# On Hirth Ring Couplings: Design Principles Including the Effect of Friction

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## Abstract

**:**

## 1. Introduction

- ○
- accuracy, in terms of relative positioning between the parts connected together;
- ○
- stiffness of the system, thanks to the steady self-center geometry of the teeth;
- ○
- power, in terms of torque transmission;
- ○
- resistance to wear, thanks to a wide support surface of the teeth and the use of special alloyed steels.

## 2. Standard Calculation

_{u}(N) is applied on the mean radius R

_{m}(mm) of the G point (center of gravity) of the tooth section and can be evaluated by Equation (1) as a function of the external torque T (Nmm) to be transmitted:

_{b}(MPa), approximating the trapezoidal section at the base by a rectangular one, as reported in Equation (3), where z is the number of teeth:

_{b_ref}(MPa) are reported in Table 1, based on the assumption of a minimum root radius (r > 0.3 mm [14]).

_{ref}(MPa) in Table 1.

_{u}(N) (Equation (1)). F

_{u}generates, in turn, the axial force F

_{a}(N) (Equation (2)). This axial force F

_{a}must be absorbed by preloading devices, such as disk springs, hydraulic pistons or bolts. The required pre-load F

_{v−a}(N) is calculated, introducing a safety factor ν (Equation (5)):

_{va}(N) is high enough. The varying loads on the tooth flanks result merely in a slightly irregular distribution of the pressure preload in the tooth root cross-section generating a maximum pressure p

_{max}(MPa) according to Equation (6). In this equation (see Figure 4) A

_{z}(mm

^{2}) is the effective tooth flank area, D and d (mm) are the outer and the inner diameters of the teeth, d

_{L}(mm) is the fixing hole diameter, n

_{b}is the number of bolts in the teeth surface, r (mm) is the tooth root radius, s (mm) is the crown clearance, z the number of the teeth and η

_{z}is the load bearing percentage (0.65 for milled teeth or 0.75 for grinded teeth). Under compression, with a high enough F

_{va}load and with no transmission of torque, this load is equally distributed on both faces of each tooth. Conversely, when transmitting the torque T, the pressure rises on one face of the tooth and drops down on the other. The maximum pressure p

_{max}(MPa) is calculated as follows:

_{m}(mm) (see Figure 4, Equation (7) and Table 2).

_{4}(W. Nr. 1.7225) or 34Cr

_{4}(W. Nr. 1.7033). Their production cycle is based on the following operations: turning, drilling the fixing holes and pins, milling the teeth, inductive hardening of the tooth area (hardening in the range 52 … 60 HRC), rough grinding of the teeth, finishing grinding of the teeth and of the reference surfaces, dimensional checking, finishing the pin holes, final checking and production of measuring report. An example of a measuring report is shown in Table 3.

## 3. Improved Calculation

#### 3.1. Effect of Friction

_{u,μ}(N) is reduced if the pre-load remains unchanged. Otherwise, a higher axial pre-load becomes necessary to achieve the same transmission. The coefficient of friction μ = tan(ρ) in compression couplings can be evaluated by experimental tests, such as those reported in [24,25], and can may range between 0.1 and 0.3 for smooth surfaces.

_{p}) with respect to the value calculated by Equation (6)) and, then, to plot it in the diagram of Figure 6. In the case of negligible friction (μ = 0) the difference between Equation (6) and the numerical investigation is almost equal to zero.

#### 3.2. Self-Centering Capability

_{a}(N) has to be high enough to move all the weight related to the rotary table on the bearing surface. The calculation of the self-centering force F

_{c}(N), which is not provided in the technical or in the scientific literature, will be shown in the following.

_{c}should recreate the (a) condition). As a consequence, the space between the two rings is not constant on the 360° span and furthermore an angular misalignment γ (°) of the teeth also occurs: this is variable on the 360° span.

_{max}(mm) for the displacement of the center of the rotating ring, in order to ensure the correct position (Equation (11)). The maximum misalignment γ for θ = 90° (see Figure 9c) is considered for processing.

_{max}.

_{c}(N) must be high enough to overcome all the friction forces, and to consequently move the rotating ring (along with the overall mass) towards its correct position. The self-centering force F

_{c}(N) can be calculated by means of Equation (12), where every single tooth provides a different contribution, as a function of its angular position:

_{C}equal to 0.642 for z = 360; 0.650 for z = 144; 0.664 for z = 72.

## 4. Experimental Tests

_{a}, and considering the angle of friction ρ between the teeth of 11.3° (μ = 0.2), a self-centering force F

_{c}= ~225 N is yielded by Equation (12). The frictional resistance force R

_{μ}(N) can be evaluated as the overall sum of the weights of the rotating and upper rings (i.e., the rings that must be moved into the correct position, leveraging the position of the fixed ring), multiplied by the aforementioned coefficient of friction in the bearing surface; a force R

_{μ}= ~112 N is determined (reference sketch in Figure 11). The experimental test has shown the self-centering capability: the rings were able to shift to the correct engaged position, being F

_{c}> R

_{μ}. To fully validate the correctness of the formulation, an additional external force in the radial direction F

_{r,e}(N) has been applied and measured by the same dynamometer (Figure 12). This force can be directly combined with the frictional force R

_{μ}(it can be regarded as an extra-frictional force), because it is able to work against the self-centering force Fc. In this way, the self-centering capability has been evaluated accurately, step-by-step increasing the force F

_{r,e}, until (R

_{μ}+ F

_{r,e}) equalized the available F

_{c}. The retrieved results are reported in Table 4; it can be highlighted that up to F

_{r,e}= 100 N, the self-centering force is able to move the rings into the correct position, whereas in the range between F

_{r,e}= 100 and 120 N, the self-centering capability is equal to the resistance forces: therefore, no movement occurs. As a matter of a fact, the calculated value of Fc = ~225 N (Equation (12)) is reliable since the addition of R

_{μ}and F

_{r,e}yields ~112 + 120 = 232 N.

## 5. Case Study

- The condition of the minimum torque T to be transmitted (ID8) is satisfied with the requested safety factor (SF = 4, ID11): 15,750 Nm > 3500 Nm (ID21 > ID8). The same condition also applies (obviously) to the tangential forces F
_{u}: 35,000 N > 7778 N (ID13 > ID10). - The axial force F
_{a}required for the transmission of the torque is 30,965 N (ID14): it is granted by a hydraulic piston (upon setting the pressure level of the oil, this force is fixed). - The frictional force R
_{μ}, depending on the weight of the rotary table (40,000 N, ID16) and on the coefficient of friction (~0.2, ID5), is 8138 N. The self-centering capability is verified since the condition F_{c}> R_{μ}is satisfied (22,326 N > 8138 N, ID15 > ID17). - The accuracy of the angular positioning is also verified. The friction torque on the bearing surface of the rotating ring, which depends on the frictional force R
_{μ}, on the mean sliding radius (ID18) and on the coefficient of friction (ID5), is 3825 Nm; the additional frictional torque for the rotating elements (such as rotary sealings for oil distribution, bearings and gears) is 3000 Nm (ID20). The torque T that the Hirth connection is able to provide proves to be higher than the total frictional torque T_{μ}: 15,750 Nm > 6825 Nm (ID21 > ID22).

_{b}= 0.23 MPa and τ = 0.60 MPa, which are very far from their admissible thresholds (50 MPa and 18.5 MPa respectively, Table 1).

_{a}has not been modified, provided that the system works with a hydraulic piston set at a fixed pressure level and is able to provide F

_{a}= 30,965 N (ID14). F

_{u,μ}is therefore recalculated by Equation (8), depending on the actual value of the friction angle ρ. The other forces depending on friction are also re-calculated. The results are reported in the chart of Figure 15 and in Table 6, where the double effect of the increase of the coefficient of friction is well visible; on one hand, it reduces the capability of the active (self-centering) forces and on the other, it increases the resistance (frictional) forces. In the proposed case study, the detrimental effect of rust is very clear: an increase of the coefficient of friction from its initial value of ~0.2 (for new surfaces in contact) to a value of 0.4 (or more for the rusted ones) results in the complete loss of the positioning capability of the Hirth connection for the rotary table. This in turn implies the loss of accuracy and precision in the production of the components manufactured by the transfer machine tool. When this situation occurs, the machine tool requires maintenance.

## 6. Closed-Form Determination of a Friction Threshold to Ensure Self-Centering

_{a}, and on the actual overall weight to be moved, W. When friction at the interface is lower than this threshold, the centering force is able to warrant full alignment. Conversely, when friction is above this level, due to wear or rust, the axial force is no longer sufficient to allow centering.

_{c}, is greater than the frictional force arising from the weight to be moved, R

_{μ}. This condition is highlighted in Equation (13), where μ indicates the frictional coefficient and ρ the corresponding frictional angle:

_{th}:

_{C}= 0.642 (z = 360). It can be observed that μ

_{th.}exhibits a non-linear increasing trend. The higher the axial force that can be transmitted by the third ring, the higher the frictional threshold, which even tends to unity. Remarkably high values of μ

_{th.}clearly indicate that the frictional force component can be always easily overcome by the centering force, regardless of the presence of rust or wear. In addition, the higher the weight to be moved, the lower the retrieved threshold. This outcome indicates that, when large masses have to be centered, remarkably high axial forces are required to overcome friction, even for a low frictional coefficient at the interface. Finally, it is interesting to observe that the results in the diagram below are consistent with those of the case study above. In fact, for an axial force of around 31 kN and a weight of 40 kN, the friction threshold is 0.395. This result confirms the unverified condition for centering, as μ increases up to 0.4, as shown in Table 6.

## 7. Conclusions

## Author Contributions

## Conflicts of Interest

## Nomenclature

A_{z} | Effective tooth flank area (mm^{2}) |

a’, a”, h_{G} | Geometrical features of the Hirth ring and related teeth (from Figure 3) |

c | Center misalignment between the fixed and the rotating rings (mm) |

c_{max} | Maximum center misalignment between the fixed and the rotating rings (mm) |

D | Outer diameter of the teeth (mm) |

d | Inner diameter of the teeth (mm) |

d_{L} | Fixing hole diameter (mm) |

Factor c | Factor depending on the number of teeth z (from Table 2) |

F_{a} | Axial force generated by the Hirth coupling (N, kN) |

F_{c} | Self-centering force (N, kN) |

F_{r,e} | Additional external force in the radial direction (experimental validation test) (N) |

FS | Safety factor for the total tangential force (-) |

F_{u} | Total tangential force (N, kN) |

F_{u,μ} | Actual tangential force (including friction) (N, kN) |

F_{ν-a} | Preload (N, kN), considering a safety factor ν |

h | Tooth height (mm) |

K | Reduction factor for the transmitted load (-) |

K_{C} | Coefficient for the self-centering force computation, depending on the number of teeth |

K_{p} | Percentage reduction factor for the tooth pressure (%) |

n_{b} | Number of bolts in the teeth surface |

p_{max} | Maximum pressure on tooth flank (MPa) |

R | Ring diameter (mm) |

R_{m} | Mean radius of the ring (mm) |

R_{μ} | Frictional resistance force (N, kN) |

r | Tooth base radius (mm) |

s | Crown clearance (mm) |

T | External torque to be transmitted (Nm, Nmm) |

T_{µ} | Frictional torque (Nm, Nmm) |

W | Overall weight to be moved upon self-centering (N, kN) |

z | Number of teeth (-) |

γ | Angular misalignment between the fixed and the rotating rings (°) |

γ_{max} | Maximum angular misalignment between the fixed and the rotating rings (°) |

η_{z} | Load bearing percentage (-) |

μ | Coefficient of friction (-) |

μ_{th} | Coefficient of friction upper threshold to achieve self-centering (-) |

ν | Safety factor for preload (-) |

θ | General angular coordinate over the Hirth ring (°) |

ρ | Angle of friction (°, rad) |

σ_{b} | Tooth bending normal stress (MPa) |

σ_{b_ref} | Allowable normal stress (MPa) |

τ | Shear stress due to the transmitted torque (MPa) |

τ_{_ref} | Allowable shear stress (MPa) |

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**Figure 2.**Example of Hirth connection in automotive: (

**a**) old design with radial teeth and (

**b**,

**c**) new design with frontal teeth.

**Figure 6.**Reduction factor for the tooth pressure K

_{p}as a function of the coefficient of friction µ.

**Figure 7.**(

**a**) Example of rotary table for the transfer machine tool; (

**b**) position of ring #2 before engaging phase: left—the angle has to be adjusted; right—the center has to be adjusted.

**Figure 8.**(

**a**) Ring #1 and ring #2 engaged correctly; (

**b**) before engagement: pure angular misalignment; (

**c**) before engagement: center misalignment (which also generates angular misalignment).

**Figure 9.**(

**a**) Sketch for the evaluation of the misalignment angle γ; (

**b**) practical example via CAD system and (

**c**) related trend of γ as a function of θ.

Type of Stress | Carbon Steel | Alloyed Steel Cr-Ni and Cr-Mo | ||
---|---|---|---|---|

σ_{b_ref} (MPa) | τ_{_ref} (MPa) | σ_{b_ref} (MPa) | τ_{_ref} (MPa) | |

No Shocks | 90 | 33.5 | 120 | 44.5 |

With shocks | 50 | 18.5 | 70 | 26 |

With shocks and with torsional vibration | 35 | 13 | 50 | 18.5 |

**Table 2.**Tables provided by the manufacturer “Voith-Turbo Gmbh” for the proper choice of Factor c and of the crown clearance s.

Number of Teeth z | Factor c |

12 | 0.234 |

24 | 0.114 |

36 | 0.075 |

48 | 0.056 |

60 | 0.045 |

72 | 0.037 |

96 | 0.028 |

120 | 0.022 |

144 | 0.018 |

180 | 0.015 |

240 | 0.011 |

288 | 0.009 |

360 | 0.007 |

720 | 0.003 |

Tooth Root Radius r | Crown Clearance s |

0.3 | 0.4 |

0.6 | 0.6 |

1.0 | 1.0 |

1.6 | 1.6 |

2.5 | 2.5 |

Control | Theoretical Measures | Tolerance | Results | |
---|---|---|---|---|

∅ ext. | C1 | |||

C2 | 990 mm | −0.01 to −0.03 mm | 989.971 mm | |

C3 | ||||

∅ ext. Circularity | C1 | |||

C2 | 0.005 mm | |||

C3 | ||||

∅ int. | C1 | 849 mm H7 | +0.090 mm | 849.018 mm |

C2 | ||||

C3 | 829 mm | +0.030 mm | 829.017 mm | |

∅ int. Circularity | C1 | 0.004 mm | ||

C2 | ||||

C3 | 0.004 mm | |||

Concentricity ∅ int.- ∅ ext. | C1 | 0.003 mm | ||

C2 | 0.004 mm | |||

C3 | 0.003 mm | |||

Flatness | C1 | 0.003 mm | ||

C2 | 0.002 mm | |||

C3 | 0.002 mm | |||

Parallelism C1-C2 Rotating C2 | 0° | 0.007 mm | ||

90° | 0.009 mm | |||

180° | 0.008 mm | |||

270° | 0.009 mm | |||

Parallelism C1-C3 Rotating C3 | 0° | 0.008 mm | ||

90° | 0.009 mm | |||

180° | 0.006 mm | |||

270° | 0.009 mm | |||

Concentricity C1-C2 Rotating C2 | 0° | 0.008 mm | ||

90° | 0.009 mm | |||

180° | 0.006 mm | |||

270° | 0.009 mm | |||

Concentricity C1-C3 Rotating C3 | 0° | 0.007 mm | ||

90° | 0.009 mm | |||

180° | 0.008 mm | |||

270° | 0.009 mm | |||

Indexing accuracy | 1.50’’ | |||

Hardness | HRC 54 | ±2 (HRC) | HRC 54 |

Additional External Radial Load F_{r,e} (N) | Self-Centering Capability (YES/NO) |
---|---|

20 | YES |

40 | YES |

60 | YES |

80 | YES |

100 | YES |

120 | NO |

**Table 5.**Calculation data for a rotary table of a transfer machine (Figure 7a).

ID | Description | Type of Data | Value | Units |
---|---|---|---|---|

1 | Tooth Angle | Input data (fixed for Hirth connection) | 30 | ° |

2 | 0.524 | rad | ||

3 | Friction angle | Input data | 11.5 | ° |

4 | 0.201 | rad | ||

5 | Coefficient of friction µ | Calculated: µ = tan(ρ) | 0.203 | |

6 | Maximum external tangential force to be transmitted during a milling operation | Input data | 5000 | N |

7 | Distance from the center of the tangential force (radius of the rotary table) | Input data | 700 | mm |

8 | Minimum Torque T to be transmitted | Calculated: ID6 × ID7 | 3500 | Nm |

9 | Hirth rings mean radius | Input data | 450 | mm |

10 | Tangential force F_{u} on the Hirth connection | Calculated: ID8 × 1000/ID9 | 7778 | N |

11 | Safety factor SF | Input data (reference value: 3 … 5) | 4 | |

12 | Tangential force F_{u} on the Hirth connection including SF | Calculated: ID10 × ID11 | 31,111 | N |

13 | Tangential force F_{u} used for the design | Selected by the designer on the basis of ID12 | 35,000 | N |

14 | Axial force F_{a} (including friction) | Calculated (info in the paper) | 30,965 | N |

15 | Self-centering capability F_{c} | Calculated (info in the paper) | 22,326 | N |

16 | Weight of the rotary table connected to the rotary ring | Input data | 40,000 | N |

17 | Frictional force R_{µ} | Calculated: ID16 × ID5 | 8138 | N |

18 | Mean sliding radius of the bearing surface | Input data | 470 | mm |

19 | Frictional torque on the bearing surface | Calculated: ID17 × ID18 | 3825 | Nm |

20 | Additional frictional torque (sealings, bearings, …) | Input data | 3000 | Nm |

21 | Torque T that Hirth rings can transmit | Calculated: ID13 × ID9 | 15,750 | Nm |

22 | Frictional torque T_{µ} | Calculated: ID19 + ID20 | 6825 | Nm |

Active Actions Provided by the Hirth | Resistance Actions Due to Friction | Check | |||||||
---|---|---|---|---|---|---|---|---|---|

Coefficient of Friciton μ | Angle of Friction ρ (°) | Axial Force F_{a} from Hydraulic Piston (N) | Tangential Force F_{u} (N) | Self-Centering Force F_{c} (N) | Torque Available for Correct Angular Alignment (Nm) | Frictional Force R_{μ} (N) | Frictional Torque T_{μ} (Nm) | F_{c} > R_{μ} | T > T_{μ} |

0.200 | 11.3 | 30,965 | 35,234 | 22,431 | 15,855 | 8000 | 6760 | YES | YES |

0.300 | 16.7 | 29,181 | 18,577 | 13,131 | 12,000 | 8640 | YES | YES | |

0.400 | 21.8 | 24,366 | 15,512 | 10,965 | 16,000 | 10,520 | NO | YES | |

0.500 | 26.6 | 20,445 | 13,016 | 9200 | 20,000 | 12,400 | NO | NO | |

0.600 | 31.0 | 17,190 | 10,944 | 7735 | 24,000 | 14,280 | NO | NO |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Croccolo, D.; De Agostinis, M.; Fini, S.; Olmi, G.; Robusto, F.; Vincenzi, N. On Hirth Ring Couplings: Design Principles Including the Effect of Friction. *Actuators* **2018**, *7*, 79.
https://doi.org/10.3390/act7040079

**AMA Style**

Croccolo D, De Agostinis M, Fini S, Olmi G, Robusto F, Vincenzi N. On Hirth Ring Couplings: Design Principles Including the Effect of Friction. *Actuators*. 2018; 7(4):79.
https://doi.org/10.3390/act7040079

**Chicago/Turabian Style**

Croccolo, Dario, Massimiliano De Agostinis, Stefano Fini, Giorgio Olmi, Francesco Robusto, and Nicolò Vincenzi. 2018. "On Hirth Ring Couplings: Design Principles Including the Effect of Friction" *Actuators* 7, no. 4: 79.
https://doi.org/10.3390/act7040079