Characterization of a novel rolling-sliding hybrid bearing for main shafts of high-power wind turbines

3.1. Mathematical model of the hydrodynamic bearing

The mathematical model of the hydrodynamic bearing is based on the theory of hydrodynamic lubrication. The reaction force of a conventional hydrodynamic bearing is obtained by integrating the pressure distribution $p\left(x,z\right)$ over the bearing surface. The pressure distribution must be determined by numerically solving the Reynolds equation. According to reference [22], the Reynolds equation for an incompressible fluid in a radial bearing is given by:

where: $p$ is the unknown pressure distribution function; $x=R\phi$ is circumferential coordinate; $z$ is axial coordinate; $h$ is radial clearance function: $h=h_0-X\sin \alpha -Y\cos \alpha$; $\rho$, $\mu$ are the density and dynamic viscosity of the lubricant; $U$ is the circumferential velocity distribution: $U=\frac{\omega D}{2}-{\dot{X}}_0\sin \frac{2x}{D}+{\dot{Y}}_0\cos \frac{2x}{D}$; $w$ is the rotor angular velocity; ${\dot{X}}_0$, ${\dot{Y}}_0$ are the velocities of the journal center in the $XOY$ coordinate system; $V$ is the velocity distribution in the radial direction on the surface of the journal: $V={\dot{X}}_0\cos \frac{2x}{D}+{\dot{Y}}_0\sin \frac{2x}{D}$. The coordinate system and parameter definitions of the sliding bearing are shown in Fig. 2.

The load-carrying capacity is calculated based on the pressure field distribution:

where, $p(\phi ,z)$ is oil film pressure distribution function; $R$ is the bearing radius.

The total load-carrying capacity is:

Fig. 2Sliding bearing coordinate system and parameter definitions

The frictional torque of a hydrodynamic bearing primarily depends on the sliding velocity. Therefore, the frictional torque is:

3.2. Fatigue life model of tapered roller bearings

The fatigue life of the bearing involves the calculation of the rated roller load, roller load distribution, and equivalent roller load. Under normal operating conditions, the basic reference rated life of the bearing corresponding to 90 % reliability is:

where: $C$ is the basic dynamic load rating (N); $P$ is the equivalent dynamic load (N); $n$ – operating speed (rpm).

The formula for calculating the equivalent dynamic load $P$ is as follows:

where, $X$ is the radial dynamic load coefficient and $Y$ is the axial dynamic load coefficient.

For a double-row tapered roller bearing, the basic dynamic load rating $C$ is:

where: $C_a=b_m{f}_c(L_{we}\cos \alpha {)}^{7/9}\tan \alpha Z^{3/4}{D}_{we}^{29/27}$; $b_m$ is the bearing type correlation coefficient; $Z$ is the number of rolling elements; $f_c$ is a material coefficient; $L_{we}$ is the effective length of the roller; $D_{we}$ is the roller diameter; $i$ is the number of rows, $\alpha$ is the contact angle.

Based on the basic rated life, modifications are applied to account for load magnitude, bearing speed, lubricant viscosity, and contamination levels. This yields the modified rated life, which more closely reflects the actual operating conditions of the bearing, expressed as:

where: $a_1$ is the reliability life modification factor and $a_{ISO}$ is the life modification factor.

In this study, for ease of presentation, “hybrid bearing life” specifically refers to the modified rating life of the rolling elements within the hybrid bearing under load-sharing conditions, as the sliding component does not undergo rolling fatigue.

4.1. Analysis of hydrodynamic characteristics in the hybrid bearing

The Sommerfeld number ($S$) is a dimensionless number that describes the relationship between the load borne by the bearing, the viscosity of the lubricant, the rotational speed, and other comprehensive factors relative to the geometric dimensions of the bearing. Its calculation formula is as follows:

where: $r$ is the journal radius; $\mu$ is the dynamic viscosity of the lubricating oil; $N$ is the shaft rotational speed; $P$ is the average pressure on the projected bearing area; $P=\frac{F}{L\times D}$, $F$ is the total load; $L$ is the bearing length; $D$ is the bearing diameter.

Based on lubrication characteristic analysis, a reasonable range for the length-to-diameter ratio and initial radial clearance was determined. The calculated Sommerfeld numbers under different initial oil film thicknesses are shown in Fig. 3.

As shown in Fig. 3, a significant increase in S is observed with increases in the length-to-diameter ratio and rotational speed. When the ratio increases from 1.0 to 1.4 under the same conditions, $S$ rises from 0.196 to 0.274. With a fixed initial oil film thickness of $c=$400 µm, an increase from 3 rpm to 12 rpm causes $S$ to increase from 0.0294 to 0.118. Conversely, a noticeable decrease in $S$ is observed as the initial oil film thickness increases. When $c$ expands from 200 µm to 800 µm, $S$ drops sharply from 0.118 to 0.00735.

Considering the low-speed and heavy-load operating conditions of wind turbine main shafts, to ensure that the bearing operates in the hydrodynamic lubrication regime, S should be greater than 0.05. Therefore, the initial oil film thickness should be in the range of 200-600 μm, and the length-to-diameter ratio is preferably controlled within 1.4-1.8. Taking into account the compatibility of the sliding bearing with the rolling bearing in the wind turbine main shaft, the length-to-diameter ratio should not be excessively large. In this paper, a length-to-diameter ratio of 1.4 is adopted.

Simulation parameters are based on the actual operating conditions of a 10 MW wind turbine main shaft. Since the oil supply pressure is extremely low, a value of 0 is used. The specific values are shown in Table 2.

Fig. 3Variation of Sommerfeld number with length-to-diameter ratio and rotational speed: a) c= 200 µm, b) c= 400 µm, c) c= 600 µm, d) c= 800 µm

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To determine the geometric parameters of the eccentric bearing bush structure, this study systematically analyzed the effects of different radial clearances and rotational speeds on the bearing lubrication performance. The investigation focused on the dynamic response characteristics of the maximum oil film pressure, minimum oil film thickness, eccentricity ratio, and attitude angle. The results are shown in Fig. 4.

At a fixed rotational speed, an increase in the maximum oil film pressure is observed with enlargement of the radial clearance. For example, at 8 rpm, the pressure rises from 1.79 MPa at 200 µm clearance to 3.16 MPa at 800 µm clearance. This suggests that appropriately increasing the radial clearance significantly elevates oil film pressure. Oil film formation capability and lubrication robustness are thereby enhanced. Overall load-carrying performance is improved. Under the same clearance, a decrease in oil film pressure is noted as rotational speed increases. This indicates that stronger oil film pressure can be generated at lower speeds.

Fig. 4a) Maximum oil film pressure, b) minimum oil film thickness, c) eccentricity ratio, d) attitude angle

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b)

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The minimum oil film thickness increases with higher rotational speeds. It is substantially greater than typical hydrodynamic bearing oil film thicknesses. At the same rotational speed, an increase in initial clearance from 400 µm to 600 µm results in a decrease in oil film thickness.

Regarding shaft center trajectory, under fixed clearance, the eccentricity ratio decreases as rotational speed increases. Under fixed speed, it increases with larger clearances. For instance, at 8 rpm, the ratio rises from 0.181 (200 µm) to 0.845 (800 µm). The attitude angle increases with higher speeds under the same clearance. It decreases with larger clearances at the same speed.

Given the large size and weight, friction torque and frictional power consumption were further analyzed. Their variation with rotational speed and initial clearance is shown in Fig. 5.

Fig. 5a) Bearing frictional power, b) friction torque

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b)

As shown in Fig. 5, with the increase in rotational speed, both the friction torque and frictional power consumption exhibit a significant upward trend overall. For instance, with a fixed clearance of 200 µm, when the rotational speed increases from 3 rpm to 16 rpm, the friction torque rises from approximately 3.73 kN·m to about 15.04 kN·m, while the frictional power consumption increases from approximately 1.17 kW to about 25.20 kW. This indicates that the friction is overly sensitive to changes in rotational speed, which can exacerbate bearing wear. To balance lubrication performance and friction losses, the initial radial clearance should be controlled within the range of 400-800 µm.

Taking all factors into consideration, it is determined that the sliding bearing has a length-to-diameter ratio of 1.4 and an initial radial clearance of $c=$ 400 μm. The selected radial clearance of 400 μm corresponds to a relative clearance of $\psi =c/R=$ 0.465 ‰ for the 1720 mm diameter bearing. According to ISO 12129-1:2024, the recommended relative clearance range for metal plain bearings is 0.56 ‰-3.15 ‰, while heavy-duty low-speed applications such as rolling mills typically adopt 0.5 ‰-1.5 ‰. The value of 0.465 ‰ is slightly below these ranges, which is reasonable for the ultra-low-speed conditions of 8 to 10 rpm and the heavy-load condition of 4000 kN considered in this study. Additionally, ISO 5753-1991 indicates that radial internal clearances for comparably sized rolling bearings range from 330 μm to 1450 μm, confirming the technological feasibility of 400 μm.

To further address the effect of manufacturing tolerances and assembly deviations, a sensitivity analysis was performed for radial clearances of 350 μm, 400 μm, and 450 μm under the same operating conditions (8 rpm and 10 rpm, radial load 4000 kN, $L$/$D =$ 1.4). The results are summarized in Table 3.

In all cases, the minimum oil film thickness remains above 230 μm, which is two orders of magnitude larger than typical surface roughness of approximately 1 μm. Thus, full-film lubrication is maintained across a clearance band of ± 50 μm around the nominal 400 μm value. The maximum pressure varies by less than 8 %, indicating that load-carrying capacity is not highly sensitive to moderate clearance deviations.

Moreover, the operating speed is extremely low, ranging from 3 to 16 rpm, so centrifugal forces and dynamic unbalance are negligible. Small geometric deviations do not induce significant pressure fluctuations. The pressure field is dominated by the convergent wedge geometry, which is not critically affected by moderate misalignment or roundness errors. Additionally, the shaft with a diameter of 1500 mm and the housing with an outer diameter of 1820 mm have very large cross-sections, providing high structural stiffness. Elastic deformations under the applied loads are extremely small and do not compromise the clearance distribution or load-sharing mechanism.

Consequently, within a realistic manufacturing tolerance band of ± 50 μm, which corresponds to IT7-IT8 grade for a bearing of this diameter, the hybrid bearing’s lubrication performance and load-sharing capability remain effective. The nominal clearance of 400 μm is therefore a well-justified design choice for the ultra-low-speed, heavy-load conditions of a 10 MW wind turbine main shaft.

It should be emphasized that the nominal radial clearance of 400 μm used in this study represents an ideal geometric value. Manufacturing tolerances, deviations in roundness and alignment, assembly inaccuracies, and housing deformations are not considered in the present analysis. A detailed tolerance analysis is beyond the scope of this work, which focuses on the fundamental feasibility and performance potential of the hybrid bearing under idealized nominal conditions. Therefore, the results presented herein should be interpreted as an upper-bound estimate of load-sharing capability and life extension. Future work should include tolerance analysis and experimental validation.

According to reference [23], since the main shaft bearing operates at speeds ranging from 0 to 16 rpm, this study conducts simulation analyses at radial clearances of $c \text{=}$ 400 μm and rotational speeds of 8 rpm and 10 rpm to validate the hydrodynamic effect of the sliding bearing.

Fig. 6Simulation results for pressure and film-thickness distributions (radial load F acts vertically along the positive Y-axis, with circumferential coordinate φ= 0 corresponding to the load line): a) 3D oil film pressure distribution at 8 rpm; b) 3D oil film thickness distribution at 8 rpm; c) 3D oil film pressure distribution at 10 rpm; d) 3D oil film thickness distribution at 10 rpm; e) 2D planar unwrapped pressure distribution at 8 rpm; f) 2D planar unwrapped pressure distribution at 10 rpm

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Based on the simulation analysis from Fig. 6, the results indicate that at 8 rpm, the minimum oil film thickness is approximately 242.1 µm, demonstrating that the bearing operates under a thick-film lubrication condition. Simultaneously, the maximum oil film pressure is about 1.95 MPa. Under these operating conditions, the corresponding eccentricity ratio is approximately 0.395, and the attitude angle is about 68.8°. When the rotational speed increases to 10 rpm, the minimum oil film thickness rises to 267.0 µm, indicating a further thickening of the lubricating film. The maximum oil film pressure slightly decreases to 1.89 MPa, suggesting a more gradual load distribution. Under these conditions, the bearing bush eccentricity ratio decreases to approximately 0.332, while the attitude angle increases to about 72.4°.

The oil film pressure distribution shown in Fig. 6 exhibits near-perfect circumferential symmetry. This result stems from the rigid-shaft assumption, which requires the operating speed n to be significantly lower than the first critical speed $n_{c1}$(typically $n$/$n_{c1}$ ≲ 0.3). To justify this assumption for the specific shaft-bearing system, a conservative estimate of $n_{c1}$ is provided.

The main shaft is stepped. Taking the smallest diameter (1500 mm, at the rolling bearing mounting position) as the reference diameter, and using the length-to-diameter ratio of 1.4, the shaft length is 2100 mm. The cross-sectional moment of inertia is $I=\pi d^4/64=$ 0.2485 m⁴, and the shaft mass is $m=\rho \pi d^{2}L/4=$29100 kg. For such a short-thick shaft ($L/d=$ 1.4), the Euler-Bernoulli beam model is invalid because shear deformation and rotary inertia dominate. Instead, the Timoshenko beam model and engineering experience for megawatt-scale wind turbine main shafts indicate that the first critical speed lies in the range of 50-150 rpm. Owing to the extremely large diameter and high bending stiffness, the first critical speed of this shaft is at the upper end of this range, approximately 150 rpm. This is six times the maximum operating speed of 25 rpm, well satisfying the rigid-shaft condition.

Under the rigid-shaft, purely radial loading, and ideal alignment assumptions adopted herein, the present simulations show high load-sharing ratios, reaching 98.43% at 8 rpm and 99.23% at 10 rpm. However, the influence of shaft flexibility, bending moments, misalignment, and non-uniform support conditions on the load-sharing mechanism has not been investigated and requires verification in future work. It should be emphasized that these results are obtained under nominal conditions; actual performance under more realistic conditions may differ.

Based on the above reasoning, the idealized rigid-shaft model provides an upper-bound estimate of load-sharing and life extension under nominal conditions. Further quantitative investigation using flexible-shaft models, including bending moments, misalignment, and non-uniform support conditions, is required to assess the performance under more realistic operating conditions.

The total radial load on the hybrid bearing remains constant, yet the distribution ratio between the rolling elements and the sliding bearing shell is determined by the shell's eccentric design. This passive load-sharing mechanism, resulting from the eccentric bush geometry and hydrodynamic pressure generation, enables dynamic load variation without active control. Consequently, the aforementioned eccentricity parameters are identified as key determinants in establishing the structural configuration of hybrid rolling-sliding bearing shells.

4.2. Analysis of life characteristics of the rolling-sliding hybrid bearing

The rolling bearing in the wind turbine main shaft includes the bearing inner ring, outer ring, and rolling elements. The design parameters of the tapered roller bearing are shown in Table 4.

In practical wind turbine applications, tapered roller bearings are mounted with a light axial preload achieved by axial spacer adjustment. This preload is commonly applied in large-scale wind turbine main shafts to eliminate internal radial clearance and ensure positioning accuracy. Under this preload, the internal radial clearance of the rolling bearing is effectively reduced to within ±2 μm. It should be emphasized that this clearance refers to a different physical interface than the sliding bearing’s nominal 400 μm concentric radial clearance; therefore the two clearances do not conflict. The load-sharing mechanism does not require relative radial motion between the rolling bearing inner and outer rings; it relies only on the static eccentric geometry of the sliding bearing bush, making the proposed design geometrically feasible.

The total radial load $F$ is shared between the rolling bearing and the sliding bearing as $F=F_{rolling}+F_{sliding}$. This relationship holds regardless of the rolling bearing’s internal clearance or preload. When the shaft reaches an operating speed of 8 to 10 rpm, the sliding bearing develops a full-film hydrodynamic pressure due to its eccentric bush geometry, automatically carrying a large portion of the radial load: 98.43 % at 8 rpm and 99.23 % at 10 rpm, as shown above. Consequently, $F_{rolling}$ becomes very low, namely 1.57 % and 0.77 % of the total radial load, respectively, whether the rolling bearing has a small clearance after preload or a larger clearance.

Regarding the effects of preload on friction, heat generation, equivalent dynamic load, and bearing life, the preload effect is considered negligible for the fatigue life assessment under the specific operating conditions of this study for the following reasons. First, under steady state load-sharing, the rolling bearing carries only 1.57 % or 0.77 % of the total radial load. The additional contact stress caused by preload is much smaller than the residual radial stress and therefore has a negligible impact on fatigue life. Second, the rotational speed is extremely low, ranging from 3 to 16 rpm, so the extra friction torque due to preload is very small and does not significantly affect the bearing's operating condition. Therefore, in the life calculation using ISO 281, preload is not explicitly included as a separate factor; the reported modified lives are conservative estimates without considering the minor beneficial or adverse effects of preload. A detailed preload analysis is beyond the scope of this study and is recommended for future work.

The values of each coefficient are shown in Table 5.

The coefficients listed in Table 5 are selected according to ISO 281:2007. Here, $b_m$ is the bearing type correlation coefficient, taken as 1.1 for tapered roller bearings; $f_c$ is the material and geometry coefficient, taken as 165.5 for standard bearing steel; $a_1$ is the reliability life modification factor, taken as 1 for 90 % reliability; $a_{ISO}$ is the life modification factor based on lubrication condition and contamination level. The selection of $a_{ISO}=$ 1.75 is justified as follows based on the actual operating parameters provided by our industrial partner. The lubricant is VG320 oil with a supply temperature of 50 °C. At this temperature, the kinematic viscosity $\nu$ is approximately 160 mm²/s. The bearing pitch diameter is approximately 1660 mm, and the rotational speed is 8-10 rpm. According to the reference viscosity estimation formula in ISO/TR 1281-2, the required kinematic viscosity ${\nu }_1$ for the bearing under these conditions is calculated as approximately 70 mm²/s. Therefore, the viscosity ratio $\kappa =\nu /{\nu }_1$ is approximately 2.28. The oil cleanliness is maintained at ISO 4406 class 16/13/10, which corresponds to a contamination factor ${\eta }_c$ of approximately 0.9-1.0 based on ISO 281:2007 Annex A. For a roller bearing with $\kappa \approx$ 2.28 and ${\eta }_c\approx$ 0.9-1.0, the life modification factor $a_{ISO}$ is determined as 1.75 according to the standard procedure.

The dynamic load distribution mechanism of the rolling-sliding hybrid bearing lies in the fact that during the start-stop phase of the wind turbine main shaft, both radial and axial loads are borne by the rolling bearings. Upon entering stable operation, the sliding bearing forms a complete load-bearing oil film through the hydrodynamic effect, thereby beginning to share the radial load. The key to this design is that the sliding bearing bush is eccentrically arranged, with its eccentricity ratio and attitude angle determined based on the operating condition where the bearing can independently establish full-film lubrication and thereby carry the entire radial load at the target rotational speed. Under the adopted idealized assumptions of a rigid shaft, perfect alignment, and steady-state operation, the sliding bearing can assume nearly the entire radial load, and the radial load on the rolling bearings is reduced to a very low value close to zero, i.e., nearly full unloading. Specifically, at the design condition of 8 rpm, the rolling bearing residual radial load is 62.8 kN, which is only 1.57 percent of the total radial load of 4000 kN, and the sliding bearing carries the remaining 98.43 percent. At 10 rpm, the rolling bearing residual radial load is 30.73 kN, which is only 0.77 percent of the total load, and the sliding bearing carries the remaining 99.23 percent. These values confirm nearly complete unloading under the idealized assumptions. It should be noted that in real-world applications, manufacturing tolerances, elastic deformation, preload, housing flexibility, transient operation, and changes in load direction may prevent complete radial unloading. Therefore, the reported load-sharing ratios represent an upper-bound estimate under ideal nominal conditions. Based on this mechanism, $F_{sliding}$ can be obtained through hydrodynamic simulation according to Eq. (1), thereby deriving the dynamically varying $\left(F_{rolling}\right)$. Since the equivalent dynamic load of the rolling bearings changes dynamically with the radial load-sharing state, the coefficients $X$ and $Y$ in its life calculation are also adjusted accordingly. Finally, the life comparison between the hybrid bearing and the pure rolling bearing is shown in Fig. 7, fully demonstrating the advantage of the hybrid design in enhancing lifespan.

Fig. 7Comparison of modified rating life of the rolling bearing under pure rolling and hybrid load-sharing conditions

As shown in Fig. 7, the Modified rating life of the hybrid bearing under both design conditions is significantly higher than that of the pure rolling bearing across the entire speed range. The solid blue line represents the hybrid bearing designed based on the eccentricity parameters obtained at 8 rpm; the dashed red line represents the hybrid bearing designed based on the eccentricity parameters obtained at 10 rpm; and the green line represents the modified rating life of the pure rolling bearing. Over the operating speed range of 3-16 rpm, the hybrid bearing exhibits a distinct trend: its modified rating life increases rapidly to a maximum at 8 rpm, then gradually decreases with further speed increase. In contrast, the pure rolling bearing shows a continuous and slow decline across the entire speed range. This fully demonstrates the advantage of the hybrid design in enhancing lifespan.

At 8 rpm with an initial radial clearance of 400 μm, the hybrid bearing achieves a modified rating life of 43.52 years, which is 4.9 times that of the pure rolling bearing with a life of 8.89 years. At 10 rpm, the hybrid bearing achieves a modified rating life of 35.43 years, representing a 5.0-fold improvement over the pure rolling bearing with a life of 7.11 years.

Such a remarkable lifespan enhancement arises from the optimized eccentricity design of the bearing bush, which nearly fully unloads the radial load on the rolling elements under these operating conditions. The results clearly validate the significant advantages of the dynamic load-sharing mechanism and provide a reliable reference for wind turbine bearing applications.