Crossbow Physics: Calculating Spring Constant, Tension Force & Power

[Nidum]

OK - we'll start again . Please post a good drawing of the actual set up . Include detailed dimensions and materials information for the metal bar and for the elastic string .

 

[pevi70]

This is the set-up:

I can measure the velocity of the departing arrow. The mass of the rubberband spring is unknown and not practical to measure.

I would be interested to use the velocity and the other known parameters to calculate bow characteristics including the power of the bow.

Let me know if you need more information!

 

[Nidum]

What are the cross sectional shape and dimensions of the rubber band and what material is it made from ?

Alternatively (or in addition) - what extensions of the actual rubber band do you get under different applied loads ?

 

[pevi70]

The rubberband has a round shape (rod shape) with a diameter of about 5 mm. I will post a picture of the actual bow later today.

I have no equipment to measure extensions under different loads, so I don't have or can easily collect that information.

 

[Nidum]

I will post a picture of the actual bow later today

Yes please - that would be very helpful .

 

[pevi70]

Despite the picture, quite some assumptions need to made: no recoil for example.
If you would assume that the propulsion is coming only from a mass less rubber spring, no friction exists, how would you calculate the time it would take for the arrow to cross the 17.5 cm (knowing the final speed)?

How could I approach that?

Thanks for your thoughts!

 

[Bandersnatch]

I have no equipment to measure extensions under different loads, so I don't have or can easily collect that information.

Extension of the rubber band can be measured easily enough with a ruler. If you don't have a dynamometer on hand, you can measure draw force the way it was traditionally measured - hang the crossbow on a wall, and attach weights of known mass to the spring, measuring the distance the nocking point moves for each weight (hence the traditional nomenclature for bow power - 'draw weight'), as well as the associated extension of the rubber band.
Different crossbows are then ranked by their draw weight at full draw.

 

[pevi70]

Thats simply not practical. You need a special lever to lock the spring. I can measure the speed of the arrow.

I would really appreciate it if someone could help me how to calculate the time needed to propel the arrow by the spring in an idealized environment.

 

[Bandersnatch]

?
Why would you need to lock the string? Get a bottle, fill it with sand, weigh it, then attach it to the string and measure how much it stretches. Then change the bottle for a different weight or add more bottles and repeat.

 

[pevi70]

These bows are so strong, you need a special lever to charge the spring.

 

[Bandersnatch]

If you do it by hand, sure. But here you're just hanging weights on the string. You can hang as much as it takes. I really don't see the problem here.
I really don't have any experience with stretching strings, but if it's anything like traditional bows in efficiency, then you'll need something like 50-70 kg at full draw for 60m/s bolt (which you don't need, since we just want to find out how the string extension vs draw weight looks like - the first few cm of draw should suffice).

 

[pevi70]

Here is a picture of the bow:
(The width of the wooden part of the bow is 4 cm)

 

[JBA]

Since you can measure v (velocity at the arrow separation); and, for an object accelerating from 0 to v, s= 1/2vt, and with s and v then you can determine t (time of acceleration) and subsequently the acceleration a. If of any benefit, from that using F = ma you can determine the average effective bow force. (if, as you specified, you ignore friction and recoil.)

Edit: I just saw thr bow picture and are you sure there is no flexing of the crossbar? If there is as strong a pull force as you state, it is hard for me not to expect flexing in that component.

 

[pevi70]

I think s=1/2vt only applies if the acceleration is constant.

 

[JBA]

What this will give you is the average acceleration for the entire travel, not the profile of the acceleration. For for a given succession of values regardless of linearity there exists an average value.

 

[pevi70]

OK, I thought it didnt apply. Thats great, thanks a million!

 

[Nidum]

I estimated the dimensions but they won't be far out . 150 N on each arrow .

 

[pevi70]

Thanks Nidum, what is it?
Do you agree with JBA?

 

[Nidum]

FEA deflection estimate for the bow .

If deflection had turned out to be very small then it would have confirmed what you have said about bow being just a rigid bar .

 

[pevi70]

What does that mean?
And do you agree with JBA?

 

[Nidum]

Problem is that all responders - including myself - are now a bit uncertain as to what exactly we are trying to analyse .

So I'm trying to get a definite understanding of the mechanics of this cross bow . Is it the metal spring , the rubber rope or a combination of both that provides the driving force ?

 

[pevi70]

Do you agree with JBA?

 

[Dr.D]

So I'm trying to get a definite understanding of the mechanics of this cross bow . Is it the metal spring , the rubber rope or a combination of both that provides the driving force ?

Amen!