*Mattia Marchiori – mattiamarchiori@alice.it – Master degree in Mechanical Engineering – July 2017*

# Introduction

The project wants to analyze the problems encountered on the use of a full-suspended mountain bike during some ordinary cycling situations. This kind of bicycle it’s clearly designed for off-road riding (downhill, mountain climbing, etc..), but when used on a smooth and flat road an oscillation of the saddle can be clearly felt by the rider. Another problem encountered is that a few times was necessary to substitute the rear wheel’s pin, because it was deformed (it was no more rectilinear, causing the wear of the bearing), even if the bike have been used on a normal road, getting down at most from a sidewalk.

The geometry and all the dimensions are taken directly from a real bike.

The names used in the description of the bodies, the joints and the forces are the same used in the model made in Adams.

# Objectives

The aims of the project are:

- Evaluating the oscillation of the saddle and the power dissipated by the rear damper due to pedaling on a smooth road;

- Evaluating the force on the rear axle while getting down from a sidewalk. This information could be used as an input information for the structural analysis of the rear wheel’s pin.

To achieve these two objectives, two different simulations have been created.

# The modeling problem

## Bodies

The model consists of three main bodies which are described on the following paragraphs, and they are: the bicycle, the rider, and the street.

A set of construction points has been used as reference for the building of the model.

The center of mass of the system made by the rider and the bicycle, with respect to the point of contact between the rear wheel and the street is: x= 460,1 mm (longitudinal); y= 932,9 mm (height).

### Bicycle

The bicycle consists of the following bodies, fig.1:

Fig. 1: Bicycle.

**Chassis**;

It has been created by connecting different tubes, which have been made with a CAD software and then imported. The saddle has been modeled using a box. Density is set equal to 2700 Kg/m^{3} (aluminum).

**Handlebar (fig. 2);**

It has been created directly in ADAMS using cylinders and a box where the springs of the front suspension are connected. It is considered made of Zn-AL-Mg alloy with a density of 6600 Kg/m^{3}.

**Rwheel;**

Both the rear and the front wheel have been modeled using a torus with major radius 282,95 mm and minor radius 27,95 mm (26×1.90 inches mountain bike wheels). Each one has been given a mass of 3 Kg.

**Fwheel;**

**Fwheel_pin;**

Simply modeled with a cylinder of density 7800 Kg/m^{3} (steel), it is used to connect the wheel with the springs of the front suspension.

**Susp_1 (fig. 3);**

It’s the first body that forms the rear suspension system, it’s connected to the chassis and to the rear wheel. Like Susp_2 and Susp_3 it has been created using a CAD software and then imported. All the components of the rear suspension are made of steel with a density of 7800 Kg/m^{3}.

**Susp_2 (fig. 3);**

It connects Susp_1 and Susp_3.

**Susp_3 (fig. 3);**

It consists of two rockers that are jointed to the chassis and it connects Susp_2 with the spring-dumper system of the rear suspension.

**Pedals;**

It’s a unique body of steel (7800Kg/m^{3}) made using a CAD software and then imported. To simplify the analysis, it has been modeled like a rotating shaft, without considering the rotation of the pedals, where the feet of the rider are attached, with respect to the cranks.

The whole mass of the bicycle is 16,2 Kg.

### Rider

The rider consists of 9 bodies, all made using elementary shape, fig. 4:

**Upper_body;**

**R_forearm;**

**L_forearm;**

**R_arm;**

**L_arm;**

**R_femur;**

**L_femur;**

**R_shin;**

**L_shin;**

The mass of the rider has been distributed using the information reported in [2]. Assuming a total mass of the rider of 80 kg the following results are obtained (tab.1-2):

Body main-part |
% |
Mass [Kg] |

Torso | 45 | 36 |

Head | 10 | 8 |

Arms | 15 | 12 |

Legs | 30 | 24 |

Tab.1: Body main parts.

Body sub-part |
% |
Mass [Kg] |

Arm | 60% of 1 arm | 3,6 |

Forearm | 40% of 1 arm | 2,4 |

Femur | 60% of 1 leg | 7,2 |

Shin | 40%of 1 leg | 4,8 |

Tab.2: Body sub-parts.

The right foot and the right pedals are not initially coincident, but when the simulation is started the right leg is moved to the correct position according to the joint used.

### Street

The street has been modeled, both in the first and second simulation, using boxes of different measures.

## Joints

While connecting the joints attention has been given in avoiding redundant constrains and at the end the model has 10 degrees of freedom. The name used in the code for the joints are reported in brackets.

### Pedals

The body Pedals is connected to the Chassis using a revolute joint (Rvlt_Pedals);

### Handlebar

The Handlebar is connected to the Chassis using a revolute joint (Rvlt_handlebar_chassis);

### Front suspension

The front suspension was modeled using primitive joints, fig. 4:

Fig. 4: Front suspension’s joints.

- Inline joint between Handlebar and Fwheel_pin (Inline_Ha_wh1). This joint guarantees the alignment between the Fwheel_pin center of mass and the point on the upper part of the steering tube [1];

- Another inline joint between Handlebar and Fwheel_pin (Inline_Ha_wh2). This one guaranties that the Fwheel_pin is parallel to the part of the Handlebar where the springs are attached [2];

- Inplane joint between Handlebar and Fwheel_pin (Inplane_Ha_wh). This one forces the Fwheel_pin to move parallel to the direction of the steering axes.;

### Rear suspension

The rear suspension has been created using the following joints, fig. 5:

Fig. 5: Rear suspension’s joints

- Revolute joint between chassis and Susp_1 (Rvlt_chassis_susp1) [1];

- Cylindrical joint between Susp_1 and Susp_2 (Cyl_susp1_susp2) [2];

- Spherical joint between Susp_1 and Susp_3 (Sph_susp2_susp3) [3];

- Revolute joint between Susp_3 and Chassis (Rvlt_chassis_susp3) [4];

### Wheels

The Fwheel has been connected with the Fwheel_pin using a revolute joint (Rvlt_Fwheel_pin) such as the Rwheel has been connected to Susp_1 (Rvlt_Rwheel_susp1).

### Chassis-Street

Since the rolling of the chassis is not considered during the simulation, the Chassis has been forced to remain perpendicular to the Street. This decision allows to consider the pedaling forces as external and to evaluate the saddle oscillation in the x-y plane.

For this reason, it has been used a planar joint between Chassis and the Street (Planar_chassis);

### Rider

The joints used to create the rider are:

- Spherical joints between arms and forearms (Relbow and Lelbow);

- Revolute joints between arms and Upper_body (Rshoulder and Lshoulder);

- Spherical joints between femurs and Upper_body (Rhip and Lhip);

- Spherical joints between femurs and shins (Rknee and Lknee);

The rider has been connected to the bike using the following joints:

- Revolute joints between the Pedals and the shins (Rfoot and Lfoot). The right foot is not initially attached to the pedal, because the right leg has been obtained by coping the left leg.;

- Fixed joint between the Upper_bosdy and the Chassis (Rider);

- Spherical joint between the forearms and the Handlebar (Rwrist and Lwrist).

## Forces and motions

### Springs and dampers

The rear suspension has been created using a translational spring-dumper (Rear_spring) with these characteristics:

- Stiffness: 132,28 N/mm (750 lb/inch);

- Damping coefficient: 5000 Ns/m;

- Preload: 15 mm.

The stiffness value is the one written in the real spring of the bicycle. The damping coefficient instead is taken from [2], because the real value hasn’t been found on the suspensions manufacturer website.

The front suspension instead has been made using two translational spring-dampers (RFront_spring, LFront_spring), setting the following parameters:

- Stiffness: 50 N/mm;

- Damping coefficient: 600 Ns/m;

- Preload: 10 mm.

For the same reason as before, the damping coefficient is taken from [2].

### Contacts

The contact between the road and the tires has been modeled using continuous impact modeling approach with a solid to solid contact type. Since there have been problems during simulation, the contact solver’s geometry library was switched to “Parasolid”, this has given smoother results. The same settings have been used in the rear wheel and in the front wheel.

The continuous impact parameters that have been set are, fig. 6:

Fig. 6: Continuous impact modeling parameters.

To simulate the friction between the tires and the tarmac the following values have been used, fig. 7 [2]:

Fig.7: Coulomb force parameters.

### Pedaling forces

Since the Chassis has been constrained to remain perpendicular to the street (discussed before), the pedaling forces have been applied on the Pedals like external forces. It has been decided to decompose the pedaling force of one leg in two components, fig.8:

- The effective component (PE), which is perpendicular to the crank and responsible of the torque;

- The parallel component (PP), which is parallel to the pedal and causes the tension/compression of the crank.

Fig. 8: Forces on the pedals.

Both the components are a function of the pedals angle, which is defined as the angle between a vertical axis and the crank. After analyzing the experimental data reported in [1] it has been decided to simplify the pedaling force with sinus and cosines functions as follows:

This formulation allows to have the maximum effective component when the crank is parallel to the street.

Using design variable and measures that will be discussed in the next paragraphs, the following forces have been applied on the pedals:

- Effective components:These were applied on the revolute joint of the right and left foot respectively, with direction orthogonal to the crank.

These forces were applied on the center of mass of the Pedals, with direction parallel to the crank.

The magnitude of the forces has been calculated using the formula for the power reported in [3] and the ratio between the max value of PE and PP registered in [1]. In particular the steady-speed equation proposed in [3] to calculate the power required on the rear wheel is:

Where:

- K
_{A}= 0,5 C_{d}Aρ, with C_{d}as the drag coefficient, A the frontal area of the system bike+rider and ρ the density of the air. On table 4.2 page 139 [3] in case of a utility bicycle it is suggested a value of 0,368.

- V: traveling speed;

- V
_{w}: speed of the wind, assumed equal to zero during simulation;

- m: mass of the rider + mass of the bicycle

- s: slope;

- C
_{r}: rolling resistance coefficient. On table 4.2 page 139 [3] in case of a utility bicycle it is suggested a value of 0,008.

Using a rider’s mass of 80 Kg, a bicycle’s mass of 16,2 Kg and a velocity of 20 Km/h:

That is the power required on the wheel. Assuming an efficiency of 0.95 for the chain transmission the power required to the rider is:

This means that the torque required on the pedals is equal to:

Where:

is the angular velocity of the wheel, and:

is the transmission ratio between the pedals and the rear wheel.

Being the length of the crank of 170 mm, the force required to have a constant torque (assuming that only one pedal is propelling) is:

But, as reported before, the component of the pedaling force responsible for the torque is not constant during the pedaling action. It has therefore decided to impose that the work made by the constant force during half a turn of pedal, must be equal to the work made by the PE (see definition above):

So:

Analyzing the results in [1] it has been decided to assume:

### Motions

To reproduce the motion of the bike 3 rotative motors have been used:

- Handlebar_turn;

This motor has been used to lock the steering axis, since the simulation has been computed on a straight line, so the velocity has been set equal to zero.

- Rwheel_rotation;

The motor has been placed in Rvlt_Rwheel_susp1 and it has been controlled in velocity.

- Pedals_rotation;

The motor has been placed in Rvlt_Pedals and it has been controlled in velocitylike the one I the rear wheel.

The functions used to simulate the riding will be discussed on the next paragraphs.

# Simulations and analysis of the results

As discussed in the introduction, two simulations have been made to achieve the objectives of the project. A parametric simulation technique has been used in order to make easier the setting of the different parameters of the system, so a group of design variables has been created.

### Design variables

The design variables created are:

- DV_rear_stiff: stiffness of the rear suspension’s spring [N/mm];

- DV_front_stiff: stiffness of each spring of the front suspension [N/mm];

- DV_rear_dump: damping coefficient of the rear suspension’s dumper [Ns/m];

- DV_front_dump: damping coefficient of each dumper of the front suspension [Ns/m];

- DV_wheel_rot: angular velocity of the rear wheel [deg/s];

- DV_PE: peak value of the effective force [N];

- DV_PP: peak value of the parallel force [N];

- DV_Tau: transmission ratio.

### First simulation: riding on a smooth road at 20 Km/h

The transmission of the bicycle has the following gears, (In brackets are reported the number of teeth for each gear):

- Gears on the rear wheel: 1 (28), 2 (24), 3 (22), 4 (20), 5 (18), 6 (16), 7 (14);

- Gears on the pedals: 1 (24), 2 (34), 3(42).

At 20 Km/h it is usually adopted the second gear on the pedals and seventh gear on the rear wheel, thus giving a transmission ratio of:

So, the design variable used to control the angular velocity of the rear wheel of the bicycle has been set equal to:

In order to avoid wheeling during the simulation, the motion on the motors of the rear wheel and the pedals has been applied using the following step function:

- Rwheel_rotation: step(time,0,0,3.0,DV_wheel_rot);

- Pedals_rotation: step(time,0,0,3.0,DV_wheel_rot)*DV_Tau;

Note that in the acceleration phase, the change of the transmission ratio that usually occurs hasn’t been counted.

The simulation can be run with a script (Adams Solver Commands) called “Riding_20Km_h” which is divided on the following parts:

- Deactivation of the pedaling forces and reaching of the equilibrium;

- Beginning of the simulations and activation of the pedaling forces after 0,2 s;

The validation of the model has been made with a reduction of the step size of the simulation, using GSTIFF integrator and I3 formulation.

Fig. 9: Script “Riding_20Km_h”.

Since one of the objectives of the project is to evaluate if there could be any resonance phenomena the model has been linearized around the equilibrium configuration, obtaining, fig. 10:

Fig. 10: Eigenvalues obtained from the linearization around the equilibrium configuration.

Using linear modes controls it has been found that mode number 16 is related to the rear suspension and mode number 17 is related to the front suspension. Both modes are stable because the real part of the corresponding eigenvalues is negative. It can be seen that, the mode associated with the rear suspension has a lower undamped natural frequency (2,7 Hz) than the one associated to the front suspension.

To get close to a pedaling frequency of 2,7 Hz you need to use higher transmission’s ratio (at 20 Km/h) than the one used in the model, for example fist gear on the rear wheel and second gear on the pedals (DV_Tau = 1,17), but with this kind of ratios it’s almost impossible to reach 20 Km/h because they’re usually used at much lower speed, for example on steep climb.

The results obtained from the simulation are:

**Angular velocity of the rear wheel and the pedals, (fig. 11);**

Fig.11: Angular velocity of the pedals and the rear wheel.

This graph shows the z component of the angular velocity on Rvlt_Rwheel_susp1 and Rvlt_Pedals.

**Oscillation of the saddle, (fig. 12);**

Fig.12: Oscillation of the saddle.

This measure has been made by subtracting from the y position of a marker placed in the saddle (Saddle_position), the position of the saddle at the equilibrium with respect to the origin on the ground (699,8 mm).

**Force on the rear axle, (fig. 13);**

Fig.13: x and y component of the force on the rear axle.

**Length of the rear spring, (fig. 14);**

Fig.14: Length of the rear spring during the simulation.

**Deformation velocity and force of the rear spring, (fig. 15);**

Fig.15: Force and deformation velocity on the rear suspension.

**Power dissipated by the damper, (fig. 16);**

Fig.16: Instantaneous power dissipated by the dumper.

This measure is computed by using the following formula:

Pdiss_rear_dump= DV_rear_dump (Rear_spring_def_vel/1000)^2

### Second simulation: getting down from a sidewalk

To analyze the force on the rear axle due to the getting down from a sidewalk, a 20 cm step has been created on the road used for the first simulation. Since usually this operation isn’t done at full speed (20 Km/h like the previous simulation, DV_wheel_rot=1139 deg/s) the following step function have been used for the motors:

- Rwheel_rotation:

step(time,0.0,0.0,3.0,DV_wheel_rot)-

step(x_disp,5.5e4,0,7.2e4,DV_wheel_rot/2)+

step(time,18,0,19.5,DV_wheel_rot/2);

This function allows to accelerate reaching 20 Km/h, decelerate to 10 Km/h when the bicycle is close to the road step and then accelerate again after the jump. X_disp represents the measure of the displacement along x of the bicycle.

- Pedals_rotation:

(step(time,0.0,0.0,3.0,DV_wheel_rot)-

step(x_disp,5.5e4,0,7.2e4,DV_wheel_rot/2)+

step(time,18,0,19.5,DV_wheel_rot/2))*

(step(x_disp,5.5e4,0.412,7.2e4,0.588)-

step(time,18,0,19.5,0.176))

Since the transition from 20 Km/h to 10 Km/h it’s done with variation of the transmission ratio, usually keeping second gear on the pedals and going to fourth gear on the rear wheel (transmission ratio equal to 0,588), it has been chosen to not use a design variable but to build a step function that simulate the gear changing.

Even in this case the simulation can be run with a script (Adams Solver Commands) called “Sidewalk_jump”, fig. 17, similar to the previous, but in this case it has been used GSTIFF integrator with SI2 formulation, because the I3 formulation caused excessive spikes during the jump.

Fig. 17: Script “Sidewalk_jump”.

The results obtained from the simulation are:

**Angular velocity of the rear wheel and the pedals, (fig. 18);**

Fig.18: Angular velocity of the pedals and the rear wheel.

**Force on the rear axle, (fig. 19);**

Fig.19: x and y component of the force on the rear axle.

The graph shows initially the same force of the first simulation, until the bicycle starts to decelerate and reaches the road step. A closer look to when the bicycle gets down from the sidewalk allows to understand better how the force on the rear axle evolves:

Fig. 20: Lost of contact from the front wheel and the road.

Fig. 21: The front wheel gets down from the sidewalk.

When the front wheel loses contact with the sidewalk the components of the force start increase and then reduce almost equal to zero when the front wheel reaches the road.

Fig. 22: Impact between the rear wheel and the road.

**Saddle oscillation, (fig. 23);**

Fig.23: Oscillation of the saddle.

The drop of the values when the bicycle reaches the road step it’s due to fact than the oscillation of the saddle has been calculated with the same formula of the first simulation. It can be seen that in less than two seconds the saddle reaches again the oscillation that has before the jump.

**Force on the rear suspension, (fig. 24).**

Fig.24: Force on the rear suspension.

The maximum value of the force reached on the rear suspension is almost equal to 3900 N.

# Conclusions

- On regime, the saddle shows an oscillation with a magnitude of 4 mm. The frequency oscillation results of 2,5 Hz, which is very close to the vibrating mode associated with the rear suspension. However, it must be considered that the pedaling forces used in this model are a strong approximation of the one reported in [2], so this frequency may not be reliable;

- The force on the rear axle can cause mechanical fatigue due to its oscillation. In particular the y component has an average value different from zero, instead the x component has an average value equal to zero;

- The power dissipated by the rear damper is very small compared with the one produced by the rider in steady state condition, so it is negligible.

- The maximum value of the forces on the rear axle after getting down a 20 cm sidewalk at 10 Km/h have been evaluated.

It has to be said that the way the rider is constrained to the chassis is not strictly representative of the reality, and this can cause unreliability of the results obtained. A better modeling of the rider (with attention to the way the upper body is connected to the chassis) and of the pedaling forces, can lead to more reliable results.

## References

[1] | A. Schiavon, «Analisi dei carichi di pedalata mediante pedali dinamometrici integrati in un sistema di tipo motion capture,» Università degli Studi di Padova, a.a. 2014/2015. |

[2] | E. Todaro, «http://www.multibody.net/teaching/msms/teachingmsmsstudents-projects-2016/todaro-eros/,» 2016. [Online]. |

[3] | D. G. Wilson, Bicycling Science – Third edition, The MIT Press, 2004. |