半自動平壓模切機(jī)的設(shè)計【說明書+CAD】,說明書+CAD,半自動平壓模切機(jī)的設(shè)計【說明書+CAD】,半自動,平壓模切機(jī),設(shè)計,說明書,仿單,cad Tribology International 40 (2007) Arto Design, 4 for foundation plays r 2005 Elsevier Ltd. All rights reserved. Keywords: Helical gear; Contact; Deformation; Load distribution; Modeling a realistic analysis of helical gear contact also requires given operating conditions. Gear contact ratio 12 and equal load distribution in the case of two teeth in contact finite element method with spur gears 24. Coy and Chao with 3D finite elements 8,9. This is mainly because in this case FEM contact modeling is computationally expensive and time-consuming due to the small grid size which is ARTICLE IN PRESS necessary on the gear flank surface. Different methods have been introduced to overcome this problem. Vedmar 10 separated structural and contact analysis by combining the 0301-679X/$-see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.triboint.2005.11.004 C3 Corresponding author. Tel.: +358331154442. E-mail addresses: juha.hedlundtut.fi (J. Hedlund), arto.lehtovaaratut.fi (A. Lehtovaara). information on structural deformations, such as tooth deflection. The majority of gear contact analyses within tribological studies are made on spur gears. Typically, teeth contact through the line of action is modeled as a constantly changing roller contact, whose radius, speed and load are approximated from ideal involute gear geometry in the 5 studied finite element grid size dimensions to cover the Hertzian contact. Du et al. 6 and Arafa et al. 7 later enhanced contact modeling as a part of structural analysis by using gap elements for the calculation of spur gear deformation. Only a few helical gear contact studies, which include structural deformations of the gear, have been performed 1. Introduction Helical gears are generally used in industry and their contact behavior deserves more attention to establish a realistic base for the detailed study of gear friction, wear and life. The gear contact stresses derived from tooth contact forces and geometry are very important for determining gear pitting, i.e. life performance. Tooth contact forces along the line of action depend essentially on load sharing between meshing teeth, and, therefore, (half of the single-tooth load) are often assumed. Deforma- tions are calculated according to the Hertz line contact theory, otherwise assuming rigid tooth behavior. Some studies are made by slicing the helical gear to a series of spur gears and treating these slices as spur gears 1. Finite-element-based calculation models are widely accepted for calculating structural deformations and stresses in spur and helical gears in the case of concentrated loads. In gear transmission and dynamic analyses, typical deformation studies have used the two-dimensional (2D) Modeling of helical gear contact Juha Hedlund C3 , Tampere University of Technology, Machine Available online Abstract The majority of gear tribological studies are made on spur gears. contact behavior deserves more attention to establish a realistic base modeling of helical gear contact with tooth deflection. A calculation surface profiles are constructed from gear tool geometry by simulating elements for the calculation of tooth deflection including tooth bending, contact analysis with structural analysis to avoid large meshes. Tooth load sharing between the meshing teeth, whereas contact flexibility calculation methods was also studied. 613619 with tooth deflection Lehtovaara P.O. Box 589, 33101 Tampere, Finland January 2006 However, helical gears are generally used in industry, and their detailed friction, wear and life studies. This study focuses on the model for helical gear contact analysis is introduced. Helical gear the hobbing process. The model uses three-dimensional finite shearing and tooth foundation flexibility. The model combines flexibility was found to have an essential role in contact only a minor role. The capability of different local contact ARTICLE IN Nomenclature a contact ellipse radii b contact ellipse radii b g gear width d contact deformation at center of contact E elasticity modulus E 0 reduced modulus of elasticity 2=E 0 1 2 1 C0 v 2 1 =E 1 1 C0 v 2 2 =E 2 C138 et ij difference between contact surface profiles F c calculated total contact force F ij force at surface node i,j h 1 distance from pitch line i,j indicates grid nodes K ij stiffness value at grid point (i,j) Keq ij reduced total stiffness L effective length of the line of action M g1 rotation matrix M 2g translation matrix M 21 transformation matrix M 12 transformation matrix m n normal module p contact pressure J. Hedlund, A. Lehtovaara / Tribology614 finite element method and the Weber et ij 0. (11) The FEM-based contact model takes into account structure boundaries, i.e. no half space assumption is needed. 5. Result and discussion 5.1. Contact model test case The FEM-based contact model was tested against ARTICLE IN PRESS International 40 (2007) 613619 separation s(x) must be equal to the rigid-body movement d and outside the real contact greater than d as follows: uxsxd; px40, (5) uxsx4d; px0. (6) In addition, the resulting pressure distribution must satisfy the force balance in normal direction with total force W applied on the contacting bodies. It follows: Z 1 C01 Z 1 C01 px;ydxdy W. (7) For 3D elastic contact problems the Boussinesq for- mulation can be used. The basic equation for surface pressuredeformation of surfaces in z-direction u(x) between semi-infinite solids is 14 ux 2 pE 0 Z 1 C01 Z 1 C01 px 0 ;y 0 dx 0 dy 0 x C0 x 0 2 y C0 y 0 2 q . (8) Contact problems with arbitrary undeformed surface profiles need to be solved by numerical methods. The numerical solution process is typically iterative, because pressure distribution and real contact area distribution are unknown, whereas total load, material properties and initial contact geometry are known parameters. The well-known Hertz solution to the contact problem is based on cases, where the undeformed geometry of contacting solids can be represented in general terms by two ellipsoids. The solution requires the calculation of ellipticity parameter and complete elliptic integrals. A simplified solution of the classical Hertz theory of elliptical contact solution is presented in 15. This calculation method is non-iterative and fast. The solution includes an elastic half space assumption. 4.1. FEM-based contact model Initially, geometrical overlap between the contact bodies is chosen to produce a calculation domain greater than the final contact area. The loading vector acting in the calculation domain is a combination of nodal displace- ments (overlap) and zero loads, producing non-homoge- nous FEM boundary conditions. This approach is applied to calculate force distribution over the domain. Force distribution F ij , which is located at the surface, is used for the calculation of contact stiffness values K ij in every node (i,j) as follows: K ij F ij B ij . (9) Reduced stiffness values in the grid between the contact bodies are determined as follows: Keq ij K ij1 K ij2 K ij1 K ij2 . (10) J. Hedlund, A. Lehtovaara / Tribology616 After the reduced contact stiffness is established and the initial separation of the undeformed surfaces is known, simplified Hertzs formulas 15 in the case of circular and elliptical contact. The test case dimensions and load conditions are shown in Table 1 and the results in Table 2. Both surfaces have the same material properties. The elliptical test case was chosen to evaluate the crowned spur gear contact. Mesh size was limited to 4500 elements per contact body in the test case calculation. One surface calculation domain consists of 900 nodal points. The deformed contact surface and calculation grid are shown in the case of circular contact (Fig. 4). The results show that the FEM-based contact model gives reasonable approximation of contact parameters taking into account the fairly coarse grid size. The minor semi-axis of the ellipse especially suffers from grid dimensions. Mesh size and shape have a certain effect on the results unless the mesh size is fine enough. It is obvious that the accuracy of results will decrease as the ellipticity ratio of the contact increases. Table 2 Comparison results Model a (mm) b (mm) d (mm) p 0 (GPa) Hertz/circular 0.37 0.37 18.1 3.56 FEM/circular 0.4 0.4 19.1 3.11 Table 1 Test case specifications Case Circular Elliptical E (GPa) 206 n 0.3 r 1x (mm) 12.3 12.3 r 2x (mm) 18.6 18.6 r 1y (mm) 12.3 1000 r 2y (mm) 18.6 1000 W (N) 1000 5000 Hertz/elliptical 0.27 4.13 20.7 2.11 FEM/elliptical 0.4 4.0 20.2 2.36 Helical gear contact was studied with the FZG test rig related gear data shown in Table 3. In this example, total contact ratio was over 2, which means that there are always at least two teeth pairs in contact. The test case element mesh and the mating tooth surface for pinion are shown in Fig. 5. The element mesh of the gear is equal. The calculated force distribution curves of the test case are shown in Fig. 6. The line of action is described with non-dimensional parameter c x/L, where L is the effective length of the line of action. Fig. 6 shows that the general trend in force distribution remains in the different model test cases. The area between the sharp edges near the middle represents a situation where three teeth pairs carry the total load. In the case of helical gears, this transition from two to three teeth and vice versa occurs quite smoothly. Contrary to spur gears, single tooth force is high, when all three teeth are in contact. This is because flexibility is lower at the tooth tip corners than in the middle and root area. The different test cases produce clear differences in contact force behavior. The flexibility of tooth foundation has the most crucial effect on the distribution of contact force along the line of action. Contact flexibility has less ARTICLE IN PRESS International 40 (2007) 613619 617 which makes the contact curvature change along the contact line and the line of action. Also, load sharing between the gear teeth is complicated partly because the total force is often shared between three teeth pairs. The realistic force acting on a single tooth at any location along the line of action is the basic parameter in tribological contact studies. Force distribution between meshing teeth pairs was studied in the developed model. Four different test cases were established where the model allows: (1) tooth and tooth foundation deformations with rigid contact; (2) tooth and tooth foundation deformations with contact deformation; In case (3) (4) The calcul nine solved single teeth pair vector calcul solved. Contact on distribut radius a mean by helical gear mesh, contact is more complex than in the of spur gears. The contact area has a real 3D nature, 5.2. Helical gear contact case Fig. 4. Deformed contact surface. J. Hedlund, A. Lehtovaara / Tribology tooth deformation with rigid foundation and rigid contact; tooth and contact deformation with rigid foundation. stiffness vector for a tooth pair consists of 30 ation points along the line of action. Two thousand hundred and twenty-six elements per tooth were at every calculation point. The stiffness vector of a gear pair was copied with offset to represent other in contact. The total mesh stiffness vector of a gear was obtained by summing up these single stiffness s. Finally, displacement along the line of action was ated and the contact force of a single tooth was stiffness along the line of action was calculated the Hertz line contact formula by using the force ion from test case 1. The values of tooth flank are calculated over the contact line and estimated as value. The overall reduced stiffness was obtained the iterative method, as in Eq. (10). impact, but interestingly, it shifts the force distribution curve slightly to the right in certain areas. This is because the combined contact radius is asymmetric over the pitch point. The load distribution was observed to be sensitive to stiffness properties at the start and end points of the line of action. One contact point (c C00.236), shown in Fig. 6, was chosen for a closer study. This contact point was estimated Table 3 Test case gear data m n (mm) 2.75 b g (mm) 20 b (deg) 12 z 1 26 z 2 39 e g 2.084 T (Nm) 143 Fig. 5. Element mesh of pinion tooth used in the test case. ARTICLE IN PRESS force along the line of action. Table 4 Contact specifications E (GPa) 220 n 0.3 r 1x (mm) 8.7 r 2x (mm) 23.2 r 1y (mm) 7000 r 2y (mm) 7000 with the Hertzian elliptical contact formula 16 and the FEM-based contact model. This contact point represents the situation where two teeth pairs carry the total load. The chosen contact situation was calculated with two different forces corresponding to the calculated test cases 2 and 4. The load sharing differs depending on the modeling of the tooth foundation. The radius of surface profiles was approximated with the circum circle method and these values are shown in Table 4. Some crowning was included Fig. 6. Single tooth contact J. Hedlund, A. Lehtovaara / Tribology618 in the contact line direction. At the studied contact point, different load sharing between cases 2 and 4 has only a minor effect on contact parameters (Table 5). However, force difference between the different test cases is greater at some other contact points, as shown in Fig. 6. Especially in the beginning of gear pair engagement, force difference between the different test cases may be remarkable. In the studied helical gear contact, the estimated ellipticity ratio becomes very high. This is the case even when the studied gear was rather narrow. This is the main reason why the calculated FEM results are less accurate than in the earlier contact model test case. How accurate the assumption of elliptical contact is in helical gear contact is not studied here. Future studies will determine the final capability of the used contact models. However, the FEM-based contact model has potential especially in the calculation of edge contacts, i.e. in cases which are not fully covered by analytical formulas. 6. Conclusions A calculation model for the analysis of helical gear contact is introduced. Helical gear surface profiles are constructed from gear tool geometry by simulating the International 40 (2007) 613619 hobbing process. This procedure allows deviations from ideal involute geometry. The gear pair contact line is numerically defined direct from the gear surface geometry. The model uses 3D finite elements for the calculation of tooth deflection including tooth bending, shearing and tooth foundation flexibility. The model combines contact analysis and structural analysis to avoid large meshes. The flexibility of tooth foundation was found to have an essential role in contact load sharing between the meshing teeth, whereas contact flexibility plays only a minor role. This indicates that reasonable distribution of tooth contact force along the line of action may be generated by using flexible teeth and flexible tooth foundation, but allowing rigid contact. Table 5 Calculation results Case Case 4 Case 2 Hertz FEM Hertz FEM W case (N) 1570 1723.5 a (mm) 5.7 4.5 5.8 4.5 b (mm) 0.12 0.2 0.12 0.2 d (mm) 5.6 4.94 6.1 5.27 p 0 (GPa) 1.15 1.28 1.18 1.37 The FEM-based contact model gives a reasonable approximation of contact parameters when the mesh size is fine enough. Contact shapes, such as in helical gears, require small element size, i.e. a large number of elements to avoid element dimensional distortion. However, the FEM-based contact model has potential in calculating edge contacts. References 1 Flodin A. Simulation of mild wear in helical gears. 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Simplified solution for stresses and deformations. J Lubr Technol 1983;105:1717. 16 Johnson KL. Contact mechanics. Cambridge: Cambridge University Press; 1985 451pp. ARTICLE IN PRESS J. Hedlund, A. Lehtovaara / Tribology International 40 (2007) 613619 619