Fixture Clamping Force Optimisation and its Impact on part location errors
- 格式：pdf
- 大小：1.64 MB
- 文档页数：10
International Journal of Machine Tools&Manufacture44(2004)373–382/locate/ijmatool Analysis of the effects offixture clamping sequence on partlocation errorsAnand Raghu,Shreyes N.MelkoteÃThe George W.WoodruffSchool of Mechanical Engineering,Georgia Institute of Technology,Atlanta,GA30332-0405,USAReceived10July2003;received in revised form2October2003;accepted15October2003AbstractSeveralfixture-related error sources are known to contribute to part location error,which can lead to poor part quality.In addition to typical error sources,such asfixture geometric error and elastic deformation of thefixture and part due to clamping forces,the clamping sequence used can also influence part position and orientation.In this paper,the effect of clamping sequence on workpiece location error is modeled analytically for afixture–workpiece system where all major compliance sources and fixture geometric error are considered.Part location error is quantified by the displacement of a response point on the part sur-face.An algorithmic procedure designed to understand how forces and deformations change as clamps are applied sequentially is presented.The effect of clamping sequence on part location error and locator reaction force is examined through model simula-tions and experiments via an example involving a3-2-1machiningfixture.#2003Elsevier Ltd.All rights reserved.Keywords:Fixture;Clamping sequence;Location errors1.IntroductionFixtures developed for manufacturing and assemblyserve to provide unique,accurate and repeatable posi-tioning of the part and provide sufficient work-holdingto eliminate movement of the part under machining orassembly loads.Multiple clamps are frequently used toserve the purpose of work-holding in afixture[1].Thesequence in which the clamps are applied can affect thefinal position and orientation of the part.This can leadto inaccurate part location that can have negativeconsequences for part quality.A relatively simple ana-lytical model of how clamping sequence can impactpart quality will aid in better process planning.Detailed models of the how clamping sequenceaffects part orientation and thus quality,have beenlimited.Originally,Bazrov and Sorokin[2]and laterCogun[3],presented the only analytical model ofclamping sequence where the effect of clampingsequence on the position of the part with respect to themachine tool was examined.Both simulated andexperimental results were provided.The concept ofelasticity was introduced,but no detailed description orimplementation of the models of the workpiece orfixture elasticity were presented.Furthermore,no priorwork has shed light on how the forces and deforma-tions at the contact points evolve as the clamps aresequentially applied.Chandra et al.[4]give a step-by-step procedure to model clamping sequence usingnon-linear FE models,specifically using GAP elementsto modelfixture–workpiece contact.They examine twosequences obtained by varying boundary conditionsand their effects on the surfaceflatness of a prismaticpart.The model,however,is limited to toggle clampswhere the clamping force is not constant,i.e.it is afunction of the interference between the clamp and theworkpiece.The model does not examine the use ofconstant-force work-holding devices,such as hydraulicor pneumatic clamps.Furthermore,fixture elementcompliances are not modeled and the FE model iscomputationally expensive to run.The basic problem of optimizing the sequence ofengineering operations,such as the order of move-ments of a robotic arm in bolt placement and appli-cation of welds was modeled using genetic algorithm(GA)techniques by Huang et al.[5].An elastic ÃCorresponding author.Tel.:+1-404-894-8499;fax:+1-404-894-9342.E-mail address:shreyes.melkote@(S.N.Melkote).0890-6955/$-see front matter#2003Elsevier Ltd.All rights reserved.doi:10.1016/j.ijmachtools.2003.10.015assembly variation simulation(EAVS)developed by Xie and Hsieh[6–8]based on thefinite element(FE) method yielded results for the optimization of clamp-ing sequence to minimize cycle time and assembly deformation.However,the work does not explain how the sequence is modeled and since the analysis is FE-based,it is computationally expensive to run.A ‘‘generate and test’’method presented by Tao et al.[9]iterates through a possible combination of clamp-ing sequences using a computational geometry approach and checks for force closure to establish the order.The model,however,treats the workpieceNomenclatured in;d it normal and tangential offset at the i th locator o in;o it normal and tangential force orientationsn Li;nÃLi original and intermediate locator positionD ni;D ti;D Gij normal and tangential deformation,geometric error at i th contact (x I,y I,z I),(x F,y F,z F)initial andfinal position of a point on the workpieceE i1;E i2;EÃi elastic modulus(1,fixture;2,workpiece;Ã,composite)G i1;G i2shear modulus(1,fixture;2,workpiece)m i1;m i2Poisson ratio(1,fixture;2,workpiece)P i,Q i,Q iX,Q iY normal,total tangential,X and Y component tangential loadsa i i thfixture–workpiece contact radiusR i radius of curvature for spherical tipped locatorsT i½ i th translation matrixR i½ i th rotation matrixD x,D y,D z translation of a point on the workpiece in X,Y,and Z directionsD h X;D h Y;D h Z angle of rotation of a point on the workpiece about the X,Y,and Z axesc ij;K fijflexibility and stiffness influence coefficients,respectivelyr Li;l Lifixture element radius and lengthr Cti;r Cbi radius of clamp tip and clamp base,respectivelyl Cti;l Cbi length of clamp tip and clamp base,respectivelyct,cb subscripts for clamp tip and clamp base,respectivelyK CnÃi;K CtÃi¼ðK CXÃi;K CYÃiÞi th contact stiffness in the normal(n)and tangential direction(t,where the component directions are given by subscripts X,Y);for planar contactÃis p,for spherical contactÃis sK Bni1l;K Bti1l¼ðK BXi1l;K BYi1lÞi th locator bulk stiffness in the respective directionsK Bnilc;K Btilc¼ðK BXilc;K BYilcÞi th clamp bulk stiffness in the respective directionsK Bni2;K Bti2¼ðK BXi2;K BYi2Þi th workpiece bulk stiffness in the respective directionsV Total total potential energy of the systemU Total total strain energy of the systemU C strain energy due to contact complianceU B strain energy due tofixture complianceU W strain energy due to workpiece complianceX Total work done by external forces on the systemF i;M i external and reaction forces and moments in matrix formS Yi1;S Yi2comprehensive yield strength(1,fixture;2,workpiece)A i contact area of the i thfixture elementl i coefficient of static friction at i th contactD P vector of rigid body translations and rotationsJ Jacobian matrixdI vector of deviations of the six contact pointsnÃÃL1:::nÃÃLLorientation vectors of the locator normals after loadingpÃÃL1:::pÃÃLLposition vectors of the locators after loadingR i reaction force at i th locatord total resultant displacement of point AD AX;D AY;D AZ displacement of point A in the X,Y,and Z coordinate directions374 A.Raghu,S.N.Melkote/International Journal of Machine Tools&Manufacture44(2004)373–382andfixture as rigid bodies and assumes that once a clamp is applied and the part is in equilibrium,the application of subsequent clamps does not affect the part equilibrium.Recent work by Liao and Hu[10]presents simula-tion results for various clamping sequences using a dynamic,multi-body model of aflexible workpiece and fixture ing six different sequences of four clamps,they show that there is variation in the pos-ition of the workpiece center and machined surface depending on the sequence.The order of load appli-cation is a function of time in the FE model used. However,their model does not explicitly describe the methodology of how the sequence is modeled,and does not discuss the change in reaction forces as each clamp is applied.Also,since the model is FE-based,it is com-putationally expensive to run.This paper presents an analytical model and algo-rithmic procedure that predicts the part location error as a function of the clamping sequence.Part location error is quantified by the displacement of a response point on the part surface.The method incorporates the effects offixture–workpiece contact compliance, overall compliance of thefixture–workpiece system, contact geometry,andfixture geometric error to ident-ify thefinal position and orientation of the part sub-ject to clamping forces in a given sequence.The compliance models,fixture geometric error,and the two-step part location method are only summarized here and are described in detail elsewhere[11].The focus of this paper is on modeling the clamping sequence and its effect on part location error and locator reaction force through model predictions and experiments involving a3-2-1fixture used for machin-ing of prismatic parts.2.Force-based clamping sequence model(FCSM) An analytical model that captures the effect of clamping sequence on workpiece accuracy has not been described in the framework of the complete part loading and clamping process,inclusive of system compliances andfixture geometric errors.Therefore, the focus of this paper is to present such a model and to also provide examples of the insight this model can givefixture designers and process planning engineers. To do so,the following four assumptions are laid out, where the fourth assumption is what allows modeling of the clamping sequence:–The part is in contact with all locators before any clamps are applied.–If multiple clamps are applied to one part face,they are applied simultaneously.–The new position and orientation of the part after each clamp actuation is given by the rigid body motion of the part.–The normal reactions atfixture elements opposite the clamps arefixed once the clamp opposing them is applied.The model is developed in order to capture the steps involved in part loading and clamping when thefinal position and orientation is obtained.The different steps involved in the algorithm are as follows:Step1The part is loaded into thefixture using the part location algorithm discussed later.Step2Thefirst clamp is applied and the resulting reaction forces and deformations are determ-ined using thefixture–workpiece deflectionmodel.The resulting rigid body motion of thepart due to deformations at the contact is thendetermined.Step3New information on the position of the part and the normal reaction forces at thefixtureelements opposing the applied clamp are storedfor the next step.Step4Steps2and3are repeated for each of the C clamps.Step5Thefinal position and orientation of the part is determined by the series of rigid body motionsof the part resulting from the actuation of eachof the C clamps.Fig.1shows aflow chart of the entire procedure.The effect of clamping sequence is captured by deter-mining the reaction forces that arefixed and those that are allowed to vary at each step of the clamping process.Fixed reaction forces are stored as bounds on the reactions when solving for the equilibrium reaction forces and workpiece deflection models.If the clamps are actuated simultaneously,then no reaction force is fixed.An example using a3-2-1fixture layout with two clamps is shown in Fig.2,where the clamp C1is actu-atedfirst,followed by clamp C2.For this example,the locator reactions(R4and R5)developed after C1is ap-plied are treated as known andfixed when determining the reaction force produced at locator L6due to actu-ation of clamp C2.In summary,the clamping sequence model is based on the critical assumption that the reaction force that develops remains essentially constant through the rest of the clamping sequence,which is presented inflow chart form in Fig.1.The following sections present the relevant models(part loading model,rigid body trans-formation,andfixture–workpiece compliance models) listed in theflowchart shown in Fig.1.A.Raghu,S.N.Melkote/International Journal of Machine Tools&Manufacture44(2004)373–3823753.Overview of part location modelAs noted in the previous section,the part location model is the starting point in simulating the effect of clamping sequence on part location.The part location model is developed based on the following assumptions:.The workpiece is prismatic and undergoes only rigid body motion;.The fixture has a 3-2-1layout;.Deflection and geometric errors at all the locators are known.This section provides an overview of the part location model which is discussed in detail elsewhere [11].The two-step process of part loading and clamp-ing is modeled by first determining the intermediate position and orientation of the part after the part is loaded into the fixture,but before clamps are applied (Fig.3).Then,from the intermediate position,deflec-tions at the contact points due to the application of a clamp are used to calculate the final position and orientation using rigid body transformations.A modi-fied version of the sequential method developed by Salisbury and Peters [9]is used to determine the final position and orientation after the first step of part loading (see Fig.1,Part loading model )given the fix-ture geometric errors at each locator.The nominal position and geometry of the prismatic part is described by a set of vertices in the globalcoor-Fig.1.Flow chart of the force-based clamping sequencemodel.Fig.2.Model of clamping sequence:actuation of C1followed by C2.376 A.Raghu,S.N.Melkote /International Journal of Machine Tools &Manufacture 44(2004)373–382dinate system (x i ,y i ,z i )for each side from which the positive outward-facing normal vectors for the sides are determined.The locators are described by their location in the global coordinate system,p Li ,and their orientation vector,n Li ,which is positive into the work-piece.Once the part has been loaded,the information of the new position and orientation of the part is input to the energy minimization routine (described later)to determine the deflections at the contact points under clamping loads.The new location and orientation of the locator tips after transformation by the sequential part loading algorithm is given by,p ÃÃLi ¼T 6½ R 5½ T 4½ R 3½ R 2½ T 1½ p ÃLi ð1Þn ÃÃLi ¼T 6½ R 5½ T 4½ R 3½ R 2½ T 1½ n ÃLi :ð2ÞThe resultant part translations and rotations of the workpiece due to rigid body motion (see Fig.3,Rigid body transformations )in the x,y,and z directions (D P ¼½D x ;D y ;D z ;D h X ;D h Y ;D h Z T )due to clamping force is given by:D P ¼J À1dI ð3ÞwheredI ¼½d I 1:::d I L Tð4ÞJ ¼n ÃÃL1ÁÁÁn ÃÃLL p ÃÃL1Ân ÃÃL1ÁÁÁp ÃÃL6Ân ÃÃLL!ð5ÞTherefore,if v F ¼ðx F ;y F ;z F Þrepresents the position of a point relative to the center of mass of the work-piece after the part has been loaded into the fixture,v ÃF represents the final position and orientation after load-ing and clamping:v ÃF ¼v F þB D P ð6ÞwhereB ¼1000100010z F Ày F Àz F 0x F y F Àx F2435:ð7ÞThe combined method of part loading and rigid body motion is valid in all cases.However,if there are no geometric errors,the part loading model does not pro-duce a change in position and orientation.4.Overview of fixture–workpiece compliance model This section gives an overview of the compliance models developed for the fixture–workpiece system.A prismatic workpiece that is linearly elastic everywhere is considered.The fixture elements at the points of con-tact and in the far field are also assumed to be linearly elastic.Fig.4shows the compliance modes for a rep-resentative workpiece and fixture contact.The fixture consists of L locators and C clamps that are either spherical or planar tipped.Clamps can oper-ate through either a constant force approach (hydraulic or pneumatic clamping)or constant distance approach (toggle clamp).4.1.Workpiece–fixture contact compliance model The contact compliance of the workpiece–locator/clamp system is dependent on the type of pressure distribution derived from the geometry of contact (see Fig.1,Fixture–workpiece deflection model).For planar-tipped locators the distribution is assumed to be given by that of a flat-tipped cylindrical punch indent-ing an elastic half-space [12].The resulting stiffnessinFig.3.Flowchart of the part loading method.A.Raghu,S.N.Melkote /International Journal of Machine Tools &Manufacture 44(2004)373–382377the normal direction is represented by:K Cnpi¼2E i 2ð1Àv 2i 2Þa i ð8Þwhere a i is the radius of the planar tipped element atthe contact point.For spherical tipped fixture elements,the contact phenomenon of a sphere indenting an elastic half-space gives the contact deformation of a spherical tipped element on a planar part surface [12].The contact stiff-ness in the normal direction can be linearized asK Cnsi ¼13:9516R i ðE Ãi Þ2!1=3ð9Þ1E i ¼1Àm 2i 1E i 1þ1Àm 2i 2E i 2ð10Þfor P i ranging from 0to 4000N where a least-squares fit is applied [13].In the tangential direction,the contact stiffness for a planar tipped fixture element is given by K Ctpi and K Cts for the spherical tipped element.The relationship is of the form:K Ctpi¼8G i 2a i ð2Àv i 2Þp ð11ÞK Ctsi ¼4E Ãi2Àv i 1G i 1þ2Àv i 2G i 2À1K Cnsið12Þwhere a i is the radius of the planar tipped element andm i2and G i2are the Possion ratio and shear modulus,respectively,of the workpiece at the i th contact pos-ition [12].4.2.Fixture compliance modelsIn this paper a solid workpiece is considered.There-fore,the workpiece is assumed to be perfectly rigid.However,a flexibility influence coefficient method for modeling workpiece compliance when dealing with thin-walled parts has been presented elsewhere by the authors [11].For the development of the fixture element com-pliance,the locators were modeled as short stubby beams with a cylindrical cross-section of radius,r Li and length,l Li .The stiffness in the normal direction is then given by [13],K Bni 1l ¼E i 1p r 2Lil Li:ð13ÞIn the tangential direction,deformation is assumed to be due to shear forces only and the deflection is modeled as a short,cylindrical beam fixed at one end and sheared at the other.The maximum bulk tangen-tial stiffness can be written as,K Bti 1l ¼3G i 1p r 2Li4l Li:ð14ÞThe clamp is modeled as two cylindrical elements in series,one for the clamp base,Cb ,and other for the clamp tip,Ctp ,which is screwed into the base.The normal deflection of the entire clamp is thus repre-sented by,K Bnilc ¼l Ctpi E Ctpi p r Ctpi þl CbiE Cbi p r Cbi !À1:ð15ÞIn the tangential direction,the clamp stiffness isgiven by the superposition of the Euler–Bernoulli beam bending stiffness for the clamp base and the stiffness of the clamp tip,which is identical to the tangential stiff-ness for the locator tip:K BX or Yilc ¼4l Cti3p r Cti G 2þ28l 3Cbi 3E Cb p r Cbi À1ð16Þwhere r Ctpi is the radius of the i th clamp tip,l Cti isthe length of the clamp tip,and G i2is the shear modu-lus of the tip.If the clamp geometry issignificantlyFig. pliance model for workpiece,contact region and locator.Table 1Fixture element position and orientationX (mm)Y (mm)Z (mm)L1 4.9924.520L2 4.99111.760L3148.566.30L422.31041.66L5124.91041.66L6079.7641.66C182.5127.4641.03C2153.0467.0641.03378 A.Raghu,S.N.Melkote /International Journal of Machine Tools &Manufacture 44(2004)373–382different from those considered in this paper,deflection models will need to be revised.5.Determination of workpiece deflection under applied loadsThe deflections at all points of contact are estab-lished when each clamp has been applied.The elastic deflections at this stage are obtained by minimizing the total potential energy of the workpiece–fixture system.To reach this stage,the problem is formulated as a nonlinear constrained optimization problem as described next.The total potential energy of the fixture–workpiece system,V Total ,is given by the sum of the strain energy,U Total and the work done by the external loads (clamp-ing loads)of the system,1Total .By applying the prin-ciple of minimum potential energy subject to constraints imposed by the fixturing problem,the following constrained optimization problem can be formulated [15,16].Minimize:min V Total ¼U Total þX Total ð17ÞwithU Total ¼U C þU B þU Wb :ð18ÞSubject to:Constraints:XF ¼0ð19ÞXM ¼0ð20ÞBounds:P i !0i ¼1;...;L þC ðÞð21ÞP i S yij ÀÁÁA i ðÞði ¼1;...;n Þð22ÞQ i l i P i ð23ÞðP i ¼R i Þ:ð24ÞThe solution of the nonlinear programming problem given in Eqs.(17–24)for each clamp actuation yields the resulting contact deformations and reaction forces.Note that Eq.(24)is only applied as a constraint when reaction forces develop during sequential application ofclamps.In this work,the nonlinear optimization prob-lem was solved in MATLAB 16.0.For completeness,results from prior work [11]are presented to provide an indication of the relative defor-mations that arise as a result of different fixture–work-piece variables.Note that these results do not consider the effects of clamping sequence.Table 2shows the dif-ferent fixture–workpiece variables and their respective levels for a 3-2-1fixture layout identical to that used in the experimental work presented later in this paper.The relative contributions of the fixture and workpiece variables to the total workpiece location error when a specific fixture/workpiece parameter is varied are shown in Fig.5.For instance,the relative contribu-tions change significantly compared to the baseline case when the contact compliance is varied by changing the fixture tip geometry from spherical to planar (see case B).In addition,experiments [14]have shown that a change in clamping force of 200N can produce a dis-placement of over 9l m at a critical point on the work-piece surface.Since industrial fixturing conditions involve clamping forces of the order of a few thousand Newtons [17],the workpiece location error due to elastic deformations can be quite significant.6.Results and discussionTo investigate the effect of clamping sequence on the location error of a workpiece,the model developed was applied to the following example.Table 2Fixture–workpiece variables and their levels [11]Case VariablesBaselineVariationA Clamp force200N400NB Fixture compliance Diameter,12.7mm;length,20.7mm Diameter,6.2mm;length,13.3mmC Contact compliance Planar (radius,5.9mm)Spherical (radius,35mm)D Workpiece compliance Solid workpiece 12.5mm wall thickness EGeometric variationNominalUndersized (30microns)Fig.5.Relative contribution of fixture and workpiece variables to location error [11].A.Raghu,S.N.Melkote /International Journal of Machine Tools &Manufacture 44(2004)373–3823796.1.Experimental set-upThe workpiece used was a solid aluminum block (E ¼70GPa ;m ¼0:334)with outer dimensions of 125Â150Â75mm)in a 3-2-1fixture layout with two clamps (see Fig.6).The two clamps were connected to and actuated by a single,manually operated hydraulic pump.An on/offvalve connected to clamp C 2allowed control of the clamping sequence.A clamping force of 200N per clamp was used.Planar tipped locators and clamps with a tip radius of 5.5mm and made of har-dened steel (E ¼207GPa ;m ¼0:296)were used.The locators had a nominal diameter of 14.3mm and a height of 6.6mm,while the clamps had a diameter of 24.6mm and a height of 25mm.The coordinates of the various fixture elements in the layout are given in Table 1.The mean coefficient of static friction for theworkpiece–fixture material pair in the range of forces being considered was determined (via sliding friction tests)to be 0.18[18].Reaction forces were measured at locator L6,which was instrumented with a PZT three-axis load cell (Kis-tler 9117A).Only normal reaction loads were measured since the normal deformation at the contact points is generally much larger than the tangential deformation.Workpiece location error was measured in the X,Y and Z coordinate directions at a representative response point,point A (0,125,75)(see Fig.6)using an eddy current proximity sensor (Kaman Instruments KD-2300-2SM,resolution of 0.1l m).The effects of three different clamping sequences given in Table 3were examined.6.2.Measured and predicted resultsThe workpiece location error is given by the total displacement of point A due to the application of clamping loads,d total :d total ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD 2AX þD 2AY þD 2AZ q :ð25ÞThe normal force at locator L6was measured after the application of each clamp in the specified sequence and is denoted as follows:(ða ÞF N )is the force after the application of the first clamp and (ðb ÞF N )is the force after application of the second clamp.This notation is used in the results presented later in Table 4.Initial experiments indicated that the compliance of the fixture block (see Fig.6),which was not modeled,contributed to the workpiece location error up to 3l m in the X and Y coordinate directions [14].Given the geometry of the ground steel base plate and the fact that it was rigidly bolted to the workbench,it was treated as perfectly rigid in the experiments and model simulations.Additionally,it was found that only par-tial planar contact between the fixture element and the workpiece surfaces was made due to fabrication and small misalignment errors in the experimental ing an ink-imprint method,the actual contact area was determined to be approximately 2/3of the nom-inal contact area.These two sources of error were accounted for in the model and the results are pre-sented in Table 4.Note that the values reported in the table represent the average of three experimental runs.The runs had good repeatability in both the displace-ment and force measurements.The measured displace-ments for each set of condition were within Æ1l m of the mean value,while the measured forces were within Æ10%of the mean [14].Analysis of the experimental data in the table reveals the differences in displacement and normal contact force at locator L6obtained in simultaneous versus sequential clamping.It can also be seen that thepre-Fig.6.(a)Schematic of 3-2-1fixture setup;(b)view of fixture set-up.380 A.Raghu,S.N.Melkote /International Journal of Machine Tools &Manufacture 44(2004)373–382dicted and measured displacements and forces are of the same order of magnitude.The average percentage errors in predicting the workpiece location error and the normal reaction force are 22.1%and 29.9%,respectively.It should be noted that the percentage errors tend to be very large in some cases, e.g.in sequence B(a),due to the very small magnitudes involved.The observed differences between the model and experiment are attributed mainly to a limitation of the experimental set-up that is absent in the model.The limitation consists of the use of a single,manually operated hydraulic pump and an on/offvalve to con-trol the clamping sequence.The valve was kept in the offposition,while clamp C 1was being actuated and was opened subsequently to actuate clamp C 2.The opening of the valve caused a drop in the hydraulic pressure at clamp C 1,which had to be compensated for by additional pumping.This behavior is not considered by the model.An additional source of error is the method used to account for the partial locator–work-piece contact noted earlier.The actual contact area,determined by the ink-imprint method,was approxi-mated in the model as an equivalent circular contact area.Consequently,the contact stiffness computed by the model can differ from the actual contact stiffness leading to differences in the equilibrium contact defor-mation and force.Recall that the key assumption that enabled the development of the clamping sequence model presented in this paper states that the reaction force produced at a locator opposite an actuated clamp is unaffected by the actuation of the next clamp.Examination of themeasured normal reaction force at locator L6for clamping sequence C between clamping steps (a)and (b)reveals a difference of approximately 20N.This variation in the normal contact force corresponds to a change in workpiece location that is one order of mag-nitude less than the overall deformation at locator L6.Consequently,the change in the normal contact force is considered to be small enough to validate the key assumption of constant reaction force made in this paper.Notwithstanding the observed differences between model predictions and measured workpiece location errors and location reactions,the model presented in this paper provides an efficient method for the fixture designer and/or process engineer to obtain qualitative and quantitative understanding of the impact of clamp-ing sequence on workpiece location error.In contrast to other approaches presented in the literature (e.g.rigid body,finite element modeling),the current clamp-ing sequence model includes the effects of fixture and workpiece deformation,and is fast (e.g.tens of seconds compared to hours for finite element modeling).Conse-quently,it enables the fixture designer/process engineer to quickly evaluate the relative merits of different clamping sequences.7.ConclusionsThe fixture–workpiece model presented captures the effect of clamping sequence on part location in the fix-ture.The impact of clamping sequence on part quality is quantified by the workpiece location error evaluated at a representative response point on the workpiece.ATable 4Comparison of measured and predicted results200N D total (l m)(a)F N (N)(b)F N (N)Sequence A Measured 6.7167.3NA (simultaneous)Predicted 5.3128.3NA %Error 20.923.3NA Sequence B Measured 5.9 1.3124.6(C1!C2)Predicted 8.0 2.3126.2%Error À35.6À76.9À1.3Sequence C Measured 6.3162.4141.8(C2!C1)Predicted 6.4187.6187.6%ErrorÀ1.6À15.5À32.3Table 3Clamping sequences examined Sequence Clamping force Step 1Step 2A (simultaneous)200N Apply both C1and C2NAB (C1!C2)200N Apply C1Apply C2C (C2!C1)200NApply C2Apply C1A.Raghu,S.N.Melkote /International Journal of Machine Tools &Manufacture 44(2004)373–382381。
