October 30, 2007

Fig. 1: A flying disk. |

The flying disc dates back to the ancient Indian chakram, early versions of olympic discus, and clay targets used for trapshooting. The flying disc was popularized by the toy company Wham-O with its introduction of the trademarked "Frisbee" in 1957. This remarkable feat of aerodynamics and gyroscopic stability was discovered rather than predicted. The physics of its flight is subtle and remarkable.

To first approximation, a flying disc is simply an axi-symmetric wing with an elliptical cross-section. Of course, most ordinary wings would be unstable if simply thrown through the air, but the essential mechanics of its lift are mostly ordinary. The lift on a body is described by the lift equation,

Where *A* is the cross-sectional area, ρ is
the density, *V* is the free-stream velocity, and
*C _{L}* is the coefficient of lift, a semi-empirical
constant that is a function of the shape of the object and its angle of
attack with respect to the free-stream velocity of the fluid. Typically

The coefficient of lift at zero angle of attack is
often denoted *C _{L0}*, and the increase of

Fig. 2: Lift to drag ratio versus angle of
attack in degrees. |

The drag on the flying disc is defined in a fashion similar to that of the disk. The equation is now

The only difference from the lift is that the
coefficient of lift has been replaced by the analogous coefficient of
drag. Unlike lift, drag typically depends on the angle of attack
quadratically. So now, *C _{D} = C_{D0} +
C_{Dα}α^{2}*, and the empirical values
have been found to be approximately

The lift-to-drag ratio *L/D* is shown in figure
two. For the simple lift and drag relations presented above, this ratio
is simply the ratio of the coefficient of lift to the coefficient of
drag using the empirical values. Note the maximum at around 15 degrees;
this is commonly observed in flying discs.

For a wing with an elliptical cross-section, the center of pressure due to lift is offset ahead of the center of gravity. Therefore, if one were to simply throw a flying disc, the lift would also cause a moment on the disc and cause it to flip over backwards. The key to the stability of the flying disc is its spin. The spin of the disc results in gyroscopic stability or pitch stiffness, and the greater the speed, the greater the stability.

Typically the moment due to lift and drag pressure on the disc is nearly perpendicular to the angular momentum of the spinning disc, and thus the disc experiences gyroscopic precession. The freqency of precession is

Where *M*is the moment, *I *is the moment
of inertia of the disc about its axis of symmetry, and ω is the
angular frequency of the disc's spin. This precession causes the disc to
wobble as a gyroscope wobbles when its axis of spin is perturbed from
the direction of gravity. Likewise, by spinning the disc, one trades
roll stability for pitch stiffness.

The equations of motion of the system, accounting for the external forces and moments is found to be [2]

and

The details of *F* and *M* depend on the
aerodynamics of the disc. The typical mass of a flying disc is between
90g and 175g. The lighter discs maximize duration of flight, and the
heavier discs will maximize distance thrown and minimize the effects of
wind and stray currents. It is also apparent from the equations of
motion that a greater moment of inertia* I* would also increase
stability.

It is also worth noting that the viscous no-slip
condition at the boundary of the spinning disc causes the disc to
generate some degree of vorticity. The circulation about the disc and
the free-stream flow of air past the disc causes a force in the
direction of the cross product of *V* with the angular momentum of
the disc. This is attributed to the *Magnus Effect*, which is
caused by one side of the disc percieving a higher free-stream velocity
than the other, causing a pressure gradient. This will cause a flying
disc thrown clockwise to veer to the left, which is particularly
noticable as the viscous effects become more pronounced at the end of
the flight. It is this same effect that causes a ping pong ball to
travel along a curved path when a skilled player puts spin on it with
the paddle.

Fig. 3: a) A flying disc without top grooves.
b) A flying disc with top grooves. |

The thick rim at the edge of flying disc serves multiple important purposes. First and foremost, the thick rim eases gripping and tossing the disc. Without the thick rim, throwing a flying disc would be significantly more difficult.

Additionally, the thick rim significantly increases the moment of inertia of the disc about the axis of symmetry, enhancing the stability of the disc. A flat plate without a thick rim, such as a dinner plate, has much less stability than a typical flying disc.

Finally, the cupped region on the bottom of the disc substantially increases the coefficent of drag from the vertical profile, while the horizonal profile is still somewhat streamlined. As a result, as the disc begins to fall, the cupped profile behaves like a parachute, and the horizontal component of drag dwarfs the vertical component. This allows the flying disc to be thrown much further than a ball of equivalent velocity.

Upon examing the top surface of a commercial flying disc, it is typical to find several concentric grooves. These grooves serve to trigger turbulence at the leading edge, the same principle behind a gurney flap on an airfoil . The turbulent region helps to keep the boundary layer of the flow over the top of the disc attatched to the disc, substantially increasing lift. This effect also allows the disc to be thrown at a higher angle of attack before it stalls. A stall occurs when the flow seperates from an object, causing a catastrophic decrease in lift and increase in drag. One can cause a frisbee to stall by simply throwing it at a very high angle of attack. The disc will quickly destabilize and fall to the ground.

© 2007 Anthony Scodary. 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.

[1] S. Hummel and M. Hubbard, "Identification of Frisbee Aerodynamic Coefficients using Flight Data," The International Conference on the Engineering of Sport, (Kyoto, Japan, Sept. 2002).

[2] R. D. Lorenz, "Flight Dynamics Measurements
on an Instrumented Frisbee," *Measurement Science and
Technology*, (2005)

[3] J. R. Potts and W. J. Crowther, "Visualisation of the Flow Over a Disc-wing," 9th Intl. Symp. on Flow Visualization, (2000).

[3] J. R. Potts and W. J. Crowther, "Frisbee(TM) Aerodynamics," 20th AIAA Applied Aerodynamics Conference and Exhibit, (St. Louis, Missouri, 2002).