Chapter 1: Mathematics of Vibrating Systems

Many mathematical tools can be used to analyze vibrational systems. One of the first mathematical tools a neophyte engineer learns is calculus. The basics of limits, differentiation, and integration permeate all of engineering mathematics. This chapter offers a cursory review of these topics and uses the mathematics of vibrations to demonstrate how the concepts operate. For more specific information on all these topics, consult relevant sections of this Handbook.

Many of the mathematical tools engineers use to evaluate and predict behavior, such as vibrations, require equations that have continuously varying terms. Often, there are many terms regarding the rate of change, or the rate of change of the rate of change, and so forth, with respect to some basis. For example, a velocity is the rate of change of distance with respect to time. Acceleration is the rate of change of the velocity, which makes it the rate of change of the rate of change of distance with respect to time. Determining the solutions to these types of equations is the basis of differential calculus.