate branch that exists only in the central fre-quency range;one lower branch that starts at the lower limit of the centralfrequency range and extend to the highest frequency consid-ered.

It can be shown that the higher and lower branches correspondo stable solutions while the intermediate branch corresponds ton unstable solution. When the distance between a stable and annstable branch becomes zero, a jump phenomenon occurs.

.2. Numerical method

Another possibility of solving Duffing equation is to useumerical integration algorithms in the time domain. Fig. 10hows the system’ response in the time domain when the

f 27 �N at a frequency of 2 kHz. The initial transientesponse, as well as the periodic regime response, are clearlyisible.

ig. 10. System’s response obtained with the numerical method (Runge–Kuttaecond–third order integration method) when the test structure is excited withn harmonic actuation force of 27 �N at a frequency of 2 kHz.

F. Braghin et al. / Sensors and Act

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ttitcFrotaiug(maa

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ig. 11. System’s regime response (lower graph) and corresponding spectrumupper graph).

In order to obtain the system’s frequency response function,nly the periodic regime behaviour has to be taken into account.hus, as shown in the lower graph of Fig. 11, the final part of the

ime integration is selected. As for the semi-analytical method,nly the amplitude of the main harmonic (having a frequencyqual to ωdrive) has to be taken into account.

This is done by calculating the Fourier transform (upper graphf Fig. 11) and selecting the amplitude of the main harmonic.n order to be able to determine the whole frequency responseunction (Fig. 12), the excitation frequency (ωdrive) is increaseddecreased) from one simulation to the other (with steps of0 Hz) and the initial system’s amplitude is maintained equal

o the regime amplitude determined at the previous step.

Two jumps are visible: one at approximately 2.7 kHz (calledump-up) and the other at approximately 3.8 kHz (called jump-own).

ig. 12. Nonlinear frequency response function of the test structure obtainedith the numerical method (Runge–Kutta second–third order integration method

nd 5 �s integration step).

odt

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uators A 134 (2007) 98–108 103

It should be observed that the integration step, as well ashe order of the integration method, introduce perturbations inhe function to be integrated. Increasing the order or decreas-ng the step reduces this perturbation. The first consequence ishat, as already pointed out, the numerical integration methodsannot determine unstable branches of the system’s response.urthermore, when stable and unstable branches of the system’sesponse get close to each other, the jump phenomenon mayccur earlier or later as a function of the size of the integra-ion step or of the order of the integration method. To improveccuracy, either higher order integration methods or very smallntegration step are required. Thus, high computational time issually necessary to obtain the system’s regime response withood accuracy. Several different integration methods were testedvariable step integration methods, variable order integrationethods, non-stiff and stiff integration methods) in order to

ssess the best compromise between computational time andccuracy.

The system’s regime responses, shown in Fig. 13, arebtained by using the same integration method (RK 2/3) butifferent integration step (5 and 10 �s). Since the lower stableranch determined using the two different integration steps doesot differ, it is not shown in Fig. 12. For what concerns the uppertable branch, instead, it can be noticed that the identified jump-own frequency is higher with a smaller step size. However, asxpected, the required computational time is about three timesigher.

Fig. 14, instead, shows the influence of the order of the inte-ration method: the system’s regime responses are obtainedsing the same integration step (5 �s) but integration algorithmsf different order (third and fifth order). It can be clearly seenhat, as expected, the jump-down frequency increases as the

rder of the integration method increases. This increase in jump-own frequency is however paid in terms of computational efforthat approximately doubles.

ig. 13. Nonlinear frequency response function of the test structure obtainedith the numerical method (Runge–Kutta second–third order integration method

nd different integration steps).

104 F. Braghin et al. / Sensors and Actuators A 134 (2007) 98–108

Fwm

3

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Ftsie

Fig. 16. Comparison between the nonlinear frequency response function of thetest structure obtained with the semi-analytical method (0 sub-harmonics and 5sie

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hrt

ig. 14. Nonlinear frequency response function of the test structure obtainedith the numerical method (5 �s integration step and different order integrationethod).

.3. Comparison between the two integration methods

As already said, the semi-analytical method allows to deter-ine only regime solutions (both stable and unstable) in a very

ast and efficient way while the numerical approach allows toetermine both the transient response and the stable regime solu-ion but requires a much higher computational effort. In ordero be able to compare the two integration methodologies it isherefore necessary to fix the attention on the system’s stable

egime response. Figs. 15 and 16 show the nonlinear frequencyesponse function of the test structure obtained with the semi-nalytical method (0 sub-harmonics and 5 super-harmonics) and

ig. 15. Comparison between the nonlinear frequency response function of theest structure obtained with the semi-analytical method (0 sub-harmonics and 5uper-harmonics) and the numerical method (Runge–Kutta second–third orderntegration method and 5 �s integration step) with an excitation force amplitudequal to 1.35 �N.

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uper-harmonics) and the numerical method (Runge–Kutta second–third orderntegration method and 5 �s integration step) with an excitation force amplitudequal to 2.70 �N.

he numerical method (Runge–Kutta second–third order integra-ion method and 5 �s integration step) on the same graph for twoalues of the excitation force amplitude: 1.35 and 2.70 �N.

The jump-up frequency estimated by the numerical method isigher than the one estimated by the semi-analytical method. Theeason for this strange behaviour is that the initial condition ofhe numerical integration is updated just considering the regimemplitude of oscillation of the main harmonic while the regimemplitudes of higher (lower) order harmonics are neglected andhe speed of system is always set to zero.

The jump-down frequency estimated by the numericalethod is lower than the one estimated by the semi-analyticalethod. As already pointed out, although one could improve the

umerical estimation of the jump-down frequency (by increas-ng the order of the integration method or by decreasing thentegration step), since numerical integration itself introduceserturbations, it is impossible to obtain equal jump-down fre-uencies from the two integration methods. This observation,ogether with the consideration of the much higher computa-ional costs of the numerical integration method, has led to theonclusion that, for the numerical–experimental comparison thatill be shown in the next paragraph, only the semi-analytical

ntegration method will be used.Before going on to the numerical–experimental comparison

ne last remark on the integration methods is necessary. In real-ty, perturbations are always present (vibrations of the supportingtructure, shocks, etc.). It has therefore no practical sense toefine the jump-down frequency as the frequency were stablend unstable braches coincide since the real jump-down willccur at much lower frequencies. Thus, both the semi-analytical

nd the numerical integration methods do not correctly estimatehis jump-down frequency. It would have much more sense toefine the worst working conditions of the device, thus the high-st possible perturbations and finally, form the system’s response

F. Braghin et al. / Sensors and Actuators A 134 (2007) 98–108 105

Fs

ft

4

dDOobslo

rti

isslhcobibf

5r

stt

Fig. 18. Regime response of the test structure: numerical results (star marker)and experimental results (square and circle marker). Force: 1.35 �N.

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6

ig. 17. Experimental setup used to test the dynamic behaviour of the testtructure.

unction obtained from the semi-analytical integration method,he realistic jump-down frequency.

. Experimental setup

To validate the numerical model, the test structure was pro-uced and tested using optical measurements. A Polytech laseroppler vibrometer (LDV), consisting of modules OFV512 andFV 3001, is used to measure the velocity of the proof mass. Tobtain the displacement, the velocity of the proof mass is dividedy the actuation frequency [15]. To measure the vibration of suchmall structures, special care must be taken in positioning theaser spot. To this purpose a microscope is available at the topf the test bench.

Fig. 17 shows the test bench available at STMicroelectronicesearch laboratory that was used for the experimental tests andhe connections done to power on the test structure and measurets displacement.

The data acquisition system is a PC-based system with built-n anti-aliasing filters. All tests are carried out using a steppedine excitation. Tests were repeated on three different testtructures to assess the dispersion of the results. Due to the non-inearity of the structure, the system’s response showed multiplearmonics as expected. To carry out the numerical–experimentalomparison that will be described in the following paragraph,nly the main harmonic was taken into account. This was doney acquiring the system’s response for 10 s, then by elaborat-ng the signal through a FFT (fast Fourier transform) and finallyy extracting the amplitude of the harmonic having the samerequency of the excitation force.

. Comparison between numerical and experimentalesults

Figs. 18 and 19 show the system’s regime response of the testtructure in terms of magnitude of the frequency response func-ion. Stars indicate the simulated system’s response achievedhrough the semi-analytical method while squares and cir-

ir

ig. 19. Regime response of the test structure: numerical results (star marker)nd experimental results (square and circle marker). Force: 2.70 �N.

les show the measured displacement values at increasing andecreasing actuation frequency, respectively. The differenceetween the two figures is the actuation force: Fig. 18 referso an actuation force of 1.35 �N while Fig. 19 refers to a forcef 2.70 �N.

Both the figures show that the model is able to cor-ectly predict the nonlinear systems response: the jump-up andump-down frequencies are correctly estimated as well as thencrease/decrease of the systems vibrations amplitude with actu-tion frequency.

. Nonlinear gyroscope’s prototype

The developed model was used to design a gyroscope show-ng a nonlinear behaviour along drive direction. In particular,eferring to Fig. 20, the outer folded beams (with a linear char-

106 F. Braghin et al. / Sensors and Actuators A 134 (2007) 98–108

Ft

an

frted

oaImfd

Fk

[

wssbtt

ig. 20. FEM model of the prototype gyroscope: the outer beams are non-foldedo provide a nonlinear behaviour.

cteristic curve in working conditions) were substituted by fouron-folded beams with a nonlinear characteristic curve.

Figs. 21 and 22 show the characteristic curves (reactionorce versus displacement) of the folded and non-folded beams,espectively. While the folded beam has a linear behaviour upo 30 �m, the non-folded beam’s behaviour is highly nonlin-ar. Its characteristic curve is determined through the databaseescribed in the second paragraph.

Focusing the attention on the drive direction and consideringnly rigid modes, the gyroscope’s dynamic behaviour is equiv-lent to that of the lumped parameter model shown in Fig. 23.n fact, the inner masses do not move with respect to the outer

asses. Hence, a very simple two degrees of freedom model is

ully able to describe the motion of the gyroscope along driveirection.

Fig. 21. Characteristic curve of the folded beam (kd = 0.9 N/m).

tafmv

m

F

Fm

ig. 22. Characteristic curve of the non-folded beam (k1 = 3.16 N/m,

3 = 0.11e12 N/m3).

The equations of motion can be written as follow:

m 0

0 m

] {x1

x2

}+

[2kd + 2kc + 2k1 −2kc

−2kc 2kd + 2kc + 2k1

]

×{

x1

x2

}+

{2k3x

31

2k3x32

}=

{F1

F2

}(11)

here m is the sum of the inner and outer masses; kd the lineartiffness of the folded beams along drive axis; kNL the nonlineartiffness of the non-folded beams along drive axis (k1 and k3eing the linear and cubic coefficients); kc the linear stiffness ofhe folded beams between the outer masses and F1 and F2 arehe driving forces provided by the actuators.

Since the actuation forces are in counterphase, only one vibra-ion mode is excited. Therefore, using the modal superpositionpproach, it is possible to further simplify the two degrees ofreedom lumped parameter model to a one degree of freedomodal system having the following mass and stiffness parameter

alues:

∗ = 2m, k∗ = 4kd + 4kc + 4k1, k∗NL = 4k3,

∗ = F1 − F2 (12)

ig. 23. Equivalent lumped parameter model of the designed gyroscope whileoving along drive direction.

F. Braghin et al. / Sensors and Act

Fw

p(c

m

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Fw

rs

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daw

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irh

R

ig. 24. Frequency response of the designed gyroscope at atmospheric pressureith a driving force equal to F* = 2.0 �N.

The system’s damping is computed both at atmosphericressure (r* = 1.75 × 10−5 Ns/m) and at low pressure [26 Pa]r* = 1.2 × 10−7 N/sm). The procedure used to achieve dampingoefficients is explained in [14].

The equation of motion of the modal system is

∗x + r∗x + k∗x + 4k3x3 = F∗ (13)

The semi-analytical method can be used to compute the gyro-cope’s frequency response along drive direction at atmosphericressure (Fig. 24) and at low pressure (Fig. 25). It can be clearlyeen that, with equal driving force, the achieved displacement atow pressure is much higher and the bending of the resonanceeak is more evident.

Both the inertia of the inner masses and the stiffness ofhe beam linking the inner masses to the outer ones werehosen to obtain an eigenfrequency along y-axis (sense axis)hat falls inside the bending peak of the driving frequency

ig. 25. Frequency response of the designed gyroscope at low pressure (26 Pa)ith a driving force equal to F* = 2.0 �N.

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uators A 134 (2007) 98–108 107

esponse so to obtain a dynamic amplification of the gyroscope’sensitivity.

. Conclusions

The nonlinearity introduced by the supporting beams of aest structure has been investigated both numerically (using twoifferent methodologies, a semi-analytical and a numerical inte-ration method) and experimentally. Even though a very simpleumped parameter model was used, a good agreement betweenxperimental and numerical results was obtained due to the facthat the first deformable eigenmode of the test structure is wellbove the excitation frequencies considered.

Thus, a powerful tool to predict even the nonlinear structuralynamic behaviour of translational gyroscopes has been setupnd validated. This numerical tool, already discussed in previousorks [16], is here used to design a gyroscope’s prototype.

cknowledgements

The authors wish to greatly acknowledge STMicroelectron-cs for having allowed to carry out the research and for havingealised the test structure. In particular, Ing. Marco Chrappan foraving helped in carrying out most of the work presented here.

eferences

[1] S. Iyer, Y. Zhou, T. Mukherjee, Analytical modelling of cross-axis couplingin micromechanical springs, in: Proceedings of the 1999 International Con-ference on Modeling and Simulation of Microsystems MSM99, Pittsburgh,USA, 1999.

[2] K.Y. Yasumura, Energy dissipation mechanisms in microcantilever oscilla-tors with applications to the detection of small forces, PhD Thesis, StanfordUniversity USA, 2001.

[3] R. Liu, B. Paden, K. Turner, MEMS resonators that are robust to process-induced feature width variations, J. Microelectromech. Syst. 11 (5) (2002)505–511.

[4] V. Kaajakari, T. Mattila, A. Lipsanen, A. Oja, Nonlinear mechanical effectsin silicon longitudinal mode beam resonators, Sens. Actuators A 120 (1)(2005) 64–70.

[5] W. Zhang, R. Baskaran, K.L. Turner, Effect of cubic nonlinearity on auto-parametrically amplified resonant MEMS mass sensor, Sens. Actuators A102 (2002) 139–150.

[6] C. Wang, D. Liu, R. Rosing, A. Richardson, B. De Masi, Constructionof nonlinear dynamic MEMS component models using Cosserat theory,Analog Integrated Circuits Signal Process. 40 (2) (2004) 117–130.

[7] P.F. Pai, Linear and Nonlinear Structual Mechanics, John Wiley Inter-science, New York, 2004.

[8] H. Hosaka, K. Itao, S. Kuroda, Damping characteristics of beam-shapedmicro-oscillators, Sens. Actuators A 49 (1995) 87–95.

[9] S. Reid, G. Cagnoli, D.R.M. Crooks, J. Hough, P. Murray, S. Rowan, M.M.Fejer, R. Route, S. Zappe, Mechanical dissipation in silicon flexures, in:Proceedings of the 22nd Texas Symposium on Relativistic Astrophysics atStanford University, Glasgow, UK, December 13–17, 2004.

10] K.Y. Yasumura, T.D. Stowe, E.M. Chow, T. Pfafman, T.W. Kenny, B.C.Stipe, D. Rugar, Quality factors in micron- and submicron-thick cantilevers,J. Microelectromech. Syst. 9 (1) (2000) 117–125.

11] S. Hutcherson, W. Ye, On the squeeze-film damping of microresonators inthe free-molecule regime, J. Micromech. Microeng. 14 (2004) 1726–1733.

12] J.P. Miralles, P.J. Jimenez Olivo, D.G. Peiro, A fast Galerkin method toobtain the periodic solutions of a nonlinear oscillator, Appl. Math. Comput.86 (1997) 261–282.

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13] A.H. Nayfeh, M.I. Younis, Dynamics of MEMS resonators under superhar-monic and subharmonic excitations, J. Micromech. Microeng. 15 (2005)1840–1847.

14] F. Braghin, E. Leo, F. Resta, Estimation of the damping in MEMS inertialsensors: comparison between numerical and experimental results both athigh and low pressure level, in: Proceedings of ESDA2006 8th BiennialASME Conference on Engineering Systems Design and Analysis, Torino,Italy, July 4–7, 2006.

15] D.J. Ewins, Modal Testing: Theory, Practice and Application.16] F. Braghin, E. Leo, F. Resta, Numerical and experimental analysis of

a nonlinear vibrating MEMS, in: Proceedings of IMECE2005 ASMEInternational Mechanical Engineering Congress and Exposition, Orlando,November 5–11, 2005.

iographies

rancesco Braghin was born in Milano on October 14, 1972. He graduatedn Mechanical Engineering in 1997 and received a PhD in Applied Mechanicsn 2001. In 2001 he became researcher at Politecnico di Milano, Mechanicalngineering Department. Research work mainly focuses on tyre – road andheel – rail interaction, on road and rail vehicles’ dynamics and on the control

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uators A 134 (2007) 98–108

f these mechanical systems in order to achieve better performances and higherafety standards. Moreover, his research activity deals with MEMS both with theynamics of microcomponents and with their control. In these research fields,r. Braghin has published more than 50 papers, both national and international.

lisabetta Leo was born in Mandello del Lario, Italy, on March 14 1978. Sheraduated in Mechanical Engineering in 2003 at Politecnico di Milano. Since004 she follows the Doctoral Programme in Mechanical Systems Engineeringithin Politecnico di Milano. Research work is mainly carried out in the fieldf dynamics, stability and control of mechanical systems, with applications toround vehicles (road) and micro-mechanics.

erruccio Resta was born in Bergamo, Italy, on August 29 1968. He graduatedn Mechanical Engineering in 1992 and received a PhD in Applied Mechanicsn 1996. In 1999 he became researcher at Politecnico di Milano, Mechanicalngineering Department, and in 2001 he became associate professor in theame institution. Research work is mainly carried out in the field of dynamics,

round vehicles (road and rail), large civil structures, micro-mechanics, machineibrations, in the filed of fluid and structures interaction and in mechatronics.n these subjects he is author of more than 80 papers presented at national and

nternational conferences or published on specialized reviews.