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# The Math Behind a Frisbee

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Does anyone have any information or links to information on the math behind the flight of a frisbee? I searched the forums and google and found a few decent articles but they just describe the physics without ever telling me how the math behind it works. The one post I found on here looked like it was going to be good but the guy never finished posting to it after he got back from SIGGRAPH. I''m really interested in learning about the frisbee flight and the math behind it. The_Ethernopian

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Oooooo, do I smell an Ultimate Frisbee game in the works?

-- John

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The deal here is, don't expect to find any nice, simple math. This is aerodynamics, and aerodynamics is reeeeally difficult.

Luckily, you can fake it. I suggest the following:

1. Rotational speed "a" decays exponentially. This is complete fakery.

2. Vertical air resistance is proportional to (1-tilt)(1 - (1/a)), where "tilt" is 0 for a flat frisbee and >0 otherwise. This is also complete fakery.

3. Horizontal air resistance is 0. Absolute, total fakery.

4. Acceleration is the projection of the normal vector of the frisbee onto the XY plane. Not sure if this is fakery, but I assume it is.

How appropriate. You fight like a cow.

[edited by - sneftel on May 1, 2003 5:11:12 AM]

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Seems to me if you know the physics, you should know the "math" involved.

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quote:
Original post by Hoser
Seems to me if you know the physics, you should know the "math" involved.

No, not necessarily. Physics is like knowing what the phenomena is, while the math is quanitative model describing it.

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i read a paper about a boomrang once... if a frisbee is een nearly as complicated, youre in trouble .

but if you have a decent general understanding of physics, you should be able to come up with some aceptable faking indeed.

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Haha even completey removing aerodynamics, a 3D rotating body runs you right into the moment of inertia tensor. Its no wonder no one explained the math! I dimly remember doing it in an advanced mechanics class, but its all so foggy now

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Yeah, you could go for gyroscopic effects and precession (which, of course, frisbees do) but I think a few simple rules like I described earlier would be more than adequate, given some studious parameter tweaking.

How appropriate. You fight like a cow.

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quote:

1. Rotational speed "a" decays exponentially. This is complete fakery.

2. Vertical air resistance is proportional to (1-tilt)(1 - (1/a)), where "tilt" is 0 for a flat frisbee and >0 otherwise. This is also complete fakery.

Not sure if I agree with those. I''m positive rotational speed does not decay exponentially. It''s also difficult to measure the "Vertical Air Resistance," because a frisbee flies mostly by generating lift.

Yeah, I know. You weren''t trying to be absolutely precise or anything. But I figured I''d put my 2 cents in

-- John

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well, the rotational velocity decrease would correspond to the friction of the outer edge of the frisbee with the air. The faster it turns, the more the friction. So friction force and fluid dynamics... too complicated for me. Then, if it''s exponential or not...

From my crude understanding of physics, As the frisbee rotates, the air under it is pushed towards the edges of the frisbee, due to the friction of the air with the bottom and inner edge of the frisbee and the gyroscopic effect (ever wondered why your clothes in a washing machine end up stucked to the edges?), which in turn, is pushed down due to the curvature of the edge. The more it rotates (i.e. the more it spins), the faster the air is shoveled to the edges, the better the lift. And as the frisbee falls down, the air pressure under it increase, thus creating more lift and slowing the rate of descent, but also increasing the friction and slowing the rotation, which in turn decrease the lift, ect...

Which would explain why the frisbee I''ve got have a smooth and shiny top surface, and a rough bottom surface. Arggh! Why does it have to be so complicated?

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There was a recent discussion on my ultimate disc email group about this. We were trying to figure out why a right-handers forehand will fade to the right at the end of its flight and a backhand to the left.

Anyway, someone at someone''s work has a copy of this:

Scott W. Hoenig, "On the Dynamics of a Frisbee." Honour''s thesis, Bowdoin College 1998.

I am trying to get a copy of it. You might try contacting the college www.bowdoin.edu

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quote:
Original post by sQuid
There was a recent discussion on my ultimate disc email group about this. We were trying to figure out why a right-handers forehand will fade to the right at the end of its flight and a backhand to the left.

Seems simple enough.... as the frisbee drops, it tilts forwards; and the natural precession of the axis pushes it to the right or the left, depending on which direction it''s rotating in.

How appropriate. You fight like a cow.

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quote:
Original post by oliii
From my crude understanding of physics, As the frisbee rotates, the air under it is pushed towards the edges of the frisbee, due to the friction of the air with the bottom and inner edge of the frisbee and the gyroscopic effect (ever wondered why your clothes in a washing machine end up stucked to the edges?), which in turn, is pushed down due to the curvature of the edge. The more it rotates (i.e. the more it spins), the faster the air is shoveled to the edges, the better the lift. And as the frisbee falls down, the air pressure under it increase, thus creating more lift and slowing the rate of descent, but also increasing the friction and slowing the rotation, which in turn decrease the lift, ect...

I always thought the rotation was simply to keep the thing from tilting over (like a gyroscope), while the shape of the frisbee acts like a wing of sorts, creating lift. I could be wrong of course, as I don''t actually know any of this...

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I actually wrote a frisbee simulation a few years ago, for Worldbook. If you have a copy of the Worldbook Multimedia Encyclopedia on CD-ROM for Windows, years 1999 or higher, do a search on frisbee. You'll find an interactive simulation that I wrote the physics for.

My simulation was based on the nonlinear Newton-Euler equations of motion, which are fairly complex, and an approximate aerodynamic model. I measured the moment of inertia of an actual frisbee using a trifilar pendulum to get the dynamics to be realistic.

Beyond my little plug, , I like many of the replies here. Sneftel has good advice, with the simple fake approach (though I don't know how well that fake works). And the references to theses and ultimate frisbee sites are good. There are a few links on the web that could be helpful:

Some pretty good ones:

www.afda.com/skills/physics.htm

http://www.ultimatehandbook.com/Webpages/Beginner/physics.html

Some not as good:

Graham

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

[edited by - grhodes_at_work on May 2, 2003 5:12:25 PM]

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my guess is, a frisbee thrown from the left hand (I''m a leftie) would have a natural tendancy to tilt to the right (?) because the movement of you arm would naturaly tilt the frisbee slightly when you launch it. You can do an opposite movement if you bend your wrist like mad the other way round. However, the fact that it''s rotating counter-clockwise and tilting to the right seems to keep it more stable than when it tilts to the left with the same rotational direction. Some weird-ass physics I''ve yet to comprehend.

From what I understand, the rotation of a frisbee is hardly smooth, and the gyroscpic effect helps keep it balanced. If it starts banking to the left sharply (like with a bad throw or a gust of wind), the pressure on the left side would create more lift on the left, thus making the firsbee tilt back to the right, and so on. It naturally stabilises itself, until it runs out of breath and falls like a stone. If you fire it upwards, when it comes back for you, the added speed of descent and the incline would keep the rotation and stability going.

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You''ve got to deal with gyroscopic effects, as everybody mentioned, and you also have to deal with the frisbee as an airfoil, as has also been mentioned - but in addition to that, you need to deal with the horizontal aerodynamic forces generated by the rotation! One side is moving faster relative to the air than the other side due to the rotation. If the frisbee is rotating clockwise, then it will curve to the left (I think that''s correct). Anyway, you get the point: A fresbee is an ridiculously complicated beast.

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quote:
Original post by TerranFury
You''ve got to deal with gyroscopic effects, as everybody mentioned, and you also have to deal with the frisbee as an airfoil, as has also been mentioned - but in addition to that, you need to deal with the horizontal aerodynamic forces generated by the rotation! One side is moving faster relative to the air than the other side due to the rotation. If the frisbee is rotating clockwise, then it will curve to the left (I think that''s correct). Anyway, you get the point: A fresbee is an ridiculously complicated beast.

Good point about the side force; however, the effect of this is neglible compared with the interaction between the pitching moment (torque created due to pressure differential between top and bottom surfaces) and the gyroscopic effects.

The curving of the frisbee to the left or right is a result of the frisbee turning over as it flies, which in turn is a gyroscopic precession. Think about it this way. If you throw the frisbee perfectly horizontal, it creates a lift that holds in the air. But the lift acts effectively behind the center of gravity, thus creating a torque about the center of the frisbee that tries to flip the frisbee back-over-front. But this torque is merely a change in angular momentum per unit time. If the frisbee was thrown with the left hand, the frisbee has a large angular momentum vector pointing upwards. The back-over-front torque due to lift is basically an increment in the angular momentum vector that points to the left (when you view the frisbee from the view of the thrower). Add a very long vector that points straight up added to a shorter vector that points to the left and you get a long vector that points just slightly to the left. The frisbee thus turns over to the left if thrown from the left hand. Now, once the frisbee starts turning over to the left, the left generated between the top and bottom surfaces now points slightly to the left, and thus the frisbee is now being pulled to the left by the left. The gyroscopic effect continues, and when thrown from the left hand the frisbee will typically continue to turn over to the left, and will begin pulling into a tighter and tighter left-hand-turn. If thrown by the right hand, the initial angular momentum vector is very large and points downward. The gyroscopic effect in this case causes the frisbee to turn over towards the right, and the lift now pulls the frisbee towards the right, causing a tighter and tighter turn to the right. So, the basic flight of a frisbee can be explained purely using longitudinal aerodynamic forces (lift and pitching moment, even drag can be ignored to get the basic effect) and the gyroscopic effect due only to the rate of spinning about the primary spin axis. No need to dig into the more complex aerodynamic forces (side force, yaw and rolling moment, etc.) or off-axis angular momentum.

The reason some people are able to throw frisbees that don''t turn over very much is that they apply much more spin. The more spin, the higher the initial angular momentum and the smaller the change in the angular momentum axis (which basically corresponds to the primary spin axis) and thus the less the frisbee turns over during flight.

Of course, the local wind, and ground effect, can have a dramatic role in frisbee flight, and this can lead to some strange behavior that is more difficult to simulate---but that probably can be faked pretty easily.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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quote:
Original post by Sneftel
Seems simple enough.... as the frisbee drops, it tilts forwards; and the natural precession of the axis pushes it to the right or the left, depending on which direction it''s rotating in.

Yes, that''s it exactly. I did explain this in more technical detail in my last post. I didn''t read the ultimate frisbee discussion, but perhaps the discussion there was more detailed also.

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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quote:
Original post by oliii
my guess is, a frisbee thrown from the left hand (I''m a leftie) would have a natural tendancy to tilt to the right (?) because the movement of you arm would naturaly tilt the frisbee

The tilting has to do with the spin direction, not the angle at which the frisbee is released. There are a few posts that describe the effect accurately.

But you do have a good point. Sometimes, if you work at it, you can launch the frisbee at such an angle that the lift vector is pointing in a direction opposite the direction of tilt, e.g, throw with the left hand causes a tendency to tilt to the left, but if you release it and it is already titled to the right, then the frisbee will begin flying in a rightward turning path, all the while it tilts back towards the left. I''ve seen this, and frisbees thrown in this way do have some of the more elegant flight paths!

Graham Rhodes
Senior Scientist
Applied Research Associates, Inc.

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That was an excellent explanation grhodes_at_work.

If I have understood correctly, when a righthander''s backhand is released rotating clockwise the lift force (acting slightly behind the centre of the frisbee) creates a second moment about an axis perpendicular to the direction of travel, and the gyroscopic effect then rotates the frisbee around the axis in the direction of travel. It that correct?

I imagine that some of the frisbee golf discs which sharply bank to the left for a righthander (termed overstable) are designed so the lift is centred further behind the centre of the disc. The ''understable'' ones probably have the centre of mass further forward than normal.

One more question, when a frisbee is thrown hard or into a wind they tend to flip over in the opposite direction. Is this the lift effect that Terran Fury was talking about?

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I don't think so. What I was talking about was analagous to throwing a curveball in baseball. I'm pretty sure this force points "through" the center of mass, so it wouldn't apply any torque to cause the frisbee to flip - unless maybe if the frisbee is already tilting, perhaps then it could create a torque to accelerate that tilting? This might make sense if the spin is such that the higher side of the frisbee is moving slower relative to the air and the lower part is moving faster. Hmmm.....

All this is a bit much. Is there some simple model from which all these behaviors would emerge? Something involving CFD, maybe, so we don't need to do all this analysis?

[edited by - TerranFury on May 6, 2003 10:16:02 AM]

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