# Study of the Influences of Transient Crack Propagation in a Pinion on Time-Varying Mesh Stiffness

Authors: Youmin Hu; Jikai Fan; Jin Yu
Journal: Shock and Vibration
 (2016)

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## Abstract

Cracks in a cracked gear may further propagate by a tiny length in a very short time for several reasons, such as material fatigue and load fluctuations. In this paper, this dynamic process is defined as transient propagation of cracks. This research aims to calculate the time-varying mesh stiffness of gears when transient propagation of cracks arises, which has not been extensively studied in existing literatures. The transient propagation of cracks is modelled. An improved potential energy method is proposed by incorporating the propagation model into the potential energy method. The improved method can also be utilised to calculate the mesh stiffness of gears when transient propagation of cracks arises. Different transient propagation models are considered to simulate the propagation of cracks in a short amount of time. Different deterioration levels of cracks before transient propagation and different lengths and models of transient propagation are also examined. The variation rules of mesh stiffness caused by the transient propagation of cracks are summarised. The influence of the deterioration level of cracks on mesh stiffness variation when transient propagation arises is obtained. Simulation results show that the proposed method accurately calculates time-varying mesh stiffness when transient propagation of cracks arises. Furthermore, the method improves the monitoring of further propagation of cracks in gears from the perspective of time-varying mesh stiffness.


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## Body

## 1. Introduction

Root crack, a typical fault in gear sets, may result from poor operating conditions, such as inadequate lubrication, excessive applied torque, poor specifications, material defects, and manufacturing or installation problems [1]. Undesirable dynamic responses, including serious vibration and noise, may generate once a crack appears, and this occurrence may shorten the service life of a gear [2, 3]. Therefore, reducing catastrophic accidents by improving the detection of crack faults is highly significant.Time-varying mesh stiffness, which is caused by the periodic change of working teeth pairs, is an important parameter in the gear system. On the one hand, it brings about inherited periodic internal excitations of the gear system [3]; on the other hand, time-varying mesh stiffness decreases when a gear crack appears [5]. Therefore, both crack propagation and the resultant reduction in time-varying mesh stiffness have elicited the interest of many researchers [6, 7]. The current work also belongs to this research field.In their research on crack propagation, Lewicki and Ballarini studied the influence of rim thickness on crack propagation path [8], set up different fatigue crack growth models to predict crack propagation life for thin rim teeth [9], and proposed a method to study the influence of tooth dynamic loading on crack propagation [10]. Based on linear elastic fracture mechanics, Pandya and Parey [11] adopted a 2D finite element method to study crack propagation paths in gear pairs with different contact ratios. In their recent studies [3, 12, 13], they studied the influence of backup ratio (rim thickness-to-tooth height), fillet radius, and pressure angle on crack propagation and mesh stiffness in gear pairs with a low contact ratio. The finite element method based on linear elastic fracture mechanics (LEFM) was developed to study the crack propagation path of spur gears with a high contact ratio (HCR) and with tooth root cracks for two gear parameters (backup ratio and pressure angle) [13]. Zouari et al. [14, 15] adopted a 3D finite element method (FEM) to study the effects of propagation depth, orientation, and position in the blending radius of cracks on instantaneous mesh stiffness. 2D FEM was used to analyse the variation in stress intensity factor on the tooth base according to crack depth, crack propagation angle, and crack position. The propagation process was divided into two or four stages in [6, 7, 16, 17] to study the effect of severity of crack faults on mesh stiffness. In summary, crack length is assumed to be time-invariant in all these studies. As a result, the dynamic process of crack propagation has not been revealed.In fact, according to the principle of fracture mechanics [18–20], crack arrest is a common phenomenon that occurs during crack propagation, given that cracks will propagate only if the crack driving force is greater than the crack resistance. During crack propagation, the crack driving force initially increases with the increase in crack length and then decreases. Crack arrest occurs when the driving force is lower than the resistance. For ductile materials, such as metal or alloy, when the initial crack propagates, a slow propagation stage occurs before rapid propagation. However, automatic crack arrest may appear in both slow and rapid propagation stages. The stress of the gear teeth changes rapidly for fast engaging-in and engaging-out of the gear teeth. This change may also result in the propagation and arrest of cracks in a short time, and the cracks rapidly propagate in tiny lengths.Detection of crack propagation with a tiny length in a very short time (described above) is important in condition monitoring and fault diagnosis of a device. Thus, timely detection and confirmation of further propagation of cracks are highly necessary. Crack propagation by a tiny length in a short time is defined as transient propagation of cracks (TPOC). One of the contributions of this study is the investigation of TPOC for the timely detection and confirmation of further crack propagation. Only a few studies have investigated this aspect. Moreover, in this study, crack length is assumed to be time-varying to describe TPOC. Different TPOC models are proposed to simulate crack propagation instantaneously.For the algorithm of time-varying mesh stiffness, two methods of calculating the time-varying mesh stiffness of a gear set exist [21]; these two methods are FEM and the analytical method (AM). The former is time-consuming because every gear pair must be modelled to determine the mesh stiffness of a range of gear pairs. Meanwhile, AM provides a general approach to evaluate mesh stiffness with satisfactory results [1]. The elastic deformation method has been introduced previously and has been used in other related studies [1, 22–24]. The potential energy method, a typical analytical method initially proposed by Yang and Lin, has been used to calculate the mesh stiffness of a pair of external-external spur gears [25]. Given its simple calculation process, the potential energy method has been adopted by a large number of scholars [3, 5–7, 17, 21, 26–28]. In this method, the gear tooth is usually modelled as a cantilever beam that begins at the base circle. The Hertzian energy, bending energy, and axial compressive energy corresponding to Hertzian contact stiffness, bending stiffness, and axial compressive stiffness are considered in this model. Eventually, the shear energy corresponding to shear stiffness was also considered in [26]. The potential energy method has been utilised to calculate the mesh stiffness of ring and cracked gears [27, 28], and both the mesh stiffness of a spur gear tooth with spatial crack propagation and a helical gear have been calculated with modified potential energy methods [29–31]. However, these studies regarded crack length as a constant when calculating mesh stiffness. Therefore, when a gear undergoes TPOC, the time-varying mesh stiffness cannot be calculated by existing algorithms. Although a series of different crack lengths have been used to represent crack propagation, at least two defects still exist. One is that the calculation is time-consuming because of multiple times of calculations, and the other is that only a series of discrete stiffness values for certain crack lengths can be obtained. Consequently, further propagation of cracks cannot be revealed by time-varying mesh stiffness. Instead of a constant, a propagation function is used to represent crack length in our improved potential energy method. With the improved method, the time-varying mesh stiffness when TPOC arises can be calculated, and further propagation of cracks can be detected in a timely manner.In this study, TPOC was described and defined. Different propagation models were considered and proposed to simulate TPOC. The improved potential energy method was developed by incorporating the crack propagation model into the potential energy method. Four different deterioration levels of cracked gears were assumed, which means the lengths of existing cracks before TPOC were different. The crack was assumed to further propagate by a tiny length, as represented by the propagation model in each deterioration level. Afterwards, the time-varying mesh stiffness of gears when TPOC occurred was calculated with the improved method. The mesh stiffness variations caused by different TPOCs were compared and summarised.


## 2. The Transient Propagation of Crack and Improved Potential Energy Method

Crack propagation is a gradual process in which crack length increases. If crack length is assumed to be time-invariant, the dynamic process of crack propagation cannot be revealed. If the crack length of a cracked gear is considered constant, time-varying mesh stiffness cannot be calculated when the crack propagates by a tiny length in a very short time. Thus, crack propagation cannot be identified in a timely manner. Crack length has always been regarded as a constant when the potential energy method is adopted to calculate the time-varying mesh stiffness of gears [1, 3, 5–7, 17, 18, 24, 25]. In other words, the existing potential energy method cannot be directly used to calculate time-varying mesh stiffness when TPOC occurs in a gear. Before presenting our improved potential energy method, we introduce and describe TPOC. In this paper, TPOC is defined as the situation wherein cracks further propagate by a tiny length in a very short amount of time.


### 2.1. Transient Propagation of Crack in a Pinion (Driving Gear)

#### 2.1.1. Description of TPOC

We assume that TPOC appears in a pinion (driving gear). A crack exists before TPOC, and the length of the existing crack isq1. The crack is assumed to propagate along a straight line and through the entire tooth width. The intersection angle, β, between the crack and the central line of the tooth is set at 45°. θ is the angle displacement of pinion (driving gear); the length of TPOC is represented by qθ, where θ is time-varying; therefore, qθ is time-varying. The unit of q(θ) and θ is mm and rad, respectively. In Figure 1(a), the blue line is the existing crack, and the red line is TPOC. When the crack further propagates by a length of qθ, the total length of the crack is q1+qθ. If qθ=0, then the crack has not propagated and the crack length is constant, as shown in Figure 1(b).Figure 1
Nonuniform cantilever beam model of cracked gear.
(a)
TPOC appears (b)
No TPOC


#### 2.1.2. Modelling of TPOC

Several transient propagation functions were modelled to describe TPOC. The function is shown withq(θ). The step function was used to represent the crack propagating by a tiny length instantaneously. A continuous step function was used to represent the crack propagating by several segments of tiny length instantaneously. A linear function was used to represent the crack propagating by a tiny length continuously in a very short time.In the following functions,q1a(θ) to q1d(θ) are step functions. q2(θ) is a continuous step function. q3(θ) is a linear function. Therefore, the propagation model of q1a(θ) to q1d(θ) can be adopted to investigate the effects of different propagation lengths with the same propagation model on the time-varying mesh stiffness. Moreover, the propagation model of q2(θ) and q3(θ) can be adopted to study the effects of different propagation models with the same length on the time-varying mesh stiffness. All propagation models are expressed as follows.(  1) Different Propagation Lengths with the Same Propagation Model. As is shown in Figure 2(a),(1)q1aθ=0,θ<0.02rad0.05,θ>0.02rad,q1bθ=0,θ<0.02rad0.1,θ>0.02rad,q1cθ=0,θ<0.02rad0.15,θ>0.02rad,q1dθ=0,θ<0.02rad0.2,θ>0.02rad.Figure 2
TPOC models. (a) Different propagation lengths with the same propagation model. (b) Different propagation models with the same propagation length.
(a) (b)(  2) Different Propagation Models with the Same Propagation Length. As is shown in Figure 2(b), the purple line is q2(θ), the red line is q3(θ), and the total lengths of TPOCs are the same, that is, 0.15 mm, but their propagation modes are different from each other, which can be expressed as,(2)q2θ=0,θ<0.02rad0.05,0.02rad<θ<0.04rad0.1,0.04rad<θ<0.06rad0.15,θ>0.06rad,q3θ=0,θ<0.02rad3.75θ-0.075,0.02rad<θ<0.06rad0.15,θ>0.06rad.All the transient propagation models can be plotted in Figure 4.


### 2.2. Improved Algorithm of Mesh Stiffness When TPOC Occurs in a Pinion

According to the potential energy method, the tooth can be regarded as a nonuniform cantilever beam shown in Figure1(b). d represents the distance between the contact point and the root of the tooth, and the corresponding angle is α1. hc represents the vertical distance between the tip of the crack and the tooth’s central line, which corresponds to the angle αg. Rb is the radius of base circle. α2 is the half tooth angle on the base circle. q1 is the length of existing crack. h is half of the roof chordal tooth thickness. β is the intersection angle between the crack and the central line of the tooth. The total potential energy stored in the gear system includes the following parts: the Hertzian contact energy, the bending energy, the axial compression energy, and the shear energy, which can be expressed as [7](3)Ub=F22kb=∫0dM22EIxdx,Ua=F22ka=∫0dFa22EIxdx,Us=F22ks=∫0d1.2Fb22GAxdx,where kb,ka, and ks represent the bending stiffness, the axial compression stiffness, and the shear stiffness. E is Young’s modulus. G is the shear modulus. Hence,(4)Fb=Fcos⁡α1,Fa=Fsin⁡α1,M=Fah,where Fa and Fb are the two perpendicular component forces of F and the torque M represents the bending effect of Fa.I x represents the area moment of inertia of the section where the distance from the tooth root is x; Ax represents the area of the section where the distance from the tooth root is x.For a perfect gear without crack,Ix and Ax can be obtained as(5)Ix=1122hx3L,Ax=2hxL,where hx is the distance between the point on the tooth’s curve and the tooth’s central line where the horizontal distance from the tooth root is x and L is the width of the tooth.In this case, the bending stiffness, the axial compression stiffness, and the shear stiffness can be calculated by the following [7]:(6)1kb=∫-α1α231+α2-αsin⁡αcos⁡α1-cos⁡αcos⁡α12α2-αcos⁡α2ELα2-αcos⁡α+sin⁡α3dα,(7)1ks=∫-α1α21.21+να2-αcos⁡αcos2⁡α1ELα2-αcos⁡α+sin⁡αdα,(8)1ka=∫-α1α2α2-αcos⁡αsin2⁡α12ELα2-αcos⁡α+sin⁡αdα.The fillet-foundation deflection was derived by Sainsot et al. in [32] based on the theory of [33]. It can be calculated as(9)δf=Fcos2αmLE×NufSf2+MufSf+P1+Qtan2⁡αm,where L represents the tooth width. uf and Sf are given in Figure 2. The coefficients N,M, and P can be approached by polynomial functions [32]:(10)Xihfi,θf=Aiθf2+Bihfi2+Cihfiθf+Diθf+Eihfi+Fi,where Xi, denotes the coefficients N, M, P, and Q,hfi=rf/rint, rf and rint are given in Figure 3, and the values of Ai, Bi, Ci, Di, Ei, and Fi are given in Table 1.Table 1
Values of the coefficients of (10) [4].
A i B i C i D i E i F i N - 5.574 E - 5 - 1.9986 E - 3 - 2.3015 E - 4 4.7702 E - 3 0.0271 6.8045 M 60.111 E - 5 28.1 E - 3 - 83.431 E - 4 - 9.9256 E - 3 0.1624 0.9086 P - 50.952 E - 5 185.5 E - 3 0.0538 E - 4 53.3 E - 3 0.2895 0.9236 Q - 6.2042 E - 5 9.0889 E - 3 - 4.0964 E - 4 7.8297 E - 3 −0.1472 0.6904Figure 3
Geometrical parameters for the fillet-foundation deflection [28].Figure 4
Different deterioration levels before TPOC.
(a)
Case  1 (b)
Case  2 (c)
Case  3 (d)
Case  4The stiffness with consideration of gear fillet-foundation deflection can be obtained by(11)1kf=δfF.For a cracked gear shown in Figure 1, Ix and Ax can be obtained as(12)Ix=112hx+hc3L,x≤gc1122hx3L,x>gc,Ax=hx+hcL,x≤gc2hxL,x>gc.In this study, the crack further propagates by a tiny length qθ instantaneously based on the existing crack, as shown in Figure 1(a). Therefore, the total crack length becomes q1+qθ; then,(13)hc=Rb1sin⁡α2-q1+qθsin⁡β.The crack only affects the bending stiffness and shear stiffness [17]; in this case, they change as follows: (14)1kcb=∫-αgα2121+α2-αsin⁡αcos⁡α1-cos⁡αcos⁡α12α2-αcos⁡αELsin⁡α2+sin⁡α+α2-αcos⁡α-q1+qθ/Rbsin⁡β3dα+∫-α1-αg31+α2-αsin⁡αcos⁡α1-cos⁡αcos⁡α12α2-αcos⁡α2ELα2-αcos⁡α+sin⁡α3dα,(15)1kcs=∫-αgα22.41+να2-αcos⁡αcos2⁡α1EWsin⁡α2+sin⁡α+α2-αcos⁡α-q1+qθ/Rbsin⁡βdα+∫-α1-αg1.21+να2-αcos⁡αcos2⁡α1EWα2-αcos⁡α+sin⁡αdα.The time-varying mesh stiffness of the double-tooth pair can be calculated with (6) [23]. Hence,(16)kθ=∑i=1211/kh+1/kbc1,i+1/ksc1,i+1/ka1,i+1/kf1,i+1/kb2,i+1/ks2,i+1/ka2,i+1/kf2,i.The subscripts 1 and 2 represent the pinion (driving gear) and gear (driven gear), respectively. i=1 represents the first pair of meshing teeth when two pairs of teeth mesh; i=2 represents the second pair. θ is the angle displacement of pinion (driving gear).In (13), (14), and (15), q1 is the length of existing length and q(θ) is the length of TPOC. The time-varying mesh stiffness when TPOC arises in a pinion (driving gear) can be solved with (8), (11), (14), (15), and (16).


## 2.1. Transient Propagation of Crack in a Pinion (Driving Gear)

### 2.1.1. Description of TPOC

We assume that TPOC appears in a pinion (driving gear). A crack exists before TPOC, and the length of the existing crack isq1. The crack is assumed to propagate along a straight line and through the entire tooth width. The intersection angle, β, between the crack and the central line of the tooth is set at 45°. θ is the angle displacement of pinion (driving gear); the length of TPOC is represented by qθ, where θ is time-varying; therefore, qθ is time-varying. The unit of q(θ) and θ is mm and rad, respectively. In Figure 1(a), the blue line is the existing crack, and the red line is TPOC. When the crack further propagates by a length of qθ, the total length of the crack is q1+qθ. If qθ=0, then the crack has not propagated and the crack length is constant, as shown in Figure 1(b).Figure 1
Nonuniform cantilever beam model of cracked gear.
(a)
TPOC appears (b)
No TPOC


### 2.1.2. Modelling of TPOC

Several transient propagation functions were modelled to describe TPOC. The function is shown withq(θ). The step function was used to represent the crack propagating by a tiny length instantaneously. A continuous step function was used to represent the crack propagating by several segments of tiny length instantaneously. A linear function was used to represent the crack propagating by a tiny length continuously in a very short time.In the following functions,q1a(θ) to q1d(θ) are step functions. q2(θ) is a continuous step function. q3(θ) is a linear function. Therefore, the propagation model of q1a(θ) to q1d(θ) can be adopted to investigate the effects of different propagation lengths with the same propagation model on the time-varying mesh stiffness. Moreover, the propagation model of q2(θ) and q3(θ) can be adopted to study the effects of different propagation models with the same length on the time-varying mesh stiffness. All propagation models are expressed as follows.(  1) Different Propagation Lengths with the Same Propagation Model. As is shown in Figure 2(a),(1)q1aθ=0,θ<0.02rad0.05,θ>0.02rad,q1bθ=0,θ<0.02rad0.1,θ>0.02rad,q1cθ=0,θ<0.02rad0.15,θ>0.02rad,q1dθ=0,θ<0.02rad0.2,θ>0.02rad.Figure 2
TPOC models. (a) Different propagation lengths with the same propagation model. (b) Different propagation models with the same propagation length.
(a) (b)(  2) Different Propagation Models with the Same Propagation Length. As is shown in Figure 2(b), the purple line is q2(θ), the red line is q3(θ), and the total lengths of TPOCs are the same, that is, 0.15 mm, but their propagation modes are different from each other, which can be expressed as,(2)q2θ=0,θ<0.02rad0.05,0.02rad<θ<0.04rad0.1,0.04rad<θ<0.06rad0.15,θ>0.06rad,q3θ=0,θ<0.02rad3.75θ-0.075,0.02rad<θ<0.06rad0.15,θ>0.06rad.All the transient propagation models can be plotted in Figure 4.


## 2.1.1. Description of TPOC

We assume that TPOC appears in a pinion (driving gear). A crack exists before TPOC, and the length of the existing crack isq1. The crack is assumed to propagate along a straight line and through the entire tooth width. The intersection angle, β, between the crack and the central line of the tooth is set at 45°. θ is the angle displacement of pinion (driving gear); the length of TPOC is represented by qθ, where θ is time-varying; therefore, qθ is time-varying. The unit of q(θ) and θ is mm and rad, respectively. In Figure 1(a), the blue line is the existing crack, and the red line is TPOC. When the crack further propagates by a length of qθ, the total length of the crack is q1+qθ. If qθ=0, then the crack has not propagated and the crack length is constant, as shown in Figure 1(b).Figure 1
Nonuniform cantilever beam model of cracked gear.
(a)
TPOC appears (b)
No TPOC


## 2.1.2. Modelling of TPOC

Several transient propagation functions were modelled to describe TPOC. The function is shown withq(θ). The step function was used to represent the crack propagating by a tiny length instantaneously. A continuous step function was used to represent the crack propagating by several segments of tiny length instantaneously. A linear function was used to represent the crack propagating by a tiny length continuously in a very short time.In the following functions,q1a(θ) to q1d(θ) are step functions. q2(θ) is a continuous step function. q3(θ) is a linear function. Therefore, the propagation model of q1a(θ) to q1d(θ) can be adopted to investigate the effects of different propagation lengths with the same propagation model on the time-varying mesh stiffness. Moreover, the propagation model of q2(θ) and q3(θ) can be adopted to study the effects of different propagation models with the same length on the time-varying mesh stiffness. All propagation models are expressed as follows.(  1) Different Propagation Lengths with the Same Propagation Model. As is shown in Figure 2(a),(1)q1aθ=0,θ<0.02rad0.05,θ>0.02rad,q1bθ=0,θ<0.02rad0.1,θ>0.02rad,q1cθ=0,θ<0.02rad0.15,θ>0.02rad,q1dθ=0,θ<0.02rad0.2,θ>0.02rad.Figure 2
TPOC models. (a) Different propagation lengths with the same propagation model. (b) Different propagation models with the same propagation length.
(a) (b)(  2) Different Propagation Models with the Same Propagation Length. As is shown in Figure 2(b), the purple line is q2(θ), the red line is q3(θ), and the total lengths of TPOCs are the same, that is, 0.15 mm, but their propagation modes are different from each other, which can be expressed as,(2)q2θ=0,θ<0.02rad0.05,0.02rad<θ<0.04rad0.1,0.04rad<θ<0.06rad0.15,θ>0.06rad,q3θ=0,θ<0.02rad3.75θ-0.075,0.02rad<θ<0.06rad0.15,θ>0.06rad.All the transient propagation models can be plotted in Figure 4.


## 2.2. Improved Algorithm of Mesh Stiffness When TPOC Occurs in a Pinion

According to the potential energy method, the tooth can be regarded as a nonuniform cantilever beam shown in Figure1(b). d represents the distance between the contact point and the root of the tooth, and the corresponding angle is α1. hc represents the vertical distance between the tip of the crack and the tooth’s central line, which corresponds to the angle αg. Rb is the radius of base circle. α2 is the half tooth angle on the base circle. q1 is the length of existing crack. h is half of the roof chordal tooth thickness. β is the intersection angle between the crack and the central line of the tooth. The total potential energy stored in the gear system includes the following parts: the Hertzian contact energy, the bending energy, the axial compression energy, and the shear energy, which can be expressed as [7](3)Ub=F22kb=∫0dM22EIxdx,Ua=F22ka=∫0dFa22EIxdx,Us=F22ks=∫0d1.2Fb22GAxdx,where kb,ka, and ks represent the bending stiffness, the axial compression stiffness, and the shear stiffness. E is Young’s modulus. G is the shear modulus. Hence,(4)Fb=Fcos⁡α1,Fa=Fsin⁡α1,M=Fah,where Fa and Fb are the two perpendicular component forces of F and the torque M represents the bending effect of Fa.I x represents the area moment of inertia of the section where the distance from the tooth root is x; Ax represents the area of the section where the distance from the tooth root is x.For a perfect gear without crack,Ix and Ax can be obtained as(5)Ix=1122hx3L,Ax=2hxL,where hx is the distance between the point on the tooth’s curve and the tooth’s central line where the horizontal distance from the tooth root is x and L is the width of the tooth.In this case, the bending stiffness, the axial compression stiffness, and the shear stiffness can be calculated by the following [7]:(6)1kb=∫-α1α231+α2-αsin⁡αcos⁡α1-cos⁡αcos⁡α12α2-αcos⁡α2ELα2-αcos⁡α+sin⁡α3dα,(7)1ks=∫-α1α21.21+να2-αcos⁡αcos2⁡α1ELα2-αcos⁡α+sin⁡αdα,(8)1ka=∫-α1α2α2-αcos⁡αsin2⁡α12ELα2-αcos⁡α+sin⁡αdα.The fillet-foundation deflection was derived by Sainsot et al. in [32] based on the theory of [33]. It can be calculated as(9)δf=Fcos2αmLE×NufSf2+MufSf+P1+Qtan2⁡αm,where L represents the tooth width. uf and Sf are given in Figure 2. The coefficients N,M, and P can be approached by polynomial functions [32]:(10)Xihfi,θf=Aiθf2+Bihfi2+Cihfiθf+Diθf+Eihfi+Fi,where Xi, denotes the coefficients N, M, P, and Q,hfi=rf/rint, rf and rint are given in Figure 3, and the values of Ai, Bi, Ci, Di, Ei, and Fi are given in Table 1.Table 1
Values of the coefficients of (10) [4].
A i B i C i D i E i F i N - 5.574 E - 5 - 1.9986 E - 3 - 2.3015 E - 4 4.7702 E - 3 0.0271 6.8045 M 60.111 E - 5 28.1 E - 3 - 83.431 E - 4 - 9.9256 E - 3 0.1624 0.9086 P - 50.952 E - 5 185.5 E - 3 0.0538 E - 4 53.3 E - 3 0.2895 0.9236 Q - 6.2042 E - 5 9.0889 E - 3 - 4.0964 E - 4 7.8297 E - 3 −0.1472 0.6904Figure 3
Geometrical parameters for the fillet-foundation deflection [28].Figure 4
Different deterioration levels before TPOC.
(a)
Case  1 (b)
Case  2 (c)
Case  3 (d)
Case  4The stiffness with consideration of gear fillet-foundation deflection can be obtained by(11)1kf=δfF.For a cracked gear shown in Figure 1, Ix and Ax can be obtained as(12)Ix=112hx+hc3L,x≤gc1122hx3L,x>gc,Ax=hx+hcL,x≤gc2hxL,x>gc.In this study, the crack further propagates by a tiny length qθ instantaneously based on the existing crack, as shown in Figure 1(a). Therefore, the total crack length becomes q1+qθ; then,(13)hc=Rb1sin⁡α2-q1+qθsin⁡β.The crack only affects the bending stiffness and shear stiffness [17]; in this case, they change as follows: (14)1kcb=∫-αgα2121+α2-αsin⁡αcos⁡α1-cos⁡αcos⁡α12α2-αcos⁡αELsin⁡α2+sin⁡α+α2-αcos⁡α-q1+qθ/Rbsin⁡β3dα+∫-α1-αg31+α2-αsin⁡αcos⁡α1-cos⁡αcos⁡α12α2-αcos⁡α2ELα2-αcos⁡α+sin⁡α3dα,(15)1kcs=∫-αgα22.41+να2-αcos⁡αcos2⁡α1EWsin⁡α2+sin⁡α+α2-αcos⁡α-q1+qθ/Rbsin⁡βdα+∫-α1-αg1.21+να2-αcos⁡αcos2⁡α1EWα2-αcos⁡α+sin⁡αdα.The time-varying mesh stiffness of the double-tooth pair can be calculated with (6) [23]. Hence,(16)kθ=∑i=1211/kh+1/kbc1,i+1/ksc1,i+1/ka1,i+1/kf1,i+1/kb2,i+1/ks2,i+1/ka2,i+1/kf2,i.The subscripts 1 and 2 represent the pinion (driving gear) and gear (driven gear), respectively. i=1 represents the first pair of meshing teeth when two pairs of teeth mesh; i=2 represents the second pair. θ is the angle displacement of pinion (driving gear).In (13), (14), and (15), q1 is the length of existing length and q(θ) is the length of TPOC. The time-varying mesh stiffness when TPOC arises in a pinion (driving gear) can be solved with (8), (11), (14), (15), and (16).


## 3. Simulation Results and Discussion

Different deterioration levels of existing cracks were considered to study the influences of TPOC on time-varying mesh stiffness. In each deterioration level, TPOCs with the same model but different lengths and with the same length but different models were considered, and the resultant stiffness variations were compared.


### 3.1. Different Deterioration Levels of Crack before TPOC

Different lengths of existing cracks were considered, as shown in Figure4. In Figure 4, cases 1 to 4 represent the length of the existing crack becoming increasingly large; that is, the deterioration level of the crack becomes increasingly serious. The blue line represents the existing crack; the red line represents TPOC. In cases  1 and 2, the length of the existing crack is q1. In cases  3 and 4, the length of the existing crack is q1+q2. hc before and after TPOC arises in the different cases as shown in Table 2.Table 2
h c before and after TPOC arises.
No TPOC TPOC arises Case1 h c = R b 1 sin ⁡ α 2 - q 1 sin ⁡ β h c = R b 1 sin ⁡ α 2 -q1+qθsin⁡β Case2  Case3 h c = q 2 sin ⁡ β h c = q 2 + q θ sin ⁡ β Case4According to the formula in Table2, the bending and shear stiffness when TPOC arises can be recalculated as follows:Case  1 (17) 1 k c b = ∫ - α g α 2 12 1 + α 2 - α sin ⁡ α cos ⁡ α 1 - cos ⁡ α cos ⁡ α 1 2 α 2 - α cos ⁡ α E L sin ⁡ α 2 + sin ⁡ α + α 2 - α cos ⁡ α - q 1 + q θ / R b sin ⁡ β 3 d α + ∫ - α 1 - α g 3 1 + α 2 - α sin ⁡ α cos ⁡ α 1 - cos ⁡ α cos ⁡ α 1 2 α 2 - α cos ⁡ α 2 E L α 2 - α cos ⁡ α + sin ⁡ α 3 d α , 1 k c s = ∫ - α g α 2 2.4 1 + ν α 2 - α cos ⁡ α  cos 2 ⁡ α 1 E W sin ⁡ α 2 + sin ⁡ α + α 2 - α cos ⁡ α - q 1 + q θ / R b sin ⁡ β d α + ∫ - α 1 - α g 1.2 1 + ν α 2 - α cos ⁡ α  cos 2 ⁡ α 1 E W α 2 - α cos ⁡ α + sin ⁡ α d α .Case  2 (18) 1 k c b = ∫ - α 1 α 2 12 1 + α 2 - α sin ⁡ α cos ⁡ α 1 - cos ⁡ α cos ⁡ α 1 2 α 2 - α cos ⁡ α E W sin ⁡ α 2 + sin ⁡ α + α 2 - α cos ⁡ α - q 1 + q θ / R b sin ⁡ β 3 d α , 1 k c s = ∫ - α 1 α 2 2.4 1 + ν α 2 - α cos ⁡ α  cos 2 ⁡ α 1 E W sin ⁡ α 2 + sin ⁡ α + α 2 - α cos ⁡ α - q 1 + q θ / R b sin ⁡ β d α .Case  3 (19) 1 k c b = ∫ - α 1 α 2 12 1 + α 2 - α sin ⁡ α cos ⁡ α 1 - cos ⁡ α cos ⁡ α 1 2 α 2 - α cos ⁡ α E W sin ⁡ α + α 2 - α cos ⁡ α - q 2 + q θ / R b sin ⁡ β 3 d α , 1 k c s = ∫ - α 1 α 2 2.4 1 + ν α 2 - α cos ⁡ α  cos 2 ⁡ α 1 E W sin ⁡ α + α 2 - α cos ⁡ α - q 2 + q θ / R b sin ⁡ β d α .Case  4 (20) 1 k c b = ∫ - α g α 2 12 1 + α 2 - α sin ⁡ α cos ⁡ α 1 - cos ⁡ α cos ⁡ α 1 2 α 2 - α cos ⁡ α E W s i n α + α 2 - α cos ⁡ α - q 2 + q θ / R b sin ⁡ β 3 d α , 1 k c s = ∫ - α g α 2 2.4 1 + ν α 2 - α cos ⁡ α  cos 2 ⁡ α 1 E W sin ⁡ α + α 2 - α cos ⁡ α - q 2 + q θ / R b sin ⁡ β d α .As a sample of the length of existing crack, q1 (or q1 and q2) can be set in the following 4 cases. Case  1 to case  4 represent the notion that the deterioration level becomes more serious:Case  1: whenq1=1.57Case  2: whenq1=3.13Case  3: whenq1=3.92andq2=0.78Case  4: whenq1=3.92andq2=2.35


### 3.2. Time-Varying Mesh Stiffness Simulation When TPOC Occurs

Based on the improved potential energy method, the time-varying mesh stiffness when TPOC occurs in a pinion was calculated. The main parameters are shown in Table3.Table 3
Main parameters of the gear system.
Gear type Standard involute spur teeth Material Steel Young’s modulus E = 2.068 × 1 0 11 Pa Poisson ratio ν = 0.3 Number of teeth N 1 = 19 and N2=48 Pressure angle α = 20 ° Diametral pitch P = 8 inch−1 Base radius of the gear R b 1 = 28.34 mm Width of teeth w = 18 mm Mass of the pinion m 1 = 0.96 Kg Mass of the gear m 2 = 2.88 Kg Contact ratio C r = 1.6456 Mass moment of inertia of the motor I m = 0.0021 Kg m2 Mass moment of inertia of the load I b = 0.0105 Kg m2 Mass moment of inertia of the pinion I 1 = 4.3659 × 1 0 - 4 Kg m2 Mass moment of inertia of the gear I 2 = 8.3602 × 1 0 - 3 Kg m2


#### 3.2.1. Mesh Stiffness Simulation When TPOC Occurs with the Same Model but Different Lengths

This section focuses on mesh stiffness when TPOC occurs with the model of a step function. However, the lengths are 0.05, 0.1, 0.15, and 0.2 mm. In Figure5, q(θ)=0 indicates that no TPOC occurs, and the corresponding stiffness represents the mesh stiffness of a cracked gear with the length of an existing crack.Figure 5
The influences of TPOC with different lengths but the same propagation model on mesh stiffness.q1aθ, q1bθ, q1cθ, and q1dθ represent the crack propagating with the model of a step function, which indicates that the crack has instantaneously propagated by a tiny length. The corresponding lengths are 0.05, 0.1, 0.15, and 0.2 mm. qθ=0 represents the absence of TPOC.
(a)
Case  1 (q1=1.57) (b)
Case  2 (q1=3.13) (c)
Case  3 (q1=3.92 and q2=0.78) (d)
Case  4 (q1=3.92 and q2=2.35)Figure5 shows the time-varying mesh stiffness when TPOC occurs with different lengths under the same propagation model (i.e., step function model). Figure 5(a) depicts the mesh stiffness variation when q1aθ,q1bθ,q1cθ, and q1dθ appear in case  1. Similarly, Figures 5(b), 5(c), and 5(d) depict the mesh stiffness variation when q1aθ,q1bθ,q1cθ, and q1dθ appear in cases  2, 3, and 4, respectively.Figure5(a) shows that an instantaneous increment in the length of the crack induces a sudden decrease in mesh stiffness. Indeed, large lengths of crack propagation induce large reductions in mesh stiffness. The same phenomenon appears in the other cases. After TPOC, the total length of cracks is the sum of the existing crack length and TPOC length. Furthermore, the stiffness curve represents the time-varying mesh stiffness of gears with the total length of cracks.The stiffness reduction caused by TPOC in different cases (compared with no TPOC whenq(θ)=0) is listed in Table 4. In Table 4, 0.02-/0.02+ represents the moment before/after TPOC arises and k1i(0.02-)/k1i(0.02+) represents the mesh stiffness of the moment before/after TPOC occurs with the model of q1i(θ)(i=a,b,c,d). A schematic of stiffness reduction is shown in Figure 7(a).Table 4
Mesh stiffness reduction caused by TPOC (with the model of step function but different lengths) in different deterioration levels.
Deterioration levels of crack before TP TP model Stiffness value (108 N/m) Stiffness reduction caused by TPOC (108 N/m) Case1 ⁢k1a(0.02-)=k1b(0.02-)=k1c(0.02-)=k1d(0.02-)=6.831 0 q 1 a(θ) k 1 a ( 0.02 + ) = 6.791 ↓0.04 q 1 b(θ) k 1 b ( 0.02 + ) = 6.744 ↓0.087 q 1 c(θ) k 1 c ( 0.02 + ) = 6.696 ↓0.135 q 1 d(θ) k 1 d ( 0.02 + ) = 6.649 ↓0.182  Case2 ⁢k1a(0.02-)=k1b(0.02-)=k1c(0.02-)=k1d(0.02-)=5.681 0 q 1 a(θ) k 1 a ( 0.02 + ) = 5.645 ↓0.036 q 1 b(θ) k 1 b ( 0.02 + ) = 5.598 ↓0.083 q 1 c(θ) k 1 c ( 0.02 + ) = 5.551 ↓0.13 q 1 d(θ) k 1 d ( 0.02 + ) = 5.503 ↓0.178  Case3 ⁢k1a(0.02-)=k1b(0.02-)=k1c(0.02-)=k1d(0.02-)=3.461 0 q 1 a(θ) k 1 a ( 0.02 + ) = 3.43 ↓0.031 q 1 b(θ) k 1 b ( 0.02 + ) = 3.383 ↓0.078 q 1 c(θ) k 1 c ( 0.02 + ) = 3.337 ↓0.124 q 1 d(θ) k 1 d ( 0.02 + ) = 3.293 ↓0.168  Case4 ⁢k1a(0.02-)=k1b(0.02-)=k1c(0.02-)=k1d(0.02-)=2.68 0 q 1 a(θ) k 1 a ( 0.02 + ) = 2.654 ↓0.026 q 1 b(θ) k 1 b ( 0.02 + ) = 2.607 ↓0.073 q 1 c(θ) k 1 c ( 0.02 + ) = 2.561 ↓0.119 q 1 d(θ) k 1 d ( 0.02 + ) = 2.516 ↓0.164From Table4 and Figure 7(a), we can see that, for the same length of TPOC with the model of step function, the resultant stiffness reduction becomes less gradually from case  1 to case  4. In other words, as the deterioration level of crack increases, the stiffness reduction caused by TPOC will decrease.


#### 3.2.2. Mesh Stiffness Simulation for TPOC with the Same Length but Different Models

Similar to that in Figure5, q(θ)=0 in Figure 6 indicates the absence of TPOC. The time-varying mesh stiffness when TCOP occurs with the same length but different propagation models is shown in Figure 6. In the figure, the crack instantaneously propagates by 0.015 mm in two different models: continuous step and linear function models. For TPOC with the linear function model, the length of TPOC changes continuously, and the resultant stiffness variation is also continuous and smooth. For the continuous step function model, several continuous mutations in propagation length lead to several sudden decrements in mesh stiffness. However, regardless of the TPOC model, the stiffness curve after TPOC overlaps; that is, the stiffness is equal after TPOC, when the transient propagation length remains constant.Figure 6
The influences of TPOC with the same length but different propagation models on mesh stiffness.q2θ and q3θ represent the crack propagating with the same length of 0.15mm. q2θ represents the crack with the model of a continuous step function, indicating that the crack has instantaneously propagated by several continuous segments of length. q3θ represents the crack propagating with the model of a linear function, indicating that the crack has propagated continuously and smoothly. qθ=0 represents the absence of TPOC.
(a)
Case  1 (q1=1.57) (b)
Case  2 (q1=3.13) (c)
Case  3 (q1=3.92 and q2=0.78) (d)
Case  4 (q1=3.92 and q2=2.35)Figure 7
Mesh stiffness reduction caused by TCOP models in all cases.
(a) (b)We mainly investigated TPOC with the model of a continuous step function symbolized byq2(θ). The transient propagation process is described with three segments of the step function, denoting that the crack has propagated the same small length for three incidents. The three segments of TPOC are represented with q21θ,q22θ and q23θ, respectively.”,Table5 shows the stiffness reduction caused by each segment of crack propagation of TPOC with the continuous step function model in different cases. In the table, 0.02-/0.02+ represents the moment before/after TPOC from 0 mm to 0.05 mm, 0.04−/0.04+ represents the moment before/after TPOC from 0.05 mm to 0.1 mm, and 0.06−/0.06+ represents the moment before/after TPOC from 0.1 mm to 0.15 mm. k2(0.02−), k2(0.02+), k2(0.04−), k2(0.04+), k2(0.06−), and k2(0.06+) represent the mesh stiffness of the corresponding moment. A schematic of the stiffness reduction is shown in Figure 7(b).Table 5
Mesh stiffness reduction caused by TPOC with the model of a continuous step function in different deterioration levels.
Deterioration levels of crack before TP TCP length (mm) Stiffness value (108 N/m) Reduction caused by TPOC (108 N/m) Case1 0.05 k 2 ( 0.02 - ) = 6.831 k2(0.02+)=6.791 ↓0.04  Case1 0.1 k 2 ( 0.04 - ) = 6.852 k2(0.04+)=6.813 ↓0.039  Case1 0.15 k 2 ( 0.06 - ) = 6.812 k2(0.06+)=6.781 ↓0.031  Case2 0.05 k 2 ( 0.02 - ) = 5.681 k2(0.02+)=5.645 ↓0.036  Case2 0.1 k 2 ( 0.04 - ) = 5.706 k2(0.04+)=5.673 ↓0.033  Case2 0.15 k 2 ( 0.06 - ) = 5.672 k2(0.06+)=5.641 ↓0.031  Case3 0.05 k 2 ( 0.02 - ) = 3.461 k2(0.02+)=3.43 ↓0.031  Case3 0.1 k 2 ( 0.04 - ) = 3.609 k2(0.04+)=3.581 ↓0.028  Case3 0.15 k 2 ( 0.06 - ) = 3.789 k2(0.06+)=3.763 ↓0.026  Case4 0.05 k 2 ( 0.02 - ) = 2.68 k2(0.02+)=2.654 ↓0.026  Case4 0.1 k 2 ( 0.04 - ) = 2.833 k2(0.04+)=2.808 ↓0.025  Case4 0.15 k 2 ( 0.06 - ) = 3.014 k2(0.06+)=2.992 ↓0.022Table5 and Figure 7(b) show that, per 0.05 mm of TPOC, the resultant stiffness reduction decreases from case  1 to case  4. Therefore, we conclude that as the crack deterioration level becomes increasingly serious, the stiffness reduction caused by TPOC decreases.


## 3.1. Different Deterioration Levels of Crack before TPOC

Different lengths of existing cracks were considered, as shown in Figure4. In Figure 4, cases 1 to 4 represent the length of the existing crack becoming increasingly large; that is, the deterioration level of the crack becomes increasingly serious. The blue line represents the existing crack; the red line represents TPOC. In cases  1 and 2, the length of the existing crack is q1. In cases  3 and 4, the length of the existing crack is q1+q2. hc before and after TPOC arises in the different cases as shown in Table 2.Table 2
h c before and after TPOC arises.
No TPOC TPOC arises Case1 h c = R b 1 sin ⁡ α 2 - q 1 sin ⁡ β h c = R b 1 sin ⁡ α 2 -q1+qθsin⁡β Case2  Case3 h c = q 2 sin ⁡ β h c = q 2 + q θ sin ⁡ β Case4According to the formula in Table2, the bending and shear stiffness when TPOC arises can be recalculated as follows:Case  1 (17) 1 k c b = ∫ - α g α 2 12 1 + α 2 - α sin ⁡ α cos ⁡ α 1 - cos ⁡ α cos ⁡ α 1 2 α 2 - α cos ⁡ α E L sin ⁡ α 2 + sin ⁡ α + α 2 - α cos ⁡ α - q 1 + q θ / R b sin ⁡ β 3 d α + ∫ - α 1 - α g 3 1 + α 2 - α sin ⁡ α cos ⁡ α 1 - cos ⁡ α cos ⁡ α 1 2 α 2 - α cos ⁡ α 2 E L α 2 - α cos ⁡ α + sin ⁡ α 3 d α , 1 k c s = ∫ - α g α 2 2.4 1 + ν α 2 - α cos ⁡ α  cos 2 ⁡ α 1 E W sin ⁡ α 2 + sin ⁡ α + α 2 - α cos ⁡ α - q 1 + q θ / R b sin ⁡ β d α + ∫ - α 1 - α g 1.2 1 + ν α 2 - α cos ⁡ α  cos 2 ⁡ α 1 E W α 2 - α cos ⁡ α + sin ⁡ α d α .Case  2 (18) 1 k c b = ∫ - α 1 α 2 12 1 + α 2 - α sin ⁡ α cos ⁡ α 1 - cos ⁡ α cos ⁡ α 1 2 α 2 - α cos ⁡ α E W sin ⁡ α 2 + sin ⁡ α + α 2 - α cos ⁡ α - q 1 + q θ / R b sin ⁡ β 3 d α , 1 k c s = ∫ - α 1 α 2 2.4 1 + ν α 2 - α cos ⁡ α  cos 2 ⁡ α 1 E W sin ⁡ α 2 + sin ⁡ α + α 2 - α cos ⁡ α - q 1 + q θ / R b sin ⁡ β d α .Case  3 (19) 1 k c b = ∫ - α 1 α 2 12 1 + α 2 - α sin ⁡ α cos ⁡ α 1 - cos ⁡ α cos ⁡ α 1 2 α 2 - α cos ⁡ α E W sin ⁡ α + α 2 - α cos ⁡ α - q 2 + q θ / R b sin ⁡ β 3 d α , 1 k c s = ∫ - α 1 α 2 2.4 1 + ν α 2 - α cos ⁡ α  cos 2 ⁡ α 1 E W sin ⁡ α + α 2 - α cos ⁡ α - q 2 + q θ / R b sin ⁡ β d α .Case  4 (20) 1 k c b = ∫ - α g α 2 12 1 + α 2 - α sin ⁡ α cos ⁡ α 1 - cos ⁡ α cos ⁡ α 1 2 α 2 - α cos ⁡ α E W s i n α + α 2 - α cos ⁡ α - q 2 + q θ / R b sin ⁡ β 3 d α , 1 k c s = ∫ - α g α 2 2.4 1 + ν α 2 - α cos ⁡ α  cos 2 ⁡ α 1 E W sin ⁡ α + α 2 - α cos ⁡ α - q 2 + q θ / R b sin ⁡ β d α .As a sample of the length of existing crack, q1 (or q1 and q2) can be set in the following 4 cases. Case  1 to case  4 represent the notion that the deterioration level becomes more serious:Case  1: whenq1=1.57Case  2: whenq1=3.13Case  3: whenq1=3.92andq2=0.78Case  4: whenq1=3.92andq2=2.35


## 3.2. Time-Varying Mesh Stiffness Simulation When TPOC Occurs

Based on the improved potential energy method, the time-varying mesh stiffness when TPOC occurs in a pinion was calculated. The main parameters are shown in Table3.Table 3
Main parameters of the gear system.
Gear type Standard involute spur teeth Material Steel Young’s modulus E = 2.068 × 1 0 11 Pa Poisson ratio ν = 0.3 Number of teeth N 1 = 19 and N2=48 Pressure angle α = 20 ° Diametral pitch P = 8 inch−1 Base radius of the gear R b 1 = 28.34 mm Width of teeth w = 18 mm Mass of the pinion m 1 = 0.96 Kg Mass of the gear m 2 = 2.88 Kg Contact ratio C r = 1.6456 Mass moment of inertia of the motor I m = 0.0021 Kg m2 Mass moment of inertia of the load I b = 0.0105 Kg m2 Mass moment of inertia of the pinion I 1 = 4.3659 × 1 0 - 4 Kg m2 Mass moment of inertia of the gear I 2 = 8.3602 × 1 0 - 3 Kg m2


### 3.2.1. Mesh Stiffness Simulation When TPOC Occurs with the Same Model but Different Lengths

This section focuses on mesh stiffness when TPOC occurs with the model of a step function. However, the lengths are 0.05, 0.1, 0.15, and 0.2 mm. In Figure5, q(θ)=0 indicates that no TPOC occurs, and the corresponding stiffness represents the mesh stiffness of a cracked gear with the length of an existing crack.Figure 5
The influences of TPOC with different lengths but the same propagation model on mesh stiffness.q1aθ, q1bθ, q1cθ, and q1dθ represent the crack propagating with the model of a step function, which indicates that the crack has instantaneously propagated by a tiny length. The corresponding lengths are 0.05, 0.1, 0.15, and 0.2 mm. qθ=0 represents the absence of TPOC.
(a)
Case  1 (q1=1.57) (b)
Case  2 (q1=3.13) (c)
Case  3 (q1=3.92 and q2=0.78) (d)
Case  4 (q1=3.92 and q2=2.35)Figure5 shows the time-varying mesh stiffness when TPOC occurs with different lengths under the same propagation model (i.e., step function model). Figure 5(a) depicts the mesh stiffness variation when q1aθ,q1bθ,q1cθ, and q1dθ appear in case  1. Similarly, Figures 5(b), 5(c), and 5(d) depict the mesh stiffness variation when q1aθ,q1bθ,q1cθ, and q1dθ appear in cases  2, 3, and 4, respectively.Figure5(a) shows that an instantaneous increment in the length of the crack induces a sudden decrease in mesh stiffness. Indeed, large lengths of crack propagation induce large reductions in mesh stiffness. The same phenomenon appears in the other cases. After TPOC, the total length of cracks is the sum of the existing crack length and TPOC length. Furthermore, the stiffness curve represents the time-varying mesh stiffness of gears with the total length of cracks.The stiffness reduction caused by TPOC in different cases (compared with no TPOC whenq(θ)=0) is listed in Table 4. In Table 4, 0.02-/0.02+ represents the moment before/after TPOC arises and k1i(0.02-)/k1i(0.02+) represents the mesh stiffness of the moment before/after TPOC occurs with the model of q1i(θ)(i=a,b,c,d). A schematic of stiffness reduction is shown in Figure 7(a).Table 4
Mesh stiffness reduction caused by TPOC (with the model of step function but different lengths) in different deterioration levels.
Deterioration levels of crack before TP TP model Stiffness value (108 N/m) Stiffness reduction caused by TPOC (108 N/m) Case1 ⁢k1a(0.02-)=k1b(0.02-)=k1c(0.02-)=k1d(0.02-)=6.831 0 q 1 a(θ) k 1 a ( 0.02 + ) = 6.791 ↓0.04 q 1 b(θ) k 1 b ( 0.02 + ) = 6.744 ↓0.087 q 1 c(θ) k 1 c ( 0.02 + ) = 6.696 ↓0.135 q 1 d(θ) k 1 d ( 0.02 + ) = 6.649 ↓0.182  Case2 ⁢k1a(0.02-)=k1b(0.02-)=k1c(0.02-)=k1d(0.02-)=5.681 0 q 1 a(θ) k 1 a ( 0.02 + ) = 5.645 ↓0.036 q 1 b(θ) k 1 b ( 0.02 + ) = 5.598 ↓0.083 q 1 c(θ) k 1 c ( 0.02 + ) = 5.551 ↓0.13 q 1 d(θ) k 1 d ( 0.02 + ) = 5.503 ↓0.178  Case3 ⁢k1a(0.02-)=k1b(0.02-)=k1c(0.02-)=k1d(0.02-)=3.461 0 q 1 a(θ) k 1 a ( 0.02 + ) = 3.43 ↓0.031 q 1 b(θ) k 1 b ( 0.02 + ) = 3.383 ↓0.078 q 1 c(θ) k 1 c ( 0.02 + ) = 3.337 ↓0.124 q 1 d(θ) k 1 d ( 0.02 + ) = 3.293 ↓0.168  Case4 ⁢k1a(0.02-)=k1b(0.02-)=k1c(0.02-)=k1d(0.02-)=2.68 0 q 1 a(θ) k 1 a ( 0.02 + ) = 2.654 ↓0.026 q 1 b(θ) k 1 b ( 0.02 + ) = 2.607 ↓0.073 q 1 c(θ) k 1 c ( 0.02 + ) = 2.561 ↓0.119 q 1 d(θ) k 1 d ( 0.02 + ) = 2.516 ↓0.164From Table4 and Figure 7(a), we can see that, for the same length of TPOC with the model of step function, the resultant stiffness reduction becomes less gradually from case  1 to case  4. In other words, as the deterioration level of crack increases, the stiffness reduction caused by TPOC will decrease.


### 3.2.2. Mesh Stiffness Simulation for TPOC with the Same Length but Different Models

Similar to that in Figure5, q(θ)=0 in Figure 6 indicates the absence of TPOC. The time-varying mesh stiffness when TCOP occurs with the same length but different propagation models is shown in Figure 6. In the figure, the crack instantaneously propagates by 0.015 mm in two different models: continuous step and linear function models. For TPOC with the linear function model, the length of TPOC changes continuously, and the resultant stiffness variation is also continuous and smooth. For the continuous step function model, several continuous mutations in propagation length lead to several sudden decrements in mesh stiffness. However, regardless of the TPOC model, the stiffness curve after TPOC overlaps; that is, the stiffness is equal after TPOC, when the transient propagation length remains constant.Figure 6
The influences of TPOC with the same length but different propagation models on mesh stiffness.q2θ and q3θ represent the crack propagating with the same length of 0.15mm. q2θ represents the crack with the model of a continuous step function, indicating that the crack has instantaneously propagated by several continuous segments of length. q3θ represents the crack propagating with the model of a linear function, indicating that the crack has propagated continuously and smoothly. qθ=0 represents the absence of TPOC.
(a)
Case  1 (q1=1.57) (b)
Case  2 (q1=3.13) (c)
Case  3 (q1=3.92 and q2=0.78) (d)
Case  4 (q1=3.92 and q2=2.35)Figure 7
Mesh stiffness reduction caused by TCOP models in all cases.
(a) (b)We mainly investigated TPOC with the model of a continuous step function symbolized byq2(θ). The transient propagation process is described with three segments of the step function, denoting that the crack has propagated the same small length for three incidents. The three segments of TPOC are represented with q21θ,q22θ and q23θ, respectively.”,Table5 shows the stiffness reduction caused by each segment of crack propagation of TPOC with the continuous step function model in different cases. In the table, 0.02-/0.02+ represents the moment before/after TPOC from 0 mm to 0.05 mm, 0.04−/0.04+ represents the moment before/after TPOC from 0.05 mm to 0.1 mm, and 0.06−/0.06+ represents the moment before/after TPOC from 0.1 mm to 0.15 mm. k2(0.02−), k2(0.02+), k2(0.04−), k2(0.04+), k2(0.06−), and k2(0.06+) represent the mesh stiffness of the corresponding moment. A schematic of the stiffness reduction is shown in Figure 7(b).Table 5
Mesh stiffness reduction caused by TPOC with the model of a continuous step function in different deterioration levels.
Deterioration levels of crack before TP TCP length (mm) Stiffness value (108 N/m) Reduction caused by TPOC (108 N/m) Case1 0.05 k 2 ( 0.02 - ) = 6.831 k2(0.02+)=6.791 ↓0.04  Case1 0.1 k 2 ( 0.04 - ) = 6.852 k2(0.04+)=6.813 ↓0.039  Case1 0.15 k 2 ( 0.06 - ) = 6.812 k2(0.06+)=6.781 ↓0.031  Case2 0.05 k 2 ( 0.02 - ) = 5.681 k2(0.02+)=5.645 ↓0.036  Case2 0.1 k 2 ( 0.04 - ) = 5.706 k2(0.04+)=5.673 ↓0.033  Case2 0.15 k 2 ( 0.06 - ) = 5.672 k2(0.06+)=5.641 ↓0.031  Case3 0.05 k 2 ( 0.02 - ) = 3.461 k2(0.02+)=3.43 ↓0.031  Case3 0.1 k 2 ( 0.04 - ) = 3.609 k2(0.04+)=3.581 ↓0.028  Case3 0.15 k 2 ( 0.06 - ) = 3.789 k2(0.06+)=3.763 ↓0.026  Case4 0.05 k 2 ( 0.02 - ) = 2.68 k2(0.02+)=2.654 ↓0.026  Case4 0.1 k 2 ( 0.04 - ) = 2.833 k2(0.04+)=2.808 ↓0.025  Case4 0.15 k 2 ( 0.06 - ) = 3.014 k2(0.06+)=2.992 ↓0.022Table5 and Figure 7(b) show that, per 0.05 mm of TPOC, the resultant stiffness reduction decreases from case  1 to case  4. Therefore, we conclude that as the crack deterioration level becomes increasingly serious, the stiffness reduction caused by TPOC decreases.


## 3.2.1. Mesh Stiffness Simulation When TPOC Occurs with the Same Model but Different Lengths

This section focuses on mesh stiffness when TPOC occurs with the model of a step function. However, the lengths are 0.05, 0.1, 0.15, and 0.2 mm. In Figure5, q(θ)=0 indicates that no TPOC occurs, and the corresponding stiffness represents the mesh stiffness of a cracked gear with the length of an existing crack.Figure 5
The influences of TPOC with different lengths but the same propagation model on mesh stiffness.q1aθ, q1bθ, q1cθ, and q1dθ represent the crack propagating with the model of a step function, which indicates that the crack has instantaneously propagated by a tiny length. The corresponding lengths are 0.05, 0.1, 0.15, and 0.2 mm. qθ=0 represents the absence of TPOC.
(a)
Case  1 (q1=1.57) (b)
Case  2 (q1=3.13) (c)
Case  3 (q1=3.92 and q2=0.78) (d)
Case  4 (q1=3.92 and q2=2.35)Figure5 shows the time-varying mesh stiffness when TPOC occurs with different lengths under the same propagation model (i.e., step function model). Figure 5(a) depicts the mesh stiffness variation when q1aθ,q1bθ,q1cθ, and q1dθ appear in case  1. Similarly, Figures 5(b), 5(c), and 5(d) depict the mesh stiffness variation when q1aθ,q1bθ,q1cθ, and q1dθ appear in cases  2, 3, and 4, respectively.Figure5(a) shows that an instantaneous increment in the length of the crack induces a sudden decrease in mesh stiffness. Indeed, large lengths of crack propagation induce large reductions in mesh stiffness. The same phenomenon appears in the other cases. After TPOC, the total length of cracks is the sum of the existing crack length and TPOC length. Furthermore, the stiffness curve represents the time-varying mesh stiffness of gears with the total length of cracks.The stiffness reduction caused by TPOC in different cases (compared with no TPOC whenq(θ)=0) is listed in Table 4. In Table 4, 0.02-/0.02+ represents the moment before/after TPOC arises and k1i(0.02-)/k1i(0.02+) represents the mesh stiffness of the moment before/after TPOC occurs with the model of q1i(θ)(i=a,b,c,d). A schematic of stiffness reduction is shown in Figure 7(a).Table 4
Mesh stiffness reduction caused by TPOC (with the model of step function but different lengths) in different deterioration levels.
Deterioration levels of crack before TP TP model Stiffness value (108 N/m) Stiffness reduction caused by TPOC (108 N/m) Case1 ⁢k1a(0.02-)=k1b(0.02-)=k1c(0.02-)=k1d(0.02-)=6.831 0 q 1 a(θ) k 1 a ( 0.02 + ) = 6.791 ↓0.04 q 1 b(θ) k 1 b ( 0.02 + ) = 6.744 ↓0.087 q 1 c(θ) k 1 c ( 0.02 + ) = 6.696 ↓0.135 q 1 d(θ) k 1 d ( 0.02 + ) = 6.649 ↓0.182  Case2 ⁢k1a(0.02-)=k1b(0.02-)=k1c(0.02-)=k1d(0.02-)=5.681 0 q 1 a(θ) k 1 a ( 0.02 + ) = 5.645 ↓0.036 q 1 b(θ) k 1 b ( 0.02 + ) = 5.598 ↓0.083 q 1 c(θ) k 1 c ( 0.02 + ) = 5.551 ↓0.13 q 1 d(θ) k 1 d ( 0.02 + ) = 5.503 ↓0.178  Case3 ⁢k1a(0.02-)=k1b(0.02-)=k1c(0.02-)=k1d(0.02-)=3.461 0 q 1 a(θ) k 1 a ( 0.02 + ) = 3.43 ↓0.031 q 1 b(θ) k 1 b ( 0.02 + ) = 3.383 ↓0.078 q 1 c(θ) k 1 c ( 0.02 + ) = 3.337 ↓0.124 q 1 d(θ) k 1 d ( 0.02 + ) = 3.293 ↓0.168  Case4 ⁢k1a(0.02-)=k1b(0.02-)=k1c(0.02-)=k1d(0.02-)=2.68 0 q 1 a(θ) k 1 a ( 0.02 + ) = 2.654 ↓0.026 q 1 b(θ) k 1 b ( 0.02 + ) = 2.607 ↓0.073 q 1 c(θ) k 1 c ( 0.02 + ) = 2.561 ↓0.119 q 1 d(θ) k 1 d ( 0.02 + ) = 2.516 ↓0.164From Table4 and Figure 7(a), we can see that, for the same length of TPOC with the model of step function, the resultant stiffness reduction becomes less gradually from case  1 to case  4. In other words, as the deterioration level of crack increases, the stiffness reduction caused by TPOC will decrease.


## 3.2.2. Mesh Stiffness Simulation for TPOC with the Same Length but Different Models

Similar to that in Figure5, q(θ)=0 in Figure 6 indicates the absence of TPOC. The time-varying mesh stiffness when TCOP occurs with the same length but different propagation models is shown in Figure 6. In the figure, the crack instantaneously propagates by 0.015 mm in two different models: continuous step and linear function models. For TPOC with the linear function model, the length of TPOC changes continuously, and the resultant stiffness variation is also continuous and smooth. For the continuous step function model, several continuous mutations in propagation length lead to several sudden decrements in mesh stiffness. However, regardless of the TPOC model, the stiffness curve after TPOC overlaps; that is, the stiffness is equal after TPOC, when the transient propagation length remains constant.Figure 6
The influences of TPOC with the same length but different propagation models on mesh stiffness.q2θ and q3θ represent the crack propagating with the same length of 0.15mm. q2θ represents the crack with the model of a continuous step function, indicating that the crack has instantaneously propagated by several continuous segments of length. q3θ represents the crack propagating with the model of a linear function, indicating that the crack has propagated continuously and smoothly. qθ=0 represents the absence of TPOC.
(a)
Case  1 (q1=1.57) (b)
Case  2 (q1=3.13) (c)
Case  3 (q1=3.92 and q2=0.78) (d)
Case  4 (q1=3.92 and q2=2.35)Figure 7
Mesh stiffness reduction caused by TCOP models in all cases.
(a) (b)We mainly investigated TPOC with the model of a continuous step function symbolized byq2(θ). The transient propagation process is described with three segments of the step function, denoting that the crack has propagated the same small length for three incidents. The three segments of TPOC are represented with q21θ,q22θ and q23θ, respectively.”,Table5 shows the stiffness reduction caused by each segment of crack propagation of TPOC with the continuous step function model in different cases. In the table, 0.02-/0.02+ represents the moment before/after TPOC from 0 mm to 0.05 mm, 0.04−/0.04+ represents the moment before/after TPOC from 0.05 mm to 0.1 mm, and 0.06−/0.06+ represents the moment before/after TPOC from 0.1 mm to 0.15 mm. k2(0.02−), k2(0.02+), k2(0.04−), k2(0.04+), k2(0.06−), and k2(0.06+) represent the mesh stiffness of the corresponding moment. A schematic of the stiffness reduction is shown in Figure 7(b).Table 5
Mesh stiffness reduction caused by TPOC with the model of a continuous step function in different deterioration levels.
Deterioration levels of crack before TP TCP length (mm) Stiffness value (108 N/m) Reduction caused by TPOC (108 N/m) Case1 0.05 k 2 ( 0.02 - ) = 6.831 k2(0.02+)=6.791 ↓0.04  Case1 0.1 k 2 ( 0.04 - ) = 6.852 k2(0.04+)=6.813 ↓0.039  Case1 0.15 k 2 ( 0.06 - ) = 6.812 k2(0.06+)=6.781 ↓0.031  Case2 0.05 k 2 ( 0.02 - ) = 5.681 k2(0.02+)=5.645 ↓0.036  Case2 0.1 k 2 ( 0.04 - ) = 5.706 k2(0.04+)=5.673 ↓0.033  Case2 0.15 k 2 ( 0.06 - ) = 5.672 k2(0.06+)=5.641 ↓0.031  Case3 0.05 k 2 ( 0.02 - ) = 3.461 k2(0.02+)=3.43 ↓0.031  Case3 0.1 k 2 ( 0.04 - ) = 3.609 k2(0.04+)=3.581 ↓0.028  Case3 0.15 k 2 ( 0.06 - ) = 3.789 k2(0.06+)=3.763 ↓0.026  Case4 0.05 k 2 ( 0.02 - ) = 2.68 k2(0.02+)=2.654 ↓0.026  Case4 0.1 k 2 ( 0.04 - ) = 2.833 k2(0.04+)=2.808 ↓0.025  Case4 0.15 k 2 ( 0.06 - ) = 3.014 k2(0.06+)=2.992 ↓0.022Table5 and Figure 7(b) show that, per 0.05 mm of TPOC, the resultant stiffness reduction decreases from case  1 to case  4. Therefore, we conclude that as the crack deterioration level becomes increasingly serious, the stiffness reduction caused by TPOC decreases.


## 4. Conclusions

Several TPOC models have been proposed to study the influences of TPOC on the time-varying mesh stiffness of the gear and detect the further propagation of cracks from the perspective of time-varying mesh stiffness in a timely manner. The improved potential energy method was developed by integrating the TPOC model into the potential energy method. The simulation results show that the improved potential energy method can accurately calculate the time-varying mesh stiffness when TPOC occurs. Different deterioration levels of cracks before TPOC, different propagation models, and different lengths of TPOC were considered. The following conclusions were obtained:(1)
The stiffness variation trend is similar to that in the model of TPOC which caused the variation. For example, when TPOC occurs with the model of a step function, the resultant stiffness variation is in the form of an approximate step function. When TPOC occurs with the model of a continuous step function, the resultant stiffness variation is in the form of an approximate continuous step function. When TPOC occurs with the model of a linear function, the resultant stiffness variation is smooth and continuous.(2)
For the certain deterioration level of cracks before transient propagation, large propagation lengths are associated with large variations in stiffness.(3)
When the crack propagates by the same tiny length instantaneously (e.g., TPOC occurs with the model of a step function), the resultant stiffness reduction decreases as the deterioration level of the crack increases (cases  1 to 4). When the crack continuously propagates by several segments of equal length (e.g., TPOC arises with the model of a continuous step function), for every equal length of TPOC, the resultant stiffness reduction decreases gradually from case  1 to  case 4. Therefore, we conclude that, for the same length of TPOC, the resultant stiffness reduction is related to the deterioration level of the crack; high levels of deterioration are associated with small reductions in stiffness. In other words, TPOC is easy to detect in the early stage of a gear fault.(4)
For a cracked gear, identical transient propagation lengths result in identical stiffness levels after TPOC regardless of the type of TPOC.



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*Source: 6928686-2016-12-15.xml*