Complex Notation and Symbolic Methods

Abstract

The mathematical description of the vibration of a complex structure is very complicated because of the many modes of motion in which the structure can respond. However, we are primarily interested in the periodic vibrations that are excited by harmonically varying forces and in the building up and decay of such vibrations, since the response of a vibrator to unsteady forces can be deduced from its response to harmonic forces with the aid of Fourier analysis or the Laplace transform. The harmonic sine and cosine functions are very inconvenient because of the complex addition and multiplication theorems and the complexity of all the other theorems that apply to these functions. Fortunately, it is possible to eliminate sine and cosine completely by introducing rotating vectors as the primary variables. The sine and cosine can be defined as the projections of such a vector on the vertical and horizontal axis, respectively, both being plotted as functions of the angle of this vector with the horizontal axis according to the procedure shown in Fig. 1. A condensed notation can be introduced by replacing the sinusoidal functions by rotating vectors and by considering these rotating vectors as the primary variables. The harmonic functions can then be easily reconstructed from the rotating vectors.