## 2 Materials and Methods
### Practical Framework Application
\[a_{f}(t)=a_{0}(t)\cdot e^{-k\cdot d_{f}}=a_{0}(t)\cdot 0.8607 \tag{3}\]  
Rear battery unit:  
\[a_{r}(t)=a_{0}(t)\cdot e^{-k\cdot d_{r}}=a_{0}(t)\cdot 0.6376 \tag{4}\]  
where \(a_{f}(t)\), \(a_{r}(t)\)--acceleration experienced by the frontal and rear battery unit at time \(t\) accordingly, \(\mathrm{m}/\mathrm{s}^{2}\); \(a_{0}(t)\)--a function which describes the acceleration as a function of time during a frontal crash test for an M3 category bus, \(\mathrm{m}/\mathrm{s}^{2}\); \(d_{f}\), \(d_{r}\)--distance from the impact point to the frontal and rear battery unit accordingly, \(\mathrm{m}\); \(k\)--damping coefficient that describes how the shock wave intensity decreases with distance, \(\mathrm{m}^{-1}\). The model of the acceleration pulse, preserving the character of the line segments (graph in Figure 2), can be described as follows for the case of peak acceleration \(A\):  
\[a_{o}(t)= \begin{cases}0&if\ 0\leq t<20\ \mathrm{ms}\\ \frac{A\cdot a_{i}}{(50-20)}(t-20)&if\ 20\leq t<50\ \mathrm{ms}\\ A\cdot a_{i}&if\ 50\leq t<80\ \mathrm{ms}\\ A\cdot a_{i}-\frac{A\cdot a_{i}}{(120-80)}(t-80)&if\ 80\leq t<120\ \mathrm{ms}\\ 0&if\ t\geq 120\ \mathrm{ms}\end{cases}  \tag{5}\]  
where \(a_{o}(t)\)--function which describes the acceleration as a function of time during a frontal crash test for an M3 category bus; \(t\)--variable which represents time, \(s\); \(a_{i}\)--frontal or rear battery unit acceleration coefficient (\(a_{f}(t)\) or \(a_{r}(t)\) accordingly). In the context of the provided Equation (5), the symbol { is not a mathematical operator but a notation used to define a piecewise function. A piecewise function is a function that is characterised by different expressions based on different intervals of the input variable t (time). Taking into account the typical acceleration for the M3 category mentioned above (\(A=15\ \mathrm{g}\)), the values of the front and rear battery unit accelerations (3, 4) are calculated as follows:  
Frontal battery unit:  
\[a_{f}(t)= \begin{cases}0&if\ 0\leq t<20\ \mathrm{ms}\\ \frac{15g}{30}(t-20)\cdot 0.8607&if\ 20\leq t<50\ \mathrm{ms}\\ \left(15g-\frac{15g}{40}(t-80)\right)\cdot 0.8607&if\ 80\leq t<120\ \mathrm{ms}\\ 0&if\ t\geq 120\ \mathrm{ms}\end{cases}  \tag{6}\]  
Rear battery unit:  
\[a_{r}(t)= \begin{cases}0&if\ 0\leq t<20\ \mathrm{ms}\\ \frac{15g}{30}(t-20)\cdot 0.6376&if\ 20\leq t<50\ \mathrm{ms}\\ 15g\cdot 0.6376&if\ 50\leq t<80\ \mathrm{ms}\\ \left(15g-\frac{15g}{40}(t-80)\right)\cdot 0.6376&if\ 80\leq t<120\ \mathrm{ms}\\ 0&if\ t\geq 120\ \mathrm{ms}\end{cases}  \tag{7}\]  
Thus, the peak acceleration during \(t=50\)--\(80\ \mathrm{ms}\) remains stable according to the horizontal line segment F-G (Figure 2) and is: \(a_{f}(50)=12.91\ \mathrm{g}\) and \(a_{r}(50)=9.56\ \mathrm{g}\). Intermediatevalues in characteristic moments of time t are as follows: \(a_{f}\)(30) = 4.30 g and \(a_{r}\)(30) = 3.19 g; \(a_{f}\)(100) = 6.46 g and \(a_{r}\)(100) = 4.78 g. These results show energy absorption during the frontal impact at \(A\) = 15 g.  
Bus manufacturers and engineering design teams can create their mathematical models that are an alternative to Equations (3) and (4). Still, the approach proposed in the current work already demonstrates that the acceleration in battery units depends on the location relative to the front of the bus and can be simulated. The mode index \"2-10g-L\" (Figure 1) means that unit #2 under a transverse acceleration of 10 g is directed to the left (L) side along the bus. Other designations are: 1/2/3--the unit number according to the mounting scheme (Figure 1a); 10 g/12 g--acceleration value (98.06/117.67 m/s\({}^{2}\)); L/R/F/B--acceleration direction relative to the longitudinal axis of the bus (left/right/front/back accordingly). Each of the three models has its constraints with M10 or M12 thread bolt joints and the brackets mounted to the original bus body frame, which cannot be modified due to the number of UNECE certification tests already passed by the bus originally equipped with a diesel engine.