updated on June, 2020

## Introduction

The Pro-link suspension system is one of the most common solutions adopted in the field of road and racing motorcycles. In general, the main functions of a suspension system are:

- Allowing the wheels to follow the road profile in order to maintain the maximum capability of transferring forces to the ground.
- Control the trim of the motorcycle during the riding.
- Reduce as much as possible the vertical forces and vibrations which originates from the ground in order to improve riding comfort and road holding.

Depending on the scenario in which the vehicle will be used the previous characteristics take different importance and objectives. Could be very useful have a virtual model of the suspension system which can help the designer rapidly and effectively during the design and optimization process.

## Objectives

The suspension’s behavior can be represented by the curve of the speed ratio between wheel’s vertical velocity and shock absorber’s compression velocity, against the wheel‘s vertical displacement, as well as the reduced ground stiffness curve against the wheel‘s vertical displacement. In literature and in technical practice values and trends of these curves are present. In particular, for racing motorcycles is common to see a flatter speed ratio’s curve than for the road ones in order to ensure the maximum sensibility of the rider in out-cornering conditions.

For these reasons the first section of the project is dedicated to the kinematic study of the suspension system. In particular a Pro-link suspension system mounted on a Pre moto3 racing motorcycle has been implemented in an ADAMS model. Moreover the model has been parameterized in some of its dimensions: beam length (BL), triangle side 1 (T1), triangle side 2 (T2). Subsequently the following studies have been carried out:

*Design study:*analysis of the speed ratio’s curve varying one length at a time.*Design of experiments:*analysis of the speed ratio’s curve varying all the dimensions at the same time but for a finite number of values within an interval.*Optimization:*simultaneous variation of all of the dimensions with very little step size in order to achieve a target speed ratio’s curve against vertical wheel displacement.

At this point other two sections were added in order to calculate further data useful for the following detailed design process:

- Reactions forces generated in correspondence of the joints and connections to the frame, during a stoppie landing. The values of the force during time are been obtained through another multibody code and verified experimentally.
- Study on the minimum link to frame connections’ stiffness in order to achieve a maximum variation of the ground reduced stiffness equal to 5% respect to the full rigid case. Moreover beam and triangle structural flexibility have been considered.

## The modelling problem

First of all the model has been generated starting from points which coordinates are automatically computed by the code starting from the length of the characteristic dimensions of the suspension system. The name of the dimensions and the position of the points are exposed here:

Two cylinders of a specific length were also modelled in order to add eventually the end of shock absorbers’ stroke. After the creation of all the components the joints have been imposed. They have been chosen in order to avoid any redundant constraint. They are reported here in correspondence of them insertion position:

The absence of redundant constraints is verified through Groubler’s equation:

DOF = 6*N – (5*T +5*R +4*C +4*H +3*S +3*P)

= 6*5 –(5*2 +5*1 +4*2 +3*2)= 30 – 29 = *1 DOF (swingarm rotation).*

Finally, the mono-shock was added. The curves related to its stiffness and damping (F vs def. and F vs def. vel.) were implemented in the code by the use of 2 splines which fit experimental data obtained for a Pre moto3 racing motorcycle.

## Simulations and analysis of results

Kinematic optimization:

The results obtained through the variation of the quotes of the beam, triangle side 1 and 2 are presented below. The range of variation consist of a +/- 10% with respect to the initial value which corresponds to the third iteration.

The results of the *“design of experiments” *are now presented. All the parameters have been varied simultaneously always in the same range and with a discretization equal to 3. The objective of this study is to find a configuration that is already very similar to the “theoretical” one researched by the successive optimization.

Finally an *optimization* was performed in order to find the variable’s values which are able to reproduce in the best way an objective theoretical curve. In particular a more linear behavior of the suspension system was researched.

The cost function used during the optimization was:

The script, performed at each iteration, consist in a vertical actuator that imposes a maximum rear wheel’s vertical displacement. It could be set equal to the maximum allowed by the mono shock stroke’s for example.

The speed ratio’s curve before and after optimization are now presented:

Force computation:

For the force computation the following trend of the force, applied at the wheel pivot, with respect to the time was used in order to simulate a stoppie landing: (Maximum absolute value: 4670 N)

The computed results of the forces generated in correspondence of the joints and the connections to the frame are now reported. This values can be used for an hypothetical bearing sizing and for computing the stresses generated on the frame after a stoppie landing.

Ground reduced stiffness and structural flexibility

The last section of the project consist in the calculation of the minimum stiffness of the suspension’s (to frame) connections in order to achieve a maximum variation of the reduced ground stiffness equal to 5% with respect to the completely rigid case.

Also the influence of the triangle and beam’s structural deformation was considered. To achieve this task the geometry of the components has been modify inside ADAMS to replicate in a better way the real components and then the parts were meshed in order to add the flexibility.

The spherical joints in correspondence of the connection between the suspension and the frame were substituted by some flexible bushings with zero damping and rotational stiffness but with a translational stiffness different from zero and equal along all the directions.

In order to calculate the reduced stiffness on the ground the rear pivot was lifted to a maximum height (obtained considering the shock absorber maximum stroke) and measuring the actuation force. The bushing’s stiffness was varied in order to achieve a maximum actuation force equal to 95% of the initial value.

The results can be summarized with the following table:

## Conclusion

Through the use of this model is possible to design in a quite simple way a Pro Link suspension system which own some desired requisites. Also the structural flexibility was investigated in order to provide to the designer some useful information for a more complex detailed design.

One possible future development could be the introduction of the possibility to optimize the speed ratio and the reduced ground stiffness simultaneously by letting the code to choose between some discrete spring stiffness’ values in order to obtain a speed ratio’s and a reduced stiffness’, objective curves.

## References

[1] V. Cossalter, “Motorcycle Dynamics”, 2002