Engineering Failure Analysis 48 (2015) 105–120

Contents lists available at ScienceDirect

Engineering Failure Analysis

journal homepage: www.elsevier .com/locate /engfai lanal

Fault features analysis of cracked gear considering the effectsof the extended tooth contact

http://dx.doi.org/10.1016/j.engfailanal.2014.11.0181350-6307/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel./fax: +86 24 83684491.E-mail address: mahui_2007@163.com (H. Ma).

Hui Ma ⇑, Xu Pang, Ranjiao Feng, Rongze Song, Bangchun WenSchool of Mechanical Engineering and Automation, Northeastern University, Shenyang, Liaoning 110819, PR China

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 June 2014Received in revised form 26 October 2014Accepted 17 November 2014Available online 26 November 2014

Keywords:Cracked spur gearTime-varying mesh stiffness (TVMS)Extended tooth contactFault featureInstantaneous energy

Owing to the effect of gear flexibility, the extended tooth contact occurs, which is a phe-nomenon that the incoming tooth pair enters contact earlier than the theoretical start ofcontact, and the outgoing tooth pair leaves contact later than the theoretical end of contact.Considering the effects of the extended tooth contact and tooth root crack on the time-varying mesh stiffness (TVMS), a finite element (FE) model of a spur gear pair in mesh isestablished by ANSYS software. TVMS under different crack depths at constant rated torque(60 Nm) are calculated based on the FE model. Then, a FE model of a geared rotor system isdeveloped by MATLAB software. Frequency-domain features, statistical features (Kurtosisand RMS) and instantaneous energies based on empirical mode decomposition (EMD)under different crack depths are calculated at 1000 rev/min and the corresponding mea-sured results are also performed by model experiment. The results show that consideringthe effect of extended tooth contact, TVMS of crack gear pair becomes smooth from double-tooth engagement to single-tooth engagement; the gear crack has reduced the gear bodyrigidity, which leads to the reduction of stiffness of the healthy teeth and has little effecton the vibration response. The instantaneous energy can be chosen as a distinguishing indi-cator to qualitatively diagnose the gear cracks with different levels.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Gear transmission system may induce gear fatigue crack under internal and external incentives. The research on the gearcrack simulation and diagnosis has been one of the hot issues. The key for simulation analysis is to obtain the accurate time-varying mesh stiffness (TVMS), which has a great influence on the gear pair vibration. The finite element (FE) method [1–7],analytical method (AM) [8–22] and analytical-FE approach [23–25] have been used to calculate the TVMS of gears. For AM andanalytical-FE approach, they can offer a simple and effective way to calculate the TVMS under most conditions. However, theyhave some difficulties in dealing with the extended tooth contact problem under large torsion conditions and the weakeningeffect of crack on the stiffness of the foundation (wheel body). For FE method, the calculation result is more close to actualsituation because it can consider accurate deformations of involute, transition curve and fillet foundation. In this study, FEmethod is adopted to establish the mesh model of gear pairs by considering the gear tooth crack and extended tooth contact.

Extensive efforts have been focused on gears themselves, and in fact, gear transmission systems are only a part of thewhole rotor system, and to study the dynamic characteristics of the whole rotor system better, the flexibility of shafts mustbe considered [2,6,7,17,21,26–32]. Based on FE method, Ma et al. [2] established a FE model of a geared rotor system and

106 H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120

analyzed the effects of tip relief on lateral–torsional coupling vibration responses of the system. Howard et al. [6] presented a16 degree-of-freedom (DOF) gear dynamic model which includes friction and demonstrated the major effects of a singletooth crack on the system vibration responses. Jia and Howard [7] developed a 26 DOF gear dynamic model of three shaftsand two pairs of spur gears in mesh for comparison of localized tooth spalling and damage. Chen and Shao [17] developed adynamic model of spur gear pair system having six DOF and examined the influence of gear tooth crack on the system vibra-tion response by incorporation of the developed mesh stiffness model. Wan et al. [19] developed a coupled lateral and tor-sional vibration dynamic model to simulate the vibration response of gear-rotor system with tooth crack by the FE method.Ma and Chen [21] established a four DOF dynamic model of a gear system with local defects to explore the failure mecha-nism. Parey et al. [26] established a 6 DOF nonlinear model with localized tooth defect. By using an extended FE model of atest gear-shaft-bearing system, Velex and Ajmi [27] compared the dynamic results from the formulations based on transmis-sion error with the reference solutions.

In order to diagnose the gear crack fault, many methods based on vibration signal analysis have been developed. Conven-tional methods, such as RMS, kurtosis, and power spectrum, have proved to be effective in fault diagnosis [8,16,17]. However,their main drawback is that they are based on the assumption of stationarity and linearity of the signal generation process[33]. In order to deal with non-stationary signals, many new techniques, such as time–frequency distributions [34], wavelets,higher order statistics [35] and empirical mode decomposition (EMD) [36], have received attention and obtained acceptance.Loutridis [37] presented a method for monitoring the evolution of gear faults based on the newly developed empirical modedecomposition scheme. In his another paper [38], he proposed energy-based features for gear fault diagnosis and predictionand adopted the instantaneous energy density to obtain high values when defected teeth are engaged.

In the above-mentioned researches, the analysis of the effects of the extended tooth contact [39] and the weakeningaction of wheel body which caused by tooth root crack on the vibration responses of geared rotor system are insufficient.This paper focuses on the effects of the extended tooth contact under tooth root crack on the TVMS. Moreover, the weakeningstiffness of gear-body caused by tooth crack is also considered. Then, a dynamic model of a geared rotor system with toothroot crack is established. The fault features under different crack depths are extracted by frequency-domain features, statis-tical energies and instantaneous energies based on EMD method. Moreover, the simulation results under different crackdepths are also verified by measured results in a gear test rig.

2. TVMS considering the effects of extended tooth contact and tooth root crack

2.1. Tooth profile curve equation of the involute spur gear with root crack

Before tooth profile curve equations are presented, some basic variable symbols are introduced first: m is the module, athe pressure angle of the gear pitch circle, aa the pressure angle of the gear addendum circle, aC the pressure angle of theinvolute starting point, r, rb, ra the radii of the pitch circle, base circle, addendum circle of the gear, h�a the addendumcoefficient, c⁄ the tip clearance coefficient, N the number of teeth.

The gear tooth profile which is defined by the involute curve BC and fillet curve CD is shown in Fig. 1. The x and ycoordinates of the involute profile (curve BC) are given by the following equations.

Fig. 1. Schematic of tooth profile and crack propagation: (a) tooth profile of an involute spur gear, (b) schematic of crack propagation.

H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120 107

x ¼ ri sin uy ¼ ri cos u

�; ð1Þ

u ¼ p2N� ðinvai � invaÞ; ð2Þ

where ri = rb/ cos ai; ai (aC 6 ai 6 aa) is the pressure angle of the arbitrary point at the involute curve in which aC = arccos

(rb/rC), aa = arccos (rb/ra); rC ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðrb tan a� h�am= sin aÞ2 þ r2

b

qis the radius of the involute starting point circle; invai is the

involute function and invai = tan ai � ai, inva = tan a � a.The equations of the fillet curve of the tooth profile CD are defined as follows [40]:

x ¼ r � sinðUÞ � ða1= sin cþ rqÞ � cosðc�UÞy ¼ r � cosðUÞ � ða1= sin cþ rqÞ � sinðc�UÞ

�ða 6 c 6 p=2Þ; ð3Þ

U ¼ ða1= tan cþ b1Þ=r; ð4Þ

where rq = c⁄m/(1-sin a), a1 ¼ ðh�a þ c�Þ �m� rq and b1 ¼ pm=4þ h�am tan aþ rq cos a.The crack path is assumed to be a straight line. The crack starts at the root of the driven gear and then propagates, as is

shown in Fig. 1b. The geometrical parameters of the crack (q, t, w) are presented in Fig. 1b, in which q denotes the crackdepth, t the crack propagation direction, w the crack initial position (in this paper q = 1, 2, 3 and 4 mm, t = 45�, w = 35�).

2.2. TVMS considering the effect of extended tooth contact and tooth root crack

To simplify the problem, the TVMS calculation is performed under the assumption of a plane strain problem. To improvethe computational efficiency, the analysis is performed by a 2D model with only nine teeth established in ANSYS software, inwhich the crack tip is modeled using 2D singularity elements (see Fig. 2). In the figure, the inner ring of gear 2 is fixed; theinner ring of gear 1 is loaded with a constant torque T1, which is obtained through suitable tangential forces [2]. In order toconsider the extension of the meshing period of tooth pairs under the large torque and tooth root crack condition, ninecontact pairs are established. In addition, the FE model of teeth with different crack depths are modeled respectively (seeFig. 2c).

For healthy gear pairs, TVMS under input torques of T1 = 5, 10, 30, 60, 100, 120 and 150 Nm are obtained which are shownin Fig. 3. In the figure, the zone with underpainting denotes theoretical double-tooth contact zone. It is obvious that the hand

Fig. 2. FE model of meshing gear pair: (a) meshing model of gear pair, (b) enlarged view, (c) teeth with different crack depths.

Fig. 3. TVMS under different torques.

Fig. 4. Stress nephograms at moment A: (a) T1 = 5 Nm, (b) T1 = 60 Nm, (c) T1 = 100 Nm, (d) T1 = 150 Nm.

108 H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120

over region (change over between the theoretical single zone and the double zone [3,39]) appears, which is owing to theflexibilities of the mating gears. Stress nephograms at moment A (see Fig. 3) are displayed in Fig. 4. From Figs. 3 and 4, itis clear that with the increasing input torques the single zone will be reduced by the extension of the hand over regions.For the case of gear with theoretical contact ratio between 1 and 2 and no profile modification, sometimes even three-toothengagement state may appear (see Fig. 4d).

For the gear pair with tooth root crack, TVMS with crack depths of q = 1, 2, 3 and 4 mm are calculated under T1 = 60 Nm.The results are shown in Fig. 5 which indicates that the stiffness of the cracked gear pair reduces with the increasing crackdepth. It also shows that the stiffness of the adjacent healthy mating teeth pair reduces with the increasing crack depth,

Fig. 5. TVMS with different crack depths under torque 60 Nm.

Fig. 6. Stress nephograms at moment B: (a) q = 0 mm, (b) q = 1 mm, (c) q = 2 mm, (d) q = 3 mm, (e) q = 4 mm.

H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120 109

which is because the tooth root crack weakens the rigidity of the wheel body. Stress nephograms at moment B (see Fig. 5) aredisplayed in Fig. 6. It can be seen that when q = 0, 1 and 2 mm, it is the single-tooth engagement at moment B. However,when q = 3 and 4 mm it is double-tooth engagement, which is because larger tooth root crack exacerbates the effect ofextended tooth contact (see Fig. 6c and d).

3. Vibration responses of the geared rotor system with tooth root crack

3.1. Introduction of the test rig

The one-stage reduction gearbox with crack defect (see Fig. 7) is set up to validate the theoretical results, and the param-eters of the rotor and spur gear pair are listed in Table 1. In this study, the measured results are performed under a small loadtorque condition to avoid the fluctuation of the rotating speed. The crack is manufactured near the tooth root of the drivengear, and four crack depths of 1 mm, 2 mm, 3 mm and 4 mm are obtained by wire-electrode cutting (see Fig. 7). Vibrationsignals under tooth root crack condition are measured by accelerometers whose locations are shown in Fig. 7. The samplefrequency is 10,000 Hz and the rotating speed of driving shaft is 1000 rev/min. So rotating frequencies of the driving anddriven gear are about 16.7 Hz and 12.2 Hz, which are denoted by f1 and f2, respectively, and gear meshing frequency fe isequal to 916.7 Hz, meshing frequency of the belt wheel with 32 teeth fb is 533.3 Hz.

3.2. Vibration responses analysis based on numerical simulation

In order to investigate the fault features of a cracked gear pair, it is necessary to consider the dynamic characteristics ofthe geared rotor system. Based on Ref. [2], a FE model of a geared rotor system is established according to the size of the testrig (see Fig. 7).

Dynamic model of a spur gear pair formed by gears 1 and 2 is shown in Fig. 8 where the typical spur gear meshing isrepresented by a pair of rigid disks connected by a spring-damper set along the plane of action, which is tangent to the basecircles of the gears. O1 and O2 are their geometrical centers, X1 and X2 the rotating speeds, rb1 and rb2 the radii of the gearbase circles, respectively. e12(t) denotes non-loaded static transmission error which is formed solely by the geometric devi-ation component. Assuming that the spur gear pair studied in the paper is perfect, non-loaded static transmission error e12(t)will be zero. c12(t) represents the mesh damping which is assumed to be zero. k12 is the time-varying mesh stiffness, which iscomprised of the stiffness of healthy tooth and crack tooth.

Fig. 7. Test rig of a cracked gear coupled rotor system.

Table 1Parameters of the gears and bearings.

Gears Ix = Iy (kg mm2) Iz (kg mm2) m (kg) Pressure angle (�) Pitch diameter (mm) Number of teeth N Tooth width (mm)

Gear parameters1 1247.99 4069.31 1.53 20 110 55 202 2338.84 7892.28 3.01 20 150 75 20

Bearings kxx (MN/m) kyy (MN/m) kzz (MN/m) khxhx (MN m/rad) khyhy (MN m/rad)

Bearing parametersAll 200 200 100 0.1 0.1

Fig. 8. Dynamic model of a spur gear pair.

110 H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120

The angle between the plane of action and the positive y-axis is represented by w12 defined as:

w12 ¼�aþ a12 X1 : Counterclosewiseaþ a12 � p X1 : Clockwise

�; ð5Þ

where a12 (0 6 a12 6 2p) denotes the angle between the line connecting the gear centers and the positive x-axis of the pinionand in this work, a12 = 0.

H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120 111

For considering the influence of the rotating direction of the pinion, the function r is introduced as:

r ¼1 X1 : Counterclockwise�1 X1 : Clockwise

�: ð6Þ

Each gear owns six degrees of freedom including translation in x, y and z direction and rotation about these three axes.Therefore, the gear pair has a total of 12 degrees of freedom that defines the coupling between the two shafts holding thegears. Without considering the tooth separation, backlash, friction forces at the mesh point and the position change of thepressure line, the motion equations of the gear pair are written as:

m1€x1 � k12p12ðtÞ sin w12 � c12ðtÞ _p12ðtÞ sin w12 ¼ 0m1€y1 þ k12p12ðtÞ cos w12 þ c12ðtÞ _p12ðtÞ cos w12 ¼ 0m1€z1 ¼ 0Ix1

€hx1 þ Iz1X1_hy1 ¼ 0

Iy1€hy1 � Iz1X1

_hx1 ¼ 0

Iz1€hz1 þ r� rb1k12p12ðtÞ þ r� rb1c12ðtÞ _p12ðtÞ ¼ rT1

m2€x2 þ k12p12ðtÞ sin w12 þ c12ðtÞ _p12ðtÞ sin w12 ¼ 0m2€y2 � k12p12ðtÞ cos w12 � c12ðtÞ _p12ðtÞ cos w12 ¼ 0m2€z2 ¼ 0Ix2

€hx2 þ Iz2X2_hy2 ¼ 0

Iy2€hy2 � Iz2X2

_hx2 ¼ 0

Iz2€hz2 þ r� rb2k12p12ðtÞ þ r� rb2c12ðtÞ _p12ðtÞ ¼ rT2

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:

; ð7Þ

where m1 and m2 are the mass of gears 1 and 2, and Ix1, Iy1, Iz1, Ix2, Iy2, Iz2 respectively, represent the moments of inertia aboutthe x-, y- and z-axes of gears 1 and 2. T1 and T2 are constant load torques. The term p12(t) represents the relative displacementof the gears in the direction of the action plane and is defined as:

p12ðtÞ ¼ ð�x1 sin w12 þ x2 sin w12 þ y1 cos w12 � y2 cos w12 þ r� rb1hz1 þ r� rb2hz2Þ � e12ðtÞ; ð8Þ

Eq. (7) can be rewritten in matrix form as follows:

M12€X12 þ ðC12 þ G12Þ _X12 þ K12X12 ¼ F12; ð9Þ

where X12 is the displacement vector of a gear pair. M12, K12, C12 and G12 represent the mass matrix, mesh stiffness matrix,mesh damping matrix and gyroscopic matrix of the spur gear pair, respectively. F12 is the exciting force vector of the spurgear pair. These matrices and vectors are defined in Ref. [2].

Motion equations of the whole geared rotor system can be written in matrix form as follows:

M€uþ ðC þ GÞ _uþ Ku ¼ Fu; ð10Þ

where M, K, C, G are the mass, stiffness, damping and gyroscopic matrix of the global system; u and Fu denote the displace-ment and external force vectors of the global system. All these matrixes and vectors can be obtained in Ref. [2].

In this paper, the cracked gear coupled rotor system includes two shafts, a gear pair with a tooth crack in the driven gear(gear 2) and four ball bearings. The parameters of the system are listed in Table 1 and Fig. 9, and the FE model of the geared

Fig. 9. Physical dimensions and FE model of a cracked gear coupled rotor system.

Fig. 10. TVMS introduced into the geared rotor system considering the effect of the gear crack in different mesh cycle, (a) two mesh cycles, (b) three meshcycles, (c) five mesh cycles, (d) seven mesh cycles.

112 H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120

rotor systems is also shown in Fig. 9. Material parameters of the shafts are as follows: Young’s modulus E = 210 GPa, Pois-son’s ratio v = 0.3 and density q = 7850 kg/m3. The shafts 1 and 2 are both divided into 13 elements, gears 1 and 2 are locatedat nodes 8 and 22, respectively (see Fig. 9).

In order to discuss the effect of the gear crack on the dynamic results, vibration responses under four cases (see Fig. 10)are investigated. In Fig. 10, TVMS with underpainting are part of TVMS in Fig. 5, and TVMS in other mesh cycle adopts theTVMS of healthy gear. Traditionally, TVMS are investigated in two mesh cycle (the zone with underpainting in Fig. 10a).However, TVMS of other mesh cycle are also influenced. In Fig. 10, it can be found that the TVMS is reduced during adjacenthealthy gear engagement. And the effect of the gear crack TVMS on adjacent gear increases with the increasing crack depth.Based on the FE model proposed above, vibration acceleration responses under four cases for crack depth q = 4 mm areobtained shown in Fig. 11. For visual convenience, only the local enlarged view is displayed. The figures show that these fourcases have little difference, especially for cases 2, 3 and 4, which illustrates that the reduction of gear body rigidity has littleeffect on the vibration response. In the acceleration amplitude spectra (see Fig. 11b), fe denotes the meshing frequency. Thefigures show that the amplitudes of fe and those of its harmonics (2fe, 3fe, etc.) for the cracked gear hardly change under fourcases. The interval Df between every two sideband components is exactly equal to the rotating frequency of the cracked gear.

For case 4, vibration acceleration responses under crack depths q = 1, 2, 3 and 4 mm are obtained shown in Fig. 12. Withthe growth of gear tooth crack depth, the amplitudes of the impulsive vibration signals increase. The impulses caused bytooth crack are very obvious when the crack propagates to 4 mm. Acceleration amplitude spectra are shown in Fig. 12b,which show that the amplitude of fe and those of its harmonics (2fe, 3fe, etc.) for the cracked gear hardly change comparedwith those for healthy gear, but the sidebands appear under the tooth crack condition. The interval Df between every twosideband components is equal to the rotating frequency of the cracked gear and the magnitudes of sidebands ascend with thecrack propagation along crack depth.

4. Energy-based features comparison by numerical simulation and experiment

4.1. Crack fault features based on frequency features, RMS and Kurtosis indicators

Measured acceleration amplitude spectra at the right bearing of the driving shaft in vertical direction (see Fig. 7) ofhealthy gear system and cracked gear system with crack depths of q = 1, 2, 3 and 4 mm are displayed in Fig. 13. In the testrig, the number of belt wheel is 32. Meshing frequency caused by belt wheel fb is 533.33 Hz in theory. In addition, there aresome obvious frequency components (see Fig. 13) with an interval of about 78 Hz, which may be caused by the cylindricalroller bearing (N205). For the healthy geared rotor system and cracked geared rotor system, the amplitude spectra all show

Fig. 11. Vibration responses of a cracked gear coupled rotor system: (a) full scope of waveform and zoom plot, (b) full scope of amplitude spectrum andzoom plot.

Fig. 12. Vibration responses of a cracked gear coupled rotor system: (a) full scope of waveform and zoom plot, (b) full scope of amplitude spectrum andzoom plot.

H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120 113

the meshing frequencies (914.3 Hz, 914.6 Hz, 913.8 Hz, 916.0 Hz and 915.8 Hz) and its harmonics marked by yellow1 (seeFig. 13). For the healthy geared rotor system, there are no sideband frequencies in the frequency spectra of the simulation signal(see Fig. 12b). However, due to the machining error, sideband frequencies around the meshing frequency and its harmonics aredetected in the experimental signal, which are related to the rotating frequency of the driving gear f1, for example,

1 For interpretation of color in Fig. 13, the reader is referred to the web version of this article.

Fig. 13. Measured acceleration amplitude spectra: (a) q = 0 mm, (b) q = 1 mm, (c) q = 2 mm, (d) q = 3 mm, (e) q = 4 mm.

114 H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120

930.9 Hz � fe + f1, 1811.0 Hz � 2fe � f1. For the cracked geared rotor system, frequency components of the vibration signals fromthe simulation and experiment for cracked gear system both include mesh frequency (fe), its harmonics (2fe) and sideband fre-quencies. The sideband frequency components around meshing frequency and its harmonics are different from healthy system,which are related to the rotating frequency of the driven gear, for example, for crack depth q = 1 mm, 902.3 Hz � fe � f2,

Fig. 13 (continued)

Table 2fe and its sideband frequencies, and corresponding amplitudes.

Crack depth(mm)

fe � 2f2

(Hz)Amplitude(m/s2)

fe � f2

(Hz)Amplitude(m/s2)

fe

(Hz)Amplitude(m/s2)

fe + f2

(Hz)Amplitude(m/s2)

fe + 2f2

(Hz)Amplitude(m/s2)

1 889.8 0.0378 902.3 0.0882 914.6 0.0583 928.0 0.0164 939.1 0.0142 889.0 0.0346 901.5 0.1087 913.8 0.0736 926.2 0.0082 938.2 0.01123 891.6 0.0586 903.3 0.107 916.0 0.1306 928.2 0.0578 940.4 0.06094 891.3 0.0459 903.5 0.0329 915.8 0.084 927.9 0.0436 940.3 0.0755

H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120 115

1842.0 Hz � 2fe + f2, for crack depth of q = 2 mm, 901.5 Hz � fe � f2, 1840.0 Hz � 2fe + f2. Besides, frequency components ofexperimental signal are also caused by the cylindrical roller bearing (N205), belt wheel, coupling, oil, which are not consideredin the simulation model. And vibration signals from the experiment are influenced by the electrical interference, noise andmachine errors. Moreover, in the simulation, the Rayleigh damping is selected in this paper, which also have a great influence

Table 32fe and its sideband frequencies, and corresponding amplitudes.

Crack depth(mm)

2fe � 2f2

(Hz)Amplitude(m/s2)

2fe � f2

(Hz)Amplitude(m/s2)

2fe

(Hz)Amplitude(m/s2)

2fe + f2

(Hz)Amplitude(m/s2)

2fe + 2f2

(Hz)Amplitude(m/s2)

1 1804.0 0.0687 1816.0 0.0316 1829.0 0.102 1842.0 0.0196 1853.0 0.02452 1803.0 0.0793 1815.0 0.0262 1828.0 0.1755 1840.0 0.0258 1852.0 0.03313 1807.0 0.1606 1820.0 0.0648 1832.0 0.1568 1844.0 0.0752 1856.0 0.07144 1809.0 0.1075 1820.0 0.0530 1831.0 0.2456 1844.0 0.1144 1856.0 0.0788

116 H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120

the vibration responses of the system. Therefore, there exists discrepancy in amplitude and frequency components of the vibra-tion signals from the simulation and experiment.

Meshing frequency, its harmonics and corresponding amplitudes for cracked gear are listed in Tables 2 and 3. From thetables, it can be seen that the amplitudes of 2fe + f2, 2fe + 2f2 increase with the growth of crack depth, which show greatagreement with the simulation (see Fig. 12). However, owing to the influences of many factors, such as the accuracy of invo-lute cylindrical gears, variations of bearing stiffness, friction, interference of current and noise on experiment results, fre-quency components of the system are complicated and many interference frequency components appear. The amplitudeof meshing frequency and its harmonics are not always increasing with the increasing crack depth (see Tables 2 and 3).

Statistical features which are commonly used to provide a measurement of the vibration level are widely used in mechan-ical fault detection [8,17,41]. The RMS value and Kurtosis are defined as follows [41]:

RMS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N

XN

n¼1

ðxðnÞ � �xÞ2vuut ; �x ¼ 1

N

XN

n¼1

xðnÞ; ð11Þ

Kurtosis ¼1N

PNn¼1ðxðnÞ � �xÞ4

1N

PNn¼1ðxðnÞ � �xÞ2

h i2 : ð12Þ

Statistical indicators of vibration acceleration with the increase of crack depth are shown in Fig. 14. And indicators areacquired from two types of signals (simulated signal and experiment signal), which are shown in Fig. 14a and b, respectively.In order to reflect the vibration level compared with the healthy condition, the indicators are expressed as percentages whichare defined as:

rKurtosis ¼Kurtosiscrack � Kurtosishealth

Kurtosishealth� 100%; ð13Þ

rRMS ¼RMScrack � RMShealth

RMShealth� 100%; ð14Þ

where r shows the relative change of cracked signal indicator, subscripts Kurtosis and RMS denote different statistical indi-cators, subscripts crack and health are cracked and healthy gears.

It can be seen from the Fig. 14 that for the simulated signal, RMS and Kurtosis increase with the growth of the crack depth,and RMS and Kurtosis of the experiment signal show the similar trend as that of the simulated signal. However, for the caseq = 2 mm of the experiment signal, RMS is lower than that of q = 1 mm, which is owing to the influences of many factors thathave been mentioned above.

4.2. Crack fault features comparison between simulation and experiment based on instantaneous energy

The EMD method presented by Huang et al. [36] decomposes a time-series into a finite set of oscillatory functions calledthe intrinsic mode functions (IMF). An IMF is a function which satieties two conditions: (1) the number of extreme and the

Fig. 14. Acceleration statistical indicators with different crack depths: (a) simulated signal, (b) experiment signal.

Fig. 15. First four IMFs of acceleration responses of the geared rotor system with q = 1 mm: (a) simulated signal, (b) measured signal.

Fig. 16. Simulated instantaneous energies under different crack depths: (a) q = 0 mm, (b) q = 1 mm, (c) q = 2 mm, (d) q = 3 mm, (e) q = 4 mm, (f) averagedinstantaneous energy under different crack depths.

H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120 117

number of zero crossings must either equal or differ at most by one; (2) the running mean value of the envelope defined bythe local maxima and the envelope defined by the local minima is zero. The EMD algorithm can be found in Ref. [42]. Theinstantaneous energy is defined as Ref. [37], which provides information about the time variation of energy. In this study,the instantaneous energy is used to show the existence of tooth root crack and identify the crack degree. Based on numerical

Fig. 17. Measured instantaneous energies under different crack depths: (a) q = 0 mm, (b) q = 1 mm, (c) q = 2 mm, (d) q = 3 mm, (e) q = 4 mm, (f) averagedinstantaneous energy under different crack depths.

118 H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120

simulation and experiment, the acceleration vibration waveforms and its first four IMFs at the right bearing of driving shaftunder q = 1 mm are indicated in Fig. 15, respectively. It is important to note that only the first four IMFs are presented due tosmaller amplitudes for other IMFs. For the simulated signal and its IMFs, the periodic impact features caused by tooth rootcrack are not obvious under small crack condition. For the measured signal, the impact features are also not distinct due tomany influential factors, such as the interferences related to belt drive, rigid coupling, ball bearing and lubricating oil.

For simulated and measured signals under q = 0, 1, 2, 3 and 4 mm conditions, instantaneous energies for the vibrationacceleration at right bearing of driving shaft are shown in Figs. 16 and 17. In these figures, the red points in some peaks pres-ent the impact times related to tooth root crack. Averaged instantaneous energy under different crack depths are indicated inFigs. 16f and 17f, respectively. Here, the averaged instantaneous energy is obtained by taking the average of three peaksrelated to crack period. These figures show the following fault features:

(1) For the simulated results, the obvious impacts can be observed and the impact period (0.1172–0.0354 = 0.0818 s) isequal to the rotating period (0.0818 s) of driven shaft. For the measured results, the impact period (0.1144–0.0341 = 0.0803 s) is close to the rotating period of driven shaft due to the speed slight fluctuation. Moreover, theenergy of the impact caused by crack is slightly greater than that of the interference frequency under small crack con-dition (q = 1, 2 mm) and is considerably larger than that of the interference under large crack condition (q = 3, 4 mm).

(2) The impact feature caused by tooth root crack can be enlarged by instantaneous energy, so the instantaneous energy isa sensitive indicator for the existence of tooth root crack. Moreover, it is also suitable for identifying the crack degreeby analyzing the simulated and measured signals (see Figs. 16f and 17f).

5. Conclusions

Considering the effects of the extended tooth contact and tooth root crack on the time-varying mesh stiffness (TVMS), afinite element model of a geared rotor system with tooth root crack is developed. Vibration acceleration responses underdifferent crack depths at 1000 rev/min are obtained. By using empirical mode decomposition (EMD) to deal with the simu-lated and measured signals, the instantaneous energies under different crack depths are studied. Some conclusions are sum-marized as follows:

(1) The mesh model considering the effects of extended tooth is more suitable for the practice because it can deal with themultiple teeth meshing condition under large torques and crack depths. In addition, the gear crack has reduced thegear body rigidity, which leads to the reduction of stiffness of the healthy teeth and has little effect on the vibrationresponse.

H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120 119

(2) For simulated signals, sideband frequencies, statistical features and instantaneous energy can all reflect the fault fea-tures of the cracked gear coupled rotor system. But for measured signals, it is clear that the sideband frequencies andstatistical features are quite easily disturbed. However, the instantaneous energy is a sensitive indicator to diagnosethe existence of tooth root crack and identify the crack degree.

Acknowledgments

This project is supported by Program for New Century Excellent Talents in University (Grant No. NCET-11-0078), the Fun-damental Research Funds for the Central Universities (Grant No. N130403006) and the Joint Funds of the National NaturalScience Foundation and the Civil Aviation Administration of China (Grant No. U1433109) for providing financial support forthis work. The authors also thank the anonymous reviewers for their valuable comments.

References

[1] Lin TJ, Ou H, Li RF. A finite element method for 3D static and dynamic contact/impact analysis of gear drives. Comput Methods Appl Mech Eng2007;196:1716–28.

[2] Ma H, Yang J, Song RZ, et al. Effects of tip relief on vibration responses of a geared rotor system. Proc Inst Mech Eng, Part C: J Mech Eng Sci2014;28:1132–54.

[3] Wang JD. Numerical and experimental analysis of spur gears in mesh. PhD thesis. Curtin University of Technology, Australia; 2003.[4] Tesfahunegn YA, Rosa F, Gorla C. The effects of the shape of tooth profile modification on the transmission error, bending, and contact stress of spur

gears. Proc Inst Mech Eng, Part C: J Mech Eng Sci 2010;224:1749–58.[5] Zouari S, Maatar M, Fakhfakh T, et al. Following spur gear crack propagation in the tooth foot by finite element method. J Fail Anal Prevent

2010;10:531–9.[6] Howard I, Jia SX, Wang JD, et al. The dynamic modelling of a spur gear in mesh including friction and a crack. Mech Syst Signal Process

2001;15:831–53.[7] Jia S, Howard I. Comparison of localised spalling and crack damage from dynamic modelling of spur gear vibrations. Mech Syst Signal Process

2006;20:332–49.[8] Mohammed OD, Rantatalo M, Aidanpaa J. Improving mesh stiffness calculation of cracked gears for the purpose of vibration-based fault analysis. Eng

Fail Anal 2013;34:235–51.[9] Tian XH. Dynamic simulation for system response of gearbox including localized gear faults. MS thesis. University of Alberta, Edmonton; 2004.

[10] Tian XH, Zuo MJ, Fyfe KR. Analysis of the vibration response of a gearbox with gear tooth faults. In: Proceedings of IMECE04, 2004 ASME internationalmechanical engineering congress and exposition, November 13–20, Anaheim (CA, USA); 2004. p. 785–93.

[11] Pandya Y, Parey A. Failure path based modified gear mesh stiffness for spur gear pair with tooth root crack. Eng Fail Anal 2013;27:286–96.[12] Pandya Y, Parey A. Simulation of crack propagation in spur gear tooth for different gear parameter and its influence on mesh stiffness. Eng Fail Anal

2013;30:124–37.[13] Wu S, Zuo MJ, Parey A. Simulation of spur gear dynamics and estimation of fault growth. J Sound Vib 2008;317:608–24.[14] Pandya Y, Parey A. Crack behavior in a high contact ratio spur gear tooth and its effect on mesh stiffness. Eng Fail Anal 2013;34:69–78.[15] Chaari F, Fakhfakh T, Haddar M. Analytical modelling of spur gear tooth crack and influence on gearmesh stiffness. Euro J Mech – A/Solids

2009;28:461–8.[16] Zhou XJ, Shao YM, Lei YG, et al. Time-varying meshing stiffness calculation and vibration analysis for a 16dof dynamic model with linear crack growth

in a pinion. J Vib Acoust 2012;134:011011-1–011011-11.[17] Chen ZG, Shao YM. Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth. Eng Fail Anal

2011;18:2149–64.[18] Chen ZG, Shao YM. Mesh stiffness calculation of a spur gear pair with tooth profile modification and tooth root crack. Mech Mach Theory

2013;62:63–74.[19] Wan ZG, Cao HR, Zi YY, et al. An improved time-varying mesh stiffness algorithm and dynamic modeling of gear-rotor system with tooth root crack.

Eng Fail Anal 2014;42:157–77.[20] Liang XH, Zuo MJ, Pandey M. Analytically evaluating the influence of crack on the mesh stiffness of a planetary gear set. Mech Mach Theory

2014;76:20–38.[21] Ma R, Chen YS. Research on the dynamic mechanism of the gear system with local crack and spalling failure. Eng Fail Anal 2012;26:12–20.[22] Ma H, Song RZ, Pang X, et al. Time-varying mesh stiffness calculation of cracked spur gears. Eng Fail Anal 2014;44:179–94.[23] Fernandez del Rincon A, Viadero F, Lglesias M, et al. Effect of cracks and pitting defects on gear meshing. Proc Inst Mech Eng, Part C: J Mech Eng Sci

2012;226:2805–15.[24] Fernandez del Rincon A, Viadero F, Iglesias M, et al. A model for the study of meshing stiffness in spur gear transmissions. Mech Mach Theory

2013;61:30–58.[25] Fernandez del Rincon A, Iglesias M, de-Juan A, et al. Gear transmission dynamic: Effects of tooth profile deviations and support flexibility. Appl Acoust

2014;77:138–49.[26] Parey A, Badaouib ME, Guilletb F, et al. Dynamic modelling of spur gear pair and application of empirical mode decomposition-based statistical

analysis for early detection of localized tooth defect. J Sound Vib 2006;294:547–61.[27] Velex P, Ajmi M. On the modeling of excitations in geared systems by transmission errors. J Sound Vib 2006;290:882–909.[28] Kahraman A, Ozguven HN, Houser DR, et al. Dynamic analysis of geared rotors by finite elements. J Mech Des 1992;114:507–14.[29] Rao JS, Shiau TN, Chang JR. Theoretical analysis of lateral response due to torsional excitation of geared rotors. Mech Mach Theory 1998;33:761–83.[30] Kubur M, Kahraman A, Zini DM, et al. Dynamic analysis of a multi-shaft helical gear transmission by finite elements: model and experiment. ASME J

Vib Acoust 2004;126:398–406.[31] Lee AS, Ha JW, Choi DH. Coupled lateral and torsional vibration characteristics of a speed increasing geared rotor-bearing system. J Sound Vib

2003;263:725–42.[32] Han QK, Zhao JS, Chu FL. Dynamic analysis of a geared rotor system considering a slant crack on the shaft. J Sound Vib 2012;331:5803–23.[33] Loutridis SJ. Damage detection in gear systems using empirical mode decomposition. Eng Struct 2004;26:1833–41.[34] Forrester BD. Analysis of gear vibration in the time-frequency domain. In: Proceedings of the 44th meeting of mechanical failures prevention group of

the vibration institute, Virginia Beach, VA; 3–5 April, 1989.[35] Allgood G. Detection of gear wear on the 757/767 internal drive generator using higher order statistics and wavelets. In: 15th International modal

analysis conference, Florida; 1997.[36] Huang NE, Shen Z, Long SR, et al. The empirical mode decomposition and Hilbert spectrum for nonlinear and nonstationary time series analysis. Proc R

Soc London Ser 1998;454:903–95.

120 H. Ma et al. / Engineering Failure Analysis 48 (2015) 105–120

[37] Loutridis S. Damage detection in gear systems using empirical mode decomposition. Eng Struct 2004;26:1833–41.[38] Loutridis S. Instantaneous energy density as a feature for gear fault detection. Mech Syst Signal Process 2006;20:1239–53.[39] Han Q, Wang J, Li Q. Analysis of parametric stability for a spur gear pair system considering the effect of extended tooth contact. Proc Inst Mech Eng,

Part C: J Mech Eng Sci 2009;223:1787–97.[40] Guo Z, Zhang YD. Parameterization precise modeling of involutes spur gear based on APDL. Mach Build Automat 2010;40:33–6.[41] Wu S. Gearbox dynamic simulation and estimation of fault growth. MS thesis. University of Alberta, Edmonton; 2007.[42] Cheng JS, Yu DJ, Tang JS, et al. Application of frequency family separation method based upon EMD and local Hilbert energy spectrum method to gear

fault diagnosis. Mech Mach Theory 2008;43:712–23.