Working on a giant bow to beat Allen Case!

[avcase]

Another way to look at it is that the scaled up giant bow will have the same ratio of arrow mass to stored energy as the normal size bow.

Playing around with the numbers

Normal bow: 300 grain flight arrow / 45 ft-lb = 6.67
Giant bow: 518,400 grain flight arrow / 77,600 ft-lb = 6.67

How about draw weight?  The proportion of the draw weight X draw length to stored energy for the normal size bow will be the same as the giant bow.

Giant bow draw weight = 77,600 ft-lb * (50lb * 28”/12” per foot) / 28 ft = 7,185 pounds!

In order for the giant bow to complete its launch in the same amount of time as the normal size bow, the acceleration of the arrow would have to be much higher in order to accelerate the giant flight arrow over a much longer distance, but this isn’t the case.  The acceleration of the arrow at the moment of release for the flight arrow of each bow is as follows:

Regular bow arrow acceleration:
Mass to be accelerated = 300 grain flight arrow + 200 grains virtual mass = 500 grains = .00222 slugs
Acceleration = force/mass = 50lb / .00222 slugs = 22,500 ft/s^2

Giant bow:
Mass to be accelerated = 518,400 grain fligh this arrow + 345,600 grains virtual mass = 3.83 slugs
Acceleration = 7,185 lb / 3.83 slugs = 1,874 ft/s^2

The ratio of the arrow acceleration for the regular bow to the giant bow is the same as the scale ratio of the large bow to the small bow.  In other words, it will take the giant bow 12 times longer to complete the shot compared to the regular size bow.
Length scale ratio: 80 ft/80 in = 12
Acceleration scale ratio: = 22,500 ft/s^2 / 1,874 ft/s^2 = 12


Alan

[avcase]

Regarding virtual mass, the ratio of virtual mass to stored energy should be similar for the giant 80 ft bow as it is for the 80” bow.

Virtual mass is not a perfect predictor of how fast a bow will shoot different mass arrows however. Virtual mass approximates the bow using a single degree of freedom model, but the bow has infinite degrees of freedom. The value for virtual mass will fluctuate some amount depending on whether a natural frequency is triggered in the bow limb, string, arrow, or any combination of all three.  It usually works pretty well over a limited range of arrow weights, and works better for some bow designs over others.   For example, with most of my modern footbow limbs, the virtual mass often decreases significantly going from 10ggp, to 1/2ggp. I even had an extreme example where the efficiency of one of my bows increased with a lighter arrow before sharply falling off again with an even lighter arrow!

Alan

[Badger]

   I agree virtual mass can't be regarded with any real accuracy but it does indicate an rough area of performance you might expect. I was surprised you bow actually gained performance wit a light arrow. I wonder what mechanism allowed that to happen? might be worth looking into.

[willie]

How about draw weight?  The proportion of the draw weight X draw length to stored energy for the normal size bow will be the same as the giant bow.

Giant bow draw weight = 77,600 ft-lb * (50lb * 28”/12” per foot) / 28 ft = 7,185 pounds!

Alan
Your 7185 lbs draw weight compares favorably with 7200 lbs for a similar bow modeled in woodbears spreadsheet. Your original arrow was 6GPP  (300gr @50#)?  Scaling up using this ratio giving an arrow weight of 6.2 lbs. Our often used arrow weight per lbs draw  (GPP) doesn't scale at all like the virtual mass method. It might be interesting to see how your mass/stored energy ratio correlates with distance observations.

Thanks for you insights on the degree of freedom limitations with the virtual mass method. Like Steve, I agree it wold be interesting to further examine vibration and harmonics of limbs and strings. In your extreme example  where efficiency increased before dropping, were your arrow velocities obtained from a chrono, or were bow and arrow considered together, using velocities assumed from actual arrow distances?

I hope my questions are not too pointed, this study of badgers project has brought some good concepts to light.

PS  a 12/40 th of a sec powerstroke does seem more reasonable

substantial editing to original post

[Badger]

  Willie, because of the lower acceleration rate I thing the 6# arrow would be the better choice which would be based on peek draw force as opposed to stored energy which I originally allowed for. I also think the bow would need to be scaled up from a flight bow type bow, not so much in reflex but in the limb shape. All the dynamics of it would respond much better using draw force instead of stored energy for the arrows.

[avcase]

In your extreme example  where efficiency increased before dropping, were your arrow velocities obtained from a chrono, or were bow and arrow considered together, using velocities assumed from actual arrow distances?

I was bench test shooting a real bow using a shooting machine fixture and two chronographs. The shooting machine fixture precisely releases the arrow at a pre-determined draw length. I also have a paper holder for the arrow to pass through so I can determine the quality of the arrow Flight based on the shape of the hole it makes in the paper.

I don’t think it is necessarily a good thing to see efficiency jumps like this. I’d prefer that it behaves much more predictably over a very wide range of arrow weights. I think what I am seeing has to do with a couple of natural frequencies cancelling each other out at certain arrow weights and shot speeds.

Alan

[avcase]

  Willie, because of the lower acceleration rate I thing the 6# arrow would be the better choice which would be based on peek draw force as opposed to stored energy which I originally allowed for. I also think the bow would need to be scaled up from a flight bow type bow, not so much in reflex but in the limb shape. All the dynamics of it would respond much better using draw force instead of stored energy for the arrows.

Shooting a 6-lb arrow out of the giant bow would be the equivalent of shooting about a 1/2 ggp arrow out of a 50# bow!  This will be a dry fire situation!

[Selfbowman]

Guys you know I am not a math guy! I think the problem is in the material not the mass. The wood is not 10 times as dense. It's just bigger and heavier. I also reserve the right to be wrong cause I don't know. I have said before that mass is mass but maybe not with giant bows of the same material. Arvin

[Badger]

Guys you know I am not a math guy! I think the problem is in the material not the mass. The wood is not 10 times as dense. It's just bigger and heavier. I also reserve the right to be wrong cause I don't know. I have said before that mass is mass but maybe not with giant bows of the same material. Arvin

  Arvin, scaling up the composition of the material is tricky. My thinking is that it should be much less dense to be ideal. Not even sure why i think that. Maybe in my head I am thinking if I scaled up the cell structure in the wood it would be lighter.

[DC]

Sometimes scaling things up or down just flat doesn't work. That model schooner that I posted a while back is scaled down exactly from the original and it won't float upright by itself. It needs 5 pounds of ballast hung 18" below the boat or the slightest breeze knocks it over. Sails have (basically)2 dimensions(square0 and hulls have 3(cube) and they have to be of the original design or it won't work.