Nonlinear dynamic analysis of GTF gearbox

Considering multiple nonlinear parameters, a nonlinear dynamic model of star gear-rotor-bearing transmission system of GTF (Geared Turbofan Engine) gearbox is established. The dynamic responses are obtained through Runge-Kutta numerical integration method, and the responses with the variation of input rotational speed are analyzed and illustrated. The abundant nonlinear characteristics are presented in the results, which could be guidance to avoid undesirable nonlinear motion, and provide a reference for the design and control of the transmission system in GTF gearbox.


Introduction
As the widely application of gearbox in wind turbine and aero industry, the nonlinear analysis for the gearbox, especially for those composed of planetary gears were studied. Zhou et al. [1] built dynamic model of the spur gear-rotor-bearing system with multiple nonlinear parameters, with the coupled lateral-torsional vibration considered. Tao et al. [2] developed a nonlinear model of a planetary gear system with multiple backlashes. Guo et al. [3] studied the nonlinear behavior of wedge teeth related to the bearing force of planetary gear system, and conducted the dynamic analysis of planetary gears considering bearing clearances [4], then studied the maximum load of planetary gears in the wind turbine drive system [5], and studies nonlinear behavior of planetary gear system in wind turbine under gravitational effects [6]. Kim et al. [7] proposed a new dynamics model for a gear set, which takes into account the torsional motion with time-varying meshing parameters. Bahk and Parker [8] conducted a dynamic analysis of the planetary gears, the resulting nonlinear behavior showed jumping phenomenon and subharmonic vibration at resonance. Li et al. [9] analyzed the nonlinear phenomena of the multi-stage planetary gear system. Zhao and Ji [10] proposed a nonlinear torsional model of gearbox in wind turbine multi-stage of planetary gear. Xiang et al. [11,12] studied the nonlinear dynamics of multi-stage gears, which considered the multi-clearance of the system. Hou et al. [13] established a pure rotational model of the planetary gear-rotor system of the geared turbofan engine, and conducted a nonlinear dynamic analysis to obtain the torsional vibration response between the gear and the rotor. Wang et al. [14] proposed a dynamic model of GTF gearbox with elastic support of planet carrier, then [15] established nonlinear torsional model of GTF transmission system, and obtained nonlinear characteristics under excitation of important parameters.
In this paper, a coupled lateral-torsional nonlinear dynamic model of GTF gearbox is proposed, the influence of input rotational speed on the vibration response of the system is explored and quantified, then the nonlinear behavior is studied by analyzing the states of motion, mechanisms of bifurcation, and roads to chaos. Fig. 1(a) and (b) show the physical structure and schematic diagram of star gear-rotor-bearing transmission system of GTF gearbox, which is composed of star gearing system, input shaft (the rotor between low-pressure compressor and sun gear), output shaft (the rotor that connects ring gear and fan), and bearings that support input and output rotor. The star gearing system is composed of sun gear, star gear, ring gear, and planet carrier, among which, the star gears are supported by bearings, and meshed with sun gear and ring gear, both of which are semi-floating components that splined with input and output shaft, respectively. a) Physical model b) Schematic diagram In which, , , , indicate the meshing stiffness, and damping of external and internal meshing pairs, the value of stiffness are calculated by the formulas [16]. Then the meshing forces can be calculated by the following equations:

Dynamic model of GTF gearbox
where , , , is the nonlinear function of backlash.

Mathematical model of GTF gearbox
The system has a total of 5 + 12 DOFs, where is the number of star gears, then the overall arrays of displacement can be expressed as: The dynamic equations of the transmission system in GTF gearbox can be derived as follows: where is the equivalent mass in rotational direction, is the mass of components, is rotational displacement around direction, and are translational displacement in and direction. represents the input or output torque, represents torsional damping or support stiffness, represents torsional stiffness or support stiffness.
In order to eliminate the rigid body displacement, the following dimensionless variables are introduced for rotational displacement of each member. The dimensionless time parameter can be expressed by: where, is the moment of inertia, is the radius of base circle. Then the derived dimensionless parameters can be defined as: where, is the dimensionless parameter of displacement.

Results and discussion
The basic parameters used for calculation of the nonlinear vibration response of the GTF transmission system are shown in Table 1. The vibration response of the system were conducted as the variation of input rotational speed, while other parameters are being constant. The bifurcation diagram of lateral displacement of star gear is shown in Fig. 3. Here, the input rotational speed (r/min) is control parameter, which varied between ∈ [8000, 18000]. As the input rotational speed increases, the system experiences periodic motion, quasi-periodic motion, and chaotic motion, successively, finally the system turns back to periodic motion through inverse bifurcation by period-doubling scenario. The nT-periodic motion is illustrated when < 9000, for lateral displacement of star gear, as the input rotational speed increases from = 9000 to = 10000, the peiodic motion is turned into quasi-periodic motion, while the speed further increases, the chaotic motion is indicated in the system through quasi-periodicity until the speed arrives at 11500 rpm. As the further increasing of input rotational speed, the system turns back to nT-periodic motion while the input rotational speed varying from = 11500 to = 11900, finally the system performs periodic motion as > 11900. The vibration response of the gear system in the lateral displacement of star gear were analyzed at three input rotational speeds of = 81000, = 9300, and = 10800, the results are shown in Fig. 4, from which different dynamic characteristics in various speed can be observed. When the control parameter = 8100, the system response is dominated by the meshing frequency , the amplitude of which is much bigger than the response of other orders of meshing frequency, besides the meshing frequencies, the variable bearing stiffness frequency 2 appears. The Poincare map that obtained shows a cluster of concentrated points when = 8100, while several coincident circles are illustrated in the corresponding phase diagram, therefore, the nT-periodic motion is indicated for the system. As the control parameter = 9300, although meshing frequency is still the dominant response, the multiplication frequency components of variable bearing stiffness frequencies as , 2 , 5 , 9 can be observed. The Poincare map turns to be a phase-locked loop, and a trajectory torus is presented in phase diagram. The characteristic demonstrate that the gear system turned into quasi-periodic motion. As the control parameter arrives at = 10800, a disordered set is shown in the Poincare map, and the corresponding phase diagram becomes a chaotic attractor with a random crew, thus, the chaotic motion is indicated for the system. In this situation, the meshing frequency is no longer the dominant response, which is replaced by the bearing stiffness frequencies and 5 , then the multiplication of bearing frequency components 2 , 10 also appears with more obvious amplitudes.

Conclusions
In this paper, the dynamic model of the transmission system in GTF gearbox is established, based on which the coupled bending-torsional nonlinear dynamic analysis was conducted, then the influences of input rotational speed on the nonlinear dynamic response were studied. As the input rotational speed increases, nT-periodic motion, quasi-periodic motion and chaotic motion are indicated successively, however, the system finally turns into nT-periodic motion through inverse bifurcation. The nonlinear dynamic characteristics such as appearance of combination frequency components and continuous frequency components due to the increase of input rotational speed, therefore, the system should be well controlled under certain range of input rotational speed in order to avoid the chaotic motion.