scholarly journals Computing multiple periodic solutions of nonlinear vibration problems using the harmonic balance method and Groebner bases

2015 ◽  
Vol 52-53 ◽  
pp. 529-547 ◽  
Author(s):  
Aurelien Grolet ◽  
Fabrice Thouverez
Keyword(s):  
Nonlinear Vibration ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Groebner Bases ◽  
Multiple Periodic Solutions ◽  
Vibration Problems

scholarly journals Analytical and Dynamic study of Pulled Mass Nonlinear Vibration by Two Cables using Newton's Harmonic Balance Method

2020 ◽  
Vol 2 (2) ◽  
pp. 79-86
Author(s):  
Masoud Rahmani ◽  
Ionut Cristian Scurtu ◽  
Amin Moslemi Petrudi
Keyword(s):  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
Nonlinear Vibrations ◽  
Nonlinear Differential Equations ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Good Match ◽  
Engineering Research ◽  
Solution Methods ◽  
Vibration Problems

In recent years, much research has been done on nonlinear vibrations, and analytical and numerical methods have been used to solve complex nonlinear equations. The behavior of nonlinear oscillating equations is discussed until the second order is approximated. Harmonic balance method, which itself has limitations in application. This method continues to be able to study a wider range of nonlinear differential equations. In general, nonlinear vibration problems are of great importance in physics, mechanical structures, and other engineering research. First, the equation of nonlinear vibrations governing the mass of the particle mass connect to the drawn cable is calculated and then the Newton Harmonic Balance Method is used to study the nonlinear vibrations of the set and obtain the answer and its frequency. The method (NHBM) is done with Maple software and a comparison between the results of this method with the solution methods used by other researchers is shown to be a good match.  

Finding Periodic Solutions of Forced Systems With Local Nonlinearities: A Mixed Shooting Harmonic Balance Method

2015 ◽  
Author(s):  
Frederic Schreyer ◽  
Remco Leine
Keyword(s):  
Dynamical Systems ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Shooting Method ◽  
Mechanical Systems ◽  
Harmonic Balance Method ◽  
Nonlinear Effects ◽  
Balance Method ◽  
Pros And Cons ◽  
Numerical Approaches

Several numerical approaches have been developed to capture nonlinear effects of dynamical systems. In this paper we present a mixed shooting-harmonic balance method to solve large mechanical systems with local nonlinearities efficiently. The Harmonic Balance Method as well as the shooting method have both their pros and cons. The proposed mixed shooting-HBM approach combines the efficiency of HBM and the accuracy of the shooting method and has therefore advantages of both.

scholarly journals The Spreading Residue Harmonic Balance Method for Strongly Nonlinear Vibrations of a Restrained Cantilever Beam

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Y. H. Qian ◽  
J. L. Pan ◽  
S. P. Chen ◽  
M. H. Yao
Keyword(s):  
Exact Solutions ◽  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
Nonlinear Dynamical Systems ◽  
Harmonic Balance Method ◽  
Time History ◽  
Approximate Solutions ◽  
Balance Method ◽  
Lumped Mass ◽  
Strongly Nonlinear

The exact solutions of the nonlinear vibration systems are extremely complicated to be received, so it is crucial to analyze their approximate solutions. This paper employs the spreading residue harmonic balance method (SRHBM) to derive analytical approximate solutions for the fifth-order nonlinear problem, which corresponds to the strongly nonlinear vibration of an elastically restrained beam with a lumped mass. When the SRHBM is used, the residual terms are added to improve the accuracy of approximate solutions. Illustrative examples are provided along with verifying the accuracy of the present method and are compared with the HAM solutions, the EBM solutions, and exact solutions in tables. At the same time, the phase diagrams and time history curves are drawn by the mathematical software. Through analysis and discussion, the results obtained here demonstrate that the SRHBM is an effective and robust technique for nonlinear dynamical systems. In addition, the SRHBM can be widely applied to a variety of nonlinear dynamic systems.

An Approximated Harmonic Balance Method for Nonlinear Vibration of Viscoelastic Structures

2006 ◽  
Vol 128 (3) ◽  
pp. 330 ◽  
Author(s):  
El Hassan Boutyour ◽  
El Mostafa Daya ◽  
Lahcen Azrar ◽  
Michel Potier-Ferry
Keyword(s):  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method

scholarly journals A modified harmonic balance method for solving forced vibration problems with strong nonlinearity

2020 ◽  
pp. 146134842092343 ◽  
Author(s):  
M W Ullah ◽  
M S Rahman ◽  
M A Uddin
Keyword(s):  
Forced Vibration ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Strong Nonlinearity ◽  
Algebraic Equations ◽  
Linear Algebraic Equations ◽  
Nonlinear Algebraic Equations ◽  
Vibration Problems ◽  
Nonlinear Forced Vibration

In this paper, a modified harmonic balance method is presented to solve nonlinear forced vibration problems. A set of nonlinear algebraic equations appears among the unknown coefficients of harmonic terms and the frequency of the forcing term. Usually a numerical method is used to solve them. In this article, a set of linear algebraic equations is solved together with a nonlinear one. The solution obtained by the proposed method has been compared to those obtained by variational and numerical methods. The results show good agreement with the results obtained by both methods mentioned above.

Vibration Analysis of a Shape Memory Oscillator by Harmonic Balance Method Verified Numerically

2019 ◽  
Vol 29 (03) ◽  
pp. 1930007 ◽  
Author(s):  
Rafal Rusinek ◽  
Joanna Rekas ◽  
Krzysztof Kecik
Keyword(s):  
Shape Memory ◽  
Periodic Solutions ◽  
Harmonic Balance ◽  
Primary Resonance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Multiple Time Scales ◽  
Multiple Time ◽  
System Parameters ◽  
Irregular Motion

This paper focuses on periodic solutions for a one-degree-of-freedom oscillator with a spring made of shape memory alloy (SMA). However, when periodic solutions are unstable, irregular motion is identified numerically. The shape memory spring is described by a polynomial characteristic in this model. The harmonic balance method (HBM) is employed to find periodic solutions near the primary resonance. The solutions are confronted with results obtained by the multiple time scales method and numerical simulations. Finally, the effect of system parameters and temperature on the system dynamics is discussed.

scholarly journals Multi-level residue harmonic balance method for nonlinear vibration of the beam

2021 ◽  
pp. 146134842110384
Author(s):  
Abu SMZ Hasan ◽  
M S Rahman
Keyword(s):  
Dynamic System ◽  
Nonlinear Vibration ◽  
Nonlinear Dynamic ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Nonlinear Dynamic System ◽  
Balance Method ◽  
Simply Supported Beam ◽  
Simply Supported ◽  
Multi Level

This study presents the nonlinear vibration and chaotic response of a beam subjected to harmonic excitation. The multi-level residue harmonic balance method is applied to solve the geometrically cubic nonlinear vibration of the simply supported beam. The obtained results agree well with those of the numerical integration method. The amplitude frequency response curves are presented to illustrate the nonlinear dynamic system response both for a damping and without damping model. Also, the chaotic response is examined for a simply supported beam with a nonlinear dynamic system.

scholarly journals Continuation of quasi-periodic solutions with two-frequency Harmonic Balance Method

2017 ◽  
Vol 394 ◽  
pp. 434-450 ◽  
Author(s):  
Louis Guillot ◽  
Pierre Vigué ◽  
Christophe Vergez ◽  
Bruno Cochelin
Keyword(s):  
Periodic Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Frequency Harmonic
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