Engineering Expert Witness Blog

      Last time we learned how spur gears mesh together to form a gear train and we examined a train consisting of just two gears, one being the driving gear, the other the driven gear.   Today we’ll take a look at the geometry behind the smooth functioning of modern spur gear teeth when we identify their peculiar shape to be that of an involute curve.

      The curved profile of spur gear teeth conforms to a type of mathematical curve found in geometry known as an involute.   The involute profile of a spur gear tooth is shown in red below.

      The mathematical notion of the involute was first presented in 1673 by Dutch mathematician Christiaan Huygens, in his book, Horologium Oscillatorium.   Huygens’ book presents his studies on clock pendulums and the applied mathematics he used in an effort to predict their often erratic motion on ships at sea.   His book ultimately dealt with far more than this, resulting in a treatise on the mathematical properties of the involutes of curves.

      To see how an involute curve is formed, we’ll conduct a simple experiment.    One end of string is attached with a tack to a circular object, like the yellow rod shown in the following illustration.   The other end of string has a red ball attached to it.

Forming An Involute Curve

      If we grab the ball and pull the string taught while wrapping the string around the rod, the ball’s path will form an involute curve due to the incremental shortening of the string that occurs as it wraps around the rod.

      Next time we’ll see how the involute profile of gear teeth contributes to efficient mechanical energy transmission in gear trains.

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