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In-plane and out-of-plane nonlinear size-dependent dynamics of microplates

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Abstract

The aim of the current study is to examine the in-plane and out-of-plane nonlinear size-dependent dynamics of a microplate resting on an elastic foundation, constrained by distributed rotational springs at boundaries. Employing the von Kármán plate theory as well as Kirchhoff’s hypotheses, the equations of motion for the in-plane and out-of-plane directions are derived by means of the Lagrange equations, based on the modified couple stress theory. The potential energies stored in a Winkler-type elastic foundation and the rotational springs at the edges of the microplate are taken into account. The set of second-order nonlinear ordinary differential equations, obtained via the Lagrange scheme, is recast into a double-dimensional set of first-order nonlinear ordinary differential equations with coupled terms by means of a change of variables. The linear natural frequencies of the system are obtained through use of an eigenvalue analysis upon the linear terms of the equations of motion. The nonlinear response, on the other hand, is obtained by means of the pseudo-arclength continuation method. The dynamical characteristics of the system are examined via plotting the frequency–response and force–response curves. The effect of the stiffness of the rotational and translational springs on the nonlinear size-dependent behaviour is also examined. Finally, the effect of employing the modified couple stress theory, rather than the classical theory, on the response is discussed.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

    Alireza Gholipour

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

    Hamed Farokhi

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

    Mergen H. Ghayesh

Authors
  1. Alireza Gholipour
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  2. Hamed Farokhi
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  3. Mergen H. Ghayesh
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Corresponding author

Correspondence to Mergen H. Ghayesh.

Additional information

Alireza Gholipour, Hamed Farokhi and Mergen H. Ghayesh have contributed equally to this work.

Appendix 1: Validation

Appendix 1: Validation

The frequency–response curve, from our simulations, obtained for a macroplate, is compared with that given in [34], displaying excellent agreement, as seen in Fig. 15.

Fig. 15
figure 15

The frequency–response curve of a macroplate for the generalized coordinate \(w_{1,1}\), showing an excellent agreement between the current study and that given in Ref. [34]

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Gholipour, A., Farokhi, H. & Ghayesh, M.H. In-plane and out-of-plane nonlinear size-dependent dynamics of microplates. Nonlinear Dyn 79, 1771–1785 (2015). https://doi.org/10.1007/s11071-014-1773-7

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  • DOI: https://doi.org/10.1007/s11071-014-1773-7

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