13.1: Introduction
In our daily life there are many types of periodic phenomena. The day-night rhythm, the waves on the sea, the trees swaying in the wind, a child going back and forth on a swing and the beating of our heart. Any type of periodic motion is called an oscillation. If an oscillation is of a mechanical nature, it is called a vibration.
Vibrations can be very useful, for example in a pendulum or quartz clock to keep track of time, or for generating sound tones with a speaker or musical instrument. Vibrations can also be very detrimental in the case of shaking buildings excited by earthquakes or in the uncontrolled resonant amplitude increase (flutter) of aircraft wings. It is therefore important for engineers to have good models for vibrations, both to use them for engineering vibrating devices, and for engineering methods to prevent their detrimental effects.
In this chapter we will discuss several types of vibrations that are a direct consequence of the combination of Newton’s second law and the properties of a position or velocity dependent restoring force like that of a spring or a damper. When writing down the equation of motion for such a system, according to the methods discussed in Ch. 6 , a second-order differential equation arises that we have not yet encountered before. It should be noted that analysis of vibrations can be performed using the methods of kinetics that we have already dealt with before, the main challenge in this chapter is solving the special types of differential equations that arise in vibration problems.