scholarly journals Analytical and Approximate Solutions to the Fee Vibration of Strongly Nonlinear Oscillators

2013 ◽  
Vol 05 (10) ◽  
pp. 388-392
Author(s):  
Taher A. Nofal ◽  
Gamal M. Ismail ◽  
Amal Ali M. Mady ◽  
Sayed Abdel-Khalek
Keyword(s):  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

scholarly journals Energy Method to Obtain Approximate Solutions of Strongly Nonlinear Oscillators

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Alex Elías-Zúñiga ◽  
Oscar Martínez-Romero
Keyword(s):  
Potential Energy ◽  
Analytical Solutions ◽  
Energy Method ◽  
Equations Of Motion ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
Equivalent Representation ◽  
Strongly Nonlinear ◽  
Representation Form ◽  
Strongly Nonlinear Oscillators

We introduce a nonlinearization procedure that replaces the system potential energy by an equivalent representation form that is used to derive analytical solutions of strongly nonlinear conservative oscillators. We illustrate the applicability of this method by finding the approximate solutions of two strongly nonlinear oscillators and show that this procedure provides solutions that follow well the numerical integration solutions of the corresponding equations of motion.

scholarly journals An Optimal Iteration Method for Strongly Nonlinear Oscillators

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Vasile Marinca ◽  
Nicolae Herişanu
Keyword(s):  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Nonlinear Oscillators ◽  
Approximate Solutions ◽  
High Accuracy ◽  
Restoring Force ◽  
Strongly Nonlinear ◽  
Convergence Of Approximate Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

We introduce a new method, namely, the Optimal Iteration Perturbation Method (OIPM), to solve nonlinear differential equations of oscillators with cubic and harmonic restoring force. We illustrate that OIPM is very effective and convenient and does not require linearization or small perturbation. Contrary to conventional methods, in OIPM, only one iteration leads to high accuracy of the solutions. The main advantage of this approach consists in that it provides a convenient way to control the convergence of approximate solutions in a very rigorous way and allows adjustment of convergence regions where necessary. A very good agreement was found between approximate and numerical solutions, which prove that OIPM is very efficient and accurate.

Free and Forced Vibrations of the Strongly Nonlinear Cubic-Quintic Duffing Oscillators

2017 ◽  
Vol 72 (1) ◽  
pp. 59-69 ◽  
Author(s):  
M.M. Fatih Karahan ◽  
Mehmet Pakdemirli
Keyword(s):  
Numerical Simulations ◽  
Numerical Solutions ◽  
Multiple Scales ◽  
Approximate Solutions ◽  
Response Curves ◽  
Strongly Nonlinear ◽  
Free And Forced Vibrations ◽  
Approximate Analytical Solutions ◽  
Strongly Nonlinear Oscillators ◽  
Good Agreement

AbstractStrongly nonlinear cubic-quintic Duffing oscillatoris considered. Approximate solutions are derived using the multiple scales Lindstedt Poincare method (MSLP), a relatively new method developed for strongly nonlinear oscillators. The free undamped oscillator is considered first. Approximate analytical solutions of the MSLP are contrasted with the classical multiple scales (MS) method and numerical simulations. It is found that contrary to the classical MS method, the MSLP can provide acceptable solutions for the case of strong nonlinearities. Next, the forced and damped case is treated. Frequency response curves of both the MS and MSLP methods are obtained and contrasted with the numerical solutions. The MSLP method and numerical simulations are in good agreement while there are discrepancies between the MS and numerical solutions.

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.

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