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Int J Adv Manuf Technol (2001) 17:104113 2001 Springer-Verlag London Limited Fixture Clamping Force Optimisation and its Impact on Workpiece Location Accuracy B. Li and S. N. Melkote George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Georgia, USA Workpiece motion arising from localised elastic deformation at fixtureworkpiece contacts owing to clamping and machining forces is known to affect significantly the workpiece location accuracy and, hence, the final part quality. This effect can be minimised through fixture 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 fixture subjected to quasi-static machining forces. The method uses elastic contact mechanics models to represent the fixtureworkpiece 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 32-1 type milling fixture. 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 fixture 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 fixture analysis and synthesis area have used the finite-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.melkoteme.gatech.edu the rigid-body modelling approach. Extensive work based on the FE approach has been reported 18. 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 fixture layout optimisation, and clamping force optimisation has not been addressed adequately. Several researchers have addressed fixture clamping force optimisation based on the rigid-body model 911. The rigid body modelling approach treats the fixture-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 fixturing 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 fixtureworkpiece system 15. For a relatively rigid workpiece, the location of the workpiece in the machining fixture is strongly influenced by the localised elastic defor- mation at the fixturing 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 fixture 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 fixtureworkpiece contact stiffness. In addition, the application of their algorithm for a sequence of machining loads rep- resenting a finite tool path was not discussed. Li and Melkote 19 and Hurtado and Melkote 20 used contact mechanics to Fixture Clamping Force Optimisation 105 solve for the contact forces and workpiece displacement pro- duced by the elastic deformation at the fixturing points owing to clamping loads. They also developed methods for optimising the fixture 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 fixtureworkpiece 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 32-1 fixture layout for a milling oper- ation. 2. FixtureWorkpiece Contact Modelling 2.1 Modelling Assumptions The machining fixture consists of L locators and C clamps with spherical tips. The workpiece and fixture materials are linearly elastic in the contact region, and perfectly rigid else- where. The workpiecefixture 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 fixtureworkpiece 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 fixturing point can be related as follows: F i j = k i j d i j (1) where k i j (j = x,y,z) denotes the contact stiffness in the tangential and normal directions of the local x i ,y i ,z i coordinate frame, d i j Fig. 1. A lumped-spring fixtureworkpiece contact model. x i , y i , z i , denote the local coordinate frame at the ith contact. (j = x,y,z) are the corresponding localised elastic deformations along the x i ,y i , and z i axes, respectively, F i j (j = x,j,z) represents the local contact force components with F i x and F i y being the local x i and y i components of the tangential force, and F i z the normal force. 2.2 WorkpieceFixture 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 P i acting between a spherical tipped fixture element of radius R i and a planar workpiece surface can be obtained from the closed-form Hertz- ian solution to the problem of a sphere indenting an elastic half-space. For this problem, the normal deformation D i n is given as 23, p. 93: D i n = S 9(P i ) 2 16R i (E*) 2 D 1/3 (2) where 1 E* = 1 - n 2 w E w + 1 - n 2 f E f E w and E f are Youngs moduli for the workpiece and fixture materials, respectively, and n w and n f are Poisson ratios for the workpiece and fixture materials, respectively. The tangential deformation D i t (= D i tx or D i ty in the local x i and y i tangential directions, respectively) due to a tangential force Q i (= Q i x or Q i y ) has the following form 23, p. 217: D ti t = Q i 8a i S 2 - n f G f + 2 - n w G w D (3) where a i = S 3P i R i 4 S 1 - n f E f + 1 - n w E w DD 1/3 and G w and G f are shear moduli for the workpiece and fixture materials, respectively. A reasonable linear approximation of the contact stiffness can be obtained from a least-squares fit to Eq. (2). This yields the following linearised contact stiffness values: k i z = 8.82 S 16R i (E*) 2 9 D 1/3 (4) k i x = k i y = 4 E* S 2 - n j G f + 2 - n w G w D - 1 k i z (5) In deriving the above linear approximation, the normal force P i was assumed to vary from 0 to 1000 N, and the correspond- ing R 2 value of the least-squares fit was found to be 0.94. 3. Clamping Force Optimisation The goal is to determine the set of optimal clamping forces 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 machining loads, while maintaining the fixtureworkpiece 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. 3.1 Objective Function Formulation Since the workpiece rotation due to fixturing forces is often quite small 17 the workpiece location error is assumed to be determined largely by its rigid-body translation Dd w = DX w DY w DZ w T , where DX w , DY w , and DZ w are the three orthogonal components of Dd w along the X g , Y g , and Z g axes (see Fig. 2). The workpiece location error due to the fixturing forces can then be calculated in terms of the L 2 norm of the rigid-body displacement as follows: iDd w i = (DX w ) 2 + (DY w ) 2 + (DZ w ) 2 ) (6) where ii denotes the L 2 norm of a vector. In particular, the resultant clamping force acting on the workpiece will adversely affect the location error. When mul- tiple clamping forces are applied to the workpiece, the resultant clamping force, P R C = P R X P R y P R Z T , has the form: P R C = R C P C (7) where P C = P L+1 .P L+C T is the clamping force vector, R C = n L+1 .n L+C T is the clamping force direction matrix, n L+i = cosa L+i cosb L+i cosg L+i T is the clamping force direction cosine vector, and a L+i , b L+i , and g L+i are angles made by the clamping force vector at the ith clamping point with respect to the X g , Y g , Z g coordinate axes (i = 1,2,. . .,C). In this paper, the workpiece location error due to contact region deformation is assumed to be influenced only by the normal force acting at the locatorworkpiece contacts. The frictional force at the contacts is relatively small and is neg- lected when analysing the impact of the clamping force on the workpiece location error. Denoting the ratio of the normal contact stiffness, k i z , to the smallest normal stiffness among all locators, k s z ,byj i (i = 1,. . .,L), and assuming that the workpiece rests on N X , N Y , and N Z number of locators oriented in the X g , Fig. 2. Workpiece rigid body translation and rotation. Y g , and Z g directions, the equivalent contact stiffness in the X g , Y g , and Z g directions can be calculated as k s zSO N X i=1 j iD , k s zSO N Y i=1 j iD , and k s zSO N Z i=1 j iD respectively (see Fig. 3). The workpiece rigid-body motion, Dd w , due to clamping action can now be written as: Dd w = 3 P R X k s zSO N X i=1 j iD P R Y k s zSO N Y i=1 j iD P R Z k s z SO N Z i=1 j iD 4 T (8) The workpiece motion, and hence the location error can be reduced by minimising the weighted L 2 norm of the resultant clamping force vector. Therefore, the first objective function can be written as: Minimize iP R C i w = ! 11 P R X O N X i=1 j i 2 2 + 1 P R Y O N Y i=1 j i 2 2 + 1 P R Z O N Z i=1 j i 2 2 2 (9) Note that the weighting factors are proportional to the equival- ent contact stiffnesses in the X g , Y g , and Z g directions. The components of P R C are uniquely determined by solving the contact elasticity problem using the principle of minimum total complementary energy 15, 23. This ensures that the clamping forces and the corresponding locator reactions are “true” solutions to the contact problem and yield “true” rigid- body displacements, and that the workpiece is kept in static equilibrium by the clamping forces at all times. Therefore, the minimisation of the total complementary energy forms the second objective function for the clamping force optimisation and is given by: Minimise (U* - W*) = 1 2 FO L+C i=1 (F i x ) 2 k i x + O L+C i=1 (F i y ) 2 k i y + O L+C i=1 (F i z ) 2 k i z G (10) = .l T Ql Fig. 3. The basis for the determination of the weighting factor for the L 2 norm calculation. Fixture Clamping Force Optimisation 107 where U* represents the complementary strain energy of the elastically deformed bodies, W* represents the complementary work done by the external force and moments, Q = diag c 1 x c 1 y c 1 z .c L+C x c L+C y c L+C z is the diagonal contact compliance matrix, c i j = (k i j ) - 1 , and l = F 1 x F 1 y F 1 z .F L+C x F L+C y F L+C z T is the 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. Coulombs friction law states that (F i x ) 2 +(F i y ) 2 ) #m i s F i z (m i s is the static friction coefficient). A conservative and linearised version of this nonlinear con- straint can be used and is given by 19: uF i x u + uF i y u #m i s F i z (11) 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): O F = 0 (12) O 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 fixtureworkpiece contact is strictly unilateral, the normal contact force, P i , can only be compressive. This is expressed by the following bound on P i : P i $ 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 (S y ) of the workpiece material. This upper bound is written as: P i # S y A i (i = 1, . . .,L+C) (14) where A i is the contact area at the ith workpiecefixture con- tact. The complete clamping force optimisation model can now be written as: Minimize f = H f 1 f 2 J = H .l T Ql iP R C i w J (15) subject to: (11)(14). 4. Algorithm for Model Solution The multi-objective optimisation problem in Eq. (15) can be solved by the e-constraint method 24. This method identifies one of the objective functions as primary, and converts the other into a constraint. In this work, the minimisation of the complementary energy (f 1 ) is treated as the primary objective function, and the weighted L 2 norm of the resultant clamping force (f 2 ) is treated as a constraint. The choice of f 1 as the primary objective ensures that a unique set of feasible clamping forces is selected. As a result, the workpiecefixture system is driven to a stable state (i.e. the minimum energy state) that also has the smallest weighted L 2 norm for the resultant clamping force. The conversion of f 2 into a constraint involves specifying the weighted L 2 norm to be less than or equal to e, where e is an upper bound on f 2 . To determine a suitable e,itis initially assumed that all clamping forces are unknown. The contact forces at the locating and clamping points are computed by considering only the first objective function (i.e. f 1 ). 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 in the fixture. The weighted L 2 norm of these clamping forces is computed and taken as the initial value of e. Therefore, the clamping force optimisation problem in Eq. (15) can be rewritten as: Minimize f 1 = .l T Ql (16) subject to: iP R C i w $e, (11)(14). An algorithm similar to the bisection method for finding roots of an equation is used to determine the lowest upper bound for iP R C i w . By decreasing the upper bound e as much as possible, the minimum weighted L 2 norm of the resultant clamping force is obtained. The number of iterations, K, needed to terminate the search depends on the required prediction accuracy d and ueu, and is given by 25: K = F log 2 S ueu d DG (17) where I 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 infinite set of optimum clamping forces corresponding to the infinite 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 finite number (say m) of sample points along the tool path yielding m corresponding sets of optimum clamp- ing forces denoted as P 1 opt , P 2 opt ,.,P m opt . At each sampling 108 B. Li and S. N. Melkote Fig. 4. Clamping force optimisation algorithm (used in example 1). point, the following four worst-case machining load vectors are considered: F X max = F max X F 1 Y F 1 Z T F Y max = F 2 X F max Y F 2 Z T F Z max = F 3 X F 3 Y F max Z T (18) F r max = F 4 X F 4 Y F 4 Z T where F max X , F max Y , and F max Z are the maximum X g , Y g , and Z g components of the machining force, the superscripts 1, 2, 3 of F X , F Y , and F Z stand for the other two orthogonal machining force components corresponding to F max X , F max Y , and F max Z , respectively, and iF r max i = max(F X ) 2 +(F Y ) 2 +(F Z ) 2 ). Although the four worst-case machining load vectors will not act on the workpiece at the same instant, they will occur once per cutter revolution. At conventional feedrates, the error introduced by applying the load vectors at the same point would be negligible. Therefore, in this work, the four load vectors are applied at the same location (but not simultaneously) on the workpiece corresponding to the sam- pling instant. The clamping force optimisation algorithm of Fig. 4 is then used to calculate the optimum clamping forces corresponding to each sampling point. The optimum clamping forces have the form: P i jmax = C i 1j C i 2j .C i Cj T (i = 1, . . .,m)(j = x,y,z,r) (19) where P i jmax is the vector of optimum clamping forces for the four worst-case machining load vectors, and C i kj (k = 1,. . .,C) is the force magnitude at each clamp corresponding to the ith sample point and the jth load scenario. After P i jmax is computed for each load application point, a 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 the maximum value, C max k , as given in Eq. (20): C max k # C i kj (k =