WI ! l l mm IIIIHHHI H w l H l I 133 629 THS EFFECT OF AIR DAMPING ON TRANSVERSE VIBRATIONS OF STRETCHED FILAMENTS Thesis for fhe Degree of M. S. MiCHIGAN STATE COLLEGE David Wesley Stauff 1954 IE -s- ImmmIH‘IIWWIWI‘II’IRIWHI‘IMummumu ~ 3 1293 01774 9676 LIBRARY Mlchigan State University This is to certify that the thesis entitled EFFECT OF AIR DAMPWG ON TRANSVERSE VIBRA‘HONJ OF STRETCHED FILAMENTS presented by DAVID WESLEY STAUFF has been accepted towards fulfillment of the requirements for M. 5- degree in_EHx;l__CS 13.wa Major prdfessog Date 2-8 MGM PLACE IN RETURN Box to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE use animus-m4 EFFECT OF AIR DAMPING ON THANSVERSE VIBRATIONS OF' STRETCHED FILAMENTS by David Wesley Stauff A Thesis Submitted to the School of Graduate Studies of Michigan State College of Agriculture and Aoolied Science in eartial fulfillment of the requirements for the degree of MASTER OF SCIENCE Deoartment of Physics 1954 ACKNOWLEDGMENT The author wishes to exoress his thanks to Dr. D. J. Montgomery for continued encouragement and advice. 94.] 0.23% 331302 II III IV VI TABLE OF CONTENTS INTRODUCTION THEORY APPARTUS AND PROCEDURE RESULTS CONCLUSIONS LITERATURE CITED PAGE 12 18 25 24 INTRODUCTION The first experimental determinations of the laws of vibrating strings are credited to Mersenne (1836) and Galileo (1638). A practical annlication of these laws is made in the vibroscooe, an instrument for finding the linear density of fibers and thence their equivalent diameter. The vibroscooe consists of a device whereby a filament of known length under tension is induced to vibrate transversely by an aoolied oscillatory force, and the frequency of mechan- ical resonance observed. From the values of frequency, length, tension, and other parameters, it is oossible to calculate the linear density and hence the equivalent diameter of the fiber, under the assumotion of uniform density. Gonsalves (l), anoarently the first to aooly the laws of vibrating strings to determination of linear filament density, gave the first-order stiffness correction, an anoreximation based upon the work of Seebeck (7). Montgomery (4) treated the combined effects of small stiffness and slight nonuniformities, basing his treatment on a general- ization of Rayleigh's work (5). In these treatments external dancing was neglected, so far as its effect on natural frequencies is concerned. Karrholm and Schroder (5) locked into the effect of air damping in their investigation of the bending modulus of fibers, and found that accurate results could be obtained with a resonance frequency method when the fiber diameter was large and the resonance frequency high. With diameters substantially less than twenty microns, however, it is oossible that corrections will be anareciable at the frequencies used in nractical vibrosconing. It is the nuroose of the present work to determine the existence of such corrections, and to establish the aovlicability of some of Stokes' theoretical work. THEORY For transverse vibrations of a uniform elastic string or wire of length l , cross—sectional area 5 , density p. , fixed at both ends, and under a constant tension T , the resonant frequency f; is given in the n-th mode hy (1) : —-D-— ——-—-—.T - f" 21 (’05 , h- l,1,3..... As seen from the formula, the vibrating string has discrete natural frequencies of vibration, harmonically related, each resonant frequency f; corresoonding to the n~th allowed mode of vibration. The validity of Eqn. (1) is deoendent noon certain assumotions, the first of’which is that the string is nerfectly flexible. Filaments used in the nresent work were chosen long and flexible so as to make the effect of stiffness small and under control. According to Voong and Montgomery (9), who have investigated the effect in detail, a stiffness factor can be defined as al.3(trEZ'/j‘7')%L (2) where E (dynes/cm..2) is the elastic modulus of the filament material, and I (cm.4) is the moment of inertia of the filament about the neutral axis. In the work resorted here, the largest stiffness correction was 2.2% under the assumption of circular cross-section. For our ourooses, this effect can be considered negligible. In the analysis for the stiffness correction, it was assumed that the filament was of uniform cross-section throughout its length. For actual filaments, nonuniformities certainly exist; however, it has been shown (4) that slight nonuniformities do not alter the stiffness correction in the approximation used above. Amplitudes of vibration were purposely keot small, in conformity with the assumotions of the theory. It was found that a slight change in the resonant frequency did indeed occur if the amplitude became very large and accordingly the transverse displacement of the fiber was kept below 0.1 mm. from equilibrium oosition at a mid-ooint distance of five centimeters from the end of the fiber. The most significant condition on Eqn. (1) as far as this work is concerned is the requirement that energy losses do not disturb the resonant frequency. Such losses may be traced to four sources:‘ energy transferred to sunnorts, internal damning, acoustical radiation, and air damning. The fiber is sueoorted at the too by a large brass clamo which may beconsidered to have infinite mass as compared with the mass of the fiber. Hence there are no direct energy losses to it. The tension was supplied by the weight of an aluminum foil tab cemented to the bottom of the fiber. Wire-hook weights were hung on the tab when greater tensions were required. Though energy losses to this bottom suooort were not investigated directly in this work, no movement of the tab could be observed, and it was concluded that energy dissipated in this form was negligible. Internal damning exists in the filament but its effect is small relative to that of air damping, as shown by the greatly increased amplitude in a vacuum over that in air. No investigation of the extent of acoustical radiation has been made, but it seems unlikely that there can be very much energy radiated by a fine filament at small amplitude. Air damoing is the last source of energy loss, and the one with which the present work is primarily concerned. The effect on fibers of not much less than twenty microns in diameter is negligible, but it is perhaps aooreciable when smaller fibers are employed. ”The calculation of air damning on a stretched filament.aooears not to have been worked out. Stokes (8) has worked on the theory for a related problem, however, in which he considers laminar fluid flow around an infinite cylinder in sinusoidal oscillation normal to its length. A Reynolds number calculation indicated that the flow was laminar, and on the basis of Stokes' results, the theory was worked out for the present case of a vibrating string under tension. ‘ According to Stokes, the damping force (dynes/cm.) on a unit length of a cylinder (cf. Fig. 2) of radius <1 (cm-), and density P" (gm./cm.3), vibrating in a fluid of density ‘7 (gm./cm.3) and viscosity ll (noise) is given by es: 5 9:33; 32:8 .N 8:2... _ li'ONTII. . . _ _ a i . _ _ A20; «wwzmrww m m Ammaevi >tmoom_> 730sz ll il > ll A.:o\zevx >tmzuo :2: _ 55.7.. F _ L 20:02 l ‘ +éh~za _.._..2 ._..._-_- F=—K.Mfifi‘z’KN\a-’i(fi“ <5) where f] is the displacement from the position of equilibrium, CO is the angular frequency::2nf, F4 (gm./cm.) is the mass of the gas disolaced=ft5, where [S (cm.2) is the cross-sectional area of the cylinder, and lfi and l o~__\e~. In... 20. x00. _ 55$ 230$, of. e. > con Eon...) 9: mum: .fl mooEomnm/. T wane mm>1 Oozwo 1.0m 3.. 3.5.3.. 3.er ilmzomo>xz< Li. mendo Gown—02 Lb mums-4:0 232w4d\ >m