## 2 Materials and Methods
### Practical Framework Application  
When switching from a diesel power unit (together with a gearbox) to an electric one (three battery units of 500 kg each according to the customer's technical documentation--Figure 1a), the position of the bus centre of gravity (CoG) must be preserved; otherwise, braking, handling, active, and passive safety will be affected. For the long-range BEVs, the number of battery units can be increased to four (Figure 1b), though the new CoG should be minimally changed--this is a conceptual challenge to distribute the masses evenly. A balance equation that ensures the total new mass and CoG remain consistent or within acceptable limits relative to the initial total mass can be written in the following form:  
\[m_{it}=m_{nt}=\int_{0}^{L}\rho_{i}(x)dx=\int_{0}^{L}\rho_{n}(x)dx, \tag{1}\]  
Figure 1: Layout of a 12 m long low-floor bus when switching from a diesel to an electric powertrain: (**a**) 3D model of the battery unit location; (**b**) drawing fragment (side and top view). where \(m_{it}\), \(m_{nt}\)--initial and new total mass of the bus accordingly; \(\rho_{i}(x)\), \(\rho_{n}(x)\)--linear mass density function along the longitudinal x axis of the bus, kg/m; \(L\)--bus length, m.  
\[x_{i}=x_{n}=\frac{\int_{0}^{L}x\cdot\rho_{i}(x)dx}{\int_{0}^{L}\rho_{i}(x)dx}= \frac{\int_{0}^{L}x\cdot\rho_{n}(x)dx}{\int_{0}^{L}\rho_{n}(x)dx}, \tag{2}\]  
where \(x_{i}\), \(x_{n}\)--initial and new CoG position, m.  
A similar modelling approach should be used for other coordinates (y, z), a possible direction for further research. According to R100, the test pulse applied to the battery unit shall be between the \"max curve\" and \"min curve\"--an example of a graph of longitudinal acceleration is shown in Figure 2. The corresponding values of acceleration in the intermediate points (moments of impact time) are listed in Table 1.  
In reality, for heavy buses (M3 category), typical acceleration \(A\) values during frontal crash tests range between 15 and 20 g [6; 25; 26] (measured at the impact point near the wall at the front of the bus). The energy transfer from a frontal impact will result in different acceleration values for the frontal and rear battery units due to the following concepts: (a) wave propagation: the shock wave generated by a frontal impact will travel through the bus structure, causing the front part to experience higher accelerations compared to the rear part; (b) damping and absorption: as the shock wave travels through the bus, its intensity diminishes due to damping and absorption by the structure and components; (c) distance from impact point: the farther a component is from the point of impact, the lower the peak acceleration it will experience. It is proposed to describe the difference in acceleration experienced by the frontal and rear battery units using the next mathematical model, considering such parameters \(k\) = 0.05 m\({}^{-1}\); \(d\)\({}_{f}\) = 3 m; \(d\)\({}_{r}\) = 9 m (which were determined together with the customer based on technical documentation):  
Frontal battery unit: \begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline
**Point** & **A** & **B** & **C** & **D** & **E** & **F** & **G** & **H** \\ \hline time (ms) & 20 & 50 & 65 & 100 & 0 & 50 & 80 & 120 \\ longitudinal acceleration (g) & 0 & 6.6 & 6.6 & 0 & 4 & 12 & 12 & 0 \\ transverse acceleration (g) & 0 & 5 & 5 & 0 & 2.5 & 10 & 10 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Acceleration for M3 and N3 vehicles according to R100. Figure 2: Longitudinal acceleration according to R100, points A, B, C, etc., form the limits of permissible minimum and maximum pulse values acting on battery units.