﻿ Professor Robert B. Laughlin, Department of Physics, Stanford University

# Sperry Gyrostabilizer

## Ben Shank October 31, 2007

### (Submitted as coursework for Physics 210, Stanford University, Fall 2007)

Most people who were once children are familiar with the gyroscopic effect. A spun top, provided it is spun fast enough, will sit contentedly upright or coast smoothly across the surface of a table in spite of the downward pull of gravity. When friction finally slows it, it falls. But rather than simply tipping over it will wobble increasingly rapidly before finally wiping out. What keeps it upright at first and then makes its death throes so violent is the gyroscopic effect. Any spinning body is, strictly speaking, a gyroscope. However, good gyroscopes are characterized by their large moment of inertia (I). Moment of inertia can be thought of as an object's "rotational mass." To within a constant, an object's moment of inertia is its mass times the square of its radius. This constant is largest when most of the mass is far from the center of rotation, so a thin hoop or cylindrical shell has the largest inertial moment for a given mass and radius. For this reason, commercial gyroscopes most often take these shapes.

A spinning gyroscope stores energy proportional to its moment of inertia and the square of its angular velocity ω. This can be thought of in analogy to the energy stored by any moving body. In a linear system, kinetic energy is mv2/2. In a rotating system it is Iω2/2. However the rotating system is not 'going anywhere' and is therefore a much more convenient place to store kinetic energy. Indeed the first human-made gyroscopes were used not for their gyroscopic properties, but for storing up energy from a river to grind grain or for smoothing out the energy input of a foot pedal as it turned a potter's wheel. Because the mass of a gyroscope is in motion, each bit of it carries momentum. The motion is in all different directions at once, so the net momentum is zero. However the angular momentum (L) of the device is defined as the product of its moment of inertia and its angular velocity. Again there is a parallel with linear mechanics. Instead of linear momentum p = mv, a gyroscope is described by its angular momentum L = Iω. To clarify the orientation of the gyroscope, L is usually made a vector with direction normal to the plane of rotation in the direction the thumb of your right hand would point if your fingers followed the direction of spin. Just as a large force is needed to alter the course of an object with high momentum, a large torque N is needed to change the orientation of a gyroscope with high angular momentum. Thus a top which is spinning very fast will drift across a table without tipping, resisting the lure of gravity to do so.

The similarity between force and torque goes a step further. Just as force is defined as the rate of change of linear momentum, torque is defined as the rate of change of an object’s angular momentum. However angular momentum is defined perpendicular to the direction of motion. So when you spin a top about its axis, the torque you apply is along the axis. If you were to push over a motionless top, the torque you applied would technically be perpendicular to the direction you pushed. Ordinarily this is a meaningless semantic, but when the top already has a large angular momentum, your push only causes a small change in orientation. Contrary to intuition, the top will not tip the way you pushed it, but in a perpendicular direction. This unexpected consequence of a gyroscope's "hidden" energy of rotation is called precession. The angular rate of precession &Omega is given by ω = N/L. In this equation only that component of torque perpendicular to L is counted. This is why a top that is slightly off-vertical precesses slowly, but then wobbles faster and faster as it falls and gravity becomes increasingly destabilizing.

Most gyroscopic devices make use of their tendency to maintain their orientation. With proper precautions to isolate it from its environment, a gyroscope can point in one direction for years. This is a boon to navigators, from the pilots of ships to the designers of the GPS system, and surveyors, from markers of property lines to mappers of the cosmos. However shortly before World War I, Elmer Ambrose Sperry decided to bring the gyroscope out of its gimbaled isolation and put it to work. He sought to correct wave-induced roll of ocean-going vessels. As waves rock a ship, the deck naturally tilts. This can be uncomfortable for passengers, but most people overcome sea-sickness within a few days. But it was becoming a permanent headache for naval gunners. In the first few decades of the twentieth century, naval cannons were making rapid advances in power and range. This meant that increasingly smaller angular deviations would cause them to miss their targets. Ocean waves, with a period of several seconds, seemed designed to be maximally troublesome to the targeting process. Sperry's solution involved a gigantic gyroscope, weighing several tons, in the hold of the ship. Its axis would be mounted vertically and allowed to tilt along the beam of the ship, but not side to side. As a wave began to roll the vessel, an electric motor would (attempt to) torque the axis either fore or aft, as needed. This would result in a sideways precession to counteract the impact of the wave and keep the ship upright and stable.

Although the Sperry Gyrostabilizer profoundly affected the history of nautical and aeronautical control, surprisingly few were sold in the original configuration. The US Navy purchased a few gyrostabilizers prior to WWI, but thereafter became primarily interested in gyroscopic stabilization for individual weapon platforms. A handful of luxury cruise liners purchased gyrostabilizers, but several generations of improvement would be necessary to make gyro stabilization attractive to the average captain. After the war, Sperry combined his stabilizer with various gyroscopic sensors to create the first fully automated piloting systems for ships and later for aircraft. Although run by immensely more complex electronic computers, modern autopilot systems still employ this basic control and feedback method. Because the original gyrostabilizers were very heavy and space-consuming, modern systems use a smaller gyroscope, sometimes only as a sensor, with actual pitch correction accomplished mainly by computer-controlled flaps or extendable fins. The Sperry Gyroscope Company, originally founded in 1910 to manufacture Sperry's gyrocompass, persisted under various names until 1986, when it merged with Burroughs Corporation to become Unisys. Elmer Sperry died in Brooklyn in June 1930 immensely wealthy and with over 400 patents to his name.

© 2007 Benjamin Shank. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.