夹具配置的稳健性设计@中英文翻译@外文翻译

Procedia CIRP 21 ( 2014 ) 189 194 Available online at 2212-8271 2014 Elsevier B.V. This is an open access article under the CC BY-NC-ND license (/licenses/by-nc-nd/3.0/).Selection and peer-review under responsibility of the International Scientific Committee of “24th CIRP Design Conference” in the person of the Conference Chairs Giovanni Moroni and Tullio Toliodoi: 10.1016/cir.2014.03.120 ScienceDirect24th CIRP Design ConferenceRobust design of fixture configurationGiovanni Moronia, Stefano Petroa,*, Wilma PolinibaMechanical Engineering Department, Politecnico di Milano, Via La Masa 1, 20156, Milano, ItalybCivil and Mechanical Engineering Department, Cassino University, Via di Biasio 43, 03043, Cassino, ItalyCorresponding author. 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 aect 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.c2014 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 eciency 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 eect 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 partdeviation due to geometric variation of the part-holder 9. 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 (/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 eects on machined features, such as by drillingand milling, were illustrated. A second order analysis of thelocalization error is presented by Carlson 16. The computa-tional diculties of fixture layout design have been studied withan 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 diculties 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 aect machining operations quality.In2 the theoreticalapproach is introduced, in3 a simple industrial case study ispresented, and in4 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 aect this nomi-nal parameters.2.1. Eect of locator errorsThe positions of the six locators are completely defined bytheir eighteen coordinates. It is assumed that each of these coor-dinates is aected by an error behaving independently, accord-ing to a Gaussian NparenleftBig0,2parenrightBigdistribution.The actual locator coordinates will then identify the work-piece reference frame. In particular, the zprimeaxis is constitutedby the straight line perpendicular to the plane passing throughthe actual positions of locators p1, p2and p2, the xprimeaxis is thestraight line perpendicular to the zprimeaxis and to the straight linepassing through the actual position of locators p4and p5, andfinally the yprimeaxis is straightforward computed as perpendicularto both zprimeand xprimeaxes. The origin of the reference frame can beobtained as intersection of the three planes having as normalsthe xprime, yprime, and zprimeaxes 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 Pprime0, through the formula:Pprime0=0R1pP0(2)191 Giovanni Moroni et al. / Procedia CIRP 21 ( 2014 ) 189 194 2.2. Eect 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=0R11R22R3P3P0(3)where P0=bracketleftbigxyzl 1bracketrightbigTis the nominal drill tip locationin the machine tool reference system (x, y, and z being the trans-lations along the machine tool axes, and l being the drill length),P3= 00l 1Tis 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=1 z(x) y(x) x+x(x)z(x) 1 x(x) y(x)y(x) x(x) 1 z(x)000 1(4)where theandterms 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 theandterms. In particular, lets define as=bracketleftBiggpdbracketrightBigg(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 isa6X18matrix)=111000.000000.zlzl l y 00000111.000lzlzl.000000000000.111000.000000000000.000100.1 1 1000000000.000111.010000000000.000000.000001x(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 NparenleftBig0,2pparenrightBigdistribution, andthat each term is independently distributed according to aNparenleftBig0,2dparenrightBigdistribution. 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 to2p, an the remaining diagonal192 Giovanni Moroni et al. / Procedia CIRP 21 ( 2014 ) 189 194 elements equal to2d. The final covariance matrix of is:2dy2+22d(lz)2+32p+l22d002d(3l2z)2d(lz)0022d(lz)2+32p+l22d02d(lz)2d(3l2z)00032p0002d(3l2z)2d(lz)042d002d(lz)2d(3l2z)0042d0000002d(8)This model can be adopted to simulate the error in the loca-tion and direction of the hole due to the machine tool volumetricerror.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 in2.1:P0prime=0R1pparenleftBigP0+pparenrightBigkprime=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 Pprime0and vector kprimedefine a straight line, which is nothingelse than the hole axis, as:pprime=P0prime+skprimepxprimepyprimepzprime=P0xprimeP0yprimeP0zprime+skxprimekyprimekzprime(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 pprimezis equal respectively to 0and T:sexit=P0zprime/kzprimesentrance=(P0zprimeT)/kzprime(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=vextendsinglevextendsinglevextendsingleP0prime+sentrancekprimeP0vextendsinglevextendsinglevextendsingled2=vextendsinglevextendsinglevextendsingleP0prime+sexitkprimeP0,exitvextendsinglevextendsinglevextendsingle(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:d1t/2d2t/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= 40 70 60T) andexit (P0= 40 70 0T) points of the hole; the length of thedrill (l = 60 mm); the standard deviation of t