An FEM modeling approach is used to obtain the individual proportions of the angular displacement caused by an external load at the worm wheel shaft of two worm gearboxes decomposed in individual components for gears, worm wheel shaft, worm shaft and wheel shaft bearings, and housing.

In worm-drive applications with precise transmission, the torsional stiffness of the worm gearbox is an important parameter for the positioning accuracy. Moreover, it influences the dynamic behavior of a worm gearbox. The torsional stiffness is a result of elasticities from all gearbox components along the load path. This leads to various options for an optimization of the torsional stiffness within the design process of a worm gearbox.

The influence of bearings, shafts, housing, and the toothing on the torsional stiffness is the subject of this article. Methods of finite element analysis are used for modeling a worm gearbox with varying compositions of the gearbox components and elasticities. By varying the setup of stiffnesses, the sensitivity of the torsional stiffness to the individual components of the gearbox is determined.

An investigation of the distribution of elasticities within the gearbox for multiple gearbox dimensions and gear geometries gives information on the potential of the individual gearbox components for optimizing the torsional stiffness.

1 Introduction

Positioning systems such as rotary and swivel tables or azimuth slewing drives require accurate transmission even under load, high positioning speed, and changing directions of rotation. To reduce the size of driving machines, the drivetrain of such systems often includes a gearbox. These negatively influence the positioning accuracy due to an additional elasticity that gearboxes bring, the backlash and transmission error of the gears, as well as the clearance of the bearings. A large variety of gearbox types can be found on the market today to meet the requirements of precise transmission. Among those are worm drives, which exhibit various beneficial properties such as smooth operation, optional design for self-locking, or, in case of dual lead worm gears, an adjustable backlash. An in-depth study on the positioning accuracy of different gearbox types for industrial robots can be found in [1]. There, the static and dynamic behavior of the investigated gearboxes were compared in terms of static characteristics, hysteresis, and vibration decay. The static characteristics, including the backlash of the gearbox, were highlighted to be crucial for predicting the behavior of the industrial robot.

Particularly in dynamic systems such as industrial robots, the mass and volume of the gearbox is an important design parameter. Considering the efforts for downsizing and reduction of material use, gearboxes need to become smaller while transmitting the same power and load and — especially in the case of positioning applications — having the same precision. Effectively optimizing the static characteristics of gearboxes requires an analysis of the elastic components inside the gearbox and their contribution to the overall torsional stiffness. Thereby, the main sources of elasticity and targets for the optimization could be identified.

Some of these individual deformations such as worm shaft deflection or tooth and contact deformation have been addressed in literature. Furthermore, experiments on the torsional stiffness of gearboxes have been conducted in academic and industrial research. However, studies on the distribution of the torsional stiffness across gearbox components have yet to be published.

This article presents an approach to decompose the torsional stiffness of a worm gearbox into its individual components. Calculations were conducted with the finite element method (FEM). The FEM model includes all relevant mechanical components, whereas gears, shafts, and housing are represented by three- dimensional models and the bearings by springs in the supported degrees of freedom (DOF). The decomposition was done for two industry-related worm gear geometries, bearings, and shaft designs. The investigated housing geometries are designed generically. Thus, the results of the individual proportions herein presented cannot be generalized without restriction, although, the method can be applied to other gearboxes.

2 State of the art

The torsional stiffness of a gearbox is often defined as the ratio of the applied load and the resulting angular displacement at the output shaft when the rotation of the input shaft is fixed. The displacement is a result of the chained deflections of bearings, shafts, gears, and housing. These deflections caused by individual gearbox components have different effects on the angular displacement at the output shaft. What the effects are depends on the orientations of the individual deflections as well as transformation factors such as transmission ratio or distance from bearing to tooth contact.

This implies an approach as described in [2], where the determination of the effect of the worm shaft deflections on the torsional stiffness was done by calculating the deflections and converting the respective result into an equivalent angular displacement at the output shaft. The advantage of this approach is that simplified methods can be used for calculating the deflections of the individual components.

Analytical methods for the calculation of worm shaft deflection are standardized in AGMA 6022 [3], ISO/TS 14521 [4] and DIN 3996 [5], where the worm shaft including toothing is reduced to a cylindrical beam with an equivalent diameter dbeam. According to experimental results by LUTZ [6], a good approximation of the analytical calculation is given for a diameter dbeam = 1.1df1. This value was incorporated in ISO/TS 14521 and DIN 3996. Based on a comparison between analytical results with an equivalent diameter df1 and FEM results, LANGENBECK [7] determined deviations especially for worm gears with smaller diameter quotients q. LANGENBECK also concluded from the FEM calculations, that the supporting effect of the helical worm thread against worm shaft deflection is small as removing single worm threads only slightly affected the results for deflection. NORGAUER et al. [8] brought forward a refined approach compared to the standardized methods for worm shaft deflection, proposing to include the worm tooth face area in the transverse section in the calculation of the areal moment of inertia of the worm shaft. The equivalent diameter of the cylindrical beam is then calculated based on the equivalency of areal moment of inertia of the worm shaft and the cylindrical beam model.

Several studies on the mesh stiffness of gears in general can be found in literature as it is a relevant measure for the transmission error under load, load distribution across the engaging teeth, and the gearbox dynamics. In case of worm gears, a standard for a simplified, analytical calculation method for the mesh stiffness of worm gears does not exist. In [6], the theory of thin slices is used for tooth modeling and calculation of the load contact pattern in worm gears based on influence numbers. LUTZ included in the determination of the influence numbers the stiffness of gears, shafts, bearings, and housing.

FEM models with different complexity levels are often used to analyze the load distribution and deformation of the gears under load and other coupled measures. JBILY et al. [10] also used the method of influence numbers for determining the load characteristics of worm gears. In addition, they considered the local contact deformation with the half-space theory of BOUSSINESQ. The influence numbers were calculated with FEM.

A comparable modeling process for worm gears to that presented in this work is described in [11]. Based on a calculated geometry of the gears, a three-dimensional model was generated to analyze various parameters for the design of worm gears by means of FEM.

3 Modeling strategy

When worm gearboxes are used in positioning applications, there are basically two different load scenarios to be considered. In the first scenario, rotation is initiated at the input side (worm shaft) with a static load at the output side (worm wheel shaft). An example are rotary tables in CNC machines, where, during the positioning procedure the accelerated moment of inertia, causes load at the output of the worm gearbox. The angular displacement at the input side causes a deviation between the targeted and the actual value of the position. In the second scenario, rotation is initiated at the output side by an external load after the targeted position is reached. This results in an angular displacement and a deviating actual position at the output. This configuration is found in azimuth drives of solar panels, which are exposed to wind load.

To analyze both load scenarios, FEM models of worm gearboxes with two different center distances 40 mm and 125 mm were created. Sectional views of two exemplary gearboxes of the study with size 125 mm are illustrated in Figure 1 with housing, bearings, shafts, and gears. Two different designs of the gearbox housing were investigated. A wide variety of different housing designs is used in industrial practice. Since this diversity is difficult to cover, two examples, as simple as possible but representative, were chosen for this study. The first housing design analyzed aligns closely with the shape of the gears enclosed while the other comes in the shape of a cuboid box. Both designs have a constant wall thickness. Common bearing arrangements for guiding the worm wheel shaft (x-arrangement with deep groove ball bearings) and worm shaft (x-arrangement with angular contact ball bearing or tapered roller bearings) are used. The bearing seats of the bearing flanges for all bearings are integrated in the housing. Thus, the influence on the mounting of the flanges i.e. with bolts was not considered. The worm shaft comes as one piece made from case-hardened steel, while the worm wheel rim made from bronze is mounted on a steel wheel shaft.

Assuming the second of the two described load scenarios, a load at the output side with a clamped input side of the gearboxes in Figure 1 would lead to deflections in gears, bearings, shafts, and housing. The angular displacement at the output side can be easily determined by experiments as described in [1] or FEM calculations to analyze the torsional stiffness of the gearbox. However, the proportions of the angular displacement caused by the individual components are difficult to isolate in experiments involving complete gearboxes. This gap in knowledge on the displacement proportions of different gearbox parts makes targeted optimization aimed at improving gearbox positioning accuracy and dynamics difficult.

Figure 1: Housing designs for the worm gearbox with size 125 mm and mounting face (M) for the FEM calculations: contour-close (left) and cuboid housing (right).

The strategy used here to isolate the individual proportions with FEM is the analysis of different model configurations of the worm gearbox. The principle is illustrated in Figure 2. The basic configuration (see left illustration in Figure 2) consists of the worm shaft and the worm wheel rim. The worm shaft is guided at the bearing seats in all five supported DOF. The rotational DOF around the worm axis is fixed at the input side of the worm shaft (φ = 0). The worm wheel rim is supported and torque is applied at the face where it is connected to the wheel shaft. In this case, the angular displacement is a result of structural deformation of rim, teeth, worm shaft, and the contact deformation. If the wheel shaft is included in the calculation (see right illustration in Figure 2), the additional elasticity decreases the torsional stiffness, leading to a higher angular displacement at the output compared to the basic configuration.

Figure 2: Basic model configuration M0 with gears only (left side) and configuration M1 with included wheel shaft. Torque T2 is applied to output shaft and rotation f of input shaft is fixed.

Assuming the effect of the elasticity of the additional component on the deformation of the components from the previous model configuration is negligible, the difference of the angular displacement between the two configurations can be assigned to the wheel shaft. The absolute contribution of the wheel shaft Δφ01 to the angular displacement of the complete gearbox is calculated with Equation 1.


φ0 s the angular displacement of the configuration including only the worm gear.

φ1 s the angular displacement of the configuration including worm gear and wheel shaft.

By gradually adding elasticity to the model, the respective proportions of the angular displacement and the torsional stiffness can be determined. In addition to worm gear and wheel shaft, the proportions of worm shaft bearing, wheel shaft bearing, and housing were investigated. This leads to a number of five different model configurations. Table 1 lists the five model configurations from M0 (basic configuration) to M4 (full configuration, Figure 1), the components they contain as well as the respective symbols for the stiffness and displacement components. The calculation of the absolute contribution with Equation 1 is analog for the other components, always comparing two subsequent model configurations. The absolute contribution of the worm gear is equal to the calculated angular displacement φ0.

Based on the values for the relative contribution of the individual components, their influence on the torsional stiffness can be determined. The components with the highest relative contribution also have the greatest influence on torsional stiffness. Exemplarily for the wheel shaft, the relative proportion of the wheel shaft Δφ01,rel is calculated with Equation 2. The calculation of the relative contribution with Equation 2 is analog for the other components.


Δφ01 is the absolute contribution of the wheel shaft according to Equation 1.

φ4 is the angular displacement of the full configuration M4 including all gearbox components.

The torsional stiffness of the complete gearbox is a series system of the torsional stiffnesses resulting from the individual components ct,0, ct,1, ct,2, ct,3, ct,4 (see Table 1), each of which is calculated from the torque at the output T2 and the respective absolute contribution of the individual component (see Equation 3 as example for the wheel shaft proportion ct,1). The torsional stiffness ct of the series system is equal to the ratio of output torque T2 and the angular displacement of the complete gearbox φ4 (see Equation 4)

The first scenario can be investigated analogously by applying a specific torque to the input side with a clamped output side and measuring the resulting angular displacement at the input side. The absolute contributions of the individual components are given by the difference of the angular displacements between consecutive model configurations.

Table 1: Gearbox, stiffness, and displacement components and the composition of the model configurations for the FEM calculations. Filled cells indicate an inclusion of the component in the model configuration.

3.1 Gear geometry

Two different worm gear geometries with a milled helicoid type K worm (ZK-profile) [12] were used in the presented study. The two geometries have a variation in center distance and gear ratio (see Table 2).

Three-dimensional models of the gears for the FEM calculations were generated. First, the worm tooth profile was modeled based on the work of PREDKI [13] and ORTLEB [14]. PREDKI published a procedure to determine discrete points in the longitudinal section of a worm tooth with ZK profile. This tooth profile was then used to calculate numerically the conjugated worm wheel tooth flank by simulating the tangential hobbing process. The hob geometry used is the worm tooth geometry with an enlarged tip diameter (addendum coefficient h*am0 = 1.2) and tip radius with ρa0 = 0.1mx. As an example, Figure 3 shows the discretized tooth flanks of the worm wheel for the second geometry (see Table 1) together with a section of a worm thread. The three-dimensional models for worm and worm wheel were created based on geometry points of both using functions of the commercial CAD software Siemens NX 11. The modeling procedure is fully automated by executing scripted journal files.

Figure 3: Discretized worm wheel flanks and section of the worm thread of the worm gear geometry no. 2.
Figure 4: Geometry of the worm shaft of the gearbox with size 125 mm.

The contour of the worm shaft is similar for both worm gear geometries investigated (see Figure 4). The worm model includes the worm shaft with bearing seats as well as the worm threads. The worm thread is formed by a helix with the worm tooth profile and duplicated according to the number of worm threads.

In contrast to the worm model, the worm wheel model has a simplified geometry to decrease the computational effort in the FEM. Specifically, since only a few teeth are engaged during meshing, the number of worm wheel teeth modeled is reduced. The worm wheel is divided into a toothed part and a wheel rim part that contains only the geometry of the worm wheel rim below the tooth root (see Figure 5). The surfaces of the worm wheel flanks are built through splines, defined by the calculated points of the tooth profile in each transverse section. The worm wheel tooth gap is then completed with surfaces for the tooth root, tip, and both face sides and subtracted from the worm wheel rim. The rim geometry as well as the mounting surface of wheel shaft and worm wheel is different for the two gearbox sizes (see filled surfaces in Figure 5). The principle of load transfers is an interference fit for the smaller gearbox size and a frictional connection by axially mounted screws for the larger gearbox.

Figure 5: Sectional views of the worm wheel for gearbox size 40 mm (left) and 125 mm (right) with surface marked in gray for mounting the wheel shaft.
Figure 6: Supported degrees of freedom at the original deep groove ball bearing (left) and spring model (right) with stiffness parameters.

3.2 Calculation of the bearing stiffness

Bearings of worm and wheel shaft are modeled as single springs in each of the supported DOF. The bearings each support three translational (x, y, z) and two rotational (β, γ) DOF, which requires five stiffness values cx, cy, cz, cβ, cγ for each bearing (see Figure 6). Coupled stiffness was not considered, which is why each stiffness matrix has elements only on the diagonal (Equation 5).

The bearings used for the two gearbox sizes are listed in Table 3. With the software Bearinx [15] the same shaft-bearing-systems and load scenarios as with the FEM were modeled to determine the bearing stiffness. In line with DIN 3996 [5], the forces from tooth contact were simplified to an axial, radial, and tangential force at the respective reference diameter. To guarantee reliable operation of the bearings on the worm shaft, the bearing arrangement was preloaded with an initial axial displacement. The deep groove ball bearings of the wheel shaft are initially in contact but unloaded, which is why clearance of the bearings was not considered in the FEM model of the worm gearbox.

Table 3: Overview on the investigated bearings.

Due to the non-linear characteristics of the bearings, load was applied successively in multiple steps in the actual calculation of the bearing stiffness. As an example, Figure 7 shows the axial stiffness cx of tapered roller bearings (type 31309) for the worm shaft of the gearbox of size 125 mm with respect to the input torque T1. The additional illustration shows the axial force Fxm1, radial force Frm1, and tangential force Ftm1 to the worm shaft together with the bearings designated “A” and “B.” The designation “A” refers to the bearing that supports the axial load from the tooth contact. The diagram indicates that, with increasing input torque, the axial stiffness of bearing “A” increases. The opposite is the case for bearing “B,” which in contrast to bearing “A” sees the axial load decline.

Figure 7: Non-linear characteristic of tapered roller bearings 31309 illustrated with the axial bearing stiffness cx with respect to input torque T1. Axial load Fxm1 from tooth contact is supported by bearing “A.”

This non-linear characteristic is simplified in the FEM model by using springs with a linear spring rate equal to a mean stiffness of the calculated load steps. The preload force at the worm shaft bearings is implemented with pre-compressed springs.

3.3 FEM model

The finite element analysis was conducted with the software Abaqus. The worm gearbox is modeled as an assembly of three-dimensional models of housing, gears, and worm wheel shaft. Springs represent the bearings and connect the shafts to the housing. The model configurations M0-M4 vary in their composition of gearbox components and in the defined boundary conditions.

The basic configuration M0 consists only of the gears. Since the wheel shaft is not included, the worm wheel is connected to a reference point in the worm wheel center (see Figure 2, left). The load from tooth contact is determined by performing a contact calculation within the finite element analysis. To consider load sharing in the calculations, contact was defined for all individual tooth pairs in normal and tangential direction. Due to the static conditions of the investigated worm gearboxes, boundary friction was assumed to be predominant in tooth contact. The boundary friction coefficient was calculated based on the mean Hertzian pressure according to DIN 3996 [5] for a material pairing bronze wheel (CuSn12Ni) with steel worm and a lubrication with oil based on polyalkylene glycol (PAG).

The configuration M1 additionally contains the wheel shaft, which is ideally tied to the worm wheel at the designated mounting surfaces (see Figure 5). Detailed modeling of the connection between wheel shaft and wheel was not included. The wheel shafts for the two gearbox sizes are shown in Figure 8. Depending on the load scenario, the indicated remote point is used for load application or locking of the wheel shaft when load is applied at the input side.

Figure 8: Sectional views of the worm wheel shafts for gearbox size 40 mm (left) and 125 mm (right).

Housing and shafts each are connected to a reference point centered at each bearing fit. Depending on the model configuration, the reference point of a shaft is fixed, which resembles a rigid bearing support, or the reference point is connected to the housing reference point with a spring to include the elasticity of the bearing.

Figure 9 shows the boundary conditions of the worm shaft for the configurations M2 and M3, where the stiffnesses C_A and C_B of the worm shaft bearings are considered in the calculation of the torsional stiffness. The subscripts A and B refer to the differently loaded bearings in the bearing arrangement of the worm shaft (see Figure 7). In Figure 9, the housing elasticity is excluded as the reference points of the housing are fixed. In configuration M3, the boundary conditions are set analogously for the wheel shaft to also include the wheel shaft bearings.

Figure 9: Boundary conditions of the worm shaft with included worm shaft bearings attached and rigid attachment.

To include the stiffness of the housing in the full configuration M4, the remote points of the housing are connected to the bearing fits to transfer the force of the bearings to the housing. In this configuration, the gearbox is ideally fixed at a mounting face of the housing. The mounting faces of both housing designs are illustrated in Figure 1.

The mesh density of the gearbox components is set individually. The gears are especially partitioned to get a structured mesh with linear hexahedral elements. To achieve an accurate calculation of the tooth contact, the element size of the mesh of the tooth flanks is significantly smaller than for the rest of the worm and worm wheel, respectively. The finite element mesh of worm and the teeth of the worm wheel are shown in Figure 10.

Figure 10: Finite element mesh of worm and worm wheel of geometry no. 2 (see Table 1).

4 Results

For each gearbox size, a variant with a cuboid housing and tapered roller bearings for the worm shaft was modeled as a reference. The second load scenario with a clamped input side and load application on the output side was investigated. The model parameters for boundary conditions and gearbox design are listed in Table 4. The applied torque represents the nominal torque of the gearboxes. The calculation was conducted for all model configurations M0-M4 and the angular displacement on the output side about the wheel shaft axis was determined in each case.

Table 4: Model parameters for decomposition of the angular displacement of the investigated worm gearboxes.

The results of the 125 mm gearbox model shown in Figure 11 demonstrate that as the boundaries of subsequent model configurations shift outwards to incrementally include further gearbox components, the angular displacement increases because the added components add more flexibility. It can be seen that as the model configuration increments, the angular displacement increases because more flexible gear components are included in the model. The torsional stiffness of the gearbox (full configuration M4) is calculated from the ratio of the torque load T2 = 2,000 Nm and the angular displacement φ4 = 0.152° to ct = 13,100 Nm/°. The proportions of the individual gear components in the angular displacement of the gearbox are derived from the differences between subsequent model configurations (see explanations in section 3). The decomposed angular displacement of the gearbox with its individual components is also shown and quantified in Figure 11. There is a significant difference between the components in terms of their contribution to the total angular displacement, with the gears and housing accounting for the largest shares and the wheel shaft showing the smallest impact. The higher the proportion of the overall angular displacement that an individual gearbox part accounts for, the greater is the reducing effect that component has on the torsional stiffness of the gearbox.

Figure 11: Angular displacement results for different model configurations of the reference model of the gearbox with size 125 mm and decomposition into individual components.
Figure 12: Angular displacement results for different model configurations of the reference model of the gearbox with size 40 mm and decomposition into individual components.

A different distribution was determined for the gearbox model with size 40 mm (see results in Figure 12). The torsional stiffness of the gearbox is ct = 623 Nm/°, which is significantly lower than that of the larger gearbox model. The proportion of the gears is comparable to the previous model, suggesting a substantial role of the gears in torsional stiffness generation. In contrast, the effects of housing and wheel shaft on the torsional stiffness are much smaller for the 40 mm model. The different designs and dimensions explain these results. Especially the wall thickness of the housing was, for reasons of manufacturing feasibility, not reduced in the same magnitude as the gearbox size, which results in a higher stiffness of the housing for the smaller gearbox size. The large contribution of the worm shaft bearings to the angular displacement in the 40 mm model is particularly remarkable. A reason for that is assumed to be the small lead angle γm1, which is 3.37° for the gearbox size 40 mm compared to 21.8° for the size 125 mm. This is because the small lead angle leads to a higher proportion of the tooth contact force in the axial direction of the worm shaft. Due to a lower stiffness of the worm shaft bearing in axial direction than in radial direction, the deformation coming from those bearings increases.

Based on the results, components with a high proportion of the angular deformation can be identified. For those components, an increase in the individual stiffness has the greatest effect on the torsional stiffness of the gearbox. The results in Figure 11 and Figure 12 imply distributions of the angular displacement proportion must be analyzed individually for each gearbox, to select components for an effective optimization of the torsional stiffness.

In the following, housing and worm shaft bearings are modified to affect the torsional stiffness of the worm gearbox of both sizes. Four variations of the reference gear boxes were defined by varying housing design (see Figure 1), housing wall thickness, or the type of the worm shaft bearing to analyze the effect on torsional stiffness of the gearbox compared to the reference. The calculation procedure was the same as for the reference model. The analysis of the results focuses, on the one hand, on changes of the angular displacement of the gearbox (total effect). On the other hand, changes of the absolute contribution to the angular displacement of the varied gearbox component (individual effect) are considered.

The results for the gearbox with size 125 mm are shown in Figure 13. The difference of the variation compared to the reference model (V1) is specified in the annotation on the right. The effects of changes in housing design and wall thickness on the torsional stiffness are considered in the variations V2-V4. The stiffness of the housing can either be increased by increasing the wall thickness or by changing the housing design. It can be concluded from the results shown in Figure 13, that the investigated contour-close design leads to the greatest total effects. Compared to the reference model, a reduction of 16.7% was achieved with the same wall thickness (V3). An additional increase in wall thickness to 10 mm leads to 20.6% (V4).

Figure 13: Decomposed angular displacement for variations of the 125 mm reference model.
Figure 14: Decomposed angular displacement for variations of the 40 mm reference model.

The variation V5 has the same housing as the reference model, but angular contact bearings of type 7309 are used for guiding the worm shaft. Compared to the tapered roller bearings, the angular contact bearings have a lower axial and radial stiffness. This is due to point contact between ball and bearing rings compared to line contact in the tapered roller bearing. As a result, the torsional stiffness of the gearbox also decreases, leading to a 9% higher angular displacement.

Considering the results of the variations of the gearbox model with size 40 mm in Figure 14, an increase in wall thickness and change to contour-close design lead to comparable individual effects for the housing. A numerical comparison of the effects for both gearbox sizes is given in Table 5. However, the total effect on the angular displacement is very small as the initial proportion of the housing is already small. Changes to the housing yield a maximum reduction of 3.6% in angular displacement for the contour-close design with higher wall thickness (V5) compared to the reference model.

Table 5: Comparison of individual effect and total effect for all variations of both gearbox sizes.

A contrasting picture compared to the results in Figure 13 can be seen for the variation V5 with angular contact bearing of type 7202. Expecting an increase in angular displacement, the change to a ball bearing leads to a higher torsional stiffness. In fact, the calculation of the bearing stiffness returned a higher axial stiffness and lower radial stiffness of the bearing 7202 compared to the tapered roller bearing 30202. In combination with the small lead angle, this leads to a large decrease in the angular displacement proportion of the worm shaft bearings. This highlights the importance of the axial stiffness of the worm shaft bearings for the torsional stiffness. Due to the share of 39.8% in the reference model (see Figure 12), the effect on the angular displacement of the gearbox of 17.1% is significantly higher than with changes to the housing.

A promising opportunity for increasing the bearing stiffness at the worm shaft is the application of a higher preload force. A drawback of this method is an increase in friction torque for the bearings, leading to a decrease in gearbox efficiency. To determine the effectivity of this method for increasing stiffness, an additional analysis was conducted with the reference model. The stiffnesses of the worm shaft bearings (tapered roller bearings of type 30202) for the supported DOF were calculated for different preload forces according to the procedure described in section 3.2 and applied to the springs in the FEM model. The results for the angular displacement of the gearbox with respect to the applied preload force are shown in Figure 15. The preload force of the reference model is 1,100 N. A force greater by 1,000 N leads to a reduction of the angular displacement of about 0.004° or 4.6% compared to the reference model, which is significantly higher than the reduction achieved with changes to the housing.

Figure 15: Dependence of the overall angular displacement on preload force on the worm shaft bearings.

5 Conclusion

The angular displacement caused by an external load at the worm wheel shaft of two worm gearboxes was decomposed in individual components for gears, worm wheel shaft, worm shaft and wheel shaft bearings, and housing. For this purpose, a FEM modeling approach was used to obtain the individual proportions of the angular displacement. The approach presented here has the potential to be applied to other gearbox varieties and could yield valuable insights for targeted and cost-efficient optimization in gearbox design processes. Moreover, with an extended and more detailed model, connections between the main parts of the gearbox such as shaft-hub-connections or housing screws could be included in the analysis.

Results for the two analyzed worm gearbox sizes show a very uneven distribution of the angular displacement across the gearbox components. For both, a consistently high proportion of the gears was found, highlighting their importance for the torsional stiffness of the gearbox. The proportions of other components vary greatly for both gearboxes. Especially the stiffnesses of wheel shaft and housing strongly depend on the design parameters and dimensions and need to be analyzed for each individual case. The worm shaft bearings have a significant effect on the torsional stiffness, in particular the axial stiffness of the bearing when the lead angle is small.

There are multiple approaches for optimizing the torsional stiffness of a worm gearbox. Often, these approaches have certain drawbacks, such as increased weight, higher material use, or, in the case of changes in preload force or in bearing type of the worm shaft bearing, efficiency losses. This suggests a targeted optimization of single gearbox components based on a cost-benefit analysis. However, to achieve a high effectiveness of the optimization, information about the individual stiffness proportions of the gearbox components is important. Indeed, the analysis of variations of the reference gearbox model indicate that a significant increase in stiffness of an individual gearbox component only has a small effect on the torsional stiffness of the gearbox if the original component’s proportion is small. A case point is the reference model of the 40 mm gearbox, where an increase in preload force by 1,000 N has a greater effect than an increase of the housing wall thickness by 2 mm. This effect is even many times greater, when the bearing type is changed.

The presented results are based on calculations and require additional experiments for validation. For this purpose, a test bench was designed to investigate the torsional stiffness of the herein calculated worm gears in the same way as in the FEM. In future work, the calculated results will be compared in terms of torsional stiffness of the complete gearbox. However, validation of the individual angular displacement proportions is very difficult as it requires a precise measurement of the individual displacements (worm shaft bending deflection, displacement of bearings and housings, etc.) and a transformation of these into an equivalent angular displacement at the gearbox shaft.

6 Acknowledgement

This work was supported by the German Federal Ministry of Economics and Energy (IGF 21768 N) within the framework of the Forschungsvereinigung Antriebstechnik e. V. (FVA project 930 I). 


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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 2022 at the AGMA Fall Technical Meeting. 22FTM17

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K. Daubach is with the Technische Universität Kaiserslautern; Chair of Machine Elements, Gears and Tribology.
M. Oehler is with the Technische Universität Kaiserslautern; Chair of Machine Elements, Gears and Tribology.
O. Koch are with RPTU Kaiserslautern-Landau, Chair of Machine Elements, Gears, and Tribology.