Solution of Nonlinear Hardening and Softening type Oscillators by Adomian’s Decomposition Method
الموضوعات : فصلنامه شبیه سازی و تحلیل تکنولوژی های نوین در مهندسی مکانیکبهرام گل محمدی 1 , قاسم اسعدی کردشولی 2 , علیرضا وحیدی 3
1 - مربی، دانشکده فنی مهندسی، دانشگاه آزاد اسلامی واحد سلماس
2 - مربی، دانشکده علوم پایه، دانشگاه آزاد اسلامی واحد شهر ری
3 - استادیار، دانشکده علوم پایه، دانشگاه آزاد اسلامی واحد شهر ری
الکلمات المفتاحية: Hardening and Softening Oscillator, Nonlinear, Adomian’s Method,
ملخص المقالة :
A type of nonlinearity in vibrational engineering systems emerges when the restoring force is a nonlinear function of displacement. The derivative of this function is known as stiffness. If the stiffness increases by increasing the value of displacement from the equilibrium position, then the system is known as hardening type oscillator and if the stiffness decreases by increasing the value of displacement, then the system is known as softening type oscillator. The restoring force as a nonlinear polynomial function of order three, can describe a wide variety of practical nonlinear situations by proper choosing of constant multipliers. In this paper, a spring-mass system is considered by the restoring force of the introduced type. Choosing suitable values for a, b and n, a hardening and softening type oscillators are constructed and related equations of motion are introduced as second order nonlinear differential equations. The equations are solved directly, using the Adomian’s decomposition method (ADM). In another approach, the equations are converted to systems of first order differential equations and then solved using the same method. The results show that the ADM gives accurate results in both approaches, beside it shows that converting the equation to a system of equations of lower order, tends to more accurate solutions when ADM applies.
[1]Adomian G., Applied Stochasti, Processes, Academic Press, 1983.
[2] Adomian G. Bellman R., Partial Differential Equations, D.Reidel Publishing Co., 1985.
[3] Adomian G., “Nonlinear Stochastic Systems Theory and Applications to Physics”, Kluwer, 1989.
[4] Adomian G., Solving Frontier Problems of physics: The Decomposition Method, Kluwer, 1994.
[5] Cherruault Y., Convergence of Adomin’s method, Kybernetes,Vol. 9 (2), 1988, pp. 31-38.
[6] Cherruault Y., Adomian G., Decomposition method: A new proof of convergence, Mathematical and ComputerModeling,Vol. 18(12), 1993, pp. 103-106.
[7] Babolian E.,Biazar J., Solving Concrete Examples by Adomian Method, Application mathematics And Computation, Vol. 135, 2003, pp. 161-167.
[8] Babolian E., Vahidi A.R.,AsadiCordshooliGh., Solving differential equations by decomposition Method, Application mathematics And Computation, Vol. 167, 2005, pp. 1150-1155.
[9] Wazwaz A.M., The modified decomposition method and Pade approximations for solving Thomas Fermi equations, Application mathematics And Computation, Vol. 105 ,1999, pp. 11-19.
[10] Wazwaz A.M., A comparison between Adomian decomposition method and Taylor series metod in the series solution, Application mathematics And Computation, Vol. 97,1998, pp. 37-44.
[11] BellomoN.,SarafyanD., On a Comparison between Adomian’sDecomposision Method and Picard Iteration, JournalMathematics and Analysis Application, Vol. 123, 1987, pp. 389–400.
[12] Vahidi A.R., AsadiCordshooliGh., Modifying Adomian Decomposition Method for Ordinary Differential Equations, Journal of Applied Mathematics, Vol. 3(10), 2006, pp. 49-54.
[13]Vahidi A.R., AsadiCordshooliGh., AzimzadehZ., Comparing numerical methods for the solution of the damped forced oscillator problem, Iranian Journal of Optimization, Vol. 2,2009, pp. 1-12.
[14] Vahidi A.R., Babolian E., AsadiCordshooliGh., Samiee F., Restarted Adomian’s Decomposition Method for Duffing’s Equation, InternationalJournal of Mathematics Analysis, Vol. 3(15), 2009, pp. 711-717.
[15] Vahidi A.R., Babolian E., AsadiCordshooliGh., Numerical solutions of Duffing’s oscillator problem, Indian Journal Physics, Vol. 86(4),2012, pp. 311-315.
[16] AsadiCordshooliGh., Vahidi A.R., Solutions of Duffing - van der Pol equation using Decomposition Method, Adv. Studies Theorist Physics, Vol. 5) 3(, 2011, pp. 121-129.
[17] Siddiqui A.M., Hameed M., Siddiqui B.M., Ghori Q.K., Use of Adomian decomposition method in the study of parallel plate flow of a third grade fluid, Communications in Nonlinear Science and Numerical Simulation, Vol. 15(9), 2010, pp. 2388-2399.
[18] Wu Guo-cheng, Adomian decomposition method for non-smooth initial value problems, Mathematical and Computer Modeling, Vol. 54(9-10), 2011, pp. 2104-2108.
[19] Sweilam N.H., Khader M.M., Approximate solutions to the nonlinear vibrations of multiwalled carbon nanotubes using Adomian decomposition method, Applied Mathematics and Computation, Vol. 217 (2), 2010, pp. 495-505.
[20] Duan J., ChaoluT., Rach R., Solutions of the initial value problem for nonlinear fractional ordinary differential equations by the Rach–Adomian–Meyers modified decomposition method, Applied Mathematics and Computation, Vol. 218(17), 2012, pp. 8370-8392.
[21] Mao Qibo, Free vibration analysis of multiple-stepped beams by using Adomian decomposition method, Mathematical and Computer Modelling, Vol. 54, (1-2), 2011, pp. 756-764.
[22] Srinivasan P., Nonlinear Mechanical Vibrations, New Age International Publishers, 2008