## 3 Mathematical Modeling
### Analytical Steady-State Thermal Model
For instance: the density \(\rm\rho_{a}\) is determined by means of cubic spline interpolation and the pair of input data files Temperature.m and Density.m, the specific heat at constant pressure \(\rm c_{t,a}\) is determined by means of cubic spline interpolation and the pair of input data files Temperature.m and Capacity.m, and so on. More details related to the determination of thermodynamic and physical properties of ambient air can be found in [22]. For the two side surfaces of the bus bars, the vibration-assisted convection heat transfer coefficient \(\rm h_{cS}\) is estimated by means of [21; 22]:  
\[\rm G_{F_{S}}=\frac{9.81\beta(T_{s}-T_{a})H^{3}}{\
u_{a}^{2}} \tag{7}\]  
\[\rm Nus=\left\{0.825+\frac{0.387(Gr_{S}Pr)^{1/6}}{\left[1+\left(0.492/Pr \right)^{9/16}\right]^{8/27}}\right\}^{2}\rm\ for\ Gr_{S}Pr\leq 10^{2} \tag{8}\]  
\[\rm Nus_{S}=0.68+\frac{0.67(Gr_{S}Pr)^{1/4}}{\left[1+\left(0.492/Pr\right)^{9/ 16}\right]^{4/9}}\rm\ for\ Gr_{S}Pr>10^{2} \tag{9}\]  
\[\rm Nus_{SV}=\rm Nus_{S}+0.0315\frac{2\pi f_{F}\rm yXH}{\alpha_{t,a}Nu_{S}}\rm \ for\ Gr_{S}Pr\leq 10^{2}\rm\ or\ Gr_{S}Pr>10^{2} \tag{10}\]  
\[\rm h_{cS}=\frac{Nu_{S}v_{t,a}}{H} \tag{11}\]where \(\rm{G_{TS}}\) is the Grashof number for a stationary vertical plate, \(\rm{N_{US}}\) is the Nusselt number for a stationary vertical plate calculated using Equation (8) for \(\rm{G_{TS}}\rm{Pr}\leq 10^{2}\) or Equation (9) for \(\rm{G_{TS}}\rm{Pr}>10^{2}\), \(\rm{N_{US}}\) is the Nusselt number for a vibrating vertical plate, \(\rm{f_{V}}\) is the vibration frequency in Hz, X is the vibration amplitude in m, and \(\rm{\alpha_{t,a}=k_{t,a}/(\rho_{a}c_{t,a})}\) is the thermal diffusivity of ambient air in m\({}^{2}\cdot\)s\({}^{-1}\). According to Equations (10) and (11), it is evident that the convection coefficient \(\rm{h_{CS}}\) includes the effect of horizontal transverse vibrations on free convection heat transfer between one of the two side surfaces of a bus bar and the ambient air. Modeling the effect of horizontal transverse vibrations on free convection using the empirical correlation (8) for the case when \(\rm{G_{TS}}\rm{Pr}\leq 10^{2}\) represents a generalization of the work of [PERSON] et al. from the year 2014 [21]. In Equation (10), there is a product of the vibration frequency \(\rm{f_{V}}\) and the vibration amplitude X, i.e.,  
\[\rm{v_{V}=f_{V}X} \tag{12}\]  
which represents the vibration velocity in m\(\cdot\)s\({}^{-1}\) and which will be further used to model the effect of horizontal transverse vibrations on free convection heat transfer from the top and bottom surfaces of rectangular bus bars using existing empirical correlations for forced convection. The vibration amplitudes considered here, assuming a vibration frequency of 120 Hz, are 0 m--for the case of free convection; 8.541 \(\rm{\upmu m}\)--for the case of twice the power system frequency [1]; 1 mm--for the case of the vibration class A [4]; 1.6 mm--for the case of the vibration class B [4]; 3 mm--for the case of the vibration class C [4]; and 4 mm and 5 mm--for the case of improper selection of supporting insulators and normal operating conditions [13]. The last two vibration amplitudes are chosen to match those of the experiments performed for the purpose of study [21]. The vibration frequencies considered here, assuming a vibration amplitude of 3 mm, are 0 Hz--for the case of free convection; 2.75 Hz--critical frequency for vibrations due to wind eddies [1]; 11.09 Hz--critical frequency for the vibration classes A and C [4]; 29 Hz--natural frequency determined experimentally in [21]; 120 Hz--twice the value of the power system frequency [1]; 240 Hz--twice the value of the frequency of the second order harmonic current [1,25]; and 360 Hz--twice the value of the frequency of the third order harmonic current [1,25]. The last two vibration frequencies are chosen to illustrate and quantify the potential effects of frequencies higher than twice the power system frequency.