# Mesh Phase Analysis of Encased Differential Gear Train for Coaxial Twin-Rotor Helicopter

Authors: Donglin Zhang; Rupeng Zhu; Bibo Fu; Wuzhong Tan
Journal: Mathematical Problems in Engineering
 (2019)

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## Abstract

Dynamic excitation caused by time-varying meshing stiffness is one of the most important excitation forms in gear meshing process. The mesh phase relations between each gear pair are an important factor affecting the meshing stiffness. In this paper, the mesh phase relations between gear pairs in an encased differential gear train widely used in coaxial twin-rotor helicopters are discussed. Taking the meshing starting point where the gear tooth enters contact as the reference point, the mesh phase difference between adjacent gear pairs is analyzed and calculated, the system reference gear pair is selected, and the mesh phase difference of each gear pair relative to the system reference gear pair is obtained. The derivation process takes into account the modification of the teeth, the processing, and assembly of the duplicate gears, which makes the calculation method and conclusion more versatile. This work lays a foundation for considering the time-varying meshing stiffness in the study of system dynamics, load distribution, and fault diagnosis of compound planetary gears.


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## Body

## 1. Introduction

Planetary gear transmission is widely used in aviation, automobile and other industrial fields because of its strong load-carrying capacity and small size. However, the vibration and noise caused by the dynamic meshing force and dynamic supporting force of the system affect its reliability and service life. Equally distributing the load to each planet and avoiding excessive vibration are the goals of design and manufacture of planetary gear train. The mesh phase relations between gear pairs are an important factor affecting the meshing stiffness of the system. It has a great influence on the load distribution and vibration of the system and has always been the focus of researchers.Based on an improved potential energy method, a time-varying mesh stiffness calculation model considering crack growth was proposed. The meshing stiffness of normal gear pair and gear pair with different levels of crack is calculated by traditional method, proposed method, and ANSYS method, respectively. The influence of the crack level of the gear on the dynamic response was studied by using a one-stage gear system dynamic model [1]. Cui calculated the gear tooth thickness decrements by straight-affecting line method and parabolic-affecting line method. Based on the universal tooth profile equation, the meshing stiffness of gear pairs with different crack levels was calculated, and their effects on the vibration response are studied by fault detection indicators [2]. Considering three possible contact situations, the meshing stiffness under different torques is calculated and compared with the stiffness obtained by ISO 6336. The results of internal and external meshing stiffness calculated by various models are compared [3]. Based on the potential energy principle, Chen deduced the stiffness calculation model considering the modification and calculated the stiffness of spur gears under different modification coefficients [4].Kahraman established a lumped parameter dynamic model of a single-stage planetary gear train and obtained the natural modes and forced vibration response caused by static transmission error. The influence of mesh phase relation on the dynamic characteristics of four-planet system was studied [5]. Kahraman also established a nonlinear time-varying dynamic model of planetary transmission considering tooth profile error and time-varying meshing stiffness, and analyzed the influence of errors on dynamic load-sharing coefficient [6]. Based on the harmonic decomposition of meshing forces acting on the sun-planet and ring-planet mesh, Parker and Abbasha studied the suppression of mesh phase relations on different vibration types of planetary gears. The nonlinear dynamic characteristics of system considering mesh phase difference are studied by using analytical model and finite element model [7, 8]. The results show that the rules of suppressing the vibration caused by mesh phase difference are applicable to the nonlinearity caused by contact loss of gear teeth [9].Parker defined the mesh tooth number function to describe the mesh phase relations, deduced the calculation method of the phase difference between the gear pairs, and calculated the phase difference of each mesh gear pair in the single-stage planetary gear train of OH58 helicopter main reducer when the planet gear rotates in the positive and negative directions [10]. Chen Yong studied the relationship between mesh phase difference and torsional vibration and carried out experimental research under different rotational speeds and loads. The results show that the design considering mesh phase difference can effectively reduce the vibration and noise levels of planetary gear system, and the tooth profile contact ratio has no significant effect on the vibration and noise levels in the presence of phase differences [11].Guo and Parker [12] proposed a method to accurately define the phase difference of the compound planetary gear system. By introducing the concepts of relative phase and reference datum, the mesh phase relations between any two gear pairs in a compound planetary gear train were derived. Wang [13] studied the influence of mesh phase on the vibration of spur planetary ring gear by using the superposition principle and Fourier series method and analyzed the relationship between modal characteristics and mesh phase difference. Gawande measured the noise level of the planetary gears and found that the noise level of the planetary gears with mesh phase difference was significantly lower than that of the planetary gears without mesh phase difference [14, 15].Chaari [16] established a dynamic model of a single-stage planetary gear train with consideration of the mesh phase. The dynamic responses of healthy planetary gear train and planetary gear train containing defective gear tooth were compared in time and frequency domains, and the validity of Wigner-Ville method for fault diagnosis of planetary gear train was verified. Li [17, 18] studied the mesh phase relations of two-stage planetary gears with meshed-planet gears. On this basis, the time-varying meshing stiffness was calculated, and the simulation analysis and experiment of gear tooth fault signal were compared. Hu added the mesh phase difference introduced by helical angle to that of spur gear to obtain the mesh phase difference of the helical gear, established a load sharing calculation model considering the mesh phase, and calculated the load sharing factors of the system under different phase relations. The trajectory of the sun gear [19] was obtained, which is in good agreement with the experimental results [20, 21].In these literatures, pitch points are used as reference points of mesh phase difference, and researches are mainly for the single-stage planetary gear systems. There is no specific calculation method for the mesh phase difference of the compound planetary gears with stepped-planet gears. In this paper, the starting point of the meshing is taken as the reference point to make the calculation of phase difference more concise. The calculation method of phase difference of the compound planetary gear train with stepped-planet gears and duplicate central gears is derived. The gear modification is considered in the derivation process, which makes the conclusions in this paper more versatile.


## 2. Definition of Mesh Phase Difference

The research object of this paper is an encased differential planetary gear train shown in Figure1 which can be regarded as a combination of two different subsystems: the fixed-shaft gear train with stepped-planet gears, that is, the encased stage, and the differential planetary gears is the differential stage. The input power is split into two paths as it is transmitted to the system: one is transmitted to the ring gearr1 through the encased stage and the other is transmitted to the carrierc2 and gear ringr2 through the differential stage. When the number of teeth of each gear in a planetary gear train conforms to a specific relationship, the output shaftscs andrs can be output in reverse direction at the same rotational speed. The number of stepped planetsab and planetsp isM andN, respectively.Figure 1
Configuration of encased differential planetary gear train.The gear pair sun gears1 and stepped gearai (i=1, 2, …,M) is defined ass1ai, and the definitions of other gear pairs in the system are similar. The angular velocity of the componenth is ωh (h=s1,s2,r1,r2,c2,a,b,p) and the counterclockwise is positive, and the mesh period of all gear pairs is determined by the following:(1)Ts1ai=2πωs1zs1=-2πωazaTr1bi=-2πωr1zr1=-2πωbzbTs2pj=2πωs2-ωc2zs2Tr2pj=-2πωr2-ωc2zr2i=1,2,…,M,j=1,2,…,NAs shown in Figure2, γAB represents the mesh phase difference of the mesh B relative to the mesh A [10], where 0≤γ<1 and εg is the contact ratio of the meshg. Starting from the reference position, the gear pairg is in contact at the meshing starting point for the first time at t=tg, and the mesh phase difference can be obtained. (2)γAB=dectB-tATBwhere dec(x)=x-int(x) is a fractional function and int(x) is the nearest integer less thanx.Figure 2
Schematic diagram of mesh phase difference.


## 3. Mesh Phase Relations of Encased Differential Gear Train

The calculation process of mesh phase difference of the encased differential planetary gear train is shown in Figure3.Figure 3
Flow chart of mesh phase difference calculation.The initial assembly position of the compound planetary gears is shown in Figure4. In the figure,MN is the theoretical line of action,A is the meshing starting point,P is the pitch point of the mesh,E is the point where the gear tooth exits contact,D is the intersection point of the reference tooth midline and the addendum circle, and the definitions of radius and angle are given in the Appendix.Figure 4
Initial position of the planetary gear train.
(a)
Encased stage (b)
Differential stage (c)
Relative position of duplicate sun gearsThe angle of the midline of the planet gearb1 reference tooth relative to that of the planet geara1 reference tooth is φab. In order to meet the assembly conditions, the relative angles of gearsb anda of each stepped-planet gear need to be the same. The angle of the midline of the sun gears2 reference space relative to that of the sun gears1 reference space is φs1s2.


### 3.1. Calculation of Phase Difference between Adjacent Mesh

For the sun-planet meshs1a1, the calculations follow from the geometry of Figure 4(a).(3)∠A1O2C1=∠A1O2N1-∠N1O2P1-∠P1O2C1=αaa-αa′-sa2ra-θaa-θahere(4)αaa=arccos⁡rbaraa(5)αa′=arccos⁡maza+ms1zs1/2cos⁡αaO1O2where O1O2 is the actual center distance between the sun gears1 and the stepped-planet geara1.It can be concluded that when the stepped-planeta1 rotates from the initial position by an angle φa, the meshing starting point is in contact for the first time at t=t1.(6)φa=-2πZa-2πZadec∠A1O2C12π/Za(7)t1=t0+φaωa


#### 3.1.1. Calculation of Phase Differenceγs1a1s2p1

In order to meet the assembly requirements, it is advisable to design the installation position of planet gearp1 to be the same as that of the stepped-planet geara1 circumferentially; that is, the center line O1O2 coincides with the center line O3O4 in the circumferential direction, as shown in Figure 4(b).For the sun-planet meshs2p1, geometrical relation can be derived from Figures 4(b) and 4(c). (8)∠A3O4C3=∠A3O4P3+∠P3O4D3-∠D3O4C3=αap-αs2′+φs1s2zs2zp-sp2rp-θap-θphere(9)αs2′=arccos⁡mpzp+ms2zs2/2cos⁡αpO3O4⁡(10)αap=arccos⁡rbprapwhere O3O4 is the actual center distance between the sun gears2 and the planet gearp1.When the planetp1 rotates from the initial position by an angle φp, the meshing starting point is in contact for the first time at t=t2. (11)φp=-2πZp-2πZpdec∠A3O4C32π/Zp(12)t2=t0+φpωp-ωc2The mesh phase differenceγs1a1s2p1 between gear pairss2p1 ands1a1 is(13)γs1a1s2p1=dect2-t1Ts2p1


#### 3.1.2. Calculation of Phase Differenceγs1a1r1b1

As shown in Figure4(a), B2 is the starting point of the involute of the reference gear teeth on the base circle, and the curves B2N2^ and N2A2 are calculated as follows:(14)B2N2^=rbb∗∠N2O2B2(15)N2A2=M2A2-M2N2=rar12-rbr12-O1O2sin⁡αr1′where(16)∠N2O2B2=π-φab-N2O2P2-∠B2O2D2=π-φab-αr1′-sb2rb+θbEquations (14)-(16) give(17)Lr1b=B2N2^+N2A2=rbb∗π-αr1′-sb2rb+θb+φab+rar12-rbr12-O1O2sin⁡αr1′where (18)αr1′=arccos⁡mr1zr1-mbzb/2cos⁡αr1O1O2From the initial position, the meshing starting point of gear pairr1b1 is in contact for the first time at t=t3.(19)t3=t0+Tr1b1decLr1bpbbThe mesh phase differenceγs1a1r1b1 between gear pairr1b1 and gear pairs1a1 is(20)γs1a1r1b1=dect3-t1Tr1b1


#### 3.1.3. Calculation of Phase Differenceγs2p1r2p1

Figure4(b) shows the following for length of the curves:(21)A3N3=O4A32-O4N32=rap2-rbp2(22)N3N4^=rbp∗∠N3O4N4=rbp∗π-αs2′-αr2′(23)N4A4=M4A4-M4N4=rar22-rbr22-O3O4sin⁡αr2′According to the above equations (21)-(23), we have(24)Ls2r=A3N3+N3N4^+N4A4=rap2-rbp2+rbp∗π-αs2′-αr2′+rar22-rbr22-O3O4sin⁡αr2′The phase difference between internal gear pair and external gear pair of the differential stage is(25)γs2p1r2p1=decLs2r-sbppbpwhere the tooth thickness on the base circle sbp=π/2+2xptan⁡αpmpcos⁡αp+2rbpθp.


#### 3.1.4. Calculation of Phase Differenceγs1a1r1bi

The number of teeth meshed when the sun gears1 rotates one revolution is zs1. When the sun gears1 rotates from the position of the stepped-planet geara1 to the position of the stepped-planet gearai, the angle of rotation φs1i and the number of teeth meshed zs1i are (26)φs1i=2πMi-1(27)zs1i=Zs12πφs1iFrom (26) and (27), the mesh phase difference between gear pairss1ai ands1a1 is(28)γs1a1s1ai=deczs12πφs1i=deczs1Mi-1


#### 3.1.5. Calculation of Phase Differenceγr1b1r1bi

Similarly, when the ring gearr1 rotates from the position of the planetb1 to the position of the planet gearbi, the angle of rotation φr1i and the number of teeth meshed zr1i are as follows: (29)φr1i=2π-2πMi-1(30)zr1i=Zr12πφr1iThe mesh phase difference between the gear pairsr1bi andr1b1 obtained by (29) and (30) is(31)γr1b1r1bi=decZr12πφr1i=dec-Zr1Mi-1


#### 3.1.6. Calculation of Phase Differenceγs2p1s2pj

It is known that the angle between thej-th planet gear and the 1st planet gear in the differential stage is(32)φj=2πNj-1When the sun gears2 rotates over the angle φj relative to the carrierc2, thej-th planet gear moves to the initial position of the 1st planet gear, and the time required is(33)tsj=φjωs2-ωc2Considering that the time difference caused by mesh phase difference is∆tsj=γs2p1s2pjTs2pj, the time tsj can also be expressed as(34)γs2p1s2pj+Cs2Ts2pj=tsjwhere Cs2 is an integer. Available from (1), (33) and (34), the mesh phase difference between the gear pairss2pj ands2p1 is(35)γs2p1s2pj=deczs22πφj=deczs2Nj-1


#### 3.1.7. Calculation of Phase Differenceγr2p1r2pj

Similar to the above, when the ring gear rotates2π-φj relative to the carrierc2, thej-th planet gear moves to the initial position of the 1st planet gear, and the time required is (36)trj=-2π-φjωr2-ωc2Considering that the time difference caused by mesh phase difference is∆trj=γr2p1r2pjTr2pj, we obtain the following:(37)γr2p1r2pj+Cr2Tr2pj=-2π-φjωr2-ωc2where Cr2 is an integer. The mesh phase difference between the gear pairsr2pj andr2p1 can be obtained by introducing (1) into (37). (38)γr2p1r2pj=deczr22π-φj2π=dec-zr2Nj-1


### 3.2. Calculation of Comprehensive Mesh Phase Difference

Through the above analysis, the mesh phase differences between adjacent gear pairs are obtained. In order to unify the time difference caused by mesh phase difference to the absolute time of the system, it is necessary to synthesize the adjacent phase difference to get the phase difference of each gear pair relative to the base referred mesh. In the following analysis, the meshs1a1 is selected as the base referred mesh of the system.


#### 3.2.1. Calculation of Phase Differenceγs1a1r2p1

After the contact of the meshing start point of sun-planet meshs2p1, the time tr2p1′ at which the meshing start point of ring-planet meshr2p1contact for the first time is(39)tr2p1′=γs2p1r2p1Tr2p1+ts2p1=γs2p1r2p1Tr2p1+γs1a1s2p1Ts2p1+t1Relative to the base referred meshs1a1, the mesh phase difference of the gear pairr2p1 is(40)γs1a1r2p1=dectr2p1′-t1Tr2p1=decγs2p1r2p1+γs1a1s2p1


#### 3.2.2. Calculation of Phase Differenceγs1a1r1bi

After the meshr1b1 contacts at meshing starting point, the first contact time of meshing starting point of the meshr1bi is(41)tr1bi′=γr1b1r1biTr1bi+tr1b1=γr1b1r1biTr1bi+γs1a1r1b1Tr1b1+t1The mesh phase difference of the gear pairr1bi relative to the base referred meshs1a1 is(42)γs1a1r1bi=dectr1bi′-t1Tr1bi=decγr1b1r1bi+γs1a1r1b1


#### 3.2.3. Calculation of Phase Differenceγs1a1s2pj

After the meshs2p1 contacts at meshing starting point, the first contact time of meshing starting point of the meshs2pj is(43)ts2pj′=γs2p1s2pjTs2pj+ts2p1=γs2p1s2pjTs2pj+γs1a1s2p1Ts2p1+t1The mesh phase difference of the gear pairs2pj relative to the base referred meshs1a1 is(44)γs1a1s2pj=dects2pj′-t1Ts2pj=decγs2p1s2pj+γs1a1s2p1


#### 3.2.4. Calculation of Phase Differenceγs1a1r2pj

After the meshr2p1 contacts at meshing starting point, the first contact time of meshing starting point of the meshr2pj is(45)tr2pj′=γr2p1r2pjTr2pj+tr2p1=γr2p1r2pjTr2pj+γs1a1r2p1Tr2p1+t1The mesh phase difference of the gear pairr2pj relative to the base referred meshs1a1 is(46)γs1a1r2pj=dectr2pj′-t1Tr2pj=decγr2p1r2pj+γs1a1r2p1Since the starting point of the referred gear pairs1a1 contacts at t=t1, the time at which the meshing starting point of the gear pairg contacts for the first time is(47)tg=γs1a1gTg+t1Then the mesh tooth variation function in the system can be expressed as [12](48)kgt=κgt-tg=κgt-γs1a1gTg-t1where κgτ is a periodic time-varying mesh tooth variation function of the gear pairg, τ=0 corresponds to the meshing starting point, andt is absolute time of the system.


## 3.1. Calculation of Phase Difference between Adjacent Mesh

For the sun-planet meshs1a1, the calculations follow from the geometry of Figure 4(a).(3)∠A1O2C1=∠A1O2N1-∠N1O2P1-∠P1O2C1=αaa-αa′-sa2ra-θaa-θahere(4)αaa=arccos⁡rbaraa(5)αa′=arccos⁡maza+ms1zs1/2cos⁡αaO1O2where O1O2 is the actual center distance between the sun gears1 and the stepped-planet geara1.It can be concluded that when the stepped-planeta1 rotates from the initial position by an angle φa, the meshing starting point is in contact for the first time at t=t1.(6)φa=-2πZa-2πZadec∠A1O2C12π/Za(7)t1=t0+φaωa


### 3.1.1. Calculation of Phase Differenceγs1a1s2p1

In order to meet the assembly requirements, it is advisable to design the installation position of planet gearp1 to be the same as that of the stepped-planet geara1 circumferentially; that is, the center line O1O2 coincides with the center line O3O4 in the circumferential direction, as shown in Figure 4(b).For the sun-planet meshs2p1, geometrical relation can be derived from Figures 4(b) and 4(c). (8)∠A3O4C3=∠A3O4P3+∠P3O4D3-∠D3O4C3=αap-αs2′+φs1s2zs2zp-sp2rp-θap-θphere(9)αs2′=arccos⁡mpzp+ms2zs2/2cos⁡αpO3O4⁡(10)αap=arccos⁡rbprapwhere O3O4 is the actual center distance between the sun gears2 and the planet gearp1.When the planetp1 rotates from the initial position by an angle φp, the meshing starting point is in contact for the first time at t=t2. (11)φp=-2πZp-2πZpdec∠A3O4C32π/Zp(12)t2=t0+φpωp-ωc2The mesh phase differenceγs1a1s2p1 between gear pairss2p1 ands1a1 is(13)γs1a1s2p1=dect2-t1Ts2p1


### 3.1.2. Calculation of Phase Differenceγs1a1r1b1

As shown in Figure4(a), B2 is the starting point of the involute of the reference gear teeth on the base circle, and the curves B2N2^ and N2A2 are calculated as follows:(14)B2N2^=rbb∗∠N2O2B2(15)N2A2=M2A2-M2N2=rar12-rbr12-O1O2sin⁡αr1′where(16)∠N2O2B2=π-φab-N2O2P2-∠B2O2D2=π-φab-αr1′-sb2rb+θbEquations (14)-(16) give(17)Lr1b=B2N2^+N2A2=rbb∗π-αr1′-sb2rb+θb+φab+rar12-rbr12-O1O2sin⁡αr1′where (18)αr1′=arccos⁡mr1zr1-mbzb/2cos⁡αr1O1O2From the initial position, the meshing starting point of gear pairr1b1 is in contact for the first time at t=t3.(19)t3=t0+Tr1b1decLr1bpbbThe mesh phase differenceγs1a1r1b1 between gear pairr1b1 and gear pairs1a1 is(20)γs1a1r1b1=dect3-t1Tr1b1


### 3.1.3. Calculation of Phase Differenceγs2p1r2p1

Figure4(b) shows the following for length of the curves:(21)A3N3=O4A32-O4N32=rap2-rbp2(22)N3N4^=rbp∗∠N3O4N4=rbp∗π-αs2′-αr2′(23)N4A4=M4A4-M4N4=rar22-rbr22-O3O4sin⁡αr2′According to the above equations (21)-(23), we have(24)Ls2r=A3N3+N3N4^+N4A4=rap2-rbp2+rbp∗π-αs2′-αr2′+rar22-rbr22-O3O4sin⁡αr2′The phase difference between internal gear pair and external gear pair of the differential stage is(25)γs2p1r2p1=decLs2r-sbppbpwhere the tooth thickness on the base circle sbp=π/2+2xptan⁡αpmpcos⁡αp+2rbpθp.


### 3.1.4. Calculation of Phase Differenceγs1a1r1bi

The number of teeth meshed when the sun gears1 rotates one revolution is zs1. When the sun gears1 rotates from the position of the stepped-planet geara1 to the position of the stepped-planet gearai, the angle of rotation φs1i and the number of teeth meshed zs1i are (26)φs1i=2πMi-1(27)zs1i=Zs12πφs1iFrom (26) and (27), the mesh phase difference between gear pairss1ai ands1a1 is(28)γs1a1s1ai=deczs12πφs1i=deczs1Mi-1


### 3.1.5. Calculation of Phase Differenceγr1b1r1bi

Similarly, when the ring gearr1 rotates from the position of the planetb1 to the position of the planet gearbi, the angle of rotation φr1i and the number of teeth meshed zr1i are as follows: (29)φr1i=2π-2πMi-1(30)zr1i=Zr12πφr1iThe mesh phase difference between the gear pairsr1bi andr1b1 obtained by (29) and (30) is(31)γr1b1r1bi=decZr12πφr1i=dec-Zr1Mi-1


### 3.1.6. Calculation of Phase Differenceγs2p1s2pj

It is known that the angle between thej-th planet gear and the 1st planet gear in the differential stage is(32)φj=2πNj-1When the sun gears2 rotates over the angle φj relative to the carrierc2, thej-th planet gear moves to the initial position of the 1st planet gear, and the time required is(33)tsj=φjωs2-ωc2Considering that the time difference caused by mesh phase difference is∆tsj=γs2p1s2pjTs2pj, the time tsj can also be expressed as(34)γs2p1s2pj+Cs2Ts2pj=tsjwhere Cs2 is an integer. Available from (1), (33) and (34), the mesh phase difference between the gear pairss2pj ands2p1 is(35)γs2p1s2pj=deczs22πφj=deczs2Nj-1


### 3.1.7. Calculation of Phase Differenceγr2p1r2pj

Similar to the above, when the ring gear rotates2π-φj relative to the carrierc2, thej-th planet gear moves to the initial position of the 1st planet gear, and the time required is (36)trj=-2π-φjωr2-ωc2Considering that the time difference caused by mesh phase difference is∆trj=γr2p1r2pjTr2pj, we obtain the following:(37)γr2p1r2pj+Cr2Tr2pj=-2π-φjωr2-ωc2where Cr2 is an integer. The mesh phase difference between the gear pairsr2pj andr2p1 can be obtained by introducing (1) into (37). (38)γr2p1r2pj=deczr22π-φj2π=dec-zr2Nj-1


## 3.1.1. Calculation of Phase Differenceγs1a1s2p1

In order to meet the assembly requirements, it is advisable to design the installation position of planet gearp1 to be the same as that of the stepped-planet geara1 circumferentially; that is, the center line O1O2 coincides with the center line O3O4 in the circumferential direction, as shown in Figure 4(b).For the sun-planet meshs2p1, geometrical relation can be derived from Figures 4(b) and 4(c). (8)∠A3O4C3=∠A3O4P3+∠P3O4D3-∠D3O4C3=αap-αs2′+φs1s2zs2zp-sp2rp-θap-θphere(9)αs2′=arccos⁡mpzp+ms2zs2/2cos⁡αpO3O4⁡(10)αap=arccos⁡rbprapwhere O3O4 is the actual center distance between the sun gears2 and the planet gearp1.When the planetp1 rotates from the initial position by an angle φp, the meshing starting point is in contact for the first time at t=t2. (11)φp=-2πZp-2πZpdec∠A3O4C32π/Zp(12)t2=t0+φpωp-ωc2The mesh phase differenceγs1a1s2p1 between gear pairss2p1 ands1a1 is(13)γs1a1s2p1=dect2-t1Ts2p1


## 3.1.2. Calculation of Phase Differenceγs1a1r1b1

As shown in Figure4(a), B2 is the starting point of the involute of the reference gear teeth on the base circle, and the curves B2N2^ and N2A2 are calculated as follows:(14)B2N2^=rbb∗∠N2O2B2(15)N2A2=M2A2-M2N2=rar12-rbr12-O1O2sin⁡αr1′where(16)∠N2O2B2=π-φab-N2O2P2-∠B2O2D2=π-φab-αr1′-sb2rb+θbEquations (14)-(16) give(17)Lr1b=B2N2^+N2A2=rbb∗π-αr1′-sb2rb+θb+φab+rar12-rbr12-O1O2sin⁡αr1′where (18)αr1′=arccos⁡mr1zr1-mbzb/2cos⁡αr1O1O2From the initial position, the meshing starting point of gear pairr1b1 is in contact for the first time at t=t3.(19)t3=t0+Tr1b1decLr1bpbbThe mesh phase differenceγs1a1r1b1 between gear pairr1b1 and gear pairs1a1 is(20)γs1a1r1b1=dect3-t1Tr1b1


## 3.1.3. Calculation of Phase Differenceγs2p1r2p1

Figure4(b) shows the following for length of the curves:(21)A3N3=O4A32-O4N32=rap2-rbp2(22)N3N4^=rbp∗∠N3O4N4=rbp∗π-αs2′-αr2′(23)N4A4=M4A4-M4N4=rar22-rbr22-O3O4sin⁡αr2′According to the above equations (21)-(23), we have(24)Ls2r=A3N3+N3N4^+N4A4=rap2-rbp2+rbp∗π-αs2′-αr2′+rar22-rbr22-O3O4sin⁡αr2′The phase difference between internal gear pair and external gear pair of the differential stage is(25)γs2p1r2p1=decLs2r-sbppbpwhere the tooth thickness on the base circle sbp=π/2+2xptan⁡αpmpcos⁡αp+2rbpθp.


## 3.1.4. Calculation of Phase Differenceγs1a1r1bi

The number of teeth meshed when the sun gears1 rotates one revolution is zs1. When the sun gears1 rotates from the position of the stepped-planet geara1 to the position of the stepped-planet gearai, the angle of rotation φs1i and the number of teeth meshed zs1i are (26)φs1i=2πMi-1(27)zs1i=Zs12πφs1iFrom (26) and (27), the mesh phase difference between gear pairss1ai ands1a1 is(28)γs1a1s1ai=deczs12πφs1i=deczs1Mi-1


## 3.1.5. Calculation of Phase Differenceγr1b1r1bi

Similarly, when the ring gearr1 rotates from the position of the planetb1 to the position of the planet gearbi, the angle of rotation φr1i and the number of teeth meshed zr1i are as follows: (29)φr1i=2π-2πMi-1(30)zr1i=Zr12πφr1iThe mesh phase difference between the gear pairsr1bi andr1b1 obtained by (29) and (30) is(31)γr1b1r1bi=decZr12πφr1i=dec-Zr1Mi-1


## 3.1.6. Calculation of Phase Differenceγs2p1s2pj

It is known that the angle between thej-th planet gear and the 1st planet gear in the differential stage is(32)φj=2πNj-1When the sun gears2 rotates over the angle φj relative to the carrierc2, thej-th planet gear moves to the initial position of the 1st planet gear, and the time required is(33)tsj=φjωs2-ωc2Considering that the time difference caused by mesh phase difference is∆tsj=γs2p1s2pjTs2pj, the time tsj can also be expressed as(34)γs2p1s2pj+Cs2Ts2pj=tsjwhere Cs2 is an integer. Available from (1), (33) and (34), the mesh phase difference between the gear pairss2pj ands2p1 is(35)γs2p1s2pj=deczs22πφj=deczs2Nj-1


## 3.1.7. Calculation of Phase Differenceγr2p1r2pj

Similar to the above, when the ring gear rotates2π-φj relative to the carrierc2, thej-th planet gear moves to the initial position of the 1st planet gear, and the time required is (36)trj=-2π-φjωr2-ωc2Considering that the time difference caused by mesh phase difference is∆trj=γr2p1r2pjTr2pj, we obtain the following:(37)γr2p1r2pj+Cr2Tr2pj=-2π-φjωr2-ωc2where Cr2 is an integer. The mesh phase difference between the gear pairsr2pj andr2p1 can be obtained by introducing (1) into (37). (38)γr2p1r2pj=deczr22π-φj2π=dec-zr2Nj-1


## 3.2. Calculation of Comprehensive Mesh Phase Difference

Through the above analysis, the mesh phase differences between adjacent gear pairs are obtained. In order to unify the time difference caused by mesh phase difference to the absolute time of the system, it is necessary to synthesize the adjacent phase difference to get the phase difference of each gear pair relative to the base referred mesh. In the following analysis, the meshs1a1 is selected as the base referred mesh of the system.


### 3.2.1. Calculation of Phase Differenceγs1a1r2p1

After the contact of the meshing start point of sun-planet meshs2p1, the time tr2p1′ at which the meshing start point of ring-planet meshr2p1contact for the first time is(39)tr2p1′=γs2p1r2p1Tr2p1+ts2p1=γs2p1r2p1Tr2p1+γs1a1s2p1Ts2p1+t1Relative to the base referred meshs1a1, the mesh phase difference of the gear pairr2p1 is(40)γs1a1r2p1=dectr2p1′-t1Tr2p1=decγs2p1r2p1+γs1a1s2p1


### 3.2.2. Calculation of Phase Differenceγs1a1r1bi

After the meshr1b1 contacts at meshing starting point, the first contact time of meshing starting point of the meshr1bi is(41)tr1bi′=γr1b1r1biTr1bi+tr1b1=γr1b1r1biTr1bi+γs1a1r1b1Tr1b1+t1The mesh phase difference of the gear pairr1bi relative to the base referred meshs1a1 is(42)γs1a1r1bi=dectr1bi′-t1Tr1bi=decγr1b1r1bi+γs1a1r1b1


### 3.2.3. Calculation of Phase Differenceγs1a1s2pj

After the meshs2p1 contacts at meshing starting point, the first contact time of meshing starting point of the meshs2pj is(43)ts2pj′=γs2p1s2pjTs2pj+ts2p1=γs2p1s2pjTs2pj+γs1a1s2p1Ts2p1+t1The mesh phase difference of the gear pairs2pj relative to the base referred meshs1a1 is(44)γs1a1s2pj=dects2pj′-t1Ts2pj=decγs2p1s2pj+γs1a1s2p1


### 3.2.4. Calculation of Phase Differenceγs1a1r2pj

After the meshr2p1 contacts at meshing starting point, the first contact time of meshing starting point of the meshr2pj is(45)tr2pj′=γr2p1r2pjTr2pj+tr2p1=γr2p1r2pjTr2pj+γs1a1r2p1Tr2p1+t1The mesh phase difference of the gear pairr2pj relative to the base referred meshs1a1 is(46)γs1a1r2pj=dectr2pj′-t1Tr2pj=decγr2p1r2pj+γs1a1r2p1Since the starting point of the referred gear pairs1a1 contacts at t=t1, the time at which the meshing starting point of the gear pairg contacts for the first time is(47)tg=γs1a1gTg+t1Then the mesh tooth variation function in the system can be expressed as [12](48)kgt=κgt-tg=κgt-γs1a1gTg-t1where κgτ is a periodic time-varying mesh tooth variation function of the gear pairg, τ=0 corresponds to the meshing starting point, andt is absolute time of the system.


## 3.2.1. Calculation of Phase Differenceγs1a1r2p1

After the contact of the meshing start point of sun-planet meshs2p1, the time tr2p1′ at which the meshing start point of ring-planet meshr2p1contact for the first time is(39)tr2p1′=γs2p1r2p1Tr2p1+ts2p1=γs2p1r2p1Tr2p1+γs1a1s2p1Ts2p1+t1Relative to the base referred meshs1a1, the mesh phase difference of the gear pairr2p1 is(40)γs1a1r2p1=dectr2p1′-t1Tr2p1=decγs2p1r2p1+γs1a1s2p1


## 3.2.2. Calculation of Phase Differenceγs1a1r1bi

After the meshr1b1 contacts at meshing starting point, the first contact time of meshing starting point of the meshr1bi is(41)tr1bi′=γr1b1r1biTr1bi+tr1b1=γr1b1r1biTr1bi+γs1a1r1b1Tr1b1+t1The mesh phase difference of the gear pairr1bi relative to the base referred meshs1a1 is(42)γs1a1r1bi=dectr1bi′-t1Tr1bi=decγr1b1r1bi+γs1a1r1b1


## 3.2.3. Calculation of Phase Differenceγs1a1s2pj

After the meshs2p1 contacts at meshing starting point, the first contact time of meshing starting point of the meshs2pj is(43)ts2pj′=γs2p1s2pjTs2pj+ts2p1=γs2p1s2pjTs2pj+γs1a1s2p1Ts2p1+t1The mesh phase difference of the gear pairs2pj relative to the base referred meshs1a1 is(44)γs1a1s2pj=dects2pj′-t1Ts2pj=decγs2p1s2pj+γs1a1s2p1


## 3.2.4. Calculation of Phase Differenceγs1a1r2pj

After the meshr2p1 contacts at meshing starting point, the first contact time of meshing starting point of the meshr2pj is(45)tr2pj′=γr2p1r2pjTr2pj+tr2p1=γr2p1r2pjTr2pj+γs1a1r2p1Tr2p1+t1The mesh phase difference of the gear pairr2pj relative to the base referred meshs1a1 is(46)γs1a1r2pj=dectr2pj′-t1Tr2pj=decγr2p1r2pj+γs1a1r2p1Since the starting point of the referred gear pairs1a1 contacts at t=t1, the time at which the meshing starting point of the gear pairg contacts for the first time is(47)tg=γs1a1gTg+t1Then the mesh tooth variation function in the system can be expressed as [12](48)kgt=κgt-tg=κgt-γs1a1gTg-t1where κgτ is a periodic time-varying mesh tooth variation function of the gear pairg, τ=0 corresponds to the meshing starting point, andt is absolute time of the system.


## 4. Example Calculation of Phase Difference in Encased Differential Gear Train

Table1 shows the parameters of the encased differential planetary gear train withM=6,N=6. For the convenience of processing and assembly, it is usually guaranteed that the midline of referred gear tooth or space of the duplicate gear is coincident during machining, that is, the offset angle φs1s2=0, φab=0. The mesh phase difference of all gear pairs in the system relative to the referred gear pairs1a1 is calculated as shown in Table 2.Table 1
Encased differential planetary gear train parameters.
Encased stage Differential stage Gears1 Planeta Planetb Gearr1 Gears2 Planetp Gearr2 Tooth number 57 54 18 107 38 25 88 Module (mm) 2.75 2.75 3.5 3.5 4 4 4 Pressure angle (°) 20 20 20 20 20 20 20 Modification coefficient +0.4618 +0.45 +0.5038 +0.2935 0 0 0 Base radius (mm) 73.65 69.77 29.6 175.96 71.42 46.98 165.39 Addendum radius (mm) 82.26 78.11 36.78 184.99 80 54 172.45 Center distance (mm) O 1 O 2 = 155 O 3 O 4 = 126Table 2
Comprehensive mesh phase difference.
Mesh phase differenceγ Encased stage γ s 1 a 1 s 1 a 1 = 0   γ s 1 a 1 s 1 a 2 = 0.5000  γ s 1 a 1 s 1 a 3 = 0   γ s 1 a 1 s 1 a 4 = 0.5000  γ s 1 a 1 s 1 a 5 = 0   γ s 1 a 1 s 1 a 6 = 0.5000  γ s 1 a 1 r 1 b 1 = 0.1246   γ s 1 a 1 r 1 b 2 = 0.2913  γ s 1 a 1 r 1 b 3 = 0.4579   γ s 1 a 1 r 1 b 4 = 0.6246  γ s 1 a 1 r 1 b 5 = 0.7913   γ s 1 a 1 r 1 b 6 = 0.9579   Differential stage γ s 1 a 1 s 2 p 1 = 0.1978   γ s 1 a 1 s 2 p 2 = 0.5311  γ s 1 a 1 s 2 p 3 = 0.8645   γ s 1 a 1 s 2 p 4 = 0.1978  γ s 1 a 1 s 2 p 5 = 0.5311   γ s 1 a 1 s 2 p 6 = 0.8645  γ s 1 a 1 r 2 p 1 = 0.0432   γ s 1 a 1 r 2 p 2 = 0.3765  γ s 1 a 1 r 2 p 3 = 0.7099   γ s 1 a 1 r 2 p 4 = 0.0432  γ s 1 a 1 r 2 p 5 = 0.3765   γ s 1 a 1 r 2 p 6 = 0.7099When the input shaft speed isns1=1490 (r/min), the mesh period of all gear meshes in the example system can be obtained. (49)Ts1ai=7.06×10-4sTr1bi=2.12×10-3sTs2pj=Tr2pj=1.29×10-3sWithout loss of generality, let the starting timet1=0 of the reference gear pairs1a1 be in contact. The gear pairs2pj mesh tooth variation function changes with time as shown in Figure 5, and the other gear pairs in the system can be similarly unified to the system timet.Figure 5
Mesh tooth variation function of gear pairss2pj.The time-varying meshing stiffness of each gear pair in a meshing period calculated by theoretical method or finite element method combined with the meshing time difference caused by the phase difference can be applied to the dynamic model of the system, which can accurately describe the meshing stiffness of all gear pairs in the model at any time and carry out the dynamic analysis of the gear system.


## 5. Conclusions

The accurate solution of the mesh phase difference of all gear pairs is an extremely important part of the study of the dynamics, fault diagnosis, and load sharing characteristics of compound planetary gear train. In this paper, for the encased differential compound planetary gear train widely used in coaxial twin-rotor helicopters, taking the meshing starting point as the reference point, considering the gear modification, the calculation method of mesh phase difference is studied.Firstly, the initial position of the system and the referred gear pair of the system are defined, and the time when the referred gear pair is in contact at the meshing starting point is calculated. According to the angle relations of the meshing starting points and the distance relations of the starting points on the meshing line, from the initial position, the time at which the gear pair adjacent to the base referred gear pair first contacted at the starting point is calculated, and the phase difference can be obtained by the meshing time difference between the gear pairs. The phase difference between planet gears is calculated by the relations between the number of meshing teeth and the meshing time caused by the symmetry of planetary gear train.Based on the phase difference of adjacent gear pairs, the calculation method of the phase difference of each gear pair relative to the referred gear pair is derived. The derivation process considers the processing and assembly requirements of the duplicate gear, and a specific compound planetary gear train is calculated as an example, which lays a foundation for the accurate introduction of time-varying meshing stiffness into the dynamic analysis of compound planetary gear train.



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*Source: 8421201-2019-07-25.xml*