Rope stretch/Pulley Calculations

[Irwinfletcher]

I have a pulley constrained by a single rope fixed at each end to beams in the ceiling. The beams are at different heights giving different "rope" lengths. If I apply a large load to the pulley, how do I calculate the rope loads? Do I take the stiffnesses in parrallel, like two separate springs, or in series, like two springs joined end to end. Does the pulley rotate? If it does, the rope extensions are different. If it does not rotate, the extensions are equal. The two approaches give different results and unfortunately one gives a load greater than my minimum break load. Which is right???

 

[mrMikee]

It sounds statically determinant to me so I don't think the stiffness of the ropes is important. You should be able to solve this based on the geometry of the pulley and the two supports.

The pulley will rotate and move to a point of equilibrium where the angle on each side of the pulley is the same, and the tension throughout the rope it the same. Note that the working loads of wire rope for example (you just said rope) is on the order of 6 or 7 to 1, but I'm not sure. You shouldn't be even close to the breaking strength.

If you're not absolutely sure about this get the advice of another engineer, preferably onsite, who can take a look at what you are trying to do. Check working loads and factors of safety of all components you intend to use, and don't forget to check the beams in the ceiling for the additional loads.

-Mike

 

[aggman]

You can also get additional information from CROSBY. They have manuals which also go through reeving design. If its a static system then you can solve it using simple statics so if you can't remember how to sum the forces then find a statics book. It will likely have almost an exact example in it. If the system is dynamic then you will have to take into account friction with the sheave blocks. (ref Crosby) Factors of safety as mrMikee stated will be in the neighborhood of 3-5 to 1 on a static system against the breaking strength. Dynamic systems utilize higher safty factors.

 

[JStephen]

Something not mentioned above is whether the load below moves when the pulley does. The response above is based on the applied load always being vertical. If the applied load is through a particular point, it would get a bit more complicated.

As a check, if you'll add up all the vector loads on the pulley when you're done, you should get zero in the horizontal direction and zero in the vertical direction. If you don't, something's wrong.

 

[ccw]

Hi Irinfletcher,
If I visualize your system correctly, there was a child's toy made somewhat similar back in the day. It was a deep grooved wooden wheel that rolled on a string suspended between the ends of two sticks, one for each hand. The sticks were manipulated up and down to give the wheel considerable velocity and travel along the string. At speed, the wheel had considerable gyroscopic stability and could be made to do some interesting tricks.

You can make up a physical model of your system to verify that:

1. the pulley does turn and move to an equilibrium point between beam-to-rope attachment points where:

2. the rope tension is roughly the same throughout;

3. the departure angle from vertical of the rope from the pulley is the same on both sides;

4. the tension in the rope is a tangent function of the departure angle, ie. will approach infinity for large angles approaching 90 deg.;

5. for the special case where the departure angle is zero, the rope tension is 1/2 the suspended load;

5. assumptions are that the suspended load is free to translate horizontally and vertically as JStephen stated, and that the rope has sufficient slack so that the pulley is below both attachments at equilibrium and that the pulley functions as normal.