Dynamical Modeling and Simulation of Multi-body Systems by Using Udwadia-Kalaba Theory

Received December 20, 2013; revised March 20, 2013; accepted March 27, 2013
Abstract: Laboratory experiments were conducted for falling U-chain, but explicit analytic form of the general equations of motion was not presented. Several modeling methods were developed for fish robots. However, they just focused on the whole fish’s locomotion which does little favor to understand the detailed swimming behavior of fish. Udwadia-Kalaba theory is used to model these two multi-body systems and obtain explicit analytic equations of motion. For falling U-chain, the mass matrix is non-singular. Second-order constraints are used to get the constraint force and equations of motion and the numerical simulation is conducted. Simulation results show that the chain tip falls faster than the freely falling body. For fish robot, two-joint Carangiform fish robot is focused on. Quasi-steady wing theory is used to approximately calculate fluid lift force acting on the caudal fin. Based on the obtained explicit analytic equations of motion (the mass matrix is singular), propulsive characteristics of each part of the fish robot are obtained. Through these two cases of U chain and fish robot, how to use Udwadia-Kalaba equation to obtain the dynamical model is shown in detail and the modeling methodology for multi-body systems is presented. It is also shown that Udwadia-Kalaba theory is applicable to systems whether or not their mass matrices are singular. In the whole process of applying Udwadia-Kalaba equation, Lagrangian multipliers and quasi-coordinates are not used. Udwadia-Kalaba theory is creatively applied to dynamical modeling of falling U-chain and fish robot problems and explicit analytic equations of motion are obtained. Key words: Udwadia-Kalaba equation, multi-body systems, falling U-chain, fish robot
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Introduction

The general problem of obtaining equations of motion for constrained discrete mechanical systems has been an area of considerable interest among scientists and engineers. It is also one of the central issues in multi-body dynamics. The problem has been aggressively and continuously pursued by many scientists, engineers and mathematicians since constrained motion was initially described by LAGRANGE[1]. He invented the special Lagrange multiplier method to deal with constrained motion. However, the Lagrange multiplier method relies on problem-specific approaches to determine the multipliers; it is often very difficult to find the multipliers to obtain the explicit equations of motion for systems with large numbers of degrees of freedom and a mass of non-integrable constraints. GAUSS[2] introduced a new general principle of mechanics for handling constrained motion. Gauss's Principle gives a clear description of the general nature of constrained motion through minimization of a function of the accelerations of the particles of a system. GIBBS[3] and APPELL[4] have independently
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Dynamic Modeling and Simulation of Multi-body Systems Using the Udwadia-Kalaba Theory
ZHAO Han1, ZHEN Shengchao1, 2, *, and CHEN Ye-Hwa2
1 Mechanical Engineering, Hefei University of Technology, Hefei 230009, China 2 Mechanical Engineering, Georgia Institute of Technology, Atlanta 30332, USA
CHINESE JOURNAL OF MECHANICAL ENGINEERING
Vol. 26, No. 5, 2013
DOI: 10.3901/CJME.2013.05.839, available online at ; ;
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YZHAO Han, et al: Dynamic Modeling and Simulation of Multi-body Systems Using the Udwadia-Kalaba TheoryY
* Corresponding author. E-mail: zhenshengchao@
© Chinese Mechanical Engineering Society and Springer-Verlag Berlin Heidelberg 2013
developed equations of constrained motion when the constraints satisfy D’ Alember’ s principle. In the treatise on the analytical dynamics, PARS[5] refers to the Gibbs-Appell equations as “probably the most comprehensive equations of motion so far discovered ”. But the Gibbs-Appell equations require a “lucky” choice of problem-specific quasi-coordinates and they suffer from similar problems when dealing with systems with a large number of degrees of freedom and many non- integrable constraints. DIRAC[6] used Poisson brackets, a recursive scheme for determining the Lagrange multipliers, for singular Hamiltonian systems where the constraints do not exactly depend on time. On the other hand, UDWADIA, et al[7–8], obtained a concise, explicit set of equations of motion for constrained discrete dynamic systems which lead to a simple and new fundamental view of Lagrangian mechanics. They derived the fundamental equation of motion that describes the dynamics of constrained systems from Gauss’ s principle which seems somewhat less popular than the principles of Lagrange, Hamilton, Gibbs and Appell. The equations can deal with holonomic and also non-holonomic constraints. UDWADIA, et al[9–10], observed that all the research above has used D’ Alember’ s principle as their starting point. D’ Alember’ s principle assumes that the forces of constraints are considered to be ideal and the total work done by the