Improvement on Meshing Sti ﬀ ness Algorithms of Gear with Peeling

: This paper investigates the e ﬀ ect of a gear tooth peeling on meshing sti ﬀ ness of involute gears. The tooth of the gear wheel is symmetric about the axis, and its symmetry will change after the gear spalling, and its meshing sti ﬀ ness will also change during the meshing process. On this basis, an analytical model was developed, and based on the energy method a meshing sti ﬀ ness algorithm for the complete meshing process of single gear teeth with peeling gears was proposed. According to the inﬂuence of the change of meshing point relative to the peeling position on the meshing sti ﬀ ness, this algorithm calculates its sti ﬀ ness separately. The inﬂuence of the peeling sizes on mesh sti ﬀ ness is studied by simulation analysis. As a very important parameter, the study of gear mesh sti ﬀ ness is of great signiﬁcance to the monitoring of working conditions and the prevention of sudden failure of the gear box system.


Introduction
Gearbox is one of the most important components in mechanical industry and daily life. With constant development of modern industrial technology, fault monitoring of gearbox is more and more valued in research field [1][2][3][4][5][6][7][8][9][10]. As the key component of gearbox, the study of gear fault diagnosis method [11][12][13][14] is of great significance. Meshari et al. [15] carried out an extensive study of gearbox fault and found that failure of gear tooth is one of the leading causes of gearbox fault, and that peeling is one of the common gear failures. Amarnath [16] indicated that peeling often occurs in the early stage of failure for the geared system. Song et al. [17] proposed the combination of the trivalent logic inference theory with the possibility and fuzzy theories. Cui et al. [18] proposed a new concatenated dictionary matching tracking method combining impact dictionary and step dictionary. Operating in the presence of peeling, the contact stress tends to increase enormously in the contact area of the mating teeth surface. The propagation of tooth damage causes instantaneous reduction in tooth stiffness. The vibration signal of gear transmission varies as the stiffness changes.
Subramanian [19] found that with the increase of tooth thickness and height, the gear tooth stiffness increases, which in turn reduces the radial load deflection and the vibration amplitude in the vertical direction. This shows that time-varying meshing is one of the main inner excitation sources in gearbox. Solving meshing stiffness accurately is the basic condition to research on the fault mechanism of a gear system. Lin [20] simplified finite element with Fourier function to extract the meshing stiffness of different locations, and received a square wave function of meshing stiffness. It is only applied for some specific situations. Silurian ishikawa [21] proposed a method to simplify the involute part into a

Calculate Hertz Contact Stiffness and Wheel Stiffness
When calculating the gear meshing stiffness by the energy method, it is assumed that the gear body is an isotropic elastomer. According to the law of Hertz, the elastic compression deformation in contact area of two isotropic elastomers can be approximately equivalent to that of parabolic contact. Based on the literature [34], Hertz contact stiffness of a meshing gear pair with same material is a constant on the meshing line, whose value is relatively independent of the position of meshing contact.
The Hertz contact stiffness can be calculated by the formula [27]: where E is the elastic modulus, L is axial width of the gear, and v is the Poisson's ratio. According to Formula (1), when the gear material is determined, the Hertz contact stiffness of the gear is proportional to the tooth width. The meshing stiffness of gear pair also includes wheel body stiffness K f . The formula can be expressed as [25]: The coefficients L*, M*, P* can be obtained from the formula [35] where coefficient X* means L*, M*, P*, Q*, and the other parameters can refer to Figure 1. The coefficients A, B, C, D, E and F are shown in Table 2. When calculating L*, it takes the values of A, B, C, D, E, and F of the corresponding row of L*, then, add them into Formula (3) to calculate L*.

Calculate Hertz Contact Stiffness and Wheel Stiffness
When calculating the gear meshing stiffness by the energy method, it is assumed that the gear body is an isotropic elastomer. According to the law of Hertz, the elastic compression deformation in contact area of two isotropic elastomers can be approximately equivalent to that of parabolic contact. Based on the literature [33], Hertz contact stiffness of a meshing gear pair with same material is a constant on the meshing line, whose value is relatively independent of the position of meshing contact.
The Hertz contact stiffness can be calculated by the formula [26]: where E is the elastic modulus, L is axial width of the gear, and v is the Poisson's ratio. According to Formula (1), when the gear material is determined, the Hertz contact stiffness of the gear is proportional to the tooth width. The meshing stiffness of gear pair also includes wheel body stiffness Kf. The formula can be expressed as [24]: The coefficients L*, M*, P* can be obtained from the formula [34] where coefficient X* means L*, M*, P*, Q*，and the other parameters can refer to Figure 1. The coefficients A, B, C, D, E and F are shown in Table 2. When calculating L*, it takes the values of A, B, C, D, E, and F of the corresponding row of L*, then, add them into Formula (3) to calculate L*.    Figure 2 shows the forces loaded on the tooth, where F is the meshing force at meshing point. With material mechanics, bending potential energy Ub, shear potential energy Us, and axial compression potential energy Ua is respectively [27]: where K b , K s , K a respectively represent the bending stiffness, shear stiffness, and compression stiffness.  Figure 2 shows the forces loaded on the tooth, where F is the meshing force at meshing point. With material mechanics, bending potential energy Ub, shear potential energy Us, and axial compression potential energy Ua is respectively [27]:

Calculate Bending Stiffness, Shear Stiffness, and Compression Stiffness
where Kb、Ks、Ka respectively represent the bending stiffness, shear stiffness, and compression stiffness. According to the cantilever beam theory, bending potential energy Ub, shear potential energy Us, and axial compression potential energy Ua can be calculated with:  According to the cantilever beam theory, bending potential energy Ub, shear potential energy Us, and axial compression potential energy Ua can be calculated with: The contact ratio of spur gear is between 1 and 2, so the integral meshing cycle of gear pair includes meshing intervals of single teeth pair and two teeth pair. It is necessary to study the accurate relations between meshing angle and intervals of single and double pair of teeth for meshing stiffness of integral gear pair.
In the stage of single pair of teeth meshing, Hertz stiffness, gear body stiffness, bending stiffness, shear stiffness, and compression stiffness come together into meshing stiffness Kt of a meshing teeth pair. It can be represented as: In a double pair of teeth, define the left tooth pair as the first one, i = 1; the right pair as the second, i = 2. Then, the meshing stiffness Kt of a tooth pair can be represented as:

Improvement on Meshing Stiffness Algorithms of Gear with Peeling
Ankur [33] solved the meshing stiffness of gear with peeling based on the energy method and studied the meshing stiffness of different sizes, positions, and shapes. Figure 3 presents the peeling failure meshing stiffness in the literature. According to Figure 3, the meshing stiffness algorithm model used in the literature only considers the influence of the peeling interval on the meshing stiffness, but does not consider the influence of peeling interval on the subsequent meshing process.
of integral gear pair.
In the stage of single pair of teeth meshing, Hertz stiffness, gear body stiffness, bending stiffness, shear stiffness, and compression stiffness come together into meshing stiffness Kt of a meshing teeth pair. It can be represented as: In a double pair of teeth, define the left tooth pair as the first one, i = 1; the right pair as the second, i = 2. Then, the meshing stiffness Kt of a tooth pair can be represented as:

Improvement on Meshing Stiffness Algorithms of Gear with Peeling
Ankur [33] solved the meshing stiffness of gear with peeling based on the energy method and studied the meshing stiffness of different sizes, positions, and shapes. Figure 3 presents the peeling failure meshing stiffness in the literature. According to Figure 3, the meshing stiffness algorithm model used in the literature only considers the influence of the peeling interval on the meshing stiffness, but does not consider the influence of peeling interval on the subsequent meshing process. Mohammed et al. [37] pointed out that although the gear crack would not change the shape of the tooth profile, it would affect the mesh stiffness, and the same goes for peeling. The gear model with peeling as shown in Figure 4 is established. The model divides the gear teeth into three regions. When the meshing point is located in region 1, the meshing stiffness is not affected by peeling; when Mohammed et al. [37] pointed out that although the gear crack would not change the shape of the tooth profile, it would affect the mesh stiffness, and the same goes for peeling. The gear model with peeling as shown in Figure 4 is established. The model divides the gear teeth into three regions. When the meshing point is located in region 1, the meshing stiffness is not affected by peeling; when entering area 2, due to the gear tooth profile being incomplete, there is a sudden change in meshing stiffness; in area 3, despite the complete tooth profile, due to the presence of peeling cavities, peeling failure continues to affect the subsequent meshing stiffness, and the form of influence can be analogized as a crack fault.
Symmetry 2018, 10, x FOR PEER REVIEW 6 of 14 entering area 2, due to the gear tooth profile being incomplete, there is a sudden change in meshing stiffness; in area 3, despite the complete tooth profile, due to the presence of peeling cavities, peeling failure continues to affect the subsequent meshing stiffness, and the form of influence can be analogized as a crack fault.

Establishing a Calculation Model of Peeling Failure Meshing Stiffness
As shown in Figure 5, a gear parameter model with peeling was established. Based on the fault model, the time-varying meshing stiffness of peeling gear was solved. Three assumptions were made on the model: 1. The peeling fault is rectangular, the peeling depth is the same, and the shape of the peeling is symmetrical about the central axis; 2. Before entering the peeling fault, peeling failure has no effect on mesh stiffness;

Establishing a Calculation Model of Peeling Failure Meshing Stiffness
As shown in Figure 5, a gear parameter model with peeling was established. Based on the fault model, the time-varying meshing stiffness of peeling gear was solved. Three assumptions were made on the model: 1. The peeling fault is rectangular, the peeling depth is the same, and the shape of the peeling is symmetrical about the central axis; 2. Before entering the peeling fault, peeling failure has no effect on mesh stiffness; 3. There is quantitative relation: or consider the tooth as failure. Part of the peeling fault parameters are presented in Table 3. stiffness; in area 3, despite the complete tooth profile, due to the presence of peeling cavities, peeling failure continues to affect the subsequent meshing stiffness, and the form of influence can be analogized as a crack fault.

Establishing a Calculation Model of Peeling Failure Meshing Stiffness
As shown in Figure 5, a gear parameter model with peeling was established. Based on the fault model, the time-varying meshing stiffness of peeling gear was solved. Three assumptions were made on the model: Part of the peeling fault parameters are presented in Table 3. Table 3. Part of the peeling fault parameters.
For the peeling fault, the meshing stiffness K is solved in five parts, respectively, K1, K2, K3, Kh, and Kfi, and the formulas are:

Solve Peeling Failure Meshing Stiffness
With the energy method and geometrical characteristics of the asymptote gear, the bending stiffness Kb, shear stiffness Ks, and compression stiffness Ka can be calculated with:  For the peeling fault, the meshing stiffness K is solved in five parts, respectively, K 1 , K 2 , K 3 , K h , and K fi , and the formulas are:

Solve Peeling Failure Meshing Stiffness
With the energy method and geometrical characteristics of the asymptote gear, the bending stiffness K b , shear stiffness K s , and compression stiffness K a can be calculated with: Shear module G: G = E 2(1+υ) I x -rotation inertia, A x -CSA (cross sectional area) There is a quantitative relationship: Therefore, I x , A x can be deduced: Among them, the parameters hl, hq, y, and dy can be expressed as: , bending stiffness K b , shear stiffness K s , and compression stiffness K a can be integrated with: The formula parameter is shown in Figure 2. The time-varying meshing stiffness is obtained by MATLAB programming. Figure 6 shows normal gear meshing stiffness and peeling fault meshing stiffness. As shown in Figure 6a, peeling failure occurred in the stage of one pair of tooth meshing. Before entering the fault, the peeling meshing stiffness is the same as normal meshing stiffness; at the moment of entering peeling location, the fault meshing stiffness value decreased suddenly. The formula parameter is shown in Figure 2. The time-varying meshing stiffness is obtained by MATLAB programming. Figure 6 shows normal gear meshing stiffness and peeling fault meshing stiffness. As shown in Figure 6a, peeling failure occurred in the stage of one pair of tooth meshing. Before entering the fault, the peeling meshing stiffness is the same as normal meshing stiffness; at the moment of entering peeling location, the fault meshing stiffness value decreased suddenly.
It can be seen from Figure 6b that when the angle is 17°, the meshing position leaves a peeling fault and the tooth profile is complete in the process of the subsequent meshing. So, the compressive stiffness and contact stiffness increase, which causes the peeling meshing stiffness to suddenly increase. However, due to the existence of the peeling cavity, the bending stiffness and shearing stiffness are affected; the meshing stiffness is still less than the normal meshing stiffness.

The Meshing Stiffness of Gear with Variable Peeling Parameter
In changing the size of the peeling fault, the effect of different fault size on the meshing stiffness can then be studied. It can be seen from the formula that the fault size Y2 mainly affects the division of fault interval, and has no influence on the meshing stiffness. Then, the fault depth q0 and fault width W2 are thoroughly studied in this paper.

Variable Depth Variable Meshing Stiffness
In order to study the effect of peeling depth on the time-varying meshing stiffness, the width of the peeling was fixed, the peeling depth was changed, and then variable depth of peeling failure meshing stiffness was calculated. Making a study to compare two groups of size: (1) peeling width 6 mm, peeling depth 1 mm, 1.5 mm, 2 mm, 2.5 mm and 3 mm, respectively; (2) peeling width 16 mm, peeling depth 1 mm, 1.5 mm, 2 mm, 2.5 mm, 3 mm, respectively. The tooth width of the gear model is 16 mm. For group 1, the peeling fault is a partial failure; for group 2, the peeling fault crosses transversely. Figure 7 shows the meshing stiffness curves corresponding to different peeling depths when the peeling width is 6 mm; the curves shown in the figure are normal meshing stiffness, 1 mm peeling depth meshing stiffness, 1.5 mm peeling depth meshing stiffness, 2 mm peeling depth meshing stiffness, 2.5 mm peeling depth meshing stiffness, and 3 mm peeling depth meshing stiffness from top to bottom. The meshing stiffness with a peeling width of 16 mm is shown in the Figure 8, where the curve is the same as shown in Figure 7. Through comparative study, it can be found that when It can be seen from Figure 6b that when the angle is 17 • , the meshing position leaves a peeling fault and the tooth profile is complete in the process of the subsequent meshing. So, the compressive stiffness and contact stiffness increase, which causes the peeling meshing stiffness to suddenly increase. However, due to the existence of the peeling cavity, the bending stiffness and shearing stiffness are affected; the meshing stiffness is still less than the normal meshing stiffness.

The Meshing Stiffness of Gear with Variable Peeling Parameter
In changing the size of the peeling fault, the effect of different fault size on the meshing stiffness can then be studied. It can be seen from the formula that the fault size Y2 mainly affects the division of fault interval, and has no influence on the meshing stiffness. Then, the fault depth q0 and fault width W2 are thoroughly studied in this paper.

Variable Depth Variable Meshing Stiffness
In order to study the effect of peeling depth on the time-varying meshing stiffness, the width of the peeling was fixed, the peeling depth was changed, and then variable depth of peeling failure meshing stiffness was calculated. Making a study to compare two groups of size: (1) peeling width 6 mm, peeling depth 1 mm, 1.5 mm, 2 mm, 2.5 mm and 3 mm, respectively; (2) peeling width 16 mm, peeling depth 1 mm, 1.5 mm, 2 mm, 2.5 mm, 3 mm, respectively. The tooth width of the gear model is 16 mm. For group 1, the peeling fault is a partial failure; for group 2, the peeling fault crosses transversely. Figure 7 shows the meshing stiffness curves corresponding to different peeling depths when the peeling width is 6 mm; the curves shown in the figure are normal meshing stiffness, 1 mm peeling depth meshing stiffness, 1.5 mm peeling depth meshing stiffness, 2 mm peeling depth meshing stiffness, 2.5 mm peeling depth meshing stiffness, and 3 mm peeling depth meshing stiffness from top to bottom. The meshing stiffness with a peeling width of 16 mm is shown in the Figure 8, where the curve is the same as shown in Figure 7. Through comparative study, it can be found that when the peeling width is 6 mm, as the depth of fault expands, the meshing stiffness decreases, but the decrease is small; when the peeling width is 16 mm, as the depth of fault expands, the meshing stiffness decreases, and the reduction is greater. This also shows that when the peeling width is small, the peeling depth has little effect on the meshing stiffness. As the peeling width increases, the peeling depth has a greater impact on the meshing stiffness amplitude.

Variable Width Variable Meshing Stiffness
In order to study the effect of peeling width on the time-varying meshing stiffness, the peeling depth was fixed, the peeling width was changed, and then variable width of peeling failure meshing stiffness was calculated. Peel depth is 3 mm. Peel width is 2 mm, 4 mm, 6 mm, 8 mm, 10 mm, 12 mm, 14 mm, and 16 mm. Figure 9 shows the meshing stiffness curves corresponding to different peeling widths when the peeling depth is 3 mm; the curves shown in the figure are normal meshing stiffness, 2 mm peeling width meshing stiffness, 4 mm peeling width meshing stiffness increasing to 16 mm peeling width meshing stiffness from top to bottom. From the figure, it can be seen as the peel width increases, the meshing stiffness decreases with the same peel depth. In the project, most of the peeling failures are local peeling, and horizontal peeling rarely occurs. Therefore, comparing Section 3.1 with Section 3.2, it is found that the peeling engagement stiffness is greatly affected by the peeling width, while the

Variable Width Variable Meshing Stiffness
In order to study the effect of peeling width on the time-varying meshing stiffness, the peeling depth was fixed, the peeling width was changed, and then variable width of peeling failure meshing stiffness was calculated. Peel depth is 3 mm. Peel width is 2 mm, 4 mm, 6 mm, 8 mm, 10 mm, 12 mm, 14 mm, and 16 mm. Figure 9 shows the meshing stiffness curves corresponding to different peeling widths when the peeling depth is 3 mm; the curves shown in the figure are normal meshing stiffness, 2 mm peeling width meshing stiffness, 4 mm peeling width meshing stiffness increasing to 16 mm peeling width meshing stiffness from top to bottom. From the figure, it can be seen as the peel width increases, the meshing stiffness decreases with the same peel depth. In the project, most of the peeling failures are local peeling, and horizontal peeling rarely occurs. Therefore, comparing Section 3.1 with Section 3.2, it is found that the peeling engagement stiffness is greatly affected by the peeling width, while the

Variable Width Variable Meshing Stiffness
In order to study the effect of peeling width on the time-varying meshing stiffness, the peeling depth was fixed, the peeling width was changed, and then variable width of peeling failure meshing stiffness was calculated. Peel depth is 3 mm. Peel width is 2 mm, 4 mm, 6 mm, 8 mm, 10 mm, 12 mm, 14 mm, and 16 mm. Figure 9 shows the meshing stiffness curves corresponding to different peeling widths when the peeling depth is 3 mm; the curves shown in the figure are normal meshing stiffness, 2 mm peeling width meshing stiffness, 4 mm peeling width meshing stiffness increasing to 16 mm peeling width meshing stiffness from top to bottom. From the figure, it can be seen as the peel width increases, the meshing stiffness decreases with the same peel depth. In the project, most of the peeling failures are local peeling, and horizontal peeling rarely occurs. Therefore, comparing Section 3.1 with Section 3.2, it is found that the peeling engagement stiffness is greatly affected by the peeling width, while the peeling depth has less effect on the splicing stiffness by the peeling width.

Conclusions
Aiming at the limitation in the existing meshing stiffness algorithm for gear peeling faults, based on the energy method, a gear peeling meshing stiffness algorithm was proposed in this paper. The new algorithm considers the influence of the peeling cavity on the area between the peeling position and the tooth tip. The influence form was analogized as crack fault.
The influence of exfoliation size on mesh stiffness was studied by using the new algorithm. Peeling length has little effect on mesh stiffness. With the increase of peeling depth and width, the meshing stiffness decreases gradually.
For the common local peeling failure, compared with peeling depth, peeling width has more influence on mesh stiffness.

Conclusions
Aiming at the limitation in the existing meshing stiffness algorithm for gear peeling faults, based on the energy method, a gear peeling meshing stiffness algorithm was proposed in this paper. The new algorithm considers the influence of the peeling cavity on the area between the peeling position and the tooth tip. The influence form was analogized as crack fault.
The influence of exfoliation size on mesh stiffness was studied by using the new algorithm. Peeling length has little effect on mesh stiffness. With the increase of peeling depth and width, the meshing stiffness decreases gradually.
For the common local peeling failure, compared with peeling depth, peeling width has more influence on mesh stiffness.