# Reliability Analysis of Critical Systems in A Fuel Booster Pump Using Advanced Simulation Techniques

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

_{e}indicates the elastic strain amplitude; σ

_{f}

^{′}indicates the fatigue strength coefficient; b indicates the fatigue strength index; E indicates the elastic modulus; and N

_{f}indicates fatigue life, respectively.

_{y}is the oxidation wear depth; Δh

_{m}is the wear depth of abrasive particles; Δh

_{n}is the adhesive wear depth; k

_{y}is the oxidation wear coefficient; k

_{m}is the abrasive wear coefficient; k

_{n}is the adhesive wear coefficient; s is the relative sliding distance; and p is the contact pressure, respectively.

_{max}is the allowable wear depth.

_{max}and Va

_{min}are the maximum and minimum values of the design parameters, respectively; N

_{1}and N

_{2}are the (logarithmic) life means corresponding to Va

_{max}and Va

_{min}, respectively.

## 3. Reliability Simulation

#### 3.1. LHS

#### 3.2. Finite Element Simulation

#### 3.2.1. Local Conditions of the Key Components

#### 3.2.2. Sealing Bolt

^{3}kg/m

^{3}, 2.00 × 10

^{5}MPa, and 0.32, respectively. The finite element model of the sealing bolt was established by using a parametric modeling approach with a set of geometric parameters, operational loads, material properties, and PoF model parameters. Figure 4a illustrates a random sealing bolt finite element model in which the boundary conditions have two aspects: (1) local conditions obtained from the fuel booster pump structural simulation; and (2) preload by Equation (6) applied to the cross-section of the bolt screw [40].

_{s}is the nominal ultimate strength of the bolt and A

_{s}is the cross-sectional area of the bolt. Figure 4b shows the simulation results of the von Mises strains in which the maximum von Mises strains of the sealing bolt are located at the interface between the screw and the nut, and used for calculating the fatigue life of the sealing bolt by using the Basquin model.

#### 3.2.3. Spline Shaft

^{3}kg/m

^{3}, 2.09 × 10

^{5}MPa and 0.3, respectively. Similar to the sealing bolts, the finite element model of a spline shaft was also established with design parameters related to its small spline, chamfer, smooth shaft, and large spline by using the parametric modeling approach, and illustrated in Figure 6a. In addition, Figure 6b shows an extended spline shaft model that is connected to the main shaft (made of the same material) via an internal spline. The internal spline of the main shaft is set in frictional contact with the small spline of the spline shaft, with a friction coefficient of 0.1. The boundary conditions of the spline model have two aspects: (1) local conditions obtained from the fuel booster pump structural simulation; and (2) full constraints from the main shaft.

#### 3.2.4. Graphite Ring

^{−3}MPa on the right side of the moving ring.

#### 3.2.5. Inducer

^{3}kg/m

^{3}, 7.18 × 10

^{4}MPa, and 0.33, respectively. The parametric model of the inducer is shown in Figure 10a. The boundary conditions of the graphite ring were from three aspects: (1) local conditions obtained from the fuel booster pump structural simulation; (2) Z directional constraints on the left and right ends of the inducer; and (3) fuel pressures of 0.5MPa and 0.3MPa on the front and back surfaces of the inducer, respectively.

## 4. Results and Discussions

#### 4.1. Prediction of Life Distributions

#### 4.2. Sensitivity Analysis

- (1)
- Sealing bolt

_{1}, and preload F had the greatest impacts on the mean fatigue life of the sealing bolt. Among these three parameters, b

_{1}acted as a negative exponential factor in the Basquin model. Therefore, a small increase in b

_{1}would lead to an obvious decrease in the predicted fatigue life. The parameters d and F belong to the geometrical parameters and operational loads, respectively. They exhibited strong sensitivities to the sealing bolt fatigue life because they play significant roles in calculating von Mises strains in the finite element simulations. This agrees with the observations in the literature [36,42], which also claim that the nominal diameter and the preload have a great influence on the fatigue lives of the bolts.

_{f1′}, height of nut m, and inner circle diameter of nut s also exhibited moderate sensitivities to the mean fatigue life of the sealing bolt. It can also be seen that the impact of the pre-exponential factor (i.e., σ

_{f1′}on the predicted fatigue lives was much less than that of exponential factor b), according to the characteristics of exponential law. m and s are both geometric parameters associated with the nuts. Small fluctuations in them will not cause great changes in the calculation of the maximum von Mises strains of sealing bolts as well as their fatigue lives.

_{1}has very limited influence on the mean fatigue life of the sealing bolt. This is because E

_{1}influences the fatigue reliability from two completely opposite directions. On one hand, it has a negative effect on the maximum von Mises strain calculated from the finite element simulation and therefore is positive in improving the fatigue life predicted from the Basquin model. On the other hand, E

_{1}is the pre-exponential factor in the Basquin model and plays a negative role in increasing the fatigue life. While they counteract each other, the modulus of elasticity has a very limited influence on fatigue reliability.

- (2)
- Spline shaft

_{2}and D

_{1}of the small spline had the strongest influence on the life of the spline shaft. The decrease in D

_{2}would lead to an obvious reduction in the shaft section area, whereas the increase in D

_{1}will increase the number of the teeth. Both actions will result in significantly higher stress/strain levels to reduce the life of the spline shaft, regardless of whether it is driven by either shaft fatigue or tooth wear. In [43], a similar argument was also drawn by claiming that the increase in the number of teeth plays an important role to enhance the strength of the spline shaft. Therefore, appropriate treatments on the parameters D

_{2}and D

_{1}will be the key issues to ensuring a long service life for the spline shaft.

_{f3′}was also sort of sensitive to the spline shaft life, but not as strong as the D

_{2}and D

_{1}parameters. In contrast, the elastic modulus E

_{2}of the spline shaft, oxidation wear coefficient k

_{y}, and chamfer R were significantly less sensitive to the life of the spline shaft.

- (3)
- Graphite ring

_{2}. Since the graphite ring is closely attached to the moving ring, a slight increase in d

_{2}would produce a rapid increase in the contact stress between the graphite ring and the moving ring, which would significantly accelerate the abrasive wear failure. Moreover, the abrasive wear coefficient k

_{n}and spring pressure p had a moderate sensitivity to the life of the graphite ring, probably in the following ways. The wear coefficient k

_{n}is related to the lubrication state of the contact surface, the hardness of the grinding material, and other material properties. The spring pressure p provides the driving force of wear. The sensitivities of other parameters including the Brinell hardness H, thickness h

_{2}, and elastic modulus E

_{3}were much lower and could be ignored.

_{2}is well under control in the wear life design of the graphite ring. In addition, it is also recommended that the r wear coefficient k

_{n}is reduced by controlling the lubrication, temperature, heat dissipation and viscosity of the seal, and paying attention to the selection of the spring to ensure that the spring pressure p is within a stable pressure range.

- (4)
- Inducer

_{1}exhibited the strongest sensitivity to the fatigue life of the inducer due to the fact that it is the most critical parameter in the calculation of the maximum von Mises strain, as also indicated in the literature [44]. Next, the blade thickness W

_{0}, fatigue strength coefficient σ

_{f}

_{4′}, fuel pressure at the back of blade P

_{2}, and fatigue strength index b

_{4}exhibited middle-level sensitivity to the inducer’s fatigue life. An appropriate increase in the blade thickness W

_{0}is helpful to strengthen the blade structure of the inducer and reduce the maximum von Mises strain at the blade root. Finally, the least influential parameter is the elastic modulus E

_{4}, which barely affected the inducer fatigue life.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Distribution of displacement of the four key components: (

**a**) sealing bolt; (

**b**) graphite ring; (

**c**) inducer; (

**d**) spline shaft.

**Figure 4.**Longitudinal section of a sealing bolt model: (

**a**) finite element model; (

**b**) simulation results of the von Mises strains.

**Figure 6.**The finite element model of a spline shaft: (

**a**) overall model; (

**b**) an extended model with connection to the main shaft.

**Figure 8.**Distribution of the contact parameter results of the spline shaft model: (

**a**) relative sliding distance; (

**b**) contact stress.

**Figure 9.**The graphite ring model in an oblique view: (

**a**) finite element model; (

**b**) simulation results of contact stresses.

**Figure 10.**The inducer model in a front view: (

**a**) finite element model; (

**b**) Simulation results of von Mises strains.

Key Components | Function | Failure | PoF Model |
---|---|---|---|

Sealing bolt | Tighten and seal | Fatigue | Basquin model |

Spline shaft | Transmission torque | Wear | Fretting wear model |

Fatigue | Basquin model | ||

Graphite ring | Mechanical seal | Wear | Archard wear model |

Inducer | Guide and pressurize the fuel | Fatigue | Basquin model |

No. | Parameter | Unit | Distribution | Mean | CoV |
---|---|---|---|---|---|

1 | Fatigue strength coefficient σ_{f1′} | MPa | Normal | 1.57 × 10^{3} [21] | 0.05 [21] |

2 | Fatigue strength index b_{1} | / | Normal | −0.10 [21] | 0.05 [21] |

3 | Preload F | N | Normal | 6.04 × 10^{4} [28] | 8.33 × 10^{−2} [28] |

4 | Nominal diameter d | mm | Normal | 16.0 [29] | 5.80 × 10^{−3} [29] |

5 | Inner circle diameter of nut s | mm | Normal | 24.0 [29] | 4.60 × 10^{−3} [29] |

6 | Height of nut m | mm | Normal | 8.00 [29] | 2.38 × 10^{−2} [29] |

7 | Elastic modulus E_{1} | MPa | Normal | 2.00 × 10^{5} [30] | 0.05 [20] |

No. | Parameter | Unit | Distribution | Mean | CoV |
---|---|---|---|---|---|

1 | Elastic modulus E_{2} | MPa | Normal | 2.09 × 10^{5} [31] | 0.05 [21] |

2 | Chamfer δ | mm | Normal | 0.200 [31] | 0.25 [21] |

3 | Major diameter of small spline D_{1} | mm | Normal | 20.0 [31] | 2.15 × 10^{−3} |

4 | Minor diameter of small spline D_{2} | mm | Normal | 17.5 [13] | 3.43 × 10^{−3} |

5 | Fatigue strength coefficient σ_{f2′} | MPa | Normal | 2.04 × 10^{3} [31] | 0.05 [21] |

6 | Oxidation wear coefficient k_{y} | / | Normal | 8.60 × 10^{−9} [27] | 0.05 [21] |

Order | Parameter | Unit | Distribution | Mean | Coefficients of Variation |
---|---|---|---|---|---|

1 | Thickness h_{2} | mm | Normal | 5.90 | 2.83 × 10^{−3} |

2 | Inside diameter d_{2} | mm | Normal | 30.0 | 2.30 × 10^{−4} |

3 | Elastic modulus E_{3} | MPa | Normal | 1.25 × 10^{5} | 0.05 [21] |

4 | Spring pressure p | MPa | Normal | 0.0650 | 0.05 [21] |

5 | Brinell hardness H | HB | Normal | 33.0 | 0.05 [21] |

6 | Wear coefficient k | / | Normal | 3.30 × 10^{−7} | 0.05 [21] |

Order | Parameter | Unit | Distribution | Mean | Coefficients of Variation |
---|---|---|---|---|---|

1 | Fuel pressure on the back of blade P_{2} | MPa | Normal | 0.30 | 0.05 [21] |

2 | Fuel pressure on the front of blade P_{1} | MPa | Normal | 0.50 | 0.05 [21] |

3 | Elastic modulus E_{4} | Mpa | Normal | 7.18 × 10^{4} | 0.05 [21] |

4 | Blade thickness W_{0} | mm | Normal | 1.60 | 9.36 × 10^{−3} |

5 | Fatigue strength coefficient σ_{f4′} | Mpa | Normal | 1.37 × 10^{3} | 0.05 [21] |

6 | Fatigue strength index b_{4} | / | Normal | −7.30 × 10^{−2} | 0.05 [21] |

Key Components | Assembly Surfaces |
---|---|

Sealing bolt | Outer surface of the stud Lower surface of the bolt head Upper surface of the nut data |

Spline shaft | Outer surface of the optical shaft data |

Graphite ring | Inner ring surface |

Inducer | Inner ring surface |

Material | Density (kg/m^{3}) | Elastic Modulus (MPa) | Poisson’s Ratio |
---|---|---|---|

40CrNiMoA | 7.83 × 10^{3} | 2.09 × 10^{5} | 0.30 |

9Cr18 | 7.70 × 10^{3} | 2.15 × 10^{5} | 0.26 |

Key Component | Order | Parameter | S |
---|---|---|---|

Sealing bolt | 1 | d | 1.1148 |

2 | b_{1} | −1.0689 | |

3 | F | −0.7077 | |

4 | σ_{f}_{1′} | 0.5983 | |

5 | s | 0.2806 | |

6 | m | 0.1884 | |

7 | E_{1} | 0.003 | |

Spline shaft | 1 | D_{2} | 1.8717 |

2 | D_{1} | −1.5694 | |

3 | σ_{f}_{3′} | 0.7242 | |

4 | E_{2} | −0.0683 | |

5 | k_{y} | −0.0566 | |

6 | R | 0.0054 | |

Graphite ring | 1 | d_{2} | 0.6017 |

2 | p | −0.0705 | |

3 | k_{n} | −0.0696 | |

4 | H | 0.0673 | |

5 | h_{2} | −0.0221 | |

6 | E_{3} | −0.0007 | |

Inducer | 1 | P_{1} | −4.2274 |

2 | W_{0} | 0.7147 | |

3 | σ_{f}_{4′} | 0.46 | |

4 | P_{2} | 0.4225 | |

5 | b_{4} | −0.2206 | |

6 | E_{4} | −0.0036 |

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## Share and Cite

**MDPI and ACS Style**

Luo, Y.; Dong, Y.; Li, Y.; Hu, T.; Guo, Y.; Qian, C.; Yang, Z.; Zheng, H.
Reliability Analysis of Critical Systems in A Fuel Booster Pump Using Advanced Simulation Techniques. *Materials* **2022**, *15*, 1989.
https://doi.org/10.3390/ma15061989

**AMA Style**

Luo Y, Dong Y, Li Y, Hu T, Guo Y, Qian C, Yang Z, Zheng H.
Reliability Analysis of Critical Systems in A Fuel Booster Pump Using Advanced Simulation Techniques. *Materials*. 2022; 15(6):1989.
https://doi.org/10.3390/ma15061989

**Chicago/Turabian Style**

Luo, Ying, Yuanyuan Dong, Yuguang Li, Tian Hu, Yubei Guo, Cheng Qian, Zhihai Yang, and Hao Zheng.
2022. "Reliability Analysis of Critical Systems in A Fuel Booster Pump Using Advanced Simulation Techniques" *Materials* 15, no. 6: 1989.
https://doi.org/10.3390/ma15061989