Optimization analysis on assembly interference of cylindrical roller bearings

Abstract

Distribution of loads and contact stress has a great influence on fatigue life of bearings. In view of the fact that the nonload area is ubiquitous in cylindrical roller bearings, there is a method for prolonging the fatigue life design by adjusting the working clearance of cylindrical roller bearings. In this article, optimization model of fatigue life of cylindrical roller bearings is developed with a quasi-dynamic method and calculated method of bearing life based on micro-area contact, considering the effect of assembly interference, temperature rise, and high-speed centrifugation. Aiming to the different geometry parameters and operating parameters of bearings, the effects of assembly interference on bearing fatigue life are discussed. The results show that the optimum fatigue life of roller bearings is achieved at negative working clearance with the loads distributed evenly within half ring. The optimum working clearance of roller bearings is not influenced by radial load. But with the increase in pitch diameter or decrease in rolling elements, the optimum assembly interference increases, and accelerating revolution of roller bearings will increase the optimum assembly interference as well.

Introduction

Initial clearance, one of the important design parameters of rolling bearings, influences the load-carrying properties and fatigue life of bearings directly. ¹ In the past several decades, a number of researches on roller modification have been reported,^2–4 and modification model of cylindrical roller was proposed, such as changing the fitness between roller and raceway, using different modification curves, and so on,^5,6 to make full use of roller length and improve edge stress concentration of rollers effectively. However, since cylindrical roller bearings only bear radial load, its initial clearance could not be eliminated through preloading such as ball bearings. There is still a nonloading region on cylindrical roller bearings in operation. Moreover, external factors caused by service environment and working conditions of bearings (e.g. bearing assembly, temperature rise, and centrifugal force) will make actual working clearance disagree with the initial clearance. As a result, working clearance of bearings is necessary to analyze dynamic properties and predict fatigue life of bearings accurately.

Additionally, load distribution in loading region of the cylindrical roller bearings will directly cause different fatigue life of rollers at different azimuth angles. The ISO standard algorithm which calculates fatigue life of bearings based on rated dynamic load and equivalent dynamic load could not predict the fatigue life of bearings accurately. Under this circumstance, the overall fatigue life of bearings can only be determined by fatigue life of different rollers and the rings. ⁷ To ensure prediction accuracy of fatigue life, internal load of bearings is often calculated by quasi-dynamic analysis because complete dynamic analysis will complicate the calculation significantly, ⁸ and quasi-static method based on the hypothesis of ring control theory is inapplicable to high-speed bearings for its poor calculation accuracy. ⁹ Compared to other methods, quasi-dynamic analysis can provide satisfying calculation accuracy and efficiency. Therefore, quasi-dynamic method is used most widely in practical engineering. ¹⁰

To analyze the effect of working clearance of cylindrical roller bearings on internal load-carrying properties and fatigue life and prolong fatigue life of cylindrical roller bearings by optimizing the assembly interference, this article established a mathematical model based on theoretical modeling and simulation analysis to calculate the working clearance and predict fatigue life of bearings. Through performance analysis and optimization design of assembly interference, the influence law of assembly interference of cylindrical roller bearings on the fatigue life is discussed under different structural parameters and working conditions.

Working clearance analysis and fatigue life calculation of cylindrical roller bearings

Analysis of working clearance

Calculation of working clearance of bearings involves the effects of bearing assembly, temperature rise, and centrifugal force on bearing initial clearance

$${\delta }_w={\delta }_0+\Delta {\delta }_d+\Delta {\delta }_t+\Delta {\delta }_f$$

(1)

where ${\delta }_w$ and ${\delta }_0$ are working clearance and initial clearance of the bearings, respectively; and $\Delta {\delta }_d$ , $\Delta {\delta }_t$ , and $\Delta {\delta }_f$ are clearance variation caused by assembly interference, temperature rise, and centrifugal force, respectively.

Clearance variation caused by assembly interference

As Figure 1 shows, bearing installed through interference fit suffers deformations in both inner raceway and outer raceway, which will influence its initial clearance. Therefore, $\Delta {\delta }_d$ is the sum of deformation of inner raceway ( ${\delta }_s$ ) and deformation of outer raceway ( ${\delta }_q$ )

$$\Delta {\delta }_d=-({\delta }_s+{\delta }_q)$$

(2)

Figure 1.

Sketch of interference at bearing installation: ¹ (a) interference between inner ring and shaft and (b) interference between outer ring and bearing housing.

${\delta }_s$ and ${\delta }_q$ were calculated through interference between fitting surfaces (G ₁ and G ₂) and elastic mechanical parameters of fitting surfaces. ¹¹

Clearance variation caused by temperature rise

Temperature rise causes thermal expansion of bearing elements, thus influencing the bearing clearance. Under free state, $\Delta {\delta }_t$ is determined by diameter expansion of rings and rolling elements ¹¹

$$\Delta {\delta }_{t1}=\Delta D_{\mathrm{to}}-\Delta D_{\mathrm{ti}}-2\Delta D_{\mathrm{tr}}$$

(3)

The diameter expansion of bearing elements is calculated according to the coefficient of thermal expansion of corresponding material and temperature rise. ¹¹

In addition, diameter expansion of bearing elements caused by temperature rise will also affect the fitting surface, further changing the bearing clearance. Interference of fitting surfaces caused by thermal expansion is

$$G{\prime }_2=\Delta D_{\mathrm{ts}}-\Delta D_{\mathrm{ti}},G{\prime }_1=\Delta D_{\mathrm{to}}-\Delta D_{\mathrm{th}}$$

(4)

Based on the deformation compatibility condition in $\Delta {\delta }_d$ analysis, interferences on the inner ring ( $\delta {\prime }_s$ ) and outer ring ( $\delta {\prime }_q$ ) can be calculated. Clearance variation caused by the effect of temperature rise on fitting surfaces is

$$\Delta {\delta }_{t2}=-(\delta {\prime }_s+\delta {\prime }_q)$$

(5)

Therefore, $\Delta {\delta }_t$ is calculated as

$$\Delta {\delta }_t=\Delta {\delta }_{t1}+\Delta {\delta }_{t2}$$

(6)

Clearance variation caused by centrifugal force

Under free state, $\Delta {\delta }_f$ is determined by diameter changes of the inner ring and outer ring ¹¹

$$\Delta {\delta }_{f1}=\Delta D_{\mathrm{fo}}-\Delta D_{\mathrm{fi}}$$

(7)

Diameter changes of elements caused by the rotating speed ( $\omega$ ) can be calculated from the following equation ¹¹

$$\Delta D_{\mathrm{fn}}=\frac{2{\rho }_n{R}_n^3{\omega }_n^2}{E_{n}g}$$

(8)

where $\rho$ , $\omega$ , g, R, and E are density, angular velocity, gravitational acceleration, equivalent radius, and equivalent elastic modulus, respectively. The subscript n represents bearing outer ring, bearing inner ring, shaft, and bearing housing while it fetches o, i, s, and h, respectively.

Similarly, diameter changes of elements caused by high-speed centrifugal force will influence the fitting surfaces and thereby change the bearing clearance. Interference of fitting surfaces caused by centrifugal force is

$$G{″}_2=\Delta D_s-\Delta D_i,G{″}_1=\Delta D_{f0}-\Delta D_{\mathrm{fh}}$$

(9)

Based on the deformation compatibility condition in $\Delta {\delta }_d$ analysis, it is calculated that

$$\Delta {\delta }_{f2}=-\delta {″}_s-\delta {″}_q$$

(10)

Therefore, $\Delta {\delta }_f$ can be calculated as

$$\Delta {\delta }_f=\Delta {\delta }_{f1}+\Delta {\delta }_{f2}$$

(11)

Calculation model of internal contact load

External loads and coordinates of cylindrical roller bearings are shown in Figure 2. $\mathrm{OXYZ}$ and $O_2{X}_2{Y}_2{Z}_2$ are the coordinates of outer ring and inner ring, respectively. The outer ring is fixed, and external loads are applied on the inner ring. Suppose the relative displacements between inner ring axis and out ring axis are $[\begin{array}{cccc}\hfill {\Delta }_y\hfill & \hfill {\Delta }_z\hfill & \hfill {\theta }_y\hfill & \hfill {\theta }_z\hfill \end{array}]$ while external loads $[\begin{array}{cccc}\hfill F_y\hfill & \hfill F_z\hfill & \hfill M_y\hfill & \hfill M_z\hfill \end{array}]$ are applied on the bearings.

Figure 2.

Sketch of cylindrical roller bearings and external loads.

The positional relation between roller and raceways is shown in Figure 3. $o_b{x}_b{y}_b{z}_b$ is the position coordinate of roller (coordinate center $o$ is the geometry center of roller, x-axis is parallel to the roller axis, y-axis is perpendicular to the roller axis and points to roller revolution, and z-axis is determined by right-hand rule). The transformation matrix from $\mathrm{OXYZ}$ to $o_{\mathrm{bj}}{x}_{\mathrm{bj}}{y}_{\mathrm{bj}}{z}_{\mathrm{bj}}$ (j is the jth roller) can be determined according to the azimuth angle of the roller

$${\mathbf{T}}_{\mathbf{ib}}=\left[\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \multicolumn{1}{c}{0}& \hfill \cos {\theta }_j\hfill & \hfill \sin {\theta }_j\hfill \\ \multicolumn{1}{c}{0}& \hfill -\sin {\theta }_j\hfill & \hfill \cos {\theta }_j\hfill \end{array}\right]$$

(12)

Figure 3.

Relative positions of roller slices to inner and outer raceways: (a) roller slice and outer ring and (b) roller slice and inner ring.

Slice method is used on the roller along its axis. According to the azimuth angle of the roller in fixed coordinate and position of roller slice in the roller position coordinate $o_b{x}_b{y}_b{z}_b$ , position vectors of the kth slice of jth roller as well as the inner raceway and outer raceway can be expressed as

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\mathbf{\Delta }}_{\mathbf{ojk}}^b={\mathbf{T}}_{\mathbf{ib}}·{\left[\begin{array}{ccc}\hfill x_{\mathrm{ok}}\hfill & \hfill y_{\mathrm{ok}}\hfill & \hfill R_o\hfill \end{array}\right]}^T\hfill \\ \multicolumn{1}{c}{}& -({\mathbf{T}}_{\mathbf{ib}}·\left[\begin{array}{ccc}\hfill 0\hfill & \hfill r_j\sin {\theta }_j\hfill & \hfill r_j\cos {\theta }_j\hfill \end{array}\right]+{\left[\begin{array}{ccc}\hfill x_k\hfill & \hfill 0\hfill & \hfill \frac{D_w}{2}\hfill \end{array}\right]}^T)\hfill \\ \multicolumn{1}{c}{}& {\mathbf{\Delta }}_{\mathbf{ijk}}^b=({\mathbf{T}}_{\mathbf{ib}}·\left[\begin{array}{ccc}\hfill 0\hfill & \hfill r_j\sin {\theta }_j\hfill & \hfill r_j\cos {\theta }_j\hfill \end{array}\right]+{\left[\begin{array}{ccc}\hfill x_k\hfill & \hfill 0\hfill & \hfill -\frac{D_w}{2}\hfill \end{array}\right]}^T)\hfill \\ \multicolumn{1}{c}{}& -{\mathbf{T}}_{\mathbf{ib}}·{\mathbf{T}}_{\mathbf{ir}}^T·{\left[\begin{array}{ccc}\hfill x_{\mathrm{ik}}\hfill & \hfill y_{\mathrm{ik}}\hfill & \hfill R_i\hfill \end{array}\right]}^T\hfill & \hfill \end{array}$$

(13)

where R_o , R_i , and r_j are distances from the center of outer raceway, inner raceway, and jth roller to the bearing center, respectively; x_ok , y_ok , x_ik , and y_ik are the coordinates of kth slice on the raceway, which are calculated by letting the first and second components of ${\mathbf{\Delta }}_{\mathbf{ojk}}^b$ and ${\mathbf{\Delta }}_{\mathbf{ijk}}^b$ be zero; and superscript b represents the position coordinates of roller, while subscripts o and i represent outer ring and inner ring, respectively.

According to displacement–deformation compatibility condition between each roller slice and raceway, displacements of geometry centers between roller slice and raceways will correspond coordinately to deformation of roller slice and raceways at contact points. Deformation of roller slice and raceways is calculated by the following formulation

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\delta }_{\mathrm{ojk}}={\left({\mathbf{\Delta }}_{\mathbf{ojk}}^b\right)}_3-C_k-h_{\mathrm{oj}}-\frac{{\delta }_0}{4}\hfill \\ \multicolumn{1}{c}{}& {\delta }_{\mathrm{ijk}}={\left({\mathbf{\Delta }}_{\mathbf{ijk}}^b\right)}_3-C_k-h_{\mathrm{ij}}-\frac{{\delta }_0}{4}\hfill \end{array}$$

(14)

where ${\delta }_{\mathrm{ojk}}$ and ${\delta }_{\mathrm{ijk}}$ are the deformation between roller slice and outer raceway and the deformation between roller slice and inner raceway, respectively; C_k is the modification of roller slice; h_j is the minimum thickness of the oil film, calculated by elasto-hydrodynamic lubrication (EHL) theory ¹² as shown in equation (15)

$$\begin{array}{cc}\multicolumn{1}{c}{}& h_{\mathrm{oj}}=\frac{2.65{\lambda }^{0.54}{\nu }^{0.7}{U}_{\mathrm{oj}}^{0.7}{R}_o^{\prime 0.43}{w}^{0.13}}{max{\left(Q_{\mathrm{ojk}}\right)}^{0.13}{E}_o^{\prime 0.03}}\hfill \\ \multicolumn{1}{c}{}& h_{\mathrm{ij}}=\frac{2.65{\lambda }^{0.54}{\nu }^{0.7}{U}_{\mathrm{ij}}^{0.7}{R}_i^{\prime 0.43}{w}^{0.13}}{max{\left(Q_{\mathrm{ijk}}\right)}^{0.13}{E}_i^{\prime 0.03}}\hfill \end{array}$$

(15)

where μ is the viscosity of oil, λ is the pressure–viscosity coefficient; U_j is sliding velocity between ring and jth roller; R′ and E′ are equivalent radius and elastic modulus of roller and ring, respectively; Q_jk is the contact load between slice k of roller j and raceway; and w is the slice width, w = l/m, l and m are roller length and slice number, respectively.

According to the deformation of roller slice, contact pressure between roller slice and raceways is calculated through Palmgren theory. ¹ Contact loads on slice k of roller j are calculated by the following formulation ¹

$$\begin{array}{cc}\multicolumn{1}{c}{}& Q_{\mathrm{ojk}}={\left(\frac{{\delta }_{\mathrm{ojk}}·w^{0.8}}{3.81}\right)}^{1/0.9}·\left(\pi E{\prime }_o\right)\hfill \\ \multicolumn{1}{c}{}& Q_{\mathrm{ijk}}={\left(\frac{{\delta }_{\mathrm{ijk}}·w^{0.8}}{3.81}\right)}^{1/0.9}·\left(\pi E{\prime }_i\right)\hfill \end{array}$$

(16)

Moreover, the sum of contact force on all the roller slices should balance coordinately with external loads

$$\begin{array}{cc}\multicolumn{1}{c}{}& {\displaystyle \sum _{j=1}^N}{\displaystyle \sum _{k=1}^m}{Q}_{\mathrm{ijk}}\sin {\theta }_j=F_y\hfill \\ \multicolumn{1}{c}{}& {\displaystyle \sum _{j=1}^N}{\displaystyle \sum _{k=1}^m}{Q}_{\mathrm{ijk}}\cos {\theta }_j=F_z\hfill \end{array}$$

(17)

Besides the balance and coordination between roller slice and raceways, coordinative balance should be kept between rollers and cage, as well as rings and cage. The specific load calculation method refers to Cui et al. ¹³ Combining load balance equations in bearings, the Newton–Raphson method and steepest descent method are employed to calculate the load and the axial load distribution on each roller.

Calculation model of fatigue life

Based on the slice method, the raceway (length l) was cut into m slices, as shown in Figure 4. Each slice can be viewed as a l/m wide raceway. Then, the basic rated dynamic load of each slice could be calculated as follows, ¹⁴ where $\gamma$ is a structure parameter of bearings, $\gamma =D_w/D_m$

$$Q_{\mathrm{cnjk}}=552\frac{{(1\mp \gamma )}^{\frac{29}{27}}}{{(1\pm \gamma )}^{\frac{1}{4}}}{\gamma }^{\frac{2}{9}}{D}_w^{\frac{29}{27}}{\left(\frac{l}{m}\right)}^{\frac{7}{9}}{N}^{\frac{1}{4}}n=i,o$$

(18)

Figure 4.

Roller and raceway are divided into m slices along the axis of the roller using the slice method. ¹¹

Then, the fatigue life of raceway slice is ¹⁴

$$L_{\mathrm{ijk}}={\left(\frac{Q_{\mathrm{cijk}}}{Q_{\mathrm{ijk}}}\right)}^4,L_{\mathrm{ojk}}={\left(\frac{Q_{\mathrm{cojk}}}{Q_{\mathrm{ojk}}}\right)}^{4.5}$$

(19)

$L_{\mathrm{Rojk}}$ and $L_{\mathrm{Rijk}}$ are fatigue life of roller slice when contacting with outer raceway and inner raceway, respectively. They were calculated by the same way of fatigue life of raceway slices. The basic rated dynamic load of roller slices is calculated as ¹⁴

$$q_{\mathrm{cjk}}=464{(1\mp \gamma )}^{1.324}{\left(\frac{l}{m}\right)}^{\frac{7}{9}}{D}_w^{\frac{29}{27}}$$

(20)

Considering that the unit of roller fatigue life calculated by equation (20) is the roller revolution number, it is necessary to convert roller revolution number into revolution number of the inner ring. Also, the roller revolution numbers i while the inner ring ran by one circle is

$$i=\frac{D_m}{2D_w}(1-{\gamma }^2)$$

(21)

Dividing the calculated roller fatigue life by i is the revolution number of inner ring upon roller fatigue. Therefore, the overall fatigue life of roller bearings can be expressed as ¹⁴

$$\begin{array}{cc}\multicolumn{1}{c}{L=}& [{\displaystyle \sum _{j=1}^N}{\displaystyle \sum _{k=1}^m}{L}_{\mathrm{ijk}}^{-e}+{\displaystyle \sum _{j=1}^N}{\displaystyle \sum _{k=1}^m}{L}_{\mathrm{ojk}}^{-e}+{\displaystyle \sum _{j=1}^N}{\displaystyle \sum _{k=1}^m}{\left(\frac{L_{\mathrm{Rijk}}}{i}\right)}^{-e}\hfill \\ \multicolumn{1}{c}{}& {+{\displaystyle \sum _{j=1}^N}{\displaystyle \sum _{k=1}^m}{\left(\frac{L_{\mathrm{Rojk}}}{i}\right)}^{-e}]}^{-1/e}\hfill & \hfill \end{array}$$

(22)

Optimization model of assembly interference

Optimization is to prolong the fatigue life of bearings under specific working condition to the maximum extent. The objective function is

$$\begin{array}{cc}\multicolumn{1}{c}{}& minf\left(X\right)=\hfill \\ \multicolumn{1}{c}{}& min[{\displaystyle \sum _{j=1}^N}{\displaystyle \sum _{k=1}^m}{L}_{\mathrm{ijk}}^{-e}+{\displaystyle \sum _{j=1}^N}{\displaystyle \sum _{k=1}^m}{L}_{\mathrm{ojk}}^{-e}+{\displaystyle \sum _{j=1}^N}{\displaystyle \sum _{k=1}^m}{\left(\frac{L_{\mathrm{Rijk}}}{i}\right)}^{-e}\hfill \\ \multicolumn{1}{c}{}& +{\displaystyle \sum _{j=1}^N}{\displaystyle \sum _{k=1}^m}{\left(\frac{L_{\mathrm{Rojk}}}{i}\right)}^{-e}]\hfill & \hfill \end{array}$$

(23)

Interferences of shaft and inner ring during roller bearing assembly are viewed as design variables

$$X=\left[G_2\right]$$

(24)

Since excessive differential sliding between the rolling elements and ring of high-speed rolling bearing will generate abundant friction heats which are easy to cause burning and scuffing failures of the assistant contact surface, it is necessary to control sliding of rolling elements at bearing operation. The sliding of rolling elements can be characterized by sliding rate of cage. Generally, sliding rate of cage is controlled less than 5%. The corresponding constraint is

$$g\left(X\right)=\frac{2{\omega }_c}{\frac{D_m-D_w}{D_m}{\omega }_i+\frac{D_m+D_w}{D_m}{\omega }_o}\ge 0.95$$

(25)

Independent variables in the objective function ( $L_{\mathrm{ijk}}$ , $L_{\mathrm{ojk}}$ , $L_{\mathrm{Rijk}}$ , $L_{\mathrm{Rojk}}$ , and ${\omega }_c$ ) are implicit functions of design parameters and can be calculated through quasi-dynamic model of bearings. ¹³ Considering the strong nonlinearity of quasi-dynamic equations, the direct search algorithm was used to calculate the design variables according to the objective function and constraint.

Optimization calculation and discussion of assembly interference

In this article, the optimization analysis of assembly interference was based on N1006 cylindrical roller bearings as an example. Related geometry parameters, material parameters, assembly parameters, and working conditions are listed in Tables 1–3.

Table 1.

Geometry parameters of N1006 cylindrical roller bearings. ¹⁵

Geometry parameters	Value	Geometry parameters	Value
Pitch diameter of bearings (mm)	43	No. of rollers	7
Roller diameter (mm)	7	Initial radial clearance of bearings (mm)	0.02
Roller length (mm)	7	Central land length of the roller (mm)	4
Crown radius of the roller (mm)	250	Corner radius of the roller (mm)	0.25

Table 2.

Material parameters of cylindrical roller bearings. ¹⁵

Elements	Density (kg/m3)	Elastic modulus (GPa)	Poisson’s ratio
Roller	3200	310	0.260
Rings	7750	200	0.250
Cage	1500	1.73	0.300

Table 3.

Operating parameters and installation parameters of cylindrical roller bearings.

Working conditions	Temperature (°C)				Load (N)	Velocity (r/min)	Assembly interference (μm)
	Inner ring	Roller	Outer ring	Environment			Inner ring or shaft	Outer ring or bearing housing
Normal	180	170	160	27	250	70,000	20	5
Extreme	230	205	180	27	1000	70,000	20	5

MIL-L-7808 lubrication oil was used in the testing bearings. Density and dynamic viscosity of lubrication oil are 960 kg/m³ and 0.277 Pa s, respectively. The traction coefficient $\mu$ was calculated from the following equation according to the relative sliding speed of contact surface and traction characteristics of lubricant ( ${\mu }_0=0$ , ${\mu }_{\infty }=0.10$ , ${\mu }_m=0.08$ , and $u_m=1.0\mathrm{m}/\mathrm{s}$ ) ¹⁶

$$\mu =(A+\mathrm{Bu})\exp (-\mathrm{Cu})+D$$

(26)

where u is relative sliding speed of the contact surface. A, B, C, and D are undetermined coefficients of lubricant, which are determined through the following relationship according to the traction characteristics of lubricant

$$\mu =\{\begin{array}{ccc}\hfill {\mu }_0\hfill & \hfill \hfill & \hfill u=0\hfill \\ \multicolumn{1}{c}{{\mu }_{\infty }}& \hfill \hfill & \hfill u\to \infty \hfill \\ \multicolumn{1}{c}{{\mu }_m}& \hfill \hfill & \hfill u=u_m\hfill \end{array}{\frac{\partial \mu }{\partial u}|}_{u=u_m}=0$$

Effect of working clearance on dynamic properties of cylindrical roller bearings

The working clearances of the roller bearings under normal working condition and extreme working condition were calculated as 0 and −7 μm, respectively. The negative clearance reflects that the rolling elements have suffered certain pre-deformation when neglecting external load. Performances under these two working clearances were compared to that under initial clearance to analyze the effect of working clearance on bearing performance. The average sliding ratio and rotating speed of rolling elements under three radial internal clearances are presented in Figure 5. With the increase in radial internal clearance, internal loading region of bearing will be narrowed, thus weakening the traction effect of raceway on rolling elements. As a result, average sliding ratio and rotating speed of the cage fluctuate within wider ranges.

Figure 5.

Effect of initial clearance on roller (a) average sliding ratio and (b) revolution speed.

Figure 6 is the spectrogram of interaction between cage and rolling elements under different radial internal clearances. Figure 7 is the spectrogram of interaction between cage and guide ring under different radial internal clearances. With the decrease in radial internal clearance, the rotating speed fluctuation of rolling elements weakens, thus reducing dominant frequencies of the collision between cage pockets and rolling elements. The dominant frequency of force between the cage and guiding ring, which used to balance the impact force of the rolling elements, decreases accordingly.

Figure 6.

Spectrum of the interaction between cage pocket and roller on different bearing initial clearances: (a) 20, (b) 0, and (c) −7 μm.

Figure 7.

Spectrum of the interaction between cage and guiding ring on different bearing initial clearances: (a) 20, (b) 0, and (c) −7 μm.

According to the calculation method of fatigue life mentioned above, the fatigue life under normal working condition (bearing clearance is 0 μm) and extreme working condition (bearing clearance is −7 μm) is 15,428 and 505 h, respectively, indicating that extreme load shortens the fatigue life significantly. Comparatively, if the effect of working condition on bearing clearance is not considered, the fatigue life under normal working condition and extreme working condition based on the initial clearance is 7234 and 131 h, respectively. This confirms the significance of bearing clearance in calculating fatigue life because bearing clearance can affect contact load between the rolling elements and ring as well as load distribution directly.

Optimization analysis of assembly interference

Considering the effect of working clearance on fatigue life, optimum design of bearing working clearance was proposed under fixed working condition. Based on the relationship between assembly interference and working clearance, the optimum assembly interference was calculated. Although the fatigue life of bearing under extreme working condition is far shorter than that under normal working condition, the optimum design was still calculated under normal working condition, because extreme working condition accounts for a very small proportion in the whole working cycle.

The effect of working clearance on bearing load, fatigue life, and number of rolling elements is shown in Figure 8. Smaller working clearance can contribute more even internal load distribution and increase rolling elements in the loading region. However, pre-deformation caused by negative clearance will also produce additional load on rolling elements in the loading region. Therefore, the working clearance under full loading of all rolling elements is not the optimum working clearance. Under normal working condition, the longest fatigue life is achieved at −11 μm working clearance. The corresponding relationship between assembly interference and working clearance is shown in Figure 9. Figure 9 reveals that the optimum fitting interference between shaft and the inner ring shall be designed 26 μm to achieve the best performance of the roller bearing.

Figure 8.

Effect of working clearance on loading performance and fatigue life of bearings: (a) fatigue life and maximum contact load of bearings and (b) total number of rollers in the loading zone.

Figure 9.

Assembly interference between shaft and bearing inner ring corresponding to the working clearance.

Effect of geometry parameters on the optimum assembly interference

Effect of pitch diameter on the optimum assembly interference

The effect of assembly interference and pitch diameter on fatigue life and working clearance is shown in Figure 10. With the increase in pitch diameter, the optimum assembly interference increases gradually (from 21 to 32 μm), but the corresponding working clearance reduces (from −8.5 to −13 μm). This is mainly because the revolution radius of rolling elements increases with the increase in pitch diameter, forcing rolling elements approaching to the outer ring raceway by the centrifugal force during bearing running. Thus, to maintain uniform load distribution within the loading region of bearings and prolong its fatigue life, greater assembly interference (or smaller working clearance) is needed to adjust pre-deformation of rolling elements.

Figure 10.

Effect of assembly interference and pitch diameter on fatigue life and working clearance of roller bearings: (a) fatigue life of bearings and (b) working condition of bearings.

Furthermore, Figure 10(a) shows an inversely proportional relationship between the optimum fatigue life of bearing (L ₁₀) and pitch diameter (L ₁₀ ranges from 109,200 to 27,440 h with the increase in pitch diameter). To depict the influence law of pitch diameter on the optimum fatigue life of bearing, the number of rolling elements in the loading region and load on the rolling element with maximum load under the optimum assembly interference were calculated (Table 4). Table 4 reveals that although different pitch diameters have different optimum assembly interferences, the number of rolling elements in the loading region under the optimum assembly interference is the same and load on the rolling element with maximum load is similar to each other. Since radial load on the bearing is fixed, rest rolling elements in symmetric distribution show basically same loads. According to the fatigue life calculation formula of roller bearing, $\gamma$ (the ratio of roller diameter D_w and pitch diameter of bearing D_m ) decreases with the increase in D_m , thus shortening the fatigue life of bearings.

Table 4.

Number of rollers and maximum load in the loading area while the assembly interference is optimal.

Pitch diameter of bearings (mm)	No. of rollers in the loading area	Maximum load of rollers (N)
33	3	139.2
38	3	140.5
43	3	141.5
48	3	142.1
53	3	142.4

Effect of number of rolling elements on the optimum assembly interference

The effect of assembly interference and number of rolling elements on fatigue life and working clearance is shown in Figure 11. The number of rolling elements is inversely proportional to the optimum assembly interference (optimum assembly interference ranges from 27 to 24 μm with the increase in the number of rolling elements) but proportional to fatigue life of bearings (L ₁₀) and working clearance (with the increase in the number of rolling elements, L ₁₀ ranges from 30,480 to 83,370 h and working clearance ranges from −12 to −10 μm). With the increase in the number of rolling elements, load distribution within the loading region of bearing will be changed. On one hand, there are more rolling elements to bear loads. On the other hand, the rolling element with maximum load will bear fewer loads, which will reduce deformations between rolling elements and raceway. Consequently, the fatigue life of bearing prolongs, while working clearance increases gradually.

Figure 11.

Effect of assembly interference and roller number on fatigue life and working clearance of roller bearings: (a) fatigue life of bearings and (b) working condition of bearings.

Effect of working conditions on the optimum assembly interference

Effect of radial load on the optimum assembly interference

The effect of assembly interference and radial load on fatigue life and working clearance is shown in Figure 12. It can be seen from Figure 12(a) that the optimum assembly interference is same (26 μm) under different radial loads. Thus, it can be concluded that the radial load does not influence the optimum assembly interference. In other words, the optimum working clearance of bearing is immune to the radial load. However, fatigue life of bearing (L ₁₀) drops quickly (from 109,400 to 635 h) as the radial load increases. Based on the optimization analysis of assembly interference (section “Optimization analysis of assembly interference”), load shall be distributed within half ring of the bearing upon the optimum fatigue life. This requires constant optimum working clearance under fixed pitch diameter and rolling elements. However, radial load growth will increase load on rolling elements within the loading region directly, thus shortening the optimum fatigue life significantly.

Figure 12.

Effect of assembly interference and radial load on fatigue life and working clearance of roller bearings: (a) fatigue life of bearings and (b) working condition of bearings.

Effect of rotation speed on the optimum assembly interference

The effect of assembly interference and rotating speed on fatigue life and working clearance is shown in Figure 13. Rotation speed is proportional to the optimum assembly interference (optimum assembly interference ranges from 18 to 26 μm with the increase in rotation speed) but inversely proportional to the fatigue life of bearing (L ₁₀) and working clearance (with the increase in rotation speed, L ₁₀ ranges from 31,880 to 65,460 h and working clearance ranges from −6 to −11 μm). Higher rotation speed will accelerate the revolution of rolling elements which will approach to outer raceway by the increasing centrifugal force. Under this circumstance, the rolling elements have to produce greater pre-deformation at initial state in order to maintain load distribution within half bearing ring. In other words, the optimum working clearance shall be reduced, while assembly interference to adjust working clearance shall be increased. Meanwhile, the increasing centrifugal force of rolling elements will increase contact load and contact stress between the rolling elements and outer raceway. These will shorten the fatigue life of both outer raceway and rolling elements, finally shortening the overall fatigue life of bearings.

Figure 13.

Effect of assembly interference and rotation speed on fatigue life and working clearance of roller bearings: (a) fatigue life of bearings and (b) working condition of bearings.

Conclusion

To optimize the fatigue life of roller bearing, a calculation method of assembly interference was proposed in this article based on the analysis of working clearance and fatigue life. The effect of working clearance on dynamic properties of cylindrical roller bearings and the optimum assembly interference was analyzed through a case study. Moreover, the influence law of geometry parameters and working conditions of cylindrical roller bearings on the optimum assembly interference was discussed. It concludes that

1. Working clearance growth of roller bearing will increase sliding ratio and rotating speed fluctuation of the cage;

2. The optimum fatigue life of roller bearing is achieved at negative clearance when working loads distribute evenly within half ring;

3. With the increase in pitch diameter or decrease in rolling elements, the optimum assembly interference increases, while the corresponding working clearance and optimum fatigue life (L ₁₀) decrease gradually;

4. The optimum working clearance of roller bearing is immune to radial load. However, radial load growth will shorten the optimum fatigue life (L ₁₀) quickly. In addition, accelerating revolution of roller bearing will increase the optimum assembly interference but decrease the corresponding working clearance and optimum fatigue life (L ₁₀) gradually.