Nicola Masiero – firstname.lastname@example.org
One of the most popular motorcycle in the world is the Piaggio Vespa. Born in the 1946, it remains a symbol for many generation and also today reatins its charm. In these years many models were created but for everybody the most important Vespa is the Vespa 50.
This motorcycle was born from the need to have a vehicle that could travel rough roads, and that it was possible to change the wheel quickly in case of puncture. For this purpose, Corradino d’Ascanio decided to use a kind of suspension derived from airplanes, calls “pull-arm”.
It might seem that this model has never changed over the years, but in 1977 something changed in the front suspension. In fact, since that year the Vespa has a lower dive effect during the brake because the braking torque reacts directly on suspension , thanks to the use of disc brakes where the brake force can be applied externally to the whell with the use of a brake caliper, when before 1977 it reacts on the arm of the suspension from the drum brake.
The objective of this modelling and analysis is to verify that, given the braking torque, the suspension dive is different when changing the reacting body . So the purpose is to model the Vespa, in particular the front suspension kinematics, and measure the compression of the front spring-damper element and the dive of the front part of the Vespa (the effect that the rider feels riding the motorcycle).
The modelling problem
The main problem of the modelling of the front suspension of the Vespa is the lack of parameters for the spring and damper. In particular, in order to perform a simulation faithful to reality, the stiffness and damping values would be necessary. For the first parameter it was quite simply to find some datas in internet, but the second datas are not available. So the modelling has required many test to set the spring-damper element to get a result that reflects quite well the reality.
Another important aspect is the modelling of the entire Vespa. For this purpose, it was not necessary to have a very faithful model, for example the rear suspension has not been inserted because its effect is irrilevant, but it was enough to build a body that had inertial properties faithful to the real motocycle to have the effect of load transfer during the braking.
1. Front suspension
It was modeled taking to reference the follow picture:
The most tricky part is the “disk” that connect the suspension to the structure of the steering, because this component allows to make small shifts and rotations to the upper part of the suspension, in order to allow the kinematic mechanism to move. So, for this component, i decided to use a “spherical joint” with a torsion spring between the upper part of the suspension and the “steering”.
After that, i connect the “leg” with the “steering” with a “fixed joint” because in reality this two components are the same. Between the leg and the arm there is a “cylindrical joint” for the relative rotation of these two component and between the arm and the lower part of the suspension there is a revolute joint (revolute and not cylindrical to fix the arm in the space).
To be able to apply a force between the suspension body and the wheel, it’s not enough to use only the “spring-damper element”, but is necessary to creare two different link that represent the lower part and the upper part of the suspension and connect each other with a “traslational joint”. In this way we have a physical body where the software allows to apply an external force.
The following picture show the result fot the front suspension kinematics:
For this model was used the SAE reference, where the x axis point in the direction of motion of the bike and the z axis point in the road.
As written before, the chassis of the vespa doesn’t have a rear suspension so the structure is rigid.
Two tires have been insert using the same *.tir file for the front and the rear tyres, to simulate the behaviour of the motorcycle in a flat road loaded as a *.rdf file.
Inertial values have been assigned on the links, in order to get the load transer as said before, using “design variabiles” and the entire structure was been locked in the vertical plane with a planar joint in order to avoid the fall of the Vespa during the simulation. The steering was joint with the “chassis” with a revolute joint and controlled by the use of a motion on it with the displacement setting to zero.
Simulation and Analysis
The first step is to set the motion for the Vespa and the following braking. For the first one, i decide to use “rotational joint motion” acting on the rear wheel, using a step function in order to have a gradual acceleration of the Vespa up to a speed of 45 km/h, the maximum velocity for this kind of motorbike.
For the braking i decide to use two different torque (single component force) acting on the front wheel for simulate the two different scenarios that i described. The first one, called “Torque_WheelArm” is the braking torque that reacts on the arm of the suspension system, the second “Torque_WheelSusp” is the braking torque that reacts directly on the suspension, in particular on the “Susp_inf link”. This two functions have the same torque, because i want to simulate the condition that the Vespa undergoes the same deceleration and be able to compare the kind of braking.
Initially both braking torques are deactivated in order to have only the driving torque acting and allow the Vespa to reach the operating speed. At this point, the driving torque is deactivated and the braking torque activated for a time of 2.5 [s] to osberve how braking proceeds.
The following figures show the compression trends of the front suspension and also the dive of the steering for the two cases.
The different behavior of the Vespa for the two cases is evident, showing a difference between the two models of about 0.007 [m] on the suspension and about the same on the feeling of the driver under braking. An other important thing we can observe from this measures: in the case of the reactive torque on the arm, there is an important oscillation when the brake is activated. Probably this is caused by the impulsive action of the force, but we don’t see the same oscillation in the other case. However this oscillation vanishes in a second and then the Vespa brake with the same trend.
The numerically results for the suspension compression are (distance between the upper and the lower parts of the suspension):
- Res_susp: 0,175 [m]
- Res_arm: 0,168 [m]
And for the steering dive (z quote from the road):
- Res_susp: 0,559 [m]
- Res_arm: 0,552 [m]
In the next figures i verify that the two models have the same longitudinal speed during the simulation and the control has also been extended to the angular velocity of the front wheel, where we can see the same vibration that starts at the same moment of action of the braking with the torque that react on the arm. Probably the vibration that we measure on the suspension born from the vibration of the tyre. This particular needs more considerations and simulations.
An other interesting thing that i observe watching the animation of the simulations is the apparent difference in the response of the Vespa between the two configurations. In the video you can see how the braking torque applied to the suspension arm causes a much faster compression of the suspension.
So applying the “differentiate” tool to the measures of dive i obtained the acceleration suffered by the steering during the braking.
The maximum acceleration for the two configurations are:
- Res_susp: 17.8 [m/s2] (about 2g)
- Res_arm: 46.2 [m/s2] (about 5g)
The last measures are about the longitudinal force and the longitudinal slip on the front tyre under braking, that respect the analytical “Magic Formula”:
Under braking the longitudinal slip is about 0,0279 and the force is 372,5 [N].
The goal of this analysis was to model the two different braking configurations and verify their different dive effect. The simulations, and the relative measures carried out, have shown that there is a significant difference between the two braking configuration, in terms of dive and compression. The final results is a difference on the compression and also on the dive effect, between the two simulations, of about 0.007 [m], for a mean deceleration of 0.4g.
 Fabio Fazi “La progettazione della motocicletta”, Giorgio Nada editore 2013