A Simplified Lindstedt-Poincaré Method for Saving Computational Cost to Determine Higher Order Nonlinear Free Vibrations

In order to improve the Lindstedt-Poincaré method to raise the accuracy and the performance for the application to strongly nonlinear oscillators, a new analytic method by engaging in advance a linearization technique in the nonlinear differential equation is developed, which is realized in terms of a weight factor to decompose the nonlinear term into two sides. We expand the constant preceding the displacement in powers of the introduced parameter so that the coefficients can be determined to avoid the appearance of secular solutions. The present linearized Lindstedt-Poincaré method is easily implemented to provide accurate higher order analytic solutions of nonlinear oscillators, such as Duffing and van Der Pol nonlinear oscillators. The accuracy of analytic solutions is evaluated by comparing to the numerical results obtained from the fourth-order Runge-Kotta method. The major novelty is that we can simplify the Lindstedt-Poincaré method to solve strongly a nonlinear oscillator with a large vibration amplitude.

Combination of Kharrat-Toma Transform and Homotopy Perturbation Method to Solve a Strongly Nonlinear Oscillators Equation.

This article introduces a new hybridization between the Kharrat-Toma transform and the homotopy perturbation method for solving a strongly nonlinear oscillator with a cubic and harmonic restoring force equation that arising in the applications of physical sciences. The proposed method is based on applying our new integral transform "Kharrat-Toma Transform" and then using the homotopy perturbation method. The objective of this paper is to illustrate the efficiency of this hybrid method and suggestion modified it. The results showed that the modified method is effectiveness and more accurate.

A simple cubication method for approximate solution of nonlinear Hamiltonian oscillators

A new cubication method is proposed for periodic solution of nonlinear Hamiltonian oscillators. The method is formulated based on quasi-static equilibrium of the original oscillator and the undamped cubic Duffing oscillator. The cubication constants derived from the present cubication method are always based on elementary functions and are simpler than the constants derived by other cubication methods. The present method was verified using three common examples of strongly nonlinear oscillators and was found to give reasonably accurate results. The method can be used to introduce nonlinear oscillators in relevant undergraduate physics and mechanics courses.