ballon theory

8/8/2019 ballon theory

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THEORY OF THE SPINNING BALLOONBy C. MACK {Shirley Institute, Manchester)

[Received 7 February 1957.Revise received 6 June 1957]S U M M A R YDifferential equ ations of mo tion are obtained for a flexible uniform string rotatin gwith a c ons tant large angular velocity a bout a fixed axis when the string has a con-st an t velocity along its length and air drag per u nit length is proportional to a fixedpower of the normal component of air velocity and acts in the direction of thiscom ponent. For ste ady sta te curves the equations are solved, a formal solution beingobtained for the case of zero air-drag, and an expansion for finite air-drag.

1. IntroductionTHEEE are many dynamical systems which can be represented to a goodapproximation by the above differential equations and this is especially soin textile processing e.g. in ring spinning, cap spinning , unw inding, rew ind-ing, 'doubling', and so on (and practically all yarns go through several ofsuc h processes). Th e angular velocities are of the order of 10,000 revo lutionsper m inu te and th us only the envelope of the w hirling yarn is seen. This hasth e appea rance of a balloon w hence the nam e spinning balloon is derived.How ever, the actu al curve ado pted by the y arn does no t lie in one plane and,in this respect, the system is more complicated than that of centrifugespinn ing (the theo ry of wh ich ha s been given earlier (1)). Moreover, incentrifuge spinning, air-drag is essential for stability whereas in thespinning balloon it is usually a minor factor, while the great rotationalvelocities show that gravity can safely be neglected.The properties of the spinning balloon are of considerable importancean d several solutions have been pu t forward based on various and , usu ally,appreciable simplifications of the problem (a review of these has recentlyapp eared (2)). How ever, the equations of motion of the prese nt paper arem ore acc urate a nd more general than any of these, while th e solutionsobtained are exact for zero air-drag (involving only an elliptic integralof the first kind), and, for finite air-drag, the expansions can, by takingsufficient terms, be made as accurate as desired and would seem to covermo st prac tical cases. A simple num erical method of solving the eq uation s isdescribed which gives sufficient accuracy for most purposes and theasy m pto tic solution (which m ost cases appea r to appro ach) is also given.From the differential equations of motion and their solutions the mainpro perties of spinning balloons can be deduced. These are discussed in thefinal section of the paper and illustrated by curves computed from thesolutions.

[Quart. Journ. Mech. and Applied Math., Vol. XI, Pt. 2, 1958]

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THEORY OF THE SPINTNING BAXLOON 1972. Notat ionThe 2-axis is taken as axis of revolution, distance along the curve isdenoted by s, taking dsjdz to be positive, while differentiation w ith respec tto s is denoted by a dot. The velocity of the string along its own pa th (the'wind ing' velocity) is denoted by v and ta ken positive when in the directionof s increasing. The angular velocity is denoted by a> radians per secondand is positive when right-handed about the z-axis. T denotes the tensionin the yarn at a typical point P on the curve having cylindrical coordinates(r, 8, z) with respect to axes rotating with the curve. The mass per unitlength of the string is denoted by m and the air-drag per unit length takento be k times the component of air-velocity norm al W t h e curve (the othercomponent being tangential) raised to the power and to act in the samedirection. Th e values of k and can be obtain ed from the curve of air-dragcoefficient against Reynolds number calculated for a cylinder of the samediameter as the yarn (3).Now the tangent to the curve at P is the vector

(r,rd,z) (1)in the direction of the radius, the perpendicular to the radius and thez-axis, and the z-axis itself respectively. The air velocity at P relative tothe instantaneous curve is, thus,

(0, -ro>, 0). (2)If th e direction of this air velocity mak es an angle y with the tangen t thenit follows from (1) a'nd (2) t h a t

cosy = rd. ' (3)Hence the normal component of air velocity has a magnitude Vn where

Vn = rcosiny = ra>(l r20E)*. ' (4)I t can also be seen tha t the direction of this normal component (which liesin the same plane as the air velocity and tan gen t an d is perpendicular to thelatter) is (frej{\-rWf, - { l - r 2 0 2 } * , z^/{l-r262}*). (5)3 . Equat ions of m o t i o nWe first equate the moment, about the axis, of forces on an element Ssat P to its rate of change of angular momentum. The combined momentof the tensional force and the air-drag (see (4) and (5)) is

{d(TrW)lds-kVir(l-rW)i}8s. (6)The angular momentum of the element is

m 8s[r2a)+vrW]. (7)

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198 C. MACKHence, since the winding velocity v is constant, the rate of change of thismomentum is m8svd[r2co+vr26]lds. (8)Thus we obtain the relation

d{(T-mv2)rW}lds-kVir(l rW)* = mv d{r2co)jds. (9)Integrating, and writing ds = drjf, we find, using (4), that

(Tmv2)r28 = mvcor2+kw^ f {r^+^l r202)*/r} dr. (10)Now, considering forces and acceleration in the z-direction, we have

d(Tz)lds+Wlzr0l(l-r2e2)i = mv d(vi)lds. (11)Integrating, we find that

(Tmv2)z = Kk J {r(lr202)*-zr0/r} rfr (12)where K is a co ns tan t. Again, considering forces and accelerations along thetangen t we f ind that f+moh-r = 0. (13)The most convenient form of the integral of (13) is

Tmv2 = \m w2(h 2 r2) (14)where h is a constant.W e now generalize coordinates and writeR = rjh, Z = z/h, S = sjh, (15)

fj, = 2vl(h)z-l, x = 2KI(mco2h2). (16)However, since dBjdS = r, we shall continue to use the symbol f and, forsimilar reasons, rd and z. The basic equations of motion (14), (10), and (12)then become T-mv2 = mo2h2(l-R2), (17>

(1 R2)rd = fiR+n/R, n=*\ {RUl(lr2d2)^+1)lf} dR, (18)(1 R2)z = x-^> ^ = A J {^(l-r202)1(-1)zr0/r} di?. (19)Now, since f2+rd'2+z2 = 1 (20)it follows thatThus , by integrating any two of the equations (18), (19), and (21), we canobtain a complete solution.The following rearrangement of (21) gives an altern ative form which hascertain useful properties. Now, from (18) we have

2H.n+nzjR 2+2 J (ra2/i?3) dR = 2 J {(n+fiR2)/R 2} dn= 2A J {(l- .R)rftR(lr^Kf+y/f} dR. (22)

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THEORY OF THE SPINNING BALLOON 199Again, from (19), we have

p*-2xp = 2 J (p-x) dP = ~ 2 A J {(l--R2)22-R ?(l-r2fl2)i-1V0/r} dR.(23)Substituting for z2 from (20) in (23), and combining with (22) we find th at(21) becomes

(1J?2)2?12 =( l - ^ 2 ) 2 - / L i 2 i ? 2 - x 2 + 2 J(nZ/R3) dR-q (24)where q = 2A J ri?(l-r202)^-W(l-JR 2)r0 di?. (25)The problem can be reduced to th a t offindingr and r8 from (18) and (24),for z does not appear explicitly in these two equations.4. Zero air-drag solutionWe now solve the equations of motion when k, and hence A, are zero.I t follows from (18) th at n is a constant N, say, and p a constant which canbe taken as zero without loss of generality. Instead of the parameter %it is more convenient to take one of the values of R, Rt say, for which r iszero. It then follows from (18), (19), and (21) that

x 2 = (l-Rlf- +NIRtf (26)and we can write.R 2(l- .fl2)2r2 = i?2( li?2)2-(/xi? 2+iV)2 x 2-B 2= {Rl-R2){A+B(Rl-R2)-(Rl-R*)*} (27)

where A = i^{2(li^)+/x2}iV2/i^, B =ZR\2-M . (28)Hence,.from (19) and (27), we havez\r =dZ\dR =x ^ / [ (^ i - -R 2 )M + -S (^ i - i ? 2 ) - ( -R 2 - - B 2 ) z} ] i- (29)li Z = Z1 when R = i?1, and if we write

R\-R* = Ag\ (30)then (27) becomes i?2(li?2)2r2 =AYH+Bg2-^*). (31)It follows from (30) that Agdg = -RdR; (32)hence, integrating (29) with respect to R, we haveZ = Z1-x[g-Bg3l6+(4:A-B^li0+...]. (33)

It follows from (27) that if A is positive (negative), Rt must be a maximum(minimum), and, hence, from (30) we see that g is always real, while (31)shows th a t we should take g to have th e same sign as r.We can in fact integrate (29) exactly, for writingA+bB-b2 =0 (34)

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200 C. M A C Kwe find that (29) becomes

... . Z = ZiTxiblU+b*)}* j dtl[l-{b*l(A+b*)}Sin*]K (35)However, it is more convenient to express (34) in terms of the minimumvalue of R, Rx say, which occurs when = 0. Hence-Ri-(-Ri)2 =b. (36)It follows from (36) and (28) that

B&BtfiBi+lBtf-Z-tf+N* =0, (37)and (35) becomesZ = ZrfxCi J# / [ l - C { 2 g - ( i S ) > n P , (38)

where C = l / {2+ /x 2 +^- ( i ? i ) 2 } . (39)Thus R^ can be found from (37), and hence C from (39) and,finally,(38)can be eva luated . Alternatively, if the minimum Rx is take n as param eter,M 1 can be found from (37) and then C from (39). Again, by virtue of (32),it will be seen that, if in (26), (27), (28), and (29), R1 is replaced throughoutby R'lt (26), (27), and (29) are still tru e (N .B. A and B, but not x , will hav edifferent values, however).The case N= 0 is of considerable interest, for (37) then shows th a t R^is zero and such curves therefore intersect the axis. In this case, too, wehave from (18) and (19),

dO/dZ = hd'/z = rdl(Rz) = ^Rj^Rx) =Wx- (40)Hence - 9 = (/x/x)Z-\-constant. (41)Non-zero values of N can arise even when the air-drag vanishes. Thenannular 'restraining' rings concentric with the axis, on which the yarnrubs, are used, and, for the balloon curves below such rings, Nwill havenon-zero values. In such cases 6 can only be expressed in terms of as anelliptic integral of the third kind. However, 6is not of much importancefor, though end conditions have to be considered in any given problem,this involves the tangent and hence 6, but not , in general, 0 explicitly.The above shows th at there are three essential parameters fi , N, and eitherRv or Rx, wh ich, when specified, dete rm ine the shape of the balloon cu rve,while the parameter h determines size.5. Finite air-drag solution

Here we have two additional parameters A, (see (16)), and . We nowsolve the equations of motion (18), (19), and (24) by expanding r, rd, z inpowers of g (see (30)). Thus we writei? 2 ( l - i? 2 ) 2 r 2 =A*g*[l+ aig+otig*+...], (42)

n = N-n -n 2-..., p = -

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T HE ORY OF THE SPINNING BALLOON 201where the c , %, and pi are to be determined. In fact nx and p1 can be foundimmediately from the integral expressions for n and p in (18) and (19) and,from (24), we can find ax in terms of nx and px (ifwe use (21) instead of (24)very involved expressions arise containing n2 and p 2, though when all can-cellations are m ade we ultimately obtain the same results). Eq uation (24)also justifies t h e form we ha ve adop ted for r2 in (42) tho ug h th is will becomeapparent later . The expressions for the higher ait nit pi are somewhatlengthy as is usual with non-linear equations but a fairly economical procedure is now given. F ro m (42), (43), (30), and (21) we see t h a t

B2(l W)\\= X

2(l+Pi9+Pz92+-)2m-Ag2)+AY(l+*i9+-)...), ' (44)

where sx = 2Pl, s2 = pl+2p2-AIRl+A2l(xR1)2,

s t =pl+2p1p3+2pl-(AIBl)(Pl+2p2)+Aioc2l(xR1)2,... . (45)Thus we have

i 9+..] i(?-1)/( 12+-), (46)whereh =U^- 3+UC-S)s1{s2+a-5)slll2}]-ita-l)s1AI(l-Rl) (47)Further , it is convenient to writewhere

Pi = -i/2> Pz= 3a2/8-a2/2, j33 = -5a?/16+3ai


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202 C. MACKTo evaluate p it is convenient to write

...), (52)whence

Th en we ha ve from (19), (46), and (52)Plx = {A (xU l)-1(/*^+^)/(i-^)} / (53)whence

(54)I t can be seen from (51) and (54) th a t nx and p x are imm ediately d etermined.We shall now express ax in terms of ^ and px using (24). Now it followsfrom (26), (28), and (30) that

JHence, from (24) and (42), we have

2 J [{n*-N*)lBP\ dR-q\. (55)From (30), (32), and (43), we have

R\ J {{n*-N*)IIP} dR= Rlj[{(n1g+n2g*+...)*-2N(n1g+...)}l(Rl-Ag*)*](-Agdgy= A'[mlg*+mag*+...], (56)

where 3AR*m^= 2^^, AR\m2 = 2Nniv\,5AR\mz = 2Nn3-2n1n2+2Nn1AIRl,... . (57)

Again, from (25), (46), (49), and (52) it follows thatq = 2A J [R*(l B*)*fRl-1(l-ra0a)M-nBrdl{IP(l-&)r}] dR

X

X= -(A*IRl)(q ig3+q2g*+...), (58)

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TH EO RY OF TH E SPINNING BALLOON 203where 39l = 2X(xR1)^^Rl+N)l(l-IP 1)C,

4?2/3?i = !+!, 5q3ft3q1) = * 2+v1a1+v2+AIRl.... (59)Hence, from (55), (56), and (58), we haveax = 2m 1+q1, a2 = B+2m 2+q2, a3 = 2m 3-\-q3{AjRl){2m 1-\-q1),a4 = 2m i+q i-A-(A/Rl)(2m 2+q2), g' where f(g') = 0. However, we canextend the solution in the following manner. Assuming that the turningpoin t of R nex t to i?x is R^ (the zero air-drag value of (37))we can calculate,to a first approximation, n and p by using the expansions (43) first aboutRx and then about R[. Th us we obta in a s first value s for n and p, a t R'x , thevalues Ni and P x say. To obtain a more accurate value of R^ (rememberingt h a t f = 0 at this point) we solve the equation

0 = R2(l i? 2)2 (fj.R2+N1)2R2(xPi)2- (6 2)From the appropriate root R[, say, of this equation, we can expand againto obtain better values of N and P and then repeat the process untilsufficient ac curacy has been ob taine d. This whole process is rep eate d a t th enext turning point of R, and so on, and a com plete solution can t hu s beobtained.W e can, if necessary, find a be tter ap proxim ation t o the ne xt tu rnin g po intthan R[ by solving the equation

l+a 10+ 80 2+af l '+40* = 0, (63)for rapid methods are now available for solving quartics (4).In most cases the first two or three terms of the expansions for n and pgive sufficient accuracy since air drag is usually small compared withcentrifugal force. In an y case th e ma in effect of air drag is to deflect th ecurve until it tends to lie tangentially to the local air velocity, when theair-drag normal com ponent w ill natu rally be sm all; in fact b y analogyw ith calculations m ade in t he special case ft = 0, = 2-0, (5), balloon curvesshould ultimately approach the asymptotic solution,

r6 = 1, r = z = 0, 3R= (^+3)*/x. (64)

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204 C. MA CKHe re the ya rn lies in a circle, the tension exac tly counterbalancing centri-fugal force, and th e normal co mp onent of air velocity is zero.In most pra ctica l cases a num erical solution of sufficient accuracy can besimply calculated b y taking the zero air-drag values of r and rd and, usingthem in the integrands of the integrals of (18) and (19), obtaining a goodapproximation to the correct values of rd and z and , hence, by (20), r. Ifin this process, we replace dR/r by dZjz and zdR/r by dZ we can obtainsecond approximations by repeating, treating the first (zero air-drag)value s of Z as a param eter a nd using the be tter v alues of r, rd, and z.Direct numerical integration of (18) and (19) by modified AdamsBashforth m ethods can be used, bu t the divergence of the expansions in gwhen this is grea ter t ha n a certain value seems to be reflected in large valuesfor higher differences at certain points. Again, further difficulties arise atthe turning points of R for, here, since r is small, loss of significant figuresoccurs upon evalu ating f = {1 r2d2z2}* (while sometimes the 'predicted'values of rd and i are such that (rd)2-\-z2 is greater than unity), and sometrial an d error is needed to obtain consistent values. The curve A = 2-0 ofth e figure was computed this way . I t should be noted, however, tha t, inpractice, thoug h A can be as high as 2-50, the corresponding value of B1 isusu ally less th a n 0-25. Our case A = 2-0, R x = 0-5 was in fact chosen forillustrative purposes.6. Discussion of some balloon propertiesThe main features of spinning balloon curves are of great importanceand we discuss, in this section, some qualitative and semi-quantitativeprop erties which may be deduced from our solutions. These are illustrate din our figure by curves of R against Z and of X against Z when A = 0, 1*0,and 2-0.To pre sen t th e cu rves more realistically we also give the reflection of theR, Z curve. T hus the R, Z curves give the envelope of th e rot atin g balloon,and the X, Z curve the instantaneous curve as seen by an observer at adistance when a steady illumination is provided plus a stroboscopic flashoperating once per revolution at the instant when the tangent to theinstantaneous curve at the apex lies in the plane of the paper. In thecase A = 0, th e X, Z and R, Z curves coincide.

Most spinning balloons start or finish on the axis and the properties ofsuch curves are the most impo rtan t. In th e absence of air-drag suchballoon curves are somewhat sinusoidal in appearance especially if i? x issmall (in fact as R x -* 0 the spinning balloon approaches the case of acircularly vibrating string). When R1 is not small the curves are lesssinusoidal (see th e case A = 0 in th e figure). Th e distance between successive

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THEORY OF THE SPINNING BALLOON 205zero values of R is, for small values of R lt approximately TT/V2 = 2-22... .This can be seen by letting i?x - 0 in (26), (28), (34), an d (35). W hen R x isno t sm all thi s distance is less and , in the case A = 0, R1 = 0-5 of th e figure,the value is just under 1-85.

Small values of air-drag do no t apprec iably alter th e R, Z curve from thezero case except at the points where, in the latter, R = 0. Here R is notzero b ut a finite minimum (which is pro por tiona l to A to a good approxim a-0-5 0 0 5 05 0 0 5 0-5 0 0 5

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\=o A = 10 A=20FIG. 1. Exam ples of spinning balloon curves when = 2-0. The JR, Z curves arecontinuous, the X, Z curves brokention, provided othe r param eters are the same). In the case of curves startingon the axis the second such minimum is approximately twice the first,th e thir d is three times the first and so on, until the minimum becomescomparable w ith the maxima. In the illustrated case, A = 1-0, a t th e firstminimum R = 0157, at the second R = 0-302, while th e correspond ingvalues of Z are 1-82 and 3-51. A t the first m axim um R = 0-500, Z = 0-93,and at the second maximum R = 0-518, Z = 3-74. The distance betweensuccessive minima (or maxima) is slightly smallerthough not greatly

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206 C. MA CKdifferent from the zero air-drag case (as can be seen from these figures.However, more pronounced air-drag (e.g. the case A =20 of the figure)makes these effects m ore m arke d. A t th e first minimum of th e case A =2-0,R =0-291 (just under twice that for A = 1-0) and Z = 1-68. At the nextminimum, however, R is only 0-472 and Z = 2-98. Subsequent maximaand minima are closer together and both approach the asymptotic valueof R (which, in this case of y, = 0, is, by (64), 1/V3 =0-577...). Simul-taneously rd and z approach 1 and zero respectively, and this ten dency canbe observed from th e increasing inclination of the Xo Z curve to the Z-axisand the decreasing values of f.

W hen air-drag a nd winding velocity are zero (i.e. when A =/* =0) butN is finite, the spinning balloon curve has a smaller distance betweensuccessive maxim a (or between m inima, w hich are now finite) for th e sam evalue of R1 th an the zero air-drag case, thoug h it is not easy to dem ons trateth is sim ply. This effect can be qualita tively observed from t h e curvesA = 1-0, 2-0 as Z and, consequently, n increases.The effect of winding velocity may be qualitatively different however.W hen positive it can be seen from (18) th a t it has a similar effect to th a t ofincreasing n. But, in the important case of unwinding, v (and hence fi) isnegative and it tends to reduce the effect of air-drag. It can in fact causechan ges of sign in 6. Usually in unwinding the ya rn is pulled off a sta tio na rycylindrical package through an eyelet on the cylinder axis above thepac ka ge. A t th e balloon apex ( the eyelet itself) since R -* 0, by (18), n -* 0.Therefore for small but non-zero R, rd =fj.R-j-O(R^+1). Hence rd is nega-tive here. At th e point of leaving the package it must, however, be positive.So since n increases as we go down th e curve from the apex it m ust increasesufficiently (i.e. the balloon curve must be sufficiently long or of sufficientradius or both to make 9 positive at the package). Ginzburg (6) has in factobserved tha t 6 initially increases and th en decreases bu t has not given a nyexplan ation of the phenomenon. Provided th at the values of k and arechosen to give the right air-drag at the balloon maximum radius (i? x) nogre at inac curac y will occur if the w rong value of is chosen. Thus, supposekv x and k2, 2 a r e chosen so that

*i *$ = **$ (65)where Vn is the normal air velocity at Rv Then approximately Vn =and we have , . .r , . . ,,kRjt' k^Rjh. (66)Now the main effect of air-drag is embodied in the term nxg (see (51)),w h e r e A f c j y f + v a a B i (67)

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THEORY OF THE SPINNING BALLOON 207But A = 2khftajh)2-t and x 4= (1 -^ I) , if w a n ( i /

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