## INTRODUCTION

Twist Bike Atlantic® by Dobertec® (www.dobertec.com) is an innovative bike that implements a transmission system similar to that described by Leonardo Da Vinci in the Codex Atlanticus and applied to the propulsion of paddle boats.

The fundamental characteristic of this mechanism is that it derives the motion from an alternating rotation of the pedal-lever, and not from a continuous rotation as in common bicycle cranks.

## OBJECTIVES

The aim of this project is to verify the correct working of the bike transmission, comprehensive of two functioning freewheels, then to perform a test protocol simulating the cyclist’s acceleration by different gearshift velocity ratios.

## MODELLING

The geometrical model of the bike is given by Dobertec®, so the first step of the work consists in assembling all the model parts with the appropriate joints.

Each part of the bike rigidally connected to the chassis is assembled with a Bracket Joint, instead the bodies which have a rotational motion with respect to the chassis (e.g. the wheels, the pedal-levers, etc.) are combined with a Revolute Joint. Finally the chassis is forced to remain on a vertical plane with a Planar Joint, that because our purpose is to study the behaviour of the bike in some simple situations without considering rolling and yawing.

### Freewheels

To make an appropriate simulation of the kinematics, we need to create the model of a freewheel to be inserted inside the bike transmission.

Specifically, a frewheel is a device similar to a radial bearing; its main feature is the free relative rotation on one sense (e.g. clockwise) between inner and outer ring with a low resistant torque, and instead the locked relative rotation on the other sense (e.g. counterclockwise) with full torque transmission between the two rings.

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To model our two freewheels we take as a reference those of the C.T.S.® freewheels catalogue, adapting them with some simplifications. The external ring is modelled as a rectangular section ring with chamfered edges. The internal ring is modelled similarly to the external one, but with housings for the spheres on the outer profile.

As a simplification we assume that a roller-type freewheel could be simplified by replacing the spring-loaded rollers with spring-loaded spheres, as showed in figure. This way it becomes possible to use the “Sphere to Revolved Surface Contact Force” to simulate the contact between the rolling bodies and the two rings. Moreover we decide to reduce to 4 the number of rolling bodies in each freewheel to speed up the computing time. Nevertheless the torque transmission is ensured by amplifying the paramethers of the contact forces and those of the springs, this according to the fact that non-modelled spheres would work in parallel with the modelled ones.

### Transmission

A particular attention is given to the modellation of the transmission, which is the core of this innovative bike; it consists of two counterrotating pulleys, each one controlled by one of the two pedal-levers. Both the main pulleys are attached through freewheels to a common shaft that gives the motion to the bike rear wheel.

Two cables are wrapped on each pulley, in the opposite sense one to the other. The first cable ends on an appropriate connection to the correspondant pedal-lever; the other cable passes on the so-called “little pulley” (which inverts its direction) an then finally wraps on the other pulley. This way the created powertrain makes the two pedal-levers always rotate in the opposite sense one to the other. This means that for exemple a thrust on the right pedal provides the lift of the left pedal. The three cables so far mentioned can be considered flexible (that to allow the wrapping on the pulleys) and yet inextensible due to the special composite material they are made of. Because of this last property of the cables, it is permitted to avoid modelling the cables and instead to fix a set of “Three-Body Relative Constraint” concerning the relative angular position of the pedal-levers, the main pulleys and the “little pulley”.

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While the “little pulley” is constrained to the chassis with a Revolute Joint, each main pulley is coupled to the external ring of one of the freewheels previously modelled. Then both the freewheels and a cogwheel (Z=22) are keyed on a common shaft. A bike chain joins the cogwheel to the sprocket (Z=18) of a SHIMANO NEXUS epicyclic gearshift which finally drives the bike rear wheel. So we decide to model this powertrain between the cogwheel and the bike rear wheel by defining a Gear Joint.

Due to the so-modelled system, each thrust on a pedal involves only one of the two freewheels to transmit torque to the shaft (“locked” configuration) while the other frewheel allows the non-loaded pedal to lift.

The following video may visually explain the transmission functionment.

### Tyres

To model the tyres of the bike we use the “Simple Tire” routine, because it allows the definition of the tyres by choosing just few simple parameters.

Since it’s hard to find data in literature regarding the characteristics of bicycle tyres and since they strongly depend on inflation pressure, we have to assume some values that would be sufficiently close to reality. These parameters are shown in the next picture:

Although probably these values are not entirely related to reality, it’s not a problem because the tires are not the most important component of the model.

### Pedal Forces

We choose to simulate the action of the rider legs through the application of a couple of forces (one on each pedal) which remain orthogonal to the pedal-lever during the entire pedaling.

We decide not to assume a much simpler constant value force on each pedal, but a more gradual force application through a parabolic law.

In this modelling phase the main difficulties derive from having to link the expressions of the forces to the angular position of the pedal-levers. In fact the pedal-levers pass through the same positions in both the upstroke and the downstroke, but during the downward stroke the force must be active, while in the upstroke it must not.

This problem is solved through the use of Boolean operators, which are true if the pedal-lever is in a position between the TDC and the BDC, in parallel with a maximum function between the sign of the pedal-lever speed and zero. This way the function returns the force value if the pedal-lever is in its downward stroke or zero if it is in the upstroke. Then the expression is further changed and the force application advanced slightly before the TDC (1 degree), that because we don’t want the pedal-lever to exceed the TDC position due to the pedal-lever angular momentum.

The following chart displays the trend of the right pedal-lever angular position, which varies between 0° (TDC) and 19° (BDC), and the corresponding pedal force value, which is zero in the upstroke and varies between 400N and 600N during the downstroke.

### Dummy

In order to model the cyclist we use a pre-existing dummy CAD model. The different body segments are generated within the simulation environment and constrained by appropriate relative joints. We consider a 70kg cyclist and then reconstruct the mass of each segment due to Dempster’s proportions. So these calculated masses are assigned to the segments by applying fictitious densities to the segments bodies in order to approximate the exact cyclist’s mass distribution. The following table summarizes the calculated masses and densities.

Segment | Mass [kg] | Volume [m^3] | Density [kg/m^3] |
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Head | 5,53 | 0,007866 | 703,026 |

Trunk | 35,77 | 0,03793 | 943,053 |

Arm | 1,89 | 0,003391 | 557,358 |

Frontarm | 1,12 | 0,001487 | 753,194 |

Hand | 0,42 | 0,0004531 | 926,948 |

Thigh | 6,79 | 0,010038 | 676,430 |

Shank | 3,15 | 0,005488 | 573,980 |

Foot | 0,98 | 0,001373 | 713,765 |

Finally, the dummy is constrained to the bike using Braket Joints at saddle, handlebars and pedals. This way during the simulation it will be possible to display the cyclist’s posture and pedaling.

### Aerodynamics

In order to model the drag force that the system undergoes while moving because of air resistance we use the “Vehicle Aerodinamic” routine that implements a simplified aerodynamic matrix suitable for the degrees of freedom of common land vehicles. We assume A = 1m^2 the characteristic section and c_d = 0.21 the drag coefficient of the modeled system and we set ρ = 1.225kg/m^3 the air density. The following screenshot summarizes the chosen values and the characteristic area.

## SIMULATION AND RESULTS ANALYSIS

### Freewheels functionment test

After modeling the freewheel and chosing the parameters needed to characterize the contact forces, we define a test protocol in order to verify the correct working of the freewheel model.

We decide to set the inner ring fixed to the ground and apply the following sinusoidal torque to the outer ring:

T = 20·sen(2π·t/1,6) [N]

This way we can verify both the working modes of the freewheel and in particular:

- Both the rings will be blocked during the half period of positive torque that corresponds to the so-called “Transmission mode”
- The outer ring will be free to rotate during the half period of negative torque that corresponds to the so-called “Free mode”.

After the calculation we obtain the following animation of the working freewheel, which reflects the results we expected.

In the following graphs, to analyze properly the correct working of the model in both the modes, we plot the torque transmitted from the outer ring to the inner ring and the outer ring angular speed.

During the “Transmission mode” the entire torque applied to the outer ring is transmitted to the inner ring, reproducing exactly the shape of the corresponding half-period of the torque sinusoid.

In the “Free mode” the torque transmitted should be null, but in facts there are still some irregular torque transfers. This occurs because the spheres must always remain in contact with both the rings so that the working mode change is almost instantaneous, and also because of the vibrations of the system composed by the spheres and the springs.

The transition to the next “Transmission mode” does not occur exactly at the moment when the torque returns positive, as it is first necessary to contrast and brake the rotation of the outer ring, which during the “Free mode” has acquired a certain angular momentum.

When finally the outer ring is blocked, there is a peak of torque due to the collision that comes from the locking. However this peak is not too high and is damped quickly. Then there is again the complete transmission of the torque acting on the outer ring, which means that the “Transmission mode” has been restored.

During the “Transmission mode” the outer ring speed should be null because the inner ring is blocked, but in fact a slight rotation of the outer ring occurs in the direction of the applicated torque. That is because of the non-infinite stiffness of the bodies, which then penetrate each other [see detail (a)]. This slight rotation stops when the deformation forces balance the applied torque, then it reverses its direction as the torque begins to decrease because, with the assumption of perfect elasticity of the bodies, the deformations are completely recovered.

In the following “Free mode” the outer ring is free to rotate in the direction of the applicated torque and reaches a very high angular speed due to the low inertia that characterizes the ring itself; some small angular speed fluctuations occurs during this phase, that due to the previously described torque transmissions still present during the “Free mode”.

Towards the end of the “Free mode” the outer ring is slowed down due to the fact that the torque direction is now opposite to the rotation. Then some angular speed fluctuations occurs around zero due to the spheres vibrations when the new “Transmission mode” is reached [see detail (b)]. Finally the system behaves in the way previously described in (a) due to the non-infinite stiffness of the bodies.

### Bike ride simulations

Now that the bicycle system is ready to be tested we define a test protocol to simulate the cyclist’s acceleration by different gearshift ratios. The test protocol consists of three independent simulations, each one lasting until t=10s. We also impose an initial 3m/s speed as a “One Body Initial Condition” to the bike chassis and to the dummy trunk to avoid numerical errors encountered in the application of the pedal forces at low or null chassis speed. The gear ratios selected for the three simulations are n°1 (0,527), n°4 (0,851) and n°7 (1,419). The calculations give back three different ride animations, one of wich is represented in the following videos.

The following four graphs analyze the dynamics of the bike in the three different simulations. In particular, the first graph represents in a single Cartesian plane the speed of the chassis of the three simulations. Instead the remaining three graphs (each one specific to one of three different gear ratios) represent the trend of the right pedal load force, the left pedal load force, the shaft angular velocity and the right pedal-lever angular position.

Now let us express some comments on the results so far collected:

- The chassis acceleration is obviously different by each different selected gear ratio.
- The chassis speed trend is almost linear (acceleration almost constant) except in the case of the 0,527 gear ratio by which a very slightly descending acceleration can be detected.
- The chassis speed trend presents a periodical interference caused by the periodical collisions of the freewheels spheres at the instant of the freewheel locking.
- The pedaling frequency is increasing in all the three cases.
- The magnitude of the chassis speed periodical interference is proportional to the pedaling frequency. This because the pedaling frequency is proportional to the maximum value of angular velocity of the pedal-levers, which is also proportional to the angular momentum of them and, in general, to the momentum of all the components in alternating motion. These momenta are strictly proportional to the might of the freewheels spheres collisions.
- The chassis speed is perfectly proportional to the shaft angular speed and the proportionality coefficient is their velocity ratio.
- At the 0.527 gear ratio, the pedaling frequency increases over any reasonable value for a cyclist, the increasing angular momentum of each pedal-lever involves a longer time to let the pedal force reverse the pedal lever motion and after t = 7s the forces function loses the desired shape. All these facts suggest that at this time it would require a gear shift to a harder gear ratio.
- The linear chassis speed trend is caused by the fact that the maximum value of the pedal forces is constant (600N) and not inversely proportional to the pedaling frequency as it is in reality. Besides the increasing pedal-levers angular momenta imply a progressive TDC and BDC distancing.

## CONCLUSIONS

The developed model is suitable to verify the correct operation of the transmission of Twist Bike, which may open new perspectives of innovation in the field of cycles or in that of rehabilitation, where the traditional method of pedaling is impossible due to disabilities.

The acceleration tests performed with three different speed ratios showed results compatible with our predictions, thus supporting the validity of the created model.

This project can be a starting point for further refinements through an optimization of the parameters related to the transmission, as well as through the application of forces on the pedals with a constant-power function. This way a steady-state condition can be reached in which the chassis speed remains constant and the applied torque equals the resistance torque exerted by both the road and the drag force. Thus more reliable quantitative results can be obtained, that can be compared with a traditional bicycle pedaling data.

Finally, new test protocols can be developed to simulate Twist Bike working in conditions of downhill or uphill through the application of a constant torque to the wheel, concordant or discordant with the rotation of the same. Even wind situations in favor or against Twist Bike advancement can be implemented through the Aerodinamic Vehicle routine that offers the possibility to choose the wind direction and intensity.