How would I model a stiff piece of paper dropped from a tall building?

[Qacer]

Are there any existing models or articles that I can read that discusses a stiff paper being dropped from a tall building?

I'm interested in modeling the behavior of the paper as it floats down. Eventually, I'd like to figure out how it would be affected if wind is taken into account and so on.

Thanks!

 

[CycloWizard]

This is simply a rigid slab falling through a stagnant fluid. I doubt there are any papers on your specific example, but it's a very straightforward problem. I'll write up a little something about it later if I have time. What exactly do you want to know about it - how long it takes to fall, the velocity, drag, acceleration as it falls, etc...?

 

[Gibsons]

Couldn't this get non-linear pretty quickly? I'm assuming it's aerodynamically unstable, but I don't know if that's really correct.

 

[Qacer]

Originally posted by: CycloWizard
This is simply a rigid slab falling through a stagnant fluid. I doubt there are any papers on your specific example, but it's a very straightforward problem. I'll write up a little something about it later if I have time. What exactly do you want to know about it - how long it takes to fall, the velocity, drag, acceleration as it falls, etc...?

Yep: how long it takes to fall, the velocity, drag, acceleration as it falls, etc...



The thing that is bothering me is that when the paper falls its relative motion is not constant (it starts rotating in different orientation and so on). I figured this would have some affect on its speed since in one orientation the area the paper's side would be in the field of resistance and on another side the entire paper is contributing to resistance.

 

[CycloWizard]

Originally posted by: Qacer
The thing that is bothering me is that when the paper falls its relative motion is not constant (it starts rotating in different orientation and so on). I figured this would have some affect on its speed since in one orientation the area the paper's side would be in the field of resistance and on another side the entire paper is contributing to resistance.

The only way to analytically model this would be at terminal velocity with unchanging orientation. Since the paper is rigid and there is no wind, it would pretty quickly adopt one orientation and stick with it. If you dropped it flat-side down and perfectly parallel to the earth's surface, it would fall that way (given the assumptions that you've made). If you dropped it at even the slightest angle, it would quickly reorient such that its narrow side (edge) was its leading edge to minimize drag. Again, it depends on how you want to look at the problem. If you want to consider anything remotely realistic, such as the paper flapping around, rotating, and bending as it falls, then you'd need some serious modeling background and a supercomputer to solve it.

This problem would fall under the category of aeroelasticity, which is a very complicated subject. Essentially, one would have to simultanesously solve differential equations governing the flow of air around the paper (hydrodynamic theory) and how the forces from said air cause stresses, bending, translation, and rotations within the paper (elasticity theory). The coupling of these two systems of equations makes my head hurt, as the boundary conditions would be given as the solution of coupled nonlinear partial differential equations.

 

[Qacer]

It would make my head hurt, too. I'd like to start on something simple and take steps from there. I'm going to use the classic boat on a river scenario using vectors. Only this time the river is put sideways and the boat is floating down. The wind would be the river flow.

 

[f95toli]

Why don't you start with somehing slighty simpler? E.g. a piece of paper falling in a vacuum. That would give you some idea about how to write down the equation of motions for the paper itself. This is still a tricky problem but it should be relatively straightforward to solve numerically (it should be possible to solve numerically as well, it ís just very messy).
Once you have a worlking model you can start to think about adding the effects of the air (starting with laminar flow).

A full solution of this problem which takes into account turbulence, bending of the paper etc is extremely complex and would require a very good understanding of fluid dynamics.

 

[Tarzanman]

Don't use paper. The problem with paper is that it is *very* susceptible to eddies, drag, and other 'non-ideal' effects (even at low magnitudes)... and each of these effects will change the shape and orientation of the paper as it goes through the air which means even more iterations and recalculations.

 

[Qacer]

Are there any freeware program or matlab scripts that does this already? I don't feel like re-inventing the wheel.

 

[bendixG15]

Aw, go ahead . reinvent the wheel ... if you can (everybody else has to)

 

[CycloWizard]

Originally posted by: Qacer
Are there any freeware program or matlab scripts that does this already? I don't feel like re-inventing the wheel.

Solving the simplified problem f95toli suggested as a starting point (rigid body falling in a vacuum) is pretty straightforward. Solving the entire problem as initially specified would probably require a program like FLUENT or ABAQUS, which are pretty complicated and expensive. The student version of ABAQUS is like $50, but can only solve problems that are small in scope (I think 1000 nodes is the limit). I'm not sure about FLUENT. In other words, I don't think I would recommend trying to solve the entire full-blown problem in MATLAB. It's just too complicated.

Anyway, what kind of background do you have and why are you interested in this problem? It sounds like a course project to me...

 

[Qacer]

It's actually not a course project. A friend of mine was asking for my help regarding a promotional stunt that they were going to do for a local church fair. They had an idea of dropping pamphlets / small gifts to attendees from a tethered balloon about the size of a weather balloon. They think that it would be a good way to get everyone's attention besides handing out pamphlets individually. The pamphlets would contain an advertisement to upcoming events and an entry ticket for a raffle drawing.

Anyway, they want to limit the spread of pamphlets to the carnival area, so as to not litter the surrounding streets, but at the same time, they want the pamphlets to be dispersed in a wide enough area. I know the military was/is big on this stuff for years now, but I could not find any sort of guide that gives out formulas and such.

 

[CycloWizard]

Originally posted by: Qacer
It's actually not a course project. A friend of mine was asking for my help regarding a promotional stunt that they were going to do for a local church fair. They had an idea of dropping pamphlets / small gifts to attendees from a tethered balloon about the size of a weather balloon. They think that it would be a good way to get everyone's attention besides handing out pamphlets individually. The pamphlets would contain an advertisement to upcoming events and an entry ticket for a raffle drawing.

Anyway, they want to limit the spread of pamphlets to the carnival area, so as to not litter the surrounding streets, but at the same time, they want the pamphlets to be dispersed in a wide enough area. I know the military was/is big on this stuff for years now, but I could not find any sort of guide that gives out formulas and such.

OK, then I should warn you that this is probably illegal. I used to be on a hot air balloon crew, so I know that such an event would cause you to lose your hot air balloon license. I'm not sure if the same laws would apply to an unmanned balloon.

So, for your problem, you basically want to know the balloon height and drop location such that you'll get good dispersion of the flyers but not too much dispersion. Honestly, the best way to do this would be empirically: go there a day early and try it a couple times. You've now added on flyer-to-flyer interactions, which makes the problem much more complicated still. It can, of course, still be solved (theoretically anyway), but it's much, much harder than a single piece.

edit: Actually, this problem might not be so tricky to get a pretty good, simplified model. If you know the wind speed/direction and the size of your target area, you can pretty easily calculate the height that you should drop from with a few decent assumptions:
1. The horizontal velocity of the paper will be less than or equal to the wind velocity.
2. The vertical (downward) velocity of the paper will be bounded above by the amount predicted by the rigid body falling in a vacuum. Perhaps this velocity divided by 5 (just made up, though maybe it's a reasonable guess? it will really vary depending on how many flyers you drop.) would be a good guess for the average falling velocity, though you could make this as high as 10-20 to be conservative and make sure the flyers don't scatter too much. I would choose higher numbers if the wind speed were higher just to be on the safe side.
3. The net time falling would be equal to the height of the balloon divided by the average falling velocity.
4. The maximum horizontal movement is equal to the time from #3 multiplied by the wind velocity.

So, then, the only part that's a little complicated is #2. The rest is just arithmetic.

 

[DrPizza]

I have to agree with cyclowizard. Sometimes the best way to "model" a problem is not to model it mathematically. That's why they have things like wind tunnels. Just a hunch, and probably a pretty good hunch, but I think you'd find when modeling the problem as originally stated, you would be highly succeptible to chaos.. That is, a very very very tiny perturbation from an exact state would lead to dramatically different results. But, I suspect you already know this - if you drop 2 pamphlets from that height, they're not going to land on top of each other.

btw, if I'm wrong, I'd love to know. There are a couple of places that I've seen donation jars - with a slight twist: there's a little shot glass at the bottom of a gallon or so of water. If you drop a quarter into the water and it lands in the shot glass, you get a free something or other. It'd be great to be able to hit the shot glass 100% of the time. (I'd settle for 25% of the time!)

So, simply figure out about how widely your pamphlets would be distributed by testing it. Don't forget that wind speeds above the ground are typically higher than at ground level; buildings, etc. do funny things to air flow, and that a "minor" change in atmospheric conditions (a wind gust of 15 mph) can greatly change the distribution of your pamphlets to the point of "littering." Of course, you can find something not quite as succeptible to such changes... Perhaps write your message on 15 pound bowling balls; but that leads to a few other problems

Hmmm... come to think of it, you could spend a week of spare time doing origami and folding your pamphlets into all sorts of weird shapes, including those little boxes that you puff up.

 

[CycloWizard]

I think the solution to the rigid body falling under the force of gravity in a vacuum gives the velocity equal to gt (gravity times time), which gives a total falling time tf=(2*H/g)^(1/2), where H is the height at which you drop the papers from.

I got this by taking the simple dynamic force balance F=ma=mg (since we're in a vacuum). This means acceleration a = g. Now, acceleration is the time derivative of velocity, which is the time derivative of displacement. So you just integrate gravity twice with respect to time to give the general solution h=gt^2/2+c1*t+c2, where c1 and c2 are integration constants. Apply the initial conditions (h(t=0)=H and dh/dt(t=0)=0) to get the solution h=gt^2/2+H. We are interested in the time required to reach h=0 (that is, how long does it take to fall from H to the ground), so set h=0 and solve for t. Voila! This gives a slight problem because 'h' should be defined in a negative sense, but hopefully this gets the point across.