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An analytical-finite-element method for calculating mesh stiffness of spur gear pairs with complicated foundation and crack
Engineering Failure Analysis 94 (2018) 339–353
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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
An analytical-finite-element method for calculating mesh stiffness of spur gear pairs with complicated foundation and crack
T
⁎
Qibin Wanga, Kangkang Chenb, Bo Zhaoa, Hui Mab, , Xianguang Konga a b
School of Mechano-Electronic Engineering, Xidian University, Xi'an 710071, China School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Analytical method Finite element method Crack Mesh stiffness Spur gear
Gear weight optimization is mainly through thin-web and hole in gear foundation. In this paper, an analytical-finite-element model for calculating time-varying mesh stiffness of spur gear pairs with complicated foundation (thin-web and holes) and crack is developed. The pinion and gear foundations are simulated as cylinders and modeled by three-dimensional finite element. The tooth pair is equivalent to a spring and mesh stiffness of a tooth pair is obtained by potential energy method. The spring is rigidly coupled with the pinion and gear foundations. Then, three type examples are presented to verify the model. The first type example is three spur gear pairs with solid foundation. The second type example is gear pairs with thin-web and web holes. And the third example is a gear pair with the cracked foundation. The analysis results show that the model proposed in this paper is applicable and efficient for spur gears with complicated foundation and crack. Finally, the time-varying mesh stiffness is also studied to reveal the effects of web widths, web hole radii and crack lengths.
1. Introduction Gear systems are widely used in machine tools, vehicles, ships, aerospace and other fields. For the aerospace industry, gear vibration and weight are main concerns to obtain a great design. For the time-varying mesh stiffness (TVMS), it is an important inner excitation for gear systems [1], which has great effects on vibration and noise. Gear weight optimization is mainly through thin-web and weight reduction holes in gear foundation (see Fig. 1). Therefore, TVMS model of gear pairs with a complicated foundation (thinweb and holes) is important to assess vibration characteristics of the entire system. Finite element (FE) method, analytical method and analytical-FE method have become the mainstream of mesh stiffness calculation methods [2,3]. For analytical method, potential energy method [4–8] is of vital importance. The tooth is assumed as a nonuniform cantilever beam in the potential energy method, in which shear stiffness, bending stiffness, radial compressive stiffness, foundation stiffness and Hertzian contact stiffness are considered in the total mesh stiffness. When a gear has a solid gear foundation, the compliance and stiffness of the gear foundation were studied based on the Muskhelishvili's theory [9]. In the later studies, some new researches related to gear crack [10–15], tooth pitting [16], tooth profile modification [17–19], tooth surface friction [20–22], extended tooth contact [23] (or corner contact [24]) and nonlinear Hertzian contact stiffness [20] were carried out, which make potential energy method more comprehensive. The most popular approach for analyzing TVMS of gear pairs is FE method, which has a great ability to simulate gear with different tooth profile curves and different types of gear foundation. Moreover, it has a high calculation precision. However, FE ⁎
Corresponding author. E-mail address: [email protected] (H. Ma).
https://doi.org/10.1016/j.engfailanal.2018.08.013 Received 2 May 2018; Received in revised form 5 August 2018; Accepted 13 August 2018 Available online 18 August 2018 1350-6307/ © 2018 Elsevier Ltd. All rights reserved.
Engineering Failure Analysis 94 (2018) 339–353
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Fig. 1. Gear pictures with different gear foundations.
method has a heavy computation and need to redevelop the gear model when the gear has different parameters. Wang and Howard [25] developed several finite element models for calculating the torsional mesh stiffness of spur gears. Kiekbusch et al. [26] presented two- and three-dimensional finite element models to calculate the torsional mesh stiffness and the results were used to derive a simple formula for the torsional stiffness of spur gears. Liang et al. [27] developed three different FE models to analyze the TVMS of a spur gear, in which the gear bore surface constraint was studied. Li [28–30] established an FE model of a spur gear pair with solid foundation and some effects on loading capacity, strength and mesh stiffness were also studied. Raghuwanshi and Parey [31] investigated the mesh stiffness of a spur gear with the back-side contact by FE method and the results revealed that the mesh stiffness increased with the back-side contact. Wei et al. [32] researched the effects of flank deviations on load distributions of a helical gear by FE method. In a later reports [33–35], Li studied a spur gear with thin-web by FE method and the results showed that the web had great influences on contact stress and bending stress. What's more, some researches about TVMS of gear pairs with crack have been studied by FE method [36–38]. Some researches have studied TVMS by analytical-FE method. The integral equation method [39–41] is a main analysis method, in which the TVMS was obtained by integrating the contact stiffness and bending stiffness. The contact stiffness and bending stiffness were calculated by FE method. Ma et al. [15,18] and Feng et al. [42] investigated the TVMS of gear pairs by the analytical method, in which the revised foundation stiffness in multi-tooth contact zone was analyzed by FE method. Fernandez del Rincon et al. [43,44] developed an analytical-FE model of a spur gear with tooth profile modification and pitting. Chang et al. [45] developed an analytical-FE model of a helical gear pair. The gear was modeled by FE method and the contact was modeled by analytical method. The influences of web thickness, helix angle and face width on TVMS were also studied. Abbes et al. [46] presented a dynamic model of a gear with thin-web, in which the pinion was modeled by shaft FE and the wheel by three-dimensional FE. The gear mesh was modeled by spring element. The analytical results showed that natural frequency and vibration characteristics were influenced by the thin-web and crack. In this paper, an analytical-FE model is established to study the TVMS of a gear pair with complicated foundation and crack. Cylinders are used to simulate the pinion and gear foundations and are modeled by the three-dimensional FE. The tooth pair is modeled by a spring element and the spring stiffness is equal to the mesh stiffness of a tooth pair which can be calculated by the potential energy method. The spring is coupled with the foundations of the pinion and gear. Then, a time-varying modeling method is presented and the TVMS is analyzed. Finally, the model is verified by three type cases and the effects of the web widths, web hole radii and crack lengths on TVMS are also studied. 2. Theory There are two types of mesh stiffness for a gear pair, rectilinear mesh stiffness and torsional mesh stiffness [25–27]. Rectilinear mesh stiffness is an equivalent mesh stiffness of a gear pair along the action line. Torsional mesh stiffness is defined as the ratio between a torque applied on the pinion (the gear body is fixed) and the corresponding angular displacement of the pinion body. These two types of mesh stiffness are related and can be converted to each other. In this study, the rectilinear mesh stiffness is used and the mesh stiffness represents the rectilinear mesh stiffness. Mesh stiffness of a gear pair includes tooth stiffness and gear foundation stiffness. Mesh stiffness of a tooth pair can be calculated 340
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by the potential energy method. An FE model is established for the complicated gear foundation (thin-web and holes) and crack. The mesh stiffness of the tooth pair is modeled by a spring element and coupled with the foundations of the pinion and gear. Finally, mesh stiffness of a gear pair can be got by the static analysis. 2.1. Mesh stiffness of a tooth pair A cantilever beam with the variable cross-section is mostly used to simulate the tooth. The tooth stiffness consists of bending stiffness, shear stiffness, radial compressive stiffness and Hertzian contact stiffness [47]. The curve of tooth profile starts from the dedendum circle to addendum circle. The tooth profile curve is composed of involute curve and transition curve. So the bending stiffness, shear stiffness and radial compressive stiffness can be calculated as follows:
1 = kb
φt
∫π/2
1 = ks
∫π/2
1 = ka
∫π/2
φt
φt
[cos φ1 (yφ1 − y1) − x φ1 sin φ1 ]2 dy1 dγ + dγ EIy1
1.2cos2 φ1 GAy1
dy1 dγ + dγ
sin2φ1 dy1 dγ + EAy1 dγ
∫τ
∫τ
φ1 1.2cos2 φ1
GAy2
C
φ1
C
∫τ
φ1
C
[cos φ1 (yφ1 − y2 ) − x φ1 sin φ1 ]2 dy2 dτ dτ EIy2
dy2 dτ dτ
(1)
(2)
sin2 φ1 dy2 dτ EAy2 dτ
(3)
where φt represents the operating pressure angle. φ1 is the pressure angle at the contact point. Ay1, Ay2, Iy1, Iy2, xφ1, yφ1, x1, x2, y1, y2, dy dy τc, 1 and 2 are listed in Ref. [47]. B is the tooth width. E and v denote Young's modulus and Poisson's ratio. The Hertzian stiffness kh dγ
dτ
of the tooth pairs can be calculated as follows:
kh =
πEB 4(1‐ν 2)
(4)
Therefore, the total tooth mesh stiffness can be calculated as follows:
kt =
1 1 k b1
+
1 k s1
+
1 ka1
+
1 k b2
+
1 k s2
+
1 ka2
+
1 kh
(5)
where the subscripts 1 and 2 represent the pinion and gear, respectively. 2.2. TVMS calculation method for gear with complicated gear foundation As shown in Fig. 2, TVMS model of a spur gear is developed in ANSYS software. The applied load on the tooth is perpendicular to the tooth face and along the action line. The direction of stiffness is usually same as the direction of the applied load. Therefore, the direction of the spring representing the gear tooth stiffness is along the action line. In this paper, the action line is set to a horizontal state for convenient modeling. The relative displacement of the gear center o2 in the horizontal direction is (rb1 + rb2)tanα relative to
Fig. 2. An analytical-FE model of a spur gear pair. 341
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Fig. 3. Diagram of the connected zone between the tooth and foundation.
that of the pinion center o1. The relative displacement of the gear center o2 in the vertical direction is (rb1 + rb2) relative to that of the pinion center o1. rb1 and rb2 are the base circle radii of the pinion and gear. α denotes the pressure angle. The pinion and gear foundations are simulated as cylinders and developed by Solid185 element. The radii of the cylinders are equal to the radii of the dedendum circles because the curve of the tooth profile starts from the dedendum circle. As shown in Fig. 3, on the dedendum circle, the arc curve of a tooth consists of the tooth connected zone θ1 and the dedendum zone θ2. θ1 and θ2 can be calculated as follows:
θ1 =
(πm + 4ha m tan α + 4cm cos α /(1 − sin α )) mz
(6)
θ2 =
2π − θ1 z
(7)
where m, z, ha and c represents module, tooth number, addendum coefficient and tip clearance coefficient. Then, two nodes are established in middle of the tooth contact line. On the assumption that the displacements of the nodes in the connected zone between the tooth and gear foundation are same, one node is rigidly coupled with the connected zone between the pinion tooth in mesh and pinion foundation by Mpc184 element. The other node is rigidly coupled with the connected zones between the gear tooth in mesh and gear foundation by Mpc184 element. The two nodes are connected by spring element (Combin14 element in ANSYS) and the value of the spring stiffness is set to the mesh stiffness of the tooth pair in this mesh position. Because the boundary conditions (external torque and constraint) cannot be applied directly to the axle hole. Two master nodes are established in the geometric centers of the pinion and gear, respectively. For the pinion, the master node is coupled with the all nodes on the center hole surface and all degrees of freedom except rotation direction of the master node are fully constrained. Also for the gear, the master node is coupled with the all nodes on the center hole surface and all degrees of freedom of the master node are fully constrained. An external torque is equivalent to tangential forces applied to the hole nodes of the pinion. Finally, the angle displacement Δθ of the pinion can be obtained by the static analysis. The relationship between the rectilinear mesh stiffness and the torsional mesh stiffness for gears has been established in Ref. [25]. So the mesh stiffness of a gear pair can be calculated as follows:
k=
k t∗ T = 2 2 rb1 Δθ⋅rb1
(8)
kt∗
where represents the torsional mesh stiffness. T is the applied torque and rb1 denotes the base circle radius of the pinion. Due to contact ratio of a gear pair is usually not an integer, the number of the tooth pairs in mesh changed periodically. On the other hand, the mesh stiffness of a tooth pair changes with the varying mesh position of a tooth. Therefore, the mesh position and the number of tooth pairs in mesh should be considered when calculating TVMS. As shown in Fig. 4, N1N2 denotes the theoretical action line and B1B2 represents the actual action line. C1 and C2 are the contact points in the contact zone. The N1C1 and N1C2 can be calculated as follows:
⎧ N1 C1 = r b1 (φ b + mod(Ω1 t , 2π/ z1 )) ⎨ ⎩ N1 C2 = r b1 (φ b + 2π/ z1 + mod(Ω1 t , 2π/ z1))
(9)
where ‘mod’ function obtains the modulus of Ω1t divided by 2π/z1. z1 represents the tooth number of the pinion. Ω1 is the rotating 342
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Fig. 4. Action line of a gear pair.
speed of the pinion and t is for time. φb can be expressed as follows:
φb =
N1 B1 r b1
(10)
When the N1C1 is located between the N1B1 and N1B2, the point C1 is in mesh. The corresponding node and spring will be developed and the spring stiffness is set to mesh stiffness of the tooth pair in this mesh position, otherwise the node and spring will not be developed. It is also true for the point C2. Then, the mesh stiffness of a gear pair with multi-tooth contact can be obtained by the static analysis. According to the previous theory, the TVMS model of a spur gear pair with complicated foundation can be calculated by the MATLAB and ANSYS softwares. The flowchart of the computational procedure can be listed as follows (see Fig. 5): Step 1: Determine the number of tooth pairs in mesh and the mesh positions of the tooth pairs. Step 2: Calculate the mesh stiffness of tooth pairs in mesh in MATLAB software. Mesh stiffness of a tooth pair consist of bending stiffness, shear stiffness, radial compressive stiffness and contact stiffness. Those stiffness can be calculated by potential energy method. Step 3: Develop an FE model of the gear foundation in ANSYS software. The pinion foundation and gear foundation are connected by spring element and the spring stiffness is set to the mesh stiffness of the tooth pair. Mesh stiffness of a gear pair at a certain position can be obtained according to Eq. (6). At last, the whole process is repeated in the next mesh position until one period is achieved. The TVMS of a gear pair can be obtained.
2.3. Calculation of mesh stiffness of a spur gear with cracked foundation An FE model of a spur gear with crack is established in ANSYS software and the crack is built on the gear foundation (see Fig. 6a). It is noted that the crack path is assumed a straight line at the dedendum circle. The crack propagation angle υ and the crack length q are shown in Fig. 6b. In this paper, the crack propagation angle υ is constant. The analytical-FE model and boundary conditions can refer to the Section 2.2. Based on the potential energy method, the mesh stiffness of a tooth pair can be obtained (see Section 2.1), and its value is applied on the spring element of the proposed model. After solving, the angle displacement Δθ of the pinion can be obtained and the mesh stiffness of a spur gear with cracked foundation can be calculated by the Eq. (8) at a certain meshing position. Finally, the TVMS of a spur gear with cracked foundation can be obtained.
343
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Fig. 5. A flow chart of the computational procedure.
3. Model validation Before any study can be performed with confidence, the model must be validated. Here, the model is verified by three type cases. In case 1, three healthy spur gear pairs with the solid gear foundation are established to validate the correctness of the analytical-FE model. The geometric characteristics of the healthy gear are shown in Fig. 7a. In case 2, two types of the healthy spur gear with the web, web and holes are applied to validate the applicability of the model, respectively. And the geometric characteristics of the gear with the web, web and holes are shown in Fig. 7b and Fig. 7c, respectively. In case 3, a spur gear with cracked foundation is also applied to model validation. The parameters of the spur gear pairs used in the above three type examples are listed in Table 1. 3.1. Case 1: healthy gear 3.1.1. Gear pair 2 for model validation In this part, the gear pair 2 (see Table 1) is applied to verify the proposed method. The proposed model and FE model are shown in Fig. 8a and Fig. 9a, respectively. The static characteristics are analyzed and the comprehensive deformation of two models are plotted in Fig. 8b and Fig. 9b, respectively. It is noted that the proposed model and the FE model have similar deformation distribution. The TVMS obtained from the proposed method and FE method is shown in Fig. 10 and the errors at the moments A and B are listed in Table 2 (moments A and B are the middle time of the double-tooth contact zone and single-tooth contact zone, respectively). It is obvious that the stiffness curves obtained from the two methods are similar and the maximum error at moments A and B is 1.20%, i, e. the proposed method have high accuracy for calculating the mesh stiffness. 3.1.2. Gear pair 1 for model validation Under the parameters of gear pair 1 (see Table 1), the comprehensive deformation results of the proposed and FE models are shown in Fig. 11. According to the Fig. 11, the foundation deformation obtained from two models are similar. In addition, the TVMS obtained from the proposed method and FE method is plotted in Fig. 12 and the calculation errors at the moments C and D are listed in Table 3 (moments C and D are the middle time of the double-tooth contact zone and single-tooth contact zone, respectively). It is noted that the stiffness curves obtained from the two models have similar trends and the maximum error is less than 1.5%. 3.1.3. Gear pair 3 for model validation In this part, the gear pair 3 (see Table 1) is adopted to verify the proposed method and the results obtained from the proposed 344
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Fig. 6. A model of a spur gear with cracked foundation: (a) analytical-FE model, (b) crack diagram.
method are compared with that of the formulaic method and FE method in Ref. [26]. Based on the analytical-FE model, the maximum mesh stiffness in single tooth contact zone and double teeth contact zone is listed in Table 4. Converting torsional mesh stiffness into rectilinear mesh stiffness, the maximum mesh stiffness by the formulaic method in Ref. [26] and FE method is also listed in Table 4. In the single tooth contact zone, the maximum mesh stiffness is about 1.42 × 108 N/m by the proposed method, 1.34 × 108 N/m by the formulaic method in Ref. [26] and 1.41 × 108 N/m by FE method. The relative error of the proposed method is about 5.9% compared with the method in Ref. [26] and that is 0.7% compared with the FE method. In the double teeth contact zone, the maximum mesh stiffness is about 1.93 × 108 N/m by the proposed method, 1.72 × 108 N/m by the formulaic method in Ref. [26] and 1.86 × 108 N/ m by FE method. The relative error of the proposed method is about 12.2% compared with the method in Ref. [26] and that is 3.8% compared with the FE method. The results reveal that proposed method is effective. 3.2. Case 2: healthy gear with web and holes Firstly, the proposed method is applied to the healthy gear with web (the pinion is a solid foundaiton and the gear is a web foundation) and the gear parameters refer to the gear pair 1 (see Table 1). The proposed model and FE model are shown in Fig. 13a and c, and the comprehensive deformation obtained from two models are shown in Fig. 13b and d. It is obvious that the deformation result of the proposed model is very similar to that of the FE model. The TVMS obtained from FE method and the proposed method is shown in Fig. 14. It is obvious that the variation trend of the stiffness curves matches well. Then, the proposed method is applied to the healthy gear with web and holes (see Fig. 15). The gear parameters refer to the gear pair 1 (see Table 1). The models and the comprehensive deformation obtained from two methods are shown in Fig. 15 and the TVMS obtained from the proposed method and FE method is shown in Fig. 16. It is obvious that the deformation result of the proposed model is very similar to that of the FE model. At the same time, the TVMS obtained from the proposed method has similar change law with the results from FE method and the maximum relative error of the proposed method is about 2.03%. 3.3. Case 3: gear with cracked foundation To verify the feasibility of the proposed method for cracked foundation, the spur gear pair 1 is applied and the crack length q is 1.5 mm (see Fig. 6). The TVMS obtained from the FE method and the proposed method is plotted in Fig. 17. It is obvious that change 345
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Fig. 7. Geometric parameters of the healthy gear with different foundations: (a) solid foundation; (b) web foundation; (c) web and holes foundation. Table 1 Parameters of the spur gear pairs. Parameters
Tooth number Normal module (mm) Tooth width (mm) Pressure angle (°) Addendum coefficient Tip clearance coefficient Young's modulus (Pa) Poisson's ratio Hub radius (mm) Web inner radius (mm) Web outer radius (mm) Web hole position (mm) Web hole radius (mm) Number of web holes Web width (mm) Crack angle (°) Crack length (mm) Applied torque (N·m)
Symbol
Gear pair 1
z2
z1 m B α ha c E ν rh ri ro rm rw n b υ q T
Gear pair 2
Gear pair 3
Pinion
Gear
Pinion
Gear
Pinion
52
72
19
31
23
1.75 20 20 1 0.25 2.1 × 1011 0.3 10 – – – – – – – – 200
3.2 38 20 1 0.25 2.1 × 1011 0.3 17.5 18 52 36 8 6 16 30 1.5 –
– – – – – – – – 200
Gear 23 6.0 15 20 1 0.25 2.1 × 1011 0.3 15
– – – – – – – – –
– – – – – – – – 1000
– – – – – – – – –
law of the TVMS obtained from two methods is similar and the maximum error is about 2.39%, i. e., the new method is also effective for the spur gear with cracked foundation. Besides, the analysis results reveal that the cracked foundation still influence the TVMS when the corresponding tooth is no longer in mesh. This is because the deformation of the gear foundation cannot be transferred from one side of the crack to the other side. The flexibility of the gear foundation increases and the mesh stiffness decreases. In the previous three type examples, the time consuming with the proposed procedure is about 4 min on a personal computer with 32G RAM and 8-core CPU for 20 analysis points in one period. However, the time consuming is about 10 h for 20 analysis points using the FE method in one mesh period. The proposed model is 150 times more efficient than the FE model. Above all, the proposed method not only has a high efficiency as the analytical method, but also has ability for the complicated 346
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Fig. 8. Proposed model: (a) schematic of FE model; (b) comprehensive deformation.
Fig. 9. FE model: (a) schematic of FE model; (b) comprehensive deformation.
Fig. 10. TVMS obtained from the proposed method and FE method. Table 2 Calculation error of proposed method at different moments. Method
FE method Proposed method
Moment A
Moment B
TVMS (×108 N/m)
Error (%)
TVMS (×108 N/m)
Error (%)
7.52 7.43
– 1.20
5.12 5.07
– 0.98
Note: the error denotes the relative error of the stiffness obtained from the proposed method relative to that obtained from the FE method. 347
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Fig. 11. Comprehensive deformation: (a) proposed model; (b) FE model.
Fig. 12. TVMS obtained from the proposed method and FE method. Table 3 Calculation errors of proposed method at moments C & D. Method
FE method Proposed method
Moment C
Moment D
TVMS (×108 N/m)
Error (%)
TVMS (×108 N/m)
Error (%)
2.13 2.16
– 1.41
1.70 1.69
– 0.59
Table 4 Mesh stiffness of the gear pair 3 under different methods.
FE method
Relative errors (Compared with the method in Ref. [26])
Relative errors (Compared with the FE method)
1.41 × 108 1.86 × 108
5.9% 12.2%
0.7% 3.8%
Mesh stiffness (N/m) Proposed method Single tooth contact zone Double teeth contact zone
8
1.42 × 10 1.93 × 108
Method in Ref. [26] 8
1.34 × 10 1.72 × 108
gear foundation as the FE method. It can also solve the gear foundation sharing in multi-tooth mesh zone [15,18].
4. Effects of the gear foundation parameters on gear mesh stiffness Different gear parameters like web width, web hole radius and crack length cause the variation of mesh stiffness. In this section, their influences on gear mesh stiffness are quantitatively analyzed.
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Fig. 13. Schematics of a spur gear with web: (a) proposed model; (b) comprehensive deformation of the proposed model; (c) FE model; (d) comprehensive deformation of the FE model.
Fig. 14. TVMS of gear with web obtained from the proposed method and FE method.
Fig. 15. Schematics of a spur gear with web and holes: (a) proposed model; (b) comprehensive deformation of the proposed model; (c) FE model; (d) comprehensive deformation of the FE model.
4.1. Effects of the gear web width In this case, the gear pair 1 is used and the parameters are listed in Table 1, in which the web hole radius rw = 8 mm, crack length q = 0 mm. Keeping all the other parameters unchanged, the web width b increases gradually from 4 to 16 mm with interval 4 mm. The TVMS is analyzed by the proposed method and the results are shown in Fig. 18. The results reveal that the mesh stiffness of single-tooth contact zone and double-tooth contact zone has a bigger reduction with the decreasing the web widths. The maximum mesh stiffness decreased by 39.47% when the web width b decreases from 16 mm to 4 mm. The reason for this phenomenon is that the stiffness of gear foundation decrease with the decreasing the web widths. 349
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Fig. 16. TVMS of gear with web and holes obtained from the proposed method and FE method.
Fig. 17. TVMS of a spur gear with crack obtained from the proposed method and FE method.
Fig. 18. TVMS of spur gear under different gear web widths.
4.2. Effects of the web hole radius Similarly, web hole radius rw is changing from 0 mm to 12 mm incrementally while web width b = 16 mm, crack length q = 0 mm. The TVMS of the gear pair with different web hole radii is shown in Fig. 19. The results demonstrate that the mesh stiffness is greatly influenced by the size of the web hole radius. For the case rw = 0 mm (i.e., web foundation), the maximum value of mesh stiffness is about 2.06 × 108 N/m and the TVMS varies periodically with the meshing tooth. When the web hole radius rw = 12 mm, the maximum value is about 1.57 × 108 N/m and the maximum mesh stiffness decreased by 31.2% compared with that rw = 0 mm. In addition, the results show that the TVMS curve gradually exhibits periodicity with the web hole and the periodicity becomes more and more obvious with the increasing hole radii. This is because web hole weakens the stiffness of gear foundation and the web hole has a great influence on mesh stiffness of a gear pair.
4.3. Effects of the crack length To expose the effects of the crack length, the spur gear pair 1 is used and the TVMS is plotted in Fig. 20 while the crack length q 350
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Fig. 19. TVMS of spur gear under different web hole radii.
Fig. 20. TVMS of spur gear under different crack lengths.
increases from 1.5 to 3 mm with interval of 0.5 mm. In Fig. 20, a reduction phenomenon in mesh stiffness of single-tooth contact zone and double-tooth contact zone can be observed when the cracked tooth comes into contact. The maximum mesh stiffness in the crack zone under the crack length q = 1.5 mm decreased by 2.09% compared with that in the healthy zone. When the crack length q = 3 mm, the maximum mesh stiffness in the crack zone decreased by 3.83% compared with that in healthy zone. The reason for this phenomenon is that the foundation stiffness decrease with the increasing crack lengths. Above all, the maximum mesh stiffness of the entire gear system decreases with the decreasing web widths, and it also decreases with the increasing web hole radii and crack lengths. The mesh stiffness in single-tooth contact zone and multi-tooth contact zone change with the same tendency. For the spur gear with complicated foundation, the web width has a greatest influence on the TVMS. The TVMS exhibits periodicity for the gear with the web holes. The shape of the TVMS curve is changed when the tooth with crack comes into mesh. 5. Conclusions (1) An analytical-FE model of TVMS for a spur gear with complicated foundation and crack is developed, in which the pinion and gear foundations are simulated as cylinders and modeled by three-dimensional FE. The tooth pair is equivalent to a spring and mesh stiffness of the tooth pair is calculated by potential energy method. The spring is rigidly coupled with the pinion and gear foundations. The model proposed in this paper is verified using three different type examples. The results show that the maximum relative error of the proposed method under different foundations is about 2.39% relative to the FE method and the proposed model is 150 times more efficient than the FE model. The complicated gear foundation has a great influence on TVMS of a spur gear system. Besides, the proposed method can solve the problem of the gear foundation sharing in multi-tooth mesh zone. (2) Effects of the web width, web hole radius and crack length on TVMS of a spur gear pair are also studied. The results reveal that the maximum value of the mesh stiffness of the spur gear decreases with the decreasing web width, the increasing web hole radius and crack length. The maximum mesh stiffness decreased by 39.47% when the web width b decreases from 16 mm to 4 mm. The maximum mesh stiffness decreased by 31.2% when the web hole radius rw increases from 0 mm to 12 mm and TVMS curve gradually exhibits periodicity as the web hole radius increasing. There is a distinct reduction of the stiffness when a cracked tooth is going to mesh.
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Acknowledgement This work was supported by the National Natural Science Foundation of China [grant numbers 51605361, 51505357]; the Fundamental Research Funds for the Central Universities of China [grant numbers N170308028, N160313004]; and the Natural Science Basic Research Plan in Shaanxi Province of China [grant number 2018JQ5034]. Conflicts of interest The authors declare that they have no conflict of interest. References [1] X. Gu, P. Velex, P. Sainsot, J. Bruyere, Analytical investigations on the mesh stiffness function of solid spur and helical gears, J. Mech. Des. 137 (6) (2015) 063301. [2] H. Ma, J. Zeng, R. Feng, X. Pang, Q. Wang, B. Wen, Review on dynamics of cracked gear systems, Eng. Fail. Anal. 55 (2015) 224–245. [3] X. Liang, M.J. Zuo, Z. Feng, Dynamic modeling of gearbox faults: a review, Mech. Syst. Signal Process. 98 (2018) 852–876. [4] Z. Chen, Y. Shao, Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth, Eng. Fail. 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Shao, Dynamic features of a planetary gear system with tooth crack under different sizes and inclination angles, J. Vib. Acoust. 135 (3) (2013) 031004. [11] Z. Chen, J. Zhang, W. Zhai, Y. Wang, J. Liu, Improved analytical methods for calculation of gear tooth fillet-foundation stiffness with tooth root crack, Eng. Fail. Anal. 82 (2017) 72–81. [12] Z. Chen, W. Zhai, Y. Shao, K. Wang, G. Sun, Analytical model for mesh stiffness calculation of spur gear pair with non-uniformly distributed tooth root crack, Eng. Fail. Anal. 66 (2016) 502–514. [13] H. Ma, R. Feng, X. Pang, R. Song, B. Wen, Effects of tooth crack on vibration responses of a profile shifted gear rotor system, J. Mech. Sci. Technol. 29 (10) (2015) 4093–4104. [14] H. Ma, X. Pang, J. Zeng, Q. Wang, B. Wen, Effects of gear crack propagation paths on vibration responses of the perforated gear system, Mech. Syst. Signal Process. 62-63 (2015) 113–128. [15] H. Ma, X. Pang, R. Feng, J. Zeng, B. Wen, Improved time-varying mesh stiffness model of cracked spur gears, Eng. Fail. Anal. 55 (2015) 271–287. [16] Y. Lei, Z. Liu, D. Wang, X. Yang, H. Liu, J. Lin, A probability distribution model of tooth pits for evaluating time-varying mesh stiffness of pitting gears, Mech. Syst. Signal Process. 106 (2018) 355–366. [17] Z. Chen, Y. Shao, Mesh stiffness calculation of a spur gear pair with tooth profile modification and tooth root crack, Mech. Mach. Theory 62 (2013) 63–74. [18] H. Ma, J. Zeng, R. Feng, X. Pang, B. Wen, An improved analytical method for mesh stiffness calculation of spur gears with tip relief, Mech. Mach. Theory 98 (2016) 64–80. [19] Q. Wang, Y. Zhang, A model for analyzing stiffness and stress in a helical gear pair with tooth profile errors, J. Vib. Control. 23 (2) (2017) 272–289. [20] H. Ma, M. Feng, Z. Li, R. Feng, B. Wen, Time-varying mesh characteristics of a spur gear pair considering the tip-fillet and friction, Meccanica 52 (7) (2017) 1695–1709. [21] A. Saxena, A. Parey, M. Chouksey, Effect of shaft misalignment and friction force on time varying mesh stiffness of spur gear pair, Eng. Fail. Anal. 49 (2015) 79–91. [22] L. Han, L. Xu, H. Qi, Influences of friction and mesh misalignment on time-varying mesh stiffness of helical gears, J. Mech. Sci. Technol. 31 (7) (2017) 3121–3130. [23] H. Ma, X. Pang, R. Feng, R. Song, B. Wen, Fault features analysis of cracked gear considering the effects of the extended tooth contact, Eng. Fail. Anal. 48 (2015) 105–120. [24] W. Yu, C.K. Mechefske, Analytical modeling of spur gear corner contact effects, Mech. Mach. Theory 96 (2016) 146–164. [25] J. Wang, I. Howard, The torsional stiffness of involute spur gears, Proceedings of the Institution of Mechanical Engineers Part C-Journal of Mechanical Engineering Science 218 (2004) 131–142. [26] T. Kiekbusch, D. Sappok, B. Sauer, I. Howard, Calculation of the combined torsional mesh stiffness of spur gears with two- and three-dimensional parametrical FE models, Stroj Vestn-J Mech E 57 (2011) 810–818. [27] X. Liang, H. Zhang, M.J. Zuo, Y. Qin, Three new models for evaluation of standard involute spur gear mesh stiffness, Mech. Syst. Signal Process. 101 (2018) 424–434. [28] S. Li, Finite element analyses for contact strength and bending strength of a pair of spur gears with machining errors, assembly errors and tooth modifications, Mech. Mach. Theory 42 (1) (2007) 88–114. [29] S. Li, Effects of machining errors, assembly errors and tooth modifications on loading capacity, load-sharing ratio and transmission error of a pair of spur gears, Mech. Mach. Theory 42 (6) (2007) 698–726. [30] S. Li, Effects of misalignment error, tooth modifications and transmitted torque on tooth engagements of a pair of spur gears, Mech. Mach. Theory 83 (2015) 125–136. [31] N.K. Raghuwanshi, A. Parey, Effect of back-side contact on mesh stiffness of spur gear pair by finite element method, in: N.K. Gupta, M.A. Iqbal (Eds.), Plasticity and Impact Mechanics, 2017, pp. 1538–1543. [32] J. Wei, W. Sun, L. Wang, Effects of flank deviation on load distributions for helical gear, J. Mech. Sci. Technol. 25 (7) (2011) 1781–1789. [33] S. Li, Effects of centrifugal load on tooth contact stresses and bending stresses of thin-rimmed spur gears with inclined webs, Mech. Mach. Theory 59 (2013) 34–47. [34] S. Li, Contact stress and root stress analyses of thin-rimmed spur gears with inclined webs, J. Mech. Des. 134 (5) (2012) 051001. [35] S. Li, Centrifugal load and its effects on bending strength and contact strength of a high speed thin-walled spur gear with offset web, Mech. Mach. Theory 43 (2) (2008) 217–239. [36] Y. Pandya, A. Parey, Crack behavior in a high contact ratio spur gear tooth and its effect on mesh stiffness, Eng. Fail. Anal. 34 (2013) 69–78. [37] Y. Pandya, A. Parey, Simulation of crack propagation in spur gear tooth for different gear parameter and its influence on mesh stiffness, Eng. Fail. Anal. 30 (2013) 124–137. [38] S.X. Jia, I. Howard, Comparison of localised spalling and crack damage from dynamic modelling of spur gear vibrations, Mech. Syst. Signal Process. 20 (2006)
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Engineering Failure Analysis journal homepage: www.elsevier.com/locate/engfailanal
An analytical-finite-element method for calculating mesh stiffness of spur gear pairs with complicated foundation and crack
T
⁎
Qibin Wanga, Kangkang Chenb, Bo Zhaoa, Hui Mab, , Xianguang Konga a b
School of Mechano-Electronic Engineering, Xidian University, Xi'an 710071, China School of Mechanical Engineering and Automation, Northeastern University, Shenyang 110819, China
A R T IC LE I N F O
ABS TRA CT
Keywords: Analytical method Finite element method Crack Mesh stiffness Spur gear
Gear weight optimization is mainly through thin-web and hole in gear foundation. In this paper, an analytical-finite-element model for calculating time-varying mesh stiffness of spur gear pairs with complicated foundation (thin-web and holes) and crack is developed. The pinion and gear foundations are simulated as cylinders and modeled by three-dimensional finite element. The tooth pair is equivalent to a spring and mesh stiffness of a tooth pair is obtained by potential energy method. The spring is rigidly coupled with the pinion and gear foundations. Then, three type examples are presented to verify the model. The first type example is three spur gear pairs with solid foundation. The second type example is gear pairs with thin-web and web holes. And the third example is a gear pair with the cracked foundation. The analysis results show that the model proposed in this paper is applicable and efficient for spur gears with complicated foundation and crack. Finally, the time-varying mesh stiffness is also studied to reveal the effects of web widths, web hole radii and crack lengths.
1. Introduction Gear systems are widely used in machine tools, vehicles, ships, aerospace and other fields. For the aerospace industry, gear vibration and weight are main concerns to obtain a great design. For the time-varying mesh stiffness (TVMS), it is an important inner excitation for gear systems [1], which has great effects on vibration and noise. Gear weight optimization is mainly through thin-web and weight reduction holes in gear foundation (see Fig. 1). Therefore, TVMS model of gear pairs with a complicated foundation (thinweb and holes) is important to assess vibration characteristics of the entire system. Finite element (FE) method, analytical method and analytical-FE method have become the mainstream of mesh stiffness calculation methods [2,3]. For analytical method, potential energy method [4–8] is of vital importance. The tooth is assumed as a nonuniform cantilever beam in the potential energy method, in which shear stiffness, bending stiffness, radial compressive stiffness, foundation stiffness and Hertzian contact stiffness are considered in the total mesh stiffness. When a gear has a solid gear foundation, the compliance and stiffness of the gear foundation were studied based on the Muskhelishvili's theory [9]. In the later studies, some new researches related to gear crack [10–15], tooth pitting [16], tooth profile modification [17–19], tooth surface friction [20–22], extended tooth contact [23] (or corner contact [24]) and nonlinear Hertzian contact stiffness [20] were carried out, which make potential energy method more comprehensive. The most popular approach for analyzing TVMS of gear pairs is FE method, which has a great ability to simulate gear with different tooth profile curves and different types of gear foundation. Moreover, it has a high calculation precision. However, FE ⁎
Corresponding author. E-mail address: [email protected] (H. Ma).
https://doi.org/10.1016/j.engfailanal.2018.08.013 Received 2 May 2018; Received in revised form 5 August 2018; Accepted 13 August 2018 Available online 18 August 2018 1350-6307/ © 2018 Elsevier Ltd. All rights reserved.
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Fig. 1. Gear pictures with different gear foundations.
method has a heavy computation and need to redevelop the gear model when the gear has different parameters. Wang and Howard [25] developed several finite element models for calculating the torsional mesh stiffness of spur gears. Kiekbusch et al. [26] presented two- and three-dimensional finite element models to calculate the torsional mesh stiffness and the results were used to derive a simple formula for the torsional stiffness of spur gears. Liang et al. [27] developed three different FE models to analyze the TVMS of a spur gear, in which the gear bore surface constraint was studied. Li [28–30] established an FE model of a spur gear pair with solid foundation and some effects on loading capacity, strength and mesh stiffness were also studied. Raghuwanshi and Parey [31] investigated the mesh stiffness of a spur gear with the back-side contact by FE method and the results revealed that the mesh stiffness increased with the back-side contact. Wei et al. [32] researched the effects of flank deviations on load distributions of a helical gear by FE method. In a later reports [33–35], Li studied a spur gear with thin-web by FE method and the results showed that the web had great influences on contact stress and bending stress. What's more, some researches about TVMS of gear pairs with crack have been studied by FE method [36–38]. Some researches have studied TVMS by analytical-FE method. The integral equation method [39–41] is a main analysis method, in which the TVMS was obtained by integrating the contact stiffness and bending stiffness. The contact stiffness and bending stiffness were calculated by FE method. Ma et al. [15,18] and Feng et al. [42] investigated the TVMS of gear pairs by the analytical method, in which the revised foundation stiffness in multi-tooth contact zone was analyzed by FE method. Fernandez del Rincon et al. [43,44] developed an analytical-FE model of a spur gear with tooth profile modification and pitting. Chang et al. [45] developed an analytical-FE model of a helical gear pair. The gear was modeled by FE method and the contact was modeled by analytical method. The influences of web thickness, helix angle and face width on TVMS were also studied. Abbes et al. [46] presented a dynamic model of a gear with thin-web, in which the pinion was modeled by shaft FE and the wheel by three-dimensional FE. The gear mesh was modeled by spring element. The analytical results showed that natural frequency and vibration characteristics were influenced by the thin-web and crack. In this paper, an analytical-FE model is established to study the TVMS of a gear pair with complicated foundation and crack. Cylinders are used to simulate the pinion and gear foundations and are modeled by the three-dimensional FE. The tooth pair is modeled by a spring element and the spring stiffness is equal to the mesh stiffness of a tooth pair which can be calculated by the potential energy method. The spring is coupled with the foundations of the pinion and gear. Then, a time-varying modeling method is presented and the TVMS is analyzed. Finally, the model is verified by three type cases and the effects of the web widths, web hole radii and crack lengths on TVMS are also studied. 2. Theory There are two types of mesh stiffness for a gear pair, rectilinear mesh stiffness and torsional mesh stiffness [25–27]. Rectilinear mesh stiffness is an equivalent mesh stiffness of a gear pair along the action line. Torsional mesh stiffness is defined as the ratio between a torque applied on the pinion (the gear body is fixed) and the corresponding angular displacement of the pinion body. These two types of mesh stiffness are related and can be converted to each other. In this study, the rectilinear mesh stiffness is used and the mesh stiffness represents the rectilinear mesh stiffness. Mesh stiffness of a gear pair includes tooth stiffness and gear foundation stiffness. Mesh stiffness of a tooth pair can be calculated 340
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by the potential energy method. An FE model is established for the complicated gear foundation (thin-web and holes) and crack. The mesh stiffness of the tooth pair is modeled by a spring element and coupled with the foundations of the pinion and gear. Finally, mesh stiffness of a gear pair can be got by the static analysis. 2.1. Mesh stiffness of a tooth pair A cantilever beam with the variable cross-section is mostly used to simulate the tooth. The tooth stiffness consists of bending stiffness, shear stiffness, radial compressive stiffness and Hertzian contact stiffness [47]. The curve of tooth profile starts from the dedendum circle to addendum circle. The tooth profile curve is composed of involute curve and transition curve. So the bending stiffness, shear stiffness and radial compressive stiffness can be calculated as follows:
1 = kb
φt
∫π/2
1 = ks
∫π/2
1 = ka
∫π/2
φt
φt
[cos φ1 (yφ1 − y1) − x φ1 sin φ1 ]2 dy1 dγ + dγ EIy1
1.2cos2 φ1 GAy1
dy1 dγ + dγ
sin2φ1 dy1 dγ + EAy1 dγ
∫τ
∫τ
φ1 1.2cos2 φ1
GAy2
C
φ1
C
∫τ
φ1
C
[cos φ1 (yφ1 − y2 ) − x φ1 sin φ1 ]2 dy2 dτ dτ EIy2
dy2 dτ dτ
(1)
(2)
sin2 φ1 dy2 dτ EAy2 dτ
(3)
where φt represents the operating pressure angle. φ1 is the pressure angle at the contact point. Ay1, Ay2, Iy1, Iy2, xφ1, yφ1, x1, x2, y1, y2, dy dy τc, 1 and 2 are listed in Ref. [47]. B is the tooth width. E and v denote Young's modulus and Poisson's ratio. The Hertzian stiffness kh dγ
dτ
of the tooth pairs can be calculated as follows:
kh =
πEB 4(1‐ν 2)
(4)
Therefore, the total tooth mesh stiffness can be calculated as follows:
kt =
1 1 k b1
+
1 k s1
+
1 ka1
+
1 k b2
+
1 k s2
+
1 ka2
+
1 kh
(5)
where the subscripts 1 and 2 represent the pinion and gear, respectively. 2.2. TVMS calculation method for gear with complicated gear foundation As shown in Fig. 2, TVMS model of a spur gear is developed in ANSYS software. The applied load on the tooth is perpendicular to the tooth face and along the action line. The direction of stiffness is usually same as the direction of the applied load. Therefore, the direction of the spring representing the gear tooth stiffness is along the action line. In this paper, the action line is set to a horizontal state for convenient modeling. The relative displacement of the gear center o2 in the horizontal direction is (rb1 + rb2)tanα relative to
Fig. 2. An analytical-FE model of a spur gear pair. 341
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Fig. 3. Diagram of the connected zone between the tooth and foundation.
that of the pinion center o1. The relative displacement of the gear center o2 in the vertical direction is (rb1 + rb2) relative to that of the pinion center o1. rb1 and rb2 are the base circle radii of the pinion and gear. α denotes the pressure angle. The pinion and gear foundations are simulated as cylinders and developed by Solid185 element. The radii of the cylinders are equal to the radii of the dedendum circles because the curve of the tooth profile starts from the dedendum circle. As shown in Fig. 3, on the dedendum circle, the arc curve of a tooth consists of the tooth connected zone θ1 and the dedendum zone θ2. θ1 and θ2 can be calculated as follows:
θ1 =
(πm + 4ha m tan α + 4cm cos α /(1 − sin α )) mz
(6)
θ2 =
2π − θ1 z
(7)
where m, z, ha and c represents module, tooth number, addendum coefficient and tip clearance coefficient. Then, two nodes are established in middle of the tooth contact line. On the assumption that the displacements of the nodes in the connected zone between the tooth and gear foundation are same, one node is rigidly coupled with the connected zone between the pinion tooth in mesh and pinion foundation by Mpc184 element. The other node is rigidly coupled with the connected zones between the gear tooth in mesh and gear foundation by Mpc184 element. The two nodes are connected by spring element (Combin14 element in ANSYS) and the value of the spring stiffness is set to the mesh stiffness of the tooth pair in this mesh position. Because the boundary conditions (external torque and constraint) cannot be applied directly to the axle hole. Two master nodes are established in the geometric centers of the pinion and gear, respectively. For the pinion, the master node is coupled with the all nodes on the center hole surface and all degrees of freedom except rotation direction of the master node are fully constrained. Also for the gear, the master node is coupled with the all nodes on the center hole surface and all degrees of freedom of the master node are fully constrained. An external torque is equivalent to tangential forces applied to the hole nodes of the pinion. Finally, the angle displacement Δθ of the pinion can be obtained by the static analysis. The relationship between the rectilinear mesh stiffness and the torsional mesh stiffness for gears has been established in Ref. [25]. So the mesh stiffness of a gear pair can be calculated as follows:
k=
k t∗ T = 2 2 rb1 Δθ⋅rb1
(8)
kt∗
where represents the torsional mesh stiffness. T is the applied torque and rb1 denotes the base circle radius of the pinion. Due to contact ratio of a gear pair is usually not an integer, the number of the tooth pairs in mesh changed periodically. On the other hand, the mesh stiffness of a tooth pair changes with the varying mesh position of a tooth. Therefore, the mesh position and the number of tooth pairs in mesh should be considered when calculating TVMS. As shown in Fig. 4, N1N2 denotes the theoretical action line and B1B2 represents the actual action line. C1 and C2 are the contact points in the contact zone. The N1C1 and N1C2 can be calculated as follows:
⎧ N1 C1 = r b1 (φ b + mod(Ω1 t , 2π/ z1 )) ⎨ ⎩ N1 C2 = r b1 (φ b + 2π/ z1 + mod(Ω1 t , 2π/ z1))
(9)
where ‘mod’ function obtains the modulus of Ω1t divided by 2π/z1. z1 represents the tooth number of the pinion. Ω1 is the rotating 342
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Fig. 4. Action line of a gear pair.
speed of the pinion and t is for time. φb can be expressed as follows:
φb =
N1 B1 r b1
(10)
When the N1C1 is located between the N1B1 and N1B2, the point C1 is in mesh. The corresponding node and spring will be developed and the spring stiffness is set to mesh stiffness of the tooth pair in this mesh position, otherwise the node and spring will not be developed. It is also true for the point C2. Then, the mesh stiffness of a gear pair with multi-tooth contact can be obtained by the static analysis. According to the previous theory, the TVMS model of a spur gear pair with complicated foundation can be calculated by the MATLAB and ANSYS softwares. The flowchart of the computational procedure can be listed as follows (see Fig. 5): Step 1: Determine the number of tooth pairs in mesh and the mesh positions of the tooth pairs. Step 2: Calculate the mesh stiffness of tooth pairs in mesh in MATLAB software. Mesh stiffness of a tooth pair consist of bending stiffness, shear stiffness, radial compressive stiffness and contact stiffness. Those stiffness can be calculated by potential energy method. Step 3: Develop an FE model of the gear foundation in ANSYS software. The pinion foundation and gear foundation are connected by spring element and the spring stiffness is set to the mesh stiffness of the tooth pair. Mesh stiffness of a gear pair at a certain position can be obtained according to Eq. (6). At last, the whole process is repeated in the next mesh position until one period is achieved. The TVMS of a gear pair can be obtained.
2.3. Calculation of mesh stiffness of a spur gear with cracked foundation An FE model of a spur gear with crack is established in ANSYS software and the crack is built on the gear foundation (see Fig. 6a). It is noted that the crack path is assumed a straight line at the dedendum circle. The crack propagation angle υ and the crack length q are shown in Fig. 6b. In this paper, the crack propagation angle υ is constant. The analytical-FE model and boundary conditions can refer to the Section 2.2. Based on the potential energy method, the mesh stiffness of a tooth pair can be obtained (see Section 2.1), and its value is applied on the spring element of the proposed model. After solving, the angle displacement Δθ of the pinion can be obtained and the mesh stiffness of a spur gear with cracked foundation can be calculated by the Eq. (8) at a certain meshing position. Finally, the TVMS of a spur gear with cracked foundation can be obtained.
343
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Fig. 5. A flow chart of the computational procedure.
3. Model validation Before any study can be performed with confidence, the model must be validated. Here, the model is verified by three type cases. In case 1, three healthy spur gear pairs with the solid gear foundation are established to validate the correctness of the analytical-FE model. The geometric characteristics of the healthy gear are shown in Fig. 7a. In case 2, two types of the healthy spur gear with the web, web and holes are applied to validate the applicability of the model, respectively. And the geometric characteristics of the gear with the web, web and holes are shown in Fig. 7b and Fig. 7c, respectively. In case 3, a spur gear with cracked foundation is also applied to model validation. The parameters of the spur gear pairs used in the above three type examples are listed in Table 1. 3.1. Case 1: healthy gear 3.1.1. Gear pair 2 for model validation In this part, the gear pair 2 (see Table 1) is applied to verify the proposed method. The proposed model and FE model are shown in Fig. 8a and Fig. 9a, respectively. The static characteristics are analyzed and the comprehensive deformation of two models are plotted in Fig. 8b and Fig. 9b, respectively. It is noted that the proposed model and the FE model have similar deformation distribution. The TVMS obtained from the proposed method and FE method is shown in Fig. 10 and the errors at the moments A and B are listed in Table 2 (moments A and B are the middle time of the double-tooth contact zone and single-tooth contact zone, respectively). It is obvious that the stiffness curves obtained from the two methods are similar and the maximum error at moments A and B is 1.20%, i, e. the proposed method have high accuracy for calculating the mesh stiffness. 3.1.2. Gear pair 1 for model validation Under the parameters of gear pair 1 (see Table 1), the comprehensive deformation results of the proposed and FE models are shown in Fig. 11. According to the Fig. 11, the foundation deformation obtained from two models are similar. In addition, the TVMS obtained from the proposed method and FE method is plotted in Fig. 12 and the calculation errors at the moments C and D are listed in Table 3 (moments C and D are the middle time of the double-tooth contact zone and single-tooth contact zone, respectively). It is noted that the stiffness curves obtained from the two models have similar trends and the maximum error is less than 1.5%. 3.1.3. Gear pair 3 for model validation In this part, the gear pair 3 (see Table 1) is adopted to verify the proposed method and the results obtained from the proposed 344
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Fig. 6. A model of a spur gear with cracked foundation: (a) analytical-FE model, (b) crack diagram.
method are compared with that of the formulaic method and FE method in Ref. [26]. Based on the analytical-FE model, the maximum mesh stiffness in single tooth contact zone and double teeth contact zone is listed in Table 4. Converting torsional mesh stiffness into rectilinear mesh stiffness, the maximum mesh stiffness by the formulaic method in Ref. [26] and FE method is also listed in Table 4. In the single tooth contact zone, the maximum mesh stiffness is about 1.42 × 108 N/m by the proposed method, 1.34 × 108 N/m by the formulaic method in Ref. [26] and 1.41 × 108 N/m by FE method. The relative error of the proposed method is about 5.9% compared with the method in Ref. [26] and that is 0.7% compared with the FE method. In the double teeth contact zone, the maximum mesh stiffness is about 1.93 × 108 N/m by the proposed method, 1.72 × 108 N/m by the formulaic method in Ref. [26] and 1.86 × 108 N/ m by FE method. The relative error of the proposed method is about 12.2% compared with the method in Ref. [26] and that is 3.8% compared with the FE method. The results reveal that proposed method is effective. 3.2. Case 2: healthy gear with web and holes Firstly, the proposed method is applied to the healthy gear with web (the pinion is a solid foundaiton and the gear is a web foundation) and the gear parameters refer to the gear pair 1 (see Table 1). The proposed model and FE model are shown in Fig. 13a and c, and the comprehensive deformation obtained from two models are shown in Fig. 13b and d. It is obvious that the deformation result of the proposed model is very similar to that of the FE model. The TVMS obtained from FE method and the proposed method is shown in Fig. 14. It is obvious that the variation trend of the stiffness curves matches well. Then, the proposed method is applied to the healthy gear with web and holes (see Fig. 15). The gear parameters refer to the gear pair 1 (see Table 1). The models and the comprehensive deformation obtained from two methods are shown in Fig. 15 and the TVMS obtained from the proposed method and FE method is shown in Fig. 16. It is obvious that the deformation result of the proposed model is very similar to that of the FE model. At the same time, the TVMS obtained from the proposed method has similar change law with the results from FE method and the maximum relative error of the proposed method is about 2.03%. 3.3. Case 3: gear with cracked foundation To verify the feasibility of the proposed method for cracked foundation, the spur gear pair 1 is applied and the crack length q is 1.5 mm (see Fig. 6). The TVMS obtained from the FE method and the proposed method is plotted in Fig. 17. It is obvious that change 345
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Fig. 7. Geometric parameters of the healthy gear with different foundations: (a) solid foundation; (b) web foundation; (c) web and holes foundation. Table 1 Parameters of the spur gear pairs. Parameters
Tooth number Normal module (mm) Tooth width (mm) Pressure angle (°) Addendum coefficient Tip clearance coefficient Young's modulus (Pa) Poisson's ratio Hub radius (mm) Web inner radius (mm) Web outer radius (mm) Web hole position (mm) Web hole radius (mm) Number of web holes Web width (mm) Crack angle (°) Crack length (mm) Applied torque (N·m)
Symbol
Gear pair 1
z2
z1 m B α ha c E ν rh ri ro rm rw n b υ q T
Gear pair 2
Gear pair 3
Pinion
Gear
Pinion
Gear
Pinion
52
72
19
31
23
1.75 20 20 1 0.25 2.1 × 1011 0.3 10 – – – – – – – – 200
3.2 38 20 1 0.25 2.1 × 1011 0.3 17.5 18 52 36 8 6 16 30 1.5 –
– – – – – – – – 200
Gear 23 6.0 15 20 1 0.25 2.1 × 1011 0.3 15
– – – – – – – – –
– – – – – – – – 1000
– – – – – – – – –
law of the TVMS obtained from two methods is similar and the maximum error is about 2.39%, i. e., the new method is also effective for the spur gear with cracked foundation. Besides, the analysis results reveal that the cracked foundation still influence the TVMS when the corresponding tooth is no longer in mesh. This is because the deformation of the gear foundation cannot be transferred from one side of the crack to the other side. The flexibility of the gear foundation increases and the mesh stiffness decreases. In the previous three type examples, the time consuming with the proposed procedure is about 4 min on a personal computer with 32G RAM and 8-core CPU for 20 analysis points in one period. However, the time consuming is about 10 h for 20 analysis points using the FE method in one mesh period. The proposed model is 150 times more efficient than the FE model. Above all, the proposed method not only has a high efficiency as the analytical method, but also has ability for the complicated 346
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Fig. 8. Proposed model: (a) schematic of FE model; (b) comprehensive deformation.
Fig. 9. FE model: (a) schematic of FE model; (b) comprehensive deformation.
Fig. 10. TVMS obtained from the proposed method and FE method. Table 2 Calculation error of proposed method at different moments. Method
FE method Proposed method
Moment A
Moment B
TVMS (×108 N/m)
Error (%)
TVMS (×108 N/m)
Error (%)
7.52 7.43
– 1.20
5.12 5.07
– 0.98
Note: the error denotes the relative error of the stiffness obtained from the proposed method relative to that obtained from the FE method. 347
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Fig. 11. Comprehensive deformation: (a) proposed model; (b) FE model.
Fig. 12. TVMS obtained from the proposed method and FE method. Table 3 Calculation errors of proposed method at moments C & D. Method
FE method Proposed method
Moment C
Moment D
TVMS (×108 N/m)
Error (%)
TVMS (×108 N/m)
Error (%)
2.13 2.16
– 1.41
1.70 1.69
– 0.59
Table 4 Mesh stiffness of the gear pair 3 under different methods.
FE method
Relative errors (Compared with the method in Ref. [26])
Relative errors (Compared with the FE method)
1.41 × 108 1.86 × 108
5.9% 12.2%
0.7% 3.8%
Mesh stiffness (N/m) Proposed method Single tooth contact zone Double teeth contact zone
8
1.42 × 10 1.93 × 108
Method in Ref. [26] 8
1.34 × 10 1.72 × 108
gear foundation as the FE method. It can also solve the gear foundation sharing in multi-tooth mesh zone [15,18].
4. Effects of the gear foundation parameters on gear mesh stiffness Different gear parameters like web width, web hole radius and crack length cause the variation of mesh stiffness. In this section, their influences on gear mesh stiffness are quantitatively analyzed.
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Fig. 13. Schematics of a spur gear with web: (a) proposed model; (b) comprehensive deformation of the proposed model; (c) FE model; (d) comprehensive deformation of the FE model.
Fig. 14. TVMS of gear with web obtained from the proposed method and FE method.
Fig. 15. Schematics of a spur gear with web and holes: (a) proposed model; (b) comprehensive deformation of the proposed model; (c) FE model; (d) comprehensive deformation of the FE model.
4.1. Effects of the gear web width In this case, the gear pair 1 is used and the parameters are listed in Table 1, in which the web hole radius rw = 8 mm, crack length q = 0 mm. Keeping all the other parameters unchanged, the web width b increases gradually from 4 to 16 mm with interval 4 mm. The TVMS is analyzed by the proposed method and the results are shown in Fig. 18. The results reveal that the mesh stiffness of single-tooth contact zone and double-tooth contact zone has a bigger reduction with the decreasing the web widths. The maximum mesh stiffness decreased by 39.47% when the web width b decreases from 16 mm to 4 mm. The reason for this phenomenon is that the stiffness of gear foundation decrease with the decreasing the web widths. 349
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Fig. 16. TVMS of gear with web and holes obtained from the proposed method and FE method.
Fig. 17. TVMS of a spur gear with crack obtained from the proposed method and FE method.
Fig. 18. TVMS of spur gear under different gear web widths.
4.2. Effects of the web hole radius Similarly, web hole radius rw is changing from 0 mm to 12 mm incrementally while web width b = 16 mm, crack length q = 0 mm. The TVMS of the gear pair with different web hole radii is shown in Fig. 19. The results demonstrate that the mesh stiffness is greatly influenced by the size of the web hole radius. For the case rw = 0 mm (i.e., web foundation), the maximum value of mesh stiffness is about 2.06 × 108 N/m and the TVMS varies periodically with the meshing tooth. When the web hole radius rw = 12 mm, the maximum value is about 1.57 × 108 N/m and the maximum mesh stiffness decreased by 31.2% compared with that rw = 0 mm. In addition, the results show that the TVMS curve gradually exhibits periodicity with the web hole and the periodicity becomes more and more obvious with the increasing hole radii. This is because web hole weakens the stiffness of gear foundation and the web hole has a great influence on mesh stiffness of a gear pair.
4.3. Effects of the crack length To expose the effects of the crack length, the spur gear pair 1 is used and the TVMS is plotted in Fig. 20 while the crack length q 350
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Fig. 19. TVMS of spur gear under different web hole radii.
Fig. 20. TVMS of spur gear under different crack lengths.
increases from 1.5 to 3 mm with interval of 0.5 mm. In Fig. 20, a reduction phenomenon in mesh stiffness of single-tooth contact zone and double-tooth contact zone can be observed when the cracked tooth comes into contact. The maximum mesh stiffness in the crack zone under the crack length q = 1.5 mm decreased by 2.09% compared with that in the healthy zone. When the crack length q = 3 mm, the maximum mesh stiffness in the crack zone decreased by 3.83% compared with that in healthy zone. The reason for this phenomenon is that the foundation stiffness decrease with the increasing crack lengths. Above all, the maximum mesh stiffness of the entire gear system decreases with the decreasing web widths, and it also decreases with the increasing web hole radii and crack lengths. The mesh stiffness in single-tooth contact zone and multi-tooth contact zone change with the same tendency. For the spur gear with complicated foundation, the web width has a greatest influence on the TVMS. The TVMS exhibits periodicity for the gear with the web holes. The shape of the TVMS curve is changed when the tooth with crack comes into mesh. 5. Conclusions (1) An analytical-FE model of TVMS for a spur gear with complicated foundation and crack is developed, in which the pinion and gear foundations are simulated as cylinders and modeled by three-dimensional FE. The tooth pair is equivalent to a spring and mesh stiffness of the tooth pair is calculated by potential energy method. The spring is rigidly coupled with the pinion and gear foundations. The model proposed in this paper is verified using three different type examples. The results show that the maximum relative error of the proposed method under different foundations is about 2.39% relative to the FE method and the proposed model is 150 times more efficient than the FE model. The complicated gear foundation has a great influence on TVMS of a spur gear system. Besides, the proposed method can solve the problem of the gear foundation sharing in multi-tooth mesh zone. (2) Effects of the web width, web hole radius and crack length on TVMS of a spur gear pair are also studied. The results reveal that the maximum value of the mesh stiffness of the spur gear decreases with the decreasing web width, the increasing web hole radius and crack length. The maximum mesh stiffness decreased by 39.47% when the web width b decreases from 16 mm to 4 mm. The maximum mesh stiffness decreased by 31.2% when the web hole radius rw increases from 0 mm to 12 mm and TVMS curve gradually exhibits periodicity as the web hole radius increasing. There is a distinct reduction of the stiffness when a cracked tooth is going to mesh.
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Acknowledgement This work was supported by the National Natural Science Foundation of China [grant numbers 51605361, 51505357]; the Fundamental Research Funds for the Central Universities of China [grant numbers N170308028, N160313004]; and the Natural Science Basic Research Plan in Shaanxi Province of China [grant number 2018JQ5034]. Conflicts of interest The authors declare that they have no conflict of interest. References [1] X. Gu, P. Velex, P. Sainsot, J. Bruyere, Analytical investigations on the mesh stiffness function of solid spur and helical gears, J. Mech. Des. 137 (6) (2015) 063301. [2] H. Ma, J. Zeng, R. Feng, X. Pang, Q. Wang, B. Wen, Review on dynamics of cracked gear systems, Eng. Fail. Anal. 55 (2015) 224–245. [3] X. Liang, M.J. Zuo, Z. Feng, Dynamic modeling of gearbox faults: a review, Mech. Syst. Signal Process. 98 (2018) 852–876. [4] Z. Chen, Y. Shao, Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth, Eng. Fail. 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