Friday, July 31, 2009

How to power a frisbee

I was wondering the other day how a free standing object like a frisbee, or a rotating space station, can be made to spin (where it counts) at variable rates in an efficient manner. By "variable rates", I mean that the thing can be made to spin fast, then slow, and then fast again, for example. By "an efficient manner", I mean a mechanism that doesn't cause significant mass loss and which therefore conserves the angular momentum of the whole system.

So the idea here is that if there is one component of the thing that's spinning in one direction (the outer hull of the frisbee) then there must be another component spinning in the opposite direction (e.g. an internal flywheel). If you use only one internal flywheel, then the only way to slow the rate the spinning is to slow the internal flywheel's spinning by braking against the outer hull that is spinning in the opposite direction. A better solution (no doubt already discovered by some gyro tinkerer from the 18th century) is to use 2 flywheels: instead of having to change the magnitude of the spinning, you change the orientation of the flywheels. Here's how it works.

The schematic is a side view of the contraption. At the center of the device we have an "engine" from which 3 arms extend. One arm is fixed to the engine and represents the thing that is to be spun (labeled "power train"): the engine will spin along with the thing it's spinning. The other 2 are affixed with identical flywheels at the ends and spin along the axes of their respective arms.

The engine controls the rate of rotation of the "power train" by two means: one, by controlling the rate at which the 2 flywheels spin, and two, by changing the angle between the arms of the flywheels. (The 3 arms always lie in a same plane relative to each other in, as we shall see, the rotating reference frame of the engine).

Using those 2 levers of control, we may employ a variety of strategies to make the "power train" spin. For example, starting at rest (top figure in drawing), we can arrange the arms of the flywheels (initially not spinning) along a straight line perpendicular to the power train (shown horizontally in the figure). The engine could then expend energy to make the two flywheels spin in opposite directions in such a way their net angular momenta cancel each other. So in this configuration as the engine stores rotational energy in the flywheels, the engine and its attached "drive train" remain at rest.

After the flywheels each have absorbed a sufficient amount of kinetic energy, their angular momenta can be transfered to the power train by drawing the arms attached to the flywheels inwards--towards the engine and therefore the axis of the power train arm. The two arms of the flywheels, each subtended at an equal angle from the horizontal, induce a rotation on the engine and the drive shaft (power train) it is affixed to such that angular momentum of the whole system is conserved (see lower figure in drawing). (Given the initial conditions of our example, the angular momentum of the whole system adds up to zero.)

For a frisbee, of course, some of the angular momentum would be dissipated into the environment (the surrounding air) in order to create lift. So in a real application, the angular momentum of the whole system is not conserved. Nevertheless, the same principles can be applied. To get a better appreciation of the dynamics of the system, it would be instructive to look at the system's Lagrangian--which doesn't seem too complicated. That's a task left to the reader or a later article.

So why do I think this is interesting? Mostly because it avoids using gears and such for directing rotational motion: instead, it uses inertial forces. Compared to the teeth of a gear, for example, inertial forces can be more evenly distributed over a larger volume of material. This suggests we could achieve greater acceleration (or deceleration) for spinning the outer of hull of our hypothetical frisbee: the achievable angular acceleration is more likely bounded by the structural limits of the whole system, than by the limits of some small component of it (e.g. a tooth in a gear).
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