Vibration analysis of a piezoelectric micromachined modal gyroscope (PMMG)

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1 IOP PUBLISHING JOURNAL OF MICROMECHANICS AND MICROENGINEERING J. Micromech. Microeng. 19 (2009) (10pp) doi: / /19/12/ Vibration analysis of a piezoelectric micromachined modal gyroscope (PMMG) Xiaosheng Wu, Wenyuan Chen, Yipeng Lu, Qijun Xiao, Gaoyin Ma, Weiping Zhang and Feng Cui Key Laboratory for Thin Film and Microfabrication Technology of Ministry of Education, National Key Laboratory of Micro/Nano Fabrication Technology, Research Institute of Micro/Nano Science and Technology, Shanghai Jiao Tong University, Shanghai , People s Republic of China wu showson@163.com and chenwy@sjtu.edu.cn Received 1 April 2009, in final form 20 August 2009 Published 22 October 2009 Online at stacks.iop.org/jmm/19/ Abstract Piezoelectric micromachined modal gyroscope (PMMG) is a novel kind of rotating rate sensor, which is based on the special thickness-shear vibrating mode of a piezoelectric body. Compared with the general vibratory micro-gyro, the PMMG has no evident mass-spring component in its structure, so it has larger stiffness and robust resistance to shake and strike. Therefore, the PMMG can be used in the high-g environment especially, such as the fuse of smart munitions. In this paper, first, the working mechanism of the PMMG is proposed. Then, two kinds of models of the PMMG are introduced, one of which is the piezoelectric rectangular parallelepiped with driving electrodes or sensing electrodes on the top and bottom surface of the body (model 1), and the other is the piezoelectric rectangular parallelepiped with concentrated masses at four corners of the top and bottom surface of the body (model 2). For the two kinds of models, the modal and harmonic analyses are conducted, and the working mode of the PMMG is obtained based on the finite element method (FEM) analysis results. It is found that for model 2 the resonance frequency of working mode is lower than that of model 1 and the vibration directivity is improved. The fabrication processes and the controlling circuits of the PMMG are detailed in this paper. The PMMG prototype test validated the results of vibration analysis of the PMMG. The introduction of concentrated masses can lower the resonance frequency and improve the vibration directivity of the working mode of the PMMG. The work in this paper provides the theoretical and experimental foundation for realizing this novel kind of micromachined gyroscope. (Some figures in this article are in colour only in the electronic version) 1. Introduction Different from the micromachined vibratory gyroscope, which has the structure of suspending springs and proof masses, some kinds of solid-state gyroscopes have no part which moves as a whole. According to their working principle, these kinds of gyroscopes can be classified into two categories. One is the non-vibratory solid-state gyroscope, in which there is absolutely no vibration, such as optical gyroscope [1] and atom gyroscope [2, 3]. The other is the vibratory solid-state gyroscope, in which there exist parts with only local vibrations, such as hemispherical resonant gyroscope [4] and surface acoustic wave gyroscope [5, 6]. Because there is no moving part and suspending structure in a solid-state gyroscope, it is robust and has higher resistance to shock and shake. Due to no vibration or existing vibration with very small amplitude and high frequency, the air-damping effect is not evident in this kind of solid-sate gyroscope, so they can work in atmospheric environment and have no special requirement of vacuum packaging. This is advantageous to increase the reliability of micro-gyro. Recently, with the development of MEMS technology, the volume of solid-state gyroscopes /09/ $ IOP Publishing Ltd Printed in the UK

2 rotation input (c) Primary vibration Induced Coriolis force Figure 1. Working principle of the PMMG. has been reduced through micromachining processes. These micromachined solid-state gyroscopes include micro optical gyroscopes [7, 8], micro atomic gyroscopes [9], micro-surface acoustic wave gyroscopes [6, 10], etc. Piezoelectric gyroscopes make use of two vibration modes of a vibratory piezoelectric body in which material particles move in two perpendicular directions respectively. When a piezoelectric gyroscope is excited into vibration in one of the two modes (the primary mode) by an applied alternating voltage and attached to a rotating body, Coriolis force will excite the other mode (the secondary mode) through which the rotation rate of the body can be detected. These piezoelectric gyroscopes include flexural vibrations in two perpendicular directions of beams and tuning forks [11 14], thicknessshear vibrations in two perpendicular directions of plates [15, 16], radial and torsion vibrations of circular cylindrical shells [17] and degenerate modes of circular disks, shells and rings [18 20]. In January of 2006, Japanese researchers, Maenaka et al, proposed a novel piezoelectric solid micro-gyroscope [21], which is named the piezoelectric micromachined modal gyroscope (PMMG) in this paper. It is found that in a higher order resonance mode, the movements of the mass elements are almost in one direction and differential on the sides of the prism, and this modal vibration can be used as the primary mode of the piezoelectric gyroscope. The work of Maenaka et al did not cover the modal and harmonic vibration analysis and its test validation in detail. In this paper, a kind of PMMG model, which has the concentrated masses in the position with the largest deformation displacement and the largest vibration velocity, is proposed. The modal and harmonic vibration of two kinds of PMMG is analyzed through simulation and experiment in this paper. In this paper, the working principle of a PMMG is detailed. Two kinds of models of the PMMG are introduced. Then the modal and harmonic vibration analyses of two models are conducted. The fabrication processes of the PMMG and driving, detecting circuit are designed and realized. The test result of the prototype of the PMMG verified the vibration analysis in this paper. The PMMG can also be used as a dual-axis rotating rate sensor. S1 R1 S2 D+ D- Polarization S4 R1 S3 Figure 2. Model 1 of the PMMG. 2. The working principle of the PMMG PZT Substrate As mentioned above, the primary vibratory mode of the PMMG is a kind of higher order resonance mode in which the movements of the mass elements are almost in one direction and differential on the sides of the prism [21]. In this resonance mode, the device is sensitive to the applied angular velocity in the perpendicular direction. This higher order resonance mode is called the working resonance mode in this paper. Figure 1 schematically shows the operation of the device. The rectangular prism is made of PZT and it is polarized along the z-axis. With excitation of the PZT prism at the working resonance mode, the differential vibration along the y-axis in the prism appears as shown in figure 1. When the angular velocity in the direction of the x-axis is input, the Coriolis force is generated according to the movement of the mass elements as shown in figure 1 (for rotation in the x- axis), resulting in compressive and tensile stresses depending on the position. These stresses differentially generate the piezoelectric voltage on the surface of the device as shown in figure 1(c). The induced piezoelectric voltage is proportional to the input angular rate. A kind of PMMG model is introduced as shown in figure 2, which was first proposed by Maenaka et al. This model is called model 1 in the paper. Piezoelectric ceramic material PZT is selected as the substrate block. On the top and bottom surfaces of the PZT block, a lot of driving electrodes and sensing electrodes are distributed. The polarization 2

3 Polarization PZT Substrate z y x u P 2 u2 2b 2c 2a Figure 3. Model 2 of the PMMG. direction of the PZT substrate is perpendicular to the surfaces with electrodes distributed. As shown in the figure, D+ and D are the driving electrodes, and the exciting voltages are applied on the D+ and D electrodes. R1 and R2 are the reference electrodes. The analysis shows that there is an evident voltage peak on the R1 and R2 electrodes in the position with the frequency of the working resonance mode, so R1 and R2 can be used for searching and tracking the working resonance mode. S1, S2, S3 and S4 are the sensing electrodes. When there is no angular input, the voltages of two adjacent electrodes, such as S1 and S2, or S3 and S4, are the same because of the symmetric structure of the piezoelectric block. If the rotation in any direction perpendicular to the modal vibration is input, the voltage of sensing electrodes is changed because of the coupling of the Coriolis effect, and then the voltages of two adjacent electrodes are not the same. Through detecting the voltage difference of two adjacent sensing electrodes, the rotation input can be quantized. On the bottom surface, the sensing and driving electrodes are distributed in the corresponding positions as on the top surface. In figure 1, we can find that for the working resonance mode, the largest nodal displacement and the largest vibration velocity exist at the positions of the four corners of the piezoelectric body s top and bottom surfaces. Therefore, in order to increase the kinetic energy of the working resonance mode and to improve the vibration directivity of the piezoelectric block, some concentrated masses m1, m2, m3 and m4 are added in the corners of the top and bottom surfaces of the piezoelectric block, as shown in figure 3, and the sensing electrodes of S1, S2, S3 and S4 in model 1 are replaced by m1, m2, m3 and m4, respectively. The concentrated masses in figure 3 are made with heavy metal, such as copper. The two models will be compared in the following analysis. In this paper, we name the model shown in figure 3 as model 2. The main research work of the paper is focused on the research of the working resonance mode of the two models of the PMMG. Piezoelectric ceramic is a commonly used material for transducers, and the requirements for the performance of piezoelectric ceramic vary for different regions of applications. For PMMG, piezoelectric ceramic is used as an actuator to excite the working resonance mode, and at the same time it is also used as a sensitive material to sense Coriolis force, so the Figure 4. Model sizes in the coordinate system and modal displacement on edge points. piezoelectric material should have larger piezoelectric constant d 33 and electromechanical coupling coefficients k 33 and k 15. According to the above requirement, piezoelectric ceramic PZT-5H is selected as a substrate material in the analysis. The PZT-5 H material constants used in this paper are listed as follows: The dielectric constant (ε r ): ε 11 = 1700, ε 33 = 1470 Material mass density (ρ): 7500 kg m 3 The piezoelectric constant (e): e 31 = 6.5 C m 2, e 33 = 23.3 C m 2, e 15 = 17.0 C m 2 The elastic constant matrix (c): c 11 = Nm 2, c 12 = Nm 2, c 13 = Nm 2, c 33 = Nm 2, c 44 = Nm 2, c 66 = Nm Modal vibration analysis of the PMMG 3.1. Finite element analysis of working modal vibration of the piezoelectric body In this section we give the results of the finite element analysis (FEA) of the working modal vibrations of the piezoelectric body of the PMMG. According to elastic mechanical theory, the working modal vibrations of the piezoelectric body can be defined as thickness-shear modes, in which the displacement component in the y-axis for the edge nodes is described as in figure 4. From the figure we can find for the working modal vibration that the displacement component is symmetric about the center of the piezoelectric body. From the working principle of the PMMG, it is concluded that the working resonance mode should have the following characteristics. (1) The movement of points in the piezoelectric block should be almost in one direction, x- axis or y-axis. (2) The moving direction of points should be perpendicular to the polarization direction of the piezoelectric block. (3) The moving direction of a point on one edge is the same as that of the corresponding point on the diagonal edge, and is opposite to that of the corresponding point on the adjacent edge. (4) Moving edges should be in the state of tension or compression. In order to search for such a kind of special vibration, the finite element software ANSYS is used to analyze the modes of the piezoelectric block. The size of the piezoelectric block is selected as 5 4 3mm 3. 3

4 Figure 5. The node displacement of the seventh mode for model 1: y-component displacement contour and the node displacement vectors on the top surface. Figure 6. The node displacement of the sixth mode for model 2: y-component displacement contour and the node displacement vectors on the top surface. For model 1, more than 40 modes are obtained. If only concerning the displacement in the x-axis direction, the modes with orders of 6 and 12 can be satisfied. If only concerning the displacement in the y-axis direction, the modes with orders of 7 and 10 can be satisfied. For the ideal working resonance mode, the points vibrate only in one direction, and there is no vibration in other two orthogonal directions. Considering displacement contours in both x-axis direction and y-axis direction, the seventh mode of the piezoelectric block is the better working resonance mode which is closest to the ideal vibration as shown in figure 4. The displacement contour of the seventh mode in the y-axis direction is given in figure 5. The resonance frequency of the seventh mode is khz. In figure 5, the nodal displacement vectors on the top surface are shown. The nodal movement in the y-axis direction is satisfied, but still there is considerable x- component for displacement vectors. As the x-component for displacement of the resonance mode is in the same direction, it does not affect the voltage difference caused by the Coriolis force on the adjacent two sensing electrodes. For model 2, the concentrated masses are added and they replace the sensing electrodes in model 1. The material of the concentrated masses is copper and the size of every concentrated mass is mm 3. If only concerning the displacement component in the x-axis direction, the modes with orders of 5 and 18 can be satisfied. If only concerning the displacement component in the y-axis direction, the modes with orders of 6 and 15 can be satisfied. Considering the displacement contours of both the x component and the y component, the sixth mode is the better working resonance mode which is closest to the ideal vibration as shown in figure 4. The displacement contour of the sixth mode in the y-axis direction is given in figure 6. The resonance frequency of the sixth mode is khz. In figure 6, the node displacement vectors on the top surface are shown. 4

5 Working resonance mode Working resonance mode Figure 7. Relation of the voltage output of the reference electrode to frequency: model 1 and model 2. Comparing the seventh mode of model 1 and the sixth mode of model 2, the resonance frequency ( khz) of model 2 is found to be much lower than that ( khz) of model 1. The maximum value for the y-component of displacement for model 2 is , and that for model 1 is , so as mentioned before, the added masses can increase the stored energy of the piezoelectric body. The increased stored energy is advantageous to enhance the Coriolis effect and further to increase the sensitivity of the gyroscope. Comparing figures 5 and 6, the displacement component in the x-axis direction of model 2 is found to be smaller than that of model 1, so the displacement directivity of PMMG modal vibration is improved because of the added concentrated masses Finite element analysis of forced vibration of the piezoelectric body In the PMMG, the driving voltages are applied on the driving electrodes D+ and D on the top surface and bottom surface of the piezoelectric block. In order to obtain vibration under the working resonance mode, the frequency of the driving voltage should be the same as that of the working resonance mode, and for example, it is khz for model 1 and khz for model 2. In principle, the piezoelectric block will vibrate with the mode corresponding to the frequency of the driving voltage. In order to validate the result of mode driving, we conducted the forced excitation analysis using finite element software ANSYS. The amplitude of driving voltage is 7.5 V, and the phase difference of the voltages on D+ and D is 180. The damping constant of the piezoelectric material is introduced with the value of The frequency of the driving voltage is scanned from 250 khz to 400 khz for model 1 (figure 7) and from 200 khz to 500 khz for model 2 (figure 7). Figures 7 and 8 give the harmonic excitation analysis result of models 1 and 2, in which the x-coordinate refers to the frequency of the driving voltage and the y-coordinate refers to the piezoelectric voltage amplitude on the reference electrode, R1 or R2. From the figure we can see that there are two peaks of the voltage of the reference electrode for model 1. The peak corresponding to the working resonance mode is in the position of around khz for model 1 and khz for model 2. The working resonance mode of model 1 is the sixth-order mode and its frequency is khz. The working resonance mode of model 2 is the seventh mode with the resonance of khz. The frequency difference of modal analysis and harmonic analysis is caused by the scaning frequency step in harmonic analysis. Figures 8 and show the displacement vectors on the top surface of the piezoelectric block at the exciting frequency of the working resonance mode of models 1 and 2, respectively. Comparing figures 5,8, 6 and 9, it is concluded that the exciting vibration of the piezoelectric block is the same as the vibrating shape of the working resonance mode. That is to say that through applying exciting voltage with the frequency of the working resonance mode on the driving electrodes, the vibration of the working resonance mode can be obtained. From figures 7 and 8 we can also see that there is a peak of piezoelectric voltage on the reference electrodes in the position of the working resonance frequency, so the R1 or R2 electrode can be effectively used to scan and track the working resonance frequency. Comparing figures 8 and 8, the displacement vectors of model 2 are found to be larger than those of model 1 in the same position, so the energy stored in model 2 is increased. In figures 7 and, some other voltage peaks occur in the position with larger frequency. These voltage peaks correspond to the higher order vibrating modes which have similar vibrating shape to the working vibration mode. That is the 12th mode of model 1 and the 10th mode of model 2, as presented in earlier modal analysis of the paper. 5

6 Figure 8. Displacement vectors on the top surface of forced vibration of the PMMG: model 1 and model 2. (g) (h) (c) (i) (d) (j) (e) (k) ( f ) (l) PZT Substrate Seed layer Photo resist Copper Conductive adhesive Figure 9. Fabrication processes of the PMMG. 4. Fabrication of PMMG For model 1 of the PMMG, the micromachined fabrication process is used to define driving electrodes, sensing electrodes and reference electrodes. In this model, the copper film is electroplated and patterned as an electrode. For model 2 of the PMMG, an additional process to fabricate concentrated masses is needed. The detailed fabrication process of the PMMG is shown in figure 9. The PZT substrate is first rinsed to remove the particles or contamination on both surfaces. Then the substrate is sputtered by the seed layer Cr/Cu (10 nm/90 nm) for the following electrodes electrodeposition process. After the double-side lithographic processes with the positive photo resist AZ P4620 (7 μm), the copper electrodes are 6

7 Figure 10. Prototypes of the PMMG: model 1 and model 2. electroplated on both sides with the thickness of 5 μm (c) (e). Then the photo resist and exposed seed layer (Cr/Cu) are removed with wet etchant, the solution of NH 4 OH(1) + H 2 O 2 (5) for Cu and the solution of K 3 Fe(CN) 6 +NaOH+ H 2 OforCr(f ). Now, the processes for model 1 of the PMMG are complete. After dicing, the prototype of model 1 can be obtained. The prototype of model 1 is shown in figure 10. The following processes are for concentrated masses of model 2 (g) (l). In order to increase the adhesion between the substrate and concentrated masses of model 2, the metal foundation structure is applied here [22, 23]. After the photo resist used for etching mask is spun on and patterned through the lithographic process, the PZT is wet etched by the solution of BHF(1) + HCl(2) + NH 4 Cl(4) + H 2 O(4). Rectangular hollows with a depth of 20 μm are formed on both sides of the substrate (g) (h). After removing the photo resist and sputtering the seed layer (Cr/Cu), the substrate is patterned through the lithographic process (i) (j) for the following electroplate of the metal foundation structure. Then the nickel layer with the thickness of 25 μm is electrodeposited, and the photo resist and the exposed seed layer are removed (k). Finally, the concentrated masses are adhered on the surface of the metal foundation structure using electric conductive adhesive. The completed prototype of model 2 is shown in figure Experiment Figure 11 shows the driving and detecting scheme of the PMMG. The digital signal processor (DSP, TMS320F2812), as a controller, sends the controlling words to the direct digital synthesizer (DDS). According to the controlling words, the DDS outputs the sine signal with expected working frequency and phase. After the band passing filter, the signal produced by the DDS is improved and is then sent to two linear proportional amplifiers, amplifiers 1 and 2, one of which is the non-inverting amplifier and the other is the inverting amplifier. Two amplified voltage signals DSP (TMS 320 F2812) Non-inverting Amplifier 1 Charge Amplifier (1) V 1 DDS (AD9833) PMMG Inverting Amplifier 2 Charge Amplifier (n) Figure 11. Driving and detecting scheme of the PMMG. were used to drive the PMMG. The resolution of frequency obtained through the DDS can reach a high level, about 0.1 Hz for AD 9833, and this is good for accurately obtaining and tracking the working resonance of the PMMG. Because the piezoelectric sensor has huge output impedance, it is necessary to introduce charge amplifiers to pick up the charge on sensing electrodes or reference electrodes. As shown in figure 11, the inputs of charge amplifiers are connected to the sensing electrodes and reference electrodes of the PMMG. The charge induced by vibration of the piezoelectric body on sensing or reference electrodes is converted to voltage signal through the charge amplifiers (1) (n) as shown in figure 11. Figure 12 shows the experimental setup and a close-up view of the device. V n 7

8 Figure 12. Picture of the experimental setup and a close-up view of the device. 3 experiment Voltage on electrode (V) simulation experiment Voltage on electrode (V) simulation without considering additional mass simulation considering additional mass Frequency of driving voltage (khz) Figure 13. Voltage output versus driving frequency of model 1. The driving voltages are applied on driving electrodes. Through changing the frequency controlling words of the DDS, the frequency of driving voltage signals is scanned with the range from 300 khz to 340 khz for model 1 and 228 khz to 270 khz for model 2. The relations of the voltage output to the driving frequency for models 1 and 2 are shown in figures 13 and 14. In the figures, the results of simulation and experiment are compared. In figure 13, two curves respectively give the voltages on reference electrodes obtained through simulation and prototype experiment for model 1. The frequency of the driving voltage is scanned from 300 khz to 340 khz. In the figure, the voltage in the simulation curve is the piezoelectric voltage on the reference electrode obtained through forced Frequency of driving voltage (khz) Figure 14. Voltage output versus driving frequency of model 2. vibration analysis of finite element, and the voltage for the experimental curve is the output of the charge amplifier, whose input is connected with the reference electrode, so the output voltage of the charge amplifier is converted from the piezoelectric charge. The piezoelectric voltage and piezoelectric charge are related by the equivalent capacitance of the piezoelectric body. Here, we are only interested in the resonance frequency, so the piezoelectric voltage and piezoelectric charge will not be investigated further. The peaks of the two curves must occur at the frequency of the working resonance mode. From the figure we can see that the resonance frequencies obtained from simulation and experiment are both at about 326 khz, which is equal to the seventh modal frequency of model 1 (figure 5). The experimental result is consistent with the modal and forced vibration analysis. 8

9 As for model 1, for model 2, the driving voltages are applied on driving electrodes and the voltage or charge output of reference electrodes is investigated. Figure 14 compares the result of the experiment and forced vibration simulation result of model 2. For model 2, it is concluded from the forced vibration analysis that the frequency of the working resonance mode is about 259 khz, which is the peak position of the simulation curve in figure 14. The experimental results show that the resonance occurs at the frequency of 254 khz, and there is about 5 khz frequency difference between the peaks of the experimental curve and simulation curve. The frequency difference is caused by the additional masses of electric conductive adhesive. There are some other small peaks on the experimental curve. These small peaks are caused by the asymmetry due to the error of the position and dimension of concentrated masses. These can be improved in the further research by increasing the fabrication precision of the PMMG. The additional masses of the electric conductive adhesive cannot be given accurately. In the simulation, we assumed the electric conductive adhesive with the height of 50 μm and conducted the harmonic analysis. The results show that the resonance frequency is reduced to khz, as shown in figure 14. That is to say that the additional masses lower the frequency of the working resonance mode. 6. Conclusion and discussion PMMG is a novel kind of solid-state angular rate sensor, which works based on the special vibration mode of the solid-state piezoelectric body. The structure of the PMMG has larger stiffness, so it is resistant to shake and strike caused by highg or other conditions. In this paper, two kinds of models of the PMMG are introduced and compared. The analysis results of mode and forced vibration show that the model with concentrated masses can lower the resonance frequency. The frequency of the working resonance mode of model 1 is khz, while that of model 2 is khz. The decreased working frequency is good for designing a driving circuit with large output voltage and diminishing the noise influence. Model 2 has better modal vibration directivity. The concentrated masses included in model 2 can increase the modal displacement of nodes, so the stored energy in model 2 is increased and the sensitivity of the PMMG can be improved further. For the fabrication processes of the PMMG, the structure of metal foundation increased the adhesion of concentrated masses and substrate. The experimental results of the PMMG prototype are consistent with the analysis, and at the same time the results of modal and forced vibration analysis are validated. During the working of the PMMG, the frequency of the working resonance mode must drift from the initial position because the temperature or other environmental parameters are varied. The reference electrodes not only can be used to search the frequency of the working resonance mode but they also can be used to track the working mode when the modal frequency drifts. The realization of working resonance modal vibration is the basis for the work of the PMMG. The following research work of the PMMG will be focused on the Coriolis effect analysis and its detection. The work in this paper provides a theoretical and experimental foundation for the optimization and further realization of the PMMG. Acknowledgment Financial support for this research from the PLA General Armament Department and National Natural Science Foundation of China ( /E051202) is gratefully acknowledged. References [1] Killian K M, Burmenko M and Holinger W 1994 High-performance fiber optic gyroscope with noise reduction Proc. SPIE [2] Scully M O and Dowling J P 1993 Quantum-noise limits to matter-wave interferometry Phys. Rev. A [3] Gustavson T L, Landragin A and Kasevich M A 2000 Rotation sensing with a dual atom-interferometer Sagnac gyroscope Class. Quantum Grav [4] Matthews A and Rybak F J 1992 Comparison of hemispherical resonator gyro and optical gyros IEEE Aerosp. Electron. Syst. Mag [5] Kurosawa M, Fukuda Y, Takasake M and Higuchi T 1998 A surface-acoustic-wave gyro sensor Sensors Actuators A [6] Lee S W, Rhim J W, Park S W and Yang S S 2007 A micro rate gyroscope based on the SAW gyroscopic effect J. Micromech. Microeng [7] Mottier P and Pouteau P 1997 Solid state optical gyrometer integrated on silicon Electron. Lett [8] Guo W, Ma H, Jin Z, Tang Y Z and Wang Y L 2002 A novel structure of passive ring waveguide resonator on silicon substrate Proc. SPIE [9] Serkland D K, Geib K M, Peake G M, Lutwak R and Rashed A 2007 VCSELs for atomic sensors Proc. SPIE [10] Sharma J N, Walia V and Gupta S K 2008 Effect of rotation and thermal relaxation on Rayleigh waves in piezothermoelastic half space Int. J. Mech. Sci [11] Soderkvist J 1991 Piezoelectric beams and vibrating angular rate sensors IEEE Trans. Ultrason. Ferroelectr. Freq. Control [12] Chou C S, Yang J W, Huang Y C and Yang H J 1991 Analysis on vibrating piezoelectric beam gyroscope Int. J. Appl. Electromagn. Mater [13] Ulitko I A 1995 Mathematical theory of the fork-type wave gyroscope Proc. IEEE Int. Frequency Control Symp. pp [14] Fujishima S, Nakamura T and Fujimoto K 1991 Piezoelectric vibratory gyroscope using flexural vibration of a triangular bar Proc. IEEE 45th Ann. Symp. on Frequency Control pp [15] Reese G M, Marek E L and Lobitz D W 1989 Three-dimensional finite element calculations of an experimental quartz resonator sensor Proc. IEEE Ultrasonics Symp. pp [16] Abe H, Yoshida T, Ishikawa T, Miyazaki N and Watanabe H 2001 Trapped-energy vibratory gyroscopes using a partially polarized piezoelectric ceramic plate Electron. Commun. Japan

10 [17] Yang J S, Fang H Y and Jiang Q 2000 A vibrating piezoelectric ceramic shell as a rotation sensor Smart Mater. Struct [18] Burdess J S and Wren T 1986 The theory of a piezoelectric disk gyroscope IEEE Trans. Aerosp. Electron. Syst [19] Burdess J S 1986 The dynamics of a thin piezoelectric cylinder gyroscope Proc. Inst. Mech. Eng [20] Loveday P W 1996 A coupled electromechanical model of an imperfect piezoelectric vibrating cylinder gyroscope J. Intell. Mater. Syst. Struct [21] Maenaka K, Kohara H, Nishimura M, Fujita T and Takayama Y 2006 Novel solid micro-gyroscope Tech. Dig. of the 19th IEEE Int. Conf. on Microelectromechanical Systems (Istanbul) pp [22] Ho C H and Hsu W 2004 Experimental investigation of an embedded root method for stripping SU-8 photoresist in the UV-LIGA process J. Micromech. Microeng [23] Cui F, Chen W Y, Zhao X L, Jing X M and Wu X S 2006 Metal foundation construction to consolidate electroplated structures for successful removal of SU-8 mould Electron. Lett

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