Effects of gear crack propagation paths on vibration responses of the perforated gear system

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Mechanical Systems and Signal Processing

Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]]

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Effects of gear crack propagation paths on vibration responsesof the perforated gear system

Hui Ma n, Xu Pang, Jin Zeng, Qibin Wang, Bangchun WenSchool of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, Liaoning, PR China

a r t i c l e i n f o

Article history:Received 10 August 2014Received in revised form23 February 2015Accepted 5 March 2015

Keywords:Time-varying mesh stiffnessPerforated gearCrack propagating pathVibration response

x.doi.org/10.1016/j.ymssp.2015.03.00870/& 2015 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ86 24 83684491;ail address: [email protected] (H. Ma).

e cite this article as: H. Ma, et al., Esystem, Mech. Syst. Signal Process.

a b s t r a c t

This paper investigates the dynamic behaviors of a perforated gear system consideringeffects of the gear crack propagation paths and this study focuses on the effects of a crackpropagating through the rim on the time-varying mesh stiffness (TVMS) and vibrationresponses. Considering the effects of the extended tooth contact, a finite element (FE)model of a gear pair is established based on ANSYS software. TVMS of the perforated gearwith crack propagating through tooth and rim are calculated by using the FE model.Furthermore, a lumped mass model is adopted to investigate the vibration responses ofthe perforated gear system. The results show that there exist three periods related to slotsof the gear body in a rotating period of the perforated gear. Gear cracks propagatingthrough tooth and rim both reduce the gear body stiffness and lead to reduction of TVMSbesides the crack tooth contact moment, and the TVMS weakening for the former is lessthan that for the latter. Moreover, the results also show that the gear crack propagatingthrough the rim (CPR) has a greater effect on vibration responses than the gear crackpropagating through the tooth (CPT) under the same crack level. Vibration level increaseswith the increasing crack depth, especially for the gear with CPR.

& 2015 Elsevier Ltd. All rights reserved.

1. Introduction

Gear pair is widely used in power transmission system and its vibration has a great influence on tooth fatigue failure. Thegear vibration is closely related to the inherent periodic internal excitations stemming from time-varying mesh stiffness(TVMS) and transmission errors. Many researchers have paid their attention to calculate the TVMS. Analytical method[1–10], finite element (FE) method [11–16] and analytical-FE approach [17,18] have been widely used to obtain the TVMS.Analytical method generally simplifies the gear tooth as a cantilever beam and provides an efficient way to determine theTVMS. However, there exist some difficulties for analytical method to deal with the extended tooth contact problem underlarge torsion conditions and the weakening effect of crack on the stiffness of the foundation (gear body). In contrast with theanalytical method, FE method is time consuming but is close to the actual condition.

Aiming the crack propagating through the tooth, based on the analytical method, many researches [1–10,19–23] studiedthe TVMS of cracked solid gear (gear without slots). In order to reduce the weight of the gear and improve transmissionefficiency, thin rim is usually designed to meet this objective. However, too thin rim may lead to bending fatigue problems,which would be catastrophic for aircraft applications [24,25]. For the thin-rimmed gear, the effects of the crack propagation

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ffects of gear crack propagation paths on vibration responses of the perforated(2015), http://dx.doi.org/10.1016/j.ymssp.2015.03.008i

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path and the rim thickness on crack paths have been performed [24–28]. The results of these studies can be used to simplifycrack propagation paths, which provides some information to calculate TVMS under different crack levels. Lewicki andBallarini [24] investigated the effect of rim thickness on gear tooth crack propagation and estimated stress intensity factorsto determine crack propagation direction. Based on linear elastic fracture mechanics, Zouari et al. [25] investigated the crackpropagation in the tooth foot of a spur gear and predicted catastrophic rim fracture failure modes. By using FE method andthe boundary element method, Kramberger et al. [26] examined the bending fatigue life of thin-rimmed spur gears of truckgearboxes. Ševčík et al. [27] estimated the effect of a constraint on a predicted crack path. Aiming at spur gears with highcontact ratio, Pandya et al. [28] carried out the researches on the crack propagation path, and analyzed the effects of thebackup radio and pressure angle on the crack propagation path.

Quasi-static analysis were mainly performed for the gear with the crack propagating through the rim [24–28], and theloaded tooth contact analysis for the healthy gear system is carried out considering the flexibility effect of the gear body[29]. Moreover, vibration responses are also calculated by considering the effects of flexible gear or thin-rimmed gear in[30,31]. Aiming at a flexible gear system, Abbes et al. [30] proposed a novel approach to investigate the dynamics behaviorsof the gear system by using the FE method combined with elastic foundation theory. Bettaieb et al. [31] compared the staticand dynamic behavior of geared transmissions including flexible components. In their model, a hybrid method is adopted,which combines classical beam elements, elastic foundations for simulating tooth contacts, and substructures derived from3D FE grids for thin-rimmed gears and their supporting shafts.

The above literatures indicate that the extensive attention has been focused on predicting the gear crack propagationpath, estimating the stress intensity factors and calculating the TVMS of solid gear. However, for the perforated gear withcrack propagating through the rim, the researches on TVMS and crack-induced vibration responses are still insufficient.Neglecting the effect of the dynamic flexibility of the gear body on the system vibration responses, the main objective of thisresearch is to achieve TVMS of a perforated gear considering the effect of the gear body by FE method, and the systemvibration responses are also evaluated by a lumped mass model. In addition, two groups of comparisons of TVMS andvibration responses are also carried out. The first group compares the healthy solid gear with healthy perforated gear, andthe second group compares the perforated gears with cracks propagating through the tooth (CPT) and rim (CPR).

2. Time-varying mesh stiffness calculation

Assuming that the cracks appear in driving gear with backup ratio (rim thickness divided by tooth height) of 0.5, crackpropagating through the tooth and rim are investigated in this section. The crack is simulated as a parabolic curve startingfrom the tooth root of driving gear (see Fig. 1). In Fig. 1, q denotes the crack depth, υ the crack propagation direction,subscripts 1 and 2 respectively denote cracks propagating through the tooth and rim, Ψ the crack initial position (in thispaper υ1¼751, Ψ¼351 for gear with CPT, υ2¼701, Ψ¼351 for gear with CPR). The basic parameters of the gear pair are listedin Table 1.

By using Plane183 element based on the plane strain assumption in ANSYS software, a 2D FE model of the gear pair isestablished to calculate the TVMS. The driving gear is respectively modeled as a healthy solid gear, a healthy perforated gearand a cracked perforated gear. The driven gear is modeled as a solid gear. Here, singularity element is used to simulate thecrack tip. In order to obtain TVMS accurately, all teeth are established, and the effect of the extended tooth contact is alsotaken into account. The accurate transition curve and involute are used for modeling the tooth geometry [22]. For theboundary conditions, the inner ring nodes of both gears are coupled with corresponding master nodes (the geometric centerof gear, namely O1 and O2), respectively, which means the inner ring nodes have the same behavior with the master node.

Fig. 1. Schematic of crack propagation path: (a) crack propagating through the tooth (CPT), (b) crack propagating through the rim (CPR).

Please cite this article as: H. Ma, et al., Effects of gear crack propagation paths on vibration responses of the perforatedgear system, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.03.008i




Table 1Basic parameters of the gear pair.

Parameters Driving gear Driven gear

Tooth number 19 48Module (mm) 3.175 3.175Tooth width (mm) 16 16Pressure angle (1) 20 20Addendum coefficient 1 1Tip clearance coefficient 0.25 0.25Young’s modulus (GPa) 206.8 206.8Poisson’s ratio 0.3 0.3Density (kg/m3) 7850 7850Hub radius (mm) 12.5 50

Fig. 2. Finite element model of the meshing gear pair.

H. Ma et al. / Mechanical Systems and Signal Processing ] (]]]]) ]]]–]]] 3

The master node of the driving gear (O1) is restrained from all degrees of freedom. And the master node of driven gear (O2)is only free to rotate around the gear axis. According to the rotation direction of the gears, a constant loaded torque in theclockwise direction is equivalent to tangential forces applied to the inner ring nodes of the driven gear. Here, it is worthnoting that the torque is applied to the driven gear based on the force balance theory. Once the gear deformation isevaluated, the rotational displacement of the master node of gear 2 is determined and the TVMS can be calculated. The FEmodel of the gear pair and two kinds of cracks with depths of 1, 2, 3 and 4 mm are displayed in Fig. 2.

Based on FE model, TVMS of healthy solid and perforated gears are obtained under torques of 10, 30, 60 and 100 Nm (seeFig. 3). It can be found that TVMS for both two types of gear pairs increase with the increasing loaded torque. Furthermore,TVMS of the solid gear pair in a single mesh cycle is the same as that of another mesh cycle (see Fig. 3a). However, for theperforated gear, three new periods marked by arrows appear, and the average meshing stiffness decreases due to three slotsin the driving gear (see Fig. 3b). Namely, TVMS of the perforated gear is smaller than that of solid gear. TVMS of gear withcrack propagating through the tooth and rim under crack depths of 1, 2, 3 and 4 mm are shown in Fig. 4. In the figure, theunderpainting zone denotes the crack tooth contact zone, which is defined as the crack tooth mesh cycle. For bettervisualization, the enlarged view of TVMS is shown in Fig. 5. Absolute change of TVMS for gears with cracks relative to TVMSof healthy perforated gear at moments A and B are determined to study the variation of the TVMS (see Table 2). In order toobserve conveniently, absolute change of TVMS of gear with cracks relative to healthy perforated gear is displayed in Fig. 6.Contour map of the comprehensive deformation of healthy perforated gear and perforated gears with cracks of q¼2 and4 mm for moments A (left column) and B (right column) are shown in Fig. 7. Figs. 4–7 illustrate some special phenomena asfollows.

(1) TVMS at moment B decrease obviously compared with those at moment A, which shows a good agreement with thecontour map of the comprehensive deformation in Fig. 7. In addition, at moment A, mesh stiffness reduction of gear with





Fig. 3. TVMS of healthy gear: (a) solid gear (without slots), (b) perforated gear.

Fig. 4. TVMS under different cases: (a) healthy and cracked gears with q¼1 mm, (b) healthy and cracked gears with q¼2 mm, (c) healthy and cracked gearswith q¼3 mm, (d) healthy and cracked gears with q¼4 mm. (Keys: healthy solid gear healthy perforated gear perforated gearwith CPR perforated gear with CPT).


CPT is larger than that of gear with CPR. On the contrary, at moment B, the reduction of the former is smaller than that of thelater (see Table 2 and Fig. 6). The reason for these phenomena is that at moment A, the effect of CPT on the weakening toothstiffness plays a dominant role; at moment B, the effect of CPR on weakening the gear body stiffness works effectively.

(2) Crack propagating through the rim influences not only the TVMS in the crack tooth mesh cycle (underpainting zonein Fig. 5), but also the TVMS in the mesh cycles without crack tooth engagement. This is mainly caused by the weakeninggear body stiffness. Far away from crack tooth mesh cycle, the reduction of TVMS becomes small and TVMS is the same asthat of the healthy gear. Namely, CPT and CPR both affect the TVMS without crack tooth engagement, and the effect of thelatter is more obvious than that of the former.





Fig. 5. Enlarged view of TVMS (Note: CPT and CPR denote crack propagating thought the tooth and rim, respectively, and the followed number representsthe crack depth, unit is millimeter.).

Table 2Absolute change of TVMS of gears with CPT and CPR relative to healthy thin-rimmed gear.

Crack depth (mm) Crack propagating through the tooth (CPT) Crack propagating through the rim (CPR)

KA (N/m) Change (N/m) KB (N/m) Change (N/m) KA (N/m) Change (N/m) KB (N/m) Change (N/m)

0 (Healthy) 1.927�108 – 2.076�108 – 1.927�108 – 2.076�108 –

1 1.908�108 0.192�107 2.033�108 0.437�107 1.910�108 0.167�107 2.009�108 0.671�107

2 1.865�108 0.626�107 1.976�108 1.004�107 1.879�108 0.485�107 1.925�108 1.515�107

3 1.712�108 2.155�107 1.831�108 2.459�107 1.842�108 0.850�107 1.760�108 3.159�107

4 1.649�108 2.778�107 1.771�108 3.057�107 1.789�108 1.375�107 1.556�108 5.202�107

Fig. 6. Absolute change of TVMS of gears with CPT and CPR relative to healthy perforated gear.


3. Dynamic model of the gear system and model validation

3.1. Dynamic model of the gear pair

Lumped mass model [2,4,5,9,11,32] and FE model [6,22,23,33–38] are widely utilized to analyze the vibration responsesof gear system. For the rigid shafts and gear bodies, a lumped mass model is commonly adopted, which owns few degrees offreedom and high efficiency. In this study, the effects of the dynamic flexibility of the gear body on the system vibrationresponses are ignored and a lumped mass model for the gear system is established to investigate the system vibrationresponses.

The gear pair is simplified as a pair of rigid disks. The mesh between gears is represented by a spring-damper in the planeof action (see Fig. 8). In Fig. 8, rb1 and rb2 are the radii of the base circles.Ω1 andΩ2 are the rotating speeds, subscripts 1 and2 denote the gear 1 (driving gear) and 2 (driven gear), respectively. c12(t) represents the mesh damping which is assumed tobe zero in this paper. k12(t) represents the TVMS obtained from the FE model of the gear system. The gears are supported byshafts and bearings, which are simplified as springs kbxi, kbyi, and viscous dampers cbxi, cbyi (i¼1, 2), and herekbx1¼kby1¼31.14 MN/m, kbx2¼kby2¼49.31 MN/m, cbxi¼cbyi¼0 N s/m. The relative position angle of gears is depicted bythe angle α12 (0rα12o2π), which is between the line connecting the gear centers and the positive x-axis of gear 1, and





Fig. 7. Contour map of the comprehensive deformation for healthy perforated gear and perforated gears with CPT and CPR of q¼2 and 4 mm for momentsA (left column) and B (right column): (a) healthy perforated gear, (b) CPT of q¼2 mm, (c) CPR of q¼2 mm, (d) CPT of q¼4 mm, (e) CPR of q¼4 mm.


α12¼0 in this work. The angle ψ12 between the plane of action and the positive y-axis is defined as

ψ12 ¼�αþα12 Ω1:Counterclosewiseαþα12�π Ω1:Clockwise

(; ð1Þ

where α is the pressure angle.





Fig. 8. Lumped mass model of the gear system.

Table 3Parameters of the gear system.

Gears Iz (kg mm2) m (kg) Bearings kbx (MN/m) kby (MN/m)

1 (Perforated gear) 136.06 0.24 1 31.14 31.141 (Solid gear) 158.66 0.292 (Solid gear) 5381.77 1.29 2 49.31 49.31


Considering the influence of the rotating direction of the driving gear, the function σ is introduced as

σ ¼1 Ω1:Counterclockwise�1 Ω1:Clockwise

(: ð2Þ

The equations of motion of the gear pair with six degrees of freedom are written as

m1 €x1�k12ðtÞp12ðtÞ sin ψ12�c12ðtÞ _p12ðtÞ sin ψ12þkbx1x1þcbx1 _x1 ¼ 0m1 €y1þk12ðtÞp12ðtÞ cos ψ12þc12ðtÞ _p12ðtÞ cos ψ12þkby1y1þcby1 _y1 ¼ 0

Iz1 €θz1þσrb1k12ðtÞp12ðtÞþσrb1c12ðtÞ _p12ðtÞ ¼ σT1

m2 €x2þk12ðtÞp12ðtÞ sin ψ12þc12ðtÞ _p12ðtÞ sin ψ12þkbx2x2þcbx2 _x2 ¼ 0m2 €y2�k12ðtÞp12ðtÞ cos ψ12�c12ðtÞ _p12ðtÞ cos ψ12þkby2y2þcby2y2 ¼ 0

Iz2 €θz2þσrb2k12ðtÞp12ðtÞþσrb2c12ðtÞ _p12ðtÞ ¼ σT2

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

; ð3Þ

wherem1 and m2 denote the masses of gears 1 and 2, and Iz1, Iz2 represent the moments of inertia about the z axis of gears 1and 2, respectively (see Table 3). T1 and T2 are constant torques. The term p12(t) represents the relative displacement of thegears in the direction of the action plane and it is defined as

p12ðtÞ ¼ ð�x1 sin ψ12þx2 sin ψ12þy1 cos ψ12�y2 cos ψ12þσrb1θz1þσrb2θz2Þ�e12ðtÞ; ð4Þ

where e12(t) is the non-loaded static transmission error. Here, assembly error, machining error and misalignments caused byshaft bending are ignored and e12(t)¼0.

Considering the combined effect of the system damping composed of the bearing, shaft and gear pair, Rayleigh-typedamping is adopted [38], which is one of the widely used viscous dampings. The motion equations of the whole system canbe written as follows:

M12€X12þC _X12þðK12þKbÞX12 ¼ F12; ð5Þ

where X12 is the displacement vector of the gear pair. M12 and K12 represent the mass and mesh stiffness matrixes of thespur gear pair, respectively. C is the damping matrix of the system. Kb denotes the supporting stiffness matrix. F12 is theexciting force vector of the spur gear pair.






3.2. Model validation

In order to verify the validity of the dynamic model, another FE model of the meshing gear pair is shown in Fig. 9. Thedriving gear and driven gear are established by shell elements (Shell181 element in ASNYS). Every node has six degrees offreedom and the inner ring nodes of both gears are coupled with the corresponding master nodes (the geometric centre ofgear, namely O1 and O2), respectively. The support stiffness on the master nodes (O1 and O2) is simulated by the spring anddamping (Combin14 element in ANSYS). The mesh between gears is represented by a spring km (Combin14 element inANSYS) in the plane of action. In this paper, km is set as the average value of TVMS. One side of the spring (node N1) iscoupled with the nodes of meshing tooth on driving gear and the other side (node N2) is coupled with the nodes of meshingtooth on driven gear. Model parameters of the meshing gear pair are listed in Table 4.

The first four natural frequencies of the solid gear system calculated by lumped mass model and FE model are listed inTable 5, and corresponding mode shapes are shown in Fig. 10. Based on the comparison and error analysis, it is clear thatresults of the lumped mass model agree well with those of the FE model.

Natural frequencies of the solid gear system reflecting flexible gear body vibration are shown in Fig. 11. The results showthat the lowest natural frequency of the system reflecting the flexible gear body vibration is 15,265.0 Hz, which is largerthan the natural frequencies of the system reflecting the rigid gear body vibration. It also means that the presented dynamicmodel is feasible within the concerning frequency range.

Based on the lumped mass model and finite element model, the first four natural frequencies of the perforated gearsystem are listed in Table 6, and corresponding mode shapes are showed in Fig. 12. Natural frequencies reflecting the flexiblegear body vibration in perforated gear system are exhibited in Fig. 13. The results of perforated gear system are similar to

Fig. 9. Finite element model of the meshing gear pair.

Table 4Model parameters of the meshing gear pair.

Parts Element types Geometric parameters Material parameters

Gear pair Shell181 See Table 1 See Table 1Support stiffness Combin14 kbx1¼kby1¼31.14 MN/m, kbx2¼kby2¼49.31 MN/m,

cbxi¼cbyi¼0 Ns/m (i¼1, 2)Mesh stiffness Combin14 km¼287 MN/m (solid gear), km¼193 MN/m (perforated gear)

Table 5Natural frequencies of the solid gear system.

Modes Lumped mass model (Hz) Finite element model (Hz) Error (%) Corresponding mode shapes

f1 907.4 904.8 0.29 Coupled translational–torsional vibrationf2 984.0 981.8 0.22 Translational vibration of driven gearf3 1361.6 1353.0 0.64 Coupled translational–torsional vibrationf4 1649.2 1643.5 0.35 Translational vibration of driving gear





Fig. 10. Mode shapes of the solid gear system: (a) f1, (b) f2, (c) f3, (d) f4.

Fig. 11. Mode shapes of solid gear system reflecting the flexible gear body vibration: (a) the lowest mode shape (15,265.0 Hz), (b) the second-lowest gearbody mode shape (15,403.0 Hz).


those of solid gear system, the frequencies obtained by lumped mass model agree well with those obtained by FE model,and the lowest frequency reflecting the gear body vibration is 13,331.9 Hz, which is also larger than the naturalfrequencies of the system reflecting the rigid gear body vibration. Therefore, in this study, the dynamic flexibility of the





Table 6Natural frequencies of the thin-rimmed gear system.

Modes Lumped mass model (Hz) Finite element model (Hz) Error (%) Corresponding mode shapes

f10 918.5 913.7 0.53 Coupled translational–torsional vibrationf20 984.0 981.8 0.22 Translational vibration of driven gearf30 1464.7 1441.8 1.59 Coupled translational–torsional vibrationf40 1812.9 1813.7 0.04 Translational vibration of driving gear

Fig. 12. Mode shapes of the perforated gear system: (a) f10 , (b) f20 , (c) f30 , (d) f40 .


gear body is ignored and the lumped mass model is reasonable to analyze the vibration responses in the concerningfrequency range.

From the first four natural frequencies of the gear system, it can be found that natural frequencies of the system withsolid gear are lower than that of the system with perforated gear. This is because the TVMS and the mass both decrease forperforated gear system, and the mass reduction plays an important role.





Fig. 13. Mode shapes of perforated gear system reflecting the flexible gear body vibration: (a) the lowest mode shape (13,331.9 Hz), (b) the second-lowestgear body mode shape (14,208.5 Hz).


4. Effects of gear crack propagation paths on vibration responses

4.1. Vibration responses at a constant speed

Based on the lumped mass model, vibration responses at a constant speed (1000 rev/min) are analyzed in this section.Acceleration waveforms for a healthy solid gear and a healthy perforated gear are displayed in Fig. 14. In the figure, it is clearthat the acceleration amplitude of the former is smaller than that of the latter. Amplitude fluctuation marked by red dotdash line can be observed, which does not appear in the healthy solid gear. Acceleration frequency spectra of two gears inthe horizontal direction are indicated in Fig. 15. In the figure, fe (z1Ω1/60¼316.67 Hz) denotes the mesh frequency and fr(Ω1/60) the rotating frequency of the driving gear. The enlarged views of positions A and B in Fig. 15b are also presented (seeFig. 15c and d). For healthy solid gear, there are mesh frequency (fe) and its harmonics (nfe, n¼2,3,4,…) without any sidebandfrequencies. However, for the healthy perforated gear, there exists not only fe and nfe (n¼2,3,4,…), but also fr andcombination frequencies of fe and fr. Here, it is worth noting that the amplitude of fr is very small, which cannot be observedin Fig.15b. Furthermore, peak amplification phenomenon appears at the combination frequencies of fe�9fr, fe�6fr, fe�3fr, fe,feþ3fr, feþ6fr and 2fe�9fr, 2fe�6fr, 2fe�3fr, 2fe, 2feþ3fr, 2feþ6fr, which shows a law that from fe�9fr to feþ10fr, amplitude isamplified at every interval of 3fr or 4fr and the sum of these intervals is fe. This phenomenon is caused by appearance ofthree new periods related to the three slots (see Fig. 3) in a rotating period of the perforated gear. In addition, amplitude ismuch larger at 5fe than those of surrounding frequency components, which is due to 5fe is close to the natural frequency f30.

Partial frequency spectra of the cracked gear system under different crack depths are shown in Fig. 16. The amplitudes at2fe�6fr, 2fe�3fr, 2fe, 2feþ3fr and 2feþ6fr are listed in Table 7. It can be seen that amplitudes for gear system with CPR at2fe�3fr, 2fe, 2feþ3fr increase with the increasing crack depth; amplitudes for gear system with CPT also show the similarchange law except for the condition under crack depth q¼4 mm.

Statistical features are widely used in mechanical fault detection [9,21] to evaluate the vibration level. The RMS value isdefined as follows [21]:

RMS¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1N

XNn ¼ 1

ðxðnÞ�xÞ2vuut ; x¼ 1

N

XNn ¼ 1

xðnÞ: ð6Þ

In order to evaluate the vibration level for the cracked gear system, the indicator is expressed as percentage, which iswritten as

rRMS ¼RMScrack�RMShealth

RMShealth� 100%; ð7Þ

where r shows the relative change of cracked signal indicator, subscripts crack and health denote cracked and healthy gears.In addition, the percentage change of amplitude in frequency spectrum is also used to provide an evaluation of vibration

level, which is given by

rAmplitude ¼Amplitudecrack�Amplitudehealth

Amplitudehealth� 100%: ð8Þ

Percentage change of statistical indicator RMS and amplitude at 2feþ3fr are shown in Fig. 17. It is obvious that thepercentage change of amplitude presents the same trend as statistical indicator. And it also can be seen that deteriorationlevel of gear system with CPR is more serious than that of gear system with CPT under the same crack depth.





Fig. 14. Acceleration waveforms: (a) healthy solid gear system, (b) healthy perforated gear system. (For interpretation of the references to color in thisfigure legend, the reader is referred to the web version of this article.)

Fig. 15. Acceleration frequency spectra in x direction: (a) healthy solid gear system, (b) healthy perforated gear system, (c) and (d) enlarged views of A and B.


4.2. Amplitude–frequency responses

For the healthy and cracked gear systems, amplitude–frequency responses of the displacement in x direction are shownin Figs. 18 and 19, respectively. These figures indicate the following dynamic phenomena.

(1).

Plege

The first and third natural frequencies are easily excited, and the corresponding mode shapes are both coupledbending–torsional vibration. Besides these primary resonances, super-harmonic resonances of the first and thirdmodes are also excited, such as f3/2, f30/2, f3/4, f30/4, f3/8 and f30/8. In contrast with the solid gear system, theamplitude for perforated gear system is higher owing to the static flexibility of the gear body (see Fig. 18).

(2).
The amplitudes under CPT and CPR both increase with the increasing crack depth. In contrast with the vibration for theCPT, vibration responses for CPR are more severe.
ase cite this article as: H. Ma, et al., Effects of gear crack propagation paths on vibration responses of the perforatedar system, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.03.008i




Fig. 16. Partial frequency spectra of the cracked gear system under different crack depths: (a) q¼1 mm, (b) q¼2 mm, (c) q¼3 mm, (d) q¼4 mm for gearwith CPT (left column) and CPR (right column).






Table 7Double mesh frequency, its sideband frequencies and corresponding amplitudes.

Crack depth underCPT (mm)

2fe�6fr(Hz)

Amplitude(m/s2)

2fe�3fr(Hz)

Amplitude(m/s2)

2fe(Hz)

Amplitude(m/s2)

2feþ3fr(Hz)

Amplitude(m/s2)

2feþ6fr(Hz)

Amplitude(m/s2)

0 (Healthy) 533.3 1.53�10�2 583.3 4.58�10�2 633.3 6.47�10�1 683.3 4.98�10�2 733.3 4.38�10�2

1 1.24�10�2 4.67�10�2 6.52�10�1 5.56�10�2 3.68�10�2

2 8.43�10�3 4.82�10�2 6.58�10�1 6.35�10�2 2.69�10�2

3 2.93�10�3 5.06�10�2 6.62�10�1 6.72�10�2 1.83�10�2

4 3.53�10�3 5.05�10�2 6.62�10�1 6.63�10�2 1.82�10�2

Crack depth underCPR (mm)

2fe�6fr(Hz)

Amplitude(m/s2)

2fe�3fr(Hz)

Amplitude(m/s2)

2fe(Hz)

Amplitude(m/s2)

2feþ3fr(Hz)

Amplitude(m/s2)

2feþ6fr(Hz)

Amplitude(m/s2)

1 533.3 1.07�10�2 583.3 4.84�10�2 633.3 6.57�10�1 683.3 6.15�10�2 733.3 3.12�10�2

2 6.77�10�3 5.67�10�2 6.75�10�1 8.25�10�2 2.49�10�2

3 1.58�10�2 7.31�10�2 6.99�10�1 1.09�10�1 6.26�10�2

4 3.56�10�2 1.04�10�1 7.40�10�1 1.59�10�1 1.33�10�1

Fig. 17. Percentage change of statistical indicator RMS and amplitude at 2feþ3fr: (a) rRMS, (b) ramplitude at 2feþ3fr.

Fig. 18. Amplitude–frequency responses for two types of healthy gear systems in x direction.


5. Conclusions

Based on FE model of gear system, TVMS for healthy and cracked gear systems are analyzed. By using a lumped massmodel of a gear system, the effects of crack propagating through the tooth and rim on vibration responses of the gear systemare discussed. Some conclusions can be summarized as follows:

(1)

Plge

TVMS increases with the increasing loaded torque and decreases with the increasing crack depth. TVMS of theperforated gear is lower than that of the solid gear owing to the slots in gear. In addition, there exists the three new

ease cite this article as: H. Ma, et al., Effects of gear crack propagation paths on vibration responses of the perforatedar system, Mech. Syst. Signal Process. (2015), http://dx.doi.org/10.1016/j.ymssp.2015.03.008i




Fig. 19. Amplitude–frequency responses for two types of cracked gear systems in x direction: (a) crack propagating through the tooth (CPT), (b) crackpropagating through the rim (CPR).

Plge


periods for perforated gear in the rotating period. The gear body stiffness weakening due to crack propagating throughthe rim leads to the reduction of TVMS in multi-mesh cycles besides the crack tooth mesh cycle, especially when thecrack depth is large. Owing to the effect of crack propagating through the rim on gear body cannot be ignored, it isnecessary to establish accurate FE model with the whole teeth in order to obtain TVMS accurately.

(2)
There exists not only mesh frequency fe and its harmonic frequencies nfe (n¼2,3,4,…) but also rotating frequency ofdriving gear fr and combination frequencies of fe and fr for the healthy perforated gear system, which is different fromthose for the healthy solid gear. Effects of crack propagating through the rim on the system vibration are more obviousthan those of crack propagating through the tooth under the same crack depth.
The conclusions mentioned above are drawn by assuming a small gear with slots. Moreover, only the static flexibility ofthe gear body is considered for calculating the TVMS based on the plane strain assumption, and the dynamic flexibility ofthe gear body is neglected by using a lumped mass model. In future work, a 3D FE model for thin-rimmed gears will bedeveloped to calculate the TVMS with a gear crack, and the dynamic analysis under different crack conditions will also becarried out by adopting suitable modal reduction methods. Moreover, the effects of out-of-plane modes and associatedexcitation frequencies on the vibration responses of a flexible gear system with a rim crack will also be investigated.

Acknowledgments

This project is supported by Program for New Century Excellent Talents in University (Grant no. NCET-11-0078), theFundamental Research Funds for the Central Universities (Grant no. N130403006 and N140301001) and the Joint Funds ofthe National Natural Science Foundation and the Civil Aviation Administration of China (Grant no. U1433109) for providingfinancial support for this work. The authors also thank the anonymous reviewers for their valuable comments.

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Effects of gear crack propagation paths on vibration responses of the perforated gear system

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