Strojniški vestnik - Journal of Mechanical Engineering 51(2005)9, 560-569 UDK - UDC 534.01:517.9 Izvirni znanstveni članek - Original scientific paper (1.01) Viskozno dušene prečne vibracije osno gibajoče se strune Viscously Damped Transverse Vibrations of an Axially-Moving String Nikola Jakšič- Miha Boltežar V tem prispevku predstavljamo analizo delovanja viskoznega dušenja na prečna nihanja osno gibajoče se strune. Analizirani linearni model viskoznega dušenja je zapisan v obliki b1wt + b2 vfw. Najprej smo rešili gibalno enačbo lastnega nihanja - linearno parcialno diferencialno enačbo. Nato smo analizirali vplive vrednosti koeficientov viskoznega dušenja b1 in b2 na lastne frekvence in na odziv sistema pri lastnih nihanjih. Pokazali smo, da je potrebno vrednosti koeficientov izbrati previdno, da bi se izognili fizikalno neustreznim odzivom. © 2005 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: enačbe diferencialne, enačbe hiperbolične, nihanja dušena, strune gibajoče) In this paper the linear viscous-damping mechanism acting on an axially-moving string is analyzed. The analyzed damping model is in the form b1wt + b2vfw. The equation of motion, i.e., the linear partial differential equation, of the free, transverse vibrations of the strings span is solved first. Then the influence of the coefficients b1 and b2 on the natural frequencies and the free responses is studied. It was found that the values of the coefficients should be carefully selected in order to avoid physically unrealistic responses. © 2005 Journal of Mechanical Engineering. All rights reserved. (Keywords: partial differential equations, hyperbolic equations, free damped vibrations, moving string systems) 0 UVOD 0 INTRODUCTION Modeliranje osno premikajočih se struktur je deležno nezmanjšane pozornosti že zadnjih 50 let. Pri tem gre za modeliranje verig v verižnih gonilih [1], modeliranje lista pri žagah [2], ali pa modeliranje vej pri jermenskih pogonih [3]. Obsežen pregled modeliranja jermenov in jermenskih pogonov do leta 1992 je podan v prispevku [3]. Problematika osno gibajočega se modela strune je obravnavana v številnih virih, naj izpostavimo le nekatere: [4] do [10]. Prispevek [1] analizira prečna nihanja verig verižnih gonil. Analiza je pokazala, da se amplituda nihanja točke na sredini veje verige poveča, če se poveča osna hitrost potovanja verige, če upoštevamo le dw/dt del modela viskoznega dušenja. Enak pojav sta odkrila avtorja v prispevku [2]. Prispevek [3] ga na kratko povzame. Jermeni, ki so v rabi, so v glavnem zahtevne nekovinske strukture, pri katerih je modeliranje disipacije energije praktično neizogibno. Uporabimo The modelling of moving continua has received constant attention over the past 50 years; studies have included the modelling of the vibrations of transmission chains [1], the modelling of band saws [2], and the modelling of belts [3]. A comprehensive review of the modelling of belts and belt drives up until 1992 is presented in [3]. There are many papers dealing with the problem of the moving string: [4] to [10]. The paper of Mahalingam [1] deals with the transverse vibrations of power-transmission chains. It was reported that the mid-span amplitude of the chain span, when kinematically excited at one end of the span, increases when the chain’s axial velocity increases if only the dw/dt part of the chain’s velocity is taken into account. The same phenomenon is reported in [2] and recapitulated in [3]. The modelling of energy dissipation in the form of viscous damping, or another more sophisticated energy-dissipation or rheological model, is 560 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)9, 560-569 lahko viskozni model disipacije energije ali pa katerega od bolj izpopolnjenih reoloških modelov ali modelov disipacije energije. Modeli, ki popisujejo viskoelastične lastnosti jermenov, se že uporabljajo pri modeliranju vej jermenov ([9] do [14]). Viskozni model dušenja lahko služi kot ekvivalentni model disipacije energije še posebej v primerih kompozitnih struktur. Tu se lahko skriva fizikalno ozadje raziskovanega viskoznega modela. Čeprav so bolj izpopolnjeni materialni modeli že vstopili v domeno modeliranja vej jermenov, pa pojav, opisan v delih [1] in [2], še ni bi deležen večje pozornosti. Slednje je cilj tega prispevka. practically unavoidable when dealing with modern belts. These are mainly complex non-metal structures, where viscoelasticity plays an important role. The modelling of viscoelastic properties is presented in [9] to [14]. Viscous damping can provide a suitable way of equivalent energy-dissipation modelling, par-ticularly when dealing with composite structures, and this can provide a physical background for the viscous model under consideration. Although more sophisticated material models have been introduced for belt modelling, the vis-cous-damping model used in [1] and [2] has not been analyzed in detail. Such an analysis is the aim of this paper. 1 GIBALNA ENAČBA PREČNIH NIHANJ OSNO GIBAJOČE SE STRUNE 1 THE EQUATION OF MOTION OF THE TRANSVERSE VIBRATIONS OF AN AXIALLY-MOVING STRING Prečni odmik strune od statične ravnovesne lege popisuje koordinata w = w(x,t) (sl. 1). Predpostavimo, da se struna osno giblje z osno hitrostjo v = v(t). Diferencial koordinate prečnega pomika podaja enačba (1). Uporabimo jo za izpeljavo hitrosti strune v prečni smeri, enačba (2): The transverse vibrations of a string are repre-sented by the transverse displacement w = w(x,t) (Fig. 1). Let us suppose that the string is moving with an axial velocity v = v(t). The differential of the displacement, Eq. (1), is used to deduce the transverse velocity of the string, Eq. (2): 8w 8w dw-—dx+-----dt 8x dt (1) dw w dw dx 8w dw dw dt dx dt dt dx dt (2). Tako zapišemo diferencialna operatorja: Hence, the differential operators can be deduced as 2 = v+ in/and Dt dx dt Dt Gibalno enačbo prečnih nihanj (4) osno gibajoče se strune izpeljemo z uporabo 2. Newtonovega zakona: 2 a2 a2 a2 .a v —2 + 2v------+ 2 + v— dx dxdt dt dx (3). The equation of motion (4) can be deduced from Newton’s second law for a differentially small section of the string, and written as: ky A,r Sl.1. Osno gibajoča se struna Fig. 1. The axially-moving string Viskozno dušena premikajoča se struna - Viscous Damped Moving String 561 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)9, 560-569 Dt 2 d2w(x,t) Dw(x,t) dx2 Dt (4), kjer popisujejo: r gostoto gradiva strune, A velikost prečnega prereza strune, P natezno obremenitev strune in d koeficient viskoznega dušenja. Leva stran enačbe (4) pomeni vztrajnostno silo v sistemu. Prvi člen na desni strani iste enačbe popisuje aktivno silo zaradi natezanja strune. Drugi člen popisuje silo viskoznega dušenja. Z uporabo diferencialnih operatorjev (3) dobimo gibalno enačbo v obliki: where r is the density of the string material, A is the area of the cross-section of the string, P is the tension force and d is the viscous damping coefficient. The left-hand side of Equation (4) represents the iner-tial forces of the system. The first part of the right-hand side of the same equation stands for the active force due to the string’s tension, and the second part stands for the viscous damping force acting on the string. By considering the differential operators, Eq. (3), the equation of motion can be rewritten as: <92w 2d2w d2w d2w dw P—-rAv —-2rAv-----rA—-d--- dx 2 dx 2 dxdt dt 2 dt dv \dw r A — + dv — dt dx (5). Upoštevaje nespremenljivo osno hitrost strune, dv/dt = 0, ter z deljenjem enačbe s konstanto rA, dobimo gibalno enačbo v obliki: Taking the constant velocity into account, dv/dt = 0, and dividing the equation by rA, the equation becomes: (c2 2 d2w d2w d2w v )—-2v dx dxdt dt Namen tega prispevka je analizirati gibalno enačbo lastnega nihanja osno gibajoče se strune, če imata koeficienta b različne vrednosti. Tako dobimo enačbo, ki nas zanima: dw dw b------bv— dt dx 0 (6). P d — ; b =----- r A r A The aim of this paper is to analyze the equation of motion for different values of b. Hence, the equation of interest is: (c2-v2) d2w dx2 2v d w d w dxdt dt 2 dw dw b------b v— = 0 dt dx (7), kjer so b1 > 0, b2> 0 in 0< v < c, kar predpostavlja podkritično osno hitrost strune. where b1> 0, b2> 0 and 0< v < c applying the subcritical string’s axial velocity. 2 ANALITIČNA REŠITEV GIBALNE ENAČBE PREČNIH NIHANJ OSNO GIBAJOČE SE STRUNE 2 AN ANALYTICAL SOLUTION OF THE EQUA-TION OF MOTION FOR THE TRANSVERSE VI-BRATIONS OF AN AXIALLY-MOVING STRING Enačbo (7) preslikamo v drugo kanonično obliko s preslikavo: z = ax+bt h = gx + dt Če se hočemo znebiti mešanega odvoda v enačbi (7), morajo parametri preslikave zasesti naslednje vrednosti: a = 1, b = 0, g = v/(c2 - v2) in d = 1, preslikavo zapišemo kot: z = x h = xv/(c2 Tako lahko gibalno enačbo zapišemo v drugi kanonični obliki kot: 22 <32wc2 d2w 2 22 -b v Equation (7) can be transformed into the second canonical form by the transformation: ab g #0 (8). To get rid of the second mixed derivative in Eq. (7) the values of the transformation parameters should be a = 1, b = 0, g = v/(c2 - v2) and d = 1, and the transformation can be written as: v2) + t (9). The equation of motion is thus given in the second canonical form as: (c -v2) 2 dz v2 dh2 dw v \dw -----------2—2b2+b1 — = 0 dz [c -v Jdh (10). 562 JakšičN. - Boltežar M. Strojniški vestnik - Journal of Mechanical Engineering 51(2005)9, 560-569 Enačbo (10) rešujemo z Euler-Fourierjevim Equation (10) can be solved by using the nastavkom, ki ločičasovno in krajevno spremenljivko: Euler-Fourier approach of variables separation, w(z,h) = W(z)T(h) (11). Nastavek (11) vnesemo v enačbo (10) in By putting the supposed solution (11) into delimo z nastavkom. Tako lahko enačbo (10) zapišemo (10) and dividing the latter by the solution itself, Eq. kot: (10) can be rewritten as: 22 W" W c2 T (c -v)-------b2v— =------ + W W c 2 -v 2 T —2 b+b c -v T = -w1 (12). In na tej podlagi sestavimo dve navadni And thus two ordinary differential equa- diferencialni enačbi: tions are formed: (c2-v2)W"-b2vW' + w2W = 0 (13) T & + w2T = 0 (14). c T && + v b +b 2 2 2 1 c -v Prvo rešujemo navadno diferencialno The ordinary differential equation (13) should enačbo (13), da bi določili izraz za parameter w. be solved first in order to get the expression for the param- Rešitev predpostavimo v obliki W(z) = Celz in dobimo eter w. The solution is assumed to be in the form W(z) = karakteristični polinom: Celz, and the characteristic polynomial is formed as: 2 b2 v l +-----2 = 0 (15) s koreni with the roots \ 17---------\2----------2---1 1 bv If bv i . w 1 l1,2=7222±Jh22 -42—2 =za %±ig % (16), 2 c - v \ !y c - v ) c -v 2 kjer so a = f2 v / (c 2 - v 2 ) , b = w2/ (c 2 - v 2 ) in where a= bv / (c 2- v 2 ) , b = w2/ (c 2- v 2 ) and g % = Jb % - (a %/2)2 . Izraz pod korenom enačbe za g % je g % = Jb % - (a/2)2 . The expression under the square vedno pozitiven, če le velja v < c. Ta izraz lahko root in g % is non-negative as long as v < C2 This zapišemo tudi kot w2 > ( 22 ) c2v-v2 , njegovo veljavo expression can also be rewritten as w2 > ( b22 ) 2v-v2. pa ocenimo na podlagi enačbe (20). Its validity can be verified on the basis of Eq. (20). Rešitev enačbe (13) je linearna kombinacija The solution of (13) is a linear combination rešitve obeh korenov: of both roots: W(z ) = ea %L/2 (C1 cos g % z + C2 sin g % z ) (17), kjer sta C1 in C2 konstanti. Funkcija (17) mora zadostiti where C1 and C2 are constants. The function (17) robnim pogojem: must satisfy the boundary conditions w(0,t) = 0 => W(0)T(t) = 0 => W(0) = 0 = C1 w(L,t)=0 => W(L)T(Lv/(c2-v2) + t) = 0 => W(L) = 0 = C2ea %L/2sing %L (18), iz česar izhaja: which yields sing % = ^ g %k= L =^ a % ; = 1, , 3,"' (1). Iz enačbe (19) dobimo izraz za wk; k = 1,2,3,... It is straightforward to derive an expression for wk; k = 1,2,3,... from Eq. (19). Viskozno dušena premikajoča se struna - Viscous Damped Moving String 563 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)9, 560-569 4= (c2 -v2) k p v ( b (20). L2 c2 -v2 2 Nabor navadnih diferencialnih enačb (14) With respect to the time function Tk(t), the lahko, glede na funkcijo Tk(t), zapišemo kot: set of the ordinary differential equations (14) can be rewritten as: 2 I 2 - 2 c c (21). Pri reševanju diferencialne enačbe (21) The solution is assumed to be in the form uporabimo nastavek v obliki Tk (h) = Ck eXkh in Tk (h) = Ck ekh , and the characteristic polynomial is dobimo karakteristični polinom: formed as: \+ —(b2-b1)+b1 c 2 2 1 c -v 2 0 s koreni " Ü----------------- with the roots (b2-b1)+b1 2 22 v /7 7 N 7 , 2c -v —(b2-b1)+b1 -4»k—— c c " L V«2-4k (22) (23), kjer sta a = \(b2-b1) + b1 in /? = Če velja: c2 2 2 2 J A> 4 a where a = vc22(b2 -b1) + b1 and/?k = «k c2c-2v2 . If the relation: 2 1c2-v2f v2 2 »k>-----2------2----2b2+b1 , (24), 4 c c -v potem imamo opravka z nihanjem, torej s podkritičnim is satisfied, then vibrations exist and the system is dušenjem. Če pa neenačbi (24) ni zadoščeno, nihanja underdamped. Otherwise, the system is overdamped ni in opredelimo dušenje kot kritično ali nadkritično. or critically damped. 2.1 Podkritično dušenje 2.1 The underdamped system V tem primeru je izrazu (24) zadovoljeno. In this case Expression (24) is satisfied. The Rešitev karakterističnega polinoma (22) zapišemo kot: roots of the characteristic polynomial (22) are: /L =-ac + iy k 2 i k (25), kjer je fk = Jßk-(a/2)2 . Rešitev diferencialne where fk = Jßk-(a/2)2 . The solution of the ordi-enačbe (21) zapišemo kot: nary differential equation (21) can be written as: Tk (h) = e^h/2 (A cos(ykh) + B sin(fkh)) (26). Iz enačbe (26) vidimo, da je parameter fk enak lastni frekvenci dušenega nihanja: It can be seen in Eq. (26) that the parameter yk is equal to the damped natural frequency: adk = Yk k2p2 1c2 L2 4c2 Odziv k-tega načina izpeljemo kot: 12 v2 (b2-b1)x kpx -12 12[v2 (b2-b1)+c2 b1]t wk (x, t) = e c sin-------e c (b2-b1)2-b2 ; k =1,2,3, The k-th mode is as follows: (27). kcos (»dk (-----2 + t) +Bksin a>dk(-----2 + t) xv c -v (28). 564 JakšičN. - Boltežar M. Strojniški vestnik - Journal of Mechanical Engineering 51(2005)9, 560-569 2.2 Kritično dušenje V primeru kritičnega dušenja velja /y 2 2 { 2 \ b = a2/4 oziroma w2 =14cc-2v\c2v-v2b2+b1\ , kar pomeni, da ima karakteristični polinom (22) dve enaki realni rešitvi: 2.2 The critically damped system In this case the equality b = a2/ 4 or w2 = 14 c2-2v2\c-v b2 +bA is established. This means that the characteristic polynomial (22) has two identical roots. li 1v2 2 c 2(b2-b1) + b1 1 „ — a 2 (29). Rešitev diferencialne enačbe (21) zapišemo The solution of the ordinary differential equation (21) can be written as: kot: in odziv k-tega načina kot: T(h) = (A + Bh)e- and the expression for the k-th mode as: wk(x,t) 12 v2(b2-b1)x kpx -12 12[v2(b2-b1)+c2b1]t e c sin-------e c L xv A+B------- + t (30) (31). kk\ 2 2 ' c - v 2.3 Nadkritično dušenje 2.3. The overdamped system V primeru nadkritičnega dušenja se nihanja In the case of the overdamped system the ne pojavijo ker velja bk 0, b2 > 0 in 0 < v < c pričakujemo, da bo model viskoznega dušenja, kot model disipacije energije v sistemu, vplival na lastne frekvence dušenega nihanja strune na način, da se bodo omenjene lastne frekvence zmanjšale ob povečanju vrednosti koeficientov viskoznega dušenja. Druga domneva govori o tem, da se amplituda odziva sistema ne bi smela zmanjševati počasneje s povečanjem osne hitrosti strune. 3 RESULTS AND DISCUSSION Assuming that b1 > 0, b2 > 0 and 0 < v < c, one can expect that the viscous-damping energy-dissipation model would influence the natural frequencies of the vibrations in such a way that the frequencies would not increase with the increasing value of the coefficient of the viscous damping model. The second assumption is that the response’s amplitude should not decrease more rapidly with the string’s increasing axial velocity. Viskozno dušena premikajoča se struna - Viscous Damped Moving String 565 Strojniški vestnik - Journal of Mechanical Engineering 51(2005)9, 560-569 3.1 Vpliv koeficientov viskoznega duŠenja na lastno frekvenco prečnega nihanja strune Nihanje strune s svojo lastno frekvenco se pojavi le pri podkritičnem dušenju. Enačba (27) lastne frekvence dušenega nihanja je sestavljena iz dveh delov. Prvi del je enak lastni frekvenci nedušenega nihanja osno gibajoče se strune. Na drugi del pomembno vpliva viskozno dušenje. Pričakujemo, da se bo lastna frekvenca dušenega nihanja zmanjšala glede na lastno frekvenco nedušenega nihanja, iz česar izhaja matematično formuliran pogoj: 3.1 The influence of the viscous damping coeffi-cients on the natural frequency of the damped transverse string vibrations Let us focus first on the natural frequen-cies whose values should not increase if the damp-ing also increases. It is clear that the expression for the natural frequency (27) is made up of two parts. The first part is actually the undamped natural fre-quency of the moving string, and the second part is influenced by the damping mechanism. Since the damping should decrease the natural frequency the following relation must be satisfied: (b2-b1)2-b^0 (35). Rešitev tega pogoja je ploskev, ki jo podaja neenačba: The solution of relation (35) is a half-plane defined by: b > c+v le ta pa je omejena s premico: and bounded by the straight line: c +v b =k1b2 (36), (37). Koeficient strmine premice k1 doseže The coefficient k1 reaches its minimum najmanjšo vrednost pri v = 0, k1 (v = 0) = 0, in največjo value at v = 0, k1 (v = 0) = 0, and its maximum value in vrednost v mejnem primeru v -> c, k1 (v -> c) = 0,5. the limit case of v -> c, k (v -> c) = 0.5. The border Mejo med nihanjem strune in nadkritičnim between the underdamped and overdamped sys- obnašanjem slednje za vsak način posebej izpeljemo tems’ response for each mode can be deduced from iz enačbe w k4 2 (22b2+b1\ (gl. 2.2).: 1 wk2 =14 c22v{2v2b2+b1 I , see subsection 2.2.: fk (b2 ) k2 c2 -v2 -b2v2 +cb22v2 +\2 — (c2 -v2) (38). Enačba (38) doseže svojo najmanjšo st v točki bk =bk =2k Lp-> asimptotično približuje enačbi (37). vrednost v točki b =b = 2kpc2-v2 in se Equation (38) reaches its minimum at point v2 . It also asymptotically ap- bk =bk2 =2k LpVc2-proaches Eq. (37). 3.2 Vpliv koeficientov viskoznega dušenja na amplitudo odziva strune Druga domneva govori o tem, da se amplituda odziva sistema ne bi smela zmanjševati počasneje s povečanjem osne hitrosti strune v. Izrazi k-tega načina so podani v enačbah (28), (31) in (34). Prvo moramo najti vrh lastne oblike z zanemaritvijo faznega zaostajanja točk vzdolž dolžine strune, ki je skrito v koordinati h, kot primer naj bo enačba (28). Lastno obliko okarakterizira izraz: 3.2 The influence of the viscous damping coefficients on the response amplitude In contrast, we expect that the response’s amplitudes would not decrease more rapidly with the string’s increasing axial velocity, v. The expressions for the k-th mode are given in Equations (28), (31) and (34). The maximum of the mode shape should be found first By neglecting the different phase lags of the different points along the string’s length which are hidden in the coordinate h, e.g., see Eq. (28), the mode shapes are governed primarily by the following expression: 566 JakšičN. - Boltežar M. Strojniški vestnik - Journal of Mechanical Engineering 51(2005)9, 560-569 _ ^(b2-b1)x kpx Wk(x) = e2 2 sin----- ; L k = 1,2,3,... (39), iz katerega lahko sklepamo o legi največje amplitude nihanja strune zaradi koordinate lege v odvisnosti od predznaka izraza b - b1 . Če je b - b1 > 0, sledi, da se vrh pojavi na zadnjem polvalu lastne oblike, ko je xk = (2k–1)L/(2k), ter če je b - b < 0, sledi, da se vrh pojavi na prvem polvalu lastne oblike, ko je x = L/(2k). Amplitudo odziva, glede na enačbe (28), (31) in (34), dominantno opredeljuje enačba: which applies a different shape-maximum position, as the expression b2 - b1 can have different signs. In the case of b - b1 > 0 the shape maximum is found at xk = (2k–1)L/(2k), and in the case of b - b < 0 the shape maximum is found at xk = L/(2k). The response’s amplitude is, according to Equations (28), (31) and (34), governed by the expression: Amp(w(xk, t)) = exp 2c2 (v[b2 -b1]x-[v2 (b2 -b1) + c2b1]t) (40), kjer x lahko zavzame dve vrednosti x = (2k–1)L/(2k) ali xk = L/(2k), glede na predznak izraza b - b1. Eksponentna funkcija je monotona, zato zadošča analiza eksponenta. Zahtevamo, da se eksponent ne povečuje s povečevanjem hitrosti strune, kar matematično zapišemo kot: where x can take only two values, xk = (2k–1)L/(2k) or xk = L/(2k), depending on the sign of b2 – b1. The exponential function is monotonous, so an analysis of the exponent is sufficient. The demand for de-creasing amplitudes with the string’s increasing axial velocity can be mathematically formulated as: dv v[b2 -b1]xk -[v2 (b2 -b1) + c2 b1]t) = (xk - 2vt)(b2 -b1) < 0 (41). Poglejmo si oba primera vrednosti b - b1. Če je b - b1 > 0, sledi x = (2k–1)L/(2k) in xk - 2vt < 0, kar velja za poljuben, dovolj velik t. Dovolj velik čas, ki zadovolji vse načine, dobimo, ko gre k -> oo , je t = L/(2v), kar zadovolji enačbo (41).Če pa je b - b1 < 0, sledita xk = L/(2k) in xk - 2vt > 0, kar pa je mogoče zagotoviti le za nekatere načine in za omejen čas. To pa v splošnem ne velja, saj je lastnih oblik neskončno veliko in ker lahko t zasede poljubne vrednosti. Sledi, da rešitev b - b < 0 ni fizikalno sprejemljiva. Če povzamemo, dobimo sistem treh neenačb, od katerih sta pomembni le prvi dve: b1 > k1 b2 Two different cases are possible again. In the first case, when b - b1 > 0, then x = (2k–1)L/(2k) and xk - 2vt < 0, which is true for sufficiently large t. The largest mode, k -> oo, would give a time large enough, t = L/(2v), to satisfy the condition in Eq. (41). In the second case, when b - b < 0, then xk = L/(2k) and xk - 2vt > 0, which can only be met for some nodes and for a limited amount of time, and cannot be met for all of the nodes. For this reason, the second case is considered to be unrealistic. When summing together all of the constraints: k1 c + v