Page 20 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 1.1 Harmonic Motion

Figure 1.1-2. Harmonic motion as a projection of a point moving on a
circle.
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. 1.1-2. With the
angular speed of the line 0-p designated by (o, the displacement x can be written
as
x= A s\n o )t (1.1-2)
The quantity o) is generally measured in radians per second, and is referred to as
the circular frequency Because the motion repeats itself in 27t radians, we have
the relationship
2tt
0) = = 277/ (1.1-3)
where r and / 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. (1.1-2). Using the dot notation for the derivative, we
obtain
X = (i)A cos (ot = (oA sin {o)t tt/2)

X = A sin (ot = o)M sin {(ot + tt) (1.1-5)
Thus, the velocity and acceleration are also harmonic with the same frequency of
oscillation, but lead the displacement by 77/2 and tt radians, respectively. Figure
1.1-3 shows both time variation and the vector phase relationship between the
displacement, velocity, and acceleration in harmonic motion.
Examination of Eqs. (1.1-2) and (1.1-5) reveals that
X = -o)^x (11-6)

so that in harmonic motion, the acceleration is proportional to the displacement
and is directed toward the origin. Because Newton’s second law of motion states

^The word circular is generally deleted, and o> and / are used without distinction for frequency.