Analytical method for calculating the nominal tooth root stress in worm gear shafts

According to the state of the art, worm gears are designed due to the softer bronze-material of the worm wheel primarily against damage of the wheel because of wear, pitting or breakage of the teeth. The aim of research in the last decades is to optimize the load capacity on the worm wheel by using higher-strength bronze alloys, cast iron or steel to increase the transmission capacity of worm gears. This development may lead to an increased number of damage cases on the worm shaft. In literature, documented cases of tooth segment fractures of the worm shaft can already be found. Since the worm is only designed against deflection according to the state of the art, there is a need for a method to calculate the material stress in worm shafts. This paper presents an analytical method based on the nominal stress approach for calculating bending, compression, and shear stresses in the tooth root of worm shafts to close this gap. The stresses resulting from different load distributions in the tooth contact due to assembly deviations are calculated with the presented method and compared with results from the Finite Element Method.

one gear stage, high transmissible torques and smooth-running properties.Due to the high sliding ratio and the high temperatures occurring in tooth contact, the worm shaft is usually made of case-hardened steel and the worm wheel of a copper-tin alloy to avoid scuffing damage.The design of worm gears is primarily carried out on the side of the wheel to prevent excessive wear and pitting of the tooth flanks as well as tooth fracture due to the softer material.The worm shaft on the other hand is designed according to the state of the art in DIN 3996 [1] and ISO/TS 14521 [2] only against deflection to prevent displacements of the contact pattern and increase of local loads on the tooth flanks [1,3].Further investigations in [4,5] aim to improve this calculation method.In ANSI/AGMA 6022-C93 [6], an additional design aspect provides the calculation of the bending stress in the worm shaft with a highly simplified approach.
Addressing the damage on worm shafts, only the formation of scores and cracks on the tooth flank surfaces is studied in detail in [7,8].However, fractures of the worm teeth are observed in various research projects [9][10][11][12][13] as side effect during other investigations.Photographs of the damages as documented in the literature are shown in Fig. 1.The fractures are attributed to low material quality in [9], to contamination with water in [11] and to the influence of inductive hardening of the worms' surface layer in [13].In [10,12], the damage is observed when operating with standard bronze and without unusual external influences.
Past and current efforts to enhance the wheel material by using higher strength bronze alloys [12] or to substitute bronze with cast iron [9,14] and steel [15,16] aim to increase the transmissible torque at the wheel.This development may lead to the strength of the worm shaft constraining the performance.To close this gap in the state of the art, the authors devote this paper to the development of a method for calculating the material stress in the tooth root of the worm shaft.

State of the art
For designing helical gears against fracture at the tooth root, the maximum local stress σF is usually compared with the tooth root stress limit σFG which is calculated from the permissible stress σFP by considering stress concentration, higher load capacity for a limited number of load cycles, notch sensitivity, surface roughness and the influence of the tooth dimensions on the tooth bending strength.According to ISO 6336-3 [17], it is assumed that exceeding the permissible bending stress results in damage of the gear.The safety factor SF is calculated according to Eq. 1.The gearbox is assumed to be operated safely with SF being equal to or greater than SFmin.In DIN 3990-1 [18] it is recommended to select the minimum safety factor SFmin depending on the accuracy of the used calculation method and specific for different applications.In ISO 6336-3 [17], the maximum local tooth root stress σF is obtained by multiplying the nominal tooth root stress σF0 with various stress correction factors.The nominal tooth root stress σF0 is calculated according to Eq. 2 by the ratio of the nominal tangential load Ft to the product of the facewidth b and the normal module mn.Various influences on the bending stress such as the gears tooth form factor YF, the stress concentration factor YS, the factor Yβ for the irregular load distribution along the contact lines in helical gears, the rim thickness factor YB and the deep tooth factor YDT for high precision gears are additionally considered.
When designing worm wheels according to DIN 3996 [1], the maximum shear stress τF is compared with the shear stress limit τFG from which a safety factor SF is calculated.This method is also based on a nominal stress approach by obtaining τF from the ratio of the tangential force Ftm2 to the product of the facewidth b2H and the axial modulus mx multiplied by similar stress correction factors.
The FKM guideline [19] offers a comprehensive calculation method for assessing the strength of components with any geometry under both static and dynamic load by performing in depth analysis with the Finite Element Method (FEM).Since this procedure is usually very time-consuming, it can only be applied to a small number of components with reasonable effort.The permissible stress of any component at the most stressed location depends on the material and the respective quality, the manufacturing process, the technological size, the surface roughness, the hardness of the component as well as the experienced load over time.Specific studies on the strength of worm shafts and the influences are hardly available.Reference values for the roughness of the worm shaft are given in DIN 3996 [1] while detailed measurements are provided in [12,[20][21][22].Data of measured surface hardness is found in [20,22,23].At the current state of the art, the strength of worm shaft geometries cannot be determined.

Stress calculation in worm gears
According to ANSI/AGMA 6022-C93 [6], the design of worm shafts against deflection and bending stress is based on the model of a circular bending beam.The maximum bending stress Sb is compared with the permissible stress depending on the material.Stress concentrations in the worm thread due to the tooth rounding as well as the load distribution and the geometric influence on the worm shafts stiffness remains unattended.
The stress concentration in grooved shafts can be calculated using the method described in DIN 743-2 [24].Due to the helical geometry of the worm thread, this method is not suitable for calculating the local bending stress.
To calculate the stress in the tooth root fillet, an approach based on the slice theory was formulated in [25].Bending, compression and shear stresses are calculated in each loaded slice along the contact line.The method does not consider the distribution of stress in the tooth root under point load.
In experimental studies carried out in [26], the bending of the worm and wheel teeth were measured with an optical interferometer and an empirical equation for calculating the material stress in the tooth root based on the measurements was derived.The influence of deviation-related contact patterns on the tooth deformation and tooth root stress was investigated.
To calculate the tooth root stress in both worm and wheel, in [27] an approach based on the finite element method was developed.The calculations were performed with a constant line load to investigate the influence of geometry parameters on the stress and to approximate the distribution in the tooth root with equations.
Investigations of material stress in worm gears were carried out in [28,29] for both the worm and the wheel teeth using FEM.The investigations were application-related and did not aim to develop a calculation method for the stress in the worm.Further investigations in [10,30,31] using FEM were focused exclusively on the material stress in the worm wheels.

Load distribution in worm gears
The load distribution in the tooth contact of worm gears depends on the contact points between the flanks.In ISO/TR 10828 [32], a method for calculating the contact lines in unloaded condition is presented.The basis for the Software SNETRA by the German Research Association for Drive Trains (FVA) for calculating load patterns under load was provided in [33].By simulating the manufacturing process and considering assembly deviations, the load distribution can be predetermined during the design process.The analytical approximation of the stiffness matrix of all contact points allows the time-efficient calculation of a wide variety of different geometric parameters.The availability of the contact load distribution is the basic requirement for calculating the material stresses in the bodies.

Stress calculation in helical spur gears
An extended calculation approach for the tooth root stress in helical spur gears based on DIN 3990-3 [34] was developed in [35] by considering the load distribution on the tooth flanks and the supporting effect of the unloaded width of the tooth.The distribution of the bending stress in the tooth root is calculated using approximate equations developed in [36] based on numerical investigations of straight cantilever plates.

General stress in worm shafts
Worm shafts experience various types of stress during operation.On the one hand, stresses occur in the tooth root due to the load transmission like a helical spur gear.The stress distribution in the tooth root is different due to the curvature of the teeth around the worm shaft axis and the constant supporting effect of the teeth on both sides of the contact line due to the continuous helical geometry.On the other hand, bending stresses occur in the tooth gap of the worm, since it is usually designed with a large distance between the bearings and relatively small diameters.Because of the helical geometry of the tooth gap, the stress cannot be calculated according to DIN 743-2 [24] since there is no similar groove provided.According to [37], the tooth root of external gears experience bending, compression and shear.The stress in the tooth root and in the cross section of the shaft is shown schematically in Fig. 2.
The ability to accurately calculate stress in worm shafts offers the potential to derive a simplified yet validated calculation method.First, the stresses in the tooth root of a worm shaft with standard reference geometry according to DIN 3996 [1] and DIN 3975-1 [38] are calculated at a defined operating point for a deviation-free toothing using FEM.The gearing and load data of the investigated geometry as well as the contact pattern including the lines of contact and the load distribution as calculated with SNETRA [33] are summarized in Table 1.The load distribution on several tooth flanks due to the overlap ε is already considered in the contact pattern calculation.[mm] 160

FEM-Model for calculating the stress in the tooth fillet
To build the FEM model, a toothed segment of the worm shaft is separated and clamped with a fixed support at the cut surfaces.The load is applied along an imprinted contact line on the tooth flank according to the calculated load distribution as seen in Table 1.For this purpose, the line load F as calculated with SNETRA is converted to n point loads for every node along the contact line in the FEM.Assuming that contact forces are always acting normal to the contact surface, the line load F is divided into its components Fx, Fy and Fz by calculating the direction of the normal vector in each point along the contact line.The FEM model is shown in Fig. 3.The model is solved for 10 equally distributed contact lines of the contact pattern as shown in Table 1, starting with the line in point A of the first contact.The evaluation of the results is further carried out in the point of the maximum principal stress σ1.The components of the stress tensor s are evaluated along paths intersecting this point in the circumferential (path β), the tooth root rounding (path βrf1) and the normal direction into the component (path t) as shown in Fig. 3.The maximum principal stress σ1 resulting from passing through the entire contact pattern is shown in Fig. 4. It can be observed that the maximum stress is not caused by either the largest lever arms when contact happens at the tip diameter, or the largest line loads near the root diameter.Fig. 5, 6 and 7 show the evaluated stress components along the paths as defined in Fig. 3.
According to Fig. 5, the normal stresses σy and σz and the shear stress τxy represent the dominant components.The maximum bending stress σy,max occurs in the tooth root radius at βrf1 = 60°as seen in Fig. 6 which corresponds to the assumption of the location at the 30°tangent according to ISO 6336-3 [17].The maximum of all stress components

Analytical calculation model
To calculate the stress in the tooth root of worm shafts, an initial approach based on the calculation of the nominal stresses following the method according to ISO 6336-3 [17], DIN 3990-3 [34] and the extended calculation method in [35] is presented.Further, equations for the calculation of the nominal bending, compression and shear stresses in the tooth root are derived.The calculation model is based on the discretization of the line load F along each contact line.The stress distributions resulting from each individual point load Fi are summarized to calculate the total stress distribution in the worm tooth root.The method is shown in Fig. 8.
The dismantling of the point load Fi into its components Fx,i, Fy,i and Fz,i at any point PF,i = (xi | yi | zi) along the contact line is performed by obtaining the direction of the normal vector nP,i at PF,i on the surface of the flank.The force vector components Fx,i and Fy,i need to be rotated in every point PF,i by the position angle βF,i so that Fx,i 0 points in tangential and Fy.I 0 in radial direction.The force components are rotated according to Eqs. 3 and 4.
F y;i 0 = F y;i cos ˇF;i + F x;i sin ˇF;i The force component Fz always points in axial direction and does not need to be converted.Figure 9 shows the rotation of the force components of a single point load Fi K Fig. 8 Calculation model for discrete load application and stress distribution in point PF,i.The radius ri of the force application point corresponds to the distance of the point PF,i to the worm axis.
Equations are formulated for the types of loads in the tooth root of the meshing flank of an external gear according to [37].The calculation of the nominal stress is based on a worm shaft with ZA flank profile and geometry according to DIN 3975-1 ( [38], see Fig. 10).
Following DIN 3990-3 [34], the nominal bending stress σb,nom is related to the tooth thickness sfγ at the root circle diameter df1 in the effective direction of the forces Fx 0 and Fy 0 at the angle γF.
The tooth thickness sfx in the axial section is determined approximately according to Eq. 5.The tooth thickness sfγ is calculated for every force application point by Eq. 6.
To calculate the bending Moment Mb, the effective lever arm hFe is adopted from DIN 3990-3 [34] and modified for the worm geometry.It represents the distance between df1 and the point of intersection of the centre of the tooth with the force line resulting from Fy 0 and Fz in the axial section.The effective lever arm is calculated according to Eq. 7.

K
The nominal bending stress σb,nom resulting from the tangential and axial force components Fx 0 and Fz is calculated according to Eq. 8. b The nominal compressive stress σd,nom is calculated using the radial force Fy 0 according to Eq. 9 and is related to the tooth thickness sfx at the root diameter in the axial section.
The nominal shear stress τs,nom is calculated from the forces in tangential and axial directions Fx 0 and Fz in relation to the tooth root thickness sfγ at the root circle diameter df1 in the direction of the force according to Eq. 10.

Numerical calculation of the distribution of force and moment on a curved plate
To calculate the nominal stress, the distribution of the bending moment as well as the reaction forces at the root of the tooth need to be considered.For this purpose, numerical simulations using FEM are carried out on a plate model according to [36] but with modified geometry.The presented plate geometry corresponds to the standard reference geometry of a worm shaft according to DIN 3996 [1] and DIN 3975-1 [38].The plate is located at the centre of the worm tooth segment and is loaded with a point load F = 1 N at the point PF = (xF | yF | zF).The lever arm h is the distance of the point PF to the clamping at the root circle diameter.The model is shown in Fig. 11.The response variables Mx, Fy and Fz are distributed at the clamping under load.The integral of the distribution of the reaction moment is equal to the product of force F and lever arm h.The curvature of the plate compared to the model of a straight spur gear tooth reduces the reaction moment in the tooth root due to the supporting effect of the non-loaded areas of the worm.The support factor of the curved geometry is considered via the function KS.The reaction moment is calculated according to Eq. 11.
To determine the supporting effect KS, the root circle diameter df1 of the plate is varied up until the curvature is equal to a uncurved plate for Fz = 1 N and h = 0.9.The values are calculated from the difference of the nominal moment and the integral of the reaction moment of the numerical solution.
The increase of the supporting effect with decreasing diameter and thus the reduction of the bending moment is shown in Fig. 12.The function KS is approximated by Eq. 12.
Figure 13 shows the model of a straight tooth with a contact line that does not extend over the entire width of the flank.According to [35], the length and position of the contact line change the distribution of the bending moment.When analysing the worm geometry, no case distinction is necessary since there are always identical supporting widths to both sides of the contact line for each possible position on the worm flank as well as no free-standing tooth ends.

Approximate function for the distribution of force and moment
The force moment distributions on the curved plate are approximated with the equation of the standard normal distribution.The distribution function D is defined in Eq. 13.
In function D, the parameter p1 resembles the maximum value, p2 the opening width and βF the angular position of the force application point according to Fig. 9.The distributions resulting from the numerical plate model are evaluated as an example for the bending moment DMx (see Fig. 14) and the shear force DFz (see Fig. 15) under load with F = 1 N and different lever arms h.The lever arm h is normalized with df1 corresponding to 0 and da1 to 1.
The change of the parameters p1 and p2 of the distribution function D depending on the lever arm h are approximated Fig. 14 Distribution of the moment reaction Mx in the tooth root calculated with the plate model with 2nd degree polynomials with the parameters k1, k2 and k3 according to Eq. 14.
The variation of the parameters p1 and p2 depending on the lever arm h is shown in Fig. 16 for the moment distribution DMx and in Fig. 17 for the axial reaction force distribution DFz.

Calculation of the stress distribution
The individual stress distribution σb,nom,D,i in the tooth root resulting from each individual load Fi along a contact line of the contact pattern is calculated according to Eq. 15. using Eqs.8, 12 and 13.The stress distributions are calculated for the gear geometry and load as shown in Table 1.The calculation is performed for a deviation-free assembly, a displaced worm in axial direction by a = -0.2mm and a displacement in width direction by b = -0.2mm as defined in [33].The load patterns calculated with SNETRA are shown in Fig. 18.

b;nom;D
Applying the presented calculation method, the results of the nominal stress components σb,nom, σd,nom and τs,nom for each contact pattern are shown in Fig. 19.Since the stress concentration in the tooth root rounding is not considered, all stress components are normalized for comparison to the respective maximum value of the deviation-free assembly.The normalisation is carried out exemplarily for the bending stress σb,norm according to Eq. 17.Although the stress concentration effect in the worm tooth root rounding is not included in the calculation, the amount of increase of the stress level caused by assembly deviations is expected to be accurate.It is observed that the contact patterns of the deviating assemblies do impact the maximum values of the stress components with the shear stress being the most affected with an increase of up to 72%.

Validation and discussion
As shown in Fig. 20, the analytical calculated bending stress distribution for σb,nom is corresponding with the FEM-results for the stress component σy when compared qualitatively, although the contours of the distributions differ slightly.
Since the stress concentration is not considered, all stress values are normalized to the respective maximum value.The reason for the divergence is due to the stress component σy from the FEM calculation being similar to the bending stress due to its direction on the worm tooth root but does not correspond exactly to the same value.It is not possible to determine the amount of bending stress in the 3D FEM analysis because of the complex triaxial stress state.

Summary and conclusion
The presented calculation method allows design engineers do determine the expected nominal stress in the tooth root of worm shafts based on the load distribution in the tooth contact.The results show qualitatively good accuracy compared to the results calculated with FEM while the calculation time becomes faster and thus more efficient.This enables a fast and easy comparison of the stress overload between different worm geometries for different operating points.However, at the current stage of development, no statement about the actual stress values is possible since the stress concentration effect of the tooth root rounding is not In addition, the lack of experimental of permissible stresses of worm shaft geometries does not allow calculate the of strength.
The subject of future investigations can therefore be the stress occurring in worm tooth root by either specific calculation using FEM or tation of existing values in accordance ISO 6336 and DIN [34].To provide a strength verification, location experiencing the highest stress value required.
the worm is rotating and locations along the shaft stationary with respect to the axis position, it is not possible to directly identify a main stressed location from the calculation results.It is conceivable to derive periodic stress curves of the locations the worm shaft.Like the stress concentration effect in the root of the shaft, the concentration of helicoidal tooth space experiencing bending stress remains unknown.The adaptation of the method according DIN 743-2 [24] in combination with the determination of new stress concentration factors for helical grooves offers potential.Finally, experimental investigations of the permissible stresses of different worm shaft geometries, both under load in the tooth contact area as well under bending stress, are necessary to calculate the strength of worm shafts.

Nomenclature
The nomenclature is shown in Table 2. Poisson ratio of worm and wheel [-]

Fig. 2
Fig. 2 Schematic stress in the tooth root and in the transverse section of a worm shaft

Fig. 3 Fig. 4
Fig. 3 FEM-Model for calculating the stress distribution in the tooth root under line load

Fig. 7
Fig. 7 Stress components calculated with FEM along path t

Fig. 10 Fig. 9
Fig. 10 Geometric parameters of the worm shaft tooth with ZA flank profile

Fig. 13
Fig. 13 Supporting and nonsupporting tooth width on helical gear and worm shaft

Fig. 15 Fig. 16 Fig. 17 16 )Fig. 18
Fig.15 Distribution of the force reaction Fz in the tooth root calculated with the plate model

Fig. 19 Fig. 20
Fig. 19 Calculated nominal bending stress σb,nom, nominal compressive stress σd,nom and nominal shear stress τs,nom for the contact patterns calculated with SNETRA: a) deviation-free assembly, b) axial displaced worm and c) width displaced worm.All values of the stress components are normalized to the corresponding max.value of the deviation-free assembly

Table 1
Gear geometry, load data, contact lines and load distribution on the tooth flanks as calculated with SNETRA