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XS

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ARRAA

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himdskiQdQ[b

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dc(

0d

Sensors and Actuators A 171 (2011) 199– 206

Contents lists available at SciVerse ScienceDirect

Sensors and Actuators A: Physical

jo u rn al hom epage: www.elsev ier .com/ locate /sna

upport loss for beam undergoing coupled vibration of bending and torsion inocking mass resonator

iong Wang1, Dingbang Xiao ∗, Zelong Zhou, Zhihua Chen, Xuezhong Wu, Shengyi Lichool of Mechatronics Engineering and Automation, National University of Defense Technology, Changsha, China

r t i c l e i n f o

rticle history:eceived 25 April 2011eceived in revised form 24 August 2011ccepted 29 August 2011vailable online 7 September 2011

a b s t r a c t

Rocking mass resonator is widely used to design various sensors and actuators, which is a dual-axialsymmetry resonator with high sensitivity. Qsupport is the dominant energy loss mechanism influencingits high sensitivity. The anchor types and support loads applied to attachment points of rocking massresonator are analyzed. Then support loss is simplified as a model with a beam attached to a finite

eywords:ocking mass resonatorupport lossuality factor

thickness plate at its end. The general formulations for power radiated into support structure are given.An accurate analytical model of support loss for rocking mass resonator has been developed and verifiedby experiments. When the thickness of resonator is 240 �m, the measured Q can achieve a value of 589.1;while the thickness of resonator is reduced to 60 �m, the measured Q can achieve a value more than 8500.The derived model is general and might be applicable to various micro beam resonators and anchor types,providing significant insight to design of high-Q rocking mass devices.

. Introduction

Rocking mass resonator is a dual-axial symmetry resonator withigh sensitivity, consisting of four slender beams attached to a rock-

ng mass in the middle [1]. Rocking mass resonator has a plungingode and two uniform rocking modes; it is thus widely used to

esign high performance gyroscope [1–5], multiple axis stress sen-or [6], five-axis motion sensor [7], and so on. Higher Q is one of theey problems in those resonators. For a micro resonator operatingn air, there coexist several energy loss mechanisms. The measured

is mainly the combination of these loss mechanisms, such as airamping loss Qair, support loss Qsupport, thermoelastic damping lossted, surface loss Qsurface, and the remaining damping effects Qother

8]. For a resonator operating in vacuum, air damping loss Qair cane omitted. The measured Q can be expressed as

1Q

= 1Qsupport

+ 1Qted

+ 1Qsuface

+ 1Qother

(1)

hermoelastic loss Qted is intrinsic material damping that occurs as

result of thermal energy dissipation due to elastic deformation.ted has been reported to limit the Q-factor of vacuum packagedyroscopes to values from 100,000 to 200,000 [9], while Qsurface

∗ Corresponding author. Tel.: +86 731 84576373; fax: +86 731 84574963.E-mail addresses: wangxiong0081@sina.com (X. Wang),

ingbangxiao@yahoo.com.cn (D. Xiao), zhouzelong2005@163.com (Z. Zhou),zhnudt804@163.com (Z. Chen), xzwu@nudt.edu.cn (X. Wu), syli@nudt.edu.cnS. Li).

1 Tel: +86 731 84576373.

924-4247/$ – see front matter © 2011 Elsevier B.V. All rights reserved.oi:10.1016/j.sna.2011.08.023

© 2011 Elsevier B.V. All rights reserved.

is negligible due to the large surface-to-volume ratio of rockingmass resonators [10]. Qother captures remaining damping effectsestimated around 250,000 [11]. Qsupport is due to support loss whichcould be as lower than 10,000 depending on the anchor type andmaterial [12]. Qsupport is thus the dominant energy loss mecha-nism, and has been paid some attentions. Hao presented analyticalmodels for support loss in clamped–free and clamped–clampedmicro beam resonators, only with in-plane flexural vibrations [13].She also developed analytical model for support loss in both side-supported and center-supported micro disk resonators [10], basedon the model with in-plane flexural vibrations. Judge provided ana-lytical model of support loss for MEMS and NEMS resonators inthe limits of thick and thin support: cantilevered beams and dou-bly fixed beams [14]. However, his method cannot be accuratelyused in torsional vibrations. Chouvion developed models to pre-dict vibration transmission and support loss in ring-based MEMSsensors based on Judge’s model [15]. These methods cannot be useddirectly to solve support loss for rocking modes in rocking mass res-onator, because the beams undergo coupled vibration of bendingand torsion.

This paper is thus to provide an accurate analytical model of sup-port loss for rocking modes in rocking mass resonator. The rockingmodes for the resonator are equivalent to a superposition of rockingmode and torsional mode of an equivalent paddle, which has rock-ing mass and two beams in line. The support loads of the attachment

points and anchor types of the resonator are analyzed. Then sup-port loss is simplified as a model with a beam attached to rigid plateat its end. The general formulations applied to the model for powerradiated into the plate are given. Support loss analytical models for

200 X. Wang et al. / Sensors and Actuators A 171 (2011) 199– 206

F 1st ps

tsie

2

2

mioa

sFvolasspb

Fm

ig. 1. The fundamental modes and structural parameters of the resonator: (a) thetructural parameters.

he rocking mode and torsional mode of the equivalent paddle areolved, respectively. Then support loss analytical model for rock-ng mode of rocking mass resonators is obtained and validated byxperiments.

. Support structure analysis of rocking mass resonator

.1. Support loads analysis of rocking mass resonator

The plunging mode and two uniform rocking modes of rockingass resonator are illustrated in Fig. 1. When the resonator vibrates

n its rocking modes, two beams in line vibrate as bending, and thether two perpendicular beams vibrate as torsion at the same time,s shown in Fig. 1(b) and (c).

Based on vibration analysis for rocking modes of the resonator,upport loads of the attachment points are also described inig. 2(a), without considering the gravitation. When the resonatoribrates in rocking modes, it can be equivalent to a superpositionf the torsional mode and rocking mode of an equivalent rectangu-ar paddle, as shown in Fig. 2(b) and Fig. 2(c). There are two loadst the end of bending beams, such as the shear force normal to

ubstrate Fz, and bending moment about the axis parallel to sub-trate edge Mb. There is only one torsional moment about the axiserpendicular to the edge of substrate Mt, at the end of torsionaleams.

ig. 2. The attachment loads of rocking mode for the resonator and the equivalent paddle:ode of the paddle.

lunging mode, (b) the 2nd rocking mode, (c) the 3rd rocking mode and (d) related

When the resonator vibrates in rocking modes, the four beamsvibrate as coupled vibration of bending and torsion. Accordingly,the bending angle and the torsional angle at the end of those beamsare equal, denoted by �0. These attachment loads can be scaled by�0 [16] as

Mt = GIp�0

l1(2)

Mb = −EIy(4l1 + 3l2)�0

l21(3)

Fz = −6EIy(l1 + l2)�0

l31(4)

Iz = 112�hl42 + 1

12�h3l22 (5)

where E is Young’s modulus, G is shear modulus, � is density. h isthe thickness of vibratory structure, l1 is the length of beams, l2 is

the length of center supporting. Ip is the polar moment of inertia ofbeams’ cross-section, Iy is the moment of inertia of beams’ cross-section, and Iz is moment of inertia of vibratory structure, withoutthe mass of beams considered.

(a) attached loads of the resonator, (b) rocking mode of the paddle and (c) torsional

X. Wang et al. / Sensors and Actuators A 171 (2011) 199– 206 201

2

iawreoTvtV

V

F1Viftvtc

ctrs

3

3

amfimibwta

c

Fig. 3. The simplified support model and its main structural parameters.

.2. Limits of thick and thin support structure

The energy lost from micro resonators into support structures summarized as two limiting cases: one case that can be treateds plates, and the other case that acts as semi-infinite elastic mediaith effectively infinite thickness. The former is applicable to MEMS

esonators, while the latter is appropriate for NEMS devices. Thenergy lost from the resonator into surrounding structure consistsf the network done by the resonator at the attachment point.he dimensions of the attachment point are small compared toibration wavelengths in substrate at the resonant frequency ofhe resonators. Shear waves propagate in elastic solids at a speeds

s =√

E

2(1 + �)�(6)

or single crystal silicon material, elastic modulus E is.31 × 1011 Pa, Poisson’s ratio � is 0.28, and density � is 2330 kg/m3.s is thus obtained as 4686.38 m/s. Wave length of the shear waves

n the solid is �s, �s = Vs/f. For silicon resonator, given vibratingrequency of the resonator is 5 kHz, �s is 0.9373 m. Commonly, thehickness of substrate hp is several millimeters order of magnitude,iz. hp � �s, while still being much greater than the thickness h ofhe resonator. Accordingly, the supports of rocking mass resonatoran be treated as finite thickness plates.

For the case of finite thick support of rocking mass resonator, weonsider a semi-infinite plate with the thickness that need not behe same as the resonator itself. Using the plate-edge admittanceesults first reported by Eichler [17], we derive analytical expres-ions of support loss, which are also applicable to other resonators.

. The support loss mechanism of rocking mass resonator

.1. The support loss mechanism of rocking mass resonator

Support loss for the resonator is simplified as a model with beam attached to rigid plate support at its end. The simplifiedodel and its main structural parameters are shown in Fig. 3. The

our beams of the resonator vibrate as coupled vibration of bend-ng and torsion, and their vibrating frequencies are equal to rocking

odes of the resonator. In ideal situation, the mass of rocking masss much larger than that of beams and can be assumed as a rigidody without distortion, which is used to transfer kinetic energyithout energy loss. When the resonator vibrates at original posi-

ion, the total energy is translated into kinetic energy of the beamsnd rocking mass.

In such cases, the effects of vibrating resonator on substratean be modeled as harmonic point forces and moments acting at

Fig. 4. Semi-infinite plate support and applied loads to the attachment point.

support structure. An assumption is given: all the energy propa-gated into support structure would not be reflected, viz., all theenergy that reaches the support is considered lost. The estimatedQ-factor is thus a lower bound. The Q-factor is the ratio of vibrationenergy of the resonator to the energy lost; the reciprocal of Q is theloss factor ı [18]

ı = 1Q

= �U

2�U= ˘

ωU(7)

where U is the total energy lost per cycle of oscillation, due to allapplicable loss mechanisms, U = 2�˘/ω. U is the total vibrationenergy of oscillation. is the total net power flow out of the res-onator, and the average power transmitted to the support is simply,

= 1/2Re(F·V). F is a vector of the point loads, and V is the corre-sponding vector of the harmonic linear and angular velocities at theattachment point.

3.2. Power flow into the support structure

Consider shear forces normal to the plate Fz, bending momentsabout the axis parallel to the plate edge Mb, and torsional momentsabout the axis perpendicular to the plate edge Mt, as shown in Fig. 4.Note that three other loads not considered, are also possible: forcesnormal to the plate edge, shear forces parallel to the plate, andbending moments about the axis perpendicular to the plate. Thestiffness of the plate with respect to the latter three loads is muchgreater than the stiffness with respect to the former three. For mostresonator geometries, support loss due to the latter loads can beassumed to be negligible compared with the loss due to the loadsconsidered here.

The admittance at edge of the plate was first formulated inintegral form by Eichler. The elements of matrix Y are given asclosed-form integrals by Kauffmann [19]. These integrals have beensolved in closed form by Su and Moorhouse [20]. The point mobilitymatrix relates the normal angular velocity ˝n, tangential angu-lar velocity ˝t, and transverse linear velocity Vz of the attachmentpoint to the applied loads by the expression[˝b˝tVz

]= iω

[��w

]= 1√

�phpD[Y]

[MbMtFz

]

= 1√�phpD

[y11k2 0 0

0 y22k2 y23k0 y32k y33

] [MbMtFz

](8)

where D is the plate stiffness, D = Ehp3/12(1 − �2), and k is the free

wave number, k = [ω(�hp/D)1/2]1/2. The corresponding coefficientshave been calculated, for � = 0.3, which were calculated in [20]; for

� = 0.28, which were calculated in [15], as shown in Table 1. Thematerial in our experiments is n-type (1 0 0) single crystal silicon,Poisson ratio � = 0.28, so these coefficients can be determined.

202 X. Wang et al. / Sensors and Actuators A 171 (2011) 199– 206

Table 1The values of coefficients for Y.

Coefficients � = 0.30 � = 0.28

Re(y ) = Re(y ) 0.21645 0.22172

˘

˘

˘

˘

˘

˘

Wti[ri

4

atdir

4p

c

k

Tt

ω

U

wt

Et

11 22

Re(y23) = Re(y32) −0.29149 −0.28546Re(y33) 0.46198 0.45735

The resulting expressions for power radiated into the plate are

Fz =√

3(1 − �2)y33F2

z

h2p

√E�

(9)

Mb = 6(1 − �2)y11ωM2

b

Eh3p

(10)

Mt = 6(1 − �2)y22ωM2

t

Eh3p

(11)

Mb and Fz = ˘Fz + ˘Mb (12)

Mb and Mt = ˘Mb + ˘Mt (13)

Mt and Fz = ˘Fz + ˘Mt + [12(1 − �2)]3/4 y23ω1/2MtFz

�1/4E3/4hp2/5

(14)

hen each load is considered individually, Eqs. (9)–(11) apply forhree isolated load conditions. Eq. (12) is different from Eq. (13)n [14]. The reason is that it is wrong in velocity expressions in14] compared with Eq. (4) in [20]. However, the off-diagonal termsesult in an additional contribution that the total power is in Eq. (14)f both torsional moment and shear force are present.

. Predictions of support loss

The rocking modes of the resonator have been equivalent to superposition of the rectangular paddle. The rocking mode andorsional mode of the equivalent paddle are considered in detail toetermine their support loss. For each case, the analytical model

s built. Finally, support loss calculation for rocking modes of theesonator is carried out numerically.

.1. Support loss predictions for rocking mode of the equivalentaddle

The spring constant of rocking mode for the equivalent paddlean be expressed as [16]

r = Ewh3

l1

[12

(l2l1

)2

+ l2l1

+ 23

](15)

he resonant frequency and vibration energy of rocking mode ofhe paddle are expressed, respectively as

r =√krIz

(16)

r = 14�whl1ω

2�20 l

22 + 1

12�hl42ω

2�20 (17)

here vibration energy is scaled by �0, the arbitrary amplitude ofhe vibration mode shape.

Given w ≥ h, the radiated power can be found by substitutingqs. (12), (16) and (17) into Eq. (7), and support loss factor is foundo be

1Qr

= 18

√1 − �2y33�2 3/2

3/2(3wl1 + l22)l41

√w√l1

h2

h2p

+ 124

(1 − �2)y11�2

(3wl1 + l22)l21

w

l1

h3

h3p

(18)

Fig. 5. Support loss for rocking mode of the equivalent paddle.

where = 1/2(l2/l1)2 + l2/l1 + 2/3, � = l1 + 12, � = 4l1 + 3l2, = (l24 + h2l22). It is different from Eq. (18) in [11], becausethere is no cross term depending on Fz and Mb, as in Eqs. (9)–(14).It can be seen from Eq. (18) that, for hp is large relative to h, theshear force term dominates, and the effect of bending moment atthe support may be neglected.

For � = 0.28, w = 80, l1 = 2390, l2 = 2220, a plot of support loss forrocking mode in the paddle is shown in Fig. 5. Three curves areshown relating Q-factor to the thickness of the support hp. Whenthe beams are 40 �m thick with the thickness hp increasing from500 to 1000 �m, Q-factor increases from almost 10,000 to 50,000.While the beams are 20 �m thick, Q-factor increases from almost50,000 to 190,000.

4.2. Support loss predictions for torsional mode of the equivalentpaddle

The spring constant of torsional mode for the paddle can beexpressed as [16]

kt = 2GIpl1

(19)

The resonant frequency and vibration energy of torsional mode ofthe paddle are expressed as

ωt =√ktIz

(20)

Ut = 112�whl1(w2 + h2)ω2�2

0 + 112�hl42ω

2�20 (21)

where the energy is scaled by �0.Given that the cross section of beams is rectangle, and w ≥ h,

so Ip = ˇwh3, is the function of w/h [21]. The torsional radiatedpower can be found by substituting Eqs. (11), (20) and (21) into Eq.(7), and support loss factor is found to be

1Qt

= ˘

ωU= 3

2(1 − �)y22ˇ

w3l1 + wl1h2 + l42

w

l1

h3

h3p

(22)

This is different from Judge’s results. He pointed out that, the tor-sion mode involving only torsional moment Mt applied to substrate,resulted in predicted support loss which was many orders of mag-nitude less than the measured values. Comparing Eq. (22) with Eq.

(18), the contribution to power flow for torsional vibration may bethe similar order with bending vibration.

For � = 0.28, w = 80, l1 = 2390, l2 = 2220, a plot of support loss fortorsional mode in the paddle is shown in Fig. 6. Three curves are

X. Wang et al. / Sensors and Actuators A 171 (2011) 199– 206 203

sfoohTt

4r

aer

k

Tt

ω

U

w

Fr

Fig. 6. Support loss for torsional mode of the equivalent paddle.

hown relating Q-factor to the thickness of support plate hp. Q-actor of torsional mode has similar increasing tendency to thatf the rocking mode. However, Q-factor of torsional mode is 2-rders of magnitude larger than that of the rocking mode. For

= 20, hp = 1000, Q-factor can reach even more than 4,000,000.hese curves can provide a good estimate of support loss for MEMSorsional mirrors.

.3. Support loss predictions for rocking modes of rocking massesonator

The rocking modes of the resonator have been equivalent to superposition of the torsional mode and rocking mode of thequivalent rectangular paddle. Accordingly, the spring constant ofocking modes of the resonator can be expressed as

= 2GIpl1

+ Ewh3

l1

[12

(l2l1

)2

+ l2l1

+ 23

](23)

he resonant frequency and vibration energy of rocking mode forhe resonator are expressed, respectively as

=√k

Iz(24)

= 112�hω2�2

0

[3wl1l22 + wl1(w2 + h2) + l42

](25)

here the energy is also scaled by the amplitude �0.

ig. 7. Support loss dependent on the length l1 and l2 for rocking modes of theesonator.

Fig. 8. Support loss for rocking modes of rocking mass resonator: (a) when w ≥ h,support loss curves dependent on hp and (b) when w < h, support loss curves depen-dent on hp .

Given w ≥ h, the radiated power of the resonator can be found bysubstituting Eqs. (11), (12), (24) and (25) into Eq. (7), and supportloss factor is found as Eq. (26); given w < h, Ip = ˇw3h, support lossfactor is found as Eq. (27)

1Qtotal

= 14

√(1 − �2)y33�2 3/2(

(ˇ/(1 + v)) + )3/2

l41�

√w√l1

h2

h2p

+ 112

(1 − �2)y11�2

((ˇ/(1 + v)) + )l21�

w

l1

h3

h3p

+ 3(1 − �)y22ˇ2

(1 + v) ((ˇ/(1 + v)) + )�w

l1

h3

h3p

(26)

1Qtotal

= 14

√(1 − �2)y33�2 3/2

((ˇ/(1 + v)) (w2/h2) + )3/2l41�

√w√l1

h2

h2p

+ 112

(1 − �2)y11�2

((ˇ/(1 + v)) (w2/h2) + )�l21

w

l1

h3

h3p

+ 3(1 − �)y22ˇ2

(1 + v) ((ˇ/(1 + v)) + (h2/w2) )�h

l1

w3

h3p

(27)

where � = 3wl1l22 + l24 + w3l1 + wl1h2. It can be seen from Eq. (26)

that, for hp is large enough relative to h, the shear force term dom-inates, and the effect of bending moment and torsional moment atsupport structure may be neglected. Support loss is largely correl-ative to the length l1 and l2.

204 X. Wang et al. / Sensors and Actuators A 171 (2011) 199– 206

Fig. 9. The mode measuring system.

rockin

rswib

TT

Fig. 10. The rocking mass resonator prototype and frequency response: (a)

For � = 0.28, hp = 1000, w = 80, h = 20, a plot of support loss forocking modes in the resonator is shown in Fig. 7. Four curves are

hown relating Q-factor to the length l1 and l2. Q-factor increasesith l1 increasing, while decreases with l2 increasing. The increas-

ng is on the similar order with the decreasing. Both l1 and l2 muste considered cautiously to improve Q-factor.

able 2he predicted f and Q, measured f and Q for several rocking mass resonator prototypes.

Prototype # Prototype batch h/hp (�m) Predicted f (Hz) M

f

04 A 240/1000

05 A 240/1000 12468.3

08 A 240/1000

03 B 120/1000

05 B 120/1000 6856.6

11 B 120/1000

02 C 60/1000

03 C 60/1000 3416.5

05 C 60/1000

g mass resonator prototype; (b) frequency response curve of the prototype.

For � = 0.28, w = 80, l1 = 2390, l2 = 2220, support loss for the res-onator vibrating in rocking modes is shown in Fig. 8. When w ≥ h,

four curves are shown relating Q-factor to the thickness of sup-port plate hp in Fig. 8(a), while w ≥ h, three curves are shown inFig. 8(b). For hp = 1000 and h = 20, Q-factor is almost 160,000; whilefor h = 240, Q-factor even decreases to less than 1000. These curves

easured f (Hz) Predicted Q Measured Q

1 f2 Q1 Q2

12345.7 12370.6 141.9 381.812202.2 12216.9 829.4 537.7 589.112256.1 12218.8 510.7 394.1

6795.2 6797.2 2343.2 2955.36740.0 6792.4 3694.0 3547.4 3574.96785.0 6788.1 3469.4 3572.73206.3 3171.9 7008.3 4647.43208.7 3110.6 11439.0 8443.8 8581.03200.0 3158.1 5958.6 7624.0

Actua

cr

5

tapwsfbdfpaTt

sTimuQpa

tgrs

foatm

bita

ttrecstiss

6

f

l

mo

[

[

[

[

[

[

[

[

[

[

X. Wang et al. / Sensors and

an provide a good order-of-magnitude estimate of support loss forocking mass resonator.

. Experimental verification and discussions

The analytical results are compared with experimental resultso demonstrate the validity of the analytical model. Experimentsre conducted on several rocking mass resonator prototypes. Theserototypes are fabricated on the same single crystal silicon wafersith different thick beams by reducing thickness technology. Mea-

urements of the frequency and Q-factor are obtained by sweepingrequency of the harmonic drive signal and using half-powerandwidth method. The electrostatic actuation and capacitanceetection method are used to measure vibratory frequency and Q-actor. The measuring setup is composed of vacuum chamber, DCower, Agilent 33250A waveform generator, frequency responsenalyzer FRA 5087, and mode measuring board, as shown in Fig. 9.he mode measuring board and prototypes measured are put intohe vacuum chamber.

The prototypes are designed with a part at each corner of centerupporting to be measured conveniently, as shown in Fig. 10(a).he prototypes are measured with a lower pressure less than 10 Pan vacuum chamber, and a frequency response curve of prototypes

easured is shown in Fig. 10(b). Because each resonator has twoniform rocking modes, the Q values are measured and denoted by1 and Q2, respectively. The Q values are measured by using half-ower bandwidth method, and they are only the referenced valuest a certain degree.

All the results measured are compared with theoretical predic-ions, as shown in Table 2. The measured results show the sameeneral trend as predicted values of Qtotal. Though the measuredesults are lower than predictions in all cases, they indicate that aignificant amount of vibration energy is radiated into the support.

Note that it is not expected that the experimental data shouldall directly on the theoretical values, since the measurement is onef total Q, including contributions of thermalelastic loss, surface lossnd other loss. The measured results thus represent a lower boundhat measured data would be expected to approach when other loss

echanisms are negligible.Besides, the silicon resonator and the Pyrex base plate are

onded together by coating epoxy resin. Bad coating uniformity willnduce serious energy loss. Some Q values are thus much smallerhan other Q values of the resonators with the same dimensionsnd in the same batch, and Q1 of 4# prototype is one of the cases.

By comparing all the experimental results, and commenting onhe utility in various thicknesses of the prototypes, we concludehat all the cases indicate that the dominant loss mechanism inocking mass resonator may be radiation into the support. Thexpressions for power flow given in Section 3.2, could be appli-able for other beam resonators for which the attachment to theupport structure acts essentially as a point source for vibration inhe support. The plate support model is appropriate for situationsn which a resonator is attached to the edge of a plate like supporttructure, with thickness that is small enough compared with thehear wavelength in the material at the operation frequency.

. Conclusions

The general formulations applied to the simplified beam modelor power radiated into the plate support are developed.

An accurate analytical model has been developed for support

oss for rocking mode of rocking mass resonator.

When the thickness of prototypes is fabricated as 240 �m, theeasured Q can achieve a value of 589.1, less than the predicted Q

f 829.4; while the thickness of prototypes is reduced to 60 �m, the

[

[

tors A 171 (2011) 199– 206 205

measured Q can achieve a value more than 8500, less than the pre-dicted Q of 11,439. These results indicate that a significant amountof vibration energy is radiated into the support, and the validityof the analytical model has also been verified by the experimentalresults.

The derived model is general and applicable to various microbeam resonators and supports, providing significant insight todesign of high-Q rocking mass resonator devices.

Acknowledgements

The authors would like to thank the Laboratory of Microsystem,National University of Defense Technology, China, for equipmentaccess and technical support. This work was supported by NationalNatural Science Foundation of China (Grant No: 51005239 and51175506).

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iographies

iong Wang, born in 1981, is currently a PhD candidate in Micro Systemaboratory of College of Mechatronics Engineering and Automation, Nationalniversity of Defense Technology, China. He received his ME degree in mechan-

cal engineering from National University of Defense Technology and BEegree in mechanical engineering from Wuhan University, China, in 2006 and004, respectively. His research interests include micro-structure and inertiaensors.

ingbang Xiao, born in 1983, is currently a instructor in Micro System Labora-

ory of College of Mechatronics Engineering and Automation, National University ofefense Technology, China. He received his PhD degree and BE degree in mechani-al engineering from National University of Defense Technology, China, in 2009 and003, respectively. His research interests include mechatronics engineering, MEMS,nd nano-technology.

tors A 171 (2011) 199– 206

Zelong Zhou, born in 1988, is currently a ME candidate in Micro System Labora-tory of College of Mechatronics Engineering and Automation, National Universityof Defense Technology, China. He received his BE degree in instrument science andtechnology from Huazhong University of Science and Technology, China, in 2008.His research interests include micro-sensors and MEMS technology.

Zhihua Chen, born in 1963, is currently a professor in National University of DefenseTechnology, China. He received his PhD degree in mechanical engineering fromNational University of Defense Technology, China, in 2000. His research interestsinclude mechatronics engineering and MEMS.

Xuezhong Wu, born in 1965, is currently a professor and PhD candidate supervisorin National University of Defense Technology, China. His research interests include

mechatronics engineering, MEMS, and nano-technology.

Shengyi Li, born in 1946, is currently a professor and PhD candidate supervisorin National University of Defense Technology, China. His main research interestsinclude mechatronics engineering, advanced manufacture, and MEMS.