Free vibration analysis of multi-cracked micro beams based on Modified Couple Stress Theory

Document Type: Full Length Article

Authors

1 Assistant Professor, Mechanical & Energy Engineering, Shahid Beheshti University, A.C., Tehran, Iran

2 Ph.D. Candidate, Faculty of Mechanical & Energy Engineering, Shahid Beheshti University, A.C., Tehran, Iran

10.22064/tava.2019.89997.1113

Abstract

In this article, the size effect on the dynamic behavior of a simply supported multi-cracked microbeam is studied based on the Modified Couple Stress Theory (MCST).  At first, based on MCST, the equivalent torsional stiffness spring for every open edge crack at its location is calculated; in this regard, the Stress Intensity Factor (SIF) is also considered for all open edge cracks. Hamilton’s principle has been used in order to achieve the governing equations of motion of the system and associated boundary conditions are derived based on MCST. Then the natural frequencies of multi-cracked microbeam
are analytically determined. After that, the Numerical solutions have been presented for the microbeam with two open edge cracks. Finally, the variation of the first three natural frequencies of the system is investigated versus different values of the depth and the location of two cracks and the material length scale parameter. The obtained results express that the natural frequencies of the system increase by increasing the material length scale parameter and decrease by moving away from the simply supported of the beam and node points, in addition to increasing the number of cracks and cracks depth.

Highlights

  • The size effect on vibration behavior of a multi-cracked microbeam is studied.
  • The governing equations are derived based on the Modified Couple Stress Theory.
  • The equivalent stiffness of the crack is calculated considering the SIF.
  • The natural frequencies of multi-cracked microbeam are determined analytically

Keywords

Main Subjects


[1] R.A. Coutu Jr, P.E. Kladitis, L.A. Starman, J.R. Reid, A comparison of micro-switch analytic, finite element, and experimental results, Sensors and Actuators A: Physical, 115 (2004) 252-258.

[2] V.R. Mamilla, K.S. Chakradhar, Micro machining for micro electro mechanical systems (MEMS), Procedia materials science, 6 (2014) 1170-1177.

[3] S. Kong, S. Zhou, Z. Nie, K. Wang, The size-dependent natural frequency of Bernoulli–Euler micro-beams, International Journal of Engineering Science, 46 (2008) 427-437.

[4] S.K. Park, X.L. Gao, Bernoulli–Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, 16 (2006) 2355.

[5] M.H.F. Dado, O. Abuzeid, Coupled transverse and axial vibratory behaviour of cracked beam with end mass and rotary inertia, Journal of sound and vibration, 261 (2003) 675-696.

[6] K.S. Al-Basyouni, A. Tounsi, S.R. Mahmoud, Size dependent bending and vibration analysis of functionally graded micro beams based on modified couple stress theory and neutral surface position, Composite Structures, 125 (2015) 621-630.

[7] X. Li, L. Li, Y. Hu, Z. Ding, W. Deng, Bending, buckling and vibration of axially functionally graded beams based on nonlocal strain gradient theory, Composite Structures, 165 (2017) 250-265.

[8] N. Shafiei, M. Kazemi, L. Fatahi, Transverse vibration of rotary tapered microbeam based on modified couple stress theory and generalized differential quadrature element method, Mechanics of advanced materials and structures, 24 (2017) 240-252.

[9] Y.-L. Zhang, J.-M. Wang, Exact controllability of a micro beam with boundary bending moment, International Journal of Control, (2017) 1-9.

[10] J. Fang, J. Gu, H. Wang, Size-dependent three-dimensional free vibration of rotating functionally graded microbeams based on a modified couple stress theory, International Journal of Mechanical Sciences, 136 (2018) 188-199.

[11] A. Babaei, M.R.S. Noorani, A. Ghanbari, Temperature-dependent free vibration analysis of functionally graded micro-beams based on the modified couple stress theory, Microsystem technologies, 23 (2017) 4599-4610.

[12] E. Taati, N. Sina, Multi-objective optimization of functionally graded materials, thickness and aspect ratio in micro-beams embedded in an elastic medium, Struct Multidisc Optim, 58 (2018) 265-285.

[13] M. AkbarzadehKhorshidi, M. Shariati, Buckling and postbuckling of size-dependent cracked microbeams based on a modified couple stress theory, Journal of Applied Mechanics and Technical Physics, 58 (2017) 717-724.

[14] M. Akbarzadeh Khorshidi, M. Shariati, A multi-spring model for buckling analysis of cracked Timoshenko nanobeams based on modified couple stress theory, Journal of Theoretical and Applied Mechanics, 55 (2017) 1127-1139.

[15] A.S.Y. Alsabbagh, O.M. Abuzeid, M.H. Dado, Simplified stress correction factor to study the dynamic behavior of a cracked beam, Applied Mathematical Modelling, 33 (2009) 127-139.

[16] B. Panigrahi, G. Pohit, Effect of cracks on nonlinear flexural vibration of rotating Timoshenko functionally graded material beam having large amplitude motion, Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 232 (2018) 930-940.

[17] M. Soltanpour, M. Ghadiri, A. Yazdi, M. Safi, Free transverse vibration analysis of size dependent Timoshenko FG cracked nanobeams resting on elastic medium, Microsystem Technologies, 23 (2017) 1813-1830.

[18] S.D. Akbas, Free vibration of edge cracked functionally graded microscale beams based on the modified couple stress theory, International Journal of Structural Stability and Dynamics, 17 (2017) 1750033.

[19] N.N. Huyen, N.T. Khiem, Frequency analysis of cracked functionally graded cantilever beam, Vietnam Journal of Science and Technology, 55 (2017) 229.

[20] R.K. Behera, A. Pandey, D.R. Parhi, Numerical and experimental verification of a method for prognosis of inclined edge crack in cantilever beam based on synthesis of mode shapes, Procedia Technology, 14 (2014) 67-74.

[21] A. Rahi, Crack mathematical modeling to study the vibration analysis of cracked micro beams based on the MCST, Microsystem Technologies, 24 (2018) 3201-3215.

[22] A. Mofid Nakhaei, M. Dardel, M.H. Ghasemi, Modeling and frequency analysis of beam with breathing crack, Archive of Applied Mechanics, 88 (2018) 1743-1758.

[23] C. Fu, Y. Wang, D. Tong, Stiffness Estimation of Cracked Beams Based on Nonlinear Stress Distributions Near the Crack, Mathematical Problems in Engineering, 2018 (2018).

[24] A. Khnaijar, R. Benamar, A new model for beam crack detection and localization using a discrete model, Engineering Structures, 150 (2017) 221-230.

[25] A. Greco, A. Pluchino, F. Cannizzaro, S. Caddemi, I. Caliò, Closed-form solution based Genetic Algorithm Software: Application to multiple cracks detection on beam structures by static tests, Applied Soft Computing, 64 (2018) 35-48.

[26] K.V. Nguyen, Q. Van Nguyen, Element stiffness index distribution method for multi-crack detection of a beam-like structure, Advances in Structural Engineering, 19 (2016) 1077-1091.

[27] U. Andreaus, P. Baragatti, P. Casini, D. Iacoviello, Experimental damage evaluation of open and fatigue cracks of multi‐cracked beams by using wavelet transform of static response via image analysis, Structural Control and Health Monitoring, 24 (2017) e1902.

[28] S. Ghadimi, S.S. Kourehli, Multi cracks detection in Euler-Bernoulli beam subjected to a moving mass based on acceleration responses, Inverse Problems in Science and Engineering, 26 (2018) 1728-1748.

[29] K. Zhang, X. Yan, Multi-cracks identification method for cantilever beam structure with variable cross-sections based on measured natural frequency changes, Journal of Sound and Vibration, 387 (2017) 53-65.

[30] H. Chouiyakh, L. Azrar, K. Alnefaie, O. Akourri, Vibration and multi-crack identification of Timoshenko beams under moving mass using the differential quadrature method, International Journal of Mechanical sciences, 120 (2017) 1-11.

[31] N.T. Khiem, N.T.L. Khue, Change in mode shape nodes of multiple cracked bar: I. The theoretical study, Vietnam Journal of Mechanics, 35 (2013) 175-188.

[32] N.T. Khiem, L.K. Toan, N.T.L. Khue, Change in mode shape nodes of multiple cracked bar: II. The numerical analysis, Vietnam Journal of Mechanics, 35 (2013) 299-311.

[33] S. Caddemi, I. Caliò, F. Cannizzaro, A. Morassi, A procedure for the identification of multiple cracks on beams and frames by static measurements, Structural Control and Health Monitoring, 25 (2018) e2194.

[34] M. Shoaib, N.H. Hamid, M.T. Jan, N.B.Z. Ali, Effects of crack faults on the dynamics of piezoelectric cantilever-based MEMS sensor, IEEE Sensors Journal, 17 (2017) 6279-6294.

[35] N.T. Khiem, D.T. Hung, A closed-form solution for free vibration of multiple cracked Timoshenko beam and application, Vietnam Journal of Mechanics, 39 (2017) 315-328.

[36] F. Cannizzaro, A. Greco, S. Caddemi, I. Caliò, Closed form solutions of a multi-cracked circular arch under static loads, International Journal of Solids and Structures, 121 (2017) 191-200.

[37] H.I. Yoon, I.S. Son, S.J. Ahn, Free vibration analysis of Euler-Bernoulli beam with double cracks, Journal of mechanical science and technology, 21 (2007) 476-485.

[38] T.V. Lien, N.T. Đuc, N.T. Khiem, Mode Shape Analysis of Multiple Cracked Functionally Graded Timoshenko Beams, Latin American Journal of Solids and Structures, 14 (2017) 1327-1344.

[39] S.S. Rao, Vibration of continuous systems, Wiley Online Library, 2007.