1、04FTMS1Stress Analysis of Gear Drives Basedon Boundary Element Methodby: D. Vecchiato, University of Illinois at ChicagoTECHNICAL PAPERAmerican Gear ManufacturersAssociationStress Analysis of Gear Drives Based on BoundaryElement MethodDaniele Vecchiato, University of Illinois at ChicagoThe statement
2、s and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractThe stress analysis is performed as a part of TCA (Tooth Contact Analysis) for a gear drive.Unlike the existing approaches, the p
3、roposed one does not require application of commercial codes (likeANSYSorABAQUS)forderivationofcontactmodelanddeterminationofcontactandbendingstresses. Thecontacting model is derived directly by using the equations of tooth surfaces determined analytically. Theboundary element approach allows to red
4、uce substantially the number of nodes of themodel. Determinationof stresses caused by applied load is obtained directly for the applied contacting model for any position ofmeshing.The developed approach is illustrated by stress analysis of helical gears with modified geometry.Copyright 2004American
5、Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2004ISBN: 1-55589-837-81Stress Analysis of Gear Drives Based on Boundary Element Method Daniele Vecchiato, Ph.D. student University of Illinois at Chicago, Department of Mechanical and Industrial Engine
6、ering Supervisior: Professor Faydor L. Litvin E-mail: danieleuic.edu 1. Introduction Stress analysis of gear drives is still a challenging and important problem in design, even if it has been the subject of extensive research for more than hundred years. The history of research in this area is an at
7、tractive picture that shows tremendous progress: simple models like a beam used at the beginning are now substituted by models based on discrete presentation of tooth surfaces involving systems of hundred of thousands of equations to be solved by powerful computers. However, getting the advantages o
8、f the new technique of 21st century, we have to pay credit to the pioneers that have made the first steps in stress analysis of gears and whose works have inspired us, the descendants. We begin the history of development of stress analysis by the name of Professor William H. Lewis 1, that has develo
9、ped in 1892 the most important model for analysis of bending stresses of spur gears. Professor Lewis has determined the critical sections of a gear tooth by inscribing a parabola that is tangential to the root fillet (Fig. 1). The parabola represents the section of an uniform-strength beam, and the
10、load is considered as applied at the vertex of the parabola. The model has been improved by application of semi-empirical factors based on testing, photoelastic experiments etc. Being concerned with the problem of contact stresses (as one of the main reasons of pitting), the next steps were directed
11、 at application of the Heinrich Hertz theory 1. Application of the Hertz theory to gear drives involves the contact of two cylinders with parallel axes (Fig. 2) with the following considerations: (a) the cylinders are loaded Figure 2. Hertzian model for determination of contact stresses for spur gea
12、rs Figure 1. Lewis model for determination of bending stresses for 1 along the normal to the tooth flank; (b) the length of the cylinders is the length of the line of contact; (c) the radii of the cylinders are the radii of curvature of tooth flanks determined at chosen contact positions. Also in th
13、is case, successive improvement of the model by application of semi-empirical factors was necessary to obtain results in good agreement with experiments. We have to pay credit to AGMA, ISO and other organizations that have developed systematic Standards for computation of bending and contact stresse
14、s based on the ideas of Lewis and Hertz, but complemented such approaches with application of coefficients based on experimental tests thoroughly performed 2. The revolutionary step of stress analysis of solids happened with application of Finite Element Method (FEM) and later with Boundary Element
15、Method (BEM). The BEM is the main topic of this paper. Both methods are based on discrete presentation of tooth surfaces by a large number of elements, say tens of thousands for FEM and thousands for BEM. Fig. 3 shows a broken view of the tooth model applied for BEM. If will be shown below (see Nume
16、rical Example) that application of BEM enables to reduce substantially the number of elements in comparison with FEM. It is shown in the numerical example represented in the paper that the number of elements is 2,550 and 15,744, respectively. Application of FEM for stress analysis may be a very expe
17、nsive procedure in terms of time. In general, it requires application of a Pre-Processor computer program, for instance ANSYS, for preparation of the numerical contacting model. Then the model has to be processed by a FEM program, for instance ABAQUS, where theory of elasticity is applied to obtain
18、tens of thousands of equations that relate the elastic deformation of finite elements with applied forces. Since it is usually required to investigate stresses and formation of bearing contact for at least one angular pitch, several applications of FEM are necessary for different angles. In each cas
19、e, solution of a new model is necessary. It will be shown in this paper that it is possible to reduce substantially the time required for: (i) derivation of the contacting model of gear tooth surfaces, and (ii) stress analysis. This is achieved: (i) by application of an automatic acting pre-processo
20、r (for derivation Figure 3. Illustration of discrete presentation of tooth surface by boundary elements Figure 4. Illustration of FEM model produced by pre-processor 2 of contacting tooth models) that uses analytical representation of tooth surfaces, and (ii) by application of a new automatic BEM ap
21、proach for stress analysis that uses only a single numerical model for all considered angular positions. The ideas of application of pre-processor and BEM are illustrated by the example of stress analysis of a helical gear drive with modified tooth surface equations (See section 4). The purposes of
22、modification (of the gear drive) are avoidance of edge contact, reduction of noise caused by transmission errors, and improvement of bearing contact. The results have been compared with those obtained by a conventional approach based on application of ABAQUS program, 4. The comparison shows a good a
23、greement of obtained data and confirms a substantial reduction of time of computations. 2. Computational Procedure by Application of FEM: Overview and Improved Approach Description of Procedure The FEM contacting model is formed by the pre-processor (section 3) and is illustrated in Fig. 4. Fig. 5 s
24、hows: pinion elastic model (Fig. 5(a), gear elastic model (Fig. 5(b), and the model of the contact interface formed by two contacting surfaces of driving sides that are tangent each to other (Fig. 5(c). The FEM contacting model of the pinion (the gear) is represented by many bricks with reference po
25、ints at corners (nodes) (Fig. 6). The bricks are obtained by discrete presentation of tooth surfaces and tooth interior volume. The contact interfaces are represented by the nodes of the bricks that lie on the contact surfaces. Determination of the contact and bending stresses requires solution of t
26、he contacting model that is performed automatically, for instance by ABAQUS, in two stages as follows: Stage 1 A guess of the extension of the contact area has to be done. The contact area includes a certain number of nodes. Figure 5. Illustration of the contacting model formed by: (a) pinion elasti
27、c model, (b) gear elastic model, (c) interface model Figure 6. Illustration of finite elements: (a) brick element, (b) tetrahedral (pyramidal) element 3 Figure 7. Illustration for determination of compliance between nodes NS and NM Separate consideration of the elastic models of the pinion and the g
28、ear allows to determine the compliance of the included nodes. Considering as given two nodes SN and MN of the surface of the pinion (gear), compliance of MN with respect to SN is defined as the elastic deformation, measured at MN , caused by a unit force applied at SN (Fig. 7). Compliance of all con
29、tact nodes may be represented in a table defined as compliance matrix (Appendix A). The term compliance matrix may be referred in the literature as matrix of coefficients of influence or, in more mathematical terms, as to the Green Operator. Compliance matrix may be obtained by experiments or, more
30、conveniently, by application of BEM or FEM. For this reason, we might identify Stage 1 as the solution of the elastic problem. Stage 2 Stage 2 is based on application of conditions of contact of surfaces (Appendix B). Generally, these conditions state that: (a) surfaces cannot penetrate each to the
31、other, and (b) contact forces will arise at the contact area and deform the surfaces preventing penetration. Contact forces should be directed primarily towards compression of surfaces, and may have eventually a tangential component given by friction. For instance, Fig. 8 illustrate the contact of t
32、wo cross sections of half cylinders: Fig. 8(a) shows the undeformed configuration wherein cylinder surfaces are overlapping, and Fig. 8(b) shows the deformed configuration and the resulting distribution of contact pressure. Note that this Stage requires relations between applied forces and displacem
33、ents. These relations are represented by the compliance matrices defined at Stage 1. Compliance matrix is the link between Stages 1 and 2. We may define Stage 2 as solution of the interface problem. Complete Solution Solution of elastic problem followed by solution of interface problem constitutes t
34、he full solution of the contact problem. Full solution may require more than one iteration of Stages 1 and 2, since the contact area that is initially guessed at Stage 1 may be larger or smaller than the actual one, and therefore Figure 8. Geometry of two contacting half cylinders being under pressu
35、re: (a) undeformed, (b) deformed 4 has to be adjusted. This approach, with some variations depending on the considered FEM computer program, constitutes the basic procedure for the solution of a generic contacting model. Disadvantages and Improvement of Approach for Application to Gear Drives The ge
36、neral approach described above may be an advantageous one in the case that the analysis has to be performed for the given problem only once. For the case of gear drives, the analysis has to be performed at various angular positions. In the existing approach, this requires derivation of various model
37、s and repetition of steps described above for all of the angular positions of the contacting models. However, repeated derivation of compliance matrices increases substantially the time of solution. In the proposed approach, the compliance matrices for the pinion and the gear are computed in two ste
38、ps: Step 1. Compliance matrices are computed only once, for a reference position of the pinion (gear). Step 2. Based on reference compliance matrices (see Step 1), compliance matrices for other angular positions may be computed by their transformation that is similar to “rotation“. Such transformati
39、on is based on orientation of the pinion (gear) that is indicated by the angle of rotation with respect to reference position. This yields the following main advantages: (a) a single contact model may be applied instead of a set of contact models required for the solution of contact problems for var
40、ious angular positions; (b) the solution of the elastic problem for various angular positions is obtained just by “rotation“ of the compliance matrix. The approach proposed in the paper has as well other advantages: (i) The solution of the elastic problems for tooth surfaces represented discretely i
41、s based on Boundary Element Method (BEM) instead of Finite Element Method (FEM), see Appendix A. Then the number of elements (bricks) may be reduced substantially. In the numerical example considered in the paper (section 4), the number of 17,744 is reduced to 2,550. (ii) A more simple algorithm for
42、 the solution of contact problem, based on the Signorini boundary conditions, is proposed (Appendix B). The Signorini algorithm does not take into account conditions of friction and plasticity that might be ignored for many cases. (iii) It is shown that the contacting model may be determined automat
43、ically, using the analytical representation of tooth surfaces and applying the pre-processor (section 3). The same may be done for automatic presentation of the results of computation of contact and bending stresses obtained as functions of angular positions. This requires development of post-proces
44、sor program. Development of such pre- and post-processors enables a significant reduction of time required for the analysis and visualization of obtained results. (iv) The output of the developed computer program includes (in addition to the results in item (iii): (a) formation of the path of contac
45、t, and (b) the function of transmission errors for a loaded gear drive. Investigation of formation of the bearing contact for several cycles of meshing allows discovering and avoidance of areas of severe contact stresses. Determination of function of transmission errors for a loaded and misaligned g
46、ear drive is important for the analysis of noise of a misaligned gear train. The numerical example represented in the paper covers the stress analysis for a single tooth, but for a modified helical gear drive. The results obtained have been compared with the FEM approach based on application of ABAQ
47、US. Comparison of computer time is included (see numerical example in section 4). 5 3. Pre-Processing of the Numerical Model Wherein a numerical method like BEM or FEM is applied for the solution of a structural problem, a numerical model is required for representation of the object assigned for inv
48、estigation. The information required for the description of a numerical model (formed by nodes, elements, and bodies) may be organized as follows: (i) Description of all nodes represented as point coordinates in the applied Cartesian reference system. (ii) Each element is represented as a set of nod
49、es. (iii) Description of the set of bodies wherein each of the bodies is defined by a set of elements. (iv) Indication of how the bodies are constrained and how the loads are applied. In the early days of stress analysis, the information above was generated manually. Computerization of required information for the procedure of computation for gear drives is based on application of special pre-processors with the following features: (i) all the nodes that belong to the tooth surfaces (including the fillet) are represented analytically; (ii) the nodes of the interior are repr
