Influence of finite element model, load-sharing and load distribution on crack propagation path in spur gear drive

Journal Pre-proofs Influence of finite element model, Load-sharing and load distribution on crack propagation path in spur gear drive Rama Thirumurugan, N. Gnanasekar PII: DOI: Reference:

S1350-6307(19)31117-3 https://doi.org/10.1016/j.engfailanal.2020.104383 EFA 104383

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Engineering Failure Analysis

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1 August 2019 3 January 2020 7 January 2020

Please cite this article as: Thirumurugan, R., Gnanasekar, N., Influence of finite element model, Load-sharing and load distribution on crack propagation path in spur gear drive, Engineering Failure Analysis (2020), doi: https:// doi.org/10.1016/j.engfailanal.2020.104383

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Influence of finite element model, Load-sharing and load distribution on crack propagation path in spur gear drive Rama Thirumurugan a and N. Gnanasekar b a Professor, Department of Mechanical Engineering, Dr. Mahalingam College of Engineering and Technology, Pollachi, Tamilnadu, India b Assistant Professor, Department of Mechanical Engineering, P.A. College of Engineering and Technology, Pollachi, Tamilnadu, India Abstract The tooth fracture is one of the main failure modes in spur gear drives which is due to the bending fatigue load acting on the spur gear tooth during power transmission. The fatigue load initiates the crack at the tooth root region and propagate along the least resistance path which leads to complete tooth/ rim fracture. The crack propagation path is influenced by the magnitude of the stress intensity factor (SIF) and its location along the face width at the crack front. Generally, the prediction of crack propagation studies was carried out using a three-dimensional (3D) finite element (FE) model in literature with uniform / parabolic load distribution on the contact line along the face width. An attempt has been made to explore the finite element (FE) model and the load distribution on the SIF for the given crack size. Two different 3D models namely three teeth sector model (TTSM) and Single tooth sector model (STSM) are developed with the initial crack with the size of 1mm at the location of maximum principal stress in the fillet region. The load is moved from lowest point of tooth contact (LPTC) to highest point of tooth contact (HPTC) in the case of STSM whereas in TTSM the loads are applied simultaneously to the leading and trailing teeth whenever there is a double pair contact established as per the gear kinematics. The results are compared with the published results and found that the STSM and TTSM predict a deeper crack propagation path. The results of the FE models are compared with the single tooth fatigue load test results and found that the simulation results with actual load distribution predicting almost similar crack path as that obtained from the experiment

Keywords: Stress intensity factors; Finite element model; Experimental test; single tooth and three teeth spur gear; three-dimensional crack propagation

1. Introduction Spur gears are widely used as power transmission gear drives in automobiles, aerospace, machinery and in domestic appliances. In spur gear drives, the common failures modes are in the tooth root fillet region or in the active surface of contact. This may be due to excessive service loads and improper operating conditions which initiate the crack leads to complete tooth fracture or pitting failure [1-4]. Among these two-failure modes, tooth breakage is a major mode of failure due to the repeated fatigue loading on the gear tooth which leads to the formation of crack at the tooth root fillet region [5-7]. Hence, the detection of a crack in advance on spur gear drives which allows planned shutdown in order to prevent the sudden failures and ensure the safe operation [2]. The fatigue failure of tooth breakage is divided into three main stages as follows: (i) crack-initiation, (ii) crack-propagation period and (iii) final failure [8-11]. Kumar et al. [12] used the American Gear Manufacturing Association (AGMA) approach to predict the crack initiation due to pitting and bending fatigue in the spur gear tooth. They observed that the FE results for contact and bending stresses had a reasonable agreement with AGMA results. The linear elastic fracture mechanics (LEFM) was widely adopted to find the crack growth behavior at the tooth root fillet region by calculating the values of SIF for mode I (KI), mode II (KII) and mode III (KIII) using finite element method [13]. Pehan et al. [14] performed crack propagation studies for different crack initiation locations using a 3D FE spur gear model by applying uniform load distribution along the face width. Glodez et al. [15] conducted the experimental test to investigate the effect of uniform load distribution for different loading positions along the face width to measure the fatigue crack growth in the spur gear tooth root. They noted that the load distribution along the face width influenced significantly the crack growth in the tooth root. Pehan et al. [16] considered the uniform load distribution with different load positions along the face width on a 3D single tooth spur gear FE model to measure the maximum tensile stresses and deformation at the tooth tip. The crack propagation was modeled along with crack depth from the SIF of each crack front nodal point in the path of the higher magnitude of strain energy release rate. Lewicki et al. [17] predicted the 3D crack propagation path for different initial crack locations on a split gear tooth. The crack path was studied for constant load distribution along the face width to simulate tooth load. Ghaffari et al. [18] conducted the crack initiation and propagation path studies on the developed 3D single tooth spur gear FE model by including the effect of friction. They also investigated the effect of uniform load distribution with different load cases and crack path trajectories. Also, they have proposed a method to delay the crack propagation rate of tooth root crack by using the composite patches at both ends of the gear tooth surface. Li et al. [19] investigated the effect of three kinds of crack types on the spur gear pairs by using the Uniform Distribution of Load (UDL) with the time-varying mesh stiffness model. They

established the analytical and FE model of single tooth cracked spur gear to study the meshing characteristics while the crack was propagating along tooth width. Chen and Shao [20] investigated the influences of tooth root crack on the mesh stiffness for different crack lengths and depth using the analytical model. They simulated the tooth root crack for constant load along tooth width to find the crack propagation for the single tooth spur gear. Later they proposed the general analytical mesh model with including the effect of gear tooth errors. This proposed model established to investigate the influence of tooth profile modification, torque, and gear tooth root crack on the total mesh stiffness, load sharing and loaded static transmission errors [21]. Yu et al. [22] presented the analytical investigation on influences of spatial crack propagation for uniform load distribution on time-varying gear mesh stiffness and LSR. Verma et al. [23] investigated the effect of crack on the time-varying mesh stiffness and crack propagation behavior for constant load case along the face width of a single tooth spur gear using extended FE method. Podrug et al. [24] developed the 2D three teeth spur gear FE model to study the effect of the moving load model on the crack initiation and propagation behavior. Sánchez et al. [25] developed an LSR model among couples of teeth in contact for standard and high contact ratio spur gear pairs. They calculated the bending stresses with the corresponding critical loading conditions. Ma et al. [26] conducted the reviewed literature on prediction of the crack propagation path, time-varying mesh stiffness and calculation of vibration response. Chen et al. [27] computed a time-varying mesh stiffness in the spur gear for the web and slot type gear foundation by using the analytical FE method. The uniform load was considered along the face width to simulate the crack propagation along with the web and rim thickness in the spur gear. They noted that the web thickness was mainly an influencing factor than rim thickness in the variation of mesh stiffness. Thirumurugan et al. [28] compared LSR estimated by the different 2D FE models such as (a) Single-tooth sector model (STSM), (b) Single-tooth full-rim model (STFM), (c) Three-teeth sector model (TTSM) and (d) Three-teeth full-rim model (TTFM). The study has been extended with a single-pair contact model (SPCM) and a multi-pair contact model (MPCM) to estimate the LSR. They reported that both SPCM and MPCM yields good result based on LSR values. A similar study has been extended by them to explore the effect of gear parameters on the LSR, and LSR based contact and bending stresses [29]. In most of the research work, the prediction of crack propagation has been done by considering the normally reported load case (i.e., UDL) along the direction of face width without considering load shared between pairs. Authors[30] explored the variations of three modes of SIFs for STSM at the initial crack tip of tooth root were presented for the four types of load conditions such as case1: UDL along the face width without considering LSR, case 2: UDL along the face width by including the LSR, case 3: Actual load distribution along the face width without considering LSR, case 4: Actual load distribution along the face width by including the LSR in the previous study and concluded that, among the four load types, the influences of LSR on the actual load distribution model (type 4) yielded a reasonable result in the prediction of three modes of SIFs at the initial crack tip nodes of 3D FE STSM.

In this work, the influences of LSR on the actual load distribution model was calculated for TTSM of spur gear from the line of LPTC to the line of HPTC along the face width. The variation of KI, KII along the crack length and the predicted crack propagation path were validated with available literature results. A comparative study was made to find the variation of SIFs along the crack length and crack propagation path for both STSM and TTSM of spur gear by applying actual load distribution with LSR at the HPSTC line. Further, the experimental test was conducted on standard spur gear to predict the maximum tooth load in the tangential direction and also the crack propagation was measured at the region of gear tooth root up to the final fracture of the tooth. 2. Comparative study on 3D spur gear FEM 2.1 Spur gear geometry models The 3D FE STSM and TTSM were developed for the geometrical data given in Table.1 [14] using the ABAQUS finite element code. The linear brick element with eight-node (C3D8) was adopted to discretize a 3D spur gear model. The active profile meshed with 29 elements and the face width meshed with 28 elements. High strength alloy steel 16MnCr5 material with modulus of elasticity (E) = 2.1x105 N/mm2 and Poisson’s ratio () = 0.3 is considered for the analysis. The geometric model of STSM and TTSM are shown in Fig.1 (a) and (b) respectively. Table 1 Geometrical data of the spur gear (Ref. Pehan et al. [14]) Gear Parameter

Symbol

Gear

Module (mm)

m

4.5

Number of teeth

Z

39

Engagement angle on the pitch diameter (0)

α

24

Gear face width (mm)

B

28

Addendum modification coefficient

x

0.06

Addendum

ha

1 module

Dedendum

hf

1.35 module

The FE model was developed for both STSM and TTSM. The sector surfaces of STSM (Fig.2(a)) were constrained in the normal direction and the actual loads distributed along the face width are moved from contact line of LPTC (line along with Node no1) to contact line of HPTC (line along Node no 30). Whereas, the sector surfaces of TTSM (Fig.2(b)) were constrained in the normal direction and the actual loads distributed along the face width are moved from contact line of LPTC (line along with Node no1) to contact line of HPTC (line along Node no 30) of the middle tooth. At the same time, the corresponding adjacent tooth also was loaded while the load was moved at the double pair contact region.

2.2 Spur gear moving load model In actual gear meshing operation, the amplitude and position of the load vary on the active profile surface of the tooth. As the gear rotates, the load will move from LPTC to HPTC. Hence the deflection and tooth bending stress vary accordingly. Due to variation in deflection, the stiffness of the tooth during a mesh cycle also varies. The equivalent mesh stiffness of a tooth pair in contact is directly proportional to the load shared by the pair, while multi pairs are in contact. Similar way the load distribution along the contact line depends on the stiffness of each point along the contact line. Hence it is important to calculate the load shared between the pairs and the load distribution along the contact line to estimate the actual stress induced at the fillet during power transmission. The individual stiffness of the gear (g) and pinion (p) tooth in pair 1 and pair 2 (kp1, kp2, kg1, and kg2) have been calculated as the ratio between the load applied and deflection due to the load. The equivalent stiffness of a pair 1 and 2 are calculated using Eqs. (1) and (2) [28]. kp1 × kg1

Kequ1 =

kequ2 =

(kp1 × kg1) kp2 × kg2

(kp2 × kg2)

(1)

(2)

While two pairs are in contact load shared by a pair is quantified by calculating LSR. It is the ratio between the stiffness of a pair and the total stiffness of the pairs that are in contact. (Eq. (3) and (4) [28]). LSR1 = LSR2 =

kequ1

(kequ1 × kequ2) kequ2

(kequ1 × kequ2)

(3) (4)

The predicted LSR values were plotted against the non-dimensional parameter X/m from HPTC to LPTC for both STSM and TTSM (Fig. 3). The LSR is obtained as unity between the lowest point of single tooth contact (LPSTC) to HPSTC. The load distribution along the contact line depends on the stiffness of each and every point along the contact line. The stiffness (Kij) of a point(i) on a contact line(j) is calculated by applying a known load at that point and measuring the deflection (δij) due to that load (Eq. (5)). The LDR of a nodal point along the contact line is defined as the ratio of stiffness of that node to the total stiffness of all nodes of the corresponding contact lines. The LDR was calculated using

Eq. (6) [30]. The actual load at that nodal point i was calculated using the corresponding LDR using Eq. (7). The actual load distribution by considering LSR was calculated from the HPTC line to the LPTC line using Eq. (8). There is not much difference in the value of LSR calculated along the path of contact by using STSM and TTSM, the only marginal difference is noted during the double pair contact (Fig.3).

Fig. 1. Spur gear geometry (a) STSM (b) TTSM.

Fig. 2. Moving load models with the effect of LSR (a) STSM (b) TTSM.

Kij =

Fn

(5)

δij

LDRij =

Kij ∑29 Kij i=1

(6)

(Fact) ij = Fn . LDRij

(7)

Fact with LSR = (Fact)ij . LSRj

(8)

Fig. 3. LSR versus contact position: A comparison between TTSM and STSM

The LSR calculated for both the models are shown in Figs. 4(a) and (b). It observed that the resulting actual load distribution with LSR is parabolically distributed along the face width for both STSM and TTSM. But, the TTSM yield maximum load and it is spread wide at the mid of face width while the load is moving from contact line at HPSTC to contact line at LPSTC when compared to STSM as shown in Figs. 5(a) and (b).

Fig. 4. Actual load distribution with LSR from HPTC line to LPTC line (a) STSM (b) TTSM

Fig. 5. Actual load distribution with LSR from HPTC line to LPTC line in contour plot (a) STSM (b) TTSM

2.4 Evaluation of SIFs at the crack initiation stage The stress analysis was performed to find the maximum principal stress location at the tooth root for both STSM and TTSM while the actual load is moved from the contact line at LPTC to the contact line at HPTC. The maximum principal stress was found in the middle region of the tooth root along the face width for both the models. The edge crack was initiated in the region of tooth root along the face width where the maximum principal stress was the maximum as shown in Fig. 6. This edge crack was modeled for the crack length of 1 mm in the direction that is perpendicular to the maximum principal stress plane at the tooth root as shown in Fig. 7. The quarter-point finite element was used around the crack front tip to attain the quality of mesh to simulate r-1/2 stress singularity and r1/2 displacement variation.

Fig. 6. Position of maximum principal stress at the tooth root region

Fig. 7. Finite element model of (a) STSM and (b) TTSM with the initial crack of 1 mm at tooth root region

The values of KI, KII, and KIII at the crack tip nodes along the face width was calculated by using linear elastic fracture mechanics (LEFM) theory. The influence of actual load distribution with LSR was studied with STSM and TTSM on SIFs at the crack tip. The variation of SIFs was measured at the crack tip nodes by applying actual load distribution with LSR along the face width and it is shown in Figs. 8 to 13. The variation of KI for STSM and TTSM while the load is moved from the contact line at HPTC to the contact line at LPTC is as shown in Figs. 8 and 9. It can be noted that the variation of KI in the 3D plot was parabolic for both STSM and TTSM and it was maximum at the middle region of tooth root for both the models. The KI predicted by the TTSM is 9.2% more than the STSM due to the higher magnitude load distributed at the HPSTC line when compared to the STSM (Fig.4).

Fig. 8. Variation of KI at the crack tip along face width from HPTC to LPTC for actual load distribution with LSR in 3D plot: (a) STSM and (b) TTSM

Fig. 9. Variation of KI at the crack tip along face width from HPTC to LPTC for actual load distribution with LSR in Contour plot: (a) STSM and (b) TTSM

The magnitude of KII changes from the negative to positive while the load is moved from HPTC to LPTC also it is observed that the magnitude is approximately zero when the load is applied near LPSTC. This trend is common for both the models which are shown in Figs. 10 and 11. The magnitude of the KII value is high in the STSM compared to the TTSM while the load is applied near the LPSTC region; this may be attributed to the boundary condition that makes the STSM as rigid and allows the sliding. Whereas the magnitude is less in the case of TTSM due to the availability of material to resists the sliding.

Fig. 10. Variation of KII at the crack tip along face width from HPTC to LPTC for actual load distribution with LSR (3D plot): (a) STSM and (b) TTSM

Fig. 11. Variation of KII at the crack tip along face width from HPTC to LPTC for actual load distribution with LSR (Contour plot): (a) STSM and (b) TTSM

The KIII becomes zero in the middle region node of the crack tip for both STSM and TTSM when the load is moved from the contact line at HPTC to the contact line at LPTC as shown in Figs. 12 and 13. The magnitude of the KIII at the front side nodes at the crack tip is gradually increasing from the maximum negative value to the positive value when the load is moved towards LPTC. On the other hand, the magnitude of the KIII at the backside nodes at the crack tip is gradually decreasing from the maximum positive value to the negative value when the load is moved towards LPTC. The trend is common for both models.

Fig. 12. Variation of KIII at the crack tip along face width from HPTC to LPTC for actual load distribution with LSR (3D plot): (a) STSM and (b) TTSM

Fig. 13. Variation of KIII at the crack tip along face width from HPTC to LPTC for actual load distribution with LSR (Contour plot): (a) STSM and (b) TTSM

2.5 Modeling of fatigue crack propagation The crack propagation procedure was developed using Richard 3D mixed-mode criterion with the initial crack length of 1 mm at the tooth root region. The crack kinking angle (φo) and twisting angle (ψo) were calculated by using Eq. (9) and (10) [31] to determine the direction of the new crack increment (∆a) based on predicted values of KI, KII, and KIII. The value of KI and KII was played a vital role in the orientation of the crack kinking angle whereas twisting angle orientation is influenced by KIII.

[ [

|KII| |𝐾𝐼𝐼| 𝜑0 = ∓ 140 . ― 700. |KI| + |KII| + |KIII| |𝐾𝐼| + |𝐾𝐼𝐼| + |𝐾𝐼𝐼𝐼| 0

(

|KIII| |KIII| 𝜓0 = ∓ 78 . ― 330. |KI| + |KII| + |KIII| |KI| + |KII| + |KIII| 0

(

)] 2

)]

(9)

2

(10)

Where φo > 0 for KI ≥ 0 and KII < 0, φo < 0 for KII > 0, and also ψo > 0 for KI ≥ 0 and KIII < 0, ψo < 0 for KIII > 0 In the simulation of crack propagation, the incremental crack length (∆a) was gradually increased from 0.1 mm (for a crack length up to 4 mm) to 0.5 mm (for crack length from 6 mm to final crack length of 7.8 mm). The crack propagation became unstabled condition if stress intensity factor KI exceeds plane strain fracture toughness of 2650 MPa mm0.5 in the core region. Hence the crack propagation procedure was stopped once it is reached [14]. As discussed in section 2.4, the SIF (KI) was maximum at the crack tip nodes in the middle region of a tooth root when the actual load was applied with the effect of LSR at the

contact line at HPSTC on both STSM and TTSM. The distribution of actual load with LSR along the face width was symmetrical about the midplane of face width this leads to the large kinking angle at this section for both STSM and TTSM. The developed crack propagation procedure using 3D FEM of STSM was validated with the result available in [14] by applying uniform load distribution without LSR along the face width. The validated crack propagation procedure was extended to investigate the influences of the number of teeth included in the FE model (STSM and TTSM) on the crack propagation path. The same study has been extended by applying actual parabolic load distribution with LSR at HPSTC for both STSM and TTSM. In addition, the three-dimensional variation of KI, KII, and KIII along the face width was explored and reported. From the results plotted in Figs. 14 and 15, it can be observed that the results of the present FE model have good agreement with the results reported in [14] for the same boundary conditions and loading. The variation is marginal when the crack size is below 1.5mm. But due to the parabolic load distribution, the KI value is decreasing for both models for the given crack length. There is not much deviation is observed between the results of STSM and TTSM when the crack size is over 6mm.

Fig. 14. Comparison between predicted KI of STSM and TTSM with [14]

Fig. 15. Comparison between predicted KII of STSM and TTSM with [14]

The variations of SIFs KI, KII, and KIII along the face width for every incremental crack length are shown in Fig.16 to 18. An incremental increase in crack length increases the magnitude of KI at all nodes in the crack tip. When compared to the crack length of 1mm the value of the KI for the crack length of 8mm is approximately 4.25 times more (Fig. 16). Overall there is a decrease in trend for KII is observed for the incremental increase in crack length (Fig. 17). Both the trends of KI and KII are symmetrical about the midplane of the face width is observed, whereas the trends of the KIII is anti-symmetric about the midplane of the face width. Overall there is a decrease in trend in the values of KIII at the crack tip nodes in the backside region and an increase in trend in the values of KIII at the crack tip nodes in front side region are observed from the Fig.18.

Fig. 16. The 3D variation of KI at the crack tip nodes along face width of TTSM for crack length

Fig. 17. The 3D variation of KII at the crack tip nodes along face width of TTSM for crack length

Fig. 18. The 3D variation of KIII at the crack tip nodes along face width of TTSM for crack length

Fig. 19. Comparison between the predicted crack propagation path of STSM and TTSM with [14]

Further, the crack propagation path was compared between the literature result [14] and the present work. It shows good agreement between predicted FEA results and the results reported in [14] (Fig. 19). It also noted that when the actual parabolic distribution of loads with LSR is applied on STSM, the crack path deviates deeper when compared to the uniform load case. It deviates much deeper when the actual parabolic distribution of loads with LSR is applied to TTSM. 2. Experimental investigation on the standard spur gear A pair of 20 teeth standard spur gear was used in the experimental investigation of the crack path simulation study. The geometric parameters of the gears considered in this study are given in Table.2. The gear was made of EN353 steel with Young's modulus E = 2.1 x 105 MPa and Poisson's ratio (μ) = 0.3. The surface of spur gear was hardened up to 0.5 mm depth through carburizing heat treatment process and the core material of gear should remain tough to carry the bending load.

Table 2 Geometrical data of the spur gear Gear Parameter

Symbol

Gear

Module (mm)

m

3

Z

20

α

20

Gear face width (mm)

B

15

Fillet radius

rfillet

0.38 module

Addendum

ha

1 module

Dedendum

hf

1.35 module

Number of teeth Engagement angle on the pitch diameter

(0)

In order to compare the fracture path under maximum static loading conditions, a single tooth load test rig was developed to apply the tangential load along the face width at the HPSTC line. The single tooth load test was performed using MTS tensile testing machine. The arrangements are shown in Fig.20. The load from the MTS tensile testing machine to the spur gear tooth was transmitted by the vertical loading rod. The tip of the loading rod was given with the same curvature as that the matting gear so that the line contact is established between the loading rod and the gear tooth surface at HPSTC.

Fig. 20. The Single tooth bending test in UTM tensile machine

The static tests were conducted to predict the tooth static load capacity by applying a tangential static load at the HPSTC line along the face width. It is found that the tooth was failed at 43.2 kN load. Then, the fatigue tests were conducted on the spur gear tooth with the load 22.5 kN with a loading frequency of 30 Hz and the stress ratio as R=0.1 using MTS Landmark fatigue testing machine. The fracture path trajectory is measured at the midsection of the tooth and the same is shown in Fig.21.

Fig. 21. Failure path after static and fatigue test: (a) Static test sample-I (b) Static test sample-II (c) Fatigue test sample-I and (d) Fatigue test sample-II

2.3 Numerical Investigation on standard spur gear using FEM The procedure mentioned in section 2.3 was applied to calculate the actual load distribution with LSR from HPTC to LPTC along the face width for the FE TTSM. The distribution of actual load with the effect of LSR was plotted along the face width by applying a tangential load of 43.2 kN from the contact line HPTC to LPTC as shown in Fig. 22. It can be observed that the plot is symmetrical about the midplane of the face width and the load is maximum in this region for all contact lines.

Fig. 22. Actual load distribution with LSR from HPTC line to LPTC line for 20 teeth TTSM

The procedure of crack propagation as discussed in section 2.4 was applied to find out the crack path to final failure for considering actual load with LSR at the contact line of HPSTC. The crack of 0.5 mm was initiated at the tooth root region in a plane perpendicular to maximum principal stress. The crack propagation analysis was carried out into 24 number of load steps with different crack lengths using the ABAQUS finite element code as shown in Fig. 23.

Fig. 23. Crack propagation paths for actual load distribution with LSR on 3D TTSM: (a) 5th load step (b) 10th load step (c) 16th load step and (d) last load step

The crack propagation path from experimental static and fatigue tests and FEM results are compared in Fig. 24. The FEM results for both uniform and the actual load distributions are predicting a much deeper propagation path when compared to the experimental static test findings. Whereas the crack path predicted by the FEM has a close agreement with the experimental fatigue test results. The small deviations observed may be due to the geometry of the gear and boundary conditions considered for the FEM.

Fig. 24. Comparison of crack propagation path in X-Y plane for 20 teeth spur gear pair of TTSM

4. Conclusions A detailed study has been made in this work to explore the effect of the finite element model in predicting the SIFs for fracture study. Two different FE models are explored in this work namely the normally reported STSM and the TTSM with adjacent tooth loads. The models are validated with the results reported in [14] by considering the same geometry, boundary conditions, and loading and found that there is a good agreement between the results predicted and published. The effect of load sharing and adjacent tooth loads are also explored and found that the actual load on STSM and TTSM predicting deeper crack propagation path when compared to the uniform loading. The results of the FE simulation carried out for the standard spur gear drive with 20 teeth 200 pressure angle were compared with the experimental investigations and showed that the simulation results with actual load distribution predicting almost similar crack path as that obtained from the experiment.

References 1. Z. Chen, J. Zhang, W. Zhai, Y. Wang, J. Liu, Improved analytical methods for calculation of gear tooth fillet-foundation stiffness with tooth root crack, Eng. Fail. Anal. 82 (2017) 72-81. 2. O.D. Mohammed, M. Rantatalo, J.O. Aidanpää, Improving mesh stiffness calculation of cracked gears for the purpose of vibration-based fault analysis, Eng. Fail. Anal. 34 (2013) 235–251. 3. T. Eritenel, R.G. Parker, An investigation of tooth mesh nonlinearity and partial contact loss in gear pairs using a lumped-parameter model, Mech. Mach. Theory 56 (2012) 28–51. 4. N. Gnanasekar, R. Thirumurugan, A. Arunmuthukumar, M. Azmalkhan, Prediction of tooth root failure under static loading condition for EN353 steel spur gear, IOP conf. series: Materials Science and Engg. 402(2018) 1-10. 5. J.P. Jing, G. Meng, Investigation on the failure of the gear shaft connected to extruder, Eng. Fail. Anal. 15 (2008) 420–9. 6. H. Ma, X. Pang, R. Feng, R. Song, B. Wen, Fault features analysis of cracked gear considering the effects of the extended tooth contact, Eng. Fail. Anal. 48 (2015) 105–120. 7. R. Thirumurugan, G. Muthuveerappan, Prediction of load sharing based HCR spur gear stresses and critical loading points using artificial neural networks, Int. J. Computer Application in Tech. 47 (2013) 14–22. 8. P.J.L. Fernandes. Tooth bending fatigue failures in gears, Eng. Fail. Anal. 1996;3(3):219–225. 9. Z. Hu, J. Tang, J. Zhong, S. Chen, Frequency spectrum and vibration analysis of high speed gearrotor system with tooth root crack considering transmission error excitation, Eng. Fail. Anal. 60 (2016) 405–441. 10. Z. Dayi, L. Shuguo, L. Baolong, H. Jie, Investigation on bending fatigue failure of a micro-gear through finite element analysis, Eng. Fail. Anal. 31 (2013) 225–235. 11. V.G. Sfakiotakis, N.K. Anifantis, Finite element modeling of spur gearing fractures, Finite Elem. Anal. Des. 39 (2002) 79–92. 12. P. Kumar, H. Hirani, A. Agrawal, Fatigue failure prediction in spur gear pair using AGMA approach, Mater. Today 4 (2017) 2470–2477. 13. A. Belsak, J. Flasker, Detecting cracks in the tooth root of gears, Eng. Fail. Anal. 14 (2007) 14661475. 14. S. Pehan, J. Kramberger, J. Flasker, B. Zafosnik, Investigation of crack propagation scatter in a gear tooth’s root, Eng. Fract. Mech. 75 (2008) 1266–1283. 15. S. Glodez, S. Pehan, J. Flasker, Experimental results of the fatigue crack growth in a gear tooth root, Int. J. Fatigue 20 (1998) 669–675. 16. S. Pehan, T.K. Hellen, J. Flasker, S. Glodez, Numerical methods for determining stress intensity factors vs. crack depth in gear tooth roots, Int. J. Fatigue 19 (1997) 677–685. 17. D.G. Lewicki, A.D. Sane, R.J. Drago, P.A. Wawrzynek, Three-dimensional gear crack propagation studies. NASA, Technical Memorandum (USA), 1998-208827, 1998. 18. M.A. Ghaffari, E. Pahl, S. Xiao, Three dimensional fatigue crack initiation and propagation analysis of a gear tooth under various load conditions and fatigue life extension with boron/epoxy patches, Eng. Fract. Mech. 135 (2015) 126–146. 19. Z. Li, H. Ma, M. Feng, Y. Zhu, B. Wen, Meshing characteristics of spur gear pair under different crack types, Eng. Fail. Anal. 80 (2017) 123–140. 20. Z. Chen, Y. Shao, Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth, Eng. Fail. Anal. 18 (2011) 2149–2164.

21. Z. Chen, Y. Shao, Mesh stiffness calculation of a spur gear pair with tooth profile modification and tooth root crack, Mech. Mach. Theory 62 (2013) 63–74. 22. W. Yu, Y. Shao, C.K. Mechefske, The effects of spur gear tooth spatial crack propagation on gear mesh stiffness, Eng. Fail. Anal. 54 (2015) 103–119. 23. J.G. Verma, S. Kumar, P.K. Kankar, Crack growth modeling in spur gear tooth and its effect on mesh stiffness using extended finite element method, Eng. Fail. Anal. 94 (2018) 109–120. 24. S. Podrug, D. Jelaska, S. Glodez, Influence of different load models on gear crack path shapes and fatigue lives, Fatigue Fract. Engg. Mater. Struct. 31 (2008) 327–339. 25. M.B. Sánchez, M. Pleguezuelos, J.I. Pedrero, Approximate equations for the meshing stiffness and the load sharing ratio of spur gears including Hertzian effects, Mech. Mach. Theory 109 (2017) 231–249. 26. H. Ma, J. Zeng, R. Feng, X. Pang, Q. Wang, B. Wen, Review on dynamics of cracked gear systems, Eng. Fail. Anal. 55 (2015) 224–245. 27. K. Chen, Y. Huangfu, H. Ma, Z. Xu, X. Li, B. Wen, Calculation of mesh stiffness of spur gears considering complex foundation types and crack propagation paths, Mechanical Systems and Signal Processing, 130 (2019), 273-292. 28. R. Thirumurugan, G. Muthuveerappan, Maximum fillet stress Analysis based on load sharing in normal contact ratio spur gear drives, Int. J. Mech. Design Struct. Machines 38 (2010) 204–226. 29. R. Thirumurugan, G. Muthuveerappan, Critical loading points for maximum fillet and contact stresses in normal and high contact ratio spur gears based on load sharing ratio, Int. J. Mech. Based Des. Struct. Machines 39 (2011) 118–141. 30. T. Rama, N. Gnanasekar, Investigation of the effect of load distribution along the face width and load sharing between the pairs in contact on the fracture parameters of the spur gear tooth with root crack, Eng. Fail. Anal. 97 (2019) 518–533. 31. H. Hosseini-Toudeshky, M.A. Ghaffari, B. Mohammadi, Finite element fatigue propagation of induced cracks by stiffeners in repaired panels with composite patches, Compos. Struct. 94 (2012) 1771–1780.

Highlights of the present work Title: Influence of finite element model, Load-sharing and load distribution on crack propagation path in spur gear drive.

Authors: Rama Thirumurugan and N.Gnanasekar 

The load distribution based on LSR has been calculated.

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The stiffness based approach was used to calculate LDR and LSR.

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Two different FE models such as STSM &TTSM were discussed

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Results of 3D-FE models are compared with experimental and published results

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STSM and TTSM predict deeper crack propagation path.