# A New Finite Element Analysis Model to Estimate Contact Stress in Ball Screw

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Theoretical Analysis

#### 2.1. Assumptions

#### 2.2. Theoretical Study of Ball Screw

## 3. FEA

#### 3.1. Subject Configuration and Material Property

#### 3.2. Boundary Conditions

#### 3.2.1. Contact Theory Based Model

#### 3.2.2. Contact Condition Applied Model

#### 3.3. Results and Discussion

## 4. Conclusions and Future Works

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$Z$ | The number of balls between shaft and nut |

$P$ | Axial load |

$Q$ | Normal load |

$\alpha $ | Contact angle |

$\lambda $ | Lead angle |

$a$ | Semi major axis of contact ellipse |

$b$ | Semi minor axis of contact ellipse |

${R}_{x}$ | Equivalent radius in the rolling direction |

${R}_{y}$ | Transversal equivalent radius |

$k$ | Radii ratio |

${d}_{w}$ | Ball diameter |

${d}_{m}$ | Pitch circle diameter of balls |

${f}_{s}$ | Curvature parameter for the shaft race |

${f}_{n}$ | Curvature parameter for the nut race |

${E}^{*}$ | Equivalent elastic modulus |

${E}_{s}$ | Elastic modulus of shaft |

${E}_{b}$ | Elastic modulus of ball |

${E}_{n}$ | Elastic modulus of nut |

${\nu}_{s}$ | Poisson’s ratio of shaft |

${\nu}_{b}$ | Poisson’s ratio of ball |

${\nu}_{n}$ | Poisson’s ratio of nut |

${\sigma}_{max}$ | Max contact pressure |

$\sigma $ | Local contact pressure |

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**Figure 5.**Contact ellipse configuration: (

**a**) Shaft; (

**b**) Ball (ball shaft contact); and (

**c**) Ball (ball nut contact).

**Figure 8.**Equivalent stress (MPa) results of contact theory based model (shaft): (

**a**) Type1; (

**b**) Type2; (

**c**) Type3; (

**d**) Type4; and (

**e**) Type5.

**Figure 9.**Equivalent stress (MPa) results of contact theory based model (ball): (

**a**) Type1; (

**b**) Type2; (

**c**) Type3; (

**d**) Type4; and (

**e**) Type5.

**Figure 11.**Equivalent stress (von Mises) results of contact condition applied model: (

**a**) Pure penalty; (

**b**) Augmented Lagrange.

Parameter | Value |
---|---|

The number of balls between shaft and nut | 42 |

Contact angle | 49.0495° |

Ball diameter | 2 mm |

Pitch circle diameter of balls | 10.3 mm |

Curvature parameter | 0.515 |

Parameter | STS440C | SUJ2 | |
---|---|---|---|

Shaft | Nut | Ball | |

Elastic modulus (GPa) | 200 | 210 | |

Poisson’s ratio | 0.283 | 0.28 | |

Yield strength (MPa) | 1280 | 1176 | |

UTS (MPa) | 1750 | 1274 |

Type | Ball-Shaft Contact | Ball-Nut Contact |
---|---|---|

Max. contact stress (MPa) | 1178.2 | 1058.2 |

Type | Ball-Shaft Contact | Ball-Nut Contact |
---|---|---|

Semi major axis (mm) | 0.4611 | 0.4442 |

Semi minor axis (mm) | 0.0411 | 0.0475 |

Types | Contact Surface Mesh Size (mm) | Other Mesh Size (mm) | Max. Equivalent Stress of Shaft (MPa) | Max. Equivalent Stress of Ball (MPa) |
---|---|---|---|---|

Type1 | 0.016 | 0.32 | 644.69 | 545.04 |

Type2 | 0.008 | 0.16 | 703 | 659.16 |

Type3 | 0.004 | 0.08 | 714.54 | 672.62 |

Type4 | 0.002 | 0.04 | 713.74 | 672.83 |

Type5 | 0.001 | 0.02 | 713.15 | 662.7 |

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**MDPI and ACS Style**

Shin, G.-H.; Hur, J.-W. A New Finite Element Analysis Model to Estimate Contact Stress in Ball Screw. *Appl. Sci.* **2022**, *12*, 4713.
https://doi.org/10.3390/app12094713

**AMA Style**

Shin G-H, Hur J-W. A New Finite Element Analysis Model to Estimate Contact Stress in Ball Screw. *Applied Sciences*. 2022; 12(9):4713.
https://doi.org/10.3390/app12094713

**Chicago/Turabian Style**

Shin, Geon-Ho, and Jang-Wook Hur. 2022. "A New Finite Element Analysis Model to Estimate Contact Stress in Ball Screw" *Applied Sciences* 12, no. 9: 4713.
https://doi.org/10.3390/app12094713