# Analytical Model of a Multi-Step Straightening Process for Linear Guideways Considering Neutral Axis Deviation

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## Abstract

**:**

## 1. Introduction

## 2. Straightening System of Linear Guideway

#### 2.1. Straightening Principle

#### 2.2. Geometrical and Material Models

## 3. Analytical Model of Multi-Step Straightening Process

#### 3.1. Straightening Moment

#### 3.2. Mechanism of Residual Stress and Strain

#### 3.3. Analytical Model of the Second Straightening Process

## 4. Numerical Simulation of the Multi-Step Straightening Process

#### 4.1. Analytical Model of the Straightening Process

#### 4.2. Finite Element (FE) Simulation of the Multi-Step Straightening Process

## 5. Results

#### 5.1. Experimental Schemes of Straightening Process and Deflection Measurement

#### 5.2. Results

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 9.**FE model of the multi-step straightening process with respect to linear guideways: (

**a**) the 3D numerical model of the straightening process; (

**b**) meshing and boundary conditions in the compressive direction of straightening process; (

**c**) meshing and boundary conditions in the tensile direction of straightening process.

**Figure 10.**Analysis of simulation results with respect to the multi-step straightening process of linear guideways: (

**a**) deformation of linear guideway after the straightening process; (

**b**) longitudinal stress curve obtained by the defined path.

**Figure 11.**Theoretical results and simulation results of longitudinal stress distribution during the multi-step straightening process with respect to the opposite straightening directions: (

**a**) stress curves with respect to first loading process; (

**b**) residual stress curves with respect to first unloading process; (

**c**) stress curves with respect to second loading process; (

**d**) permeation between workpiece and indenter during the multi-step straightening process.

**Figure 12.**Theoretical results and simulation results of longitudinal stress distribution during the multi-step straightening process with respect to the same straightening directions: (

**a**) stress curves with respect to first loading process; (

**b**) residual stress curves with respect to first unloading process; (

**c**) stress curves with respect to second loading process; (

**d**) permeation between workpiece and indenter during multi-step straightening process.

**Figure 16.**Superficial strain of the sample workpiece: (

**a**) superficial strain on tensile surface; (

**b**) superficial strain on compressive surface.

Material | GCr15 |
---|---|

Density (kg/m^{3}) | 7810 |

Young’s modulus (E/Mpa) | 190,089 |

Hardening modulus (${E}^{\prime}$/Mpa) | 3800 |

Poison’s ratio | 0.3 |

Yield stress (${\sigma}_{s}$/Mpa) | 402 |

Elastic-limit strain (${\epsilon}_{t}$) | 0.004 |

Specifications (mm) | LG15.5 × 14.5 × 400 |

Distance between two clamps ($2{L}_{s}$/mm) | 300 |

Initial deflection (${\delta}_{0}$/mm) | 1.581 |

Measurement range | ±3 mm |

Measurement accuracy | 1/1024 mm |

Linear error | ±1% |

Repeatability error | 0.2 μm |

First Straightening Step | First Unloading | Second Straightening Step | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Theor. | FEA | Expt. | Theor. | FEA | Expt. | Theor. | FEA | Expt. | ||

Tensile region | Strain (10^{−3}) | 7.0 | 6.5 | 7.5 | 4.8 | 5.0 | 5.2 | 5.5 | 5.7 | 6.0 |

Stress (Mpa) | 408 | 382 | ─ | 278 | 286 | ─ | 405 | 407 | ─ | |

Compressive region | Strain (10^{−3}) | 7.5 | 7.8 | 8.0 | 4.5 | 4.1 | 5.1 | 5.5 | 5.8 | 5.9 |

Stress (Mpa) | 409 | 421 | ─ | 230 | 216 | ─ | 405 | 408 | ─ |

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**MDPI and ACS Style**

Zhang, Y.; Lu, H.; Ling, H.; Lian, Y.; Ma, M.
Analytical Model of a Multi-Step Straightening Process for Linear Guideways Considering Neutral Axis Deviation. *Symmetry* **2018**, *10*, 316.
https://doi.org/10.3390/sym10080316

**AMA Style**

Zhang Y, Lu H, Ling H, Lian Y, Ma M.
Analytical Model of a Multi-Step Straightening Process for Linear Guideways Considering Neutral Axis Deviation. *Symmetry*. 2018; 10(8):316.
https://doi.org/10.3390/sym10080316

**Chicago/Turabian Style**

Zhang, Yongquan, Hong Lu, He Ling, Yang Lian, and Mingtian Ma.
2018. "Analytical Model of a Multi-Step Straightening Process for Linear Guideways Considering Neutral Axis Deviation" *Symmetry* 10, no. 8: 316.
https://doi.org/10.3390/sym10080316