Cable length calculation

[Doga07]

Hello everybody,

I believe that I will find experts here who have already solved a similar topic. I need to calculate the length of the wire/cable via proximity sensor. I used a proximity sensor because it has to be placed inside of the machine and there is no physical space for the encoder. The sensor can give 20 pulses for 1 revolution and it measure pulses from pulley's rotation with diameter D1 (in the attachment). The wire/cable has a very tight grip on the pulleys since they are wrapped three times.(same shaft, three pulleys behind each other firmly connected) so no slip at all.

The one thing I am not sure, is that I probably need to take account the cable/wire diameter for the best/ better result. If I want to calcute the lenght/distance of the cable axis /center of the cable I need to add half of the measured cable diameter?(D1+D2/2)

Because if I have for example 6000 revolutions (120 000 pulses) for one production cycle and D1 =0,318319 m so the pulley circumference is 1m.

Total lenght without taking account the wire diameter is 6000* bare pulleys circumference= 6000m

if i take account the wire diameter for example 8 mm the center of the wire traveled 6000*(bare pulleys circumference+0,5*wire diameter)=6075m

the difference is quitebig. The wire diameter may vary (different production different wire diameter.) My question is if my idea is right or complete bull**it.

If you have any other ideas please share, I hope my description is sufficient enough. In the attachment there is a sketch of the pulley
s.

Thank you

 

[drbitboy]

The analysis seems reasonable (6k * (1m + 8mmπ/2) ≈ 6075m), but it has an untested assumption: that the central layer* of the wire neither stretches nor compresses as the wire lays on the pulley.

* a layer of the wire is the locus of all points at a fixed distance from the pulley's rotation axis.

All of the effects will be difficult to model (see below), but an empirical approach may be sufficient: put the wire in the system, run the pulley for 100, or 10,000, rotations, and measure the resulting length of wire. Do it several times as well, because the measurement/system noise ("common cause variation") is probably as important to know as the mean value. The biggest question is, what is the required level of accuracy?

TL;DR

The wire material will exhibit elastic strain (∆Length/Length) under stress (Force/Area) stretches under tension (and compresses under compression). The ratio of stress to strain is called the Young's Modulus of the wire material.

1) When a length of wire is laying on the pulley, the outside surface of the wire (farthest from the pulley) will probably stretch (∆L/L > 0), but will the inside surface of the wire (that is on the pulley) compress (∆L/L < 0) an equal amount, and the center of the wire neither stretch nor compress (∆L/L = 0), as assumed by the model in Post #1.

1.1) I think the only thing we can safely say is that, assuming adjacent cross sections perpendicular to the centerline of the wire do not intersect within the wire, the ∆L/L of the outside surface will be more positive than the ∆L/L of the inside surface. Maybe the inside surface ∆L/L is 0, and if it is, then you do not need to add any portion of the wire diameter to the pulley diameter. So the question is, what layer* of the wire has zero strain (∆L/L = 0)?

2) When the wire is being pulled around the pulley, it may be under some tension, in which case it will be is a stretched state (∆L/L > 0) relative to how it will be when it is measured.

3) The wire diameter is not constant i.e. has some variation as mentioned by OP; I would there is some mean diameter for any single wire type that could be used, but if ∆L/L = 0 at the surface of the pulley, then it does not matter.

4) Does if the surface of wire against the pulley compress when pressed against the pulley?

5) The wire may have a covering such as electrical insulation, which may make the all of the behaviors above even more complex.

6) Is it possible for any of these forces to compress or stretch the wire material past its elastic limit, essentially changing the wire material?

Have you searched for research of any of these these behaviors e.g. "winding model compression stretching"

 

[VaMike]

I suspect you will find the diameter of the wire is moot. Pretty sure what happens is that as the wire just touches the first pulley, it's perfectly tangent to the diameter. Therefore as the pulley keeps pulling the straight wire on this tangent, the diameter of the pulley should equal the linear length of wire that has entered on the tangent for 1 revolution. Like any wire counter, counting is done on the linear pull, not some deformed radius while spooling.
Unless or course you have invented the illusive wire stretcher.

Jmho

 

[lfe]

In addition to what has been said, steel cables wear out the pulleys so the diameters will vary over time.
Furthermore, if what really matters is the accuracy of the cable position, it is essential that the machine periodically perform a zeroing sequence, since an error will always accumulate, more or less.

 

[drbitboy]

The wire/cable has a very tight grip on the pulleys since they are wrapped three times.(same shaft, three pulleys behind each other firmly connected) so no slip at all.

If there is "no slip," then the innermost "layer" of the wire cannot compress, which means the pulley circumferential travel is the length, and @VaMike's description is how things should be measured.

 

[lfe]

In my time as a mechanical designer I designed a system with a steel cable to drag a train of kiln cars, up to about 50 tons in weight, and the pulley was not cylindrical but slightly conical, which is the way to ensure that the cable does not slip. , the cable also made 3 turns on the pulley, but everything worked through position detectors on the track. The revolutions of the pulley were not taken into account at all.

 

[VaMike]

Follow up here. I mistated some geomtry terms. The wire entering the first pulley is tangent to the circumference of the first pulley. The linear length of the wire that enters the pulley should equal the circumference of the first pulley for one revolution (pi x D) Error would be very difficult to nail down. Plus minus 1 percent would be acceptable for spooling wire at local wire supplier. Maybe not so much spooling an expensive material. Proxy repeat matters plus scan times of inputs.
Your plan will certainly work as long as you understand errors and it falls within you spec.

 

[arpus4KM]

counting would best be done by a main driver of the system, whichever it is. or by a secondary pully/encoder that is attached to the cable as it's pulled through, since the wear on the diameter of your capstans will eventually get worse over time, and i'm sure they aren't the highest priority to replace when they do get a little worn, and nobody wants to recalibrate every week due to a .25mm loss of material over a rough production run on them.


easier to put your encoder or prox on the driver of the system. and/or on a separate tensioned wheel in a straight section. then there is no need to account for weird variances that can be caused by wrapping around something with wire rope that conforms to shapes.