How do I build pulley systems with 3 or more pulleys?

I do seem to miss a thing here. I want a network of 3 or more pulleys, where the shaft of one pulley is driven by a motor while the others drive some inertias via their shafts (in the following examples there are free rotational ends instead of inertias). I have studied the "best practices for pulley networks" example and read all the documentation that is out there, but I still encounter this problem:

Modeling the network with "ideal" pulleys without slip works fine. However, if I change the belts to "V-belts", the solver is not able to solve it anymore. Either it does not converge, or it can not solve the algebraic equations, etc. Or another one: The simulation is successfull, however, while the driven pulley shaft is rotating, the belt ends of said pulley do not receive any translational motion (meaning there is "total" slip, right?). I attached a sample file.

As you can see, the pulleys/belts slip in an absolute way. This is also true, if I remove one spring/damper couple. The springs have a small initial deformation (-0.005m) as recommended, and I am using an ideal torque source instead of velocity source, also as recommended. What am I doing wrong? I should add that I do care about the wrap angle and either calculate it or copy it from the transmission's datasheet.

Please help!

Kind regards,

Felix

[Felix Schönig]

After a lot of trial and error, I have one hypothesis:

If the number of pulleys is even, all pulleys can have "ends move in opposite direction". If the number of pulleys is odd, one pulley (or probably an odd number of pulleys) need to be set to "ends move in same direction".

While I can picture this to be valid for some real life examples, it is very confusing, since with wrap angles that are not close to 180° or 360° (as in a rope drum), the "ends move in same direction" does not make a lot of sense to me.

Answer 1

[Jocie Kluger]

Hi Felix,

As you mention, try changing the “Belt direction” in one of the pulleys from “Ends move in opposite direction” to “Ends move in same direction”. Then the simulation shows belts with translational motion and little slip.

In the original model, the issue is that even though the springs are initialized in tension, the force in the belt ends of each pulley remains trivial (even at t= 0 during initialization). This issue is unique to having an odd number of pulleys in a loop.

It was tricky to understand the issue. One way to think about it is considering the simplified model below: the Simscape solver requires that the forces sum to 0 at the nodes: i.e. for pulleys 0, 1, and 2, the model has F0A = -F1B, F0B= -F2A, and F2B= -F1A. If all of the Belt pulleys use “Ends move in opposite direction”, then the additional constraining equations provided by the block are F0A= -F0B, F1A= -F1B, and F2A= -F2B. This set of 6 equations only has the trivial solution of all forces = 0. If Belt Pulley2 uses “Ends move in same direction”, then the last equation is replaced by F2A=F2B, and the system has a nontrivial solution for the forces.

Hope this helps,

Jocie

[Felix Schönig]

Hello Jocie,

thank you for your answer. This is what I figured out as well, with one pulley set to same direction it works, however only with spring rates in a certain range. The second part of my problem however remains unsolved: The "slipping" pulleys itself. I attached another sample file, this time containing 5 pulleys (which is the actual problem I am trying to solve).

• one is set to "same direction", the others rotate with "opposite direction"
• tensioners added again
• both torque and angular velocity sources are in the model, one is always commented out
• for the angular velocity path I used a ramp, because a step or constant also seems to cause solver problems

Run the simulation and you will notice:

Pulley 1 is driven as intended, ports A and B translate to translational velocity accordingly.

Pulley 2 receives the correct translational velocities at B and outputs them at A, however, the shaft's rotational velocity is close to zero. The belt just slips through, and this is true for all the pulleys 2-5.

Only the driven pulley 1 is rotating, the others almost stand still. What is blocking them? Lowering the angular velocity or torque on the source does not do anything.

Thanks in advance!

Kind regards

Felix

[Jocie Kluger]

Hi Felix,

Actually, I need to correct the reasoning for why one pulley needs the "same" belt direction when 3 pulleys are connected in a loop. The issue is an inconsistency in the velocities as constrained by the block and at the nodes, not the forces. Extending the logic, a loop with 5 pulleys actually needs 3 pulleys to have "Ends move in same direction" and 2 pulleys to have "Ends move in opposite direction."

Attached is a model with the 5 pulleys that is an extension of the original model with 3 pulleys. I started to adjust the parameters to match pulleys.slx. This model uses workspace parameters K and D for the belt stiffness and damping, and works for K= 1e5 N/m and D= 100 Ns/m. Some numerical issues were resolved by increasing the Belt springs initial deformation from 0.005 mm to 0.005 m, which makes sense since the system needs sufficient belt tension. The model had some numerical issues while adjusting the parameters to match Prim. No. 2 and 3, so those are not implemented in the attached model. The system is especially sensitive to the wrap angle of those pulleys.

I am still investigating these remaining numerical issues, but wanted to give you an update.

Best,

Jocie

[Felix Schönig]

Hi Jocie,

thank you so much already! It is good to know that you are on it and investigating the matter, I was almost getting grey hairs in the past week when trying to solve this, mostly with trial and error. Especially the case of the sensitivity to wrap angle changes was troubling. Your solution so far is already helping a lot and I am looking forward to the solution and/or correct handling of those numerical belt pulley issues. Thank you for checking up on this!

Kind regards,

Felix

[Felix Schönig]

Hey Jocie,

it´s been a week now, are there any updates on the numerical problems?

These instabilities seem weird, maybe you should forward this problem to the programming/development department? As we have seen, there is much more to pulley networks than is written in the 'best practices for pulley networks' guide. I think an update of the documentation part would be in order.

I am still encountering solver problems while using wrap angles which are definitely possible (used a real world example). For some reason, increasing the number of "same direction" pulleys allows the simulation to run successfully in some case, however, the results dont make a lot of sense then (pulleys are rotating, belts are standing still => high relative velocity). Also, the position (or the wrap angle) of the belt pulley that is chosen as "same direction" seems to have some influence, it´s not just about the number of "same direction" pulleys.

Would you be so kind to give an update on your progress with the matter? Is this something you´ll be talking about with the responsible department?

Kind regards,

Felix

[Jocie Kluger]

Hi Felix,

Try the attached model, which works more robustly. I believe it matches the pulley parameters in the original model. Let me know if you encounter issues for different parameters/wrap angles.

It helps initialization to add an element that smooths the initial torque input to the loop during start-up. The smoothing element may be an inertia or damper added to the Torque source. For the velocity source, increasing the belt stiffness, workspace parameter K from K= 1e4 N/m to K= 1e6 N/m resolves the initialization issue. I’m checking with the solver team why that is the case for the velocity source.

I am fairly certain a loop with 5 pulleys needs exactly 3 pulleys with the “same direction” and 2 pulleys with the “different direction” to avoid the high relative velocity issue. Once the pulleys have been adjusted this way, it’s also important to check that the springs in the tensioners (a.k.a. belts) are oriented the right way. You can do this by using the Simscape Results Explorer. If any of the springs in the tensioners show a sign change, swap the tensioner connections.

By the way, I’m a library developer. I’ve been discussing this issue with the solver team and discussing potential future updates to the Best Practices Pulley page with the documentation team. We may continue this thread for this question. For formal support on future questions, it's best to contact MathWorks technical support.

Best,

Jocie

Answer 2

[J Chen]

I think you need to put some mass in the system for non-ideal pulley. The pulleys output forces and torques at the terminals. The equations of motion need forces and masses.

[Felix Schönig]

Good idea, but it didnt help. I added a mass to the belts (between pulley and spring/damper couple) and it got me the same results: no velocity at pulley end A or B.