This paper outlines the vibrational motion of a nonlinear system with a spring of linear stiffness. Homotopy perturbation technique (HPT) is used to obtain the asymptotic solution of the governing equation of motion. The numerical solution of this equation is obtained using the fourth order Runge-Kutta method (RKM). The comparison between both solutions reveals high consistency between them which confirms that, the accuracy of the obtained solution using aforementioned perturbation technique. The time history of the attained solution is represented through some plots to reveal the good effect of the different parameters of the considered system on the motion at any instant. The conditions of the stability of the attained solution are presented and discussed.
Nayfeh, A.H. and Mook, D.T. (1979) Nonlinear Oscillations. Wiley, New York.
Minorsky, N. and Krieger, R.E. (1974) Nonlinear Oscillations. Huntington, New York.
Nayfeh, A.H. (1973) Perturbation Methods. Wiley, New York.
Nayfeh, A.H. (1985) Problems in Perturbation. Wiley, New York.
Awrejcewicz, J., Andrianov, I.V. and Manevitch, L.I. (1998) Asymptotic Approaches in Nonlinear Dynamics (New Trends and Applications). Springer, Berlin. https://doi.org/10.1007/978-3-642-72079-6
El-Dib, Y.O. and Moatimid, G.M. (2018) On the Coupling of the Homotopy Perturbation and Frobenius Method for Exact Solutions of Singular Nonlinear Differential Equations. Nonlinear Science Letters A, 9, 220-230.
He, J.H. (1999) Homotopy Perturbation Technique. Computer Methods in Applied Mechanics and Engineering, 178, 257-262 https://doi.org/10.1016/S0045-7825(99)00018-3
He, J.H. (2002) Modified Lindstedt-Poincaré Methods for Some Strongly Nonlinear Oscillations, Part I—Expansion of a Constant. International Journal of Non-Linear Mechanics, 37, 309-314. https://doi.org/10.1016/S0020-7462(00)00116-5
He, J.H. (2008) Max-Min Approach to Nonlinear Oscillator. International Journal of Nonlinear Sciences and Numerical Simulation, 9, 207-210. https://doi.org/10.1515/IJNSNS.2008.9.2.207
Ganji, S.S., Ganji, D.D., Karimpour, S. and Babazadeh, H. (2009) Applications of He’s Homotopy Perturbation Method to Obtain Second-Order Approximations of the Coupled Two-Degree-of-Freedom Systems. International Journal of Nonlinear Sciences and Numerical Simulation, 10, 303-312. https://doi.org/10.1515/IJNSNS.2009.10.3.305
Yildirim, A. (2010) Determination of Periodic Solutions for Nonlinear Oscillators with Fractional Powers by He’s Modified Lindstedt-Poincaré Method. Meccanica, 45, 1-6. https://doi.org/10.1007/s11012-009-9212-4
El-Dib, Y.O. (2017) Homotopy Perturbation for Excited Nonlinear Equations. Science and Engineering Applications, 2, 96-108. https://doi.org/10.26705/SAEA.2017.2.1.96-108
El-Dib, Y.O. (2017) Multiple Scales Homotopy Perturbation Method for Nonlinear Oscillators. Nonlinear Science Letters A, 8, 352-364.
El-Dib, Y.O. (2018) Periodic Solution and Stability Behavior for Nonlinear Oscillator Having a Cubic Nonlinearity Time-Delayed. International Annals of Science, 5, 12-25. https://doi.org/10.21467/ias.5.1.12-25
Bayat, M., Pakar, I. and Bayat, M. (2015) Nonlinear Vibration of Mechanical Systems by Means of Homotopy Perturbation Method. Kuwait Journal of Science, 42, 64-85.
He, J.H. (2004) Comparison of Homotopy Perturbation Method and Homotopy Analysis Method. Applied Mathematics and Computation, 156, 527-539. https://doi.org/10.1016/j.amc.2003.08.008
Biazar, J. and Ghazvini, H. (2008) Numerical Solution for Special Non-Linear Fredholm Integral Equation by HPM. Applied Mathematics and Computation, 195, 681-687. https://doi.org/10.1016/j.amc.2007.05.015
Shaher, M., Erjaee, G.H. and Alnasr, M.H. (2009) The Modified Homotopy Perturbation Method for Solving Strongly Nonlinear Oscillators. Computers and Mathematics with Applications, 58, 2209-2220. https://doi.org/10.1016/j.camwa.2009.03.082
Öziş, T. and Akçı, C. (2011) Periodic Solutions for Certain Non-Smooth Oscillators by Iterated Homotopy Perturbation Method Combined with Modified Lindstedt-Poincaré Technique. Meccanica, 46, 341-347. https://doi.org/10.1007/s11012-010-9312-1
He, J.H. (2012) Homotopy Perturbation Method with an Auxiliary Term. Abstract and Applied Analysis, 2012, Article ID: 857612. https://doi.org/10.1155/2012/857612
Ghasemi, S.E., Zolfagharian, A. and Ganji, D.D. (2014) Study on Motion of Rigid Rod on a Circular Surface Using MHPM. Propulsion and Power Research, 3, 159-164. https://doi.org/10.1016/j.jppr.2014.07.003
Hosen, M.A. (2014) Approximate Solutions of the Equation of Motion’s of the Rigid Rod Which Rocks on a Circular Surface without Slipping. Ain Shams Engineering Journal, 5, 895-899. https://doi.org/10.1016/j.asej.2014.01.005
Belndez, A., Mndez, D., Belndez, T., Hernndez, A. and lvarez, M.L. (2008) Harmonic Balance Approaches to the Nonlinear Oscillators in Which the Restoring Force Is Inversely Proportional to the Dependent Variable. Journal of Sound and Vibration, 314, 775-782. https://doi.org/10.1016/j.jsv.2008.01.021
El-Dib, Y.O. and Moatimid, G.M. (2019) Stability Configuration of a Rocking Rigid Rod over a Circular Surface Using the Homotopy Perturbation Method and Laplace Transform. Arabian Journal for Science and Engineering, 44, 6581-6591. https://doi.org/10.1007/s13369-018-03705-6
Enayati, S.G., Azimi, M. and Jouya, M. (2017) Application of Modified Homotopy Perturbation Method and Amplitude Frequency Formulation to Strongly Nonlinear Oscillators. NTMSCI, 5, 66-82. https://doi.org/10.20852/ntmsci.2017.127
Shen, Y. and El-Dib, Y.O. (2020) A Periodic Solution of the Fractional Sine-Gordon Equation Arising in Architectural Engineering. Journal of Low Frequency Noise, Vibration and Active Control. https://doi.org/10.1177/1461348420917565
El-Dib, Y.O. and Elgazery, N.S. (2020) Effect of Dractional Derivative Properties on the Periodic Solution of the Nonlinear Oscillations. Fractals. https://doi.org/10.1142/S0218348X20500954
Ji, Q.-P., Wang, J., Lu, L.-X. and Ge, C.-F. (2020) Li-He’s Modified Homotopy Perturbation Method Coupled with the Energy Method for the Dropping Shock Response of a Tangent Nonlinear Packaging System. Journal of Low Frequency Noise, Vibration and Active Control. https://doi.org/10.1177/1461348420914457
Gilat, A. (2013) Numerical Methods for Engineers and Scientists. Wiley, Hoboken.
El-Dib, Y.O. (2018) Stability of a Strongly Displacement Time-Delayed Duffing Oscillator by the Multiple Scales-Homotopy Perturbation Method. Journal of Applied Mathematics and Computational Mechanics, 4, 260-274.