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Abstract

Planter gear system is one of the critical components of various industrial transmission systems. In general, the ring gear is elastically fixed with the gearbox. The gearbox materials and their assembly relationships will affect the support stiffness of the ring gear and system vibrations. In this paper, a multi-body dynamic model for a planetary gear system with the elastic support of ring gear is developed to discuss the influence of the radial support stiffness of ring gear on the system vibrations. The planet bearings are also considered in the multi-body dynamic model. The rotational speed of the planet gear and carrier from the simulation and theoretical results are compared to validate the developed multi-body dynamic model. The influences of the radial support stiffness of the ring gear, carrier moment, and sun gear speed on the time- and frequency-domain vibrations of the planetary gear system are analyzed. The results denote that the waveform and amplitude of the time-domain vibration of the ring gear are greatly affected by the radial support stiffness of ring gear as well as the peak frequency amplitude and its sidebands. The peak frequency in the spectrum of ring gear is slightly affected by the radial support stiffness. It indicates that this study can give some guidance for the vibration control approaches for the planetary gear systems.

Keywords

Support stiffness ring gear vibrations planetary gear

Introduction

Planetary gear system is one of the critical components of various industrial transmission systems, such as aero-engines and wind turbine gearboxes. The dynamic performances of the planetary gear system have a large influence on vibrations of the transmission systems. Moreover, the manufacturing errors and faults in the planetary gear system will greatly affect the vibrations of the transmission systems.^1–3 Thus, vibration analysis of the planetary gear system is very helpful for understanding the vibrations and noise of the transmission systems.

Many works have developed various methods including the lumped parameter modeling, finite element (FE) modeling, multibody dynamic (MBD) modeling, and experimental methods for studying the vibrations of planetary gear systems.^4,5 For example, Ambarisha and Parker⁶ and Kahraman et al.⁷ introduced lumped-parameter and FE models to study the vibrations of a planetary gear system. Lethé et al.,⁸ Halsen et al.,⁹ Xing and Moan,¹⁰ Helsen et al.,¹¹ Jin et al.,¹² and Li et al.¹³ established different MBD models to analyze the vibrations of the planetary gearbox system. Guo and Parker¹⁴ presented a nonlinear dynamic model considering the bearing clearance and tooth backlash to discuss the nonlinear vibrations. Guo and Parker¹⁵ developed a lumped-parameter model considering bearing clearances, back-side contact, tooth separation, and tooth wedging to analyze the translational vibrations of planetary gear system. Kim et al.¹⁶ studied the influences of the bearing deformations on the time-varying contact ratios and pressure angles of the planetary gear system as well as the vibrations. Chen et al.^17,18 proposed a new method to study the effect of ring gear flexibility on the dynamic performance of planetary gear transmission set with a thin ring gear rim. Although various analytical, FE, and MBD methods have been utilized to describe the vibrations of planetary gear system, the influence of the support stiffness of ring gear on the vibrations of planetary gear system was not discussed in the above literature. However, in practice, the support stiffness of ring gear has a great influence on the system vibrations.¹⁹ On the other hand, Wu and Parker¹⁹ and Liu et al.²⁰ studied the support stiffness of the ring gear on its vibrations. They only formulated the single ring gear models. The dynamic model for the planetary gear system considering both the support stiffness of the ring gear and planet bearings was not reported in the literature. Thus, the support stiffness of ring gear should be modeled for vibration analysis of the planetary gear systems, which is the goal of this paper.

In this paper, an MBD model for a planetary gear system with the elastic support of the ring gear is developed to discuss the influence of the radial support stiffness of the ring gear on the system vibrations. The planet bearings with the radial clearance are also considered in the MBD model. The meshing stiffness and damping for the meshing gears and bearing components are considered in the MBD model. A Coulomb friction model is utilized to model the frictions between the mating components of the system. A commercial MBD analysis software is applied to solve the developed MBD model. The rotational speed of the planet gear and carrier from the simulation and theoretical results are compared to validate the developed MBD model. The influences of the radial support stiffness of the ring gear, carrier moment, and sun gear speed on the time- and frequency-domain vibrations of the planetary gear system are analyzed.

Problem description

A planetary gear system with the elastic support of the ring gear and planet bearings is plotted in Figure 1. In general, the ring gear is elastically fixed with the gearbox. The gearbox materials and the assembly relationship will affect the radial support stiffness of the ring gear. As the results in Wu and Parker,¹⁹ the dynamics of the ring gear can be greatly influenced by the radial support stiffness of the ring gear. The dynamics of the ring gear will affect the system vibrations.

Figure 1.

Schematics of a planetary gear system.

To analyze the influence of the radial support stiffness of the ring gear on the vibrations of the system, a radial stiffness ratio is defined as

η = \frac{K_{r i}}{K_{r}}

(1)

where K_r is the reference radial stiffness and K_r _i is the designed radial stiffness. The simulation cases are given in Table 1, where n_in is the input speed of the sun gear and M_p is the moment of the carrier. Moreover, the circumferential torsion stiffness is 1.67 × 10⁵ N/mm.

Table 1.

Radial stiffness ratios in the simulation cases.

K_r (N/mm)	Stiffness ratio η
1.65 × 10⁶	0.5
3.3 × 10⁶	1
6.6 × 10⁶	2
13.2 × 10⁶	4

Model formulation

The dynamic model for a planetary gear system is depicted in Figure 2. The model includes three planet gears, three planet bearings, one carrier, one ring gear, and one sun gear. The input speed and the moment are applied on the sun gear and carrier, respectively. The contacts between the gears and those in the planet bearings are modeled too. The mesh stiffnesses between the gears from equations (2) and (3) are considered in the MBD model as well as the contact stiffness between the planet-bearing components. The planet gears, planet bearings, carrier, ring gear, and sun gear are considered as rigid bodies. The discrete radial support stiffness and circumferential torsion stiffness of the ring gears are considered in the MBD model as shown in Figure 2.

Figure 2.

A dynamic model for a planetary gear system.

Based on the potential energy principle, the total equivalent mesh stiffness between the sun and planet gear is calculated by^21–23

\frac{1}{K_{eo}} = \frac{1}{K_{b1}} + \frac{1}{K_{s1}} + \frac{1}{K_{a1}} + \frac{1}{K_{b2}} + \frac{1}{K_{s2}} + \frac{1}{K_{a2}} + \frac{1}{K_{h}}

(2)

where k_b, k_s, k_a, and k_h are bending, shear, axial compressive, and Hertzian contact stiffnesses, respectively. In equation (2), the subscript 1 denotes the sun gear and subscript 2 denotes the planet gear. The total equivalent mesh stiffness between the planet gear and ring gear is as follows^21–23

\frac{1}{K_{ei}} = \frac{1}{K_{b1}} + \frac{1}{K_{s1}} + \frac{1}{K_{a1}} + \frac{1}{K_{f1}} + \frac{1}{K_{b2}} + \frac{1}{K_{s2}} + \frac{1}{K_{a2}} + \frac{1}{K_{h}}

(3)

where k_f is the fillet-foundation stiffness. In equation (3), the subscript 1 denotes the planet gear and subscript 2 denotes the ring gear. The details for the stiffnesses in equations (2) and (3) are given in Tian,²¹ Chen and Shao,^22,23 and Ma.^24,25 In these references, Chen and Shao^22,23 proposed the method for the involvement of the tooth fillet-foundation in the gear mesh stiffness calculation based on the potential energy principle and then applied to the gear mesh stiffness calculation of railway locomotive gear transmissions.^26,27 The bending stiffness K_b is

\frac{1}{K_{b}} = \int_{0}^{d} \frac{{(x \cos α_{1} - h \sin α_{1})}^{2}}{E I_{x}} d x

(4)

where x, h, α₁, and d are the distance between the mesh point and the a piece with the width of dx, the distance between the mesh point and the tooth center line, the pressure angle at the applied load point, and the distance between the mesh point and the root circle, respectively, and E and I_x are the Young’s modulus and the area moment of inertia where the distance between the section and the acting point of the applied force is x. The shear stiffness K_s is

\frac{1}{K_{s}} = \int_{0}^{d} \frac{1.2 \cos^{2} α_{1}}{G A_{x}} d x

(5)

where G_x and A_x are the shear Young’s modulus and the area of the section where the distance between the section and the acting point of the applied force is x. The axial stiffness K_a is

\frac{1}{K_{a}} = \int_{0}^{d} \frac{\sin^{2} α_{1}}{E A_{x}} d x

(6)

The Hertzian contact stiffness K_h is

K_{h} = \frac{4 (1 - ν^{2})}{π Ew}

(7)

where v is the Poisson’s ratio and w is the tooth width. Here, the calculation method for the K_f is complex and is given in Chen and Shao.²² The contact damping for the gears is given by²⁸

c_{max} = 2 ζ \sqrt{\frac{K I_{1} I_{2}}{I_{1} R_{2}^{2} + I_{2} R_{1}^{2}}}

(8)

where ζ is the damping ratio, which is 0.1, K is the mesh stiffness, I₁ and I₂ are the inertia of the meshing gears, and R₁ and R₂ are the radii of the meshing gears.

The contact stiffness between the ball and races is as follows^29,30

K_{br} = {(\frac{π^{2} e^{2} E_{eq}^{2} ε}{4.5 τ^{3} \sum ρ})}^{0.5}

(9)

where E_eq is equivalent modulus of elasticity, ∑ρ is the curvature sum, e is the elliptical eccentricity parameter, and τ and ε are the first and second kind of elliptical integral. The details for the parameters in equation (9) are discussed in Harris and Kotzalas.²⁹ Furthermore, the bearing damping is written as

c_{br} = λ K_{br}

(10)

where the range of λ is from 0.25 to 2.5 × 10⁻⁵.^31,32

Moreover, the governing equation of the MBD model is defined as

M \ddot{q} + (C + C^{'}) \dot{q} + (K + K^{'}) q = T + E + F

(11)

where M, K, and C are the mass, support stiffness, and support damping matrices for the MBD model of the planetary gear system, respectively;

K^{'}

and

C^{'}

are the contact stiffness and damping matrices, respectively; q is the generalized coordinates vectors for the system; T is the external system load vector; E is the fault excitation vector; and F is the internal system load vector. Moreover, the friction force between the gears is included in the MBD model. The friction force is included in the internal system load vector F.

Results and discussion

The simulation cases are given in Table 2, where n_in is the input speed of the sun gear and M_p is the moment of the carrier. The dynamic model in “Model formulation” section is solved in a commercial MBD analysis software MSC. ADAMS. The influence of the radial support stiffness of the ring gear is given in the following sections. Tables 3 and 4 give the geometrics for the planetary gear system and planet bearing, respectively. According to the parameters, the meshing stiffness and damping between sun and planet gears are 4.71 × 10⁵ N/mm and 86.62 N s/mm, respectively, and those between the planet and ring gears are 9.28 × 10⁵ N/mm and 157.59 N s/mm, respectively. The contact stiffness and damping between the ball and inner race are 2.49 × 10⁷ N/mm and 6228 N s/mm, respectively, and those between the ball and outer race are 1.94 × 10⁷ N/mm and 4858 N s/mm, respectively. The time-varying mesh stiffness is calculated in equations (2) and (3). In this paragraph, only the maximum meshing stiffness of the single tooth meshing case is given. The meshing stiffness for the multi-tooth meshing cases is determined by the contact relationships between the teeth of gears in the MBD model. This method means that a rectangular profile of total meshing stiffness between the gears is used in the MBD model.

Table 2.

Input speed and applied moment used in the simulation cases.

Simulation cases	Input speed n_in (r/min)	Applied moment M_p (Nm)
1	4000	50
2	6000	50
3	8000	50
4	6000	25
5	6000	100

Table 3.

Geometrics for the planetary gear system.

	Planet	Sun	Ring
Normal module (mm)	1.5	1.5	1.5
Number of teeth	19	32	83
Normal pressure angle (°)	22.5	22.5	22.5
Pitch radius (mm)	28.5	48	117
Face width (mm)	30	30	30
Root radius (mm)	26.68	45.11	124.5
Number of planet gears	3

Table 4.

Geometrics for the planet bearing.

Parameters	Value
Ball radius (mm)	3
Pitch radius (mm)	12.65
Diametric clearance (mm)	0.01
Number of ball	8
Number of rows	1

Table 5.

Peak frequencies and their amplitudes for case 1.

Peak frequency f (Hz)		Amplitude (m/s²)
Frequency	Value	η₁ = 0.5	η₂ = 1	η₃ = 2	η₄ = 4
f_m – 2f_s^(r) – 8f_c	821	–	0.3573	–	–
f_m – 3f_s^(r)	839	–	–	0.214	0.218
f_m – 2f_s^(r) – 3f_c	862	0.2772	–	–	–
f_m – 2f_s^(r)	903	–	0.245	0.2626	–
f_m – f_s^(r) – 3f_c	922	0.4507	–	–	–
f_m – 2f_s^(r) – f_c	932	–	–	–	0.2481
f_m – f_s^(r) – f_c	949	0.3105	0.2132	–	–
f_m – 2f_s^(r) – f_c	982	–	0.236	–	–
f_m – f_s^(r) + 2f_c	995	–	–	0.2382	–
f_m – f_c	1016	1.897	1.425	0.9162	0.4816
f_m + f_c	1041	1.488	1.072	0.5347	0.2551
f_m + 6f_c	1098	0.3856	0.2141	–	–
f_m + 6f_c	1109	–	–	0.2064	–
f_m + f_s^(r) + 2f_c	1134	0.3318	–	–	–
f_m + f_s^(r) + 4f_c	1150	–	–	–	0.1594
f_m + 2f_s^(r)	1161	0.3629	–	–	–
f_m + f_s^(r) + 6f_c	1175	–	0.2398	–	–

Remark: f_s^(r), f_o^(r), f_c, f_m, f_cg, f_b^(s), f_o, and f_i are the rotational frequency of the sun gear, rotational frequency of the planet gear, spin frequency of the carrier, meshing frequency between the sun and planet gears, rotational frequency of the planet bearing cage, spin frequency of the planet bearing ball, ball passing outer race frequency of planet bearing, and ball passing inner race frequency of planet bearing, respectively.

Model validation

To validate the developed model, the rotational speed of the planet gear and carrier from the simulation and theoretical results are compared. Here, the input speed of the sun gear is 387 r/min, and the moment applied on the carrier is 350 Nm. Figure 3 gives the simulation rotational speed of the planet gear #1 and carrier. In Figure 3, the simulation mean values for the planet gear # 1 and carrier are 1779.84 and 1116.28 r/min, respectively. According to the theortical method in Lei et al.,³³ their theoretical values are 1779.84 and 1116.28 r/min, respectivley. The above discussions may show some valiation for the developed modeling approach. Note that the fluctuation of the calculation speed of the component can be observed. The fluctuation maybe caused by some clearances or gaps between the gear pairs or in the rolling bearings. Although the fluctuation occurs in the calculation speed, the mean value of the calculation speed is a constant value as shown in Figure 3. Thus, the mean value of the calculation speed from the simulation results is compared with the theoretical results.

Figure 3.

The simulation rotational speed of (a) planet gear #1 and (b) carrier.

The influence of the radial support stiffness on the vibrations

Time-domain vibrations

Figure 4 gives the influences of the stiffness ratios on the time-domain accelerations of the ring gear under the simulation case 1 (n_in = 4000 r/min and M_p = 50 Nm). It depicts that the radial support stiffness has a large influence on the profile and amplitude of the acceleration for the ring gear. The maximum amplitude will decrease with the radial support stiffness. In Figure 4, the maximum acceleration amplitudes of the ring gear with different stiffness ratios are 68.08, 57.54, 48.67, and 40.68 m/s², respectively.

Figure 4.

Time-domain accelerations of the ring gear for simulation case 1: (a) η₁ = 0.5, (b) η₂ = 1, (c) η₃ = 2, and (d) η₄ = 4.

To evaluate the radial support stiffness of the ring gear on the time-domain vibration, three statistical parameters, such as maximum, peak to peak (PTP), and root mean square (RMS) values, are utilized. Figure 5 introduces the influence of the radial support stiffness on the RMS, maximum, and PTP values of the accelerations of the ring gear for different moments. As shown in Figure 5, the input speed is 6000 r/min. Figure 6 describes the influence of the radial support stiffness on the RMS, maximum, and PTP values of the vibrations of the ring gear for different input speeds. As shown in Figure 6, the moment is 50 Nm. As plotted in Figures 5 and 6, the maximum, PTP, and RMS values always decrease with the increase of the stiffness ratio; they always increase with the increase with the carrier moment when the input speed of the sun gear is fixed; and they always increase with the increase with the input speed of the sun gear when the carrier moment is a constant.

Figure 5.

The influence of the radial support stiffness on the (a) RMS, (b) maximum, and (c) PTP values of the accelerations of the ring gear for different moments.

Figure 6.

The influence of the radial support stiffness on the (a) RMS, (b) maximum, and (c) PTP values of the accelerations of the ring gear for different input speeds.

Furthermore, the influence of the radial support stiffness on the orbit plots of the ring gear and planet gear #1 is given in Figures 7 and 8, respectively. Since the orbit plot profiles of three planet gears are similar, only the orbit plot for planet gear #1 is shown. The orbit ranges of three planet gears are still analyzed. The orbit ranges of the system bodies are described by four range parameters a, b, c, and d. The influence of the radial support stiffness on the above four range parameters is plotted in Figure 9. It gives that the orbits of ring and planet gears are greatly influenced by the radial support stiffness, especially for that of ring gear. The orbit ranges of ring and planet gears decrease with the radial support stiffness as shown in Figure 9. The orbits for three planet gears are different, which may be produced by the clearances or elastic ring foundations, etc. As shown in Figure 9(b) to (d), although the behaviors of planet gear #1 are different from those of planet gear #2 and #3 when η is increasing, their differences are very small; the reason may be that the meshing positions of the three planet gears are different due to the vibrations caused by the clearances of the planet bearings, clearances between the gears, and the elastic support of ring gear, etc.

Figure 7.

The influence of the radial support stiffness on the orbit plots of the ring gear for (a) η = 0.5, (b) η = 1, (c) η = 2, and (d) η = 4.

Figure 8.

The influence of the radial support stiffness on the orbit plots of the planet gear #1 for (a) η = 0.5, (b) η = 1, (c) η = 2, and (d) η = 4.

Figure 9.

The influence of the radial support stiffness on the orbit ranges of (a) ring gear, (b) planet gear #1, (c) planet gear #2, and (d) planet gear #3.

Frequency-domain vibrations

Figure 10 gives the influence of the radial support stiffness on the frequency-domain vibrations of the ring gear for simulation case 1 (n_in = 4000 r/min and M_p = 50 Nm). In Figure 10(a), for η₁ = 0.5, a peak frequency f_m – f_c = 1016 Hz is observed, and there are some sidebands around the peak frequency, which include f_m – 2f_s^(r) – 3f_c = 862 Hz, f_m – f_s^(r) – 3f_c = 922 Hz, f_m – f_s^(r) – f_c = 949 Hz, f_m + f_c = 1041 Hz, f_m + 6f_c = 1098 Hz, f_m + f_s^(r) + 2f_c = 1134 Hz, and f_m + 2f_s^(r) = 1161 Hz. In Figure 10(b), for η₂ = 1, a peak frequency f_m – f_c = 1016 Hz is also observed, and the sidebands are f_m – 2f_s^(r) – 8f_c = 821 Hz, f_m – 2f_s^(r) = 903 Hz, f_m – f_s^(r) – f_c = 949 Hz, f_m + f_c = 1041 Hz, f_m + 6f_c = 1098 Hz, and f_m + f_s^(r) + 6f_c = 1175 Hz. In Figure 10(c), for η₃ = 2, a peak frequency f_m – f_c = 1016 Hz is observed too, and the sidebands include f_m – 3f_s^(r) = 839 Hz, f_m – 2f_s^(r) = 903 Hz, f_m – f_s^(r) + 2f_c = 995 Hz, f_m + f_c = 1041 Hz, and f_m + 6f_c = 1109 Hz. In Figure 10(d), for η₄ = 4, a peak frequency f_m – f_c = 101 Hz is also observed, and the sidebands are f_m – 3f_s^(r) = 842 Hz, f_m – 2f_s^(r) – f_c = 932 Hz, f_m – f_s^(r) – 3f_c = 932 Hz, f_m + f_c = 1041 Hz, and f_m + f_s^(r) + 4f_c = 1150 Hz. Table 5 gives comparisons of the peak frequency, sidebands, and their amplitudes for different stiffness ratio cases. Although the four stiffness ratio cases have a same peak frequency, as given in Table 5, their amplitudes are very different. The amplitudes for the above four cases are 1.897, 1.425, 0.9162, and 0.4861 m/s², respectively. It denotes that the peak frequency amplitude decreases with the radial support stiffness.

Figure 10.

Frequency-domain accelerations of the ring gear for simulation case 1: (a) η₁ = 0.5, (b) η₂ = 1, (c) η₃ = 2, and (d) η₄ = 4.

Since the spectrum waveforms for different simulation cases are similar and their differences are the peak frequencies, the spectrum waveforms for the simulation cases 2, 3, 4, and 5 are not shown as follows. However, their peak frequencies, sidebands, and relative amplitudes will be discussed in the next paragraphs.

Table 6 plots the influence of the radial support stiffness on the frequency-domain vibrations of the ring gear for simulation case 2 (n_in = 6000 r/min and M_p = 50 Nm). For η₁ = 0.5, a peak frequency f_m – f_c = 1532 Hz is observed, and the sidebands include f_m – f_s^(r) – 2f_c = 1405 Hz, f_m + f_c = 1569 Hz, f_m + f_s^(r) = 1641 Hz, and f_m + f_s^(r) + 3f_c = 1701 Hz. For η₂ = 1, a peak frequency f_m – f_c = 1532 Hz is also observed, and the sidebands are f_m – f_s^(r) – f_c = 1426 Hz, f_m + f_c = 1569 Hz, f_m + f_s^(r) = 1641 Hz, and f_m + f_s^(r) + 3f_c = 1701 Hz. For η₃ = 2, a peak frequency f_m – f_c = 1532 Hz is observed too, and the sidebands include f_m – f_s^(r) – 2f_c = 1405 Hz, f_m – 3f_c = 1491 Hz, f_m + f_c = 1569 Hz, f_m + f_s^(r) = 1641 Hz, and f_m + f_s^(r) + 3f_c = 1701 Hz. For η₄ = 4, a peak frequency f_m + f_c = 1569 Hz is also observed, and the sidebands are f_m – f_s^(r) – 2f_c = 1405 Hz, f_m – 3f_c = 1491 Hz, f_m – f_c = 1532 Hz, f_m + f_s^(r) = 1641 Hz, and f_m + f_s^(r) + 3f_c = 1701 Hz. Table 6 shows that the peak frequencies of the first three stiffness ratio cases are at f_m – f_c, but that of the fourth stiffness ratio case is at f_m + f_c. Although the first three stiffness ratio cases have a same peak frequency, as given in Table 6, their amplitudes are very different. The peak frequency amplitudes for the above first three cases are 2.209, 1.166, and 1.281 m/s², respectively, and that for the fourth case is 0.5774 m/s². Similarly, it shows that the peak frequency amplitude decreases with the radial support stiffness.

Table 6.

Peak frequencies and their amplitudes for case 2.

Peak frequency f (Hz)		Amplitude (m/s²)
Frequency	Value	η₁ = 0.5	η₂ = 1	η₃ = 2	η₄ = 4
f_m – f_s^(r) – 2f_c	1405	0.2636	–	0.1732	0.1604
f_m – f_s^(r) – f_c	1426	–	0.1458	–	–
f_m – 3f_c	1491	–	–	0.192	0.1324
f_m – f_c	1532	2.209	1.166	1.281	0.4915
f_m + f_c	1569	1.866	1.140	1.181	0.5774
f_m + f_s^(r)	1641	0.3265	0.1291	0.1946	0.1750
f_m + f_s^(r) + 3f_c	1701	0.2382	0.1417	0.1875	0.0935

Table 7 plots the influence of the radial support stiffness on the frequency-domain vibrations of the ring gear for simulation case 3 (n_in = 8000 r/min and M_p = 50 Nm). For η₁ = 0.5, a peak frequency f_m – f_c = 2041 Hz is observed, and the sidebands are f_m – 3f_s^(r) – 4f_c = 1565 Hz, f_m – 3f_s^(r) = 1669 Hz, f_m – 3f_c = 1963 Hz, f_m + f_c = 2090 Hz, f_m + f_s^(r) + 4f_c = 2299 Hz, and f_m + f_s^(r) + 8f_c = 2392 Hz. For η₂ = 1, a peak frequency f_m – f_c = 2041 Hz is also observed, and the sidebands include f_m – 3f_s^(r) – 4f_c = 1565 Hz, f_m – 3f_s^(r) = 1669 Hz, f_m – f_s^(r) – 5f_c = 1778 Hz, f_m – f_s^(r) = 1932 Hz, f_m + f_c = 2090 Hz, f_m + f_s^(r) + 4f_c = 2299 Hz, and f_m + f_s^(r) + 8f_c = 2392 Hz. For η₃ = 2, a peak frequency f_m – f_c = 2041 Hz is observed too, and the sidebands are f_m – 3f_s^(r) – 4f_c = 1565 Hz, f_m – 3f_s^(r) = 1669 Hz, f_m – f_s^(r) – 5f_c = 1778 Hz, f_m – f_s^(r) = 1932 Hz, f_m + f_c = 2090 Hz, and f_m + f_s^(r) + 8f_c = 2392 Hz. For η₄ = 4, a peak frequency f_m – f_c = 2041 Hz is also observed, and the sidebands include f_m – 3f_s^(r) – 4f_c = 1565 Hz, f_m – 3f_s^(r) = 1669 Hz, f_m – f_s^(r) = 1932 Hz, f_m + f_c = 2090 Hz, f_m + f_s^(r) + 4f_c = 2299 Hz, and f_m + f_s^(r) + 8f_c = 2392 Hz. Although the four stiffness ratio cases have a same peak frequency, as given in Table 7, their amplitudes are very different. The peak frequency amplitudes for the above four cases are 1.597, 1.795, 1.181, and 0.8976 m/s², respectively. Similarly, it shows that the peak frequency amplitude decreases with the radial support stiffness.

Table 7.

Peak frequencies and their amplitudes for case 3.

Peak frequency f (Hz)		Amplitude (m/s²)
Frequency	Value	η₁ = 0.5	η₂ = 1	η₃ = 2	η₄ = 4
f_m – 3f_s^(r) – 4f_c	1565	0.3073	0.2806	0.1954	0.2026
f_m – 3f_s^(r)	1669	0.2796	0.1814	0.2134	0.1347
f_m – f_s^(r) – 5f_c	1778	–	0.2528	0.192	–
f_m – f_s^(r)	1932	–	0.2318	1.281	0.146
f_m – 3f_c	1963	0.3247	–	–	–
f_m – f_c	2041	1.597	1.795	1.181	0.8976
f_m + f_c	2090	1.500	1.109	0.1875	0.5382
f_m + f_s^(r) + 4f_c	2299	0.2117	0.1907	–	0.0874
f_m + f_s^(r) + 8f_c	2392	0.2999	0.1434	–	0.1009

Table 8 plots the influence of the radial support stiffness on the frequency-domain vibrations of the ring gear for simulation case 4 (n_in = 6000 r/min and M_p = 25 Nm). For η₁ = 0.5, a peak frequency f_m – f_c = 1532 Hz is observed, and the sidebands include f_m – 2f_s^(r) – 2f_c = 1311 Hz, f_m – f_s^(r) – 2f_c = 1411 Hz, f_m – 4f_c = 1445 Hz, f_m + f_c = 1569 Hz, f_m + 4f_c = 1656 Hz, and f_m + f_s^(r) + 2f_c = 1697 Hz. For η₂ = 1, a peak frequency f_m – f_c = 1532 Hz is also observed, and the sidebands are f_m – 2f_s^(r) – 2f_c = 1311 Hz, f_m – f_s^(r) – 2f_c = 1411 Hz, f_m + f_c = 1569 Hz, f_m + 4f_c = 1656 Hz, and f_m + f_s^(r) + 2f_c = 1697 Hz. For η₃ = 2, a peak frequency f_m – f_c = 1532 Hz is observed too, and the sidebands include f_m – 2f_s^(r) – 2f_c = 1311 Hz, f_m – f_s^(r) – 2f_c = 1411 Hz, f_m + f_c = 1569 Hz, and f_m + f_s^(r) + 2f_c = 1697 Hz. For η₄ = 4, a peak frequency f_m – f_c = 1532 Hz is also observed, and the sidebands are f_m – 2f_s^(r) – 2f_c = 1311 Hz, f_m – f_s^(r) – 2f_c = 1411 Hz, f_m + f_c = 1569 Hz, f_m + 4f_c = 1656 Hz, and f_m + f_s^(r) + 2f_c = 1697 Hz. Although the four stiffness ratio cases have a same peak frequency, as given in Table 8, their amplitudes are very different. The peak frequency amplitudes for the above four cases are 1.87, 1.887, 1.408, and 0.6242 m/s², respectively. Similarly, it shows that the peak frequency amplitude decreases with the radial support stiffness.

Table 8.

Peak frequencies and their amplitudes for case 4.

Peak frequency f (Hz)		Amplitude (m/s²)
Frequency	Value	η₁ = 0.5	η₂ = 1	η₃ = 2	η₄ = 4
f_m – 2f_s^(r) – 2f_c	1311	0.3628	0.2176	0.2517	0.1167
f_m – f_s^(r) – 2f_c	1411	0.2927	0.2245	0.1411	0.1088
f_m – 4f_c	1445	0.3385	–	0.1057	–
f_m – f_c	1532	1.8700	1.8870	1.408	0.6242
f_m + f_c	1569	1.4050	1.432	0.997	0.478
f_m + 4f_c	1656	0.3053	0.275	–	0.098
f_m + f_s^(r) + 2f_c	1697	0.2780	0.261	0.1253	0.1669

Table 9 plots the influence of the radial support stiffness on the frequency-domain vibrations of the ring gear for simulation case 2 (n_in = 6000 r/min and M_p = 100 Nm). For η₁ = 0.5, a peak frequency f_m – f_c = 2041 Hz is observed, and the sidebands are f_m – 2f_s^(r) – 2f_c = 1311 Hz, f_m – f_s^(r) – 2f_c = 1411 Hz, f_m – 4f_c = 1445 Hz, f_m + f_c = 1569 Hz, f_m + 4f_c = 1656 Hz, f_m + f_s^(r) + 2f_c = 1697 Hz, and f_m + 2f_s^(r) = 1748 Hz. For η₂ = 1, a peak frequency f_m – f_c = 2041 Hz is also observed, and the sidebands include f_m – 2f_s^(r) – 2f_c = 1311 Hz, f_m – f_s^(r) – 2f_c = 1411 Hz, f_m – 4f_c = 1445 Hz, f_m + f_c = 1569 Hz, f_m + 3f_c = 1625 Hz, f_m + 4f_c = 1656 Hz, and f_m + f_s^(r) + 2f_c = 1697 Hz. For η₃ = 2, a peak frequency f_m – f_c = 2041 Hz is observed too, and the sidebands are f_m – 2f_s^(r) – 2f_c = 1311 Hz, f_m – f_s^(r) – 2f_c = 1411 Hz, f_m + f_c = 1569 Hz, f_m + 3f_c = 1625 Hz, f_m + 4f_c = 1656 Hz, and f_m + f_s^(r) + 2f_c = 1697 Hz. For η₄ = 4, a peak frequency f_m – f_c = 2041 Hz is also observed, and the sidebands include f_m – 2f_s^(r) – 2f_c = 1311 Hz, f_m – f_s^(r) – 2f_c = 1411 Hz, f_m + f_c = 1569 Hz, f_m + 4f_c = 1656 Hz, and f_m + f_s^(r) + 2f_c = 1697 Hz. Although the four stiffness ratio cases have a same peak frequency, as given in Table 9, their amplitudes are very different. The peak frequency amplitudes for the above four cases are 2.368, 1.922, 1.495, and 0.7283 m/s², respectively. Similarly, it shows that the peak frequency amplitude decreases with the radial support stiffness.

Table 9.

Peak frequencies and their amplitudes for case 5.

Peak frequency f (Hz)		Amplitude (m/s²)
Frequency	Value	η₁ = 0.5	η₂ = 1	η₃ = 2	η₄ = 4
f_m – 2f_s^(r) – 2f_c	1311	0.3798	0.2167	0.2187	0.2627
f_m – f_s^(r) – 2f_c	1411	0.2483	0.1954	0.2603	0.1811
f_m – 4f_c	1445	0.3385	0.1744	0.1950	–
f_m – f_c	1532	2.3680	1.9220	1.4950	0.7283
f_m + f_c	1569	2.0300	1.4670	1.1880	0.5714
f_m + 4f_c	1656	0.2809	0.2682	0.1913	0.1487
f_m + f_s^(r) + 2f_c	1697	0.2134	0.2490	0.1784	0.1565
f_m + 2f_s^(r)	1748	0.3083	–	–	–

In this section, the sidebands around the peak frequency are clearly observed in the spectrum of the ring gear. The sidebands around the peak frequency include f_m – f_s^(r) – 2f_c, f_m + f_c, f_m + f_s^(r), f_m + f_s^(r) + 3f_c, f_m – f_s^(r) – f_c, f_m + f_s^(r), f_m + f_s^(r) + 3f_c, and f_m – 3f_c. It shows that the vibrations of planetary gear system are mainly caused by the meshing vibration between sun gear and planet gear, the rotational vibrations of sun gear, and the spin vibrations of carrier. It seems that the passing effects of planet gears will mainly modulate the amplitude of meshing vibration of planetary gears.

Conclusions

This paper develops a MBD model for a planetary gear system with an elastic ring gear support to discuss the influence of the radial support stiffness of the ring gear on the system vibrations. The rotational speed of the planet gear and carrier from the simulation and theoretical results are compared to validate the developed MBD model. The influences of the radial support stiffness of the ring gear, carrier moment, and sun gear speed on the time- and frequency-domain vibrations of the planetary gear system are analyzed. The following conclusions are given as

The waveform and amplitude of the time-domain vibration of the ring gear are greatly affected by the radial support stiffness of the ring gear as well as the peak frequency amplitude and its sidebands.

The maximum, PTP, and RMS values always decrease with the increase of the stiffness ratio; they always increase with the increase with the carrier moment when the input speed of the sun gear is fixed, and they always increase with the increase with the input speed of the sun gear when the carrier moment is a constant.

The peak frequency (f_m – f_c) in the spectrum of the ring gear is slightly affected by the radial support stiffness. The peak frequency amplitude decreases with the radial support stiffness. The sidebands around the peak frequency include f_m – f_s^(r) – 2f_c, f_m + f_c, f_m + f_s^(r), f_m + f_s^(r) + 3f_c, f_m – f_s^(r) – f_c, f_m + f_s^(r), f_m + f_s^(r) + 3f_c, and f_m – 3f_c.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The author(s) received financial support from the National Natural Science Foundation of China (No. 51605051), Chongqing Research Program of Basic Research and Frontier Technology (No. cstc2017jcyjAX0202), and Chongqing Special Key R&D Projects for Technology Innovation of Key Industries (No. cstc2017zdcy-zdyfX0101) for the research, authorship, and/or publication of this article.

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Effect of the radial support stiffness of the ring gear on the vibrations for a planetary gear system

Abstract

Keywords

Introduction

Problem description

Model formulation

Results and discussion

Model validation

The influence of the radial support stiffness on the vibrations

Time-domain vibrations

Frequency-domain vibrations

Conclusions

Footnotes

Declaration of conflicting interests

Funding

References