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Dynamics analysis of timoshenko perforated microbeams under moving loads

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Abstract

This paper aims to present a modified continuum mathematical model capable on investigation of dynamic behavior and response of perforated microbeam under the effect of moving mass/load for the first time. A size-dependent finite element model with non-classical shape function is exploited to solve the mathematical model and obtain the dynamic response of perforated Timoshenko microbeams under moving loads. To that end, first, equivalent material and geometrical parameters for perforated beam are developed, based on the regular squared perforation configuration. Second, both the stiffness and mass property matrices including the microstructure effect based on modified couple stress theory and Timoshenko first-order shear beam theory are derived for two-node finite element using new shape function. After that, the interaction between the load and beam is modeed and unified with the equation of motion of the beam incorporating mass inertia effects of moving load. The developed procedure is validated and compared. Effects of perforation parameters, moving load velocities, inertia of mass, and the microstructure size parameter on the dynamic response of perforated microbeam structures have been investigated in a wide context. The achieved results are helpful for the design and production of MEMS structures such as frequency filters, resonators, relay switches, accelerometers and mass flow sensors, with perforation.

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Authors and Affiliations

  1. Department of Mechanical Engineering, Karabuk University, Karabuk, Turkey

    Ismail Esen

  2. Mechanical Design & Production Department, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt

    Alaa A. Abdelrahman

  3. Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, Saudi Arabia

    Mohamed A. Eltaher

  4. Faculty of Engineering, Zagazig University, Zagazig, Egypt

    Mohamed A. Eltaher

Authors
  1. Ismail Esen
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  2. Alaa A. Abdelrahman
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  3. Mohamed A. Eltaher
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Appendices

Appendix

Appendix A

\({{\varvec{N}}}_{{\varvec{w}}}\left({\varvec{x}}\right)\) and \({{\varvec{N}}}_{{\varvec{\phi}}}({\varvec{x}})\) in Eq. (31) are

$$N_{w} \left( x \right) = \left[ {\begin{array}{*{20}c} {N_{w1} } & {\begin{array}{*{20}c} {N_{w2} } & {\begin{array}{*{20}c} {N_{w3} } & {N_{w4} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]$$
$$N_{\phi } \left( x \right) = \left[ {\begin{array}{*{20}c} {N_{\phi 1} } & {\begin{array}{*{20}c} {N_{\phi 2} } & {\begin{array}{*{20}c} {N_{\phi 3} } & {N_{\phi 4} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right]$$
(A.1)

where

$$N_{w1} = 1 + \frac{1}{{\left( {1 + \Phi } \right)}}\left[ {2\left( \frac{x}{L} \right)^{3} - 3\left( \frac{x}{L} \right)^{2} - \Phi \left( \frac{x}{L} \right)} \right],$$
$$N_{w2} = \frac{L}{{2\left( {1 + \Phi } \right)}}\left[ {2\left( \frac{x}{L} \right)^{3} - \left( {4 + \Phi } \right)\left( \frac{x}{L} \right)^{2} + \left( {2 + \Phi } \right)\left( \frac{x}{L} \right)} \right],$$
$$N_{w3} = \frac{1}{{\left( {1 + \Phi } \right)}}\left[ {3\left( \frac{x}{L} \right)^{2} - 2\left( \frac{x}{L} \right)^{3} + \Phi \left( \frac{x}{L} \right)} \right],$$
(A.2)
$$N_{w4} = \frac{L}{{2\left( {1 + \Phi } \right)}}\left[ {2\left( \frac{x}{L} \right)^{3} + \left( {\Phi - 2} \right)\left( \frac{x}{L} \right)^{2} - \Phi \left( \frac{x}{L} \right)} \right]$$
$$N_{\phi 1} = - \frac{6}{{L\left( {1 + \Phi } \right)}}\left( \frac{x}{L} \right)\left( {1 - \frac{x}{L}} \right)$$
$$N_{\phi 2} = \left[ {1 - \frac{3}{{\left( {1 + \Phi } \right)}}} \right]\left( \frac{x}{L} \right)\left( {1 - \frac{x}{L}} \right),$$
(A.3)
$$N_{\phi 3} = \frac{6}{{L\left( {1 + {\Phi }} \right)}}\left( \frac{x}{L} \right)\left( {1 - \frac{x}{L}} \right),$$
$$N_{\phi 4} = \left[ {1 - \frac{3}{{\left( {1 + \Phi } \right)}}\left( {1 - \frac{x}{L}} \right)} \right]\left( \frac{x}{L} \right)$$

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Esen, I., Abdelrahman, A.A. & Eltaher, M.A. Dynamics analysis of timoshenko perforated microbeams under moving loads. Engineering with Computers 38, 2413–2429 (2022). https://doi.org/10.1007/s00366-020-01212-7

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