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河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 1 河南理工大學(xué) 本 科 畢 業(yè) 設(shè) 計(jì) 論 文 英 文 翻 譯 院 系部 機(jī)械與動(dòng)力工程學(xué)院 專業(yè)名稱 機(jī)械設(shè)計(jì)制造及其自動(dòng)化 年級(jí)班級(jí) 2002 級(jí) 3 班 學(xué)生姓名 李太平 指導(dǎo)教師 姜無疾 日 期 2006 年 6 月 7 日 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 2 Efficient prediction of workpiece fixture contact forces using the rigid body model Michael Yu Wang1 Diana M Pelinescu2 1Department of Automation and Computer Aided Engineering The Chinese University of Hong Kong Shatin N T Hong Kong 2Department of Mechanical Engineering University of Maryland College Park MD 20742 USA ABSTRACT Prediction of workpiece fixture contact forces is important in fixture design 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 INTRODUCTION 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 3 Fixture 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 Xi an 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 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 4 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 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 5 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 Coulomb s 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 MODEL 2 1 Fixture elements For 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 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 6 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 contact Within 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 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 7 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 Coulomb s law For a locator i contacting the workpiece at position i the contact force 1r and moment exerted on the workpiece is represented as where Clamps 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 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 8 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 1 1 are also defined accordingly 2 3 Coulomb s friction law A simple Coulomb s friction law is applied to the tangential forces such that for every locator contact and clamp contact respectively with corresponding friction coefficients and i j 2 4 The force equations Suppose 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 as With indicating the intensity vector of the unknown passive forces at all contacts 3 THE METHOD OF MINIMUM NORM SOLUTION 3 1 The minimum norm principle For 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 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 9 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 as Thus 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 Coulomb s friction law In addition it is required that so the clamping forces are applied 0 j 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 2 WUT similar sense but under the simpler provision of rigid body contact 3 2 Solution procedures 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 10 A 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 CLAMPING Eq 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 solution 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 11 When 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 and0 iz 1 22n izyxiii 22 ccyx 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 as directly in terms of the minimum norm generalized inverse of matrix which is also known as the left pseudo inverse 17 1 TH 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 forces However 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 ha The first term is the specific solution of Eq 20 and the the second term is 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 12 said to be the homogeneous solution According to the linear algebra the specific solution is a projection of the minimum norm solution defined as by the projection matrix The homogenous HHPT1 solution is the other orthogonal projection given as Thus 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 c component is generated at the contacts to balance the clamping force s only while the homogenous solution component is to solely maintain the h 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 release It is possible that the minimum norm principle yields a solution with contact forces to vanish at a locator i e This situation is 0 iiiyxz 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 clamp In 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 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 13 its normal force the frictional forces at the clamp would not exist cchz i e This is evident from fact that the contact normal is 0cyx orthogonal to the contact tangent plane or A clamp cannot 0 cTCh 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 FORCES 5 1 Sequential loadings in fixture From 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 loads As 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 河南理工大學(xué)本科畢業(yè)設(shè)計(jì)英文翻譯 2006 屆 注 14 system of Eq 14 should be described more precisely as follows Let s 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 1 knh 1 KQ The contact forces a in response to this instantiating load only would satisfy The resultant contact forces of all the sequential loadings are given as Thus 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 VALIDATION The 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 clampi