# Analysis of Contact Deformations in Support Systems Using Roller Prisms

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Research Problem

#### 2.2. Reaction Forces Determination

#### 2.3. Finite Element Model for Contact Analysis

## 3. Results and Discussion

_{0}is determined as follows:

_{mean}is:

_{e}= 250 GPa and the Poisson ratio was ν = 0.28. In terms of elasticity, the material tensile diagram is rectilinear with a steep inclination angle, the tangent value of which, in the adopted reference frame, corresponds to a Young’s modulus of E = 210 GPa. The steeply sloping waveform is valid until:

- Stage 1—initial shaft pressure on roller the with a force of R = 150 N, “zero” position;
- Stage 2—rotation of the shaft by an angle of +4.5°, a gradual increase in the pressure force to a value of R = 4000 N;
- Stage 3—rotation of the shaft by an angle of −4.5° under the pressure of the force R = 4000 N, “zero” position;
- Stage 4—rotation of the shaft by a further angle of −4.5° under the pressure of the force R = 4000 N;
- Stage 5—rotation of the shaft by an angle of +4.5° and a gradual decrease in the pressure force to R = 0 N.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Finite element model of the analyzed crankshaft with the adopted numbering of the supported main journals.

**Figure 3.**Distribution of the reaction forces guaranteeing zero value of deformations on the main pins of the crankshaft of a Buckau Wolf R8DV-136 motor. Diagrams of the coordinate systems: (

**a**) Cartesian and (

**b**) polar.

**Figure 4.**The cooperation model adopted for the analysis of surface pressures and stresses at the contact point of the rolling prism roller with the main shaft journal.

**Figure 5.**Scheme of the shaft–roller section against the background of both wheels, showing the shaft and the roller (

**a**). The adopted model of cooperation between the shaft section and a 1-mm thick V-block (

**b**).

**Figure 6.**Division with finite elements with a side of 0.01 mm, near the contact of the shaft with the support roller.

**Figure 7.**Calculation results of the finite element model, showing reduced stresses in the shaft and support roller, according to the von Mises hypothesis, for the case where a shaft and support roller are made of steel, with a reaction force R = 1253 N.

**Figure 8.**Calculations, showing reduced stresses in the shaft and support roller according to the von Mises hypothesis, for the case of a steel shaft and a support roller made of aluminum–bronze, under a reaction force of R = 1253 N.

**Figure 10.**Stages of simulation calculations when shifting the shaft journal over a support roller over a distance of 5 mm.

**Figure 11.**Distribution of permanent stress values in the surface layer of the steel shaft journal, not burdened with outline irregularities, after the outline has been turned over the steel supporting roller.

**Figure 12.**Distribution of permanent stress values in the surface layer of a steel shaft journal with a profile free of outline irregularities after the outline has been turned over a support roller made of aluminum–bronze.

**Figure 13.**Comparison of the profile of the steel shaft journal, which is not burdened with outline irregularities, with the profiles after the outline has turned over the steel support roller and the support roller made of aluminum–bronze.

**Figure 15.**Actual measured outline of the roundness of journal No. 10 developed over a length equal to 471 mm, which corresponds to the circumference of a wheel with a nominal diameter of the measured main shaft journal of 150 mm.

**Figure 16.**Profiles: not burdened with irregularities and actual one around coordinate x = 200 mm (from Figure 15), transformed into coordinate x = 0.

**Figure 17.**Distribution of permanent stress values in the surface layer of the steel shaft journal, with the actual profile of journal No. 10, after this profile has been turned over the steel support roller.

**Figure 18.**Distribution of permanent stress values in the surface layer of the steel shaft journal with actual journal profile No. 10, after this profile has been turned over the support roller made of aluminum–bronze.

**Figure 19.**Comparison of the actual profile of a steel shaft with the profiles after the actual outline has been turned over a support roller made of steel and a support roller made of aluminum–bronze.

**Figure 20.**Graphs of von Mises stresses remaining in the steel shaft journal as a function of the depth of their occurrence, for the “zero” position at the reaction force of R = 4000 N for the journal profile not burdened with outline irregularities and the actual one, after turning these outlines on a steel support roller and a support roller made of aluminum–bronze.

**Table 1.**Summary of the results of calculations of the maximum stresses and the depth of their occurrence for the shaft journal (SJ) and the steel roller support (RS), under a reaction force of R = 1253 N.

Quantity | Unit | Hertz Contact Theory | Finite Element Model | Relative Error % |
---|---|---|---|---|

Hertz contact stress | MPa | 3.4 | 359.9 | 5.4 |

Max. shear stress RS | MPa | 114.2 | 110.7 | 3.1 |

Max. shear stress SJ | MPa | 114.2 | 108.6 | 4.8 |

Max. von Mises stress body RS | MPa | 215.2 | 206.4 | 4.6 |

Max. von Mises stress body SJ | MPa | 215.2 | 203.4 | 5.4 |

Max. shear stress location RS | μm | −117.8 | −120.0 | 1.8 |

Max. shear stress location SJ | μm | −117.8 | −120.0 | 1.8 |

Max von Mises stress location RS | μm | −102.8 | −110.0 | 6.5 |

Max von Mises stress location SJ | μm | −102.8 | −110.0 | 6.5 |

**Table 2.**Summary of the calculation results of the maximum stresses and the depths of their occurrence for the steel shaft journal (SJ) and the aluminum–bronze roller support (BRS), after a reaction force of R = 1253 N.

Quantity | Unit | Hertz Contact Theory | Finite Element Model | Relative Error % |
---|---|---|---|---|

Hertz contact stress | MPa | 309.8 | 300.6 | 3.0 |

Max. shear stress BRS | MPa | 93.03 | 93.7 | 0.7 |

Max. shear stress SJ | MPa | 93.03 | 91.9 | 1.2 |

Max. von Mises stress body RS | MPa | 168.5 | 168.5 | 0.0 |

Max. von Mises stress body SJ | MPa | 175.3 | 171.5 | 2.2 |

Max. shear stress location BRS | μm | −144.6 | −150.0 | 3.6 |

Max. shear stress location SJ | μm | −144.6 | −152.0 | 4.8 |

Max von Mises stress location BRS | μm | −135.1 | −130.0 | 3.8 |

Max von Mises stress location SJ | μm | −126.2 | −128.0 | 1.4 |

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**MDPI and ACS Style**

Nozdrzykowski, K.; Grządziel, Z.; Dunaj, P.
Analysis of Contact Deformations in Support Systems Using Roller Prisms. *Materials* **2021**, *14*, 2644.
https://doi.org/10.3390/ma14102644

**AMA Style**

Nozdrzykowski K, Grządziel Z, Dunaj P.
Analysis of Contact Deformations in Support Systems Using Roller Prisms. *Materials*. 2021; 14(10):2644.
https://doi.org/10.3390/ma14102644

**Chicago/Turabian Style**

Nozdrzykowski, Krzysztof, Zenon Grządziel, and Paweł Dunaj.
2021. "Analysis of Contact Deformations in Support Systems Using Roller Prisms" *Materials* 14, no. 10: 2644.
https://doi.org/10.3390/ma14102644