Specification of a Caliper Disc Brake

by Shane Moore

Submitted : Fall 2012

This project concerns the operation of a disc brake on a motor vehicle and determining the optimal diameter of the brake and the size of the rotor to optimize braking torque. The diameter and the size of the rotor can be broken down to the radius from the center of the rotor to the outer edge and the radius from the center of the rotor to the area contacted by the brake pad respectfully. An expression for the total torque can be derived from these two radii, the angle they make, the coefficient of friction between the pad and the disc, and the maximum pressure that the pad exerts on the disc. The total torque can be then found by integrating the differential torque over the frictional contact area. The equation dA=rdrd(-O-) was a clear hint that this was a polar integration problem. After setting up the appropriate integral with respect to the total torque, the problem can be made much simpler by removing constants from within the integral and multiplying them at the end of the problem. Solving the integral, u, Ri, and Pmax are constants and can be removed from within the integral and you are left with the simple integral rdrd(-O-) from Ri to Ro and then from 0 to A. You ended up with an expression of torque in terms of u, Pmax, Ro, Ri and A as request in problem (a).

With help from Dr. Rimbey, the solution to problem (b) was found by determining an inner radius that would maximize torque if the other expression in the equation from (a) were considered fixed. Setting u, Ri, Pmax, and A equal to k, Ri can be found in terms of Ro by taking the derivative with respect to Ri. After taking the derivative, set the solution equal to zero and solve for Ri, neglecting the negative value since Ri is a length. Applying the second derivative test to this solution of Ri ensured that this solution was really a max value.. The second derivative resulted with my solution being less than zero if Ri was greater than zero, verifying that it was a maximum value. Solution (c) was a simple substitution of my equations I found in (a) and (b) and the values chosen in the problem. Plugging the equation from (b) into (a) resulted in an answer for Ro. Ri was found by simply plugging my answer for Ro into the equation from (b). Lastly, the diameter was found by multiplying each radius by two. Something that one could learn from this assignment is how the physics behind the performance of brakes is derived from Calculus.

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