### Analysis of Nonlinear Dynamic Behaviour of Nanobeam resting on Winkler and Pasternak Foundations

#### Abstract

Dynamic modeling of nanobeam under stretching and two-parameter foundation effects result in nonlinear equations that are very difficult to find exact analytical solutions. In this study, variation iteration method is used to develop approximate analytical solutions to nonlinear vibration analysis of nanobeam under the effects of stretching and Winkler and Pasternak foundations. The governing equation of motion for the nanotube was derived based on Euler-Bernoulli beam theory. The developed approximate analytical solutions for the governing equation are validated the results of other methods of analysis, are also used to carry out effects of some model parameters on the dynamic behaviour of the nanobeam. These analytical solutions can serve as a starting point for a better understanding of the relationship between the physical quantities in the problems as it provides clearer insights to understanding the problems in comparison with numerical methods.

#### References

M.G. Sobamowo, Thermal analysis of longitudinal fin with temperature-dependent properties and internal heat generation using Galerkin’s method of weighted residual. Applied Thermal Engineering 99 (2016) 1316–1330.

M. Rafei., D. D. Ganji, H. Daniali, H. Pashaei. The variational iteration method for nonlinear oscillators with discontinuities. J. Sound Vib. 305, 614–620 (2007)

V. Marinca, N. Herisanu. A modified iteration perturbation method for some nonlinear oscillation problems. Acta Mech. 184, 231–242 (2006)

S. S. Ganji, D. D. Ganji, D. D., H. Ganji, Babazadeh, Karimpour, S.: Variational approach method for nonlinear oscillations of the motion of a rigid rod rocking back and cubic-quinticduffing oscillators. Prog. Electromagn. Res. M 4, 23–32 (2008)

S. B. Tiwari., B. N. Rao, N. S. Swamy, K. S. Sai, H. R. Nataraja. Analytical study on a Duffing harmonic oscillator. J. Sound Vib. 285, 1217–1222 (2005)

R. E. Mickens. Periodic solutions of the relativistic harmonic oscillator. J. Sound Vib. 212, 905–908 (1998)

Y. Z. Chen and X. Y. Lin. A convenient technique for evaluating angular frequency in some nonlinear oscillations. J. Sound Vib. 305, 552–562 (2007)

Ö. Civalek,.. Nonlinear dynamic response of MDOF systems by the method of harmonic differential quadrature (HDQ). Int. J. Struct. Eng. Mech. 25(2), 201–217 (2007)

T. C. Fung. Solving initial value problems by differential quadrature method. Part 1: First-order equations. Int. J. Numer. Methods Eng. 50, 1411–1427 (2001)

R. E. Mickens. Mathematical and numerical study of the Duffing-harmonic oscillator. Journal of Sound Vibration 244(3), 563–567 (2001)

C. W. Lim and B. S. Wu. A new analytical approach to the Duffing-harmonic oscillator. Phys. Lett. A 311(5), 365–377 (2003)

H. Hu, H. and Tang, J. H. Solution of a Duffing-harmonic oscillator by the method of harmonic balance. Journal of Sound Vibration 294(3), 637–639 (2006)

C. W. Lim, B. S. Wu, and W. P. Sun. Higher accuracy analytical approximations to the Duffing harmonic oscillator. Journal of Sound Vibration 296(4), 1039–1045 (2006)

H. Hu, H. Solutions of the Duffing-harmonic oscillator by an iteration procedure. Journal of Sound Vibration 298(1), 446–452 (2006)

S. J. Liao, S. J. The Proposed Homotopy Analysis Technique for the Solution of Nonlinear Problems,Ph. D. dissertation, Shanghai Jiao Tong University (1992)

S. J. Liao and Y. A. Tan, Y. A general approach to obtain series solutions of nonlinear differential equations, Studies Appl. Math. 119(4), 297–354 (2007)

S. J. Liao, An approximate solution technique not depending on small parameters: a special example. Int. J. Non-Linear Mech. 30(3), 371–380 (1995)

J. K. Zhou: Differential Transformation and its Applications for Electrical Circuits. Huazhong University Press: Wuhan, China (1986)

C. K. Chen and S.H. Ho. Application of Differential Transformation to Eigenvalue Problems. Journal of Applied Mathematics and Computation, 79 (1996), 173-188.

Ü, Cansu, and O. Özkan, Differential Transform Solution of Some Linear Wave Equations with Mixed Nonlinear Boundary Conditions and its Blow up. Applied Mathematical Sciences Journal, 4(10) (2010), 467-475.

M-J Jang, C-L Chen, Y-C Liy: On solving the initial-value problems using the differential transformation method. Appl. Math. Comput. 115 (2000): 145-160.

M. K¨oksal, S. Herdem: Analysis of nonlinear circuits by using differential Taylor transform. Computers and Electrical Engineering. 28 (2002): 513-525.

I.H.A-H Hassan: Different applications for the differential transformation in the differential equations. Appl. Math. Comput. 129(2002), 183-201.

A.S.V. Ravi Kanth, K. Aruna: Solution of singular two-point boundary value problems using differential transformation method. Phys. Lett. A. 372 (2008), 4671-4673.

F. Ayaz: Solutions of the system of differential equations by differential transform method. Appl. Math. Comput. 147: 547-567 (2004)

S-H Chang, I-L Chang: A new algorithm for calculating one-dimensional differential transform of nonlinear functions. Appl. Math. Comput. 195 (2008), 799-808

S. Momani, V.S. Ert¨urk: Solutions of non-linear oscillators by the modified differential transform method. Computers and Mathematics with Applications. 55(4) (2008), 833-842.

S. Momani, S., 2004. Analytical approximate solutions of non-linear oscillators by the modified decomposition method. Int. J. Modern. Phys. C, 15(7): 967-979.

M. El-Shahed: Application of differential transform method to non-linear oscillatory systems. Communic. Nonlin. Scien. Numer. Simul. 13 (2008), 1714-1720.

S. K. Lai, C. W. Lim, B. S. Wu. Newton-harminic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators. Applied Math. Modeling, vol. 33, pp. 852-866, 2009.

H. Rafiepour, S. H. Tabatabaei and M. Abbaspour. A novel approximate analytical method for nonlinear vibration analysis of Euler-Bernoulli and Rayleigh beams on the nonlinear foundation. Arab J. Sci Eng., 2014.

J. K. Zhou: Differential Transformation and its Applications for Electrical Circuits. Huazhong University Press: Wuhan, China (1986)

S.H. Ho, C.K. Chen: Analysis of general elastically end restrained non-uniform beams using differential transform. Appl. Math. Modell. 22 (1998), 219-234

C.K. Chen, S.H. Ho: Transverse vibration of a rotating twisted Timoshenko beams under axial loading using differential transform. Int. J. Mechanical Sciences. 41 (1999), 1339-1356.

S. K. Lai, C. W. Lim, B. S. Wu. Newton-harminic balancing approach for accurate solutions to nonlinear cubic-quintic Duffing oscillators. Applied Math. Modeling, 33(2009), 852-866.

A. Fernandez. On some approximate methods for nonlinear models. Appl Math Comput., 215(2009). :168-74.

He JH. Approximate analytical solution for seepage flow with fractional derivatives in porous media, Comput.Meth. Appl. Mech. Eng., 1998, 167(1-2): 57-68

He JH. Variational iteration method - a kind of non-linear analytical technique: Some examples, Int. J. Nonl. Mech., 1999; 34:699-708.

He J. H. Variational iteration method - Some recent results and new interpretations, J. Computat. Appl. Math., 2007, 207(1) 3-17

He J. H, Wu XH, Variational iteration method: New development and applications, Comput. Meth. Appl. Mech. Eng, 2007,54(7-8): 881-894

He J. H. Some asymptotic methods for strongly nonlinear equations, Int. J. Mod. Phys. B, 2006; 20: 1141- 1199

He J. H. Asymptotic methods for solitary solutions and compactons, Abstract and Applied Analysis, 2012; 916793

Rafieipour H, Lotfavar A, Masroori A, Mahmoodi E, Application of Laplace Iteration method to Study of Nonlinear Vibration of laminated composite plates, Latin American Journal of Solids and Structures, 2013: 10(4) :781-795

He J. H.A short remark on fractional variational iteration method, Phys. Lett. A, 2011, 375(38) 3362-3364

Hesameddini E, Latifizadeh H. Reconstruction of Variational Iteration Algorithms using the Laplace Transform, Int. J. Nonlin. Sci. Num., 2009, 10(11-12): 1377-1382

Wu G. C, Laplace transform overcoming principal drawbacks in application of the variational iteration method to fractional heat equations, Thermal Science, 2012, 16(4): 1257-1261

He J. H. Comment on "Variational Iteration Method for Fractional Calculus Using He's Polynomials" , Abstract and Applied Analysis, 2012, 964974

He J. H. An Approximation to Solution of Space and Time Fractional Telegraph Equations by the Variational Iteration Method , Mathematical Problems in Engineering, 2012, 394212