# Concept Design and Development of an Electric Go-Kart
## 2 Materials and Methods
### Vehicle Power Train
This leads to a rotational speed of \(n_{\text{w}}=\omega_{\text{w}}/(2\pi)=124.3/(2\pi)=19.785\ \text{s}^{-1}=1187\ \text{min}^{-1}\). To reduce the rotational speed from the motor to the wheel, a simple chain drive is presumed. The transmission ratio should be \(i=n_{\text{m}}/n_{\text{w}}=4600/1187=3.875=z_{\text{w}}/z_{\text{m}}\). \(z_{\text{w}}\) stands for the number of teeth on the rear shaft chain gear, and \(z_{\text{m}}\) stands for the number of teeth on the motor chain gear. If the motor chain gear is chosen with 13 teeth, then the rear shaft chain gear should have \(z_{\text{w}}=3.875\cdot 13=50\). The chain that could be used to drive the vehicle has a pitch of at least 8 mm, for \"reason\" of the minimum pitch diameter of the small chain gear, following the diameter of the sleeve for the motor shaft. Such a chain gear would have a pitch diameter \(r_{\text{2w}}=4/\sin(0.5\cdot 360/50)=63.7\ \text{mm}=0.0637\ \text{m}\). The force acting on the chain gear on the rear shaft is \(F_{ch}=1600\cdot 0.9/\left(124.3-0.0637\right)=181.9\ \text{N}\). Based on the wheel loads, the moment and torque diagrams were calculated and drawn. In this paper, they are not detailed, i.e., shown, for the sake of brevity. The equations for design of the shaft were based on dynamic strength conditions according to [PERSON], ASME, [PERSON], [PERSON], and others found in [29]. The stress components are firstly transformed into equivalent mean and amplitude stresses according to the [PERSON] theory (HMH, or distortion energy density theory):  
\[ \begin{gathered}\sigma_{\text{a}}^{\text{ekv}}=\sqrt{\left(a_{ \text{k}}^{\text{b}}\sigma_{\text{a}}^{\text{pb}}+a_{\text{k}}^{\text{a}} \sigma_{\text{a}}^{\text{a}}/0,85\right)^{2}+3\left(a_{\text{k}}^{\text{t}} \tau_{\text{a}}^{\text{t}}\right)^{2}},\\ \sigma_{\text{m}}^{\text{ekv}}=\sqrt{\left(a_{\text{k}}^{\text{b }}\sigma_{\text{m}}^{\text{pb}}+a_{\text{k}}^{\text{a}}\sigma_{\text{m}}^{ \text{a}}\right)^{2}+3\left(a_{\text{k}}^{\text{t}}\tau_{\text{m}}^{\text{t}} \right)^{2}}.\end{gathered}  \tag{12}\]  
In Equation (12), for calculation of equivalent amplitude and mean stress values, the \(a_{\text{k}}^{\text{b}}\) stands for the stress concentration factor at bending, the upper index \"t\" stands for torsion, and \"a\" for axial load case. Further on, \(\sigma_{\text{a}}^{\text{b}}\) is the bending induced normal stress amplitude, \(\sigma_{\text{a}}^{\text{a}}\) is the axial load induced stress amplitude, \(\tau_{\text{a}}^{\text{t}}\) is the torsion induced stress amplitude, while all stress components with lower index \"m\" are mean values of, respectively, depicted stress components. The Soderberg criterion is chosen as the simplest and most conservative condition in the form of:  
\[\frac{\sigma_{\text{a}}^{\text{ekv}}}{\sigma_{-1}^{\text{d}}}+\frac{\sigma_{ \text{m}}^{\text{ekv}}}{\sigma_{\text{T}}}=\frac{1}{f_{\text{s}}}.\ \ \frac{\sigma_{\text{a}}^{\text{ekv}}}{k_{1}k_{2}\sigma_{-1}^{\text{d}}}+\frac{ \sigma_{\text{m}}^{\text{ekv}}}{k_{1}k_{2}\sigma_{\text{T}}}=\frac{1}{f_{\text{ s}}}. \tag{13}\]  
In the Soderberg criteria (13), on the left-hand side is the form without an explicit member, which takes into account the influence of the surface roughness and size of the part, which are introduced into the right-hand side as \(k_{1}\) and \(k_{2}\), respectively, [37]. \(\sigma_{-1}^{\text{d}}\) stands for inverse dynamic bending strength [29; 37], \(\sigma_{\text{T}}\) stands for yield strength, and \(f_{\text{s}}\) the chosen safety factor. These dynamic strength conditions use the stress components transformed into equivalent stresses, as shown in Equation (12). It is one of the best suited explicit equations for the case of combined non-proportional stress components (i.e., one stress component changing with time, differently from other stress components).