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

This paper describes the application of the harmonic balance parameter identification method to beam energy harvesters. The method is applied to weakly nonlinear and nonlinear, bistable fixed-free piezoelectric beams with tip masses. It is shown that only one measurement is required to identify parameters even though the systems are continuous. In addition, an experimental method of determining the number of restoring force coefficients required to accurately model the systems is presented. The harmonic balance parameter identification method is extended to account for multiple concurrent frequencies in order to identify parameters of weakly nonlinear systems. Finally, parameters are identified for two experimental energy harvesters. Good agreement is shown between the experimental data and the identified parameters using simulations and closed form solutions.

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A Modified Newton–Harmonic Balance Approach to Strongly Odd Nonlinear Oscillators

Journal of Vibration Engineering & Technologies ◽

10.1007/s42417-019-00176-3 ◽

2019 ◽

Vol 8 (5) ◽

pp. 721-736 ◽

Cited By ~ 2

Author(s):

Baisheng Wu ◽

Weijia Liu ◽

Huixiang Zhong ◽

C. W. Lim

Keyword(s):

Harmonic Balance ◽

Nonlinear Oscillators ◽

Balance Approach

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The VIM-Padé technique for strongly nonlinear oscillators with cubic and harmonic restoring force

Journal of Low Frequency Noise Vibration and Active Control ◽

10.1177/1461348418813612 ◽

2018 ◽

Vol 38 (3-4) ◽

pp. 1272-1278 ◽

Cited By ~ 2

Author(s):

Junfeng Lu ◽

Li Ma

Keyword(s):

Nonlinear Oscillators ◽

Restoring Force ◽

Strongly Nonlinear ◽

Strongly Nonlinear Oscillators

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

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.430.14 ◽

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.

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Accurate numerical solutions of conservative nonlinear oscillators

Nonlinear Engineering ◽

10.1515/nleng-2014-0009 ◽

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.

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Isolated Response Curves in a Base-Excited, Two-Degree-of-Freedom, Nonlinear System

Volume 6: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

10.1115/detc2015-46106 ◽

2015 ◽

Cited By ~ 10

Author(s):

J. P. Noël ◽

T. Detroux ◽

L. Masset ◽

G. Kerschen ◽

L. N. Virgin

Keyword(s):

Nonlinear System ◽

Harmonic Balance ◽

Global Analysis ◽

Normal Modes ◽

Nonlinear Normal Modes ◽

Basins Of Attraction ◽

Response Curves ◽

Restoring Force ◽

Nonlinear Normal ◽

Two Degree Of Freedom

In the present paper, isolated response curves in a nonlinear system consisting of two masses sliding on a horizontal guide are examined. Transverse springs are attached to one mass to provide the nonlinear restoring force, and a harmonic motion of the complete system is imposed by prescribing the displacement of their supports. Numerical simulations are carried out to study the conditions of existence of isolated solutions, their bifurcations, their merging with the main response branch and their basins of attraction. This is achieved using tools including nonlinear normal modes, energy balance, harmonic balance-based continuation and bifurcation tracking, and global analysis.

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