## 3 Bearing and Power Loss Analysis  
As an essential mechanical element in a power transmitting system, bearings are installed to enable two adjacent parts to relatively rotate in different angular velocities. While delivering rotating power through shafts and gears carrying loads, bearings must support them against loads. Bearings are generally composed of rolling elements (balls or rollers), inner and outer races. While the inner and outer races of a bearing rotate, rolling and sliding occur between the surfaces of the rolling elements and races. Furthermore, for radial roller bearings with flanges, roller end-race flange sliding frictions take place by thrust loads. These physical phenomena yield friction power losses in the mechanical aspect. In the aspect of fluid mechanics, as a form of viscous friction, viscosity of ATF yields power losses in bearing rotations. Practically, bearing power loss is a complicated outcome involving mechanical, viscous, thermal and other factors, and it is difficult to not only predict the influences of the individual factors but also form theoretical models. Hence, research [1; 2; 3; 4] for predicting bearing power losses has proposed empirical models including critical parameters. In this research, the power loss models proposed by [PERSON] and [PERSON] [1] are adapted for predicting bearing power losses, since the models are composed of mechanical and spin power losses which can be analyzed straight forwardly with the results of the kinematic and structural analyses. Furthermore, the power loss models are represented as simple functions including mainly applied loads and rotational speeds along with less constant factors defining bearing types, lubricant conditions and other operations.  
[PERSON] and [PERSON] [1] identified generally two contributions to bearing power losses, which are mechanical power loss \(P_{m}\) (load dependent) and spin power loss \(P_{s}\) (load independent), and proposed empirical formulations of bearing friction torques and power loss for rolling element bearings supporting loads in the axial and radial directions. Here, it should be noted that rolling bearing friction torques predominantly result from applied loads and lubricant viscous frictions. Furthermore, an influence of roller end-race flange sliding friction is included for radial roller bearings carrying axial loads. Since the purpose of this study is to predict bearing power losses resulted by thrust loads acting on bearings, the selected formulations are used for the analysis. Formulations of bearing power loss \(P_{b}\) for thrust cylindrical roller bearings are given as  
\[P_{b}=P_{m}+P_{s}\, \tag{8}\]  
\[P_{m}=1.5705\times 10^{-7}F_{a}d_{m}n\, \tag{9}\]  
\[P_{s}=\begin{array}{cc}1.047\times 10^{-11}f_{o}(v_{o}n)^{2/3}d_{m}^{3}n& \text{for}\ v_{o}n\geq 2000\\ 167.52\times 10^{-11}f_{o}d_{m}^{3}n&\text{for}\ v_{o}n<2000\end{array}\, \tag{10}\]  
where \(F_{a}\), \(d_{m}\), \(n\) and \(v_{o}\) denote an axial load, pitch diameter of a bearing, rotational speed and lubricant kinematic viscosity, respectively. \(f_{o}\) represents a value of lubrication type covering grease, oil bath, and oil jet for thrust cylindrical roller bearing. Here, it should denote that the Equation (9) contains a power loss term for thrust loads since roller end-race flange friction is not involved in the case of the bearing type.