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Modeling a Pulley

  1. Apr 19, 2015 #1
    Dear friends, I am a naval modelist and as i am retired due to health problems I use my project ob building a sail boat model as a red line to learn about the diverse technologies I am confronted on this path. So my goal is not that of finishing the project of my sail boat, but to learn ans study and practice the topics i am meeting on this path. here a pretty old foto I made of my model to gat a sense about how it will look like:


    Now the opening of the sail, means the rotation of the boom to which the sail is connected to on the bottom side, this boom is capable to rotate around the Z-Axis, the vertical axis.


    Now it is my goal to implement in my model the same kind of pulley as can be seen from a sail boat, the Endeavour, called J-Class from the beginning of last century. This boat had a hull length of about 35 to 40 yards!

    Now serious "experts" in building model sailboats wrote to me that due to the friction in the sheaves of the blocks in a model that is 1:20 times smaller, will not be operable. None of the methods existing this day allow to support a displacement of the sheet by 8400 mm in my model, this is the name for the rope that goes through the pulley and limits how much the sail can open, means rotating the boom about the vertical Z axis by increasing the angle between the boom and the hull center line!

    I have deviced a so called "sheet control system", pretty complex, here a block diagram I developed to be able to talk about my systems concept:


    I also have other objectives, like the one to get the most energy efficient implementation of the system. Now as you all are aware of it is best to start modelling a relatively simple part of my system. So I decided to model the pulley as seen on the photo from the original sailboat!


    So I did start by drawing the discs that make up the pulley on the photo! lets start from the bottom right:

    The sheet comes over the deck from the from part of the boat, where it goes under deck to the drum that is part of the winch. For this first part of the modeling effort I simply assume the winch will make sure sheet length is made available without a tension coming over the sheet is required.

    The sheet passes through the pulley on the deck and goes up to the first pulley connected to the boom of the sail. from here it goes to the block that contains 2 discs and from it to the rear disc connected to the boom and back to the second disc of the rear block on the deck. From their it gets to a second disc connected to the boom is invisible behind the one on the drawing and then back to another disc that is equally invisible and located on an opposite side of the deck. The the sheet goes back to the from of the boat and under deck, where it is fixed connected to the hulls hardware! So this end of the sheet is fixed!


    This drawing shows how the friction between a pulley and the rope depends on the tension to which the rope is exposed, here the tension is coming from the wind blowing into the sail and passing this pressure to the sheet over the 3 pulleys connected to the boom! But also on the size of the angle it is embracing the pulley. her the link to wikipedia regarding the Euler-Eytelwein equation!

    Now you have the background information for my request of help!

    I want to start modeling a single drum embraced by a rope and have the friction computed. The I want to represent the pulley on the photo and my drawing by using 7 instances of this first model. my objective is, in a first step, to be able to simulate the model to see by studying the plots how the friction is impacted over a range of pull forces applied to the rope. I want to see the same way how the parameters like "d" and "D" impact the resulting friction. To accomplish this I want to apply the methods coming from the complex dynamic systems by using a software tool called "Berkeley Madonna", so that at a later step i can translate this into the modeling environment "Modelica", of which their are a variety of free tools available under "OpenModelica.org", I am starting using a trial version of "Dymola" from Dassault. At the end I want to use "SystemModeler" from Wolfram Software together with Mathematica.

    As I plan to do this in an environment driven by Physics I will apply the Methodology of System Physics. system Physics, originally developed as a method for teaching physics in a modern way by research under the title "Karsruher Physics Course", Professor Werner Mauer from the University "ZAHW Winterthur" in Switzerland has started the last decades to evolve this into System Physics and uses it to teach in his Physics courses at the University he teaches.


    Werner Maurer somewhere defines System Physics as a methodology to address physics from the perspective of the modern complex dynamics system approach by using those tools to model the subject he teaches in his course and have his students apply the tools of the complex dynamic systems arena, like Berkeley Madonna for example, to model a work assignment. This way his students can verify what they are learning in the physics course by applying it to the modeling tool, simulate the model and verify that the results from the simulation correctly reflect what they have just learned. This way students not only learn how to apply mathematical formulas to solve assignments, but to verify that by the models they generate their understanding is really correct and how to apply it! The above flow chart, made using Berkeley Madonna shows the basic elements when applying modern methods from the science of complex dynamic systems. The basic structure consists of recipients and flows that add to the content of a recipient of that take substance from the recipient. You can see the full analogy the the fluids dynamics as mentioned in a thread here in the forum about 2 water tanks!

    SD stands for "system dynamics". So taking the analogy from the fluids in the 2 tanks thread in this form, water is flowing from tank 1 to tank 2, here Vol 1 and Vol 2, marked with a red box.

    To explain the "law of resistance" and "law of capacitance" I like to use a table that shows for the different kind of areas in physics what those term mean. I get this from material in english language available at this site:


    Here you can see what magnitudes, titled "Quantity" in the table reflect the different areas of physics as teached in a bachelor course of physics, the related symbol and the units. The Potential reflects the nature of the second element. So i.e. mass in combination with the earth gravity field leads to the weight or potential energy. Mass in combination with velocity leads to Momentum. Equivalently for each of the seven kind of quantities there is an associated potential. Just as a side comment. The same equations, relations are found in the different areas of physics, uncovering by this perspective a common property across apparently completely different fields in physics!

    Lets take what in system Physics applies to Mechanics, the Momentum "p"!

    The equation for the Momentum P = Mass * Velocity = m * v.


    i do not have neither the ability as I am just learning it myself, nor can this be the topic of this post and threat to teach System Physics methodology and concepts. I just want to brief you and give you access to more resources on the subject to be able to have a chance for support! Basically I can say the in mechanics the content of the recipient is "Momentum" as defined by the above formula and the flow in and out is "momentum-flow-strength". This happens to be force F. I do refer to the material available at the site my link to the Karlsruher Physics Course, where this is explained in a lengthy way in english!

    The final concept related to system physics is the energy part of it. Professor Werner Maurer also calls it "process energy" and uses a waterfall as an example for it! The water falling from its starting point further up has a higher potential energy than the water after it has passed the water fall, where the amount of potential energy is lower. The difference between the 2 is the process energy gained from the water falling. We use this process energy in hydrodynamic plants that generate electrical energy by converting the process energy available as kinetic energy to rotate turbines to generate electricity!

    Now I want to present you with my first trial to model a pulley llnked to the image of the pulley and the forces on it!


    All I did in this image was to draw for every disc in the pulley a recipient and for the rope through that pulley arrows reflecting the rope! As you can see by the question marks on the violet colored globes, those are still empty. In here I will have to place the equations. What is till missing completely are those brownish globes and the curved lines that you can see on the flow chart above. Those brownish globes are used to contain the formula that reflects the relation and the source from which a parameter of the equations in the violet globe on the flow arrows gets its value from! So this way, until not all the question marks on the violet globes disappear, there are still undefined parameters in the equations they contain.

    This I consider a wonderful way to handle a complex system design! The graphics feedback in the flow chart not only makes it easy to keep adding dependencies, but it also helps to keep track of eventually uncomplete definitions of parameters in equations!

    Now I have said that each recipient in this flow chart represents one of the 7 drums of the pulley and each of this recipients stands for the model of each individual drum!


    This flow chart is my first trial to generate a model of an individual drum by reflecting the red colored forces on the graphic of a pulley i have made available above and on which the Euler-Eytelwein equations are based.

    Now finally my problem and question:

    If the Pulley recipient in the last image represents a physical disc, then its content is "Momentum". The flows into that recipent is the pull tension coming from the sail over the sheet, a force: Fine.

    The other arrow pointing into the recipient is the friction, I guess and whose direction is opposite to the pull force from the sail! What I am searching for is the balance, were the force coming into the recipient from the pull of the sail is equal to the friction. or better expressed the sum of this 2 forces equals "0". If a pulling force on the sheet is the result from wind blowing into the sail, then this equality would reflect the wind speed at which the force the sail pulling the sheet equals the friction in the pulley. any wind faster then this one would be able to overcome the friction is the pulley!

    But at this condition the "velocity" = "0" and as a consequence the content of momentum in the recipient equal "0". So the balance only happens, when the momentum content of the recipient is empty!

    But what value has to be taken for the mass? The rope is considered to be free of mass!


    May be this "fluid image" often used in system physics as a visual mean to reflect physical situation, as the same principle apply to any of the 7 quantities listed above!

    The image shows in detail a recipient and uses it, as I do need to do it, to represent a mechanical system! The mass represent the cross cut area of the recipient, the velocity the filling level are fluid level in the recipient. You can also see that in system physics, here translational mechanics the reference Momentum is given by earth! In my thinking right now when my pulley is in balance the recipient filling level is equal to the surrounding earth level! Still so, what would be the mass in my case?

    I hope here in the forum is anybody that can help me! I am trying to get hold of a school book about systems physics, as I obviously am missing something!
  2. jcsd
  3. Apr 22, 2015 #2

    jack action

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    First, your post is way too long and way too complex to understand it clearly.

    From what I understood, you are using the wrong model. If you assume you have no mass in the system, then your model shouldn't have mass involved. The Wkin is the kinetic energy (= ½pv = ½mv²), which you cannot have since it is massless. What you will have is friction, which will translate into heat (= friction force X displacement). That is where you will have an energy loss.

    If there is no motion (v= 0), then a model based on kinetic energy is not useful either. In static version, the disc (or more probably the rope) will compress under the normal force; That could be modeled into a spring for which the energy stored is ½kx².

    Although, I'm not entirely sure of what you are trying to do, it seems to be more complicated than it needs to be.
  4. Apr 22, 2015 #3
    The problem is that you require the background information to understand my question! This is why i presented that i am building a model sailboat that will have a pulley like the one the original sailboat Endeavour has! I have been warned by experts that such a pulley will not work in a model due to side effects from having the scale 20:1, were the friction prevents the sail from opening! So I am trying to model the pulley as part of my system!

    Now I want to apply system physics, a way to view on physics that is different from the way it is usually done in physics. It uses the methodology coming from the complex system dynamics and applies many of the concepts known from quantum physics to the classic physical topics! So in mechanics you have to deal with momentums and momentums flows! Now the problem I am facing is exactly what you addressed in your response: no definable mass and no velocity, ergo no momentum!

    Now I am guessing you are not familiar with system physics, right? As a consequence of expecting that many readers also have not heard about system physics either, I am giving the links were additional information can be found!

    Why is the use of system physics so important to me? I plan to model my whole concept of a sheet control system for a radio controlled sailboat model using the modeling and simulation language Modelica, which is based on the same principles of system physics and of complex dynamic systems! Modelica makes it possible not only to model multi domain systems, as my system has, but it also uses the model scheme form the methodology of system dynamics that allows to create acausal models.

    I also do guess you might have never heard about acausal models and of Modelica either. This is why my thread does provide information to give you my reader the chance to know what I am dealing with and offer you the possibility to ask questions!

    I am a retired person and so my objective in my hobby, naval modeling, is to use a project of building a model of a sailboat as the red line to get in touch with many technologies. I am on this project more then a decade and so over time I have dealt with all the classical technologies you face when building a model ship, here you can read my report of a build from scratch! Quite a number of years ago I also engaged in electronics and self build electronics evangelizing its use in naval modeling. I started my professional career long ago as what is called a "Field Application Engineer" in a US semiconductor company. So as many careers you start technically and then move towards managerial positions, so my skills in mathematics and electronics have eroded and in the last 4 decades extensive developments have taken place both in electronics and mathematics, also due to the fact that the performance PCs offer on the desktop have made discrete and numerical approaches feasible! So taking the task to model my sheet control system using Modelica as a language that will be used in the environment offered by Wolfram Software SystemModeler and adding to this the capabilities of the Wolfram language and the software tool Mathematica Opens the door for a whole bunch of activities, experiments, studies and researches that I love to do!

    So I hope this explains to you why I want to to model the pulley and its individual sheaves using system physics and to explain why it is so key to me to be able to model my pulley this way!

    All your explanations about why either kinetics or momentum that implies mass does not apply here I do fully agree, so I have no clue how to make a model! What I need to accomplish using system physics is to identify at which pulling force applied to the rope there is an equilibrium between the force pulling at the rope and the friction generated in the 7 sheaves the pulley contains!

    I plan to apply the Euler-Eytelwein-Formula that gives an equation that puts the friction and the force pulling from the rope in a relation depending on the angle the rope embraces the sheave and geometric data of the sheaves! I know and I have downloaded myself a master degree work of a student about the friction between a rope and cylindric bodies also analyzing the diverse sailor knots. He used in his thesis the finite element method!
    Last edited: Apr 22, 2015
  5. Apr 22, 2015 #4

    jack action

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    If there's no mass and velocity, you cannot call it a dynamic model.

    The fact is that in your model there is a mass (it's the boom of the sail). You just assume that the mass of the rope is such a little mass that it will have negligible effect compare to the boom. One of your «discs» should have that model implemented. All the others will be connected to it and the mass of the boom will have repercussion on them.

    Although I'm not particularly familiar with Modelica, the modeling you are trying to do is purely based on mathematical expressions where there are x equations with x unknowns. But you still need to use the appropriate equations that represents adequately the physics behind every system component.

    If you use such a model with dynamics in mind, you either will have to input a force or a velocity function at the sail and your program will give you the response your looking for (which I'm not sure what it is).

    If you are looking for a simple static equilibrium, such a program will be kind of useless as simple free body diagrams will give you the only possible answer you can get (still a system with x equations and x unknowns, but no derivative or integral involved).
  6. Apr 22, 2015 #5


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    No friction either without mass.
  7. Apr 22, 2015 #6

    jack action

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    Actually, I would say «no friction without matter (substance)»; you don't need a mass (property of the substance), only a normal force.
  8. Apr 22, 2015 #7


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    The original J-boat pulleys likely had only sleeve bearings, but your detailed drawing shows what looks like a ball bearing. What do you actually want to model?

    You use the phrase "red line" in a strange way. This usually means a limit to some sort of behavior. "Don't cross the red line, or I will shoot at you," sort of thing. You seem to be using this phrase to mean something more like "a learning vehicle," or something similar. Perhaps you might want to consider your terminology here.

    Your project seems to be exceedingly ambitious in that you have set yourself a physical problem, but then you have also set limits on how you will allow it to be solved, all the while admitting that you don't really understand the physics of the whole system. This seems to be unduly limiting if your have any interest at all in your original objective of learning how the system works.
  9. Apr 22, 2015 #8


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    Your focus on "system physics" seems to have obscured your vision as to what is really important in terms of the pulley friction. The Euler friction equation you mention will describe the friction between the rope and the pulley roller. It is usually safe to assume that there is no slipping at that interface, but the friction you need to worry about, the one that will cause your system to bind up and fail, is the friction between the roller and the axle. Whether it is a simple sleeve bearing or a rolling element bearing as shown in your detail drawing, this is what will make or break your system. The Euler friction model is not relevant here. To model this friction properly, we need to know whether you are talking about a sleeve or a rolling element.

    Looking at the forest, it seems you might not have seen a tree!
  10. Apr 24, 2015 #9
    Dear friends, thanks for the qualified responses. I will try to respond to each remark!

    Lets start explaining what i do mean with read line. This is may be a case were translating terms too literally has the result of not expressing what is meant! With read line I mean a guiding road along whose path I meet the most diverse technologies to engage with! I hope this explains my use of this term!



    This picture shows visually the elements used with the Euler-Eytelwein-Formula! if you read my previous post you can see that the friction in the axis is minor, only between 0.03 to 0.05 percent of the total friction. I plan to use ball-bearings! key to the friction is the angle "beta" expressing the angle with which the rope embraces the sheave! No slipping is assumed in the Euler-Eytelwein-Formula, as you can see there is no differentiation of the friction between a slipping condition and a none slipping condition!


    This image gives you indications of how the friction effect of a rope around a cylindrical object has been recognized since centuries and is used until today!

    @mheslep: This is the great aspect of physics versus pure mathematics! Your statement not only expresses why I am blocked right now, I fully agree in principle to what you write, but the physical world unfortunately shows and proves that there is actually friction. And this friction is mayor. I have received elsewhere a response from a skipper in the real world that owns a sailboat that has a pulley in the path of his rope. he confirms that the friction is a mayor issue when handling the sailboat! It is not just the friction between the rope and the sheave, but also between the rope and the lateral walls of the blocks! This advice has made clear that when designing the blocks for my model I will make sure that as far as possible any friction between the rope and the lateral faces of the block is not taking place! This is specially critical, as the blocks will have to handle situations where the boom of the sail is orthogonal to the hull and so the sheaves will have an attitude close to horizontal!

    @jack action: I have chosen to respond to you as the last of my appreciated readers of the thread, because it might be the most complex one to be answered and may be some of the other responses can be involved!

    My focus on System Physics is due to the fact hat it is centered around using the complex dynamics methodology and the principles of the modeling language and environment of Modelica. Without understanding both enough to judge why they both represent a foundation of system physics, and simply taking it as a fact that I plan to use Modelica as the modeling language, the discussion is missing its bases. Complex system dynamics has not to be confused with "dynamics" as understood as a term in physics! This methodology has been developed in the context of finance sciences and is not only used in engineering, read about the markets when investigating about Dymola, one of the Modelica environments, but also in the study of corporate management. It is basically a method that makes it possible to model the complexity of real world, for which i can relate you to 2 university courses available for free from the MIT vis its OpenCourseware offerings. Engineering Dynamics and System Dynamics , the latter being a course offered by the Sloan Institute from the MIT, the inventors of the methodology! Also famous is the application of this methodology by the Club of Rome in its famous study about "Limits of Growth". So equilibrium is basically the goal of people searching for a sustainable future of our human society!

    I did investigate also the well known software tool "Matlab" and its toolboxes being "Simulink" a key tool. i had to investigate alternatives, as matlab used to be available only for commercial customers at prohibitive cost or for students while at the university. Matlab and the like model and simulate using what is called the "causal" objects and which is closely related to the way controls are described. It starts from some inputs, those are processed in what is called a "plant" generating an output. A feedback loop allows the control circuit to adapt to external noise and to changes. Should a model be used in the way that what used to be output becomes the input and what were the inputs are the new outputs, the original model is useless. So this relation from input to output is why it is called a causal object. An often used example to explain this is a simple circuit, where a DC power source is applied to a DC motor, voltage and current being the inputs and the torque generated by the DC motor is the output. Her i want to indicate that this is called a multi domain modeling and simulation, as we have an electromagnetical part in the model and a rotational mechanical part combined. Would we add to this an analysis of the heat flow in this same system we would already have three domains involved.

    So in my research for alternatives I found Maple and Maplesim from Maplesoft. This products are equally prohibitive in cost and only available for commercial customers and students. maplesoft has not changed this until now, but mathworks has made available Home Edition licensing. With MapleSim I had my first encounter with Modelica! The I learned about Wolfram Softwares offering of Mathematica and of the SystemModeler, which is also based on Modelica for modeling and simulation. But Wolfram Software does offer non commercial licenses and so i decided to embrace their product offering and as a consequence the language Modelica. Now Modelica allows the use of so called "acausal" components. When you search for this in the tutorial informations available for MapleSim you also find the example of the DC motor circuit. So when you replace the power source in the circuit presented above by a display of the current and voltage instrument and you apply a physical torque to the DC motor, that same model using an acausal model of the DC motor converts a DC motor to a generator and you can plot the voltages visible in the instrument applied to the circuit! So acausal models have a special beauty in that they are much better reusable, as the direction from Input and Output can be inverted. So that is the reason why the models that can be generated using Modelica are called acausal!

    Searching in Youtube for tutorials about Modelica I found an excellent video tutorial series about Modelica from a professor Werner Maurer from a university in Switzerland, Winterthur, the ZAHW Winterthur. He used Berkeley Madonna as a software tool to present the concepts of Modelica and referred to videos about System Physics used in the context of his teaching of physics to engineering students at his university. So I learned about system Physics, its close relationship to Modelica and being a way to view Physics that works with having the students model their home assignment work using system dynamics modeling and either Berkeley madonna and/or Modelica, here the free trial version of Dymola! So here I found "the link" between physics and modeling that I required to model my sheet control system, a multi domain system strongly interacting with the physical world, as it is typical for a sailboat.

    Now researching further about "System Physics" I learned that it is based on didactical research work being done at the university of Karlsruhe and where you can find extensive information on a website called "Karlsruher Physics Course" in english language. Now, as you are all aware I am sure, even the classical mechanics can be viewed mathematically from multiple perspectives. There is the classical historical view as it is presented for example in the famous physics book from Tipler, here a link to it. Then I learned about a fascinating course of theoretical mechanics teached at a german university, unfortunately only in german available as video lectures and material related, where the content of physics in a bachelor study is presented using topological manifolds. Also I am sure you are aware of Lagrange view and Hamiltons view. Each of this perspectives that uses a different set of mathematical tools might open new windows in research efforts. So in the same context system Physics is a view on physics that interestingly leads to find that the most diverse fields in physics can have their rules expressed by the same mathematical equations. You will find more details researching the link to the Karlsruher Physics Course!

    Allow me to close this answer technically by presenting what is called a fluids image of a mechanical concept and in which I am guessing right now is hidden for me yet the solution to my lack of knowledge:


    As you can see in this picture is the momentum of a mechanical system contained in the recipient. The recipients geometry and that of its content reflects the equation for the momentum:

    p = m * v

    and that of the kinetic energy:

    Wkin = P * v/2

    But what is key to note is that velocity vector shown in blue starts from the environmental level given by earth! So the content with a velocity of "0" would just mean that the momentum content of the recipient has the same level as the environment earth! The force filling momentum into the recipient is called in system physics terms also a momentum pump, as it extracts momentum from the infinite resource earth into the recipient. same applies to the kinetics energy! Kinetics energy in this model is "0" when the volume of the recipient is at the same level as the environment.


    To add in my efforts to understand myself and to respond to you my dear readers and repliers, the fluid image of an inelastic collision should extent this principle! You can see that the equilibrium case is a "special case" when the velocity = zero! This image representing such an inelastic collision shows an object represented by the left recipient colliding inelastically with an object on the right. As you can see a movement in the opposite direction as the first object leads to a momentum content below the earth level.

    So I close now my post of today saying that I will put my effort into grasping how to model my individual sheaves and promising that as soon as I find out how are i have made some substantial advance I will present it here. Should, what I would highly appreciate any of you dear readers have something to help I will be monitoring this thread and respond! I am aware that there are many ways to Rome, but it is may objective to model my sheet control system using Modelica and I am not aware of a more adequate path to achieve this then by applying system physics to model the physical behaviour. And even more, I find the learning of using system physics a worth effort to do, as I am also studying the mathematics required to be one day able to to understand physics by applying all the mathematical methods mentioned. Even more, I have downloaded the master thesis of a student that dealt with modeling the nautical nodes applied to ropes using the finite elements methodology!
  11. Apr 24, 2015 #10


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    Helmutt, look again please at your friction problem, using your image.

    The Euler friction equation describes the friction developed between the rope and the rotating pulley. It does not describe the axle friction. In your case, the tension difference is due to two effects: (1) inertia of the pulley, and (2) axle friction. With ball bearings, the axle friction will be small and the pulley inertia possibly a more significant term, depending on the acceleration of the rope.

    Look at the center picture below. It is clear here that the force difference generates a torque that is transferred directly to the ground. That is NOT your case.
  12. Apr 24, 2015 #11


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    This is a continuation of my previous response; I was interrupted for a few minutes.

    If you will draw a proper free body diagram of the pulley and the rope segment in contact with it, the external forces acting on that FBD will include:
    1) Rope force to the left;
    2) Rope force to the right;
    3) Distributed tangential force on the inner surface of the pulley (axle friction force);
    4) Radial force from the axle.
    The first three of these produce torques about the axle (the 4th produces no torque because it acts through the axle center line).

    What this shows is that the distributed force between the rope and the pulley does not enter into the motion of this system; it is an internal force pair (a pair because it acts on the pulley in one direction and on the rope in the opposite direction). It is this irrelevant internal pair that is described by the Euler equation.

    What you need is an expression for rolling element ball bearing axle friction. For this, I would refer you to Rolling Bearing Analysis by Harris (1965), p. 446.

    All the Dymola in the world will accomplish nothing if you use the wrong model.
  13. Apr 24, 2015 #12
    Sorry to contradict you! here a link to an article about the Euler-Eytelwein-Formula, in english also called the "Capstan Equation"!

    I am showing the second paragraph in the page to which I deliver the linK.

    Because of the interaction of frictional forces and tension, the tension on a line wrapped around a capstan may be different on either side of the capstan. A small holding force exerted on one side can carry a much larger loading force on the other side; this is the principle by which a capstan-type device operates.

    Specially that part that says that a "small holding force exerted on one side can carry a much larger loading force on the other side!




    bccfc7022dfb945174d9bcebad2297bb.png being the angle the rope embraces the sheave in radian

    66059b8c636bc4dbcf897bae0e12b2b6.png being the static friction coefficient

    Fh being the pulling holding force and
    Fz being the force from the load!

    Unfortunately i did not find an equivalent wikipedia site in english to the german version I am using! But in this article it is written that the holding force required to hold a load decreases fast with the increase of the angle with which the rope embraces the cylindric body! it gives as an example a steel cylindrical body were an embracing angle of 360° of a steel rope already reduces the holding force to just 40% of the load! from the same source I have the data that the impact of the bearing of the sheave is just 3% in case of a ball bearing and can increase to 5%. In this context the axle friction is closed to negligible and using a 3% value will be at least initially acceptable!

    As I mentioned, it is a mistake that is not encountered with people familiar with naval ships or naval models, but often with people not familiar with this topic! Here a link to a PDF archive that is even more explicit!


    This copy of the first page of the pdf archive I have supplied the link to is more then adequate to not only illustrate the impact of the friction in the individual sheaves of the pulley I plan to model. The archive goes on and even presents a simple pulley with 2 sheaves!

    What I am still in doubt with is a comment that the diameter of the cylindrical object, in my case the sheave is not relevant! Here the equation I have found at the german version of the wikipedia about the Euler-Eytelwein is more accurate. from practical experience it is well known that if the diameter of a sheave is too small then the rope will not make the sheave turn and so it will not work as an element of the pulley!

    But back to basics. I want to know how to apply Euler-Eytelwein in the context of the System Physics!
    Last edited by a moderator: Apr 17, 2017
  14. Apr 24, 2015 #13


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    Can't help those who will not be helped. Have a ball with "system physics"!!!
  15. Apr 24, 2015 #14


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    Let me clear up one point for you. The Euler friction equation is not news to most folks at PF. I first encountered it as a college freshman 58 years ago, and I've used it off and on, as needed, through the years. I try to avoid using it where it is irrelevant.

    This is not so very foreign that your elaborate explanations are required. In my own case, I am an accomplished sailor, and I did my doctoral work many years ago on the hull design for heavy displacement sail boat like a J-boat. I do understand how the boom sheet system works, so no need to lecture on that point.

    If you look at the information you gave, it is for a CAPSTAN, but that is not the same thing as a PULLEY.

    In this case, BASICS has to mean physics, not fancy computer programs. Your insistence on "System Physics" reminds me of a man with several new hammers who wants to try them all out by driving a box of screws.
  16. Apr 24, 2015 #15
    Sadly, not everyone has your background and as a consequence I get to read stuff like there is no friction between the sheave and the rope! You might have noticed that I am not an english native speaker so I used the term "capstan" as the Euler-Eytelwein formula was also named this way in the Internet and I have been doing the effort to search for an english site of wikipedia addressing the Euler-Eytelwein formula! So never claimed it to be the same as a pulley! You are welcome to prove that you are more proficient in either spanish or german, as to my best effort I am trying to express myself in english! So lets let emotions and assumption of disqualifying one another outside as this is neither the intention of either of us I do assume. So mutual respect and the believe that everyone is doing his best effort to exchange ideas and concepts, even if it is across language and cultural borders! i do never write for just a single person in any post I place in the Internet and the responses by some are a prove that not everyone might be as proficient as you are my dear OldEngr63, roughly 5 years older then I am! Now, being an accomplished sailor I am more then surprised that instead of being so aggressive to me you are not lecturing mheslep and jack action for whom I mainly wrote and supplied the data about the Euler-Eytelwein-Formula that builds on a wisdom about 4 centuries old!

    But having been so aggressive with me, I may remind you why system physics is of importance to me? I did explain this in my first post to this thread! it is the view on physics and mechanics that best fits to my intention to model as a first part of my sheet control system the pulley. With all your wisdom apparently you have never heard about system physics and apparently you are not even willing to look into it, but you are willing to criticize me, who explained why this is of importance to me!

    I am posting here in a forum about physics! So asking questions about a respected branch in physics is in line with the objectives of this forum. So either you have something constructive to add value to this thread or you simply enjoy being offensive with a person coming from another cultural background! Excuse me dear old OldEngr63 but your personal offending response, your arrogance to somebody not being a native speaker and who is doing his best effort to supply information leaves a poor picture of yourself!
  17. Apr 24, 2015 #16

    jack action

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    Even though the Capstan equation can give a value for the friction force, this can only be a maximum force. The true force will be one that can be resisted. In your case the torque produced by the friction force of the rope will be equal to the torque of the rolling resistance of the bearing.

    To better imagine the phenomena, you can take for example trying to push your hand across a fixed piece of wood: You will feel a certain resistance due to friction. Now put the piece of wood on a slab of ice and the resistance will be a lot less. Why? Because now the piece of wood slides on the ice before the friction force between your hand and the piece of wood can increase to its maximum value. Your hand will not move with respect to the piece of wood and the force will be equal to the friction force between the piece of wood and the ice.

    That was the point that @OldEngr63 was trying to make.

    I do understand what system physics is all about as one of my teacher in my university has developed a similar theory and it was part of my mechanical engineering program. He was an electrical engineer and I always felt it was more a tool for an electrical engineer to relate problems from other fields to equations and concepts he understands better. I, on the opposite, tend to model electrical devices as mechanical devices to better see them. This is probably why I never really used this theory since then (over 15 years ago).

    The point is, you should understand the relevant equations before «translating» them into system physics. Do a free body diagram for every component and develop the equations for them ([itex]\sum F = ma[/itex]; [itex]\sum T = I\alpha[/itex]) for every axis (x, y, z). I think this will help you a lot in your quest (It certainly won't hurt).
  18. Apr 24, 2015 #17


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    Hi Hellmut

    Of course there is nothing "paradoxical" about size of pulley being irrelevant. The Euler-Eytelwein relates the maximum difference in tension that can be obtained on either side of the pulley, or cylindrical object, due to the amount of wrap, by using only the angle of contact of the rope with the object. Big pulley, small pulley - using the same coefficient of friction between the rope and pulley, the ratio of Sn to Sz will always be the same. Of course, the absolute values of Sn and Sz will be different.

    The size of the cylindrical object comes into play in several instances. The number of wraps needed for a winch, the holding force around a non-rotating cylindrical object to balance a much larger force. For no slip, a lessor number of wraps is needed for a larger radius. And the friction through a jammed pulley.

    Which brings us to the equation you have seem to have overlooked. Namely the moment equation.
    Mn = Rb P

    Rb is the radius of the cylindrical object.
    The moment might be supplied by a motor for a winch or capstan, or even a hand turning a lever. For tying up the rope to a pier to dock your boat, the moment is supplied by the torsion of the pillar and the earth.

    For the pulley, for nice rotation, one wants the moment supplied by the bearing to be much much less than that due to the Capstan equation. If the bearing moment is greater than, the pulley will not rotate and either there will be undesirable rope slippage, jamming.

    In other words, one wants the tension on both sides of the pulley to be as close in value as possible, and NOT near anywheres close to that given by the capstan equation.
  19. Apr 25, 2015 #18
    Fine, experiments verifying the results will tell what is the truth! Still so, as I have the firm intention to model my sheet control system using the language Modelica. I want to be able to model the sheaves and the pulley as part of this sheet control system! As good practice when modeling is always comparing the result from simulation using a model and the results from experiments. This is why I am pursuing in paralel to have the model environment to get data from real hardware and verify the results. Its called hardware-in-the-loop and software-in-the loop!

    Additionally as a resource to investigate I do have the master thesis about the friction between a rope and a sheave and in different kind of nautical nodes. So my suggestion is to let the discussion of the applicability of the Euler-eytelwein-Formula out. Just have to see how to model the sheave and combining 7 instances of the sheave to model the pulley using in a first step the scheme from System Physics and then to pass this to Modelica. Thanks.
  20. Apr 25, 2015 #19


    Staff: Mentor


    In post #1, you show a picture of a sailboat's mainsheet system. There are several blocks, arranged to give a 4:1 mechanical advantage.

    In post #9 you show a picture of a bollard, and speak of the capstan equation. Yes, the capstan equation models bollards.

    A block is far from a capstan, and the capstan equation should not be used to model blocks. An ideal capstan does not rotate. An ideal block rotates with zero friction.

    It is true that line-block systems with high mechanical advantage can have significant friction. The properties of the line may contribute as much as the properties of the blocks. Simple equations are given in Wikipedia.

  21. Apr 25, 2015 #20
    Thanks for the valuable source, I will study it! You should be aware of, that "experts" in building sailboats have considered my implementation of the pulley, 6:1 advantage from the sailboat Endeavour to be inoperable in a model due to friction that makes it impossible for the sail to open by pulling rope through such a pulley due to friction in a model of 1.20 scale! This is why I am spending the special effort to capture the friction!

    Purpose in the context of a much more complex system I want to model and simulate is to verify it is operatable, so i want to identify what the minimum wind strength blowing into the sail has to be so that there is an equilibrium between the friction of the rope in the 6 advantage pulley and the pulling force from the sail!

    As with any model, simulation will be compared to measured data to verify the models. I am more than happy if I can prove the 1:20 scale of the model does not prevent the model from operating adequately. But there are also friction components in the pulley resulting from friction between the rope an the side walls of the blocks! Reports from "real skippers" of real boats with y pulley have confirmed the sometimes higher force required to overcome friction!


    The 2 sheaves in the drawing on the right side are actually each 2, one on each side of the deck and on the boom. The block to the right on the deck has 2 sheaves!


    This very simplified drawing shows the 2 orthogonal extreme positions the boom of a sail can take and does so by showing that I plan to have the main and the foque sail to have a boom. Now the angles the blocks and the rope through the seven sheaves of the pulley have, differ if the boom is above the center line of the hull or if it is in a 90° position. The greater the angle between the centerline and the boom is the more the rope will display friction with the side walls of the blocks! Also the angles with which the rope embraces the sheaves will vary a lot!
  22. May 12, 2015 #21
    Hi friends, this topic is exactly what I love about my dealings with physics in the context of my naval modelism in which I run into the issue of friction in a pulley!

    After spending a lot of time studying the issue, sorry, I know I am not efficient in my way to deal with it, I have run into an sometimes highly emotional feedback! The result is, that I will probably use the finite element methodology to compute the friction is the sheaves of the pulley within the mathematica software tool, so that the results from the finite element computation of the friction is passed and assigned to the proper parameters in the Modelica model of the sheaves by using the tool SystemModeler, a Modelica modeling and simulation tool from Wolfram software.

    The reason for doing so is that the ≤ sign in the Euler-Eytelwein formula in my case is inadequate, as the uncertainty expressed in the formula by this sign is in the order of magnitude relevant to respond what strength of wind is required to pull the rope through the pulley through its pressure into the sail!

    But seeing the contradiction of the experts in sailboats and sailboat models and the physics, I decided to contact the institute of physics at the university Ludwig Maximilian Universität in Munich to get an advice from them. Allow me to summarize the responds I received, as I guess it can be of interest to you my dear readers!

    I was told that in an idealized view from the physics standpoint the friction takes place only in the bearing of the sheaves between the disc and the axis around which it turns. In reality, and so inline with what the experts from the practical view point express, be it experts in sailboats or in model sailboats, there is a loss of pull in the rope due to friction and this is what the Euler-Eytelwein-Formula expresses, but unfortunately not in an equation with an = sign, but with an ≤ symbol!

    So using the finite element computation methodology and in this case based on the Master-Thesis to which I have supplied the link above will allow me to compute the friction in the sheaves of the pulley, to link its computational results dynamically to the proper parameters in my Modelica generated models within Mathematica to the SystemModeler simulation environment!

    I tell you that this by sure will take me a year or so to achieve this! And I just mean the implementation of the finite element computation of the friction. This is a 2-way interaction between the Mathematica tool and the SystemModeler tool. As to be able to generate the plots that will allow me to analyze the results has parameters that will be varied over certain ranges within the simulation of the models within SystemModeler change the parameters of the finite element computation that will take place within Mathematica!

    Having expressed what I have just done in writing I consider it mandatory to express key considerations that I want you to be aware of to judge my way to proceed!

    1. All this modeling and simulation finally depends on the quality of the used models!

    This will require to define experiments to verify and improve the quality of the models. Here the other aspect of this project I have been dealing with stays relevant! "Hardware-in-the-Loop", short HiL, and "Software-in-the-Loop", short SiL, will require to have the experiments to generate data that is the fed back into the Mathematica and SystemModeler environments to use solvers to fine tune the equations in the models to be as close as possible to the data generated through experiments!

    2. All this does not have any impact whatsoever on the results of this modeling and simulation efforts.

    I do not exclude any possible outcome of this efforts. It might prove that the pulley is inadequate to be used in a sailboat model at a 1.20 scale as planned due to excessive friction of the rope in the sheaves of the pulley. It might prove that the friction between the rope and the cylindric surface of the sheaves is irrelevant and that only the ball-bearing I am using adds friction to the equation:



    Here 2 pictures of the sheave of which I will be using 7 in my pulley! Its diameter is 20 mm! In the center you can see the ball bearing I have inserted there! This sheaves will have to be integrated into blocks as can be seen in the photo from the original Endeavour. An important information I have received from real sailboat skippers with experience in navigating sailboats with pulleys is, that the lateral friction between the rope and the block adds substantially to the friction! Now the aluminium used for making the sheaves offers a very reduced friction between the Aramid rope with 0.9 mm diameter I will be using. So I will have to ensure that the wooden lateral faces of the blocks have as little as possible contact with the rope! The higher radius of the lateral borders between which the rope will be placed should prove to be a good protection! Those lateral borders of the sheave to the rope will rotate with the sheave and so hopefully limit the friction added!

    3. Finding the ideal implementation of the pulley to achieve the lowest possible wind velocity to achieve the equilibrium between the friction and the pulling force, or to say it in other words, the tension of the rope around the sheaves that equals the friction generated in the sheaves of the pulley!

    Besides that obviously it will be interesting to compare the results of applying the Euler-Eytelwein-Formula using the equal sign to the results from the finite elements computations method, verified by comparing real data with the results from models. How close or how distant will the result be from the maximum value resulting from the Euler-Eytelwein-Formula!

    Buts also of high interest to me will be to find out how much the result can be influenced from design decisions in the implementation of the elements of the pulley!


    This drawing allows me to present to you my dear readers, as poor and as of limited use the drawing is, what parameters are available for me to "play" with! Allow me to expand on this!

    The boom of the main sail in my sailboat model has a length of 1 meter. So where along the rear end of the boom I do place the blocks containing the sheaves shown in this drawing has an effect on the angle with which the rope embraces the sheaves! You can imagine, the more I move the bottom right 2 sheaves to the right the smaller will be the angle with which the rope embraces this 2 sheaves, one on each side of the deck of the sailboat!

    Similarly, also the placement of the other sheaves impacts the angle with which the rope embraces the sheaves and, as assumed the friction generated, not by the bearings, but between the rope and the cylindric surface within the sheaves! Of interest will also be to understand the impact of the length of the ropes from which the 2 heaves on the right side are connected to the boom! The pulling force on the rope within the pulley is the result of the forces from the boom to the 3 sheaves that are connected to the boom and also to the ones connected to the deck! This means that the pulling force resulting from the wind blowing into and around the sail are fed into the pulley at different locations!

    Finally, and here I am not sure, as the photos do not reveal enough details, but I am guessing that not all of the sheaves used on the original Endeavour have the same diameter. Why is this so and how can by this means the impact of the pulley be positively influenced!

    I like to point to a comment made by the physics scientist I did talk to at the Munich university. he said that their goal was not just to teach students the laws of physics and how to apply mathematics to them, but to learn to apply that knowledge to resolve real world problems, not in an ideal environment, but in the real one! remember that according to him this helps to explain the difference between the responses I have received from physics in the physics forums and from sailboat experts in the world of real sailboats and of sailboat models.

    The controversial that you have been able to see by some of the comments here in this forum, just as an example, make the project I am dealing with just much more fascinating and attractive to me! Just to make sure I do not get more comments like the one I got from OldEngr63. I am not lecturing anybody nor am I saying expressively or implicitly that somebody lucks intellect or knowledge! I am simply sharing openly and driven by the fascination of my project my work and reflections around the project!
  23. May 12, 2015 #22


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    We will all be looking forward to seeing your results.
  24. May 12, 2015 #23
    Me too! Lets hope I will stay alive and mentally healthy long enough to achieve my goals!
  25. May 16, 2015 #24
    Hellmut, I think the problem of reducing the scale of a J-Class sheeting system is not one of friction in the classic sense. In the real world of modeling, the problem is not the pulley system, but the sheet (rope) itself. The sheet material has memory and resistance to bending. In low wind force on the sails, notice that the rigging is hanging loose with the sheets not linear. You need to forget about physics for the moment and do a hardware search for the most supple and light weight line you can find that can carry the loads in the system. With this task completed you can model the system.

    Friction in the system must be kept as low as possible in all components including boom vangs, sheave attachment points and of course the pulley bearings.
  26. May 16, 2015 #25
    Hellmut, one more point. Weight of the blocks must be a light as mechanically possible.
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