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Research Papers

Internal Energy Transfer in Dynamical Behavior of Slightly Curved Shear Deformable Microplates

[+] Author and Article Information
Mergen H. Ghayesh

School of Mechanical, Materials, and
Mechatronic Engineering,
University of Wollongong,
Wollongong, NSW 2522, Australia
e-mail: mergen@uow.edu.au

Hamed Farokhi

Department of Mechanical Engineering,
McGill University,
Montreal, QC H3A 0C3, Canada
e-mail: hamed.farokhi@mail.mcgill.ca

Gursel Alici

School of Mechanical, Materials,
and Mechatronic Engineering,
University of Wollongong,
Wollongong, NSW 2522, Australia
e-mail: gursel@uow.edu.au

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received February 25, 2015; final manuscript received August 5, 2015; published online November 13, 2015. Assoc. Editor: Ahmet S. Yigit.

J. Comput. Nonlinear Dynam 11(4), 041002 (Nov 13, 2015) (11 pages) Paper No: CND-15-1051; doi: 10.1115/1.4031290 History: Received February 25, 2015; Revised August 05, 2015

This paper investigates the internal energy transfer and modal interactions in the dynamical behavior of slightly curved microplates. Employing the third-order shear deformation theory, the microplate model is developed taking into account geometric nonlinearities as well as the modified couple stress theory; the initial curvature is modeled by an initial imperfection in the out-of-plane direction. The in-plane displacements and inertia are retained, and the coupled out-of-plane, rotational, and in-plane motion characteristics are analyzed. Specifically, continuous models are developed for kinetic and potential energies as well as damping and external works; these are balanced and reduced via Lagrange's equations along with an assumed-mode technique. The reduced-order model is then solved numerically by means of a continuation technique; stability analysis is performed by means of the Floquet theory. The possibility of the occurrence of modal interactions and internal energy transfers is verified via a linear analysis on different natural frequencies of the system. The nonlinear resonant response of the system is obtained for the cases with internal energy transfer, and energy transfer mechanisms are analyzed; as we shall see, the presence of an initial curvature affects the system dynamics substantially. The importance of taking into account small-size effects is also shown by discovering this fact that both the linear and nonlinear internal energy transfer mechanisms are shifted substantially if this effect is ignored.

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References

Rembe, C. , and Muller, R. S. , 2002, “ Measurement System for Full Three-Dimensional Motion Characterization of MEMS,” J. Microelectromech. Syst., 11(5), pp. 479–488. [CrossRef]
Ghayesh, M. H. , Farokhi, H. , and Amabili, M. , 2013, “ Nonlinear Behaviour of Electrically Actuated MEMS Resonators,” Int. J. Eng. Sci., 71, pp. 137–155. [CrossRef]
Younis, M. I. , Abdel-Rahman, E. M. , and Nayfeh, A. , 2003, “ A Reduced-Order Model for Electrically Actuated Microbeam-Based MEMS,” J. Microelectromech. Syst., 12(5), pp. 672–680. [CrossRef]
De, S. K. , and Aluru, N. R. , 2004, “ Full-Lagrangian Schemes for Dynamic Analysis of Electrostatic MEMS,” J. Microelectromech. Syst., 13(5), pp. 737–758. [CrossRef]
Liu, S. , Davidson, A. , and Lin, Q. , 2004, “ Simulation Studies on Nonlinear Dynamics and Chaos in a MEMS Cantilever Control System,” J. Micromech. Microeng., 14(7), pp. 1064–1073. [CrossRef]
Sulfridge, M. , Saif, T. , Miller, N. , and Meinhart, M. , 2004, “ Nonlinear Dynamic Study of a Bistable MEMS: Model and Experiment,” J. Microelectromech. Syst., 13(5), pp. 725–731. [CrossRef]
Zhang, W. M. , and Meng, G. , 2007, “ Nonlinear Dynamic Analysis of Electrostatically Actuated Resonant MEMS Sensors Under Parametric Excitation,” IEEE Sens. J., 7(3), pp. 370–380. [CrossRef]
Duwel, A. , Candler, R. N. , Kenny, T. W. , and Varghese, M. , 2006, “ Engineering MEMS Resonators With Low Thermoelastic Damping,” J. Microelectromech. Syst., 15(6), pp. 1437–1445. [CrossRef]
Mestrom, R. M. C. , Fey, R. H. B. , van Beek, J. T. M. , Phan, K. L. , and Nijmeijer, H. , 2008, “ Modelling the Dynamics of a MEMS Resonator: Simulations and Experiments,” Sens. Actuators, A, 142(1), pp. 306–315. [CrossRef]
Bergers, L. I. J. C. , Hoefnagels, J. P. M. , Delhey, N. K. R. , and Geers, M. G. D. , 2011, “ Measuring Time-Dependent Deformations in Metallic MEMS,” Microelectron. Reliab., 51(6), pp. 1054–1059. [CrossRef]
Haghighi, H. S. , and Markazi, A. H. D. , 2010, “ Chaos Prediction and Control in MEMS Resonators,” Commun. Nonlinear Sci. Numer. Simul., 15(10), pp. 3091–3099. [CrossRef]
Farokhi, H. , and Ghayesh, M. , “ Size-Dependent Behaviour of Electrically Actuated Microcantilever-Based MEMS,” Int. J. Mech. Mater. Des. (in press).
Gholipour, A. , Farokhi, H. , and Ghayesh, M. , 2014, “ In-Plane and Out-Of-Plane Nonlinear Size-Dependent Dynamics of Microplates,” Nonlinear Dyn., 79(3), pp. 1771–1785. [CrossRef]
Ghayesh, M. H. , and Farokhi, H. , 2015, “ Nonlinear Dynamics of Microplates,” Int. J. Eng. Sci., 86, pp. 60–73. [CrossRef]
Farokhi, H. , and Ghayesh, M. H. , 2015, “ Nonlinear Dynamical Behaviour of Geometrically Imperfect Microplates Based on Modified Couple Stress Theory,” Int. J. Mech. Sci., 90, pp. 133–144. [CrossRef]
Ghayesh, M. H. , Farokhi, H. , and Alici, G. , 2015, “ Subcritical Parametric Dynamics of Microbeams,” Int. J. Eng. Sci., 95, pp. 36–48. [CrossRef]
Das, K. , and Batra, R. C. , 2009, “ Pull-in and Snap-Through Instabilities in Transient Deformations of Microelectromechanical Systems,” J. Micromech. Microeng., 19(3), p. 035008. [CrossRef]
Slava, K. , Bojan, R. I. , David, S. , Shimon, S. , and Harold, C. , 2008, “ The Pull-In Behavior of Electrostatically Actuated Bistable Microstructures,” J. Micromech. Microeng., 18(5), p. 055026. [CrossRef]
Lam, D. C. C. , Yang, F. , Chong, A. C. M. , Wang, J. , and Tong, P. , 2003, “ Experiments and Theory in Strain Gradient Elasticity,” J. Mech. Phys. Solids, 51(8), pp. 1477–1508. [CrossRef]
McFarland, A. W. , and Colton, J. S. , 2005, “ Role of Material Microstructure in Plate Stiffness With Relevance to Microcantilever Sensors,” J. Micromech. Microeng., 15(5), p. 1060. [CrossRef]
Fleck, N. A. , Muller, G. M. , Ashby, M. F. , and Hutchinson, J. W. , 1994, “ Strain Gradient Plasticity: Theory and Experiment,” Acta Metall. Mater., 42(2), pp. 475–487. [CrossRef]
Zhao, J. , Zhou, S. , Wang, B. , and Wang, X. , 2012, “ Nonlinear Microbeam Model Based on Strain Gradient Theory,” Appl. Math. Modell., 36(6), pp. 2674–2686. [CrossRef]
Akgöz, B. , and Civalek, Ö. , 2013, “ A Size-Dependent Shear Deformation Beam Model Based on the Strain Gradient Elasticity Theory,” Int. J. Eng. Sci., 70, pp. 1–14. [CrossRef]
Salamat-Talab, M. , Shahabi, F. , and Assadi, A. , 2012, “ Size Dependent Analysis of Functionally Graded Microbeams Using Strain Gradient Elasticity Incorporated With Surface Energy,” Appl. Math. Modell., 37(1–2), pp. 507–526.
Mohammadi, H. , and Mahzoon, M. , 2013, “ Thermal Effects on Postbuckling of Nonlinear Microbeams Based on the Modified Strain Gradient Theory,” Compos. Struct., 106, pp. 764–776. [CrossRef]
Farokhi, H. , Ghayesh, M. H. , and Amabili, M. , 2013, “ Nonlinear Dynamics of a Geometrically Imperfect Microbeam Based on the Modified Couple Stress Theory,” Int. J. Eng. Sci., 68, pp. 11–23. [CrossRef]
Ghayesh, M. H. , Farokhi, H. , and Amabili, M. , 2014, “ In-Plane and Out-of-Plane Motion Characteristics of Microbeams With Modal Interactions,” Compos. Part B: Eng., 60, pp. 423–439. [CrossRef]
Ghayesh, M. H. , Amabili, M. , and Farokhi, H. , 2013, “ Three-Dimensional Nonlinear Size-Dependent Behaviour of Timoshenko Microbeams,” Int. J. Eng. Sci., 71, pp. 1–14. [CrossRef]
Hashemi, S. H. , and Samaei, A. T. , 2011, “ Buckling Analysis of Micro/Nanoscale Plates Via Nonlocal Elasticity Theory,” Physica E, 43(7), pp. 1400–1404. [CrossRef]
Sharma, J. N. , and Sharma, R. , 2011, “ Damping in Micro-Scale Generalized Thermoelastic Circular Plate Resonators,” Ultrasonics, 51(3), pp. 352–358. [CrossRef] [PubMed]
He, L. , Lou, J. , Zhang, E. , Wang, Y. , and Bai, Y. , 2015, “ A Size-Dependent Four Variable Refined Plate Model for Functionally Graded Microplates Based on Modified Couple Stress Theory,” Compos. Struct., 130, pp. 107–115. [CrossRef]
Wang, B. , Zhou, S. , Zhao, J. , and Chen, X. , 2011, “ A Size-Dependent Kirchhoff Micro-Plate Model Based on Strain Gradient Elasticity Theory,” Eur. J. Mech. A. Solids, 30(4), pp. 517–524. [CrossRef]
Ramezani, S. , 2012, “ A Shear Deformation Micro-Plate Model Based on the Most General Form of Strain Gradient Elasticity,” Int. J. Mech. Sci., 57(1), pp. 34–42. [CrossRef]
Tahani, M. , Askari, A. R. , Mohandes, Y. , and Hassani, B. , 2015, “ Size-Dependent Free Vibration Analysis of Electrostatically Pre-Deformed Rectangular Micro-Plates Based on the Modified Couple Stress Theory,” Int. J. Mech. Sci., 94–95, pp. 185–198. [CrossRef]
Roque, C. M. C. , Ferreira, A. J. M. , and Reddy, J. N. , 2013, “ Analysis of Mindlin Micro Plates With a Modified Couple Stress Theory and a Meshless Method,” Appl. Math. Modell., 37(7), pp. 4626–4633. [CrossRef]
Asghari, M. , 2012, “ Geometrically Nonlinear Micro-Plate Formulation Based on the Modified Couple Stress Theory,” Int. J. Eng. Sci., 51, pp. 292–309. [CrossRef]
Thai, H.-T. , and Choi, D.-H. , 2013, “ Size-Dependent Functionally Graded Kirchhoff and Mindlin Plate Models Based on a Modified Couple Stress Theory,” Compos. Struct., 95, pp. 142–153. [CrossRef]
Ansari, R. , Gholami, R. , Faghih Shojaei, M. , Mohammadi, V. , and Darabi, M. A. , 2015, “ Size-Dependent Nonlinear Bending and Postbuckling of Functionally Graded Mindlin Rectangular Microplates Considering the Physical Neutral Plane Position,” Compos. Struct., 127, pp. 87–98. [CrossRef]
Das, K. , and Batra, R. C. , 2009, “ Symmetry Breaking, Snap-Through and Pull-In Instabilities Under Dynamic Loading of Microelectromechanical Shallow Arches,” Smart Mater. Struct., 18(11), p. 115008. [CrossRef]
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]
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]
Ouakad, H. M. , 2013, “ An Electrostatically Actuated MEMS Arch Band-Pass Filter,” Shock Vib., 20(4), pp. 809–819. [CrossRef]
Mohammad, T. , and Ouakad, H. , 2014, “ Static, Eigenvalue Problem and Bifurcation Analysis of MEMS Arches Actuated by Electrostatic Fringing-Fields,” Microsyst. Technol.
Medina, L. , Gilat, R. , Ilic, B. , and Krylov, S. , 2014, “ Experimental Investigation of the Snap-Through Buckling of Electrostatically Actuated Initially Curved Pre-Stressed Micro Beams,” Sens. Actuators, A, 220, pp. 323–332. [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]
Medina, L. , Gilat, R. , and Krylov, S. , 2012, “ Symmetry Breaking in an Initially Curved Micro Beam Loaded by a Distributed Electrostatic Force,” Int. J. Solids Struct., 49(13), pp. 1864–1876. [CrossRef]
Yang, F. , Chong, A. C. M. , Lam, D. C. C. , and Tong, P. , 2002, “ Couple Stress Based Strain Gradient Theory for Elasticity,” Int. J. Solids Struct., 39(10), pp. 2731–2743. [CrossRef]
Park, S. K. , and Gao, X. L. , 2006, “ Bernoulli–Euler Beam Model Based on a Modified Couple Stress Theory,” J. Micromech. Microeng., 16(11), p. 2355. [CrossRef]
Ma, H. M. , Gao, X. L. , and Reddy, J. N. , 2008, “ A Microstructure-Dependent Timoshenko Beam Model Based on a Modified Couple Stress Theory,” J. Mech. Phys. Solids, 56(12), pp. 3379–3391. [CrossRef]
Gao, X. L. , Huang, J. X. , and Reddy, J. N. , 2013, “ A Non-Classical Third-Order Shear Deformation Plate Model Based on a Modified Couple Stress Theory,” Acta Mech., 224(11), pp. 2699–2718. [CrossRef]
Amabili, M. , 2004, “ Nonlinear Vibrations of Rectangular Plates With Different Boundary Conditions: Theory and Experiments,” Comput. Struct., 82(31–32), pp. 2587–2605. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Schematic representation of an initially slightly curved shear deformable microplate

Grahic Jump Location
Fig. 2

Variations of the dimensionless natural frequencies of the out-of-plane motion of the microplate with the amplitude of the initial curvature

Grahic Jump Location
Fig. 5

Force-response curves of the system for the generalized coordinates w1,1, w3,1, w3,3, and u2,1; Ω = 29.1033 and A0/h = 0.36

Grahic Jump Location
Fig. 4

Frequency-response curves of the system for the generalized coordinates w1,1, w3,1, w3,3, and u2,1, obtained via modified couple stress and classical continuum theories; f1 = 12.00 and A0/h = 0.36

Grahic Jump Location
Fig. 3

Frequency-response curves of the system for the generalized coordinates w1,1, w3,1, w3,3, and u2,1; f1 = 12.00, A0/h = 0.36, and ω1,1 = 24.8747

Grahic Jump Location
Fig. 6

Frequency-response curves of the system for the generalized coordinates w1,1, w3,1, w3,3, and u2,1; f1 = 11.00, A0/h = 0.535, and ω1,1 = 28.0375

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Fig. 10

Frequency–response of a macroplate for the w1,1 mode: solid line and symbols represent the results obtained by the numerical simulations employed in the present study and those given in Ref. [51], respectively

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Fig. 9

Frequency-response curves of the system for the generalized coordinates w1,1, w3,1, w3,3, and u2,1; f1 = 14.50, A0/h = 1.00, and ω1,1 = 39.3379

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Fig. 8

The effect of forcing amplitude on the frequency-response curves of the system for the generalized coordinates w1,1, w3,1, w3,3, and u2,1; A0/h = 0.535 and ω1,1 = 28.0375

Grahic Jump Location
Fig. 7

Frequency-response curves of the system for the generalized coordinates w1,1, w3,1, w3,3, and u2,1, obtained via modified couple stress and classical continuum theories; f1 = 11.00 and A0/h = 0.535

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