# The Optimal Braking

Marini Davide – mdavide90@gmail.com – Degree in Mechanical Engineering
Villotta Matteo – villotta.matteo@gmail.com – Degree in Mechanical Engineering

## Introduction

Figure 1: Example of a forward flip

The aim of this analysis is to investigate the characteristics of a bicycle’s optimal braking and to analyze which are the most important elements a rider should take into account in order to achieve it.

An optimal braking, in fact, is a synonymous of safety for a leisure rider but also a synonymous of time gained for an athlete during a bicycle race.

This analysis in particular  wants to model, at first, an ideal optimal braking and, then, wants to investigate how it can be achieved in a more practical way.

Many other elements related to the braking, like the forward flip, will be described and different conditions of the asphalt will be taken into account.

## Objectives

The purpose of this work has been, first of all, to model all the characteristics of an ideal optimal braking where the minimum space required to stop the bicycle is achieved. This simulations have been obtained imposing a specific angular speed to the wheels. The kinematic and dynamic quantities obtained from the software have been compared with the results calculated with an analytical description of the same problem in order to validate the model and to do some considerations.

Then, the problem of the forward flip has been investigated in detail and, in particular, the most important factors which can modify the forward flip limit conditions (in particular the rider position on the bicycle) have been studied quantifying their influence.

At the end, a series of simulations applying a specific torque to the wheels have been performed in order to obtain a more realistic description of what happens during a bicycle braking. From this point of view, the effects of different torques applied to the wheels and the effect of different braking distribution (between the front and the rear wheel) have been taken into account with the objective of obtaining the optimal combination of the factors that leads to the lowest space required to stop the bicycle. This simulations will be presented with respect to different conditions of the asphalt (wet and dry).

## The modelling problem

The aim of our model was to obtain a kinematical and dynamical description of a bicycle with a rider which could take into account all the effects that occur during a typical braking in a straight road. All the parts, except the dummy, were modeled by our own in a Catia environment.

The bicycle has been modeled as a series of rigid bodies: chassis, handlebar, suspension tube, saddle, rear and front rims, and rear and front tires. All these parts have been carefully assembled in order to obtain a mechanism with only five DOF: horizontal and vertical  movement of the chassis, in plane rotation of the chassis and the rotation of the two wheels. The decision to divide the chassis from the handlebar, from the front and from the saddle has been taken in order to obtain a more precise distribution of the masses (using different mass properties) and to make it possible to add, later, the effect of the front suspension which requires a relative movement between the front and the suspension tube.

Figure 2: Bicycle assembled

Figure 3: Dummy assembled

To assemble the bicycle the following joints have been applied to the parts:

• Planar joint between chassis and global fixed to ground: this joint constrains the model to remain in the plane.
• Bracket joint between the handlebar and the chassis.
• Bracket joint between the saddle and chassis.
• Bracket joint between the suspension tubes and the chassis
• Revolute joints between the chassis and the tires
• Bracket joints between the rims and the tires

Regarding the inertial properties of the parts, it has been applied, as a first modeling attempt, a density that is typical of the alloy to all the metallic components and a density typical of the rubber to the tires. Then, to obtain a more precise model, these values of densities have been updated to obtain, for each single component (Chassis, Saddle, Front etc..), a mass that is really similar to that of a modern MTB.

The dummy, as well, has been modeled as a series of rigid bodies (Head and Neck, Thorax, Pelvis, Arms, Forearms, Hands, Tights, Shanks and Foots ,comprising the shoes). These CAD parts have been taken from a CAD library and have been scaled in order to make it possible to match them with the bicycle designed. The single body components have been assembled in order to obtain an isostatic mechanism and to permit, later, to easily modify the position of the dummy: one of the aims of the analysis was to study the effects of different positions of the dummy on the bike. The following joints have been used:

• Bracket joint between the saddle and the pelvis
• Bracket joint between the pelvis and the thorax
• Bracket joint between the thorax and the head
• Bracket joints between the hands and the handlebar
• Cylindrical joint between the thorax and the arms
• Spherical joint between the arm and the forearm
• Revolute joints between hands and forearms

The lower limbs were modeled with a similar structure:

• Cylindrical joints between the pelvis and the tights
• Spherical joints between the tights and the shanks
• Revolute joints between the shanks and the foots.

Here there’s the Grubler’s equation as a verification of the system’s DOF:

As expected the mechanism has only five DOF and there aren’t redundant constraints which could lead to problems when solving.

Regarding the mass properties of the dummy, in order to use values as close as possible to the reality, the Dempster Model for the body anthropometry has been taken into account. This model describes the body as a series of segments and provides the mass properties as a function of the overall mass of the subject (85 Kg in this case) and the overall height of the dummy (185 cm).

 Body segment Mass [Kg] Volume [m3] Density [kg/m3] Head 6.715 0.008 839.375 Thorax+Pelvis 43.437 0.047 924.1914894 Arm 2.295 0.002 1147.5 Forearm 1.36 9.66E-04 1407.721768 Hand 0.51 3.00E-04 1700 Thigh 8.245 0.014 588.9285714 Shank 3.825 0.004 956.25 Foot 1.19 0.001 1190

The forces between the wheels and the asphalt have been modeled using the so called simple tire model of Virtual.Lab. This element allows to model the forces including: lateral force, normal or vertical force and longitudinal force. The values used to model the tires are shown in the figure below:

Tire parameters

Figure 4: Tire parameters

## Analysis of a perfect braking

In order to achieve the perfect braking, one of the most important factors, is the interaction between the wheels and the road. In order to minimize the space required by the vehicle to stop, the deceleration imposed to the vehicle has to be maximized. In this analysis a series of brakings will be studied, while supposing that the rider has the ability to maintain the tires at their limiting conditions in terms of adherence.

### Theoretical analysis

A global balance of the forces acting on a bicycle is here presented. During  the numerical analysis, the drag force (FD) won’t be taken into account: this approximation makes it possible to simplify the model without affecting the accuracy of the results, since this force is very small for a bicycle, given the small values of the exposed area and speed reached by the vehicle.

Figure 5: Forces equilibrium

From the above figure, the following relations can be obtained:

During a braking, the deceleration, causes a load transfer from the rear to front wheel. The dynamic loads on the wheels becomes:

Given the values of the front friction coefficient μf and the rear friction coefficient μr, the overall braking force at the slipping limit can be obtained:

From the equation written above it can be noticed that, in order to maximize the overall braking force (and therefore the deceleration), given the values of the dynamic loads on the wheels, the friction coefficients have to be maximized. These coefficients (in vehicle’s dynamic problems) depend on two factors: the first one is the type of surfaces placed in contact (e.g. rubber with dry tarmac, rubber with wet tarmac, rubber with gravel, etc.) and the second one is the rotational slip coefficient (S) defined as follows:

Given the surfaces placed in contact it can be considered a value of the friction coefficient (e.g. 0.4 for rubber with wet tarmac, 0.8 for rubber with dry tarmac) that, of course, will give the longitudinal force in the case of a complete slipping condition. This friction coefficient, in the simple tire model, is a function of the rotational slip as it can be seen in the following figure:

Figure 6: Friction coefficient with respect to rotational slip

To reach the maximum value of the friction coefficient is necessary to maintain a condition in which S =0.2 (S=0.2 during an acceleration, S=-0.2 during braking). An expert driver might be able to maintain the rotational slip at a value very close to 0.2, whereas an average driver usually obtains values of S quite lower than 0.2.

It has to be stressed out that reaching the highest value of the friction coefficient is quite complex in a real situation since, this condition, is very close to the wheel lock event. This situation is not desirable since it increases the space required to stop the vehicle and the difficulty to control the vehicle itself.

### The modeling problem: controlling the vehicle

Given the models of the bicycle and the rider, in order to simulate a perfect braking condition, the rotational slip has to be kept equal to -0.2. In order to do this, the rotational speed of the wheels has been controlled and, in particular, two joint velocity drivers were used to impose the angular speed of the wheels with respect to the chassis. The first joint velocity driver acts on the revolute joint between the rear wheel and the chassis, and the second one acts between the front wheel and front fork. From the previous formula of S  the value of ω can be obtained as ω=(S+1)*V/R.

Since both the wheel radius and S are known, the angular speed of the wheels can be immediately controlled when it is known the vehicle speed V. Pay attention to the fact that the wheels angular speed has to be controlled constantly during the simulation and  this requires to know the value of the vehicle speed at each simulation step. In order to do this, a couple of sensor axis systems have been used. These axis systems are capable of measuring quantities (position, speed, torque, etc) while the simulation is running and to use this results as an input for the same simulation. In particular, one axis system attached to ground and another one attached to the vehicle have been used and, by doing so, the speed of the vehicle with  respect to the ground along the forward speed axis has been measured. Knowing this information, the formula written above, has been implemented inside the velocity drivers in order to impose the exact value of the angular speed that leads to a rotational slip coefficient equal to -0.2.

### Results: the perfect braking

In this section we’re going to analyze the most important quantities that describe a perfect brake. Here, as an example, a simulation with both the front and the rear friction coefficients equal to 0.2 is reported. First of all the vehicle speed is presented:

Figure 7: Speed with respect to time

As it can be seen in the figure, the vehicle’s speed decreases linearly from the initial value (10 m/s) to 0 m/s as expected since the total braking force and the deceleration are constant. Pay attention to the fact that, as the speed reaches values very close to zero, the linearity is lost. This can be explained considering that, in this phase of the simulation, the dynamic behavior changes as the vehicle comes to a complete stop. To better understand what is happening in this phase the behavior of both the system and the software when modeling friction forces at low speed should be analyzed in detail. This analysis though are beyond the aim of this project.

The following figure represents the deceleration of the vehicle; this quantity, as expected, is quite constant for almost the entire braking phase of the simulation.

Figure 8: Acceleration with respect to time

Observing more carefully the figure, it can be noticed that, at the very beginning of the simulation and when the vehicle is almost still, the acceleration isn’t constant. Concerning the latter of these two moments, it has already been stressed out that in this phase further analysis would be required. At the very beginning of the simulation the presence of some spikes of acceleration can be noticed. They are caused by the fact that, when the simulation starts, the bicycle isn’t perfectly positioned on the road. Through a manual positioning of the bicycle, infact, it wasn’t possible  to obtain the exact contact points for the wheels on the road. From this point of view, the first tenths of a second are required to reach an equilibrium position. Please note that both this effects don’t affect the accuracy of the results since they both concern a very small period of the simulation and their influence is therefore neglectable.

Figure 9: Forces acting on the wheel with respect to time

The graphs above represent the values of the normal and longitudinal forces during the braking (again with both the friction coefficients equal to 0.2). The link between this two quantities, exposed during the theoretical analysis, can be seen clearly in these figure: note that during the entire simulation  F=1.05μN as it was supposed from the tire modeling. This observation, makes it possible to consider this  as an optimal braking: this ideal rider is capable to maintain the wheels at their limits during the entire braking and, therefore, the best performances in terms of time and space required to brake are reached.

The analysis of the normal forces clearly shows the effect of the load transfer: during the braking phase, infact, the normal load is increased on the front wheel and decreased on the rear wheel. When the speed reaches zero this effect isn’t present anymore, and so the force on both the front and rear wheel becomes equal to the static value (this value depends on the total weight and the position of the center of gravity).

The main goal of this part of the project is to validate, with the numerical analysis, the analytical model that describes the optimal braking. As already written above, the equilibrium of the forces in the horizontal and vertical direction are:

From these expressions the deceleration can be obtained as a function of the rear and front friction coefficients:

and also the braking force of the front and the rear wheels, with respect to the total braking force

these equations underline that the deceleration depends on geometric values (the position of the center of gravity) and the friction coefficients. Considering our situation (p = 1060.755 mm, b = 506.832 mm, h = 1119 mm) the above equations can be represented in the following graph:

Figure 10: Theorical values for the decelleration and the braking forces

With the LMS model previously described, the analytical model was intended to be validated using numerical simulations. In order to do this, some values of the friction coefficients have been chosen and they have been introduced into the simulation.  Then, the quantities obtained from these analysis were compared with the analytical values. The following tables summarize the results:

 Deceleration [m/s2] μfront μrear Analytical Virtual Lab Difference 0.171 0 0.981 0.970 -1.17% 0.136 0.05 0.981 0.973 -0.83% 0.064 0.15 0.981 0.980 -0.14% 0.000 0.24 0.981 0.981 -0.04% 0.290 0 1.962 1.962 -0.02% 0.200 0.2 1.962 1.935 -1.40% 0.110 0.4 1.962 1.936 -1.30% 0.001 0.64 1.962 1.942 -1.04% 0.378 0 2.943 2.907 -1.23% 0.326 0.2 2.943 2.904 -1.34% 0.274 0.4 2.943 2.902 -1.40% 0.222 0.6 2.943 2.914 -1.00% 0.170 0.8 2.943 2.914 -1.00% 0.445 0 3.924 3.911 -0.34% 0.422 0.2 3.924 3.849 -1.91% 0.400 0.4 3.924 3.882 -1.08% 0.378 0.6 3.924 3.879 -1.14% 0.355 0.8 3.924 3.883 -1.05%
 Ff/F [%] Fr/F [%] μfront μrear Analytical Virtual Lab Difference Analytical Virtual Lab Difference 0.171 0 100.0 100.0 0.00% 0.00 0.00 0.00% 0.136 0.05 79.2 79.0 -0.26% 20.84 21.04 0.98% 0.064 0.15 37.5 37.3 -0.62% 62.51 62.74 0.37% 0.000 0.24 0.0 0.0 0.00% 100.00 100.00 0.00% 0.290 0 100.0 100.0 0.00% 0.00 0.00 0.00% 0.200 0.2 68.9 68.8 -0.19% 31.12 32.25 3.63% 0.110 0.4 37.8 37.7 -0.26% 62.24 62.34 0.16% 0.001 0.64 0.4 0.4 -0.29% 99.59 99.59 0.00% 0.378 0 100.0 100.0 0.00% 0.00 0.00 0.00% 0.326 0.2 86.3 86.3 0.06% 13.71 13.66 -0.37% 0.274 0.4 72.6 72.6 0.10% 27.43 27.36 -0.26% 0.222 0.6 58.9 59.0 0.20% 41.14 41.03 -0.28% 0.170 0.8 45.1 45.2 0.22% 54.86 54.76 -0.18% 0.445 0 100.0 100.0 0.00% 0.00 0.00 0.00% 0.422 0.2 95.0 95.2 0.19% 5.01 4.83 -3.56% 0.400 0.4 90.0 90.2 0.25% 10.02 9.79 -2.29% 0.378 0.6 85.0 85.3 0.34% 15.04 14.75 -1.91% 0.355 0.8 80.0 80.2 0.34% 20.05 19.78 -1.35%
 Time required to brake [s] μfront μrear Analytical Virtual Lab Difference 0.171 0 10.194 10.879 6.72% 0.136 0.05 10.194 10.770 5.65% 0.064 0.15 10.194 10.635 4.33% 0.000 0.24 10.193 10.548 3.48% 0.290 0 5.097 5.375 5.46% 0.200 0.2 5.097 5.290 3.79% 0.110 0.4 5.097 5.250 3.01% 0.001 0.64 5.097 5.222 2.46% 0.378 0 3.398 3.547 4.39% 0.326 0.2 3.398 3.512 3.36% 0.274 0.4 3.398 3.494 2.83% 0.222 0.6 3.398 3.526 3.77% 0.170 0.8 3.398 3.588 5.59% 0.445 0 2.548 2.635 3.40% 0.422 0.2 2.548 2.619 2.77% 0.400 0.4 2.548 2.745 7.71% 0.378 0.6 2.548 2.767 8.58% 0.355 0.8 2.548 2.774 8.85%

The above tables clearly show that the model is capable of representing with high accuracy the physical problem previously described. In particular it can be noticed that the difference between numerical and analytical values in terms of deceleration and forces is very small: 1.91% maximum for the deceleration and 3.63% maximum for the forces.

Higher differences are present in the time required to brake the vehicle, to justify these differences though it can be recalled what has already been told regarding the linearity of the speed in the last part of the braking: the analytical model doesn’t present the non-linearity so the time in this model is inferior than what the numerical analysis would calculate.

As a conclusion it can be said that our numerical model is capable to fulfill the aim of this part of the analysis: by using LMS Virtual Lab it has been possible to validate the proposed  analytical model.

## Forward flip

As  described in the previous chapter, increasing the value of the friction coefficients at the front and at the rear wheel has allowed  to obtain higher values of deceleration and lower values of the space required to stop the vehicle. In this section the limit of this procedure, caused by the forward flip of the vehicle, will be analysed. As commonly known, braking with a bicycle, can lead to a critical condition in which the vehicle and the rider flip forward rotating around the front wheel. This event can cause a catastrophic fall (see the video below) if the rider isn’t expert enough to balance his riding position. In this section it will be studied how to increase the forward flip limit in order to achieve higher decelerations without falling.

### The modeling problem: the front suspensions and different riding position

First of all, to make the model more accurate, a front suspension has been introduced on the vehicle. To do this, the bracket joint connecting the handlebar with the suspensions tubes has been replaced with a translational joint, which allows the translation of the suspensions elements along their axis. Furthermore a Translational Spring Damper Actuator (TSDA) has been introduced to simulate the stiffness and the damping of the suspensions. The parameters used to model the TSDA were obtained by searching on the web typical values used by suspension manufacturers for a MTB.

Figure 11: TSDA paremetres

This model has been used to investigate the limit conditions of forward flip and, then, other improvements have been made in order to increase that limit. First of all, the possibility to move backward the thorax has been taken into account and, then, the movement of the entire rider has been considered. In the first step only  the thorax orientation has been changed: this modification represents a common practice for riders since at the beginning of the braking usually they move from a crouched position to a more upright one. The second step was to completely move the rider by moving backward the position of the pelvis until the limit of the seat is reached: this movement influences the position of the thorax, the arms and legs, that are moved slightly backward. All this movements have been possible since the dummy was modeled using a series of parts connected in a proper way when assembling the model.

### Results: the stoppie

To evaluate the limit condition of flip forward the deceleration has been increased (by increasing the friction coefficients) until the point in which the vehicle started flipping forward, but doidn’t complete it. This happens since the vehicle’s speed becomes null before the rider could fall. This situation is represented in the following video:

This operation has been performed for different configurations: in the first one the rider had a crouched position, in the second the thorax was moved to a more upright position and in the third one the pelvis was moved backward.

The reduction of the space required to stop the vehicle can be seen in the following table:

 Configuration Space required to stop the vehicle [m] Difference Standard vehicle with TSDA 11.56 Thorax upright 11.48 0.69% Pelvis backward 10.97 5.37%

Taking the first configuration as reference, it can be noticed that both the second and the third configurations cause a reduction in the space required to stop the vehicle. The second one though doesn’t lead to a big improvement, this can be justified observing that in this configuration the center of gravity moves backward, but at the same time it moves upward and this leads to opposite effects. In the third configuration instead a good improvement has been reached. From this point of view it is possible to came to the conclusion that the common practice to move backward the position of the weight has a significant influence in the braking.

## Optimization of braking torque distribution

The previous results, obtained with a speed control, describe an ideal braking where the wheels are capable to reach the maximum value of the friction force through the contact with the asphalt. In a real case, however, the friction force is the result of the action of the brakes, which produce a braking torque, and it is controlled by the rider. From this point of view, it becomes really difficult to obtain a perfect braking which actually gives the maximum value of the friction force the tires can provide.

The aim of this analysis is to study the effects of 1) different torques (intended as the sum of the front and rear braking torque) and 2)different distributions of braking torque between the front and the rear wheel on the overall space required to stop the bicycle in order to get the optimal combination of the two factors.

From a theoretical point of view it is important to distinguish between the braking torque required by the rider and the effective braking torque that is applied to the wheels during the braking. The required torque can be defined as the torque that the rider would produce on the wheel applying a force to the brake levers. But this required torque is equal to the effective torque only if it is small enough to be balanced by the torque produced on the wheel by the friction force acting on the tire as the result of the contact with the asphalt.

Writing the equilibrium of the rotation of the wheel:

Where Tbrakes is the effective braking torque acting on the wheel, Ffriction is the friction force and rwheel is the radius of the wheel.  Analysing the previous equation it is possible to say that: if the torque produced by the brakes increases, at a certain point, the wheel is locked. When this situation occurs, the torque produced by the brakes is balanced only by the torque produced by the friction force. This friction force depends on the normal force acting on the wheel and on the friction coefficient which has the behaviour described in the section “The Modeling Problem”.

This means that, as the braking force applied by the rider increases, the rotational slip increases as well, and, if it becomes higher than 0.2, the wheel is blocked since the friction coefficient decreases reaching a constant value: in this case the effective braking torque reaches a constant value and doesn’t follow the behaviour of the required braking torque. From this point of view it becomes really important to investigate what is the best required braking torque and what it would be its better distribution between the front and the rear wheel.

In order to perform the analysis, a series of different values of required torque, and the effects of 4 different braking distribution (20% front/80% rear, 40% front / 60% rear, 60% front/40% rear, 80% front/20% rear ) have been analyzed in order to compute the overall space required to stop the bicycle. This evaluation has been carried out considering, at first, a wet condition of the ground (μ=0.4) and, then, a dry condition (μ=0.8), where the problem of the forward flip occurs.

Regarding the modelling problem, in order to apply a desired value of the torque on the wheels, two RSDA have been applied to the revolute joints between the chassis and the wheels. With each of these RSDA we wanted to apply a constant torque to the wheels (the two wheels can have different values of the torque). A saturation function was implemented to maintain the torque constant except when the wheel’s angular speed is close to zero. This solution allows the vehicle to reach a complete stop, otherwise if a constant value had been applied all the time, once the bicycle had been stopped the constant torque would have accelerated it backwards. The saturation function is:

The formula written above clearly shows that the torque is constant for all the values of angular speed except those very close to zero. The torque applied to each RSDA has been defined as a fraction of the total braking torque (sum of front and rear braking torque) and it has been set as a parameter in the software. In order to manage such a great number of simulations and results, the Solution Manager of LMS Virtual Lab has been used and the output data have been analyzed in Matlab. In particular, a series of design tables have been defined which made possible to apply all the values of the torques and all the values of the braking distributions that we wanted to investigate. The first analysis have been carried out considering a friction coefficient of 0.4, corresponding to a wet condition of the asphalt. The overall results are shown in the graph below, where, the abscissa describes the braking distribution (from 20% front- 80% rear on the left to 80% front-20% rear on the right) and the ordinate displays the corresponding values of the space. The plot, of course, is parametric with respect to the overall braking torque.

Figure 12: Spaces required to brake

This first analysis clearly shows how, for each single value of the overall torque, the space required to stop the bicycle decreases as the amount of torque applied to the front wheel increases. This behaviour is related to the load transfer which occurs during the braking. What happens is that, even for the lowest value of the overall torque (140 Nm), when the rider brakes with 20% front  and 80% rear, the rear wheel is locked since it is discharged by the load transfer. The front wheel, instead, is very far from being locked since it has a higher value of the normal force (due to the same reason), so its rotational slip S is quite lower than 0.2. From a practical point of view, there’s a huge difference between the required torque and the effective torque applied to the rear wheel while they are the same at the front.

Then, considering again the same value of overall braking torque and increasing the fraction of torque applied to the front, the graph shows a sensible decrease in the space required to stop the bicycle. This can be easily described since the front wheel receives greater values of braking torque without being stopped so it increases the friction force since it increases the value of μ and the value of N due to the greater load transfer.  When the amount of torque applied to the front wheel grows up to the 80% of the overall braking torque, what happens is that even the front is blocked and so, for both the front and the rear wheel the friction coefficient reaches 0.4. When this situation occurs, an increase in the overall torque applied (for example moving to 180Nm or 190 Nm) won’t produce any positive effect on the space required to stop the bicycle since both the wheels have reached the maximum value of the friction force they can produce.

This behavior becomes clearer when studying the conduct of the slip coefficient for every braking just analyzed. In order to do this we plotted, for each single braking, the graph which describes the behavior of the front and the rear slip coefficient over time. As an example, when the overall torque is equal to 140 Nm and the braking distribution is 60% rear and 40% front, the following graphs have been obtained:

Figure 13: Slip coefficients with respect to time-example braking

Then, using Matlab, the values corresponding to the section of the braking where the slip coefficient is quite constant (see figure above) have been selected and the average value of these sections has been computed. This results can be plotted in a graph which describes how the value of the slip coefficient varies while the braking moves from the rear to the front. The results are shown below (parametric with respect to the torque).

Figure 14: Rear and Front slip coefficients in every single braking

From the analysis of the graphs it appears clear how the rear wheel has a slip coefficient that remains quite constant and close to one during all the analysis. The front wheel, instead, when considering the lower values of torque (140 Nm, 150 Nm or 160 Nm) increases its slip while moving from 20% front-80% rear  to 60% front- 40% rear (remaining lower than 0.2 which is the value corresponding to the peak of the friction coefficient in the simple tire model) and then has a huge increase (reaching values close to one) when moving to 80 % front – 20 % rear. This is related to the fact that even the front starts slipping.

If the higher values of the torque are taken into account (180Nm or 190Nm) what appears is that even for 60% front- 40% rear, the front wheel starts slipping since the values of the rotational slip reaches 0.8 or 0.95.

Then, another simulation has been carried out, investigating what happens when the braking torque is completely applied to the front. The results are shown in the graph below where, for the same values of torque considered in the previous analysis, the effects of a braking  distribution more focused on the front (70/30, 80/20 90/10 100/0) have been studied.

Figure 15: Space required to brake-brakings with front

This graph shows how, from 80/20 to 90/10, nothing changes in the space required to brake since, for each value of the torque, both the rear and the front wheel remain locked and therefore the overall braking force remains the same. Moving to a braking completely applied to the front (100/0) it is possible to see that, for each value of the torque, the space required increases. This fact can be easily explained since the front wheel remains locked but the rear wheel is in a condition of pure rolling as it doesn’t receive any braking torque.

As a conclusion it is possible to say that, when a wet condition of the asphalt occurs, the optimal braking can be reached with a braking distribution which is comprised between 75 % front -25% rear and 80% front –20% rear (when applying an overall torque lower than 180 Nm). Moving to a braking more related to the front (values higher than 80% front) would drive to the locking of the front and this leads to  a strong reduction in the possibility to control the bike. Another important effect to emphasise is that, when braking with a distribution comprised in the optimal range, the effect of the overall torque becomes not so important  since there isn’t a great difference between the braking spaces related to 140 Nm and the space related to 170Nm: when looking for a perfect braking in wet conditions it is not important the effort the rider applies to the brake levers but his ability to calibrate the distribution between the front and the rear.

In order to provide a complete analysis of the optimal braking characteristics, also the dry condition of the asphalt (μ=0.8) has been taken into account. In this case the risk of forward slip occurs. The following graph, as usual, describes the effect of different overall torques and different braking distributions on the space required to stop the bicycle.

Figure 16: Space required to brake-dry condition

This graph, as expected, shows a clear decrease in the space as the braking becomes more focused on the front for the same reason described in the previous analysis. What has to be noticed here is that, when increasing the value of the torque (for example growing up to 180 Nm) not all the braking distributions can be obtained since the problem of forward flip occurs: this is evident from the graph where it is possible to notice that if the torque is greater than 160 Nm and the braking is more distributed in the front the riders falls so there’s not solution.

As done in the previous case with wet asphalt, the behaviour of the rotational slip is presented.

Figure 17: Rear and Front Slip coefficients-dry condition

The rear slip coefficient, as expected, is always proximal to one which means that the rear wheel is slipping in every simulation taken into account. The front wheel, instead, increases the value of the rotational slip while remaining below the limit of 0.2 so it maintains some rolling: in this case the limit of the braking is due to the forward flip and not to the creeping of the wheel. The following video represents the situation just described in which the rear wheel is locked while the front wheel still rotates:

As a conclusion it is possible to say that in a dry condition of the asphalt, an optimal braking can be reached using values of torque quite low (150 Nm or 160 Nm) and putting more effort in the front brake since brakings with a distribution of 80% front – 20% rear can be obtained without falling. Using higher values of torque requires a braking distribution less focused on the front (avoiding the forward flip) but this leads to greater distances required to stop the bicycle.

## Conclusion

The numerical models implemented in Virtual.Lab were capable of satisfying all the goals prefixed at the beginning of this project. In particular, the results obtained with the first model are compatible with those calculated with the analytical one. This demonstrates that the model effectively works as intended and that it can be used to analyze in great detail the behavior of a bicycle when braking at the limit condition of slippage.

The analysis conducted at the limit of the flip forward gave interesting information on how to increase the braking performances. Further analysis could be conducted regardin how to obtain the best possible position in order to maximize the deceleration.

Finally, the last model considered, describing the effects of different torques and braking distributions, gave an interesting analysis about which are the value that can drive to the best performances.

An interesting future development of this study it would be to model the braking torque acting on the wheel as the effect of the contact of the braking pads to the rims.

Future developments of these models could implement the contact between the braking pads and the rims to better analyze the influence of this component during braking.