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Mathematical Modeling And Analysis Of Forced Vibrations In Micro-Scale Microstretch Thermoelastic Simply Supported Beam

Authors: Partap, Geeta; Nitika Chugh;

Mathematical Modeling And Analysis Of Forced Vibrations In Micro-Scale Microstretch Thermoelastic Simply Supported Beam

Abstract

{"references": ["A. C. Eringen, Linear theory of micropolar Elasticity, Journal of\nMathematics and Mechanics, 909-923 (1966).", "A. C. Eringen, Microcontinuum Field Theories. I, Foundation and Solids,\nSpringer -Verlag, New York (1999).", "W. Nowacki, Dynamic Problems of Thermoelasticity, Pwn-Polish\nScientific Publishers, Warszawa, Poland (1975).", "A. C. Eringen, Theory of thermo-microstretch elastic solids, International\nJournal of Engineering Sciences, 28, 1291-1301 (1990).", "C. Zener, Internal friction in solids. I, Theory of internal friction in reeds,\nPhysical Review, 52, 230-235 (1937).", "R. Lifshitz, M.L. Roukes, Thermoelastic damping in micro and\nnanomechanical systems, Physical Review B, 61, 5600-5609 (2000).", "Y. Sun, D. Fang, A.K. Soh, Thermoelastic damping in micro-beam\nresonators, International Journal of Solids Structure, 43, 3213-3229\n(2006).", "S. Prabhakar, S. Vengallatore, Theory of thermoelastic damping in\nmicromechanical resonators with two-dimensional heat conduction,\nJournal of Microelectro Mechanical Systems, 17, 494-502 (2008).", "X. Guo, Y.B. Yi, S. Pourkamali, A finite element anaylsis of thermoelastic\ndamping invented MEMS beam resonators, International Journal of\nMechanical Sciences, 74, 73-82 (2013).\n[10] D. Grover, Transverse vibrations in micro-scale viscothermoelastic beam\nresonators, Archive of Applied Mechanics, 83(2), 303-314 (2013).\n[11] D. Grover, Viscothermoelastic Vibrations in micro-scale beam resonators\nwith linearly varying thickness, Canadian Journal of Physics, 90, 487-496\n(2012).\n[12] D. Grover, Damping in thin circular viscothermoelastic plate resonators,\nCanadian Journal of Physics, 93(12), 1597-1605 (2015).\n[13] J. N. Sharma, D. Grover, Thermoelastic vibrations in micro-/nano-scale\nbeam resonators with voids, Journal of Sound and Vibrations, 330,\n2964-2977 (2011).\n[14] B. Yanping, H. Yilong, Static deflection analysis of micro-cantilevers\nbeam under transverse loading, Recent Researchers in Circuits, Systems,\nElectronics,Control and Signal Processing, 17-21 (2010).\n[15] H. W. Lord, Y. Shulman, The generalized dynamical theory of\nthermoelasticity, Journal of the Mechanics and Physics of Solids, 15,\n299-309 (1967).\n[16] S. S. Rao, Vibration of Continuous Systems, John Wiley and Sons, New\nJersey, USA (2007).\n[17] R. S. Dhaliwal, A. Singh, Dynamic coupled thermoelasticity, Hindustan\nPublication Corporation, New Delhi, India (1980).\n[18] R. V. Churchill, Operational Mathematics, McGraw Hill, New York,\nUSA (1972).\n[19] J. W. Brown, R. V. Churchill, Complex Variables and Applications,\nMcGraw Hill, New York, USA (1996).\n[20] G. Partap, N. Chugh, Deflection analysis of micro-scale microstretch\nthermoelastic beam resonators under harmonic loading, Applied\nMathematical Modelling, 46, 16-27 (2017)."]}

The present paper deals with the flexural vibrations of homogeneous, isotropic, generalized micropolar microstretch thermoelastic thin Euler-Bernoulli beam resonators, due to Exponential time varying load. Both the axial ends of the beam are assumed to be at simply supported conditions. The governing equations have been solved analytically by using Laplace transforms technique twice with respect to time and space variables respectively. The inversion of Laplace transform in time domain has been performed by using the calculus of residues to obtain deflection.The analytical results have been numerically analyzed with the help of MATLAB software for magnesium like material. The graphical representations and interpretations have been discussed for Deflection of beam under Simply Supported boundary condition and for distinct considered values of time and space as well. The obtained results are easy to implement for engineering analysis and designs of resonators (sensors), modulators, actuators.

Keywords

simply supported., deflection, Residue theorem, exponential load, Laplace transforms, Microstretch

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