Analysis of Rigid-Flexible Coupled Collision Force in a Variable Load Offshore Wind Turbine Main Three-Row Cylindrical Roller Bearing

Abstract

In response to the limitations and one-sidedness of the simulation results of a rigid three-row cylindrical roller bearing for an offshore wind turbine main shaft under constant-load conditions, this paper proposes a simulation analysis method under variable loads. A contact mechanics model and a flexible body model are established, and the rigid-flexible coupled treatment is applied to the bearing’s inner and outer ring and cages. Based on variable load conditions, the theoretical speeds, simulated speeds, and acceleration responses of the pure rigid model and the rigid-flexible coupled model are compared, and the model is validated. Finally, the dynamic and transient responses reveal the time-varying characteristics of bearing loads and stress distribution patterns under different driving speeds and contact friction coefficients in the rigid-flexible coupled model. The conclusions are as follows: the rotational error of the rigid model is 1.67 to 3.76 times greater than that of the rigid-flexible coupled model, and the acceleration trend of the rigid-flexible coupled model is more stable with smaller speed fluctuations. The average forces on the thrust roller and cages increase with the driving speed, while those on the radial roller, cages, and inner ring decrease with the driving speed. The average force on the near-blade end cage is approximately 1.19 to 1.59 times that of the far end. The average force on the roller and cages significantly decreases with decreasing friction coefficient, with a reduction ranging from 50.08% to 76.41%. The maximum stress of the bearing increases with increasing driving speed. The novel simulation method for a rigid-flexible, coupled, three-row cylindrical roller bearing model under variable load conditions proposed in this paper can more accurately simulate the dynamic response of offshore wind turbine main shaft bearings during service. The results obtained in this paper provide highly valuable guidance for the research and design of offshore wind turbine main shaft bearings.

1. Introduction

Wind energy, as one of the most promising and dynamic renewable energy sources, boasts numerous advantages such as significant potential for capacity expansion, a secure means of energy acquisition, and inexhaustible supply. Typically characterized by large-scale mechanical equipment, offshore wind turbines operate amidst alternating loads and harsh natural environments, necessitating components with adequate hardness, strength, corrosion resistance, and impact resistance [1]. As the core component of a wind turbine, the performance of the main shaft bearing directly affects the system’s transmission efficiency and reliability. This is especially critical under alternating loads with variable impact, which can easily lead to fatigue failure. Particularly vulnerable to fatigue failure under variable load impacts, incidents related to the quality of main bearing components have surged in recent years, resulting in substantial economic losses for the wind energy industry and jeopardizing the safety of professionals. Therefore, studying the dynamic response characteristics of the main shaft bearing during operation to accurately reveal the causes of fatigue damage is of significant importance for improving the structural design level of wind turbines [2,3].

Chen Yan et al. [4] established a rigid-flexible coupled model of a wind turbine generator system in ADAMS, investigating its vibration characteristics. The results indicate that the primary vibration mode of the wind turbine tower is bending mode, and the developed rigid-flexible coupled model can simulate the aeroelastic characteristics of the wind turbine generator system. Liu Xiangyang et al. [5] developed a finite-element model of the wind turbine main bearing system with flexible structures and proposed a bearing optimization method considering the flexibility of the support structure. The results show that this model can obtain a more accurate distribution of bearing loads. P. Göncz et al. [6] proposed a computational model for determining the dynamic load capacity of a large three-row roller bearing, providing a method for calculating the internal stress field and maximum equivalent stress values of different types of rollers and raceways. Cao Xu et al. [7] optimized the structure of an offshore wind turbine double-row tapered roller bearing using the finite-element software Romax DESIGNER R17, effectively reducing the peak stresses at the edges of roller-raceway contact. Zhang Shenglin et al. [8] discussed the modal energy distribution of rigid and flexible body wind turbine transmission systems, indicating that flexible supports to some extent reduce the load fluctuations of the power transmission system and increase both axial and radial bearing loads. Zhao Rongzhen et al. [9], using ANSYS 16.0 and ADAMS 2016, proposed a simulation method for calculating the dynamic performance of a rigid-flexible coupled multi-body system at the low-speed shaft end of a wind turbine generator, providing computational support for bearing selection and determination of the overall system parameters. Ma Defu et al. [10] conducted dynamic performance simulations of a constant-load wind turbine main bearing under rigid-flexible coupled conditions, obtaining the time-varying curves of collision forces between rollers and cages, and identifying the critical locations of forces on the inner ring under two operating conditions: startup and steady-state operation. Zhang Juan et al. [11] studied the dynamic characteristics of a 1.5 MW wind turbine main shaft spherical roller bearing under different conditions using ADAMS, showing that the maximum internal interaction force occurs during emergency braking, followed by the speed change stage, and is minimal during the startup to steady-state operation phase. Fan Hengming [12] investigated the influence of the number of rollers and upsetting moment on the contact load of a rigid-flexible coupled double-row tapered roller bearing, revealing that the variation of the upsetting moment increases the number of rollers that are simultaneously supported by the bearing, and, considering roller inclination, contributes to more accurate research. Martin Cardaun et al. [13] studied the effect of yaw misalignment on loads over a period of more than 20 years for a wind turbine generator, obtaining a method to predict load spectra under actual conditions. Wei Zhao et al. [14] modified the material parameters, friction coefficient, and other structural parameters of the wind turbine main shaft bearing, and conducted a dynamic response analysis on it. The results show that increasing the inner ring thickness can improve the stress reliability of the main bearing, while increasing parameters such as outer ring thickness, roller length, interference fit density between the bearing and stationary shaft, and friction coefficient can decrease the stress reliability of the bearing.

For three-row cylindrical roller bearings, relevant studies have also been conducted by scholars. Wang Mingwei et al. [15] calculated the distribution of contact loads of a three-row cylindrical roller bearing under compound loads. The research results show that appropriately increasing the diameter of the third row of rollers can reduce the contact stress and subsurface stress borne by the third row of rollers and the raceway surface. Jia Xianzhao et al. [16] proposed a method using ANSYS nCode DesignLife to predict the fatigue life of a three-row cylindrical roller bearing, providing simulation and visualization references for the life calculation of this type of bearing. Wang Jing [17] obtained the distribution law of contact stress along different generatrices of rolling elements for a three-row cylindrical roller bearing using the finite-element method, providing a feasible approach for the optimization design of a large wind turbine main bearing. Ge Haotian [18] used genetic algorithms to optimize the three-row cylindrical roller bearing of shield tunneling machine main shaft, obtaining a higher life value for the main bearing structure.

In addition to ADAMS and ANSYS, in Europe and the United States, Samcef Wind Turbines (SWT) are frequently employed for dynamic simulation of wind power transmission systems. Scholar Sanem Evren and others from Sabanci University conducted research utilizing SWT and achieved a series of outcomes [19,20].

In published studies, simulations of spindle bearing operating conditions often involve constant loads, with a scarcity of related results simulating variable loads using actual load spectrum data obtained from wind field measurements. Furthermore, the analysis of rigid-flexible coupled multi-body system dynamics is a result of the development and extension of multi-body system dynamics. Compared to pure rigid-body model analysis, virtual simulation results of rigid-flexible coupled models will be closer to actual operating conditions, providing more realistic and comprehensive information for optimizing system design [21]. In order to investigate the time-varying characteristics of contact force and the distribution of equivalent stress of a three-row cylindrical roller bearing under variable loads at different operating condition parameters, a method for simulating variable load conditions and establishing a rigid-flexible coupled model based on 20.5 years of wind data from an actual offshore wind field is proposed. This aims to ensure that the simulation results better match the actual offshore environment. The simulated dynamic response results of the main shaft bearing during operation will provide efficient guidance for the research and design of offshore wind turbine main shaft bearings. Subsequent sections will analyze the dynamic and transient responses of the three-row cylindrical roller bearing [22].

2. Model Architecture

Before proceeding with the formal model architecture, two assumptions are made at this stage:

(1)

Dry neo-Hertzian contacts are assumed between the rollers and raceway grooves. The expressions stated for these dry contacts should be referenced to Hertz or Harris. These are not the case in practice; the contacts are lubricated. So, the Hertzian expressions given for contact loads are approximate and only apply for highly loaded contacts (EHL).

(2)

The following analysis is conducted under the assumption of isothermal conditions. However, in practical scenarios, temperature variations are inevitable due to frictional heat generation. Nonetheless, since this paper focuses on the mechanical or dynamic response in terms of contact forces, deformations, and rotational speeds, the aforementioned assumption is made. It is worth noting that in future research, investigating thermal analysis of the main shaft bearing could be an important direction to explore. Furthermore, under high loads and shear, thermally lubricated roller bearings exhibit certain characteristic behaviors under transient conditions. As mentioned by Mohammadpour in the literature [23], these characteristics are crucial for gaining a more detailed understanding of the dynamic properties of roller bearings under transient conditions.

In addition, it is well-established that under highly loaded contact conditions, the damping effect of lubricants can be considered negligible and is only represented by friction. This has been mentioned and proven in numerous previous studies [24,25].

This paper primarily focuses on analyzing the dynamic performance response of wind turbine main shaft bearings under varying load conditions. Under varying load conditions, the contact load on the contact surface does not remain in a heavy load response state for extended periods. Time-varying contact loads will result in different damping and stiffness conversion efficiencies. Under highly loaded conditions fully, the damping constant will be 100% fully converted into stiffness. In lighter load conditions, this proportion will decrease, but thereinafter dynamic calculations will still be carried out according to the model.

2.1. Contact Mechanics Model of the Three-Row Cylindrical Roller Bearing

2.1.1. Roller Force Analysis

As depicted in Figure 1, the front thrust roller and rear thrust roller of the three-row cylindrical roller bearing experience an axial force denoted as F_a and an upsetting moment denoted as M, while the radial roller bears a radial force denoted as F_r. (“Front” indicates the proximal blade end, and “rear” indicates the distal blade end, consistently throughout the following text.)

2.1.2. Rigid Contact Mechanics Model

Let ψ denote the angular position of any radial roller in a three-row cylindrical roller bearing. The normal approach distance ${\epsilon }_{\theta }^3$ between the radial roller and the inner ring groove contact is given by:

$${\epsilon }_{\theta }^3={\epsilon }_r\cos \theta -\frac{1}{2}{\delta }_r$$

(1)

where ${\epsilon }_r$ represents radial displacement and ${\delta }_r$ represents radial clearance. The contact force of the radial roller at the random angular position on the inner raceway groove can be expressed as [26]:

$$F_{\theta }^3={\kappa }_r{\left({\epsilon }_r\cos \theta -\frac{1}{2}{\delta }_r\right)}^{\frac{10}{9}}$$

(2)

where ${\delta }_r$ is the load deformation constant between the radial roller and the radial groove.

In a three-row cylindrical roller bearing used in a wind turbine main shaft, the front and rear thrust rollers bear the axial force and upsetting moment transmitted by the impeller. The force acting on the inner ring of the bearing will result in certain displacement. At this point, the normal approach amount of the contact surface between the front and rear rollers and the groove can be expressed as:

$${\epsilon }_{\theta }^1=-\frac{1}{2}{d}_1\gamma \cos \theta -\frac{1}{2}{\delta }_a-{\epsilon }_a$$

(3)

$${\epsilon }_{\theta }^2=\frac{1}{2}{d}_2\gamma \cos \theta -\frac{1}{2}{\delta }_a+{\epsilon }_a$$

(4)

where ${\epsilon }_a$ represents axial displacement, ${\delta }_a$ denotes axial clearance, $d_1$ signifies the pitch circle diameter of the front thrust roller, $d_2$ represents the pitch circle diameter of the rear thrust roller, and $\gamma$ represents angular displacement.

The loads borne by the front thrust roller and the rear thrust roller at any angular position are as follows:

$$F_{\theta }^1={\kappa }_1{\left(-\frac{1}{2}{d}_1\gamma \cos \theta -\frac{1}{2}{\delta }_a-{\epsilon }_a\right)}^{\frac{10}{9}}$$

(5)

$$F_{\theta }^2={\kappa }_2{\left(\frac{1}{2}{d}_2\gamma \cos \theta -\frac{1}{2}{\delta }_a+{\epsilon }_a\right)}^{\frac{10}{9}}$$

(6)

where ${\kappa }_1$ represents the deformation constant of the front thrust roller and raceway load and ${\kappa }_2$ represents the deformation constant of the rear thrust roller and raceway load.

Thus, the balance equation set for three-row cylindrical roller bearing is established as:

$$\left\{\begin{array}{c}{F}_r-{\displaystyle \sum _{\theta =0}^{2\pi }{F}_{\theta }^3\cos \theta }=0\\ F_a-{\displaystyle \sum _{\theta =0}^{2\pi }\left(F_{\theta }^1-F_{\theta }^2\right)}=0\\ M-\frac{1}{2}\left[{\displaystyle \sum _{\theta =0}^{2\pi }\left(d_1{F}_{\theta }^1-d_2{F}_{\theta }^2\right)\cos \theta }\right]=0\end{array}\right.$$

(7)

By simultaneously solving the equations, the displacement and load of each row of rollers can be determined.

2.2. Theory of a Flexible Body

In traditional collision dynamics research, rigid-body collision dynamics models have been widely employed to analyze collision processes. However, with the advancement of science and technology, many objects exhibit significant flexible characteristics during collision processes, such as the flexible joints of robots, the flexible structures of spacecraft, a wind turbine main bearing, etc. [27,28]. In such cases, using rigid-body models for analysis can result in significant errors, or even lead to analysis failure. Therefore, investigating the effects of rigid-flexible coupled conditions on the collision dynamics of flexible multi-body systems becomes particularly important. Figure 2 illustrates the rigid-body movement, rotation, and deformation of a flexible body P_r (r = 1, 2, 3, …, n) in arbitrary directions in space.

The floating coordinate system used to describe the motion of the flexible body is denoted as O_e − e_xe_ye_z. Relative to the inertial coordinate system, this coordinate system is capable of undergoing certain motions within a defined range, such as spatial translation or spatial rotation. The inertial coordinate system is denoted as O_g − g_xg_yg_z. The point b₀ is located within P_r, and after the deformation of the flexible body, it can move to position b. At this point, the deformation of b₀ relative to b is represented as Δ, and it can be described using modal coordinates:

$$\Delta =J_k{\Phi }_b$$

(8)

where ${\Phi }_b$ is the deformation mode matrix of the flexible body, $J_k$ is the generalized coordinate used for the deformation of the flexible body, β₀ is the relative position of b₀ in the flexible body P_r, and ${\tau }_0$ is the relative position vector. Then, according to the vector operator, the operation can be obtained:

$${\tau }_b={\tau }_0+E\left({\rho }_0+{\Phi }_b{J}_k\right)$$

(9)

where E represents the relative rotation transformation matrix and ${\tau }_b$ represents the relative position vector of b₀ after deformation. The velocity vector and acceleration vector of point b can be obtained by differentiating Equation (9).

2.3. Equivalent Treatment of the Coupled Multi-Body Contact Model for a Three-Row Cylindrical Roller Bearing

The thrust cage, radial cage, and interaction between the inner and outer ring all have mutual effects in a three-row cylindrical roller bearing. Taking into account the rigid motion of these components, as well as the coupled relationship between the flexible deformation and the contact between the radial roller and thrust roller, an equivalent coupled multi-body contact model for a three-row cylindrical roller bearing is shown in Figure 3.

Among them, the inertial coordinate system O_g′ − g_x1 g_p2 g_z3 is located at the center of the three-row cylindrical roller bearing. O_j − j_x j_y j_z is located on the cage, while the body-fixed coordinate system O_i − i_x i_y i_z is located at the center of the inner ring wall thickness. The contact interactions between various components in the three-row cylindrical roller bearing can be defined using the function F_impact:

$$F_{impact}=\left\{\begin{array}{l}0,\begin{array}{c} \end{array}\sigma >{\sigma }_0\\ k{\left({\sigma }_0-\sigma \right)}^{\xi }-spline\cdot c\dot{\sigma },\begin{array}{c} \end{array}\sigma \le {\sigma }_0\end{array}\right.$$

(10)

In the equation, F_impact represents the contact collision force between parts, σ is the actual distance between the two parts, σ₀ is the critical distance of collision, $\dot{\sigma }$ is the rate of change of distance between the two parts over time, ξ denotes the contact index, k represents the contact stiffness, and c denotes the damping factor.

As shown in Figure 4, the nonlinear rigid-flexible coupled contact model of the main shaft thrust roller is depicted. The establishment of the thrust roller contact model is similar to that of the radial roller contact model, and further elaboration on this aspect is omitted here.

The three-row cylindrical roller bearing is lubricated with grease. When analyzing the contact forces between the roller and inner/outer ring, as well as the collision forces on the cage, the influence of the oil film thickness between the roller and the inner/outer raceway on the axial and radial stiffness and damping of the bearing cannot be ignored. As shown in Figure 5, an equivalent consideration will be made for the axial and radial contact stiffness and damping of the three-row cylindrical roller bearing under grease lubrication.

In Figure 5, K_e represents the radial equivalent contact stiffness, which is calculated as follows [29]:

$$K_e=\frac{1}{\frac{1}{K_{oil}}+\frac{1}{K_r}}=\frac{1}{0.13C{F}_r{ }^{-0.13}+B\ln F_r+A+B}$$

(11)

$$A=\frac{8.16}{E^{\prime }Zl}\left[\frac{1}{\pi }\ln 0.39E^{\prime }Zl(R_1+r)+1.15-0.5\ln \frac{8.16r{R}_2}{E^{\prime }(R_2-r)Zl}\right]$$

(12)

$$B=-6.68\frac{1}{E^{\prime }Zl}$$

(13)

$$C=\left\{{\left(\frac{{\eta }_0\omega l_i\sin \alpha }{2}\right)}^{0.7}{\left[r+r{\left(\frac{r}{R_1+r}\right)}^2\right]}^{0.43}+{\left({\eta }_0\omega l_i\sin \alpha \right)}^{0.7}{\left[r-r{\left(\frac{r}{R_1+r}\right)}^2\right]}^{0.43}\right\}\cdot 2.65{\delta }^{0.54}{\left(\frac{E}{1-{\mu }^2}\right)}^{-0.03}{\left(\frac{4.08}{Zl\cos \alpha }\right)}^{-0.13}$$

(14)

K_oil and K_r represent the oil film stiffness and contact stiffness, respectively; C_oil and C_r denote the oil film damping and contact damping, respectively; G′ denotes the storage module; G″ denotes the loss module; F_r stands for the radial force endured by the three-row cylindrical roller bearing; E′ represents the elastic modulus; Z is the number of rollers; l is the effective contact length of the roller; R₁ is the radius of the inner raceway of the bearing; R₂ is the radius of the outer raceway of the bearing; r is the radius of the roller; δ is the elastic approach amount; μ is the Poisson’s ratio of the material; α is the contact angle; ω is the angular velocity; and η₀ denotes the dynamic viscosity of the grease.

The calculation method for the equivalent damping C_e is shown in Equation (5) as described in reference [10]:

$$C_e=\frac{1}{\frac{1}{C_1}+\frac{1}{C_2}}=\frac{C_1{C}_2}{C_1+C_2}$$

(15)

wherein the damping C₁ and C₂ between the roller and the inner and outer raceway, respectively, are:

$$C_1=\frac{27.4l^{0.805}{Z}^{0.805}{F}_r{ }^{0.195}{R}_{x1}^{1.5}}{{(R_1+\frac{r}{R_1+r})}^{1.05}{(1-\frac{r}{R_1+r})}^{1.695}{(1+\frac{r}{R_1+r})}^{1.05}{E}^{\prime }{ }^{-0.045}{\lambda }^{0.81}{\eta }_0{ }^{0.05}{n}_i^{1.05}{r}^{0.645}}$$

(16)

$$C_2=\frac{27.4R_{x2}^{1.5}{l}^{0.805}{Z}^{0.805}{F}_r{ }^{0.195}}{{(R_1+\frac{r}{R_1+r})}^{1.05}{(1-\frac{r}{R_1+r})}^{1.695}{(1+\frac{r}{R_1+r})}^{1.05}{E}^{\prime }{ }^{-0.045}{\lambda }^{0.81}{\eta }_0{ }^{0.05}{n}_i^{1.05}{r}^{0.645}}$$

(17)

where n_i represents the inner ring or inner race speed, λ denotes the lubricating oil pressure viscosity coefficient, and R_x is the equivalent curvature radius at the contact point.

This study focused solely on the modeling and parameter determination of the rigid-flexible coupled multi-body contact dynamics simulation for a wind turbine main bearing. The establishment of the model in the virtual prototype and the determination of the required parameters should be carried out according to the principles described in this section. The analysis was conducted using a three-row cylindrical roller bearing manufactured by Luoyang Xinqianglian Slewing Bearings Co., Ltd. (Luoyang, China), with the model number 130.60.2160.03-F22. The main structural parameters of the three-row cylindrical roller bearing are shown in Table 1, and the model is depicted in Figure 6.

The specific parameter values and material values for the spindle bearing can be found in Table 1 and Table 2, respectively.

3. Three-Row Cylindrical Roller Bearing Rigid-Flexible Coupled Multi-Body Contact Dynamics Simulation Model

3.1. Establishment of a Rigid-Flexible Coupled Simulation Model

This study selected the three-row cylindrical roller main bearing of a 12 MW offshore wind turbine as the research object. Based on the principles described in Section 1, a rigid-flexible coupled multi-body contact dynamic simulation model of the main bearing was established using virtual prototype technology. The elastic deformation of the inner and outer ring, thrust cage, and radial cage was taken into account.

As shown in Figure 7, modal analysis was conducted using the ANSYS 2020 R2 Workbench platform to flexibilize the thrust bearing, radial bearing, and inner and outer ring. The *.MNF files were then imported into ADAMS View 2019 for simulated assembly. To simplify the computational workload during simulation, the first and second inner ring were treated integrally. The first six modes represent rigid-body degrees of freedom, which are deactivated by the system by default. The natural frequencies obtained through simulation calculations are listed in Table 3. It was observed during the simulation process, as shown in Table 3, that the influence of higher-order modes on the results was relatively small. Therefore, the modal truncation was set to 20 modes.

The equivalent physical model of the main transmission chain system, constructed based on the spatial relationship between the main spindle bearing and various transmission components, is illustrated in Figure 8.

According to the equilibrium conditions during wind turbine operation, the force and moment balance equations shown in Equation (18) can be derived.

$$\left\{\begin{array}{c}{\displaystyle \sum F_x=F_1-F_{g3}=0}\\ {\displaystyle \sum F_y=F_2+G-F_{g4}+F_{g5}=0}\\ {\displaystyle \sum M_{CZ}=F_2\times H_1-F_{g5}\times \left(H_2+H_3\right)-G\times H_2=0}\end{array}\right.$$

(18)

3.2. Simulation Environment Configuration

Wind data collected from a 12 MW offshore wind turbine at a certain wind farm for 20.5 years were utilized. However, due to factors such as time cost control and manageable risks, the simulation duration in this study cannot synchronize with the 20.5-year data from the wind farm. In order to simulate the dynamic response of the three-row cylindrical roller bearing after actual loading conditions more realistically, an algorithm was utilized to optimize and integrate the entire process data into 32 sets of data. This resulted in obtaining 32 sets of transient M_x, M_y, M_z, F_x, F_y, and F_z data for the main spindle’s three-row cylindrical roller bearing, along with the respective time proportions for each data set.

As illustrated in Figure 9 and Figure 10, the simulation time for the ADAMS virtual prototype was set to 8 s, with 1000 steps. Based on the time proportion of each dataset, instantaneous wind force variation time points within the 8 s simulation period were identified. Six spline function curves were then created to represent these variations. Each data unit, spanning 32 sets of mutations within the 8 s simulation duration, generated general force vectors. The Cubic Fitting Method (CUBSPL) function was employed to import the six data units, which were applied to the inner ring mass points of the three-row cylindrical roller bearing to simulate transient wind force effects. The aforementioned varying loads imposed on the main spindle bearing served as a fundamental load condition throughout all subsequent simulations, aiming to render the simulated scenarios closer to actual operational conditions.

Based on material Poisson’s ratio, Young’s modulus, and Contact Characteristics, a total of 1447 contacts were established for the components in the model. In the “Contact Characteristics”, the EXponent was set to 1.5, damping to 70, stiffness to 1.0 × 10⁵, and Dmax to 0.1. Coulomb friction was enabled, with static translational velocity set to 10, frictional translational velocity to 100, static friction coefficient to 0.2, and dynamic friction coefficient to 0.1. Here, the “cmd” command stream incorporates relevant oil film parameters for simulation environment setup, aiming to streamline the calculations. The outer ring of the three-row cylindrical roller bearing was set as fixed, while the inner ring was driven at a constant speed. To ensure the simulation results closely approximated real-world conditions, a rotational pair was configured at the inner ring mass points. The rotational pair object was set between the inner ring and distant blocks, with all six degrees of freedom of the blocks constrained. Consequently, when the inner ring is driven by the power source, it can freely descend under gravity onto the roller, thereby enhancing the mechanical contact realism between the components.

The following section describes a comparative analysis of simulation results conducted between the rigid-flexible coupled and pure rigid-body models of the three-row cylindrical roller bearing. The aim is to investigate the dynamic response characteristics of the bearing’s different contact surfaces under varying load conditions that are more representative of real-world scenarios. This analysis will provide guidance for the design and manufacturing of offshore wind turbine main bearing.

4. Dynamic Response Analysis of Three-Row Cylindrical Roller Bearing

In actual wind farms, under constant speed driving, the spindle bearings of wind turbine units do not exhibit significant fluctuations in component speeds during normal operation, or the speed should remain linearly time-varying over a relatively extended period (this paper did not take into account situations such as “Emergency Stop and Crow Bar events” that could result in the sudden cessation of service of the spindle bearings). However, when using a pure rigid-body model for simulation, the resulting speeds and collision forces of the components within the main bearing exhibit sensitive responses and significant data fluctuations that do not align with reality. Therefore, in order to obtain more accurate and reliable simulation results, this study aims to compare and explore the dynamic response results of a three-row cylindrical roller bearing under a variable load condition with a constant driving speed of 30 r/min using both a rigid model and a rigid-flexible coupled model.

4.1. Response Comparison of Rigid-Flexible Coupled Models

Figure 11, Figure 12 and Figure 13 illustrate the comparison of average rotational speed and acceleration between the rigid model and the rigid-flexible coupled model (with both inner and outer ring as well as the cage being flexible) for the radial roller, front thrust roller, and rear thrust roller of the three-row cylindrical roller bearing. The average rotational acceleration of the roller can reflect the extent of impact from various high-value collision forces during operation. The lower the numerical value, the smaller the speed fluctuation of the component, which is more in line with actual working conditions.

Taking the radial roller as an example and observing Figure 11a,b, the average rotational acceleration of the rigid model’s radial roller is 27,592.855 deg/s², with a maximum value of 1,740,000 deg/s². In the rigid-flexible coupled model, the average rotational acceleration of the radial roller is 24,329.844 deg/s², with maximum accelerations of 1,245,884 deg/s², all significantly lower than those of the rigid model. Clearly, the acceleration trend of the rigid-flexible coupled model is smoother. Moreover, comparing the two figures, it is evident that the rigid-flexible coupled model exhibits smaller fluctuations in its velocity–time curve and is closer to real working conditions compared to the rigid model.

Furthermore, by observing Figure 11a,b, it is evident that there is a distinct periodic behavior in the velocity of individual radial rollers during the operation of the bearing. For the rigid radial roller, the velocity decreases first within 1 s–3 s before rising again, and then maintains a relatively constant average speed of 530.540 r/min during 3–5 s, completing one cycle. Analysis suggests that the roller not only rotates and revolves around the inner raceway within the bearing but also experiences periodic gravitational work or overcomes gravitational work during revolution due to the presence of gravity in the spatial field. The energy loss or gain associated with this effect couples with the rotational speed of the roller, resulting in periodic variations in their rotational speed. As shown in Figure 11b, the radial roller in the rigid-flexible coupled model also exhibits the aforementioned periodic pattern. However, the descending or ascending trend in their velocity response curves is more gradual, and the lowest point at 200 r/min is evidently higher than the speed of 0 r/min for the radial roller in the rigid model (representing the occurrence of sticking of roller in the raceway), thus better reflecting the operating conditions of actual roller motion.

As shown in Figure 12 and Figure 13, the velocity curves of the thrust rollers in the rigid model exhibit multiple instances of abrupt response, while the velocity curves of the thrust rollers in the rigid-flexible coupled model transition smoothly. The same variation patterns in average rotational acceleration for the two models are also reproduced in the acceleration response of the front and rear thrust roller. It can be inferred that the greatest advantage of incorporating flexibility and coupled treatment in the simulation of mechanical components lies in the fact that instant contact impacts between components do not result in sensitive or excessive responses in key representative parameters. This will make the simulation results more closely resemble actual operating conditions.

Observing the initial position of the curves in Figure 11a,b, during the 0–0.5 s spindle bearing startup phase, both acceleration and velocity exhibit significant fluctuations due to the sudden change in driving torque. This situation is unavoidable in both actual and simulated rigid or rigid-flexible coupled models. During the startup phase, the substantial fluctuation in roller velocity is primarily attributed to the circumferential collisions between the cages. Moreover, due to environmental conditions such as wind force and unit gravity, or structural factors such as roller manufacturing errors and clearances between the cage and roller, the contact impacts between the cage and roller are uneven. For a three-row cylindrical roller, during startup or phases with frequent spindle speed variations, both radial and thrust rollers are prone to cause uneven loading on the cage, leading to damage or fatigue fracture. By flexibly treating key components of the spindle’s three-row cylindrical roller bearing, a more realistic dynamic contact response of the spindle bearing under variable load conditions has been achieved, facilitating subsequent analysis.

4.2. Model Verification

Based on the kinematic relationships among the components of the bearing, the kinematic characteristics between bearing parts can be derived. According to reference [30], it can be inferred that when the roller undergoes pure rolling on the raceway, kinematic parameters such as thrust and radial roller rotational speed can be calculated using the average values obtained earlier as simulation data and comparing them with the calculated results, as shown in Table 4. It can be observed from Table 4 that the kinematic simulation results of the modeled system have relatively small errors compared to theoretical calculations. Specifically, the errors of the rigid model are 1.67 to 2.96 times larger than those of the rigid-flexible coupled model overall. The rigid-flexible coupled model used meets the basic requirements for kinematic characteristic analysis and has been validated.

4.3. Bearing Contact Force Analysis

The following analysis investigated the impact of various operational parameters (such as driving speed and friction coefficient) on the collision force between roller and the cage, as well as the contact force response between the roller and the inner ring, under variable load conditions. The variable loading was applied from the front end towards the rear end.

4.3.1. Different Driving Speeds

As per Table 5, based on the actual speed range of offshore wind turbines, constant speed intervals of 20, 25, and 30 revolutions per minute (r/min) were set to investigate the varying impact trends of different spindle drive speeds on the internal forces of the rigid-flexible coupled three-row cylindrical roller bearing. Radial roller-cage collision force, front thrust roller-cage collision force, rear thrust roller-cage collision force, and radial roller-inner ring contact force were selected as key indicative parameters for analysis. In Table 5, the acquisition of friction coefficients was based on the values provided by Luoyang Xinqianglian Slewing Bearings Co., Ltd. for the internal component mating of their manufactured three-row cylindrical roller bearings. The values for both dynamic and static friction coefficients represent an optional range, which the authors rounded, and the coefficient values were set in a gradient manner in the design drawings. These values were then utilized as input boundary conditions for the simulation in both this section and the following one.

As shown in Figure 14, the average force exerted on the rear thrust roller by the cage significantly increases as the speed increases from 20 r/min to 30 r/min. Within the simulation duration of 8 s, the average force on the roller-cage interface at 30 r/min is 67.8 N, approximately 1.44 times greater than the 46.8 N observed at 20 r/min. The maximum collision force occurs at a speed of 20 r/min, reaching approximately 3400 N.

As depicted in Figure 15, the increasing trend of average force on the front thrust roller and the cage is similar to that of the rear thrust roller, both significantly rising with increasing rotational speed. Under the simulation conditions of 30 r/min within 8 s, the average force exerted on the roller and the cage is 108 N, approximately 1.94 times the 55.6 N recorded at 20 r/min. The maximum collision force between the front thrust roller and the front thrust cage is approximately 2000 N, also occurring at a rotational speed of 20 r/min. A comparison between Figure 14 and Figure 15 reveals that the average force exerted by the front thrust roller on the cage is approximately 1.19 to 1.59 times greater than that exerted by the rear thrust roller.

Observing the initial bearing startup from 0 s to 1 s, there is a period of “blank stage” where the rear thrust roller and cage experience no collision forces at various speeds. Visualizing the entire simulation process through animation in simulation software reveals that during the initial startup, the rear thrust roller and cage remain stationary for approximately 0.9 s and do not rotate normally with the inner ring. In contrast, the front thrust roller and cage are in normal service during this period. Analysis suggests that this situation is not only related to the initial variable load values set in the accompanying table but also to the differences in the forces acting on the rear and front thrust rollers during startup, ultimately resulting in the observed “blank stage” period depicted in Figure 14. Under variable load conditions, the rear thrust roller experiences a different compression from the flexible inner ring compared to the front thrust roller. The front thrust roller bears a greater variable load, resulting in an overall higher force between the front roller and its corresponding cage. A more detailed and comprehensive analysis of these forces will be presented in the next section.

As shown in Figure 16, the trend of average force on the radial roller and radial cage is the opposite to that of the thrust roller. The average force decreases as the rotational speed increases. Within a simulated time of 8 s, under conditions of 20 r/min, the average force on the roller and cage is 52.7 N, approximately 1.56 times that of 33.7 N under 30 r/min conditions. The maximum collision force between the radial roller and cage occurs at the initial stage of bearing startup. This intense collision force during the startup stage persists across different speeds. Furthermore, as inferred from Figure 16 regarding the bearing startup phase, the maximum collision force is largely unaffected by variations in bearing speed.

The analysis suggests that the different variations in collision forces between the thrust roller and radial roller with their respective cages are due to the unique geometric structure of the three-row cylindrical roller bearing. In rotating machinery, changes in speed have the greatest impact on the centrifugal force experienced by individual components. It is evident that as the speed increases, the centrifugal force on the radial roller also increases. Under variable load conditions, the contact between the radial roller and inner ring is somewhat reduced, resulting in a decrease in the effect of variable loads and affecting the contact force with the radial cage. The response of the thrust roller is exactly the opposite: the higher the speed, the greater the centrifugal force, leading to stronger contact force between the thrust roller and thrust cage. Furthermore, the wind force acts on the main bearing from the near-blade end to the far blade end, i.e., “from front to rear”. To ensure the simulation in this study aligned closely with reality, the direction of variable loads was also set as “from front to rear”. Based on these operating conditions, the thrust roller and cages at the front and rear ends experienced different loads. Overall, the front thrust cage experienced greater forces. Therefore, when producing a three-row cylindrical roller bearing for a wind turbine main bearing, the safety and reliability of the cage at the near-blade end should be fully considered.

As shown in Figure 17, the contact force between the inner ring and the radial roller exhibits periodic variations due to the influence of environmental gravity. Observing the time-varying curves of the contact force under three different rotational speeds, it is easy to deduce that lower speeds result in longer contact durations between the radial roller and the inner ring. The graph intuitively demonstrates that the curve period at 20 r/min is noticeably longer than that at 30 r/min, indirectly validating the model’s accuracy. With increasing rotational speed, the average force on the inner ring from the radial roller significantly decreases. Within the 8 s simulation period, the average force on the roller against the cage at 20 r/min is 5533.63 N, approximately 1.09 times that of the 5048.09 N force at 30 r/min. This confirms the conclusion mentioned earlier regarding the significant influence of centrifugal force on the changing trend of forces between the cage and the roller.

4.3.2. Different Contact Friction Coefficients

As shown in Table 6, the ranges of dynamic and static friction coefficients were set as four groups: 0.025 and 0.05; 0.05 and 0.1; 0.075 and 0.15; 0.1 and 0.2. The aim was to explore the influence of gradient-changing friction coefficients between components on the variation trend of internal forces in the rigid-flexible coupled three-row cylindrical roller bearing. The grouping of friction coefficients into four sets with gradient variations aligns with practical operating conditions and facilitates the smooth operation of debugging the simulation model. The collision forces between the radial roller and the radial cage, the front thrust roller and the cage, and the rear thrust roller and the cage were selected as key parameters for analysis.

As shown in Figure 18, Figure 19 and Figure 20, the average forces on the radial roller and radial cage, as well as the front and rear thrust roller and thrust cages, exhibit a similar increasing trend, which decreases as the dynamic and static friction coefficients decrease. Specifically, the average force on the radial roller decreases by 56.19%, the average force on the front thrust roller decreases by 76.41%, and the average force on the rear thrust roller decreases by 50.08%. Clearly, whether in the radial or axial direction, the variation in friction coefficient has a significant impact on the collision forces between the roller and the cages.

Analysis suggests that the force exerted by the roller on the cage still originates from the force exerted by the inner ring on the roller. According to the friction force formula f = μF_n, it is easy to see that when F_n is constant or consistent, the dynamic friction coefficient μ directly affects the power driving the inner ring to rotate the roller, thus indirectly causing changes in the force between the roller and the cage. A comparison between Figure 19 and Figure 20 reveals that the average force on the front roller decreases more significantly under variable load conditions compared to the rear roller. This pattern also reflects the sensitivity of the force response between the front roller and the cage to operating parameters and structural parameters. Additionally, under conditions of variable loads and low speeds, gentler friction driving forces optimize the ability of components to resist complex variable forces, significantly reducing the probability of damage and failure of internal bearing components, thus enhancing the reliability of the main bearing in offshore wind turbines. Therefore, in the actual production of a three-row cylindrical roller bearing for offshore wind turbines, a method for reducing the internal contact friction coefficient should be fully incorporated into the design process. The current mainstream and more optimal methods of friction reduction primarily include [31]:

(1)

Coatings: Different types of coatings are applied to the surfaces of friction pairs, such as graphite-like carbon (GLC) and diamond-like carbon (DLC). GLC exhibits excellent self-lubricating properties, a low friction coefficient, and corrosion resistance. DLC, on the other hand, offers outstanding characteristics such as high hardness, high load-bearing capacity, superior wear resistance, and excellent chemical inertness.

(2)

Improved lubrication: The addition of rare earth elements to lubricants can enhance their performance in terms of extreme pressure and anti-friction properties, such as small amounts of nano CeF₃, LaF₃, and so forth.

(3)

Roller profile optimization: Optimization algorithms can be utilized for roller profile modification, such as particle swarm optimization, genetic algorithms, etc.

(4)

Rational design of mechanical structures incorporating buffering and damping mechanisms internally.

4.4. Bearing Transient Response Analysis

As shown in Figure 21, the model is imported into finite-element simulation software to conduct transient structural analysis, aiming to obtain the stress distribution of the wind turbine main spindle bearing and the variation of bearing stress under different driving speeds. The simulation settings should closely resemble real-world operating conditions. The outer ring of the main spindle bearing is fixed to the ground, while the inner ring is set as a rotational pair with transient structural loads applied. These loads include rotational speed (30 r/min), standard Earth gravity, bearing loads, and torque. Due to the presence of nearly a thousand contact points in the wind turbine main shaft bearing and its 1:1 scale interaction modeling, optimizations were made to the roller contacts and local refinements were applied to critical areas of the overall mesh to avoid high simulation time costs and insufficient computational power. The transient structural simulation results are shown in Figure 21.

Figure 22 and Figure 23 depict the transient structural equivalent stress simulation results of the three-row cylindrical roller bearing rollers and cages. The red marking indicates the far-wind turbine hub end, and it is evident from both figures that the force distribution on the bearing is uneven. The maximum stress value for the front thrust rollers (1.5197 × 10⁸ Pa) and front thrust cages (1.1756 × 10⁸ Pa) is significantly higher than that for the radial rollers, cage, and rear thrust rollers and cage. Furthermore, both rollers and cages exhibit a linear variation in stress values, following the pattern: front > radial > rear. This pattern also reflects the trend of contact collision forces between rollers and cages discussed in the previous section’s dynamic simulation, which aligns with the actual operating conditions of the wind turbine main bearing and indirectly validates the model.

Figure 24 depicts the stress distribution cloud map of the cage, rollers, and inner ring concealed beneath the outer ring of the bearing. It can be observed that the maximum stress value (1.6368 × 10⁸ Pa) of the three-row cylindrical roller bearing is concentrated at the geometric edge of the thrust rollers. Additionally, significant stress distribution is also present at the geometric edge of the cage. This indicates that the edge stress of the rollers should be taken into full consideration in the design of the main bearing. Employing relevant algorithms to shape the rollers is one effective approach to prolonging the lifespan of a megawatt wind turbine main bearing.

Figure 25 represents the stress distribution cloud map of the outer ring. It can be observed that the contact stress on the outer ring primarily occurs at the positions where the radial rollers, thrust rollers, radial cage, and thrust cages make contact with the outer ring. From the stress distribution in the figure, it can be inferred that the average stress values in the contact area between the outer ring and the front thrust rollers range from 8.3256 × 10⁶ Pa to 1.0119 × 10⁷ Pa. The contact force level between the outer ring and the rear thrust rollers is even lower. As the outer ring is spatially fixed, the torque from the wind turbine hub is transmitted to the inner ring, compressing components such as the inner ring, front rollers, and cage against the outer ring. This causes the rear rollers and cage to be somewhat distanced from the raceway, leading to inconsistent stresses on the raceway of the outer ring.

In order to investigate the variation of the maximum contact stress of a three-row cylindrical roller bearing with an increasing gradient of driving speed, the “transient structural loads” were set with increasing speeds (10, 20, 30 r/min). The results, as shown in Figure 26, indicate that the maximum contact stress of the three-row cylindrical roller bearing increases with the increasing speed. Under the operating condition of 30 r/min, the maximum stress value of the bearing reaches 1.6368 × 10⁸ Pa, which is higher than the maximum stress values at 20 r/min and 10 r/min.

As analyzed earlier, with the increase in speed, both the force on the thrust roller and the force on the cage increase, both in terms of magnitude and average value. Stress refers to the distribution of internal or surface forces within an object caused by external applied forces, describing the object’s degree of response to external forces. Therefore, the variation in stress values here also reflects to some extent the variation pattern of the forces mentioned earlier.

5. Conclusions

This paper addresses the limitations and one-sidedness of the simulation results of the rigid three-row cylindrical roller bearing for offshore wind turbine main shaft under constant-load conditions. To overcome these issues, a bearing contact mechanics model and a flexible body model were established, and a rigid-flexible coupled treatment was applied to the inner and outer ring as well as the cage of the bearing. Based on load conditions closer to reality, the theoretical velocities and simulated velocities, as well as the acceleration responses of the rigid model and the rigid-flexible coupled model, were compared and analyzed to validate the proposed models. Dynamic response analysis and transient response analysis were conducted on the rigid-flexible coupled model of the three-row cylindrical roller bearing, focusing on elucidating the contact force characteristics and stress distribution patterns of different contact surfaces within the bearing.

This study innovatively applied variable load conditions as a simulation global environment and introduced a novel “distant blocks” contact pair configuration. These innovations led to a more realistic mechanical contact between components and simulation results that closely resemble real-world scenarios. Flexible treatments were applied to all components of the main shaft bearing except for the rollers, and a rigid-flexible coupling operation was performed on the model, further enhancing the accuracy of the main shaft bearing’s dynamic response. Lastly, the study investigated the effects of gradient changes in operational and structural parameters on the main shaft bearing, providing efficient theoretical guidance for the application of three-row cylindrical roller bearings as main shaft bearings in existing offshore wind turbine designs. The conclusions drawn from this study are as follows:

(1)

Based on actual load spectrum data collected from offshore wind turbines, a method for simulating variable load conditions acting on the bearing center is proposed, enabling the simulation to closely resemble real operating conditions.

(2)

Comparing the rigid model with the rigid-flexible coupled model, the acceleration trends of the thrust and radial cages are more stable in the latter, with smaller velocity fluctuations. The overall rotor speed error of the rigid model is 1.67 to 3.76 times greater than that of the rigid-flexible coupled model. Using the rigid-flexible coupled model for dynamic response analysis is a superior analytical approach.

(3)

The average force on the thrust rollers and cage increases with increasing driving speed, while the average force on the radial rollers and cage decreases with increasing driving speed. The average force on the inner ring and radial rollers decreases with increasing driving speed.

(4)

The average force on the cage near the wind blade tip is approximately 1.19 to 1.59 times greater than that at the far end. Stress distribution maps of the bearing also indicate higher stress values in the front cage and rollers. When producing a three-row cylindrical roller bearing for a wind turbine main shaft, the safety and reliability of components near the wind blade tip should be fully considered.

(5)

The average force on the rollers and cage significantly decreases with a decrease in the internal friction coefficient of the bearing, by 50.08% to 76.41%. In the design of a three-row cylindrical roller bearing, methods such as coatings, lubrication, and roller modification should be employed to reduce the friction coefficient as much as possible.

(6)

Both rollers and cage exhibit a linear change in stress values, with the specific order being: front > radial > rear. The maximum stress value of the bearing increases with increasing driving speed.