Drag coefficient of 2012 Game Piece
 

This year shooting is going to play a huge role in the game and I was wondering if anyone has calculated the drag coefficient for the game pieces yet? This would be a great value to be able to incorporate to increase accuracy. Any help is appreciated.

Re: Drag coefficient of 2012 Game Piece
 

Anyone live near one of those indoor skydiving places?

Re: Drag coefficient of 2012 Game Piece
 

There's one in Hollywood.

Re: Drag coefficient of 2012 Game Piece
 

I think over the next week I'm going to work with my physics teacher and the bleachers at our school to do an experiment.

Re: Drag coefficient of 2012 Game Piece
 

The indoor skydiving is a dead end because the force of drag is directly proportional to the speed of the game piece. (I can't imagine that the speeds will approach that of skydiver at terminal velocity) The coeffient of drag can be determined mathmatically, look up the equation, or experimentally if you can find a room whose air density is similar to the playing field and if you know the velocity that your game piece will be launched at.

Re: Drag coefficient of 2012 Game Piece
 

Just do experimentation. Just forget about the drag and actually launch it and compare it to the prediction and the real result to find the margin of error and you are good to go.

Re: Drag coefficient of 2012 Game Piece
 

Wait.. the drag coefficient of the game piece?

I thought it can be assumed 0 as it's the "drag coefficient" (not sure if right there, but c1 * v * r + c2 * v^2 * r^2.. with the first term ~= 0) of air that matters (c1 = 3.1e-4, c2=0.85 at STP)

Re: Drag coefficient of 2012 Game Piece
 

If you decide to include drag in your calculations, you turn the algebraic equations of projectile motion into differential equations and differential equations are complicated. This is because while the acceleration of the ball in simple projectile motion depends only on the time since launch, if you include drag it depends upon the velocity, and velocity also depends on acceleration -- see how it might get complicated?

It is pretty easy to calculate the magnitude of the force though, and check what the magnitude is.

Drag Force=.5*density*velocity^2*Area*Cd

For a sphere, a Cd of .5 is a pretty good. Entire books have been written on drag, but FIRST robots tend to operate pretty well on the back of a napkin.

The projected area of the ball should be about .34 square feet. That is, the area of a circle with a diameter of 8".

The density of air at sea level in the standard atmosphere is .00237 slugs/ft^3.

Velocity will obviously depend, but I'd say 30 ft/s is probably a pretty decent upper bound.

If you do the math, you'll find that the force of drag on the ball at that speed would be about .36 lbs, and it'll go down fairly quickly, if the ball is traveling at 15 ft/s, it goes down to .09 lbs.

Re: Drag coefficient of 2012 Game Piece
 

Sorry, during my fluid mechanics lectures on drag I must have been designing a gearbox in my mind. FRC season will do that.

Is the viscosity calculation equally correct here?

Re: Drag coefficient of 2012 Game Piece
 

In air, anything at FIRST relevant velocities is at such a low Reynolds number that viscosity is not a concern (inertial drag hardly is).

yeah but im using that to calculate drag ... feel like it's easier... not sure if it is completely correct, though.

Re: Drag coefficient of 2012 Game Piece
 

It is actually not a dead end. The question is what is the coefficient of drag (not drag force).

Fdrag = 1/2 * rho * A * v^2 * Cd

You can easily find out what rho is at your indoor skydiving place by using the temperature, ambient pressure, humidity, and a standard air chart.

A is easy

Fdrag is the weight of the ball as long as you can get it to hover.

v can be read form the indoor skydiver operator controls (and you can measure it inside the tunnel).

Then all you have to do is solve the equation for Cd.


While it is theoretically possible using the skydiving place, the are a lot of books that have characterize the Cd of spheres with many surface textures. I'm sure the wind tunnel would be unnecessary.

Re: Drag coefficient of 2012 Game Piece
 

I just did some back of the envelope aerodynamic calculations for the game piece this year and realized we're RIGHT on a near asymptote when trying to predict drag on a ball. (Hey, I'm bored at work with nothing else to do.)

I estimated the Reynolds number to be around 120000 (assuming around 10 m/s) and the relative roughness factor to be around 4.9 *10^-3 (I assume 1 mm dimples with an 8 inch ball, best I can estimate without having the game piece in my posession).

Then I found this little jewel (see attachment):

As you can see, the roughness (epsilon/D) line for 5*10^-3 has a nice little vertical right around Re=120000.

This doesn't take into account any spin, mind you (and I really don't feel like getting into Magnus Effect forces right now).

Enjoy!

P.S. In my opinion, the dodgeball couldn't be any more imperfectly designed if consistancy is desired.

Re: Drag coefficient of 2012 Game Piece
 

I should also note that 120000 is a rather high Reynolds number. 10 m/s is extremely fast for this kind of game. When I was doing the calculations, I was still stuck in Aim High mode, but looking back, Re should probably be about half that, and a C_D of 0.5 should be a decent estimation.

If you want an even better estimation, you can use C_D = 0.09015*log(Re) + 0.06924

Where:
C_D = Drag Coefficient
Re = Reynolds Number = rho*V*D/mu
rho = Density of Air
V = Velocity
D = Diameter of Ball
mu = Dynamic Viscosity

This is fine as long as the Reynolds number stays below about 8*10^4

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.

Re: Drag coefficient of 2012 Game Piece
 

Alright, well if we go the way of a look up table, is there some way you can put a "line of best fit" equation to the data points, but using three variables, x,y, AND z for distance angle and exit velocity. Just get all of your data points then fit a equation to it? Does anyone know of any software or method to do this? Or is it even possible? (As I am not too fimilar with 3-Dimentional graphing)?

This way, the robot can plug in ANY distance and pick an angle that gives a good entry angle to the basket, then get a initial velocity back.

Re: Drag coefficient of 2012 Game Piece
 

This is what we've done for the most part. We have a program that takes an angle and tells you the hang time, ball velocity as it leaves the launcher, and the entrance angle to the hoop. We then took that and did some logic to determine that the steepest angle entrance angle to the hoop with the shortest hang time within the capabilities of the robot is likely the best shot. It seems to produce excellent results but it's tough to say for sure. We're adding code to plot the results next to see if visibly they look good.

Honestly, now that we have this simulation running I'm waiting for one of you pros to produce a really good equation for ball trajectory that takes into account drag and I'll toss it in there to see if it produces better results.

I'm not sure how much farther to take it though as I gather most everyone on my team thinks the better approach is to build the robot and fire a million shots through it to calibrate what it can do. I don't disagree with that approach but I do feel the math is very helpful is us understanding what velocities and angles are appropriate, etc.

Re: Drag coefficient of 2012 Game Piece
 

I didn't read most of the thread, but as for the original query -

Air resistance for this piece is close to negligible. For a regular basketball, it would be negligible, but the mass of this game piece is a bit light, so air resistance may play some role. If so, you merely need to increase the initial velocity slightly. Because Air resistance decreases time in the air when the movement is horizontal in 3D space, while increasing time in the air when moving vertically downward in 3D space.

So you should use simple kinematics of projectile motion and fine-tune it by adding a small number to the initial velocity.