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

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3070
Author(s):  
Chein-Shan Liu ◽  
Yung-Wei Chen
Keyword(s):  
Nonlinear Term ◽  
Computational Cost ◽  
Nonlinear Oscillators ◽  
Analytic Solutions ◽  
Free Vibrations ◽  
Higher Order ◽  
Van Der Pol ◽  
Strongly Nonlinear ◽  
Two Sides ◽  
Strongly Nonlinear Oscillators

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.

Computation of the adiabatic invariant of the symmetric nonlinear oscillator $$d^2y \over dt^2+Q(\epsilon t)y^n=R(\epsilon t)y^m$$

1998 ◽  
Vol 9 (2) ◽  
pp. 187-194
Author(s):  
J. HU
Keyword(s):  
Nonlinear Oscillator ◽  
Nonlinear Oscillators ◽  
Adiabatic Invariant ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

In a recent paper, the author showed that for certain symmetric bisuperlinear equations, cosine-like boundary behaviours will not yield symmetric solutions [1]. In this paper, we attack the adiabatic invariant problem by showing that, for these strongly nonlinear oscillators, the adiabatic invariant is intimately related to z′(0;∈) for a family of solutions.

scholarly journals Iterative Multistep Reproducing Kernel Hilbert Space Method for Solving Strongly Nonlinear Oscillators

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Banan Maayah ◽  
Samia Bushnaq ◽  
Shaher Momani ◽  
Omar Abu Arqub
Keyword(s):  
Hilbert Space ◽  
Series Solution ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Nonlinear Oscillators ◽  
Strongly Nonlinear ◽  
Runge Kutta Method ◽  
Strongly Nonlinear Oscillators ◽  
Space Method ◽  
Good Agreement

A new algorithm called multistep reproducing kernel Hilbert space method is represented to solve nonlinear oscillator’s models. The proposed scheme is a modification of the reproducing kernel Hilbert space method, which will increase the intervals of convergence for the series solution. The numerical results demonstrate the validity and the applicability of the new technique. A very good agreement was found between the results obtained using the presented algorithm and the Runge-Kutta method, which shows that the multistep reproducing kernel Hilbert space method is very efficient and convenient for solving nonlinear oscillator’s models.

Stochastic Averaging of Strongly Nonlinear Oscillators Under Combined Harmonic and Wide-Band Noise Excitations

2008 ◽  
Vol 130 (5) ◽  
Author(s):  
Y. J. Wu ◽  
W. Q. Zhu
Keyword(s):  
Probability Density ◽  
Wide Band ◽  
Nonlinear Oscillators ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stationary Response ◽  
Strongly Nonlinear ◽  
Band Noise ◽  
Wide Band Noise ◽  
Strongly Nonlinear Oscillators

Physical and engineering systems are often subjected to combined harmonic and random excitations. The random excitation is often modeled as Gaussian white noise for mathematical tractability. However, in practice, the random excitation is nonwhite. This paper investigates the stationary response probability density of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations. By using generalized harmonic functions, a new stochastic averaging procedure for estimating stationary response probability density of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations is developed. The damping can be linear and (or) nonlinear and the excitations can be external and (or) parametric. After stochastic averaging, the system state is represented by two-dimensional time-homogeneous diffusive Markov processes. The method of reduced Fokker–Planck–Kolmogorov equation is used to investigate the stationary response of the vibration system. A nonlinearly damped Duffing oscillator is taken as an example to show the application and validity of the method. In the case of primary external resonance, based on the stationary joint probability density of amplitude and phase difference, the stochastic jump of the Duffing oscillator and P-bifurcation as the system parameters change are examined for the first time. The agreement between the analytical results and those from Monte Carlo simulation of original system shows that the proposed procedure works quite well.

scholarly journals Analytical approximate technique for strongly nonlinear oscillators problem arising in engineering

2012 ◽  
Vol 51 (4) ◽  
pp. 351-354 ◽  
Author(s):  
Y. Khan ◽  
H. Latifizadeh ◽  
H. Rafieipour ◽  
E. Hesameddini
Keyword(s):  
Nonlinear Oscillators ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators
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