1、2000FTM7 o, An Analytical - FEM Tool for the Design and Optimization of Aerospace Gleason Spiral Bevel Gears A by: C. Gorla, E Rosa F. Rosa; P. G. Schiannini Politecnico di Milano, Dipartimento di Meccanica, Italy Outside cone distance Addendum Dedendum Pitch cone angle Face angle of blank Root angl
2、e Hand of spiral (LH o RH) Pinion normal chordal thickness (due to pinion generation method) Mean spiral angle 1. Introduction Due to the weight reduction requirements, aerospace transmissions structures generally show high deflection under the applied loads, and such deflection, which influence the
3、 contact pattern of mating gear teeth, must be taken into account in the design and optimization of gears. A tool that analyses the contact at the beginning of the design process can be time and money saving, with respect to the traditional development procedures based on tests. A design tool based
4、both on analytical algorithms and on FEM models generated automatically has been developed for Gleason Spiral Bevels gears, that are widely used in aerospace transmissions. As a first step, the conjugate surfaces theory is applied at the generation of the tooth flank surfaces, by simulating the manu
5、facturing process of the pinion and the gear defined by a Gleason summary. The simulation considers the real tool data and the machine settings. Then an analytical Tooth Contact Analysis is performed in order to determine the theoretical contact points on the flank surfaces versus the meshing positi
6、on. The misalignments can be introduced in the model as given data and are therefore taken into account. The Hertz theory is applied to calculate the axes of the theoretical contact ellipses without considering load sharing. In the following step, the information derived by the contact analysis is u
7、sed in order to automatically generate finite elements models of the gear pair: in particular, on the basis of the theoretical contact pattern, the tooth flank surface mesh can be adapted to the simulation of tooth contact. The automatically generated FEM models also include the main geometric param
8、eters of the gear blank (web and rim thickness, hub diameter) and misalignments that must be calculated separately, on the basis of the characteristics of the supporting structure. The final simulation by means of FEM models takes into account of the load sharing between tooth pairs. 2. Cutting meth
9、od The spiral bevel gear sets considered in this paper are cut with the Spread Blade / Fixed Setting Method. During the generation of the gear, the concave and the convex tooth flanks are generated simultaneously by means of a tool with alternate inside and outside cutting blades (Spread-blade metho
10、d). The pinion is generated following the Fixed Setting method. A tool with internal blades is used to simulate the generation of the convex flank, and another tool with external blades is used to generate the driving concave flank. In this way the two flanks are generated independently with two dif
11、ferent machine settings. The space width is usually controlled by means of the stock allowance indexing. 3. Input data The input data for the developed software comes directly from the Gleason summary. Table 1 reports the needed gear set data. n, N I Number of teeth F I Face width Another set of dat
12、a is used to define the tool geometry. The gear tool is defined by the cutter diameter, the outside and inside pressure angle, the point width and the blade edge radius. Two tools are used to generate the pinion; the first tool is similar to the gear tool, so the two data set are analogous, while th
13、e other tool with only external blades is defined by the cutter point diameter, the outside blade angle, the point width and the blade edge radius. The last set of data contains the machine settings, listed in Table 2. 1 Machine number Machine root angle Machine center to back Sliding base Eccentric
14、 angle Cradle angle Ratio of roll Table 2. Machine settings. Notes Usually zero for the gear, while the pinion has a little offset in order to optimize the bearing contact. Distance between the machine center and the pitch cone apex. This angle and the machine constant (k2) are related to the radial
15、 (distance between the cardle center and the tool center, called b), which define the tool position on the cardle. Cradle test roll / Work test roll The pinion machine settings are defined by two of this set of data, one for each cut. Si Smi SSi 4. Generation method Rigidly connected to the gear bei
16、ng generated Rigidly connected to the machine frame Rigidly connected to the generating surface Reference frames Theoretical basis for the generation process The method used to simulate the generation of the gear is based on the work of Litvin 8. The tool surface in SSi is described by means of the
17、following equation: O where (u;.,O:) are the surface variables. During its motion, the tool surface generates a family of surfaces: where pj is the motion parameter. From Eq. 2 its possible to compute the normal vector for each surface of the family: and its unit normal vector: (3) (4) O Since the g
18、enerated wheel surface is the envelope to the family of the tool surfaces, it must be tangent to each surface of this family; analytically this condition is expressed by means of the Equation of Meshing: -Zj -rji n, v, =O (5) where cf is the sliding velocity of the tool with respect to the wheel bei
19、ng generated. Equations 5 and 2 (projected in Si) define the gear tooth geometry. Pinion Equation of Meshing calculation On the basis of the briefly summarized theory, in this paragraph the procedure followed to determine the equation of the active portion of the pinion tooth flank surface is shown.
20、 2 Step 1. First of all, the equation of the family of the tool surface (CF) will be derived. This family is generated during the generation motion of the tool and its equation in the reference frame is: where (PF is the tool rotation and cutter radius. is the average Step 2. The normal vector of th
21、e surface is derived by means of the following expression: (7) Starting from this expression the unit normal vector can be derived: Step 3. The sliding velocity can be computed as: (9) + aF where vm(i) is the velocity of a point M considered rigidly connected to the tool surface CF and Vm(i) is the
22、velocity of the same point M considered rigidly connected to the tooth flank surface C1. The vector Vrn(i) is computed by means of the following expression +al -aF where Llm, 1 is the generating surface angular velocity and the vector rrn(i) is expressed by equation (6). This angular velocity is rel
23、ated to the pinion blank angular velocity by means of a polynomial of the 4th order, in order to consider the Modified Roll effects in further developments. Since axes x,() and z1 do not intersect but cross each other, the velocity of the pinion point M can be computed as: -U where O,(,O, is the dis
24、tance between the pitch cone apex and the machine center (calculated considering the corrections of the machine tool setting), and Qk, is the pinion angular velocity projected in reference frame Sm(,). The expression of the sliding velocity vm(i) can now be calculated. - -aF1 Step 4. The Equation of
25、 Meshing can now be derived by means of its definition: Equation 12 can be solved analytically with respect to u; , but it is not printed here due to its length. This equation can be used for the concave and the convex flank, using the correspondent parameter. The Equation of Meshing for the other t
26、ooth flank portions (root) has been derived following analogous procedu res. The system of equations composed by the Equation of Meshing and the family of the tool surface in the reference frame S1 define the pinion tooth active portion. 3 5. Tooth Contact Analysis Y 1,!1 m . , TCA input data and re
27、ference frames A P.cos(y+or) z1,ii o 1,1l Fig. 1. Misalignment sketch Zr.Zti?i o 11?1 -+ G.cos(y!) The TCA is performed following Litvin method. In the TCA algorithm, pinion and gear tooth flanks geometry is defined by means of the analytical expressions derived in the preceding paragraphs. Sf Sfcl,
28、 The misalignments are taken into account as they are defined by “The Gleason Works” (Fig. 1). defined in the generation process) fixed reference frame rigidly connected to Sf, with the origin coincident with the Dinion Ditch cone vertex The values of these misalignments can be calculated by means o
29、f a separate FEM structural analysis of the gear housing, shafts and bearings. Sf(2) The following reference frames are defined in order to introduce misalignments: and z-axis coincideni with the pinion axis. rigidly connected to Sf, with the origin I SI I rigidly connected to the pinion (previously
30、 I defined in the generation process) t-t- S2 rigidly connected to the gear (previously coincident with the gear pitch cone vertex and z axis coincident with the gear axis. Y1 i- Go; Fig. 2. TCA reference frames with misalignment Table 3. Pinion and gear position vectors and unit normal vectors. Tab
31、le 3 reports the symbolic expressions used to describe pinion and gear geometry. In the TCA algorithm a set of pinion angular positions is considered. For each of these positions, the developed software “rotates” the gear searching for a position where pinion and gear tooth flanks are in tangency. T
32、he point of tangency is the contact point; in this way a relationship between pinion and gear 4 rotation is also established; such relationship takes Contact Analysis softwares. The model is realized into account the misalignments. considering MSC/Nastran as the FEM solver and utilizes eight nodes 3
33、D brick elements with a mesh The expressions listed in Table 3 are projected in the that can be modified and refined acting on several reference frame Sf by means of the following parameters. equations: The result expected by the finite element model is an adequate calculation of the bending root st
34、ress taking into account the tooth load sharing. The effects of surface corrections can be introduced because they are considered in the software for the -(I) f (isFcp,l)=M,l)lM,(,),(l)r(,) (1 3) In order to determine the tangency condition, the following system of vector equations has to be solved:
35、 This system leads to five independent scalar equations in six unknowns; so it is possible to choose a pinion position (fixing the parameter ,) and to calculate the correspondent values of the other parameters. The main results obtained from this analysis are the path of contact and the transmission
36、 error that show the gear set meshing behavior. The following step is to calculate the principal curvatures and to apply the Hertz Theory in order to determine the contact ellipse dimensions and location that will be used to define the finite element mesh. 6. Development of the Finite Element Model
37、Objectives and criteria The main goal of the developed tool is to automatically generate FEM models of the meshing gears, in order to perform a quick comparison among various solutions in the first design steps, without the need of manual generation of FEM models with the aid of solid modeling packa
38、ges. Therefore, at the present stage, the tool is not meant either as a substitute of more accurate FEM models, generated specifically for each single case, or of an experimental development of the gear set. The software for the definition of the Finite Element Model automatically generates the mesh
39、 of the gear set, in the desired meshing position, on the basis of the data calculated by the Generation and the Tooth generation oi the tooth surfaces. The algorithm takes into account the misalignments and their influence on the position of the contact area, likewise the TCA software: there is an
40、exact match between the matrices that transform coordinate systems in both the algorithms. The meshing is simulated by modeling three teeth, the rim and the web for gear and pinion, in order to take into account their compliance and the load sharing. The contact is simulated by means of gap elements
41、 with facing nodes, and therefore an approximation is introduced in the determination of the local compliance. Geometry of the model The model for the non-linear analysis of a meshing gear set is defined by assembling the following sub- models: - three tooth pairs describing the region of meshing -
42、the gear and pinion blanks (rim and web) - the gap elements Teeth. Pinion and gear tooth are replicated three times and each is modified in the contact area on the basis of the TCA results, in correspondence of the chosen meshing position. The tooth model consists of five parts: 1) tooth body 2) con
43、cave flank root fillet 3) convex flank root fillet 4) rim portion 5) web portion Tooth body. This portion is delimited by the convex and the concave flanks, the top-land surface, the toe and heel surfaces, and by a parabolic shaped surface below. The parabolic shape of this surface has been chosen i
44、n order to reduce the elements distortion, especially in the fillet area. Fillets. This portion is delimited laterally by the low boundary of the tooth body and by a surface defined by the cones generatrices that correspond to an 5 angular pitch for each transverse section. The boundary is delimited
45、 upside by the fillet and root surfaces and downside by a parabolic shaped surface. The parameters that can be varied in order to influence the shape and dimensions of the elements are: - number of tooth nodes along height, thickness and length, - element spacing bias (not in length direction), - sh
46、ape of the parabolic surfaces trough the coefficients of their equations. Rim portion. The boundary of the rim portion included in the tooth portion is determined by the neighboring surfaces of the tooth body and fillet, and in the lower part by a cone defined by the rim dimension: the number of ele
47、ment along the thickness can be varied as a parameter. Web portion. Its upside boundary is determined by the lower boundary of the rim portion, while the downside one is a cylindrical surface defined by the hub diameter. Front-side and rear-side surfaces are cone sectors, which define a conic plate
48、of the desired thickness and orientation. Other parameters are the number of nodes along the radial direction and their relative spacing. Fig. 3. Tooth model Gear blanks The rims and the webs are solids of revolution generated by the rotation, around the gear axis, of their axial sections. The varia
49、ble parameters are the number of nodes in circumferential direction and their relative spacing (bias). Through these parameters it is possible to increase the dimension of the elements gradually away from the meshing zone, getting a lower global number of elements in the model. Contact model The contact is modeled by means of Gap elements, the definition of which requires the presence of two facing nodes. The ideal point of contact and the O 6 tangent plane to the contacting surfaces have been defined by TCA, while the contact ellipse major axis and its direct
