let./vol. 51 - št./no. S/05 - str./pp. 73-130 zvezek/issue 478 STROJNIŠKI VESTNIK JOURNAL OF MECHANICAL ENGINEERING cena BOO SIT 9 770039 248001 ISSN DQ39-24BQ Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 73 Vsebina - Contents Vsebina - Contents Strojniški vestnik - Journal of Mechanical Engineering letnik - volume 51, (2005), številka - number 2 ISSN 0039-2480 Izhaja mesečno - Published monthly Razprave Papers Praček, S., Jakšič, D.: Odvijanje preje z navitka - Praček, S., Jakšič, D.: Yarn unwinding from packages obravnava kinematičnih in dinamičnih - a discussion on the kinematic and dynamic lastnosti preje 74 properties of yarn Štubna, I., Trnik, A.: Izrazi za popis upogibnega Štubna, I., Trnik, A.: Equations for the Flexural Vibration nihanja palice nespremenljivega prereza 90 of a Sample with a Uniform Cross-Section Vekteris, V.: Prehodni pojavi pri postopku Vekteris, V.: Transient Phenomena in the Grinding brušenja 95 Process Biluš, I., Škerget, L., Predin, A., Hriberšek, M.: Biluš, I., Škerget, L., Predin, A., Hriberšek, M.: Ex- Eksperimentalno numerična analiza perimental and numerical analyses of the kavitacijskega toka okoli lopatičnega profila 103 cavitational flows around a hydrofoil Poročila Reports Papotnik, A.: Kotiček za tehniko in tehnologijo v Papotnik, A.: Small Corner for Technics and Technol- vlogi razpoznavanja in razvijanja nadarjenosti ogy in Function Identification and the Devel- predšolskega otroka 119 opment of Talents in Children under School Age Čudina, M.: Darilo Fakulteti za strojništvo v Čudina, M.: Donation to the Faculty of Mechanical Ljubljani 123 Engineering in Ljubljana Strokovna literatura Professional Literature Iz revij 125 From Journals Osebne vesti Personal Events Doktorati, magisteriji, diplome 127 Doctor’s, Master’s and Diploma Degrees Navodila avtorjem 128 Instructions for Authors 73 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 UDK - UDC 677.072/.073:531.3 Izvirni znanstveni članek - Original scientific paper (1.01) Odvijanje preje z navitka - obravnava kinematičnih in dinamičnih lastnosti preje Yarn unwinding from packages - a discussion on the kinematic and dynamic properties of yarn Stanislav Praček - Danilo Jakšič Obravnavamo odvijanje preje z navitkov, kar je ključnega pomena pri številnih tekstilnih procesih. Izpeljemo zelo splošen sistem diferencialnih enačb, ki opisujejo gibanje preje med odvijanjem. Opisan je fizikalni pomen posameznih členov, ki nastopajo v enačbah, s posebnim poudarkom na navideznih silah v vrtečem se koordinatnem sistemu. Prikažemo tudi, kako lahko v kvazistacionarnem približku sistem enačb numerično rešimo in dobimo sliko preje v prostoru med odvijanjem. © 2005 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: gibanje preje, lastnosti kinematične, lastnosti dinamične, enačbe diferencialne) We discuss yarn unwinding from packages, which is of chief importance in many textile processes. We derive a very general system of differential equations that describe the motion of the yarn during unwinding. We discuss the physical meaning of individual terms in the equations with special emphasis on virtuals forces, which appear in rotating coordinate systems. We also show how the equations can be numerically solved in the quasistationary approximation in order to obtain an image of yarn in space during unwinding. © 2005 Journal of Mechanical Engineering. All rights reserved. (Keywords: yarn motion, kinematic properties, dynamic properties, partial differential equations) 0 UVOD 0 INTRODUCTION Nihanja mehanske napetosti, do katerih prihaja med odvijanjem preje z navitka, povzročajo številne težave in lahko vplivajo na učinkovitost tekstilnega postopka in na kakovost končnega izdelka. Ta nihanja so še posebej opazna pri vzdolžnem odvijanju, pri katerem je navitek nameščen nepremično, preja pa se hitro odvija in teče stran od navitka v smeri njegove osi. Pomembno je, da poiščemo obliko navitka, pri kateri bo gibanje preje takšno, da bo napetost v preji majhna in čim bolj enakomerna. Odvijanje preje bomo obravnavali s teoretičnega vidika. Izpeljali bomo sistem diferencialnih enačb, ki opisuje gibanje preje med odvijanjem. Enačbe bomo izpeljali z najmanjšim številom privzetkov, tako da bodo čim bolj splošne. Pri tem bomo dali velik poudarek fizikalnim Oscillations of tension in yarn, which appear when the yarn is unwinding from a package, cause many problems and can degrade the efficiency of the textile process and the quality of the end product. These oscillations are particularly strong in axial unwinding, where the package is stationary and the yarn is being withdrawn in the direction of package axis. It is thus important to find the optimum shape of the package for which the motion of the yarn will be such that the yarn tension will be small and as steady as possible. The unwinding will be discussed from the theoretical point of view. We will derive a system of differential equations that provides a description of the yarn motion during the unwinding. We will derive these equations using a minimal set of assumptions, so that the resulting equations will retain their generality. An 74 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 razlagam posameznih matematičnih izrazov in členov, ki se v njih pojavljajo: namen prispevka ni le razviti računski formalizem, temveč tudi bralcu omogočiti razumevanje bistvenih fizikalnih dejavnikov, ki lahko vplivajo na končni rezultat. Zato bomo bolj izdatno, kakor je sicer v navadi, spregovorili o opisu preje kot krivulji v prostoru in o njeni parametrizaciji, o podobnosti s hidrodinamičnim problemom toka tekočin (razlika med lokalnim in substancialnim odvodom), o uporabi Newtonovega zakona pri razsežnih telesih in o tem, kako opišemo napetostno stanje v enorazsežnih telesih. Na koncu bomo pokazali še, kako zapišemo enačbe v vrtečem se koordinatnem sistemu, opisali bomo navidezne “sistemske” sile, ki jih pri tem dobimo, ter izpeljali pogoj, ki ga dobimo zaradi privzetka nerazteznosti preje. 1 OPIS PREJE IN KINEMATIČNE LASTNOSTI Pri obravnavi gibanja preje običajno zanemarimo prečno razsežnost preje. Z drugimi besedami, mislimo si, da je preja neskončno tanka in da jo zato lahko obravnavamo kot enorazsežni predmet. Takšni približki so v mehaniki pogosti: tudi kovinske žice, strune glasbenih inštrumentov, elastike, vlakna in podobne dolge, vendar tanke predmete, obravnavamo kot idealno tanka telesa. Napaka, ki jo s takšnih približkom storimo, je zanemarljivo majhna. Poleg tega privzamemo, da je preja neraztezna: to pomeni, da zanemarimo raztezke v preji, vendar pa kljub temu upoštevamo napetost v preji. Dokazano je bilo, da vodi ta približek le k majhni napaki pri običajno uporabljeni preji [1]. Enorazsežno telo opišemo kot krivuljo v prostoru. Najlaže jo podamo v parametrizirani obliki: vsako točko na krivulji podamo z njenim krajevnim vektorjem r(s), pri čemer je s parameter, s katerim oštevilčimo točke na krivulji. Pri opisu preje je najbolj naravna in pripravna parametrizacija z ločno dolžino, kar pomeni, da je s dolžina preje med izbrano točko r(s) in izbranim izhodiščem. Pri odvijanju preje z navitkov je najbolj primerna izbira izhodišča vodilo, skozi katerega prejo vlečemo z nespremenljivo hitrostjo (sl. 1). Prejo odvijamo s hitrostjo V skozi vodilo O, ki je tudi izhodišče koordinatnega sistema. Točka Dv je točka dviga, to je točka, v kateri preja zapusti emphasis will be given to the physical interpretations of the mathematical expressions and of the various terms that appear in them: the purpose of this paper is not only to develop a formalism for the calculations, but also to lead to an understanding of the key physical elements that have a direct influence on the final result. For this reason we will amply describe details that are usually neglected, such as the description of the yarn as a curve in space and its parametrisation, the similarity with the hydrodynamical problem of liquid flow (the difference between the local and the substantial derivative), the use of Newton’s law for a description of the extended bodies and the description of the elastic state of one-dimensional objects. We will show how the transition to the rotating cylindrical coordinate system is accomplished and how this leads to virtual “system” forces. The condition resulting from the non-extensibility of the yarn will also be given. 1 DESCRIPTION OF THE YARN AND THE KINEMATIC PROPERTIES The lateral dimensions of the yarn are usually neglected in descriptions of yarn motion. In other words, we consider the yarn to be infinitely thin, so that it can be described as a one-dimensional object. Such approximations are common in mechanics, they are used for objects such as metal wires, the strings of musical instruments, fibers and other long, but thin objects, that can be considered as ideally thin. This approximation leads to a small error that can be safely neglected. We furthermore assume that the yarn is not extensible: by this we mean that any elongation is neglectable; however, we do take into account the tension in the yarn. It has been shown that this assumption leads to only a small error with commonly used yarns [1]. A one-dimensional body can be mathematically described as a space curve. It can be most conveniently given in parametric form: each point on the curve is described by its coordinates (radius vector) r(s), where s is some parameter used to number the points on the curve. For yarns the most natural and useful parametrisation is given by its arc length. Parameter s is then the length of yarn between the chosen point r(s) and the origin. A very convenient choice of the origin for yarn-unwinding problems is the guide through which the yarn is being pulled away with constant velocity (Fig. 1). The yarn is being withdrawn with unwinding speed V through a guide O, which also serves as the origin of the coordinate system. Point Dv is the lift-off point, i.e., the point where the yarn leaves the surface Odvijanje preje z navitka - Yarn Unwinding from Packages 75 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 Sl. 1. Odvijanje preje z valjastega navitka Fig. 1. Yarn unwinding from a cylindrical package površino navitka in naprej ustvarja balon. Točka Od je točka, kjer se preja začne odvijati in drseti po navitku. Kot f je kot navijanja preje na valjasti navitek. Vektor k je tangentni vektor na prejo v točki odvijanja. Zanimalo nas bo gibanje preje, torej časovno spreminjanje lege krivulje r(s) v prostoru. Zato uvedemo še dodaten parameter t, ki podaja trenutek, ob katerem ima krivulja obliko r(s,t=konst). Rečemo lahko tudi, da je funkcija r(s,t) pri izbranem stalnem času t “trenutna slika preje v prostoru”, kakor bi jo posneli s fotoaparatom. Časovni potek odvijanja preje zato podamo kot dvoparametrično vektorsko funkcijo r(s,t). Pri kinematiki odvijajoče se preje naletimo na podobno težavo kakor pri hidrodinamičnem problemu toka tekočin [2]. Problema sta si podobna v tem, da imamo tudi tukaj opravka s prejo, ki “teče” proti vodilu vzdolž svoje “struge”, ki jo v nekem trenutku t podaja funkcija r(s,t=t). Dejansko sta pri preji sočasno prisotni dve različni gibanji: “tok” preje, ki jo vlečemo proti vodilu (to gibanje je v vsaki točki tangentno na krivuljo r(s,t=t)), ter spreminjanje “struge” same, zaradi odvijanja preje z navitka. Lega preje je, kakor rečeno, odvisna tako od ločne doline s kakor od časa t: r=r(s,t). Iz zaporednih opazovanj lege pri eni in isti ločni dolžini dobimo lokalni časovni odvod, ki ga označimo z dr/ dt. Ta odvod pa ni enak hitrosti preje! Enak je spreminjanju “struge” preje, ne upošteva pa dejstva, da preja “teče po strugi” s hitrostjo odvijanja V. of the packages to form the balloon. Point Od is the unwinding point, where the yarn starts to slide on the surface of the package. Angle f is the angle of winding on the package. Vector k is the tangent vector on the yarn at the unwinding point. We are interested in yarn motion, i.e., the time variation of the position r(s) of the yarn in space. We therefore introduce an additional parameter t, which gives us a moment in time when the form of the yarn is r(s,t=const). In other words, the function r(s,t) at fixed time t is a “snapshot of the yarn in space” as taken by a camera. The process of yarn unwinding is therefore described using a two-parameter vector function r(s,t). In a kinematic description of yarn unwinding we face a similar problem to that in the hydrodynamic problem of liquid flow [2]. Both problems are similar in that the yarn also “flows” in the direction of the guide along its “riverbed”, which at time t is given by the function r(s,t=t). In fact there are two simultaneous motions of the yarn: the “flow” of the yarn being pulled through the guide (this motion is at every point tangent to the curve r(s,t=to)) and the time variation of the “riverbed” itself due to the unwinding of the yarn from the package. The position of a point on the yarn depends on two parameters: the arc length s and the time t: r=r(s,t). If we observe the position of the yarn at a fixed arc length we obtain a local time derivative, denoted by dr/ft. This derivative, however, is not equal to the yarn velocity. Instead, it is equal to the velocity of the changes of the “riverbed”, but it does not take into account that the yarn “flows within the riverbed” with the unwinding speed V. 76 Praček S. - Jakšič D. Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 Če hočemo izmeriti hitrost izbrane točke preje, ne smemo preje opazovati pri nespremenljivem s, temveč se moramo vzdolž “toka” gibati skupaj s prejo. Gibanje odseka preje opiše funkcija r(s(t),t), pri čemer je s(t) ločna dolžina obravnavanega odseka preje ob času t. Lega tega odseka je odvisna samo od časa. Hitrost dobimo kot totalni odvod, v=d(r(s(t),t)/dt. V hidrodinamiki je takšen odvod znan kot substancialni odvod. Z lokalnim odvodom dr/dt ga povežemo z uporabo pravil diferencialnega računa: dv(s(t),t) dt Sedaj upoštevamo privzetek, da je preja neraztezna. Če prejo odvijamo z odvijalno hitrostjo V, ki se s časom ne spreminja, potem za vsak kratek odsek preje velja s(t)=s - Vt. Predznak minus dobimo zato, ker prejo vlečemo v smeri vodila, zato se ločna dolžina s(t) izbranega odseka zmanjšuje s časom linearno proti nič. V poljubni točki na preji tedaj velja ds/dt=- V. Zapišemo torej: dr dr df,=d~t Ker je parametrizacija z ločno dolžino naravna parametrizacija, je 3r/3s enotski tangentni vektor k na prejo v dani točki. Končna enačba za hitrost se torej glasi: dr v = — dt Prvi člen opisuje “spreminjanje struge”, drugi člen pa tangentni “tok” preje proti vodilu. Na tem mestu moramo poudariti, da izraz (3) nikakor ne pomeni, da je tangentna komponenta hitrosti enaka -V v vseh točkah preje, saj lahko tudi člen 3r/3t vsebuje komponento v tej smeri. Medtem ko je tangenta komponenta hitrosti po definiciji naloge enaka V pri vodilu, je tangentna komponenta zagotovo enaka nič v točki odvijanja Od, kjer se preja ravno začne premikati. Vpeljemo lahko abstraktni operator totalnega časovnega odvoda D, ki sledi gibanju točke na preji: If one wants to measure the velocity of a chosen point on the yarn, one should not observe the yarn at fixed s, but should instead follow the “flow” of the yarn. The motion of a short segment of the yarn is described by the function r(s(t),t), where s(t) is the arc-length of the segment at time t. The position of the segment is a function of time only. The velocity can be obtained using a total derivative, v=d(r(s(t),t))/ dt. Such a derivative is known in hydrodynamics as a substantial derivative. It can be related to the local derivativedr/dt using the chain rule of calculus: dv dvds dt + d~sdf, (1). We now take into account that the yarn was assumed inextensible. If the yarn is withdrawn with an unwinding speed V that is constant with time, then for any short segment of the yarn we have s(t)=so -Vt. We obtain a minus sign because the yarn is being pulled in the direction of the guide, so that the arc-length s(t) to a given segment decreases with time linearly toward zero. Therefore we have ds/dt=-V at any point of the yarn. We can then write: V dv ds (2). As arc-length parametrisation is a natural parametrisation of a curve, the derivative dr/ds is equal to the unit tangent vector k to the yarn at a given point. The final expression for the velocity of a segment is therefore: dr df, Vk (3). The first term describes the changing “riverbed” and the second term gives the tangential “flow” of the yarn along the riverbed in the direction of the guide. At this point we should emphasize that the expression (3) in no way implies that the tangential component of the velocity is equal to - V at all points on the yarn, because the term dr/8t can also contain a component along this direction. Indeed, while the tangential component of the velocity is from the definition of the problem equal to V at the guide, it is clearly equal to zero at the unwinding point Od where the yarn just starts to move. We can introduce an abstract total time derivative operator D, which follows the motion of a point along the yarn: D = dt d dt V d ds (4). Če operator D uporabimo na krajevnem vektorju r, dobimo izraz (2). Hitrost gibanja odseka By applying the operator D on a radius vector r, we obtain expression 1. The velocity of the yarn segment Odvijanje preje z navitka - Yarn Unwinding from Packages 77 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 preje je torej v=Dr. Pričakujemo torej, da bomo dobili pospešek odseka preje, če operator D uporabimo dvakrat na krajevnem vektorju: a=D 2r. Dobimo: is therefore v=Dr. We can expect that the acceleration of a yarn segment is given by applying the operator D twice on the radius vector: a=D2r. We obtain: D 2v = torej d dt & dt? d2r dt? V ds 2V d2 dsdt or v PL ds* IV Ta rezultat lahko preverimo tudi z neposrednim izračunom pospeška brez uporabe operatorja D. 2 DINAMIKA: NEWTONOV ZAKON ZA ODSEK PREJE dk dt V dk ds (5), (6). We can ascertain that this is in fact the correct expression for acceleration by a direct calculation without using the operator D. 2 DYNAMICS: NEWTON’S LAW FOR A YARN SEGMENT Gibanje preje bomo opisali v inercialnem opazovalnem sistemu, v katerem velja Newtonov zakon F=ma, kjer so F sila na telo, a pospešek, m pa masa telesa. Newtonov zakon je zapisan v obliki, ki je uporabna za obravnavo gibanja snovnih delcev (na primer atomov), togih teles (krogel, planetov itn.), ter za opis gibanja težišča deformljivih teles kot celote. Pri preji, ki je deformljivo telo, nas ne zanima, kako se giblje preja kot celota, temveč kako se spreminja oblika preje same. Zato prejo v mislih razrežemo na (infinitezimalno) kratke odseke dolžine 8s in uporabimo Newtonov zakon za vsak odsek posebej. Oglejmo si najprej, katere sile delujejo na naš sistem, torej na kratek odsek preje, katerega eno krajišče je v točki r(s), drugo krajišče pa v točki r(s+8s). Očitno je odsek izpostavljen sili težnosti F=mg (m je masa odseka preje, g pa težnostni pospešek). Izkaže se, da je sila težnosti zanemarljiva v primerjavi z drugimi silami, zato jo zanemarimo [3]. Če se odsek premika po zraku, nanj deluje tudi sila zračnega upora F : We will describe yarn motion in an inertial observation frame where Newton’s law F=ma is valid. Here, F is the force acting on the body, a is the acceleration and m is the mass of the body. This form of Newton’s law is appropriate for describing the motion of material particles (such as atoms), rigid bodies (balls, planets etc.) and for describing the motion of the center of mass of deformable bodies. The yarn is deformable and we are not interested in how it moves as a whole. We are interested instead in the changes of the form of the yarn. We therefore divide the yarn into (infinitesimally) short segments of length ds and apply Newton’s law to each segment individually. First we need to determine which forces act on our system, i.e., on the short segment of yarn whose one extremity is r(s) and other extremity is r(s+ds). The force of gravity Ft=mg (m is the mass of the segment and g the gravitational acceleration) acts on the segment. It turns out that the effect of gravitation can usually be neglected in comparison to the other forces [3]. If the segment is moving through the air, there is also a contribution from the air drag force Fzr: cupv2 S (7), kjer so: c koeficient zračnega upora,/? gostota zraka, v pravokotna komponenta hitrosti, S pa čelni prerez odseka preje ([4] in [3]). Reynoldsevo število je višje od 1000 v tistih delih preje, kjer je hitrost največja, in nižje drugod. Kljub temu uporabimo kvadratni zakon zračnega upora na celotni dolžini preje, saj velja v tistih delih preje, kjer je učinek zračnega upora največji. where cu is the coefficient of air drag, r is the density of the air, vn is the normal component of the velocity and S is the frontal area of the yarn segment ([4] and [3]). The Reynolds number is higher than 1000 on the parts of the yarn with the highest velocity and smaller elsewhere. Nevertheless, we use the law of quadratic air drag on the entire length of the yarn because it is valid on those parts of the yarn where it has the largest effect. 78 Praček S. - Jakšič D. Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 v tej smeri narašča s increasing s _^ f - zračni upor f - air drag force T(S) r(s + Ss) ^rdr l J- gs\s+da Sl. 2. Sila na odsek preje Fig. 2. Forces acting on a yarn segment Na kratek odsek preje pa neposredno delujeta tudi preostala kosa preje na obeh straneh obravnavanega odseka (sl. 2), zato na vsako krajišče deluje neka sila. Ti sili sta posledica notranjega napetostnega stanja zaradi natezne obremenjenosti preje, podobno kakor pri napeti elastiki. Na odsek preje delujejo sile zaradi napetosti in sila zračnega upora. V trorazsežnih telesih (kontinuih) napetostno stanje opišemo z napetostnim tenzorjem, v enorazsežnem telesu, kakršna je preja, pa zadostuje skalarna količina, imenovana napetost T. Ta pove, kakšna sila deluje na rob enorazsežnega telesa zaradi deformacij in ima enoto sile [N]. Definiramo jo z enačbo: In addition to these obvious forces, there are also forces imparted on the segment by the remaining yarn on each side of the segment, see Fig. 2, so that there is an additional force on each extremity of the segment. These two forces are a consequence of the internal elastic state due to elastic strain on the yarn, similar to the case of a stretched elastic band. Forces of tension and air friction force act on a short yarn segment. In three-dimensional continuum bodies the stress state is given by the stress tensor, whereas in one-dimensional bodies such as the yarn, a single scalar quantity is sufficient. This quantily is called the tension, and it is denoted by T. The tension is numerically equal to the force that acts on an extremity of a one-dimensional body due to deformations and it has the same unit as force, i.e., Newton [N]. It is defined with the equation: F = Tk (8), kjer sta F sila na rob obseka preje, vektor k pa where F is the force acting on the extremity of the tangentni vektor na prejo v točki prijemališča sile segment, vector k is a tangent vector on the yarn at F, torej v robni točki. Sila na rob preje v točki r(s) the point of the application of the force F, i.e., at the je: extremity. The force on the extremity at r(s) is therefore: sila na drugi rob v točki r(s+Ss) pa: -T(s)k(s) (9), and the force on the other extremity at r(s+Ss) is T(s + 5s)k(s + Ss) (10). Drugi Newtonov zakon za odsek preje zato Newton’s second law for a segment of yarn zapišemo kot: can be expressed as ma = T(s + Ss)k(s + Ss) T(s)k(s) + Fzr (11). Masa odseka je m=f&s, kjer je p linearna The mass of the segment is m=pbs, where p gostota (masa na enoto dolžine preje), silo F pa is the mass of yarn per unit length. The force F can Odvijanje preje z navitka - Yarn Unwinding from Packages 79 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 zapišemo kot F =f <5s, kjer je f linearna gostota sile zračnega upora (torej sila zračnega upora na enoto dolžine preje). Zato enačbo (11) delimo z 5s in naredimo limiti proti infinitezimalno kratki dolžini odseka, 5s—»0: T {s + <5,s)k(s + 6 s) be written as F =f <5s, where f is the linear density zr zr zr of the air drag force (the air drag force per unit length). By dividing Equation (11) by 8s and going to the limit of an infinitesimally short segment length, 8s—»0, we obtain: pa = lim i5s->0 Limita v zgornjem izrazu je po definiciji odvod funkcije T(s)k(s) po ločni dolžini s. Končni rezultat, torej gibalna enačba za infinitezimalno kratek odsek preje, je: d T(s)k(s) Ss (12). The limit in this expression is, by definition, the arc-length derivative of the function T(s)k(s). The final result, the equation of motion of an infinitesimally short segment of yarn, is then pa(S)=-(T(S)k(S))+fzr(S) (13). Če uporabimo rezultat za pospešek, (6), jo lahko zapišemo tudi v obliki: Using the expression (6) for the acceleration, the equation of motion can also be put in the form P d2r 2V dk dt V ,2dk\ = d ds I ds Tk +fzr (14). 3 PREHOD V VRTEČI SE VALJNI KOORDINATNI SISTEM 3 TRANSITION TO A ROTATING CYLINDRICAL COORDINATE SYSTEM Pri odvijanju preje z navitka ustvarja preja “balon” (slika 1): preja se z veliko kotno hitrostjo vrti okoli osi Z in v eni periodi oriše rotacijsko telo z enim ali več “trebuhi”. Oblika rotacijskega telesa - balona, se v času ene periode vrtenja spremeni le malo. Celotno gibanje krivulje r(s) lahko torej razstavimo na dve gibanji z različnima značilnima časoma. Prvo gibanje je vrtenje krivulje okoli osi Z in ima kratek značilni čas (reda 2n/(Q, kjer je ro kotna hitrost: to je čas, v katerem se odvije en ovoj niti). Drugo gibanje je spreminjanje oblike balona in ima dolg značilni čas (ta je velikostnega reda časa, v katerem se odvije ena plast). Takšen razcep je primeren predvsem za navitke (oziroma plasti), pri katerih je število ovojev veliko, torej predvsem za natančno navite navitke. Recimo, da ima ena plast navitka okoli 100 ovojev niti. Tedaj se oba značilna časa razlikujeta za dva velikostna reda. Imamo torej opravka z dvema gibanjema na zelo različnih časovnih merilih, zato je smiselno, da takšen razcep izrecno upoštevamo v naših enačbah. Pri navitkih z manjšim številom ovojev je razcep manj uporaben in naloge se je bolje lotiti z neposrednim numeričnim reševanjem enačbe (14), kar pa je izjemno težko. The unwinding yarn forms a “balloon” (Fig. 1): the yarn rotates with a high angular velocity around the Z axis and with one turn it defines the contour of a rotational body with one or several balloons. The shape of the rotational body - the balloon - changes only a little in one period of the motion. The motion of the curve r(s) can therefore be decomposed into two separate motions with two different characteristic times. The first motion is the rotation of the rigid curve around the Z axis. It has a short characteristic time 2jt/a>, where a> is the angular velocity. In this time one loop of the yarn is unwound. The second motion corresponds to the time-varying form of the balloon. This has a long characteristic time, of the order of the time in which one layer of the yarn is unwound from the package. Such a decomposition makes sense only for packages (or layers) where the number of loops in a layer is high, i.e., for precision-wound packages. Let one layer have 100 loops of yarn. Then both characteristic times differ by two orders of magnitude and it is beneficial to take this decomposition explicitly into account in our equation of motion. In packages with a smaller number of loops such a decomposition is less useful and the problem is best approached by directly numerically solving the Equation (14); however, this is a very difficult task. 80 Praček S. - Jakšič D. Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 Sl. 3. Valjni koordinatni sistem Fig. 3. Cylindrical coordinate system Vsaka točka ima lastno trojico osnovnih vektorjev e ,e ,ez. Najprej se iz kartezičnega koordinatnega sistema preselimo v valjnega. Ta je bolj primeren za probleme, v katerih obstaja simetrijska os. Počasi spreminjajoče se rotacijsko telo, balon, dejansko ima takšno simetrijsko os, zato bo obravnava hitrega dela gibanja (vrtenja) v takšnem koordinatnem sistemu lažja. V tem koordinatnem sistemu točko opišemo s koordinatami p (oddaljenost točke od osi z), polarnim kotom f in višino točke z, kakor je prikazano na sliki 3. Spremembo zapišemo z naslednjimi enačbami: To each point there corresponds a different triplet of basis vectors e , e , ez. We first affected the change from a Cartesian to a cylindrical coordinate system. A cylindrical coordinate system is more appropriate for problems that possess a symmetry axis. The slowly deforming rotational body, the balloon, does have such an axis, therefore the fast motion (rotation) can be handled more easily in this coordinate system. In the cylindrical system the position of a point is given by coordinates p (the distance from the Z axis), the polar angle f and the height z, as shown in Fig. 3. The transformation can be expressed using the following equations: = \/ x? + y2,

dej, d(f> ki sta osnovna lastnost valjnih koordinatnih sistemov in ju dobimo, če naredimo infinitezimalno Later we will need the following two relations SI" (18), ep, which are a basic property of cylindrical coordinate systems and can be obtained by performing an infinitesimal Odvijanje preje z navitka - Yarn Unwinding from Packages 81 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 zavrtitev koordinatnega sistema okoli osi z za kot 8f. Z njima lahko tudi dokažemo, da velja: Tu pika nad simbolom označuje parcialni odvod po času, d/dt. Hitro gibanje je vrtenje preje okoli osi z kot celote. To pomeni, da se pri tem gibanju polarni koti vseh točk spremenijo za enak kot na enoto časa. To zapišemo kot: Tu smo predpostavili, da je kotna hitrost vrtenja a, stalna. Poudariti moramo, da je kot f (s,t) polarni kot točke v inercialnem valjnem sistemu (p, f,z), kot 0 (s,t) pa polarni kot točke znotraj vrtečega se koordinatnega sistema (p, f,z). Če točka v vrtečem se sistemu “miruje” (6>je stalen), potem se točka v inercialnem sistemu enakomerno vrti okoli osi z s kotno hitrostjo ca. Če se kotna hitrost s časom spreminja, moramo zgornjo enačbo popraviti in dobimo: Razvidno je, od kod izvirajo posamezni členi: 1. člen v je, kakor je rečeno, relativna hitrost gibanja točke P v vrtečem se koordinatnem sistemu; 2. člen wxr je hitrost kroženja točke P okoli osi Z s trenutno kotno hitrostjo w(t); 3. člen -Vk je hitrost, ki jo ima točka P na preji zaradi tega, ker prejo vlečemo skozi vodilo. rotation of the coordinate system around the z axis. These relations can be used to derive two useful formulas: (19) 4>eP (20). Here the dot above a symbol denotes a partial derivative with respect to time, d/dt. The fast motion corresponds to the rotation of the yarn as a whole around the z axis. In other words, the polar angle of each point on the yarn changes by the same amount per unit time. This can be expressed as: (21). Here we assumed that the angular velocity of rotation, a, is constant. We point out that the angle f (s,t) is the polar angle of the point in the inertial cylindrical system (p, f,z), while the point 9(s,t) is the polar angle of the point in the rotating cylindrical system (p,0,z). If a point is “motionless” in the rotating frame (i.e., if 6 is constant), then such a point rotates in the inertial system with a constant angular velocity co around the z axis. If the angular velocity varies, the previous equation has to be modified to read: The origin of the different terms is fairly clear: 1. the term vr is the relative velocity of the point P in the rotating coordinate system; 2. the term wxr is the velocity of the circular motion of point P around the Z axis with momentary angular velocity w(t); 3. the term - Vk is the velocity of the point P because the yarn is withdrawn through the guide. oep d

Mdt 4>{s,t) = ut + 0(s,t) 4>{s,t)= 0j(T)dT + 6(s,t) (22). Hitrost v =Dr in pospešek a = D2r izračunamo z neposrednim odvajanjem krajevnega vektorja (17). Podroben izračun lahko bralec najde v viru [5], tu pa bomo navedli le končni rezultat. Uvedemo vektor relativne hitrosti vrel = rer + rOef + zez: to je hitrost počasnega (relativnega) gibanja v hitro vrtečem se koordinatnem sistemu. Hitrost točke na preji lahko potem zapišemo kot: Velocity v=Dr and acceleration a = D2r have to be calculated by explicit differentiation of the expression for the radius vector (17). A detailed account of this calculation can be found in Ref [5], here we will only provide the final result. We introduce the relative velocity vector vrel = rer + r^ef + z ez : this is the velocity of the slow (relative) movement within the rapidly rotating coordinate system. The velocity of a point on the yarn can then be expressed as: Dr = vrei + u) x r — Vk (23). and 82 Praček S. - Jakšič D. Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 arel = (r& & Če uvedemo še relativni pospešek (2rd + rd)eg +zez, lahko z njim rq2 )er & Introducing the relative acceleration = (r&& - rq&2 )er + (2r& q& + rq&&)eq + &z& ez we can write zapišemo pospešek kot [5]: the acceleration in the form [5]: = arei+2wxvrei—21/o;xr'+a;x(wxr)+d;xr—2l/vrei'+Vr2r" (24). Tu znak ' pomeni parcialni odvod po ločni dolžini s, torej d/ds. Uvedemo lahko operator totalnega časovnega odvoda D, ki sledi gibanju izbrane točke znotraj vrtečega se koordinatnega sistema: Here the suffix ' denotes the partial derivative with respect to the arc length s, i.e. d/ds. We can introduce a formal operator of the total time derivative D, which follows the motion of the chosen point within the rotating coordinate system: D 9\ V (25). To, da operator sledi gibanju izbrane točke znotraj vrtečega se koordinatnega sistema, pomeni, da pri odvajanju izraza (21) ali (22) po času zanemarimo člen z a. Tako dosežemo, da opravek odvajanja po času deluje znotraj vrtečega se koordinatnega sistema (p, 0,z), namesto v inercialnem sistemu (p, f,z), na kar smo opomnili z oznako (p, 6,z) pri operatorju za odvajanje v izrazu (25). Tako dobimo na primer: Dr = V«,! S2r = arei Dobljena izraza se od ustreznih izrazov za Dr (enačba (3)) in za D2r (enačba (5)) razlikujeta v tem, da se v njima pojavljata relativna hitrost in pospešek namesto absolutne hitrosti in pospeška. Z uporabo operatorja D lahko pospešek zapišemo v krajši obliki: By requiring that the operator follows the motion within the rotating coordinate system we mean that when calculating the time derivative of the expressions (21) or (22) we should not take into account the term in w. In this way we ensure that the time derivation applies within the rotating coordinate system (p,q,z), instead of in the inertial system (p,f,z). As a reminder we write the subscript (p, q,z) after the time derivative in the expression (25). For example, we thus obtain: Vk, 2Vk + V 2r". (26). These expressions differ from the expressions for Dr (Eq. (3)) and D2r (Eq. (5)) in that they involve relative velocity and acceleration in place of their absolute counterparts. Using the D operator we can write the acceleration in a compact form: L>2r + 2ux (2)r) + wx(wxr)+wxr (27). Gibalno enačbo lahko tedaj zapišemo v obliki [5]: Finally, the equation of motion can be put in the following form [5] d p(D"r+2u x Sr+u x (w x r)+w x r) = —(Tk)+fzr (28). To je iskana enačba gibanja preje. Prvi člen na levi pomeni relativni pospešek, podobno kakor običajni drugi odvod po času v inercialnih koordinatnih sistemih. Naslednji trije členi so sistemske (navidezne) sile, ki se pojavijo v neinercialnih vrtečih se sistemih: 1.-yo20) in manjša med odvijanjem v smeri proti prednji strani navitka (f <0). Na obeh robovih navitka se kotna hitrost spremeni dokaj naglo: ko točka odvijanja doseže prednji rob navitka, se zveča, ko doseže zadnji rob navitka, se zmanj ša. Zato sistemska sila -pa x r kaže v smeri Coriolisove sile, ko je točka dviga preje na sprednjem robu navitka, ko se kotna hitrost co povečuje. Vektor kotnega pospeška m je na sliki narisan črtkano. Ko pa je točka dviga preje na zadnjem robu navitka, ta sila kaže v nasprotni smeri od Coriolisove sile, saj se tedaj kotna hitrost co zmanjšuje in kaže vektor kotnega pospeška a v nasprotni smeri kakor na sprednjem robu navitka. Na sredini navitka, ko so razmere navidezno ustaljene in se s časom spreminjajo le počasi, te sile ni. (To seveda velja le za plasti z velikim številom ovojev, torej za natančno navite navitke.) Omenjena sila ima vpliv na gibanje preje na robovih navitka, kjer se kot navijanja obrne. Tako hitra sprememba kota navijanja povzroči naglo spremembo kotne hitrosti a>, kar pomeni, da je kotni pospešek a velik. Zato je tudi sistemska sila -paxr velika in spremeni dinamiko preje. Iz navidezno ustaljenih razmer pridemo tedaj v prehodno območje, ko se gibanje preje naglo spreminja. Na robovih lahko zato prihaja do nestabilnosti v obliki balona, nitka se lahko zagozdi in pretrga. 2. -pa>x(a>xr) is directed radially outward: this is the well-known centrifugal force. -paxr describes a system force in rotating frames, where the angular velocity changes with time (on Earth this force is negligible). The system forces that act on a short segment of yarn are shown in Fig. (4). The centrifugal and Coriolis forces are well known; however, we would like to emphasize the role of the system force -pa x r, which was not taken into account in the literature, to the best of our knowledge. Typical cylindrical packages consist of layers of yarn with an alternating winding angle f: in one layer this angle is positive and in the next layer it is negative [6]. The steady-state angular velocity co depends on the winding angle, and in a simple approximation it is given by [7]: (29). The angular velocity is approximately constant in the middle of the package: it is higher during unwinding toward the rear end of the package (f >0) and lower during unwinding toward the front end of the package (f <0). It changes rather abruptly at the edges of the package: it increases when the unwinding point reaches the front end of the package and it decreases when it reaches the rear end of the package. Therefore, the system force -pa x r directed in the same direction as the Coriolis force when the lift-off point is near the front edge of the package, where the angular velocity co is increasing. The angular acceleration vector co is depicted in the figure by a dashed arrow. On the other hand, when the lift-off point is on the rear edge of the package, the direction of this force is opposite to the direction of the Coriolis force since the angular velocity co is then decreasing and the angular acceleration vector co has the opposite direction as it has on the front edge of the package. In the middle part of the package, where the conditions are quasi-stationary and hardly change with time, this force is not present (This is of course only true for layers with a large number of loops, i.e., for precision-wound packages.) This force exerts an influence on the yarn motion near the edges of the package, where the winding angle changes. Such a sudden change of the winding angle leads to a rapid change of the angular velocity co, which means that the angular acceleration a is high. For this reason the system force -pa x r is also substantial and it can modify the dynamics of the yarn motion Near the edges the quasi-stationary state changes to a transient state when the yarn motion is changing rapidly. This can lead to instability of the balloon form, the yarn can jam and then break V cos (p c 1 — sin (S> 84 Praček S. - Jakšič D. Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 CoriolisWa. Coriolis sredobežna centrifuga! p U> X T iSota sredobežna is \ N^ centrifuga! a) Sprednji rob a) Front edge h) Zadnji rob h) Rear edge Sl. 4. Sistemske sile na kratek odsek preje Fig. 4. System forces acting on a segment of the yarn Na kratek odsek preje delujejo sredobežna, Centrifugal, Coriolis and -pebxr forces act Coriolisova sila in sila -pd> x r. on a segment of the yarn. 4 POGOJ NERAZTEZNOSTI 4 CONDITION FOR INEXTENSIBILITY Vzeli smo, da je preja neraztezna, s čimer smo imeli v mislih, da smemo zanemariti elastični raztezek preje. Poglejmo, kaj to pomeni z matematičnega vidika. Izberimo si dve bližnji točki na preji, A s parametrom s in B s parametrom s+ds. Ker je parameter s ločna dolžina od izhodišča koordinatnega sistema, ki smo ga postavili na vodilo, je dolžina preje med točkama A in B kar ds. Točki A in B povezuje vektor dr=rA-rB, tako da je razdalja med točkama A in B enaka dr|. V limiti ds—>0 je razdalja med točkama enaka dolžini preje, ki povezuje točki, zato velja: We have assumed that the yarn is inextensible in the sense that we can neglect the elastic elongation in the yarn. We now show what this means from a mathematical point of view. We choose two nearby points on the yarn: A with parameter s and B with parameter s+ds. As the parameter s is the arc-length from the origin of the coordinate system, chosen in the guide, the length of yarn between points A and B is ds. Points A and B can be joined by a vector dr=rA-rB so that the distance between points A and B equals |dr|. In the limit ds—»0 the distance between the points is equal to the length of yarn between the points, so that: \Sr\ = Ss (30). To lahko zapišemo tudi kot |3r/3s|=1, s pomočjo zveze x-x=|x||x| pa dobimo izraz [8]: This can also be written as |3r/ds|=1. Using the relation x-x=|x||x| we obtain the expression [8]: r' • r' = 1 (31). Odvajamo izraz (16) po parametru s: We calculate the arc-length derivative of expression (16): pep de d(t> p'ep + p(j)'e,i, + z'ez, (32) Odvijanje preje z navitka - Yarn Unwinding from Packages 85 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 in izračunamo skalarni produkt, iz enačbe(31) pa dobimo iskani pogoj nerazteznosti: 5 NUMERIČNO SIMULIRANJE ODVIJANJA PREJE Tri komponente vektorske enačbe (28) in skalarna enačba (34) skupaj sestavljajo sistem štirih nelinearnih diferencialnih enačb za štiri neznane funkcije: p,6,z, ki opisujejo obliko in gibanje preje, ter napetost T v preji. Problem bo popolnoma določen, če podamo še robne in začetne pogoje. Pogosto je preja gosto vzporedno navita na navitkih. Tedaj se (v vrtečem se valjnem opazovalnem sistemu) razmere le malo spremenijo v času odvijanja enega navoja in smemo uporabiti navidezno ustaljeni približek. V tem primeru časovne odvode v gibalni enačbi zanemarimo, vso časovno odvisnost pa prenesemo na robne pogoje (ker se točka dviga počasi premika po navitku). Začetnih pogojev v tem primeru sploh ne potrebujemo. Prvi robni pogoj je, da gre preja skozi vodilo v izhodišču, kar zapišemo kot r(s=0)=0, ali p(0)=0, #(0)=0, z(0)=0. V resnici si lahko robni pogoj za 6 izberemo poljubno, vendar je 9=0 najbolj praktična izbira. V točki dviga mora preja biti zvezna in ne sme biti prelomljena. Od tod sledita pogoja o zveznosti: Z indeksoma + oziroma - tukaj označujemo točko tik za, oziroma tik pred točko dviga. Nazadnje velja še, da je preja v točki dviga tangenta na navitek, kar zapišemo kot: Gibalne enačbe moramo integrirati numerično. Pri tem uporabljamo strelsko metodo, then we calculate the scalar product in Eq. (31). We obtain the inextensibility condition in the form: 5 NUMERICAL SIMULATION OF YARN UNWINDING The three components of the vector equation Eq. (28) and the scalar equation Eq. (34) constitute a system of four nonlinear differential equations for four unknown functions: p,q,z, that describe the form and motion of the yarn and the tension T. The problem will be fully defined if the boundary and initial conditions are known. The yarn is often densely wound in parallel on the package. In this case the conditions (as observed in the rotating cylindrical system) hardly change in the time required for unwinding one loop and the quasi-stationary approximation applies. In this case we can neglect all the time derivatives in the equation of motion and transfer the time dependence to the changing boundary conditions (because the lift-off point slowly moves on the surface of the package). Knowing the initial conditions is not necessary since we have reduced the calculation to a boundary-value problem. The first boundary condition is given by the fact that the yarn is withdrawn through the guide, which can be mathematically expressed as r(s=0)=0, or equivalently p(0)=0, q (0)=0, z(0)=0. In fact the boundary condition for q can be arbitrary, but we choose q (0)=0 for convenience. In the lift-off point the curve has to be continuous with a continuous first derivative. This gives two conditions of continuity: (35). Indices + and - denote a point on the yarn just before and just after the lift-off point. Finally, we take into account that at the lift-off point the yarn is tangential to the package, which gives: (36). The equations of motion have to be integrated numerically. For this we use the shooting method that we r'.r' = (p') 2 +PW + (^) =1 (33). Če izračunamo odvod po ločni dolžini By calculating the arc-length derivative of enačbe(21), dobimo '=&, zato smemo tudi zapisati: Equation (21) we obtain '=&, so we can also write r'.r' = (p')2 +pW + (V) 2 = l (34) r0 = s5v) = r(s = sDv) in r (s = s5v) = r 0 = 4v)¦ P (s = SDv) = 0. 86 Praček S. - Jakšič D. Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 ki jo bomo sedaj opisali. V izhodišču s=0 si izberemo začetne približke za odvode p’, f’ in z’ in za napetost T (izkaže se, da te štiri količine niso med seboj neodvisne, zato zadostuje, da si izberemo le dve, na primer p’ in T). Nato gibalne enačbe integriramo, dokler ne zadenemo navitka; ustavimo se lahko na primer tedaj, ko je koordinata z enaka koordinati z točke dviga. Strel je uspešen, če v končni točki v okviru vnaprej izbrane natančnosti velja p(sDv )=c in p’(sDv )=0, kjer je c polmer valja. Če nismo “zadeli”, moramo račun ponoviti pri ustrezno popravljenih začetnih vrednostih za r’ in T. Numerični postopek smo izvedli z uporabo numeričnih rutin iz zbirke numeričnih napotkov [9]. Diferencialne enačbe smo integrirali z Runge-Kuttovo metodo, pri streljanju pa smo uporabili Powellovo metodo. Pri iskanju optimalne oblike navitka moramo določiti ne le obliko balona, temveč moramo rešiti še problem drsenja preje po navitku, ki ga tu nismo opisali. Zaradi oprijemanja preje in zaostale napetosti preje v navitku kratek odsek preje drsi po navitku in prihaja do trenja, namesto da bi se preja vzdignila v balon takoj v točki odvijanja. V drsečem delu preje zaostala napetost pade na vrednost napetosti v balonu v točki dviga. Tudi ta naloga se prevede na reševanje sistema diferencialnih enačb s streljanjem, rešitve pa nato zlepimo v točki dviga z uporabo zgoraj zapisanih pogojev o zveznosti. Oblika navitka določa robne pogoje v točki odvijanja (Od na sliki 1). Optimiranje navitkov poteka tako, da ponavljamo celoten račun za različne oblike navitkov in iščemo takšno obliko, ki vodi k najmanjši napetosti v preji. Metoda optimiranja je najbolj učinkovita za natančno navite navitke z gostim navitjem, pri katerih imajo prehodni pojavi na robovih navitka majhen učinek na celotno dinamiko odvijanja. 6 PRIMER IZRAČUNA Na sliki 5 sta prikazana dva pogleda na “balon”, ki smo ga dobili po zgoraj opisanem numeričnem postopku. Slika ustreza poenostavljenemu računu, pri katerem nismo upoštevali drsenja preje po navitku, v polni meri pa so upoštevani vplivi sredobežne in Coriolisove sile ter zračnega upora. Takšen izračun je odvisen od now describe. At the origin s=0 we choose starting approximations for the derivatives p’, f’, z’ and for the tension T (it turns out that these four quantities are not mutually independent and we only need to set two quantities, for example p’ and T). We integrate the differential equations until we “hit” the package; the stopping condition can, for example, be that the current coordinate z is equal to the coordinate z of the lift-off point A shot is successful if at the final point the equations p(sDv)=c (c is the package radius) and p(sDv)=0 are fulfilled within some predetermined numerical accuracy. If we “missed”, we need to repeat the calculation for suitably modified starting values of p’ and T. We implemented the numerical procedure using numerical routines from the Numerical Recipes library [9]. The differential equations are integrated using the Runge-Kutta method and the shooting is done using the Powell method. For optimizing the package construction we have to determine not only the shape of the balloon, but also the sliding motion of the yarn on the surface of the package, which we have not described in this paper. Due to stiction and the residual tension of the yarn in the package, a short segment of yarn slides on the surface of the package and it rubs against it, instead of immediately lifting off in the balloon at the unwinding point. In this part of the yarn the residual tension of the yarn in the package is reduced to the value of tension in the balloon at the lift-off point. The problem of sliding motion can also be reduced to solving a system of differential equations using the shooting method The solutions are then glued together at the lift-off point using the conditions of the continuity that we described, while the construction of the package determines the boundary conditions at the unwinding point (Od on Fig. 1). The process of optimization involves repeating the calculations for different package designs and searching for the design that gives the least possible tension in the yarn. The optimisation method is most efficient for precision-wound packages with dense layers, where the transient effects at the package edges have a small effect on the overall dynamics of the unwinding process. 6 EXAMPLE OF A CALCULATION Figure 5 represents two views of the “balloon” that we calculated using the numerical methods described above. The figure corresponds to a simplified calculation which does not take into account the sliding motion of yarn on the surface of the package. It does, however, fully take into account the effects of the centrifugal and Coriolis forces, as well as the effect of the air drag. Such Odvijanje preje z navitka - Yarn Unwinding from Packages 87 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 Sl. 5. Dva pogleda na “balon”: črna krivulja pomeni trenutno lego preje, siva ploskev pa je rotacijska ploskev, ki jo preja orise v eni periodi vrtenja okoli osi. Obe sliki sta zaradi boljšega prikaza skrčeni za faktor štiri vzdolž osi vrtenja. Fig. 5. Two views of the “balloon”: the black curve shows the current position of the yarn, while the gray surface is the surface of revolution, that the yarn generates in one periode of its rotational motion around the axis. Both figures are scaled with a ratio of one fourth in the direction of the axis for reasons of clarity. enega samega brezrazsežnega parametra (p , brezrazsežnega koeficienta zračnega upora) in o o d enega robnega pogoja (navpične razdalje zDv, na kateri leži točka dviga). Parameter po je enak [8]: a calculation depends on a single dimensionless parameter (po, the dimensionless coefficient of air friction) and on one boundary condition (the vertical distance zDv to the lift-off point). The parameter po is [8] p0 = 8cdcu rzrak / mpr (37), kjer so: c polmer navitka, d premer preje, c koeficient zračnega upora, p k gostota zraka in M u j linearna gostota preje. Izbrali smo si parameter po =4 in razdaljo z=12. Dobljena rotacijska ploskev (“balon”) ima trebuh, ki nastane zaradi sredobežne sile. Kakovostno podobno sliko bi dobili tudi z uporabo preprostega modela, v katerem zanemarimo Coriolisovo silo in zračni upor. Učinek teh sil pa je v resnici znaten, kar je razvidno iz oblike krivulje, ki pomeni trenutno sliko preje. Kot 9 se v spodnjem delu krivulje močno spremeni in krivulja se ovija okoli balona. Račun, pri katerem Coriolisove sile ne bi upoštevali, bi dal krivuljo, ki leži v ravnini (6=konst). Tako bi podcenili dolžino preje, ki ustvarja balon, velika pa bi bila tudi napaka v izračunani napetosti T. 7 SKLEP Izpeljali in utemeljili smo sistem parcialnih diferencialnih enačb, ki opisuje gibanje preje. Enačbe veljajo za gibanje preje med poljubnim tekstilnim postopkom in so povsem splošne. V drugem delu smo se osredotočili na odvijanje preje z vzdolžnega where c is the radius of the package, d the diameter of yarn, cu the coefficient of air friction, rair the density of air and myarn the linear density of the yarn. We chose po =4 and z=12. The surface of revolution thus obtained (the “balloon”) has a belly-shaped protrusion due to the centrifugal force. A qualitatively similar picture could be obtained using a simpler model that neglects the Coriolis force and the air drag. Nevertheless, the effect of these forces is significant, as one can see from the form of the curve that represents the snap-shot of the yarn in motion. The angle q undergoes a rapid change in the bottom part, where the curve spirals around the balloon. A calculation which does not take into account the Coriolis force and the air drag would give a plane curve (q =const). In this case the length of the yarn would be underestimated and the error in the calculated value of the tension T would also be significant. 7 CONCLUSION We have derived and justified a system of partial differential equations that describes the motion of the yarn. The derived equations are general and can be used to describe the yarn motion in any textile process. In the second part we focused on the over-end 88 Praček S. - Jakšič D. Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 74-89 navitka in zapisali enačbe v vrtečem se koordinatnem sistemu, ki je bolj primeren za nadaljnjo obravnavo pri dejanskih primerih. Enačbe lahko na primer zapišemo v navidezno ustaljenem približku in jih rešimo numerično s strelsko metodo. V tem primeru se geometrijska oblika in način navitja preje na navitek kaže samo v robnih pogojih, zato je reševanje preprosto. Na ta način lahko izračunamo napetost v preji za poljubno zamišljene navitke, kar je v veliko pomoč pri iskanju navitka optimalne oblike za izbran tekstilni postopek. ZAHVALA Za strokovne nasvete in pomoč se avtorja zahvaljujeta izr. prof. Viliju Bukovšku. Zahvaljujeva se tudi doc. dr. Blažu Vratanarju s Fakultete za gradbeništvo in geodezijo, ki je tragično preminil. unwinding of yarn from an axial package and we cast the equations in a form that is suitable for solving real problems by transforming them to a rotating coordinate system. The equations can be simplified using the quasi-stationary approximation and solved using the shooting method. In this case the geometry of the package and the type of winding appear in the boundary conditions and numerical solving is tractable. In this manner one can calculate the yarn tension for an arbitrary package design, which is helpful in optimizing the package shape for a chosen textile process. ACKNOWLEDGMENT The authors wish to acknowledge assist. prof. Vili Bukovšek for technical advice and help. We also obliged to doc. dr. Blaž Vratanar from Faculty for Civil and Geodetic Engineering, who tragically deceased. 9 LITERATURA 9 REFERENCES [1] Fraser, W.B. (1992) The effect of yarn elasticity on an unwinding ballon. J. Text. Inst, 83 603-613. [2] Kuščer, I., A. Kodre (1994) Matematika v fiziki in tehniki. DMFA. [3] Praček, S., D. Jakšič (2002) Teorija odvijanja preje z navitka - Robni pogoji in sila zračnega upora. Tekstilec, 45(7-8) 175. [4] Roberson, J.A., C.T. Crowe (1980) Engineering fluid dynamics. Houghton Mifflin Company, Boston, druga izdaja. [5] Praček, S., D. Jakšič (2002) Teorija odvijanja preje z navitka - Izpeljava gibalnih enačb. Tekstilec, 45(5-6) 119. [6] De Barr, A.E., H. Catling (1976) Manual of Cotton Spinning, Volume Five. Butterworth. [7] Praček, S. (2002) Modifikacija dinamike odvijanja preje. Doktorsko delo, Univerza v Ljubljani, Naravoslovnotehniška fakulteta, Oddelek za tekstilstvo. [8] Fraser, W.B., T.K. Ghosh, S.K. Batra (1992) On unwinding yarn from cylindrical package. Proc. R. Soc. Lond. A, 436 479-498. [9] Press, W.H., S.A. Teukolsky, W.T. Vetterling, B.P Flannery (1992) Numerical recipes in C: The art of scientific computing. Cambridge University Press, druga izdaja. Naslov avtorjev: dr. Stanislav Praček prof.dr. Danilo Jakšič Naravoslovnotehniška fakulteta Univerze v Ljubljani Aškerčeva 12 1000 Ljubljana stane.pracek@ntftex.uni-lj.si danilo.jaksic@guest.arnes.si Authors’ Address: Dr. Stanislav Praček Prof.Dr. Danilo Jakšič Naravoslovnotehniška fakulteta Univerze v Ljubljani Aškerčeva 12 1000 Ljubljana stane.pracek@ntftex.uni-lj.si danilo.jaksic@guest.arnes.si Prejeto: Received: 19.5.2003 Sprejeto: Accepted: 2.12.2004 Odprto za diskusijo: 1 leto Open for discussion: 1 year Odvijanje preje z navitka - Yarn Unwinding from Packages 89 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 90-94 UDK - UDC 681 892:534 Pregledni znanstveni članek - Review scientific paper (1.02) Izrazi za popis upogibnega nihanja palice nespremenljivega prereza Equations for the Flexural Vibration of a Sample with a Uniform Cross-Section Igor Štubna - Anton Trnik V prispevku je predstavljen kratek pregled že znanih izrazov za popis upogibnega nihanja, uporabljenih za določitev Youngovega modula in hitrosti zvoka. Predstavljen je tudi nov izraz, ki velja za vztrajnost kroženja in vpliv strižnih sil z izrazom i2[2(1 + m) / k](84 y / 8t28x2), v katerem je iz polmer vrtenja prereza, m je Poissonovo razmerje in k je oblikovni faktor, ki ga je uvedel Timošenko. Krivulje porazdelitve kažejo zelo dobro ujemanje splošno uporabljanega Timošenkovega izraza in novega izraza, ki sta ga razvila Štubha in Majernik. © 2005 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: upogibno nihanje, enačbe diferencialne, izraz Timošenkov, momenti upogibni) A short review of the known equations of flexural vibration used for determining the Young’s modulus and sound velocity is presented, as well as a new equation that accounts for the rotary inertia and the influence of the shear forces with the term i2[2(1 + m) / k](84y /8t28x2), where iz is the radius of gyration of the cross-section, m is Poisson’s ratio, and k is the shape coefficient introduced by Timoshenko. The dispersion curves show a very good fit between the commonly accepted Timoshenko’s equation and the new equation derived by Štubha and Majernik. © 2005 Journal of Mechanical Engineering. All rights reserved. (Keywords: flexural vibration, partial differential equation, Timoshenko’s equation, bending moments) 0 INTRODUCTION The most convenient type of vibration used for measurement is a flexural vibration. It is easy to excite it, and the magnitude of the vibration is sufficiently high. The resonant frequency of the flexural vibration is smaller than the resonant frequency of the longitudinal or torsional vibration of a sample of the same length and cross-section. These properties of flexural vibration make it preferable for measuring the elastic modulus (or velocity of sound propagation) at elevated temperatures. The theory of the flexural vibration of prisms and rods is based on deriving and then solving a partial differential equation of vibration for the sample. The exact solution of a three-dimensional form of the equation is extremely difficult. Fortunately, the mathematical approach to the solution of the vibration of a sample with a simple and symmetrical form can be simplified, and a reasonably exact solution can be obtained. For this reason, only the vibration of the sample with a simple uniform cross-section (circular or rectangular) serves for a measurement of the elastic parameters of solid materials. In this paper a short review of the equations of flexural vibration commonly used for a determination of the Young’s modulus or sound velocity, as well as the new equation, is presented. 1 THEORY OF FLEXURAL VIBRATION The simplified partial equation of flexural vibration of beams with a uniform cross-section is derived on the basis of the following assumptions ([1] and [2]): a) The amplitude of vibration is small. b) The mass element in the direction of vibration is in equilibrium (see Fig. 1), i.e.: 90 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 90-94 x T Fig. 1. Bending line, forces and moments effecting the mass element pSdx2 = (T + dT)-T = —dx 8t dx (1) where r is the density of the beam material, S is the area of the cross-section, T is the shear force, t is time and x, y are coordinates. c) The equation of the elastic line holds: dx 2 (2) where M is the bending moment, E is the Young’s modulus and J is the moment of inertia of the cross-section around the axis parallel with the z-axis. d)The relationship between the shear force and the bending moment has the form: dM dx = T (3). Eliminating the shear force T from Eq. (1) with the help of Eqs. (2) and (3) we obtain: d t2 dx4 (4) where c0=yE/p is the sound velocity (i.e., the velocity of the longitudinal wave propagation in the sample), iz = *JJ/S is the radius of gyration of the cross-section. Eq. (4) describes the vibrational motion of the sample with a sufficient exactness only when the ratio l/d > 20, where l is the length of the sample and d is the diameter of the cylindrical sample or thickness of the prismatic sample in the direction of vibration. The solution of Eq. (4) is the function: y = ym exp ja\ t± (5) where j = -1, co = 2n c/ l is the angular frequency, c is the phase velocity of the flexural wave and l is the wavelength. Substituting Eq. (5) into Eq. (4) we obtain: 2p (6). In Eq. (4) we anticipated only a displacement motion of the mass element in the direction of the y-axis. In the case of a fundamental mode vibration of a short sample (in which l/d < 20) the rotation of the mass element around the axis parallel with the z-axis must be taken into account. The rotation of the mass element must also be accounted for in the case l/d > 20 when the sample vibrates at a higher mode because the sample is divided into short parts by knots. The rotary motion of the mass element is described as (see Fig. 1): d2 (dy rJdx dt2 {dx Tdx + M -(M + dM ) =Tdx dM dx (7). dx If we derive Eq. (7) according to x and eliminate T and M by means of Eqs. (1) and (2) we obtain an equation that includes the Rayleigh’s correction (see e.g., [3]): d2y dt2 +c02iz2 d4y dx4 d4y dt2dx2 0 (8). Izrazi za popis upogibnega nihanja - Equations for the Flexural Vibration 91 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 90-94 Substituting Eq. (5) into Eq. (8) we obtain: c c 2n we obtain Timoshenko’s equation [1] by the sequential elimination of the values of M, T, y and c: 1 + 4tt2| z (9). d2y 2-2^y i 2 (1 ) ^y Kcs dt (13) As we can see from Fig. 2, the curves of the functions (6) and (9) correspond to the curve of function (15) only for a long wavelength. Another step in the agreement between theory and experiment was made by Timoshenko [1], who proposed a correction for the effect of shear forces. Timoshenko made a hypothesis according to which the angle between the tangent to the elastic line and the x-axis is the sum: where cs=JGTp and p = 2(1 + h)/k, and where ju = (E/2G)-1 is Poisson’s ratio. Timoshenko’s equation describes the flexural vibration of the sample with a circular or square cross-section very well and in accordance with experimental results. For samples with a different form of cross-section Pickett proposed equation [4]: dx = V + X (10) ^y 22 0, it is necessary to change the coefficient from (1+p) in Eq. (17) to p. Then: lim—= lim 2n(iz ) l->0 l->0 1 + 4n2p(iz /l)2 (18), as can be seen in Fig. 2. After substituting p into Eq. (16) we obtain the equation: 1 (19) c = 2n\i 1+4^l which gives a result very close to the curve of Eq. (15), see Fig. 2. We obtain the equation for phase velocity (19) from the new equation derived by Štubna and Majernfk [6]: d y 22 d y 2 d y — + ci —- ip------- dt2 0zc?x4 z dx2dt2 which we obtain in the same way as Eq. (16) by using p instead of (1+p). The solution for the differential equation of flexural vibration (20) can also be written in the form of a function of the type: y(x, y) = Y(x)Q(t) = [a sinh ax + f3 cosh ax +y sin bx + S cos bx] exp(jcot) (21) where: a = CO c0 -+ +ti b= c0 + 2 + f4~ + {iz 0>) The values for the bending moment and the shear force are: M = -EJ ~d2Y co2' _dx + Yp c0_ exp(jcot) (23) T = -EJ d3Y dY dx 3 dx co2 c0. exp(jcot) (20) which together with the solution of Eq. (21) and its derivation with respect to x make it possible to compile the frequency equation for given boundary conditions. 2 CONCLUSION The simplified Eq. (4) suffices for flexural waves with a long wavelength (i/l < 0.03). For this case Eqs. (8) and (16) give identical results, but they are more complicated. For flexural waves with a shorter wavelength (i/l > 0.03) Eq. (13) or Eq. (20) must be used. The dispersion curves show a very good agreement between the commonly accepted Timoshenko’s equation (13) and the new equation (20) derived by Štubna and Majernik. Acknowledgement This work was supported by grant VEGA 1/0279/03. Izrazi za popis upogibnega nihanja - Equations for the Flexural Vibration 93 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 90-94 3 REFERENCES [1] Timoshenko, S. P. (1955) Vibration problems in engineering, D. Van Nostrand Co., New York 1955 [2] Brepta, R., M.Prokopec (1972) Propagation of mechanical waves and shocks in solid bodies. Academia, Prague. [3] Lamb, H. (1960) The dynamical theory of sound. Gos. izd. fiz-mat literatury, Moskva. [4] Schreiber, E., O. Anderson, N. Soga (1973) Elastic constants and their measurement. McGraw-Hill Book Co., New York. [5] Kuzmenko, V. A. (1962) Zavodskaya Laboratoria, 28, No. 6, 726-731. [6] Štubna, I., V. Majernik (1998) Acustica - Acta Acustica, 84, No. 6, 999-1001. Authors’ Address: Doc.Dr. Igor Štubna Anton Trnik Constantine the Philosopher University Physics Department A Hlinku 1 SK-949 74 Nitra, Slovakia istubna@ukf.sk Prejeto: Sprejeto: Odprto za diskusijo: 1 leto 12.2.2004 2.12.2004 Received: Accepted: Open for discussion: 1 year 94 Štubna I. - Trnik A. Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 95-102 UDK - UDC 621.923:539.3 Pregledni znanstveni članek - Review scientific paper (1.02) Prehodni pojavi pri postopku brušenja Transient Phenomena in the Grinding Process Vladas Vekteris Lokalne stične premike, ki so posledica elastičnih deformacij orodja in obdelovanca, smo proučevali z uporabo končnih elementov in s preskusi. Izdelali smo grafične in analitične prikaze rezalne sile s harmoničnimi in stohastičnimi elementi. Predhodno objavljene raziskave stika med brusilnim kolesom in obdelovancem temeljijo na predpostavki, da predstavljajo stične deformacije neposredno funkcijo normalnih in tangencialnih sil, ki med postopkom brušenja delujejo na kolo oz. njegova zrna, brez upoštevanja obrabe in lomljenja zrn. V prispevku smo, s pomočjo metode končnih elementov in preskusa krožnega polirnega brušenja z velikimi hitrostmi, opisali postopek raziskave in grafično prikazali prehodne pojave in vzorec uničenja brusilnih zrn znotraj stične zone kot posledico impulznih obremenitev. Prej omenjeno metodo lahko uporabimo pri prožnih, togih, plastičnih in drugih nelinearnih materialih, ki jih obdelujemo z brušenjem. Uporabo nelinearnih lastnosti materiala, modul prostornine in modul pomika v stični coni med brusilnim kolesom (vrsta 24A12TICM28K5) in obdelovancem (jeklo 45), smo pri preskusu simulirali z uporabo tri-parametričnih elementov. S predstavljeno metodo lahko izračunamo prehodne napetosti in deformacije med krožnim polirnim brušenjem z velikimi hitrostmi ter proučujemo uničenje brusilnih zrn na dvorazsežnem modelu z nedoločenimi mejami in nelinearlnimi značilnostmi. Namen predstavljene razsikave je določitev vpliva prehodnih pojavov na sestavo rezalne sile med postopkom brušenja. © 2005 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: postopek brušenja, napetosti, deformacije, simuliranje, analize eksperimentalne) Local contact displacements resulting from the elastic deformation of the tool and the blank, were studied using the finite-element and by experiments. Graphical and analytical expressions for the cutting force, with harmonical and stochastic components, were obtained. Previously published research on the behaviour of the contact between the grinding wheel and the workpiece has been based, on the assumption that the contact deformations represent a direct function of both the normal and the tangent forces acting on the wheel or its grains during the grinding process, without taking into account the attrition and breaking of the grains. This paper covers the procedure for researching and graphically representating transient processes and the pattern of the abrasive grains’ destruction within the contact zone under impulse loads, which is based on the method of finite elements and the results of a high-speed circular plunge-grinding experiment. The above-mentioned method can be applied to elastic, inelastic, plastic and other nonlinear materials machined by grinding. To introduce the nonlinear properties of the material in the experiment, the modulus of the volume and the modulus of the shift in the contact zone between the grinding wheel (grade 24A12TICM28K5) and the workpiece (steel 45) are simulated by three-parametric elements. The presented method makes it possible to calculate transient stress and deformations during high-speed circular plunge grinding and to study the destruction of abrasive grains in a two-dimensional medium with indefinite boundaries and nonlinear characteristics. The present research is aimed at finding out the influence of transient phenomena on the structure of the cutting force during the process of grinding. © 2005 Journal of Mechanical Engineering. All rights reserved. (Keywords: grinding process, stress, strain, simulation, experimental analysis) 95 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 95-102 0 INTRODUCTION Factory-wide automation, the increase in machining precision and operational concentration, as well as the intensification of cutting processes, and other important factors for increasing the output and efficiency of adaptive production, constitute the objective rules of the development of technological equipment. On the whole, these trends in the development of industrial production bring forth new problems when developing grinding equipment, particularly of the spindle systems based on the intensification of cutting processes. The intensification of cutting processes is one of the basic methods of scientific and technical progress in the machine-tool building industry. An increase in the grinding speed up to 60 m/s (instead of 30 to 35 m/s) has drastically increased the efficiency of grinding equipment. Nowadays, there are all the necessary grounds for applying grinding speeds od up to 100 to 120 m/s ([1] to [3]). Despite this, a simple increase in the cutting speed by increasing the grinding wheel’s velocity will not produce a tangible effect unless all the grinding system’s reserves are used together, particularly the radial and circular feeds. Reference [4] shows that an increase in the circular feeding velocity of circular grinders is particularly effective when CBN grinding wheels are used. In such cases of high-speed grinding, as well as in cases of normal-velocity grinding, the quality of the work surface increases in proportion to the reduction of the cutting force. A large number of abrasive grains per unit time take part in the metal-cutting process during a high-speed grinding operation. This results in a decrease in the depth of cut-offs per abrasive grain and, consequently, in a lower stress on the grain, thus reducing its rate of wear. At present the relative speed of the tool and the workpiece in metal machining is considered to be in the range from 25 to 500 m/s ([1] to [3]). Information is rather scarce about the phenomena occurring under such heavy-duty velocity and stress conditions. Here, theoretical physical investigations indicate that high-velocity grinding is characterized by the occurrence of the temperature field ([5] and [6]) and the field of forces during the grinding process. At the present time there is a lot of activity to simulate the properties of abrasive tools with a particular grain and cutting-edge microgeometry in order to develop grinding wheels with new structures to operate under n-fold load and allow functional cutting speeds of up to 300-500 m/s ([1] to [3]). However the phenomena that take place during the interaction of the two elements with particular stochastic properties are still insufficiently studied. This includes the characteristics of the force field, generated during high-velocity cutting, and those of the field’s stochastic components. To make use of all the specific advantages of high-speed grinding it is necessary to clearly understand the mechanism of the wheel and workpiece interaction in the contact zone. A number of researchers studied local elastic deformations in the contact zone between the grinding wheel and the workpiece by applying different approaches and methods. Reference [7] provides a review of this research. According to this research the deformation in the contact zone under the effect of normal and tangent forces is determined by the elastic properties of the tool and the elastoplastic properties of the workpiece. The local elastic displacements of the abrasive grains inside an abrasive tool, caused by normal and tangent forces, are transferred to the adjacent grains thtough intergranular contacts (directly or through the binder). The intensity of these displacements depends on the geometry of an abrasive grain, the stress value, the amount and the properties of the intergranular contacts. It is common ([1], [2] and [7]) knowledge that abrasive grains have a random shape and geometry, they are also randomly oriented during the production of the abrasive tool, and the grains differ considerably from each other as regards shape, size, thickness and the number of binding ties ([2] and [7]). Because of this their displacement in the normal direction and the rotation in the tangent direction, resulting from the shock of their interaction with the billet, contributes to the activation of vibration in the cutting zone. The pulse stress waves generated in this zone spread over the material of the abrasive grains and binder ([1] and [2]). For this reason the material particles in the cutting zone vibrate at a very high frequency and produce a certain effect on the system’s state and the chip-grinding process. This paper presents a method for calculating the transient stress and the deformations in the contact zone between the grinding wheel and the workpiece, and the destruction of abrasive grains in the two-dimensional medium with indefinite boundaries and nonlinear characteristics. The 96 Vekteris V. Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 95-102 suggested method and the program designed on the basis thereof ([16] and [17]) were verified with experimental data obtained during the process of grinding a steel workpiece (steel 45) with a grinding wheel (grade 24A12IICM28K5), which made it possible to substantiate the structure of the cutting force. 1 STRESS AND STRAIN In order to understand the mechanism of the grinding processes and to determine the degree of system strain let us discuss local (contact) shifts resulting from the elastic deformation of the tool and workpiece during the penetration of the tool into the workpiece. Let us assume that the tools with determined geometry are not deformed whereas the grinding system undergoes eccentric deformation, though the cutting section later undergoes local thermoelastic deformations. The machining of materials using a tool with a stochastic micro-geometry is associated with local transient deformation at the contact of the interaction and the deformation of the grinding system ([1] and [8]). In the case of a rigid system the contact deformation changes the shape of the interacting bodies ([1], [8] and [9]). If r is the radius of the non-deformed tools, then the change in the curvature determined by the force Fij per unit of width and acting upon the contact will be as follows [10]: --- = % (1) ri ri Cnlk where r is the curvature of the deformed tool; ln is the contact; C is a constant depending on the elastic properties of the tool (Cn =pEi/16(1-v 2 )), where nI is Poisson’s ratio. The dependence of the elasticity modulus on the temperature of a tool with a ceramic binder according to [11] is expressed with the exponential dependence Ei = E0exp(aTT), where E0 is the modulus of elasticity at room temperature (E0=(50...100).103 MPa), Ei is the modulus of elasticity at higher temperatures, aT is a constant dependent on temperature (aT =(3...6)-10-4), and T is the temperature. Then ri, =ri(1 + Fij/Cnlk2), where lk=(1 + 1q *)-Jri,t0~; q* = vi / vj, vi is the speed of the grinding wheel, vj is the speed of the work piece, and t0 is the real value of the depth of cut in one revolution. The transient force field Fij is an unknown parameter in these expressions; it determines the degree of strain in the system. Control of the value of strain also guarantees appropriate control of the elasticity, the vibration resistance and the damping ability of the system. The shaping process strain and its field of forces are determined by the normal and tangent voltages caused by the changing characteristics of the integrating elements (instrument and part). The elastic displacements of abrasive grains at the point of interaction contact were determined by calculation and by experiment [7]. However, neither the strained state [12] nor the beginning of the transient processes with accompanying fracture of abrasive grains under the effect of impulse loads (Fig. 1) were observed. This can be explained by complicacy on account of lots of factors, particularly during the non-linear behavior of the instrument’s and the part’s material. The nonlinear behavior of the material can be simulated by rheological equations on the basis of a three-parametrical model, including the model of Kelvin-Foight and spring (elastic materials, materials with Poisson’s ratio and plastic flow). It is known from [7] that it is the removable layer that possesses the greatest elasticity, which is why it can be modulated as a spring and as a plastic flow of metal, it can also be modulated as a model of Kelvin-Foight. In this case the bulk modulus will be represented by a rigid spring, and the modulus of shear by a dashpot. Then the relation between the stress and the strain can be expressed as follows [13]: s=[C]{} +[C]{} i =1|[C]0 e - {e ( x}dx (2). Where s is the stress; C, C0,and Ci are the matrixes characterising the material properties; x is the integration variable; and e is the strain. The application of the principle of conformity [13] makes it possible to automatically calculate the ratio of the stresses and strains_(i.e., to calculate the values of the matrixes \C , \C0\, and Ci ). The transient grinding process with the resulting ractures of abrasive grains during an impulse load (Fig. 1) belong to the type of problems for which no analytical solutions can be found. Therefore, in this case a widely known method of finite elements ([14] to [17]) to structurally idealize the continuous medium, to evaluate the rigidity of elements through the following node movement and Prehodni pojavi pri postopku brušenja - Transient Phenomena in the Grinding Process 97 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)2, 95-102 ab c d Fig. 1. Basic types of destruction of abrasive grains: a - rotation and wear of grinding grains; b - crack-formation; c - destruction of grains with the separation of large particles; d - pull-out of grains from the binder; 1 - abrasive grain; 2 - binder of the grinding wheell;