Introduction

In this chapter, we examine systems of rigid bodies. Our goal is simply to discover how our previous developments can be used to obtain the equations of motion for these systems. As you might imagine, the equations of motion can be very complex, and judicious component selections from the balance laws are often needed to extract the equations of motion. This is illustrated with the example of a rate gyroscope.

The presentations here are limited in scope and we do not have the opportunity to discuss many interesting systems featuring several rigid bodies such as the dualspin spacecraft, bicycles, gyrocompasses, and the Dynabee in detail. As discussed in, a dual-spin satellite has the ability to reorient itself in an environment where the resultant moment on the satellite is negligible. This ability has been used in communications satellites and was employed in the Galileo spacecraft. This spacecraft was launched in 1989 and some 6 years later began its orbits of the planet Jupiter. These orbits were designed so that the spacecraft could capture data on some of the largest moons of Jupiter; Galileo's mission was a remarkable success. The Dynabee (or Rollerball) was invented in the early 1970s by Archie Mishler and features spinning a rotor to speeds in excess of 5000 rpm by carefully rotating an outer casing (housing). This novel gyroscopic device is discussed in an exercise at the end of this chapter.

Introduction

In this chapter, we establish Lagrange's equations for a system of particles by starting with the balances of linear momentum for each of the particles. Our derivation is based on the results presented in Chapter 15 of Synge and Griffith. We supplement their work with a discussion of constraints and potential energies. To examine the geometry inherent in Lagrange's equations of motion for the system of particles, we use the construction of a representative single particle by Casey. All the work presented in this chapter emphasizes the equivalence of Lagrange's equations of motion for a system of particles and the balances of linear momenta. For completeness, a brief discussion of the principle of virtual work, D'Alembert's principle, Gauss' principle of least constraint, and Hamilton's principle are also presented in Section 4.11. The chapter closes with a discussion of a canonical form of Lagrange's equations of motion in which time-independent integrable constraints are present.

For many specific problems, we can obtain Lagrange's equations by merely calculating the kinetic and potential energies of the system. This approach is used in most dynamics textbooks, and neither the construction of a single particle nor the components of force vectors are mentioned. Indeed, once we establish Lagrange's equations we can also ignore the explicit construction of the single particle. However, for many cases – which are not possible to treat using the approach adopted in most dynamics textbooks – we find that the use of Synge's and Griffith's representation of Lagrange's equations of motion allows us to tremendously increase the range of application of Lagrange's equations.

The writing of this book started more than a decade ago when I was first given the assignment of teaching two courses on rigid body dynamics. One of these courses featured Lagrange's equations of motion, and the other featured the Newton–Euler equations. I had long struggled to resolve these two approaches to formulating the equations of motion of mechanical systems. Luckily, at this time, one of my colleagues, Jim Casey, was examining the elegant works of Synge and his co-workers on this topic. There, he found a partial resolution to the equivalence of the Lagrangian and Newton–Euler approaches. He then went further and showed how the governing equations for a rigid body formulated by use of both approaches were equivalent. Shades of this result could be seen in an earlier work by Greenwood, but Casey's work established the equivalence in an unequivocal fashion. As is evident from this book, I subsequently adapted and expanded on Casey's treatment in my courses. My treatment of dynamics presented in this book is also heavily influenced by the texts of Papastavridis and Rosenberg. It has also benefited from my graduate studies in dynamical systems at Cornell in the late 1980s. There, under the guidance of Philip Holmes, Frank Moon, Richard Rand, and Andy Ruina, I was shown how the equations governing the motion of (often simple) mechanical systems featuring particles and rigid bodies could display surprisingly rich behavior.

Introduction

One of the main goals of this book is to enable the reader to take a physical system, model it by using particles or rigid bodies, and then interpret the results of the model. For this to happen, the reader needs to be equipped with an array of tools and techniques, the cornerstone of which is to be able to precisely formulate the kinematics of a particle. Without this foundation in place, the future conclusions on which they are based either do not hold up or lack conviction.

Much of the material presented in this chapter will be repeatedly used throughout the book. We start the chapter with a discussion of coordinate systems for a particle moving in a three-dimensional space. This naturally leads us to a discussion of curvilinear coordinate systems. These systems encompass all of the familiar coordinate systems, and the material presented is useful in many other contexts. At the conclusion of our discussion of coordinate systems and its application to particle mechanics, you should be able to establish expressions for gradient and acceleration vectors in any coordinate system.

The other major topics of this chapter pertain to constraints on the motion of particles. In earlier dynamics courses, these topics are intimately related to judicious choices of coordinate systems to solve particle problems. For such problems, a constraint was usually imposed on the position vector of a particle. Here, we also discuss time-varying constraints on the velocity vector of the particle.

Introduction

In this chapter, the balance law F = ma for a single particle plays a central role. This law is then used to examine models for several physical systems ranging from planetary motion to a model for a roller coaster. Our discussion of the behavior of these systems predicted by the models relies heavily on numerical integration of the equations of motion provided by F = ma, and it is presumed that the reader is familiar with the numerical integration of ordinary differential equations.

Two of the most important types of forces featured in many applications are conservative forces and constraint forces. For the former, the gravitational force between two particles is the prototypical example, whereas the most common constraint force in particle mechanics is the normal force. It is crucial to be able to properly formulate and represent conservative and constraint forces, and we will spend a considerable amount of time discussing them in this chapter. In contrast to most texts in dynamics, here we consider friction forces to be types of constraint forces.

For most applications, exact (or analytical) solutions are not available and recourse to numerical methods is often the only course of action. In validating these solutions, any conservations that might be present are crucial. To this end, conservations of momentum and energy are discussed at length and we also show (with the help of two examples) how angular momentum conservation can often be exploited.

Introduction

One of the key features of the rigid body dynamics problems that we will shortly examine is the presence of a variable axis of rotation. This is one of the reasons for the richness of phenomena in rigid body dynamics. It is also a reason why this subject is intimidating. To quote the mechanician Louis Poinsot (1777–1859), from, “… if we have to consider the motion of a body of sensible shape, it must be allowed that the idea which we form of it is very obscure.” In this chapter, several representations of rotations are discussed that will enable us to establish both a clear picture of rigid body motions and straightforward proofs of several major results. To this end, many results on two key kinematical quantities for rigid bodies, rotation tensors and their associated angular velocity vectors, are discussed in considerable detail.

The subject of rotations in rigid body dynamics has a wonderful history, a wide range of interesting results, and an impressive list of contributors. Here, however, space limits the presentation of only the handful of results that are most relevant to our purposes. From a historical perspective, much of what is presented was established by Leonhard Euler (1707–1783) in his great works on rigid body dynamics that started to appear in the 1750s. The foundations Euler established were built upon by such notables as Cayley, Gauss, Hamilton, and Rodrigues in the early part of the 19th century.

Introduction

In this chapter, several examples of systems of particles are discussed. We pay particular attention to how the equations of motion for these systems are established by use of Lagrange's equations. The examples we discuss are classical and range from simple harmonic oscillators to dumbbell satellites and pendula. Our goals are to illuminate the developments of the previous chapter and to present representative examples.

Examples that are closely related to the ones we discuss can be found in many dynamics texts. Most of these texts use alternative formulations of Lagrange's equations of motion that do not readily accommodate nonconservative forces. Here, because we have established an equivalence between Lagrange's equations of motion and the balances of linear momenta, we are easily able to incorporate nonconservative forces such as dynamic Coulomb friction. This chapter closes with a brief discussion of some recent works on the dynamics of systems of particles.

Harmonic Oscillators

We first consider simple examples involving a system of two particles. The system shown in Figure 5.1 is the first of several related systems that we discuss in this section.

Referring to the figure, we see that a particle of mass m1 is connected by a spring of stiffness K1 and unstretched length L1 to a fixed support. It is also connected by a spring of stiffness K2 and unstretched length L2 to a particle of mass m2.