轴承座机械加工工艺及夹具设计【粗镗φ240孔】【说明书+CAD】
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本 科 毕 业 设 计 (论 文)英 文 翻 译院 (系部):专业名称:年级班级:学生姓名:指导教师日 期: Efficient prediction of workpiece-fixture contact forces using the rigid body model Michael Yu Wang1 , Diana M. Pelinescu21Department of Automation and Computer-Aided EngineeringThe Chinese University of Hong Kong, Shatin, N.T., Hong Kong2Department of Mechanical EngineeringUniversity of Maryland, College Park, MD 20742, USAABSTRACTPrediction of workpiece-fixture contact forces is important in fixturedesign since theydefine the fixture stability during clamping and strongly influence workpiece accuracyduring manufacturing. This paper presents a solution method for predicting the normal and frictional contact forces bet-ween workpiece-fixture contacts. The fixture and workpiece are consid-ered to be rigid bodies, and the model solution is solved as a constrained quadratic optimization by applying the minimum norm principle. The model reveals some intricate properties of the passive contact forces, including the potential of a locator release and the history dependency during a seq-uence of clamping and/or external force loading. Model predictions are shown to be in good agreement with known results of an elastic-contact model prediction and experimental measurements. This presented method is conceptually simple and computationally efficient. It is particularly useful in the early stages of fixture design and process planning.1 INTRODUCTIONFixture design is a practical problem and is crucial to product manufac- turing.In particular, the positioning and form accuracy of the workpiece being machined might be highly influenced by the contact forces between the work- piece and the fixture elements of locators and clamps. Localized contact forces cancause elastic /plastic deformation of the workpiece at the contact regions. This can contribute heavily to workpiece displacement and surface marring. On the other hand, insufficient contact forces may lead toslippage. This research work is supported in part by the US National Science Foundation (grants DMI- 9696071 and DMI-9696086), the ALCOA Technical Center (USA), the Hong Kong Research Grants Council (Earmarked Grant CUHK4217/01E), the Chin- ese University of Hong Kong (Direct Re-search Grant 2050254), the Ministry of Education of China (a Visiting Scholar Grant at the Sate Key Laboratory of Manufacturing Systems in Xian Jiaotong University), and the Natural Science Foundation of China (NSFC) (Young Overseas Investigator Collaboration Aw- ard 50128503) or separation of the workpiece from a locator during the manu- facturing process. Frictional forces at the workpiece-fixture contacts may help prevent workpiece from slipping and therefore act as holding forces. Their presence, however, increases the complexity of fixture analysis and design. Therefore, it is of significant help to provide the fixture designers with good knowledge of the contact forces based on an efficient engineering analysis. This would al-low the designers to be able to determine the best fixturing scheme that would minimize product quality error 1. The essential requirement of fixturing concerns with the kinematic concepts of localization and force closure, which have been extensively studied in recent years. There are several formal methods for fixturekinematic analysis based on the assumptions of rigid workpiece and fixture and frictionless workpiece fixture contacts 2, 3. Conventional fixture design procedures have been described in traditional design manuals 4, while feature-based, geometric- reasoning, or heuristic approaches have also been employed in automated fixture design schemes 5, 6, 7.For the analysis of workpiece-fixture contact forces a comprehensive approach is to consider the workpiece-fixture system as an elastic system. This system can be analyzed with a finite element model 8, 9, 10, 11. Such a model is often sensitive to the boundary conditions. It also results in a large sizemodel and requires high computational effort. Thus, this approach is not suited for the early stages of design of fixture layout and clamping schemes. The modeling complexity may be reduced if quasi-static loading conditions are assumed and a local elastic/ plastic contact model is used at each workpiece- fixture contact 12, 13. In using the principle of minimum total complementary energy 14, the geometric compatibility of workpiece-fixturedeformation is maintained without resorting to any empirical force-deformation relation such as the meta-functions used in 15.system usually is statically indetermin- ate,especially in the the presence of friction 16, 9. It is not unusual in the literature that the frictional forces are ignored so that the issue of static indeterminacy is avoided, in spite of the significant impact that the frictional forces can make.In this paper we present a solution method for theprediction of workpiece- fixture contact forces based onthe rigid body model and Coulomb friction model. The method is based on the application of the minimumnorm principle with frictional forces as constraints. As a result, it yields a unique solution for the contact forces without requiring computationally intensive numerical procedures. The paper focuses on two areasof discussions contributing to the general understanding of workpiece fixturing. (1) It is shown that the minimum norm solution of the workpiecefixture contact system can be regarded as a special form of the minimum energy principle. The proposed method gives a quick estimate of the contact forces without the need of a deformation model of the workpiece-fixture contact. When compared with experiment data and results of another approach, the prediction accuracy of the rigid body model approach is considered reasonable. This indicates that the proposed method might be particularly useful in the early stages of fixture layout and clamping scheme design. (2) The second focused discussion of the paper is the concept of history dependency of the frictional contact forces. The fixture contact forces are considered reactive forces to applied forces on the workpiece. When a friction constraint is active as defined by Coulombs law, the minimum norm solution reveals that the reactive frictional contact forces will depend on the sequence in which the external and/or clamping forces are applied on the workpiece. This history dependency may have a strong implication in work piece clamping especially when multiple clamps are applied.2 THE CONTACT SYSTEM MODEL2.1 Fixture elementsFor the purpose of analysis of workpiece-contact forces in this paper, the basic elements of a fixture are classified into passive and active types as locators and clamps. Here, a locator is referred to as a component to provide a kinematic constraint (position and/or rotation) on the workpiece. A locators represents a passive element. It includes the conventional locator pins or buttons that are used essential for a unique localization of the workpiece withrespect to a fixture reference frame. A support of a movable anvil that is sometimes used for providing additional rigidity to the workpiece is also treated as a locator for the purpose. A support is usually actuated by spring force (pop-up support), screw thread (jack support), or by hydraulics. In all cases, it is engaged only after workpiece localization and is locked into place once it makes contact with the workpiece, transforming it into a passive element. A clamp is represented as a force applied on the workpiece to provide a complete restraint of the workpiece against any external forces on the workpiece. Clamps are typically engaged manually or pneumatically. Clamping forces are said to be active elements, so as the external forces. These fixture elements are illust- rated in Fig. 1.2.2 Frictional contactWithin the framework of rigid body model, we describe each workpiece-fixture contact with a point contact model with Coulomb friction for clarity 2, 3. As shown in Fig. 2, the frictional contact produces three force components on the workpiece, with force intensities ( z , x , y ) for the normal and tangent directions respectively. Here, the inward surface unit normal of the workpiece is represented by n , while t and b represent two orthogonal tangent unit vectors. The tangential forces are due to friction as defined by Coulombs law. For a locator i contacting the workpiece at position i , the contact force and moment exerted on the workpiece is represented aswhereClamps are also defined similarly as point contacts. A clamp j is located at rj along the surface unit normal nj . It also exerts force and moment on the workpiece. However, the normal and the tangential clamping forces are considered in a different way. The normal clamping force is an active force and is treated as given. The tangential clamping forces are frictional forces that usually cannot be controlled in clamp actuation. They may have to be considered as unknowns and to be solved for. Thus, the clamping force and moment exerted on the workpiece is given as where denotes the clamping force intensity ( 0), and hn , j, ht , j and hb , j are also defined accordingly.2.3 Coulombs friction lawA simple Coulombs friction law is applied to the tangential forces such that for every locator contact and clamp contact respectively with corresponding friction coefficients and .2.4 The force equationsSuppose that the fixture has n locators and m clamps. Let Q represent all external force (and its moment) vectors applied on the workpiece. Then, the static equilibrium equation of the workpiece is given asWithindicating the intensity vector of the unknown passive forces at all contacts.3 THE METHOD OF MINIMUM NORM SOLUTION3.1 The minimum norm principleFor a general three-dimensional workpiece its fixture would must have at least 6 locators and one clamp, i.e., n 6 and m 1. In the presence of friction, the fixture system represented by Eq.11 is statically undeterminate. If _ clamps are simultaneously applied, there exist (3n+2m) unknown intensities of the reaction forces at all locator and clamp contacts in the equilibrium equation (Eq. 11). Within the framework of the rigid body model the workpiece -fixture contact problem is solved by invoking the principle of minimum norm 17. This principle essentially states that of all possible equilibrium forces for a rigid body subjected to prescribed loading, the unique force solution compatible to the equilibrium renders a minimum force norm. This is mathematically described asThus, the contact force solution is represented by a quadratic minimization with equality and inequality constraints. The linear equality constraints of Eq.15 describe the equilibrium state. The inequality constraints of Eq.16 maintain that the workpiece fixture contacts are passive and unilateral, while Eq.17 and Eq.18 define the tangential forces to obey Coulombs friction law. In addition, it is required that so the clamping forces are applied always inward to the workpiece. It should be pointed out that the minimum norm principle is equivalent to the principle of minimum complementary energy for an elastic contact system 13, 14, if we consider it to be linear and with contact elasticity defined by a compliance matrix W. In that case, the complementary energy is defined by . Thus, the minimum norm principle provides a solution in a similar sense but under the simpler provision of rigid body contact.3.2 Solution proceduresA standard optimization routine may be used for the numerical solution of Eq.14 as a quadratic minimization with linear equality constraints and nonlinear inequality constraints, for example, the popular MATLAB system. For a typical case of practice, e.g., n=6 and m=1, it is usually takes less than a few seconds to obtain a solution on a common 1GHz PC. Another numerical approach, as often used in a robotic grasping analysis 18, is to approximate the friction cones of the nonlinear inequality constraints Eq.17 and Eq.18) with polyhedral convex cones 19. This will replace the nonlinear constraints with a number of linear ones. The polyhedral approximation of the friction cones results in a minimum norm solution system with linear equality constraints and lower bounds on variable. Thus, a standard quadratic programming method could be used for efficient solution. Typically, it is sufficient to use a 4-12 sided polyhedra for an sufficiently accurate result 13, 19. Practically, this appro- ximation method does not offer significant computational advantage since the number of locators and clamps in an industrial fixture is relatively small, typically in a total of 7-12.4 CONTACT FORCES IN CLAMPINGEq.14 deals with a general case of multiple loads of clamping and external forces applied on the workpiece simultaneously. Under the unilateral and/or frictional inequality constraints, the minimum norm principle would reveal a number of intricate properties of the solution. For conceptual clarity we shall first examine the case of a single clamp in the fixture and without any external loads, i.e., m=1 and Q=0. A understanding of the special properties is essential for obtaining a complete solution for the general workpiece-fixture system. In particular, the following situations are examined: (1) the minimum-norm generalized inverse solution, (2) internal contact forces, (3) a locator release, (4) frictional forces at the clamp, and (5) the potential of history dependency of the contact forces.4.1 The specific solutionWhen the workpiece is considered to subject to a single clamp only m=1 and Q=0, the equilibrium equations become If all locators generate reactive forces and all frictional forces of the locators and the clamp are within their respective friction cones, i.e., and , for the clamp, then it is said that all the inequality constraints are inactive. In this case, the minimum norm solution for Eq.19 is easily obtained asdirectly in terms of the minimum-norm generalized inverse of matrix ,which is also known as the left pseudo-inverse 17. This is the specific solution to the linear system (Eq.19), which is effectively unconstrained.It is well known that the unconstrained linear system attains its minimum norm with the specific solution and its homogeneous solution vanishes 17. The system of contact forces is essential linear in this case where at each contact its normal contact force exists and its friction forces lie strictly inside the friction cone. From an optimization point of view, it can be said that the solution satisfies the Kuhn-Tucker (K-T) conditions as a minimum point.4.2 Internal contact forcesHowever, when any of the locators becomes nonreactive (i.e., zi=0) and/or the limit friction is reached at a locator or the clamp, one or more inequality constraints become active. Then, the solution to Eq.19 with all relevant constraints has to be solved as a minimum norm solution 17,i.e., min/ a/ , with a numerical procedure as described above. So the minimum-norm solution is in the form of (21) The first term is the specific solution of Eq.20, and the the second term is said to be the homogeneous solution. According to the linear algebra, the specific solution is a projection of the minimum-norm solution defined asby the projection matrix . The homogenous solution _ is the other orthogonal projection given asThus, in using the common terminology of robotics, the homogenous component shall be referred to as theinternal forces among the locators and clamps. In reaction to the clamping force represented by , the specific solution component is generated at the contacts to balance the clamping force only, while the homogenous solution component is to solely maintain the unilateral and frictional contact constraints. The constraint satisfaction is achieved at the cost of increasing the contact force intensities. Internal forces in the fixture are passive forces as a result of a reaction to the applied load, unlike those of a multi-fingered hand which could be actively controlled and arbitrarily specified.4.3 Locator releaseIt is possible that the minimum-norm principle yields a solution with contact forces to vanish at a locator, i.e. . This situation is called locator release, since this locator does not generate any reaction forces to the given load. In the presence of friction, this is especially possible, even in the case of minimally required kinematic localization of six locators. In other words, a clamp or an external load may render one or more locators to release, creating a potential situation of locator lift-off. These situations are undesirable in practice.4.4 Frictional forces at the clampIn Eq.19 the unknown contact forces include the frictional forces at the clamp contact. In the case that the only loading is from this clamp itself with its normal force , the frictional forces at the clamp would not exist, i.e.,. This is evident from fact that the contact normal is orthogonal to the contact tangent plane, or .A clamp cannot generate friction forces for itself. However, friction forces could be generated by other clamping or external forces. This is related to the issue of history dependency of contact forces discussed next.5 HISTORY DEPENDENCY OF FRICTIONAL FORCES5.1 Sequential loadings in fixtureFrom an operation point of view workpiece fixturing may have five basic steps : (1) stable workpiece resting under gravity, (2) accurate localization, (3) support reinforcement, (4) stable clamping, and (5) external force application. These steps have strong precedence conditions. When a workpiece is placed into a fixture, it must first assume a stable resting against the gravity. Then, the locators should provide accurate localization. Next, support anvils (if any) are moved in place, and finally clamps are activated for the part (or force-closure) immobilization. The part location must be maintained in the process of instant- tiating clamps without workpiece lift-off.5.2 Loading history and pre-loadsAs discussed in Section 4.2, an instantiating clamping load or external load may render an inequality constraint active and cause internal forces among the contacts. The (Eq.14) equilibrium system becomes nonlinear. Thus, the linear superposition principle would not apply for this load with any other clamping or external load that is applied at another time. The contact forces reactive to this load will become preload forces for the contacts when another load is applied later. In other words, the contact force solution for an instantiating (clamping and/or external) load depends on the contact forces that are already in existence. The contact forces may depend on their history.When the potential of history dependency is considered, the contact force system of Eq.14 should be described more precisely as follows. Lets denote the existing contact forces by _ and the next applied load is a clamping load , an external load , or both if they are applied simultaneously. The contact forces a in response to this instantiating load only would satisfy The resultant contact forces of all the sequential loadings are given asThus, the total contact forces may depend on the specific sequence in which the clamps and external forces are loaded on the workpiece. Practically, hydraulic or pneumatic clamps may provide for simultaneous clamping, while manual clamps are generally loaded individually. Considering the potential of history dependency (or sequence dependency), even when simultaneous clamping is possible, it is not practically reliable.6 MODEL VALIDATIONThe numerical solution using the rigid body model approach is carried out for a fixture system previously studied 13. The fixture consists of a 4-2-1 localization scheme for a rectangular workpiece. Two hydraulic clamps are used to apply equal clamping force simultaneously. The details of the fixture set-up and force measurement procedure were fully described in 13. Three different clamping normal loads were tested. An elastic contact model is used in the approach of 13 for the contact force prediction with the use of the minimum total complementary energy principle. The prediction results of our model are compared with the model prediction and the experimental results reported in 13.For three different clamping forces, Table 1 shows a comparison of the normal forces for each locatorworkpiece contact 13. The prediction of our model is quite agreeable with the elastic model prediction and the measured data. Considering the fact that the minimum norm solution of the rigid body model does not incorporate any knowledge of the elastic properties and contact geometries of the workpiece, locators and clamps, the prediction results would be very useful especially for fixture performance evaluation during fixture planning and layout design. It is also worthy pointing out that when the loading forces are not applied simule taneously, the predictedcontact forces would be different as a history dependency phenomenon. In fact, it is observed that the contact friction at one locator will reach its Coulomb friction limit. In Table 2, results of force pre- diction based on our model are given for three different loading sequences: (1) The gravity load, the first clamp and the second clamp are all applied simul- taneously (S1). (2) The gravity weight is applied first, then the two clamps are activated simultaneously. (3) The loads are applied in a sequence of the gravity weight first, then the first clamp, and finally the second clamp (S3). Although the differences among the predicted locator normal forces for all cases of clamping loads are relatively small, they verify that the frictional reaction forces are indeed dependent of the loading sequence.7 CONCLUSIONA rigid body model for the prediction of workpiece fixture contact forces under clamping and external loads is presented. At each workpiece-fixture contact, Coulomb friction model is used along with the unilateral constraint. The resulting system is a static equilibrium with a set of inequality constraints. The minimum norm principle is invoked to solve the overall model as a constrained quadratic optimization problem. It yields a unique solution of the contact forces if the equilibrium exists. Good agreements were shown for an example fixture system between the predicted results of this model and previous results of an elastic contact model prediction and experimental measurements.This proposed model reveals a number of important properties of the passive contact forces. In particular, it shows that the frictional contact forces may depend on the sequence of clamping and external loading. It might be possible that a locator will not generate any reactive forces, rendering its force release or a possible lift-off. The sequence of clamping and/or external force application may have a strong implication in fixture analysis and design.The proposed model and solution procedure are conceptually simple and more efficient than a model based on elastic contact properties of the workpiece fixture system. 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IEEE Trans. on Robotics and Automation, Vol.17, No.6, 2001, pp. 833 841用刚性坯件有效预测工件夹具的夹紧力麦克逊,黛安娜 中国香港大学自动化和计算机辅助工程研究所 香港机械工程研究所美国马里兰大学 MD 20742 摘要在夹具设计中工件夹具的控制力的有效预测是非常重要的，因为他们在稳定时确定夹具的稳定性而在制造的时候影响工件准确性。这篇论文介绍了一种为预知正常情况的解决方法和在工件- 夹具连络之间的摩擦力力量呈现的解决方法。夹具和工件被考虑是刚性坯件，而且样板的解决应该是应用最小的基准原则来解决如何强迫的二次最佳化。模型显示固定的夹紧力的一些复杂的性质,包括定位器的潜在性释放和在负担时段的夹紧顺序或外部压迫力的载入。模型预测被用于表明与松紧带- 加紧模型的结果和已知的实验的测量一致。这种表示方法概念简单和计算简洁有效。在夹具设计和程序计划的早期阶段它是特别地有用。简介夹具设计是一个实践问题并且对产品制造起决定性的作用。特别是工件的定位和形式的准确性正被以机器制造的高度化和被夹紧在工件和定位的螺丝钳的夹具元件之间的力量影响。局部夹紧力可以引起在加紧区域的工件表面发生弹性/ 塑性的毁坏。这可以很有效地防止工件的位移和表面损毁。另一方面，不充分的夹紧力可能工件的攻击的滑移。这个研究工作在美国国家的科学研究所(授与DMI-9696071 和DMI-9696086)中被证明,ALCOA 技术中心(美国),香港的中国香港大学研究所在制造业的程序期间允许会议 (指定了授予CUHK4217/01E)，(直接的关于- 搜寻授予2050254),中国(一个访问学者授予在西安交通大学中使心满意足制造业的系统主要实验室)的教育部,和来自一个定位器的工件的中国 (NSFC)(年轻的海外调查员合作奖赏 50128503) 或分离的天然科学基础。摩擦力在工件- 夹具加紧中可能帮助防止工件滑移，因此应当作为固定力。然而，他们的出现将增加夹具分析和设计的困难。因此,它是为一个具有夹紧力的专业知识的夹具设计者提供重要辅助的一项有效的工程分析。这将是一个所有- 低点设计者会决定最好的夹具图表并会将产品质量错误减到最少的方案1.近几年来夹具的结构和局部的运动精度的需求和封闭力被广泛的学习和研究。这里有很多有关基于假定刚性工件和夹具的假定运动学的分析和无摩擦的工件夹具夹紧2,3的方法.传统的夹具设计程序在传统的设计手册中被描述4,当以特征为基础的时候,几何学的-推论,或启发式接近有在自动化的固定物中被雇用设计方案5,6,7 .对于工件- 夹具夹紧的分析的方式要把工件- 夹具系统视为一个有柔性的系统。这个系统能与之一起分析成一个有限的元素模型8,9,10,11。如此一个模型对边界情况是敏感的。它也能造成大的模型而且需要高的计算的要求。因此，这种方法并不适合夹具设计方案初期制定和夹具图制定。如果假定类似的- 静电载入情况，而且一条局部的松紧带/塑料连络模型在每工件- 夹具夹紧被用，靠模切复杂可能被减少12,13. 在使用最小的总补充的能源原则方面 14, 那几何学的工件- 夹具毁坏的相容性不同，被用的例如如此任何功能的，完全跟据经验的关系而被维护在15.在夹具设计过程中，这种夹具模型对夹具的形成和计算是非常的有用的。一种简单地解决方法是假定刚性模型，这里的工件和夹具假定完全是刚性的1,2,3。在夹具的初期设计和计划中，这种刚性模型可以稳定减少模型的复杂性和提高其有用的潜力。一个通常解决的问题是工件夹具系统的静力分析，特别是在摩擦力存在的情况下16,9。在文献中，它不是例外的，忽略摩擦力和避免发生静电，尽管摩擦力可以产生重要冲力。在这篇论文中，基于刚性模型和库仑摩擦模型，我们提出了一种关于工件夹具加紧力的解决方法。这种方法以最小量的使用如限制为基础的摩擦力的基准原则。结果，它产生了一种为加紧力不需要计算的数字的程序的独特的解决方法。这篇论文把重心集中在工件夹具的二个方面进行讨论以辅助其一般理解。(1)一般显示工件夹具夹紧系统的最小基准解决能被视为一项特别的最小能源形式则。被计划的方法给没有工件- 夹具夹紧的一个毁坏模型，需要对加紧力量有一个快的估计。当另外的一个方式的数据和结果与经验数据相比较的时候，刚性模型方式的预言准确性被认为合理的。在夹具表面加紧和定位方案设计的早阶段中这种指示被计划的方法可能是特别地有用的。(2)这篇论文讨论的第二个焦点是摩擦力加紧力的辅助时段的观念。夹具夹紧力是应用工件上的力量考虑过的反动量。当一个摩擦限制是活跃的如库仑的法律所定义的时候, 最小的基准解决显示反动的摩擦力夹紧力将会仰赖外部及定位力量在工件上被应用的序列。当多样的螺丝被应用的时候,辅助时段可能有工件尤其定位的强烈含意。2 夹紧系统模型2.1 夹具元件在这篇论文中提出了工件- 夹具加紧力的分析，夹具的基本元件被分为消极的和活跃的两种类型如定位器和螺丝钳。在这里，一个定位器被称为一个为工件提供一个运动学的限制 (移动和/或旋转) 。一个定位器表现为一种消极的元件。 它包括传统的定位器大头针或被用作为必要的关于一个夹具叁考价值的工件的一个独特的局限加紧。有时作为提供附加的刚性工件的可动铁砧也被认为一个定位器。一种支撑通常被弹性力(向上支持), 螺丝钉线 (定位支撑) 促使, 或藉着液压学。在所有的情况，它是唯一的一种在工件局限之後而且进入工件制造连接的地方一次之内被锁的情况, 它转换进一种消极的元件之内。一个螺丝钳被表示在细工件上被应用提供对抗工件上的任何外部力量的工件的完全抑制的力量。典型地螺丝钳被用手或由气动作用预紧。定位力被称为活跃元素, 如此同样地外部的力量。这些夹具元件被在图1中举例说明。2.2 摩擦力夹紧 在刚性模型的结构里面，我们用一个点夹紧模型来描述每个工件- 夹具连接，2，3. 如图2所示，摩擦力夹紧产生在工件上的三个力量成份, 随着加紧力强度( z，x, y) 分别在常态和接触的方向延伸。这里, 工件的常态被 n 表现为内心表 面单位,而 t 和 b 表现为二个直角的接触单位矢量。 切线的力量是由于库仑准则所定义的摩擦力。因为定位器在位置 i 加紧工件,加紧力和夹紧力矩在工件上被表现为那里螺丝钳也被同样地定义如点连接。螺丝钳 j 沿着表面的单位常态 nj 位於 rj 。 它也发挥工件上的力量和时间性。然而，需考虑常态和切线的定位力以一种不同的方式。正常的定位力是被当做活跃的力量并且被给予。切线的定位力是摩擦力，其以通常不能够被控制在螺丝钳刺激中。他们被认为可能必须是未知的和被解决方法。因此，定位力量在工件上发挥被当作哪里是指定位力强度 (0), 和hn , j, h ， j ， ht ， j 和 ,也被因此定义。2.3 库仑的摩擦定律假如夹具有 n 个定位器和 m 螺丝钳。让 Q 表现所有的外部夹紧力(和它的力矩) 在工件上被应用的矢量。然后，假定工件是静态平衡由于在所有的夹紧指示中显示未知的钝态的强烈矢量。3 最小基准的解决方法3.1 最小的基准原则对于一个一般的立体的工件它的夹具必须会至少有 6个自由度和一个螺丝钳,也就是 和 . 在加紧之前摩擦，例11被表示位稳定的夹具系统。如果 螺丝钳同时被应用, 在平衡相等中的所有定位器和螺丝钳夹紧那里存在 (3n+2m) 强烈的未知反应力量 (例11)。 在刚性模型结构工件- 夹具夹紧问题里面被解决被唤起的最小的基准原则 17.这项原则本质上是指那些可能平衡的力量，因为一个刚性模型主题规定载入,对平衡的独特力量解决相容产品提出一个最小的力量基准。 这种算术被描述为因此，夹紧力解决被用一个二次的最小限度减到平等和不平等限制。例 15的线平等限制描述为平衡州。例 16 的不平等限制是指维持工件夹具夹紧以及消极方面,而例 17 和例 18 定义切向力服从库仑的摩擦定律。 除此之外, 如果定位力总是指向工件的内心，那么它是必需的。如果我们考虑切线它应该被指出，而且由于在那种情况下服从母体 W. 定义的夹紧弹力，补充的能源被定义，对于相等的最小补充的能源原则，最小的基准原则为一个有柔性的夹紧系统。因此，类似的最小的基准原则提供解决除了刚性模型连接的比较简单的装备。3.2 解决过程举例14来说明一个标准的最佳化常式可能用平行线的限制和非线性不平等限制对数字的解决使用过程如一个二次的减到最小限度流行的MATL
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