STATIC, DYNAMIC AND ACOUSTICAL PROPERTIES OF SANDWICH COMPOSITE MATERIALS. Zhaohui Yu. Certificate of approval: Distinguished University Professor

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1 STATIC, YNAMIC AN ACOUSTICAL PROPERTIES OF SANWICH COMPOSITE MATERIALS Ecept here reference is made to the ork of others, the ork described in this dissertation is my on or as done in collaboration ith my advisory committee. This dissertation does not include proprietary or classified information. Zhaohui Yu Certificate of approval: George T. Floers Professor Mechanical Engineering Malcolm J. Crocker, Chair istinguished University Professor Mechanical Engineering Hareesh Tippur Professor Mechanical Engineering ZhongYang (Z.-Y.) Cheng Assistant Professor Materials Engineering Joe F. Pittman Interim ean Graduate School

2 STATIC, YNAMIC AN ACOUSTICAL PROPERTIES OF SANWICH COMPOSITE MATERIALS Zhaohui Yu A issertation Submitted to the Graduate Faculty of Auburn University in Partial Fulfillment of the Requirements for the egree of octor of Philosophy Auburn, Alabama May 0, 007

3 STATIC, YNAMIC AN ACOUSTICAL PROPERTIES OF SANWICH 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 epense. The author reserves all publication rights. Signature of Author ate of Graduation iii

4 VITA Zhaohui Yu, daughter of Shuiyong Yu and Limin Xu, as born on July 5, 976, in Qingdao, Shandong Province, China. She graduated from the No. Middle School of Jimo, Qingdao in 995. She entered Ocean University of Qingdao, China in September 995, and graduated ith Bachelor of Science degree in Electrical Engineering and a Minor iploma in Softare Engineering in July 999. She orked as a signal processing engineer in HuaYi Building Materials Company of Qingdao from September 999 to November 00. She entered Graduate School, Auburn University, in January 00. She married to Yuquan Li on July, 000. iv

5 ISSERTATION ABSTRACT STATIC, YNAMIC AN ACOUSTICAL PROPERTIES OF SANWICH COMPOSITE MATERIALS Zhaohui Yu octor of Philosophy, May 0, 007 (B.S., Ocean University of Qingdao, 999) Typed Pages irected by Malcolm J. Crocker Sandich composite materials have been idely used in recent years for the construction of spacecraft, aircraft, and ships, mainly because of their high stiffness-toeight ratios and the introduction of a viscoelastic core layer, hich has high inherent damping. One of the main objects of this research is the measurement and estimation of the bending stiffness and damping of sandich structures. Knoledge of the elastic properties of the core and face sheet of the sandich structures is indispensable for the analysis and modeling of sandich strictures. Hoever, traditional methods to determine the elastic properties are not suitable for the core, hich is usually brittle, and the face sheets, hich 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 v

6 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 as used. The damping is an important property used for the analysis of the acoustical behavior of the sandich 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 sandich structures and to simulate their acoustical behavior using the statistical energy analysis method (SEA). While solving vibroacoustic problems, FEM and SEA are commonly used. Hoever, common vibroacoustic problems involve a very large number of modes over a broad frequency range. At high frequencies these modes become both epensive 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 sandich structures ith polyurethane foamfilled honeycomb cores and sandich structures ith 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; hile 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-eight ratios and vi

7 great resistance to ater absorption, and ill not sell, crack, or split on eposure to ater. vii

8 ACKNOWLEGMENTS I ould like to epress my sincere appreciation and thanks to my advisor Auburn University distinguished professor, r. Crocker for his guidance and support during my studies and research. I am grateful for the considerable assistance provided by the other committee members, r. Floers, r. Tippur and r. Cheng. I appreciate outside reader, r. 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 ant to epress my special thanks to my parents and my husband for their support and love. viii

9 Journal used: Journal of Sound and Vibration Computer softare used: Microsoft Word 00 i

10 TABLE OF CONTENTS LIST OF TABLES... iii LIST OF FIGURES...v CHAPTER BACKGROUN.... Introduction.... Objective.... Structure of honeycomb panels....4 Organization of issertation...6 CHAPTER STATIC PROPERTIES CHARACTERIZATION...8. Introduction...8. Four-point bending method...8. Tisting method Calculation of static bending stiffness of sandich beam....5 Eperimental results and analysis Specimen Four-point bending method Tisting method Calculation of static bending stiffness of sandich beam Summary and analysis... 8

11 .6 Finite element method model Introduction FEM in panel problems Element Types Finite element modeling of sandich structures Comparison of results from eperiments and theoretical analyses... 0 CHAPTER THEORY FOR SANWICH BEAMS.... Literature revie.... Theory of sandich structure...5. Boundary conditions Wave numbers Least Squares Method amping measurement methods...5 CHAPTER 4 YNAMIC PROPERTIES CHARACTERIZATION Eperiments Steps Set up Samples Analysis of eperimental results Frequency response functions Boundary conditions ynamic bending stiffness amping i

12 4. Conclusions...77 CHAPTER 5 SOUN TRANSMISSION LOSS OF SANWICH PANELS Classical sound transmission analysis Mass la sound transmission theory The effect of panel stiffness and damping The coincidence effect Critical frequency Sound transmission coefficient and transmission loss at coincidence Literature revie of the Sound Transmission Loss of Sandich Panels Statistical energy analysis model (SEA) Prediction of Sound Transmission through Sandich Panels using SEA Simulation using SEA softare AutoSEA Revie of sound transmission measurement technique: to-room method Eperiments of TL and simulations using AutoSEA Summary and conclusions of eperiments of TL and simulations using AutoSEA...04 CHAPTER 6 SUMMARY AN CONCLUSIONS...05 REFERENCES...09 ii

13 LIST OF TABLES Table The dimensions and densities of the specimens tested for their static stiffness... Table Static properties obtained from the four-point bending and tisting methods...8 Table Static bending stiffness obtained from to ays...9 Table 4 Load-deflection relations for Beam from the eperiments and FEM analysis. Table 5 Load-deflection relations for Plate E from the eperiments and FEM analyses.. Table 6 Boundary conditions for ends of beam...45 Table 7 Geometrical and material parameters for sandich beam...48 Table 8 Values of α n for particular boundary conditions...5 Table 9 Geometry and density of the sandich beam ith foam filled honeycomb core.6 Table 0 Geometry and density of the sandich beam ith foam core...6 Table Natural frequencies for different measurements on Beam F and their corresponding bending stiffness Table Natural frequencies of Beam C for the three different beam boundary conditions...67 Table Comparison of static stiffness measured by four-point bending method and to stiffness limits from dynamic characterization...70 Table 4 Comparison of shear modulus of the core measured by tisting method and that from the dynamic stiffness curve for Beam G...74 iii

14 Table 5 amping ratio of Beam C...76 Table 6 Geometrical parameters of panels under study...97 Table 7 Reverberation times (s) of the receiving room ith different panels...99 iv

15 LIST OF FIGURES Figure Sandich panel ith a honeycomb core... Figure A: Nome honeycomb core, B: irregular aluminum honeycomb core...4 Figure Corrugation process used in honeycomb manufacture...5 Figure 4 Geometry and dimensions of the four-point bending test...9 Figure 5 To principal direction sandich panel strip for the four-point bending eperiments and its equivalent plate representation...0 Figure 6 Loading scheme for the pure tisting test... Figure 7 Core, face sheet and entire sandich structure...4 Figure 8 eflection of beam ith four-point bending method...5 Figure 9 Out-of-plane shape ith un-deformed edge of beam ith four-point bending method...6 Figure 0 In-plane shear stress contour for beam ith four-point bending method...7 Figure Out-of-plane deflection contour of plate ith the tisting method...8 Figure eformed plate ith un-deformed edge of plate ith the tisting method...9 Figure In-plane shear stress contour for plate ith the tisting method...0 Figure 4 Bending of composite bar or panel by (a) bending and (b) shearing of the core layer...5 Figure 5 Ecitation of a beam and resulting forces and moments. imensions and material parameters for the laminates and core are indicated...7 v

16 Figure 6 Elastic properties and area density of sandich structure...7 Figure 7 Particular boundary conditions...44 Figure 8 Wave numbers for beam [6]...47 Figure 9 ecay rate method used to determine damping...54 Figure 0 Modal bandidth method to determine damping...55 Figure Poer balance method to determine damping...56 Figure Eperimental steps to determine some properties of sandich structures...57 Figure Set Up Using shaker...59 Figure 4 Set Up Using hammer...60 Figure 5 Honeycomb sandich composite structures; (a) foam-filled honeycomb core, (b) composite beam...6 Figure 6 Closed-cell foam core...6 Figure 7 FRF for the sandich beam F for free-free boundary condition...6 Figure 8 FRF for the sandich beam G for free-free boundary condition Figure 9 FRF for the sandich beam E for free-free boundary condition...65 Figure 0 Bending stiffness of the sandich beam C for three different boundary conditions...68 Figure Bending stiffness for the sandich beam C Figure ynamic stiffness for beams C and...7 Figure ynamic stiffness for core in to principal directions...7 Figure 4 Bending stiffness for the sandich beam G...7 Figure 5 Bending stiffness for the sandich beam E, F and H...75 Figure 6 amping ratio of sandich beam C...77 vi

17 Figure 7 The coincidence effect...8 Figure 8 Idealized plot of transmission loss versus frequency...8 Figure 9 Schematic of the poer flo in three-coupled systems using SEA...89 Figure 40 Sound transmission loss model using the AutoSEA softare...94 Figure 4 Set up for the to reverberation room sound transmission loss measurement method...95 Figure 4 Measurements of TL, mass la and simulation of AutoSEA for panel A...00 Figure 4 Measurements of TL, mass la and simulation of AutoSEA for panel B...00 Figure 44 Measurements of TL, mass la and simulation of AutoSEA for panel C...0 Figure 45 Measurements of TL, mass la and simulation of AutoSEA for panel...0 Figure 46 Measured TL for panels A and B...0 Figure 47 Measured TL for panels C and...0 Figure 48 Measurements of TL and simulation by AutoSEA for panel E...04 Figure 49 Measured TL for panels A, B, C, and E...04 vii

18 CHAPTER BACKGROUN. Introduction Applications for sandich structures are steadily increasing. The term sandich panel here refers to a structure consisting of to thin face plates bonded to a thick and lighteight core. The face plates are typically made of aluminum or some composite laminate. The core can be a lighteight foam or a honeycomb structure. These types of sandich structures having a high strength to eight ratio have been used by the aircraft industry for over 70 years. Hoever, during the last decade, various types of lighteight 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 lighteight vehicles could be considerable in reducing fuel consumption and increasing load capacity. Hoever, there are also certain constraints like passenger comfort, safety and costs for ne types of vehicles. Passenger comfort requires lo noise and vibration levels in any type of vehicle. In addition to ne 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 ne designs to achieve the required noise levels. Lighteight structures often have poor acoustical and dynamic

19 properties. In addition, the dynamic properties are often frequency dependent. In order to avoid noise problems in lighteight vehicles, it is essential that the main structure-borne sound sources are as eakly coupled as possible to the supporting structure foundation and to adjoining elements. With respect to material and construction costs, sandich structures can compare very favorably ith other lighteight materials like aluminum. The number of applications for sandich panels is steadily increasing. One reason for the groing 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 ell as different plate structure are ell understood, although still under development.. Objective As discussed in the previous section, this thesis is mainly concerned ith the static, dynamic and acoustical properties of sandich structures. In particular, emphasis is placed on the study of the dynamic response of structural beams to acoustical ecitation. The thesis includes three main parts:. Characterization of the static properties of sandich beams and model the foam-filled honeycomb sandich structures by the finite element method (FEM),. Study of the vibration response of sandich structures and characterization of the dynamic properties of sandich beams,

20 . Analysis of the sound transmission loss of sandich panels and modeling of foam-filled honeycomb sandich structures using the Statistical Energy Analysis (SEA) method.. Structure of honeycomb panels A honeycomb panel is a thin lighteight plate ith a honeycomb core ith heagonal cells. Layered laminates are bonded to both sides of the core as shon in Figure. Each component is by itself relatively eak and fleible. When incorporated into a sandich panel the elements form a stiff, strong and lighteight structure. The face sheets carry the bending loads and the core carries the shear loads. In general, honeycomb cores are strongly orthotropic. Figure Sandich panel ith a honeycomb core The types of core materials in the panels used for the measurements presented here are either Nome or aluminum. Nome is an aramid fiber paper dipped in phenolicresin ith a lo shear modulus and lo shear strength. A typical thickness of the

21 Nome honeycomb panels investigated is 0 mm ith the thickness of the face sheet or laminate being beteen mm. The eight per unit area is of the order of kg/m unit area. Each laminate consists of -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 beteen the to laminates to give the required bending stiffness for the entire beam. The bending stiffness of the core itself is in general very lo. The bending stiffness of sandich materials is frequency dependent. The cells in the core give it an orthotropic structure. The dynamic characteristics should be epected to be different in each direction. The to main in-plane directions and are defined in Figure. The shape of the honeycomb cells of a typical aluminum core is generally very irregular hich makes it impossible to describe its geometry in a simple ay. Nome cores have very regular shapes as compared to aluminum-cores. A B Figure A: Nome honeycomb core, B: irregular aluminum honeycomb core 4

22 Honeycomb cores, hich ere developed starting in the 940 s primarily for the aerospace industry, have the greatest shear strength and stiffness-to-eight ratios, but require special care to ensure adequate bonding of the face sheets to the core since such cores are hollo. The standard heagonal honeycomb is the basic and most common cellular honeycomb configuration, and is currently available in all metallic and nonmetallic materials. Figure illustrates the manufacturing process, and the L (ribbon direction) and W (transverse to the ribbon) directions of the heagonal 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 sandich applications. Figure Corrugation process used in honeycomb manufacture As discussed in the previous chapters, sandich structures ith foam-filled honeycomb cores have some advantages over pure honeycomb cores. By filling foam in the honeycomb cells, not only the longitudinal cell alls but also the foam can carry the 5

23 uniaial load. So the foam is epected to reduce the discontinuities in elastic properties possessed by ith pure honeycomb cores. Also the foam can make the fabrication of sandich structures easier than those made ith pure honeycomb cores. Another important advantage of foam-filled honeycomb cores is their improved damping and shear strength properties..4 Organization of issertation This dissertation contains the results of the present research investigation into the current objectives. The research as performed in the Sound and Vibration Laboratory of the epartment of Mechanical Engineering at Auburn University. The results reported are divided into four major parts. Chapter described the static properties of foam-filled honeycomb sandich beams. Orthotropic plate theory is introduced and to methods, the four-point bending method and the tisting method, are used to measure the shear modulus, Young s modulus and other properties of sandich structures. The finite element method (FEM) as used to simulate the response of the beams and panels ith the four-point bending method and the tisting method. A thorough derivation of the beam theory of the sandich structure is given in Chapter. The theoretical model as derived using Hamilton's principle. The general dynamic behavior of sandich structures is discussed. Chapter also describes the measurements, hich include eperiments and the analysis of results. In the eperiments, measurements on foam-filled honeycomb sandich beams ith different configurations ere performed and finally the conclusions ere dran from the analysis of the results. 6

24 Chapter 4 is devoted to the study of sound transmission through sandich panels. This chapter starts ith a brief introduction of the classical theory of sound transmission loss. Then the previous research on sound transmission through sandich panels is revieed. The sound transmission loss of several panels ith different thickness of core and face sheet as measured by the to room method. Simulations of the sound transmission loss ere conducted using AutoSEA. Eperimental results for sound transmission loss are presented as ell. The summary and conclusions dran from this research are given in Chapter 5. 7

25 CHAPTER STATIC PROPERTIES CHARACTERIZATION. Introduction In this ork, an eperimental study on the bending stiffness of sandich beams as conducted. An eperimental procedure to measure the bending stiffness of sandich beams as 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 sandich structure. In addition, special plate tisting tests have been used for finding the in-plane shear modulus and Poisson s ratios of core, face sheet and entire sandich structures. Using these elastic properties, the bending stiffness of entire sandich structures as calculated in to different ays. The static behavior of the entire sandich structures as simulated using the finite element method (FEM).. Four-point bending method The four-point bending arrangement used in this ork is shon in Figure 4. The beams ere simply supported at the ends of the central span d and half of the total load P as applied at the ends of the beams, separately. The loading points ere 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 d. The central span, therefore, is in a state of pure bending. Measurement of the load P versus central deflection response enables the calculation of the Young s moduli E E of the core, 0 face sheet and entire sandich structure, by means of the equation 8

26 4bh P = 0 dd S, () d 0 here P is the total load on the ends, is the central span, is the central deflection, d is the outer span and h is the thickness as shon in Figure 4. And S is the compliance along the direction parallel to the length of the beam. S S = S = S = = E E in principal direction, in principal direction. () Figure 4 Geometry and dimensions of the four-point bending test The eact location at hich the loads are applied is largely arbitrary, ecept that the to outer spans must be equal. 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 ork, a micrometer as used to measure the beam deflections. The micrometer could be read to the nearest 0.00 mm. 9

27 Both the principal directions, and, of the four-point bend specimens, as shon in Figures 5 ere considered. Since the core pf the sandich beams used in the measurements is orthotropic, several etensive revies of equivalent plate models in the literature [-4] ere consulted. Figure 5 To principal direction sandich panel strip for the four-point bending eperiments and its equivalent plate representation. Tisting method 0

28 Figure 6 Loading scheme for the pure tisting test The tisting method configuration used in this ork is shon in Figure 6. The plates ere simply supported at the three quadrant corners and the load P as applied at the other corner of the plates. Measurement of the load P versus central deflection response enables calculation of shear modulus G, Poisson's ratios ν, ν of the 0 core, face sheet and entire sandich structure, by means of the equation: 4 t P = 0 SG L, () here S G ( m n) S 8m n S + mn( m + n) S + ( m n ) S, = (4) mn 66 and ν ν S =, S =, S = =, S66 = E E E E G and m = cosα and n = sinα, here α is the angle beteen the principal direction and coordinate. For various material orientations, Equation (4) can be used to evaluate, α = 0 o, α = 45 o, S G = S66, S = ( S S ). G (5) Compliances and S have been obtained from the four-point beam bending S o o tests. By performing tisting tests, ith α = 0 and = 45, compliances and S α S66 can be determined. Combining these results, the equivalent Poisson's ratios are obtained

29 S S ν = = ν S S, ν. (6) The shear modulus is also obtained since S is knon from the equation S 66 =. G G 66 The load and supporting points for the tisting tests ere located 5 mm from the corners of the plate specimens. The amount of overhang as not found to be critical. The method for measuring the displacement of the center of the plate, 0, as the same as ith the four-point bending tests. The samples tested ere meter square plates..4 Calculation of static bending stiffness of sandich beam Once e kno the properties of core and face sheet, the bending stiffness per unit idth of the beam b is b Ectc tc tl tl = + El + tctl +, (7) here t and t are the thicknesses of the core and face sheets (laminate), c l respectively. E is the effective modulus of the core and E is the effective Young s c modulus of the face sheet (laminate) and they can be calculated using Equation (8). l for for core face E E c c sheets c E = c c ν ν c E = c c ν ν E l for for l E = l l ν ν principal direction, principal direction, for principal direction or. (8)

30 Then the bending stiffness of the entire sandich beams, in to principal directions, and, can be calculated by Equation (9): s s = = E E ctc ctc t + El c tl t + El c tl + t + t ctl ctl tl + tl + for for principal principal direction direction,. (9) Alternatively, the bending stiffness per unit idth of the entire sandich beams can be calculated directly from the modulus of the sandich beams using the four-point bending method and the tisting method (see section. and.) as follos: s s s E = s E = ( t + t ) c l s s ( ν ν ) ( t + t ) c l s s ( ν ν ) for for principal principal direction direction,. (0).5 Eperimental results and analysis.5. Specimen The specimens used in the eperiments included to beams of honeycomb core filled ith foam in to principal directions, one beam made of a face sheet, one square o o plate of core ith α = 0, one square plate of core ith α = 45 and to entire sandich beams cut ith the -coordinate oriented in to the principal directions. All the beams ere uses ith the four-point bending method and all the plate ere used in tisting method. The dimensions and densities of the specimens are listed in Table : Specimen content Thickne Length Width ensity irection name ss (mm) (m) (mm) Core Face sheet

31 (kg/m ) (kg/m ) Beam A Core N/A Beam B Core N/A Beam C Face sheet N/A 6 () Beam Beam E Entire sandich Entire sandich o Plate F Core N/A 6 N/A α = 0 o Plate G Core N/A 6 N/A α = 45 Plate H Plate I Entire sandich Entire sandich o N/A 6 N/A α = 0 o N/A 6 N/A α = 45 Table The dimensions and densities of the specimens tested for their static stiffness The pictures of core, face sheet and entire sandich are shon in Figure 7. Figure 7 Core, face sheet and entire sandich structure 4

32 .5. Four-point bending method Beams A, B, C, and E ere measured for their static stiffness using the fourpoint bending method. For beam A, the slope of the load-deflection curve and the properties derived are given by Equations (). 4bh slope = 5.8, c d d S c 6 for core E = c = Pa. () S For beam B, the slope of the load-deflection curve and the properties derived are given by Equations (). 4bh slope =.4, c d d S c 6 for core E = c = Pa. () S For beam C, the slope of the load-deflection curve and the properties derived are given by Equations (). Since the face sheet (laminate) is isotropic in the to in-plane l l principal directions E. E = 4bh slope = 5., l d d S l l 0 for face sheet E = E = = Pa. () l S For beam, the slope of the load-deflection curve and the properties derived are given by Equations (4). 5

33 4bh slope = , s d d S s 0 for sandich E = s = Pa. (4) S For beam E, the slope of the load-deflection curve and the properties derived are given by Equations (5). 4bh slope = 984.5, s d d S s 0 for sandich E = s = Pa. (5) S.5. Tisting method Plates F, G, H and I ere measured under the configuration of tisting method. For plate F, the slope of the load-deflection curve and the properties derived are given by Equations (6). o 4 h α = 0 slope = 09.66, c S a 66 for core G Pa. (6) c 7 = c = S66 For plate G, the slope of the load-deflection curve and the properties derived are given by Equations (7). α = 45 for o core slope ν ν c c S = S c c ( S S ) S = S c c c c 6 ν h = 0., c a = 0.6 =.5,. (7)

34 For plate H, the slope of the load-deflection curve and the properties derived are given by Equations (8). α = 4 h o 0 slope s S66a = , for sandich G Pa. (8) s 0 = s =.45 0 S66 For plate I, the slope of the load-deflection curve and the properties derived are given by Equations (9). α = 45 o slope for sandich s ν s ν s s ( S S ) h S = S S = S s s s c ν a = , = 0., s = 0.. (9).5.4 Calculation of static bending stiffness of sandich beam From Equations (-5), e kno 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 follos. for core E 6 Ec = 9. 0 Pa for principal direction, c = c ν c ν E 6 Ec = Pa for principal direction, c = c ν c ν for face sheets E l E l = l ν l ν = Pa for principal direction or. (0) 7

35 Then using Equation (9), e can calculate the bending stiffness of the entire sandich beams: s Ec tc tc tl tl = + El + tctl + = N. m for principal direction, s Ectc tc tl tl = + El + tctl + = N. m for principal direction. () The bending stiffness per unit idth of the entire sandich beams can be calculated directly from the modulus of the sandich beams using four-point bending method and the tisting method as follos: ( t + t ) s s E c l = = s s ( ν ν ) ( t + t ) 5.8 s s E c l = = 5.5 s s ( ν ν ).5.5 Summary and analysis N. m for principal direction, N. m for principal direction. () After calculations from the measurements of the four-point bending method and tisting method, the static properties for the core and entire sandich structures ere obtained as follos: Item E [ Pa] E [ Pa] ν ν G [ Pa] Core Face sheet 4.86E Entire sandich Table Static properties obtained from the four-point bending and tisting methods 8

36 The bending stiffness per unit idth of the entire sandich beams ere calculated in to ays, one from the properties of the core and face sheet, ν ν c c l c c E E, E,, and the other directly from the properties of the entire sandich beams, ν ν s s s s E E,,. All the properties ere obtained from the fourpoint bending and tisting methods. The comparison of the results from these to ays (Equation and ) is given in Table : Bending stiffness per unit idth of the entire sandich beams From properties, ν ν c c l c c E E, E,, [N. m] From properties, ν ν s s s s E E,, [N. m] s s Table Static bending stiffness obtained from to ays Some conclusions can be dran from Tables and :. The shear modulus of the core and that of entire sandich structure have very similar values, hich means the core bears most of the shear and there is almost no shear in the face sheet.. The results of the measurements are reasonably accurate and the bending stiffnesses per unit idth of the entire sandich beams obtained from to approaches are in good agreement. 9

37 . The measurement methods described in this chapter are useful for the determination of the static elastic properties of the core, face sheets and entire sandich structure..6 Finite element method model.6. Introduction Finite element methods have only become of significant practical use ith the introduction of the digital computer in the 950'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. Hoever, 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 hich are in good agreement ith solutions found by eact methods [5]. The finite element method has been perhaps most idely used for the solution of structural problems in the aerospace industry, one eample being the design of the static strength of ings. Hoever, FEM is also idely used in other branches of engineering. In civil engineering it is used for eample in the design of dams and shell structures. It can also be applied to such diverse problems as heat conduction and fluid flo. It should also be noted that the method is not confined to linear problems but can also be used in non-linear structural problems here large deformations or creep and plastic deformation occur [6]..6. FEM in panel problems There are several ays of representing a structure in a series of finite elements. Each method has its advantages and disadvantages. The main difficulty to overcome is 0

38 the representation of an infinite number of connection points by a finite number. Turner et al [7] ere 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, hich at first does not seem completely convincing, has been given a firm foundation by Zienkieicz [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 hich yields approimate numerical solutions to difficult boundary value problems. This method of analysis can treat nonlinear problems ith irregular boundary shapes and mied conditions. The basic concept of finite element modeling is to replace the solid body to be analyzed ith a netork of finite elements [8]. The elements are solid elements hose 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 follos [74]: ) The structure is divided into a finite number of the elements by draing a series of imaginary lines.

39 ) The elements are considered to be interconnected at a series of modal points situated on their boundaries. ) 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 no the unknons for the system. The displacement function no defines the state of strain in each element and together ith 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 hich ill result in equilibrium of the boundary stresses ith the distributed loading. A stiffness relationship can no be determined for each element relating the nodal forces to the nodal displacements. This method is useful for the analysis of complicated structures hich cannot be approached easily using classical modal analysis. It is most useful for static problems and for dynamic problems at lo frequencies. The analysis is limited at high frequencies in much the same ay as classical modal analysis. In classical modal analysis, the number of modes hich must be included becomes very large at high frequencies. In the case of finite elements the number of elements hich must be included also becomes very large at high frequencies. It is desirable to have at least four elements per bending avelength. In this research, the FEM technique as used to approimate the bending stiffness of sandich panels, and to predict the load-deflection response of sandich panel specimens. The finite element softare package utilized as ANSYS. Finite element

40 models for the eperimental test method configurations discussed in section.6 have been created using the pre-processor in ANSYS. The basic concepts of these models are presented in this chapter, hile results of the models along ith a comparison ith eperimental and analysis results is shon in the folloing section. Each of the finite element models created for the different test configurations in this thesis ere developed ith the computer aided design pre-processor in ANSYS. The models created in ANSYS ere:. Entire sandich beam ith the four-point bending model for the principal direction. Entire sandich beam ith the four-point bending model for the principal direction o. Entire sandich plate ith the pure tisting model ( α = 0 ) The geometries of these models ere chosen to be the same as those used in the actual eperiments conducted on the specimens in Table..6. Element Types The SHELL99 (linear layered structural shell element) may be used for layered applications of a structural shell model. The SOLI46 (8-noded, - layered solid element) is essentially a - version of the layered SHELL99 element type designed to model thick layered shells or solids. SOLI46 is recommended rather than SHELL99 for calculating interlaminar stresses, primarily because multiple SOLI46 elements can be stacked to allo through-the-thickness deformation slope discontinuities. The SHELL99 element as developed assuming that the shear disappears at the top and bottom surfaces of the element, hile the SOLI46 element does not use such an assumption. In the

41 SOLI46 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 SOLI46 elements stacked through the thickness are recommended [9]. Solid elements ere used to model the sandich structure panels to simulate the behavior of loaddeflection and shear stress ith the four-point bending and tisting methods. Hoever, shell elements ould likely have sufficed to simulate the load-deflection behavior of the entire sandich structures ith the four-point bending and tisting methods, because accurate stresses ere not required for the current investigation..6.4 Finite element modeling of sandich structures Four-point Bending Method Modeling The sandich beam, hich is ith the four-point bending method, is modeled using the elements of SOLI46. The geometry of the beam is similar as that of Beam in Table and the properties of the three layers of the beam-face sheet, core and face sheets ere obtained from the results of measurements of the four-point bending and tisting methods as given in section.5. The out-of-plane deflection contour is shon in the folloing figure: 4

42 Figure 8 eflection of beam ith four-point bending method The out-of-plane deformation shape ith an un-deformed edge is shon in the folloing figure: 5

43 Figure 9 Out-of-plane shape ith un-deformed edge of beam ith four-point bending method The in-plane shear stress contour is shon in the folloing figure: 6

44 Figure 0 In-plane shear stress contour for beam ith four-point bending method From figure 0, e can see that no shear stresses act over the central span. The central span, therefore, is in state of pure bending, hich is the advantage of using the four-point bending method instead of the three-point bending method. Tisting Method Modeling The sandich plate, hich is ith the tisting method, is modeled using the elements of SOLI46. The geometry of the plate is similar to that of Plate H in Table and the properties of the three layers of the beam-face sheet, core and face sheet ere obtained from the results of measurements made ith the four-point bending and tisting methods as described in section.6. The out-of-plane deflection contour is shon in the folloing figure: 7

45 Figure Out-of-plane deflection contour of plate ith the tisting method The out-of-plane deformation shape ith un-deformed edge is shon in the folloing figure: 8

46 Figure eformed plate ith un-deformed edge of plate ith the tisting method The in-plane stress contour is shon in the folloing figure: 9

47 Figure In-plane shear stress contour for plate ith the tisting method.6.5 Comparison of results from eperiments and theoretical analyses Table 4 shos the load-deflection relations for Beam obtained from the eperiments and FEM analysis: Load (N) eflection measured in eperiments (m) eflection calculated by FEM (m)

48 Table 4 Load-deflection relations for Beam from the eperiments and FEM analysis Table 5 shos the load-deflection relations for Plate E from the eperiments and FEM analysis: Load (N) eflection measured in eperiments (m) eflection calculated by FEM (m) Table 5 Load-deflection relations for Plate E from the eperiments and FEM analyses

49 CHAPTER THEORY FOR SANWICH BEAMS. Literature revie There are a large number of papers and publications on the dynamic properties of sandich structures. Already in 959, Hoff [9] concluded that there as an abundance of theoretical ork in the field. Some of the basic theories are no also summarized in tetbooks. To of the basic eamples are the books by Zenkert [0] and Whitney []. The bending of sandich 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 sandich structures. Reference is often made to the Timoshenko [] and Mindlin [] models. There are various types of Finite Element Models. One of the first fundamental orks on the bending and buckling of sandich plates as published by Hoff [9]. In the paper, Hamilton's principle is used to derive the differential equations governing the bending and buckling of rectangular sandich panels subjected to transverse loads and edgeise compression. Many of basic ideas introduced by Hoff form the basis for many subsequent papers on bending of sandich plates. Another classical paper as published by Kurtze and Waters in 959 [4]. The aim of the paper as the development of a simple model for the prediction of sound transmission through sandich 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 beteen to limits. The high frequency asymptote is determined by the bending

50 stiffness of the laminates. The model introduced by Kurtze and Waters as later somehat improved by ym and Lang [5]. A more general description of the bending of sandich beams is given by Nilsson [6]. In this model, the laminates are again described as thin plates. Hoever, the general ave 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 sandich plates. Some boundary conditions and their influence on the bending stiffness of a structure ere later discussed by Sander [7]. Guyader and Lesueur [8] investigated the sound transmission through multilayered orthotropic plates. The displacement of each layer is described based on a model suggested by Sun and Whitney [9]. A considerable computational effort is required. Renji et al derived a simple differential equation governing the apparent bending of sandich panels [0]. The model includes shear effects. Compared ith measured results, this model overestimates the shear effects. In particular, in the high frequency region, the predicted bending stiffness is too lo. A modified Mindlin plate theory as suggested by Lie []. The influence of some boundary conditions is also discussed in a subsequent paper []. Again a considerable computational effort is required. Maheri and Adams [], used the Timoshenko beam equations to describe fleural vibrations of sandich 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. ue to the frequency dependence of sandich, the solutions

51 ith four unknon ill agree very ell for lo frequency. With increasing frequency, the result normally disagrees strongly ith measured vibrations. The main ork on sandich structures has been made on conventional foam-core structures ith various face sheets. Little ork has been done on the dynamics of honeycomb panels. In 997, Saito et al [4] presented an article on ho 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 [5] proposed a - 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 sandich panels. For eample, a finite element vibration analysis of composite beams based on Hamilton's principle as presented by Shi and Lam [6]. A standard FEM code as used by Cummingham et al. [7] to determine the natural frequencies of curved sandich panels. The agreement beteen predicted and measured natural frequencies is found to be very good. There are certainly a large number of methods available hich describe describing the vibration of sandich panels. Hoever, the aim of this dissertation is the formulation of simple but sufficiently accurate differential equations governing the apparent bending of sandich beams and plates. Boundary conditions should also be formulated for the calculation of eigen-frequencies and modes of vibration. The models should allo simple parameter studies for the optimization of the structures ith respect 4

52 to their acoustical performance. The aim is also to describe a simple measurement technique for determining some of the material parameters of composite beams.. Theory of sandich 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 shon in Figure 4. This figure shos the deflection of a beam due to pure bending (a) and due to shear in the core hich is the thickest layer (b). Figure 4 Bending of composite bar or panel by bending (a), shearing of the core layer (b) and vibration aves in high frequency range (c). The total lateral displacement,, of a sandich 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 γ. Figure 4 (a) and (b) shos the bending of composite bar by 5

53 bending of the hole sandich structures and shearing of the core layer. Figure 4 (c) shos the vibration aves of the hole sandich structures in high frequency range. The relationship beteen, and γ is given by = + γ. () For a honeycomb beam the lateral displacement can be found hen 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 eample derived and discussed in references [, 8-]. According to Hamilton's principle the time integral of the differential beteen U the potential energy per unit length, T the corresponding kinetic energy per unit length and A the potential energy induced per unit length by eternal and conservative forces is an etremum. In mathematical terms ( U T + ) A ddt = δ 0 () In deriving the equations governing the lateral displacement of the structure shon in Figure 5, symmetry is assumed. Some properties are shon in Figure 6. The identical laminates have a Young's modulus, bending stiffness, density El ρ l and thickness t. The core has effective shear stiffness G, its Young's modulus E, its l e c equivalent density ρ and its thickness t. For thick core the parameter G is not c c e necessarily equal to the shear stiffness G as suggested by Timoshenko [4]. The core itself is assumed to have no stiffness or a very lo stiffness in the -direction. In the y- direction, the core is assumed to be sufficiently stiff to ensure that the laminates move in phase ithin the frequency range of interest. 6

54 Figure 5 Ecitation of a beam and resulting forces and moments. imensions and material parameters for the laminates and core are indicated. El ρ l = tcρc tl ρ G l c E c ρc µ + Mass per unit area beam is Figure 6 Elastic properties and area density of sandich structure According to paper of Nilsson [6], the bending stiffness per unit idth of the 7

55 t / c = y Ecdy + 0 t / + t c t / c l y E dy = l Ectc tc t + El l c l + t t tl +. (4) In general, E l >> E c. The bending stiffness of one laminate is E l t l =. (5) The mass moment of inertia per unit idth is defined as + = tc / tl ρ t t t I ρ ρ l, (6) hile the mass per unit area is 0 c c ( ) c l y y dy = + ρ + + l tctl t µ = t ρ + t ρ. (7) c c l l According to Hamilton's principle, as defined in equation (4), the kinetic and potential energies of the structure must be given as functions of the displacement of the beam given by, and γ as in equation (). 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 entire beam is U per unit area due to pure bending of the U =. (8) The potential energy U per unit area due to pure bending of the to laminates is U γ =. (9) The potential energy U per unit area due to shear deformation of the core is 8

56 γ The total potential energy per unit idth is thus U = G e t c. (0) L γ Ud = 0 0 c L ( U + U + U ) d = + + Gt γ d. () The kinetic energy of the honeycomb panel consists of to parts, the kinetic energy T per unit area due to vertical motion of the beam T = µ, () t and the kinetic energy T per unit area due to the rotation of a section of the beam T = I ρ. () t This gives the entire kinetic energy per unit idth of the beam as L ( T + T ) d = L µ ρ d. (4) Td = + I 0 0 t t The total potential energy for the conservative eternal forces according to Figure 6 is L L [ F ( L) F ( ) M ( L) + M ( 0) ] = pd + F M, L Ad = pda [ ] 0 here F is the force per unit idth, M is the moment per unit idth and p is the eternal dynamic pressure on the beam. The moments and forces are defined in Figure 6. By using the definition of γ, equation (), and by inserting equations (), (4) and (5) into the variational epression () the result is 9 (5)

57 [ ]. 0 0 = dt M F pddt t I t G t ddt L c e δ δ µ γ γ δ ρ (6) The integration over time is from to, and over the length from 0 to L. Using Equation () and epressing 0 t t γ as function of and gives [ ]. 0 0 = dt M F pddt t I t G t ddt L c e δ δ µ δ ρ (7) According to standard procedures e obtain [ ]. 0 0 = + + dt M F p t t I t t Gt ddt L c δ δ δ δ δ µ δ δ δ δ δ ρ (8) Integrating by parts gives = dt M F ddt p ddt t I d t I ddt t d t ddt G t ddt G t dt G t ddt dt ddt dt dt ddt dt L L t t t t c e c e L c e L L L L δ δ δ δ δ δ µ δη µ δ δ δ δ δ δ δ δ δ δ ρ ρ 40

58 (9) When assuming that the displacement is defined so that 0 y δ for, and for 0 t and t = t ( ) 0 = y y δ δ, Equation (9) is reduced to = dt dt M dt F G t ddt G t t I ddt p t G t L L L c e c e c e δ δ δ δ µ δ ρ (40) Since the epression should be equal to zero for δ and 0 δ, e obtain five sets of equations hich must be satisfied. The first to brackets in Equation (40) give to differential equations governing the displacement of the beam as epressed by and. These are = + + p t t G c e µ, (4) and 0 = + + ρ G t t I c e. (4) Using these to equations, eliminating by using the simplification and, the equation governing is given by: ( k t Ae i = ω ) ) ( k t Be i = ω 4

59 ( ) ( ) t I p G t p t I t G t t G t I t I c e c e c e + = ρ ρ ρ ρ µ µ µ µ (4) Eliminating instead gives the corresponding equation for as ( ) p G t p t I t G t t G t I t I c e c e c e + = µ µ µ µ ρ ρ ρ (45) The shear angle γ can be shon to satisfy the same differential equation as in Equation (45). They are the same results that obtained by Nilsson in [6]. Each one of the last three integrals in Equation (40) must also be equal to zero. For these conditions to be satisfied, it follos that at the boundaries of the beam the folloing conditions must be satisfied 0, 0, 0,, 0,, = = = = = = or or M or F t G c e (46) for =0 and =L. These equations satisfy the boundary conditions for a beam. For free vibration, there are no eternal forces and moments. Then the boundary conditions for to ends can be epressed from Equation (46) as 4

60 0 0 0, 0, = = = = = = or or or t G c e (47) Nilsson obtained the same boundary condition equations in [6]. Using the ave Equations (4) and (44), together ith the si boundary conditions, the displacements and can be determined. For free vibrations the eternal 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 Equation (4) are reduced to 0, = t G t t c e µ µ (48) hich is the ave equation for the bending of beams neglecting the moment of inertia. The corresponding equation for, Equation (4) is reduced to = t G t t c e µ µ (49) As seen from the equations the epressions and satisfy the same differential equation for zero eternal eciting force.. 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 (4) through (47). For each boundary condition, certain requirements for the 4

61 displacement and the angular displacement as ell as forces and bending moments must be considered. Figure 7 Particular boundary conditions The three boundary conditions for a beam shon in Figure 7 are summarized in Table 6. Clamped ends Simply supported ends = 0 = 0 = 0 = 0 = 0 = 0 44

62 Free ends t I = ρ 0 = = 0 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. Hoever, hen shear is considered, as in Table 6, there is a difference beteen the natural frequencies for these to conditions. The natural frequencies for the clamped boundaries are the loest. This is due to the fact that shear in the beam is induced by the clamped boundaries to a clamped boundaries to a larger etend as compared to the case ith free boundaries. The apparent bending stiffness for the clamped beam is therefore somehat loer than for the beam ith free ends..4 Wave numbers In Sections. and., the ave equation and the boundary conditions for honeycomb beams ere determined. The equation governing the free fleural vibrations of a honeycomb beam is in Section. given by Equation (4) for ( ) = 0 p ( ) = t I t G t t G t I t I c e c e µ µ µ µ ρ ρ ρ (50) Neglecting losses in the structure, the bending and shear stiffnesses, and are real quantities. By assuming a solution G e ( ). ~ k t Ae i ω 45

63 the ave number k must satisfy k 6 I ρ µ 4 + ω = 0. I 4 ρ k ω µ + µ + I ρ G t e c k ω + G t e c k 4 µ ω (5) This equation has the si solutions k = ± κ, ± iκ, ± iκ here the κ variables are real quantities if the losses are neglected. If the stiffness is defined as = ( + iη) and G = G ( + iη) shon in Figure 8. 0, losses can be included. The absolute values of the ave numbers are 0 46

64 Figure 8 Wave numbers for beam [6]. Solid line real root of the ave number and ashed and dotted lines purely imaginary roots ash-dotted line real root Upper asymptote corresponds to the bending of one laminate Loer asymptote corresponds to the bending of the entire beam The material and geometrical parameters describing the beam are given in Table 7. The loer of the to parallel lines in Figure 9 represents the ave number corresponding to pure bending of the entire beam. The upper line represents the ave 47

65 number for pure bending of one of the identical laminates. The parallel lines define the lo- and high-frequency limits for the ave number κ for the first propagating mode. In the mid-frequency region, shear and rotation become important. As these effects increase, the ave number deviates from the loer asymptote and shifts toards the upper one. For high frequencies, the ave numbers of the structure adjust to the asymptote for the ave number for the bending aves propagating in one of the identical laminates. L b t l t c ρ l ρ c µ (m) (m) (mm) (mm) (kg/m ) (kg/m ) (kg/m ) G (MPa) (GPa) e E l Table 7 Geometrical and material parameters for sandich beam The ave number k for a propagating ave is defined as 4 µω k =, (5) here 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 µ and ith ave number, for the first mode of propagating bending aves. k Considering this definition, the agreement ith the asymptotes confirms that the bending stiffness for lo frequencies is determined as the bending stiffness of the entire beam given in Equation (4). For high frequencies the bending stiffness is determined by the bending of the laminates only, as given by Equation (5). The bending stiffness given in Equations (4) and (5) ill give the limiting values for the calculated bending stiffness. 48

66 The transition of the bending stiffness beteen the to asymptotes is determined by shear and rotation in the core. The dotted and the dashed lines in Figure 8 represent the purely imaginary roots given by κ and κ and correspond to the near field solutions or the evanescent aves for the in-phase motion of the laminates. The constant value for lo frequencies, the dotted line, is determined by the thickness of the core. For increasing frequencies, this curve approaches the limit determined by the ave number for evanescent aves in one of the identical laminates. ecreasing the thickness of the core ill increase the constant value in the loest frequency range. The other near field solution closely follos the asymptote for the bending of the entire beam for lo frequencies. In summary the limiting values for the ave numbers are / 4 ( ) µω µω + limκ = ; limκ =, (5) f 0 f / 4 limκ f 0 µω = / 4 ; limκ f I ρω = /, (54) / GH µω limκ = ; limκ =, (55) f 0 f or κ e see a minimum at a certain frequency f p, here κ shifts from being entirely comple to real and describes a propagating ave. As the frequency increases, κ approaches zero for / 4 f = f p here f p Getc =. (56) π I ρ 49

67 Belo this frequency f p, κ is imaginary, and describes evanescent aves. For higher frequencies κ is real and describe propagating aves..5 Least Squares Method In the previous section the ave equation governing the displacement of a honeycomb beam as derived. Based on this differential equation, ave 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 knon. The dynamic properties of a composite beam are not alays ell defined. This is due to the fact that the elements of the assembled structure perform differently. Hoever, the main dynamic properties of a composite beam can be determined from measurements of the first fe natural frequencies hen the structure is freely suspended. The ave number k for the first propagating bending mode is 4 µω k =, (57) here is the apparent bending stiffness of the structure. Consequently is defined as the bending stiffness of a simple homogeneous beam, hich at a certain frequency has the same dynamic properties as the honeycomb structure. By inserting the definition for k in the ave equation (5) the result becomes G t µ / e c / + / = ω 0, (58) 50

68 if the moment of inertia is neglected. In the lo frequency range, or as ω 0, the first part of the equation dominates hy. The bending stiffness is consequently determined by pure bending of the beam. In the high frequency range, hen ω. 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 ith the results discussed in the previous sections. For a composite beam, the ave number k for fleural aves is defined by Equation (57). For a beam ith boundary conditions ell 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 epression / 4 µ 4π f n κ nl =, L = α n ; n=,,,(59)here α n for a n beam ith various boundaries is given in Table 8. Boundary conditions n 4 5 n>5 f n Free-free Clamped-clamped Free-clamped Simply supported α n π + π / n α n π π / n α n n π Table 8 Values of α n for particular boundary conditions 5

69 Measurements reveal the fundamental natural frequency of the beam. By arranging Equation (59), the apparent bending stiffness n for mode n having the natural frequency f is for a beam of length L and mass per unit area µ, given by n n 4 4π fn µ L = for n =,,... 4 α n. (60) The bending stiffness of a composite beam is strongly frequency dependent as given by equation (60). Since most of the parameters are unknon, only the frequency dependent parameters are preserved and the equation is reritten as here A / B / + C = 0, (6) f f A = Getc Getc ; B = ; C = / / µ π µ π. (6) For non-metallic materials, Young's modulus can ehibit frequency dependence as discussed in, for eample, Reference [] and demonstrated in References [] and []. Hoever, ithin the frequency range of interest, here up to 4 khz, the parameters, and G in Equation (6) 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 A / B / Q = i i + i C, (6) i f i f i here is the measured bending stiffness at the specific frequency f. The i i constants A, B and C are chosen to give the minimum of Q. The shear modulus G c and 5

70 the bending stiffness and can be determined, once the parameters A, B and C are predicted. With knon constants A, B and C the dynamic parameters can be determined G = C /, / = πbµ e t c /. (64) The parameters are predicted from the MATLAB program..6 amping measurement methods Basically there are four measures of damping, the loss factor η, the quality factor Q, the damping ratio ζ, and the imaginary part of the comple modulus. Hoever, they are related to each other. The loss factor or damping ratio is used in measurements: η = πw = Q = ζ = C C c = E E = tanφ. (65) Here and W are the dissipated and total poers 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 comple modulus. Many references present revies of damping measurements [5-9]. Generally, there are three sorts of eperimental methods. ecay 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 ecitation by a force in a given frequency band, the ecitation 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 5

71 plot, as shon in Figure 9. One of the disadvantages of this method is that the effect of noise is considerable. Figure 9 ecay rate method used to determine damping here δ is the decay rate. Modal bandidth method δ = ln, (66) A n m An + m δ ζ =, (67) ( π ) + δ With the frequency response function (log magnitude-time plot or Nyquist diagram), the modal bandidth method, also called as half-poer point method is the most common form used to determine the damping (shon in Figure 0). 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 sandich structures, the loss factor is frequency dependent. In this ork, the same 54

72 eperimental set up as used to determine the dynamic stiffness and to estimate the damping for the sandich structures. Figure 0 Modal bandidth method to determine damping f f ζ =,. (68) f n here and f are the closest frequencies at hich the poer is dropped db from f that at the mode frequency. Poer balance method The SEA method is based on the relationship beteen the input poer and the dissipated poer. So the loss factor can be determined by measuring the input poer and the total energy of a modal subsystem (shon in Figure ). Assuming a stationary 55

73 poer input at a fied location, the input poer must be equal to the dissipated poer under steady state conditions. The disadvantage of this method is that it is relatively hard to determine the input poer and the total energy of a modal subsystem. Figure Poer balance method to determine damping W in ζ =, W tot (69) here W is the input poer and W is the total energy of a modal subsystem. in tot 56

74 CHAPTER 4 YNAMIC PROPERTIES CHARACTERIZATION 4. Eperiments 4.. Steps The steps used to determine the dynamic stiffness and static properties of entire beams and face plates are as follos: Measure natural frequencies f n Calculate dynamic bending stiffness, n, for mode n Calculate damping for natural frequencies f n etermine parameters A, B and C ra dynamic stiffness curve Calculate dynamic propertiesg e,, Figure Eperimental steps to determine some properties of sandich structures These steps provide an interesting approach to determine some parameters of sandich structures. 4.. Set up Since honeycomb plates are typically anisotropic, measurements are performed on beams representing the to main in-plane directions of the plate. For 57

75 materials tested in to directions, the structure and the results are given a subscript of or to indicate the orientations of the beam. The tests have been made for different boundary conditions. ue to the lo mass of the material ~.5 kg/m an accelerometer ould have a certain influence on the vibrations of the beam. The vibration measurements ere therefore made ith a laser vibrometer to avoid transducer contact. The frequency response function, FRF, as determined to give the natural frequencies for the beam. Based on the frequency response function, the loss factor or damping as also determined by the Modal Bandidth Method. The eperiment set up is shon in Figure. The set up includes: a B&K Pulse System (Sound & Vibration Multi-analyzer), laser vibrometer, sample, shaker and poer amplifier. The sample shon in the figure has simply supported ends. The other boundaries such as free-free and clamped-clamped ere tested as ell. The B&K Pulse system gives the output of the frequency response function of the beam hen it is ecited by hite noise. The advantage of set up is that high frequencies can be ecited. But one of the disadvantages is that mass is added to the lighteight structure by the probe or the needle of the shaker. Since the sample is very light eight, a little eight 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 is shon in Figure 4, hich includes a B&K pulse system, laser vibrometer, modal hammer and sample. The modal hammer gives an impulse to the 58

76 sample. The dynamic signal analyzer provides the output of the frequency response function of the beam hen it is ecited by the impulse given by modal hammer. The advantage of the set up is that no mass is added to the lighteight structure so there is no mass loading problem. The disadvantage is that it is hard to high frequency vibration ith the modal hammer. In measurements made, these to set ups ere combined to obtain the natural frequencies in the frequency band of interest. Sound & Vibration Multianalyzer Pulse system Laser vibrometer V(t) Sample Shaker Force transducer F(t) White noise Poer amplifier Figure Set Up Using shaker 59

77 Sound & Vibration Multi-analyzer Pulse system F(t) V(t) Laser Vibrometer Modal Hammer Sample (b) Figure 4 Set Up Using hammer 4.. Samples The composite sandich structures ith honeycomb cores and sandich structures ith closed-cell foam cores, used in the eperiments, are shon in Figures 5 and 6. The face sheets are made ith oven cloth impregnated ith a resin, and the core is a lighteight foam-filled honeycomb structure and closed-cell foam. 60

78 The geometry and density of the sample beams used in the eperiments are shon in the Tables 9 and 0. Beam No. Beam A Beam B Beam C Beam Content Honeycomb Honeycomb Entire Entire core filled core filled sandich sandich ith foam ith foam beam ith beam ith foam filled foam filled honeycomb honeycomb core core Length (m) Thickness(t) (m) Width (m) ensity (kg/m ) ensity (kg/m ) irection Face sheet N/A N/A Single sheet Single sheet Table 9 Geometry and density of the sandich beam ith foam filled honeycomb core. Beam No. Beam E Beam F Beam G Beam H Entire Entire Entire Entire Content sandich sandich sandich sandich 6

79 beam ith beam ith beam ith beam ith foam core foam core foam core foam core Length (m) Thickness(t) (m) Width (m) ensity (kg/m ) ensity (kg/m ) Table 0 Geometry and density of the sandich beam ith foam core. Figure 5 Honeycomb sandich composite structures; (a) foam-filled honeycomb core, (b) composite beam 6