Harmonic Motion

Oscillatory motion may repeat itself regularly, as in the balance wheel of a watch, or display considerable irregularity, as in earthquakes. When the motion is repeated in equal intervals of time T, it is called period motion. The repetition time t is called the period of the oscillation, and its reciprocal, ,is called the frequency. If the motion is designated by the time function x(t), then any periodic motion must satisfy the relationship .

Harmonic motion is often represented as the projection on a straight line of a point that is moving on a circle at constant speed, as shown in Fig. 1. With the angular speed of the line o-p designated by w , the displacement x can be written as

(1)

Figure 1 Harmonic Motion as a Projection of a Point Moving on a Circle

 

The quantity w is generally measured in radians per second, and is referred to as the angular frequency. Because the motion repeats itself in 2p radians, we have the relationship

(2)

where t and f are the period and frequency of the harmonic motion, usually measured in seconds and cycles per second, respectively.

The velocity and acceleration of harmonic motion can be simply determined by differentiation of Eq. 1. Using the dot notation for the derivative, we obtain

(3)

(4)