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 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.

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 Nonlinear Vibration Response of a Rectangular Tube with a Flexible End and Non-Rigid Acoustic Boundaries

2016 ◽  
Author(s):  
Y. Y. Lee
Keyword(s):  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Vibration Response ◽  
Balance Method ◽  
Tube Length ◽  
Shift Parameter ◽  
Acoustic Problem ◽  
Rectangular Tube ◽  
Multi Level

This paper addresses the analysis for the nonlinear vibration response of a rectangular tube with a flexible end and non-rigid acoustic boundaries. This is a further work of the linear structural acoustic problem in a well-known acoustic book. In fact, the acoustic boundaries of an enclosed space sometimes are non-rigid and the structural vibration responses are nonlinear. These two points are the focuses of this paper. The multi-level residue harmonic balance method is applied to this nonlinear structural acoustic problem. The results obtained from the multi-level residue harmonic balance method and numerical method are generally in good agreement. The effects of excitation magnitude, tube length, and phase shift parameter etc. are examined.

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.  

Nonlinear vibration analysis of forced response for rubbing problems using the automatic differential frame

2021 ◽  
pp. 095440892110396
Author(s):  
Tianyuan Liu ◽  
Lei Sun ◽  
Yonghui Xie
Keyword(s):  
Nonlinear Vibration ◽  
Harmonic Balance ◽  
Vibration Analysis ◽  
Automatic Differentiation ◽  
Jacobian Matrix ◽  
Harmonic Balance Method ◽  
Degree Of Freedom ◽  
Balance Method ◽  
Time Marching ◽  
Marching Method

The multi-harmonic balance method is widely applied to obtain the forced responses of nonlinear systems undergoing rubbing problems. Despite large-scale time savings compared with the time marching method, it suffers from the complicated derivations of the Jacobian matrix. To solve this problem, this paper focuses on applying the automatic differentiation frame to the multi-harmonic balance method to implement the nonlinear vibration analysis of systems subjected to the rub phenomena. By establishing computational graph and utilizing the automatic differentiation process, tedious works such as the derivations of the complicated analytical expressions of the Jacobian matrix are avoided, which guarantees the efficiency and applicability of the presented method. A single-degree-of-freedom system with nonlinear force in the form of cubic is used to verify the accuracy of the method, and numerical analysis results reveal that the method is accurate enough compared with the time marching method. Furthermore, for the purpose of application, the responses of two common friction models, which are of great concern in some practical engineering fields, including a two-degree-of-freedom system containing a friction damper and a rotor disk system with circumferential rubbing, are obtained utilizing the presented approach. The influences of several model parameters on their responses are investigated as well. Numerical investigations demonstrate that the automatic differential solution framework developed in this research for solving nonlinear vibration equations has high accuracy and eliminates the need for a complicated partial derivative analytical formula derivation.

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