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If car or any a driving motor vehicle is moving with very high speed, the lateral force will be acting on four different tire which considerably increases the probability for vehicle to get into critical situation. So, for high lateral acceleration with much speed, the equation for cornering will be different as compared with low speed. Therefore, in order to prevent the lateral accelerations the tires must be developed with slip angels and lateral forces which is present in four different tires. While cornering time, when the tire is to develop lateral force, all tires will exhibit lateral slip during rolling. So the angle between the direction of travelling and heading is called as a slip angle α.
2.0 Discussion and analysis
2.1 Basic relationship between slip angle and lateral force
The lateral force (Fy) is also called as cornering force it is hen the camber angle come to zero. For any given load on the tire, the cornering forces is increasing corresponding to slip angle (Biral et al 2018). At very low slip angle (nearly up to 8 degree), there is a linear relationship ,and the cornering force equation is defined as
Fy=Cα *α
Where, Cα =the proportionality constant as cornering stiffness
The proportionality constant i.e. cornering stiffness is explained as the curve slope of Fy versus α when α is equal to zero. When the slip angle is positive it produces a negative force (in left direction) on the tires, which implies that Cα is to be negative and when the cornering stiffness is of negative slope then the Cα will take positive value.
There are some factors in which cornering stiffness is dependent on many variables such as the size of the tire and the types (basis construction or radial), cord angles, width of the tires, number of plies and the treads (Braden et al 2019). For any tire combination there are mainly two factors that affect cornering stiffness i.e the load and the inflation pressure. While the speed of the tires is not so much strongly influences to the cornering forces which is produced by the tires.
Since, there is strong dependence of the tires load the cornering resources might be elaborate by using the coefficient of cornering which is developed by dividing the stiffness of cornering by the vertical load (Cunha et al 2018). Therefore the coefficient of cornering (C.Cα) is defined by the formula as :
C.Cα= Cα/Fα
The coefficient of cornering usually have largest value at very low loads ,which is contiousuly diminishing as its load reaches to its rated value i.e The rim & tire load. At full load (100% load),the coefficient of cornering is ranging at 0.2.
2.2 Mathematical tire model
It is very well known that, tires are very complex and composite which consists of many different layers of materials as shown in the figure. A tire is very in homogeneity .As a result, tire behavior is affected by more than just the tire's molecular materials, qualities and structure. As a result, approximations are made in order to develop empirical tire models there are a few
There are three basic models that can be used to obtain a greater understanding of
Under what conditions do tire deflection, forces, and footprint behavior occur?
Parameters of cornering: the elastic foundation model (Kopf et al 2020). The string pattern and the beam model are two different types of models. Regardless, any of them. These models are capable of addressing the complexities of a situation. By employing a physical tire, genuine information can be achieved by empirical values of stiffness.
In the elastic foundation model, every small unit is thought to act independently of the others (Figure 3). This model is the smallest of the three because each constituent functions as a basic spring that is autonomous of the others. It's worth noting that the lateral force assessed at the axle is equal to the value below the lateral force statistical distribution (Levy et al 2019). This confirms the theory that, despite its simplicity, this model can be quite effective in predicting and showing diverse tire behaviour.The elastic foundation model also provides for discontinuities in disturbance distribution and tire centre line slope. While on the other side, the string model and the lateral displacement is hold by against the tension in between the elements. This elastic foundation model also allowable for the discontinuity in slope but the discontinuity in displacement is not permissible (McRae et al 209). String model is just similar to the beam model, where single element has diverse effect in the neighboring elements. Since, in beam model, every single element creates the bending moments which are acting on the neighboring elements.
2.3 What is FSAE TTC ?
The FSAE TTC or in long form Formula SAE Tire Test Consortium was created for giving good condition tire data to the teams of FSAE for use in their racetrack analysis, design and installation. At Research Facility of Calspan's Tire, data on 10 various tire architectures is currently being monitored and allocated to all associated parties. For this research, we examine the background of the FSAE TTC, including its founding, organization, and ongoing operations. Specification of tire testing will be discussed, including the numerous alternatives and limits that were taken into account while creating the tire test matrix. Finally, there is an evaluation of the uniform data (Moore et al 2017 ). This document includes a description of all transmit power as well as an outline of how FSAE groups can use the data.
The flat -belt test machine for the tire was used for FSAE TTC tire test and this testing machines is the nature based first high load, high speed, six-constituent flat-belt .Its has following capacities and are listed below:
In 2004, Motorsports conference and exhibition, it has been discussed about analyzing the FSAR tires at the Calspan potential and that knowledge of the FSAE tire test combination was arise. Firstly the students had a keen interest in finding the data for FSAE transports which was the most primary technique vacant faced by their teams because very limited data was available for one or tire construction. Secondly, the testing was more expensive as it is more in specialized form. So the pooling resources across numerous FSAE projects would be the most feasible approach for candidates to gain approach to a big volume of high-aspect of tire data.
2.3.1 Operating Condition of FSAE
The majority of FSAE cars weigh between 500 and 700 pounds (1790 to 3210N). Several teams use mechanical down force devices, which can add up to 220 lb. (900 N) to total mechanical stresses at high speeds. The largest single tire loads observed on FSAE cars which is projected to be roughly 450 pound and based on the aerodynamic distribution, static distribution of weight, and exchanges loads is experienced during acceleration, braking, and turning (1800 N). This FSAE TTC will chose to analyses up to a standard load of 460 lb. (1980 N). According to an poll of teams of FSAE, tire pressure level were in between the 12-15 psi (72 to 101 k Pa) range. According to Good tire, 12 psi (83 k Pa) is the recommended pressure for their tires. Therefore we have chosen to take the tire pressures between 10 and 15 psi (56 and 110 k Pa).
Calspan also contains heaters to enhance the temperature difference of the test room as well as the temp of the belt- highway but further it is critical because the bearing air that supports the tire weight and roadway draws heat from the belt (Murua et al 2019). To prevent this impact, belt warmers were used during the tests. At the FSAE competition, a test rate of 42.2 k ph is chosen as a normal speed that also provided suitable temperatures. For the Post-test pyrometer, the values are ranging from 115 to 152 degrees Fahrenheit (44 to 65 degrees Celsius), which the students of FSAE teams who participated in the tests claimed was around what they're viewing on their automobiles.
2.3.2 Capabilities of Calspan Tire Machine & Setup
Despite Calspan TIRF's extensive tire testing capabilities, the compact FSAE tires posed a great challenge to the building's engineers (Nandu et al 2018). The first issue would be whether the machine's head, where is the tire is fixed, have adequate travelling to produce the appropriate tire loads.
However, with the 25.5cm tires, an adaptor plate was put on the heads to allow the wheel to be installed just below head while still allowing enough travel above the pretty tough and
Figure 8 shows this plate that how the wheel's spin axis is underneath the primary axis of the head (Pelisser et al 2019). As a result of the inability to send drive/brake torques through to the adaptor plate, the (25.5 cm) tires will be limited to restless rolling tests only.
2.4 Analysis of Coefficient of Friction, slip angle and cornering stiffness
A unit less ratio of friction force to normal force is known as the coefficient of friction. It is widely acknowledged that the friction force generated is not proportional to the contact surface area (Semenuk et al 2021). Some of the implications of the coefficient of friction decreasing with speed should indeed be examined and when there is increment in slip angle, the back of the footprint begins to move, resulting in a decreased coefficient of friction. As a result, at a small slip angle, lateral force will reach its maximum value.
If the travel diffraction is different from heading the wheel i.e the angular displacement of the wheel is slightly different with respect to the path of the tire which it is following, the angle of slip (α) produces another component called as lateral force represented as Fy.Now the lateral for Fy will be acting at the central point of wheel direction (Swamy et al 202) .Also it must be marked that the slip angle is not similar to steering angle.
From elastic model foundation, there is a relationship between the lateral force and under steer angle i.e.
Where ,Fv =Normal Force
c=stiffness
d=displacement of tire centre line
l= Contact path length
And thus lateral force is directly equivalent to slip angle and maximum forces can be calculated by the formula
Where dm is the maximum center line displacement of the tire.
Also, the cornering stiffness and the lateral force can be estimated by the formula given below:
The cornering stiffness is defined on the basis of radian as shown in below equation:
This cornering stiffness is used to find the slope on lateral force curve. It is usually around 5 to 6 times higher than camber stiffness for modern bias tires. Different components of lateral forces i.e Fs ,FD & Fy can be calculated by the relationship :
Where, FD=Drag force
FS =Component of the force perpendicular to
Travel direction
μR =Rolling resistance
3.0 Conclusion
The establishment of FSAE is make high quality tire data in large scale. With the combining values of cornering stiffness captured by results gives Vehicle stability enhancement in the area of chassis and other automatic vehicles. The main objective of this research is to integrate the data processing, mathematical model and analyzing in real world. By this condition, the values of the components in designing phase, car or vehicle sets can be adjusted. When it comes to rubber tires, however, this is far from the case. The temperature rises over the ideal value as the tire is applied the brakes further, and the coefficient of friction starts to fall. Similarly, when the speed rises, the temperature goes up, and the coefficient of friction begins to fall after reaching its maximum value.
Refrence list
Journal
Biral, F. and Piccinini, M., PATH PLANNING AND CONTROL OF SELF-DRIVING VEHICLES AT THE LIMITS OF HANDLING.2018
Braden, D.P., 2019. Reader Approval (Doctoral dissertation, California State University, Sacramento).
Cunha, J.R.F.C.D., 2018. Desenvolvimento dos sistemas de suspensão e direção para veículos do tipo formula SAE.
Kopf, L.F., 2020. Dynamic analysis of the WESMO FSAE car’s suspension (Doctoral dissertation, The University of Waikato).
Levy, A. and Potter, J.J., 2019. Design of the WUFR-19 FSAE Suspension.
McRae, J. and Potter, J.J., 2019. Design Considerations of an FSAE Steering System.
Moore, D., Developing a Tire Model for Computational Simulators.2017
Murua, A.M., Bistue, G., Rubio, A. and Gonzalez, J., A Direct Yaw Moment Control logic for an electric 2WD FSAE using an error cube PD controller.2019
Nandu, P.V., 2018. Woods Tire Model (Doctoral dissertation, The University of Texas at Arlington).
Paul, T., 2019. Design of Chassis, Impact Attenuator, Suspension and Aerodynamic Systems of a Formula SAE Car (Doctoral dissertation).
Pelisser¹, A., Kawamoto¹, A., Vatanabe, W. and Namba, V., 2019. A Gaussian process model for tires in combined slip case.
Semenuk, M., 2021. Torque Vectoring Development for Formula Student Vehicle.
Swamy, V.S., Shivayogi, H.K. and Mathivanan, N.R., 2020, April. Selection of Optimal Tire and Design Optimisation of Steering System for a Formula Student Race Car through Tire Data
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