Study on structural parameter design and life analysis of large four-point contact ball bearing channel

Zhu Liang; Tang Zhanqi

Journal & Issue Details

Applied Mathematics and Nonlinear Sciences

Journal Details

Format: Journal
eISSN: 2444-8656
First Published: 01 Jan 2016
Publication timeframe: 2 times per year
Languages: English

Large rolling bearings are widely used in engineering and construction machinery, such as tower cranes, excavators and concrete pump trucks. On such occasions, the bearing should have a large bearing capacity and high reliability [1, 2]. In addition, large rolling bearings are also used in heavy equipment such as metallurgy and mining. Such bearings have a poor working environment, large load, high temperature, and large dust, which requires bearings to have a large bearing capacity, good sealing and high-temperature resistance [3–5]. Large rolling bearings are also used in offshore equipment, such as offshore platforms, various port cranes and deck cranes. It requires large rolling bearings to have high safety, good anti-corrosion, sealing and reliability [6]. Large rolling bearings are also used in wind turbine equipment and military equipment such as radar, antenna and airborne vehicles [7, 8]. Therefore, it is very necessary to design the channel structure and study the service life of a large four-point contact ball bearing.

Four-point contact ball bearings are used more and more in the aviation transmission system, automobile transmission systems, precision machinery and other aspects because of their advantages of many balls, large bearing capacity, two-way axial load, small axial space and high limit speed [9, 10]. With the progress of technology, users put forward higher requirements for the accuracy, performance, service life and reliability of bearings. The structural design of the cage directly affects the service performance of the bearing [11, 12]. S. Zupn, Z. pre and others believe that the flexible support of a large four-point contact ball bearing and the deformation of its ferrule have a great impact on the load distribution of the bearing. In this paper, two different models are established. After comparing the results, it is found that the deformation of flexible support and ferrule has a great impact on the load distribution of the bearing [13]. Jose et al. Assuming that the inner and outer rings of the bearing were purely rigid and considering that there was only contact local deformation between the rolling element and the ferrule, using Hertz contact theory and spatial coordinate transformation, the balance equation of a single row four-point contact bearing was derived to calculate the load distribution and contact angle distribution of the bearing [14]. The use of 50K in the finite element method is to replace the rolling element with a special super element with the deformation characteristics of the contact load between the rolling element and the channel. Hence we can calculate the deformation and load distribution of the three-row thrust roller bearing [15]. Moxse et al. established the full model of single row four-point contact bearing and support structure. Considering the deformation of the bearing ring and its support structure, finite element analysis software was used to truly simulate the load distribution and contact angle of the bearing [16]. Kania et al. [17] Established the finite element simplified models of angular contact ball bearing, four-point angular contact ball bearing, double row four point angular contact ball bearing and three-row cylindrical roller combined bearing respectively by using the super element method to analyse the mechanical properties of various bearings. In the analysis process, he considered the deformation of bearing support structure and ferrule and the influence of bearing connecting bolts on the mechanical properties of bearings. The research on the fatigue life of large rolling bearings is generally based on the analysis of rolling contact fatigue life. F. Based on the Lagrange method, the relationship between the contact force and the contact stress in the direction of X W Z and the contact stress on the surface is calculated by using the Lagrange method [18]. R. J. S proposed a new rolling contact fatigue analysis method by comprehensively using the elastic-plastic finite element method and the dangerous plane multiaxial fatigue crack generation model, and analysed the rolling contact fatigue life of railway wheel-rail contact [19]. Abbas et al. [20] Used elastodynamics and optimisation methods to optimise the fatigue life of ordinary bearings. Glode et al. [21] proposed a static strength and fatigue life analysis model of large rotary table bearing. First, they established the static balance equation of large rotary table bearing, solved the load distribution of rolling element and the maximum load of rolling element and channel contact, and calculated the static strength of bearing by using Hertz contact theory, Then, according to the mechanical properties of the bearing ring material, the stress life model and strain life model are used to analyse the fatigue life of the channel and compared with the calculation results. The disadvantage of this method is that it does not consider the influence of the bearing support groove and the deformation of the bearing ring on the load distribution on the rolling element.

First, the channel structure parameters of a large four-point contact ball bearing are optimised by optimisation, and its fatigue life is analysed. At present, the fatigue life of rolling bearings is analysed according to the standard load rating life formula [22], but this formula is established based on ordinary bearings made of steel and does not consider the impact of bearing materials and their attribute changes on bearing life. There is a certain error in using this formula to calculate the life of large rolling bearings.

Simplified bearing fatigue model and four-point finite element method

2.1

Structure and simplified model of four-point contact ball bearing

The geometric structure of four-point angular contact ball bearing is shown in Figure 1(a) [23]. The inner diameter of the bearing is D_i, the outer diameter is d0, the diameter of the distribution circle of the rolling element is D_M, and the diameter of the rolling element is D_W. R_I and R₀ are the radius of curvature of the inner and outer ring channels of the bearing respectively. The bearing channel is composed of four channel surfaces, of which ci1 is the curvature centre of the upper channel of the inner ring, C_i2 is the curvature centre of the lower channel of the inner ring, C₀₁ is the curvature centre of the upper channel of the outer ring, and C₀₂ is the curvature centre of the lower channel of the outer ring, β is the initial contact angle of four-point angular contact ball bearing. The super-element method simplifies the structure of the bearing rolling element. As shown in Figure 1(b), each rolling element is replaced by eight rigid rod elements and two nonlinear spring elements. The two ends of one spring yellow are the curvature centre C_i1 of the upper channel of the inner ring and the curvature centre C₀₂ of the lower channel of the outer ring, and the two ends of the other spring are the curvature centre C_i2 of the lower channel of the inner ring and the curvature centre C₀₁ of the upper channel of the outer ring. C_i1 is connected with the upper channel of the inner ring through two rigid rods, C_i2 is connected with the lower channel of the inner ring through two rigid rods, C₀₁ is connected with the upper channel of the outer ring through two rigid rods, and C₀₂ is connected with the lower channel of the outer ring through two rigid rods. The four-point angular contact ball bearing deforms under the action of external load, and its contact angle $β^{'}$ is the included angle between the nonlinear spring and the radial bearing, as shown in Figure 1(c) [24].

Shows the structural diagram of four-point angular contact ball bearing, (a) geometric structure of four-point angular contact ball bearing; (b) The super element method simplifies the structure of bearing rolling element; (c) Angle structure between nonlinear spring and radial bearing

The simplified finite element model of a large four-point angular contact ball bearing mainly uses the nonlinear spring connecting the centre points of curvature of two channels to simulate the compression of rolling elements [25, 26]. Therefore, the nonlinear spring only acts when in tension to simulate the compression of the bearing rolling element, which can well simulate the mechanical characteristics and geometric structure changes of the bearing under the action of load. As shown in Figure 2, under the action of load, the distance between the compression channels of the rolling element close to the curvature centre of the channel becomes larger. At this time, the compression of the rolling element can be simulated as the tension of the nonlinear spring connecting the curvature centre of the channel, The load-deformation curve of the spring in tension is defined according to the load-deformation relationship when the rolling element and the channel contact are compressed [27]. This method can not only analyse the load on the rolling element but also analyse the change of the contact angle between the rolling element and the channel through the displacement of the centre points C_i1, C_i2, C₀₁ and C₀₂ of the channel curvature.

Shows the bearing force model of four-point angular contact shaft

The simplified finite element model of four-point angular contact ball bearing is shown in Figure 3.

2.2

Fatigue life analysis method

General steps of stress life analysis method: first, calculate the isotropic stress, maximum and minimum stress, average stress and stress amplitude at the fatigue part of parts by theoretical method or finite element method, and then calculate the equivalent stress, equal effect stress amplitude and equal effect average stress by distortion energy theory; second, the reverse bending stress is calculated by selecting different fatigue analysis models in Table 1. Finally, the fatigue life is calculated according to the life calculation formula [28].

Table 1

Calculation of stress in opposite direction

Fatigue model	Calculation formula
Gerber model	$σ_{e r} = σ_{a} / [1 - (σ_{m} / σ_{x})]$
Modified Gerber model	$σ_{e r} = σ_{a} / [1 - (σ_{m} / σ_{n})]$
SAE model	$σ_{e r} = σ_{a} / [1 - (σ_{m} / σ_{y})]$
Soderberg model	$σ_{e r} = σ_{a} / [1 - (σ_{m} / σ_{y})]$

Table 2

Midpoint data in S-N diagram

Fatigue model	σ_f	N_f	σ_e	N_e
Toughness and rigidity ( $σ_{u} \leq 98.5$ MPa)
Gerber theory	0.9σ_u	10³	Kσ_u/2	10⁶
Modified Gerber theory	0.9σ_u	10³	Kσ_u/2	10⁶
SAE theory	σ_u+50000	1	Kσ_u/2	10⁶
Soderberg theory	0.9σ_u	10³	Kσ_u/2	10⁶
Toughness and rigidity ( $σ_{u} \geq 98.5$ MPa)
Gerber theory	0.9σ_u	10³	Kσ_u/3	10⁸
Modified Gerber theory	0.9σ_u	10³	Kσ_u/3	10⁸
SAE theory	σ_u	1	Kσ_u/3	10⁸
Soderberg theory	0.9σ_u	10³	Kσ_u/3	10⁸

The average amplitude of stress and strain can be calculated by the following formula: 1 $\begin{aligned} σ_{m} = \frac{σ_{max} + σ_{min}}{2} \end{aligned}$ 2 $\begin{aligned} σ_{a} = \frac{σ_{max} - σ_{min}}{2} \end{aligned}$ 3 $\begin{aligned} ε_{m} = \frac{ε_{max} + ε_{min}}{2} \end{aligned}$ 4 $\begin{aligned} ε_{a} = \frac{ε_{max} - ε_{min}}{2} \end{aligned}$

σ_m, σ_a, ε_m, ε_a are all are stress.

Von-Mises equivalent stress calculation formula is: 5 $\begin{aligned} σ_{e q} = \frac{1}{\sqrt{2}} \cdot \sqrt{{(σ_{x} - σ_{y})}^{2} + {(σ_{y} - σ_{z})}^{2} + {(σ_{x} - σ_{a z})}^{2} + 6 (τ_{x y}^{2} + τ_{y z}^{2} + τ_{x z}^{2})} \end{aligned}$

Equivalent force amplitude $σ_{a . e q}$ 6 $\begin{aligned} σ_{a e q} = \frac{1}{\sqrt{2}} \cdot \sqrt{{(σ_{a x} - σ_{a y})}^{2} + {(σ_{a y} - σ_{a z})}^{2} + {(σ_{a x} - σ_{a z})}^{2} + 6 (τ_{a x y}^{2} + τ_{a v z}^{2} + τ_{a x z}^{2})} \end{aligned}$

Among $σ_{a x} > σ_{a y} > σ_{a z}$ , subscript indicates three axial directions of micro elements in dangerous parts of parts. among σ_n is the tensile limit, σ_y is the yield limit.

The average equivalent stress is: 7 $\begin{aligned} σ_{m e q} = \frac{1}{\sqrt{2}} \cdot \sqrt{{(σ_{m x} - σ_{m y})}^{2} + {(σ_{m v} - σ_{m z})}^{2} + {(σ_{m x} - σ_{m z})}^{2} + 6 (τ_{m x y}^{2} + τ_{m z z}^{2} + τ_{maxz}^{2})} \end{aligned}$ in $σ_{m x} > σ_{m y} > σ_{m z}$ .

The maximum principal strain criterion is to use the maximum strain amplitude of dangerous parts of parts ξ_a1 calculate the fatigue life by Manson coffin formula [29]: 8 $\begin{aligned} ε_{a 1} = \frac{σ_{f}^{'} - σ_{m}}{E} {(2 N_{f})}^{b} + ε_{f}^{'} {(2 N_{f})}^{c} \end{aligned}$

The maximum shear strain criterion is to calculate the maximum shear strain of the dangerous part $γ_{max}$ , i.e.: 9 $\begin{aligned} γ_{a max} = max (ε_{a 1} - ε_{a 2}, ε_{a 2} - ε_{a 3}, ε_{a 1} - ε_{a 3}) \end{aligned}$

The fatigue life is calculated by Manson’s coffin formula according to the maximum shear strain: 10 $\begin{aligned} γ_{a max} = (1 + v) \frac{σ_{f}^{'} - σ_{m}}{E} {(2 N_{f})}^{b} + (1 + v_{p}) ε_{f}^{'} {(2 N_{f})}^{c} \end{aligned}$

The maximum strain criterion is to calculate the equivalent strain of dangerous parts of parts by using the Von-Mises principle $ξ_{a}$ , namely: 11 $\begin{aligned} {\bar{ε}}_{a} = \frac{1}{\sqrt{2} (1 + u)} \sqrt{{(ε_{x} - ε_{y})}^{2} + {(ε_{y} - ε_{z})}^{2} + {(ε_{x} - ε_{z})}^{2} + \frac{3}{2} (γ_{x y}^{2} + γ_{y z}^{2} + γ_{x z}^{2})} \end{aligned}$

Then use the Manson-Coffin formula to calculate the fatigue life of parts: 12 $\begin{aligned} {\bar{ε}}_{a} = \frac{σ_{f}^{'} - σ_{m}}{E} {(2 N_{f})}^{b} + ε_{f}^{'} {(2 N_{f})}^{c} \end{aligned}$

Fatigue damage accumulation theory includes Miner linear cumulative damage theory and cortan Dolan [24] modified damage theory. Miner linear cumulative damage theory is generally used for accumulation in the early stage of component design, because Miner linear cumulative damage theory is relatively simple, practical and convenient, but cortan Dolan modified damage theory considers the interaction between various groups of stresses, Therefore, the accuracy of calculating the service life of parts by the modified damage theory is higher than that by miner’s linear cumulative damage theory [30].

The total damage rate D calculated by Miner linear cumulative damage theory is as follows: 13 $\begin{aligned} D = \sum_{1}^{i} D_{i} = \sum_{1}^{i} \frac{n_{i}}{N_{i}} \end{aligned}$

The calculation formula of Cortan-Dolan modified damage theory to calculate the life N_g is as follows: 14 $\begin{aligned} N_{g} = \frac{N_{1}}{\sum α_{i} {(σ_{i} / σ_{1})}^{d}} \end{aligned}$

Where N₁ is the component under the maximum constant amplitude load stress σ₁ fatigue life, α is the ratio of the action times of group I constant amplitude load stress to the action times of total load, and D is the material constant [31].

Channel parameter design of four-point contact ball bearing

The structural parameters of double row four-point angular contact ball bearing are shown in Figure 4. The influence of channel spacing x on the distribution of load on the two-channel rolling body is analysed. When the distance between the two channels of the bearing decreases, it is conducive to improve the bearing capacity of the bearing. However, with the decrease of channel spacing, the H-Side decreases, the stiffness of the side begins to decrease, and the deformation will increase under the action of load. The influence is analysed by establishing a simplified finite element model [32].

Structure diagram of double row four point contact bearing

3.1

Analysis of finite element calculation results of bearing

The calculation results of six groups of finite elements are shown in Table 3. According to the analysis results, when the distance between the two channels of the bearing is 105 mm, the maximum load on the first channel rolling element is 101.097 kn, the maximum load on the second channel rolling element is 137.313 kn, and the ratio of the maximum load of the first channel to the second channel rolling element is 0.7499. When the distance between the two channels of the bearing is 65 mm, the maximum load on the rolling element of the first channel is 107.778, the maximum load on the rolling element of the second channel is 126.754, and the ratio of the maximum load of the first channel to the rolling element of the second channel is 0.8503. When the bearing channel spacing gradually decreases from 105 mm to 65 mm, the maximum contact load on the first channel rolling element gradually increases from 103.097 kn to 107.778 kn, the maximum contact load on the second channel rolling element gradually decreases from 137.383 kn to 126.754 kn, and the ratio of the maximum value of the first channel load to the maximum value of the second channel load gradually increases from 0.7504 to 0.85029, which indicates that when the distance between the two channels of the bearing decreases, The closer the load is distributed between the first channel and the second channel.

Table 3

Load comparison of two channels with different channel spacing

X/mm	105	97	89	81	73	65
1-Qmax/kN	101.097	103.847	105.77	106.65	107.809	107.778
2-Qmax/kN	137.313	137.211	133.59	131.39	129.510	126.754
Level	0.7499	0.7568	0.79175	0.81170	0.83244	0.85029

In the process of reducing the channel spacing, the thickness of the retaining edge between the two channels will be reduced accordingly, which will reduce the stiffness of the retaining edge. Under the action of external load, the deformation of the retaining edge between the two channels will increase, which will reduce the mechanical properties and service life of the bearing. The maximum deformation of the retaining edge of each model is shown in Table 4. When the bearing channel spacing is 105 mm, the maximum deformation of the retaining edge is 0.1714 mm, when the bearing channel spacing is 81 mm, the maximum deformation of the retaining edge is 0.1572 mm, and when the bearing channel spacing is 65 mm, the deformation of the retaining edge is 0.1903 mm. It can be seen from the data in the table that when the bearing channel spacing gradually decreases from 105 mm to 81 mm, since the load on the first channel rolling element tends to be consistent with the load on the second channel rolling element, the load on the first channel rolling element increases, the load on the second channel rolling element decreases, and the deformation of the retaining edge between the first channel and the second channel gradually decreases from 0.1714 mm to 0.1572 mm. When the bearing channel spacing gradually decreases from 81 mm to 65 mm, Although the load on the first channel rolling element continues to be consistent with the load on the second channel rolling element, and the maximum load of the second channel decreases, the deformation of the retaining edge between the two channels increases from 0.1572 mm to 0.1903 mm due to the reduction of the stiffness of the retaining edge between the two channels.

Table 4

Deformation of retaining edge at different channel spacing

Spacing/mm	65	73	81	89	97	105
Deformation/mm	0.1903	0.1617	0.1572	0.1601	0.1648	0.1714

3.2

Optimise bearing channel spacing

The Newton difference method is used to fit the difference between different channel spacing and retaining edge deformation, and the channel spacing at the minimum deformation of retaining edge is obtained. The Newton interpolation fitting curve of different channel spacing and retaining edge deformation is shown in Figure 5. From the fitting curve, it can be seen that the corresponding channel spacing is 78 mm.

The bearing channel spacing x = 80 mm is selected, and the finite element simplified model of large rolling bearing is used for modelling and analysis. The results are that the maximum load of the first channel is 107.777 kn and the maximum load of the second channel is 129.922 kn. The ratio of the maximum load of the first channel to the maximum load of the second channel is 0.8295. The deformation of the retaining edge between channels is shown in Figure 5. At this time, the maximum deformation of the retaining edge is 0.1569 mm. To sum up, it is reasonable to take 80 mm as the spacing between the two channels of the bearing.

3.3

Parameter optimisation of channel curvature radius coefficient

15 $\begin{aligned} f = \frac{r}{D_{w}} \end{aligned}$

Where r is the channel radius and D_w is the rolling element diameter.

The influence of groove radius of curvature coefficient on bearing performance mainly includes: contact stress, friction torque, lubrication performance (oil film thickness). Since the rotating speed of large rolling bearings is low, generally no more than 10 R/min, and grease lubrication is adopted, the influence on lubrication is not considered here. When the bearing rotates, the steel ball makes a complex three-dimensional motion. That is, the rotation motion, revolution motion and spin sliding of the steel ball, so the friction in the rolling bearing is a complex tribological system. For heavy-duty low-speed rolling bearings with solid lubrication or grease lubrication, due to their low speed, the lubricating oil film cannot be formed, and there is certain friction between the rolling body and the channel. Its manifestation is the friction torque on the rotation axis of the bearing. The factors affecting the bearing friction torque include the micro slip between the rolling body and the channel, the elastic lag of the material at the contact between the rolling body and the channel, and the spin during the revolution of the rolling body, To consider the friction torque formed at the contact between the rolling element and the inner and outer rings of the bearing respectively by these three factors, the following assumptions should be made [33]:

In the process of contact with the inner and outer ring channels, the rolling body pure rolls with one channel and rolls and spins with the other channel.

Micro slip only appears on the channel contact surface where the rolling element and the channel are pure rollings, and the elastic hysteresis of the material at the contact between the rolling element and the channel appears on the inner and outer ring channels in contact with the rolling element at the same time.

The sliding friction coefficient between the rolling element and the channel is a fixed value, which is independent of the contact stress between them.

Because there is a certain speed difference in the contact area during the contact between the rolling body and the channel, there is not only pure rolling but also sliding in the contact area. The following equilibrium equation can be obtained: 16 $\begin{aligned} \int_{0}^{1} (1 - Y^{3}) H (2 - H) (1 - K_{3} Y^{2}) d Y = 0 \end{aligned}$

Among Y = y/a

17 $\begin{aligned} H = \frac{K_{1} (γ^{2} - Y^{2})}{\sqrt{1 - Y^{2}}} \end{aligned}$ 18 $\begin{aligned} K_{1} = \frac{π a^{3} b E^{'}}{3 Q D_{W}^{2} μ} \end{aligned}$ 19 $\begin{aligned} K_{3} = \frac{2 a^{2}}{D_{w}^{2}} \end{aligned}$

Q – contact load of rolling element, N; E ‘– equivalent elastic modulus, MPa; μ – Sliding friction coefficient between rolling element and channel, μ = 0.1; a. B – respectively the long and short half axis dimensions of the contact ellipse area between the rolling element and the channel, mm; D_w – diameter of rolling element, mm; γ – Unknown parameter, quantity to be found [34].

The resistance caused by micro sliding is: 20 $\begin{aligned} F_{m} = K_{2} \int_{0}^{1} (1 - Y^{2}) H (2 - H) d Y \end{aligned}$

Among: 21 $\begin{aligned} K_{2} = \frac{3 μ Q}{2} \end{aligned}$

The rolling resistance Fm generated by micro sliding acts on the bearing rolling element. At this time, there are two situations: one is that the micro sliding of the rolling element in contact with the channel appears on the contact surface with the outer channel; Second, the micro slip of the contact between the rolling element and the channel appears on the contact surface with the inner channel. Therefore, there are two forms when the rolling resistance generated by micro sliding is transformed into the friction torque of the bearing [24].

First, micro slip occurs on the contact surface between the rolling element and the outer channel. At this time, there are: 22 $\begin{aligned} T_{m o} = \frac{F_{m} (R_{i} + D_{W} \cos α)}{2} \end{aligned}$

First, micro slip occurs on the contact surface between the rolling element and the inner channel. At this time, there are: 23 $\begin{aligned} T_{m i} = \frac{F_{m} R_{i}}{2} \end{aligned}$

Where R_i is the radius of the contact circle between the rolling element and the inner channel [35].

To sum up, if the groove radius of curvature coefficient is too large, the bearing contact stress is large, and the bearing life is reduced. If it is too small, the total friction resistance moment will increase and the bearing friction will increase proportionally. Normalise the contact stress and friction resistance moment and draw them on the same figure. As shown in Figure 6, the intersection of the two curves is the optimal groove curvature radius coefficient. The best value is 0.524.

Atigue life analysis

The stress life and strain life analysis are used to analyse the fatigue life of the bearing after changing the bearing structure. After changing the bearing structure, the maximum contact load between the rolling element and the channel is 130.5 kN, the maximum contact pressure stress between the rolling element and the inner ring channel is 3148.5 MPa, the maximum contact pressure stress between the rolling element and the outer ring channel is 3100 MPa, and the maximum equivalent stress on the contact subsurface between the rolling element and the outer ring is 1976.7 MPa. The maximum shear strain of the contact subsurface between the rolling element and the outer ring is 12.24×10⁻³ mm. The fatigue life of the bearing after changing the bearing structure is shown in Table 5.

Table 5

Bearing life

		Life calculation method
		σ-N	ε-N
N (2Nf)	[·10⁶cyc]	12.190	7.26
L	[·10⁴cyc]	21.818	12.91

The bearing life calculated by the stress life model is 21.818×10⁴r. The bearing life calculated by the strain life model is 12.91×10⁴r. The calculation result of the stress life model is 1.699 times that of strain life.

Conclusion

Firstly, the structure and simplified model of four-point contact ball bearing are introduced, and the analysis method of fatigue life is introduced. The channel parameters of four point contact ball bearing are designed and optimised. First, the bearing finite element calculation results are analysed, then the bearing channel spacing is optimised, and finally, the parameters of the channel curvature radius coefficient are optimised. After the design and optimization of the channel parameters of four point contact ball bearing, the fatigue life is analysed, The conclusions are as follows: (1) for double row four-point contact ball bearing, the smaller the channel spacing is, the more uniform the load is in the two channels, which can reduce the maximum load of the bearing, but the deformation of the retaining edge between the two channels will increase after the channel spacing is reduced to a certain extent; (2) For ball bearings, when the load on the rolling element is constant, the maximum contact compressive stress between the rolling element and the channel, the maximum equivalent stress and the maximum shear strain on the secondary surface of the channel decrease with the decrease of the curvature radius coefficient of the channel; (3) For the optimised bearing life, the result of the model calculation is significantly higher than that of the first case of strain calculation.

Figures & Tables

Deformation of retaining edge at different channel spacing

Spacing/mm	65	73	81	89	97	105
Deformation/mm	0.1903	0.1617	0.1572	0.1601	0.1648	0.1714

Calculation of stress in opposite direction

Fatigue model	Calculation formula
Gerber model	σer=σa/[1−(σm/σx)]
Modified Gerber model	σer=σa/[1−(σm/σn)]
SAE model	σer=σa/[1−(σm/σy)]
Soderberg model	σer=σa/[1−(σm/σy)]

Bearing life

		Life calculation method
		σ-N	ε-N
N (2Nf)	[·10⁶cyc]	12.190	7.26
L	[·10⁴cyc]	21.818	12.91

Midpoint data in S-N diagram

Fatigue model	σ_f	N_f	σ_e	N_e
Toughness and rigidity (σu≤98.5 MPa)
Gerber theory	0.9σ_u	10³	Kσ_u/2	10⁶
Modified Gerber theory	0.9σ_u	10³	Kσ_u/2	10⁶
SAE theory	σ_u+50000	1	Kσ_u/2	10⁶
Soderberg theory	0.9σ_u	10³	Kσ_u/2	10⁶
Toughness and rigidity (σu≥98.5 MPa)
Gerber theory	0.9σ_u	10³	Kσ_u/3	10⁸
Modified Gerber theory	0.9σ_u	10³	Kσ_u/3	10⁸
SAE theory	σ_u	1	Kσ_u/3	10⁸
Soderberg theory	0.9σ_u	10³	Kσ_u/3	10⁸

Load comparison of two channels with different channel spacing

X/mm	105	97	89	81	73	65
1-Qmax/kN	101.097	103.847	105.77	106.65	107.809	107.778
2-Qmax/kN	137.313	137.211	133.59	131.39	129.510	126.754
Level	0.7499	0.7568	0.79175	0.81170	0.83244	0.85029

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Applied Mathematics and Nonlinear Sciences

Study on structural parameter design and life analysis of large four-point contact ball bearing channel

Zhu Liang and
Zhu Liang and
Tang Zhanqi
Tang Zhanqi

Published Online: 15 Dec 2022

Volume & Issue: AHEAD OF PRINT

Page range: -

Received: 04 May 2022

Accepted: 08 Jul 2022

Journal & Issue Details

Applied Mathematics and Nonlinear Sciences

PDF Preview

Article

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Figures & Tables

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Deformation of retaining edge at different channel spacing

Calculation of stress in opposite direction

Bearing life

Midpoint data in S-N diagram

Load comparison of two channels with different channel spacing

References

Recent Articles

Plan your remote conference with Sciendo

Applied Mathematics and Nonlinear Sciences

Study on structural parameter design and life analysis of large four-point contact ball bearing channel

Zhu Liang andZhu Liang andTang ZhanqiTang Zhanqi

Published Online: 15 Dec 2022

Volume & Issue: AHEAD OF PRINT

Page range: -

Received: 04 May 2022

Accepted: 08 Jul 2022

Journal & Issue Details

Applied Mathematics and Nonlinear Sciences

PDF Preview

Article

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Figures & Tables

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Fig. 5

Fig. 6

Deformation of retaining edge at different channel spacing

Calculation of stress in opposite direction

Bearing life

Midpoint data in S-N diagram

Load comparison of two channels with different channel spacing

References

Recent Articles

Plan your remote conference with Sciendo

Zhu Liang and
Zhu Liang and
Tang Zhanqi
Tang Zhanqi