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 ns<2 350 r/min, but chaos motion state is dominant in high speed region when ns>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 xs, displacement in vertical direction is ys, rotational motions of the ring, carrier, sun, and planets are denoted by θh, h=r, c, s, p1, p2, …, pn, where n indicates the number of planets: There exists θr=0 since the ring gear is fixed on gearbox. The character rh, h=s, r, p1, p2, …, pn, 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 kx, ky, cx and cy 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; γsr 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; φi 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 bc is a character length; Js, and Jc are the inertia of sun gear, the i-th planet gear and the carrier, respectively; Ms 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 zs=15, , zr=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 ns r/min is regarded as bifurcation parameter.
3.1 Bifurcation diagram with ns
The bifurcation diagram of the system’s motion state with ns 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 ns that is made according to the above four steps.
Fig. 2 Bifurcation diagrams of system’s motion state (ns: 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 ns=2 350 r/min and it just appears non-periodic when the range of ns is from 2 189 r/min to 2 285 r/min temporarily. After ns=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 ns=2 245 r/min: (a) Poincare map; (b) Relative displacement curve; (c) Phase diagram
Fig. 4 Period-2 motion when ns=2 290 r/min: (a) Poincare map; (b) Relative displacement curve; (c) Phase diagram
Fig. 5 Period-4 motion when ns=2 330 r/min: (a) Poincare map; (b) Relative displacement curve; (c) Phase diagram
Fig. 6 Period-8 motion when ns=2 343 r/min: (a) Poincare map; (b) Relative displacement curve; (c) Phase diagram
Fig. 7 Period-16 motion when ns=2 345.5 r/min: (a) Poincare map; (b) Relative displacement curve; (c) Phase diagram
Fig. 8 Chaos motion when ns =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 ns<2 350 r/min, but chaos motion state is dominant in high speed region ns> 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.
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(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