## 2 Modeling of Internal Pressure for a Single Factor
### Oxygen Supply System
#### 2.5.5 Heat Transfer from Infiltrated Air  
The carriage has structural gaps, such as weld seams, doors, and windows, through which air can leak. The heat transferred by the air leaking through these airtight gaps is as follows:  
\[Q_{5}= \begin{cases}q_{g}h_{i},P_{i}\geq P_{e}\\ q_{g}h_{nt},P_{i}<P_{e}\end{cases}  \tag{31}\]  
where \(q_{g}\) is the mass flow rate of the air leaking through the carbody, \(h_{n}\) is the specific enthalpy of the fresh air, and \(h_{i}\) is the specific enthalpy of the internal air. \begin{table}
\begin{tabular}{c c c c c} \hline  & \(F_{j}\) & \(k_{j}\)**Standstill** & \(k_{j}\) Max & \(k_{j}\)**Average** \\  & (**m\({}^{2}\)**) & (**W**/**m\({}^{2}\)**-**K**) & (**W**/**m\({}^{2}\)**-**K**) & (**W**/**m\({}^{2}\)**-**K**) \\ \hline Roof & 68 & 1.16 & 2.22 & 2.11 \\ Floor & 78 & 1.16 & 2.22 & 2.11 \\ Side Wall 1 & 47.5 & 1.16 & 2.22 & 2.11 \\ Side Wall 2 & 47.5 & 1.16 & 2.22 & 2.11 \\ Window 1 & 8 & 1.16 & 2.22 & 2.11 \\ Window 2 & 8 & 1.16 & 2.22 & 2.11 \\ \hline \end{tabular}
\end{table} 
Table 1: Carbody structure parameters [25]. 2.5.6. Heat Production by the Heating System  
In the train heating system, the electric heaters within the air conditioning units and the cabin floor electric heaters are divided into primary heating units and auxiliary heating units [23], with the total heating capacity expressed as:  
\[Q_{6}=P_{AC}+P_{FH} \tag{32}\]  
2.5.7. Modeling Internal Pressure Fluctuations Caused by Temperature Changes  
The relationship between the heat in the carriage and the temperature can be derived from the specific heat capacity formula:  
\[Q(t)=c_{i}(t)m_{i}(t)\frac{dT_{i}(t)}{dt} \tag{33}\]  
By discretizing Equation (33), it can be derived as follows:  
\[T_{i}(k+1)=T_{i}(k)+\frac{Q_{1}(k)+Q_{2}(k)+Q_{3}(k)+Q_{4}(k)+Q_{5}(k)+Q_{6}(k) }{c_{i}(k)m_{i}(k)f_{s}} \tag{34}\]  
By substituting Equations (25) to (32) into Equation (34), the internal temperature at time \(k+1\) can be expressed as follows:  
\[T_{i}(k+1)=\left\{ \begin{array}{c}T_{i}(k)+\frac{\sum\limits_{j=1}^{6}k_{j} \bar{F}_{j}[T_{i}(k)-T_{e}(k)]+n\phi_{r}(Q_{rs}+Q_{it})+2500-P_{EH}(k)+q_{g}(k) h_{i}(k)+q_{n}(k)h(k)_{n}+q_{O}(k)h_{O}(k)-q_{w}(k)h_{n}(k)}{c_{i}(k)m_{i}(k)f_{s}}\\ T_{i}(k)+\frac{\sum\limits_{j=1}^{6}k_{j}\bar{F}_{j}[T_{i}(k)-T_{e}(k)]+n\phi_{ r}(Q_{rs}+Q_{it})+2500-P_{EH}(k)+q_{g}(k)h_{i}(k)+q_{n}(k)h_{n}(k)+q_{O}(k)h_{O}(k)-q_{w}(k)h _{i}(k)}{c_{i}(k)m_{i}(k)f_{s}}\\ \end{array} \right. \tag{35}\]  
Based on Equation (35), a coupled mapping relationship between the temperature inside the carriage and the internal pressure, along with the partial pressure of oxygen, can be established. The internal pressure \(P_{iT}\) resulting from temperature changes is determined using the ideal gas law:  
\[P_{iT}(k)=\frac{m_{i}RT_{i}(k)}{M_{i}V} \tag{36}\]  
3. Modeling of Internal Pressure Fluctuations under Multi-Factor Coupling  
3.1. Modeling  
When the EMU train passes through extreme tunnels, the external pressure fluctuations generated are superimposed on the baseline tunnel pressure changes. Under the disturbance of external pressures, the switching states of the train's air ducts, the dynamic characteristics of airtight gaps in the carriage, carbody deformation, and frequent thermal exchanges due to the temperature difference between the inside and outside of the carriage, as well as the intermittent operation of heating and oxygen supply systems, all contribute to the multi-factor coupling effect. This results in a strongly nonlinear characteristic of the internal pressure wave transmission system. Based on the internal pressure fluctuation models under the single factors of air ducts, airtight gaps, carbody deformation, oxygen supply, and temperature, this study investigates the mechanism of internal pressure fluctuations and establishes a mathematical model for external-internal pressure transmission in this nonlinear system.