Vol.:(0123456789) 1 3 Engineering with Computers https://doi.org/10.1007/s00366-020-01212-7 ORIGINAL ARTICLE Dynamics analysis of timoshenko perforated microbeams under moving loads Ismail Esen 1  · Alaa A. Abdelrahman 2  · Mohamed A. Eltaher 3,4 Received: 9 October 2020 / Accepted: 28 October 2020 © Springer-Verlag London Ltd., part of Springer Nature 2020 Abstract This paper aims to present a modifed continuum mathematical model capable on investigation of dynamic behavior and response of perforated microbeam under the efect of moving mass/load for the frst time. A size-dependent fnite 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, frst, equivalent material and geometrical parameters for perforated beam are developed, based on the regular squared perforation confguration. Second, both the stifness and mass property matrices including the microstructure efect based on modifed couple stress theory and Timoshenko frst-order shear beam theory are derived for two-node fnite element using new shape function. After that, the interaction between the load and beam is modeed and unifed with the equation of motion of the beam incorporating mass inertia efects of moving load. The developed procedure is validated and compared. Efects of perforation parameters, moving load velocities, iner- tia 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 flters, resonators, relay switches, accelerometers and mass fow sensors, with perforation. Keywords Perforated microbeam · Moving load · Modifed couple stress theory · Dynamic response · MEMS · Finite element model 1 Introduction Over past years, the increase of using nanotechnology and development of novel nanoscale materials have escorted to an increasing interest in micromechanical modeling of sol- ids. This has triggered concern in non-classical continuum mechanics theories, those include microstructure and length scale efects, [60]. To account nanoscale efect, diferent modifed continuum theories have been proposed, such as, nonlocal integral and diferential of elasticity of [39, 40], couple stress theory [68, 75, 100], strain gradient theory Mindlin [76, 82] and surface elasticity theory [50, 51]. In addition to energy equivalent method [30, 78, 79, 87, 102], doublet mechanics [2, 33, 46], quantum mechanics (QM) [47], molecular dynamics (MD) Rapaport and Rapaport [86], are theories proposed to consider the size dependence of carbon nanotubes (CNTs). Based on the modifed coupled stress theory, Yang et al. [103] proved that the size scale of microstructure can be clarifed by couple stress theory, where the full curvature tensor is employed as deformation measures and also the conventional strain measures. McFarland and Colton [73] presented the role of material microstructure in plate stif- ness with relevance to micro-cantilever sensors. Park and Gao [84] provided a variational formulation for modifed couple stress theory (MCST) utilizing principles of mini- mum total potential energy. Ma et al. [71] captured the size- scale efect of Timoshenko microbeam on bending and axial deformation by employing MCST and Hamilton’s principle. Reddy [89] studied the impact of material size scale on static * Mohamed A. Eltaher mohaeltaher@gmail.com 1 Department of Mechanical Engineering, Karabuk University, Karabuk, Turkey 2 Mechanical Design & Production Department, Faculty of Engineering, Zagazig University, P.O. Box 44519, Zagazig, Egypt 3 Mechanical Engineering Department, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah, Saudi Arabia 4 Faculty of Engineering, Zagazig University, Zagazig, Egypt