Udwadia–Kalaba Equation for Constrained Mechanical Systems: Formulation and Applications
Abstract There are many achievements in the field of analytical mechanics, such as Lagrange Equation, Hamilton’s Principle, Kane’s Equation. Compared to Newton–Euler mechanics, analytical mechanics have a wider range of applications and the formulation procedures are more mathematical. However, all...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2018-12-01
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| Series: | Chinese Journal of Mechanical Engineering |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s10033-018-0310-x |
| _version_ | 1818519677724262400 |
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| author | Xiao-Min Zhao Ye-Hwa Chen Han Zhao Fang-Fang Dong |
| author_facet | Xiao-Min Zhao Ye-Hwa Chen Han Zhao Fang-Fang Dong |
| author_sort | Xiao-Min Zhao |
| collection | DOAJ |
| description | Abstract There are many achievements in the field of analytical mechanics, such as Lagrange Equation, Hamilton’s Principle, Kane’s Equation. Compared to Newton–Euler mechanics, analytical mechanics have a wider range of applications and the formulation procedures are more mathematical. However, all existing methods of analytical mechanics were proposed based on some auxiliary variables. In this review, a novel analytical mechanics approach without the aid of Lagrange’s multiplier, projection, or any quasi or auxiliary variables is introduced for the central problem of mechanical systems. Since this approach was firstly proposed by Udwadia and Kalaba, it was called Udwadia–Kalaba Equation. It is a representation for the explicit expression of the equations of motion for constrained mechanical systems. It can be derived via the Gauss’s principle, d’Alembert’s principle or extended d’Alembert’s principle. It is applicable to both holonomic and nonholonomic equality constraints, as long as they are linear with respect to the accelerations or reducible to be that form. As a result, the Udwadia–Kalaba Equation can be applied to a very broad class of mechanical systems. This review starts with introducing the background by a brief review of the history of mechanics. After that, the formulation procedure of Udwadia–Kalaba Equation is given. Furthermore, the comparisons of Udwadia–Kalaba Equation with Newton–Euler Equation, Lagrange Equation and Kane’s Equation are made, respectively. At last, three different types of examples are given for demonstrations. |
| first_indexed | 2024-12-11T01:27:20Z |
| format | Article |
| id | doaj.art-35fe6fbbb299417ba1f80259a2eff536 |
| institution | Directory Open Access Journal |
| issn | 1000-9345 2192-8258 |
| language | English |
| last_indexed | 2024-12-11T01:27:20Z |
| publishDate | 2018-12-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Chinese Journal of Mechanical Engineering |
| spelling | doaj.art-35fe6fbbb299417ba1f80259a2eff5362022-12-22T01:25:28ZengSpringerOpenChinese Journal of Mechanical Engineering1000-93452192-82582018-12-0131111410.1186/s10033-018-0310-xUdwadia–Kalaba Equation for Constrained Mechanical Systems: Formulation and ApplicationsXiao-Min Zhao0Ye-Hwa Chen1Han Zhao2Fang-Fang Dong3School of Automotive and Transportation Engineering, Hefei University of TechnologyThe George W. Woodruff School of Mechanical Engineering, Georgia Institute of TechnologySchool of Mechanical Engineering, Hefei University of TechnologySchool of Mechanical Engineering, Hefei University of TechnologyAbstract There are many achievements in the field of analytical mechanics, such as Lagrange Equation, Hamilton’s Principle, Kane’s Equation. Compared to Newton–Euler mechanics, analytical mechanics have a wider range of applications and the formulation procedures are more mathematical. However, all existing methods of analytical mechanics were proposed based on some auxiliary variables. In this review, a novel analytical mechanics approach without the aid of Lagrange’s multiplier, projection, or any quasi or auxiliary variables is introduced for the central problem of mechanical systems. Since this approach was firstly proposed by Udwadia and Kalaba, it was called Udwadia–Kalaba Equation. It is a representation for the explicit expression of the equations of motion for constrained mechanical systems. It can be derived via the Gauss’s principle, d’Alembert’s principle or extended d’Alembert’s principle. It is applicable to both holonomic and nonholonomic equality constraints, as long as they are linear with respect to the accelerations or reducible to be that form. As a result, the Udwadia–Kalaba Equation can be applied to a very broad class of mechanical systems. This review starts with introducing the background by a brief review of the history of mechanics. After that, the formulation procedure of Udwadia–Kalaba Equation is given. Furthermore, the comparisons of Udwadia–Kalaba Equation with Newton–Euler Equation, Lagrange Equation and Kane’s Equation are made, respectively. At last, three different types of examples are given for demonstrations.http://link.springer.com/article/10.1186/s10033-018-0310-xUdwadia–Kalaba EquationMechanical systemConstraintMoore–Penrose inverse |
| spellingShingle | Xiao-Min Zhao Ye-Hwa Chen Han Zhao Fang-Fang Dong Udwadia–Kalaba Equation for Constrained Mechanical Systems: Formulation and Applications Chinese Journal of Mechanical Engineering Udwadia–Kalaba Equation Mechanical system Constraint Moore–Penrose inverse |
| title | Udwadia–Kalaba Equation for Constrained Mechanical Systems: Formulation and Applications |
| title_full | Udwadia–Kalaba Equation for Constrained Mechanical Systems: Formulation and Applications |
| title_fullStr | Udwadia–Kalaba Equation for Constrained Mechanical Systems: Formulation and Applications |
| title_full_unstemmed | Udwadia–Kalaba Equation for Constrained Mechanical Systems: Formulation and Applications |
| title_short | Udwadia–Kalaba Equation for Constrained Mechanical Systems: Formulation and Applications |
| title_sort | udwadia kalaba equation for constrained mechanical systems formulation and applications |
| topic | Udwadia–Kalaba Equation Mechanical system Constraint Moore–Penrose inverse |
| url | http://link.springer.com/article/10.1186/s10033-018-0310-x |
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