Approximate Analytical Methodology for the Restricted Three-Body and Four-Body Models Based on Polynomial Series

The restricted three-body problem (R3BP) and restricted four-body problem (R4BP) are modeled based on the rotating frame. The conservative autonomous system for the R3BP and nonautonomous system with period parametric resonance due to the fourth body are derived. From the vibrational point of view,...

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Main Authors:	Ying-Jing Qian, Xiao-Dong Yang, Lei-Yu Yang, Wei Zhang
Format:	Article
Language:	English
Published:	Hindawi Limited 2016-01-01
Series:	International Journal of Aerospace Engineering
Online Access:	http://dx.doi.org/10.1155/2016/9747289

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spelling	doaj-6ae3ddab4abf42b1a7f80fd980b9f62f2020-11-25T00:09:00ZengHindawi LimitedInternational Journal of Aerospace Engineering1687-59661687-59742016-01-01201610.1155/2016/97472899747289Approximate Analytical Methodology for the Restricted Three-Body and Four-Body Models Based on Polynomial SeriesYing-Jing Qian0Xiao-Dong Yang1Lei-Yu Yang2Wei Zhang3Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, ChinaBeijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, ChinaBeijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, ChinaBeijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical Structures, College of Mechanical Engineering, Beijing University of Technology, Beijing 100124, ChinaThe restricted three-body problem (R3BP) and restricted four-body problem (R4BP) are modeled based on the rotating frame. The conservative autonomous system for the R3BP and nonautonomous system with period parametric resonance due to the fourth body are derived. From the vibrational point of view, the methodology of polynomial series is proposed to solve for these problems analytically. By introducing the polynomial series relations among the three directions of motion, the three-degree-of-freedom coupled equations are transferred into one degree-of-freedom containing the full dynamics of the original autonomous system for the R3BP. As for the R4BP case, the methodology of polynomial series combined with the iterative approach is proposed. During the iterative approach, the nonautonomous system can be treated as pseudoautonomous equation and the final polynomial series relations and one-degree-of-freedom system can be derived iteratively.http://dx.doi.org/10.1155/2016/9747289
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Ying-Jing Qian Xiao-Dong Yang Lei-Yu Yang Wei Zhang
spellingShingle	Ying-Jing Qian Xiao-Dong Yang Lei-Yu Yang Wei Zhang Approximate Analytical Methodology for the Restricted Three-Body and Four-Body Models Based on Polynomial Series International Journal of Aerospace Engineering
author_facet	Ying-Jing Qian Xiao-Dong Yang Lei-Yu Yang Wei Zhang
author_sort	Ying-Jing Qian
title	Approximate Analytical Methodology for the Restricted Three-Body and Four-Body Models Based on Polynomial Series
title_short	Approximate Analytical Methodology for the Restricted Three-Body and Four-Body Models Based on Polynomial Series
title_full	Approximate Analytical Methodology for the Restricted Three-Body and Four-Body Models Based on Polynomial Series
title_fullStr	Approximate Analytical Methodology for the Restricted Three-Body and Four-Body Models Based on Polynomial Series
title_full_unstemmed	Approximate Analytical Methodology for the Restricted Three-Body and Four-Body Models Based on Polynomial Series
title_sort	approximate analytical methodology for the restricted three-body and four-body models based on polynomial series
publisher	Hindawi Limited
series	International Journal of Aerospace Engineering
issn	1687-5966 1687-5974
publishDate	2016-01-01
description	The restricted three-body problem (R3BP) and restricted four-body problem (R4BP) are modeled based on the rotating frame. The conservative autonomous system for the R3BP and nonautonomous system with period parametric resonance due to the fourth body are derived. From the vibrational point of view, the methodology of polynomial series is proposed to solve for these problems analytically. By introducing the polynomial series relations among the three directions of motion, the three-degree-of-freedom coupled equations are transferred into one degree-of-freedom containing the full dynamics of the original autonomous system for the R3BP. As for the R4BP case, the methodology of polynomial series combined with the iterative approach is proposed. During the iterative approach, the nonautonomous system can be treated as pseudoautonomous equation and the final polynomial series relations and one-degree-of-freedom system can be derived iteratively.
url	http://dx.doi.org/10.1155/2016/9747289
work_keys_str_mv	AT yingjingqian approximateanalyticalmethodologyfortherestrictedthreebodyandfourbodymodelsbasedonpolynomialseries AT xiaodongyang approximateanalyticalmethodologyfortherestrictedthreebodyandfourbodymodelsbasedonpolynomialseries AT leiyuyang approximateanalyticalmethodologyfortherestrictedthreebodyandfourbodymodelsbasedonpolynomialseries AT weizhang approximateanalyticalmethodologyfortherestrictedthreebodyandfourbodymodelsbasedonpolynomialseries
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Approximate Analytical Methodology for the Restricted Three-Body and Four-Body Models Based on Polynomial Series

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