J. Cent. South Univ. (2012) 19: 1543-1547

DOI: 10.1007/s11771-012-1174-3

Stability of motion state and bifurcation properties of planetary gear train

LI Tong-jie(李同杰)^1,2, ZHU Ru-peng(朱如鹏)¹, BAO He-yun(鲍和云)¹, XIANG Chang-le(项昌乐)³

1. College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics,Nanjing 210016, China;

2. College of Engineering, Anhui Science and Technology University, Fengyang 233100, China;

3. School of Mechanical and Vehicular Engineering, Beijing Institute of Technology, Beijing 100081, China

? Central South University Press and Springer-Verlag Berlin Heidelberg 2012

Abstract: A nonlinear lateral–torsional coupled vibration model of a planetary gear system was established by taking transmission errors, time varying meshing stiffness and multiple gear backlashes into account. The bifurcation diagram of the system’s motion state with rotational speed of sun gear was conducted through four steps. As a bifurcation parameter, the effect of rotational speed on the bifurcation properties of the system was assessed. The study results reveal that periodic motion is the main motion state of planetary gear train in low speed region when n_s<2 350 r/min, but chaos motion state is dominant in high speed region when n_s>2 350 r/min, The way of periodic motion to chaos is doubling bifurcation. There are two kinds of unstable modes and nine unstable regions in the speed region when 1 000 r/mins<3 000 r/min.

Key words: planetary gear train; nonlinear dynamical model; stability of motion state; bifurcation properties

1 Introduction

Planetary gear systems have been widely used in engineering owing to their advantages, such as small space required, large ratio of transmission and high efficiency. The dynamic model of a planetary gear system is complicated and inherently nonlinear owing to the multiple clearances, transmission errors and time varying meshing stiffness. Over the past three decades, the dynamics of planetary transmissions has been paid much attention, but almost all of the published studies on this field were only focused on their linear vibration property [1-5]. Literatures about nonlinear dynamics of a planetary transmission system are still limited. AUGUST and KASUBA [6] numerically computed dynamic responses to mesh stiffness variations for planetary gears with three sequentially phased planets, and the results showed the dynamic impact of mesh stiffness variation on dynamic response, tooth loads, and load sharing among planets. LIN and PARKER [7] derived simple formulate that allows designers to suppress particular instabilities by properly selecting contact ratios and mesh phasing of a planetary gear set. AMBARISHA and PARKER [8] proved the correctness of lumped parameter model for a planetary gear set by comparing calculation result of the lumped parameter model with the finite element model. KAHRAMAN [9] took the possibility of tooth separation in a planetary gear system into account and used a step function in nonlinear model to distinguish the tooth contact and the tooth separation roughly. LIN and PARKER [10] analytically identified the operating conditions leading to parametric instability and obtained the boundaries separating stable and unstable conditions. SUN and HU [11] established a lateral-torsional coupled model considering multiple backlashes, time-varying mesh stiffness, error excitation and sun-gear shaft compliance, and the solutions were determined by using harmonic balance method from the equations in matrix form.

Motion stability and bifurcation are the important targets for the dynamic research of nonlinear system. The bifurcation property of a system can provide important information for a designer, such as stable parameter region, proper working speed, and the way to chaos. However, there is still less study on this aspect. The objective of this work is to analyze the stability of motion state and the bifurcation properties of a planetary gear train’s motion state with rotational speed based on a nonlinear lateral–torsional coupled vibration model with multiple backlashes, time-varying mesh stiffness and error excitation.

2 Dynamic model of system

In this work, the planetary gear system concerned is a single-stage 2K-H type planetary gear, as shown in  Fig. 1. The system consists of a sun gear (S), a ring gear (R), a planet gears p and a carrier (C). The translational freedom of sun gear and rotational freedoms of every part are considered in this model. The lateral displacement of sun gear in horizontal direction is x_s, displacement in vertical direction is y_s, rotational motions of the ring, carrier, sun, and planets are denoted by θ_h, h=r, c, s, p₁, p₂, …, p_n, where n indicates the number of planets: There exists θ_r=0 since the ring gear is fixed on gearbox. The character r_h, h=s, r, p₁, p₂, …, p_n, stand for the base circle radius for the sun, ring, planets, and  means the radius of the circle passing through the planet centers for the carrier, respectively. Parameters , , and denote the backlash, the time-varying mesh stiffness, the viscous damping and the static transmission error of the gear-planet gear-i pair, respectively; , , and denote the corresponding terms in the ring gear-planet gear-i pair, respectively. This nomenclature is also applied to other physical parameters of the corresponding mesh pair. Parameters k_x, k_y, c_x and c_y denote the bending stiffness and damping coefficient of the sun gear’s flexible shaft in the horizontal and vertical directions, respectively.

Fig. 1 Dynamic model of planetary gear set

2.1 Time-varying mesh stiffness

For spur gears, rectangular waves are often used to approximate mesh stiffness alternating between d and d+1 pairs of teeth in contact [10]. Expanding rectangular waves in Fourier series and taking the first harmonic term of the system’s mesh frequency, the time-varying mesh stiffness in corresponding mesh pair can be assumed as

              (1)

where  and are mean values of the stiffness in the corresponding meshing pair, respectively; and  are time-varying amplitudes of the stiffness in the corresponding meshing pair; andare initial phases of the stiffness in the corresponding meshing pair; ω is the system’s mesh frequency.

2.2 Static transmission errors

The static transmission error is one of the primary excitations of planetary gear train’s vibration and noise. According to Ref. [12], the static transmission errors in the corresponding mesh pair and can be expressed as

               (2)

where and are amplitudes of the static transmission error in the corresponding meshing pair; and  are initial phases of the static transmission error in the corresponding meshing pair; γ_sr denotes the phase between the sun-planet and ring-planet mesh error for a given planet.

2.3 Nonlinear displacement function about backlash

According to Ref. [11], the nonlinear displacement function about backlash is

                     (3)

where X denotes the relative gear mesh displacement and b denotes half backlash in the corresponding mesh pair.

2.4 Equations of motion

In order to eliminate rigid body motion and simplify equations of the system, N (in this work, N=2n) dimension equivalent displacements coordinate  and  are introduced according to the model shown in Fig. 1.

      (4)

where α is the pressure angle; φ_i means the position angle of the i-th planet gear with respect to axis x.

The dimensionless lateral–torsional motion equations of the planetary gear system can be established by using the Lagrange principle and equivalent displacements coordinate (Eq. (4)), and introducing some dimensionless parameters:

(5)

where         

; ; , and

Parameter b_c is a character length; J_s, and  J_c  are the inertia of sun gear, the i-th planet gear and the carrier, respectively; M_s is the mass of sun gear.

3 Numerical results and discussion

The following parameters are adopted in the numerical simulation of the planetary gear set shown in Fig. 1: gear module m=5.0 mm; pressure angle α=20°; the static transmission error  μm; teeth number of the corresponding gear z_s=15, , z_r=63; the number of planet gear in every stage n=3, mean values of the stiffness in the corresponding meshing pair =0.825 6×10⁹ N/m, =1.06× 10⁹ N/m, input power of the system P=200 kW, and the rotational speed of sun gear n_s r/min is regarded as bifurcation parameter.

3.1 Bifurcation diagram with n_s

The bifurcation diagram of the system’s motion state with n_s can be made through the following steps:

Step 1: Obtain the asymptotically stable numerical solutions of system by using the fixed-step fourth order Runge-Kutta method (ODE45) to solve Eq. (5).

Step 2: Let the Poincare section of         system is, where τ=mod(2π/ω) denotes the value of the i-th state variable which has a determined phase in a cycle of external excitation force, and the system’s Poincare map at a determined speed can be obtained.

Step 3: Change the speed constantly and repeat the Step 2 in order to obtain the system’s Poincare map at different speeds.

Step 4: Assemble all of the Poincare maps calculated in Step 3 from low speed to high speed.

Figure 2 shows the bifurcation diagram of the system’s motion state with n_s that is made according to the above four steps.

Fig. 2 Bifurcation diagrams of system’s motion state (n_s: bifurcation parameter)

3.2 Way to chaos

It can be seen from Fig. 2 that periodic motion is the system’s main motion state before n_s=2 350 r/min and it just appears non-periodic when the range of n_s is from 2 189 r/min to 2 285 r/min temporarily. After n_s=2 350 r/min, it appears chaos through period doubling bifurcation. Figures 3-8 show the process of period doubling bifurcation. The three sub-maps in each map are the Poincare map, the relative displacement curve and the phase diagram, respectively.

Fig. 3 Period-1 motion when n_s=2 245 r/min: (a) Poincare map; (b) Relative displacement curve; (c) Phase diagram

Fig. 4 Period-2 motion when n_s=2 290 r/min: (a) Poincare map; (b) Relative displacement curve; (c) Phase diagram

Fig. 5 Period-4 motion when n_s=2 330 r/min: (a) Poincare map; (b) Relative displacement curve; (c) Phase diagram

Fig. 6 Period-8 motion when n_s=2 343 r/min: (a) Poincare map; (b) Relative displacement curve; (c) Phase diagram

Fig. 7 Period-16 motion when n_s=2 345.5 r/min: (a) Poincare map; (b) Relative displacement curve; (c) Phase diagram

Fig. 8 Chaos motion when n_s =2 400 r/min: (a) Poincare map; (b) Relative displacement curve; (c) Phase diagram

3.3 Unstable speed region

According to the figures shown in above section, the system’s bifurcation points of motion state can be determined, which are 2 070, 2 100, 2 148, 2 278, 2 299, 2 340 and 2 343 r/min. The working speed that is close to those bifurcation points will be unstable, because a small disturbance of the working speed will lead to the sudden change of the motion state of system. This kind of unstable mode can be called motion state instability.

When the speed is close to 1 480 r/min, the vibration intensity of system is unstable because the vibration amplitude changes from small to large although the motion state is still stable. The situation is same when the speed is 1 670 r/min because system’s vibration amplitude changes from large to small at this speed. This kind of unstable mode can be called vibration intensity instability.

In a word, there are two kinds of unstable    modes and nine unstable regions when the speed region is 1 000 r/mins<3 000 r/min.

4 Conclusions

1) Periodic motion is the planetary gear train’s main motion state in low speed region n_s<2 350 r/min, but chaos motion state is dominant in high speed region n_s> 2 350 r/min. The way of periodic motion to chaos is doubling bifurcation.

2) There are two kinds of unstable modes, which are motion state unstable mode and vibration intensity unstable mode, and nine unstable regions when the speeds are 1 480, 1 670, 2 070, 2 100, 2 148, 2 278, 2 299, 2 340 and 2 343 r/min.

References

[1] HIDAKA T. Analysis of dynamic tooth load on planetary gear [J]. Bulletin of the Japanese Society of Mechanical Engineers, 1980, 23(176): 315-322.

[2] PARKER R G, AGASHE V, VIJAYAKAR S M. Dynamic response of a planetary gear system using a finite element/contact mechanics model [J]. ASME, Journal of Mechanical Design, 2000, 122(9): 304-310.

[3] LIN J, PARKER R G. Analytical characterization of the unique properties of planetary gear free vibration [J]. ASME, Journal of Vibration and Acoustics, 1999, 121(7): 316-321.

[4] SAADA A, VELEX P. An extended model for the analysis of the dynamic behavior of planetary trains [J]. ASME, Journal of Mechanical Design, 1995, 117(6): 241-247.

[5] KAHRAMAN A. Planetary gear train dynamics [J]. ASME, Journal of Mechanical Design, 1994, 116(3): 713-720.

[6] AUGUST R, KASUBA R. Torsional vibration and dynamic loads in a basic planetary gear system [J]. Journal of vibration, Acoustics, Stress, and Reliability in Design, 1986, 108(7): 348-353.

[7] LIN J, PARKER R G. Mesh stiffness variation instabilities in two-stage gear system [J]. Journal of Vibration and Acoustics, 2001, 124(6): 316-321.

[8] AMBARISHA V K, PARKER R G. Nonlinear dynamics of planetary gears using analytical and finite element models [J]. Journal of Sound and Vibration, 2007, 302(1): 577-595.

[9] KAHRAMAN A. Load sharing characteristics of planetary transmissions [J]. Mechanism and Machine Theory, 1994, 29(8): 1151-1165.

[10] LIN J, PARKER R G. Planetary gear parametric instability caused by mesh stiffness variation [J]. Journal of Sound and Vibration, 2002, 249(1): 129-145.

[11] SUN Tao, HU Hai-yan. Nonlinear dynamics of a planetary gear system with multiple clearances [J]. Mechanism and Machine Theory, 2003, 38(5): 1371-1390.

[12] PARKER R G. A physical explanation for the effectiveness of planet phasing to suppress planetary gear vibration [J]. Journal of Sound and Vibration, 2000, 236(4): 561-573.

(Edited by DENG Lü-xiang)

Foundation item: Project(50775108) supported by the National Natural Science Foundation of China

Received date: 2011-03-23; Accepted date: 2011-06-30

Corresponding author: ZHU Ru-peng, Professor, PhD; Tel: +86-25-84892500; E-mail: rpzhu@nuaa.edu.cn