STATIC, DYNAMIC AND ACOUSTICAL PROPERTIES OF SANDWICH COMPOSITE MATERIALS Except where reference is made to the work of others, the work described in this dissertation is my own or was done in collaboration with my advisory committee. This dissertation does not include proprietary or classified information. ____________________________ Zhaohui Yu Certificate of approval: ____________________________ ____________________________ George T. Flowers Malcolm J. Crocker, Chair Professor Distinguished University Professor Mechanical Engineering Mechanical Engineering ____________________________ ____________________________ Hareesh Tippur ZhongYang (Z.-Y.) Cheng Professor Assistant Professor Mechanical Engineering Materials Engineering __________________ Joe F. Pittman Interim Dean Graduate School STATIC, DYNAMIC AND ACOUSTICAL PROPERTIES OF SANDWICH COMPOSITE MATERIALS Zhaohui Yu A Dissertation Submitted to the Graduate Faculty of Auburn University in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy Auburn, Alabama May 10, 2007 iii STATIC, DYNAMIC AND ACOUSTICAL PROPERTIES OF SANDWICH COMPOSITE MATERIALS Zhaohui Yu Permission is granted to Auburn University to make copies of this dissertation at its discretion, upon request of individuals or institutions and at their expense. The author reserves all publication rights. _____________________________ Signature of Author _____________________________ Date of Graduation iv VITA Zhaohui Yu, daughter of Shuiyong Yu and Limin Xu, was born on July 25, 1976, in Qingdao, Shandong Province, China. She graduated from the No. 1 Middle School of Jimo, Qingdao in 1995. She entered Ocean University of Qingdao, China in September 1995, and graduated with Bachelor of Science degree in Electrical Engineering and a Minor Diploma in Software Engineering in July 1999. She worked as a signal processing engineer in HuaYi Building Materials Company of Qingdao from September 1999 to November 2001. She entered Graduate School, Auburn University, in January 2002. She married to Yuquan Li on July 23, 2000. v DISSERTATION ABSTRACT STATIC, DYNAMIC AND ACOUSTICAL PROPERTIES OF SANDWICH COMPOSITE MATERIALS Zhaohui Yu Doctor of Philosophy, May 10, 2007 (B.S., Ocean University of Qingdao, 1999) 132 Typed Pages Directed by Malcolm J. Crocker Sandwich composite materials have been widely used in recent years for the construction of spacecraft, aircraft, and ships, mainly because of their high stiffness-to- weight ratios and the introduction of a viscoelastic core layer, which has high inherent damping. One of the main objects of this research is the measurement and estimation of the bending stiffness and damping of sandwich structures. Knowledge of the elastic properties of the core and face sheet of the sandwich structures is indispensable for the analysis and modeling of sandwich strictures. However, traditional methods to determine the elastic properties are not suitable for the core, which is usually brittle, and the face sheets, which are usually very thin. A set of special techniques has to be used to estimate the elastic properties of these materials. The dynamic bending stiffness of such materials vi is difficult to measure because it depends on frequency unlike ordinary non-composite materials. A simple measurement technique for determining the material parameters of composite beams was used. The damping is an important property used for the analysis of the acoustical behavior of the sandwich structures, especially for the characterization of the sound transmission loss. An interesting fact is that damping can be measured as a byproduct in the procedure of the measurements of dynamic stiffness. Another main object of the research is to analyze the sound transmission loss of sandwich structures and to simulate their acoustical behavior using the statistical energy analysis method (SEA). While solving vibroacoustic problems, FEM and SEA are commonly used. However, common vibroacoustic problems involve a very large number of modes over a broad frequency range. At high frequencies these modes become both expensive to compute and highly sensitive to uncertain physical details of the system. Many processes involved in noise and vibration are statistical or random in nature. So SEA is suitable for the high frequency problems such as vibroacoustic problems. The materials used in the research include sandwich structures with polyurethane foam- filled honeycomb cores and sandwich structures with closed-cell polyurethane foam cores. Foam-filled honeycomb cores possess mechanical property advantages over pure honeycomb and pure foam cores. The honeycomb structure enhances the stiffness of the entire structure; while the foam improves the damping. Closed-cell polyurethane foam is CFC-free, rigid, and flame-retardant foam. Both foam-filled honeycomb and closed-cell foam cores meet the requirements of many aircraft and aerospace manufacturers. Also, foam-filled honeycomb and closed-cell structures have high strength-to-weight ratios and vii great resistance to water absorption, and will not swell, crack, or split on exposure to water. viii ACKNOWLEDGMENTS I would like to express my sincere appreciation and thanks to my advisor Auburn University distinguished professor, Dr. Crocker for his guidance and support during my studies and research. I am grateful for the considerable assistance provided by the other committee members, Dr. Flowers, Dr. Tippur and Dr. Cheng. I appreciate outside reader, Dr. Cochran for his time and valuable suggestions to my dissertation. My appreciation also goes to my friends and colleagues at Auburn University. Last and not least, I want to express my special thanks to my parents and my husband for their support and love. ix Journal used: Journal of Sound and Vibration Computer software used: Microsoft Word 2002 x TABLE OF CONTENTS LIST OF TABLES........................................................................................................... xiii LIST OF FIGURES ...........................................................................................................xv CHAPTER 1 BACKGROUND ......................................................................................1 1.1 Introduction..........................................................................................................1 1.2 Objective..............................................................................................................2 1.3 Structure of honeycomb panels............................................................................3 1.4 Organization of Dissertation................................................................................6 CHAPTER 2 STATIC PROPERTIES CHARACTERIZATION...................................8 2.1 Introduction..........................................................................................................8 2.2 Four-point bending method..................................................................................8 2.3 Twisting method ................................................................................................10 2.4 Calculation of static bending stiffness of sandwich beam.................................12 2.5 Experimental results and analysis......................................................................13 2.5.1 Specimen................................................................................................... 13 2.5.2 Four-point bending method....................................................................... 15 2.5.3 Twisting method ....................................................................................... 16 2.5.4 Calculation of static bending stiffness of sandwich beam........................ 17 2.5.5 Summary and analysis .............................................................................. 18 xi 2.6 Finite element method model.............................................................................20 2.6.1 Introduction............................................................................................... 20 2.6.2 FEM in panel problems............................................................................. 20 2.6.3 Element Types .......................................................................................... 23 2.6.4 Finite element modeling of sandwich structures ...................................... 24 2.6.5 Comparison of results from experiments and theoretical analyses........... 30 CHAPTER 3 THEORY FOR SANDWICH BEAMS ..................................................32 3.1 Literature review................................................................................................32 3.2 Theory of sandwich structure.............................................................................35 3.3 Boundary conditions ..........................................................................................43 3.4 Wave numbers ...................................................................................................45 3.5 Least Squares Method........................................................................................50 3.6 Damping measurement methods........................................................................53 CHAPTER 4 DYNAMIC PROPERTIES CHARACTERIZATION ...........................57 4.1 Experiments .......................................................................................................57 4.1.1 Steps.......................................................................................................... 57 4.1.2 Set up ........................................................................................................ 57 4.1.3 Samples..................................................................................................... 60 4.2 Analysis of experimental results........................................................................63 4.2.1 Frequency response functions................................................................... 63 4.2.2 Boundary conditions ................................................................................. 66 4.2.2 Dynamic bending stiffness........................................................................ 69 4.2.3 Damping.................................................................................................... 76 xii 4.3 Conclusions........................................................................................................77 CHAPTER 5 SOUND TRANSMISSION LOSS OF SANDWICH PANELS.............78 5.1 Classical sound transmission analysis ...............................................................78 5.1.1 Mass law sound transmission theory ........................................................ 79 5.1.2 The effect of panel stiffness and damping ................................................ 81 5.1.3 The coincidence effect .............................................................................. 81 5.1.4 Critical frequency...................................................................................... 84 5.1.5 Sound transmission coefficient and transmission loss at coincidence...... 84 5.2 Literature review of the Sound Transmission Loss of Sandwich Panels...........85 5.3 Statistical energy analysis model (SEA)............................................................88 5.3.1 Prediction of Sound Transmission through Sandwich Panels using SEA 88 5.3.2 Simulation using SEA software AutoSEA ............................................... 93 5.4 Review of sound transmission measurement technique: two-room method .....94 5.5 Experiments of TL and simulations using AutoSEA.........................................97 5.6 Summary and conclusions of experiments of TL and simulations using AutoSEA......................................................................................................................104 CHAPTER 6 SUMMARY AND CONCLUSIONS ...................................................105 REFERENCES ...............................................................................................................109 LIST OF TABLES Table 1 The dimensions and densities of the specimens tested for their static stiffness ...13 Table 2 Static properties obtained from the four-point bending and twisting methods.....18 Table 3 Static bending stiffness obtained from two ways .................................................19 Table 4 Load-deflection relations for Beam D from the experiments and FEM analysis .31 Table 5 Load-deflection relations for Plate E from the experiments and FEM analyses ..31 Table 6 Boundary conditions for ends of beam.................................................................45 Table 7 Geometrical and material parameters for sandwich beam....................................48 Table 8 Values of n ? for particular boundary conditions .................................................51 Table 9 Geometry and density of the sandwich beam with foam filled honeycomb core.61 Table 10 Geometry and density of the sandwich beam with foam core............................62 Table 11 Natural frequencies for different measurements on Beam F and their corresponding bending stiffness. ...............................................................................65 Table 12 Natural frequencies of Beam C for the three different beam boundary conditions...................................................................................................................67 Table 13 Comparison of static stiffness measured by four-point bending method and two stiffness limits from dynamic characterization..........................................................70 Table 14 Comparison of shear modulus of the core measured by twisting method and that from the dynamic stiffness curve for Beam G...........................................................74 xiii xiv Table 15 Damping ratio of Beam C...................................................................................76 Table 16 Geometrical parameters of panels under study...................................................97 Table 17 Reverberation times (s) of the receiving room with different panels .................99 xv LIST OF FIGURES Figure 1 Sandwich panel with a honeycomb core ...............................................................3 Figure 2 A: Nomex honeycomb core, B: irregular aluminum honeycomb core .................4 Figure 3 Corrugation process used in honeycomb manufacture..........................................5 Figure 4 Geometry and dimensions of the four-point bending test .....................................9 Figure 5 Two principal direction sandwich panel strip for the four-point bending experiments and its equivalent plate representation ..................................................10 Figure 6 Loading scheme for the pure twisting test...........................................................11 Figure 7 Core, face sheet and entire sandwich structure ...................................................14 Figure 8 Deflection of beam with four-point bending method..........................................25 Figure 9 Out-of-plane shape with un-deformed edge of beam with four-point bending method........................................................................................................................26 Figure 10 In-plane shear stress contour for beam with four-point bending method..........27 Figure 11 Out-of-plane deflection contour of plate with the twisting method ..................28 Figure 12 Deformed plate with un-deformed edge of plate with the twisting method......29 Figure 13 In-plane shear stress contour for plate with the twisting method......................30 Figure 14 Bending of composite bar or panel by (a) bending and (b) shearing of the core layer............................................................................................................................35 Figure 15 Excitation of a beam and resulting forces and moments. Dimensions and material parameters for the laminates and core are indicated....................................37 xvi Figure 16 Elastic properties and area density of sandwich structure.................................37 Figure 17 Particular boundary conditions..........................................................................44 Figure 18 Wave numbers for beam [16]............................................................................47 Figure 19 Decay rate method used to determine damping ................................................54 Figure 20 Modal bandwidth method to determine damping..............................................55 Figure 21 Power balance method to determine damping...................................................56 Figure 22 Experimental steps to determine some properties of sandwich structures........57 Figure 23 Set Up 1 Using shaker .......................................................................................59 Figure 24 Set Up 2 Using hammer ....................................................................................60 Figure 25 Honeycomb sandwich composite structures; (a) foam-filled honeycomb core, (b) composite beam....................................................................................................62 Figure 26 Closed-cell foam core........................................................................................63 Figure 27 FRF for the sandwich beam F for free-free boundary condition.......................63 Figure 28 FRF for the sandwich beam G for free-free boundary condition. .....................64 Figure 29 FRF for the sandwich beam E for free-free boundary condition.......................65 Figure 30 Bending stiffness of the sandwich beam C for three different boundary conditions...................................................................................................................68 Figure 31 Bending stiffness for the sandwich beam C. .....................................................70 Figure 32 Dynamic stiffness for beams C and D...............................................................71 Figure 33 Dynamic stiffness for core in two principal directions .....................................72 Figure 34 Bending stiffness for the sandwich beam G......................................................73 Figure 35 Bending stiffness for the sandwich beam E, F and H........................................75 Figure 36 Damping ratio of sandwich beam C ..................................................................77 xvii Figure 37 The coincidence effect.......................................................................................83 Figure 38 Idealized plot of transmission loss versus frequency ........................................83 Figure 39 Schematic of the power flow in three-coupled systems using SEA..................89 Figure 40 Sound transmission loss model using the AutoSEA software...........................94 Figure 41 Set up for the two reverberation room sound transmission loss measurement method........................................................................................................................95 Figure 42 Measurements of TL, mass law and simulation of AutoSEA for panel A......100 Figure 43 Measurements of TL, mass law and simulation of AutoSEA for panel B ......100 Figure 44 Measurements of TL, mass law and simulation of AutoSEA for panel C ......101 Figure 45 Measurements of TL, mass law and simulation of AutoSEA for panel D......101 Figure 46 Measured TL for panels A and B ....................................................................102 Figure 47 Measured TL for panels C and D ....................................................................102 Figure 48 Measurements of TL and simulation by AutoSEA for panel E.......................104 Figure 49 Measured TL for panels A, B, C, D and E ......................................................104 1 CHAPTER 1 BACKGROUND 1.1 Introduction Applications for sandwich structures are steadily increasing. The term sandwich panel here refers to a structure consisting of two thin face plates bonded to a thick and lightweight core. The face plates are typically made of aluminum or some composite laminate. The core can be a lightweight foam or a honeycomb structure. These types of sandwich structures having a high strength to weight ratio have been used by the aircraft industry for over 70 years. However, during the last decade, various types of lightweight structures have also been introduced in the vehicle industry. This trend is dictated by demands for higher load capacity for civil and military aircraft, reduced fuel consumption for passenger cars, increased speed for passenger and navy vessels of catamaran types and increased acceleration and deceleration for trains to increase their average speeds. The environmental impact of lightweight vehicles could be considerable in reducing fuel consumption and increasing load capacity. However, there are also certain constraints like passenger comfort, safety and costs for new types of vehicles. Passenger comfort requires low noise and vibration levels in any type of vehicle. In addition to new materials being introduced, certain types of trains and fast passenger vessels are being built of aluminum. This means that traditional solutions developed for steel constructions must be replaced by completely new designs to achieve the required noise levels. Lightweight structures often have poor acoustical and dynamic 2 properties. In addition, the dynamic properties are often frequency dependent. In order to avoid noise problems in lightweight vehicles, it is essential that the main structure-borne sound sources are as weakly coupled as possible to the supporting structure foundation and to adjoining elements. With respect to material and construction costs, sandwich structures can compare very favorably with other lightweight materials like aluminum. The number of applications for sandwich panels is steadily increasing. One reason for the growing interest is that today it is possible to manufacture high quality laminates for many applications. The material used in the laminates is often glass reinforced plastic (GRP). The composition of a laminate and thus its material parameters can be considerably important in the manufacturing process. Various types of core materials are commercially available. The techniques for bonding core materials and laminates as well as different plate structure are well understood, although still under development. 1.2 Objective As discussed in the previous section, this thesis is mainly concerned with the static, dynamic and acoustical properties of sandwich structures. In particular, emphasis is placed on the study of the dynamic response of structural beams to acoustical excitation. The thesis includes three main parts: 1. Characterization of the static properties of sandwich beams and model the foam-filled honeycomb sandwich structures by the finite element method (FEM), 2. Study of the vibration response of sandwich structures and characterization of the dynamic properties of sandwich beams, 3. Analysis of the sound transmission loss of sandwich panels and modeling of foam-filled honeycomb sandwich structures using the Statistical Energy Analysis (SEA) method. 1.3 Structure of honeycomb panels A honeycomb panel is a thin lightweight plate with a honeycomb core with hexagonal cells. Layered laminates are bonded to both sides of the core as shown in Figure 1. Each component is by itself relatively weak and flexible. When incorporated into a sandwich panel the elements form a stiff, strong and lightweight structure. The face sheets carry the bending loads and the core carries the shear loads. In general, honeycomb cores are strongly orthotropic. Figure 1 Sandwich panel with a honeycomb core The types of core materials in the panels used for the measurements presented here are either Nomex or aluminum. Nomex is an aramid fiber paper dipped in phenolicresin with a low shear modulus and low shear strength. A typical thickness of the 3 Nomex honeycomb panels investigated is 10 mm with the thickness of the face sheet or laminate being between 0.3-0.7 mm. The weight per unit area is of the order of 3 kg/m 2 unit area. Each laminate consists of 3-5 different layers bonded together to give the best possible strength. The laminates are not necessarily symmetric and are usually orthotropic. The core acts as a spacer between the two laminates to give the required bending stiffness for the entire beam. The bending stiffness of the core itself is in general very low. The bending stiffness of sandwich materials is frequency dependent. The cells in the core give it an orthotropic structure. The dynamic characteristics should be expected to be different in each direction. The two main in-plane directions 1 and 2 are defined in Figure 2. The shape of the honeycomb cells of a typical aluminum core is generally very irregular which makes it impossible to describe its geometry in a simple way. Nomex cores have very regular shapes as compared to aluminum-cores. A B Figure 2 A: Nomex honeycomb core, B: irregular aluminum honeycomb core 4 Honeycomb cores, which were developed starting in the 1940?s primarily for the aerospace industry, have the greatest shear strength and stiffness-to-weight ratios, but require special care to ensure adequate bonding of the face sheets to the core since such cores are hollow. The standard hexagonal honeycomb is the basic and most common cellular honeycomb configuration, and is currently available in all metallic and nonmetallic materials. Figure 3 illustrates the manufacturing process, and the L (ribbon direction) and W (transverse to the ribbon) directions of the hexagonal honeycomb. In this process, adhesive is applied to the corrugated nodes, the corrugated sheets are stacked into blocks, the node adhesive cured, and sheets are cut from these blocks to the required core thickness. The honeycomb cores are suitable for both plane and curved sandwich applications. Figure 3 Corrugation process used in honeycomb manufacture As discussed in the previous chapters, sandwich structures with foam-filled honeycomb cores have some advantages over pure honeycomb cores. By filling foam in the honeycomb cells, not only the longitudinal cell walls but also the foam can carry the 5 6 uniaxial load. So the foam is expected to reduce the discontinuities in elastic properties possessed by with pure honeycomb cores. Also the foam can make the fabrication of sandwich structures easier than those made with pure honeycomb cores. Another important advantage of foam-filled honeycomb cores is their improved damping and shear strength properties. 1.4 Organization of Dissertation This dissertation contains the results of the present research investigation into the current objectives. The research was performed in the Sound and Vibration Laboratory of the Department of Mechanical Engineering at Auburn University. The results reported are divided into four major parts. Chapter 2 described the static properties of foam-filled honeycomb sandwich beams. Orthotropic plate theory is introduced and two methods, the four-point bending method and the twisting method, are used to measure the shear modulus, Young?s modulus and other properties of sandwich structures. The finite element method (FEM) was used to simulate the response of the beams and panels with the four-point bending method and the twisting method. A thorough derivation of the beam theory of the sandwich structure is given in Chapter 3. The theoretical model was derived using Hamilton's principle. The general dynamic behavior of sandwich structures is discussed. Chapter 3 also describes the measurements, which include experiments and the analysis of results. In the experiments, measurements on foam-filled honeycomb sandwich beams with different configurations were performed and finally the conclusions were drawn from the analysis of the results. 7 Chapter 4 is devoted to the study of sound transmission through sandwich panels. This chapter starts with a brief introduction of the classical theory of sound transmission loss. Then the previous research on sound transmission through sandwich panels is reviewed. The sound transmission loss of several panels with different thickness of core and face sheet was measured by the two room method. Simulations of the sound transmission loss were conducted using AutoSEA. Experimental results for sound transmission loss are presented as well. The summary and conclusions drawn from this research are given in Chapter 5. CHAPTER 2 STATIC PROPERTIES CHARACTERIZATION 2.1 Introduction In this work, an experimental study on the bending stiffness of sandwich beams was conducted. An experimental procedure to measure the bending stiffness of sandwich beams was used. The technique includes the standard four-point bending tests of beam specimens to assist in the evaluation of the in-plane Young?s modulus of the core, face sheet and entire sandwich structure. In addition, special plate twisting tests have been used for finding the in-plane shear modulus and Poisson?s ratios of core, face sheet and entire sandwich structures. Using these elastic properties, the bending stiffness of entire sandwich structures was calculated in two different ways. The static behavior of the entire sandwich structures was simulated using the finite element method (FEM). 2.2 Four-point bending method The four-point bending arrangement used in this work is shown in Figure 4. The beams were simply supported at the ends of the central span and half of the total load 2 d P was applied at the ends of the beams, separately. The loading points were located 5 mm from the corners of the plate specimens. The advantage of this bending test is that normal stresses, but not shear stresses, act over the central span . The central span, therefore, is in a state of pure bending. Measurement of the load versus central deflection response enables the calculation of the Young?s moduli of the core, face sheet and entire sandwich structure, by means of the equation 2 d P 0 w 21 EE 8 0 2 21 3 3 4 w Sdd bh P x ? ? ? ? ? ? ? ? = , (1) where P is the total load on the ends, is the central span, is the central deflection, is the outer span and is the thickness as shown in Figure 4. 2 d 0 w 1 d h And is the compliance along the direction parallel to the length of the beam. x S 2.direction principalin 1 1,direction principalin 1 2 22 1 11 E SS E SS x x == == (2) Figure 4 Geometry and dimensions of the four-point bending test The exact location at which the loads are applied is largely arbitrary, except that the two outer spans must be equal. 1 d In the measurement of the deflection, the force required to actuate the deflection measurement device must be kept small relative to the applied loads to assure a pure bending state. In this work, a micrometer was used to measure the beam deflections. The micrometer could be read to the nearest 0.001 mm. 9 Both the principal directions, 1 and 2, of the four-point bend specimens, as shown in Figures 5 were considered. Since the core pf the sandwich beams used in the measurements is orthotropic, several extensive reviews of equivalent plate models in the literature [1-4] were consulted. Figure 5 Two principal direction sandwich panel strip for the four-point bending experiments and its equivalent plate representation 2.3 Twisting method 10 Figure 6 Loading scheme for the pure twisting test The twisting method configuration used in this work is shown in Figure 6. The plates were simply supported at the three quadrant corners and the load was applied at the other corner of the plates. Measurement of the load versus central deflection response enables calculation of shear modulus , Poisson's ratios P P 0 w 12 G 2112 , ?? of the core, face sheet and entire sandwich structure, by means of the equation: 0 2 3 3 4 w LS t P G ? ? ? ? ? ? ? ? = , (3) where () () ( ) ,282 66 2 22 22 2 12 22 11 2 SnmSnmmnSnmSnmmnS G ?+++???= (4) and , 1 ,, 1 , 1 12 66 2 21 1 12 12 2 22 1 11 G S EE S E S E S =?=?=== ?? and ?cos=m and ?sin=n , where ? is the angle between the principal direction 1 and x coordinate. For various material orientations, Equation (4) can be used to evaluate (5) ().2,45 ,,0 1222 66 SSS SS G G ?=?= == o o ? ? Compliances and have been obtained from the four-point beam bending tests. By performing twisting tests, with and , compliances and can be determined. Combining these results, the equivalent Poisson's ratios are obtained 11 S 22 S o 0=? o 45=? 66 S 12 S 11 . , 21 11 22 12 21 22 12 ?? ? S S S S = =? (6) The shear modulus is also obtained since is known from the equation 12 G 66 S 12 66 1 G S = . The load and supporting points for the twisting tests were located 5 mm from the corners of the plate specimens. The amount of overhang was not found to be critical. The method for measuring the displacement of the center of the plate, , was the same as with the four-point bending tests. The samples tested were 0 w 5.05.0 ? meter square plates. 2.4 Calculation of static bending stiffness of sandwich beam Once we know the properties of core and face sheet, the bending stiffness per unit width of the beam is b D ? ? ? ? ? ? +++= 3 2 212 3 2 23 l lc lc l ccb t tt tt E tE D , (7) where and are the thicknesses of the core and face sheets (laminate), respectively. is the effective modulus of the core and is the effective Young?s modulus of the face sheet (laminate) and they can be calculated using Equation (8). c t l t c E l E .21 1 ,2 1 ,1 1 2112 1 2112 2 2 2112 1 1 ordirectionprincipalfor E Esheetsfacefor directionprincipalfor E E directionprincipalfor E E corefor ll l l cc c c cc c c ?? ?? ?? ? = ? ? ? ? ? ? ? ? = ? = (8) 12 Then the bending stiffness of the entire sandwich beams, in two principal directions, 1 and 2, can be calculated by Equation (9): .2 3 2 212 ,1 3 2 212 3 2 23 2 22 3 2 23 1 11 directionprincipalfor t tt tt E tE D directionprincipalfor t tt tt E tE D l lc lc l ccs l lc lc l ccs ? ? ? ? ? ? ? ? +++= ? ? ? ? ? ? ? ? +++= (9) Alternatively, the bending stiffness per unit width of the entire sandwich beams can be calculated directly from the modulus of the sandwich beams using the four-point bending method and the twisting method (see section 2.2 and 2.3) as follows: () () () () .2 112 2 ,1 112 2 2112 3 2 22 2112 3 1 11 directionprincipalfor ttE D directionprincipalfor ttE D ss lc s s ss lc s s ?? ?? ? + = ? + = (10) 2.5 Experimental results and analysis 2.5.1 Specimen The specimens used in the experiments included two beams of honeycomb core filled with foam in two principal directions, one beam made of a face sheet, one square plate of core with , one square plate of core with and two entire sandwich beams cut with the x-coordinate oriented in two the principal directions. All the beams were uses with the four-point bending method and all the plate were used in twisting method. o 0=? o 45=? The dimensions and densities of the specimens are listed in Table 1: Density Specimen name content Thickne ss (mm) Length (m) Width (mm) Core Face sheet Direction 13 14 (kg/m 3 ) (kg/m 3 ) Beam A Core 6.35 0.61 25.4 163 N/A 1 Beam B Core 6.35 0.61 25.4 163 N/A 2 Beam C Face sheet 0.325 0.61 25.4 N/A 2161 1(2) Beam D Entire sandwich 7 0.61 25.4 163 2161 1 Beam E Entire sandwich 7 0.61 25.4 163 2161 2 Plate F Core 6.35 0.5 N/A 163 N/A o 0=? Plate G Core 6.35 0.5 N/A 163 N/A o 45=? Plate H Entire sandwich 6.35 0.5 N/A 163 N/A o 0=? Plate I Entire sandwich 6.35 0.5 N/A 163 N/A o 45=? Table 1 The dimensions and densities of the specimens tested for their static stiffness The pictures of core, face sheet and entire sandwich are shown in Figure 7. Figure 7 Core, face sheet and entire sandwich structure 2.5.2 Four-point bending method Beams A, B, C, D and E were measured for their static stiffness using the four- point bending method. For beam A, the slope of the load-deflection curve and the properties derived are given by Equations (11). 81.5 3 4 11 2 21 3 = c Sdd bh slope , 6 11 1 1005.9 1 ?== c c S Ecorefor Pa. (11) For beam B, the slope of the load-deflection curve and the properties derived are given by Equations (12). 34.3 3 4 22 2 21 3 = c Sdd bh slope , 6 22 2 1020.5 1 ?== c c S Ecorefor Pa. (12) For beam C, the slope of the load-deflection curve and the properties derived are given by Equations (13). Since the face sheet (laminate) is isotropic in the two in-plane principal directions . ll EE 21 = 22.5 3 4 11 2 21 3 = l Sdd bh slope , 10 11 12 1086.4 1 ?=== l ll S EEsheetfacefor Pa. (13) For beam D, the slope of the load-deflection curve and the properties derived are given by Equations (14). 15 03.9185 3 4 11 2 21 3 = s Sdd bh slope , 10 11 1 100449.1 1 ?== s s S Esandwichfor Pa. (14) For beam E, the slope of the load-deflection curve and the properties derived are given by Equations (15). 15.9184 3 4 22 2 21 3 = s Sdd bh slope , 10 22 2 100448.1 1 ?== s s S Esandwichfor Pa. (15) 2.5.3 Twisting method Plates F, G, H and I were measured under the configuration of twisting method. For plate F, the slope of the load-deflection curve and the properties derived are given by Equations (16). 66.109 3 4 0 2 66 3 == aS h slope c o ? , 7 66 12 1003.8 1 ?== c c S Gcorefor Pa. (16) For plate G, the slope of the load-deflection curve and the properties derived are given by Equations (17). () . 126.0 ,22.0 ,15.3 3 2 45 21 11 22 12 22 12 21 2 1222 3 ? ? ? ? ? ? ? == =?= = ? = c c c c c c c cc S S S S corefor aSS h slope ?? ? ? o (17) 16 For plate H, the slope of the load-deflection curve and the properties derived are given by Equations (18). ,80.27097 3 4 0 2 66 3 == aS h slope s o ? 10 66 12 1045.1 1 ?== s s S Gsandwichfor Pa. (18) For plate I, the slope of the load-deflection curve and the properties derived are given by Equations (19). () ? ? ? ? ? ? ? == =?= = ? = .3.0 ,3.0 ,92.7509 3 2 45 21 11 22 12 22 12 21 2 1222 3 s s s s c s s ss S S S S sandwichfor aSS h slope ?? ? ? o (19) 2.5.4 Calculation of static bending stiffness of sandwich beam From Equations (11-15), we know the properties of the core and face sheets. According to Equation (8), the effective Young?s modulus of the core and face sheets can be calculated as follows. for core 6 2112 1 1 1031.9 1 ?= ? = cc c c E E ?? Pa for principal direction 1, 6 2112 2 2 1035.5 1 ?= ? = cc c c E E ?? Pa for principal direction 2, for face sheets 10 2112 1 1009.5 1 ?= ? = ll l l E E ?? Pa for principal direction 1 or 2. (20) 17 Then using Equation (9), we can calculate the bending stiffness of the entire sandwich beams: 53.400 3 2 212 3 2 23 1 11 = ? ? ? ? ? ? ? ? +++= l lc lc l ccs t tt tt E tE D N . m for principal direction 1, 45.400 3 2 212 3 2 23 2 22 = ? ? ? ? ? ? ? ? +++= l lc lc l ccs t tt tt E tE D N . m for principal direction 2. (21) The bending stiffness per unit width of the entire sandwich beams can be calculated directly from the modulus of the sandwich beams using four-point bending method and the twisting method as follows: () () 28.335 112 2 2112 3 1 11 = ? + = ss lc s s ttE D ?? N . m for principal direction 1, () () 25.335 112 2 2112 3 2 22 = ? + = ss lc s s ttE D ?? N . m for principal direction 2. (22) 2.5.5 Summary and analysis After calculations from the measurements of the four-point bending method and twisting method, the static properties for the core and entire sandwich structures were obtained as follows: Item 1 E [ Pa] 2 E [ Pa] 12 ? 21 ? G 12 [ Pa] Core 9.05?10 6 5.20?10 6 0.22 0.126 8.03?10 7 Face sheet 4.86E?10 10 4.86?10 10 0.213 0.213 1.67?10 10 Entire sandwich 1.0449?10 10 1.0448?10 10 0.3 0.3 8.025?10 7 Table 2 Static properties obtained from the four-point bending and twisting methods 18 The bending stiffness per unit width of the entire sandwich beams were calculated in two ways, one from the properties of the core and face sheet and the other directly from the properties of the entire sandwich beams . All the properties were obtained from the four- point bending and twisting methods. The comparison of the results from these two ways (Equation 2 and 3) is given in Table 3: cclcc EEE 2112121 ,,,, ?? ssss EE 211221 ,,, ?? Bending stiffness per unit width of the entire sandwich beams From properties cclcc EEE 2112121 ,,,, ?? [N . m] From properties ssss EE 211221 ,,, ?? [N . m] s D 11 335.28 335.25 s D 22 400.53 400.45 Table 3 Static bending stiffness obtained from two ways Some conclusions can be drawn from Tables 2 and 3: 1. The shear modulus of the core and that of entire sandwich structure have very similar values, which means the core bears most of the shear and there is almost no shear in the face sheet. 2. The results of the measurements are reasonably accurate and the bending stiffnesses per unit width of the entire sandwich beams obtained from two approaches are in good agreement. 19 20 3. The measurement methods described in this chapter are useful for the determination of the static elastic properties of the core, face sheets and entire sandwich structure. 2.6 Finite element method model 2.6.1 Introduction Finite element methods have only become of significant practical use with the introduction of the digital computer in the 1950's. Ideally an elastic structure should be considered to have an infinite number of connection points or to be made up of an infinite number of elements. However, it is found that if a structure is represented by a finite, although normally large number of elements, solutions for the static or dynamic behavior of the structure may be obtained which are in good agreement with solutions found by exact methods [5]. The finite element method has been perhaps most widely used for the solution of structural problems in the aerospace industry, one example being the design of the static strength of wings. However, FEM is also widely used in other branches of engineering. In civil engineering it is used for example in the design of dams and shell structures. It can also be applied to such diverse problems as heat conduction and fluid flow. It should also be noted that the method is not confined to linear problems but can also be used in non-linear structural problems where large deformations or creep and plastic deformation occur [6]. 2.6.2 FEM in panel problems There are several ways of representing a structure in a series of finite elements. Each method has its advantages and disadvantages. The main difficulty to overcome is 21 the representation of an infinite number of connection points by a finite number. Turner et al [7] were perhaps the first to advance the concept of finite elements. Their concept attempts to overcome this difficulty by assuming the real structure to be divided into elements interconnected only at a finite number of modal points. At these points some fictitious forces, representative of the distributed stresses actually acting on the elements boundaries, are supposed to be introduced. This procedure, which at first does not seem completely convincing, has been given a firm foundation by Zienkiewicz [6]. Many engineering problems in solid mechanics are essentially impossible to solve using analytical solution techniques. To solve these types of problems, stress analysts have sought other methods. FEM has become a popular technique which yields approximate numerical solutions to difficult boundary value problems. This method of analysis can treat nonlinear problems with irregular boundary shapes and mixed conditions. The basic concept of finite element modeling is to replace the solid body to be analyzed with a network of finite elements [8]. The elements are solid elements whose properties duplicate the material they replace. These elements are then connected by nodes. As the size of the finite elements become smaller and smaller, the method yields results that are more closely related to those obtained from a rigorous mathematical analysis. The procedure for dividing a structure into finite elements can be described as follows [74]: 1) The structure is divided into a finite number of the elements by drawing a series of imaginary lines. 22 2) The elements are considered to be interconnected at a series of modal points situated on their boundaries. 3) Functions are then chosen to define the displacement in each element (and sometimes additional properties such as slope, moment, etc.) in terms of the displacement at the modal points. 4) The modal displacements are now the unknowns for the system. The displacement function now defines the state of strain in each element and together with any initial strains and the elastic constant relates it to the stress at the element boundaries. 5) A system of forces is determined at the nodes which will result in equilibrium of the boundary stresses with the distributed loading. A stiffness relationship can now be determined for each element relating the nodal forces to the nodal displacements. This method is useful for the analysis of complicated structures which cannot be approached easily using classical modal analysis. It is most useful for static problems and for dynamic problems at low frequencies. The analysis is limited at high frequencies in much the same way as classical modal analysis. In classical modal analysis, the number of modes which must be included becomes very large at high frequencies. In the case of finite elements the number of elements which must be included also becomes very large at high frequencies. It is desirable to have at least four elements per bending wavelength. In this research, the FEM technique was used to approximate the bending stiffness of sandwich panels, and to predict the load-deflection response of sandwich panel specimens. The finite element software package utilized was ANSYS. Finite element models for the experimental test method configurations discussed in section 2.6 have been created using the pre-processor in ANSYS. The basic concepts of these models are presented in this chapter, while results of the models along with a comparison with experimental and analysis results is shown in the following section. Each of the finite element models created for the different test configurations in this thesis were developed with the computer aided design pre-processor in ANSYS. The models created in ANSYS were: 1. Entire sandwich beam with the four-point bending model for the principal direction 1 2. Entire sandwich beam with the four-point bending model for the principal direction 2 3. Entire sandwich plate with the pure twisting model ( ) o 0=? The geometries of these models were chosen to be the same as those used in the actual experiments conducted on the specimens in Table 1. 2.6.3 Element Types The SHELL99 (linear layered structural shell element) may be used for layered applications of a structural shell model. The SOLID46 (8-noded, 3-D layered solid element) is essentially a 3-D version of the layered SHELL99 element type designed to model thick layered shells or solids. SOLID46 is recommended rather than SHELL99 for calculating interlaminar stresses, primarily because multiple SOLID46 elements can be stacked to allow through-the-thickness deformation slope discontinuities. The SHELL99 element was developed assuming that the shear disappears at the top and bottom surfaces of the element, while the SOLID46 element does not use such an assumption. In the 23 24 SOLID46 formulation, effective thickness-direction properties are calculated using thickness averaging. The result is that the interlaminar stresses are relatively constant though the element thickness. To calculate accurate interlaminar stresses, multiple SOLID46 elements stacked through the thickness are recommended [9]. Solid elements were used to model the sandwich structure panels to simulate the behavior of load- deflection and shear stress with the four-point bending and twisting methods. However, shell elements would likely have sufficed to simulate the load-deflection behavior of the entire sandwich structures with the four-point bending and twisting methods, because accurate stresses were not required for the current investigation. 2.6.4 Finite element modeling of sandwich structures Four-point Bending Method Modeling The sandwich beam, which is with the four-point bending method, is modeled using the elements of SOLID46. The geometry of the beam is similar as that of Beam D in Table 1 and the properties of the three layers of the beam-face sheet, core and face sheets were obtained from the results of measurements of the four-point bending and twisting methods as given in section 2.5. The out-of-plane deflection contour is shown in the following figure: Figure 8 Deflection of beam with four-point bending method The out-of-plane deformation shape with an un-deformed edge is shown in the following figure: 25 Figure 9 Out-of-plane shape with un-deformed edge of beam with four-point bending method The in-plane shear stress contour is shown in the following figure: 26 Figure 10 In-plane shear stress contour for beam with four-point bending method From figure 10, we can see that no shear stresses act over the central span. The central span, therefore, is in state of pure bending, which is the advantage of using the four-point bending method instead of the three-point bending method. Twisting Method Modeling The sandwich plate, which is with the twisting method, is modeled using the elements of SOLID46. The geometry of the plate is similar to that of Plate H in Table 1 and the properties of the three layers of the beam-face sheet, core and face sheet were obtained from the results of measurements made with the four-point bending and twisting methods as described in section 2.6. The out-of-plane deflection contour is shown in the following figure: 27 Figure 11 Out-of-plane deflection contour of plate with the twisting method The out-of-plane deformation shape with un-deformed edge is shown in the following figure: 28 Figure 12 Deformed plate with un-deformed edge of plate with the twisting method The in-plane stress contour is shown in the following figure: 29 Figure 13 In-plane shear stress contour for plate with the twisting method 2.6.5 Comparison of results from experiments and theoretical analyses Table 4 shows the load-deflection relations for Beam D obtained from the experiments and FEM analysis: Load (N) Deflection measured in experiments (m) Deflection calculated by FEM (m) 1 0.0001089 0.0001087 2 0.0002177 0.0002175 3 0.0003266 0.0003262 4 0.0004355 0.0004351 30 31 5 0.0005444 0.0005440 6 0.0006532 0.0006526 Table 4 Load-deflection relations for Beam D from the experiments and FEM analysis Table 5 shows the load-deflection relations for Plate E from the experiments and FEM analysis: Load (N) Deflection measured in experiments (m) Deflection calculated by FEM (m) 1 0.0001089 0.0001088 2 0.0002177 0.0002175 3 0.0003266 0.0003263 4 0.0004355 0.0004351 5 0.0005444 0.000544 6 0.0006532 0.0006526 Table 5 Load-deflection relations for Plate E from the experiments and FEM analyses 32 CHAPTER 3 THEORY FOR SANDWICH BEAMS 3.1 Literature review There are a large number of papers and publications on the dynamic properties of sandwich structures. Already in 1959, Hoff [9] concluded that there was an abundance of theoretical work in the field. Some of the basic theories are now also summarized in textbooks. Two of the basic examples are the books by Zenkert [10] and Whitney [11]. The bending of sandwich beams and plates is often described by means of some simplified models. Often a variational technique is used to derive the basic equations governing the vibrations of the sandwich structures. Reference is often made to the Timoshenko [12] and Mindlin [13] models. There are various types of Finite Element Models. One of the first fundamental works on the bending and buckling of sandwich plates was published by Hoff [9]. In the paper, Hamilton's principle is used to derive the differential equations governing the bending and buckling of rectangular sandwich panels subjected to transverse loads and edgewise compression. Many of basic ideas introduced by Hoff form the basis for many subsequent papers on bending of sandwich plates. Another classical paper was published by Kurtze and Waters in 1959 [14]. The aim of the paper was the development of a simple model for the prediction of sound transmission through sandwich panels. The thick core is assumed to be isotropic and only shear effects are included. Using this model, the bending stiffness of the plate is found to vary between two limits. The high frequency asymptote is determined by the bending 33 stiffness of the laminates. The model introduced by Kurtze and Waters was later somewhat improved by Dym and Lang [15]. A more general description of the bending of sandwich beams is given by Nilsson [16]. In this model, the laminates are again described as thin plates. However, the general wave equation is used to describe the displacement in the core. The influence of boundary conditions is not discussed. The model is used for the prediction of the sound transmission loss of sandwich plates. Some boundary conditions and their influence on the bending stiffness of a structure were later discussed by Sander [17]. Guyader and Lesueur [18] investigated the sound transmission through multilayered orthotropic plates. The displacement of each layer is described based on a model suggested by Sun and Whitney [19]. A considerable computational effort is required. Renji et al derived a simple differential equation governing the apparent bending of sandwich panels [20]. The model includes shear effects. Compared with measured results, this model overestimates the shear effects. In particular, in the high frequency region, the predicted bending stiffness is too low. A modified Mindlin plate theory was suggested by Liew [21]. The influence of some boundary conditions is also discussed in a subsequent paper [22]. Again a considerable computational effort is required. Maheri and Adams [23], used the Timoshenko beam equations to describe flexural vibrations of sandwich structures. In particular variations of the shear coefficient is discussed for obtaining satisfactory results. Common for many of those references is that the governing differential equations derived are of the 4th order. Due to the frequency dependence of sandwich, the solutions 34 with four unknown will agree very well for low frequency. With increasing frequency, the result normally disagrees strongly with measured vibrations. The main work on sandwich structures has been made on conventional foam-core structures with various face sheets. Little work has been done on the dynamics of honeycomb panels. In 1997, Saito et al [24] presented an article on how to identify the dynamic parameters for aluminum honeycomb panels using orthotropic Timoshenko beam theory. They used a 4th order differential equation and determined the dynamic parameters comparing their theories to frequency response measurements. Chao and Chern [25] proposed a 3-D theory for the calculation of the natural frequencies of laminated rectangular plates. The paper also includes a long reference list, each reference being classified according to the method used. Various finite element methods are often proposed for describing the vibration of sandwich panels. For example, a finite element vibration analysis of composite beams based on Hamilton's principle was presented by Shi and Lam [26]. A standard FEM code was used by Cummingham et al. [27] to determine the natural frequencies of curved sandwich panels. The agreement between predicted and measured natural frequencies is found to be very good. There are certainly a large number of methods available which describe describing the vibration of sandwich panels. However, the aim of this dissertation is the formulation of simple but sufficiently accurate differential equations governing the apparent bending of sandwich beams and plates. Boundary conditions should also be formulated for the calculation of eigen-frequencies and modes of vibration. The models should allow simple parameter studies for the optimization of the structures with respect to their acoustical performance. The aim is also to describe a simple measurement technique for determining some of the material parameters of composite beams. 3.2 Theory of sandwich structure The bending of a honeycomb panel cannot be described by means of the basic Kirchoff thin plate theory. The normal deflection of a honeycomb panel is primarily caused not only by bending but also by shear and rotation in the core. A honeycomb panel could be compared to a three-layered panel as shown in Figure 14. This figure shows the deflection of a beam due to pure bending (a) and due to shear in the core which is the thickest layer (b). Figure 14 Bending of composite bar or panel by bending (a), shearing of the core layer (b) and vibration waves in high frequency range (c). The total lateral displacement, w, of a sandwich beam is a result of the angular displacement due to bending of the core as defined by ? and the angular displacement due to shear in the core ? . Figure 14 (a) and (b) shows the bending of composite bar by 35 bending of the whole sandwich structures and shearing of the core layer. Figure 14 (c) shows the vibration waves of the whole sandwich structures in high frequency range. The relationship between w, ? and ? is given by ?? += ? ? x w . (23) For a honeycomb beam the lateral displacement w can be found when the differential equations governing the motion of the structure are determined. The differential equation can be determined using Hamilton's principle. This basic principle is for example derived and discussed in references [12, 28-33]. According to Hamilton's principle the time integral of the differential between U the potential energy per unit length, T the corresponding kinetic energy per unit length and the potential energy induced per unit length by external and conservative forces is an extremum. In mathematical terms A ( ) ?? =+? 0dxdtATU? (23) 36 In deriving the equations governing the lateral displacement of the structure shown in Figure 15, symmetry is assumed. Some properties are shown in Figure 16. The identical laminates have a Young's modulus , bending stiffness , density l E 2 D l ? and thickness . The core has effective shear stiffness , its Young's modulus , its equivalent density l t e G c E c ? and its thickness . For thick core the parameter is not necessarily equal to the shear stiffness G as suggested by Timoshenko [34]. The core itself is assumed to have no stiffness or a very low stiffness in the x-direction. In the y- direction, the core is assumed to be sufficiently stiff to ensure that the laminates move in phase within the frequency range of interest. c t e G Figure 15 Excitation of a beam and resulting forces and moments. Dimensions and material parameters for the laminates and core are indicated. areaunit per Mass 2 llcc tt ??? += ccc EG ? ll E ? Figure 16 Elastic properties and area density of sandwich structure According to paper of Nilsson [16], the bending stiffness per unit width of the beam is 37 ? ? ? ? ? ? +++=+= ?? + 3 2 212 22 3 2 23 2/ 2/ 2 2/ 0 2 1 l lc lc l cc tt t l t c t tt tt E tE dyEydyEyD lc c c . (24) In general, . The bending stiffness of one laminate is cl EE >> 12 3 2 ll tE D = . (25) The mass moment of inertia per unit width is defined as () ? ? ? ? ? ? ? ? +++== ? + 32 23 2/ 0 2 3 2 212 2 l lc lc l cc tt ttt ttt dyyyI lc ? ? ? ? , (26) while the mass per unit area is llcc tt ??? 2+= . (27) According to Hamilton's principle, as defined in equation (24), the kinetic and potential energies of the structure must be given as functions of the displacement of the beam given by w, ? and ? as in equation (23). The total potential energy of a honeycomb beam is due to pure bending of the entire beam, bending of both laminates and shear in the core. The potential energy per unit area due to pure bending of the entire beam is 1 U . 2 1 2 11 ? ? ? ? ? ? ? ? = x DU ? (28) The potential energy per unit area due to pure bending of the two laminates is 2 U . 2 22 ? ? ? ? ? ? ? ? = x DU ? (29) The potential energy per unit area due to shear deformation of the core is 3 U 38 . 2 1 2 2 ? ce tGU = (30) The total potential energy per unit width is thus () .2 2 1 0 2 2 2 2 1 0 321 ??? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? =++= L c L dxGt x D x DdxUUUUdx ? ?? (31) The kinetic energy of the honeycomb panel consists of two parts, the kinetic energy per unit area due to vertical motion of the beam 1 T 2 1 2 1 ? ? ? ? ? ? ? ? = t w T ? , (32) and the kinetic energy per unit area due to the rotation of a section of the beam 2 T 2 2 2 1 ? ? ? ? ? ? ? ? = t IT ? ? . (33) This gives the entire kinetic energy per unit width of the beam as () ??? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? =+= LL dx t I t w dxTTTdx 0 22 0 21 2 1 ? ? ? . (34) The total potential energy for the conservative external forces according to Figure 16 is (35) () () () ()[],00 0 00 1212 L LL MFwpwdxMLMwFLwFpwdxaAdx ??? ?+=+??+=? ??? [] where F is the force per unit width, M is the moment per unit width and p is the external dynamic pressure on the beam. The moments and forces are defined in Figure 16. By using the definition of ? , equation (22), and by inserting equations (31), (34) and (35) into the variational expression (23) the result is 39 [].0 2 2 0 22 2 2 2 2 1 =??? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?+ ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?? dtMFwpwdxdt t I t w tG x D x D dxdt L ce ??? ? ?? ?? ? ? (36) The integration over time is from to , and over the length x from 0 to L. Using Equation (23) and expressing 0 t 1 t ? as function of ? and x w ? ? gives [].0 2 2 0 222 2 2 2 2 2 1 =??? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ? ?? dtMFwpwdxdt t I t w x w tG xx w D x D dxdt L ce ??? ? ?? ?? ? ? (37) According to standard procedures we obtain [].0 2 0 2 2 2 2 21 =?? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ?? dtMwFwp tt I t w t w x w x w Gt xx w xx w D xx Ddxdt L c ???? ???? ? ?? ? ? ??????? ? (38) Integrating by parts gives .0 222 22 0 0 2 2 2 2 2 2 0 2 2 3 3 2 0 2 2 2 3 3 4 4 2 0 2 2 3 3 2 0 2 2 2 2 2 1 0 1 1 0 1 0 =+? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ?? ???????? ????? ????? ????? dtMwF wdxdtpdxdt t Idx t Idxdt t w wdx t w dxdt x w tGdxdt xx w wtGdt x w wtG dxdt xx w Ddt xx w Ddxdt xx w wD dt xx w wDdt xx w x w D x dxdtDdt x D L L t t t t cece L ce L LL L ??? ? ? ?? ? ??????? ??? ? ??? ? ?? ? ?? ? ? ? ? ?? ?? ? ?? ? ?? 40 (39) When assuming that the displacement is defined so that 0?y? for , and for 10 tandtt = () 0? ? ? ? ? ? ? ? ? = ? ? x y x y ? ? , Equation (39) is reduced to .02 2 2 2 2 0 2 2 2 0 2 2 21 0 2 2 3 3 2 2 2 2 2 3 3 2 2 2 1 2 2 3 3 4 4 2 2 2 = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ?+ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?? ?? dt xx w x w D dtM xx w D x D dtF xx w D x w tGw dxdt x w tG t I xx w D x D dxdtp t w xx w D xx w tGw L L L ce ce ce ?? ?? ?? ? ?? ? ??? ?? ? ?? ? ? (40) Since the expression should be equal to zero for ?? and 0?w? , we obtain five sets of equations which must be satisfied. The first two brackets in Equation (40) give two differential equations governing the displacement of the beam as expressed by w and ? . These are 02 2 2 3 3 4 4 2 2 2 =? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? p t w xx w D xx w tG ce ? ?? , (41) and 02 2 2 2 2 3 3 2 2 2 1 = ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ??? ? x w tG t I xx w D x D ce . (42) Using these two equations, eliminating ? by using the simplification and , the equation governing w is given by: (xkti x Aew ? = ? ) )(xkti x Be ? = ? ? 41 () () .2 222 2 2 2 2 21 4 4 2 2 4 4 1 22 4 21 24 6 2 6 6 12 t w IptG x p DD t w I t w x w DtG tx w tGIDD tx w ID x w DD ce cece ? ? ?? ? ? += ? ? + ? ? ? ? ? ? ? ? ? ? + ? ? + ?? ? ++? ?? ? + ? ? ? ?? ?? ? ??? (43) Eliminating instead gives the corresponding equation for w ? as () .2 222 3 3 2 4 4 2 2 4 4 1 22 4 21 24 6 2 6 6 12 x p tG x p D t I tx DtG tx tGIDD tx ID x DD ce cece ? ? + ? ? ?= ? ? + ? ? ? ? ? ? ? ? ? ? + ? ? + ?? ? ++? ?? ? + ? ? ? ? ? ? ? ?? ?? ?? ? ?? (45) The shear angle ? can be shown to satisfy the same differential equation as ? in Equation (45). They are the same results that obtained by Nilsson in [16]. Each one of the last three integrals in Equation (40) must also be equal to zero. For these conditions to be satisfied, it follows that at the boundaries of the beam the following conditions must be satisfied ,0,0 ,0,2 ,0,2 2 2 2 2 21 2 2 3 3 2 = ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? =?= ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? == ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? x w or xx w orM xx w D x D worF xx w D x w tG ce ? ? ?? ? ? (46) for x=0 and x=L. These equations satisfy the boundary conditions for a beam. For free vibration, there are no external forces and moments. Then the boundary conditions for two ends can be expressed from Equation (46) as 42 00 ,02 ,02 2 2 2 2 21 2 2 3 3 2 = ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? = ? ? ? ? ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? x w or xx w or xx w D x D wor xx w D x w tG ce ? ? ?? ? ? (47) Nilsson obtained the same boundary condition equations in [16]. Using the wave Equations (43) and (44), together with the six boundary conditions, the displacements and w ? can be determined. For free vibrations the external pressure p is equal to zero and for honeycomb panels the moment of inertia can be assumed to be very small. Using these assumptions, the resulting equations for w Equation (43) are reduced to ,0 2 2 2 2 1 4 4 22 4 1 21 6 6 2 = ? ? ? ? ? ? ? ? ? ? + ? ? ? ?? ? ? ? ? ? ? ? ? ? + + ? ? t w Dx w tG tx w D DD x w D ce ? ? (48) which is the wave equation for the bending of beams neglecting the moment of inertia. The corresponding equation for ? , Equation (43) is reduced to .0 2 2 2 2 1 4 4 22 4 1 21 6 6 2 = ? ? ? ? ? ? ? ? ? ? + ? ? ? ?? ? ? ? ? ? ? ? ? ? + + ? ? tDx tG txD DD x D ce ???? ? ? (49) As seen from the equations the expressions ? and satisfy the same differential equation for zero external exciting force. w 3.3 Boundary conditions The beam must satisfy certain boundary conditions at each end. Boundary conditions such as simply supported, clamped and free can be defined by means of the Equations (43) through (47). For each boundary condition, certain requirements for the 43 displacement and the angular displacement w x w ? ? as well as forces and bending moments must be considered. Figure 17 Particular boundary conditions The three boundary conditions for a beam shown in Figure 17 are summarized in Table 6. Clamped ends Simply supported ends 0= ? ? x w 0=? 0=w 0 2 2 = ? ? x w 0= ? ? x ? 0=w 44 Free ends 2 2 2 2 1 t I x D ? ? = ? ? ?? ? 0 2 2 = ? ? x w 0= ? ? x ? Table 6 Boundary conditions for ends of beam The natural frequencies for a clamped and a free beam are identical, according to the Euler theory. However, when shear is considered, as in Table 6, there is a difference between the natural frequencies for these two conditions. The natural frequencies for the clamped boundaries are the lowest. This is due to the fact that shear in the beam is induced by the clamped boundaries to a clamped boundaries to a larger extend as compared to the case with free boundaries. The apparent bending stiffness for the clamped beam is therefore somewhat lower than for the beam with free ends. 3.4 Wave numbers In Sections 3.2 and 3.3, the wave equation and the boundary conditions for honeycomb beams were determined. The equation governing the free flexural vibrations of a honeycomb beam is in Section 3.2 given by Equation (43) for ( ) 0=p () .0 222 4 4 2 2 4 4 1 22 4 21 24 6 2 6 6 12 = ? ? + ? ? ? ? ? ? ? ? ? ? + ? ? + ?? ? ++? ?? ? + ? ? ? t w I t w x w DtG tx w tGIDD tx w ID x w DD cece ? ??? ? ?? (50) Neglecting losses in the structure, the bending and shear stiffnesses , and are real quantities. By assuming a solution 1 D 2 D e G () .~ xkti x Aew ?? 45 the wave number must satisfy x k .0 22 2 4 1 2 1 422 11 22 4 1 2 6 2 =+ ? ? ? ? ? ? ? ? ?+ ? ? ? ? ? ? ? ? ++?? ? ? ? ? ???? ? ? ? D I D ktGk D tGI D D kI D D kD xcex ce xx (51) This equation has the six solutions 321 ,, ??? iik x ???= where the ? variables are real quantities if the losses are neglected. If the stiffness is defined as ( )?iDD += 1 0 and ()?iGG += 1 0 , losses can be included. The absolute values of the wave numbers are shown in Figure 18. 46 Figure 18 Wave numbers for beam [16]. Solid line ? real root of the wave number and Dashed and dotted lines ? purely imaginary roots Dash-dotted line ? real root Upper asymptote ? corresponds to the bending of one laminate Lower asymptote ? corresponds to the bending of the entire beam The material and geometrical parameters describing the beam are given in Table 7. The lower of the two parallel lines in Figure 19 represents the wave number corresponding to pure bending of the entire beam. The upper line represents the wave 47 number for pure bending of one of the identical laminates. The parallel lines define the low- and high-frequency limits for the wave number 1 ? for the first propagating mode. In the mid-frequency region, shear and rotation become important. As these effects increase, the wave number deviates from the lower asymptote and shifts towards the upper one. For high frequencies, the wave numbers of the structure adjust to the asymptote for the wave number for the bending waves propagating in one of the identical laminates. L (m) b (m) l t (mm) c t (mm) l ? (kg/m 3 ) c ? (kg/m 3 ) ? (kg/m 2 ) (MPa) e G (GPa) l E 1.2 0.1 0.5 10 1264 137.6 2.64 80 55 Table 7 Geometrical and material parameters for sandwich beam The wave number for a propagating wave is defined as x k x x D k 2 4 ?? = , (52) where is the apparent bending stiffness of the beam. The apparent bending stiffness is defined as the bending stiffness of an equivalent orthotropic beam of mass x D x D ? and with wave number , for the first mode of propagating bending waves. Considering this definition, the agreement with the asymptotes confirms that the bending stiffness for low frequencies is determined as the bending stiffness of the entire beam given in Equation (24). For high frequencies the bending stiffness is determined by the bending of the laminates only, as given by Equation (25). The bending stiffness given in Equations (24) and (25) will give the limiting values for the calculated bending stiffness. x k 48 The transition of the bending stiffness between the two asymptotes is determined by shear and rotation in the core. The dotted and the dashed lines in Figure 18 represent the purely imaginary roots given by 2 ? and 3 ? and correspond to the near field solutions or the evanescent waves for the in-phase motion of the laminates. The constant value for low frequencies, the dotted line, is determined by the thickness of the core. For increasing frequencies, this curve approaches the limit determined by the wave number for evanescent waves in one of the identical laminates. Decreasing the thickness of the core will increase the constant value in the lowest frequency range. The other near field solution closely follows the asymptote for the bending of the entire beam for low frequencies. In summary the limiting values for the wave numbers are () , 2 lim;lim 4/1 21 21 2 1 4/1 1 2 0 1 ? ? ? ? ? ? + = ? ? ? ? ? ? = ??? DD DD D ff ?? ? ?? ? (53) ,lim;lim 2/1 1 2 2 4/1 1 2 0 2 ? ? ? ? ? ? ? ? = ? ? ? ? ? ? = ??? D I D ff ? ? ?? ? ? (54) , 2 lim; 2 lim 4/1 2 2 3 2/1 2 0 3 ? ? ? ? ? ? = ? ? ? ? ? ? = ??? DD GH ff ?? ?? (55) or 2 ? we see a minimum at a certain frequency , where p f 2 ? shifts from being entirely complex to real and describes a propagating wave. As the frequency increases, 2 ? approaches zero for p ff = where ? ? I tG f ce p 2 1 = . (56) 49 Below this frequency , p f 2 ? is imaginary, and describes evanescent waves. For higher frequencies 2 ? is real and describe propagating waves. 3.5 Least Squares Method In the previous section the wave equation governing the displacement of a honeycomb beam was derived. Based on this differential equation, wave numbers, natural frequencies and modes of vibration can be determined for different boundary conditions. For the response of a beam to be calculated, all the material parameters of the beam must be known. The dynamic properties of a composite beam are not always well defined. This is due to the fact that the elements of the assembled structure perform differently. However, the main dynamic properties of a composite beam can be determined from measurements of the first few natural frequencies when the structure is freely suspended. The wave number for the first propagating bending mode is x k x x D k 2 4 ?? = , (57) where is the apparent bending stiffness of the structure. Consequently is defined as the bending stiffness of a simple homogeneous beam, which at a certain frequency has the same dynamic properties as the honeycomb structure. By inserting the definition for in the wave equation (51) the result becomes x D x D x k 02 2 2/1 1 2/3 2/1 =?+ ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? DDD D DtG xx xce ?? , (58) 50 if the moment of inertia is neglected. In the low frequency range, or as 0?? , the first part of the equation dominates why . The bending stiffness is consequently determined by pure bending of the beam. 1 DD x ? In the high frequency range, when ??? . For high frequencies, the laminates are assumed to move in phase. In this frequency range, the bending stiffness for the entire beam is equal to the sum of the bending stiffnesses of the laminates. This result agrees with the results discussed in the previous sections. 2 2DD x ? For a composite beam, the wave number for flexural waves is defined by Equation (57). For a beam with boundary conditions well defined, the bending stiffness can be determined by means of simple measurements. The natural frequencies- for a beam of length-L are given by the expression x k n f ;, 4 4/1 22 n xn n n L D f L ? ?? ? = ? ? ? ? ? ? = n=1, 2, 3? ,(59)where n ? for a beam with various boundaries is given in Table 8. Boundary conditions n 1 2 3 4 5 n>5 Free-free Clamped-clamped n ? 4.73 7.85 11.00 14.14 17.28 2/?? +n Free-clamped n ? 1.88 4.69 7.85 11.00 14.14 2/?? ?n Simply supported n ? 3.14 6.28 9.42 12.56 15.7 ?n Table 8 Values of for particular boundary conditions n ? 51 Measurements reveal the fundamental natural frequency of the beam. By arranging Equation (59), the apparent bending stiffness for mode n having the natural frequency is for a beam of length ? and mass per unit area xn D n f L ? , given by ...3,2,1 4 4 422 == nfor Lf D n n xn ? ?? . (60) The bending stiffness of a composite beam is strongly frequency dependent as given by equation (60). Since most of the parameters are unknown, only the frequency dependent parameters are preserved and the equation is rewritten as ,0 2/12/3 =?+? CDD f B D f A xxx (61) where .2; 2 ; 2 2 2/1 1 2/1 DC tG B D tG A cece === ???? (62) For non-metallic materials, Young's modulus can exhibit frequency dependence as discussed in, for example, Reference [12] and demonstrated in References [32] and [33]. However, within the frequency range of interest, here up to 4 kHz, the parameters , and in Equation (62) are assumed to be constant for the structures investigated. Using the measured data the constants A, B and C can be determined by means of the least squares method. The quantity Q is defined by 1 D 2 D G , 2 2/12/3 ? ? ? ? ? ? ? ? ? ?+?= i xixi i xi i CDD f B D f A Q (63) where is the measured bending stiffness at the specific frequency . The constants A, B and C are chosen to give the minimum of Q. The shear modulus and xi D i f c G 52 the bending stiffness and can be determined, once the parameters A, B and C are predicted. With known constants A, B and C the dynamic parameters can be determined 1 D 2 D ./2 ,2/ 2/1 2 ce tBG CD ??= = (64) The parameters are predicted from the MATLAB program. 3.6 Damping measurement methods Basically there are four measures of damping, the loss factor ?, the quality factor Q, the damping ratio ? , and the imaginary part of the complex modulus. However, they are related to each other. The loss factor or damping ratio is used in measurements: ?? ? ? tan 2 2 1 2 = ? ?? ===== E E C C QW D c . (65) Here D and W are the dissipated and total powers in one cycle of vibration, C and C c are the damping coefficient and the critical damping, 'E and "E are the real and imaginary parts of the complex modulus. Many references present reviews of damping measurements [35-39]. Generally, there are three sorts of experimental methods. Decay rate method This method can be used to measure the damping of a single resonance mode or the average of a group of modes in a frequency band. The structure is given an excitation by a force in a given frequency band, the excitation is cut off, the output of the transducer is passed through a band pass filter and then the envelope of the decay is observed. The damping ratio can be calculated from the slope of the envelope of the log magnitude-time 53 plot, as shown in Figure 19. One of the disadvantages of this method is that the effect of 54 noise is considerable. Figure 19 Decay rate method used to determine damping mn n A = ln 1 ? , (66) Am + () 22 2 ?? ? ? + = , (67) where ? is the decay rate. se function (log magnitude-time plot or Nyquist diagram Modal bandwidth method With the frequency respon ), the modal bandwidth method, also called as half-power point method is the most common form used to determine the damping (shown in Figure 20). This method applies only to the determination of the damping of a single mode. The shortcoming of this method is that the repeatability is, in general, rather poor. The loss factor or the damping can therefore only be estimated, based on averages of several measurements. For sandwich structures, the loss factor is frequency dependent. In this work, the same experimental set up was used to determine the dynamic stiffness and to estimate the damping for the sandwich structures. Figure 20 Modal bandwidth method to determine damping , 2 12 ff ? =? . (68) n f where and are the closest frequencies at which the power is dropped 3 dB from od d on the relationship between the input power and the dissipa 1 f 2 f that at the mode frequency. Power balance meth The SEA method is base ted power. So the loss factor can be determined by measuring the input power and the total energy of a modal subsystem (shown in Figure 21). Assuming a stationary 55 power input at a fixed location, the input power must be equal to the dissipated power under steady state conditions. The disadvantage of this method is that it is relatively hard to determine the input power and the total energy of a modal subsystem. Figure 21 Power balance method to determine damping tot in W =? , W2 (69) where is the input power and is the total energy of a modal subsystem. in W tot W 56 CHAPTER 4 DYNAMIC PROPERTIES CHARACTERIZATION 4.1 Experiments 4.1.1 Steps The steps used to determine the dynamic stiffness and static properties of entire beams and face plates are as follows: 57 Figure 22 Experimental steps to determine some properties of sandwich structures These steps provide an interesting approach to determine some parameters of sandwich structures. 4.1.2 Set up Since honeycomb plates are typically anisotropic, measurements are performed on beams representing the two main in-plane directions of the plate. For Calculate dynamic bending stiffness, , for mode n n D Determine parameters A, B and C Draw dynamic stiffness curve x D Calculate dynamic propertiesG 21 ,, DD e Measure natural frequencies n f Calculate damping for natural frequencies n f materials tested in two directions, the structure and the results are given a subscript of 1 or 2 to indicate the orientations of the beam. The tests have been made for different boundary conditions. Due to the low mass of the material ~ kg/m5.2 2 an accelerometer would have a certain influence on the vibrations of the beam. The vibration measurements were therefore made with a laser vibrometer to avoid transducer contact. The frequency response function, FRF, was determined to give the natural frequencies for the beam. Based on the frequency response function, the loss factor or damping was also determined by the Modal Bandwidth Method. The experiment set up 1 is shown in Figure 23. The set up includes: a B&K Pulse System (Sound & Vibration Multi-analyzer), laser vibrometer, sample, shaker and power amplifier. The sample shown in the figure has simply supported ends. The other boundaries such as free-free and clamped-clamped were tested as well. The B&K Pulse system gives the output of the frequency response function of the beam when it is excited by white noise. The advantage of set up 1 is that high frequencies can be excited. But one of the disadvantages is that mass is added to the lightweight structure by the probe or the needle of the shaker. Since the sample is very light weight, a little weight added can influence the result to some degree. The other disadvantage is that some resonances may be missed since the probe of the shaker effects the configuration of the beam in some cases. Set up 2 is shown in Figure 24, which includes a B&K pulse system, laser vibrometer, modal hammer and sample. The modal hammer gives an impulse to the 58 sample. The dynamic signal analyzer provides the output of the frequency response function of the beam when it is excited by the impulse given by modal hammer. The advantage of the set up 2 is that no mass is added to the lightweight structure so there is no mass loading problem. The disadvantage is that it is hard to high frequency vibration with the modal hammer. In measurements made, these two set ups were combined to obtain the natural frequencies in the frequency band of interest. Sound & Vibration Multi- analyzer Pulse system V(t) Laser vibrometer Sample Force transducer F(t) Shaker White noise Power amplifier Figure 23 Set Up 1 Using shaker 59 Sound & Vibration Multi-analyzer Pulse system F(t) V(t) Laser Vibrometer Modal Hammer Sample (b) Figure 24 Set Up 2 Using hammer 4.1.3 Samples The composite sandwich structures with honeycomb cores and sandwich structures with closed-cell foam cores, used in the experiments, are shown in Figures 25 and 26. The face sheets are made with woven cloth impregnated with a resin, and the core is a lightweight foam-filled honeycomb structure and closed-cell foam. 60 61 The geometry and density of the sample beams used in the experiments are shown in the Tables 9 and 10. Beam No. Beam A Beam B Beam C Beam D Content Honeycomb core filled with foam Honeycomb core filled with foam Entire sandwich beam with foam filled honeycomb core Entire sandwich beam with foam filled honeycomb core Length (m) 0.6 0.6 0.6 0.6 Thickness(t) (m) 0.00635 0.00635 0.00705 0.00705 Width (m) 0.0254 0.0254 0.0254 0.0254 Density (kg/m 3 ) 162.73 162.73 362.331 362.331 Density (kg/m 2 ) 1.03334 1.03334 2.554 2.554 Direction 1 2 1 2 Face sheet N/A N/A Single sheet Single sheet Table 9 Geometry and density of the sandwich beam with foam filled honeycomb core. Beam No. Beam E Beam F Beam G Beam H Content Entire sandwich Entire sandwich Entire sandwich Entire sandwich beam with foam core beam with foam core beam with foam core beam with foam core Length (m) 0.457 0.612 0.854 0.608 Thickness(t) (m) 0.0256 0.0256 0.0256 0.0256 Width (m) 0.029 0.029 0.028 0.030 Density (kg/m 3 ) 343.148 292.3406 334.5601 268.3411 Density (kg/m 2 ) 8.9218 7.6009 8.6986 6.9769 Table 10 Geometry and density of the sandwich beam with foam core. Figure 25 Honeycomb sandwich composite structures; (a) foam-filled honeycomb core, (b) composite beam 62 Figure 26 Closed-cell foam core 4.2 Analysis of experimental results 4.2.1 Frequency response functions Some typical measured frequency response functions for beams C, E, F and G with free edges are shown in Figures 27, 28, 29 and 30. For low frequencies, the peaks are easily resolved. For increasing frequencies, the peaks are not easily separated. This is due to the hammer excitation. It is difficult to excite the higher modes using an impact hammer. The peaks representing the natural frequencies for the beams are marked in the figures. The unmarked peaks result from the mounting of the beams and twisting modes. FRF for Beam F -150 -140 -130 -120 -110 -100 -90 -80 -70 0 500 1000 1500 2000 Frequency [Hz] Relat i ve L e ve l [d B] Figure 27 FRF for the sandwich beam F for free-free boundary condition. 63 FRF for Beam G -7.00E+01 -6.00E+01 -5.00E+01 -4.00E+01 -3.00E+01 -2.00E+01 -1.00E+01 0.00E+00 1.00E+01 2.00E+01 0.00E+00 5.00E+02 1.00E+03 1.50E+03 2.00E+03 2.50E+03 3.00E+03 Frequency [Hz] Rel a t i ve L e ve l [d B] Figure 28 FRF for the sandwich beam G for free-free boundary condition. 64 FRF for Beam E -7.00E+01 -6.00E+01 -5.00E+01 -4.00E+01 -3.00E+01 -2.00E+01 -1.00E+01 0.00E+00 1.00E+01 0.00E+00 5.00E+02 1.00E+03 1.50E+03 2.00E+03 2.50E+03 3.00E+03 Frequency [Hz] Relat i ve L e v e l [d B] Figure 29 FRF for the sandwich beam E for free-free boundary condition. The natural frequency values vary slightly for different measurements, although the repeatability is rather good. Four measurements on the same beam are compared in Table 11. The set-ups were disassembled between measurements. Measurement 1 Measurement 2 Measurement 3 Measurement 4 r f [Hz] D [Nm] r f [Hz] D [Nm] r f [Hz] D [Nm] r f [Hz] D [Nm] 294 6771.4 296 6863.8 295 6817.5 296 6863.8 705 5046.2 707 5074.9 706 5060.5 706 5060.5 1194 3767.8 1197 3786.7 1195 3774.1 1196 3780.4 1683 2739.5 1686 2749.2 1684 2742.7 1685 2746.0 Table 11 Natural frequencies for different measurements on Beam F and their corresponding bending stiffness. 65 The relative error, , in the bending stiffness is equal to where is the uncertainty in the recorded eigenfrequency of the beam. Consequently the error is fairly large in the low frequency region. For high frequencies, the result is less sensitive to this type of variation. The table reveals some variations in the bending stiffness in the low frequency range. For increasing frequencies the differences are negligible. DD /? ff /2? f? D? 4.2.2 Boundary conditions In the measurements, the boundary conditions were found to have some effect. For a clamped beam, shear is induced at the boundaries, thus rendering the beam more flexible as compared to a beam with free ends. The natural frequencies of a clamped beam are consequently lower than the corresponding natural frequencies for the same beam with free ends. Hence, the frequency-dependent stiffness is affected by the boundaries. The natural frequencies for the three special boundary conditions are shown in the Table 12. This is due to the fact that shear in the beam is induced by the clamped boundaries to a larger extend as compared to the case with free boundaries. The apparent bending stiffness for the clamped beam is therefore somewhat lower than that for the beam with free ends as shown in Figure 30. Boundary conditions Free-free Simply supported- simply supported Clamped-clamped Mode Frequencies (Hz) 1 119 53.7 118 2 317.5 206.5 308.5 66 67 3 583 432.5 562 4 883 711 864 5 1206.5 1024 1177.5 6 1539 1335 1498 7 1890 1676 1835 8 2246 2011 2175 9 2806.1 2542.2 2789 10 3247.3 2963.8 3228 Table 12 Natural frequencies of Beam C for the three different beam boundary conditions. 10 0 10 1 10 2 10 3 10 4 0 50 100 150 200 250 300 350 400 450 bending stiffness for beam frequency [Hz] bend i n g s t i f f nes s [ N m ] Figure 30 Bending stiffness of the sandwich beam C for three different boundary conditions. Measurement points *, and : bending stiffness for the natural frequencies of the beam with free, simply supported and clamped ends, respectively. + o Solid, dashed and dotted curves: calculated static bending stiffness of the beam with free, simply supported and clamped ends, respectively. 68 4.2.2 Dynamic bending stiffness The dynamic bending stiffness for the sandwich Beam C is shown in Figure 31. The bending stiffness is frequency dependent and decreases with increasing frequency. The bending stiffness at low frequency is dominated by the bending stiffness of the entire beam. At high frequency, the bending stiffness is dominated by the bending of the face plates only. 10 0 10 1 10 2 10 3 10 4 0 50 100 150 200 250 300 350 400 bending stiffness for beam frequency [Hz] bend i n g s t i f f nes s [ N m ] 69 Figure 31 Bending stiffness for the sandwich beam C. Measurement points *: bending stiffness for the natural frequencies. Curves -: calculated static bending stiffness, based on material parameters for face plates and core. The static stiffness of the entire beam and the static stiffness of the faceplate are determined by use of the four-point bending method. In the low frequency range, when 1 D 2 D 0?? , the apparent dynamic stiffness is close to . The bending stiffness is consequently determined by pure bending of the beam. In the high frequency range, when 1 D ??? , and then the apparent dynamic stiffness is close to . The face plates are assumed to move in phase. In this frequency range, the bending stiffness for the entire beam is equal to the sum of the bending stiffness of the two face plates. The comparison is as shown in Table 13: 2 2D Static stiffness by four-point method Two stiffness limits from dynamic characterization 1 D [Nm] 2 D [Nm] 0?? [Nm] ??? [Nm] 344 24 345.2 48.6 Table 13 Comparison of static stiffness measured by four-point bending method and two stiffness limits from dynamic characterization. The dynamic stiffness plots in the two principal directions are shown in Figure 32: 70 10 0 10 1 10 2 10 3 10 4 0 50 100 150 200 250 300 350 400 450 bending stiffness for beam frequency [Hz] be nd i n g s t i f f n es s [ N m ] Figure 32 Dynamic stiffness for beams C and D From Figure 32, we can see that the stiffness in the two principal directions is in very good agreement. The reason is that the stiffness of entire sandwich structures is dominated by the face sheet while the stiffness of the face sheets for the sandwich structures, usually, is much higher than that of the core. 71 The dynamic bending stiffness for the core in two directions is shown in Figure 33. 10 0 10 1 10 2 10 3 10 4 1 2 3 4 5 6 7 bending stiffness for beam frequency [Hz] bendi n g s t i f f nes s [ N m ] Figure 33 Dynamic stiffness for core in two principal directions 72 Since the core is orthotropic, the stiffness is different in the two principal directions. But the effect of orthotropy on the stiffness of the whole sandwich structures is very small. 10 0 10 1 10 2 10 3 10 4 1000 2000 3000 4000 5000 6000 7000 8000 bending stiffness for beam frequency [Hz] ben di ng s t i f f n es s [ N m ] Figure 34 Bending stiffness for the sandwich beam G. The dynamic bending stiffness for Beam G is shown in Figure 34. Since Beam G is the longest beam among the beam samples E, F, G and H, eight natural frequencies could be measured using the hammer method but only four or five could be measured for the others. So this implies that the longer is the beam, the more 73 natural frequencies and the lower is the first natural frequency and the more accurate stiffness curve that can be obtained. c G by twisting method c G from dynamic characterization 5.189?10 7 Pa 5.226?10 7 Pa Table 14 Comparison of shear modulus of the core measured by twisting method and that from the dynamic stiffness curve for Beam G. From Table 14, it can be seen that the shear modulus of the core measured by twisting method and that from dynamic characterization are in good agreement. 74 10 0 10 1 10 2 10 3 10 4 0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 bending stiffness for beam frequency [Hz] be ndi ng s t i f f nes s [ N m ] Figure 35 Bending stiffness for the sandwich beam E, F and H. Red line: Beam E; Green line: Beam F; Blue line: Beam H The dynamic bending stiffness curves for beams E, F, and G are shown in Figure 35. Note that the order of the area density for beams E, F and G is from the smallest to the largest is E>F>G. From this plot, such conclusions can be drawn that the stiffness increases if the area density increases. 75 76 4.2.3 Damping The damping ratio of beam C is given in this table: Natural frequency (Hz) Damping ratio (%) 53.7 0.315 206.5 0.367 432.5 0.471 711 0.492 1024 0.479 1335 0.466 1676 0.5 2011 0.557 2542.2 0.689 2963.8 0.81 Table 15 Damping ratio of Beam C The damping for beam C is shown in this figure. damping of sandwich Beam C with 1/4 in thickness and single face sheet 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 500 1000 1500 2000 2500 3000 3500 Frequency (Hz) D a m p i ng ra t i o ( % ) Figure 36 Damping ratio of sandwich beam C This figure proves that the damping for the sandwich structures under investigation is very high compared to the traditional materials such as aluminum. 4.3 Conclusions Several theoretical models for the determination of the dynamic bending stiffness of sandwich beams were reviewed. Nilsson?s sixth-order differential equation model was discussed. A simple measurement technique for determining the material parameters of composite beams was used. The experimental results show that this technique can be used to determine the dynamic stiffness of composite sandwich beams. 77 CHAPTER 5 SOUND TRANSMISSION LOSS OF SANDWICH PANELS The sound transmission loss (TL), or sound reduction index, is a measure of the sound insulation provided by a wall or other structural element. Sound transmission loss (TL) of a panel is given by TL ? ? ? ? ? ? ? ? = T I W W lg10 , (70) where is sound power incident on the panel and is sound power transmitted through the panel. Since the sound transmission loss is dependent on the frequency of the sound, it is usually reported for each octave band or each one-third octave band. This number indicates the noise insulation capability of the panel. The sound transmission loss is often an important consideration in the analysis and design of partitions or panels separating adjoining spaces in industry, housing and various types of vehicles. I W T W 5.1 Classical sound transmission analysis The classical sound transmission analysis theories are described by Wilson in the book [80]. The sound transmission coefficient is the fraction of the sound power transmitted through a wall or barrier. Thus: I T W W =? , (71) 78 where ? is the sound transmission coefficient. Comparing the definitions of transmission loss and sound transmission coefficient, we have ? ? ? ? ? ? = ? 1 lg10TL , (72) or . (73) 10/TL 10 ? =? 5.1.1 Mass law sound transmission theory If the panel is thin, that is, the panel thickness is small compared with a wavelength of sound in air, and if the panel stiffness and damping may be neglected, then the sound transmission is mass-controlled. The mass law transmission loss equation based on theoretical considerations is: , 2 cos 1lg10 1 lg10TL 2 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? += ? ? ? ? ? ? = c m ? ?? ? (74) where )Hz(2)rad/s( f?? = =sound frequency m = panel mass density per unit surface area , 2 kg/m ? = angle of incidence, ? = air density, c = speed of sound in air. If the angle of incidence is equal to zero, the above equation becomes: ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? += 2 0 2 1lg10TL c m ? ? , (75) 79 the normal incidence mass law TL for approximating the transmission loss of panels with incident sound in the mass-controlled frequency. o 0 When studying transmission loss between two rooms, the sound source may produce a reverberant field on the source side. Thus, the incident sound may strike the wall at any angle between 0 and . For field incidence, it is assumed that the angle of incidence lies between 0 and . This results in a field incidence transmission loss of about o 90 o 72 dB5TLTL 0 ?= . (76) The following additional changes are made in the mass law equation. Converting the frequency of the sound from radians per second to hertz, where ),Hz(2)ad/s( fr ?? = (77) using typical values for the speed of sound in air and air density ( )elyapproximatc skg/m400 2 ?=? , (78) and assuming that 1 2 >> c m ? ? , (79) We obtain the field incidence mass law equation ( ) dB.47lg20TL ?= fm (80) Based on the above equation for the mass-controlled frequency region, the transmission loss of a panel increases by 6 dB per octave. In addition, a doubling of panel thickness or doubling of panel mass per unit area should produce a 6 dB increase in transmission loss at a given frequency. While the above equations are useful for prediction of material behavior, the transmission loss of actual structural elements should be confirmed by laboratory or field testing whenever possible. 80 5.1.2 The effect of panel stiffness and damping Low-frequency sound transmission is governed largely by panel bending stiffness. Transmission of somewhat higher frequency sound is governed by panel resonances. The sound transmission loss of the panel may be considered mass controlled when responding to frequencies above twice the lowest resonance frequency, but below the critical frequency. The sound transmission coefficient equation for panels with significant bending stiffness and damping is given by Ver and Holmer (1971) as follows: , sin 1 2 cossin 2 cos 1 1 2 4 42 2 4 42 ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? += mc B c m mc B c m ?? ? ???? ? ?? ?? (81) where it is assumed that the thickness of the panel is small compared with the wavelength of the incident sound, and B = panel bending stiffness ( mN? ), = panel surface density ( ), m 2 kg/m ? = composite loss factor (dimensionless). 5.1.3 The coincidence effect Sound in panels and other structural elements can be converted into vibration and propagated as bending waves, longitudinal waves, and transverse waves. Bending waves are of particular importance because of the coincidence effect. Figure 37 shows a panel with an airborne sound wave of wavelength ? and incident at angle? . Assume bending wave of wavelength B ? is excited in the panel. The propagation velocity of bending 81 waves is frequency dependent, with higher frequencies of vibration corresponding to higher propagation velocities. The coincidence effect occurs when ,arcsin * ? ? ? ? ? ? ? ? == B ? ? ?? (82) where the asterisk (*) is used to indicate coincidence and is called the coincidence angle. When wave coincidence occurs, the sound pressure on the surface of the panel is in phase with the bending displacement. The result is high-efficiency energy transfer from airborne sound waves in the source space to bending waves in the panel, and thence to airborne sound waves in the receiving space. Efficient sound energy transfer from air space to air space is obviously undesirable from a noise control standpoint. Figure 38 is an idealized plot of transmission loss versus frequency, showing the stiffness-controlled, resonance-controlled, mass-controlled, and coincidence- controlled regions for a panel. Note that the transmission loss curve dips substantially at frequencies beyond the critical frequency due to the coincidence effect. * ? 82 Figure 37 The coincidence effect 83 Figure 38 Idealized plot of transmission loss versus frequency f (Octave scale) Stiffness controlled Resonance controlled Mass controlled Coincidence controlled 6 dB/octave slope T L ( d B ) Coincidence dip 5.1.4 Critical frequency It can be seen from Equation (82) that the coincidence effect cannot occur if the wavelength of airborne sound ? is greater than B ? , the bending wavelength in the panel. The lowest coincidence frequency, called the critical frequency, occurs at the critical airborne sound wavelength , (84) B ??? == * where critical frequency * * ? c f = , (85) which corresponds to grazing incidence ( ) o 90=? . For any frequency above the critical frequency, there is a critical angle at which coincidence will occur. * ? Unfortunately, the critical frequency often falls within the range of speech frequencies, thereby limiting the effectiveness of some partitions intended to provided privacy and prevent speech interference. 5.1.5 Sound transmission coefficient and transmission loss at coincidence The coincidence effect depends on the characteristics of the plate or panel and on the airborne sound wave. Coincidence occurs when 1 sin 4 42 = mc B ?? . (86) Substituting the above condition in the equation for the sound transmission coefficient of a plate results in the coincidence condition sound transmission coefficient, 2 * * 2 cos 1 1 ? ? ? ? ? ? + == c m ? ??? ?? , (87) 84 where ? is the loss factor of the panel. and the coincidence condition transmission loss [72] . cos 1lg20 2 cos 1lg10 1 lg10 , 2 cos 1 1 * 2 * * * 2 * * ? ? ? ? ? ? ? ? += ? ? ? ? ? ? ? ? += ? ? ? ? ? ? == ? ? ? ? ? ? ? ? ? ? ? ? ? ? + == c fm c m TLTL c m ? ??? ? ??? ? ? ?? ? ?? (88) Based on the above equations, we should predict and (89) 1 * =? ,0 * =TL for undamped panels (i.e., if loss factor 0=? . There is some damping, however, in all construction materials. Note that the above transmission loss equation is based on theoretical behavior of an infinite plate. The actual constraints of windows and walls may produce a different response to sound waves. It is recommended that the behavior of actual materials be verified by laboratory or field tests. 5.2 Literature review of the Sound Transmission Loss of Sandwich Panels The purpose of research on the sound transmission loss of sandwich structures is to improve the sound transmission loss without compromising the stiffness to density ratio so that the critical frequency of the panel can be raised out of the audio frequency range. As early as 1959, Kurtze and Watters designed some kinds of sandwich plates and analyzed their acoustical behavior [40]. They inserted a soft core between two thin face sheets to introduce shear waves in the middle frequency range. If the shear wave speed is less than the speed of sound in air, the critical frequency is then shifted to higher frequencies, which avoids locating the critical frequency in the range of interest. But one of the shortcomings in their model is that they assumed that the core material is 85 86 incompressible, which is impossible in reality for soft core. Although their experimental results agree with the theoretical estimation of the anti-symmetric motion, their model cannot predict symmetric motion. In the work of Ford, Lord and Walker, they assumed the polyurethane foam core to be a compressible material, and they studied both the anti-symmetric and symmetric modes of vibration [41]. Smolenski and Krokosky corrected the energy expression used in [42] and investigated the influence of the core material properties on the critical frequency due to the symmetric mode [43]. Dym and Lang derived five equations of motion, representing the symmetric and anti-symmetric vibration modes, for sandwich panels with identical face sheets in [44]. By using their five equations, TL can be calculated for both the symmetric and anti- symmetric cases. Lang and Dym presented optimal TL properties for sandwich panels in [44] and they found that by increasing in the stiffness of the core the coincidence effect caused by the symmetric vibration mode could be eliminated, but the anti-symmetric coincidence effect would still occur at a low frequency. Since in some cases, the core of sandwich structures is orthotropic and the face sheets are not identical, Dym and Lang expanded their theoretical model to include these cases in [46]. One of the improvements is assumption of damping in both the face sheets and the core [46]. Ordubadi and Lyon studied the effect of orthotropy on the sound transmission through plywood panels [48]. They presented an analytical expression for the TL of such panels. The most important contribution of the study of Narayanan and Shanbhag is that they found the transmission loss is more sensitive to the variation of the core shear 87 parameter than to the change of other parameters. However, in their models, they did not consider symmetric modes. And also they did not integrate the TL results calculated by their model at some particular angles of incidence to obtain the field incidence representation. Moore and Lyon developed an analytical model for sandwich panels with isotropic and orthotropic cores [50]. They considered both symmetric and anti-symmetric modes. They further developed a design approach which lowers the double wall resonance frequency to below the frequency band of interest, and shifts the critical frequency to higher frequencies. Wang, Sokolinsky, Rajaram and Nutt derived expressions to predict the TL in infinitely wide sandwich panels using two models, (1) the consistent high-order approach, and (2) the two-parameter foundation model [52]. In both the models, the TL is calculated using a decoupled equation which represents the symmetric and anti- symmetric motions. They compared their numerical prediction with experimental results. The consistent high-order approach is more accurate, while the two-parameter fundamental model is more convenient. The TL of multi-layer panels has also been analyzed by some other researchers. Guyander and Lesueur studied the equation of motion, the modal density and the TL of viscoelastic and orthotropic multi-layer plates [53-55]. They used both plane wave and reverberant sound excitations to study the TL. Panneton and Atalla developed a three- dimensional finite element model to predict the TL through a multi-layer system made of elastic, acoustic and porous-elastic media [56]. The three-dimensional Biot theory was used to model the porous-elastic medium. However, at low frequencies (lower than 100 Hz, and sometimes even 200 Hz), the predicted behavior is completely incorrect. For higher frequencies, the model is only useful for unbonded plates. Kurra and Arditi used the ASTM and ISO standards to measure the sound transmission loss of multi-layered plates [57, 58]. Uris and Estelles studied the sound transmission of multi-layered sandwich plates using different configurations of polyurethane and polystyrene layers [59]. They found that multi-layered sandwich plates possess better sound transmission loss, and the coincidence effect is not as obvious as with three-layer sandwich plates. These observations result from the fact that the multi-layered plates are much thicker and the increased surface densities help to increase the sound transmission loss. 5.3 Statistical energy analysis model (SEA) 5.3.1 Prediction of Sound Transmission through Sandwich Panels using SEA SEA method was first developed by Lyon and others in the 1960?s and later Lyon gave the detailed description of theory and application about SEA in [66]. Crocker and Price used SEA to predict the sound transmission loss of isotropic single-layered panels [81]. Here we used the same theoretical model to predict the sound transmission loss of sandwich panels. The subsystems and energy flow relationship are illustrated schematically in Figure 40. The source room and the receiving room are the first and third subsystems, and the panel under study is the second subsystem. Here the two rooms are assumed to be reverberant. This means that the sound pressure level measured in each room is the same at any position in that particular case. is the power input from loudspeakers in the in 1 ? 88 source room, is the power dissipated in the i-th subsystem, and is the power flow from the i-th to the j-th subsystem. diss i ? ij ? Figure 39 Schematic of the power flow in three-coupled systems using SEA If only the source room is excited using loudspeakers and there is no other power input to the other subsystems, the power balance equations can be expressed as: , (90) 131211 ?+?+?=? dissin , (91) 23212 ?+?=? diss . (92) 13233 ?+?=? diss The power dissipated in a system in a specified frequency band is related to the energy stored in the system, , through the internal loss factor i E i ? , namely, , (93) ii diss i E ??=? where ? is the center frequency of the frequency band. The power flow between subsystems and is i j ? ? ? ? ? ? ? ? ?=? j j i i iijij n E n E n?? , (94) 89 where n i and n j are the modal densities of subsystems i and j; ? ij is the coupling loss factor from subsystem i to subsystem j. The equation (95) must be satisfied. i j ji ij n n = ? ? , (95) 33 2 2 air i i c V n ? ? = and similar to , (96) j n where is the room volume. i V Applying Equations (94) to (90)-(92) gives the power balance of the partition in a frequency band with the center frequency ?: 23222 2 2 1 1 22112 ?????? EE n E n E n += ? ? ? ? ? ? ? ? ?=? , (97) 33 1 1 33113 ???? E n E n ==? . (98) Note that generally the sound pressure level in the source room is significantly greater than that in the receiving room, 33 1 /En En 1 /<< , so the term can be neglected in Equation (95). The subsystems are coupled and the coupling loss factors ? 3 /En 3 23 and ? 23 are related to the sound radiation efficiency ? rad : ?? ?? ??? ? s rad rad c 2 2321 === , (99) where rad ? is the acoustic radiation loss factor of the panel and the subscript 2? means that 2 rad ? ? represents the one-sided radiation efficiency. Thus Equations (91) and (92) can be reduced: 90 , 2 21 2 2 2 ? ? ? ? ? ? ? ? + = rad rad n E n E ?? ? (99) . 32313 232131 3 ??? ?? ++ + = EE E (100) Substituting Equations (99) into (100), gives the ratio of energy between two rooms, . 32313 232131 3 ??? ?? ++ + = EE E (101) Assuming the two rooms are large enough, , and ignoring high order of loss terms, 231 , nnn >> . 13 3 3 1 3 1 ? ? += n n E E (102) The coupling loss factor 13 ? due to non-resonant transmission is obtained from [66], ? ? ? ? ? ? +?= ? ? 1 101310 4 log10TLlog10 V cA airp , (103) where is the effective area of the panel. p A Then the TL can be expressed, ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?= 3 1 3 1 31 10 4 log10TL n n E E V cA airp ?? . (104) Since the source room sound field is assumed to be reverberant, the energy stored in it is expressed by the pressure p 1 , and volume V 1 : 2 1 2 11 / cVpE ?>=< . (105) 91 The mechanical energy stored in the panel is expressed by its velocity and mass: ps AvE ?>=< 2 2 , (106) where ? s and A p are the surface density and area of the panel. Combining the results above, the averaged squared velocity of the panel is obtained 2 2 2 rad 21 1rad2 2 sp nV v nA ? ? ? ??? ? 1 p c < > <>=? ? ? + . (107) Thus the power radiated by the resonant modes into the receiving room is 22 23 vcA radp ? ??=? . (108) Similarly, 13 13 1 WE??= . The coupling loss factor ? 13 due to non-resonant random incidence mass-law transmission is obtained from [66] 10 13 RI 10 1 10log TL 10log 4 p Ac V ? ? ?? =? + ? ?? ? 2 a , (109) 2 RI 10 10 TL 10log 10log ln(1 )a? ?=? + ? ? , (110) where / 2 s ac???= , and TL RI is called the random incidence transmission loss. Substituting (110) in (109), ? 13 can be derived as 23 22 13 32 22 1 ln 1 4 p s s Ac Vc ? ? ? ? ?? ? ? ? =+ ? ? ? ? . (111) ? ? ? ? ? ? +=? 22 22 2 2 1 13 4 1ln )( c l cAp s s p ? ?? ?? ? . (112) In the source room, the sound power incident on the dividing partition of area is: p A pinc A c p ?4 2 1 >< =? . (13) 92 Below the critical frequency, the sound transmission loss of a finite panel is more controlled by the contribution of those modes that have their resonance frequencies outside the frequency band of the excitation signal than by those with resonance frequencies within that band. So taking into account both the forced and resonant panel motions, the transmission coefficient can be approximated by ? ? ? ? ? ? ++?? + ? = ?+? ? = 22 22 2 21 2 2 2 1 2 1323 4 1ln )(2 4/ 1 c cA c V n n cA s s p rad srad rad p inc ? ?? ?? ? ? ??? ? ? ? ? ? ? . (114) Then, the sound transmission loss TL can be calculated from TL 10log(1/ )?= . (130) 5.3.2 Simulation using SEA software AutoSEA AutoSEA is an interactive vibro-acoustics simulation tool based on the SEA method. In order to calculate the sound transmission loss and the radiation efficiency of a panel, two virtual rooms and a panel must be created in AutoSEA, as shown in Figure 41. The two virtual rooms were assumed each to have a diffuse sound field, and to have identical volumes as do the two real rooms in the Sound and Vibration Laboratory. A random sound source was used to generate the white noise generation in the source room. The panel under investigation has clamped boundary conditions. To create a sandwich panel in AutoSEA, the material properties of the face sheets and the core are required separately. The face sheets of a sandwich panel must only be isotropic while the core can be orthotropic. The material properties assumed for the face sheets and the core are listed in Appendix B. 93 Figure 40 Sound transmission loss model using the AutoSEA software. 5.4 Review of sound transmission measurement technique: two-room method There are several methods which have been devised for the measurement of transmission loss. The most widely used method consists of the use a transmission suite. 94 Figure 41 Set up for the two reverberation room sound transmission loss measurement method The two-room method was used to determine the sound transmission loss of panels in the two room suit in the Sound and Vibration Laboratory at Auburn University. The two- room suite consists of two reverberation rooms with the panels under investigation mounted in a window in the walls between the two rooms, as shown in Figure 42. The four edges of the panels under investigation were clamed on a window between two rooms. The volumes of the source room and receiving room are both equal to 51.15 m 3 . The area of the panels under test is 0.36 m 2 . These two reverberation chambers are 95 vibration-isolated from each other and the ground to reduce environmental noise. Each room has two walls made of wood with fiberglass filled in between them. The two rooms are also separated from each other using fiberglass. Springs are installed under the rooms to reduce the mechanical noise between the rooms and the ground. It should be noted that the transmission suite method cannot be used exactly ?in the field? since the sound fields are rarely sufficiently diffuse. So the sound absorption in each room should be taken into account. By assuming that the rooms are designed to minimize sound transmission paths other than through the test specimen and the panel under test is the only path that the sound travels through, the sound transmission loss measured using the two-room method is given by ? ? ? ? ? ? ? ? +?= R p Rs A A LL lg10TL , (131) where = space averaged sound pressure level in the source room s L = space averages sound pressure level in the receiving room R L = area of the panel under test p A = equivalent absorption area of the receiving room R A and 2 2 10ln24 R R cT V A = , (132) where and V 2R T 2 are the reverberation time and volume of the receiving room [74]. So transmission loss can be expressed as, 96 10ln24 log10 10ln24 log10 2 2 2 2 V cTA NR V cTA LLTL RpRp RS +=+?= , (133) where is the noise reduction between these two rooms. NR 5.5 Experiments of TL and simulations using AutoSEA Table 15 lists the properties of the panel whose TL was measured by two room method. Sandwich panels under study Panel Core thickness (mm) Face sheet thickness (mm) Density (kg/m 3 ) Surface density (kg/m 2 ) A 6.35 0.35 362.33 2.55 B 6.35 0.7 525.88 4.07 C 12.7 0.7 345.19 4.86 D 25.4 0.7 248.28 6.65 Aluminum panels under study Panel Thickness (mm) Surface density (kg/m2) E 6.35 17.15 Table 16 Geometrical parameters of panels under study 97 98 First of all, the reverberation times in the receiving room were measured for each panel using a B&K PULSE system. Table 16 shows the reverberation times measured for one-third octave frequency bands from 80 Hz to 8 kHz for Panel A ~ D. Center frequency (Hz) Panel A Panel B Panel C Panel D 80 0.380 0.375 0.421 0.377 100 0.632 0.544 0.557 0.506 125 0.493 0.395 0.352 0.360 160 0.636 0.659 0.699 0.634 200 0.578 0.740 0.736 0.698 250 0.903 0.818 0.856 0.860 315 1.048 1.086 1.034 1.043 400 1.240 1.162 1.155 1.111 500 1.264 1.254 1.309 1.345 630 1.316 1.407 1.324 1.344 800 1.411 1.449 1.473 1.430 1000 1.326 1.364 1.368 1.338 1250 1.238 1.248 1.224 1.247 1600 1.132 1.157 1.121 1.146 2000 1.028 1.036 1.016 1.016 2500 0.947 0.916 0.947 0.937 3150 0.846 0.809 0.865 0.830 4000 0.778 0.759 0.780 0.747 5000 0.698 0.688 0.690 0.713 6300 0.633 0.639 0.629 0.652 8000 0.556 0.545 0.555 0.538 Table 17 Reverberation times (s) of the receiving room with different panels Then a steady white noise from the generator in the Pulse system was provided to two loudspeakers in the source room. An air jet nozzle was used to increase the noise level in the high frequency region since the noise from the two loudspeakers is not enough in the high frequencies. With the loudspeakers and air jet, we can assume that the white noise is generated by these sound sources. The sound pressure levels in the two rooms were measured using two B&K microphones, type 4188, whose optimized frequency response range is from 8 Hz to 12.5 kHz. For the panels listed in Table 16, the measurements of the sound pressure levels in both the source and receiving rooms ( and ) were repeated eight times by putting the two microphones at eight randomly selected positions. The spatial averages were calculated to obtain the noise reduction (NR). Note that the background noise in the source and receiving rooms was also measured and subtracted from and . The sound transmission loss was then calculated using Equation (133). S L R L S L R L Figures 42, 43, 44 and 45 compare the measured transmission loss of panels A, B, C and D, with the results simulated using AutoSEA software in which the panel damping was set to either 2% or 5%. By comparing the two simulated cases with the measured 99 result, it can be seen that the simulated result with damping of 5% is closer to the measured result. That means the damping of these sandwich panels is close to 5% or loss factor is close to 10%, or at least damping is greater than 2%. 0 5 10 15 20 25 30 35 40 100 1000 10000 Frequency (Hz) TL ( d B ) Measured AutoSEA with 2% damping AutoSEA with 5% damping Mass Law AutoSEA with measured damping values Figure 42 Measurements of TL, mass law and simulation of AutoSEA for panel A 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 100 1000 10000 Frequency (Hz) TL ( d B ) Measured AutoSEA with 2% damping AutoSEA with 5% damping Mas s Law Figure 43 Measurements of TL, mass law and simulation of AutoSEA for panel B 100 0 5 10 15 20 25 30 35 40 45 100 1000 10000 Frequency (Hz) TL ( d B ) Measured AutoSEA with 2% damping AutoSEA with 5% damping Mass Law Figure 44 Measurements of TL, mass law and simulation of AutoSEA for panel C 0 5 10 15 20 25 30 35 40 45 100 1000 10000 Frequency (Hz) TL ( d B ) Measured AutoSEA with 2% damping AutoSEA with 5% damping Mass Law Figure 45 Measurements of TL, mass law and simulation of AutoSEA for panel D 101 0 5 10 15 20 25 30 100 1000 10000 Frequency (Hz) TL ( d B ) Panel A Panel B Figure 46 Measured TL for panels A and B 0 5 10 15 20 25 30 100 1000 10000 Frequency (Hz) TL ( d B ) Panel C Panel D Figure 47 Measured TL for panels C and D Figure 46 shows the TL for panels A and B. The TL increases when the thickness of the face sheet increases. Figure 47 shows the TL for panels C and D. The TL increases when the thickness of the core increases. 102 One aluminum panel E was simulated using AutoSEA shown in Figure 48. Panel E has the same thickness with panel A. From Figure 49, it can be seen that the critical frequency is above 6 kHz, which is an advantage for such sandwich panels since the human hearing tends to be more sensitive to sound in the range of approximately 2-6 kHz. Figure 49 shows the comparison of the TL of sandwich panels A, B, C and D and aluminum panel E. It is seen that the sound transmission loss values of the sandwich panels are smaller than that of the aluminum panel. This implies that the surface mass density still dominates the overall transmission loss behavior of these panels. The foam- filled honeycomb sandwich design does not demonstrate any advantage of sound transmission over heavier metal counterpart, although the sandwich structures have higher damping. That means such a foam-filled honeycomb sandwich design must be modified if it is to obtain higher sound transmission loss. But considering the light weight materials, the sandwich structures can be used as a good substitute for heavier metal panels when the weight and static bending strength are important factors for design. TL Aluminum 0.25 inch 0 5 10 15 20 25 30 35 100 1000 10000 Frequency (Hz) TL ( d B ) measured autosea 1% autodea 0.01% 103 Figure 48 Measurements of TL and simulation by AutoSEA for panel E 0 5 10 15 20 25 30 35 100 1000 10000 frequency (Hz) TL (dB ) panel A panel B panel C panel D aluminum 0.25 inch Figure 49 Measured TL for panels A, B, C, D and E 5.6 Summary and conclusions of experiments of TL and simulations using AutoSEA The four sandwich panels with the same original honeycomb have similar acoustical performances in the frequency range 125~8000Hz. Increasing the thickness either of the face sheets or core increases the first resonance frequency. Increasing the thickness in the core improves the damping. The estimated transmission losses from SEA agree well with the experimental values. The foam filled honeycomb core sandwich panel has high damping, greater than 2% and close to 5%. 104 105 CHAPTER 6 SUMMARY AND CONCLUSIONS The materials we used include the sandwich structures with honeycomb core filled with foam and the sandwich structures with close-cell foam core. We purchased the foam filled honeycomb core panels, close-cell foam core panels and face sheets according to our needs. The professor, Dr. Vaidya, at Birmingham made the sandwich panels with the materials we provided. The author improved the existing methods and used them to study the sandwich materials. The sandwich materials we designed were proved to have good stiffness and damping properties and fine sound transmission behavior. From the measurements using the four-point bending method and twisting method, the static properties for the core and entire sandwich structures were calculated. The finite element method was used to model the static response of the sandwich beams with four-point bending method and twisting method. Dynamic properties of thin sandwich structures with honeycomb cores were discussed in this dissertation. The structure was considered anisotropic in the theoretical models. The dynamic parameters of the laminates and the core are usually not known. For a bonded honeycomb structure though, the mass per unit area, the dimensions of the entire structure and the laminates are easily determined. Hamilton's principle was used to derive a six order wave equation governing the bending stiffness of a sandwich structure. Measurements gave the first few natural frequencies for the beams and from these results their corresponding equivalent bending stiffnesses at the natural frequencies could be determined. Using the least squares method on the bending stiffnesses at the natural frequencies, the bending stiffness is determined in the entire frequency range. The results for the frequency dependent bending stiffness curve allow the determination of the Young's modulus of the bonded laminates and the shear modulus of the core. The value of bending stiffness in the low frequency region give the Young's modulus for the bonded laminates and the slope of the decreasing stiffness reveals the shear modulus of the core. For the structures the necessary dynamics parameters can be now estimated. The wave number for the honeycomb structure can be determined based on the six order wave equation. The wave number equation has the six solutions of the form 321 ,, ??? iik x ???= . The real solution 1 ? represents a propagating wave. In the low frequency region, the characteristics of the honeycomb structure are determined by the characteristics of the entire bonded structure. The core serves as a spacer between the laminates, and in the mid frequency region it provides shear. In the high frequency region the wave number plot confirms that the vibration of a structure is determined by the laminates only. The imaginary solution 2 ?i represents an evanescent wave which becomes real and propagating above a certain frequency. This change denotes the frequency for which the moment of inertia becomes of importance for the calculations. The third solution 3 ?i is determined by the core thickness and is constant in the low 106 107 frequency region and for increasing frequencies adjusts to the asymptote for the wave number of the laminates. The boundary conditions for the honeycomb structures are shown to influence on the dynamic properties with decreased effect of shear when tightening the boundary conditions. The theory presented provides a good estimate for the bending stiffness of traditional sandwich elements with foam cores. The analysis of the sound transmission loss of foam-filled honeycomb sandwich panels was carried out. A theoretical model was used for the statistical energy analysis (SEA) method. A parameter study was made for the sound transmission loss. It can be concluded that an increase in the core thickness will increase the critical frequency. Measurements on five different panels, including one isotropic aluminum panel and four sandwich panels, were carried out using the two-room method. Predictions of sound transmission loss were conducted using AutoSEA software. The measured results agree quite well with the results predicted using AutoSEA. The surface densities of the sandwich panels are much smaller than those of the aluminum panels. The measured results show that the overall sound transmission loss values of sandwich panels are smaller than the aluminum panel. This implies that the surface density still dominates the overall transmission loss behavior of these panels. 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