Modeling And Dynamic Behavior Analysis Of The Power-split Planetary Gear Train

Planetary gears are widely used in aerospace,ships,wind turbines because of their compactness,high efficiency,excellent bearing capacity and stability.The planetary gear set is a multiple parametric excitation system.Internal excitations including stiffness excitation,error excitation and teeth separation excitation are the main factors of gear system vibration.In addition,vibration and noise are the main factors that influence the reliability and life of the mechanical system.Therefore,it is necessary to inveatigate the dynamic characteristics of the planetary gear systems to reduce the vibration and noise.Most of the dynamic models established by designers are purely rotational models and two-dimensional translational-rotational models.The axial deformations are ignored in those models.The three-dimensional dynamic model can describe the dynamic characteristics of the system more accurately.As a form of planetary gears,the power-split planetary gears are more competent due to its distinguishing structure that the power is experienced by two or multiple stages simultaneously.However,the relative displacements and internal excitations of the system are more complex.The modeling and dynamic characteristic analyses of the system are difficult.In this paper,the dynamic modeling and dynamic characteristics analysis of the single-stage planetary gear and the power-split planetary gear transmission system are conducted respectively.The following several aspects are concluded in this paper.The three-dimensional dynamic model of a single-stage planetary gear set with consideration of the three-dimensional translation and axial rotation is established.The dynamic equations of motion are deduced.The modal properties of the single-stage planetary gear set are investigated by solving the corresponding characteristic equation.The results indicate that vibration modes of the single-stage planetary gear set can be categorized as four classes:the rigid body mode,the axial translational-rotational mode,the radical translational mode and the planet mode.The influence of planet number,pressure angle and mass on natural frequencies is further detected.The results show that the natural frequencies of planet modes decrease with the increase of planet mass while they are immune to the mass change of central members.Natural frequencies of other modes play a decreasing tendency when the member's mass increases.Only the natural frequency of the second planet mode increases when the pressure angle become larger,the other natural frequencies decrease or remain constant when the pressure angle increases.The natural frequencies with multiplicity 1 and 2 vary monotonically with the planet number.Those investigations lay a foundation for the three-dimensional modeling of the power-split planetary gears.The design principle of improving the multiple-stage planetary gear set by realizing power-split is proposed in the paper.The power-split structure can disperse the transmission path and enhance the capacity of planetary gears.The multi-stage planetary gears are composed by 2K-H differential planetary gears and the quasi planetary gear.The power is input by the sun gear of the 2K-H differential planetary gear train.It is split into the second stage by the carrier and ring of the first 2K-H differential planetary gear and then the power is also split by the second stage.Finally,the power is output by the ring of the quasi planetary gear train.The structure that the input power is experienced by two or multiple stages simultaneously makes the system more compact.The three-dimensional translational-rotational dynamic model of two-stage power-split planetary gear sets with consideration coupling between stages is established.The nonlinear equations of motion of system are derived.Then,the natural frequencies and vibration modes are predicted by the modal analysis method.According to the analyses,vibration modes of the two-stage power-split planetary gear set are categorized as five classes:the rigid body mode,the two-stage axial translational-rotational mode,the two-stage radical translational mode,the 1st stage and 2nd stage planet mode.It is revealed that the stages coupling exist in the axial translational-rotational mode and the radical translational mode while the planet mode is immune to the stages coupling in the power-split planetary gears.Subsequently,the typical vibration graphs are derived.The unique properties of these vibration modes are then mathematically proved.The dynamic characteristics of single stage planetary gear and two-stage power-split planetary gear set are analyzed respectively under the three-dimensional dynamic model.The multiple scale method is developed to analyze the dynamic characteristics of the two-stage power-split planetary gears.The general displacements matrix and coefficient matrix with stage-modal coordinate transformation are deduced are compared.The results show that the three-dimensional dynamic model developed by the paper is correct.The mesh frequency of the system is controlled by the speed of the motor.The dynamic response of the system is tested under some cases.The analysis results indicate that the resonance can be deduced by meshing stiffness of any stage.Primary resonance and harmonic resonance are occurred with the fundamental frequency of meshing frequency.Hence,the assumption of ignoring the external motivation and the transformation of expanding the mesh stiffness to Fourier harmonics are reasonable.The research work of this paper provides guidance to the modeling and dynamic characteristics analysis of the single-stage and multistage power-split planetary gears.The conclusions provide theoretical foundation for vibration and noise reduction.It also has great theoretical and practical application value.