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T‘Iu'b Illllllllll’lIIIHIII!!!”lllllllHUlllIHIUHIIIIHHHIIHI 1293 00908 5790 This is to certify that the thesis entitled NDDELING OF DRIVELINE NOISE USING STATISTICAL ENERGY ANALYSIS presented by NELSON SCOI'I‘ EMERY has been accepted towards fulfillment of the requirements for MASTEBS__degree in MECH- ENG- Date Wl2% [99/ 0—7639 MS U is an Affirmative Action/Equal Opportunity Institution r k Michigan State University LIBRARY PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. #_____________________————— ‘ DATE DUE DATE DUE DATE DUE V M MSU Is An Affirmative Action/Equal Opportunity Institution cmeH.‘ MODELING OF DRIVELINE NOISE USING STATISTICAL ENERGY ANALYSIS By Nelson Scott Emery A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1991 62V— 212’: .55 ABSTRACT MODELING OF DRIVELINE NOISE USING STATISTICAL ENERGY ANALYSIS By Nelson Scott Emery Noise and vibration levels of mechanical systems are historically modeled using differential equation techniques. These modeling techniques have problems describing systems with large numbers of modes. Statistical Energy Analysis is a useful technique for predicting noise and vibration levels for systems with large numbers of modes. Software was developed to model complex multi-modal systems using Statistical Energy Analysis. The utility of the software is demonstrated with the development of an automobile noise model. The computer predictions are compared with experimental results. The noise and vibration levels computed by the software are typically within 10 dB of the experimental results for the modeled components. DEDICATION To my parents, Bill and Rachel Emery iii ACKNOWLEDGEMENTS The work which went into this research was only possible with the help of many people. General Motors corporation has been extremely helpful throughout the project. Special thanks to Joe Wolf, Phil Oh, and Steve Rhode at GM’s Systems Engineering for there technical and financial support. I am also indebted to Sue Carruthers, Bell Elsesser, Dean Marple, and Fred Patterson of GM Proving Grounds for their help in obtaining experimental data. I wish to thank the peOple of NASA’s Jet Propulsion Lab for being extremely helpful in answering all my questions regarding VAPEPS. Of course, none of this would have been possible without the efforts of my my advisor, Dr. Clark Radcliffe. For his guidance, support, and general insight into the world of engineering I am truly grateful. A word of special thanks goes to Matt and Paula Brach for their friendship, support, and encouragement. Finally, I would like to thank all the guys in the lab for their friendship and patience. iv TABLE OF CONTENTS LIST OF TABLES ............................................................................. vi LIST OF FIGURES ........................................................................... viii NOMENCLATURE ........................................................................... x INTRODUCIION ............................................................................. 1 STATISTICAL ENERGY ANALYSIS ..................................................... 2 THE MODELING PROCESS ................................................................ 7 MODELING AUTOMOTIVE DRIVELINE NOISE ...................................... 8 LABORATORY MEASUREMENT OF AUTOMOBILE RESPONSE ................. 17 RESULTS ...................................................................................... 19 CONCLUSIONS .............................................................................. 23 APPENDD( A: VARPS ....................................................................... 25 APPENDIX B: VARPS VERIFICATION AND EXAMPLE MODEL ................. 65 LIST OF REFERENCES ..................................................................... 73 LIST OF TABLES Table 1 Automobile Model Statistical Energy Analysis Element Classification ......... 10 Table 2 Automobile Model Statistical Energy Analysis Connector Classification ...... 11 Table A-1 VARPS Supported Elements ........................................................ 25 Table A-2 VARPS Supported Connectors ...................................................... 25 Table A-3 VARPS Model Inputs ................................................................ 25 Table A—4 VARPS Model Database Format .................................................... 27 Table A-5 VARPS Database Node Summary .................................................. 29 Table A-6 AVOL and AVOL2 Element Parameters ............................................ 30 Table A-7 FPLATE Element Parameters ........................................................ 32 Table A-8 BEAM Element Parameters .......................................................... 34 Table A-9 FPL_AVOL and FPL_AVOL2 Connector Parameters ........................... 37 Table A— 10 AVOL_FPL and AVOL_FPL Connector Parameters ............................. 40 Table A-ll FPL_FPL Connector Parameters .................................................... 41 Table A-12 Matrices for evaluating FPL_FPL loss factors .................................... 44 vi Table A-13 FPL_BEAM Connector Parameters ................................................. 47 Table A- 14 FPL_PLB Connector Parameters ................................................... 55 Table A-15 Frequency Node Parameters ......................................................... 62 Table A- 16 Power Node Parameters .............................................................. 63 Table A-17 Energy Node Parameters ............................................................. 64 Table B-1 Acoustic Volume Parameters for Volume-Plate-Volume Model ................. 67 Table B—2 Plate Parameters for Volume-Plate-Volume Model ............................... 67 Table B-3 The Model Database .................................................................. 68 Table B-4 Volume-Plate-Volume Example Run ............................................... 69 vii LIST OF FIGURES Figure 1 Acoustic Modal Density of a Typical Automobile Interior versus Frequency.. 2 Figure 2 Simplified Conceptual SEA Model of Vibration and Sound Transmission from a Coarse Road to Vehicle Interior ........................................................ 4 Figure 3 Schematic of the Statistical Energy Analysis Model showing general element layout ...................................................................................... 9 Figure 4 Automobile Statistical Energy Analysis Model showing energy storage and power connection between them ........................................................ 9 Figure 5 RMS Engine Accelerations versus Frequency - Used as element model energy input ........................................................................................ 19 Figure 6 Automobile Driveline Model Interior Response - Experimental and Predicted Values — Run 1 ........................................................................... 20 Figure 7 Automobile Driveline Model Over Car Volume Response - Run 1 ............. 21 Figure 8 Automobile Driveline Model Over Car Volume Response - Run 9 ........... 22 Figure 9 Automobile Driveline Model Interior Response - Run 9 ......................... 22 Figure 10 Automobile Driveline Model Front of Dash Response -Run 9 .................. 23 Figure A-l FPL_FPL Connector Diagram ....................................................... 41 Figure A-2 FPL_BEAM Connector Diagram .................................................... 48 viii Figure A-3 FPL_PLB Connector Diagram ...................................................... 55 Figure B—l Volume-Plate-Volume Example System Conceptual Model .......................................................... 6 6 Figure B-2 Statistical Energy Analysis Model for Volume-Plate-Volume Model ........... 66 Figure B-3 Plate - Predicted RMS Acceleration (VARPS and VAPEPS) .................... 72 Figure B-4 VolA and VolB - Predicted Sound Pressure Levels (VARPS and VAPEPS).. 72 ix NOMENCLATURE = mean-square pressure, expected value (NZ/m4) = mean-square velocity, expected value (m2/secz) = loss and coupling loss factor matrix (dimensionless) = FPL_FPL coupling loss factor intermediate calculation variable = plate surface area (m2) = surface area of door (m2) = surface area of batch (m2) = plate surface area (m2) = plate i surface area (m2) = surface area of roof (m2) = body panels combined surface area (m2) = surface area of windows (m2) = beam cross sectional area (m2) = FPL_FPL coupling loss factor intermediate calculation variable .-. FPL_FPL coupling loss factor intermediate calculation variable = FPL_FPL coupling loss factor intermediate calculation variable = speed of sound (m/sec) = FPL_BEAM coupling loss factor intermediate calculation variable .-. FPL_BEAM coupling loss factor intermediate calculation variable = speed of sound in air (m/sec) = beam flexural wave speed (m/sec) Cfl Cl Cl Clb C’ff elf: Cli czp Ct Ctb = element i flexural wave speed (tn/sec) = longitudinal wave speed (m/sec) = plate longitudinal wave speed (tn/sec) = beam longitudinal wave speed (ID/sec) = flexural coupling loss factor = torsional coupling loss factor = longitudinal wave speed of elementi (tn/sec) = plate longitudinal wave speed (tn/sec) = torsional wave speed (tn/sec) = beam torsional wave speed (m/sec) = FPL_FPL coupling loss factor intermediate calculation variable = determinant of matrix i .-. inner diameter beam with circular cross section (m) -.— outer diameter beam with circular cross section (m) = FPL_FPL coupling loss factor intermediate calculation variable = beam Young’s modulus (GPa) = subsystem total energy (joules-sec/rad) = subsystem total energy (ioules-sec/rad) = plate Young’s modulus (GPa) = FPL_FPL coupling loss factor intermediate calculation variable = center frequency (Hz) = critical frequency (Hz) = FPL_BEAM coupling loss factor calculation constant .—. shear modulus (GPa) = angle between plate and beam in FPL_BEAM = beam shear modulus (GPa) = plate or beam thickness (m) xi hequiv hhatch hroof hwind "door "lb "fp "hatch nj (0)) = plate thickness of door (m) = body panels total plate thickness (m) = plate thickness of hatch (m) = plate thickness (m) = plate thickness of roof (m) = plate thickness of windows (m) = radius of gyration (m) = beam radius of gyration (m) = plate radius of gyration (m) = beam torsional radius of gyration (m) = beam length (m) = FPL_FPL joint length (m) = plate length (m) = structure mass (kg) = beam mass (kg) = plate structural mass (kg) = structural mass (kg) = Mode count per frequency band (modes) = modal density (modes-sec/rad) = modal density (modes-sec/rad) = acoustic element modal density (modes-sec/rad) = modal density of an acoustic element (modes-sec/rad) = modal density of door (modes-sec/rad) = beam flexural modal density (modes-sec/rad) .-_- plate flexural modal density (modes-SCC/rad) = modal density of hatch (modes-sec/rad) = modal density of subsystem “j” (modes-sec/rad) xii "roof "5 nrb "tot r bi R equiv r hw Rrad 100 tb va xb = modal density of roof (modes-sec/rad) = modal density of a structural element (modes-sec/rad) = structural element modal density = beam torsional modal density = body panels total modal density (modes-sec/rad) = modal density of windows (modes-sec/rad) = plate perimeter edge length (m) = number of connection points in FPL_PLB connector = power dissipated by subsystem “i” (Watts) = power input to subsystem “i” (Watts) 6‘"? = power transferred from subsystem j to “k” (Watts) -.-. plate perimeter (m) = Quality factor = wave number ratio (dimensionless) = bending rigidity of element i = equivalent radius (m) = ratio of beam thickness to beam width (dimensionless) = plate radiation resistance (kg-rad/sec) = FPL_FPL coupling loss factor intermediate calculation variable .—_ FPL_FPL coupling loss factor intermediate calculation variable = FPL_FPL coupling loss factor intermediate calculation variable = Reverberation time (sec) = acoustic volume (m3) = plate width (m) = FPL_BEAM coupling loss factor intermediate calculation variable = FPL_BEAM coupling loss factor intermediate calculation variable = FPL_BEAM coupling loss factor intermediate calculation variable xiii xc = FPL_BEAM coupling loss factor intermediate calculation variable 21, = beam impedance zbf = beam flexural impedance zbm = beam moment impedance 2b: = beam torsional impedance zp = plate impedance zpf = plate flexural impedance zpm = plate moment impedance Greek Letters (1 = (f/fc)1/2 (dimensionless) [3 = plate edge condition (dimensionless) Af = frequency band centered at f (Hz) 7 = angle between plate and beam in FPL_BEAM n = loss factor nag = acoustic to structural element coupling loss factor hi = thickness of element i map) = subsystem i loss factor (dimensionless) Tlij = i to j coupling loss factor npb = plate to beam coupling loss factor nsa = structure to acoustic element coupling loss factor Kb = beam wave number (m'l) K; = element i wave number (m‘l) Kp = plate wave number (ml) 2.3 = wavelength of sound in acoustic volume (m) 2c = wavelength of sound at critical frequency (m) u = angle between plate and beam in FPL_BEAM (degrees) 9 = angle between two coupled plates (degrees) xiv Pa Pb Pt Plb Pp c7rad = density (kg/m3) = density of air (kg/m3) = beam mass density (kg/m3) .-_ element i mass density (kg/m3) = beam lineal density (kg/m3) = plate mass density (kg/m3) = plate radiation efficiency = band center frequency (rad/sec) = damping ratio (dimensionless) XV W The customer’s perception of automobile quality is directly linked to vehicle interior noise amplitude and quality. Noise amplitudes are quantified by sound pressure levels. Noise quality is related to the sound spectrum. Sound level measurements such as A- weighted sound pressure levels attempt to adjust these levels to be more representative of human hearing perception. In 1990, the Toyota Motor Company introduced the Lexus LS400 with an advertising campaign focusing on the automobile’s interior sound quality. Although other vehicles had similar low sound pressure levels, the Lexus was chosen quietest (Consumer Reports, June 1990) based on hearing perception. Automobile noise is typically modeled using ordinary differential equation models based on structural analysis. Finite element analysis, a differential-equation-based modeling technique, models a system's natural frequencies and mode shapes. The number of modes present in complex systems becomes large at high frequencies. For example, the modal density of an acoustic volume increases at an exponential rate as frequency increases (Figure 1). When analyzing systems with large numbers of modes, differential equation methods require many calculations, are sensitive to small variations in model parameters, and generate large amounts of data. Due to the limits of differential equation methods, vehicle noise problems are difficult to analyze with limited information available from early design specifications and are typically addressed in the latter stages of the design process when production prototypes are available for testing. Statistical Energy Analysis is a method of modeling steady state energy storage and power flow in systems with large numbers of modes per frequency bandwidth. For complex systems, predictions of sound pressure levels as a function of frequency can be made early in the design process because Statistical Energy Analysis is not sensitive to design details. Since Statistical Energy Analysis produces results in the frequency domain, both noise amplitude and quality are predicted. Modal Density of Typical Automobile Interior 30 g 25— 0 '8 20* E 2.? 15— i a 10- 2 5_ 0 . 1 . . . a 1 r 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Frequency (Hz) Figure 1: Acoustic Modal Density of a Typical Automobile Interior versus Frequency Software was developed to evaluate structural systems using Statistical Energy Analysis (See Appendices). As an example of the software’s capabilities, a model is built to predict the noise level spectrum of a typical automobile interior. Vehicle noise is generally broken into three components: road, wind, and driveline. This study focuses on determining the requirements for Statistical Energy Analysis (SEA) modeling of driveline noise for a typical automobile. A model of a typical automobile which emphasizes driveline noise is developed and evaluated. In addition, an experiment to determine model parameters and to verify the modeling process was conducted and the results are reported. W Statistical Energy Analysis is a method for modeling energy storage and power flow in vibrational and acoustical systems. Elements store energy. Connectors transfer energy between elements. External sources such as turbulent flow, acoustic excitation, and mechanical vibration provide energy to the system (Lyon 1975, 10; Beranek 1971, 297). The mean squared velocities or accelerations of structural elements and the mean squared pressure levels of acoustic elements can be calculated from element energy levels. Statistical Energy Analysis was developed as a ‘simple’ approach to modeling complex multi-modal systems (Ungar 1967, 626). SEA is robust in that it is not sensitive to design details. This is an advantage over other methods because the engineer is often required to make response predictions at a point in the design process when little detail is available. SEA provides a framework for modeling based on fundamental parameters such as average panel thickness and damping (Lyon 1975, 4). There are two general categories in an SEA model: energy storage and energy transfer (Lyon 1975, 12). Energy is stored in model elements. Energy transfer includes energy lost to element internal damping and energy transferred between elements. Details of SEA theory are best illustrated with the simple vehicle acoustic energy transmission model shown below in Figure 2. This figure shows a conceptual model of coarse road noise transmitted into the vehicle interior. The conceptual model shown will be used to predict acoustic response of the vehicle’s interior. Boxes 1-3 represent three vehicle subsystem models: the tires, the suspension/body structure and the interior acoustic volume. The variables, P, represent power flows into, through, and dissipated by each subsystem. The power flow of each vehicle subsystem must satisfy a steady-state flow balance. Pin1=de31+P12 +1013 (1a) P912: dissz'P12+P23 (1b) Pin3=Pdiss3'P23'P13 (1°) where: Phi = power input to subsystem “i” ‘6", l Pdiss‘, = power dissipated by subsystem P ,- = power transferred from subsystem “j” to “k” In this example, P12 represents the structural power flow from the tire structure to the suspension and body structure while [’3 represents the airborne acoustic power directly to the vehicle interior. The steady-state power dissipated in each subsystem, P453”, in a 1 rad/sec band centered at frequency to can be written in terms of that subsystem’s energy in that band, band frequency, and internal loss factor. Pain.- = 031115.- I (2) where: a) = band center frequency (rad/sec) ni(to) = subsystem loss factor (dimensionless) E;((u) = subsystem total energy (joules-sec/rad) P in1 sz P1113 1 P12 2 P23 3 . Body - 1 - Tire Suspension Interior P13 Pdissl Pdissz Pdiss3 Figure 2: Simplified Conceptual SEA Model of Vibration and Sound Transmission from a Coarse Road to Vehicle Interior The ability to compute the steady-state energy in a l rad/sec band centered at frequency, to, (Crocker and Price 1969, 472) is a central feature of SEA theory developed by Lyon, Scharton and Newland in 1962-1968 (Lyon 1975, 7-10). The theory states that the steady-state power transfer in such a nan'ow frequency band is such that the energies in each mode of both systems are equalized in that frequency band. E j E, ij =wnjtnj _"— (3) n j n, where: P ,- = power transferred from subsystem “j” to “k” (u = band center frequency (rad/sec) njk(co) = subsystem coupling loss factor (dimensionless) E ,- (co) = subsystem total energy (joules-sec/rad) nj (0)) = modal density of subsystem “j” (modes-sec/rad) At high frequencies where the modal density is high in any system’s spectrum, the above result is important because it will be used to compute the subsystem RMS pressure or velocity implied by the system’s total energy. At these frequencies of high modal density, there is no comparable way of predicting subsystem response from the consideration of individual modes because of the model and numerical accuracies required in modal analysis. The result in equation (3) is therefore uniquely valuable at high frequencies but has been shown to apply to low frequencies as well. Even the assumption of relatively weak coupling between subsystems in the original derivation has been shown to be unnecessary (Lyon 1975, 8) and the result is valid for strong linear and/or non-linear coupling (Newland 1965, 199). Modal densities can be accurately estimated from geometrical parameters and/or measured by counting resonances in laboratory verification studies. In fact, obtaining accuracy in modal density estimates at any frequency is typically much easier than obtaining the same degree of accuracy in the computed properties of individual modes such as natural frequencies in that frequency range. The system of linear equations giving the energies, Ell-(w), in terms of the input powers, Pin}, can be found by substituting equations (2) and (3) into equation (la-c) and rearranging. The result is: E, 1 P,“ [N] E, = ('6 13,2 (4) E3 Pu, where: l' n1 ( n1 \\ '1 (T11 + 7112 + 7113) '111 —) “1113(— "1 l "211 [N] = (-n12) n2+n1 - +1123 -n23 — "2 K ’13)) n n ("7113) (‘le3) [713 + 1113(4) 4’ 7123(1)] _ n3 3 The above equations can be solved for steady-state subsystem energy as a function of frequency given values for the loss factors, 1]. The mean-square velocities for vibrating subsystems and mean-square pressures for acoustic subsystems can be directly related to steady-state subsystem energies, E ,- . (v2) = (iyi‘j for structures (5a) m 2 (p2) = [Lg—)5). for acoustic systems (5b) where: = mean-square velocity, expected value (m2/sec2) = mean-square pressure, expected value (NZ/m4) Ej = E ,- (to) = subsystem total energy m = structure mass (kg) p = fluid density (kg/m3) c = speed of sound (m/sec) V = acoustic volume (m3) The loss factor values required to compute the above steady-state mean-square response variables can be either estimated using conventional modeling methods and/or measured once prototype subsystems are available. The subsystem loss factor, 11,-, is simply the reciprocal of the electrical engineer’s quality factor, Q, or twice the mechanical engineer’s damping ratio, 2;. Typical values for the loss factor, 11,-, are between 0.001 and 0.1. The coupling loss factor, 1],}, is also related to other familiar modeling parameters. The coupling loss factor for acoustic power flow between two rooms is the transmission loss of the wall familiar to acoustic engineers while the coupling loss factor between a plate and an acoustic volume can be directly related to the plate’s radiation efficiency, 0,04. It can be seen that the estimation of SEA parameters is no more difficult than the estimation of more conventional modeling parameters such as structural mass, stiffness and damping. WW Creating an SEA model requires model subsystem identification, subsystem type classification, and subsystem parameter quantification. Model subsystems are elements and connectors. Elements are classified as acoustic volumes, flat plates, beams, etc. Connector classifications are: acoustic volume to flat plate, flat plate to beam, flat plate to flat plate, etc. Subsystem parameters include volumes, surface areas, plate thicknesses, damping ratios, and reverberation times. The model is evaluated over a specified frequency range. The frequency range is typically in 1/3 octave bandwidths. Power inputs are specified explicitly as functions of frequency or implicitly by specifying the average response of one or more elements. W Modeling the response of an automobile interior to driveline vibration illustrates the application of the modeling methodology and software. An SEA model consists of energy storage elements and the power flow paths between them. The energy source for the driveline model is the engine. The acoustic response of the interior is the desired model output. The interior and engine are the first model elements identified. The model building methodology is to determine how energy is transferred from the engine to the interior. Energy storing subsystems, elements, and the paths between them, connectors, are identified by tracing the power flow paths from the engine to the interior. Energy is transmitted from the engine to the interior through both structureborne and airborne paths. One example of a structureborne path is from the engine to frame structure to floor pan to interior. An example of an airborne path is from the engine to under car volume to floor pan to interior. Identifying these two paths creates three model elements: frame structure, floor pan, and under car volume (Figure 3). In addition, five model connectors are created: engine to frame structure, frame structure to floor pan, floor pan to interior, engine to under car volume, and under car volume to floor pan. The remaining model elements and connectors are identified in the same manner (Figure 4). The body panels element required in the model combines the effects of the windshield, hatch glass, door glass, and roof into a single element. Modeling these components individually would have created 4 to 5 elements and 5 or more connectors. Since all of these elements can be modeled as flat plates, the model size is reduced by combining their effects. 14““ 90".. o i. (10) INTERIOR Schematic of the Statistical Energy Analysis Model showing general element . ee . I.m cum . v M ou U m . Vmw m m .Ak U“. “V P m “We Vim %C E l D( USP Or. 1 O o..y em .150 V "W e¢.B 9U D m.“ wee. O 9. 009 O a '90. av afix: H .. as. Xxx ) m0 RnnL Man t WAO 0N w ocv an m. oo .99.... AKQQQQWQ & coco Irrkw e m .$ F Automobile Statistical Energy Analysis Model showing energy storage and power connection between them Ifigne4 10 SEA element and connector classification of the 10 elements and 13 connectors is the next phase of the modeling process. For this example, only a few element types were to be developed: acoustic volumes, flat plates, and beams. The associated connector types developed are: acoustic volume to flat plate, flat plate to acoustic volume, flat plate to flat plate, flat plate to beam, and flat plate to parallel beam. Tables 1 and 2 show the model element and connector classifications respectively. Table 1: Automobile Model Statistical Energy Analysis Element Classification Automobile Model Element Element IIlype ‘ engine compartment acoustic volume under car volume over car volume interior engine flat plate hood front of dash body panels floor pan frame structure beam Element parameters vary depending on the element type. For acoustic volumes, the volume, surface area, edge length, and reverberation time are required. The plate parameters are thickness, length, width, perimeter edge length, mass, longitudinal wave speed, Young’s modulus, and damping ratio. The beam parameters are length, width, thickness, mass, longitudinal wave speed, Young’s modulus, shear modulus, and damping ratio. For beams with circular cross sections, width and thickness are replaced by inner and outer diameters. The damping ratios and reverberation times are used in loss factor calculations. Element masses are used in calculating element energy levels and RMS velocities or accelerations. 1 1 Table 2: Automobile Model Statistical Energy Analysis Connector Classification Automobile Model Connectors {Tonnector T engine to engine compartment flat plate engine to under car volume to hood to over car volume acoustic volume body panels to interior front of dash to interior floor pan to interior engine compartment to hood acoustic volume engine compartment to front of dash to under car volume to floor pan flat plate over car volume to body panels hood to front of dash flat plate to lat plate engine to frame flat plate to beam floor pan to fame flat plate to parallel beam Connector parameters are required to compute coupling loss factors. Acoustic volume to flat plate and flat plate to acoustic volume connectors require the plate edge condition, B. B is defined as (Lyon 1975, 300): 1 simply supported B = 2 clamped - clamped «l2 realistic cases (6) For flat plate to flat plate connections, the angle between the two plates and the length of the connection joint must be known. For flat plate to beam, two angles which describe the position of the beam relative to the plate are required. Flat plate to parallel beam is modeled as a ‘point connection,’ with the number of connection points specified. The frame structure properties are determined from an equivalent rectangular beam structure which is 12 converted to an equivalent pair of circular beams for final model evaluation because only flat plate to circular beam connection models were available. The parameters of the system are used to evaluate the three SEA values: modal densities, coupling loss factors, and loss factors. While modal densities and coupling loss factors are related to system and element geometry, loss factors must be derived from experimentally determined values such as acoustic absorption coefficient, reverberation time, and damping ratio. Parameter accuracy depends on the model, the modeler’s knowledge of the system being modeled, and the modeler’s experience developing SEA models. The body panels element parameter evaluation is more involved than the evaluation of the other elements. The body panels element includes the windshield, roof, rear window, and door glasses, and is to be modeled as a single flat plate element. The individual components of the body panels element are made of glass and steel which have the same longitudinal wave speed (Lyon 1975, 282), but do not have the same Young’s modulus. In addition, the element’s components do not have equivalent plate thicknesses. Parameter evaluation is facilitated by examining the related model equations. The equation for calculating the modal density of a flat plate is (Lyon 1975, 282): ,, = 3.5.4 (7) hCl where: A = plate surface area (m2) h = plate thickness (m) c: = plate longitudinal wave speed (tn/sec) The only paths to and from the body panels connect to acoustic volumes. The equation for the plate to volume coupling loss factor which characterizes energy transfer between elements is (Lyon 1975, 300): 13 “u = Rind/(om: (8) where: Rmd = plate radiation resistance (kg-rad/sec) ms = plate structural mass (kg) 0) = band center frequency (rad/sec) The volume to plate coupling loss factors are found from the relationship (Lyon 1975, 300): n... = mans/n.) (9) where n; and na are the structural and acoustic volume modal densities respectively. Acoustic waves in a fluid travel at the same speed regardless of frequency (Crocker and Kessler 1982, 63-66). In contrast, high frequency bending waves in a structure like a panel travel at faster speeds than low frequency waves. The transfer of energy between a volume and a panel is maximized at the frequency where acoustic waves in the volume and bending waves in the panel have the same propagation velocity. This frequency is the critical frequency of the panel (Beranek 1971, 275), c2 = __ 10 1.8138h c, ( ) f. where: c = speed of sound (m/sec) h = plate thickness (m) C] = plate longitudinal wave speed (m/sec) The radiation resistance, Rmd, is evaluated as follows (Maidanak 1962, 817-818): Rm, = Apm where: and with 14 '[g.(f/f.)+(PA./A)g.(f/f.)]fi. f .[1- f./f]’i. f > f., 2.6 = wavelength of sound at critical frequency (m) 2.3 = wavelength of sound in acoustic volume (m) = density of air (kg/m3) = edge condition [See equation (6)] = plate surface area (m2) 'etbu'o = plate perimeter edge length (m) c = speed of sound in air (tn/sec) N5 = center frequency (Hz) 1 = plate length (m) w = plate width (m) 2 gl(f/fc)= 1 Q f>§fl 1— 21D 1 l— 2 82(f/fc) = {( 0% ) l( 2-I- (JO/(2 3 a)]+ or} 43(1—(1): (4/14)(1 — 2.12) / a(1 - «2);. f < -1-f. (11) (12) (13) 15 “(mi 04) The surface area appears in calculations of the modal density (7) and the radiation resistance (11). Equation (11), the radiation resistance equation, affects the calculated flow of energy between the plate and surrounding acoustic volumes. It is desired to model the combined effects of the individual components. Therefore the component areas are summed to generate a total plate surface area, AT. The plate mass is used in calculating the loss factor (8) and in calculating the plate’s RMS velocity (5a). To model combined plate effects, the individual component masses are summed. The plate thickness is used in evaluating the modal density (7) and in evaluating the critical frequency (10). The definition of modal density is n = N/Af, where N is the total number of modes in the frequency bandwidth Af. It is observed that the total number of modes in the body panels element is equal to the sum of the modes of the individual elements (Lyon 1975, 288). Therefore: n”, = nw+ nmf+ th+nm, (15) where nm, "winds nmof, "hatch: ndoo, are the total,window, roof, hatch glass, and door glass modal densities respectively. The relationship for composite modal density is derived here by substituting equation (7) into equation (15) and noting that c: is the same for steel and glass, = 3’41 =£[E+E+A~m‘+ifl] (16) hm hm! where Awind, Aroof. Ahatcha Adam. hwind. hmof. hhatch» and hdoor are the windshield. roof. hatch, and door, surface areas and thicknesses respectively. hequiv is the equivalent l6 thickness for the composite element which will yield the element’s total modal density. Solving for hequiv yields: equiv = AT (17) AMT-d + Aroor + Ahatch + Adoor hm'nd hm] hlnlch hdoor The plate perimeter , length , and width are used in calculating the radiation resistance (11). The perimeter edge length models the edge effects of the plate. Since each component of the body panels element is considered to act independently, the perimeter is taken as the sum of the individual component edge lengths. The length and width are defined as the square root of AT. The Young’s modulus does not enter into any of the calculations for this model. However, should Young’s modulus be required it can be calculated as an area weighted average. Acoustic volume parameters for the over car and under car volumes are evaluated from vehicle geometry. Since the over car and under car volumes do not have definite boundaries, they could be considered infinite in volume and surface area. To practically model the system these values must be finite; thus volume boundaries are defined. The under car volume boundaries are defined as the road (ground), the car underside, and the imaginary edge created by extending the car’s lower outside edge to the ground. The car top, the car top’s mirror image 0.6m above the car, and the edges created, form the over car volume. The length 0.6m is chosen because the experimental measurements of sound pressure level were taken 0.3m above the car. The volumes, surface areas, and edge lengths can be estimated from the defined volume boundaries. Reverberation times and damping ratios are determined from experimental measurements or estimates based on engineering experience since theoretical models do not l7 exist. Since this model is of a production vehicle, some of these values could be measured. It has been suggested that these parameters be determined by removing elements from systems, providing excitation to these elements, removing excitation, and measuring the element response decay rate. For a complex system, this can be difficult. In the driveline model, removal of the body panels element is virtually impossible. It is also difficult to separate acoustical elements from systems. In general, the effects of neighboring elements cannot be eliminated from reverberation time measurements. For the driveline model, only the reverberation time of the interior was measured. The remaining reverberation times and damping ratios were estimated based on experience. Since these values were not accurately determined, they were adjusted to match the model results more closely with experimental results. Two purposes are served by making such adjustments. First, it provides a way to evaluate the validity of the model itself. For example, when adjusting the frame structure damping ratio, it was found that a large damping ratio was required to produce reasonable results. This fact led to the discovery that the frame structure had been incorrectly modeled. The second benefit is that potential values for future models are determined. a101,; 01 u a 10.; O a 0 O: _l_’lu Data were collected from a typical automobile operating in first gear at 3000 RPM with a 1001b tractive force on a dynamometer. These conditions were chosen to emphasize the noise created by the driveline. The majority of the car’s interior was removed to facilitate acquisition of data. The car’s dash board, steering wheel, head liner, rear seat lateral cushion, and hatch carpeting were not removed. Acceleration data were measured using PCB accelerometers model 303A, with a Kistler Pieztron Coupler 5122. A-weighted sound pressure levels were taken using B&K 4144 omnidirectional microphones with a B&K WB1057 microphone amplifier. The data were recorded on a TEAC RD-l 1 1T PCM Data Recorder with a TEAC TZ-314FA Antialiasing 18 Filter Unit. For each set of data, a thirty second sample was taken and then averaged to give an RMS acceleration or pressme value for each measurement location. The desired result was to characterize the steady state energy content of the ten model elements. For each structural element (i.e. floor pan, hood, body panels, etc.) several acceleration measurements where taken. For example, eight accelerometer readings where taken on the hood, front of dash (toe pan), roof, door glass, windshield, and hatch glass. For the floor pan, a larger number of measurements were taken to get a more accurate sample of the element’s response. The floor pan was broken into five sections with six to eight measurements taken for each section. For the sound pressure measurements, two to four measurements were taken for each volume. The interior and engine compartment were each measured in two locations. Four measurements were taken for the under car and over car volumes. The raw vibration and pressure data were analyzed using a B&K 2133 spectrum analyzer and reduced to power spectrums in 1/3 octave bandwidths. The acceleration data were calibrated in dB re 20 microvolts, with 1 volt equal to 1 g. Sound pressure levels were calibrated in standard A-weighted dB format. For the purpose of SEA, it is desired to have a characteristic energy spectrum for each element. With the collected data, several sets of data for each element were available. To reduce this data to a single spectrum for each element, it is necessary to average the data. First, the acceleration data were reduced to characteristic acceleration plots for each structural element. The data were first converted to voltages (or g’s since 1 g equals 1 volt). Next, the root mean square of the data for a given element at each frequency was calculated to give a characteristic value for each structural element. The RMS acceleration values are used as inputs and expected outputs for the driveline model. A similar procedure was followed for reducing the sound pressure data. The data were first converted to pressures, roOt mean squared values calculated, and then converted to dB. l9 ENGINE Acceleration, g’s (dB) 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 Frequency (Hz) Figure 5: RMS Engine Accelerations versus Frequency — Used as element model energy input RESULTS The model was evaluated from 500 to 5000 Hz at 1/3 octave bandwidth intervals using the SEA modeling software developed . For the driveline model, energy enters the system via the engine. The engine input is modeled by specifying the engine’s energy response spectrum which is available from experimental data (Figure 5). The results of the model analysis are modal densities, loss factors, coupling loss factors, and steady state energy levels. The fust analysis of the model used estimates of damping and reverberation times based on engineering experience. For the interior, the first run modeled values are within 7 dB of the RMS experimental values (Figure 6). Considering the lack of model refinement, the modeled interior response is quite satisfying. However, the results of the entire model must be analyzed before the model can be judged correct. The interior models a pressure I- Mean of(2) -model I I"\ I \ _— . I ' "’ T ~~~L-'-’ ------- -“ Sound Presmre Level (dBA) l—,_,—‘| o a. w 5 5 5 5 5 5 5 5 5 5 5 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 Frequency (Hz) Figure 6: Automobile Driveline Model Interior Response - Experimental and Predicted Values - Run 1 response that is very close to the measured values, while the results of the over car volume model are very different from those measured (Figure 7). Due to the severe discrepancy in the model and experimental results of some elements, an attempt to improve the model results was made. The parameters of damping ratio, reverberation time, and the size of the over car volume were altered in an attempt to improve the results. Even with a reverberation time 1.0x106 seconds a difference of 8 to 25 dB between modeled values and experimental values existed. This led to the discovery that the hood to over car volume connector had not been included in the model database. Run nine included the hood to over car volume connector in the model database and good results were obtained with an over car volume reverberation time of 50 seconds. This reverberation time is rather large, but it should be noted that the theoretical model used for modeling the over car volume is not reperesentative of the actual over car volume. The over car volume will not have the modal 21 interaction that a room or automobile interior might have. For run nine, the interior response has decreased slightly, but is within 10 dBA of the RMS experimental values (Figure 9). The remaining model element results were typically within the range of the experimental values (Figure 10). I r Meanof(4) — model I I Sound Pressure Level (dBA) .. 110.3 I ~— “~’ -” 500' 630' 800' 1000' 1250' 1600' 2000' 2500' 3150' 4000' 5006 Frequency (Hz) Figure 7: Automobile Driveline Model Over Car Volume Response - Run 1 22 I :- Meanof(4) - model I J I 12"’F“‘t“"l“‘r"""“'T”’T“‘r--+--—} l v l t 110 dB Sound Pressure Level (dBA) l ‘ 500' 630' 800' 1000' 1250' 1600' 2000' 2500' 3150' 4000' 50001 Frequency (Hz) Figure 8: Automobile Driveline Model Over Car Volume Response - Run 9 l’ Meanof(2) -rnodel .. l ”/1~\‘- r _,._.——-1_ ‘ P 1_ -- [I ~...-.—-— " —--F\“" 2 T ‘~“—-—~- *3; '2‘ Ch .3 1 .. IlOdB ‘2 uh 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 Frequmcy (Hz) Figure 9: Automobile Driveline Model Interior Response - Run 9 23 P Mean of (8) " model " 10 dB Acceleration. g's (dB) I \ I \ : i i i 1 1 1 i 1 i i 500 630 800 1000 1250 1600 2000 2500 3150 4000 5000 Frequency (Hz) Figure 10: Automobile Driveline Model Front of Dash Response - Run 9 W The ability to model automobile drive line noise has been demonstrated. Although a rough model was used, relatively accurate values were predicted for most elements. The modeled values were generally in the range of the experimental values. The reverberation times and damping ratios which were not determined experimentally have reasonable values based on engineering experience, with the exception of the over car volume. There are many possible explanations for the large over car volume reverberation time. It is possible that a path to the volume was not identified. Perhaps the contribution of unmodeled exhaust noise is large, or the theoretical model used for the volume does not accurately model the volume. Further investigation into this problem and its solutions is recommended. Refinement of the model involves breaking the system into more elements, identifying new flow paths, and developing new theoretical elements to model automotive 24 components. Elements such as the body panels and floor pan can be divided into and modeled as multiple elements. Creating more elements will create more flow paths. Elements such as engine do not necessarily fit into the categories of beam, volume, or flat plate; thus new element models need to be developed. SEA provides a method for modeling the high frequency response of automotive systems. The model developed uses rough estimates of vehicle parameters similar to values that would be available in the early stages of vehicle design. The modeled vehicle response results are typically within the range of the experimental values and predict the general trends of most elements. This model incorporated a very limited range of model elements and connectors. With the development of new element and connector models, refinement of the model should produce vastly improved results. The high frequency analysis capabilities of SBA and the low fiequency range of finite element methods could be combined to provide an acoustic model which represents the full frequency range of an interior’s acoustic response. 25 W INTRODUCTION VARPS, Vibro Acoustic Response Prediction System, evaluates user defined system models using Statistical Energy Analysis. A model database file in ASCII format is created from VARPS element models, connector models, and model input nodes using a text editor such as EDT. VARPS supported elements and connectors are shown in Tables A-1 and A-2. The model input nodes are shown in Table A-3. VARPS outputs modal densities, loss factors, coupling loss factors, and RMS responses in tabular form. Table A-1: VARPS Supported Elements em en 300115th Table A-2: VARPS Supported Connectors II I' aCOUSth [0 ID 80008110 ID t0 t0 Table A-3: VARPS Model Inputs 26 GENERAL MODEL DATA BASE ENTRY Each VARPS model database entry is called a node. A VARPS model node has the four general fields: type, name, location, and parameters. The general form of a data base node is: type nwne location parameters type must be one of the types shown in Tables A-l, A-2, and A-3. name is a user specified node name used in displaying the final results. location describes the elements topological position in the model. For each element, location is a single, unique integer. For a connector, location is two integer numbers separated by a comma. The first number indicates the element from which energy flows and the second indicates the element to which energy flows. For example, the may 1,2 indicates a connector from element 1 to element 2. It is important that the user be sure that the from and to elements exist and are of the type specified by the connector type. The software does not currently check connectors for correct element reference, thus incorrectly designating the location will cause the program to give incorrect results. The number and type of parameters vary depending on the element. CREATING A MODEL Creating a VARPS model consists of dividing the system being modeled into elements and connectors, determining model power inputs, classifying the elements and connectors as VARPS supported types, and determining the required model parameters. (See individual element and connector write ups below.) After determining the required model data, the model data base is built using any text editor. File names must have a “dat” file extension. The order of data base entries is elements, connectors, frequency vector, and power/energy inputs (Table A-4). Elements must be in numerical order with element 1 appearing before element 2, element 2 appearing before element 3, etc. Connectors must be listed after the elements they connect are listed. The model must include one frequency 27 vector and one or more energy/power inputs. (See Model Inputs for details on frequency, power, and energy inputs.) Table A-4: VARPS Model Database Format Elements Connectors Frequency Power or Energy Inputs Data Base Rules 1. One or more spaces separate database fields. Tabs must not be used anywhere in the database. A single comma separates field values. . For node entries exceeding 80 characters or for node entries being continued on the next line, an ‘-’ is placed at the end of the line being continued. In the database, Elements must be listed in order with respect to spatial location. For example, element 1 must appear before element 2. Element 1 appearing after element 2 will cause errors in model evaluation. . Elements must be listed before they are used in any connectors. File names must have “dat” as the file extension. i.e. ‘model.dat’. 28 EVALUATING THE MODEL Type “@VARPS” at the system prompt to begin model evaluation. VARPS will prompt for a file name. Enter the name of the data base file to be evaluated. VARPS will generate modal densities, loss factors, coupling loss factors, and RMS responses in tabular form. The response data is produced in standard sound pressure levels for volumes and in RMS accelerations for structural elements. To quit VARPS, type “EXIT" at the file name prompt. 29 DATA BASE NODES - SUMMARY Table A-5: VARPS Database Node Summary Ava: nwne location volume,swface area,edge length,- reverberation time AVOL2 name location vqume,swface area,edge length,- reverberation time FPLATE name location thickness,surface area ,mass,- edge length,longitudinal wave speed,damping ratio, length,width,- Young’s modulus BEAM name location l,length,mass,Young’s modulus,- shear modulus,longitudinal wave speed,damping ratio,width,thickness BEAM name location 2 ,leng th,mass,Young 's modulus,- shear modulus,longitudinal wave speed,damping ratio,inner diameter,outer diameter FPL_AVOL name to from B FPL_AVOL2 name to from B AVOL_FPL name to from B AVOL_FPL2 name to from B FPL_FPL name to from 0Joint length FPL_BEAM name to from 7,11 m3 name to from number of connection points FREQ name location f1 ,1? f3 f4 15 , ...... POW name location POW(fl ),POW(f2), POW(f3), ..... ENRG name location ENRG(f1),ENRG(f2)..... 30 ELEMENT MODELS AVOL and AVOL2 (Sheet 1 of 2) Table A-6: AVOL and AVOL2 Element Parameters Name Units volume m3 surface area m2 edge length m reverberation time seconds DATA BASE ENTRY FORMAT AVOL name location volume,sutface area,edge length,- reverberation time or AVOL2 name location volume,swface area,edge length,- reverberation time DATA BASE ENTRY EXAMPLE AVOL VOLA l 0.125,].5,6.0,1.77 or AVOL2 VOLA 1 0. 125,1.5,6.0,1.77 DESCRIPTION AVOL and AVOL2 model acoustic volumes such as rooms or car interiors. Both element models require the same database entries. AVOL2 models the element identically to the way in which VAPEPS, an SEA modeling package available from NASA, models an acoustic volume. The modal density, n, is (Lyon 1975, 281): 3 1 AVOL and AVOL2 (Sheet 2 of 2) ”Ween. c 2c 8c (A-l) where: f = band center frequency (Hz) V = volume (m3) A = volume surface area (m2) 1 = perimeter edge length (m) c = speed of sound of fluid in volume (m/sec) AVOL2 ignores the second and third terms of equation (A-l). This is acceptable when frequency is high or a large volume is being evaluated. The loss factor for an acoustic volume can be modeled using the reverberation time, TR (Lyon 1975, 264): n = 22/ 2 (A-2) where f is defined as before. 3 2 FPLATE (Sheet 1 of 2) Table A-7: FPLATE Element Parameters Name Units thickness m surface area m2 mass kg edge length m longitudinal wave speed m/second damping ratio dimensionless length m widlth m Young’s modulus GPa DATA BASE ENTRY FORMAT FPLATE name location thickness ,surface area ,mass,- edge length,longitudinal wave speed,damping ratio,length,width,Young's modulus DATA BASE ENTRY EXAMPLE FPLATE PLATE 2 0.0075,0.25,5.08125,2.0,5082.3,0.00005,- 0.5,0.5,70.0 DESCRIPTION FPLATE models flat plate elements such as walls or floor pans. The modal density, n, is (Lyon 1975, 282): hc, (A-3) where: A = plate surface area (m2) h = plate thickness (m) c; = longitudinal wave speed (tn/sec) 3 3 FPLATE (Sheet 2 of 2) For steel, aluminum, and glass the longitudinal wave speed is 5181.6 m/sec (Lyon 1975, 282). The loss factor of a flat plate is modeled as a function of the plate’s damping ratio, C (Lyon 1975, 264). T] = 2.01; (A-4) 3 4 BEAM (Sheet 1 of 3) Table A-8: BEAM Element Parameters Name Units type rectangular = I circular = 2 length m mass kg Young’s modulus GPa shear modulus GPa longitudinal wave speed m/second damping ratio dimensionless width or inner diameter m thickness or outer diameter m DATA BASE ENTRY FORMAT Rectangular Cross Section BEAM name location 1,1ength,mass,Young’s modulus,shear modulus,,- longitudinal wave speed,damping ratio,width,thickness Circular Cross Section BEAM name location 2, length, mass,Young’s modulus,shear modulus, longitudinal wave speed,damping ratio,inner diameter,outer diameter DATA BASE ENTRY EXAMPLE BEAM FRAME 3 2,1.7272,94.29,190.0,75.0,5181.67,- l.5,2.77,2.8787 3 5 BEAM (Sheet 2 of 3) DESCRIPTION BEAM models structural elements such as frame structures. A BEAM element can have a solid rectangular or a circular cross section. Circular cross sections may be tubular or solid. For both cross sections, length, mass, Young’s modulus, Shear modulus, longitudinal wave speed, and critical damping ratio are specified. For rectangular cross sections, the beam width and thickness are specified. Inner and outer diameters are specified for circular cross sections with the inner diameter being 0.0 for solid cross sections. The modal density equation is (Lyon 1975, 288-289; VAPEPS): l + 21_ n = 1/kc,(u c, (A-S) where l = the beam length (m) k = radius of gyration (m) c; = longitudinal wave speed (m/sec) c, = torsional wave speed (tn/sec) 0) = band center frequency (rad/sec) The first term in equation (A-5) is the flexural modal density and the second term is the torsional modal density. c, is calculated from the following equation for circular cross sections (V APEPS): 9 = G/P (A-6) where: G = shear modulus (Pa) p = mass density (kg/m3) For rectangular cross sections (V APEPS): r 9’ (A-7) where: J = polar moment of inertia (m4) 3 6 BEAM (Sheet 3 of 3) p = beam mass density (kg/m3) and n=Ga with :1 evaluated as follows (V APEPS): '1 = (Afb’h-wc')C2 where A x), is the beam cross sectional area and rhw is the ratio of the beams thickness to its width. If rhw < 1.0, fhw = 1.0/rm... C1 and C2 are evaluated as follows (V APEPS): ‘ 'O.2901,0.141 1.0 < r,” < 2.0 O.7603,0.194 2.0 < r," < 6.0 > 0.9101,0.2537 6.0 < r,w < 10.0 10,0333 10.0 < r,w J (A-8) C,,C2 =< The internal loss factor is (Lyon 1975, 264): 11 = 2. 0C (A-9) where C is the critical damping ratio. Hi... .4. g. . 11! 37 CONNECTOR MODELS FPL_AVOL and FPL_AVOL2 (Sheet 1 of 3) Table A-9: FPL_AVOL and FPL_AVOL2 Connector Parameters Name Units B(plate edge condition) dimensionless DATA BASE ENTRY FORMAT FPL_AVOL name to from [3 or FPL_AVOL2 name to from [3 DATA BASE ENTRY EXAMPLES FPL_AVOL FPAN_INTER 3,4 1.41412 or FPL_AVOL2 FPAN_INTER 3,4 1.41412 DESCRIPTION FPL_AVOL and FPL_AVOL2 model flat plate to acoustic volume connections such as walls to rooms or floor pans to automobile interiors. FPL_AVOL2 provides results matching VAPEPS, an SEA modeling package available from NASA. The model inputs are connector name, connectivity, and B. The connector name is user selected and does not affect model evaluation. Connectivity is input as two integers separated by a comma. The first number is the flat plate location and the second is the acoustic volume location. Suggested values for B, the plate edge condition, are (Lyon 1975.300): 1 simply supported [3 = 2 clamped - clamped «5 realistic cases (A-10) 3 8 FPL_AVOL and FPL_AVOL2 (Sheet 2 of 3) Other required parameters are obtained from the associated FPLATE and AVOL element database entries. W The coupling loss factor is (Lyon 1975.300): 11:: = rad/(om: (A-11) where Tlsa indicates the coupling loss factor from the structural element to the acoustic volume element. Rmd, is the radiation resistance, to is the band center frequency in radians and ms is the plate structural mass. For FPL_AVOL (Maidanak 1962, 818): R, = AW. *« [UM + (mails ll - f c/ f ]-%’ and for FPL_AVOL2 (VAPEPS): '- 2. our _ "ZAP R, ad = AP pa ca II: 1 (Pr Avc/ Ap)82(f / f c)]Ba :(I/x.)i + (mafia .ll-fc/fl'i sin"(a)]fl. where 81(f/fc) = 0. (4/n‘)(1 - 2a2)/a(1- a2)? '[0».x./A,.>gl(f/f.) + (etc/Ap)g2(f/fc)]fi. f < f2 f=fi Y f>ch ffl f<§fcL 1 f>5fl , (A-12) L (A-13) (A-14) 39 FPL_AVOL and FPL_AVOL2 (Sheet 3 of 3) and 82(f/fc)=' with {(1— a2)ln[(1+ a)/(1- or)] + 20:} 41:2(1- a2); (A- l 5) a = (f/f.)i (A-16) where: p = density of air (kg/m3) C = speed of sound (tn/sec) f = band center frequency (Hz) fc = critical frequency (Hz) and fk, the wave number frequency, is the frequency at which —;-ka * dm,g = 1.0. ka is the acoustic wave number and davg, the average plate dimension, is davg = (1+ w)/2.0. I and w are the plate’s length and width respectively. The critical frequency and wave number frequency are (Beranek 1971,270): 2 f = __c__ ‘ 1.8138hpc, (M7) and c f k = Mm“? (A-18) where: hp = plate thickness (m) c; = plate longitudinal wave speed (tn/sec) 4 O AVOL_FPL and AVOL_FPL2 (Sheet 1 of 1) Table A-10: AVOL_FPL and AVOL_FPL Connector Parameters Name Units “Plate edge condition) dimensionless DATA BASE ENTRY FORMAT AVOL_FPL name location B 01' AVOL_FPL2 name location [3 DATA BASE ENTRY EXAMPLE AVOL_FPL VOLA_PLATE 1,2 1.4142 or AVOL_FPL2 VOLA_PLATE 1,2 1.4142 DESCRIPTION Modeling the connectivity between an acoustic volume and a flat plate requires the same parameters as the acoustic volume to flat plate connector. It is calculated using the relationship (Lyon 1975, 300): n... = moms/n.) (A-19) where n; and no are the structural and acoustical modal densities respectively. 4 l FPL_FPL (Sheet 1 of 6) Table A-1 1: FPL_FPL Connector Parameters Name Units 6(angle between plates) degrees 1(joint length) m DATA BASE ENTRY FORMAT FPL_FPL name location Ojoint length DATA BASE ENTRY EXAMPLE FPL_FPL HOOD_FOD 6,8 45.0,0.25 (2) (1) 9 Figure A-l: FPL_FPL Connector Diagram. 4 2 FPL_FPL (Sheet 2 of 6) DESCRIPTION FPL_FPL models flat plate to flat plate connections (Figure A-l). The angle between the plates, 9, and the length of the joint, I, are needed to model the connection. Other required parameters are obtained from the FPLATE elements this connector links. The coupling loss factor equations obtained from VAPEPS source code are: _ 21(tao) 12 " nKlA” (A-20) where: l = joint length (m) Apl = plate 1’s surface area (m2) K1 = plate 1’s acoustic wave number which is evaluated as follows: K _ [@T’Z 1 ’ C hl 11 (A-21) where: (u = the band center frequency (rad/sec) h] = plate 1 thickness (m) c; 1 = plate 1 longitudinal wave speed (m/sec) too is evaluated as follows: [00 = (If + tb)/3.0 (A-22) where: (|(det1/det2)|)2r * (psi) 2.0 (A-23) and 2 - tb = (|(det3/det4)|) r * (ps1) 2.0 (A-24) with 4 3 FPL_FPL (Sheet 3 of 6) r = Kz/K1 (A-26) where K2, the wave number of plate 2, is evaluated using equation (A-21). pSi = rzrb /rb 2 t (A-27) rhand r51, the bending rigidity of plates 1 and 2 respectively, are evaluated from: Eh? Tb. = — ' 12.0 (A-28) and detl = |matrix1| (A-29) det2 = Imatrix2| (A-30) det3 =Imatrix3| (A-31) det4 = |matrix4| (A-32) where matrix] , matrixZ, matrix3, and matrix7 are shown in Table A-12. 44 (Sheet 4 of 6) FPL_FPL ....~+m .5.6+_..~+m: .56+Q .56+Q Raw I 93600 A«.O\a\.uv I a¢ I 95600 m0.0 + aG I unvaOO—A«.U\n\0v I 0.; maflvuflfi .7 04 m0 .0 + :Nvuv& + 0.: I. wining ..6.6+LI .2I66 .5.6+6.~ 5.2.66 .5 .6 + 665:3 .5 .6 + 6 653% I .5 .6 + 6 A .56 + 6A1 3+” .56+QI .56+Q .56+Q .A6 I 580 Acuiiov I A6 I 530 .5 .6 + H6; I 332: A325 + on .5 .6 + H3636 + 6.; I meshes. .5.6+..I .5.~+66 .56+6.~ .5.~+66 . .5.6+6.N\.a¢ .5.6+6.~ .5.6+6.~ .56+6.~I ..U+Q .5.6+HU+Q_ .5.6+< .5.6+< goveooAsuEAuv I 2530 .5 .6 + onaoofie.6\§uv I 6 .L Raveuo + 6 A .5 .6 + 23.5 + 6.: I «5.603 .56+LI .2I66 .5.6+6.H .5.~+66 . .5 .6 + 6.N\.=i .5 .6 + 6 65.2? .5 .6 + 64 .5 .6 + 6 AI ..U+m .5.6+0 com 36532 ”N“ -< 2an 4 5 FPL_FPL (Sheet 5 of 6) The matrix variables are: A = [1.0 + p—2hz—clz]cos(6) Pl "161] B = [(1. 0 + m] cos(9) — cos(6)] cos(6) + 1.0 Pr hrcn C = bet(1)[sin(9)]2 D = [1.0 + mjcosm — 0) Prhrcn E = [(1.0 + m]cos(n — 6) - cos(1t - 0)]cos(1t — 0) + 1.0 PI "1011 F = bet(1)[sin(tt - 9)]2 and cflplhl bet(2) = Ctzpzhz where: Cfi = flexural wave speed of plate i (m/sec) c1; = longitudinal wave speed of plate i (tn/sec) (A-33) (A-34) (A-35) (A-36) (A-37) (A-38) (A-40) 4 6 FPL_FPL (Sheet 6 of 6) p,- = the mass density of plate i (kg/m3) h,- = thicknesses of plate i (m) 0 = joint angle (degrees) 4 7 FPL_BEAM (Sheet 1 of 8) Table A-13: FPL_BEAM Connector Parameters Name Units 7(angle between plate and degrees beam - See Figure A-2) u (angle between plate and degrees beam - See Figure A-2) DATA BASE ENTRY FORMAT FPL_BEAM name location 7, It DATA BASE ENTRY EXAMPLE FPL_BEAM ENG_FRAME 1 ,3 0.0,90.0 48 FPL_BEAM (Sheet 2 of 8) Figure A-2: FPL_BEAM Connector Diagram DESCRIPTION FPL_BEAM models the connection between flat plates and beams attached at acute angles (Figure A-2). In addition to the parameters obtained from the FPLATE and BEAM element models, the angles 7 and It must be specified. The equations for the coupling loss factor were obtained from VAPEPS source code. The loss factor is evaluated by 4 9 FPL_BEAM (Sheet 3 of 8) (2.0C3C4)[real(zp)] (A-41) where: (0 = band center frequency (rad/sec) mb = beam mass (kg) nu, = beam torsional modal density (modes/Hz) nfl, = beam flexural modal density (modes/Hz) ' 2,, = plate impalance with: l Va’kbctb (A-42) where: l =beam length (m) nfl): a) = band center frequency (rad/sec) kb = beam radius of gyration (m) cu, = beam longitudinal wave speed (tn/sec) with: 703+ D} 4.0 (A43) for circular cross sections. And kb = hb 412.0 (A-44) for rectangular cross sections. where: Do = beam outer diameter (m) D,- = beam inner diameter (m) hb = beam thickness (m) 5 0 FPL_BEAM (Sheet 4 of 8) ntb = 2.01/C‘b (A'45) where: l = beam length (m) Cgb = beam torsional wave speed (m/sec) with: Ctb = \IGb/pb (A-46) where: G}, = beam shear modulus (Pa) p1, = beam density (kg/m3) 2 where: 2;, = beam flexural impedance zp = plate impedance with: ify = 90.0° orp. = o.o°, then Zb=be else 1.0 2 2 2b = 2b +x 22b \/1.0+xaz( ' a f) (A-48) where: Zbr = beam torsional impedance zbf = beam flexural impedance with: be = M00 -1.0i) cfl’ (A-49) where: plb = beam linear density (kg/m) kb = beam radius gyration (m) 5 1 F PL_BEAM (Sheet 5 of 8) with: czb = beam longitudinal wave speed (In/sec) Ctb =beam torsional wave speed (m/sec) PM = Pbeb (A-SO) where: n, = beam density (kg/m3) A ,1, = beam x-sectional area (m2) and kb evaluated by Equations (A-43) and (A-44) and cu, evaluated by equation (A—46). and __ 2 lb! " plbktbctb where: with: (A-Sl) Plb = beam linear density (kg/m) kw = beam torsional radius of gyration Ctb =beam torsional wave Speed (m/sec) k _ 4.0k,J ‘ #80 (A-52) for circular cross sections. And ktb = ll— Axb (A—53) for rectangular cross sections. beam cross sectional area (m2) where: A xb = and: t1 = (Afbrh'f')C2 (A-54) the ratio of the beams thickness to its width. If rhw < 1.0, rhw = 1.0/rm... where: "hw = 5 2 FPL_BEAM (Sheet 6 of 8) and C1 and C2 are evaluated as follows: r0.2901,o.141 1.0 < r,” < 2.0‘ 0.7603,0.194 2.0 < r,w < 6.0 > 0.9101,0.2537 6.0 < r,“ <1o.0 \1.o,0.333 10.0 < r,” C,,C2 =4 and = [cos(y) cos(tt)]2 + sin2(y) [cos(v)<=os(u)]2 53m .05 26m 98 Ram can»: 9 .68. .68m .0668 0:: 8.5 6H AN+§V 09¢. .bm: B 39: H6: - x38 0E .5 gm umn.6HQmeo “Hugo 60¢ 66 ame oozav Aeeo.mHmz¢xmv oeecmHHm mmmm> House mmm<>® ""Hx<>HoI mmcHH OH vaeao.mHmzwme¢Hm NHo>m Hem HNVH6.H ~.H madqm ¢H0> qum Ho>¢ ee.H.o.m.m.H.m~H.o m mHo> ~H0>¢ o.oe.m.o.m.o .moooo.o.m.HmHm.o.N.mNHmo.m.mm.o.meoo.o m ma E.H.o.o.m.H.mmH.o H 50> 305 was: Laud—6830. 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