Natural Characteristics of The Herringbone Gear Transmission System

According to the structure characteristics of herringbone gear transmission, a more realistic dynamic model of the transmission system is built in consideration of the inner excitation, herringbone gears axial positioning and sliding bearing etc. The natural frequencies of the system are calculated, and the vibration mode is divided into symmetric vibration modes and asymmetric vibration modes. The time history of system dynamic force is obtained by solving the dynamic model. The effects of the connection stiffness of left and right sides of herringbone gears and axial support stiffness on natural characteristics are discussed.


Introduction
Gear transmission is one of the most extensively adopted transmission modes; herringbone gear is favorably characterized by high bearing capacity, steady operation and small axial force[ [1]], having been extensively applied to heavy load device, noise is associated with comfort and reliability of equipment directly. Therefore, proper establishment of transmission system analysis model to accurately analyze its dynamic characteristic shall be the basis to improve transmission system performance.
Scholars have made remarkable achievement in terms of dynamic characteristic of gear transmission system. Feng set up herringbone gear pair analysis model by three-dimensional finite element method to carry out analysis of contact behavior and modality of gear pair [ [2]]. Wang established dynamical model for bending-torsion-axis coupling of herringbone gear based on lumped parameter method， then the influence of various excitations and gear tooth corrections on dynamic characteristic of herringbone gear are analyzed [ [3]]. Wang built a 12-DOF nonlinear herringbone gear transmission model, the effect of the meshing stiffness and corner mesh impact on vibration characteristics under multi-load were studied [ [4]]. Zhao proposed a parabola modification method for herringbone gears, and examples show that path of contact and transmission errors of optimized gears are preferable distinctly [ [5]]. Considering that axial component of herringbone gear is identical in size and opposite in direction in theory, therefore, some researches on building of analysis model have ignored axial vibration of herringbone gear, which is reduced to spur gear, however, for herringbone gear with larger width-diameter ratio, such reduction is likely to bring about bigger error in calculation.
The paper has built the dynamic model of herringbone gear transmission, in which the axial positioning of herringbone gear and inner excitation are considered. The natural characteristic of herringbone gear transmission system is analyzed.  In the process of dynamical model building, left hand and right hand gears in gear reducer are taken as helical gear, two gears adopt axis section connection with stiffness, with bending-torsion-axis coupling vibration model for herringbone gear built based on the relationship shown in assembly drawing of gear reducer, as shown in fig.1, in which X and Y are transverse vibration respectively, Z is axial, p denotes input end, g indicates output end, k mi in the fig represents meshing stiffness (in which i=1,2 indicates helical gear pair on left and right sides respectively). Every gear adopt sliding bearing support, in order to embody asymmetry of oil film stiffness[ [6]], the paper adopts four stiffness coefficients to describe oil film stiffness of sliding bearing, in which k pix and k piy represent bearing support stiffness at input end; k gix and k giy represent bearing support stiffness at output end. Every gear has four degrees of freedom, namely translational degree of freedom along X, Y and Z directions and torsional degree of freedom around its center Generalized displacement vector of the system is expressed as Where: k m is meshing stiffness of gear and cm is meshing damping of gear Dynamic mesh force of gear pair is Model is converted into matrix form (3) Where: M , C , K -mass matrix, damping matrix and stiffness matrix X -displacement vector P -generalized force vector

Herringbone gear meshing stiffness
Herringbone gear pair is split into two symmetric helical gear pairs as shown in fig.2(a). One meshing period is divided into several parts, with finite element method adopted to calculate helical gear pair stiffness, in which meshing stiffness curve in one period is as shown in fig.2  Finite element method is adopted to calculate bending, torsional and axial stiffness of intermediate connecting axis to obtain that lateral bending stiffness of intermediate axis section of driving gear is k px =k py =1.92×10 9 N/m, axial stiffness is k pz =3.1×10 9 N/m, and torsional stiffness is k p =1.89×10 10 N· m/rad, while lateral bending stiffness of intermediate axis section of driven gear is k gx =k gy =1.376×10 9 N/m, axial stiffness is k gz =1.1×10 10 N/m, torsional stiffness is k g =2.7×10 10 N· m/rad.

Dynamic characteristic of herringbone gear transmission system
3.1. Inherent characteristic of herringbone gear transmission system Inherent frequency of system is obtained by solution, as shown in table 2. Mode of vibration of transmission system is as shown in fig.3, the abscissa in the figures corresponds to every degree of freedom of system in turn, namely degree of freedom in dynamical model {X i }, ordinate indicates vibration amplitude relevant to every degree of freedom. It is observed from fig that mode of vibration of transmission system may be divided into two categories, one is symmetrical mode and the other one is asymmetrical mode, in which the first, third, fourth and sixth order mode of vibration is symmetrical mode of vibration for pinion gear p 1 , p 2 and bull gear g 1 and g 2 , other modes of vibration are not fully symmetrical, in which the second order is axial vibration mode of pinion gear p 1 and p 2 , the fifth order is axial vibration mode of bull gear, the seventh order is lateral vibration mode of bull gear g 1 and g 2 , the ninth and eleventh orders are lateral vibration mode of pinion gear p 1 and p 2 , and the thirteenth and fourteenth orders are torsional vibration mode of gear.   Fig.4 shows time domain course and frequency spectrum of dynamic load of gear box bearing along direction Y, in which the mean value of dynamic load of bearing at input end is 8,192.3N, being on the contrary to positive direction defined in model and negative value; in addition, gravity action on gear at input end is consistent with dynamic load component on bearing along direction Y, being mutual superposition. Gravity of gear at output end is opposite to dynamic load component on bearing along direction Y, therefore they offset each other, with mean value smaller than that at input end, being 6385.4N.

Dynamic load of system
For frequency spectrum, dynamic load on both ends generated larger peak value at mesh frequency and 2-multiple-frequency, since mesh frequency (1,233Hz) of gear pair and inherent frequency (1,507Hz) of lateral vibration of transmission system are similar, accordingly, fundamental harmonic is to directly transferred to bearing and become main part. The peak value at mesh frequency of bearing at input end is maximum, being 14.32N.

Influence of axial positioning stiffness on dynamic characteristic
In herringbone gear transmission system, pinion gear is free of positioning in axial direction as a general rule; while, the entire system is subject to axial constraint by axial positioning on bearing thrust face of bull gear. The paper made a calculation of inherent frequency of transmission system in case that axial positioning stiffness is 0.5×k z , 0.75×k z , 1.5×k z and 2×k z , from which it is observed that axial positioning stiffness only exerts an influence on inherent frequency (axial vibration mode of pinion gear) at second order and inherent frequency (axial vibration mode of bull gear) at fifth order of the system, having no influence on other inherent frequencies. The inherent frequency at second order and fifth order of system varies with axial positioning stiffness as shown in fig.6, from which it is observed that inherent frequencies at two orders are on the increase with the increase of positioning stiffness and that the difference is that positioning stiffness is to constrain pinion gear along axial direction in a indirect way by meshing spring after constraining bull gear, therefore, variation in stiffness is to exert insignificant influence on pinion gear, inherent frequency of axial vibration of pinion gear is less dependent on variation of positioning stiffness. Relatively speaking, inherent

Conclusion
The paper set up dynamic model of the herringbone gear system, and natural characteristics of the gear transmission system are analyzed, some conclusions are found: (1) the natural characteristic of transmission system may be divided into symmetrical modes and asymmetrical modes. The dynamic load of the bearing contains significant mesh frequency component.
(2) Variation in intermediate connection stiffness of herringbone gear is only to influence asymmetrical modes. With the increase of stiffness value, there is no influence on symmetrical modes, but other inherent frequencies are on the increase.
(3) Axial positioning stiffness of gear is only to influence axial mode of vibration in system. With the increase of axial positioning stiffness, fluctuation of dynamic load of axial support for gearbox is to gradually intensify.

ACKNOWLEDGMENTS
Thanks must be given to Xinjiang University for giving me a good surrounding, and Chinese National Natural Science Foundation (51665054) and Scientific and Technological Training Project (QN2016BS0082) for giving the author a good financial backing to finish the most important part of this research.