Chinese Journal of Mechanical Engineering >

2018 , Vol. 31 >Issue 6: 106 - 106

DOI: https://doi.org/10.1186/s10033-018-0310-x

Review

Udwadia–Kalaba Equation for Constrained Mechanical Systems: Formulation and Applications

  • Xiao-Min Zhao ,
  • Ye-Hwa Chen ,
  • Han Zhao ,
  • Fang-Fang Dong
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  • 1. School of Mechanical Engineering, Hefei University of Technology, Hefei 230009, China;
    2. The George W. Woodruf School of Mechanical Engineering, Georgia Institute of Technology, Atlanta 30332, USA;
    3. School of Automotive and Transportation Engineering, Hefei University of Technology, Hefei 230009, China

Received date: 2017-03-08

  Online published: 2019-07-23

Supported by

Supported by National Natural Science Foundation of China (Grant No. 51705116), Anhui Provincial Science and Technology Major Project of China (Grant No. 17030901036), and Fundamental Research Funds for the Central Universities of China (Grant Nos. JZ2018HGBZ0096, JZ2018HGTA0217, JZ2018HGTB0261)

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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.

Key words： Udwadia–Kalaba Equation; Mechanical system; Constraint; Moore–Penrose inverse

Cite this article

Xiao-Min Zhao , Ye-Hwa Chen , Han Zhao , Fang-Fang Dong . Udwadia–Kalaba Equation for Constrained Mechanical Systems: Formulation and Applications[J]. Chinese Journal of Mechanical Engineering, 2018 , 31(6) : 106 -106 . DOI: 10.1186/s10033-018-0310-x

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