This paper considers the modelling of wind turbine main bearings using analytical models. The validity of simplified analytical representations used in existing work is explored by comparing main-bearing force reactions with those obtained from higher-fidelity 3D finite-element models. Results indicate that there is good agreement between the analytical and 3D models in the case of a non-moment-reacting support (such as for double-row spherical roller bearings), but the same does not hold in the moment-reacting case (such as for double-row tapered roller bearings). Therefore, a new analytical model is developed in which moment reactions at the main bearing are captured through the addition of torsional springs. This latter model is shown to significantly improve the agreement between analytical and 3D models in the moment-reacting case. The new analytical model is then used to investigate load characteristics, in terms of forces and moments, for this type of main bearing across different operating points and wind conditions.

Wind energy provides an important and growing contribution to the European energy market, with 205 GW installed as of 2019 – accounting for 15 % of consumed electricity

With turbines moving further offshore and a need to bring costs down, reducing operation and maintenance costs, which can be as high as 35 % of the total lifetime costs of a project, is becoming increasingly important for wind farm operators

One turbine component with relatively high failure rates and associated downtime is the main bearing (MB). MBs are becoming recognised as an important component for which failures need to be better understood and reliability improved

Wind turbine drivetrains and MBs in particular are specific to individual turbine designs. As such, it is beneficial to understand in as much generality as possible how existing simple representations may be used to study MB load response without focusing on any design case (since this would reduce the generality of results). In order to move in this direction, it is necessary to work through levels of modelling complexity, understanding at each stage how well a given model represents the next in the chain. This approach also develops knowledge about which effects can be adequately captured at a given level of model complexity, helping inform decisions with respect to model selection for specific applications.

This paper considers an important step in the overall modelling chain starting with existing 2D, orthogonally independent, simply supported models and looking to compare with higher-fidelity models which are closer to representing real-world wind turbine MBs. The strongest assumptions in the existing models are independence of horizontal and vertical planes (from a load perspective) and simply supported load reactions (i.e. the bearing does not support moment loads). Therefore, this work seeks to compare their performance with more realistic models that remove one or both of these assumptions. More explicitly, 3D finite-element (FE) modelling removes the 2D and orthogonality assumptions. With respect to simple versus other supports, MBs for wind turbines have two “types” of reaction behaviour in general: those that support forces only and not moments (e.g. double-row spherical roller bearings, DSRBs) and those that support both forces and moments (e.g. double-row tapered roller bearings, DTRBs). Three-dimensional FE models which have reaction behaviours that emulate each of the two types are therefore considered. Hence, the overarching goal of this paper is to explore the following question:

Section 2 summarises previous work undertaken in this area. Section 3 then introduces the higher-fidelity 3D models which are used to compare with analytical model outputs. Section 4 presents the results of the comparison, with Sect. 5 then extending the analytical model to include moment reactions at the MB. In Sect. 6 the new analytical model is used to study load behaviours for this bearing type. Finally, Sect. 7 discusses some practicalities surrounding the application of these models before Sect. 8 presents the conclusions of this work.

Despite having received less attention than other drivetrain components, there have been a number of high-quality research papers which include modelling and analysis of wind turbine MBs.

In addition to the analyses outlined above, work has also been undertaken in which simple drivetrain representations are used to study general characteristics of MB loads and their relationship to the incident wind field

Analytic model for single-main-bearing set-up in one plane. The full model consists of two such representations, one in each of the horizontal and vertical planes

The equation system for the SMB drivetrain set-up is statically determinate and can be solved by balancing the moments about the gearbox, giving

Parameters for all models.

While models and results in

In order to assess the effectiveness of the simple analytical models used thus far, two FE models of the SMB system were created in ANSYS. The FE models were designed to be general and do not seek to represent any particular bearing specifically but rather the global behaviour of different bearing types: one designed to behave like a DSRB (non-moment-reacting) and the other to behave like a DTRB (does support moment loads). Likewise, the rest of the drivetrain system such as the shaft and gearbox connections remains both general and similar for the two different bearing types to create a like-for-like study. The models were subjected to the same hub loading as the analytical models outlined in the previous section, with bearing support reaction forces outputted and compared with those from the analytical model. Both FE models share dimensions with the SMB analytical model. The FE models themselves still remain relatively simple, with relevant behaviours captured without the modelling of individual rolling elements, as described below. To aid reproducibility, input and output value examples for all models are given in Table

The DSRB FE model was created with three separate bodies, referred to here as the shaft, the bearing, and the bearing housing (see Fig.

This additional aspect of modelling will be considered in future work as progressively more complex representations are implemented.

A fixed support was added to the base of the bearing housing to represent the connection to the bedplate, and the connection between the low-speed shaft and the gearbox was modelled by three body-to-ground spring connections in the horizontal, vertical, and axial directions. Appropriate equivalent stiffness values of the connections between the low-speed shaft and the gearbox were determined with the use of Romax Technology software. The stiffness values, along with model dimensions, can be found in TableThe three-dimensional finite-element model with double-row spherical-roller-bearing-type reaction behaviour.

The DTRB FE model was created with two separate bodies, referred to here as the shaft and the bearing–bearing housing (see Fig.

The three-dimensional finite-element model with double-row tapered-roller-bearing-type reaction behaviour.

Internal contact conditions and load distributions around the bearing circumference are important (and non-linear) aspects of bearing behaviour. However, the SMB analytical model being studied is not designed to go to this level of detail, instead outputting the reaction forces at (or equivalently the loads applied to) the MB. As such, the simplified FE representations for DSRB and DTRB bearings outlined above are considered reasonable for the following reasons: in the DSRB case, DSRBs are self-aligning and hence provide force but not moment reactions across the bearing, and as such, the reaction force required to balance the system should remain the same irrespective of the spring properties, with only displacement magnitudes effected, since the system is determinate; in the DTRB case, the system supports moments through opposite force reactions over the two bearing rows in addition to providing an overall force reaction. Consequently, non-linear contact properties of the rollers will influence the share between force and moment reactions at the MB. However, the non-linearity present in line contact rollers

Including tapered and cylindrical cases.

is only slight, with an exponent of 1.11The analytical model presented in Sect.

Figure

The analytical model reaction force results were then compared with the DTRB FE model, with the results displayed in Fig.

The above comparisons suggest that the orthogonal independence and simple support assumptions made in the analytical model still allow for valid force outputs when representing a DSRB. However, the results also show that the analytical model has significantly overestimated the force reactions for the DTRB system. This motivates the derivation of a new analytical model to try and emulate the positive results seen in the DSRB case for moment-reacting DTRBs. Such a model is developed in the following section.

In order to allow moment reactions at the MB, torsional springs were added to the fixed bearing support in both planes of the analytical model. Thus, a new analytical model was created, displayed in Fig.

The two deflection models can then be decoupled again to show the two mechanisms causing deflection in the shaft: bending of the beam due to the applied moment and rotation about the MB support due to spring support (gearbox) compression or extension. As the deflection mechanisms and equation derivation process are similar for the overturning moment and spring reaction moment on the system, only the equations and deflection mechanisms for the overturning moment are presented here. The two deflection mechanisms for the decoupled model with overturning moment and rotor weight are shown in Fig.

Deflection mechanisms for deflection model 1 under some applied moment

Calculating

Force balance model for the whole system.

Having derived the relevant equations for a new analytical model with moment reaction capabilities, it is then necessary to determine appropriate spring-stiffness values in each plane. These were estimated using the FE DTRB model. The body-to-ground springs representing the shaft connection to the gearbox were removed from the model and four nodes selected: one at the bedplate connection and one at the top of the bearing housing for the vertical plane and one on both sides of the bearing housing at points of mid-height and mid-thickness for the horizontal plane. Known moments were then applied about the horizontal and vertical axes separately and the displacement of the nodes recorded. The angle of rotation about the midpoint of the vertical nodes was calculated and used to determine the vertical-axis spring stiffness via the standard spring equation (Eq.

Node selection within the bearing housing for estimating torsional-spring stiffness in the vertical plane.

Node selection within the bearing housing for estimating torsional-spring stiffness in the horizontal plane.

RMSE results in this case are plotted in Fig.

Presented results imply that the analytical SMB model used in

The mean radial loading for the analytical DSRB model in the previous study showed high sensitivity to shear exponent, with the low-shear exponent wind files resulting in larger radial loading. Plotted low-shear results lay between 400 and 500 kN; similarly, high-shear exponent plotted results were between around 200 and 300 kN. The mean loads within each CPLS remained fairly constant, with small standard deviations, and TI had some effect on the results, with higher TI resulting in slightly higher loading.

Mean radial force and moment results for the analytical DTRB model are shown in Fig.

Considering moment reactions, the magnitude increases with increasing mean wind speeds, and the high-shear cases contribute to larger moment loading compared to low-shear cases. There are also sensitivities to TI in both shear exponent cases.

The analytical DSRB peak radial loads presented in

The previous sections have outlined how the original SMB analytical model can be extended to recreate moment-reacting behaviour at the MB. It is worth considering the practicalities of this approach given that determining torsional-spring-stiffness values requires access to an FE model. Two pertinent questions related to this are therefore as follows: (1) if one requires an FE model in the first place, why cannot all analysis be undertaken using it instead of the simplified representations proposed here? (2) Is it practical to assume that an FE model will be available in general? With respect to the first question, there are two main considerations which imply that simplified models will likely be necessary. First, as has been touched upon, analysis over large numbers of load cases and/or turbines becomes infeasible for high-complexity models due to processing power requirements. In addition, any MB load model which might be embedded within existing aeroelastic software would likewise need to be computationally efficient (see e.g.

To be clear, the detailed and high-quality models used in existing work and outlined in Sect.

This paper considers the question of whether analytical models can be used to effectively evaluate load reactions for 3D main-bearing support configurations with either moment-reacting or non-moment-reacting behaviours. The results of comparisons with 3D FE drivetrain models, designed to exhibit the relevant load reaction properties, indicate that the existing single-main-bearing analytical model can well represent bearing reaction forces in the non-moment-reacting case (e.g. double-row spherical roller bearings). However, it was also shown to be unsuitable for cases where a support has moment-reacting capabilities (e.g. double-row tapered roller bearings). Therefore, a second analytical model was created through the addition of torsional springs to represent a bearing which supports moments as well as forces. Spring stiffnesses were found for this model using a static analysis of the FE model. Outputs from the new analytical model were compared with the moment-reacting 3D model, with results indicating that it offers a greatly improved tool for analysis in the moment-reacting case. The developed model was then used to consider mean and peak forces and moment reactions for this type of bearing across a range of operating conditions.

Table

Hub loading inputs and corresponding model outputs at various time steps. Quantities are in the DNV GL Bladed reference frame.

A shaft sensitivity analysis was carried out to explore the effect of shaft thickness and therefore stiffness on the MB reaction force results for each model. The implemented models are assumed to have solid shafts; however, main shafts typically have small boreholes throughout the centre to allow for the passing of electrical cables. Therefore, thicknesses of 100 %, 75 %, and 50 % were compared to conservatively cover typical main shaft thicknesses and ensure the solid-shaft assumption does not impact the results of this work. Results are plotted below for the analytical DTRB, FE DSRB, and FE DTRB models, respectively. These results, in which only very small deviations can be seen, indicate that shaft thickness appears to have a minimal effect on model accuracy.

Computer code and data used throughout this paper will be made available on request. Please contact the lead author for more information.

JS was the main researcher and also led the writing of the manuscript. EH and AKA supported model development, results analysis, and manuscript preparation.

The authors declare that they have no conflict of interest.

This article is part of the special issue “WindEurope Offshore 2019”. It is a result of the WindEurope Offshore 2019, Copenhagen, Denmark, 26–28 November 2019.

The authors would like to acknowledge extensive support throughout this work from Onyx Insight, specifically Robin Elliott, Rhys Evans, and Evgenia Golysheva.

This research has been supported by the Engineering and Physical Sciences Research Council (grant no. EP/L016680/1).

This paper was edited by Charlotte Bay Hasager and reviewed by Fabian Schwack, Ron A. J. van Ostayen, and one anonymous referee.