Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique

Mohamed Abd Elkhalek; Tharwat Osman; Mohamed Saad Matbuly

doi:doi:10.11648/j.engmath.20190301.15

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Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique

Mohamed Abd Elkhalek, Tharwat Osman, Mohamed Saad Matbuly

Published in Engineering Mathematics (Volume 3, Issue 1)

Received: 1 June 2019 Accepted: 8 July 2019 Published: 17 July 2019

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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.

Published in	Engineering Mathematics (Volume 3, Issue 1)
DOI	10.11648/j.engmath.20190301.15
Page(s)	19-29
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2019. Published by Science Publishing Group

Keywords

Cracked Nanobeam, Free Vibration, Euler–Bernoulli Theory, Timoshenko Theory, Differential Quadrature Method

References

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APA Style

Mohamed Abd Elkhalek, Tharwat Osman, Mohamed Saad Matbuly. (2019). Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique. Engineering Mathematics, 3(1), 19-29. https://doi.org/10.11648/j.engmath.20190301.15

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Mohamed Abd Elkhalek; Tharwat Osman; Mohamed Saad Matbuly. Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique. Eng. Math. 2019, 3(1), 19-29. doi: 10.11648/j.engmath.20190301.15

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Mohamed Abd Elkhalek, Tharwat Osman, Mohamed Saad Matbuly. Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique. Eng Math. 2019;3(1):19-29. doi: 10.11648/j.engmath.20190301.15

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@article{10.11648/j.engmath.20190301.15,
  author = {Mohamed Abd Elkhalek and Tharwat Osman and Mohamed Saad Matbuly},
  title = {Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique},
  journal = {Engineering Mathematics},
  volume = {3},
  number = {1},
  pages = {19-29},
  doi = {10.11648/j.engmath.20190301.15},
  url = {https://doi.org/10.11648/j.engmath.20190301.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20190301.15},
  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.},
 year = {2019}
}

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TY - JOUR
T1 - Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique
AU - Mohamed Abd Elkhalek
AU - Tharwat Osman
AU - Mohamed Saad Matbuly
Y1 - 2019/07/17
PY - 2019
N1 - https://doi.org/10.11648/j.engmath.20190301.15
DO - 10.11648/j.engmath.20190301.15
T2 - Engineering Mathematics
JF - Engineering Mathematics
JO - Engineering Mathematics
SP - 19
EP - 29
PB - Science Publishing Group
SN - 2640-088X
UR - https://doi.org/10.11648/j.engmath.20190301.15
AB - 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.
VL - 3
IS - 1
ER -

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Author Information

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

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APA Style

Mohamed Abd Elkhalek, Tharwat Osman, Mohamed Saad Matbuly. (2019). Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique. Engineering Mathematics, 3(1), 19-29. https://doi.org/10.11648/j.engmath.20190301.15

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ACS Style

Mohamed Abd Elkhalek; Tharwat Osman; Mohamed Saad Matbuly. Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique. Eng. Math. 2019, 3(1), 19-29. doi: 10.11648/j.engmath.20190301.15

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AMA Style

Mohamed Abd Elkhalek, Tharwat Osman, Mohamed Saad Matbuly. Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique. Eng Math. 2019;3(1):19-29. doi: 10.11648/j.engmath.20190301.15

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@article{10.11648/j.engmath.20190301.15,
  author = {Mohamed Abd Elkhalek and Tharwat Osman and Mohamed Saad Matbuly},
  title = {Vibrations Analysis of Cracked Nanobeams Using Quadrature Technique},
  journal = {Engineering Mathematics},
  volume = {3},
  number = {1},
  pages = {19-29},
  doi = {10.11648/j.engmath.20190301.15},
  url = {https://doi.org/10.11648/j.engmath.20190301.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20190301.15},
  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.},
 year = {2019}
}

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