Modified harmonic balance method for solving strongly nonlinear oscillators

2019 ◽  
Vol 22 (3) ◽  
pp. 353-375 ◽  
Author(s):  
Nazmul Sharif ◽  
Abdur Razzak ◽  
M. Z. Alam
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Nonlinear Oscillators ◽  
Balance Method ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

scholarly journals Application of a modified rational harmonic balance method for a class of strongly nonlinear oscillators

2008 ◽  
Vol 372 (39) ◽  
pp. 6047-6052 ◽  
Author(s):  
A. Beléndez ◽  
E. Gimeno ◽  
M.L. Álvarez ◽  
D.I. Méndez ◽  
A. Hernández
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Nonlinear Oscillators ◽  
Balance Method ◽  
Strongly Nonlinear ◽  
Strongly Nonlinear Oscillators

A Global Residue Harmonic Balance Method for a Class of Nonlinear Jerk Equation

2013 ◽  
Vol 353-356 ◽  
pp. 3324-3327
Author(s):  
Xin Xue ◽  
Pei Jun Ju ◽  
Dan Sun
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Nonlinear Oscillators ◽  
Solution Procedure ◽  
High Accuracy ◽  
New Approach ◽  
Strongly Nonlinear ◽  
Accuracy Comparison ◽  
Comparison Of The Results ◽  
Strongly Nonlinear Oscillators

A new approach, namely the global residue harmonic balance, was developed to determine the accurately approximate periodic solution of a class of nonlinear Jerk equation containing velocity times acceleration-squared and velocity. Unlike other improved harmonic balance methods, all the forward harmonic residuals were considered in the present approximation to improve the accuracy. Comparison of the results obtained using this approach with the exact one and the existing results reveals that the high accuracy, simplicity and efficiency of the presented solution procedure. The method can be easily extended to other strongly nonlinear oscillators.

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.

RATIONAL ENERGY BALANCE METHOD TO NONLINEAR OSCILLATORS WITH CUBIC TERM

2013 ◽  
Vol 06 (02) ◽  
pp. 1350019 ◽  
Author(s):  
M. Daeichin ◽  
M. A. Ahmadpoor ◽  
H. Askari ◽  
A. Yildirim
Keyword(s):  
Energy Balance ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Nonlinear Problems ◽  
Nonlinear Oscillators ◽  
Initial Guess ◽  
Second Order ◽  
Balance Method ◽  
Strong Nonlinearity ◽  
Novel Approach

In this paper, a novel approach is proposed for solving the nonlinear problems based on the collocation and energy balance methods (EBMs). Rational approximation is employed as an initial guess and then it is combined with EBM and collocation method for solving nonlinear oscillators with cubic term. Obtained frequency amplitude relationship is compared with exact numerical solution and subsequently, a very excellent accuracy will be revealed. According to the numerical comparisons, this method provides high accuracy with 0.03% relative error for Duffing equation with strong nonlinearity in the second-order of approximation. Furthermore, achieved results are compared with other types of modified EBMs and the second-order of harmonic balance method. It is demonstrated that the new proposed method has the highest accuracy in comparison with different approaches such as modified EBMs and the second-order of harmonic balance method.

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