Dynamic Simulation of Cracked Spiral Bevel Gear Pair Considering Assembly Errors

Abstract

The tooth root crack fault is a common fault type of the spiral bevel gear pair (SBGP). Affected by the strong bearing capacity, the early crack fault of the SBGP cannot be found in time. In this study, a finite element (FE) model of the SBGP is established and assembled through the tooth contact analysis. The maximum tooth root stress is analyzed considering the variation of assembly errors. Meanwhile, this study simulates the tooth root crack fault of the bevel pinion with different crack degrees. The initial position of the crack is located where the maximum tooth root stress appears. The time-varying mesh stiffness (TVMS) of the SBGP considering different degrees of the pinion tooth root crack fault is obtained. The TVMS and the non-load transmission error are brought into a hybrid FE dynamic model, and steady responses are solved. Based on this, the sensitivities of various statistical indicators for identifying the tooth root crack fault of SBGP under the influence of assembly errors are verified. This paper can provide the necessary theoretical basis for the analysis and diagnosis of tooth root crack faults in the SBGP transmission system.

1. Introduction

The transmission system of the SBGP has a strong load-bearing capacity, and minor cracks are not easy to be found in time, resulting in broken teeth after the cracks expand to the strength limit. From the example of the broken tooth fault shown in Figure 1, at the tooth root of the active flank, the crack is initiated in the middle position along the tooth width. The propagation path of the crack has different characteristics in different directions. Kim et al. [1] and Park [2] pointed out that in the tooth thickness direction, the crack propagates toward the gear foundation first, and when the crack propagates beyond the tooth centerline, the propagation direction turns to the root of the opposite flank. In the tooth width direction, the crack propagations towards the heel and toe directions have different performance characteristics. For the toe side, the propagation direction of the crack almost does not change along the propagation direction of the early crack. For the heel direction, the propagation direction keeps unchanged at first, then after a certain degree, the crack propagation direction turns towards the top of the tooth. Spievak et al. [3] proved this phenomenon by the principle of fracture mechanics. Based on this, Levicki et al. [4] studied the effect of moving the tooth load on the crack propagation path of bevel gears. Vukelic et al. [5] studied the fractured SBGP of a shipyard crane. They pointed out that gear crack failure may be caused by misalignments. Ural et al. [6] used the fracture mechanics method to accurately simulate the crack propagation direction of the SBGP compared with the actual broken tooth morphology of the OH-58 Kiowa Helicopter. In this paper, the FE method of the SBGP is established, and the influences of the assembly errors on the tooth root stress of the SBGP are analyzed. Meanwhile, based on the crack propagation law proposed by the previous literature, the crack fault at the tooth root of the pinion is simulated and the TVMS curves of the SBGP under different crack degrees considering the assembly errors are calculated.

The complex space surface structure of the SBGP makes it different from the spur gear pair or the helical gear pair, which can obtain the TVMS by the energy method [8,9]. The general research on the TVMS of the SBGP are mainly divided into the loaded tooth contact analysis method [10,11,12,13] and the FE method. The loaded tooth contact analysis method simplifies the gear local contact relationship and is obtained by the linear programming method [14]. The efficiency of this method is much higher than that of the FE method. Through this method, scholars have carried out research on gear pair machining parameter optimization [15,16,17], machining error influence analysis [18], assembly error influence analysis [19], system-level modelling [20] and so on. However, there were few studies about gear tooth fault simulation and tooth bending stress analysis by using this method. Due to the robustness of the contact analysis of the FE method, scholars have conducted many studies on the static and dynamic characteristics of the SBGP based on it. Bibel et al. [21] analyzed the tooth surface contact stress of SBGP by the FE method. Argyris et al. [22] analyzed the contact stress and bending stress of the SBGP by the FE method. Gonzalez-Perez et al. [23] established a FE model of the SBGP by reverse engineering, which well restored the designer’s intention. They simulated the tooth surface contact pattern and contact stress accurately through the FE model. Hou et al. [24] proposed the vector form intrinsic FE method, and based on this method, the static contact stress and bending stress of the SBGP were analyzed. They pointed out that when dealing with dynamic problems and strong nonlinear problems, the efficiency of the proposed model will be greatly improved. Gan et al. [25] studied the temperature distribution of the SBGP under the mixed electrohydrodynamic lubrication state based on the FE method. Fu et al. [26] established a FE model of the SBGP and analyzed the influence of input torque on the tooth bending stress, tooth surface contact stress and transmission error. Hou et al. [27] analyzed the contact stress and tooth bending stress changes of the SBGP considering different torques and different modulus by the FE method. They pointed out that under the influence of edge contact, the contact stress of the tooth surface will surge at the edge. Wang et al. [28] simulated the dynamic contact and impact phenomena of an SBGP based on an FE model. The maximum resonance frequency of the system was analyzed. Guo et al. [29] analyzed the tooth surface contact stress distribution before and after tool optimization by the FE method. This method can be used to eliminate edge contact and reduce the maximum contact stress under heavy load conditions. Wang et al. [30] established a coupled thermo-elastic 3D FE model to improve the thermal behavior of SBGP by adjusting the machine setting parameters. Yang et al. [31] considered the broken tooth fault of the SBGP, the TVMS was calculated by the FE method, and the dynamic response of the system is obtained by bringing it into the pure torsional dynamic model.

Through the above survey, it is found that the FE method is a common method used to analyze the bending stress and the contact stress of the SBGP. Hence, this paper analyzes the tooth root stress considering the variation of assembly errors to locate the initial crack position. Meanwhile, the TVMS curve of the SBGP under the pinion tooth crack fault condition is obtained by introducing a simple space crack into the FE model. Through the dynamic analysis, the system responses under different crack degrees are obtained. Based on this, the sensitivity of some statistical indicators for monitoring the crack fault of SBGP under the influence of assembly errors is verified.

2. FE Model Description

2.1. Tooth Contact Analysis Considering the Assembly Error

In this section, the teeth surfaces of the bevel pinion and the bevel gear are derived by the generation method and the tile method, respectively. The blank parameters can be seen in

Table 1. Blanks parameters.

Parameter	Bevel Pinion	Bevel Gear
Tooth number z1/z2	17	81
Modulus m (mm)/Shaft angle Σ (°)/Mean spiral angle β (°)	5.6/90/35
Direction of rotation	Left-handed	Right-handed
Face width b (mm)	60
Mean cone distance R (mm)	201.741
Pitch angle δ (°)	11.8530	78.1470
Root angle δf (°)	10.9321	76.1944
Face angle δa (°)	13.8056	79.0679
Addendum height ha (mm)	6.8480	2.6720
Dedendum height hf (mm)	3.7250	7.9010

and the machine tooth setting parameters of the bevel pinion and the bevel gear can be seen in

Table 2. Machine tool settings of bevel pinion.

Parameter	Concave	Convex
Cutter point radius r01 (mm)	226.47	230.86
Pressure angle α01 (°)	−18.75	21.25
Root fillet radius ρ01 (mm)	1	1
Machine center to back X1 (mm)	−5.237	8.545
Sliding base XB1 (mm)	15.217	13.183
Blank offset E1 (mm)	3	3.5
Radial distance Sr1 (mm)	197.762	209.634
Machine root angle γm1 (°)	7.7936	10.05
Cradle angle q1 (°)	71.6258	67.3267
Tilt Angle i (°)	2.7343	3
Swivel angle j (°)	286.2802	227.2839
Velocity ratio i1	4.6799	5.2574

and

Table 3. Machine tool settings of bevel gear.

Parameter	Value
Cutter point radius r02 (mm)	229.975(concave)/227.225(convex)
Pressure angle α02 (°)	−19(concave)/21(convex)
Root fillet radius ρ02 (mm)	1.6
Machine center to back X2 (mm)	0
Sliding base XB2 (mm)	0
Blank offset E2 (mm)	0
Radial distance Sr2 (mm)	200.091
Machine root angle γm2 (°)	76.1944
Cradle angle q2 (°)	69.3682
Velocity ratio i2	1.0212

, respectively.

Through the local synthesis method [32,33], the tooth surface points r₁ and r₂, and the tooth surface normal directions n₁ and n₂ of the pinion and the gear are obtained, respectively. The detail of the tooth surface derivation can refer to Appendix A. The following coordinate transformations are used to assemble the two gears into the same mounting coordinate system:

$$\{\begin{array}{l}{r}_{\mathrm{h}i}\left({\psi }_{\mathrm{c}i},s_i,{\theta }_i\right)=M_{\mathrm{h}i}{r}_i\left({\psi }_{\mathrm{c}i},s_i,{\theta }_i\right)\\ n_{\mathrm{h}i}\left({\psi }_{\mathrm{c}i},s_i,{\theta }_i\right)=L_{\mathrm{h}i}{n}_i\left({\psi }_{\mathrm{c}i},s_i,{\theta }_i\right)\end{array},$$

(1)

where i = 1, 2 represent the pinion and the gear, respectively; M_hi is the transformation matrix from the cutter head coordinate system to the blank coordinate system; L_hi is the first three-order sub-matrix of M_hi; ψ_ci, s_i and θ_i are the rotation angle of the cradle, the distance from the point on the theoretical tool tip to the tool profile, and the rotation angle of the cutter head, respectively. M_hi can be expressed as:

$$\{\begin{array}{l}{M}_{\mathrm{h}1}=TRSZ(\Delta A_{\mathrm{p}})·ROTZ(-{\phi }_1)\\ M_{\mathrm{h}2}=TRSY(\Delta E)·ROTY(\Delta \Sigma -\Sigma )·TRSZ(\Delta A_{\mathrm{g}})·ROTZ\left({\phi }_2\right)\end{array}$$

(2)

where ΔE is the offset error; ΔΣ is the shaft angle error; ΔA_p is the pinion axial error; ΔA_g is the gear axial error. The assembly errors are expressed in Figure 2. φ₁ and φ₂ are the initial installation angles of the pinion and the gear, respectively; Σ is the shaft angle.

In the assembly coordinate system O-x_h-y_h-z_h, the two meshing gear teeth should have collinear normal vectors and the same coordinate values at the contact point:

$$\{\begin{array}{l}{r}_{\mathrm{h}1}\left({\psi }_{\mathrm{c}1},s_1,{\theta }_1,{\phi }_1\right)=r_{\mathrm{h}2}\left({\psi }_{\mathrm{c}2},s_2,{\theta }_2,{\phi }_2\right)\\ n_{\mathrm{h}1}\left({\psi }_{\mathrm{c}1},s_1,{\theta }_1,{\phi }_1\right)=n_{\mathrm{h}2}\left({\psi }_{\mathrm{c}2},s_2,{\theta }_2,{\phi }_2\right)\\ \left(\frac{\partial r_1}{\partial s_1}\times \frac{\partial r_1}{\partial {\theta }_1}\right)·\frac{\partial r_1}{\partial {\phi }_1}=0\\ \left(\frac{\partial r_2}{\partial s_2}\times \frac{\partial r_2}{\partial {\theta }_2}\right)·\frac{\partial r_2}{\partial {\phi }_2}=0\end{array}$$

(3)

There are seven independent equations in the above equation system, including eight unknown parameters. Given the rotation angle of the pinion ψ₁, other variables can be determined. Then, the contact trajectory and the non-load transmission error (NLTE) of the SBGP in the rotation range of ψ₁ = (−π/z₁, π/z₁) are obtained. The transmission error between the two gears can be expressed as:

$$\Delta \varphi ={\varphi }_2-{\varphi }_{20}-\left({\varphi }_1-{\varphi }_{10}\right)\frac{z_1}{z_2}$$

(4)

where ϕ₁₀ and ϕ₂₀ are the initial rotation angles of the pinion and the gear, respectively; ϕ₁ and ϕ₂ are the real-time rotation angles of the pinion and the gear, respectively; z₁ and z₂ are the numbers of teeth of the pinion and the gear, respectively.

It is known that for the SBGP, the NLTE is one of the main sources of vibration and noise. The smaller the NLTE is, the smoother the gear system operates. Based on the parameters of the SBGP listed in Ref. [34], the NLTEs and contact trajectories of the SBGP under different assembly error states are illustrated in Figure 3. Influenced by assembly errors, both the NLTE values and the contact trajectories of the SBGP have changed. For the parameters of the SBGP in this study, the larger the value of ΔE, ΔΣ and ΔA_g become, the larger the value of NLTE becomes, and the smaller the value of ΔA_p becomes, the larger the value of NLTE becomes. Therefore, in the process of assembling the SBGP, it is necessary to control these errors to obtain a smaller value of ΔE, ΔΣ and ΔA_g, and a larger value of ΔA_p to ensure that the NLTE values are as small as possible.

2.2. FE Modelling of the SBGP

The FE model of the SBGP is illustrated in Figure 4.

Since the FE model established in this paper is a five-tooth model and the actual gears foundation structures are not fully established (see Figure 4), a master node is used to constrain the nodes on the inner hub and the rigid surfaces of both sides of the pinion. The nodes on the bottom surface and the rigid surfaces of the gear are fully constrained. Five contact pairs are established among the mating tooth surfaces by the Conta 174 and Target 170 element pairs. A mesh cycle is divided into 20 positions. At each position, the rotation of the SBGP must meet the requirement of the transmission ratio; meanwhile, the pinion must rotate with an angle to close the NLTE. The nonlinear contact iteration is performed to complete the static analysis of the gear pair. After that, the equivalent static rotation angle θ_p1 of the master node along the rotation direction is obtained. The mesh stiffness value of the SBGP can be obtained by the following equation:

$$k=\frac{T}{{\theta }_{\mathrm{p}1}{\mathrm{r}}_{\mathrm{p}}^2}$$

(5)

where T is the driving torque applied on the pinion; r_p is the pitch circle radius of the SBGP. It is worth noting that the TVMS obtained in this paper is entirely caused by the contact deformation of the SBGP, so the changing of NLTE should be considered additionally in the subsequent dynamic analysis.

The TVMS of the SBGP in the healthy state can be seen in Figure 5.

2.3. Tooth Root Crack Fault Simulation

According to the results of Ref. [3], it can be seen that on the active tooth side, the crack originated at the midpoint along the tooth width of the tooth root. The crack propagation can be explained in two directions. In the tooth thickness direction, the crack expands to the foundation first, and after reaching the middle of the tooth, the expanding direction turns towards the root of the opposite flank. In the tooth width direction, the early cracks propagate parallel to the tooth width direction (see the shaded part in Figure 6). In this paper, the early tooth root crack fault of the pinion is simulated through the node replacement method (see Figure 7). Through the tooth root stress analysis, the position where the maximum tooth root stress of the pinion occurs is obtained. The crack initial position is usually happening where the maximum tooth root stress appears due to the lack of strength. The distance from the toe end to the crack initial position is set as λ. According to the simplified fault morphology, taking the initial position as the intermediate position, on the tooth root surface, the width of the crack along the tooth width direction is δ. The crack depth is specified as χ. The crack width gradually narrows with depth.

The flow chart of the FE modelling process is shown in Figure 8.

3. Results and Discussion

3.1. Influence of the Assembly Errors on the Tooth Root Stress Distribution

The direct cause of tooth root crack fault is that the root stress exceeds the material bearing limit. Therefore, the position where the crack initiates should be where the root stress is the largest. Affected by the assembly errors, the maximum tooth root stress value and its position are volatile. In order to explore the influences of assembly errors on the maximum root stress value and its position, the following research is carried out.

After the static analysis, the von Mises stress cloud contour of the pinion in the error-free condition is shown in Figure 9. Based on the mesh layer where the maximum von Mises stress of the tooth root is located (the red line in the figure), all nodes on the red line are extracted and the relative von Mises stress is obtained. After three meshing cycles (a single tooth goes through the entire cycle of meshing in and out), the contour plots of the stress variation of the selected nodes under different assembly error conditions are analyzed below.

3.1.1. Offset Error

As shown in Figure 10, with the increase in ΔE, the maximum von Mises stress of the tooth root of the pinion increases gradually. When ΔE raises in the positive direction, the maximum stress value increases faster, which indicates that to avoid the excessive maximum stress value of the tooth root of the pinion, it should avoid the too large positive value of ΔE. The position of the maximum root stress will gradually change towards toe with the increase in ΔE because the contact trajectory will gradually move towards toe with the increase in ΔE. In general, to reduce the maximum value of the root stress of the pinion, the variation of ΔE should not be too large.

3.1.2. Shaft Angle Error

The influence of ΔΣ on the maximum root stress value and its position is shown in Figure 11. It can be seen from the figure that with the gradually increase in ΔΣ, the maximum root stress shows a gradually increasing trend. However, in the range of ΔΣ varying from −0.1° to 0.1°, the position variation of the maximum root stress is not obvious, which indicates that the position of the maximum root stress is not sensitive to the changing of ΔΣ.

3.1.3. Pinion Axial Error

The influence of ΔA_p on the maximum root stress value and its position is illustrated in Figure 12. It can be seen from the figure that the smaller the value of ΔA_p, the greater the maximum root stress. However, no matter how much ΔA_p changes, it does not affect the position where the maximum root stress occurs.

3.1.4. Gear Axial Error

Figure 13 illustrates the influence of ΔA_g on the maximum root stress value of the pinion and its position along the tooth width. It can be seen from the figure that with the increase in ΔA_g, the position of the maximum tooth root stress of the pinion moves towards the toe direction. However, within the range of ΔAg = −0.1 mm to 0.1 mm, the maximum tooth root stress value only increased by 6.5 MPa. This indicates that the maximum root stress of the pinion is insensitive to the change in ΔA_g.

In general, among all the assembly errors, ΔE has the greatest influence on the position of the maximum root stress of the pinion, whereas ΔA_g has the least. Within the scope of ensuring the reasonable root stress of the mating gear, the changes of ΔE, ΔΣ and ΔA_p should be especially controlled. The smaller the value of ΔE and ΔΣ are, the better it is to obtain a small root stress value. The larger the value of ΔA_p becomes, the better it is to obtain a small root stress value. No matter how the assembly error changes in this study, the maximum root stress always appears around the middle of the tooth width and deviates from the toe side. This shows that the root crack fault of the pinion is most likely to occur around this zone.

3.2. Crack Fault Analysis of the SBGP

3.2.1. TVMS Due to the Crack Fault

The early cracks propagated nearly parallel to the tooth width direction. In this study, the tooth root crack of the SBGP is divided into four working cases. The crack cross-section morphology is shown in Figure 14. Through the above results of the root stress distribution in the healthy state, it is assumed that the initial crack of the tooth root occurs at the position of the maximum stress (λ = 29.2 mm). Case 1 represents the initial crack. Case 2 and Case 3 represent the further aggravation of the crack fault, in which the crack morphology increases in both length δ and depth χ. Finally, Case 4 indicates that the crack has crossed the centerline of the tooth and has begun to turn its direction and propagate towards the root of the opposite side. The location and the morphology parameters of the four cases are listed in

Table 4. Crack location and morphology parameters.

	Case 1	Case 2	Case 3	Case 4
λ (mm)	29.2	29.2	29.2	29.2
δ (mm)	4.43	8.46	12.48	16.51
χ (mm)	1.78	4.45	6.23	8.02

and Figure 14.

Based on the method described in Section 2, Figure 15a shows the TVMS curves of the SBGP under different degrees of crack faults. It can be seen from the figure that when a tooth root crack fault occurs, two meshing cycles of the TVMS curves are affected. Figure 15b illustrates the initial gaps of all five tooth pairs over two meshing cycles. It is worth noting that the root crack fault is artificially implanted at the tooth root of the pinion of tooth pair 3. Here, the two meshing cycles are divided into three parts according to the change of the tooth pair with the smallest initial gap. The first part is when tooth pair 2 has the smallest absolute initial gap value. During this period, tooth pair 1 exits the meshing state, the absolute value of the initial gaps of tooth pair 3 decreases gradually, and the absolute value of the initial gaps of tooth pair 2 increases gradually. For tooth pair 3, the crack is not directly below the contact ellipse currently, so the TVMS is less affected by the crack fault. When the absolute value of the initial gaps of tooth pair 3 among all the tooth pairs is the smallest, tooth pair 2 gradually exits the meshing state, and tooth pair 4 gradually enters the meshing state. At this time, the contact force is mostly endured by tooth pair 3, and the contact ellipse is above the root crack. Therefore, the TVMS in this period is mostly affected by the crack fault. The stiffness value of Case 4 in this region can reach a maximum relative difference of 5.45% compared with the healthy condition. When the minimum initial gap exists on the tooth pair 4, the tooth pair 3 is about to leave or has already left the meshing state, and the tooth pair 5 is about to enter or has entered the meshing state. During this period, the contact force endured by tooth pair 3 decreases gradually, and the contact ellipse leaves the tooth surface above the crack gradually. Therefore, the TVMS values of crack fault condition in this region tend to the curves of the healthy state.

3.2.2. Dynamic Simulation

A schematic of the SBGP and shafts system is established (see Figure 16). In this system, the driving torque and the slave torque are applied on the drive node and the slave node, respectively. The dimensions of the input shaft and the intermediate shaft, the supporting stiffnesses and damping coefficients of the gear, the supporting bearings B1, B2, B3, and B4, and the mass of the gear are referenced in Ref. [34].

The undamped equations of motions of the SBGP are as follows:

$$\{\begin{array}{l}{m}_{\mathrm{b}1}{\ddot{x}}_{\mathrm{b}1}+k_{\mathrm{b}1x}{x}_{\mathrm{b}1}+k_{\mathrm{b}1\mathrm{b}2}({\delta }_{\mathrm{b}1\mathrm{b}2}^{\mathrm{T}}{q}_{\mathrm{b}1\mathrm{b}2}-NLTE){\delta }_{\mathrm{b}1\mathrm{b}2}^1=0\hfill \\ m_{\mathrm{b}1}{\ddot{y}}_{\mathrm{b}1}+k_{\mathrm{b}1y}{y}_{\mathrm{b}1}+k_{\mathrm{b}1\mathrm{b}2}({\delta }_{\mathrm{b}1\mathrm{b}2}^{\mathrm{T}}{q}_{\mathrm{b}1\mathrm{b}2}-NLTE){\delta }_{\mathrm{b}1\mathrm{b}2}^2=0\hfill \\ m_{\mathrm{b}1}{\ddot{z}}_{\mathrm{b}1}+k_{\mathrm{b}1z}{z}_{\mathrm{b}1}+k_{\mathrm{b}1\mathrm{b}2}({\delta }_{\mathrm{b}1\mathrm{b}2}^{\mathrm{T}}{q}_{\mathrm{b}1\mathrm{b}2}-NLTE){\delta }_{\mathrm{b}1\mathrm{b}2}^3=0\hfill \\ I_{\mathrm{b}1x}{\ddot{\theta }}_{\mathrm{b}1x}+k_{\mathrm{b}1\theta x}{\theta }_{\mathrm{b}1x}+k_{\mathrm{b}1\mathrm{b}2}({\delta }_{\mathrm{b}1\mathrm{b}2}^{\mathrm{T}}{q}_{\mathrm{b}1\mathrm{b}2}-NLTE){\delta }_{\mathrm{b}1\mathrm{b}2}^4=T_{\mathrm{b}1}\hfill \\ I_{\mathrm{b}1y}{\ddot{\theta }}_{\mathrm{b}1y}+k_{\mathrm{b}1\theta y}{\theta }_{\mathrm{b}1y}+k_{\mathrm{b}1\mathrm{b}2}({\delta }_{\mathrm{b}1\mathrm{b}2}^{\mathrm{T}}{q}_{\mathrm{b}1\mathrm{b}2}-NLTE){\delta }_{\mathrm{b}1\mathrm{b}2}^5=0\hfill \\ I_{\mathrm{b}1z}{\ddot{\theta }}_{\mathrm{b}1z}+k_{\mathrm{b}1\theta z}{\theta }_{\mathrm{b}1z}+k_{\mathrm{b}1\mathrm{b}2}({\delta }_{\mathrm{b}1\mathrm{b}2}^{\mathrm{T}}{q}_{\mathrm{b}1\mathrm{b}2}-NLTE){\delta }_{\mathrm{b}1\mathrm{b}2}^6=0\hfill \end{array},$$

(6)

$$\{\begin{array}{l}{m}_{\mathrm{b}2}{\ddot{x}}_{\mathrm{b}2}+k_{\mathrm{b}2x}{x}_{\mathrm{b}2}+k_{\mathrm{b}1\mathrm{b}2}({\delta }_{\mathrm{b}1\mathrm{b}2}^{\mathrm{T}}{q}_{\mathrm{b}1\mathrm{b}2}-NLTE){\delta }_{\mathrm{b}1\mathrm{b}2}^7=0\hfill \\ m_{\mathrm{b}2}{\ddot{y}}_{\mathrm{b}2}+k_{\mathrm{b}2y}{y}_{\mathrm{b}2}+k_{\mathrm{b}1\mathrm{b}2}({\delta }_{\mathrm{b}1\mathrm{b}2}^{\mathrm{T}}{q}_{\mathrm{b}1\mathrm{b}2}-NLTE){\delta }_{\mathrm{b}1\mathrm{b}2}^8=0\hfill \\ m_{\mathrm{b}2}{\ddot{z}}_{\mathrm{b}2}+k_{\mathrm{b}2z}{z}_{\mathrm{b}2}+k_{\mathrm{b}1\mathrm{b}2}({\delta }_{\mathrm{b}1\mathrm{b}2}^{\mathrm{T}}{q}_{\mathrm{b}1\mathrm{b}2}-NLTE){\delta }_{\mathrm{b}1\mathrm{b}2}^9=0\hfill \\ I_{\mathrm{b}2x}{\ddot{\theta }}_{\mathrm{b}2x}+k_{\mathrm{b}2\theta x}{\theta }_{\mathrm{b}2x}+k_{\mathrm{b}1\mathrm{b}2}({\delta }_{\mathrm{b}1\mathrm{b}2}^{\mathrm{T}}{q}_{\mathrm{b}1\mathrm{b}2}-NLTE){\delta }_{\mathrm{b}1\mathrm{b}2}^{10}=0\hfill \\ I_{\mathrm{b}2y}{\ddot{\theta }}_{\mathrm{b}2y}+k_{\mathrm{b}2\theta y}{\theta }_{\mathrm{b}2y}+k_{\mathrm{b}1\mathrm{b}2}({\delta }_{\mathrm{b}1\mathrm{b}2}^{\mathrm{T}}{q}_{\mathrm{b}1\mathrm{b}2}-NLTE){\delta }_{\mathrm{b}1\mathrm{b}2}^{11}=0\hfill \\ I_{\mathrm{b}2z}{\ddot{\theta }}_{\mathrm{b}2z}+k_{\mathrm{b}2\theta z}{\theta }_{\mathrm{b}2z}+k_{\mathrm{b}1\mathrm{b}2}({\delta }_{\mathrm{b}1\mathrm{b}2}^{\mathrm{T}}{q}_{\mathrm{b}1\mathrm{b}2}-NLTE){\delta }_{\mathrm{b}1\mathrm{b}2}^{12}=T_{\mathrm{b}2}\hfill \end{array},$$

(7)

where m_bi, I_bix, I_biy and I_biz are the mass and moments of inertia of the gear i (i = 1,2 represents the bevel pinion and bevel gear, respectively). k_bix, k_biy, k_biz, k_biθx, k_biθy and k_biθz are the supporting stiffness of gear i in six directions, respectively. k_b1b2 is the TVMS of the SBGP and NLTE is the NLTE of the SBGP accordingly. Considering the NLTE in the system means that only when the dynamic projected displacement of the SBGP exceeds the value of NLTE, the contact of the SBGP can occur. The derivation of the projection displacement vector of the SBGP δ_b1b2 can be seen in Ref. [34]. The superscript k of δ^k_b1b2 denotes the k-th element of vector δ_b1b2. q_b1b2 is the displacement vector consisting of 12 degrees of freedom of the two gears. Due to the introduction of the shaft components, the driving torque T_b1 originally added to the rotation direction of the pinion is transferred to the drive node of the input shaft. At the same time, the slave torque T_b2 originally added in the rotation direction of the gear is transferred to the slave node of the intermediate shaft. The schematic of the grouping of the system stiffness matrix and the damping matrix is illustrated in Figure 17.

The damping matrices of the input shaft-bevel pinion structure (with footnote “1”) and the damping matrices of the intermediate shaft-bevel gear structure (with footnote “2”) are described as:

$$C_i={\alpha }_i·M_i+{\beta }_i·K_i, (i=1, 2)$$

(8)

where α_i and β_i are the damping coefficients of the structure i; M_i and K_i are the mass matrix and stiffness matrix of the structure i, which are solved by the Timoshenko beam theory. The damping of the supporting bearings is considered as contact values. The mesh damping of the SBGP is described as:

$$C_{\mathrm{b}1\mathrm{b}2}=2\xi \sqrt{\frac{k_{\mathrm{b}1\mathrm{b}2}}{1/m_{\mathrm{b}1}+1/m_{\mathrm{b}2}}},$$

(9)

where ξ denotes the damping ratio of the gear pair, m_b1 and m_b2 are the mass of the bevel pinion and the bevel gear, respectively.

The system level equation of motions can be constructed as the matrix form:

$$M\ddot{x}+C\dot{x}+Kx=F$$

(10)

In the simulation process, being as the excitation source, k_b1b2 and NLTE are interpolated according to the time series relationship to obtain the mesh stiffness and NLTE values under each Newmark integration step. After solving for a whole sampling time, the dynamic response of the system is obtained.

3.2.3. Response Analysis Due to the Crack Fault of the SBGP

Figure 18 and Figure 19 illustrate the time-domain waveforms and amplitude spectrums of the pinion in the healthy condition and cracked Case 4 condition, respectively. It can be seen from the figures that in the healthy state, the acceleration time-domain waveforms of all meshing cycles are identical, and the components in the amplitude spectrums are only the meshing frequency f_m and its harmonics. Under the crack fault condition, every time the pinion revolts one period T_bp, the faulty tooth participates in meshing for two mesh periods. There are fluctuations at intervals of T_bp in the time-domain waveform of Case 4. The fault causes the mutation waveform of three meshing cycles. In the faulty amplitude spectrum, under the influence of the crack fault, sidebands with the rotation frequency of the pinion f_bp as the interval appear on both sides of the meshing frequency and its harmonics.

3.2.4. Sensitivity Analysis of Statistical Indicators under Healthy Condition

Based on the acceleration signal of bearing B3 in the y-direction, the time-domain statistical indicators are validated for detecting the crack fault and its severity. The statistical indicators [35] can be seen in

Table 5. Time-domain statistical indicators.

Name	Equation	Name	Equation
Average (A)	$\overline{x}=\frac{1}{N}{\displaystyle \sum _{n=1}^N}x(n)$	Crest (C)	$C=\frac{x_{\mathrm{p}}}{x_{\mathrm{rms}}}$
Standard deviation (STD)	${\sigma }_x=\sqrt{\frac{1}{N-1}{\displaystyle \sum _{n=1}^N}{[x(n)-\overline{x}]}^2}$	Impulse (I)	$I=\frac{x_{\mathrm{p}}}{\overline{x}}$
Square mean root (SMR)	$x_{\mathrm{r}}={\left(\frac{1}{N}{\displaystyle \sum _{n=1}^N\sqrt{\left|x\left(n\right)\right|}}\right)}^2$	Clearance (L)	$L=\frac{x_{\mathrm{p}}}{x_{\mathrm{r}}}$
Root mean square (RMS)	$x_{\mathrm{rms}}=\sqrt{\frac{1}{N}{\displaystyle \sum _{n=1}^N}{x}^2(n)}$	Peak-to-peak (PP)	$\mathrm{PP}=\max \left(x(n)\right)-\mathrm{m}\mathrm{in}\left(x(n)\right)$
Peak (P)	$x_{\mathrm{p}}=\max |x(n)|$	Skewness (S)	$S=\frac{{\displaystyle \sum _{\mathrm{n}=1}^N}{[x(n)-\overline{x}]}^3}{(N-1){\sigma }_x^3}$
Waveform (W)	$W=\frac{x_{\mathrm{rms}}}{\overline{x}}$	Kurtosis (K)	$K=\frac{{\displaystyle \sum _{\mathrm{n}=1}^N{[x(n)-\overline{x}]}^4}}{(N-1){\sigma }_x^4}$

.

It can be seen from the variation trends of multiple time-domain statistical indicators (see Figure 20) that when a tooth crack fault occurs, the values of the indicators A, SMR, P, C, I and L have monotonic changing trends relative to the health condition. So, these indicators can be used to monitor the severity of crack degrees.

Figure 21 illustrates the change of the frequency-domain indicators for the acceleration signal of bearing B3 in the y-direction under different crack degrees. The name and the expression of each indicator in the frequency domain can be referred to in

Table 6. Frequency-domain statistical indicators.

Name	Equation	Name	Equation
F12	$F_{12}=\frac{1}{K}{\displaystyle \sum _{k=1}^K}s(k)$	F18	$F_{18}=\sqrt{{\displaystyle \sum _{k=1}^K{f}_k^{2}s(k)}/{\displaystyle \sum _{k=1}^{K}s(k)}}$
F13	$F_{13}=\sqrt{\frac{1}{K-1}{\displaystyle \sum _{k=1}^K}{\left[s(k)-F_{12}\right]}^2}$	F19	$F_{19}=\sqrt{{\displaystyle \sum _{k=1}^K{f}_k^{4}s(k)}/{\displaystyle \sum _{k=1}^K{f}_k^{2}s(k)}}$
F14	$F_{14}={\displaystyle \sum _{k=1}^K{\left[s(k)-F_{12}\right]}^3}/(K-1)F_{13}^3$	F20	$F_{20}={\displaystyle \sum _{k=1}^K{f}_k^2}s(k)/\sqrt{{\displaystyle \sum _{k=1}^{K}s}(k){\displaystyle \sum _{k=1}^K{f}_k^4}s(k)}$
F15	$F_{15}={\displaystyle \sum _{k=1}^K{\left[s(k)-F_{12}\right]}^4}/(K-1)F_{13}^4$	F21	$F_{21}=\frac{F_{17}}{F_{16}}$
F16	$F_{16}={\displaystyle \sum _{k=1}^K{f}_k}s(k)/{\displaystyle \sum _{k=1}^{K}s}(k)$	F22	$F_{22}={\displaystyle \sum _{k=1}^K{\left(f_k-F_{16}\right)}^{3}s(k)}/(K-1)F_{17}^3$
F17	$F_{17}=\sqrt{\frac{1}{K-1}{\displaystyle \sum _{k=1}^K}{\left(f_k-F_{16}\right)}^{2}s(k)}$	F23	$F_{23}={\displaystyle \sum _{k=1}^K{\left(f_k-F_{16}\right)}^{4}s(k)}/(K-1)F_{17}^4$

.

As can be seen from Figure 21, when there is a crack fault at the tooth root of the pinion, the values of the frequency-domain indicators F12, F14, F15, F17, F18, F19, F22 and F23 have monotonic changes compared with the healthy ones, which indicates that the above frequency-domain indicators can clearly distinguish the severity of the crack degrees in the SBGP system. Therefore, these frequency-domain indicators can be used to diagnose root crack faults in the SBGP system.

3.2.5. Crack Fault Detection under the Influence of the Offset Errors

Assembly errors are unavoidable during the SBGP operation. To explore the sensitivity of the upper-mentioned indicators in diagnosing the crack fault of the SBGP under the influence of assembly errors, the following analysis is performed.

Through the analysis of Section 3.1, it is known that the offset error affects the position of the maximum tooth root stress the most. Therefore, the following discussion deals with the effectiveness of the indicators for diagnosing the crack fault under the influence of the offset error. First, the crack severities are still classified as four cases. The crack sizes remain unchanged (i.e., δ and χ follow the values of Case 1–Case 4 in Table 4). Under the influence of the offset error, the maximum root stress position has changed, so the position parameter of the cracks λ has changed (see

Table 7. Crack location considering various offset errors.

	ΔE = −0.1 mm	ΔE = −0.05 mm	ΔE = 0 mm	ΔE = 0.05 mm	ΔE = 0.1 mm
λ (mm)	27.2	28.0	29.2	30.4	30.8

).

The TVMS curves are shown in Figure 22. First, it can be seen from the healthy condition diagram (see Figure 22a) that as ΔE changes from −0.1 mm to 0.1 mm, the TVMS curves of the SBGP show an overall upward trend. Compared with the condition of ΔE = −0.1 mm, the maximum stiffness value increased by 5.09% and the minimum stiffness value increased by 3.47% under the condition of ΔE = 0.1 mm.

The effect of crack degrees on TVMS is analyzed for different offset errors. For ΔE varies from −0.1 mm to 0.1 mm, compared to the health condition, the max/min relative differences under various crack degrees are listed in

Table 8. The max/min relative differences of TVMS under various crack degrees and offset errors.

	ΔE = −0.1 mm	ΔE = −0.05 mm	ΔE = 0 mm	ΔE = 0.05 mm	ΔE = 0.1 mm
	Relative Error Max/Min (%)
Case 1	0.02/1.31	0.03/1.20	0.03/1.02	0.07/0.95	0.09/1.34
Case 2	0.14/2.13	0.18/2.06	0.35/2.06	0.40/2.22	1.23/1.63
Case 3	0.30/3.46	0.31/3.50	0.75/3.75	0.71/4.14	1.79/3.60
Case 4	0.60/5.23	0.65/5.34	1.25/5.89	1.29/6.52	2.53/5.97

. It can be seen from the table that with the increase in ΔE, the max/min relative differences values show an overall upward trend as the crack severity increases, which indicates that a higher offset error value may aggravate the meshing state of the SBGP with crack fault.

The system response is obtained by substituting the above TVMSs and the corresponding time-varying NLTE into the proposed dynamic model. It is already known that six time-domain indicators and eight frequency-domain indicators are sensitive to the tooth root crack fault of SBGP under the assembly error-free condition. They are A, P, SMR, C, I, and L in the time-domain, and F12, F14, F16, F17, F18, F19, F20, F21, F22 and F23 in the frequency-domain. Based on this, we will continue to analyze the sensitivity of these indicators to the SBGP tooth root crack fault under the influence of the offset error. According to the acceleration signal in the y direction of the bearing B3, the sensitivity of these indicators is obtained.

In Figure 23 and Figure 24, under the interference of offset error, the time-domain indicators A, P, C, I and L, and the frequency-domain indicators F12, F18 and F19 still maintain a good judgment threshold for fault information, so these indicators can be used as the indicators for diagnosing crack faults of the SBGP in the presence of offset error.

4. Conclusions

In this paper, the meshing model of the spiral bevel gear pair (SBGP) is established by the finite element (FE) method. The maximum tooth root stress value of the pinion and its position are studied considering the influence of assembly errors. The time-varying mesh stiffness (TVMS) of the SBGP under different degrees of tooth root crack fault of the pinion is derived. Meanwhile, the sensitivity of different statistical indicators to the crack fault of the pinion is verified by the dynamic simulation of the SBGP system. Through the analysis, some conclusions are summarized as follows:

(1)

Through static analysis, the value of the maximum tooth root stress and its position is discussed considering the assembly errors. It is found that the position of the maximum tooth root stress appears in the middle of the tooth width. The value is influenced by the assembly errors. To avoid excessive tooth root stress of the pinion, the changes in errors ΔE, ΔΣ and ΔA_p should be strictly controlled. A smaller ΔE, ΔΣ, and a larger ΔA_p are preferred to ease the pinion tooth root stress.

(2)

The dynamic response of the SBGP with the pinion tooth root crack fault is obtained by introducing the faulty TVMS curve as the excitation. When a crack fault occurs on the pinion, every time the pinion revolves one cycle, the faulty tooth participates in meshing for two meshing periods. There are fluctuations at intervals of the pinion rotation period in the time-domain waveform. Each time the faulty tooth participates in meshing, there are three meshing cycle mutations in the time-domain waveform. In the faulty amplitude spectrum, under the influence of the crack fault, sidebands with the rotation frequency of the pinion as the interval appear on both sides of the meshing frequency and its harmonics.

(3)

Through the analysis of statistical indicators. The sensitive indicators for identifying the root crack of the pinion are obtained. They are the A, P, SMR, C, I, and L in the time-domain, and F12, F14, F16, F17, F18, F19, F20, F21, F22 and F23 in the frequency-domain. These indicators can be used to monitor and diagnose crack faults in the SBGP system under the assembly error free condition. Moreover, under the interference of offset error, the time-domain indicators A, P, C, I and L, and the frequency-domain indicators F12, F18 and F19 still maintain a good judgment threshold for fault information, so these indicators can be used as the indicators for diagnosing crack faults in the presence of offset errors.

In the future, the research will focus on the simulation of the real tooth root crack fault morphology. The loaded tooth contact analysis will be carried out to solve the efficiency issues. Moreover, an experiment will be carried out in the future to validate the fault indicators for diagnosing the tooth root crack fault of SBGP.