Calculation method for the static carrying curve of double-row different-diameter ball slewing bearings

Abstract

A method for calculating the static carrying curve of a double-row different-diameter ball slewing bearing was proposed. The relationship between the internal maximum rolling element load of each row and the combined external axial load and tilting moment load of the slewing bearing was established using the deformation compatibility and force equilibrium conditions. The rolling element load distribution range parameters of the main and auxiliary raceways of the double-row different-diameter ball slewing bearing were used as input invariables, and the corresponding external load combinations of the axial and tilting moment loads of the slewing bearing were obtained. These external load combinations were plotted in the coordinate system to obtain the static carrying curve of the slewing bearing. The obtained static carrying curve was compared with that calculated using the finite element method for verification. Finally, the influences of detailed design parameters such as the raceway groove radius coefficient, raceway contact angle, and rolling element diameter on the carrying capacity of the double-row different-diameter ball slewing bearing were analyzed based on the carrying curves. As the groove radius coefficient increases from 0.515 to 0.530, or the contact angle increases from 50° to 65°, the carrying capacity of the slewing bearing decreases. As the rolling element diameter increases from 0.90 times the initial diameter to 1.05 times the initial diameter, the carrying capacity of the slewing bearing increases.

Introduction

Slewing bearings are key components of many mechanical systems and are widely used in heavy machinery, such as cranes, excavators, stackers, and shield machines. As a special rolling bearing with a large size that carries combined heavy loads, slewing bearings have attracted considerable attention in recent years. Some researchers have focused on modeling methods for slewing bearings and used calculation models to analyze the factors affecting the load-carrying performance. He et al.^1,2 investigated the effects of the mesh size of the finite element model and the material parameters on the analysis of the maximum contact load, deformation, and load distribution of a single-row four-point contact ball slewing bearing. Zhang et al.³ established elastic mechanics-based numerical models for a double-row four-point contact ball slewing bearing with rigid and flexible rings. The effects of the initial contact angle, coefficient of the groove curvature radius, and clearance on the maximum contact force were analyzed. Kania et al.⁴ analyzed the contact phenomena in a single-row four-point contact ball slewing bearing and found that an increase in the contact angle led to displacement of the contact zone toward the raceway edge. Modification of the raceway profile by introducing a transition curve was discussed to eliminate the pressure concentration at the raceway edge. Spiewak⁵ proposed a two-stage analysis strategy for a double-row four-point ball slewing bearing, which included analysis of the local ball–raceway contact and global load distribution in terms of the static carrying capacity. The calculations were performed using a finite element model. Yu et al.⁶ studied the load distribution of a single-row four-point contact ball slewing bearing. The retainer force was considered in the force equilibrium of the balls. The results showed that the retainer reduced the number of contact balls in the slewing bearing; thus, the maximum contact load increased compared to the case without consideration of the retainer. Chen et al.⁷ studied the effects of the supporting structure and bolt connection on the fatigue life and carrying capacity of a double-row four-point contact ball slewing bearing using finite element models. Potocnik et al.⁸ presented a carrying capacity calculation method for a double-row four-point contact ball slewing bearing based on the vector description method for the bearing geometry. The influences of multirow geometry and a predefined irregular geometry of the bearing on the raceway contact load distribution was analyzed. Li and Jiang⁹ proposed a method for verifying the strength of a three-row roller slewing bearing using a mixed finite element model. Both the structural strength of the rings and contact strength of the raceway were analyzed. Li¹⁰ analyzed the effects of detailed design parameters such as the axial clearance, contact angle, and roller semi-cone angle on the carrying capacity of a double-row tapered roller slewing bearing by solving a system of equilibrium equations.

Some researchers have focused on the influence of the machining error of slewing bearings on the load-carrying capacity. Heras et al.^11,12 proposed a calculation method for the load distribution in a single-row four-point contact ball slewing bearing considering the ball preload, manufacturing errors, and ring flexibility. The mechanical model was built by formulating and minimizing the potential energy of the bearing to consider ring deformations. Aithal et al.¹³ studied the influence of manufacturing errors on the rolling element load distribution of a single-row four-point contact ball slewing bearing using the finite element method. It was observed that the size error of the ball and waviness error of the raceway were the two factors influencing the load distribution. Starvin et al.¹⁴ analyzed the effect of manufacturing tolerances on the load-carrying capacity of large-diameter ball bearings based on a finite element model. A Monte Carlo simulation was used to allocate tolerances to the balls and raceways. Gao et al.¹⁵ studied the effects of raceway geometry parameters on the carrying capability and service life of a single-row four-point contact ball slewing bearing by solving a set of nonlinear equations. Lostado et al.¹⁶ built a finite element model of a double-row tapered roller bearing to compute the contact stresses over bearing raceways, and a theoretical model and contact pressure sensors were used to adjust and validate the model. Lostado et al.¹⁷ combined regression trees with a finite element model to predict contact pressures at any point on an axle with a preloaded double-row tapered roller bearing for any load condition on the wheel. Daidie et al.¹⁸ presented a simplified three-dimensional finite element analysis of the load distribution and contact angle variation in a single-row four-point contact ball slewing bearing; the rolling elements were modeled as nonlinear traction springs.

Researchers have also focused on the analysis, calculation, and experimental testing of slewing bearings. Liu et al.¹⁹ built a finite element model of a slewing bearing to estimate the load distribution. Static loading experiments were performed by mounting strain gauges on the inside circumference of the inner ring to determine the load distribution. The numerical results and experimental data were compared. Chen et al.²⁰ proposed a method for measuring the contact force in a single-row four-point contact ball slewing bearing. The deformation under the ball–raceway contact and displacement of the end face of the load-bearing ring were measured. The ball–raceway contact force was determined using the finite element method based on the measured data. Hrcek et al.²¹ established a finite element model of a single-row four-point contact ball slewing bearing to determine its axial stiffness characteristics. The axial stiffness was experimentally measured to verify the results of the finite element analysis. The mathematical relationship between the deformation constant and geometric parameters such as the pitch diameter, rolling element diameter, and contact angle was determined using a genetic algorithm.

Correct type selection is a prerequisite to ensure the safe and reliable application of slewing bearings, and the carrying capacity is an important indicator for the type selection. Calculation of the carrying curve of a slewing bearing has become a popular research topic in recent years. Aguirrebeitia et al.²² presented a calculation method for the acceptance surface of a single-row four-point contact slewing bearing based on Sjovall and Rumbarger's equations. Aguirrebeitia et al.²³ established a geometric interference model of a three-row roller slewing bearing. The acceptance curve in the load space was derived using the model. Glodez et al.²⁴ established static equilibrium equations for a single-row four-point contact ball slewing bearing using a vector-based approach. The static carrying capacity curves of the slewing bearings with and without clearance were calculated. Abasolo et al.²⁵ developed selection curves for a single-row four-point contact ball slewing bearing. The curves considered two possible static failure types: ball–raceway contact failure and bolted joint failure. Potocnik et al.⁸ calculated the static carrying capacity curve of a double-row four-point contact ball slewing bearing considering irregular geometry. The effects of the clearance and predefined irregular geometry of the rings on the static capacity of the bearing were discussed. Li²⁶ calculated the static carrying capacity surface of a double-row four-point contact ball slewing bearing by solving a system of equilibrium equations. The effects of detailed design parameters such as the clearance, groove curvature, and contact angle on the carrying capacity of the slewing bearing were analyzed. Aguirrebeitia et al.²⁷ developed a multiparametric finite element model for a single-row four-point contact ball slewing bearing. The acceptance surface of the slewing bearing was calculated using the model. Kania et al.^28,29 proposed a method for computing the carrying capacity of a slewing bearing based on a finite element model. The method considered the flexural–torsional flexibility of the bearing rings as well as the flexibility and number of clamping bolts.

The calculation methods for the carrying curve^22–26 take the displacement of the loaded bearing ring as the unknown variable and establish equilibrium equations for the bearing ring displacement. The displacements are obtained by solving the equilibrium equations for all load directions. Carrying curve calculations require repeated numerical solution calculations of the equilibrium equations, which require additional computations. Although the finite element analysis method for the carrying curve^27–29 replaces the rolling elements with nonlinear truss elements to reduce the computation scale, numerous iterative computations are required to obtain the points comprising the static carrying capacity curve.

Currently, relevant research is in the theoretical exploration stage, and the methods are impracticable for engineering applications. The focus of this study is to derive a calculation method for the static carrying curve of a slewing bearing using a simple and convenient process for actual engineering applications. Using the rolling element load distribution range parameter of the two rows of raceways, analytical expressions for the external load combinations of the axial load and tilting moment load of the slewing bearing were obtained. Although the method based on the load distribution range parameter does not consider the change in the contact angle of the rolling element after the slewing bearing is loaded and the axial clearance of the bearing cannot be included in the model, it does not require the solution of nonlinear equations. Thus, the proposed model has the advantage of convenient engineering implementation.

Slewing bearings mainly sustain the combined action of an axial load and tilting moment load during operation; therefore, the carrying capacity index of the slewing bearing should be able to indicate the ability to sustain the combined action of the axial load and tilting moment load simultaneously. The position of a point in a two-dimensional coordinate space can represent two numbers corresponding to the axial load and tilting moment load; therefore, using a curve in a two-dimensional coordinate system to represent the carrying capacity of the slewing bearing is a feasible method.

Double-row ball slewing bearings with different diameters are commonly used in practical engineering, as shown in Figure 1. These consist of an inner ring, outer ring, rolling elements of the main raceway, and rolling elements of the auxiliary raceway. The rolling element of the main raceway, which has a larger diameter, mainly sustains the combined action of the axial and tilting moment loads, whereas the rolling element of the auxiliary raceway, which has a smaller diameter, only sustains the action of the tilting moment load. This type of slewing bearing has a different up-down raceway structure, and its mechanical modeling method is much more typical.

Figure 1.

Structure of the double-row different-diameter ball slewing bearing.

In this study, a method for calculating the static carrying curve of a double-row different-diameter ball slewing bearing is derived, and the static carrying curve is calculated for a real case. The obtained carrying curve is compared with that calculated using the finite element method for verification. The main contributions of this study are summarized as follows.

The relationship between the maximum rolling element load of the main raceway and that of the auxiliary raceway is established.

The external load combination of the axial and tilting moment loads expressed by the maximum rolling element loads of the slewing bearing is derived.

A finite element model for calculating the static carrying curve of a double-row different-diameter ball slewing bearing is proposed.

The parameters affecting the carrying capacity of the double-row different-diameter ball slewing bearing are identified.

The remainder of this article is organized as follows. The second section presents the derivation of the mechanical model for the slewing bearing using deformation compatibility and force equilibrium conditions. The calculation method for the carrying curve of the slewing bearing is introduced in third section. The static carrying curve of a double-row different-diameter ball slewing bearing is calculated in the fourth section. The influences of the design parameters on the carrying capacity of the slewing bearing are analyzed in the fifth section based on the carrying curves. Finally, conclusions are presented in the sixth section.

Mechanics modeling

The static carrying capacity of a slewing bearing depends directly on the rolling element loads. These are the external load combinations that the slewing bearing can withstand as the maximum rolling element contact stress reaches the allowable contact stress of the raceway material. Therefore, the relationship between the external and rolling element loads of the slewing bearing must be established. As the external loads of the slewing bearing are given, the calculation of the rolling element load distribution is a statically indeterminate problem. With the aid of deformation compatibility and force equilibrium conditions, a mechanical model of the slewing bearing is established in this study and the relationship between the external and rolling element loads is determined.

For convenience in defining the position of each rolling element on the circumference of the slewing bearing, a polar coordinate system is established in the radial plane of the slewing bearing with the center of the slewing bearing as the coordinate origin. The polar axis of the coordinate system passes through the center of the most heavily loaded rolling element, and the angular position is represented by $\psi$ . Then, for each row of rolling elements, the angular position of each rolling element can be expressed as

$$\left[\psi ={\displaystyle \frac{2\pi }{Z}i-\pi }\right]$$

(1)

where i is the sequence number of the rolling element ( $i=1,2,\dots ,Z$ ), and Z is the quantity of rolling elements in the main or auxiliary raceway.

Deformation compatibility conditions

The outer ring of the slewing bearing is installed on the installation platform of the host machine and immovably fixed, whereas the inner ring of the slewing bearing is combined with the rotating parts. The inner ring of the slewing bearing sustains the combined action of the axial load and tilting moment load as the host machine operates. The inner ring of the slewing bearing produces displacements relative to the fixed outer ring under the action of external loads. In Figure 2, the corresponding axial displacement and angular displacement of the inner ring are ${\delta }_a$ and $\theta$ , respectively, under the combined action of an external axial load ( $F_a$ ) and tilting moment load ( $M$ ). Displacements of the inner ring cause contact loads between the rolling elements and the two raceways. The direction of contact load $Q_1$ between the rolling element and main raceways is represented as contact direction 1, and the direction of contact load $Q_2$ between the rolling element and auxiliary raceways is represented as contact direction 2. Thus, indices 1 and 2 denote contact directions 1 and 2, respectively.

Figure 2.

Displacements of the inner ring under combined loads.

Within a sufficiently small scope, the contact area between the rolling element and raceway can be regarded as the contact area between two parallel planes, as shown in Figure 3. For contact direction 1, the angular position of the most heavily loaded rolling element is set as ${\psi }_1=0$ . Then, the displacement along the axial direction at the contact center between the inner ring raceway and rolling element caused by the inner ring displacement at an arbitrary angular position ${\psi }_1$ is given as follows:

$$\left[\overline{A{A}^{\mathrm{\prime }}}={\delta }_a+{\displaystyle \frac{1}{2}\theta \cdot d_{m1}\cdot \cos {\psi }_1}\right]$$

(2)

where $d_{m1}$ is the distribution diameter of the rolling elements of the main raceway.

Figure 3.

Normal contact deformation.

The total contact deformation along the normal direction of contact between the rolling element and inner and outer ring raceways in contact direction 1 at an arbitrary angular position, ${\psi }_1$ , is $\overline{AB}$ , namely:

$$\left[{\delta }_{\psi 1}=\left({\delta }_a+{\displaystyle \frac{1}{2}\theta \cdot d_{m1}\cdot \cos {\psi }_1}\right)\cdot \sin {\alpha }_1\right]$$

(3)

where ${\alpha }_1$ is the contact angle between the rolling element and the main raceway.

As shown in equation (3), the maximum contact deformation occurs at the angular position of ${\psi }_1=0$ is largest:

$$\left[{\delta }_{max1}=\left({\delta }_a+{\displaystyle \frac{1}{2}\theta \cdot d_{m1}}\right)\cdot \sin {\alpha }_1\right]$$

(4)

The rolling element load distribution range parameter of the main raceway is defined as follows:

$$\left[{\epsilon }_1={\displaystyle \frac{1}{2}\left(1+{\displaystyle \frac{2{\delta }_a}{\theta \cdot d_{m1}}}\right)}\right]$$

(5)

Hence, from equations (3), (4), and (5), the contact deformation at each angular position, ${\psi }_1$ , can be obtained:

$$\left[{\delta }_{\psi 1}={\delta }_{max1}\left[1-{\displaystyle \frac{1}{2{\epsilon }_1}(1-\cos {\psi }_1)}\right]\right]$$

(6)

According to the Hertz contact theory, the relationship between the contact load, Q, and elastic deformation, $\delta$ , is given as follows ³⁰ :

$$\left[Q=K_n{\delta }^{\frac{3}{2}}\right]$$

(7)

where $K_n$ is the total load–deformation constant between the rolling element and the inner and outer ring raceways.

Using equations (6) and (7), the rolling element load, $Q_{\psi 1}$ , at each angular position, ${\psi }_1$ , along contact direction 1 can be obtained as follows:

$$\left[Q_{\psi 1}=Q_{max1}{\left[1-{\displaystyle \frac{1}{2{\epsilon }_1}(1-\cos {\psi }_1)}\right]}^{\frac{3}{2}}\right]$$

(8)

Similarly, the total contact deformation along the normal direction of the contact between the rolling element and inner and outer ring raceways in contact direction 2 at each angular position, ${\psi }_2$ , is given as follows:

$$\left[{\delta }_{\psi 2}=\left(-{\delta }_a+{\displaystyle \frac{1}{2}\theta \cdot d_{m1}\cdot \cos {\psi }_2}\right)\cdot \sin {\alpha }_2\right]$$

(9)

The contact deformation in contact direction 2 is largest at the angular position of ${\psi }_2=\pi$ :

$$\left[{\delta }_{max2}=\left(-{\delta }_a+{\displaystyle \frac{1}{2}\theta \cdot d_{m2}}\right)\cdot \sin {\alpha }_2\right]$$

(10)

The rolling element load distribution range parameter of the auxiliary raceway is defined as follows:

$$\left[{\epsilon }_2={\displaystyle \frac{1}{2}\left(1-{\displaystyle \frac{2{\delta }_a}{\theta \cdot d_{m2}}}\right)}\right]$$

(11)

Hence, from equations (9), (10), and (11), the contact deformation at each angular position, ${\psi }_2$ , can be obtained:

$$\left[{\delta }_{\psi 2}={\delta }_{max2}\left[1-{\displaystyle \frac{1}{2{\epsilon }_2}(1-\cos {\psi }_1)}\right]\right]$$

(12)

The rolling element load, $Q_{\psi 2}$ , at each angular position, ${\psi }_2$ , along contact direction 2 is obtained as follows:

$$\left[Q_{\psi 2}=Q_{max2}{\left[1-{\displaystyle \frac{1}{2{\epsilon }_2}(1-\cos {\psi }_2)}\right]}^{\frac{3}{2}}\right]$$

(13)

The following can be obtained by eliminating $\frac{2{\delta }_a}{\theta }$ in equations (5) and (11):

$$\left[{\epsilon }_2={\displaystyle \frac{1}{2}+{\displaystyle \frac{d_{m1}}{d_{m2}}\left({\displaystyle \frac{1}{2}-{\epsilon }_1}\right)}}\right]$$

(14)

Substitution of equations (4) and (10) into equation (7) yields the following:

$$\left[{\displaystyle \frac{Q_{max1}}{Q_{max2}}={\displaystyle \frac{K_{n1}{\delta }_{max1}^{\frac{3}{2}}}{K_{n2}{\delta }_{max2}^{\frac{3}{2}}}={\displaystyle \frac{K_{n1}{\left[\left({\delta }_a+{\displaystyle \frac{1}{2}\theta \cdot d_{m1}}\right)\cdot \sin {\alpha }_1\right]}^{\frac{3}{2}}}{K_{n2}{\left[\left(-{\delta }_a+{\displaystyle \frac{1}{2}\theta \cdot d_{m2}}\right)\cdot \sin {\alpha }_2\right]}^{\frac{3}{2}}}}}}\right]$$

(15)

Substitution of equations (5) and (11) into equation (15) yields the following:

$$\left[{\displaystyle \frac{Q_{max1}}{Q_{max2}}={\displaystyle \frac{K_{n1}{({\epsilon }_1\cdot d_{m}1\cdot \sin {\alpha }_1)}^{\frac{3}{2}}}{K_{n2}{({\epsilon }_2\cdot d_{m}2\cdot \sin {\alpha }_2)}^{\frac{3}{2}}}}}\right]$$

(16)

Then, the following expressions can be obtained:

$$\left[Q_{max1}={\displaystyle \frac{K_{n1}{({\epsilon }_1\cdot d_{m}1\cdot \sin {\alpha }_1)}^{\frac{3}{2}}}{K_{n2}{({\epsilon }_2\cdot d_{m}2\cdot \sin {\alpha }_2)}^{\frac{3}{2}}}{Q}_{max2}}\right]$$

(17)

$$\left[Q_{max2}={\displaystyle \frac{K_{n2}{({\epsilon }_2\cdot d_{m2}\cdot \sin {\alpha }_2)}^{\frac{3}{2}}}{K_{n1}{({\epsilon }_1\cdot d_{m1}\cdot \sin {\alpha }_1)}^{\frac{3}{2}}}{Q}_{max1}}\right]$$

(18)

Force equilibrium conditions

According to equations (8) and (13), the inner ring sustains the action of all rolling element loads $Q_{\psi 1}$ and $Q_{\psi 2}$ inside the slewing bearing and sustains the action of external working loads $F_a$ and M outside the slewing bearing. The inner ring is in equilibrium under the combined action of the internal and external loads.

The equilibrium equation for the axial load is given as follows:

$$\left[F_a={\sum }_{{\psi }_1=-\pi }^{{\psi }_1=+\pi }{Q}_{\psi 1}\cdot \sin {\alpha }_1-{\sum }_{{\psi }_2=-\pi }^{{\psi }_2=+\pi }{Q}_{\psi 2}\cdot \sin {\alpha }_2\right]$$

(19)

The equilibrium equation for the tilting moment is

$$\left[M=\sum _{{\psi }_1=-\pi }^{{\psi }_1=+\pi }{Q}_{\psi 1}\cdot \sin {\alpha }_1\cdot {\displaystyle \frac{1}{2}{d}_{m1}\cdot \cos {\psi }_1+{\sum }_{{\psi }_2=-\pi }^{{\psi }_2=+\pi }{Q}_{\psi 2}\cdot \sin {\alpha }_2\cdot {\displaystyle \frac{1}{2}{d}_{m2}\cdot \cos {\psi }_2}}\right]$$

(20)

To facilitate the calculation, the summation in the above two equations can be written in an integral form as follows:

$$[F_a=Z_1\cdot Q_{max1}\cdot J_a({\epsilon }_1)\cdot \sin {\alpha }_1-Z_2\cdot Q_{max2}\cdot J_a({\epsilon }_2)\cdot \sin {\alpha }_2]$$

(21)

$$\left[M={\displaystyle \frac{1}{2}{d}_{m1}\cdot Z_1\cdot Q_{max1}\cdot J_m({\epsilon }_1)\cdot \sin {\alpha }_1+{\displaystyle \frac{1}{2}{d}_{m2}\cdot Z_2\cdot Q_{max2}\cdot J_m({\epsilon }_2)\cdot \sin {\alpha }_2}}\right]$$

(22)

where

$$\left[J_a(\epsilon )={\displaystyle \frac{1}{2\pi }{\int }_{-\pi }^{+\pi }{\left[1-{\displaystyle \frac{1}{2\epsilon }(1-\cos \psi )}\right]}^{\frac{3}{2}}d\psi }\right]$$

(23)

$$\left[J_m(\epsilon )={\displaystyle \frac{1}{2\pi }{\int }_{-\pi }^{+\pi }{\left[1-{\displaystyle \frac{1}{2\epsilon }(1-\cos \psi )}\right]}^{\frac{3}{2}}\cos \psi d\psi }\right]$$

(24)

The relationship between the maximum rolling element loads, $Q_{max1}$ and $Q_{max2}$ , and the external loads, $F_a$ and M is established using equations (21) and (22).

Carrying capacity curve

If the restrictive values of $Q_{max1}$ and $Q_{max2}$ are given and continuous values of parameters ${\epsilon }_1$ and $\epsilon {}_2$ are defined under the condition satisfying equation (14), the static carrying curves of the main and auxiliary raceways can be obtained, as shown in the schematic diagram in Figure 4. The working loads, $F_a$ and M of the slewing bearing should simultaneously satisfy the carrying capacity requirements of the main and auxiliary raceways. In other words, they should fall within the shaded area of the diagram. Thus, the carrying curve is composed of segment $AB$ of the carrying curve of the auxiliary raceway and segment $BC$ of the carrying curve of the main raceway. Therefore, the coordinates of the intersection points A, B, and C must first be determined by plotting the static carrying curve.

Figure 4.

Schematic diagram of the carrying curve.

Calculation of the carrying curve

Allowed rolling element load

Slewing bearings usually operate under overloading conditions. This type of working condition can easily cause plastic contact deformation of the raceway surface. To avoid the influence of plastic contact deformation of the raceway surface on the slewing performance, the total permanent deformation at the center of the most heavily loaded rolling element–raceway contact should be limited to approximately 0.0001 times the rolling element diameter. ³¹ The double-row different-diameter ball slewing bearing uses a steel ball as the rolling element. According to the experimental results reported by Qiu et al., ³² the allowable contact stress $[{\sigma }_{max}]$ between the steel ball and raceway surface of a slewing bearing made of 42CrMo should be limited to 3600 MPa. The contact stress between the most heavily loaded rolling element and the raceway surface is used to calculate the static carrying curve. In other words, various combinations of the axial and tilting moment loads are calculated to obtain a contact stress level between the most heavily loaded rolling element and raceway that is equal to the limit value.

The Hertz contact theory establishes the relationship between the contact load and contact stress. For the point contact between the rolling element and raceway surface, the maximum contact stress at the center of the contact area is given as follows:

$$\left[{\sigma }_{max}={\displaystyle \frac{1}{\pi n_a{n}_b}{\left[{\displaystyle \frac{3}{2}{\left({\displaystyle \frac{{\sum }^{\rho }}{\eta }}\right)}^{2}Q}\right]}^{\frac{1}{3}}}\right]$$

(25)

where $n_a$ and $n_b$ are the dimensionless semi-major axis and dimensionless semi-minor axis of the contact ellipse, respectively, $\eta$ the composite elastic constant of the two elastic contact bodies, Q the contact load, and $\sum \rho$ the sum of the curvatures of the two contact bodies. The sum of the curvatures of the rolling element and inner raceway can be expressed as follows:

$$\left[\sum {\rho }_i={\displaystyle \frac{1}{D_w}\left(4-{\displaystyle \frac{1}{f_i}+{\displaystyle \frac{2\gamma }{1-\gamma }}}\right)}\right]$$

(26)

where $D_w$ is the rolling element diameter, $f_i$ the groove radius coefficient of the inner raceway, and $\gamma =\frac{D_w\cos \alpha }{d_m}$ .

The sum of the curvatures of the rolling element and outer raceway can be expressed as follows:

$$\left[\sum {\rho }_e={\displaystyle \frac{1}{D_w}\left(4-{\displaystyle \frac{1}{f_e}+{\displaystyle \frac{2\gamma }{1-\gamma }}}\right)}\right]$$

(27)

where $f_e$ is the groove radius coefficient of the outer raceway.

Because the allowable contact stress $[{\sigma }_{max}]$ at the contact center between the rolling element and raceway surface is given, the allowable load of the most heavily loaded rolling element is obtained as follows:

$$\left[Q_{max}={\displaystyle \frac{2}{3}{\left({\displaystyle \frac{\eta }{\sum \rho }}\right)}^2(\pi n_a{n}_b[{\sigma }_{max}{]}^3)}\right]$$

(28)

Calculation procedure for the carrying curve

The points on the static carrying curve should ensure that the main and auxiliary raceways meet the carrying capacity requirements. Therefore, the allowable load $[Q_{max1}]$ of the most heavily loaded rolling element of the main raceway and the allowable load $[Q_{max2}]$ of the most heavily loaded rolling element of the auxiliary raceway should first be calculated according to equation (28). The procedure for calculating the carrying curve is as follows:

Step 1: The intersection point of the static carrying curve of the auxiliary raceway and the vertical axis is calculated. Because the load of the most heavily loaded rolling element of the auxiliary raceway is equal to the allowable load $[Q_{max2}]$ , the corresponding load $Q_{max1}$ of the most heavily loaded rolling element of the main raceway can be obtained using equation (17):

$$\left[Q_{max1}={\displaystyle \frac{K_{n1}{({\epsilon }_1\cdot d_{m1}\cdot \sin {\alpha }_1)}^{\frac{3}{2}}}{K_{n2}{({\epsilon }_2\cdot d_{m2}\cdot \sin {\alpha }_2)}^{\frac{3}{2}}}\cdot [Q_{max2}]}\right]$$

(29)

By setting $Q_{max2}=[Q_{max2}]$ and substituting equations (14) and (29) into equations (21) and (22), the following can be obtained:

$$\left[F_a=Z_1\cdot {\displaystyle \frac{K_{n1}{({\epsilon }_1\cdot d_{m1}\cdot \sin {\alpha }_1)}^{\frac{3}{2}}}{K_{n2}{({\epsilon }_2\cdot d_{m2}\cdot \sin {\alpha }_2)}^{\frac{3}{2}}}\cdot [Q_{max2}]\cdot J_a({\epsilon }_1)\cdot \sin {\alpha }_1-Z_2\cdot [Q_{max2}]\cdot J_a\left[{\displaystyle \frac{1}{2}+{\displaystyle \frac{d_{m1}}{d_{m2}}\left({\displaystyle \frac{1}{2}-{\epsilon }_1}\right)}}\right]\cdot \sin {\alpha }_2}\right]$$

(30)

$$\left[\begin{array}{c}M={\displaystyle \frac{1}{2}{d}_{m1}\cdot Z_1\cdot {\displaystyle \frac{K_{n1}{({\epsilon }_1\cdot d_{m1}\cdot \sin {\alpha }_1)}^{\frac{3}{2}}}{K_{n2}{({\epsilon }_2\cdot d_{m2}\cdot \sin {\alpha }_2)}^{\frac{3}{2}}}\cdot [Q_{max2}]\cdot J_m({\epsilon }_1)\cdot \sin {\alpha }_1+}}\\ {\displaystyle \frac{1}{2}{d}_{m2}\cdot Z_2\cdot [Q_{max2}]\cdot J_m\left[{\displaystyle \frac{1}{2}+{\displaystyle \frac{d_{m1}}{d_{m2}}\left({\displaystyle \frac{1}{2}-{\epsilon }_1}\right)}}\right]\cdot \sin {\alpha }_2}\end{array}\right]$$

(31)

For $F_a=0$ , the value ${\epsilon }_{1A}$ of ${\epsilon }_1$ corresponding to the intersection point A of the static carrying curve of the auxiliary raceway and the vertical axis can be obtained from the numerical solution of equation (30). The value of M can be obtained by substituting ${\epsilon }_{1A}$ into equation (31), which is the vertical coordinate of the intersection point A of the carrying curve of the auxiliary raceway and the vertical axis.

Step 2: The intersection point between the static carrying curves of the auxiliary and main raceways is calculated. Because the load of the most heavily loaded rolling element of the main raceway is equal to the allowable load $[Q_{max1}]$ , the corresponding load $Q_{max2}$ of the most heavily loaded rolling element of the auxiliary raceway can be obtained using equation (18):

$$\left[Q_{max2}={\displaystyle \frac{K_{n2}{({\epsilon }_2\cdot d_{m2}\cdot \sin {\alpha }_2)}^{\frac{3}{2}}}{K_{n1}{({\epsilon }_1\cdot d_{m1}\cdot \sin {\alpha }_1)}^{\frac{3}{2}}}\cdot [Q_{max1}]}\right]$$

(32)

By setting $Q_{max1}=[Q_{max1}]$ and substituting equations (14) and (32) into equations (21) and (22), the following can be obtained:

$$\begin{array}{rl}& F_a=Z_1[Q_{max1}]J_a({\epsilon }_1)\sin {\alpha }_1-\\ & Z_2{\displaystyle \frac{K_{n2}{({\epsilon }_2\cdot d_{m2}\cdot \sin {\alpha }_2)}^{\frac{3}{2}}}{K_{n1}{({\epsilon }_1\cdot d_{m1}\cdot \sin {\alpha }_1)}^{\frac{3}{2}}}\cdot [Q_{max1}]\cdot J_a\left[{\displaystyle \frac{1}{2}+{\displaystyle \frac{d_{m1}}{d_{m2}}\left({\displaystyle \frac{1}{2}-{\epsilon }_1}\right)}}\right]\cdot \sin {\alpha }_2}\end{array}$$

(33)

$$\begin{array}{rl}& M={\displaystyle \frac{1}{2}{d}_{m1}\cdot Z_1\cdot [Q_{max1}]\cdot J_m({\epsilon }_1)\cdot \sin {\alpha }_1+}\\ & {\displaystyle \frac{1}{2}{d}_{m2}\cdot Z_2\cdot {\displaystyle \frac{K_{n2}{({\epsilon }_2\cdot d_{m2}\cdot \sin {\alpha }_2)}^{\frac{3}{2}}}{K_{n1}{({\epsilon }_1\cdot d_{m1}\cdot \sin {\alpha }_1)}^{\frac{3}{2}}}\cdot [Q_{max1}]\cdot J_m\left[{\displaystyle \frac{1}{2}+{\displaystyle \frac{d_{m1}}{d_{m2}}\left({\displaystyle \frac{1}{2}-{\epsilon }_1}\right)}}\right]\cdot \sin {\alpha }_2}}\end{array}$$

(34)

When the right side of equation (30) is set equal to the right side of equation (33), the value ${\epsilon }_{1B}$ of ${\epsilon }_1$ corresponding to the intersection point B between the static carrying curves of the auxiliary and main raceways can be obtained from the numerical solution. The values of $F_a$ and M can be obtained by substituting ${\epsilon }_{1B}$ into equations (33) and (34), respectively, which are the coordinates of the intersection point B between the static carrying curves of the auxiliary and main raceways.

Step 3: The intersection point between the static carrying curve of the main raceway and the horizontal axis is calculated. For $M=0$ , the value ${\epsilon }_{1C}$ of ${\epsilon }_1$ corresponding to the intersection point C between the static carrying curve of the main raceway and the horizontal axis can be obtained by the numerical solution of equation (34). The value of $F_a$ can be obtained by substituting ${\epsilon }_{1C}$ into equation (33), which is the horizontal coordinate of the intersection point C between the carrying curve of the main raceway and the horizontal axis.

Step 4: The static carrying curve of the auxiliary raceway is calculated based on the carrying capacity of the auxiliary raceway. As ${\epsilon }_1$ takes a continuous value in the set $[{\epsilon }_{1A},{\epsilon }_{1B}]$ and ${\epsilon }_2$ takes a value following ${\epsilon }_1$ according to equation (14), the corresponding values of $F_a$ and M can be obtained using equations (30) and (31), respectively, which constitute the static carrying curve of the auxiliary raceway.

Step 5: The static carrying curve of the main raceway is calculated based on the carrying capacity of the auxiliary raceway. As ${\epsilon }_1$ takes a continuous value in the set $[{\epsilon }_{1B},{\epsilon }_{1C}]$ and ${\epsilon }_2$ takes a value following ${\epsilon }_1$ according to equation (14), the corresponding values of $F_a$ and M can be obtained using equations (33) and (34), respectively, which constitute the static carrying curve of the main raceway.

After completing the calculation of segments $AB$ and $BC$ of the carrying curves, the region around intersection point B is smoothed, and the carrying curve of the double-row different-diameter ball slewing bearing can be obtained.

Case calculation

The proposed calculation method for the carrying curve of the double-row different-diameter ball slewing bearing was implemented using the Matlab programming environment. The parameters of the double-row different-diameter ball slewing bearing used in a shipyard crane are listed in Table 1. The coordinate data of the points on the static carrying curve were obtained using the proposed calculation program. The static carrying curve was plotted in the coordinate system using the coordinate data.

Table 1.

Parameters of the slewing bearing.

Parameter name	Value
Main raceway	$D_{w1}$ (mm)	45
	$Z_1$	173
	$d_{m1}$ (mm)	2990
	$f_{i1}$	0.525
	$f_{e1}$	0.525
	${\alpha }_1$ (°)	60
Auxiliary raceway	$D_{w2}$ (mm)	39.6875
	$Z_2$	196
	$d_{m2}$ (mm)	2980
	$f_{i2}$	0.525
	$f_{e2}$	0.525
	${\alpha }_2$ (°)	60

To verify the correctness of the established algorithm and calculation program in this study, the results calculated by the algorithm were compared with those calculated using commercial ANSYS finite element analysis software. To reduce the scale of the finite element analysis, a mixed model composed of solid elements and spring elements was adopted for the double-row different-diameter ball slewing bearing assembly. The balls were replaced by extension spring elements. The two nodes of the spring element were located at the center of curvature of the two grooves, as shown in Figure 5.

Figure 5.

Locations of spring elements; (a) contact direction 1 and (b) contact direction 2.

To obtain a high-quality mesh, the outer ring gear teeth, which are irrelevant to the load-carrying capacity of the raceway, were omitted. Detailed structures such as chamfers and sealing strip installation grooves were ignored. The rolling element was replaced by a nonlinear spring element, COMBIN39, whose stiffness curve was calculated using equation (7). The rings were meshed using the tetrahedral element SOLID92. All degree of freedom constraints were applied to the lower bottom surface of the outer ring. To facilitate the application of the axial load and tilting moment load, a loading auxiliary node was established at the height center of the slewing bearing, which was meshed using the mass element MASS21. The auxiliary node was then coupled with all of the nodes on the upper end face of the inner ring to form a rigid region. Finally, the axial load, $F_a$ , and tilting moment load, M were applied to the auxiliary node. A schematic of the finite element model is shown in Figure 6. The geometric and mesh models of the slewing bearing were established using ANSYS software, as shown in Figure 7. The finite element model after applying the boundary conditions is shown in Figure 8.

Figure 6.

Finite element modeling scheme.

Figure 7.

Models of the slewing bearing; (a) geometric model and (b) mesh model.

Figure 8.

Finite element model of slewing bearing.

Different combinations of the axial and tilting moment loads were applied to the models. When the calculated maximum rolling element load of the main raceway reached $[Q_{max1}]$ or the maximum rolling element load of the auxiliary raceway reached $[Q_{max2}]$ , a point on the carrying curve was obtained. The static carrying curve calculated using finite element analysis software was built by connecting all of the neighboring plotted points.

The static carrying curves calculated by the proposed method and the finite element method are shown in Figure 9. The results show that the curves obtained using the two methods are very similar. The reason for the slight difference between the two is that the proposed method neglects the change in the contact angle of the rolling element after the slewing bearing is loaded.

Figure 9.

Calculated carrying curves.

Analysis and discussion

As the design parameters change, the load-carrying capacity of the slewing bearing changes accordingly. Therefore, the position of the corresponding carrying curve in the coordinate system will also change. To study the influence of the detailed design parameters on the static carrying capacity of a double-row different-diameter ball slewing bearing, different values of the raceway groove radius coefficient, rolling element diameter, and raceway contact angle were considered, and the corresponding static carrying curves were calculated.

The effect of the raceway groove radius coefficient on the static carrying capacity of the double-row different-diameter ball slewing bearing is shown in Figure 10. When the raceway groove radius coefficient increases from 0.515 to 0.530, the carrying capacity of the slewing bearing decreases. With an increase in the groove radius coefficient, the downward trend in the carrying capacity becomes more gradual. According to equation (28), a reduction in the raceway groove radius coefficient will increase the value of the allowable rolling element load $[Q_{max}]$ , and thus the allowable external loads, $F_a$ and M will increase accordingly. It can be observed that the carrying capacity of the slewing bearing can be improved when the raceway groove radius coefficient is reduced.

Figure 10.

Effect of the groove radius coefficient.

The effect of the rolling element diameter on the static carrying capacity of the double-row different-diameter ball slewing bearing is shown in Figure 11. The results show that when the rolling element diameter increases from 0.90 times the initial diameter to 1.05 times the initial diameter, the carrying capacity of the slewing bearing increases accordingly. With an increase in the rolling element diameter, the increasing trend of the carrying capacity remains largely unchanged. According to equation (28), an increase in the rolling element diameter will increase the value of the allowable rolling element load $[Q_{max}]$ , and thus the allowable external loads, $F_a$ and M will increase accordingly. It can be observed that the carrying capacity of the slewing bearing can be improved by increasing the rolling element diameter.

Figure 11.

Effect of the rolling element diameter.

The effect of the contact angle on the static carrying capacity of the double-row different-diameter ball slewing bearing is shown in Figure 12. As shown in Figure 12, when the contact angle increases from 50° to 65°, the carrying capacity of the slewing bearing increases accordingly. With an increase in the contact angle, the increasing trend of the carrying capacity remains basically unchanged. According to equations (19) and (20), an increase in the contact angle will increase the axial component, $Q_{\psi }\cdot \sin \alpha$ , of the rolling element load; thus, the allowable external loads, $F_a$ and M will increase accordingly. It is observed that the carrying capacity of the slewing bearing can be improved by increasing the contact angle.

Figure 12.

Effect of the raceway contact angle.

Conclusion

Based on the static modeling of a double-row different-diameter ball slewing bearing, a calculation method for the static carrying curve was derived. The influences of detailed design parameters on the carrying capacity of the double-row different-diameter ball slewing bearing were analyzed based on the static carrying curve. The following conclusions could be drawn:

When the groove radius coefficient increases from 0.515 to 0.530, the carrying capacity of the slewing bearing decreases accordingly.

When the rolling element diameter increases from 0.90 times the initial diameter to 1.05 times the initial diameter, the carrying capacity of the slewing bearing increases accordingly.

When the contact angle increases from 50° to 65°, the carrying capacity of the slewing bearing increases accordingly.

Therefore, decreasing the raceway groove radius coefficient, increasing the rolling element diameter, and increasing the contact angle are advantageous for enhancing the carrying capacity of a double-row different-diameter ball slewing bearing.