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Research Papers

Coupled Nonlinear Dynamics of Geometrically Imperfect Shear Deformable Extensible Microbeams

[+] 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

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received December 8, 2014; final manuscript received August 10, 2015; published online November 13, 2015. Assoc. Editor: Daniel J. Segalman.

J. Comput. Nonlinear Dynam 11(4), 041001 (Nov 13, 2015) (10 pages) Paper No: CND-14-1312; doi: 10.1115/1.4031288 History: Received December 08, 2014; Revised August 10, 2015

This paper aims at analyzing the coupled nonlinear dynamical behavior of geometrically imperfect shear deformable extensible microbeams based on the third-order shear deformation and modified couple stress theories. Using Hamilton's principle and taking into account extensibility, the three nonlinear coupled continuous expressions are obtained for an initially slightly curved (i.e., a geometrically imperfect) microbeam, describing the longitudinal, transverse, and rotational motions. A high-dimensional Galerkin scheme is employed, together with an assumed-mode technique, in order to truncate the continuous system with an infinite number of degrees of freedom into a discretized model with sufficient degrees of freedom. This high-dimensional discretized model is solved by means of the pseudo-arclength continuation technique for the system at the primary resonance, and also by direct time-integration to characterize the dynamic response at a fixed forcing amplitude and frequency; stability analysis is conducted via the Floquet theory. Apart from analyzing the nonlinear resonant response, the linear natural frequencies are obtained via an eigenvalue analysis. Results are shown through frequency–response curves, force–response curves, time traces, phase-plane portraits, and fast Fourier transforms (FFTs). The effect of taking into account the length-scale parameter on the coupled nonlinear dynamic response of the system is also highlighted.

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References

Figures

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Fig. 1

Schematic representation of an initially curved shear deformable extensible microbeam

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Fig. 2

Frequency–response curves of the system: (a) and (b) the first generalized coordinate of the transverse motion and rotation, respectively; (c) the second generalized coordinate of the longitudinal motion. Solid and dashed lines represent the stable and unstable solutions, respectively; f1 = 1.15 and A0 = 0.2

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Fig. 3

The details of the dynamics of the system of Fig. 2 at  = 1.0290 ω1; (a)–(c) time trace, phase-plane portrait, and FFT of the q1 motion, respectively.

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Fig. 4

Comparison between the frequency–response curves of the system obtained via the modified couple stress and classical theories: (a), (b) the first generalized coordinate of the transverse motion and rotation, respectively; (c) the second generalized coordinate of the longitudinal motion. f1 = 1.15 and A0 = 0.2; l = 17.6 μm for the modified couple stress theory and l = 0 for the classical theory. Solid and dashed lines represent the stable and unstable solutions, respectively.

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Fig. 5

Frequency–response curves of the system: (a), (b) the first generalized coordinate of the transverse motion and rotation, respectively; (c) the second generalized coordinate of the longitudinal motion. Solid and dashed lines represent the stable and unstable solutions respectively; f1 = 2.85 and A0 = 0.4.

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Fig. 6

Comparison between the frequency–response curves of the system obtained via the modified couple stress and classical theories: (a), (b) the first generalized coordinate of the transverse motion and rotation, respectively; (c) the second generalized coordinate of the longitudinal motion. f1 = 2.85 and A0 = 0.4; l = 17.6 μm for the modified couple stress theory and l = 0 for the classical theory. Solid and dashed lines represent the stable and unstable solutions, respectively.

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Fig. 7

Frequency–response curves of the system for several amplitudes of the initial imperfection, A0: (a), (b) the first generalized coordinate of the transverse motion and rotation, respectively; (c) the second generalized coordinate of the longitudinal motion; f1 = 2.15

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Fig. 8

Force–response curves of the system: (a), (b) the first generalized coordinate of the transverse motion and rotation, respectively; (c) the second generalized coordinate of the longitudinal motion. Solid and dashed lines represent the stable and unstable solutions respectively; Ω = 1.0650ω1 and A0 = 0.2.

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Fig. 9

Force–response curves of the system: (a), (b) the first generalized coordinate of the transverse motion and rotation, respectively; (c) the second generalized coordinate of the longitudinal motion. Solid and dashed lines represent the stable and unstable solutions respectively; Ω = 0.9650ω1 and A0 = 0.4.

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Fig. 10

Force–response curves of the system for different frequency ratios, Ω/ω1: (a), (b) the first generalized coordinate of the transverse motion and rotation, respectively; (c) the second generalized coordinate of the longitudinal motion. A0 = 0.4.

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Fig. 11

Force–response curves of the system for several amplitudes of the initial imperfection, A0: (a), (b) the first generalized coordinate of the transverse motion and rotation, respectively; (c) the second generalized coordinate of the longitudinal motion. Ω = 1.02ω1.

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