Positions of rollers and races in a bearing under a constant unidirectional radial load. 

Contexts in source publication

Context 1

... nQ c f 1⁄4 2 π δ m ð 1 Þ where δ and Q are the normal displacement and the normal contact force between the roller and the raceway, respectively. m is the mass of the roller, and n is equal to 1.11 for the case of line contacts. This high frequency motion can be fi ltered out by an equilibrium constraint for the special cases, such as the current study, where the position of the inner raceway and the outer raceway are fi xed with respect to one another [14]. Thus, the radial and axial position of the loaded rollers between the races is fi xed for any orbital position and can be calculated from the quasi-static solution associated with any orbital position of the rollers; see Fig. 1. Considering the fact that the radial load and the rotational speed are kept constant in each simulation, the present study employs the equilibrium constraint in the radial direction for the loaded rollers in order to remove the high frequency motions, thus increasing the computational speed. This increase in the computational speed occurs because the Newton equations of motion are reduced to just orbital direction for each roller. The details of these constraint equations can be found in Ref. [14]. It is also assumed that the rollers and the cage are allowed to rotate only about their primary axes. Therefore, the Euler equations reduce to only one equation for each roller and the cage, i.e. rotational motion about their primary axis of rotation. It is believed that these assumptions about the direction of rotation of the cage and the rollers have negligible effects on the fi nal results of the present study because of the assumed geometry of the cage discussed in Section 3.1. Another assumption made in the present study, compared to the fully dynamic models, is about neglecting the radius of the contact curvature in the calculation of the traction forces. This assumption has been widely used in the literature and is also believed to have a negligible effect on the results. Available literature contains different approaches for calculating the traction forces in the dynamic models of rolling bearings, but their application is restricted to a limited range of operating conditions. This limited range is due to the signi fi cant effect of contact pressure on the lubricant viscosity and the deformation of the contact surfaces. For example, in the case of radially-loaded cylindrical roller bearings appropriate formulas should be used for the different regimes of lubrication as the roller passes through the loading zone; see Fig. 1. This is, in fact, due to the variation of contact pressure between the rollers and the raceways in different locations within the loading zone. Therefore, in order to simulate a realistic dynamic behavior of the bearing elements, it is important to fi gure out the appropriate lubrication regime for each contact as the rollers travel in orbital direction through the loading zone. In this study, a map of lubrication regimes, presented by Winer and Cheng [45] is utilized to fi nd the appropriate lubrication regime for each roller-race contact; see Fig. 2. This map is based on the different studies available in the literature [46] about different lubrication regimes in line contacts. Fig. 2 depicts four different lubrication regimes that may exist in the lubricated line contacts [45,47]. As shown, the elastohydrodynamic lubrication (EHL, elastic solid – piezoviscous) – which is widely used for the roller bearings in the literature – covers only a small region on the map of the lubrication regimes. Thus, depending on the operating condition, other regimes of lubrication may exist in the lubricated line contacts. Winer and Cheng [45] introduced two dimensionless variables g v (dimensionless viscosity parameter) and g E (dimensionless elasticity parameter) in order to determine the governing lubrication regime of a line contact. These parameters are de fi ned as ...

Context 2

... nQ c f 1⁄4 2 π δ m ð 1 Þ where δ and Q are the normal displacement and the normal contact force between the roller and the raceway, respectively. m is the mass of the roller, and n is equal to 1.11 for the case of line contacts. This high frequency motion can be fi ltered out by an equilibrium constraint for the special cases, such as the current study, where the position of the inner raceway and the outer raceway are fi xed with respect to one another [14]. Thus, the radial and axial position of the loaded rollers between the races is fi xed for any orbital position and can be calculated from the quasi-static solution associated with any orbital position of the rollers; see Fig. 1. Considering the fact that the radial load and the rotational speed are kept constant in each simulation, the present study employs the equilibrium constraint in the radial direction for the loaded rollers in order to remove the high frequency motions, thus increasing the computational speed. This increase in the computational speed occurs because the Newton equations of motion are reduced to just orbital direction for each roller. The details of these constraint equations can be found in Ref. [14]. It is also assumed that the rollers and the cage are allowed to rotate only about their primary axes. Therefore, the Euler equations reduce to only one equation for each roller and the cage, i.e. rotational motion about their primary axis of rotation. It is believed that these assumptions about the direction of rotation of the cage and the rollers have negligible effects on the fi nal results of the present study because of the assumed geometry of the cage discussed in Section 3.1. Another assumption made in the present study, compared to the fully dynamic models, is about neglecting the radius of the contact curvature in the calculation of the traction forces. This assumption has been widely used in the literature and is also believed to have a negligible effect on the results. Available literature contains different approaches for calculating the traction forces in the dynamic models of rolling bearings, but their application is restricted to a limited range of operating conditions. This limited range is due to the signi fi cant effect of contact pressure on the lubricant viscosity and the deformation of the contact surfaces. For example, in the case of radially-loaded cylindrical roller bearings appropriate formulas should be used for the different regimes of lubrication as the roller passes through the loading zone; see Fig. 1. This is, in fact, due to the variation of contact pressure between the rollers and the raceways in different locations within the loading zone. Therefore, in order to simulate a realistic dynamic behavior of the bearing elements, it is important to fi gure out the appropriate lubrication regime for each contact as the rollers travel in orbital direction through the loading zone. In this study, a map of lubrication regimes, presented by Winer and Cheng [45] is utilized to fi nd the appropriate lubrication regime for each roller-race contact; see Fig. 2. This map is based on the different studies available in the literature [46] about different lubrication regimes in line contacts. Fig. 2 depicts four different lubrication regimes that may exist in the lubricated line contacts [45,47]. As shown, the elastohydrodynamic lubrication (EHL, elastic solid – piezoviscous) – which is widely used for the roller bearings in the literature – covers only a small region on the map of the lubrication regimes. Thus, depending on the operating condition, other regimes of lubrication may exist in the lubricated line contacts. Winer and Cheng [45] introduced two dimensionless variables g v (dimensionless viscosity parameter) and g E (dimensionless elasticity parameter) in order to determine the governing lubrication regime of a line contact. These parameters are de fi ned as ...