The Effects of the Planet–Gear Manufacturing Eccentric Errors on the Dynamic Properties for Herringbone Planetary Gears

: In the presence of manufacturing errors, the dynamic properties of herringbone planetary gear train (HPGT) can be altered from the originally designed properties to have undesired behavior. In this paper, by considering the herringbone gear actual structure characteristics, manufacturing eccentric errors of members (i.e., carrier and gears) and tooth profile errors of gears, time ‐ varying meshing stiffness, bearing deformation, and gyroscopic effect, a novel lateral– torsional–axial coupling dynamic model for the herringbone planetary gear system is formulated by using the lumped ‐ parameter method, which is able to be employed in the dynamic feature analysis of the HPGT with an arbitrary number of planets and different types of manufacturing errors. By applying the variable ‐ step Runge–Kutta algorithm, the dynamic response of a HPGT system is studied for cases with and without planet–gear eccentric error excitations. The dynamic contact forces of gears and bearings are analyzed for the two cases in time and frequency domains, respectively. Moreover, the effect of the planet–gear eccentricity on the vibration accelerations of the HPGT system is also discussed. The obtained results indicate that manufacturing error excitations such as the planet–gear eccentricity have a pronounced influence on the dynamic behavior of the HPGT system.


Introduction
Owing to the characteristics of power split flow, compactness, and high torque-to-weight ratio, planetary gear train (PGT) is extensively employed in various industrial applications, such as in aerospace, automobiles, wind turbines, and nuclear power plants. In contrast to spur or helical gears, herringbone gears possess many benefits, including smoother transmission, lower axial force, and greater transmission torque. Consequently, the herringbone planetary gear system has also been used in the power train of heavy machinery such as aerospace engines, and long-wall shearers. The existence of inevitable processing errors in the manufacturing process of a drive train can significantly affect the reliability and durability of herringbone planetary gears. Nevertheless, there have been few related reports on the manufacturing error impacts on the dynamic properties of herringbone planetary gears. For the sake of achieving reliable and quiet operation, it is vital and necessary to formulate a dynamic model of herringbone planetary gears for the analysis of manufacturing error effects on dynamic properties.
There have been several investigations into the PGT dynamic properties. Lin and Parker constructed a spur PGT dynamic model to examine the modal characteristics [1,2]. Guo established an analytical model of spur PGTs with bearing backlash nonlinearity and tooth wedging [3]. Zhao and Ji proposed a nonlinear torsional model of multi-stage spur PGTs applied to a wind turbine gearbox for the analysis of nonlinear vibration characteristics [4]. Kahraman presented an analytical model for a stage of helical PGT for the study of its dynamic behavior [5]. Based on the lumpedparameter and finite element theories, Tatar et al. constructed a three-dimensional analytical model of helical planetary geared rotor sets by considering gyroscopic effects to investigate the dynamic behavior [6]. Bu incorporated journal bearings to a dynamic model of herringbone planetary gears to study their modal properties [7]. Sondkar and Kahraman investigated the free and forced vibration characteristics of double-helical planetary gears [8]. Chaari considered the eccentric and profile errors and examined the manufacturing error impacts on the spur PGT dynamic properties [9]. Xu studied the influences of pin error and stiffness in wind turbine gearbox on the PGT load distribution performance [10]. Ren et al. conducted a study of the manufacturing error effects on dynamic performance and load distribution properties for HPGT [11][12][13][14]. Zhu et al. explored the impact of meshing-frequency and run-out errors on dynamic load distribution among different planets for encased differential gear sets [15].
However, there has been no research on the impact of manufacturing errors such as the planetgear eccentricity on the dynamic properties for herringbone planetary gears. In this paper, by considering the actual structural features of herringbone gears, a novel transverse-axial-torsional coupling dynamic analytical model for herringbone planetary gears involving manufacturing errors is formulated by applying the lumped-parameter method. The dynamic features of a HPGT system are then numerically studied, and the impacts of manufacturing errors such as the planet-gear eccentricity on the dynamic performance are examined.

Components and Drive Principle
The schematic of the components and drive principle of the herringbone planetary gear is shown in Figure 1, which are composed of a sun gear s, a planet-carrier c, a right ring gear r1, a left ring gear r2, and N uniform planet gears p. Here, the sun gear connects with the high-speed input shaft, while the carrier is linked with the low-speed output shaft. The sun gear is subjected to the input torque and eventually the power is transmitted to the external load by the carrier as an output. The planets and sun adopt the herringbone gears, whereas the left and right stationary ring gears use two internal helical gears possessing opposite helical angles.  Figure 2 shows the developed transverse-axial-torsional coupling dynamic model of the herringbone planetary gear set. All the gears (i.e., the sun, two rings, and planets) and the planetcarrier are considered to be rigid bodies. Bearing components are modeled by linear springs between their housings and bodies. Linear springs applied along the action line denote gear mesh interactions. The damping, gravity and friction impact are neglected, and the tilting motion is ignored. Each member possesses four degrees of freedom (DOFs), including one axial translation, two lateral translations, and one rotation. As demonstrated in Figure 2, there are three kinds of reference frames established, namely (1) the static reference frame OXYZ, (2) the dynamic reference frame Oxyz rotating around the coordinate origin O together with the carrier, and (3) the dynamic reference frame Oixiyizi rotating with the carrier, whose origin Oi is at the ith planet-gear's center, and xi-, yi-axis are, respectively, in the radial and tangential direction.

Dynamic Model
Translational coordinates xi, yi, zi (i = c, r1, r2, s) are respectively assigned to the carrier, right ring gear, left ring gear, and sun gear. Rotational coordinates are given by

Component Acceleration Analysis in HPGT
To conduct the dynamic analysis, the acceleration analysis of each component for herringbone planetary gears should be first carried out. The generalized coordinates of herringbone planetary gears are established in the dynamic reference frames rotating along with the carrier. However, the accelerations of component centroids used in the dynamic analysis should be the absolute acceleration, so the absolute acceleration expressed in the dynamic coordinate systems should be derived. The relationship among the reference frames is displayed in Figure 3. Let G be the assumed centroid. Since the origin of the static reference frame {i', j', k'} and the origin of the dynamic reference frame {i, j, k} coincide at point O, the radius vectors of G are both expressed by r in these two reference frames and . The components in the dynamic reference frame of the vector r are, respectively, x•i, y•j and z•k, and those in the static reference frame are, respectively, X•i', Y•j' and Z•k'. From Figure 3, the radius vector can be obtained as cos sin , cos sin , ', ', ' , , ', ', ' The absolute acceleration of G can be written as The absolute acceleration of G in the dynamic reference frame can be derived as , , where c ω means the angular velocity of the carrier.

Component Equivalent Displacements in HPGT
Figures 4-6 respectively show the dynamic models of the external (sun-planet i) and internal (carrier-planet i) mesh pair, and carrier-planet i pair. In the derivation of the dynamic equations of the entire HPGT system, the relative displacements between the components in HPGT need to be obtained first. According to the motion and deformation relationships between components as shown in Figures 4-6, the relative displacements can be deduced [16][17][18][19][20][21].   (1) The equivalent deformation L spi  of the ith external mesh on the left side is written as (3) The equivalent deformation 2 L r pi  of the ith left-side internal mesh is given by of the ith right-side internal mesh is written as (5) The radial relative deflection xcpi  between the ith planet and the carrier is determined by The tangential relative deflection ycpi  between the ith planet and the carrier is expressed as The axial relative deflection zcpi  between the ith planet and the carrier is given by

Equations of Motion
Based on the dynamic models of each mesh pair as displayed in Figures 4-6, Newton's Second Law of Motion in non-inertial coordinate frames [2] is employed to acquire the equations of motion for herringbone planetary gears. The carrier's motion equations can be given by In a similar manner, the equations of motion of the other components for herringbone planetary gears can be derived.
The right ring gear's motion equations are represented by The left ring gear's motion equations can be expressed as The sun gear's motion equations are given by The ith planet-gear's motion equations can be derived as where mg and Jg (g = s, p, c, r) represent the mass and rotational inertia of member g, respectively. N refers to the plane number. kn and knz are the x-or y-direction and z-direction support stiffness of member n (n = s, p, c, r1, r2), respectively. knt (n = r1, r2) respectively means the torsional support stiffness of the right and left ring gear.

Error Excitation
This paper considers the manufacturing eccentric errors of each member (i.e., carrier and each gear) as well as tooth profile errors of each gear. These manufacturing errors are projected to the contact lines of the left-side and right-side meshing pairs of the HPGT system, respectively, and finally the cumulative meshing error gained by the superposition of eccentric and tooth profile error of components at the left-and right-side action lines can be written as [11,12].
1, 2, , g ω g p s c  is the angle velocity of member g; t is time. m ω is the system meshing angular frequency, Tm is the system meshing period.

Numerical Calculation Approach
Since the numerical integration method is widely used, which is suitable for solving any type of nonlinear differential equations, and in engineering practice, the differential equations of gear system dynamics are generally difficult to obtain accurate analytical solutions, the numerical methods have been widely applied in solving the dynamic equations of complex gear systems, where the Runge-Kutta method is the main numerical integration method.
In the present paper, the system response is solved by using the variable-step Runge-Kutta algorithm [11,12]. To apply Runge-Kutta method to solve Equation (16), first, the descending order processing of second order differential equations is needed to transform the governing Equation (16)

U(t) (F(t) [ ] U(t) ([ ] [ (t)] [ ]) U(t))
Introducing the state vector   Equation (19) can be written as a matrix form of first-order ordinary differential equations while calculating, based on the above Equation (21), the ODE solver in MATLAB is used to solve the system equation.

Numerical Simulations
The parameters of a herringbone planetary gear with both stationary rings and three equally spaced planets shown in Figure 1 are given in Tables 1 and 2. The variable-step Runge-Kutta method described in Section 5 is employed for solving the system Equation (16). The dynamic responses such as the vibration accelerations of the components of the HPGT system are compared for two cases where the planet-gear eccentric error excitations are present and absent. Furthermore, the dynamic contact forces of gear and bearing for the HPGT system are also compared between the results with and without planet-gear eccentric error excitations in the time domain and frequency domain, respectively.

Item /(N•m −1 ) Left Ring Right Ring Sun Planet Carrier
x-direction To investigate the impacts of planet-gear eccentricity Ep1 on the HPGT dynamic features, the planet-gear eccentric error is assumed to be Ep1 = 100 μm, while the other manufacturing errors are not considered. The planet-gear eccentricity is defined to be the deviation between the realistic and theoretical rotational center as depicted in Figure 7 which shows a schematic of the planet-gear eccentricity.  Figure 8 illustrates the time-domain responses of the dynamic meshing force on the right-side planet-sun gear pair, which is represented as  (5) and (18)), for two cases of with and without planet-gear eccentric error. For the sake of clarity, just one side of the dynamic meshing forces of a part of the time interval for the HPGT is demonstrated in Figure 8. It can be observed from Figure 8 that because the input torque of the system Tin = 100 KN•m is constant, the dynamic meshing forces of the right-side meshing pair possess no low-frequency components for both models with and without the planet-gear eccentric error excitations. The average values of the dynamic meshing forces for the models with and without the planet-gear eccentric error excitations are approximately 57 KN as shown by the red dot-dash lines in Figure 8a,b. A comparison of Figure 8a,b indicates that the meshing force fluctuation amplitude from the model with the planet-gear eccentric error excitation Ep1 = 100 μm is obviously larger than that without error excitations. That is to say, the planet-gear eccentric error excitation increases the meshing force amplitude significantly in contrast to the model without manufacturing error excitations. The similar tendency is also able to be discovered for the meshing forces on one side of internal (planet-ring) meshing pair. In short, the planet-gear eccentric error excitation prominently increases the dynamic meshing force fluctuation amplitude for herringbone planetary gears.   Figure 9a,b, it is noted that the variation amplitudes of dynamic mesh forces with the planet eccentricity excitation are remarkably greater than those without error excitations. Thus, the manufacturing error excitations such as the planet eccentricity enhance the meshing force fluctuation.  Figure 10 illustrates the variation of the bearing forces of some main components involving the carrier, sun and planet in the HPGT system with gear mesh time (denoted as Fc, Fs, and Fp1, respectively). For the model with the planet-gear eccentric error excitation, as displayed in Figure  10a-c, it is observed that dynamic bearing forces for each component (carrier, sun and planet-gear) fluctuate prominently and periodically. The variation of the planet-gear dynamic bearing force is the most significant in each component of the system, and the variation of the sun gear bearing force is relatively smaller, owing to better flexible support of the sun gear. Figure 10d Figure 11 illustrates the variations of the accelerations in x-, y-, z-, and u-directions for some main components of HPGT (i.e., the carrier, sun and planet) with gear mesh time (denoted as aix, aiy, aiz, aiu (i = c, s, p1), respectively), which reflect the vibrations of main system components involving the carrier, sun, and planet in the corresponding directions, respectively, for the case of the planet eccentricity Ep1 = 100 μm, while Figure 12 shows the results for the case of without error excitations. For the model with the planet-gear eccentric error excitation, the time-domain vibration accelerations at each degree of freedom of system components behave the fluctuation up and down around the horizontal zero axis; the component vibration accelerations in the z-direction (i.e., axial direction) are pronouncedly smaller than in the lateral directions (i.e., x, y-direction), induced by the symmetrical tooth structure of herringbone gear, making the axial forces of each component smaller. The carrier vibration accelerations in the horizontal, vertical, axial, and torsional directions are smaller than those in the corresponding directions of the sun and planet, possibly owing to the larger carrier inertia as given in Table 1. The vibration accelerations in x-, and y-directions of the planet-gear are larger than those in the corresponding directions of other components such as the carrier and sun gear, particularly the vibration acceleration in the y-direction (i.e., tangential direction) of the planet-gear is the greatest, due to the larger force transmitting power in the planet tangential direction. In the vibration accelerations in the torsional direction of each component, the sun acceleration is greater, with the maximal amplitude of 300 m•s −1 , and that of the planet-gear is also relatively large, with great amplitude of 275 m•s −1 . Figure 11. Variations of the vibration accelerations of the main components from the model in the presence of the planet-gear eccentric error excitation Ep1 = 100 μm with gear mesh time. (a,d,g,j) the carrier accelerations in x-, y-, z-, and u-directions, represented as acx, acy, acz, and acu, respectively; (b,e,h,k) the sun accelerations in x-, y-, z-, and u-directions, represented as asx, asy, asz, and asu, respectively; (c,f,i,l) the planet accelerations in x-, y-, z-, and u-directions, represented as ap1x, ap1y, ap1z, and ap1u, respectively. Figure 12 exhibits the variations of the accelerations for some main components of HPGT (i.e., the carrier, sun and planet) with gear mesh time (denoted as aix, aiy, aiz, aiu (i = c, s, p1), respectively), which reflect the vibrations of main system components containing the carrier, sun, and planet in the corresponding directions, respectively, for the case of the absence of manufacturing error excitations. A comparison of Figure 11a-c with Figure 12a-c indicates that in HPGT system, the vibration acceleration amplitudes in the x-direction of each component for the model without errors become evidently smaller than those for the model with the planet-gear eccentric error excitation, particularly the amplitude variation of vibration accelerations in the x-direction of the planet-gear is the most distinct in the two models with and without error excitations, on account of the planet eccentricity excitations. In the meantime, it is also seen that the vibration acceleration amplitudes in the ydirection of each member in the absence of error excitations displayed in Figure 12 are also noticeably smaller than those in the presence of the planet eccentricity excitation displayed in Figure 11, probably because of the planet-gear eccentricity excitation. The same tendency is also found for the vibration acceleration amplitudes in the u-direction (i.e., torsional direction) of each component. From Figures 11g,h,j and 12g,h,j, it can be observed that for the model with the planet-gear eccentric error excitation, the vibration accelerations in the z-direction of the herringbone sun, herringbone planet, and carrier are small but nonzero; while for the model in the absence of error excitations, the z-direction vibration accelerations of the herringbone sun, planet, and carrier disappear, similar to the spur planetary gears. This means that manufacturing error excitations are the vibration source of the axial direction and directly impact the axial vibration of each member in the HPGT system.

Vibration Accelerations of the Components
Through comparing the results obtained from the model in the presence of the planet eccentricity shown in Figure 11 and those in the absence of the error shown in Figure 12, it is seen that manufacturing error excitations such as the planet eccentricity   . (a,d,g,j) the carrier accelerations in x-, y-, z-, and udirections, expressed as acx, acy, acz, and acu, respectively; (b,e,h,k) the sun accelerations in x-, y-, z-, and u-directions, expressed as asx, asy, asz, and asu, respectively; (c,f,i,l) the planet accelerations in x-, y-, z-, and u-directions, expressed as ap1x, ap1y, ap1z, and ap1u, respectively.

Conclusions
In this study, by considering the HPGT actual structure characteristics, manufacturing eccentric errors of components (i.e., each gear, and carrier), tooth profile errors of gears, gear tooth timevarying meshing stiffness and bearing deflections, different from the two-dimensional model, a novel and generalized three-dimensional lumped-parameter dynamic model of herringbone planetary gears has been given for studying the dynamic feature of the HPGT system in the presence of arbitrary number of planets and different types of manufacturing errors. The effects of manufacturing errors such as the planet eccentricity on the HPGT dynamic features were investigated. The main conclusions are given below.
(1) Manufacturing errors such as the planet eccentricity prominently affect the HPGT dynamic features, and the manufacturing error excitations significantly increase the fluctuations of the dynamic meshing forces, dynamic bearing forces, and vibrations of components of herringbone planetary gears. This investigation provides the new effective idea and methodology for the prediction and analysis of the dynamic feature of complex herringbone planetary gear systems with other type of error faults, and also offers the theoretical foundation for the error fault diagnosis and dynamic optimization of herringbone planetary gears in the next step. Our ongoing investigation will focus on the studies of the effects of system stiffness on herringbone planetary gear dynamic response.
Author Contributions: F.R. presented the HPGT system dynamic model, analyzed the results, and wrote this paper; A.L. analyzed some simulation results and reviewed the manuscript; G.S. modified the first draft of the manuscript and guided the contribution; X.W. revised the manuscript and gave some helpful and valuable suggestions; N.W. analyzed some simulation results. All authors have read and agreed to the published version of the manuscript.

Acknowledgment:
The authors would like to express their gratitude to A/Prof. Jinchen Ji in the University of Technology Sydney in Australia for helping modify the manuscript, especially polishing the language, and they are also very grateful to the editors and reviewers for their valuable comments and suggestions.