Closed-Form Expression for the Exact Period of a Nonlinear Oscillator Typified by a Mass Attached to a Stretched Wire

2011 ◽  
Vol 3 (6) ◽  
pp. 689-701
Author(s):  
Malik Mamode
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Harmonic Balance Method ◽  
Elliptic Functions ◽  
Oscillatory System ◽  
Balance Method ◽  
Closed Form Expression ◽  
Form Expression ◽  
Polynomial Potential ◽  
Exact Analytical Expression

AbstractThe exact analytical expression of the period of a conservative nonlinear oscillator with a non-polynomial potential, is obtained. Such an oscillatory system corresponds to the transverse vibration of a particle attached to the center of a stretched elastic wire. The result is given in terms of elliptic functions and validates the approximate formulae derived from various approximation procedures as the harmonic balance method and the rational harmonic balance method usually implemented for solving such a nonlinear problem.

Steady State Solutions to a Forced Nonlinear Oscillator: A Comparison of Results Using a Generalized Harmonic Balance Method and by Numerical Integrations

1993 ◽  
Author(s):  
Ben Noble ◽  
Julian J. Wu
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Initial Point ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Original Equation ◽  
Long Time ◽  
Steady State Solutions ◽  
Generalized Harmonic Balance Method

Abstract Steady state solutions for nonlinear dynamic problems are interesting because (1) the long time behaviors of many problems are of practical concern, and, (2) these behaviors are often difficult to predict. This paper first presents a brief description of a generalized harmonic balance method (GHB) for steady state solutions to nonlinear problems via a nonlinear oscillator problem with a quadratic nonlinearity. Using this approach, steady state solutions are obtained for problems with several parameters: damping, nonlinearity and frequency (subharmonic, superharmonic and primary resonance). These results, plotted in time evolution curves and phase diagrams are compared with those obtained by numerically integrating the original differential equations. The effect of initial conditions on long time solutions is discussed. This investigation indicates that (1) the GHB steady state is an excellent approximate solution to that of the original equation if such a solution is numerically stable, and (2) the GHB steady state simply indicates a region of instability when the numerical solution to the original equation, using a point in that region as the initial point, is unstable.

Oscillations in an x(2n+2)/(2n+1) Potential via He’s Frequency-Amplitude Formulation

2009 ◽  
Vol 64 (12) ◽  
pp. 877-878 ◽  
Author(s):  
Abd Elhalim Ebaid
Keyword(s):  
Periodic Solution ◽  
Fractional Order ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Harmonic Balance Method ◽  
Solution Procedure ◽  
Present Approach ◽  
Balance Method

A recent technique, known as He’s frequency-amplitude formulation approach, is proposed in this letter to obtain an analytical approximate periodic solution to a nonlinear oscillator equation with potential of arbitrary fractional order. The solution procedure of the present approach is very simple and more convenient in comparison with the harmonic balance method

scholarly journals The modified Lindstedt–Poincare method for solving quadratic nonlinear oscillators

2020 ◽  
pp. 146134842097975
Author(s):  
Ismot A Yeasmin ◽  
MS Rahman ◽  
MS Alam
Keyword(s):  
Analytical Solution ◽  
Small Parameter ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Harmonic Balance Method ◽  
Nonlinear Oscillators ◽  
Balance Method ◽  
Algebraic Equations ◽  
Related Parameter ◽  
Nonlinear Algebraic Equations

Recently, an analytical solution of a quadratic nonlinear oscillator has been presented based on the harmonic balance method. By introducing a small parameter, a set of nonlinear algebraic equations have been solved which usually appear among unknown coefficients of several harmonic terms. But the method is not suitable for all quadratic oscillators. Earlier, introducing a small parameter to the frequency series, Cheung et al. modified the Lindstedt–Poincare method and used it to solve strong nonlinear oscillators including a quadratic oscillator. But due to some limitations of both parameters, a changed form of frequency-related parameter (introduced by Cheung et al.) has been presented for solving various quadratic oscillators.

scholarly journals Global Residue Harmonic Balance Method for a System of Strongly Nonlinear Oscillator

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Huaxiong Chen ◽  
Wei Liu
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Order Approximation ◽  
Harmonic Balance Method ◽  
High Accuracy ◽  
Balance Method ◽  
Strongly Nonlinear ◽  
Improve Accuracy ◽  
Residual Errors ◽  
Strongly Nonlinear Oscillator

In this paper, the global residue harmonic balance method is applied to obtain the approximate periodic solution and frequency for a well-known system of strongly nonlinear oscillator in engineering. This method can improve accuracy by considering all the residual errors in deriving each order approximation. With this procedure, the expressions of the higher-order approximate solution and corresponding frequency for the considered system can be determined easily. The comparison of the obtained results with previously existing and corresponding exact solutions shows the high accuracy and efficiency of the method.

scholarly journals The Elliptic Harmonic Balance Method for the Performance Analysis of a Two-Stage Vibration Isolation System with Geometric Nonlinearity

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Weilei Wu ◽  
Bin Tang
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Vibration Isolation ◽  
Harmonic Balance Method ◽  
Elliptic Functions ◽  
Balance Method ◽  
Damping Force ◽  
Two Stage ◽  
Isolation System ◽  
Vibration Isolation System

This study develops a modified elliptic harmonic balance method (EHBM) and uses it to solve the force and displacement transmissibility of a two-stage geometrically nonlinear vibration isolation system. Geometric damping and stiffness nonlinearities are incorporated in both the upper and lower stages of the isolator. After using the relative displacement of the nonlinear isolator, we can numerically obtain the steady-state response using the first-order harmonic balance method (HBM1). The steady-state harmonic components of the stiffness and damping force are modified using the Jacobi elliptic functions. The developed EHBM can reduce the truncation error in the HBM1. Compared with the HBM1, the EHBM can improve the accuracy of the resonance regimes of the amplitude-frequency curve and transmissibility. The EHBM is simple and straightforward. It can maintain the same form as the balancing equations of the HBM1 but performs better than it.

An analytical technique for solving quadratic nonlinear oscillator

2017 ◽  
Vol 13 (3) ◽  
pp. 424-433 ◽  
Author(s):  
Md. Helal Uddin Molla ◽  
Md. Abdur Razzak ◽  
M.S. Alam
Keyword(s):  
Energy Balance ◽  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Design Methodology ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Energy Balance Method ◽  
Content Type ◽  
Analytical Technique ◽  
Present Technique

Purpose The purpose of this paper is to present an analytical technique, based on the He’s energy balance method (an improved version recently presented by Khan et al.), to obtain the approximate solution of quadratic nonlinear oscillator (QNO). Design/methodology/approach This oscillator (QNO) is used as a mathematical model of the human eardrum oscillation. Findings It has been shown that the results by the present technique are very close to the numerical solution. Originality/value The results obtained in this paper are compared with those obtained by Hu (harmonic balance method) and Khan et al. The result shows that the method is more accurate and effective than harmonic balance as well as improved energy balance methods.

Application of the harmonic balance method to a nonlinear oscillator typified by a mass attached to a stretched wire

2007 ◽  
Vol 302 (4-5) ◽  
pp. 1018-1029 ◽  
Author(s):  
A. Beléndez ◽  
A. Hernández ◽  
T. Beléndez ◽  
M.L. Álvarez ◽  
S. Gallego ◽  
...  
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Oscillator ◽  
Harmonic Balance Method ◽  
Balance Method
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