Main Discussion Area > Flight Bows
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 isnt 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:
--- Quote ---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!
--- End quote ---
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.
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