Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain
In this paper, a double pendulum model is presented with unilateral rigid constraint under harmonic excitation, which leads to be an asymmetric and non-smooth system. By introducing impact recovery matrix, modal analysis, and matrix theory, the analytical expressions of the periodic solutions for un...
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| Format: | Article |
| Language: | English |
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MDPI AG
2019-07-01
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| Series: | Symmetry |
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| Online Access: | https://www.mdpi.com/2073-8994/11/7/886 |
| _version_ | 1798004834375303168 |
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| author | Xiuying Guo Gang Zhang Ruilan Tian |
| author_facet | Xiuying Guo Gang Zhang Ruilan Tian |
| author_sort | Xiuying Guo |
| collection | DOAJ |
| description | In this paper, a double pendulum model is presented with unilateral rigid constraint under harmonic excitation, which leads to be an asymmetric and non-smooth system. By introducing impact recovery matrix, modal analysis, and matrix theory, the analytical expressions of the periodic solutions for unilateral double-collision will be discussed in high-dimensional non-smooth asymmetric system. Firstly, the impact laws are classified in order to detect the existence of periodic solutions of the system. The impact recovery matrix is introduced to transform the impact laws of high-dimensional system into matrix. Furthermore, by use of modal analysis and matrix theory, an invertible transformation is constructed to obtain the parameter conditions for the existence of the impact periodic solution, which simplifies the calculation and can be easily extended to high-dimensional non-smooth system. Hence, the range of physical parameters and the restitution coefficients is calculated theoretically and non-smooth analytic expression of the periodic solution is given, which provides ideas for the study of approximate analytical solutions of high-dimensional non-smooth system. Finally, numerical simulation is carried out to obtain the impact periodic solution of the system with small angle motion. |
| first_indexed | 2024-04-11T12:29:44Z |
| format | Article |
| id | doaj.art-05ca81c38ac64d76ab7d323a59d3d669 |
| institution | Directory Open Access Journal |
| issn | 2073-8994 |
| language | English |
| last_indexed | 2024-04-11T12:29:44Z |
| publishDate | 2019-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Symmetry |
| spelling | doaj.art-05ca81c38ac64d76ab7d323a59d3d6692022-12-22T04:23:47ZengMDPI AGSymmetry2073-89942019-07-0111788610.3390/sym11070886sym11070886Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid ConstrainXiuying Guo0Gang Zhang1Ruilan Tian2Department of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050000, ChinaDepartment of Mathematics and Information Science, Hebei Normal University, Shijiazhuang 050000, ChinaDepartment of Mechanics Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, ChinaIn this paper, a double pendulum model is presented with unilateral rigid constraint under harmonic excitation, which leads to be an asymmetric and non-smooth system. By introducing impact recovery matrix, modal analysis, and matrix theory, the analytical expressions of the periodic solutions for unilateral double-collision will be discussed in high-dimensional non-smooth asymmetric system. Firstly, the impact laws are classified in order to detect the existence of periodic solutions of the system. The impact recovery matrix is introduced to transform the impact laws of high-dimensional system into matrix. Furthermore, by use of modal analysis and matrix theory, an invertible transformation is constructed to obtain the parameter conditions for the existence of the impact periodic solution, which simplifies the calculation and can be easily extended to high-dimensional non-smooth system. Hence, the range of physical parameters and the restitution coefficients is calculated theoretically and non-smooth analytic expression of the periodic solution is given, which provides ideas for the study of approximate analytical solutions of high-dimensional non-smooth system. Finally, numerical simulation is carried out to obtain the impact periodic solution of the system with small angle motion.https://www.mdpi.com/2073-8994/11/7/886non-smooth high-dimensional systemasymmetric systemimpact periodic solutionimpact recovery matrixnon-smooth analytic solution |
| spellingShingle | Xiuying Guo Gang Zhang Ruilan Tian Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain Symmetry non-smooth high-dimensional system asymmetric system impact periodic solution impact recovery matrix non-smooth analytic solution |
| title | Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain |
| title_full | Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain |
| title_fullStr | Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain |
| title_full_unstemmed | Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain |
| title_short | Periodic Solution of a Non-Smooth Double Pendulum with Unilateral Rigid Constrain |
| title_sort | periodic solution of a non smooth double pendulum with unilateral rigid constrain |
| topic | non-smooth high-dimensional system asymmetric system impact periodic solution impact recovery matrix non-smooth analytic solution |
| url | https://www.mdpi.com/2073-8994/11/7/886 |
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