Abstract
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.
Highlights
The Lindstedt-Poincaré method is a well-known perturbation technique to seek analytic solutions of weakly nonlinear oscillators
To reduce the computational cost in the traditional Lindstedt-Poincaré method and He’s modified method, we proposed a new analytic method based on a linearization technique executed on the nonlinear differential equation and applied the perturbation method to find the frequency and analytic solution
Instead of the squared frequency being multiplied on the second-order differential term in the traditional Lindstedt-Poincaré method, we divided the whole equation by ω2 on both sides, which can eliminate the second-order differential terms appearing in the sequential perturbed equations
Summary
The Lindstedt-Poincaré method is a well-known perturbation technique to seek analytic solutions of weakly nonlinear oscillators. Some traditional analytical techniques were used to treat nonlinear differential equations, which are usually applied to solve the nonlinear oscillators involving small parameters. The Lindstedt-Poincaré method is applied to the original nonlinear differential equation and a sequence of linear differential equations is derived to determine the analytic solutions. The Lindstedt-Poincaré method employed a recursion series of linear ordinary differential equations (ODEs) to determine the analytic solution. Given a zeroth-order fundamental solution x0(t) in Equation (16) for the function x(t), we can linearize the cubic term x3(t) in Equation (1) around x0(t) by x.. We develop a linearized Lindstedt-Poincaré method to seek the higher order analytic solutions of Equation (1) through Equation (25).
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