Arrow Speed & Energy vs Mass

[Del the cat]

Ever wondered exactly how velocity and energy vary as you change arrow weight?
If you enjoy good graph read on.
I did some tests using a small crossbow for repeatabilty, having made a bolt with a hollow tubular end so I could load it up with shot.
Rather than post pics her it easier to give you the link.
Bowyer's Diary
You can see the set up and the resulting graph (left click on the graph to see it bigger).
I was a bit surprised that the energy didn't drop off, despite going up to 4.46 ounces, from a 50# bow with v short draw.
Del

[Badger]

     Del, this is an interesting topic. For quite some time now I have used a very heavy arrow to find out how much energy a bow is storing instead of plotting a force draw curve. After having done this little experiment your next logical topic to learn on might be " virtual mass". This allows you t predict the speed of any weight arrow as well as the energy storage. Your bow may establish a virtual mass of say 180 grains and this will remain relatively constant with any arrow even though the energy imparted to an arrow might go from a low of 15# to a high of 44# say for a 50# bow. Depending on the arrow weight. If you are interested I will show you how to calculate " virtual mass"  Steve

[Del the cat]

Yeah, I'd be interested to see your 'virtual mass' calcualtion.
Ovbviously my little test is only true for that particular (Aluminium Alloy) bow, but it wiil be interesting to  take the prod off and measure it's actual mass.
Presumably 'virtual' ('effective' might also be a good word to use) mass correlates to the materal and design of the bow as well as it's actual mass.
I sometimes wish my maths and physics went a bit further, but on the other had my 30 odd years as an Electronics Designer has taught me that you gotta have the practical feel first and foremost.
Cheers
Del

[Badger]

 Del, I never went past basic arithmatic so not much math needed here. The first step is to find out how much energy the bow stores, You basicaly did this when you found the point that the energy leveled off.
   Lets say you determine your energy storage from a 50# bow drawing 28#@ 47#
If the bow was 100% efficient it would shoot a 500 grain arrow at about 205 fps, in reality say the bow shoots at 167 fps, you simply find out how much arrow mass could go 167 fps with 47# energy behind it. In this case if the bow were 100% efficient it could shoot a 760 grain arrow at 167fps. You simply subtract the 500 from the 760 and it will give you a virtual bow mass of 260 grains.
    You can use this to predict the arrow speed with any weight arrow as it remains fairly constant. If you decide to shoot a 150 grain flight arrow for instance you would add the 260g to the 150g for a total of 410 grains and find out how fast 47# could move 410 grains. In this case you could expect about 227 fps from your bow.
    A flight bow concentrates more on efficiency or low virtual mass because it is shooting very light arrows.

[Del the cat]

Hmm I'm a tad confused, I need some method and explanation.
So when you say I've determined the energy storage from my 47# @ 28"
Do you mean using a V heavy arrow and chrono to get a figure in Joules?

You then say " If the bow was 100% efficient it would shoot a 500 grain arrow at about 205 fps.
How do you calculate this figure? Or is it from a look up table?

Sorry if I'm being thick.
Del

[Pat B]

Where does arrow material fit into this? I know(from visual observation and not a chrono) that a 600gr cane arrow appears to fly faster than the same weight sourwood shoot arrow when shot from the same bow. I'm assuming that the cane arrow has a faster recovery time thus allowing it to maintain more speed during or after paradox than the sourwood arrow.  I've never measured the amount of energy imparted by either at the target and wouldn't know how to anyway. 

[Badger]

  Pat, the speed is calculated right out the end of the bow before any other forces can react on it.
  Del, I am usung a kinetic energy chart to convert arrow speed and arrow mass to kinetic energy. You can also do it just measuring a speed of a very heavy arrow like 5 ounces or so and then adding on an estimate for virtual mass based on bow design. Or you can measure the force draw curve and figure it out. I just finished a series of 5 short bows from 48" to 54" all were built for the purpose of low virtual mass. No numbers on them yet but I will post the results once I work them all out. It sounds a lot more complicated than it really is. You can also look at the virtual mass in the form of a fraction. Say you have a virtual mass of 200 and the arrow you will be shooting weighs 150. Your efficiency fraction will be 150/350 for instance. If your arrow weighed 800 grains your fraction would be 800/1000.
I will try and do a better job of explaining this and post it later.

[Del the cat]

@Pat... That's why I used the crossbow, to avoid the flexing arrow thing .
@Badger cheers, I sort of get the idea. My heads still a bit fuzzy with this cold (and being a cat doesn't help )
Del

[Prarie Bowyer]

OOOoooooo cool.  I haven't read it all yet.  But it's giving me an idea for a linear program to maximize something while minimizing something else. 

Basicaly using excell you create an equation with changeable variables and some constants.  .  . say draw weight, and arrow length.  Then "optimize" the equation by minimizing or maximizing it.  Result is the ideal grain weight for the goal.  May not be possible.  I need to think about it some and finish the article.

Love the crossbow.  Have you heard of "New World Arbolast".  They sell NICE crossbows at the Renfairs stateside.

[artcher1]

You're correct Pat, cane/bamboo arrows are faster that wood arrows. For whatever reasons.......Art

[Del the cat]

@Prarie Bowyer
Yeah I wish I could do clever stuff with excel. It took me long enough to plot 2 graphs on the same sheet .
The techical stuff is fun for those times when you can't can't play with the real wood .
That bow pistol is about 35-40 years old now! It still comes in hand for test work or entertaining visitors who've never seen such stuff, even the little kids can manage to aim it and pull the trigger while I keep hold of the tip of the butt.
I'm toying with the idea of making a nice target crossbow, I made a laminated prod many years ago and I have a big baulk of hardwood fro the stock too.
So many things to make!
I shall google New World Arbolast.
Cheers
Del

[Del the cat]

You're correct Pat, cane/bamboo arrows are faster that wood arrows. For whatever reasons.......Art

That's what's so great about this game. There is so much stuff to look into, could probably spend a lifetime just looking into arrow behaviour.
I love the way natural materials behave.
Del

[Prarie Bowyer]

@Prarie Bowyer
Yeah I wish I could do clever stuff with excel. It took me long enough to plot 2 graphs on the same sheet .
The techical stuff is fun for those times when you can't can't play with the real wood .
That bow pistol is about 35-40 years old now! It still comes in hand for test work or entertaining visitors who've never seen such stuff, even the little kids can manage to aim it and pull the trigger while I keep hold of the tip of the butt.
I'm toying with the idea of making a nice target crossbow, I made a laminated prod many years ago and I have a big baulk of hardwood fro the stock too.
So many things to make!
I shall google New World Arbolast.
Cheers
Del

If you go to options>addins> enable addins> Solver, analysis VBA and analysis tool pack and camparative statics.  check them all. 

Then you can begin to do basic econometric analysis. 

IE:  Say you have Draw weights in one column, arrow speed in another collmn, limb length in another, arrow weight in another you could go to the data tab click analysis, click "regression" then select arrow speed in the first dialoge and the other factors in the second dialogue.  Click any graphs you want and the result will be something resuembling an equation that will project with in a range of error, what an arrow speed will be given various inputs.  So if you had limb length , limb type (wood), design (recurve or long bow as dummy variables), draw weight and arrow weight as known variables say for a project, then you would know where arrow speeds are likley to fall.  There are statistic acrobatics necessary to minimize the error term that inflates the range of error but it's a start. 

Given the equation you could also use the solver and a variant given constraints to give you the best arrow speeds given some fixed input via a linear program.

You would need to keep data for all that stuff or obtain it for at least 30 points.  T-tests and various descriptive statistics are readily available there also with out needing to hand calculate all the averages.

Ah well fun stuff.  I have leaves to rake.

[Del the cat]

Cheers PB. I will have a shufti and create some data...
I might even be able to find a use for it at work so I can learn in works time
Del