Читать книгу Engineering Acoustics - Malcolm J. Crocker - Страница 40

So far we have examined the displacement y of a point. Note that, when the displacement is in the OY direction, we say it is positive; when it is in the opposite direction to OY, we say it is negative. Displacement, velocity, and acceleration are really vector quantities in mathematics; that is, they have magnitude and direction. The velocity v of a point is the rate of change of position with time of the point x in m/s. The acceleration a is the rate of change of velocity with time. Thus, using simple calculus:

(2.4)

and

(2.5)

Equations are plotted in Figure 2.4.

Figure 2.4 Displacement, velocity, and acceleration.

Note, by trigonometric manipulation we can rewrite Eqs. (2.4) and (2.5) as (2.6) and (2.7):

(2.6)

and

(2.7)

and from Eq. (2.3) we see that a = −ω² y.

Equations tell us that for simple harmonic motion the amplitude of the velocity is ω or 2πf greater than the amplitude of the displacement, while the amplitude of the acceleration is ω² or (2πf)² greater. The phase of the velocity is π/2 or 90° ahead of the displacement, while the acceleration is π or 180° ahead of the displacement.

Note we could have come to the same conclusions and much more quickly if we had used the complex exponential notation. Writing

then

and