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Frontiers of Mechanical Engineering

Front. Mech. Eng.    2014, Vol. 9 Issue (2) : 177-190     https://doi.org/10.1007/s11465-014-0288-8
RESEARCH ARTICLE |
Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM
M. R. AKBARI1,D. D. GANJI2,*(),A. MAJIDIAN3,A. R. AHMADI3
1. Department of Civil Engineering and Chemical Engineering, University of Tehran, Tehran, Iran
2. Department of Mechanical Engineering, Babol University of Technology, Babol, Iran
3. Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran
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Abstract

In the present paper, three complicated nonlinear differential equations in the field of vibration, which are Vanderpol, Rayleigh and Duffing equations, have been analyzed and solved completely by Algebraic Method (AGM). Investigating this kind of equations is a very hard task to do and the obtained solution is not accurate and reliable. This issue will be emerged after comparing the achieved solutions by numerical method (Runge-Kutte 4th). Based on the comparisons which have been made between the gained solutions by AGM and numerical method, it is possible to indicate that AGM can be successfully applied for various differential equations particularly for difficult ones. The results reveal that this method is not only very effective and simple, but also reliable, and can be applied for other complicated nonlinear problems.

Keywords Algebraic Method (AGM)      Angular Frequency      Vanderpol      Rayleigh      Duffing     
Corresponding Authors: D. D. GANJI   
Issue Date: 22 May 2014
 Cite this article:   
M. R. AKBARI,D. D. GANJI,A. MAJIDIAN, et al. Solving nonlinear differential equations of Vanderpol, Rayleigh and Duffing by AGM[J]. Front. Mech. Eng., 2014, 9(2): 177-190.
 URL:  
http://journal.hep.com.cn/fme/EN/10.1007/s11465-014-0288-8
http://journal.hep.com.cn/fme/EN/Y2014/V9/I2/177
Fig.1  Chart of the obtained solution by AGM
Fig.2  Chart of the first derivative for the obtained solution by AGM
Fig.3  Resulted phase plane for example1 by AGM.
t/s01020304050
u (t)Num. Rk 450.20.0756030.0145714-0.00544708-0.0075291-0.0047593
u ˙(t)Num. Rk 450.00.06328810.004930730.02498010.008496810.00109830
Tab.1  Obtained numerical solution of Eq. (31) based on the given physical values
Fig.4  Comparing the obtained solutions by AGM and numerical method
Fig.5  Comparing the first derivative of the obtained solutions by AGM and numerical method
Fig.6  Comparing the related phase planes of the achieved solutions by AGM and numerical method
Fig.7  Difference the obtained solutions by AGM and numerical method
Fig.8  Difference the first derivative of the obtained solutions by AGM and numerical method
Fig.9  Charts of the obtained solution and the locus of maximum displacement points
Fig.10  Achieved results for the vibrational velocity and the locus of the maximum vibrational velocity points
Fig.11  Resulted charts of the vibrational acceleration equation and its related locus
Fig.12  Chart of angular frequency in terms of initial vibrational amplitude
Fig.13  Variation of damping ratio in terms of the initial amplitude of vibration (A)
Fig.14  Chart of the obtained solution by AGM
Fig.15  Chart of the first derivative for the obtained solution by AGM
Fig.16  Resulted phase plane for example 2 by AGM
t0816243240
u(t)Num. Rk 450.1-0.0451040.01990540-0.008598120.0036300-0.00149353
u(t)Num.Rk 450.00.0076792-0.006629590.00452472-0.002646440.001440892
Tab.2  Results of numerical solution based on the given physical values in the specified domain
Fig.17  Comparison between the achieved solutions by AGM and numerical method
Fig.18  Comparing the first derivative of the obtained by AGM and numerical method
Fig.19  Comparing the related phase planes of the achieved solutions by AGM and numerical method
Fig.20  Difference of the obtained solutions by AGM and numerical method
Fig.21  Difference of the first derivative of the obtained solutions by AGM and numerical method
Fig.22  Chart of the obtained solution by AGM
Fig.23  Chart of the first derivative for the obtained solution by AGM
Fig.24  The resulted phase plane for the presented Duffing equation by AGM.
t0816243240
u(t)Num. Rk 450.150.132085-0.35225820.40210454-0.260445640.004382949
u(t)Num. Rk 450.00-0.0081056-0.03040860.09030328-0.131689450.12219430
Tab.3  Results of numerical solution based on the given physical values in the specified domain
Fig.25  Comparison between the achieved solutions by AGM and numerical method
Fig.26  Comparing the first derivative of the obtained by AGM and numerical method
Fig.27  Comparing the related phase planes of the achieved solutions by AGM and numerical method
Fig.28  Difference the obtained solutions by AGM and numerical method
Fig.29  Difference the first derivative of the obtained solutions by AGM and numerical method
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