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Int J Adv Manuf Technol (2001) 17:104113 2001 Springer-Verlag London LimitedFixture Clamping Force Optimisation and its Impact onWorkpiece Location AccuracyB. Li and S. N. MelkoteGeorge W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Georgia, USAWorkpiece motion arising from localised elastic deformationat fixtureworkpiece contacts owing to clamping and machiningforces is known to affect significantly the workpiece locationaccuracy and, hence, the final part quality. This effect can beminimised through fixture design optimisation. The clampingforce is a critical design variable that can be optimised toreduce the workpiece motion. This paper presents a newmethod for determining the optimum clamping forces for amultiple clamp fixture subjected to quasi-static machiningforces. The method uses elastic contact mechanics modelsto represent the fixtureworkpiece contact and involves theformulation and solution of a multi-objective constrainedoptimisation model. The impact of clamping force optimisationon workpiece location accuracy is analysed through examplesinvolving a 32-1 type milling fixture.Keywords: Elasticcontactmodelling;Fixtureclampingforce; Optimisation1.IntroductionThe location and immobilisation of the workpiece are twocritical factors in machining. A machining fixture achievesthese functions by locating the workpiece with respect to asuitable datum, and clamping the workpiece against it. Theclamping force applied must be large enough to restrain theworkpiece motion completely during machining. However,excessive clamping force can induce unacceptable level ofworkpiece elastic distortion, which will adversely affect itslocation and, in turn, the part quality. Hence, it is necessaryto determine the optimum clamping forces that minimise theworkpiece location error due to elastic deformation whilesatisfying the total restraint requirement.Previous researchers in the fixture analysis and synthesisarea have used the finite-element (FE) modelling approach orCorrespondenceandoffprintrequeststo:DrS.N.Melkote,George W. Woodruff School of Mechanical Engineering, GeorgiaInstitute of Technology, Atlanta, Georgia 30332-0405, USA. E-mail:shreyes.melkotethe rigid-body modelling approach. Extensive work based onthe FE approach has been reported 18. With the exceptionof DeMeter 8, a common limitation of this approach is thelarge model size and computation cost. Also, most of the FE-based research has focused on fixture layout optimisation, andclamping force optimisation has not been addressed adequately.Several researchers have addressed fixture clamping forceoptimisation based on the rigid-body model 911. The rigidbody modelling approach treats the fixture-element and work-piece as perfectly rigid solids. DeMeter 12, 13 used screwtheory to solve for the minimum clamping force. The overallproblem was formulated as a linear program whose objectivewas to minimise the normal contact force at each locatingpoint by adjusting the clamping force intensity. The effect ofthe contact friction force was neglected because of its relativelysmall magnitude compared with the normal contact force. Sincethis approach is based on the rigid body assumption, it canuniquely only handle 3D fixturing schemes that involve nomore than 6 unknowns. Fuh and Nee 14 also presentedan iterative search-based method that computes the minimumclamping force by assuming that the friction force directionsare known a priori. The primary limitation of the rigid-bodyanalysis is that it is statically indeterminate when more thansix 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 theelasticity of the fixtureworkpiece system 15. For a relativelyrigid workpiece, the location of the workpiece in the machiningfixture is strongly influenced by the localised elastic defor-mation at the fixturing points. Hockenberger and DeMeter 16used empirical contact force-deformation relations (called meta-functions) to solve for the workpiece rigid-body displacementsdue to clamping and quasi-static machining forces. The sameauthors also investigated the effect of machining fixture designparameters on workpiece displacement 17. Gui et al 18reported an elastic contact model for improving workpiecelocation accuracy through optimisation of the clamping force.However, they did not address methods for calculating thefixtureworkpiece contact stiffness. In addition, the applicationof their algorithm for a sequence of machining loads rep-resenting a finite tool path was not discussed. Li and Melkote19 and Hurtado and Melkote 20 used contact mechanics toFixture Clamping Force Optimisation105solve for the contact forces and workpiece displacement pro-duced by the elastic deformation at the fixturing points owingto clamping loads. They also developed methods for optimisingthe fixture layout 21 and clamping force using this method22. However, clamping force optimisation for a multiclampsystem and its impact on workpiece accuracy were not coveredin these papers.This paper presents a new algorithm based on the contactelasticity method for determining the optimum clamping forcesfor a multiclamp fixtureworkpiece system subjected to quasi-static loads. The method seeks to minimise the impact ofworkpiece motion due to clamping and machining loads onthe part location accuracy by systematically optimising theclamping forces. A contact mechanics model is used to deter-mine a set of contact forces and displacements, which are thenused for the clamping force optimisation. The complete prob-lem is formulated and solved as a multi-objective constrainedoptimisation problem. The impact of clamping force optimis-ation on workpiece location accuracy is analysed via twoexamples involving a 32-1 fixture layout for a milling oper-ation.2.FixtureWorkpiece Contact Modelling2.1Modelling AssumptionsThe machining fixture consists of L locators and C clampswith spherical tips. The workpiece and fixture materials arelinearly 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 forceis assumed to be constant during machining. This assumption isvalid when hydraulic or pneumatic clamps are used.In reality, the elasticity of the fixtureworkpiece contactregion is distributed. However, in this model development,lumped contact stiffness is assumed (see Fig. 1). Therefore, thecontact force and localised deformation at the ith fixturingpoint can be related as follows:Fij= kijdij(1)where kij(j = x,y,z) denotes the contact stiffness in the tangentialand normal directions of the local xi,yi,zicoordinate frame, dijFig. 1. A lumped-spring fixtureworkpiece contact model. xi, yi, zi,denote the local coordinate frame at the ith contact.(j = x,y,z) are the corresponding localised elastic deformationsalong the xi,yi, and ziaxes, respectively, Fij(j = x,j,z) representsthe local contact force components with Fixand Fiybeing thelocal xiand yicomponents of the tangential force, and Fizthenormal force.2.2WorkpieceFixture Contact Stiffness ModelThe lumped compliance at a spherical tip locator/clamp andworkpiece contact is not linear because the contact radiusvaries nonlinearly with the normal force 23. The contactdeformation due to the normal force Piacting between aspherical tipped fixture element of radius Riand a planarworkpiece surface can be obtained from the closed-form Hertz-ian solution to the problem of a sphere indenting an elastichalf-space. For this problem, the normal deformation Dinisgiven as 23, p. 93:Din=S9(Pi)216Ri(E*)2D1/3(2)where1E*=1 n2wEw+1 n2fEfEwand Efare Youngs moduli for the workpiece and fixturematerials, respectively, and nwand nfare Poisson ratios forthe workpiece and fixture materials, respectively.The tangential deformation Dit(= Ditxor Dityin the local xiand yitangential directions, respectively) due to a tangentialforce Qi(= Qixor Qiy) has the following form 23, p. 217:Dtit=Qi8aiS2 nfGf+2 nwGwD(3)whereai=S3PiRi4S1 nfEf+1 nwEwDD1/3and Gwand Gfare shear moduli for the workpiece and fixturematerials, respectively.A reasonable linear approximation of the contact stiffnesscan be obtained from a least-squares fit to Eq. (2). This yieldsthe following linearised contact stiffness values:kiz= 8.82S16Ri(E*)29D1/3(4)kix= kiy=4E*S2 njGf+2 nwGwD1kiz(5)In deriving the above linear approximation, the normal forcePiwas assumed to vary from 0 to 1000 N, and the correspond-ing R2value of the least-squares fit was found to be 0.94.3.Clamping Force OptimisationThe goal is to determine the set of optimal clamping forcesthat will minimise the workpiece rigid-body motion due to106B. Li and S. N. Melkotelocalised elastic deformation induced by the clamping andmachining loads, while maintaining the fixtureworkpiece sys-tem in quasi-static equilibrium during machining. Minimisationof 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.3.1Objective Function FormulationSince the workpiece rotation due to fixturing forces is oftenquite small 17 the workpiece location error is assumed to bedetermined largely by its rigid-body translation Ddw= DXwDYwDZwT, where DXw, DYw, and DZware the three orthogonalcomponents of Ddwalong the Xg, Yg, and Zgaxes (see Fig. 2).The workpiece location error due to the fixturing forces canthen be calculated in terms of the L2norm of the rigid-bodydisplacement as follows:iDdwi =(DXw)2+ (DYw)2+ (DZw)2)(6)where i i denotes the L2norm of a vector.In particular, the resultant clamping force acting on theworkpiece will adversely affect the location error. When mul-tiple clamping forces are applied to the workpiece, the resultantclamping force, PRC= PRXPRyPRZT, has the form:PRC= RCPC(7)wherePC= PL+1. . .PL+CTistheclampingforcevector,RC= nL+1. . .nL+CTis the clamping force direction matrix,nL+i= cosaL+icosbL+icosgL+iTis the clamping force directioncosine vector, and aL+i, bL+i, and gL+iare angles made by theclamping force vector at the ith clamping point with respectto the Xg, Yg, Zgcoordinate axes (i = 1,2,. . .,C).In this paper, the workpiece location error due to contactregion deformation is assumed to be influenced only by thenormal force acting at the locatorworkpiece contacts. Thefrictional force at the contacts is relatively small and is neg-lected when analysing the impact of the clamping force on theworkpiece location error. Denoting the ratio of the normalcontact stiffness, kiz, to the smallest normal stiffness among alllocators, ksz, by ji(i = 1,. . .,L), and assuming that the workpiecerests on NX, NY, and NZnumber of locators oriented in the Xg,Fig. 2. Workpiece rigid body translation and rotation.Yg, and Zgdirections, the equivalent contact stiffness in theXg, Yg, and Zgdirections can be calculated askszSONXi=1jiD, kszSONYi=1jiD, and kszSONZi=1jiDrespectively (see Fig. 3). The workpiece rigid-body motion,Ddw, due to clamping action can now be written as:Ddw=3PRXkszSONXi=1jiDPRYkszSONYi=1jiDPRZkszSONZi=1jiD4T(8)The workpiece motion, and hence the location error can bereduced by minimising the weighted L2norm of the resultantclamping force vector. Therefore, the first objective functioncan be written as:Minimize iPRCiw=!11PRXONXi=1ji22+1PRYONYi=1ji22+1PRZONZi=1ji222(9)Note that the weighting factors are proportional to the equival-ent contact stiffnesses in the Xg, Yg, and Zgdirections.The components of PRCare uniquely determined by solvingthe contact elasticity problem using the principle of minimumtotal complementary energy 15, 23. This ensures that theclamping 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 staticequilibrium by the clamping forces at all times. Therefore, theminimisation of the total complementary energy forms thesecond objective function for the clamping force optimisationand is given by:Minimise (U* W*) =12FOL+Ci=1(Fix)2kix+OL+Ci=1(Fiy)2kiy+OL+Ci=1(Fiz)2kizG(10)= .lTQlFig. 3. The basis for the determination of the weighting factor for theL2norm calculation.Fixture Clamping Force Optimisation107where U* represents the complementary strain energy of theelastically deformed bodies, W* represents the complementarywork done by the external force and moments, Q = diagc1xc1yc1z. . . cL+CxcL+CycL+Cz is the diagonal contact compliancematrix, cij= (kij)1, and l = F1xF1yF1z. . . FL+CxFL+CyFL+CzTis thevector of all contact forces.3.2Friction and Static Equilibrium ConstraintsThe optimisation objective in Eq. (10) is subject to certainconstraints and bounds. Foremost among them is the staticfriction constraint at each contact. Coulombs friction law statesthat(Fix)2+ (Fiy)2) # misFiz(misis the static friction coefficient).A conservative and linearised version of this nonlinear con-straint can be used and is given by 19:uFixu + uFiyu # misFiz(11)Since quasi-static loads are assumed, the static equilibriumof the workpiece is ensured by including the following forceand moment equilibrium equations (in vector form):OF = 0(12)OM = 0where the forces and moments consist of the machining forces,workpiece weight and the contact forces in the normal andtangential directions.3.3BoundsSince the fixtureworkpiece contact is strictly unilateral, thenormal contact force, Pi, can only be compressive. This isexpressed by the following bound on Pi:Pi$ 0(i = 1, . . ., L + C)(13)where it is assumed that normal forces directed into theworkpiece are positive.In addition, the normal compressive stress at a contact cannotexceed the compressive yield strength (Sy) of the workpiecematerial. This upper bound is written as:Pi# SyAi(i = 1, . . .,L+C)(14)where Aiis the contact area at the ith workpiecefixture con-tact.The complete clamping force optimisation model can nowbe written as:Minimize f =Hf1f2J=H.lTQliPRCiwJ(15)subject to: (11)(14).4.Algorithm for Model SolutionThe multi-objective optimisation problem in Eq. (15) can besolved by the e-constraint method 24. This method identifiesone of the objective functions as primary, and converts theother into a constraint. In this work, the minimisation of thecomplementary energy (f1) is treated as the primary objectivefunction, and the weighted L2norm of the resultant clampingforce (f2) is treated as a constraint. The choice of f1as theprimary objective ensures that a unique set of feasible clampingforces is selected. As a result, the workpiecefixture system isdriven to a stable state (i.e. the minimum energy state) thatalso has the smallest weighted L2norm for the resultantclamping force.The conversion of f2into a constraint involves specifyingthe weighted L2norm to be less than or equal to e, where eis an upper bound on f2. To determine a suitable e, it isinitially assumed that all clamping forces are unknown. Thecontact forces at the locating and clamping points are computedby considering only the first objective function (i.e. f1). Whilethis set of contact forces does not necessarily yield the lowestclamping forces, it is a “true” feasible solution to the contactelasticity problem that can completely restrain the workpiecein the fixture. The weighted L2norm of these clamping forcesis computed and taken as the initial value of e. Therefore,the clamping force optimisation problem in Eq. (15) can berewritten as:Minimize f1= .lTQl(16)subject to: iPRCiw$ e, (11)(14).An algorithm similar to the bisection method for findingroots of an equation is used to determine the lowest upperbound for iPRCiw. By decreasing the upper bound e as muchas possible, the minimum weighted L2norm of the resultantclamping force is obtained. The number of iterations, K, neededto terminate the search depends on the required predictionaccuracy d and ueu, and is given by 25:K =Flog2SueudDG(17)where I denotes the ceiling function. The complete algorithmis given in Fig. 4.5.Determination of Optimum ClampingForces During MachiningThe algorithm presented in the previous section can be usedto determine the optimum clamping force for a single loadvector applied to the workpiece. However, during millingthe magnitude and point of cutting force application changescontinuously along the tool path. Therefore, an infinite set ofoptimum clamping forces corresponding to the infinite set ofmachining loads will be obtained with the algorithm of Fig. 4.This substantially increases the computational burden and callsfor a criterion/procedure for selecting a single set of clampingforces that will be satisfactory and optimum for the entire toolpath. A conservative approach to addressing these issues isdiscussed next.Consider a finite number (say m) of sample points alongthe tool path yielding m corresponding sets of optimum clamp-ing forces denoted as P1opt, P2opt, . . ., Pmopt. At each sampling108B. Li and S. N. MelkoteFig. 4. Clamping force optimisation algorithm (used in example 1).point, the following four worst-case machining load vectorsare considered:FXmax= FmaxXF1YF1ZTFYmax= F2XFmaxYF2ZTFZmax= F3XF3YFmaxZT(18)Frmax= F4XF4YF4ZTwhere FmaxX, FmaxY, and FmaxZare the maximum Xg, Yg, and Zgcomponents of the machining force, the superscripts 1, 2, 3 ofFX, FY, and FZstand for the other two orthogonal machiningforcecomponentscorrespondingtoFmaxX, FmaxY, and FmaxZ,respectively, and iFrmaxi = max(FX)2+ (FY)2+ (FZ)2).Although the four worst-case machining load vectors willnot act on the workpiece at the same instant, they will occuronce per cutter revolution. At conventional feedrates, the errorintroduced by applying the load vectors at the same pointwould be negligible. Therefore, in this work, the four loadvectorsareappliedatthesamelocation(butnotsimultaneously) on the workpiece corresponding to the sam-pling instant.The clamping force optimisation algorithm of Fig. 4 is thenused to calculate the optimum clamping forces correspondingto each sampling point. The optimum clamping forces havethe form:Pijmax= Ci1jCi2j. . . CiCjT(i = 1, . . .,m)(j = x,y,z,r)(19)where Pijmaxis the vector of optimum clamping forces for thefour worst-case machining load vectors, and Cikj(k = 1,. . .,C)is the force magnitude at each clamp corresponding to the ithsample point and the jth load scenario.After Pijmaxis computed for each load application point, asingle set of “optimum” clamping forces must be selected fromall of the optimum clamping forces found for each clamp fromall the sample points and loading conditions. This is done bysorting the optimum clamping force magnitudes at a clampingpoint for all load scenarios and sample points and selectingthe maximum value, Cmaxk, as given in Eq. (20):Cmaxk# Cikj(k = 1,. . .,C)(20)Once this is complete, a set of optimised clamping forcesPopt= Cmax1Cmax2. . . CmaxCTis obtained. These forces must beverified for their ability to ensure static equilibrium of theworkpiecefixture system. Otherwise, more sampling points areselected and the aforementioned procedure repeated. In thisfashion, the “optimum” clamping force, Popt, can be determinedfor the entire tool path. Figure 5 summarises the algorithm justdescribed. Note that although this approach is conservative, itprovides a systematic way of determining a set of clampingforces that minimise the workpiece location error.6.Impact on Workpiece LocationAccuracyIt is of interest to evaluate the impact of the clamping forcealgorithm presented earlier on the workpiece location accuracy.The workpiece is first placed on the fixture baseplate in contactwith the locators. Clamping forces are then applied to pushthe workpiece against the locators. Consequently, localiseddeformations occur at each workpiecefixture contact, causingthe workpiece to translate and rotate in the fixture. Sub-sequently, the quasi-static machining load is applied causingadditional motion of the workpiece in the fixture. The work-piece rigid-body motion is defined by its translation Ddw= DXwDYwDZwTand rotation Duw= DuwxDuwyDuwz)Tabout the Xg,Yg, and Zgaxes (see Fig. 2).As noted earlier, the workpiece rigid-body motion arisesfrom the localised deformation, di= dixdiydizT, at each fixturingpoint. Assuming that ri= XiYiZiTdescribes the positionvector of the ith locating point relative to the workpiece centreof mass, the coordinate transformation theorem can be used toexpress diin terms of the workpiece translation, Ddw= DXwDYwDZw, and workpiece rotation, Duw= DuwxDuwyDuwzT,as follows:di= (R1i)TR(Duw)ri+ Ddw ri(21)where R1idenotes the rotation matrix describing the orientationof the local xi, yi, zicoordinate frame at the ith contact relativeto the global coordinate frame and R(Duwc) is a rotation matrixFixture Clamping Force Optimisation109Fig. 5. The algorithm used in example 2.Fig. 6. The workpiecelocation errorDrmduetoclampingandmachining.defining the orientation of the workpiece fixed coordinate framerelative to the global coordinate frame.Assuming that workpiece rotation within the fixture due toclamping, Duw, is small, R(Duw) can be approximated as:R(Duw) =F1DuwzDuwyDuwz1DuwxDuwyDuwx1G(22)Equation (21) can now be rewritten as:di= (R1i)TBiq(23)whereBi=31 0 00ZiYi0 1 0 Zi0Xi0 0 1YiXi04isthetransformationmatrixobtainedafterre-arrangingEq. (21), and q = DXwDYwDZwDuwxDuwyDuwzTis the work-piece rigid-body motion vector due to clamping and machining.The unilateral nature of the workpiecefixture contact impliesthat no tensile forces are possible at a workpiecefixture con-tact. Hence, the contact forces Fi= FixFiyFizTat the ith fixtur-ing point can be related to dias follows:Fi=HKidi, dzi. 00,otherwise(24)where dzi= (ez)Tdiis the net normal deformation at the ithcontact due to clamping and machining loads, and dzi. 0implies that the net deformation is compressive while a nega-tive sign represents tensile deformation; Ki= diagkixkiykiz isthe ith contact stiffness matrix expressed in the local coordinateframe, and ez= 0 0 1Tis a unit vector.Since hydraulic/pneumatic clamps are assumed in this study,clamping force intensities in the normal direction remain con-stant under external machining loads. Therefore, Eq. (24) mustbe modified for the clamping points as:Fi= FixFiyPiT(25)where Piis the clamping force at the ith clamping point.Letting fEdenote a 6 1 vector of the external machiningforces and weight, and combining Eqs (23)(25) with the staticequilibrium equation, the following set of equations are obtain-ed:OL+Ci=1F(R1i)Firi (R1i)FiG+ fE= 0(26)where denotes the cross-product operation. The workpiecerigid-body motion due to clamping and machining, q, can beobtained by solving Eq. (26).The workpiece location error vector, Drm= DrXmDrYmDrZmT(see Fig. 6), can now be computed as follows:Drm= Bmq(27)where rm= XmYmZmTis the position vector of the machiningpoint with respect to the workpiece centre of mass, andBm=31 0 00ZmYm0 1 0 Zm0Xm0 0 1YmXm04110B. Li and S. N. MelkoteFig. 7. The workpiece fixture configuration used in the simulationstudy. L1L6, the workpiece fixture locator contacts; Xg, Yg, Zg, theglobal coordinate frame.7.Simulation WorkThe algorithm presented earlier was used to determine theoptimum clamping forces and its impact on the workpieceaccuracy for two example cases:1. A single point force applied to the workpiece.2. A sequence of quasi-static milling loads applied to the work-piece.A 32-1 fixturing scheme shown in Fig. 7 was used to locateand hold a prismatic block of 7075-T6 aluminium (127 mm 127 mm 38.1 mm). The assumed layout for the spherical-tipped hard steel locators/clamps is given in Table 1. The staticcoefficient of friction for the workpiecefixture material pairwas taken to be 0.25.The instantaneous end milling forces were computed usingthe EMSIM program developed at the University of Illinois26 for the machining conditions given in Table 2. Forexample (1), the instantaneous machining forces were appliedto the workpiece at the point (109.2 mm, 25.4 mm, 34.3 mm).The initial and optimum clamping forces were computed usingthe algorithm described in Fig. 4 and are given in Tables 3and 4.The algorithm shown in Fig. 5 was also tested. The millingof a 25.4 mm slot was simulated using EMSIM, with the cutstartingat(0.0 mm,25.4 mm,34.3 mm)andendingat(127.0 mm, 25.4 mm, 34.3 mm) (see Fig. 8). The machiningTable 1. Fixture element position.FixtureType of elementX (mm)Y (mm)Z (mm)tipL1Spherical12.7127.019.05L2Spherical114.3127.019.05L3Spherical127.063.519.05L4Spherical114.312.70.0L5Spherical12.712.70.0L6Spherical63.5114.30.0C1Spherical73.70.019.05C2Spherical0.063.519.05C3Spherical76.276.238.1Note: L1L6 are locators; C1C2 are clamps. The tip radius is 5 mm.Table 2. Machining conditions.ConditionFeedAxial depthRadial depthSpindle(mm/tooth)(mm)(mm)speed(r.p.m.)10.20323.8125.466020.30483.8125.466030.40643.8125.4660End mill: 25.4 mm diameter, 45 helix angle; 10 radial rake angle;92.08 mm long; 4 flute coated carbide.Table 3. Objective function value example 1.Machining forceAlgorithmIntialOptimumparametersclampingclampingforceforceFx(N)Fy(N)Fz(N)d (N)KWeightedWeightedL2normL2norm(N)(N)1298.95776.06314.004.4487347.6289.52283.561105.67442.104.4487503.0392.13443.691194.51509.964.4487561.9452.7Note: Machining force is applied at (109.2 mm, 25.4 mm, 34.3 mm).Table 4. Optimum clamping forces example 1.ConditionClamping force intensities (N)1(346.3, 211.2, 645.5)Initial clamping force2(470.9, 332.3, 885.7)3(561.8, 345.3, 1028.6)1(335.8, 66.1, 679.2)Optimum clamping force2(433.9, 104.5, 928.4)3(487.9, 99.3, 1104.7)Fig. 8. The end milling process simulated for example 2.Fixture Clamping Force Optimisation111Table 5. Worst-case machining loads for example 2.Fx(N)Fy(N)Fz(N)FXmax393.40963.66406.66FYmax236.131116.58421.46FZmax331.521077.31449.05Frmax283.561105.67442.10conditions used were the same as condition 2 in Table 2. Thefour worst-case machining load vectors were computed fromthe simulated milling force data and are given in Table 5. Fivesampling points whose coordinates are listed in Table 6 wereselected for the simulation. The optimum clamping force vec-tor, Pijmax(= Pixmax, Piymax, Pizmax, Pirmax), was calculated for eachsampling point and load vector application, and the final opti-mum clamping force, Popt, was determined using the sortingalgorithm discussed earlier.8.Results and DiscussionThe convergence of the optimum clamping force algorithm forexample 1 is plotted in Fig. 9. For the fixture set-up assumedin this example (see Fig. 7), the weighted L2norm of theresultantclampingforceshastheformiPRCiw=(PRX/2)2+ (PRY)2+ (PRZ/3)2). The results reveal that the optimumclamping forces under the stated machining conditions have aTable 6. Optimum clamping forces example 2.Machining forcepijmaxOptimum clamping forceFinal weightedapplication(N)L2norm (N)point (mm)p1xmax(644.9, 312.2, 904.7540.8p1ymax(680.6, 168.8, 1010.2)507.6(21.2, 25.4, 34.3)p1zmax(738.5 341.1, 954.5)594.9p1rmax(716.3, 193.1, 1026.8)531.7p2xmax(479.2, 102.2, 972.7)415.9p2ymax(583.5, 178.5, 936.5)463.1(42.4, 25.4, 34.3)p2zmax(529.3, 120.5, 1022.9)448.1p2rmax(538.5, 145.3, 997.0)451.7p3xmax(386.3, 52.9, 918.7)365.9p3ymax(412.2, 75.0, 911.1)374.6(63.5, 25.4, 34.3)p3zmax(450.0, 100.8, 945.4)400.0p3rmax(433.9, 104.5, 928.4)392.1p4xmax(338.2, 126.0, 873.2)359.4p4ymax(307.3, 192.1, 930.6)395.9(84.7, 25.4, 34.3)p4zmax(349.5, 171.4, 957.1)402.1p4rmax(332.0, 186.1, 954.8)404.3p5xmax(230.1, 0.0, 863.9)310.1p5ymax(265.5, 15.2, 958.1)346.2(105.9, 25.4, 34.3)p5zmax(276.0, 0.0, 973.7)352.7p5rmax(277.3, 5.1, 979.6)354.8Popt(738.5, 341.1, 1026.8)608.2Fig. 9. Covergence characteristics of the algorithm for example 1.much lower weighted L2norm compared to the initial clampingforce intensities. The initial clamping forces are those obtainedat the start of the algorithm by minimising the complementaryenergy of the workpiecefixture system.The workpiece location errors due to clamping and the pointload are given in Table 7. The results show that the workpiecerotation is small. The reduction in error at the machining pointranges from 13.1% to 14.6%. In this case, the improvementis not very large for all three machining conditions becausethe initial clamping force values determined from the minimis-ation of the complementary potential energy are close to theoptimum clamping forces.ThealgorithmofFig. 5wasusedinthesecondexample where a sequence of milling loads was appliedto the workpiece. The optimum clamping forces, Pijmax(=Pixmax, Piymax, Pizmax, Pirmax), corresponding to each sample pointare given in Table 6, along with the final optimum clampingforce, Popt. The weighted L2norm of the initial and optimumclamping forces are plotted in Fig. 10, and at each samplingpoint, the weighted L2norm of Pixmax, Piymax, Pizmax, and Pirmaxare plotted.The results indicate that since every component of Poptisthe maximum among all the corresponding clamping forcemagnitudes, it has the maximum weighted L2norm. However,as shown in Fig. 10, if the maximum component at eachclamping point is used for the initial clamping force, then thecorresponding set of clamping forces, Pini, has a considerablylarger weighted L2norm than Popt. Hence, Poptis an improvedsolution for the complete tool path.The above simulation results indicate that the approachpresented in this paper can be used to optimise the clampingforces. In comparison to the initial clamping force intensity,this approach will reduce the weighted L2norm of the resultantclamping force, and will therefore improve the workpiecelocation accuracy.112B. Li and S. N. MelkoteTable 7. Workpiece location error reduction example 1.Resultant workpiece motion, qError vector at the machiningError vector magnitude,Error(103mm, 103degree)point DrmiDrmireduction (%)(103mm)(103mm)1(10.6 16.2 4.5 2.9 2.8 3.5)(12.0 17.1 2.6)21.1Initial clamping force2(13.3 22.5 5.7 4.6 4.7 3.6)(14.2 24.0 2.6)28.13(16.5 25.4 6.8 4.6 4.6 5.2)(18.5 26.9 3.7)32.81(7.8 15.8 9.1 1.3 4.2 3.3)(7.7 13.8 9.0)18.213.7Optimum clamping2(9.4 21.7 12.0 1.2 6.7 3.7)(8.7 19.1 11.5)24.014.6force3(13.4 25.2 9.6 3.0 3.2 5.4)(14.4 23.6 6.8)28.513.1Fig. 10. Comparison of initial and optimum objective function valuesfor example 2.9.ConclusionsThe paper presented a new method for determining the optimumclamping forces for a multiple clamp fixtureworkpiece systemsubjected to quasi-static machining loads. The algorithm forclamping force optimisation is based on a contact mechanicsmodel of the fixtureworkpiece system and seeks to minimisethe weighted L2norm of the resultant clamping force applied tothe workpiece, and therefore the workpiece location error. Theoverall model is formulated as a bi-objective constrained optimis-ation problem and solved using the e-constraint method. Thealgorithm was illustrated via two simulation examples involvinga 32-1 type milling fixture with two clamps.Future work will address optimisation of the fixtureworkpiecesystem in the presence of dynamic loads where the inertia,stiffness, and damping effects play an important role in determin-ing the workpiecefixture system response characteristics.References1. J. D. 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