Desgn for Relablty: Case Studes n Manufacturng Process Synthess Y. Lawrence Yao*, and Chao Lu Department of Mechancal Engneerng, Columba Unversty, Mudd Bldg., MC 473, New York, NY 7, USA * Correspondng author: Tel: -854-887, Fax: -854-334, emal: yly@columba.edu Keywords: Response surface methodology (RSM), propagaton of error (POE), robust desgn Abstract: The task of manufacturng process desgn s to determne a set of process parameters for a manufacturng task. The desgn s normally carred out to optmze process performance, such as, to mnmze dscrepancy from a prescrbed shape or to maxmze producton effcency. An optmal desgn, however, s often not the most robust or relable desgn because nevtable uncertantes n nput varables may cause unacceptable devatons from the optmal pont. Robust desgn s therefore needed to acheve compromse between optmzaton and robustness. Such a methodology s llustrated n case studes nvolvng process desgn of laser formng of sheet metal, n whch a set of parameters ncludng laser scannng paths, laser power, and scannng speed s determned for a prescrbed shape. Response surface methodology s used as an optmzaton tool. The propagaton of error technque s bult nto the desgn process as an addtonal response to be optmzed va desrablty functon and hence make the desgn robust. The results are also valdated expermentally.. Introducton Compared wth conventonal formng technques, laser formng of sheet metal does not requre hard toolng or external forces and hence, can ncrease process flexblty and reduce the cost of the formng process when low to medum producton volume s concerned. Many efforts have been made on studyng the mechansms and modelng of the process. Magee, et al. (998) revewed lteratures avalable up to 998. More recently, selected ssues related to extendng laser formng to more practcal applcatons started beng addressed. For nstance, repeated scannng s necessary to obtan the magntude of deformaton that practcal parts requre, and hence coolng effects durng and between consecutve scans become crtcal (Cheng & Yao, ). A vast majorty of work on laser formng ncludng the ones mentoned above can be consdered as solvng the drect problem. Such a problem s typcally formulated based on physcal laws such as heat transfer and elastcty/plastcty theores. To apply the laser formng process to real world problems, however, the nverse problem needs to be addressed. Solvng the nverse problem analytcally or numercally s dffcult, f not mpossble, for the followng reasons. Frstly, no physcal laws are readly avalable to establsh governng equatons leadng to nverse soluton. Secondly, to manpulate ether the soluton to the drect problem or the underlyng dfferental equatons leadng to a soluton to the nverse problem s also mpossble because of the complexty nvolved. Thrdly, whle the soluton to the drect problem s unque, the soluton to the nverse problem s certanly mult-valued. Gven the understandng that numercal or analytcal solutons to the nverse problem are less lkely, emprcal and heurstc approaches have been attempted. A genetc algorthm (GA) based approach was proposed by Shmzu, (997) as an optmzaton engne to solve the nverse problem of the laser formng process. In hs study, a set of arbtrarly chosen heat process condtons for a dome shape was encoded nto strngs of bnary bts, whch evolve over generatons followng the natural selecton scheme. One of the mportant process parameters, heatng path postons, was assumed gven. To apply GA, t s necessary to specfy crossover rate and mutaton rate but ther selecton suffers from lack of rgorous crtera. The objectve of ths paper s to develop a more systematc and relable methodology to solve the nverse problem n laser formng for a class of shapes. A response surface methodology (RSM) based approach s attempted as an optmzaton tool. Dscrete desgn varables are properly dealt wth n the optmzaton process. Propagaton of error (POE) technque s bult nto the desgn process as an addtonal response to be optmzed va a desrablty functon and hence make the desgn more robust. Experments and at places fnte element method (FEM) are used to enable and valdate the optmzaton process. The proposed approach s appled to two cases to demonstrate ts valdty.. Problem Descrpton As shown n Fg., rectangular sheet metal s to be formed nto 3D shapes by parallel laser rradaton. The 3D shape can be vewed as shape generated by a D generatrx n the y-z plane extruded n the x drecton Therefore for ths class of 3D shapes, the nverse desgn problem can be treated as a D curve desgn problem.
Fg. Schematc of a class of shapes to be laser formed, lnear parallel scannng paths on a rectangular plate. As shown n Fg., the parameters needs to be determned nclude number of scannng paths, N, postons of laser scannng paths d, laser powers p, beam scannng velocty v and laser spot dameter D b. The approach presented n ths paper, however, s not restrcted to monotonc cases. A non-monotonc generatrx can be smlarly dealt wth usng dfferent beam spot szes for dfferent paths. The objectve functon s to mnmze the dfference between a possble soluton shape and the prescrbed shape, that s Mnmze: h = k = ( z z ) / k () s p where z s and z p are the z coordnates of correspondng ponts on the generatrx of the possble soluton shape and the prescrbed shape, respectvely, and k s the number of ponts. It wll be seen that values of the objectve functon h, n fact, serves as responses n the optmzaton process. In addton, f a pont n the possble soluton shape has a smaller z value than the correspondng pont of the prescrbed shape, the dstance s defned as negatve; otherwse t s postve. When the sum of the dstances s postve, the objectve (or response) s consdered postve; otherwse, t s negatve. The fnal desgn of a product requres not only to be optmal, but also robust, namely, nsenstve to the varaton of nput varables. In laser formng process, the achevable accuracy of formng s lmted by numerous uncertantes. Hennge, et al. (997) nvestgated nfluencng uncertantes n laser formng based on analyzng error propagaton and found varatons n power and couplng coeffcent are most nfluental factors on the varaton of deformaton. In ths paper, the nfluence of laser power on the robustness of the optmal desgn wll be addressed n a robust desgn phase based on desrablty consderaton. In ths study, square plates of sze 8 8. 89 mm are used. The materal used s AISI low carbon steel. The laser system used s a 5W CO laser. In all the experments, the laser beam dameter s set to 4 mm, the beam movng velocty s kept constant at 5mm/s. A coordnate measurng machne (CMM) s used to measure the coordnates of the deformed plates. 3. RESPONSE SURFACE METHODOLOGY AND OPTIMAL DESIGN Response surface methodology (RSM) s a collecton of statstcal and mathematcal technques useful for developng, mprovng, and optmzng process (Myers, et al. 995). Applcatons of RSM comprse two phases. In the frst phase the response surface functon s based on a factoral desgn, approxmated by a frst-order regresson model (Eq. ), and complete wth steepest ascent/descent search, untl t shows sgnfcant lack of ft wth experment data. After reachng the vcnty of the optmum, the second phase of the response surface functon s approxmated by a hgher order regresson functon such as a second-order one shown n Eq. 3. T yˆ = b + b x + ε () T T yˆ = b + b x + x b x + ε (3) T where ŷ and x = [ x, x,..., x n ] are estmated response and decson varable vector, respectvely, ε s the fttng error whch s assumed to be normally dstrbuted, and b o, b, and b are coeffcents determned usng the least square method. If one or more of the decson varables are ntegers, the optmum problem s consdered as a dscrete problem, whch can be dealt wth methods such as the branch-and-bound method (Taha, 987). The steepest lne search starts wth the ntal desgn ponts of q nteger varables and n-q non-nteger varables. The next teraton starts by arbtrarly choosng a desgn pont from one of the q ntegers varables, the ths remanng
q- desgn ponts of nteger varables of next movement are determned based on the coeffcent from Eq. and the branch-and-bound method. After reachng the vcnty of optmum, the response surface s approxmated by a hgher regresson order wth q ntegers determned from prevous teraton and n-q non-nteger factors. At ths stage, the problem can be solved as a regular one wth n-q contnuous varables. 4. DESIRABILITY AND ROBUST DESIGN Desrablty based robust desgn s a tool to fnd controllable factor settngs that optmze the objectve yet mnmze the response varaton of the desgn (Kraber, et al., 996). The transmtted varaton of responses from nput varables can then be reduced by movng the optmal soluton to a flatter part of the response surface. The varaton transmtted to the response can be determned by the error propagaton equaton POE = whereσˆ n n yˆ yˆ yˆ / ˆ σ ( y = σ + σ j + σ e ) (4) x = x < j x j y s the model-predcted standard devaton of the response or known as propagaton of error (POE), the varance of decson varable x, σ k e = ε j / j= df σ j s the covarance between x and x j, and σ s σ e s the resdual varance,, df s the resdual degree of freedom. To reduce varance n the response, POE (Eq. 4) should be mnmzed therefore can be treated as an addtonal response bult nto the desgn process. The smultaneous optmzaton of several responses (n ths case, y n Eq. 3 and ˆσ y n Eq. 4) s the essence of the desrablty based robust desgn (Kraber, et al., 996 and Derrnger et al., 98). For each response ŷ, a desrablty functon D ( ŷ ) assgns a value between and to the possble values of ŷ, wth D ( ŷ ) = representng a completely undesrable value of ŷ and D ( ŷ ) = representng the deal response value. The ndvdual desrabltes are then combned usng the geometrc mean, whch gves the overall desrablty D D y D y D y / m = ( ( ˆ ) ( ˆ )... ( ˆ )) m m (5) where m s the number of responses. The maxmum of D represents the hghest combned desrablty of the responses. Dependng on whether a partcular response ŷ s to be maxmzed, mnmzed, or assgned to a target value, dfferent desrablty functons D ( ŷ ) are to be used. Let L, U and T be the lower, upper, and target values desred for response ŷ, where L T U. If a response s of the target s best, ts desrablty functon s expressed as: r yˆ L L yˆ T T L s yˆ U (6) D ( yˆ ) = T yˆ U T U yˆ > U or yˆ < L where the exponents r and s determne how strctly the target value s desred. The desrablty functon can be defned smlarly f a response s to be mnmzed or maxmzed. As seen from Eqs. 3 and 4, ŷ s a contnuous functon of x, t follows that D and D are pecewse, contnuous functon of x. The above numercal optmzaton problem reduces to a general non-lnear problem. However, as seen from equaton (5) and (6), the dervatve of D s not contnuous. Therefore drect search methods need to be appled to fnd the optmal value of D. Downhll smplex method (Mller, ) s chosen n ths study. An mplementaton of the algorthm s Desgn-Expert by Stat-Ease, Inc, whch s used n the paper. 5. APPROACH TO BENDING ANGLE ATTAINMENT In the steepest ascent/descent search and RSM process, a large number of experments are requred to obtan bendng angles under dfferent condtons, whch s tme consumng and costly and thus poses a serous lmtaton to the method. If the total deformaton of a sheet generated by the parallel laser scans can be obtaned by summng deformatons generated by these scans, a much smaller number of experments wll suffce. In other
words, f deformatons caused by scans at dfferent d (Fg. ) can be consdered ndependent each other, only experments wth sngle scannng paths are needed. Ths s expermentally and numercally valdated by Lu & Yao (). Fg., adopted from Cheng & Yao. (), s for the square plate under dfferent laser power level at scannng velocty of 5 mm/s. The scannng was done at y= and the bendng angle s assumed to be the same f scannng s done at a non-zero y value. A lne s ftted through the data ponts to allow nterpolaton n between. 3..8 Bendng angle (degree).4..6..8.4 v=5mm/s D b =4mm. 5 55 6 65 7 75 8 Power (W) Fg. Expermental results of bendng angle nduced by a sngle scan on a square plate (Cheng & Yao, ). 6. RESULTS AND DISCUSSIONS 6. Case : In ths case, the desrable shape (Fg. 3) s gven n terms of the followng process parameters: number of scan paths N=6 and laser power p =p=7w. The scan paths are equally spaced. The way the prescrbed shape s specfed s to facltate n ths frst case comparson of desgn result wth the prescrpton. The task here s to fnd power p and number of scan paths N to mnmze the dfference between the shape formed usng the found condton and that usng the prescrbed condton (Eq. ). Optmal desgn To apply RSM, an ntal desgn pont, N=4 and p=6w, s arbtrarly chosen. The correspondng ntal shape s shown n Fg. 3. A two-level factoral desgn s conducted wth half wdth N= and p=3w. As outlned n Secton 5, bendng angles under the factoral desgn condtons are obtaned from nterpolatng the expermental results shown n Fg.. To mmc the formng process repeatablty characterstcs, normally dstrbuted random numbers are generated and added to each bendng angle value. The standard devaton of the random numbers are chosen as the same as that shown n Fg. n the form of error bars. A frst-order regresson model s ftted based on the factoral desgn and the drecton of the steepest descent s determned from the coeffcents of the regresson model. At each movement along the steepest descent drecton, the response obtaned from regresson model s compared wth that based on the expermental result to examne f ths model s stll vald. The percentage dscrepances of the comparson at the ntal pont and frst movement along the path are.4% and.%, respectvely, whch are consdered to ndcate that the drecton of steepest descent path s vald at these ponts. The responses h at these ponts are -.36 and -.98, respectvely accordng to Eq.. After the next movement along the path (N=6 and p=664w), however, the percentage dscrepancy ncreases to 5.87%, whch ndcates that the drecton s no longer vald at ths pont (a tolerance of 5% s chosen as a lack of ft). Hence, another -level factoral desgn s conducted based on ths pont and a new steepest descent path s calculated. The new ntal pont (N=6 and p=664w) and the next pont along the new path (N=7 and p=696w) have responses h = -.5 and., respectvely. The change of sgn s clearly ndcatve of overshootng, that s, the possble soluton shape s bent more than the prescrbed shape. Ths ndcates that the optmum condton s n the vcnty of the last movement and normally a 3-level factoral desgn needs to be consdered. In ths case, however, the desgn varable N s subject to nteger constrant and the soluton must le on ether N=6 or 7. Therefore the branch-and-bound approach s appled. Three-level sngle-factor (p) desgns are conducted separately at N=6 and N=7. The quadratc equaton for N=6 s found as 5 h = 3.57 p.3933p +. (7) Solvng the equaton for zero response gves the optmal solutons p=7w for N=6 and p=67w for N=7. Ths ndcates that multple solutons are possble. An addtonal objectve functon, such as, one mnmzes producton tme screens out the soluton (p=67w for N=7). The optmzaton result (N=6 and p=7w) agrees very well wth the prescrbed value, (N=6 and p=7w). The optmzaton process towards the prescrbed shape s shown n Fg. 3, whch ncludes prescrbed shape, ntal desgn, some ntermedate shapes and fnal desgn. Robust desgn To make the desgn more robust propagaton of error (POE) s calculated based on Eqs. 4 and 7, 4 and assumed power standard devaton σ p =6W as POE = 4.9 p. 36. The response h s scaled to a desrablty functon D ranged from [,] accordng to Eq. 6 because the response problem s of the target s the
Fg. 3 Case Desgn evoluton towards the prescrbed shape best type, where T = (target), L s set as.5, and U +.5 to ensure the robust desgn s not off the optmal soluton too much. The POE s also scaled to a desrablty functon D accordng to Eq. 7 because the POE problem s of the mnmzaton type, where T s set as.4 (for p=65w) and U.68 (for p=7w) accordng to Eq. 7. The overall desrablty D s then expressed as the geometry mean of D and D usng Eq. 7. As seen n Fg. 4, the overall desrablty functon D s a contnuous, nonlnear, pecewse functon, and the maxmum value of desrablty,.4, s found at p=698w, where the POE value s lower than that at the optmal soluton (p=7w), and the response value s.3. At the optmal soluton, the desrablty s about.33 and response s obvously zero. The robust desgn therefore balances between the response and POE. In ths case, the robust desgn does not dffer much from the optmal soluton but the desgn process s generally applcable. 6. Case : In ths case, the desred shape s prescrbed n terms of a generatrx that s a second order y y polynomal z = 4( ) + ( ), as shown n Fg. 5, ts curvature ncreases wth y. It s obvous that evenly spaced 4 4 scannng paths are no longer approprate, ths resultng n a large number of desgn varables and makng the RSM based optmal desgn less feasble. As seen from Fg. 5, however, the curvature of the gven profle decreases monotoncally. Snce the trend of spacng between adjacent laser paths s closely related to the curvature of the prescrbed shape, the followng control functon s proposed to relate all d s, m d = 4( ) (8) N where d specfes the poston of the th laser path, m s the desgn varable to be determned, and N s number of laser scan paths. In ths case, N s set as 7. The problem, therefore, becomes to determne the value of laser power p and exponent m to acheve the gven profle. The same process as n the prevous case starts wth arbtrarly choosng an ntal desgn at m=. and p=65w wth half wdth of. and 3W, respectvely. The response functon s found as h=.58-5.8m+.73p-.3m -.4* -6 p +.5mp. Snce mult-solutons correspondng to zero response. Thus, another objectve functon, POE, s used to obtan the most desred soluton. Suppose the varatons n m and p are. and 6W, respectvely, the POE s constructed as POE=(9.38* -8 p -6.87* -5 p-3.* -5 mp+.9* -3 m+.m +.).5, n whch the resdual varance of.75 s ncluded. h and POE are agan scaled to desrablty functons ranged between [,] wth h constraned between [-.5,.5]. The overall desrablty D s then calculated. The desrablty n two-dmensonal cont-our s plotted n Fg. 9. As seen from Fg. 9, the most desrable value s.83 correspondng to p=69w and m=.7. Wth ths m value and Eq. 8, laser path postons, d s, are calculated and plotted n form of crosses n Fg. 5, along wth the fnal robust soluton and prescrbed shape. As expected, laser path locatons become coarser wth decreasng curvature of the prescrbed shape. The profle based on the robust soluton agrees wth the prescrbed profle. Laser formng experments were conducted under the condton determned by the robust desgn and the results were plotted n Fg. 5. As seen, the expermental result shows a good ft wth the predcted one. 7. CONCLUSIONS It s shown that the proposed optmal and robust desgn schemes are feasble and effectve for the class of shapes consdered. Integer desgn varables are effectvely dealt wth by usng standard methods such as the
. Response h POE (σ p =6W) Desrablty Desrablty.4..4..8.3 Response (mm) -. -. -.3 Desrablty Robust Soluton p=698w POE (mm).6.. Desrablty -.4 RSM Soluton p=7w.4. 65 66 67 68 69 7 7 Power (W) Fg. 4 Relatonshp of response, POE and desrablty. (Standard devaton n laser power σ = W ). z (mm) 7 6 5 4 3 - Prescrbed Shape Prescrbed Shape Curvature Robust Soluton (m=.7, p=69w) Laser Path Poston Expermental Result 3 4 y (mm) 5. 4.9 4.8 4.7 4.6 4.5 Prescrbed Shape Curvature (* -3 /mm) Fg. 5 Comparson of the robust soluton and the prescrbed shape (case ). branch-and-bound method and by ntegratng wth RSM. To reduce the number of desgn varables, the laser path postons are specfed by proper selecton of control functons. The predcted results agree wth the expermental ones. REFERENCES Cheng, J., and Yao, Y.L., (), Coolng effects n multscan laser formng, J. of Manufact. Process, Vol. 3, pp. 6-7. Derrnger, G., and Such, R., (98), "Smultaneous optmzaton of several response varables,'' Journal of Qualty Technology, Vol., pp. 4-9. Hennge, T., Holzer, S., and Vollertsen, F., (997), On the workng accuracy of laser bendng, Journal of Materals Processng Technology, Vol. 7, pp. 4-43. Kraber, S.L., and Whtcomb, P.J., (996), Robust desgn-reducng transmtted varaton, 5 th Annual Qualty Congress, Indanapols, IN. Lu,C., and Yao, Y.L., (), Optmal and robust desgn of laser formng process, F-5, Transacton of NAMRC/SME. Magee, J., Watkns, K.G., and Steen, W. M., (998), Advances n laser formng, J. of Laser Applcatons, Vol., No. 6, pp. 35-46. Mller, R.E., (), Optmzaton Foundatons and Applcatons, John Wley & Sons, New York. Myers, R.H., and Montgomery, D.C., (995), Response Surface Methodology, John Wley & Son, New York. Shmzu, H., (997), A heatng process algorthm for metal formng by a movng heat source, M.S. thess, M.I.T. Taha, H.A., (987), Operatons Research, Macmllan Publshng Company, New York. 6 4 p 6