Nonlinear resonant response of imperfect extensible Timoshenko microbeams
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Nonlinear resonant response of imperfect extensible Timoshenko microbeams Hamed Farokhi . Mergen H. Ghayesh
Received: 29 March 2015 / Accepted: 20 June 2015 Ó Springer Science+Business Media Dordrecht 2015
Abstract This paper investigates the nonlinear sizedependent dynamics of an imperfect Timoshenko microbeam, taking into account extensibility. Based on the modified couple stress theory, the nonlinear equations of motion for the longitudinal, transverse, and rotational motions are derived via Hamilton’s energy method. A high-dimensional finite degree-offreedom system of ordinary differential equations is obtained by the application of the Galerkin scheme. This set of equations is solved through use of the pseudo-arclength continuation method. A stability analysis is conducted via use of the Floquet theory. The resonant motion characteristics of the microbeam are examined by plotting the frequency-response and force-response curves. The effect of system parameters on the resonant response of the system is highlighted. Keywords Timoshenko microbeam Nonlinear dynamics Timoshenko beam theory Modified couple stress theory
H. Farokhi Department of Mechanical Engineering, McGill University, Montreal, QC H3A 0C3, Canada e-mail: [email protected] M. H. Ghayesh (&) School of Mechanical, Materials and Mechatronic Engineering, University of Wollongong, Wollongong, Australia e-mail: [email protected]
1 Introduction Microscale continuous elements can be found in a large class of electromechanical devices and engineering components. Among them, microbeams are present, for example in microactuators, biosensors, microswitches, and electrostatically excited microactuators (Azizi et al. 2013; Krylov et al. 2011; Li et al. 2008; Yu et al. 2012; Ghayesh et al. 2013c; Farokhi and Ghayesh 2015b; Ghayesh and Farokhi 2015; Gholipour et al. 2014; Farokhi and Ghayesh 2015a). The experimental investigations discovered that microbeams display size-dependent deformation behaviour; classical continuum theories cannot predict this behaviour. The necessity of taking into account size effects resulted in the advent of new continuum theories, namely the strain gradient and modified couple stress theories, so as to investigate the sizedependent deformation phenomenon. The linear and nonlinear size-dependent motion of Euler–Bernoulli microbeams has been examined by several authors in the literature and is still of interest today (Farokhi et al. 2013a, b). Starting with the linear aspects, Kong et al. (2008) examined the size-dependent natural frequencies of an Euler–Bernoulli microbeam. Akgo¨z and Civalek (2011, 2013) investigated the free oscillations and buckling of a microbeam, based on both the strain gradient and modified couple stress theories, respectively. Asghari et al. (2010a) examined the size-dependent behaviour
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of functionally graded microbeams employing the modified couple stress theory. S¸ ims¸ ek (2010) analyzed the motion characteristics of an embedded microbeam under the action
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