As mentioned, the brakes system plays a big part in the stability of the car while driving. Braking force should scale linearly with driver input, which aids in the driver's ability to understand the limits of the car. Temperature stability is also important, as brake temperatures often rise during an event as energy is put into the system. Changes in brake temperature have a significant impact on deceleration, and selecting components and materials that retain temperature helps with consistency in brake performance and reduces driver error.

The most demanding task the brakes system would have to be capable of is a full lock up, so this is the condition I based my calculations on. To achieve this, the force provided by the brakes must exceed the longitudinal force provided by the tire, and this was the underlying premise in my analysis. 

Knowing that I needed to study longitudinal tire forces before beginning the brake studies, I created a vehicle force model. Tires grip varies with normal load, so an important detail here was finding the normal load on the tires at different points during a competitive event. In addition to the static weight of the vehicle, downforce also contributes an additional amount of normal load on the tires, and weight transfer causes changes in the load distribution. I calculated weight transfer by using static weight, center of gravity position and wheel base, which enabled me to find total individual axle weights. Next, I created a model for downforce, which varies with speed and supplies additional vertical load. Adding this to the weight transfer model created a more detailed normal load quantity, and would allow me approximate tire forces by scaling a known tire friction coefficient.

The caliper is possibly the most critical part of the brake system. It houses hydraulic pistons, which are responsible for clamping down on the brake rotor and supplying the deceleration forces. One factor I needed to analyze was how different caliper piston dimensions affected the required brake pressure, which is an important metric for later design choices.

Since I now knew the longitudinal force provided by the tire at various speeds, I could set up a simple torque balance problem to find the required caliper force to cause lockup. Using my tire forces found in the previous step, I applied this force at the radius of the tire, representing the tire torque about the wheel center. Next, I set up a brake torque calculation using a variable known as effective rotor radius. This is the location on the brake rotor where the caliper clamps down, and can be adjusted depending on rotor design. To solve for lock up brake force, I divided the tire torque by the effective rotor radius, which gave me a required brake force. Since brakes apply deceleration forces by using friction, I converted frictional brake force to caliper piston normal force by dividing by the brake pad friction coefficient. This is another variable to consider downstream, as a higher friction coefficient will mean less required brake force to achieve a given amount of deceleration.

Finally, solving for brake pressure was done by dividing normal force by the piston area in the caliper. Since each caliper candidate had different piston sizes, each had a different required pressure to achieve the required force. The required brake pressures serve as an important characteristic with which to analyze the caliper later in the design cycle. It can be thought of as a link between driver effort and performance, where higher pressures correlate to more driver effort.

Now that I had derived a way to calculate both vehicle forces and caliper pressures, I could construct a more comprehensive deceleration model. This model would help me to quantify later design choices such as pedal ratio and master cylinder sizing. 

I created two separate variables: theoretical deceleration and tire limited deceleration. To create the former, I creates a brake pedal force versus brake torque plot by factoring mechanical leverage from pedal ratio, hydraulic leverage from master cylinders and braking friction force. These are variables I would work with more after calipers were chosen, but I put in placeholders to get an initial read on how the brake system was shaping up. In a similar manner to the caliper pressure calculations, I could now calculate brake force as a function of applied pedal force. 

Next, I created a speed versus tire torque plot, which allowed me to see how downforce affected the tire's ability to grip the road. This used a weight transfer model with downforce, so I was able to plot maximum available tire force versus speed.

I plotted the theoretical and tire limited deceleration against each other, and to a certain point, the graphs are the same. The point where they diverge indicated where braking forces exceed the maximum grip of the tire, allowing me to verify my earlier lock up calculations, and also begin to understand how downstream design decisions affected this lock up point.

Another parameter I wanted to visualize was maximum deceleration, so I added a third variable: deceleration in G's, which was calculated using braking forces. This third variable gave me a way to visualize the maximum deceleration allowed by the tires, which would aid later validation and driver performance optimization. 

There are many potential caliper candidates for FSAE purposes, so the first step I took was narrowing down the available choices to ones that passed the most critical constraints. The first constraint was based on wheel sizing. The team runs 10" diameter wheels, which places constraints on the outer diameter of the calipers, which sits inside the wheel. Additionally, the location of the brake fluid inlet needed to be in a position that did not interfere with the rest of the wheel assembly and could be easily accessed without disassembling. In the end, six potential candidates met the initial criteria and will be referred to as calipers A, B, C, D, E and F, with A and B being the incumbent choices for front and rear respectively.

Now that I had created a standardized metric for system compatibility in brake pressure, I used this in combination with other caliper properties to decide on a final candidate. I consulted the wheel packaging team in addition to considering braking metrics, and created a decision matrix with several categories: weight, size, required lockup pressure, available pad options, and cost. After taking inventory of tools and stock, I weighted these criteria, and scored each caliper. Caliper C and F scored the highest, owing to their pad options and size.

The master cylinder is the component responsible for supplying hydraulic leverage, which is necessary due to the high pressures involved in slowing the vehicle. It works in conjunction with the brake pedal ratio (which supplies mechanical leverage), so the combination of these variables would set the required foot force from the driver. To set this target, the team set up a foot force testing rig, and asked the drivers to apply the maximum force they could reliably produce in a racing event. Using my deceleration model from before, I was able to iterate through combinations of master cylinder bore sizes and pedal ratios to achieve the target foot force. This enabled me to select master cylinders and also provide a pedal ratio that the pedal box designer could use.