A pathway towards energy efficiency classes for gearboxes related to superefficiency

Open Access
01.12.2025
Originalarbeiten/Originals

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Verfasst von: T. Lohner; C. Paschold
Erschienen in: Engineering Research | Ausgabe 1/2025

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Abstract

Der Artikel geht auf die Konstruktionskriterien von Getrieben ein und betont Effizienz, Leistungsdichte und Geräuschreduzierung. Es stellt das Konzept der Energieeffizienzklassen vor und untersucht die Auswirkungen verlustarmer Technologien auf die Reduzierung von Getriebeverlusten. Insbesondere hebt die Studie das Potenzial synthetischer Getriebeöle und verlustarmer Getriebe hervor, die Energieeffizienz deutlich zu steigern und sogar eine Supereffizienz zu erreichen. Die Forschung diskutiert auch die Herausforderungen und zukünftigen Arbeiten, die erforderlich sind, um diese Technologien in praktische Getriebe umzusetzen.

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1 Introduction

The main design criteria of gearboxes are (i) efficiency and heat balance, (ii) power density and load capacity, and (iii) noise, vibration, and harshness. The underlying physical principles can influence each other, so a compromise is required based on the design aims. The product lifecycle analysis superimposes the main design criteria [1, 2]. In a circular economy, a sustainable design accounts for re-usage, recycling, and repairing [3]. The energy dissipation during the use phase is significant for many gearboxes [4, 5]. Standards and target values for energy efficiency can help classify products and accelerate technological developments.

The accumulated energy dissipation W_L during the use phase of a gearbox results from the sum of total power loss P_L per operating point i and its corresponding time share t:

$$W_{L}={\sum }_{i=1}^{n}P_{L,i}\cdot t_{i}$$

(1)

According to ISO/TR 14179‑2 [6], the total power loss P_L of a gearbox consists of power losses of gears (G), bearings (B), seals (S), and other components (X). Gear and bearing power losses are subdivided into no-load (0) and load-dependent (P) power losses:

$$P_{L}=P_{LG0}+P_{LGP}+P_{LB0}+P_{LBP}+P_{LS}+P_{LX}$$

(2)

The load-dependent gear power loss P_LGP can be expressed as the frictional power P_f of the tribological gear contact integrated over t_c, representing the time of a single contact line during its transition through the gear contact area. The frictional power P_f is calculated by multiplying the values of the normal load F_N, the coefficient of friction µ, and the sliding velocity v_s:

$$\begin{aligned}&P_{LGP}&&=\frac{\varepsilon _{\gamma }}{t_{c}}\cdot {\int }_{t=0}^{t_{c}}P_{f}\left(t\right)dt\\&&&=\frac{\varepsilon _{\gamma }}{t_{c}}\cdot {\int }_{t=0}^{t_{c}}F_{N}\left(t\right)\cdot \mu \left(t\right)\cdot v_{s}\left(t\right)dt\end{aligned}$$

(3)

P_LGP often makes up the largest share of P_L for gearboxes with high torque [7].

There is an enormous potential to reduce load-dependent gear power losses by implementing low-loss technologies. Holmberg and Erdemir [8] found that approx. 23% of the world’s total energy consumption originates from tribological contacts. Out of these 23%, up to 40% can be saved by implementing new tribology technologies. These technologies comply largely with the principles of green tribology [9].

The lubricant greatly influences the coefficient of friction µ and, therefore, the load-dependent gear power loss P_LGP according to Eq. 3. Compared to the base oil viscosity, its type and molecular structure have a superordinate impact [10, 11]. Within [12], Michaelis introduced a lubricant factor X_L, which acts as a relative value referring the mean coefficient of gear friction of a gear base oil to a mineral oil as a reference. According to Schlenk [13], X_L is 1.0 for a mineral oil, 0.8 for a polyalphaolefin, 0.7 for a non-water-soluble polyglycol, 0.6 for a water-soluble polyglycol, 1.5 for a traction oil, and 1.3 for a phosphoric acid ester. For the X_L of polyglycols, Schlenk [13] suggests a calculation equation depending on the sum speed at the gear pitch point. Based on power loss measurements at the FZG efficiency test rig, Hinterstoißer et al. [14] found an average reduction of the mean coefficient of gear friction by 36% for polyalphaolefin and 46% for polyether compared to mineral oil.

Polyglycols and glycerols with functional water content as candidates for green lubrication have shown superlubricity with coefficients of friction of less than 0.01 [15‐18]. For water-containing polyglycols, Yilmaz et al. [19] measured superlubricity with cylindrical gears for a wide range of practical operating conditions. Based on this, Yilmaz [20] modifies the equation for the lubricant factor X_L of Schlenk [13] for water-containing polyglycols and suggests a value of X_L = 0.2 for simplicity. With the same lubricants, Sedlmair et al. [21] measured a remarkably high efficiency of up to 99.6% in a dip-lubricated two-stage gearbox of a battery electric vehicle. The increase in efficiency due to using a water-containing polyglycol corresponds to a total power loss saving of up to 74% compared to a polyalphaolefin.

The gear geometry strongly influences the normal load F_N and sliding velocity v_s in the gear contact and, therefore, the load-dependent gear power loss P_LGP according to Eq. 3. When integrating Eq. 3 over the tooth flank contact area with coordinates x and y instead of time, it reads:

$$\begin{aligned}&P_{LGP}=&&\frac{1}{p_{et}}\cdot {\int }_{y=0}^{b}{\int }_{x=A}^{E}f_{N}\left(x,y\right)\cdot \mu \left(x,y\right)\\&&&\cdot v_{s}\left(x,y\right)dxdy\end{aligned}$$

(4)

For the sake of simplicity, µ across the gear mesh is often considered as a constant mean coefficient of gear friction µ_mz. By additionally considering the input power P_In, Eq. 4 then reads:

$$\begin{aligned}&P_{LGP}=&&P_{In}\cdot \mu _{mz}\\&&&\cdot \underset{H_{VL}}{\underbrace{\frac{1}{p_{et}}\cdot {\int }_{y=0}^{b}{\int }_{x=A}^{E}\frac{f_{N}\left(x,y\right)}{F_{bt}}\cdot \frac{v_{s}\left(x,y\right)}{v_{tb}}dxdy} }\end{aligned}$$

(5)

Wimmer [22] defines the local geometric tooth power loss factor H_VL (see Eq. 5), which can be derived numerically from a loaded tooth contact analysis, e.g., in RIKOR [23‐25]. By assuming a simplified load distribution along the path of contact and without consideration of the tooth width, analytical solutions of the integral in Eq. 5 were obtained, corresponding to the geometric tooth power loss factor H_V [22, 26, 27]. Wimmer [22] allocates the tooth power loss factors H_V and H_VL to different efficiency levels. Accordingly, a non-efficient gear geometry relates to H_V|H_VL ≥ 0.20, a less efficient gear geometry to 0.15 ≤ H_V|H_VL < 0.20, and an efficient gear geometry to 0.10 ≤ H_V|H_VL < 0.15. A very efficient gear geometry shows values of H_V|H_VL < 0.10. Velex and Ville [28] describe an analytical approach to calculate the load-dependent gear power loss considering the influences of tooth profile modifications. Fernandes et al. [29] compare different approaches for tooth power loss factors.

Low-loss gear geometries relate to involute gear geometries with the gear mesh concentrated around the pitch point [22, 30]. Characteristic features are a decreased normal module, transverse contact ratio, and increased normal pressure angle. Hinterstoißer et al. [14, 31] designed low-loss gear geometries for the 6^th gear of a passenger car manual transmission and for a gear stage of an industrial transmission. Based on the reference geometries and similar load-carrying capacities, the geometric tooth power loss factor H_v is reduced by up to 78% and 57%, respectively. It should be noted that a change in gear geometry also affects the coefficient of gear friction. Farrenkopf et al. [32] used a thermal elastohydrodynamic simulation to analyze a low-loss gear geometry and found that the thermal reduction in fluid friction is very small.

A reduction in load-dependent gear power loss can reduce the cooling requirements. This, in turn, introduces the possibility of applying on-demand-orientated lubrication methods. At the same time, on-demand-orientated lubrication methods reduce no-load gear power losses due to a reduction in lubricant interaction with rotating gears [33]. Reduced quantity lubrication can be achieved by simply decreasing the immersion depth for dip lubrication [34] or the oil volume flow rate for injection lubrication [35]. Minimum quantity lubrication can be realized by supplying minimal oil volume flow rates through a continuous air stream. Höhn et al. [36] show its potential to reduce the no-load power loss by supplying mineral oil at only 28.1 ml/h to a gear contact, but also its limitations in terms of thermal load limit and load-carrying capacity. Drop-on-demand lubrication is a novel method for oil supply using single droplets [37]. Mirza et al. [38] show its applicability for gear lubrication. Hinterstoißer et al. [14] and Yilmaz et al. [35] show that for on-demand orientated lubricated gears, the higher resulting gear bulk and contact temperatures can reduce the load-dependent gear power loss due to lower contact oil viscosity.

Suitable methods are required to classify energy dissipation and the potential of technologies for gearboxes. The European Union Regulation 2017/1369 [39] sets a framework for energy labeling for a specific energy-related product group. The label corresponds to a closed scale using letters for energy efficiency classes from A to G, with seven colors from dark green to red. When a label is introduced, no product should fall in class A. Only after at least ten years should most products fall into that class. Labels are rescaled when further technological development can be expected, and many products sold fall into the energy efficiency classes. For example, for refrigerating appliances, an energy efficiency index as relative energy efficiency in percentage is used to define energy efficiency classes [40]. It relates an annual energy consumption to an annual energy reference consumption of a refrigerating appliance. Although there are component-specific power loss test methods [41‐45], no such documented classifications exist for product groups of gearboxes.

The literature review shows that implementing low-loss technologies can significantly increase the energy efficiency of gearboxes. This potential study analyzes different gear geometries, gear oils, and lubrication methods using an energy efficiency index and mean energy efficiency. Superefficiency is allocated to the highest potential setup of low-loss technologies and its dependence on the particular gearbox and the underlying operating cycle is discussed. This publication is based on presentations at the 9th International Tribology Conference 2023 [46] and 24th International Colloquium Tribology 2024 [47].

2 Methods and materials

This section describes the methods and materials used in this calculation study. First, the simulation program for calculating the gearboxes’ efficiency and heat balance is introduced (cf. Sect. 2.1). Second, the FZG gearbox as the main object of investigation, and the technologies under consideration are described (cf. Sect. 2.2). Third, the considered industrial gearbox is presented (cf. Sect. 2.3).

2.1 Simulation program

The study uses the established simulation program WTplus to calculate individual gearbox components’ power losses according to Eq. 2 and, subsequently, the gearbox efficiency. Moreover, the simulation program has been developed to investigate the heat balance of gearbox systems. A steady-state oil temperature can be determined by calculating the heat dissipation via the housing according to ISO/TR 14179‑2 [48] and comparing it to the sum of all power losses. Furthermore, WTplus can calculate component temperatures in gearboxes for transient operation conditions using the thermal network method [49], similar to [50, 51].

WTplus contains various validated calculation models for predicting power losses and heat transfers in gearboxes. In particular, the calculation models for the technologies considered in this study were derived by Schlenk [13] and Doleschel [52] for the mean gear coefficient of friction, by Ohlendorf [27] and Wimmer [22] for the tooth power loss factor and by Mauz [53] for no-load gear power loss. Most of the literature on which WTplus is based is available in German. The relevant calculation models are described in the appendix to make it easier for the reader to understand.

2.2 FZG gearbox

The main object of investigation is the test gearbox of the FZG efficiency test rig in Fig. 1, which is a modified version of the FZG back-to-back test rig [54]. It consists of the gearbox housing with front and top cover, pinion and wheel shaft, four cylindrical roller bearings of type NU406-XL-M1, and two radial shaft seals of size 30 × 47 × 7. The gearbox housing has a volume of 2.75 liters. The center distance of the pinion and wheel shaft is 91.5 mm. The considered rotational direction of the gears is depicted in Fig. 1.

Fig. 1

FZG gearbox of the efficiency test rig

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2.2.1 Gears

Three different cylindrical gear geometries from Hinterstoißer et al. [14] are considered. A conventional high contact ratio (HCR) gear is used as a reference. Moreover, a moderate low-loss (mLL) and an extreme low-loss (eLL) gear are considered. Figure 2 shows renderings and transverse section geometry plots of the considered gear geometries. Table 1 lists their geometry data.

Fig. 2

Renderings and transverse section geometry plots of the considered gears in the FZG gearbox: (a) high contact ratio (HCR) gear, (b) moderate low-loss (mLL) gear, and (c) extreme low-loss (eLL) gear

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Table 1

Geometry calculation data of the considered gear geometries in the FZG gearbox [14]

Gear	High contact-ratio (HCR)		Moderate low-loss (mLL)		Extreme low-loss (eLL)
Normal pressure angle α_n in °	16.0		27.0		36.0
Normal module m_n in mm	2.302		1.920		1.810
Base diameter d_1,2 in mm	74.2	99.8	66.5	90.0	60.8	81.0
Reference diameter d_1,2 in mm	78.3	105.3	77.8	105.3	77.9	103.8
Pitch diameter d_w1,2 in mm	78.0	105.0	77.8	105.2	78.4	104.6
Tip diameter d_a1,2 in mm	85.0	111.1	81.8	108.7	80.6	106.5
Addendum mod. coeff. x_1,2	0.094	−0.220	0.059	−0.097	0.183	0.168
Number of teeth z_1,2	29	39	34	46	39	52
Helix angle β at pitch circle in °	31.5		33.0		25.0
Transverse contact ratio ε_α	2.10		1.10		0.65
Overlap ratio ε_β	1.27		2.10		2.08
Face width b in mm	17.6		23.3		28.0
Center distance a in mm	91.5		91.5		91.5
Tooth power loss factor H_V	0.226		0.094		0.049

Compared to the HCR gear with a transverse contact ratio ε_α over just over 2, the low-loss gears feature much smaller values with ε_α of just over 1 for mLL and significantly below 1 for eLL. The normal pressure angle α_n is increased considerably from 16.0 to 27.0° and 36.0° respectively, and the normal module m_n is decreased from 2.302 to 1.920 mm and 1.810 mm respectively. The face width b is increased from 17.6 to 23.3 mm and 28.0 mm, respectively, to achieve a load-carrying capacity similar to the HCR gear. The overlap ratio ε_β of the low-loss gears is adjusted to approx. 2 for favorable excitation behavior.

Due to the low-loss design of the mLL and eLL gear, the geometric tooth power loss factor H_V is reduced from 0.226 for the HCR gear to 0.094 and 0.049 for the mLL and eLL gear. Hence, according to Wimmer [22], the HCR gear is a “non-efficient” gear, whereas the mLL and eLL gears are “very efficient.” Note that H_V is directly proportional to the load-dependent gear power loss acc to Eq. 5.

For all gears, axially ground tooth flanks with an arithmetic mean roughness of Ra = 0.2 µm are considered, and a surface structure factor of X_OS = 0.5 [55] is used accordingly.

2.2.2 Gear oils

Five different gear oils with a common kinematic viscosity of approx. 10 mm²/s at 100 °C are considered: mineral oil MIN10, polyalphaolefin PAO10, polyglycol PG10, polyether PE10 and water-containing polyglycol PAGW09. Table 2 shows the viscosity and density calculation data considered. The kinematic viscosities ν at 40 °C and 100 °C, viscosity index (VI), and density ρ at 15 °C for MIN10, PAO10, PG10, and PE10 are from Hinterstoißer et al. [14, 31] and for PAGW09 from Yilmaz et al. [19]. The pressure-viscosity coefficients α_p at 40 °C and 100 °C for MIN10, PAO10, and PG10 are based on Gold et al. [56]. The α_p values of PE10 are assumed to be the same as those of PG10. The value of α_p (40 °C) for PAGW09 is from Yilmaz et al. [18], while the value of α_p (100 °C) is estimated by assuming the same temperature dependency as for PG10.

Table 2

Viscosity and density calculation data of the considered gear oils [14, 18, 19, 31, 56]

Gear oil	MIN10	PAO10	PG10	PE10	PAGW09
ν (40 °C) in mm²/s	94.5	63.7	46.0	42.6	45.7
ν (100 °C) in mm²/s	9.8	9.9	10.0	10.1	9.2
Viscosity index (VI)	77	140	212	235	189
ρ (15 °C) in kg/m³	885	847	1040	1009	1115
α_p (40 °C) in GPa^-1	18.92	12.92	12.09	11.91	6.26
α_p (100 °C) in GPa^-1	13.21	9.25	8.52	8.55	4.31

The mean coefficients of gear friction µ_mz of the considered gear oils were derived from power loss measurements at the FZG efficiency test rig similar to test method FVA 345 [41, 42] with a wide spread of operating points with varying load, circumferential speed and oil temperature [19, 31]. Test gears of FZG-type C_mod made of 16MnCr5 with an arithmetic mean roughness Ra of approximately 0.2 µm were used. Based on a regression analysis, the tribosystem-specific parameters μ_ref,s, α_s, β_s, μ_ref,f, α_f, β_f, and γ_f in Table 3 for Eqs. 14 and 15 were obtained. Due to a reduced number of derived mean coefficient of gear friction values for PAGW09, regression coefficients are unavailable [19]. Therefore, the regression parameters α_s, β_s, α_f, β_f, and γ_f of PG10 are used, while μ_ref,s, and μ_ref,f were manually adjusted to match the measurements of Yilmaz et al. [19].

Table 3

Mean coefficient of gear friction calculation data of the considered gear oils [31] derived from gear power loss test FZG-E-C/0,5:20/5:9/40:120 according to FVA 345 [41, 42]

Gear oil	MIN10	PAO10	PG10	PE10	PAGW09
μ_ref,s	0.0673	0.0533	0.0533	0.0414	0.0107
α_s	0.32	0.35	0.47	0.56	0.47
β_s	−0.14	−0.20	−0.26	−0.30	−0.26
μ_ref,f	0.0421	0.0310	0.0249	0.0227	0.0027
α_f	0.15	0.19	0.45	0.16	0.16
β_f	−0.15	−0.13	−0.18	−0.04	−0.04
γ_f	0.15	0.15	0.32	0.36	0.36

2.2.3 Lubrication methods

The three different lubrication methods visualized in Fig. 3 are considered. Dip lubrication (DL) with an immersion depth of the pinion e₁ of 3∙m_n is used as a reference. Also, DL with an increased immersion depth of pinion and wheel e of d_a / 2 is considered. Moreover, minimum quantity lubrication (MQL) with an oil volume rate of 28 ml/h supplied by a continuous air stream, as used in [35, 36], is investigated.

Fig. 3

Visualization of the considered lubrication methods in the FZG gearbox exemplified by the HCR gear: (a) dip lubrication (DL) with an immersion depth of e = d_a / 2, (b) DL with an immersion depth of e₁ = 3 ∙ m_n and (c) minimum quantity lubrication (MQL) with an oil volume rate $\dot{V}$= 28 ml/h

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Table 4 shows the immersion depths of pinion e₁ and wheel e₂ as well as the oil volume rate $\dot{V}$ of the considered lubrication methods for the considered gears.

Table 4

Calculation data of the considered lubrication methods in the FZG gearbox

	Immersion depth of pinion e₁ in mm HCR \| mLL \| eLL	Immersion depth of wheel e₂ in mm HCR \| mLL \| eLL	Oil volume rate $\dot{V}$ in ml/h
DL with e = d_a / 2	42.52 \| 40.90 \| 40.30	55.57 \| 54.35 \| 53.25	–
DL with e₁ = 3∙m_n	6.90 \| 5.76 \| 5.43	19.95 \| 19.21 \| 18.38	–
MQL	–	–	28

2.2.4 Calculation procedure

Within the calculation study of the FZG gearbox, the gear geometry, the gear oil, and the lubrication method are varied according to Sect. 2.2.1 and 2.2.2, and 2.2.3. The combination of PAGW09, eLL, and MQL is considered to be the highest potential technological setup of low-loss technologies.

The operating points and cycles considered are similar to the gear power loss test method FVA 345 [41, 42]. A complete operating cycle OC|_{{35.3…302.0}Nm} according to Table 5 is defined by a fully parametric combination of the wheel rotational speed n₂ = {87; 174; 348; 870; 1444; 2609; 3479} min⁻¹, pinion torque T₁ = {35.3; 94.1; 183.4; 302.0} Nm and oil temperature ϑ_Oil = {40; 60; 90; 120} °C with t = 5 min per operating point. The different gear geometries result in different circumferential speeds v_t,C and Hertzian pressures at the pitch point p_H,C for a given n₂ and T₁.

Table 5

Considered variation of operating variables in the complete operating cycle of the FZG gearbox

n₂ in min⁻¹		87	174		348	870	1444		2609	3479
v_t,C in m/s	HCR	0.47	0.95		1.91	4.78	7.93		14.34	19.11
	mLL	0.47	0.95		1.91	4.79	7.95		14.37	19.16
	eLL	0.47	0.95		1.90	4.76	7.90		14.29	19.04
T₁ in Nm		35.3		94.1		183.4		302.0
p_H,C in N/mm²	HCR	530		865		1210		1550
	mLL	375		615		855		1100
	eLL	320		530		740		950
ϑ_Oil in °C		40		60		90		120

In addition to the complete operating cycle, operating cycles with different weightings of operating points with loads are investigated in Sect. 4. Thereby, operating points with T₁ = 302.0 Nm, T₁ = 183.4 Nm, and T₁ = 94.1 Nm are successively removed (OC|_{{35.3…183.4}Nm}, OC|_{{35.3…94.1}Nm}, OC|_{35.3}Nm), or operating points with T₁ = 302.0 Nm, T₁ = 183.4 Nm, T₁ = 94.1 Nm, and T₁ = 35.3 Nm are considered load-specifically (OC|_{35.3}Nm, OC|_{94.1}Nm, OC|_{183.4}Nm, OC|_{302.0}Nm).

2.3 Industrial gearbox

Additionally to the FZG gearbox as the main object of investigation, a single-stage industrial gearbox [57] with cylindrical gears is considered. The gearbox is equipped with bearings of type 23226 CC/W33 and has a seal with a diameter of 120 mm on the input shaft and a seal with a diameter of 130 mm on the output shaft. Heat dissipation via the fundament does not occur for the considered use case.

Besides a reference (REF_IG) gear, a moderate low-loss (mLL_IG) and an extreme low-loss (eLL_IG) gear with similar load-carrying capacity are available [57]. Table 6 lists the geometry data. The geometric tooth power loss factor H_V is 0.118 for the REF_IG gear, 0.080 for the mLL_IG gear, and 0.051 for the eLL_IG gear. Hence, according to Wimmer [22], the REF_IG gear is an “efficient” gear, whereas the mLL_IG and eLL_IG gears are “very efficient.”

Table 6

Geometry calculation data of the considered gear geometries in the industrial gearbox [57]

Gear	Reference (REF_IG)		Moderate low-loss (mLL_IG)		Extreme low-loss (eLL_IG)
Normal pressure angle α_n in °	20.0		25.0		35.0
Normal module m_n in mm	6.0		5.0		4.0
Number of teeth z_1,2	26	58	32	71	40	89
Helix angle β at pitch circle in °	13.0		12.0		12.0
Transverse contact ratio ε_α	1.51		1.11		0.80
Overlap ratio ε_β	1.43		1.99		2.98
Face width b in mm	120		150		180
Center distance a in mm	265		265		265
Tooth power loss factor H_V	0.118		0.079		0.051

The reference gear oil of the industrial gearbox is a mineral oil of ISO viscosity grade (VG) 320. To study the power loss influence of the gear oils considered for the FZG gearbox (cf. Sect. 2.2.2), Eq. 16 is applied with a normalized averaged mean coefficient of friction $\overline{\mu }$_mz/$\overline{\mu }$_mz|MIN10 derived in Sect. 3.1.1 used as lubricant factor X_L. The dynamic viscosity is calculated based on ν(40 °C) = 320 mm²/s and the values for VI and ρ in Table 2 for the considered gear oils MIN_IG, PAO_IG, PG_IG, PE_IG, and PAGW_IG.

As the lubrication method, dip lubrication with an oil level of 69 mm below the axis is considered for the industrial gearbox in a horizontal mounting position, resulting in immersion depths for the pinion e₁ of 20.65 mm for REF_IG, 18.30 mm for mLL_IG, and 16.50 mm for eLL_IG, and immersion depths for the wheel e₂ of 118.94 mm for REF_IG, 116.35 mm for mLL_IG, and 115.55 mm for eLL_IG.

Within the calculation study of the industrial gearbox, the gear geometry and gear oil are varied, as described above. An exemplarily operating cycle with a pinion torque of T₁ = 12,732 Nm and a pinion rotational speed of n₁ = 230 min^-1 is considered with an annual operating time of 6000 h/a. The resulting input power is P_In = 307 kW. For the considered operating condition, no external cooling unit is required to avoid thermal load limits. The combination of PAGW_IG and eLL_IG is considered to be the highest potential technological setup of low-loss technologies.

3 Results

The structure of the results section refers to the FZG gearbox in Sect. 3.1 and industrial gearbox in Sect. 3.2 and shows the influence of the considered low-loss technologies. The focus of the description of the results is on load-dependent gear power losses.

3.1 FZG gearbox

The calculation study on the FZG gearbox power loss was performed with the validated simulation program WTplus (cf. Sect. 2.2). First, results on the mean coefficient of gear friction µ_mz of the considered gear oils are shown in Sect. 3.1.1. Second, calculation results on the load-dependent gear power loss P_LGP and gear bulk temperature ϑ_M are addressed for individual operating points in Sect. 3.1.2 and 3.1.3. Third, results on the accumulated energy dissipation W_L and energy efficiency index EEI are presented in Sect. 3.1.4 and 3.1.5.

3.1.1 Mean coefficient of gear friction

For calculation of the mean coefficient of gear friction of MIN10, PAO10, PG10, PE10, and PAGW09, the oil-specific parameters μ_ref,s, α_s, β_s, μ_ref,f, α_f, β_f, and γ_f in Table 3 are used in Eqs. 14 and 15.

Figure 4a shows the calculated mean coefficient of gear friction µ_mz over the relative lubricant film thickness λ_rel,C for v_t,C = {0.5; 1; 2; 5; 8.3; 15; 20} m/s, p_H,C = {566; 924; 1290; 1655} N/mm² and ϑ_Oil = {40; 60; 90; 120} °C. The values of λ_rel,C are calculated using Eq. 13 without considering the thermal correction of Murch and Wilson [58] according to FVA 345 [41]. The mineral oil MIN10 shows the highest mean coefficients of gear friction, followed by the polyalphaolefin PAO10, the polyglycol PG10, the polyether PE10, and the water-containing polyglycol PAGW09, which outperforms the other gear oils and shows superlubricity in the gear contact for most operating points.

Fig. 4

Calculated mean coefficient of gear friction µ_mz over relative lubricant film thickness λ_rel,C for a wide spread of operating points with varying p_H,C, v_t,C, and ϑ_Oil (a) and averaged mean coefficient of friction $\overline{\mu}$_mz(λ_rel,C) allocated to boundary lubrication (λ_rel,C < 0.7), mixed lubrication (0.7 ≤ λ_rel,C < 2) and fluid film lubrication (λ_rel,C ≥ 2) (b)

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To analyze the trend of µ_mz over λ_rel,C, Fig. 4b shows the averaged mean coefficient of friction $\overline{\mu }$_mz(λ_rel,C) allocated to boundary lubrication with λ_rel,C < 0.7, mixed lubrication with 0.7 ≤ λ_rel,C < 2 and fluid film lubrication with λ_rel,C ≥ 2 [55]. In general, an increasing trend of µ_mz with decreasing λ_rel,C is present due to increasing solid load portion and higher solid coefficient of friction in the gear contact.

Considering all operating points and normalizing the averaged mean coefficient of gear friction of the gear oils to MIN10 gives the normalized averaged mean coefficient of gear friction $\overline{\mu }$_mz/$\overline{\mu }$_mz|MIN10. Table 7 classifies the derived values into the lubricant factor X_L, as suggested by Schlenk [13] and Yilmaz [20]. In general, a good correlation is found. Considering $\overline{\mu }$_mz/$\overline{\mu }$_mz|MIN10, PAGW09 has, on average, only approximately 10% of the mean coefficient of gear friction of MIN10.

Table 7

Derived normalized averaged mean coefficient of gear friction $\overline{{\mu}}$_mz/$\overline{{\mu}}$_mz|MIN10 for gear oils and classification into lubricant factor X_L suggested by Schlenk [13] and Yilmaz [20]

Gear oil	$\overline{\mu }$_mz/$\overline{\mu }$_mz\|MIN10	X_L from [13, 20]
MIN10	1.00	1.0
PAO10	0.70	0.8
PG10	0.61	0.7
PE10	0.48	0.6
PAGW09	0.10	0.2

Figure 5 shows the calculated mean coefficient of gear friction µ_mz for the FZG gearbox dependent on the gear oils MIN10, PAO10, PG10, PE10, and PAGW09 and gears HCR, mLL, and eLL for the lubrication methods DL with e = d_a / 2, DL with e₁ = 3 ∙ m_n and MQL. An operating point with P_In = 37.3 kW at moderate load and speed with T₁ = 183.4 Nm, n₂ = 1444 min⁻¹, and ϑ_Oil = 90 °C is considered.

Fig. 5

Mean coefficient of gear friction µ_mz dependent on gear oil and gear for lubrication methods DL with e = d_a / 2 (a), DL with e₁ = 3 ∙ m_n (b), and MQL (c) for operating point T₁ = 183.4 Nm, n₂ = 1444 min⁻¹, and ϑ_Oil = 90 °C of FZG gearbox

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The influence of the gear oils on µ_mz is represented by the relationships shown in Fig. 4. As the gear mesh conditions depend on the gear geometry, µ_mz differs for the considered gear geometries. Particularly, the decrease of p_H,C (cf. Table 5) of the mLL and eLL gear compared to the HCR gear reduces µ_mz. The lubrication method has a subordinate influence on µ_mz. However, due to the decreasing convective heat transfer from DL with e = d_a / 2 to DL with e₁ = 3 ∙ m_n to MQL, the gear bulk temperature ϑ_M (cf. Sect. 3.1.3) increases. According to Eq. 11, an increasing value of ϑ_M is associated with a thermal reduction of µ_mz, particularly pronounced for operating points with high load and speed.

3.1.2 Load-dependent gear power loss

Figure 6 shows the calculated load-dependent gear power loss P_LGP for the FZG gearbox dependent on the gear oils MIN10, PAO10, PG10, PE10, and PAGW09 and gears HCR, mLL, and eLL for the lubrication methods DL with e = d_a / 2, DL with e₁ = 3 ∙ m_n and MQL. From the considered operating cycle (cf. Sect. 2.2.4), an operating point with P_In = 37.3 kW at moderate load and speed with T₁ = 183.4 Nm, n₂ = 1444 min⁻¹, and ϑ_Oil = 90 °C is selected.

Fig. 6

Load-dependent gear power loss P_LGP dependent on gear oil and gear for lubrication methods DL with e = d_a / 2 (a), DL with e₁ = 3 ∙ m_n (b), and MQL (c) for operating point T₁ = 183.4 Nm, n₂ = 1444 min⁻¹, and ϑ_Oil = 90 °C of FZG gearbox

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As for the mean coefficient of friction µ_mz in Sect. 3.1.1, the lubrication method has an overall subordinate influence on P_LGP. Only the thermal reduction of µ_mz according to Eq. 11, and therefore, P_LGP plays a role, particularly for operating points with higher load and speed. The gear oil and the gear geometry have a superordinate influence on P_LGP. Table 8 shows the numerical values of P_LGP for DL with e₁ = 3 ∙ m_n in Fig. 6b.

Table 8

Numerical values for the load-dependent gear power loss P_LGP for DL with e₁ = 3 ∙ m_n in Fig. 6b

P_LGP in W	MIN10	PAO10	PG10	PE10	PAGW09
HCR	362.4	251.2	206.6	151.3	36.4
mLL	128.7	90.4	67.1	57.4	10.2
eLL	63.5	44.9	31.7	29.7	4.5

The influence of the gear geometry is mainly represented by the geometric tooth power loss factor H_V (cf. Sect. 2.2.1). Depending on the considered gear oil, a similar reduction of P_LGP by 62…72% from HCR to mLL gear and 80…88% from HCR to eLL gear applies for each gear oil. The influence of the gear oil is represented by its mean coefficient of gear friction µ_mz (cf. Sect. 3.1.1). For the HCR gear, the reduction of P_LGP is 31% from MIN10 to PAO10, 43% from MIN10 to PG10, 58% from MIN10 to PE10, and 90% from MIN10 to PAGW09.

3.1.3 Gear bulk temperature

Figure 7 shows the calculated gear bulk temperature ϑ_M for the FZG gearbox dependent on the gear oils MIN10, PAO10, PG10, PE10, and PAGW09 and gears HCR, mLL, and eLL for the lubrication methods DL with e = d_a / 2, DL with e₁ = 3 ∙ m_n and MQL. From the considered operating cycle (cf. Sect. 2.2.4), an operating point with P_In = 148 kW at high load and speed with T₁ = 302.0 Nm, n₂ = 3479 min⁻¹, and ϑ_Oil = 90 °C is selected to result in high bulk temperatures.

Fig. 7

Gear bulk temperature ϑ_M dependent on gear oil and gear for lubrication methods DL with e = d_a / 2 (a), DL with e₁ = 3 ∙ m_n (b), and MQL (c) for operating point T₁ = 302.0 Nm, n₂ = 3479 min⁻¹, and ϑ_Oil = 90 °C of FZG gearbox

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The gear bulk temperature ϑ_M is directly related to P_LGP and the lubrication method. As P_LGP decreases strongly from MIN10 to PAO10 to PG10 to PE10 to PAGW09 and from HCR to mLL to eLL (cf. Sect. 3.1.2), the values of ϑ_M decrease strongly. For DL with e₁ = 3 ∙ m_n in Fig. 7b, ϑ_M decreases from 136 °C for MIN10 and HCR to 100 °C for MIN10 and eLL, to 98 °C for PAGW09 and HCR, and to 91 °C for PAGW09 and eLL. The more effective convective heat transfer for DL with e = d_a / 2 compared to DL with e₁ = 3 ∙ m_n is visible by comparing Fig. 7a and 7b. For MQL in Fig. 7c, the convective heat transfer is minimal, and ϑ_M increases to very high values. For the HCR gear with MIN10, PAO10, and PG10, ϑ_M exceeds the annealing temperature of approximately 160 °C of typical case-hardened steel.

3.1.4 Accumulated energy dissipation

While Sect. 3.1.1 and 3.1.2, and 3.1.3 focus on specific operating points under load, this section evaluates the accumulated energy dissipation W_L of the FZG gearbox according to Eq. 1 for the complete operating cycle defined in Sect. 2.2.4. It covers a large spread of operating points from partial load and full load with an equal weight of 5 min per operating point. Note that operating points with ϑ_M > 160 °C are excluded from the evaluation of W_L for each lubrication method. Concerning all operating conditions, this includes 43 individual operating points, equal to 0.9%. 18 of those operating points with ϑ_M > 160 °C can be assigned to the technological setup with MIN10, HCR, and MQL alone. Other technological setups reach ϑ_M > 160 °C at the same or fewer operating points. As the EEI accumulates the dissipation energy of the individual operating points, a uniform basis for comparison must be created for all technological setups, which is why all technological setups are reduced by the 18 operating points.

Figure 8 shows the calculated accumulated energy dissipation W_L for the FZG gearbox dependent on the gear oils MIN10, PAO10, PG10, PE10, and PAGW09 and gears HCR, mLL, and eLL for the lubrication methods DL with e = d_a / 2, DL with e₁ = 3 ∙ m_n and MQL. Thereby, besides P_LGP, all other power losses of the FZG gearbox according to Eq. 2 summing up to P_L are included.

Fig. 8

Accumulated energy dissipation W_L dependent on gear oil and gear for lubrication methods DL with e = d_a / 2 (a), DL with e₁ = 3 ∙ m_n (b), and MQL (c) for the complete operating cycle of FZG gearbox (operating points with ϑ_M > 160 °C are excluded for each lubrication method)

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The general trends of the gear oil and gear geometry influence in Fig. 8 resemble those shown for P_LGP in Fig. 6. The similarity in trends follows the significant impact of the gear oil and gear on load-dependent power loss portions. In contrast, the lubrication method significantly affects no-load power loss portions. It can be seen by comparing Fig. 8a to c, as the magnitude of W_L reduces from DL with e = d_a / 2 to DL with e₁ = 3 ∙ m_n to MQL. The highest and lowest values of W_L for the complete operating cycle of the FZG gearbox are 9.6 MJ for MIN10, HCR, and DL with e = d_a / 2 and 2.5 MJ for the highest potential technological setup with PAGW09, eLL, and MQL.

To understand the portion of no-load related accumulated energy dissipation per gear oil and gear, Table 9 shows for the FZG gearbox the percentage ratio W_L of no-load (0) and load-dependent (P) portion for the lubrication methods DL with e = d_a / 2, DL with e₁ = 3 ∙ m_n, and MQL.

Table 9

Percentage W_L ratio of no-load (0) and load-dependent portion (P) dependent on the gear oil and gear for lubrication methods DL with e = d_a / 2, DL with e₁ = 3 ∙ m_n, and MQL in Fig. 6

W_L0/W_LP in %		MIN10	PAO10	PG10	PE10	PAGW09
DL e = d_a / 2	HCR	10.1	11.8	13.6	14.7	22.3
	mLL	14.9	16.6	19.0	19.2	23.8
	eLL	17.4	18.9	21.3	21.3	24.2
DL e₁ = 3∙m_n	HCR	4.7	5.7	6.5	7.2	12.1
	mLL	7.5	8.6	9.9	10.1	13.3
	eLL	9.2	10.2	11.6	11.7	13.8
MQL	HCR	4.0	4.9	5.4	6.1	9.6
	mLL	6.4	7.3	8.1	8.3	10.7
	eLL	7.7	8.6	9.4	9.5	11.0

For each lubrication method, the ratio W_L0 / W_LP increases from MIN10 to PAO10 to PG10 to PE10 to PAGW09 and from HCR to mLL to eLL. The ratio W_L0 / W_LP increase is related to decreasing load-dependent power losses. Furthermore, the ratio W_L0 / W_LP decreases from DL with e = d_a / 2 to DL with e₁ = 3 ∙ m_n to MQL as the no-load power loss decreases. Hence, the ratio W_L0 / W_LP for MIN10, HCR, and MQL is minimal with a value of 4.0% and maximal for PAGW09, eLL, and DL with e = d_a / 2 with a value of 24.2%.

3.1.5 Energy efficiency index

Based on the accumulated energy dissipation W_L in Sect. 3.1.4, an energy efficiency index EEI similar to [39] can be defined for the FZG gearbox:

$$EEI=\frac{W_{L}}{W_{L,ref}}\cdot 100{\%}.$$

(6)

The reference accumulated energy dissipation W_L,ref refers to the gear HCR, the gear oil MIN10, and the lubrication method DL with e₁ = 3 ∙ m_n. If a technological setup results in a lower value W_L than W_L,ref, the EEI is smaller than 100%. If a technological setup results in a higher value W_L than W_L,ref, the EEI is higher than 100%.

Figure 9 shows the calculated energy efficiency index EEI for the FZG gearbox dependent on the gear oils MIN10, PAO10, PG10, PE10, and PAGW09 and gears HCR, mLL, and eLL for the lubrication methods DL with e = d_a / 2, DL with e₁ = 3 ∙ m_n, and MQL. Based on Fig. 9b with the reference EEI of 100%, the variation of the gear oil for the gear HCR shows EEI values of 79% for PAO10, 70% for PG10, 63% for PE10, and 38% for PAGW09. This trend is mainly due to the reduction of µ_mz and, therefore, P_LGP (cf. Sect. 3.1.2). The variation of the gear geometry for the gear oil MIN10 shows EEI values of 64% for mLL and 52% for eLL. This decrease in the EEI is mainly due to the reduction of H_V and, therefore, P_LGP (cf. Sect. 3.1.2). The minimum EEI for DL with e₁ = 3 ∙ m_n of 34% is found when combining eLL and PAGW09.

Fig. 9

Energy efficiency index EEI dependent on gear oil and gear for lubrication methods DL with e = d_a / 2 (a), DL with e₁ = 3 ∙ m_n (b), and MQL (c) for the complete operating cycle of FZG gearbox (operating points with ϑ_M > 160 °C are excluded for each lubrication method)

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The influence of the lubrication method can be seen by comparing the EEI in Fig. 9b with DL with e = d_a / 2 (Fig. 9a) and MQL (Fig. 9c). MQL can reduce the EEI to a minimum of 31% relating to the highest potential technological setup with PAGW09, eLL, and MQL. For technologies with an EEI > 50%, limited convective heat transfer of MQL results in critical gear bulk temperatures of ϑ_M > 160 °C (cf. Sect. 3.1.3).

3.2 Industrial gearbox

The FZG gearbox and its considered operating cycle are academic. Therefore, the influence of the considered low-loss technologies is transferred to the industrial gearbox described in Sect. 2.3. Fig. 10 shows the calculated energy efficiency index EEI (a) and steady-state oil temperature ϑ_Oil (b) for the industrial gearbox dependent on the gear oils MIN_IG, PAO_IG, PG_IG, PE_IG, and PAGW_IG and gears REF_IG, mLL_IG, and eLL_IG. The reference accumulated energy dissipation W_L,ref refers to the gear geometry REF_IG and the gear oil MIN_IG. For the considered operating cycle with a constant operating point and an operating time of 6000 h/a, there is a potential to reduce the EEI to 44% for the highest potential technological setup with eLL_IG and PAGW_IG. This EEI decrease correlates to a W_L reduction from 67 to 30 GJ/a and a steady-state oil sump temperature ϑ_Oil reduction from 86 to 53 °C.

Fig. 10

Energy efficiency index EEI (a) and steady-state oil temperature ϑ_Oil (b) dependent on gear oil and gear for an operating cycle of an industrial gearbox

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4 Discussion

The results of this study are discussed in terms of the minimum energy efficiency index in Sect. 4.1, maximum energy efficiency in Sect. 4.2, and superefficiency and energy efficiency classes in Sect. 4.3. The study is reflected in Sect. 4.4.

4.1 Minimum energy efficiency index

The results on the energy efficiency index EEI of the FZG gearbox in Sect. 3.1.5 show that the minimum EEI is 31% for the highest potential technological setup with PAGW09, eLL, and MQL. Due to very low energy dissipation, critical gear bulk temperatures can be avoided even at high input power despite minimum quantity lubrication.

The minimum EEI depends on the considered operating cycle. Figure 11a shows the EEI of the reference technological setup with MIN10, HCR, and DL with e₁ = 3 ∙ m_n and the highest potential technological setup with PAGW09, eLL, and MQL over different operating cycles for the FZG gearbox. The complete operating cycle OC|_{{35.3…302.0}Nm} refers to Sect. 3.1.5. Based on this, operating points with T₁ = 302.0 Nm, T₁ = 183.4 Nm, and T₁ = 94.1 Nm are successively removed to obtain operating cycles with different weightings of operating points with load, i.e. OC|_{{35.3…183.4}Nm}, OC|_{{35.3…94.1}Nm}, and OC|_{35.3}Nm. Thereby, the ratio W_L0 / W_LP changes (cf. Table 9), influencing the relevance of individual technologies. PAGW09 and eLL significantly affect the load-dependent energy dissipation and, therefore, operating points with a higher load. MQL has a significant influence on the no-load energy dissipation. This influence can be seen in Fig. 11a, as the minimum EEI of the highest potential technological setup gradually decreases from 51.6 to 27.9% with increasing weighting of operating points with load.

Fig. 11

Energy efficiency index EEI of reference and highest potential technological setup for FZG gearbox for operating cycles (OC) with different weighting of operating points with load (a) and relevance of the individual low-loss technologies of the highest potential technological setup

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Figure 11b illustrates the relevance of individual low-loss technologies of the highest potential technological setup with PAGW09, eLL, and MQL for operating cycles with different weightings of operating points with load. MQL is most relevant for operating points with low load. Its relevance decreases with increasing weighting of operating points with high load. In contrast, PAGW09 and eLL show increasing relevance with increasing weighting of operating points with load and a similar trend, with PAGW09 having a greater relevance than eLL. This trend can be understood when comparing the normalized averaged mean coefficient of friction $\overline{\mu }$_mz / $\overline{\mu }$_mz|MIN10 for PAGW09 of 0.10 (cf. Table 7) with the geometric tooth power loss factor ratio H_V (eLL) / H_v (HCR) of 0.22 (cf. Sect. 2.2.1). Accordingly, the potential of µ_mz and, therefore, of the gear oil to reduce P_LGP is higher than that of H_V and, therefore, of the gear geometry (cf. Eq. 8). Note that when using a gear oil enabling superlubricity like PAGW09, the additional reduction of EEI by using the low-loss gear eLL is comparable low and vice versa.

Similar relationships are found for the results on the industrial gearbox in Sect. 3.2. The minimum EEI is 44% when the highest potential technological setup with PAGW_IG, eLL_IG, and MQL is considered.

4.2 Maximum energy efficiency

While the energy efficiency index EEI is a relative value related to a defined reference, the energy efficiency η generally relates the energy output W_Out to the energy input W_In:

$$\eta =100{\%}\cdot \frac{W_{Out}}{W_{In}}=100{\%}\cdot \left(1-\frac{W_{L}}{W_{In}}\right).$$

(7)

Equation 7 can be evaluated per operating point, for multiple operating points, or operating cycles.

Figure 12a illustrates the mean efficiency $\overline{\eta }$ of the reference technological setup with MIN10, HCR, and DL with e₁ = 3 ∙ m_n and the highest potential technological setup with PAGW09, eLL, and MQL over the same operating cycles OC|_{{35.3…302.0}Nm}, OC|_{{35.3…183.4}Nm}, OC|_{{35.3…94.1}Nm}, and OC|_{35.3}Nm considered in Fig. 11. As the minimum EEI in Sect. 4.1, the efficiency depends on the considered operating cycle and increases with increasing weighting of operating points with load. The calculated mean efficiency $\overline{\eta }$ for the complete operating cycle increases from 98.3% for the reference technological setup with HCR, MIN10, and DL with e₁ = 3 ∙ m_n to 99.5% for the highest potential technological setup with eLL, PAGW09, and MQL. This increase of approx. 1.2% also applies approximately for operating cycles with a reduced weighting of operating points with load. The results correlate with the experimental findings of Fernandes et al. [29], who found an efficiency increase of up to 1% when combining a polyglycol with a low-loss gear.

Fig. 12

Mean efficiency $\overline{\eta}$ of reference and highest potential technological setup for FZG gearbox for operating cycles (OC) with a different weighting of operating points with load (a) and mean, maximal, and minimum efficiency of reference and highest potential technological setup for FZG gearbox for load-specific operating cycles (OC)

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Figure 12b shows the mean, maximal, and minimal efficiency of the reference technological setup with MIN10, HCR, and DL with e₁ = 3 ∙ m_n and the highest potential technological setup with PAGW09, eLL, and MQL for load-specific operating cycles OC|_{35.3}Nm, OC|_{94.1}Nm, OC|_{183.4}Nm, and OC|_{302.0}Nm. The mean efficiency $\overline{\eta }| _{{T_{1}}}$ is higher compared to Fig. 12a, as only individual loads are considered in the operating cycles, which increases the relevance of the low-loss technologies PAGW09 and eLL for operating cycles at high loads. The overall maximum efficiency η_max for the highest potential technological setup with PAGW09, eLL, and MQL is even 99.9% for individual operating points at T₁ = 183.4 Nm and T₁ = 302.0 Nm.

For the results on the industrial gearbox in Sect. 3.2, an increase of the efficiency η from 99.0 to 99.6% is calculated for the highest potential technological setup with PAGW_IG and eLL_IG.

4.3 Superefficiency and energy efficiency classes

The discussion in Sects. 4.1 and 4.2 shows that the considered low-loss technologies can significantly increase the energy efficiency of the gearbox. For the highest potential technological setups, the calculated values of EEI and $\overline{\eta }$ in Table 10 can be allocated to the FZG gearbox and industrial gearbox and interpreted as the border to superefficiency.

Table 10

Allocation of EEI and $\overline{\eta}$ to superefficiency of the FZG and industrial gearbox with underlying operating cycles (OC) considering the highest potential technological setups

	FZG gearbox				Industrial gearbox
OC	OC\|_{35.3}Nm	OC\|_{{35.3…94.1}Nm}	OC\|_{{35.3…183.4}Nm}	OC\|_{{35.3…302.0}Nm}	Industrial gearbox
EEI in %	51.6	41.7	33.9	27.9	44.3
$\overline{\eta }$ In %	99.1	99.3	99.4	99.5	99.6

Both the minimum EEI and maximum efficiency in Table 10 depend on the considered gearbox and underlying operating cycle. Hence, the values are not generalizable.

For component-specific analysis, operating points of the FZG gearbox based on Sect. 2.2.4 might be explicitly selected and measured to obtain data for a representative operating cycle of a practical gearbox. For example, the established WLTC [59, 60] could be used as an input for a vehicle model, resulting in an applicable operating cycle for transmissions in light vehicles. Simulation programs, like WTplus in this study, can be used to analyze practical gearboxes specifically. Validation is possible by prototype measurements.

The definition of energy efficiency classes for gearboxes can be discussed based on the results of this study. The energy efficiency classes can be based on the energy efficiency index EEI related to the framework for energy labeling of the European Union Regulation 2017/1369 [39]. A reference energy dissipation W_L,ref might be developed for specific product groups based on a calculation formula that considers parameters describing the general gearbox structure. Energy labels can be assigned to the EEI. The letter A can be related to superefficiency.

4.4 Reflection

In this section, methods and results are reflected, and the limitations of the study are presented.

This study is based on calculations with validated methods (cf. Sect. 2.1). However, calculation uncertainties can result according to the accuracy of individual methods. Furthermore, the potentials of the considered low-loss technologies related to gear oil, gears, and lubrication methods were successfully investigated individually (cf. Sect. 1) but not in combination. Hence, the highest potential technological setup with PAGW, eLL, and MQL requires testing and validation.

Introducing the highest potential technological setups into practical gearboxes requires further technological development. In some cases, the reference’s gearbox design likely has to be rethought to implement one or all of these technological setups. For low-loss gears, the face width usually has to be increased to achieve a similar load-carrying capacity as a reference. Furthermore, the noise, vibration, and harshness behavior has to be considered [31]. For water-containing polyglycols with superlubricity in gear contacts [19], challenges like water evaporation and material incompatibilities have to be tackled [15]. For MQL, the required load-carrying capacity has to be considered [36], probably by combining adaptive on-demand lubrication methods with monitoring [61, 62].

The study focused on load-dependent gear power losses P_LGP of cylindrical gears. It is possible to extend the study to other gear types. Furthermore, low-loss technologies might also apply to bearings, seals, and other components, reducing bearing power losses P_LB, seal power losses P_LS, or other power losses P_LX. Hence, even lower energy efficiency indexes and higher energy efficiencies than those obtained in this study might be possible. Otherwise, the EEI and energy efficiency also depend heavily on where the system boundary is drawn for balancing since these depend on the overall energy dissipation caused by the sum of all power losses P_L (cf. Eq. 2). When considering other power losses P_LX, a wider system boundary would mean more considered components, higher absolute power losses P_L and higher energy dissipation. So, if auxiliary units and oil coolers were included in the balancing, the values for the EEI would consequently be higher. However, when using MQL, for instance, an oil pump is actually mandatory. By aligning the system boundary with the gearbox housing, this study did not consider the energy required for this. Further work has to precisely define how to consider other power losses P_LX and where to draw the system boundary to evaluate gearbox systems with different technological setups for practical use.

5 Conclusion

The following conclusions can be drawn from this potential study on increasing the energy efficiency of gearboxes:

Synthetic gear oils, particularly when enabling superlubricity, and low-loss gears can significantly reduce load-dependent power losses.
Low load-dependent power losses enable minimum quantity lubrication, which can subsequently further reduce the no-load power loss.
The highest potential technological setup refers the water-containing polyglycol, extreme low-loss gear, and minimum quantity lubrication. When using the extreme low-loss gear, the effect of the water-containing polyglycol is significantly less prominent, and vice versa. Therefore, implementing just one of the two low-loss technologies can already lead to significant advantages in practical applications.
For the considered operating cycles, superefficiency can be defined by EEI < 27.9% and $\overline{\eta }$ > 99.5% for the FZG gearbox and by EEI < 44.3% and $\overline{\eta }$ > 99.6% for the industrial gearbox.
The minimum EEI and maximum efficiency depend on the considered gearbox and underlying operating cycle.

Future work might focus on defining energy efficiency classes for gearboxes as a common standard. Thereby, reference energy dissipations must be developed for specific product groups, and labels must be assigned, e.g., to the EEI. The letter A can be related to superefficiency.

6 Nomenclature

The symbols are shown in Table 11.

Table 11

Symbols

Symbol	SI-Unit	Name
$\alpha$	$-$	Regression-parameter for Eqs. 14 and 15
$\alpha _{n}$	${^{\circ}}$	Normal pressure angle
$\alpha _{p}$	$m^{2}/N$	Pressure-viscosity-coefficient
$\beta$	$-$	Regression-parameter for Eqs. 14 and 15
$\beta _{b}$	${^{\circ}}$	Helix angle at base diameter
$\gamma$	$-$	Regression-parameter for Eq. 14
$\varepsilon$	$-$	Contact ratio
$\varepsilon _{\alpha }$	$-$	Transverse contact ratio
$\varepsilon _{\beta }$	$-$	Overlap ratio
$\varepsilon _{\gamma }$	$-$	Total contact ratio
$\eta _{M0}$	$Pa\cdot s$	Dyn. viscosity at bulk temp. and ambient pressure
$\eta _{R}$	$Pa\cdot s$	Dyn. reference viscosity
$\vartheta$	${^{\circ}}C$	Temperature
$\lambda _{rel}$	$-$	Relative film thickness
$\mu$	$-$	Coefficient of friction
$\mu _{mz}$	$-$	Mean coefficient of gear friction
$\mu _{sl}$	$-$	Sliding friction coefficient acc. to [63]
$\mu _{EHL}$	$-$	Sliding friction coefficient in full-film conditions acc. to [63]
$\nu$	$m^{2}/s$	Kinematic viscosity
$\xi$	$-$	Fluid load portion
$\rho$	$kg/m{^{3}}$	Density
$\rho _{red}$	$m$	Reduced radius of curvature
$\Upphi _{bl}$	$-$	Weighting factor for the sliding friction coefficient acc. to [63]
$a$	$m$	Center distance
$a_{0\ldots 4}$	$-$	Variable acc. to appendix
$b$	$m$	Tooth width
$d$	$m$	Diameter
$e$	$m$	Immersion depth
$f_{N}$	$N/m$	Line load
$h$	$m$	Oil level
$h_{0}$	$m$	Central film thickness
$m_{n}$	$m$	Normal module
$n$	$s^{-1}$	Rotational speed
$p_{et}$	$m$	Base pitch
$p_{H}$	$N/m^{2}$	Hertzian pressure
$p_{R}$	$N/m^{2}$	Reference pressure
$t$	$s$	Time share
$t_{c}$	$s$	Time of a single contact line during its transition through the gear contact area
$u$	$-$	Gear ratio
$v_{\Upsigma }$	$m/s$	Sum velocity
$v_{s}$	$m/s$	Sliding velocity
$v_{tb}$	$m/s$	Tangential velocity at base diameter
$v_{R}$	$m/s$	Reference velocity
$x$	$-$	Addendum modification coefficient
$z$	$-$	Number of teeth
$A$	$m$	Start of engagement
$E$	$m$	End of engagement
$D$	$-$	Parameter for rotational direction
$F_{bt}$	$N$	Load at base circle
$F_{N}$	$N$	Normal load
$G_{sl}$	$-$	Variable acc. to [63]
$H_{V\left(L\right)}$	$-$	(Local geometric) tooth power loss factor
$P_{In}$	$W$	Input power
$P_{L}$	$W$	Power loss
$P_{f}$	$W$	Frictional power
$\dot{Q}_{out}$	$W$	Heat flow emitted
$Ra$	$m$	Arithmetic mean roughness
$T_{drag}$	$Nm$	Frictional moment of drag losses, churning, splashing etc
$T_{rr}$	$Nm$	Rolling frictional moment
$T_{seal}$	$Nm$	Frictional moment of seals
$T_{sl}$	$Nm$	Sliding frictional moment
$T_{H}$	$Nm$	Hydraulic loss torque
$\dot{V}$	$m^{3}/s$	Oil volume rate
$W_{L}$	$J$	Energy dissipation
$X_{Ca}$	$-$	Profile modification coefficient
$X_{L}$	$-$	Lubricant factor
$X_{OS}$	$-$	Surface structure factor
$X_{S}$	$-$	Lubrication coefficient

The indices are shown in Table 12.

Table 12

Index

Index	Name
0	No-load
1,2	Pinion (1), Wheel (2)
a	Tip/outer
f	Fluid
i	Inner
ref	Reference
s	Solid
Sh	Shaft
B	Bearing
C	Pitch Point C
M	Bulk/tooth mass
P	Load
S	Sealing
X	Other

Acknowledgements

The authors gratefully acknowledge Karsten Stahl for providing the opportunity to conduct the research at the Gear Research Center (FZG) and for the discussions on the topic.

Conflict of interest

T. Lohner and C. Paschold declare that they have no competing interests.

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Metadaten
Anhänge
Literatur

Titel: A pathway towards energy efficiency classes for gearboxes related to superefficiency
Verfasst von: T. Lohner
C. Paschold
Publikationsdatum: 01.12.2025
Verlag: Springer Berlin Heidelberg
Erschienen in: Engineering Research / Ausgabe 1/2025
Print ISSN: 0015-7899
Elektronische ISSN: 1434-0860
DOI: https://doi.org/10.1007/s10010-024-00768-w

Appendix

The load-dependent gear power losses P_LGP are calculated similarly to Eq. 5 considering the geometric tooth power loss factor H_V according to Wimmer [22]:

$$P_{LGP}=P_{In}\cdot \mu _{mz}\cdot H_{V}$$

(8)

With:

$$\begin{aligned}&H_{V}=&&\frac{\pi \cdot \left(u+1\right)}{z_{1}\cdot u\cdot \cos \left(\beta _{b}\right)}\cdot \left(a_{0}+a_{1}\cdot \left| \varepsilon _{1}\right| +a_{2}\cdot \left| \varepsilon _{2}\right| +a_{3}\right.\\&&&\left. \vphantom{(a_{0}+a_{1}\cdot \left| \varepsilon _{1}\right| +a_{2}\cdot \left| \varepsilon _{2}\right| +a_{3}} \cdot \left| \varepsilon _{1}\right| \cdot \varepsilon _{1}+a_{4}\cdot \left| \varepsilon _{2}\right| \cdot \varepsilon _{2}\right)\end{aligned}$$

(9)

Table 13 shows the general formulation of the factors a₀ to a₄ in Eq. 9, with l, m, and n being the contact ratios rounded up to whole numbers:

$$\begin{aligned}&\varepsilon _{1}\in \left[l-1;l\right];\varepsilon _{2}\in \left[m-1;m\right];\varepsilon _{\alpha }\in \left[n-1;n\right]\\&\;\text{with}\:l,m,n\in \mathbb{Z}\end{aligned}$$

(10)

Table 13

General formulation of the factors a₀ to a₄ for calculations of the geometric tooth power loss factor

	Ε_α ≤ 1	Ε_α > 1 ε₁ < 0 ∨ ε₂ < 0	Ε_α > 1 ε₁ > 0 ∧ ε₂ > 0 $l+m=n$	Ε_α > 1 ε₁ > 0 ∧ ε₂ > 0 $l+m=n+1$
a₀	$0$	$0$	$\frac{2\cdot l\cdot m}{n}$	$\frac{2\cdot \left(l\cdot m-n\right)}{\left(n-1\right)}$
a₁	$0$	$1$	$\frac{l\cdot \left(l-1\right)-m\cdot \left(m-1\right)-2\cdot l\cdot m}{n\cdot \left(n-1\right)}$	$\frac{l\cdot \left(l-1\right)+m\cdot \left(m-1\right)-2\cdot \left(m-1\right)\cdot n}{n\cdot \left(n-1\right)}$
a₂	$0$	$1$	$\frac{-l\cdot \left(l-1\right)+m\cdot \left(m-1\right)-2\cdot l\cdot m}{n\cdot \left(n-1\right)}$	$\frac{l\cdot \left(l-1\right)+m\cdot \left(m-1\right)-2\cdot \left(l-1\right)\cdot n}{n\cdot \left(n-1\right)}$
a₃	$\frac{1}{\varepsilon _{\alpha }}$	$0$	$\frac{2\cdot m}{n\cdot \left(n-1\right)}$	$\frac{2\cdot \left(m-1\right)}{n\cdot \left(n-1\right)}$
a₄	$\frac{1}{\varepsilon _{\alpha }}$	$0$	$\frac{2\cdot l}{n\cdot \left(n-1\right)}$	$\frac{2\cdot \left(l-1\right)}{n\cdot \left(n-1\right)}$

The mean coefficient of gear friction µ_mz is calculated using the approaches according to Doleschel [52] and Schlenk [13]. Doleschel [52] considers the lubrication regime by applying the load-sharing concept:

$$\mu _{mz,\textit{Doleschel}}=\xi _{C}\cdot \mu _{mz,f}+\left(1-\xi _{C}\right)\cdot \mu _{mz,s}$$

(11)

$$\xi _{C}=\begin{cases} 1-\left(1-X_{OS}\cdot \lambda _{rel,C}\right)^{2}, & \lambda _{rel,C}< 2\\ 1, & \lambda _{rel,C}\geq 2 \end{cases}$$

(12)

$$\lambda _{rel,C}=\frac{h_{0,C}}{0.5\cdot \left(Ra_{1}+Ra_{2}\right)}$$

(13)

The central film thickness h_0,C is calculated according to Ertel/Grubin [64, 65] and thermally corrected according to Murch and Wilson [58]. The mean fluid (f) and solid (s) coefficient of friction are calculated as follows:

$$\mu _{mz,f}=\mu _{ref,f}\cdot \left(\frac{p_{H,C}}{p_{R}}\right)^{{\alpha _{f}}}\cdot \left(\frac{v_{\Upsigma ,C}}{v_{R,f}}\right)^{{\beta _{f}}}\cdot \left(\frac{\eta _{M0}}{\eta _{R}}\right)^{{\gamma _{f}}}$$

(14)

$$\mu _{mz,s}=\mu _{ref,s}\cdot \left(\frac{p_{H,C}}{p_{R}}\right)^{{\alpha _{s}}}\cdot \left(\frac{v_{\Upsigma ,C}}{v_{R,s}}\right)^{{\beta _{s}}}$$

(15)

The parameters μ_ref,f, α_f, β_f, γ_f, μ_ref,s, α_s and β_s are tribosystem-specific, i.e., specific to a particular lubricant and a tooth flank surface, and can be derived from a gear power loss test FZG-E-C/0,5:20/5:9/40:120 according to FVA 345 [41, 42].

Compared to Doleschel [52], Schlenk’s [13] approach does not consider the lubrication regime and is, therefore, more simplistic. It determines the mean coefficient of friction µ_mz by a small set of input parameters:

$$\begin{aligned}&\mu _{mz,\textit{Schlenk}}=&&0.048\cdot \left(\frac{F_{bt}/b}{v_{\Upsigma ,C}\cdot \rho _{red,C}}\right)^{0.2}\cdot {\eta _{oil}}^{-0.05}\\&&&\cdot \left(\frac{Ra_{1}+Ra_{2}}{2}\right)^{0.25}\cdot X_{L}\end{aligned}$$

(16)

$$X_{L}=\begin{cases} 1.0, & \text{for mineral oils}\\ 0.8, & \text{for polyalphaolefins and esters}\\ 0.75\cdot \left(6/v_{\Upsigma ,C}\right)^{0.2}, & \text{for polyglycols}\\ 1.3, & \text{for phosphoric esters}\\ 1.5, & \text{for traction oils} \end{cases}$$

(17)

As the oil viscosity η_Oil is strongly dependent on temperature and significantly influences µ_mz, its calculation results are consequently affected by choice of temperature in calculating η_Oil. Usually, the oil temperature ϑ_Oil is considered when calculating the oil viscosity to determine µ_mz since it is a measure that is easy to determine in practice. However, ϑ_Oil usually represents a mean measure of the gearbox. Depending on the operating point, the gear bulk temperature ϑ_M differs significantly from ϑ_Oil, as does η_Oil. So, to increase the predictive quality of the calculation, the gear bulk temperature ϑ_M, according to Oster ([66]; Eq. 18) and the extension of Otto ([67]; Eq. 19), is used in this study to calculate the oil viscosity used for determining µ_mz.

$$\vartheta _{M}=\vartheta _{Oil}+7400\cdot \left(\frac{P_{LGP}}{a\cdot b}\right)^{0.72}\cdot \frac{X_{S}}{1.2\cdot X_{Ca}}$$

(18)

The profile modification coefficient X_Ca is calculated according to ISO 6336‑1 [48] and ISO/TS 6336-21 [68]. The lubrication coefficient X_S is calculated with reference to the wheel as follows:

$$0.3\leq X_{S}=0.35\cdot \left(\frac{e_{2}}{d_{a2}}\right)^{-D}\leq 3.7$$

(19)

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. A constant value of X_S = 3.7 is used for minimum quantity lubrication, with negligible heat dissipation by cooling oil.

The no-load gear power losses are calculated according to Mauz [53], who derived an extensive set of formulas from comprehensive experimental investigations. Generally, a hydraulic loss torque T_H is calculated per gear (1, 2), consisting of a churning loss torque and a squeezing loss torque. The oil viscosity, circumferential speed of the gears, gears and gearbox dimensions, and immersing gear surface in operation are considered. Additionally, wall distance factors are used at the oil inlet and outlet side, as well as a module factor and an oil volume factor. Adding up the shares of the hydraulic loss torques of both gears, T_H1 and T_H2, gives the total hydraulic loss torque for a gear stage. The overall hydraulic power loss of a gear stage P_LG0 reads:

$$P_{LG0}=2\cdot \pi \cdot \left(T_{H1}\cdot n_{1}+T_{H2}\cdot n_{2}\right)$$

(20)

The rolling bearing power losses P_LB are calculated according to the current SKF approach [63] considering “oil bath lubrication,” which calculates the total loss torque of an individual rolling bearing by summing up the rolling frictional moment T_rr, sliding frictional moment T_sl, frictional moment of seals T_seal, and frictional moment of drag and churning losses T_drag:

$$\begin{aligned}&P_{LB}=\\&\underset{P_{LBP}}{\underbrace{2\cdot \pi \cdot \left(T_{rr}+\underset{T_{sl}}{\underbrace{G_{sl}\cdot \overset{\mu _{sl}}{\overbrace{\left(\Upphi _{bl}\cdot \mu _{bl}+\left(1-\Upphi _{bl}\right)\cdot \mu _{EHL}\right)} } } }\right)\cdot n} }\\&+\underset{P_{LB0}}{\underbrace{2\cdot \pi \cdot \left(T_{seal}+T_{drag}\right)\cdot n} }\end{aligned}$$

(21)

The single shares are calculated by a comprehensive set of equations, considering various operating parameters such as load, rolling bearing type, rotational speed, immersion depth of the rolling bearing, or the oil viscosity. The approach is well documented by SKF [63]. It should be mentioned that for calculating the sliding coefficient of friction μ_sl, the base oil classification for the sliding friction coefficient in fluid film lubrication μ_EHL as described in [63] is used. Still, one modification is made compared to the SKF approach, which concerns calculating the drag loss torque T_drag. The calculation approach calculates the drag loss torque of the rolling bearing T_drag by considering an oil level at the rolling bearing h_B measured from the inner diameter of the bearing outer ring. According to SKF’s approach [63], an oil level below the bearing results in a drag loss torque of 0. However, a minimum oil level h_B at the rolling bearings is considered in this study to prevent the no-load rolling bearing power losses from becoming 0, even if the actual immersion depth of the rolling bearing e_B is 0:

$$h_{B}=\begin{cases} e_{B}-\frac{\left(d_{Ba}-d_{Bi}\right)}{6}, & h_{B}\geq \frac{d_{Ba}-d_{Bi}}{4}-\frac{d_{Ba}-d_{Bi}}{6}\\ \frac{d_{Ba}-d_{Bi}}{4}-\frac{d_{Ba}-d_{Bi}}{6}, & h_{B}< \frac{d_{Ba}-d_{Bi}}{4}-\frac{d_{Ba}-d_{Bi}}{6} \end{cases}$$

(22)

The sealing power losses P_LS are only speed-dependent and are calculated according to [69]:

$$P_{LS}=7.69\cdot 10^{-6}\cdot {d_{sh}}^{2}$$

(23)

When determining a steady-state oil temperature, it is iteratively calculated by finding the equilibrium between the overall power losses P_L of the gearbox and the heat dissipation via the housing for a specific operating condition. The heat dissipation via the housing is calculated according to ISO/TR 14179‑2 [6], considering the housing dimensions, oil volume, environmental airflow and temperature, and material properties. The gearboxes do not have oil coolers to consider. Note that the power losses P_L and the heat dissipation via the housing $\dot{Q}_{out}$ strongly depend on the oil temperature ϑ_Oil.

$$P_{L}\left(\vartheta _{oil}\right)\overset{!}{=}\dot{Q}_{out}\left(\vartheta _{oil}\right)$$

(24)

The oil temperature ϑ_Oil has to be varied iteratively until $\dot{Q}_{out}$ matches P_L. Since the power losses result from the calculation with the “old” ϑ_Oil, the power losses are calculated again using the “new” ϑ_Oil. Again, this is iteratively repeated until the power losses between iterations are smaller than a given threshold. For further information, Paschold et al. [49] explain this iterative calculation approach in detail.

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Symbol	SI-Unit	Name
\(\alpha\)	\(-\)	Regression-parameter for Eqs. 14 and 15
\(\alpha _{n}\)	\({^{\circ}}\)	Normal pressure angle
\(\alpha _{p}\)	\(m^{2}/N\)	Pressure-viscosity-coefficient
\(\beta\)	\(-\)	Regression-parameter for Eqs. 14 and 15
\(\beta _{b}\)	\({^{\circ}}\)	Helix angle at base diameter
\(\gamma\)	\(-\)	Regression-parameter for Eq. 14
\(\varepsilon\)	\(-\)	Contact ratio
\(\varepsilon _{\alpha }\)	\(-\)	Transverse contact ratio
\(\varepsilon _{\beta }\)	\(-\)	Overlap ratio
\(\varepsilon _{\gamma }\)	\(-\)	Total contact ratio
\(\eta _{M0}\)	\(Pa\cdot s\)	Dyn. viscosity at bulk temp. and ambient pressure
\(\eta _{R}\)	\(Pa\cdot s\)	Dyn. reference viscosity
\(\vartheta\)	\({^{\circ}}C\)	Temperature
\(\lambda _{rel}\)	\(-\)	Relative film thickness
\(\mu\)	\(-\)	Coefficient of friction
\(\mu _{mz}\)	\(-\)	Mean coefficient of gear friction
\(\mu _{sl}\)	\(-\)	Sliding friction coefficient acc. to [63]
\(\mu _{EHL}\)	\(-\)	Sliding friction coefficient in full-film conditions acc. to [63]
\(\nu\)	\(m^{2}/s\)	Kinematic viscosity
\(\xi\)	\(-\)	Fluid load portion
\(\rho\)	\(kg/m{^{3}}\)	Density
\(\rho _{red}\)	\(m\)	Reduced radius of curvature
\(\Upphi _{bl}\)	\(-\)	Weighting factor for the sliding friction coefficient acc. to [63]
\(a\)	\(m\)	Center distance
\(a_{0\ldots 4}\)	\(-\)	Variable acc. to appendix
\(b\)	\(m\)	Tooth width
\(d\)	\(m\)	Diameter
\(e\)	\(m\)	Immersion depth
\(f_{N}\)	\(N/m\)	Line load
\(h\)	\(m\)	Oil level
\(h_{0}\)	\(m\)	Central film thickness
\(m_{n}\)	\(m\)	Normal module
\(n\)	\(s^{-1}\)	Rotational speed
\(p_{et}\)	\(m\)	Base pitch
\(p_{H}\)	\(N/m^{2}\)	Hertzian pressure
\(p_{R}\)	\(N/m^{2}\)	Reference pressure
\(t\)	\(s\)	Time share
\(t_{c}\)	\(s\)	Time of a single contact line during its transition through the gear contact area
\(u\)	\(-\)	Gear ratio
\(v_{\Upsigma }\)	\(m/s\)	Sum velocity
\(v_{s}\)	\(m/s\)	Sliding velocity
\(v_{tb}\)	\(m/s\)	Tangential velocity at base diameter
\(v_{R}\)	\(m/s\)	Reference velocity
\(x\)	\(-\)	Addendum modification coefficient
\(z\)	\(-\)	Number of teeth
\(A\)	\(m\)	Start of engagement
\(E\)	\(m\)	End of engagement
\(D\)	\(-\)	Parameter for rotational direction
\(F_{bt}\)	\(N\)	Load at base circle
\(F_{N}\)	\(N\)	Normal load
\(G_{sl}\)	\(-\)	Variable acc. to [63]
\(H_{V\left(L\right)}\)	\(-\)	(Local geometric) tooth power loss factor
\(P_{In}\)	\(W\)	Input power
\(P_{L}\)	\(W\)	Power loss
\(P_{f}\)	\(W\)	Frictional power
\(\dot{Q}_{out}\)	\(W\)	Heat flow emitted
\(Ra\)	\(m\)	Arithmetic mean roughness
\(T_{drag}\)	\(Nm\)	Frictional moment of drag losses, churning, splashing etc
\(T_{rr}\)	\(Nm\)	Rolling frictional moment
\(T_{seal}\)	\(Nm\)	Frictional moment of seals
\(T_{sl}\)	\(Nm\)	Sliding frictional moment
\(T_{H}\)	\(Nm\)	Hydraulic loss torque
\(\dot{V}\)	\(m^{3}/s\)	Oil volume rate
\(W_{L}\)	\(J\)	Energy dissipation
\(X_{Ca}\)	\(-\)	Profile modification coefficient
\(X_{L}\)	\(-\)	Lubricant factor
\(X_{OS}\)	\(-\)	Surface structure factor
\(X_{S}\)	\(-\)	Lubrication coefficient

	Ε_α ≤ 1	Ε_α > 1 ε₁ < 0 ∨ ε₂ < 0	Ε_α > 1 ε₁ > 0 ∧ ε₂ > 0 \(l+m=n\)	Ε_α > 1 ε₁ > 0 ∧ ε₂ > 0 \(l+m=n+1\)
a₀	\(0\)	\(0\)	\(\frac{2\cdot l\cdot m}{n}\)	\(\frac{2\cdot \left(l\cdot m-n\right)}{\left(n-1\right)}\)
a₁	\(0\)	\(1\)	\(\frac{l\cdot \left(l-1\right)-m\cdot \left(m-1\right)-2\cdot l\cdot m}{n\cdot \left(n-1\right)}\)	\(\frac{l\cdot \left(l-1\right)+m\cdot \left(m-1\right)-2\cdot \left(m-1\right)\cdot n}{n\cdot \left(n-1\right)}\)
a₂	\(0\)	\(1\)	\(\frac{-l\cdot \left(l-1\right)+m\cdot \left(m-1\right)-2\cdot l\cdot m}{n\cdot \left(n-1\right)}\)	\(\frac{l\cdot \left(l-1\right)+m\cdot \left(m-1\right)-2\cdot \left(l-1\right)\cdot n}{n\cdot \left(n-1\right)}\)
a₃	\(\frac{1}{\varepsilon _{\alpha }}\)	\(0\)	\(\frac{2\cdot m}{n\cdot \left(n-1\right)}\)	\(\frac{2\cdot \left(m-1\right)}{n\cdot \left(n-1\right)}\)
a₄	\(\frac{1}{\varepsilon _{\alpha }}\)	\(0\)	\(\frac{2\cdot l}{n\cdot \left(n-1\right)}\)	\(\frac{2\cdot \left(l-1\right)}{n\cdot \left(n-1\right)}\)

Springer Professional

Abstract

Publisher’s Note

1 Introduction

2 Methods and materials

2.1 Simulation program

2.2 FZG gearbox

2.2.1 Gears

2.2.2 Gear oils

2.2.3 Lubrication methods

2.2.4 Calculation procedure

2.3 Industrial gearbox

3 Results

3.1 FZG gearbox

3.1.1 Mean coefficient of gear friction

3.1.2 Load-dependent gear power loss

3.1.3 Gear bulk temperature

3.1.4 Accumulated energy dissipation

3.1.5 Energy efficiency index

3.2 Industrial gearbox

4 Discussion

4.1 Minimum energy efficiency index

4.2 Maximum energy efficiency

4.3 Superefficiency and energy efficiency classes

4.4 Reflection

5 Conclusion

6 Nomenclature

Acknowledgements

Conflict of interest

Publisher’s Note

Appendix