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Flanking transmission of acousto-vibrational energy between adjacent acoustic cavities Wakefield, Clair William 1973

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c THE FLANKING TRANSMISSION OF ACOUSTO-VIBRATIONAL ENERGY BETWEEN ADJACENT ACOUSTIC CAVITIES by CLAIR WILLIAM WAKEFIELD B.A.Sc, University of British Columbia, 1971 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER-OF APPLIED SCIENCE in the Department of Mechanical Engineering We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1973 In presenting th is thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Univers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make i t f ree ly ava i lab le for reference and study. I fur ther agree that permission for extensive copying of th is thes is for scho la r ly purposes may be granted by the Head of my Department or by his representat ives. I t i s understood that copy-ing or pub l ica t ion of th i s thes is for f i nanc ia l gain sha l l not be allowed without my wr i t ten permission. CLAIR WILLIAM WAKEFIELD Department of Mechanical Engineering The Univers i ty of B r i t i s h Columbia Vancouver 8, Canada ABSTRACT The effectiveness of partitions erected to , among other things, provide acoustic insulation between adjacent cavities in buildings, ships and aero-space structures is often limited by the presence of acoustical flanking paths. These paths, which can be provided by air gaps, ventilation ducts, etc., or by continuous walls or floors, allow sound and vibrational energy to pass be-tween adjacent cavities without suffering the attenuation of the primary acoustic barrier (the partition). Techniques for the measurement of flanking transmission in existing.structures and for its theoretical prediction have to date been awkward and imprecise, (the measurement technique requir-ing the erection of a second barrier to cover the barrier under test) and limited in range of application, (the theory only applying at frequencies for which the panel responses are mass controlled). The work described here was directed at removing these limitations on the present ability to measure and predict the effects of flanking transmission as they act to reduce the noise insulation attainable between two adjacent acoustic cavities. A cross correlation - Fourier transform technique was em-ployed to measure the contributions of individual airborne flanking paths to the total- sound field in a receiving cavity. It was dis-covered that cross correlation between two microphones, one in each of the source and receiving rooms, could not yield useful informa-tion about individual flanking paths because of the strong correla-tion of natural modes common to both cavities. This problem was overcome by replacing the source room microphone signal with the in-put signal to the noise source. The common room modes could not then correlate since only the receiving cavity microphone signal contained the mode components. This alteration, however, meant that only the relative magnitudes of the various flanking path transmission spectra could be obtained since the source room microphone signal was no longer available to provide a reference spectrum. However, for purposes of determining which flanking path contributes most to the receiving room sound field, relative magnitudes of their transmission spectra are all that is required. The altered technique allowed measurement of the trans-mission spectrum of an induced airborne path. Agreement with the measured difference in sound pressure levels with and without the flanking path, was very good. A relatively new structural dynamics technique known as Statistical Energy Analysis (SEA) was used to predict the noise re-duction between two adjacent cavities, the boundaries of which were structurally coupled. SEA is based on an analogy to conductive heat transfer. Therefore, i t becomes more accurate at higher fre-quencies as the wave fields in the acoustic cavities become more diffuse (have more spatially uniform energy densities). The SEA model developed here, then applied over all but the lower end of the experimental panel response range. This lower limit corresponds to the breakdown of diffuse wave field conditions in the smaller of the two experimental cavities. The model allowed the effects of the variation of panel internal damping and bending stiffness upon noise reduction to be investigated and was successful in predicting the noise reduction between two alu-minum walled model rooms (for two different partition thicknesses) to within 2 dB over most of the frequency range 400 to 20,000 Hz. The results of the above two experiments and the corres-ponding SEA values of noise reduction, showed that, except at low frequencies where panel response is predominantly mass controlled, the increasing of primary barrier (partition) surface density does l i t t l e to increase the noise reduction between cavities bounded and coupled by relatively lightweight, resilient walls. Increased panel internal or joint damping, however, increased the noise reduc-tion at all frequencies but especially those near panel coincidences (when flexural wave speed in a panel equals the acoustic v/ave speed). V TABLE OF CONTENTS Chapter Page 1. INTRODUCTION 1 1.1 The Effects of Flanking Transmission . . . . . . . 1 1.2 Present Methods for the Measurement and Prediction of Flanking Transmission 5 2. ANALYTICAL METHODS FOR FLANKING TRANSMISSION PREDICTION 8 2.1 Scope and Purpose of the Analytic Work 8 2.2 Statistical Energy Concepts 10 2.3 The Prediction of Noise Reduction in the Presence of Structure-borne Flanking Transmission Using SEA . . . . 23 2.3.1 The SEA Model . . 23 2.3.2 The Power Balance Equations . . . . . . . . 30 3. THE EVALUATION OF THE PARAMETERS CONTAINED IN THE SEA SOLUTION FOR NOISE REDUCTION 38 3.1 The Modal Densities of the SEA Elements 38 3.1.1 Introduction 38 3.1.2 The Modal Density of an Acoustic Cavity 39 3.1.3 The Modal Density of Panels . 40 3.2 The Non-Resonant Mass Law Transmission Coefficient 41 3.3 Evaluation of the Structural Coupling Loss Factors n 4 j 2 ' n 4 j 8 and n 2 } 8 4 7 3.3.1 The Selection of a Suitable Theory of Wave Transmission 47 3.3.2 The Derivation of T. ? (as performed by Bhattacharya) . . . »f 49 Chapter Page 3.3.3 The Derivation of g 57 3.3.4 The Derivation of T~ 8 and the Coupling Loss Coefficients. . . 59 3.3.5 Experimental Verification of the Flexural Wave Transmission Coefficients T. 9 and ^ 4 , 8 • • • : 6 1 3.4 The Coupling Between a Diffusely Resonant Panel and an Acoustic Field . , 69 3.4.1 The Case of Free Waves in an Infinite Panel 69 3.4.2 Finite Panel - Acoustic Field Coupling . . . 73 3.4.3 The Radiation Resistance of a Finite Panel in Single Mode Vibration ............. 77 3.4.4 The Radiation Resistance of a Reverber-ant Finite Panel 78 3.4.5 The Effects of Panel Boundary Near Fields Upon Radiation Resistance 81 3.4.6 Panel - Acoustic Field Coupling Loss Coefficients 82 3.5 The Evaluation of the Loss Factors of Panels and Cavities 83 3.5.1 Introduction 83 3.5.2 The Internal (Material) Damping of Panels. . 84 3.5.3 Air Absorption in the Cavities •. 84 3.5.4 Structural Junction Damping 89 3.5.5 Radiation Damping 105 3.5.6 Dissipation of Cavity Soundfields by Mass Law Transmission to the Model Exterior . . . 105 3.5.7 Total Loss Factors of SEA Elements 106 vii Chapter Page 4. COMPARISON-OF STATISTICAL ENERGY AND EXPERIMENTAL VALUES OF NOISE REDUCTION BETWEEN A PARTICULAR PAIR OF STRUCTURALLY COUPLED CAVITIES . . 107 4.1 The Experimental Flanking Transmission Suite and the Measurement of Noise Reduction . . . . . . . 107 4.2 Experimental Noise Reduction Results: Comparison with SEA 112 4.2.1 Comparison: Experiment and SEA 112 4.2.2 Parameter Variation and the Application of the Statistical Energy Model to Other Structures . 120 5. DETECTION AND MEASUREMENT OF AIRBORNE FLANKING TRANSMISSION 126 5.1 Introduction 126 5.2 The Cross Correlation - Fourier Transform Technique 127 5.3 The Experimental Method in Model Rooms 140 5.4 Direct Path Transmission Loss 144 5.5 Transmission Spectrum of an Induced Airborne Flanking Path 148 6. CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY 152 6.1 Conclusions 152 6.2 Suggestions for Further Study 156 FOOTNOTES . • 159 BIBLIOGRAPHY 164 APPENDIX A - The Derivation and Verification of the Consistency Relationship 169 APPENDIX B - The Evaluation of Flexural Wave Transmission Coefficient g 173 APPENDIX C - The Coincidence Phenomena in Finite Panels 179 APPENDIX D - Computer Program for Calculation of SEA NOISE REDUCTION 184 vi i i LIST OF FIGURES Figure Page 1.1 Airborne and Structure-borne Flanking Transmission Paths 3 13 2.1 Two Coupled Elements 2.2: Three Coupled Elements (Sound Transmission Between Rooms) 21 2.3 Schematic SEA Model of Two Structurally Coupled Cavities 25 2.4 The Simplified SEA Model (5 resonant elements) 28 2.5 Power Flow Diagram for the SEA Model of Two Structurally Coupled Cavities 31 3.1 Bhattacharya's Plate - Tie - Plate Model 48 3.2 Plexiglas Model for Measurement of x^ ^ and x^ g 62 3.3 Experimental Layout for Measurement of x^ ^ and x^ g . . . 63 3.4 Accelerometer Positions for Measurement of x. 0 and T 4 ' 2 6 4 3.5 Cross Correlation Function of Band Limited Random Noise (4 KHz Octave Band) 6 5 3.6 Wave Coincidence in an Infinite Panel 0^ 3.7 Single Mode Vibration of a Finite Panel 74 3.8 The Distribution of Acoustically Slow Panel Modes in Wave Number Space 79 3.9 The Total Absorption of Sound in Air 88 3.10 A Typical "Pop" Riveted Juntion Between Sidewalls and Partition 91 3.11 The Location of the Edge Node of Flexural Modes in a Panel with Riveted Edges . 94 rx Figure Page 3.12 Structural Junction Damping Measurement: Panel Edge Rigidly Clamped 96 3.13 Structural Junction Damping Measurement: Panel Edge Riveted as in Experimental Flanking Suite . 98 3.14 Structural Junction Damping Measurement: Panel Edge Riveted as in Experimental Flanking Suite . . . . . . . . 98 3.15 Experimental Layout for the Measurement of Structural Junction Damping . . . . . . 100 3.16 The Loss Factor of the Clamped Edge Panel and the Increase in Loss Factor with Change to Riveted Edge 103 4.1 The Experimental Structure-Borne Flanking Transmission Suite 109 4.2 Schematic of the Flanking Transmission Suite and the Accompanying Instrumentation for Measurement of Noise Reduction ''0 4.3 Experimental and SEA Values of Noise Reduction for the Case of the 1/32" Aluminum Partition 4.4 Experimental and SEA Values of Noise Reduction for the Case of the 1/8" Aluminum Partition 114 4.5 Noise Reduction with 1/8" Aluminum Partition and (1) 1/4", (2) 1/16" Aluminum Sidewalls 121 4.6 Noise Reduction Between the Mechanical Room and a Suite of a Concrete Apartment Building (6" concrete floor and (1) 6", (2) 3" concrete walls) 123 4.7 Noise Reduction Between the Engine Room and a Berth of a Towboat (1/2" steel hull and deck, 1/4" bulkheads) For Two Values of Internal Damping 124 5.1 Cross Correlation Between Two Microphones in the Free Field 129 5.2 The Ideal Correlogram Obtained Between the Two Microphones of Figure 5.1 129 X Figure Page 5.3 Cross Correlation Between Two Microphones in Resonant Cavities . . . . . . . . . . . . . . 136 5.4 Cross . Correlogram Obscured by Room Mode Correlation 137 5.5 Schematic of Air-Borne Flanking Transmission Suite and Its Accompanying Instrumentation 139 5.6 The Experimental Airborne Flanking Transmission Suite 141 5.7 The Experimental Airborne Flanking Transmission Suite . . . 141 5.8 Correlograms With and Without Induced Airborne Flanking Path . . . . . . . . . . . . . . . . . . . . . . 145 5.9 Transmission Loss of a 1/8" Aluminum Panel . . . . . . . 147 5.10 Normalized Cross-Power Spectra, With and Without In-duced Airborne Flanking Path 150 5.11 Increase in SPL (AL) at Receiving Room Microphone Position Due to the Induced Airborne Flanking Path 151 B. l The Co-ordinate System and Positive Displacement Directions for Evaluation of g 173 C l Erroneous Conceptions of the Coincidence Phenomenon in Finite Panels 180 C. 2 The Coincidence Phenomenon in Finite Panels 182 LIST OF SYMBOLS area of panel total surface area of a room bending rigidity acoustic wavespeed in air flexural wavespeed in panel group velocity longitudinal wavespeed in panel trace speed of sound wave on panel th total energy of i ' element in narrow bandwidth Aw average modal energy of i t h element frequency (Hz) critical coincidence frequency bandwidth of Fourier Transform functions of f/f „ cc panel thickness impulse response of a linear system complex frequency response of a linear system /=T~ acoustic intensity acoustic wavenumber flexural wavenumber in panel i length of tie plate (2) in Bhattacharya's Model total perimeter of a cavity = 4(L + L + L ) sound pressure level distance attenuation constant of air (ft"'') mass per unit area of panel i total panel mass modal density of i element at angular frequency co x component of mode number y component of mode number number of significant correlation data points noise reduction acoustic pressure longitudinal wavenumber in i ^ panel acoustic power cross correlation function between shaker input and accelerometer position i cross correlation function auto correlation function cross-power spectrum of x(t) and y(t) power spectrum of x(t) total surface area of a room correlation function averaging time transmission loss of a panel reverberation time sampling period of correlation function volume of i t h cavity acoustic velocity <y > space averaged mean square panel ve loc i t y W upper frequency l i m i t of band l im i ted noise x( t ) time varying input signal X(w) Four ier transform of x( t ) y ( t ) time varying output s ignal Y(co) Four ier transform of y ( t ) Z acoust ic rad ia t ion impedance a(sect ion 3.3.2) K ? / K 4 afsect ion 3.4.4) ( f / f c c ) 1 / 2 a(sect ion 3.5.3) distance attenuation of a i r (dB/1000 f t . ) a average absorption coe f f i c i en t 3 time attenuation constant of sound in a i r (sec" 6( T ) Dirac de l ta funct ion e. acoust ic energy density in i cav i ty th r\- to ta l loss fac to r of i element n- • coupl ing loss factor from to j*"* 1 element n. e x t loss fac tor from i element to model ex te r io r r\j . t in terna l loss fac tor of i t n element j t n junct ion loss factor of i element T i t Q t to ta l loss fac to r of panel 6 angle of incidence of sound waves X a acoust ic wavelength in a i r A c c c r i t i c a l coincidence wavelength in panel f l exu ra l wavelength TT. . power f low from i to j element r d i s s i thTT.. , power d iss ipa ted in i element XIV i r ^ n power supplied to i element by external sources mass density of air a T time delay T. j flexural wave amplitude transmission coefficient from i t h to j t h panel Tp transmission coefficient of panel mass law transmission coefficient i ( X p X 2 ) cross correlation of flexural wave field on a panel ,x2) cross correlation of pressure field at a panel surface w angular frequency ACKNOWLEDGEMENTS The author would like to express his sincere thanks to his joint supervisors, Drs. A. J. Price and T. E. Siddon. Their guidance and company throughout the periods of research and prepara tion of this work have made them enjoyable ones. The author also extends thanks to Mr. K. D. Harford of Acoustical Engineering, who was so cooperative during the final days of preparation. This work was supported by Grants from the National Research Council of Canada and the Defense Research Board of Canada. 1 CHAPTER I INTRODUCTION 1.1 The Effects of Flanking Transmission Acoustical flanking transmission refers to the transmission of sound energy from one acoustic cavity to another, by paths which skirt (or flank) the primary barrier which has been erected to provide acoustic (as well as thermal and visual) insulation between the two cavities. Such "flanking paths" may be provided by air gaps resulting from poor design or construction of the barrier (partition) itself, or by services such as heating and ventilation ducts or electrical outlets. These are called airborne flanking paths. Flanking paths are also provided by structural elements such as continuous walls or floors which are common to both cavities and supply mechanical coupling between them. These are known as structure-borne flanking paths. Here energy is absorbed from the sound field of the source cavity by the continuous wall or floor, and is transmitted in the form of flexural wave fields into the receiving room where i t is converted back into sound energy by radiation. Figure 1.1 depicts two typical adjoining rooms between which exist both types of flanking paths. Flanking transmission is a problem in all situations in which low noise and/or vibration environments must be provided to protect people or delicate instrumentation. In architectural 2 structures, noise from speech or music and noise and vibration from building mechanical systems can be transmitted by both types of paths. This can greatly reduce the noise reduction attainable between adjacent rooms and thus interfere with sleep and reduce privacy and work efficiency. Noise reduction (NR) is simply the difference (in decibels) between the sound pressure levels of the source and receiving cavities, L and L : p l P2 N R = L p - L p do ( l . D = IO L09 (£-./£ 2) dB where £^ and £^ are the reverberant acoustic energy densities in the source and receiving cavities respectively. In marine and aero-space structures, noise and vibration created by the propulsion system and by boundary layer turbulence propagates easily through the lightly damped metal hulls and skins. These resilient structures are particularly susceptible to structure-borne flanking transmission. This type of flanking is in general harder to control and is of more far reaching concern because i t is akin to structural vibration transmission problems of all types. In situations where more acoustic insulation is needed between cavities, the tendency is to increase the transmission loss of the dividing partition by increasing its weight or damping capacity. Although this can be effective, in most cases acoustic flanking transmission limits the maximum insulation possible. The Transmission Loss (TL) of a partition is defined as the ratio (in decibels) of the acoustic intensity incident oh the wall (I. ) to that transmitted from the J inc other side d t r a n s ) . ;•• : T L = 10 L o g ( J l n c / l t r w l > dB (1.2) .''= 10 Log ( 1/Tp) d B Here T p is the transmission coefficient of the partition. To illustrate the limiting effect of flanking transmission upon noise reduction consider ttatwo adjoining rooms of Figure 1.1. Suspended Ceiling -Path A inc --, Duct trans Figure 1.1 Airborne and Structure-borne Flanking Transmission Paths 4 Flanking path A is an airborne path through the ventilation duct. Path B is a structure-borne path through the continuous floor slab. Since acoustic power equals acoustic intensity times the area of transmission, the following power balance equation can be writted between the two rooms: T F c a ( e r £ z ) A p + c aT A A A £, + c a T B A B £ ( = c a R A T Z £ 2 (1.3) where x ^ , x ^ and Xg represent the transmission coefficient of the partition, path A and path B respectively. "• ^" The first term on the left hand side of Equation (1.3) is the net power transmitted from the source to the receiving room through the partition. The second and third terms on the left represent the powers transmitted by flanking paths A and B respectively. For simplicity the structure-borne path (B) has been assigned an effective transmission coefficient Xg and an effective area of transmission Ag. The term on the right hand side is the power absorbed in,the receiving room which has a total surface area AT and an average sound absorption coefficient 2 a . Solving Equation (1.3) for the ratio of energy'densities e 1 / e 2 gives: £ , / £ , = S A t 1 t X r k r (1.4) 5 Noting that the second term in the numerator is generally negligible and applying Equation (1.1) the noise reduction is obtained. N R = 10 L o q ( — — - ^ ^ 7 — — - ) d B (1.5) 3 \ TpAp + T A A A + T B A B / Equation (1.5) shows that even though a partition may have a low transmission coefficient T and hence a high transmission loss (see Equation (1.2)), the noise reduction between the two rooms i t separates could be substantially reduced by a flanking path having a high transmission coefficient. 1.2 Present Methods for the Measurement and Prediction of Flanking  Transmission The transmission loss to be expected from a particular wall con-struction can be predicted with increasing accuracy by each of the following methods (a) Mass Law [1], (b) Field Incidence [2] and (c) Statistical Energy Analysis (SEA) [3]. The noise reduction to be expected when such a wall is installed in a building, ship, or aircraft is not as easily calculated. The building codes of several countries state that for concrete buildings the sound insulation of a wall after installation is 2 dB lower than the insulation of the same wall measured in the lab (with no flanking paths) [4]. This is a rule of thumb which roughly accounts for the relatively low levels of structure-borne flanking transmission that exist between rooms of heavy, well 6 damped concrete buildings. Such a blanket rule can not be applied to lighter and less well damped structures such as aircraft, ships, and wood framed buildings in which the levels of structure-borne flanking transmission are generally much greater and have a large variability dependent on damping and structural configuration. Zaborov [5] calculated the increase in noise level in concrete walled rooms due to structure-borne flanking transmission. Diffuse sound field conditions were considered to exist in the rooms and transmission coefficients describing the flow of energy from sound field to wall and wall to wall were derived. All the concrete walls were assumed to be thick enough to have * piston radiation characteristics over the entire frequency range of interest. In other words, the study was confined to frequencies well above the critical coincidence frequency (see Section 3.4 ) at which the response of the walls is mass controlled. For thick concrete structures this includes all frequencies above about 200 to 300 HZ. However for gypsum board and plywood walls, ship hulls and bulkheads, etc., resonances and coincidence effects control the response and hence the sound radiation over large parts of the audible frequency range. Since Zaborov's work dealt only with the region of mass controlled wall response, the damping capacity of the walls was not a crucial factor and was therefore not included in the analysis. However, when a wall or panel being considered is resonant and coincident in the frequency range of interest, the values of damping assumed are critical to the response level predicted. * A piston radiator is one in which all parts of the radiating surface move in phase. The amplitude of the surface motion is then mass controlled. 7 A theory is then needed that applies above and below the coincidence frequency i f the effects of structure-borne flanking transmission are to be predicted in lighter walled elastic structures such as aircraft, boats and wood frame buildings. This requires that the acoustic radiation characteristics of finite panels be known throughout the audible range, and that reasonable estimates of the total damping capacity of each panel and acoustic cavity be made. At present the measurement of acoustic flanking transmissions in existing structures is an awkward and difficult task. It usually involves the construction of a secondary barrier to cover the primary barrier between cavities. By comparing the increase in noise reduction caused by the secondary barrier with its previously measured transmission loss, an estimate of the total contribution of flanking transmission to the receiving cavity sound field can be made. However, no information about individual flanking paths can be gained from such an experimental procedure, and without such detailed information i t is difficult to recommend treatments to reduce, flanking transmission. An experimental method is needed which can detect the contributions of individual flanking paths so that corrective efforts can be directed at those which are most critical. This thesis presents a theoretical analysis and an experi-mental technique which overcome these two limitations on our present ability to evaluate the effects of flanking transmission upon the noise reduction between adjacent cavities. 8 CHAPTER 2 ANALYTIC METHODS FOR FLANKING TRANSMISSION PREDICTION 2.1 Scope and Purpose of the Analytic Work In this and the next two chapters an analytic method based on Statistical Energy Analysis will be developed to predict the effects of structure-borne flanking transmission on the noise reduction attainable between two adjacent cavities. The method is also capable of handling airborne flanking. The emphasis here, however, is on the development of a model which will produce some basic understanding of how structural parameters such as panel material, thickness and damping capacity effect the amount of structural flanking observed. The ideal culmination of a study of structure-borne noise transmission would be the establishment of guiding principles, which, i f followed during the construction of buildings,ships or aircraft, would result in a minimization of this transmission. However, con-sidering the enormous variety of structural configurations in which this type of sound transmission is a problem, and the strong dependence of solid wave transmission and sound radiation efficiency upon details of the structure's geometry, materials and construction technique, the guiding principles that could be established from a study of any one particular type of structure would be so general as to be useless. It is not the aim of this work to make such "handbook recommendations," since the data base available could not support 9 them. The purpose is to study the effects of structural parameter variation on flanking transmission, and, in doing so, test the ability of a relatively new analytic technique, Statistical Energy Analysis, to predict the steady state dynamic response of a fairly complex multi-degree of freedom system. Provided the statistical energy model of the particular test structure is successful, the arguments and assumptions used in its development can be applied to the modelling of other structures about which there is more realistic concern. Once good agreement with experiment has been assured, computer simulation can be used to observe the effects of parameter variation within the bounds of the statistical energy model. Some practical recommendations may be made from the results of such simulation but care must be taken not to overstep their legitimate range of application. To aid in broadening this range the acousto-mechanical system modeled here was made as generally representative as possible. The system consists of a hollow rectangular box, divided into two adjacent cavities by a full partition. It forms a basic unit in architectural, marine and aerospace structures (see Figure 2.3). Although this unit may rarely occur in the isolated form seen in Figure 2.3, but more often as part of some larger structure, i t is the transmission of sound and vibration from a space housing a sound source to the immediately adjacent spaces which is most critical. Hence study of the basic unit is justified. 10 2.2 Statistical Energy Concepts In the last decade the need to produce aerospace structures capable of withstanding the severe vibration environments to which they are subjected, by their engines and by turbulent boundary layer forces, has led to the development of a structural dynamics technique which has become known as Statistical Energy Analysis or SEA. The basis of this technique were established in the early 1960's by Lyon and Maidanik [6], Smith [7], Ungar and Scharton [8] and others. SEA permits the calcu-lation of both the spatially and temporally averaged steady state response of complex structures to random loading and the flow of energy between coupled resonant acousto-mechanical elements. It does so with much less effort and computation than would be required i f classical or even Finite Element methods were to be used. The price paid for the use of this effort saving technique is the loss of the ability to predict spatial variations of response within a particular acousto-mechanical element. However, i f the systems to which SEA is applied satisfy the criteria on which the method is based, then such spatial variations will, by necessity, be minimal. Acousto-mechanical "elements" here refer to physical entities such as plates, beams or acoustic cavities which are the basic energy exchanging units of Statistical Energy Analysis. The technique predicts a steady state response level for each element which is assumed to be uniform throughout that element. To illustrate the advantages of SEA in the analysis of multi-degree of freedom systems subject to broad band excitation, consider the calculation of the resonant response of a finite plate to a random pressure field by the classical differential equation technique. It would be necessary to determine the mode shapes of all the natural plate modes having their resonances within the frequency range of interest subject to the boundary conditions of the plate. The response of each of these modes to the prescribed excitation would then be calculated from the differential equation of plate motion and these superimposed to give the total plate response. This technique is feasible at low frequencies where only the fundamental mode and its fi r s t few harmonics are of interest since these low order mode shapes are quite easily predicted and superimposed. However, as the frequency range of interest extends upwards, the resonant modes become more numerous and their mode shapes become more complex and more sensitive to the boundary conditions and geometry of the plate [9]. Hence a complete mathematical description of the geometry, boundary conditions and elastic properties of the plate and its adjoining elements would be needed to predict these mode shapes accurately. It is not hard to picture how the complexity of such a pro-cedure would increase as frequency was extended over the audible range and beyond, especially i f multiple element systems were considered. Statistical Energy Analysis avoids these formidable problems by dealing, not with individual modal displacements or velocities, but with average modal energies. Elastic elements can store energy in their various natural modes of vibration. Each energy storing element can be considered to be composed of a system of single degree of freedom oscillators each with its own resonant frequency [10]. If the wave fiald in the element is diffuse then all the oscillators (or natural modes) which have their resonant frequencies within any particular narrow frequency band Aw, will have the same energy of vibration [11]. Since element modes, adjacent in frequency, are coupled by damping and scattering, their energies tend to equalize within a narrow frequency band even though the wave field may not be totally diffuse [12]. The oscillators representing the modes of an element are assumed to be linear so that significant coupling can only exist among the modes of an element i f the modes are of similar frequencies [13]. Therefore the energies of the modes of an element can only be considered to be equal within a narrow frequency band. Given that within an narrow frequency band Aw, the modes of an element do have equal energies, then the average "modal energy" of the modes in that band can be defined as: _ E(ACJ) tm - " r ~ r ~ — (2.1) where n(w) = the "modal density" of the element at frequency w; i.e. the number of modes resonant in a unit bandwidth centered on w E(Aw)= the total resonant energy possessed by the element in the frequency band Aw. It is the difference between the modal energies of coupled resonant elements in a particular frequency band that ultimately A diffuse wave field is one in which the waves travel in all directions with equal probability and therefore the energy density is constant throughout the element. 13 d e t e r m i n e s t h e amount a n d t h e s e n s e o f t h e p o w e r f l o w b e t w e e n t h e m . The m o d a l d e n s i t y a n d h e n c e t h e w a v e f i e l d d i f f u s i v i t y o f m o s t r e s o n a n t a c o u s t o - m e c h a n i c a l e l e m e n t s i n c r e a s e s w i t h f r e q u e n c y ( w i t h t h e e x c e p t i o n o f t h e p l a t e w h o s e m o d a l d e n s i t y d e p e n d s o n l y o n i t s d i m e n s i o n s a n d b o u n d a r y c o n d i t i o n s ) s o t h a t t h e c o n c e p t o f a n a v e r a g e m o d a l e n e r g y b e c o m e s m o r e l e g i t i m a t e a t h i g h e r f r e q u e n c i e s . C o n v e n i e n t l y t h e n , a s c l a s s i c a l a n d o t h e r m e t h o d s o f a n a l y s i s become c u m b e r s o m e , S t a t i s t i c a l E n e r g y A n a l y s i s b e g i n s t o r e a l i z e i t s f u l l p o t e n t i a l . Two C o u p l e d E l e m e n t s C o n s i d e r t h e f l o w o f e n e r g y b e t w e e n t h e t w o s e t s o f m o d e s , r e s o n a n t i n t h e n a r r o w f r e q u e n c y b a n d A w , i n e l e m e n t s 1 a n d 2 o f F i g u r e 2 . 1 . d i s s , F i g u r e 2 . 1 S c h e m a t i c P o w e r F l o w D i a g r a m f o r Two C o u p l e d E l e m e n t s 14 Several assumptions must be made about the coupling which exists between individual modes of the set in element 1 and each mode of the set in element 2 before a power balance can be written between the two elements. These are [14]: (a) The coupling must be linear. (b) It must be conservative (neither supplying nor extracting mechanical energy). However in almost all real mechanical coupling situations, some energy is dissipated. More recent works [ 14a ]. have shown that this does not limit the applicability of SEA to any extent i f the coupling losses are not great. (c) The coupling between an individual mode of one set and each mode of the other set must be approximately equal. Recalling that only a narrow frequency band is considered, i t is reasonable to assume that within such a band an individual mode of one set will encounter and couple with only one type of mode from the other set and that these couplings will be of about equal strength. (d) In order that TT-J ^ (the net rate of energy flow between the two elements in a frequency bandwidth of 1 rad/sec, centered on co) be simply equal to the sum of two constituent energy flows TT -j ^  and TT^  ^ , i t is necessary that the forces driving the two mode sets in that bandwidth be uncorrected and have spectra that are flat over the frequency band Aco. This is the case in most diffuse field sound and vibration problems 15 since usually only one element is directly excited, and energy flows from i t into adjoining elements, there setting up wave fields. The transmitted forces which act to set up these secondary wave fields are generally delayed and randomized in phase to such an extent [15] that the two forcing functions are quite dissimilar at any instant. These conditions assured, the net power flow between elements 1 and 2 in a 1 rad/sec bandwidth centered on w can be written as [16]: (2.2) In writing this power flow equation, the "coupling loss factors" n-j 2 a n d n2 i a r e defined. They are the fractions of the total element energies in bandwidth Aw, E-j or E 2, which are transmitted to the adjoining element with each cycle of vibration. The evaluation of coupling loss factors between pairs of coupled resonant elements is often the critical activity in the application of SEA to the study of new systems. See sections 3.2 and 3.3 and 3.4. Fortunately i f the resonant modes of the coupled elements meet the criteria of equal energy and equal coupling with modes of the adjoining element within a narrow frequency band Aw, then the following consistency relationship between the two complementary coupling loss factors is seen to hold [17]: 16 (2.3) (See Appendix A for a derivation and a verification of this relationship.) This relationship is particularly useful since the modal densities n^  and n 2 are usually calculated easily and therefore i f one coupling loss factor is known, the other may be calculated directly. Further i t is clear from Equations (2.2) and (2.3) that i f the two elements have equal total energies E^  and E^ in a particular narrow frequency band, the net power flow will be from the element with the smaller modal density to that with the larger. the coupling loss factor n.2 -j in Equation(2.2),the net power flow becomes: and introducing the modal energies E and E from Equation (2.1) as ni-j m2 the independent dynamic variables, Equation (2.4) becomes: Using the consistency relationship, Equation (2.3), to replace (2.4) (2.5) Equation (2.5) expresses the underlying principal of SEA; that within a narrow frequency band centered on w, the net power flow between two coupled resonant elements is proportional to the difference 17 in their modal energies at that frequency and that the sense of this flow is from the element with the higher modal energy to that with the lower [18]. A clearer understanding of the principal of SEA can be obtained by considering the close analogy between the phenomena of energy transmission between diffusely resonant elements as described by Equation (2.5) and that of conductive heat transfer. Suppressing the frequency dependence of Equation (2.5), i t is seen that the following analogies can be made: (a) power flow (TT-J 2) to heat flow. (b) modal energy (Effl) to temperature. (c) the product of coupling loss factor and modal density (n-j 2 n-|) t o t n e product of conductive heat transfer coefficient and area of interface. By writing a simple power balance equation for each of elements 1 and 2, the steady state energy level of each element can be determined. However, as Figure 2.1 shows, there are power flows other than that between the two elements which effect their overall energy levels. There is the possibility of energy being supplied to each element by a source external to the system and there is always the loss of energy through dissipating mechanisms. From the power flow schematic, Figure 2.1, the following power balance equations can be written [19]: 18 (2.6) " i n , CO 7. E . + 0) (2.7) (2.8) 1« Hi (2.9) where TT. and TT. are the rates of energy flow into elements 1 and 2 1 2 respectively (in a bandwidth of 1 rad/sec centered on co) from sources external to the system and TT. . and TT,. are the rates of energy J di ss-j disSg 3 J dissipation in elements 1 and 2 respectively (in a 1 rad/sec. bandwidth). Also n-j and are the "loss factors" of elements 1 and 2 which give the fraction of the total element energy that is dissipated with each cycle. If, for example, element 1 was a flexible panel hung in a reverberant room (element 2), then would express the porous boundary absorption of the room and its air absorption, and n-j would include the hysterisis loss of the panel and any edge damping present. The coupling loss factor n-j 2 W 0 L | l d D e based on the acoustic radiation capabilities of the panel (see section 3.4). It should be noted that the coupling loss factors, for example n-| 2' are strictly only defined when there is zero energy in the receiving element (here element 2). To illustrate this point Equations (2.6) and (2.8) are combined to give: 19 'tt/n, + ^mz ~ T j i s s , + TTdissi . (2.10) Thus all the power supplied to the elements is dissipated (since we deal with steady state conditions). The power input to element 1, the panel, must therefore equal the total rate of energy removal from i t . This can be expressed as follows: (2.11) where Rtot 1 S the total damping factor of the panel. It is composed of an internal damping term R. . and a radiation damping term R„ int raa The panels space averaged mean square velocity is represented by <v >. Since the total energy E-j of a panel of total mass M in a particular narrow frequency band is given by: = M <Vp*} , (2.12) Equation (2.11) becomes: and R - tot E l M TTin, - 77 " > (2.13) TT. - 'Mnt -I + r* q ^ 1 . (2.14) ,n> AA M 20 In terms of loss factor n-| and coupling loss factor n-j 2 Equation (2.14) becomes: TTin, = WK|, E, + ( J ^ E , . (2.15) This is only equivalent to the power balance (Equation (2.7)) when the energy in element 2 is zero. Hence n-| 2 1 S strictly only defined when E 2 is zero. However in most real situations only one element is supplied with power by an external source, for example a shaker driving the hanging panel, so that E » E and m o is approximately equal to its value when E 2 = 0. Three Coupled Elements As another preliminary example, consider the transmission of sound between two adjacent rooms separated by a homogeneous panel [20]. An illustration of the system and its schematic power flow diagram are seen in Figure 2.2. Here the exterior walls are considered to be infinitely s t i f f so that they cannot exchange energy with the sound fields. Therefore no flanking transmission takes place. Infinitely sti f f walls Source Room (1) Receiving Room (3) - Panel (2) n 1,3 — H f n d i ss 1 n i n | (1) n l , 2 (2) n2,3 (3) ? ndi n i n , S S , diss. Figure 2.2 Three Coupled Elements (Sound Transmission Between Rooms) From the power flow schematic the following power balance equations can be written: TTin, ~ "Hats*, + ^ 1 , 2 . + TT|,3 > (2 E j ^ 3 » (2 (2 + cu E z E3 " 3 ? ( 2 22 ISS: ^ , 3 (2.20) u 3 n 3 VI, n 3 .(2.21) The coupling loss factor n 2 3 is based on panel radiation and is equal to n 2 1• The consistency relationship, Equation (2.3), is used to evaluate the coupling loss factor n-j 2 i n terms of the more easily determined n 2 -|-The power term TT^  ^  represents the non-resonant mass law transmission of energy through the panel. This energy is not stored temporarily in the wave field of the panel but passes directly through by virtue of panel modes which are resonant outside the frequency band presently being considered. The mass law coupling loss factor n-j 3 is derived in section 3.2. Equations (2.17), (2.19) and (2.21) can be solved for any one of the three steady state element energy levels E-j, E 2 or E^ in terms of an arbitrary input power to one of the elements. In this case power is only supplied to the source room (element 1) so that 1. = TT. =0. Alternately the equations can be solved for the ratio 1 3 of energies in the source and receiving rooms E-J/E-J and hence be made to yield the noise reduction between the two rooms. 23 2.3 The Prediction of Noise Reduction in the Presence of Structure- borne Flanking Transmission Using SEA 2.3.1 The SEA Model Several authors have recently used Statistical Energy Analysis to predict the transmission loss of partitions of various configurations. This method has been applied to single panels [21] and double panels with an air gap [22] by Price and Crocker, to double panels coupled by tie plates [23] by Bhattacharya, Crocker and Price and to multiple panels [24] by Mullholland, Price and Parbrook. The results have been more than satisfactory. In the case of the single panel the agreement with experiment was to within ±2 dB over the frequency range 400 to 10,000 Hz when measured values of panel radiation resistance were used. The present work utilizes SEA to study the effects of struc-ture-borne flanking transmission on the noise reduction attainable between two adjacent acoustic cavities, the walls of which are struc-turally coupled. Structure-borne noise is most successful at reducing sound insulation in structures which are light and resilient and hence can easily interact, (i.e. exchange energy), with a sound field. The structures should not be heavily damped, so that vibrational energy is able to accumulate and propagate some distance within them. These criteria are certainly met in some architectural and in most aerospace and marine structures. In the above investigations of partition transmission loss, the cavity walls, except for the partition, were assumed to be infinitely rigid (see Figure 2.2). They did not accept energy from 24 the source cavity sound field or give up energy to the receiving room sound field. The only energy transmission was into, out of and through the partition. However, in the present case the cavity sound fields exchange energy with the cavity side walls as well as with the partition. There is also energy exchange between the cavity sidewalls and the partition, and between the sidewalls themselves. New coupling loss factors had to be obtained to describe the flow of energy between the coupled walls and partition. These are derived in section 3.3. The SEA model considered here consisted of a light rwalled rectangular box separated into two cavities by a partition. It is schematically depicted in front and end views in Figure 2.3. All exterior walls, that is all walls except the partition, were considered to be of the same material and thickness. The partition could be of a different material and thickness and could be located at any position thereby generating two cavities of any desired volume ratio. The elements (panels and cavities) were numbered 1 through 13 as shown in Figure 2.3. Analyzing a system with 13 different elements would be an enormously cumbersome task even when using SEA. Therefore two simplifications have been made to reduce the number of unique elements. Figure 2.3 The Schematic SEA Model of Two Structurally Coupled Cavities The model cavities have been considered to be square in end view. This induces a considerable simplification since i t means that the four sidewalls of each cavity (elements 4, 5, 6 and 7 in the source cavity and 8, 9, 10 and 11 in the receiving cavity) have the same shape and area and therefore the same modal density (see section 3.1). Since the sound field in each cavity is considered diffuse and since the coupling of the four identical sidewalls of each cavity to their neighbouring soundfields and panels is the same, then i t follows that the four source cavity sidewalls have equal energies in any particular bandwidth AOJ as do the four sidewalls of the receiving cavity. Having equal modal densities and equal total energies, the sidewalls of each cavity must then have equal modal energies in any narrow handwidth Aw by Equation (2.1). Therefore by Equation (2.5), there can be no net power flow between any two sidewalls of the same room. The four sidewalls of each cavity can therefore be referred to collectively as one resonant element (having four times the energy storage and transmission capacity of one sidewall alone.) Accordingly the source and receiving cavity sidewalls are known collectively as elements 4 and 8 respectively. The number of unique elements is therefore reduced from thirteen to seven by considering the model cavities to be square in end section. The heuristic value of the SEA model is l i t t l e reduced by this simplification. The second simplification involves the source and receiving cavity endwalls, elements 13 and 12 respectively. They have not been considered as resonant, energy storing elements in this analysis. However, they do permit the mass law transmission of sound energy through them to the model exterior (as do all exterior walls) and this is included in the energy dissipation terms of the source and receiv-ing cavity sound fields (see section 3.2). The space exterior to the model is considered to be an energy sink so that any sound energy emitted from the model boundaries, either by mass law transmission or by panel radiation, is lost. The latter radiation loss is accounted for by applying a suitable loss factor to the energy of the cavity sidewalls (see section 3.4). This limitation of the participation of the endwalls in the SEA model was felt to be justified because the endwalls represent only about 1/6 of the total wall area of their cavities. They there-27 fore account for only about 1/6 of the energy which, after being extracted from the source cavity sound field for example, is available for structural transmission into the receiving room. Furthermore this energy, in the form of a flexural wave field in the endwall (element 13) of the source cavity, must suffer the attenuation of two structural junctions before any of i t can reach boundaries of the receiving room and there be radiated as sound. However flexural waves excited in the source cavity sidewalls (of which there are four) encounter only one structural junction before entering the boundaries of the receiving cavity. Therefore i t is unlikely that the increase in accuracy to be gained by treating the endwalls (elements 12 and 13) as resonant energy storing elements would justify the ensuing increase in complexity of the model. The increase in complexity referred to is due in part to the need to write and solve two more power balance equations. However the greatest difficulty arises with the necessity, when considering flexural wave transmission through more than one structural junction, of dealing with the phenomenon of "mode transformation" [25]. When a flexural wave in a panel impinges normally upon a discontinuity such as one of the panel junctions of this model, not only flexural butlongitudinal and distortional waves are transmitted and reflected (see section 3.3). Mode transformation refers to this transfer of wave * Distortional waves are "distortions" of the travelling flexural waves which suffer rapid exponential decay with distance from the point of generation [25a]. They are poor radiators of sound [25b^. energy from one type of solid wave to another. It is the interaction between flexural and longitudinal waves which is of importance here. Figure 2.4 shows the simplified SEA model (now having only five unique resonant elements [circled]) and the three flexural wave amplitude transmission coefficients which are included within the bounds of the model (T^ 2 ' T4 8 a n c* T2 8^ ' T DIIIII Mlllllllllll III © 4 > 8 © Figure 2.4 The Simplified SEA Model (5 Resonant Elements) Such a coefficient T. gives the amplitude of the flexural wave transmitted into panel j when a unit amplitude flexural wave impinges normally on the junction of panels 4, 8 and 2 from panel i . These three coefficients are derived in section 3.3. To illustrate why "mode transformation" should be accounted for when i t is desired to trace the transmission of flexural wave energy through more than one structural junction, the following situation is considered. A flexural wave is generated in the end-wall (element 13 of Figure 2.4) by the source room sound field. It impinges on the junction between endwall and sidewall (element 4) where i t generates transmitted flexural and longitudinal waves in the sidewall. When these two secondary waves in turn reach the junction of sidewalls 4 and 8 and the partition 2, they both generate, among other things, flexural waves in sidewall 8 and partition 2. These tertiary flexural waves then radiate sound into the receiving room. Therefore to predict accurately how much of the energy extracted from the source room sound field by the endwall 13 reaches the receiving room sound field by structural paths, i t is necessary that the panel elements involved have the ability to store and exchange the energy of both flexural and longitudinal wave fields. This requires that there be two unknown wave field energies associated with each of these elements. Unless these two energies could be expressed as fractions of the total element energy, there would be more unknown energies than power balance equations making their explicit solution impossible. In addition to this complication, several extra wave amplitude transmission coefficients would have to be evaluated i f the endwalls were to be considered as resonant energy storing elements. Mode transformation occurs at all structural junctions. To overcome the complications imposed by this phenomenon upon the description of power flow between the panel elements which are con-sidered resonant, that is, elements 4, 8 and 2, a final simplifying assumption has been made. Any energy possessed by a wave after i t has propagated through more than one structural junction is neglected. Therefore any flexural wave energy that is generated by the impingement of a longitudinal wave on a junction is neglected since the longitudinal wave must itself have been generated by a previous impingement. In explanation of this last statement i t should be noted that panel longitudinal waves cannot generate or be generated by a sound field, and owe their existence to the action of flexural waves at panel discontinuities. Since panel longitudinal waves cannot themselves produce sound and since flexural wave energy generated by longitudinal waves at junctions is to be neglected, only flexural wave energy need be considered in writing the power balance equations for each resonant element of the SEA model. However longitudinal and also distortional waves must be considered when determining the flexural wave amplitude transmission coefficients between panel elements. 2.3.2 The Power Balance Equations Power is supplied to the model by a suitably wide band noise source located in the source cavity. The diffuse sound field thus produced in the source cavity impinges on the cavity walls and induces diffuse flexural wave fields in them. The energy stored in these flexural wave fields is continually being depleted by: (a) sound radiated back into the source cavity and to the model exterior. 31 (b) structural transmission to the walls of the receiving cavity. (c) internal and structural junction damping. The relative strengths of the energy supplying and depleting mechanisms acting on a particular panel or acoustic cavity determine its steady state energy level. The schematic power flow diagram for the SEA model of two structurally coupled cavities (now reduced to 5 unique elements) is shown in Figure 2.5. ^ing ^disSg Figure 2.5 Power Flow Diagram for the SEA Model of Two Structurally Coupled Cavities 32 As in section 2.2 the terms IT. (i = 1,2,3,4,8) represent i possible power flows into elements i from sources external to the system. The total power loss suffered by element i due to all dissipa-tive mechanisms acting on i t is represented by the term TT^- . The i terms Tr-j 2» 4' ^2 3 a n c* ^8 3 r e P r e s e n t power transferred from sound field to panel or vice versa by virtue of the coupling of acoustic waves and panel flexural waves (see section 3.4). The terms ir- 0, TT« 0 and TT^  g represent the flow of flexural wave energy from one panel to another and again g stands for the power transmitted between cavities due to the mass law (non resonant) response of the partition. With the aid of Figure 2.5 power balance equations can be written for each of the model's five unique resonant elements as fol1ows: ~ TTdt»t ~ ~ 4 ^ + 4 % 2 + 7T 2 ) 3 (2.23) •TTih3 = T T A « 3 -TT2t3 -4 - 7 T M - 7T,)3 ( 2 > 2 4 ) ^Tf«4 = ^ a u s * ~ TT|>+ + TT4>2. + 7T 4 > 8 (2.25) ^Ina - ^ J ' . M S + ^ . s - TTa.e ~~ (2.26) 33 The only external source of power here is a sound source in the source cavity (element 1). Therefore ir. = 0. i n2,3,4,8 When expressed in terms of the total element energies (E.), loss factors (n-)» coupling loss factors (n-.) and modal densities (n.j) described in section 2.2 (all defined for a bandwidth of 1 rad/sec centered on w), Equations (2.22) to (2.26) become: O = u> O = O . = o> ^ 4 O = Et + fc"?i|3n. _ i v\, (2.27) n, n» '4- VU *3 9 (2.28) CO (2.29) 'E, E> n, n4_ E2" E 4 E n4 n 8 (2.30) If 7*. n- (2.31) 34 These five simultaneous equations in five unknown element energies can be solved for any of these energies in terms of the arbitrary input power IT. . They have been solved here for the i n 1 ratio of total energy of the source cavity to that of the receiving cavity E-j/E3- Cramer's rule was used in part to find the numerator terms of E-j and E^ and the ratio of these was taken yielding E-|/Eg> the denominator terms of both energies being equal to the determinant of the coefficients of Equations (2.77) to (2.31). (2.34) where the consistency relationship, Equation (2.3), has been used to replace coupling loss coefficients which are not obtainable The resulting expression is given by Equations (2.32) -to with those that are; for example ry n2 n8 -j and ~ n 8 3 respectively. 1,2 and r\ 3,8 have been replaced by E 3 (2.32) where : A * = 33 1M n, 1M ( S V . + V ^ I M / V J2s n 8 X I . = 4 % 4 1zt *U »I 36 and where: Equation (2.32) expresses the ratio of total energies in a 1 rad/sec band between the two cavities. However, noise reduction is defined as the ratio of acoustic energy densities, so that each cavity's total energy must be divided by its volume. Then in decibel notation: N R = 10 / o 3 l o ( E , V 3 / E 3 V , ) a B , ( 2 . 3 . 5 ) dlB . (2.36) 37 Now that an expression for the noise reduction between source and receiving cavity of the SEA model has been obtained, the various modal densities, coupling loss factors and loss factors which appear in Equations(2.33) and(2.34) must be determined. These are the critical activities in the application of SEA to the study of new vibratory systems. There are three types of coupling loss factors involved. They are: 1 . that between a reverberant panel and a sound field which is governed primarily by the panel's radiation resistance. 2. that between orthogonally joined panels which depends on the severity of the impedance mismatch offered to flexural waves by the panel joint. 3. that between source and receiving cavity due to the mass law transmission of energy through the partition. Chapter 3 will be devoted to the development of these various parameters beginning with the more straight forward modal densities. 38 CHAPTER 3 THE EVALUATION OF THE PARAMETERS CONTAINED IN THE SEA SOLUTION FOR NOISE REDUCTION 3.1 The Modal Densities of the SEA Elements 3.1.1 Introduction The modal density of an elastic element such as a panel or air cavity is defined as the number of natural modes of vibration which the element possesses per unit of angular frequency. If, as assumed here, the elements are excited to diffuse wave field conditions, then every one of the element's natural modes within the excitation band-width will be resonant. Therefore the larger an element's modal density, the greater is its capacity to store energy in a reverberant wave field since, within a narrow bandwidth at least, each resonant mode has been assumed to possess the same energy. This trend manifests itself in the basic equation of SEA which gave the power flow between two coupled elements 1 and 2. If the elements 1 and 2 possess the same total energy (E^=E2) in a particular narrow bandwidth, then the net power flow in that band (2.4) 39 will be from the element with the lower modal density to that with the higher. A modal resonance occurs in a mechanical element when a wave travels a closed path within the element and returns to its starting position in phase with itself giving rise to a standing wave. The condition for such an occurrence is that the path length travelled in the element by a wave of a particular frequency be equal to an integer multiple of the wavelength at that frequency. 3.1.2 The Modal Density of an Acoustic Cavity There are three categories of natural modes in a rectangular cavity. These are termed axial, tangential and oblique modes according * to whether their wave number has components in one, two or all three co-ordinate directions. A new higher order mode occurs each time the acoustic wavelength decreases enough to permit the inclusion of another half wavelength in any cavity dimension. The derivation of the modal density expression for a cavity is then simply based on cavity mode wave number and cavity dimensions and can be found in most standard texts [26], [27]. The exact modal density of a hard-walled rectangular cavity of volume V, total surface area A t Q t, and total perimeter L = ZK'-x+1-y'H-z) is: Wave number t is given by ui/c where to is the angular frequency of the wave and c its propagation velocity. It expresses the spatial frequency of the waveform in radians per unit of distance. 40 V w 2 A t o t u 8-rr cl L modes radian (3.1) + 16 TT C a The three terms represent the contributions from oblique, tangential and axial modes respectively. For large cavities the oblique mode term alone gives a good estimate but for the small model rooms to be tested, 50 and 80 ft , at least the first two terms should be used. 3.1.3 The Modal Density of Panels The expression for the modal density of flexural modes in a panel is essentially the same as the tangential (two dimensional) mode term of the cavity modal density. However the panel boundary conditions must be considered and the flexural wave speed used. For a panel of thickness h and longitudinal wave speed c^ the flexural wave speed is given by [28]: 60 h c L 2. IT (3.2) If the panel edges are clamped causing zero flexural displacement there, corresponding to zero acoustic displacement at the walls of a cavity, then: r\(cj) = Ap 60 41 8 i r c} V3 Ap 47T h cx modes to o\ radian Beranek [29] gives twice this value for simply supported panels. This larger value is believed to be a good estimate for all boundary conditions at all but very low frequencies and is therefore used in this work. ZlT h C x The modal densities of the source and receiving cavities, n-j and n^ and of the panels n 2, n^, and are then calculated from Equations (3.1) and (3.4) respectively. 3.2 The Non Resonant Mass Law Transmission Coefficient As mentioned in Section 2.2 in explanation of the power flow diagram (Figure 2.2), TT^  ^  represents power which is transmitted from source to receiving cavity by virtue of the mass law response of the partition. This dominates the total panel response at low frequencies for which the panel impedance can be considered to be purely reactive. The transmission coefficient of a plane wave incident on a panel at an angle 6 from the normal is defined as the ratio of the transmitted to the incident acoustic intensity and is given by [30]: In purely reactive mass law response, the panel acts as i f its bending stiffness B and damping factor n were both zero. Application of these two conditions reduces Equation (3.5) to: (3.6) where M_ is the panel surface mass density and p c the specific acoustic s a a impedance of air. Since the transmission coefficient depends on angle of incidence i t must be averaged over the possible range of such angles.For the case of a diffuse sound field this range is 0 <_ 6 <_ TJ- radians; each angle being equally likely. The normal component of intensity from a wave incident on a unit area of the panel at angle 9 is I. cose. By r . i n c J integrating Tf/||_^*inc c o s 9 o v e r a hemisphere with center on this unit area, the total transmitted intensity is attained. In a similar fashion 43 the average value of the mass law transmission coefficient T ^ ( 6 ) is obtained as follows: S L TL.. (©) cose sin© de Ce' cose sin e de . Cose Sin 0 de P* cose sin 0 de The integral in the denominator is simply: ^ ""cose sm e de = o To evaluate the integral in the numerator, note that i t is of the form: n o w : i ( l + K 2 c o s z e ) = c o s e s m e (3.7) therefore : Cos © sin © d© 44 and : >L sin 8 cose de 1 + K 2 cos*e J _ f e L ({(i + ^cos^e) 2.K2 X 1 + K 2 cos z e 1 IK2 ln( l + K a cos*e ) 1 , IT? , n 1 + K' 1 + KxcosKeL Therefore ML 03 M s s i " ©L In 1* (3. Now that the mass law transmission coefficient of intensities has been evaluated, i t must be related to the coupling loss coefficient n-j 3 which gives the fraction of the energy of the diffuse sound field in cavity 1 which reaches cavity 3 per unit of angular frequency by mass law transmission. The power which is transmitted into the receiving cavity 3 is given by the transmitted acoustic intensity times the area of the partition separating the cavities. This transmitted power must equal the rate of energy removal from the source cavity 1 by virtue of n-j 3. Accordingly the following power balance equation is written: 45 = * w A P • ( 3 ' 9 ) Here E-j,the total energy of the source cavity in a unit bandwidth, can be replaced by e^ V^  where is the energy density in the band. The intensity created by a sound wave of energy density e impinging normally on a surface is simply eca. By averaging over all possible angles of incidence, i t can be shown [31] that the acoustic intensity created at a surface by a diffuse sound field of energy density e is ^ p3. Therefore ; J-int - "4 ' (3-1°) and E i = e,V, = Equation (3.9) then becomes: 4 V, Solving for the coupling loss factor: (3.11) tr«XA5 A p C o , 'inc * V< - Ap 4 6 In terms of the transmission loss of the partition: T L = lOloSlo ( l / T M u ) (3.13) the mass law coupling loss factor is expressed as: 10 l o M , , 3 = - T I + > ° H . ( ^ § ) < 3 . « > A coupling loss factor like that of Equation (3.12) is used to predict the energy dissipation suffered by the sound field of the receiving cavity due to mass,law transmission to the exterior of the model. This contributes to the total loss factor of the cavity, r\y It will be referred to as 113 e x t since i t represents energy transmission from cavity 3 to the model exterior. The coupling loss factor ri^ e x t c a n ^ e considered to act as a loss factor in this case since the receiving element, the model exterior, is considered to have zero energy density (an energy sink). Then: Y) ~ T : a ^ x V C t - ( 3 . 1 5 ) where A^ e x t l s the total area of the five exterior walls of cavity 3. 3.3 Evaluation of the Structural Coupling Loss Factors o, g andji^g 3.3.1 The Selection of a Suitable Theory of Wave Transmission One of the simplifications made when the statistical energy model of two structurally coupled acoustic cavities was formulated in Section 2 . 3 . 1 , was that any energy possessed by a panel wave,after the wave had been transmitted through more than one structural junction would be neglected. In view of this simplification i t was shown that only the transmission of flexural wave energy had to be accounted for in the statistical energy model. However in order to determine the transmission coefficients of flexural waves accurately, the presence of longitudinal and distortional waves must also be accounted for. Several authors have developed theories of wave transmission through structural joints aimed at the evaluation of structure-borne noise and vibration in buildings, ships and aircraft. See T. Kihlman [ 3 2 ] , V.I. Zaborov [ 3 3 ] , S.V. Budrin and A.S. Nikiforov [34] and M.C. Bhattacharya [ 35 ] . Kihlman gives flexural transmission coefficients through a cross joint of four semi-infinite plates averaged over all angles of incidence. Zaborov treats several configurations of finite building elements giving flexural transmission coefficients derived without consideration of longitudinal waves. Neither of these authors presented the derivation of their coefficients, making difficult the extension of their theories to deal with the present model. Budrin and Nikiforov determined the transmission and reflection coefficients of flexural and longitudinal waves when unit amplitude flexural and longitudinal waves respectively were incident normally 48 upon a cross joint of semi-infinite plates. The general cross joint could be reduced to a tee, elbow or line joint by setting the appropriate plate thicknesses to zero in the analysis. This basic work was extended by Bhattacharya to treat the case of a light weight finite tie plate coupling two infinite plates. Bhattacharya's plate arrangement models a double wall connected by resilient metal studs and is shown in Figure 3.1. The plate numbering scheme of the present model has been adopted here. (6) (2) x9(xr y2U) (10) '10 tx 10 Figure 3.1 Bhattacharya's Plate-tie-Plate Model 49 Because of the similarity between Bhattacharya's configuration and the resonant panel elements of the present model (see Figures 2.3 and 2.4), and the availability of the derivation of his coefficients, Bhattacharya's analysis was utilized and extended in the derivation of the three flexural wave transmission coefficients required here. These are T4 2' T4 8 a n d T2 8' In order to illustrate the method, the calculation of the coupling loss factor 2> a s Performed by Bhattacharyas will now be outlined. By extension of these calculations n. o will be evaluated. 1,0 Finally in Appendix B, n.2 3 will be obtained by considering a unit amplitude flexural wave originating in plate 2. 3.3.2 The Derivation of 2 (As Performed by Bhattacharya) A sound field (in the present case, that of the source cavity) is considered to generate a unit amplitude flexural wave in plate 4. The wave impinges normally upon the junction 4-2-8 from the semi-infinite plate 4 and generates: (a) reflected flexural and longitudinal waves as well as a distortional flexural wave in plate 4, (b) incident, reflected and distortional flexural waves and also incident and reflected longitudinal waves in finite plate 2, (c) incident flexural and longitudinal waves and distortional flexural waves in semi-infinite plates 8, 6 and 10. 50 The energy of waves induced in plates 6 and 10 in this way, is neglected in the statistical energy model, but here these waves must be considered in order to predict the transmission coefficient into the finite plate 2. The co-ordinate system for the flexural, y, and longitudinal, u, displacements of each plate is shown in Figure 3.1 where junction A (4-2-8) is taken as the origin. The equations governing the two types of plate displacements y^ and u^  at the junctions A and B (i = 2,4,6,8,10) are written as: y 4 = G x p ( j k 4 x 4 ) + B4exp(-j k4x4) + C 4 ex P (k 4x 4) (3.16) y2 = A 2 exp(-jk 2x z) + B 2eK P ( j k 2x^ (3.17) + ^ x p C - M a ) + F * e x p ( - k 2 J l + k 2 x 2 ) y8 = B8 e * p ( j k8x8) + C8 e xpC - kgXa ' ) (3.18) y « - B 6 e x p ( - j k^Xc) + exp ( k< X 4 ) ( 3 . 1 9 ) % = B,0exp(j klox10) + Cl0 exp(-k10x/0) (3.20) U 4 = H 4 expfj \>4.X4) (3.21) (3.22) 51 U 8 = G 8 e x P ( j p $ x & ) (3.23) U 6 = G 6 exp ( - j p 6 x 6 ) ( 3 2 4 ) U l 0 . = G , 0 e x p ( j p,0 X l 0 ) (3.25) where the time dependence of displacement given by the factor e~Jco^ has been suppressed and where : ^2 ~ ^ n e a mPlitude of the flexural wave moving towards junction A in plate 2. B.j(i=4,2,8) •= amplitude of the flexural wave moving away from junction A in plate i . B. (i=6,10) = amplitude of the flexural wave moving away from junction B in plate i . C. j(i = 2,4,6,8,10)= amplitude of the distortional flexural wave in plate i . G. .(i=2,6,8,10) = amplitude of the longitudinal wave leaving junctions in plate i . H. (i=2,4) = amplitude of the reflected longitudinal wave in plate i . Since a unit amplitude flexural wave was considered incident on junction A, the amplitudes Bi(i=2,4,6,8,10) are in fact the transmission and reflection coefficients of the various induced flexural waves. The flexural and longitudinal wave numbers and p^  are given by: 52 2 VJ CO (3.26) and (3.27) placement equations (3.16 to 3.25) must satisfy the following boundary conditions: (a) continuity of linear displacement, (b) continuity of angular displacement, (c) sums of forces in the x and z directions must be zero, (d) sum of bending moments about the y axis must be zero. The application of these boundary conditions results in nine equations at each junction giving in all a set of 18 equations in 18 complex variables (the wave amplitudes). These equations will not be listed here in their entirety nor will their generation be examined further [36]. However, the complete derivation of the transmission coefficient g is given in Appendix B. The reduction and solution of the 18 equations to yield the flexural and longitudinal wave amplitudes in plate 2 as performed by Bhattacharya and the evaluation of the flexural wave transmission coefficient x^ g by the author, will now be outlined. As in the present statistical energy model, the four side-walls (4,8,6,10) of Bhattacharya's model were of the same thickness 53 h,and material. Therefore Bhattacharya was able to adopt the following simplifying notation: S , « exp(-jf> ai) J S 2 = e x p ( j p 2 j l ) * = E ^ P * / 0 * * * ; b = E 4 h 4 p 4 / D A k * c = p a k * / D 4 k 4 ; <* = k t / k 4 After a fairly lengthy process of variable elimination Bhattacharya reduced the original 18 equations to the following sets of two and four equations in the amplitudes of the longitudinal and flexural waves in plate 2 respectively. fc(l-j) = G t ( z - a - 2 j ) + H 2 ( 2 + a - 2 j ) (3.26) 0 = G2 (2+ a+2j) + H z S , ( 2 - a - 2 j ) (3.27) 4 = A 2 ( c - 4 o c - jc ) + B 2 ( c + 4<x-jc} -C z[c-j ( c + 4 « ) ] - r 3 Fj,[c + j(4-o(-c)] (3.28) O = j A 2 ( » + 2 b ) - j B 2 ( ) - 2 b ) ~ C 4 ( i - j a b ) ' + r 3 F a(i + j 2 b > ) (3.29) o ;= ir\Ax(\-zb) - j r x B z ( i + 2b) - r 3 C 2 ( i + j ^b ) + F2 ( i - j a b ) O = r, A 2(c + ao(' + j2of) + rz B2(c-2cx-j2<?c) To facilitate the explicit solving of Equations 3.28 to 3.31 for the flexural wave amplitudes A2, B2, C 2 and F 2, the following approximations were made. If the tie plate 2 is light compared to the side plates 4,8,6 and 10 as was the case with Bhattacharya's model then b » 1 and r 3 « 1. By applying these inequalities to Equations (3.28) to (3.31), the following matrix of complex coefficients was obtained: A, C + 4-c< - j c C - j ( C + 4 « ) 0 4 j 2 . b j 2 b O 0 - j z b r , - j 2 b r 2 - j z b r3 - j 2 b 0 O 0 The solution of this set of simultaneous equations for B A9, C9 and F 9 gives: (3.30) (3.31) 55 Q -j — — (3.32) Z r,(C + 2o<)20-j) - r z [ c z - 4 o < C - 4 o C z - j ( c z + 4 o < C - 4 e C z ) J * l ( C - J 2 ° 0 Bj, (3.33) r, ( c + 2<x) [cto-rfl - 2tt(rt»jrQ] Ba (3.34) r,(c+2tx) p 1 - g « * ( ' » J ) B 2 (3 3 5 ) C + 2.« It will now be demonstrated that the approximations b » 1 and r 3 « 1 and therefore the results expressed by Equations (3.32) to (3.35) applied to the models tested experimentally in this work. The model rooms tested (see section 4.1) had side plates of 1/16" aluminum and were separated by a partition (plate 2) of 1/32" aluminum in one case and 1/8" aluminum in the other. The case which deviated furthest from Bhattacharya's assumption of a light partition, that i s , the 1/8" partition case, will be considered. (a) b = where  E+ h+ h D2 = E - z I i ^ i ' ~ E 2 1^/12 pev unit length, , , JO_ _ CO 56 6J | u P4 = r ; K= ^ j = 0.0/04' , Z 7T* h a 3 / i f * ' The longitudinal wave speed in aluminum, c £, is 17,000 ft/sec. 4.6 x / o * .'. b — so that even at the highest frequency of interest, 20,000 Hz, b = 336. » 1 (b) r 3 = e ' k a , w h e r e £ = 4 . 0 *f. Then at the lowest frequency under consideration, about 200 Hz, (set by the limit for diffuse field conditions in the small model rooms), we have: - 2 0 57 -9 2.0 x 10 « 1 Therefore although one of the experimental partitions used in the present work had twice the surface density of the sidewalls, the inequalities r^ « 1 and b >> 1 s t i l l hold. Since a unit amplitude flexural wave was considered incident on junction A from plate 4, the flexural wave transmission coefficient from plate 4 to plate 2 is simply: r4 = i M 4 l 3.3.3 The Derivation of 8 Bhattacharya was not concerned with the coefficient x ^ g as he was interested in wave transmission between the two sides of a double wall connected by metal studs. Of the nine displacement equations generated by applying the boundary conditions at junction A, four can be found which contain the flexural wave amplitudes Bg, Cg, B^  and C^ . These can be solved to give Bg in terms of the flexural and longitudinal wave amplitudes in plate 2, In matrix form the four equations are: B8 c8 1 1 -1 -1 » J - i * J -1 1 * J i 1 - 1 - i 1 58 1 J j[l+a(G 2-H 2)J l+c (M 2 -B 2 + + r3F2) (3.37) Solving for B g and neglecting terms in which r^ is a factor in accord with r^ « 1 yields: (3.38) where from Equations (3.26) and (3.27) in which s 1 = e~ J P2 £, s 2 = e d p2 A i t is easily seen that: G - - H 2 = — T -=; p ; ~ ' (3.39) 4 [ a c o s p z JL + Z sin ? t i j + j aLct 5 m ~ ^ c o s P i AJ Finally upon substituting the expressions for amplitudes A2 and C 2 from Equations (3.33) and (3.34), Bg can be written as: a /• \ c Bo _ r, (c+ 2.o<) (3.40) 59 and ^ 4 , 8 = l B s | (3.41) 3.3.4 The Derivation of TQ g and the Coupling Loss Coefficients The third flexural wave transmission coefficient required in this analysis, T 2 g> is derived in full in Appendix A. This was done by considering a unit amplitude flexural wave to be normally incident on junction A from plate 2. The result was: t 2 , 8 = l Bs| 2*c(q.i-z)(j+  4-«a-£c*- 2c(3-a) + j[4(*a + c (a-i) - 8oc] Having now available expressions for the various transmission coefficients of flexural wave amplitudes, i t remains to incorporate these into energy transmission expressions, that is the coupling loss factors 2> n4 8 a n c* n2 8' 1 S desired to find the fraction of the flexural wave energy incident on a unit length of structural junction per unit of angular frequency, which is transmitted through the junction as flexural energy. The ratio of the power carried into a junction by the incident flexural wave to the power carried away by the transmitted flexural wave will give the desired coupling loss coefficient [37]. The power associated with a flexural wave is given by the product of its kinetic energy density and its group velocity (the velocity at which energy propagates through a medium). The maximum kinetic energy per unit area of a panel supporting 2 2 a flexural wave of amplitude A is: 1/2 m^u A where nip is the mass per unit area of the panel. The group velocity of flexural waves is given by, „ dCf 7ThcA " c 3 " «* + * ~ T W ~ - C f + WaT = 2 f Therefore the coupling loss coefficient per unit length of junction between panels 2 and 8 becomes: vn a coz \ B8\Z c f R m8 c> a T 2 % Similarly: 4.2 ™ 4 C"f4 2 ' L - ~ ~ L4,8 61 3.3.5 Experimental Verification of the Flexural Wave Transmission Coefficients T . 0 and X / , 0 In order to gain confidence in the flexural wave transmission coefficients calculated in section 3.3, an experimental technique based on cross correlation was developed to measure these coefficients. A plexiglas plate-tie-plate model similar to Bhattacharya's (Figure 3.1) was suspended with springs from a heavy frame. The semi-infinite side plates were represented by 4' x 4' xl/8" plexiglas sheets and the tie plate (partition) by a 4' x 2' x 1/16" sheet (see Figure 3.2). Plexiglas can be bonded very strongly without increasing the size of a joint appreciably and hence allowed a good approximation of an ideal butt joint to be obtained. Figure 3.3 shows schematically the experimental layout. Side plate 4 was excited into random vibration by Bruel and Kjaer Mini shaker applied at a point, and the attenuation of the spreading flexural waves by the structural joints was measured with the aid of an accelerometer and a PARI01A Correlation Function Computer. As will be discussed in detail in Chapter 5, cross correlation allows waves which are spreading directly out from the source (shaker), and hence s t i l l bear a temporal resemblance to the random input force which created them, to be distinguished from the myriad of often reflected and standing waves which are always present in vibrating finite bodies. If an accelerometer was used to directly measure the acceleration levels at points 4A and 8A on plates 4 and 8 in Figure 3.4, i t would give the resultant levels due to all flexural waves passing through those positions. 62 Figure 3.2 Plexiglas Model for Measurement of x^  2 a n d t ^ ) 8 Cross correlation is simply the computation of the time averaged product of two time varying signals, here, the shaker input and the accelerometer signal. The cross correlation function is obtained by computing these time averaged products with various amounts of time delay imposed on the shaker input signal before i t is sampled. When the time delay equals the travel time of flexural waves from shaker to accelerometer, the two signals will be coherent; their product will there-fore be large, causing a peak in the cross correlation function. The height of the peak depends on the degree of coherence (considering 1/16' Plexiglass 1/8" Plexiglass General Radio Random Noise Generator Type 1390-B Power Amp. Altec 1594A Accelerometer B + K 4344 S B + K Mini Shaker Type 4810 B + K Preamp Type 2623 Measuring Amp. B + K Type 2606 yea Oscilloscope Tektronik type 5103N xct) 4 PAR 101A Correlation Function Computer Octave Band Filter B'+ K Type 1614 Figure 3.3 Experimental Layout for Measurement of x 4 2 and Figure 3.4 Accelerometer Positions for Measurement of T„ 0 and x, 0 amplitude and phase) that exists between the properly delayed shaker input signal and the waves a r r i v i n g at the accelerometer. The r a t i o of the peak heights of the cross correlation functions computed between the shaker input signal (octave band limited random noise) and the accelerometer signals from each of points 4A and 8A then gave the displacement amplitude attenuation suffered by flexu r a l waves i n t r a v e l l i n g from point 4A to point 8A as a function of frequency. Since the shaker input was composed of a random mixture of sinusoids, the attenuation of displacement and ve l o c i t y amplitudes was the same as that measured for acceleration amplitudes. A t yp ica l cross co r re la t ion funct ion as i t appeared on the osc i l loscope is seen in Figure 3.5. During i t s computation the shaker was fed with random noise f i l t e r e d through the 4000 Hz octave band. I t d isplays the envelope shape charac te r i s t i c of cross co r re la t i on funct ions of a l l band l im i ted noise. The narrower the bandwidth the braoder the cor re la t ion envelope. For th is reason octave bands bel 500 Hz were not considered as the pos i t i ons , (on the time delay a x i s ) , of t he i r cor re la t ion peaks were ambiguous. F i g u r e 3.5 C r o s s C o r r e l a t i o n F u n c t i o n o f Band L i m i t e d Random N o i s e (4 KHz O c t a v e Band) The junction impedance change, however, was not the only mechanism working to attenuate flexural waves as they travelled out from the shaker. Plexiglas has an internal damping coefficient of 0.002 so that the high frequency waves, at least, suffered noticeable attenuation from this mechanism while travelling from position 4A to position 8A. Also, since the shaker provided a point source in the two dimensional medium of the plates, the wave intensities, in the far field, decreased linearly with increasing distance from the shaker. These two mechanisms, and also any dispersion effects, were accounted for simply by repeating the above procedure to obtain the ratio of cross correlation peak heights between the source signal and the two points 4A* and 8A* of Figure 3.4. These points Were the same distances from the shaker as points 4A and 8A respectively but did not span the junction. The source was located far enough from the edges of plate 4 so that any waves reflected from these edges would reach points 4A* and 8A* considerably later than their direct travelling counterparts. The attenuation measured between these two points was then equal to that incurred between points 4A and 8A by internal damping and divergence only. The ratio of the attenuation measured between points 4A and 8A to that measured between points 4A* and 8A* then gave the true attenuation of flexural waves by the structural junction. The flexural wave transmission coefficient X* 0 was calculated as follows from the double amplitudes of the cross correlation function peaks, r .-(i = 4A,8A,4A*,8A*) between the shaker input signal (s) and s, 1 the signal from each of the four accelerometer positions: 67 1 $>4A rs,eA ^ 4 A* * Attenuation Attenuation A similar procedure was used to measure x^ 2- T n e ratio of peak heights of the cross-correlation functions between the shaker input signal and the accelerometer signals from points 4A and 2A was found, and normalized to exclude the effects of internal damping and wave divergence. The transmission coefficients were examined in octave bands to see i f any frequency dependence existed although the theory, excluding resonances caused by the finite length of plate 2, predicted none. No definite trend was found. The values given here have then been averaged over all octave bands from 500 to 16,000 Hz. Agreement with Theory - x^ 2 Note that the transmission coefficients obtained by the above experimental method are not influenced by finite plate resonances since only the outgoing waves and immediate reflections can correlate with the source input signal. The reflections can be avoided by proper choice of imposed time delay range and accelerometer positions. Therefore in order to compare the experimental results with theory, the effects of resonances must be depressed in the theory. Bhattacharya Attenuation = Attenuation* = 68 [38] has in effect done this while deriving a simplified expression 2 2 for |B2| (or x^ 2) a n a" hence a simpler coupling loss factor. The expression for |B2| can be written from Equation (3.32) as: | B , I* = Z(c+Zot)* ; ( 3 > 4 7 ) where 9 = k2^2- This was integrated over one period, excluding the resonant part of the response, which occurs at intervals of approximately 9 = Tr. The result was: <*> = « > = T< where when both panels are of the same material. Therefore, in the present case in which h 4 = 2h2, The experimental value of x^ 2 averaged over the six octave bands and two sets of accelerometer positions was: 4,2 " 1 * 8 , 4 * l i i± - 0 . 3 2 5 The excellent agreement here is likely due more to coincidence than to the merit of the experimental technique but i t does suggest that the method is basically sound. The experimental value of g averaged over the six octave bands and four sets of accelerometer positions was found to be: 0 . 4 8 8 The theoretical value of g obtained from Equation (3.41) by ignoring the minima created by resonances of plate 2, is: <-r4,8> = 0.69 In both cases the measured transmission coefficient has been less than what the theory predicted for the ideal butt joint. This is at least partly due to damping in the plexiglas joints, the ideal joints being lossless. In any event, the agreement is certainly good enough to i n s t i l l confidence in the flexural wave transmission coeffic-ients that have been derived. 3.4 The Coupling Between a Diffusely Resonant Panel and an Acoustic  Field 3.4.1 The Case of Free Waves in an Infinite Panel To gain an understanding of how energy is transferred from a reverberant sound field to a resonant finite panel and vice versa, 70 it is best to first consider the case of an infinite panel exposed to infinite plane acoustic waves as depicted in Figure 3.6. Figure 3.6 Wave Coincidence in an Infinite Panel Acoustic waves (of wavelengthAa) incident at an angle 0 to the panel normal, have trace wavelengths on the panel surface equal toWcosS and therefore trace wave speeds of c /cose. Hence i f the a sound field is diffuse, trace waves move over the panel surface at speeds ranging from the acoustic wave speed c, when the wavefronts a are normal to the panel, to near infinite speeds when 8 approaches TT/2. The trace waves excite forced flexural waves of all frequencies 1/2 in the panel. The flexural wave speed increases with f , see 71 Equation (3.2), so that at a certain frequency characteristic of the This frequency is called the critical coincidence frequency of the panel since i t is the lowest frequency at which the trace wave speed can equal the flexural wave speed. When this occurs large amplitudes flexural waves are generated in the panel, and there is excellent coupling between panel and sound field. Coincidence can occur at all frequencies above the critical frequency since the trace wave speed can have any value above c a. However the strength of the coincidence effect decreases with increasing frequency due to the increasing mass impedance of the panel [39]. incidence can be found by equating the trace wave speed at that angle to the flexural wave speed. panel, the flexural wave speed c^ equals the acoustic wavespeed c . T a The coincidence frequency of a panel for any angle of C tr fc = f3 CJ TT h cz cosxe Wz (3.48) For critical coincidence, 6 = 0, so that: ^3 Cl 7T h CJI (3.49) 72 This same critical coincidence frequency marks a transition in the ability of an infinite panel to radiate sound. In contrast to the previous case of forced excitation of an infinite panel by a diffuse sound field, in which the panel responded to sound waves of all frequencies but responded much more vigourously at and above critical coincidence, i t will be shown that an infinite panel excited into modal vibration cannot radiate any sound to the far field below critical coincidence. Consider an infinite panel excited by an external source which is far removed from the part of the panel being observed. At frequencies for which the flexural wave speed is less than the acoustic wave speed, that is, below critical coincidence, there is time for the acoustic pressure variations generated over the surface of the panel to establish a pure flow of air from the compressed regions to the rarefied regions. The pressure field is effectively neutralized and there is no radiation to the far field. It can be shown [40] that tha acoustic radiation impedance per unit area of such a plate is given by: Therefore for f < f the impedance is imaginary (a pure mass reactance) and the acoustic pressure generated by the panel forms a wattless near field which suffers rapid exponential decay with distance from the panel surface [41]. For f > f the impedance is real and therefore represents a radiation resistance. Sound power is radiated off to the far field. As frequency increases the radiation impedance approaches the specific impedance of the medium, p c . a a At the critical coincidence frequency the radiation impedance would become infinite i f such a thing as an infinite lossless panel existed. [42] 3.4.2 Finite Panel - Acoustic Field Coupling It has been shown that in order for a free flexural wave on an infinite panel to couple with an acoustic field and hence radiate sound, i t must have a.wavelength longer than that of an acoustic wave of the same frequency, that i s , i t must be acoustically fast [43]. This is not true,however,in the realistic case of finite panels. Consider the simply supported panels of Figure 3.7 in which two dimensional standing wave modes exist [44]. The dashed lines indicate panel nodal lines. If the natural mode under consideration has flexural wave velocity c^ = i c ^ x + jc^ , the x and y components of which are both less than the acoustic wave speed then the acoustic short circuiting found in the case of the infinite panel, occurs in both co-ordinate directions. The acoustic pressures generated by the quarter wave cells are effectively cancelled by the neighbouring cells of opposite sign everywhere except in the corners as shown in Figure 3.7(a). If the jnode has one component of its flexural wave velocity which is greater than the acoustic wave speed, then the quarter wave cell cancellation Figure 3.7 Single Mode Vibration of a Simply Supported Panel 75 does not occur in that component direction, and the panel radiates from quarter wave cell strips along the edges parallel to the supersonic direction as shown in Figure 3.7(b). These mode types are both classed as acoustically slow, the former being termed piston modes and the latter, strip modes. Finally, i f the mode is such that both components of its flexural wave velocity are supersonic, then acoustic short circuiting cannot occur at all and the panel radiates efficiently from its entire surface. Such a mode is therefore called a surface mode and is termed acoustically fast (see Figure 3.7(c)). The radiation resistance of a finite panel supporting a surface mode is the same as that of an infinite panel above critical coincidence. It should be noted that although critical coincidence occurs at the same frequency in finite and infinite panels (of the same material and thickness), there is a substantial difference between the two phenomena. In infinite panels coincidence occurs when free travelling waves in the air match up with free travelling flexural waves in the panel. But free travelling waves cannot exist in a finite panel unless its internal damping is so high that the wave energy which reaches the panel boundaries, and is reflected back into the panel, is negligible. It is the matching of standing waves in the air with standing waves (natural modes) in the panel which causes coincidence in finite surround-ings. This led Bhattacharya and Crocker [45] to propose that a finite panel can only be driven to coincidence by an acoustic field i f the panel is backed by a cavity of some sort which can support standing acoustic waves. An explanation of how this matching of standing waves actually takes place can be found in Appendix C. The radiation resistance of a panel supporting a particular piston or strip mode depends upon the distance, in terms of acoustic wavelengths, separating the effectively radiating regions. If the panel, of dimensions £, h, is large enough that l,h » X /2, then the four a quarter wave cells of piston modes are uncoupled as are the two strip radiators of strip modes. The total acoustic power radiated from the panel is then simply the sum of the radiated powers of the four quarter wave cells or the two strip radiators. If the panel dimensions are not much greater than half an acoustic wavelength, then the radiating cells are coupled and distinction must be made between modes which have A L O n even and odd numbers of half wavelengths (n = ^ — , n = T — ) in the two x Afx y A f y co-ordinate directions. Three types of coupled piston modes exist depending on whether (n x, n ) are (odd, odd), (odd, even), or (even, even) [46]: (a) If n and n are both odd (see Figure 3.7a), the four x y corner monopoles are coupled in phase and their combined radiation is 16 times that of each monopole alone. (b) If n x is odd and n^ even, or vice versa, the radiation is that of two dipoles in phase which is four times that of each dipole alone. (c) If both n and n are even, the total panel radiation is x y that of a quadrupole. Dipole and quadrupole radiation resistances are second and third order respectively to that of a monopole. Therefore i t is justifiable to neglect the radiation of even-odd and even-even piston, modes unless of course no odd-odd modes are active in a panel. 77 Two types of coupled s t r i p modes ex i s t depending on whether n ( for a "y" s t r i p mode) i s odd ( in phase) or even (ant i -phase) . Since A the f lexura l wave ve loc i t y component along the s t r i p d i rec t i on i s supersonic, each const i tuent quarter wave c e l l i s decoupled from the rest and the rad ia t ion i s that of a s t r i p monopole. I f the two s t r i p monopoles are in phase, they radiate four times the power that e i ther would alone. I f they are anti-phase they form a s t r i p d ipo le , the radiated power of which i s second order to that of a s t r i p monopole and can therefore be neglected when s t r i p monopoles are present. 3.4.3 The Radiat ion Resistance of a F i n i t e Panel in Single Mode Vibrat ion Maidanik [47] derived expressions for the rad ia t ion resistance ( in to ha l f space) of a simply supported panel in s ing le mode v ib ra t ion in the various frequency ranges that have been discussed here. The basic expression for rad ia t ion res is tance was obtained e a r l i e r by Lyon and Maidanik [48] by computing the power flow between the sound f i e l d of a room and a pane l , considering the room and panel modes to be o s c i l l a t o r s . The power f low was then re lated to the rad ia t ion resistance by the def in ing equation: P a M = < V p > R,ad(co) > ( 3 . 5 1 which appl ies when the panel v ib ra t ion i s e i ther that of a s ing le mode or 2 of a reverberent f i e l d . Here <Vp > is the s p a t i a l l y averaged mean square panel ve loc i t y . Lyon and Maidanik were able to show that the rad ia t ion resistance due to a single panel mode is given by: R r « i C w > = (^f^&lT f t * - * * * ) 2») 4 * . (3. where ^(x-j, x 2) and $(x-j, x 2) are the cross correlations of. the pressure field at the panel surface and the flexural wave field of the panel respectively. 3.4.4 The Radiation Resistance of a Reverberant Finite Panel To appreciate how Maidanik used the expressions from Equation (3.52) for single mode radiation resistance to deal with the highly reverber-ant panels necessary for the application of SEA, consider the distribution of acoustically slow modes in wave number space as shown in Figure 3.8. This figure applies to panel modes, the frequency of which satisfies: k a L ? ^ k a h >^ 1 that is, when the panel dimensions are not small compared to an acoustic wavelength. The corner monopoles of piston modes and the strip monopoles of strip modes are therefore uncoupled and radiate as monopoles and strip radiators respectively. In real rooms with a smallest dimension of 8 feet, this type of radiation would occur down to about 200 Hz. For the model rooms tested in this work, the lower limit is more like 400 Hz. Figure 3.8 The D is t r ibu t ion of Acous t i ca l l y Slow Panel Modes in Wavenumber Space Returning to Figure 3.8, i t i s seen that wave numbers correspond-ing to x and y s t r i p modes l i e along arcs A and C respect ive ly and those corresponding to piston modes ( k f > /2k a) l i e on arc B. By summing the con-t i rbu t ions of the modes along arcs A , B and C, the to ta l acoust ic power radiated below coincidence by the panel i s by de f i n i t i on [49]: P* " I ( ^ A R^ ( A ) + I ^) R^(C) + R^d(B) (3-53) A C C B From the assumption of equal modal energies; basic to SEA: < V P > A ~ < V P ) B « ( V P )C «** ( V P) 80 Then by assuming a fairly large modal density along the entire kf are and by extending the summation of Equation (3.53) to integration, Maidanik obtained the following expressions for the radiation resistance into half space of a reverberant finite panel with 1/2 k I, 1/2 k h > 1: a a cc I cc where L. ^ , "f > ~ f cc fc(f/fcc) * 0 " T ) " 1 {(I - *»•) ln[(i+ocV0-«)] + 2oc} ^ c c=- c r i t i c a l coincidence wavelength in +Vi£ panel . The radiation resistances of finite and infinite panel are the same above coincidence as can be seen by comparing the value given above for f > f and that given by Equation (3.50). However the phenomenona responsible for above coincidence coupling of a finite reverberant panel to a half space and to a reverberant cavity (as in the case of two adjacent rooms) differ significantly. As will be shown in Appendix C, this difference could mean an over-estimation of the radiation resistance of the interior panel surfaces of the statistical energy model by Equations (3.54) for f >^ f . This in turn would mean an under-estimation of the noise reduction at those frequencies. 81 3.4.5 The Effects of Panel Boundary Near Fields upon Radiation Resistance The radiation resistance expressions given by Equations (3.54) are only totally valid for simply supported panels. Panel boundaries in general introduce near field effects consisting of flexural waves of distor-tion which decay exponentially with distance from the boundary. Such near fields can make noticeable contributions to panel radiation resistance especially at low frequencies where the panel mode contribution is minimal. The near field contribution decreases with increasing frequency as the near field itself collapses into the boundary [50]. Simply supported panels, because of the zero moment condition at their boundaries, do not support near fields and hence their radiation resistances can be predicted from considerations of modal radiation alone from Equations (3.54). For f « f , a panel with clamped-clamped edges (zero rotation) has a radiation resistance, due to its near field, which is twice that of a simply supported panel of equal area [51]. Similar to most real panels, those of the experimental model tested here had boundary conditions some-where between the two extremes represented by simply supported and clamped-clamped. Therefore, in the analysis, the values of radiation resistance for f < f c c/2 have been multiplied by a factor of 1.5. One further alteration has been made to Equations (3.54) for f < f . This was because all panel surfaces which face into the interior of the statistical energy model are forced, at their edges, to radiate into quarter space instead of half space. Below coincidence, reverberant panel radiation is dominated by radiation from quarter wavelength wide strip monopoles along the panel edges; therefore, the radiation resistance of all interior facing panel surfaces will be increased by a factor of two for cc 82 3.4.6 Panel - Acoustic Field Coupling Loss Coefficients It now remains to incorporate the reverberant panel radiation resistance into an expression for the panel - acoustic field coupling loss factor. This coupling loss factor is defined as the fraction of the maximum panel kinetic energy which is dissipated per radian through acoustic radiation. Equation (3.51) gave the acoustic power radiated by a panel as: - Rrad (Vp) (3.51) Therefore, the panel energy dissipated per radian through radiation is P /to. a 2 The maximum kinetic energy of the panel is M<v > where M is the total panel 2 mass and <Vp > is the space averaged mean square panel velocity. The coupling loss factor due to panel acoustic radiation is therefore: Y) . = = ' (3.55) l r * * /.iM A/A . r i J K A 0)iV\ (vP2) By using this result in conjunction with Equations 3.54, the coupling loss factors ^4 -|» 3 a n c* ^2 l ^ = r i 2 3^ ' i a v e ^ e e n e v a l u a t e d . Expressions like that of Equation (3.55) have also been used to calculate panel loss factors due to radiation to the exterior of the statistical energy model (see section 3.5.5). Since the model exterior has been considered to be an energy sink which cannot return energy to the model, nra(j» when associated with an exterior panel surface, becomes a loss factor rather than a coupling loss factor. Radiation resistance need not be doubled for f < f „ in these cases since exterior cc panel surface radiation is not confined to quarter space along the panel edges. 3.5 The Evaluation of the Loss Factors of Panels and Cavities 3.5.1 Introduction In this analysis where the quantity sought, namely the deviation of the noise reduction between acoustic cavities from mass law values, owes its existence to the ability of the various elements, panels and cavities, to store (and exchange) energy in the form of resonant wave fields, the amount of damping acting on the elements is crucial. Unfortunately, the accurate prediction of damping in all but the most simple acousto-mechanical systems is an impractical, difficult task. Some of the damping mechanisms occurring in the present model are well documented such as the internal damping of the panel material and the air absorption in the rooms [52, 53, 54, 55], Others are unique to the particular system, such as structural junction damping. These latter must be estimated by applying general trends established through theoretical arguments, or by conducting reverberation time tests on similar elements which are isolated from the model as a whole, so that they cannot exchange energy with other elements but have boundary conditions duplicating as closely as possible those found in the model. There are five types of damping accounted for in the SEA model about which quantitative information must be gained. They are: 1. internal damping of panels (material damping), 2. air absorption in cavities, 3. structural junction damping, 4. radiation damping, 5. mass law transmission from cavity sound fields to model exterior. These will now be discussed in turn. 3.5.2 The Internal (Material) Damping of Panels Most metals and structural materials have internal loss factors which are relatively independent of wave amplitude and frequency within the ranges of sub fatigue limit stress and audio frequencies. For the aluminum panels used in the models tested here, the internal damping factor, that i s , the fraction of the total panel energy which is dissipated per cycle, is 10"^ [56]. Therefore this value is used for the internal components of the total panel loss factors: n 2 » n 4 and n 8 • int int int 3.5.3 Air Absorption in the Cavities Normally in small enclosures, in which the mean free path between boundary reflections is short, the attenuation of sound due to air absorption is negligible compared to boundary absorption. If the room boundaries are very hard however, as in the aluminum model, air 85 absorption can be the major contributor to what l i t t l e attenuation the sound field suffers. Sabine's reverberation time formula corrected for air absorption [57] is a good place to begin the calculation of the loss factor to be applied to the energy of the diffuse sound field in the reception room. In English units this is calculated from: 0.049 V To = ~T~Z T T S*C-' (3.56) where V and S are the room volume and total surface area and a is the average surface absorption.coefficient. The attenuation constant of air m, in units of •ft"'', dictates the rate of decay of sound energy with distance x according to: D(x) = D 0 e ~ m x ^ (3-57) The loss factor of a room can be related to its reverberation time by recalling that reverberation time is the time required for sound pressure level (SPL) to drop by a factor of 106 or by 60 dB. The decay of SPL with time can be expressed by: where 3 is the attenuation constant in units of sec -^. 86 At t = T R : f> ( T R ) = or Taking logarithms we f i nd : 13.818 . ' — S e c Tft (3.59) Loss fac tor n i s the f rac t ion of the rooms energy d iss ipated per rad ian; therefore: yj = _ ^ A - « ( 3 6 0 ) Subst i tu t ing for B from Equation (3.59) ^ " vf r"r" (3-61) Combining Equations (3.56)and(3.61), the loss fac tor of the rece iv ing room can be expressed by: YL. = ~ r ^ a - 1 0,62) The ef fec ts of boundary and a i r absorption are seen to add together to give the to ta l loss fac tor . In the usual est imation of reverberat ion t ime, the boundary absorption coe f f i c i en t a s accounts for losses due to both surface porosi ty and wall f lexure induced by the pressure f i e l d in the room. In the case of an aluminum panel , surface porosi ty i s inconsequential so that a l l the rated absorption of the ma te r ia l , given as 0.05 for Aluminum at 1200 Hz, [58], i s due to the in ternal d i ss ipa t i on of f l exu ra l wave energy. But the t rans fe r of room sound f i e l d energy to the bounding panels has already been accounted for by the coupling loss coe f f i c ien ts -j, n 8 3 and n 2 -j as has the d i ss ipa t i on of the resu l t i ng f l exu ra l wave energy in the panels. There-f o re , only the a i r absorption can t ru l y be a t t r ibuted to the recept ion room volume and the loss fac tor no reduces to : J i n t Z.2 4 m 1 7 9.5 m < 3mt f .043 -f-Values of the attenuation exponent of a i r , m, were obtained from a set of curves g iv ing attenuation in db/1000 f t versus frequency on a log- log p lo t for various values of r e l a t i v e humidity [59]. C l a s s i c a l a i r absorption as wel l as the molecular re laxat ion absorption of the major resonating molecules were considered in the computing of these curves. See Figure 3.9. The r e l a t i v e humidity in the lab was f a i r l y steady at 30% during the days of experimentation. 88 i 10 10* 103 10* ' 10 5 f/P (Hz/ohn) Figure 3.9 The Total Absorption of Sound in A i r 89 Since the attenuation of sound in air is an exponential function of distance from the source, see Equation (3.57), the drop expressed in dB/ft is simply 1/1000 times the drop expressed in dB/1000 ft. Therefore, i f the attenuation in dB/1000 ft as given in Figure 3.9 [60] is denoted by a, the attenuation constant m defined by Equation(3.57)is given by: !CT3<X = \o\o3lo(e m) Z.303 ^ m - j o * ' * (3.64) Substituting this expression for m into Equation(3.63),the loss factor applied to the receiving room sound field due to air absorption, n.o > J i n t becomes: )0 = 0.0414 — . rad"1 (3.65) «3int . -f 3.5.4 Structural Junction Damping The energy dissipation mechanisms available to a completely free or simply supported panel are internal damping and acoustic radiation. By applying more rigid boundary conditions, considerably more damping can be introduced. Depending on their nature, the structural junctions forming the panel boundaries can contribute damping of the order of the radiation damping, or they can dominate the panel damping entirely. 90 Experimental values of the total internal damping factor, i.e., material and junction damping, of a 1/8" aluminum panel clamped in a rigid frame vary between 0.005 to 0.01 over a frequency range from 100 to 10,000 Hz [61]. Note that this is 50 to 100 times the value due to material damping acting alone (n^^ = 10"^). At the time the test model was being built, the difficulties inherent in the prediction of real structural junction damping were not appreciated by the author so that the criteria for the model's construction did not include a simple junction construction from the point of view of damping estimation. The criteria applied were that the model be light-weight and resilient so that i t could be easily excited by sound^that i t be capable of fairly easy partial disassembly to allow for partition changes, and that the structural junctions differ as l i t t l e as possible from the ideal butt joints for which the flexural wave transmission coefficients were derived. It was decided to use aluminum "Pop" rivets to fasten the aluminum sheets to 3/4" x 3/4" x 1/16" aluminum angle, the lightest available, as typified in Figure 3.10. A narrow strip of plasticene was used to prevent leakage between the rooms and to the exterior of the model. Two major damping mechanisms characteristic of point fastened joints act here. The first is due to interface slip between panel and angle in the vicinity of the rivet and to hysterisis loss from bending of the rivets. The second arises from the relative normal motion induced between panel and angle in the inter-rivet regions when flexural waves impinge on the joint. This relative motion squeezes air in and out 91 Figure 3.10 Typical Riveted Joint in the Experimental Model of the gap between panel and angle, causing the viscous dissipation of energy in the air. This mechanism has been termed "air pumping" [62]. The relative importance of these two mechanisms certainly depends on such parameters as rivet tightness and spacing, and the width of the aluminum angle material. It also depends, the author believed, on the rigidity of the joint with respect to angular rotation. Ungar and Carbonell [63] found that the latter "air pumping" mechanism was the dominant source of damping in a system, intended to model aircraft skin and stringers, consisting of thin aluminum plates to which flat beams had been riveted to act as stiffeners. It is 92 believed that the slip-hysterisis mechanism was found to be unimportant in this case because the stiffening beams were comparatively free to ride on the panel surface and had no external restraints on their rotation. This resulted in the creation of only small bending stresses in the rivets and small shearing forces between panel and beam. In the present model however, where the riveted panel-angle joints supply the coupling between pairs of large orthogonal panels, the resistance offered by one panel to angular rotation of the joint by flexural waves in the other panel of the pair is substantial. The resisting torques must act primarily through the rivets, thereby generating considerable bending stress and shearing action in their immediate vicinities. Therefore, i t is believed that both these mechanisms should be included in any attempt at theoretical or semi-empirical prediction of the damping of light resilient structures caused by point fastened junctions which offer considerable resisting torque to their rotation. Here the air pumping mechanism has been considered to be of less importance than the slip-hysteresis mechanism because construction peculiar-ities made i t impossible to assure contact between panel and angle, especially in the inter-rivet regions; i t was found [64] that when beam and plate were not in direct contact (as when separated by thin washers), the beams caused a negligible increase in plate damping. Theoretical [65] and empirical [66] studies have been made of air pumping damping. The latter work gives a procedure for its estimation based on reverberation tests of a number of steel and aluminum panel-beam systems. This procedure was applied to the present model but did not yield large enough damping values at high frequencies. Since the beams and plates of the systems 93 tested to establish the above procedure were in direct contact, unlike those of the present model, the procedure should have over-estimated the air pumping damping in the model. The failure of the empirical air pumping procedure to over-estimate'the junction damping, in as much as the predicted noise reduction was not greater than that measured between the two experimental rooms, indicates that another, more powerful, damping mechanism (slip-hysterisis) is controlling. In an effort to retain the purely theoretical nature of this analysis, riveted junction damping values were estimated as functions of frequency from theoretical arguments. Although these results were finally discarded in favour of experimental values, the arguments developed give some insight into the nature of the phenomena responsible for the measured damping, and hence will be outlined briefly. Damping due to bending and slipping at riveted joints has been found to vary non-1inearly with the bending stress amplitude in the joint [67]. Over the low load range, which certainly includes acoustically induced loads, this damping has been seen to vary quite uniformly with the second power of the load amplitude [68]. It was then necessary to estimate the bending stress amplitudes to which the riveted joints were subjected by the reverberant flexural wave fields in the panels, and apply a damping factor proportional to the square of those amplitudes. The spectrum of displacement amplitudes of a clamped-clamped panel excited by a diffuse sound field was obtained from statistical energy arguments by Price and Crocker [69]. The corresponding spectrum of bending stress amplitudes in the panel was then estimated from the linear bending theory of plates. The riveted junction 94 damping distribution was then obtained by choosing an arbitrarily small init i a l damping factor for the lowest frequency modes of interest, and applying to this a weighting distribution consisting of the square, of the bending stress amplitude spectrum. This approach was an approximation because, as i t was later realized, the rivets were not necessarily exposed to the full bending stress amplitudes generated by the various natural panel modes, but could have been subjected to a range of values of cyclic bending stress from near zero to near peak amplitude, depending on their positions relative to the edge node of the particular panel mode. As shown in Figure 3.11, the standing flexural wave nodes in regions near a rivet were located quite near the rivet. They were not considered to be located exactly at the rivet because the angle itself Figure 3.11 The Location of the Edge Node of Flexural Modes in a Panel With Riveted Edges was light enough to flex somewhat. The curvature, and hence the bending stress, induced at the rivet was therefore larger (for a given wave amplitude) for higher frequency modes which had their first antinode nearer the rivet. This trend of increased damping with decreased flexural wave length was also felt to apply to the air pumping mechanism since this damping could only occur in the regions covered by the aluminum angle, and, the shorter the wavelength, the greater was the portion of that wavelength which could be covered by the angle. The above arguments were used to modify the initial approach based on the bending stress amplitude spectrum. They also formed the basis of another attempt at quantitative description of riveted junction damping, in which the ratio of the energy possessed by the portion of a standing flexural wave beneath the angle material to the energy possessed by its entire wavelength was found, and a relatively high damping factor applied to the former. Although these methods could be made to yield fairly acceptabl results, they were crude approximations at best and both required that a guess be made at an i n i t i a l , low frequency value of damping. Also, because of the highly unique nature of structural junction damping, i t was questionable whether the theory developed for the present joints could be legitimately applied to other point fastened joints. For these reasons i t was decided to obtain the riveted junction damping values through experimentation. Figure 3.12 Structural Junction Damping Measurement: Panel Edge Rigidly Clamped Measurement of the Damping Capacity of Riveted Junctions A control experiment was conducted in which: (a) A 1/16" aluminum panel was clamped rigidly along one of its edges to a heavy frame as shown in Figure 3.12. A diffuse flexural wave field was generated in the panel and the rate of decay of the field's acceleration level was recorded in 1/3 octave bands at three positions on the panel's surface. (b) The formerly clamped edge was riveted, in a fashion typical of the flanking suite joints, to one arm of a length of 97 3/4" x 3/4" x 1/16" aluminum angle, the other arm of which was damped rigidly to the frame (see Figures (3.13) and (3.14)). The acceleration decay measurements were then repeated. Ideally the difference between the panel damping factors measured in parts (a) and (b) above, would have given the damping due to the length of riveted joint. In reality some error was introduced due to the inability to exert complete control over the other damping mechanisms present. To exert complete control over acoustic radiation damping, the panel would have to be placed in a vacuum. Only in this way could i t be assured that any change in panel radiation resistance which might accompany the change from clamped edge to riveted edge conditions, would have no effect upon the total panel damping. Even i f a vacuum chamber was available, its use would create a second problem while solving the f i r s t , since air pumping damping depends strongly on air pressure. This is why riveted aerospace structures which exhibit sufficient damping on the ground sometimes fail to do so in flight. Panel radiation resistance below critical coincidence was expected to be greater for the clamped edge case than the riveted edge case since riveted edge conditions lie somewhere between the two extremes; clamped and simply supported. Therefore the damping that was here attributed to the riveted joint will actually be less than the true value by an amount equal to the radiation damping lost in going from clamped to riveted edge conditions. Also, in order for the exact riveted joint damping to be measured, the original clamped edge would have to behave as a lossless boundary. F i g u r e s 3.13 & 3.14 S t r u c t u r a l J u n c t i o n Damping Measurement: Pa n e l Edge R i v e t e d as In E x p e r i m e n t a l F l a n k i n g S u i t e 99 This is of course impossible and whatever damping the clamped edge con-tributes is lost in the change to the riveted edge and therefore further adds to the under-estimation of the true riveted joint damping. This inherent error can be more easily compensated for since typical values of the internal damping factor for a clamped-clamped aluminum panel are known to vary between 0.005 and 0.01 throughout the 100 to 10,000 Hz range [70]. Then since the purely material damping of aluminum is 0.0001, the above damping can be almost entirely attributed to the clamped boundaries. Therefore a damping factor of the order of 0.005 has been added to the total damping of each panel to account for lost clamped edge damping. In spite of these inaccuracies, experimental values of the joint damping are s t i l l more credible than those predicted by the theoretical arguments presented earlier. A schematic of the experimental layout is shown in Figure 3.15. A counterweight was attached to the top corner of the panel (which was cantilevered from its single fixed edge) so that no shear stresses, which did not exist in the actual model, would be applied to the rivets due to the cantilevering. The counterweight should have had l i t t l e or no effect on the results of the experiment, since i t was used during both parts of the control. Two methods of panel excitation were tried: impulsive excitation with a steel ball pendulum, and steady state excitation with a mechanical shaker fed with random noise. The impulsive excitation tests gave smaller values of riveted junction damping than the shaker excitation tests, Heavy Support!n Frame Clamped/Riveted Edge Accelerometer B + K 4344 S //////////// Preamplifier B + K 2623 Mini Shaker B + K4810 Counterweight Measuring Amp. B + K 2606 Power Amp. Altec 1594A 200-2500HZ 3.150-20,000 Hz 1/3 Octave Filter B + K 1614 Tape Recorder Uher Figure 3.15 Experimental Layout For the Measurement of Structural Junction Damping. General Radio Random Noise Generator 1390-B Level Recorder B + K 2305 1/4 or 1/8 speed o o 101 especially at the higher frequencies. These results tend to confirm those obtained by Price and Crocker [70] in which the decay records of impulsively induced panel acceleration showed a transition in slope from high to lower values much earlier than did the decay records of panel acceleration induced by a mechanical shaker. Of the phenomenon, the authors explained that impulsive excitation was observed to produce higher acceleration levels in the lower order modes than in the higher order modes. The opposite trend was observed for the case of a shaker exciting the panel with white noise. Since the decay rates of low order panel modes are much less than those of high order modes, i t was suggested that in the case of impulsive excitation, energy was being transferred from low to high order modes and thereby prolonging the lives of the latter. A reciprocal phenomenon could not occur in the case of shaker excitation since the higher energy, high order modes decay faster than the lower energy, low frequency modes. Granted that this mode coupling hypothesis is correct, then only the initial portions of the higher frequency impulsive excitation decay records express the true decay history of those frequencies, the rest being distorted through coupling with lower frequency modes. For this reason, and because the steady state flexural wave fields established in the test panel by shaker excitation prior to the decay are more akin to the fields encountered in the flanking suite panels than are the inherently transient fields caused by impulsive excitation, only the shaker excitation results have been considered here. The panel acceleration decay records of the 200 Hz to 2500 Hz 1/3 octave bands were recorded directly on the level recorder at a paper 102 speed of 30 mm/sec, while those of the 3150 Hz to 20,000 Hz 1/3 octave bands were first tape recorded and then played back into the level recorder at 1/4 or 1/8 speed so that the decay rates would not exceed the maximum pen writing speed. The reverberation times of the panel accelerations, that i s , the time required for the acceleration level to decrease by 60 dB, obtained from the decay records of the clamped and riveted edge cases were converted to panel loss factors using Equation (3.61) as derived in section 3.5.3: 2 * 2 ^ = : T i ? . ( 3 -The loss factor of the clamped edge panel and the increase in panel loss factor which accompanied the change from clamped to riveted edge condition are shown in Figure 3.16. The peak in the clamped edge panel loss factor (curve (a)) centered on 10,000 Hz is due to the increased acoustic radiation of the panel at and above the critical coincidence frequency (8,000 Hz for 1/16" aluminum). The gradual increase in loss factor with decreasing frequency below 2000 Hz is believed to be due to the action of some amplitude dependent damping mechanism in the clamped joint, upon the mass controlled response of the panel which increases with decreasing frequency. The loss factor increase due to the riveted edge (curve (b)) shows a strong peak near panel critical coincidence as well. This supports the earlier conclusion that slip-hysterisis was the dominant Figure 3.16 The Loss Factor of the Clamped Edge Panel and the Increase in Loss Factor With Change to Riveted Edge 104 damping mechanism acting in the riveted joints since this type of damping increases with the square of the panel flexural wave amplitudes. The increased flexural amplitudes which accompanied panel coincidence there-fore resulted in a further increase inpanel loss factor due to the slip-hysterisis mechanism. The effects of the riveted joints upon the loss factors of the 1/8" and 1/32" aluminum partitions tested were accounted for by adding to the existing loss factor a distribution similar to that of curve (b) of Figure 3.16 but shifted on the frequency axis to match the peak with the particular panel coincidence region. A correction factor had to be applied to the riveted junction loss factor distributions to account for the fact that the 1/16" test panel was only clamped and riveted on one of its four sides, while the walls of the flanking transmission suite were riveted on all sides. Heckl [72] claims that, i f the wave field in a panel is truly diffuse, the total damping capacity of a particular type of linear discontinuity on the panel depends only upon the length of discontinuity exposed to the wave field. This is analogous to the placement of sound absorbing material in the diffuse sound field of a room. The increase in panel loss factor to be expected when all four edges are riveted can then be found by multiplying the increase measured when one edge was riveted by the ratio of the panel perimeter to the single riveted panel edge length. This ratio was 4.4 in the case of the test panel. Returning to the analogy of the reverberant room, the larger the room's volume, the smaller is the increase in its loss factor than can be obtained by adding a unit area of absorption. Therefore, the total increase in 105 the loss factor of each flanking suite panel, due to its four riveted edges, v/as obtained by multiplying the corresponding increase found for the test panel by the perimeter to area ratio of the panel in question, and normalizing this with the perimeter to area ratio of the test panel as follows: (3.66) 3.5.5 Radiation Damping The loss of energy from the sidewalls 4 and 8 through acoustic radiation to the model exterior was discussed in section 3.4.6 where the coupling loss factor between a reverberent panel and an acoustic half space was developed. These loss factors will be known as n 4 e x t and n8 ext a n d a r e c a l c u l a t e d from Equation (3.5 5) 3.5.6 Dissipation of Cavity Soundfields by Mass Law Transmission to  the Model Exterior The loss of energy from cavity soundfields by this mechanism was discussed in section 3.2 where the mass law coupling coefficient n-j 3 was derived. The loss coefficient from the receiving room to exterior, n3 ext' ^ s calculated i f the same way from Equation (3.15). Source room losses do not enter into the expression for noise reduction, Equation (2.36). 106 3.5.7 Total Loss Factors of SEA Elements Summing the effects of the various damping mechanisms acting on each resonant element, the total element loss factors are obtained as follows: 107 CHAPTER 4 COMPARISON OF STATISTICAL ENERGY AND EXPERIMENTAL VALUES OF NOISE REDUCTION BETWEEN A PARTICULAR PAIR OF STRUCTURALLY COUPLED CAVITIES 4 . 1 The Experimental Flanking Transmission Suite and the Measurement  of Noise Reduction As has been mentioned earlier in section 3.5.4, the main criteria used in the selection of a suitable flanking transmission suite design were: (a) that the material be lightweight, resilient and have a low internal damping factor so that i t could be easily excited into diffuse flexural vibration by a diffuse sound field. Aluminum sheet was chosen. (b) that the panels be joined in a manner which gave as much rigidity as possible while permitting fairly easy removal of one sidewall and the partition so that the later could be changed. (c) that the joints should resemble as closely as possible the ideal butt joints for which the flexural wave transmission coefficients were derived. The rectangular experimental cavities were to have no openings other than two small holes for the insertion of the microphone. This necessitated that the last sidewall be attached and the partition secured to i t without having access to the interior of the cavities. Welding 108 was therefore eliminated as a possible joining technique (it did not comply with criteria (b) either). Bolting also would have been difficult without access to both sides of the work. This left rivets and sheet metal screws as possible fasteners. Aluminum "Pop" rivets were chosen for their superior grip and ease of application. The flanking transmission suite was then constructed as follows (see Figure 4.1). The four continuous sidewalls were 4' x 8' panels of 1/16" aluminum; the two endwalls were 4' x 4' panels of 1/16" aluminum; and the partitions were 4' x 4' panels of 1/32" and 1/8" aluminum in the fir s t and second experiment respectively. The panels were joined with the aid of 3/4" x 3/4" x 1/8" aluminum angle stock and 1/8" diameter Pop rivets as was shown in Figure 3.10. The rivets had an average spacing of 3". A narrow strip of plasticene was applied to all joints to make them airtight. The resulting model was placed on an open wooden frame so that i t could radiate freely from all of its exterior surfaces. Four 2" square neoprene pads were used to isolate the model from the frame. A schematic diagram of the flanking transmission suite and the equipment used to measure the noise reduction between its two rooms is shown in Figure 4.2. The sound source was a University Sound ID-60 horn driver which was suspended resiliently in an upper corner of the source room as shown. By placing the source in a corner, the maximum number of room modes were excited because every room mode has a pressure antinode there [73]. The horn driver was supplied with the amplified signal from a General Radio random noise generator. Both horn driver and noise generator were capable of output up to 20,000 Hz. 109 Figure 4.1 The Experimental Flanking Transmission Suite A Bruel and Kjaer (B + K), 1/2" microphone, type 4133, was used to measure the sound pressure level (SPL) in the two rooms of the flanking transmission suite. The microphone was lowered into the model rooms on a light aluminum boom through small centrally located holes in the top panel of each room (see Figure 4.2). Aluminum plugs fitted with set screws and set in the access holes, permitted the microphone to be located at any level. The booms could be rotated through 360° to give a full range of possible microphone positions. Receiving Room (3) 3'-h" Mic. B & K 4133 University Sound ID-60 Horn Driver Source Room (1) i or i Aluminum 32 ' 8 5' B & K 5078 Frequency Analyzer •Synchronized Sweep B & K 2606 Measuring Amp. B & K 2305 Level Recorder Figure 4.2 Schematic of the'Flanking Transmission Suite and the Accompanying Instrumentation for Measurement of Noise Reduction. m The microphone signals went to a B + K type 2606 measuring amplifier where they were filtered, through use of the amplifier's external f i l t e r mode, with a B + K type 5078 frequency analyzer. In order to best duplicate the unit bandwidth nature of SEA calculations, the frequency analyzer was set for maximum octave selectivity. Such filtering applied a 3 dB attenuation to frequencies which were 6% of an octave bandwidth away from the centre frequency of the octave band in question. This then represented 12% constant percentage bandwidth filtering. The SPL's were recorded on a B + K type 2305 level recorder which was synchronized with the sweep of the frequency analyzer. The random noise generator and its power amplifier were adjusted to give source room SPL's of over 90 dB in some of the mid frequency 12% bandwidths to assure both dominance over any background noise in the unsoundproofed lab and substantial excitation of the aluminum panels. Source and receiving room SPL's were recorded over the 200 to 20,000 Hz range at five different positions in each room. The spatial variation in SPL within each room was very slight for frequencies greater than 1000 Hz indicating that truly diffuse field conditions existed in the rooms above this frequency. The upper limit of 20,000 Hz, although unnecessary as far as noise control is concerned, was required since in the interest of strong panel excitation, the panels had to be kept thin. The frequency range then had to extend above the highest critical coincidence frequency, 16,00 Hz, associated with the thinnest panel, 1/32" aluminum. The difference between the averaged (over five positions) SPL's in the source and receiving rooms gave the desired noise reduction. 112 4.2 Experimental Noise Reduction Results: Comparison with SEA 4.2.1 Comparison: Experiment and SEA The noise reduction between the two rooms of the flanking transmission suite was measured in the manner described in section 4.1 for the 1/32" aluminum and 1/8" aluminum partition cases. The results are shown in Figures 4.3 and 4.4 respectively along with the corresponding values of SEA noise reduction obtained from solutions of Equation (2.36). The computer program used in the evaluation of Equation (2.36) can be found in Appendix D. Also shown in Figures 4.3 and 4.4 are the mass law noise reductions which would be measured i f all the exterior walls of the SEA model were infinitely s t i f f (not able to. exchange resonant energy with the room sound fields) and i f the partitions responded as limp masses. According to the mass law, which has been a standard indicator of the sound insulating ability of walls in the past, the noise reduction between two given rooms increases with the square of the mass per unit area of their dividing partition. The absolute values of these mass law noise reduction curves cannot be considered accurate to within less than 2-3 dB since their evaluation depended upon the assigning of a total loss factor to the receiving room sound field under the hypothetical conditions mentioned above; this has been represented by the air absorption loss factor TI3 . j n t (section 3.5.2) and a uniform wall absorption coefficient of 0.01. The difference in levels however, between the mass law noise reductions for the 1/8" and 1/32" partition cases can be legitimately compared to the corresponding difference between the measured noise 40 S t a t i s t i c a l Energy Analysis — o —, Experiment • Mass Law 16000 Experimental and SEA Values of Noise Reduction for the Case of the 1/8" Aluminum P a r t i t i o n 115 reductions. Such a comparison illustrates the effects of resonant energy transmission through the partition and via the coupled sidewalls. The fourfold increase in panel surface density from the 1/32" 2 to the 1/8" partition case resulted in a 12 dB (10 log (4) ) increase in mass law noise reduction. The measured increase falls far short of this value over the entire frequency range. At relatively low frequencies, below the f i r s t panel coincidence, where the response of the panels is mass controlled and panel radiation resistance is small, the 1/8" partition affected a 6 to 8 dB increase in noise reduction. At higher frequencies where panel response becomes dominated by coincidence effects and panel radiation resistance is high, the levels of noise reduction provided by the two partitions are practically the same. Except in the immediate vicinities of the two panel coincidence dips, the two experimental curves are within 1 dB of each other as are the two curves predicted by SEA. The transmission of resonant acousto-vibrational energy is here too strong for the net noise reduction to be noticeably affected by changes in partition surface density. The limiting effect of structure-borne flanking transmission on noise reduction is thus well illustrated. The statistical energy theory results agreed quite well with both measured noise reductions (within 2 dB) over most of the frequency range 300 to 20,000 Hz. However some regions of poorer agreement exist and explanations of these discrepancies will now be offered. 116 (a) Substantial ripple is observed in the experimental results below about 1400 Hz, particularly in the 1/32" partition case (Figure 4.3). This is felt to be caused by the preferential excitation of natural modes in one or the other of the rooms. SEA cannot reflect variations in element frequency response due to individula modal resonances. One of the basic assumptions of SEA is that enough modes are resonant in each bandwidth to allow a meaningful average response level to be obtained, that is, diffuse wave field conditions are assumed to exist. Diffuse conditions are assumed to exist in a room when the bandwidth being used to measure the SPL spectrum within i t contains at least ten resonant modes. Therefore the wider the bandwidth, the lower the frequency at which diffuse field conditions can exist. In order to simulate as closely as possible the unit bandwidth nature of SEA, the narrowest possible bandwidth was used in the noise reduction measurements. This was a 12% constant percentage bandwidth. By computing the modal density of the smaller and therefore more critical model room from Equation 3.2, i t is found that a minimum octave band centre frequency of 1150 Hz is required before there will be ten modes in 12% of the corresponding octave bandwidth. This minimum frequency coincides quite well with the smoothing out of the ripple in the experimental noise reduction curves. In the non diffuse frequency range a natural mode occurring in one cavity may have no corresponding mode of similar freqnency in the other cavity. Therefore a strong mode in the source cavity may result in a peak in noise reduction while one in the receiving cavity may result in a dip. Note that the variations in level between adjacent data 117 points have the same sense in both the 1/8" and 1/32" p a r t i t i o n cases though the variations are more pronounced in the l a t e r . This supports the idea that the r i p p l e results from cavity modal effects rather than from those of the p a r t i t i o n s . As the modal densities of the c a v i t i e s increase with frequency, see Equation 3.1 } the experimental noise reduction curves smooth out. (b) The onset of coincidence greatly increases the coupling between the panels and the cavity sound f i e l d s thereby increasing the e f f i c i e n c i e s of sound transmission through the p a r t i t i o n and of structural flanking transmission from source to receiving cavity v i a the coupled panels. P a r t i t i o n and sidewall coincidences are then marked by sharp dips in noise reduction. The strength of these dips i s governed by the damping present at the p a r t i c u l a r frequency and the panel radiation resistance. The s t a t i s t i c a l theory predicts much more severe dips than were measured experimentally f o r a l l coincidences expect that of the 1/32" p a r t i t i o n at 16,000 Hz by which frequency the a i r absorption has greatly increased. There are several factors which contribute to t h i s disagreement. 1. SEA computes noise reduction on a unit frequency basis while the frequency analyzer used i n the experiments gave a minimum f i l t e r bandwidth of 12% of an octave bandwidth. The averaging of the SEA results over comparable bandwidths would round of the c o i n c i -dence dips somewhat. 2. Since panel resonant response increases sharply at coincidence, the panel damping assumed i s most crucial at these frequencies. If some of the damping i s amplitude dependent, as the riveted j o i n t damping has proved to be, the value chosen w i l l be even more critical. Even though the joint damping values used in the theory were obtained from experiment as described in section 3.5, there were several possible sources of error: (i) Precise control could not be exerted over the experiments. (ii) The very short reverberation times found at high frequencies especially when the test panel edge was riveted, resulted in near vertical decay records and hence made precise slope measurement difficult, ( i i i ) The transformation of the damping information gained from the test panel into values applicable to the various flanking suite panels was based on the author's interpretation and a statistical argument due to Heckl [74]. As explained in section 3.5, the error from these sources is expected to cause an underestimation of the panel joint damping and hence an underestimation of noise reduction at coincidence frequencies. An attempt was made to correct for some of the error arising from the fi r s t source above by increasing the internal damping of each panel to 0.005, an average value for clamped aluminum panels [75]. As is explained in Appendix C, the acoustic radiation from a finite panel into a closed cavity above the critical coincidence frequency depends on the coupling of standing waves in the panel with standing waves in the cavity. The panel modes require displacement nodes at the boundaries while the cavity modal pressure must have antinodes there. The cavity pressure and 119 panel displacement can therefore only be in phase over the central region of the panel. The radiation resistance of the flanking suite panels above coincidence is therefore expected to have values less than those predicted by Equations (3.54) which dealt with finite panel radiation into a half space. (c) Above the second coincidence dip in both Figures 4.3 and 4.4 the theoretical noise reduction begins to increase rapidly and with increasing frequency would eventually attain near mass law values. The experimental noise reduction, however, begins to level off at these fre-quencies. This is felt to be due to leakage along one edge of the partition. Since in the construction of the flanking suite the last sidewall had to be affixed without access to the suite's interior, i t was impossible to assure a perfect seal along the corresponding partition edge. It could also be caused by a high electrical noise levels at these very high frequencies which approach the limits of equipment response. 120 4.2.2 Parameter Variation and the Application of the Statistical  Energy Model to Other Structures By variation of physical parameters of two structurally coupled cavities within the bounds of the SEA model,insight into the effects of structure-borne flanking transmission in other structures can be gained. Consider the following three examples: (a) The effects of sidewall thickness and hence bending rigidity variation are illustrated in Figure 4.5 for the case of the standard aluminum flanking suite with 1/8" partition. The suite has been considered to have 1/4" sidewalls. It is seen that although the lower critical coincidence frequency of the 1/4" sidewalls (2000 Hz as opposed to 8000 Hz for 1/16") has allowed the noise reduction to begin earlier its high frequency asymptotic approach to the mass law curve, the 1/4" sidewalls give inferior noise reduction at the more critical low frequencies. The author offers the following explanation of this phenomenon. The impedance change imposed on the continuous sidewalls by their junction with the partition is much less severe when the sidewalls are twice as thick as the partition than in the reverse case. Therefore the flexural wave transmission coefficient g will be much greater between the 1/4" source and receiving room sidewalls. Considerably more flexural wave energy will then be transmitted into the 1/4" receiving room sidewalls resulting in lower values of noise reduction. F i g u r e 4 ^ N o i s e R e d u c t i o n w i t h 1/8" Aluminum P a r t i t i o n and ( 1 ) 1 / 4 " , ( 2 ) 1/16" Aluminum S i d e w a l l s . 122 As a second example consider a concrete apartment building and the noise reduction between its top floor mechanical room and the suite directly below. The floor slab is 6" concrete and the exterior walls are 6" concrete in one case and 3" in the other. Figure 4.6 shows that the choice of wall thickness (if noise control considerations took top priority) would depend upon the operating frequencies of the main mechanical room noise and vibration sources. If these frequencies lay below the critical coincidence fre-quency of 6" concrete (120 Hz) then use of the 3" walls would result in more reduction of structural flanking transmission due to acoustically induced wall vibration. If the operating frequencies were well above 120 Hz, the 6" walls would give more noise reduction than the 3". However, i f the vibration sources were not isolated from the floor slab, and excited strong flexural waves in i t , then the 6" walls would likely give the greater noise reduction at all frequencies since they would accept less flexural wave energy from the slab. Figure 4.7 shows the noise reduction between the engine room and an adjacent berth aboard a steel hulled towboat (1/2" hull and deck, 1/4" bulkheads). In one case the steel surfaces were bare and in the other, they have had a visco-elastic damping layer applied to their interior surfaces which increased their internal damping capacity by a perhaps unrealistic factor of ten from 0.005 to 0.05. No damping was attributed to their FIGURE 4.6 Noise Reduction Between the Mechanical Room and a Sui te in a Concrete Apartment Bui ld ing (6" concrete f l oo r and (1) 6 " , (2) 3" concrete w a l l s ) . CO FIGURE 4.7 N o i s e R e d u c t i o n B e t w e e n t h e E n g i n e Room a n d a B e r t h o f T o w b o a t ^ (V s t e e l h u l l a n d d e c k , %" s t e e l b u l k h e a d ) F o r Two V a l u e s o f I n t e r n a l Damping m 'nt. welded joints. The damping layer increased the noise reduction by as much as 8 dB in the regions controlled by coincidence effects. Much less improvement was attained at frequencies for which the responses of the steel panels were to some degree mass controlled, that is, those frequencies for which the noise reduction with resonant energy transmission most nearly equals the mass law noise reduction. This latter result is to be expected since true mass controlled response is independent of damping. 126 CHAPTER 5 DETECTION AND MEASUREMENT OF AIRBORNE FLANKING TRANSMISSION 5.1 Introduction The acoustical engineer is often called upon to increase the noise reduction between adjacent acoustic cavities, for example, between a typing pool space and private offices or between the mechanical room and a nearby suite in an apartment building. There generally exist several flanking paths, structure-borne and airborne, between the two cavities. It would be very advantageous i f the engineer could detect the sound reaching the receiving cavity by the various flanking paths and by the direct path through the primary barrier, and could determine the relative importance of each path to the total sound field. As was demonstrated in Chapter 1, a small airborne flanking path (or leak) can, because of its very high transmission coefficient, severely limit the noise reduction attainable between two cavities. Such paths are often not obvious to the eye or the ear but once they are located their elimination can result in a noise reduction increase comparable to that to be gained by doubling the weight of the primary barrier and at much less cost. The vital question here is how can the acoustic signals which enter the receiving cavity by the direct and various flanking paths be distinguished from one another, and from the myriad of reflected sound which makes up the cavity's reverberant field? A highly direc-tional microphone could isolate the signal from a particular path from a good part of the reverberent sound but the path's location would fi r s t have to be discovered. The powerful mathematical technique of correlation analysis has recently been made available to the acoustical engineer for application in the field by the development of small, relatively inexpensive special purpose computers capable of real time correlation function computation. This technique allows signals which have time histories (or frequency content) similar to a reference signal to be extracted from a noisy or reverberant background of signals which have no such similarity. Correlation is therefore ideally suited for the identification of acoustic signals entering a reverberant cavity by various paths. The application of correlation analysis to measurement of flanking transmission will now be described. 5.2 The Cross Correlation - Fourier Transform Technique The correlation function computed between two time varying sig nals indicates the degree of similarity existing between the two signals as a function of time. It does so by taking a time average of the product of the two signals when one of the signals is delayed by a time "T" relative to the other. The two signals can originate from two different sources, two microphones for example, or they can come from the same source in which case the signal is multiplied by a time delayed version of itself. If the two signals x(t) and y(t) have different sources then a "cross correlation function" is obtained as follows: 1 2 8 (5.1) where T is the averaging time. If the two signals have the same source then an "auto correlation function" results: mean values and standard deviations do not vary with time, then the values of the correlation functions of Equations (5.1) and (5.2) do not depend on time but only on the time delay T imposed between them. signals are retained in the correlation function itself. For example, the auto correlation function of a sine wave is a cosine wave of the same frequency. of a source of wide band random noise on opposite sides of a partition as shown in Figure 5.1. The signals x(t) and y(t) received from microphones A and B respectively will have effectively no similarity or coherence except when the signal x(t) is delayed by an amount x equal to the difference between the acoustic travel time from the source to microphone A and that from the source to microphone B by one of the paths shown. The correlogram (cross correlation function (5.2) If the signals being correlated are stationary, that i s , i f their Frequency components which are present in both of the correlated Consider two microphones, A and B, placed in the sound field 129 Figure 5.1 Cross Correlation Between Two Microphones in the Free Field o • I — O Time Delay x Figure 5.2 The Ideal Correlogram Obtained Between the Two Microphones of Figure 5.1 130 versus time delay) of these two signals which would be obtained under ideal free field conditions is shown in Figure 5.2. The fi r s t and second peaks correspond to the arrival of sound at microphone B by direct transmission through the partition, and by diffraction over the top of the partition respectively. Under these ideal (free field) conditions, cross correlation between two micro-phone signals can then disclose the existence of a direct transmission path and various airborne flanking paths around an acoustic barrier provided the cross correlation peaks are sufficiently separated in time delay to be distinguished from one another. Structure-borne flanking paths, however, cannot be so easily identified from a cross correlation between microphone signals. This is because when a wall or floor of the receiving cavity is excited by structure-borne energy coming from the source cavity, i t radiates (above its critical coincidence frequency) from its entire surface (surface modes). Therefore the radiated sound reaches the receiving microphone over a wide range of delay times and the resulting cross correlation peak is spread over a correspondingly wide range. Because no great amount of coherent sound arrives at any one delay time, the correlation peak is also very weak and hence hard to distinguish. Correlation between an accelerometer placed on a receiving cavity wall or floor and the source cavity microphone could indicate the levels of excitation of each wall or floor due to direct structural transmission from the source room. However, another problem arises due to the dispersive nature of flexural waves. The speed of flexural 1 /2 waves in a plate increases wi-th f ' (see Equation 3.2) so that the 131 farther a packet of flexural waves travels in a wall or floor, the less is its temporal resemblance to the acoustic wave packet which generated i t . For these reasons, only airborne flanking paths will be considered in the rest of this chapter. Returning to Figure 5.2, the heights of the cross correlation peaks give an indication of the overall intensities of the acoustic signals arriving by the two paths (see section 3.3.5). However more useful information about the distribution of acoustic power with frequency within the signal from a particular path can be gained by Fourier transforming the section of the correlogram containing the peak due to that path. In the following text upper case letters will be used to donate the Fourier transforms of time functions. The Fourier transform of any time varying signal gives its amplitude-phase spectrum. The Fourier transforms of the auto and cross correlation functions, which are averaged products of two time varying signals, give the power spectra of these signals. The transform of the auto correlation function of Equation (5.2) is the power spectrum of x(t): R X X M = ± j T r x x(r) e i w r d r (5.3) Since r (x) is an even function about x = 0 (that is r ( x ) = r (-x)), AA XX XX then: 132 Similarly the transform of the cross correlation function, Equation (5.1), is the cross-power spectrum of x(t) and y(t). Since r Xy(x) is neither even nor odd, then: R X u H ~ rr S r x u ( c o s co-z: + z s i n c d T ) eta: (5.5) If stationary random variables x(t) and y(t) are the input and output respectively of a stable linear system, then i t can be shown [76] that the complex frequency response H(w) (or transfer function), of the linear system is given by the ratio of the cross-power spectrum of x(t) and y(t) to the power spectrum of x(t): Consider the case in which white noise (a signal having an even distribution of power across an infinite bandwidth) is the input x(t) to a linear system. The auto correlation of x(t) is then the Dirac delta function, that is, an impulse at x> 0: = six) = s(o) , T = O ' - . ( 5 > 7 ) o , o . Taking the Fourier transform of Equation (5.7) gives the power spectrum RVY(w) as follows: 133 (5.8) where X(w) is the Fourier transform of the input signal x(t) and the asterisk denotes the complex conjugate. If, however, as is generally the case in practice, x(t) is bandlimited and contains no components with angular frequencies greater than W, then the auto correlation is no longer an impulse. It instead decays away from T = 0 in a (sin x/x) manner. Equations (5.7) and (5.8) become in this case [77]: where U denotes a step function. If white noise is the input x(t) to a linear system with out-put y(t), the cross correlation gives the impulse response of the system h(t): (5.9) and (5.10) h i t ) (5.11) which upon Fourier transformation becomes: (5.12) 134 where Y(to) is the Fourier transform of the system output y(t). Under the present conditions (white noise input and linearly responding system) the output spectrum Y(ui) is given by the product of the input spectrum and the system frequency response [78]: Y(u>) = H M X(u>) . (5.13) Therefore equation (5.12) becomes: R * y M = X * M H M X M (5.14) If once again x(t) is bandlimited, then the cross correlation function r Xy(x) is given by the system impulse response h(t) convolved with (W/7r)(sin Wt/Wt), and the cross-power spectrum R v v ( w ) is given xy by the system frequency response H ( w ) multiplied by X*(w)X(w)[U(w+W) - U(u-w)]. The complications which arise here when the input is bandlimited were avoided by Schomer [79] by noting that the frequency response can be obtained by dividing equation (5.14) by equation (5.8). This results in the cross-power spectrum of x(t) and y(t) being normalized with the power spectrum of x(t). R x x M This is the result stated earlier in Equation (5.6). The finite bandwidth effects cancel leaving the true system frequency response. Schomer used such a method to measure the transmission loss of panels hung in an anechoic chamber by cross correlating between two microphones, one on each side of the panel. He claimed that a normalizing spectrum superior to RYV(w) could be obtained from the AA cross correlation of the two microphone signals taken with the test panel memoved. This was said to nullify the effects of sound source directionality. It would also cancel the effects of air damping and spherical spreading which would make the acoustic pressure measured by the receiving side microphone less than that measured by the source side microphone. These latter effects would be attributed to panel transmission loss i f RVY(w) was used as the normalizing spectrum. The A A two cross correlations, with and without the test panel, were truncated to include only the peak due to the direct transmission path through the panel. Additional zero value data points were appended to these truncated sections (see section 5.3). The ratio of the Fourier transforms of these two sections gave the frequency response of the panel. It was attempted in this work to use a method similar to Schomer's to measure the transmission spectra of airborne flanking paths between two model rooms separated by a partition (see Figure 5.3) A microphone was located in each room and a wideband source placed in line with the two microphones. Since when studying flanking transmission in the field or in model rooms there is no procedure analogous to "removal of the panel," the power spectrum of the source signal was to be used as the normalizing spectrum. 136 2" Fibreglass 1/8" Aluminum Panel Mic.2 Mic.T Noise Source Mi c. Amp Mic. Amp x(t) y(t) 1" Plywood Correlator i — r x y( w) Figure 5.3 Cross Correlation Between Two Microphones in Resonant Cavities It was soon discovered that Schomer had gained an unappreciated advantage by testing his panels in an anechoic chamber. The observed cross correlation functions consisted at all delay times of large amplitude sinusoids which completely obscured the distinct correlation peaks that were expected to occur at delay times corresponding to the arrival of direct and flanking path signals at the receiving room microphone. The auto correlations of the source room signal were like-wise obscured (see Figure 5.4). These sinusoids were traced to the strong correlation of natural room modes. In the case of the cross -20 -30 5 6 7 8 9 Time Delay T (msec) Figure 5.4 Cross Correlogram Obscured by Room Mode Correlation 10 CO 138 correlation functions the existence of natural modes of approximately the same frequency in both rooms caused a sinusoid of that frequency to dominate the correlogram at a l l delay times. In the case of auto c o r r e l a t i o n , exactly the same modal frequency components were contained in both correlated signals so that these components correlated even more strongly. I was concluded that measurements of the type described by Schomer could only be performed successfully in an anechoic chamber or in the free f i e l d where no natural room modes ex i s t . Therefore, a clear need existed for a technique which could overcome the problem of room mode correlation and hence be used in real room situations. Since most pairs of adjoining rooms have at least one and often two common dimensions, mode correlation can be expected to occur i n almost a l l cases. Even i f the rooms have very d i f f e r e n t shapes, the auto correl a t i o n function would s t i l l be obscured by the correlation of source room modes. I t was necessary to eliminate the room mode contributions from one of the two signals being correlated. F i l t e r i n g was unpractical because many modes were present. The source room microphone was therefore removed and the input signal to the random noise source was taken as x(t) i n the correlation function computation as shown in Figure 5.5. The room modes, being then present only in the receiving signal y ( t ) , could not correlate and the peaks corres-ponding to the dir e c t path and induced flanking paths became e a s i l y recognizable. However the substitution of the input signal to the noise source for the source room microphone signal placed l i m i t a t i o n s on the information that could be gained from the resulting cross-power ROOM 2 4 ' 7 I" M IC. ^ 1 / 8 " ALUMINUM PANEL MIC. AMP Yd) MIC. A M P X(t) CORRELATOR PAR O A r xy(T) I P L Y W O O D TRUMPET DRIVER 2 " FIBERGLASS POWER R A N D O M A M P NOISE G E N . F F T P A C K A G E F Figure 5.5 Schematic of Airborne Flanking Transmission Suite and Its Accompanying Instrumentation spectra. The power spectrum computed from the auto correlation of the noise source input signal could not be used as a normalizing spectrum since i t had not been shaped by the noise source's frequency response. The cross-power spectrum of the noise source input signal and the receiving room microphone signal could not there-fore y i e l d the absolute sound pressure levels i n the receiving room due to the d i r e c t and various flanking paths. I t could however give the r e l a t i v e levels of the contributions made by these paths, and i f corrected f o r the frequency response of the noise source, y i e l d the shape of the transmission spectrum of each. I t i s therefore possible even i n reverberent surroundings to discover which paths are contributing most to the receiving room sound f i e l d i n each frequency band. I f the cross-power spectrum i s not normalized with the source signal power spectrum, the f i n i t e bandwidth effects w i l l be present. However, i f the random noise source and loudspeaker are chosen so that they can produce a sound f i e l d at frequencies up to 20,000 Hz, these effects w i l l be well above the normal frequencies of interest in noise control problems. 5.3 The Experimental Method i n Model Rooms The experimental flanking transmission suite i s shown in Figures 5.6 and 5.7 and the schematic of the suite and i t s accompanying instrumentation i n Figure 5.5. F i g u r e 5.6 The E x p e r i m e n t a l A i r b o r n e F l a n k i n g T r a n s m i s s i o n S u i t e F i g u r e 5.7 The E x p e r i m e n t a l A i r b o r n e F l a n k i n g T r a n s m i s s i o n S u i t e 142 A box of dimensions 4'0" x 4'0" x 8'0" was constructed out of 1" thick plywood and divided into source (1) and receiving (2) rooms 3 of volumes 65 and 49.2 ft respectively by a 1/8" aluminum panel. The edges of the dividing panel were clamped rigidly onto a frame of 3/4" x 3/4" x 1/8" aluminum angle. The rooms were lined over approximately 75% of their area with 2" fibreglass to reduce initial reflections, which due to the small size of the model rooms, could have possibly arrived at the receiving room microphone early enough for their correlation peaks to have overlapped those of the direct flanking paths. The whole structure was vibration isolated from the supporting floor by four neoprene pads placed at the box corners. The noise source was a University Sound ID-60 horn driver whose input was supplied by a General Radio 1390-B random noise generator. The receiving room microphone was suspended in line with the horn driver, being 7" from the partition and 24.5" from the driver. The microphone was placed so close to the partition again to assure an adequate difference between the direct path length and the panel edge flanking path lengths so that their correlation peaks would be well separated on the time delay scale. This problem of overlap of information from two or more paths in the correlogram is one of the major limitations of this method. In larger rooms the separation between peaks will be proportionately larger so that the problem should diminish. Also, i f peak overlap is suspected, the peaks concerned can be separated simply by changing the position of the noise source or the receiving microphone. The input signal to the horn driver and the signal from the receiving microphone were amplified and fed into a PAR 101A real time correlator as x(t) and y(t) respectively. The PAR 101A allowed a wide range of computational and precomputational delay periods. The correlation function was computed at 100 increments of time delay over the computational range chosen. The width of the computational delay range T, then determined the sampling rate and hence the frequency resolution. The value of T used in these experiments was 1 msec giving a sampling period AT = T/100 or 10 usee. Therefore, at 10,000 Hz (the highest frequency of interest), one cycle was composed of 10 time delay increments. The distortion at this frequency is only a few percent and decreases for lower frequencies. 5 The sampling rate used was 1/AT = 10 Hz. The Nyquist rate, that is, the lowest sampling rate which will prevent aliasing of the Fourier transform in the frequency domain, is twice the highest frequency component in the signal. Here the highest frequency produced is 20,000 Hz so that the Nyquist rate is 40 KHz, less than the sampling rate being used. The precomputational delay allows the computation interval to be shifted along the time delay range. In this way the same high value of resolution can be maintained while examining correlation peaks occurr-ing over a wide range of time delays. Two experiments were conducted to see how well the cross correlation - Fourier transform technique could predict the frequency responses of acoustic transmission paths. 144 5.4 Direct Path Transmission Loss With the entire perimeter of the aluminum partition sealed with plasticine and neoprene tape, a correlogram was generated by computing 8 cross correlation functions using 1 msec computational periods over the delay range 0 - 8 msec. See Figure 5.8a. The direct distance from horn driver to microphone was 24.5" and corresponded to an acoustic travel time of 1.81 msec. However, i t was found by placing the microphone right up to the horn driver,that the driver itself had an inherent lag in response of 0.3 msec. The sum of these two delays is 2.21 msec and this is seen to correspond well with the large (n-shaped) peak in the correlogram. This peak then corresponds to the direct transmission path through the partition. The correlogram was truncated at 3.0 msec so that information about later arriving reflections etc. would not be included. The first three 1 msec sections of the correlogram were then digitized into 100 points each on a Gradicon graphical digitizer. An additional number of zero value data points were appended to these 300 points to give 2048 (2^) points in a l l . This was done because a discrete Fourier transform calculated from a correlogram must be based on a set of data points larger than the set of significant correlation points (here 300) to prevent aliasing of the correlogram in the time domain [80]. Such a data set gave an upper frequency limit of 12,500 Hz to the discrete Fourier transform. This was computed using a Fast Fourier Transform (FFT) package [81] on an IBM 360 computer. The bandwidth Figure 5.8 Correlograms With and Without Induced Airborne Flanking Path 146 Af of such a discrete Fourier transform i s given by: Ai = H s , ( 2 N AT K where N i s the number of s i g n i f i c a n t correlation data points. With a AT of 10 usee and using 300 points, the bandwidth obtained was 166 Hz. The cross-power spectrum of the two signals was obtained i n graphical form from the FFT computation. Since the source signal was the horn driver input, the cross power spectrum had to be corrected for the driver's frequency response to obtain the true shape of the transfer function of the d i r e c t and other paths. The horn driver's response had been previously obtained from tests i n an anechoic chamber. The re s u l t i n g frequency response or transfer function of the dir e c t path through the 1/8" aluminum p a r t i t i o n i s given i n the more f a m i l i a r form of a transmission loss i n Figure 5.9. The dashed l i n e shows the quantitatively correct mass law transmission loss f o r 1/8" aluminum. The correlation - Fourier transform resu l t has been drawn on top of the mass law curve i n order to compare spectrum shapes. A comparison of absolute values can not be made. Except at low f r e -quencies, the shapes of the two curves are in good agreement. The transmission loss measured i n the model rooms i n the standard manner i s also shown i n Figure 5.9. 50 CQ -o iA O o 1/5 1/1 •r— E to c ca i~ l — 30 10 O -_ © — : N O R M A L INCIDENCE 1/3 OCTAVE MEASUF C R O S S C O R R E L A MASS LAW B4ENTS TION • • n r Frequency (Hz) 150 300 1000 3000 10,000 •P. Figure 5.9 Transmission Loss of a 1/8" Aluminum Panel 148 5.5 Transmission Spectrum of an Induced Airborne Flanking Path The second experiment involved the introduction of a known airborne flanking path to allow correlograms with and without the path present to be obtained. The difference between the cross-power spectra obtained from the two correlograms was then compared with the difference in sound pressure levels measured by the receiving room micro-phone with and without the flanking path present. This allowed a check of the qualitative accuracy of the correlation technique. The flanking path was produced by removing a section of the neoprene tape used to seal the front edge of the partition clamping frame. This opened a small air gap 7" long, 1.7" deep and about 1/16" wide which generated an airborne flanking path 5.62 ft long from horn driver to receiving microphone. As before the correlograms consisted of eight 1 msec computational periods covering the delay range 0 to 8 msec. These are shown in Figure 5.8. With the air gap closed, only the direct path peak at 2.2 msec and various weak reflections were evident. However, with the air gap open, a new strong peak was seen at 5.3 msec. This time delay plus the 0.3 msec horn driver lag just equals the acoustic travel time over 5.62 f t . The decaying periodic nature of the flanking peak reflects the nature of its path, the air gap. The strong periodicity of the peak corresponds to a frequency of 3400 Hz. The air gap had a depth of 1.7" and therefore a half wavelength frequency of 3900 Hz. This frequency would be reduced i f an appropriate end correction could be applied to the slit- l i k e air gap. The air gap therefore acted like a band pass fil t e r . The two correlograms were again digitized into 100 points for each 1 msec of time delay but this time over the entire range 0 to 8 msec. Therefore the number of significant points N became 800 and the bandwidth of the FFT was reduced to 62.5 Hz according to Equation (5.16). Additional zero value data points were added to bring the total again to 2048. The normalized cross-power spectra, with and without flanking, are shown in Figure 5.10. The graphical FFT outputs were averaged in 1/3 octave bands to facilitate comparison of the difference between them with the directly measured difference in sound pressure levels. Since the interest here is in the difference between two cross-power spectra, the spectra need not be corrected for the horn driver frequency response. The difference between the spectra of Figure 5.10 is the increase in sound pressure level at the receiving room microphone position due to the induced flanking path. This is plotted in Figure 5.11 along with the increase measured directly in 1/3 octave bands. The agreement is shown to be very good. Both transmission spectra reflect the filtering action of the narrow gap with its pass band centered on 3400 Hz. One would expect, i f anything, that the directly measured increase might be marginally greater than that measured by correlation due to the reflection of sound arriving via the flanking path. This was not observed, possibly because of the absorptive material used. 15 10 CO -o — —A — — .1/3 O C T A V E MEASU - ° ; C O R R E L A T I O N TECH ,j" "•••^"'••"•"^1 KtlrltlN 1 0 INIQUE li o \ 9 ^ ' — A '/ N^1 \ 0 t Y ** u • 300 1000 3000 10,000 Figure 5.11 Frequency (Hz) • Increase in SPL (AL) at Receiving Room Microphone Due to the Induced Airborne Flanking Path 152 CHAPTER 6 CONCLUSIONS AND SUGGESTIONS FOR FURTHER STUDY 6.1 Conclusions The work that has been described here had two objectives: 1. To develop a theory of the flanking transmission of acousto-vibrational energy between structurally coupled cavities which would apply both above and below panel critical coinci-dence. 2. To develop an experimental technique that could detect and measure the contributions of individual airborne flanking paths to the total sound field of a receiving room. A relatively new structural dynamics technique, Statistical Energy Analysis, which is based on analogy to thermal processes, was applied to the first problem with satisfactory results. The noise reduction between two structurally coupled cavities made of riveted aluminum panels (1/32" partition and 1/16" sidewalls) was predicted to within 2 dB over most of the frequency range 300 to 20,000 Hz. Some-what poorer agreement was obtained at two points in the spectrum. The firs t was due in part to an overestimation of the coupling between finite panel and finite acoustic cavity at and just above the panel critical coincidence frequency. The second was believed due to high frequency leakage between experimental cavities. 153 It was observed that when the surface density of the partition used in the experimental model was increased by a factor of four (1/8" partition), the increase in noise reduction was 5 to 8 dB for frequencies below the first panel coincidence (4000 Hz) and was insignificant for frequencies above this value where coincidence effects controlled the response of the panels. These increases are far below the 12 dB which according to the mass law should accompany a fourfold increase in partition surface density. This large discrepancy is due to resonant energy transmission from source to receiving cavity via the coupled sidewalls and the partition. This demonstrates the extent to which structure-borne flanking transmission can limit the noise reduction benefits to be obtained by increasing the surface density of primary acoustic barriers. This noise reduction ceiling is particularly restrictive in the case of relatively lightweight and resilient structures. The SEA model predicted the marginally increased noise reduction that was obtained with the 1/8" aluminum partition with an accuracy similar to that observed for the 1/16" aluminum case. This good agreement establishes that SEA can predict quite accurately the space averaged steady state response of relatively com-plex acousto-mechanical systems even when the coupling between some of the elements of such systems is not conservative (as with the riveted joints). This encourages the application of SEA to other point fastened (and other real jointed) acousto-mechanical systems with light but moderately non conservative element coupling. The critical activity in such applications of SEA will no doubt prove to be the 154 determination of the amount of damping contributed by the various types of mechanical junctions. The SEA model developed here permits, for the fi r s t time, the evaluation of the quantitative effects of the variation of panel damping capacity (internal and boundary) upon noise reduction. It was observed (Figure 4.7) that structum-borne flanking transmission can be reduced considerably, particularly in the parts of the spectrum controlled by panel coincidence,by increasing the internal damping capacity of the sidewalls. The SEA model also allows the effects upon noise reduction of variation of partition to sidewall bending stiffness ratio to be investigated. By increasing this ratio, the transmission of flexural wave energy between source and receiving room sidewalls is decreased since the impedance change presented to flexural waves in the sidewalls by the sidewall-partition junction is increased. However, by thickening the partition, its noise reduction coincidence dip is shifted to a lower frequency (see Figures 3.3 and 3.4) which could possibly coincide with a dominant component of the source room noise spectrum. Also, as is shown in Figure 4.5, an increase in sidewall thickness will increase noise reduction above the partition coincidence frequency, but will decrease i t considerably at lower frequencies. Some consideration should therefore be given to the frequency content of the noise being designed against before the physical parameters of the constituent panels (surface density, bending stiffness and damping) are chosen. 155 The second problem, that of identification of airborn flanking paths and the measurement of their contributions to the receiv-ing room sound field, was approached with a cross correlation -Fourier transform technique. The results of the two experiments (direct path transmission loss and SPL increase due to an induced air-borne flanking path) indicate that this technique can be used success-fully to obtain the spectral shapes and relative magnitudes of acoustic signals passing between adjacent rooms by various airborne paths. Once the relative contributions of these paths to the receiving room sound field are known, noise control work can be optimized. Provided that the acoustic travel times by the various paths are sufficiently varied, the cross correlation peaks can be truncated and Fourier transformed separately to yield their relative transmission spectra. If overlapping of cross correlation peaks in the time delay domain occurs, a change in location of the noise source or the microphone could separate them. Also by obtaining correlograms for two different microphone positions, and noting the time delay shift of the correlation peaks between them, the general locations of the various flanking paths can be determined by a procedure analogous to triangulation. The technique as i t was executed in this work made use of a FFT program and a large computer. But portable, special purpose, low cost computers are now available which perform discrete Fourier transforms in real time. The use of such a real time Fourier Analyzer in conjunction with the correlation computer would allow the entire procedure to be carried out in the field. This would represent a large time saving since the slow process of correlogram digitization would be eliminated. 6.2 Suggestions for Further Study SEA Because of the small size of the experimental rooms, the SEA model could not be considered accurate below about 1000 Hz. The receiving room sound field was not diffuse below this frequency. Investigation is needed into the applicability of SEA to low modal density systems. Any work which would increase our understanding of real structural joint damping would expose new acousto-mechanical systems to investigation by SEA. A better description of diffuse panel acoustic radiation into a finite cavity is also needed. Cross Correlation - Fourier Transformation The major limitation of the correlation technique developed here for use in reverberant rooms (or any other resonant system) is its inability to give the absolute values of the direct and various flanking path contributions to the receiving room sound field. It was recently realized however, that this limitation could be overcome by using the cross-power spectrum obtained between the horn driver input signal and a source room microphone as a normalizing spectrum. As D before the cross-power spectrum R Xy( u) between the white noise input to the horn driver input x(t), and the receiving room microphone 157 signal y^(t) arriving by a particular path is found by Fourier trans-forming only the part of the correlogram pertaining to that path. Following Equation (5.12) this gives: R R M =* YR(*>) X*(co) (6.1) 0 where, YR(<o) = X ( « ) H ( « ) i n W H(w)p»iv ( 6 . 2 ) Thus the cross-power spectrum may be written in terms of the frequency response of both the noise source and path: R x y M = X * ( C 0 ) H H i m e r t t M p < ^ (6.3) If now the horn driver input and the source room micro-phone signal are cross correlated, the normalizing spectrum can be obtained by Fourier transforming the appropriate section of the correlogram. R ' X 1 | M = Y , ( « ) **(*>) (6.4) w h e r e > Y s(w) = K M H M j - , w (6.5) 158 (6.6) The frequency response of a particular path can then be realized as the ratio of these two cross-power spectra. Dividing Equation (6.3) by Equation (6.6) as follows: to be manually corrected for after the cross-power spectra were calculated, now will cancel out. Finite bandwidth effects will also cancel. Experiments in real rooms will hopefully soon be carried out to check the validity of this improved technique. H ( W ) P 4 (6.7) (6.8) The frequency response of the horn driver, which before had 159 FOOTNOTES [1] L.L. Baranek, ed., Noise and Vibration Control, New York: McGraw-Hill (1971), p. 282. [2] Ibid., p. 285. [3] M.J. Crocker and A.J. Price, "Sound Transmission Using Statistical Energy Analysis," J. Sound Vib., Vol. 9, (1969), pp. 469-486. [4] V.I. Zabarov, "Calculation of Sound Insulation of Barrier Constructions in Buildings with Regard to Flanking Transmission," J. Sound Vib., Vol. II, No. 2 (1970), p. 274. [5] Ibid., pp. 263-274. [6] R.H. Lyon and Gideon Maidanik, "Power Flow between Linearly Coupled Oscillators," J. Acoust. Soc. Am., Vol. 34, No. 5, (May 1962), pp. 623-639. [7] P.W. Smith, Jr., "Response and Radiation of Structural Modes Excited by Sound," J. Acoust. Soc. Am., Vol. 34, No. 5, (May 1962), pp. 690-647. [8] E.E. Ungar and T.D. Scharton, "Analysis of Vibration Distributions in Complex Structures," Shock and Vibration Bulletin, No. 36, Part 5, (1967), pp. 41-53. [9] Ibid., p. 41. [10] Ibid., p. 43. [11] Lyon and Maidanik, "Power Flow," pp. 628-629. [12] Baranek, Noise, p. 299. [13] Lyon and Maidanik, "Power Flow," p. 629. [14] Ungar and Scharton, "Analysis," p. 43. [14a] R.H. Lyon and T.D. Scharton, "Vibrational-Energy Transmission in a Three Element Structure," J.Acoust. Soc. Am., Vol. 38, (1965), pp. 253-261. [15] Baranek, Noise, p. 299. 160 [16] Lyon and Scharton, "Vibrational Energy," p. 254. [17] R.H. Lyon and E. Eichler, "Random Vibration of Connected Structures," J. Acoust. Soc. Am., Vol. 36, (1964), p. 1344. [18] Baranek, Noise, p. 299. [19] Crocker and Price, "Sound Transmission," p. 471. [20] Ibid., p. 472. [21] Ibid., pp. 469-486. [22] A.J. Price and M.J. Crocker, "Sound Transmission through Double Panels Using Statistical Energy Analysis," J. Acoust. Soc. Am., Vol. 47, (1970), pp. 683-693. [23] M.J. Crocker, M.C. Bhattacharya and A.J. Price, "Sound and Vibration Transmission through Panels and Tie-Beams Using Statistical Energy Analysis," ASME, Journal of Engineering for Industry, No. 70 WA/DE-2, (August 1970), pp. 1-7. [24] K.A. Mulholland, A.J. Price and H.0. Parbrook, "Transmission Loss of Multiple Panels in a Random Incidence Field," J. Acoust. Soc. Am., Vol. 43, (1968), pp. 1432-1435. [25] M.C. Bhattacharya, K.A. Mulholland and M.J. Crocker, "Coincidence Effect with Sound Waves," University of Liverpool, (1969), p. 54. [25a] Eugen Skudrzyk, Simple and Complex Vibratory Systems, New York: ASME, (1959), p. 386. [26] L.E. Kinsler and A.E. Frey, Fundamentals of Acoustics, New York: John Wiley and Sons,(1950), p. 419. [27] Beranek, Noise, pp. 289 and 348. [28] Kinsler and Frey, Fundamentals, p. 80. [29] Beranek, Noise, p. 289. [30] Ibid., p. 281. [31] Kinsler and Frey, Fundamentals, p. 401. [32] T. Kihlman, "Sound Transmission in Building Structures of Concrete," J. Sound Vib. , Vol. 11, No. 4, (1970), pp. 435-445. [33] Zabarov, "Calculation of Sound," pp. 263-274. [34] S.V. Budrin and A.S. Nikiforov, "Wave Transmission Through Assorted Plate Joints," Soviet Physics - Acoustics, Vol. 9, No.4, (April-June, 1964), pp. 333-336. 161 [35] Bhattacharya, Mulholland and Crocker, "Coincidence Effect," pp. 45-70. [36] M.C. Bhattacharya, "The Transmission and Radiation of Acousto-Vibrational Energy," Ph.D. Thesis, Department of Building Science, University of Liverpool, (1969), p. 85. [37] Bhattacharya, Mulholland and Crocker, "Coincidence Effect," p. 63. [38] Bhattacharya, "Transmission and Radiation," (thesis), p. 95. [39] Ibid., p. 59. [40] Skudrzyk, Simple and Complex, p. 374. [41] Ibid., p. 375. [42] Ibid., p. 377. [43] Ibid., p. 379. [44] Crocker and Price, "Sound Transmission," p. 470. [45] M.C. Bhattacharya and M.J. Crocker, "Forced Vibration of a Panel and Radiation of Sound into a Room," Acustica, Vol. 22,(1969-1970), pp. 275-295. [46] G. Maidanik, "Response of Ribbed Panels to Reverberant Acoustic Fields," J. Acoust. Soc. Am., Vol. 34, No. 6, (June 1962), p. 817. [47] Ibid., p. 813. [48] Lyon and Maidanik, "Powerflow," p. 635., [49] Maidanik, "Response of Ribbed Panels," p. 817. [50] Skudrzyk, Simple and Complex, p. 402. [51] P.W. Smith, Jr., "Coupling of Panel and Sound Vibration below the Critical Frequency," J. Acoust. Soc. Am., Vol. 36, No. 8, (August 1964), p. 1526. [52] Berarek, Noise, p. 453. [53] E. Eichler, "Thermal Circuit Approach to Vibrations in Coupled Systems and the Noise Reduction of a Rectangular Box," J. Acoust. Soc. Am., Vol. 37, No. 6, (June 1965), p. 1001. [54] Beranek, Noise, p. 242. L.B. Evans, H.E. Bass and L.C. Sutherland, "Atmospheric Absorption of Sound: Theoretical Predictions," J. Acoust. Soc. Am., Vol. 51, No. 5, (May 1972), p. 1574. Baranek, Noise, p. 453. Ibid., p. 241. CM. Harris, ed. Handbook of Noise Control, New York: McGraw-H i l l , (1957), p Evans, Bass and Sutherland, "Atmospheric Absorption," p. 1574. Loc. cit. Crocker, Bhattacharya and Price, "Sound and Vibration," p. 4. G. Maidanik, "Energy Dissipation Associated with Gas-Pumping in Structural Joints," J. Acoust. Am,,Vol. 40, No. 5, (1966), pp. 1064-1072. E.E. Ungar and J.E. Carbonell, "Oh Panel Vibration and Damping Due to Structural Joints," AIAA Journal, Vol. 4, No. 8, (Aug. 1966), pp. 1385-1390. Ibid., p. 1386. Maidanik, Energy Dissipation, pp. 1064-1072. Ungar and Carbonell, "On Panel Vibration," pp. 1385-1390. S.H. Crandall, Random Vibrations, Cambridge, Massachusetts: MIT Press, (1971), p. 101. D.J. Mead and E.J. Richards, ed., Noise and Acoustic Fatigue in  Aeronautics, London: John Wiley and Sons, (1968), p. 374. Crocker and Price, "Sound Transmission," p. 477. Crocker, Bhattacharya and Price, "Sound and Vibration," p. 4. M.J. Crocker and A.J. Price, "Damping in Plates," J. Sound Vib., Vol. 9, No. 3, (1969), pp. 501-508. M.A. Heckl, "Measurement of Absorption Coefficients on Plates," J. Acoust. Soc. Am., Vol. 34, (1962), pp. 803-808. Baranek, Noise, p. 209. [74] Heckl, "Measurement of Absorption Coefficients," pp. 803-808. [75] Crocker, Bhattacharya and Price, "Sound and Vibration," p. 4. [76] A.A. Winder, Introduction to Acoustical Space-Time Information  Processing, Washington, D.C: ONR Report, ACR-63, (1963), p. 99. [77] P.D. Schomer, "Measurement of Sound Transmission Loss by Combining Correlation and Fourier Techniques," J. Acoust. Soc. Am., Vol. 51, No. 4, (April 1972), p. 1129. [78] Crandall, Random Vibrations, p. 82. [79] Schomer, "Measurement of Sound Transmission," p. 1129. [80] Ibid., p. 1130. [81] R. Rackl, "CTFT-Correlation Processing with Fourier Transforms," Thesis:Department of Mechanical Engineering, University of British Columbia, (May 1972), pp. 1-17. [82] Lyon and Eichler, "Random Vibration," p. 1344. [83] Bhattacharya and Crocker, "Forced Vibration," p. 287. [84] Loc. cit. [85] Ibid., p. 283. 164 BIBLIOGRAPHY Books Beranek, Leo L., ed. Noise and Vibration Control. New York, McGraw-Hill, 1971. Crandall, Stephen H. Random Vibrations. 2 Volumes, Cambridge, Mass., M.I.T. Press, 1958 and 1963. and W.P. Mark. Random Vibrations in Mechanical Systems. New York and London, Academic Press, 1963. Downing, John J. Modulation Systems and Noise. Englewood Cliffs, New Jersey, Prentice-Hall, 1964. Gerlach, Albert A. Theory and Applications of Statistical Wave-Period  Processing. Vol. II, New York, London and Paris, Gordon and Breach Science Publishers, Gillott, Jack E. Clay in Engineering Geology. Amsterdam, London and New York, Elsevier Publishing, 1968. Greenspon, Joshua E., ed. 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Smith, T. et al_., eds. Building Acoustics. British Acoustical Society, Vol. 2, New Castle, Oriel Press, 1971. Skudrzyk, Eugen. Simple and Complex Vibratory Systems. Pennsylvania, Pennsylvania State University Press, 1968. Trapp, W.J., and D.M. Forney. Acoustical Fatigue'im Aerospace Structures. Syracuse, N.Y., Syracuse University Press, 1965. Volterra, Enrico, and E.C. Zachmanoglov. Dynamics of Vibrations. Columbia, Ohio, C.E. Merrill Books, 1965. Winder, A.A. Introduction to Acoustical Space-Time Information  Processing. U.S. Government Printing Office, Washington, D.C, ONR Report ACR-63, 1963. Journal Articles, Theses and Bulletins Bhattacharya, M.C. "The Transmission and Radiation of Acousto-Vibrational Energy." Ph.D. Thesis, Department of Building Science, University of Liverpool, [1969]. , R.W. Guy and M.J. Crocker. "Coincidence Effect with Sound Waves in a Finite Plate." J. Sound Vib., Vol. 18, No. 2 [1971], pp. 157-169. , and M.J. Crocker. "Forced Vibration of a Panel and Radiation of Sound into a Room." Acustica, Vol. 22 [1969-1970], pp. 275-295. , K.A. Mulholland and M.J. Crocker. "Coincidence Effect with Sound Waves." Department of Building Science, University of Liverpool, [1969]. Budrin, S.V. and A.S. Nikiforov. "Wave Transmission through Assorted Plate Joints." Soviet Physics-Acoustics, Vol. 9, No. 4 [April-June, 1964J, pp. 333-336. 166 Burd, A.N. "The Measurement of Sound Insulation in the Presence of Flanking Paths." J. Sound Vib., Vol. 7 [1968], pp. 13-26. Clarkson, B.L. "Structural Aspects of Acoustic Loads." ACARDograph 65, NATO Advisory Group for Aeronautical Research and Development, [Sept. 1960], p. 32. Crocker, M.J. and A.J. Price. "Damping in Plates." J. Sound Vib., Vol. 9, No. 3, [1969], pp. 501-508. , M.C. Bhattacharya and A.J. Price. "Sound and Vibration Transmission through Panels and Tie Beams Using Statistical Energy Analysis." ASME Journal of Engineering for Industry, Paper No. 70-WA/DE-2TAugust, 1970J, pp, 1-7. and A.J. Price. "Sound Transmission Using Statistical Energy Analysis." J. Sound Vib., Vol. 9 [1969], pp. 469-486. Eichler, Ewald. "Thermal Circuit Approach to Vibrations in Coupled Systems and the Noise Reduction of a Rectangular Box." J_. Acoust. Soc. Am., Vol. 37, No. 6 [June 1965], pp. 995-1007. Evans, L.B., H.E. Bass and L.C. Sutherland. "Atmospheric Absorption of Sound: Theoretical Predictions." J. Acoust. Soc. Am., Vol. 51, No. 5 [May 1972], pp. 1565-1575. Heckl, Manfred A. "Measurement of Absorption Coefficients on Plates." J. Acoust. Soc. Am., Vol. 34 [1962], pp. 803-808. . "Wave Propagation on Beam-Plate Systems." J. Acoust. Soc. Am., Vol. 33, No. 5 [May, 1961], pp. 640-643. Kihlman, T. "Sound Transmission in Building Structures of Concrete." J. Sound Vib., Vol. 11, No. 4 [1970], pp. 435-445. Lyon, R.H. "Random Noise and Vibration in Space Vehicles." Shock and  Vibration Information Centre, U.S. Dept. of Defence, Mono-graph SVM-1 L1967], pp. 1-69. - and E. Eichler. "Random Vibration of Connected Structures." J. Acoust. Soc. Am., Vol. 36 [1964], p. 1344. and Gideon Maidanik. "Power Flow Between Linearly Coupled Oscillators." J. Acoust. Soc. Am., Vol. 34, No. 5 [May 1962], pp. 623-639. and T.D. Scharton. "Vibrational-Energy Transmission in a Three Element Structure." J. Acoust. Soc. Am., Vol. 38, No [1965], pp. 253-261. 167 Maidanik, Gideon. "Energy Dissipation Associated with Gas-Pumping in Structural Joints." J. Acoust. Soc. Am., Vol. 40, No. 5 [1966], pp. 1064-1072. . "Response of Ribbed Panels to Reverberant Acoustic Fields." J. Acoust. Soc. Am., Vol. 34, No. 6 [June 1962], pp. 809-826. Mulholland, K.A., A.J. Price and H.O. Parbrook. "Transmission Loss of Multiple Panels in a Random Incidence Field." J. Acoust. Soc. Am., Vol. 43 [1968, pp. 1432-1435. Nash, W.A. and H.G. Kizner. "Experimental Determination of Transfer Functions of Beams and Plates by Cross-Correlation Techniques." Pretlove, A.J. and A. Craggs. "A Simple Approach to Coupled Panel-Cavity Vibrations." J. Sound Vib., Vol. 11, No. 2 [1970], pp. 207-215. Price, A.J. and M.J. Crocker. "Sound Transmission Through Double Panels Using Statistical Energy Analysis." J. Acoust. Soc. Am., Vol. 47 [1970], pp. 683-693. Rack!, R. "CIFT-Correlation Processing with Fourier Transforms." Department of Mechanical Engineering, University of British Columbia, [May 1972], pp. 1-17. Schomer, P.D. "Measurement of Sound Transmission Loss by Combining Correlation and Fourier Techniques." J. Acoust. Soc. Am., Vol. 51, No. 4 (part 1) [April 1972], pp. 1127-1141. "Signal Correlators and Fourier Analyzer. Models 100A, 101A, 102." Princeton Applied Research Corporation Manual [1969]. Smith, P.W. Jr. "Coupling of Sound and Panel Vibration below the Critical Frequency." J. Acous. Soc. Am., Vol. 36, No. 8 [Aug. 1964], pp. 1516-1520. . "Response and Radiation of Structural Modes Excited by Sound." J. Acoust. Soc. Am., Vol. 34, No. 5 [May 1962], pp. 640-647. Ungar, E.E. and J.R. Carbonell. "On Panel Vibration and Damping Due to Structural Joints." AIAA Journal, Vol. 4, No. 8 [Aug. 1966], pp. 1385-1390. 168 Ungar, E.E. and T.D. Scharton. "Analysis of Vibration Distributions in Complex Structures." Shock and Vibration Bulletin, No. 36, Part 5 [1967], pp. 41-53. Zabarov, V.I. "Calculation of Sound Insulation of Barrier Constructions in Buildings with Regard to Flanking Transmission." J. Sound  Vib., Vol. 11, No. 2 [1970], pp. 203-274. 169 APPENDIX A THE DERIVATION AND VERIFICATION OF THE CONSISTENCY RELATIONSHIP (Equation 2.3) The derivation of the consistency relationship: NA ^AB = ^6 >]BA > (2-3) as performed by Lyon [82], will now be given. Starting with the assump-tion of reciprocity of coupling between two modes of different elements: ^ i j ~ ^ J i > ( A . I ) where A. . is the coupling coefficient between modes i and j , the power flows from mode set A to mode set B (of elements A and B respectively) and vice versa are expressed as: PA B = N A E „ A <{>ij PBA = N A <Pji where N^  and Ng are the total number of modes of sets A and B within the frequency interval Af. Using Equation (A.I) to eliminate one of A., or A., from equations (A.2) and (A.3) and combining the latter two equations we have: (A.2) (A.3) 170 FAB —. P B A Therefore: 6 The energy coupling loss coefficients between sets A and B are defined as: Substituting these expressions into Equation (A.4) gives; A - f _ _A_f Introducing the modal densities of elements A and B: \ - 'Jk • n -  Nb i t is found that: M « = " M B A (A.4) (A.5) A verification of the consistency relationship for the case of structural coupling between the 1/32" aluminum partition and a 1/16" aluminum 171 receiving room sidewall will now be presented: This will also provide a check on the accuracy of one of the structural coupling coefficients derived in section 3.3. Because of the symmetry of the tee junctions (A and B in Figure 3.1), the flexural amplitude transmission coefficients x^ 2 and Tg 2 are equal. Therefore the approximate expression for x^ 2 presented 8,2" ' 5 in section 3.3.5 can be used here for x T 2 - — - * 8 CJZ m 8 Cfe and m 8 h / * 2. <f2 8 =• 0.04-42 The modal density of a panel is given by Equation (3.4) as: P 2TT K C A • 2^ 8 = A 8 - 0 .37 1 = 0 . 3 7 * 0 . 0 4 4 2 =» 0 . 0 1 6 3 The calculation of g from the expression for g derived in Appendix B, gives for the case of the 1/32" partition: 0.01048 Similar agreement is obtained for other partition/sidewall thickness ratios lending considerable credence to the consistency relationship as i t applies to the coupling loss coefficients between mechanically coupled plates. 172 173 APPENDIX B THE EVALUATION OF FLEXURAL WAVE TRANSMISSION COEFFICIENT x0 0 In this case all the plates comprising the junction are considered to be semi-infinite in order to simplify the analysis. This simplifica-tion has proved to be justifiable since the effects of resonances in plate 2 when i t was considered finite (as with 2) w e r e evident only at frequencies below the lower limit for diffuse field conditions in the experimental model. The expression obtained here for x^ g is therefore valid over the range of frequencies for which the statistical model itself is valid. The junction A and the positive displacement directions for each of its three plates are shown in Figure B.l Figure B.l The Co-ordinate System and Positive Displacement Direc-tions for Evaluation of x^ g 174 A unit amplitude flexural wave is considered to be normally incident on the junction from plate 2. This gives rise to reflected and distortional flexural waves and a reflected longitudinal wave in plate 2. In each of plates 4 and 8 there will be transmitted and distortional flexural waves and a transmitted longitudinal wave. Con-sidering all these waves to be sinusoidal except the distortional waves which decay exponentially with distance from the junction at which they were created, and suppressing the time dependence e - 1 6 0^ which is of no consequence here, the amplitudes of flexural and longitudinal waves in the plates may be written as: \)z = exp ( jk z x 2 ) + B 4 e*p( -Jk 2 x a ) + exp (k 2 * z ) JJ4- = exp(j + C4- e * P ( ~ k * X 4 ) <fe = B 8 exp(j kax&) + C 8 e x p ( - k 8 % 8 ) u4 =  G4 e x P ( j P4 **) U a = G8 e x p ( j pgXg) The displacements given by Equations (B.l) to (B.6) are subject to the following boundary conditions at the junction A (x^ = x^ = Xg = 0): (a) There must be continuity of linear displacements; (b) There must be continuity of Angular Displacements; 175 (c) The sums of the forces in the X and Z directions must equal zero. (The co-ordinate system used for the forces and moments is also shown in Figure 8.1); (d) The sum of the bending moments about the Y axis must equal zero; Z M Y = ~ P 2 ^ + - D 4 l % + D . J L . - o By applying these conditions to the displacement equations at x^ = x^ = Xg = 0 the following nine equations are generated: B 8 + C 8 =• G%\ (B.7) B 4 4- C 4 = - G 2 , (B .8) . • 1 + B z + C 2 = G 4 , (B.9) G + = - G 8 , (B.io) 176 M j - j B x + q ) = k 4 ( j e 4 - c 4 ) , ( B > 1 1 ) k z ( j - o B ^ + c i ) = k 8 ( j B 8 - C 6 ) > (B.12) P x k \ ( i + j B a + C2) - j E 4 . h 4 p 4 6 4 - j E 8 h 8 p 8 G 8 = 0 , ( B . i 3 ) P# k 4 (-j B 4 - C 4 ) + D 8 k 3 8 ( - j B 8 - C8) + j E ^ ^ G a " =o,(B.i4) - O x ^ f - l - B ^ + C O 4- D 4k 4 ( - B 4 . + C 4 ) + D 8kV - B 6 + C 8 ) =0-.(B.15) Making use of Bhattacharya's notation: and noting that plates 4 and 8 are of the same material and thickness in this analysis so that: ^4= kB ; E 4 = E 8 'j p 8 • h 4 = h 8 cmd D 4 - D & Equations (B.ll) to (B.15) become: * . ( j - J B * + C2) = j B 4 - C*. , (B.16) * ( J - J B 2 + CO = j B 8 - C 8 , ( B J 7 ) ""j + j B a + C 2 . - 0 b C G 4 - + ^ 8 ) = = - 0 , (B.18) - j ( B 4 + B 8 ) " + 0 - j a G ^ O , (B.19) C (|+. B j t - O - ( B 4 + B 8 ) + C 4 + C 8 = 0 . (B.20) 177 The nine equations, (B.7) to (B.10) and (B.16) to (B.20), will be reduced to four in amplitudes Bg, Cg, B 2 and B 4 and these will be solved for Bg. Substituting G2 = B g + Cg from Equation (B.7) into (B.19) we have: Equation (B.10) states that G^  = -Gg. Applying this to Equation (B.18) we have: C 2 = j ( l - B 2). Upon substituting this expression for C ? into Equations (B.16), (B.17) and (B.18) we obtain: - j B 4 - C 4 + j B 8 fa.-l) + C 8 (jOL-l) = 0 (B.21) J 2 0 C ( l - Ba) (B.22) J 2 * ( l - BO ~ j B 8 ~ C 8 (B.23) 0 (B.24) Equations (B.7) and (B.8) when added yield: C 4 = Be + C8 + B 4 When this is substituted into Equations (B.21), (B.22) and (B.24) the following are obtained: = O 178 (B.25) B 4 0 + j)+ B 8 + C 8 - j 2 * ( I - = O (B.26) = O (B.27) The Equations (B.23),(B.25), (B.26) and (B.27) will now permit an explicit solution for Bg to be found by applying Cramer's rule to the matrix of their coefficients: Ba c 8 B 4 » 1 O - j 2 o C i+jO- i ) jo- 0 O 1 .1 J-20C j 2 * -2 0 - 2 This yields upon separation into real and imaginary parts: ZoiC (a+ 2 ) ( j+ 1) B 8 4o<a-6c<-2c(3-a) + j [ 4 - c x a . + c(a-i) -0ot] (B.28) and (B.29) 179 APPENDIX C THE COINCIDENCE PHENOMENA IN FINITE PANELS The occurrence of wave coincidence in finite panels can be under-stood in two ways. It can be regarded as the limiting case of infinite panel coincidence in which internal panel damping is so high that flexural waves reflected from the boundaries are too weak to form standing waves. The second possibility involves the matching of standing flexural waves in the panel with standing acoustic waves. This would require that at least one side of the finite panel be backed by a finite cavity so that standing acoustic waves could develop. The mode numbers of cavity, (1, m, n), and panel, (q, r ) , indicate the number of half cycles of standing wave acoustic pressure and flexural displacement, respectively, existing in each co-ordinate direction for a particular mode. Critical coincidence would then occur when a (o, m, n), that i s , an axial or tangential cavity mode grazing the panel surface, f i r s t matched its trace wavelength with a similar mode in the panel. Such a situation is shown in Figure C l in which a flexible panel is backed by a rigid walled cavity. The panel displacement for the critical coincidence mode, considered here to be an axial mode, is shown as curve (a); the displacement nodes occur at the rigid boundaries. The pressure signature of an ideally coinciding axial cavity mode is shown as curve (b). The standing wave pressure and displacement signatures, (a) and (b) are in phase so that coincidence coupling could apparently occur. 180 Panel Figure C l Erroneous Conceptions of the Coincidence Phenomenon in Finite Panels Subsequent higher frequency panel coincidences occur when the component of the cavity mode number which is normal to the panel, "£", takes on the values I = 1,2,3 that is , when the cavity wave vector no longer grazes the panel surface [83]. Coincidence in finite panels occurs then at discrete frequencies corresponding to either the true matching of a cavity modal wavelength with a panel modal wavelength or the matching of the trace wavelength of a cavity mode with the same. Infinite panel coincidence, in contrast, occurs continuously in frequency above the critical coincidence value as the angle of wave incidence on the panel varies from grazing to normal incidence. 181 In reality the ideal coupling between panel and cavity modes shown in Figure C l (a) and (b) cannot occur, for in order for pressure and displacement to be in phase, the standing wave pressure signature must have nodes at the cavity walls. This is impossible with any real rigid walls. The realistic pressure signature of an axial cavity mode of the same frequency is shown in Figure C l (o). It is 90° out of phase with the panel displacement and therefore the panel mode could not drive the cavity mode, nor vice versa. Yet coincidence phenomena are observed in finite panels excited by sound waves. To discover how this is possible, consider the three conditions shown by Bhattacharya and Crocker [84] to be necessary to produce a maximum acoustic velocity potential, that is, critical coincidence, in the cavity of a panel-cavity system like that of Figure C l . 1. Bmn = maximum qr 2' W q , r = W 3 - V m , n =•<•>••• Here to and to m are the modal frequencies of the panel and q,r o,m,n ^ r cavity respectively, to is the frequency of sound waves incident on the panel from the outside (the forcing frequency) and Bmn is the coupling qr coefficient between panel and cavity modes. The mathematical analysis [85] shows that the coupling coefficient assumes its maximum value when the components of the panel and cavity mode numbers differ by unity, that i s , when m = q ± 1 and n = r ± 1. This does not contradict conditions (2) and (3) 182 since i t is possible for there to be more or less flexural half wave-lengths across the panel length than acoustic half wavelengths across the same length of cavity, while the two waves have the same frequency. The two wave speeds would simply have to differ by a compensating amount. In Figure C.2, a situation is shown in which m = q - 1; considering an axial cavity mode to be grazing the panel in the y direction. / Cavi ty Mode y (1, m, n,) = '/ (0,10, 0) / /. Panel Mode M ( q , >) = U l , 0) +90° Phase Di fference Figure C.2 The Coincidence Phenomenon in Finite Panels 183 This shows that although the cavity pressure and panel displace-ment are 90° out of phase at the boundaries, they are predominantly in phase in the central region, and quite good coupling can occur there. Thus coincidence coupling can occur in a modified fashion in finite panels driven by sound waves i f the panel is backed by a cavity. Note that here critical coincidence does not occur at exactly the frequency predicted by Equation 3.49 for infinite panels because the flexural wave speed must be slightly greater or less than the acoustic wavespeed for maximum coupling. This deviation increases as the panel dimensions and hence the numbers of half wavelengths across the panel and cavity decrease. The coupling at and above the critical coincidence frequency between the reverberant acoustic field in a cavity and the reverberant flexural wave field of a finite panel bounding that cavity cannot be expected to be as strong as that between a finite reverberant panel and an acoustic half space, since, as just seen, the modal panel dis-placement and cavity pressure are only truly in phase along the panel centre line. The expression (Equations 3*54) for the radiation resistance above critical coincidence of a finite reverberant panel into half space, which is the same as for an infinite panel, likely over-estimates the radiation resistance of the same panel into a finite cavity. This then results in an over-estimation of the flanking transmission and the resonant partition transmission between adjacent structurally coupled cavities at frequencies for which panel response is controlled by coincidence phenomena. The noise reduction between the two cavities is accordingly under-estimated at these frequencies. HrrXlNUiA u - rKUOKrtnmitu rvjr\ o c n i m u i : T N L U U P I I U I I ^g^ S .CCM3ILE R E A L L 2 , L X < 4 > , L Y { 4 ) , E 2 , F 4 , E S , H 2 , H 4 , H 8 , M 2 » M 4 , V 8 , A 2 , A 4 * A B * R H Q , + F t C M F G A , V I , V 3 , A VAL F 3 , A T O T 3 , 0 2 , T D 8 , K 2 * K 4 , K 8 , A L P H A * P I • C A I R 1 1 F r • + P2 » P4 , P3 » C L 2 » C l . A , C L 8 , C B 2 f C R 4 , C B 8 t A , B , C , E X N U 1 3 , N U 1 3 , N 1 , N 2 , N 3 , N 4 , + N 8 , C A ftS,SQ R T , S I N t C 0 S t A L O G 1 0 , A L G G , R G , RH , JH , R M , J M , M R 4 2 » N U 4 2 , R E » J E 2 R E AL M B 48 , NU4P * 3 P i JP , R Q t JO t !"l B 2 9 , NU2 3 » I AM AI P.* F C C ( 4 ) t D U M C L ( 4 ) , + OUMH< 4 ) , LAMt C (4.) , A R E A ( 4 ) , P E R ( 4 ) , D I F F , A L F , R R A D ( 4 ) + M U 2 1 , NU2 3 , N U 4 1 , NU8 3 , L F 2 , H F 2 , L F 4 , H F 4 , N ' L I 2 , N U 3 , N U 4 , + NU 4 T , N ' J 3 T , T l , T 2 , T 3 , T * , T 5 , T 6 , T 7 , T 8 , R A T I 0 * N R , R F , J F , D U M M ( 4 ) , N U 8 * N U 2 T , , A R S I N N U F ( 4 ) , N U 3 T * - J 5 R E A L T H R E E / 3 . G O Q O / * Z E R O / 0 . 0 0 0 0 / , O N E / 1 « . X ' O O / R E A L L 2 Y , L 4 X * L 4 Y , L 8 X , L 8 Y R E A L T 9 . T 1 0 * T l ! , T 1 2 , T H L , N U M T L , D E N T L * T L , M L T L , M L N R , D B N U 4 8 6 7 8 RE AL N U 3 F . X T * N U M T L 4 , D E N 7 L 4 , T L 4 , E X 3 E XT , A3 EX T» D I S3 » R E A L N R A T , OR A T T R A T * N R W F , D I S 4 , 0 I S 8 , N U 4 E X T , N U 8 E X T , RE A L K A P P A ,N I N T , D I S 2 , N U J T N ( 3 ) t F A C T O R ( 3 ) * M , A L F A I R D E L N U ( 3 ) , F C C 4 * N U R I V { 3 ) A T 0 T 1 9 1 0 •' R E A L W , 0 ( 3 ) * G A H R E D { 3 ) * F R E D ( 3 ) , G A M ( 3 ) »L A M B ( 3 ) , \ ' U A P ( 3 ) , D D 2 , D D 4 C O M P L E X RI , R ? . , R 2 1 t C E X P , C M P L X , S 2 » C G » C H , C M , B A 2 t E E l t F F 1 , Z Z T C C O E F , + C F 3 4 2 , 3 4 3 , C P , C Q , B 2 8 11 12 13 1 0 I N T E G E R I , M I N F , M A X F RE A l ( 5 T 1 0 ) L 2 , M 2 , M 4 * H 2 , H 4 , W , D D 2 , D D 4 F O R M A T ( 9 F 8 . 5 ) 14 15 16 1 3 • R E A D J 5 , 1 3 ) E 2 , E 4 , C L 2 * C L 4 F O R M A T ( 4 E 1 0 . 3 ) •RE A3 ( 5 , 1 5 ) V I , V 3 » A 2 , A 4 , A 8 1 7 18 1 9 1 5 2 5 F O R M A T ( & F 1 0 . 2 ) R E A D ( 5 , 2 5 ) L 2 Y , L4X , L4 Y , L 8 X , L 8 Y F O R M A T ( 5 F 7 . 3 1 2 0 21 22 3 0 Rt AO( 5 , 3 0 ) N U 2 , N U 4 , N U 8 , T H L F O R M A T ( 4 F 6 o 4 ) HS = H4 2 3 2 4 2 5 CL 3 = C L 4 E3 = E 4 D2 - E 2 * ( H 2 * * A / 1 2 . 0 ) 2 6 2 7 28 D4 •= E 4 * ( H 4 * * 4 / 1 2 . 0 ) •08 = D4 M8 = M4 29 3 0 31 AT D T I = 2 . 0 * A 2 + 4 . C * A 4 A T 0 T 3 = 2 . 0 * A 2 + 4 . 0 * A 8 PI = 2 . 0 * A R S I N ( 1 . 0 ) 32 3 3 3 4 RH 0 = G . . C 7 55 C A I R = 1 1 3 0 . 0 WR I T E { 6 , 33 ) 35 3 6 3 7 3 3 F O R M A T { 1 H 1 f 3 X , • F R E Q ' » l l X f • M L N R * , 1 G X , * N R W F « , 1 0 X , IF = 2 * * 1 1 . 0 / 1 0 . 0 ) R E A D ( 5 , 5 Q ) M I N F , M A X F • N R • , / / ) 3 8 3 9 4 0 . 5 0 I C O F O R M A T ( 2 1 1 0 ) F = M I N F / I F F = F * I F 4 i C IF < F . G T . M A X F ) GO TO 9 9 9 D E F I N E F R E Q U E N C Y V A R Y IMG P A R A M E T E R S 42 4 3 c CM EGA = 2 « 0 * P I * F P2 = O M E G A / C L ? " 4 4 4 5 4 6 P4 = O M E G A / C L 4 P8 = P4 C B 2 = .SORT < O M F G A * H 2 * C L 2 / < 2 . 0 * S Q R T ( T H R E E )) ) 4 7 4 8 4 9 C B 4 .= SORT ( 0 M E G A * H 4 * C L 4 / ( 2 . C 1 * S Q R T ( T H R E E )) ) C B 8 = C B 4 K2 = C M 5 G A / C B 2 50 K4 = OMEGA / C 134 51 KR = K 4 5 2 AL PHA = K2 / K 4 53 A = ( E 2 * H 2 * ? 2 ) / 1 0 4 * ( K 4 * * 3 ) ) 5 4 3 = E 3 * P 8 * H 3 / ( n 2 * K 2 * * 3 ) 5 5 C •= ( 0 2 * I K 2 * * 2 ) ) / { D 4 * ( K 4 * * 2 ) ) 5 6 5 7 5 8 R l = C M P L X ( Z b ' F 0 , - K 2 * L 2 ) R l = C E X P l R l ) R2 = C M P L X ( Z E P 0 , K 2 * L 2 ) 5 9 6 0 61 R2 = C E X P l R2> R2 1 = . R 2 / R 1 52 = C M P L X fZE-PO , P 2 * L 2 ) 62 C c S2 = C E X P l S2 ) E V A L U A T E S T R U C T U R A L C O E F F NU42 63 6 4 6 5 RG = C + 2 * A L P H A RH •= ( C + 2 * A L P H A ) * * 2 JH = - 1 * RH . 6 6 6 7 6 8 RM = C * * 2 - 4 * A L P H A * C - 4 * JM = - l * { C * * 2 + 4 * A L P H A * C CG = C K P L X ( R G . Z E R O ) A L P H A * * 2 - 4 * A L P H A * * 2 ) 6 9 7 0 71 CH = C V P L X ( R H , J H ) CM = C V P L X ( R M , J M ) 34 2 = 2 * C G * R 1 / ( R 1 * C H - R 2 * C M ) 72 7 3 c MB42 = C A B S l B42.) ML)42 = M 2 * C B 2 * M B 4 2 * * 2 / < M 4 * C 8 4 ) E V A L U A T E S T R U C T U R A L C O E F F NU48 7 4 7 5 76 RE = - 2 * A * S I N l P2 * L 2 ) J E = 8 * C 0 S I P 2 * L 2 ) - 2 * A * RF = 4 # ( A * C 0 S I P 2 * L 2 ) + 2 * S I N I P 2 * L 2 ) S I N l P 2 * L 2 ) ) 7 7 7 8 7 9 J F = A * (A * S I N I P 2 * L 2 ) - 4 * EE 1 = C M P L X (P-E , J E ) F F 1 = C M P L X I R F , J F ) C O S l P 2 * L 2 ) ) 8 0 81 82 ZZ - ( E E 1 / F F I ) * (A / 4 . 0 ) C C O E F = C M P L X I C , — 2 * A L P H A ) C F B 4 2 = ( R 2 1 * C C O E F / (C + 2 * A L P H A ) - 1 . 0 ) * C / 2 . 0 8 3 8 4 8 5 B4 8 = 1 . 0 0 0 0 0 0 + Z Z + C F B 4 2 * B 4 2 MB 43 = C A B S l B 4 8 ) \<;j43 = M 8 * C B 8 * M 8 4 8 * * 2 / I M 4 * C B 4 ) c c E V A L U A T E S T R U C T U R A L C O E F F N U 2 8 86 RP = 2 . 0 * A L P H A * C * IA + 2 . 0 ) 5 7 88 8 9 JP •= RP CP = C M P L X ( R P , J P ) PQ = 4 . 0 * A L P H A * A - 6 „ 0 * A L P H A - 2 . 0 * C * ( 3 . 0 - A ) 9 0 91 9 2 J Q = 4 . 0 * A L P H A * A + C * < A - l . ' i ) -CQ = CMPLX I R Q , J Q ) 32 8 = C P / C O 8 . 0 * A L P H A 9 3 9 4 c MB 28 = CAB S l 8 2 8 ) NIJ2 3 = M 8 * f B 8 * W B 2 8 * * 2 / ( M 2 * C B 2 1 . c c c E V A L U A T E P A N E L TO ROOM C O U P L I N G CO E F F S 9 5 9 6 9 7 L A M AI R = .C A I R / F DO 5 5 0 1=1 , 3 , 1 D U M C L U ) = C L 2 9 8 9 9 1 0 0 D U M C L 1 2 ) = C L 4 D U M C L 1 3 ) = C L 8 OU.MHl 1 ) = H2 186. 101 102 10 3 .104 105 106 OU MH( 2 ) = H4 DU MH ( 3 ) = H3 0( 1) = 002 0 ( 2 ) = C 04 0( 3) = 004 FC C ( I) = SORT ( THREE ) * C A IR * * 2 / ( P I *DUMCL ( I) *DUMH{ I) ) 10 7 10 8 10 9 LA ••ICC { I ) = C AIR/FCC ( I ) • • LX ( 1) = L2 LY ( I) - L2V 110 111 112 LX ( 2) = L4 X L Y ( 2 ) = L4Y LX ( 3 ) = L8X 113 114 11 5 LY{3) = L3Y AR EA( I ) = LX( I ) * L Y ( I ) PE R(I ) = 2 . 0 * ( L X U >+LY{I ) ) 116 117. 118 42 0 DIFF = F - FCC( I) I F ( O I F F ) 4 2 0 , 4 4 0 , 4 6 0 CONTINUE 119 120 121 ALF = SORT IF/FCC ( I ) ) IF<F.GT.FCC( I J / 2 . 0 ) GO TO 425 Gl = ( 4.0/ P I * * / * )*{ 1 .0-2.0*ALF**2 ) / ( ALF*SQRT( 1.0 - A L F * * 2 ) ) 122 123 124 42 5 . GO TO 4 30 CONTINUE Gl = ZERO 125 126 430 CONTINUE G2 = (( 1.0/(2. 0*PI ) ) * * 2 ) * { ( 1.0 - A L F * * 2 ) * A L 0 G ( ( 1.O + ALF ) /(1.0-ALF) ) + + 2.0*ALF )/SQRT(( l . G - A L F * * 2 ) * * 3 1 127 128 RR.AO( I ) = AR E A { I )*RHO*CAIR * ( ( L AMCC ( I ) *L AM AI R /AREA ( I ) ) *2 . 0* + (F/FCC( I ) ) *G1. + ( P E R ( I )*LAMCC(I)/AREA( I ) ) * G 2 ) GO TO 500 129 130 440 CONTINUE R R A O U ) = A R E A ( I ) * R H O * C A I R * { S O R T ( L X ( I ) / L A M C C ( I ) ) + S O R T I L Y ( I ) / + LAMCC ( M ) ) 131 13 2 133 4 6 0 50 0 GO TO 500 RRAZMI) = AREA( I }*RHO*C A I R*SQRT ( F/ ( F-FCC< I ) ) ) CONTINUE 134 135 136 OUMM( 1 ) = M2 DUI*M{ 2 ) = M4 0UMM(3) = M3 137 138 139 IF (F.GE„FCC( I ) ) GO TO 520 RR A1( I ) = 3.0*RRAD( I ) 520 CONTINUE 140 c N U F ( I ) = RRADCI )/( OMEGA* DUMM ( I ) *.A R EA ( I ) ) L c INCREASE IN PANEL DAMPING DUE TO RIVETED JUNCTIQNS( EXPERIMENT ) 141 142 c I F ( F . L E . C . 7 5 * F C C ( I ) ) GO TO 523 IF ( F . L E . l . 25*FCC - l I )) GO TO 525 143 144 145 IF ( F . L E. 2. 0 0 * FCC •( I )) GO TO 52 7 DE LNU( I ) .= 3.0 GO TO 5 30 146 147 148 523 52 5 OE LNU ( I ) = 1.0 GO TO 530 OELNU(I) = 1 . 0 + 3 . 8 * ( 2 . 0 * F - 1 . 5 * F C C ( I ) )/FCC( I ) 149 150 151 52.7 530 GO TO 5 30 OELNU(I) = 3.0 + 1 . 8 * ( 2 . 0 * F C C ( I ) - F ) / ( O . 7 5 * F C C ( I ) ) OS L 10 ( I ) = DEL N U ( I ) *0 . 0 0 1 152 15 3 154 5 5 0 N U J T N ( I ) = 6.65*DELNU( I ) * P E R ( I ) / A R E A ( I ) CO NTIMUE MU21 = NIJF ( 1 ) XfcSX 1 5 5 1 5 6 1 5 7 N U 2 3 = NU2 1 NU4 1 = NUF (2 ) N U S 3 = N U F ( 3 ) C C E V A L U A T E L O S S C O E F F S C 1 5 8 1 5 9 1 6 0 561 F C C 4 = S G R T f T H R E E ) * C A I R * * 2 7 ( P I * C L 4 * H 4 ) IF ( F . G E . F C C 4 ) GO T O 5 6 3 N U 4 E X T = M U 4 I / 2 . 0 161 1 6 2 1 6 3 N U 8 E X T = N U 3 3 / 2 . 0 GO TO 55 5 5 6 3 N J 4 E X T = NU41 1 6 4 1 6 5 1 6 6 N U 8 E X T = NU8 3 56 5 C O N T I N U E DI S2 = NU2 + N U J T N ( 1 ) 1 6 7 1 6 8 DI S4 = NU4 + N U 4 E X T + N U J T N ( 2 ) D I S S = NUB + N U 8 E X T + N U J T N { 3 ) C C A I R A B S O R P T I O N IN ROOM 3 r 1 6 9 tr A V A L F 3 -- 0 . 0 0 5 1 7 0 171 1 7 2 IF ( F . G E . 9 C 0 . ) GO TO 5 6 0 A L F A I R = ( 0 . 2 * 1 0 * * ( 0 . 6 7 * A L Q G 1 0 ( F / 2 0 0 , » ) ) * 0 . 0 C 1 GO TO 56 6 1 7 3 1 7 4 1 7 5 5 6 0 A L F A I R = { 0 . 8 6 * 1 0 * * U « 7 5 * A L G G 1 0 < F / 9 0 0 . ) ) ) * G . 0 0 1 5 6 6 C O N T I N U E M = 0 © 2 3 03 * A L F A IR 1 7 6 NO 3 = 2 . 2 * < A V A L F 3 * A T 0 T 3 + 4 . 0 * M * V 3 ) / ( 0 . < H 9 * F * V 3 ) C C E V A L U A T E MASS LAW C O U P L I N G C O E F F N U 1 3 1 7 7 1 7 8 C A V A L F 3 = 0 , 0 1 NU MTL = (S I N ( T H L ) * 0 M E G A * M 2 / ( 2 * P H 0 * C A I R ) )**2 1 7 9 1 8 0 D E N T L = A L O G ( ( l o O + ( O M E G A * M 2 / ( 2 * R H O * C A I R ) ) * * 2 ) / + ( i o O + (COS < T H L ) * C M E G A * M 2 / ( 2 * R H 0 * C A I R ) ) * * 2 ).) TL - A L 0 G 1 C M N I J M T L / D E N T L ) 181 1 8 2 183 ML T L = 1 0 . 0 * T L -MLNR = V | . T L - 1 0 . 0 * A L 0 G 1 C ( A 2 / ( A V A L F 3 * A T 0 T 3 + 4 * M * V 3 > ) E X N U 1 3 = A L 0 G 1 0 < A 2 * C A I R / ( 4 . 0 * V I * C M E G A ) ) - T L 1 8 4 N U 1 3 = 1 0 * * E X N U 1 3 C C E V A L U A T E MASS LAW C O U P L I N G C O E F F N U 3 E X T 185 1 8 6 C A3 E X T = A 2 + 4 . 0 * A 8 NU MTL 4= (S I N ( T H L ) * 0 M E G A * M 4 / ( 2 * R H 0 * C A I R ) ) * * 2 1 8 7 1 8 8 DE N T L 4 = A L 0 G ( ( 1 . 0 + ( G M E C - A * M 4 / < 2 * R H 0 * C A 1ft ) } * * 2 ) / + ( 1 . 0 + ( C 0 S ( T H L ) * 0 M E G A * M 4 / ( 2 * R H 0 * C A I R ) ) * * 2 ) ) T L 4 = A L 0 G 1 0 ( N U f T L 4 / 0 E N T L 4 ) 1 8 9 1 9 0 1 9 1 E X 3 E X T = A L O G 1 0 ( A 3 E X T * C A 1 R / ( 4 . D * V 3 * O M E G A ) ) - T L 4 • N U 3 E X T •= 1 0 * * E X 3 E X T D I S 3 = M U 3 E X T + NU3 C C C A L C U L A T E MOCAL D E N S I T I E S C 1 9 2 19 3 19 4 N2 = 0 . 2 7 6 * A 2 / ( H 2 * C L 2 ) N4 = G . 2 7 . 6 * A 4 / ( H 4 * C L 4 ) NS = 0 < , 2 7 6 * A 8 / ( H 8 * C L 8 ) 1 9 5 1 9 6 N l = V 1 * O M F G A * * 2 / ( 2 . 0 * P I * * 2 * C A I R * * 3 ) + A T O T 1 * O M E G A / ( 8 . 0 * P I * C A I R * * 2 " ) ' N3 = V 3 * 0 M E G A * * 2 / ( 2 . 0 * P I * * 2 * C A I R * * 3 ) + A T OT 3 * 0 M E G A / ( 8 . 0 * P I * C AI P * * 2) C c r G R O U P I N G C O U P L I N G A M D L O S S C O E F F S FOR E A C H E L E M E N T 198 19 9 2 0 0 L N'J 2 T = DI S 2 +*) U 2 1 + NU 2 3+ 4 * N i.j 4 2 * N 4 / N 2 + 4 * NU 2 8 NU 3 T = 01 S3 + N U 1 3 * N1 / N3 + N U 2 3 * N2 / N 3 + 4 * N U 8 3 * N 8 / N 3 N U 4 T = D I S 4 + NU4 1+NU42+NU48 NU 8 T =01 S8 -+NU28*N2 / N 8 +NU3 3 + N U 4 8 * N 4 / N 8 c c c F U R T H E R G R O U P I N G S 201 20 2 2 0 3 TI = N U 4 T * N U 8 T - < N 4 / N 8 ) * N U 4 8 * * 2 T2 = N U 2 T * N U 3 T - { N 2 / N 3 ) * N U 2 3 * * 2 . T3 = 4 * < N U 4 2 * N U 4 8 * N 4 / N 2 + N U 2 8 * N U 4 T ) 2 0 4 .20 5 T4 = N U 8 3 * N U 2 3 * N 2 / N 3 + N U 3 T * N U 2 8 * N 2 / N 8 T5 = 4 * N U 4 2 * ( ( N - U 2 3 * N U 8 3 * N 8 / N 3 ) * ( N U 4 8 * N 4 / N 8 + N U 4 T * N U 2 8 * N 2 / - M N B * N U 4 2 ) ) + N U 3 T * { N U 4 2 * N U 8 T * M 4 / N 2 + N U 2 8 * N U 4 8 * N 4 / N 8 ) -20 6 20 7 + 4 * N U 3 3 * N U 4 2 * N U 8 3 * N 4 * N 8 / < N 2 * N 3 ) ) T 6 = N U 2 1 * N U 2 3 * N 2 / N 1 + N U 2 T * N U 1 3 T7 = N U 2 1 * N U 3 3 * N 2 / N 1 - N U 2 8 * N U 1 3 * N 2 / N 8 20 8 T3 = 4 * N U 4 2 * ( ( 4 * N U 8 3 * N U 4 1 * N 4 / N i ) * ( N U 2 8 + N U 2 T * N U 4 8 / ( 4 * N U 4 2 ) ) + + ( N U 2 3 * N U 4 I * N 4 / N 1 ) * ( N U 8 T + N U 4 8 * N U 2 8 * N 2 / ( N 8 * N U 4 2 ) ) - N U 1 3 * ( N U 8 T * + N U 4 2 * N 4 / N 2 + N U 2 8 * N U 4 8 * N 4 / N 8 ) ) 2 0 9 r C J 9 = 4 . 0 * N U 2 T * N U 4 T * ( N U S 3 * * 2 K - N 8 / N 3 E V A L U A T I C N O F N O I S E R E D U C T I O N 2 1 0 211 c R A T I O =. ( T 1 * T 2 - T 3 * T 4 - T 5 - T 9 ) / ( T 1 * T 6 + T 3 * T 7 + T 8 ) NR = 1 0 . 0 * A L O G I O ( R A T I O ) - I G . G * A L 0 G 1 0 ( V 1 / V 3 ) c c c C A L C U L A T E D I R E C T P A N E L T R A N S M I S S I O N W I T H O U T F L A N K I N G 2 1 2 2 1 3 NR AT = ( D I S 3 + N U 8 3 ) * ( I D I S 2 + 2 * N U 2 1 » * N U 3 T - ( N U 2 3 * * 2 ) * N 2 / N 3 ) -+ 4 * ( 0 I S 2 + 2 * N U ? 1 ) * ( N U 8 3 * * 2 ) * N 8 / N 3 OR A T = ( D I S 8 + N U 8 3 ) * ( ( N U 2 1 * * 2 1 * N 2 / N l + ( D I S 2 + 2 * N U 2 1 ) * N U 1 3 J 2 1 4 2 1 5 c R A T = N R A T / D R A T NR WF = 1 0 . 0 * A L 0 G 1 0 ( R A T ) - 1 0 . 0 * A L O G 1 0 ( V 1 / V 3 ) c /-P R I N T R E S U L T S 2 1 6 W R I T E ( 6 , 6 0 0 ) F , K L N R , N R W F , N R , N U J T N < 2 ) - . D I S4 , N U 3 , DI S 3 , N U 4 8 2 1 7 2 1 8 2 1 9 6 0 0 9 9 9 F O R M A T { IH , F 1 0 . 3 » 2 X , 8 ( G 1 2 . 4 , 2 X ) ) GO TO 1 0 0 C O N T I N U E 2 2 0 S T O P 221 END S C A T A 

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