## 2 Materials and Methods
### Calculation Method  
The 25 design parameters are influenced by the bicycle geometries, the addition of a battery and motor, and the cyclist. The values are determined for the six electric bicycle configurations and a conventional bicycle. The bicycle and cyclist are simplified to geometric shapes (cylinders, torus, cuboids, and spheres) to calculate the inertia tensors in the local centres of gravity. The shapes form a part (cyclist, motor, battery, framework, fork, handlebar, and wheel) and the parts form assemblies (front wheel, rear wheel, front frame, and rear frame). In the case of pedal power assistance, the mass values of the battery is added to the rear frame and the mass values of the motor is added to the front wheel, rear wheel, or rear frame assembly, depending on the motor placement. Table 2 presents an overview of the parts for the seven bicycle configurations. The inertia tensors are calculated in the centres of gravity of the assemblies. The centre of gravity of the rear frame assembly depends on multiple components (cyclist, motor, battery, and framework) and is calculated afterward. Therefore, the inertia tensor of the shapes are initially calculated in an external, uninfluenced point and for every configuration translated to the centre of gravity of the assembly with a derived formula of the parallel axis theorem (10). The formula is derived as follows. To translate the inertia tensor of a geometric shape from the local centre of gravity \(\left\{l\right\}\) to an external point \(\left\{o\right\}\), the parallel axis theorem is utilized, given by Equation (2). \[I_{o}=I_{l}-m(p_{lo}\,{p_{lo}}^{T})+m({p_{lo}}^{T}\,p_{lo})\,I_{3} \tag{2}\]  
Wherein, \(I_{o}\) is the inertia tensor in \(\left\{o\right\}\), \(I_{l}\) is the inertia tensor in \(\left\{l\right\}\), \(p_{lo}\) is the place vector from \(\left\{o\right\}\) to \(\left\{l\right\}\), and \(m\) is the mass. \(I_{o}\) is translated from the external point \(\left\{o\right\}\) to the centre of gravity of a part \(\left\{c\right\}\) given by Equation (3) and presented in Figure 4.  
\[Ic=Il-m\,p_{lc}\,{p_{lc}}^{T}+{m_{lc}}^{T}\,p_{lc}\,I_{3} \tag{3}\]  
\(p_{lc}\) is expressed in function of \(p_{lo}\) and \(p_{co}\) given by Equation (4). \[p_{lc}=p_{lo}-p_{co} \tag{4}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Bicycle 1** & **Bicycle 2** & **Bicycle 3** & **Bicycle 4** & **Bicycle 5** & **Bicycle 6** & **Bicycle 7** \\  & & & & & & & \\ \hline Rear wheel & wheel & wheel & wheel + & wheel + & wheel + & wheel & wheel & wheel \\ assembly & wheel + & wheel + & motor & wheel & wheel & wheel & wheel \\ wheel & motor & wheel & wheel & wheel & wheel & wheel & wheel \\ assembly & motor & motor & wheel & wheel & wheel & wheel & wheel \\ \hline Rear frame & framework & framework & Framework & framework & framework & framework & framework & \\ assembly & + cyclist + & + cyclist + & + cyclist + & + cyclist + & + cyclist + & + cyclist + & framework \\ battery & battery & battery & battery & battery & battery + & battery + & + cyclist \\ motor & motor & motor & motor & motor & motor & motor \\ \hline Front & handlebar & Handlebar & Handlebar & Handlebar & handlebar+ & handlebar+ & Handlebar+ & Handlebar \\ assembly & + fork & + fork & + fork & + fork & fork & fork & fork & + fork \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parts of every assembly for the seven bicycle configurations. \(I_{l}\) is expressed in function of \(I_{o}\) given by Equation (5). \[ \begin{array}{c}Ic=Il-m(p_{lo}\ p_{lo}\ ^{T}-p_{lo}p_{co}\ ^{T}-p_{co}p_{lo}\ ^{T}+p_{co}p_{co}\ ^{T})+m(p_{lo}\ ^{T}p_{lo}-p_{lo}\ ^{T}p_{co}-p_{co}\ ^{T}p_{lo}+p_{co}\ ^{T}p_{co})I_{3}\\ =I_{l}-m(p_{lo}\ ^{T}p_{lo})+m(P_{lo}\ ^{T}p_{lo})\ I_{3}-m(-P_{lo}\ p_{co}\ ^{T}-P_{co}\ ^{T}+P_{co}\ ^{T})+m(-2\ p_{lo}\ ^{T}\ p_{co}+P_{co}\ ^{T}\ ^{T}\ p_{co})\ I_{3}\\ =I_{o}-m(-p_{lo}p_{co}\ ^{T}-p_{co}p_{lo}\ ^{T}+p_{co}\ ^{T})+m(-2p_{lo}\ ^{T}p_{co}+p_{co}\ ^{T}p_{co})I_{3}\end{array}