So far, we have been looking at the displacement of a vibrating object as a measure of its vibration amplitude. The displacement is simply the distance from a reference position, or equilibrium point. In addition to varying displacement, a vibrating object will experience a varying velocity and a varying acceleration. Velocity is defined as the rate of change of displacement, and in the English system is usually measured in units of inches per second. Acceleration is defined as the rate of change of velocity, and in the English system, is usually measured in units of G, or the average acceleration due to gravity at the earth's surface.
The displacement of a body undergoing simple harmonic motion is a sine wave as we have seen. It also turns out (and is easily proved mathematically), that the velocity of the motion is sinusoidal. When the displacement is at a maximum, the velocity will be zero because that is the position at which its direction of motion reverses. When the displacement is zero (the equilibrium point), the velocity will be at a maximum. This means that the phase of velocity waveform will be displaced to the left by 90 degrees compared to the displacement waveform. In other words, the velocity is said to lead the displacement by a 90-degree phase angle.
Remembering that acceleration is the rate of change of velocity, it can be shown that the acceleration waveform of an object undergoing simple harmonic motion is also sinusoidal, and also that when the velocity is at a maximum, the acceleration is zero. In other words, the velocity is not changing at this instant. Then, when the velocity is zero, the acceleration is at a maximum -- the velocity is changing the fastest at this instant. The sine curve of acceleration versus time is thus seen to be 90 degrees phase shifted to the left of the velocity curve, and therefore acceleration leads velocity by 90 degrees.
These relationships are shown here:
Note here that the acceleration is 180 degrees out of phase with the displacement. This means the acceleration of a vibrating object is always in the opposite direction to the displacement!
It is possible to define another parameter that is the rate of change of acceleration, and it is called "jerk". Jerk is what you feel when your car comes to a stop if you maintain a constant brake pedal pressure. It is really the sudden cessation of the deceleration. Elevator manufacturers are interested in measuring jerk, for it is the variation in acceleration that elevator passengers are especially sensitive to.
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