Worm gear drives are characterized by a simple design, which allows the realization of a high gear ratio within one stage. Furthermore, they are characterized by low vibration and noise behavior. For these reasons, they are used both as power transmissions and servo drives in various drive solutions. The allowable load and the lifetime of the gearbox is usually limited by wear on the softer worm wheel. The stiffness and the associated worm shaft deflection is considered as an influence factor on the wear as well as the noise, vibration, harshness (NVH) behavior of worm gear drives.
According to the current state of the art, the worm shaft deflection can be calculated according to AGMA 6022, DIN 3996 and ISO/TR 14521.
In this paper, the current calculation status for worm shaft deflection is discussed. The underlying experimental results for the calculation of the worm shaft deflection according to DIN 3996 and ISO/TR 14521 are analyzed. A new approach for the worm shaft deflection calculation is developed. Therefore, an analytical model for the bending stiffness of a worm shaft is developed. The model was validated through various FEM simulations. As a result, a new calculation method for the equivalent bending diameter of a worm as well as the formulas for the calculation of the worm shaft deflection are presented.
The developed calculation method details the current state of the art, thus providing a basis for more optimized worm gear design. Furthermore, with this calculation, it is now possible to calculate the bending stiffness of overhung worm shafts as well as worms of reduced tooth thickness which are usually used in crossed helical gear boxes.
The new calculation method is presented within this paper and compared to the current state of the art for calculating the worm shaft deflection according to AGMA 6022, DIN 3996 and ISO/TR 14521.
Worm gear drives are an established part of the used transmission concepts in the drive train sector. They are used as power transmissions, e.g. in escalators, elevators, and conveyor belts as well as servo-drives in precision applications. In these applications, the advantages of a high transmission ratio in small installation space, low-noise operation, and the possibility of self-breaking and self-locking outweigh the disadvantages of reduced efficiency.
Typically, a worm gear drive consists of a case-hardened worm paired with a bronze worm wheel, as shown in Figure 1. With worm gears, as well as with other types of gears, there is a continuous increase of power density through new materials, coatings, lubricants, or higher manufacturing qualities.
This increase results in higher tooth forces. Since the shape of the worm is slim compared to the worm wheel, the higher tooth forces lead to a higher deflection of the worm shaft, which can have negative consequences for the wear load capacity, the efficiency, as well as the NVH behavior.
3 State of the art
The continuous increase of the required power density in worm gears over the last decades has primarily been realized through advancements in bronze materials, lubricants, and manufacturing qualities.
Currently-used material-lubricant systems show only limited wear ,  and best efficiencies , , . The use of cast iron wheels   or steel wheels ,  is an existing part of current research. The higher strength of iron compared to bronze leads to further increase in the permissible load and thus to higher tooth forces within the worm toothing.
The tooth forces cause the worm shaft to bend. A high worm shaft deflection can cause interference. This interference can result in transmission deviations, higher noise levels, increased wear, and pitting damage. For this reason, the calculation of the worm shaft deflection is part of the design process of worm-gear drives. The calculation according to AGMA 6022 , ISO/TS 14521 , and DIN 3996  is used for this purpose. It should be mentioned that the calculation according to ISO/TS 14521 is mostly identical to DIN 3996. All three calculations are based on the model of a bending beam, which is loaded under the assumptions according to Euler-Bernoulli. Simplifications are made here, which sometimes lead to large deviations between calculated and observed values. Equation 1 shows the maximum worm shaft deflection calculated according to AGMA 6022 .
According to DIN 3996 or ISO/TS 14521 the maximum worm shaft deflection can be calculated according to Equation 3.
The formula considers the friction in the tooth contact. According to Niemann and Winter , the radial load can be calculated with the following formula:
Using Equation 4, Equation 3 can be simplified and both Equations 1 and 3 can be brought to the following form:
For AGMA 6022, the moment of inertia is defined according to Equation 6.
For ISO/TS 14521 the moment of inertia is defined according to Equation 7.
This corresponds to a symmetrically mounted cylinder, which is loaded in the middle with the radial force FR. The difference between the calculation according to AGMA 6022 and ISO/TS 14521 is the used moment of inertia I. According to AGMA 6022, the root diameter of the worm df 1 is used as an alternative diameter for the bending beam. The calculation according to ISO/TS 14521 uses 1,1 · df 1 .
The alternative diameter is based on test results from Lutz . He carried out tests with worm shafts at the Gear Research Centre (FZG) determining the deflection caused by radial forces. The aim of the tests was to experimentally determine the supporting effect of the worm gear toothing on the worm shaft.
In the test, the worm shafts were symmetrically and flexibly mounted on two blocks. The test setup is shown in Figure 2. The radial force was applied to the worm shaft at defined levels using a pulsator. Then the worm shaft deflection was measured. Based on the test results, the correction factor of 1,1 · df 1 was developed.
At the same time, Langenbeck  investigated the deflection of worm shafts using FEM. His conclusion was that the axial force of the toothing can be ignored with regard to the deflection. Langenbeck also showed that using the root diameter as equivalent bending diameter of worm shafts with a small diameter quotient q, the deviation between FEM and the analytical solution increases.
The results of Lutz  and Langenbeck  suggest the toothing has an influence on the worm shaft deflection, which is not adequately considered in the present state of the art calculations.
The currently used calculation approaches for worm shaft deflection and bending stiffness of worm shafts are simple and efficient calculation approaches. Because of the simplification, they are limited in resolution and accuracy. It has to be considered that the Euler-Bernoulli approach is generally valid for slim bars. In literature, the length-to-width ratio of L/W > 10 can be found as a limit for the applicability of the model. However, this length-to-width ratio is generally not reached for worm shafts.
Furthermore, the standardized calculation approaches are only valid for worm shafts with toothing between the bearing points. A calculation of overhung worm gear toothings, which are commonly used within crossed helical gear sets, is not possible. In addition, there is no consideration of the shaft design next to the toothing area. When using the root diameter as an equivalent bending diameter, the supporting effect of the worm toothing is ignored. Therefore, this calculation represents the extreme case on the safe side.
The replacement diameter correction according to Lutz , which is included in the calculation method of ISO/TS 14521 , considers this supporting effect. However, the difference in calculation from reality grows with increasing geometrical difference between the calculated worm shaft and the tested gears by Lutz .
In addition to the calculation of the worm shaft deflection, calculation systems such as RIKOR , LDP , ROMAX  and others allow the calculation of the stiffness of complete gearboxes with different transmission stages. For the calculation of this overall transmission stiffness, the stiffness of individual machine elements, such as the bending stiffness of shafts, are necessary for the calculation of the stiffness of the whole system. The current calculations of the bending stiffness of worm shafts do not meet this requirement.
4 Approach of the investigation
The investigation is carried out in a four-stage process, shown in Figure 3. In a first step, the suitability of the finite element method for calculating the worm shaft deflection is determined. Within this investigation, the necessary simulation parameters are defined. Therefore, the bending lines of the experimentally tested worm shafts by Lutz  are compared to the simulation results. After the results confirmed the suitability of FEM calculation and the simulation parameters are determined, a parameter study is carried out using 15 worm gear toothings without shaft geometry.
Within the study, parameters are specifically varied in order to test the influence of individual toothing factors on the bending stiffness of the worm gear toothing. The developed analytical model for calculating the bending stiffness of worm gear toothings is compared with the calculation of a CAD software and the simulation results of the FEM calculations based on the moment of inertia, the bending line of the worm toothings and the equivalent bending diameter.
Through the detailed parameter variation carried out within the described investigations, influencing toothing factors can be quantified, and an equivalent bending diameter of the worm gear toothing can be determined. For this, a refinement of the developed analytical bending model is necessary. Finally, the developed calculation model is validated using experimental results from Lutz  as well as FEM calculations of worm gear shafts.
5 Applicability test of FEM
The existing calculation methods for calculating the worm shaft deflection were analyzed for the standard reference gear according to ISO/TS 14521 . The boundary conditions of the test setup of Lutz were used. The worm shaft was also calculated using the finite element method. The shaft of the standard reference gear and the resulting bending lines are shown in Figure 4.
The bearings were assumed flexible regarding bending as within the test setup. The calculations according to DIN 3996  and ISO/TS 14521  show a good correlation with the test results for both considered loads. The reason for this is that this worm shaft was tested by Lutz , and the correction factor for the equivalent diameter was derived from his results. The calculation according to AGMA 6022  uses the root diameter as equivalent bending diameter. Thus, the calculated bending lines differ constantly from the test results. This calculation is on the safe side.
The bending lines of the FEM calculations are very similar to the experimental values. In the context of the simulations, the locally different shaft dimensions can be considered by using a fine meshing. The increase in bending stiffness due to larger shaft diameters at the bearing points can be seen at the edges of the bending line.
The calculations show two main results. As the standard reference gear was used for the development of the correction factor by Lutz , the calculation by ISO/TS 14521  fits very well. Using the calculation according to AGMA 6022 , the bending lines show a lack of accuracy. With untested worm shaft geometries, the calculation of ISO/TS 14521  shows significant differences to FEM calculations.
On the other hand, the results show the suitability of the finite element method for the precise calculation of the worm shaft deflection. As part of the calculations, parameters such as element size and boundary conditions for optimal simulation results were obtained, which were subsequently used in the simulations described in the following.
6 Toothing parameters for the parameter variation
The aim of the calculations is to determine the influence of different toothing parameters on the worm shaft deflection. The following toothing parameters were varied: tooth thickness, tooth height, worm gear size, pressure angle, size factor, and number of worm threads. The variation was based on the standard reference gear of ISO/TS 14521 . Table 1 shows the variation parameter and the resulting transverse section of the worm. The ID indicates the center distance as well as the changed parameter. For example, A100_H080 is the same toothing as A100_Stand with a reduced tooth height of 80 percent. Following that, A100_ALF15 is the geometry of the standard reference gear A100_Stand with a decreased pressure angle.
The toothing parameters of the standard reference gear can be found in ISO/TS 14521 . A further worm gear was derived from the geometry of the standard reference gear by scaling the center distance and the module. This was done in a way so the relative geometry parameters are the same. The percentage changes in tooth thickness, tooth height, and the diameter quotient relate to the values of the standard reference gear.
7 Analytical model for the equivalent bending diameter and first bending results
An improved calculation method for the worm shaft deflection can be achieved by calculating the tooth-dependent bending stiffness of the worm gear toothing. With knowledge of the stiffness, the toothing section of the wormshaft can be calculated using an alternative bending diameter, and the stiffness of the entire worm shaft can be calculated based on the local stiffness of the individual shaft sections.
The following model was developed as part of the investigations. Figure 5 shows the transverse section of a two-threaded worm and the approximated tooth area that is used by the described analytical model.
The calculation model determines the moment of inertia as a combined moment of the root diameter and the moment of the worm gear toothing. The total moment of inertia can be calculated according to Equation 8.
The relevant parameters for the calculation are shown in Figure 6.
The moment of inertia of the tooth root is defined according to Equation 9.
Based on the toothing factors, a trapezoid is calculated. This trapezoid is the basis for the moment of inertia for the toothing. The moments of inertia of the trapezoid for both relevant axes are calculated according to Equations 10 and 11.
The distance between the main inertia axes and the axis of the worm shaft is calculated according to Equation 14.
The moment of inertia of the toothing is calculated according to Equations 15. The factors a and b depend on the number of threads of the worm. Due to the thread number, the angle between the teeth change. The position and the resulting coordinate transformation of the moment of inertia of a single tooth, as well as the change of the effective distance of the tooth for the total moment of inertia is considered by the factors a and b.
Given the total moment of inertia, the analytical equivalent bending diameter can be calculated according to Equation 18.
Table 2 shows the difference of the tooth area in transverse section calculated by the CAD system and the tooth area calculated by the herein presented analytical model. Moreover, the different moments of inertia are displayed.
The maximum difference in toothing area in transverse section and the resulting difference in moment of inertia of the toothing is 9,5 %. The mean deviation for the investigated toothings is 3,3 %.
Figure 7 shows the calculated maximum deflection δm of the investigated worm toothings. The deflection was calculated according to Euler-Bernoulli as well as with additional consideration of the shear deformation according to Timoshenko. The moment of inertia and the area in the transverse section was calculated by CAD software and with the introduced analytical model. Furthermore, the deflection calculated with FEM is shown.
The results show the calculation according to Euler-Bernoulli as well as the calculation with additional consideration of the shear deformation differ from the results of the FEM calculations. This confirms the mentioned calculation limits. The calculation according to Timoshenko with consideration of the shear deformation tends to show results closer to FEM than the Bernoulli model.
The pressure angle and the tooth height have a minor effect on worm shaft deflection compared to tooth thickness. As Langenbeck  stated, the diameter quotient q has a high influence on the bending.
The results of this comparison show the approach for calculating the worm shaft deflection described in the previous chapters is capable of assessing the influence factors from the considered geometry parameters qualitatively as the results show the same tendencies than the FEM calculations. However, there are some deviations in the absolute numbers compared to the FEM calculations that are used as reference. These deviations show the limitations of the currently used approaches and confirm an effect of the twisting worm gear toothing on the worm shaft deflection.
An additional method for modifying the stiffness values based on the calculation results is presented in the following.
8 Refinement of the approach for calculating the worm gear deflection
The results of Section 7 show the support or notch effect of the worm gear toothing on the shaft deflection cannot be determined with a purely analytical approach based on only the transverse section of the toothing. Figure 8 shows the ratio of the maximum deflection of the worm shaft according to the FEM in relation to the analytically calculated deflection for the varying toothing parameters. To facilitate the comparison, dimensionless factors were defined for each varied parameter:
The four dimensionless factors represent efficiently the most influencing tooth parameters on the bending stiffness of a worm gear toothing. In the diagrams in Figure 8, the ratio of the maximum deflection of the FEM calculation and the presented analytical calculation is shown for each factor. The slope of the best fit line indicates the variation of the tooth parameter, which is represented by the dimensionless factor and considered well by the analytical model.
The variation of the tooth height as well as of the pressure angle is well considered by the analytical calculation model, since the slope of the best-fit line is small and the deflection values vary slightly. The variation of the tooth thickness as well as the lead angle or the diameter quotient leads to larger deviations between FEM results and analytical calculation.
The variation of the tooth thickness shows that thinner tooth thicknesses are considered better than thicker tooth thicknesses. The calculations show a large influence of the lead angle on the bending stiffness of the toothing. It is also shown the deflection of the one threaded worm variant is outside of the systematic and cannot be calculated correctly with the model.
A reason for this can be found in the difference of the center of the main deviation axis of the moment of inertia of the worm gear tooth. The exception is shown by comparing a single with a seven-threaded worm. The transverse section of both are shown in Figure 9. The main axis of the moment of inertia are shown in green and red, the axis of the worm with a white cross. For the seven-threaded worm, the main axes of inertia and the worm axis are identical. This is the case for every worm except for a one threaded worm.
Using the bending beam model with considered shear bending, the maximum deflection values calculated by FEM can be used to calculate an equivalent bending diameter. Figure 10 shows the ratio of this equivalent diameter according to the FEM calculations (dFEM) and the equivalent diameter according to the analytical calculation model (drep). Through the research process, a dependency of the resulting equivalent bending diameter and the relationship between toothing area and root area. Therefore, a further dimensionless parameter was developed. It is defined by Equation 23.
Considering the results shown in Figure 10, a linear correlation between the correct equivalent bending diameter determined by FEM calculations and the replacement diameter drep can be found for all worm shafts except of single-threaded ones. This correlation can be described as follows:
9 Validation of the developed calculation model
Figure 11 shows the maximum worm shaft deflection of the examined worm toothing variants.
The comparison of the results shows the presented calculation model takes better account of the influence of different toothing parameters on the worm shaft deflection than the calculation according to ISO/TS 14521  and AGMA 6002 . Furthermore, there is a good alignment between the results of the analytical calculation and the results of the calculation using FEM. In general, the results of the analytical calculation described herein are on the safe side.
The advantage of the presented analytical calculation model is the calculation of an equivalent bending diameter, which can be used when calculating bending beams with different diameters. This means the stiffness of the worm toothing as idle part of the worm shaft can be taken into account in complete gearbox systems.
Figure 12 shows the bending line of the standard reference gear of the ISO/TS 14521 .
The bending line starts at the middle of the bearing position. In addition to the real worm shaft geometry, the analytical shaft geometry with the corrected equivalent bending diameter dcor instead of the toothing is shown. The calculation was done using the RIKOR  program system. As shown in Figure 4, there is a good correlation between the results of the FEM calculation and the test results. At the same time, the presented analytical calculation approach used within RIKOR shows a good agreement with the experimental results.
Figure 13 shows the bending lines of a tested worm shaft by Lutz  with a center distance of a = 65 mm. Contrary to the calculation of the standard reference gear, there is only a small difference between the calculated equivalent bending diameter of the toothing and the subsequent shaft diameters of the worm shaft.
As for the results regarding the standard reference gear, the resulting bending line and maximum worm shaft deflection of the analytical calculation model closely resembles the FEM calculation and the experimental results.
In this paper, the currently available approaches for calculating the worm shaft deflection of worm gears are presented. In some cases, the comparison of the different calculation approaches and experimental tests show large deviations between existing calculation models and experimental results.
The results of the investigations show the finite element method is suitable for a detailed calculation of the worm shaft deflection.
An analytical calculation model for the calculation of the equivalent bending diameter of worm gear toothings was developed. The comparison of experimental results and results gained from FEM calculations with results of analytically carried out calculations with the calculation approach presented within this paper show the supporting effect of the toothing cannot be completely described analytically. Using the bending beam model and taking the shear deformation according to Timoshenko into account, a correction factor was derived.
The results gained by using the corrected equivalent bending diameter match well with the experimental as well as with FEM results. Through the investigation, the calculation results showed that a one-threaded worm is best calculated using the root diameter as equivalent bending diameter. Because of this, the calculation according to ISO/TS 14521  and AGMA 6022  provides good results in general for one-threaded worms.
The results also can be used for the calculation of the shaft deflection of high reduction crossed helical gear sets. In this setup, the gear set consists usually of a worm with involute flank profile and a helical gear.
It is now possible to take the bending stiffness of worm toothings into account in calculations of complete gearbox systems. Furthermore, overhung worms that are widely used in practical application can now be calculated.
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Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented October 2020 at the AGMA Fall Technical Meeting. 20FTM06