Failure investigation of a marine diesel engine timing gear

Engineering Failure Analysis 107 (2020) 104203

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Failure investigation of a marine diesel engine timing gear Yunbo Yuana, Zhiyong Wanga, Donghua Wanga, Lieyi Donga, Wanyou Lia, ⁎ Yibin Guoa, a

T

College of Power and Energy Engineering, Harbin Engineering University, Harbin, Heilongjiang Province, China

ARTICLE INFO

ABSTRACT

Keywords: Timing gear Fatigue fracture Torsional vibration Alternating load Inclusion

In this paper, a fracture failure of a marine diesel engine timing gear in a chemical tanker was investigated in detail using both experimental analysis and numerical analysis. The experimental analysis includes macroscopical inspection, Scanning Electron Microscope analysis, Energy Dispersive Spectroscopy analysis, metallographic investigation and mechanical property investigation. The numerical analysis includes torsional vibration analysis and finite element method (FEM) simulations. In the torsional vibration analysis, the coupling influence of the associated structures on the timing gear system was taken into account, not only analyzing the fault gear itself. In the FEM simulations, two models, one is with inclusion and the other is noninclusion, were constructed. Both investigations indicate that the fracture is a fatigue fracture and the initial cracks are caused by the inclusion. Further investigation should pay attention to finding a way to overcome the problem of fatigue fracture.

1. Introduction The fractures had occurred many times at the area of timing gearwheel tooth of the engine timing system in a type of 6000 ton chemical tankers during their service time. A timing gear system of diesel engine is one of the most important drive systems, which controls the work of valve mechanism and fuel system to ensure the proper operation of diesel engine [1–3]. The torsional vibration excitation from the crankshaft front-end and the load excitations from valve mechanism and fuel system are simultaneously applied on the timing gear train and they are all alternating. If these excitations are too large, the working stability of timing gear train might be deteriorative to such an extent as to cause the fatigue failure of timing gear system [4]. Hence, it is very important to find a solution to the fracture problem of the timing gear. There are a lot of studies involving gear failures available. While studying the gear fatigue, numerical simulations of finite element method (FEM) and fatigue characterization experiments are used extensively. Among the FEM simulation studies, the contact fatigue damage analysis in shot-peened gears was carried out by Guaglian et al. [5]. Akata et al. [6] focused on studying mechanical properties of the gears manufactured using different methods. Zhang et al. [7] investigated the influence rules of tooth back thickness and tooth root fillet radius on the stress concentration and the bending fatigue failure and Olsson et al. [8] investigated the influence of residual stress on the distribution of tooth stress. Among the fatigue characterization experimental studies, a test comparison of bending fatigue strength of carburized and nitrided gears for industrial applications was studied by Conrado et al. [9]. Besides standard tests, they carried out the X-ray measurements to determine the residual stress caused by combinations of different steels and thermo-chemical treatments. Sun gears of agricultural vehicles damaged by pitting during endurance tests were analyzed to highlight the key aspects of the morphology and the evolution of pitting damage on sun gears by Terrin et al. [10]. Aslantas and

⁎

Corresponding author at: College of Power and Energy Engineering, Harbin Engineering University, Harbin 150001, China. E-mail addresses: [email protected] (Y. Yuan), [email protected] (Y. Guo).

https://doi.org/10.1016/j.engfailanal.2019.104203 Received 3 October 2017; Received in revised form 6 September 2019; Accepted 22 September 2019 Available online 22 October 2019 1350-6307/ © 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. General view of the studied timing gear system.

Tasgetiren [11] studied spur gear pitting formation and performed a series of experiment study to determine the pitting formation life. All researchers above mentioned mainly focus on the deficiencies of gear and only analyze the gear itself. However, since the timing gear system is connected to the crankshaft shafting, the valve mechanism and the fuel system, the dynamic responses of those associated structures are coupled with the timing gear system. Hence, later in the numerical analysis while studying the timing gear system, the influence of the associated structures was taken into account. In this paper, the fracture of a marine diesel engine timing gear in a 6000 ton chemical tanker was investigated using both numerical analysis and experimental analysis to find the causes of the fracture. The experimental analysis included the macroscopical inspection, the Scanning Electron Microscope (SEM) analysis, the Energy Dispersive Spectroscopy (EDS) analysis, the metallographic investigation, the carbonization examination and the mechanical property investigation. The numerical analysis included two steps. The first step was to use the torsional vibration analysis of the timing gear system with the associated structures to obtain the static load and the alternating load applying on the fault timing gear. The second step was to use FEM simulations of the fault timing gear under the loads above to obtain the maximum static bending stress and the maximum alternating bending stress on tooth root of the fault timing gear. 2. Tooth fracture The investigated fractures occurred at the area of gearwheel tooth of the engine timing system in a type of 6000 ton chemical tanker (two tankers were built, named Tanker A and Tanker B). The general view of the engine timing system is shown in Fig. 1. It consists of six spur gears, namely the crankshaft timing gear, the timing gearwheel, the pinion, the idle gear, the fuel camshaft gear, and the valve camshaft gear. The main propulsion diesel engines in two chemical tankers are both the type of 8L27/38. The rated speed and the rated power are 800 r/min and 2720 kW, respectively. Normally, the engine runs at the rotation speed of 650 r/min, for it is the most economic working condition. The material of the timing gearwheel is 15CrNi6E. The fracture occurred three times in the timing gearwheel during the service of Tanker A. Twice were tooth fracture and another time was most teeth surfaces fraying due to meshing. The failure history of the timing gearwheel of Tanker A is listed in table 1 and the fractured timing gearwheel for the second time is presented in Fig. 2(a). For Tanker B, one tooth broke after 30 months service as shown in Fig. 2(b). The tooth fracture of timing gears for both tankers occurred at the area of gear tooth root. The following failure analysis was performed with the timing gearwheel of Tanker B for its fault gear specimen was well preserved. Table 1 Timing gear failure history of Tanker A. Trial voyage

Date Description

2012.1.14

Failure No. 1

2

3

2013.6.24 5 teeth fracture

2014.3.18 2 teeth fracture

2014.11.16 Some teeth frayed

2

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Fig. 2. Fractured timing gearwheels: (a) of Tanker A and (b) of Tanker B.

3. Experimental analysis To find out the causes of the fracture, experimental analysis, including macroscopical inspection, SEM and EDS analysis, metallographic investigation, carbonization examination and mechanical property investigation, was executed. Fig. 3 shows the overview of the studied fracture and the specimens cutting from the fracture. Five specimens were cut from the fracture to carry out SEM and EDS analysis (specimen 1), metallographic investigation (specimen 2), carbonization examination (specimen 4) and mechanical investigation (specimen 5), where specimen 3 was a backup specimen. 3.1. Macroscopical inspection and SEM analysis The macroscopical view and the SEM micrograph of the fracture surface are presented in Fig. 4. The studied fracture was a typical conchoidal fracture, as shown in Fig. 4(a) and 4(b). Crack initiated at the lower left of tooth root according to the run of the crack (the directions that arrows point to). To further seek the crack initiation and also to further identify the fracture type, SEM analysis was carried out using specimen 1. As shown in Fig. 4(c), it can be seen that the crack initiation started on the left of the fracture surface, about 4 mm away from the rim. What’s more, there were obvious fatigue striations that occupied about 80% of the fracture surface area, which was typical features of High Cycle Fatigue (HCF) fracture. Therefore, the initial judgment is that the tooth failure was highly probable a HCF fracture. 3.2. Further SEM analysis and EDS analysis Further SEM analysis was carried out to identify the possible causes of the fracture. Under close magnification as shown in Fig. 5(a), many particles with diameter of around 1–10 μm were found at the area of the crack initiation. Furthermore, an agglomeration was found near the tooth root, as shown in Fig. 5(b). Such an agglomeration was covered by a deflected tool mark, which was probably a manufacturing deficiency. EDS analysis was conducted to estimate the compositions of the particle shown in Fig. 5(a) and the agglomeration shown in Fig. 5(b). The results (energy spectra) are presented in Fig. 6. It can be seen in Fig. 6(a) that Ti, Al and O took up a large proportion in the particle. This phenomenon clearly indicates that the particle was typical non-metallic inclusions (TiO2 and Al2O3). On the contrary, the major composition within the agglomeration was Fe as shown in Fig. 6(b).

Fig. 3. Overview of the studied timing gear fracture: (a) overview of fracture and (b) cutting styles of specimens. 3

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Fig. 4. Macro- and micro- graphs of the studied fracture: (a) on-site photo; (b) macroscropical view and (c) SEM micrograph.

Fig. 5. Morphology of crack initiation: (a) particle; (b) agglomeration and (c) tool mark.

Fig. 6. Energy spectra obtained by EDS: (a) energy spectrum of the particle shown in Fig. 5(a) and (b) energy spectrum of the agglomeration shown in Fig. 5(b).

3.3. Metallographic investigation and carbonization examination Metallographic investigation and carbonization examination were separately conducted using specimen 2 and specimen 4 by an Optical Microscope to check the microstructure. The results of carbonized layer and the metallographic structures of tooth tip are presented in Fig. 7 and Fig. 8, respectively. As shown in Fig. 7, the thickness of the carburized layer was about 1.00 mm and the thickness of the decarburized layer was about 0.43 mm. Acicular martensite was found in the carburized and decarburized layer marked as Zone A and Zone B in Fig. 8. The amount of lath martensite increased gradually as it extended inward, and mixed acicular and lath martensite was found in Zone C. Inside the carburized layer of the tooth tip, typical mixture of martensite and bainite was clearly shown in Fig. 8 marked as Zone D. The metallographic structures near the tooth root are depicted in Fig. 9. Only relevant results at about 0.90 mm depth are shown here. Mixed acicular and lath martensite could be found close to the rim (near the crack initiation) of tooth root as shown in Fig. 9(a), while bainite-based microstructure was found close to the middle of the tooth root. No remarkable differences were detected between the crack initiation zone at tooth root and the undamaged zone at tooth tip. Therefore, the fracture failure of the timing gear was not resulted from the material defects.

4

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(a)

(b)

(c)

1020 μm 430 μm 200 μm

200 μm

200 μm

Fig. 7. Carburized layer of tooth tip: (a) left side; (b) middle and (c) right side.

Zone B

Zone A

Zone D

20 μm Zone C

Zone C

20 μm Zone D

Zone B

Zone A

200 μm

20 μm

20 μm Fig. 8. Metallographic structure near tooth tip.

(a)

(b)

20 μm

20 μm

Fig. 9. Metallographic structure near tooth root: (a) close to the rim of the specimen and (b) close to the middle of the specimen.

3.4. Mechanical property investigation Both the quasi-static compressive tests and surface hardness tests were carried out to obtain the mechanical properties at the area of the fracture tooth. The compressive yield strength was measured using specimen 5 by a universal testing machine. The Vickers hardness was measured by using a Vickers hardness tester and the schematic diagram of detailed test positions are depicted in Fig. 10. The relevant results are given in table 2. The results show that the average hardness of the side away from the crack initiation (the inner side in Fig. 10) was greater than the side close to the crack initiation (the outer side in Fig. 10). Moreover, the compressive yield strength was about 1100 MPa and met the design requirements.

5

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8

11

7

10

6

9

5

3

4

1

2

20 μm Fig. 10. Schematic diagram of hardness test positions. Table 2 Material properties of the fracture tooth. Compressive yield strength (MPa)

Surface hardness (HV1/15)

1100

The side near crack initiation

The side away from crack initiation

467

511

3.5. Discussion on experimental results The experimental analysis results indicate that the crack initiation of the fracture tooth was located at the area of tooth root with inclusion. The machining scratch on the tooth root deflected at the location of the agglomeration. All these demonstrate that the origin of the tooth fracture in Tanker B came from the initial manufacturing deficiency at the tooth root. Moreover, the experimental analysis results show that the studied fracture was a typical HCF fracture. During the running period of diesel engine, under the alternating stress, the fatigue fracture started from the initial manufacturing deficiency at tooth root and developed until the tooth was broken. 4. Torsional vibration analysis To better understand the cause of the gear fracture, the numerical analysis was conducted. First, the torsional vibration analysis of the timing gear system with the associated structures was carried out to obtain the static load and the alternating load applying on the timing gearwheel. Further, FEM simulations were conducted, which will be discussed in Section 5. 4.1. Torsional vibration model To obtain the actual loads acting on the timing gearwheel during service, a lumped parameter model including the whole timing gear system, the fuel camshaft, the valve camshaft, the crankshaft and the propulsion shafting was established, as shown in Fig. 11. Inertias 1–10 represent the crankshaft, the crankshaft timing gear, the propeller, the timing gearwheel (the fault timing gear), the pinion, the idle gear, the fuel camshaft gear, the fuel camshaft, the valve camshaft gear and the valve camshaft, separately. According to the two degrees of freedom dynamic model of spur gear pair [12] and the torsional model of parallel transmission system [13], the torsional vibration equation of the system in Fig. 11 is given as follows: (1)

J¨ + C + K = T

where the angular displacement vector φ, the inertia matrix J, the stiffness matrix K and the damping matrix C are given as follows:

= { 1,

2,

3,

4,

5,

6,

7,

8,

9,

T 10 }

J = diag(J1, J2, J3, J4 , J5, J6, J7, J8, J9, J10) K = Kshaft + Kmesh

C = Cabs + Cmesh where Kshaft is the torsional stiffness matrix of shaft section, Kmesh is the torsional meshing stiffness matrix of gear pairs, Cabs is the 6

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Fig. 11. Lumped parameter model of timing gear system and its associated camshaft and main shaft systems.

absolute damping matrix, Cmesh is the torsional meshing damping matrix of gear pairs. Detailed expressions are given in Appendix A. T is the force vector and will be discussed in detail in Section 4.2. The meshing stiffness and the meshing damping of each gear pair are calculated by the formula in Ref. [14]. All meshing stiffness is considered to be constant and equal to the corresponding average value. 4.2. Dynamic boundary conditions To obtain the actual loads acting on the timing gearwheel in the period of service precisely, it is important to give the boundary conditions as accurate as possible. The timing gear train is affected by the torsional vibration of the crankshaft and the load torques resulting from camshafts carrying the cam-valve mechanisms [2,3]. The interface between the crankshaft and the timing gear train can be expressed as a kinematic boundary condition, that is the instantaneous speed fluctuation of flywheel [2,15]. The on-the-spot measured instantaneous speed fluctuation of flywheel was applied on the inertia 2, as shown in Fig. 12. The torques of the fuel camshaft and the valve camshaft during the operation of diesel engine are important boundary conditions of the timing gear system. In this study, to make the analysis better understood, a model with single degree of freedom is chosen to calculate the torques caused by the fuel camshaft and the valve camshaft. Ignoring the friction, the normal pressure force Fzf on each single fuel cam is mainly composed of three components, namely, the elastic force Fpf of plunger spring, the inertial force Fmf of plunger and the fuel pressure force Fsf [4,16]. The torque Tfuel on each single

Fig. 12. On-the-spot measured speed fluctuation of flywheel: (a) time-domain and (b) frequency-domain. 7

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fuel cam can be written as follows:

Tfuel = Fzf

dp 2 dhi dh d 2h dhi = (Fpf + Fmf + Fsf ) i = 2p + m1 w 2 2i + Kp (hi0 + hi ) d d 4 d d f f

Fs

f

Fp

Fm

(2)

where all symbols are given in Table B.1 of Appendix B. Ignoring the friction, the normal pressure force Fzg on each single valve cam is mainly composed of five components, namely, the inertial force Fmg1 of tappet and mandrill, the inertial force Fmg2 of valve and spring, the elastic force Fpg of valve spring, the inertial force Fmg3 of rocker and the gas pressure force Fsg [4,16]. The torque Tgas of each single valve cam can be written as follows: dh

dh

Tgas = Fzg d = (Fpg + Fmg1 + Fmg2 + Fmg3 + Fsg ) d

(

= iKv (h 0 + h) + aT (mT1 + mT 2 ) + iaT m v + g Fp

g Fm1

ms 3

g

Fm2

)+a

IR Tl 2 T

+ ip

dK 2 4 g

g

dh d

(3)

Fs

Fm3

where all symbols are detailed in Table B.1 of Appendix B. From Eqs. (2) and (3), the torques acting on the single fuel cam and the single valve cam can be obtained. Future, the torques on the inertia 8 and the inertia 10 were obtained from the vectorial sum of the torques acting on eight single fuel cam and valve cam, respectively. 4.3. Static load on gearwheel under rated speed In terms of the fuel cam and the valve cam of single cylinder, the torques can be expanded as the series of the working frequency as follows: 0 Tfuel = T fuel +

T fuel sin(

t+

fuel )

=1

0 Tgas = Tgas +

Tgas sin(

t+

=1

(4)

gas )

(5)

where are the mean components of torques of crankshaft, T fuel, Tgas are the corresponding amplitudes of the order component and fuel , gas are the corresponding initial phases. The static torque applying on the timing gearwheel is the sum of the mean torque from the fuel camshaft and the valve camshaft. Under rated speed, the static load can be calculated as follows (the transmission efficiency of gear is taken as 0.98): 0 T fuel ,

Ff =

0 Tgas

Tf r4

=

×

0 0 (T fuel + Tgas )

0. 5mn Z4

=

1 0 . 982

× (8 × (985. 2 + 42. 06)) 0. 5 × 7 × 58

× 103 = 42152(N)

where Tf is the static torque applying on the timing gearwheel, r4 is the base radius of the timing gearwheel, efficiency of gear, mn is the gear normal module and Z4 is the number of teeth.

(6) is the transmission

4.4. Alternating load on gearwheel under working speed Substituting the on-the-spot measured instantaneous speed and the torques of camshafts and using the Newmark time integration method, Eq. (1) was solved. Then, the torsional angle 2 of the crankshaft gear and the torsional angle 4 of the timing gearwheel can be obtained. Hence, the alternating load acting on the tooth flank of the timing gearwheel can be given as follows:

Fac =

k2,4 (r2 2 r4 Tac = r4 0.5mn Z4

4)

(7)

where Tac is the alternating torque acting on the timing gearwheel. The results are given in Fig. 13, it can be seen that the load on the timing gearwheel is a periodical alternating load with a series of frequency components. The first three dominant frequency components correspond to the 3. 5th , 4th and 0. 5th order of diesel engine. Among them, the 3. 5th and 4th order components are dominated by the crankshaft and the 0. 5th order component is dominated by the camshaft. 5. FEM simulations of tooth root stress In this section, to future investigate the causes of the fracture of the timing gearwheel, FEM was used to obtain the stress level in tooth root under the loads obtained by the torsional vibration analysis in Section 4. Both the static bending stress of tooth root under 8

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(a) 4000

(b) 1800

3000

1600 1400 1200

1000

Force (N)

Force (N)

2000

0 -1000

800 600

-2000

400

-3000 -4000

1000

200 0

0.05

0.1

Time (s)

0

0.15

0

2

4

6

Engine Order

8

10

12

Fig. 13. Alternating load on timing gearwheel: (a) time-domain and (b) frequency-domain.

rated speed and the alternating bending stress of tooth root under working speed were investigated. 5.1. Allowable bending stress and allowable alternating stress Currently, the most popular tooth-root strength evaluation standards are ISO and AGMA [17]. In this study, the ISO standard is chosen to calculate the allowable tooth-root bending strength. Following the ISO, the allowable bending stress FP of tooth root in the studied timing gear is given as [18,19] FP

=

1120 × 1 FE YNT Y relT YR relT YX = × 1. 004 × 0. 941 × 0. 99 = 698. 37(MPa) SFmin 1. 5

(8)

where FE is the bending strength of materials, YNT is the life factor for tooth root stress, Y relT is the relative notch sensitivity factor, YR relT is the relative surface factor, YX is the size factor relevant to tooth root strength, and SF min is the minimum required safety factor for tooth root stress, separately. Under the working speed of 650 r/min, the allowable alternating bending stress fac of the timing gearwheel is defined as 8.75% f [20], where f is the static bending stress of the timing gearwheel under the rated speed of 800 r/min. f will be discussed later in Section 5.2. 5.2. Static bending stress on tooth root under rated speed Two FEM models of the fault timing gear were set up. One is with inclusion defects and the other is without inclusion defects, as shown in Fig. 14. A close-up of the inclusion region is also shown in Fig. 14(b). The timing gearwheel is made of 15CrNi6E with Young’s modulus E = 2.1 × 105 MPa and Poisson’s ratio = 0.3. For the inclusion region, its Young’s modulus was set as E = 2.35 × 105 MPa . As shown in Fig. 15, the static load was applied uniformly at the upper edge of the single engagement area and 6degreet constraints were put on all the inner edge and all nodes on each flank. The static load obtained in Section 4.3 is 42152 N. The Von-Mises stress results of the fault timing gear under rated speed are given in Fig. 16. The resultsshow that for the noninclusion model, the maximum stress occurs in the tooth root region with symmetrical distribution along the tooth thickness. The

Fig. 14. FEM models of teeth: (a) non-inclusion and (b) with inclusion. 9

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Fig. 15. Boundary conditions and applied force.

(a)

(b)

Fig. 16. Von-Mises stress results: (a) non-inclusion and (b) with inclusion.

maximum stress is 476.25 MPa. For the inclusion model, the maximum stress moves to the lower limb of the inclusion region with non-symmetrical distribution along the tooth thickness. The maximum stress increases about 15.2% up to 548.43 MPa. Two main factors causing the stress increasing are the higher hardness and the thin edge structure in the inclusion region and the hardness factor plays a dominant role. But, for either model, the maximum stress obtained is much lower than the allowable bending stress FP of 698.37 MPa. Therefore, under the static load in the rated speed, the timing gearwheel cannot be broken. 5.3. Alternating bending stress on tooth root under working speed Here, the alternating load and the static load obtained in Section 4 were simultaneously applied on the tooth of both FEM models.

Fig.17. Maximum stress on tooth root during one operation cycle. 10

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The transient analysis was carried out using ANSYS. The obtained Von-Mises stress at the maximum stress point on tooth root is shown in Fig. 17. It can be seen that the stress obtained of both FEM models during one operation cycle are lower than the allowable bending stress of tooth root. As obtained before, f , the static bending stress of tooth root under rated speed, is 476.25 MPa. The alternating bending stress of tooth root with non-inclusion model is 39.05 MPa (8.20% f ), which is far lower than the allowable alternating bending stress fac (8.75% f ). However, with inclusion model, the alternating bending stress of tooth root is 44.86 MPa (9.42% f ), which is much higher than the allowable alternating bending stress. The results indicate that eventually, a fatigue failure will occur at the timing gearwheel with inclusion as its alternating bending stress is already considerably higher than the allowable value. 6. Conclusions In this paper, the fracture failure of a marine diesel engine timing gear in a chemical tanker was investigated to find out the causes of the fracture. First, the experimental investigation was taken to study the fracture on the timing gearwheel tooth in Tanker B. The investigation includes the macroscopical inspection, the SEM and EDS analysis, the metallographic investigation, the carbonization examination and the mechanical property investigation. All test results show that the origin of tooth fracture in Tanker B came from the initial manufacturing deficiency at the tooth root. Under the alternating stress, the fatigue fracture started from the initial manufacturing deficiency at tooth root and developed until the tooth broken. Further, a numerical analysis was carried out to better understand the causes of the fracture. The numerical analysis includes the torsional vibration analysis and FEM simulations. The torsional vibration analysis is to obtain the actual static load and the actual alternating load applying on the timing gearwheel. In the torsional vibration analysis, the coupling influence of the associated structures on the timing gear system was taken into account. The FEM simulations are to obtain the stress level of the timing gearwheel. Two FEM models, one is with inclusion defects and the other is non-inclusion defects, were created. The results from the numerical analysis show that regardless of the FEM models, under the static load in the rated speed, the maximum stress obtained was much lower than the allowable bending stress. The results also show that eventually, a fatigue failure will occur at the timing gearwheel with inclusion as its alternating stress is considerably higher than the allowable value. Both investigations indicate that the fracture is a fatigue fracture and the initial cracks are caused by the inclusion. Further investigation should pay attention to finding a way to reduce the stress level at the area of tooth root to overcome the problem of fatigue fracture. Declaration of Competing Interest The authors declare that there is no conflict of interest regarding the publication of this paper. Acknowledgement This research is funded by the National Natural Science Foundation of China (Grant No. 51805106), China Postdoctoral Science Foundation (Grant No. 2015M581433), Postdoctoral Science Foundation of Heilongjiang Province (Grant No. LBH-Z15038) and Fundamental Research Funds for the Central Universities (Grant No. 3072019CFM0306). Appendix A The torsional stiffness matrix Kshaft of shaft section is

k1,2

Kshaft =

k1,2 k1,2 k1,2 + k2,3 k2,3

k2,3 k2,3 k4,5 k4,5

k4,5 k4,5 0 k 7,8 k7,8

k 7,8 k 7,8 k 9,10 k 9,10

where kn 1, n denotes the stiffness between inertias n 1 and n . The torsional meshing stiffness matrix Kmesh of gear pairs is

11

k9,10 k 9,10

(A.1)

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0 m r22 k2,4

m r2 r4 k 2,4

0 m r42 k 2,4

m r2 r4 k 2,4

m r52 k5,6

Kmesh =

m r5 r6 k5,6

m r5 r6 k5,6 m r62 (k5,6

m m ) + k6,7 + k6,9 m r6 r7 k6,7

m r6 r7 k6,7

m r6 r9 k6,9

m r72 k6,7

0 m r92 k6,9

m r6 r9 k6,9

(A.2)

0 where denotes the meshing stiffness between gears n1 and n2 , rn denotes the base radius of the gear n . The absolute damping matrix Cmesh is

knm1, n2

Cabs = diag(cabs1, 0, cabs3, 0, 0, 0, 0, cabs8, 0, cabs10) where cabsn denotes absolute damping of inertia n . The torsional meshing damping matrix Cmesh of gear pairs is

0 m r22 c2,4

m r2 r4 c2,4

0 m r2 r4 c2,4

m r42 c2,4 m r52 c5,6

Cmesh =

m r5 r6 c5,6

m r5 r6 c5,6 m r62 (c5,6

m m ) + c6,7 + c6,9 m r6 r7 c6,7

m r6 r7 c6,7

m r6 r9 c6,9

m r72 c6,7

0 m r6 r9 c6,9

m r92 c6,9

(A.3)

0 where cnm1, n2 denotes the meshing damping between gears n1 and n2 . Appendix B See Table B.1. Table B1 Variable parameters in Eqs. (2) and (3). i dK h0 aT mT2 ms lT dp

Kp

Rocker arm length ratio The diameter of the valve disk Pre-compression of valve spring Acceleration of cam Mass of tappet Mass of valve spring Length of rocker arm near push rod Diameter of plunger

p Kv h mT1 mv IR

m1

Speed of cam Stiffness of plunger spring

hi hi0

Cylinder pressure Stiffness of valve spring Lift of valve Mass of push rod Mass of valve Inertial of rocker arm Cam angle Mass of plunger

Lift fuel injection cam Pre-compression of plunger spring

Appendix C. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.engfailanal.2019.104203.

References [1] J.B. Heywood, Internal Combustion Engine Fundamentals, McGraw-Hill, New York, 1988. [2] Martin Sopouch, Wolfgang Hellinger, H. Hans Priebsch, Simulation of engine's structure borne noise excitation due to the timing chain drive, SAE Technical Paper 01 (2002) 0451. [3] A. Rivola, M. Troncossi, Dynamic analysis of a motorbike engine timing system: experimental and numerical investigation of the geartrain, Mech. Syst. Signal Process. 48 (1) (2014) 325–338. [4] R. Stone, Introduction to Internal Combustion Engines, Palgrave Macmillan, 2012.

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