*Eros Todaro – eros.todaro@studenti.unipd.it – Master degree in Mechanical Engineering
updated July 2016 *

**INTRODUCTION**

** **During the last 12 months I discovered how beautiful was to go out for a bike ride, just to relax and have some physical activity in order to feel healthy.

I live in a beautiful hilly region of Italy in the north of Treviso’s province, so last summer I started to practice up and downhill with my bike and I loved it, that’s the reason why I decided to focus my modelling project on mountain bike.

With this modelling project I’m trying to recreate a typical bike jump, trying to find the best damping ratio and stiffness for the suspensions in order to land correctly.

**OBJECTIVES **

The purpose is to model and simulate a mountain bike, and its rider jumping over a knoll, in particular to evaluate the deformation of the bicycle focusing on the dynamic properties of the system in order to land correctly after the jump.

**THE MODELLING PROBLEM**

The height of the jump that was simulated in the treatment is about** 2 m** overall and it begins after a short climb.

In order to model the vehicle 20 rigid bodies were used (maximum number for the student version of the software). The main parts are :

- Bike frame (1 body)
- Front suspension (3 bodies)
- Rear suspension (6 bodies)
- Front wheel (1 body)
- Rear wheel (1 body)
- Biker (7 bodies)
- Road (1 body)

*Front Suspension and handlebar*

The classic front suspension is made by two spring dampers acting in parallel on the steering axis, the trail angle is about ε=14°.

The stiffness of the suspension was estimated assuming that applying the whole rider’s mass on the front the spring displacement would have been about 1cm.

The damping was estimated by a trial and error procedure until the final configuration of the vehicle reached a realistic configuration.

Stiffness = 20kN/m (total front stiffness = 40kN/m)

Damping = 600 Ns/m (total front damping = 1200 Ns/m)

That would have bring to a 70 kN/m equivalent stiffness on the front but running some simulation seemed a bit too much so a 40 kN/m equivalent stiffness was adopted.

The vertical reduced stiffness at the front contact point is : Kf = K/cos²ε = 41224 N/m

* Detail of the front suspension of the bicycle*

About the steering motion, two coaxial cylinders were made (the external one is a part of the frame and the internal one is the steering bar) and they are linked by a rotational joint. The steering bar is linked to the handlebar where the rider exerts his action by his hands that are modelled with two spherical joints.

*Rear Suspension*

The alloy frame of the vehicle is linked to the rear suspension by a spring damper that insists on a rod which shape is studied in order to guarantee (for small angles of rotation) a translational movement for the free end of the rod where the spring exerts its action.

The rear suspension is a swingarm one : the lower rod links with the upper one thanks to a rigid pin that connects them on the sides of the rear wheel, and the upper arm is connected to the spring damper by another rod which is connected to the frame with a rotational joint.

The properties of the rear spring damper are estimated assuming as follows :

Stiffness = 40kN/m (assumed equal to the equivalent front stiffness, as two times the single front spring stiffness)

Damping = 5000 Ns/m, this parameter was estimated by a trial and error procedure after simulating.

*Detail of the rear suspension of the bicycle*

* *

* Tires*

As it regards the tires, the front and the rear model are the same one, with a 29” diameter (mostly used in competitions nowadays), it is a torus made of rubber (found in Adams library) and its weight is about 2 kg. The Adams tyre model was not used because it is an advanced feature not available in the student edition.

Something very important for the project are the contact conditions between tires and ground, whose parameters were set this way :

These friction conditions are similar to a contact between a tire and wet cement.

The radial tire stiffness was ignored because it is about 10 times higher than the suspension one.

These conditions are put between the tires and the road in order to guarantee a pure rolling motion.

*Motion*

As it regards the motion of the bike, a constant rotation speed was set on the rear tire :

Being the tire diameter of 29”, assuming 2″ for the tire radial width (1″ each side of the diameter) that condition can be seen as a forward motion :

1 inch = 0.0254 m

Φ (external tire diameter) = 31 * 0.0254 = 0.7874 m

The tire circumference is : C = π*Φ = 2.47 m

So the motion can be seen as: 1000d*time = 6.8714 m/s ≈ **24.7 km/h** (pure rolling assuntion)

*Rider*

The whole bike weight is about 11kg (alloy frame) whereas the rider mass is 71 kg . The center of mass of the whole system is not far from the rider’s one assuming that the bike is well balanced.

The rider is modelled by many cylinders interconnected between them by spherical, rotational and translational joints.

The mass of each part was estimated starting from the following table and changing the values a little bit in order to manage simple numbers :

*[2] Reference table for estimating body parts masses.*

Assumed masses for each body part

**Legs** : two cylinders are joined by a rotational joint corresponding to the knee, the whole mass of the two legs is 20kg.

**Body** : a box body was used to model the body and a torsional spring (applied on a rotational joint) was introduced between it and the legs to simulate the human pelvis stiffness. (Kt=1800 Nm/rad). The aggregate mass of the body including the head is 35 kg.

**Arms** : the rider’s arms are independent and made of two parts (arm and forearm) joined by a spherical joint to simulate the elbow. The arms are joined to the body by a couple of spherical joint as shoulders are. There is no degree of freedom regarding the (local) vertical axis rotation for the body so the shoulders are horizontally aligned. In fact that choice has been made thinking about the flexible proprieties of human body, that allows us to rotate the shoulders keeping the pelvis locked. Such a complicate system should have been extremely complicate to model, anyway the previous simplification is not compromising at all the simulation. The arms’ mass is 8 kg each.

The whole rider is locked to the frame by fixed joints on his foot, so the legs are fixed at the base in order to simplify the system. During the simulation the rider keeps his legs almost fixed because the motion is given to the rear tire without simulating the rider pedalling in order to simplify the problem (in fact that is not the aim of the modelling process).

*Detail of the rider’s body with all the joints used to model him. Only one rotational joint was used to model rider’s knees because the legs are made by two only bodies : one is for the upper part of the two legs (black in the picture) and the other part is for the lower part of the two legs (yellow in the picture)*

**Simulations and analysis of results**

*2D Simulation*

As told, the main motion condition of the system is given to the rear tire and it allows the bike to reach 23 km/h speed before the jump. Initially a planar joint was put between the frame and the ground in order to study a two-dimensional problem focusing on the suspensions and ignoring the motion out of the symmetry plane of the bike. By that way the steering motion was locked and no forces were acting on the handlebar.

The real velocity of the bike is reported in the following graph :

*Vehicle velocity during the simulation*

The rear tire initially slips until it reaches the established velocity just before the beginning of the climb, during which the bike slows down a bit from 23.4 to 21 km/h more or less. Then during the jump the velocity magnitude rises up a bit and the most high value for it is reached just after the landing : in fact the rear tire touches the ground before the front one, and the ramp reaction is along the x direction (main motion direction).

During that simulation it was interesting to evaluate the deformation of the suspensions and the forces exchanged between the spring dampers and the bike frame.

Here there are some interesting measures regarding the suspension configurations during the motion:

*Deformation of the front spring in 2D model*

*Front suspension force for 2D case (single spring)*

* 3D Simulation*

Afterwards the planar joint was removed and the system became unstable due to the freedom of the roll angle degree. The bike trend was to start rolling after a few meters, so it felt down with the rider on the side because of the lateral displacement of the center of mass.

The problem has been resolved introducing a basic motion condition on the steering wheel.

As known, in order to reduce the right roll angle (for example), a steering in the same direction is necessary (the rider can not move his center of mass so he has a passive role about that) so the condition that allows the bike to keep the equilibrium (roll angle near 0 degrees) is :

Mt = Kt*Φ

Where Mt represents the required torque on the steering axis, Φ is the roll angle of the vehicle and Kt is an arbitrary constant chosen in order to allow to the roll angle function to stay around 0°.

In this case the appropriate constant seems to be :

Kt=0.11

The constant was determined by trial and error until a good configuration was found. Higher values make the vehicle nervous and lower values make it unable to keep the equilibrium condition with high wavings around the vertical position.

Here there are some interesting measures taken during the 3D simulation, obtained deactivating the planar joint that fixes the roll angle at 0° :

The maximum roll angle is about 3° just after the jump but as the picture shows the rider is able to reduce it fastly just after landing.

At the real beginning of the simulation the rear tire slips a bit so the roll angle starts to rise, as well as the attitude angle. The proportional control system is well working so the bike keeps the equilibrium but the vehicle does not reach the climb perfectly orthogonal with the road (seen from above) and that contributes in having a 4° roll angle at the end of the simulation. * Roll angle function in 3D model*

*Deformation of the front spring in 3D model*

*Front Suspension force for 3D case (single spring)*

In particular it’s evident that there are 2 force peaks, the first one is at the beginning of the climb and the maximum force value is about 800 N, whereas the second peak is an impulsive force up to 3000N ≈ 300kg.

Considering the second spring acting in parallel the impulsive force exchanged during the landing shock is about 8 times the whole system’s mass.

In the end there is a comparison between the single spring damper on the front suspension of the 2D case and the 3D one :

*Comparison between 2D and 3D model Force diagram*

For the rear suspension the Force values are bigger than the front ones because there is just one spring damper instead of two :*Deformation of the rear spring in 3D model*

*Rear suspension force for 3D case (single spring)*

**Front and rear loads
**

A third simulation was runned where the original condition is a 3D motion (the planar joint is deactivated) but there is no jump, in order to focus more on how front and rear load are variating.

In order to calculate those component the relative rotational joints were selectioned (front and rear wheel pin) and the Y component of the force was calculated.

At the beginning of the simulation the bike falls down onto the road due to an imperfect coupling between road and tire, so that’s the reason why the rear suspension (blue curve) has a peak just some instants before the front one (red curve). In fact the rear touches the ground much earlier than the front due to the backward position of the center of mass.

The result of this simulation is reported in the picture below :

*Comparison between front and rear load during the forward motion of the vehicle. *

The peaks shown in the photo are related to the beginning contact between the tires and the ground.

From this analysis the average values are about :

Nf = 500 N on the rear tire

Nr = 320 N on the front tire

The length between front wheel pin and rear one is about 1.2 m and the whole mass of the system is 82 kg, it’s easy then to calculate the CM position using that simple relation :

*Longitudinal position of the center of mass for 2 wheeled vehicle*

b = (Nr * P)/mg = (320 * 1200)/ 820 = 480 mm

a = p – b = 1200 – 480 = 720 mm

The longitudinal position of the center of mass is now determined.

**Conclusion **

The vehicle responds in a predictive way at the ground solicitations, and the system seems to be well damped, in particular during the landing it takes just a few seconds to come back at the original setting. The influence of the rider is however much more important in reality than in this model, simple and approximated because of the complexity of the human body.

During a jump the rider use to move the center of mass in an unpredictable way, so that it would be extremely difficult to simulate it, so the result is heavily approximated but seems to be realistic as it regards the dynamics of the motion.

*Overall view of the model including the bike, the rider and the road*

*Video grab of the jump *

*Video grab of the jump seen from above*

At the beginning of the motion the bike slips and the attitude angle increases a bit until the slip keeps existing, then the vehicle is able to carry on a straight motion, whereas the angle between the velocity vector and the road horizontal width is not 90°.

**References**

[1] Vittore Cossalter, “Motorcycle dynamics”

[2] www.blog.swimmershop.it