*Francesco Collini – francesco.collini.1@studenti.unipd.it*

*updated on June, 2020*

## Introduction

At EUROBIKE 2018 ceramicspeed, a daneish company known in the cycling world for its high end bearings and shifting components, announced DRIVEN, a revolutionary shifter that promises to reach 99% efficiency and better aerodynamics performance. The following figure shows a CAD drawing of a mock up of DRIVEN’s rear end:

Notice that unlike traditional shifters there is no chain and derailleurs and the cassette is completely different to the standard ones as it grows radially. The transmission is guaranteed by the contact of 12 ceramic bearings located on the rear pinion that engages the cassette to reduce frictions: in traditional shifters there are up to 40 sliding friction points (e.g. in a 54 teeth front 32 teeth rear configuration) while in this solution there are only 2 rolling contact points (1 in the front and one in the rear). The main challenge of this solution is the shifting; this is allowed by splitting in 2 halves the rear pinion . With reference to the following video:

https://www.youtube.com/watch?v=uNBiuOt_zi8&feature=youtu.be

let master be the part that begins the shift and slave the other one. When the shifting signal is sent to the rear pinion an appropriate mechanism, modeled in this work as a simple translational joint with a motion that guarantees the movement of the pinion , moves the half that is not engaged with the cassette (master) above the gear required. Once the first tooth of the master half of the pinion engages the cassette on the new gear the rotational speed of the rear wheel will be determined by the new speed ratio. Only when the last tooth of the slave pinion leaves the old gear the slave half can move to complete the shift.

## Objectives

The objectives of this work are:

- the simulation of a shift from 32 tooth gear to a 30 tooth gear;
- the comparison of HHT I3 and GSTIFF I3, GSTIFF SI2 solvers tuning the tolerance and analytical results given by the speed ratios of the gears;
- evaluate the influence of the stiffness of the stiffness of the contacts on the solution;
- the quantification of the efficiency of this solution considering contacts between the cassette and the bearings non frictionless. Efficiency is calculated as:

where P_shaft is power measured at the shaft and P_wheel the power measured at the wheel, i.e. at the cassette.

## The modelling problem, simulations and analysis of results

### Cad modelling

The geometry of the model was created in PTC Creo v 6.0.

The model is made of the following components:

- cassettte with 2 gears, 32 and 30 teeth respectively:

- master and slave pinion halfs:

- 12 bearings for the shaft pinion:

- shaft

#### Cassette profile and CAD implementation

The profile of the Cassette was generated following these steps:

- extrusion of a cylindrical surface of diameter coincident to the pitch circle diameter;

- drawing of a single tooth on a flat surface following the convention adopted in the book “”, summed in the following picture;

- wrapping of the signle tooth about the cylindrical surface;
- generation of the boundary blend of a single tooth;
- thickening of the single tooth wrapped around the cylindrical surface, used as the mid plane;
- geometrical series of n = z Cassette (32 or 30 in the model) about the axis of the cylinder to reproduce the whole cassette;

### ADAMS CAD import

The PTC CREO .asm is saved as a STEP and imported in ADAMS, with out any sort of issues, each part of the assembly is divided with the same cryteria as in PTC CREO. Before moving on with the set up of the simulation the material properties of the bodies are defined:

- shaft: ADAMS CFRP;
- slave and master pinion: ADAMS alluminum alloy;
- bearings: ADAMS steel;
- cassette: alluminum

### Connectors and motors definition

The following connectors have been defined:

- revolute joint between shaft and ground:
- body i: shaft
- body j: ground
- rotational axis: rotational axis of the shaft

A rotational motor is applied to this joint to introduce the rotation of the axis at a constant speed of 10 deg/s

- translational joints between shaft and pinions:
- body i: master/slave pinions:
- body j: shaft
- translational axis: rotational axis of the bearings

- revolute joints between each bearing and the master/slave pinions:
- body i: bearing
- body j: master/slave pinion
- rotational axis: rotational axis of the bearings

- revolute joint between cassette 32 and ground:
- body i: cassette 32
- body j: ground
- rotational axis: rotational axis of the cassette

Translational motors are applied at these joints to introduce the

- fixed joint between cassette 30 and cassette 32:
- body i: cassette 32
- body j: cassette 30

### Forces and contacts

The forces defined are:

- torque force acting on the revolute joint between the cassette and the ground against the rotation of the cassette to simulate the wheel resistance;
- contacts between the 12 bearings and the 32 and 30 teeth cassette. To speed up the operations they were not defined through the GUI, but written down in the .cmd file of the model. The definition of a contact is in the form:

The value of K will be properly tuned to reach convergence if necessary. The values of c, dmax and e are the ADAMS default ones. The friction forces are modelled with the implementation of ADAMS of the Coulmb’s model. Stiction and friction transition velocity are the dafault ones, the static and dynamic friction coefficients are 0.23 and 0.16 respectively (for greasy alluminum, greasy steel contacts);

- Measuring the angular velocity about the spinning axis of the bearings in a simulation where the revolute joint between the bearings and the master (or slave) moving part is ideal (i.e. no stiff-damper spring applied to the revolute joint) shows the following trend:

To solve the issue (the other bearings show similar trends) a torsion spring is applied at each revolute joint with null stiffness coefficient and 1 E-6 Nmms/deg, that gives:

### Solvers comparison

The model is solved with the following solvers:

- GSTIFF I3
- GSTIFF SI2
- HHT I3

and they are compared in terms of accuracy of the output of the wheel speed (same as the cassette speed), varying the tolerance, with respect to the analytical results computed with the speed ratios of the gear system: given the angular velocity of the shaft:

the number of teeth (i.e. the number bearings for the shaft pinion) of the pinions:

the angular velocity of the wheel is:

- if the 32 teeth cassette is engaged:

- if the 30 teeth cassette is engaged:

The tolerance is increased up to a core i7 4700mq, 12 GB of ram laptop maximum performances. The simulations are performed with the ADAMS default value of k, stiffness of the contacts, of 1E4 N/mm.

GSTIFF I3

In the following table are summed up the results obtained with GSTIFF I3:

#### GSTIFF SI2

In the following table are summed up the results obtained with GSTIFF SI2:

#### HHT I3

In the following table are summed up the results obtained with HHT I3:The following figure shows the convergence of the numerical solution to the analytical one, calculated with the solvers shown and tolerances discussed before:

Given the computational cost, not taken into account in this work, it looks reasonable to use HHT I3 solver with a tolerance of 1 E-8. The trends of the angular velocities computed with such solver and tolerance are shown in the following figure:

As a comparison in the following figures are shown the best and worst solutions respectively:

### Continuous impact modeling stiffness influence

According to the continuous impact modeling implementation in ADAMS the normal component of the impact force can be quantified as:

The penalization is exact as k approaches infinity, as there is no penetration, but large values of k may lead to numerical failure. Using HHT I3 solver with a tolerance of 1 E -8 several simulations have been performed to investigate the influence of k on the velocity of the wheel during the shift simulated. The results are summed up in the following table:

The following figure shows the dependency of the solutions on the stiffness K:

that shows how higher stiffnesses lead to poorer results affected by bigger spikes as shown in the following pictures:

### Efficiency quantification of the DRIVEN shifter:

The efficiency of the shifter is:

where:

- P_shaft is the power of the motor of the shaft and is computed as

with omega_shaft = 10 [deg/s], set by the rotational motor applied at joint between the shaft and M_shaft the ground and being the moment measured at the same motor;

- P_wheel is the power measured at the wheel, computed as:

with omega_wheel being the measured velocity of the wheel, i.e.:

and M_wheel = 10 [Nmm]

The simulation is run using the solver HHT I3 solver with a tolerance of 1 E -10 and with the stiffness of the contacts set to k = 1 E2 N/mm. The analytical results are:

The simulated power has the trend shown in the figure:

The computed values of P_shaft and P_wheel have the following averages:

Looking at the diagrams of the measured power vs the analytical results, and at the values of the previous figure it is clear that the computational power of the laptop isn’t sufficient to compute a reasonable value for the efficiency of the shifter.

## CONCLUSIONS

The simulations developed show:

- the performances of the solution of the HSTIFF solver that results to be much faster and accurate if compared to GSTIFF I3 and GSTIFF SI2;
- that the laptop used for this work has not enough computational performances to compute with sufficient accuracy the efficiency of the shifter;

Possible improvement are the introduction of the double clutch mechanism to allow the shifting in every position of the cassette.

## REFERENCES

[1] Carlo Negri – CATENE E LORO APPLICAZIONI – EDITORE ULRICO HOEPLI MILANO, https://digit.biblio.polito.it/3593/13/Catene_PARTE_I.pdf, https://digit.biblio.polito.it/3593/14/Catene_PARTE_II.pdf

[2] www.ceramicspeed.com

[3] Bowden & Tabor, “The Friction and Lubrication of Solids,” Oxford University Press, 2001