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(M. Da Lio, F. Biral) |
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(M. Da Lio, F. Biral) |
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These researches have been carried out in co-operation with Aprilia racing team and in particular with eng. Luca Fabbri |
Studying motorcycle dynamics is quite a hard problem both because of its complex mathematical descriptions and because this kind of vehicles is naturally unstable. In fact, without control they fall to one side due to capsize mode.
Designers would find it very useful to know in advance how motorcycle performances change in portions of real tracks with regard to their principal geometrical and inertial parameters.
Besides, it would be very important to measure performances independently of the driver's skills, but only with regard to intrinsic vehicle characteristics.
With this aim in mind, we have developed a method (called "optimal manoeuvre") and a software (named Handling) which can measure motorcycle manoeuvrability and handling on portions of real tracks, independently of the driver's skill and changing the principal vehicle characteristics.
Although often used, the concepts of handling and manoeuvrability don't have a unique and rigorous definition yet. Thus, we think that the meaning of these two terms have to be cleared up.
We define a vehicle "handy", when it is easy to drive. This involves the driver's judgement, who may find different vehicles more or less easy to drive.
On the other hand, a vehicle is said to have good manoeuvrability characteristics when is able to follow complex manoeuvres fast. This correspond to the ability of a vehicle to complete a given manoeuvre as fast as possible without exceeding existing physical limitations, like tyre adherence or road borders, but without considering the physical and mental pilot effort.
With the term manoeuvre we mean a generic motion from an initial to a final position (both supplied) trying to follow a given portion of a track.
Unlike handling, manoeuvrability is thus defined without involving driver skill and for this reason is an intrinsic vehicle property.
With easier words, we could say that manoeuvrability is a natural characteristic typical of the vehicle, very important to define its dynamics qualities and able to say if the vehicle in object is better with respect to another one chosen as reference, apart from pilot judgement.
From the definitions expressed above, itís evident that a good manoeuvrability itís a necessary condition, but not sufficient to have also good handling behaviours.
From a rigorous mathematical point of view, either manoeuvrability or handling can be measured by means of a penalty function, defined as an integral in the interval of time T during which the given manoeuvre is completed:
The integrand function depends from the state x and inputs u and is properly formulated in order require a velocity as high as possible and at the same time to penalise the vehicle states in which this is near the physical limits (of adherence, road borders etc.). In case, the integrand function also involves some more terms regarding a cost of acting due to the driver, we would have a measure of handling.
To evaluate manoeuvrability, as the integral above expressed, the vehicle motion between the initial and final states needs determining. Of course infinite solutions are possible and from a mathematical point of view they derived from different values of the inputs u, while from a physical point of view they correspond to different trajectories completed to reach the final position. Among these the most efficient one will be chosen, that is the one which is identified by the minimum value for the integral index I previously defined. This is the best result the supplied vehicle is able to perform in order to accomplish the desired task.
Thus, the minimum value of the integral index I, which a vehicle can get during a given manoeuvre, is the measure of its manoeuvrability and it differs from vehicle to vehicle.
Itís like to have the best (o better a perfect driver) who always chooses the best trajectory (the optimal manoeuvre) among the possible ones and in order to run along the supplied track and using at his best the dynamical characteristics of the vehicle and the whole width of the road.
We have developed a software for simulations, named handling, which solves the optimal manoeuvre problem (from a mathematical point of view is a problem of optimal control between the given initial state and a final one). One of its final outputs is the measure of manoeuvrability granted the vehicle geometrical and inertial characteristics.
The software is based on a mathematical simplified model of a motorcycle, which describes accurately its gross motion. Even if simple it requires a system of 27 differential equations with boundary conditions (initial and final state). The trajectory is forced to stay inside whatever strip we like which represents a realistic portion of a track.
The results, we are going to show below, derive from the analysis of the behaviour of a racing motorcycle on some curves of Mugello racing track (Italy, is one of the world motorcycle championship tracks). In particular the two curves S. Donato and Luco Poggio Secco.
The following picture shows the optimal trajectory determined with the optimal manoeuvre method and which corresponds to the best possible performance with the given vehicle in the portion of track in object.
In the next graph, we compare the telemetric data of the velocity measured on the front and rear wheel of a motorcycle, (which are different because of slip) and the one we obtained with the optimal manoeuvre method.
Instead, this last image shows lateral forces and thrust developed by motorcycle tyres.
Of course using this software makes possible (at least as tendency) to measure the effect in changing some design parameter. For example, in the picture displayed below the different velocity of two different motorcycles is shown (the one with the black curve is obviously faster).
The optimal manoeuvre method turned out to be an excellent tool to measure motorcycle manoeuvrability independently form driver skill. The results are realistic, in spite of a mathematical model, which only describes the gross motion. At the moment we are developing a new software simulating the optimal manoeuvre, and based on a more detailed mathematical model which includes more bodies and more sophisticated tyre models.
In conclusion we would like to point out that this method can be applied to every kind of vehicle not only to motorcycles, provided the mathematical model which describes the vehicle motion is given.
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