銷軸軸端的槽、面銑成組夾具設(shè)計(jì)【三維UG】【含圖紙及及檔全套】

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Tel.: +39-02-2399-8530; fax: +39-02-2399-8585. E-mail address: stefano.petropolimi.itAbstractThe paper deals with robust design of fixture configuration. It aims to investigate how fixture element deviations and machine tool volumetricerrors affect machining operations quality. The locator position configuration is then designed to minimize the deviation of machined featureswith respect to the applied geometric tolerances.The proposed approach represents a design step that goes further the deterministic positioning of the part based on the screw theory, and may beused to look for simple and general rules easily applicable in an industrial context.The methodology is illustrated and validated using simulation and simple industrial case studies.c ? 2014 The Authors. Published by Elsevier B.V.Selection and peer-review under responsibility of the International Scientific Committee of “24th CIRP Design Conference” in the person of theConference Chairs Giovanni Moroni and Tullio Tolio.Keywords:Tolerancing; Error; Modular Fixture.1. IntroductionWhen a workpiece is fixtured for a machining or inspectionoperation, the accuracy of an operation is mainly determined bythe efficiency of the fixturing method. In general, the machinedfeature may have geometric errors in terms of its form and posi-tion in relation to the workpiece datum reference frame. If thereexists a misalignment error between the workpiece datum ref-erence frame and machine tool reference frame, this is knownas localization error 1 or datum establishment error 2. Alocalization error is essentially caused by a deviation in the po-sition of the contact point between a locator and the workpiecesurface from its nominal specification. In this paper, such a the-oretical point of contact is referred to as a fixel point or fixel,and its positioning deviation from its nominal position is calledfixel error. Within the framework of rigid body analysis, fixelerrors have a direct effect on the localization error as defined bythe kinematics between the workpiece feature surfaces and thefixels through their contact constraint relationships 3.The localization error is highly dependent on the configu-ration of the locators in terms of their positions relative to theworkpiece. A proper design of the locator configuration (orlocator layout) may have a significant impact on reducing thelocalization error. This is often referred to as fixture layout op-timization 4.A main purpose of this work is to investigate how geomet-ric errors of a machined surface (or manufacturing errors) arerelated to main sources of fixel errors. A mathematic frame-work is presented for an analysis of the relationships among themanufacturing errors, the machine tool volumetric error, andthe fixel errors. Further, optimal fixture layout design is speci-fied as a process of minimizing the manufacturing errors. Thispaper goes beyond the state of the art, because it considers thevolumetric error in tolerancing. Although the literature demon-strates that the simple static volumetric error considered hereis only a small portion of the total volumetric error, a generalframework for the inclusion of volumetric error in tolerancingis established.There are several formal methods for fixture analysis basedon classical screw theory 5,6 or geometric perturbation tech-niques 3. In nineties many studies have been devoted to modelthe part deviation due to fixture 7. Sodenberg calculated a sta-bility index to evaluate the goodness of the locating scheme 8.The small displacement torsor concept is used to model the partdeviationduetogeometricvariationofthepart-holder9. Con-ventional and computer-aided fixture design procedures havebeen described in traditional design manuals 10 and recent lit-erature 11,12, especially for designing modular fixtures 13.A number of methods for localization error analysis and reduc-tion have been reported. A mathematical representation of thelocalization error was given in 14 using the concept of a dis-placements screw vector. Optimization techniques were sug- 2014 Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http:/creativecommons.org/licenses/by-nc-nd/3.0/).Selection and peer-review under responsibility of the International Scientifi c Committee of “24th CIRP Design Conference” in the person of the Conference Chairs Giovanni Moroni and Tullio Tolio190 Giovanni Moroni et al. / Procedia CIRP 21 ( 2014 ) 189 194 gested to minimize the magnitude of the localization error vec-tor or the geometric variation of a critical feature 14,15. Ananalysis is described by Chouduri and De Meter 2 to relatethe locator shape errors to the worst case geometric errors inmachined features. Geometric deviations of the workpiece da-tum surfaces were also analyzed by Chouduri and De Meter2 for positional, profile, and angular manufacturing tolerancecases. Their effects on machined features, such as by drillingand milling, were illustrated. A second order analysis of thelocalization error is presented by Carlson 16. The computa-tionaldifficultiesoffixturelayoutdesignhavebeenstudiedwithan objective to reduce an overall measure of the localization er-ror for general three dimensional (3D) workpieces such as tur-bine air foils 1,4. A more recent paper shows a robust fixturelayout approach as a multi-objective problem that is solved bymeans of Genetic Algorithms 17. It considers a prismatic andrigid workpiece, the contact between fixture and workpiece iswithout friction, and the machine tool volumetric error is notconsidered.About the modeling of the volumetric error, several modelshave been proposed in literature. Ferreira et al. 18,19 haveproposed quadratic model to model the volumetric error of ma-chines, in which each axis is considered separately, thoghetherwith a methodology for the evaluation of the model parame-ters. Kiridena and Ferreira in a series of three papers 2022discuss how to compensate the volumetric error can be mod-eled, the parameters of the model evaluated, and then the er-ror compensated based on the model and its parameters, for athree-axis machine. Dorndorf et al. 23 describe how volu-metric error models can help in the error budgeting of machinetools. Finally, Smith et al. 24 describe the application of vol-umetric error compensation in the case of large monolithic partmanufacture, which poses serious difficulties to traditional vol-umetric error compensation. Anyway, it is worth noting that allthese approaches are aimed at volumetric error compensation:generally volumetric error is not considered for simulation intolerancing.In previous papers a statistical method to estimate the po-sition deviation of a hole due to the inaccuracy of all the sixlocators of the 3-2-1 locating scheme was developed for 2Dplates and 3D parts 25,26. In the following, a methodologyfor robust design of fixture configuration is presented. It aimsto investigate how fixel errors and machine tool volumetric er-ror affect machining operations quality. In 2 the theoreticalapproach is introduced, in 3 a simple industrial case study ispresented, and in 4 some simple and general rules easily ap-plicable in an industrial contest are discussed.2. Methodology for the simulation of the drilling accuracyTo illustrate the proposed methodology, the case study ofa drilled hole will be considered. The case study is shown inFig. 1. A location tolerance specifies the hole position. Threelocators on the primary datum, two on the secondary datum,and one on the tertiary determine the position of the workpiece.Each locator has coordinates related to the machine tool refer-ence frame, represented by the following six terns of values:p1(x1,y1,z1)p2(x2,y2,z2)p3(x3,y3,z3)p4(x4,y4,z4)p5(x5,y5,z5)p6(x6,y6,z6)(1)Fig. 1. Locator configuration schema.The proposed approach considers the uncertainty source inthe positioning error of the machined hole due to the error inthe positioning of the locators, and the volumetric error of themachine tool. The final aim of the model is to define the ac-tual coordinates of the hole in the workpiece reference system.The model input includes the nominal locator configuration, thenominal hole location (supposed coincident with the drill tip)and direction (supposed coincident with the drill direction), andthe characteristics of typical errors which can affect this nomi-nal parameters.2.1. Effect of locator errorsThe positions of the six locators are completely defined bytheireighteencoordinates. Itisassumed thateachof thesecoor-dinates is affected by an error behaving independently, accord-ing to a Gaussian N?0,2?distribution.The actual locator coordinates will then identify the work-piece reference frame. In particular, the z?axis is constitutedby the straight line perpendicular to the plane passing throughthe actual positions of locators p1, p2and p2, the x?axis is thestraight line perpendicular to the z?axis and to the straight linepassing through the actual position of locators p4and p5, andfinally the y?axis is straightforward computed as perpendicularto both z?and x?axes. The origin of the reference frame can beobtained as intersection of the three planes having as normalsthe x?, y?, and z?axes and passing through locators p4, p6andp1respectively. The formulas for computing the axis-directionvectors and origin coordinates from the actual locators coordi-nates are omitted here, for reference see the work by Armillottaet al. 26.The axis-direction vectors and origin coordinates define anhomogeneous transformation matrix0Rp27, which allows toconvert the drill tip coordinate expressed in the machine toolreference frame P0to the same coordinates expressed in theworkpiece reference frame P?0, through the formula:P?0=0R1pP0(2)191 Giovanni Moroni et al. / Procedia CIRP 21 ( 2014 ) 189 194 2.2. Effect of machine tool volumetric errorTo simulate the hole location deviation due to the drillingoperation, i.e. to the volumetric error of the machine tool, theclassical model of three-axis machine tool has been considered27. It will be assumed the drilling tool axis is coincident withthe machine tool z axis, so that, in nominal conditions and at thebeginning of the drilling operation, its tip position can be de-fined by the nominal hole location and the homogeneous vectork = 0010T. The aim is to identify the position errorpof the drill tip in the machine tool reference system, and thedirection error dof the tool axis. According to the three-axismachine tool model it is possible to state that:p=0R11R22R3P3 P0(3)where P0=?xyz l1?Tis the nominal drill tip locationin the machine tool reference system (x, y, and z beingthe trans-lations alongthe machinetool axes, and l being thedrill length),P3= 00 l1Tis the drill tip position in the third (zaxis) reference system, and0R1,1R2,2R3are respectively theperturbed transformation matrices due to the perturbed transla-tion along the x, y, and z axes. These matrices share a similarform, for example:0R1=1z(x)y(x)x + x(x)z(x)1x(x)y(x)y(x)x(x)1z(x)0001(4)where the and terms are the translation and rotation errorsalong and around the x, y, and z axes (e.g. z(x) is the rotationerror around the z axis due to a translation along the x axis).Considering three transformation matrices, there are eighteenerror terms. These errors are usually a function of the volu-metric position (i.e. the translations along the three axes), butif the volumetric error is compensated, their systematic com-ponent can be neglected and they can be assumed to be purelyrandom with mean equal to zero. Developing Eq. (3) leads tovery complex equations; for example,dx=x(x) + x(y) z(x)(y(y) + y) y(z) (z(x) + z(y) x(y)y(x) + z(y)y(x) x(z)(y(x)y(y) + z(x)z(y) 1) l(y(x) + y(y) + x(z)(z(x) + z(y) x(y)y(x) + x(y)z(x) y(z)(y(x)y(y)+ z(x)z(y) 1) + (z(z) + z)(y(x) + y(y)+ x(y)z(x)(5)However, volumetric errors in general should be far smallerthan translations along the axes, so only the first order com-ponents of Eq. (3) are usually significant. Finally, Assumingthe drilling tool axis coincide with the z axis, Eq. (3) can alsocalculate the direction error dby substituting P0= P3= k.If only the first order components are considered, it is possi-ble to demonstrate that pand dare linear combination of the and terms. In particular, lets define as =?pd?(6)the six-elements vector containing pand dstaked. ApplyingEq. (3), neglecting terms above the second order, it is possibleto demonstrate that (please note that, due format constraints, inEq. (7) the dots . indicate that a row of the matrix is bro-ken over more lines, so the overall linear combination matrixappearing here is a 6 X 18 matrix) =111000.000000.z lz lly00000111.000l zl zl.000000000000.111000.000000000000.000100.111000000000.000111.010000000000.000000.000001x(x)x(y)x(z)y(x)y(y)y(z)z(x)z(y)z(z)x(x)x(y)x(z)y(x)y(y)y(z)z(x)z(y)z(z)= Cd(7)Now, lets assume that each term is independently dis-tributed according to a Gaussian N?0,2p?distribution, andthat each term is independently distributed according to aN?0,2d?distribution. It is then possible to demonstrate 28that follows a multivariate Gaussian distribution, with nullexpected value and covariance matrix which can be calculatedby the formula CCT, where is the covariance matrix of d,which happens to be a diagonal 18 X 18 matrix with the firstnine diagonal elements equal to 2p, an the remaining diagonal192 Giovanni Moroni et al. / Procedia CIRP 21 ( 2014 ) 189 194 elements equal to 2d. The final covariance matrix of is:2dy2+22d(l z)2+32p+l22d002d(3l2z)2d(l z)0022d(l z)2+32p+l22d02d(l z)2d(3l2z)00032p0002d(3l2z)2d(l z)042d002d(l z)2d(3l2z)0042d0000002d(8)This model can be adopted to simulate the error in the loca-tionanddirectionoftheholeduetothemachinetoolvolumetricerror.2.3. Actual location of the manufactured holeNow, it is possible to simulate the tip location and directionaccording to the model described in 2.2, and to transform itinto the workpiece reference frame as described in 2.1:P0?=0R1p?P0+ p?k?=0R1p(k + d)(9)With this information it is possible to determine the entranceand exit location of the hole in the workpiece reference system.Point P?0and vector k?define a straight line, which is nothingelse than the hole axis, as:p?= P0?+ sk?px?py?pz?=P0 x?P0y?P0z?+ skx?ky?kz?(10)where p is a generic point belonging to the line and s R isa parameter. Defining T as the plate thickness, it is possible tocalculate the values of s for which p?zis equal respectively to 0and T:sexit= P0z?/kz?sentrance= (P0z? T)/kz?(11)These values of s substituted in Eq. (10) yield respectivelythe coordinates of the exit and entrance point of the hole.Finally, it is possible to calculate the distances between thetwo exit and entrance points of the drilled and nominal holes:d1=?P0?+ sentrancek? P0?d2=?P0?+ sexitk? P0,exit?(12)where P0,exitis the nominal location of the hole exit point. Theaxis of the drilled hole will be inside location tolerance zoneof the hole if both the distances calculated by Eq. (12) will belower than the half of the location tolerance value t:d1 t/2d2 t/2(13)3. Case study resultsThe model proposed so far has been considered to identifythe expected quality due to locator configuration, given a ma-chine tool volumetric error. To identify which is the optimalone an experiment has been designed and results have been an-alyzed by means of analysis of variance (ANOVA) 29.Because the aim of the research regards only the choice oflocators positions, most of the model parameters can be keptconstant. The constant parameters include: the nominal sizeof the plate (100 x 120 x 60 mm); the standard deviation ofthe random errors in locator positioning ( = 0.01 mm); thenominal location of the entrance (P0= 407060T) andexit (P0= 40700T) points of the hole; the length of thedrill (l = 60 mm); the standard deviation of the machine toolaxes positioning errors (p= 0.01 mm) and of their rotationalerrors (d= 0.01); the location tolerance value (t = 0.1 mm);the plate thickness (T = 60 mm). Each locator has insteadbeen left free to change in order to evaluate its influence on thedrilling accuracy; candidate configurations will be introducedin the next paragraphs, together with their impact discussion.By substituting the simulation parameters indicated so far inEq. (8) the following covariance matrix is yielded (all valuesare in mm2):68000.180.18006300.180.18000300000.180.1800.01000.180.18000.010000000.003 105(14)The considered performance indicator is the fraction of con-forming parts generated by a specific locator configuration, i.e.the fraction of parts for which both of the inequalities in Eq.(13) hold.The conforming fraction has been evaluated tentimes for each experimental condition, for each evaluation tenthousand workpieces have been simulated. Of course, highervalues of this performance indicator are preferable.The ANOVA analysis has worked efficiently, with its hy-potheses correctly verified. The main effect plot in Fig. 2 sum-193 Giovanni Moroni et al. / Procedia CIRP 21 ( 2014 ) 189 194 Fig. 2. Main effect plot for the fraction of conforming workpieces.marizes the results, which are described in depth in the follow-ing paragraphs.3.1. Impact of p1, p2and p3locator configurationThe p1, p2and p3locators define the part z?axis, so theyhave been indicated in Fig. 2 as “z locator configuration”. Toevaluate their impact on the hole accuracy three candidate con-figurations have been considered. The first one (“max area”)tries to cover as much as possible the surface of the workpiecetouched by the locators themselves. The second one (“barycen-tric”) has the barycenter of the locators coincident with the holeposition, but with an area coverage smaller than the max areaconfiguration. The last one (“non barycentric”) has the samearea coverage of the barycentric one, but is far from the hole.Please note that the plate equilibrium has been neglected in thisfirst analysis.The ANOVA suggests that the best condition is the onein which the area coverage is maximum, and that given thesame area coverage, having a barycentric distribution is prefer-able. The impact of the z locator configuration is very relevant,changing the conforming fraction of about 10%.3.2. Impact of p4and p5locator configurationThe p4and p5locators define the part x?axis. Two factorshave been considered for them: their height with the hypothesisthat they have the same one (“x locator height” in Fig. 2), andtheir positions in the y direction (“x locator configuration”).Three candidate heights have been considered, 5, 30 and 55mm. It seems that it is slightly better to have the locators placedat the lower height, but the impact is quite small (about 1%).The impact of the x locator configuration, accounting forabout the 20% of the conforming fraction, is far more relevant.Similarly to the z locator configuration, three configurationshave been considered. The first one (“max distance”) maxi-mizes the distance between the two locators. The second one(“barycentric”) has the barycenter of the two locators in corre-spondence of the hole axis, but with a distance smaller than the“max distance” one. The last one (“non barycentric”) has thesame distance of the barycentric, but it has the barycenter ofthe two locators fa