# Investigation of Dynamic Load Sharing Behavior for Herringbone Planetary Gears considering Multicoupling Manufacturing Errors

Authors: Fei Ren; Jinchen Ji; Guofu Luo; Shaofu Zhao; Liya Zhao; Guiqin Shi; Xiaoling Wu; Ning Wang
Journal: Shock and Vibration
 (2021)

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## Abstract

In this study, based on the lumped-parameter theory and the Lagrange approach, a novel and generalized bending-torsional-axial coupled dynamic model for analyzing the load sharing behavior in the herringbone planetary gear train (HPGT) is presented by taking into account the actual structure of herringbone gears, manufacturing errors, time-dependent meshing stiffness, bearing deflections, and gyroscopic effects. The model can be applied to the analysis of the vibration of the HPGT with any number of planets and different types of manufacturing errors in different floating forms. The HPGT equivalent meshing error is analyzed and derived for the tooth profile errors and manufacturing eccentric errors of all components in the HPGT system. By employing the variable-step Runge–Kutta approach to calculate the system dynamic response, in conjunction with the presented calculation approach of the HPGT load sharing coefficient, the relationships among manufacturing errors, component floating, and load sharing are numerically obtained. The effects of the combined errors and single error on the load sharing are, respectively, discussed. Meanwhile, the effects of the support stiffness of the main components in the HPGT system on load sharing behavior are analyzed. The results indicate that manufacturing errors, floating components, and system support stiffness largely influence the load sharing behavior of the HPGT system. The research has a vital guiding significance for the design of the HPGT system.


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## Body

## 1. Introduction

Herringbone planetary gear train (HPGT) possesses the advantages in power split, compact structure, large transmission ratio, high transmission efficiency, strong carrying capacity, and small axial force, and thus it has been widely applied in the transmission systems of wind turbines, nuclear power plants, aircrafts, ships, and vehicles. However, in the presence of the inevitable manufacturing errors, elastic deformations, and so on, the HPGT cannot achieve the uniform load distribution among different transmission paths. On the one hand, this would lose the advantages of herringbone planetary transmission, and on the other hand, this would lead to overload, vibration, and noise of some components, which affect the service life, stability, and reliability of the gear set [1]. Consequently, it is of practical significance to construct a refined dynamic load sharing model of herringbone planetary gears to develop an in-depth understanding of load distribution characteristics resulting from the inevitable manufacturing errors.Some studies were devoted to investigating the dynamic behavior and load sharing of the planetary gear train (PGT). Lin and Parker established a spur PGT dynamic model to analyze its modal properties [2, 3]. Zhao and Ji developed a torsional dynamic model of the multistage spur PGTs of a wind turbine gearbox for the analysis of its nonlinear dynamic characteristics [4]. Zhang and Zhu investigated the planet meshing phase effect on load sharing characteristics of herringbone gear transmission system [5]. Mo et al. studied the load distribution behavior of compound planetary powertrains applied in wind turbine gearbox [6]. Singh developed the load sharing behavior analysis model of planetary gear transmission [7, 8]. Ligata et al. [9] and Bodas and Kahraman [10] examined the static load sharing behaviors of the planetary transmission. Zhu and Xu researched the impact of flexible pin shaft stiffness on load sharing features of planetary gearbox used in wind turbines [11]. Ren et al. investigated the effects of manufacturing errors on the dynamic behaviors and floating characteristics of herringbone planetary gears [1, 12, 13]. Li et al. conducted the analysis of load sharing and reliability prediction of helicopter planetary gear transmission [14]. Kang and Kahraman presented the dynamic features of the double-helical gear pair adopting theoretical and experimental methods [15]. Sondkar and Kahraman studied the dynamics of double-helical planetary gears [16].Despite the extensive investigations on the PGT dynamics, there is little research on the effects of multicoupling manufacturing errors and system stiffness on the load sharing behavior of the HPGT system. Meanwhile, the presence of multicoupling manufacturing errors can induce the axial dynamic force for the HPGT system. Therefore, the axial degree of freedom (DOF) of each component needs to be considered in the HPGT dynamic modeling. In addition, the HPGT system has its own specific characteristics. Specifically, due to the assembly limitations, the internal ring gear needs to adopt two separate helical gears. As a result, there are the characteristics of both helical and herringbone gears on both sides in HPGT. Thus, these specific characteristics should be considered in the dynamic modeling of the HPGT system.Meanwhile, the load sharing coefficient is one of the key design parameters of the HPGT system, whose reasonable selection noticeably affects its dynamic characteristics. In the past, the load sharing coefficient was chosen in terms of the empirical value in the design manual, whereas in this paper, a refined HPGT dynamic model will be proposed to calculate the HPGT load sharing coefficient more accurately. It is expected that this research provides an alternative and novel methodology for accurately selecting the load sharing coefficient and thus has important scientific and engineering significances in applications.In this paper, based on the structure features of herringbone gears, a novel refined dynamic load sharing model for the HPGT considering manufacturing errors is presented by using the lumped-parameter method. The load sharing behavior of an example HPGT is numerically simulated, and the effects of manufacturing errors and system support stiffness on the dynamic load sharing behavior are analyzed.


## 2. Composition and Operation Principle of the HPGT System

Figure1 shows the structural and kinematic diagram of herringbone planetary gear transmissions, which comprises a herringbone sun gear, two ring gears (i.e., left and right ring gears), N herringbone planet gears, and a carrier. As demonstrated in Figure 1, the symbols s, r2, r1, pi, and c indicate the sun gear, ring gears, planet gear, and carrier, respectively. The sun gear s is taken as the input component, carrier c is taken as the output member, and the two ring gears are taken as the stationary components. By considering assembly, the ring gear adopts two helical gears with contrary helix angles. In the HPGT system, there are two kinds of meshing pairs, namely, internal mesh (i.e., planet-left ring mesh and planet-right ring mesh) and external mesh (i.e., sun-planet mesh).Figure 1
Kinematic sketch of herringbone planetary gears.


## 3. Dynamic Model of the HPGT System

### 3.1. Dynamic Model

Figure2 shows the three-dimensional (3D) bending-torsional-axial coupled dynamic model for the HPGT system in consideration of both sides of herringbone gears and the axial DOF of each component, which differentiate the developed dynamic model from the existing models of two-dimensional (2D) PGT. For the sake of simplicity, only one planet gear is displayed in Figure 2. The floating component is represented by the low radial support stiffness. In the previous investigations on herringbone gears, it was assumed that the axial forces between two helical gears with opposite rotation directions canceled each other. Accordingly, the axial vibration was ignored and the herringbone gear was simplified to a 2D plane model. As a matter of fact, since there exist manufacturing errors, the dynamic performance of left and right gear pairs cannot be identical. To more accurately demonstrate the dynamic behaviors of herringbone gears, the actual structural properties of herringbone gears on both sides, the axial vibrations of the central member, and each planet are considered in the present dynamic model as depicted in Figure 2. In assembly, the ring gear utilizes two helical gears with the opposite helix angles. Consequently, each component (i.e., sun gear, left and right ring gears, carrier, and each planet gear) has four DOFs including one rotation and three translations, which are denoted as xj, yj, zj, θj, j = s, r2, r1, c and ξi, ηi, ρi, θi, i = 1, 2, …, N, respectively. The frames of reference are illustrated in detail in Figures 2–5, where the superscripts R and L indicate the right and left sides of the component, respectively. In order to facilitate the calculation, each angular displacement θ is converted into an equivalent linear displacement u in the tangential direction of the corresponding base circle:(1)us=rs⋅θs,ur2=rr2⋅θr2,ur1=rr1⋅θr1,uc=rc⋅θc,ui=ri⋅θi,i=1,2,…,N,where rs, rr2, rr1, ri are, respectively, the base circle radius of the sun gear, left and right ring gears, and the ith planet gear, while rc is the distance from the carrier rotation center to the planet gear center. Hence, there are a total of 16 + 4N DOFs in the proposed model shown in Figure 2. They can be expressed as(2)U=Uc,Ur2,Ur1,Us,U1,U2,…,UNT,where Uc=xc,yc,zc,ucT, Ur2=xr2,yr2,zr2,ur2T, Ur1=xr1,yr1,zr1,ur1T, Us=xs,ys,zs,usT, and Ui=ξi,ηi,ρi,uiTi=1,⋯,N.Figure 2
The refined coupled dynamic model of the HPGT system.Figure 3
Dynamic model of theith planet-sun gear pair.Figure 4
Dynamic model of theith left and right ring-planet gear pairs.Figure 5
Dynamic model of theith planet-carrier pair.In the dynamic load sharing model shown in Figure2, kspi, kr1pi, and kr2pi are the meshing stiffness for the ith sun-planet pair, right ring-planet pair, and left ring-planet pair, respectively. ks, kc, kr1, kr2, and kp are the bearing support stiffness of the sun gear, carrier, right and left ring gears, and planet gear, respectively. kt is the torsional support stiffness. Here, the subscript spi stands for sun-planet gear i pair, the subscript r1pi indicates right ring-planet gear i pair, and the subscript r2pi represents left ring-planet gear i pair. The symbol pi denotes the ith planet gear, and symbols s, c, r2, and r1 are of the same meanings as those demonstrated in Figure 1.According to the presented HPGT dynamic model in Figure2, the equivalent displacements of the ith planet-sun gear mesh, the ith planet-left (right) ring gear mesh, and the ith planet-carrier pair can be derived as [12, 17, 18](3)μspiLR=−xssinϕsi⋅cosβb+yscosϕsi⋅cosβb+zs⋅sin±βb+us⋅cosβb−ξisinαt⋅cosβb−ηicosαt⋅cosβb−ρi⋅sin±βb+ui⋅cosβb+espiLRt,μr21piLR=−xr21sinϕri⋅cosβb+yr21cosϕri⋅cosβb−zr21⋅sin±βb+ur21⋅cosβb+ξisinαt⋅cosβb−ηicosαt⋅cosβb+ρi⋅sin±βb−ui⋅cosβb+er21piLRt,μξcpi=−ξi+xc⋅cosϕi+yc⋅sinϕi,μηcpi=−ηi−xc⋅sinϕi+yc⋅cosϕi+uc,μρcpi=ρi−zc,where μspiLR indicates the equivalent microdeformation of the ith planet-sun gear pair on the left (right) side. μr21piLR indicates the equivalent microdeformation of the ith planet-left (right) ring gear pair. μξcpi,μηcpi,μρcpi are the relative deformations in the ξ-, η-, and ρ-directions of the ith carrier-planet pair, respectively. espiLRt represents the static transmission error of the ith planet-sun gear pair on the left (right) side. er21piLRt represents the static transmission error of the ith planet-left (right) ring gear pair. Here, “+” refers to the left-side sun-planet gear i pair (left ring-planet gear i pair) and “−” for the right-side sun-planet gear i pair (right ring-planet gear i pair) in symbol “±.” βb expresses the base helix angle. ϕsi=ϕi−αt,ϕri=ϕi+αt, where ϕi denotes the ith planet assembly position angle and αt stands for the transverse pressure angle.The meshing forces can be expressed as(4)Fr2piL=kr2piL⋅μr2piL,Fr2piL=kr1piR⋅μr1piR,FspiL=kspiL⋅μspiL,FspiR=kspiR⋅μspiR,where FspiL and FspiR are the meshing forces for the ith planet-sun gear pair on the left side and right side, respectively. Fr2piL and Fr2spil are the meshing forces of the ith planet-left and right ring gear pair, respectively.


### 3.2. Equations of Motion of the HPGT

#### 3.2.1. Equations of Motion for the Herringbone Sun Gear

Figure3 presents the dynamic model of the ith planet-sun gear pair in the HPGT system in isometric view. Based on D'Alembert's principle, the equations of motion for herringbone sun gear can be obtained as(5)msx¨s−2ωcy˙s−ωc2xs−∑i=1NFspiR+FspiL⋅cosβb⋅sinφsi+ksx⋅xs=0,msy¨s+2ωcx˙s−ωc2ys+∑i=1NFspiR+FspiL⋅cosβb⋅cosφsi+ksy⋅ys=0,msz¨s+∑i=1NFspiR+FspiL⋅sinβb+ksz⋅zs=0,Jsrs2u¨s+∑i=1NFspiR+FspiL⋅cosβb+kst⋅us−Tsrs=0,where ms is the sun gear mass; Js is the sun gear rotational inertia; xs, ys, and zs are the vibration microdisplacements in x-, y-, and z-direction, respectively. us is the torsional microdisplacement about z-direction for the sun gear, and Ts indicates the external torque acting on the herringbone sun gear.


#### 3.2.2. Equations of Motion for the Left Helical Ring Gear

Figure4 presents the dynamic model of the ith left ring- (right ring-) planet gear pair in the HPGT system in isometric view. According to the dynamic model of the internal meshing pair shown in Figure 4 and D’Alembert’s principle, the equations of motion for the left ring gear can be written as(6)mrx¨r2−2ωcy˙r2−ωc2xr2−∑i=1NFr2piL⋅cosβb⋅sinφri+kr2x⋅xr2=0,mry¨r2+2ωcx˙r2−ωc2yr2+∑i=1NFr2piL⋅cosβb⋅cosφri+kr2y⋅yr2=0,mrz¨r2+∑i=1NFr2piL⋅sinβb+kr2z⋅zr2=0,Jrrr2u¨r2+∑i=1NFr2piL⋅cosβb+kr2t⋅ur2=0,where mr is the ring gear mass and Jr is the ring gear rotational inertia. xr2, yr2, and zr2 are the vibration microdisplacements in x, y, and z directions for the left ring gear, respectively. ur2 is the torsional microdisplacement about z-direction for the left ring gear.


#### 3.2.3. Equations of Motion for the Right Helical Ring Gear

The equations of motion for the right ring gear can be deduced as(7)mrx¨r1−2ωcy˙r1−ωc2xr1−∑i=1NFr1piR⋅cosβb⋅sinφri+kr1x⋅xr1=0,mry¨r1+2ωcx˙r1−ωc2yr1+∑i=1NFr1piR⋅cosβb⋅cosφri+kr1y⋅yr1=0,mrz¨r1−∑i=1NFr1piR⋅sinβb+kr1z⋅zr1=0,Jrrr2u¨r1+∑i=1NFr1piR⋅cosβb+kr1t⋅ur1=0,where xr1, yr1, and zr1 are, respectively, the vibration microdisplacements in x-, y-, and z-directions for the right ring gear. ur1 refers to the torsional microdisplacement about the z-direction for the right ring gear.


#### 3.2.4. Equations of Motion for the Carrier

Figure5 displays the dynamic model of the ith planet-carrier pair in the HPGT system. By applying D'Alembert's principle, the equations of motion for the carrier can be obtained as(8)mcx¨c−2ωcy˙c−ωc2xcx¨c−2ωcy˙c−ωc2xc+∑i=1Nkpξ⋅μξcpi⋅cosφi−kpη⋅μηcpi⋅sinφi+kcx⋅xc=0,mcy¨c+2ωcx˙c−ωc2yc+∑i=1Nkpξ⋅μξcpi⋅sinφi+kpη⋅μηcpi⋅cosφi+kcy⋅yc=0,mcz¨c−∑i=1Nkpρμρcpi+kcz⋅zc=0,Jcrc2u¨c+∑i=1Nkpημηcpi+kct⋅uc+Tcr=0,where mc is the carrier mass; Jc is the carrier rotational inertia; and xc, yc, and zc are the vibration microdisplacements in the x-, y-, and z-direction for the carrier. uc is the torsional microdisplacement about the z-direction for the carrier. Tc refers to the load torque acting on the carrier.


#### 3.2.5. Equations of Motion for theith Herringbone Planet Gear

According to Figures2–5, the equations of motion for the ith herringbone planet gear can be given by(9)mpξ¨i−2ωcη˙i−ωc2ξi−FspiR+FspiL⋅sinαt⋅cosβb+Fr1piR+Fr2piLb⋅sinαt⋅cosβb−kpξ⋅μξcpi=0,mpη¨i+2ωcξ˙i−ωc2ηi−FspiR+FspiL⋅cosβbcosαt−Fr1piR+Fr2piL⋅cosβb⋅cosαt−kpη⋅μηcpi=0,mpρ¨i−FspiR⋅sinβb+FspiL⋅sinβb+Fr1piR⋅sinβb−Fr2piL⋅sinβb+kpρ⋅μρcpi=0,Jprp2u¨i+FspiR+FspiL⋅cosβb−Fr1piR+Fr2piL⋅cosβb=0,where mp is the planet gear mass and Jp is the planet gear rotational inertia. ξi, ηi, and ρi mean the vibration microdisplacements in the ξ, η, and ρ directions for the ith herringbone planet gear, respectively. ui indicates the torsional microdisplacement about the ρ-direction for the sun gear.By assembling the above equations of motion for each member, the matrix form of herringbone planetary gear transmissions withN planet gears can be written as(10)M⋅U¨t+ωc⋅G⋅U˙t+Kb+Kmt−ωc2⋅KΩ⋅Ut=Tk+Tt,where Ut indicates the generalized displacement column vector, [M] indicates the generalized inertial matrix, [Kb] refers to the supporting stiffness matrix, [Km] means the time-dependent meshing stiffness matrix, [G] represents the gyroscopic matrix, [KΩ] denotes the centripetal stiffness matrix, {Tk} expresses the excitation matrix caused by meshing stiffness and manufacturing error, and {T(t)} denotes the vector of external excitation forces.


## 3.1. Dynamic Model

Figure2 shows the three-dimensional (3D) bending-torsional-axial coupled dynamic model for the HPGT system in consideration of both sides of herringbone gears and the axial DOF of each component, which differentiate the developed dynamic model from the existing models of two-dimensional (2D) PGT. For the sake of simplicity, only one planet gear is displayed in Figure 2. The floating component is represented by the low radial support stiffness. In the previous investigations on herringbone gears, it was assumed that the axial forces between two helical gears with opposite rotation directions canceled each other. Accordingly, the axial vibration was ignored and the herringbone gear was simplified to a 2D plane model. As a matter of fact, since there exist manufacturing errors, the dynamic performance of left and right gear pairs cannot be identical. To more accurately demonstrate the dynamic behaviors of herringbone gears, the actual structural properties of herringbone gears on both sides, the axial vibrations of the central member, and each planet are considered in the present dynamic model as depicted in Figure 2. In assembly, the ring gear utilizes two helical gears with the opposite helix angles. Consequently, each component (i.e., sun gear, left and right ring gears, carrier, and each planet gear) has four DOFs including one rotation and three translations, which are denoted as xj, yj, zj, θj, j = s, r2, r1, c and ξi, ηi, ρi, θi, i = 1, 2, …, N, respectively. The frames of reference are illustrated in detail in Figures 2–5, where the superscripts R and L indicate the right and left sides of the component, respectively. In order to facilitate the calculation, each angular displacement θ is converted into an equivalent linear displacement u in the tangential direction of the corresponding base circle:(1)us=rs⋅θs,ur2=rr2⋅θr2,ur1=rr1⋅θr1,uc=rc⋅θc,ui=ri⋅θi,i=1,2,…,N,where rs, rr2, rr1, ri are, respectively, the base circle radius of the sun gear, left and right ring gears, and the ith planet gear, while rc is the distance from the carrier rotation center to the planet gear center. Hence, there are a total of 16 + 4N DOFs in the proposed model shown in Figure 2. They can be expressed as(2)U=Uc,Ur2,Ur1,Us,U1,U2,…,UNT,where Uc=xc,yc,zc,ucT, Ur2=xr2,yr2,zr2,ur2T, Ur1=xr1,yr1,zr1,ur1T, Us=xs,ys,zs,usT, and Ui=ξi,ηi,ρi,uiTi=1,⋯,N.Figure 2
The refined coupled dynamic model of the HPGT system.Figure 3
Dynamic model of theith planet-sun gear pair.Figure 4
Dynamic model of theith left and right ring-planet gear pairs.Figure 5
Dynamic model of theith planet-carrier pair.In the dynamic load sharing model shown in Figure2, kspi, kr1pi, and kr2pi are the meshing stiffness for the ith sun-planet pair, right ring-planet pair, and left ring-planet pair, respectively. ks, kc, kr1, kr2, and kp are the bearing support stiffness of the sun gear, carrier, right and left ring gears, and planet gear, respectively. kt is the torsional support stiffness. Here, the subscript spi stands for sun-planet gear i pair, the subscript r1pi indicates right ring-planet gear i pair, and the subscript r2pi represents left ring-planet gear i pair. The symbol pi denotes the ith planet gear, and symbols s, c, r2, and r1 are of the same meanings as those demonstrated in Figure 1.According to the presented HPGT dynamic model in Figure2, the equivalent displacements of the ith planet-sun gear mesh, the ith planet-left (right) ring gear mesh, and the ith planet-carrier pair can be derived as [12, 17, 18](3)μspiLR=−xssinϕsi⋅cosβb+yscosϕsi⋅cosβb+zs⋅sin±βb+us⋅cosβb−ξisinαt⋅cosβb−ηicosαt⋅cosβb−ρi⋅sin±βb+ui⋅cosβb+espiLRt,μr21piLR=−xr21sinϕri⋅cosβb+yr21cosϕri⋅cosβb−zr21⋅sin±βb+ur21⋅cosβb+ξisinαt⋅cosβb−ηicosαt⋅cosβb+ρi⋅sin±βb−ui⋅cosβb+er21piLRt,μξcpi=−ξi+xc⋅cosϕi+yc⋅sinϕi,μηcpi=−ηi−xc⋅sinϕi+yc⋅cosϕi+uc,μρcpi=ρi−zc,where μspiLR indicates the equivalent microdeformation of the ith planet-sun gear pair on the left (right) side. μr21piLR indicates the equivalent microdeformation of the ith planet-left (right) ring gear pair. μξcpi,μηcpi,μρcpi are the relative deformations in the ξ-, η-, and ρ-directions of the ith carrier-planet pair, respectively. espiLRt represents the static transmission error of the ith planet-sun gear pair on the left (right) side. er21piLRt represents the static transmission error of the ith planet-left (right) ring gear pair. Here, “+” refers to the left-side sun-planet gear i pair (left ring-planet gear i pair) and “−” for the right-side sun-planet gear i pair (right ring-planet gear i pair) in symbol “±.” βb expresses the base helix angle. ϕsi=ϕi−αt,ϕri=ϕi+αt, where ϕi denotes the ith planet assembly position angle and αt stands for the transverse pressure angle.The meshing forces can be expressed as(4)Fr2piL=kr2piL⋅μr2piL,Fr2piL=kr1piR⋅μr1piR,FspiL=kspiL⋅μspiL,FspiR=kspiR⋅μspiR,where FspiL and FspiR are the meshing forces for the ith planet-sun gear pair on the left side and right side, respectively. Fr2piL and Fr2spil are the meshing forces of the ith planet-left and right ring gear pair, respectively.


## 3.2. Equations of Motion of the HPGT

### 3.2.1. Equations of Motion for the Herringbone Sun Gear

Figure3 presents the dynamic model of the ith planet-sun gear pair in the HPGT system in isometric view. Based on D'Alembert's principle, the equations of motion for herringbone sun gear can be obtained as(5)msx¨s−2ωcy˙s−ωc2xs−∑i=1NFspiR+FspiL⋅cosβb⋅sinφsi+ksx⋅xs=0,msy¨s+2ωcx˙s−ωc2ys+∑i=1NFspiR+FspiL⋅cosβb⋅cosφsi+ksy⋅ys=0,msz¨s+∑i=1NFspiR+FspiL⋅sinβb+ksz⋅zs=0,Jsrs2u¨s+∑i=1NFspiR+FspiL⋅cosβb+kst⋅us−Tsrs=0,where ms is the sun gear mass; Js is the sun gear rotational inertia; xs, ys, and zs are the vibration microdisplacements in x-, y-, and z-direction, respectively. us is the torsional microdisplacement about z-direction for the sun gear, and Ts indicates the external torque acting on the herringbone sun gear.


### 3.2.2. Equations of Motion for the Left Helical Ring Gear

Figure4 presents the dynamic model of the ith left ring- (right ring-) planet gear pair in the HPGT system in isometric view. According to the dynamic model of the internal meshing pair shown in Figure 4 and D’Alembert’s principle, the equations of motion for the left ring gear can be written as(6)mrx¨r2−2ωcy˙r2−ωc2xr2−∑i=1NFr2piL⋅cosβb⋅sinφri+kr2x⋅xr2=0,mry¨r2+2ωcx˙r2−ωc2yr2+∑i=1NFr2piL⋅cosβb⋅cosφri+kr2y⋅yr2=0,mrz¨r2+∑i=1NFr2piL⋅sinβb+kr2z⋅zr2=0,Jrrr2u¨r2+∑i=1NFr2piL⋅cosβb+kr2t⋅ur2=0,where mr is the ring gear mass and Jr is the ring gear rotational inertia. xr2, yr2, and zr2 are the vibration microdisplacements in x, y, and z directions for the left ring gear, respectively. ur2 is the torsional microdisplacement about z-direction for the left ring gear.


### 3.2.3. Equations of Motion for the Right Helical Ring Gear

The equations of motion for the right ring gear can be deduced as(7)mrx¨r1−2ωcy˙r1−ωc2xr1−∑i=1NFr1piR⋅cosβb⋅sinφri+kr1x⋅xr1=0,mry¨r1+2ωcx˙r1−ωc2yr1+∑i=1NFr1piR⋅cosβb⋅cosφri+kr1y⋅yr1=0,mrz¨r1−∑i=1NFr1piR⋅sinβb+kr1z⋅zr1=0,Jrrr2u¨r1+∑i=1NFr1piR⋅cosβb+kr1t⋅ur1=0,where xr1, yr1, and zr1 are, respectively, the vibration microdisplacements in x-, y-, and z-directions for the right ring gear. ur1 refers to the torsional microdisplacement about the z-direction for the right ring gear.


### 3.2.4. Equations of Motion for the Carrier

Figure5 displays the dynamic model of the ith planet-carrier pair in the HPGT system. By applying D'Alembert's principle, the equations of motion for the carrier can be obtained as(8)mcx¨c−2ωcy˙c−ωc2xcx¨c−2ωcy˙c−ωc2xc+∑i=1Nkpξ⋅μξcpi⋅cosφi−kpη⋅μηcpi⋅sinφi+kcx⋅xc=0,mcy¨c+2ωcx˙c−ωc2yc+∑i=1Nkpξ⋅μξcpi⋅sinφi+kpη⋅μηcpi⋅cosφi+kcy⋅yc=0,mcz¨c−∑i=1Nkpρμρcpi+kcz⋅zc=0,Jcrc2u¨c+∑i=1Nkpημηcpi+kct⋅uc+Tcr=0,where mc is the carrier mass; Jc is the carrier rotational inertia; and xc, yc, and zc are the vibration microdisplacements in the x-, y-, and z-direction for the carrier. uc is the torsional microdisplacement about the z-direction for the carrier. Tc refers to the load torque acting on the carrier.


### 3.2.5. Equations of Motion for theith Herringbone Planet Gear

According to Figures2–5, the equations of motion for the ith herringbone planet gear can be given by(9)mpξ¨i−2ωcη˙i−ωc2ξi−FspiR+FspiL⋅sinαt⋅cosβb+Fr1piR+Fr2piLb⋅sinαt⋅cosβb−kpξ⋅μξcpi=0,mpη¨i+2ωcξ˙i−ωc2ηi−FspiR+FspiL⋅cosβbcosαt−Fr1piR+Fr2piL⋅cosβb⋅cosαt−kpη⋅μηcpi=0,mpρ¨i−FspiR⋅sinβb+FspiL⋅sinβb+Fr1piR⋅sinβb−Fr2piL⋅sinβb+kpρ⋅μρcpi=0,Jprp2u¨i+FspiR+FspiL⋅cosβb−Fr1piR+Fr2piL⋅cosβb=0,where mp is the planet gear mass and Jp is the planet gear rotational inertia. ξi, ηi, and ρi mean the vibration microdisplacements in the ξ, η, and ρ directions for the ith herringbone planet gear, respectively. ui indicates the torsional microdisplacement about the ρ-direction for the sun gear.By assembling the above equations of motion for each member, the matrix form of herringbone planetary gear transmissions withN planet gears can be written as(10)M⋅U¨t+ωc⋅G⋅U˙t+Kb+Kmt−ωc2⋅KΩ⋅Ut=Tk+Tt,where Ut indicates the generalized displacement column vector, [M] indicates the generalized inertial matrix, [Kb] refers to the supporting stiffness matrix, [Km] means the time-dependent meshing stiffness matrix, [G] represents the gyroscopic matrix, [KΩ] denotes the centripetal stiffness matrix, {Tk} expresses the excitation matrix caused by meshing stiffness and manufacturing error, and {T(t)} denotes the vector of external excitation forces.


## 3.2.1. Equations of Motion for the Herringbone Sun Gear

Figure3 presents the dynamic model of the ith planet-sun gear pair in the HPGT system in isometric view. Based on D'Alembert's principle, the equations of motion for herringbone sun gear can be obtained as(5)msx¨s−2ωcy˙s−ωc2xs−∑i=1NFspiR+FspiL⋅cosβb⋅sinφsi+ksx⋅xs=0,msy¨s+2ωcx˙s−ωc2ys+∑i=1NFspiR+FspiL⋅cosβb⋅cosφsi+ksy⋅ys=0,msz¨s+∑i=1NFspiR+FspiL⋅sinβb+ksz⋅zs=0,Jsrs2u¨s+∑i=1NFspiR+FspiL⋅cosβb+kst⋅us−Tsrs=0,where ms is the sun gear mass; Js is the sun gear rotational inertia; xs, ys, and zs are the vibration microdisplacements in x-, y-, and z-direction, respectively. us is the torsional microdisplacement about z-direction for the sun gear, and Ts indicates the external torque acting on the herringbone sun gear.


## 3.2.2. Equations of Motion for the Left Helical Ring Gear

Figure4 presents the dynamic model of the ith left ring- (right ring-) planet gear pair in the HPGT system in isometric view. According to the dynamic model of the internal meshing pair shown in Figure 4 and D’Alembert’s principle, the equations of motion for the left ring gear can be written as(6)mrx¨r2−2ωcy˙r2−ωc2xr2−∑i=1NFr2piL⋅cosβb⋅sinφri+kr2x⋅xr2=0,mry¨r2+2ωcx˙r2−ωc2yr2+∑i=1NFr2piL⋅cosβb⋅cosφri+kr2y⋅yr2=0,mrz¨r2+∑i=1NFr2piL⋅sinβb+kr2z⋅zr2=0,Jrrr2u¨r2+∑i=1NFr2piL⋅cosβb+kr2t⋅ur2=0,where mr is the ring gear mass and Jr is the ring gear rotational inertia. xr2, yr2, and zr2 are the vibration microdisplacements in x, y, and z directions for the left ring gear, respectively. ur2 is the torsional microdisplacement about z-direction for the left ring gear.


## 3.2.3. Equations of Motion for the Right Helical Ring Gear

The equations of motion for the right ring gear can be deduced as(7)mrx¨r1−2ωcy˙r1−ωc2xr1−∑i=1NFr1piR⋅cosβb⋅sinφri+kr1x⋅xr1=0,mry¨r1+2ωcx˙r1−ωc2yr1+∑i=1NFr1piR⋅cosβb⋅cosφri+kr1y⋅yr1=0,mrz¨r1−∑i=1NFr1piR⋅sinβb+kr1z⋅zr1=0,Jrrr2u¨r1+∑i=1NFr1piR⋅cosβb+kr1t⋅ur1=0,where xr1, yr1, and zr1 are, respectively, the vibration microdisplacements in x-, y-, and z-directions for the right ring gear. ur1 refers to the torsional microdisplacement about the z-direction for the right ring gear.


## 3.2.4. Equations of Motion for the Carrier

Figure5 displays the dynamic model of the ith planet-carrier pair in the HPGT system. By applying D'Alembert's principle, the equations of motion for the carrier can be obtained as(8)mcx¨c−2ωcy˙c−ωc2xcx¨c−2ωcy˙c−ωc2xc+∑i=1Nkpξ⋅μξcpi⋅cosφi−kpη⋅μηcpi⋅sinφi+kcx⋅xc=0,mcy¨c+2ωcx˙c−ωc2yc+∑i=1Nkpξ⋅μξcpi⋅sinφi+kpη⋅μηcpi⋅cosφi+kcy⋅yc=0,mcz¨c−∑i=1Nkpρμρcpi+kcz⋅zc=0,Jcrc2u¨c+∑i=1Nkpημηcpi+kct⋅uc+Tcr=0,where mc is the carrier mass; Jc is the carrier rotational inertia; and xc, yc, and zc are the vibration microdisplacements in the x-, y-, and z-direction for the carrier. uc is the torsional microdisplacement about the z-direction for the carrier. Tc refers to the load torque acting on the carrier.


## 3.2.5. Equations of Motion for theith Herringbone Planet Gear

According to Figures2–5, the equations of motion for the ith herringbone planet gear can be given by(9)mpξ¨i−2ωcη˙i−ωc2ξi−FspiR+FspiL⋅sinαt⋅cosβb+Fr1piR+Fr2piLb⋅sinαt⋅cosβb−kpξ⋅μξcpi=0,mpη¨i+2ωcξ˙i−ωc2ηi−FspiR+FspiL⋅cosβbcosαt−Fr1piR+Fr2piL⋅cosβb⋅cosαt−kpη⋅μηcpi=0,mpρ¨i−FspiR⋅sinβb+FspiL⋅sinβb+Fr1piR⋅sinβb−Fr2piL⋅sinβb+kpρ⋅μρcpi=0,Jprp2u¨i+FspiR+FspiL⋅cosβb−Fr1piR+Fr2piL⋅cosβb=0,where mp is the planet gear mass and Jp is the planet gear rotational inertia. ξi, ηi, and ρi mean the vibration microdisplacements in the ξ, η, and ρ directions for the ith herringbone planet gear, respectively. ui indicates the torsional microdisplacement about the ρ-direction for the sun gear.By assembling the above equations of motion for each member, the matrix form of herringbone planetary gear transmissions withN planet gears can be written as(10)M⋅U¨t+ωc⋅G⋅U˙t+Kb+Kmt−ωc2⋅KΩ⋅Ut=Tk+Tt,where Ut indicates the generalized displacement column vector, [M] indicates the generalized inertial matrix, [Kb] refers to the supporting stiffness matrix, [Km] means the time-dependent meshing stiffness matrix, [G] represents the gyroscopic matrix, [KΩ] denotes the centripetal stiffness matrix, {Tk} expresses the excitation matrix caused by meshing stiffness and manufacturing error, and {T(t)} denotes the vector of external excitation forces.


## 4. Multicoupling Manufacturing Error Analysis

This study considers the manufacturing eccentric errors for each component (i.e., the carrier and each gear) and tooth profile errors of each gear. These tooth profile errors and eccentric errors are transformed into the line of action (LOA) of the gear pair, respectively, and the equivalent displacements of the errors can be acquired by superposition.


### 4.1. Equivalent Displacements of Eccentric Errors

Figure6 illustrates the projection relation of manufacturing eccentric errors for herringbone planetary gear set on the line of action, where Es and βs, respectively, represent the manufacturing eccentric error and its initial phase of the sun gear. e refers to the equivalent meshing error on the meshing line. ω denotes the angular velocity of each component. φi means the position angle of the ith planet gear, and φi=2πi−1/N, where N is the planet number. Here, the subscripts s, r1, r2, pi, and c have the same meanings as those exhibited in Figure 1. n and w refer to the internal and external mesh, respectively. α is the pressure angle, and αn=αw=αt.Figure 6
Equivalent meshing error on the line of action in the HPGT system.In view of Figure6 and the relative motion state of each component, the equivalent meshing error of sun-gear manufacturing eccentric error and initial phase Es,βs on the external meshing line can be obtained as(11)eEsi=−Essinωs−ωct+βs+αt−φi.Similarly, the equivalent meshing error of planet-gear manufacturing eccentric error and initial phaseEpi,βpi on the external and internal meshing line can, respectively, be given by(12)ewEpi=−Episinωp−ωct+βpi+αt,enEpi=Episinωp−ωct+βpi+αt.The equivalent meshing error of left ring-gear manufacturing eccentric error and initial phaseEr2,βr2 on the internal meshing line can be denoted as(13)eEr2i=Er2Lsin−ωct+βr2−αt−φi.The equivalent meshing error of right ring-gear manufacturing eccentric error and initial phaseEr1,βr1 on the internal meshing line can be derived as(14)eEr1i=Er1Rsin−ωct+βr1−αt−φi.The equivalent meshing error of carrier manufacturing eccentric error and initial phaseEc,βc on the external and internal meshing line can, respectively, be expressed as(15)ewEci=Ecsinβc−φi⋅cosαt,enEci=−Ecsinβc−φi⋅cosαt.The equivalent cumulative meshing errors generated by the superposition of the eccentric errors of each component for the internal and external meshing lines on the left and right sides of herringbone planetary gear transmission can, respectively, be expressed as(16)eEr2piL=Er2L⋅sin−ωc⋅t+βr2−αt−φi+Epi⋅sinωp−ωc⋅t−αt+βpi−Ec⋅sinβc−φi⋅cosαt,eEr1piR=Er1R⋅sin−ωc⋅t+βr1−αt−φi+Epi⋅sinωp−ωc⋅t−αt+βpi−Ec⋅sinβc−φi⋅cosαt,eEspiL=−Es⋅sinωs−ωc⋅t+βs+αt−φi−Epi⋅sinωp−ωc⋅t+αt+βpi+Ec⋅sinβc−φi⋅cosαt,eEspiR=−Es⋅sinωs−ωc⋅t+βs+αt−φi−Epi⋅sinωp−ωc⋅t+αt+βpi+Ec⋅sinβc−φi⋅cosαt,where eEr2piL, eEr1piR, eEspiL, and eEspiR, respectively, refer to the equivalent displacements of manufacturing eccentric errors for the ith internal and external meshing pair on the left and right sides of HPGT. Es,Epi,Ec,Er2L, and Er1R, respectively, denote the amplitudes of manufacturing eccentric errors for component g (g = s, pi, c, r2, and r1). βs, βpi, βc, βr2, and βr1, respectively, represent the initial phases of manufacturing eccentric errors for component g (g = s, pi, c, r2, and r1). ωs, ωp, and ωc, respectively, mean the angular velocity for component g (g = s, p, and c).


### 4.2. Equivalent Displacements of Tooth Profile Errors

This paper also considers the tooth profile errors, which directly act on the engagement surface of the gear tooth. The tooth engagement is a dynamic interaction between two or more gear teeth. The tooth profile error on the line of action has something to do with the tooth profile errors of the two gears in the gear pair. The profile errors on the meshing tooth surface need to be converted to the meshing line of the gear pair, which are generally performed by using a simple harmonic function according to the meshing frequency. Figure7 displays the tooth profile error sketch.Figure 7
Schematic of tooth profile error.The formulas of the tooth profile errors transformed into the external (sun-planet) meshing line are deduced as(17)E¯spiL=Espi⋅sinωmt+γsi⋅Tm,E¯spiR=Espi⋅sinωmt+γsi⋅Tm.Similarly, the computation formulas of tooth profile errors transformed into the internal (ring-planet) meshing line are derived as(18)E¯r2piL=Er2piL⋅sinωmt+γsi+γsr⋅Tm,E¯r1piR=Er1piR⋅sinωmt+γri+γsr⋅Tm,where E¯spiL,E¯spiR,E¯r2piR, and E¯r1piR, respectively, indicate the equivalent displacements of the tooth profile errors for the ith external and internal meshing pair on the left and right sides of the HPGT. Espi, Er2piL, and Er1piR, respectively, indicate the amplitudes of tooth profile errors for the ith sun-planet pair, ith left and right ring-planet pairs. γsi, γri, and γsr, respectively, denote the ith sun-planet pair phase, ith ring-planet pair phase, and the phase difference between the internal and external mesh. ωm means the meshing angular frequency. Tm is the meshing cycle in HPGT.


### 4.3. Multicoupling Manufacturing Errors

In herringbone planetary gears, the excitation of the gear mesh resulting from the gear error is mainly exhibited in the displacement change along the direction of the line of action. The displacement is the superposition of the equivalent displacements of tooth profile error and manufacturing eccentric error of each component. The superpositions of tooth profile error and manufacturing eccentric error on the internal and external meshing lines on the left and right sides are referred to here as multicoupling manufacturing errors, which are denoted byespiL, espiR, er2piL, and er1piR. The computational formulas of espiL, espiR, er2piL, and er1piR are given by(19)espiL=eEspiL+E¯spiL,espiR=eEspiR+E¯spiR,er2piL=eEr2piL+E¯r2piL,er1piR=eEr1piR+E¯r1piR.Therefore, the HPGT system manufacturing error excitations can be deduced as(20)espiL=−Es⋅sinωs−ωc⋅t+βs+αt−φi−Epi⋅sinωp−ωc⋅t+αt+βpi+Ec⋅sinβc−φi⋅cosαt+Espi⋅sinωmt+γsi⋅Tm,espiR=−Es⋅sinωs−ωc⋅t+βs+αt−φi−Epi⋅sinωp−ωc⋅t+βpi+αt+Ec⋅sinβc−φi⋅cosαt+Espi⋅sinωmt+γsi⋅Tm,er2piL=Er2L⋅sin−ωc⋅t+βr2−αt−φi+Epi⋅sinωp−ωc⋅t−αt+βpi−Ec⋅sinβc−φi⋅cosαt+Er2piL⋅sinωmt+γsi+γsr⋅Tm,er1piR=Er1R⋅sin−ωc⋅t+βr1−αt−φi+Epi⋅sinωp−ωc⋅t+βpi−αt−Ec⋅sinβc−φi⋅cosαt+Er1piR⋅sinωmt+γri+γsr⋅Tm.


## 4.1. Equivalent Displacements of Eccentric Errors

Figure6 illustrates the projection relation of manufacturing eccentric errors for herringbone planetary gear set on the line of action, where Es and βs, respectively, represent the manufacturing eccentric error and its initial phase of the sun gear. e refers to the equivalent meshing error on the meshing line. ω denotes the angular velocity of each component. φi means the position angle of the ith planet gear, and φi=2πi−1/N, where N is the planet number. Here, the subscripts s, r1, r2, pi, and c have the same meanings as those exhibited in Figure 1. n and w refer to the internal and external mesh, respectively. α is the pressure angle, and αn=αw=αt.Figure 6
Equivalent meshing error on the line of action in the HPGT system.In view of Figure6 and the relative motion state of each component, the equivalent meshing error of sun-gear manufacturing eccentric error and initial phase Es,βs on the external meshing line can be obtained as(11)eEsi=−Essinωs−ωct+βs+αt−φi.Similarly, the equivalent meshing error of planet-gear manufacturing eccentric error and initial phaseEpi,βpi on the external and internal meshing line can, respectively, be given by(12)ewEpi=−Episinωp−ωct+βpi+αt,enEpi=Episinωp−ωct+βpi+αt.The equivalent meshing error of left ring-gear manufacturing eccentric error and initial phaseEr2,βr2 on the internal meshing line can be denoted as(13)eEr2i=Er2Lsin−ωct+βr2−αt−φi.The equivalent meshing error of right ring-gear manufacturing eccentric error and initial phaseEr1,βr1 on the internal meshing line can be derived as(14)eEr1i=Er1Rsin−ωct+βr1−αt−φi.The equivalent meshing error of carrier manufacturing eccentric error and initial phaseEc,βc on the external and internal meshing line can, respectively, be expressed as(15)ewEci=Ecsinβc−φi⋅cosαt,enEci=−Ecsinβc−φi⋅cosαt.The equivalent cumulative meshing errors generated by the superposition of the eccentric errors of each component for the internal and external meshing lines on the left and right sides of herringbone planetary gear transmission can, respectively, be expressed as(16)eEr2piL=Er2L⋅sin−ωc⋅t+βr2−αt−φi+Epi⋅sinωp−ωc⋅t−αt+βpi−Ec⋅sinβc−φi⋅cosαt,eEr1piR=Er1R⋅sin−ωc⋅t+βr1−αt−φi+Epi⋅sinωp−ωc⋅t−αt+βpi−Ec⋅sinβc−φi⋅cosαt,eEspiL=−Es⋅sinωs−ωc⋅t+βs+αt−φi−Epi⋅sinωp−ωc⋅t+αt+βpi+Ec⋅sinβc−φi⋅cosαt,eEspiR=−Es⋅sinωs−ωc⋅t+βs+αt−φi−Epi⋅sinωp−ωc⋅t+αt+βpi+Ec⋅sinβc−φi⋅cosαt,where eEr2piL, eEr1piR, eEspiL, and eEspiR, respectively, refer to the equivalent displacements of manufacturing eccentric errors for the ith internal and external meshing pair on the left and right sides of HPGT. Es,Epi,Ec,Er2L, and Er1R, respectively, denote the amplitudes of manufacturing eccentric errors for component g (g = s, pi, c, r2, and r1). βs, βpi, βc, βr2, and βr1, respectively, represent the initial phases of manufacturing eccentric errors for component g (g = s, pi, c, r2, and r1). ωs, ωp, and ωc, respectively, mean the angular velocity for component g (g = s, p, and c).


## 4.2. Equivalent Displacements of Tooth Profile Errors

This paper also considers the tooth profile errors, which directly act on the engagement surface of the gear tooth. The tooth engagement is a dynamic interaction between two or more gear teeth. The tooth profile error on the line of action has something to do with the tooth profile errors of the two gears in the gear pair. The profile errors on the meshing tooth surface need to be converted to the meshing line of the gear pair, which are generally performed by using a simple harmonic function according to the meshing frequency. Figure7 displays the tooth profile error sketch.Figure 7
Schematic of tooth profile error.The formulas of the tooth profile errors transformed into the external (sun-planet) meshing line are deduced as(17)E¯spiL=Espi⋅sinωmt+γsi⋅Tm,E¯spiR=Espi⋅sinωmt+γsi⋅Tm.Similarly, the computation formulas of tooth profile errors transformed into the internal (ring-planet) meshing line are derived as(18)E¯r2piL=Er2piL⋅sinωmt+γsi+γsr⋅Tm,E¯r1piR=Er1piR⋅sinωmt+γri+γsr⋅Tm,where E¯spiL,E¯spiR,E¯r2piR, and E¯r1piR, respectively, indicate the equivalent displacements of the tooth profile errors for the ith external and internal meshing pair on the left and right sides of the HPGT. Espi, Er2piL, and Er1piR, respectively, indicate the amplitudes of tooth profile errors for the ith sun-planet pair, ith left and right ring-planet pairs. γsi, γri, and γsr, respectively, denote the ith sun-planet pair phase, ith ring-planet pair phase, and the phase difference between the internal and external mesh. ωm means the meshing angular frequency. Tm is the meshing cycle in HPGT.


## 4.3. Multicoupling Manufacturing Errors

In herringbone planetary gears, the excitation of the gear mesh resulting from the gear error is mainly exhibited in the displacement change along the direction of the line of action. The displacement is the superposition of the equivalent displacements of tooth profile error and manufacturing eccentric error of each component. The superpositions of tooth profile error and manufacturing eccentric error on the internal and external meshing lines on the left and right sides are referred to here as multicoupling manufacturing errors, which are denoted byespiL, espiR, er2piL, and er1piR. The computational formulas of espiL, espiR, er2piL, and er1piR are given by(19)espiL=eEspiL+E¯spiL,espiR=eEspiR+E¯spiR,er2piL=eEr2piL+E¯r2piL,er1piR=eEr1piR+E¯r1piR.Therefore, the HPGT system manufacturing error excitations can be deduced as(20)espiL=−Es⋅sinωs−ωc⋅t+βs+αt−φi−Epi⋅sinωp−ωc⋅t+αt+βpi+Ec⋅sinβc−φi⋅cosαt+Espi⋅sinωmt+γsi⋅Tm,espiR=−Es⋅sinωs−ωc⋅t+βs+αt−φi−Epi⋅sinωp−ωc⋅t+βpi+αt+Ec⋅sinβc−φi⋅cosαt+Espi⋅sinωmt+γsi⋅Tm,er2piL=Er2L⋅sin−ωc⋅t+βr2−αt−φi+Epi⋅sinωp−ωc⋅t−αt+βpi−Ec⋅sinβc−φi⋅cosαt+Er2piL⋅sinωmt+γsi+γsr⋅Tm,er1piR=Er1R⋅sin−ωc⋅t+βr1−αt−φi+Epi⋅sinωp−ωc⋅t+βpi−αt−Ec⋅sinβc−φi⋅cosαt+Er1piR⋅sinωmt+γri+γsr⋅Tm.


## 5. Calculation Approach of the HPGT Load Sharing Coefficients

In a HPGT, due to the influence of elastic deflections and manufacturing errors of each member, the load borne by each planet gear is not equal. The load sharing coefficient is usually utilized to represent the load distribution of the system. The larger the load sharing coefficient is, the more unbalanced the load distribution among different planet gears is. Based on the proposed dynamic load sharing model of HPGT, the load sharing coefficientsLSCr2pikL,LSCr1pikR, LSCspijL, and LSCspijR of the ith planet-ring and planet-sun meshes on the left and right sides in each meshing cycle are expressed as(21)LSCr2pikL=N⋅Fr2pikLmax∑i=1NFr2pikLmax,LSCr1pikR=N⋅Fr1pikRmax∑i=1NFr1pikRmax,LSCspijL=N⋅FspijLmax∑i=1NFspijLmax,LSCspijR=N⋅FspijRmax∑i=1NFspijRmax,where i=1,2,…,N.j and k are the meshing cycle numbers. N is the planet number. F indicates the dynamic meshing force in the meshing cycle which can be calculated by equation 4.The load sharing coefficientsLSCr2piL, LSCr1piR, LSCspiL, and LSCspiR of the ith internal and external meshes on the left and right sides of HPGT in a system period can be obtained as(22)LSCr2piL=1+LSCr2pikL−1max,LSCr1piR=1+LSCr1pikR−1max,LSCspiL=1+LSCspijL−1max,LSCspiR=1+LSCspijR−1max.The system load sharing coefficientsLSCr2pL, LSCr1pR, LSCspL, and LSCspR of the internal and external meshes on the left and right sides of herringbone gear in a system cycle are written as(23)LSCr2pL=max1≤i≤NLSCr2piL,LSCr1pR=max1≤i≤NLSCr1piR,LSCspL=max1≤i≤NLSCspiL,LSCspR=max1≤i≤NLSCspiR.Therefore, the load sharing coefficientLSC in a system period is denoted as(24)LSC=maxLSCr2pL,LSCr1pR,LSCspL,LSCspR.


## 6. Numerical Simulations

In this paper, the HPGT system with three planet gears as demonstrated in Figure1 is numerically simulated and analyzed. The basic parameters of the set are given in Tables 1 and 2.Table 1
Parameters of the HPGT system.
SubjectCarrierPlanetSunLeft ringRight ringTooth number—17235757Helix angle (deg.)—25252525Tooth width (mm)—360360170170Normal pressure angle (deg.)—20202020Mass (kg)5091525750450450Equivalent moment of inertiaJ/r2 (kg)5724420663546546Material20CrMnMoInput torque (kN·m)100Input speed (r/min)100Table 2
Stiffness parameters of the HPGT system.
Subject (N/m)u-directionx-directiony-directionz-directionCarrier—1e91e91e9Planet—1e91e91e9Sun—1e91e91e9Left ring1e101e101e101e10Right ring1e101e101e101e10


### 6.1. Influence of Manufacturing Errors on HPGT Load Sharing

In order to investigate the impacts of manufacturing errors such as the carrier eccentric error on the HPGT system load sharing coefficient, it is assumed that only the carrier eccentric errorEc is variable, while the tooth profile errors and the eccentric errors of other components are constant. Figure 8 exhibits a schematic diagram of the planet-carrier eccentric error Ec. Figure 9 shows the variation of the system load sharing coefficient LSC with the carrier eccentric error Ec in different floating configurations when manufacturing errors of the other elements are constant as 20 μm. In Figure 8, α represents the result for the model without floating, γ is the result for the model with the sun gear float, and △ indicates the result for the model with the floating sun gear and carrier. From Figure 9, it can be seen that the carrier eccentric error Ec has prominent effects on the load sharing behavior of the HPGT system. That is, the load distribution amongst different planet gears becomes worse with the increase of error Ec. Meanwhile, it is also found that the floating configurations of the HPGT have an appreciable impact on the load sharing performance. For the model with nonfloating, the load sharing coefficient LSC is larger, meaning at this time, there exists a more serious load distribution imbalance amongst different planet gears in HPGT. In contrast with the result for the model with the single floating as displayed by the black solid line γ in Figure 9, the combined floating is better. In other words, the load distribution among different planet gears is more even and balanced under the combined floating configurations as exhibited by the red double dot-dashed line △ in Figure 9. In short, the planet-carrier eccentric error Ec and floating configurations significantly affect the HPGT system load sharing performance, and as the planet-carrier eccentric error Ec goes up, the HPGT system load sharing coefficient LSC becomes larger. Floating can reduce the HPGT load sharing coefficient; that is, when the floating form is adopted, the load distribution amongst different planet gears will become more uniform. Moreover, the combination of floating is better than the single floating, and this shows a good agreement with Ren's study [11].Figure 8
Schematic diagram of the planet-carrier eccentric error.Figure 9
Relationships between the HPGT system load sharing coefficient (LSC) and the planet-carrier eccentric errorEc in different floating configurations.Figure10 illustrates the variation of the load sharing coefficients of each planet gear of the external mesh with the meshing time under multiple manufacturing errors, where LSPspiL(R) (i = 1, 2, …, N) denotes the ith planet-gear load sharing coefficient of the external mesh at the left (right) side as described in formula (22) and N is the planet number; here N = 3. The tooth profile errors and eccentric errors of each component are assumed to be 20 μm. It can be observed from Figure 10 that the load sharing coefficient of each planet gear in HPGT shows a certain periodic change. This is mainly because the error excitation is a simple harmonic function. By comparing Figures 10(a) and Figure 10(b), it can be found that the change curves of the load sharing coefficients of the corresponding planet gear i of the left-side and right-side external mesh are the same. It can also be seen that, under multiple manufacturing errors, the example HPGT system load sharing coefficient is 1.15.Figure 10
The variation of the planet-gear load sharing coefficients under multiple manufacturing errors. (a) The left-side external mesh LSCspiL. (b) The right-side external mesh LSCspiR.
(a)(b)In this subsection, it is assumed that only a single manufacturing error such as the sun-gear eccentric errorEs or the planet-gear eccentric error Ep1 exists, while the other error values are zero. Figure 11 demonstrates the variation of the load sharing coefficient of each planet gear at the right-side external mesh in the HPGT system with the meshing time, where (a) and (b) are the results from the model with the sun-gear eccentric error Es=20μm and the planet-gear eccentric error Ep1=20μm, respectively. From Figure 11, it can be observed that, in these two models with the sun-gear eccentricity and planet-gear eccentricity, the planet-gear load sharing coefficients of the right-side external mesh fluctuate periodically with the meshing time. The system load sharing coefficient can be obtained by the maximum values in each planet-gear load sharing coefficient changing curve. The HPGT system load sharing coefficients from the model with the sun-gear eccentricity Es=20μm and the planet-gear eccentricity Ep1=20μm are 1.10 and 1.043, respectively. This shows, in the HPGT system, the load distribution situation among different planet gears under the sun-gear eccentric error excitation Es=20μm is even worse and more uneven than that under the planet-gear eccentric error excitation Ep1=20μm. This is possible because the example HPGT system is more sensitive to the sun-gear eccentric error excitation. By comparing Figures 10 and 11, it can be found that the example HPGT system load sharing coefficient LSC under multiple manufacturing errors shown in Figure 10 is larger than that under a certain single error shown in Figure 11. The manufacturing error impact on the load sharing coefficient exhibits a cumulative effect, and this has a good agreement with Ren’s investigation [12].Figure 11
The variation of the planet-gear load sharing coefficients LSCspiR of the right-side external mesh in the HPGT. (a) The model with the sun-gear eccentric error Es=20μm. (b) The model with the planet-gear eccentric error Ep1=20μm.
(a)(b)


### 6.2. Influence of System Support Stiffness on HPGT Load Sharing

To investigate the effects of the support stiffness of the central member (i.e., sun gear, ring gears, and carrier) on the HPGT system load sharing coefficient LSC, it is assumed that only the support stiffness of the sun gear, ring gears, and carrier is changed while the other parameters are fixed. Figure12 demonstrates the variation of system load sharing coefficient LSC with the support stiffness of the center member, where the blue solid line α denotes the variation of the carrier support stiffness kc, the red double dot-dash line β denotes the variation of ring gear support stiffness kr, and the black dashed line γ denotes the variation of sun gear support stiffness ks. From Figure 12, it is found that the support stiffness of the central member, such as the sun gear support stiffness ks, the ring-gear support stiffness kr , and the carrier support stiffness kc, significantly affects the load sharing performance. When the supporting stiffness of the central component ranges from 0 to 1.5 × 1010 N/m and gradually increases, the HPGT system load sharing coefficient LSC is very sensitive to the supporting stiffness of the central component in the initial stage and increases rapidly with an increase in the supporting stiffness of the central component. When the supporting stiffness of the central member is greater than 1.5 × 1010 N/m and increases further, the system load sharing coefficient increases at a slow pace. After the support stiffness of the central member increases to a certain extent, the system load sharing coefficient changes insignificantly and approaches a fixed value. In addition, when the sun gear, carrier, and ring gear have the same support stiffness value, the system load sharing coefficient LSC under the variation of the sun gear support stiffness ks is the smallest and is the largest under the carrier support stiffness kc, while it is in-between under the ring gear support stiffness kr. In other words, when the sun gear, ring gear, and carrier have the same support stiffness value, the floating of the sun gear gives the best results and the floating of the carrier is the worst. This may be due to the fact that the sun gear inertia is the smallest, and the floating is more sensitive and easier to achieve.Figure 12
Relationships between the support stiffness for the central element in the HPGT and the system load sharing coefficient.


## 6.1. Influence of Manufacturing Errors on HPGT Load Sharing

In order to investigate the impacts of manufacturing errors such as the carrier eccentric error on the HPGT system load sharing coefficient, it is assumed that only the carrier eccentric errorEc is variable, while the tooth profile errors and the eccentric errors of other components are constant. Figure 8 exhibits a schematic diagram of the planet-carrier eccentric error Ec. Figure 9 shows the variation of the system load sharing coefficient LSC with the carrier eccentric error Ec in different floating configurations when manufacturing errors of the other elements are constant as 20 μm. In Figure 8, α represents the result for the model without floating, γ is the result for the model with the sun gear float, and △ indicates the result for the model with the floating sun gear and carrier. From Figure 9, it can be seen that the carrier eccentric error Ec has prominent effects on the load sharing behavior of the HPGT system. That is, the load distribution amongst different planet gears becomes worse with the increase of error Ec. Meanwhile, it is also found that the floating configurations of the HPGT have an appreciable impact on the load sharing performance. For the model with nonfloating, the load sharing coefficient LSC is larger, meaning at this time, there exists a more serious load distribution imbalance amongst different planet gears in HPGT. In contrast with the result for the model with the single floating as displayed by the black solid line γ in Figure 9, the combined floating is better. In other words, the load distribution among different planet gears is more even and balanced under the combined floating configurations as exhibited by the red double dot-dashed line △ in Figure 9. In short, the planet-carrier eccentric error Ec and floating configurations significantly affect the HPGT system load sharing performance, and as the planet-carrier eccentric error Ec goes up, the HPGT system load sharing coefficient LSC becomes larger. Floating can reduce the HPGT load sharing coefficient; that is, when the floating form is adopted, the load distribution amongst different planet gears will become more uniform. Moreover, the combination of floating is better than the single floating, and this shows a good agreement with Ren's study [11].Figure 8
Schematic diagram of the planet-carrier eccentric error.Figure 9
Relationships between the HPGT system load sharing coefficient (LSC) and the planet-carrier eccentric errorEc in different floating configurations.Figure10 illustrates the variation of the load sharing coefficients of each planet gear of the external mesh with the meshing time under multiple manufacturing errors, where LSPspiL(R) (i = 1, 2, …, N) denotes the ith planet-gear load sharing coefficient of the external mesh at the left (right) side as described in formula (22) and N is the planet number; here N = 3. The tooth profile errors and eccentric errors of each component are assumed to be 20 μm. It can be observed from Figure 10 that the load sharing coefficient of each planet gear in HPGT shows a certain periodic change. This is mainly because the error excitation is a simple harmonic function. By comparing Figures 10(a) and Figure 10(b), it can be found that the change curves of the load sharing coefficients of the corresponding planet gear i of the left-side and right-side external mesh are the same. It can also be seen that, under multiple manufacturing errors, the example HPGT system load sharing coefficient is 1.15.Figure 10
The variation of the planet-gear load sharing coefficients under multiple manufacturing errors. (a) The left-side external mesh LSCspiL. (b) The right-side external mesh LSCspiR.
(a)(b)In this subsection, it is assumed that only a single manufacturing error such as the sun-gear eccentric errorEs or the planet-gear eccentric error Ep1 exists, while the other error values are zero. Figure 11 demonstrates the variation of the load sharing coefficient of each planet gear at the right-side external mesh in the HPGT system with the meshing time, where (a) and (b) are the results from the model with the sun-gear eccentric error Es=20μm and the planet-gear eccentric error Ep1=20μm, respectively. From Figure 11, it can be observed that, in these two models with the sun-gear eccentricity and planet-gear eccentricity, the planet-gear load sharing coefficients of the right-side external mesh fluctuate periodically with the meshing time. The system load sharing coefficient can be obtained by the maximum values in each planet-gear load sharing coefficient changing curve. The HPGT system load sharing coefficients from the model with the sun-gear eccentricity Es=20μm and the planet-gear eccentricity Ep1=20μm are 1.10 and 1.043, respectively. This shows, in the HPGT system, the load distribution situation among different planet gears under the sun-gear eccentric error excitation Es=20μm is even worse and more uneven than that under the planet-gear eccentric error excitation Ep1=20μm. This is possible because the example HPGT system is more sensitive to the sun-gear eccentric error excitation. By comparing Figures 10 and 11, it can be found that the example HPGT system load sharing coefficient LSC under multiple manufacturing errors shown in Figure 10 is larger than that under a certain single error shown in Figure 11. The manufacturing error impact on the load sharing coefficient exhibits a cumulative effect, and this has a good agreement with Ren’s investigation [12].Figure 11
The variation of the planet-gear load sharing coefficients LSCspiR of the right-side external mesh in the HPGT. (a) The model with the sun-gear eccentric error Es=20μm. (b) The model with the planet-gear eccentric error Ep1=20μm.
(a)(b)


## 6.2. Influence of System Support Stiffness on HPGT Load Sharing

To investigate the effects of the support stiffness of the central member (i.e., sun gear, ring gears, and carrier) on the HPGT system load sharing coefficient LSC, it is assumed that only the support stiffness of the sun gear, ring gears, and carrier is changed while the other parameters are fixed. Figure12 demonstrates the variation of system load sharing coefficient LSC with the support stiffness of the center member, where the blue solid line α denotes the variation of the carrier support stiffness kc, the red double dot-dash line β denotes the variation of ring gear support stiffness kr, and the black dashed line γ denotes the variation of sun gear support stiffness ks. From Figure 12, it is found that the support stiffness of the central member, such as the sun gear support stiffness ks, the ring-gear support stiffness kr , and the carrier support stiffness kc, significantly affects the load sharing performance. When the supporting stiffness of the central component ranges from 0 to 1.5 × 1010 N/m and gradually increases, the HPGT system load sharing coefficient LSC is very sensitive to the supporting stiffness of the central component in the initial stage and increases rapidly with an increase in the supporting stiffness of the central component. When the supporting stiffness of the central member is greater than 1.5 × 1010 N/m and increases further, the system load sharing coefficient increases at a slow pace. After the support stiffness of the central member increases to a certain extent, the system load sharing coefficient changes insignificantly and approaches a fixed value. In addition, when the sun gear, carrier, and ring gear have the same support stiffness value, the system load sharing coefficient LSC under the variation of the sun gear support stiffness ks is the smallest and is the largest under the carrier support stiffness kc, while it is in-between under the ring gear support stiffness kr. In other words, when the sun gear, ring gear, and carrier have the same support stiffness value, the floating of the sun gear gives the best results and the floating of the carrier is the worst. This may be due to the fact that the sun gear inertia is the smallest, and the floating is more sensitive and easier to achieve.Figure 12
Relationships between the support stiffness for the central element in the HPGT and the system load sharing coefficient.


## 7. Verifications of Theoretical Results

In order to verify the validity of theoretical calculation in the present study, the approaches of comparing the obtained research results in the paper with the existing literature are mainly adopted to validate the correctness of the proposed model and method. Influence curves of carrier eccentricityEc on the HPGT LSC shown in Figure 9 in this research have a similar tendency and changing law with Mo’s investigation in [19], and both of them show a good agreement. The LSC curves of the left and right sides shown in Figure 10 are the same, which is consistent with Mo’s study in [19]. Moreover, all the variation curves of LSC of each planet gear in Figures 10 and 11 in the present investigation exhibit the periodically changing trend, and just these are relatively consistent with [19, 20]. Meanwhile, from Figures 10 and 11, it is obvious that the HPGT system LSC under multicoupling errors is larger than that under single error, which is consistent with [19]. As exhibited in Figure 12, the variation tendencies and laws of the system LSC curves with the central member (i.e., sun, ring, and carrier) are consistent with [21].In summary, comparing the theoretical calculation results in the study with the existing literature, both of them are in a good agreement. In other words, the reliability and correctness of theoretical calculation are validated.


## 8. Conclusions

Based on the actual structure feature of herringbone gears and by considering multicoupling manufacturing errors, time-dependent meshing stiffness, bearing deformation, and gyroscopic effect, a refined bending-torsional-axial coupled dynamic model for HPGT has been developed to study the load sharing performance. The influences of manufacturing errors and system support stiffness on dynamic load sharing behavior were numerically investigated. The main conclusions are summarised as follows:(1)
Manufacturing errors such as the planet-carrier eccentric errorEc and component floating have palpable influences on the load sharing behavior of the HPGT system. The load sharing coefficient increases significantly with the increase of manufacturing errors. The system load sharing coefficient is significantly smaller when using the component floating, and the load distribution among planet gears tends to be more uniform under a combined floating configuration. Hence, adopting component floating and lessening manufacturing errors exert a vital role in the load sharing of the HPGT system. And the HPGT load sharing coefficient when the errors act in common is larger than that when the error acts alone. The influence of the error on the load sharing coefficient has a cumulative effect.(2)
The support stiffness of the central member noticeably affects the HPGT load sharing behavior. When the support stiffness of the central member is gradually increased, the load sharing coefficient is also increased in the initial stage. As the support stiffness of the central component is further increased, the increase of the load sharing coefficient becomes slower. After the support stiffness increases to a certain extent, the system load sharing coefficient LSC no longer changes significantly and tends to be a fixed value. This will provide a theoretical foundation for reasonably selecting the support stiffness of the central component.



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*Source: 5511817-2021-07-10.xml*