# Stress-Based Weibull Method to Select a Ball Bearing and Determine Its Actual Reliability

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## Abstract

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## Featured Application

**Based on the catalogue L**

_{10}life, the paper let us to select the ball bearing that corresponds to the analyzed application, both formulation and steps are given. Moreover, when the actual application conditions are different of those at which the ball bearing was designed, the proposed method let practitioners to determine the real L-life (or reliability) that corresponds to the actual conditions, a step by step application and formulation is presented. Additionally, this methodology can be applied to any field or application where the input variables are either contact stress or principal stress values.## Abstract

## 1. Introduction

_{10}life percentile, that is obtained through tests performed on static loads and at a certain rotation speed [6]. Therefore, machine designers and people who use machines need to know the actual reliability that corresponds to their own application. In various articles, Erwing Zarestsky explained how the L

_{10}life equation has been analyzed in different models such as the Waloddi Weibull fatigue life model, Lundberg-Palmgren model, Ionnides-Harris model, and Zaretsky model [7,8,9]. In 1962, The Lundberg-Palmgren life equation has been incorporated into the International Organization for Standardization (ISO). Over the years, the formula has been questioned for various reasons, one of which is that it was obtained by testing bearings of the same model and under laboratory conditions. The aim of this article is not to criticize the formula, but rather, to make use of it in a new methodology to obtain the reliability in a current ball bearing application. The actual reliability must be different from the reliability offered in the L

_{10}life equation.

_{10}life of a ball bearing, while Section 2.2 describes the steps to determine the contact area of a ball bearing. In Section 2.3, the steps to determine the principal stresses are given. Section 2.4 shows how to determine the Weibull shape β and scale η parameters, and Section 2.5 shows the steps to determine the actual reliability of the selected ball bearing. In Section 3, an application of the new methodology is presented. Finally, in Section 4, the conclusions are given.

## 2. Hertz Contact Generalities

_{x}, σ

_{y}, σ

_{z}and the maximum shear stress τ

_{max}that are generated below the bearing surface [13]. Note that at the point where τ

_{max}occurs, the cracking occurs in the outer race and that generates the failure. Therefore, the analysis is performed on the outer race of the ball bearing. The analysis is as follows.

#### 2.1. Steps to Determine the Contact Stresses and the L_{10} Life of a Ball Bearing Subsection

_{d}load at which the ball bearing will be subjected. By using the radial (F

_{r}) and axial (F

_{a}) forces determined in the static shaft analysis, with the X radial and Y axial ball bearing coefficients given in the selected ball bearings’ catalogues, determine the equivalent radial load p-value from Equation (1) as:

_{d}-value in Equation (2) is determined in the static phase analysis and, is done based on the applied loads that correspond to the point where the ball bearing is going to be mounted.

_{d}addressed in step 1 must be lower than the catalogues’ dynamic load (C). If C is lower than P

_{d}, another type of ball bearing must be selected. Similarly, the rotation speed at which the ball bearing is going to be mounted must be lower than the one given in the catalogue. Additionally, the P

_{d}-value has to be lower than the basic static load (C

_{o}) of the catalogue.

_{10}life of the ball bearing. By using C from step 2 and P

_{d}from step 1, the ball bearing L

_{10}life is given as [2,17]:

_{10}represents the expected number of cycles at which 10% of the ball bearings will fail.

#### 2.2. Steps to Determine the Contact Area of the Ball Bearing

_{ax}and r

_{ay}are the radius of the ball in the x and y direction, respectively, then r

_{ax}= r

_{ay}. Moreover, r

_{bx}is the curvature radius from the center of the ball bearing to the race of the exterior ring, and r

_{by}is the curvature radius of the race of the exterior ring (see Figure 2). As seen in Figure 1, a ball bearing has two races, one in the exterior ring and one in the inner ring. Note that the analysis is performed over the race of the outer ring due to fact that the distribution of the load between the race of the exterior ring and the ball is higher than between the race of the inner ring and the ball [19].

_{B}is the ball diameter. Therefore, the race of the exterior ring r

_{by}radius is given as:

_{r}as:

_{e}as:

_{r}and the elliptical parameter k

_{e}in the corresponding elliptical functions.

_{r}, in such a way that if 1 ≤ α

_{r}≤ 100, the equations from the left column of Table 1 [12] must be selected. In contrast if 0.01 ≤ α

_{r}≤ 1, the equations from the right column of Table 1 must be selected. From Table 1, also note that based on the α

_{r}value and from the x and y axes of the generated contact ellipse, the a and b values are determined.

_{a}is the Poisson coefficient of the ball, and v

_{b}is the Poisson coefficient of the outer ring. Similarly, E

_{a}is the elasticity module of the ball, and E

_{b}is the elasticity module of the exterior ring.

_{y}) and (D

_{x}) diameters of the ellipse given as:

_{y}> D

_{x}, then a = D

_{y}/2 and b = D

_{x}/2, in contrast if D

_{x}> D

_{y}, then a = D

_{x}/2 and b = D

_{y}/2.

#### 2.3. Steps to Determine the Contact Principal Stresses Values

_{max}value, which occurs in the contact point between the ball and the race of the exterior ring and is given as:

_{x}, σ

_{y}, σ

_{z}, and τ

_{max}values, which are given as:

_{x}, σ

_{y}, y σ

_{z}values are given.

_{x}, Ω

_{y}, Ω’

_{x}, Ω’

_{y}y Δ parameters, which are given as:

_{a}, v

_{b}, E

_{a}, and E

_{b}values are the Poisson coefficient and elasticity module values used in step 8. In addition, A and B are the constant values that depend on the curvature ratio of the ball and inner race.

_{ax}, r

_{ay}, r

_{bx}y r

_{by}were determined in step 4. Now, let’s present the steps to determine the Weibull parameters.

#### 2.4. Steps to Determine the Weibull Shape and Scale Parameters

_{1}and minimum σ

_{3}stresses values, the Weibull scale η and shape β parameters are determined as follows.

_{1}and σ

_{3}values [17].

#### 2.5. Steps to Determine the Actual Reliability of the Selected Ball Bearing

_{10}life value of step 3, with the addressed Weibull β

_{use}and η

_{use}parameters, as well as with the Weibull supplier η

_{cat}parameter.

_{10}life and the η

_{use}parameter. By using the L

_{10}life of step 3 and the β and η

_{use}parameter of step 12, determine the reliability of the ball bearing as:

_{use}life for which R(t) = 0.90. Using the η

_{use}and β parameters estimated in step 12 with R(t) = 0.9 in Equation (32), the L

_{use}life value is given as:

_{cat}value that corresponds to the L

_{10}life. Using the β value in step 12 with the L

_{10}life in step 3, and R(t) = 0.90 from Equation (32), the catalogue scale parameter is given as:

## 3. Application of the Proposed Method

^{6}N mm/s. Additionally, in the intermediate shaft and through the gears, the initial speed of 188 rad/s is reduced to 94 rad/s, while the power of 8.95 × 10

^{6}N mm/s remains constant. The intermediate shaft is shown in Figure 5. This shaft is made of AISI 1020 steel with a shaft’s diameter (d

_{S}) of 45 mm. The designed shaft will rotate with an angular speed of 94 rad/s. In addition, while the pass diameter of gear B is 127 mm, for gear C it is 76.2 mm. Moreover, the pression angle between these gears is 20°, as shown in Figure 6.

_{c}= −95,212.76 N mm. On the other hand, as shown in Figure 6, the radial and tangential forces that are acting on the gears depend on the pression angle of Φ = 20°.

_{B}= 63.5 mm and r

_{C}= 38.1 mm, then numerically:

_{D}= 2415.6 N. Additionally, since the inner race is the one that is rotating, then V = 1, and consequently, from Equation (2) the designed load is:

_{d}= 2415.6 N, then from the SKF catalogue the selected ball bearing is the 6009 SKF type ball bearing with a bore diameter of 45 mm. The related ball bearing characteristics are given in Figure 8 and in Table 2.

_{10}life of the ball bearing is:

_{by}is unknown, then from Equation (7) by using the compliance value of 0.52, the r

_{by}value is ${r}_{by}=d{R}_{r}=\left(8.731\text{}\mathrm{mm}\right)\left(0.52\right)=4.54\text{}\mathrm{mm}$, therefore, $\frac{1}{{R}_{y}}=\frac{1}{4.365\text{}\mathrm{mm}}-\frac{1}{4.54\text{}\mathrm{mm}}=0.00883{\mathrm{mm}}^{-1}$ implying ${R}_{y}=113.24\mathrm{mm}.$ Moreover, since $\frac{1}{R}=0.20052{\mathrm{mm}}^{-1}+0.00883{\mathrm{mm}}^{-1}=0.20935{\mathrm{mm}}^{-1}$, then the total curvature is $R=4.77\mathrm{mm}$.

_{r}is:

_{e}is:

_{e}< 100, the equations of Table 1 are selected.

_{a}= v

_{b}= 0.30 and its elasticity module E

_{a}= E

_{b}= 200 GPa of the material, the effective elasticity module is:

_{max}value is:

_{x}, σ

_{y}, σ

_{z}, values, and the maximum shear τ

_{max}value, as well as the depth at which τ

_{max}occurs are as follows.

_{x}, Ω

_{y}, Ω’

_{x}, Ω’

_{y}and Δ parameters used to determine the corresponding σ

_{x}, σ

_{y}y σ

_{z}values are:

_{1}and minimum contact stress value in Equation (30), the use Weibull scale parameter is [22]:

_{10}life of 765.77 × 10

^{6}rev and the Weibull scale parameter, the reliability of the ball bearing is estimated to be:

_{10}value does not represent the applied stress, (it represents the expected life at the conditions at which the ball bearing was designed). In addition, the η

_{use}value does not represent the strength conditions at which the ball bearing was designed. Therefore, the estimated reliability of R(t) = 0.4485 does not represent the life that corresponds to the actual environment at which shaft 2 operates. Therefore, to determine the actual reliability we must proceed with the next steps.

_{10(use)}life that represents the actual conditions at which shaft 2 operates is given from Equation (33) as ${L}_{10\left(use\right)}={\eta}_{use}\left(\sqrt[\beta ]{-\mathrm{ln}\left(0.90\right)}\right)=\mathrm{156,862,111.04}\text{}rev$.

_{10(use)}value represents the environmental actual stress, it is necessary to determine the Weibull scale parameter that represents the strength of the selected ball bearing. It is determined as follows.

_{cat}value that represents the strength of the ball bearing is determined from Equation (34) using the catalogue L

_{10}life. It is η

_{cat}= $\frac{{L}_{10}}{\sqrt[\beta ]{-\mathrm{ln}\left(0.90\right)}}$ = 4,442,538,996.4 rev.

_{cat}value, the actual reliability is as follows.

_{10}offered by the manufacturer due to the fact that it was analyzed with different loads than those used by the manufacturer. Since the actual reliability is different, it can be said that this methodology can be used to obtain the reliability in cases where the bearing is subjected to different load magnitudes and is required to know its current reliability.

## 4. Conclusions

- The environment in which the catalog life of ball bearings is determined does not correspond to its use in practice.
- Based on the contact stresses generated between the ball and the race, the proposed method allows us to determine the current life of the ball bearing, which represents the conditions of use.
- In determining the current reliability of the ball bearing, the Weibull distribution of two parameters is used where the parameters are directly determined from the contact stresses.
- The efficiency of the method to determine the life of the ball bearing is based on the fact that the catalog η value represents the resistance of the ball bearing material.
- The proposed method can be applied to determine the reliability for different bearing applications.

## Author Contributions

## Funding

## Conflicts of Interest

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Range of the Curvature’s Radius | ||
---|---|---|

Property | (1 ≤ α_{r} ≤ 100) | (0.01 ≤ α_{r} ≤ 1) |

Geometry | ||

Ellipticity ratio | ${k}_{e}={\alpha}_{r}{}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\pi $}\right.}$ | ${k}_{e}={\alpha}_{r}{}^{\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$\pi $}\right.}$ |

Elliptical integer of first order $\mathcal{F}$ | $\mathcal{F}=\frac{\pi}{2}+\left(\frac{\pi}{2}-1\right)\mathrm{ln}{\alpha}_{r}$ | $\mathcal{F}=\frac{\pi}{2}-\left(\frac{\pi}{2}-1\right)\mathrm{ln}{\alpha}_{r}$ |

Elliptical integer of second order $\mathcal{E}$ | $\mathcal{E}=1+\frac{\pi -2}{2{\alpha}_{r}}$ | $\mathcal{E}=1+\left(\frac{\pi}{2}-1\right){\alpha}_{r}$ |

Bearing Type | Capped Single Row Deep Groove Ball Bearings |
---|---|

Bore Diameter (d) | 45 mm |

Outer Diameter (D) | 75 mm |

Width (B) | 16 mm |

Dynamic Load Rating (C) | 22,100 N |

Static Load Rating (C_{o}) | 14,600 N |

Max Speed | 1047 rad/s |

Max. Shaft Shoulder Dia. Inner (U_{i}) | 67.8 mm |

Min. Housing Shoulder Dia., Outer (U_{o}) | 54.7 mm |

Chamfer radius (r) | 1 mm |

Ball Quantity | 13 |

Ball Diameter (d_{b}) | 8.731 mm |

Material | 52,100 Chrome Steel |

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**MDPI and ACS Style**

Villa-Covarrubias, B.; Piña-Monarrez, M.R.; Barraza-Contreras, J.M.; Baro-Tijerina, M.
Stress-Based Weibull Method to Select a Ball Bearing and Determine Its Actual Reliability. *Appl. Sci.* **2020**, *10*, 8100.
https://doi.org/10.3390/app10228100

**AMA Style**

Villa-Covarrubias B, Piña-Monarrez MR, Barraza-Contreras JM, Baro-Tijerina M.
Stress-Based Weibull Method to Select a Ball Bearing and Determine Its Actual Reliability. *Applied Sciences*. 2020; 10(22):8100.
https://doi.org/10.3390/app10228100

**Chicago/Turabian Style**

Villa-Covarrubias, Baldomero, Manuel R. Piña-Monarrez, Jesús M. Barraza-Contreras, and Manuel Baro-Tijerina.
2020. "Stress-Based Weibull Method to Select a Ball Bearing and Determine Its Actual Reliability" *Applied Sciences* 10, no. 22: 8100.
https://doi.org/10.3390/app10228100