Tooth Fracture Detection in Spiral Bevel Gears System by...

Research ArticleTooth Fracture Detection in Spiral Bevel Gears System byHarmonic Response Based on Finite Element Method

Yuan Chen,1,2 Rupeng Zhu,1 Guanghu Jin,1 and Yeping Xiong2

1College of Mechanical and Electrical Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China2Engineering and Environment, University of Southampton, Boldrewood Innovation Campus, Southampton SO16 7QF, UK

Correspondence should be addressed to Rupeng Zhu; [email protected]

Received 11 August 2017; Revised 11 October 2017; Accepted 19 October 2017; Published 3 December 2017

Academic Editor: Zhixing Cao

Copyright © 2017 Yuan Chen et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Spiral bevel gears occupy several advantages such as high contact ratio, strong carrying capacity, and smooth operation, whichbecome one of the most widely used components in high-speed stage of the aeronautical transmission system. Its dynamiccharacteristics are addressed by many scholars. However, spiral bevel gears, especially tooth fracture occurrence and monitoring,are not to be investigated, according to the limited published issues. Therefore, this paper establishes a three-dimensional modeland finite element model of the Gleason spiral bevel gear pair. The model considers the effect of tooth root fracture on the systemdue to fatigue. Finite element method is used to compute the mesh generation, set the boundary condition, and carry out thedynamic load.The harmonic response spectra of the base under tooth fracture are calculated and the influence of main parameterson monitoring failure is investigated as well. The results show that the change of torque affects insignificantly the determination ofwhether or not the system has tooth fracture. The intermediate frequency interval (200Hz–1000Hz) is the best interval to judgetooth fracture occurrence. The best fault test region is located in the working area where the system is going through meshing.Thesimulation calculation provides a theoretical reference for spiral bevel gear system test and fault diagnosis.

1. Introduction

Spiral bevel gears are one of the most important componentsof the aeronautical transmission system. Due to the harshworking environment, failure occurs sometimes, and themost common situation is resonance failure.The failure of thespiral bevel gear transmission system often leads to seriousaccidents; therefore, accurate detection, the positioning ofthe fault, and eliminating hidden danger have very importantsignificance in improving the operating efficiency of the gearsystem.

Most studies were related to spiral bevel gear modelingand tooth contact analysis (TCA); Tsai and Chin [1] devel-oped mathematical modeling of the tooth surface geometryfor bevel gear pairs based on the basic gearing kinematics andinvolute geometry along with the tangent planes geometry.Litvin et al. [2–6] proposed an integrated computerizedapproach for spiral bevel gear drives and simulated meshand contact stress analysis; they reduced the magnitude of

transmission errors for reduction of noise and vibration andfound severe contact stresses areas to increase the enduranceof the gear drives; the proposed results were proved by themanufacturing and test of prototypes. Sheveleva et al. [7, 8]developed a modified computer program for tooth contactanalysis (TCA) for the most general case and providedcomputer codes for contact path, bearing contact, and contactpressure distribution. Simon [9] presented computer aidedtooth contact analysis in mismatched spiral bevel gears anddiscussed the influence of relative position errors of meshingpinion on tooth contact.

About spiral bevel gear fault detection, Zakrajsek et al.[10] applied a variety of gear fault detection techniques toinvestigate the growth and propagation of the fault; therelationship between the system running time and piniondamage was specified as well. Dempsey et al. [11] developedthe diagnostic tool by collecting vibration and oil debrisdata from fatigue tests performed in Glenn spiral bevelgear fatigue rigs. Decker and Lewicki [12] discussed the

HindawiJournal of Control Science and EngineeringVolume 2017, Article ID 3169172, 8 pageshttps://doi.org/10.1155/2017/3169172

https://doi.org/10.1155/2017/3169172

2 Journal of Control Science and Engineering

Mean cone circle

Outer cone circle

Inner cone

L1

O

j0jk

O1

S0

Sk

m

Virtual cutting gear

Figure 1: Pitch surface unfolded map.

damage detection effectiveness of the different metrics anda comparison of effects of different accelerometer locationsby spiral bevel pinion running test. Ural et al. [13] predictedthe crack shape and fatigue life for a spiral bevel piniongear by using computational fracture mechanics, which wasbased on linear elastic fracture mechanics theories combinedwith finite element method. However, the detection in thefrequency domain and the influence of excitation were notprovided in these papers.

In the area of spiral gear dynamics analysis, Li and Hu[14] analyzed the axial–lateral–torsional coupled spiral bevelgeared system theoretically and the dynamic behavior of thesystem was investigated by numerical method. Yinong et al.[15] studied the effect of the asymmetric mesh stiffness on the8-DOF spiral bevel gear transmission system. Feng and Song[16] investigated the effects of the dynamicmeshing force anddynamic transmission error.

Therefore, through system vibration signal to determinethe occurrence of tooth fracture, thus eliminating the fault,is a meaningful work. In this paper, a pair of Gleasonspiral bevel gears is mathematically modeled and a three-dimensional solid model is generated as well. Finite elementmethod is applied to analyze the harmonic response ofthe system; deformation amplitude and phase of the baseare calculated. Therefore some tooth fracture features andmain parameters’ influences are analyzed to assist the faultrecognition.

2. System Modeling

2.1. Three-Dimensional Precise Solid Modeling. Typical spiralbevel gears applied in aeronautical transmission system aremanufactured by Gleason face hobbing process [17]. Thetooth profile of the spiral bevel gear is a spherical involute,and its tooth profile is circular, and the tooth surface is acomplex three-dimensional surface.Therefore, themethod ofmodeling the cylindrical gear tooth profile is not suitable for

the spiral bevel gear. By adding the auxiliary involute to thetooth line equation, the accuracy of the tooth surface couldbe increased.

In the spherical coordinate system, the spherical involuteequation on one side of outer space width is

𝑟 = 𝑅𝜃 = 𝛿𝑓 + 𝑡 (𝛿𝑎 − 𝛿𝑓)

𝜑 = arccos (cos 𝜃/ cos 𝛿𝑏)sin 𝛿𝑏 − arccos( tan 𝛿𝑏

tan 𝜃 ) ,(1)

where 𝛿𝑏 is base angle; 𝛿𝑓 is root angle; 𝛿𝑎 is tip angle; 𝑅 issphere radius, that is, outer cone distance.

The spherical involute equation on the other side of outerspace width is

𝑟 = 𝑅𝜃 = 𝛿𝑓 + 𝑡 (𝛿𝑎 − 𝛿𝑓) ;

𝜑 = arccos (cos 𝜃/ cos 𝛿𝑏)sin 𝛿𝑏 − arccos( tan 𝛿𝑏

tan 𝜃 )

− (360𝑧 − 𝜙𝑏)

(2)

here, 𝑧 is tooth number; 𝜑𝑏 is tooth thickness base angle.When spiral bevel gears are in cutting processing, tooth

pitch surface unfolded drawing is shown in Figure 1.The distance between cutting tool circle 𝑂1 and cone

circle 𝑂 is depicted as

𝐿1 = √𝑅𝑚2 + 𝑅𝑑2 − 2𝑅𝑚𝑅𝑑 sin𝛽𝑚, (3)

where 𝛽𝑚 is mean spiral angle; 𝑅𝑚 is mean cone distance; 𝑅𝑑is the radius of cutting tool.

Journal of Control Science and Engineering 3

(a) Healthy model (b) Tooth fracture model

Figure 2: Spiral bevel gear assembly model.

In circle 𝑂1, offset angle 𝑆0 of outer cone circle and 𝑆𝑘 ofinner cone circle are

𝑆0 = 180 − arccos𝐿1 + 𝑅𝑑2 − 𝑅2

2𝐿1𝑅𝑑𝑆𝑘 = 180 − arccos

𝐿1 + 𝑅𝑑2 − (𝑅 − 𝐵)22𝐿1𝑅𝑑 ;

(4)

here, 𝐵 is tooth thickness.In circle 𝑂, offset angle 𝑗0 of outer cone circle and 𝑗𝑘 of

inner cone circle are

𝑗0 = arctan𝑅𝑑 sin 𝑆0

𝐿1 + 𝑅𝑑 cos 𝑆0𝑗𝑘 = arctan

𝑅𝑑 sin 𝑆𝑘𝐿1 + 𝑅𝑑 cos 𝑆𝑘 .

(5)

According to the spherical geometrical relationship,spherical angle 𝑞 corresponding to plane angle 𝑗 could be

𝑞0 = 𝑗0sin 𝛿

𝑞𝑘 = 𝑗𝑘sin 𝛿 ;

(6)

here, 𝛿 is reference cone angle.Thus, 𝑞0𝑘 = 𝑞𝑘 − 𝑞0, that is, the rotation angle between

outer cone and inner cone in the spherical coordinate system.The spherical involute equation on one side of inner space

width could be derived by substituting 𝑟 to (R-B) and addingthe value of 𝜑 to 𝑞0𝑘 in (1). Similarly, involute equation on theother side of inner space width could be deduced as well.

In order to improve the accuracy of the three-dimensional model, in this paper, a number of equallyspaced auxiliary spherical involute lines are inserted in thedirection of the tooth line. Substitute (R-B) to (R-0.1nB) in(1) and (4), and 𝑛 is followed by 1, 2, 3 . . . 9 . . ., then involuteequation could be rederived.

Based on the above-mentioned involute equation, keypoints of the curve are established by “law command” inCATIA. Then “spline command” is applied to connect thesekey points for the tooth profile involute curve, and “multisec-tion surface command” is used to generate the tooth surface.Moreover, tooth space ismodeled by “split command,” which

Table 1: System parameters.

Active gear Driven gearModulus 6Pressure angle (∘) 20Tooth number 15 46Tooth width (m) 0.44 0.44Shaft angle Σ(∘) 90Mean spiral angle (∘) 35

is based on these tooth surfaces. Finally, with the Booleansubtraction calculation of the solids, the parametricmodelingof the spiral bevel gear could be successfully achieved, whichverifies the correctness of the design method [18].

In addition, due to long-term operation in the high-speedcircumstance, tooth root is prone to fatigue fracture. Thispaper simulates the fractured tooth and completes the healthyand fracture spiral bevel gear assembly model, as shown inFigure 2; main parameters are listed in Table 1.

2.2. Finite Element Meshing and Pretreatment Process. Thegeometric model in CATIA is introduced into ANSYS work-bench 18.1, so finite element analysis model is obtained. Asfriction contact analysis is a nonlinear problem, in orderto save the computing resources and improve calculationaccuracy, hexahedral meshing method is applied for calcula-tion, and then the grid is refined in contact areas. The gridsize of the noncontact region of this model is defined as5mm, and the contact area is defined as 1mm, which ensuresthe rationality of the mesh and the maximum efficiency. Atthe same time, in order to ensure the convergence of thecalculation results, the meshing model is firstly calculated,and then the mesh size of the contact area is graduallyreduced. If the difference between calculation results issmall after multiple attempts, the result is convergent andacceptable. Figure 3 shows the meshing model of spiral bevelgear pair; the results tend to converge when the numberof nodes is 151443 and the number of elements is 47982 inhealthymodel; the number of nodes is 151749 and the numberof elements is 47728 in tooth fracture model.

After mesh is complete, the model is processing multiplesettings, that is, pretreatment process:

(1) Material definition: the model is set according tosteel properties, the active and driven gears belong to


(a) Healthy model (b) Tooth fracture model

(c) Contact area of healthy model (d) Contact area of tooth fracture model

Figure 3: Spiral bevel gear meshing.

the same material, Young’s modulus is 2.1 × 1011 Pa,Poisson’s ratio is 0.3, and the density is 7850 kg/m3.

(2) Contact definition: the calculation of the spiralbevel gear pair is surface-to-surface contact problembetween the elastomers. It is of great importance toselect contact surface and target surface; the inap-propriate selection will lead to excessive penetration,affecting the accuracy of the solution. In general,when the convex and concave surfaces contact, theconcave surface should be set as the target surface, sothe spiral bevel gear outer tooth surface is defined asthe target surface. In addition, the amount of pene-tration between the two contact surfaces depends onthe normal contact stiffness. If the normal contactstiffness is too large, it will increase solution iterationnumber, which may lead to nonconvergence; if thenormal contact stiffness is too small, the penetra-tion between nodes could be too large, resulting inmodel instability. Based on the above analysis, thispaper firstly uses the small normal contact stiffnesscoefficient and then gradually increases until analysisresults deviation is so small. The results show that thebest normal contact stiffness coefficient is 1.0 whenthe normal contact stiffness coefficient is set from0.001, 0.01, 0.1, 1.0, and 1.1.

(3) Rotational pair definition: the inner surfaces of thegears are set as reference surfaces, and then therotational centers of reference surfaces are created.Finally, a pair of rotational gears is defined.

(4) Solver definition: the general equation for harmonicresponse analysis is

[𝑀] {�̈�} + [𝐶] {�̇�} + [𝐾] {𝑋} = {𝐹} , (7)

where [𝑀], [𝐶], and [𝐾] are the mass matrix, damp-ing matrix, and stiffness matrix of the system. Matrix[𝐹] is external excitation and is equal to 𝐹0 cos(𝜔𝑡).The time-varying meshing stiffness, meshing linedisplacement, and dynamic meshing force can beregarded as periodic format and can be expanded inFourier series under fundamentalmeshing frequency.In this paper, dynamic meshing force is simulatedas the excitation load, the torque is loaded in thesinusoidal form, the sweep frequency is set from 0to 2000Hz, initial phase angle is 0∘, and solutionintervals are 100.

(5) Boundary conditions and load setting: degree of free-dom (DOF) is released only in rotational direction,and the value of torque is 1200N⋅m,which is graduallyapplied on the active gear.

3. Fault Diagnosis Analysis

3.1. Tooth Fracture Detection. The system is solved by theaugmented Lagrangian method in ANSYS and the first sixnatural frequencies of the system are obtained, as shown inTable 2. As illustrated in the table, the natural frequency of thetooth fracture model decreases in each order; the frequencydeviation of the fourth and sixth order is relatively large.

The base of the driven gear is the optimum positionfor deformation vibration detection sensor and accelerationvibration detection sensor so that the base is also set inresponse to the output in finite element method. Figure 4shows the calculated deformation and acceleration harmonicresponse maps.

Here, in order to separate ideal monitoring frequency,three frequency intervals are defined to describe the sweepfrequency, that is, low frequency interval (below than 200Hz), intermediate frequency interval (200Hz to 1000Hz),and high frequency interval (greater than 1000Hz) [19].


Table 2: Natural frequency.

Order Healthy tooth model frequency (Hz) Fracture tooth model frequency (Hz) Deviation (Hz)(1) 18.4 12.0 6.4(2) 168.0 140.8 27.2(3) 206.5 147.3 59.2(4) 678.1 571.6 106.5(5) 1061.2 1042.4 18.8(6) 1543 1367.3 175.7

0.015

0.01

0.005

0

Def

orm

atio

n am

plitu

de (m

)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

HealthyCrack

(a) Deformation amplitude

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

180

160

140

120

100

80

60

40

20

0

Phas

e ang

le (∘

)

HealthyCrack

(b) Deformation phase angle

HealthyCrack

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

12

10

8

6

4

2

0

Acce

lera

tion

(m/Ｍ

2)

×104

(c) Acceleration amplitude

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

180

160

140

120

100

80

60

40

20

0

Phas

e ang

le (∘

)

HealthyCrack

(d) Acceleration phase angle

Figure 4: Harmonic response spectrum of spiral bevel gear.

In low frequency interval, the tooth fracture model hasa significant peak in the deformation amplitude spectrum,while the healthy tooth model has a valley in this interval.However, in the acceleration amplitude spectrum, the peaksof two models are equivalent.

In intermediate frequency interval, the tooth fracturemodel does not show an obvious peak in deformation and

acceleration amplitude spectrum, while the peak value ofhealthy tooth model is very obvious. And there is no otherinterference signal in this interval and adjacent interval, soit can be effective in monitoring frequency interval for toothfracture detection.

In high frequency interval, the acceleration amplitudespectrum is peaked at about 1050Hz in fracture model, and


0.014

0.012

0.01

0.008

0.006

0.004

0

0.002

Def

orm

atio

n am

plitu

de (m

)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

1200 N·m1100 N·m1000 N·m

900 N·m800 N·m


0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

180

160

140

120

100

80

60

40

20

0

Phas

e ang

le (∘

)

800 N·m


Figure 5: Influence of torque on harmonic response of spiral bevel gear.

the peak of healthy tooth model is four times that of thefracture model.

Comparedwith Figures 4(a) and 4(c), several peaks of thefracture are in the vicinity of the sidebands of the healthy,and for each corresponding peak, the fracture is less than thehealthy; as is shown in Figure 4(a), the peaks also graduallyreduce with the increase of the frequency, so high frequencyinterval cannot easily detect fracture tooth signal. However,acceleration signal is mainly concentrated in intermediateand high frequency intervals.

Compared with Figures 4(b) and 4(d), it can be seenthat deformation phase spectrum and acceleration phasespectrum are corresponding to the contrary. And during thesweep process, the phase will be reversed between 0∘ and180∘ in different frequencies. In low frequency interval, thehealthy and the fracture coincide and separate from eachother quickly, so it is difficult to judge whether the tooth isfractured. The phase mutation inversion is most pronouncedat about 600Hz in the intermediate interval, and it is depictedin both the deformation and acceleration phase spectra. Inaddition, phase reversal at 1400Hz in high frequency intervalis also relatively obvious.

In brief, low frequency, intermediate frequency, and highfrequency interval sections all have tooth fracture charac-teristic signal; however, the intermediate frequency interval(200Hz–1000Hz) is the best interval for detection amongothers.

3.2. Influence of Torque on Harmonic Response of Spiral BevelGear. In order to investigate the effect of torque (dynamicmeshing force) and other excitations on the harmonicresponse of the spiral bevel gear system, tooth fracturemodel is target object, and the torque of different sizes is

loaded without changing the other pretreatment process, soexcitation’s influence on tooth fracture detection could beobtained. The harmonic response of each load is calculated,respectively, and the result is shown in Figure 5. It can beseen from Figure 5(a) that the torque magnitude has a greatinfluence on the peak value in low frequency interval, andthe peak value increases with the torque obviously. However,the torque magnitude has little impact on the intermediatefrequency and high frequency intervals. As can be seen fromFigure 5(b), the phase spectrum does not change with theincrease of torque; that is, torque does not affect the harmonicresponse phase. This conclusion successfully verifies theexperimental test in [12].

3.3. Influence of Measuring Point Position on HarmonicResponse of Spiral Bevel Gear. In order to improve the accu-racy of the test, the influence of the measuring point positionon the harmonic response of the spiral bevel gear systemis studied. The responses of the different measuring pointsare calculated without changing the other pretreatment, sothe best monitoring area could be found. The position ofthe test points is shown in Figure 6, where point A isthe measuring point near the working area. The harmonicresponse spectra of each measurement point are calculated,respectively, and the results are shown in Figure 7. It canbe seen from Figure 7(a) that the peak of point C is moreobvious in the low frequency interval, and the peak of point Ais prominent in the intermediate frequency interval, whereasthe vibration of each point is not visible at high frequencyinterval. According to Figure 7(b), each point has obviousphase mutation inversion at about 600Hz (intermediatefrequency interval) and 1400Hz (high frequency interval).Based on the above analysis and the conclusion in Section 3.1,


Figure 6: Measuring points position.

0.14

0.12

0.1

0.08

0.06

0.04

0.02

0

Def

orm

atio

n am

plitu

de (m

)

0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

Measuring point AMeasuring point BMeasuring point C

Measuring point D


0 200 400 600 800 1000 1200 1400 1600 1800 2000

Frequency (Hz)

180

160

140

120

100

80

60

40

20

0

Phas

e ang

le (∘

)

Measuring point AMeasuring point BMeasuring point C

Measuring point D


Figure 7: Harmonic responses of spiral bevel gear under different measuring points.

it can be seen that the measuring point near the working areais the best test point when the system is monitored underintermediate frequency interval.

4. Conclusion

Based on the meshing principle and gear cutting process, thetooth involute equation of the bevel gear is obtained, and theparametric spiral bevel gear three-dimensionalmodel and thefinite element model are established.

The analysis results draw the following conclusions:(1) Compared with the healthy tooth model, natural

frequency in each order decreases in tooth fracture model,and the natural frequency deviation of the fourth and sixthorder is relatively large. Although the low frequency, inter-mediate frequency, and high frequency interval section havetooth fracture signals, the intermediate frequency interval

(200Hz–1000Hz) is the best interval to determine whetherthe tooth is fractured.

(2) The effect of torque on the harmonic response of thespiral bevel gear is quite little, and the change of excitationdoes not affect the fractured tooth monitoring.

(3)Theamplitude of the vibration signal near theworkingarea is relatively strong, and it is the bestmeasuring point areafor detecting the fractured tooth signal.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The work described in this paper was fully supported bythe National Natural Science Foundation of PRC (Grant nos.


51375226 and 51475226); Postgraduate Research and PracticeInnovation Program of Jiangsu Province; China ScholarshipCouncil’s support for joint research with Professor YepingXiong in University of Southampton.

References

[1] Y. C. Tsai and P. C. Chin, “Surface geometry of straight andspiral bevel gears,” Journal of Mechanisms, Transmissions, andAutomation in Design, vol. 109, no. 4, pp. 443–449, 1987.

[2] F. L. Litvin, A. Fuentes, Q. Fan, and R. F. Handschuh, “Com-puterized design, simulation of meshing, and contact and stressanalysis of face-milled formate generated spiral bevel gears,”Mechanism and Machine Theory, vol. 37, no. 5, pp. 441–459,2002.

[3] F. L. Litvin, A. Fuentes, and K. Hayasaka, “Design, manufac-ture, stress analysis, and experimental tests of low-noise highendurance spiral bevel gears,”Mechanism and Machine Theory,vol. 41, no. 1, pp. 83–118, 2006.

[4] A. Fuentes, F. L. Litvin, B. R. Mullins, R. Woods, and R. F.Handschuh, “Design and stress analysis of low-noise adjustedbearing contact spiral bevel gears,” Journal ofMechanicalDesign,vol. 124, no. 3, pp. 524–532, 2002.

[5] I. Gonzalez-Perez, A. Fuentes, and K. Hayasaka, “Analyticaldetermination of basic machine-tool settings for generationof spiral bevel gears from blank data,” Journal of MechanicalDesign, vol. 132, no. 10, Article ID 101002, 2010.

[6] J. Argyris, A. Fuentes, and F. L. Litvin, “Computerized inte-grated approach for design and stress analysis of spiral bevelgears,” Computer Methods Applied Mechanics and Engineering,vol. 191, no. 11-12, pp. 1057–1095, 2002.

[7] G. I. Sheveleva, A. E. Volkov, and V. I. Medvedev, “Algorithmsfor analysis of meshing and contact of spiral bevel gears,”MechanismandMachineTheory, vol. 42, no. 2, pp. 198–215, 2007.

[8] F. L. Litvin, G. I. Sheveleva, D. Vecchiato, I. Gonzalez-Perez,and A. Fuentes, “Modified approach for tooth contact analysisof gear drives and automatic determination of guess values,”ComputerMethods AppliedMechanics and Engineering, vol. 194,no. 27-29, pp. 2927–2946, 2005.

[9] V. Simon, “Computer simulation of tooth contact analysis ofmismatched spiral bevel gears,” Mechanism and Machine The-ory, vol. 42, no. 3, pp. 365–381, 2007.

[10] J. Zakrajsek, R. Handschuh, andH. Decker, “Application of faultdetection techniques to spiral bevel gear fatigue data,” 1994.

[11] P. J. Dempsey, R. F. Handschuh, and A. A. Afjeh, “Spiralbevel gear damage detection using decision fusion analysis,” inProceedings of the 5th International Conference on InformationFusion, pp. 94–99, July 2002.

[12] H. Decker and D. Lewicki, “Spiral bevel pinion crack detectionin a helicopter gearbox,” 2003.

[13] A. Ural, G. Heber, P. A. Wawrzynek, A. R. Ingraffea, D. G.Lewicki, and J. B. C. Neto, “Three-dimensional, parallel, finiteelement simulation of fatigue crack growth in a spiral bevelpinion gear,” Engineering Fracture Mechanics, vol. 72, no. 8, pp.1148–1170, 2005.

[14] M. Li and H. Y. Hu, “Dynamic analysis of a spiral bevel-gearedrotor-bearing system,” Journal of Sound and Vibration, vol. 259,no. 3, pp. 605–624, 2003.

[15] L. Yinong, L. Guiyan, and Z. Ling, “Influence of asymmetricmesh stiffness on dynamics of spiral bevel gear transmissionsystem,”Mathematical Problems in Engineering, vol. 2010, Arti-cle ID 124148, 13 pages, 2010.

[16] Z. Feng and C. Song, “Effects of geometry design parameters onthe static strength and dynamics for spiral bevel gear,” Inter-national Journal of Rotating Machinery, vol. 2017, Article ID6842938, 8 pages, 2017.

[17] Q. Fan, “Computerized modeling and simulation of spiralbevel and hypoid gears manufactured by Gleason face hobbingprocess,” Journal of Mechanical Design, vol. 128, no. 6, pp. 1315–1327, 2006.

[18] K. Shi, J. Xia, and C. Wang, “Design of noncircular bevelgear with concave pitch curve,” Proceedings of the Institution ofMechanical Engineers, Part C: Journal ofMechanical EngineeringScience, vol. 227, no. 3, pp. 542–553, 2012.

[19] W. Levine,The Control Handbook, CRC Press, Boca Raton, Fla,USA, 1996.

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