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大红鹰HX-40型变速箱机械加工工艺规程及钻孔装置设计带机械图
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DOI 10.1007/s00170-003-1796-6ORIGINAL ARTICLEInt J Adv Manuf Technol (2005) 25: 409419Nicholas Amaral Joseph J. Rencis Yiming (Kevin) RongDevelopment of a finite element analysis tool for fixturedesign integrityverification and optimisationReceived: 15 March 2003 / Accepted: 11 May 2003 / Published online: 25 August 2004 Springer-Verlag London Limited 2004Abstract Machining fixtures are used to locate and constraina workpiece during a machining operation. To ensure that theworkpiece is manufactured according to specified dimensionsand tolerances, it must be appropriately located and clamped.Minimising workpiece and fixture tooling deflections due toclamping and cutting forces in machining is critical to machiningaccuracy. An ideal fixture design maximises locating accuracyand workpiece stability, while minimising displacements.The purpose of this research is to develop a method for mod-elling workpiece boundary conditions and applied loads duringa machining process, analyse modular fixture tool contact areadeformation and optimise support locations, using finite elementanalysis (FEA). The workpiece boundary conditions are definedby locators and clamps. The locators are placed in a 3-2-1 fixtureconfiguration, constraining all degrees of freedom of the work-piece and are modelled using linear spring-gap elements. Theclamps are modelled as point loads. The workpiece is loadedto model cutting forces during drilling and milling machiningoperations.Fixture design integrity is verified. ANSYS parametric de-sign language code is used to develop an algorithm to auto-matically optimise fixture support and clamp locations, andclamping forces, to minimise workpiece deformation, subse-quently increasing machining accuracy. By implementing FEAin a computer-aided-fixture-design environment, unnecessaryand uneconomical “trial and error” experimentation on the shopfloor is eliminated.Keywords FEA Finite Element Analysis Fixture OptimisationN. Amaral (u)V-Engine Manufacturing Engineering,Ford Motor Company, Powertrain Operations,21500 Oakwood Boulevard, Dearborn, MI 48124-4091 USAE-mail: namaralFax: +1-313-2486734J.J. Rencis Y. RongWorcester Polytechnic Institute,100 Institute Road, Worcester, MA, 01609-2280 USA1 IntroductionMachining fixtures are used to locate and constrain a work-piece during a machining operation. To ensure that the work-piece is manufactured according to specified dimensions andtolerances, it must be appropriately located and clamped. Pro-duction quality depends considerably on the relative position ofthe workpiece and machine tools. Minimising workpiece and fix-ture tooling deflections due to clamping and cutting forces inmachining is critical to machining accuracy. The workpiece de-formation during machining is directly related to the workpiece-fixture system stiffness. An ideal fixture design maximises locat-ing accuracy, workpiece stability, andstiffness, while minimisingdisplacements.Traditionally, fixtures were designed by trial and error, whichis expensive and time consuming. Research in flexible fixtur-ing and computer-aided-fixture-design (CAFD) has significantlyreduced manufacturing lead-time and cost. The purpose of thisresearch is to develop a computer-aided tool to model workpieceboundary conditions and applied loads in machining.The majority of finite element analysis (FEA) research con-ducted in fixture design considers workpiece boundary condi-tions to be rigid and applied loads to be concentrated. In all caseswhere friction is considered, rigid Coulomb friction is assumed.Cutting tool torque, which results in a trend of workpiece ro-tation, is not considered. Clamping forces are considered to beconstant point loads.This study acknowledges that workpiece boundary condi-tions are deformable and influence the global stiffness of theworkpiece-fixture system. The boundary conditions of the work-piece, the locators, are modelled as multiple springs in parallelattached to the actual workpiece-fixture contact area on the sur-face of the workpiece. Also, tangential and normal stiffness com-ponents of the boundary conditions are not assumed to be equalas in rigid Coulomb friction, but are assigned independently. Inapplying loads representative of the machining operation, torque,axial and transverse loads due to feeding are considered. An in-410depth discussion of the work presented herein can be found inAmaral 1.In this study, both the finite element analysis and optimisa-tion are conducted in ANSYS. Within the analysis, a workpieceis imported in initial graphics exchange specification (IGES) for-mat. Material properties, element type, and real constants aredefined. The workpiece is meshed and boundary conditions andloads are applied. The model is then solved and results are re-trieved parametrically, and support locations, clamp locations,and clamping forces are optimised tominimise workpiece deflec-tion 1. The advantage of the method developed herein is that anexternal software package for optimisation is not required, thuscompatibility between two packages is not a concern.2 Literature reviewPrinciples of fixture design and preceding FEA research in fix-ture design are discussed. Although some research has been con-ducted in fixture design, a comprehensive finite element modelthat accurately represent applied boundary conditions and loadshas not been developed. Tables 1 and 2 summarise the precedentresearch conducted on FEA and fixture design.Table1. Literature survey of workpiece modelsReferenceWorkpiece modelMaterialElement typeTypeE (Pa)Lee and Haynes 2Steel homogeneous6.91080.3U/A*3-D solid 8-node brickIsotropic linear elasticPong et al. 3Aluminium homogeneous6.910100.3U/A3-D solid 10-node tetrahedral;Isotropic linear elasticANSYS SOLID92Trappey et al. 5Aluminium homogeneous6.910100.30.3U/AIsotropic linear elasticCai et al. 6Steel2.110110.3U/A2-D 4-node rectangular element;Isotropic linear elasticMSC NASTRAN QUAD4Kashyap and DeVries 7Aluminium homogeneous6.910100.3U/A3-D solid tetrahedral elementsIsotropic linear elastic*U/A: unavailableTable2. Literature survey of boundary conditions and loadingReferenceFixture component modelSteady-state load modelLocatorsClampsDrillingMillingLee and Haynes 2Rigid area constrain,U/A*U/ANormal and shear point loadsRigid coulomb frictionPong et al. 33-D spring-gap interface element,N/A*Normal point loadsN/ARigid coulomb frictionTrappey et al. 53-D solid deformable constraintsPoint loadsNormal point loadsNormal and shear point loadsCai et al. 6Rigid point constraintsN/ANormal point loadsNormal and shear point loadsKashyap and DeVries 7Rigid point constraintsPoint loadsNormal point loadsNormal and shear point loads*U/A: unavailable*N/A: not applicableLee and Haynes 2 used FEA to minimise workpiece deflec-tion. Their workpiece was modelled as linear elastic, howeverfixture tooling was modelled as rigid. Their objective functionincluded the maximum work done by clamping and machiningforces, the deformation index, and the maximum stress on theworkpiece. Their study considers the importance of part defor-mation with respect to the necessary number of fixturing elem-ents and the magnitude of claming forces 3. Coulombs law offriction was used to calculate the frictional forces the workpiece-fixture contact points. The machining forces were applied atnodal points. Manassa and DeVries 4 conducted similar re-search to that of Lee and Haynes 2, but modelled fixturingelements as linear elastic springs.Pong et al. 3 used spring-gap elements with stiffness, sep-aration, and friction capabilities to model elastic workpieceboundary conditions. Three-dimensional tetrahedral elementswere used to mesh the finite element model of the solid work-piece. All contacts between the workpiece and the fixture wereconsidered to be point contacts and machining forces were ap-plied sequentially as point loads. The positions of locators andclamps, and clamping forces were considered design variablesfor optimisation. Trappey et al. 5 developed a procedure forthe verification of fixtures. FEA was used to analyse the stress-strain behaviour of the workpiece when machining and clamping411forces were applied. A mathematical optimisation model wasformulated to minimise workpiece deformation with a feasiblefixture configuration.Cai et al. 6 used FEA to analyse sheet metal deforma-tion and optimised support locations to minimise resultantdisplacements. Kashyap and DeVries 7 used FEA to modelworkpiece and fixture tool deformation, and developed an op-timisation algorithm to minimise deflections at selected nodalpoints by considering the support and tool locations as designvariables.A summary of research on FEA and fixture design optimi-sation is shown in Table 3. The majority of research conductedin finite element analysis and fixture design optimisation, re-sulted in the development of a mathematical algorithm. Pong etal. 3 used the ellipsoid method to optimise support locationsand minimise nodal deflection. Trappey et al. 5 used an exter-nal software package, GINO 8, to optimise support locationsand clamping forces. Cai et al. 6 used a sequential quadraticprogramming algorithm in an external FORTRAN based soft-ware package, VMCON, to perform a quasi-Newton non-linearconstrained optimisation of N-2-1 support locations to minimisesheet metal deflection. Kashyap and DeVries 7 developed a dis-crete mathematical algorithm for optimisation.Table3. Literature survey of optimisation analysisReferenceOptimization analysisMethodObjective functionSoftware packagePong et al. 3Ellipsoid methodNodal deflectionN/A*Trappey et al. 5Non-linear mathematical algorithmNodal deflectionGINO 8Cai et al. 6Sequential quadratic programming algorithmNodal deflection normal to sheet metal surfaceVMCON 9Kashyap and DeVries 7Discrete mathematical algorithmNodal deflectionN/A*N/A: not applicableFig.1.Fixturedesignanalysismethodology3 Fixture design analysis methodologyThe flowchart in Fig. 1 is a summary of the fixture design analy-sis methodology developed and used in this work. In summary,workpiece IGES geometry is imported from the solid modellingpackage, the workpiece model is meshed, boundary conditionsare applied, the model is loaded, representative of a machiningoperation, the model is solved, and then boundary conditions areoptimised to minimise workpiece deflections.3.1 Workpiece modelThe workpiece model is the starting point of the analysis. Thisresearch currently limits the workpiece geometry to solids withplanar locating surfaces. Some workpiece geometry may containthin-walls and non-planar locating surfaces, which are not con-sidered in this study.GeometryTheworkpiecemodel,createdinPro/ENGINEERor other solid modelling software is exported to ANSYS in IGESformat with all wireframes and surfaces. IGES is a neutral stan-dard format used to exchange models between CAD/CAM/CAEsystems. ANSYS provides two options for importing IGES412Table4. Workpiece and locator material propertiesMaterialE (Pa) (kg/m3)y(Pa)WorkpieceAISI 12122.0101178610.2952.3108LocatorsAISI 11442.0101178610.2956.7108files, DEFAULT and ALTERNATE. The DEFAULT option al-lows file conversion without user intervention. The conversionincludes automatic merging and creation of volumes to pre-pare the model for meshing. The ALTERNATE option usesthe standard ANSYS geometry database, and is provided forbackward compatibility with the previous ANSYS import op-tion. The ALTERNATE option has no capabilities for automat-ically creating volumes and modes imported through this trans-lator require manual repair through the PREP7 geometry tools.To select the options for importing an IGES file, the IOPTNis used. See Appendix A in 1 for a detailed description ofimplementation.Material properties The workpiece material in this studyis homogenous, isotropic, linear elastic and ductile; this is con-sistent with the material properties of most metal workpieces.The material selected is SAE/AISI 1212 free-machining grade(a)carbon steel with Youngs modulus, E = 30106psi Poissonsratio, = 0.295, and density, = 0.283 lb/in3, and hardness of175 HB. Although SAE1212 steel was selected for use in thisstudy because it is commonly used and is a benchmark materialfor machinability, any material could be used for the workpieceby simply changing the isotropic material properties in ANSYS.Table 4 lists the material properties selected in this study for theworkpiece and locators.3.2 Meshed workpiece modelAn 8-node hexahedral element (SOLID45), with three degreesof freedom at each node, and linear displacement behaviouris selected to mesh the workpiece. SOLID45 is used for thethree-dimensional modelling of solid structures. The elementis defined by eight nodes having three degrees of freedom ateach node: translations in the nodal X, Y, and Z directions. TheSOLID45 element degenerates to a 4-node tetrahedral configu-ration with three degrees of freedom per node. The tetrahedralconfiguration is more suitable for meshing non-prismatic geom-etry, but is less accurate than the hex configuration. ANSYSrecommends that no more than 10% of the mesh be comprisedof SOLID45 elements in the tetrahedral configuration. For a de-tailed description of the element type selection process, referto 1.3.3 Boundary conditionsLocators and clamps define the boundary conditions of the work-piece model. The locators can be modelled as point or areacontact and clamps are modelled as point forces.LocatorsPoint contact. The simplest boundary condition is a pointconstraint on a single node. A local coordinate system (LCS),referenced from the global coordinate system origin, is createdat the centre of each locator contact area, such that the z-axisnormal to the workpiece locating surface. The node closest tothe centre of the local coordinate system origin is selected andall three translational degrees of freedom (ux, uy, and uz) areconstrained. The point constraint models a rigid locator with aninfinitesimally small contact area.Tomodellocatorstiffnessandfrictionatthecontactpoint,a3-D interface spring-gap element is placed at the centre of the LCS.The element is connected to existing nodes on the surface of theworkpiece and to a fully constrained copied node offset from theworkpiece surface in the z-direction of the local coordinate sys-tem, i.e., perpendicular to the surface. Figure 2 is a model of theCONTAC52element usedtorepresent a linearelastic locator.Area contact. To model a rigid locator with a contact area,multiple nodes are fixed within the contact area. An LCS is cre-ated on the workpiece surface at the centre of the locator contactarea. For a circular contact area, a cylindrical LCS is created andnodes are selected at 0 r rL. For a rectangular contact area,a Cartesian LCS is created and nodes are selected at 0 x xLand 0 y yL. All three translational degrees of freedom (ux,uy, and uz) of each of the nodes are constrained. This model as-sumes rigid constraints, however in reality locators are elastic.A more accurate representation of the elastic locators con-sists of multiple ANSYS CONTAC52 elements in parallel.Nodes are selected within the locator contact area and are copiedoffset perpendicular to the locating surface. Each selectednode isconnected to the copied node sequentially with the CONTAC52element. Figure 3 shows the contact area model with multiplespring-gap elements in parallel used to represent a linear elasticlocator. It is important to note, that the user is constrained to thenumber of nodes within the specified contact area, when attach-ing the CONTAC52 elements. It is possible that there could bea different number of elements modelling each locator, becauseof the number of associated nodes within the contact area. Thus,the element normal and tangential stiffness, which is specifiedin the real constant set would vary. For this reason, multiple realconstant sets must be created for the CONTAC52 element, andthen assigned accordingly when creating elements in a specifiedlocal coordinate system.In Fig. 4, the method for obtaining the normal and tangentialstiffness for a locator is shown. The stiffness divided by the totalFig.2. CONTAC52 element used to model point contact for locators 10413Fig.3. CONTAC52 elements in parallel, used to model area contact forlocators 10Fig.4. Normal and tangential stiffness for locatornumber of springs is assigned accordingly to each spring-gapelement, in the real constant set. A point load is applied to thethree-dimensional finite element model of the real locator, nor-mal to the contact area to determine the normal stiffness. A pointload is applied tangent to the contact area of the real locator todetermine the tangential or “sticking” stiffnessof the locator. Thestiffness values are then assigned to the CONTAC52 elements.Clamps The clamps are used to fully constrain the work-piece once it is located. It is common to use multiple clamps andclamping forces that are generally constant for each clamp. Theclamping force, Fclis applied through either a toggle mechan-ism or a bolt mechanism, which lowers a strap that comes intocontact with the workpiece. Although friction is just as import-ant in clamping as it is in locating, it is not modelled at theclamp contact area due to limitations in ANSYS. In order tomodel friction, a comprehensive three-dimensional model of theentire workpiece-fixture system is required, with contact and tar-get surfaces defined at the fixture-workpiece contact areas. Theclamping forces are modelled in ANSYS as point loads on nodesselectedeither within a rectangular area for a clamp strap or a cir-cular area on the workpiece surface for a toggle clamp. Bothclamps may also be modelled with a single point load at the cen-tre of the clamp contact area.3.4 LoadingThe two machining operations, milling and drilling, are dis-cussed. The purpose of this research is not to accurately modelthe machining process, but to apply the torque and forces that aretransferred through the workpiece in machining, to determine thereactions at the boundary conditions of the workpiece. The de-sired result of the load model is the trend of rotation from theapplied torque of the cutting tool, and translation, due to axialfeeding of the workpiece and transverse motion of the table inmilling.Drilling The forces in a drilling operation include a torque,T, to generate tool rotation, shear force, V, created by tool rota-tion at the cutting edge contact for chip removal, and an axialload, P, due to feeding. The forces in drilling are time and pos-ition dependent and oscillatory due to cutter rotation, since thecutting edge of the tool is not in constant contact with the work-piece at a particular location. The cutting force increases mono-tonically during tool entry and then approaches steady-state.Fluctuations in the cutting force are due to cutting tool toothdistribution during rotation. In this study, the torque and thrustforces in feeding are applied as steady-state loads, since initialtool entry is not considered. In previous FEA fixture design re-search, loads were applied as a steady-state. Also neglected wascutting tool torque and subsequently the workpiece deflectionsdue to the trend of rotation in the fixture. An initial attemptto model the distributed loading using a number of point loadsapplied at key points was unsuccessful, due to limitations inANSYS. The model consisted of placing key points on a localcoordinate system created on the machining surface of the work-piece. The key points were located at exact R, , and Z positionson the cutting tool perimeter. At each key point forces were ap-plied to model a drilling operation. The torque was modelledwith tangential forces placed at the outer radius of the cuttingtool contact area. The tangential couple forces were decomposedinto global X and Y components. The axial load was modelledby applying forces at each key point in the global Z direction.The reason this model failed is that the key points created onthe workpiece surface are geometric entities and are not part ofthe finite element mesh, i.e., key points are not nodes. Due tothis limitation in ANSYS, the point load model was modifiedto apply loads at existing nodes on the workpiece surface. Fig-ure 5 shows the modified load model for drilling. Notice thatnode i is slightly offset from the cutting tool perimeter. Becausea node may not exist in the exact location specified by R, , andZ, the node closest to that location in the local coordinate sys-tem is selected and forces are applied as point loads with globalX, Y, Z components. The user may minimise the distance be-tween a specified coordinate location and an existing node by414Fig.5. Drilling load modelincreasing the mesh density. The nodes are selected at equivalent intervals on or near the cutting tool perimeter. At each selectednode, global X and Y components of the tangential couple force,Ftiand axial load component, Fciare applied. The applied torqueis equal to the sum of the tangential forces multiplied by thecutting tool radius, r.FtiXand FtiYare the global X and Ycompo-nents, respectively, of the tangential force, Fti. Fciis equal to thetotal axial load, Fc, divided by the number of nodes over which itis applied.A simplified model entails the use of a single point force nor-mal to the surface of the workpiece to model the cutting toolaxial load and a couple to model the applied torque. A study wasconducted to determine whether multiple point forces appliedalong the cutting tool perimeter are actually necessary to modelthe axial load and assess the validity of the simplified modelMilling The loading in a milling operation involves an axialload, a transverse load due to the linear feeding of the workpiece,a torque to generate tool rotation, which is transmitted throughthe workpiece, and shear force in the cutting area. Figure 6 isthe loading model for end milling. The end milling model is thesame as the drilling model, with the transverse load added. Be-cause the objective of the analysis is to determine the maximumresultant displacements and equivalent stresses in the workpieceduring the operation and tool entry are not considered, only theFig.6. Milling load modelaverage steady-state load magnitude is addressed. In this study,the cutting forces are applied as steady-state loads. In previousFEA research, forces in milling were traditionally modelled assteady-state single point loads and torque was neglected. Theaxial load due to feeding can be applied as multiple point loadson the cutting tool perimeter or as a single point load. The trans-verse load, Ftri, is applied as a single point load at the centre ofthe cutting tool.4 Fixture design optimisationIn order to minimise workpiece deformation and maximise lo-cating accuracy, the boundary conditions (support locations andclamp location, and clamping force magnitude) of the model areoptimised. The object of optimisation is to maximise machiningaccuracy by minimising workpiece deformation. The locatorssatisfy two functional requirements: (1) Locate and stabilise theworkpiece, and (2) Serve as supports to minimise workpiecedeflections. The optimisation analysis attempts to satisfy bothfunctional requirements with a single design parameter, the pos-ition of the locators on the workpiece surface.The optimisation analysis is performed in ANSYS 5.6.2. TheANSYS program offers two optimisation methods to accommo-date a wide range of optimisation problems. The subproblem ap-proximation method is an advanced zero-order method that canbe efficiently applied to most engineering problems. The first-order method is based on design sensitivities and is more suitablefor problems that require high accuracy. For both the subproblemapproximation and first-order methods, the program performsa series of analysis-evaluation-modification cycles. That is, ananalysis of the initial design is performed, the results are evalu-ated against specified design criteria, and the design is modifiedas necessary. This process is repeated until all specified criteriaare met. In addition to the two optimisation techniques avail-able, the ANSYS program offers a set of strategic tools that canbe used to enhance the efficiency of the design process. For ex-ample, a number of random design iterations can be performed.The initial data points from the random design calculations canserve as starting points to feed the optimisation methods men-tioned above.The design variables, state variables, and objective functionare referred to as the optimisation variables. In an ANSYS op-timisation, these variables are represented by user-named vari-ables called parameters. The user must identify which param-eters in the model are design variables (DVs), which are statevariables (SVs), and which is the objective function.The analysis file is an ANSYS input file that contains a com-plete analysis sequence (preprocessing, solution, and postpro-cessing). It must contain a parametrically defined model, usingparameters to represent all inputs and outputs, which will be usedas DVs, SVs, and the objective function. The loop file residesin the working directory and is used by the control file to buildthe model. The control file initialises the design variables, de-fines the feasible design space, optimisation analysis method,and looping controls, and executes the optimisation analysis415(Looman D, ANSYS Corporate Technical Support Center, 2001,personal communication).A loop is one pass through the analysis cycle. Output for thelast loop performed is saved on file Jobname.OPO. An optimisa-tion iteration is one or more analysis loops which result in a newdesign set. Typically, an iteration equates to one loop. How-ever, for the first order method, one iteration represents morethan one loop. The optimisation database contains the currentoptimisation environment, which includes optimisation variabledefinitions, parameters, all optimisation specifications, and accu-mulated design sets.This database can be saved to Jobname.OPTor resumed at any time in the optimiser 10.DVs are independent quantities that are varied in order toachieve the optimum design. Upper and lower limits are speci-fied to serve as “constraints” on the design variables. The designvariables in the optimisation are locator and clamp positions,and clamping force. SVs are quantities that constrain the design.They are also known as “dependent variables”, that are functionsof the design variables. A state variable may have a maximumand minimum limit, or it may be “single sided”, having onlyone limit. The state variable is the von Mises effective stress.The objective function is the dependent variable that you are at-tempting to minimise. It should be a function of the DVs, that is,changing the values of the DVs should change the value of theobjective function. The objective function is the maximum resul-tant displacement in the model. Table 5 lists all the optimisationvariables used in this study.A design set is simply a unique set of parameter values thatrepresents a particular model configuration. Typically, a designset is characterised by the optimisation variable values, however,all model parameters, including those not identified as optimisa-tion variables, are included in the set. A feasible design is onethat satisfies all specified constraints on the SVs as well as con-straints on the DVs. If any one of the constraints is not satisfied,the design is considered infeasible. The best design is the onethat satisfies all constraints and produces the minimum objectivefunction value. If all design sets are infeasible, the best design setTable5. Optimisation variablesDesign variablesPosition of locatorsLocator 1 (X1, Y1, Z1)Locator 2 (X2, Y2, Z2)Locator 3 (X3, Y3, Z3)Locator 4 (X4, Y4, Z4)Locator 5 (X5, Y5, Z5)Locator 6 (X6, Y6, Z6)Position of clampsClamp 1 (X1, Y1, Z1)Clamp 2 (X2, Y2, Z2)Clamping force magnitudeClamp 1 (Fcl1)Clamp 2 (Fcl2)State variablesVonmises effective stress(VONMISES)Objective functionMaximum resultant displacement(DMAX)is the one closest to being feasible, irrespective of its objectivefunction value 10.Because there are a finite number of positions where themodular tooling can be fastened to the base plate, the optimisa-tion algorithm is discrete. There are also geometric constraintson the locators and clamps. For example, although it would beideal to position the primary reference plane supports directlyunder the applied load during machining, since the forces wouldbe transferred directly through the support and the bending mo-ment would be zero, it is impractical in some instances, such asin the drilling of a through hole, because of interference with thesupport. For maximum workpiece stability and locating accuracythe supports on the primary reference plane should be placed asfar apart as possible. However, to minimise workpiece deforma-tion, the supports should be placed as close to the loads normalto the primary surface as possible. The support locations are op-timised where workpiece deflections are minimised and locatingaccuracy is highest. Locating accuracy, workpiece stability, andworkpiece deformations are all affected by the support locationsand contribute to the overall fixture stiffness and subsequently,the machining accuracy 3.5 ANSYS optimisationstudyA sample optimisation analysis shown in Fig. 7 was conductedto demonstrate the validity of the ANSYS parametric design lan-guage (APDL) batch code. As mentioned in the fixture designanalysis methodology section in Part I, the optimisation analy-sis is used to minimise the maximum resultant displacement inthe workpiece, by optimising support locations, clamp locations,and clamping force magnitudes. The same 3-2-1 fixture configu-ration used for the workpiece in the loading study, was used asthe initial configuration in the optimisation analysis. The algo-rithm for selectinginitial support locations is explicitly describedin the loading study. Three feasible design sets resulted from theoptimisation analysis. The results are listed inTable 6. Design set1 is the initial fixture configuration. Design set 2 is the optimisedconfiguration given a limited design space, as shown in Fig. 7.Design set 3 is the optimised configuration given an extendeddesign space. The design space for the optimisation analysis re-sulting in design set 2 is shown in Fig. 7 as a dashed square. Thedesign space for the optimisation analysis resulting in design set3 was extended to include the entire surface on each referenceplane. The von Mises stress at each support location is com-pared to the yield stress of the workpiece material, AISI 1212Steel, y= 58015 psi, to ensure that the material does not ex-hibit plastic deformation during machining. The von Mises stressis treated as a state variable and is not allowed to exceed theworkpiece material yield strength.The von Mises stresses at the locators on the secondary andtertiary reference planes (SEQV1, SEQV2, and SEQV3) varybetween design sets due to their position and the magnitudeof the clamping forces. Notice that on the primary referenceplane, the von Mises stresses (SEQV4, SEQV5, and SEQV6)remain relatively constant, since the axial thrust force magni-416Fig.7ad. Benchmark fixture design configurations a Tertiary referenceplane, b Secondary reference plane, c Primary reference plane, d Isometricview of 221 blocktude is constant. The clamping force is increased to 249 lbf indesign set 2 from 100 lbf in design set 1. In design set 3, it isonly increased to 112 lbf. The maximum resultant displacementwas subsequently reduced by 8.4%, from 1.47103in. (designTable6. ANSYS problem optimization analysis resultsOptimizationVariableDesign set 1Design set 2Design set 3variabletype(feasible)(feasible)(feasible)SEQV1SV1.51107Pa1.51108Pa2.24108PaSEQV2SV6.39107Pa1.16108Pa7.21107PaSEQV3SV3.50106Pa2.94108Pa2.05106PaSEQV4SV1.56108Pa1.89108Pa1.81108PaSEQV5SV1.71108Pa1.43108Pa1.70108PaSEQV6SV3.34108Pa3.25108Pa3.09108PaFCL1DV444.8 N1.107103N498.2 NFCL2DV444.8 N1.107103N498.2 NDMAXOBJ0.0373 mm0.0341 mm0.0370 mmset 1) to 1.34103in (design set 2). In design set 3, the opti-mised fixture configuration did not vary significantly from theinitial configuration. The maximum resultant displacement wasonly reduced by 0.75% from 1.47103in to 1.46103in.In design set 2, note that the locators on the primary referenceplane (4, 5, and 6) were moved closer to the centre of the plane tominimise deflections due to the applied axial load. The locatorson the secondary and tertiary reference planes were moved up tominimise deflections due to the applied torque.It is obvious that without some knowledge base in fixture de-sign, the optimisation analysis is meaningless. An initial fixtureconfiguration must be provided. If all of the supports are initiallyplaced at the global coordinate system origin, for example, theoptimisation analysis will not result in a feasible design set. Theuser must also specify the design space, by selecting the rangeof values for the design variables. It is more appropriate to de-clare the entire surface on each reference plane as feasible designspace, but the analysis is more time intensive than if the designspace is limited to a smaller range of values.6 Industrial optimisationcase studyAn industrial case study was conducted to validate the fixturedesign analysis method developed in this study. The workpiecemodel isa simplified die castaluminium brake calipertaken fromDelphi Automotive Systems. The model is simplified to protectproprietary features and dimensions. The locators are placed ina 3-2-1 configuration. Three locators are placed on the primaryreference plane (one on the bottom of the caliper and two di-rectly below the slide bushing holes). Two locators are placedon the secondary reference plane, which is on the side of thecaliper, and one locator is placed on the tertiary reference plane,directly behind the cylinder bore at the centre of the cylinder.The configuration is shown in Fig. 8. The clamps are placeddirectly opposite the locators on each reference plane, so thatthe clamping force is transferred directly through the workpieceto the locator without generating any bending moments. Be-cause the tertiaryreference plane is perpendicular to the directionof applied loading, no clamp is necessary opposite the locator.A list of brake caliper model parameters and results is listedin Table 7. Table 8 lists the locator and clamp positions in mil-417Fig.8. Simplified brake caliper modellimetres relative to the global coordinate system origin. DelphiAutomotive Systems provided the initial fixture configuration,clamping force magnitude, machining forces, and locator stiff-ness values. The locators were modelled with multiple ANSYSCONTAC52 spring-gap elements in parallel, attached to a circu-lar contact area at specified fixturing points on the brake caliper.The loading is representative of a boring operation.The maximum resultant displacement in the preloaded work-piece model is 0.0032 mm, and increases slightly to 0.0036 mmin the fully loaded workpiece model, thus it is evident that thepreloading due to clamping is the major contribution to the re-sultant displacement throughout the machining operation. Thedisplacement near the cylinder bore increases significantly, by asmuch as 100%, but does not exceed the maximum resultant dis-placement in the preloaded workpiece model. Figures 9 and 10are the resultant displacement and von Mises stress plots, respec-tively, for the preloaded model (clamping loads, no machiningTable7. Brake caliper model parameters and resultsElement typeANSYS SOLID454-node tetrahedralMesh typeFree tetrahedralWorkpiece material type6061-T6 aluminiumLocator material typeAISI 1144 steelLocator normal stiffness1.75105N/mmLocator tangential stiffness1.75104N/mmYoungs modulus, E7.01010PaWorkpiece material yield strength, y1.7108PaPoissons ratio, 0.35Coefficient of static friction, 0.61Thrust force, Fc249.1 NTorque, T18865 NmmSEQV17.67105PaSEQV25.95105PaSEQV37.40105PaSEQV41.31105PaSEQV52.66105PaSEQV64.11105PaClamping force, FCL1200 NClamping force, FCL2200 NClamping force, FCL3200 NClamping fForce, FCL4200 NDMAX0.0036 mmTable8. Optimised brake caliper locator and clamp positionsInitial configuration (mm)Optimized configuration (mm)LocatorXYZXYZ137.9517.00-89.5037.9517.00-89.50237.9517.0089.5037.9517.0089.503133.8548.000.00133.8548.000.00478.4217.51-76.0078.4217.51-76.005126.8417.51-76.00126.8417.51-76.0060.000.000.000.000.000.00ClampXYZXYZ137.95-10.00-89.5037.95-10.00-89.50237.95-10.0089.5037.95-10.0089.503141.8527.000.00141.8527.000.004102.6117.5176.00102.6117.5176.00Optimized experimentallyFig.9. Pre-loaded brake caliper resultant displacement (mm) contour plotFig.10. Pre-loaded brake caliper von Mises stress (MPa) contour plot418loads). Figures 11 and 12 are the resultant displacement and vonMises stress plots, respectively, for the loaded model.There is a stress concentration at the bottom of the cylinderbore, as shown in Fig. 12, during machining due to bending mo-ments generated by the thrust force. The maximum von Misesstress occurs at the contact area of clamp 3, located opposite lo-cator 3 on the primary reference plane.An optimisation analysis was conducted to validate the op-timisation tool developed in ANSYS. Because the fixture con-figuration for the caliper has been optimised experimentally, thedesired result of the optimisation analysis is that ANSYS willproduce the same fixture configuration. As expected, the supportlocation optimisation resulted in the same fixture configuration.However, ANSYS further reduced the maximum resultant dis-placement in the workpiece by minimising the clamping forcemagnitude. The clamping force was reduced to 100 N, subse-Fig.11. Loaded brake caliper resultant displacement (mm) contour plotFig.12. Loaded brake caliper von Mises stress (MPa) contour plotTable9. Optimised brake caliper resultsOptimizationVariableInitialOptimizedvariabletypeconfigurationconfigurationSEQV1SV7.67105Pa4.72105PaSEQV2SV5.95105Pa1.98105PaSEQV3SV7.40105Pa3.68105PaSEQV4SV1.31105Pa0.68105PaSEQV5SV2.66105Pa1.41105PaSEQV6SV4.11105Pa4.11105PaFCL1DV200 N100 NFCL2DV200 N100 NFCL3DV200 N100 NFCL4DV200 N100 NDMAXOBJ0.0036 mm0.0025 mmquently reducing the maximum resultant displacement by 31%to 0.0025 mm. The von Mises stresses at the supports, which arelocated directly opposite the clamps, were also reduced signifi-cantly as shown in Table 9. The von Mises stress at locator 6,SEQV6 remained the same, since locator 6 is not reacting tothe clamping forces, but rather to the applied machining loads,which remained constant.7 ConclusionsIn this study a finite element model was developed for fixturedworkpiece boundary conditions and applied loads in machin-ing using ANSYS 5.6.2. As opposed to preceding finite elem-ent analysis research in fixture design, in this study, boundaryconditions modelled as both area and point constraints were con-sidered to determine whether a single point constraint model isappropriate. Only Pong et al. 3 modelled boundary conditionsto be elastic and deformable, but this research only consideredelastic point constraints. His research does not specify whetheran elastic area constraint model was considered.A more accurate representation of machining loads was alsodeveloped. The load model developed in this study includestorque, which is neglected in all preceding research. Distributedand concentrated loading is considered in this study, whereas inprevious research all machining forces are applied as single pointloads.Because the model boundary conditions and loads are ap-plied parametrically, APDL code can be used for solid modelswith planar locating surfaces and user defined (1) support lo-cations, (2) clamp locations, (3) clamping force magnitude,(4) cutting tool location, (5) axial load, (6) transverse load, and(7) torque magnitude.The following analysis specific conclusions are realisedbased on the research conducted throughout this study:Workpiece elements. The SOLID45, 8-node brick element, issuitable for meshing prismatic geometry. The SOLID45, 4-nodetetrahedral element is not as accurate as the brick element,
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