research article detection and localization of...

Research ArticleDetection and Localization of Tooth Breakage Fault onWind Turbine Planetary Gear System considering GearManufacturing Errors

Y. Gui,1,2 Q. K. Han,1 Z. Li,1 and F. L. Chu1

1 State Key Laboratory of Tribology, Department of Mechanical Engineering, Tsinghua University, Beijing 100084, China2Department of Mechanical Engineering, Academy of Armored Force Engineering, Beijing 100072, China

Correspondence should be addressed to F. L. Chu; [email protected]

Received 18 May 2014; Revised 23 July 2014; Accepted 27 July 2014; Published 19 August 2014

Academic Editor: Ahmet S. Yigit

Copyright © 2014 Y. Gui et al. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Sidebands of vibration spectrum are sensitive to the fault degree and have been proved to be useful for tooth fault detectionand localization. However, the amplitude and frequency modulation due to manufacturing errors (which are inevitable in actualplanetary gear system) lead tomuchmore complex sidebands.Thus, in the paper, a lumped parameter model for a typical planetarygear system with various types of errors is established. In the model, the influences of tooth faults on time-varying mesh stiffnessand tooth impact force are derived analytically. Numerical methods are then utilized to obtain the response spectra of the systemwith tooth faults with and without errors. Three system components (including sun, planet, and ring gears) with tooth faults areconsidered in the discussion, respectively. Through detailed comparisons of spectral sidebands, fault characteristic frequencies ofthe system are acquired. Dynamic experiments on a planetary gear-box test rig are carried out to verify the simulation results andthese results are of great significances for the detection and localization of tooth faults in wind turbines.

1. Introduction

Planetary gear systems have been widely used in wind powersystems because of the advantages of compact structure, largecarrying capacity, and high transmission efficiency [1]. Inrecent years, the tooth faults that have occurred in planetarygear systems have brought numerous troubles to wind powerplants [2–4]. Therefore, fault diagnosis of the planetary gearsystem is of great significant to the safe operation of windturbine.

Tooth pitting, spalling, cracking, and breakage are someof the common fault modes that have occurred in windturbine planetary gear systems. Detection and localization ofthose faults are of great significance. Vibration based diag-nosis is one of the most effective and frequently used healthmonitoring technologies. In recent years, much attention hasbeen paid on analyzing the vibration signal in time andfrequency domains. To name a few, Lei et al. [5] extractedtwo diagnostic parameters based on the examination of

vibration characteristics of a planetary gearbox in bothtime and frequency domains. Experiments revealed that theproposed diagnostic parameters performed better than otherparameters. Lin and Zuo [6] introduced a method basedon the Morlet wavelet time-frequency analysis. The methodwas found more effective in detection of tooth cracks thantwo other types of discrete wavelet transform. Liu et al.[7] carried out fault diagnosis based on the local meandecomposition (LMD) method. The method was suitable forobtaining instantaneous frequencies in frequency domains.

The above-mentioned studies are helpful for the faultdetection of planetary gear system. In addition, the side-bands of vibration signals also attract the attention of manyresearchers, because the sidebands contain abundant infor-mation which is useful for the detection and localizationof faults in planetary gear systems. McFadden and Smith[8] found that the modulation sidebands of the planetarygear system are typically asymmetric and the frequency

Hindawi Publishing CorporationShock and VibrationVolume 2014, Article ID 692347, 13 pageshttp://dx.doi.org/10.1155/2014/692347

2 Shock and Vibration

with largest amplitude is the multiple of the carrier rotationfrequency. Based on the research, McFadden [9] presenteda method to calculate the time domain average of thevibration signal and the method for the separation of planetvibration and sun gear vibration is extracted. Feng and Zuo[10] deduced the relationship between fault characteristicfrequencies and rotating frequencies of the components ina planetary gear system. The complicity of sidebands andfault characteristic frequencies were verified by simulationsand experiments. Chaari et al. [11] modeled a planetary gearsystem which contained the fault modes of tooth pittingand cracking. The dynamic responses of the fault systemsand a healthy gear system were compared in frequencydomains. Chaari et al. [12] also studied the influence of toothpitting and cracking on the sun gear vibration spectrumbased on a planetary gear model. Sidebands were found tobe induced because of amplitude modulation. Cheng et al.[13, 14] estimated the degree of gear pitting and crackingby comparing the dynamic response of a fault system witha health system. A model containing tooth spalling andcracking was developed by Jia and Howard [15]. His researchindicated that the amplitude and phase modulation of thevibration signal could indicate the difference between toothspalling and cracking effectively.

However, various gear manufacturing errors were notconsidered in the above-mentioned investigations. Typicalgear errors, such as eccentricities, pitch-line run-out errors,tooth spacing, and indexing errors, are inevitable in actualplanetary gear system. Ligata et al. [16] pointed out that theaforementioned errors of a planetary gear set might causemodulations to the sidebands of the system. Inalpolat andKahraman [17] developed a nonlinear time-varying dynamicmodel to predict modulated sidebands of a planetary gearsystem, and both the numerical and experimental resultsshowed that the sidebands were greatly changed due to theamplitude and frequency modulations caused by gear errors.As the gear errors can influence the sidebands significantly[16, 17], the fault characteristics might be greatly changedafter considering the gear errors. Therefore, it is necessary totake the influence of errors into consideration to increase theaccuracy and precision of the tooth faults diagnosis.

In this paper, a lumped parameter model of a typicalplanetary gear system is established.The influences of typicaltooth faults on the time-varying mesh stiffness and toothimpact forces are derived analytically. Numerical methodsare utilized to obtain the response spectra of the system withtooth faults with and without gear errors. Three system com-ponents (including sun, planet, and ring gears) with toothfaults are considered, respectively, in the discussion.Throughdetailed comparisons on spectral sidebands, fault charac-teristic frequencies of the system are acquired. Dynamicexperiments on a planetary gear-box test rig are carried outto verify the simulation results.

2. Dynamic Model of Planetary Gear System

Figure 1 shows the schematic diagram of the dynamic modelfor a planetary gear system. The system composes of sun

Planet 1

Planet 2

Sun

Ring

x2

kp2

y2

ksx

usksy

ks1

u1

ks2

u2ksu

kp1kr1

uc

xc xr xs

x1

krx

ur

y1

kru

kcu

ycysyr

kr2

o

j

Carrier

kcy

kcx 𝜑pii

Figure 1: The dynamic model for a planetary gear system.

gear (𝑠), carrier (𝑐), ring (𝑟), and three planets (𝑝). Bear-ings are considered as linear springs and gear meshes areconsidered as linear springs acting on the lines of action.Each component has three degrees of freedom, transverse (𝑥),longitudinal (𝑦), and torsional degrees (𝑢).

The reference coordinate system rotates with the carrierand the origin of the coordinate coincides with the centerof the carrier. The circumferential positions of planets arespecified by the fixed angles𝜑𝑝𝑖 (Figure 2)measured relativelyto the rotating coordinate system with 𝜑𝑝1 = 0.

The global equation of motion for the system can besimplified as [18, 19]

Mq + Ω𝑐Gq + [K𝑏 + K𝑒 (𝑡) − Ω2

𝑐KΩ] q

= T (𝑡) + F𝑊 (𝑡) + F𝑒 (𝑡) ,(1)

where q represents the vector of the degrees of freedom

𝑞 = {𝑥𝑐, 𝑦𝑐, 𝑢𝑐, 𝑥𝑟, 𝑦𝑟, 𝑢𝑟, 𝑥𝑠, 𝑦𝑠, 𝑢𝑠,

𝑥1, 𝑦1, 𝑢1, . . . , 𝑥𝑁, 𝑦𝑁, 𝑢𝑁}𝑇.

(2)

Ω𝑐 is the rotating speed of the carrier. Due to the unstablewind conditions, frequent low speed start and emergencybrakes, the rotating speed is often nonstationary. However, ifthe concerned time period is small, the work condition of thewind turbine could be considered as quasi-stable [20]. Thus,the premise of constant rotating speed is used in this paper.

Shock and Vibration 3

ksx

ksy

us

xs

kp𝑖

ksp𝑖

kp𝑖

ui

yi

xi

𝜑p𝑖

ys

𝛼sesp𝑖 (t)

Figure 2: A sun-planet mesh.

M,G represent themass and gyroscopicmatrices, respec-tively, and are expressed by

M =

[

[

[

[

[

[

[

[

[

[

[

M𝑐 0 0 0 0 0 ⋅ ⋅ ⋅ 0

0 M𝑟 0 0 0 0 ⋅ ⋅ ⋅ 0

0 0 M𝑠 0 0 0 ⋅ ⋅ ⋅ 0

0 0 0 M1 0 0 ⋅ ⋅ ⋅ 0

0 0 0 0 M2 0 ⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. 0

0 0 0 0 0 0 ⋅ ⋅ ⋅ M𝑁

]

]

]

]

]

]

]

]

]

]

]

,

M𝑗 =[

[

[

[

[

𝑚𝑗 0 0

0 𝑚𝑗 0

0 0

𝐼𝑗

𝑟2𝑗

]

]

]

]

]

, 𝑗 = 𝑐, 𝑟, 𝑠, 1, . . . , 𝑁

G =

[

[

[

[

[

[

[

[

[

[

[

G𝑐 0 0 0 0 0 ⋅ ⋅ ⋅ 0

0 G𝑟 0 0 0 0 ⋅ ⋅ ⋅ 0

0 0 G𝑠 0 0 0 ⋅ ⋅ ⋅ 0

0 0 0 G1 0 0 ⋅ ⋅ ⋅ 0

0 0 0 0 G2 0 ⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. 0

0 0 0 0 0 0 ⋅ ⋅ ⋅ G𝑁

]

]

]

]

]

]

]

]

]

]

]

,

G𝑗 = [

[

0 −2𝑚𝑗 0

2𝑚𝑗 0 0

0 0 0

]

]

, 𝑗 = 𝑐, 𝑟, 𝑠, 1, . . . , 𝑁,

(3)

where𝑚𝑗 and 𝑟𝑗 are the mass and the base radius of part 𝑗, 𝐼𝑗is the inertial moment of part 𝑗 with respect to its rotationalaxis.

K𝑏 is the bearing stiffness matrices and are expressed by

K𝑏 =

[

[

[

[

[

[

[

[

[

K𝑐𝑏 0 0 0 ⋅ ⋅ ⋅ 0

0 K𝑟𝑏 0 0 ⋅ ⋅ ⋅ 0

0 0 K𝑠𝑏 0 ⋅ ⋅ ⋅ 0

0 0 0 0 ⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. 0

0 0 0 0 ⋅ ⋅ ⋅ 0

]

]

]

]

]

]

]

]

]

,

K𝑗𝑏 = [

[

𝑘𝑗𝑥 0 0

0 𝑘𝑗𝑦 0

0 0 𝑘𝑗𝑢

]

]

, 𝑗 = 𝑐, 𝑟, 𝑠,

(4)

where 𝑘𝑗𝑥, 𝑘𝑗𝑦, 𝑘𝑗𝑢 represent the bear supporting stiffness longthe three degrees of freedom.

The centripetal stiffness matrix due to the carrier rotationis denoted by KΩ. Consider

KΩ =

[

[

[

[

[

[

[

[

[

[

[

KΩ𝑐 0 0 0 0 ⋅ ⋅ ⋅ 0

0 KΩ𝑟 0 0 0 ⋅ ⋅ ⋅ 0

0 0 KΩ𝑠 0 0 ⋅ ⋅ ⋅ 0

0 0 0 KΩ1 0 ⋅ ⋅ ⋅ 0

0 0 0 0 KΩ2 ⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. 0

0 0 0 0 0 0 KΩ𝑁

]

]

]

]

]

]

]

]

]

]

]

,

KΩ𝑗 = [

[

𝑚𝑗 0 0

0 𝑚𝑗 0

0 0 0

]

]

, 𝑗 = 𝑐, 𝑟, 𝑠, 1, . . . , 𝑁.

(5)

The time-varying mesh stiffness matrix is represented byK𝑒(𝑡). The matrices are expressed as

K𝑒 (𝑡) =

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[

∑K𝑖𝑐1

0 0 K1𝑐2

K2𝑐2

⋅ ⋅ ⋅ K𝑁𝑐2

0 ∑K𝑖𝑟1

0 K1𝑟2

K2𝑟2

⋅ ⋅ ⋅ K𝑁𝑟2

0 0 ∑K𝑖𝑠1

K1𝑠2

K2𝑠2

⋅ ⋅ ⋅ K𝑁𝑠2

K1𝑐2

K1𝑟2

K1𝑠2

K1 0 ⋅ ⋅ ⋅ 0

K2𝑐2

K2𝑟2

K2𝑠2

0 K2 ⋅ ⋅ ⋅ 0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

. 0

K𝑁𝑐2

K𝑁𝑟2

K𝑁𝑠2

0 0 ⋅ ⋅ ⋅ K𝑁

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

]

, (6)


where

K𝑖𝑟1= 𝑘𝑟𝑝𝑖

(𝑡)[

[

[

sin2𝜑𝑟𝑝𝑖 − cos𝜑𝑟𝑝𝑖 cos𝛼𝑟 sin𝜑𝑟𝑝𝑖− cos𝜑𝑟𝑝𝑖 cos𝜑𝑟 cos2𝜑𝑟𝑝𝑖 cos𝜑𝑟𝑝𝑖

sin𝜑𝑟𝑝𝑖 cos𝜑𝑟𝑝𝑖 1

]

]

]

,

K𝑖𝑐1= 𝑘𝑝𝑖

[

[

1 0 − sin𝜑𝑝𝑖0 1 cos𝜑𝑝𝑖

− sin𝜑𝑝𝑖 cos𝜑𝑝𝑖 1

]

]

,

K𝑖𝑠1

= 𝑘𝑠𝑝𝑖(𝑡)

[

[

[

sin2𝜑𝑠𝑝𝑖 − cos𝜑𝑠𝑝𝑖 sin𝜑𝑠𝑝𝑖 − sin𝜑𝑠𝑝𝑖− cos𝜑𝑠𝑝𝑖 sin𝜑𝑠𝑝𝑖 cos2𝜑𝑠𝑝𝑖 cos𝜑𝑠𝑝𝑖

− sin𝜑𝑠𝑝𝑖 cos𝜑𝑠𝑝𝑖 1

]

]

]

,

K𝑖𝑐2= 𝑘𝑝𝑖

[

[

− cos𝜑𝑝𝑖 sin𝜑𝑝𝑖 0

− sin𝜑𝑝𝑖 − cos𝜑𝑝𝑖 00 −1 0

]

]

,

K𝑖𝑠2= 𝑘𝑠𝑝𝑖

(𝑡)[

[

sin𝜑𝑠𝑝𝑖 sin𝛼𝑠 sin𝜑𝑠𝑝𝑖 cos𝛼𝑠 − sin𝜑𝑠𝑝𝑖− cos𝜑𝑠𝑝𝑖 sin𝛼𝑠 − cos𝜑𝑠𝑝𝑖 cos𝛼𝑠 − cos𝜑𝑠𝑝𝑖

− sin𝛼𝑠 − cos𝛼𝑠 1

]

]

,


(𝑡)[

[

− sin𝜑𝑟𝑝𝑖 sin𝛼𝑟 sin𝜑𝑟𝑝𝑖 cos𝛼𝑟 sin𝜑𝑟𝑝𝑖cos𝜑𝑟𝑝𝑖 sin𝛼𝑟 − cos𝜑𝑟𝑝𝑖 cos𝛼𝑟 − cos𝜑𝑟𝑝𝑖

sin𝛼𝑟 cos𝛼𝑟 −1

]

]

,

K𝑖 = K𝑖𝑐3+ K𝑖𝑟3+ K𝑖𝑠3, K𝑖𝑐3=[

[

𝑘𝑝𝑖0 0

0 𝑘𝑝𝑖0

0 0 0

]

]

K𝑖𝑠3= 𝑘𝑠𝑝𝑖

(𝑡)[

[

[

sin2𝛼𝑠 cos𝛼𝑠 sin𝛼𝑠 − sin𝛼𝑠cos𝛼𝑠 sin𝛼𝑠 cos2𝛼𝑠 − cos𝛼𝑠− sin𝛼𝑠 − cos𝛼𝑠 1

]

]

]

,


(𝑡)[

[

[

sin2𝛼𝑟 − cos𝛼𝑟 sin𝛼𝑟 − sin𝛼𝑟− cos𝛼𝑟sin𝛼𝑟 cos2𝛼𝑟 cos𝛼𝑟− sin𝛼𝑟 cos𝛼𝑟 1

]

]

]

,

(7)

where 𝑘𝑗𝑝𝑖(𝑡) (𝑗 = 𝑠, 𝑟) is the meshing stiffness between thepart 𝑗 and the 𝑖th planet. 𝑘𝑝𝑖 is the bear supporting stiffnessof the 𝑖th planet in the transverse and longitudinal direction.𝛼𝑗 is the pressure angle of part 𝑗. 𝜑𝑠𝑝𝑖 = 𝜑𝑝𝑖 − 𝛼𝑠 and 𝜑𝑟𝑝𝑖 =𝜑𝑝𝑖 + 𝛼𝑟, 𝜑𝑝𝑖 is illustrated in a sun-planet mesh (Figure 2). 𝑖 =1, 2, . . . , 𝑁,𝑁 is the number of the planet gears.

T(𝑡) represents the force matrix and can be expressed by

T (𝑡) = [0, 0,

𝑇𝑐

𝑟𝑐

, 0, 0,

𝑇𝑟

𝑟𝑟

, 0, 0,

𝑇𝑠

𝑟𝑠

, 0, . . . , 0]

𝑇

, (8)

where 𝑇𝑗 (𝑗 = 𝑐, 𝑠, 𝑟) is the external torques applied on part 𝑗.𝑟𝑗 is the base radius of part 𝑗.

F𝑊(𝑡) is the impact force caused by faults, the mathemat-ical expression of which will be given in Section 3.

F𝑒(𝑡) is the excitation force induced by manufacturingerrors and can be written as

F𝑒 (𝑡) = [0, F𝑐, F𝑠, F1, F2, . . . , F𝑁] , (9)

where

F𝑐 = [

𝑁

∑

𝑖=1

𝑘𝑟𝑝𝑖(𝑡) 𝑒𝑟𝑖 (𝑡) sin (𝜑𝑟𝑝𝑖) ,

−

𝑁

∑

𝑖=1

𝑘𝑟𝑝𝑖(𝑡) 𝑒𝑟𝑖 (𝑡) cos (𝜑𝑟𝑝𝑖) , −

𝑁

∑

𝑖=1

𝑘𝑟𝑝𝑖(𝑡) 𝑒𝑟𝑖 (𝑡)]

𝑇

,

F𝑠 = [

𝑁

∑

𝑖=1

𝑘𝑠𝑝𝑖(𝑡) 𝑒𝑠𝑖 (𝑡) sin (𝜑𝑠𝑝𝑖) ,

−

𝑁

∑

𝑖=1

𝑘𝑠𝑝𝑖(𝑡) 𝑒𝑠𝑖 (𝑡) cos (𝜑𝑠𝑝𝑖) , −

𝑁

∑

𝑖=1

𝑘𝑠𝑝𝑖(𝑡) 𝑒𝑠𝑖 (𝑡)]

𝑇

,

F𝑛 = {

𝑁

∑

𝑖=1

−𝑘𝑟𝑝𝑖(𝑡) 𝑒𝑟𝑖 (𝑡) sin (𝛼𝑟) + 𝑘𝑠𝑝𝑖 (𝑡) 𝑒𝑠𝑖 (𝑡) sin (𝛼𝑠) ,

𝑁

∑

𝑖=1

𝑘𝑟𝑝𝑖(𝑡) 𝑒𝑟𝑖 (𝑡) sin (𝛼𝑟) − 𝑘𝑠𝑝𝑖 (𝑡) 𝑒𝑠𝑖 (𝑡) cos (𝛼𝑠)

𝑁

∑

𝑖=1

𝑘𝑟𝑝𝑖(𝑡) 𝑒𝑟𝑖 (𝑡) − 𝑘𝑠𝑝𝑖

(𝑡) 𝑒𝑠𝑖 (𝑡)}

𝑛 = 1, 2, . . . , 𝑁

(10)

𝑒𝑗𝑖(𝑡) (𝑗 = 𝑠, 𝑟) is the errors between the part 𝑗 and the 𝑖thplanet and can be written as

𝑒𝑠𝑖 (𝑡) = 𝐸𝑝𝑖𝑠(𝑡) − 𝐸𝑠𝑝𝑖

(𝑡) + 𝑒𝑠𝑝𝑖(𝑡)

𝑒𝑟𝑖 (𝑡) = 𝐸𝑟𝑝𝑖(𝑡) − 𝐸𝑝𝑖𝑟

(𝑡) + 𝑒𝑟𝑝𝑖(𝑡) ,

(11a)

where 𝐸𝑗𝑝𝑖(𝑡) and 𝐸𝑝𝑖𝑗(𝑡) (𝑗 = 𝑠, 𝑟) are errors associated withrotational frequencies of gears. A unified mathematical for-mula which is able to express those errors (gear eccentricities,pitch-line run-out errors, tooth spacing, and indexing errors)was extracted by Inalpolat and Kahraman [17],

𝐸𝑗𝑝𝑖(𝑡) = 𝐸𝑗 sin[

𝑓mesh𝑧𝑗

𝑡 + 𝜀𝑗 − 𝜓𝑗𝑖] ,

𝐸𝑝𝑖𝑗(𝑡) = 𝐸𝑝𝑖

sin[𝑓mesh𝑧𝑝

𝑡 + 𝜀𝑝𝑖± 𝛼𝑗] ,

(11b)

where 𝐸𝑗 and 𝐸𝑝𝑖are the amplitudes of these errors, 𝑓mesh

is mesh frequency, 𝑧𝑗 and 𝑧𝑝 are tooth numbers of thecomponent 𝑗 and planet, 𝜀𝑗 and 𝜀𝑝𝑖

are phase angles, 𝜓𝑗𝑖 isrelative position angle, and the arithmetic sign of the pressureangle 𝛼𝑗 is plus for 𝑗 = 𝑟 and minus for 𝑗 = 𝑠.

In (11a), 𝑒𝑠𝑝𝑖and 𝑒𝑟𝑝𝑖

are gear transmission errors.Inalpolat and Kahraman [17] believed that those errors were


amplitude and frequency modulated by 𝐸𝑗𝑝𝑖(𝑡) and 𝐸𝑝𝑖𝑗(𝑡).

Therefore, another equation is able to express the geartransmission errors as given by Inalpolat and Kahraman [17],

𝑒𝑠𝑝𝑖(𝑡) = 𝐴 𝑠𝑝𝑖

𝑒𝑠𝑝𝑖sin [𝐵𝑠𝑝𝑖 (𝑡) 𝑓mesh𝑡 + 𝑧𝑠𝜑𝑝𝑖 + 𝜙𝑠𝑝𝑖]

𝑒𝑟𝑝𝑖(𝑡) = 𝐴𝑟𝑝𝑖

𝑒𝑟𝑝𝑖sin [𝐵𝑟𝑝𝑖 (𝑡) 𝑓mesh𝑡 + 𝑧𝑟𝜑𝑝𝑖 + 𝛾𝑠𝑟 + 𝜙𝑟𝑝𝑖] ,

(11c)

where

𝐴 𝑠𝑝𝑖(𝑡) = 1 + 𝛽𝑠 sin [

𝑓mesh𝑧𝑠

𝑡 + 𝜙𝛽𝑠𝑖]

+ 𝛽𝑝𝑖 sin[𝑓mesh𝑧𝑝

𝑡 + 𝜙𝛽𝑝𝑖]

𝐵𝑠𝑝𝑖(𝑡) = 1 +

𝛽𝑠 sin [

𝑓mesh𝑧𝑠

𝑡 +𝜙𝛽𝑠𝑖]

+𝛽𝑝𝑖 sin[

𝑓mesh𝑧𝑝

𝑡 +𝜙𝛽𝑝𝑖]

(11d)

𝐴𝑟𝑝𝑖(𝑡) = 1 + 𝛽𝑟 sin(

𝑓mesh𝑧𝑐

𝑡 + 𝜙𝛽𝑟𝑖)

+ 𝜗𝑝𝑖 sin[𝑓mesh𝑧𝑝

𝑡 + 𝜙𝜗𝑝𝑖 + 𝜋]

𝐵𝑟𝑝𝑖(𝑡) = 1 +

𝛽𝑟 sin [

𝑓mesh𝑧𝑐

𝑡 +𝜙𝛽𝑟𝑖]

+𝜗𝑟𝑖 sin[

𝑓mesh𝑧𝑝

𝑡 +𝜙𝜗𝑝𝑖

+ 𝜋] ,

(11e)

where 𝜙𝑗𝑝𝑖(𝑗 = 𝑠, 𝑟) is the phase angle of 𝑒𝑗𝑝𝑖(𝑡), 𝛾𝑠𝑟 is

the phase angle between 𝑒𝑠𝑝𝑖(𝑡) and 𝑒𝑟𝑝𝑖

(𝑡), 𝛽𝑠, 𝛽𝑝𝑖, 𝛽𝑟, 𝜗𝑝𝑖are amplitude coefficients, 𝜙𝛽𝑠𝑖, 𝜙𝛽𝑝𝑖, 𝜙𝛽𝑟𝑖, 𝜙𝜗𝑝𝑖 are the phaseangles of amplitude modulation, 𝛽𝑠, 𝛽𝑝𝑖, 𝛽𝑟, 𝜗𝑟𝑖 are amplitudecoefficients, and 𝜙𝛽𝑟𝑖, 𝜙𝜗𝑝𝑖, 𝜙𝛽𝑟𝑖, 𝜙𝜗𝑝𝑖 are the phase angles offrequency modulation. The research results by Conry andSeireg [21] showed that the peak-to-peak amplitude of 𝑒𝑠𝑝𝑖(𝑡)might experience up to 4–6% variation due to the influenceof 𝐸𝑠𝑝𝑖(𝑡) and 𝐸𝑝𝑖𝑠(𝑡) even with the modest error magnitudes.

3. Modeling of Tooth Faults

In this section, the typical tooth fault of tooth breakage ismodeled to illustrate the influence of tooth faults on gearmesh. The tooth breakage can be modeled by the reductionof the time-varying mesh stiffness and the generation ofadditional impact forces (F𝑤(𝑡)).

3.1. The Influence of Tooth Breakage on Time-Varying Stiff-ness. As shown in Figure 3, a tooth is often considered asa cantilever beam with varying section when calculating

rb

𝜑i

ri

di

xi

𝛾i

F

A BX

h

Figure 3: The schematic diagram of a cantilever beam.

deformation. Deformation consists of five parts: bending,compressive, shear, fillet-foundation, and contact deforma-tions.

The total deformation of the point 𝐵 is expressed by

𝛿𝐵𝑖 = 𝑢𝐵 sin 𝛾𝑖 + V𝐵 cos 𝛾𝑖 + 𝑤𝐵 sin 𝛾𝑖 + 𝛿𝑓 + 𝛿𝑐, (12)

where 𝑢𝐵, V𝐵, 𝑤𝐵, 𝛿𝑓, 𝛿𝑐 represent bending, compressive,shear, fillet-foundation, and contact deformations, respec-tively, which can be calculated by the method used in [12],𝛾𝑖 (see Figure 3) defines the angle between the tooth normalforce and the tooth centerline. The stiffness of one gear toothat point 𝐵 is calculated as 𝑘𝑖 = 𝐹/𝛿𝐵𝑖.The total stiffness of onepair of meshing tooth is obtained by 𝑘pair = (𝑘1×𝑘2)/(𝑘1+𝑘2)

according to the serial relation. The stiffness when double-teeth meshing is 𝑘 = 𝑘pair1 + 𝑘pair2 according to the parallelrelation.

The parameters of the studied planetary gear system arelisted in Table 1. Young’smodulus𝐸 = 2.06×10

11 Pa, Poisson’sratio 𝜐 = 0.3. Figure 4 shows the mesh stiffness of the sungear and ring gear with planet 1 (𝑘𝑠𝑝1 and 𝑘𝑟𝑝1). The planet 1with and without tooth breakage is considered and the meshstiffness is seriously affected by tooth breakage.

3.2. The Impact Force Generated by Tooth Breakage. Theimpact force is generated during the approaching and separa-tion of themeshing teeth [22]. As one gear tooth is broken, thenormal gear meshing process is interrupted. When the nextgear tooth is going to mesh, additional impacts would occurbetween the driven and driving gears. The dynamic responseof the system will be influenced by the additional impactforce.Thus, only the change ofmesh stiffness could not reflectthe effect of tooth breakage upon the dynamic response of theplanetary gear system. In this section, the additional toothimpact forces are modeled.

The additional impact force F𝑊𝑝𝑓(𝑡) generated by thetooth breakage of the planet gear can be expressed as (planet1 is assumed to be the fault gear)

F𝑊𝑠1 (𝑡) = [0, 0, F𝑤𝑠 (𝑡) , 0, F𝑤1 (𝑡) , 0, 0]𝑇

F𝑊𝑟1 (𝑡) = [0, F𝑤𝑟 (𝑡) , 0, 0, F𝑤1 (𝑡) , 0, 0]𝑇

,

(13)


Table 1: Parameters of the planetary gear system.

Parameters Sun gear Ring Carrier PlanetsNumber of tooth 17 79 — 31Module (mm) 3 3 — 3Face width (mm) 35 35 — 35Mass (kg) 1.75 3.552 7 0.979Mass moment of inertia (kg/m2) 1 3 4.56 0.544Bearing support stiffness (N/m) 𝐾𝑃 = 𝐾𝑠𝑥 = 𝐾𝑠𝑦 = 𝐾𝑟𝑥 = 𝐾𝑟𝑦 = 𝐾𝑐𝑥 = 𝐾𝑐𝑦 = 2 × 10

8

Torsional stiffness (N/m) 𝐾𝑟𝑢 = 109; 𝐾𝑠𝑢 = 𝐾𝑐𝑢 = 2 × 10

7

Pressure angle (∘) 20∘ 20∘ — 20∘

Helix angle (∘) 0∘ 0∘ — 0∘

×108

4.5

4

3.5

3

2.5

2

1.5

1

0.5

00.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98

HealthyBreakage

Stiff

ness

(N/m

)

Time (s)

(a)×108

5

4

3

2

1

0

Stiff

ness

(N/m

)

1.26 1.28 1.3 1.32 1.34 1.36 1.38

Time (s)

HealthyBreakage

(b)

Figure 4: The mesh stiffness of planet 1 with and without toothbreakage: (a) planet-sun gear pair, (b) planet-ring gear pair.

where

F𝑤𝑠 (𝑡) = [𝐹𝑠1 (𝑡) sin (𝜑𝑠𝑝1) , −𝐹𝑠1 (𝑡) cos (𝜑𝑠𝑝1) , −𝐹𝑠1 (𝑡)]𝑇

F𝑤𝑟 (𝑡) = [𝐹𝑟1(𝑡) sin(𝜑𝑟𝑝1), −𝐹𝑟1(𝑡) cos(𝜑𝑟𝑝1), 𝐹𝑟1(𝑡)]𝑇

F𝑤1 (𝑡) = [−𝐹𝑟1 (𝑡) sin (𝛼𝑟) + 𝐹𝑠1 (𝑡) sin (𝛼𝑠) , 𝐹𝑟1 (𝑡) sin (𝛼𝑟)

+𝐹𝑠1 (𝑡) cos (𝛼𝑠) , 𝐹𝑟1 (𝑡) − 𝐹𝑠1 (𝑡)]𝑇,

(14)

where F𝑊𝑗1(𝑡) represents the impact force vector of gear 𝑗(𝑗 = 𝑠, 𝑟), 𝐹𝑗1(𝑡) (𝑗 = 𝑠, 𝑟) denotes the impact force and can beexpressed as

𝐹𝑗1 (𝑡) = {

𝐶𝑗1 (𝑡) 𝑡𝐼𝑏𝑗1< 𝑡 < 𝑡𝐼𝑓𝑗1

0 else,(15)

where 𝑡𝐼𝑏𝑗1 and 𝑡𝐼𝑓𝑗1represent the beginning and finishing

time of the impact force of gear 𝑗. The relationship between𝐶𝑟(𝑡) and 𝐶𝑠(𝑡) is expressed by 𝐶𝑟(𝑡) = 𝐻(𝑡)𝐶𝑠(𝑡). Becausethe impact acceleration will be generated,𝐻(𝑡) > 1 when thefault toothmeshes with ring gear and𝐻(𝑡) < 1when the faulttooth meshes with sun gear.

The additional impact forceF𝑊𝑟𝑓(𝑡) generated by the toothbreakage of the ring gear could be expressed by

F𝑊𝑟1 (𝑡) = [0, F𝑤𝑟1 (𝑡) , F𝑤𝑠1 (𝑡) , 0, F𝑤1 (𝑡) , 0, 0]𝑇

F𝑊𝑟2 (𝑡) = [0, F𝑤𝑟2 (𝑡) , F𝑤𝑠2 (𝑡) , 0, 0, F𝑤2 (𝑡) , 0]𝑇

F𝑊𝑟3 (𝑡) = [0, F𝑤𝑟3 (𝑡) , F𝑤𝑠3 (𝑡) , 0, 0, 0, F𝑤3 (𝑡)]𝑇

,

(16)

where

F𝑤𝑟𝑖 (𝑡) = [𝐹𝑟𝑖(𝑡) sin(𝜑𝑟𝑝𝑖), −𝐹𝑟𝑖(𝑡) cos(𝜑𝑟𝑝𝑖), −𝐹𝑟𝑖(𝑡)]𝑇

F𝑤𝑠𝑖 (𝑡) = [𝐹𝑠𝑖(𝑡) sin(𝜑𝑠𝑝𝑖), −𝐹𝑠𝑖(𝑡) cos(𝜑𝑠𝑝𝑖), −𝐹𝑠𝑖(𝑡)]𝑇

F𝑤𝑖 (𝑡) = [−𝐹𝑟𝑖 (𝑡) sin (𝛼𝑟) + 𝐹𝑠𝑖 (𝑡) sin (𝛼𝑠) ,

𝐹𝑟𝑖 (𝑡) sin (𝛼𝑟) + 𝐹𝑠𝑖 (𝑡) cos (𝛼𝑠) , 𝐹𝑟𝑖 (𝑡) − 𝐹𝑠𝑖 (𝑡)]𝑇,

(17)

where F𝑊𝑟𝑖(𝑡) represents the impact forcematrix when planet𝑖 meshes with ring gear, 𝐹𝑠𝑖(𝑡) and 𝐹𝑟𝑖(𝑡) are the pulsatingimpact forces, the mathematical expressions of which aresimilar to (15) and could be expressed by

𝐹𝑗𝑖 (𝑡) = {

𝐶𝑗𝑖 (𝑡) 𝑡𝐼𝑏𝑗𝑖< 𝑡 < 𝑡𝐼𝑓𝑗𝑖

0 else.(18)

Similarly, there are several coefficients𝐻𝑗𝑖(𝑡) used to describethe relation among𝐶𝑗𝑖(𝑡).The starting time of𝐹𝑠𝑖(𝑡) and𝐹𝑟𝑖(𝑡)varies with 𝑖.


Frequency (Hz)

×10−3

109876543210

33 33.5 34 34.5 35 35.5 36 36.5

Am

plitu

de

fm

esh

(a)

×10−3

8

7

6

5

4

3

2

1

031 32 33 34 35 36 37 38

Am

plitu

de

Frequency (Hz)

Without breakageBreakage

fm

esh

fm

esh−3(f

p+fc)

fm

esh−2(fp+fc)

fm

esh−(f

p+fc)

fm

esh+(f

p+fc)

fm

esh+2(fp+fc)

fm

esh+3(f

p+fc)

(b)

0

1

2

3

4

5

6×10−3

28 30 32 34 36 38 40

Am

plitu

de

Frequency (Hz)

fm

esh−3

(fs−fc)

fm

esh−2

(fs−fc)

fm

esh

fm

esh+2

(fs−fc)

fm

esh+3

(fs−fc)


fm

esh−(f

s−fc)

fm

esh+(f

s−fc)

(c)

×10−3

Am

plitu

de

Frequency (Hz)

fm

esh−3fc

fm

esh−2f

c

fm

esh−fc

fm

esh

fm

esh+fc

fm

esh+2f

c

fm

esh+3fc


5.55

4.54

3.53

2.52

1.51

0.50

33.5 34 34.5 35 35.5 36

(d)

Figure 5: The acceleration spectrum of the ring gear in longitudinal direction in the system with and without tooth breakages (withouterrors): (a) without breakage, (b) planet gear with and without tooth breakages, (c) sun gear with and without tooth breakages, (d) ring gearwith and without tooth breakages.

The impact force generated by the tooth breakage of thesun gear is F𝑊𝑠𝑓(𝑡), the mathematical expression of which isthe same as that of F𝑊𝑟𝑓(𝑡). However, the values for beginningand finishing times of 𝐹𝑗𝑖(𝑡) are different.

The research results by Yao and Wei [23] showed thatthe action time of impact force is usually about 5–10% ofone mesh cycle (𝑇𝑚). Wang et al. [24] pointed out that themaximal impact force induced by tooth breakage is abouttwice the averagemesh force.Thus, we can assume 𝑡𝐼𝑓𝑗𝑖−𝑡𝐼𝑏𝑗𝑖 =𝑇𝑚/10 and 𝐶𝑗𝑖(𝑡) = 2𝐹𝑗𝑝𝑖

= const.

4. The Dynamic Response of Planetary GearSystem without Errors

Physical parameters of the studied planetary gear system arelisted in Table 1. The ring gear is fixed and there are threeplanets rotating around the sun gear. The speed of the inputpart (carrier) is 2.77 rad/s (Ω𝑐) and the load of the output part(sun gear) is 200N⋅m. The rotating frequency of the carrier

(𝑓𝑐), sun gear (𝑓𝑠), planets (𝑓𝑝), and the mesh frequency(𝑓mesh) can be calculated by

𝑓𝑐 =Ω𝑐

2𝜋

,

𝑓𝑠 =𝑧𝑠 + 𝑧𝑟

𝑧𝑠

𝑓𝑐,

𝑓𝑝 =

𝑓𝑠𝑧𝑠 − 𝑓𝑐 (𝑧𝑠 + 𝑧𝑝)

𝑧𝑝

,

𝑓mesh = 𝑓𝑐𝑧𝑟.

(19)

Obviously, these frequency values are 𝑓𝑐 = 0.441Hz, 𝑓𝑠 =2.49Hz, and 𝑓𝑝 = 0.68Hz, respectively, and the meshfrequency is 𝑓mesh = 34.9Hz.

The errors 𝑒𝑠𝑝𝑖 , 𝑒𝑟𝑝𝑖 , 𝐸𝑝𝑖 , 𝐸𝑠, 𝐸𝑟 are set to be zero. Thedynamic equation of the model is solved by the NEWMARKmethod. Figure 5(a) shows the acceleration spectrum ofthe ring gear in longitudinal direction (without breakage).The spectrum is occupied by the mesh frequency and no


×10−3

10987654321033 33.5 34 34.5 35 35.5 36 36.5

Am

plitu

de

Frequency (Hz)

fm

esh−4fc(f

mes

h−fc−2f

p)

fm

esh−2f

c−fp

fm

esh−3fc(f

mes

h−2f

p)

fm

esh−fc−fp

fm

esh−2f

c

fm

esh−fp

fm

esh−fc

fm

esh−fp+fc

fm

esh

fm

esh+fp−fc

fm

esh+fc

fm

esh+fp

fm

esh+2f

c

fm

esh+fc+fp

fm

esh+3fc(f

mes

h+2f

p)

fm

esh+2f

c+fp

fm

esh+4fc(f

mes

h+fc+2f

p)

(a)

×10−3

7

6

5

4

3

2

1

035 35.5 36 36.5 37 37.5

Am

plitu

de

Frequency (Hz)


fm

esh

fm

esh+fp−fc

fm

esh+fc

fm

esh+fp

fm

esh+2f

c

fm

esh+fc+fp

fm

esh+3fc(f

mes

h+2f

p)

fm

esh+2f

c+fp

fm

esh+4fc(f

mes

h+fc+2f

p)

fm

esh+3fp(f

mes

h+3fc+fp)

fm

esh+5fc(f

mes

h+2f

c+2f

p)

fm

esh+fc+3fp

fm

esh+3fc+2f

p(f

mes

h+6fc)

fm

esh+2f

c+3fp

(b)

35 35.5 36 36.5 37 37.5

0.045

0.04

0.035

0.03

0.025

0.02

0.015

0.01

0.005

Am

plitu

de

Frequency (Hz)


fm

esh

fm

esh+fs−5fc

fm

esh+fc

fm

esh+fs−4fc f

mes

h+2f

c

fm

esh+fs−3fc f

mes

h+3fc

fm

esh+fs−2f

c fm

esh+4fc

fm

esh+fs−fc

fm

esh+5fc

fm

esh+2f

s−6fc f

mes

h+fs

fm

esh+2f

s−5fc

fm

esh+fs+fc

(c)

×10−3

12

14

10

8

6

4

2

033 33.5 34 34.5 35 35.5 36 36.5

Am

plitu

de

Frequency (Hz)


fm

esh−4fc

fm

esh−3fc

fm

esh−2f

c

fm

esh−fc

fm

esh

fm

esh+fc

fm

esh+2f

c

fm

esh+3fc

fm

esh+4fc

(d)

Figure 6:The spectrum of the system with and without tooth breakages (with errors): (a) without breakage, (b) planet gear with and withouttooth breakages (the right side of 𝑓mesh), (c) sun gear with and without tooth breakages (the right side of 𝑓mesh), (d) ring gear with and withouttooth breakages.

Table 2: Parameters of the parallel-stage gear.

Level Stage1 Stage2Gear Input Output Input OutputNumber of tooth 73 21 66 23

sidebands are found around the mesh frequency. Then toothbreakage is added to the planet gear, sun gear, and ring gearsuccessively, and the acceleration spectrum (the ring gearin longitudinal direction) is showed in Figures 5(b)–5(d).Compared with the spectrum of the normal system (withoutbreakage), modulation sidebands appeared in the spectrumof the breakage system and the peaks of the sidebands areat the frequencies of 𝑓mesh ± 𝑛(𝑓𝑝 + 𝑓𝑐), 𝑓mesh ± 𝑛(𝑓𝑠 − 𝑓𝑐)

and 𝑓mesh ± 𝑛𝑓𝑐 (𝑛 = 1, 2, . . .) successively. Therefore, themesh frequency 𝑓mesh is modulated by those frequencies,which are caused by the influence of tooth breakage on meshstiffness and additional impact force. So, the sidebands of theplanetary gear system are comparatively simple if the errorsare not included in the system.

5. The Dynamic Response of Planetary GearSystem with Errors

In order to display the modulation effect of errors morereasonably, a formula was given by Inalpolat and Kahraman[17] to express the acceleration response of the system

𝑎 (𝑡) =

𝑁

∑

𝑖=1

𝑤𝑖 (𝑡) (𝐶𝑖𝑘𝑠𝑝𝑖(𝑡) 𝛿𝑠𝑝𝑖

(𝑡) + 𝐷𝑖𝑘𝑟𝑝𝑖(𝑡) 𝛿𝑟𝑝𝑖

(𝑡)) , (20)

where 𝐶𝑖 and 𝐷𝑖 are constants, 𝛿𝑠𝑝𝑖(𝑡) and 𝛿𝑟𝑝𝑖(𝑡) are definedas

𝛿𝑠𝑝𝑖(𝑡) = 𝑦𝑠 cos𝜑𝑠𝑝𝑖 − 𝑥𝑠 sin𝜑𝑠𝑝𝑖 − 𝑥𝑖 sin𝛼𝑠 + 𝑢𝑠 + 𝑢𝑖 − 𝑒𝑠𝑖 (𝑡) ,

𝛿𝑟𝑝𝑖(𝑡) = 𝑦𝑟 cos𝜑𝑟𝑝𝑖 − 𝑥𝑟 sin𝜑𝑟𝑝𝑖 − 𝑥𝑖 sin𝛼𝑟 + 𝑢𝑟 − 𝑢𝑖 + 𝑒𝑟𝑖 (𝑡)

(21)

𝑥𝑗, 𝑦𝑗, 𝑢𝑗 (𝑗 = 𝑠, 𝑟, 𝑖) can be obtained by the dynamic model.


Loading motor Loader Parallel

gear boxPlanetary gear box

Planetary gear box

Parallel gear box

Drive motor

Frequency converter

Speed multiplierSpeed reducer

Figure 7: The test bench of the planetary gear system.

Acceleration sensor

(a)

(b) (c)

Figure 8: The acceleration sensor and breakage tooth: (a) acceler-ation sensor position, (b) planet gear with tooth breakage, and (c)ring gear with tooth breakage.

𝑤𝑖(𝑡) is a weighting function for a planet gear positionedat the angle of 𝜑𝑝𝑖 and 𝑤𝑖(𝑡) can be defined as

𝑤𝑖 (𝑡) = 𝑊(𝑡 −

𝜑𝑝𝑖

𝑓𝑐

)𝑈𝑖 (𝑡) , (22)

where 𝑊(𝑡) is a Hanning function used to represent theinfluence of rolling planets on fixed accelerometer (scrolling

×10−3

3.5

3

2.5

2

1.5

1

0.5

33.5 34 34.5 35 35.5 36 36.5

Am

plitu

de

Frequency (Hz)

fm

esh

Figure 9: The acceleration spectrum of the normal system.

effect) and 𝑈𝑖(𝑡) is the unit step function. Both of them aregiven as follows:

𝑊(𝑡) =

1

2

−

1

2

[cos (2𝜋𝑁𝑓𝑐𝑡)]

𝑈𝑖 (𝑡) =

∞

∑

𝑛=1

{𝑢 [𝑡 − (

(𝑛 − 1)𝑁 + 𝑖 − 1

𝑁

)𝑇𝑐]

−𝑢 [𝑡 − (

(𝑛 − 1)𝑁 + 𝑖

𝑁

)𝑇𝑐]} ,

(23)

where 𝑢(𝑡 − 𝑎) = 1 for 𝑡 > 𝑎 and 𝑢(𝑡 − 𝑎) = 0 for 𝑡 < 𝑎,𝑁 isthe number of planets. In this study,𝑁 is equal to 3 and 𝑖 is 1,2, and 3.

Run-out errors of the planets, sun gear, and ring gearare taken into account and 𝑒𝑠𝑝𝑖

= 0.6 𝜇m, 𝑒𝑟𝑝𝑖 = 0.3 𝜇m,𝐸𝑝𝑖

= 10 𝜇m, and 𝐸𝑠 = 𝐸𝑟 = 15 𝜇m. According to Conry’sstudy [21], modulation coefficients are set as 𝛽𝑠 = 0.033,𝜙𝛽𝑠𝑖 = 2(𝑖−1)𝜋/3, 𝛽𝑝𝑖 = 0.034, 𝜙𝛽𝑝𝑖 = 2(𝑖−2)𝜋/3, 𝜗𝑝𝑖 = 0.033,𝜙𝜗𝑝𝑖 = 2(𝑖 − 2)𝜋/3, 𝛽𝑟 = 0.035, 𝜙𝛽𝑟𝑖 = 2(𝑖 − 1)𝜋/3 and allother modulation coefficients are set to be zero.The dynamicequation of the model is solved to get the 𝛿𝑠𝑝𝑖(𝑡) and 𝛿𝑟𝑝𝑖(𝑡),then acceleration is calculated by (20). Figure 6(a) shows thesystem acceleration spectrum of a normal system (withoutbreakage). Compared with Figure 5(a), complex sidebandsappeared around the mesh frequency due to the existenceof errors. They are symmetrically arranged on both sidesof the mesh frequency. The sidebands of the system haveundergone tremendous changes when errors are taken intoconsideration. The phenomenon is bound to influence thefault characteristic frequencies of the system components.

Figure 6(b) shows the acceleration spectrum of the sys-tem with and without tooth breakage on planet 1. Since themesh frequency 𝑓mesh is modulated by the frequencies ofweighting function (which characteristic frequencies are𝑚𝑓𝑐)and the frequencies of 𝑛(𝑓𝑝 + 𝑓𝑐) (caused by the influenceof tooth breakage on mesh stiffness and additional impactforce), the amplitudes of the tooth breakage system are muchlarger than those of the normal (without breakage) system atthe frequencies of 𝑓mesh ± 𝑚𝑓𝑐 ± 𝑛(𝑓𝑝 + 𝑓𝑐) (𝑚, 𝑛 = 1, 2, . . .).Therefore, the fault characteristic frequencies of the planetgear are 𝑓𝑝𝑓 = 𝑓mesh ±𝑚𝑓𝑐 ± 𝑛(𝑓𝑝 + 𝑓𝑐) and can be simplifiedas 𝑓mesh ± 𝑚𝑓𝑐 ± 𝑛𝑓𝑝 (𝑚, 𝑛 = 1, 2, . . .).


×10−3

54.54

3.53

2.52

1.51

0.5

32 33 34 35 36 37

Am

plitu

de

Frequency (Hz)


(a)

×10−3

2.5

2

1.5

1

0.5

032

35 35.5 36 36.5 37

Am

plitu

de

Frequency (Hz)


fm

esh

fm

esh+fc

fm

esh+fp

fm

esh+2f

c

fm

esh+fc+fp

fm

esh+3fc(f

mes

h+2f

p)

fm

esh+2f

c+fp

fm

esh+4fc(f

mes

h+fc+2f

p)

fm

esh+3fp(f

mes

h+3fc+fp)

fm

esh+5fc(f

mes

h+2f

c+2f

p)

fm

esh+fc+3fp

(b)

×10−3

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

035 35.5 36 36.5 37 37.5

Am

plitu

de

Frequency (Hz)


fm

esh

fm

esh+fp−fc

fm

esh+fc

fm

esh+fp

fm

esh+2f

c

fm

esh+fc+fp

fm

esh+3fc(f

mes

h+2f

p)

fm

esh+2f

c+fp

fm

esh+4fc(f

mes

h+fc+2f

p)

fm

esh+3fp(f

mes

h+3fc+fp)

fm

esh+fc+3fp

fm

esh+3fc+2f

p(f

mes

h+6fc)

fm

esh+5fc(f

mes

h+2f

c+2f

p)

(c)

×10−3

2.22

1.81.61.41.21

0.80.60.40.2

28 28.5 29 29.5 30

Am

plitu

de

Frequency (Hz)


fm

esh

fm

esh+fp−fc

fm

esh+fc

fm

esh+fp

fm

esh+2f

c

fm

esh+fc+fp

fm

esh+3fc(f

mes

h+2f

p)

fm

esh+2f

c+fp

fm

esh+4fc(f

mes

h+fc+2f

p)

fm

esh+3fp(f

mes

h+3fc+fp)

fm

esh+fc+3fp

fm

esh+3fc+2f

p(f

mes

h+6fc)

fm

esh+5fc(f

mes

h+2f

c+2f

p)

(d)

Figure 10: The spectrum of the system with and without tooth breakage on planet at different speeds and loads: (a) load = 200N⋅m, inputspeed = 2.74 rad/s, (b) zoom the right side of (a), (c) load = 229N⋅m, input speed = 2.74 rad/s, and (d) load = 229N⋅m, input speed = 2.22 rad/s.

Likewise, the amplitudes of the sun gear tooth breakagesystem are found to be larger than those of the normal system(without breakage) at the frequencies of 𝑓mesh ± 𝑛𝑓𝑐 ± 𝑚𝑓𝑠

(𝑚, 𝑛 = 1, 2, . . .) in Figure 6(c). The main reason is thatthe mesh frequency 𝑓mesh is modulated by the frequenciesof weighting function (𝑚𝑓𝑐) and the frequencies of 𝑛(𝑓𝑠 −𝑓𝑐) (caused by the influence of tooth breakage on meshstiffness and additional impact force). Therefore, the faultcharacteristic frequencies of the sun gear are 𝑓mesh ± 𝑚𝑓𝑐 ±

𝑛(𝑓𝑠 − 𝑓𝑐) and can be as 𝑓𝑠𝑓 = 𝑓mesh ± 𝑛𝑓𝑐 ± 𝑚𝑓𝑠 (𝑚, 𝑛 =

1, 2, . . .).Similarly, because themesh frequency𝑓mesh is modulated

by the frequencies of the weighting function𝑚𝑓𝑐 and the fre-quencies of 𝑛𝑓𝑐 (caused by the influence of tooth breakage onmesh stiffness and additional impact force), the amplitudesof the ring gear tooth breakage system are larger than thoseof the normal system (without breakage) at frequencies of𝑓mesh ± 𝑛𝑓𝑐 ± 𝑚𝑓𝑐 (𝑚, 𝑛 = 1, 2, . . .), as shown in Figure 6(d).Therefore, the fault characteristic frequencies of the ring gearare 𝑓𝑟𝑓 = 𝑓mesh ± 𝑛𝑓𝑐 ± 𝑚𝑓𝑐 and can be simplified as 𝑓𝑟𝑓 =𝑓mesh ± 𝑛𝑓𝑐 (𝑛 = 1, 2, . . .).

6. Experimental Validations

Figure 7 shows the test bench of a planetary gear system.The drive motor is controlled by frequency converter andprovides power for the system. The speed of the systemis slowed down by the speed reducer and speeded up bythe speed multiplier. A loading motor continuously pumpshigh-pressure oil into the loader to induce a workload. Thestructure and the size of the speedmultiplier are proportionalto the gearbox of a real wind turbine and it consists ofa planetary and a parallel gear boxes. The parallel gearbox is composed of two pairs of parallel-stage gearing. Theparameters of the planetary gear system are listed in Table 1and those of the parallel-stage are listed in Table 2. Thestructure style and parameter setting of the speed reducer arethe same as those of the speed multiplier.

As shown in Figure 8(a), an acceleration sensor ismounted on the planetary gear box of the speed multiplier.The tooth breakage of planet gear and ring gear are added tothe planetary gear system separately, as shown in Figures 8(b)and 8(c). Vibration signals are collected for a normal mode


×10−3

54.54

3.53

2.52

1.51

0.5

33.5 34 34.5 35 35.5 36

Am

plitu

de

Frequency (Hz)


fm

esh−3fc

fm

esh−2f

c

fm

esh−fc

fm

esh

fm

esh+fc

fm

esh+2f

c

fm

esh+3fc

(a)

×10−3

8

7

6

5

4

3

2

1

033.5 34 34.5 35 35.5 36 36.5

Am

plitu

de

Frequency (Hz)


fm

esh−3fc

fm

esh−2f

c

fm

esh−fc

fm

esh

fm

esh+fc

fm

esh+2f

c

fm

esh+3fc

(b)

×10−4

12

10

8

6

4

2

27 27.2 27.4 27.6 27.8 28 28.2 28.4 28.6 28.8 29

Am

plitu

de

Frequency (Hz)


fm

esh−3fc

fm

esh−2f

c

fm

esh−fc

fm

esh

fm

esh+fc f

mes

h+2f

c

fm

esh+3fc

(c)

Figure 11: The spectrum of the system with and without tooth breakage on ring gear at different loads and speeds: (a) load = 200N⋅m, inputspeed = 2.77 rad/s, (b) load = 229N⋅m, input speed = 2.77 rad/s, and (c) load = 229N⋅m, input speed = 2.24 rad/s.

(without breakage) and the two fault modes. The samplingfrequency is 16384Hz.

Figure 9 shows the acceleration spectrum of the nor-mal system (without breakage). The drive motor speed is156.24 rad/s and the input speed of the speed multiplier is2.77 rad/s.Therefore, the mesh frequency is 34.9Hz. Figure 9shows that sidebands appeared around the mesh frequency.The results indicate the existence of errors in the system.

6.1. Planet Gear with Tooth Breakage. Figure 10 shows theacceleration spectrum of the system with and without toothbreakage on planet.

In Figure 10(a), the load and the speed of the systemare 200N⋅m and 2.74 rad/s, respectively. Therefore, the meshfrequency is 34.87Hz. Compared with the amplitudes ofthe normal system (without breakage), those of the toothbreakage system change significantly. In order to make itmore clear, the right side of Figure 10(a) is zoomed in,as shown in Figure 10(b). All amplitudes increase at thefrequencies of 𝑓mesh ± 𝑚𝑓𝑐 ± 𝑛𝑓𝑝 (𝑚, 𝑛 = 1, 2, . . .), except forthe frequencies of 𝑓mesh + 2𝑓𝑐 and 𝑓mesh + 𝑓𝑐 + 3𝑓𝑝.

Then, the load is increased to 229N⋅m and the speedremains the same. The differences of amplitudes between thenormal system and the tooth breakage system are increasedat the frequencies of 𝑓mesh ± 𝑚𝑓𝑐 ± 𝑛𝑓𝑝 (𝑚, 𝑛 = 1, 2, . . .), asshown in Figure 10(c).

Finally, the load is kept unchanged at a value of 229N⋅mand the input speed is decreased to 2.22 rad/s. Therefore, themesh frequency is changed to 27.86Hz.The amplitudes of thetooth breakage system are also larger than those of the normalsystem at the frequencies of𝑓mesh±𝑚𝑓𝑐±𝑛𝑓𝑝 (𝑚, 𝑛 = 1, 2, . . .)significantly, as shown in Figure 10(d).

Figure 10 verified the correctness of the simulation resultsin Section 5.The fault characteristic frequencies of the planetgear with tooth breakage are 𝑓𝑝𝑓 = 𝑓mesh ±𝑚𝑓𝑐 ± 𝑛𝑓𝑝 (𝑚, 𝑛 =1, 2, . . .).

6.2. Ring Gear with Breakage. Figure 11 shows the accelera-tion spectrum of the system with and without tooth breakageon the ring gear.

In Figure 11, the load and the speed of the system are200N⋅m and 2.77 rad/s, respectively; therefore, the mesh


frequency is 34.9Hz, similar with the method used inSection 6.1. The amplitudes of the acceleration spectrum arecompared between the normal system and the tooth breakagesystem. The fault characteristic frequencies of 𝑓mesh ± 𝑛𝑓𝑐

(𝑛 = 1, 2, . . .) are further confirmed by the experiment results.The change law of amplitudes with loads and speeds of

the normal and the tooth breakage systems is same as thelaw described in Section 6.1. The conclusion can be obtainedthrough Figures 11(a)–11(c).

7. Conclusions

Detection and localization of tooth fault on wind turbineplanetary gear system are investigated numerically and exper-imentally in this paper. Through detailed comparisons onboth numerical and experimental results, the fault character-istic frequencies can be obtained after considering the gearmanufacturing errors.

(1) The sidebands of the planetary gear system becomemuch more complex when manufacturing errors aretaken into consideration.

(2) The fault characteristic frequencies of planet gears,sun gear and ring gear with tooth breakage are 𝑓𝑝𝑓 =𝑓mesh ± 𝑚𝑓𝑐 ± 𝑛𝑓𝑝, 𝑓𝑠𝑓 = 𝑓mesh ± 𝑛𝑓𝑐 ± 𝑚𝑓𝑠 and𝑓𝑟𝑓 = 𝑓mesh ± 𝑛𝑓𝑐, respectively.

(3) As the fault characteristic frequencies for the sun,planet, and ring gears are different, so the toothbreakage fault could also be located bymonitoring thevalues of these frequencies.

The above results are useful for the detection and local-ization of tooth faults in actual planetary gear system of windturbines.

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

Acknowledgments

The research work described in this paper was supported byNatural Science Foundation of China under Grant 51335006,Tsinghua University Initiative Scientific Research Programunder Grant 2011Z08137, and the State Key Laboratory of Tri-bology Initiative Research ProgramunderGrant SKLT11A02.

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