FSAE Brake Design
Accomplishments:
- Use fundamental physics equations to size brake rotor
- Use heat transfer equations, MATLAB, and shop testing to build thermal model of rotor temperature
- Use FEA software to ensure strength of rotor under braking loads
- Source and assemble brake system
Performance Goals:
- Lock all four wheels as required by FSAE rules
- Provide 2.0 g braking required by lap-time simulations
- Meet weight schedule requirements:
- Durable at elevated temperature from braking loads
The brake system is a critical safety feature of the vehicle which must also provide sufficient deceleration performance at elevated temperatures and high braking loads. The brake system includes the driver’s braking force, the brake pedal geometry, master cylinders, bias bar, brake lines, caliper, brake pads, and rotor. Several of these parameters have been constrained by the existing USC Racing platform, leaving the caliper selection and rotor design to dictate the performance of the brake system.
Rotor Sizing:
When the driver applies the brakes, the car is decelerated by the friction force between the tires and the road. Greater deceleration requires higher friction force. "Maximum possible deceleration occurs at the maximum coefficient of friction between the tires and the road. This happens just as the tires are about to skid” (Puhn). This is the moment before the wheels lock, after which deceleration decreases as the static friction between the tire and the road becomes kinetic. Therefore, the 2.0 g deceleration goal must be achieved at the instant the wheels lock. With the weight on the front wheels due to longitudinal weight transfer from 2.0 g deceleration given by W_f = 678 lbs, tire coefficient of friction μ_t = 1.46, and tire radius R = 8 in, the braking torque at the tire required to lock the front left wheel, T_t , is calculated as 335 ft-lbs by:
T_t = 2 × W_f × μ_t × R (Equation 1)
The brake system must then supply a torque at the caliper, Tcal , equivalent to Tt (335 ft-lbs) to lock the wheels during the 2.0 g deceleration. Tcal is given by:
T_cal = F_clamp × μ_pad × r_eff (Equation 2)
Figure 1. Free body diagram shows caliper force (F_clamp green), creating a torque on the rotor (T_cal brown), which is equivalent to the torque at the tire (T_t red).
where F_clamp is the clamping force produced by the caliper, μ_pad , is the brake pad friction coefficient, and ref is the effective rotor radius (the radial distance from the rotor center to the assumed point of contact at the brake pad centerline). The value of r_eff is variable, and the average value of μ_pad is dependent on the specific pad compounds compatible with a given caliper. F_clamp is the product of the caliper piston area, A_piston , and the hydraulic pressure in the brake lines, P_hydraulic . The hydraulic pressure is dependent on several brake system parameters as shown below:
P_hydraulic = F_driver × k_bias × k_pedal / A_mc (Equation 3)
where F_driver is the force applied by the driver (approximated as 100 lbs), k_bias is the front/rear bias bar setting (0.76), k_pedal is the pedal ratio (4.0 is used as a rule of thumb start point from The Braking Handbook), and A_mc is the master cylinder area (0.307 in^2 taken from existing USC Racing vehicles). The parameters for P_hydraulic are constrained by the pedal box geometry and hydraulic components. Therefore, the remaining variables in the brake system design are related solely to the caliper and rotor: ref , A_piston , and μ_pad . These three variables are linked to each other because each brake pad is caliper specific, and each caliper has a maximum possible rotor size. The rotor size constraint is defined by the caliper which must be positioned within the wheel such that no point on the caliper is less than 5.00 mm to the inside of the wheel as advised by OZ Racing.
The AP Racing CP4227-S0 caliper with its respective AP Racing CP4226D27-RX brake pads requires a minimum rotor diameter of 6.63in which fits within the packaging constraints of the wheel package and provides sufficient caliper torque due to the high piston area and brake pad coefficient of friction. The final design of the rotor is 7.50in in diameter, which ensures clearance for the hub and provides a larger factor of safety for the brake lock requirement. The maximum hydraulic pressure at this increased 7.50in diameter is approximately 866 psi, far below the brake line and caliper manufacturer specification of 1200 psi.
Figure 2. AP Racing CP4227-2S0 and 7.50in Rotor. Note the caliper is positioned at the minimum 5.00 mm distance from the inner barrel of the wheel.
Based on weight targets, the rotor must have a mass no greater than 0.42 kg, though the initial rotor shown in Figure 2 has a mass of 0.46 kg. This basic rotor was designed with the maximum 7.5in outer diameter, and an inner diameter just large enough (with safety margin) to allow the brake pads to make full contact with the rotor. Decreasing the rotor diameter significantly is not feasible due to hub packaging constraints, and the rotor thickness of 4 mm is prescribed by AP Racing; thus, mass reduction to meet the 0.42 kg target must come from rotor area reduction (as seen in Figure 4 with the drilled hole pattern). The drilled holes drop the rotor mass to 0.40 kg, as well as aid in the dissipation of gases which cause brake fade.
Thermal Analysis:
To ensure the rotor can withstand the heat absorption under braking and continue operating as expected, two scenarios are considered. The first is an emergency braking scenario where it is conservatively assumed that 100% of the vehicle’s kinetic energy is transferred to heat as it comes to a complete stop from its gearing limited top speed of 69 mph (Equation 2):
1/2m_C × Δv^2 = m × R × c_p × ΔT (Equation 4)
where m_C is the mass on the front left wheel under maximum deceleration (155 kg) and Δv^2 is the change in vehicle speed, m_R is the weight of the rotor (0.40 kg), c_p is the specific heat capacity of steel (420 J/kg°C), and ΔT is the change in rotor temperature. Solving for ΔT yields a final rotor temperature of approximately 400°C, allowing a 1.5 Factor of Safety before reaching the 600°C limit (determined by steep decline in steel yield stress beyond 600°C and brake pad maximum operating temperatures).
The second heat absorption scenario considers repeated heat absorption and convective cooling to simulate rotor temperature driving about an autocross track. An iterative MATLAB script uses Equation 1 above for determining rotor temperature increase while braking and Equation 2 below to determine rotor temperature decrease during straight sections of the track where the brakes are not used:
h × A × t(T_R-T_∞) = m × R × c_p × ΔT (Equation 5)
where h is the convective heat transfer coefficient for forced air flow (turbulent flow over a flat disc), A is the surface area of the rotor, t is the time in seconds of the non-braking section, TR is the rotor temperature entering the corner, and T_∞ is the air temperature. Using track data from previous USC Racing vehicles, realistic input velocities and time intervals are determined. The script indicates a maximum average temperature of 406°C at the end of the several lap endurance event. Despite the several laps of heat generated from braking, the temperature does not significantly exceed the approximate 400°C temperature from the emergency braking scenario. This is largely a result of the lower velocity change in an autocross circuit (rarely exceeding 50 mph and never reaching a complete stop), which decreases heat energy into the rotor by Δv^2 . Additionally, straight sections of the track allow for sufficient convective cooling to negate the temperature increases.
Figure 3. Plot of Rotor Temperature vs. Time in a Multi-lap Endurance Event Simulation