Solvable vs integrable

In this chapter we will consider n coupled and generally nonlinear differential equations written in the form dxi/dt = Vi(x1,…,xn). Since Newton's formulation of mechanics via differential equations, the idea of what is meant by a solution has a short but very interesting history (see Ince's appendix (1956), and also Wintner (1941)). In the last century, the idea of solving a system of differential equations was generally the ‘reduction to quadratures’, meaning the solution of n differential equations by means of a complete set of n independent integrations (generally in the form of n – 1 conservation laws combined with a single final integration after n – 1 eliminating variables). Systems of differential equations that are discussed analytically in mechanics textbooks are almost exclusively restricted to those that can be solved by this method. Jacobi (German, 1804–1851)systematized the method, and it has become irreversibly mislabeled as ‘integrability’. Following Jacobi and also contemporary ideas of geometry, Lie (Norwegian, 1842–1899) studied first order systems of differential equations from the standpoint of invariance and showed that there is a universal geometric interpretation of all solutions that fall into Jacobi's ‘integrable’ category.

Chasle's theorem states that the general motion of a rigid body can be represented as a superposition of a translation following any point in the body and a pure rotation about that point. Kinematics studies are concerned only with the description of that motion. The developments in this chapter will disclose how the motion is related to the force system acting on the body. The resultant force may be regarded intuitively as the net tendency of the force system to push a body, so it may be expected to be related to the translational effect. Similarly, the resultant moment may be considered to be the net rotational effect. We shall confirm and quantify these expectations in the following presentation for general spatial motion, and then specialize the derived principles for the case of planar motion.

Fundamental Principle

Newton's laws govern the motion of a particle. A rigid body may be treated as a collection of particles whose motions are not independent. In the first part of this chapter, we shall derive the basic kinetics principles for rigid-body motion. The foundation for these derivations is Newton's second law, which describes inertial effects, and the third law, which describes the nature of the force system.

Basic Model

From a philosophical perspective, we initially recognize the atomic nature of matter by considering a body to consist of N particles having mass mi .

Our emphasis thus far has been on development of basic principles for treating the kinematics and kinetics of rigid-body motion. Regardless of whether we employed a Newtonian or Lagrangian formulation, it was usually necessary to account for constraints associated with the way in which the system is supported. Sometimes the goal was to characterize the force system required to produce a specified motion, as when the reactions must be evaluated. Other situations required the determination of conditions that are satisfied during the motion, as typified by the task of deriving differential equations of motion. In this chapter we will formulate and solve the equations of motion governing the rotational motion of a rigid body. Because the angular momentum in spatial motion is usually not aligned with the instantaneous axis of rotation, a portion of the rotational effect does not coincide with that axis. Such phenomena are exploited in gyroscopes, whose theory will be introduced here. However, we may learn much about the nature of dynamical responses by beginning with studies of simpler, yet more common, systems that display comparable effects.

Free Motion

One of the first types of spatial motion treated in basic physics and engineering courses on mechanics is projectile motion, whose study is devoted to the determination of the motion of the center of mass. In contrast, the manner in which the body rotates about its center of mass is seldom discussed in a fundamental course.

All who model mechanical systems are aware of the unique demands such activity places on conceptual abilities. We must characterize the manner in which numerous individual components interact, and select the appropriate physical laws applicable to each. No one tells us which variables are important. In complicated situations, a multitude of approaches are likely to be available. Thus, an important aspect of training students in this area is developing a level of experience in identifying the salient aspects of a system. They must learn to identify the pathways by which the basic parameters characterizing the inputs may be connected to the desired information representing the solution. In other words, a basic hallmark of the study of engineering dynamics is problem solving.

Some instructors believe that engineers learn by example. If that statement is true, it is only because an engineer is problem-oriented. One of the most prominent features of this textbook is its wealth of examples and homework problems. I have tried to select systems for this purpose that are recognizable as being relevant to engineering applications, yet sufficiently simplified to enable one to focus on the many facets entailed in implementing the associated theoretical concepts. One example of my approach may be found in the development of the method of Lagrangian multipliers. Some texts employ rather simple systems to illustrate this topic. In contrast, Example 3 in Chapter 7 employes Lagrangian multipliers to obtain the equations of motion for a rolling disk in arbitrary motion.

This chapter develops some basic techniques for describing the motion of a particle. Each description is based on a different set of coordinates. The set best suited to a particular situation depends on a variety of factors, but a primary consideration is whether the coordinates naturally fit known aspects of the motion. At the end of this chapter, we will examine situations where more than one of these descriptions may be employed beneficially.

Path Variables – Intrinsic Coordinates

The idea that the motion of a point should be described in terms of the properties of its path may not seem to be obvious. However, this is precisely how one thinks when using a road map and the speedometer and odometer of an automobile. This type of description is known as path variables, or less commonly as intrinsic coordinates, because the basic parameters that are considered to change are associated with the properties of the path. The terms tangent and normal components are also used because those are the primary directions, as we shall see. We assume that the path is known. The most fundamental variable for a specified path is the arclength s along this curve, measured from some starting point to the point of interest. As shown in Figure 2.1, measurement of s requires statement of positive sense along the path. Thus, negative s means that the point has receded, rather than advanced, along its path.

The constraints imposed on the motion of a system enter the Newton–Euler formulation of the equations of motion in two ways. The kinematical relations must account for the restrictions imposed on the motion, while the kinetics principles must account for the reaction force (or moment) associated with each constraint. When the system consists of more than one body, the need to account individually for the constraints associated with each connection substantially enhances the level of difficulty.

The Lagrangian formulation we shall develop in this chapter takes a different view of systems. The principles are based on an overview of the system and its mechanical energy (kinetic and potential). In contrast, the Newtonian equations of motion are time derivatives of momentum principles. Another, and perhaps the most important, difference is that the reactions exerted by supports will usually not appear in the Lagrangian formulation. This is a consequence of the fact that the reactions and the geometrical description of the system are two manifestations of the same physical feature. It is from the kinematical perspective that we shall begin our study.

Generalized Coordinates and Degrees of Freedom

Suppose the reference location of a system is given. (Such a location might be the starting position or the static equilibrium position.) We must select a set of geometrical parameters whose value uniquely defines a new position of the system relative to the initial position.

Introduction

The subject of dynamics is concerned with the relationship between the forces acting on a physical object and the motion that is produced by the force system. Our concern in this text shall be situations in which the classical laws of physics (i.e., Newtonian mechanics) are applicable. For our purposes, we may consider this to be the case whenever the object of interest is moving much more slowly than the speed of light. In part, this restriction means that we can use the concept of an absolute (i.e. fixed) frame of reference, which will be discussed shortly.

A study of dynamics consists of two phases: kinematics and kinetics. The objective of a kinematical analysis is to describe the motion of the system. It is important to realize that this type of study does not concern itself with what is causing the motion. A kinematical study might be needed to quantify a nontechnical description of the way a system moves, for example, finding the velocity of points on a mechanical linkage. In addition, some features of a kinematical analysis will always arise in a kinetics study, which analyzes the interplay between forces and motion. A primary objective will be the development of procedures for applying kinematics and kinetics principles in a logical and consistent manner, so that one may successfully analyze systems that have novel features. Particular emphasis will be placed on three-dimensional systems, some of which feature phenomena that you might not have encountered in your studies thus far.