Equations of Motion
If the position, or displacement, of an object undergoing simple harmonic motion is plotted versus time on a graph as shown above, the resulting curve is a sine wave,or sinusoid, and is described by the following equation:
where d = instantaneous displacement,
D = maximum, or peak, displacement
= angular frequency, = 2f
t = time
This is the same curve that the sine function from trigonometry generates, and it can be considered the simplest and most basic of all possible repetitive wave forms. The mathematical sine function is derived from the relative lengths of the sides of a right triangle, and the sine wave is actually a plot of the value of the sine function versus angle. In the case of vibration, the sine wave is plotted as a function of time, but one cycle of the waveform is sometimes considered to equal 360 degrees of angle. More will be said about this when we consider the subject of phase.
The velocity of the motion described above is equal to the rate of change of the displacement, or in other words how fast its position is changing. The rate of change of one quantity with respect to another can be described by the mathematical derivative, as follows:
where v = instantaneous velocity.
Here we see that the form of the velocity function is also sinusoidal, but because it is described by the cosine, it is displaced by 90 degrees. We will see the significance of this in a moment.
The acceleration of the motion described here is defined as the rate of change of the velocity, or how fast the velocity is changing at any instant:
where a = instantaneous acceleration.
Note here also that the acceleration function is displaced by an additional 90 degrees, as indicated by the negative sign.
If we examine these equations, it is seen that the velocity is proportional to the displacement times the frequency, and that the acceleration is proportional to the frequency squared times the displacement. This means that at a large displacement and a high frequency, very high velocities result, and extremely high levels of acceleration would be required. For instance, suppose that a vibrating object is undergoing 0.1 inch of displacement at 100 Hz. The velocity equals displacement times frequency, or
Acceleration equals displacement times frequency squared, or
a = 0.1 x 10000 = 1000 inches per second per second.
One G of acceleration equals 386 inches per second per second, so this acceleration is
Now, see what happens if we raise the frequency to 1000 Hz:
Thus, we see that in practice, high frequencies can not be associated with high displacement levels.
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