The calculation of the temperature on a disc of a disc brake system analytically

The brakes have been existed since the beginning of the automotive history as it is required to stop the motion of the vehicle whenever it is needed. The braking operation, in other words stopping the vehicle is done by friction created by pushing a material against the other. This operation is done by pushing the shoes against a drum which rotates with wheels and cover the shoes in some applications and by pushing the pads against a disk rotates with the wheel in most of the other applications. During a braking the kinetic energy of the vehicle is converted to heat created by the friction of the pad and disc surfaces. Therefore the temperatures of the discs and pads reach very high levels and this leads to thermal stresses. Thermal stresses may cause cracks on the disc surfaces. In this thesis it was aimed to derivate a one dimensional model with some assumptations to calculate the maximum temperatures the discs can reach in disc brake systems. In the early chapters the aim of the thesis has been explained briefly and the studies about the subject has been summarised. After the summary of the literature about the subject the variety of the brake systems used in the vehicles have been illustrated and it has been explained how the disc brake systems became widespread. Then the working principles of disc brakes have been explained, the precaution actions against the possibility of a leakage in hydraulic systems have been mentioned and the different design methods to be able to dissipate the heat generated from the disc and the pad to the air has been illustrated. The advantages and the disadvantages of drilled discs and slotted discs against each other has been explained. The materials used for pads and discs have been stated. The forces exerted by the pad to the disc and exerted by the disc to the pad are internal forces when the whole vehicle is considered as the free body. Therefore a direct relation between the pressure exerted by the pad to the disc and the decelaration of the vehicle cannot be created. It can be seen that the external forces those lead to stop the vehicle are the frictional forces acting on the contact point of the wheels to the ground. Hence, to be able to relate the decelaration of the vehicle to the pressure applied by the pads a model has been established on the braking dynamics on a vehicle. In the model the vehicle body without the wheels, the front wheels and the rear wheels have been chosen as three separate free body diagrams. In total nine independent equations have been obtained and solved. The model also has provided the conditions to keep the wheels rotating without slipping during a braking process. After the numeric value ranges in the real life for the dynamic parameters have been introduced a thermal model has been established. The thermal analysis part of the thesis has some assumptations as the dynamic model. First of all the heat transfer via conduction has been ignored and the problem has been modelled as a one dimensional model. The temperatures in the linear and radial directions have been considered as identical. Secondly the coefficient of friction between the pad and the disc has been considered as constant although it can vary regarding the speed and the temperature. Another assumption has been that the pressure applied by the pad to the disc is uniform. Another assumption can be considered the choice of the heat convection coefficient. It has been assumed that the heat convection coefficient is constant during the process. As many articles in the literature use the same approach it has been considered the use of an avarage heat convection coefficient would be more suitable to be able to compare the results with the data of the other articles. Also the effects of the vents have been represented in the heat convection coefficient. The one dimensional conservation of energy equations has been formed specific for the problem. The approach while solving the equation has been the control mass method. A differential area has been defined and the equations for the time interval while the differential area is under the pad and for the time iterval while the differential area is not under the pad have been solved seperataly. As the time increases the valid equation may change a software has been required so that the initial conditions for different time intervals can be assigned. The solution algorithm includes two different steps running repeadetly one after the other. Firstly the general solution for the time interval which the differential area is under the pad is solved by using the initial condition. The temperature at the time which the differential area is at the end of the pad becomes the initial condition of the second equation, the equation valid for the time interval which the differential area is not under the pad. The temperature value of the equation at the end of this time interval becomes the inital condition for the first equation again as the differential area gets under the pad again. By calculating the temperature values against different time values a chart can be drawn which shows the temperature against time. The software has been used to calculate the temperatures for different applications which were defined in one of the articles. After the calculation the charts have been drawn and the charts have been compared with the ones in the article. As the real life applications include the effect of the design and the influence of the vents and the method used in the thesis represents these effects in the heat convection coefficient the results have been shown with also suitable overall heat convection coefficients which fit the real life conditions. As the final result of the comparisons it has been seen that the analytically calculated temperatures are between the maximum and the minimum temperatures of the numerically calculated results. After the analytical method was validated with the data obtained from the articles the method has been used to see the situation in the real life cases. The investigation shows that for the worse applications the design must be changed as it will provide more heat convection. In the appendix section the MATLAB code used to calculate the temperatures is given. The code includes the calculation algorithm and the parts which avoid the calculation for the complex time values.