The theory of vibration which is presented in the previous chapters relates to an idealised vibrating system having a finite number of degrees of freedom. This chapter is concerned mainly with continuous systems, that is, systems with an infinity of degrees of freedom. First it will be shown how receptances may be defined for continuous systems, then it will be shown how composite systems, whether continuous or discrete, may be analysed by the use of matrices. Finally, it will be shown how the theory developed for systems with a finite number of degrees of freedom may be applied to the vibration of continuous systems.

Throughout this chapter only undamped vibration will be discussed. But the reader will observe that much of the analysis may be generalised to cover damped vibrating systems. Throughout this chapter, therefore, all displacements and forces will be harmonic, and will have the same frequency and phase.

The receptances of continuous systems

The receptances of a discrete system—that is, a system with a finite number of degrees of freedom—were defined in § 2.2. Although they were first introduced as elements of the receptance matrix, their fundamental meaning is such that the receptance αxy gives the displacement at the coordinate x due to a harmonic force at the coordinate y.