New method of angle error measurement in angular artifacts using minimum zone flatness plane

Transcription

1 Alied Mechanics and Materials Subitted: ISSN: , Vols , Acceted: doi:0.408/ Online: Trans Tech Publications, Switzerland New ethod of angle error easureent in angular artifacts using iniu zone flatness lane Muhaad Ilyas Khan, a, Shuyuan Ma,b Mechatronic Centre, School of Mechanical Engineering, Beijing Institute of Technology, Beijing, China. a ilyas75@gail.co, b bitc@bit.edu.cn Keywords: angle easureent error, flatness lane, iniu zone ethod, genetic algoriths Abstract. Angle error easureent is one of the iortant icons in orientation tolerances. This aer resents new ethod of angle error easureent in angular artifacts based on flatness lanes. The flatness lane has been develoed based on iniu zone ethod. The ethod is sile and no exensive equients like interferoeter or autocolliator and index table are required. One ris, having four sides has been resented as an exale to show the effectiveness of the roosed ethod. Results indicate that new ethod is very effective to easure angle error in angular artifacts. Introduction In engineering, easureent /verification and insection occuy a roinent lace and cannot be ignored as the coonents are qualified and sent to assebly line based on easureent reort. To easure diensions and tolerances recisely; etrology engineers try to irove the revious ethods and device new ethods keeing easureent tie, easureent accuracy and easureent cost as a ajor target. Keeing these three araeters as a ajor objective fast easureent instruents like co-ordinate easuring achines (CMM, interferoeters, gonioeters, autocolliators etc are develoed. Most of the easuring equients are equied with latest technology ensuring the accuracy of easureent. To easure the angle error in angular artifacts; various techniques are used deends uon the accuracy required. To easure angle error in the angular artifacts for exale in ris faces, two instruents known as autocolliator or interferoeters and rotation device for exale index table are required. Pris is laced on the index table, reference easureent is taken using autocolliator and then index table is rotated to required degree to bring the second face of the ris and another reading is taken fro autocolliator, the difference between the two readings is the angle error in the ris two faces. The easureent rocedure stated above has soe sources of errors fro index table and autocolliator. Errors in index table is easily quantified but autocolliator is a ajor source of errors induction and very difficult to quantify, these errors include [], error in autocolliator easureent due to non-flat faces, non-orthoganlity of autocolliator axis with reference to the ris face, eccentricity in ris ounting, edge effect etc. For ris angle error easureent, assue olygon is laced in such a way that the y-axis of the autocolliator is laced arallel to the y-axis of index table, then x-axis of autocolliator is sensitive axis and accordingly x-z lane is the easured lane and z-axis is height of the surface at each oint. According to [], autocolliator deterine angle error at both ends of the olygon face (taking edges in to account even if the flatness in ost of the area is good. Another alternative to autocolliator is interferoeter [] which work on several definitions, the ost easy and vital is to coute the tilt of the best lane through a easured surface but interferoeter again take into account the edges into consideration but relatively less sensitive to edges as coared to autocolliator. Edges are very difficult to control in anufacturing rocess. Autocolliators with larges angular easureent range will occuy ore edges and will lead to different results as has been studied in [3, 4]. All rights reserved. No art of contents of this aer ay be reroduced or transitted in any for or by any eans without the written erission of Trans Tech Publications, (ID: , Pennsylvania State University, University Park, USA-/05/6,::00

2 998 Frontiers of Manufacturing Science and Measuring Technology IV In order to address the above flaw in angle easureent, another easureent ethod is roosed to easure the angle error in angular artifacts. Flatness lane is develoed for each face and the angle between the two flatness lanes is the angle error between the two faces. The flatness lane has been develoed using iniu zone ethod [5]. The subject ethod does not take the edges into consideration and calculate the flatness lane by taking all the ris face (surface contour into account. Flatness lane develoent using iniu zone ethod cannot be aroached using linear atheatical techniques and can be handled by non-linear techniques using soe otiization tool. Different otiization techniques, for exale genetic algorith [6-8] and article swar otiization [9] has been exloited successfully in the ast for flatness error. Real coded genetic algoriths have been exloited as otiization tool for the develoent of flatness lane and lane coefficients in this aer. The aer rovided the easureent rincile, algorith for develoent of flatness lane, exeriental work, results and discussions and finally conclusions are drawn. Measureent rincile The angle error between olygon faces can be deterined by develoing the flatness lane for each face. The olygon is laced between the reference surfaces (like the reference surface and reference fixture shown in Fig. and surface contours are extracted for all faces using three diensional easureent equient e.g. co-ordinate easuring achine (CMM or roundness tester. The flatness lane for each face is develoed by deterining the lane coefficients. The lane coefficients are used to deterine the angle error in angular artifacts. Fig. shows the flatness lane of two faces, face and face of ris having flatness lane F and F resectively. Let N and N are the two noral on F and F lanes resectively, the angle between the two norals is the angle error (α between face and face. Siilarly, flatness lanes are develoed for all surfaces and angle error is calculated between faces. y-axis F Reference fixture F F 90 ±α ±α3 N 90 ±α 90 ±α4 3 N 4 90 ±α 90 ±α 90 ±α 3 90 ±α3 z-axis x-axis Reference surface Position Position α Fig. Angle error between two faces of ris based on flatness lane Consider a set of data oints; P i (x i, y i, z i, where (i =,, 3.n extracted fro a ris face. According to definition of flatness based on iniu zone ethod [5], consider one of the lane is

3 Alied Mechanics and Materials Vols z = ax + by + d ( The distance between the data oints P i (x i, y i, z i, (where i =,, 3.n, and the lane above is D i, then zi axi byi d D = ( i + a + b i =,,3... n If f is the iniu searation, then ( f = in(ax( D i in( D i (3 In Eq.3, f is the flatness error based on the iniu zone ethod, Eq.3 ensure that all the data oints lies between the two lanes with least ossible distance. Inserting Eq. in Eq.3, lane coefficients (a, b, are required to be deterined to satisfy Eq.3. In Eq.3, f is known as a cost function in otiization terinology and required to be iniized. Plane coefficients (a, b are subjected to the constraints [0] a b 0.00 (4 The lane coefficient (a, b are the essential araeters to colete the flatness lane and deterine the angle error between the olygon faces. Let (a, b and (a, b are the lanes coefficients of two faces of olygon having flatness lane F and F resectively then the flatness lane equations are: F : z = ax + b y + d F : z = ax + b y + d (5 In Eq.5, d and d are the offset fro the origin in flatness lane F and F resectively. The angle between the two faces of olygon, α can deterined using calculus forula [], where α is in radians. a a + b b + c c α = cos (6 a + b + c a + b + c As the angle of interest is the angle between y-z lanes of F and F, let α is the angle between the y-z lanes of F and F, so Eq.6 can be odified as: b b + c c α = cos (7 b + c b + c In Eq.6 and 7, c and c are the coefficients of z and z resectively and c. c = c = c =. Eq. 7 shows the angle error between the two faces (having flatness lanes F and F of ris in y-z lane as shown in Fig.; the angle error for the other faces can be calculated in the sae way by develoing the flatness lane equation for each ris face.

4 000 Frontiers of Manufacturing Science and Measuring Technology IV Algoriths for develoent of flatness lane The objective of the algoriths is to develo the flatness lane by deterining the lane coefficients (a, b and calculate the angle error between the ris faces. Custoized algoriths has been develoed in MATLAB using basic genetic algorith araeters suorted by grahic user interface (GUI which load the data in secific forat and deterine the flatness error f, flatness lane coefficients (a, b and calculate angle errors between the ris faces according to Eq.7. The rogra flow for using genetic algoriths otiization techniques are as follows:. Matrix of rando nubers r is generated as initial oulation, P n, i.e. P n = rando( r ij l where, = (as there are two variables a and b fro Eq.3 and l=60 (l is the oulation size. The atrix P n of rando nubers r is in noralized for and required to be un-noralized according to the constraints of the cost function Eq.4 [] ij ax in rij + = ( (8 in In Eq.8, ax and in corresond to the axiu and iniu values of a and b fro Eq.4. After un-noralization the initial oulation is P un, and can be written as: P un = ij l ( (9 In Eq.9, atrix P un is the roosed initial oulation of (a, b, the iniu searation, f (also known as cost function in otiization terinology is calculated fro P un and exeriental data by using Eq.3.. Selection rate is set to 50% i.e. 50% of flatness error values are selected and rest of the 50% are thrown away, then airing has been erfored to generate arents. Both selection and airing for arent generation has been erfored according to roulette wheel ethod [3]. Elitis [4] has been introduced to overcoe the inherited roble in roulette wheel ethod in the selection rocess as soe tie the best value of cost function (flatness error is not selected by roulette wheel ethod and sooth convergence do not occur. Elitis coy the best value of the cost function (flatness error along with their corresonding lane coefficients and included in the next generation for sooth convergence. 3. The arents are ated according to blend crossover [3]. Let P and P are arent chroosoes, the offsring s chroosoes Pʹ and Pʹ are exressed as: P = P + ζ ( P P P = P ζ ( P P ζ = ( + ε u ε (0 In Eq.0, u is a rando generated nuber for each gene with unifor distribution in interval (0, and Ԑ is a local basic GA araeter and set to 0.5. After blend crossover, offsring s are added to the oulation. 4. After blend crossover, utation has been erfored. Non-unifor utation is selected as a utation oerator as the subject utation is equied with local fine tuning caabilities and is exloited in the roosed algoriths [5]. Let is selected in soe rando fashion fro t s = (,...,,..., chroosoes; and assue that utated vector after utation is 0 k t+ s = (,...,,..., 0 k t+, the utated vector in s can be deterined by Eq. and. 0 = = + ( t, ( t, ax in (

5 Alied Mechanics and Materials Vols In Eq., ax and in corresond to the axiu and iniu values of lane coefficients (a, b fro Eq.4, the first art of the equation is alicable if soe rando generated nuber v is 0 and second art is alicable if v is. If t and gen ax are the current and axiu generations resectively, λ is a rando nuber; q is degree of non-unifority then: ( t gen q ax ( t, y = y ( λ. ( In Eq., (t, y is a function and return a value in the (0, y range. In the roosed algorith gen ax is set to 00 and q is set to [3]. After non-unifor utation, selected chroosoes are relaced with utated chroosoes. Non-unifor utation is the last ste in genetic algoriths; the cost function is deterined and ranked in ascending order, the first generation is coleted and roceeds for the next generation, i.e. ju to ste above, the loo continue and ceases after gen ax is achieved. The utation rate is set to 0%. Exeriental work Roundness testers fro Mitutoyo, equied with ultidiensional caabilities were exloited for three diensional easureents. First of all roundness tester were adjusted according to the technical and environental requireents. The reference fixture were ounted on the reference surface (table to of the roundness tester and adjusted in both x-y axis. The contour of the first face of ris was extracted (as shown in Fig. by lacing the ris in osition and then laced on osition for the second face and siilarly surface contours data were extracted for other faces. The extracted data for four faces is given in aendix A. Results and discussions In the roosed algoriths, real coded genetic algorith has been exloited. The flatness lane equation is coleted by the lane coefficients (a, b. The flatness error value based on iniu zone ethod corresonds to the values of flatness lane coefficients (a, b, so the correct value of the flatness error ensure the correct value of lane coefficients (a, b. The roosed angle error easureent ethod exloits flatness lane coefficients to calculate the angle error between the ris faces (Eq.7. In order to verify the authenticity of the algoriths for flatness error and lane coefficients, data fro [7] and data sale to sale 3 fro [8] were tested on the roosed algoriths and the results were in colete agreeent to those of other researchers, the roosed algoriths results are suarized in Table. Table Proosed algorith results, data fro [7] and Aendix A data of sale to 3 fro [8] Corresonding flatness lane equation Literature Flatness error[] [ z = ax + by + d ] [7] z = x y + d [8] z = x y + d [8] z = x y + d [8] z = x y + d A cube ris of selected for easureent of angular error between faces. Data oints were extracted for four faces. The data oints (contour of ris faces are suarized in aendix A. The surface contour data of four faces (aendix A were loaded in the algoriths through custoized develoed MATLAB GUI, the flatness error, lane coefficients (a, b and corresonding lane equations were deterined and are suarized in Table.

6 00 Frontiers of Manufacturing Science and Measuring Technology IV Face No Table Flatness lane equation based on iniu zone ethod (MZM Corresonding flatness lane equation a b [ z = ax + by + d ] z = x y + d z = x y + d z = x y + d z = x y + d4 The angle errors between the faces were calculated based on flatness lane (using lane coefficients given in Table by using Eq.7 and results were then coared with conventional technique by using autocolliator. The results are suarized in Table 3. Table 3 Angle errors between ris faces based on flatness lane and coarison to autocolliator Angle error between indicated faces Autocolliator result [arc sec] Flatness lane result using MZM [arc sec] Face - (α Face -3 (α Face 3-4 (α Face 4- (α The results of the develoed codes for the flatness error and flatness lane are in colete agreeent with other ethods as shown in Table. The easureent of angle errors are calculated fro Table data and suarized in Table 3. The results fro autocolliator and flatness lane based on MZM are coared and the results are in good agreeent as shown in Table 3. Conclusions The new ethod is based uon flatness lane and lane coefficients to calculate the angle errors between the faces. The ethod exloits iniu zone ethod to develo the flatness lane, so all the oints of surface are taken into consideration. The new ethod reduces the insection tie, not costly as there is no use of exensive equients like autocolliator and index table. The roosed ethod is based on flatness lane, so the algorith also hold good for flatness error easureent. Acknowledgent This research work was suorted by National Natural Science Foundation of China (Grant No Aendix A Pris exeriental data easureent data [x, y and z are in ] No s of data oints x y Face (z Face (z Face 3(z Face 4(z

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