Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes
This paper extends the Wittrick-Williams (W-W) algorithm for hybrid dynamic stiffness (DS) models connecting any combinations of line and point nodes. The principal novelties lie in the development of both the DS formulation and the solution technique in a sufficiently systematic and general manner....
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| Format: | Article |
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MDPI AG
2021-12-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/1/57 |
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| author | Xiang Liu Shaoqi Qiu Suchao Xie Jnan Ranjan Banerjee |
| author_facet | Xiang Liu Shaoqi Qiu Suchao Xie Jnan Ranjan Banerjee |
| author_sort | Xiang Liu |
| collection | DOAJ |
| description | This paper extends the Wittrick-Williams (W-W) algorithm for hybrid dynamic stiffness (DS) models connecting any combinations of line and point nodes. The principal novelties lie in the development of both the DS formulation and the solution technique in a sufficiently systematic and general manner. The parent structure is considered to be in the form of two dimensional DS elements with line nodes, which can be connected to rigid/spring point supports/connections, rod/beam point supports/connections, and point connections to substructures. This is achieved by proposing a direct constrain method in a strong form which makes the modeling process straightforward. For the solution technique, the W-W algorithm is extended for all of the above hybrid DS models. No matrix inversion is needed in the proposed extension, making the algorithm numerically stable, especially for complex built-up structures. A mathematical proof is provided for the extended W-W algorithm. The proposed DS formulation and the extended W-W algorithm are validated by the FE results computed by ANSYS. This work significantly extends the application scope of the DS formulation and the W-W algorithm in a methodical and reliable manner, providing a powerful eigenvalue analysis tool for beam-plate built-up structures. |
| first_indexed | 2024-03-10T03:32:13Z |
| format | Article |
| id | doaj.art-c14ee11bbe2841e5b4adce83bd40a1ca |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T03:32:13Z |
| publishDate | 2021-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-c14ee11bbe2841e5b4adce83bd40a1ca2023-11-23T11:53:28ZengMDPI AGMathematics2227-73902021-12-011015710.3390/math10010057Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point NodesXiang Liu0Shaoqi Qiu1Suchao Xie2Jnan Ranjan Banerjee3Key Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic and Transportation Engineering, Central South University, Changsha 410075, ChinaKey Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic and Transportation Engineering, Central South University, Changsha 410075, ChinaKey Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic and Transportation Engineering, Central South University, Changsha 410075, ChinaSchool of Mathematics, Computer Science and Engineering, City University of London, London EC1V 0HB, UKThis paper extends the Wittrick-Williams (W-W) algorithm for hybrid dynamic stiffness (DS) models connecting any combinations of line and point nodes. The principal novelties lie in the development of both the DS formulation and the solution technique in a sufficiently systematic and general manner. The parent structure is considered to be in the form of two dimensional DS elements with line nodes, which can be connected to rigid/spring point supports/connections, rod/beam point supports/connections, and point connections to substructures. This is achieved by proposing a direct constrain method in a strong form which makes the modeling process straightforward. For the solution technique, the W-W algorithm is extended for all of the above hybrid DS models. No matrix inversion is needed in the proposed extension, making the algorithm numerically stable, especially for complex built-up structures. A mathematical proof is provided for the extended W-W algorithm. The proposed DS formulation and the extended W-W algorithm are validated by the FE results computed by ANSYS. This work significantly extends the application scope of the DS formulation and the W-W algorithm in a methodical and reliable manner, providing a powerful eigenvalue analysis tool for beam-plate built-up structures.https://www.mdpi.com/2227-7390/10/1/57dynamic stiffness methodline nodespoint nodesLagrange multipliermodal analysisWittrick-Williams algorithm |
| spellingShingle | Xiang Liu Shaoqi Qiu Suchao Xie Jnan Ranjan Banerjee Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes Mathematics dynamic stiffness method line nodes point nodes Lagrange multiplier modal analysis Wittrick-Williams algorithm |
| title | Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes |
| title_full | Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes |
| title_fullStr | Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes |
| title_full_unstemmed | Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes |
| title_short | Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes |
| title_sort | extension of the wittrick williams algorithm for free vibration analysis of hybrid dynamic stiffness models connecting line and point nodes |
| topic | dynamic stiffness method line nodes point nodes Lagrange multiplier modal analysis Wittrick-Williams algorithm |
| url | https://www.mdpi.com/2227-7390/10/1/57 |
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