## 2 Modelling the Brake Pressure Response  
A typical pneumatic brake circuit is shown in Figure 1. It consists of a compressor, an air filter, a compressed-air reservoir, a pedal valve, pipes and four brake chambers, and so on. This pneumatic brake circuit consists of four subcircuits, which are front brake circuit, rear brake control circuit, rear brake circuit, and parking brake circuit. When the driver treads on the pedal, the pedal valve opens and the compressed air flows into the front and rear brake chambers through the brake control valve and pipe. The rear brake chamber is relatively farther from the air tank, and then a relay valve is used to decrease the time delay. In these four subcircuits, the similarity is that all of the subcircuits use long pipes to transmit the air. Moreover, the time delay of a pipe is the key point that should be first solved. The operation pressure is generally 0.6-0.8 MPa for a pneumatic brake system, and the diameter of a transmission pipe is 6-16 mm. When analysing the brake circuit, we can study the subcircuit one by one. In this study, we mainly focused on the calculation method of the time delay caused by a transmission pipe; thus, we will simplify the system into one circuit to describe the modelling method clearly.  
Figure 1: Schematic of the pneumatic brake circuit of a bus.  
Although the brake control valve has a complex internal structure, the air flows through it quickly and its volume is small, so it can be thought of as lumped parameter components. The main effects of the brake control valve on the braking process are its flow-rate characteristics and the time delay of the mechanism. Therefore, when modelling the brake system, the brake control valve can be dealt with as an orifice [22]. For a pneumatic transmission circuit, the flow-rate characteristics of the circuit depend mainly on the smallest component and its upstream components; the downstream components have little effect [23]. In this paper, we mainly focused on the modelling and solving method of the pipe. Conversely, compared with the pipe, the sonic conductance of the control valve is smaller. Therefore, the valve can be considered as an orifice with a constant effective area.  
Thus, the simple circuit shown in Figure 2 is adopted as the research object in the present paper, which is composed of an air source, two orifices, a pipe, and a brake chamber. In the actual application, orifice 1 depends on the control valve and orifice 2 is the inlet orifice of the brake chamber. According to ISO6358 [24], the flow-rate characteristics of the components can be measured by the isothermal chamber (ITC) discharge method, and the time delay of the brake control valve can be obtained according to ISO19973 [25]. The resultant flow-rate characteristics of the components in a circuit can be calculated by referring to [23; 26].  
The following assumptions are used to facilitate the analysis model when modelling the PBS [27; 28]:  
1. The working fluid (air) of the system follows all ideal gas laws;
2. There is no leakage from the fittings or the brake chamber;
3. The airflow in the pipe is a one-dimensional flow, and the parameters in each section are uniform.