8. N. V. Smirnov and I. V. Dunin-Barkovskii, A Course in the Theory of Probabilities and Mathematical Statistics for Technical Applications [in Russian], Nauka, Moscow (1965).

9. E. S. Venttsel', The Theory of Probabilities [in Russian], Nauka, Moscow (1969). i0. V. I. Kovpak, "The question of predicting the residual life of metallic materials,"

Probl. Prochn., No. i0, 95-99 (1981). ii. I. L. Mirkin, L. P. Trusov, and R. P. Zaletova, "The micropicture of long-term failure

of alloys of different classes," in: Physical and Chemical Fundamentals of the High Tem- perature Strength of Metallic Materials [in Russian], Izd. Akad. Nauk SSSR, Gor'kii (1971), pp. 17-23.

12. E. I. Krutasova, The Reliability of the Metal of Power Equipment [in Russian], Energoiz- dat, Moscow (1981).

13. Industry Standard 108.901.102--78. Boilers, Turbines, and Piping. Methods of Determin- ing the High Temperature Strength of Metals [in Russian], Introduced January i, 1981.

14. V. K. Adamovich, Ya. F. Fridman, M. B. Revzyuk, and A. V. Stanyukovich, "A comparison of methods of extrapolation of stress--rupture strength," Probl. Prochn., No. ii, 26-29 (1975).

RESONANT FREOUENCIES IN THE TRANSVERSE VIBRATIONS OF MAGNETIC TAPE REELS

E. S. Umanskii, N. S. Shidlovskii, V. V. Kryuchkov, and D. G. Kofto

UDC 539.3/5:678

A tape transport mechanism may undergo considerable vibration under working conditions (with accelerations up to 40 g) [i], which are transmitted to the tape reels. If there is resonance between the frequency of these vibrations and the transverse Vibrations of the reels the transverse displacements or bending of the latter may exceed the levels permissible from the viewpoint of tape motion and damage the reels.

This makes it necessary to establish the range of natural transverse-vibration frequen- cies for the reels in relation to factors such as the tape tension during winding, the tempera- ture, and the angular velocity.

We used reels of 1-4406-6 tape, wbich is made of polyethyleneterephthalate (PETP),which is widely used in engineering. The reels were wound with a constant tension H in the range of forces from 0.5 to 2 N. We also examined the dependence of the resonant frequencies on the geometrical dimensions of the reel, i.e., we determined the transverse-vibration frequencies for various outside radii for reels wound on a core of diameter 90 mm.

The resonant frequencies were determined with a special equipment based on a VEDS-100B electrodynamic tester (Fig. i).

The vibrating system consists of the vibrator i, whose working table 2 bears the object, namely the reel of PETP magnetic tape 3, wound on the core 4. The vibrations of the working table 2 are excited by an electronic system consisting of the sine-wave generator 5, a unit automatically maintaining the vibration parameters 6, the power amplifier 7, and the power supply for the magnetizing coil 8. The vibration frequency was monitored with a Ch3-33 digi- tal frequency-meter connected to the generator 5.

Unit 6, working with the piezoceramic transucer 9, maintains a constant displacement amplitude w or effective acceleration aef in the working table as the frequency varies. The second piezoceramic transducer i0 is connected to the parameter-measurement unit ii. Both transducers are fixed to the working table.

The resonant frequency corresponding to the maximum displacement amplitude at the edge of the reel was determined by two methods. In the first, the source of the signal was an electrodynamic microphone 12 attached to the edge of the reel and connected to the SI-19B oscilloscope. The second method employed an induction transducer consisting of the output coil 13, receiving coil 14, and core 15 made of electromagnet steel together with the B5-8 power supply.

Kiev Polytechnical Ins titute7--Transi~ate-d--from P~bfemy Prochnos6~7,-No~.-5~- pp_ 8-i--8~,-- May, 1983. Original article submitted November 17, 1982.

686 0039-02316/83/1505-0686507.50 �9 1984 Plenum Publishing Corporation

15 i

f, Hz ] ...... !

I I000 "

koo ~/ 600- - -

0,5 CO /.5 M,N a

L7 /,~' k9 b

Fig. 1 Fig. 2

Fig. i. Electromechanical test scheme for examining the dynamic characteristics of magnetic tape reels in transverse vibration.

Fig. 2. Dependence of the first resonant transverse-vibration frequency of a magnetic tape reel on the tension during winding (a) and on the outside relative radius (b).

In both methods one varies the frequency smoothly and uses the SI-19B oscilloscope to determine the maximum amplitude at the edge of the reel and the corresponding resonant fre- quency. IIere the effective acceleration in the core 4 was kept constant at aef = i0 m/sec 2 throughout the working range.

The preliminary experiments consisted in measuring the displacements of points at var- ious radii a~d showed that the perturbations introduced by the coil 8 as a localized mass (the weight of the coil was not more than lO -2 N) were very small and could be neglected.

The previously calibrated induction transducer enabled us to measure the displacements of various points in the plane of the reel directly, which is particularly important in examin- ing the form of the vibrations and in constructing the resonance curves.

To eliminate the effects of self-heating on the resonance frequency, a special fan sys- tem was fitted. The temperature of the reel during the experiments was monitored with the copper--Constantanthermocouple 16, with one of the junctions in the body of the coil and the other in the thermostat 17 at O~ X1~e voltage between the junctions was measured with a V7-21 digital millivoltmeter using a thermocouple switch.

The points in Fig. 2 show the results obtained by this method.

~le dependence of the first resonant frequency fr on the tape tension (Fig. 2a) is gov- erned by two factors. On the one hand, raising the tension increases the radial rigidity of the reel (consolidates the tape), which means that the resonant frequency is increased throughout the range. On the other hand, the radial compressive forces in the median plane increase with the tension, which reduces the resonant frequency. For H = 0.5 = 0.15 N, the first factor predominates, and there is a nearly linear relationship between the resonant frequency and the tension. At higher values, the consolidation stabilizes and the second factor becomes more important, with a substantial reduction in the rate of rise in the frequency.

One assumes that further increase in the force would result in a reduction in the reso- nant frequency as a result of the radial compressive forces.

Figure 2b shows the resonant frequency as a function of the relative outside radius k = Rb/R a (R b is the outside radius of the reel and R a is the radius of the core).

As would be expected, the frequency falls as the outside radius increases, and in the range of dimensions and forces considered, the rate of change in frequency is almost indepen-

dent of the tape tension.

Rayleigh's method was used to determine the first resonant frequency for these reels of polyethyleneterephthalate tape wound on rigid cores. The reel was considered as a circular cylindrically orthotropic plate rigidly clamped on the inner edge but free at the outer edg4 and loaded by radial forces acting in the median plane. According to Rayleigh's method, the square of the first frequency in this system is as follows:

687

O,

LO 1,2 i,, 1,6 L8

Fig. 3. Fonn of the transverse vibra- tions of a magnetic tape reel: i) H =

0.65 N; 2) H = 1.25 N; 3) H - 2.00 N (points from experiment; solid line by calculation from (2) with VSr " 0.3),

Rb Rb ~( I 1 '~ 1 1--~12 X &:' " X [ O"-w, ,, O.w O"-w ,l'(r) '&.:' "-] } g,,rdr , (1) [ r = ~ N(r) l-oT'rd,-';- D(r) E,(r) r{-TT-_, ) zv,v 3r... 0," (-EZ I dr T I' ~b , r g j

R~ R a ~ Ra

where D(r ) = ~ h 3 / [ 1 2 G ~ - V~r)] i s the r educed b e n d i n g r i g i d i t y o f the r e e l ; N ( r ) , f o r c e a c t - ing i n the median p l a n e ; w ( r ) , d e f l e c t i o n f u n c t i o n ; ,~(r) = E 0 / E r ( O r ) , a n i s o t r o p y f u n c t i o n ; P~, Rb, internal and external radii of the reel, correspondingly; EO, Er, elastic moduli in the circumferential and radial directions, correspondingly; Or, radial compressive stress; r, current radius; h, reel thickness; and VSr, Poisson's ratio.

Experiments on compressing a packet cut directly from a shaped reel show that the radial modulus of elasticity is very much dependent on Or, where the relationship can be represented with adequate accuracy by a linear function.

The value of E 8 was determined by a dynamic method using forced longitudinal oscilla- tions of a tape specimen with an added mass [2, 3].

The form of the oscillations was taken as the deflection function for the circular ring plate under a static uniformly distributed load:

w (r| = Wo [.qt, ___ R" q- 2 (R = - - <z ---- 2) (R = - - ~1 ~) . - 4c~R=ln 1y'R - - 8 q q n ~1 RI, (2 )

whe re

I - V . r - - ' , ( I -!- ~,,.) (R ~ - - - t l n R }

~ I q - v,~ r q - { I - - "%r) I')~ :

R ==R~/Rb; ~ r/R~.

Figure 3 shows the deflection form calculated from (2) and also experimental results on the displacements of various points from the relative radius 0 = r/R{~ for reels wound with a constant tension. The calculations agree satisfactorily with the experiments, which shows that this form of deflection is applicable to tile dynamic problem.

Analytical calculations from (I) are laborious, so numerical methods were used.

This method enabled us to estimate the first transverse resonant frequency of a ring reel with allowance for the effects of ~a~toc~ such as tl~ t~q~q~erature ol th~ body ol the reel, the centrifugal loads, and the residual-stress relaxation. The radial compr~,ssive force N(r) appearing in (]) is found by solving the boundary-value problem [or the residual stresses in winding a disk.

Reasonably reliable methods have been developed for calculating residual radial and cir- cumferential stresses as arising during winding [4-7] and also for calculating the additional stresses arising from the effects of temperature and the rotation of the disk obtained by windin Z a thin film on a core [5, 6, 8, 9]. For this purpose one can also use some experi- mental methods of determining residual stresses, in particular, the introduction of thin mea- surement plates into the body [7].

Therefore, the soJution for the resonant frequency is preceded by calculation of the re- sidual stresses, which are dependent on the above factors.

688

fr' H Z C-"-'--"-~---~ -- - -

~176

I i/ /i, oo

,oo ~ L .......... r ........... i

500 1 23 +s J ~, %

a

20 i

t _ i o

I ~ ' a : ' I i I o ,'jg JV w, rad / sec

b

Fig. 4. Dependence of the first resonant frequency of a magnetic tape reel on the winding tempera- ture (a) and on the angular veloc-

ity (b).

We now consider the method of calculating the first resonant frequency for a reel .of PETP magnetic tape for various temperatures and angular velocities.*

A turn of tape is thin by comparison with the outside radius of the reel, so we take the body of the reel as a continuous medium having cylindrical anisotropy with a variable modulus of elasticity in the radial direction, which gives us a differential equation for the radial displacement :

: , ( - ) < dr" "}- " ~ " ' 7 I . . . . U I i" r - - �9 " ~ 7 +v0, , _ v ~ , I ' - %,

+) ., ( i d,b 4 .t- i _ - - ~:~ (~ - - ~'~D r + 7 1 . "q- ~ - - ~r dr f %'Or g E I t "

= % - - r k - "

at' r I '%-~ %,% , d~ r ' \ ' ~- ' d r 7 ' ' ) ( [ 3 ) dr ~,__ " ~ r ar ' VorC~o

where u is the radial displacement; H, tension force, dependent on the radius; F, cross-sec- tional area of the tape; ~8, ~r, coefficients of linear expansion of the reel in the circum- ferential and radial directions, correspondingly; y, density of the winding material; m, angu- lar velocity of the reel; and T, change in temperature of the wound reel by comparison with the tape temperature on winding.

This equation was solved by a finite-difference method using a standard Gauss procedure. The following boundary conditions were incorporated. At the inner edge (r = P~), the dis- placement of the reel is equal to the displacement of the core, while at the outer edge (r = R b) o r = O; the stresses at any point on the radius can be derived from the following expres-

sions :

r , , , , " " , '7 <=r -+- ,.o,. + =o;'l j

mj --= ~ _ v ~ l i t %, + q: { T -~- L" o F - - re . �9 (4)

As the zeroth approximation we took the stresses at each nodal point along the radius, obtained by solving the corresponding linear problem [7] with E r = const; in the first ap- proximation, for each node we used the relation

(~) = Eo / (A -- B~ r)

to find the value of the function 4, where A and B are constants determined by experiment.

Then (3) was solved, and the stresses were determined from (4) and the procedure was repeated. Usually, seven to nine iterations were sufficient to give a solution differing

little from the previous one.

The resulting solution scarcely differs from that in the quasilinear formulation, in which the dependence of o r on s is approximated by two straight lines. This solution agrees

*N V. Skripnik and Yu. B. Yashmanov participated in the calculations.

689

well with the experimental data obtained in determining the radial stresses by introducing thin steel plates [7, 8].

The stress fields were determined with the following geometrical and mechanical param- eters for the reels: R a = 45 ram; h = 6.25 mm; F = 0.231 ram2; E0 = 5300 MPa; A = 620 MPa; B : 820; ~er = 0.3; m e = 1.3-10 -5 deg-1; m r = 5.0"10 -5 deg-1; and 7 = 1.41"10 -2 MN/m 3.

When the levels of the radial stresses had been found, and consequently

N (r) ----- ar (r) h,

the integrals in (i) were calculated, e,g., by Simpson's method.

The solid lines in Fig. 2 correspond to the theoretical calculation performed by the above method. Note that the calculated value for the first resonant frequency exceeds the observed value by not more than 8%.

Therefore, a reasonably simple energy method enables one to estimate the first resonant frequency for the transverse vibrations of a reel of magnetic tape with allowance for the ef- fects of various working factors.

Figure 4 shows the effects of winding tempera.ture and angular velocity at various tem- peratures on the resonant transverse-vibration frequency for a reel of 1-4406-6 tape. It is evident that with a given winding conditions (H = const; T = 20~ any further rise in tem- perature produces an appreciable increase in the resonant frequency (Fig. 4a). This is due to ~he additional compressive stresses in the radial direction and, consequently, the in- crease in the rigidity of the reel in that direction.

Increase in the angular velocity causes the converse effect. In that case there are addi- tional tensile radial stresses, and the rigidity in the radial direction is reduced, so the frequency is correspondingly lowered.

1.

2.

3.

4.

5.

6.

7.

8.

9.

LITERATURE CITED

E. N. Travnikov, Mechanisms in Magnetic-Recording Apparatus [in Russian], Tekhnika, Kiev (1976). E. S. Umanskii, V. V. Kryuchkov, N. S. Shidlovskii, et al., "A study of the dynamic char- acteristics of magnetic tapes and tape bases in longitudinal vibrations," Tekh. Sredstv Svyazi, Ser. OT, No. 2, 74-80 (1977). E. S. Umanskii, V. V. Kryukhkov, and N. S. Shidlovskii, "A study of the damping capaci- ties of magnetic tapes and their bases in longitudinal vibrations," in: Energy Dissipa- tion in Mechanical-System Vibrations [in Russian], Naukova Dumka, Kiev (1980), pp. 310- 314. G. G. Portnov and A. I. Beil', "A model incorporating the property nonlinearity in a semifinished product in force analysis for composite winding," Mekh. Polim., No. 2, 231-240 (1977) . V. L. Biderman, I. P. Dimitrenko, V. I. Polyakov, and N. A. Sukhova, "Residual-stress determination in the manufacture of fiberglass rings," ibid., No. 5, 892-898 (1969). V. P. Nikolaev and V. M. Indenbaum, "Residual-stress calculation for wound components containing fiberglass," ibid., No. 6, 1026-1030 (1970). E. S. Umanskii, V. V. Kryuchkov, and V. A. Rakovskii, "Determination of the state of stress in a magnetic tape reel," Probl. Prochn., No. 3, 83-85 (1978). ~. S. Umanskii, V. V. Kryuchkov, and N. S. Shidlovskii, "Estimating the effects of the temperature factor on the carrying capacity of a magnetic tape reel," ibid., No. 8, 62- 65 (1981). E. S. [~nanskii, V. V. Kryuchkov, N. S. Shidlovskii, et al., "Estimating the integrity of a magnetic tape reel working in start-stop mode," Tekh. Sredstv Svyazi, Set. OT, No. 2, 82-86 (1980).

690