scholarly journals Accurate approximate solution to nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable

2008 ◽  
Vol 77 (6) ◽  
pp. 065004 ◽  
Author(s):  
A Beléndez ◽  
E Gimeno ◽  
E Fernández ◽  
D I Méndez ◽  
M L Alvarez
Keyword(s):  
Approximate Solution ◽  
Nonlinear Oscillators ◽  
Restoring Force

He's MAX–MIN METHOD FOR THE RELATIVISTIC OSCILLATOR AND HIGH ORDER DUFFING EQUATION

2009 ◽  
Vol 23 (32) ◽  
pp. 5915-5927 ◽  
Author(s):  
R. AZAMI ◽  
D. D. GANJI ◽  
H. BABAZADEH ◽  
A. G. DVAVODI ◽  
S. S. GANJI
Keyword(s):  
Approximate Solution ◽  
Nonlinear Equation ◽  
Nonlinear Problem ◽  
Nonlinear Oscillators ◽  
Minimal Solution ◽  
High Order ◽  
Duffing Equation ◽  
Relativistic Oscillator

This paper implements He's max–min method to solve accurately strong nonlinear oscillators. Maximal and minimal solution thresholds of a nonlinear problem can be easily found, and an approximate solution of the nonlinear equation can be easily deduced using He Chengtian's interpolation, which has a millennia history. Some typical examples are employed to illustrate its validity, effectiveness, and flexibility.

Nonlinear Oscillators with a Power-Form Restoring Force: Non-Isochronous and Isochronous Case

2013 ◽  
Vol 430 ◽  
pp. 14-21
Author(s):  
Ivana Kovacic
Keyword(s):  
Kinetic Energy ◽  
Constant Coefficient ◽  
Nonlinear Term ◽  
Nonlinear Oscillators ◽  
Equation Of Motion ◽  
Constant Amplitude ◽  
Restoring Force ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
First Case

This work is concerned with single-degree-of-freedom conservative nonlinear oscillators that have a fixed restoring force, which comprises a linear term and an odd-powered nonlinear term with an arbitrary magnitude of the coefficient of nonlinearity. There are two cases of interest: i) non-isochronous, when the system has an amplitude-dependent frequency and ii) isochronous, when the frequency of oscillations is constant (amplitude-independent). The first case is associated with the constant coefficient of the kinetic energy, while the frequency-amplitude relationship and the solution for motion need to be found. To that end, the equation of motion is solved by introducing a new small expansion parameter and by adjusting the Lindstedt-Poincaré method. In the second case, the condition for the frequency of oscillations to be constant is derived in terms of the expression for the position-dependent coefficient of the kinetic energy. The corresponding solution for isochronous oscillations is obtained. Numerical verifications of the analytical results are also presented.

Accurate numerical solutions of conservative nonlinear oscillators

2014 ◽  
Vol 3 (4) ◽  
Author(s):  
Najeeb Alam Khan ◽  
Khan Nasir Uddin ◽  
Khan Nadeem Alam
Keyword(s):  
Differential Equation ◽  
Plasma Physics ◽  
Nonlinear Oscillator ◽  
Numerical Solutions ◽  
Arbitrary Order ◽  
Nonlinear Oscillators ◽  
Higher Order ◽  
Algebraic Equations ◽  
Restoring Force ◽  
Arbitrary Power

AbstractThe objective of this paper is to present an investigation to analyze the vibration of a conservative nonlinear oscillator in the form u" + lambda u + u^(2n-1) + (1 + epsilon^2 u^(4m))^(1/2) = 0 for any arbitrary power of n and m. This method converts the differential equation to sets of algebraic equations and solve numerically. We have presented for three different cases: a higher order Duffing equation, an equation with irrational restoring force and a plasma physics equation. It is also found that the method is valid for any arbitrary order of n and m. Comparisons have been made with the results found in the literature the method gives accurate results.

Generalization of the Krylov-Bogoliubov Method for Nonlinear Oscillators

2015 ◽  
Vol 801 ◽  
pp. 3-11 ◽  
Author(s):  
Livija Cveticanin
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Nonlinear Oscillators ◽  
Original Method ◽  
Strong Nonlinearity ◽  
First Order ◽  
Numerical Example ◽  
The Difference ◽  
First Order Differential Equations

In this paper the Krylov-Bogoliubov method for solving nonlinear oscillators is considered. Based on the original method, developed for the oscillator with small nonlinearity, a generalization is made to oscillators with strong nonlinearity. After rewriting the equation into two first order differential equations, the averaging procedure is introduced. Truly nonlinear differential equations are investigated where the linear term does not exist nor the linearization of the equation is possible. Solution is assumed in the form of the Ateb-function. After averaging the approximate solution for the oscillator is obtained. A numerical example is tested. It is shown that the difference between the analytical approximate solution and the exact numerical one is negligible.

scholarly journals Harmonic balance approaches to the nonlinear oscillators in which the restoring force is inversely proportional to the dependent variable

2008 ◽  
Vol 314 (3-5) ◽  
pp. 775-782 ◽  
Author(s):  
A. Beléndez ◽  
D.I. Méndez ◽  
T. Beléndez ◽  
A. Hernández ◽  
M.L. Álvarez
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Oscillators ◽  
Restoring Force

scholarly journals Parameters Approach Applied on Nonlinear Oscillators

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Najeeb Alam Khan ◽  
Fatima Riaz ◽  
Nadeem Alam Khan
Keyword(s):  
Natural Frequency ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Nonlinear Oscillators ◽  
Solution Procedure ◽  
Power Exponent ◽  
Restoring Force ◽  
Arbitrary Power ◽  
Collocation Points ◽  
Approximate Frequency

We applied an approach to obtain the natural frequency of the generalized Duffing oscillatoru¨+u+α3u3+α5u5+α7u7+⋯+αnun=0and a nonlinear oscillator with a restoring force which is the function of a noninteger power exponent of deflectionu¨+αu|u|n−1=0. This approach is based on involved parameters, initial conditions, and collocation points. For any arbitrary power ofn, the approximate frequency analysis is carried out between the natural frequency and amplitude. The solution procedure is simple, and the results obtained are valid for the whole solution domain.

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