Calculating tension

[Insteel]

I have a situation that requires expertize beyond my capabilities. I need to know how to calculate the individual tensions in each strand and then find the bearing loads. the setup is that I have two wheels one driven the other free, each with ten grooves (with an included angle of 45 degrees), the strand (1/2" dia) first enters the driven wheel, (under 540 lbs of tension) makes a 180 degree wrap and then enters the free wheel (see the attached file) makes a 180 degrees wrap and enters the 2nd groove of the driven makes a 180 degree wrap a then enters the 2nd groove of the free wheel. this continues until the strand leaves the 5th groove of the free wheel under tension of 18,000 lbs wraps around the turn a round wheel (wheel is free) 180 degrees and the enters the 6th groove of the free wheel. it then makes successive 180 degrees wraps of the driven and free wheel until it leaves the driven wheel at 540 lbs of tension. I would greatly appreciate any help in solving this problem.

 

[MikeHalloran]

One of your college texts probably has a derivation showing why v-belts need to stretch in order to transmit power. That would be a good place to start.



Mike Halloran
Pembroke Pines, FL, USA

 

[dvd]

It's all about friction. Web search on ["belt friction" & exponential]. If you can't figure it out from there, call me.

 

[Insteel]

Gentlemen,

This is not a Belt but more like wire rope. I understand the friction part of of a pulley, T1/T2 = e(uB), but the tension cannot not be equal in each strand and do you sum the the total tensions to get the total load the bearings see.

 

[unclesyd]

This setup looks strangely like a synthetic fiber hot draw toe line. We have two of them only they are not threaded up as yours line.
I have seen only one book that has this type apparatus and I don't have a clue as to the name or author.

If there is any stretch in you line the calculations get tricky. We have a 4::1 draw ratio between our gear stands.

 

[Insteel]

This is a cable wire Stress releiving line. There is some strech, but accures in the last 5 wraps of the wheels due to the different diameters between the 1st 5 wraps and the last 5 wraps. I just need to be pointed in the right direction. There is one wheel that is driven (max rpm 100) and one wheel is free to trun. The idea is that you wrap the wheels 5 time to prevent slippage before subjecting it to the stress relieving process (and 18,000 lbs) then wrap it 5 times on a larger diameter section of the wheel to elongate it approx. 1.0125%. This is supposed to make the strand hold its tension for years while bonded into a concrete structural member.

 

[dvd]

why wouldn't T1/T2 = e(uB) and T2/T3 = e(uB) and T3/4= e(uB) and so on? You start calculating from one end and progress through the system.

 

[Insteel]

Thats basically what I did (T1/T2 = e(uB/sin(alpha/2)) etc.
But when I get to the 5th wrap on the payoff side the tension is something like 75k not the 18k.
I would assume that the tension around the free wheeling wheel would be the same ( if it goes in at 500 lbs it must must be 500 lbs on the other strand as well because there is no friction to be accounted for). Is this assume correct?

 

[dvd]

There would be bearing friction losses around the free-wheel. This would require slightly different tension between in-bound and out-bound sides. But if you neglected that you would have a system that has 5 complete turns around the pulley, with 540 lb on the in-bound and 18,000 lb on the out-bound side.

Where did you get you coefficient of friction values? You probably need to conduct an experiment to determine an accurate number.

I don't have any reference texts in front of me right now, but where did you get the sin(alpha/2) term in the exponent from?

 

[Insteel]

The coefficient value was given to me based on some old calcutions done 30 yrs ago. I will look into the coefficient value and confrim it. The sin(alpha/2) comes from the formula for V belt tension where alpha is the angle of the groove the V belt rides in. Since we are looking for the strand to be wedge in the groove to prevent slippage (so I am told)

 

[chicopee]

Here is an alternative reference and that is "Cranes and Derricks" by Shapiro. That books has a chapter that will give you estimates of tensions in Sheeve/Cable systems found in mobile cranes--highly applicable to your situation