Mechanism and Machine Theory 129 (2018) 261–278 Contents lists available at ScienceDirect Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory Research paper A revised time-varying mesh stiffness model of spur gear pairs with tooth modiﬁcations Yanning Sun a, Hui Ma a,b,∗, Yifan Huangfu a, Kangkang Chen a, LinYang Che a, Bangchun Wen a a School of Mechanical Engineering and Automation, Northeastern University, Shenyang, Liaoning 110819, PR China Key Laboratory of Vibration and Control of Aero-Propulsion Systems Ministry of Education of China, Northeastern University, Shenyang, Liaoning 110819, PR China b a r t i c l e i n f o Article history: Received 28 January 2018 Revised 3 April 2018 Accepted 2 August 2018 Keywords: Tooth modiﬁcations Spur gear pairs Time-varying mesh stiffness Revised analytical model Thin slice method a b s t r a c t Based on the thin slice assumption, a revised time-varying mesh stiffness (TVMS) model of spur gear pairs with tooth modiﬁcations is developed. The spur gear is divided into many individual slices along tooth width, and considering the revised ﬁllet-foundation stiffness, the nonlinear contact stiffness, the extended tooth contact and the tooth proﬁle errors, the stiffness of each slice gear pair is ﬁgured out. According to the relationship between the deformation and the total stiffness in mesh period, the TVMS of spur gear pairs can be worked out. Meanwhile, relative to the ﬁnite element (FE) method, the errors of the proposed method under different modiﬁcation quantities are discussed. The proposed method is more accurate than those previous methods but there are still some errors. Taking the FE model as a benchmark, the TVMS is further revised based on a simple model updating technique. Based on the revised model, the effects of the tooth width and torque on mesh stiffness are also studied. The result shows that based on the proposed method, the TVMS under any given modiﬁcation quantities in a suitable range can be calculated accurately. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction Tooth modiﬁcation is an important method to reduce vibration and noise, which can be carried out by tooth proﬁle and lead crowning modiﬁcations. In recent years, many scholars have studied the vibration and noise behaviors of gears with tooth modiﬁcations [1–9]. Lin and He  developed a ﬁnite element (FE) method to calculate the transmission error of gear transmission systems with machining errors, assembly errors and modiﬁcations. Wang et al.  and Maatar and Velex  analyzed the effects of gear modiﬁcations on the contact and dynamic characteristics of a gear pair. Bruyère et al.  investigated the transmission errors of modiﬁed gears and obtained the optimal proﬁle modiﬁcations which can minimize the transmission error ﬂuctuations. Velex et al.  proposed a design criterion for tooth modiﬁcations minimizing dynamic tooth loads. They also pointed out that the transmission error ﬂuctuations are related to the tooth modiﬁcations . Bahk and Parker  studied the effects of tooth proﬁle modiﬁcations on the vibration of spur planetary gears. Ma et al. Abbreviations: 3D, Three-dimensional; CW1∼CW9, Nine modiﬁcation quantities for the gear pair 1; CL1∼CL8, Eight modiﬁcation quantities for the gear pair 2; FE, Finite element; FEM, Finite element model; PAM, Proposed analytical model; TVMS, Time-varying mesh stiffness. ∗ Corresponding author at: School of Mechanical Engineering and Automation, Northeastern University, Shenyang, Liaoning 110819, PR China. E-mail address: firstname.lastname@example.org (H. Ma). https://doi.org/10.1016/j.mechmachtheory.2018.08.003 0094-114X/© 2018 Elsevier Ltd. All rights reserved. 262 Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 Nomenclature Cq C Ca dz E Epc Lead crowning quantity Tooth proﬁle error in mesh position caused by tip relief along horizontal direction (x-direction in Fig. 2b) Amount of proﬁle modiﬁcation Width of each piece spur gear pair Young’s modulus Tooth errors of each slice gear caused by lead crowning relief along horizontal direction (x-direction in Fig. 2b) Epi Total tooth proﬁle error of the ith tooth pair of each slice gear pair along line of action Eri Static transmission error of the ith tooth pair of each slice gear pair at meshing point F Total meshing force Fi (i = 1, 2) Meshing force of ith meshing tooth pair Fn (n = 1∼N) Total meshing force of the nth piece tooth j Number of meshing position ka , kb , ks Axial compressive stiffness, bending stiffness and shear stiffness kn TVMS of the nth piece tooth ktooth Total mesh stiffness of meshing teeth pairs kitooth Stiffness of the ith tooth pair k∗h Hertzian contact stiffness kih Nonlinear Hertzian contact stiffness of the ith tooth pair kmean Mean mesh stiffness of gear pairs kit1 , kit2 Tooth stiffness of the ith tooth pair and subscripts 1 and 2 denote the driving and driven gears, resepectively. kf Stiffness of ﬁllet-foundation K Total mesh stiffness of spur gear pairs with tooth modiﬁcations L Width of the tooth La Length of proﬁle modiﬁcation lsfi Load-sharing ratio of the ith tooth pair q Number of tooth pair N Number of slice gear pair R Radius of the lead crowning circular curve Tl Torque applied to the driving gear Z Number of tooth zn Coordinate of each sliced spur gear pair along the axial direction (z-direction in Fig. 3c) Greek symbols λ1 , λ2 (i = 1, 2) Coeﬃcients of the ﬁllet-foundation stiffness, subscripts 1 and 2 denote the driving gear and driven gear, respectively λk TVMS correction coeﬃcient ν Poisson’s ratio  proposed a mesh stiffness model for proﬁle shifted gears with addendum modiﬁcations and tooth proﬁle modiﬁcations, and determined the optimum proﬁle modiﬁcation curve under different amounts of tooth proﬁle modiﬁcations. The vibration characteristics of gear transmission systems will be signiﬁcantly affected by the time-varying mesh stiffness (TVMS) [10–13], which will change with tooth modiﬁcations. In the earlier study, the TVMS of healthy spur gear pairs can be evaluated by analytical method on the basis of the potential energy method in elastic mechanics [14–18]. However, the stiffness of the spur gear with tooth modiﬁcations or tooth faults is diﬃcult to be determined accurately by this analytical method, and the effects of gear errors are usually ignored. Subsequently, based on the analytical method, Chen and Shao  took the gear errors into account and developed a TVMS calculation model, in which the relationship between the mesh deformation and the total mesh stiffness in mesh period is determined. Furthermore, they also proposed a more general calculation method for both healthy and tooth root crack cases, and developed an analytical model to calculate the mesh stiffness of spur gear pairs with non-uniformly distributed tooth root crack, and the dynamic simulation of spur gears with tooth root crack propagating along tooth width and crack depth is also carried out [20–22]. The effects of extended tooth contact (ETC) on TVMS should not be ignored in TVMS calculation [23,24] and Ma et al. [25–28] established an improved TVMS model for healthy gear pairs, cracked spur gears and gear pairs with tip relief, in which the transition curve, revised ﬁllet-foundation stiffness, nonlinear contact stiffness and ETC were considered. Liu et al.  studied the mesh stiffness of a gear pair with tooth proﬁle modiﬁcation by analytical method and determined the optimal modiﬁcation amount. Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 263 With the development of computer technique, the application of the FE method is becoming more and more widely. And the ﬂexibility of gears, tooth errors and modiﬁcations can be easily considered in the ﬁnite element model (FEM). So the TVMS can be acquired by applying a meshing force on the theoretical engagement position of gear pairs [30–32] or employing contact elements between the two contact faces of gears [33–35]. Ma et al. [36,37] developed an FEM of a spur gear pair to calculate TVMS, and they also studied the inﬂuences of tip reliefs on the vibration responses. Based on the FE method, Li [38,39] developed a model to calculate tooth surface contact stress and tooth root bending stress of spur gear pairs, in which manufacturing errors, setting errors and tooth modiﬁcations were all considered. And the inﬂuences of misalignment error, tooth modiﬁcations especially lead crowning relief and transmission torque on mesh characteristics were further studied . Compared with the analytical method on the basis of potential energy method, the FE method is more accurate, but it is also relatively time-consuming. In order to combine the accuracy of FE method and the eﬃciency of analytical method, an analytical-FE method is put forward which adopts a linear FE method to calculate the deformation (shearing and bending) of the tooth and the whole gear body and employs a non-linear analytical formulation to compute the local contact deformation near the contact point [41–44]. Fernandez et al. [41,42] researched the tooth proﬁle modiﬁcations of spur gears and planetary gears by the analytical-FE method. Taking the proﬁle errors into account, a non-linear model of spur gears is developed based on an analytical-FE approach and the inﬂuences of the tip relief lengths on mesh stiffness were studied . In order to calculate TVMS of gear pairs with lead crowning relief, tooth shape deviations or alignment errors, the thin slice method was developed [44–51]. That is, the tooth is divided into many individual slices along tooth width and each slice can be regarded as a spur gear pair with rather small thickness. Supposing that the meshing deformation of the sliced spur gear pair which has the minimum lead crowning quantity is the total mesh deformation of gear pairs, Wang et al.  evaluated the TVMS of spur gear pairs with lead crowning relief and analyzed the vibration responses by the thin slice method. Based on this method, an analytical model was also developed to calculate TVMS and contact stress of helical gear pairs with tooth proﬁle errors . Considering tooth shape deviations and alignment errors, Ajmi and Velex  presented a novel method to calculate deﬂections and load distributions on solid spur gears and helical gears, in which the elastic couplings (also known as elastic convective effects) were also considered. Generally, modiﬁcations include tooth proﬁle modiﬁcations and lead crowning relief, moreover, tip relief is the most common in tooth proﬁle modiﬁcations and lead crowning relief with circular curve is mostly used. The analytical method for mesh stiffness calculation of spur gears with tip relief has a higher accuracy because it can consider the inﬂuences of nonlinear contact stiffness, ETC and revised ﬁllet-foundation stiffness . However, for lead crowning relief, the analytical method can be challenging due to the different amounts of modiﬁcation along the tooth width. Based on the slice method, this paper recommends a revised TVMS model of spur gear pairs with tooth modiﬁcations including simultaneously tip relief and lead crowning relief, which upgrades the overall calculation accuracy by improving the TVMS calculation precision of each slice gear. The proposed method is also veriﬁed by the FE method. A further revision for the TVMS is carried out by comparing TVMS obtained from the proposed analytical method (PAM) with that from the FE method. In addition, the effects of tooth widths and transmission torques on mesh stiffness are also discussed. 2. Mesh stiffness model for spur gear pairs with tooth modiﬁcations 2.1. Analytical model 2.1.1. TVMS calculation for heathy spur gear pairs The total mesh stiffness of meshing teeth pairs can be expressed as: ktooth = q kitooth = q i=1 1 i=1 kih 1 + 1 kit1 + 1 kit2 , (1) where q denotes the number of tooth pairs in engagement; kitooth is the stiffness of the ith tooth pair; kih denotes the nonlinear Hertzian contact stiffness which can be elaborated in Fig. 1; k∗h denotes the Hertzian contact stiffness without nonlinearity, which can refer to Ref. ; kit1 and kit2 denote the stiffness of teeth. Subscripts 1, 2 denote the driving and the driven gears, and superscript i denotes the ith tooth pair (see Fig. 2a); E represents Young’s modulus; ν denotes Poisson’s ratio; L is the tooth width and F is the total meshing force; Fi is the meshing force of the ith meshing tooth pair; lsfi is the load sharing ratio of the ith meshing tooth pair. The tooth stiffness kit1 and kit2 can be expressed as: kit1 = 1 1 kib1 + 1 kis1 + 1 kia1 , kit2 = 1 1 kib2 + 1 kis2 + 1 kia2 , (2) where the bending stiffness kb , the shear stiffness ks and the axial compressive stiffness ka can be written as: 1 = kb α β 2 [cosβ (yβ − y1 ) − xβ sinβ ] dy1 (cosβ (yβ − y2 ) − xβ sinβ )2 dy2 dγ + dτ , π E Iy1 dγ E Iy2 dτ αC 2 (3) 264 Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 Fig. 1. Calculation ﬂowchart of nonlinear Hertzian contact stiffness of healthy gear pair. Fig. 2. Schematic of tooth modiﬁcations for each slice spur gear: (a) schematic of meshing gear pair, (b) schematic of tooth with modiﬁcations, (c) enlarged view of tooth with modiﬁcations. Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 265 Fig. 3. Schematic of the spur gear with tooth modiﬁcations. (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.) α β 1.2cos2 β dy1 1.2cos2 β dy2 dγ + dτ , π GAy1 d γ GAy2 dτ αC 2 α β 1 sin2 β dy1 sin2 β dy2 = dγ + dτ , π ka E A d γ y1 αC E Ay2 dτ 2 1 = ks (4) (5) E where α C is the pressure angle of the involute starting point; G = 2(1+ v ) is the shear modulus; β is the operating pressure dy1 dy2 angle; and α is the pressure angle. xβ , yβ , y1 , y2 , dγ , dτ , Iy1 , Iy2 , Ay1 andAy2 can be found in Appendix A and Ref. . The stiffness of ﬁllet-foundation kf can be expressed as: 2 1 cos2 β ∗ uf = L kf EL Sf +M ∗ u f Sf + P (1 + Q tan β ) , ∗ ∗ 2 (6) where the detailed expressions of uf , Sf , L∗ , M∗ , P∗ andQ∗ are elaborated in Ref. . Based on the above equations, the total mesh stiffness kn of two meshing teeth can be redeﬁned and kn of each slice spur pair can be expressed as: kn = 1/ ( 1 λ1 kf1 + 1 + ktooth 1 λ2 kf2 ), (7) where λ1 and λ2 are the correction coeﬃcients of the ﬁllet-foundation stiffness which can refer to Ref. . 2.1.2. TVMS calculation for spur gear pairs with tooth modiﬁcations The schematic of spur gears with tooth modiﬁcations is shown in Fig. 3. Following assumptions are made to calculate mesh stiffness: 266 Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 (1) The tooth in Fig. 3a can be divided into N individual slices (see Fig. 3b) along tooth width where N is the total slice number. The top view is shown in Fig. 3c, and the convective (or coupling) effects among individual thin tooth pieces are ignored. (2) The quantity of modiﬁcation is so small that teeth contact at the theoretical action plane. (3) Each slice gear pair can be equivalent to a spur gear pair with tooth errors, and its TVMS can be calculated by Eq. (7). As is shown in Fig. 3, the parts of the green dashed and dot line and the red dot line represent the tip relief and the lead crowning relief, respectively. The linear tip relief (described by the length of proﬁle modiﬁcation La , the amount of proﬁle modiﬁcation Ca ) and the arc lead crowning relief (described by lead crowning quantity Cq ) are adopted. Tooth proﬁle error in mesh position along horizontal direction (x-direction in Fig. 2b) in tip relief part can be deﬁned as: C = Ca ( u s ) , u ∈ [0, La ], La (8) where u is the vertical distance between ideal mesh position and starting point of tip relief (see Fig. 2b); s is the index of tip relief curve and s = 1 in this paper. As for lead crowning relief part, the tooth errors of each slice gear along horizontal direction (x-direction in Fig. 2b) can be expressed as: Epc = R − L L R2 − zn 2 , zn ∈ [− , ], 2 2 (9) where zn is the coordinate of the each slice gear along the axial direction (z-direction in Fig. 3c); R is the radius of the lead crowning circular curve of the gear (see Fig. 3), which can be calculated as follows: R= (L/2 )2 + Cq 2 2Cq . (10) The total tooth proﬁle errors of the ith tooth pair of each slice spur gear along line of action can be expressed as: Epi = (C + Epc ) cos β . (11) Considering the tooth modiﬁcations, the mesh stiffness due to teeth deformation can be expressed as: ktooth = F Er2 − Epi , where Epi = min(Ep1 , Ep2 ), (12) where Er2 is the static transmission error ignoring the ﬁllet-foundation deformation and it can refer to Ref. ; superscript 1 or 2 denotes the 1st or 2nd tooth pair. Based on Refs. [19,26], load sharing ratio lsfi (i = 1 or 2) of the gear with tooth modiﬁcation can be calculated by: ls f1 = 1 − ls f2 , ls f2 = k2tooth (Er2 − Ep2 ) F , (13) Substituting Eq. (12) into Eq. (7), the stiffness of each slice gear pair with tooth modiﬁcations can be obtained. Based on the above analysis, the total mesh stiffness can be obtained as follows [19,46]: F· K= F+ N n=1 N n=1 kn , (14) kn Epc Considering the nonlinear contact stiffness, the revised ﬁllet-foundation stiffness and the ETC, the calculation procedure of the TVMS is shown in Fig. 4. BC (see Fig. 2a) denotes the single-tooth contact region; Fn is the force of the nth slice gear pair; rb1 is the radius of base circle of driving gear; Tl is the torque and detailed parameters about ETC can be found in Ref. . 2.2. Model veriﬁcation Based on the basic parameters of a spur gear pair (deﬁned as gear pair 1, see Table 1), a three-dimensional FE model considering tooth modiﬁcations is established in ANSYS software (see Fig. 5). In the model, the gears are discretized using Solid185 element, the contact between teeth are simulated using contact pairs (Targe170 and Conta174 elements), as shown in Fig. 6. In the ﬁgure, the master node of the driven gear (O2 ) is completely constrained. Only the rotational degree of freedom around the gear axis direction of the master node of driving gear (O1 ) is retained. Based on the rotation direction of the gears, a constant loaded torque Tl in the right-handwise direction is applied to the master nodes of the driving gear. According to the deformation of gears, the mesh stiffness can be acquired by the rotational angular displacement of the master node of the driving gear. Nine cases (CW1-CW9) for different lead crowning relief parameters are listed in Table 2. It should be noted that La and Ca are all zero under these nine cases. The TVMS obtained from the proposed analytical method (PAM) is compared Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 Fig. 4. Calculation procedure of TVMS of a spur gear pair with tooth modiﬁcations. Table 1 Parameters of gear pair 1. Parameters Driving gear/ driven gear Parameters Driving gear/ driven gear Number of teeth Z Module m (mm) Tooth width L (mm) Hub radius rint (mm) Torque load Tl (Nm) 30, 25 2 20 5.5, 4.5 100 Pressure angle α (°) Addendum coeﬃcient ha ∗ Tip clearance coeﬃcient c∗ Young’s modulus E (GPa) Poisson’s ratio ν 20 1 0.25 210 0.3 267 268 Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 Fig. 5. Schematic of the spur gear with tooth modiﬁcation. Fig. 6. Schematic of the FEM. Table 2 Nine cases for different modiﬁcation quantities of the gear pair 1. Modiﬁcation quantities Cq (μm) Modiﬁcation quantities Cq (μm) Modiﬁcation quantities Cq (μm) CW1 CW2 CW3 0 2.5 5 CW4 CW5 CW6 7.5 10 12.5 CW7 CW8 CW9 15 17.5 20 with those from the FEM and Wang’s model . Wang’s model belongs to the traditional analytical model which neglects the effects of some inﬂuence factors (such as extended tooth contact and nonlinear Hertz contact) so that the model is not accuracy enough. The TVMS obtained from the three models (PAM, FEM and Wang’s model) under CW3 and CW5 are shown in Fig. 7. Some errors between the PAM and the FEM, Wang’s model and the FEM under double- and single-tooth contact regions (positions A and B) are listed in Table 3. It is clear that taking the revised ﬁllet-foundation stiffness, the nonlinear contact stiffness and the ETC into account, PAM is more accurate to calculate the TVMS but there are still some errors. The error between the PAM and the FE method is relatively small and the maximum is about 9.8% but the maximum error between Wang’s method and the FE method is as high as 30.8%. Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 269 Fig.7. TVMS using different methods: (a) CW3, (b) CW5. Table 3 Errors between the PAM and FEM, Wang’s model and the FEM under double- and single-tooth contact regions. Modiﬁcation quantities TVMS (× 108 N/m) FEM PAM Error (%) Wang’s model Error (%) CW3 kA kB kA kB 1.782 1.466 1.633 1.327 1.954 1.564 1.786 1.457 9.7 6.7 9.4 9.8 2.330 1.477 1.909 1.273 30.8 0.8 16.9 4.1 CW5 Fig. 8. TVMS considering the tooth modiﬁcations: (a) PAM, (b) FEM. 3. A further revision for improving the calculation accuracy 3.1. Revision of PAM under lead crowning relief The TVMS obtained from PAM and FEM is shown in Fig. 8, which shows the TVMS error between two methods. To further improve the accuracy of the PAM, a revision method is proposed. The mean square error function between the PAM and the FEM can be deﬁned as: φ (λ ) = (λkA−PAM − kA−FEM )2 + (λkB−PAM − kB−FEM )2 , (15) where λ is the independent variable of this function; kA-PAM and kB-PAM denote the TVMS from the PAM at meshing positions A and B, respectively; kA-FEM and kB-FEM denote the TVMS from the FEM at meshing positions A and B, respectively. Based on the minimum mean square error, the TVMS correction coeﬃcient λk under CW1, CW3, CW5, CW7 and CW9 are plotted in Fig. 9 (red circles). The best ﬁtting curve of λk and Cq is ﬁgured out by polynomial interpolation method and 270 Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 Fig. 9. The best ﬁtting curve for correction coeﬃcient λk. . (For interpretation of the references to color in this ﬁgure, the reader is referred to the web version of this article.) Table 4 TVMS errors compared with the FEM. Modiﬁcation quantities Mesh stiffness (× 108 N/m) FEM PAM before revision Error (%) PAM after revision Error (%) CW1 kA kB kmean kA kB kmean kA kB kmean kA kB kmean kA kB kmean kA kB kmean kA kB kmean kA kB kmean kA kB kmean 2.096 1.682 2.015 1.922 1.560 1.843 1.782 1.466 1.717 1.701 1.385 1.635 1.633 1.327 1.572 1.588 1.285 1.525 1.543 1.247 1.483 1.514 1.212 1.451 1.479 1.181 1.420 2.157 1.688 2.076 2.050 1.624 1.982 1.954 1.564 1.895 1.866 1.509 1.815 1.786 1.457 1.740 1.713 1.409 1.673 1.646 1.364 1.608 1.584 1.321 1.549 1.526 1.282 1.494 2.9 0.4 3.0 6.7 4.1 7.5 9.7 6.7 10.4 9.7 8.9 11.0 9.4 9.8 10.7 7.9 9.6 9.7 6.7 9.4 8.4 4.6 8.9 6.8 3.2 8.5 5.2 2.116 1.650 2.037 1.940 1.537 1.872 1.801 1.442 1.748 1.706 1.379 1.656 1.630 1.330 1.587 1.578 1.298 1.536 1.527 1.266 1.493 1.489 1.241 1.455 1.449 1.217 1.417 1.0 1.9 1.1 1.0 1.5 1.6 1.1 1.6 1.8 0.3 0.4 1.3 0.9 0.2 1.0 0.6 1.0 0.7 1.0 1.5 0.7 1.7 2.4 0.3 2.0 3.0 0.2 CW2 CW3 CW4 CW5 CW6 CW7 CW8 CW9 J Note: kmean is the mean stiffness which can be calculated by: kmean = j=1 J kj , where J is the number of mesh positions. it can be written as Eq. (16): λk = p1Cq3 + p2Cq2 + p3Cq + p4 , (16) where p1 = − 2.931 × 10−5 , p2 = 1.408 × 10−3 , p3 = − 1.803 × 10−2 and p4 = 0.981 are the coeﬃcients of this polynomial function. Compared with the FEM, the errors before and after revision are shown in Table 4. Calculation procedure of correction coeﬃcient λk is shown in Fig. 10. The correction coeﬃcient λk is calculated based on the calculation procedure under CW2, CW4, CW6 and CW8 (see Fig. 9, black triangles), which agree well with best ﬁtting curve. The TVMS under those modiﬁcation quantities is shown in Fig. 11. Those ﬁgures show that TVMS obtained from the PAM and the FEM are similar. So it is reasonable to improve the calculation accuracy by λk . Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 Fig. 10. Calculation procedure of correction coeﬃcient λk . Fig. 11. TVMS obtained from different methods: (a) FEM, (b) PAM before revision, (c) PAM after revision. 271 272 Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 Table 5 Parameters of gear pair 2. Parameters The driving/driven gear Parameters The driving/driven gear Tooth number Z Module m (mm) Tooth width L (mm) Hub radius rint (mm) Torque load Tl (Nm) 20, 30 4 40 11.7, 38.3 98 Pressure angle α (°) Addendum coeﬃcient ha ∗ Tip clearance coeﬃcient c∗ Young’s modulus E (GPa) Poisson’s ratio ν 20 1 0.25 212 0.3 Table 6 Eight modiﬁcation quantities for driving gear. Modiﬁcation quantities Ca (μm) La (mm) Cq (μm) CL1 CL2 CL3 CL4 CL5 CL6 CL7 CL8 0 0 0 0 0 0 8 8 0 0 0 0 0 0 0.6 0.6 0 2 4 6 8 10 0 10 Fig. 12. TVMS under different modiﬁcation quantities: (a) PAM, (b) FEM. 3.2. Revision of PAM under lead crowning relief, tip relief and combination of both To further verify the proposed method, the gear pair in Ref.  is adopted, in which only the tooth of the driving gear is modiﬁed. And the detailed gear parameters are listed in Table 5 (deﬁned as gear pair 2). The TVMS obtained from the PAM and the FEM under different modiﬁcation quantities (CL1–CL6, see Table 6) is shown in Fig. 12a and Fig. 12b, respectively. The correction coeﬃcient λk under CL1-CL6 and the best ﬁtting curve are plotted in Fig. 13. The correction coeﬃcient increases with the increasing Cq . When Cq = 10 μm, the maximum error is about 13.1% before revision while about 4.9% after revision (see Table 7). The formula of λk can be expressed as: λk = 1.130 × 10−3 · Cq2 + 3.115 × 10−3 · Cq + 0.977, (17) The TVMS under CL1, CL6, CL7 and CL8 is shown in Fig. 14. It indicates that the PAM can consider the combined effects of tip relief and lead crowning relief on TVMS and accurately ﬁgure out TVMS under any given modiﬁcation quantities. 4. Effects of different parameters on TVMS The transmitting load will also affect the contact behavior between a mating toot pair under a certain amount of modiﬁcation. And tooth widths may inﬂuence the validity of slicing principle. In this section, the effects of tooth widths and torques on TVMS of spur gear pair 1 are discussed. Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 273 Fig. 13. The best ﬁtting curve of correction coeﬃcient λk . Table 7 Mesh stiffness obtained from PAM and FEM. Modiﬁcation quantities Mesh stiffness (× 108 N/m) FEM PAM before revision Error (%) PAM after revision Error (%) CL1 kA kB kmean kA kB kmean kA kB kmean kA kB kmean kA kB kmean kA kB kmean 7.357 5.031 6.781 6.277 4.491 5.813 5.525 4.052 5.140 5.033 3.713 4.706 4.739 3.474 4.411 4.464 3.258 4.193 7.770 5.027 7.133 6.445 4.455 6.001 5.517 3.999 5.174 4.830 3.629 4.557 4.301 3.321 4.080 3.881 3.061 3.695 5.6 0.1 5.2 2.6 0.8 3.2 0.1 1.3 0.7 0.1 2.3 3.2 9.2 4.4 7.5 13.1 6.0 11.9 7.608 4.922 6.969 6.348 4.388 5.928 5.550 4.023 5.213 4.999 3.756 4.723 4.649 3.590 4.383 4.335 3.419 4.143 3.4 2.2 2.8 1.1 2.3 2.0 0.5 0.7 1.4 0.7 1.2 0.4 1.9 3.3 0.6 2.9 4.9 1.2 CL2 CL3 CL4 CL5 CL6 Fig. 14. TVMS under different modiﬁcation quantities: (a) PAM after revision, (b) FEM. 274 Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 Fig. 15. TVMS of the spur gear pairs with modiﬁcation under different L: (a) PAM after revision, (b) FEM. Table 8 Mesh stiffness obtained from FEM and PAM after revision. L (mm) Mesh stiffness (× 108 N/m) PAM after revision FEM Error (%) 20 kA kB kmean kA kB kmean kA kB kmean kA kB kmean 1.630 1.330 1.585 2.776 2.328 2.704 3.634 3.109 3.543 4.304 3.738 4.202 1.633 1.331 1.572 2.744 2.232 2.638 3.656 2.970 3.511 4.445 3.619 4.283 0.2 0.1 0.8 1.2 4.1 2.5 0.6 4.5 0.9 3.3 3.2 1.9 40 60 80 4.1. Effects of tooth widths on TVMS Based on the proposed method and FE method, the TVMS under different tooth widths (L = 20 mm, 40 mm, 60 mm and 80 mm) and the modiﬁcation quantity CW5 is shown in Fig. 15. Two methods all show that the magnitude of mesh stiffness increases with the increase of L, the TVMS obtained from the revised PAM agrees well with that from the FEM and the maximum error is only 4.5%, and kmean is more accurate (maximum error is about 2.5%, see Table 8). This also suggests that the proposed method is also valid for wide-faced spur gears. 4.2. Effects of torques on TVMS TVMS under different transmission torques (Tl = 50 Nm, 100 Nm, 150 Nm and 200 Nm) and the modiﬁcation quantity CW5 is shown in Fig. 16. The ﬁgure shows that TVMS obtained from PAM agrees well with that from the FEM. The maximum error under double-tooth contact region is about 4.8%, and maximum error of mean stiffness is about 5.5% (see Table 9). The stiffness increases with the increasing Tl and increases slowly under larger torques. In addition, the region of single-tooth engagement becomes smaller. That is to say, the increasing torque Tl enhances the effects of the ETC on mesh stiffness. 5. Conclusions Based on the thin-slice method, a previously developed analytical model for healthy spur gear pairs is further improved to consider the effects of tooth modiﬁcations including tip relief and lead crowning relief. The proposed model is veriﬁed and revised by comparing with the ﬁnite element model (FEM), and the TVMS correction coeﬃcient is ﬁgured out based on the minimum mean square error. The revision process is elaborated by two cases with different modiﬁcation types (case 1: only lead crowning relief for driving and driven gear; case 2: lead crowning relief, tip relief and combination of both only for driving gear). The effects of tooth widths and torques on mesh stiffness are also discussed. Some detailed conclusions are summarized as follows: (1) Under case 1 before revision, the errors between two models are very small and the maximum error before revision is about 11% at Cq = 7.5 μm where Cq denotes the lead crowning quantity. However, the maximum error after revision is Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 275 Fig. 16. TVMS of the spur gear pairs with modiﬁcation under different Tl : (a) PAM after revision, (b) FEM. Table 9 Mesh stiffness obtained FEM and PAM after revision. Tl (Nm) Mesh stiffness (× 108 N/m) PAM after revision FEM Error (%) 50 kA kB kmean kA kB kmean kA kB kmean kA kB kmean 1.388 1.164 1.316 1.630 1.330 1.546 1.733 1.399 1.650 1.790 1.437 1.710 1.454 1.164 1.393 1.633 1.327 1.572 1.735 1.427 1.676 1.804 1.488 1.747 4.8 0 5.5 0.2 0.2 1.7 0.1 2.0 1.6 0.8 3.5 2.1 100 150 200 about 3.0% at Cq = 20 μm. Under case 2, the maximum error before revision is about 13.1% at Cq = 10 μm. However, the maximum error after revision is about 4.9% at Cq = 10 μm. (2) The proposed analytical method (PAM) is also valid for wide-faced spur gears. Under different tooth widths, the TVMS obtained from the revised PAM is in good agreement with that from the FEM and the maximum error is only 4.5%. Under different torques, the TVMS obtained from the PAM is also in good agreement with that from the FEM and the maximum error is 5.5%. With the increasing torque, the region of single-tooth engagement becomes smaller, i.e., the large torque will enlarge the region of the extended tooth contact. At the end, it should be noted that the FEM is the prerequisite to determine the correction coeﬃcient in order to make sure that the proposed analytical method can yield accurate results. Thus it is inevitable that the computation time will increase using the proposed method. However, once the correction coeﬃcient is determined, the calculation eﬃciency will be greatly improved for the studied gear pair in the optimizing process of relief parameters. In our future work, we will focus on the improvement of calculation accuracy by considering the effects of elastic couplings among thin slices or using analytical-FE method. Acknowledgment This project is supported by the National Natural Science Foundation (Grant no. 11772089), the Fundamental Research Funds for the Central Universities (Grant nos. N170308028, N160313004 and N160312001). Appendix A. The gear tooth is modeled as a nonuniform cantilever beam on root circle in this paper (see Fig. A.1). Radii of the base circle rb and root circle rf of the gear can be expressed as: rb = 1 1 mZ cosα , rf = mZ − (h∗a + c∗ )m, 2 2 (A.1) 276 Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 Fig. A.1. Geometric model of the gear proﬁle. where m is the module, Z is the number of teeth, α is the pressure angle of the gear pitch circle, ha ∗ is the addendum coeﬃcient and c∗ is the tip clearance coeﬃcient. If the gear is a standard spur gear with ha ∗ = 1, c∗ = 0.25 and α = 20° Tooth proﬁle curve can be divided into four parts: addendum curve AB, involute curve BC, transition curve CD and dedendum curve DE (see Fig. A.1). The transition curve is cut out by the cutter tip. From the perspective of the machining process of a gear tooth, transition curve is the tooth proﬁle between the involute starting point and the root circle, and no matter how many teeth the gear has, the transitional part will always exist. The shape of the transition curve is directly dependent on the shape of the cutter tip. When the shape of the cutter tip is ordinary ﬁllet, the transition curve equations are [25,52]: x1 = r × sin( ) − (a1 /sin γ + rρ ) × cos(γ − ) , ( α ≤ γ ≤ π /2 ), y1 = r × cos( ) − (a1 /sin γ + rρ ) × sin(γ − ) (A.2) where r is the radius of the pitch circle; = (a1 / tan γ + b1 )/r, a1 = (h∗a + c∗ ) × m − rρ , b1 = π m/4 + h∗a m tan α + rρ cos α ; rρ = c∗ m/(1 − sin α ). Equations of involute curve are expressed as follows: x2 = rb [(τ + θb )cosτ − sinτ ] , (αC ≤ τ ≤ αa ), y2 = rb [(τ + θb )sinτ + cosτ ] (A.3) where τ (α C ≤ τ ≤ α a ) is the pressure angle of the arbitrary point at the involute curve, in which αC = arccos(rb /rC ),αa = arccos (rb /ra ) is the pressure angle of the involute starting point and the addendum circle, respectively; rC = (rb tan α − h∗a m/sin α )2 + rb2 is the radius of the involute starting point circle. θ b is the half tooth angle on the base circle of the gear, θb = 2πZ + inv α . β is the operating pressure angle and the coordinate of contact point (τ = β ) can be expressed as follows: xβ = rb [(β + θb )cosβ − sinβ ] , yβ = rb [(β + θb )sinβ + cosβ ] (A.4) Iy 1 , Ay 1 , Iy 2 , Ay 2 represent the area moment of inertia and the cross-sectional area, respectively, which can be calculated by Eqs. (A.5) and (A.6): Ay1 = 2x1 L, Ay2 = 2x2 L, (A.5) 2 3 2 x L, Iy2 = x32 L, 3 1 3 (A.6) Iy1 = dy1 dγ and dy2 dτ can be calculated as follows: dy1 = dγ α1 sin α1 / tan γ +b1 r tan2 γ (1 + tan2 γ ) α1 cos γ α1 / tan γ + b1 + sin γ − 2 r sin γ Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278 α1 α1 / tan γ + b1 − + rρ cos γ − sin γ r dy2 = rb (τ + θb ) cos dτ τ, α1 1 + tan2 γ 1+ r tan2 γ 277 (A.7) (A.8) References  T.J. Lin, Z.Y. He, Analytical method for coupled transmission error of helical gear system with machining errors, assembly errors and tooth modiﬁcations, Mech. Syst. Signal Process. 91 (2017) 167–182.  S.S. Wang, C.C. Zhu, C.S. Song, et al., Effects of gear modiﬁcations on the dynamic characteristics of wind turbine gearbox considering elastic support of the gearbox, J. Mech. Sci. Technol. 31 (3) (2017) 1079–1088.  M. Maatar, P. Velex, Quasi-static and dynamic analysis of narrow-faced helical gears with proﬁle and lead modiﬁcations, J. Mech. Des. 119 (1997) 474–480.  J. Bruyère, X. Gu, P. Velex, On the analytical deﬁnition of proﬁle modiﬁcations minimising transmission error variations in narrow-faced spur helical gears, Mech. Mach. Theory 92 (2015) 257–272.  P. Velex, M. Chapron, H. Fakhfakh, et al., On transmission errors and proﬁle modiﬁcations minimising dynamic tooth loads in multi-mesh gears, J. Sound Vibr. 379 (2016) 28–52.  P. Velex, J. BruyeR̀e, D.R. Houser, Some analytical results on transmission errors in narrow-faced spur and helical gears: inﬂuence of proﬁle modiﬁcations, J. Mech. Des. 133 (2011) 031010-1-11.  C.J. Bahk, R.G. Parker, Analytical investigation of tooth proﬁle modiﬁcation effects on planetary gear dynamics, Mech. Mach. Theory 70 (2013) 298–319.  H. Ma, X. Pang, R.J. Feng, et al., Evaluation of optimum proﬁle modiﬁcation curves of proﬁle shifted spur gears based on vibration responses, Mech. Syst. Signal Process. 70,71 (2016) 1131–1149.  J. Wei, A.Q. Zhang, D.T. Qin, et al., A coupling dynamics analysis method for a multistage planetary gear system, Mech. Mach. Theory 110 (2017) 27–49.  X.H. Liang, M.J. Zuo, Z.P. Feng, Dynamic modeling of gearbox faults: a review, Mech. Syst. Signal Process. 98 (2018) 852–876.  H. Ma, J. Zeng, R.J. Feng, et al., Review on dynamics of cracked gear systems, Eng. Fail. Anal. 55 (2015) 224–245.  J. Wei, P.X. Bai, D.T. Qin, et al., Study on vibration characteristics of fan shaft of geared turbofan engine with sudden imbalance caused by blade off, J. Vibr. Acoust. 140 (2018) 041010-1-14.  H. Ma, Z.W. Li, M J Feng, et al., Time-varying mesh stiffness calculation of spur gears with spalling defects, Eng. Fail. Anal. 66 (2016) 166–176.  D.C.H. Yang, J.Y. Lin, Hertzian damping, tooth friction and bending elasticity in gear impact dynamics, J. Mech. Transm. Autom. Des. 109 (1987) 189–196.  X.H. Tian, Dynamic Simulation for System Response of Gearbox Including Localized Gear Faults M.S. thesis, University of Alberta, Edmonton, 2004.  S. Wu, M.J. Zuo, A. Parey, Simulation of spur gear dynamics and estimation of fault growth, J. Sound Vibr. 317 (2008) 608–624.  P. Sainsot, P. Velex, O. Duverger, Contribution of gear body to tooth deﬂections–a new bidimensional analytical formula, J. Mech. Des. 126 (2004) 748–752.  F. Chaari, T. Fakhfakh, M. Haddar, Analytical modeling of spur gear tooth crack and inﬂuence on gear mesh stiffness, Eur. J. Mech. A/Solids 28 (2009) 461–468.  Z.G. Chen, Y.M. Shao, Mesh stiffness calculation of a spur gear pair with tooth proﬁle modiﬁcation and tooth root crack, Mech. Mach. Theory 62 (2013) 63–74.  Z.G. Chen, J. Zhang, W.M. Zhai, et al., Improved analytical methods for calculation of gear tooth ﬁllet-foundation stiffness with tooth root crack, Eng. Fail. Anal. 82 (2017) 72–81.  Z.G. Chen, W.M. Zhai, Y.M. Shao, et al., Analytical model for mesh stiffness calculation of spur gear pair with non-uniformly distributed tooth root crack, Eng. Fail. Anal. 66 (2016) 502–514.  Z.G. Chen, Y.M. Shao, Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth, Eng. Fail. Anal. 18 (2011) 2149–2164.  W.N. Yu, C.K. Mechefske, Analytical modeling of spur gear corner contact effects, Mech. Mach. Theory 96 (2016) 146–164.  Q.K. Han, J. Wang, Q. Li, Analysis of parametric stability for a spur gear pair system considering the effect of extended tooth contact, Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 223 (2009) 1787–1797.  H. Ma, R.Z. Song, X. Pang, et al., Time-varying mesh stiffness calculation of cracked spur gears, Eng. Fail. Anal. 44 (2014) 179–194.  H. Ma, X. Pang, R.J. Feng, et al., Improved time-varying mesh stiffness model of cracked spur gears, Eng. Fail. Anal. 55 (2015) 271–287.  H. Ma, J. Zeng, R.J. Feng, et al., An improved analytical method for mesh stiffness calculation of spur gears with tip relief, Mech. Mach. Theory 98 (2016) 64–80.  H. Ma, M.J. Feng, Z.W. Li, et al., Time-varying mesh characteristics of a spur gear pair considering the tip-ﬁllet and friction, Meccanica 52 (7) (2017) 1695–1709.  H. Liu, C. Zhang, C.L. Xiang, et al., Tooth proﬁle modiﬁcation based on lateral- torsional-rocking coupled nonlinear dynamic model of gear system, Mech. Mach. Theory 105 (2016) 606–619.  O.D. Mohammed, M. Rantatalo, J. Aidanpaa, Improving mesh stiffness calculation of cracked gears for the purpose of vibration-based fault analysis, Eng. Fail. Anal. 34 (2013) 235–251.  S. Zouari, M. Maatar, T. Fakhfakh, et al., Three-dimensional analyses by ﬁnite element method of a spur gear: effect of cracks in the teeth foot on the mesh stiffness, J. Fail. Anal. Prevent. 7 (2007) 475–481.  H. Ma, R.Z. Song, X. Pang, et al., Fault feature analysis of a cracked gear coupled rotor system, Math. Probl. Eng. 2014 (2014) 832192.  I. Howard, S.X. Jia, J.D .Wang, The dynamic modeling of a spur gear in mesh including friction and a crack, Mech. Syst. Signal Process. 15 (2001) 831–853.  S. Jia, I. Howard, Comparison of localised spalling and crack damage from dynamic modelling of spur gear vibrations, Mech. Syst. Signal Process. 20 (2006) 332–349.  J.D. Wang, Numerical and Experimental Analysis of Spur Gears in Mesh Thesis, Curtin University of Technology, Australia, 2003.  H. Ma, J. Yang, R.Z. Song, et al., Effects of tip relief on vibration responses of a geared rotor system, Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 228 (2014) 1132–1154.  H. Ma, X. Pang, Q.B. Wang, et al., Commentary on effects of tip relief on vibration responses of a geared rotor system, Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 23 (11) (2017) 2159–2169.  S.T. 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 (2007) 88–114.  S.T. 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 (2007) 698–726.  S.T. 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.  A. Fernandez del Rincon, F. Viadero, M. Iglesias, et al., A model for the study of meshing stiffness in spur gear transmissions, Mech. Mach. Theory 61 (2013) 30–58. 278 Y. Sun et al. / Mechanism and Machine Theory 129 (2018) 261–278  A. Fernandez del Rincon, M. Iglesias, A. de-Juan, et al., Gear transmission dynamic: effects of tooth proﬁle deviations and support ﬂexibility, Appl. Acoust. 77 (2014) 138–149.  M. Iglesias, A. Fernandez del Rincon, A. Magdalena De-Juan, et al., Planetary gear proﬁle modiﬁcation design based on load sharing modelling, Chin. J. Mech. Eng. 28 (2015) 810–820.  M. Iglesias, A. Fernandez del Rincon, A. De-Juan, et al., Advanced model for the calculation of meshing forces in spur gear planetary transmissions, Meccanica 50 (2015) 1–26.  J.D. Smith, in: Gear Noise and Vibration, Marcel Dekker, Cambridge, 2003, pp. 35–40.  Q.B. Wang, P. Hu, Y.M. Zhang, et al., A model to determine mesh characteristics in a gear pair with tooth proﬁle error, Adv. Mech. Eng. 2014 (2014) 751476.  Q.B. Wang, Y.M. Zhang, A model for analyzing stiffness and stress in a helical gear pair with tooth proﬁle errors, J. Vibr. Control 23 (2) (2017) 272–289.  M. Ajmi, P. Velex, A model for simulating the quasi-static and dynamic behaviour of solid wide-faced spur and helical gears, Mech. Mach. Theory 40 (2005) 173–190.  X. Gu, P. Velex, P. Sainsot, et al., Analytical investigations on the mesh stiffness function of solid spur and helical gears, J. Mech. Des. 137 (2015) 063301–1–7.  Q.B. Wang, B. Zhao, Y. Fu, et al., An improved time-varying mesh stiffness model for helical gear pairs considering axial mesh force component, Mech. Syst. Signal Process. 106 (2018) 413–429.  M.J. Feng, H. Ma, Z.W. Li, et al., An improved analytical method for calculating time-varying mesh stiffness of helical gears, Meccanica 53 (2018) 1131–1145.  Z. Guo, Y.D. Zhang, Parameterization precise modeling of involutes spur gear based on APDL, Mach. Manuf. Res. 39 (2010) 33–36 (In Chinese).