KEY FEATURES

The key features of this chapter are the use of perturbation theory to solve weakly non-linear problems, the notion of phase space, the Poincaré–Bendixson theorem, and limit cycles.

In reality, most oscillating mechanical systems are governed by non-linear equations. The linear oscillation theory developed in Chapter 5 is generally an approximation which is accurate only when the amplitude of the oscillations is small. Unfortunately, non-linear oscillation equations do not have nice exact solutions as their linear counterparts do, and this makes the non-linear theory difficult to investigate analytically.

In this chapter we describe two different analytical approaches, each of which is successful in its own way. The first is to use perturbation theory to find successive corrections to the linear theory. This gives a more accurate solution than the linear theory when the non-linear terms in the equation are small. However, because the solution is close to that predicted by the linear theory, new phenomena associated with non-linearity are unlikely to be discovered by perturbation theory! The second approach involves the use of geometrical arguments in phase space. This has the advantage that the non-linear effects can be large, but the conclusions are likely to be qualitative rather than quantitative. A particular triumph of this approach is the Poincaré–Bendixson theorem, which can be used to prove the existence of limit cycles, a new phenomenon that exists only in the non-linear theory.

KEY FEATURES

The key features of this chapter are the angular momentum principle and conservation of angular momentum. Together, the linear and angular momentum principles provide the governing equations of rigid body motion.

This chapter is essentially based on the angular momentum principle and its consequences. The angular momentum principle is the last of the three great principles of multiparticle mechanics that apply to every mechanical system without restriction. Under appropriate conditions, the angular momentum of a system (or one of its components) is conserved, and we use this conservation principle to solve a variety of problems.

Together, the linear and angular momentum principles provide the governing equations of rigid body motion; the linear momentum principle determines the translational motion of the centre of mass, while the angular momentum principle determines the rotational motion of the body relative to the centre of mass. In this chapter, we restrict our attention to the special case of planar rigid body motion. Three-dimensional motion of rigid bodies is considered in Chapter 19.

THE MOMENT OF A FORCE

We begin with the definition of the moment of a force about a point, which is a vector quantity. The moment of a force about an axis, a scalar quantity, is the component along the axis of the corresponding vector moment.

Definition 11.1 Moment of a force about a pointSuppose a force F acts on a particle P with position vector r relative to an origin O.

KEY FEATURES

The key features of this chapter are the existence of small oscillations near a position of stable equilibrium and the matrix theory of normal modes. A simpler account of the basic principles is given in Chapter 5.

Any mechanical system can perform oscillations in the neighbourhood of a position of stable equilibrium. These oscillations are an extremely important feature of the system whether they are intended to occur (as in a pendulum clock), or whether they are undesirable (as in a suspension bridge!). Analogous oscillations occur in continuum mechanics and in quantum mechanics. Here we present the theory of such oscillations for conservative systems under the assumption that the amplitude of the oscillations is small enough so that the linear approximation is adequate. A simpler account of the theory is given in Chapter 5. This treatment is restricted to systems with two degrees of freedom and does not make use of Lagrange's equations. Although the material in the present chapter is self-contained, it is helpful to have solved a few simple normal mode problems before.

The best way to develop the theory of small oscillations is to use Lagrange's equations. We will show that it is possible to approximate the expressions for T and V from the start so that the linearized equations of motion are obtained immediately. The theory is presented in an elegant matrix form which enables us to make use of concepts from linear algebra, such as eigenvalues and eigenvectors.

KEY FEATURES

The key features of this chapter are the rules of vector algebra and differentiation of vector functions of a scalar variable.

This chapter begins with a review of the rules and applications of vector algebra. Almost every student taking a mechanics course will already have attended a course on vector algebra, and so, instead of covering the subject in full detail, we present, for easy reference, a summary of vector operations and their important properties, together with a selection of worked examples.

The chapter closes with an account of the differentiation of vector functions of a scalar variable. Unlike the vector algebra sections, this is treated in full detail. Applications include the tangent vector and normal vector to a curve. These will be needed in the next chapter in order to interpret the velocity and acceleration vectors.

VECTORS AND VECTOR QUANTITIES

Most physical quantities can be classified as being scalar quantities or vector quantities. The temperature in a room is an example of a scalar quantity. It is so called because its value is a scalar, which, in the present context, means a real number. Other examples of scalar quantities are the volume of a can, the density of iron, and the pressure of air in a tyre. Vector quantities are defined as follows:

Definition 1.1 Vector quantityIf a quantity Q has a magnitude and a direction associated with it, then Q is said to be a vector quantity.

KEY FEATURES

The key concepts in this chapter are the velocity and acceleration of a particle and the angular velocity of a rigid body in planar motion.

Kinematics is the study of the motion of material bodies without regard to the forces that cause their motion. The subject does not seek to answer the question of why bodies move as they do; that is the province of dynamics. It merely provides a geometrical description of the possible motions. The basic building block for bodies in mechanics is the particle, an idealised body that occupies only a single point of space. The important kinematical quantities in the motion of a particle are its velocity and acceleration. We begin with the simple case of straight line particle motion, where velocity and acceleration are scalars, and then progress to three-dimensional motion, where velocity and acceleration are vectors.

The other important idealisation that we consider is the rigid body, which we regard as a collection of particles linked by a light rigid framework. The important kinematical quantity in the motion of a rigid body is its angular velocity. In this chapter, we consider only those rigid body motions that are essentially two-dimensional, so that angular velocity is a scalar quantity. The general three-dimensional case is treated in Chapter 16.

STRAIGHT LINE MOTION OF A PARTICLE

Consider a particle P moving along the x-axis so that its displacement x from the origin O is a known function of the time t.

KEY FEATURES

The key features of this chapter are generalised coordinates and configuration space, the derivation and use of Lagrange's equations, the Lagrangian, and the connection between symmetry of the Lagrangian and conservation principles.

Lagrange's equations mark a change in direction in our development of mechanics. Building on the work of d'Alembert, Lagrange devised a general method for obtaining the equations of motion for a very wide class of mechanical systems. In earlier chapters we have used conservation principles for this purpose, but there is no guarantee that enough conservation principles exist. In contrast, Lagrange's method is completely general and is not restricted to problems soluble by conservation principles. The method is so simple to apply that it is quite possible to solve complex mechanical problems whilst knowing very little about mechanics! However, the supporting theory has its subtleties.

Lagrange's equations also mark the beginning of analytical mechanics in which general principles, such as the connection between symmetry and conservation principles, begin to take over from actual problem solving.

CONSTRAINTS AND CONSTRAINT FORCES

A general mechanical systemS consists of any number of particles P1, P2, …, PN. The particles of S may have interconnections of various kinds (light strings, springs and so on) and also be subject to external connections and constraints. These could include features such as a particle being forced to remain on a fixed surface or suspended from a fixed point by a light inextensible string.

A principal objective of any theory of fluid motion is the prediction of the spread of matter or “tracer” within the fluid. The problem is trivial for the fluid particles themselves in steady flow: they follow streamlines. It is nontrivial if the motion is time dependent, or if the tracer is dissolved in the fluid but diffusing through it. The time dependence of general interest is turbulence. The next four chapters develop a coherent framework for considering inhomogeneous and nonstationary turbulence, with elaboration in detail for the homogeneous, stationary and incompressible case, excluding and including tracer diffusion. Applications to the spread of phytoplankton are of special interest to oceanographers; these marine organisms are modeled as reacting tracers having nonlinear reaction rates. Absolute dispersion is considered first. This is the problem of predicting the path of a single fluid particle, or the path of the centroid of a cluster of particles, in turbulent flow. Turbulence being conceived as a random process, the problem is the prediction of the probability distribution function or pdf for the particle path. The mathematical difficulty is the closure of the infinite heirarchy of moments of the nonlinear kinematics, that is, the relating of certain high-order moments of particle displacement to low-order moments. There are any number of workings of this task in the literature, most of which close at second order, that is, second moments are related to first.