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**Nonlinear vibration analysis of functionally graded nanobeam using homotopy perturbation method.**
*(English)*
Zbl 07407119

Summary: 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.

### MSC:

74G10 | Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics |

74H45 | Vibrations in dynamical problems in solid mechanics |

74E30 | Composite and mixture properties |

74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |

74B20 | Nonlinear elasticity |

### Keywords:

homotopy perturbation method; Lindstedt-PoincarĂ© method; analytical solution; nonlocal nonlinear free vibration; functionally graded nanobeam
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\textit{M. Ghadiri} and \textit{M. Safi}, Adv. Appl. Math. Mech. 9, No. 1, 144--156 (2017; Zbl 07407119)

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