Due to substantial sliding in tooth contact, friction losses are an issue in worm gears, but a simulation tool can be used to study the behavior between two interacting tribological measures – friction and wear – during the running-in of worm gears.

The load capacity of worm gears strongly depends on the size of the contact pattern. Worm wheels are often manufactured by using an oversized hob, which results in a relatively small initial contact pattern. Wear on the worm wheel with a softer material during the running-in process increases the contact pattern and thereby the load capacity. For the investigation of the continuous change of friction in the tooth contact during that process, a tribological simulation program is used. With a simplified model of the EHL-tooth contact, boundary as well as fluid friction are calculated locally, and the tooth efficiency is evaluated. The included wear model associates abrasive wear with solid friction energy occurring in the tooth contact and allows a time-dependent simulation by considering the wear-modified tooth flank in the tribological calculation.

The simulative results are compared with experimental wear studies on the running-in of worm gears. Since various values are determined in the simulation model, the comparison covers different aspects to verify the model. However, for measurement reasons, a comparison takes place on the macro scale. The tooth friction is reflected by the measured efficiency of the gearbox on the test bench. Wear is on one hand a directly measured value, and on the other hand, it changes the geometry of the tooth flank and influences and thereby the unloaded kinematics of the gears. Both aspects are considered for a verification of the wear calculation.

1 Introduction

Worm gears are transmission elements used in various applications for power transmission or precision gears. They offer a high load capacity, smooth operation, as well as high-gear ratios in a single stage. Due to substantial sliding in tooth contact, friction losses are an issue in worm gears. In case of the commonly used combination of a bronze wheel with a steel worm, friction is accompanied by wear on the worm wheel when an insufficient lubricant film leads to solid body contact. As a result, the tribological system of the tooth contact in worm gears is subjected to changes, in particularly within the first phase of the gear’s lifetime (running-in).

This phase is generally characterized with severe wear on the worm wheel, leading to an adjustment of the micro and macro geometry of the tooth flank. In terms of micro geometry, plastic deformation of surface asperities yields a reduction of the surface roughness and an increase of the load capacity of the surface. On the macroscale, a determining factor for the load capacity is the contact pattern of the gears, which comprises all contact lines of worm and worm wheel of the entire gear-meshing cycle. The load capacity is influenced by the size and the position of the contact pattern on the worm wheel tooth flank. Both are determined by several aspects of the manufacturing processes of worm and worm wheel, the gear’s assembly, and the load conditions. Worm gears are often manufactured with an oversized hob that has a larger reference diameter than the worm [1], which leads to a relatively small initial contact pattern. The position of the contact pattern is also affected in the manufacturing process by parameters as swivel angle of the hob axis [1]. Moreover, the contact pattern is a result of the alignment of the gears in the assembly. Thus, a colored ink is often used for illustration of the idle contact pattern to check the alignment. Optimization of the alignment and thereby the idle contact pattern can be achieved by adjusting the wheel position along its axis. The load contact pattern is again different from the idle contact pattern as deformations of teeth, worm shaft, bearings, and other gearbox components lead to a variation of the tooth contact.

In the event of a wear-related change of the macro geometry of the wheel tooth flank during running-in, the contact pattern develops from its initial state, given by the manufactured geometry, alignment, and the load conditions of the worm gear. For worm gears, this initial wear process with an increase of the contact pattern size is tolerated to achieve a maximum load capacity and to prevent fatigue wear (pitting). The intensity of this process and the results for the tribological system as surface roughness depend on various aspects as operating conditions, lubricant, initial surface roughness, and the gear geometry. The tribological system after the running-in, in turn, influences friction and wear for the following steady wear phase.

Many studies on running-in were conducted by means of tribometers. BLAU [2] analyzed in his work various experimental results from tribometers in terms of the progression of the coefficient of friction during running- in. He derived from these results a reduced number of characteristic curves by which the change of the coefficient of friction is represented in many cases.

The effect of different running-in load conditions on the coefficient of friction and the efficiency of spur gears was experimentally investigated in [3]. SOSA measured surface profiles of the tooth flanks during the tests and analyzed them in terms of wear at the asperities of the surface. Based on a calculation of the contact pressure and subsurface stress, wear caused by plastic deformations of the asperities was simulated and compared to measured surface profiles.

WEISEL [4] experimentally analyzed wear on worm gears with an incomplete contact pattern during running- in. An empirical approach for the wear calculation in this phase was derived from the correlation between wear, load, and the ratio of the size of the contact pattern to the maximum size.

Simulative approaches for wear prediction are an effective instrument during the design process of gears, as they offer a significant reduction of wear tests during the gear-design process. However, due to the complexity of wear with many influencing variables, wear models such as the widely used ARCHARD model [5] require a parameter that has to be determined from experiments. In this context, JBILY et al. [6] developed a numerical wear calculation of worm gears based on the wear model of ARCHARD [5] with a locally defined wear coefficient that depends on a dimensionless gap height. SHARIF et al. [7] implemented a modification of the model of ARCHARD [5] for local wear simulation in worm gears. A dependence on the sliding speed as well as on the local lubricant film thickness and surface roughness was added to the original model.

As friction and wear interact with each other, a combined analysis of these tribological quantities is necessary. Therefore, a physical-based simulation model for friction in worm gears is coupled with the energetic wear model of FLEISCHER [8]. This model correlates boundary friction energy with volumetric wear from abrasion and includes wear-determining factors such as contact pressure, coefficient of friction, and sliding distance. The wear simulation is designed as an iterative procedure, which is initiated with a wear calculation for the initial manufactured geometry of the gears. To consider the effect on wear on the tribological calculations, the wear-modified geometry of the worm wheel tooth is used in all subsequent calculation steps. As digital twins experience a growing importance in engineering fields, the presented model can be potentially applied during the design process of worm gears. Potential issues that can be addressed with the model are the analysis of the running-in for different conditions to obtain an efficient running-in process. Moreover, it can be used to calculate the wear rate after the running-in, which is particularly relevant for the lifetime of the worm gearbox.

2 Tribological simulation of worm gears

The complexity of the tribological conditions in the tooth contact in worm gears is a result of a complex geometry and kinematic of the gears. Thus, the tribological conditions as load, sliding, or entraining velocity vary significantly across the meshing area. Under realistic operating conditions, the state of friction in the lubricated tooth contact of worm gears can be considered as mixed friction. Depending on the tribological conditions, the local contact situation varies between an entire separation of the tooth flanks by a lubricant film (fluid friction) on the one side, and contact of both metallic surfaces occurs at single asperities (boundary friction) on the other side. For that reason, a valid calculation of total tooth friction or wear requires a local analysis of these conditions.

Within studies on the tribological conditions of worm gears, a simulation tool was developed to determine reliably the coefficient of friction in the tooth contact of worm gears [9]. The tool is based on the work of PREDKI [10] and BOUCHÉ [11]. The simulation procedure includes an analysis of the contact lines, local velocities and radii of curvature for approximating the tooth contact by a simplified model with pairs of rollers. Based on the simplified model, a locally resolved calculation of the lubricating gap height is conducted by using approximate equations for the elasto-hydrodynamic contact. The simulation tool is supplemented by an external software for worm gear design SNETRA for the calculation of the flank pressure [12] .

In a first step, the geometry of the gears is analyzed. The worm tooth flank is described analytically according to [13]. The discrete geometry of the worm wheel is determined in the external software SNETRA [12] by simulating the manufacturing process. By using either an ideal or an oversized hob, the size of the initial contact pattern of worm and wheel is influenced.

Due to line contact between worm and worm wheel, the meshing area of the tooth contact is discretized in lines of contact points of multiple mesh positions. The contact points divide each line in segments. The contact points are determined with an unloaded tooth contact analysis (TCA). For this purpose, worm and worm wheel are paired together in a common coordinate system. Load sharing is considered by including multiple worm wheel teeth in the TCA. The algorithm behind the TCA is equality of the position vectors of worm and worm wheel, Equation 1.

Here, r2,i represents the point with subscript i of the discretized wheel tooth flank as a function of the wheel rotation angle Δϕ2 . According to [13] the geometry of the worm tooth flank r1 the worm tooth flank geometry is given as a function of the radius parameter u, the angle parameter v and the related mesh position angle ϕ1 . Solving Equation 1 gives the contact point on the worm flank and the required angle of rotation Δϕ2 for each relevant point of the worm wheel in a discrete mesh position. Contact lines of a single mesh position consist of contact points with minimal angle of rotation. Figure 1 shows exemplarily contact lines of a worm gear with incomplete contact pattern in the initial state.

Figure 1: Contact lines on the worm wheel obtained with the tooth contact analysis for a worm gear with initially small contact pattern (center distance a = 32 mm, number of threads in worm z1 = 1, number of wheel teeth z2 = 39, axial module mx = 1.25 mm).

With mixed friction, load is distributed on the surface asperities of the tooth flanks and the lubricant. Boundary friction depends on the proportion of the load directly applied to the metallic surfaces of the tooth flanks. This proportion is quantified by a solid contact ratio ψ. It becomes 0 in case of pure fluid friction and 1 in case of pure boundary friction. Further, it is dependent on the surface topography and, due to mechanical deformation in contact, also on the material properties. Therefore, real measured three- dimensional surfaces of the tooth flanks of worm and worm wheel together with the respective material properties are used to obtain results related to real gears. With an outsourced contact simulation based on the half-space theory [9], a relationship between the lubricating gap height and the solid body contact ratio was derived. The relationship of a dimensionless gap height λ and solid body contact ratio can be approximated by an analytical equation, Equation 2 [14], and thereby efficiently integrated into the tribological simulation. The parameters a and b are determined by fitting the equation to the calculated data of the contact simulation. The dimensionless gap height λ is calculated with the local minimum gap height hmin and the 3D root-mean-square roughness of worm Sq1 and worm wheel Sq2 , Equation 3.

The solid body contact ratio is determined for each contact point of the meshing area with Equation 2. It is used as a weighting factor to divide friction in boundary friction and fluid friction, from which a mean coefficient of friction of the mixed friction regime is calculated. In this context, the tribological simulation is supplemented by experimental data for the boundary friction (twin-disc tribometer tests). In these tribometer tests, contact pressures, slide-to-roll-ratio and the lubricant temperature were varied to obtain maps of the coefficient of boundary friction for different operating conditions. A low entraining speed was set to achieve conditions in the boundary friction region. For the calculation of the fluid friction, data for the lubricant was obtained in a high-pressure viscometer. Based on the lubricant data and with the assumption of a pseudoplastic fluid behavior, the internal friction of the lubricant is calculated with the model according to BAIR and WINER.

2.1 Power losses of worm gears

The power losses PV in worm gears are composed of losses in gears, bearings, seals and churning losses, see Equation 4. These can be categorized in load-dependent and load-independent losses [15]. The load- dependent losses include the friction losses in tooth PVZ,P and bearing contact PVL,P. Invariant with the load are churning losses at the gears PVZ,0 and bearings PVL,0 as wells as the losses that emerge in seals PVD and other components of the worm gearbox PVX.

With the tribological simulation described in section 2, the local friction force can be calculated for each contact point between worm and worm wheel. Including the distances of the contact points to the respective axis of rotation, the friction torque for each meshing position can be determined by integrating the friction forces over all contact lines. By balancing the friction torque and the input and output torque, the power loss due to friction in the tooth contact is calculated. The method of calculating losses using local quantities is described in [11] and was also used in [13].

For all other loss sources in the gearbox, state-of-the-art methods are used for the simulation. With the help of dimensionless parameters of fluid mechanics such as the Reynolds number, the hydraulic losses of the gears are calculated according to the method from CHANGENET et al. [16]. For the bearing losses, the empirically based method of the bearing manufacturer SKF [17] is used. Here, the individual loss components of rolling and sliding friction between rolling elements and raceways, the friction of the seals and the hydraulic losses due to oil drag and churning are included. The power loss of the contacting shaft seals at the gearbox input and output are determined using the method presented by ENGELKE in [18].

Whereas the load-dependent losses vary periodically due to the changing contact situation for each meshing position, the load-independent losses are constant according to the chosen calculation methods. The procedure for determining the transmission power loss of worm gears is described in detail by OEHLER et al. in [19].

2.2 Wear calculation

In tribological contacts, frictional energy is generated by the relative movement of the two contacting surfaces (sliding). Boundary friction in particular is a result of elastic and plastic deformations of surface asperities. Under these conditions, abrasive wear manifests where various wear mechanisms as cutting by harder surface asperities or fatigue due to repeated plastic deformations could be involved [20].

Independent of the wear mechanism, frictional energy is considered to be the physical basis for material damage. Thus, by using an energetic approach for wear calculation in combination with the physical-based simulation model for friction described in Section 2, valid results for a wide range of gear geometries as well as tribological conditions were expected.

The energetic approach of FLEISCHER [8] associates abrasive wear in tribological contacts with frictional energy from solid body contact. This energy is, for the most part, dissipated as heat energy. As an effect of the mechanical deformations, a part of the frictional energy is stored irreversibly in a local volume of material as lattice defects. If the locally stored energy reaches a critical level of energy, fracture of material and wear debris are the result. The critical energy level is represented by the wear energy density eR*, which is the main parameter of the energetic wear model. According to [8] the wear energy density is given by the ratio of frictional energy WR and the volumetric wear removal Vv, Equation (5).

Due to the good feasibility of the measurement of frictional energy and volumetric wear removal, the wear energy density is often determined with experimental data in related literature [21],[22],[23]. However, no available data for volumetric wear and frictional energy related to the running-in phase were available, which is why the wear energy density was differently determined (see Section 5).

In the simulation, wear is calculated locally based on Equation 5. Results of local tooth friction are given by the tribological simulation for each contact point of the discretized tooth contact. Rearranging Equation 5 and calculating frictional energy with friction force FR and sliding distance sR yields in:

In the tribological simulation, load is represented by a local line load wb, which is applied on each segment of the contact line with an individual length bH. As the tooth contact is modeled as a HERTZ’ian contact between two cylinders, this leads to a HERTZ’ian contact area with the length of the segment bH and twice the half width aH (see Figure 2). Since abrasive wear is caused mainly by solid body friction, only the proportion of the load from solid body contact is considered. This proportion is given locally by the solid body contact ratio ψ. The wear relevant friction force FR is calculated with Equation 7, where μGr is the local boundary friction coefficient.

During gear meshing, each contact point carries load for a certain time tc, while the two contacting surfaces of the tooth flanks slide relative to each other with the sliding velocity vg. The contact time tc depends on the half width aH and the velocity v2bn of the contact point when passing over the contact area. The velocity v2bn is the component of the sliding velocity in normal direction to the contact line [10], [11]. The sliding distance sR of each contact point during contact is calculated according to Equation 8.

With the simplification of the local wear volume as a cuboid volume with the HERTZ’ian contact area A = 2aH bH, the height represents the local wear height δv and calculated using Equation 9. As mainly the worm wheel is subjected to wear, only the macro geometry of the worm wheel tooth is modified by the calculated wear. Here, each point is modified individually by a tangential displacement with the respective calculated wear height.

This modification does not affect the micro geometry of the worm wheel surface. To consider the effects on the microstructure in the tribological calculations, surfaces of the wheel flank were also measured after half the time of the running-in test in the experiment and evaluated with the contact simulation in terms of solid body contact ratio (see Section 2).

Figure 2: Contact line segment with the dimensions of the HERTZ’ian contact area aH and bH, the sliding velocity vg and the velocity v2bn normal to the contact line.

3 Simulative results for friction and wear during running-in

A tribological simulation was conducted for a worm gear box with the same geometry and material of the gears as in the experimental investigations. The initial contact pattern was comparable in both cases, by which a comparable change of the contact pattern and also of the tribological conditions was expected within the simulation. A comparison of the contact patterns is conducted in Section 4. The wear energy density was specified as constant with
eR* = 1.1 1013 J/m3, as it gives a good approximation to the experimental results in section 4.

The expansion of the contact pattern is indicated Figure 3a-3d, which show the area of the worm wheel tooth flank affected by wear for four stages of the simulation. The color map of the wear distribution after the first calculation step (see Figure 3a) in particular reflects the local differences of wear on the wheel flank caused by locally different tribological conditions and flank pressure in the tooth contact. Points with the highest accumulated wear heights are at the initial position of the contact pattern since they are exposed to wear from the very first step. With the number of calculation steps, the accumulated wear on the wheel increases and the contact pattern expands. After 1,200 calculation steps (Figure 3d), wear is evident almost on the entire wheel flank, and the contact pattern is almost completely developed.

The effect of an increasing contact pattern is a reduction of the mean flank pressure as the load is carried by a larger contact area. The local HERTZ’ian pressure distributions pH on the wheel flank for the same calculation steps as in Figure 3 are illustrated in Figure 4. There, the maximum values of the pressure distribution decrease with an increasing number of load cycles. Due to the discrete calculation of wear and discrete wear modification of the wheel flank, pressure peaks occur that lead to an increasing pressure maximum from Figure 4c to Figure 4d. However, for the most part of the local contact points, the contact pressure decreases significantly (Figure 4d).

Figure 3: Accumulated wear height dv on the worm wheel for the indicated calculation stages of the wear simulation for a worm gear during running-in phase (a = 32 mm, i = 39, n1 = 150 min-1, T2 = 32 Nm). In each illustration, the entry side is on the left.
Figure 4: HERTZ’ian pressure distribution pH on the worm wheel for the indicated calculation stages of the wear simulation for a worm gear during running-in phase (a = 32 mm, i = 39, n1 = 150 min-1, T2 = 32 Nm). In each figure, the entry side is on the left.

With more points included in the tooth contact, the distribution of the tribological conditions within the tooth contact changes as well. As an example, the distribution of the lubricant film height hFilm for two stages is illustrated in Figure 5. Due to a relatively low input speed n1 = 150 min-1 in the simulation, the film heights are relatively small in general. In both stages, minimum film heights occur in the center of the tooth in width direction, which is because of a low entraining speed and poor hydrodynamic effects at this location [13]. For the larger contact pattern, however, after 400 calculation steps, significant more points with larger film heights are included in the tooth contact.

Figure 5: Film height distribution hfilm on the wheel tooth flank in the first calculation step (left) and after 400 calculation steps (right). In each figure, the entry side is on the left.

Furthermore, calculating wear locally with the energetic approach leads to higher wear for contact points with a higher level of frictional energy during the meshing. In the simulation, this imbalance is corrected by wear, by which the flank pressure moves gradually from points with a higher level to points with a lower energy level. In case of a balanced energy level, wear is equal for all contact points within a single mesh position of the gears. In total, these effects influence the overall tooth friction losses of the gears and the wear progression significantly.

Results of the maximum wear height across all points of the wheel tooth flank, the mean flank pressure, and total frictional energy (per wheel tooth and load cycle) progression with respect to load cycles are shown in Figure 6. To indicate the expansion of the contact pattern, the percentage proportion of the respective contact area relative to the maximum contact area is specified. The progressions correspond to a typical running-in behavior of worm gears. The first phase is assigned to severe wear with a maximum wear rate, which continuously decreases before reaching a steady wear phase with a linear wear progression.

Figure 6: Calculated wear height, mean HERTZ’ian pressure and total frictional energy per wheel tooth and load cycle progression during running-in of a worm gear (a = 32 mm, i = 39, T2 = 32 Nm, n1 = 150 min-1) after 600 calculation steps. The percentage proportion of the respective contact area related to the theoretical maximum contact area is indicated on the wear progression. Gray bars represent the proportion of contact points above the pitting resistance for contact stress for the used wheel material (520 MPa [25]).

At the same time, the mean pressure as well as the total frictional energy decreases and approaches asymptotically to a constant level. The gray bars indicate for discrete calculation steps the proportion of the calculated contact points, which have a higher contact pressure pH than the pitting resistance for contact stress of 520 MPa, that is given for the used material in [25]. With the pressure distributions from Figure 4 and the proportion of contact points with critical contact pressure, the risk for fatigue wear and pitting could be estimated, which significantly decreases along the running-in process here. Fatigue wear is not only dependent on the contact stress but also on the number of load cycles. For that reason, operation time with critical contact stress and high risk for fatigue wear during running-in should be minimized. This could be achieved on the one hand with incrementing the load to the desired load level during running-in and/or with wear-promoting running-in conditions. In the given worm gear setting with an input speed n1 = 150 min-1, the hydrodynamic lubricant film formation is relatively weak. This leads to a higher solid body contact ratio and more intense abrasive wear than a running-in with a higher input speed. Simulations and gear tests with an input speed of n1 = 1,500 min-1 indicated a significant higher duration of the running-in. Wear-promoting conditions can also be achieved by using oil with a lower viscosity, which also influences the lubricant film formation.

4 Experimental setup

For the experiment of the running-in, worm gears with a center distance a = 32 mm and a gear ratio i = 39 were used. The profile flank form of the worm is a milled helicoid type K [24]. For the worms, steel 16MnCr5 (1.7131/AISI 5110) was used, and the gearing was finished with a grinding process. The worm wheels were milled from continuously casted bronze CuSn12Ni-GC (UNS C91700) with an oversized hob. For an input speed n1 = 1,500 min-1, the nominal output torque of the worm gear is 16 Nm. The maximum output torque is specified with 50 Nm. A polyglycol based oil (ISO VG 220) with practical relevance for worm gears was used for oil sump lubrication. The geometrical parameters as well as the test conditions of the validation test are presented in Table 1. The mean contact stress in Table 1 is calculated with ISO 14521 [25] and does not represent the initial contact stress of the gear test with incomplete contact pattern.

Table 1: Data of the test gear and test conditions.

The worm gear test rig with the concept of electrical wiring is shown schematically in Figure 7 together with the test worm gearbox with attached encoder on the worm wheel shaft. The tested worm gearbox (5) is driven at the worm shaft by a servo motor (1). The load on the tested worm gearbox is applied on the output shaft with a second servo motor (11) and a planetary gearbox (10), which is used for reduction of the torque. Various couplings are used for torque transmission (2), (4), (8). Torque and speed are measured at the input (3) and output (9) of the worm gearbox to determine the overall efficiency by evaluating the power losses in the gearbox and relating it to the input power. Two incremental encoders (6) attached to input and output shaft measure the rotation angles of the worm and worm wheel.

Figure 7: Worm gear test rig: schematic design and worm gearbox.

The contact pattern was adjusted by painting multiple wheel tooth flanks and meshing of the worm gear under low load conditions. In this way, the paint layer is removed in regions of contact and illustrates the actual contact pattern. The initial contact pattern after the assembly and alignment is shown on the right side in Figure 8. The idle contact pattern of the simulation, indicated with the distances between the tooth flanks from the tooth contact analysis, is shown in Figure 8 on the left side. The comparison of both shows a good correlation of the calculated and the measured contact pattern.

Figure 8: Comparisons of initial contact patterns from the simulation (left side; idle contact pattern with distances between flanks d) and from the gear test after the assembly (right side). In each figure, the entry side is on the left.

During the loaded test, the worm gear unit is driven under constant conditions for input speed n1 and output torque T2. At regular time intervals, wear is monitored by measuring the circumferential backlash under low load conditions and calculating the change to the initially measured backlash. Here, an efficient method was used to determine the backlash for numerous meshing positions of the gears over one revolution of the worm wheel allowing variations of wear across all teeth of the worm wheel to be monitored. Figure 9 shows measured data of the backlash increase relative to the initial backlash of the unworn worm gear from two different points in time of the validation test (see Section 5). The measurements over a full revolution of the worm wheel show a variation of the backlash increase with a periodic behavior, which is dominated by the mesh frequency. The variations within a single meshing period indicate varying abrasive wear along the contact path. The method is based on the single-flank-test for measuring the kinematic error of gears [26]. From the results of the change in backlash for all tested meshing positions, single wear heights for each tooth are calculated and averaged. The entire wear measurement and evaluation procedure is described in detail in [27]. Therefore, a mean wear height progression with respect to time is obtained, which allows the analysis of the running-in characteristics of the gears under the given operating conditions.

Figure 9: Results of the measurement of the backlash increase for two different time steps of the gear test.

5 Comparison of simulative and experimental results

The input for the wear simulation was configured according to the testing configuration in the experimental investigations on running-in to obtain a good comparability of experimental and simulative results. The only unspecified parameter is the wear-energy density, which is required to adjust the energetic wear model. Thus, the available experimental data of the wear progression was used to fit the simulative data with a suitable wear energy density.

Simulations of the running phase were conducted for different wear energy densities, and the resulting wear progression was compared to the measured data of the experiments. Therefore, the wear height was determined with the same method as in the experiments by calculating the backlash increase within the tooth contact analysis (see Section 2).

In this context, results for the unloaded kinematic error of the tooth contact analysis and wear measurement in the experiment are shown in Figure 10. Both represent, respectively, the kinematic error for the final geometry of the gears of the simulation and experiment. Due to an identical geometry of all worm wheel teeth in the simulation, the kinematic error is calculated only for a single meshing period in the tooth contact analysis. Contrary to that, the measurement was conducted for all meshing periods within one revolution of the worm wheel. A section of the measured course is depicted in the diagram on the right side of Figure 10. Since the measurement is not only influenced by the gear geometry, but also by other components such as bearings and shafts, the measured course includes various components with frequencies that are different to the meshing frequency. Moreover, signal variations from the encoders are included in the measured signal. Therefore, the data was filtered by cutting-off frequencies greater than four times the meshing frequency, by which a smoothed curve is obtained. The kinematic error from the calculation and from the filtered measurement data show comparable characteristics in terms of shape as well as of the peak-to- valley amplitude. As the kinematic error is influenced by the tooth geometry, this implies a comparable tooth geometry of the simulation and the experiment after running-in.

Figure 10: Calculated kinematic error from the tooth contact analysis for a single meshing period (left side) and a section of the measured kinematic error from the experiment representing two meshing periods (right side).

A comparison of the results of the wear progression is shown in Figure 11. The diagram includes two simulated wear progressions and the measured data points of the experiment. In the first section, the wear model with the lower wear energy density eR* = 1.1   1013 J/m3 approximates the measured wear progression well. At a load cycle number of approximately 3,000, the deviation to the experimental data increases significantly because of a different wear rate. According to the energetic wear model, see Equation 5, this deviation is either a result of a different frictional energy in simulation and experiment or caused by a change of the wear energy density during running-in. Assuming the second case, the wear-energy density must increase to obtain a lower wear rate and a better approximation of the experimental data. A reason  for this could be the decreasing mean flank pressure (see Figure 6), as in studies of BOLEY [21], a correlation of the mean pressure and wear-energy density was determined in block-on-ring tribometer tests for a comparable combination of material. Therein, the wear-energy density increases with a decreasing mean pressure, which is in good accordance with the observations made in here.

Figure 11: Comparison between simulative results with different model parameters eR* and wear measurement for a worm gear during running-in (a = 32 mm, i = 39, n1 = 150 min-1, T2 = 32 Nm). The black markers represent a computed wear progression with a gradually increasing model parameter.
Figure 12: Comparisons of final contact patterns from the simulation (left side) and from the gear test (right side). In each figure, the entry side is on the left.

Consequently, a wear simulation was carried out with a progressive characteristic of the wear energy density by fitting sectionally the model parameter to the experimental wear progression. The settings for the wear energy density in five load cycle intervals are given in Table 2. The resulting wear progression is represented in Figure 13 by black markers and approximates the entire experimental wear progression well. The maximum value of the parameter is eR* = 1.7 1014 J/m3, which is more than 10 times larger than the initial value eR* = 1.1 1013 J/m3. The most significant changes occur within a relatively short period of time in the first three intervals. Table 2 indicates only small changes of the mean flank pressure at the same time in the simulation. This leads to the conclusion that the change in wear energy density is, if at all, only partly due to a decreasing mean flank pressure.

Figure 13: Results of the gearbox efficiency with respect to the number of load cycles for the simulation and the experiment. Vertical lines represent the time period with significant changes of the wear energy density (see Table 2).
Figure 13: Results of the gearbox efficiency with respect to the number of load cycles for the simulation and the experiment. Vertical lines represent the time period with significant changes of the wear energy density (see Table 2).

Furthermore, the final contact patterns after from the simulation and from the experiment were compared with each other (Figure 12). Both refer to the last calculated and measured wear height (Figure 11), respectively, after approximately 20,000 load cycles or 86.6 hours of load operation. Regarding the location and the contour of the wear pattern, a good correlation of the calculated and the experimental results pattern can be determined. From Figure 11, it can be seen that the worm gear reached the steady-wear phase in the gear test. This implies a completed running-in of the worm gear for the specified load, after which this specified load can be applied to the worm gearbox. This is assuming the operating conditions also meet other requirements for the load capacity, as steady-wear is not necessarily associated with a completed contact pattern. A significant increase in load leads to another phase of load adaption of the gears with an additional running-in.

In addition to wear and friction in tooth contact, power losses of bearings, seals, and the churning losses were calculated in the simulation as described in Section 2.1. The settings for the bearings, seals, and lubrication were the same as in the experiment to obtain comparable conditions. For all load-independent losses, these conditions remain unchanged with wear of the gears as they are only dependent on the speed and the lubricant properties. Degradation of the lubricant with wear particles was not considered here. Since the distribution of the normal and friction force in the tooth contact changes with wear, the load-dependent losses in the bearing are affected by that. However, according to the calculated results, these changes are negligible compared to those in the tooth contact. In Figure 6, the effect of wear on the frictional energy was already indicated. Corresponding results of the overall gearbox efficiency to the wear simulation with a progressive model parameter and the experiment from Figure 11 are shown in Figure 13. The reason for the unsteady progression of the experiment is the discontinuous operation of the gearbox due to the regular wear measurement. As a result, at the beginning of every load test, the whole system requires time to reach a steady-state operation, for example, regarding temperature. Though the measured data indicate the gearbox efficiency increases within the running-in of the gears, a decomposition of the overall losses into its individual components is not possible. Nevertheless, it can be assumed with good approximation this increase is due to a reduction in tooth friction.

The same characteristic can be observed for the simulation. Regarding the efficiency at the beginning and at the end of the running-in, both methods give comparable results. Contrary to the simulative progression, the efficiency in the experiment remains on an almost constant level for approximately 3,000 load cycles. After that, the efficiency rapidly increases — comparable to the simulative progression — before reaching an almost constant efficiency level.

This delayed increase of the efficiency cannot be represented by the tribological simulation as the tooth contact and tooth friction change continuously with an increasing contact pattern from the very beginning. A clear reason for the deviations at the beginning and the delayed increase cannot be given, because many components of the gearbox have an influence here. Moreover, the evolution of the microstructure as wells as the contact pattern was not continuously measured and could vary from the conditions in the simulation. Even though there is a growing deviation in the efficiency progression within the first interval of Table 2, the wear progressions of simulation (eR* = 1.1 J/m3) and experiment are in good accordance. This leads to the assumption that the difference in boundary friction energy is relatively small. Otherwise, a constant wear-energy density would have led to higher deviations already within the first interval.

Analyzing the experimental efficiency progression in Figure 13 together with the intervals for the progressive model parameter indicates a correlation of the changes in efficiency and wear-energy density. Within the time period with the most significant changes of the efficiency, the wear-energy density substantially changes as well. For illustrative purposes, two lines representing the limits of this time period according to Table 2 are added in Figure 13. This implies major changes of the tribological conditions in the tooth contact that lead to a reduction of tooth friction and to an increase of the wear-energy density.

6 Conclusion and outlook

In this article, a simulation tool for studies on the behavior of the two interacting tribological measures friction and wear during running-in of worm gears was presented. Wear is calculated based on frictional energy from solid body contact by using the energetic wear model of FLEISCHER. The iterative simulation procedure allows a transient analysis of local friction and wear by considering the change of the tooth flank geometry in the following calculation step.

Results for a worm gear with a relatively small initial contact pattern were used to describe the characteristic effects on the tooth contact within the simulation of the running-in phase. A main effect is the expansion of the contact pattern, which, on the one hand, leads to a reduction of the mean pressure load. On the other hand, additional contact points are included in the tooth contact, which change the distribution of parameters as the lubricant film height and influences, thereby tooth friction. Together with the effect of load being transferred to contact points with a lower frictional energy load due to the energetic wear model, a reduction of the total tooth friction energy was observed for the analyzed worm gear.

Based on experimental data for the wear progression during running-in with identical test conditions, the wear energy density was determined. The results indicate the wear-energy density changes, since a progressive wear energy density leads to the best approximation. Moreover, the final geometry from the simulation gives results for the kinematic error, which are comparable to those of the measured data. As wear was here only measured by integral values (circumferential backlash), a local measurement of wear for validation of the local wear calculation in the simulation will be an objective for future work.

Regarding the overall efficiency of the gearbox, the setting with a progressive model parameter resulted in a good accordance at the beginning and at the end of the running-in test. Although the simulation and experiment indicate a significant and comparable increase in efficiency, the behavior of the physical test with a delayed increase cannot be reproduced in the simulation model.

As a conclusion, more effects apart from a continuous change of the macro geometry need to be considered for a reliable calculation of wear and friction during the running-in phase. This includes, in particular, a description of the effects on the micro geometry of the tooth flank surfaces, as a change of the microstructure is currently considered with solid body contact ratio equations related to measured surfaces from discrete points in time. Since the significant changes of the tribological conditions during running-in also affect the wear-energy density, a thorough analysis and description of the influences on the wear energy density is required. A precise description of the wear-energy density is particularly important if a wear simulation is to be carried out without additional experimental investigations.


This work was supported by the German Federal Ministry of Economics and Energy (IGF 19699 N) within the framework of the Forschungsvereinigung Antriebstechnik e. V. (FVA project 503 III). For the evaluation of the measured surface data the software Digital Surf – MountainsMap was used. 


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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 November 2021 at the AGMA Fall Technical Meeting. 21FTM02