Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-23T07:23:45.522Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear Vibration Analysis of Functionally Graded Nanobeam Using Homotopy Perturbation Method

Part of: Elastic materials Dynamical problems Material properties given special treatment Equilibrium (steady-state) problems

Published online by Cambridge University Press:  11 October 2016

Majid Ghadiri  and
Mohsen Safi
Majid Ghadiri*
Affiliation:
Faculty of Engineering, Department of Mechanics, Imam Khomeini International University, Qazvin, Iran
Mohsen Safi
Affiliation:
Faculty of Engineering, Department of Mechanics, Imam Khomeini International University, Qazvin, Iran
*
*Corresponding author. Email:[email protected] (M. Ghadiri)
Get access

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 methodLindstedt-Poincare methodanalytical solutionnonlocal nonlinear free vibrationfunctionally graded nanobeam

MSC classification

Secondary: 74G10: Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) 74H45: Vibrations 74E30: Composite and mixture properties 74B99: None of the above, but in this section
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 1 , February 2017 , pp. 144 - 156
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Ansari, R. and Sahmani, S., Bending behavior and buckling of nanobeams including surface stress effects corresponding to different beam theories, Int. J. Eng. Sci., 49(11) (2011), pp. 1244–1255.CrossRefGoogle Scholar
[2] Eringen, A. C., On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves, J. Appl. Phys., 54(9) (1983), pp. 4703–4710.CrossRefGoogle Scholar
[3] Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, John Willey and Sons, New York, 1979.Google Scholar
[4] Nash, C. and Sen, S., Topology and Geometry for Physicists, Academic Press, London, 1983.Google Scholar
[5] He, J. H., A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Int. J. Non-Linear Mech., 35(1) (2000), pp. 37–43.CrossRefGoogle Scholar
[6] He, J. H., A new approach to nonlinear partial differential equations, Commun. Nonlinear Sci. Numer. Simulation, 2(4) (1997), pp. 230–235.CrossRefGoogle Scholar
[7] Liao, S. J., An approximate solution technique not depending on small parameters: a special example, Int. J. Non-Linear Mech., 30(3) (1995), pp. 371–380.CrossRefGoogle Scholar
[8] He, J. H., Homotopy perturbation technique, Comput. Meth. Appl. Mech. Eng., 178 (1999), 257262.CrossRefGoogle Scholar
[9] Foda, M. A., Influence of shear deformation and rotary inertia on nonlinear free vibration of a beam with pinned ends, Comput. Structures, 71(6) (1999), pp. 663–670.CrossRefGoogle Scholar
[10] Ramezani, A., Asghar, A., Aria, A. and Javad, A., Effects of rotary inertia and shear deformation on nonlinear free vibration of microbeams, J. Vibration Acoustics, 128(5) (2006), pp. 611–615.CrossRefGoogle Scholar
[11] Nazemnezhad, R. and Shahrokhhosseini, H., Nonlocal nonlinear free vibration of functionally graded nanobeams, Composite Structures, 110 (2014), pp. 192–199.CrossRefGoogle Scholar
[12] Evensen, D. A., Nonlinear vibrations of beams with various boundary conditions, AIAA J., 6(2) (1968), pp. 370–372.CrossRefGoogle Scholar
[13] Pirbodaghi, T., Ahmadian, M. T. and Fesanghary, M., On the homotopy analysis method for non-linear vibration of beams, Mech. Res. Commun., 36(2) (2009), pp. 143–148.CrossRefGoogle Scholar
[14] Akbarzade, M. and Ganji, D. D., Coupled method of homotopy perturbation method and variational approach for solution to nonlinear cubic-quintic Duffing oscillator, Adv. Theor. Appl.Mech., 3(7) (2010), pp. 329–337.Google Scholar
[15] Bayat, M., Iman, P. and Mahdi, B., Analytical study on the vibration frequencies of tapered beams, Latin American J. Solids Structures, 8(2) (2011), pp. 149–162.CrossRefGoogle Scholar
[16] Ahmadian, M. T., Mojahedi, M. and Moeenfard, H., Free vibration analysis of a nonlinear beam using homotopy and modified lindstedt-poincare methods, J. Solid Mech., 1(1) (2009), pp. 29–36.Google Scholar
[17] Moeenfard, H., Mahdi, M. and Ahmadian, M. T., A homotopy perturbation analysis of nonlinear free vibration of Timoshenko microbeams, J. Mech. Sci. Tech., 25(3) (2011), pp. 557–565.CrossRefGoogle Scholar
[18] Poorjamshidian, M., Sheikhi, J., Mahjoub-Moghadas, S. and Nakhaie, M., Nonlinear vibration analysis of the beam carrying a moving mass using modified homotopy, J. Solid Mech., 6(4) (2014), pp. 389–396.Google Scholar
[19] Yazdi, A. A., Applicability of homotopy perturbation method to study the nonlinear vibration of doubly curved cross-ply shells, Composite Structures, 96 (2013), pp. 526–531.CrossRefGoogle Scholar
[20] Sedighi, H. M., Hamid, M., Shirazi, Kourosh H. and Zare, J., An analytic solution of transversal oscillation of quintic non-linear beam with homotopy analysis method, Int. J. Non-Linear Mech., 47(7) (2012), pp. 777–784.CrossRefGoogle Scholar
[21] Sedighi, H. M. and Daneshmand, F., Nonlinear transversely vibrating beams by the homotopy perturbation method with an auxiliary term, J. Appl. Comput. Mech., 1(1) (2014), pp. 1–9.Google Scholar
[22] He, J. H., Modified Lindstedt-Poincare methods for some strongly non-linear oscillations: Part I: expansion of a constant, Int. J. Non-Linear Mech., 37(2) (2002), pp. 309–314.CrossRefGoogle Scholar
[23] He, J. H., Modified Lindstedt-Poincare methods for some strongly non-linear oscillations: Part II: a new transformation, Int. J. Non-Linear Mech., 37(2) (2002), pp. 315–320.CrossRefGoogle Scholar
[24] Reddy, J. N., Nonlocal theories for bending, buckling and vibration of beams, Int. J. Eng. Sci., 45(2) (2007), pp. 288–307.CrossRefGoogle Scholar
[25] Lestari, W. and Hanagud, S., Nonlinear vibration of buckled beams: some exact solutions, Int. J. Solids Structures, 38(26) (2001), pp. 4741–4757.CrossRefGoogle Scholar
[26] Singh, G., Sharma, A. K. and Rao, G. V., Large-amplitude free vibrations of beamsa discussion on various formulations and assumptions, J. Sound Vibration, 142(1) (1990), pp. 77–85.CrossRefGoogle Scholar
[27] Zhu, R., Ernian, P., PETER Chung, W., Cai, X. L., Liew, Kim M. and Buldum, Alper, Atomistic calculation of elastic moduli in strained silicon, Semiconductor Sci. Tech., 21(7) (2006), pp. 906–911.CrossRefGoogle Scholar
[28] Ogata, S., Li, J. and Yip, S., Ideal pure shear strength of aluminum and copper, Science, 298(5594) (2002), pp. 807–811.CrossRefGoogle Scholar