═════════════════Frequently Asked Question═════════════════
Author: Alexander aka dll (madtuning.ru; live4race.ru)
The lack of lively discussion in the previous post made me think that few people understood what I wrote using my car as an example. Here, I will try to explain everything and provide abstract examples. Who will manage to understand it all? =))))
This will help you:
1) Understand how the brake system works
2) Accurately determine what you don't like about your brakes
3) Communicate intelligently when discussing a brake system
4) Decide which modifications work for you to achieve goals
5) Select the right components and understand how they will work together
6) Maintain balance between axles
What does a braking system consist of?
1) Pedal assembly, this is a lever that increases the force created by your foot (Pedal ratio).
2) Master brake cylinder (Master Cylinder)
3) Brake lines
4) Valves to maintain balance. The braking system may have these valves between the master cylinder and calipers: Residual pressure valve, proportioning valve, combination valve, load-sensing valve or restrictor.
5) Calipers
6) Brake pads
7) Brake rotors
-=So let's start with basics (physics)=-
Braking force
This is the torque created by the effective radius of the brake rotor, the pressure applied to the brake pad and the coefficient of friction between the pad and rotor. This is the force that slows down the wheel along with the tire. The main components affecting braking force are how tightly the pads press against the rotors and how far from the center of the hub this force is applied. Thus, the larger the size of the brake rotor, the further away from the center of the wheel is the pressure applied and thus we increase braking force (lever effect). This is like when you need to unscrew a rusted bolt, the longer the wrench (lever) the easier it is.
The recommended braking force is calculated with the following formula:
Tsr = CTP * (tire rolling radius)
the tire-road friction coefficient can be quite hard to calculate, it may range from 0.1 on ice to 1.4 on a dry race track with slicks. If you do not know this value, use one.
Remember to take weight transfer into account as the rear end is unloaded and the front is loaded during braking.
Front:
CTPf = μ*WBTf / 2
WBTf = Wt*((1-Xcg/Rat)+(μ*Ycg/Rat))
Rear:
CTPr = μ*WBTr / 2
WBTr = Wt - WBTf
Where
Tsr — recommended braking force (kg)
CTP — tire-road friction force (kg)
CTPf — front tire-road friction force (kg)
CTPr — rear tire-road friction force (kg)
μ — tire-road friction coefficient (use 1)
VSp — vertical force acting on both front tires (kg)
VCz — vertical force acting on both rear tires (kg)
Wm — weight of the vehicle (kg)
Hcg — distance from the front axle to the center of gravity of the machine (cm)
WB — wheelbase (cm)
Ycg — distance from the ground to the center of gravity of the machine (cm)
After careful calculations, we can understand how powerful brakes we need and what factors this force depends on:
— It does not depend on speed at all
— Can change depending on tire quality, road surface quality, weather conditions
— Depends on wheel size (do you think anyone who installs huge wheels or huge brakes ever calculates them together? =)
— Depends on the weight of the car, ground clearance, and wheelbase; indeed, the lighter and lower the vehicle is, the less transfer of weight affects braking.
Compression force
The force with which the caliper presses brake pads against the disc is measured in kilograms; this force is created by pressure in the braking system multiplied by the area of the pistons (caliper without bracket) or 2*times the area of the pistons (caliper with bracket), measured in kg/cm^2. To increase compression force, you either need to change the system's pressure or enlarge the piston surface. Changing the brake pad material does not affect compression force.
It is calculated using the following formula:
CZ = Pd*Ap
Where
CZ — Compression force (kg)
Pd — Pressure created by the master cylinder (kg/cm^2)
Ap — Effective area of pistons (for a caliper with bracket it's 2*times piston surface area)
So now we can calculate the braking force produced:
SBp = CZ * µL * Re
Where
SBp — Produced braking force (kg)
CZ — Compression force (kg)
µL — Coefficient of friction between pad and disc
Re — Effective radius of brake disk (from center of hub to center of pad)
Coefficient of friction
This is an indicator of the friction force between brake rotor and pad. The higher the coefficient, the greater the friction force; for stock pads it varies from 0.3 to 0.4, while for racing ones from 0.5 to 0.6. 'Hard' pads have a lower friction coefficient but wear less. In contrast, 'soft' pads have high friction and wear out faster. Most brake pads have a temperature-dependent friction coefficient; thus, race pads need heating whereas street pads lose their properties at such temperatures.
Thermal capacity
I hope that it's no secret to anyone that brakes stop the car by converting kinetic energy into heat. Therefore, the heavier your vehicle is and the faster you drive, the more heat it should dissipate to avoid overheating fluid, rotors, and burning pads. A rotor’s ability to dissipate heat depends on its weight and how well it cools.
Kinetic energy formula for a moving car:
K = (Wm * Vm^2) / 2
Where
K — kinetic energy (J)
Wm — weight of the vehicle (kg)
Vm — speed of the vehicle (m/s)
Nothing new here, we perfectly understand that brake selection depends on how heavy your car is and/or how fast you drive. And from driving courses, you should remember (for those who didn't buy licenses=), doubling your speed quadruples your braking distance; this is kinetic energy in action.
Temperature increase formula during braking:
Tp = ((Kd-Kp) / (417 * Wd)) + Tw
Where
Tp - Temperature after braking (C)
Kd - Kinetic energy before braking (J)
Kp - Kinetic energy after braking (J)
Wb - Weight of brake disks (total) (kg)
Tv - Temperature of brake disks before braking (C)
Let's take my car as an example, braking before T2 in Myachkovo =)
Car weight - 1220 kg
Weight of disks - 33.5 kg (front 12 kg, rear 4.75 kg)
Speed on straight - 177 km/h (49.17 m/s)
Speed before T2 - 70 km/h (19.44 m/s)
Temperature of brake disks before braking - 25 C
Kd = (1220 * 49.17^2) / 2 =1474826J
Kp = (1220 * 19.44^2) / 2 = 230669 J
Tp = ((1474826-230669) / (417*33.5)) + 25 =
114 C
So after such braking the temperature of the disks will be approximately 114 degrees. Let's compare with your results? =) For simplicity, you can just say the weight of the car and the total weight of all brake disks.
Alright, let's pause on physics for now and move to a more theoretical part.
There are three things that brakes should do to stop the car:
1) Press the pads sufficiently against the disk
2) Produce enough braking force to lock wheels on any surface
3) Have sufficient mass and cooling of disks to dissipate heat generated by kinetic energy.
All these together should provide excellent feedback.
Brake Pedal Assembly
As we discussed, to brake, a driver must move fluid and create pressure. The master cylinder moves the fluid to create enough force for the pads to press against the disk.
With the pedal you activate the brakes, also it serves as a lever that increases the applied force. This is called 'pedal ratio'.
Usually we apply from 22 to 45 kg of force on the brake pedal to actively decelerate.
As an example in race cars without power assist this effort is about 35 kg, for cars with power assist it's around 22 kg. 45 kg is too much, the pedal will be very stiff.
Pedal ratio A/B
as we see, the higher this ratio the more force is transmitted to the master cylinder. But remember one thing, increasing this ratio also increases the pedal travel.
For cars with power assist this ratio is usually about 4-4.5. For cars without power assist it's from 6 to 7.
Therefore removing the booster with a stock pedal is not a correct option =)
Hydraulics
As I've already mentioned, to press pads against the disk requires fluid movement and pressure in the circuit. These laws are governed by hydraulics (Pascal’s law).
Ideally you want sufficient pad clamping force at minimum pedal travel.
Force applied to the master cylinder creates pressure in the circuit. Pressure is force applied to the master cylinder piston divided by its cylinder area. Therefore, the smaller the area of the cylinder, the higher the pressure.
Pressure in system = Sp / Ap
Where
PP - Piston area of the master cylinder (cm^2)
I will give an example of my stock GTM (cylinder 0.875") at a force of 500 kg
System pressure = 500 / 3.87 = 129 kg/cm^2
And with the master cylinder (cylinder 1")
System pressure = 500 / 4.91 = 101 kg/cm^2
From this, it follows that the higher the pressure, the stronger the brake pads are pressed against the disk, thus increasing the braking force. But this does not mean that if we want powerful brakes, we should install a small master cylinder. Here another factor comes into play - movement. Since liquid is incompressible, any movement of the master cylinder leads to the movement of pistons in the calipers. This movement in hydraulics is called displacement. It is calculated as the product of piston stroke by its area. Measured in cm^3
Displacement = PP * SP
Where
PP - Piston area (cm^2)
SP - Stroke of the master cylinder (cm)
Let's calculate it again for my car's stock GTM (0.875) and a stroke of 3 cm
Displacement = 3.87 * 3 = 11.61 cm^3
And for the master cylinder (cylinder 1") with a stroke of 3 cm
Displacement = 4.91 * 3 = 14.73 cm^3
Here we see an opposite situation, the smaller the piston area, the less displaced volume at the same pedal travel (thus more pedal travel).
Now let's move on to a general analysis of the system, we know that the braking system is closed and therefore pressure is transmitted throughout the system in equal values. Also, besides the master cylinder, there are calipers with pistons (for calculations, the total area of all pistons is used)
This means that the pressure created by the master cylinder drives all the pistons in the system. Since the piston area in the caliper is larger than the master cylinder's, according to hydraulic laws, the force output from the caliper increases significantly.
The higher this ratio of effort, the less force needs to be applied to the pedal (and more pedal travel) to achieve the same result.
Calculate the amplifying factor can by formula
Sz = (Sp * Ps) / Pg
Where
Sz - Force exerted by the caliper (kg)
Sp - Force applied to the master cylinder piston (kg)
Ps - Effective area of pistons (for a caliper with a lever, this is 2 times the area of pistons)
Pg - Piston area of the master cylinder (cm^2)
For example, I take my car again (cylinder 0.875") =)
Sz = (500 * 10.17 * 4) / 3.87 = 5255.8 kg
And with the master cylinder (cylinder 1")
Sz = (500 * 10.17 * 4) / 4.91 = 4142.6 kg
From this it follows that we can increase the compression force by increasing the area of caliper pistons or reducing the master cylinder piston area.
But it's not so simple. Don't forget about another factor - movement. Unfortunately, playing with cylinder areas, we change pedal travel. For example, reducing the master cylinder reduces the amount of displaced fluid - you need to work more on the pedal to compensate for this (pressure will not increase until the pad presses against the disk). The same applies when increasing the caliper piston area (with one master cylinder).
Calculate the stroke of the caliper:
Hp = (SP * Pg) / Ps
Where
Hp - Stroke of the caliper piston (cm)
SP - Master cylinder piston movement (cm)
Pg - Piston area of the master cylinder (cm^2)
Ps - Effective area of pistons (for a caliper with a lever, this is 2 times the area of pistons) (cm^2)
And how can it be without an example? =) My stock car (cylinder 0.875"), master cylinder stroke 3 cm:
Hp = (3 * 3.87) / 40.68 = 0.29 cm
And the cylinder (1")
Xp = (3 * 4,91) / 40,68 = 0,36 cm
From this we see that if you do not want to change the pedal stroke, then when changing the brake pad area (installing large brakes), you should also consider the master cylinder. And vice versa.
Master Cylinder
This is the heart of the entire braking system. It is activated by pressing on the pedal; initially, the piston moves fluid through the system until the pads come into contact with the disc, then once the system becomes closed, pressure begins to build up creating braking force. From this it can be seen that the harder you press on the pedal the higher the braking force.
The main parameters of the master cylinder are the diameter and stroke of the piston. Master cylinders typically have diameters ranging from 0.625" to 1.5" and strokes ranging from 2.5 cm to 3.81 cm. Matching these parameters to recommended ones for your vehicle is key to good performance. It's worth remembering that with one pedal effort, a smaller master cylinder will give higher pressure but will displace less fluid. Also the larger the stroke of the master cylinder, the more fluid it can displace but it will require a longer pedal travel. A better result can be achieved by calculating a compromise between pedal travel and pressure for your vehicle.
Pressure regulators
— Residual Pressure Valve (RPV)
Needed to maintain set pressure in the system (for disc brakes 0.14 kg/cm^2, for drum brakes 0.70 kg/cm^2).
There are a couple of reasons to use these valves.
1) Only for drum brakes so that the return spring doesn't pull the pads too far from the drum creating extra pedal travel on subsequent braking.
2) For disc brake systems where the master cylinder is below the calipers (some race cars and hot rods). Without such a valve, fluid would leak back to the master cylinder making the pedal spongy and again increasing its travel.
If you are converting from drum brakes to discs — be sure to remove these valves from your system.
— Hold-off Valve
Since there is a return spring on rear drum brakes as described above, drum brakes require more travel for the pad to reach the drum than self-adjusting disc brakes where the pad is always in contact with the disc. A metering valve (placed in the front circuit) prevents pressure creation in the front braking circuit until it reaches a set value in the rear (typically up to 5-10 kg/cm^2) allowing the drum pads to approach the drum.
If you are converting from drum brakes to discs — be sure to remove such valves from your system.
— Proportioning Valve (PBV)
As we wrote above, during braking weight shifts forward. Since braking force should distribute proportionally with load weight (where there's more weight, there's more braking force), the front-to-rear brake balance must be maintained. For example, during hard braking up to 85% of the weight is on the front of the vehicle. On a correctly adjusted system, both the front and rear brakes lock almost simultaneously. It is usually installed between the master cylinder and rear circuit to reduce pressure in the rear circuit at the initial moments of braking. Note that pressure in the rear circuit does not always remain lower than the front one; this valve changes the rate of pressure increase. In front brakes, upon pressing on the brake, it will be created faster than in the rear.
Stock valves are non-adjustable but there are racing versions with which you can adjust the braking balance on a modified braking system.
If you install an adjustable valve, don't forget to remove the stock one!
— Combined
Used on most stock vehicles with drum-disc systems. Combines metering and proportioning valves.
Brake pads
Here it all depends on quality and material. Where not worth skimping. On good branded pads the coefficient of friction is always indicated! It's designated by two letters. The first indicates the cold pad coefficient of friction, the second hot.
C = up to 0.15.
D= 0.15 to 0.25.
E= 0.25 to 0.35.
F= 0.35 to 0.45.
G= 0.45 to 0.55,
H= over 0.55.
For example
Ferodo DS2500 — FF
Hawk HPS — FF
Hawk HP+ — GG
Some stock option — FE, meaning it will brake worse when hot than cold.
------------------------------------------------------------------------
So. I have talked about many nuances in the braking system. So how do you actually create a proper braking system from scratch?
Let's go step by step
1) Try to use brake discs of appropriate size for dissipating kinetic energy of your vehicle (after deceleration from maximum speed to 0, temperature should not exceed 540°C).
If you plan on racing, when calculating the dissipated heat, use a disc temperature before braking equal to 260°C.
2) Choose the stiffest and strongest caliper (to minimize deformation under brake disc compression). Use a caliper with maximum possible effective piston area.
3) Calculate the recommended braking force
4) Calculate the recommended maximum pressure in the braking system
Rd = Ts / (μL * Aa * Re)
Where
Rd — Recommended pressure created by the master cylinder (kg/cm^2)
Ts — Recommended braking force (kg)
μL — Coefficient of friction between pad and disc
Aa — Effective piston area (for a caliper with a bridge, this is 2 times the area of the pistons)
Re — Effective radius of the brake rotor (from the center of the hub to the center of the pad)
5) Estimate how sensitive you want your pedal to be. For example, for sports use, you might choose 35 kg for aggressive braking.
6) Decide whether you want brakes with or without a booster.
A booster is needed if, for instance, there's no possibility to achieve sufficient pedal travel and required force with the chosen components, or you can't install a pedal with high leverage ratio, or your vehicle is VERY heavy.
7) Determine the pedal ratio, size of master cylinder, and (if installed) vacuum booster gain coefficient.
We know what pressure you need to create, and how stiff you want the pedal. We have three components (or two), through which this pressure can be created. Maybe some components you don't want to change in your car, such as the brake pedal. So its value can remain fixed while playing with other components.
— Pedal ratio
Can range from 3 to 7. When choosing, consider several factors: is there enough space for installation, will the pedal hit the floor before full travel of the master cylinder. And don't forget, the higher the ratio, the more travel and mushiness the pedal has.
— Calculate the force with which the pedal shaft will press on the master cylinder
For example, if you want to achieve maximum braking force at 35 kg pedal pressure, and your pedal ratio is 4.5, then the force applied to the master cylinder would be 35*4.5 = 157.5 kg. If you are using a booster, multiply it by the boost coefficient.
— Choosing the right size of the master cylinder
Now that we know the recommended pressure and the force exerted by the pedal shaft, we can calculate the size of the master cylinder
Aa = F / Rd
Where
Aa — Area of the master cylinder piston (cm^2)
F — Force applied to the master cylinder piston (kg)
Rd — Recommended pressure created by the master cylinder (kg/cm^2)
Suppose we need 65 kg/cm^2 pressure, and can apply a force of 157.5 kg on the master cylinder
Aa = 157.5 / 65 = 2.45 cm^2
Convert to typical master cylinder inch specifications
Diameter of the master cylinder in (in) = (2 * (sqrt(2.45/3.14))) / 2.54 = 0.695 in
So we would need a cylinder 11/16 inches, one of the smallest. Don't forget to ensure it will displace enough fluid.
If not sufficient, you'll have to increase the size of the master cylinder, meaning more force must be applied to the piston to create adequate pressure. If not changing pedal ratio — the only solution is installing a booster.
8) Check the amount of displaced fluid for the selected components. Ensure that pedal travel is sufficient to create compression force.
9) Calculate the braking force with the components you have chosen and compare it to the recommended value
Voilà! :) Now, if you've managed all this, you know how to build your braking system or what changes to make!
I hope the article will be very useful and I am happy to answer any of your questions and help with calculations if you can provide the initial data:
-Brake booster coefficient
-Pedal ratio
-Master cylinder diameter
-Front piston diameter + (their quantity and type of caliper)
-Rear piston diameter + (their quantity and type of caliper)
-Force on the pedal
-Car weight
-Tire-road friction coefficient
-Distance from front axle to car center of gravity
-Wheelbase
-Distance from ground to car center of gravity
-Pad-disk friction coefficient
-Effective radius of brake disc (from hub center to pad center) front
-Effective radius of brake disc (from hub center to pad center) rear
-Tyre rolling radius (total wheel diameter/2)
-Master cylinder stroke
In general, with my car, the force applied to the pedal sufficient for wheel lock-up is only 15kg =) While a typical setup for race cars is 35kg and more =)
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