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1、附录#附录1#英文原文#NATURAL FREQUENCY VEERING IN PLANETARY GEARS#ABSTRACT# To achieve noise and vibration reduction in planetary gear applications, key design parameters are often varied to avoid resonances,optimize load distribution, and reduce weight. In the plots of natural frequencies versus planetary g

2、ear parameters,veering phenomena occur when two eigenvalue loci approach each other, but then abruptly veer away. The importance of veering is manifested in the dramatic changes in the vibration modes of the veering natural frequencies and the consequent impact on response.This work analytically cha

3、racterizes the rules of eigenvalue veering in planetary gears.The coupling factors between two close eigenvalue loci are approximated by perturbation analysis. Special veering patterns were obtained using the unique properties of planetary gear vibration modes.The results are illustrated by an examp

4、le. Key design parameters were investigated, and generalized guidance is provided for tuning planetary gear natural frequencies.# I. INTRODUCTION# Noise and vibration reduction are critical concerns in planetary gear applications. During the design process, system parameters are varied to evaluate a

5、lternative design choices,avoid resonances,optimize load distribution,and reduce weight. It is important to characterize the effects of parameter variations on the natural frequencies and vibration modes for effective vibration tuning. In planetary gear dynamic models （Fig. 1）, the key design parame

6、ters include the mesh stiffnesses, support/bearing stiffnesses, component masses, and moments of inertia. Some plots of natural frequencies versus planetary gear parameters are presented by Cunliffe et al. 1, Botman 2, Kahraman 3,4, and Saada and Velex 5.# Natural frequency plots in these studies, e

7、specially Ref. 3, show natural# #frequency veering phenomena when two eigenvalue loci approach each other as a parameter is varied, but then abruptly veer away like two similar charges repelling （point B in Fig. 2a）. The vibration modes of the veering eigenvalues are strongly coupled and undergo dra

8、matic changes in the veering neighborhood.The phenomenon has been studied extensively 69, but it has not been explored in planetary gears.Eigenvalue veering is also related to mode localization that can occur when disorder is introduced into nominally symmetric systems like turbine blades, space ant

9、ennae, multispan beams, and other structures 6. In the case of especially sharp veering, it is sometimes difficult to distinguish between intersection and veering just by observing eigenvalue plots.Curve veering/crossing complexity obstructs the tracing of eigenvalue loci under parameter changes. Al

10、so, when multiple curves veer or intersect close together （Fig. 3）, strong modal coupling and the associated operating condition response changes that occur are not identifiable from frequency loci plots.# The objectives of this work were twofold. The first was to analytically derive simple rules th

11、at predict eigenvalue veering in planetary gears. The second was to use the veering results, along with previously developed modal properties and eigensensitivity analysis, to define more fully the influence of model parameters on free vibration and to give guidance for tuning natural frequencies.Li

12、n and Parker 10,11 analytically characterized the unique, highly structured properties of planetary gear natural frequency spectra and vibration modes. They also provided simple, closed-form expressions for the sensitivities of natural frequencies and vibration modes to design parameters 12. These a

13、nalytical results provide the necessary foundation for the present study of veering rules in planetary gears. The veering rules yield concrete conclusions expressed in simple forms for when two approaching eigenvalues veer or cross. The importance of the veering rules is to identify those ranges in

14、which small changes in design parameters can dramatically change the vibration modes and consequently the response. The results are illustrated on a benchmark planetary gear（the model parameters and natural frequencies are given in Tables 1 and 2,respectively） used in a helicopter power train.# The

15、lumped parameter model derived in Ref. 10 is used here. The model is applicable for general epicyclic gears with N planets. Each component has two translational and one rotational degree of freedom （DOF） in planar motion,so the system has L=3（N-3） DOF. Numerical results presented are for fixedring c

16、onfigurations, and L=3（N-2） in this case. The associated eigenvalue problem is#II. VEERING/CROSSING CRITERION#A method for detecting eigenvalue veering/crossing in general dynamic systems is introduced here. Point B in Fig. 2a is an example of veering. When two eigenvalue loci veer away, their loci curvatures indicate the abruptness of curve direction changes. Perkins and Mote 7 proposed a vee