## Introduction

In ropeways systems, such as the ones used in ski resorts, having a high flow of people is really important, to do so you can either use larger cabins or increment the speed of the cable system.

For this reason modern ropeways are almost all detachable, this means that at the stations the cabin/chairlift is detached from the portant and traent cable (which maintains a constant speed but does a longer travel) to allow the users to sit/unsit in a safe and comfortable way as the system slows down.

In order to do so the cabin is connected to the cable trough a grip mechanism composed by two claws that, thanks to a spring, hold firmly the cable. There are various configurations possible for this mechanism but the working principle is the same for all of them.

When the cabin reaches the station the claws are opened through a guide which wins the spring force opening the claws, as this happens the cabin is unconstrained from the cable and its movement starts to be controlled by the station guides. These presents a series of wheels that brake the cabin until it reaches the speed set for sitting/unsitting. The wheels are also able to accelerate the system to the cable speed when exiting the station, in this way the cabin can reconnect with the cable and continue its journey.

My work started from the analysis of different videos of the system in action, from this videos I tried to learn as much as possible about the working principle and the physical and geometrical quantities that I had to replicate in my model to match with the real system.

Hopefully there is a forum of ropeway lovers where I found a pdf file[1] with some geometrical details about the mechanism and other part of the systems as the cabin.

After having recreated the CAD I have created the model in Adams and I have started several analysis in order to reach realistic results in terms of kinematic output by adjusting the unknown parameters in the system.

After the model reproduced properly the real system I have extracted the forces between the mechanism and the station guide.

## Objectives

My project aims to extimate the forces that are generated during the detachment phase of the system, with focus on the forces exchanged between the guides and the wheels on the claw.

I have also conducted a sensitivity study to investigate the effect of the claws spring characteristics (stiffness and preload) and cable speed on the force acting on the upper guide.

Knowing this forces is important in order to design properly the mechanism and prevent its failure.

## The modelling problem

There are several bodies composing my model, as there are contacts involved in the interaction between some of them there are more than one kinematic chain. I will go trough a description of each kinematic chain talking about its bodies, joints, motion and forces (springs, contacts).

**Cable**

In the real system the cable is a steel wire of about 60 mm diameter, this was reproduced by three rigid cylinders of the same diameter with total length of 1.5 m, the reason for the three cylinder is purely estetical, in order to enable the possibility of using different color and evidentiate the displacement relative to the claws.

This cylinder is then attached to another cylinder (stiffness for traent cable, SFTC) which gives the translational Z motion to the cable, this allows the translational Y DOF to the cable, obtained by placing a translational joint between the two, a spring reproduces the shear stiffness of the cable, the value for this stiffness was set to 2000 N/mm.

Also the torsional stiffness of the cable is considered, but is attached to the claw and not to the cable to neglect any sliding in the contact which can bypass the effect of the cable stiffness on the system, the value was set to Kt = 5E6 Nmm/rad, as this stiffness is attached to the claw I have used a step function that brings the stiffness to 0 when the cable is detached.

The motion for the cylinder is set by General motion, the definition of the motions sets also the constraints for the SFTC part: The speed along Z, is set equal to VZ_cable=5 m/s which is a common value for ropeways, then I’ve set a displacement law for Y using a step function, in order to reproduce the cabin hitting the ground from the top.

These two parts build up the first kinematic chain of the sistem, which has 1 DOF, given by the Y traslation of the cable.

**Grip Mechanism & Cabin**

The sources for making the cad model and getting the dimensions of the system where a YouTube video from Leitner[2] which shows the maintenance procedure of the system, providing an overall look of the mechanism but not showing technical details.

The second source was a pdf file found thanks to an online forum of ropeways lovers: funivie.org, the pdf shows the technical drawing of a Grip Mechanism assembly with some quotes, from this file I’ve created the 3D model of the two claws using Solidworks.

The other components of the mechanism were directly generated inside the Adams environment, because they were simple geometrical entities like cylinders or spheres, another reason was that when importing parasolid .x_t files in Adams the program doesn’t recognize the geometrical entities (like cylinders) and so it is not possible to use analytical formulations for contacts.

The simplified model of the mechanism can be seen in the picture below together with the traent cable

The stiffness of the spring was derived by analytical expressions which links stiffness to the dimensions of an helicoidal spring[3], starting from the dimensions found in the pdf I have derived the stiffness of each spring, the two original spring were then reduced to a single spring of double stiffness.

Using the data I have obtained a total stiffness of about 500 N/mm for the claw spring.

To model the contact I have placed three cylinders of 2 mm radius on the contact surface of the claws, one for the upper and two for the lower (placed at ±45°). The formulation for the contact was set to Cylinder to Cylinder with Coulomb friction to avoid sliding. I have measured the gripping force of the claws on the cable considering only its normal component, which for equilibrium is also equal to the Fx of the Upper claw cylinder (the one on the left).

Also the cabin has been modeled following the quotes found in the pdf file, a simple model was created in Adams using cylinders, as in the real model the cylinders are steel pipes, the density was modified to reach a reasonable weight for the cabin, which was set to 872 kg.

The person inside the cabin were modeled by nine 160cm high cylinders equally distributed, the density was adjusted in order to get an approximate weight of 73kg for each person, giving a total mass of 660 kg to the passengers.

These bodies build up the second kinematic chain of the model, the constraint used are:

- A revolute joint between the two claws, allowing the rotation around Z
- A revolute joint between the lower claw and the cabin, allowing rotation around X, this joint has also friction in order to prevent the cabin to enter into an oscillation regime when slowed down.
- A fixed joint between the cabin and the passengers
- Then there are two general constraint equation that set AX and AZ equal to zero for the lower claw, regarding the latter I have also taken into account the torsional stiffness of the cable, this lumped stiffness was attached straight to the lower claw in order to not involve the contact and prevent sliding.

This kinematic chain has a total of 6 DOFs.

**Station (ground part)**

The station consists in a series of guides that have the function of releasing the claws, sustain and guide the cabin during the detached phase.

The modeling of the station was the critical point where I’ve focused and went through several reviews.

The first approach I tried was to model these guides as planes fixed to the ground, the distance between the upper and lower guide planes was derived by the pdf file stated above, while the inclination of the “invite” planes was set to a reasonable value.

The choice of using planes rather than solid entities permitted to use an Adams suggested formulation for contacts: Sphere to Plane.

However the simulations results using this simplified approach shown a force profile for the upper guide which was full of spikes, at first I tried to get rid of those spikes by changing the quantities that could have been related to them (such as stiffness and damping for spring and contacts involved) but the results were still unsmooth.

The final solution was to model the guides through splines profile in order to get a smoother geometry, the model was created in Solidworks.

In this way the contact between the front and the rear wheels of the claw is a Solid to Solid contact, because the Wheels are actually part of the Lower claw part (and so fixed to it) the contact was set to have no friction in order to reproduce the real system where the wheels are free to rotate.

The same procedure applies to the contact between the upper claw and the upper guide, also here i have used a solid to solid contact formulation, on the upper claw I have placed a sphere in order to have a smooth geometry in contact with the guide, considering that together with the Y translation the point of contact also moves along X axis and rotates around Z.

There are also other guides that control the X translation and the Z rotation of the claws, these contacts are modeled using planes and a sphere placed in correspondance of the rear wheel of the claw, allowing to use Sphere to Plane contact formulation.

Howewer this was still not sufficient to get good results and the use of Parasolid for contacts was necessary.

**Spinning Wheels**

The last bodies of the model are a series of 14 spinning wheels that have the function of slowing down the cabin after the detachment, this wheels were modeled through cylinders of R=15 cm with a spacing between the wheels of 25 cm.

The wheels create a kinematic chain, they consist in two bodies: the cylinder previously discussed is connected to another cylinder (wheel guide). Starting from the ground we have a translational joint between the ground and the wheel guide along the Y axis, then the wheel guide is connected to the wheel through a revolute joint with rotation axis parallel to Z. Each wheel has 1 DOF, giving a total number of 14 DOFs for the spinning wheels all together.

The motion of the revolute joint is defined by the law shown:

This brings the speed of the cabin to the final speed by 10% steps.

The wheels interact with the lower claw through a contact between the wheel and the geometry wheel_contact_zone of the lower claw, I had to use also in this case Solid to Solid formulation with Coulomb friction, the normal force which produces the Friction force is given by a spring placed along the translational joint between wheel_guide and ground.

## Simulations and analysis of results

During the development of the project I have carried out several analysis mainly to adjust the parameters and get a realistic behaviour for the system. The final outcome is a 9 seconds simulation which starts with the Cabin coming at full speed to the station, hitting the station, being detached and finally slowed down to the final speed. I have also included a shorter 4 seconds simulation (Landing_Detachment) that stops briefly after the detachment and doesn’t include the slowing phase.

The simulations were carried out using SI2 solver and Parasolid for contacts. The convergence was found with solver error equal to 1e-3. I have used a step size of 1e-3 which also in this case was the size for the convergence of results.

The measure I’ve extracted from the system are:

- Claw Grip, defined as the X component of the contact force of the upper claw contact cylinder
- F_rear_wheel: vertical reaction force of the lower guide on the rear wheel
- F_front_wheel: vertical reaction force of the lower guide on the front wheels
- F_upper_guide: Reaction magnitude of the upper guide
- Torque_wheel1: Braking torque of the first wheel
- AX_Cabin: X angle of the cabin
- VZ_claw: Speed along Z axis of the lower claw

Here you can see some videos of the model during the simulation which help understanding how it works.

**AX Cabin**

The Ax cabin changes accordingly to the slowing of the system, this because the cabin acts as a pendulum. Thanks to the friction in the revolute joint the sinusoidal regime is suppressed as you can see from second 7 to the end of simulation. The peak angle reached is about 10° and is a result that is comparable to the one you can have in a real system [4]. You can clearly see this quantity in the first video.

**VZ Claw**

This measure is useful to see how effective is the braking wheel system in my model, the results shown that the wheels are able to slow down the cabin and follow the rule I have set for the braking. Also this quantity is clearly visible in the first and second videos.

**Torque on the 1st wheel**

It is interesting also to see how much torque the motor has to apply in order to keep the rotational speed set to the wheel. The results shows a peak of 9e6 Nmm of torque at the first contact, the torque then decreases to zero when the cabin reaches the wheel speed. You can also see a negative torque, the explanation is that the wheel spacing is such that there is a fraction of time in which two wheels are together in contact. As the speed of the upcoming wheel is lesser the previous is actually pushing the claw and so it provides a negative torque.

**Claw Grip**

The grip of the two claws goes very quickly to a constant value, which is kept until the detachment where it goes to zero. There is a zone in which the grip has a transient peak. This happens as the claw impacts the lower guide, the reason is due to the high peak of force received to the lower claw and to the inertia of the bodies which wins for a brief time the stiffness of the spring. In the second video this can be clearly seen.

**Reaction of the lower guide**

The reaction forces is divided into front and rear wheels. The geometry of the model and the forces applied by the wheels have the result of distributing the load of the cabin all on the front wheels after the initial contact.

Both forces have a peak when the cabin impacts the station (F=175 kN for Front and F= 30 kN for Rear).

After the detachment the mean value of the force is given by the mass of the system plus the force on the upper guide given by the compression of the spring. However you can see some periodic increments in the force, these are given by the impact between the wheels and the claw. You can clearly see how the duration of this peaks and their spacing is increasing in time (first graph), this because the system is slowing down, while using the Xcm of lower claw as indipendent variable (second graph) the spacing and duration is kept constant.

**F Upper Guide**

The force on the upper guide is also slightly influenced by the wheels, but the interest is on its peak value, which is of F=5688 N.

The shape is linked to the geometry chosen for the guide, the first “parabolic” increase is in correspondence of the constant inclination of the guide, when the upper claw reaches the spline profile the deformation of the spring increases its velocity and the force becomes larger reaching its peak.

I’ve also investigated the sensitivity of the Upper guide force regarding the characteristics of the spring (stiffness and preload).

Regarding stiffness the effect is to increment a little the peak value, for the three values tested (normal stiffness K=500 N/mm, +50% K=750 N/mm, -50% K=250 N/mm) the F peak changes linearly with a slope of 1,75 [N/N/mm], also the force after the transitory increases linearly with the stiffness.

The effect of the preload of the spring (which I have set to 6, 8, 10 kN) is similar but it changes also the force prior the peak also anticipating the contact between the claw and the guide. Regarding this latter aspect there is no linearity, this because the first contact time depends on the Y of the wheel on the top of the upper claw, which is dependent to the rotation of the claw and this quantity depends to the penetration depth of the contact, which is not linear dependent to the force applied according to the contact penalty formulation. The slope of the preload effect is 256 [N/kN]

Finally I checked the effect of a lower speed of the system, also in this case there is a reduction in the peak value, I have reported the results for two speeds (5 m/s, 4 m/s, 3 m/s). The slope of the speed sensitivity is 600 [N/m/s]

## Conclusion

My model seems to replicate quite well, at least qualitatively, some of the aspects of a real ropeway, such as the behavior of the cable-claw system and the action of the slowing wheels on the cabin.

The aim of the project to estimate the forces acting on the guides has been fullfilled, also the sensitivity studies for the upper guide force have showed interesting results.

In particular the presence of linear dependencies can simplify further optimization studies. To check the linearities I have used the superposition principle. I guessed the force in the case of having stiffness equal to 750 N/mm and preload equal to 10 kN at the normal 5 m/s speed, the force resulting assuming the validity of the principle is of 6640 N and if the linearities are true should be the same as the one obtained through a new simulation. Running a simulation with the same parameters I have obtained a peak force of 6705 N, the error between the two is lesser than 1% so I can conclude that the system responds linearly to stiffness, preload and speed in the range considered for these values.

However I have focused only on one geometry possible for the guide, while in a future development also a variation in the shape of the upper guide should be considered. To do so efficiently I started with the plane modeling of the guide which would have permitted an easy variation of the orientation but the problems encountered with the contacts made me switch to a solid spline profile which is not that easy to parametrize.

Another interesting aspect can be to provide a better model to the cable, determining an accurate value for the lumped stiffnesses, which in my model were based on the observation of the displacements of real systems and modified to reproduce such displacements in my model. The cable should also be modeled considering all its degrees of freedom and not only vertical displacement and rotation around its axis.

Finally it would be interesting, after having developed a new cable model, to remove the AX and AY constraints to the claws to see how these two assumptions have influenced the final results. To do so you must also change the contact modeling I have used because simply adding these dofs in my model provided unrealistic behaviour.

## References

[1] http://www.provinz.bz.it/service/resdownload.aspx?source=VIA-UVP&ID=8BD21E8FBFA3E689E050960A25327617

[2] https://www.youtube.com/watch?v=BJRgCllt53c

[3] Domenico Gentile, “Principi e Metodologie delle Costruzioni di Macchine.”, Slide UniCas, 2011/2012.