J. Cent. South Univ. (2020) 27: 2280-2290

DOI: https://doi.org/10.1007/s11771-020-4449-0

Influence of support stiffness on vibrations of a planet gear system considering ring with flexible support

LIU Jing(刘静)^{1, 2}, PANG Rui-kun(庞瑞琨)², LI Hong-wu(李洪武)³, XU Jin(许晋)³

1. State Key Laboratory of Mechanical Transmission, Chongqing University, Chongqing 400044, China;

2. College of Mechanical Engineering, Chongqing University, Chongqing 400044, China;

3. China North Vehicle Research Institute, Beijing 100081, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020

Abstract: Planet gear systems (PGSs) are key components of transmission mechanisms. Structural and material characteristics of gearbox and shaft can affect the support stiffness and vibrations of PGSs. The ring gear flexibility should affect the vibrations of PGSs too. However, most previous work did not completely consider the effects of the ring gear flexibility on the vibrations of PGSs and flexible supports of ring and sun gears. Thus, this paper presents a flexible-rigid coupling multi-body dynamic (FMBD) model for a PGS with the flexible supports and ring gear flexibility. A finite element model of ring gear is established to formulate the ring gear flexibility. The influences of clearance and damping of planet bearings on the vibrations of PGS are considered. The effects of flexible supports and ring gear flexibility on the vibrations of PGS under different moment and speed conditions are studied. The statistical parameters and peak frequencies of PGS from the proposed FMBD and previous rigid multi-body dynamic (RMBD) models are compared. The results denote that the flexible support has a great effect on the vibrations of PGS. This paper can provide some guidance for the support structure design and vibration control for PGSs.

Key words: planet gear; flexible-rigid coupling; flexible supports; ring gear flexibility; vibrations

Cite this article as: LIU Jing, PANG Rui-kun, LI Hong-wu, XU Jin. Influence of support stiffness on vibrations of a planet gear system considering ring with flexible support [J]. Journal of Central South University, 2020, 27(8): 2280-2290. DOI: https://doi.org/10.1007/s11771-020-4449-0.

1 Introduction

Planet gear systems (PGSs) are widely used in various mechanical equipment. The PGSs have high transmission ratio and high transmission efficiency. A large vibration in the PGSs may cause different failures including cracks and pits in the gears and bearings [1-3]. Therefore, vibration studies of PGSs should be necessary. The supports of PGSs may greatly affect the vibrations of PGSs. It is of great significance to understand the effect of the supports on the vibrations of PGSs.

In previous works, various methods including the centralized parameter method, finite element (FE), rigid multi-body dynamic (RMBD), and experimental methods were presented to study the vibrations of PGSs [4, 5]. LIU et al [6] established a dynamic model to analyze the vibrations of a pitting gear system. THORET-BAUCHET et al [7] established a model of a PGS to analyze the influence of a pitting on the dynamic behaviours of the PGS. WANG et al [8, 9] formulated a generalized dynamic model of gear driveline system to understand the interaction between the hypoid gear pair and propeller shaft. LIN et al [10] proposed a FMBD model considering the ring gear flexible deformation to study the effect of gyro response on the vibrations of a PGS. KAHRAMAN et al [11] established a torsion model of PGS and proposed a calculation method of natural frequency for analyzing the inherent characteristic of a PGS. YU et al [12] established a new dynamic model of a cylindrical gear pair with localized spalling defects to assessment the effects of tooth defects on the vibration response of a PGS. GUO et al [13] developed a centralized parameter model with the bearing clearance, backside contact, pitch and width of the gear to analyze the vibrations of a PGS. XIAO et al [14] established a torsion dynamic model to analyze the vibrations of a PGS. BODAS et al [15] studied the effects of manufacturing errors of carrier and gear on the static uneven load coefficient of a PGS by using a FE model. LIU et al [16] established a dynamic model with a locally imperfection rigid rotor bearing system considering additional edge deformation. XUE et al [17] studied the effect of the flexible ring gear on the vibrations of a PGS by using FE method.

Most previous researches only studied the effects of single rigid ring gear, rigid planet gear or planet bearing on the vibrations of PGSs. Some studied a ring gear flexibility FE model with rigid support [18]. LI et al [19] studied an RMBD model with flexible support without considering ring gear flexibility. In practice, both the ring gear flexibility and flexible support have great effect on the vibrations of PGSs, which cannot be ignored. Thus, this paper presents a flexible-rigid coupling multi-body dynamic (FMBD) model for a PGS with the flexible supports and ring gear flexibility. A finite element model of the ring gear is established to formulate the ring gear flexibility. The Hertzian theory is used to calculate the meshing stiffness and damping of gears. The effects of flexible supports and ring gear flexibility on the vibrations of PGS are studied under different load and input speed conditions. The statistical parameters and peak frequencies of PGS from the proposed FMBD and previous rigid multi-body dynamic (RMBD) models are compared. The results denote that the flexible support has a great effect on the time- and frequency-domain vibrations of PGS. This paper can provide some guidance for the support structure design and vibration control for PGSs.

2 Problem description

The meshing forces and radial loads can cause the flexible deformation of the ring gear during the working conditions of the PGS, which cannot be described by the RMBD model. The flexible deformation of the ring gear can produce the unacceptable vibrations and imbalance of the PGS. The finite element method is used to calculate the flexibility of the ring gear. The finite element model of the ring gear is developed by using a commercial software ANSYS. The dynamic model of the PGS is established by a commercial software ADAMS.

Figure 1 shows a FMBD model of a PGS with the ring gear flexibility and flexible support of sun and ring gears. The model has three planet bearings. The bearings are used to connect the planet gears and the carrier; the outer of bearings are linked to the planet gears while the inner of bearings are linked to the carrier. The planet gears are simplified to the internal mesh stiffness between the bearing and the outer ring. According to the research [20], the flexible support of ring gear affects the vibration of PGS. The flexible support of the ring can be determined by the support stiffness and number of the support positions between the ring gear and the gearbox. In practice, the sun gear floats in the radial direction; and its displacement is limited by the radial flexible support. The flexible supports of ring and sun gears are considered in this paper, where K_r1 and K_r2 are the support stiffness of ring and sun gears respectively.

Figure 1 Schematics of a PGS with ring gear flexibility and flexible supports

Here, K_r1 and K_r2 distribute from 1×10⁶ N/mm to 5×10⁶ N/mm. The FMBD and RMBD models with different K_r1 and K_r2 are simulated under different input speeds (n_in) and moments (M_p). The input speed and moment are applied on the sun gear and carrier, respectively. The time-domain and frequency-domain results are analyzed. Table 1 shows n_in and M_p in this paper.

Table 1 Simulation conditions of ring and sun gears

3 Model formulation

The FMBD model for a PGS is depicted in Figure 1. The FMBD model includes a sun gear with the flexible support, a ring gear flexibility with the flexible support, a carrier, three planet gears and three IBC NU202 planet bearings. The dynamic performance and rotational accuracy of the PGS can be greatly affected by the bearing clearance. Therefore, the bearing clearance is defined as 20 μm. The geometric parameters of the components are listed in Tables 2 and 3.

According to the potential energy principle, the meshing stiffness between the ring and planet gears (K_eo) is calculated by [21-23]:

      (1)

where K_b, K_s, K_a and K_h are the bending, shear, axial compressive, and Hertz contact stiffness, respectively. In Eq. (1), the subscripts 1 and 2 denote the sun and planet gears, respectively. The meshing stiffness between the planet and sun gears (K_ei) is calculated by [16-18]:

 (2)

where K_f is the fillet-foundation stiffness. In Eq. (2), the subscripts 1 and 2 represent the planet and ring gears, respectively. The details in Eqs. (2) and (3) are given in CHEN et al [24], TIAN [21], and MA et al [25, 26]. In Eqs. (1) and (2), K_b, K_s, K_a and K_h are calculated by:

             (3)

                      (4)

                         (5)

                            (6)

where h is the distance between the mesh point and gear midline; x is the distance between the mesh point and a region of width of dx on the gear; α₁ is the defunct angle of contact force; d is the distance between the mesh point and root of gear; E and G are the elastic and shear modulus, respectively; I_x and A_x are the area moment of inertia and the area of section at a distance of x between the mesh point and the section, respectively.

The contact damping of the gears (c_max) is calculated by [27]:

                   (7)

where ζ is 0.1; K is the meshing stiffness; I₁ and I₂ are the rotational inertias of meshing gears, respectively; R₁ and R₂ are the radii of two gears, respectively.

The contact stiffness (K_br) between the rollers and bearing race is [28, 29]:

                              (8)

where δ is expressed as:

              (9)

Table 2 Geometric parameters of gears

Table 3 Geometric parameters of planet bearings

where E₁ and E₂ are the elastic modulus; v₁ and v₂ are Poisson ratios; Q is the load; l is the equivalent contact length of roller. Furthermore, the contact damping of bearing is given as:

                            (10)

where λ is from 0.25 to 2.5×10^-5 [30, 31].

Moreover, the governing equation of the FMBD model is represented by:

         (11)

where M is the mass; K and K′ are the flexible support matrice and contect stiffness matrice for PGS, respectively; C and C′ are the support damping matrice and contect damping matrice for PGS, respectively; q is the vector of generalized displacement for PGSs; T, E and F are the external system load vector, fault excitation vector and internal system vector, respectively. Moreover, F also includes the friction force. The meshing stiffness and damping between sun and planet gears are 4.71×10⁵ N/mm and 86.62 N·s/mm, respectively. The meshing stiffness and damping between ring and planet gears are 9.28×10⁵ N/mm and 157.59 N·s/mm, respectively. The contact stiffness and damping between the roller and bearing race are 4.35×10⁵ N/mm and 8.7 N·s/mm, respectively.

4 Results and discussion

The commercial software MSC. Adams (for developing the FMBD model and solving the results) and ANSYS (for developing FE model of ring gear) are used in this paper to study the influence of the support stiffness on the RMBD and FMBD models. The FE model of ring gear is meshed by using the tetrahedron elements as shown in Figure 2. The element length is defined as 1 mm. The material model of the ring gear is defined as a linear elastic one.

To limit that the radial deformations d_r1 and d_r2 of the ring and sun gears are less than 1 mm, the flexible support K_r in the FMBD and RMBD models is assumed to be from 1×10⁶ N/mm to 5×10⁶ N/mm in the studied conditions. The support stiffness of ring and sun gears are calculated by:

                               (12)

                            (13)

                          (14)

when F_cr is the critical force; D and d are the outside and inside diameters of ring and sun gear, respectively; I is the moment of inertia.

Figure 2 Flexible ring gear meshed by using tetrahedron elements

The effects of flexible supports on the vibration of PGS are discussed. The results of FMBD and RMBD models are compared to analyze the effect of ring gear flexibility on the vibrations of PGS.

4.1 Model validation

To ensure the accuracy of results, the correctness of FMBD model is validated. The simulation and theoretical angular velocities are compared when n_in is 1000 r/min (6000°/s). The time-domain simulation results are shown in Figure 3. The planet gears and carrier’s angular velocity are 1780°/s and 1116°/s. Although the angular velocity fluctuates due to the bearing clearance, gear backlash, or elastic deformations, it is relatively stable one. The simulation results are basically consistent with the theoretical results, which can validate the FMBD model.

4.2 Effects of flexible supports on time-domain vibration

The radial acceleration of ring gear is used to describe the vibrations of PGS here. To analyze the effect of the ring gear flexibility on the time-domain vibration, three statistic parameters of the radial acceleration of ring gear, such as the maximum (MAX), peak to peak (PTP) and root mean square (RMS), are introduced. The simulation results of FMBD and RMBD models are analyzed. The values used in the studied flexible support cases are defined by the practical work conditions of the PGS. Figures 4 to 8 describe the effects of the flexible supports (K_r1 and K_r2) on the MAX, PTP and RMS. The MAX, PTP and RMS increase with n_in. n_in is 3500 r/min; and M_p values are 500, 1000 and 1500 N·m in Figures 9 to 10. In Figures 4 to 8, the RMS, MAX, and PTP distributions are similar for different simulation conditions; the RMS decreases with the increment of K_r2; the effect of K_r1 on the RMS is small. The maximum values of MAX and PTP are observed when K_r1 is about 4×10⁶ N/mm and K_r2 is from 2×10⁶ N/mm to 4×10⁶ N/mm. The distributions of RMS, MAX and PTP for the FMBD and RMBD models are also similar. But, the RMS, MAX, and PTP values of FMBD model are much larger than those of RMBD model. The time-domain results indicate that the vibrations of PGS are affected by both K_r1 and K_r2. The effect of K_r2 is bigger than that of K_r1. The vibration of FMBD model is larger than that of RMBD model under the same n_in and M_p. Thus, the ring gear flexibility should be considered in the dynamic modeling of PGS.

Figure 3 Simulation angular velocity of sun gear (a), planet gear (b), and carrier (c)

4.3 Effects of flexible supports on frequency- domain vibrations

To analyze the effects of flexible supports and ring gear flexibility on the frequency-domain vibrations, the frequency of radial acceleration of ring gear is used here.

The simulation results of FMBD and RMBD models are analyzed. Figures 9 and 10 describe the effects of flexible supports (K_r1 and K_r2) on the frequencies of PGS. Table 5 shows the peak frequencies (The frequency corresponding to the peak value in the spectrum diagram) of FMBD and RMBD models with different flexible support cases when n_in=3000 r/min, M_p=500 N·m. In Figure 9, K_r2 is 1×10⁶ N/mm, and K_r1 is 1×10⁶ N/mm, 2×10⁶ N/mm, 3×10⁶ N/mm, 4×10⁶ N/mm, and 5×10⁶ N/mm. Two peak frequencies f_m-f_s=723 Hz and 2f_m-2f_s-2f_c=1417 Hz of FMBD and RMBD models are observed. In Figure 10, K_r1 is 1×10⁶, K_r2 is 1×10⁶, 2×10⁶, 3×10⁶, 4×10⁶ and 5×10⁶ N/mm, the same peak frequencies f_m-f_s=723 Hz and 2f_m-2f_s- 2f_c=1417 Hz of FMBD and RMBD models are observed. Here, f_m is the meshing frequency of ring and planet gears; f_s is the rotational frequency of sun gear; and f_c is the rotational frequency of carrier [32, 33]. The relative spectra are similar for different flexible support cases. The maximum values of peak frequencies f_m-f_p and 2f_m-2f_s-2f_c are observed when K_r1=1×10⁶ N/mm and K_r2=2×10⁶ N/mm or K_r2=1×10⁶ N/mm and K_r1=2×10⁶ N/mm. The peak frequency amplitudes decrease with the increment of K_r1 and K_r2. The effect of K_r1 on the peak frequency is small. The spectrum waves of FMBD and RMBD models are similar. But the peak frequency amplitudes of FMBD model are much larger than those of RMBD model. The frequency-domain results indicate that the vibrations of PGS are affected by both K_r1 and K_r2. The effect of K_r2 is bigger than that of K_r1. The vibrations of FMBD model are larger than those of RMBD model under the same n_in and M_p. Thus, the ring gear flexibility should be considered in the dynamic modeling of PGS.

Figure 4 Effects of flexible supports (K_r1 and K_r2)on time-domain vibrations when n_in=2500 r/min and M_p=500 N·m:

Figure 5 Effects of flexible supports on time-domain vibration when n_in=3000 r/min and M_p=500 N·m:

Figure 6 Effects of flexible supports on time-domain vibration when n_in=3500 r/min and M_p=500 N·m:

Figure 7 Effects of flexible supports on time-domain vibration when n_in=3500 r/min and M_p=1000 N·m:

Figure 8 Effects of flexible supports on time-domain vibration when n_in=3500 r/min and M_p=1500 N·m:

5 Conclusions

In this paper, a FMBD model of PGS with the flexible supports of ring and sun gears is established. The effects of the ring gear flexibility and flexible supports on the vibrations of PGS are discussed. The analysis results may give a new method to decrease the vibrations and improve the quality of PGSs. The conclusions are as follows.

Figure 9 Effect of flexible support of ring gear (K_r2) on frequency-domain vibrations when n_in=3000 r/min, M_p=500 N·m, and K_r1=1×10⁶ N/mm:

Table 4 Characteristic frequencies of PGS when n_in=3000 r/min

1) The flexible supports of sun and ring gears can affect the vibrations of PGSs. In general, the vibration magnitude of PGS decreases with the increment of the flexible supports. The effect of the flexible support of ring gear is bigger than that of sun gear. The results show that the effect of the input speed on the vibrations of PGS is bigger than that of the moment.

2) The ring gear flexibility can affect the vibrations of PGS too. The FMBD model’s statistic parameters, such as RMS, MAX, and PTP, are bigger than those of RMBD model. The magnitude of the vibrations from FMBD model is larger than that from RMBD model according to the RMS. The effects of the flexible supports on the vibration waveforms from FMBD and RMBD models are similar.

Figure 10 Effect of flexible support of sun gear (K_r1) on frequency-domain vibrations when n_in=3000 r/min, M_p=500 N·m, and K_r2=1×10⁶ N/mm:

Table 5 Peak frequencies of FMBD and RMBD models with different flexible support cases when n_in=3000 r/min, M_p=500 N·m

3) The FMBD and RMBD models have the same peak frequencies f_m-f_s and 2f_m-2f_s-2f_c. The amplitude of peak frequency in the spectrum is affected by the flexible supports. The changes of peak frequencies for the FMBD and RMBD models caused by different flexible support cases are similar. The FMBD model’s peak frequency amplitude is bigger than that of RMBD model.

In the current study, the simulation results show that FMBD and RMBD models have a big difference on time-domain and frequency-domain vibrations. The purpose of this paper is to study the influence factors-support stiffness and flexible deformation of ring gear. The confirmatory experiment will be done in in the future study.

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(Edited by YANG Hua)

中文导读

柔性齿圈支承刚度对行星齿轮系统振动特性的影响规律

摘要：行星齿轮系统是传动机构的关键部件之一。齿轮箱和传动轴的结构和材料特性会影响行星齿轮系统的支承刚度和振动特性。齿圈的柔性也会影响行星轮系的振动特性。然而，当前的研究大多没有同时考虑齿圈的柔性以及齿圈和太阳轮支承刚度对行星轮系振动特性的影响规律。本文建立了考虑弹性支承的柔性齿圈行星齿轮系统的刚柔耦合多体动力学模型。采用有限元方法，计算获得了齿圈的柔性变形。此外，该模型还考虑了轴承游隙及阻尼对行星齿轮系统振动特性的影响。研究了不同力矩和转速工况下弹性支承和齿圈的柔性对行星齿轮系统振动特性的影响规律。对所提出的刚柔耦合模型和已有的刚性多体动力学模型的获得的计算结果进行了对比分析。结果表明，弹性支承的柔性齿圈对行星齿轮系统的振动特性影响较大。本文的研究成果对行星齿轮系统支承结构的设计和振动控制具有一定的指导意义。

关键词：行星轮；刚柔耦合；弹性支承；齿圈柔性；振动特性

Foundation item: Projects(51605051, 51975068) supported by the National Natural Science Foundation of China

Received date: 2020-03-05; Accepted date: 2020-05-22

Corresponding author: LIU Jing, PhD, Professor; Tel: +86-13658335960; E-mail: jliu@cqu.edu.cn; ORCID: https://orcid.org/0000- 0003-2323-3475