Open Access
 Issue Mechanics & Industry Volume 23, 2022 19 13 https://doi.org/10.1051/meca/2022019 01 August 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## 1 Introduction

Cylindrical roller bearing mainly plays the role of rotation and support in the engine [1]. The cylindrical roller bearing with circular raceway has a greater risk of skidding under high-speed and low-load conditions [2]. It will generate too much heat, leading to the sliding damage of the bearing and further causing the engine to fail to run properly. To avoid effectively the bearing sliding damage [3], the outer raceway of the bearing is improved from a circular shape to a triple-lobe shape. When the bearing is mounted with interference, the radial clearance [4] relative to the low point circle of the outer triple-lobe raceway becomes negative. Three effective preload areas are formed in the circumferential direction of bearing, and the contact load between the rollers and the inner raceway increases, thereby reducing the risk of skidding.

Deng et al. [17] established the nonlinear dynamic differential equations of cylindrical roller bearing with triple-lobe raceway and analyzed the influences of the different raceway structures, working conditions and outer ring mounted angles on the cage slip ratio. However, when the bearing rotates, the change in the number of loaded rollers and the contact load is not considered, and the preload state of the bearing is not analyzed.

In this paper, considering the radial load and moment conditions, a quasi-static model of cylindrical roller bearing with triple-lobe raceway is established and an improved Newton-Raphson method to solve the numerical model is proposed. For the two states of single-pressed and double-pressed, the contact load distribution of bearing with triple-lobe raceway is analyzed, and the effects of mounting radial clearance, outer triple-lobe raceway waveform value, radial load and rotational speed on inner raceway maximum contact load and preload are investigated. At the same time, the bearing temperature trend predicted by the preload characteristic is verified by the test. It shows the numerical model is effective.

## 2 Numerical model of cylindrical roller bearing with triple-lobe raceway

### 2.1 Assumptions

To facilitate the analysis of bearing preload characteristics, the numerical model adopts the following assumptions:

• The outer ring is fixed in space and the inner ring rotates at a constant speed with a fixed axis.

• Each part of the bearing is rigid and only local contact deformation is considered.

• The numerical model ignores the effects of lubrication, friction, heat and cage on the bearing.

• The linear velocity of the contact point of the roller with the inner raceway and the outer raceway is equal to the linear velocity of the raceway at that point.

### 2.2 Outer raceway triple-lobe curve

The triple-lobe wave curve is periodic in the circumferential direction. To establish the numerical model of cylindrical roller bearing with triple-lobe raceway, the equation of triple-lobe is proposed. The schematic diagram of the triple-lobe profile is shown in Figure 1.

The polar coordinate equation of the triple-lobe curve is:(1)

 Fig. 1The schematic diagram of the triple-lobe profile.

### 2.3 Force analysis of bearing with triple-lobe raceway

The cylindrical roller bearing with triple-lobe raceway will generate internal contact loads at the three low point areas of the waveform after mounting. During the bearing operation, the maximum contact load position is always changing between one position (roller is facing the low point of the triple-lobe raceway) and the other position (roller moves half of the roller position angle from the low point of the triple-lobe raceway). The load state that the roller is facing the low point of the triple-lobe raceway is defined as “single-pressed”, The load state that the roller moves half of the roller position angle from the low point of the triple-lobe raceway is defined as “double-pressed”. The contact loads of single-pressed and double-pressed states are two extreme loads. The contact load of the transition region is between the two extreme contact loads.

The preload distribution of cylindrical roller bearing with triple-lobe raceway before loading in the single-pressed and double-pressed states is shown in Figure 2. When the bearing is subjected to an external load, the internal contact load will change. The preload distribution of cylindrical roller bearing with triple-lobe raceway after loading in the single-pressed and double-pressed states is shown in Figure 3.

In Figure 3, O 1 and O 2 are the center points of the inner ring before loading and after loading, F r is the radial load, δ r is the radial displacement of the inner ring after loading, ψ j is the position angle of the jth roller, ω i is the inner ring angular speed, G r is the radial clearance relative to the low point circle of the outer triple-lobe raceway, which can be expressed as(2)

Assuming that the number of rollers is Z, the position angle [18] of the jth roller is:(3)when the roller rotates with the inner ring, the position angle of the roller will change, and the value of ψ 1 needs to be adjusted in the numerical model.

 Fig. 2Preload distribution of the cylindrical roller bearing with triple-lobe raceway before loading: (a) single-pressed state; (b) double-pressed state.
 Fig. 3Preload distribution of the cylindrical roller bearing with triple-lobe raceway after loading: (a) single-pressed state; (b) double-pressed state.

#### 2.3.1 Force analysis of roller

When the bearing is subjected to the radial load F r , moment M and high-speed centrifugal load, the inner ring will happen a radial displacement δ r and a misalignment angle θ relative to the outer ring [19], and the contact load at each roller position will vary. Considering the profile modification of rollers, the slicing method [20] is used in the numerical model. The roller effective length is L we , the number of slices is N, and the roller slice thickness is λ. The schematic diagram of the bearing contact model is shown in Figure 4.

The force balance equations [21] of the jth roller are:(4)

The centrifugal force of the jth roller is(5)

The contact load and moment of the jth roller on the inner raceway and outer raceway are:(6)

The contact slice loads of the kth slice of the jth roller on the inner raceway and outer raceway are:(7)

The contact elastic deformations of the kth slice of the jth roller on the inner raceway and outer raceway are:(8)

The relationship between the contact deformation of the jth roller relative to the inner raceway, the contact deformation of the jth roller relative to the outer raceway and the inner ring radial displacement δ r is:(9)

The relationship between the misalignment angle of the jth roller relative to the inner raceway, the misalignment angle of the jth roller relative to the outer raceway, and the inner ring misalignment angle θ is:(10)

 Fig. 4Schematic diagram of bearing contact model: (a) bearing cross-section geometry and roller force; (b) schematic diagram of slicing method; (c) schematic diagram of contact deformation.

#### 2.3.2 Overall force analysis of bearing

When the bearing is subjected to the radial load F r and moment M, the external load is shared between each roller and inner raceway. The contact load between the roller and the inner raceway is accumulated to form the overall force balance equations of bearing [22,23], it is expressed as(11)

### 2.4 Solution of the numerical model

Substituting equation (5) through equation (10) into equation (4) and equation (11), it forms an equation matrix [24] with , , δ r , θ totaling (2Z+2) unknowns, and the matrix is defined as [F (x  (t))]. The matrix [F (x  (t))] is expressed as(12)

The unknown matrix [x  (t)] is expressed as(13)

Taking the derivative of the matrix [F (x  (t))] to the unknown matrix, the (2Z+2)×(2Z+2) order Jacobian matrix can be obtained, which is defined as .

To obtain good convergence, a variable correction coefficient h(0 < h < 1) is introduced based on the Newton-Raphson method. The improved Newton-Raphson method [25] is expressed as(14)

The solution procedure of the numerical model is shown in Figure 5.

1) The improved Newton-Raphson method has a faster calculation speed, but it is still sensitive to the initial value. The initial value setting needs to first solve the initial radial displacement δ r (0) and initial misalignment angle θ  (0) of the inner ring according to bearing design parameters, radial load, moment and rotational speed.(15) (16)

If θ  (0) ≤ 0, set θ  (0) = 1 × 10−10.

2) According to the initial radial displacement δ r (0) and initial misalignment angle θ  (0) of the inner ring, further estimate the contact deformation and misalignment angle of the jth roller relative to the outer raceway.(17)

If , set .(18)

3) Bring the initial value into the equation matrix [F (x  (t))] and Jacobian matrix to calculate the new initial value by equation (14), and start the iterative calculation. It will stop until the calculation error Δx = |x  (t+1) − x  (t)| is less than the allowable error ϵ, and the number of iteration steps is less than the number of set iteration steps S. When it does not converge after reaching the number of set iteration steps S, the iterative calculation is performed again by reducing the correction coefficient h.

 Fig. 5The solution procedure of numerical model.

## 3 Analysis and discussion

The comparison between the improved Newton-Raphson method and the original Newton-Raphson method is shown in Table 2. The improved Newton-Raphson method has better convergence than the original Newton-Raphson method.

Since the effect of lubrication is ignored in the numerical model, the Hamrock and Dowson theory is used to further analyze the lubricating oil film thickness. Under the above conditions, the average temperature monitored by the bearing test is 123.5 °C, so the MIL-PRF-23699 lubricating oil properties at 123.5 °C are used. The results of contact deformation and oil film thickness are shown in Table 3. It can provide a reference for future scholars to supplement the numerical model.

Table 1

Bearing parameter.

 Fig. 6The contact load distribution: (a) single-pressed state; (b) double-pressed state.
Table 2

Comparison between the improved Newton-Raphson method and the original Newton-Raphson method.

Table 3

The results of contact deformation and oil film thickness.

### 3.1 Effect of mounting radial clearance on maximum contact load and preload of the inner raceway

 Fig. 7The maximum contact load variation of the inner raceway and the number of loaded rollers variation under the different mounting radial clearances: (a) single-pressed state; (b) double-pressed state.

### 3.2 Effect of outer raceway waveform value on maximum contact load and preload of the inner raceway

 Fig. 8The maximum contact load variation of the inner raceway and the number of loaded rollers variation under the different outer raceway waveform values: (a) single-pressed state; (b) double-pressed state.

 Fig. 9The maximum contact load variation of the inner raceway and the number of loaded rollers variation under the different radial loads: (a) single-pressed state; (b) double-pressed state.

### 3.4 Effect of rotational speed on maximum contact load and preload of the inner raceway

 Fig. 10The maximum contact load variation of inner raceway and the number of loaded rollers variation under the different inner ring rotational speeds: (a) single-pressed state; (b) double-pressed state.

## 4 Verification

The cylindrical roller bearing with triple-lobe raceway mainly reduces the internal sliding frictional heat through preload, preventing the bearing from overheating and damage under high-speed and light-load conditions. When the bearing preload is insufficient or excessive, the bearing temperature will increase. Therefore, the temperature of the cylindrical roller bearing with triple-lobe raceway can also reflect the bearing preload.

We average the two states in Figure 7 to obtain the preload characteristics and predict the temperature trend of bearing under the different mounting radial clearances, which is shown in Figure 11. As the mounting radial clearance increases, the predicted temperature first decreases and then increases.

We average the two states in Figure 8 to obtain the preload characteristics and predict the temperature trend of bearing under the different outer raceway waveform values, which is shown in Figure 12. As the outer raceway waveform value increases, the predicted temperature first decreases and then increases.

To verify the predicted temperature trend, bearing tests with different mounting radial clearances and outer raceway waveform values are carried out to monitor the bearing temperature. The basic parameters of the bearing are shown in Table.1, the variable parameters of the test bearing are shown in Table 4, and the layout of the tester is shown in Figure 13. The test uses two preloaded ball bearings as support bearings. The lubricant type is MIL-PRF-23699, the lubrication method is jet lubricated, the oil flow is 1.5 L/min, and the inlet oil temperature is 75 °C. Bearing temperature is measured to use two similar size cylindrical roller bearings with triple-lobe raceway per test, and the average temperature of bearing in the stable phase is used as the final temperature due to the large amount of bearing temperature data. The load on the two support bearings is 200 N, and the shaft rotational speed is 40,000 r/min. Under the current test conditions, the bearing temperature under the different mounting radial clearances is shown in Figure 14, and the bearing temperature under the different outer raceway waveform values is shown in Figure 15. The test temperature trend is consistent with the predicted temperature trend of the preload characteristics. It shows that the larger preload, the preload absence or the more loaded rollers will generate a lot of heat and cause the temperature to rise.

 Fig. 11The preload characteristics and temperature trend prediction of bearing under the different mounting radial clearances.
 Fig. 12The preload characteristics and temperature trend prediction of bearing under the different outer raceway waveform values.
Table 4

Variable parameters of the test bearing.

 Fig. 13Layout of the tester.
 Fig. 14Bearing temperature under the different mounting radial clearances: (a) Time point temperature in the stable phase; (b) average temperature.
 Fig. 15Bearing temperature under the different outer raceway waveform values: (a) Time point temperature in the stable phase; (b) average temperature.

## 5 Conclusions

In the paper, the quasi-static numerical analysis model of cylindrical roller bearing with triple-lobe raceway is established and solved by the improved Newton-Raphson method. The effects of mounting radial clearance, outer raceway waveform value, radial load and rotational speed on the maximum contact load and preload of cylindrical roller bearing with triple-lobe raceway are investigated. The temperature trend of the test is consistent with the predicted temperature trend of preload characteristics. It shows that the larger preload, the preload absence or the more loaded rollers will generate a lot of heat and cause the temperature to rise.

• A quasi-static analysis model of cylindrical roller bearing with triple-lobe raceway is established, and an improved Newton-Raphson method to solve the model is proposed.

• According to the radial load and rotational speed conditions of the bearing, the double-pressed state needs to be considered in designing the outer raceway triple-lobe structure to ensure the bearing preload always exists.

• In the design of the mounting radial clearance value of cylindrical roller bearing with triple-lobe raceway, it is necessary to consider first the double-pressed state to ensure the bearing preload always exists.

• In the design of the lower limit of the outer raceway waveform value, it is necessary to consider the single-pressed state to prevent the bearing overheating, and in the design of the upper limit of the outer raceway waveform value, it is necessary to consider the double-pressed state to ensure the bearing preload always exists.

## Nomenclature

r : Radius of triple-lobe curve base circle

ee : Triple-lobe curve waveform value

G r : Bearing radial clearance

D w : Roller diameter

d i : Inner raceway diameter

m w : Mass of the roller

d m : Diameter of the pitch circle

L we : Roller effective length

λ : Thickness of roller slice

C: Roller convexity drop amount

M : Moment

δ r : Radial displacement of inner ring

θ : Misalignment angle of inner ring

ψ : Roller position angle

ω i : Inner ring angular speed

ω mj : Revolution angular speed of the roller j

T : Contact moment

F Cj : Centrifugal force of roller j

K n : Bearing contact stiffness

Z : Number of rollers

N : Number of slices

: Deformations at the roller/inner raceway contact

: Deformations at the roller/outer raceway contact

: Misalignment angle at the roller/inner raceway

: Misalignment angle at the roller/outer raceway

K : Contact stiffness

x  (t) : Unknown matrix

[F (x  (t))]: Matrix of equations

: Derivation of equations matrix

h: Variable correction coefficient

Subscripts

i : Inner raceway

o : Outer raceway

j : Rolling element index

k : Slice index

t : Iteration index

Matrix notations

[]: Matrix

[] T : Transposed matrix

[]−1 : Inversed matrix

## Acknowledgments

The authors gratefully acknowledge the financial support by an independent special fund from the China Aviation Engine Corporation (ZZCX-2018-048) in Beijing, China.

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Cite this article as: Q.J. Yu, J. Chi, P. Gong, H.W. Jiang, L.W. Zhan, L.L. Xue, Study on contact load and preload characteristics of cylindrical roller bearings with triple-lobe raceway, Mechanics & Industry 23, 19 (2022)

## All Tables

Table 1

Bearing parameter.

Table 2

Comparison between the improved Newton-Raphson method and the original Newton-Raphson method.

Table 3

The results of contact deformation and oil film thickness.

Table 4

Variable parameters of the test bearing.

## All Figures

 Fig. 1The schematic diagram of the triple-lobe profile. In the text
 Fig. 2Preload distribution of the cylindrical roller bearing with triple-lobe raceway before loading: (a) single-pressed state; (b) double-pressed state. In the text
 Fig. 3Preload distribution of the cylindrical roller bearing with triple-lobe raceway after loading: (a) single-pressed state; (b) double-pressed state. In the text
 Fig. 4Schematic diagram of bearing contact model: (a) bearing cross-section geometry and roller force; (b) schematic diagram of slicing method; (c) schematic diagram of contact deformation. In the text
 Fig. 5The solution procedure of numerical model. In the text
 Fig. 6The contact load distribution: (a) single-pressed state; (b) double-pressed state. In the text
 Fig. 7The maximum contact load variation of the inner raceway and the number of loaded rollers variation under the different mounting radial clearances: (a) single-pressed state; (b) double-pressed state. In the text
 Fig. 8The maximum contact load variation of the inner raceway and the number of loaded rollers variation under the different outer raceway waveform values: (a) single-pressed state; (b) double-pressed state. In the text
 Fig. 9The maximum contact load variation of the inner raceway and the number of loaded rollers variation under the different radial loads: (a) single-pressed state; (b) double-pressed state. In the text
 Fig. 10The maximum contact load variation of inner raceway and the number of loaded rollers variation under the different inner ring rotational speeds: (a) single-pressed state; (b) double-pressed state. In the text
 Fig. 11The preload characteristics and temperature trend prediction of bearing under the different mounting radial clearances. In the text
 Fig. 12The preload characteristics and temperature trend prediction of bearing under the different outer raceway waveform values. In the text
 Fig. 13Layout of the tester. In the text
 Fig. 14Bearing temperature under the different mounting radial clearances: (a) Time point temperature in the stable phase; (b) average temperature. In the text
 Fig. 15Bearing temperature under the different outer raceway waveform values: (a) Time point temperature in the stable phase; (b) average temperature. In the text

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