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

Abstract

Rings with Hirth couplings are primarily used for the accurate positioning of axial-symmetric components in the machine tool industry and, generally, in mechanical components. It is also possible to use Hirth rings as connection tools. Specific industries with special milling and grinding machines are able to manufacture both tailor made and standard Hirth rings available on stock. Unfortunately, no international standard (for instance ISO, DIN or AGMA) is available for the production and the design of such components. In the best-case scenario, it is possible to find simplified design formulae in the catalogue of the suppliers. The aim of this work is to provide some accurate formulae and computational methods for design to provide better awareness on the limitations and the potential of this type of connection. The work consists of five parts: (i) a review of the base calculation derived mainly from the catalogues of manufacturers; (ii) an improved calculation based on a new analytical method including the friction phenomenon; (iii) an experimentation run for validating the method; (iv) a case study applied to a machine tool; and, (v) a closed form formulation to determine an upper threshold for friction, thus ensuring the Hirth coupling regular performance.

1. Introduction

Rings with Hirth couplings (Figure 1) are based on the concept of couplings achieved by frontal teeth connections.

The concept was invented and patented by a German engineer, Carl Albert Hirth in 1928 [1]. These couplings allow obtaining very high levels of:

○

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.

For these reasons, Hirth connections are successfully used in many mechanical applications: machine tools, turning tables, transfer units, turbo-chargers, robotics, ship building, and, recently, also in the automotive industry. With regard to the automotive industry, it is possible to highlight the example in Figure 2, dealing with the replacement of the standard splined shaft connection by a frontal teeth connection in the wheel hub. In conventional systems, the drive torque is transferred between the wheel bearing and the axle journal by means of radial splines, which are affected by a certain amount of clearance. Loads that occur in day-to-day driving conditions produce a risk of looseness beyond the normal mounting clearance, resulting in comfort reduction and unpleasant noises and, in some cases, the failure of the bearing. On the contrary, the self-centering axial gear teeth are placed on the axle journal and fixed by the central screw. The connection remains completely clearance-free in the gear teeth during the entire service life. Proceeding this way, it is possible to achieve up to 10% weight reduction per wheel (resulting in fuel consumption reduction and higher dynamic performance of the vehicle), a simplified assembly operation, a clearance free coupling and, above all, up to 150% of the torque transmission capability.

Hirth couplings, also regarded as curvic couplings, are widely used for torque transmission between mating discs in turbine machinery and are not covered by international standards (for instance ISO, DIN or AGMA). Their application makes it possible to fulfil an accurate centering, as well as an improved reliability and good structural stability. For instance, Reference [2] is focused on the structural and contact analysis of a curvic coupling in a gas turbine. In this study, a numerical model was developed to investigate the stress distribution and the contact evolution at different stages of the turbine service, in particular during preload, warm-up, speed-up, and running. The effects of coupling bolt preload, as well as of torsional load and of environmental temperature were also investigated. Other studies [3,4,5] have dealt with tooth contact aspects and with applications to turbo-engines in the aerospace industry [6,7].

A further interesting application of Hirth couplings, operating as servo-actuators, is in rotary tables for automatic machines and transfer machine tools [8]. These tables, fixed at predetermined angular positions, must be able to be fully constrained, thus resisting against high tangential forces. At the same time, they must have the capability of transmitting high torque, relying on tooth contact, rather than on friction couplings. Moreover, the Hirth coupling must ensure the accomplishment of strict specifications regarding the precise positioning of the rotating table, from the points of view of its angular position and especially its centering with respect to the machine axis, after rotation. The importance of correct alignment is also highlighted in [9,10], dealing with a similar issue.

Most papers in the field of transfer machine tools or of multiple-axis automatic machines are focused on modal or dynamic analyses, such as [11,12]. This research trend is also confirmed by a recent study [13] that deals with the development and validation of a multibody dynamic model of a tool-spindle-bearing system, to be incorporated into a five-axis machine with rotary-tilting spindle heads. Conversely, the topic of the stress peak at the tooth root has not been investigated so far. Moreover, no experimental validation has ever been conducted and published in the scientific literature on numerically computed stress peaks in Hirth couplings. A further unexplored point is the relationship between the stress peak and the actual response of the Hirth connection that may be affected by several factors, such as the force acting on each tooth, the contact area, or the friction affecting the device.

The subject of the present paper is to provide an original analytical model for the Hirth coupling design with regard to the peak stress estimation and to the presence of friction. As remarked above, issues of novelty arise from the lack of similar models in the literature, focused on the actual estimation of the stress maximum value for structural assessment. A further issue of novelty arises from the inclusion of the effect of friction (between the mating teeth or on the bearing surface) in the algorithm. In fact, this is usually disregarded in the standard procedure, but its effect deserves to be considered, as it may be even detrimental for safety and accuracy. The importance of taking friction into account is also emphasized in [2], where the developed curvic coupling finite element model accounts for this item. The present study, as a further evolution, proposes an experimentally validated closed form analytical approach that can be more efficiently applied for industrial design purposes and to properly arrange scheduled maintenance.

2. Standard Calculation

Matzke [14] provided some technical data able to give a rough overview of the calculation of Hirth rings. He mainly focused his investigation on the “nominal stress” in the tooth. Based on the nomenclature of Figure 3, the total tangential force F_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:

$${\mathrm{F}}_{\mathrm{u}}=\frac{\mathrm{T}}{{\mathrm{R}}_{\mathrm{m}}}$$

(1)

Neglecting the effect of friction, for a tooth angle of 60° (in Hirth connection this angle is fixed to this value) the axial force generated by the coupling is given by Equation (2):

$${\mathrm{F}}_{\mathrm{a}}={\mathrm{F}}_{\mathrm{u}}·\tan \left(\frac{\mathsf{\pi }}{6}\right)$$

(2)

The axial force can be produced, for example, by means of one or more bolts that must be accurately dimensioned [15,16,17,18,19]. The tooth is calculated against bending stress σ_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:

$${\mathsf{\sigma }}_{\mathrm{b}}=\frac{6·\frac{{\mathrm{F}}_{\mathrm{u}}}{\mathrm{z}}·{\mathrm{h}}_{\mathrm{G}}}{\mathrm{L}·{\left(\frac{{\mathrm{a}}^{\prime }+{\mathrm{a}}^{″}}{2}\right)}^2}\le {\mathsf{\sigma }}_{\mathrm{b}\_\mathrm{ref}}$$

(3)

The reference values for the bending stress σ_{b_ref} (MPa) are reported in

Table 1. Allowable stresses (σ_{b_ref} and τ_{_ref}) for carbon steel and for alloyed steels for base radius r > 0.3 mm [14,20].

Type of Stress	Carbon Steel		Alloyed Steel Cr-Ni and Cr-Mo
	σb_ref (MPa)	τ_ref (MPa)	σb_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

, based on the assumption of a minimum root radius (r > 0.3 mm [14]).

Technical German literature [20] provides a further formula in Equation (4) that is able to relate the torque T (Nmm) to the shear stress τ (MPa) to be compared to the allowable values τ_ref (MPa) in Table 1.

$$\mathsf{\tau }=\frac{16·\mathrm{T}}{\mathsf{\pi }·{\mathrm{D}}^3\left[1-{\left(\frac{\mathrm{d}}{\mathrm{D}}\right)}^4\right]}\le {\mathsf{\tau }}_{\mathrm{ref}}$$

(4)

The maximum (peak) bending stress at the tooth base depends on the stress distribution along the tooth and on the root radius r (mm) at the base. The values reported in [14,20] are able to provide nominal values only, without considering the local stress concentration. These simplified formulae have, anyway, the advantage of having been fully experimentally validated. In the most recent years, the development and improvements of finite element analysis have made it possible to achieve a more accurate estimation of the actual peak stress in mechanical components under defined hypothesis [21,22]. These methods can also be suitable for frontal teeth couplings [2,5,23]. The limitation in this field is related to the lack of experimental campaigns on real components, in order to relate the stress peak to the actual response of the Hirth connection. This is likely to be affected by several factors: the force acting on each tooth, the contact area, the holes for the installation, the friction between the teeth, the heat treatment process (case hardened teeth, not always with the same hardness), the hardness gradient on the teeth and the mechanical residual stresses due to the grinding operations.

Besides the analysis of Matzke, the technical literature on Hirth rings provides some more data for the designer, which are reported below. The calculation method is based on the assumption that the external torque T (Nmm) generates the tangential force F_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)):

$${\mathrm{F}}_{\mathrm{v}-\mathrm{a}}={\mathrm{F}}_{\mathrm{a}}·\mathsf{\nu }\hspace{1em}\hspace{1em}\hspace{1em}\mathsf{\nu }=1.8\dots 3$$

(5)

When compressed together, the teeth support each other if the preload F_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 (mm²) 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:

$$\{\begin{array}{l}{\mathrm{p}}_{\max }=\frac{{\mathrm{F}}_{\mathrm{v}-\mathrm{a}}+{\mathrm{F}}_{\mathrm{a}}}{{\mathrm{A}}_{\mathrm{z}}}\\ {\mathrm{A}}_{\mathrm{z}}=\left(\mathrm{D}-\mathrm{d}-\frac{{\mathrm{n}}_{\mathrm{b}}·{\mathrm{d}}_{\mathrm{L}}^2}{\mathrm{D}+\mathrm{d}}\right)·\left[\frac{\mathsf{\pi }}{4}\left(\mathrm{D}+\mathrm{d}\right)-1.155·\mathrm{z}·\left(\mathrm{r}+\mathrm{s}\right)\right]·{\mathsf{\eta }}_{\mathrm{z}}\end{array}$$

(6)

Concerning the geometry, the manufacturers offer a fixed parametric rule, in order to calculate the tooth height h (mm) in correspondence of the mean radius R_m (mm) (see Figure 4, Equation (7) and

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

).

$$\mathrm{h}=\mathrm{c}·\mathrm{D}-\left(2·\mathrm{r}+\mathrm{s}\right)$$

(7)

Finally, concerning Hirth ring manufacturing, the most widely used materials are alloyed steels for quenching and tempering with Cr and Mo, namely 42CrMo₄ (W. Nr. 1.7225) or 34Cr₄ (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. Measuring report of Hirth rings engagements.

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

.

The high precision and costs of these components allow an indexing accuracy of +/−2’’ (arcsec), a repeat accuracy <0.001 mm, a self-centering capability, a high wear resistance and a long-term service life.

3. Improved Calculation

3.1. Effect of Friction

The effect of friction between the mating teeth is not taken into account in the standard procedure. This effect is quite easy to be implemented and it is very important since it is detrimental to safety: according to Equation (8), in the presence of friction (angle of friction ρ (°)), with the same axial force, the actual transmission load F_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.

$$\{\begin{array}{l}{\mathrm{F}}_{\mathrm{u},\mathsf{\mu }}=\frac{{\mathrm{F}}_{\mathrm{a}}}{\tan \left(\frac{\mathsf{\pi }}{6}+\mathsf{\rho }\right)}\\ {\mathrm{F}}_{\mathrm{u},\mathsf{\mu }}={\mathrm{F}}_{\mathrm{u}}·\mathrm{K}\hspace{1em}\hspace{1em}\mathrm{K}=\frac{\tan \left(\frac{\mathsf{\pi }}{6}\right)}{\tan \left(\frac{\mathsf{\pi }}{6}+\mathsf{\rho }\right)}\end{array}$$

(8)

In Figure 5 the reduction factor for the transmitted load (K, Equation (8)) is reported as a function of the angle of friction.

The pressure between the teeth is also affected by the coefficient of friction, but with a lower effect. By non-linear finite element analyses (non-linear due to sliding elements in contact), it is possible to calculate the percentage reduction factor for the tooth pressure (K_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

It is well known that Hirth connections have self-centering capability. This feature is particularly important in all the applications in which the rings must be disengaged during their life. A very common example is the turntables of the machine tools. In a multi-station round transfer machine the sequence of two actions takes place: (i) the table is fixed with the Hirth rings engaged, when the machining operations are running (the rigid connection ensures the accomplishment of the strict geometrical tolerances of the machined parts); (ii) after the end of the machining tasks in one station, the table must rotate to the next station: for this purpose, the connections must be disengaged. Following the alignment with the next station, the rotary table must then be engaged and constrained again to let new machining operations start. Repeatability, accuracy and self-centering capability are necessary to produce parts compliant with very strict geometrical tolerances. Figure 7a explains the aforementioned cycle with regard to a triple Hirth connection: ring #1 is fixed in the machine bench and it is the reference for the accurate position (it does not move). Ring #2 is connected to the rotary table, so it moves (rotates) with the table (the table must return always in the same position with respect to the machine bench). Ring #3 is responsible for engaging/disengaging and for accurate position. When ring #3 moves up, the table is disengaged and can be rotated from station to station by a motor and a gearbox. Once the table gets close to the expected position, the motor is stopped in a pre-positioning area (as the allowance in the gears of the gearbox does not allow for a precise positioning), so ring #3 can move down. During its movement from the top to the bottom, since the teeth are conical from exterior to interior (Figure 3), ring #3 initially fits the reference ring #1 (so, ring #3 is aligned with ring #1). Afterwards, it fits the movable ring #2 that is therefore taken in the correct position (so, ring#2 is also aligned with ring #1). For the correct positioning of ring #2, the axial force F_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.

Three possible scenarios before the engaging phase have to be considered (see Figure 7b): (i) the fixed and the rotating ring have the same center and just have a different angular position; (ii) the fixed and the rotating ring have different centers and the same angular position; (iii) a combination of (i) and (ii).

Different configurations and alignments of teeth are highlighted in Figure 8: in case (a) the teeth of the fixed and rotating rings are engaged: they are all aligned and the space between the rings is constant (the upper ring is not represented); in case (b) before the engagement of the upper ring, there is an angular misalignment of the rotating ring. Since the rings have the same center, the space between the two rings is constant. In case (c) before the engagement of the upper ring, the two centers are not aligned (the self-centering force F_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.

The conditions shown in Figure 8a,b (cases a and b) do not deserve any particular analytical evaluation. Conversely, the effect of the center misalignment c (mm) (Figure 8c, case c) requires a detailed analysis. According to Figure 9, the trend of the misalignment angle γ (°) around the 360° (θ between 0° and 360°) span is yielded by Equation (9):

$$\{\begin{array}{l}\mathrm{r}=\sqrt{{\mathrm{R}}^2+{\mathrm{c}}^2-2·\mathrm{R}·\mathrm{c}·\cos \left(\mathsf{\theta }\right)}\\ \mathsf{\gamma }=\arcsin \left[\frac{\mathrm{c}}{\mathrm{r}}·\sin \left(\mathsf{\theta }\right)\right]\end{array}$$

(9)

As an example, for a ring diameter R = 450 mm, a center displacement c = 1 mm, when the angle θ is 29.5°, the angular misalignment γ is 0.063° (~4’); for θ = 89.5°, the angular misalignment γ is 0.127° (~8’). These results have also been checked by CAD system (Figure 9b). The whole trend is reported in Figure 9c for the interval 0–90°, provided that the function is symmetric every 90°.

In order to achieve the correct engagement, the following condition in Equation (10) must be fulfilled (the maximum misalignment γ must be within the angular step of the teeth):

$${\mathsf{\gamma }}_{\max }<\frac{360°}{\mathrm{z}}$$

(10)

Combining Equations (9) and (10), it is possible to work out the upper threshold c_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.

$${\mathrm{c}}_{\max }=\mathrm{r}·\sin \left(\frac{360°}{\mathrm{z}}\right)\approx \mathrm{R}·\sin \left(\frac{360°}{\mathrm{z}}\right)$$

(11)

During the design phase, clearances and gaps in bearings, sealings and gears must accomplish this condition and the sum of all the assembly tolerances must be within c_max.

Unfortunately, the condition reported in Equation (11) is not sufficient to warrant the actual occurrence of self-centering; an additional force analysis and assessment is necessary. As a matter of fact, the self-centering force F_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:

$${\mathrm{F}}_{\mathrm{c}}=4·{\displaystyle \sum _{\mathrm{i}=0}^{\mathrm{z}/4}\frac{{\mathrm{F}}_{\mathrm{a}}}{\mathrm{z}·\tan \left(\frac{\mathsf{\pi }}{6}+\mathsf{\rho }\right)}}·\sin \left(\mathrm{i}·\frac{360°}{\mathrm{z}}\right)={\mathrm{K}}_{\mathrm{C}}·\frac{{\mathrm{F}}_{\mathrm{a}}}{\tan \left(\frac{\mathsf{\pi }}{6}+\mathsf{\rho }\right)}$$

(12)

with K_C equal to 0.642 for z = 360; 0.650 for z = 144; 0.664 for z = 72.

4. Experimental Tests

Experimental tests on a triple Hirth ring connection with a mean diameter of 900 mm and z = 360 have been performed in order to check the proposed formulas (Figure 10).

At first, the static coefficient of friction μ in the bearing surface of the rotating ring (see Figure 7a) has been evaluated, in order to get its actual value. Thanks to a dynamometer, the radial force to be applied to move the rotating ring (merely supported on its bearing surface, without engagement) has been detected. The static coefficient of friction μ has therefore been calculated (five repeated tests) as the ratio between the radial force and its weight. The value of the static coefficient of friction μ is equal to 0.204 +/− 0.012; μ ~0.2 was consequently used for the calculation. Regarding the own weight of the upper ring (310 N) as the axial pre-loading force, F_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. Experimental verification of the self-centering capability.

Additional External Radial Load Fr,e (N)	Self-Centering Capability (YES/NO)
20	YES
40	YES
60	YES
80	YES
100	YES
120	NO

; 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 case study proposed in this section refers to the rotary table of a big transfer machine tool, which has a Hirth ring connection with the same geometrical dimensions of that tested in the previous section. The present work moves from some issues related to the actual positioning of the rotary table (with concern regarding the loss of self-centering capability) after several years of work. After disassembling the rotary table, it was found that a lot of rust was present both in the teeth and in the bearing surface of the rotating ring (Figure 13). The presence of rust, as highlighted in [24,25], significantly increases the coefficient of friction, therefore reducing performance and self-centering capability. In this section, the aforementioned analytical predictive model is applied and the influence of the coefficient of friction on the actual performance is also studied.

The data for the calculation are reported in

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 Fu on the Hirth connection	Calculated: ID8 × 1000/ID9	7778	N
11	Safety factor SF	Input data (reference value: 3 … 5)	4
12	Tangential force Fu on the Hirth connection including SF	Calculated: ID10 × ID11	31,111	N
13	Tangential force Fu used for the design	Selected by the designer on the basis of ID12	35,000	N
14	Axial force Fa (including friction)	Calculated (info in the paper)	30,965	N
15	Self-centering capability Fc	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 Rµ	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 Tµ	Calculated: ID19 + ID20	6825	Nm

.

Analyzing the data in Table 5, based on force balance, it is possible to highlight that the following conditions are fulfilled in the presence of a coefficient of friction μ = ~0.2 (ID5).

• 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).

Concerning the stress analysis, it is easy to run this by Equation (3) or (4), with the teeth geometry reported in Figure 14 and considering the full torque capability T = 15,750 Nm. The equations above yield: σ_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).

Finally, the effect of the coefficient of friction is shown. According to [24,25], the coefficient of friction is strongly affected by the presence of the rust, which makes it increase. For this reason, a sensitivity analysis has been performed. The axial force F_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. Sensitivity analysis for different values of the friction coefficient.

			Active Actions Provided by the Hirth			Resistance Actions Due to Friction		Check
Coefficient of Friciton μ	Angle of Friction ρ (°)	Axial Force Fa from Hydraulic Piston (N)	Tangential Force Fu (N)	Self-Centering Force Fc (N)	Torque Available for Correct Angular Alignment (Nm)	Frictional Force Rμ (N)	Frictional Torque Tμ (Nm)	Fc > Rμ	T > Tμ
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

, 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

The outcome of the previous analysis suggests the existence of an upper threshold for friction. For a fixed Hirth geometry, this depends on the applied axial force, F_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.

Considering Equation (12), the friction threshold can be determined by imposing that the centering force, F_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:

$${\mathrm{F}}_{\mathrm{c}}={\mathrm{K}}_{\mathrm{c}}·\frac{{\mathrm{F}}_{\mathrm{a}}}{\tan \left(\frac{\mathsf{\pi }}{6}+\mathsf{\rho }\right)}\ge {\mathrm{R}}_{\mathsf{\mu }}=\mathsf{\mu }·\mathrm{W}=\mathrm{tg}\left(\mathsf{\rho }\right)·\mathrm{W}$$

(13)

Some simple algebraic operations yield the solution in terms of the upper threshold for the frictional coefficient, μ_th:

$${\mathsf{\mu }}_{\mathrm{th}.}=\frac{-\frac{\sqrt{3}}{3}\left({\mathrm{K}}_{\mathrm{c}}{\mathrm{F}}_{\mathrm{a}}+\mathrm{W}\right)+\sqrt{\frac{1}{3}{\left({\mathrm{K}}_{\mathrm{c}}{\mathrm{F}}_{\mathrm{a}}+\mathrm{W}\right)}^2+{4\mathrm{WK}}_{\mathrm{c}}{\mathrm{F}}_{\mathrm{a}}}}{2\mathrm{W}}$$

(14)

This threshold is plotted in Figure 16, as a function of the applied axial force, for different overall weights to be moved for K_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

This paper presents a methodology for the precise calculation of the forces generated in Hirth ring connections. The role of friction is taken into account and is demonstrated to be fundamental for the correct calculation of this type of component. Simplified formulae in scientific or technical literature are not sufficiently accurate to describe the performance of Hirth rings in full detail. Starting from a practical example that occurred in a working machine tool that lost the accuracy of its positioning capability, some more advanced design formulae have been proposed. These novel equations have also been experimentally verified. Friction conditions are likely to change during the lifespan of a machine tool, mainly due to wear, rust and corrosion. Therefore, the designer needs to use a reliable design method for Hirth connection calculations for long-term life. The developed model, which incorporates the effect of friction, is very important to predict the capability of the connection, and also indicates the related friction thresholds to achieve self-centering and adequate torque transfer.