喜歡這套資料就充值下載吧。。。資源目錄里展示的都可在線預(yù)覽哦。。。下載后都有，，請放心下載，，文件全都包含在內(nèi)，，【有疑問咨詢QQ:1064457796 或 1304139763】 ==============================================喜歡這套資料就充值下載吧。。。資源目錄里展示的都可在線預(yù)覽哦。。。下載后都有，，請放心下載，，文件全都包含在內(nèi)，，【有疑問咨詢QQ:1064457796 或 1304139763】 ==============================================

Res Eng Des (1989) 1:69-73 Technical Note Research in E gineedng eslgn 1989 Springer-Verlag New York Inc. On the Role of Geometry in Mechanical Design Vadim Shapiro Herb Voelcker* The Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, New York, USA A complete design usually specifies a mechanical system in terms of component parts and assembly relationships. Each part has a fully defined nominal or ideal form and well defined material properties. Tolerances are used to permit variations in the form and properties of the components, and are used also to permit variations in the assembly relationships. Thus the geometry and material properties of the system and all of its pieces are fully defined (at least in principle). Henceforth we shall focus on geome- try and, for reasons that will become evident, will not deal with materials despite their obvious impor- tance. Mechanical systems specified in the manner just described meet functional specifications that ap- peared initially as design goals. The process of de- sign can be thought of as generating the geome- tryMthe breakdown into components with coarsely specified geometry, and then the detailed specification of the component forms and fitting re- lationships. Design seems to proceed through si- multaneous refinement of geometry and function I. An important line of design research seeks sci- entific models for this refinement process and sys- tematic procedures for improving and perhaps auto- mating it. At present we have tools for dealing with two widely separated stages of the refinement process. For single parts, function is usually specified through loads on pieces of surface (e.g. a force distribution over a support surface, a flow rate through an orifice, a radiation pattern over a cool- ing fin); specification of the solid material that pro- vides a carrier for the pieces of surface may be viewed as a constrained shape optimization pro- cess. Also with the Computer Science Department, GM Research Laboratories, Warren, Michigan, USA * Reprint requests: The Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA At the higher level of unit functionality, where one deals with springs, motors, gear boxes, heat exchangers, and the like, geometry usually is ab- stracted into real numbers if acknowledged at all, and function is cast in terms of ordinary differen- tial or algebraic equations (for heat flow, motor torque as a function of field current, and so forth). Systems of such equations describe the composite functionalism of networks of functional units. There is a big gap between these islands of un- derstanding, and intermediate stages of abstrac- tion are needed which acknowledge the partial ge- ometry and spatial arrangement topology of subassemblies. Broadly speaking, geometry is far- ing badly in contemporary design research; many investigators either sweep it under the carpet or deal with it syntactically, e.g. through features defined in ad hoc ways. Clearly we need more sys- tematic ways to address the relationship between geometry and function, and we suggest below some initial steps toward this goal. Energy Exchange as a Mechanism for Modeling Mechanical Function Mechanical artifacts interact with their environ- ments through spatially distributed energy ex- changes, and we argue below that mechanical func- tionalism can be modeled in terms of these exchanges. The initial cast of the argument draws heavily on seminal work by Henry Paynter 2. We shall regard mechanical artifacts as systems that range from single solids or fluid streams, which usually are the lowest level of natural system that exhibit important properties of mechanics, to com- plex assemblies of solids and streams. A closed boundary, which may be physical or conceptual, is a distinguishing characteristic of a system: the sys- tem lies within (and partially in) the boundary, the environment lies outside, and interaction occurs 70 Shapiro 8S : the boundary of S; V : a spatial region containing S whose complement is the environment; 8 V : the boundary of V. S may coincide with V, and 8S and 8 V are closed surfaces (usually 2-mainfolds) in E 3. We distinguish S from V because S may be partially or wholly un- known (recall that this note is about design) but boundable by a known V. The principle of continuity of energy applies at all levels of system abstraction. If no energy is gen- erated by the system, then O_f dV fsv Pnd(SV)= fv Ot + fvgdV. (1) The surface integral on the left describes the total energy flux (instantaneous power) through the boundary; P is a generalized Poynting vector de- scribing the instantaneous rate at which energy is transported per unit area, and n is the normal at a point in the boundary 8 V. On the right, Oe/Ot is the (volumetric) density of energy stored in the system, and g is the rate of energy loss or dissipation. A system interacts with its environment by ex- changing energy through its physical boundary: for example, by radiating energy stored in the system over a portion of its area, or by providing support to an external mating part and thereby inducing stor- age of deformation energy in the system. The sub- sets of the physical boundary over which such ex- changes occur will be called (following Paynter) energy ports. If s is the physical boundary subset (piece of surface) associated with the i tu port, then P nd i fv dV+fvgdV (2a) where sl C 8S. (2b) Thus the total energy flux through the boundary is a sum of signed fluxes through the ports. We note that a boundary subset si may belong to several ports, and that body forces, such as those induced by gravitational and magnetic fields, may be accom- odated by taking S as the associated port. Geometrical and Functional Refinement in the Limit The left side of Eq. (2a) specifies energy exchanges through the systems ports and requires that the flux vector(s) and port geometries be known. The terms on the right cover internal energy (re)distribution and/or dissipation. The physical effects implied by these terms depend on the energy regime(s) and the geometry of the system; there may be rigid body motion, elastic or plastic deformation, temperature redistribution, and so forth. Mathematical evalua- tion requires the solution of 3-D boundary- and/or initial-value problems. Very marked simplifications ensue if one as- sumes that 1) the ports are spatially localized and idealized so that the integrals on the left of Eq. (2a) may be evaluated individually to yield terms Pi, and 2) internal energy storage and dissipation are simi- larly localized in disjoint discrete regions, thereby permitting the right-hand integrals to be decom- posed into sums of local integrals which may be evaluated individually. With these assumptions, Eq. (2a) may be rewritten Z e, = 2-07 + Gk i j k (3) where Pi is the power through the i h discrete port, Ej is the instantaneous energy stored in the jth dis- crete region, and Gk is the dissipation rate in the k th discrete region. A limiting form of this refinement (or discretization, or-in Paynters terminology- reticulation) is a Dirac-delta limit wherein the ports shrink to spots of zero area and the volumetric regions shrink to point masses, idealized resistors, and the like. Equation (3) is the basis for Paynters energy bond diagrams, or bond graphs. It describes a sys- tem that may transfer, transform, store, and dissi- pate energy through elements whose geometry has been refined into a few real numbers-the spatial positions of the discrete ports and lumped regions (which generally are not carried in bond-graph rep- resentations), and integral characterizations of the discrete ports and regions (for example the value, in kilograms, of a point mass). This higher view enables one to analyze the dynamics of the idealized (discretized) system, but one can deduce little about the geometry of feasible distributed (i.e., real) systems from such analyses; essentially all ge- ometry must be induced. Apparently we have gone too far, i.e., have thrown away too much geometry. Shapiro .O) O O (a) .,. -o. (O) : 0 ) (b) (c) (d) Fig. 1. Design of a simple bracket. Toward an Appropriate Role for Geometry We would like to step back from the limiting refine- ment just discussed, where all notions of form have been lost, and include in the problem some continu- ous geometry-but not the full-blown field problem covered by Eq. (I) unless this is unavoidable. We shall suggest below three principles governing the interaction of form and function that we believe will yield geometrically well defined (but not necessarily optimum) designs. A simple but common example drawn from practice-design of a bracket-will motivate the discussion (Fig. 1). The design begins with three holes of known di- ameter and configuration that are to be carried by an unknown solid (Fig. la); these mate with other parts (two screws and a pivot pin). Bosses are created to contain the holes (Fig. lb) because of concern about interference with other components passing between the holes. Finally the holes and bosses are bound together into a single part as in Figs. lc and ld, with the final shape being governed by criteria for clearance, strength, weight, and aes- thetic and manufacturing simplicity. Two simple but important inferences may be drawn from the example. Firstly, the initial holes (plus some implied constraint surfaces in the third dimension) are the brackets energy ports; they are fully specified geometrically and specify by implica- tion what the bracket is to do-maintain the relative position of ports whose geometry admits rotational motion. In principle the associated energy regimes (force, torque:elasticity) can be fully specified as well, but in practice they are often only implied or understood. Secondly, the remaining geometry is discretionary but constrained by requirements that the holes be bound into a connected solid, that Fig. 2. Position-fixing character of the bracket. the solid not interfere with other components, and so forth. We note that, at the single-component level of the bracket, shape optimization usually does require solution of the full 3-D field problem covered by Eq. (2a). From this example and others we induce: Principle 1. A systems function is determined by its energy ports, which are generally subsets of its physical boundary, and the energy regimes oper- ating on those ports; both should be fully defined. The remaining geometry of the system is discretion- ary provided that 1) it admits at least one physical realization of the system that satisfies the port spec- ifications, and 2) other external constraints, e.g. on overall size, are met. Principle 2. Energy exchanges within a system al- ways may be represented independently of geome- try, e.g. via bond graphs. Figure 2 shows the position-fixing capabilities of the bracket represented (nonuniquely) by ideal springs attached to the locally rigid ports. This rep- resentation of the brackets partial functionalism as- sumes ideally elastic behavior, and this assumption should be checked, e.g. by finite-element analysis, as the brackets final shape is being determined. Figure 3 shows a slightly more complicated sys- tem-an indicator that senses pressure via an orifice (port) of known geometry, and displaces a rotary indicator correspondingly. The output indicator is a port because we require that it be able Input Output V/ Support Port Fig. 3. A pressure measuring system. 72 Shapiro it may be replaced with other, arbitrarily elaborate arrangements of idealized elements hav- ing the same input/output functionalism plus other paths that terminate internally. Equation (4) provides the rationale for Principle 2. The essential idea is that the port i P n dsi = 2 -ot- + Z Gk (4) i j k flow on the left of Eq. (2a) may be handled inter- nally (the right-hand integrals in Eq. (2a) in many ways. If we are assured by Principle I, or simply assume, that internal solutions exist, then we may reticulate the internal geometry and deal with inte- gral quantities as in Eq. (3). Principle 3. Principles 1 and 2 must hold for all subsystems defined on combinatorial decomposi- tions of a system. Principle 3 provides means for the simultaneous refinement of geometry and function. It enables complicated systems to be decomposed recursively into functional subsystems provided that one de- fines the ports as one proceeds. The limiting combi- natorial refinement is single parts, and at this level one must solve the field problem of Eq. (2a) to ob- tain complete geometric specifications. Concluding Remarks The thoughts above are aimed at finding means to establish for geometry an appropriate formal role in a theory of mechanical design. It seems obvious to us that geometry should have such a role, but the work needed to establish it has barely begun. EpilogueRRemarks on Features This work grew out of a several-month effort to characterize geometric features in a formal man- nerwan effort that largely failed. The effort was motivated by the fact that mechanical design and manufacturing are often discussed and done in terms of features, but there are no agreed views on what features are or do 4. (Slots, fibs, webs, and shafts, are typical features; all involve geometry in one way or another.) We began with a conjecture: A geometric feature may be defined as a geometric idealization of a port for energy exchange in a defined regime. (This no- tion is appealing because it implies that a systems feature-set specifies all of the geometry needed to define the systems interactions with its environ- ment; the remaining geometry is determined by constraints and optimization.) We then proceeded to show that the conjecture is formally consistent in design, manufacturing, and inspection applications. In machining, for example, geometric features may be associated with the boundary of the removed material; the energetic process is machining itself, whose dynamics are reasonably well understood in a macroscopic sense. Clamping features may be de- fined primarily through elastic energy storage, in- spection features through the energetic exchange involved in the measurement process, etc. But as our explanations grew increasingly contrived and our difficulties with solid and other non-surface fea- tures mounted, we began to sense that features could not be defined in any universal system other than a purely syntactic system. Currently we believe that features are simply in- formation structures that represent, often in para- metric form, known solutions to local problems. While a syntactic structure can be imposed on them, their underlying semantics can vary widely and need not involve particular kinds of geometry, or indeed any geometry at all. However, if a feature is to be used properly, a feature-context must be suppliedmthe technical conditions and criteria that led to the solution the feature represents. Given the feature-context (e.g., as domain knowledge in a de- signers head) and appropriate reasoning power to adapt the solution to the current problem, features can be very effective; their popularity among hu- man designers attests to this. Recent work by Duffey and Dixon 5 illustrates that features can be used in automatic design when feature-contexts and appropriate reasoning power are provided. (The handling of features by Duffey and Dixon seems ad hoc, but ad-hocery may be intrinsic if our current permissive view of features is correct.) Features can be dangerous when used without their contexts and appropriate reasoning power, as nonsense designs produced by certain au- tomatic design systems illustrate. Finally, we wish to point out that the character- Shapiro & Voelcker: Geometry in Mechanical Design 73 ization of features as known solutions to local problems places strong constraints on schemes for combining features to make new features. A feature combination makes sense only if it can be shown to be a valid solution to a well defined local problem. But even determining the domain of the combina- tion problem as a function of its component do- mains may prove very difficult. Acknowledgments. The work reported here was supported in part by the General Motors Corporation under its corporate fel- lowship program, and by the National Science Foundation under grant MIP-87-19196. References 1. Alexander, C., Notes on the Synthesis of Form, Harvard Uni- versity Press, 1964 2. Paynter, H.M., Analysis and Design of Engineering Systems, MIT Press, 1961 3. Ulrich, K.T. and Seering, W.P., Conceptual Design: Syn- thesis of System of Components, 1987 ASME Winter Annual Meeting, PED Vol. 25 4. Report of the Workshop on Features in Design and Manufac- turing, February 26-28, 1988 University of California, Los Angeles 5. Duffey, M.R. and Dixon, J.R., Automating extrusion de- sign: a case study in geometric and topological reasoning for mechanical design, Computer-Aided Design, Vol. 20, No. 10, pp. 589-596, December 1988