Asymptotic analysis of strongly nonlinear oscillators

1985 ◽  
Vol 6 (5) ◽  
pp. 409-415 ◽  
Author(s):  
Dai Shi-qiang
Keyword(s):  
Asymptotic Analysis ◽  
Nonlinear Oscillators ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

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

scholarly journals Escape time from potential wells of strongly nonlinear oscillators with slowly varying parameters

2005 ◽  
Vol 2005 (3) ◽  
pp. 365-375 ◽  
Author(s):  
Jianping Cai ◽  
Y. P. Li ◽  
Xiaofeng Wu
Keyword(s):  
Elliptic Function ◽  
Nonlinear Oscillators ◽  
Oscillatory System ◽  
Escape Time ◽  
Potential Well ◽  
Jacobian Elliptic Function ◽  
Potential Wells ◽  
Varying Parameters ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

The effect of negative damping to an oscillatory system is to force the amplitude to increase gradually and the motion will be out of the potential well of the oscillatory system eventually. In order to deduce the escape time from the potential well of quadratic or cubic nonlinear oscillator, the multiple scales method is firstly used to obtain the asymptotic solutions of strongly nonlinear oscillators with slowly varying parameters, and secondly the character of modulus of Jacobian elliptic function is applied to derive the equations governing the escape time. The approximate potential method, instead of Taylor series expansion, is used to approximate the potential of an oscillation system such that the asymptotic solution can be expressed in terms of Jacobian elliptic function. Numerical examples verify the efficiency of the present method.

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