Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

SAGE Publications - Tập 29 Số 1-2 - Trang 428-439 - 2023
J.R. Banerjee1, S. О. Papkov2, Thuc P. Vo3, Isaac Elishakoff4
1City, University of London, London, UK
2Sevastopol State University, Sevastopol, Russia
3La Trobe University, Melbourne, Australia
4Florida Atlantic University, Boca Raton, FL, USA

Tóm tắt

Several models within the framework of continuum mechanics have been proposed over the years to solve the free vibration problem of micro beams. Foremost amongst these are those based on non-local elasticity, classical couple stress, gradient elasticity and modified couple stress theories. Many of these models retain the basic features of the Bernoulli–Euler or Timoshenko–Ehrenfest theories, but they introduce one or more material scale length parameters to tackle the problem. The work described in this paper deals with the free vibration problems of micro beams based on the dynamic stiffness method, through the implementation of the modified couple stress theory in conjunction with the Timoshenko–Ehrenfest theory. The main advantage of the modified couple stress theory is that unlike other models, it uses only one material length scale parameter to account for the smallness of the structure. The current research is accomplished first by solving the governing differential equations of motion of a Timoshenko–Ehrenfest micro beam in free vibration in closed analytical form. The dynamic stiffness matrix of the beam is then formulated by relating the amplitudes of the forces to those of the corresponding displacements at the ends of the beam. The theory is applied using the Wittrick–Williams algorithm as solution technique to investigate the free vibration characteristics of Timoshenko–Ehrenfest micro beams. Natural frequencies and mode shapes of several examples are presented, and the effects of the length scale parameter on the free vibration characteristics of Timoshenko–Ehrenfest micro beams are demonstrated.

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Tài liệu tham khảo

10.1007/s00419-011-0565-5

Anderson MS, 1986, NASA Technical Memorandum, 87669, 1

10.1016/j.euromechsol.2015.02.005

10.1016/S0020-7683(98)00283-2

10.1016/j.ijengsci.2010.09.025

10.1016/j.ijsolstr.2011.03.006

10.1007/s00707-018-2162-y

10.1016/S0921-5107(97)00281-X

10.1016/S0045-7949(96)00326-4

10.1016/S0045-7949(03)00211-6

10.4203/ctr.11.2

10.1142/S0219455415400088

10.1088/0957-4484/19/34/345703

10.1557/JMR.1999.0554

10.1557/JMR.2001.0146

Cosserat E, Cosserat F (1909) Theory of deformable bodies. Scientific Library A. Hermann and Sons. Hermann et Fils. Paris.

10.1016/j.ijmecsci.2014.08.019

Elishakoff I, 2020, Handbook of Timoshenko-Ehrenfest Beam and Uflyand-Mindlin Plate Theories

10.1177/1081286519856931

Eringen AC, 1966, Journal of Mathematics and Mechanics, 15, 909

10.1016/0020-7225(72)90070-5

10.1063/1.332803

10.1016/j.ijmecsci.2020.105501

10.1016/j.ijengsci.2013.03.001

10.1016/j.mechrescom.2014.06.003

10.1016/0956-7151(94)90502-9

10.1016/j.compositesb.2013.02.021

10.1016/0141-1195(83)90108-0

Koiter W, 1964, I & II. Philosophical Transactions of the Royal Society of London B, 67, 17

10.1016/j.ijengsci.2007.10.002

10.1016/j.ijengsci.2008.08.008

10.1016/S0022-5096(03)00053-X

10.1016/S0304-3991(03)00077-9

10.1016/j.ijmecsci.2018.03.035

10.1016/j.compstruct.2014.01.019

10.1557/JMR.1995.0853

10.1016/j.jmps.2008.09.007

10.1007/BF02327219

10.1007/BF00248490

10.1007/BF00253946

10.1016/j.ijleo.2012.06.025

10.1177/1077546319892470

10.1080/15376494.2014.984090

10.1088/0960-1317/16/11/015

10.1021/ac035048k

Pipes LA, 2014, Applied Mathematics for Engineers and Physicists

10.1016/j.ijengsci.2007.04.004

10.1016/j.jmps.2011.06.008

10.1016/j.compstruct.2014.02.010

10.1016/S1359-6454(98)00153-0

10.1016/j.compstruct.2017.06.040

10.1080/14786442108636264

10.1080/14786442208633855

10.1007/BF00253945

10.1016/j.physe.2012.10.020

10.1016/0045-7949(93)90334-A

10.1061/(ASCE)0733-9445(1983)109:1(169)

10.1093/qjmam/24.3.263

10.1016/S0020-7683(02)00152-X

10.1080/15376494.2016.1143989

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SAGE Publications

Tập 29 Số 1-2

428-439

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