Investigating the vibration of cracked micro-cantilever beam with concentrated mass and rotary inertia based on MCST

Document Type : Research Article


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

2 Associate Professor, Faculty of Mechanical & Energy Engineering, Shahid Beheshti University, Tehran, Iran



The current paper proposes the vibrational behavior of a cracked micro-cantilever Euler-Bernoulli beam with a concentrated mass and its moment of inertia at the free-end boundary condition based on the Modified Couple Stress Theory (MCST). We model the open-edge crack and calculate its stiffness. We also consider The Stress Intensity Factor (SIF). Using Hamilton’s principle, the associated boundary conditions followed by the system’s dynamic equations are derived based on MCST. Afterward, the natural frequencies of the cracked micro-cantilever beam are semi-analytically determined. In the numerical results, we obtain the first three natural frequencies of the system versus various parameters containing the crack depth and location changes and the material length scale parameter with the different mass ratios. The results are verified with similar previous research. The calculated results indicate that increasing the crack depth, approaching the crack location to the concentrated mass and the node points, and increasing the mass ratio cause a decrease in frequencies. However, increasing the material length scale parameter causes an increase in the natural frequencies due to raising the total strain energy of the system.


  • A cracked micro-cantilever beam with concentrated mass and its inertia is modeled.
  • The governing equations and boundary conditions are obtained based on MCST.
  • Material length scale parameter for the beam and SIF for the crack are considered.
  • Effect of mass ratio with inertia is investigated on the natural frequencies.
  • Changing the depth and location of the crack is considered


Main Subjects

[1] R.A. Coutu, 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] C.-Y.L. Yu-Hsiang Wang, Che-Ming Chiang, A MEMS-based Air Flow Sensor with a Free-standing Microcantilever Structure, sensors, (2007).
[4] W. Zhang, G. Meng, H. Li, Adaptive vibration control of micro-cantilever beam with piezoelectric actuator in MEMS, The International Journal of Advanced Manufacturing Technology, 28 (2005) 321-327.
[5] M. Ghavami, S. Azizi, M.R. Ghazavi, On the dynamics of a capacitive electret-based micro-cantilever for energy harvesting, Energy, 153 (2018) 967-976.
[6] M. Chen, S. Zheng, Size-dependent static bending of a micro-beam with a surface-mounted 0–1 polarized PbLaZrTi actuator under various boundary conditions, Journal of Intelligent Material Systems and Structures, 28 (2017) 2920-2932.
[7] B. Akgöz, Ö. Civalek, Strain gradient elasticity and modified couple stress models for buckling analysis of axially loaded micro-scaled beams, International Journal of Engineering Science, 49 (2011) 1268-1280.
[8] A.C.M.C. F. Yang, D.C.C. Lam, P. Tong, Couple stress based strain gradient theory for elasticity, International Journal of Solids and Structures 39,  2731–2743, (2002).
[9] W.H.M. Christian Liebold, Applications of Strain Gradient Theories to the Size Effect in Submicro Structures incl. Experimental Analysis of Elastic Material Parameters, Bulletin of TICMI, Vol. 19, No. 1, 45–55, (2015).
[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] R.P. Joseph, Wang, B., Samali, B., Size-dependent stress intensity factors in a gradient elastic double cantilever beam with surface effects, Archive of Applied Mechanics, (2018).
[12] 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 (2016) 240-252.
[13] 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.
[14] X.-L.G. S. K. Park, A new Bernoulli-Euler beam model based on a modified couple stress theory, Journal of Micromechanics and Microengineering, Volume 16, Number 11, (2006).
[15] E. Taati, N. Sina, Static Pull-in Analysis of Electrostatically Actuated Functionally Graded Micro-Beams Based on the Modified Strain Gradient Theory, International Journal of Applied Mechanics, 10 (2018) 1850031.
[16] A. Khnaijar, R. Benamar, A new model for beam crack detection and localization using a discrete model, Engineering Structures, 150 (2017) 221-230.
[17] 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.
[18] 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.
[19] 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.
[20] A. Mofid Nakhaei, M. Dardel, M. Hassan Ghasemi, Modeling and frequency analysis of beam with breathing crack, Archive of Applied Mechanics, (2018).
[21] 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) 1-12.
[22] B.A. Zai, M. Park, S.-C. Lim, J.-W. Lee, R.A. Sindhu, Structural Optimization of Cantilever Beam in Conjunction with Dynamic Analysis, in:  Proceedings of the Computational Structural Engineering Institute Conference, Computational Structural Engineering Institute of Korea, 2008, pp. 397-401.
[23] 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, (2017) 1127.
[24] K. Kutukova, S. Niese, J. Gelb, R. Dauskardt, E. Zschech, A novel micro-double cantilever beam (micro-DCB) test in an X-ray microscope to study crack propagation in materials and structures, Materials Today Communications, 16 (2018) 293-299.
[25] R.P. Joseph, C. Zhang, B.L. Wang, B. Samali, Fracture analysis of flexoelectric double cantilever beams based on the strain gradient theory, Composite Structures, 202 (2018) 1322-1329.
[26] Ş.D. Akbaş, 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.
[27] Ş.D. Akbaş, Forced vibration analysis of cracked nanobeams, Journal of the Brazilian Society of Mechanical Sciences and Engineering, 40 (2018).
[28] S. Rajasekaran, H.B. Khaniki, Free vibration analysis of bi-directional functionally graded single/multi-cracked beams, International Journal of Mechanical Sciences, 144 (2018) 341-356.
[29] 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.
[30] M. Shoaib, N.H. Hamid, M. Tariq Jan, N.B. Zain Ali, Effects of Crack Faults on the Dynamics of Piezoelectric Cantilever-Based MEMS Sensor, IEEE Sensors Journal, 17 (2017) 6279-6294.
[31] A. Rahi, Crack mathematical modeling to study the vibration analysis of cracked micro beams based on the MCST, Microsystem Technologies, (2018).
[32] A. Rahi, H. Petoft, Free vibration analysis of multi-cracked micro beams based on Modified Couple Stress Theory, Journal of Theoretical and Applied Vibration and Acoustics, 4 (2018) 205-222.
[33] P. Laura, J. Pombo, E. Susemihl, A note on the vibrations of a clamped-free beam with a mass at the free end, Journal of Sound and Vibration, 37 (1974) 161-168.
[34] J.S. Wu, T.L. Lin, Free vibration analysis of a uniform cantilever beam with point masses by an analytical-and-numerical-combined method, Journal of Sound and Vibration, 136 (1990) 201-213.
[35] M.H. Ghayesh, A. Farajpour, A review on the mechanics of functionally graded nanoscale and microscale structures, International Journal of Engineering Science, 137 (2019) 8-36.
[36] S. S.Rao, Vibration of Continuous Systems, wiley, john wiley and sons, 2007.