資源目錄里展示的全都有，所見即所得。下載后全都有,請放心下載。原稿可自行編輯修改=【QQ：401339828 或11970985 有疑問可加】 Int J Adv Manuf Technol (2001) 17:104–113 ? 2001 Springer-Verlag London Limited Fixture Clamping Force Optimisation and its Impact on Workpiece B. Li and S. N. Location Melkote Accuracy George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Georgia, USA Workpiece motion arising from localised elastic deformation the rigid-body modelling approach. Extensive work based on at ?xture–workpiece contacts owing to clamping and machining forces is known to affect signi?cantly the workpiece location accuracy and, hence, the ?nal part quality. This effect can be minimised through ?xture design optimisation. The clamping force is a critical design variable that can be optimised to reduce the workpiece motion. This paper presents a new method for determining the optimum clamping forces for a multiple clamp ?xture subjected to quasi-static machining forces. The method uses elastic contact mechanics models to represent the ?xture–workpiece contact and involves the formulation and solution of a multi-objective constrained optimisation model. The impact of clamping force optimisation on workpiece location accuracy is analysed through examples involving a 3–2-1 type milling ?xture. Keywords: Elastic contact modelling; Fixture clamping force; Optimisation 1. Introduction The location and immobilisation of the workpiece are two critical factors in machining. A machining ?xture achieves these functions by locating the workpiece with respect to a suitable datum, and clamping the workpiece against it. The clamping force applied must be large enough to restrain the workpiece motion completely during machining. However, excessive clamping force can induce unacceptable level of workpiece elastic distortion, which will adversely affect its location and, in turn, the part quality. Hence, it is necessary to determine the optimum clamping forces that minimise the workpiece location error due to elastic deformation while satisfying the total restraint requirement. Previous researchers in the ?xture analysis and synthesis area have used the ?nite-element (FE) modelling approach or Correspondence and offprint requests to: Dr S. N. Melkote, George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332-0405, USA. E-mail: shreyes.melkote@me.gatech.edu the FE approach has been reported [1–8]. With the exception of DeMeter [8], a common limitation of this approach is the large model size and computation cost. Also, most of the FE- based research has focused on ?xture layout optimisation, and clamping force optimisation has not been addressed adequately. Several researchers have addressed ?xture clamping force optimisation based on the rigid-body model [9–11]. The rigid body modelling approach treats the ?xture-element and work- piece as perfectly rigid solids. DeMeter [12, 13] used screw theory to solve for the minimum clamping force. The overall problem was formulated as a linear program whose objective was to minimise the normal contact force at each locating point by adjusting the clamping force intensity. The effect of the contact friction force was neglected because of its relatively small magnitude compared with the normal contact force. Since this approach is based on the rigid body assumption, it can uniquely only handle 3D ?xturing schemes that involve no more than 6 unknowns. Fuh and Nee [14] also presented an iterative search-based method that computes the minimum clamping force by assuming that the friction force directions are known a priori. The primary limitation of the rigid-body analysis is that it is statically indeterminate when more than six contact forces are unknown. As a result, workpiece displace- ments cannot be determined uniquely by this method. This limitation may be overcome by accounting for the elasticity of the ?xture–workpiece system [15]. For a relatively rigid workpiece, the location of the workpiece in the machining ?xture is strongly in?uenced by the localised elastic defor- mation at the ?xturing points. Hockenberger and DeMeter [16] used empirical contact force-deformation relations (called meta- functions) to solve for the workpiece rigid-body displacements due to clamping and quasi-static machining forces. The same authors also investigated the effect of machining ?xture design parameters on workpiece displacement [17]. Gui et al [18] reported an elastic contact model for improving workpiece location accuracy through optimisation of the clamping force. However, they did not address methods for calculating the ?xture–workpiece contact stiffness. In addition, the application of their algorithm for a sequence of machining loads rep- resenting a ?nite tool path was not discussed. Li and Melkote [19] and Hurtado and Melkote [20] used contact mechanics to Fixture Clamping Force Optimisation 105 i i i i j i solve for the contact forces and workpiece displacement pro- duced by the elastic deformation at the ?xturing points owing to clamping loads. They also developed methods for optimising the ?xture layout [21] and clamping force using this method [22]. However, clamping force optimisation for a multiclamp system and its impact on workpiece accuracy were not covered in these papers. This paper presents a new algorithm based on the contact elasticity method for determining the optimum clamping forces for a multiclamp ?xture–workpiece system subjected to quasi- static loads. The method seeks to minimise the impact of workpiece motion due to clamping and machining loads on the part location accuracy by systematically optimising the clamping forces. A contact mechanics model is used to deter- mine a set of contact forces and displacements, which are then used for the clamping force optimisation. The complete prob- lem is formulated and solved as a multi-objective constrained optimisation problem. The impact of clamping force optimis- ation on workpiece location accuracy is analysed via two examples involving a 3–2-1 ?xture layout for a milling oper- ation. 2. Fixture–Workpiece Contact Modelling 2.1 Modelling Assumptions The machining ?xture consists of L locators and C clamps with spherical tips. The workpiece and ?xture materials are linearly elastic in the contact region, and perfectly rigid else- where. The workpiece–?xture system is subjected to quasi- static loads due to clamping and machining. The clamping force is assumed to be constant during machining. This assumption is valid when hydraulic or pneumatic clamps are used. In reality, the elasticity of the ?xture–workpiece contact region is distributed. However, in this model development, lumped contact stiffness is assumed (see Fig. 1). Therefore, the contact force and localised deformation at the ith ?xturing point can be related as follows: Fj = kj dj (1) i i i j i i i i j i i x y i i i z i i 2 1/3 i n 2 2 w f = + E and E are Young’s moduli for the workpiece and ?xture materials, respectively, and and are Poisson ratios for i i i t tx ty i and y tangential directions, respectively) due to a tangential i i i i f w = + (3) i 1/3 i f w a = + 2 1/3 i k = 8.82 (4) ?1 4 2 ? 2 ? i i i x y z (j = x,y,z) are the corresponding localised elastic deformations normal force. 2.2 Workpiece–Fixture Contact Stiffness Model The lumped compliance at a spherical tip locator/clamp and workpiece contact is not linear because the contact radius varies nonlinearly with the normal force [23]. The contact deformation due to the normal force Pi acting between a spherical tipped ?xture element of radius Ri and a planar workpiece surface can be obtained from the closed-form Hertz- ian solution to the problem of a sphere indenting an elastic n given as [23, p. 93]: 16Ri(E*) where E* Ew Ef the workpiece and ?xture materials, respectively. x y 8ai Gf Gw where 4 Ef Ew and Gw and Gf are shear moduli for the workpiece and ?xture materials, respectively. A reasonable linear approximation of the contact stiffness can be obtained from a least-squares ?t to Eq. (2). This yields the following linearised contact stiffness values: 9 E* Gf Gw In deriving the above linear approximation, the normal force Pi was assumed to vary from 0 to 1000 N, and the correspond- ing R2 value of the least-squares ?t was found to be 0.94. 3. Clamping Force Optimisation Fig. 1. A lumped-spring ?xture–workpiece contact model. xi, yi, zi, The goal is to determine the set of optimal clamping forces denote the local coordinate frame at the ith contact. that will minimise the workpiece rigid-body motion due to 106 B. Li and S. N. Melkote localised elastic deformation induced by the clamping and Yg, and Z g directions, the equivalent contact stiffness in the machining loads, while maintaining the ?xture–workpiece sys- tem in quasi-static equilibrium during machining. Minimisation of the workpiece motion will, in turn, reduce the location error. This goal is achieved by formulating the problem as a multi- objective constrained optimisation problem, as described next. s Xg, Yg, and Zg NX z i i=1 s s directions can be calculated as NY NZ z i z i i=1 i=1 respectively w (see Fig. 3). The workpiece rigid-body motion, 3.1 Objective Function Formulation d , due to clamping action can now be written as: T Since the workpiece rotation due to ?xturing forces is often dw = P R X P R Y P R Z (8) w w w w T w w w quite small [17] the workpiece location error is assumed to be determined largely by its rigid-body translation d = [ X Y Z ] , where X , Y , and Z are the three orthogonal s kz N X i k s z N Y i s kz NZ i components of d w along the Xg, Yg, and Z g axes (see Fig. 2). i=1 i=1 i=1 The workpiece location error due to the ?xturing forces can The workpiece motion, and hence the location error can be reduced by minimising the weighted L2 norm of the resultant then be calculated in terms of the L2 displacement as follows: norm of the rigid-body clamping force vector. Therefore, the ?rst objective function w d = (( w 2 X ) + ( w 2 Y ) + ( w 2 Z ) ) (6) can be written as: R 2 R 2 R 2 where denotes the L2 norm of a vector. Minimize R C w = P X + P Y + P Z In particular, the resultant clamping force acting on the NX NY NZ R R R R T workpiece will adversely affect the location error. When mul- tiple clamping forces are applied to the workpiece, the resultant C X y Z PR = R P (7) i=1 i i=1 i i=1 i (9) C C C L+1 L+C T Note that the weighting factors are proportional to the equival- ent contact stiffnesses in the Xg, Yg, and Zg directions. where P C = [P . . .P ] is the clamping force vector, The components of P R are uniquely determined by solving R = [n . . .n T ] is the clamping force direction matrix, C n C L+i L+1 = [cos L+i L+C cos L+i cos L+i T ] is the clamping force direction the contact elasticity problem using the principle of minimum total complementary energy [15, 23]. This ensures that the cosine vector, and L+i , L+i , and L+i are angles made by the clamping forces and the corresponding locator reactions are clamping force vector at the ith clamping point with respect “true” solutions to the contact problem and yield “true” rigid- to the Xg, Yg, Z g coordinate axes (i = 1,2,. . .,C). body displacements, and that the workpiece is kept in static In this paper, the workpiece location error due to contact equilibrium by the clamping forces at all times. Therefore, the region deformation is assumed to be in?uenced only by the minimisation of the total complementary energy forms the normal force acting at the locator–workpiece contacts. The second objective function for the clamping force optimisation frictional force at the contacts is relatively small and is neg- and is given by: i z s z i lected when analysing the impact of the clamping force on the workpiece location error. Denoting the ratio of the normal rests on NX, NY, and NZ number of locators oriented in the Xg, Minimise (U* ? W*) = = . T Q 1 2 L+C i=1 i 2 i x kx + L+C i=1 i 2 i y ky + L+C i=1 i 2 i z kz (10) Fig. 2. Workpiece rigid body translation and rotation. Fig. 3. The basis for the determination of the weighting factor for the L2 norm calculation. Fixture Clamping Force Optimisation 107 where U* represents the complementary strain energy of the other into a constraint. In this work, the minimisation of the 1 1 1 L+C L+C L+C x y z x y z i i ?1 1 1 1 L+C L+C L+C T j j x y z x y z i 2 i 2 i i i i i i i i T f . Q Minimize f = = (15) elastically deformed bodies, W* represents the complementary work done by the external force and moments, Q = diag vector of all contact forces. 3.2 Friction and Static Equilibrium Constraints The optimisation objective in Eq. (10) is subject to certain constraints and bounds. Foremost among them is the static friction constraint at each contact. Coulomb’s friction law states x y s z s A conservative and linearised version of this nonlinear con- straint can be used and is given by [19]: x y s z Since quasi-static loads are assumed, the static equilibrium of the workpiece is ensured by including the following force and moment equilibrium equations (in vector form): F = 0 (12) M = 0 where the forces and moments consist of the machining forces, workpiece weight and the contact forces in the normal and tangential directions. 3.3 Bounds Since the ?xture–workpiece contact is strictly unilateral, the normal contact force, Pi, can only be compressive. This is expressed by the following bound on P :i Pi 0 (i = 1, . . ., L + C) (13) where it is assumed that normal forces directed into the workpiece are positive. In addition, the normal compressive stress at a contact cannot exceed the compressive yield strength (Sy) of the workpiece material. This upper bound is written as: y i where Ai is the contact area at the ith workpiece–?xture con- tact. The complete clamping force optimisation model can now be written as: f2 PC w subject to: (11)–(14). 4. Algorithm for Model Solution The multi-objective optimisation problem in Eq. (15) can be solved by the -constraint method [24]. This method identi?es one of the objective functions as primary, and converts the complementary energy (f ) is treated as the primary objective function, and the weighted L norm of the resultant clamping force (f ) is treated as a constraint. The choice of f as the in the ?xture. The weighted L norm of these clamping forces T R R C w as possible, the minimum weighted L norm of the resultant 1 2 m primary objective ensures that a unique set of feasible clamping forces is selected. As a result, the workpiece–?xture system is driven to a stable state (i.e. the minimum energy state) that also has the smallest weighted L2 norm for the resultant clamping force. The conversion of f2 into a constraint involves specifying the weighted L2 norm to be less than or equal to , where is an upper bound on f2. To determine a suitable , it is initially assumed that all clamping forces are unknown. The contact forces at the locating and clamping points are computed by considering only the ?rst objective function (i.e. f1). While this set of contact forces does not necessarily yield the lowest clamping forces, it is a “true” feasible solution to the contact elasticity problem that can completely restrain the workpiece is computed and taken as the initial value of . Therefore, the clamping force optimisation problem in Eq. (15) can be rewritten as: Minimize f1 = . Q (16) C w An algorithm similar to the bisection method for ?nding roots of an equation is used to determine the lowest upper clamping force is obtained. The number of iterations, K, needed to terminate the search depends on the required prediction accuracy and , and is given by [25]: K = log2 (17) where ??? denotes the ceiling function. The complete algorithm is given in Fig. 4. 5. Determination of Optimum Clamping Forces During Machining The algorithm presented in the previous section can be used to determine the optimum clamping force for a single load vector applied to the workpiece. However, during milling the magnitude and point of cutting force application changes continuously along the tool path. Therefore, an in?nite set of optimum clamping forces corresponding to the in?nite set of machining loads will be obtained with the algorithm of Fig. 4. This substantially increases the computational burden and calls for a criterion/procedure for selecting a single set of clamping forces that will be satisfactory and optimum for the entire tool path. A conservative approach to addressing these issues is discussed next. Consider a ?nite number (say m) of sample points along the tool path yielding m corresponding sets of optimum clamp- opt opt opt 108 B. Li and S. N. Melkote to each sampling point. The optimum clamping forces have i the form: Pjmax i i = [C1j i C2j . . . C i Cj T ] (i = 1, . . .,m) (j = x,y,z,r) (19) where P jmax is the vector of optimum clamping forces for the i four worst-case machining load vectors, and Ckj (k = 1,. . .,C) i jmax max max i max max max T opt 1 2 C fashion, the “optimum” clamping force, P , can be determined is the force magnitude at each clamp corresponding to the ith sample point and the jth load scenario. single set of “optimum” clamping forces must be selected from all of the optimum clamping forces found for each clamp from all the sample points and loading conditions. This is done by sorting the optimum clamping force magnitudes at a clamping point for all load scenarios and sample points and selecting k Ck Ckj (k = 1,. . .,C) (20) Once this is complete, a set of optimised clamping forces veri?ed for their ability to ensure static equilibrium of the workpiece–?xture system. Otherwise, more sampling points are selected and the aforementioned procedure repeated. In this for the entire tool path. Figure 5 summarises the algorithm just described. Note that although this approach is conservative, it provides a systematic way of determining a set of clamping forces that minimise the workpiece lo