Volume 9, Issue 1
Nonlinear Vibration Analysis of Functionally Graded Nanobeam Using Homotopy Perturbation Method

Majid Ghadiri & Mohsen Saf

Adv. Appl. Math. Mech., 9 (2017), pp. 144-156.

Published online: 2018-05

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  • Abstract

In this paper, He’s homotopy perturbation method is utilized to obtain the analytical solution for the nonlinear natural frequency of functionally graded nanobeam. The functionally graded nanobeam is modeled using the Eringen’s nonlocal elasticity theory based on Euler-Bernoulli beam theory with von Karman nonlinearity relation. The boundary conditions of problem are considered with both sides simply supported and simply supported-clamped. The Galerkin’s method is utilized to decrease the nonlinear partial differential equation to a nonlinear second-order ordinary differential equation. Based on numerical results, homotopy perturbation method convergence is illustrated. According to obtained results, it is seen that the second term of the homotopy perturbation method gives extremely precise solution.

  • Keywords

Homotopy perturbation method, Lindstedt-Poincare method, analytical solution, nonlocal nonlinear free vibration, functionally graded nanobeam.

  • AMS Subject Headings

74G10, 74H45, 74E30,74B99

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-9-144, author = {}, title = {Nonlinear Vibration Analysis of Functionally Graded Nanobeam Using Homotopy Perturbation Method}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {9}, number = {1}, pages = {144--156}, abstract = {

In this paper, He’s homotopy perturbation method is utilized to obtain the analytical solution for the nonlinear natural frequency of functionally graded nanobeam. The functionally graded nanobeam is modeled using the Eringen’s nonlocal elasticity theory based on Euler-Bernoulli beam theory with von Karman nonlinearity relation. The boundary conditions of problem are considered with both sides simply supported and simply supported-clamped. The Galerkin’s method is utilized to decrease the nonlinear partial differential equation to a nonlinear second-order ordinary differential equation. Based on numerical results, homotopy perturbation method convergence is illustrated. According to obtained results, it is seen that the second term of the homotopy perturbation method gives extremely precise solution.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.2015.m899}, url = {http://global-sci.org/intro/article_detail/aamm/12141.html} }
TY - JOUR T1 - Nonlinear Vibration Analysis of Functionally Graded Nanobeam Using Homotopy Perturbation Method JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 144 EP - 156 PY - 2018 DA - 2018/05 SN - 9 DO - http://doi.org/10.4208/aamm.2015.m899 UR - https://global-sci.org/intro/article_detail/aamm/12141.html KW - Homotopy perturbation method, Lindstedt-Poincare method, analytical solution, nonlocal nonlinear free vibration, functionally graded nanobeam. AB -

In this paper, He’s homotopy perturbation method is utilized to obtain the analytical solution for the nonlinear natural frequency of functionally graded nanobeam. The functionally graded nanobeam is modeled using the Eringen’s nonlocal elasticity theory based on Euler-Bernoulli beam theory with von Karman nonlinearity relation. The boundary conditions of problem are considered with both sides simply supported and simply supported-clamped. The Galerkin’s method is utilized to decrease the nonlinear partial differential equation to a nonlinear second-order ordinary differential equation. Based on numerical results, homotopy perturbation method convergence is illustrated. According to obtained results, it is seen that the second term of the homotopy perturbation method gives extremely precise solution.

Majid Ghadiri & Mohsen Saf. (2020). Nonlinear Vibration Analysis of Functionally Graded Nanobeam Using Homotopy Perturbation Method. Advances in Applied Mathematics and Mechanics. 9 (1). 144-156. doi:10.4208/aamm.2015.m899
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