car braking system
TRANSCRIPT
CAR BRAKING SYSTEM
A MATHEMATICAL STUDY
MECHANICS OF BRAKE PEDALS
The brake pedal is nothing more than a mechanical lever that amplifies the force of the driver.
n is called a multiplier which is simply what we call the pedal ratio.
PASCAL’S PRINCIPAL OF HYDRAULICS
Pressure is transmitted undiminished in an enclosed static fluid.
This means that any externally applied pressure is transmitted to all parts of the enclosed fluid, making possible a large multiplication of force (hydraulic press principle)
MASTER CYLINDER• They are an integral component in the brake system and are
responsible for sending the correct amount of pressure and balance to the brake calipers.
• In automotive engineering, the master cylinder is a control device that converts non-hydraulic pressure (commonly from a driver's foot) into hydraulic pressure. This device controls slave cylinders located at the other end of the hydraulic system
PEDAL RATIOPedal Ratio = X divided by YPedal ratio is the overall pedal length or distance from the pedal pivot to the center of the pedal pad divided by the distance from to the pivot point to where the push rod connects. The optimal pedal ratio is 6:1 on a disc/drum vehicle without vacuum or other assist method.
This means that if a driver applies 32 kg on a break pedal, it will then be amplified to:
6×32kg= 192 kg which is ~ 1920 N
MASTER CYLINDERWhen the force obtained above is applied to the back of the master cylinder, the force is transferred into the brake fluid.
If the a master cylinder has a bore of 2.5 cm diameter then it will have a surface area:
A = π × r2 = π × (1.27) 2 = 5.03
the piston surface area ~ 5.03 cm2
If you divide the output force of 1920 N from the pedal by the surface area of the piston, you would get: 0.00053
= 3622641.5 Nm-2
~ 3.62 × 106 Pa
If a driver applies 32 kg then we would expect 3.62 × 106 Pa at the port of the master cylinder.
AREA AND PRESSURE GRAPH OF AREA V.S PRESSURE
We know that pressure is inversely proportional to area as shown in the diagram to the right.
this means that if we reduce the surface area of the piston by reducing the size of the master cylinder bore we would then get more pressure. Note that the output from the pedal stays the same.
• Let’s assume the master cylinder bore is 2 cm
• Then the piston surface area ~ 2.84 cm2
Dividing the output force of 1920 N from the pedal by the surface area of the piston, we would get: 0.00028
= 6857142.9 Nm-2
~ 6.9× 106 Pa which is almost double the one with the first
one of about doubled surface area.
FORCE TRANSFER TO THE CALIPERSThe ratio between the caliper and master cylinder is a function of the net effective caliper piston bore area divided by the bore area of the master cylinder. To compare these ratios and do the calculation, you must start with the total piston area of the pistons in one side of one caliper.
Let’s assume these values belowA front brake set using four piston calipers with 4.5cm diameter each
Two pistons will have a net bore area of 31.0 cm2
The bore area of the master cylinder 2.84 cm2 (smaller surface area)
To calculate the leverage ratio between the caliper and master cylinder:
l.r = = 10.9 : 1 this is ~ 11:1
We can then calculate the driver’s leverage which is equal to
the Pedal Ratio x (the Caliper Piston Bore to Master Cylinder ratio).
CALCULATIONS FOR THE DRIVER’S LEVERAGE
The Pedal Ratio = 6:1
the Caliper Piston Bore to Master Cylinder ratio = 11:1
For a car with the ratios approximate to the ones above,
The driver’s leverage = 6×(11/1) = 66:1
You can substitute any number of piston bore combinations with master cylinder sizes with any pedal ratio to determine the driver’s actual brake leverage.
The same procedure above could be used to compare the front leverage ratio to the rear leverage ratio on any given car, this tells us the front to rear static bias capability of the car.
Let’s calculate the force per unit area that reaches the caliper from the output force of the driver’s foot.
OVERALL TRANSFERRED FORCE TO THE CALIPERS • If we have an input force of 320 N• And our driver’s leverage ratio is 66:1• if a driver applies 32 kg on a break pedal, it will then be amplified to: 66 × 320N = 21120 N
If two pistons on one the sides a caliper have a net surface area of 31.0 cm2 the overall pressure applied to the pad by only two pistons
= = 6.8 × 106 Nm-2
However this is not the end to the breaking process. When the pads have leaned against the Rotor there is friction.
DID WE FORGET ANYTHING??
• Well, remember that there exist what we call the fluid pressure.
• Apart from the pressure initiated by the drivers foot, the brakes are also affected by the fluid pressure which adds up to the total pressure that gets to the brake pads!
• Pressure at Pads= net fluid pressure + net pressure initiated by the driver’s foot
PRACTICE
• 4. In a newly designed car with a hydraulic braking system, a force of 85 N is applied to one of the master cylinders, which has an area of 8.1 cm2. The master cylinder is connected to one brake piston, which exerts a force of 296 N. What is the area of the brake piston?
