Power Flow and Efficiency Analysis of High-Speed Heavy Load Herringbone Planetary Transmission Using a Hypergraph-Based Method

applied sciences

Article Power Flow and Eﬃciency Analysis of High-Speed Heavy Load Herringbone Planetary Transmission Using a Hypergraph-Based Method

Yun Wang 1, Wei Yang 2,*, Xiaolin Tang 2, Xi Lin 2 and Zhonghua He 1

1 State Key Laboratory of Mechanical Transmission, College of Mechanical Engineering, Chongqing University, Chongqing 400044, China; [email protected] (Y.W.); [email protected] (Z.H.) 2 State Key Laboratory of Mechanical Transmission, College of Automotive Engineering, Chongqing University, Chongqing 400044, China; [email protected] (X.T.); [email protected] (X.L.) * Correspondence: [email protected]; Tel.: +86-236-511-2031

Received: 19 July 2020; Accepted: 19 August 2020; Published: 24 August 2020

Abstract: The existing research problem is that the hypergraph-based method does not comprehensively consider the power loss factors of the mechanical transmission system, especially the existing windage power loss. In the power flow diagram, it is difficult to express and calculate the power loss separately, based on the power loss mechanism. Furthermore, the hypergraph-based methods purposed by some researches are not suitable for high-speed and heavy-load conditions. This paper presents a new hypergraph-based method for analyzing the transmission efficiency of complex planetary gear trains. In addition, a formula for calculating transmission efficiency is derived. The power loss model is established for a high-speed heavy-load herringbone planetary transmission pair considering several sources of power losses such as gear meshing friction, windage, and bearing friction. Then, the efficiency of a two-stage herringbone planetary transmission system is calculated using the proposed method. Furthermore, the influence of input speed, input power, and lubrication state on transmission efficiency is investigated. Finally, the proposed method is verified by comparing calculated values with values published in the literature.

Keywords: herringbone gear; planetary transmission; transmission efficiency; power flow; hypergraph

1. Introduction

1.1. Motivation and Challenge The herringbone planetary gear train is widely used in high-speed heavy-duty transmissions owing to its simple and compact structure and several other advantages such as split path power transmission, high power efficiency, and large transmission range [1–3]. As the power transmitted by the gear train increases, large vibrations can increase the temperature of the machine and along with other power losses can lead to wear, which reduces the working life and reliability of the machine [4–6]. For these reasons, power loss mechanisms of high-speed heavy-duty helical gear transmission are widely studied [7–12]. Models of tooth surface friction were previously established to accurately predict power losses associated with gear meshing. However, most models do not consider the influence of other sources of power loss in gear trains. As a consequence, analytical expressions of transmission efficiency for any type of planetary gear train may be higher than the actual transmission efficiency. Most graph-based methods for analyzing transmission efficiency define the efficiency of each planetary gear stage as a fixed value according to meshing power loss, which is then used to calculate the overall

Appl. Sci. 2020, 10, 5849; doi:10.3390/app10175849 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 5849 2 of 25 transmission eﬃciency of the planetary gearbox. However, this is not consistent with real-world scenarios. To precisely model the power loss and eﬃciency of a planetary gear transmission system, various loss factors should be considered.

1.2. Literature Review To analyze gear meshing losses, Marques et al. [13] proposed a dynamic lumped mass model of a four-degree-of-freedom (4-DOF) gear system, and two separate formulas for friction coefficient were used to calculate friction power losses for gears of different geometries under various working conditions. However, the influence of different lubrication conditions on gear transmission efficiency was not been considered in the study. Diez-Ibarbia et al. [14] studied the influence factor of friction factor on the efficiency of spur gears with tip reliefs, calculated them using the average friction factor along the mesh cycle, and then compared with values obtained using an enhanced friction factor formula based on the fundamentals of elastohydrodynamic lubrication, but the study does not consider the effect of temperature on the elastohydrodynamic lubrication. Fernandes et al. [15] used experimental data to derive the average friction factor of meshing gear pairs, and discussed the relationship between friction coefficient and the type of lubricating oil. Petrishchev et al. [16] designed a machine for testing power losses and then experimentally determined the variation in gear frictional losses during each individual tooth engagement period. Shao et al. [17] calculated the frictional power loss on the tooth surface and meshing efficiency of a planetary gear train by considering elastic deformation of the flexible ring gear and dynamic friction in the contact region, but did not consider the influence of different lubrication conditions on gear transmission efficiency. Wang et al. [18] proposed a method for calculating gear meshing efficiency using data from a gear testing machine. Recently, graph-based methods have become a major research focus [19]. Graph-based methods can be used to calculate the transmission efficiency of epicyclic gear trains, and effectively remove the need to manually determine the power flow direction, which can be cumbersome and error-prone. Liu et al. [20] proposed a method for designing automatic transmission configuration schemes based on the lever analogy. In addition, they established a process for synthesizing multi-row multi-speed automatic transmissions; whereas a three-row mixed connection was the preferred connection, both two-row and three-row connections were demonstrated. Gomà i Ayats et al. [21] presented a hypergraph-based approach for the kinematic analysis of complex gear trains. The hypergraphs were used to aid in the understanding of the inter-relation between the various branches of the mechanism and to develop the kinematic equations. Laus et al. [22] proposed a method for calculating the efficiency of complex gear trains using graph and screw theory and established a one-to-one mapping relation between friction and efficiency, but the efficiency calculation considers the tooth surface friction power loss, ignoring other power losses. Chen et al. [23] used graph theory to model a 2-DOF planetary gear transmission and a constraint analysis and the virtual power ratio were used to predict the efficiency of the system. Similarly, Li et al. [24] performed motion, torque, and power balances of a single-stage planetary gear transmission, plotted the power flow diagram of a 2K-H planetary gear transmission, and derived a power distribution law for the closed planetary gear transmission system. Finally, Yang et al. [25] proposed a power flow analysis method based on matrix operations to determine power flow and transmission efficiencies of multi-stage planetary gear transmission systems.

1.3. Scientific Contribution Energy saving and reducing consumption are crucial for sustainable development of economies, and efficiency is an important performance indicator of machinery. The primary objective of this work is to develop a power flow diagram for modeling and analyzing the efficiency of planetary transmission systems. Herein, we put forward hypergraphs for analyzing power loss mechanisms within the Appl. Sci. 2020, 10, 5849 3 of 25

Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 27 planetary gear transmission structure. Then, precise modeling is combined with the graph-based 1.4. Outline of the Paper method to analyze the overall efficiency of a planetary gear transmission system. The rest of this paper is organized as follows. Hypergraphs of the planetary gear train are 1.4.detailed Outline in Section of the Paper 2. In Section 3, we introduce formulas for calculating transmission efficiency by followingThe rest the of power this paper flow. is A organized power flow as follows. diagram Hypergraphs of the herringbone of the planetary gear planetary gear train drive are system detailed is inpresented Section2 .in In Section Section 34., weIn introduceSection 5, formulas the prop forosed calculating method transmissionis used to analyze e fficiency power by followingloss and thetransmission power flow. efficiency A power flowof a diagramtwo-stage of theherrin herringbonegbone gear gear transmission planetary drive system system under is presented various in Sectionlubrication4. In states, Section input5, the proposedspeeds, and method input is power. used to Finally, analyze the power main loss conclusions and transmission of the estudyfficiency are ofsummarized a two-stage in herringbone Section 6. gear transmission system under various lubrication states, input speeds, and input power. Finally, the main conclusions of the study are summarized in Section6. 2. Loss Mechanism Hypergraph of Single-Stage Planetary Gear Transmission 2. Loss Mechanism Hypergraph of Single-Stage Planetary Gear Transmission 2.1. Basic Structure of Planetary Transmission System 2.1. Basic Structure of Planetary Transmission System As shown in Figure 1, the herringbone gear is comprised of two congruent helical gears with As shown in Figure1, the herringbone gear is comprised of two congruent helical gears with opposite tooth helical angle, with left-handed teeth on one side, right-handed teeth on the other side, opposite tooth helical angle, with left-handed teeth on one side, right-handed teeth on the other side, and a tool withdrawal groove in the middle. The helical angles of two rows of gears are the same size, and a tool withdrawal groove in the middle. The helical angles of two rows of gears are the same size, but with forces acting in opposite directions, and thus offset axial forces. but with forces acting in opposite directions, and thus offset axial forces.

(a) (b)

Figure 1.1. Photographs of herringbone tooth drive. (a) Structure of herringbone gear;gear; ((b)) herringboneherringbone gear transmission.transmission.

The structure of the herringbone gear planetary transmissiontransmission system is illustrated in Figure 22a,b.a,b. TheThe systemsystem consistsconsists ofof threethree planetaryplanetary gearsgears and powerpower isis ﬁrstfirst splitsplit acrossacross thethe planetaryplanetary gears,gears, thenthen transmittedtransmitted throughthrough anan additionaladditional sunsun gear,gear, whichwhich convergesconverges onon thethe internalinternal gears.gears. TheThe sunsun geargear andand planetaryplanetary gearsgears of of the the system system have ha herringboneve herringbone teeth teeth and theand inner the inne ringr gear ring engages gear engages with two with internal two helicalinternal gears. helical The gears. herringbone The herringbone gear eliminates gear eliminates axial forces, axial as forces, described as described above. above.

2.2. Loss Mechanism of Single-Stage Planetary Gear Transmission

2.2.1. Hypergraph of Single-Stage Planetary Gear Transmission The structure of the single-stage planetary gear transmission structure is shown in Figure2, including the frame, sun gear, planetary gear, ring gear, bearing surface contacts, and teeth surface contacts. The topology of the system is presented as a hypergraph in Figure2d, in which numbered vertices represent components of the gear system and edges represent kinematic pairs. Appl. Sci. 2020, 10, 5849 4 of 25 Appl. Sci. 2020, 10, x FOR PEER REVIEW 4 of 27

(a) (b)

FigureFigure 2.2. SchematicSchematic representation representation of herringbone of herringbone planetary planetary gear transmission gear transmission system. (a ) Herringbonesystem. (a) Herringboneplanetary gear planetary structure; gear (b )structure; cross-sectional (b) cross-sectional view of planetary view gear;of planetary (c) schematic gear; (c of) schematic transmission of transmissionstructure; (d) structure; hypergraph (d) of hypergraph power loss of mechanism power loss (1, mechanism 2, 3, and 4 denote(1, 2, 3, the and frame, 4 denote sun the gear, frame, planetary sun gear,gear, planetary and ring gear, gear, respectively; and ring gear, R1 respectively;,R2, and R3 represent R1, R2, and bearing R3 represent contacts bearing and G contacts2-3 and G and2-4 represent G2-3 and Gtooth2-4 represent surface contacts,tooth surface respectively). contacts, respectively).

2.2. LossA single-stage Mechanism of planetary Single-Sta gearge Planetary transmission Gear canTransmission be represented as a diamond-shaped hypergraph with four connected nodes. Each node corresponds to one main branch of the gear train including 2.2.1.the carrier, Hypergraph ring gear, of Single-Stage sun gear, and Planetary frame. The Gear sun Transmission gear and frame, planet gear and carrier, and ring gear and frame are connected by bearings, denoted R , R , and R , respectively. In addition, a teeth The structure of the single-stage planetary gear 1transmission2 3 structure is shown in Figure 2, surface contact link exists between the sun gear and planetary gear and between the planetary gear including the frame, sun gear, planetary gear, ring gear, bearing surface contacts, and teeth surface and ring gear. contacts. The topology of the system is presented as a hypergraph in Figure 2d, in which numbered vertices2.2.2. Power represent Loss components Mechanism ofof Single-Stagethe gear system Planetary and edges Gear represent Transmission kinematic System pairs. A single-stage planetary gear transmission can be represented as a diamond-shaped hypergraph with Asfour shown connected in Figure nodes.2d, powerEach node is transmitted corresponds to theto one helical main gear branch planetary of the drive gear via train the including sun gear, thewhich carrier, is in directring gear, contact sun with gear, the and frame. frame. The The frame sun isgear incontact and frame, with planet the bearings gear and of the carrier, planetary and ring gear and ring gear, resulting in bearing power losses P , P , and P . Both the sun gear-planetary gear gear and frame are connected by bearings, denotedR1 R1,R2 R2, and R3R3, respectively. In addition, a teeth surfacemeshing contact pair and link ring exists gear-planetary between the gearsun gear meshing and planetary pair are in gear contact and at between the teeth the surface, planetary and gear gear andpower ring losses gear.P G1 and PG2 are generated. Finally, power is transmitted via the ring gear to the output Appl. Sci. 2020, 10, 5849 5 of 25 shaft. Power losses can be divided into two categories: power loss due to bearing contact and power loss due to gear meshing. Gears of the planetary gear transmission system are mounted on a rotating shaft, which is typically supported by a roller bearing and ball bearing. Power loss due to friction of the roller bearing is

4 PR = 1.05 10− nM (1) × where n is bearing speed and M is bearing friction torque. Power loss associated with gears meshing in the lubricating oil-gas mixture, including power loss due to friction on the tooth surface and windage power loss, is

PG = Pm + Pω (2) where Pm is tooth surface friction power loss and Pω is windage power loss.

2.2.3. Structural Kinematics of Single-Stage Planetary Gear Transmission The gear ratio of sun gear 2 and planetary gear 3 is

z Number of teeth of wheel 3 H = 3 = (3) ±z2 ±Number of teeth of wheel 2 where zp is the number of planetary gear teeth and zs is the number of sun gear teeth. Figure3 is kinematic relationship of a single-stage planetary gear. For each single-stage planetary gear transmission in the system hypergraph, the Willis equation can be obtained as

ω ω H = 2 − 1 (4) ω ω 3 − 1 For a single-stage planetary gear transmission system,

T1 + T2 + T3 = 0 (5)

P2 + P2 + P3 = 0 (6) P ω 3 = H 3 (7) P2 − ω2 P 1 H ω 1 = − 1 (8) P3 H ω2 where T, P and ω represent torque, power, and angular velocity, respectively; subscripts 1, 2, and 3 denoteAppl. Sci. the 2020 planet, 10, x FOR carriers, PEER REVIEW sun gear, and planet gear, respectively. 6 of 27

FigureFigure 3. Kinematic3. Kinematic relationship relationship between between gear gear 2, gear2, gear 3, and3, and carrier carrier 1. 1.

3. Efficiency Analysis of Herringbone Planetary Gear Power loss along the transmission path in the herringbone planetary transmission is mainly owing to gear meshing transmission power loss and bearing contact power loss. The main factors that contribute to gear transmission power loss are illustrated in Figure 4. Herein, an accurate mathematical model of the gear transmission power loss is established and will lay the theoretical foundation for the subsequent power flow and efficiency analysis.

Figure 4. Main factors affecting gear transmission power loss.

3.1. Bearing Contact Power Loss Under heavy load and high-speed conditions, the herringbone planetary drive relies on the line contact between the rollers of the cylindrical roller bearing and raceway to withstand large radial loads. However, sliding and rolling contact between the cylindrical roller and inner and outer raceway surfaces, as well as viscosity of the lubricant and contact between the cylindrical roller and bearing cage, generate friction torque. Because the bearings were under high-speed and heavy-load conditions, the calculation model of friction torque introduced by SKF is more accurate. The amount of friction torque will depend on the type of bearing and its geometry. The parameters of the bearing are listed in Table 1.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 27

Appl. Sci. 2020, 10, 5849 6 of 25 Figure 3. Kinematic relationship between gear 2, gear 3, and carrier 1.

3. Efficiency Efficiency Analysis of Herringbone Planetary Gear Power loss along the transmission path in the herringbone planetary transmission is mainly owing to geargear meshingmeshing transmission transmission power power loss loss and and bearing bearing contact contact power power loss. loss. The The main main factors factors that thatcontribute contribute to gear to transmission gear transmission power losspower are illustratedloss are illustrated in Figure4 .in Herein, Figure an 4. accurate Herein, mathematical an accurate mathematicalmodel of the gear model transmission of the gear power transmission loss is established power loss and is willestablished lay the theoretical and will lay foundation the theoretical for the foundationsubsequent for power the subsequent flow and effi powerciency flow analysis. and efficiency analysis.

FigureFigure 4. 4. MainMain factors factors affecting aﬀecting gear gear transmission power loss. 3.1. Bearing Contact Power Loss 3.1. Bearing Contact Power Loss Under heavy load and high-speed conditions, the herringbone planetary drive relies on the line Under heavy load and high-speed conditions, the herringbone planetary drive relies on the line contact between the rollers of the cylindrical roller bearing and raceway to withstand large radial contact between the rollers of the cylindrical roller bearing and raceway to withstand large radial loads. However, sliding and rolling contact between the cylindrical roller and inner and outer raceway loads. However, sliding and rolling contact between the cylindrical roller and inner and outer surfaces, as well as viscosity of the lubricant and contact between the cylindrical roller and bearing raceway surfaces, as well as viscosity of the lubricant and contact between the cylindrical roller and cage, generate friction torque. Because the bearings were under high-speed and heavy-load conditions, bearing cage, generate friction torque. Because the bearings were under high-speed and heavy-load the calculation model of friction torque introduced by SKF is more accurate. The amount of friction conditions, the calculation model of friction torque introduced by SKF is more accurate. The amount torque will depend on the type of bearing and its geometry. The parameters of the bearing are listed of friction torque will depend on the type of bearing and its geometry. The parameters of the bearing in Table1. are listed in Table 1. Table 1. Main parameters of cylindrical roller bearing.

Parameter R1 S1 S2 KL0 KZ0 Krs VM0 Value 1.48 10 6 0.16 1.5 10 3 0.65 5.1 6 10 8 1.2 10 4 × − × − × − × −

In Table1, R1 is a geometric constant based on bearing type; S1 is a constant based on bearing type; S2 is a constant based on bearing type and seal; Kr0, KL0, and KZ0 are geometric constants based on bearing type; and VM0 is the drag loss variable. The frictional moment of the cylindrical roller bearing is [26–28]

M0 = φriφrsMr0 + Ms0 + Mse0 + Md0 (9) Appl. Sci. 2020, 10, 5849 7 of 25

where M0 is total friction torque of the bearing, Mr0 is rolling friction torque of the bearing, Ms0 is sliding friction torque of the bearing, Mse0 is friction torque of the bearing seal ring, Md0 is friction torque due to eddy current drag and splash, φri is the heat reduction factor, and φrs is the lubricating oil backﬁll reduction factor. A single row of cylindrical roller bearings (SKF bearing series) that are not fully loaded is adopted in the study with no seal ring structure; therefore, the eﬀect of the frictional moment of the seal ring is not considered, Mse0 = 0.

3.1.1. Rolling Friction Moment of Cylindrical Roller Bearing The rolling friction moment of the cylindrical roller bearing is [26–28]   = (µ )0.6  Mr0 Gr0 nz  2.41 0.31  Gr0 = R1dm Fr (10)  2Ttanϑ  Fr = d1cos β where nz is bearing speed, u is dynamic viscosity, Gr0 is the rolling friction variable of the bearing, dm is the pitch diameter of the bearing, ϑ is pressure angle of meshing gear, and Fr is radial load of the bearing.

3.1.2. Sliding Friction Moment of Cylindrical Roller Bearing The sliding friction moment of roller bearing can be calculated as [26–28] ( Ms0 = Gs0µs0 0.9 (11) Gs0 = S1dm Fa + S2dmFr where Gs0 is the sliding friction variable of bearing, µs0 is the sliding friction coeﬃcient, and Fa is axial load of the bearing.

3.1.3. Drag Friction Moment of Cylindrical Roller Bearing In a high-speed heavy-duty gearbox, frictional torque of the bearing due to drag caused by viscosity of the lubricating oil has the largest inﬂuence on total frictional torque, regardless of whether the bearing is partially or wholly immersed in lubricating oil in the gearbox. Without considering internal dimensions of the gearbox or other nearby components, the frictional moment due to viscous drag of the cylindrical roller bearing is [26–28]  4 2  Md0 = 10VM0Kr0Bd n  m z (12)  KL0Kz0(d+D) 12  Kr0 = D d 10− − × where VM0 is the drag loss variable, D is outer diameter of the bearing, d is inner diameter of the bearing, and B is width of the bearing.

3.1.4. Reduction of Friction Coefficient Due to Heat and Backfill When the bearing is fully lubricated, an oil film forms, but is comprised of only a small amount of the total volume of lubricating oil. The rest of the oil flows out of raceway and produces backflow, which generates shear heating. In addition, when the bearing operates at high speeds, the rolling element repeatedly passes over the raceway and a large amount of lubricating oil is extruded, reducing the amount of lubricating oil covering the bearing. In both cases, the viscosity of the lubricating oil and the oil film thickness of the rolling friction component decrease, which can be expressed as [26,27] Appl. Sci. 2020, 10, 5849 8 of 25

  = 1  φri 9 1.28 0.64  1+1.84 10− (ndm) ν  × 1 (13)  φrs = r  K υn(d+D) KZ  rs 2(D d) e − where Krs and Kz are geometric constants and ν is kinematic viscosity of the oil.

3.1.5. Analysis of Total Power Loss of Cylindrical Roller Bearing The power loss of cylindrical roller bearing is

4 PR = 0.001M ωz = 1.047 10− M nz (14) 0 · × 0 · where ωz is angular velocity of the bearing.

3.2. Power Loss Due to Gear Meshing Frictional power losses associated with the gear surface can be divided into sliding friction power loss and rolling friction power loss. The total losses are related to the lubrication condition (friction coeﬃcient of the gear surface), gear speed, and load acting on the gear tooth surface. Moreover, windage power loss is related to the viscosity of the lubricating oil, gear speed, gear radius, gear width, and so on.

3.2.1. Calculation of Gear Frictional Power Loss The internal and external gears aﬀect relative sliding speeds of tooth surfaces during operation. As shown in Figure5, the theoretical meshing segment is N N and the actual meshing segment is B B . Appl. Sci. 2020, 10, x FOR PEER REVIEW 1 2 9 of 27 1 2 The point of intersection of the centerline and mesh line when two gears are meshing is P. At contact point2. K, theWith sliding boundary speed lubrication, of gear 1 gear relative teeth to must gear be 2 separated is [29] by boundary lubrication in order to separate the tooth flanks. However, absolutely smooth tooth flanks do not exist, and for this ! reason, almost all gears work in a hybrid lubrication state.1 1 V12 = (ω1 + ω2)l = ω1z1 + l (15) 3. In the hybrid lubrication state, the elastic oil film will eitherz1 be zin2 a mixed or boundary-lubricated state. Mixed lubrication refers to the combination of fluid lubrication and boundary lubrication, where ω thatis angular is, the normal velocity; loadz isis shared number by of the gear fluid teeth; lubrication subscripts film and 1 and the 2boundary denote gearfilm, 1and and the gear 2, respectively;friction and forcel is theincludes distance fluid fromdamping meshing and boundary point to friction node P components..

FigureFigure 5. 5.Schematic Schematic diagramdiagram of of external external helical helical gear gear transmission. transmission.

BecauseAccuratelyV12 = 0estimating at node P the, P Nminimum2 represents oil film the positivethickness direction. between tooth Power profiles before is andan important after the node, Pf+ andbasisP f-for, are analyzing not the the same. lubrication Friction state loss of near gears. node TheP calculationis [29] formula of the minimum oil film thickness is as follows: . . . . . ℎ = 1.6𝛼 (𝑢𝜈) 𝐸 𝑅 𝛿 (17)

where ve is the the suction speed of the meshing point, Re is the equivalent radius of curvature of the

meshing point, and Ee is the equivalent elastic modulus. The viscosity of lubricating oil is affected by temperature and pressure; the relationship is as follows: 𝜇=𝜇𝑒 (18) loglog(𝑢+𝑎) =𝑏−𝑐logT (19)

where p is lubricating oil pressure and a, b, and c are constants related to lubricating oil. The film thickness ratio is an important parameter of the lubrication state of the gear friction pair. When the film thickness ratio coefficient λ < 1.4, it is the boundary lubrication state; when 1.4 < λ < 3, it is the mixed lubrication state; and when λ > 3, it is the elastohydrodynamic lubrication state. ℎ λ = (20) 𝑅 +𝑅 Considering the empirical formula proposed by Benedict and Kelley [29], which involves more comprehensive factors and better practical application effect, this empirical formula is used for the friction factor in this paper. Appl. Sci. 2020, 10, 5849 9 of 25

! 1 1 P f (l) = F f V12 = f FNω1z1 + l (16) ± · ± z1 z2 where f is the tooth surface sliding friction coeﬃcient and FN is normal force of the tooth surface. Gear drives typically have three lubrication states:

1. Under the EHL (Elastohydrodynamic lubrication) state, friction surfaces of the meshed teeth are completely separated by a viscous fluid film. The film thickness is determined by relative sliding speed of the tooth surfaces. Pressure generated by the fluid film balances the external load. 2. With boundary lubrication, gear teeth must be separated by boundary lubrication in order to separate the tooth flanks. However, absolutely smooth tooth flanks do not exist, and for this reason, almost all gears work in a hybrid lubrication state. 3. In the hybrid lubrication state, the elastic oil film will either be in a mixed or boundary-lubricated state. Mixed lubrication refers to the combination of fluid lubrication and boundary lubrication, that is, the normal load is shared by the fluid lubrication film and the boundary film, and the friction force includes fluid damping and boundary friction components.

Accurately estimating the minimum oil film thickness between tooth profiles is an important basis for analyzing the lubrication state of gears. The calculation formula of the minimum oil film thickness is as follows: 0.6 0.7 0.03 0.43 0.13 hmin = 1.6α (uνe) Ee Re δ− (17) where ve is the the suction speed of the meshing point, Re is the equivalent radius of curvature of the meshing point, and Ee is the equivalent elastic modulus. The viscosity of lubricating oil is affected by temperature and pressure; the relationship is as follows: αp µ = µ0e (18) loglog(u + a) = b clogT (19) − where p is lubricating oil pressure and a, b, and c are constants related to lubricating oil. The film thickness ratio is an important parameter of the lubrication state of the gear friction pair. When the film thickness ratio coefficient λ < 1.4, it is the boundary lubrication state; when 1.4 < λ < 3, it is the mixed lubrication state; and when λ > 3, it is the elastohydrodynamic lubrication state.

hmin λ = q (20) 2 + 2 Ra1 Ra2

Considering the empirical formula proposed by Benedict and Kelley [29], which involves more comprehensive factors and better practical application eﬀect, this empirical formula is used for the friction factor in this paper. ! ! 50 29.66δ f = 0.0127 lg (21) 50 39.37s uV V2 − f s r where sf is the gear surface roughness (root mean square roughness); δ is the unit load and Vs and Vr are the instantaneous sliding speed and rolling speed of the meshing point, respectively. From the perspective of the end face of the helical gear, the helical gear can be regarded as the superposition of an inﬁnite number of thin-walled spur gears rotating a helix angle. When the front face wheel of the helical gear enters into engagement and disengages, its engagement is exactly the same as that of a pair of spur gears. To simplify the calculation, we assume that the gear teeth are subjected to equal loading during meshing. When contact ratio is a εr a + 1, the load acting on the ≤ ≤ tooth is FN/a. As shown in Figure6, N1N2 includes all meshed segments of the helical gear drive, B1B2 is a meshed segment of the tooth, N2B1 and B2N1 are the a + 1 meshed segments, and node P is located Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 27

50 29.66𝛿 𝑓 = 0.0127( )lg( ) (21) 50 − 39.37𝑠 𝑢𝑉𝑉

where sf is the gear surface roughness (root mean square roughness); δ is the unit load and Vs and Vr are the instantaneous sliding speed and rolling speed of the meshing point, respectively. From the perspective of the end face of the helical gear, the helical gear can be regarded as the superposition of an infinite number of thin-walled spur gears rotating a helix angle. When the front face wheel of the helical gear enters into engagement and disengages, its engagement is exactly the same as that of a pair of spur gears. To simplify the calculation, we assume that the gear teeth are subjected to equal loading during meshing. When contact ratio is a ≤ εr ≤ a + 1, the load acting on the tooth is FN/a. As shown in Figure 6, N1N2 includes all meshed segments of the helical gear drive, B1B2 is a meshed segment of the tooth, N2B1 and B2N1 are the a + 1 meshed segments, and node P is located inside meshed tooth segment a. For Pf±(l), friction power loss in the whole meshed segment can be obtained by taking the average integral value of the meshing segment [28]. Appl. Sci. 2020, 10, 5849 10 of 25 1 𝑃 = 𝑃 (𝑙)𝑑𝑙 (22) 𝑃 ± inside meshed tooth segment a. For Pf (l), friction power loss in the whole meshed segment can be ± obtained by taking the average integral value of the meshing segment [28]. Integrating Equation (17) along N1N2, we obtain

() ZN2 1 1 3 = ( ) 𝑃 = 𝑃𝑑𝑙P+𝑃f Pf 𝑑𝑙l dl (22) 𝑃 4 Pbt ± (N1) Integrating Equation (17) along N N , we3 obtain (23) + 𝑃1 𝑑𝑙2 + 𝑃𝑑𝑙 4   (P l1) 0 P l l  − Rbt− R btR 2 R2  1  3 1 1 1 − 3  P f = P  4 P f dl + P f dl+ P f +dl + 4 P f +dl bt  = 𝑓−𝐹𝜔𝑧( + )(𝑃− +2𝑙 −𝑃𝑙 +2𝑙 −𝑃𝑙) (23) l24𝑃 (P𝑧 l ) 𝑧 0 Pbt l2 − − bt− 1 − = 1 1 + 1 2 + 2 + 2 4P f FNω1z1 z z Pbt 2l1 Pbtl1 2l2 Pbtl2 where Pbt is normal btpitch of the end1 face.2 − − where Pbt is normal pitch of the end face.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 27

′ ′ ′ ′ ′ 1 1 ′ Figure𝑃 6.=𝐹Schematic∙𝑉 =± diagram𝑓 𝐹 𝜔 of′ 𝑧 helical′ − gear engagement.𝑙 𝑓±Figure(𝑙) 6.𝑓 Schematic21 diagram𝑁 of2 helical2 ′ gear engagement.′ (24) 𝑧1 𝑧2 Similar to the meshing analysis for the external gear, friction power loss in the internal gear can The transmissionThe transmission eﬃ ciencyefficiency calculation calculation of of an an internal internal helical helical gear is gear similar is similarto the calculation to the calculation for for externalbeexternal calculated meshing. meshing. as [29] TheThe force force diagram diagram of a of pair a pairof internal of internal helical helicalgears is presented gears is presentedin Figure 7. inThe Figure 7. The slidingsliding speed speedV V’21 ofof gear gear 2 relative 2 relative to gear to gear 1 is [29] 1 is [29] ′021 1 ′ ′ ′ ′ 1 1 ′2 ′2 ′ ′ ′2 ′ ′ 𝑃𝑓 = 𝑓 𝐹𝑁𝜔2𝑧2 − 𝑃𝑏𝑡 +2𝑙1 −𝑃𝑏𝑡𝑙1 +2𝑙2 −𝑃𝑏𝑡𝑙2 (25) 4𝑃′ 𝑧′ 𝑧′ ! 𝑏𝑡 1 2 1 1 P0 = F0 V0 = f 0F0 ω0 z0 l0 (24) f (l) f · 21 ± N 2 2 z − z ± 10 20

FigureFigure 7. Schematic 7. Schematic diagram diagram ofof internal gear gear transmission. transmission.

3.2.2. Rolling Friction Power Loss When a pair of teeth engage at point K from node P, the relative rolling speed at the engagement point is [28] 𝑉 =[𝜔(𝑃 −𝑙 +𝑙) +𝜔(𝑙 −𝑃 +𝑙)]×10 (26)

The average rolling speed VTM is obtained by integrating the instantaneous speed along the meshing line N1K.

𝐵1𝐾 𝐵1𝐾 −3 𝑉 𝑑𝑙 [𝜔1(𝑃𝑏𝑡 −𝑙2 +𝑙) +𝜔2(𝑙1 −𝑃𝑏𝑡 +𝑙)] ×10 𝑑𝑙 0 𝑇 0 (27) 𝑉𝑇𝑀 = = 𝐵1𝐾 𝐵1𝐾 According to Crook [30], an empirical formula can be derived from experimental data on the EHL of rolling friction. 0.09ℎ𝑏 𝐹= (28) cos 𝛽

where h is the oil film thickness, b is the tooth width, and β is the helical gear helix angle.

𝑃 =𝐹∙𝑉 (29) According to [31], oil film thickness h is Appl. Sci. 2020, 10, 5849 11 of 25

Similar to the meshing analysis for the external gear, friction power loss in the internal gear can be calculated as [29] ! 1 1 1 2 2 2 P0f = f 0FN0 ω20 z20 Pbt0 + 2l10 Pbt0 l10 + 2l20 Pbt0 l20 (25) 4Pbt0 z10 − z20 − −

3.2.2. Rolling Friction Power Loss When a pair of teeth engage at point K from node P, the relative rolling speed at the engagement point is [28] 3 VT = [ω (P l + l) + ω (l P + l)] 10− (26) 1 bt − 2 2 1 − bt × The average rolling speed VTM is obtained by integrating the instantaneous speed along the meshing line N1K.

R B K R B Kn o 1 V dl 1 [ω (P l + l) + ω (l P + l)] 10 3dl 0 T 0 1 bt 2 2 1 bt − VTM = = − − × (27) B1K B1K

According to Crook [30], an empirical formula can be derived from experimental data on the EHL of rolling friction. 0.09hb F = (28) cos β where h is the oil ﬁlm thickness, b is the tooth width, and β is the helical gear helix angle.

PVTM = F VTM (29) · According to [31], oil ﬁlm thickness h is

0.57 0.4 0.71 3.07α κ (νVTM) h = (30) E0.03χ0.11 where α is the pressure-viscosity coeﬃcient of the lubricant, κ is the radius of curvature of the tooth proﬁle, χ is the load factor, and E is the comprehensive elastic modulus. From ANSI/agma6011-i03 [32], the formula for estimating the meshing power loss for a high-speed gear transmission is " # z1 + z2 Pm = (22 0.8αn)0.01P1 (31) − z1z2

3.2.3. Calculation of Power Loss Due to Gear Transmission Windage As free space inside the gearbox is reduced during power transmission, the contact area between the surrounding air and a rotating body, such as a gear moving at a higher rotational speed, increases relative to the volume reduction, resulting in windage. This increases the relative motion between rotating parts, which is accompanied by agitation of the lubricating oil. Furthermore, windage heats the inside of the gearbox, which can lead to the formation of oil mist. Heating, agitation, and other phenomena become more serious as the gear rotational speed increases. According to Anderson et al. [33], the gear windage power loss can be calculated as

4 P = 1.047 10− T n (32) × · Moreover, torque T can be expressed as ! ρω2R5 b 0.09 T = 1 + 2.3 (33) 2 R Re0.2 where Re is the Reynolds number, which can be expressed as Appl. Sci. 2020, 10, 5849 12 of 25

ρωR2 Re = (34) µ Thus, the windage power loss calculation model can be obtained. ! b P = C 1 + 2.3 ρ0.8n2.8R4.6µ0.2 (35) w 1 R eq eq

8 C = 2.04 10− (36) 1 × ρ + 34.25ρair ρeq = (37) 35.25 µ + 34.25µair µeq = (38) 35.25 where b is the tooth width, µeq is the dynamic viscosity of the air–oil mixture, µeq is the air–oil equivalent density, ρair is the air density, µair is the air dynamic viscosity, and R is the pitch radius.

4. Power Flow Diagram of Herringbone Gear Planetary Drive System

4.1. System Power Flow The two-stage herringbone planetary transmission system is illustrated in Figure8. A schematic representation of the structure is depicted in Figure8a and the corresponding power flow diagram is shownAppl. inSci. Figure2020, 10, 8x b.FOR The PEER power REVIEWflow diagram uses the following conventions: components13 of 27 are represented by a solid circle, bearing contacts are represented by a hollow circle, and teeth contacts are shownsun as agear, hollow 3 represents triangle. the In first-stage the first diagram, transmission 1 represents planet carrier, the frame, 4 is the 2 denotesfirst-stage the ring first-stage gear, 4″ sun gear, 3represents represents the the secondary first-stage sun transmission gear, 5 is the secondary planet carrier, planetary 4 is gear, the first-stage 6 represents ring the gear, secondary 4” represents ring gear, and power is output via the secondary ring gear. Hollow circles PR1, PR2, and PR3 represent the secondary sun gear, 5 is the secondary planetary gear, 6 represents the secondary ring gear, bearing contact power losses; hollow triangles PG2-3, PG3-4, PG4”-5, and PG5-6 represent gear meshing and power is output via the secondary ring gear. Hollow circles P ,P , and P represent bearing power losses; and subscript 2–3 indicates that gear 2 and gear 3 areR1 meshed.R2 R3 contact power losses; hollow triangles PG2-3,PG3-4,PG4”-5, and PG5-6 represent gear meshing power losses; and subscript 2–3 indicates that gear 2 and gear 3 are meshed.

(a) (b) (c)

FigureFigure 8. Double-stage 8. Double-stage herringbone herringbone planetary planetary gear gear transmission transmission system. system. (a) Schematic (a) Schematic drawing; drawing; (b) (b) powerpower ﬂow flow diagram; diagram; ((cc)) graphicalgraphical representation representation of offixed ﬁxed rotary. rotary.

4.2. System4.2. System Equations Equations AccordingAccording to the to lawthe law of conservation of conservation of of energy, energy, basic basicpower power balance equations equations at atthe the nodes nodes are are 𝑃 =𝑃 +𝑃 ⎧   𝑃 P=𝑃IN = P2+𝑃IN + PR1  ⎪  P = P + P 𝑃 =𝑃2IN +𝑃32 +𝑃G2 3 ⎪  −  P𝑃32 ==𝑃P43 + P+𝑃G3 4 + P63  −  P43 = P4 4 + PR2 (39) ⎨ 𝑃 =𝑃 +𝑃00 (39)  P = P + P  400 4 5400 G400 5 ⎪𝑃 =𝑃 +𝑃 +𝑃−  P = P + P + P  54𝑃00 +𝑃G5 6 =051 65 ⎪  −  P51 + PR3 = 0 ⎩ 𝑃 =𝑃 +𝑃 −𝑃  Pout = P + P P The transmission efficiency of the system 63is equa65l −to theR4 ratio of output power to input power, which can be calculated as follows: 𝑃𝐼𝑁 −𝑃𝑅1 𝑃2𝐼𝑁 η = = × 100% (40) IN −2 𝑃𝐼𝑁 𝑃𝐼𝑁

𝑃32 𝑃32 η = = × 100% (41) 2−3 𝑃32 +𝑃𝐺2−3 𝑃2𝐼𝑁

𝑃43 η = × 100% (42) 3−4 𝑃43 +𝑃𝐺3−4 𝑃 𝑃 4′′4 4′′4 η = = × 100% (43) 4−4′′ 𝑃 +𝑃 𝑃 4′′4 𝑅2 43

𝑃 𝑃 54′′ 54′′ η = = × 100% (44) 4′′−5 𝑃 +𝑃 𝑃 54′′ 𝐺4−5 4′′4 Appl. Sci. 2020, 10, 5849 13 of 25

The transmission eﬃciency of the system is equal to the ratio of output power to input power, which can be calculated as follows:

PIN PR1 P2IN ηIN 2 = − = 100% (40) − PIN PIN ×

P32 P32 η2 3 = = 100% (41) − P32 + PG2 3 P2IN × − P43 η3 4 = 100% (42) − P43 + PG3 4 × − P P η = 400 4 = 400 4 100% (43) 4 400 + − P400 4 PR2 P43 × P P = 5400 = 5400 η400 5 100% (44) − P54 + PG4 5 P4 4 × 00 − 00 P65 + P63 P65 + P63 η5 6 = = 100% (45) − P65 + PG5 6 + PR3 + PR4 P54 + PR4 × − 00 5. Case Study In a 1-DOF planetary gear train with six links, as shown in Figure8, input power turns the sun gear and the power is subsequently split among the planetary gears (3, 4, 5, 6), and then transmitted via the ring gear to the output shaft. The rotational speed of each axis is listed in Table2. The main parameters of the high-speed heavy-load herringbone gear transmission system are listed in Table3. The rated speed of the gear system is 10,000 rpm, and the rated power is 1000 Kw.

Table 2. Rotational speed of each gear shaft (rpm).

n1 n2 n3 n4 n5 n6 0 4920 4640 1190.4 1344 429.6 − −

Table 3. Main parameters of high-speed heavy-duty herringbone gear transmission system.

Gear Parameter Pinion Gear

z2 = z4” = 35 Number of teeth z4 = z6 = 97 z3 = z5 = 31 Normal modulus (mm) 7 Normal pressure angle (◦) 20 Helix angle β (◦) 22 Tooth width (mm) 80 Elastic modulus (Gpa) 210 Poisson’s ratio 0.3 Root mean square roughness (um) 0.1 Face addendum coeﬃcient 1 Face radial coeﬃcient 0.25 Ambient temperature (K) 323 Ambient viscosity (Pa.s) 0.08 Bearing inner diameter (mm) 110 Bearing outer diameter (mm) 170 Bearing width (mm) 38 Bearing clearance (mm) 0.06 Input speed (rpm) 4920 Input power (kW) 424

5.1. Power Flow Analysis Ignoring Power Loss Power ﬂow through the system can be described as follows. Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 27

Table 4. Transmission efficiency when power transmitted along the transmission path. Appl. Sci. 2020, 10, 5849 14 of 25 Efficiency (%) Number of ηIN-2 η2-3 η3-4 η4-5 η5-6 PlanetaryHypothesis GearsP = 1: IN  3  99.98 PIN =98.83P2IN = P32 99.09 98.89 99.23  4  99.98 P =98.46P + P 98.82 98.52 98.97  32 43 63 (46) 5  99.98P = P 98.08= P = P 98.55 98.16 98.71  43 400 4 5400 65 6  99.98 97.71 98.29 97.79 98.46  Pout = P63 + P65

5.2. PowerTable Flow4 shows Analysis the Consideringtransmission Power efficiency Loss when power is transmitted along the transmission path. The transmission efficiencies of planetary transmission systems with 3, 4, 5, and 6 planetary gearsThe are ﬂowshown diagram on the of line power graphs loss presented along the transmissionin Figure 10, pathand iswere shown calculated in Figure as9 96.53%, and the 95.45%, results of94.37%, the transmission and 93.32%, e ﬃrespectively.ciency calculations are presented in Table4.

FigureFigure 9. 9.Flowchart Flowchart of of gear gear drive drive power power loss loss calculation calculation process. process. Table 4. Transmission eﬃciency when power transmitted along the transmission path.

Eﬃciency (%) η η η η η Number of Planetary Gears IN-2 2-3 3-4 4-5 5-6 3 99.98 98.83 99.09 98.89 99.23 4 99.98 98.46 98.82 98.52 98.97 5 99.98 98.08 98.55 98.16 98.71 6 99.98 97.71 98.29 97.79 98.46

Table4 shows the transmission e fficiency when power is transmitted along the transmission path. The transmission efficiencies of planetary transmission systems with 3, 4, 5, and 6 planetary gears are shown on the line graphs presented in Figure 10, and were calculated as 96.53%, 95.45%, 94.37%, and 93.32%, respectively.(a) (b) Transmission efficiencies of each transmission element along the transmission path as well as the total efficiencies are presented in Figure 11. As shown in Figure 11a, different power losses occur in different components of the planetary gear transmission system. The transmission efficiency from gear 2 to gear 3 and from gear 4 to gear 5 is lower because the sun gear meshes with multiple planet gears. Therefore, wind resistance loss and tooth surface friction loss will be large. As shown in Figure 11b, each transmission element produces some power loss and the total power loss is the sum of power losses of single elements. Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 27

Table 4. Transmission efficiency when power transmitted along the transmission path.

Efficiency (%) Number of ηIN-2 η2-3 η3-4 η4-5 η5-6 Planetary Gears 3 99.98 98.83 99.09 98.89 99.23 4 99.98 98.46 98.82 98.52 98.97 5 99.98 98.08 98.55 98.16 98.71 6 99.98 97.71 98.29 97.79 98.46

Table 4 shows the transmission efficiency when power is transmitted along the transmission path. The transmission efficiencies of planetary transmission systems with 3, 4, 5, and 6 planetary gears are shown on the line graphs presented in Figure 10, and were calculated as 96.53%, 95.45%, 94.37%, and 93.32%, respectively.

Appl. Sci. 2020, 10, 5849 15 of 25 Figure 9. Flowchart of gear drive power loss calculation process.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 27

Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 27 (a) (b)

Figure 10. Variation of efficiency along the transmission path using hypergraph method. (a) Three planetary gears; (b) four planetary gears; (c) five planetary gears; (d) six planetary gears.

Transmission efficiencies of each transmission element along the transmission path as well as the total efficiencies are presented in Figure 11. As shown in Figure 11a, different power losses occur in different components of the planetary gear transmission system. The transmission efficiency from gear 2 to gear 3 and from gear 4 to gear 5 is lower because the sun gear meshes with multiple planet gears. Therefore, wind resistance(c) loss and tooth surface friction loss will be( dlarge.) As shown in Figure 11b, each transmission element produces some power loss and the total power loss is the sum of FigureFigure 10. 10.Variation Variation of of e ﬃefficiencyciency along along thethe transmissiontransmission pathpath usingusing hypergraphhypergraph method. ( (aa)) Three Three power losses of single elements. planetaryplanetary gears; gears; (b (b)) four four planetary planetary gears; gears; ( c(c)) ﬁve five planetary planetary gears;gears; ((dd)) sixsix planetaryplanetary gears.

1.000Transmission efficiencies of each transmission 3 planets element1.000 along the transmission path as well as 3 planets 4 planets 0.992 the total efficiencies are presented in Figure 11. As shown in Figure 11a, different power losses 4 planets occur 0.995 5 planets 5 in different components of the planetary gear 6 planetstransmission0.984 system. The transmission efficiency planets from 6 planets gear0.990 2 to gear 3 and from gear 4 to gear 5 is lower because 0.976the sun gear meshes with multiple planet gears. Therefore, wind resistance loss and tooth surface friction0.968 loss will be large. As shown in Figure 11b,0.985 each transmission element produces some power loss0.960 and the total power loss is the sum of Efficiency Efficiency Efficiency 0.952 power0.980 losses of single elements. 0.944 0.975 1.000 3 planets 0.9361.000 3 planets 4 planets 0.9280.992 4 planets 0.9700.995 5 planets Gear2 Gear3 Gear4 Gear5 5 Gear6 Gear2 Gear3 Gear4 Gear5 Gear6 6 planets 0.984 planets Power transmission path Power transmission path 6 planets 0.990 0.976 (a) 0.968 (b) 0.985 0.960

Figure 11. Transmission efficiency along the power transmission Efficiency path. (a) Efficiency of each Efficiency Figure 11. Transmission efficiency along the power transmission0.952 path. (a)Efficiency of each transmission 0.980transmission element; (b) efficiency along the transmission path. element; (b) efficiency along the transmission path. 0.944 0.975 0.936 ItIt can can be be noticed noticed that that tooth tooth surface surface friction friction efficiency efficiency decreases decreases and and windage windage efficiency efficiency increases increases as 0.928 speedas speed0.970 increases, increases, and bearingand bearing friction fric efficiencytion efficiency decreases decreases very slowly.very slowly. Figure Figure 12 shows 12 shows that 4.2 that per 4.2 cent per of Gear2 Gear3 Gear4 Gear5 Gear6 Gear2 Gear3 Gear4 Gear5 Gear6 thecent total of lossthe total is owing loss tois bearingowing to friction, bearing 5.3 friction, percent 5.3 of thepercent total of loss the is total owing loss to windage,is owing to and windage,90.5 per and cent Power transmission path Power transmission path of90.5 the per total cent loss of isthe owing total toloss tooth is owing friction to tooth with anfriction input with speed an ofinput 4920 speed rpm. of It 4920 further rpm. shows It further that 3.9shows per cent that of3.9 the per total cent lossof(a )the is owingtotal loss to is bearing owing fricition,to bearing 15.1 fricition, per cent 15.1 of per the(b )cent total of lossthe total is owing loss is to windage,owingFigure to and windage, 11. 81 Transmission per and cent 81 of per the efficiency cent total of loss thealong is total owing the loss power to is toothowing transmission friction to tooth with path.friction an (a input) withEfficiency speedan input of 9348eachspeed rpm. of Y9348 Diab transmissionrpm. studies Y Diab the element; powerstudies loss( bthe) efficiency inpower high-speed alongloss in th gearshighe transmission-speed based gears on path. experiments. based on experiments. Analysing the Analysing distribution the ofdistribution each power of loss,each 46power per centloss, of46 theper totalcent of loss the is total owing loss to is windageowing to [windage34]. Therefore, [34]. Therefore, in the gear in transmissionthe gearIt can transmission be system, noticed distribution thatsystem, tooth distribution surface of each friction power of each losseffi powerciency is related lossdecreases tois therelated structureand towindage the andstructure efficiency working and conditionsincreases working ofconditionsas the speed gear increases, system. of the gear Di andff erentsystem. bearing input Different fric conditionstion efficiency input will conditions decreases also leasd will very to also the slowly. dileasdfferent Figure to the power 12different shows loss distribution. thatpower 4.2 lossper distribution.cent of the total loss is owing to bearing friction, 5.3 percent of the total loss is owing to windage, and 90.5 per cent of the total loss is owing to tooth friction with an input speed of 4920 rpm. It further shows that 3.9 per cent of the total loss is owing to bearing fricition, 15.1 per cent of the total loss is owing to windage, and 81 per cent of the total loss is owing to tooth friction with an input speed of 9348 rpm. Y Diab studies the power loss in high-speed gears based on experiments. Analysing the distribution of each power loss, 46 per cent of the total loss is owing to windage [34]. Therefore, in the gear transmission system, distribution of each power loss is related to the structure and working conditions of the gear system. Different input conditions will also leasd to the different power loss distribution. Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 27

bearing fricion windage tooth surface friction bearing fricion windage tooth surface friction 0.965 1.0

0.8 Appl. Sci. 2020, 10, 5849 16 of 25 Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 27 0.6 tot tot bearing fricion windage tooth surface friction 0.960 bearing fricion windage tooth surface friction 0.965

1.0 P/P P/P 0.4 0.8

0.2 0.6 tot tot 0.960 P/P 0.0 P/P 0.4 0.955 5100 5950 6800 7650 8500 9350 5100 5950 6800 7650 8500 9350 Ω [rpm] Ω [rpm] 0.2 sun sun

0.0 (a) 0.955 (b) 5100 5950 6800 7650 8500 9350 5100 5950 6800 7650 8500 9350 Ω [rpm] Ω [rpm] Figure 12. Power loss distributionsun according to the rotational speed for ansun input power of 424 Kw. (a) Power loss distribution according(a) to the rotational speed; (b) partially( benlarged) view of the figure. FigureFigure 12.12. Power loss loss distribution distribution according according to tothe the rota rotationaltional speed speed for foran input an input power power of 424 of Kw. 424 ( Kw.a) Under high-speed(aPower) Power loss loss distribution and distribution heavy-load according according to conditions,the to the rotational rotational speed; speed;owing (b) (partiallyb) to partially the enlarged relatively enlarged view view of thehigh of thefigure. ﬁgure.sliding speed of the meshing tooth Undersurface,Under high-speedhigh-speed the instantaneous andand heavy-loadheavy-load temperature conditions, owing owing rise to to the thedirectly relatively relatively affects high high sliding sliding the internal speed speed of of the temperature the of the gear system.meshingmeshing As toothtooth shown surface, in the the Figure instantaneous instantaneous 13, the temperature temperature higher rise the rise directly directlytooth affects surface aﬀects the the internal temperature, internal temperature temperature the of smaller the power loss,of thebecause the gear gear system. system. the viscosityAs As shown shown in of in Figure the Figure lubricatin 13, 13 the, the high higherger oilthe the decreasestooth tooth surface surface with temperature, temperature, the increasing the the smaller smaller temperature.the the powerpower loss, loss, becausebecause thethe viscosityviscosity ofof thethe lubricatinglubricating oil decreases with the increasing temperature. temperature.

42 42

39 50 ℃ 39 50 ℃ 90 ℃ 36 90 ℃ 36 ) 33 Kw ) 33 30

Kw 30 27 24 27 21 Power Loss ( 24 18 21 15 Power Loss ( 4800 5600 6400 7200 8000 8800 9600 18 Ω [rpm] sun 15

Figure 13.4800 Comparison 5600 of power 6400 loss 7200as functions 8000 of Ωsun 8800 at 50 °C 9600and 90 °C. Ω [rpm] 5.3. Results and Discussion sun

Figure 13. Comparison of power loss as functions of Ωsun at 50 ◦C and 90 ◦C. 5.3.1. Analysis of Transmission Efficiency under Different Lubrication Conditions Figure 13. Comparison of power loss as functions of Ωsun at 50 °C and 90 °C. 5.3. Results and Discussion Power losses of the gear system under three possible lubrication conditions are shown in Figure 5.3.1.14. With Analysis boundary of Transmission lubrication, Epartialfficiency surface under co Dintactfferent occurs Lubrication between Conditionsrough tooth surfaces and the 5.3. Results andfriction Discussion coefficient is highest. As relative density and relative dynamic viscosity of the lubricating oil increase,Power windage losses of loss the will gear also system increase. under three possible lubrication conditions are shown in Figure 14. 5.3.1. AnalysisWith boundaryofIn theTransmission EHL lubrication, state, the viscousEfficiency partial surfacefluid membrane under contact occursDifferent completely between Lubricationseparates rough tooththe two surfacesConditions gear tooth and the surfaces, friction coetherebyfficient greatly is highest. reducing As relative the friction density coefficient. and relative The dynamicmixed lubrication viscosity ofstate the occurs lubricating when oil only increase, part Powerwindage oflosses the contact lossof the will surface gear also increase.is system under the under EHL condition three possib and otherle areaslubrication of the contact conditions surfaces areare in shown the in Figure 14. With boundaryboundary lubricationlubrication, state. partial surface contact occurs between rough tooth surfaces and the Transmission efficiency of the gear system is highest under the EHL condition, followed by the friction coefficientmixed lubrication is highest. state, As and relative is lowest density under the and boundary relative lubrication dynamic state. viscosity As the number of the of lubricating oil increase, windage loss will also increase. In the EHL state, the viscous fluid membrane completely separates the two gear tooth surfaces, thereby greatly reducing the friction coefficient. The mixed lubrication state occurs when only part of the contact surface is under the EHL condition and other areas of the contact surfaces are in the boundary lubrication state. Transmission efficiency of the gear system is highest under the EHL condition, followed by the mixed lubrication state, and is lowest under the boundary lubrication state. As the number of Appl. Sci. 2020, 10, 5849 17 of 25

In the EHL state, the viscous fluid membrane completely separates the two gear tooth surfaces, thereby greatly reducing the friction coefficient. The mixed lubrication state occurs when only part of the contact surface is under the EHL condition and other areas of the contact surfaces are in the boundary lubrication state. TransmissionAppl. Sci. 2020, 10 e, xffi FORciency PEER REVIEW of the gear system is highest under the EHL condition, followed18 of 27 by the mixed lubrication state, and is lowest under the boundary lubrication state. As the number of planetary planetary gears increases, power losses increase and transmission efficiency drops, as shown in gears increases,Figure 15. power losses increase and transmission efficiency drops, as shown in Figure 15.

EHL(Elastohydrodynamic lubrication) 25,000 ML(Mixed lubrication) BL(Boundary lubrication) ) w

( 20,000

15,000 Power Loss

10,000

3456

Number of Planets (a) 6000 EHL(Elastohydrodynamic lubrication) 5000 ML(Mixed lubrication) BL(Boundary lubrication) ) w 4000 ( 3000

2000 Power Loss 1000

0 3456

Number of Planets (b) Figure 14. Cont. Appl. Sci.Appl.2020 Sci., 10 2020, 5849, 10, x FOR PEER REVIEW 19 of 27 18 of 25 Appl. Sci. 2020, 10, x FOR PEER REVIEW 19 of 27 35,000 Tooth Surface Meshing Loss(EHL) 35,000 WindageTooth Surface power Meshing loss (EHL) Loss(EHL) 30,000 ToothWindage Surface power Meshing loss (EHL) Loss(ML)

) 30,000 WindageTooth Surface power Meshing loss (ML) Loss(ML) w ) ( 25,000 TWindageooth Surface power Meshing loss (ML) Loss(BL) w (

25,000 WindageTooth Surface power Meshing loss (BL) Loss(BL) 20,000 Windage power loss (BL) 20,000 15,000 15,000

Power Loss 10,000 Power Loss 10,000 5000 5000

3456 3456 Number of Planets Number(c) of Planets (c) Figure 14. Power losses of herringbone planetary gear transmission system under different FigurelubricationFigure 14. Power 14. conditions.Power losses oflosses herringbone(a) Powerof herringbone loss planetarydue to planetary tooth gear friction; transmissiongear (btransmission) windage system power system under loss; under (c di) totalﬀ erentdifferent power lubrication conditions.loss.lubrication (a) Power conditions. loss due(a) Power to tooth loss friction; due to tooth (b) windagefriction; (b power) windage loss; power (c) totalloss; ( powerc) total power loss. loss. 1.000 1.000 0.995 0.995 BL3 0.990 BL4BL3 0.990 BL5

BL4 BL6 0.985 BL5 ML3BL6 0.985 ML4ML3 Efficiency 0.980 ML5ML4

Efficiency ML6 0.980 ML5 EHL3ML6 0.975 EHL4EHL3 0.975 EHL5EHL4 EHL6EHL5 0.970 Gear2 EHL6 Gear4 Gear6 0.970 Gear2 Gear4 Gear6 Power Transfer Path Appl. Sci. 2020, 10, x FOR PEER REVIEW 20 of 27 Power (Transfera) Path (a) 1.00

BL3 0.98 BL4

BL5 BL6 0.96 ML3 ML4 ML5

Efficiency 0.94 ML6 EHL3 EHL4 0.92 EHL5 EHL6 Gear2 Gear4 Gear6 Power Transfer Path (b)

FigureFigure 15. Transmission 15. Transmission effi efficiencyciency under under didifferentfferent lubrication lubrication conditions. conditions. (a) Efficiency (a)Effi ciencyof each of each transmissiontransmission element element along along the the transmission transmission path; path; ( (bb)) efficiency efficiency along along the thetransmission transmission path. path.

5.3.2. Variation of Transmission Efficiency with Input Speed The input speed was varied while the input torque remained constant and the transmission efficiency of the herringbone planetary transmission system was analyzed. Variation of power loss with input speed is shown in Figure 16. Frictional power losses on the tooth surface increased as the rotational speed increased, and the windage power loss increases, suggesting that, when the rotational speed increases, the increase in resistance due to the surrounding air is greater than the increase of frictional power loss. The bearing power loss is different at different positions. The power loss at bearing 2 is the largest. With the speed increment, the total power loss at the bearing increases. As shown in Figure 17, the gear meshing efficiency slowly decreases as the rotational speed increases. However, windage losses rapidly increase, suggesting windage power loss is an essential consideration for gear systems operating at high rotational speeds.

Appl. Sci. 2020, 10, 5849 19 of 25

5.3.2. Variation of Transmission Efficiency with Input Speed The input speed was varied while the input torque remained constant and the transmission efficiency of the herringbone planetary transmission system was analyzed. Variation of power loss with input speed is shown in Figure 16. Frictional power losses on the tooth surface increased as the rotational speed increased, and the windage power loss increases, suggesting that, when the rotational speed increases, the increase in resistance due to the surrounding air is greater than the increase of frictional power loss. The bearing power loss is different at different positions. The power loss at bearing 2 is the largest. With the speed increment, the total power loss at the bearing increases. As shown in Figure 17, the gear meshing efficiency slowly decreases as the rotational speed increases. Appl. Sci. 2020, 10, x FOR PEER REVIEW 21 of 27 However, windage losses rapidly increase, suggesting windage power loss is an essential consideration for gear systems operating at high rotational speeds.

55 3 planets 4 planets 50 5 planets

) 6 planets 45 Kw

( 40 35 30

Power Loss Power 25 20 15 4800 5600 6400 7200 8000 8800 9600 10400 Ω [rpm] sun (a)

10 3 planets 9 4 planets 8 5 planets

) 6 planets 7 Kw

( 6 5 4 3 Power Loss Power 2 1 4800 5600 6400 7200 8000 8800 9600 Ω [rpm] sun (b)

Figure 16. Cont. Appl. Sci. 2020, 10, 5849 20 of 25 Appl. Sci. 2020, 10, x FOR PEER REVIEW 22 of 27

350 4920 rpm 5412 rpm 5904 rpm 300 6396 rpm 6888 rpm 250 7380 rpm 7872 rpm 8364 rpm 200 8856 rpm 9348 rpm 150 Power Loss (w)

100

PR1 PR2 PR3 PR4 Bearing location (c) 1100 3 planets 4 planets 1000 5 planets 6 planets 900

800

700

Power (w) Loss 600

500

400 4800 5600 6400 7200 8000 8800 9600 Ω [rpm] sun (d)

Figure 16.FigureVariation 16. Variation of powerof power loss loss withwith input input speed. speed. (a) Frictional (a) Frictional power loss power on tooth loss surface; on tooth (b) surface; windage power loss; (c) bearing power loss at different positions; (d) bearing friction power loss. (b) windageAppl. Sci. 2020 power, 10, x loss;FOR PEER (c) bearingREVIEW power loss at diﬀerent positions; (d) bearing friction power23 of 27 loss.

3 planets 0.985 4 planets 5 planets 0.980 6 planets 0.975 0.970 0.965 0.960 0.955 0.950

Efficiency 0.945 0.940 0.935 0.930 0.925 4800 5600 6400 7200 8000 8800 9600 Ω [rpm] sun (a) 0.999 Figure 17. Cont. 3 planets 0.998 4 planets 5 planets 0.997 6 planets 0.996 0.995 0.994 Efficiency Efficiency 0.993 0.992 0.991 0.990 4800 5600 6400 7200 8000 8800 9600 Ω [rpm] sun (b)

Figure 17. Variation of efficiency loss with input speed. (a) Gear meshing efficiency loss; (b) windage efficiency loss.

5.3.3. Variation of Transmission Efficiency with Input Power Various power inputs were analyzed under the same input speed, and the transmission efficiency of the herringbone gear planetary transmission system was calculated. As shown in Figure 18, power loss due to friction on the tooth surface increases rapidly, whereas the power loss of the bearing increases, and the power loss at the bearings 2 and 4 is higher than the power loss at locations 2 and 3. Frictional power loss is greatly affected by the friction factor and load on the tooth surface. As gears rotate lubricant is flung off the gear teeth in small oil droplets owing to the centrifugal force acting on the lubricant. These lubricant droplets create a fine mist of oil that is suspended inside the gear housing. The effect of this oil mist is an increase in ‘friction resistance’ on the gears, and hence an increase in the power consumption. With the input torque increases, the expulsion of the oily atmosphere from the tooth spaces as the gear teeth come into engagement creates turbulence within the gearbox and increases the power consumption. As the power increases, the torque also increases, resulting in larger loads on the tooth surface and an increase in gear tooth frictional power Appl. Sci. 2020, 10, x FOR PEER REVIEW 23 of 27

3 planets 0.985 4 planets 5 planets 0.980 6 planets 0.975 0.970 0.965 0.960 0.955 0.950

Efficiency 0.945 0.940 0.935 0.930 0.925 4800 5600 6400 7200 8000 8800 9600 Appl. Sci. 2020, 10, 5849 Ω [rpm] 21 of 25 sun (a) 0.999 3 planets 0.998 4 planets 5 planets 0.997 6 planets 0.996 0.995 0.994 Efficiency Efficiency 0.993 0.992 0.991 0.990 4800 5600 6400 7200 8000 8800 9600 Ω [rpm] sun (b)

Figure 17.FigureVariation 17. Variation of effi ofciency efficiency loss loss with with input input speed. (a) ( aGear) Gear meshing meshing efficiency effi loss;ciency (b) windage loss; (b ) windage efficiencyefficiency loss. loss. 5.3.3. Variation of Transmission Efficiency with Input Power 5.3.3. Variation of Transmission Efficiency with Input Power Various power inputs were analyzed under the same input speed, and the transmission Variousefficiency power of the inputs herringbone were analyzed gear planetary under transm theission same system input was speed, calculated. and As the shown transmission in Figure efficiency of the herringbone18, power loss gear due planetary to friction transmissionon the tooth surfac systeme increases was rapidly, calculated. whereas As the shown power inloss Figure of the 18, power Appl. Sci. 2020, 10, x FOR PEER REVIEW 24 of 27 loss due tobearing friction increases, on the and tooththe power surface loss at the increases bearings 2 rapidly, and 4 is higher whereas than the the power power loss at loss locations of the bearing 2 and 3. Frictional power loss is greatly affected by the friction factor and load on the tooth surface. increases,loss. Moreover, andAs the gears the power transmissionrotate loss lubricant at the ef isficiency bearingsflung off of the the 2 gear and herringbone teeth 4 is in higher small planet oil than dropletsary the transmission power owing to loss the system atcentrifugal locations slowly 2 and 3. Frictionalincreasesforce power as actingthe lossinput on isthe power greatly lubricant. increases. aff Theseected lubricant by the droplets friction create factor a fine and mist load of oil on that the is suspended tooth surface. inside the gear housing. The effect of this oil mist is an increase in ‘friction resistance’ on the gears, and hence an increase in the power consumption. With the input torque increases, the expulsion of the 50 P 3 planets R1 oily48 atmosphere from the tooth spaces as the gear teeth240 come into engagement creates turbulence 46 4 planets PR2 44 within the gearbox 5 planets and increases the power consumption.220 As thePR3 power increases, the torque also

） 42 6 P increases,40 resultingplanets in larger loads on the tooth surface and200 an increaseR4 in gear tooth frictional power Kw 38

（ 36

34 180 32 30 160 28 26 140 24 Power Loss (w)

Power Loss Power 22 120 20 18 100 16 14 80 12 400 450 500 550 600 650 700 750 800 850 450 500 550 600 650 700 750 800 850 Input power (Kw) Input Power (Kw) (a) (b)

3 3 planets 50 planets 0.976 4 planets 4 planets 5 planets 45 5 planets ） 6 planets 0.968 6 planets Kw 40 （

0.960 35

30 0.952 Efficiency 25 0.944 Power Loss 20 0.936 15 400 450 500 550 600 650 700 750 800 850 900 450 500 550 600 650 700 750 800 850 Input Power (Kw) Input power (Kw) (c) (d)

FigureFigure 18. 18.Variation Variation ofof powerpower loss loss and and transmission transmission efficiency eﬃciency with with input input power. power. (a) Gear (a )tooth Gear tooth frictionalfrictional power power loss; loss; ( (bb)) bearingbearing power power loss; loss; (c () ctotal) total power power loss; loss; (d) total (d) total transmission transmission efficiency. eﬃciency.

5.4. Algorithm Verification The proposed method was verified by comparing calculated efficiency values with previously published values. In [24], the transmission efficiency of a transmission system with five planetary gears and an input rotational speed of 12,300 rpm was calculated as 93.39%, versus 93.1% using the method proposed in this paper. Thus, our approach can correctly estimate the transmission efficiency of a herringbone planetary gear transmission system, which was slightly lower in this paper because we comprehensively considered power losses, whereas previously, only power losses associated with gear meshing were considered. The power loss of the planetary transmission system in the literature [3,35] was calculated based on the method proposed in this paper, and the following results were obtained. As shown in the Figure 19, when the input torque is 1000 N·m, the method of calculation of tooth surface friction power loss in this paper is compared with the experimental results in the literature [35], and the total power loss of the gear transmission system is compared with the experimental results in the literature [3]. It can be concluded that the magnitude of power loss is the same, and as the speed increases, the trend of power loss is also consistent. Appl. Sci. 2020, 10, 5849 22 of 25

As gears rotate lubricant is flung off the gear teeth in small oil droplets owing to the centrifugal force acting on the lubricant. These lubricant droplets create a fine mist of oil that is suspended inside the gear housing. The effect of this oil mist is an increase in ‘friction resistance’ on the gears, and hence an increase in the power consumption. With the input torque increases, the expulsion of the oily atmosphere from the tooth spaces as the gear teeth come into engagement creates turbulence within the gearbox and increases the power consumption. As the power increases, the torque also increases, resulting in larger loads on the tooth surface and an increase in gear tooth frictional power loss. Moreover, the transmission efficiency of the herringbone planetary transmission system slowly increases as the input power increases.

5.4. Algorithm Verification The proposed method was verified by comparing calculated efficiency values with previously published values. In [24], the transmission efficiency of a transmission system with five planetary gears and an input rotational speed of 12,300 rpm was calculated as 93.39%, versus 93.1% using the method proposed in this paper. Thus, our approach can correctly estimate the transmission efficiency of a herringbone planetary gear transmission system, which was slightly lower in this paper because we comprehensively considered power losses, whereas previously, only power losses associated with gear meshing were considered. The power loss of the planetary transmission system in the literature [3,35] was calculated based on the method proposed in this paper, and the following results were obtained. As shown in the Figure 19, when the input torque is 1000 N m, the method of calculation of tooth surface friction power · loss in this paper is compared with the experimental results in the literature [35], and the total power loss of the gear transmission system is compared with the experimental results in the literature [3]. It can be concluded that the magnitude of power loss is the same, and as the speed increases, the trend ofAppl. power Sci. 2020 loss, 10 is, alsox FOR consistent. PEER REVIEW 25 of 27

4.5 5.0 exp[3] 4.0 pred 4.5 exp[35] pred 4.0

) 3.5

) 3.5 Kw (

3.0 Kw ( 3.0 2.5 2.5

2.0 2.0 1.5 Power Loss Loss Power

1.5 Loss Power 1.0 1.0 0.5 0.0 0.5 500 1000 1500 2000 2500 3000 3500 4000 4500 1000200030004000 Ω Ω [rpm] carrier [rpm] sun (a) (b)

FigureFigure 19. 19.Power Power loss loss comparison comparison with with literature literature experimental experimental results. results. (a) Tooth(a) Tooth surface surface friction friction power loss;power (b) totalloss; power(b) total loss. power loss.

5.5.5.5. Summary Summary (1)(1) This This paper paper presented presented aa formulaformula forfor calculatingcalculating the the transmission transmission efficiency efficiency of ofa power-split a power-split planetaryplanetary gearbox. gearbox. TheThe transmissiontransmission efficiency efficiency was was calculated calculated for for a aherringbone herringbone planetary planetary gear gear transmissiontransmission system system with with di ffdifferenterent numbers numbers of planetaryof planetary gears. gears. The The transmission transmission effi ciencyefficiency was was found tofound gradually to gradually decrease decrease as power as power was transmitted was transmitte alongd along the the transmission transmission path; path; however, however, thethe rate at whichat which the transmissionthe transmission effi ciencyefficiency decreased decreased varied. varied. (2) The influence of power loss and transmission efficiency on the herringbone planetary transmission system was studied under different lubrication conditions. The results suggest that lubrication state has a large influence on transmission efficiency and selecting the optimal lubrication method can greatly reduce gear tooth frictional power loss and increase the transmission efficiency. (3) The input speed was varied and the influence of power losses and transmission efficiency of the herringbone planetary transmission system was studied. When the rotational speed increased, the gear meshing efficiency decreased more slowly than the windage efficiency. Moreover, input speed was found to have a larger influence on windage power loss than on gear tooth frictional power loss. Thus, windage power loss should not be neglected in transmission efficiency calculations, particularly when gears operate at higher speeds. (4) Finally, the input power was varied and the influence of power losses on the transmission efficiency of the herringbone planetary transmission system was analyzed. Frictional power loss at the tooth surface was found to rapidly increase with increasing input power, whereas the windage power loss hardly changed at all. Furthermore, the transmission efficiency of the gear transmission system increased with the increasing input power.

6. Conclusions In this paper, a mathematical power loss model was combined with power flow diagrams to study the efficiency of planetary gear transmission systems. A novel method for analyzing the transmission efficiency of complex planetary gear systems based on the hypergraph was introduced and the power loss model was established by considering several power losses sources such as gear meshing friction, windage, and bearing friction. The proposed method was then used to analyze power loss and transmission efficiency of a two-stage herringbone gear transmission system under different lubrication conditions, input speeds, and input power. The results suggest that all of the main loss factors should be considered in the transmission efficiency calculation; in particular, windage loss should not be neglected, particularly for gears operating at high speeds. Studying the influence of various power loss factors is necessary for developing methods to improve transmission Appl. Sci. 2020, 10, 5849 23 of 25

(2) The influence of power loss and transmission efficiency on the herringbone planetary transmission system was studied under different lubrication conditions. The results suggest that lubrication state has a large influence on transmission efficiency and selecting the optimal lubrication method can greatly reduce gear tooth frictional power loss and increase the transmission efficiency. (3) The input speed was varied and the influence of power losses and transmission efficiency of the herringbone planetary transmission system was studied. When the rotational speed increased, the gear meshing efficiency decreased more slowly than the windage efficiency. Moreover, input speed was found to have a larger influence on windage power loss than on gear tooth frictional power loss. Thus, windage power loss should not be neglected in transmission efficiency calculations, particularly when gears operate at higher speeds. (4) Finally, the input power was varied and the influence of power losses on the transmission efficiency of the herringbone planetary transmission system was analyzed. Frictional power loss at the tooth surface was found to rapidly increase with increasing input power, whereas the windage power loss hardly changed at all. Furthermore, the transmission efficiency of the gear transmission system increased with the increasing input power.

6. Conclusions In this paper, a mathematical power loss model was combined with power flow diagrams to study the efficiency of planetary gear transmission systems. A novel method for analyzing the transmission efficiency of complex planetary gear systems based on the hypergraph was introduced and the power loss model was established by considering several power losses sources such as gear meshing friction, windage, and bearing friction. The proposed method was then used to analyze power loss and transmission efficiency of a two-stage herringbone gear transmission system under different lubrication conditions, input speeds, and input power. The results suggest that all of the main loss factors should be considered in the transmission efficiency calculation; in particular, windage loss should not be neglected, particularly for gears operating at high speeds. Studying the influence of various power loss factors is necessary for developing methods to improve transmission efficiency; thus, this paper provides a useful reference for analyzing and improving gear transmission efficiency.

Author Contributions: Conceptualization, Y.W. and W.Y.; methodology, Y.W.; software, X.T.; validation, Y.W., W.Y., and X.T.; formal analysis, Y.W.; investigation, X.L.; resources, Z.H.; data curation, Y.W.; writing—original draft preparation, Y.W.; writing—review and editing, W.Y.; visualization, Y.W.; supervision, W.Y.; project administration, W.Y.; funding acquisition, W.Y. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by National Key R & D Projects [grant number No: 2018YFB2001502], and Chongqing Natural Science Foundation [grant number No: cstc2018jcyjAX0468]. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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