*2.2. Vehicle Braking System*

The principle scheme of the braking system used for simulation is shown in Figure 2.

**Figure 2.** Basic scheme of the vehicle brake system.

The input variable for the vehicle braking system is the force exerted on the brake pedal by the driver—the pedal force. The force that enters the master cylinder is multiplied by the mechanical transmission brake pedal—ratio.

$$F\_{in} = F\_{petal} \times ratio \tag{2}$$

where:

*Fpedal* is the pedal force,

*ratio* is the mechanical transmission brake pedal,

*Fin* is the force entering the brake booster.

The pressure of the master cylinder is calculated according to the equation:

$$P\_{\rm MC} = \frac{F\_{\rm MC}}{D\_{\rm MC}^2 \times \frac{\pi}{4}} \tag{3}$$

where:

*FMC* is the master cylinder output force,

*D* is the diameter.

The braking torque depends on the pressure in the brake cylinders and the temperature of the brake disc. The calculation of brake disc temperature uses the input energy from the braking torque, the rotational speed and the energy loss due to cooling. Input energy is dependent on brake torque and

wheel speed. Output energy loss due to cooling is dependent on the cooling coefficient as a function of vehicle speed, brake disc weight, specific heat of the wheel as a function of the temperature of the wheel, brake disc temperature and air temperature.

The output heat change (Δ*T*) is calculated according to:

$$
\Delta T = \frac{(E\_{in} - E\_{out})}{M\_d \times C\_p (T\_d)} \tag{4}
$$

where:

Δ*T* is output heat change,

*Ein* is input energy,

*Eout* is output energy,

*Md* is brake disc weight,

*Cp*(*Td*) is specific heat of the wheel as a function of the temperature of the wheel.

First order delay is applied to the brake booster. It is a specific time constant that depends on whether the brake pedal is pressed or released. Consequently, the brake pressure is transferred through proportioning valves to the brake cylinders of the wheels as the supply pressure, which together with other parameters generates the mechanical braking torque, as demonstrated in Figure 3. This type of brake system was used for simulation in the CarSim program.

**Figure 3.** Brake system—basic scheme of generating mechanical braking torque.
