# Influence of a Shaft Shoulder on the Torsional Load-Bearing Behaviour of Trochoidal Profile Contours as Positive Shaft–Hub Connections

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Profile Geometry

## 3. Numerical Investigations of Static Torsional Load

#### 3.1. Finite Element (FE) Model Structure

- -
- Von Mises stress σ
_{v,mises}; - -
- Maximum principal stress σ
_{1}; - -
- Contact normal stress σ
_{nn}.

#### 3.2. Load Condition in the Connection with Stepless Profile Shaft

^{2}correspond to a contact. Outside this range, the connection opens up. Shortly before the flank engages, a maximum of the maximum principal stress ${\sigma}_{1}$ is formed on the side of the flank subjected to tensile stress. Comparative stress ${\sigma}_{V,\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{s}}$ is also shown. Only the shaft was analysed numerically, as it generally fails in combination with a sufficiently thick-walled hub, as observed in this study. The curves depicted in Figure 5 served as references for the following procedure.

#### 3.3. Superimposed Influence of the Hub Edge and the Shaft Shoulder

^{2}in the fillet radius ${r}_{s}$ of the shaft shoulder right into the joint. The two extreme points in the fillet radius ${r}_{s}$ and in the area of the hub edge at z = 4 mm are clearly recognisable. Here, the shaft shoulder represents a more critical notch point.

## 4. Experimental Investigations

#### 4.1. Test Bench and Test Parameters

#### 4.2. Influence of a Shaft Shoulder on the Dynamic Transmission Capacity

_{G}= 10 million load cycles. This value has become established in the field of form-fit connections, which fail due to frictional fatigue [6]. In the following procedure, the torsional notch effect number ${\beta}_{\tau}$ could be determined from the achieved endurance limit ${\tau}_{tADK}$ by recursively applying the calculation procedure in DIN 743-Part 1 [19].

_{τ}are shown on the right. All of the hubs in described in this figure were made of normalised C45R+N.

_{τ}of the hobbed shafts, seen on the right in Figure 13. The hubs were all uniformly wire-cut.

^{2}. This comparison confirms the numerical results discussed in Section 3.3, whereby the shaft shoulder represents a relief notch from a stress mechanics point of view. Compared to the FEA results, however, there is an even greater gap here. The tribological stress in the joint provides an explanation for this. Due to the shaft shoulder, the torsional rigidity of the M6-profiled shaft is greater than that of the stepless shaft. This increased torsional rigidity leads, in less relative movement, to the joint between the shaft and hub, which is why the tribological stress is reduced at this point and also has a strength-enhancing effect.

#### 4.3. Interference between the Shaft and Hub

_{a}= 0.5‰ related to the circular tip diameter d

_{a}(see Figure 2) was also investigated. Here, too, the FE simulation was confirmed with a slight increase in the endurance limit ${\tau}_{tADK}$ by 6%, as shown in Figure 13 on the left. In line with this result, the notch coefficient β

_{τ}decreases for the oversized variant.

## 5. Epitrochoidal and Hypotrochoidal Connections with the Stepped Shaft

#### 5.1. Geometry of Specimen

_{a}xd

_{i}.

#### 5.2. Dynamic Strength under Swelling Torsion

_{τ}for each profile type for the load case of pulsating torsion.

^{2}and, thus, a slightly lower notch effect compared to the E3-profile, as can be seen on the right in Figure 16 in the comparison of the notch coefficients β

_{τ}. However, as mentioned, these results relate to the profile being connected with a stepless shaft.

^{2}resulted in one survivor (0), which reached the number of load cycles N = 10 million, and one fracture (x). This means that this load horizon is no longer in the fatigue strength range, which results in a significant reduction in the fatigue strength of ${\tau}_{tADK}$ = 90 N/mm

^{2}of the stepless shaft. The fracture of the test with a distance a = 4 mm at N = 3.1 million load cycles does not initially suggest any strength-increasing influence of the distance a either. However, the number of samples studied is not sufficient for a reliable statement in this context.

## 6. Conclusions and Outlook

_{τ}for the higher-strength shaft material. As a result, it was found that due to the frictional corrosive damage in dynamic operation, the potential of the higher-strength material cannot be utilised due to its greater sensitivity to frictional corrosion.

_{τ}and, therefore, exhibits a slightly worse dynamic torsional load-bearing behaviour. In terms of the effects that a shaft shoulder induces on the overall strength, however, the H3-profile has a slight disadvantage due to its geometry.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

$a$ | Distance of the hub edge to the shaft shoulder; |

A | Surface area; |

${d}_{a}$ | Tip diameter; |

${d}_{aN}$ | Outer diameter of the hub; |

${d}_{i}$ | Root diameter; |

$e$ | Eccentricity; |

${e}_{0}$ | Main eccentricity; |

${e}_{1}\dots {e}_{i}$ | Single eccentricity; |

FE | Finite element |

FEA | Finite element analysis |

l | Arc length; |

${l}_{F}$ | Joining length; |

${M}_{t}$ | Torsional moment; |

n | Number of corners; |

N | Number of load cycles; |

${N}_{G}$ | Limit number of load cycles; |

${Q}_{A}$ | Diameter ratio; |

$r$ | Nominal radius; |

${r}_{s}$ | Fillet radius; |

R | Stress ratio (min/max); |

s | Running coordinate; |

t | Parameter angle; |

${u}_{x,y,z}$ | Displacement in the spatial direction; |

U | Interference fit; |

x | Cartesian coordinate; |

y | Cartesian coordinate; |

${\beta}_{\tau}$ | Notch coefficient; |

µ | Coefficient of friction; |

${\sigma}_{nn}$ | Contact normal stress; |

${\sigma}_{V,\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{s}}$ | Von Mises stress; |

${\sigma}_{1}$ | Maximum principal stress; |

${\tau}_{ta}$ | Torsional stress amplitude; |

${\tau}_{tADK}$ | Fatigue strength. |

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**Figure 1.**Geometric variability of a hybrid trochoid. The form-fit degree is increased by varying the main eccentricity ${e}_{0}$ (

**top**row) and the number of corners n (

**bottom**row).

**Figure 2.**Geometry of the M6-profile used for our numerical and experimental investigations [14].

**Figure 3.**Illustration of the FE model structure with the boundary conditions, simplified for a sector for the load case of pure torsion.

**Figure 4.**Local stress increase in the joint between the shaft and hub in the area of the hub edge using the example of the equivalent stress according to Mises ${\sigma}_{V,\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{s}}$, M

_{t}= 600 Nm.

**Figure 5.**Curve of the evaluation path between two driver heads of the shaft (

**left**) and representation of the normal contact stress ${\sigma}_{nn}$, the maximum principal stress ${\sigma}_{1}$, and the equivalent stress according to Mises ${\sigma}_{V,\mathrm{m}\mathrm{i}\mathrm{s}\mathrm{e}\mathrm{s}}$ (

**right**), for ${M}_{t}$ = 600 Nm, with linear–elastic material behaviour and idealised zero clearance.

**Figure 6.**Stepless shaft with bonded outlet without a shoulder (

**left**) and profile-following fillet radius of ${r}_{s}$ = 1 mm in the shaft shoulder (

**right**).

**Figure 7.**Hub with a stepless M6-profile shaft indicating the distance a of the hub edge from the shaft shoulder and the fillet radius ${r}_{s}$.

**Figure 8.**Distribution of the maximum principal stress ${\sigma}_{1}$ for a stepped M6-profiled shaft with a significant increase in stress in the fillet radius ${r}_{s}$ of the shaft shoulder and in the area of the hub edge; distance a = 4 mm.

**Figure 9.**Axial curve of the maximum principal stress ${\sigma}_{1}$ due to the maximum stress in the rounding radius ${r}_{s}$ of the shaft shoulder (

**left**) and maximum stress ${\sigma}_{1,max}$ as a function of variable distances a of the hub edge from the shaft shoulder (

**right**); ${M}_{t}$ = 600 Nm, linear–elastic material behaviour, idealised zero clearance.

**Figure 10.**Comparison of the stress curves between two driver heads directly at the hub edge (

**left**) and in the fillet radius of the shaft shoulder (

**right**); ${M}_{t}$ = 600 Nm, linear–elastic material behaviour, idealised zero clearance, distance a = 4 mm.

**Figure 11.**Electromechanical unbalanced-mass test rig for generating alternating and pulsating torsional loads.

**Figure 12.**Specification of the hub geometry and position in conjunction with a stepless (

**left**) and stepped (

**right**) M6-profiled shaft for the dynamic component tests.

**Figure 13.**Fatigue strength ${\tau}_{tADK}$ (

**left**) and notch coefficient β

_{τ}(

**right**) of the stepless M6-profiled shaft in direct comparison with the stepped shaft and depiction of the influences of the manufacturing technology, fitting, and material.

**Figure 14.**Initial crack in the area of the friction-corrosive damaged zone (

**left**) and further growth of the friction fatigue fracture (

**centre**) of the stepless M6-profile shaft under a torsional swelling load; purely mechanical stress crack initiation in the fillet of the shaft shoulder when the hub is positioned at a distance of a = 4 mm from the shaft shoulder (

**right**).

**Figure 15.**Hypotrochoidal profile contour standardised according to DIN 3689-1 (

**left**) and epitrochoidal profile (

**right**), both with n = 3 corners and identical tip diameters of ${d}_{a}$ = 40 mm.

**Figure 16.**Fatigue strength ${\tau}_{tADK}$ (

**left**) and notch coefficient β

_{τ}(

**right**) of the H3- and E3- profiles, determined on the basis of a connection with a stepless shaft.

**Figure 17.**Influence of the shaft shoulder on the achieved number of load cycles N for a given torsional stress amplitude ${\tau}_{ta}$ for the H3-profile (

**left**) and E3-profile (

**right**); breaks are marked with “x” and run-outs are marked with “0”.

Labelling | Material of the Shaft | Manufacturing Process Used to Create the Shaft | Fitting |
---|---|---|---|

A | C45R+N | Oscillating non-circular turning | Clearance fit |

B | C45R+N | Hobbing | Clearance fit |

C | C45R+N | Oscillating non-circular turning | Interference fit |

D | 42CrMoS4+QT | Oscillating non-circular turning | Clearance fit |

Profile Type | H-Profile | E-Profile |
---|---|---|

Nominal profile radius r | 18.181 mm | 18.824 mm |

Eccentricity e | 1.818 mm | 1.177 mm |

Number of corners n | 3 | 3 |

Tip diameter d_{a} | 40.00 mm | 40.00 mm |

Root diameter d_{i} | 32.73 mm | 35.29 mm |

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**MDPI and ACS Style**

Ziaei, M.; Selzer, M.; Sommer, H.
Influence of a Shaft Shoulder on the Torsional Load-Bearing Behaviour of Trochoidal Profile Contours as Positive Shaft–Hub Connections. *Eng* **2024**, *5*, 834-850.
https://doi.org/10.3390/eng5020045

**AMA Style**

Ziaei M, Selzer M, Sommer H.
Influence of a Shaft Shoulder on the Torsional Load-Bearing Behaviour of Trochoidal Profile Contours as Positive Shaft–Hub Connections. *Eng*. 2024; 5(2):834-850.
https://doi.org/10.3390/eng5020045

**Chicago/Turabian Style**

Ziaei, Masoud, Marcus Selzer, and Heiko Sommer.
2024. "Influence of a Shaft Shoulder on the Torsional Load-Bearing Behaviour of Trochoidal Profile Contours as Positive Shaft–Hub Connections" *Eng* 5, no. 2: 834-850.
https://doi.org/10.3390/eng5020045