George wrote:Your figure is excellent, thanks for posting that!
No problem. It is part of a more complete write-up I've been doing on my free time, to summarize and teach the knowledge I've gained about disc flight, which I'll post publicly when I finish (or maybe before, for comments). I'm also going to begin simulating disc flights, and sharing that info as well. This is relative straightforward to do, I only need to code up a basic time integration (Runge-Kutta should be more than adequate).
I've been inspired by some work done at UC Davis, described here...http://mae.ucdavis.edu/~biosport/frisbee/frisbee.html
They basically have all sorts of parameters for their flight model, which they can fine tune in order to simulate the flight of almost any disc. They do a flight test with a real disc, high speed camera and calibration markers. Then they compare a simulation run with an initially guessed set of parameters to the flight measurements, and change the parameters in a manner that improves the fit, wet hair, lather, rinse, and repeat several times until the fit is very good.
George wrote:I would add to your explanation that the weight of the disc affects the disc flight in (at least) two different ways. One is the moment of inertia (which you explain)—a lighter disc of the same shape will have a smaller moment of inertia, and, if the angular velocity is the same, will have a smaller angular momentum and therefore be subject to greater precession.
Right. Moment of inertia is typically proportional to mass, and to radius-squared. A change in mass from m_i to m_f (so long it isn't also accompanied by radial shrinkage/expansion) will cause the rate of increase in turn to be slower for a given throw by the ratio m_f/m_i.
George wrote:The second manner in which the weight enters is in the angle of attack. If two discs, identical except for mass, are thrown level with the same speed and the same angular velocity, the lift on the two discs will be the same, but the force of gravity will be smaller on the lighter disc. The net upward force will therefore be greater on the lighter disc, and it will rise faster. A level disc that is rising has a negative angle of attack. (I’m considering angle of attack to be slightly different from nose angle in that angle of attack is with respect to the air whereas nose angle is with respect to the horizontal ground.) Because of its faster rise, the angle of attack of the lighter disc will be more negative (than that of the heavier disc), causing the center of pressure to move farther behind the center of the disc, and thereby creating a greater torque and more precession. When a level disc is falling, the reverse is true: the center of pressure moves forward. In airfoils, I believe this effect is largely characterized by the pitching moment coefficient, which also changes with angle of attack, but I could be wrong.
Yes, the inertia and gravity are both proportional to mass, while the aerodynamic forces depend solely on shape. So the lift will go inversely with mass.
However, there is another phenomenon which seems very important, and points to many of the sensitive trade-offs in optimal disc design and flight. For every disc, at a given angle of attack (almost always negative, and around -4 degrees according to Potts' experiments) the lift force (defined here as the aerodynamic force component normal to velocity) completely vanishes, in which case the mass cancels in the force balance along the lift direction. (Also, according to Potts, this angle of attack corresponds to the same angle that minimizes drag.)
One might think that the disc begins a parabolic trajectory if the lift force vanishes, but in reality the disc never gets to that point. Instead, there is a dynamically stable balance achieved between lift and rise rate as follows: Your hypothetical flat disc rises up and the angle of attack becomes negative. If the disc rose so quickly that the angle of attack reduces to -4 degrees or so, then the disc would stop lifting and commence leveling out . This causes the angle of attack to increase again, hence providing positive lift once again. A flat disc (ignoring any changes in nose angle during flight) can steadily rise at an angle (=angle of attack, in this instance) where a balance is achieved between lift and gravity.
(It is physical reasoning and thought experiments like this that convince me we can refine disc flight simulation models for each mold by careful comparison to disc flight.)
Jonny Potts (who founded Discwing) measured the pitching moment of a disc (I don’t think it was a golf disc) in a wind tunnel and measured the change of pitching moment with angle of attack and described it in this paper:http://www.discwing.com/research/flowOverRotate.html
His experiments were quite fun. This is recommended reading for all disc golf nerds.
George wrote:Although the initial turn and the final fade are generally thought of solely in terms of speed by disc golfers, I think the changing angle of attack (negative in the first part of the flight, and positive in the final part) plays a major role in causing the typical S-curve flight. Of course since lift increases with increasing speed, the angle of attack and the speed are closely related.
The lift force function (call it "F_l") is fairly well constrained. If you look at the figure I drafted, there is an offset "x_p" of the lift force from the center of the disc. The pitching moment is easily calculated as x_p*F_l, that is, if one knows x_p. This x_p is, however, initially a poorly constrained function of speed ("v") and angle of attack ("alpha"), i.e., x_p=x_p(v,alpha).
But here is the gold mine in the rough: the function x_p(v,alpha) contains the most essential information about how a given disc flies.
George wrote:Disclaimer: I am not an aerodynamicist but I am a physicist (and a mediocre rec masters disc golfer).
Heh, but the physics is pretty straightforward, at least at the descriptive level we are discussing. If you read the Potts flow visualization papers, you'll see that the aerodynamics part of the picture is not at all simple. I love the schematic flow planforms they artfully whipped up. Of course, these studies were also mostly descriptive, and I didn't see any attempt to derive particular flow structures or their characteristics using math.