1、I 2000FTM12 The Finite Strips Methods as an I Alternative to the Finite Elements in Gear Tooth Stress and Strain Analysis by: C. Gosselin and R. Guilbault, Laval University and P. Gagnon, National Optics Institute American Gear Manufacturers Association I 1 I TECHNICAL PAPER COPYRIGHT American Gear
2、Manufacturers Association, Inc.Licensed by Information Handling ServicesThe Finite Strips Methods as an Alternative to the Finite Elements in Gear Tooth Stress and Strain Analysis Claude Gosselin Optics Institute and R. Guilbault, Laval University and P. Gagnon, National The statements and opinions
3、contained herein are those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract The Finite Element Method is widely used in the gear industry to assess gear tooth strength and stiffness. Although automatic meshing softwar
4、e is available, the Finite Element Method still requires a significant amount of manipulation which, though acceptable as a verification tool when the design is completed, may be considered prohibitive at the design stage. The Finite Strip Method, which may be considered as a subset of the Finite El
5、ement Method, is presented as an alternative that requires limited manipulation and computer memory while offering super-fast computation along with precision comparable to that of Finite Elements. The formulation allows for easy integration to a gear tooth geometry simulation software since, beside
6、s tooth thickness distribution, several geometrical parameters are derived directly from the gear blank, and most common gear tooth geometries may be accommodated. Copyright O 2000 American Gear Manufacturers Association 1500 King Street, Suite 201 Alexandria, Virginia, 22314 October, 2000 ISBN: 1-5
7、5589-773-8 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesTHE FINITE STRIP METHOD AS AN ALTERNATIVE TO FINITE ELEMENTS IN GEAR TOOTH STRESS AND STRAIN ANALYSIS 1. Introduction Claude Gosselin, Raynald Guilbault2, Philippe Gagnod Professor, Department
8、of Mechanical Engineering Laval University, Quebec, Canada GI K-7P4 Ph.D. candidate, Department of Mechanical Engineering Laval University, Quebec, Canada GI K-7P4 Senior Opto-Mechanical Research Engineer National Optics Institute, Qubec Canada GI K-7P4 The knowledge of tooth stiffness is fundamenta
9、l in the calculation of the load sharing between meshing gear teeth, which can then lead to a precise evaluation of the contact and bending stresses. Several analytical stiffness and stress models have been presented in the past I, 2, 3, 4, 51 to solve this problem, which is compounded by the comple
10、xity of the tooth geometry. For spur and helical gear teeth, where the tooth geometry remains constant in the lengthwise direction, reasonable agreement is obtained using analytical models; for spiral-bevel and hypoid gear teeth, where the tooth is curved and wound on a cone, and tooth thickness and
11、 height varies along tooth facewidth, so far only the Finite Element Method (FEM) provides acceptable results. This paper presents the Finite Strip Method (FSM), an altemative to FEM and analytical formulations, to obtain reliable results with reduced preparation and solution time and that can be in
12、tegrated to a gear tooth geometry simulation software. Cheung 161 and, independently, Powell and Ogden 7, introduced the FSM which can be considered as a special case of the FEM: the Finite Strip is a 2-D element for the analysis of plates, based on simple polynomial functions in one direction and c
13、ontinuously differentiable smooth series in the other direction. Mawenya and Davies 8 included the effect of transverse shear to make the FSM applicable to thick, thin and sandwich plates. The use of cubic B-splines, introduced by Cheung, Fan and Wu 9, ensures C2- continuity and their local characte
14、ristics allow different boundaw conditions. O The analysis of gear teeth by the FSM, considered as thick plates which may show either uni- directional or bi-directional thickness variation and constant or non-constant depth, is the focus of this paper. To extend the application of the FSM from thin
15、to thick plates, Mindlins theory is used in the variant of the FSM presented in this paper Il. FEM deflection and stress results for the clamped- free case are compared to those obtained by the FSM and show good agreement. 2. Main Nomenclature a strip length am b strip width By fD. elasticity matrix
16、 f,“ kt stress concentration factor vector of nodal parameters of node i and function m strain displacement matrix for nodal line i and mth series term load vector for nodal line i and mth series term 9 6 W E Y V 0x-y- r 8. r-X-s finite strip stiffness matrix for terms m, n and nodal lines i, j tran
17、sverse loading per unit area transverse displacement displacement function generalized strain vector mth term of displacement function for nodal line i Poisson ratio nodal rotation about the x-y-r-A-s axis FSM thin plates with variable thickness in the longitudinal direction were treated by Uko and
18、Cusens IO. zxy-yr-xz shear stress in the xy-yz-xz plane ox-y x-y normal stress 1 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services2. Finite Strip Formulation 2.1 Strip of constant height (rectangular) A typical Finite Strip discretisation of a rectangul
19、ar plate, such as the tooth of a spur, helical or face gear, is presented in Figure 1. Strips of width b and length a are parallel to the y-axis and connected by nodal lines that lead to the definition of the displacement function 12. Strip widths are normally kept constant. Y a) Finite Strip of a r
20、ectangular plate 2, w b) spur gear tooth c) helical gear tooth d) Face gear tooth Figure 1 The displacement function 6 for a linear strip is the sum of a series of I terms: (1 ) m=I i=I where series (PL, and $2 : is a combination of polynomial Ni and Using Mindlins plate theory, the mid-plane displa
21、cement vector is (figure 2): and the vector of nodal parameters of node i for the mth function: The generalized strain vector E can be expressed as: (5) where w is the transverse displacement and , and O, the section rotations. denotes an additional rotation due to transverse shear deformation. 2 CO
22、PYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Servicesafter deformation ,?E m=i =i Figure 2. Cross-section of thick plate deformation As in the Finite Element displacement formulation, the Finite Strip stiffness matrix for functions m, n and nodal lines i, j c
23、an be written as: rz vz o vz z o 2 O0 o z(1- v) 00- The load vector for function m and nodal line i is obtained as follows: o o- O0 O0 O 1-v 2 1 -v 0- - 2d where transverse load q is a function of x and y, and can be localized or distributed. Forces and moments about axes x and y are respectively ap
24、plied using the first, second and third column terms of matrix t in equation (2). Bmk Individual Finite Strip stiffness matrices, nodal parameter vectors and load vectors are assembled in the usual manner to form the global equation: Solving this system yields the nodal parameters. Introducing the s
25、olution in equation (1) directly gives displacements for any strip. Figure 3 shows the load vectors, nodal parameters and stiffness submatrices assembly for a three strip, four node model for the clamped-free case. Solution of this global system yields the nodal parameters. Stri 1 2 3 1 2 3 1 2 3 Ha
26、n bandwidth _h_.l Figure 3: Global System for a 3 strip, 4 node model (clamped-free case). Introducing this solution into equation (1) provides the displacements for any strip. Stresses are obtained as: o= Loo o 2.2 Strip of variable height (circular) To deal with plates of variable height, such as
27、the tooth of straight bevel gears, the formulation presented above is modified such that equations are expressed in polar coordinates 12. Figure 4 presents a typical Finite Strip discretisation of a plate of variable height. Coordinates x and y are respectively replaced by r and h, strip width b is
28、along radius r, length a now corresponds to the sector angle and plate total width is the difference between outer and inner radii R, and Ri. 3 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services2. w a) Finite Strip of a circular plate b) straight-bevel g
29、ear tooth Figure 4 At this point, displacement function 6, nodal parameters vector am and elasticity matrix D are expressed as functions of rhz instead of xyz, and the generalized strain vector E is given in polar coordinates: The Finite Strip stiffness matrix in the rAz coordinate system is given b
30、y: and the load vector for a circular strip is obtained as follows: 2.3 Curved strip of variable height (helicoid of variable height and thickness) In order to analyze gear teeth of variable height and width, curved along a spiral wound on a cone 13, such as spiral bevel and hypoid gear teeth (figur
31、e 5), the following formulation uses a coordinate system akin to that of a strip of variable height, where curved axis s replaces axis x, and axis r, normal to axis s, replaces axis y. Strip width b corresponds to a position along s and height parameter a corresponds to the difference in height betw
32、een the face and root cones. Upon solution, the strip is unwound in a plane such that the stiffening effect of the tooth lengthwise curvature is not accounted for. This has been found to be a valid assumption for gear teeth where the tooth lengthwise curvature is normally small enough to be neglecte
33、d. However, the curvature of the tooth root caused by the spiral wound on a cone is accounted for by radii R, and R2, as is shown in figure 5. The displacement function 6, nodal parameters am and material elasticity matrix are now function of SIZ and deformation vector E becomes: aw - R, 1- drds (1
34、4) Ri and the load vector is given as: 4 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling ServicesThis formulation can be applied to both spiral-bevel and hypoid gear teeth, as the tooth thickness distribution at the nodal points is provided by the tooth geomet
35、ry simulation program, thereby accounting for unequal pressure angles on the concave and convex tooth flanks of hypoid gears. a) Circular Finite Strip of variable height and radii RI I b) spiral-beveVhypoid tooth 3. Parameters for the Finite Strip Solution For the solution of the FSM equations, seve
36、ral parameters are to be provided, some of which are readily available from the gear blank dimensions as given below. Three matrices respectively containing the normal tooth thickness, the lengthwise and the profilewise positions of each nodal point are also needed. For spiral-bevel and hypoid gears
37、, these can only be obtained from a gear tooth geometry simulation software as is used in the examples presented below. For all cases: R N number of nodes (5 is recommended) number of strips along the tooth 3.1 Spur, helical and face gear teeth (Figure 1): s tooth length (along helix) b strip width
38、(=-) a S N linear strip height (tooth height) a H distance between nodes (=-) R-1 3.2 Straight bevel gear tooth (Figure 4): s tooth length (along helix) b stripwidth (=-) a angular strip height (difference between S N face and root angles) a H angle between nodes (=-) R-1 hi tooth height at toe hi R
39、i inner radius (=-) a Ro outer radius (= Ri + s) 3.3 Spiral-bevel/hypoid gear teeth (Figure 5): S b a H RI R2 tooth length (along spiral) S strip width (=-) angular strip height face and root angles) angle between nodes root radius at toe root radius at heel N (difference between a (= -) R-1 Figure
40、5 5 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services4. Tooth Deformation and Stress Under Load Module (mm) In order to demonstrate the validity of the FSM, several types of gear teeth were submitted to comparison with FEM results in terms of tooth disp
41、lacement and stress under load. To cancel out tooth surface deformation in the vicinity of the applied loads, FEM results were taken on the opposite tooth flank. The FSM is insensitive to surface deformation caused by a concentrated load. 3,175 I 5,08 4.1 FEU and FSM meshes Press. Angle (O) Addendum
42、 Dedendum Figures 6 and 7 respectively show the FEM and FSM meshes for four gear tooth types, namely spur, straight-bevel, face and spiral-bevel. The geometric parameters of the gear sets are given in table 1. 20 20 3,175 mm 2,89“ 4,011 mm 3,44“ Loading, shown in figure 7, is applied as a series of
43、concentrated loads along tooth tip and totaling 1000 N in all cases. FEM models were used with the hub, as shown in figure 6, for stress results and without the hubs for the displacement results, as the FSM does not account for the hub in the current formulation. Facewidth (mm) I 25,4 Table 1. Geome
44、tric parameters 25,4 I Spur I Straight bevel I Press. Angle (O) I 25 I 20 Addendum I 2.381 mm I 5.25“ I (b) Straight-bevel (c) Face gear Dedendum I 3,969mm I 4,20 Facewidth (mm) I 20 I 35 I Cone Angle (O) I 90 I 45 I Figure 6: FEM models 6 COPYRIGHT American Gear Manufacturers Association, Inc.Licen
45、sed by Information Handling Services(b) Straight-bevel (c) Face (d) Spiral-bevel Figure 7: FSM models 4.2 Tooth deformation under load Comparison of the FEM and FSM displacements is shown in figures 8 to 11. Each figure plots the tooth deformation on the tensile (FEM-T) and compressive (FEM-C) tooth
46、 flanks as calculated using the FEM, and the FSM results (FSM) in the neutral plane of the tooth, in a section at mid- facewidth. As shown in figures 8 to 11 below, the FSM yields results comparable to those of the FEM, the maximum difference reached, at tooth tip, being 12 Yo for the Face gear memb
47、er. Table 2 lists the differences at tooth tip for the four test cases. These results also show how well the FSM follows the FEM displacement curves. Such results are obtained in less than 0,2 sec on a standard 450 MHz PC. Table 2. Differences between FSM and FEM tooth deformation Spur I Straight be
48、vel I Face I Spiral bevel +6.6 Yo I +7 Yo I +12% I -1 0% o 002 o O018 O 0016 O 0014 o O012 o O01 o O008 O 0006 O 0004 o 0002 O 12 3 4 5 6 7 8 910111: Radial Node Figure 8: Displacement - Spur gear tooth at mid- facewidth Straight Bevel Gear o. 002 0.0018 0.0016 0.0014 0.0012 0.001 0.0008 0.0006 O. 0
49、004 o. o002 O 12 3 4 5 6 7 8 9101112 Radial Node Figure 9: Displacement - Straight-bevel gear tooth at mid-facewidth 7 COPYRIGHT American Gear Manufacturers Association, Inc.Licensed by Information Handling Services0.002 0.001 8 0.0016 E 0.0014 c 0.0012 O 0.001 $ 0.0006 0 0.0004 0.0002 - E 0.0008 Face Gear O 1 3 5 7 9 11 13 15 17 19 Radial Node Figure 1 O: Displacement - Face gear tooth at mid- facewidth. Spirai-Bevel Gear 0.002 0.0018 - 0.0016 E 0.0014 c 0.0012 O o 0.001 $ 0.0006 0 0.0004 0.0002 O c E 0.0008 123456789101112 Radial Node F
