formula to calculate string length requirement 2

[lpop]

Hi,

I am an avid archer and am about to begin making my own strings for myself and others compound bows.

I am searching for a formula to help me calculate the appropriate length of string material needed to build a string that will equal a certain length after pre stretching and adding twist to it. I am hoping to eliminate trial and error and have a formula that will give me a near perfect length of bow string each time, regardless of the required length of cable needed for a particular bow.

To make a string; one wraps a single strand from a spool in a continuous loop onto the posts of a string jig, of which there are two posts. The posts of the jig are set to a particular distance apart from each other, i.e 50 inches.

A typical bow string will have 18 strands of string filament wrapped around the jig posts or 9 strands on each side of the posts. After the ends of the string is tied off to the bundle to prevent slippage, the string is then put under 300# of tension for a specific time. The non recoverable elongation of this string is approximately .005 percent after this procedure.

Then twist are added to the string in order to promote stability to the shooting system. Typically; this is anywhere between one twist for every two to three inches of finished cable; one twist equals 360 degrees of rotation.

Now, adding twists to a string or string bundle will shorten the string and pre stretching will slightly but permanently increase its length. So, it is necessary to anticipate these factors and add extra length to the string for twisting and subtract a bit for the non recoverable elongation.

I don't know how much a string will shorten per twist but I imagine that it is directly related to the diameter of the fiber and or its total diameter and the end to end length of the completed string. The diameter of the string material can vary but the one that I use is .014" and the diameter of a completed bow strings cross section contains 18 strands of this string.

What formula would tell me how much additional length is needed to give a string the proper length after stretching and twisting?

Thank you.

 

[btrueblood]

Well, there are a bunch of other factors that will affect the length of the twisted cable.

But, to find the straight length of a twisted string forming part of your cable, you need to assume it forms a helix, with some radius, and number of turns. Then, the straight length of the string needed to form that helix is

Straight Length = sqrt[(n*2*pi*R)^2 + L^2]

n = no. of turns, R is the radius of the helix (distance from centerline of helix to centerline of string), L is the overall length of the helix (length of the wound cable). If the number of fibers in the lay of the cable is more than 3 or 4, you can see that how the fibers pack (or lay in the parlance) will affect the helix radius for that fiber. Google cable, and research the variety of ways you can layup cables, to get reduced twist, or to improve fatigue life rolling on pulleys...

 

[lpop]

Thanks for the formula! This will be perfect for my purpose. Is there a way to solve for R using the number of strands and strand diameter?

 

[mohammadheydarivini]

Straight Length =n* sqrt[(2*pi*R)^2 + p^2]
n=number of turns
r=radius of helix
p=pitch distance
l=the length of twisted string
m.heydari.vini

 

[btrueblood]

Yup, same equation, given pitch = length/#turns.

 

[lpop]

I believe that I could use the cross sectional formula ?r² for R. However, how should this basic formula be modified to account for twisting and tension to ensure an accurate result?

 

[chicopee]

If you knew poisson's ratio between radial and longitudinal strains, you could calculate the reduced diameter when the string is under tension while being twisted.

 

[btrueblood]

I was going to answer similar to chicopee...but realize that once you have two or more strings in contact, they may deform in cross-section also, especially when rolling over pulleys. Becomes a very complicated problem to model. There are some cable lays that used pre-formed wires (somewhat triangular in section) to get improved bending properties, makes it even more complicated-er. At some point you end up having to cut & try, and hopefully keep detailed notes, so you can eventually come up with some empirical rules of thumb, or at least rules of big toe.

 

[lpop]

Hi btrueblood,

If you pick up on this reply; I have a question about L and Straight Length in the equation. From what points of the string are these two variables measured? Specifically, with the string stretched across two small diameter steel posts (1/4" dia.) is it measured from the inside (outside to outside of post) or the outside of the strings end loop ( the strings over all length)?

Thanks again.

 

[btrueblood]

straight length = along the centerline of the string once it gets bent. Really it would be along the center of bending, which might or might not be the centerline, depending on how distorted the string gets, how much plastic deformation occurs, etc.

L = developed axial length of the helix, i.e. pitch x # turns.

 

[rb1957]

i'm not so sure ... i'm surprised that we're not using the strain due to the tension.

lpop,
you say the strain is 0.005 percent ... this means that the strain is 0.00005, yes? (or 0.005 ??)

are you asking for the length of raw material ? sounds a little funny to me, i'd've thought that the corrections due to spiral and strain were very small.

or are you asking, knowing the final length required (for the bow), how far apart should the posts (in the jig) be ?

i suspect that the tension is applied when installed into the bow ? and that each bow requires a specific length string to work properly. maybe then the tension is a bit of a red herring ... you know the untensioned length you want, and the tension is applied in the bow, then you wouldn't need to consider it in setting up the posts ?

once the strings are wrapped around the posts, are the separate string loops isolated ? 'cause i'd expect that different strings would strength differently (adopt a different sprial). do you lay-out the string around the posts, then twist, then (i presume) glue them all together ??