1 Introduction

Roller bearings can support large radial loads and mainly used in large and medium-sized motors, gearboxes, machine tool spindles, gas turbines, and other rotating machinery. Due to poor bearing assembly, eccentric load, and complex working conditions, roller bearings are prone to misalignment of the rings. At this time, rollers are tilted with respect to the raceway. Roller bearings allow a small angular error (deflection angle) between the axis of the inner ring and the outer ring, which is usually less than 5–7 min. Therefore, the tilt of the rollers tends to result in non-uniform pressure distributions or stress concentration at the contact area of the raceway [1, 2]. Long-term operation of the bearing under non-uniform pressure will cause partial wear of the bearing, as shown in Fig. 1, which will reduce the service life of bearings and affect the dynamic performance of the rotor [3].

Fig. 1
figure 1

Serious partial wear on the raceway of rolling bearing

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As an important support structure of the rotor, the roller bearing has a significant impact on the vibration and critical rotational speed of the rotor. The experiment results in Ref. [4] show that the deformation of the ring will affect the subcritical vibration of the rotor, and it will also cause the sub-harmonic resonance at certain rotational speeds. Therefore, it is of great significance to analyze the contact characteristics and the deformation of bearing components under the inclined load. Scholars have done a lot of work on the partial wear of roller bearings. Harris [5] and Palmgren [6] established the statics and quasi-static model based on the kinematic relationship between the bearing components to analyze the contact characteristics and their influence on the bearings’ life. For the analysis of roller bearings under the inclined load, Mr. William Lutz of Sikorsky Aircraft Division first proposed the influence coefficients method, which has been widely used. Based on static analysis, Gupta [7] established a dynamic model of the bearing to analyze the contact characteristics and motion stability of roller bearings under misalignment. The earliest analysis of rolling bearings is mostly limited to theoretical analysis, and experiments are rarely used to verify the theoretical results. Kannel [8] analyzed the contact characteristics of a single roller under inclined load, and the experiment results became a reference for many theoretical analyses.

After years of development, for the analysis of the inclined load of roller bearings, based on the static model, the factors considered are also more complex, such as lubrication, oil film thickness, and thermal deformation. Duan [9] established a coupling model based on the quasi-static model and mixed lubrication model and analyzed the influence of inclined load on bearing life. However, the results are not verified by FEM or experiments. Wang [10] established a dynamic model to analyze the contact characteristics and motion stability of the roller bearings, but the dynamic model did not consider the deformation of the bearing ring under the inclined load and also lacked experimental verification. Kabus [11] proposed a novel six-degree-of-freedom frictionless quasi-static time domain roller bearing model based on high precision elastic half-space theory and used the experiment results in Ref. [12] to verify the model. However, the model only analyzes the internal contact characteristics of the bearing and does not introduce the influence of the bearing housing or rotor on the deformation of the rings.

Due to the expensive testing equipment, most theoretical analyses cannot be verified by experiments. In recent years, due to the rapid development of computers and analysis software, the finite element analysis (FEA) has become a common method to analyze the contact characteristics of roller bearings. The finite element model mainly includes the "half-space" model [13], the overall 3D model [14], and the overall 2D model [15]. The FEA results can be used as an important reference for theoretical models. Hou and Wang [16] used the strain test method to measure the contact pressure distribution between the roller and the rings in the loading zone. The obtained results are consistent with the quasi-static and FEA results. However, the inclined load is not considered. Yin [17] analyzed the plastic deformation of the roller surface under the normal wear and partial wear by experiments and pointed out that the contact pressure between the roller and the ring causes the plastic deformation. However, the effect of partial wear on the plastic deformation of rings was not analyzed. Zhang [18] analyzed the effect of axial misalignment on the contact characteristics of journal bearings by experiment and measured the micro-topography and roughness of the contact surface. Benchea [19] and Creţu [20, 21] considered the influence of the number of revolutions of the rollers on the contact characteristics and plastic deformation of the rollers under the inclined load. The theoretical results show that the edge contact pressure between the roller and the ring decreases after a period of wear. The plastic deformation of the roller edge is equivalent to re-modifying the roller to make it more suitable for the inclined load, but the results also lack experimental verification.

The studies of roller bearings deformation in Refs. [17, 19,20,21] mainly analyze the plastic deformation of rollers caused by wear, without considering the plastic deformation and topography of the contact surface of the rings and the influence of the inclined load on the geometric deformation. The bearing internal excitation caused by the complex rings deformation will have a serious impact on the service performance of rotor systems [22,23,24]. The novelty of this paper is to consider the combined effects of external inclined loads, rollers extreme positions, and wear on the deformation of the contact surface of the outer ring by combining analytical analysis, finite element analysis, and experimental test. The geometric deformation caused by external inclined load under alternating transformation of two extreme positions of the rollers is analyzed for the first time, which will cause the uniform wear between the raceways and rollers to become severe partial wear and further affects the plastic deformation and contact surface roughness of the rings. Combined with the bearing contact performance, such as contact load and contact stress distribution, the mechanism and effect of roller bearing partial wear are more comprehensively revealed.

Based on the non-Hertz contact theory, a quasi-static roller bearing model considering the inclined load is established first. Combined with the experimental data in the classical literature, the accuracy of the model is verified. Based on the quasi-static model, the contact characteristics of the bearing under inclined load are analyzed. Secondly, the finite element model of the bearing housing is established, and the mesh size and boundary conditions of the finite element model are corrected by comparing with the results of the quasi-static model. Based on the finite element model, the deformation of bearing caused by the deflection of the housing is analyzed. Further, based on a bearing-rotor test rig, the roller bearing partial wear experiment was carried out. The bearing after the experiment is disassembled to determine the wear area of the bearing. Severe wear areas of the outer ring are sampled by wire cutting, and the three-dimensional topography and surface roughness are analyzed by a laser scanning microscope. Finally, the experimental results are compared with the finite element results to verify the validity of the finite element results. The analysis method and process proposed in this paper are shown in Fig. 2.

Fig. 2
figure 2

Analysis method and process in this paper

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2 Theoretical basis

The premise of establishing the contact load model between the rollers and the raceways is to determine the shape and size of the contact surface, the elastic deformation of the two objects in contact with each other, the stress and load distribution in contact area, and the overall balance relationship. These contents are described in detail below.

2.1 Contact deformation between roller and raceways

When a roller bearing supports inclined load, the inclined load can be decomposed into radial load Fr and moment M. At this time, the bearing will produce corresponding radial parallel displacement δr and the declination angle θ, as shown in Fig. 3. Due to the different positions of rollers, the radial parallel displacement and declination angle between the inner and outer raceways are also different. Assume the angle between two adjacent rollers is Φ0, namely 360/z. The roller that supports the largest load, that is, the roller just above the load line, is defined as the No. 1 roller, and in the clockwise direction, they are the No. 2, No. 3, …, No. Z rollers. Along the circumferential direction of the bearing, the position angle of the No. 1 roller is defined as 0°, and thus, the position angle of the jth roller can be expressed as:

$$ \varphi_{j} = \frac{{2\pi \left( {j - 1} \right)}}{Z} $$
(1)
Fig. 3
figure 3

The loading condition caused by combined radial and moment loads

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The elastic contact deformation between each roller and raceways is different. Under the inclined load, due to the relative angle between inner and outer rings, the elastic contact deformations between each roller and raceways along the contact line are different. To accurately analyze the elastic contact deformation, the roller and raceways are sliced, and the number of slices is n. For example, the elastic contact deformation δj between the jth roller and the raceways can be expressed as

$$ \delta_{j} = \delta_{rj} + \delta_{\theta j} + \delta_{cj} $$
(2)

where δrj is the normal contact deformation between the jth roller and the inner and outer raceways caused by the radial load, δθj is the angular deformation when the jth roller and the inner and outer raceways are inclined due to the moment, and δcj is the deformation caused by profile modification of the jth roller.

The δrj caused by the radial load can be specifically expressed as

$$ \delta_{rj} = \delta_{r} \cos \varphi_{j} - \frac{u}{2} $$
(3)

where u is the radial clearance of the roller bearing.

The magnitude of δθj caused by the inclined load is related to the angle between the inner and outer rings, so the δθj of each slice is different, and its expression is as

$$ \delta_{\theta ijk} = \left( {x - \frac{l}{2}} \right) \cdot \tan \alpha_{ij} \quad 0 \le x \le l $$
(4)
$$ \delta_{\theta ojk} = \left( {x - \frac{l}{2}} \right) \cdot \tan \alpha_{oj} \quad 0 \le x \le l $$
(5)

where the subscript i represents the inner raceway, the subscript o represents the outer raceway, the subscript k represents the kth slice, αij is the relative deflection angle between the jth roller and the inner ring, and αoj is the relative deflection angle between the jth roller and the outer ring.

$$ \alpha_{ij} + \alpha_{oj} = \theta_{j} $$
(6)
$$ \theta_{j} = \theta \cos \varphi_{j} $$
(7)

where θ is the total relative deflection angle between the inner and outer rings under the moment. The deformation δcj caused by the roller profile modification is determined by the specific modification method. By modifying the roller, the material of the roller will be reduced, and the effect in the contact analysis is that the contact deformation is reduced, so δcj should be located on the negative semi-axis.

2.2 Non-Hertz contact of rollers and raceways

The contact between the rollers and the raceways is line contact. According to the Hertz contact theory, the surface pressure is distributed as a semi-elliptical cylinder when two parallel cylinders of equal length are in contact. However, in practice, the ideal line contact state between infinite long cylinders does not exist. Under the inclined load, the contact problem will be more complicated. At this time, the non-Hertz method needs to be used to solve the above problems. The roller and the raceway, the two elastic bodies, will produce elastic deformation and form a contact area Ω under the applied load. The schematic diagram of the line contact between the roller and the inner raceway is shown in Fig. 4.

Fig. 4
figure 4

Linear contact of roller and inner raceway

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Assume M and N are the points on the cylinder located in the X–Z plane, the distance between the two points and the Z-axis is x, and the distance between the two points before deformation is

$$ z_{1} + z_{2} = \frac{{x^{2} }}{2R} $$
(8)
$$ \frac{1}{R} = \frac{1}{{R_{1} }} + \frac{1}{{R_{2} }} $$
(9)

where R is the total radius of curvature at the contact position, and R1 and R2 are the radius of curvature of the roller and the inner raceway, respectively.

Under the inclined load, the radial load of the jth roller in contact with the inner raceway is Qij and the moment is Mij. The elastic contact deformation of the two elastic bodies between the roller and the inner raceway is δij, and there will be a contact area with a width of 2aij on the contact surface. The two elastic bodies satisfy the deformation coordination relationship in the Z-direction,

$$ \omega_{1} + \omega_{2} + z_{1} + z_{2} = \delta_{ij} $$
(10)

where ω1 and ω2 are the elastic deformations of the roller and the inner raceway at the contact point, respectively.

When the deformation is in elastic range, ω1 and ω2 can be solved by Boussinesq integration [25] in elastic mechanics, which is expressed as

$$ \omega_{1} + \omega_{2} = \frac{1}{{\pi E^{\prime } }}\iint\limits_{\Omega } {\frac{{p\left( {x^{\prime } ,y^{\prime } } \right){\text{d}}x^{\prime } {\text{d}}y^{\prime } }}{{\sqrt {\left( {x - x^{\prime } } \right)^{2} + \left( {y - y^{\prime } } \right)^{2} } }}} $$
(11)

Substituting Eq. (11) into Eq. (10), another form of deformation coordination equation is as

$$ \frac{1}{{\pi E^{\prime } }}\iint\limits_{\Omega } {\frac{{p\left( {x^{\prime } ,y^{\prime } } \right){\text{d}}x^{\prime } {\text{d}}y^{\prime } }}{{\sqrt {\left( {x - x^{\prime } } \right)^{2} + \left( {y - y^{\prime } } \right)^{2} } }}} = \delta_{ij} - z\left( {x,y} \right) $$
(12)

where z(x, y) represents the initial distance between contact surfaces at a distant point (x, y) before deformation, and z = z1 + z2. E′ is the composite elastic modulus,

$$ \frac{1}{{E^{\prime}}} = \frac{{1 - v_{1}^{2} }}{{E_{1} }} + \frac{{1 - v_{2}^{2} }}{{E_{2} }} $$
(13)

where E1 and E2 are the elastic modulus of the roller and the inner raceway, respectively. v1 and v2 are the Poisson's ratio of the roller and the inner ring, respectively.

The equilibrium equation between the contact stress p(x, y) and the external load is

$$ \iint\limits_{\Omega } {p\left( {x,y} \right)}\;{\text{d}}x{\text{d}}y = Q_{ij} $$
(14)
$$ \iint\limits_{\Omega } {y \cdot p\left( {x,y} \right)}\;{\text{d}}x{\text{d}}y = M_{ij} $$
(15)

Equations (12), (14), and (15) constitute the basic equations for the linear contact between the roller and the raceway under the inclined load. In general, the equations have not theoretical solution and can only be solved numerically.

Due to the complexity of the contact between the roller and the raceway under the inclined load, it is necessary to slice the contact area between the rollers and the raceways. The influence coefficient method is used to solve the integral equation between each strip element. As shown in Fig. 5, the contact area between the roller and the raceway is divided into n strip elements along the direction of the roller's generatrix (Y-axis). In each strip element, it is assumed that the contact stress is evenly distributed along the Y-direction and Hertz distributed along the X-direction.

Fig. 5
figure 5

Strip element

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Under the external radial load, the contact stress distributed along the X-direction of the kth slice is expressed as

$$ p_{k} = p_{0k} \sqrt {1 - \left( {{x \mathord{\left/ {\vphantom {x {a_{k} }}} \right. \kern-0pt} {a_{k} }}} \right)^{2} } $$
(16)
$$ a_{k} = \frac{2R}{{E^{\prime}}}p_{0k} $$
(17)

where ak is the contact half-width at the center of the kth element, and p0k is the maximum contact stress at the center of the kth element.

According to the influence coefficient method, the displacement generated by the stress of the kth strip element on the center of the mth strip element is

$$ \omega_{mk} = \frac{{p_{0k} }}{{\pi E^{\prime } }}\int\limits_{{ - a_{k} }}^{{a_{k} }} {\int\limits_{{y_{k} - h_{k} }}^{{y_{k} + h_{k} }} {\frac{{\sqrt {1 - \left( {x^{\prime } /a_{k} } \right)^{2} } {\text{d}}x^{\prime } {\text{d}}y^{\prime } }}{{\sqrt {x^{{\prime}{2}} + \left( {y_{m} - y_{k} - y^{\prime } } \right)^{2} } }}} } $$
(18)

where hk is the contact half-length of the kth strip element.

2.3 Overall equilibrium equations

When a roller bearing is under the inclined load, the radial load and moment of the kth strip element of the jth roller are

$$ q_{jk} = A_{jk} p_{0jk} = \frac{1}{2}\pi a_{jk} 2h_{jk} p_{0jk} = \pi a_{jk} h_{jk} p_{0jk} $$
(19)
$$ M_{jk} = q_{jk} \left( {y_{jk} - y_{jc} } \right) $$
(20)

where yjk is the center coordinate of the kth slice element of the jth roller, and yjc is the center coordinate of the jth roller.

The radial load on the jth roller is the sum of the radial loads on the n elements, and the moment on the jth roller is also the sum of the moment on the n elements,

$$ Q_{j} = \sum\limits_{k = 1}^{n} {q_{jk} } ,M_{j} = \sum\limits_{k = 1}^{n} {M_{jk} } $$
(21)

According to the relationship between the contact load of the outer raceway-roller and the contact load of the inner raceway-roller, the equilibrium equation of a single roller is

$$ \left\{ {\begin{array}{*{20}l} {Q_{ij} + F_{cj} = Q_{oj} } \hfill \\ {Q_{ij} = Q_{j} } \hfill \\ {M_{ij} = M_{oj} = M_{j} } \hfill \\ \end{array} } \right. $$
(22)

where Fcj is the centrifugal force of the single roller, which can be expressed as

$$ F_{cj} = \frac{1}{2}md_{m} \omega_{m}^{2} $$
(23)

where m is the mass of the roller, dm is the pitch diameter of the bearing, and ωm is the revolution angular velocity of the roller,

$$ \omega_{m} = \frac{{2\pi n_{m} }}{60} $$
(24)

where ωm is the revolution speed of the roller.

It can be seen from the above analysis that as long as the maximum contact stress p0jk at the center of each element on the jth roller is obtained, the contact half-width ajk, the radial load Qj, and the moment Mj on the jth roller will be determined.

Based on the above analysis, the basic equations of the contact between the roller and the raceway under the inclined load can be expressed as

$$ \left\{ {\begin{array}{*{20}l} {\pi \sum\limits_{k = 1}^{n} {a_{jk} h_{jk} p_{0jk} = Q_{j} } } \hfill \\ {\pi \sum\limits_{k = 1}^{n} {a_{jk} h_{jk} p_{0jk} \left( {y_{jk} - y_{jc} } \right) = M_{j} } } \hfill \\ {\frac{1}{{\pi E^{\prime } }}\sum\limits_{k = 1}^{n} {D_{mk} p_{0jk} = \delta_{ijk} - z_{k} \quad k = 1,2,...,n} } \hfill \\ \end{array} } \right. $$
(25)

From Eq. (25), when Qj, Mj, and zk are determined, the contact stresses p0jk, δijk, and ajk of each strip element in contact with the raceway can be solved. When the radial load and moment of each roller and raceway are determined, the overall equilibrium equations of the roller bearing can be obtained as

$$ \left\{ {\begin{array}{*{20}l} {F_{r} - \sum\limits_{j = 1}^{Z} {Q_{j} \cos \varphi_{j} } = 0} \hfill \\ {M - \sum\limits_{j = 1}^{Z} {M_{j} \cos \varphi_{j} = 0} } \hfill \\ \end{array} } \right. $$
(26)

The above equations can be solved using the Newton–Raphson iteration method. In the solution process, the radial load Fr and the deflection angle θ are known, and the contact stress between each roller and the raceway under the inclined load is obtained. Finally, the contact stress and contact deformation of each roller and the moment M of the bearing can be obtained.

2.4 Extreme position of roller in the loading zone

The classical bearing analysis theory ignores the bearing vibration caused by the discreteness of the rollers because the roller load distribution is treated as a continuous variable. When the rollers pass through the loading zone, the bearing produces periodic vibration due to the parity change of the number of rollers under pressure [26, 27]. In addition, the change of parity of the number of rollers under pressure also affects the deformation of the bearing rings in the loading zone.

As the rollers pass through the loading zone, the position of the rollers periodically changes between the two states shown in Fig. 6. The two states are the two extreme positions of the roller in the loading zone. The first extreme position is shown in Fig. 6a, the number of rollers under pressure is odd, that is, only one roller is located directly above the load line. The second extreme position is shown in Fig. 6b, the rollers under pressure are even and symmetrically distributed on both sides of the load line.

Fig. 6
figure 6

Two extreme positions of rollers in loading zone: a extreme position a and b extreme position b

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Under the inclined load, the contact deformations of the outer ring are different in the two extreme positions. When the roller rotates, the outer ring in the loading zone will be subjected to cyclic strain with the passing frequency of the roller as the period. When the roller moves from state a to state b and then back to state a, the deformation of the outer ring forms a cycle. This cyclic strain will cause elastic deformation of the outer ring and even plastic deformation when the inclined load is severe.

When the bearing operates under the inclined load, the clearance along the axis of the roller will present a phenomenon that one end is positive clearance and the other end is negative clearance. During operation, the alternating appearance of the two extreme positions of the rollers in the loading zone will cause radial runout of the bearing, which will cause the fatigue deformation of the rings. Under the inclined load, the change in the number of rollers in the loading zone will have a more serious influence on the vibration and ring deformation of the roller bearing. Later in this paper, the ring deformation under different declination angles of the two extreme states will be compared and analyzed.

2.5 Model validation and results discussion

To verify the roller bearing analysis program, the simulation results are compared with the experimental results of Kannel and Hartnett [8]. In their experiment, the two rollers are made of steel, with a diameter of 36 mm and a length of 16 mm. The specific profile modification method can be found in Ref. [8]. The profile modification in the original paper is extracted by a discrete method and used as the input condition. The maximum declination angle is 1.25 × 10−3 rad, which is equal to 4.3′. The test speed is 5000 rpm, and the applied load is 2240 N. The number of strip elements is 81, and the contact pressure distribution along the length of the roller under different declination angles is shown in Fig. 7. The theoretical results of the simulation program are basically consistent with the experimental results, which verifies the validity of the theoretical model and the number of strip elements.

Fig. 7
figure 7

The contact pressure distribution of roller with declination angle: a 0 rad, b 0.2 × 10−3 rad, c 0.86 × 10−3 rad, and d 1.25 × 10−3 rad

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Next, the model is used to analyze the contact characteristics of the test roller bearing (N209E) in the subsequent experiments under inclined load, mainly the contact pressure distribution of the maximum load roller. The number of slice elements is still 81, and the radial load includes 1 kN, 2 kN, and 3 kN. The declination angle mainly includes 0°, 0.05°, 0.1°, and 0.2°.

As can be seen from Fig. 8, as the declination angle increases from 0° to 0.2°, the maximum load roller contact pressure increases from 583 to 1819 MPa under a radial load of 1 kN, an increase of 1236 MPa. Under the radial load of 3 kN, the maximum load roller contact pressure increased from 1093 to 2631 MPa, an increase of 1538 MPa. It shows that the bearing declination will significantly increase the edge contact pressure of the rollers, and the greater the radial load, the more obvious the edge contact pressure increases. In addition, the radial load can significantly change the contact pressure distribution. When the declination angle is 0.05°, the contact length of the roller and the inner ring is 6.3 mm at 1 kN. While at 3 kN, the contact length is the effective length of the rollers.

Fig. 8
figure 8

The contact pressure distribution between maximum load roller and inner ring with declination angle: a 0°, b 0.05°, c 0.1°, and d 0.2°

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3 Analysis of contact characteristics based on FEM

To explore the contact characteristics of the bearing under the inclined load and the influence of the inclined load on the deformation of the rings, this paper uses the finite element model (FEM) to study the problem before carrying out experiment. The FEM results will serve as the link between the quasi-static model analysis results and the ring deformation experiment results. On the one hand, the optimal boundary conditions and mesh size are determined by comparing the FEM results with the theoretical model results. On the other hand, the verified FEM is used to analyze the influence of the inclined load on the deformation of the rings, and the FEM simulation results will be compared with experiment.

3.1 Finite element modeling

The finite element model parameters are consistent with the test bearing structural parameters used in the subsequent experiment, and the specific dimensions are shown in Table 1. When establishing the FEM, the radial clearance of the bearing and the cage are not considered to reduce the number of contact surfaces and mesh elements and improve the calculation efficiency. To analyze the deformation of the outer ring, the bearing sleeve is considered, and its outer diameter is 120 mm and width is 39 mm, which is consistent with the real structure size in experiment. In the FEM, the bearing sleeve is regarded as the bearing housing. The established bearing housing finite element model is shown in Fig. 9.

Table 1 The specific parameters of test bearing
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Fig. 9
figure 9

Bearing housing finite element model and its boundary conditions

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The finite element model mainly includes the roller bearing and the cylindrical sleeve outside the outer ring. The purpose of considering the sleeve is to analyze the influence of the bearing inclined load caused by the deflection of the housing on the geometric deformation of the outer ring. In experiment, the sleeve will deflect due to the deflection of the housing. Therefore, in the FEM analysis, only the bearing sleeve is considered to simplify the analysis.

The number of elements is 348145, the number of nodes is 1536843, and the minimum mesh size is 0.2 mm. The rollers are in frictional contact with the inner and outer rings, and the friction coefficient is 0.1. There is also frictional contact between the outer ring and the housing, and the friction coefficient is 0.2. The outer surface of the housing restricts translation in three directions, but allows rotation in three directions to facilitate the addition of the deflection angle of the housing. Limit the rotation of the rollers along the Y-axis. Limits the translation of the inner ring in the X- and Y-directions. The bearing load is applied to the lower half surface of the inner ring, and the joint load is applied to the outer surface of the housing, specifically the rotation around the X axis. Two paths are established, respectively, at the highest position in the Z-axis (corresponding to the extreme position b) and the lowest position in the Z-axis (corresponding to the extreme position a) of the inner surface of the outer ring to extract the deformation of the outer ring along the Z-direction.

3.2 Results and discussion

3.2.1 Contact characteristics of roller

Next, the contact characteristics of roller bearings under inclined load are mainly analyzed from the two aspects, maximum load roller contact pressure and rollers contact force distribution. For the maximum load roller contact pressure, the FEM results and the theoretical calculation results are compared at the declination angle of 0° and 0.05°, as shown in Fig. 10.

Fig. 10
figure 10

Contact pressure of the maximum load roller: a declination 0° and b declination 0.05°

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When there is no declination, the FEM results of the maximum load roller contact pressure are basically consistent with the theoretical results. For the declination condition, the FEM results of the contact pressure between the roller and the ring in the middle contact area are slightly larger than the theoretical results, but the maximum edge contact pressure is basically consistent. Both results verify the correctness of the theoretical analysis model and the number of strip elements, as well as the validity of the FEM and its boundary conditions. For further verification of the FEM, the contact load of the rollers in the loading zone under different radial loads, and compared with the theoretical results, as shown in Fig. 11. The FEM results of the contact load under different radial loads are consistent with the theoretical results, which further verify the accuracy of the FEM. Therefore, the contact and deformation of a single roller and the whole bearing can be effectively calculated using the FEM.

Fig. 11
figure 11

Contact force distribution of rollers in the loading zone without declination

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The internal contact load distribution of the roller bearing under different deflection angles is obtained by using the FEM, as shown in Fig. 12. The inclined load of the roller bearing will lead to the occurrence of the double-loading zones, which is sometimes referred to as back loading zone [28, 29]. The maximum roller contact force increases from 294.36N when at 0° declination angle to 3748.9N when at 0.4° declination angle. The declination angle is approximately linear with the contact force of the maximum load roller. When the declination angle increases to 0.4°, the contact force of the maximum load rollers in the upper and lower loading zones is approximately equal. The specific contact force of rollers is shown in Table 2. When the bearing rotates, the periodic change of the contact force in the double-loading zones will affect the dynamic characteristics of the rotor.

Fig. 12
figure 12

Contact force distribution between rollers and outer ring under different deflection angles

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Table 2 Contact force between roller and outer ring under different deflection angles (the negative sign indicates that the contact force is in the opposite direction to the radial load)
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3.2.2 Deformation of the outer ring in double-loading zones

The deformations of the outer ring with the declination angle of 0.1°, 0.2°, 0.3°, and 0.4° are shown in Fig. 13. When the outer ring is subjected to the inclined load, the inner surface of the outer ring containing the roller extreme position a and the upper half of the outer ring containing the roller extreme position b generate tilting deformation. The maximum deformation of the outer ring occurs in the upper half of the outer ring containing the extreme position b. When the radial load is 1 kN and the deflection angle is 0.1°, the ring in the upper loading zone shrinks inward by 16.259 μm, which is the maximum deformation of the outer ring in the vertical direction, and the ring in the lower loading zone shrinks inward by 16.143 μm. When the deflection angle is 0.4°, the outer ring in the upper loading zone shrinks inward by 64.805 μm, and the ring in the lower loading zone shrinks inward by 64.129 μm.

Fig. 13
figure 13

Deformation of the outer ring at the declination angle of a 0.1°, b 0.2°, c 0.3°, and d 0.4°

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Without regard to the transition filets on both sides of the inner surface of the outer ring, the effective length of the N209 roller bearing is 13 mm. Take the surface deformation of the effective length of the outer ring for normalization. Figure 14 shows the deformation of the inner surface of the outer ring when the rollers are at the two extreme positions.

Fig. 14
figure 14

Deformation of the outer ring under different declination angles at a extreme position a and b extreme position b

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As can be seen from Fig. 13, as the declination angle increases, the deformation of the inner surface of the outer ring gradually increases. For the extreme position a at the lower part of the inner surface of the outer ring, the deformation is shown in Fig. 14a. Since there is a roller directly above the load line in the loading zone, the inner surface of the outer ring at this position is in direct contact with the roller, and the surface deformation is resisted. For the extreme position b on the upper part of the inner surface of the outer ring, the deformation is shown in Fig. 14b. Since there is no roller at the highest point, the surface deformation is a pure oblique line.

The absolute deformations of the inner surface of the outer ring under different declination angles at two extreme positions are shown in Table 3. With the increase of the declination angle, the absolute deformation at two extreme positions basically shows a linear trend. When the declination angle is less than 0.05°, the absolute deformation at extreme position a is larger than that at extreme position b. When the declination angle is greater than 0.05°, the absolute deformation at extreme position a is smaller than that at extreme position b, and the difference between the absolute deformation at the two positions gradually increases.

Table 3 The absolute deformation of the inner surface of the outer ring under different declination angles (the distance between the highest point and the lowest point along the Y-direction)
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4 Experiment of bearing partial wear

4.1 Test equipment

The partial wear test of the roller bearing is mainly carried out using a bearing-rotor test rig in Fig. 15. This experiment mainly explores the partial wear mechanism of the roller bearing and analyzes the influence of the partial wear on the surface roughness and deformation of the outer ring. The structure and main body size of the test rig are shown in Fig. 15. The test rig includes three parts, rotor system, oil lubrication system, and measurement system.

Fig. 15
figure 15

The structure of the test rig and its main dimensions

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For the bearing-rotor system, the non-drive end bearing is the test bearing, N209E roller bearing. The bearing size is 45 × 85 × 19 mm, and the number of rollers is 15. The detailed dimensions are shown in Table 1. The drive end bearing adopts an angular contact ball bearing. The elastic rope is used for flexible connection between the drive motor and the coupling to reduce the influence of the vibration of the motor on the shaft. The motor and the housings were installed on a steel platform. The loading method adopts centrifugal loading caused by the unbalance of the disk, and the unbalance amount is 9 × 110 g mm. The test bearing was lubricated with the forced oil lubrication to avoid deformation caused by overheating.

The method to realize the partial wear of the bearing is to place the steel strips between the housing and the platform where the test bearing is located. The housing is inclined relative to the vertical direction, and then, the outer ring and the inner ring of the bearing are driven to form a relative deflection angle. The thickness of the used single steel strip is 0.25 mm, with a total of four strips on one side of the housing. The theoretical relationship between the thickness of the strips and the deflection angle θ between the inner and outer rings is θ = arctan(b/l), where b is the thickness of the steel strips, and l is the width of the bearing housing. Therefore, the strips will make the housing theoretically produce a deflection angle of θ = arctan(4 × 0.25/140) = 0.4°, which will theoretically produce a 0.4° angle between the axes of the inner and outer rings. However, due to the assembly tightness of the cover of the housing and the assembly clearance, the actual angle is slightly smaller than the theoretical value. The test measures the vibration of the rotating shaft and the acceleration signal of the housing. After the test, the surface topography of the disassembled bearing is observed with a 3D measuring laser microscope.

The rotational speed is 2400 rpm and runs stably for 3 h. After the test, disassemble the test bearing and observe the wear topography of the outer raceway, as shown in Fig. 16. The black marked point is used to determine the installation position of the outer ring. It can be seen that there are two clear bright bands on the raceway surface of the outer ring, which indicates that the partial wear occurs. Define the position of the marked point as 180°, and draw the outer raceway wear area. It can be seen that the outer raceway exhibits the characteristics of double-loading zone wear, which is consistent with the theory. The double-loading zone wear areas are in fixed angular ranges, which is determined by the direction of deflection angle and the direction of rotation. The wear area is not symmetrically distributed around the marked point, but is biased to one side, which is caused by the rotation of the shaft. This means that the inner ring is not directly below when it rotates, and the inner ring with the cage and rollers tends to climb up along the direction of rotation.

Fig. 16
figure 16

Wear topography of bearing outer raceway under the inclined load

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4.2 Topography measurement

To reveal the partial wear mechanism of the roller bearing in detail, wire cutting sampling was carried out on the severely wear area of the outer ring. The three-dimensional surface topography and two-dimensional surface roughness of the inner surface of the outer ring were measured using a 3D measuring laser microscope (LEXT OLS 4100) for the test specimen. Before the measurement, considering that the residue of lubricating oil and metal chips on the surface of the specimen may adversely affect the observation results, anhydrous ethanol was used to clean the measurement surface. After the measurement, the three-dimensional topography in the micro-nano-scale was analyzed by special software, and the two-dimensional roughness value of the specimen surface was measured according to the ISO 4287:1997 geometric product specification standard.

The surface of the outer raceway of the bearing after partial wear for 3 h was observed microscopically. Obvious polishing wear caused by the inclined load can be seen in Fig. 17. Use special software to extract the surface topography of the outer raceway of the test bearing along the effective width in the axial direction, as shown in Fig. 18. It can be seen that obvious plastic deformation occurred on the inner surface of the outer ring. After the partial wear occurs, the outer ring on the side with the greater contact pressure is deformed, and the inner diameter of the maximum contact pressure position decreases by about 45 μm inward. The deformation of the outer ring is mainly caused by the additional bending moment caused by the deflection of the housing, that is, the housing makes the outer ring cause elastic and plastic deformation. In addition, the surface topography peaks in the polishing wear area are lower than those at other positions, indicating that the surface roughness at the maximum contact pressure position (bright band) is reduced.

Fig. 17
figure 17

Surface topography of the outer raceway of the test bearing

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Fig. 18
figure 18

Deformation and roughness of the outer raceway in the axial direction

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To study the wear of the outer raceway in detail, the roughness of the outer raceway surface of the test bearing along the axial direction is extracted. The sampling length is 0.25 mm according to the ISO 4287:1997 standard, and a total of 5 sections are taken, which are marked as S1–S5. The starting points of each section are, respectively, S1(0 mm-no wear area), S2(4 mm), S3(8 mm), S4(polish wear area), and S5(12.5 mm-no wear area). Set three different sampling directions, marked as D1, D2, and D3, as shown in Fig. 17, and the distance between the two directions is about 0.4 mm.

In the D1 direction, the surface roughness of the five positions is shown in Fig. 19. The surface roughness peaks in the polishing wear area are about 5-10 μm, while other positions are between 0 and 15 μm. The Ra and Rz values of surface roughness of five positions in three directions were calculated, respectively, as shown in Table 4. The inclined load reduces the surface roughness, resulting in gradient distribution of the surface roughness value. The Ra values of the polishing wear area are in the range of 0.5–0.7, while the Ra values of other positions are in the range of 1.1–1.7. The Rz values of the polishing wear area are in the range of 4.2–5.0, while the Rz values of other positions are in the range of 7.0–11.0. In addition, along the D1 direction, the surface roughness of the four positions between S1 and S4 has a gradually decreasing trend. It means that the wear gradually increases along the direction where the contact pressure of the roller increases gradually, which is consistent with the theory. Due to the short wear time, this paper does not consider the effect on the wear volume, but only considers the roughness variation.

Fig. 19
figure 19

Surface roughness of different sampling positions in the D1 direction

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Table 4 Surface roughness of different sampling positions in three directions
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The surface deformation over the effective length of the outer raceway was normalized and compared with the FEM simulation results in Fig. 20. After partial wear, the inclined load causes plastic deformation of the outer ring. The experiment results are consistent with the changing trend of the FEM simulation results. In the experiment results, the inner diameter at the maximum contact pressure position decreased by about 45 μm. For the FEM simulation results, the deformation of the maximum contact pressure position is about 67 μm when the deflection angle is 0.4°. Compared with the simulation results, the experiment results are smaller, this is because the measured deformation is the plastic deformation after the test bearing is disassembled. However, the simulation result includes plastic deformation and elastic deformation. In addition, due to assembly reasons, there are gaps in components in the test rig, so the actual angle between the axes of the inner and outer rings caused by the steel strips is smaller than the theoretical declination angle of 0.4°.

Fig. 20
figure 20

Normalized deformation of the inner surface of the outer ring along the axial direction (dotted line is the test, solid line is the simulation)

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In addition, after passing through the polishing wear area, the surface deformation at the N1 position has a sudden increasing trend in Fig. 20, which is consistent with the simulation results. This is due to the contact fatigue deformation between the end of the roller and the outer raceway due to the cyclical alternation of the rollers at the two extreme positions. Therefore, the misalignment of the roller bearing assembly will cause elastic deformation of the outer raceway, and even plastic deformation in severe cases. The plastic deformation will continue to cause partial wear of the bearing during the subsequent service process.

5 Conclusions

Poor assembly and rotor operating conditions may cause the partial wear and deformation of the roller bearings. Severe partial wear will cause elastic deformation and plastic deformation of the rings, and the plastic deformation will have a serious impact on the bearing contact characteristics and the dynamics of rotor systems. Based on the influence coefficient method, a quasi-static model of the roller bearing considering the inclined load is established. The finite element model of the roller bearing housing is established and verified by the quasi-static results, and the deflection of the housing is considered. The influence of the two extreme positions of the rollers in the loading zone on the deformation of the outer ring is analyzed. Combining the analytical analysis and finite element analysis results, the roller-bearing contact performances under the inclined load are evaluated. The results show that the roller bearing under the inclined load has double-loading zones and the ring deformation is affected by rollers extreme positions. Based on a bearing-rotor test rig, the partial wear test of the roller bearing is carried out. The experiment results show that the outer ring has obvious partial wear in the double-loading zones. The morphological analysis of the wear surface is carried out using a 3D measuring laser microscope. The results show that under the inclined load, the outer ring deformation is obvious and affected by rollers position, which is consistent with the FEM results. In addition, the contact surface roughness of the outer ring is extracted, and the phenomenon of polishing wear caused by the inclined load is found, and the roughness at the polishing wear area is smaller than those at other areas, and the roughness value in contact area arises gradient distribution.

The researches carried out in this paper lay a foundation for the analysis of the vibration characteristics of rotor systems caused by bearings partial wear. For example, the surface roughness results in this paper can be used to establish an accurate roller-bearing contact model to obtain more realistic contact performances and fulcrum characteristics under the inclined load. Future research will focus on analyzing the influence of the nonlinear bearing force under rings deformation due to the inclined loads on the vibration characteristics of rotor systems.