Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 38

The motion of vibrating systems such as parts of machines, and the variation of sound pressure with time is often said to be simple harmonic. Let us examine what is meant by simple harmonic motion.

Suppose a point P is revolving around an origin O with a constant angular velocity ω, as shown in Figure 2.1.

Figure 2.1 Representation of simple harmonic motion by projection of the rotating vector A on the X‐ or Y‐axis.

If the vector OP is aligned in the direction OX when time t = 0, then after t seconds the angle between OP and OX is ωt. Suppose OP has a length A, then the projection on the X‐axis is A cos(ωt) and on the Y‐axis, A sin(ωt). The variation of the projected length on either the X‐axis or the Y‐axis with time is said to represent simple harmonic motion.

It is easy to generate a displacement vs. time plot with this model, as is shown in Figure 2.2. The projections on the X‐axis and Y‐axis are as before. If we move the circle to the right at a constant speed, then the point P traces out a curve y = A sin(ωt), horizontally. If we move the circle vertically upwards at the same speed, then the point P would trace out a curve x = A cos(ωt), vertically.

Figure 2.2 Simple harmonic motion.