Volume 3, Issue 1, June 2019, Page: 19-29
Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique
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
Received: Jun. 1, 2019;       Accepted: Jul. 8, 2019;       Published: Jul. 17, 2019
DOI: 10.11648/j.engmath.20190301.15      View  966      Downloads  222
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.
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
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