Machine Design

The third derivative of position is jerk

Ask someone to list the greatest inventions and discoveries of all time, and they are likely to come up with such things as fire, the wheel, electricity, and perhaps penicillin.

In my mind, however, one item that has to be on the list is calculus.

The primary reason engineering and science have progressed rapidly is our ability to model natural phenomena using mathematical means. We can define problems with symbols, manipulate the symbols, often without really understanding the principles behind what we are doing, and come up with equations that provide the answers we need. Of course, algebra will also let you do this, but calculus puts mathematical modeling on a dynamic footing. Where algebra provides a snapshot, calculus provides a movie.

There is a crucial divide that students must cross to comprehend calculus. It is called the delta process where, almost in Harry Potter fashion, the student passes through a wall from algebra into the mystical world of math that moves.

I will admit that although I've tried to understand the delta process time and time again, I never could get a handle on it. Luckily, it is not really necessary to understand the conceptual roots of calculus. Like most students, I learned to use calculus by rote.

This issue of our magazine is a special supplement on motion control, and where would motion control be without calculus? The most vivid recollection I have of college calculus is how it reveals the relationships between position, velocity, and acceleration. They, in turn, reveal what motion-control systems need in terms of force, torque, and power.

In fact, motion control probably provides the best examples of calculus. If you are a little rusty on the fundamentals, I'll remind you that plotting position against time gives you a graph where the slope of the curve represents velocity. Then if you plot velocity over time, the slope of that curve shows acceleration, often a key for determining power requirements. In calculus jargon, the slopes of the curves are established by a process called differentiation.

Inertial guidance, on the other hand, works backwards from acceleration. If you arrange three accelerometers on perpendicular axes, you can plot acceleration against time in each axis. The area under the acceleration curve gives you velocity against time in each direction. And the area under the velocity versus time curve gives you position. If you know where you started from, then you know where you are --- or where your missile is --- from the position plot. The calculus term for computing the area under a curve is called integration.

Interestingly, you can carry differentiation beyond acceleration. A number of years ago, engineers at General Motors were trying to find the analytical foundations for what passengers considered a comfortable ride in a vehicle. They assumed that minimizing vertical acceleration was the key, but road testing said otherwise. They found that the rate of change of acceleration, or the third derivative of position, was the key factor. This was new mathematical ground, and GM didn't know what to call the derivative of acceleration. Finally, someone suggested the term "jerk," and that is what stuck, at least for a while.

--- Ronald Khol, Editor

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