# Dynamic Characteristics of the Helical Planetary Transmission System considering the Installation Error of Shaft

Authors: Hongfei Zhai; Junjian Hou; Zhanpeng Fang; Yanqiu Xiao; Lei Yao; Liangwen Wang
Journal: Shock and Vibration
 (2022)

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## Abstract

The helical planetary transmission system (HPTS) is widely used in wind power, hydropower, and shield machine, which operating conditions are extremely complicated, and the existence of component installation errors will result in response deviations to the theoretical design. A dynamic model for slice type bending-torsional-pendulum of the HPTS is established in the lumped parameter method considering the installation error of shaft (IEOS) for the 1st planet gear, helical meshing characteristics, and bearing support characteristics. The vibration response characteristics of the HPTS under the IEOS for the 1st planet gear are studied. The results have shown that the FFT amplitudes for the horizontal vibration acceleration of the sun gear increased by 8.3%, and the axial vibration acceleration of the ring gear increased by 11.5%, respectively. The maximum magnitude increase of FFT amplitude for the vibration acceleration under a positive 0.13 mm IEOS for the 1st planet gear is the axial direction of the ring gear. The results indicate that the IEOS needs to be thoroughly introduced into the HPTS to simulate the actual operating conditions of the system.


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## Body

## 1. Introduction

Based on the inevitably various errors that existed in the installation and debugging of mechanical parts, the existence of installation errors will cause deviations in the vibration, meshing force, and dynamic response characteristics of the gear transmission system, resulting in a linear decrease in system reliability and operational stability. The helical gear planetary transmission system is widely used in wind power generation, hydropower generation, and nuclear power generation. Its operating conditions are extremely complex, and the existence of component installation errors will produce response deviations to the theoretical design system, and its dynamic characteristics are economical and reliable for the whole machine. Sex and longevity have an important impact.At present, many scholars investigated the component errors and response characteristics of the planetary gear transmission system. A dynamic model of wind turbine gearbox containing stiffness of bearing, stiffness of gear meshing, and flexibility of structure in multibody dynamic method was established by Helsen et al. [1, 2], and the dynamic responses of high-power wind turbine gearbox were studied. Zhao et al. [3] established a pure torsional dynamic model of wind turbine gearbox transmission, and the influences of transmission errors, input load, and mesh stiffness of gear pair on vibration responses of system components were studied. The effects of the position errors and eccentricity errors for planet gear on load characteristics and motion locus were researched by Velex et al. [4, 5].A bending-torsional-pendular dynamic model of spur planetary transmission system considering the influences of the elastic deformation for ring gear, gyro effect, and centripetal acceleration was established by Wang et al. [6], the influence inherent characteristics of the system were analyzed, and the results showed that the nonstability of system parameters is determined by the multiscale method.Tatar et al. [7] proposed a dynamic model of the planetary gear transmission system on the influences of the gyro effect and the rotating shaft. The studies show that the vibration modes of the helical planetary transmission system are divided into torsion-axial, lateral, and gearbox coupling. Shen et al. [8] built a pure torsional dynamic model of the planetary transmission system considering the wear of the tooth surface. The wear of the tooth surface is characterized by the static transmission error and time-varying mesh stiffness. The time-domain and frequency-domain response characteristics of the system vibration were analyzed. The results pointed out that the tooth surface wear will cause a significant difference between the time-domain and frequency-domain response of the system vibration.Liu et al. [9] established a bending-torsional-pendular coupled dynamic model for herringbone tooth planetary transmission of wind turbine gearbox considering tooth separation and tooth profile error, and the influence of tooth error on mesh forces of gear pairs and vibration responses was studied. Mbarek et al. [10] analyzed the spectrum characteristics of the planetary transmission system under different loads and changes in the meshing stiffness of gear pair, and the results showed that the grid stiffness change has a certain degree of influence on the natural frequency of the system. A translational-torsional dynamic model of the planetary transmission system considering the time-varying transmission path factors was established by Zghal et al. [11]. The time-domain and frequency-domain responses of the system vibration were investigated, and the theoretical analysis results were verified with experiments.Taheri et al. [12] presented a dynamic model of the planetary gear transmission system considering the deformation of the elastic ring gear and studied the meshing force response and vibration response characteristics of the system. The results showed that the meshing of gear pair has a strong influence on the system response. Park et al. [13] established a universal dynamic model of planetary gear transmission system considering the factors of the planet gear variability in transmission load and gear tooth deformation and studied the vibration response characteristics of the system under fault conditions.Due to the comprehensive influences of the gear production technique, machining, installation, and debugging, manufacturing and installation errors exist in the gear transmission process. Therefore, the influences of the manufacturing and installation errors for the gear on the vibration characteristics of the HPTS must not be neglected. Kahraman et al. [14, 15] represented a manufacturing error, then the influence of random tooth pitch error on the vibration behavior of spur gear pairs was analyzed, and the effect of the gear roughness and rotational speed on the galling load was studied.According to the bearing load characteristics and working conditions, Parker et al. [16, 17] established a dynamic model of the gear system considering the time-varying bearing support stiffness and studied the response characteristics of the system. Then, the vibration modal characteristics of the planetary gear transmission system containing an elastic body were investigated under the condition of the flexible ring gear. The results show that the elastic resonance phenomenon of the system will lead to fatigue or stress concentration in the core components of the transmission system.Ren et al. [18] established a dynamic model of the herringbone planetary gear system considering the factors of the gear manufacturing error, time-varying meshing stiffness, and bearing deformation, focused on the gear tooth profile error and manufacturing eccentricity error, and studied the effect of the manufacturing error on the planetary transmission system. The results show that the manufacturing error has a great influence on the vibration characteristics of the system. Liu et al. [19] studied the changes in the pressure angle and overlap ratio for the sun gear during the floating process of the planetary transmission system and explored the phase modulation through the dynamic response of the system.A lumped parameter dynamic model of a 5-slice type HPTS is established considering the factors of the IEOS for planetary gear and the time-varying meshing characteristics of gear pairs. Furthermore, the IEOS of the 1st planet gear was considered as the design variable. The effects of the IEOS on the system vibration characteristics were investigated.


## 2. Structure and Transmission Principle of HPTS

An HPTS is composed of 5 evenly distributed planet gears, carrier, ring gear, and sun gear as shown in Figure1. The carrier of the HPTS is the input part, and the sun gear of the HPTS is the output part. The parameters for the HPTS are listed in Table 1.Figure 1
Structure and transmission principle of the HPTS.Table 1
Gear parameters of the HPTS.
Gear dateRing gearSun gearPlanet gearCarrierNumber of teeth912931—Module (mm)232323—Pressure angle (°)252525—Helix angle (°)555—Mass (Kg)4400119010707140Moment of Inertia (N·m/rad)556075.4496.103730Addendum coefficient1Dedendum coefficient1.5The whole teeth of spur gear planetary transmission enter into meshing or exit meshing, which has a large meshing impact force. The gear meshing of the HPTS is an asymptotic meshing process along the space meshing contact line, causing the meshing contact is relatively stable. Meanwhile, due to the existence of a helix angle for the helical gear, the helical gear has transverse and overlap contact ratios, and the bearing capacity of the HPTS is relatively high. Therefore, the HPTS can effectively homogenize the impact load and reduce the impact of the load on the transmission system, which has good meshing stability, transmission stability, and large bearing capacity [20, 21].The existence of the IEOS will lead to the instability of the actual operating characteristics of the HPTS, and even cause the system to fail to meet the design requirements and cause unexpected damage. Based on the meshing characteristics and conditions of the internal and external gear pairs of the HPTS considering the constituent factors and structural characteristics of the slice-type gears, each transmission component of the system is divided into five equal slices according to the material mechanics’ relationship. The influence of directional stiffness and damping (horizontal direction, vertical direction, axial direction, and torsion direction) is shown in Figure2.Figure 2
The mechanical model of a 5 equal slicing type gear.Based on the ISO 1328-1-2013 cylindrical gear tooth surface deviation, ISO 6331 spur gear helical gear load capacity calculation, and GB/T 19073–2018 wind turbine gearbox design requirements, the influence of the IEOS on the meshing gear pair was considered to refer to the upper and lower deviation of the involute cylindrical gear axis angle. The axis angle deviation is set at 0.130 mm as shown in Figure3. The K represents the connection stiffness of the component. The C characterizes the connection damping of the component. The subscript stxk (styk, stzk, stk) represents the kth position for the sun gear along the x (y, z, θ) direction.Figure 3
Influence of installation shaft error on helical gear meshing gear pair.Because each component of the helical planetary transmission system is divided into 5 equal slices, according to the meshing principle of the helical planetary transmission gear pair, the gear shaft installation error factor is considered, as shown in the following equation:(1)Δespi,erpi=±KiEp,where Δespi,erpi is the IEOS of the HPTS sun-planet (ring-planet) gear pair along the meshing line direction; Ki is the error ratio corresponding to the ith slice; Ep is the deviation of the IEOS.According to the movement relationship and transmission characteristics of the transmission components, the helical gear meshing principle, and the bearing support characteristics, the dynamic model of the slice type for the HPTS is established as shown in Figure4.Figure 4
Dynamic model of slice type HPTS.Each transmission of the system is considered as a rigid body with 4 degrees of freedom (horizontalx, vertical y, axial z, and torsion θ). The center of the sun gear is set as the static coordinate origin, and the sun gear, planetary gear, and ring gear are set as static coordinate systems, respectively. The 5 moving coordinate systems are set as the each planet gear u,v,w correspondingly and the center of the each planet gear is set as the moving coordinate origin. The z-axis counterclockwise is set as positive. The subscripts s, c, r, and pi represent the sun gear, planet carrier, ring gear, and the ith planet gear. The K represents the spring stiffness of the components. The C characterizes the damping of the components. The subscript spi (rpi) represents the gear meshing of the ith sun-planet (ring-planet) gear pair.The total transmission errors of the SP and RP meshing gear pairs for the HPTS along the meshing line considering the influence of the IEOS are as follows:Sun-planet ring-planet(2)Δspi,rpi=δspi,rpi+Espi,rpi±Δespi,erpi,where Δspi,rpi is the total transmission error of the sun-planet (ring-planet) gear pair; δspi,rpi is the static transmission error of the sun-planet (ring-planet) gear pair.According to the Lagrange energy method, the motion equation of the 5-slice type helical planetary transmission system can be obtained as follows:Kinetic equations of the 1st stage sun gear(3)∑j=15Isjθ¨sj+∑j=15Csθjθ˙sj+∑k=14Cstkθ˙sk−θ˙sk+1+∑j=15∑i=1NCspijt∂Δspij∂θsjcosβb+∑j=15Ksθjθsj+∑k=14Kstkθ˙sk−θ˙sk+1+∑j=15∑i=1NKspijtΔspicosβb=Tout,∑j=15msjx¨sj+∑j=15Csxjx˙sxj+∑k=14Cstxkx˙sk−x˙sk+1+∑j=15∑i=1NCspijt∂Δspij∂xsjsinφi+αspcosβb+∑j=15Ksxjxsxj+∑k=14Kstxkxsk−xsk+1+∑j=15∑i=1NKspijtΔspijsinφi+αspcosβb=0,∑j=15msjy¨sj+∑j=15Csyjy˙sj+∑k=14Cstyky˙sk−y˙sk+1−∑j=15∑i=1NCspijt∂Δspij∂ysjcosφi+αspcosβb+∑j=15Ksyjysj+∑k=14Kstykysk−ysk+1−∑j=15∑i=1NKspitΔspicosφi+αspcosβb=0,∑j=15msjz¨sj+∑j=15Cszjz˙sj+∑k=14Cstzkz˙sk−z˙sk+1+∑j=15∑i=1NCspijt∂Δspij∂zsjsinβb+∑j=15Kszjzsj+∑k=14Kstzkzsk−zsk+1+∑j=15∑i=1NKspijtΔspijsinβb=0.Kinetic equations of the 1st stage ring gear(4)∑j=15Irjθ¨rj+∑j=15Crθjθ˙rj+∑k=14Crtθkθ˙rk−θ˙rk+1+∑j=15∑i=1NCrpijt∂Δrpij∂θrjcosβb+∑j=15Krθjθrj+∑k=14Krtθkθ˙rk−θ˙rk+1+∑j=15∑i=1NKrpijtΔrpijcosβb=0,∑j=15mrjx¨rj+∑j=15Crxjx˙rj+∑k=14Crtxjx˙rk−x˙rk+1−∑j=15∑i=1NCrpijt∂Δrpij∂xrjsinφi+αrpcosβb+∑j=15Krxjxrj+∑k=14Krtxjxrk−xrk+1−∑j=15∑i=1NKrpijtΔspijsinφij+αrpcosβb=0,∑j=15mrjy¨rj+∑j=15Cryjy˙rj+∑k=14Crtyky˙rk−y˙rk+1+∑j=15∑i=1NCrpijt∂Δrpij∂yrjcosφij+αrpcosβb+∑j=15Kryjyrj+∑k=14Krtykyrk−yrk+1+∑j=15∑i=1NKrpijtΔrpijcosφij+αrpcosβb=0,∑j=15mrjz¨rj+∑j=15Crzjz˙rj+∑k=14Crtzkz˙rk−z˙rk+1+∑j=15∑i=1NCrpijt∂Δrpij∂zrjsinβb+∑j=15Krzjzrzj+∑k=14Krtzkzrk−zrk+1+∑j=15∑i=1NKrpijtΔrpijsinβb=0.Kinetic equations of the 1st stage planet gear (ith)(5)∑j=15∑i=15Ipijθ¨pij−∑i=15∑k=14Cptikθ˙pik−θ˙pik+1+∑j=15∑i=1NCspijt∂Δspij∂θpijcosβb−∑j=15∑i=1NCrpijt∂Δrpij∂θpijcosβb−∑i=15∑k=14Kptikθpik−θpik+1−∑j=15∑i=1NKspijtΔspijcosβb−∑j=15∑i=1NKrpijtΔrpijcosβb=0,∑j=15mpiju¨pij+∑j=15Cpijuu˙pij−∑j=15Cpijut∂δpiju∂upij+∑k=14Kptikuu˙pik−u˙pik+1−∑j=15Cspijt∂Δspij∂upijsinαspcosβb+∑j=15Crpijt∂Δrpij∂upijsinαrpcosβb+∑j=15Kpijuupij−∑j=15Kpijutδpiju+∑k=14Kptikuupik−upik+1−∑j=15KspijtΔspijsinαspcosβb+∑j=15KrpijtΔrpisinαrpcosβb=0,∑j=15mpijv¨pij+∑j=15Cpijvv˙pij−∑j=15Cpijvt∂δpijv∂vpij+∑k=14Cptikvv˙pik−v˙pik+1+∑j=15Cspijt∂Δspij∂vpijcosαspcosβb−∑j=15Crpijt∂Δrpij∂vpijcosαrpcosβb+∑j=15Kpijvvpij−∑j=15Kpijvtδpijv+∑k=14Kptikvvpik−vpik+1+∑j=15KspijtΔspijcosαspcosβb−∑j=15KrpijtΔrpijcosαrpcosβb=0,∑j=15mpijw¨pij+∑j=15Cpijww˙pij+∑j=15Cpijwt∂δpijw∂wpij+∑k=14Cptikww˙pik−w˙pik+1−∑j=15Cspijt∂Δspij∂wpijsinβb−∑j=15Crpijt∂Δrpij∂wpijsinβb+∑j=15Kpijwwpij+∑j=15Kpijwtδpijw−∑j=15KspijtΔspijsinβb+∑k=14Kptikwwpik−wpik+1−∑j=15KrpijtΔrpijsinβb=0.Kinetic equations of the 1st stage carrier(6)∑j=15Icjθ¨cj+∑j=15∑i=1NCpijvt∂δpijv∂θcj++∑j=15∑i=1NKpijvtδpijv+∑j=15∑i=1NCspijt∂δpijv∂θcj+∑j=15∑i=1NCrpijt∂Δrpij∂θcj+∑k=14Cctkθ˙ck−θ˙ck+1+∑j=15∑i=1NKspijtΔspij+∑j=15∑i=1NKrpijtΔrpij+∑k=14Kctkθck−θck+1=Tin,∑j=15mcjx¨cj+∑j=15Ccxjx˙cj+∑j=15∑i=1NCpijut∂δpiju∂xcjcosφi−∑j=15∑i=1NCpijvt∂δpijv∂xcjsinφi+∑k=14Cctxkx˙ck−x˙ck+1+∑j=15Kcxjxcj+∑i=1NKpijutδpijucosφi−∑i=1NKpijvtδpijvsinφi+∑k=14Kctxkxck−xck+1=0,∑j=15mcjy¨cj+∑j=15Ccyjy˙cj+∑j=15∑i=1NCpijut∂δpiju∂ycjsinφi+∑j=15∑i=1NCpijvt∂δpijv∂ycjcosφi+∑k=14Cctyky˙ck−y˙ck+1+∑j=15Kcyjycj+∑j=15∑i=1NKpijutδpijusinφi+∑j=15∑i=1NKpijvtδpijvcosφi+∑k=14Kctykyck−yck+1=0,∑j=15mcjz¨cj+∑j=15Cczjz˙cj+∑k=14Cctzkz˙ck−z˙ck+1+∑j=15∑i=1NCpijwtz˙cj−w˙pij+∑j=15Kczjzcj+∑k=14kctzkzck−zck+1+∑j=15∑i=1NKpijwtzcj−wpij=0.The equations of motion for the above transmission components are sorted out, and the dynamic equation matrix form of the HPTS considering the influence of the installation shaft error is(7)Mq¨+Cq˙+Kq=Q,where M, C, K, and Q are system mass matrix, damping matrix, stiffness matrix, and external load matrix, respectively, and q¨, q˙, and q are system vibration acceleration matrix, vibration velocity matrix, and vibration displacement matrix, respectively. The system has 160 DOFs of which the planet carrier, sun gear, and ring gear have 60 DOFs, respectively, and the 5 evenly distributed planetary gears have 100 DOFs.


## 3. Dynamic Responses of the HPTS

The input torque of the system is 4.88e6 Nm, and the load torque is 1.18e6 Nm. The mesh stiffnesses of gear pairs are calculated from the Romax according to the ISO6336-2008 standard, and the Fourier series was used to calculate the stiffness excitation harmonics. The stiffnesses of the HPTS are listed in Table2.Table 2
Stiffnesses of the HPTS.
SubjectKx (N/m)Ky (N/m)Kz (N/m)Kθ (N/rad)Ring support3e113e111e115e11Sun bearing2.6e92.66e93.5e8—Planet bearing3.0e93.0e94.6e8—Carrier support4.8e104.8e108.4e9—Sun-planet gear meshing (N/m)8.63e9Ring-planet gear meshing (N/m)10.9e9Figures5 and 6 show the vibration acceleration responses and FFT transformations of the central position for the sun gear and ring gear of the HPTS under the positive 0.13 mm IEOS of the 1st planet gear.Figure 5
Vibration acceleration responses for sun gear under the IEOS of the 1st planet gear. (a) Horizontal direction. (b) Vertical direction. (c) Axial direction.
(a)(b)(c)Figure 6
Vibration acceleration responses ring gear under the IEOS of the 1st planet gear. (a) Horizontal direction. (b) Vertical direction. (c) Axial direction.
(a)(b)(c)As is shown in Figure5, the maximum vibration acceleration for the FFT amplitude of the sun gear under the positive 0.13 mm IEOS of the 1st planet gear is horizontal direction 3.66 m/s2, and the minimum vibration acceleration for the FFT amplitude is axial direction 1.84 m/s2. Compared with no IEOS condition, the horizontal vibration acceleration amplitude increased by 8.3%, and the axial vibration acceleration amplitude increased by 8.2%.As is shown in Figure6, the maximum vibration acceleration for the FFT amplitude of the ring gear is vertical direction 2.38 m/s2, and the minimum axial vibration acceleration for the FFT amplitude is 1.07 m/s2 considering the positive 0.13 mm IEOS of the 1st planet gear. Compared with no IEOS condition, the vibration acceleration amplitude for the vertical direction of the ring gear increased by 11.2%, and the vibration acceleration amplitude for the axial direction increased by 11.5%.In general, the vibration accelerations for the FFT responses of the sun gear and ring gear show three times the gear meshing frequency of the system. Due to the IEOS of the 1st planet gear, the FFT amplitudes in all directions for vibration acceleration response of transmission components are increased to a certain extent at the gear meshing frequency position. Compared with no IEOS of the 1st planet gear, the maximum magnitude increase of the vibration acceleration for the FFT amplitude under the IEOS condition for the 1st planet gear is the axial direction of the ring gear.The overall trend and characteristics for the influence of the planetary gear IEOS on the responses of the gear pair meshing force are consistent with the literature [9, 12]. The influence situation and characteristics of the IEOS of the 1st planet gear on the vibration accelerations for the transmission components of the system are consistent with the literature [13]. The results show that the IEOS of the 1st planet gear has a significant influence on the vibration response characteristics of the HPTS.


## 4. Conclusions

A lumped parameter dynamic model of a 5-slice type HPTS is established considering the factors of the IEOS for the 1st planet gear and the time-varying meshing characteristics of gear pairs. The characteristics of the vibration influence for the HPTS under the IEOS are analyzed. The main conclusions are as follows:(1)
Compared with no IEOS condition, the FFT amplitudes for the horizontal vibration acceleration of the sun gear increased 8.3% and the axial vibration acceleration of the ring gear increased 11.5% considering the positive 0.13 mm IEOS for the 1st planet gear, respectively.(2)
The vibration accelerations for the FFT responses of the sun gear and ring gear show three times the gear meshing frequency of the system considering a positive 0.13 mm IEOS for the 1st planet gear. The maximum magnitude increase of the vibration acceleration for the FFT amplitude under a positive 0.13 mm IEOS for the 1st planet gear is the axial direction of the ring gear. The results show that the IEOS of the 1st planet gear has a significant influence on the vibration response characteristics of the HPTS.(3)
In the modeling of a 5-slice type HPTS considering the factors of the IEOS for the 1st planet gear, the influence of the time-varying support stiffness for the bearings on the system vibration needs to be taken into account. The IEOS needs to be thoroughly introduced into the HPTS to simulate actual operating conditions.



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*Source: 3677691-2022-07-06.xml*