D E T E R M I N A T I O N O F T H E S T R E S S E D S T A T E

OF A C O I L OF M A G N E T I C T A P E

]~. S. U m a n s k i i , V . V . K r y u c h k o v , a n d V. A . R a k o v s k i i

UDC 539.3/5:678

Since magnetic tapes a r e widely used for recording information, topical problem is the determinat ion of the s t r e s sed - - s t r a ined state of a coil of magnetic tape. During the winding, radial and c i rcumferent ia l s t r e s s e s a r i s e which may affect the integri ty of the coil, cause distort ion of the recorded s ignals , etc. If these s t r e s se s a re known, the tensioning fo rce during the winding can be p rog rammed so as to obtain uni form- ly distr ibuted s t r e s s e s in all the spi ra ls of the coil of magnetic tape.

We regard the coil as a regula r mult i layered medium formed by the magnetic tape, wound compactly without gaps on a cyl indrical mandrel .

In view of the smal l thickness of a sp i ra l of magnetic tape compared with the outer radius of the coil, we may replace a spi ra l by a thin r ing. Then the distr ibution of the s t r e s s e s in the coil depends only on the running radius.

As is known, a magnetic tape is a composi te body consist ing of a po lymer base and a fe r romagne t ic l ayer applied to it [1]. The behavior of magnetic tapes under longitudinal tension is determined by the elastic proper t ies of the base. However, in compress ion of the film perpendicular to its su r face , the fe r romagnet ic l ayer has a substantial effect on the tape.

Thus, the elast ic proper t ies of the coil a re different radial ly and c i rcumferent ia l ly .

In view of the large number of layers in the coil and also of the above-mentioned considera t ions , the mult i layered medium may be replaced by a cylindrical ly anisotropic solid disk.

During the winding the tape is wound on the cyl indrical mandrel with a tension

T = To (~-) ~ ( a ~ < r , ~ b ) ,

where b, a a re the outer and inner radii of the coil , respect ive ly . element (Fig. 1) we have

d o r O r - - 0 t

d; -~ ~ = 0, (2)

where a r and a t a re the radial and the c i rcumferen t ia l tension in the coil, respect ively . With a view to the elongation of the tape the c i rcumferent ia l s t ra in of the coil is

u To ( r )n et~-~-7--+- ~ a ' (3)

the radial s t ra in is

(1)

F r o m the condition of equilibrium of a wound

e r ~ du /dr , (4)

where u is the radial displacement; 5 is the thickness of the tape; h is its width.

According to Hooke's genera l law for a cyl indrical ly or thotropic disk under conditions of plane s ta te of s t r e s s , E

(~, +'-',,~0; O r ~ 1 ~ ~trVrt

(5)

~t (e t + v,~e,.), f i t - - 1 - - ' ~ t r ' C r t

Kiev Polytechnic Institute, Kiev. Transla ted f rom Problemy Prochnost i , No. 3, pp. 83-85, March,1978. Original ar t ic le submitted December 2, 1976.

332 0 0 3 9 - 2 3 1 6 / 7 8 / 1 0 0 3 - 0 3 : ~ 2 $07.50 �9 1978 Plenum Publishing Corporat ion

Fig. 1 Fig. 1. A wound e lement .

-o,4 7

-O,~

Fig. 2

Fig. 2. S t resses in a coil of magnetic tape at different val - Ues of k.

and the relat ionship

E t / E ~ = ~ t , / v . , (6)

appl ies , where E t and E r a r e the moduli of e las t ic i ty of the coil c i rcumferen t ia l ly and radial ly , respect ively; Utr and Urt a r e Poisson~s ra t ios corresponding to the t r an sv e r sa l s t ra in upon c i rcumferen t ia l and radial s t ra in , respec t ive ly . If we subst i tute Eqs. (3) to (6) into Eq. (2), we obtain

dlu 1 du a~. u To + 7 - - . - f f - - - p - - ~ = ~ - ~ - - t [ ~ z - - v t , ( n + 1)1, dr z

where

~2 =Et/Er.

We wri te the genera l solution of the ctifferential equation (7) in the following form:

To rn+ I ~- -v t r (n + 1)' u = c x r a + c 2 r - ~ + ~ t an (n+ l)Z--[$ s

(7)

(s)

Then

tion.

f o r m

__ _ [ cxr ~'--I clr -(t3+z) Tor n ( n + I) I ]

+ re" (n + 1)" at-~- dS[-fi__--~-tv + I~+ vtr 6hEra" (n+ 1)'--l~' " p j "

(9)

The compliance of the mandre l radial ly is ~; this may be determined e i ther exper imental ly or by calcula- Then the radial displacement of the f i r s t sp i ra l in ~4nding is

u (a) = ~ , . (10)

Since the re a r e no radial s t r e s s e s on the outer boundary of the coil , the boundary conditions assume the

U = [ a , when r~---a;

t~r---~O when r = b .

(11)

Taking into account the conditions of (11) and of Eq. (9), we may write

or~h ! {~n [(_~) e- ' - - (~)~+t ] + (n + 1) [kn(~)~+ ' - - pn]} "

333

L22

~G

~z

0

-~,4 ~- -zt �9

F i g . 3

I~[, kg f/mma

F i g . 4

F i g . 3. E f f e c t of t h e d e g r e e o f a n i s o t r o p y on t h e s t a t e of s t r e s s o f t h e

c o i l .

F i g . 4 . T h e o r e t i c a l ( so l id l i n e s ) and e x p e r i m e n t a l (dots) d i s t r i b u t i o n

of r a d i a l s t r e s s e s in a c o i l of m a g n e t i c t a p e .

t J ~ = To ( n @ l ) ' - - ~ '

T h e p r e s s u r e of t h e c o i l on t h e m a n d r e l i s

ro 1 [ f ~ (k -~+t - - P + t ) + (n + 1) (k~+o+~ - - 1 )], po - - - - - ~ " (n + 1)5 _ ~

w h e r e

(13)

a (n+ 1) + 13~

-~E--t- ( kp+l - - k ~ - l ) + k~+l (fJ + ~'tr) ~ k l - f~ (~ - - ~ t~) a

k ~ b/a; p = r / a .

T h e c a s e when fi2 = (n + 1) 2 i s of i n t e r e s t on ly when f12 = 1, i . e . , when t h e e l a s t i c p r o p e r t i e s of t h e c o i l

a r e t h e s a m e in a l l d i r e c t i o n s , and t h e ~4nding i s e f f e c t e d ~4th c o n s t a n t t e n s i o n T :- T 0. T h e e q u a t i o n s d e s c r i b -

ii N t h e s t a t e of s t r e s s of an i s o t r o p i c (~2 : 1) t a p e wound on a m a n d r e l w~th c o n s t a n t t e n s i o n a r e g i v e n in [2].

If t h e t a p e i s wound oll a r i g i d m a n d r e l w~th c o n s t a n t t e n s i o n n = 0, ~ = 0, we o b t a i n f r o m (12) and (13)

(14)

w h e r e

To I [s (k -~+1 - - k I~+l) -[- k ~ l 11, Po = - --~-- "--ff:--[~-- (15)

k 13+1 (13 + v~) + [3~ - - vt , (16) Qo : k ~- l ([3 -I- vtr) + k ~+l ([3 - - v t , ) "

An analysis of the results obtained indicates that w=ith increasing outer radius of the coil the radial stresses in the coil increase while the level of the circumferential stresses decreases. It is theoretically

334

p o s s i b l e tha t t h e c i r c u m f e r e n t i a l s t r e s s e s change t h e i r s i g n , i . e . , t ha t the r e s p e c t i v e s p i r a l s of t he magne t i c t a p e beg in to fu l f i l l t he r o l e of t he m a n d r e l . F i g u r e 2 shows the change in the c i r c u m f e r e n t i a l and r a d i a l s t r e s s e s in the co i l in d e p e n d e n c e on t h e d i m e n s i o n l e s s p a r a m e t e r k which c h a r a c t e r i z e s t he o u t e r r a d i u s of the wotmd c o i l . T h e c u r v e s of t he s t r e s s d i s t r i b u t i o n in t he co i l w e r e p lo t t ed a c c o r d i n g to Eqs . (14) with/?2 =

4 and ~'tr = 0.20.

The e f fec t of t he a n i s o t r o p y on the s t a t e of s t r e s s of t he co i l can be judged by F ig . 3. The c u r v e s we re p lo t t ed a c c o r d i n g to E q s . (14) wi th k = 1.8 and Utr = 0.20.

F i g u r e 4 shows the c a l c u l a t e d and e x p e r i m e n t a l d a t a ob ta ined in d e t e r m i n i n g the r a d i a l s t r e s s e s in a co i l of m a g n e t i c t a p e 1-4414--6 wound with cons t an t t e n s i o n T~ = 100 g on a r i g i d m a n d r e l with a = 45 ram. The width of the t a p e i s h = 6.25 r a m , i t s t t z ickness i s 5 = 0.032 m m . The modu lus of e l a s t i c i t y of the co i l of m a g - ne t i c t a p e in t he r a d i a l d i r e c t i o n E r was d e t e r m i n e d by c o m p r e s s i n g a s t a c k of r e c t a n g u l a r s p e c i m e n s of the t a p e .

The e x p e r i m e n t s showed tha t f o r t he m a g n e t i c t a p e in q u e s t i o n E r = Eco ~ 110 k g f / m m 2.

The modu lus of e l a s t i c i t y of t h e co i l c i r c u m f e r e n t i a l l y i s equal to the modu lus of e l a s t i c i t y of the t a p e in l ong i t ud ina l t e n s i o n . F o r t he m a g n e t i c t a p e 1-4414--6 E t = E t e n s ~ 440 k g f / m m 2.

On the b a s i s of t he e x p e r i m e n t a l r e s u l t s , t he index of a n i s o t r o p y of a co i l of m a g n e t i c t a p e fo r p u r p o s e s of c a l c u l a t i o n was t a k e n as f12 = 4.

The c o e f f i c i e n t of t r a n s v e r s a l s t r a i n ~'rt was d e t e r m i n e d by c o m p r e s s i n g a s t a c k of r e c t a n g u l a r t a p e s p e c i m e n s and m e a s u r i n g t h e l o n g i t u d i n a l and t r a n s v e r s a l s t r a i n . A c c o r d i n g to the e x p e r i m e n t a l d a t a , " r t = 0.05 and in t h e c a l c u l a t i o n of t he s t a t e of s t r e s s of the co i l of m a g n e t i c t a p e 1-4414--6 the va lue s Vtr = f12, ~r t = 0.20 w e r e t a k e n f o r E q s . (14) and {16).

The r a d i a l s t r e s s e s in the co i l of m a g n e t i c t a p e w e r e found e x p e r i m e n t a l l y by m e a s u r i n g the f o r c e Q r e q u i r e d to pu l l out th in p l a t e s p l a c e d b e t w e e n the s p i r a l s of t h e m a g n e t i c t a p e wt~le winding" i t :

or =Q/2~hc, (17)

where f is the coefficient of friction between the plate and the magnetic tape; c is the width of the plate.

A comparison of the calculated and the experimental data shows that the dependences obtained on the basis of the model of the cylindrically anisotropic disk correspond fairly accurately to the stress distribution

in the coil of magnetic tape.

The equations obtained can therefore be used for assessing the stressed state of the coil, and also for programming the tension during the x~4nding of magnetic tape to obtain the optimum stress distribution in the

coil.

i,

2.

LITERATURE CITED

G. I. Braginskii and E. N. Timofeev, Technology of Magnetic Tapes [in Russian], Khimiya, Leningrad

(i 972). B. V. Yablonskii, "The state of stress of a multilayered structure in winding a tape on a cylinder,"

Prikl. Mekh., 7, No. 2, 130-133 (1971).

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