Nonlinear Response and Suppression of Chaos by Weak Harmonic Perturbation Inside a Triple Well Φ6-Rayleigh Oscillator Combined to Parametric Excitations

2006 ◽  
Vol 1 (3) ◽  
pp. 196-204 ◽  
Author(s):  
M. Siewe Siewe ◽  
F. M. Moukam Kakmeni ◽  
C. Tchawoua ◽  
P. Woafo
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Response ◽  
Main Attention ◽  
Response Curves ◽  
Local Bifurcations ◽  
Method Of Harmonic Balance ◽  
Averaged Equations ◽  
Harmonic Perturbation ◽  
Dynamical Properties ◽  
Suppression Of Chaos

The nonlinear response and suppression of chaos by weak harmonic perturbation inside a triple well Φ6-Rayleigh oscillator combined to parametric excitations is studied in this paper. The main attention is focused on the dynamical properties of local bifurcations as well as global bifurcations including homoclinic and heteroclinic bifurcations. The original oscillator is transformed to averaged equations using the method of harmonic balance to obtain periodic solutions. The response curves show the saddle-node bifurcation and the multi-stability phenomena. Based on the Melnikov’s method, horseshoe chaos is found and its control is made by introducing an external periodic perturbation.

Nonlinear Analysis of Shape Memory Devices With Duffing and Quadratic Oscillators

2017 ◽  
Vol 13 (1) ◽  
Author(s):  
Shantanu Rajendra Gaikwad ◽  
Ashok Kumar Pandey
Keyword(s):  
Shape Memory ◽  
Harmonic Balance ◽  
Nonlinear Response ◽  
Isothermal Condition ◽  
Nonisothermal Condition ◽  
Nonlinear Frequency ◽  
Method Of Harmonic Balance ◽  
Nonlinear Frequency Response ◽  
Balance Analysis ◽  
Analysis Of Results

In this paper, we investigate the linear and nonlinear response of shape memory alloy (SMA)-based Duffing and quadratic oscillator under large deflection conditions. In this study, we first present thermomechanical constitutive modeling of SMA with a single degree-of-freedom system. Subsequently, we solve equation to obtain linear frequency and nonlinear frequency response using the method of harmonic balance and validate it with numerical solution as well as averaging method under the isothermal condition. However, for nonisothermal condition, we analyze the influence of cubic and quadratic nonlinearity on nonlinear response based on method of harmonic balance. Analysis of results leads to various ways of controlling the nature and extent of nonlinear response of SMA-based oscillators. Such findings can be effectively used to control external vibration of different systems.

Nonlinear Free Vibration for Cross-Ply Laminated Damaged Plates with Piezoelectric Actuators

2006 ◽  
Vol 324-325 ◽  
pp. 479-482
Author(s):  
Yu Fang Zheng ◽  
Yi Ming Fu ◽  
Kai Qi
Keyword(s):  
Free Vibration ◽  
Harmonic Balance ◽  
Piezoelectric Actuators ◽  
Laminated Plates ◽  
Governing Equations ◽  
Nonlinear Free Vibration ◽  
Response Curves ◽  
Method Of Harmonic Balance ◽  
Amplitude Frequency Response ◽  
Damage Theory

On the basis of the anisotropic damage theory and piezoelectric theory, the nonlinear free vibration governing equations for cross-ply laminated damaged plates with piezoelectric actuators are established. The Galerkin procedure furnishes an infinite system of equations for time functions which are solved by the method of harmonic balance. In the numerical results, the influences of damage parameters and piezoelectric effect on the nonlinear amplitude-frequency response curves of the laminated plates are discussed, which results reveal the inherent features about the coupled mechanics and electricity.

DEGENERATE BIFURCATION ANALYSIS ON A PARAMETRICALLY AND EXTERNALLY EXCITED MECHANICAL SYSTEM

2001 ◽  
Vol 11 (03) ◽  
pp. 689-709 ◽  
Author(s):  
WEI ZHANG ◽  
PEI YU
Keyword(s):  
Normal Form ◽  
Mechanical System ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Main Attention ◽  
Zero Eigenvalue ◽  
Normal Form Theory ◽  
Local Bifurcations ◽  
Averaged Equations ◽  
Form Theory

A general parametrically and externally excited mechanical system is considered. The main attention is focused on the dynamical properties of local bifurcations as well as global bifurcations including homoclinic and heteroclinic bifurcations. In particular, degenerate bifurcations of codimension 3 are studied in detail. The original mechanical system is first transformed to averaged equations using the method of multiple scales. With the aid of normal form theory, the explicit expressions of the normal form associated with a double-zero eigenvalue and Z2-symmetry for the averaged equations are obtained. Based on the normal form, it has been shown that a parametrically and externally excited mechanical system can exhibit homoclinic and heteroclinic bifurcations, multiple limit cycles, and jumping phenomena in amplitude modulated oscillations. Numerical simulations are also given to verify the good analytical predictions.

scholarly journals Functional analysis and the method of harmonic balance

1975 ◽  
Vol 32 (4) ◽  
pp. 457-464 ◽  
Author(s):  
C. A. Borges ◽  
L. Cesari ◽  
D. A. Sánchez
Keyword(s):  
Functional Analysis ◽  
Harmonic Balance ◽  
Method Of Harmonic Balance

Analysis of Rotating Blade Forced Vibration Due to Torsional Excitation Using the Method of Harmonic Balance

2002 ◽  
Author(s):  
B. O. Al-Bedoor ◽  
A. A. Al-Qaisia
Keyword(s):  
Forced Vibration ◽  
Harmonic Balance ◽  
Approximate Solutions ◽  
Coefficient Matrix ◽  
Blade Vibration ◽  
Varying Coefficients ◽  
Rotating Blade ◽  
Method Of Harmonic Balance ◽  
Torsional Excitation ◽  
Truncated Fourier Series

This paper presents an analysis of the forced vibration of rotating blade due to torsional excitation. The model analyzed is a multi-modal forced second order ordinary differential equation with multiple harmonically varying coefficients. The method of Harmonic Balance (HB) is employed to find approximate solutions for each of the blade modes in the form of truncated Fourier series. The solutions have shown multi resonance response for the first blade vibration mode. The examination of the determinant of the harmonic balance solution coefficient matrix for stability purposes has shown that the region between the two resonance points is an unstable vibration region. Numerical integration of the equations is conducted at different frequency ratio points and the results are discussed. This solution provides a very critical operation and design guidance for rotating blade with torsional vibration excitation.

scholarly journals Resonant Stellar Orbits in Spiral Galaxies

1975 ◽  
Vol 69 ◽  
pp. 237-244
Author(s):  
P. O. Vandervoort
Keyword(s):  
Excellent Agreement ◽  
Spiral Galaxy ◽  
Harmonic Balance ◽  
Spiral Galaxies ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Analytic Solutions ◽  
Stellar Orbits ◽  
Method Of Harmonic Balance ◽  
Resonance Phenomena

This paper reviews a series of investigations of the orbits of stars in the regions of the Lindblad resonances of a spiral galaxy. The analysis is formulated in an epicyclic approximation. Analytic solutions of the epicyclic equations of motion are obtained by the method of harmonic balance of Bogoliubov and Mitropolsky. These solutions represent the resonance phenomena exhibited by the orbits in generally excellent agreement with numerical solutions.

A generalization of the method of harmonic balance

1987 ◽  
Vol 116 (3) ◽  
pp. 591-595 ◽  
Author(s):  
J. Garcia-Margallo ◽  
J.Diaz Bejarano
Keyword(s):  
Harmonic Balance ◽  
Method Of Harmonic Balance
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