Research Papers

Voltage-Induced Snap-Through of an Asymmetrically Laminated, Piezoelectric, Thin-Film Diaphragm Micro-Actuator—Part 1: Experimental Studies and Mathematical Modeling

[+] Author and Article Information
W. C. Tai

Department of Mechanical Engineering,
Virginia Polytechnic Institute
and State University,
Blacksburg, VA 24061
e-mail: wchtai@vt.edu

Chuan Luo

Department of Precision Instruments,
Tsinghua University,
Beijing 100084, China

Cheng-Wei Yang

Department of Mechanical and
Aerospace Engineering,
University of California,
Los Angeles, CA 90095-1597

G. Z. Cao

Department of Material Science and Engineering,
University of Washington,
Seattle, WA 98195-2120

I. Y. Shen

Department of Mechanical Engineering,
University of Washington,
Seattle, WA 98195-2600

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received July 24, 2017; final manuscript received February 21, 2018; published online April 17, 2018. Assoc. Editor: Matthew Brake.

J. Vib. Acoust 140(5), 051005 (Apr 17, 2018) (13 pages) Paper No: VIB-17-1335; doi: 10.1115/1.4039535 History: Received July 24, 2017; Revised February 21, 2018

A piezoelectric thin-film microactuator in the form of an asymmetrically laminated diaphragm is developed as an intracochlear hearing aid. Experimentally, natural frequencies of the microactuator bifurcate with respect to an applied bias voltage. To qualitatively explain the findings, we model the lead-zirconate-titanate (PZT) diaphragm as a doubly curved, asymmetrically laminated, piezoelectric shallow shell defined on a rectangular domain with simply supported boundary conditions. The von Karman type nonlinear strain–displacement relationship and the Donnell–Mushtari–Vlasov theory are used to calculate the electric enthalpy and elastic strain energy. Balance of virtual work between two top electrodes is also considered to incorporate an electric-induced displacement field that has discontinuity of in-plane strain components. A set of discretized equations of motion are obtained through a variational approach.

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Krylov, S. , and Seretensky, S. , 2006, “ Higher Order Correction of Electrostatic Pressure and Its Influence on the Pull-In Behavior of Microstructures,” J. Micromech. Microeng., 16(7), pp. 1382–1396. [CrossRef]
Zhang, Y. , Wang, Y. , Li, Z. , Huang, Y. , and Li, D. , 2007, “ Snap-Through and Pull-In Instabilities of an Arch-Shaped Beam Under an Electrostatic Loading,” J. Microelectromech. Syst., 16(3), pp. 684–693. [CrossRef]
Nayfeh, A. H. , Younis, M. I. , and Abdel-Rahman, E. M. , 2007, “ Dynamic Pull-in Phenomenon in Mems Resonators,” Nonlinear Dyn., 48(1–2), pp. 153–163. [CrossRef]
Krylov, S. , Ilic, B. , Schreiber, D. , Seretensky, S. , and Craighead, H. , 2008, “ The Pull-In Behavior of Electrostatically Actuated Bistable Microstructures,” J. Micromech. Microeng., 18(5), p. 055026.
Batra, R. C. , Porfiri, M. , and Spinello, D. , 2008, “ Effects of Van Der Waals Force and Thermal Stresses on Pull-In Instability of Clamped Rectangular Microplates,” Sensors, 8(12), pp. 1048–1069. [CrossRef] [PubMed]
Das, K. , and Batra, R. , 2009, “ Symmetry Breaking, Snap-Through and Pull-In Instabilities Under Dynamic Loading of Microelectromechanical Shallow Arches,” Smart Mater. Struct., 18(11), p. 115008. [CrossRef]
Das, K. , and Batra, R. , 2009, “ Pull-In and Snap-Through Instabilities in Transient Deformations of Microelectromechanical Systems,” J. Micromech. Microeng., 19(3), p. 035008.
Ouakad, H. M. , and Younis, M. I. , 2010, “ The Dynamic Behavior of Mems Arch Resonators Actuated Electrically,” Int. J. Non-Linear Mech., 45(7), pp. 704–713. [CrossRef]
Medina, L. , Gilat, R. , and Krylov, S. , 2014, “ Symmetry Breaking in an Initially Curved Pre-Stressed Micro Beam Loaded by a Distributed Electrostatic Force,” Int. J. Solids Struct., 51(11–12), pp. 2047–2061. [CrossRef]
Ouakad, H. M. , and Younis, M. I. , 2014, “ On Using the Dynamic Snap-Through Motion of Mems Initially Curved Microbeams for Filtering Applications,” J. Sound Vib., 333(2), pp. 555–568. [CrossRef]
Medina, L. , Gilat, R. , Robert Ilic, B. , and Krylov, S. , 2016, “ Experimental Dynamic Trapping of Electrostatically Actuated Bistable Micro-Beams,” Appl. Phys. Lett., 108(7), p. 073503.
Saghir, S. , and Younis, M. I. , 2016, “ An Investigation of the Static and Dynamic Behavior of Electrically Actuated Rectangular Microplates,” Int. J. Non-Linear Mech., 85, pp. 81–93. [CrossRef]
Saghir, S. , Bellaredj, M. , Ramini, A. , and Younis, M. I. , 2016, “ Initially Curved Microplates Under Electrostatic Actuation: Theory and Experiment,” J. Micromech. Microeng., 26(9), p. 095004. [CrossRef]
Medina, L. , Gilat, R. , and Krylov, S. , 2016, “ Bistable Behavior of Electrostatically Actuated Initially Curved Micro Plate,” Sens. Actuators A: Phys., 248, pp. 193–198. [CrossRef]
Jallouli, A. , Kacem, N. , Bourbon, G. , Le Moal, P. , Walter, V. , and Lardies, J. , 2016, “ Pull-In Instability Tuning in Imperfect Nonlinear Circular Microplates Under Electrostatic Actuation,” Phys. Lett. A, 380(46), pp. 3886–3890. [CrossRef]
Medina, L. , Gilat, R. , and Krylov, S. , 2017, “ Modeling Strategies of Electrostatically Actuated Initially Curved Bistable Micro Plates,” Int. J. Solids Struct., 118–119, pp. 1–13. [CrossRef]
Li, L. , and Zhang, Q.-C. , 2017, “ Nonlinear Dynamic Analysis of Electrically Actuated Viscoelastic Bistable Microbeam System,” Nonlinear Dyn., 87(1), pp. 587–604. [CrossRef]
Saif, M. T. A. , 2000, “ On a Tunable Bistable Mems-Theory and Experiment,” J. Microelectromech. Syst., 9(2), pp. 157–170. [CrossRef]
Han, J. S. , Ko, J. S. , Kim, Y. T. , and Kwak, B. M. , 2002, “ Parametric Study and Optimization of a Micro-Optical Switch With a Laterally Driven Electromagnetic Microactuator,” J. Micromech. Microeng., 12(6), p. 939. [CrossRef]
Receveur, R. A. , Marxer, C. R. , Woering, R. , Larik, V. C. , and de Rooij, N.-F. , 2005, “ Laterally Moving Bistable Mems DC Switch for Biomedical Applications,” J. Microelectromech. Syst., 14(5), pp. 1089–1098. [CrossRef]
Wagner, B. , Quenzer, H. , Hoerschelmann, S. , Lisec, T. , and Juerss, M. , 1996, “ Bistable Microvalve With Pneumatically Coupled Membranes,” Ninth Annual International Workshop on Micro Electro Mechanical Systems, Investigation of Micro Structures, Sensors, Actuators, Machines and Systems, (MEMS'96), San Diego, CA, Feb. 11–15, pp. 384–388.
Wilcox, D. L. , and Howell, L. L. , 2005, “ Fully Compliant Tensural Bistable Micromechanisms (Ftbm),” J. Microelectromech. Syst., 14(6), pp. 1223–1235. [CrossRef]
Ko, J. S. , Lee, M. G. , Han, J. S. , Go, J. S. , Shin, B. , and Lee, D.-S. , 2006, “ A Laterally-Driven Bistable Electromagnetic Microrelay,” ETRI J., 28(3), pp. 389–392. [CrossRef]
Simitses, G. J. , 1990, Dynamic Stability of Suddenly Loaded Structures, Springer, New York. [CrossRef]
Hsu, C. , 1968, “ Stability of Shallow Arches against Snap-Through Under Timewise Step Loads,” ASME J. Appl. Mech., 35(1), pp. 31–39. [CrossRef]
Lin, J.-S. , and Chen, J.-S. , 2003, “ Dynamic Snap-Through of a Laterally Loaded Arch Under Prescribed End Motion,” Int. J. Solids Struct., 40(18), pp. 4769–4787. [CrossRef]
Chen, J.-S. , and Lin, J.-S. , 2004, “ Dynamic Snap-Through of a Shallow Arch Under a Moving Point Load,” ASME J. Vib. Acoust., 126(4), pp. 514–519. [CrossRef]
Ye, Z. , 1997, “ The Non-Linear Vibration and Dynamics Instability of Thin Shallow Shells,” J. Sound Vib., 202(3), pp. 303–311. [CrossRef]
Ventsel, E. , and Krauthammer, T. , 2001, Thin Plates and Shells: Theory: Analysis, and Applications, CRC Press, Boca Raton, FL. [CrossRef]
Maurini, C. , Pouget, J. , and Vidoli, S. , 2007, “ Distributed Piezoelectric Actuation of a Bistable Buckled Beam,” Eur. J. Mech.-A/Solids, 26(5), pp. 837–853. [CrossRef]
Varelis, D. , and Saravanos, D. A. , 2006, “ Coupled Mechanics and Finite Element for Non-Linear Laminated Piezoelectric Shallow Shells Undergoing Large Displacements and Rotations,” Int. J. Numer. Methods Eng., 66(8), pp. 1211–1233. [CrossRef]
Luo, C. , Cao, G. , and Shen, I. , 2013, “ Development of a Lead-Zirconate-Titanate (PZT) Thin-Film Microactuator Probe for Intracochlear Applications,” Sens. Actuators A: Phys., 201, pp. 1–9. [CrossRef]
Luo, C. , Cao, G. , and Shen, I. , 2012, “ Enhancing Displacement of Lead-Zirconate-Titanate (PZT) Thin-Film Membrane Microactuators Via a Dual Electrode Design,” Sens. Actuators A: Phys., 173(1), pp. 190–196. [CrossRef]
Tzou, H. , and Bao, Y. , 1997, “ Nonlinear Piezothermoelasticity and Multi-Field Actuations—Part 1: Nonlinear Anisotropic Piezothermoelastic Shell Laminates,” ASME J. Vib. Acoust., 119(3), pp. 374–381. [CrossRef]
Varelis, D. , and Saravanos, D. A. , 2004, “ Coupled Finite Element for Non-Linear Laminated Piezoelectric Composite Shells With Applications to Buckling and Postbuckling Behavior,” AIAA Paper No. AIAA 2004-1715.
Kundu, C. , Maiti, D. , and Sinha, P. , 2007, “ Post Buckling Analysis of Smart Laminated Doubly Curved Shells,” Compos. Struct., 81(3), pp. 314–322. [CrossRef]
Lentzen, S. , Kłosowski, P. , and Schmidt, R. , 2007, “ Geometrically Nonlinear Finite Element Simulation of Smart Piezolaminated Plates and Shells,” Smart Mater. Struct., 16(6), p. 2265. [CrossRef]
Nanda, N. , 2010, “ Non-Linear Free and Forced Vibrations of Piezoelectric Laminated Shells in Thermal Environments,” IES J. Part A: Civ. Struct. Eng., 3(3), pp. 147–160. [CrossRef]
Pradyumna, S. , and Gupta, A. , 2011, “ Nonlinear Dynamic Stability of Laminated Composite Shells Integrated With Piezoelectric Layers in Thermal Environment,” Acta Mech., 218(3–4), pp. 295–308. [CrossRef]
Boroujerdy, M. S. , and Eslami, M. , 2013, “ Nonlinear Axisymmetric Thermomechanical Response of Piezo-Fgm Shallow Spherical Shells,” Arch. Appl. Mech., 83(12), pp. 1681–1693. [CrossRef]
Boroujerdy, M. S. , and Eslami, M. , 2014, “ Axisymmetric Snap-Through Behavior of Piezo-Fgm Shallow Clamped Spherical Shells Under Thermo-Electro-Mechanical Loading,” Int. J. Pressure Vessels Piping, 120–121, pp. 19–26.
Pasquali, M. , and Gaudenzi, P. , 2015, “ A Nonlinear Piezoelectric Shell Model: Theoretical and Numerical Considerations,” J. Intell. Mater. Syst. Struct., 27(6), pp. 724–742. [CrossRef]
Pasquali, M. , and Gaudenzi, P. , 2015, “ A Nonlinear Formulation of Piezoelectric Shells With Complete Electro-Mechanical Coupling,” Meccanica, 50(10), pp. 2471–2486. [CrossRef]
Tzou, H. , and Zhou, Y. , 1997, “ Nonlinear Piezothermoelasticity and Multi-Field Actuations—Part 2: Control of Nonlinear Deflection, Buckling and Dynamics,” ASME J. Vib. Acoust., 119(3), pp. 382–389. [CrossRef]
Zhou, Y.-H. , and Tzou, H. , 2000, “ Active Control of Nonlinear Piezoelectric Circular Shallow Spherical Shells,” Int. J. Solids Struct., 37(12), pp. 1663–1677. [CrossRef]
Tzou, H. , and Wang, D. , 2003, “ Micro-Sensing Characteristics and Modal Voltages of Linear/Non-Linear Toroidal Shells,” J. Sound Vib., 254(2), pp. 203–218. [CrossRef]
Giannopoulos, G. , Monreal, J. , and Vantomme, J. , 2007, “ Snap-Through Buckling Behavior of Piezoelectric Bimorph Beams—I: Analytical and Numerical Modeling,” Smart Mater. Struct., 16(4), p. 1148. [CrossRef]
Lee, C.-C. , Cao, G. , and Shen, I. , 2010, “ Effects of Residual Stresses on Lead-Zirconate-Titanate (PZT) Thin-Film Membrane Microactuators,” Sens. Actuators, A: Phys., 159(1), pp. 88–95. [CrossRef]
Luo, C. , Tai, W. , Yang, C.-W. , Cao, G. , and Shen, I. , 2016, “ Effects of Added Mass on Lead-Zirconate-Titanate Thin-Film Microactuators in Aqueous Environments,” ASME J. Vib. Acoust., 138(6), p. 061015. [CrossRef]
Tzou, H. , 1993, “ Piezoelectric Shells: Distributed Sensing and Control of Continua,” Solid Mechanics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands. [CrossRef]
Shen, I. , 1997, “ Active Constrained Layer Damping Treatments for Shell Structures: A Deep-Shell Theory, Some Intuitive Results, and an Energy Analysis,” Smart Mater. Struct., 6(1), pp. 89–101. [CrossRef]
Nath, Y. , Mahrenholtz, O. , and Varma, K. , 1987, “ Non-Linear Dynamic Response of a Doubly Curved Shallow Shell on an Elastic Foundation,” J. Sound Vib., 112(1), pp. 53–61. [CrossRef]
Reddy, J. N. , 2006, Theory and Analysis of Elastic Plates and Shells, CRC Press, Boca Raton, FL.
Amabili, M. , 2005, “ Non-Linear Vibrations of Doubly Curved Shallow Shells,” Int. J. Non-Linear Mech., 40(5), pp. 683–710. [CrossRef]
Dickinson, S. , and Di Blasio, A. , 1986, “ On the Use of Orthogonal Polynomials in the Rayleigh-Ritz Method for the Study of the Flexural Vibration and Buckling of Isotropic and Orthotropic Rectangular Plates,” J. Sound Vib., 108(1), pp. 51–62. [CrossRef]
Gibbs, G. P. , and Fuller, C. R. , 1992, “ Excitation of Thin Beams Using Asymmetric Piezoelectric Actuators,” J. Acoust. Soc. Am., 92(6), pp. 3221–3227. [CrossRef]
Cottone, F. , Gammaitoni, L. , Vocca, H. , Ferrari, M. , and Ferrari, V. , 2012, “ Piezoelectric Buckled Beams for Random Vibration Energy Harvesting,” Smart Mater. Struct., 21(3), p. 035021. [CrossRef]


Grahic Jump Location
Fig. 1

Schematic of the PZT thin-film micro-actuator (not to scale)

Grahic Jump Location
Fig. 2

Natural frequencies versus DC bias voltage. Some FRFs are plotted for the reference.

Grahic Jump Location
Fig. 3

Three-layer model of nonlinear shallow shells for the thin-film microactuator: (a) description of individual layers, the modulus-weighted midsurface S¯ (– – –), and geometry and (b) local coordinate system C(i) of the ith layer and global coordinate system C¯

Grahic Jump Location
Fig. 4

Sign convention of forces and moments at one boundary between the inner electrode and outer electrode domains: (a) outer electrode domain Ω+ and (b) inner electrode domain Ω−. Note that n and t denote the normal and tangential directions at a boundary.




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