J. Cent. South Univ. (2015) 22: 1676−1684 DOI: 10.1007/s11771-015-2686-4

Characteristic of involute slope modification of asymmetric spur gear

DENG Xiao-he(邓小禾), HUA Lin(华林), HAN Xing-hui(韩星会)

School of Automotive Engineering, Hubei Key Laboratory of Advanced Technology of Automotive Parts, Wuhan University of Technology, Wuhan 430070, China

© Central South University Press and Springer-Verlag Berlin Heidelberg 2015

Abstract: The meshing characteristic of asymmetric involute spur gear was studied, the equations of the geometric shape of the asymmetric gear for both sides were deduced, and the equations of contact ratio and the key points of contact were also obtained. Meanwhile, an involute slope modification method considering the effects of static transmission errors was proposed based on the meshing properties. The characteristic of the involute slope modification was analyzed by changing different modification parameters. The mesh stiffness and synthetic mesh stiffness of unmodified and modified asymmetric spur gears were investigated. Furthermore, the spectrums of synthetic mesh stiffness under different modification parameters were compared. Research results showed that the modification parameters influence the meshing performance of gear pairs, and the proposed modification method was feasible to improve the transmission performance of gear pairs with appropriate modification parameters. Key words: asymmetric gears; involute slope modification; synthetic mesh stiffness

1 Introduction

Gear drive is one of the most important transmission mechanisms in industrial machinery with the advantages of high reliability, extraordinary efficiency, and great load capacity. The gear design and analysis ability have made considerable development with the development of modern science and technology, especially in the field of the design of tooth profile , the study of mesh performance, the calculation of tooth strength and the technology of noise reduction. The symmetric involute profile is commonly used in gear profile design. Aiming not only to maintain the advantage of the transmission stability and the insensitivity to the center distance of involute profile, but also improve its bearing capacity at the same time, the academic circles have put forward the design ideas for asymmetric gear. Asymmetric gears have a better application prospect in customized gear design to meet the demands of precision design and manufacture in the future. Due to manufacturing errors and assembly errors, together with the elastic deformation, engagement deviation will exist inevitably in the gear mating progress. That deviation will influence the transmission properties. It was proven by theories and practices that merely relying on the improvement of the accuracy of

manufacturing and assembly are far from enough, but bring about more factory cost. Appropriate profile modification and axial modification will help to improve the meshing performance, enhance the load capacity and reduce the impact load. For asymmetric gears, appropriate modifications are also need to obtain better transmission performance.

In the previous published papers, the meshing characteristic and modification theories for traditional gears have been put forward. CHEN and SHAO [1] proposed a general analytical meshing stiffness model to establish the relationship between the gear tooth errors and the total mesh stiffness, wherein load sharing among different tooth pairs during meshing and static transmission errors under load were considered. BONORI et al [2] used genetic algorithms to research the optimum profile. FAGGIONI et al [3] presented a global optimization method that focused on reducing gear vibration by means of profile modifications. ZHANG et al [4] performed computer-aided design based on the contact conditions of conjugated gears and the tooth end relief with helix by finite element analysis (FEA). The effects of modification coefficients and helical angles on the transient meshing performance of the gears were investigated by the method of explicit dynamic FEA from the perspective of energy by HU et al [5]. WU et al [6] proposed a tooth profile modification approach for

Foundation item: Project(51105287) supported by the National Natural Science Foundation of China; Project(2012BAA08003) supported by the Key

Research and Development Project of New Product and New Technology of Hubei Province, China; Project(IRT13087) supported by the Progress for Innovative Research Team in University of Ministry of Education of China

Received date: 2014−03−18; Accepted date: 2014−08−01 Corresponding author: HUA Lin, Professor, PhD; Tel/Fax: +86−27−87168391; E-mail: lhua@whut.edu.cn

J. Cent. South Univ. (2015) 22: 1676−1684

1677

the helical pairs, and the static contact process was analyzed using FEA and experimental tests. LI [7] applied 3D FEM to conduct surface contact stress and root bending stress calculations for a pair of spur gears considering machining errors, assembly errors and tooth modifications. Meanwhile, many researches have been done for asymmetric gears modification. LITVIN et al [8] proposed the modified geometric model of an asymmetric spur gear drive which integrated an involute gear and a double crowned pinion, and simulated the meshing and contact process of the spur gears with proposed geometric parameters using computer-aided design method. QIN et al [9] analyzed gear model with wheel hub, web and rim from the Mindlin moderate plate theory, and made comparisons with the frequencies arising from the presented method, finite elements method, and structure modal experiment. TANG et al [10] carried out comparative study and detected profound phenomena, and their results show that there are many different kinds of tooth rattling phenomena at low speed. DENG et al [11] proposed a 3D FE model of a theoretical assembling straight bevel gear pair to analyze the contact fatigue on the tooth surface and bending fatigue in the tooth root. The loaded meshing simulations of multi-tooth FEM model were developed, and the determination of the transmission error and the maximal load distribution factor of face-gear drive under torques were given by TANG and LIU [12].

The gear modification methods were also studied [13−15]. The current modification theory for gear contains two aspects: tooth profile modification and tooth axial modification. With regard to tooth profile modification, the straight line modification, hyperbola modification and parabola modification are commonly used, whereas the involute slope modification is rarely seen for the asymmetric spur gear. So, the characteristic of the modification for this method is unclear. Therefore, this work studied the mesh characteristic of asymmetric involute spur gear with double pressure angle, and an involute slope modification method was proposed for asymmetric involute spur gears considering static transmission deviations. The characteristic of the involute slope modification was analyzed by changing different modification parameters. The comparison between the unmodified and modified asymmetric mesh stiffnesses and synthetic mesh stiffness for spur gears were made. Furthermore, the spectrums of synthetic mesh stiffness under different modification parameters were conducted. The results prove that the parameters have a certain relationship with the mesh characteristic of gear drive. Therefore, the appropriate modification proposed for asymmetric gears will improve the transmission performance.

2 Mesh characteristics of asymmetric spur

gear 2.1 Fundamental parameters of asymmetric spur gear

The primary difference between asymmetric gear with symmetric gear is the flank profile; each tooth of the former profile should be distinguished with the driving side and the coast side, which of the latter on both sides have the same profile. The schematic diagram of an asymmetric gear flank profile is shown in Fig. 1. Each tooth of the gear has two different flank profiles.

Fig. 1 Schematic diagram of an asymmetric gear flank profile

The profile participating in the meshing progress is

called the driving side, and the other side is called the coast side. The pressure angles on both sides are different, so there are different involute flank profiles and root fillets. The involute profiles form when two generate lines purely roll along the tangent of different base circles. As depicted in Fig. 2, a line rolls along the base circle with a radium of rbd and the track of an arbitrary point K on the line generates one flank profile of a teeth. Similarly, the other line rolls along the base circle with a radium of rbc, and the track of an arbitrary point P on line generates the other involute

Fig. 2 Asymmetric involute gear tooth profile forming

schematic diagram

J. Cent. South Univ. (2015) 22: 1676−1684

1678

profile. If we presume the track of point K as the driving side profile and the track of point P as the coast side profile, the parameter equations of two profiles in polar coordinate can be obtained as follows:

Profile of driving side:

kkkk

k

bdk

taninv

cos

rr

(1)

Profile of coast side:

pppk

p

bcp

taninv

cos

rr

(2)

where αk and αp are the pressure angles of driving side and the coast side, respectively; rbd and rbc are the circles of driving side and coast side, respectively.

The equations above can be transferred into Cartesian coordinate system of themselves as follows:

cossin

sincos

bb

bb

iii

iii

rry

rrx (3)

where θ should be within the range of 0−tanαa; αa is the pressure angle of the addendum circle. The subscripts i=d, c, where d represents the driving side and c represents the coast side. 2.2 Mesh characteristics of asymmetric spur gear

When asymmetric gears are in the mesh process, the base pitches of driving side and coast side should be equal, namely:

2bc1bcbcbd21bdbd , pppppp (4)

The base pitch can be calculated as

cosπb mp (5)

If we suppose the modules of two gears being the same, the pressure angle on driving side should be equal to the one on the coast side, namely:

2c1cc2d1dd , (6)

In order to keep the continuity of transmission when two gears are engaged, the contact ratio should be greater than 1. The overlap ratio of asymmetric gear pair can be obtained by

)]tan(tan)tan(tan[π2

12a2a11 iii zz (7)

where i=d and c denote the driving side and coast side, respectively.

In the mesh process of two asymmetric gears, there will be some key positions. Taking the driving side for example, as shown in Fig. 3, 2dd1NN is the common

normal line, and the mesh angles in the figure meet the equations as follows:

c22dd1dwt 1ccwt, (8)

Fig. 3 Mesh schematic diagram of asymmetric spur gear pairs

In the mesh areas, the load of each tooth changes

when the gears rotate alternately between single teeth- meshing area and double teeth-meshing area. Figure 4 is a schematic of the mesh areas. Supposing that gear 1 is the driving gear, the real mesh line for the gear pair is

,ddDA where ddBA and ddDC is the double teeth- meshing area, and ddCA is the single teeth-meshing area. Ad is the start of active profile (SAP for short), Bd is the lowest point of single tooth contact (LPSTC for short), Cd is the highest point of single tooth contact (HPSTC for short), and Dd is the end of the active profile (EAP for short).

From Fig. 4, the common normal line on the base circle can be calculated as

dwttan)( 2bdbd12d1d rrNN (9) where the mesh angle meets the equation:

)/cosarccos( wddwt aa (10) where a is the center distance before addendum modification, and aw is the real center distance.

The real length of mesh can be derived as

bdddd pDA (11)

According to the definition of base pitch, we will have the equation as

bddddd pCADB (12)

The single teeth-meshing areas satisfy the equation as

bdd )2(dddddddd pDACADBCB (13)

In accordance with Fig. 4, we have the following equations:

J. Cent. South Univ. (2015) 22: 1676−1684

1679

bdd )1()(5.0 dddddddd pCBDADCBA (14)

1a1bdd1d tanrDN (15)

a22bd tandd2 rAN (16)

dwtbd11d tanrPN (17)

dwt2bd2d tanrPN (18)

Fig. 4 Schematic diagram of gear mesh intervals

We set the point P as the origin point, and the

direction of vector d1PN as the positive orientation. The locations of the key points Bd, Cd, Ad, Dd, Nd1, Nd2 will be described as li(i=1, 2, 3, …, 6) by the distances between the key points and the reference point respectively. So, the distances can be derived as follows:

bd1adwt1db1 )tan(tan prl (19)

bda2dwt2bd2 )tan(tan prl (20)

bdddwt1db3 )1(tan prl (21)

bdddwtdb24 )1(tan prl (22)

dwt1̀bd5 tanrl (23)

dwtbd26 tanrl (24)

According to characteristics of generating involute line, the roll angles of each point with respect to the reference point P are

1bd/ rlii (25) where i=1, 2, 3, …, 6. 3 Involute slope modification of asymmetric

spur gear 3.1 Modification parameters

An involute slope modification is proposed. The

modification curve can be generated by the following method, as shown in Fig. 5. At first, we choose an original point Olocal as the start of the modification, and set a local Cartesian coordinate system xyz. And then, we take the involute curve from the start point of modification Olocal to the intersection point of the involute extension line and the tip circle as the modification curve, and rotate the modification curve by angle of ∆α at the original point Olocal. The involute curve at the new location will be the modified curve. Meanwhile, the modified profile of the tooth will be dual-involute curve. The schematic diagram involute slope modification of single tooth is shown in Fig. 5. In Fig. 5, ra is the radium of addendum circle; rj is the radium of pitch circle on modification side. We take rm as the radium of the modification circle at similar. If the distance between pitch circle and tip circle is ∆L, and the distance between the modification circle and tip circle is ∆l from Fig. 5 there are the following equations: ∆L=ra−rj, ∆l=ra−rm. We define f=∆l/∆L as the modification coefficient ratio representing the relationship between ∆L and ∆l. The modification angle ∆α and modification coefficient ratio f are the two main parameters used in this work to describe this involute slope modification. In fact, the modification angle represents the maximum modification amount and the modification coefficient ratio represents the length of modification. The involute modification for driving side profile is done in this work, the method and formulas can also be applied to the coast side profile.

Fig. 5 Schematic diagram of involute slope modification of

single tooth

3.2 Establishments of finite element models

The finite element models of asymmetric spur gear are established with and without involute slope modification. The basic geometric parameters adopted for gear are as follows. The solid models are constructed with the following design parameters: the number of the teeth Z1=Z2=19, normal module m=2.87 mm, tooth width b=16 mm, normal pressure angle of coast side flank α1=20°, normal pressure angle of driving side flank α1=22.69°, addendum coefficient ,002.1*

1a h tip

J. Cent. South Univ. (2015) 22: 1676−1684

1680

clearance coefficient .364.0*1 c The material was

20CrMnTi steel with elastic modulus E=2.07×1011 Pa and Poisson ratio μ=0.25. The applied torque T= 52250 N·mm. The solid model was discretized with 8-node hexahedron isoparametric elements. According to Eq. (7), the overlap ratio of gear pair εd=1.463. That means during the whole mesh circle, the gear pairs are between single teeth meshing areas and double single teeth meshing areas alternatively. Therefore, it is suitable to have the analytic model with five gear teeth and the inner ring for simplification.

Based on Eqs. (19−25), the angles from each key point to pitch point P are as follows: θ1=4.301°, θ2= −4.301°, θ3=14.646°, θ4=−14.646°, θ5= 20.853° and θ6=−20.853°. And then, the double tooth mesh area of engaging-out is from −14.646° to −4.301°, the single tooth mesh area ranges from −4.301° to 4.301°, and the double tooth mesh area of engaging-in is from 4.301° to 14.646°. Therefore, the theoretical rotating angle of a gear tooth from the moment of engaging-in to the moment of engaging-out can be calculated as ∆θ=θ3−θ4= 29.292°. Taking the point P as the reference point, it is suitable to choose the degree range from −17° to 15° with 0.5° intervals as the research range, which is efficient to describe the whole process of one pair of gear teeth engagement.

The finite element method for contact problem applied in this work is suitable to consider the mesh situation of multiple teeth pairs at the same time. Based on the elasticity theory, the established models can not only involve the multiple deformations of bend, shear, contact and compression, but also contain the state of stress and strain as well when several teeth pairs are in the mesh process at the same time. The established finite element models are shown in Fig. 6. The results of contact zone when the mesh position at the pitch point is illustrated in Fig. 7.

Fig. 6 Finite element models

4 Calculation results and discussions

The mesh stiffness indicates the capacity to resist deformation when the gear pair is engaged. Because of the alternation of single tooth contact and double tooth

Fig. 7 Contact zone diagram when mesh position at pitch point contact, mesh stiffness is time-varying in practice. The mesh stiffness is an internal excitation for gear drive and is regard as the time-varying coefficient in the terms of elastic force in dynamic formula. Therefore mesh stiffness is a parametric excitation for the gear vibration, which is one of important parameters in the research of gear vibration. Within the elastic ranges, mesh stiffness denotes the proportional coefficient of an applied load and the displacement, namely, the force needed by a unit displacement. If we suppose that the deformation of a single gear tooth is )2 ,1( jj under the force of F, where j=1 represents the driving gear and j=2 the driven gear, the single tooth mesh stiffness can be calculated as

jj

Fk

(26)

The gear pair mesh synthesizing stiffness can be

obtained as

21

21

21sum kk

kkFK

(27)

4.1 Mesh stiffness under different modification

coefficient ratio f When the torque (T=52250 N·mm) was applied, the

modified angle ∆α=0.5°, the mesh stiffness of gears under different modification coefficient ratios with and without modification is shown in Fig. 8, where Fig. 8(a) shows single tooth mesh stiffness curves (STMS for short) and Fig. 8(b) shows gear pair mesh synthesizing stiffness (GPMSS for short). The three key mesh areas are illustrated by A, B and C. Through the analysis based on Fig. 8, following results can be obtained,

1) Comparing the modification gear pairs and unmodified gear pairs in the single teeth mesh area B, the STMS without modification is lager than with modification situation (Fig. 8(a)). The highest point of STMS curve is in the degree of 0° when the gear pair meshes in the pitch point. However, the GPMSS with unmodified in region B is the lowest level in the whole

J. Cent. South Univ. (2015) 22: 1676−1684

1681

mesh circle (Fig. 8(b)). 2) Within the double teeth mesh areas A and C, the

STMS ascends from zero in a−b segment when engaging- in, and descend to zero again in c−d segment when engaging-out (Fig. 8(a)), the value of STMS in these areas is lower than other areas. As shown in Fig. 8(b), the value level of GPMSS in the same areas is the highest, because there are gear teeth nearby sharing the load.

3) Under the same modification angle, different modification lengths influence the mesh stiffness differently. The modification amount increases with the modification coefficient ratio. When ∆α=0.5°, the STMS decreases as the modification coefficient ratio increases in region A and region C. While in region B, the quantity changes scarcely, except for the extreme circumstance f=1 (Fig. 8(a)). With regard to GPMSS, it also decreases as the modification coefficient ratio increases in region A and region C. Particularly, the mesh synthesizing stiffness in region A or region C is lower than it in region B when f>1/2, as shown in Fig. 8(b). Therefore, the modification coefficient has an influence on mesh stiffness in double teeth mesh areas, and the stiffness will decrease when the modification length increases.

Fig. 8 Mesh stiffness for driving gear with and without

modification under different modification coefficient ratios:

(a) Single tooth mesh stiffness; (b) Gear pair mesh synthesizing

stiffness

4.2 Mesh stiffness under different modification angle ∆α When the torque (T=52250 N·mm) was applied, the

modified coefficient ratios f=1/3, f=1/2, f=2/3 and f=1, the mesh stiffness of gears under different modification angles with and without modification in the whole mesh process of a single tooth are shown in Fig. 9, where Figs. 9(a)−(d) are STMS and Figs. 9(e)−(h) is GPMSS. The three key mesh areas are illustrated by A, B and C. Through the analysis based on Fig. 9, the following results can be obtained,

1) In Figs. 9(a)−(d), the STMS of modified gear is larger in region B than in regions A and C. In double teeth mesh areas, the modification curves decreases with the modification angle increasing when the modification coefficient ratios is the same value.

2) In Figs. 9(e)−(h), the GPMSS varies different with different modification angles. When f=1/3, the GPMSS in region A or region C is larger than that in region B, and the GPMSS is at higher level compared with other modification angles in the whole mesh process. When f=1/3, f=2/3 and f=1, the GPMSS in region B is larger than that in region A or C. With the same modification coefficient ratio, the profile styles of stiffness cures of different modification angles are similar. The modification angle influences the peak value of the stiffness curves, and the stiffness curves decrease with the modification angles increasing in the modification areas. 5 Spectral characteristic of stiffness

excitation

There are sudden changes in the curve of GPMSS when gear pairs are in contact according to previous results, which will arouse substantial stiffness excitation. Since the stiffness excitation has a feature of periodicity, which can be described as periodic functions like a square wave, it can be expanded into the Fourier series to obtain its spectral signature. It will help to analyze the stiffness excitation in the frequency domain. In allusion to GPMSS curves mentioned above, Fourier spectrums of GPMSS under different modification parameters can be obtained in Fig. 10 and Fig. 11.

The Fourier spectra of actuation with and without modification under different modification coefficient ratios when the modification angle are 0.5° is shown in Fig. 10. In Fig. 10, the abscissa is frequency, the ordinate is amplitude, and every 5 datum are within a set of data with same frequency. According to the schematic, it can be concluded as follows.

1) The GPMSS excitation is a composition of multiple harmonic waves. At the points of the fundamental, the first, the fourth and the fifth harmonic

J. Cent. South Univ. (2015) 22: 1676−1684

1682

Fig. 9 Mesh stiffness for driving gear under different modification angles: (a, b, c, d) Single tooth mesh stiffness; (e, f, g, h) Gear pair

mesh synthesizing stiffness

J. Cent. South Univ. (2015) 22: 1676−1684

1683

Fig. 10 Fourier spectrum of GPMSS with and without

modification under different modification coefficient ratios

frequency, the stiffness excitation has relatively large amplitudes without modification. As the frequency rises up, the corresponding amplitudes are smaller enough to dismiss. The amplitudes of the first six harmonics with modification are smaller than those without modification, which can be concluded that involute modification can weaken the stiffness excitation during engagement.

2) With the same modification angles, different modification amounts have different impacts on stiffness excitation. When modification angle ∆α=0.5°, as the modification coefficient ascends, the amplitudes of the fundamental, the first, and the forth harmonic, descend. The more the decline of the modification coefficient ratio, the more the decrease of the main amplitudes. For the fifth harmonic, the amplitude of the curves is larger when f=1 and f=2/3.

Figure 11 reveals the Fourier spectrum of actuation under different modification angles when the modification coefficients are constants, and it can be concluded as follows.

Different modification coefficient ratios result in different ranges of harmonic frequencies. When f=1/3, the amplitudes of the first six harmonics are relatively large, except for the one at the fourth harmonic frequency. Under the other three coefficient ratios in Fig. 11, the fourth and the fifth harmonics have less contribution to the whole excitation.

The modification angle affects the spectral amplitude of GPMSS. As the modification angle becomes larger, the amplitudes of the first two harmonics decrease, which contribute the most for the whole actuation. When f=1, the second harmonic spectral

Fig. 11 Fourier spectra of GPMSS under different modification angles: (a) f=1/3; (b) f=1/2; (c) f=2/3; (d) f=1

J. Cent. South Univ. (2015) 22: 1676−1684

1684

amplitude reduces as the modification angle increases, whereas the one with other modification coefficient ratios varies scarcely. 6 Conclusions

1) Different modifications have different effects on mesh stiffness of gear pairs, and appropriate modifications can help to get better involute profile to decrease the sudden changes of stiffness curves.

2) In the frequency domain of the GPMSS spectrum, if the frequency ingredients are relatively simple and corresponding amplitudes are small, the synthesizing stiffness excitation is feeble. With the modification proposed, the amplitudes of dominating harmonics can be impaired, availing to lower the stiffness excitation, which can be a guidance to the study on noises and vibrations of gears.

3) The involute slope modification presented can help to obtain appropriate mesh stiffness curves, providing a new thought for gears modification. References [1] CHEN Zai-gang, SHAO Yi-min. Mesh stiffness calculation of a spur

gear pair with tooth profile modification and tooth root crack [J].

Mechanism and Machine Theory, 2013, 62: 63−74.

[2] BONORI G, BARBIERI M, PELLICANO F. Optimum profile

modifications of spur gears by means of genetic algorithms [J].

Journal of Sound and Vibration, 2012, 331: 4825−4829.

[3] FAGGIONI M, SAMANI S F, BERTACCHI G, PELLICANO F.

Dynamic optimization of spur gears [J]. Mechanism and Machine

Theory, 2011, 46: 544−557.

[4] ZHANG He, HUA Lin, HAN Xin-hui. Computerized design and

simulation of meshing of modified double circular-arc helical gears

by tooth end relief with helix [J]. Mechanism and Machine Theory,

2010, 45: 46−64.

[5] HU Yu-mei, SHAO Yi-min, CHEN Zai-gang. Transient meshing

performance of gears with different modification coefficients and

helical angles using explicit dynamic FEA [J]. Mechanical Systems

and Signal Processing, 2011, 25: 1786−1802.

[6] WU Yong-jun, WANG Jian-jun, HAN Qin-kai. Static/dynamic

contact FEA and experimental study for tooth profile modification of

helical gears [J]. Journal of Mechanical Science and Technology,

2012, 26: 1409−1417.

[7] LI Shu-ting. Finite element analyses for contact strength and bending

strength of a pair of spur gears with machining errors, assembly

errors and tooth modifications [J]. Mechanism and Machine Theory,

2007, 42: 88−114.

[8] LITVIN F L, LIAN Qi-ming, KAPELEVICH A L. Asymmetric

modified spur gear drives: reduction of noise, localization of contact,

simulation of meshing and stress analysis [J]. Computer Methods in

Applied Mechanics and Engineering, 2000, 188: 363−390.

[9] QIN Hui-bin, LU Ming, SHE Yin-zhu, WANG shi-ying, LI

Xiang-peng. Modeling and solving for transverse vibration of gear

with variational thickness [J]. Journal of Central South University,

2013, 20: 2124−2133.

[10] TANG Jin-yuan, HU Ze-hua, WU Li-juan, CHEN Si-yu. Effect of

static transmission error on dynamic responses of spiral bevel gears

[J]. Journal of Central South University, 2013, 20: 640−647.

[11] DENG Song, HUA Lin, HAN Xing-hui, HUANG Song. Finite

element analysis of contact fatigue and bending fatigue of theoretical

assembling straight bevel gear pair [J]. Journal of Central South

University, 2013, 20: 279−292.

[12] TANG Jin-yuan, LIU Yan-ping. Loaded multi-tooth contact analysis

and calculation for contact stress of face-gear drive with spur

involute pinion [J]. Journal of Central South University, 2013, 20(1):

354−362.

[13] SMITH J D. Gear noise and vibration [M]. New York: Marcel

Dekker, 2003.

[14] LITVIN F L, EGELI A. Handbook on face gear drives with a spur

involute pinion [R]. Chicago: The Boeing Company, 2000.

[15] BARONE S, BORGIANNI L, FORTE P. Evaluation of the effect of

misalignment and profile modification in face gear drive by a finite

element meshing simulation [J]. Journal of Mechanical Design, 2004,

126(5): 916−924.

(Edited by YANG Hua)