THE FOURIER SERIES USED IN ANALYSE OF THE CAM MECHANISMS FOR THE SHOEMAKING MACHINES (PART I)

Transcription

1 ANNALS OF HE UNIVERSIY OF ORADEA FASCICLE OF EXILES, LEAHERWORK HE FOURIER SERIES USED IN ANALYSE OF HE CAM MECHANISMS FOR HE SHOEMAKING MACHINES (PAR I) IOVAN-DRAGOMIR Ala, DRIȘCU Maraa, Gheorghe Asach echcal Uversty, extle, Leather ad Idustral Maagemet Faculty, Str.Prof.Dr.Doc. Dmtre Magero, r.8, 75, Ias, Romaa, E-Mal: Correspodg author: Iova-Dragomr, Ala, E-mal: Abstract: A computer asssted procedure for the cematc aalyss of the mechasm of a cam s essetal makg a certa type of research operatos. hey maly refer to the optmzato of operatos rug o specfc machery, or to the re-desg of the mechasm, order to make the mechasm dgtal. hs aalyss seems eve more mportat, whe we cosder the fact that most of the maches used shoe dustry owadays use a cam mechasm. he paper s devded two parts. I frst part, t s elaborated a method of fdg of a fucto G(x), belogg to a Fourer seres, whch approxmates the umercal values {x, y }, wth the bggest accuracy. Fdg the fucto that approxmates the most accurately the data set, for the posto parameters of the follower S(ω), () wll lead to a complete kematc ad dyamc aalyss of the cam mechasm. hese values repeat wth = π perod. I secod part, the method s tasted usg MatCAD work sessos whch allow a umercal ad graphcal aalyss of the mathematcal relatos volved, order to test the reablty of the method. he set of expermetal data are resulted after measurg a cam mechasm of a mache used shoemakg. Key words: Fourer seres, cam, mache, shoemakg,. INRODUCION Cam mechasms are volved costructo of may maches for shemakg producto, as are: lastg maches, steachg maches ad so oe. he aalyss of a cam mechasm meas determg these posto parameters of the follower, as well as determg ts momevet law. akg to accout the fact that the follower always makes a alteratve-perod move, we wll mathematcally model the posto parameters of the follower (ts movemet S ad the rotato agle, ) by usg Fourer seres (fg. ). o ths am, the paper presets theoretcal aalyss of: Fourer seres, umercal calculus methods for a defte tegral ecessary for the determato of the Fourer coefcets, establshg optmzato crtera order to obta the fucto that best approxmates the posto parameters of the follower. 67

2 ANNALS OF HE UNIVERSIY OF ORADEA FASCICLE OF EXILES, LEAHERWORK. GENERAL INFORMAION. Cam mechasms Cam mechasms [] are dvded two groups: plaar ad spatal. I case of the plaar mechasms, the cam ad the follower roll also, have the movemets the same pla or parallel plas. If the movemets of the elemets of the mechasm are dfferet plas, ths s a spatal mechasm. Fg. : Cam mechasm he mechasm preseted fg. s composed of: pla cam wth groove (blateral cotacts), oscllatg follower roll ad the framework 3. he cotact betwee the follower ad the cam s made by a roll, whch go the groove of the cam. he cematc aalyss of a cam mechasm ca be made usg the follow methods [5]: the method of the cematc charts, the method of the vectoral equatos ad the method of the trasformato of the cam mechasm the mechasm wth lower pars.. Fourer seres ad fuctos From the specal lterature oe kows that a perodcal fucto f(x):[,] wth values R ca be approxmated wth Fourer [], [3] seres, accordg to the relato: F( x) a ( x) cos b ( x) s () From ths relato oe ca otce that, the Fourer seres are a fte sum of trgoometrc fuctos s, cos wth argumets creasg arthmetcal progresso ad multpled by a ad b. o these oe adds the term dvded by. he forms of the coefcets of the the Fourer, [], [3], [4], [6], are: a f( x)d x () 68

3 ANNALS OF HE UNIVERSIY OF ORADEA FASCICLE OF EXILES, LEAHERWORK a f( x) cos ( x ) (3) b f( x) s ( x) (4) Where s a umber havg values from ad. For the perod s =*π, the, a ad b are the Fourer seres coeffcets ad, accordg wth specal lterature, they have the form: a f( x)d x (5) b b f( x) s( x) f( x) s( x) (6) (7) ad Fourer seres wll be: F( x) a cos( x) b s( x) hs s a fte sum of trgoometrcal fuctos s, cos multpled by coeffcets a ad b calculated for dfferet values of the argumet x. For a fte terms umber of the seres gve by relato (8), Fourer fuctos wll obta. Hece, a Fourer fucto wll have the followg form: f G( x) a cos( x) b s (9) ( x) = where: f represets the umber of terms of the seres; s a varable havg values from to f; 69 (8)

4 ANNALS OF HE UNIVERSIY OF ORADEA FASCICLE OF EXILES, LEAHERWORK, a ad b are the Fourer seres coeffcets calculated wth the relatos (5), (6), (7) for the rage [, f]. NOE he umber f, defg the Fourer fucto form, s choosg as fucto of covergece crtero of the seres, so that the fucto G(x) approxmates the fucto F(x) wth the best accuracy. 3. RESULAES 3. Method of approxmato for a perodcal expermetal data set I the most stuatos, a data set {y, x } ca ot be approxmated wth a elemetary aalytcal fucto f(x). For ths reaso, computer asssted desg, oe apples to moder mathematcal methods provdg methods of fdg of the fuctos whch drve to the approxmato of the data set. I ths paper, oe presets a method of fdg of the fucto G(x), belogg to a Fourer seres, whch approxmates the umercal values {x, y } wth the bggest accuracy. hese values repeat wth = π perod. he work routes of the preset methodology are: - calculato of the Fourer seres coeffcets; - executo of the tegral calculus usg the trapezes method; - determg the Fourer fucto. Calculato of the Fourer seres coeffcets From the relato (), (3), (4) or (5), (6), (7) a oe observes that, for the coeffcets, a ad b obtag, oe use a tegral calculus from a aalytcal fucto f(x) or f(x)cos(x) ad f(x)s(x). As log as the values of these fuctos are ukow, the tegral calculus wll apply to umercal methods of determato of a fte tegral. Accordg to these, the umercal value of the fte tegral represets the area of the feld lmted by axs Ox, fucto g(x) ad the segmets of bee le: x=a ad x=b, havg the followg relato: a b g( x)d x g x h () where: x values of the depedet varable for the rage [a, b]; g(x ) - values of the fucto g(x) the pots x ; h the dstace betwee two cosecutve values x, x + lke: h =x +-x 7

5 ANNALS OF HE UNIVERSIY OF ORADEA FASCICLE OF EXILES, LEAHERWORK Fg.: he area of the feld lmted by axs Ox, fucto g(x) ad the segmets of bee le: x=a ad x=b Executo of the tegral calculus usg the trapezes method I the case of the perodcal data set wth perod π, whose values are obtag from to, for example, the umercal data for the tegral calculato wll be: a= b=π h =π/36 g(x )=Y. he professoals cosder that the trapezes formula s the most accurate oe for the tegral calculato, that ca be foud the specal lterature. Based o ths methodology, for a data set Y determed the pots x =,,, 3..36, the relato (7), ca be re- wrtte as: S h 35 = Y Y 36 Y () ) From the above presetato, the relatos for the coeffcets, a ad b determato, the relatos used are: = Y Y 36 Y () a 7 35 = Y cos r Y Y 36 (3) 7

6 ANNALS OF HE UNIVERSIY OF ORADEA FASCICLE OF EXILES, LEAHERWORK where: b 7 36 = Y s r (4) Y are expermetal umercal values; Φr agle belogg to the rage [,π], rad. Determg the Fourer fucto I ths route, oe determes the umber of terms of the seres whch wll approxmate wth the bggest accuracy the route of pots {x, y } obtaed the sessos of the gettg of data, whch wll determe Fourer fucto. o ths am, oe calculates wth the relato: F f = a cos r b s r (5) the values of the Fourer fucto for a fte varable umber f of terms ad oe selects that f for whch the dfferece betwee the tal values Y ad F are mmum. It results that the selecto crtero of the umber of terms of the seres, ad therefore of the Fourer fucto s: Y - F < ε where ε s depedet of the applcato ature. he elaborate method wll be use MathCAD sessos, order to aalyse cam mechasms. he set of expermetal data are resulted after measurg a cam mechasm of a mache used shoemakg. REFERENCES [] D.B.Marghtu - edtor, Mechacal Egeer s Hadbook, ACADEMIC PRESS, USA,. [] H. Dym, H.P. Mckea, Fourer Seres ad Itegrals, Academc Pr., 986. [3] James Ward Brow, Rull V., Fourer Seres, Churcll, McGraw_hll,. [4] Bors Osleker, Fourer seres orthogoal polyomals, Mathematcal Revews,. [5] A. Iova-Dragomr, Elemete de proectare a mașlor d dustra cofecțlor d pele, Edtura PERFORMANICA, 3. [6] Cocea, M., ad Buzescu L. F., Coferţă teraţoală de comucăr ştţfce, eholog Modere, caltate, restructurare, Chşău- Iaş, 5. [7], D.F. Crotoru, A. Dragomr, Utlaje petru dustra cofecţlor d pele, Ias, 3. (6) 7