Course Introduction(Undergraduate Program):Dynamics of Multibody Systems（Bilingual）
Source: This web site Publish date: 2015-10-28 10:25:00
Course ID: 411536
Course Name:
Dynamics of Multibody Systems
Semester: 4
Class hours / Credits: 32/2
Text book: 《Dynamics of Multibody Systems》Second Edition, Jens Wittenburg, Springer, 2008
Introduction:
It is the main purpose of this course to introduce the basic principles of rigid body kinematics and kinetics in three-dimensional space with compact mathematical notation. In addition, a formalism based on the graph theory is described in detail which substantially simplifies the task for investigations into the dynamics of multibody systems.
The subject of dynamics is conveniently divided into two topics. Kinematics is the study of motion, that is, of the evolution of the position of geometric objects over time, without regard for the cause of the motion. Kinetics is the study of the relation between forces and motion. The basis of kinetics is Newton's three laws of motion.
Dynamics is the oldest and best established field of mathematical physics. It is based on relatively few, relatively simple, basic concepts, all of which should be familiar to the readers of this course. Thus one would suspect that the subject is an easy one to understand and apply, and, indeed, this is true for the class of two-dimensional problems normally considered in an introductory engineering course in dynamics. In 2-D problems, the motion is easy to visualize and the required mathematical tools and manipulations are minimal. A few decades ago, the dynamics of interest in engineering applications was in fact mostly concerned with 2-D motion of rigid bodies in simple machines. In many engineering situations of current interest, however, the dynamics problems that arise are very complicated and, in particular, involve three-dimensional motion. Noteworthy examples are the motion of robotic manipulators and the motion of aerospace vehicles. Many such problems are difficult to visualize and require lengthy mathematical analysis. This analysis must be based on a sound understanding of the basic concepts of dynamics.
There are two basic types of problems in dynamics. The first problem is: given the motion of a body, what are the forces acting on it (or more precisely, what systems of forces can produce the given motion)? For example, suppose that the path of the manipulator portion of a multilink robot is specified so as to perform some useful function. Dynamic analysis would then be used to determine the forces required to produce this path. The second type of problem in dynamics is the reverse of the first: given the forces, what is the motion. As an example, it might be desired to know the motion of an airplane for a set of specified forces acting on it. Of course there are also situations of "mixed" type: some of the forces and some aspects of the motion may be known and the remaining information is to be determined.
There are six chapters of this course. In Chap. 1 the reader is made familiar with symbolic vector and tensor notation which is used throughout this book for its compact form. In order to facilitate the transition from symbolically written equations to scalar coordinate equations matrices of vector and tensor coordinates are introduced. Transformation rules for such matrices are discussed, and methods are developed for translating compound vector-tensor expressions from symbolic into scalar coordinate form. For the purpose of compact formulations of systems of symbolically written equations matrices are introduced whose elements are vectors or tensors. Generalized multiplication rules for such matrices are defined. In Chap. 2 on rigid body kinematics direction cosines, Euler angles, Bryan angles and Euler parameters are discussed. The notion of angular velocity is introduced, and kinematic differential equations are developed which relate the angular velocity to the rate of change of generalized coordinates. In Chap. 3 basic principles of rigid body dynamics are discussed. The definitions of both kinetic energy and angular momentum leads to the introduction of the inertia tensor. Formulations of the law of angular momentum for a rigid body are derived from Euler’s axiom and also from d’Alembert’s principle. Because of severe limitations on the length of the course only those subjects are covered which are necessary for the later chapters. Other important topics such as cyclic variables or quasicoordinates, for example, had to be left out. In Chap. 4 some classical problems of rigid body mechanics are treated for which closed-form solutions exist. Chapter 5 which makes up one half of the book is devoted to the presentation of a general formalism for the dynamics of systems of rigid bodies. Kinematic relationships, nonlinear equations of motion, energy expressions and other quantities are developed which are suitable for both numerical and nonnumerical investigations. The uniform description valid for any system of rigid bodies rests primarily on the application of concepts of graph theory. This mathematical tool in combination with matrix and symbolic vector and tensor notation leads to expressions which can easily be interpreted in physical terms. The formalism leads to simple expressions of such an explicit form that an automatic generation by a general-purpose computer program is possible. It is one of the goals of this chapter to enable the reader to write a program for dynamics simulations of a large class of engineering multibody systems. The usefulness of the formalism is demonstrated by means of some illustrative examples of nontrivial nature. Chapter 6 deals with phenomena which occur when a multibody system is subject to a collision either with another system or between two of its own bodies. Instantaneous changes of velocities and internal impulses in joints between bodies caused by such collisions are determined. The investigation reveals an interesting analogy to the law of Maxwell and Betti in elastostatics.