CHAPTER 5 - SMALL OSCILLATIONS

Abstract

Publisher Summary A common form of motion of mechanical systems is small oscillations of a system about a position of stable equilibrium. This chapter reviews the simplest case, that is, a system with only one degree of freedom. It discusses free oscillations in one dimension. The frequency is a fundamental characteristic of the oscillations and is independent of the initial conditions of the motion. The chapter discusses the oscillations of a system on which a variable external force acts. These are called forced oscillations. The energy of a system executing forced oscillations is naturally not conserved because the system gains energy from the source of the external field. The chapter discusses the oscillations of systems with more than one degree of freedom; for example, damped oscillations are discussed in the chapter. The theory of forced oscillations under friction is entirely analogous to that given in forced oscillations without friction. The entire theory of small oscillations is based on the expansion of the potential and kinetic energies of the system in terms of the coordinates and velocities, retaining only the second-order terms. The equations of motion are then linear and in this approximation, it is called linear oscillations. Although such an expansion is entirely legitimate when the amplitude of the oscillations is sufficiently small, in higher approximations called anharmonic or non-linear oscillations, some minor but qualitatively different properties of the motion appear.