Study on Thermo-mechanical Modeling and Analysis of High-speed Angular Contact Ball Bearings Under Oil-jet Lubrication

2.1 The Quasi-static Model of Bearing

The ACBB model used in this study is based on the quasi-static model of ball bearings [20,21]. The effects of centrifugal forces and gyroscopic moments are included. Fig. 1(a) shows the five-degrees-of-freedom (DOF) model of an ACBB with external loads and corresponding inner ring displacement, respectively.

Fig. 1

Global and local loading and displacements

In the figure, α₀ denotes to the nominal contact angle of the ACBB.

Fig. 1(b) shows the cross-section of ACBB with local coordinates. Here, the inner ring and ball displacements are indicated, respectively, by

Fig. 2 shows the free-body diagram of a ball where the ball equilibrium equations can be established [20]. α_i, α_e represent the operating inner and outer contact angles. D_a is the ball diameter. λ_i, λ_e indicate the traction coefficients for the gyroscopic moment which are set to λ_i = 0, and λ_e = 2 for outer raceway control [11]. F_c, M_g are the centrifugal force and gyroscopic moment. ω_m, ω_R, β denote the ball orbital speed, and ball rotational speed in rad/s, and the pitch angle of the ball [21]. By solving the ball equilibrium equations, the ball displacement vector in Eq. (4) can be estimated. The inner ring force equilibrium can be written as

Fig. 2

Contact load, centrifugal force, and gyroscopic moments

where the inner ring contact load at the i-th ball is defined as

and [Rϕ] is the transformation matrix from local frame to global frame [15]. The contact loads between the ball and races can be calculated using Herzian contact theory as

where K_i, K_e denote the contact stiffness constants between the ball and inner race, and the ball and outer race, respectively. δ_i, δ_e represent the contact compression between the ball and races. If any one of these compressions becomes negative, the contact compression is assumed to be zero.

This study implements a constant force preloading condition, which treats the external loads as inputs, while the displacement vector of the inner ring is the unknown variable to be determined. To calculate the displacement vector for an ACBB, it is imperative to consider the force equilibrium between the balls and the inner ring. Due to nonlinearity of these equations, an iterative approach is necessary to solve the equilibrium equations. The calculation methodology for the entire model is depicted in Fig. 3, which relies on the iterative Newton–Raphson method. The calculation procedure encompasses iterative loops for the global equilibrium equations of the inner ring and for the equilibrium equations of each ball. Upon achieving global equilibrium solution, the contact forces and angles between the balls and races are estimated for subsequent calculations of running torque and heat generation.

Fig. 3

Calculation procedure for the quasi-static model of ACBB

2.2 The Fit-up Model

When an ACBB is mounted to a shaft, the internal diametral clearance (P_d) may change due to ball-race contact loads, fit-up condition between the shaft and inner ring and housing and outer ring, centrifugal expansions of rotating rings, and temperature changes [14-17] as shown conceptually in Fig. 4. The estimation of the change of diametral clearance of an ACBB is performed based on the thick-ring theory [16,19].

Fig. 4

Definition of diametral clearance change in CRBs

The estimation of diametral clearance change comes with estimating the total deformation of the bearing inner ring, outer ring, and the balls. The changes in bearing geometry are calculated by combining the diametral displacements from each race and the change of diametral displacement from the ball. The total diametral deformation (u_TOT) is the combination of the deformation due to surface pressures (u_SP), centrifugal expansions of rotating rings (u_CE), and temperature change (u_TC) as outlined in Eq. (9):

Therefore, the change in diametral clearance is given by

Here IR, OR denote as the inner ring and outer ring, respectively. With a new diametral clearance, the internal geometry of ACBB will change and causes changes in contact loads and heat generation.

The deformation due to surface pressures (u_SP) is observed in cold to hot mounted fittings and from the ball-race contact loads. In a thick-walled cylinder scenario, this deformation manifests when the cylinder with uniform wall thickness is exposed to evenly distributed internal and external pressures. Such deformation is symmetrical along the axis of the cylinder and is presumed not to impact its length [14]. Importantly, this deformation sums up the effect of pressures stemming from ball-race contact loads and the fittings of bearings to shafts and housings.

The deformation due to centrifugal expansion (u_CE) is associated with rotating rings originates from the centrifugal expansion effect observed in rotating shaft and ACBB inner ring. This phenomenon is primarily due to the centrifugal force generated as the inner ring and shaft operate at high velocities, which softens some of the compressive stress induced by interference fits. Thus, this reduction in compressive stress leads to a decrease in the inner ring-to-shaft contact stress, with the rate of reduction accelerating as the rotational speed increases [17]. Such a dynamic property can cause the inner ring-to-shaft operating fit to transit from a tight to a loose state, potentially resulting in fretting wear, increased shaft vibration, and rotor eccentricity [15].

The deformation due to temperature variations (u_TC) is a result of changes from ambient to operating temperatures within the inner ring, outer ring, and balls. This temperature change induces linear expansion, altering the components’ diameter and affecting the diametral clearance of the ACBB. To accurately predict the operating temperatures within ACBBs, a TNM is utilized, as elaborated in section 2.4. This methodology enables precise estimations of temperature-induced dimensional alterations, critical for evaluating their influence on ACBB performance.

2.3 Heat Generation Model

The heat generation in ACBB is obtained from the ACBBs’ running torque-based model on the friction sources presented by SKF [23], in which the friction torque of ACBB with respect to bearing axis comprises three components: sliding friction torque (M_sl), rolling friction torque (M_rr), and viscous drag friction torque (M_ν). In this study, the friction coming from any interactions in cage and seal is not considered.

The distribution of heat generation on the inner race and outer race can be calculated as

After calculating the rates of heat due to friction torque, it is possible to determine the bearing temperatures using a TNM. In this analysis, the contact parameters of the most heavily loaded ball are considered, and the heat generations for each race represent the average heat generation of that specific race.

Sliding friction torque (M_sl) is characterized by two primary sources: macro sliding and micro-sliding. Macro sliding stems from the conformity of contact influenced by the macro-geometrical features of the components and spinning motion inherent in their operation. Micro-sliding, on the other hand, results from the geometrical distortions that occur due to elastic deformation between the ball and races [23]. The sliding friction torque is defined as the combination of torque from spinning friction [21,25-29] and the differential sliding frictions [25,28,29] that occur within the ball-race contacts of ACBBs. It is noted that under high-speed operational conditions, the sliding friction is prone to increase.

Rolling friction losses are a constant factor in rolling contacts, regardless of whether they are dry or lubricated [23]. The origins of rolling friction torque (M_rr) can be traced back to two main factors: the effects of elastohydrodynamic lubrication (EHL) between the balls and races [21,23,25], and elastic hysteresis losses within the materials [23,29]. These factors contribute to the creation of rolling resistance that acts to resist motion between the balls and races. Rolling friction torque has the unique characteristic of potentially decreasing at higher speeds. This decrease is particularly noticeable at very high rotational speeds, where increased temperatures can alter the behavior of the lubricant within the bearing. These changes may lead to phenomena such as inlet shearing, caused by oil reverse flow, and lubrication starvation [23]. Both conditions can significantly reduce the rolling resistance in ball-race contacts.

This study develops a viscous drag friction torque (M_v) model designed for ACBB under oil-jet lubrication conditions. In such a bearing system, lubricated with a mixture of lubricant and air, balls travel in an orbital path around the bearing's axis, navigating through the lubricant-filled space. This motion necessitates overcoming the viscous drag force (F_ν) generated by the lubricant, which becomes significant at very high speeds [25,31]. However, estimating viscous friction for ACBBs under oil-jet lubrication presents analytical challenges, primarily due to the difficulty in precisely determining the lubricant volume within the bearing cavity. Crucial factors such as the oil density and viscous drag coefficient are highly sensitive to variations in this volume. In this study, the effective density of the lubricant is deduced by calculating the lubricant cavity ratio [32], and the viscous drag friction coefficient is derived from a sequence of CFD simulations.

2.4 Thermal Network Model (TNM)

The TNM is a representative method used to analyze the temperature distribution of thermal systems, consisting of a series of nodes and resistances, similar to an electrical circuit network.

Each node represents a specific component of the system, and a node's temperature represents the temperature associated with its component. Nodes are connected to their neighboring nodes through thermal resistances, and heat transfer occurs in proportion to the temperature difference between two adjacent nodes and in inverse proportion to their thermal resistance ($$ \dot{Q}=\Delta T/R$$). In this study, the nodes on the surface as well as the center of system components were defined. Fig. 5 and Table 1 show the locations and the detailed information of the nodes, respectively.

Fig. 5

Node arrangement in TNM

Table 1

Node numbers and locations in TNM

Governing equations need to be established to predict the temperature at each node. This study focuses on the steady-state bearing temperature during normal operation. The governing equation for each node can be determined from the energy balance, as shown in Fig. 6. The following is the energy equation for node i:

Fig. 6

Energy balance for node i

Here, the subscripts i and j are node indices, and the subscripts air and oil denote ambient air and lubricant oil, respectively. N_b is the number of neighboring nodes of node i. R_ij, R_i−air, and R_i−oil represent the thermal resistances between node i and node j, between node i and air, and between node i and oil, respectively.

Eq. (13) indicates that the sum of all heat transferred to node i and the heat generated at node i becomes zero when steady state is achieved. Each term represents, in order, heat transfer from node j, heat transfer from air, and heat transfer from oil. Eq. (13) can be rewritten as Eq. (14), where the temperatures T_i and T_j are placed on the left-hand side, and the remaining terms corresponding to source terms are on the right-hand side. By solving these equations, the temperatures of the bearing components can be obtained.

The heat generated in a bearing is transferred to the bearing parts such as ball, inner ring, outer ring, shaft, and housing, and eventually transferred to surrounding air and lubricant oil. During this process, conductive and convective heat transfer occurs, in the solid structure and at fluid-solid interfaces, respectively. The conductive heat transfer rate is affected by contact conditions. Therefore, thermal resistances can be classified into conductive resistance, convective resistance, and contact resistance, based on the heat transfer mechanism. The formulas for calculating thermal resistances can be found in the literature on thermal network [30,32]. The process of predicting the temperature of a bearing system using a TNM is shown in Fig. 7.

Fig. 7

Calculation procedure of thermal network model

2.5 Calculation Procedure for Integrated Model

The procedure for integrating the quasi-static model with the TNM is delineated in Fig. 8. It commences with gathering requisite inputs, encompassing dimensions of the bearing, shaft, and housing, external loads, rotational speed, oil properties, oil flow rate for oil jet lubrication, initial temperatures of the bearing and oil, and fit conditions between the shaft-inner ring and housing-outer ring. Subsequently, the bearing model, incorporating fit-up conditions, computes the heat generated by the inner and outer races. Following this, the TNM computes the temperature of system components. As bearings expand, alterations in ACBB clearance, material properties, and fluid properties occur, modifying the ACBB’s characteristics. These modifications are then fed-back into the ACBB model and the TNM for subsequent iterations of results. The process iterates until achieving the desired accuracy in calculations.

Fig. 8

Block diagram for thermo-mechanical model

4.1 Change of Diametral Clearance in ACBB

Table 4 displays tabulated results on the changes in the components of the investigated ACBB. Three scenarios are examined: a) without fit-up and TNM, focusing solely on a basic ACBB model that ignores all fit-up conditions and temperature changes. b) with fit-up but without TNM, adding the effects of tightly fitting parts together but keeping temperatures of the bearing parts at ambient level, and oil at given conditions. c) with both fit-up and TNM, integrating all models.

The comparison across these scenarios shows significant variations in diametral clearance of ACBB. Without any fit-up or TNM, the clearance remains unchanged. When fit-up is applied, the clearance tends to decrease because the inner ring is more affected by the combined effects of deformation from surface pressures and centrifugal expansion than the outer ring, which is mainly altered by surface pressure alone. The final scenario, which combines fit-up and TNM, results in greater expansion of all bearing components due to heating. This outcome is believed to reflect real-world behavior more precisely, underscoring the importance of accounting for both fit-up adjustments and operating temperature considerations to estimate ACBB characteristics.

4.2 Change of Contact Loads

Fig. 13 depicts the contact loads on both races of the most heavily loaded ball across various rotational speeds. These contact loads increase as the operating diametral clearance of the ACBB nears a negative value, indicating an interference fit between the balls and races. This indicates that without incorporating fit-up and TNM, the contact load outcomes (Q_i: 510.63 N, Q_e: 703.61 N) (red) differ significantly from the scenario where both are applied (Q_i: 472.23 N, Q_e: 665.14 N) (green), highlighting their critical importance for realistic applications.

Fig. 13

Max contact loads variation with fit-up and thermal effects

4.3 Change of Bearing Stiffness

Fig. 14 illustrates how fit-up and thermal changes influence the stiffness of ACBB. When neither fit-up adjustments nor thermal changes from TNM are considered, the results are derived solely from the quasi-static model of ACBB. However, introducing only the fit-up model under cold-mounted conditions leads to a gradual increase in stiffness. This is attributed to the operating clearance of ACBB becoming negative, as discussed in section 4.1, thereby creating higher contact loads. When both fit-up and thermal changes from TNM are considered, an additional notable effect is observed. In this scenario, all factors that deform the ring are considered, resulting in lower stiffness than others.

Fig. 14

Stiffness change with fit-up and thermal effects

4.4 Change of Heat Generation

Fig. 15 presents the findings on heat generation at the position of the most heavily loaded ball, showing a pattern different from the discussions in sections 4.2 and 4.3. Upon reviewing the overall results for either the total or the max-loaded ball’s heat generation, it becomes apparent that the outcomes do not vary as substantially as the contact loads and stiffness. Contrary to the behaviors observed in contact loads and stiffness, the findings related to the heat generation of ACBB are completely opposite. When the internal clearance turns negative, its orbital speed drops, leading to a decrease in drag friction. This observation highlights the complex interplay between fit-up adjustments, thermal effects, and rotational speed in influencing ACBB’s heat generation. Detailed results of the total heat generation and the distribution of heat generation across both races are provided in Table 5.

Fig. 15

Effect of fit-up and thermal changes on heat generation

Table 5

Heat generation with different conditions