Local Calculation of Load-Dependent Gear Power Losses of Cylindrical Gears

The main design criteria of gearboxes are (i) power density and load capacity; (ii) noise, vibration, and harshness; and (iii) efficiency and heat balance. Ever since it has been a goal to meet these criteria, but due to the severe need to reduce worldwide CO₂ emissions, optimizing efficiency and prolonging the use phase is more relevant than ever [1–3]. A key influencing factor on power density, load capacity, and efficiency is the temperature of the gearbox and its machine elements, including the lubricant. The temperature distribution resulting in operation is determined by the gearbox power losses and its dissipation. For efficient, economical, and sustainable gearbox design, accurate calculation models for usage in the design phase of gearboxes are essential.

Figure 1 illustrates the tooth contact of a cylindrical gear and relevant contact parameters in a simplified representation to calculate the load-dependent gear power losses P_LGP.

The integral with the line loads f_N and the sliding speeds v_s can be solved analytically or numerically. By assuming a simplified load distribution along the path of contact without considering the tooth width b, analytical solutions of the integral in Eq. (4) result in the geometric tooth power loss factor H_V [7–9]. This simple analytical calculation model considers the nominal gear macro-geometry. In practice, gears feature a specific tooth stiffness and have manufacturing deviations. They usually feature profile form modifications customized for their particular use case to ensure optimal meshing under load by smoothening the occurring contact pressures to optimize the load capacity and prevent excitations during operation to reduce noise, vibration, and harshness. When a accurate calculation is demanded, the integral in Eq. (4) can be solved numerically, according to Wimmer [8]. The solution forms the so-called local-geometric tooth power loss factor H_VL, derived from a loaded tooth contact analysis (LTCA), e.g., in RIKOR [10–13].

As for the mean coefficient of friction µ_mz, calculation models are considered mesh-averaged, usually represented by simple analytical equations with few necessary input data, which are generally applicable and usually empirical [14–18]. Since the mean coefficient of friction µ_mz is considered a constant value over the tooth contact area, the easy-to-implement calculation models usually refer to a characteristic point within the path of contact, often the pitch point C. Therefore, experimentally derived mesh-averaged calculation models for the mean coefficient of friction µ_mz implicitly take into account the meshing conditions of the underlying test gears. An increasing deviation of the gear geometry from that of the test gears leads to an error in the application of the calculation models. Latest technologies to reduce load-dependent gear power losses, like tribological coatings [19], reduced flank surface roughness [20], surface texturing [21], or lubricants with a low shear resistance [22] can only be considered to a certain extent by mesh-averaged calculation models.

Besides mesh-averaged calculation models for a mean coefficient of friction µ_mz, mesh-local calculation models exist, considering local contact parameters to calculate a coefficient of friction µ. As shown in Fig. 2, a further distinction for mesh-local calculation models can be made into models that average the tooth contact and calculation models that resolve the tooth contact using tribo-simulations.

Contact conditions can be considered mesh-locally and contact-locally using an LTCA combined with transient tribo-simulations. For elastohydrodynamic lubrication (EHL) models, the contact-local mechanisms within the highly loaded lubricated tooth contact can be extensively examined, considering back-coupling of hydrodynamics and elastic surface deformation [23]. The level of detail is very high, which can build a deep understanding of the mechanisms and support improving friction models. Still, the application requires many input parameters, boundary conditions, and a higher computing time. This approach is used particularly in the scientific and research fields [24–26] for investigating selected operating conditions. Farrenkopf et al. [24] used a transient TEHL (thermo-elastohydrodynamic lubrication) contact simulation to mesh-locally and contact-locally calculate the load-dependent gear power loss P_LGP. Calculation effort can be massively reduced by considering the tooth contact with contact-averaged calculation models. As contact-averaged calculation models are mesh-local per se, local contact conditions such as pressure, speeds, curvatures, and profile changes can still be considered. Even the influence of surface coatings can be taken into account. Contact-average calculations can be derived from tribo-simulation [27] but are often based on experimental tribometer testing [28–34].

In summary, different calculation models are available with varying detail and effort depending on the underlying goal and need for calculating the load-dependent gear power losses P_LGP. So are calculation models of a mean coefficient of friction µ_mz suitable for calculations in an early gear design phase, for map calculations or gearbox system analysis, where little input data are available or an efficient calculation is a priority. Calculation models for the mean coefficient of friction µ_mz are extensively examined and, therefore, well established. On the other hand, highly sophisticated mesh-local and contact-local calculation models using tribo-simulation models with extraordinary detail exist to study the rolling-sliding contact in gears and increase the understanding of tribological mechanisms but demand high computational effort and a high amount of modeling parameters. As an efficient middle way, mesh-local, contact-averaged calculation models are suitable. Still, they are rarely employed to calculate load-dependent gear power losses P_LGP [28,35], mainly because there is no comprehensive validation and comparison of the calculation models with established calculation models and measurement results. This publication addresses this gap by systematically comparing state-of-the-art mesh-averaged calculation models with the selected mesh-local, contact-averaged calculation models and validating results with measurements from highly loaded lubricated cylindrical steel gears. A statistical evaluation is conducted to provide a clear assessment.

This section describes the materials and methods used. First, the objects of investigation are introduced in Sec. 2.1. Second, the calculation models to determine the load-dependent gear power losses P_LGP are described in Sec. 2.2.

This study uses measurements from a twin-disk tribometer (Sec. 2.1.1) to evaluate calculation-averaged friction models for the local coefficient of friction. Different gear geometries (Sec. 2.1.2) are considered when comparing load-dependent gear power losses using contact-averaged and mesh-averaged calculation models with measurements. Different gear oils (Sec. 2.1.4) and operating conditions (Sec. 2.1.5) are considered.

The FZG twin-disk tribometer pairs two identical cylindrical disks with a diameter of d_1,2 = 80 mm and a raceway width of b = 5 mm, made of case-hardened steel 16MnCr5 (Sec. 2.1.3). The geometrical properties and relevant parameters of the considered disk contact are shown in Fig. 3.

Mayer [32] conducted extensive tests on an FZG twin-disk tribometer and measured coefficients of friction for different gear oils and operating conditions by varying the normal load F_N, rotational speeds of both disks ω_1,2, and the oil injection temperature ϑ_oil. The resulting coefficients of friction μ and disk bulk temperatures ϑ_M are used as this study's experimental basis to verify and evaluate selected calculation models for the coefficient of friction µ (Sec. 3.1). The gear oils and operating conditions considered are listed in Secs. 2.1.4 and 2.1.5.

Polished and ground disks are considered to evaluate the friction model for fluid film lubrication and mixed lubrication. The polished disks relate to an arithmetical mean roughness of Ra_1,2 = 0.03 μm, a root mean square roughness of Rq_1,2 = 0.04 μm, and a mean roughness depth of Rz_1,2 = 0.27 μm. The ground disks relate to a mean roughness of Ra_1,2 = 0.27 μm, a root mean square roughness of Rq_1,2 = 0.35 μm, and a mean roughness depth of Rz_1,2 = 2.08 μm. The roughness values were obtained by Mayer [32] from tactile measurements in the circumferential direction of the disc raceway with a measurement length of 4 mm and a cutoff wavelength of 0.8 mm.

This study considers different gear geometries from Hinterstoißer [36] to evaluate the application of mesh-local calculation models on gears and compare them to the established mesh-averaged calculation models using a mean coefficient of friction µ_mz (Sec. 3.2). The selection of gears includes the test gear of type C_mod, a high contact ratio (HCR) gear, a moderate low-loss (mLL) gear, and an extreme low-loss (eLL) gear. The C_mod gear has been used in numerous research projects, and comprehensive measurement results are available for the load-dependent power loss P_LGP. It represents a balanced gear geometry with a transverse contact ratio of ε_α = 1.44. The HCR gear is based on a practical gear and was scaled by Hinterstoißer [36] to a center distance of a = 91.5 mm while maintaining relevant gear parameters that influence the geometric tooth power loss factor H_V, such as the transverse contact ratio of ε_α = 2.10 or the helix angle of β = 31.5 deg. Hinterstoißer [36] derived the mLL and eLL gears from the HCR gear, focusing on reducing the geometric tooth power loss factor H_V and, therefore, they particularly feature a smaller transverse contact ratio ε_α of around 1 for the mLL gear and significantly below 1 for the eLL gear.

Hinterstoißer [36] conducted extensive tests on the FZG efficiency test rig using the named gears and measured load-dependent gear power losses P_LGP for various gear oils and surface roughnesses. The considered surface roughnesses for the gears are according to measurements of Hinterstoißer [36] and relate to an arithmetical mean roughness of Ra_1,2 = 0.19 μm, a root mean square roughness of Rq_1,2 = 0.24 μm, and a mean roughness depth of Rz_1,2 = 0.87 μm. The roughness values were obtained by Hinterstoißer [36] from tactile measurements along the profile line in the center of the tooth with a measurement length of 4.8 mm and a cutoff wavelength of 0.8 mm. Additionally, an internal gear (IG) from Schudy [37] is considered in a case study (Sec. 3.3).

The detailed gears' geometry, including profile modifications, is listed in Table 1.

The considered material of the disks and gears is 16MnCr5 case-hardened steel with an elastic modulus of E = 210000 N/mm², a Poisson's ratio of ν = 0.3, a specific thermal capacity of c_p = 465000 J/(kg·K), a density of ρ = 7850 kg/m³, and a thermal conductivity of λ = 35 W/(m·K) according to Mayer [32].

According to Habchi and Bair [38], for case-hardened steels, a noticeably lower value of the thermal conductivity of λ = 21 W/(m·K) is present at the surface compared to the inner material. This is based on findings of previous studies [38–40] that show that the thermal conductivity depends on the alloy, the thermal treatment, and the hardness of the material. Nevertheless, the material properties of Mayer [32] are used for consistency since they were used when he originally derived his calculation model containing material-dependent model parameters.

Three different gear oils with a common kinematic viscosity of approximately 10 mm²/s at 100 °C are considered in this study: a mineral oil (MIN10), a polyalphaolefin (PAO10), and a polyether (PE10). Table 2 shows their viscosity and density data. The kinematic viscosities ν at 40 °C and 100 °C, viscosity index (VI), and density ρ at 15 °C for MIN10, PAO10, and PE10 are from Mayer [32]. The pressure–viscosity coefficients α_p at 40 °C and 100 °C for MIN10 and PAO10 are based on the study by Gold et al. [41]. The α_p values of PE10 are calculated accordingly, assuming a PG base oil. Additionally, the lubricant factor X_L, according to Schlenk, is listed per oil. Specific oil data depending on the single calculation models are listed in Sec. 2.2.

The measurements by Mayer [32] on the FZG twin-disk tribometer include two discrete Hertzian pressures of p_H = {600; 1200} N/mm² and two oil injection temperatures of ϑ_oil = {40; 100} °C for each oil and polished disks. For all combinations of temperature and pressure, Mayer [32] varied the rotational speeds ω_1,2 of the disks to obtain friction curves with slide-to-roll ratios (SRRs) of {0; 0.02; 0.05; 0.11; 0.16; 0.22; 0.35; 0.50; 0.67} for sum velocities v_Σ of {1; 2; 4; 8; 16} m/s. Thus, a single set of measurements per oil covers 172 operating conditions, mainly including the conditions in the tooth contact of gears. For ground disks, Mayer [32] investigated the same operating conditions for the mineral oil MIN10. The calculations in Sec. 3.1 are based on these operating conditions.

Hinterstoißer [36] examined the C_mod gear based on the test procedure of FVA 345 [42], adding additional speed levels. For an oil temperature of ϑ_oil = 90 °C, this includes the pinion input torques of T_1,Cmod = {94; 183; 302} N·m and pinion input speeds of n_1,Cmod = {130; 261; 522; 1305; 2166; 3914; 5218} min⁻¹. This results in circumferential speeds of v_t,C = {0.5; 1; 2; 5; 8.3; 15; 20} m/s. For the pinion input torque of T_1,Cmod = 183.4 N·m, different oil temperatures of ϑ_oil = {40; 60; 90; 120} °C are investigated. For the low oil temperature of ϑ_oil = 40 °C, the high pinion input speeds of n_1,Cmod = {3914; 5218} min⁻¹ are excluded.

Following this scheme, Hinterstoißer transferred the operating conditions for equivalent Hertzian pressures p_H,C, and circumferential speeds v_t,C to the HCR gear. By adding one additional input torque level, this results in pinion input torques of T₁ = {107; 208; 343; 423} N·m and pinion input speeds of n₁ = {128; 249; 498; 1224; 2024; 3698; 4909} min⁻¹. Due to similar gear ratios, Hinterstoißer retained the pinion input torques T₁ and pinion input speeds n₁ of the HCR gear for the mLL and eLL gears. Table 3 summarizes the different pinion input torques T₁ with their corresponding Hertzian pressures p_H,C, and local-geometric tooth power loss factors H_VL.

For the case study of the internal gear, operating conditions according to the test procedure of FVA 345 [42] with equivalent Hertzian pressures p_H,C, and circumferential speeds v_t,C of the internal gear (IG) to a C_mod gear are considered.

The calculation study includes two objects of investigation: disks (Sec. 2.1.1) and gears (Sec. 2.1.2). The calculation study regarding the disks focuses on verifying and evaluating different contact-averaged friction models, which are then used to calculate load-dependent gear power losses P_LGP. Considering the calculation of load-dependent gear power losses P_LGP, different calculation levels I, II, and III are defined based on the used calculation model for the tooth power loss factor (H_V or H_VL) and the friction model for the coefficient of friction µ.

Table 4 summarizes the defined calculation levels I, II, and III and the used values for the coefficient of friction μ and the geometric tooth power loss factors H_V and H_VL.

This study uses selected mesh-averaged friction models to determine the mean coefficient of friction µ_mz (Sec. 2.2.1) for the calculation levels I and II and selected contact-averaged friction models (Sec. 2.2.2) to determine the mesh-local coefficient of friction µ.

The parameters µ_f,R, α_f, β_f, γ_f, µ_s,R, α_s, and β_s are tribosystem specific, i.e., specific to a particular oil and a tooth flank surface. They can be derived from a gear power loss test of the type FZG-E-C/0.5:20/5:9/40:120 according to FVA 345 [42] under consideration of either the tooth power loss factor H_V or the local-geometric tooth power loss factor H_VL. The choice of the tooth power loss factor influences the derived parameters. The parameters used in this study for Doleschel’s [18] model are listed in the Appendix (Table 12).

The lubricant factor X_L depends on the oil and can be taken from Table 2, according to Schlenk [15].

The coefficient of friction µ is calculated using mesh-local, contact-averaged friction models such as the directly solvable regression models according to Xu et al. [27] and Klein [30], and the iteratively solvable models relying on a rheological model according to Mayer [32] and Arana et al. [28].

Xu et al. [27] developed an analytical regression model for the coefficient of friction μ based on a multiple linear regression of ca. 10000 numerical tribo-simulations from the combination of typical tooth contact parameter ranges in gears such as temperature ϑ, radii of curvature R, velocity v, surface roughness Rq, or Hertzian pressure p_H. Xu et al. [27] derived regression parameters for gear oil comparable to the MIN10 used in this study.

Klein [30] developed an analytical regression model for the coefficient of friction μ, based on measurements of the coefficient of friction μ on an FZG twin-disk tribometer varying the pressure p, velocity v, temperature ϑ, and surface roughness Rz. Within his investigations, he additionally varied the grinding structure of the surface and considered it by a roughness structure factor in his calculation model. Klein [30] derived his model parameters by investigating the gear oils MIN10 and an ISO VG 100 polyalphaolefine.

In contrast to the directly solvable regression models by Xu et al. [27] and Klein [30], the semi-analytical calculation models of Mayer [32] and Arana et al. [28] use rheological models that describe the contact-averaged shear stress τ in relation to the shear rate $γ˙$ and viscosity η in highly loaded EHL contacts. Relying on such a rheological model requires an iterative solution since the contact viscosity η is temperature dependent, and the temperature T depends on the coefficient of friction μ and vice versa. Figure 4 shows a general flowchart for the iterative calculation of the mesh-local, contact-averaged coefficient of friction µ used in the calculation models of Mayer [32] and Arana et al. [28]. For each iteration i, the calculated contact temperature T_i is compared with the contact temperature of the last iteration T_i−1. The solution converges when the difference falls below a defined threshold of X = 0.001 K.

Since a limiting shear stress τ_lim less than or equal to zero is physically meaningless, the frequently used value of a minimum limiting shear stress τ_lim,0 = 5 N/mm² is used. The contact temperature T is calculated according to the established temperature model by Blok [54]. Regarding mixed friction, Mayer [32] used the model according to Doleschel [18] (Eqs. (8)–(10)). The model parameters used for the Mayer’s [32] model are listed in the Appendix (Table 13).

Another comprehensive calculation model was proposed by Arana et al. [28]. Besides Yilmaz et al. [22], they used the Ree–Eyring rheological model, which relies on a critical shear stress τ_c. Their model also limits the calculated shear stress to a pressure-dependent limiting shear stress τ_lim. They determined the critical shear stress τ_c and limiting shear stress τ_lim from an oil-specific and pressure-dependent limiting-stress coefficient Λ. Since the Mayer limiting shear stress model has a more specific description of the limiting-stress τ_lim for the oils under consideration (Sec. 2.1.4), the limiting shear stress model from Mayer [32] was also used in the calculations with Arana et al.’s [28] model. As with implementing Mayer's [32] model, a minimum limiting shear stress τ_lim,0 = 5 N/mm² was considered. Arana et al. [28] initially calculated a mean contact inlet temperature according to Olver [55] to calculate the film thickness and the mean contact temperature by adding an average flash temperature rise and internal heating of the oil film. According to Olver [30], the thermal model was derived using a particular “mini-disc” machine and considers, among other parameters, transient thermal resistances, which must be found from a thermal model of the underlying system. It is unclear how exactly Arana et al. used Olver's [55] model in their application on gears. Therefore, within the implementation of the model of Arana et al. [28] in this study, the temperature model by Blok [54], the same as in Mayer [32], was used to calculate the contact temperature. Since Arana et al. used the same calculation models as Mayer [32] to calculate the pressure–viscosity according to Barus [56] using the modulus equation [57] and temperature–viscosity according to Vogel et al. [58–60], the regression parameters according to Mayer [32] were used. Regarding mixed friction, Arana et al. [28] used a modified mixed friction model based on the study by Diab et al. [44]. The model parameters used for Arana et al.’s [28] model are listed in the Appendix (Table 14).

Table 5 gives a summary of the considered calculation models and their parameter-specific submodels.

Most calculation models of the coefficient of friction under mixed friction documented in the literature consider the solid coefficient of friction µ_s as a constant, which is either assumed or derived from measurements. Only partly, the solid coefficient of friction is considered variable depending on the actual operating condition [18,66]. Literature does not offer corresponding established calculation models for the solid coefficient of friction.

Figure 5 illustrates the procedure in a flowchart. After evaluating different calculation models, the following model from the study by Doleschel [18] was found to provide good results:

A Hertzian pressure of p_H,R = 1000 N/mm², and a sum velocity of v_Σ,R = 8 m/s was used as reference values for the de-dimensionalization. Furthermore, only values with ξ < 0.5 and μ_s < 0.12 were considered for the regression. These data preprocessing effectively excludes outliers and extreme values, leading to a more robust regression.

Since the derived solid coefficient of friction µ_s depends on the underlying model for the fluid load portion ξ, Eq. (18) is not generalizable and has to be parameterized in the context of the later used friction model. Table 6 summarizes values for the calculation models of Mayer [32] and Arana et al. [28].

For the calculation of the disk contact in Sec. 3.1, the friction models were compared with measurements [31], and the corresponding measured disk bulk temperature was used. When applying the friction models to gears in Secs. 3.2 and 3.3, measurements of bulk temperatures are not available and, therefore, have to be calculated.

The parameter D depends on the rotational direction. A value of D = 0.75 is used in this study according to the considered rotational direction.

For the statistical evaluation of the calculation data against measurements, the coefficient of determination R², the root mean square error (RMSE), and the normalized root mean square error (NRMSE) are used.

This section refers to applying friction models to the twin-disk contact (Sec. 2.1.1) and comparing calculated coefficients of friction µ_calc with measured coefficients of friction µ_exp for different gear oils (Sec. 2.1.4) under fluid film and mixed lubrication. Selected friction models are then used to calculate the load-dependent gear power losses P_LGP,calc according to the calculation levels I, II, and III (Sec. 2.2) for different gear geometries (Sec. 2.1.2) and gear oils (Sec. 2.1.4). The results are compared with measured load-dependent gear power losses P_LGP,exp for a comprehensive evaluation. Finally, a case study considering an internal gear geometry is conducted.

Figure 6 qualitatively compares calculated friction curves using Mayer’s [32] model with measurements for MIN10 and polished disks (Ra = 0.03 µm) across all operating conditions by Mayer [32], showing good agreement for most conditions.

Figure 7 shows the corresponding parity plot and the residual plot by plotting the calculated coefficients of friction in a scatter plot against the measured values. In the parity plot, the matching results from calculation and measurement are located at the angle bisector. The dashed lines represent a deviation of −10% and +10%. The calculation results shown are for full film lubrication (λ_rel ≥ 2), using Mayer’s [32] model of the gear oils MIN10, PAO10, and PE10. Based on the comparison, a statistical evaluation of the contact-averaged friction models has been performed. Table 7 summarizes the results for all considered contact-averaged friction models and oils. Since the contact-averaged friction models of Mayer [32] and Arana et al. [28] are strongly determined by the limiting shear stress τ_lim of the oil, it is assumed that a good formulation for their calculation plays a decisive role in a good description of the coefficient of friction µ.

Overall, better statistical values are observed for the mineral oil MIN10 than for synthetic oils PAO10 and PE10. This trend agrees with the findings of Zander et al. [71], who investigated different calculation models for bearing power losses. Regarding the different calculation models, the semi-analytical models of Mayer [32] and Arana et al. [28] show better agreement with the measurements compared to Xu et al. [27] and Klein [30].

Figure 8 shows the derived solid coefficient of friction µ_s,exp from measurements for MIN10 according to Sec. 2.2.3 against a constant value and a calculated prediction µ_s,calc according to Eq. (18) and Table 6. By describing the solid coefficient of friction µ_s as a function of Hertzian pressure p_H and sum velocity v_Σ according to Sec. 2.2.3, a significantly better agreement with measurements can be achieved, as confirmed by the statistical evaluation in Table 8. A statistical evaluation has been performed only concerning the friction models of Mayer [32] and Arana et al. [28], since Xu et al. [27] and Klein [30] do not explicitly consider a solid coefficient of friction µ_s.

Figure 9 compares the results of the coefficient of friction µ for polished disks (unfilled marker), which represent operating points mainly under fluid film lubrication, with the coefficients of friction µ for ground disks (filled marker), which represent operating conditions mainly under boundary and mixed lubrication. The values for the relative film thickness, calculated according to Mayer [32], are presented in Fig. 10 to facilitate a more accurate categorization of the lubrication regime. For the polished disks, 91% of the operating points exhibit a relative film thickness of λ_rel ≥ 2, indicating that they operate under fluid film lubrication. In contrast, for the rough discs, 83% of the operating points show a relative film thickness of λ_rel < 2, suggesting they operate under mixed and boundary lubrication.

In general, the quality of the models for mixed lubrication in Table 9 is consistent with the results for full film lubrication from Table 7. The semi-analytical models proposed by Mayer [32] and Arana et al. [28] demonstrate a superior degree of alignment with the experimental data compared to the models by Xu et al. [27] and Klein [30] for the mixed friction.

The load-dependent gear power losses P_LGP are calculated using the different calculation levels I, II, and III according to Eqs. (5)–(7) from Sec. 2.2. The measured values are determined using Eq. (6) by considering H_VL and measured mean coefficients of friction µ_mz,exp from Hinterstoißer [36].

The surface power density ψ, corresponding to the product of the integral's local values in Eq. (7), can be visualized using a level III model. Figure 11 shows the surface power density ψ for the mLL gear using the mesh-local, contact-averaged Arana et al.’s [28] model. The considered operating condition with MIN10 refers to an input power of P_In = 44.4 kW with a moderate load of T₁ = 208 N·m and rotational speed of n₁ = 2038 min⁻¹, resulting in a Hertzian pressure of p_H,C = 922 N/mm² and a circumferential speed of v_t,C = 8.3 m/s. The calculated bulk temperature is ϑ_M = 93.2 °C for an oil temperature of ϑ_Oil = 90 °C, and the load-dependent gear power loss is P_LGP.III = 155 W.

The individual slices in Fig. 11 with dot-marked calculation points represent the individual contact lines resulting from 12 individual mesh positions and 50 calculation points along the tooth width. The letters A to E mark the first point of tooth contact on the path of contact (A) and the last point of tooth contact on the path of contact (E), the pitch point (C), the lowest point of single-tooth contact (B), and the highest point of single-tooth contact (D).

The distribution of the surface power density ψ along the path of contact is zero at C and highest at A end E. This is due to the sliding speed v_s, which is also zero at C and linearly increases toward A and E. The pronounced surface power density ψ in the middle of the tooth width b results from the present lead crowning C_β of the considered mLL gear. The lead crowning C_β leads to a decreasing line load f_N toward the edges of the tooth width. The sliding speed v_s and the line load f_N additionally influence the coefficient of friction µ, which, in turn, influences the surface power density ψ in the tooth contact area. Figure 11 shows the potential of level III models to reflect and analyze profile form and flank form modifications and local contact conditions in the calculation of load-dependent gear power losses P_LGP.

Figure 12 compares the calculated load-dependent gear power losses P_LGP,calc using calculation levels I, II, and III with measured values P_LGP,exp [36] for the C_mod gear and MIN10, PAO10, and PE10 (Sec. 2.1.4) analogous to Fig. 7. The statistical evaluation of Fig. 12 is summarized in Table 10. In the statistical evaluation, the calculation model of Schlenk [15] was combined with the local-geometric tooth power loss factor H_VL as another calculation level II model in addition to the calculation level I variant. The Doleschel regression parameters were derived from Hinterstoißer [36] using the local-geometric tooth power loss factor H_VL. Therefore, this study considers the Doleschel model in calculation level II. The following is observed:

• When comparing the results of the Schlenk [41] model for the calculation levels I and II, both the R² and the NRMSE for calculation level II have better values for all oils considered. This is because the local-geometric tooth power loss factor H_VL considers local influences in level II, increasing the calculation quality.

• The overall best agreement is found for Doleschel’s [18] model using calculation level II. This result is reasonable since this model is a tribosystem-specific regression fit to the underlying measured values.

• Comparing the results for the different oils, overall better statistical values are observed for the mineral oil MIN10 compared to the synthetic oils PAO10 and PE10, similar to the findings in Sec. 3.1. This trend can be observed for calculation levels I and II as well as calculation level III.

• When only considering the synthetic oil PE10, it can be observed that the deviations for calculation level III are higher than for calculation levels I and II. This deviation is due to the quality of the input data for calculation level III, which can only deliver results as good as its input data.

Figure 13 compares the calculated load-dependent gear power losses P_LGP using calculation levels I, II, and III with measured values [36] for the HCR, mLL, and eLL gears (Sec. 2.1.2) and MIN10. The overall results of the statistical evaluation in Table 11 show that a good agreement between calculation and measurements for MIN10 is also found when considering different gear geometries:

• For the HCR gear, the calculation is in excellent agreement with the measurement for all calculation levels, with the best agreement for the calculation level III.

• Again, when comparing the results of Schlenk’s [41] model for the calculation levels I and II, it can be observed that level II shows better values. This is due to the consideration of the actual mesh-local contact parameters by the local-geometric tooth power loss factor H_VL.

• A high degree of agreement can be observed for calculation level III for all gear geometries. This proves the applicability of calculation level III to practical gears.

Overall, the results of this section show the capability of calculation level III with mesh-local, contact-averaged friction models to reliably calculate load-dependent gear power losses P_LGP for a wide range of operating conditions. This is particularly the case for the oil MIN10. For the synthetic oil PE10, the uncertainties in the calculation are more significant at all calculation levels and friction models. The potential of calculation level III can be seen in their application to different practical gear geometries. In the case of the practical gear geometry HCR, the best agreement can even be found for calculation level III using Mayer’s [32] and Arana et al.’s [28] models. In this case, the deviation in local contact parameters compared to a test gear geometry can be considered here, confirming that calculation level III can provide accurate results for a specific gear geometry.

A particular gear geometry is present for internal gears, where local contact parameters significantly differ from external gears, and calculation level III is expected to provide more accurate results.

It has been shown that the calculation level III with mesh-local, contact-averaged friction models is well applicable to calculating the load-dependent gear power loss P_LGP. They provide accurate calculation results when applied to practical gears, which can strongly differ from the characteristics of test gears used to derive mesh-averaged friction models. When considering internal gears, for instance, their contact parameters along the path of contact significantly differ from those of external gears. Figure 14 shows the contact velocities v_1,2 and sum velocities v_Σ, the slide-to-roll ratio (SRR), and the radii of curvature R_1,2 and R_red along the path of contact g_α for the test gear of type C_mod and an internal gear IG for an exemplary operating condition with a circumferential speed at the pitch point of v_t,C = 2 m/s. For comparability, the planet's gear geometry of the IG gear is identical to the gear geometry of the C_mod pinion (Sec. 2.1.2, Table 1).

When comparing the contact velocities v, the gradient of the sum velocity v_Σ for the IG is higher than for the C_mod gear. Regarding the SRR, the IG shows significantly lower values along the path of contact. Another fundamental difference is the trend of the reduced radius of curvature R_red in the tooth contact. Since internal gears exhibit a conformal tooth contact rather than a convex tooth contact, as for external gears, the reduced radius of curvature R_red increases by the square along the path of contact. This increase leads to significantly higher values of the reduced radius of curvature R_red for internal gears than external gears, resulting in more favorable friction characteristics, as Schmid et al. have shown in detail in Ref. [72].

Based on Tables 10 and 11 in Sec. 3.2, one friction model of calculation level II and one friction model of calculation III is chosen to calculate the load-dependent gear power losses P_LGP,II and P_LGP,III of the IG. Equivalent operating conditions to FVA 345 regarding the Hertzian pressure p_H and circumferential speed at the pitch point v_t,C (Sec. 2.1.5) are considered. In Fig. 15, the calculated load-dependent gear power loss P_LGP,II using the mesh-averaged friction model of Doleschel [18] is used as a reference against which the calculated load-dependent gear power losses P_LGP,III using the mesh-local, contact-averaged friction model of Arana et al. [28] are compared to. The NRMSE of C_mod gear for MIN10 from Table 10 is considered as a measure of uncertainty and visibly displayed as a colored scatter band. In the absence of a calculation model specifically for the bulk temperature of internal gears, the Oster [67] model (Sec. 2.2.5) was used to calculate the bulk temperatures of the IG. The evaluation of all considered operating conditions shows that, on average, the calculated load-dependent gear power losses P_LGP,III with the calculation level III using the mesh-local, contact-averaged friction model of Arana et al. [28] are 30% lower than the calculated load-dependent gear power losses P_LGP,II of level II using the mesh-averaged friction model of Doleschel [18]. This influence is solely due to a lower calculated coefficient of friction µ, which results from the more favorable friction conditions of the IG. It has to be proven by experimental investigations that the current level II calculation models do not sufficiently take into account the mesh-local conditions of internal gears and therefore calculate load-dependent gear power losses conservatively.

This study investigated mesh-averaged and mesh-local friction models classified into calculation levels I to III to calculate the load-dependent gear power loss P_LGP. While mesh-averaged friction models, used in calculation levels I and II, implicitly assume a gear geometry, mesh-local friction models consider the actual contact parameters. Different local friction models were compared with twin-disk tribometer measurements, considering a highly loaded EHL contact for various oils. These local friction models were applied in calculation level III to determine the load-dependent gear power losses P_LGP,III for various oils and gear geometries. Calculation results were compared to measured load-dependent gear power losses P_LGP,exp for a wide range of operating conditions. Based on the results, the following conclusions are made:

1. The calculation results of coefficients of friction for the disk contact agree well with measurements for fluid film and mixed lubrication for Arana et al.’s [28] and Mayer’s [32] model.

2. When the mesh-local friction models, derived from twin-disk tribometer tests, are applied to gears at calculation level III, a good agreement with measured load-dependent gear power losses is observed.

3. Irrespective of the calculation level and model, the results for the synthetic oils show a higher deviation from the measured values compared to the mineral oil.

4. When the solid coefficient of friction µ_s is described as a function of contact parameters such as pressure p and velocity v, better agreement with measurements is observed.

5. Calculation level III effectively considers mesh-local contact parameters. This allows mesh-local analysis and optimization of tooth flank topographies for load-dependent gear power loss.

6. Depending on the application of calculation level III, such as internal gears, significant differences in load-dependent gear power loss occur compared to the established calculation level I and II.

Future research should consider current observations on models for mixed lubrication, such as the one mentioned in Ref. [46]. Furthermore, reliable thermal and frictional predictions require accurate rheological characterization of synthetic oils and improved methods for calculating the bulk temperature ϑ_M. In addition, it is essential to conduct experimental investigations to validate the predicted lower load-dependent gear power loss in internal gears as determined by calculation level III. By addressing these areas, future work will improve the accuracy and reliability of calculation models of load-dependent gear power losses, ultimately contributing to efficient, economical, and sustainable gearboxes.

The presented results are based on the research project FVA no. 686/II, self-financed by the Research Association for Drive Technology e.V. (FVA). The authors thank the FVA and the project committee members for their sponsorship and support.

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.

Tables 12–14.

$a$ =

center distance (mm)

$b$ =

tooth width (mm)

$d$ =

diameter at reference circle (mm)

$p$ =

pressure (N/mm²)

$t$ =

time (s)

$v$ =

velocity (m/s)

$z$ =

number of teeth

$A$ =

first point of tooth contact on the path of contact (mm)

$B$ =

lowest point of single-tooth contact (mm)

$C$ =

pitch point (mm)

$D$ =

highest point of single-tooth contact (mm)

$E$ =

last point of tooth contact on the path of contact | Young's modulus (⁠$mm|N/mm2$⁠)

$P$ =

power (W)

$R$ =

roughness|radius of curvature (⁠$μm|mm$⁠)

$T$ =

absolute temperature|torque (⁠$K|Nm$⁠)

$X$ =

temperature threshold|observed values (⁠$K$⁠)

$Y$ =

observed values

$cp$ =

specific heat capacity (J/(kg·K))

$da$ =

diameter at tip circle (mm)

$db$ =

diameter at base circle (mm)

$dw$ =

diameter at pitch circle (mm)

$ff$ =

contact line resolved friction force (N/mm)

$fN$ =

line load (N/mm)

$gα$ =

path of contact (mm)

$h0$ =

central film thickness (µm)

$mn$ =

normal module (mm)

$pet$ =

transverse base pitch (mm)

$tc$ =

time of a contact line during its transition through the tooth contact area (s)

$vtb$ =

tangential speed at base diameter (m/s)

$vt,C$ =

circumferential speed (pitch point) (m/s)

$vΣ$ =

sum velocity (m/s)

$x1,2$ =

addendum modification coefficient

$CA$ =

tip relief (µm)

$Cα$ =

profile crowning (µm)

$Cβ$ =

lead crowning (µm)

$CHα$ =

face profile angle (µm)

$Fbt$ =

load at base circle (N)

$Ff$ =

friction force (N)

$FN$ =

normal load (N)

$HV$ =

tooth power loss factor

$HVL$ =

local-geometric tooth power loss factor

R =

radius of curvature (mm)

$Rred$ =

reduced radius of curvature (mm)

$Tfla$ =

flash temperature (K)

$TM$ =

flash temperature (K)

$XCa$ =

profile modification coefficient

$XL$ =

lubricant factor

$XOS$ =

surface structure factor

$XS$ =

lubrication coefficient

$Ra$ =

arithmetical mean roughness (µm)

$Rq$ =

root mean square roughness (µm)

$Rz$ =

mean roughness depth (µm)

$VI$ =

viscosity index

$n,m$ =

fitting parameter

$x,y$ =

coordinates in the tooth contact area (mm)

$U,G,W$ =

dimensionless parameters

$αn$ =

pressure angle (deg)

$αp$ =

pressure–viscosity coefficient (⁠$m2/N$⁠)

$α,β,γ$ =

fitting parameter Eqs. (11) and (12)

$β$ =

helix angle (deg)

$γ˙$ =

shear rate (⁠$1/s$⁠)

$εα$ =

transverse contact ratio

$εβ$ =

overlap ratio

$εγ$ =

total contact ratio

$η$ =

dynamic viscosity (⁠$Pa⋅s$⁠)

$ηM0$ =

dynamic viscosity at bulk temperature and ambient pressure (⁠$Pa⋅s$⁠)

$ϑ$ =

temperature (⁠$∘C$⁠)

$λ$ =

thermal conductivity (⁠$W/(m⋅K)$⁠)

$λrel$ =

relative film thickness

$μ$ =

coefficient of friction

$ν$ =

kinematic viscosity (mm²/s)

$ξ$ =

fluid load portion

$ρ$ =

density (kg/m³)

$τ$ =

shear stress (N/mm²)

$τc$ =

critical shear stress (N/mm²)

$τlim$ =

limiting shear stress (N/mm²)

$ψ$ =

surface power density (W/mm²)

$ω$ =

rotational speed (rad/s)

$Λ$ =

limiting-stress pressure coefficient

0 =

no-load|ambient pressure

1, 2 =

pinion, wheel

calc =

calculated

exp =

experimental/measured

f =

fluid/frictional

i =

counting variable

m =

mean

mz =

mean gear

s =

sliding

B =

bearing

C =

pitch point

G =

gear

H =

Hertzian

I, II, III =

calculation level indicator

In =

input

L =

loss

P =

load dependent

R =

reference

RMS =

root mean squared

S =

seal

X =

other components