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Sound transmission characteristics of sandwich panels with a truss lattice core Moosavi Mehr, Ehsan 2016

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Sound Transmission Characteristics of Sandwich Panelswith a Truss Lattice CorebyEhsan MoosavimehrBSc. Mechanical Engineering, Sharif University of Technology, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of Applied ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Mechanical Engineering)The University of British Columbia(Vancouver)January 2016c© Ehsan Moosavimehr, 2016AbstractSandwich panels are extensively used in constructional, naval and aerospace struc-tures due to their high stiffness and strength-to-weight ratios. In contrast, soundtransmission properties of sandwich panels are adversely influenced by their loweffective mass. Phase velocity matching of structural waves propagating within thepanel and the incident pressure waves from the surrounding fluid medium lead tocoincidence effects (often within the audible range) resulting in reduced impedanceand high sound transmission. Truss-like lattice cores with porous microarchitec-ture and reduced inter panel connectivity relative to honeycomb cores promise thepotential to satisfy the conflicting structural and vibroacoustic response require-ments. This study combines Bloch-wave analysis and the Finite Element Method(FEM) to understand wave propagation and hence sound transmission in sandwichpanels with a truss lattice core. Three dimensional coupled fluid-structure finiteelement simulations are conducted to compare the performance of a representativeset of lattice core topologies. Potential advantages of sandwich structures with alattice core over the traditional shear wall panel designs are identified. The signifi-cance of partial band gaps is evident in the sound transmission loss characteristicsof the panels studied. This work demonstrates that, even without optimization,significant enhancements in STL performance can be achieved in truss lattice coresandwich panels compared to a traditional sandwich panel employing a honeycombcore under constant mass constraint.iiPrefaceThis dissertation is ultimately based on two analysis methods: Bloch wave analysisand finite element analysis. The Bloch wave analysis code was written primarilyby A. Srikantha Phani and the author. The finite element analysis was done by theauthor using the commercial finite element package Abaqus.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Sources of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Quest for Light and Stiff Structures . . . . . . . . . . . . . . 11.3 Vibroacoustic Response of a Light and Stiff Design . . . . . . . . 21.4 Improving the STL in Sandwich Panels . . . . . . . . . . . . . . 71.5 Truss-Lattice Core Structures . . . . . . . . . . . . . . . . . . . . 121.6 STL for a Sandwich Panel . . . . . . . . . . . . . . . . . . . . . 131.6.1 Experimental Measurement . . . . . . . . . . . . . . . . 131.6.2 Analytical Modeling . . . . . . . . . . . . . . . . . . . . 141.6.3 Bloch Wave Analysis . . . . . . . . . . . . . . . . . . . . 151.6.4 Finite Element Analysis (FEA) . . . . . . . . . . . . . . . 151.7 Research Objectives and Thesis Outline . . . . . . . . . . . . . . 161.7.1 Research Objectives . . . . . . . . . . . . . . . . . . . . 18iv1.7.2 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . 182 Elastic Wave Propagation in Truss-Lattice Panels . . . . . . . . . . 202.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 Bloch Wave Propagation in Periodic Structures . . . . . . . . . . 222.3 Sandwich Panels . . . . . . . . . . . . . . . . . . . . . . . . . . 302.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . 303 Numerical Simulation of Sound Transmission through Sandwich Pan-els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1 Transmission of Sound through Panels: Physics and GoverningEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.1.1 Structural Model . . . . . . . . . . . . . . . . . . . . . . 333.1.2 Acoustic Model . . . . . . . . . . . . . . . . . . . . . . . 333.1.3 Coupling Structural and Acoustic Models . . . . . . . . . 353.1.4 Finite Element Implementation . . . . . . . . . . . . . . 353.2 FEM Procedure and Validation . . . . . . . . . . . . . . . . . . . 363.2.1 Validation of Structural Response Modeling . . . . . . . . 363.2.2 Validation of Acoustic Modeling . . . . . . . . . . . . . . 373.2.3 Structural-Acoustic Modeling in FEA . . . . . . . . . . . 403.3 Truss-Lattice Core Sandwich Panel Modeling . . . . . . . . . . . 493.4 Static Stiffness Comparison . . . . . . . . . . . . . . . . . . . . . 513.4.1 Why Not Constant Stiffness? . . . . . . . . . . . . . . . . 513.5 Results of the FEA Simulatoins of the Sound Transmission Problem 524 Main Results and Discussion . . . . . . . . . . . . . . . . . . . . . . 544.1 A Closer Look at the STL Curves . . . . . . . . . . . . . . . . . 544.2 Sound Transmission Class and Speech Interference Level . . . . . 614.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . 644.4.1 Contributions . . . . . . . . . . . . . . . . . . . . . . . . 654.4.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . 66Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68vA Finite Element Calculation of Effective Stiffness Moduli . . . . . . . 75B Sound Interference Level . . . . . . . . . . . . . . . . . . . . . . . . 78viList of TablesTable 3.1 List of material properties and model specifications used for FEA 48Table 4.1 The sound transmission class for the panels in Figure 2.2 andthe monolithic panel. . . . . . . . . . . . . . . . . . . . . . . 61Table 4.2 Incremental reductions in preferred-octave speech interferencelevel (PSIL) and speech interference levels (SIL) calculated fromthe STL curves for each geometry using the formulae: ∆PSIL=STL500+STL1000+STL20003 and ∆SIL =STL500+STL1000+STL2000+STL40004 .Note that PSIL= Lp500+Lp1000+Lp20003 and SIL=Lp500+Lp1000+Lp2000+Lp40004where Lp is the sound pressure level. All panels have identicalmass. See Appendix B for a more detailed discussion. . . . . . 63Table A.1 First order estimates of effective moduli of the lattice core topolo-gies studied as a function of relative density (ρ¯). a, h, t and l arerespectively the radius of the strut, height of the core, thicknessof the wall for a hexagonal honeycomb and the length of thehexagonal honeycomb. . . . . . . . . . . . . . . . . . . . . . 77viiList of FiguresFigure 1.1 Schematic of diffuse field sound transmission loss in sand-wich panels. At low frequencies structural resonances andanti-resonances create minima and maxima, respectively. Notethat anti-resonances can yield STL above the mass law. Highfrequency wave propagation regime is characterized by coin-cidence effects. . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.2 Schematic of diffuse field sound transmission loss in sand-wich panels. At low frequencies structural resonances andanti-resonances create minima and maxima, respectively. Notethat anti-resonances can yield STL above the mass law in anarrow frequency region, a fact used in modern acoustic meta-materials [45]. High frequency wave propagation regime ischaracterized by coincidence effects. Double wall (mass-spring-mass) resonance is usually high for incompressible cores withhigh through-thickness modulus. . . . . . . . . . . . . . . . . 6viiiFigure 2.1 The Kagome lattice (b) made by tessellating the unit cell (a)along x and y. Typical finite element model of the unit cell of alattice (kagome core) sandwich panel with partitioned degreesof freedom labelled. Displacements at nodes labelled as r, l, band t respectively correspond to the right, left, bottom, and topdegrees of freedom. Double subscripts such as lb represent theleft-bottom etc. Note that nodes on some edges of the modelare not labelled for the sake of clarity. q is a displacement vec-tor of the partitioned degrees of freedom that are representedin the subscript. A reference unit cell having a point A, withposition vector p and a displacement vector q(p) is shown in(b). Another point B, similar to point A in an arbitrary unit cellhas a position vector r and a displacement vector q(r). Thisunit cell is identified by the label (n1,n2), which represents n1unit cell translations along x and n2 translations along y fromthe reference unit cell. Using Floquet-bloch theorem, displace-ments of this arbitrary unit cell can be represented in terms ofdisplacements of the reference unit cell [8] . . . . . . . . . . 27Figure 2.2 Selected sandwich panel core designs studied: (a) Kurtze andWatters panel, (b) Tetrahedral core, (c) Double pyramidal core,(d) Pyramidal core, (e) Kagome core and (f) Hexagonal Hon-eycomb core. All panels have identical face sheets, height andmass. Panel size 1.5 m × 1.5 m × 0.038 m. Individual strutsof all cores are of circular cross section. Strut dimensions arechosen to maintain same mass across all cores based on theirrespective relative density, see Table A.1. portion of the toppanel is removed to show the core. . . . . . . . . . . . . . . . 28ixFigure 2.3 Dispersion curves of the sandwich panels for different corescalculated using Bloch theory. Curves associated with fourtruss-lattice cores and the hexagonal core are shown above: (a)Tetrahedral, (b) Double pyramidal, (c) Pyramidal, (d) Kagomeand (e) Hexagonal. The wave vector locus followed in calcu-lating the dispersion curve for each topology is shown in (f)where the irreducible part of the first Brillouin zone is shownas a shaded region. The following values for the symmetrypoints are used: Γ = (0,0), X = ( piLx ,0), M = (piLx, piLy ), where,Lx and Ly are respectively the length and width of the unit cell.Note that all cores have identical mass and the differences inthe dispersion behaviour is due to the variations in topologygoverned dynamic stiffness. The long wave length asymptotesnear the origin (Γ point) indicate the group velocities relatedto the stiffness. We observe that the steepest dispersion curve(first branch) associated with hexagonal core in (e) suggest thatit is the stiffest topology in transverse shear. At the other ex-treme Kagome core in (d) exhibits lowest stiffness for trans-verse shear wave propagation. ry =uyrms√u2zrms+u2xrms, where uxrms,uyrms and uzrms are the root mean squared average displace-ments along the x, y, and z axes with y axis taken as the normalto the face sheets. . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 3.1 A volumeV of fluid with surface area S and the normal vectorsas defined . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Figure 3.2 El-Raheb’s a) arch configuration and b) unit cell for the curve [22] 37Figure 3.3 Results of El-Raheb’s prediction of the steady-state responseof an arch to a harmonic excitation using the transfer matrixmethod[22] . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.4 Reproduced results of El-Raheb’s prediction of the steady-stateresponse of an arch to a harmonic excitation using FEA inAbaqus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39xFigure 3.5 Velocity intensity (solid line) using the transfer matrix methodfor the arch configuration. The two other curves are not dis-cussed in this dissertation. . . . . . . . . . . . . . . . . . . . 39Figure 3.6 Velocity intensity results from our FEA simulations [22] forthe arch configuration . . . . . . . . . . . . . . . . . . . . . . 40Figure 3.7 Sound transmission suite configuration considered in this study 41Figure 3.8 Diffuse field interaction implemented in a finite element model.The incident face of the panel (FE model with nodes shown asfull circles) is imagined to be subjected to excitation from pointsources distributed on a hemispherical surface. The phase ofplane waves emanating from each point source are uncorre-lated [16]. Fluid forces due to incident pressure field are di-rectly applied on the nodes of the structural FE model. Fluidloading effects are neglected. . . . . . . . . . . . . . . . . . . 42Figure 3.9 The panel and fluid medium configuration for FEA . . . . . . 44Figure 3.10 Sound transmissiom loss through a truss-like panel: derivedfrom transfer matrix method (—) and experiment (- - -) [23] . 45Figure 3.11 Mesh convergence and results of the FEA simulations of thesound transmission loss through the truss-like panel . . . . . . 46Figure 3.12 The setup and mesh for the Kurtze and Watters panel . . . . . 47Figure 3.13 Velocity versus frequency found experimentally and core con-figuration for Kurtze and Watters [40] . . . . . . . . . . . . . 47Figure 3.14 Velocity versus frequency calculated from FEA frequency anal-ysis by knowing the standing wavelengths at natural frequencies 48Figure 3.15 a) Convergence with respect to mesh size for the Kagome corelattice. b) Convergence with respect to the number of datapoints for the Kagome core lattice . . . . . . . . . . . . . . . 50Figure 3.16 Comparison of relative static stiffness of the panels studied.The stiffness of each panel is divided by that of the monolithicpanel stiffness, taken as the reference. Stiffness is evaluatedat the central point of each panel when all edges are simplysupported. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52xiFigure 3.17 A comparison of third-octave band averaged STL response ofthe Kurtze and Watters panel with five other lattice topologies.Four frequency points in each octave-band are used for the pur-poses of averaging. The stiffest panel (see Figure 3.16) hashighest STL in the low frequency region governed by staticstiffness. All panels have the same mass. Topology governedvariations in dynamic stiffness and degree of connectivity be-tween face sheets manifest in the STL response. . . . . . . . . 53Figure 4.1 Sound transmission loss characteristics of a sandwich panelwith a tetrahedral core. Deformation shapes of the panel atstructural resonances (denoted by R) and deformation shapesof the panel and the pressure field at coincidence frequencies(denoted by C) are also shown at each significant frequency.Pressure fields associated with enhanced STL due to partialwave band gaps in the high frequency region (marked as Bin STL curve) are also shown. Note that the pressures in thefluid medium on the transmitted side are reduced by 2 orders ofmagnitude between C1 and B above. The solid black straightline corresponds to the mass law line. . . . . . . . . . . . . . 55Figure 4.2 Sound transmission loss characteristics of a sandwich panelwith a double pyramidal core. Deformation shapes of the panelat structural resonances (denoted by R) and deformation shapesof the panel and the pressure field at coincidence frequencies(denoted by C) are also shown at each significant frequency.The solid black straight line corresponds to the mass law line. 56xiiFigure 4.3 Sound transmission loss characteristics of a sandwich panelwith a pyramidal core. Deformation shapes of the panel atstructural resonances (denoted by R) and deformation shapesof the panel and the pressure field at coincidence frequencies(denoted by C) are also shown at each significant frequency.Pressure fields associated with enhanced STL due to partialwave band gaps in the high frequency region (marked as Bin STL curve) are also shown. Note that the pressures in thefluid medium on the transmitted side are reduced by 2 orders ofmagnitude between C1 and B above. The solid black straightline corresponds to the mass law line. . . . . . . . . . . . . . 57Figure 4.4 Sound transmission loss characteristics of a sandwich panelwith a Kagome core. Deformation shapes of the panel at struc-tural resonances (denoted by R) and deformation shapes of thepanel and the pressure field at coincidence frequencies (de-noted by C) are also shown at each significant frequency. Pres-sure fields associated with enhanced STL due to partial waveband gaps in the high frequency region (marked as B in STLcurve) are also shown. Note that the pressures in the fluidmedium on the transmitted side are reduced by 2 orders ofmagnitude between C1 and B above. The solid black straightline corresponds to the mass law line. . . . . . . . . . . . . . 58Figure 4.5 STL characteristics of a sandwich panel with a hexagonal core.Deformation shapes of the panel at structural resonances (de-noted by R) and deformation shapes of the panel and the pres-sure field at coincidence frequency (denoted by C) are alsoshown. The solid black straight line corresponds to the masslaw line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59xiiiAcknowledgmentsI would like to thankfully acknowledge my supervisor, Dr. A. Srikantha Phani forhis excellent supervision, generous support and the productive meetings and dis-cussions during the course of this project. He helped me explore different areas ofstructural vibrations and acoustics from which emerged the main objectives for thisproject. I thank my colleagues in the Dynamic and Applied mechanics Laboratory(DAL), in particular Behrooz Yousefzadeh, for the healthy discussions and for cre-ating a friendly research atmosphere. I also want to thank Prof. Murray Hodgsonto whom I owe my knowledge of acoustics. This dissertation is dedicated to myloving parents, Masoudeh Rahimi and Hassan Moosavimehr, for being there withme at every step of my life. Without these people believing in me, this work wouldhave never been possible. Financial support from Natural Sciences and Engineer-ing Research Council of Canada (NSERC) is thankfully acknowledged.xivChapter 1Introduction1.1 Sources of NoiseThis study focuses on the acoustic and structural performance of sandwich panelsand investigates the potential ways to improve their acoustic performance by alter-ing the conventional core designs. This is especially of importance in aerospacedesigns where there are high levels of broadband noise (100 Hz to 6 kHz) [53]and vibration, and it is crucial to keep the mass of the structure as low as possi-ble. Tewes et al. [64] summarizes the sources present in an aircraft as being: 1) jetmixing noise, 2) turbo machinery noise, 3) turbulent boundary layer noise, and 4)noise radiated in the cabin because of the structural vibrations.1.2 The Quest for Light and Stiff StructuresThe constant trend towards material efficiency and ecologically friendly engineer-ing asks for structural designs which are as lightweight as possible without havingto compromise the load bearing capabilities. In the field of aerospace, these char-acteristics translate into less material and fuel consumption, less air pollution andlower dynamic loads on joints and connections. Sandwich panels have proven tobe promising alternatives for the conventional structural panel designs. As a case inpoint, Palumboa and Klos [55] found that using graphite epoxy honeycomb panelsinstead of the conventional stiffened aluminum panels in air-craft sidewall designs1would result in 35% less structural weight and 60% thinner panels.Sandwich panels comprise three elements: two solid face sheets, separated bya relatively lighter and softer core. As a result, the effective moment of inertia ofthe section highly increases, relative to the case of unseparated face sheets. Thisphenomenon, known as the sandwich effect, can significantly enhance the bendingstiffness and strength of the panels and lead to more weight-efficient structures.The face sheets bear the bending loads while the main functionality of the core isto act as a spacer between the two face sheets and assure the structural integrityof the panel by bearing the shear loads. Therefore, it is required that the corehave sufficient compressive and shear stiffness without adding much weight to thestructure.Nowadays, the sandwich configuration is widely used in everyday-life exam-ples such as corrugated card board boxes, as well as in advanced civil, naval andaerospace structures. Depending on their application, sandwich panels are madewith different core and face materials. The face sheets bear most of the structuralload on the panel. Therefore, they are often chosen to be much denser and stifferthan the core material. Depending on the application, the face-sheet material canrange from plasterboard (as in studded walls in buildings) to metals and compositematerial (such as glass fiber, carbon fiber and Kevlar) used in airframe designs.At the same time, there are various configurations available to choose from for thecore material [2, 72]. Static and dynamic characteristics of each configuration needto be studied before a structural panel is designed.1.3 Vibroacoustic Response of a Light and Stiff DesignDespite their many structural benefits, the acoustic performance of light and stiffpanels may not be as desirable, primarily because of their lightness which translatesinto little mass impedance to dynamic loads. The high bending stiffness to massratio also means that the speed of bending waves in these panels matches the speedof sound waves in the acoustic medium at frequencies within the audible range (20Hz to 20 kHz). This phenomenon, known as the acoustic coincidence [13], leadsto increased levels of acoustic radiation, which further deteriorates the acousticperformance.2As the sandwich panels are increasingly being used as structural elements incivil, aerospace and naval engineering it is important to understand their soundtransmission behavior. The transmission characteristics of any panel in general canbe quantified using a parameter called the Sound Transmission Loss (STL). TheSTL is defined as the ratio of the acoustic power that is incident on the panel to thepower transmitted to the other side of the panel:STL= 10log10(Wi/Wt). (1.1)Figure 1.1 shows a schematic view of a STL curve for a typical monolithicpanel. It can be seen that the general trend can be roughly estimated with thetwo dashed lines. The left one which is due to the static stiffness of the panelis called the stiffness line and the one that governs the high frequencies is calledthe mass law line. In between these two lines, a number of troughs and peaksare shown which correspond to resonances and anti-resonances of the mountedpanel, respectively. Such resonant behavior is due to the constructive or destructiveinterference of the structural waves in the panel that are induced by the acousticexcitation and the reflections from the panel’s boundaries. As the frequency getsmany times larger than fundamental frequency of the panel, (the first trough) thepeaks and troughs become less prominent. This feature arises as a result of twoeffects which will be explained: (a) high frequency attenuation of the reflectionsand (b) modal overlap.(a) As the frequency increases, the wavelengths get smaller and most of theenergy of the structural waves gets damped before they reach the boundaries. As aresult, the reflections are not as significant [28].(b) If a system with well-separated natural frequencies is excited at a frequencyclose to any of its natural modes, the response would be the sum of a strong reso-nant response corresponding to that mode plus a smaller contribution from all othermodes. The frequency range around each response peak over which the responsefunction drops by 3 dB (half the peak value) is called the ‘half-power bandwidth’corresponding to that peak. It is a characteristic of many plate and shell structuresthat the half-power bandwidth expands with frequency, and the modal density (thenumber of natural modes per unit frequency) increases with frequency. This leads3to the modal overlap at frequencies multiple times larger than the fundamental fre-quency of the panel. It means that the response at each frequency is comprisedof contributions from more than one mode. With the increase of the excitationfrequency, the effect of the modal overlap gets stronger and eventually the systembecomes non-‘reverberant’ [27].The two effects described above result in less prominent peaks and troughs tothe point that the reflections become negligible and the waves start to propagatefreely, as if there are no boundaries to the panel. At this stage, the STL of thepanel is primarily governed by the wave propagation behavior of the panel andthe type and speed of waves that travel through the panel at each frequency. Forthe sandwich panels that we studied (Figure 1.2) this transition happens at 1000Hz, approximately. The transition frequency is subject to change depending on thestiffness, mass per unit area and material and structural damping of the panel. Thistransition frequency is lower for limp panels than light and stiff panels. Here wediscuss three phenomena that can happen after this transition frequency:1) Acoustic coincidence: It happens when the tracing wavenumber of the acous-tic excitation becomes equal to the speed of traveling waves in the structure [13].As a result, the two waves become phase-matched and strongly coupled, and sig-nificantly higher amount of energy will be transmitted through the panel. Thedifference between this phenomenon and the conventional resonance effect is thatthe former is a wavenumber matching between the fluid and structural domainswhile in the latter it is the frequency of excitation that matches one of the structuralresonance frequencies.2) Bandgaps: can be due to the spacial periodicity of the core or the existenceof local resonators. The first case, generally results in wider bandgaps than thesecond one. Bandgaps will be discussed in more detail in Section 1.4.3) Double-wall resonance: In the case of double-wall partitions, additionaltroughs, associated with the natural modes of the air entrapped inside the closed-cell cavities, are expected [54]. These modes are referred to as double-wall res-onance frequencies. In the first double-wall resonance frequency (also called themass-air-mass resonance), the air inside the cavity acts as a spring between the twoface sheets whose deformations are out-of-phase with each other. In this case, thecontribution of the air inside the cavity to the amount of sound transmitted through4Frequency (Hz)STL (dB)Wave PropagationResonantCoincidenceMass Law (6 dB/Octave)Anti-resonancesResonancesStiffness Law(-6 dB/Octave)Figure 1.1: Schematic of diffuse field sound transmission loss in sandwichpanels. At low frequencies structural resonances and anti-resonancescreate minima and maxima, respectively. Note that anti-resonancescan yield STL above the mass law. High frequency wave propagationregime is characterized by coincidence effects.the panel is called the air-borne sound transmission. Xin and Lu [71] found theair-borne sound transmission to be negligible for sandwich panels with cellularcores.Apart from the annoyance that the audible noise inside the cabin and in theenvironment can cause, the corresponding structural-borne vibration can be detri-mental from another aspect: it is a known cause of initiation and propagation ofcracks inside the aircraft structure [12]. Part of the noise inside the cabin is gener-ated externally as either vibration or noise, transmitted via a variety of paths, andthen radiated acoustically into the cabin. These are classified as ”structure-borne”noise. Aircraft noise is primarily a result of the combination of turbulence in the5!"#$%#&'()*+,-./0)*12-345#)6"78494:;7&<#=7&4&:>7;&';1#&'#?4==)04@)*A)12BC':45#-24&1D48E&:;F"#=7&4&'#=<#=7&4&'#=.:;GG&#==)04@*FA)12BC':45#-H7%IJ#F34JJ<#=7&4&'#Figure 1.2: Schematic of diffuse field sound transmission loss in sandwichpanels. At low frequencies structural resonances and anti-resonancescreate minima and maxima, respectively. Note that anti-resonances canyield STL above the mass law in a narrow frequency region, a fact usedin modern acoustic metamaterials [45]. High frequency wave propaga-tion regime is characterized by coincidence effects. Double wall (mass-spring-mass) resonance is usually high for incompressible cores withhigh through-thickness modulus.boundary layer on the outer surface of the cabin, and engine noise transmitted ei-ther through the airframe structure or through the atmosphere and then throughstructural radiation. Noise can also be generated acoustically and propagated byairborne paths such as the noise from air–conditioning/pressurization system andassociated ducting [62].Traditional methods for controlling structure-borne noise include vibration iso-lation of the sources and the use of tuned vibration absorbers. Airborne noise can6be reduced by absorption [1] or through the use of barrier materials, both of whichwill add to the mass of the structure. These approaches, as well as implement-ing Helmholtz resonators inside the structural panels [48], and choosing micro-perforated face sheet on the transmission side [32], have been previously studied indetail. In this study, the effect of the core configuration on structural wave propaga-tion and therefore sound radiation of sandwich panels is studied. This work dealswith sandwich panels with two face sheets and a spatially periodic truss latticecore. Lattice cores, with an open micro architecture offer many multi-functionalfeatures not available in a sandwich panel made using a regular closed-cell hexag-onal honeycomb core [24, 33, 66, 68]. A detailed review of lattice materials canbe found in [29]. Our principal concern here is the acoustic properties, in partic-ular, sound transmission loss (STL) properties of such advanced multi-functionalsandwich panels.1.4 Improving the STL in Sandwich PanelsIn order to improve vibration and noise attenuation a variety of methods exists.Noise control treatment can be added to the surface of the face sheets to reduce thenoise and vibration transmission. Conventional methods rely on the acoustic masslaw Section 1.3 or the addition of an absorptive layer. The first method, results inheavy materials, which defeats the purpose of using sandwich panels in the firstplace. For example, a crude way to increase the STL of a panel by 6 dB is to dou-ble its mass per unit area. The second method, leads typically to thick materials,since to be efficient, the thickness of the absorptive layer should be in the sameorder of magnitude as the acoustic wavelength [64]. Often both methods are un-satisfactory to improve the low-frequency vibro-acoustic response for lightweightapplications. Sound insulation treatments are applied to the structural envelopesof vehicles such as cars, trains and aircraft. Examples of these treatments includecomposite layers of materials such as fibrous mats and plastic foam sheets, cov-ered by very flexible cover sheets. These secondary layers are known by vehiclemanufacturers as ‘trim’s. Multilayer sandwich constructions, consisting of an im-pervious sheet of plastic or rubber-based material (in the core) separated from theface sheets by a layer of fibrous or poroelastic material, are also used to increase7the transmission loss and to decrease the radiation efficiency of the structural shellsin vehicle compartments [28].In 2005, Tewes [64] proposed and tested a new ‘active trim’ design for aerospacepartitions in which the trim panel was mounted on the surface of the structure us-ing a set of piezo-electric actuators. Thus, the dynamic response of the trim panelcould be reduced with the correct active acoustic and structural control. For, tonaland 3rd-octave band averaged1 noise, up to 20 dB and 10 dB increase in STL wasobserved respectively.State of the art on STL properties of sandwich panels has been reviewed in [15].Early pioneering studies by Lindsay [43] on transmission of obliquely incidentcompressional waves in stratified layered media (alternating fluid-solid layers oralternating layer of two fluids) of infinite extent have revealed the Bragg reflectionmechanism whereby destructive interference leads to selective reflection of wavesof wavelength nλ = 2l sinθ where θ is the angle of incidence measured from thehorizontal plane of the incident face of the medium, and l is the effective distancebetween two solid layers and n is an integer. Thompson [65] extended these stud-ies to the general case of a laminated medium of alternating layers of solids usingSnell’s law. The influence of lamination or spatial periodicity is to introduce acous-tic wave filtering effects, particularly in the form of acoustic stop bands (partial orcomplete2.) Such band gaps are expected to be present in a sandwich panel with atruss lattice core. More recently, acoustic metamaterials [11] using local resonanceconcepts have been shown to be acoustically favourable by achieving STL higherthan the conventional mass law. Mass law (see Figure 1.2) states that STL increasesby 6dB for every doubling of the panel mass at a given frequency, or, doubling offrequency for the same mass [14].One can approach sandwich panel as a double wall construction with the coreserving as the intervening medium connecting the two walls. London’s theory [46,47] on transmission of reverberent sound through double walls suggests that hon-eycomb or other non-absorbent cellular structures having no cell walls in a direc-1This concept will be explained in Ch.3.2Waves with frequencies falling within a complete band gap are forbidden to travel in all direc-tions of incidence. In a partial band gap waves are forbidden only in certain directions or angles ofincidence.8tion perpendicular to the face sheets do not result in an increased STL. Similarly,air coupled walls having no solid sound transmitting paths are extremely effectivewith a high STL. The orientation of honeycomb cell walls perpendicular to the facesheets deteriorates their STL performance. London’s theory also suggests that theinsertion of sound absorbing materials is effective only for light walls. In contrastto honeycomb panels, the connectivity due to a truss lattice core has two noticeabledifferences. First, the connection between the struts of the core and the face sheetsis at discrete points as opposed to a line connectivity with a honeycomb core. Sec-ond, individual cells are open in a truss lattice core whereas individual hexagonalcells of a honeycomb are closed by the face sheets, see Figure 2.2. For maximumbending stiffness the orientation of honeycomb cells is such that the axis of theprism is along the normal to the face sheets as shown in Figure 2.2(f). Regardlessof the core topology sandwich panels possess double wall resonances, such as themass-spring-mass resonance in which the core acts as a spring connecting the twoface sheets (masses), and coincident frequencies associated with the phase velocitymatching of the acoustic waves with structural waves.Three principal approaches can be observed in the literature on the design ofsandwich panels from the STL perspective. The three resulting panel designs arereferred to as shear panel [40], coincident panel [36, 69], and mode-cancellingpanel [52]. The shear panel design by Kurtze and Watters[40] uses an incompress-ible core soft in shear, with shear wave speed less than the acoustic wave speed,in order to favour shear waves rather than bending wave propagation in the panel.The coincidence frequency is shifted to higher frequencies outside the range ofinterest (400-5000 Hz for transmission of intelligible speech, for example), thusextending the mass law region in Figure 1.2. The sandwich panel deforms in aglobal bending mode at low frequencies and with the increasing frequencies thisdeformation evolves into a local one involving the bending of single face sheetsas the core is subjected to pure shear. The normal deflections of the face sheetdue to incident pressure waves are transformed into core shear waves in the shearwall design. In a symmetric sandwich construction, both symmetric (dilatational)waves and anti-symmetric (flexural) waves can propagate. Core compressibilitygoverns the coincidence associated with dilatational waves and double wall reso-nances. By increasing the core stiffness dilatational wave coincidence frequencies9are increased but flexural wave coincidence frequencies are lowered [20, 21, 30].The requirement of an incompressible core soft in shear leads one to consider ma-terials such as rubber and plastics which unfortunately have other disadvantagessuch as added weight, fire hazard etc. However, periodic structures in the form ofpanels with connecting bridges between the skins have been shown to fulfill thisrequirement by Kurtze and Watters [40]. In general, anisotropic cores are advanta-geous to meet the requirements of high incompressibility and low transverse shearstiffness in shear wall design.The coincidence wall design [36, 69] employs a core of high stiffness in com-pression so that the dilatational coincidence frequency is placed well above thefrequency region of interest. In contrast with the shear wall, the shear stiffness ofthe core is also high. Consequently, coincidence associated with anti-symmetric(flexural wave) motions are excited within the frequency range of interest, but themotions of the panel are damped using attached layers of damping. Achievingsimultaneously high stiffness and damping, without adding significant mass, is achallenge with this design.In the mode-cancelling panel design, the symmetric and anti-symmetric flexu-ral motions of a symmetric sandwich panel are cancelled in a desired frequencyrange by placing the double-wall resonance (mass-spring-mass resonance) fre-quency of the panel below the frequency range of interest by using a relativelysoft core in compression. The anti-symmetric (flexural motion) coincidence liesat a frequency outside the range of interest. Consequently, within the frequencyrange the symmetric and anti-symmetric modes for the sound transmitting face ofthe panel cancel out and higher than mass-law STL is anticipated. Typically, thehexagonal honeycomb is oriented such that the axis of prismatic cells is perpendic-ular to the normal direction (thickness direction) of face-sheets, so that core is com-pressible. This orientation leads to low stiffness in transverse (through-thickness)compression, high shear stiffness for wave propagation along the cells, but lowshear stiffness for wave propagation across the cells. A concern with the mode-cancelling panel is the sacrifice in stiffness for a gain in STL performance sincethe core is made to be soft in compression undermining the sandwich effect whichrequires constant face sheet separation.Sandwich panels with a truss lattice core offer significant potential. They are10generalized anisotropic with gradient elasticity effects. Ultralight truss lattice corescan be fabricated with rapidly advancing manufacturing methods, see [51, 60] forexample. In view of the limitations of the existing panel designs and the potentialoffered by truss lattice cores, this work is aimed to address the following questions:(i) For the same panel mass what is the influence of different lattice core topologieson the stiffness and STL properties? (ii) Can one achieve a panel with high STLwithout sacrificing stiffness and without adding significant mass in the form ofdamping or absorption treatment? To address these questions we use here a finiteelement model which takes into account the full three dimensional fluid-structureinteraction effects for a systematic comparison of STL characteristics.In the example described in the beginning of the chapter, Palumboa and Klos[55] found that by removing parts of the core that supported supersonic waves, thespeed of the bending waves decreased. This reduced the radiation efficiency of thepanel. As a result, the STL of the panel was improved (up to 7 dB improvement)by making periodic cut-outs inside the core.In 2013, Claeys et al. investigated the potential of stop bands, created by in-terference and by local resonances, to suppress wave motion in certain frequencyranges. Interference stop bands function on the basis of Bragg scattering: destruc-tive interference between the waves transmitting in the structure and the wavesreflected from irregularities in the structure results in low transmission of energyand attenuated structural response [39]. Applying resonators on the surface of thestructure introduces a stop band at the resonance frequency of the resonator. Ingeneral, the resonance stop band shows stronger attenuation and the frequencies ofthe stop band are easier to manipulate; however, the width of the stop bands is lim-ited and can be increased by increasing the damping coefficient of the resonator. Ina later publication, Claeys et al. successfully designed and manufactured an acous-tic enclosure using the concept of local resonators. Their experiments showed thatthe addition of local resonators had increased the insertion loss of the enclosure atthe frequency ranges that were expected from their Bloch wave analyses [10].Recently, Song et al. [63] investigated STL improvement by the use of localresonators for a numerical example of a honey comb sandwich panel. However,they simplified the problem by homogenizing the properties of the core. It wasfound that the addition of local resonators improved the overall performance of the11panel.In conclusion, resonance stop bands show a high potential to improve the vi-brational, and by extension vibro-acoustic, behavior of periodic panels. They areeasily tunable to the frequencies of interest, however their application is limitedby two factors: 1) The attenuation frequency bands are narrow in width meaningthey are not efficient for wide-band noise problems such as the case of aircrafts.2) Because of their complex and nontrivial geometry the manufacturing techniquesare currently limited to rapid prototyping techniques such as metal powder Selec-tive Laser Sintering (SLS), which is currently not applicable in industrial scales.Therefore, in this thesis only the effect of interference stop bands caused by Braggscattering in the periodic truss-lattice panels is considered.1.5 Truss-Lattice Core StructuresThe basic idea behind a sandwich construction is that by removing material fromparts that bear little stress, significant material and weight can be saved. There-fore, cellular materials are good candidates for the core material, thanks to theirlightness.Evans et al. [24] compared FCC aluminum lattice to aluminium foams of dif-ferent densities and found that lattice materials have superior moduli compared tofoam materials. In another study, Wallach [67] compared the stiffness and strengthof a particular lattice material (called the Lattice Block) with a range of differentmetal foams with different relative density and showed the stiffness and strengthof the lattice were far superior in specific directions and comparable in the others[67].In 2001, Deshpande et al. [19] studied the strength and stiffness of the octet-truss lattice made from an aluminum casting alloy and found that properties of thisstretch dominated structure compare favorably with the corresponding propertiesof metallic foams.Wicks and Hutchinson [70] compared the optimized truss-lattice cores and op-timized honeycomb cores. The comparison showed that the honeycomb sandwichpanels optimized for a specific stiffness were slightly lighter than their truss-latticecore counterparts. However the truss-lattice core has other advantages that need to12be considered. These materials can also be designed to efficiently bear loads andconduct heat. The open lattice design also allows for passages of fluid to extractheat. It also adds more space for distributing deformations that will enhance theenergy absorption capability of the lattice [67]. Because of their periodicity, trusslattice cores have dynamic properties which are of particular importance when itcomes to sound and vibration control. Two of these dynamic characteristics ofperiodic structures are bandgaps and partial bandgaps. Bandgaps/partial bandgapsare frequency ranges through which no/limited wave vectors can propagate. Thephysics behind bandgaps and their potential application in quiet sandwich panelsis discussed in Section STL for a Sandwich PanelThere are a few different parameters defined for quantifying the acoustic perfor-mance of sandwich panels. Parameters such as STL, transmission coefficient,Sound Reduction Index (R), insertion loss and Sound Transmission Class (STC)are used in different acoustics applications but the STL is the one that is mostwidely used and, therefore, is the focus of this thesis. In this section, various formsof deterministic and probabilistic models and numerical solutions to evaluate STLare described, together with their merits and weaknesses.1.6.1 Experimental MeasurementA number of experimental methods exist for measuring the STL of a panel. Thefirst method involves the use of an impedance tube, however, it is not discussedhere because it can only measure the STL for waves that are normally incident onthe panel and it cannot capture the effects of the boundaries in an actual full sizepanel. The most accurate and widely-used way of measuring the STL of a panelis using a sound transmission suite 3. Depending on the standard being used, thereare different configurations considerable for the sound transmission suite, but whatall these configurations have in common are two adjacent rooms with an openingin-between for mounting the panel. Sound waves are generated on one side of thepanel (source room) and the amount of sound energy that is transmitted through3Refer to ASTM E90-9 or ISO 10140-5 standards for more information.13the panel to the other side (receiver room) determines the STL for the panel. Forexample, in ASTM E90-09 both rooms need to be reverberant and have enoughvolume and sufficient area of diffusers to ascertain field diffusivity on both sidesof the panel. However, SAE J1400 also caters for the case of a reverberant sourceroom adjacent to an anechoic receiver room [7] which we will later use as thebaseline for our Finite Element Method (FEM) modelling.1.6.2 Analytical ModelingMathematical models for simple STL problems such as planar infinite monolithicsingle and double partitions [46, 47], as well as, highly idealized sandwich panelshave been around for decades now.Kurtze and Watters[40] were among the first to study STL through sandwichpanels. By considering the effect of shear waves on the velocity dispersion offlexural waves in the panel, they were able to model sandwich panels with highstiffness-through-thickness (negligible dilatational deformations) and showed thatit is possible to increase the coincidence frequency by decreasing the core shearstiffness. They did not model the sound transmission problem but were able toconfirm the expected increase in coincidence frequency and improved STL, in theirexperiments. Moore and Lyon [52] extended this study to flexible cores by simpli-fying the deformations as a superposition of symmetric and anti-symmetric linearmodes. With these assumptions, they were able to calculate the STL for an infinitesandwich panel and propose a new panel design known as the mode-cancellingpanel. However, comparison with their experimental results for honeycomb panelsdoes not show good agreement. Our FEA simulations reveal some of the possiblereasons for this disagreement: 1) their analysis is for infinite panels, however, inthe experiment the panels are baffled and this results in standing-wave resonancesand anti-resonances which change the STL curves; 2) linearly varying deforma-tions are assumed through the thickness of the panel, although, FEA shows this isnot adequate and higher-order functions are needed; 3) they assume homogeneous(smeared) material properties for the honeycomb core and the validity of smearedmodels is restricted to frequencies for which the wavelengths are larger than theunit cell sizes [28], for the honeycomb panel tested these frequencies are below141000 Hz according to the FEA simulations. This discussion reveals the difficul-ties in finding an accurate analytical solution to the problem of sound transmissionthrough sandwich structures.1.6.3 Bloch Wave AnalysisAs described in Section 1.4 sandwich panels with truss-lattice cores are the focusof this study. These panels can be categorized as periodic structures, meaning thatby repeating a selected unit cell along specific directions the whole structure of thepanel can be constructed. The wave propagation equations in periodic structuresmay be simplified by making use of their intrinsic periodicity. Mead [50] used thistechnique to identify the high-radiation supersonic wave propagation zone in plateswith regular stiffening. This method is explained in more detail in Chapter Finite Element Analysis (FEA)The exact details of the Finite Element Modeling (FEM) carried out in this thesiswill be explained in Chapter 3. Here, only its merits and limitations are discussedin comparison with other computational techniques.In principle, it is possible to construct precise, detailed mathematical modelsof the sandwich panels coupled with the fluid medium, and to apply FEA to esti-mate the linear steady-state dynamics response to external harmonic excitation, atall points at each frequency. This makes FEA a relatively simple, straight-forwardand therefore popular approach for modeling sound transmission through complexstructures. There are two major limitations to this method, however. First being thecomputational cost; the size of the finite elements used, needs to be considerablysmaller than the minimum structural wavelength, in any component. As a result,the model size increases with frequency to a power of three for 3D problems [27].One way to reduce the required number of elements is by using the Spectral FiniteElement Method [59]. In this method, the shape functions for different elementsare solved for according to the dynamic loads associated with harmonic oscilla-tions at each frequency. Accordingly, the assembled mass and stiffness matriceswill also be frequency-dependent. As a result, in most cases of steady-state dy-namic problems further mesh refinement may not be required when the excitation15frequency increases.The second limitation concerns the predictive uncertainty of high-frequencyvibration response. We inevitably lack precision in modeling damping and jointflexibility. Dynamic properties vary infinitesimally from day to day and with op-erational conditions. Lower-order mode shapes and natural frequencies are ratherinsensitive to small changes in these structural details, which can nevertheless quitesignificantly alter the high-frequency modes. Hence, there exists irreducible uncer-tainty concerning the high-order natural modes and frequencies as a result of thelack of complete and precise knowledge of the dynamic properties of any modeledsystem. In addition, the precise form of the excitation forces is rarely known. Insuch cases, it is not appropriate to use FEA. Instead, alternative modeling philoso-phies and techniques should be used such as the Statistical Energy Analysis.After a careful consideration of the rate of change of the response of the panelversus frequency, no signs of a stochastic response could be identified which obvi-ates the need for a statistical approach. In fact, as will be explained in Chapter 3,the whole behaviour of the panel could be captured using 81 points throughout thefrequency range. As a result, for the range of frequencies and the panels that westudied FEA proves to be a reliable option.1.7 Research Objectives and Thesis OutlineStructure-borne sound transmission poses a fundamental challenge to be addressedin the design of lightweight and stiff structural sandwich panels used in aeronau-tical, automotive and naval structures. A sandwich panel can be viewed as a mul-tilayered structure comprising two or more thin face sheets of high strength andstiffness separated by a thick low density core. An ideal core material should havehigh out-of-plane stiffness to maintain constant separation between the face sheetsand high transverse shear stiffness of the core to prevent relative in-plane sliding ofthe face sheets [2, 72]. This work deals with sandwich panels with two face sheetsand a spatially periodic truss lattice core. Lattice cores, with an open micro archi-tecture offer many multifunctional features not available in a sandwich panel madeusing a regular closed-cell hexagonal honeycomb core [24, 33, 66, 68]. A detailedreview of lattice materials can be found in [29]. Our principal concern here is the16acoustic properties, in particular, sound transmission loss (STL) properties of suchadvanced multi functional sandwich panels.By virtue of reciprocity principle one observes that a panel that responds poorlyto sound excitation also radiates sound poorly. Hence, panels with high soundtransmission loss (STL) also radiate poorly [14]. When the bending wavelength ofa panel (λb) matches the incident acoustic wavelength (λ ) the panel radiates soundefficiently. This undesirable coincidence phenomenon, a spatial analog of tempo-ral resonance phenomenon familiar in structural vibrations, is a phase matchingcondition due to the phase velocity matching between the structural waves prop-agating within the panel and the acoustic pressure waves incident on (or radiatedfrom) the panel. The critical frequency of a panel is proportional to the ratio ofmass per unit area and the bending stiffness [14, 28]. Hence, structurally efficientlight and stiff panels with low critical frequencies are acoustically poor [55]. Sim-ilarly, a limp and heavy panel is acoustically excellent but structurally useless inan aircraft. Due to their high stiffness and low mass, sandwich panels have lowcritical frequencies compared to a monolithic panel of the same mass, and hencesuffer from acoustic problems [17]. This problem is more acute with compos-ite sandwich panels, requiring acoustic liner treatments and poroelastic materialbased acoustic absorbers [1]. The added weight of acoustic treatments can under-cut the lightweight advantage of sandwich construction. Recent availability of coretopologies, particularly of truss lattice type, offer the possibility of improving STLproperties of a sandwich panel. Moreover, the ability to fabricate such truss latticearchitectures using composite materials offers the potential to tailor their vibroa-coustic properties, structural wave dispersion in particular, without compromisingon the structural requirement of high stiffness to mass ratio.Despite its many structural benefits, the high stiffness to mass ratio impairs asandwich panel’s ability to reduce noise and vibration. Also, supersonic structuralwaves start propagating in the panels at frequencies often within the audible range.This type of wave radiates noise much more efficiently, which results in even worseacoustic performance. For sandwich panels with homogeneous cores, there areonly few control parameters that determine the wave propagation behaviour of thepanel. The same control parameters will directly influence and often significantlydeteriorate the static structural performance of the panel. On the other hand, panels17with truss lattice cores have several geometrical parameters that can control thewave propagation, as well as the static stiffness and toughness of the panels. Thisstudy focuses on sandwich panels with truss-lattice cores, establishes a frameworkfor evaluating their structural and acoustic performance and assesses their benefitsand potential applications.The conventional computational approach to assess the acoustic performanceof a panel is through numerical modeling of the whole structure, consisting ofthousands of unit cells, and a large volume of fluid surrounding it. Solving suchmodels would require significant time and computational cost and would take daysto complete. However, in the wave propagation zone, (Figure 1.2), it is possible tomake use of the periodicity of the structure and reduce the model size to a singleunit cell. To the best of the author’s knowledge, in 3D this procedure has only beendone for sandwich panels with very simple core geometries, such as orthogonalrib-stiffeners [63].1.7.1 Research ObjectivesAfter a review of the literature, two objectives for this thesis are identified:1. To develop a computationally efficient numerical framework to evaluate theacoustic and vibrational analysis of 2D periodic sandwich panels2. To compare the actual performance of these panels with the existing designsand make suggestions for improvements1.7.2 Thesis OutlineChapter 2 develops a versatile framework for studying the propagation of wavesin periodic panels using the Floquet-Bloch theorem and Finite Element Modeling(FEM) in a semi-computational approach.In Chapter 3, the Finite Element Analysis (FEA) of the sound transmissionproblem will be explained in detail. The details of the model constructed in Abaqusand the governing equations that need to be solved computationally are defined.Case studies are also presented and the convergence requirements are investigated.Chapter 4 discusses the results obtained from Chapters 2 and 3, compares thepanels from both acoustic and structural view points and summarizes the findings18from this study. The final outcomes are stated and the developed methods arecarefully critiqued. Finally, the future areas of research and the potential extensionsto this project are defined.19Chapter 2Elastic Wave Propagation inTruss-Lattice Panels2.1 IntroductionAs mentioned in the previous chapter, at high frequencies (multiple times the fun-damental resonance frequency of the panel) reflections from the boundaries be-come negligible and the modal density increases. As a result, traveling waves startto propagate in the panel. Therefore, the STL of a finite panel at high-frequencies,primarily depends upon the wave propagation properties of the panel. Finite Ele-ment modeling of a sandwich panel with a truss-lattice core requires a significantlylarge number of elements, which will result in costly computations that can takedays to finish. However, thanks to the intrinsic periodicity in truss-lattice structuresthis model can be reduced to a single unit cell by considering the correct force anddisplacement boundary conditions. This will significantly reduce the number ofelements and therefore the computation time. In this chapter we will develop aframework which combines FEA and Bloch wave analysis to give us a full pictureof the wave propagation characteristics of almost any periodic sandwich structureinfinite in extent. These characteristics will be interpreted to qualitatively evaluatethe structure-borne noise in finite sandwich panels at high frequencies at whichwaves start propagating (Section 1.3).The problem of wave transmission in a monolithic plate was studied decades20ago [57]. Three wave types are conceivable in a monolithic plate; the actual vibra-tions of the plate in the general case can be considered as the sum of these threewaves. The first type of wave, called the longitudinal wave, involves harmonic ex-tension and contraction of the sections in the plane of the plate itself. When thickerplates are studied the Poisson effects become considerable; as a result, the in-planedeformations are accompanied by relatively small out-of-plane deformations lead-ing to a quasi-longitudinal type of wave motion [28]. The transverse (shear) wavesare the second possible type of waves in a plate. Just as the name implies, the sec-tions of the plate are displaced transversely and there are no in-plane deformationsinvolved. The phase velocities of the quasi-longitudinal and shear waves are non-dispersive (independent of frequency) with the shear wave speed being about 0.6of the longitudinal wave speed for most structural materials [28]. The last wavetype is called the bending wave. Unlike the first two wave types, for this one, thein-plane deformations and transverse deflections are coupled by means of the rota-tions in the cross-section of the plate. It is worth recalling that in a pure bendingdeformation the cross-sectional planes will remain plane before and after the defor-mation. Solving for the harmonic solution to the wave equation, the bending wavephase speeds are found to be equal to cb = ω1/2(D/m)1/4, ω being the frequency,D the bending stiffness and m the mass per unit area of the plate. It can be seen thatthe bending waves show dispersive behavior with their phase velocity increasingwith frequency. Bending waves are of the greatest significance in the process offluid-structure interaction at audio frequencies. This is because they involve sub-stantial transverse displacement which can effectively disturb the fluid region, alsobecause of the coincidence effects facilitating the energy exchange [28].When studying the sandwich panels with homogeneous cores (such as foamand rubber), the discontinuities across the section of the panel require more param-eters to be involved in the solution. Kurtze and Watters [40] studied only the wavetype in which the top and bottom faces exhibited in-phase motion; known as anti-symmetric modes (flexural waves). Moore and Lyon [52] also included the effectof out-of-phase motion of the face sheets known as symmetric modes (breathingwaves), but they assumed that the deformations change as a linear function in thethickness direction which is not accurate considering that the sandwich panels thatthey studied had relatively soft core material. Frostig and Baruch [31] introduced21a rigorous approach to incorporate higher-order deformations inside the core butthe study was limited only to sandwich beams and not sandwich panels. Recently,Liu and Bhattacharya [44] conducted a thorough study of the propagation and dis-persion of waves in sandwich panels by transforming the wave equations in eachcomponent of the panel to a Hamiltonian system and then solving the system ofequations using a transfer matrix approach. Shorter [61] applied the spectral finiteelement method to study the propagation of waves in viscoelastic laminates andfound that several types of wave are in play other than the ones described here fora monolithic panel.The problem becomes more complex when we consider periodic cores. Thediscontinuities in the core can significantly complicate solving the wave transmis-sion problem inside the sandwich panel; however, there are methods to study thisproblem based on the fact that these structures are periodic. El-Raheb [22] studiedthis problem for truss-lattice sandwich beams using the transfer matrix method.The method that we are applying is based on Bloch Wave propagation [5]. Phaniet al. [56] used this technique to study the propagation of waves in two-dimensionaltruss-lattice structures. Xin and Lu [71] employed this method to study the trans-mission of sounds through rib-stiffened sandwich panels. In this chapter we willapply Bloch Wave propagation to sandwich panels with truss-lattice cores and in-troduce a platform we developed in MATLAB to simplify this process.2.2 Bloch Wave Propagation in Periodic StructuresAn infinite two dimensional sandwich panel with a truss lattice core can be visual-ized as a periodic structure obtained by repeating a single unit cell in two directions.Wave propagation in such lattice structures has been studied in solid state physicsusing Bloch theory, see [4] and Chapter 8 in [3] for a formal proof of Bloch’s the-orem. In-plane wave propagation in two dimensional lattice materials using Blochtheory has been reported in [56] and Bloch theory has been applied to periodicstructures earlier, see [42, 49, 50, 59] for example. Before proceeding to Bloch’stheorem, it is worth reviewing relevant concepts from solid state physics. A latticecan be visualized as a collection of points, called lattice points, and these points arespecified by basis vectors, not necessarily of unit length. Upon selecting a suitable22unit cell, the entire direct lattice can be obtained by tessellating the unit cell alongthe basis vectors ei, i = 1,2. In the context of a finite element model of the unitcell of a sandwich panel, as shown in Figure 2.1, lattice points are the nodes of thefinite element model.With reference to the chosen unit cell of a two dimensional lattice, such as thatshown in Figure 2.1, let the integer pair (n1,n2) identify any other cell obtainedby n1 translations along the e1 direction and n2 translations along the e2 direction.The point in the cell (n1,n2), corresponding to the jth point in the reference unitcell, is denoted by the vector r = r j+n1e1+n2e2. According to Bloch’s theoremthe displacement at the jth point in any cell identified by the integer pair (n1,n2) inthe direct lattice basis of a two dimensional lattice is given byq(r) = q(r j)ek·(r−r j) = q(r j)e(k1n1+k2n2). (2.1)Here, k1 = δ1+ iµ1 and k2 = δ2+ iµ2 represent the components of the wavevectork along the e1 and e2 vectors, that is k1 = k · e1 and k2 = k · e2. The real part δ andthe imaginary part µ are called the attenuation and phase constants, respectively.The real part is a measure of the attenuation of a wave as it progresses from oneunit cell to the next. For waves propagating without attenuation, the real part iszero and the components of the wave vector reduce to k1 = iµ1 and k2 = iµ2. Theimaginary part or the phase constant is a measure of the phase change across oneunit cell.It is convenient to define a reciprocal lattice in the wavevector space (k-space)such that the basis vectors of the direct and reciprocal lattice satisfy:ei · e∗j = δi j (2.2)where ei denote the basis vectors of the direct lattice and e∗j denote the basis ofreciprocal lattice, δi j is the Kronecker delta function and the symbol · denotes thescalar or dot product. For a two-dimensional lattice the subscripts i and j take theinteger values 1 and 2.The wavevectors can be expressed in terms of the reciprocal lattice basis e∗i .Since the reciprocal lattice is also periodic, one can restrict the wavevectors to cer-23tain regions in the reciprocal lattice called Brillouin Zones [5]. The wavevectorsare restricted to the edges of the irreducible part of the first Brillouin zone to ex-plore band-gaps, since the band extrema almost always occur along the boundariesof the irreducible zone [3, 39].The equations of motion for the unit cell of a two dimensional lattice can beexpressed as:Mq¨+Kq = f . (2.3)where the matrices M , K denote the mass and stiffness matrix of the unit cell,respectively. The vectors q, q¨ and f respectively denote the displacements, accel-erations and force vectors corresponding to the degrees of freedom of the system.Having formulated the equations of motion of a unit cell, the propagation of planarharmonic waves at a radial frequency ω within the entire lattice can be investigatedby invoking Bloch’s theorem. The equations of motion in Equation 2.3 follow as,[−ω2M+K]q = f or Dq = f , D = [−ω2M+K] (2.4)where the dynamic stiffness D reduces to the static stiffness at zero frequency.In order to simplify these equations and solve the wave equation the q matrixis partitioned into the components corresponding to the boundary nodes (edgesand corners of the unit cell) and the components corresponding to the degrees offreedom of the internal nodes. The By virtue of Bloch’s theorem the followingrelationships between the displacements, q, and forces, f , are obtained:qr = ek1ql, qt = ek2qb,qrb = ek1qlb, qrt = ek1+k2qlb, qlt = ek2qlbf r =−ek1 f l, f t =−ek2 f b,f rt + ek1 f lt + ek2 f rb+ ek1+k2 f lb = 0.(2.5)where the subscripts l, r, b, t and i respectively denote the displacements corre-sponding to the left, right, bottom, top and internal nodes of a generic unit cell, asshown in Figure 2.1. The displacements of the corner nodes are denoted by doublesubscripts: for example, lb denotes the left bottom corner.24Using the above relationships one can define the following transformation:q = T q˜,T =I 0 0 0Iek1 0 0 00 I 0 00 Iek2 0 00 0 I 00 0 Iek1 00 0 Iek2 00 0 Ie(k1+k2) 00 0 0 I, q˜ =qlqbqlbqi .(2.6)where q˜ denote the displacements of the nodes in the Bloch reduced coordinates.Now substitute the transformation given by Equation 2.6 into the governing equa-tions of motion in Equation 2.4 and pre-multiply the resulting equation with T H toenforce force equilibrium [42]. One obtains the following governing equations inthe reduced coordinates:D˜q˜ = f˜ , D˜ = T HDT , f˜ = T H f . (2.7)where the superscript H denotes the Hermitian transpose. For a plane wave propa-gating without attenuation in the x−y plane, the propagation constants along the xand y directions are k1 = iµ1 and k2 = iµ2. For free wave motion ( f = 0) the aboveequation can be written in the frequency domain to give the following eigenvalueproblem,D˜(k1,k2,ω)q˜ = 0. (2.8)Any triad (k1,k2,ω) obtained by solving the eigenvalue problem in Equation 2.8represent a plane wave propagating at frequency ω .In the characteristic equation of the eigenvalue problem defined by Equation 2.8there exist three unknowns: the two propagation constants k1, k2 which are com-plex in general and the frequency of wave propagation ω which is real since the25matrix Dr in the eigenvalue problem is Hermitian. At least two of the three un-knowns have to be specified to obtain the third. For wave motion without atten-uation the propagation constants are purely imaginary of the form k1 = iµ1 andk2 = iµ2. In this case one obtains the frequencies of wave propagation as a solutionto the linear algebraic eigenvalue problem defined in Equation 2.8 for each pair ofphase constants (µ1,µ2). The solution is a surface, called the dispersion surface, inthe ω− k1− k2 coordinates. There exist as many surfaces as there are eigenvaluesof the problem in Equation 2.8. If two surfaces do not overlap each other then thereis a gap along the ω axis in which no wave motion occurs. This gap between dis-persion surfaces is called the band-gap in the solid-state physics literature [39] andthe stop band in structural dynamics [49]. For all frequencies on the phase constantsurface, wave motion can occur and hence the frequency range occupied by thesesurfaces is a pass band. Furthermore, the normal to the phase constant surface atany point gives the Poynting vector or group velocity, and this indicates the speedand direction of energy flow [5].The procedure described below was followed to compute dispersion curves ofthe sandwich panels:1. Construct the finite element model of a unit cell of the lattice and label theedge degrees of freedom, as shown in Figure 2.1. ABAQUS[16] finite ele-ment package was used to generate the mass and stiffness matrices.2. The ABAQUS generated matrices are exported into a MATLAB environ-ment to solve the eigenvalue problem in Equation 2.8 for a specified pathΓ−X −M−Γ along the edges of the irreducible part of the first Brillouinzone portrayed in Figure 2.1(f).26ttbbl rltlbltlbrbrtrbrtXYZ(a) (b)Figure 2.1: The Kagome lattice (b) made by tessellating the unit cell (a) alongx and y. Typical finite element model of the unit cell of a lattice (kagomecore) sandwich panel with partitioned degrees of freedom labelled. Dis-placements at nodes labelled as r, l, b and t respectively correspond tothe right, left, bottom, and top degrees of freedom. Double subscriptssuch as lb represent the left-bottom etc. Note that nodes on some edgesof the model are not labelled for the sake of clarity. q is a displace-ment vector of the partitioned degrees of freedom that are representedin the subscript. A reference unit cell having a point A, with positionvector p and a displacement vector q(p) is shown in (b). Another pointB, similar to point A in an arbitrary unit cell has a position vector rand a displacement vector q(r). This unit cell is identified by the label(n1,n2), which represents n1 unit cell translations along x and n2 transla-tions along y from the reference unit cell. Using Floquet-bloch theorem,displacements of this arbitrary unit cell can be represented in terms ofdisplacements of the reference unit cell [8]27(a) (b)(c) (d)(e) (f)Figure 2.2: Selected sandwich panel core designs studied: (a) Kurtze andWatters panel, (b) Tetrahedral core, (c) Double pyramidal core, (d) Pyra-midal core, (e) Kagome core and (f) Hexagonal Honeycomb core. Allpanels have identical face sheets, height and mass. Panel size 1.5 m× 1.5 m × 0.038 m. Individual struts of all cores are of circular crosssection. Strut dimensions are chosen to maintain same mass across allcores based on their respective relative density, see Table A.1. portionof the top panel is removed to show the core.28(a) (b)0 5 10Ω (kHz)  Γ X M Γlog(ry)−3−2−10120 5 10Ω (kHz)  Γ X M Γlog(ry)−2−10123(c) (d)0 5 10Ω (kHz)  Γ X M Γlog(ry)−4−20240 5 10Ω (kHz)  Γ X M Γlog(ry)−3−2−1012(e) (f)0 5 10Γ X M ΓΩ (kHz)  −3−2−101Figure 2.3: Dispersion curves of the sandwich panels for different cores cal-culated using Bloch theory. Curves associated with four truss-latticecores and the hexagonal core are shown above: (a) Tetrahedral, (b) Dou-ble pyramidal, (c) Pyramidal, (d) Kagome and (e) Hexagonal. The wavevector locus followed in calculating the dispersion curve for each topol-ogy is shown in (f) where the irreducible part of the first Brillouin zoneis shown as a shaded region. The following values for the symmetrypoints are used: Γ= (0,0), X = ( piLx ,0), M = (piLx, piLy ), where, Lx and Lyare respectively the length and width of the unit cell. Note that all coreshave identical mass and the differences in the dispersion behaviour isdue to the variations in topology governed dynamic stiffness. The longwave length asymptotes near the origin (Γ point) indicate the group ve-locities related to the stiffness. We observe that the steepest dispersioncurve (first branch) associated with hexagonal core in (e) suggest that itis the stiffest topology in transverse shear. At the other extreme Kagomecore in (d) exhibits lowest stiffness for transverse shear wave propaga-tion. ry =uyrms√u2zrms+u2xrms, where uxrms, uyrms and uzrms are the root meansquared average displacements along the x, y, and z axes with y axistaken as the normal to the face sheets.292.3 Sandwich PanelsThe sandwich panels selected for comparison are shown in Figure 2.2. Four trusslattice core sandwich panels Figure 2.2 are compared with Kurtze and Watters’panel. All panels have solid face sheets made of square steel sheets of length 1.5 mand thickness 3.8 mm. All panels have the same mass and size. Effective propertiesof the core topologies are compared in Table A.1. It can be observed that the tradi-tional hexagonal honeycomb has the highest transverse shear modulus and Young’smodulus for a given relative density ρ¯ , defined as the ratio of the densities of theporous core to that of the density of the parent material. Further, it may be ob-served that the tetrahedral core has the highest transverse Young’s modulus amongtruss lattices followed by double pyramidal and pyramidal (equal) and Kagomecore. It should be mentioned that the Kagome core as defined in this work doesnot contain truss elements on the top and bottom faces to accommodate solid facesheets. In regards to shear modulus, hexagonal honeycomb has the highest valueand Kagome has the lowest for a given relative density. These marked differencesin effective macroscopic elastic moduli arising from differences in topologies ofthe core will manifest later in the static stiffness properties as well as the wavepropagation response.2.4 Results and DiscussionDispersion curves for wave propagation are computed for each panel in Figure 2.2using the procedure described in Section 2.2. For each point along the locusΓ− X −M − Γ the eigenvalue problem in Equation 2.8 is solved to obtain thepropagating frequencies associated with a propagation direction specified by thewavevector components along the locus. The results are shown in Figure 2.3 forfrequency up to 10 kHz. For the purposes of sound transmission waves with signif-icant out-of-plane (along the normal to the face sheets of the panel) displacementcomponents are significant. Thus, it is useful to quantify the contribution of wavesin a given direction, or frequency, along the normal direction to the face sheets. Aparameter ry is introduced to this end and its value is calculated for each point onthe dispersion curve and shown as a contour plot. ry =uyrms√u2zrms+u2xrms, where uxrms,uyrms and uzrms are the root mean squared average displacements along the x, y, and30z axes with y axis taken as the normal to the face sheets. The higher the magnitudeof ry, the higher the out-of-plane displacement magnitude. It can be noticed thatthe number of dispersion branches (or modal density) is highest in Figure 2.2(d)for Kagome core whereas the hexagonal honeycomb in Figure 2.2(e) exhibits neardispersionless features. This suggests that it is the stiffest structure, but the same isproblematic from a coincidence perspective. Recognizing that the long wavelengthasymptote of the first branch (shear wave) is related to the group velocity, whichfor a given mass depends on the transverse shear modulus, we can conclude thatKagome core has the least slope and hence the coincidence of acoustic waves in thefluid with the transverse shear waves (first branch) in the panel occurs at a higherfrequency compared to the hexagonal honeycomb. No complete band gaps exist.However, partial band gaps are evident for the tetrahedral core in Figure 2.2(a)in the 5 kHZ region and at a much lower frequency region (between 2 kHz and3.5 kHZ) for the Kagome core in Figure 2.2(d). In contrast, the dispersion curvesfor the double pyramidal lattice in Figure 2.2(b) do not show any partial band gapswithin 10 kHz region. As we shall see in Chapter 4, these partial band gaps providesignificant enhancements to the STL.31Chapter 3Numerical Simulation of SoundTransmission through SandwichPanelsIn this capter, Finite Element Analysis (FEA) of the sound transmission problemis explained. The governing equations that need to be solved computationally andthe details of the model constructed in Abaqus are defined. A step-by-step de-velopment and validation of the modeling procedure for calculating the structuralresponse and STL of sandwich panels is presented, together with the checks to ver-ify the convergence of the results. The results will be presented for the geometriesshown in Figure Transmission of Sound through Panels: Physics andGoverning EquationsTransmission of sound involves two types of media; a solid structure which is inthis case a fairly complex sandwich panel, and a medium of fluid which is usuallyair. Each region of the fluid or solid medium has its own specific material law, andneeds to be discretized in finite elements with appropriate size and aspect ratio.Also, the interaction between different regions needs to be defined at the interfaces.32V, Sn(r)n(r)n(r)Figure 3.1: A volume V of fluid with surface area S and the normal vectorsas defined3.1.1 Structural ModelIn general, the dynamics of a deformable solid medium is governed by the equa-tions of elasticity. For linear elastic materials, these equations consist of Newton’ssecond law (Equation 3.1), Hooke’s law (Equation 3.2) and the strain-displacementrelation (Equation 3.3):∇ ·σ +F = ρ u¨ (3.1)σ =C : ε (3.2)ε =12[∇u+(∇u)T ] (3.3)where σ is the Cauchy stress tensor, ε is the infinitesimal strain tensor, C is thestiffness tensor, F is the external force per unit volume (body force) and ρ is themass density. Here : is the inner product of the two tensors (summation over re-peated indices is implied as in A : B = Ai jBi j). These equations can be simpli-fied depending on the dimensionality (shell, beam etc.) of the geometries that arestudied. Refer to Chapter 12 of [58] for more information on how FEA makes itpossible to solve these equations in a complex structure.3.1.2 Acoustic ModelConsidering the conservation of mass for an element inside the fluid volume (Fig-ure 3.1) with pressure P(r, t), particle velocity V(r, t) and density ρ(r, t), leads to33the continuity equation:∂ρ∂ t=−∇.(ρV) (3.4)Also, writing Newton’s second law for the same volume we obtain:−∮S(PndS) =∫V(ρdVdtdV ) (3.5)where the left hand side is the resultant external force and the right hand side repre-sents the acceleration of the volume’s center of mass. Applying Gauss’s divergencetheorem to the left hand side of the equation, and linearizing the total differentialon the right hand side (only valid for small oscillations) Euler’s equation follows:ρ∂V∂ t=−∇P (3.6)Also, assuming small perturbations in pressure, a linear material law may be usedfor the fluidP= c2ρ. (3.7)where c is the speed of sound. Now, by taking the second derivative of Equa-tion 3.7 with respect to time and applying linearized versions of Equation 3.4 andEquation 3.6, we find the first-order approximation with respect to the perturba-tions in pressure, density and particle velocity:∂ 2P∂ t2= c2∂ 2ρ∂ t2=−c2ρ0∇.(∂V∂ t ) = c2∇2P. (3.8)Hence, we obtain the wave equation:∂ 2P∂ t2= c2∇2P (3.9)Substituting a time-harmonic solution P(r, t)=P0+ p(r)e jωt+φ we arrive at Helmholtzequation [37]. (Here P0 represents the atmospheric air pressure, p the amplitudeof the pressure changes, and ω and φ are the frequency and phase of the pressurechange in space.)∇2p+ k2p= 0 , (3.10)34where k = ωc is the wave number. Since p is the only unknown, solving for thepressure level amplitude is only a matter of knowing the boundary conditions, forinstance, knowing the pressure levels at all boundary surfaces [16].3.1.3 Coupling Structural and Acoustic ModelsModeling a fluid-structure interaction problem requires simultaneously solving forthe structural dynamics equations (3.1, 3.2, 3.3), the fluid domain equation (ei-ther Equation 3.9 or Equation 3.10) and the following equation which defines theinteraction between the two domains:∂P∂n=−ρ ∂2un∂ t2, (3.11)where un indicates the displacement of the solid surface along the direction n nor-mal to the surface of the structure and P is the pressure acting on the surface of thepanel face sheet on the incident side.3.1.4 Finite Element ImplementationPartitioning the fluid and solid regions into a finite number of elements makes itpossible to discretize the material law equations and form a system of equationswith degrees of freedom of each element as the unknowns. Solving such systemof equations for a sufficiently large number of elements with appropriate aspectratios, will give us the values for the degrees of freedom at each point inside thedomain. For linear solid elements, the displacements of the vertices (nodes) aredefined as the degrees of freedom; the reader is referred to Chapter 12 of [58] for amore detailed demonstration of this procedure. In the acoustic domain the pressureamplitude p is taken as the only degree of freedom [26].As mentioned before, the discretized fluid and solid domain equations are cou-pled by the interaction Equation 3.11 and need to be solved simultaneously. Thefaces of the acoustic elements in contact with the flexible solid medium are boundto match the deformations in the solid medium. Therefore, the transverse displace-ments of the plate elements produce volumetric acoustic excitation in the fluid.Likewise, the sound pressure inside the acoustic elements at the fluid-solid inter-35face represents the force excitation acting on the corresponding solid elements.See [28] or [37] for more details. In this study commercial FE package ABAQUSis used to accelerate computation of STL calculations. Other techniques, suchas boundary element methods and spectral methods, can also be used. Statisti-cal effects can be significant in high frequency regime [17], however they are notsignificant in the present study.3.2 FEM Procedure and ValidationTwo-dimensional realizations of real-world problems are not always realistic butbecause of their significantly lower computational cost and ease of visualizationthey are studied first. Later in this section we will find that 2D FEM shows promis-ing results for less complex problems such as vibration analysis for a sandwichbeam. However, for the analysis of more complex cases such as the actual soundtransmission through sandwich panels, a 2D model is not adequate and a 3D modelis needed to capture the features of the model that extend in three dimensions. Thepurpose of this section is to present validation of the modeling procedure using theresults already available in the literature.3.2.1 Validation of Structural Response ModelingEl-Raheb (1997) [22] calculated the frequency response of a polycarbonate trusslattice arch using a transfer matrix method. Figure 3.2 shows a typical arch thatcan be studied using this method. The corners of cells at the ends are connected toground with springs representing the flexibility of the supports. The lattice is ex-cited at the top members with point forces equivalent to a pressure loading and thesteady-state dynamic response is calculated. Abaqus steady-state dynamics FEAwas used to simulate the results for harmonic excition with the same loading aswas just described. B23 elements (two-dimensional beams with cubic formula-tion) were used for the structural members.Figures 3.3 and 3.4 show the steady state response at different frequencies forEl-Raheb [22] and our FEA simulations. We can see that the deformation patternsare the same for the faces; however, excessive deformations are observed in thecore elements. One possibility is that different deformation scale factors are used36Figure 3.2: El-Raheb’s a) arch configuration and b) unit cell for the curve [22]by El-Raheb [22] for face and core elements. To study the dynamic response overa wide range of frequencies an intensity function is defined:Irms = 20log10(vrms/v0) , v0 = 5×10(−5) mm/sFigure 3.5 and Figure 3.6 present the velocity intensity versus frequency resultsfor El-Raheb’s transfer matrix method, and our FEA simulations, which show goodagreement.3.2.2 Validation of Acoustic ModelingIn order to make sure that we are modeling the acoustic medium and the fluidboundaries correctly, we reproduced the results from a case study from Fahy [27].This is a two-dimensional analysis of the sound field in a reverberation cham-ber, modeled after the one built at Philips Research Laboratories in Eindhoven.AC2D8R (two-dimensional reduced-integration acoustic elements with 8 nodes)elements were used with the walls modeled as being perfectly rigid. The naturalfrequencies and mode-shapes from our FEA were compared with the results from[27]. The relative error for the natural frequencies is about 2% for the first fourmodes of the reverberation chamber. The slight difference may be because of thedifference in air density. We have taken air density to be 1.2 kg/m3; the value usedin [27] is not mentioned. The pressure distribution patterns are similar to the ones37Figure 3.3: Results of El-Raheb’s prediction of the steady-state response ofan arch to a harmonic excitation using the transfer matrix method[22]38Figure 3.4: Reproduced results of El-Raheb’s prediction of the steady-stateresponse of an arch to a harmonic excitation using FEA in AbaqusFigure 3.5: Velocity intensity (solid line) using the transfer matrix method forthe arch configuration. The two other curves are not discussed in thisdissertation.39Figure 3.6: Velocity intensity results from our FEA simulations [22] for thearch configurationin the book.Thus far, we are able to correctly model the structural and acoustic aspectsof the problem separately. Next, we will analyze the coupled structural acousticsystem. We will compare the results of our simulations with the numerical andexperimental ones that are already available in the literature and discuss the differ-ences.3.2.3 Structural-Acoustic Modeling in FEAIn this section, the general procedure for Finite Element modeling of the soundtransmission problem is explained and the results for two example cases are com-pared with the results available in the literature. As mentioned in Chapter 1, thereare different configurations considerable for a sound transmission suite. The oneconsidered in this study comprises a reverberant chamber (incident field) adjacentto an anechoic chamber (transmitted field) (Figure 3.7).The sound transmission problem is simulated in Abaqus as follows:• The diffuse field in the reverberant room is simulated using Abaqus DiffuseField interaction property. Figure 3.8 illustrates how this diffuse field load-40ReverberationChamberAnechoicChamberTest PanelFigure 3.7: Sound transmission suite configuration considered in this studying can be created in Abaqus: An equally-spaced N×N grid formed on theloaded surface is projected on an imaginary hemisphere which has the stand-off point as its center and the source point on its perimeter. Each node onthis final grid will be considered a source for planar waves with equal in-tensity vectors directed towards the standoff point and with random phases.In our applications results for N = 90 for 2D acoustic models and N = 60for 3D models proved to show divergence with respect to the diffuse fieldnodal grid points i.e the results did not change as we increased N. Also, thenet acoustic excitation force on the panel was calculated at different frequen-cies. It is non-zero at lower frequencies in which the characteristic acousticwavelength is comparable to the length of the structure; however, it rapidlyconverges to zero as the frequency increases.• The anechoic chamber is modeled using non-reflective interaction propertyon the outside surface of the fluid medium.• Surface-to-surface ties are used for implementing the fluid-structure interac-tion.• The incident power Pi is found from the reference pressure for the diffusefield interaction. The transmitted power Pt is calculated by integrating thereal part of the intensity vector (active intensity vector) over the external sur-41Source PointN  rowsN columnsStandoff PointSurface to be LoadedFigure 3.8: Diffuse field interaction implemented in a finite element model.The incident face of the panel (FE model with nodes shown as full cir-cles) is imagined to be subjected to excitation from point sources dis-tributed on a hemispherical surface. The phase of plane waves emanat-ing from each point source are uncorrelated [16]. Fluid forces due toincident pressure field are directly applied on the nodes of the structuralFE model. Fluid loading effects are neglected.face of the fluid volumeS . Equation (3.12) gives the relation for calculatingthe TL [7].TL= 10log10(PiPt) = 10log10(p2Ap4ρaCa∫S I.da) (3.12)Here are some general guidelines about the sizes that need to be set during themodeling:• At each frequency the structural mesh size in each member should be so finethat the minimum wavelength in the structure is many times (at least 6 times)larger than the mesh size.42• The distance between the non-reflective surface as the surface of the struc-ture should not be much smaller than the acoustic wavelength, 1/2 is a ratiorecommended in Abaqus manual [16]• The surface-to-surface tie between the fluid and structure requires that themesh size on the slave surface (fluid boundary) be smaller or equal to themesh size on the master surface (structure boundary).At this point, we consider reproducing the results from two STL case studiesfrom the literature [23, 40]. The objective is to find under which circumstancestwo-dimensional Finite Element models are relevant and whether it is worthwhileto consider 3D modeling.El-Raheb and Wagner 1997Sound transmission through a 2D truss-like partition is calculated with FEA andcompared with the experimental and numerical results from transfer matrix method.The configuration of the panel and the surrounding fluid is shown in Figure 3.9.The same elements that were described in sections 3.2.1 3.2.2 are used for struc-tural and fluid media (B23 and AC2D8R). The nodes on the faces of the truss-lattice structure are tied to the two semi-circular fluid domains at the interfaces.Non-reflective boundary conditions are applied at the circular boundaries. A dif-fuse field excitation is generated at the lower interface with the fluid using thediffuse field interaction property in Abaqus.Although they are within the same range and show similar trends (increasing),it can be seen (Figure 3.11) that our FE model is not in good agreement with El-Raheb’s transfer matrix results. It is worth noticing that the analytical model inFigure 3.10 is not in good agreement with the experimental one and fails to capturesome of the features that the FE model can do (e.g., the coincidence dip near 4000Hz.)In this example, the results were in agreement with the experiments. Using a2D model is appropriate here because the panels studied are extrusions of a 2Dtruss. As a result, in panels which are much more compliant to bending momentsin the plane of the corrugation than outside the plane of corrugation; hence, the out-of-plane deformations become negligible and a 2D approximation may be applied.43Figure 3.9: The panel and fluid medium configuration for FEAHowever, a 2D model is no more adequate for panels which have similar bendingstiffness in different directions or have features extending in three dimensions.Removing the fluid region on the incidence side of the panel did not change theresults significantly; therefore, from this point only the fluid region on the trans-mission side of the panel will be modeled. Also, we knew from [28] that the fluidloading effects are insignificant unless the fluid is a liquid.Kurtze and Watters (1959)Kurtze and Watters [40] introduced the idea of shifting the coincidence to higherfrequencies by designing cores that are much softer in shear than in compressionand tension. This concept is discussed in more detail in Chapter 1. The panelsthey tested were made of 1-mm thick steel face sheets. At the core, pieces ofwood fiber-board were arranged in such a way that the stiffness in planes parallel44Figure 3.10: Sound transmissiom loss through a truss-like panel: derivedfrom transfer matrix method (—) and experiment (- - -) [23]to the face of the sandwich panels is about 100 times less than their transversestiffness. In order to satisfy these requirements, orthogonal material propertieswere chosen for the core when setting up the FEA model. S8R (8-node reduced-integration shell element) and C3D20R (20-node reduced integration hexahedralsolid element) elements were used for the shell and core members respectively,and AC3D20R for the acoustic medium. The core and face sheet nodes were tiedtogether (Figure 3.12).A list of the material properties and model specifications used for FEA areshown in Table 3.1.Transverse wave speeds vt( fn) versus frequency Figure 3.13 can be experimen-tally determined if we know the natural frequencies fn and the wave length of thestanding waves at each mode λn for a finite sandwich beam as follows:vt( fn) = λn× fn (3.13)Using the same procedure for an identical sandwich beam in FEA, the velocity dis-persion curves can be derived (Figure 3.14), which show good agreement with theexperimental results (Figure 3.13) verifying the fact that correct material properties45Figure 3.11: Mesh convergence and results of the FEA simulations of thesound transmission loss through the truss-like panelwere assumed for the core material.The panel tested for sound transmission loss measurements was constructedsuch that Figure 3.13.a represents one cross-section and Figure 3.13.b another, per-pendicular to the first. As a result, the changes in both directions in the plane ofthe panel are equally important and a two-dimensional model cannot satisfy all therequirements. We will see in Section 3.5 that the results based on a 3D analysisshow good agreement with the experimental results.46XYZFigure 3.12: The setup and mesh for the Kurtze and Watters panelFigure 3.13: Velocity versus frequency found experimentally and core con-figuration for Kurtze and Watters [40]47Figure 3.14: Velocity versus frequency calculated from FEA frequency anal-ysis by knowing the standing wavelengths at natural frequenciesTable 3.1: List of material properties and model specifications used for FEAFace sheet and core (Steel)Density 7800 Kg/m3Young’s modulus 200 GPaPoisson’s ratio 0.249Structural damping 0.02Face sheet dimensionsThickness 1 mmLength 1.5 mWidth 1.5 mCore propertiesHeight (h) 38 mmRelative density (ρ¯) 0.0568483.3 Truss-Lattice Core Sandwich Panel ModelingThe same modeling procedure is repeated for the geometries in Figure 2.2. Theonly difference is that the core is constructed using B33 (3D beam elements withcubic formulation) elements. The convergence of all models is studied in twostages:1. Convergence with respect to mesh size: first, for a limited number of datapoints throughout our frequency range of interest, we model the sound trans-mission problem using two different cell sizes and we verify that the STLdifference is equal to or less than 2 dB (which is barely noticeable by thehuman auditory system [25]) (Figure 3.15.a).2. The number of data points: in order to make sure that enough points arechosen throughout our frequency range of interest, we take two different setsof data points and verify that the STL curves with 3rd octave-band averaging(3rd O.B.A.) show less than 2 dB difference [25] (Figure 3.15.b).49(a)102 103 104102030405060Frequency (Hz)STLd (dB)  10 mm15 mm(b)102 103 1040102030405060Frequency (Hz)STLd (dB)  81 points81 points 3rd−O.B.A.41 points41 points 3rd−O.B.A.Figure 3.15: a) Convergence with respect to mesh size for the Kagome corelattice. b) Convergence with respect to the number of data points forthe Kagome core lattice503.4 Static Stiffness ComparisonFor the same panels as described in the previous section, the stiffnesses are com-pared using a criterion used by Moore and Lyon [52]. A uniform pressure is appliedon the top surface of a sandwich panel simply supported at its edges. The deflectionof the center point of the top face is measured. In order to be able to compare theirstiffnesses the panels should share the same surface area. The stiffness associatedwith the panel is proportional to the ratio of the applied pressure to the center pointdeflections. In Figure 3.16, the stiffnesses of the panels relative to the monolithicpanel are plotted. The monolithic panel has the same mass and planar dimensions;however, it is a solid sheet of steel and is therefore thinner than the sandwich pan-els. It shows that the truss-lattice core panels can be up to 26 times stiffer than theKurtze and Watters’ panel and up to 170 times stiffer than a monolithic panel ofthe same mass, size and material.3.4.1 Why Not Constant Stiffness?Taking into account the significant difference between the static stiffness of differ-ent panels, one might question the reason behind taking panels with equal mass. Inan actual application, alternative panel designs should all satisfy a set of structuralrequirements including the stiffness; so the question is: why are we not consideringpanels sharing the same stiffness with different masses?The purpose of this study is to investigate the effect of the core topology on theSTL of a sandwich panel. In order to do so, one needs to isolate all other factorsthat influence the sound transmission characteristics of the panel. In the range thatwe are discussing, the STL is directly related to the mass of the structure. The masslaw, which gives a rough estimate of the STL, states that by doubling the mass ofa structure its STL will increase by 6 dB approximately. If one wishes to keep thestiffnesses constant; he also needs to change the mass of the structures for theirstiffnesses to become equal. The result would be panels with significantly differ-ent weights. According to the mass law the main factor determining the STL forthese panels would be their mass differences not the difference in their topologies.Therefore, in this study we chose to keep the mass of these structures the same.51Relative Panel Stiffness100101102Monolithic Kurtze-Watters Tetrahedral Double Pyramidal Pyramidal  Kagome Hexagonal Figure 3.16: Comparison of relative static stiffness of the panels studied. Thestiffness of each panel is divided by that of the monolithic panel stiff-ness, taken as the reference. Stiffness is evaluated at the central pointof each panel when all edges are simply supported.3.5 Results of the FEA Simulatoins of the SoundTransmission ProblemResults for diffuse-field STL simulations for the four geometries in Figure 2.2 areplotted in Figure 3.17 along with the results for Kurtze and Watters’ panel and amonolithic panel of the same mass and material. All sandwich panels have ex-actly the same mass, thickness and planar dimensions as the Kurtze and Watters’panel. We can see that the monolithic panel generally follows the mass law line.The panels with truss-lattice cores exhibit multiple peaks and troughs arising fromstructural resonances, coincidence frequencies and partial band gaps. The overallperformance of the panels appears to be very dependent on the geometry used as52Frequency (Hz)102 103 104STLd (dB)01020304050607080 Kurtze-WattersTetrahedralDouble PyramidalPyramidalKagomeMonolithicHexagonalFigure 3.17: A comparison of third-octave band averaged STL response ofthe Kurtze and Watters panel with five other lattice topologies. Fourfrequency points in each octave-band are used for the purposes of aver-aging. The stiffest panel (see Figure 3.16) has highest STL in the lowfrequency region governed by static stiffness. All panels have the samemass. Topology governed variations in dynamic stiffness and degree ofconnectivity between face sheets manifest in the STL response.the core material. Topology governed variations in dynamic stiffness and degree ofconnectivity between face sheets manifest in the STL response. These features willbe discussed in more detail in Chapter 4.53Chapter 4Main Results and DiscussionIn Chapter 2, the dispersion curves for the four truss-lattice sandwich panels il-lustrated in Figure 2.2 were calculated (Figure 2.3) using Bloch Wave analysis.The dispersion curves indicate the type and wavenumber of the propagating wavesat each frequency. In Chapter 3, the STL curves and stiffness of the panels werecomputed using FEA. In this chapter, the results of the two analysis methods arecombined to give a deeper insight into the problem of sound transmission throughsandwich panels. A discussion of the main results is presented, followed by aconclusion of the contributions of this research and suggestions for future work.4.1 A Closer Look at the STL CurvesFigure 3.17 shows STL curves computed for the geometries in Figure 2.2 usingthe same procedure as described in Chapter 3. Third-octave band averaged STLis calculated by using four frequency points in each octave-band. All six panelsshare the same span (1.5 m ×1.5 m), face thickness, mass and height as the Kurtzeand Watters’ panel. The only difference is the core topology. The STL curve fora monolithic panel of the same mass, material (stainless steel) and span as thesandwich panels was also included for comparison.Consider the STL curves for each lattice panel shown in Figure 4.1 to Fig-ure 4.5. The tetrahedral panel exhibits structural resonances (labelled as R1, R2etc.) and coincidence (labelled as C1) in Figure 4.1. Significantly, a very high54Figure 4.1: Sound transmission loss characteristics of a sandwich panel witha tetrahedral core. Deformation shapes of the panel at structural res-onances (denoted by R) and deformation shapes of the panel and thepressure field at coincidence frequencies (denoted by C) are also shownat each significant frequency. Pressure fields associated with enhancedSTL due to partial wave band gaps in the high frequency region (markedas B in STL curve) are also shown. Note that the pressures in the fluidmedium on the transmitted side are reduced by 2 orders of magnitudebetween C1 and B above. The solid black straight line corresponds tothe mass law line.55Figure 4.2: Sound transmission loss characteristics of a sandwich panel witha double pyramidal core. Deformation shapes of the panel at structuralresonances (denoted by R) and deformation shapes of the panel and thepressure field at coincidence frequencies (denoted by C) are also shownat each significant frequency. The solid black straight line correspondsto the mass law line.56Figure 4.3: Sound transmission loss characteristics of a sandwich panel witha pyramidal core. Deformation shapes of the panel at structural res-onances (denoted by R) and deformation shapes of the panel and thepressure field at coincidence frequencies (denoted by C) are also shownat each significant frequency. Pressure fields associated with enhancedSTL due to partial wave band gaps in the high frequency region (markedas B in STL curve) are also shown. Note that the pressures in the fluidmedium on the transmitted side are reduced by 2 orders of magnitudebetween C1 and B above. The solid black straight line corresponds tothe mass law line.57Figure 4.4: Sound transmission loss characteristics of a sandwich panel witha Kagome core. Deformation shapes of the panel at structural reso-nances (denoted by R) and deformation shapes of the panel and thepressure field at coincidence frequencies (denoted by C) are also shownat each significant frequency. Pressure fields associated with enhancedSTL due to partial wave band gaps in the high frequency region (markedas B in STL curve) are also shown. Note that the pressures in the fluidmedium on the transmitted side are reduced by 2 orders of magnitudebetween C1 and B above. The solid black straight line corresponds tothe mass law line.58Figure 4.5: STL characteristics of a sandwich panel with a hexagonal core.Deformation shapes of the panel at structural resonances (denoted byR) and deformation shapes of the panel and the pressure field at coinci-dence frequency (denoted by C) are also shown. The solid black straightline corresponds to the mass law line.59value of STL can be observed around 5 kHz region, as expected from the presenceof a partial band gap in Figure 2.2(a) in the same frequency region. This band gapphenomena are evident when the pressure field values on the transmitted side arecompared in the figures labelled as C1 and B in Figure 4.1. A reduction in trans-mitted pressure by two orders of magnitude can be verified. This enhancement inthe STL exceeds the 6 dB/octave improvement expected from mass law. Moreover,this improvement is a broad band phenomenon in contrast with local resonance insonic crystals observed earlier [45].The panel with a double pyramidal lattice core in Figure 4.2 has only the anti-symmetric coincidence (labelled as C1) and shows slightly higher than mass lawtowards the end of the frequency range where it has a partial band gap. The doublewall resonance and the symmetric coincidence for this panel happen to be at higherfrequencies than our range of interest. STL curves of the panel with a pyramidalcore in Figure 4.3 show a coincidence and enhanced STL due to a partial bandgap around 4 kHz region. Again, two orders of magnitude reduction in transmittedpressure can be noted. Kagome core has the least transverse shear stiffness andhence the coincidence C1 in Figure 4.4 in STL curve is delayed to higher frequen-cies. Also, this panel exhibits enhanced STL due to a partial band gap around 3kHZ regions besides structural resonances in the low frequency region, manifest-ing as a narrow band minima in the STL curve. Finally, the stiffest panel withhexagonal honeycomb core exhibits coincidence at a low frequency. The shallowcoincidence dip is due to the finite size of the panel compared to the wavelength.Coincidence is undesirable since transmitted sound power is high within the audi-ble range.STL curves of all the panels are compared in Figure 3.17. In the high fre-quency region the tetrahedral core is superior whereas double pyramidal seemsbetter in low frequency (200 Hz–600 Hz) region. Overall, Kagome core seems tocompare or exceed the performance of the shear panel (Kurtze-Watters) and mono-lithic panel. Pyramidal core is inferior to both shear panel and monolithic panelin the mid frequency (300 Hz — 2kHz) region. The Pyramidal and Tetrahedralsandwiches follow almost the same pattern. They both have two coincidence fre-quencies, the first one being anti-symmetric and the second one being symmetric.They exhibit higher than mass law STL in the range of their respective partial band60gaps observed in Figure 2.3.Overall, we conclude that STL performance higher than the shear panel andmonolithic panel are achievable using truss lattice cores. Note that no optimizationhas been performed. With optimization of strut thickness and length for a giventopology and material it is possible to tailor the wave propagation response andhence STL properties. Other measures such as enhancing damping by filling theopen cell space provided by the truss lattices with a sound absorbing material ordamping enhancement measures can also be considered. It is also possible to haveporous face sheets.4.2 Sound Transmission Class and Speech InterferenceLevelIn an actual panel design scenario one needs to consider the external noise spec-trum and decide which frequency ranges have higher noise power and are moreimportant to isolate. Here, we also assess single-number comparison criteria. Thefirst criterion that is widely known in building acoustics is the sound transmissionclass. This number is derived from comparing the measured STL curves with a ref-erence curve. According to this criterion Hexagonal and Tetrahedral panels haveinferior sound transmission characteristics (Table 4.1).Table 4.1: The sound transmission class for the panels in Figure 2.2 and themonolithic panel.Panel name Sound Transmission ClassMonolithic 34Hexagonal 12Kagome 31Pyramidal 30Double Pyramidal 32Tetrahedral 20Kurtze and Watters 34In the context of aircraft interior noise, speech interference level (SIL) or pre-ferred octave speech interference level (PSIL) are widely used indexes, refer to61Appendix B for more information on this topic. One can relate STL values to theincremental reductions, that is, reduction (in dB) below the incident power, de-noted by a prefix ∆. Large incremental reductions indicate low values of SIL orPSIL, and hence better acoustic performance. Differences on the order of 2dB orabove in SIL or PSIL values are significant. A comparison of different panels inTable 4.2 indicates that when high frequency response is included, that is ∆SIL isrelied upon, Kagome is the superior core and hexagonal honeycomb and tetrahe-dral cores have 3 dB lower values compared to Kagome. Once the high frequencyresponse is excluded, that Lp4000 or STLp4000 values are excluded, then Kagomeis still superior comparable to shear panel. As will be seen shortly, Kagome ismuch stiffer than shear panel of Kurtze and Watters or monolithic panel. However,one must exercise care in using single metrics such as SIL or PSIL, when there ismarked frequency dependent changes in STL. Such single numerical metrics maybe misleading. For example, double pyramidal is superior in the frequency range200 Hz–500 Hz in Figure 3.17.A concern with acoustically superior panels is their lack of stiffness. Stiffnessobtained by from the central deflection under a uniformly distributed pressure load-ing applied on each panel under simply supported edge boundary condition on allfour sides is calculated. The results are shown in Figure 3.16 which demonstratethat the truss-lattice core sandwich panels can be more than six times stiffer thanKurtze and Watters panel. Clearly, the traditional honeycomb panel is the stiffestfrom a static stiffness perspective. However, lattice cores provide an attractive de-sign which balance the structural and acoustic requirements of high stiffness andhigh STL.62Table 4.2: Incremental reductions in preferred-octave speech interferencelevel (PSIL) and speech interference levels (SIL) calculated fromthe STL curves for each geometry using the formulae: ∆PSIL =STL500+STL1000+STL20003 and ∆SIL=STL500+STL1000+STL2000+STL40004 . Note thatPSIL = Lp500+Lp1000+Lp20003 and SIL =Lp500+Lp1000+Lp2000+Lp40004 where Lp isthe sound pressure level. All panels have identical mass. See AppendixB for a more detailed discussion.Sandwich panel core ∆PSIL(dB) ∆SIL (dB)Monolithic 40 45Kurtze and Watters 42 45Tetrahedral 33 43Double Pyramidal 39 44Pyramidal 36 44Kagome 42 46Hexagonal 40 434.3 DiscussionThe Kagome lattice is not an interconnected network of trusses; hence, the shearstiffness is negligible in comparison with the three other lattice geometries. Sand-wich panels with cores made of the latter geometries will have 5-6 times the stiff-ness of the existing quiet panel designs. The stiffer we make a panel while main-taining the total mass, the wider will the resonance-dominated zone on the STLcurve become. Softer cores such as the Kagome and Kurtze and Watters’ resultin none or fewer number of troughs in the range below 1000 Hz. Also, at higherfrequencies (> 1000 Hz) a softer core can prevent coincidence using a mechanismexplained by Kurtze and Watters [40]. If the stiffness of the core in the thicknessdirection decreases or the surface area of the face sheets in a unit-cell that is notsupported by lattice struts increases, the face sheets will start to vibrate indepen-dent of the core and one another. These vibrations resemble an assembly of tinyloudspeakers propagating the vibrations of the panel in air. Hence, the closer thestruts are spaced on the surface of the face sheets the stronger the motions of thedifferent parts on the panel are coupled and the loud-speaker modes can be avoided.Also, as Moore and Lyon [52] proposed for each type of wave a correspond-63ing coincidence frequency is expected. For bending-type waves the stiffer thepanel is in bending, the lower the coincidence frequency would be. For breathing-type waves, the stiffer the core in thickness direction, the higher the double wall-resonance frequency and the corresponding coincidence frequency. The intrinsicdirectionality in the material properties of lattices gives us much more freedom inmaking the a specific direction stiffer or softer.The partial band gaps involved in this study functioned on the basis of Braggscattering. One approach to qualitatively compare the band gap frequencies is bylooking at the length scales. In order for Bragg scattering to happen, the wavelengthof the traveling waves needs to be of the same order of magnitude as separationdistance between adjacent unit cells. Also in general, wavelengths decrease withfrequency increase. Hence, one way to decrease the partial band gap frequencycan be by increasing the separation between adjacent lattice units. This argumentexplains why the band gap frequency is lowest for the Kagome lattice and highestfor the Double Pyramidal one.With proper design, the partial band gaps and directionality in stiffness cansatisfy our acoustics and noise control requirements.4.4 Concluding RemarksSandwich panels of same mass but made from different truss lattice cores havebeen systematically compared with traditional sandwich and with hexagonal hon-eycomb core and shear panels for their stiffness and STL properties. Characteristicdifferences in wave propagation response, such as partial band gaps, have lead tocorresponding differences in STL response. This study has demonstrated that, evenwithout optimization, truss lattice panels offer simultaneously promising stiffnessand acoustic (STL) response properties in comparison with the traditional designs.Kagome geometry emerges as the stiffest possible geometry with a simultaneouslyhigh STL. Further work can proceed along the analytical model development usingBloch modes to represent the structural wave response and fabrication, experimen-tal confirmation of the observed differences among the panels, and optimization ofcore parameters.644.4.1 ContributionsThe quest for material efficiency and ecologically sustainable engineering designsdefies the conventional noise and vibration control solution which was increasingthe mass of the structure. In addition, more restrictive noise and vibration standardsask for novel solutions in this area. Here are the main contributions of the presentwork towards this goal:1. A versatile method was developed for the analysis of propagation of wavesin infinite truss-lattice sandwich panels and in general any periodic structureextending in 2D. This semi-analytical method is based on FEM and Floquet-Bloch theorem which means it is sufficient to model only one unit cell of thesystem, which considerably reduces the computational cost. The developedcode is compatible with the node data, and stiffness and mass matrices outputfrom Abaqus which makes it an even more versatile tool for rapid analysisof wave propagation in basically any 2D-periodic structure for which theunit cell can be modeled in Abaqus. By post processing the outputs of thisanalysis one can obtain the dispersion curves. These curves indicate at eachwavenumber, the frequencies that can propagate, as well as, the mode shapesof the corresponding traveling waves. This information is of significancewhen it comes to the analyzing the structure-borne sound transmission.2. Using acoustic FEA it was shown that the results of the wave analysis canactually be interpreted to give a qualitative view of the high-frequency STLcurves before the significantly more time and computation consuming FEAsare carried out. From the dispersion curves one can tell the stop-bands andpartial band-gap frequencies as well as the frequencies for supersonic wavepropagation. Also, a color representation of the out-of-plane motion of thepanels reveal which branches are contributing the most to the acoustic radia-tion. This outcome is of practical importance for designing panels that haveoptimum performance at a specific frequency range of interest.3. A comparison between the structural and acoustic behavior of the exampletruss-lattice core panel designs studied and an existing quiet panel designproved that by proper design truss lattice cores can be up to 6 times stiffer65than the existing quiet panel of the same mass and thickness without sacri-ficing much of the acoustic performance4.4.2 Future WorkIn this stage, the following list can be suggested as the works that have not beencompleted so far and the future paths this project can pursue:1. Although we tried to verify our analysis in different stages with the exist-ing results and findings available in the literature, one important aspect thatcould be included in future stages is the experimental verification of theseresults. As a first step, the dispersion curves for a test panel can be calcu-lated by performing a modal test and finding the standing wave lengths andtheir corresponding resonance frequencies. Next, the actual STL of the panelcan be tested in a sound transmission suite and can be compared to the FEcalculation.2. One possibility that has not yet been added to the Bloch wave analysis iscalculating the structural response by post processing the Bolch analysis out-puts. This has previously been applied to simpler structures with a point loadexcitation. It can also be extended to find the panels’ response to acousticexcitations [41].3. The computational efficiency of the FEM-Bloch analysis relative to acousticFEM simulations suggests that topology-optimization methods can be imple-mented with the purpose of designing sandwich panels with optimal acousticperformance.4. The FEM-Bloch analysis can also be used as the starting point for a study ofdependence of the dynamic behavior of periodic panels on geometrical andmechanical parameters in a model. Understanding the effect of parameterssuch as the separation distance between the lattices or the ratio of the massof the core to the mass of the face sheets can prove helpful in designing apanel with a more desirable acoustic performance.665. Although this research did not consider surface treatments, the findings arehelpful in designing phononic crystals to be used as treatments on the surfaceof the panels.6. The potential improvements regarding having perforations in the face sheeton the transmission side of the panels need to be evaluated in conjunctionwith lattice-core panels.7. The acoustic and structural evaluations conducted in this study were for per-fect geometries meaning that all cells in a lattice core are perfectly identical.However, both the manufacturing of these cores and bonding them to the facesheets are prone to numerous imperfections. In the most extreme cases miss-ing or damaged struts, and soldering or bonding defects are expected. To beable to apply these findings confidently to an actual engineering design it isrecommended to incorporate the effect of defects in a future model.67Bibliography[1] J. Allard and N. Atalla. Propagation of Sound in Porous Media: ModellingSound Absorbing Materials 2e. Wiley, 2009. → pages 7, 17[2] H. G. Allen and B. G. Neal. 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They showed that the stiffness scales proportionalto the relative density ρ¯ for stretch dominated lattices whereas it scales as ρ¯2 forbending dominated lattices. Deshpande et al. [18] analyzed the criteria for a struc-ture to be strching dominated. A sufficient criterion is given by Calladine’s [6] 3Dgeneralization of Maxwell’s static determinacy theorem in two dimensions:b−3 j−6≤ 0Here a FE unit cell analysis approach outlined in [38, 67] is applied to find theeffective stiffness of the lattice cores studied in this thesis and scaling laws are es-tablished. A similar approach was applied to study the effective material propertiesof regular honeycombs [34], Octet lattice [19] and a fully triangulated truss-latticespecimen (JAMCORP Lattice Block) [67], and showed good agreement (13% av-erage discrepancy) with the experimental results in [68]. The approach can besummarized as:1. A cuboid unit cell of the full lattice with orthogonal basis vectors is taken.Having a cuboid unit cell significantly simplifies the next step. A beam ele-ment model of the structure is created in ABAQUS. Members that are shared75between two cells are assigned one-half of their cross section area; membersthat are shared between four cells are assigned one-quarter of their cross-section area. 40 B33 elements are created on each member of the unit cell.2. The unit cell, aligned with the Cartesian axes, is subjected to periodic bound-ary conditions, such that the rotations of the corresponding nodes on oppo-site boundary surfaces are equal and the nodes on each boundary plane re-main in plane after the deformation. The latter interactions are accessiblein ABAQUS CAE under coupling and linkage modules respectively. Forstretch-dominated deformations the bending stiffness is negligible; there-fore, the rotational coupling is not crucial.3. Displacement boundary conditions corresponding to each stress state are ap-plied to the boundaries of the unit cell. This approach requires an accuratejudgement of the boundary conditions. A detailed example is discussed inchapter 3 of [67].4. The stresses and consequently the stiffness moduli can now be calculatedfrom the resulting reaction forces on each surface boundary.The results are presented in Table A.1. All the FE results reported are based onlinear small strain approximations. For the Tetrahedral lattice, Gxz and Gzx havebending-dominated deformations and are not included in this table. The resultsfor out-of-plane moduli of the pyramidal lattice (Eyy, Gxy, Gzy, Gyx, Gyz) could beverified with Guo et al. [35].76Lattice geometry Relative density (ρ¯) Effective Young’s moduli Effective shear moduliXZYExxE =EzzE = 0.0247ρ¯GxyE =GzyE = 0.0988ρ¯ρ¯ = 2√2(ah)2 EyyE = 0.4444ρ¯GyxE =GyzE = 0.1111ρ¯ExxE =EzzE = 0.1250ρ¯GxyE =GzyE = 0.2500ρ¯ρ¯ = 8√2(ah)2 EyyE = 0.2500ρ¯GyxE =GyzE = 0.1250ρ¯GxzE =GzxE = 0.1250ρ¯ExxE =EzzE = 0.0625ρ¯GxyE =GzyE =ρ¯ = 2√2(ah)2 EyyE = 0.2500ρ¯GyxE =GyzE = 0.1250ρ¯GxzE =GzxE = 0.0625ρ¯ExxE =EzzE = 0GxyE =GzyE =ρ¯ = 2√2(ah)2 EyyE = 0.4444ρ¯GxzE =GzxE = 0GyxE =GyzE = 0.1111ρ¯ρ¯ = 2√3( tl )EyyE = ρ¯GxyE =GzyE =ρ¯3(1+ν)Table A.1: First order estimates of effective moduli of the lattice core topolo-gies studied as a function of relative density (ρ¯). a, h, t and l are respec-tively the radius of the strut, height of the core, thickness of the wall fora hexagonal honeycomb and the length of the hexagonal honeycomb.77Appendix BSound Interference LevelThe speech interference level (SIL) was introduced by Beranek for studying thecharacteristics of aircraft cabin noise. It represents the level of masking of thespeech by surrounding noise. As a method simpler and faster than ArticulationIndex (AI), it is commonly used in situ for assessing the influence of noise onspeech communication. According to ANSI S12.65-2006, the SIL is given by:SIL=Lp500+Lp1000+Lp2000+Lp40004(B.1)Here Lps are the octave-band averaged sound pressure levels. Webster introduceda modified criterion called preferred-octave speech interference level (PSIL) whichis also commonly used for aircraft noise characterization. PSIL does not take intoaccount the 4 kHz center frequency:PSIL=Lp500+Lp1000+Lp20003(B.2)The quantities ∆SIL and ∆PSIL are the absolute value of the changes in theSIL and PSIL of the incident noise after getting transmitted inside a chamber. Forexample, if the exterior noise has SIL = C0 the interior noise is going to haveSIL = C0−∆SIL as a result of the transmission loss. Next, we will explain how∆SIL and ∆PSIL are derived having the STL curves. Let indices i and t indicate theproperties of the incident and transmitted sides. The SIL on the transmitted sidecan be written as:78SILt =Lp500,t +Lp1000,t +Lp2000,t +Lp4000,t4=(Lp500,i−STL500)+(Lp1000,i−STL1000)+(Lp2000,i−STL2000)+(Lp4000,i−STL4000)4=SILi− STL500+STL1000+STL2000+STL40004(B.3)where STL values are interpolated on the STL curves. Therefore, the absolutevalue of the change in the SIL of the sound after transmission through a partitionis:∆SIL=STL500+STL1000+STL2000+STL40004(B.4)Same procedure can be applied for ∆PSIL to give:∆PSIL=STL500+STL1000+STL20003(B.5)79


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