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几何在机械设计中的作用几何在机械设计中的作用一个完整的设计通常指定一个机械系统的组成部分和装配关系。每个部分都有一个完全定义的名义或理想的形式和良好定义的材料属性。在组件的形式和属性中通常允许存在公差变化,在组装关系中也允许。因此系统的几何和材料特性和它的所有部分都被完全定义（至少在原则上） 。从此我们应该关注几何，原因显而易见，因为我们不会应用一种材料尽管它重要性很明显。机械系统中指定刚刚描述的方式满足作为最初的设计目标的功能规范，设计的过程可以被认为是“生成几何”分解为指定几何的组件，然后是组件的详细规格形式和装配关系。设计似乎是通过细化几何和功能的同时进行。 设计研究中的一个重点需要这个细化过程的科学模型，和提高或者使其自动化的系统过程。目前我们有处理两个广泛的分离阶段细化过程的工具。一方面，函数通常是通过加载指定的表面（例如在支撑面上的力的分布，通过一个孔板的流量，散热片的辐射模式等） ；固体材料提供表面碎片载体的规范可能被视为一个约束形状优化的过程。在 “单元功能” 这个高层次上， 人们需要处理弹簧,马达,齿轮箱、热交换器,等等, 这时几何通常被抽象成实数，函数变成了常微分或者代数方程（例如热流,电动机转矩作为励磁电流的函数等等） ，这种方程组描述网络的复合功能主义的功能单元。在这些“理解的岛屿”中有一个巨大的差距，抽象的中级阶级需要用来使部分几何和拓扑组件的空间安排被承认。一般来说，几何在当代设计研究中进展情况严重，许多调查人员不是将其“扫地毯下”就是生搬硬套的使用，通过“特征”来定义为特别的方法。显然我们需要更多系统的方法来解决几何和函数之间的关系,下面我们提出了一些达到这个目标的基本步骤的建议。能量交换作为机械函数建模的机制能量交换作为机械函数建模的机制机械构件通过空间分布式能源与环境交互交流, 我们认为在这种机械功能主义下可以在这些交流中建模的。 这个争论的初始想法很大程度上吸引了亨利在该参数上的开创性工作。 我们应该把机械构件看成是范围从单一的固体或液体流变化的系统， 这通常是表现出重要的力学属性的最低级的自然系统。一个封闭的物理或概念式的边界， 它是一个系统中很难被区分的特色：系统位于（或者部分位于）边界之内，而环境位于边界之外，它们通过边界进行交流。我们通过以下进行区分：S：讨论中的物理系统S：S 的边界V：一个包含 S 的空间区域，它的补充是环境V：V 的边界S 和 V 可能是一致的，S 和V 在3E里面有相近的表面（通常在两个方面） ， 我们能从 V 中区分出 S 是因为 S 部分或者完全是未知的，但是可以被一个已知的 V 绑定（重申一下这篇文章是关于设计的） 。能量的连续性的原则适用于所有级别的系统抽象。 如果这个系统没有生成新的能源，那么gdVdVtVndPVVV)(左边的曲面积分描述了总能量通量通过边界(瞬时功率)；P 是一个广义描述瞬时速率的矢量，也就是能量在单位量，n 是V 边界上的常数，在右边，t 是能量存储在系统的(体积)密度，g是能量损失的速度或耗散度。系统通过与其物理边界交换能量来与环境相互作用，例如，通过在这个区域的一部分中辐射系统中存储的能量，或者通过对外部配合零件提供支持， 从而诱导系统中储存的变形能量等， 在这种交流上的物理边界的子集将被称为能源接口。如果iS是联合thi的物理边界的子集（表面的部分） ，那么gdVdVtndsPVViisi（2a）这时有Ssi. (2b)因此,通过边界的总能量流量是通过端口的被标记的总流量，我们注意到一个边界子集iS可能属于多个港口，这个主体强调，比如那些被重力和磁场引起的，可能会把S当成相关的接口。在限制下的几何和功能细化在限制下的几何和功能细化方程式 2a 的左边指定了通过系统的能源交换端口，而且要求通量向量和端口的几何图形已知。 右边的术语涵盖能量的(重新)分布或者耗散。这些条款所带来的生理效应取决于能源政权和系统的几何；可能会有刚体运动,弹性或塑性变形，温度再分配，等等。数学评价需要解决三维边界和/或初值问题。非常显著的简化随之而来，如果假设 1) 端口空间本地化和理想化以至于方程式的左边的积分。(2a)可能是iP单独评估产生的条件，2)内部能量储存和耗散也同样在不相交的离散地区本地化， 从而允许右边积分被分解成当地的积分可能单独评估。根据这些假设，方程式（2a）可能被改写为kiGtP(3)iP是通过thi离散端口的力，jE是存储在thj离散区域的瞬时能量，kG是在thk离散区域里的耗散率。这种改进的限制形式（或者离散化，或者 Paynter 的术语网状物）是一种“Dirac-delta 限制” ，这时端口会缩小到零面积，体积缩小到临界点还有理想化的电阻等。方程式（3）是 Paynter 能源交换或者债券图表的基础，它描述了一个系统,可以转让、转换、存储和释放能量通过元素的几何被提炼成几个实数离散空间位置的港口和集中的地区 （通常不进行债券图表交涉） ，积分离散端口和空间的特征(例如在公斤中的“价值”,作为一个质点)。这种更高的观点使人能够分析这个理想化的(离散)系统的动力学，但我们可以从这些分析中推断出可行的几何分布(也就是说,实际的)系统，基本上所有的几何必须诱导。显然我们已经走得太远,也就是说，几何被我们扔掉太多。几何的适当角色几何的适当角色我们愿意退一步从限制细化讨论,已经失去了所有形式的观念,包括在一些连续的几何问题,但被方程式（I）覆盖的不成熟的领域，除非这是不可避免的。 我们建议以下三个原则管理形式和功能之间的相互作用,我们相信将产生几何定义良好(但不一定是最优)的设计，一个简单但常见的例子来自实践设计的支架,将使讨论升级(图1)。这个设计始于三个已知直径和配置但是被一种未知的固体阻挡的孔(图 la)；这样跟其他部分的配合(两个螺丝和一个枢轴销)。因为担心其他组件在孔之间传递，于是创建包含这些孔的圆框(图 1b),最后,这些孔和圆框聚在一起形成如图1c和1d中的一个单独的部分，最后的形状是由间隙标准,强度、重量、审美以及简单的拼合组成。从例子中可以得到两个可能的简单但是很重要的推论。首先， 第一个孔(加上一些隐含约束表面在第三维度)是支架的能源端口； 他们完全指定的几何和指定含蓄的支架是做什么保持端口的相对位置，它的几何承认旋转运动。原则上相关能源政权(力、扭矩:弹性)可以完全指定为好,但实际上他们往往只有隐含或“理解” 。第二， 剩下的几何可自由支配,但是要求相关的孔被绑定到一个链接的固体上， 这个固体不会干扰其他组件,等等。 根据方程式(2)， 我们注意到，在单组分水平支架的形状优化通常并不需要完整的 3 d 领域问题的解决方案。从这个例子和相关的因素我们推导出：原则 1：系统的“功能”是由其能量端口决定的,这通常是物理边界的子集,而且能量通过这些端口进行操作；两者都应该被完全定义。剩下的几何系统可自由支配地提供：1)承认至少有一个满足港口规格的系统的物理实现；2)其他外部约束,如在总体规模上等等，都被达到。原则 2：一个系统中的能源交易总是能独立地用几何表示,如能量交换的债券图表。图 2 显示了支架代表理想弹簧连接到本地刚性端口的定位功能(非唯一)。这表示支架部分的机能主义假定理想的弹性行为,以及这种假设应该检查，例如，通过有限元分析,支架的最终形态被确定。图 3 显示了一个稍微复杂一点的系统： 一个感觉压力通过一个已知孔(端口)的几何形状的指示器， 这个指示器取代了相应的旋转指示器。输出指示器是一个端口，因为我们要求它能够做的工作环境， 如克服指定定义范围的旅行限制扭矩，因而其几何必须定义。该系统还支持第三个端口。系统的主要功能是代表内部压力/扭矩变压器和旋转弹簧显示为键合图元素风格的 Ulrich and Seering3，但这种解释不是独一无二的， 这表示它可能被替换为其他任意精心安排的理想化的元素，它相同的输入/输出功能主义加上其他路径终止内部。方程(4)为原则 2 提供了理由。基本原理是端口kjisGtEndsPi流向左边的方程式(2)可能是在许多方面处理内部(在方程式(2)右边的积分)。如果我们保证原则 1，或者只是假设，内部解决方案存在，那么我们可能如方程式(3)所示用内部几何的网状来处理积分数量。原则 3：原则 1 和 2 必须持有一个系统的所有子系统在组合分解上的完整定义。原则 3 提供了手段的同时细化几何和功能。 它使复杂的系统分解递归到功能性子系统提供一个港口的定义是一个收益。 限制组合优化是单一的部分,在这个层次上必须解决方程式（2）领域问题，获得完整的几何规格。结束语结束语上面的想法旨在寻找几何方法建立一个适当的正式角色机械设计的理论。 很明显,几何应该这样一个角色,但是工作需要建立它才刚刚开始。结语结语评价功能评价功能这个工作了数月用来描述几何特征的努力,基本上以失败告终。这个努力是出于这样一个事实:机械设计和制造常常讨论和完成的“特点”,但没有达成一致意见“是”或“做”什么功能4。(槽、打网,轴,典型特征;所有以这样或那样涉及几何的方式。)我们开始于一个猜想:几何特征可以被定义为一个几何理想化的端口定义为能量交换机制。 (这个概念是有吸引力的,因为它意味着系统的特性指定所需的所有几何定义系统与其环境的互动过程中,剩余的几何形状是由约束和优化)。 然后我们开始正式表明,猜想是一致的设计、制造、检验的应用程序。在加工方面,例如,几何特性可能与删除的边界有关材料;精力充沛的过程加工本身的动力学在宏观意义上相当清楚。夹紧功能可以定义主要通过弹性能量储存、检验特性通过测量所涉及的能量交换过程,等等。但正如我们解释我们的困难与固体和其他没有表面的功能安装,我们开始意识到功能不能被定义在任何通用系统除了纯粹的语法系统。目前我们认为,功能只是代表的信息结构,通常以参数形式,解决当前的问题。虽然语法结构可以强加给他们,他们的基本语义可以相差很大,不需要涉及特定种类的几何形状,或者任何几何。然而,如果一个功能正常使用,必须提供 feature-context一种技术条件和标准,它是解决方案的功能代表。鉴于 feature-context(如作为设计师知识的领域)和适当的推理能力适应当前问题的解决方案,功能可以非常有效的;他们的支持率在人类设计师证明了这一点。最近达菲和迪克逊5的成果说明当提供 feature-contexts 和相应的推理能力，特性可用于自动设计。(达菲和迪克逊对特性的处理似乎很特别，但如果我们当前视图的功能是正确的， “ad-hocery”也可能内在被宽容。)没有他们的环境和适当的推理能力，而只有某些自动设计系统说明上无意义的设计时，功能可能是危险的。最后,我们想指出特性表征的“已知的解决当地问题”的地方强烈限制方案结合特性使新特性。 功能组合意义只有它可以证明是一个有效的解决方案一个定义良好的当前的问题。 但即使是决定组合的问题作为一个函数的域的组件域可能非常困难。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 & Voelcker: Geometry in Mechanical Design through the boundary. We distinguish the follow- ing: S : the physical system under discussion; 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 & Voelcker: Geometry in Mechanical Design 71 O :;.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 & Voelcker: Geometry in Mechanical Design to do work on the environment, e.g. overcome specified restraining torques over a defined range of travel, and hence its geometry must be defined. The system also has a third support port. The systems primary function is represented internally by a pres- sure/torque transformer and a rotary spring which are shown as bond graph elements in the style of Ulrich and Seering 3, but this representation is not unique; 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 a