ATTENUATION OF SOUND IN SOME OVERLOADED ABSORBERS BY A PULSE TECHNIQUE Thesis for H» Dist“ of M. S. MICHIGAN STATE UNIVERSITY Jay H..WoIkowisky 1961 This is to certify that the thesis entitled ATTENUATION OF SEND IN SCME OVERLOADED ABSORBERS BY A PULSE TECHNIQUE :v — m—umm . ‘ presented by JAY H. Homowxsxy ' ‘ has been accepted towards fulfillment of the requirements for msrsa OF SCIENCE degree in APPLIED MECHANICS Major professor Date May S, 1961 0-169 LIBRARY Michigan State University ABSTRACT ATTENUATION OF SOUND IN SOME OVERLOADED ABSORBERS BY A PULSE TECHNIQUE by Jay H. Wolkowisky The main purpose of this study is to investigate the sound absorption properties of some rubber—like and elastic materials as a function of the frequency and of the stress. A predetermined static stress is applied to a specimen through long transmitter rods, by means of a universal testing machine. A stress pulse is produced which travels through the specimen. By means of strain gages, the strain pulse can be recorded before it enters and after it leaves the specimen. The input and output pulses are represented by a finite Fourier series. By using this series in a Fourier Integral the amplitude dis- tribution function for each of the input and output pulses is obtained. Calculating the ratio of the mean squared (over small frequency bands) amplitude distribution functions for the input and output pulses, a measureof the energy transmission is obtained. By varying the static load applied to the specimen, the dependence of the energy transmission on the static stress is obtained. The results are plotted as histograms. These show that the more rubbery a material is the more the sound absorption properties change with stress. Also that at certain frequencies there is resonance phenomenon for some of the materials tested. ATTENUATION OF SOUND IN SOME OVERLOADED ABSORBERS BY A PULSE TECHNIQUE By Jay H. Wolkowisky A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Applied Mechanics 1961 .7 ./ ./-' !/ / ACKNOWLEDGEMENTS Iwish to express my gratitude to Dr. C. A. Tatro, who originally suggested the problem and for his assistance in solving the experimental problems encountered. I am also grateful for the advice given to me by other members of the faculty. Sincere thanks are also due to Mr. J. W. Hoffman and the Division of Engineering Research for their cooperation and support of this work. I also wish to express my gratitude to Mr. D. H. Yen for his valuable assistance in devising the program for the. "Mistic" elec- tronic computer. I would also like to thank Mr. R. W. Jenkins and Mr. E. A. Thompson for their assistance in building the apparatus. ii CONTENTS LIST OF FIGURES LIST OF SYMBOLS CHAPTER I INTRODUCTION AND REVIEW OF PAST WORK CHAPTER II DESCRIPTION OF EXPERIMENT CHAPTER III ANALYSIS OF THE DATA CHAPTER IV RESULTS AND CONCLUSIONS BIBLIOGRAPHY APPENDIX CALCULATION OF THE FOURIER INTEGRAL TRANSFORM (AMPLITUDE DISTRIBUTION FUNC TION) iii Page iv vi ll 29 33 34 Figure 1 Figure 2 Figure 3 Figure 4 Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 10 11 l2 l3 I4 15 16 17 LIST OF FIGURES M EC HANICAL APPARA T US GAGE PLACEMENT FUNCTIONAL BLOCK DIAGRAM FOR EACH BRIDGE SCHEMATIC DIAGRAM SHOWING METHOD OF TRIGGERING SOME TYPICAL INPUT AND OUTPUT STRAIN PULSES SYMBOLIC DIAGRAM OF A STRAIN PULSE SYMBOLIC DIAGRAM FOR AN INPUT OR OUTPUT FUNCTION F(t) AMPLITUDE DISTRIBUTION FUNCTION FOR WHITE FELT (1500 lb.) AMPLITUDE DISTRIBUTION FUNCTION FOR WHITE FELT (4000 lb.) AMPLITUDE DISTRIBUTION FUNCTION FOR NEOPRENE (1500 lb.) AMPLITUDE DISTRIBUTION FUNCTION FOR NEOPRENE (4000 lb.) AMPLITUDE DISTRIBUTION FUNCTION FOR NYLON (15001b.) AMPLITUDE DISTRIBUTION FUNCTION FOR NYLON (4000 lb.) SYMBOLIC DIAGRAM OF HOW Rf IS CALCULATED MEAN SQUARED AMPLITUDE DISTRIBUTION RATIO FOR WHITE FELT (1500113) MEAN SQUARE AMPLITUDE DISTRIBUTION RATIO FOR WHITE FELT (4000 lb.) MEAN SQUARED AMPLITUDE DISTRBUTION RATIO FOR NYLON (1500 lb.) iv Page 10 12 l3 13 16 l7 l8 19 20 21 22 23 24 25 Figure 18 Figure 19 Figure 20 LIST OF FIGURES CONTINUED MEAN SQUARED AMPLITUDE DISTRIBUTION RATIO FOR NYLON (4000 lb.) MEAN SQUARED AMPLITUDE DISTRIBUTION RATIO FOR NEOPRENE (1500 lb.) MEAN SQUARED AMPLITUDE DISTRIBUTION RATIO FOR NEOPRENE (4000 lb.) Page 26 27 28 A'WI=A(f) *ul'uln :0 ("I- H<1 Sb < LIST OF SYMBOLS Amplitude Distribution Function (Fourier Integral Transform) ordinate for amplitude distribution function for output pulse six by six transformation matrix ith coefficient of the finite Fourier series representa- tion of f(t) ordinate for amplitude distribution function for input pulse flit-IE) strain pulse (input or output) 0 .4. t éL frequency (cycles/sec) indices (integers) temporal wave length of f(t) number of terms of finite Fourier Series incident average power per unit area reflected average power per unit area mean squared amplitude distribution ratio for frequency band centered at f time elastic wave velocity of first medium elastic wave velocity of second medium mean density of first medium mean density of second medium 21“ vi CHAPTER I INTRODUCTION AND REVIEW OF PAST WORK This investigation was inspired by the need to select a material exhibiting good sound isolation properties while under severe compres- sive stresses. The material is to be used in isolating tensile specimens from the load holders of a testing machine in Acoustic Emission studies (Ta 60). The present analysis attempts to determine the sound deadening power of a material while under a load. In other words, the material is being tested under the same conditions as would be experienced by it in actual practice. The analysis is based solely on the comparison of the input and output shapes of a transient stress wave. This transient stress wave is measured before it enters and after it leaves the specimen (the method of measurement will be explained in the next chapter). There- fore, since the pulse is not actually being measured in the specimen, what is being measured is the combined effects of two important mech- anisms of energy dissipation. These two mechanisms will now be explained. They are first, the loss of energy of the incident pulse due to reflection at the interfaces; and second, the attenuation due to inter- nal friction. Reflection at the interfaces: Looking at the equation (Li 60), r V ‘1 31-12.}. ' /’1V1 r , fV i 1+72—V-2— I II 1 *UI “OI I I h. I d it is seen that if jfZVZ >>fiv1 or fivl "bi/'ZVZ. then Pro; Pi, or practically all the energy is reflected. Also, it is seen from this equation that if 5V2 .2 f’iVl none of the energy is reflected. If the media are well matched, ([V values are close together) then considerable energy transmission takes place. If the media are badly matched (IDV values differ greatly), then there is a corresponding poor energy transmission. So it is clear that the quantity/V, called the "specific acoustic resistance", controls the transmission of energy ' at an interface from one medium to another. Therefore, for a good iso- lator, it would be advantageous to have the specific acoustic resistances differ greatly. Internal friction: There is at present no satisfactory theory of internal friction in solids, and more experimental data are required. Internal friction in solids may be produced by several different mech- anisms, and although these all result in the mechanical energy being transformed into heat, two different dissipative processes are involved. These two processes are roughly analogous to the viscosity losses and thermal conduction losses in the transmission of sound waves through fluids. These processes will not be discussed further, since a study of the mechanism of internal friction was not the main objective of this investigation. In this analysis the effects of reflection and internal friction will not be considered separately. The total effect of these two mechanisms will be lumped together and this tOtal energy loss will be analysed. The methods which have been used previously to measure energy dissipation in solids may be divided into several classes, these are: 1. Free vibration methods 2. Resonance methods 3. Wave—propagation methods This investigation can be classified with the wave propagation methods, but yet the method of analysis is quite different from what has been done before. 1. Free Vibration Methods. This method is most suitable for materials which are linear. At a given frequency of oscillation the period and logarithmic decrement of free oscillations can be measured. From these measurements its mechanical properties and behavior can be found. 2. Resonance Methods. This method is based on the principle that if an oscillating force, whose amplitude is fixed but whose frequency can be varied’is applied to a material, the amplitude of the resulting vi— bration passes through a maximum at a frequency which is known as the resonant frequency of the system. The value of this resonant frequency depends on the elastic properties of the system, while the width of the resonance peak gives measure of the dissipative forces which are present. 3. Wave-Propagation Methods. When a stress wave is propagated through a solid, and the solid is not perfectly elastic, some of the energy of the stress wave is dissipated as it passes through the medium. The attenuation can be measured and from known relationships (Ko 53) a measure of the internal friction can be determined. The experiment. used in this investigation is based on the same principle but the method of analysis to determine the relative energy dissipation at various fre- quencies is striCtIYa mathematical one. Work on the propagation of low frequency longitudinal waves in filaments has been mainly concerned with the dynamic behavior of rubber-like materials and high polymers. Some of these investigations have been done by Ballou and Silverman (Ba 44), Nolle (No 48), Ballou and Smith (Ba 49), Hillier and Kolsky (HiK 50), and Hillier (Hi 50). Another technique using the propagation of waves has been to produce a Short pulse of high-frequency oscillation and measure its time of transit and its attenuation as it passes back and forth along the specimen. This method is similiar to the principle used in radar. Ivey, Mrowia, and Guth (Iv 49) have used this technique to work with rubber specimens. . More recently Auberger and Rinehart (Au 61) have used an electrosonic pulse technique developed by Hughes (Hu 49) to investigate the attenuation of stress waves in plastics. All these investigations have been primarily concerned with obtaining data on internal friction. This investigation differs slightly in that its main purpose is not so much to obtain quantitative results but to explain a possible method of analyzing an attenuated stress pulse and to obtain an over-all picutre of sound absorption as a function of frequency without analyzing the details responsible for it. As a conse- quence, all results have been represented graphically. CHAPTER II DESCRIPTION OF EXPERIMENT The experimental apparatus is very Simple and is depicted in Fig. 1. The stress wave is produced by a short steel striker rod (a) which falls under its own weight and hits the upper end of a one inch diameter circular steel bar (b).The striker rod is guided in its fall by a pipe (c) which has an inside diameter approximately the same size as the outside diameter of the striker rod. The striker has a light nylon line ((1) attached to its top end so that it can be pulled to the desired height again for the experiment to be repeated. The stress wave travels down the upper bar, passes through the specimen (e) and travels into the lower steel bar (f)‘The specimens are all about 1/8” thick. The upper bar is held in place by a collar (g) which is attached to the stationary head of the testing machine. All the con- tact surfaces between the collar and the upper bar are separated with a soft material so that there will be as little interference as possible with the stress wave as it travels down the upper bar. Two sets of type A-8 strain gages are used. One set (h) to measure the stress wave before it enters the specimen and the other set (i) to measure it after it has left the specimen. Each set consists of four gages. Two are used to measure the lateralstrain and are placed 1800 apart on the bar. The other two in the set are used to measure the longitudinal strain and are also placed 1800 apart. The four strain gages in each set are hooked up in a bridge circuit so that the strain measured will be a sum of the four individual strains. The wiring is described in Figs. 2, 3, 4. The strain gage set on the upper bar is approximately 5 one striker bar length from the specimen so that the stress wave will have completely passed through these strain gages before the reflection has a chance to interfere. The set of strain gages on the lower bar are placed approximately one striker bar length from its lower end so that the reflections from the welded joint do not interfere as the stress wave passes through these gages. The upper bar is quite long (three feet) since it is desirable to have the strain gages far from the struck end of the bar. This is so that the transient end effects produced when the striker hits the bar will die out before they reach the strain gages and therefore not interfere with the main pulse. A crystal (j) is used to trigger the first oscilloscope. This is placed about seven inches above the top set of strain gages. An external trigger is needed for the first oscilloscope since the entire pulse had to be recorded. The external trigger serves the purpose of triggering the sweep before the stress wave actually reaches the strain gages. The sweep of the second oscilloscope is triggered from the first for the same reason. As can be seen from Fig. l the lower bar is attached to the movable lower head of the testing machine. The compressive load is applied to the specimen when the lower head of the testing machine moves up. The oscilloscopes used were Tektronix units with Dumont Type cameras attached to photograph the pulses which are projected on the oscilloscope screen. Now for a summary of the experiment. The testing machine applies the predetermined static load to the specimen. The striker hits the upper bar, the stress wave travels down the bar to the first set of strain gages and the resulting strain pulse is photographed from the screen of the first oscilloscope. The wave continues through the specimen and the resulting strain pulse is picked up by the second set of gages and photographed. These input and output photographs are the data to be analyzed. The experiment was done with sheets of nine different materials: white felt, neoprene, nylon, silicon rubber, teflon, saran, clay impregnated bakelite, red rubber, and hard board. These materials were thought to have good sound absorption properties and also meet other design criteria. Each of these materials were tested under three different static loads: 500 lbs. , 1500 lbs. , 4000 lbs. These loads correspondto static stress in the specimens of G35psi. .1910 psi. , and 5100 psi. respectively. Due to the fact that the analysis of the data is such a lengthly process, only three of the materials (white felt, neoprene, nylon) were analyzed under loads of 1500 lbs. and 4000 lbs. each. The data not processed is on file'with the Applied Mechanics Department, .Michigan State Unive rsity . * Stah'ona rg upper fiend .9.- Ir 1 , ' ‘ \ .' / . , . I ' ' ' A , . ‘ ' ’ I I I ’ - o , l I _ , . I / . ‘ . 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SIB-r237"; Ty p e E D1 as ‘— 4 in S CHE MA 77 C DIAGRAM SHOWING f/GUPE e METHOD 0 F TFP/GGEP/NG CHAPTER III ANALYSIS OF THE DATA A set of data will be considered as the two photographs of the input and output pulses from one run of the eXperiment; four sets of such photographs are shown in Fig. 5. A pulse is represented in Fig. 6 by f(t). The pulse for 0 1’: t .4. L will be represented by a finite Fourier series. In doing this the wiggles of the pulse for t > L will be neglected. These parts of the pulse, represented by dotted lines in Fig. 5, are due to dispersion of components of higher frequencies than were of interest here. This approximated pulse will be called f(t). It has been Shown (Wh 54) that a given function can be represented by a sum of sine terms: _ . 4ft . 21ft sin (n-l)’n’t f(t) - b151n-1_J—.(..b2 Sin L + bn L which will take given values for (n - 1) given equally spaced values of the argument t; say L __ 2L ___ (n - l) L f(—I-1—)... f1. f(——n—) __ f2 . f —-————fi-——— _ fn-l where f1, f2 ..... fn-l are given numbers, and the coefficients are given by __ _2_ . i'fl’ . ZiTr . (n- Diff] bi" n{f151n—fi- +f2 Sln n + fn_151n——-—?1————( .J For our case n - 7 and hence 6 __ Z . i'TI’t ‘ f(t) ._ bi Sln L , . (1) i=1 where ll 12 INPUT OUTPUT 201/ sec—1 I- 1.11) .L. ;5——3_— millivolts - - FIGURE 5 * Some Typical INPUT and OUTPUT Strain Pulses 13 K MQDDC Q MQDOC I COR corCEE LMSMU. HM THEE: .CmZQ :Etm mm A13 EmLOOd OCOCEBW m :4 Imubst BLOOESWW ”'I f II. C «2.; CE :1 ct l4 6 ___ Z “x . jl’TT bi _ -7— : fj Sin—T (2) J21 The six equally spaced ordinates (fj) of f(t) are measured directly from the photographs using a two-dimensional measuring microscope. The Michigan State University "Mistic" electronic computer was used to calculate the bi's from equation (2). A completely new computer pro- gram had to be devised for this operation and is on file with the Applied Mechanics Department, Michigan State University. If one thinks of the input pulse as a transient excitation of the system and the output pulse as the transient response, then one can treat the system almost as if itwere a vibration problem. Since these input and output pulses are ”random" (not a single monochromatic wave) then they can be represented as the sum of an infinite number of separate monochromatic waves. The function that tells "how much" of each separate wave is contained in the input and output pulses is just the Fourier integral transform (or amplitude distribution function) of these two pulses. When the Fourier integral transform of these input and output pulses are taken they will be represented as functions of frequency. The pulses are now in a form in which they can be compared so that the results obtained will also be functions of the frequency. This is what was desired. The method for comparing the transformed input and out— put pulses will be explained later. The equation for the Fourier integral will be developed. Since the same general method of using the transform will apply to both the input and output curves we can represent both of them by F(t) 0f, t 400. As shown symbolically in Figure 7, the Fourier Integral representation 15 is then no F(t) Z I A'(C4),).. sin cat dog 0 where no A'(Cd) 2:. E- ( F(t) sinwt dt '0" .' / o This reduces to 6 ' i . , _b. (- l) i . 2L 24111. A (cu) : A(f) :: ““2 Z 1 z 2 (3) In”. i.14(fL)-i The details of the above calculation and method of plotting are given in the Appendix. These curves for A(f) are plotted in Figs. 8—13 for white felt, nylon, and neoprene at 1500 lbs. and 4000 lbs. each. The transformed input and output curves are now compared in such a manner as to give the relative energy loss as a function of the frequency. This is done by taking the ratio of the mean squared (over small frequency bands) amplitude distribution function for the input and output pulse. This ratio will be called the ”mean squared amplitude distribution ratio". If Rf is this ratio for a frequency band "f", then {L- / 3’ 1a.2 ’ 1 1 "' ‘ n ~ '- I Rf - n "X Z / 1“ CJ I . “r- n 1 ': I This is shown for a frequency band centered at fin Figure .14. For this investigation a frequency band width of 5000 cycles/sec. and subintervals of width 1000 cycles/sec. was used. 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