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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）
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(wú)疾
日 期: 2006年6月7日
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
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 [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 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
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 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 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 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 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 .
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
Withindicating 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, 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 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 procedures
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
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., 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 as
directly 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 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)
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 as
by the projection matrix . The homogenous 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 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 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 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 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 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 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. 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 c