Re: Drag coefficient of 2012 Game Piece
 

I knew someone would bring up the drag crisis, that's what I get for keeping it simple I suppose...

To remain at the low spot requires pretty spectacular attention to surface roughness. We tunnel tested several models as part of one of my aero labs, and indistinguishable (by fingers) differences in surface roughness will knock you out of the crisis. Seeing as these are low quality foam balls produced in the thousands by robots, I don't think there is much to worry about.

(Where did that chart come from? I've never seen Cd v. Re with how the drag crisis moves as a function of roughness before, that's a nice chart!)

Re: Drag coefficient of 2012 Game Piece
 

The chart is in A Brief Introduction to Fluid Mechanics. I'm really not sure if drag is entirely predictable just because robots may rough them up a bit. Unfortunately, drag might be a real issue too. At 5 m/s, the drag force is right at around 1 N (and weight is 3.1 N). That's pretty non-negligible.

Re: Drag coefficient of 2012 Game Piece
 

The best solution may just be to have a look-up table to be honest. I've already derived equations of motion, and you end up with a system of second-order nonlinear equations which would have to be solved numerically. It's something probably a little too advanced to teach high-schoolers how to do, and it may also be a little too much for the CPU to handle (with any reasonable accuracy).

For the curious:
m * p_dd = -K * sqrt(p_d^2 + h_d^2) * p_d
-m * h_dd = m*g + K * sqrt(p_d^2 + h_d^2) * h_d

where:
m = mass of ball
p = x-distance
h = negative y-distance (coordinate frame unit vector pointed down. positive h is downward)
K = rho * S * C_D/2
rho = density of air
S = Cross-sectional area of ball
C_D = Drag Coefficient
?_d = ? dot (as in first time-derivative)
?_dd = ? double-dot (as in second time-derivative)

It can also be expressed in terms of Speed, V, and flight path angle, gamma as a single order system of nonlinear eqations.

m * V_d = -m * g * sin(gamma) - K * V^2
-m * V * gamma_d = m * g * cos(gamma)

Expressing that way is "prettier," but is less useful.

Re: Drag coefficient of 2012 Game Piece
 

I'm not sure what you're doing, but I'll readily admit my grasp on Boundary Layers isn't spectacular (one of those classes you can ride the curve to victory). Is that an assumed velocity profile? If it is I doubt it works, because the velocity profile varies with theta. Boundary Layers typically assumes small Re where you have nice laminar or creeping flow. Take a look at this .gif, on the backside the flow is going to seperate with a ball moving in air at any noticeable speed.

If it's one of those experimentally determined equations for Cd, then it is probably okay.

Re: Drag coefficient of 2012 Game Piece
 

You shouldn't need to calculate any viscosities (unless you're calculating the kinematic viscosity, but it's not really necessary to do so).

Re: Drag coefficient of 2012 Game Piece
 

IF my calculations are correct, then when I play with the drag coefficient in relationship with out tests today, then it comes out at about .15. This is with us approximating the speed and angle of the throw. In the morning I will try and truly get the speed and angle.

Re: Drag coefficient of 2012 Game Piece
 

Also... that's for a SMOOTH CYLINDER. Gets more complicated with a sphere, as you've added another dimension. Not to mention that the game pieces aren't smooth, they have little bumps which will cause the turbulent section of the boundary layer to engulf the entire ball... will reduce overall CD but increase effective aerodynamic diameter for all other calculations.

http://www.mcoscillator.com/data/cha...yer-sphere.jpg

Re: Drag coefficient of 2012 Game Piece
 

Looks great

For anyone else interested I'm working on a trajectory simulator and solver in octave/matlab..

Hopefully I can push it out in the morning..

Re: Drag coefficient of 2012 Game Piece
 

Here's some preliminary simulation results

I'm using the drag coefficient of 0.15 as stated by Spen.M.P.

Using the magnus equation from http://en.wikipedia.org/wiki/Magnus_...all_in_the_air with a lift coefficient of 0.2 (some one could test?)



Though I'm kinda sceptical of the Wikipedia formula....... it seems like the formula is just the drag formula with a lift coefficient....

Re: Drag coefficient of 2012 Game Piece
 

Drag and Lift (and a couple other aerodynamic things) do use the same ideal formula, it's dynamic pressure and the coefficient. Some have extra terms of velocity (such as pitching moment and fundamental stability derivatives) and such, but all involve the same basic terms.

Also, if you're counting on the "lift" from that .20 coefficient, make sure you are including your second term in solving for your drag coefficient. Creating lift takes energy, that energy loss manifests in the same way that drag does. So it's just rolled right on into it.

http://www.grc.nasa.gov/WWW/k-12/airplane/dragco.html

Re: Drag coefficient of 2012 Game Piece
 

Here is a good sim for checks on your calculations: http://phet.colorado.edu/en/simulati...jectile-motion

Just plug in your air resistance, though I wish someone could tell me the equations behind it.

Re: Drag coefficient of 2012 Game Piece
 

When I get a chance, I hope soon, i will update my equations to include the magnus effect and see how it changes everything. Currently, I got the .15 drag coefficient with the ball spinning minimally, but it was spinning. .15 is a good place to start though. If anyone else run's some tests by throwing the balls, I would love to know how far they go, the speed of the ball initially, and the angle it was thrown at. If I had a little bit more data, I could more accurately figure it out.

Re: Drag coefficient of 2012 Game Piece
 

I will try to remember to put them up later. This is using some of the fairly simple ones... and almost all of it is already included in this thread, actually.

But, if you want to look them up yourself:

Without air resistance:

• Basic projectile trajectory (horizontal speed is constant, accelerated downward at rate of G until impact)
• Assumes no lift is generated (lift can't exist without drag :eek: )

With air resistance:

• Still assumes no lift is generated :(
• Drag is calculated using the basic drag formula (the one found on wikipedia), from your input of C_D
• The air density/viscosity (yes, they're linked) is calculated using the standard atmosphere model (that's why altitude is a required input).
• Air is assumed to be calorically perfect and molecules are assumed to be point-masses with no individual volume (this is why these simple equations are never going to be right).

And, to put them all together, you need to use a touch of vector calculus (sounds a lot more scary than it is) to find the instantaneous (and therefore all) velocities... the overall velocity will be reduced by your drag force. This will reduce both the horizontal and vertical components, and your vertical component will still be constantly affected by gravity.

To get CD, use the magnus effect stuff to first find your CL based on your spin rate (a hint, spinning at 2*Pi radians/second will give you the most lift from magnus effects), then use that CL to calculate the second term of your CD, use the chart/equations posted that included the roughness and talked of the unfortunate asymptote in the chart to find your first term, sum them to get your total CD, and then use that in the equations. Or assume no spin and just use the first term... that's all you can do in the sims I've seen posted so far. I might be feeling very generous and make a matlab code with all this in it if anyone would be interested. (And then it's up to you to add it into whatever programming methods you are using... I'm not doing all the work, after all...).

Oh yeah, don't forget... even if you include all of this stuff, it will still be wrong. Significantly. Your best bet is to try and simulate a simple trajectory for calculations; and to add in a scaling factor that will allow you to adjust it. Usually, a linear factor will handle most of the errors assuming you're using the same projectile and not changing your altitude by more than 50m during flight.

And, if this seems overwhelming... just remember... this IS rocket science. Very basic rocket science... but still in the ballpark.

Re: Drag coefficient of 2012 Game Piece
 

This is all true. In the end, like I've mentioned before, I think it's a little much for kids at a high school level to grasp, and may be even more difficult for the onboard computer to calculate. I really think this year will be yet another year for lookup tables. To be honest, the density isn't going to change much and therefore neither will viscosity. However, because there are still too many unknowns (ball roughness, ball uniformity, etc.), you're not going to be able to find a reliable analytic solution. We're shooting balls into a hoop. We don't have millions of dollars on the line on designing a new 787.

Re: Drag coefficient of 2012 Game Piece
 

Absolutely. I try to make it clear each time I leave a post on this topic...

No matter how much you apply aerodynamics and physics to this problem... you will NOT come up with a solution that is significantly more accurate than using a basic projectile motion (assuming no drag or lift) path.

Even if the balls WERE to remain perfectly constant... Aerodynamics is more of an art than a science when you get to the point of trying to have a computer predict how it's going to work out.