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Research Papers

Resonance Responses of Geometrically Imperfect Functionally Graded Extensible Microbeams

[+] Author and Article Information
Mergen H. Ghayesh

School of Mechanical Engineering,
University of Adelaide,
Adelaide, SA 5005, Australia
e-mail: mergen.ghayesh@adelaide.edu.au

Hamed Farokhi

Department of Mechanical Engineering,
McGill University,
Montreal, QC H3A 0C3, Canada

Alireza Gholipour, Maziar Arjomandi

School of Mechanical Engineering,
University of Adelaide,
Adelaide, SA 5005, Australia

Shahid Hussain

School of Mechanical, Materials
and Mechatronic Engineering,
University of Wollongong,
Wollongong, NSW 2522, Australia

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received June 9, 2016; final manuscript received November 3, 2016; published online March 9, 2017. Assoc. Editor: Stefano Lenci.

J. Comput. Nonlinear Dynam 12(5), 051002 (Mar 09, 2017) (12 pages) Paper No: CND-16-1265; doi: 10.1115/1.4035214 History: Received June 09, 2016; Revised November 03, 2016

This paper aims at analyzing the size-dependent nonlinear dynamical behavior of a geometrically imperfect microbeam made of a functionally graded (FG) material, taking into account the longitudinal, transverse, and rotational motions. The size-dependent property is modeled by means of the modified couple stress theory, the shear deformation and rotary inertia are modeled using the Timoshenko beam theory, and the graded material property in the beam thickness direction is modeled via the Mori–Tanaka homogenization technique. The kinetic and size-dependent potential energies of the system are developed as functions of the longitudinal, transverse, and rotational motions. On the basis of an energy method, the continuous models of the system motion are obtained. Upon application of a weighted-residual method, the reduced-order model is obtained. A continuation method along with an eigenvalue extraction technique is utilized for the nonlinear and linear analyses, respectively. A special attention is paid on the effects of the material gradient index, the imperfection amplitude, and the length-scale parameter on the system dynamical response.

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Figures

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Fig. 1

Schematic representation of an extensible initially curved functionally graded Timoshenko microbeam subject to a transverse distributed harmonic excitation load

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Fig. 2

Young's modulus variations with respect to the thickness of the FG microbeam for different values of n (EAl = 70 GPa and ESiC = 427 GPa)

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Fig. 3

Poisson's ratio variations with respect to the thickness of the FG microbeam for different values of n (ʋAl = 0.3 and ʋSiC = 0.17)

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Fig. 4

The fundamental dimensionless linear natural frequency of the transverse motion with respect to the dimensionless length-scale parameter ls for different values of A0

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Fig. 5

The fundamental dimensionless linear natural frequency of the transverse motion with respect to the dimensionless initially imperfection A0 for ceramic (n = 0), n = 2, n = 5, and metal; ls = 0.2 is considered for all the cases

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Fig. 6

Frequency-response curves of the initially curved functionally graded extensible microbeam: (a) and (b) the maximum and minimum amplitudes of the first generalized coordinate of the transverse motion, (c) the maximum amplitude of the first generalized coordinate of the rotational motion, and (d) the minimum amplitude of the first generalized coordinate of the axial motion. n = 2, ls = 0.2, A0 = 0.2, f1 = 6.0, and ζ = 0.0125. Solid and dashed lines represent the stable and unstable solutions, respectively.

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Fig. 7

Frequency-response curves of the initially curved functionally graded extensible microbeam: (a) and (b) the maximum amplitudes of the first generalized coordinate of the transverse and rotational motions, respectively, and (c) the minimum amplitude of the first generalized coordinate of the axial motion. n = 2, ls = 0.2, A0 = 0.4, f1 = 6.0, and ζ = 0.0125. Solid and dashed lines represent the stable and unstable solutions, respectively.

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Fig. 8

Frequency-response curves of the initially curved functionally graded extensible microbeam: (a) and (b) the maximum amplitudes of the first generalized coordinate of the transverse and rotational motions, respectively, and (c) the minimum amplitude of the first generalized coordinate of the axial motion. n = 2, ls = 0.2, A0 = 0.4, f1 = 10.0, and ζ = 0.0125. Solid and dashed lines represent the stable and unstable solutions, respectively.

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Fig. 9

Force-response curves of the initially curved functionally graded extensible microbeam: (a) and (b) the maximum and minimum amplitudes of the first generalized coordinate of the transverse motion. n = 2, ls = 0.2, A0 = 0.2, Ω/ω1 = 1.02, and ζ = 0.0125. Solid and dashed lines represent the stable and unstable solutions, respectively.

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Fig. 10

Force-response curves of the initially curved functionally graded extensible microbeam: (a) and (b) the maximum and minimum amplitudes of the first generalized coordinate of the transverse motion. n = 2, ls = 0.2, A0 = 0.5, Ω/ω1 = 0.85, and ζ = 0.0125. Solid and dashed lines represent the stable and unstable solutions, respectively.

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Fig. 11

Frequency-response curves of the initially curved functionally graded extensible microbeam: (a) and (b) the maximum and minimum amplitudes of the first generalized coordinate of the transverse motion, for different values of material gradient indices (ceramic (n = 0), n = 2, n = 5, and metal); ls = 0.2, A0 = 0.2, f1 = 6.0, and ζ = 0.0125

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Fig. 12

Frequency-response curves of the initially curved functionally graded extensible microbeam: (a) and (b) the maximum and minimum amplitudes of the first generalized coordinate of the transverse motion, for different values of initial imperfections (A0 = 0.2, 0.3, 0.4, and 0.5); n = 2, ls = 0.2, f1 = 6.0, and ζ = 0.0125

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Fig. 13

Frequency-response curves of the initially curved functionally graded extensible microbeam: (a) and (b) the maximum and minimum amplitudes of the first generalized coordinate of the transverse motion, for different values of dimensionless length-scale parameter (ls = 0.0, 0.2, 0.4, and 0.6); n = 2, A0 = 0.2, f1 = 6.0, and ζ = 0.0125

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