## 4 Results and Discussion
### FEM-Based Verification of the Proposed Analytical Model  
Verification of the accuracy of the proposed analytical model by applying the FEM to one of the examples from the experiments of [21] is the next thing to be performed. Accordingly, Figure 3 shows four steady-state temperature distributions over a two-dimensional representation of a rectangular bus bar whose area is equal to that of the plate fin 0.0004 m \(\times\) 0.025 m considered in [21]. These temperature distributions are generated for the problem formulations from Figure 1a,b (taking into account the non-adiabatic assumption), vibration amplitudes of 0, 1, 1.6, and 3 mm (free convection and vibration classes A, B, and C, respectively), a vibration frequency of 29 Hz, and a temperature difference of \(\rm{T_{s}-T_{a}=T_{cp}-T_{a}=343.157-313.157=30K}\). These parameters were chosen to match the experimental data from [21]. Figure 3a shows the two-dimensional represen \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \multirow{2}{*}{**W**} & \multirow{2}{*}{**H**} & \multirow{2}{*}{**f\({}_{\text{V}}\)**} & **I\({}_{\text{cp}}\) for** & \multirow{2}{*}{**I\({}_{\text{cp}}\) for** & **I\({}_{\text{cp}}\) for** & **I\({}_{\text{cp}}\) for** & **I\({}_{\text{cp}}\) for** & **I\({}_{\text{cp}}\) for** & **I\({}_{\text{cp}}\) for** & **I\({}_{\text{cp}}\) for** \\  & & **X = 0 m** & & **\(\mu\)m** & & **X = 1 mm** & **X = 1.6 mm** & **X = 3 mm** & **X = 4 mm** & **X = 5 mm** \\ \cline{3-10}  & & & **BUSBAR b** & **BUSBAR b** & **BUSBAR b** & **BUSBAR b** & **BUSBAR b** & **BUSBAR b** & **BUSBAR b** \\ \hline
**m** & **m** & **Hz** & **A** & **A** & **A** & **A** & **A** & **A** & **A** \\ \hline \multicolumn{10}{l}{Bus bars installed horizontally with a vertical major axis and adiabatic top and bottom surfaces} \\ \hline
0.0004 & 0.02 & 50 & 44.0 & 44.1 & 47.1 & 48.8 & 52.6 & 55.2 & 57.6 \\ \hline
0.0004 & 0.025 & 29 & 53.5 & 53.5 & 55.5 & 56.7 & 59.2 & 61.0 & 62.8 \\ \hline
0.0005 & 0.02 & 59 & 49.3 & 49.3 & 53.2 & 55.5 & 60.4 & 63.7 & 66.8 \\ \hline
0.0005 & 0.025 & 40 & 59.9 & 59.9 & 62.9 & 64.6 & 68.5 & 71.2 & 73.7 \\ \hline \multicolumn{10}{l}{Bus bars installed horizontally with a vertical major axis and non-adiabatic top and bottom surfaces} \\ \hline
0.0004 & 0.02 & 50 & 44.3 & 44.4 & 47.4 & 49.1 & 52.9 & 55.4 & 57.8 \\ \hline
0.0004 & 0.025 & 29 & 53.9 & 53.9 & 55.8 & 57.0 & 59.5 & 61.3 & 63.0 \\ \hline
0.0005 & 0.02 & 59 & 49.7 & 49.7 & 53.6 & 55.9 & 60.7 & 64.0 & 67.1 \\ \hline
0.0005 & 0.025 & 40 & 60.3 & 60.3 & 63.3 & 65.0 & 68.9 & 71.6 & 74.1 \\ \hline \hline \end{tabular}  
* For aluminum alloy 5052-O bus bars, \(\rm{k_{s}=1,T_{a}=40^{\circ}C,T_{cp}=70^{\circ}C}\), \(\epsilon=0.35\), \(\alpha=0\), and \(\rm{f=60 Hz}\). b Values calculated using the updated version of the BUSBAR.m program.  
\end{table} 
Table 2: Bus bar ampacity as a function of vibration amplitude at different natural frequencies a tation of the geometry (that is, solid) from Figure 1a,b with the associated finite element (FE) mesh. The outputs of the BUSBAR.m program used as inputs in COMSOL 4.3 to generate the temperature distributions from Figure 3 are as follows:  
(i) \(\mathrm{Q_{Hg,v}=1706547.953\ W\cdot m^{-3},\ h_{cS}=8.388\ W\cdot m^{-2}\cdot K ^{-1},\ h_{cT}=11.615\ W\cdot m^{-2}\cdot K^{-1},\) and \(\mathrm{h_{cB}=5.192\ W\cdot m^{-2}\cdot K^{-1}--for the case of free convection. (ii) \(\mathrm{Q_{tg,v}=1828981.716\ W\cdot m^{-3},\ h_{cS}=9.228\ W\cdot m^{-2}\cdot K ^{-1},\ h_{cT}=11.61\ W\cdot m^{-2}\cdot K^{-1},\) and \(\mathrm{h_{cB}=5.19\ W\cdot m^{-2}\cdot K^{-1}--for the case of vibration class A. (iii) \(\mathrm{Q_{tg,v}=1908493.466\ W\cdot m^{-3},\ h_{cS}=9.737\ W\cdot m^{-2}\cdot K ^{-1},\ h_{cT}=11.615\ W\cdot m^{-2}\cdot K^{-1},\) and \(\mathrm{h_{cB}=5.192\ W\cdot m^{-2}\cdot K^{-1}--for the case of vibration class B.