Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique
Engineering Mathematics
Volume 3, Issue 1, June 2019, Pages: 19-29
Received: Jun. 1, 2019; Accepted: Jul. 8, 2019; Published: Jul. 17, 2019
Views 86      Downloads 17
Authors
Mohamed Abd Elkhalek, Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt
Tharwat Osman, Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt
Mohamed Saad Matbuly, Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, Zagazig, Egypt
Article Tools
Follow on us
Abstract
This work concerns with free vibration analysis of cracked nanobeam problems. Based on Eringen's nonlocal elasticity theory, the governing equation of Euler–Bernoulli and Timoshenko nanobeams, are derived. It is assumed that strain at a certain point is a function of the strains at all points within the influence domain. The cracked beam is modeled as multi-segments connected by a rotational spring located at the cracked sections. This model promotes discontinuities in rotational displacement due to bending which is proportional to bending moment transmitted by the cracked section. Polynomial based differential quadrature method is employed to solve the problem. Derivatives of the field quantities are approximated as a weighted linear sum of the nodal values. For different supporting cases, the boundary conditions are directly substituted in the equation of motion, such that the problem is reduced to that of linear homogeneous algebraic system. This suggested numerical scheme accurately determined angular frequencies of the problem. A comparative study is tabulated to compare the obtained results with the previous ones. Further, a parametric study is introduced to investigate the influence of crack locations, crack severity and the nonlocal scale parameter on the obtained results. The obtained results recorded that frequency values decrease with the increasing of both of crack severity and the nonlocal scale parameter. The results of the proposed scheme may be applied for structural health monitoring.
Keywords
Cracked Nanobeam, Free Vibration, Euler–Bernoulli Theory, Timoshenko Theory, Differential Quadrature Method
To cite this article
Mohamed Abd Elkhalek, Tharwat Osman, Mohamed Saad Matbuly, Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique, Engineering Mathematics. Vol. 3, No. 1, 2019, pp. 19-29. doi: 10.11648/j.engmath.20190301.15
Copyright
Copyright © 2019 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
References
[1]
Iijima, S., Helical microtubules of graphitic carbon. nature, 1991. 354(6348): p. 56.
[2]
Eringen, A. C., Nonlocal polar elastic continua. International journal of engineering science, 1972. 10(1): p. 1-16.
[3]
Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. Journal of applied physics, 1983. 54(9): p. 4703-4710.
[4]
Eringen, A. C., Nonlocal continuum field theories. 2002: Springer Science & Business Media.
[5]
Eringen, A. C. and D. G. B. Edelen, On nonlocal elasticity. International Journal of Engineering Science, 1972. 10(3): p. 233-248.
[6]
Aydogdu, M., A general nonlocal beam theory: its application to nanobeam bending, buckling and vibration. Physica E: Low-dimensional Systems and Nanostructures, 2009. 41(9): p. 1651-1655.
[7]
Reddy, J., Nonlocal theories for bending, buckling and vibration of beams. International Journal of Engineering Science, 2007. 45(2-8): p. 288-307.
[8]
Wang, C. M., Y. Y. Zhang, and X. Q. He, Vibration of nonlocal Timoshenko beams. Nanotechnology, 2007. 18(10): p. 105401.
[9]
Behera, L. and S. Chakraverty, Free vibration of Euler and Timoshenko nanobeams using boundary characteristic orthogonal polynomials. Applied Nanoscience, 2014. 4(3): p. 347-358.
[10]
Wu, L.-Y., et al., Vibrations of nonlocal Timoshenko beams using orthogonal collocation method. Procedia Engineering, 2011. 14: p. 2394-2402.
[11]
Eltaher, M., A. E. Alshorbagy, and F. Mahmoud, Vibration analysis of Euler–Bernoulli nanobeams by using finite element method. Applied Mathematical Modelling, 2013. 37(7): p. 4787-4797.
[12]
Beni, Y. T., A. Jafaria, and H. Razavi, Size effect on free transverse vibration of cracked nano-beams using couple stress theory. International Journal of Engineering-Transactions B: Applications, 2014. 28(2): p. 296-304.
[13]
Hasheminejad, S. M., et al., Free transverse vibrations of cracked nanobeams with surface effects. Thin Solid Films, 2011. 519(8): p. 2477-2482.
[14]
Loghmani, M. and M. R. Hairi Yazdi, An analytical method for free vibration of multi cracked and stepped nonlocal nanobeams based on wave approach. Results in Physics, 2018. 11: p. 166-181.
[15]
Roostai, H. and M. Haghpanahi, Vibration of nanobeams of different boundary conditions with multiple cracks based on nonlocal elasticity theory. Applied Mathematical Modelling, 2014. 38(3): p. 1159-1169.
[16]
Loya, J., et al., Free transverse vibrations of cracked nanobeams using a nonlocal elasticity model. Journal of Applied Physics, 2009. 105(4): p. 044309.
[17]
Sourki, R. and S. Hoseini, Free vibration analysis of size-dependent cracked microbeam based on the modified couple stress theory. Applied Physics A, 2016. 122(4): p. 413.
[18]
Sourki, R. and S. Hosseini, Coupling effects of nonlocal and modified couple stress theories incorporating surface energy on analytical transverse vibration of a weakened nanobeam. The European Physical Journal Plus, 2017. 132(4): p. 184.
[19]
Bahrami, A., A wave-based computational method for free vibration, wave power transmission and reflection in multi-cracked nanobeams. Composites Part B: Engineering, 2017. 120: p. 168-181.
[20]
Wang, K. and B. Wang, Timoshenko beam model for the vibration analysis of a cracked nanobeam with surface energy. Journal of Vibration and Control, 2015. 21(12): p. 2452-2464.
[21]
Torabi, K. and J. Nafar Dastgerdi, An analytical method for free vibration analysis of Timoshenko beam theory applied to cracked nanobeams using a nonlocal elasticity model. Thin Solid Films, 2012. 520(21): p. 6595-6602.
[22]
Soltanpour, M., et al., Free transverse vibration analysis of size dependent Timoshenko FG cracked nanobeams resting on elastic medium. Microsystem Technologies, 2017. 23(6): p. 1813-1830.
[23]
Civalek, Ö. and B. Akgöz, Free vibration analysis of microtubules as cytoskeleton components: nonlocal Euler-Bernoulli beam modeling. Scientia Iranica. Transaction B, Mechanical Engineering, 2010. 17(5): p. 367.
[24]
Osman, T., et al., Applied and Computational Mathematics Analysis of cracked plates using localized multi-domain differential quadrature method. Vol. 2. 2013. 109-114.
[25]
Ragb, O., L. Seddek, and M. Matbuly, Iterative differential quadrature solutions for Bratu problem. Computers & Mathematics with Applications, 2017. 74(2): p. 249-257.
[26]
Zong, Z., K. Y. Lam, and Y. Y. Zhang, A multidomain Differential Quadrature approach to plane elastic problems with material discontinuity. Mathematical and Computer Modelling, 2005. 41(4): p. 539-553.
[27]
Civalek, Ö. and Ç. Demir, Bending analysis of microtubules using nonlocal Euler–Bernoulli beam theory. Applied Mathematical Modelling, 2011. 35(5): p. 2053-2067.
[28]
Shu, C. and H. Du, Implementation of clamped and simply supported boundary conditions in the GDQ free vibration analysis of beams and plates. International Journal of Solids and Structures, 1997. 34(7): p. 819-835.
[29]
Shu, C., Differential Quadrature and Its Application in Engineering. 2000: Springer Science & Business Media.
[30]
Matbuly, M., O. Ragb, and M. Nassar, Natural frequencies of a functionally graded cracked beam using the differential quadrature method. Applied mathematics and computation, 2009. 215(6): p. 2307-2316.
[31]
Nassar, M., M. S. Matbuly, and O. Ragb, Vibration analysis of structural elements using differential quadrature method. Journal of Advanced Research, 2013. 4(1): p. 93-102.
ADDRESS
Science Publishing Group
1 Rockefeller Plaza,
10th and 11th Floors,
New York, NY 10020
U.S.A.
Tel: (001)347-983-5186