Graduate Theses and Dissertations Graduate College 2011 Flaw detetion in Multi-layer, Multi-material Composites by Resonane Imaging: utilizing Airoupled Ultrasonis and Finite Element Modeling Rihard Livings Iowa State University Follow this and additional works at: http://lib.dr.iastate.edu/etd Part of the Aerospae Engineering Commons Reommended Citation Livings, Rihard, "Flaw detetion in Multi-layer, Multi-material Composites by Resonane Imaging: utilizing Air-oupled Ultrasonis and Finite Element Modeling" (2011). Graduate Theses and Dissertations. 10216. http://lib.dr.iastate.edu/etd/10216 This Thesis is brought to you for free and open aess by the Graduate College at Iowa State University Digital Repository. It has been aepted for inlusion in Graduate Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please ontat digirep@iastate.edu.
Flaw detetion in multi-layer, multi-material omposites by resonane imaging: Utilizing Airoupled Ultrasonis and Finite Element Modeling by Rihard Andrew Livings A thesis submitted to the graduate faulty in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Major: Aerospae Engineering Program of Study Committee: Vinay Dayal, Major Professor Dave K. Hsu Dan J. Barnard Thomas J. Rudolphi Iowa State University Ames, Iowa 2011 Copyright Rihard Andrew Livings, 2011. All rights reserved.
ii TABLE OF CONTENTS LIST OF FIGURES LIST OF TABLES ABSTRACT v xi xii CHAPTER 1. THESIS INTRODUCTION 1 CHAPTER 2. AIR-COUPLED ULTRASONIC TESTING 3 Introdution 3 Literature Review 4 Material Desription 4 Problems with Traditional Methods 5 Initial ACUT Inspetion 8 SiC Tile Charaterization 10 Resonane Pattern Superposition 13 New ACUT Sanning Method 15 Flaw Detetion Using New ACURI Method 16 Future Work 28 Conlusion 30 Referenes 30 CHAPTER 3. ANALYTICAL EQUATION OF THE TRANSVERSE VIBRATION OF A HEXAGONAL PLATE 33 Introdution 33 Literature Review 33 Solution Proposed by Kazkowski 34
iii Examination of Solution 39 Kazkowski s Solution 40 FEM Solution 42 ACURI Solution 43 Examination of Solution Formulation 44 Conlusion 47 Referenes 48 CHAPTER 4. ANSYS MODELING 49 Introdution 49 Air-Coupled Ultrasoni Resonane Imaging Models 49 Kazkowski Plate Setion Models 56 Symmetri Modeling 58 Modal Analysis 60 Harmoni Analysis 76 Conlusion 81 Referenes 82 APPENDIX A1 FULL DEVELOPMENT OF KACZKOWSKI S SOLUTION 83 A1.1 Free Boundary Conditions 84 A1.2 Fixed Boundary Conditions 84 A1.3 Symmetri Boundary Conditions Used by Kazkowski 85 A1.4 Symmetri Boundary Conditions from Appendix A2 88 APPENDIX A2 DEVELOPMENT OF BASIC PLATE EQUATIONS 92 A2.1 Equilibrium 92
iv A2.2 Differential Equations of Transverse Defletion 93 A2.3 Mid-Plane Stress 94 A2.4 Z Component of Mid-Plane Fores 96 A2.5 Boundary Conditions 98 APPENDIX A3 MATLAB CODE 100
v LIST OF FIGURES FIGURE 1.1. 5 Depition of the materials and layers of the Multi-layer, Multi-material Composite. FIGURE 1.2. 6 Plot of the approximate signal losses in db based on the refletion oeffiients for eah interfae and the transmission oeffiients for eah layer. The thikness of eah layer is proprietary and hene is not shown in the figure. Courtesy of Frank Margatan, CNDE, Iowa State University. FIGURE 1.3. 7 Immersion UT pulse-eho inspetion plot of voltage versus time. FIGURE 1.4. 8 Preliminary through sans of a sample layup at the frequenies of 120 khz and 225 khz. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. FIGURE 1.5. 9 Comparison of preliminary through san of a sample layup with defets at a frequeny of 120 khz with an immersion san at a frequeny of 2.25 MHz. The Air-Coupled image is produed by taking the amplitude maximum of the reeived waveform over the entire time gate. FIGURE 1.6. 10 Comparison of through transmission sans of a layup with no known defets and a bare SiC tile at 120 khz. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. FIGURE 1.7. 11 Frequeny san to detet strong resonanes over the bandwidth of the 100 khz transduer. FIGURE 1.8. 12 Frequeny san to detet strong resonanes over the bandwidth of the 225 khz transduer. FIGURE 1.9. 13 Different resonane patterns obtained from a through transmission san of a bare SiC tile at 120 khz. The images are produed by taking the amplitude maximum of the reeived waveform over the time sub-gate.
vi FIGURE 1.10. 14 Different resonane patterns obtained from a through transmission san of a bare SiC tile at 225 khz. The images are produed by taking the amplitude maximum of the reeived waveform over the time sub-gate. FIGURE 1.11. 17 Layout of Panel A showing the approximate size and loation of the erami tiles. The box indiates the portion of the san onsidered in the images of Panel A. FIGURE 1.12. 18 C-san images of Panel A at 120 khz (left) and 225 khz (right). The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. FIGURE 1.13. 18 C-san images of the SiC tile in Panel A at 120 khz (left) and 225 khz (right) with resonane patterns highlighted. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. FIGURE 1.14. 19 Layout of Panel B showing the approximate size and loation of the erami tiles. The approximate size, shape, and loations of the 6 simulated disbonds are shown. The disbonds are loated between different layers in the layup as indiated by the right part of the figure. FIGURE 1.15. 20 C-san of Panel B at 120 khz. (a) Resonane image of the bare SiC with 120 khz resonane pattern outlined. (b) 120 khz san of full layup panel with six engineered disbands outlined. () 120 khz san of full layup panel with six engineered disbands and the resonane patterns outlined. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. FIGURE 1.16. 22 C-san of Panel B at 225 khz. (a) Resonane image of the bare SiC with 225 khz resonane pattern outlined. (b) 225 khz san of full layup panel with six engineered disbands outlined. () 225 khz san of full layup panel with six engineered disbonds and the resonane patterns outlined. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate.
vii FIGURE 1.17. 24 Layout of Panel C showing the approximate size and loation of the erami tiles. The approximate size, shape, and loations of the 30 defets are shown. The defets are loated between different layers in the layup as indiated by the right part of the figure. FIGURE 1.18. 25 C-san of Panel C. (a) 120 khz san of full lay-up with Alumina tiles outlined. (b) 225 khz san of full lay-up with Alumina tiles outlined. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. FIGURE 1.19. 26 C-san of Panel C at 120 khz. (a) Resonane image of the bare SiC with 120 khz resonane pattern outlined. (b) 120 khz san of full layup panel with engineered defets outlined. () 120 khz san of full layup panel with engineered defets and the differene in the resonane patterns outlined. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. FIGURE 1.20. 27 C-san of Panel C at 225 khz. (a) Resonane image of the bare SiC with 225 khz resonane pattern outlined. (b) 225 khz san of full layup panel with six engineered disbands outlined. () 225 khz san of full layup panel with engineered defets and the differene in the resonane patterns outlined. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. FIGURE 1.21. 29 Results from preliminary image proessing on Panel B. FIGURE 2.1. 35 Several polygonal figures with Cartesian oordinate systems and new oordinate systems used in the development of the analytial equation. FIGURE 2.2. 36, 83 The regular hexagon and the triangular setion used in the development of the analytial equation. FIGURE 2.3. 38 Plot of the Kazkowski oeffiients for the vibration of a freely held hexagonal plate as a funtion of line length. FIGURE 2.4. 39 Plot of the Kazkowski oeffiients for the vibration of a freely held hexagonal plate as a funtion of line length.
viii FIGURE 2.5. 40 Results from Kazkowski s solution with fixed boundaries. FIGURE 2.6. 41 Results from Kazkowski s solution with free boundaries. FIGURE 2.7. 42 FEM modal results for a 1/12 setion of the hexagonal plate with fixed boundary. FIGURE 2.8. 43 FEM modal results for a 1/12 setion of the hexagonal plate with free boundary. FIGURE 2.9. 44 ACURI san results of SiC tile at 100 khz, 120 khz, 225 khz. The images are produed by taking the amplitude maximum of the reeived waveform either over the entire time gate or a smaller sub-gate. FIGURE 3.1. 50 Here are the results from a modal analysis on the SiC model with fixed boundary onditions. There are several resonane images and orresponding resonane frequenies displayed. FIGURE 3.2. 51 Here are the results from a modal analysis on the SiC model with free boundary onditions. There are several resonane images and orresponding resonane frequenies displayed. FIGURE 3.3. 52 Here are the results from a modal analysis on the GCG model with fixed boundary onditions. There are several resonane images and orresponding resonane frequenies displayed. FIGURE 3.4. 54 Here are the results from a modal analysis on the Full Lay-up Model with fixed boundary onditions. There are several resonane images and orresponding resonane frequenies displayed. Here only the S2 glass layer is vibrating. FIGURE 3.5. 55 Here are the results from a modal analysis on the Full Lay-up Model with fixed boundary onditions. There are several resonane images and orresponding resonane frequenies displayed. Here only the GCG layer is vibrating.
ix FIGURE 3.6. 56 Mathing resonane images from ACURI and FEM modal results. FIGURE 3.7. 57 FEM modal results for a 1/12 setion of the hexagonal plate with fixed boundary. FIGURE 3.8. 57 FEM modal results for a 1/12 setion of the hexagonal plate with free boundary. FIGURE 3.9. 59 Square plate and possible symmetri setions. FIGURE 3.10. 60 Hexagonal plate and possible symmetri setions. FIGURE 3.11. 61 A omparison of the natural resonane modes of symmetry models to the resonane mode of the full plate model. A) Full Plate Model @ 30670 Hz. B) Half point-point Setion Model @ 30700 Hz. C) Half side-side Setion Model @ 30690 Hz. D) One Third Setion Model @ 30669 Hz. E) One Fourth Setion Model @ 30669 Hz. F) One Sixth Setion Model @ 30668 Hz. G) One Twelfth Setion Model @ 30667 Hz. FIGURE 3.12. 61 A omparison of the natural resonane modes of symmetry models to the resonane modes of the full plate model. A) Full Plate Model @ 113926 Hz. B) Half point-point Setion Model @ 113943 Hz. C) Half side-side Setion Model @113935 Hz. D) One Fourth Setion Model @ 113926 Hz. FIGURE 3.13. 62 A omparison of the natural resonane modes of symmetry models to the resonane modes of the full plate model. A) Full Plate Model @ 113209 Hz. B) Half Triangular Setion Model @ 113911 Hz. C) Half Retangular Setion Model @ 113116 Hz. D) One Fourth Retangular Setion Model @ 112939 Hz. FIGURE 3.14. 62 A omparison of the natural resonane modes of symmetry models to the resonane modes of the full plate model. A) Full Plate Model @ 139994 Hz. B) Half Triangular Setion Model @ 140033 Hz. C) Half Retangular Setion Model @ 139991 Hz. D) One Fourth Triangular Setion Model @ 139990 Hz. E) One Fourth Retangular Setion Model @ 139973 Hz.
x FIGURE 3.15. 69 A graphial explanation of possible symmetry ondition of a Full Plate Model where the lines represent symmetry lines. The blue lines are symmetri and the red lines are antisymmetri. A) 1/12 setion with symmetri-antisymmetri boundaries. B) The 1/12 setion mirrored symmetrially into a full plate. C) Possible interior boundary onditions of a Full Plate. FIGURE 3.16. 77 2D Gaussian pressure distribution applied to the harmoni models. FIGURE 3.17. 77 Harmoni results for the full plate and all of the symmetri setions with fixed boundary onditions and a frequeny of 84400 Hz. FIGURE 3.18. 78 Harmoni results for the full plate and all of the symmetri setions with free boundary onditions and a frequeny of 84400 Hz. FIGURE 3.19. 79 Harmoni results for the full plate and all of the symmetri setions with fixed boundary onditions and a frequeny of 84400 Hz. FIGURE 3.20. 80 Harmoni results for the full plate and all of the symmetri setions with free boundary onditions and a frequeny of 84400 Hz. FIGURE A2.1 94 Fores per unit length of the mid-plane. FIGURE A2.2 96 A plate setion under defletion. The mid-plane fore exhibits a z- omponent.
xi LIST OF TABLES TABLE 3.1. 64-65 Table of resonane frequenies between 0 and 100 khz for a hexagonal plate with fixed boundary onditions and mathed frequenies of symmetri models. TABLE 3.2. 65-68 Table of resonane frequenies between 0 and 100 khz for a hexagonal plate with free boundary onditions and mathed frequenies of symmetri models. TABLE 3.3. 70-71 Table of resonane frequenies between 0 and 100 khz for a square plate with fixed boundary onditions and mathed frequenies of symmetri models. TABLE 3.4. 71-75 Table of resonane frequenies between 0 and 100 khz for a square plate with free boundary onditions and mathed frequenies of symmetri models.
xii ABSTRACT Cerami tiles are the main ingredient of a multi-material, multi-layered omposite being onsidered for the modernization of tank armors. The high stiffness, low attenuation, and preise dimensions of these uniform tiles make them remarkable resonators when driven to vibrate. Defets in the tile, during manufature or after usage, are expeted to hange the resonane frequenies and resonane images of the tile. The omparison of the resonane frequenies and resonane images of a pristine tile/lay-up to a defetive tile/lay-up will thus be a quantitative damage metri. By examining the vibrational behavior of these tiles and the omposite lay-up with Finite Element Modeling and analytial plate vibration equations, the development of a new Nondestrutive Evaluation tehnique is possible. This study examines the development of the Air-Coupled Ultrasoni Resonane Imaging tehnique as applied to a hexagonal erami tile and a multi-material, multi-layered omposite.
1 CHAPTER 1. THESIS INTRODUCTION Multi-layer, multi-material omposites with erami tiles at their heart are being onsidered for next generation armors. The multiple layers of different materials, inluding materials with high attenuation, make traditional Nondestrutive Evaluation (NDE) diffiult. Due to their brittle nature and high frature energy the erami tiles at the enter of these omposites provide the main ballisti resistane of the armor. The mehanial and geometri properties of the tiles also provide another differene from the other materials in the lay-up, namely that the high stiffness, low attenuation, and preise dimensions of these uniform tiles make them remarkable resonators when driven to vibrate. Any defets in the tile, during manufature or after usage, are expeted to hange the resonane frequenies and resonane images of the tile. Likewise, any defets in the lay-up will affet the reeived signal. Thus the omparison of the resonane frequenies and resonane images of a pristine tile/lay-up to a defetive tile/lay-up will be a quantitative damage metri. The purpose of this study was to develop the Air-Coupled Ultrasoni Resonane Imaging (ACURI) tehnique. The development of this new tehnique was performed by examining the vibrational behavior of the erami tiles and the omposite lay-up with Air-Coupled Ultrasoni Testing (ACUT), analytial plate vibration equations, and Finite Element Modeling (FEM). The seond hapter of this thesis desribes all of the ACUT analysis performed on the erami tiles and the lay-ups in the proess of developing this new ACURI tehnique. All testing setups and their results are presented along with the onlusions drawn from the results. The third hapter presents the development of an analytial equation to desribe the transverse vibration of a hexagonal plate and its implementation. The results from the
2 implementation of this equation are ompared with numerial (FEM) and experimental (ACURI) results and the validity of the equation is examined. The fourth hapter of this thesis presents and desribes all of the Finite Element Modeling performed on the erami tile and the omposite lay-up. The results from the FEM annot be ompletely presented due to the shear amount of the images and data, however, any and all signifiant data, will be presented and disussed. This hapter also examines the auray of using symmetri setions to redue problem size/omplexity and omputation time.
3 CHAPTER 2. AIR-COUPLED ULTRASONIC TESTING Introdution Ultrasoni testing has long been used in Nondestrutive Evaluation (NDE) for flaw detetion and haraterization. Most of the ultrasoni testing tehniques have used water or some other liquid ouplant due to their relatively low attenuation. This allows for a high amount of energy to be transferred into a sample. The advent of omposites and other ouplant sensitive materials, as well as their subsequent integration into many industries (suh as the aerospae industry) neessitated the development and use of non-ontat NDE methods. Aousti non-ontat methods have historially been very diffiult to use due to the enormous loss of energy at the air-solid interfae and the ineffiient transfer of aousti energy through air. Over the last two deades, more effiient transduers were developed for the generation and reeption of air-borne ultrasound, thus enabling the non-ontat, nonontaminating inspetion of omposite laminates and honeyomb strutures widely used in the aerospae industry. [1.13] Sine its ineption, Air-oupled Ultrasoni Testing (ACUT) has been treated as a low frequeny version of water-oupled ultrasonis, and its development and uses are desribed in the referenes [1.3-1.19]. Beause of this treatment almost no tehniques have been developed that utilize the strengths of ACUT. Trying to use the tehniques developed for Water-Coupled inspetion with ACUT has provided less than spetaular results. This hapter explores a new testing method for ACUT that utilizes resonane imaging.
4 Literature Review It is an undeniable fat that some materials, espeially omposites strutures, are ouplant sensitive and require some sort of non-ontat inspetion systems. The 1995 paper by Grandia and Fortunko [1.1] was one of the first to desribe the apabilities of a ommerially available air-oupled system and disus potential appliations. Bhardwaj further desribes the air-oupled systems (transduers and analyzers) in his 2001 paper [1.2]. The development of several different aspets (different tehniques, limitations, and apabilities) of this nasent yet maturing branh of NDE is hroniled in the referenes [1.3-1.19]. Migliori and Darling [1.20] were the first to explore resonant ultrasound spetrosopy in 1995 to determine the natural modes of a given sample. This was done by shaking the sample ultrasonially in short bursts then reording the amplitude of the ring-down vibration of the sample. Cabrera et al [1.21] in 2003 used this tehnique to detet raks and other flaws in samples. They demonstrated that a defet in a sample will hange the amplitude and/or frequeny of the natural modes of the sample. Material Desription A new omposite material is being onsidered for the modernization of tank armors. This new omposite is omposed of several layers of different materials and will thus be referred to as a multi-layer, multi-material omposite. The overall lay-up is depited in Fig.1.1. The layers from the bottom up are: IM7 arbon-fiber/epoxy (graphite), hexagonal Silion Carbide (SiC) tiles, IM7 arbon-fiber/epoxy, rubber, and glass-fiber/epoxy (S2 glass).
5 FIGURE 1.1. Depition of the materials and layers of the Multi-layer, Multi-material Composite. The layers are glued together with two sheets of FM94 film adhesive. Eah layer is omposed of a different material from its neighboring layers, and eah material has very different mehanial and aoustial properties. Several samples of this material and its onstituent parts were provided for this study. Multiple SiC tiles, both hexagon and square, four lay-ups, three full lay-ups with embedded defets and one graphite-erami-graphi (GCG) lay-up, and small samples of the graphite, rubber, and glass layers were provided. The SiC material has an elasti modulus of 428.28 GPa, a Poison s ratio of 0.166, and a density of 3058.4 Kg/m^3. Problems with Traditional Methods When it omes to traditional NDE Ultrasoni Testing (UT), suh as through transmission inspetion and pulse-eho inspetion, this type of material offers some diffiulties, namely high attenuation and impedane mismathes at the layer interfaes.
6 FIGURE 1.2. Plot of the approximate signal losses in db based on the refletion oeffiients for eah interfae and the transmission oeffiients for eah layer. The thikness of eah layer is proprietary and hene is not shown in the fgure. Courtesy of Frank Margatan, CNDE, Iowa State University. Fig. 1.2 shows a plot of the approximate signal loss in deibels (db) as a funtion of thikness alulated as the signal passes through the material. The signal loss is alulated very simply by using material properties obtained from Water-Coupled UT (immersion sanning), whih ditate the refletion oeffiients of eah layer interfae and the transmission oeffiients through eah layer. It an be seen from the figure that the signal loss inreases drastially with inrease in frequeny. It is also noteworthy to mention that Air-Coupled UT operates below the frequenies plotted and Water-Coupled UT typially operates above. The results of an immersion pulse-eho inspetion attempted on a multi-layer, multimaterial omposite is shown in Fig. 1.3. The signal is very onvoluted due to all of the layers whih make it diffiult to analyze. When the ultrasoni wave strikes the interfae between layer 1 and layer 2, part of the wave is refleted bak into layer 1 and part of the wave is transmitted into layer 2. The wave that was refleted bak into layer 1 strikes the interfae between layer 1 and the oupling medium and part of the wave is refleted while the other part is transmitted. This means that part of the original refleted wave is refleted bak into the material. This
7 refleted wave just keeps bouning between the two interfaes and transmits part of its energy eah time it strikes an interfae. Meanwhile the wave transmitted into layer 2 strikes the interfae between layer 2 and layer 3, again there is refletion and transmission and the proess of the wave bouning between the two interfaes of layer 2 is initiated. This happens in eah and every layer and eah transmission at eah interfae starts another series of signals bouning. The refleting waves in eah layer yield a very onvoluted signal and the result is Fig. 1.3. FIGURE 1.3. Immersion UT pulse-eho inspetion plot of voltage versus time. Traditional immersion sans suffer drasti signal loss in through transmission inspetions and are fored to operate at the very low end of their operating frequenies. In pulse-eho inspetions, the signal is so ompliated that no useful information an be easily extrated. These fats, ombined with the neessity of a water tank large enough to ontain the sample, make the immersion san diffiult to perform and an unviable option as a fieldable NDE tehnique. With that in mind, and the fat that Fig. 1.2 indiates the operating frequenies of ACUT experiene muh less signal loss through the lay-up, Air-Coupled Ultrasound is the obvious hoie for ontinued ultrasoni inspetion of this material.
8 Initial ACUT Inspetion The preliminary Air-Coupled inspetion of one of the samples yielded some surprising results. Fig. 1.4 shows two separate through sans of the same sample at the frequenies of 120 khz and 225 khz. The most notieable things from these sans are the very prominent hexagon shapes and the very bright dots that form, what has been termed to be, snowflake patterns entered on the hexagon shapes. There is only one omponent of the lay-up that ould aount for the presene of these hexagon shapes in the above sans. These hexagonal patterns are the same size, shape, loation, and orientation as the SiC tiles in the enter of the lay-up. The snowflake patterns are entered on the hexagon shapes and appear to line up with the hexagons as if the hexagon shape and the snowflake image were part of the same pattern. Beause of this, it is safe to assume the hexagon and the snowflake patterns are aused by the vibration of the erami tiles. FIGURE 1.4. Preliminary through sans of a sample layup at the frequenies of 120 khz and 225 khz. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. If it is true that the snowflake patterns are aused by the vibration of the erami tiles, then they must be resonane patterns of the tiles. If they are resonane patterns of the tiles, then they should vary with both material and frequeny. A loser inspetion of the images in Fig. 1.4 shows that the snowflake patterns for the 120 khz san are different for the different
9 materials. The SiC tile has more of a spoke-like pattern while the alumina tiles have more of a separated dot pattern, whih is signifiantly different. When omparing the 120 khz san image to the 225 khz san image (the SiC tile speifially), it an be seen that the pattern is signifiantly different at the different frequenies. Sine the snowflake patterns vary with both material and frequeny it is safe to assume that the results from the preliminary through sans are due to the resonane of the erami tiles embedded in the lay-up. FIGURE 1.5. Comparison of preliminary through san of a sample layup with defets at a frequeny of 120 khz with an immersion san at a frequeny of 2.25 MHz. The Air-Coupled image is produed by taking the amplitude maximum of the reeived waveform over the entire time gate. In a quik omparison of a preliminary through san using Air-Coupled UT at a frequeny of 120 khz and an Immersion UT through san at a frequeny of 2.25 MHz (Fig. 1.5) it is obvious that the six flaws are more easily deteted with the Immersion san. The differenes in the tile materials, however, ould not be deteted with Immersion UT. This shows that the different inspetion methods have different strengths and give different information.
10 SiC Tile Charaterization The first step in developing an NDE tehnique is to haraterize the SiC tile. Two different types of sans were performed on the SiC tile in order to better understand the behavior of the tile and to searh for several strong resonanes, their frequenies, and their visual patterns. The first tehnique used to haraterize the tiles is to perform a through transmission san on the tile, where both the transmitter and reeiver are sanning. This is exatly the same type of inspetion used in the full lay-ups. Fig 1.6 shows a omparison between a through transmission san of the bare SiC tile and a full lay-up with no known defet. In these images blue is low amplitude while red is high amplitude. Sine the plate is vibrating the high and low amplitude regions will alternate in sign, and hene olor, in a ylial manor. Beause of this the olors are interhangeable. This is true for all of the ACUT images in this study. As an be seen, the resonane patterns math quite well when damping from the other layers is taken into aount (and the olors reversed). FIGURE 1.6. Comparison of through transmission sans of a layup with no known defets and a bare SiC tile at 120 khz. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate.
11 The seond tehnique used to haraterize the SiC tiles uses a onstant frequeny sine wave (CW) signal, whih was impinged on the bare tile at low voltage and hene low amplitude. The amplitude of the reeived wave was stored, then the frequeny was inreased by a speified amount, df, and the whole proess was repeated. This tehnique identifies all of the natural frequenies of the tile, and is very similar to the method used by Cabrera et al [1.20]. Fig. 1.7 and 1.8 show the results from this type of san. The blue plot is the transduer amplitude through air at eah frequeny. The red plot is the amplitude of the reeived signal from the tile. In this ase the transmitter is impinging on the enter of the tile and the reeiver is far enough away from the tile to hopefully apture most of the surfae. FIGURE 1.7. Frequeny san to detet strong resonanes over the bandwidth of the 100 khz transduer. The 100 khz plot shows quite a few high amplitude resonanes. These high amplitude resonanes ould be used to detet flaws after the resonane images for eah resonant peak are determined. It is interesting to note here that the quality fators (Q value), whih measure how well a system osillates, for these sans are in exess of 10,000. The highest Q value observed
12 is ~47,000. This means that these SiC tiles are remarkable resonators when driven to vibrate, and explains why the resonane patterns an be seen through the entire lay-up. FIGURE 1.8. Frequeny san to detet strong resonanes over the bandwidth of the 225 khz transduer. Also notie that there are many more resonant peaks within the 225 khz transduer bandwidth than within the 100 khz transduer bandwidth. This is beause the number of resonant modes a given shape is able to support inreases (as well as the omplexity of the resonane patterns) with inreasing frequeny. Having multiple resonane peaks within the transduer bandwidth means that eah of the resonane peaks will be exited when the first type of sans disussed above is used. This auses a superposition of resonane patterns whih is disussed in the next setion. These sans were performed at low voltages so as not to damage the transduers. Unfortunately the voltage used was not high enough to overome the damping effets from the other layers when they were added to the tiles and no signal was reeived. It was onjetured that a tone-burst system would be able to rereate the effets of the CW system but in small
13 enough bursts that more voltage ould be put through the transduers without damaging them and thus a signal would be strong enough to propagate through the lay-up. No redible results were obtained using the tone-burst system. It seems that at this time, using the urrently available transduers, isolating a frequeny with whih to san the lay-up is not possible. An Air-Coupled pulse-eho inspetion was also attempted on a SiC tile without suess. This is due to the long ring-down time and noise level of the transduer. When the bak-wall eho reahes the transduer the transduer is still ringing from the initial pulse. Normally these two signals ould be separated with a Fast Fourier Transform (FFT) (assuming the bak-wall eho is not the same frequeny as the initial pulse) but in this ase the signal from the bak-wall is of lower amplitude than the transduer noise. Resonane Pattern Superposition The images in this study are made by applying a time gate to the reeived signal and plotting the maximum amplitude of the waveform inside this time gate. Smaller sub-gates an also be applied to the signal so that images an be produed that orrespond to speifi portions of the waveform. This proess allows for the observation of how the vibrations of the sample hange with time. FIGURE 1.9. Different resonane patterns obtained from a through transmission san of a bare SiC tile at 120 khz. The images are produed by taking the amplitude maximum of the reeived waveform over the time sub-gate.
14 FIGURE 1.10. Different resonane patterns obtained from a through transmission san of a bare SiC tile at 225 khz. The images are produed by taking the amplitude maximum of the reeived waveform over the time sub-gate. Fig. 1.9 shows several distint resonane patterns observed in the bare SiC tile with through transmission at a frequeny of 120 khz. Likewise, Fig 1.10 shows several distint resonane patterns observed in the bare tile with through transmission at a frequeny of 225 khz. The different images in eah figure ome from the same san but from different times in the reeived signal. This shows that as the tile is allowed to ring down, it experienes several different resonant modes. The two images from the 120 khz san are atually a repeated mode. Transduers have a finite, non-zero bandwidth that enompasses many resonanes whih our at very speifi, disrete frequenies. This means that eah and every one of the resonanes that is within the transduer bandwidth is exited by the transduer during a san of type one disussed above. When a multitude of these resonanes are exited at the same time the resonane images are superimposed in the san results. It may also lead to onstrutive and destrutive interferene. This, in part, aounts for the omplexity of the resonane images observed. Consider the impliations of Fig. 1.7 and 1.8 on the results from the through transmission sans shown in Fig. 1.9 and 1.10 respetively. This superposition an be reversed
15 somewhat by stepping through the reeived signal, from the through san of type one desribed above, with a small time-gate. This indiates that resonane modes of different frequenies will travel through the material at different speeds whih would mean there are different phase veloities and that the material is dispersive. It is not as simple as that however. Beause of the large refletion oeffiient of the SiC-air interfae even a short ultrasoni pulse will produe a long ring down. The presene of multiple resonanes within the bandwidth of the transduer will produe a superposition, yet there will be dominant resonane modes that may need to die down before the other resonane modes beome apparent. New ACUT Sanning Method As mentioned previously, traditional ACUT tries to mimi the inspetion tehniques and apabilities of Immersion UT, but due to the high attenuation of the ambient air and the inredible impedane (ρ: density * wave veloity) mismath between the material and air, pulse-eho inspetion is extremely improbable with ACUT. Pith-ath inspetion is usually unavailable as well beause of the inhomogeneous nature of the materials with whih ACUT tends to operate (this does not hold true for surfae waves). Through transmission is the only inspetion tehnique that an be used to any great affet with ACUT. Traditional ACUT has both drawbaks and benefits when ompared to Immersion UT. The main drawbaks are: the diffiulty in getting energy into the sample (due to the attenuation and impedane mismath problems desribed above) and the minimum detetable flaw size is dependent on the wavelength of the partiular frequeny in the partiular material (the same holds true for immersion), and with the low frequenies used in ACUT, this makes the
16 minimum flaw size fairly large (~1in in graphite/epoxy at 120 khz). The benefits inlude the inredible penetration depth due to lower attenuation through materials at lower frequenies and the fat that this type of inspetion is non-ontat. Based on this evidene, a new tehnique ould be reated (for this problem speifially and possibly for many others) that would utilize the strengths of ACUT. The preliminary sans (Fig. 1.4) demonstrated that the erami tiles vibrate so strongly that their resonanes an be seen through all of the other layers and they atually suppress the vibrations of the other layers and fore them to vibrate with the same pattern as the tiles. It is proposed that sine these tiles vibrate so strongly that their resonanes are learly visible and the minimum flaw size is fairly large, these resonanes be used to detet and haraterize flaws in the lay-up during the manufaturing proess or after possible damage during ombat or handling. Comparing the resonane patterns of the erami tiles in a lay-up with no defet to those of lay-ups with known defets at several frequenies should provide a good damage metri with whih to develop this tehnique. Any flaws or damage in the tiles themselves should result in a hange in the resonane pattern or the resonane frequeny of the tile whih will make the flaws and damage apparent. Likewise, any damage or flaws in the path of the propagating wave will affet the appearane of the resonane amplitude, or eliminating high amplitude regions of the resonane pattern. This tehnique is alled Air-Coupled Ultrasoni Resonane Imaging (ACURI). Flaw Detetion Using ACURI Method The question for this new flaw detetion method is whether or not flaws an be deteted and haraterized in a multi-layer, multi-material omposite lay-up using ACUT. Three
17 samples were used to explore this question. These three samples will be referred to as Panel A, Panel B, and Panel C and have the following harateristis: Panel A has the rough dimensions of 14 in x 15 in x 1.5 in, is omposed mostly of hexagonal alumina tiles with one hexagonal SiC, and has no known defet in the materials or the lay-up; Panel B has the rough dimensions of 16 in x 18 in x1.5 in, is omposed mostly of hexagonal alumina tiles with six omplete hexagonal SiC tiles, and has six simulated disbonds; finally, Panel C has the rough dimensions of 21 in x 25 in x 1.5 in, is omposed of ten omplete hexagonal alumina tiles and eighteen omplete hexagonal SiC tiles, and has 30 embedded defets, 12 inlusions and 18 simulated disbonds. All of the tests performed on the three panels are ACUT through sans with both the transmitter and reeiver sanning together (C-san). The transduers used were Quality Material Inspetion (QMI) <http://www.qmi-in.om/> spherially foused probes with enter frequenies of ~120 khz and ~225 khz with bandwidths of ~25%. The transduers were held ~2 in from both the front and bak surfaes. The signal was impinged on the graphite side of the lay-up and was reeived on the S2 glass side of the lay-up. FIGURE 1.11. Layout of Panel A showing the approximate size and loation of the erami tiles. The box indiates the portion of the san onsidered in the images of Panel A.
18 The approximate size and loation of both the hexagonal alumina tiles and the solitary hexagonal SiC tile of Panel A are shown in Fig. 1.11. The square outline over part of the figure outlines the area of interest for this panel and the san images of Panel A are ropped to this approximate size. The sans of Panel A, for both frequenies mentioned above, are shown in Fig. 1.12. The SiC tile an be easily distinguished from the alumina tiles due to a differene in the resonane patterns of the tiles. In the 120 khz image, the SiC tile exhibits a series of high amplitude dots that form a spoke-like pattern where the alumina tiles show a series of separated dots that, while showing the same symmetry as the SiC tile, form a different pattern. FIGURE 1.12. C-san images of Panel A at 120 khz (left) and 225 khz (right). The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. FIGURE 1.13. C-san images of the SiC tile in Panel A at 120 khz (left) and 225 khz (right) with resonane patterns highlighted. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate.
19 In the 225 khz image, both the alumina tiles and the SiC tile have the same (or very similar) resonane patterns, however, the amplitude of the resonane patterns is different for the different material, and the SiC tile an still be easily distinguished. Sine there are no known defets over the SiC tile in this panel (Panel A), these resonane images will be used as a basis for omparison for the panels with defets to determine whether the defets an be visually deteted. The resonane patterns have been outlined in Fig. 1.13 to make omparisons easier. The 225 khz pattern is too intriate to use the entire pattern and sine the defets are primarily over the enter of the tiles only the inner part of the pattern will be used. FIGURE 1.14. Layout of Panel B showing the approximate size and loation of the erami tiles. The approximate size, shape, and loations of the 6 simulated disbonds are shown. The disbonds are loated between different layers in the layup as indiated by the right part of the figure. The approximate size and loation of both the hexagonal alumina tiles and the hexagonal SiC tiles of Panel B are shown in Fig. 1.14. The approximate size, shape, and loations of the six engineered disbonds are also indiated. The disbonds are square, entered over the enter of the SiC tiles, and ome in the sizes of 0.5 in, 1.5 in, and 2.5 in. They were engineered by utting the appropriately sized and shaped holes in both layers of the film
20 adhesive between two layers of the lay-up as indiated in the figure. This gives void disbonds instead of kissing disbonds, whih makes it more diffiult for an ultrasoni signal to pass through. As indiated in the figure, the two rows of disbonds are loated between different material layers. The top row, labeled as row 1, has the disbonds between the rubber and S2 glass layers while the bottom row, labeled as row 2, has the disbonds between the graphite and rubber layers. The defets in Panel B were reated to simulate the effet of the inlusion of air during the manufature of the lay-up. The square disbond shape is merely for ease of fabriation and serves no sientifi purpose. The hoie of the 0.5 in, 1.5 in, and 2.5 in sizes for these disbonds is not due to any mehanial signifiane, but merely to test the detetion limits of this tehnique. We an term these sizes as small, medium, and large, based on detetion apabilities in omposites, for the purpose of detetion. FIGURE 1.15. C-san of Panel B at 120 khz. (a) Resonane image of the bare SiC with 120 khz resonane pattern outlined. (b) 120 khz san of full layup panel with six engineered disbands outlined. () 120 khz san of full layup panel with six engineered disbands and the resonane patterns outlined. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. Fig. 1.15 shows the san results for Panel B as well as the undamaged SiC tile from Panel A for the frequeny of 120 khz. In Fig. 1.15-b the approximate size, shape, and loation
21 of the disbonds are outlined. A asual glane at Fig. 1.15-b shows that the 1.5 in and 2.5 in disbonds are learly visible as dark loudy areas in both the top and bottom rows in this san. Furthermore, the 0.5 in disbond in the top row an be deteted via a disruption of the resonane pattern when ompared to the pattern of the tile with no known defet (Fig. 1.15-a) due to a large hange in the amplitude and larity of the resonane image. A slightly loser inspetion of the image in Fig 1.15-b shows that a signifiant differene between the top and bottom rows an be observed. This means the differene in the disbond loations an be distinguished with through transmission due to these resonane patterns. In Fig. 1.15- the resonane patterns have been highlighted for easier omparison. If the top row of images is visually ompared to the bottom row, the differenes in the resonanes are apparent. The bottom 0.5 in defet pattern mathes very losely with the pattern of the tile with no known defet (Fig. 1.15-a) while the top 0.5 in defet pattern has a signifiantly different amplitude and seems to have been disrupted by something. Comparison of the 1.5 in defet patterns shows that they are basially the same as the 0.5 in defet patterns, that they show the same relation, but with a very low amplitude region in the approximate loation of the disbonds. The top 1.5 in defet pattern is fairly haoti and even though some of the resonane pattern is detetable it has been disrupted by the disbond signifiantly more than the bottom pattern. The top 2.5 in defet pattern is very haoti and there is no detetable resonane pattern though the pattern is evident in the bottom row. The pattern in the bottom row mathes the pattern of the tile with no known defet quite well (though muh lower amplitude). There is also a seondary pattern (outlined in yellow-orange), though what auses this is unknown. It is lear from these omparisons that 5 out of 6 of the disbonds are detetable with ACURI at a frequeny 120 khz and that there is a signifiant and notieable differene
22 between the resonane images of the disbonds loated between the rubber and glass layers and the disbonds loated between the graphite and the rubber layers. FIGURE 1.16. C-san of Panel B at 225 khz. (a) Resonane image of the bare SiC with 225 khz resonane pattern outlined. (b) 225 khz san of full layup panel with six engineered disbands outlined. () 225 khz san of full layup panel with six engineered disbonds and the resonane patterns outlined. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. Fig. 1.16 shows the same two panels shown in Fig. 1.15, but using a 225 khz transduer instead of a 120 khz transduer. Again the first image (Fig. 1.16-a) shows the resonane pattern of the tile in the lay-up with no known defets, the seond image (Fig. 1.16-b) outlines the six engineered defets, and the third (Fig. 1.16-) highlights some of the resonane patterns. A quik glane at the san image shows that all 6 of the disbonds an be seen. Also evident is the differene between the top and bottom rows, namely that two of the three bottom disbonds have disernable resonane patterns while only one of the top disbonds has a disernable resonane pattern. Looking at the 0.5 in disbonds, the top and bottom patterns are very similar and math well with the no defet pattern, however, the high amplitude enter, learly visible in the no defet pattern, is not present. This indiates that both of the 0.5 in disbonds are detetable. The
23 resonane pattern is still detetable for the bottom 1.5 in disbond, but not for any of the rest of the disbonds. From these omparisons it is evident that 6 out of 6 of the disbonds are detetable with ACURI at a frequeny of 225 khz and that there is a signifiant and notieable differene between the resonane images of the disbonds loated between the rubber and glass layers and the disbonds loated between the graphite and rubber layers. All six of the disbonds were easily detetable by omparing the defetive resonane patterns with the non-defetive patterns. Moreover, a differene in flaw loation (whih layers the flaw is between) an also be seen by omparing resonane patterns. A possible explanation for the different disbond loations affeting the resonane patterns differently lies in the way in whih the disbonds were engineered. Essentially there is a void for eah disbond. Consider the vibrational displaement of the layers. The SiC tiles have the largest displaement (due to the vibration) of all of the layers. If the void/disbond is loated between the seond graphite layer and the rubber layer, the displaement of the graphite layer due to the SiC vibration may be larger than the thikness of the void/disbond and the displaement an be transferred to the subsequent layers. If the void/disbond is loated between the rubber layer and the S2-glass layer, the displaement of the rubber layer due to the vibration of the SiC tile may be smaller than the thikness of the void/disbond and the displaement annot be transferred to the S2-glass layer. This is beause the rubber has a low modulus and high damping that will absorb a signifiant portion of the displaement of the graphite layer.
24 FIGURE 1.17. Layout of Panel C showing the approximate size and loation of the erami tiles. The approximate size, shape, and loations of the 30 defets are shown. The defets are loated between different layers in the layup as indiated by the right part of the figure. The approximate size and loation of both the hexagonal alumina tiles and the hexagonal SiC tiles of Panel C are shown in Fig 1.17. The approximate size, shape, and loations of the 30 engineered defets are also indiated. The materials inluded in the lay-up (mold release and grease) and their loations are indiated in the figure and as indiated the inlusions are irular, entered over the enter of the Si tiles, and ome in the sizes of 0.5 in and 1 in. The disbonds are irular with eah group of three entered over the enter of the SiC tile and are made by inluding two irular layers of kapton of different thiknesses. Inluding two layers of kapton will reate a kissing disbond that will transmit the signal when in ompression but not in tension. As indiated in the figure, the six olumns have two different sizes of defets and the defets are loated between different material layers. The defets are between the SiC tiles and the graphite layer in the first and fourth olumns, labeled as olumn 1, the seond and fifth olumns, labeled as olumn 2, have the defets between the graphite and rubber layers, and the third and sixth olumns, labeled as olumn 3, have the defets between the rubber and S2-glass layers. The defets in Panel C were reated to simulate the effet of
25 different types of possible inlusions during the manufature of the lay-up and disbonds that may our. Again the sizes are arbitrary and are merely to test the detetion limits. FIGURE 1.18. C-san of Panel C. (a) 120 khz san of full lay-up with Alumina tiles outlined. (b) 225 khz san of full lay-up with Alumina tiles outlined. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. Figure 1.18 shows the results from two resonane sans of Panel C, the first at 120 khz and the seond at 225 khz. A omparison of all of the alumina tiles outlined in the figure shows a drasti differene in the amplitude and larity of the resonane patterns, espeially in the 225 khz san. Neither the sans of Panel A or Panel B show a similar issue. This indiates some deviation in the manufature or handling with respet to Panels A and B. ACUT signals are sensitive to variations in the lay-up not aused by engineered defets, variations suh as dry pathes or delaminations. These variations an loud or mask the atual engineered defets, whih means that the presene of both engineered and aidental defets ould make flaw detetion diffiult with this tehnique. Even though any results from Panel C annot be
26 onsidered valid, a omparison of Panel A and Panel C yields some indiations of flaw detetion. FIGURE 1.19. C-san of Panel C at 120 khz. (a) Resonane image of the bare SiC with 120 khz resonane pattern outlined. (b) 120 khz san of full layup panel with engineered defets outlined. () 120 khz san of full layup panel with engineered defets and the differene in the resonane patterns outlined. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. Fig. 1.19 shows the san results for Panel C as well as the undamaged SiC tile from Panel A for the frequeny of 120 khz. In Fig 1.19-b the approximate size, shape, and loation of the defets are outlined. Although the san does not have very defined or onsistent resonane patterns on the SiC tiles, due to a possible problem in the lay-up during the manufature, two SiC tiles show indiations of defets. Fig 1.19- highlights six dark, loudy, irular shapes in the fourth and fifth tiles of the bottom row of SiC tiles. These six dark areas orrespond with six of the simulated kapton disbonds. The lak of indiation of any other defet is probably due to problems with the lay-up.
27 FIGURE 1.20. C-san of Panel C at 225 khz. (a) Resonane image of the bare SiC with 225 khz resonane pattern outlined. (b) 225 khz san of full layup panel with six engineered disbands outlined. () 225 khz san of full layup panel with engineered defets and the differene in the resonane patterns outlined. The images are produed by taking the amplitude maximum of the reeived waveform over the entire time gate. Fig. 1.20 shows the san results for Panel C as well as the undamaged SiC tile from Panel A for the frequeny of 225 khz. In Fig 1.20-b the approximate size, shape, and loation of the defets are outlined. Again the san shows signs of some sort of unknown problem. Fig 1.20- highlights differenes in the resonane patterns between Panel A with no defet and Panel C. It is easier to highlight the differenes beause of the omplexity of the resonane pattern. The highlighted differenes in the top two rows of SiC tiles orrespond with the grease and mold release inlusions and eleven of the twelve are deteted beause the high amplitude enter in the resonane pattern is absent. The high amplitude enter is again absent in the first three resonane patterns of the bottom row of the SiC tiles. This indiates a flaw of some kind but does not indiate the three simulated disbonds. The resonane patterns in the fourth and fifth SiC tiles in the bottom row are ompletely disrupted with possible indiations of the irular disbonds. The sixth resonane pattern in the bottom row also shows evidene of disruption in loations that orrespond with the loations of the disbonds.
28 Although the sans of Panel C did not produe very lear images and any onlusions an only be treated as onjetures, there is good evidene that the flaws would have been detetable in a better onstruted sample. Eleven of the twelve inlusions and six of the eighteen disbonds were detetable with some ertainty. Future Work The work presented in this thesis is only a beginning in the study of the ACURI tehnique and more work needs to be done to validate the results and onlusions of this study. More experiments need to be performed on a variety of samples with different types of defets and more modeling with FE should be performed to simulate different defets. Image proessing should also be performed to determine whether the differenes in resonane images between a non-defetive tile/lay-up and a tile/lay-up with known defets an be deteted with an automated or semi-automated program. Some initial image proessing was performed as part of this study. The resonane image of the SiC tile in Panel A (with no known defet) and the resonane image from one of the SiC tiles in Panel B (with six engineered disbonds) were ompared. Several easily identifiable high amplitude regions on the image with no defet were seleted along with the orresponding regions on the image with known defet. From the relative positions of these seleted points a transform matrix was reated and applied to the known defet image in order to rotate and align it with the no defet image. A ross-orrelation was then performed to ensure that the images were properly aligned. The images were then subtrated and divided in several ombinations to see if the defets were apparent.
29 FIGURE 1.21. Results from preliminary image proessing on Panel B. The results from the initial attempt at image proessing to detet the flaws in Panel B at 120 khz are shown in Fig. 1.21. The 1.5 in and 2.5 in disbonds are learly visible as bright green shapes. There are no shapes apparent for the 0.5 in disbonds, but there is some indiation of an image differene. There is also an apparent differene between the top and bottom rows. This image proessing tehnique has orroborated the onlusions drawn from visual omparison in the setion Flaw Detetion Using Proposed Method. Sine the images show a strong periodiity the next image proessing tehnique to try utilizes 2D FFT s. Essentially the FFT of the no defet image is subtrated from the FFT of the known defet image with some preproessing to redue the noise of the images. Any residual data present after this subtration proess should indiate the presene of a defet and possibly allow for flaw haraterization.
30 Conlusion The multi-layer, multi-material omposite armor onsidered in this study provides some diffiulties for many traditional NDE tehniques. The erami tiles at the enter of this omposite material have speial properties that an and should be utilized when reating an optimized detetion tehnique. These SiC are remarkable resonators when driven to vibrate, and there are many distint natural resonanes with distintive resonane patterns within the bandwidth of the transduers used. Several of these resonanes are very strong and have a resonane sharpness, or Q fator, omparable to or above that of quartz. It has been shown that these resonane patterns an be used to detet and haraterize flaws. All six of the engineered disbonds in Panel B were detetable along with information indiating between whih two layers the disbonds were loated. Even with the poor san results from Panel C eleven of the twelve inlusions and six of the eighteen disbonds were detetable. The ACURI tehnique developed in this study shows great promise for this material speifially and for several different types of materials in general. More work needs to be done by sanning samples ontaining different defets, using frequeny sans (w sans at speifi frequenies), and using image proessing to detet flaws. Referenes 1.1. Grandia, W., Fortunko, C., NDE Appliations of Air-Coupled Ultrasoni Transduers, Proeedings, IEEE Ultrasonis Symposium, 1995 1.2. Bhardwai, M., Non-Contat Ultrasound: The Final Frontier in Nondestrutive Testing and Evaluation, Enylopedia of Smart Materials, 2001 John Wiley & Sons, New York
31 1.3. Hsu, D. K., Barnard, D. J., Peters, J. J., Dayal, V., Development of Nondestrutive Inspetion Methods for Composite Repair, Proeedings, Review of Progress in Quantitative Nondestrutive Evaluation, 2003, Vol. 657, pp. 1019-1025 1.4. Peters, J., Kommareddy, V., Liu, Z., Fei, D., Hsu, D., Non-ontat Inspetion of Composites Using Air-Coupled Ultrasound, Proeedings, Review of Progress in Quantitative Nondestrutive Evaluation, 2003, Vol. 657, pp. 973-980 1.5. Holland, S. D., Teles, S. V., Chimenti, D. E., Quantitative Air-Coupled Ultrasoni Materials Charaterization with Highly Foussed Aousti Beams, Proeedings, Review of Progress in Quantitative Nondestrutive Evaluation, 2004, Vol. 700, pp. 1376-1381 1.6. Hsu, D. K., Barnard, D. J., Peters, J. J., Polis, D. L., Appliation of Air-Coupled in NDE of omposite Spae Strutures, Proeedings, Review of Progress in Quantitative Nondestrutive Evaluation, 2004, Vol. 700, pp. 851-858 1.7. Hsu, D. K., Kommareddy, V., Barnard, D. J., Peters, J. J., Dayal, V., Aerospae NDT Using Piezoerami Air-Coupled Transduers, Proeedings, World Conferene on NDT, 2004 1.8. Hsu, D. K., Peters, J. J., Barnard, D. J., Kommareddy, V., Dayal, V., Experiene with Air-Coupled Ultrasoni Inspetion of Aerospae Composites, Review of Progress in Quantitative Nondestrutive Evaluation, 2004 1.9. Kommareddy, V., Peters, J. J., Dayal, V., Hsu, D. K., Air-Coupled Measurements in Composites, Proeedings, Review of Progress in Quantitative Nondestrutive Evaluation, 2004, Vol. 700, pp. 859-866 1.10. Peters, J. J., Barnard, D. J., Hsu, D. K., Development of a Fieldable Air-Coupled Ultrasoni Inspetion System, Proeedings, Review of Progress in Quantitative Nondestrutive Evaluation, 2004, Vol. 700, pp. 1368-1375 1.11. Peters, J. J., Dayal, V., Barnard, D. J., Hsu, D. K., Resonant Transmision of Air- Coupled Ultrasound Through Metalli Inserts in Honeyomb Sandwih Strutures, Proeedings, Review of Progress in Quantitative Nondestrutive Evaluation, 2005, Vol. 760, pp. 1026-1032 1.12. Barnard, D. J., Hsu, D. K., Development of Pratial NDE Methods for Composite Airraft Strutures, Proeedings, Review of Progress in Quantitative Nondestrutive Evaluation, 2006, Vol. 820, pp. 1019-1026 1.13. Hsu, D. K., Nondestrutive Testing using Air-Bourne Ultrasound, Ultrasonis, 2006, Vol. 44, sup. 1, pp. 1019-1024
32 1.14. Hsu, D. K., Barnard, D. J., Inspeting Composites with Airborne Ultrasound : Through Thik and Thin, Proeedings, Review of Progress in Quantitative Nondestrutive Evaluation, 2006, Vol. 820, pp. 991-998 1.15. Dayal, V, Hsu, D. K., Kite, A. H., Air-Coupled Ultrasound-A New Paradigm in NDE, in Proeedings, ASME 2007 International Mehanial Engineering Congress and Exposition, Seattle, WA, 2007, Vol. 13, pp. 153-155 1.16. Hsu, D. K., Barnard, D. J., Dayal, V., NDE of damage in Airraft Flight Control Surfaes, Proeedings, Review of Progress in Quantitative Nondestrutive Evaluation, 2007, Vol. 894, pp. 975-982 1.17. Im, K. H., Chang, M., Hsu, D. K., Song, S. J., Cho, H., Park, J. W., Kweon, Y. S., Sim, J. K., Yang, I. Y., Feasibility on Ultrasoni Veloity using Contat and Non-Contat Nondestrutive Tehniques for Carbon/Carbon Composites, Proeedings, Review of Progress in Quantitative Nondestrutive Evaluation, 2007, Vol. 894, pp. 1013-1020 1.18. Dayal, V., Hsu, D. K., Kite, A. H., Damage Modelling in Composites for NDE, Proeedings, Review of Progress in Quantitative Nondestrutive Evaluation, 2008, Vol. 975, pp. 927-933 1.19. Dayal, V., Hsu, D. K., Air-oupled Ultrasound for the NDE of Composites, World Journal of Engineering, 2010, Sup. 2, pp. 381 1.20. Migliori, A., Darling, T. W., Resonant Ultrasound Spetrosopy for Materials Studies and Non-destrutive Testing, Ultrasonis, 1995, Vol. 34, Issues 2-5, pp. 473-476 1.21. Cabrera, M., Castell, X., Montoliu, R., Crak Detetion System Based on Spetral Analysis of a Ultrasoni Referene Signals, in IEEE International Conferene on Aoustis, Speeh, and Signal Proessing, 2003 Proeedings, 2003, Vol. 2, pp. 605-6
33 CHAPTER 3. ANALYTICAL EQUATION OF THE TRANSVERSE VIBRATION OF A HEXAGONAL PLATE Introdution Analytial equations desribing the transverse defletion and vibration of retilinear and triangular plates are well known and have been a topi of inquiry for the past hundred years. These equations are fairly straight forward and solvable. Conversely, the equations desribing the transverse defletion and vibration of higher order polygonal shapes are diffiult to formulate and to solve. A solution to this problem was proposed in 1961 and has been aepted by the aademi ommunity. During the ourse of this researh the use of the 1961 analytial solution was attempted. The results were ompared with the experimental results and the results from a numerial solution, and it was observed that there are some inherent limitations with the analytial solution. The analytial results and the inherent limitations observed are presented and disussed in this hapter. Literature Review In 1820 Navier presented a paper to the Frenh Aademy of Siene on the solution of a simply supported retangular plate in the form of a double trigonometri series [2.4]. In his 1899 paper to the Frenh Aademy of Siene, M. Maurie Lévy presented a solution to the biharmoni equation that desribes the equilibrium of a retangular plate [2.6]. This solution is an alternate solution to that proposed by Navier. It suggests a form of a single trigonometri series and was readily aepted by the aademi ommunity.
34 Kazkowski utilizes Lévy s proposed solution in a 1961 paper to analytially determine the transverse vibration of regular polygonal plates by applying the solution to a triangular setion of the plate reated by the plate symmetry lines [2.2]. The results from the analysis performed by Kazkowski was used by Leissa in his 1969 publiation [2.1] and subsequently in several papers and publiations suh as K. Chauney Wu s 1991 paper, Free Vibration of Hexagonal Panels Simply Supported at Disrete Points, whih analyses hexagonal plates for spae raft. Solution Proposed by Kazkowski In his 1961 paper [2.2], Von Zbigniew Kazkowski presents and explores a unique approah to the solution of the transverse vibration of regular polygonal figures (Fig. 2.1). He proposed that applying the well known equation for the transverse vibrational displaement to a triangular setion of the plate makes the solution for shapes with more than four sides easier to obtain. The triangular setions are hosen by using the symmetry lines of the polygon, and when the solution for the triangular setion is alulated the solution for the full plate an be found by mirroring the triangular solution. The development of Kazkowski s solution is given briefly below and is given in its entirety in Appendix A1. To test the validity and auray of this analytial model, it will be ompared to the results of a Finite Element Model (FEM: numerial model) and to the results of Air-Coupled Ultrasoni Resonane Imaging (ACURI: experiment).
35 FIGURE 2.1. Several polygonal figures with Cartesian oordinate systems and new oordinate systems used in the development of the analytial equation. Kazkowski onsiders an isotropi plate with a regular n-sided polygonal shape and thikness h. The plate is on an elasti base with Winkler oeffiient K and has a timeindependent lamping fore, N, applied evenly on the entire periphery and normal to the edges. Thin, ideal, elasti plate theory is valid. The equation for the transverse vibrational displaement, w, for the natural frequenies of the plate takes the well-known form: D 2 2 w(x, y) + N 2 w(x, y) (μω 2 K)w(x, y) = 0 (2.1) Where D is the plate bending stiffness, µ is the plate area mass, and 2 is the Laplaian operator. He then applies this equation to the triangular setion desribed in Fig.2.2.
36 FIGURE 2.2. The regular hexagon and the triangular setion used in the development of the analytial equation. The differential equation (2.1) an be diffiult to solve. Kazkowski utilizes a solution proposed by M. Lévy for bi-harmoni equations. The simple series (2.2), fulfills (2.1) when the funtion Y m (η) satisfies the ODE (2.3) with the substitutions (2.4) and (2.5). w(x, y) = m=1,3,5, Y m (η)sin (mπξ) (ξ = x a, η = y a) (2.2) d 4 Y m dη 4 2γ m d 2 Y m dη 2 + δ m 2 Y m = 0 (2.3) γ m = (mπ) 2 1 Λ m 2, δ m 2 = (mπ) 4 1 2 Λ Ω m 2 m4 (2.4) Λ = Na2 2π 2 D, Ω = μω2 K π 4 D a4 (2.5) The general integral of the differential equation (2.1), whih fulfills the symmetri boundary ondition on the line x = a/2 (due to the sine funtion) has the following form: w(x, y) = m=1,3,5, (A m osh(mπφ m η) + B m sinh(mπφ m η) + C m osh(mπψ m η) + D m sinh(mπψ m η)) sin(mπξ) (2.6) φ m = 1 Φ m 2, ψ m = 1 Ψ m 2 (2.7) Φ = Λ Λ 2 + Ω, Ψ = Λ + Λ 2 + Ω (2.8)
37 Of the four oeffiients in the above equation, two an be alulated by onsidering the boundary onditions on the edge y=0. This edge is where the lamping fore is applied. Aording to Kazkowski the oeffiients for the free and fixed boundary onditions are defined, respetively, as the following: A m = C m = 0 C m = A m, D m = φ m B (2.9) ψ m m Where the oeffiients A m and C m, in the ase of a freely held plate, have most likely been hosen not beause they are the solutions to the boundary onditions but beause the plate is not symmetri about the x-axis and the even funtions of y are disarded. The oeffiients for the fixed plate ome from the solutions for the boundary onditions on the edge y=0 as desribed in Appendix A1. These oeffiient equations an then be ombined with the equations resulting from the boundary onditions on the line y=2βx to expliitly solve for the oeffiients. Kazkowski goes on to formulate the boundary onditions for the line y=2βx by implementing the new oordinate system below, x = s a 2βa 2βa n, y = s + n a, ξ = σ 2βζ, η = 2βσ + ζ, σ = s, ζ = n (2.10) This rotates the axes so they are normal and tangent to the line y=2βx just as the original oordinate system was normal and tangent to the line y=0. Doing this makes the boundary equations easier to handle. He uses this and some mathematial manipulation to obtain equations for the boundary onditions, but does not expliitly solve for the four oeffiients of the differential equation.
38 Applying the proposed solution to the four boundary onditions yields a system of homogeneous equations that an be solved expliitly. The two boundary onditions on the line y=2βx yield equations that are hyperboli funtions of the line length. This means that the equations are nonlinear in the tangent oordinate (s) of the line but linear in the oeffiients. Beause of this, we were able to solve the system of equations using Singular Value Deomposition at disrete values of s, the results of whih are shown in Fig 2.3 and 2.4. FIGURE 2.3. Plot of the Kazkowski oeffiients for the vibration of a freely held hexagonal plate as a funtion of line length. Due to the hyperboli nature of two of the four oeffiient equations, all four oeffiients vary drastially with inreasing side length as is evidened by the plots. If the determinant of the system is alulated for eah of the values of s, it an be found that the system only has exat solutions for the value s=0. With this onsideration the transverse vibration an be alulated over the fae of the plate setion.
39 FIGURE 2.4. Plot of the Kazkowski oeffiients for the vibration of a freely held hexagonal plate as a funtion of line length. Kazkowski obtains the fundamental frequeny of a freely bound plate with the following equation: ω = Xπ2 D NR2 1 + R 2 μ Xπ 2 D KR2 X 2 π 4 D (2.11) Where R omes from (Fig 2.1) and X = 0.529 for a hexagon. This gives the fundamental frequeny of a freely bound hexagonal Silion Carbide (SiC) plate with the following dimensions as: R = 0.0508m, D = 0.25373MPa, μ = 58.2625 kg m 2 (2.12) Examination of Solution f fundamental = 21.248 khz In order to examine the validity and auray of the solution proposed by Kazkowski, three ases are ompared. The first ase takes the equations from Kazkowski s paper and plots
40 the results using Matlab. The seond ase uses the results from a FEM modal analysis performed using ANSYS, whih extrats the natural modes and frequenies. The third ase uses results from ACURI testing of the hexagonal SiC tile modeled, as desribed in Chapter 1. Kazkowski s Solution In his paper Kazkowski fails to explain how to determine any resonane frequenies above the fundamental. Beause of this the only test frequenies available are those from the FEM results. These were put into the analytial equation and the results were plotted. The first image in the following two plots is the fundamental frequeny provided in Kazkowski s paper. FIGURE 2.5. Results from Kazkowski s solution with fixed boundaries.
41 FIGURE 2.6. Results from Kazkowski s solution with free boundaries. Fig. 2.5 and 2.6 show the results from Kazkowski s equation for several frequenies. In these images blue is low amplitude while red is high amplitude. Sine the plate is vibrating the high and low amplitude regions will alternate in sign, and hene olor, in a ylial manor. Beause of this the olors are interhangeable. It is easy to see from these two figures that Kazkowski s solution does not vary with frequenies above those of the fundamental frequeny. This suggests that the solution does not fully artiulate the transverse vibration of the hexagonal plate and that it is possible that this solution is only valid for the predition of the fundamental frequeny.
42 FEM Solution Sine Kazkowski s solution only onsiders a ase where both interior edges are subjet to a symmetri boundary ondition the FE model used similar boundary onditions. The results from the modal analysis are presented in Fig. 2.7 and Fig. 2.8. In these images blue is negative displaement while red is positive displaement. Again sine the plate is vibrating the high and low amplitude regions will alternate in sign, and hene olor, in a ylial manor. Beause of this the olors are interhangeable. FIGURE 2.7. FEM modal results for a 1/12 setion of the hexagonal plate with fixed external boundary. The analytial solution does not math well, or at all, with the numerial FEM solution. Only the first resonane pattern and frequeny from the FEM results find a math in the analytial solution and that is the fundamental mode. Though the fundamental resonane pattern mathes well between the analytial solution and the FEM solution, the fundamental resonane frequeny differs by ~22% between the two for the free boundary ondition.
43 FIGURE 2.8. FEM modal results for a 1/12 setion of the hexagonal plate with free external boundary. ACURI Solution The ACURI testing of the shape and material modeled with Kazkowski s equation and FEM ould not ompletely repliate the onditions in the two models. There is no way to ensure the periphery of the plate is ompletely free or fixed. Also there is no way to ensure that all of the lines of symmetry of the plate are symmetri boundaries. Furthermore, the speifi frequenies predited by FEM and used in the Kazkowski model ould not be impinged on the plate. Instead several transduers with bandwidths were employed, none of whih were able to apture the fundamental frequeny of the plate.
44 FIGURE 2.9. ACURI san results of SiC tile at 100 khz, 120 khz, 225 khz. The images are produed by taking the amplitude maximum of the reeived waveform either over the entire time gate or a smaller sub-gate. Fig. 2.9 shows the results from several ACURI sans at several frequenies. In these images blue is low amplitude while red is high amplitude. Again sine the plate is vibrating the high and low amplitude regions will alternate in sign, and hene olor, in a ylial manor. Beause of this the olors are interhangeable. The analytial solution does not math with any of the observed ACURI results of the plate, though several of the FEM and RCURI results math. Examination of Solution Formulation The above omparison of the analytial solution, numerial solution, and experimental results demonstrates that the solution developed by Kazkowski does not aurately desribe the vibrational behavior of a regular hexagonal plate. An examination of the formulation of Kazkowski s solution shows some differenes from the plate equation derivation detailed in literature as well as some disrepanies. Compare equation (A1.1) and equation (A2.20). D 4 w(x, y) + N 2 w(x, y) (μω 2 K)w(x, y) = 0 (A1.1)
45 D 4 w N 2 w 2N 2 w x x 2 xy N 2 w x y y y 2 (µω2 K)w = 0 (A2.20) The mid plane fore term in the equation used by Kazkowski is signifiantly different from the equation presented in the literature. Kazkowski never disusses the fore N exept to say that it is the lamping fore. Sine N is the lamping fore it is applied normal to the periphery of the full plate. The top of the plate was free and the bottom was subjet to the Winkler Stiffness, K. When a setion of the plate is onsidered, as in Fig 2.2b, the fore N should be applied solely in the positive y diretion on the line y=0. All boundary onditions on the lines x=a/2 and y=2βx will take into aount the resultant fores from the lamping fore applied to the other sides of the full plate. It an be argued that Eq. A1.1 is just a speial ase of Eq. A2.20 where the equation is applied on a thin film (N xy =0) and the stress is homogeneous (N x =N y ). If this were the ase then N ould not be applied to the Laplaian of w as Kazkowski does in Eq. A1.1 beause the middle term 2 w x y would still be present in the equation. The first boundary ondition used by Kazkowski is from the symmetry line x=a/2. He states that sine the ondition on the line is symmetri, the solution an only ontain solutions that are even funtions of x with respet to the line and thus retains the sin(mπξ) term. The free boundary ondition on the line y=0 seems to be proessed in a similar manor. Kazkowski assumes that sine the plate boundary is free, it is not symmetri about the line and the even funtions of y an be disarded. This idea will dupliate the ondition stated in the first line of Eq. 2.7. There is nothing about the oeffiients B m and D m. This is not the typial treatment of a free boundary ondition in literature and applying the equations for a free boundary does not yield the same oeffiient equations. (See Appendix A2).
46 The fixed boundary onditions used by Kazkowski on the line y=0 seem to be onsistent with the fixed boundary ondition disussed in literature. The symmetri boundary onditions used by Kazkowski on the line y=2βx are w = 0 ζ ζ=0 2 w = 0 ζ ζ=0 (A1.10a) (A1.11a) The fist ondition, Eq (A1.10a), mathes with the ondition in literature, Eq (A1.14a). The seond ondition, Eq (A1.11a), differs by (2-ν) in the seond term. w = 0 n y=2βx V n = Q n + M nt = 0 w t 3 + (2 ν) 3 w n 3 n t 2 = 0 y=2βx (A1.14a) (A1.15a) These differenes between the boundary onditions used by Kazkowski and those used in literature ould drastially affet the oeffiients of the displaement equation and the results. The inauray of kazkowski s solution may be partly due to the first order nature of his analysis. The proposed solution is trigonometri in x and hyperboli in y. This would give a displaement with periodiity in the x diretion and exponential inrease in the y diretion. This seems inonsistent with the results from experiment and numerial simulation as well as basi physis models of vibration as all three of these show periodiity in both the x and y diretion. Though it is true that ertain ombinations of hyperboli funtions reprodue periodiity, from these onsiderations it seems that any proposed solution should be trigonometri in nature.
47 expressed as The defletion funtion developed by Kazkowski for a freely held plate an be w(x, y) = m=1,3,5, (2.13) B m sinh mπ a 2 + ρω2 y + D D m sinh mπ a 2 ρω2 y sin mπx D a The funtion is only square root frequeny dependent in the y diretion. This ombined with the hyperboli nature in the y diretion gives a defletion funtion that does not vary drastially with frequeny as any plate vibration equation that desribes the natural resonanes should. Conlusion Kazkowski proposed a novel approah to the derivation of the analytial expression of the transverse vibration of polygonal plates by using symmetri setions. The solution proposed works perfetly if the formulation of the differential equation of the transverse vibration and the boundary onditions is valid, but until now no one has ompared the results from this solution to experimental results. Closer inspetion of the proposed solution results for a hexagon has shown that the results do not vary with frequeny or approximate the different resonane modes of the hexagonal plate very aurately. The results from the analytial solution do not math well with the numerial FEM solution or the experimental ACURI solution. The reason for this may be due to several disrepanies found during the examination of Kazkowski s solution formulation. The only part of the proposed solution that orresponds with the numerial model and the experimental results is the predition of the fundamental frequeny, whih varies 22% from the numerial model and is only a first order approximation.
48 Even though Kazkowski s solution has been widely aepted by the aademi ommunity the formulation and solution are inaurate. More work needs to be done in the development of this analytial equation if this type of solution is to be used to aurately model the transverse vibration of regular polygonal shapes. Referenes 2.1. Leissa, A. W., Vibration of Plates, NASA SP-160, 1969 2.2. Kazkowski, Z., Stabilität und Eigenshwingungen einer Platte von der Form eines regelmäϐigen Polygons, 1961, Ingenieur-Arhiv, Vol. 15, pp.103-109 2.3. Mansfield, E. H., The Bending and Strething of Plates, 1964, Pergamon Press Ltd., Oxford England 2.4. Timoshenko, S. P., Woinowsky-Krieger, S., Theory of Plates and Shells, 1959, MGraw-Hill Book Company, In., New York 2.5. Timoshenko, S. P., Goodier, J. N., Theory of Elastiity, 1970, MGraw-Hill Book Company, In., New York 2.6. Lévy, M. M., Sur L Équilibre Élastique d une Plaque Retangulaire, 1899, Comptes Rendus des Séanes de L Aadémie des Sienes, Vol. 129, pp. 535-539 2.7. Laurie, S. A., Vasiliev, V. V., The Biharmoni Problem in the Theory of Elastiity, 1995, Gordon and Breah Publishers, ISBN 2-88449-054-X 2.8. Yu, Y. Y., Vibrations of Elasti Plates, 1996, Springer-Verlag New York, In., ISBN 0-887-94514-8 2.9. Gorman, D. J., Vibration Analysis of Plates by the Superposition Method, 1999, World Sientifi Publishing Co. Pte. Ltd. 2.10. Soedel, W., Vibrations of Shells and Plates, 2004, Marel Dekker, New York 2.11. Qatu, M. S., Vibrations of Laminated Shells and Plates, 2004, Elsevier Ltd.
49 CHAPTER 4. ANSYS MODELING Introdution Finite Element Modeling (FEM) is a very useful and powerful tool for the numerial solution of a great many engineering problems. The problems range from deformation and stress analysis to wave propagation and omposite analysis to fluid and heat flow. For this researh, the FEM program ANSYS was used to numerially model the natural and fored vibrations of the multi-layer, multi-material omposite disussed in Chapter 2 as well as the polygonal plate disussed in Chapter 3. It is ommon pratie, when modeling with FEM, to utilize the symmetry of the problem in order to redue the omplexity of the model and redue the omputation time. During the ourse of this researh, the question arose of whether this ommon pratie gives aurate results, when ompared to a full model and to experiments. This hapter desribes the models used in Chapter 2 and Chapter 3 and presents the results from the analyses performed on the models. This hapter will also explore the auray of symmetri models. Air-Coupled Ultrasoni Resonane Imaging Models In this researh several FE models have been made to numerially simulate the vibrational response of the different materials of the multi-layered, multi-material omposite disussed in Chapter 2, as well as several different lay-ups. The speifis of eah material and
50 the results of eah model are desribed briefly below. The models are 4 inhes from side to side. The model for the SiC tile is made with the ANSYS element solid187. This element is a higher order 3-D, 10-node element with quadrati displaement behavior. The material has an elasti modulus of 428.28 GPa, a Poison s ratio of 0.166, and a density of 3058.4 Kg/m^3. The boundary onditions for this model are either fixed or free and are applied solely to the periphery of the hexagonal plate. Sine the researh in Chapter 2 deals with vibrations a modal analysis was performed to look at the natural resonane modes and frequenies of the hexagonal SiC tiles. Figures 3.1 and 3.2 show some of the resonane modes and frequenies extrated from the model with a modal analysis. FIGURE 3.1. Here are the results from a modal analysis on the SiC model with fixed boundary onditions. There are several resonane images and orresponding resonane frequenies displayed.
51 FIGURE 3.2. Here are the results from a modal analysis on the SiC model with free boundary onditions. There are several resonane images and orresponding resonane frequenies displayed. There is a multitude of repeated resonane modes that appear in both the fixed and free models. A repeated resonane mode ours when there is more than one mode with the same frequeny to at least one deimal plae. The resulting resonane patterns will be different, in this ase the resonane pattern is rotated slightly, but the frequenies are the same. The resonant images from the fixed model orrespond better with ACURI results from the bare SiC tile. This is not surprising beause it is diffiult to simulate a truly free boundary, also the boundary ondition will be fixed, to some degree, inside any type of lay-up. Beause of these two onsiderations, only fixed boundary onditions will be used in subsequent models. The model for the Graphite-Carbon-Graphite (GCG) onsists of three volumes, whih are glued together so that the oinident areas beome a single area. The enter volume is exatly the same as the SiC model, and the other volumes have the same shape and size. The two outside volumes are models of the arbon-fiber/epoxy lay-up. These volumes are made
52 with the ANSYS element solid185. This element is a 3-D, 8-node element with layered struture apabilities. The material has a density of 1580.0 Kg/m^3 and the following physial properties: E x = 181 GPa E y = 10.3 GPa E z = 10.3 GPa ν xy = 0.15 ν yz = 0.10 ν xz = 0.15 G xy = 5.50 GPa G yz = 6.41 GPa G xz = 5.50 GPa There are 24 layers and the thikness of eah layer is approximately 132 µm. The fiber diretion follows the lay-up [ [0,+45,90,-45] 3 ] sym. Fixed boundary onditions are applied solely to the periphery of the hexagonal plate. Figure 3.3 shows some of the resonane modes and frequenies extrated from the model with a modal analysis. FIGURE 3.3. Here are the results from a modal analysis on the GCG model with fixed boundary onditions. There are several resonane images and orresponding resonane frequenies displayed.
53 The same resonane modes appear in this modal extration as appeared in the SiC modal extration. This indiates that the SiC tile is the strongest vibrator of the three layers and that it either overpowers the other two volumes or damps out their vibrations. There are some important differenes, however. Many of the resonane patterns appear with a slightly smaller amplitude and the resonant frequenies have shifted downward. All of this is to be expeted from the damping harateristis of adding layers to the SiC tile. Also note that the repeated resonanes from the SiC model are now separated. The model for the full lay-up onsists of five volumes whih are glued together so that the oinident areas beome a single area. Two volumes are added to the GCG model; a rubber layer right next to the GCG and a S2 glass-fiber/epoxy layer next to the rubber layer. The rubber volume is made with the ANSYS element solid187. This is the same as the element used for the SiC layer and allows for large defletion. The material has the following generi rubber properties: an elasti modulus of 2.412 MPa, a Poison s ratio of 0.33, and a density of 1079 Kg/m^3. The S2 glass-fiber/epoxy volume is made with the ANSYS element solid185, whih is the same element used for the arbon-fiber/epoxy layers. The material has a density of 2000.0 Kg/m^3 and the following physial properties: E x = 55 GPa E y = 16 GPa E z = 16 GPa ν xy = 0.28 ν yz = 0.10 ν xz = 0.28 G xy = 7.6 GPa G yz = 4.52 GPa G xz = 7.6 GPa There are 96 layers and the thikness of eah layer is approximately 132 µm. The fiber diretion follows the lay-up [ [ [ [0,+45,90,-45] 3 ] sym ] sym ] sym. Fixed boundary onditions are applied solely to the periphery of the hexagonal plate. Figures 3.4 and 3.5 show some of the resonane modes and frequenies extrated from the model with a modal analysis.
54 FIGURE 3.4. Here are the results from a modal analysis on the Full Lay-up Model with fixed boundary onditions. There are several resonane images and orresponding resonane frequenies displayed. Here only the S2 glass layer is vibrating. In both of these figures, the left side of eah image is the S2 glass-fiber/epoxy side of the full model and the right side of eah image is the GCG side of the full model. The images of the modal results in Fig. 3.4 indiate that only the S2 glass-fiber/epoxy layer vibrates while only the GCG setion vibrates in the images of Fig. 3.5. This indiates that the rubber layer ats like an impedane layer and effetively separates the vibrations from the S2 glass-fiber/epoxy layer and the GCG setion. The same resonane modes appear in this modal extration as appeared in the SiC modal extration, but the resonane images are slightly damped and the frequenies are shifted slightly. Several of the resonane images in Fig. 3.5 math well with the ACURI results (Fig. 3.6) whih indiates that both the rubber layer and the S2 glassfiber/epoxy layer only at as damping layers to the SiC Tile vibration and do not add resonane patterns to the resulting resonane images during ACURI sanning.
55 FIGURE 3.5. Here are the results from a modal analysis on the Full Lay-up Model with fixed boundary onditions. There are several resonane images and orresponding resonane frequenies displayed. Here only the GCG layer is vibrating. In order to test the auray of these FE models mathes between the ACURI san results and the modal results of the SiC model were made. Fig. 3.6 shows several mathes made between the ACURI and FEM modal results. Several of the ACURI images an be diretly mathed with a single FEM mode at a frequeny lose to the enter frequeny and within the bandwidth of the transduer. Other of the ACURI images require a superposition of FEM modes in order to make a math. These results indiate that the FE model aurately desribes the behavior of the SiC tile.
56 FIGURE 3.6. Mathing resonane images from ACURI and FEM modal results. Kazkowski Plate Setion Model The model for the plate setion is made with the ANSYS element solid187. This element is a higher order 3-D, 10-node element with quadrati displaement behavior. The model is roughly 2 in by 1.5 in by 0.75 in. The material has an elasti modulus of 428.28 GPa, a Poison s ratio of 0.166, and a density of 3058.4 Kg/m^3. The boundary onditions for this model are either fixed or free for the exterior edge of the plate setion and symmetri or antisymmetri on the interior edges. Sine the researh in Chapter 2 deals with natural vibrations a modal analysis was performed to look at the natural resonane modes and frequenies of the polygonal plate setion. Figures 3.7 and 3.8 show some of the resonane modes and frequenies extrated from the model with a modal analysis.
57 FIGURE 3.7. FEM modal results for a 1/12 setion of the hexagonal plate with fixed external boundary. FIGURE 3.8. FEM modal results for a 1/12 setion of the hexagonal plate with free external boundary.
58 Symmetri Modeling The model for the material onsidered in this study is omplex with a large number of nodes. Models of anything larger than a full lay-up the same shape and size as one of the hexagonal tiles with an appropriately small mesh size are simply too large to analyze easily. Modal and harmoni analyses are very simple analyses that usually take no time at all to run. With the full lay-up this simple analysis an take hours. A more omplex analysis, suh as a waveform impingement, will take muh longer. An analysis of this type was performed using a omputer luster on a symmetri quarter of the SiC tile model. Even with this small model and a relatively oarse mesh, the analysis took four days to omplete. It is beause of these onsiderations; namely the omplexity of modeling multiple layers and several tiles at a time, the omplex analyses, and the long omputation time, that the validity of symmetri modeling was onsidered. In order to test the auray of modeling with symmetri setions two different polygonal plates, a square and a hexagon, were divided along the symmetry lines of the polygons as shown in Fig 3.9 and Fig 3.10. All possible symmetri setions are onsidered in this analysis. The setions of the polygons are subjeted to a modal analysis and a harmoni analysis and the results ompared to the results of the full models. A modal analysis shows the natural resonane modes and frequenies of the models. Comparing the natural resonanes of the plate setions to those of the full plate will determine whether the plate setions have the same behavior as the full plate. The harmoni analysis will show the response of the model to a sine input at speifi frequenies. Comparing the responses of the plate setions to the response of the full plate will determine whether the plate setions respond to a given impingement in
59 the same manner as a full plate. These two analyses an indiate whether a symmetri model an be used to rereate a full model without loss of auray. The boundary onditions on the plate periphery and on the orresponding areas of the plate setions are either fixed (zero x,y, and z displaement) or free (no onstraints). For the areas of the plate setions that would normally be in the interior of the full plate (interior areas) the possible boundary onditions are symmetri and antisymmetri. For a mathematial explanation of these boundary onditions see Appendix A.2. Combinations of the symmetri and antisymmetri boundary onditions on the interior areas of the plate setions should repliate all possible interations that our in the interior of the full plate. FIGURE 3.9. Square plate and possible symmetri setions.
60 FIGURE 3.10. Hexagonal plate and possible symmetri setions. All of the full plate ANSYS models reated for this analysis are approximately 4 inhes in length from side to side and 0.75 inhes thik, and the symmetri setion models are sized relative to the full model. The element type and material properties are the same as those listed for the SiC tile model desribed under the Air-Coupled Ultrasoni Resonane Imaging Models setion. Modal Analysis A modal analysis of the plates and plate setions extrats the natural modes of the models for the given material and boundary onditions. The extrated modes were visually ompared and like modes were grouped together.
61 FIGURE 3.11. A omparison of the natural resonane modes of symmetry models to the resonane mode of the full plate model. A) Full Plate Model @ 30670 Hz. B) Half point-point Setion Model @ 30700 Hz. C) Half sideside Setion Model @ 30690 Hz. D) One Third Setion Model @ 30669 Hz. E) One Fourth Setion Model @ 30669 Hz. F) One Sixth Setion Model @ 30668 Hz. G) One Twelfth Setion Model @ 30667 Hz. FIGURE 3.12. A omparison of the natural resonane modes of symmetry models to the resonane modes of the full plate model. A) Full Plate Model @ 113926 Hz. B) Half point-point Setion Model @ 113943 Hz. C) Half side-side Setion Model @113935 Hz. D) One Fourth Setion Model @ 113926 Hz.
62 FIGURE 3.13. A omparison of the natural resonane modes of symmetry models to the resonane modes of the full plate model. A) Full Plate Model @ 113209 Hz. B) Half Triangular Setion Model @ 113911 Hz. C) Half Retangular Setion Model @ 113116 Hz. D) One Fourth Retangular Setion Model @ 112939 Hz. FIGURE 3.14. A omparison of the natural resonane modes of symmetry models to the resonane modes of the full plate model. A) Full Plate Model @ 139994 Hz. B) Half Triangular Setion Model @ 140033 Hz. C) Half Retangular Setion Model @ 139991 Hz. D) One Fourth Triangular Setion Model @ 139990 Hz. E) One Fourth Retangular Setion Model @ 139973 Hz.
63 The mathing of resonane modes was performed by looking for distintive features, suh as high and low amplitude regions, in the resonane images of the full plate model and eah symmetry model. The size, shape, and loations of these distintive features were then ompared to math the resonane images. It is important to keep in mind that in this FE analysis red is positive displaement and blue is negative displaement, but in a sinusoidal vibration positive and negative displaement hange positions ylially. Hene the blue and red olors an be interhanged; they essentially represent the maxima in a resonane image. Figures 3.11 through 3.14 demonstrate how the extrated modes are ompared and grouped. The omparisons made in Figures 3.11 and 3.12 were relatively easy to make when small disrepanies suh as olor variation and rotation of the resonane pattern were taken into aount. The omparisons in Figures 3.13 and 3.14, on the other hand, were somewhat more diffiult to make as not only were there olor variations and rotations to onsider but some of the maximas have hanged position slightly due to the onstraints of the geometry of a given plate setion. The frequeny range of this study was from 0-300 khz. This was hosen to apture both the fundamental frequenies and frequenies where ACURI an operate. For the sake of brevity, the tables below list only the natural resonane modes between 0 and 100 khz. The resonane modes of the full plate model are listed by frequeny and all the orresponding resonane modes from the plate setions are listed for eah full plate resonane long with the speifi model and the interior boundary onditions.
64 Hexagon Resonane Modes, Fixed BC Full Model Symmetri Models Frequeny (Hz) Frequeny (Hz) Model Interior BC's 30669.7505 30700.3255 half p-p sym 30690.1668 half s-s sym 30669.3626 third sym-sym 30669.3918 fourth sym-sym 30668.3595 sixth sym-sym 30666.7199 twelfth sym-sym 55021.7374 55103.0331 half p-p antisym 55064.9021 half s-s sym 55018.6634 fourth sym-antisym 55021.8769 55097.6985 half p-p sym 55080.0526 half s-s antisym 55024.2001 fourth antisym-sym 75712.4516 72720.445 half p-p antisym 72716.6723 half s-s sym 72712.0422 fourth sym-antisym 72712.4803 72719.642 half p-p sym 72718.2008 half s-s antisym 72712.7531 fourth antisym-sym 79798.3981 79944.816 half p-p antisym 79904.0709 half s-s antisym 79799.4161 fourth antisym-antisym 79798.4261 79944.6592 half p-p sym 79879.0981 half s-s sym 79798.0312 fourth sym-sym 88235.2725 88414.5946 half p-p sym 88356.0798 half s-s sym 88233.9126 third sym-sym 88235.5451 fourth sym-sym
65 89915.4626 88231.1792 sixth sym-sym 88225.3267 twelfth sym-sym 89917.7606 half p-p antisym 89916.8434 half s-s antisym 89915.4577 third antisym-antisym 89915.4912 fourth antisym-antisym 89915.4382 sixth antisym-antisym 89915.4305 twelfth antisym-antisym TABLE 3.1. Table of resonane frequenies between 0 and 100 khz for a hexagonal plate with fixed boundary onditions and mathed frequenies of symmetri models. Hexagon Resonane Modes, Free BC Full Model Symmetri Models Frequeny (Hz) Frequeny (Hz) Model Interior BC's 18616.373 18619.9092 half p-p sym 18618.8201 half s-s sym 18616.4489 fourth sym-sym 18616.3715 18620.4733 half p-p antisym 18618.1244 half s-s antisym 18616.3804 fourth antisym-antisym 27346.8839 27350.1151 half p-p sym 27349.093 half s-s sym 27346.8937 third sym-sym 27346.9318 fourth sym-sym 27346.8909 sixth sym-sym 27346.8837 twelfth sym-sym 35064.4102 35083.5299 half p-p sym 35075.125 half s-s antisym 35064.3862 third sym-sym 354.599806 fourth antisym-sym 35064.3812 sixth sym-sym 35064.0982 twelfth sym-antisym 42353.6231
66 54116.5348 54116.5352 54132.6444 54132.6633 58986.4291 58986.4298 59671.8844 59671.8898 69656.3551 42364.6056 half p-p antisym 42359.4873 half s-s sym 42353.6163 third antisym-antisym 42653.7621 fourth sym-antisym 42353.6449 sixth antisym-antisym 42353.6087 twelfth antisym-sym 54116.8813 half p-p antisym 54116.7333 half s-s antisym 54116.5405 fourth antisym-antisym 54116.8924 half p-p sym 54116.7579 half s-s sym 54116.5388 fourth sym-sym 54154.5037 half p-p antisym 54148.2321 half s-s sym 54132.867 fourth sym-antisym 54155.0178 half p-p sym 54147.8749 half s-s antisym 54132.8515 fourth antisym-sym 58987.3549 half p-p antisym 58987.0318 half s-s sym 58986.4393 fourth sym-antisym 58987.3534 half p-p sym 58986.9903 half s-s antisym 58986.4445 fourth antisym-sym 59708.5694 half p-p sym 596954.6375 half s-s sym 59672.1037 fourth antisym-antisym 59712.5782 half p-p antisym 59691.9689 half s-s antisym 59672.6776 fourth sym-sym
67 80048.8065 84360.8609 84360.9537 84636.7707 84987.2839 84987.3289 85791.0826 69656.495 half p-p sym 69656.4419 half s-s sym 69656.3552 third sym-sym 69656.3568 fourth sym-sym 69656.3541 sixth sym-sym 69656.3546 twelfth sym-sym 80052.1805 half p-p sym 80050.9068 half s-s antisym 80048.8238 third sym-sym 80048.8591 fourth antisym-sym 80048.7882 sixth sym-sym 80048.7413 twelfth sym-antisym 84438.4958 half p-p sym 84402.0214 half s-s antisym 84362.2506 fourth antisym-sym 84438.515 half p-p antisym 84402.8692 half s-s sym 84361.8734 fourth sym-antisym 84639.668 half p-p antisym 84638.5617 half s-s sym 84636.7706 third antisym-antisym 84636.7973 fourth sym-antisym 84636.752 sixth antisym-antisym 84636.7121 twelfth antisym-sym 85050.6614 half p-p antisym 85031.8971 half s-s antisym 84988.1029 fourth antisym-antisym 85052.1805 half p-p sym 85031.059 half s-s sym 84988.0893 fourth sym-sym 85863.1278 half p-p sym 85842.007 half s-s sym
68 87722.195 87722.1897 85790.8721 third sym-sym 85791.7468 fourth sym-sym 85790.9244 sixth sym-sym 85790.4653 twelfth sym-sym 877251353 half p-p sym 87724.0727 half s-s sym 87722.2293 fourth sym-sym 87725.298 half p-p antisym 87723.9262 half s-s antisym 87722.2211 fourth antisym-antisym TABLE 3.2. Table of resonane frequenies between 0 and 100 khz for a hexagonal plate with free boundary onditions and mathed frequenies of symmetri models. Several observations an be made about the hexagonal models from the data presented in the tables above. 1. Every single resonane mode in the full plate model an be repliated with the symmetry models. 2. The frequenies of the resonane modes from the symmetry models have a variane from the resonane frequenies of the full plate that is less than 0.3% of the resonane frequeny. This means the symmetry models aurately repliate both the resonane pattern and the resonane frequeny of the full plate. 3. For the symmetry onditions sym-antisym and antisym-sym, with both the fixed and free boundary onditions, the resonane images of the one sixth and the one third models do not appear among the resonane images of the full model. 4. Of all of the symmetri models, only the half models have resonane modes that math with every single resonane mode of the full model. This means that attempting to model a struture with symmetry models any smaller than one half the size of the full
69 model will not give a full solution. This is due to the fat that a smaller model annot model the ombinations of interior boundary onditions that our in a full model. This is shown in Fig 3.15 below. Observe the symmetry onditions on the symmetry lines of the full plate speified when using a small symmetry model (in this ase a one twelfth model). Changing the ondition on just one of the lines of symmetry yields a possibility that the small symmetry model annot dupliate but that an be modeled with a larger symmetry model ( 1/2 or 1/4) or the full model itself. FIGURE 3.15. A graphial explanation of possible symmetry ondition of a Full Plate Model where the lines represent symmetry lines. The blue lines are symmetri and the red lines are antisymmetri. A) 1/12 setion with symmetri-antisymmetri boundaries. B) The 1/12 setion mirrored symmetrially into a full plate. C) Possible interior boundary onditions of a Full Plate. 5. Both the half point-point model and the half side-side model have resonane modes that math the resonane modes of the full model but both of the interior boundary onditions (symmetri and antisymmetri) must be used. This means that the idea of using a smaller symmetri model to redue the omputation time is faulty sine two of the half models will have to be run to repliate the results from an analysis of the full
70 model. This is true at least for this type of analysis. If an analysis was required in whih only a symmetri interior boundary need be onsidered, then a half model would redue the omputation time. It seems that the best model, both for auray and omputation time, for the hexagonal plate is the Full Plate model. Square Resonane Modes, Fixed BC Full Model Symmetri Models Frequeny (Hz) Frequeny (Hz) Model Interior BC's 28840.8035 28908.5353 half tri Sym 28834.616 half re Sym 28834.8514 fourth tri sym-sym 28810.4538 fourth re sym-sym 28819.842 Eighth sym-sym 51511.8833 51511.1326 half re Antisym 51441.4541 fourth re antisym-sym 51512.6166 51487.7026 half re Sym 51440.5873 fourth re sym-antisym 69096.9259 69097.1229 half re Antisym 69091.2126 fourth re antisym-sym 69097.1031 69094.8371 half re Sym 69091.0701 fourth re sym-antisym 69847.0634 70119.8839 half tri Sym 69836.3458 half re Antisym 69823.5377 fourth tri sym-sym 69746.0942 fourth re antisym-antisym 69767.0002 Eighth antisym-sym 80197.0795
71 81222.6659 86738.4232 95544.3988 95550.2627 80592.1918 half tri Antisym 80162.8529 half re Sym 80166.4387 fourth tri antisym-antisym 80041.9618 fourth re sym-sym 80069.4801 Eighth sym-antisym 81597.0163 half tri Sym 81202.6928 half re Sym 81079.1823 fourth re sym-sym 81186.2997 fourth tri sym-sym 81115.2768 Eighth sym-sym 86741.9071 half tri Antisym 86738.0156 half re Antisym 86737.8797 fourth tri antisym-antisym 86736.3265 fourth re antisym-antisym 86737.782 Eighth antisym-antisym 95869.8707 half tri Sym 95536.8782 half re Sym 95519.516 fourth tri antisym-sym 95377.898 fourth re sym-antisym 96212.4797 half tri Antisym 95528.0827 half re Antisym 95509.9986 fourth tri sym-antisym 95376.5839 fourth re antisym-sym TABLE 3.3. Table of resonane frequenies between 0 and 100 khz for a square plate with fixed boundary onditions and mathed frequenies of symmetri models. Square Resonane Models, Free BC Full Model Symmetri Models Frequeny (Hz) Frequeny (Hz) Model Interior BC's 13125.0478 13137.9604 half tri Sym 13124.1726 half re Antisym 13124.0035 fourth tri sym-sym 13121 fourth re antisym-antisym
72 19253.5843 21475.3778 30569.9849 30570.4431 47394.8394 48480.8517 48480.8765 48782.6398 48783.3643 13120.9022 eighth antisym-sym 19259.2258 half tri Antisym 19253.56 half re Sym 19253 fourth tri antisym-antisym 19252.3477 fourth re sym-sym 19247.0786 eighth sym-antisym 21483.3875 half tri Sym 21475.2961 half re Sym 21474.9901 fourth tri sym-sym 21473.5668 fourth re sym-sym 21473.8388 eighth sym-sym 30621.2278 half tri Antisym 30569.5954 half re Sym 30567.5506 fourth tri sym-antisym 30560.0403 fourth re sym-antisym 30594.932 half tri Sym 30567.3163 half re Antisym 30567.4681 fourth tri antisym-sym 30560.0247 fourth re antisym-sym 47395.2907 half tri Sym 47394.7932 half re Antisym 47394.8115 fourth tri sym-sym 47395 fourth re antisym-antisym 47384.9207 eighth antisym-sym 48480.8078 half re Antisym 48480.5894 fourth re antisym-sym 48480.8254 half re Sym 48480.5894 fourth re sym-antisym 48780.4983 half re Antisym 48769.8338 fourth re antisym-sym
73 50549.2038 53931.7643 54475.3399 58104.2338 58156.7116 62591.291 74681.4437 48780.6333 half re Sym 48769.8214 fourth re sym-antisym 50652.3 half tri Sym 50545.5822 half re Sym 50542.5802 fourth tri sym-sym 50522.7034 fourth re sym-sym 50528.1003 eighth sym-sym 53931.9833 half tri Antisym 53931.7474 half re Sym 53932 fourth tri antisym-antisym 53931.6636 fourth re sym-sym 53902.5021 eighth sym-antisym 54583.0686 half tri Antisym 54470.0764 half re Antisym 54470 fourth tri antisym-antisym 54450 fourth re antisym-antisym 54455 eighth antisym-antisym 58208.2665 half tri Sym 58100.6492 half re Antisym 58098.4384 fourth tri sym-sym 58080 fourth re antisym-antisym 58081.953 eighth antisym-sym 58157.584 half tri Anisym 58156.6564 half re Sym 58157 fourth tri antisym-antisym 58156.3307 fourth re sym-sym 58148.6589 eighth sym-antisym 62591.4391 half tri Sym 62591.2842 half re Sym 62591.2878 fourth tri sym-sym 62591.2447 fourth re sym-sym 62591.2525 eighth sym-sym
74 74682.9791 75361.7414 75361.7652 81560.6294 83483.0954 85604.5706 88067.8954 88069.9365 74848.1448 half tri Sym 74674.5888 half re Sym 74671.9474 fourth tri antisym-sym 74627.0347 fourth re sym-antisym 74954.1332 half tri Antisym 74676.0474 half re Antisym 74668.3296 fourth tri sym-antisym 74627.0747 fourth re antisym-sym 75361.3942 half re Antisym 75360.282 fourth re antisym-sym 75361.468 half re Sym 75360.2799 fourth re sym-antisym 81739.1611 half tri Antisym 81554.8963 half re Sym 81549 fourth tri antisym-antisym 81514.0965 fourth re sym-sym 81513.5414 eighth sym-antisym 83667.2195 half tri Sym 83477.3177 half re Sym 83469.4256 fourth tri sym-sym 83433.4492 fourth re sym-sym 83441.5059 eighth sym-sym 85617.1737 half tri Antisym 85603.5556 half re Antisym 85605 fourth tri antisym-antisym 85600 fourth re antisym-antisym 85601 eighth antisym-antisym 88304.0052 half tri Sym 88061.6559 half re Sym 88058.3053 fourth tri antisym-sym 88010.3732 fourth re sym-antisym
75 98300.806 99301.1668 99301.2124 88333.5406 half tri Antisym 88065.6762 half re Antisym 88057.7387 fourth tri sym-antisym 88010.3101 fourth re antisym-sym 98698.5347 half tri Sym 98284.9238 half re Antisym 98275.7085 fourth tri sym-sym 98193 fourth re antisym-antisym 98210.2637 eighth antisym-sym 99300.0083 half re Antisym 99295.0079 fourth re antisym-sym 99299.903 half re Sym 99295.0052 fourth re sym-antisym TABLE 3.4. Table of resonane frequenies between 0 and 100 khz for a square plate with free boundary onditions and mathed frequenies of symmetri models. Several observations an be made about the square models from the data presented in the tables above. 1. Every single resonane mode in the Full Plate model an be repliated with the symmetry models. 2. The frequenies of the resonane modes from the symmetry models have a variane from the resonane frequeny of the full plate that is between ~0% and 1% of the resonane frequeny. Also, Fig 3.13 and Fig 3.14 show several ases where the resonane patterns of the full square plate and the patterns of the symmetry models where very diffiult to math. This means the symmetry models do a deent but flawed job of repliating both the resonane patterns and the resonane frequenies of the full plate.
76 3. A multitude of resonane modes from several of the symmetri models with both fixed and free boundary onditions do not appear among the resonane modes of the full plate model. 4. Of all of the symmetri models only the half retangular model has resonane modes that math every resonane mode of the full plate model. This means that only the half retangular model ould potentially be used to redue the model omplexity without a signifiant loss of fidelity. 5. Only by using both of the interior boundary onditions (symmetry and antisymmetry) on the half retangular model an all of the resonane modes of the full plate model be mathed. This means that the idea of using a smaller symmetri model to redue the omputation time is faulty sine two of the half models will have to be run to repliate the results from an analysis of the full model. This is true at least for this type of analysis. If an analysis was required in whih only a symmetri interior boundary need be onsidered, then a half model would redue the omputation time. It seems that the best model, both for auray and omputation time, for the hexagonal plate is the Full Plate model. Harmoni Analysis A harmoni analysis applies a pressure or fore that varies sinusoidally with time at a speifi foring frequeny and extrats the results of a fored vibration. For this partiular analysis the Gaussian pressure distribution desribed in Fig 3.15 is applied to the models. The pressure distribution is entered on the origin of all of the models.
77 FIGURE 3.16. 2D Gaussian pressure distribution applied to the harmoni models. This analysis was performed with 37 distint frequenies ranging between 80 and 300 khz and the results for the first frequeny is shown in Fig. 3.17-3.20. FIGURE 3.17. Harmoni results for the full plate and all of the symmetri setions with fixed boundary onditions and a frequeny of 84400 Hz. Sine this analysis is a fored response it is not enough that the results from the symmetri setions math deently or losely with the results from the full plate, they must math very
78 losely or exatly in order for symmetri modeling to be onsidered valid. If the symmetri setions annot repliate the fored response of a full model then they annot be used to aurately model a struture and its responses to stimuli. FIGURE 3.18. Harmoni results for the full plate and all of the symmetri setions with free boundary onditions and a frequeny of 84400 Hz. The results from the harmoni analysis performed on the hexagonal plate and all of the orresponding setions at a frequeny of 84400 Hz are presented in Fig. 3.17 and 3.16. These are merely sample results presented for visual omparison. The fixed boundary analysis demonstrated that very few of the results from the symmetri setions math with the results from the full model. Only two of the symmetry
79 models have any results that math with the results of the full model. The fourth model with symmetri-antisymmetri interior boundary onditions is able to math two of the results from the full model. The half side-side model with a symmetri interior boundary is able to math many, but not all of the results from the full model. The free boundary analysis also demonstrated that very few of the results from the symmetri setions math with the results from the full model. Only nine of the twenty symmetry models have any results that math with the results of the full model, but no model has more than eight mathes. It is lear from this analysis that the only model of a hexagonal plate that an fully repliate the fored response is a full plate model. FIGURE 3.19. Harmoni results for the full plate and all of the symmetri setions with fixed boundary onditions and a frequeny of 84400 Hz.
80 FIGURE 3.20. Harmoni results for the full plate and all of the symmetri setions with free boundary onditions and a frequeny of 84400 Hz. The results from the harmoni analysis performed on the square plate and all of the orresponding setions at a frequeny of 84400 Hz are presented in Fig. 3.19 and 3.20. These are merely sample results presented for visual omparison. The fixed boundary analysis demonstrated that very few of the results from the symmetri setions math with the results from the full model. Only three of the symmetry models have any results that math with the results of the full model. The eighth model with symmetri-symmetri interior boundary onditions is able to math seven of the results from the full model. The fourth triangular model with symmetri-symmetri interior boundary onditions is able to math two of the results from the full model. The fourth retangular model
81 with symmetri-symmetri interior boundary onditions is able to math three of the results from the full model. The free boundary analysis also demonstrated that very few of the results from the symmetri setions math with the results from the full model. The eighth model with symmetri-symmetri interior boundary onditions is able to math eight of the results from the full model. The fourth triangular model with symmetri-symmetri interior boundary onditions is able to math five of the results from the full model. The fourth retangular model with symmetri-symmetri interior boundary onditions is able to math one of the results from the full model. It is lear from this analysis that the only model of a square plate that an fully repliate the fored response is a full plate model. Conlusion Using a struture s natural symmetry has long been thought to be a valid way to redue a model s size, omplexity, and omputation time. In order to test the auray of modeling with symmetri setions two different polygonal plates, a square and a hexagon, were divided along the symmetry lines of the polygons, and were subjeted to a modal and a harmoni analysis. These two analyses an indiate whether a symmetri model an be used to rereate a full model without loss of auray. The results from the modal analysis show that only half plate models an fully repliate all of the natural responses of the full plate, both the resonane frequenies and the resonane patterns. However, in order to repliate all of the resonanes two half models must be run; one with a symmetri interior boundary, and one with an antisymmetri interior boundary. The
82 neessity of running two separate models negates the time saved by modeling with a symmetri setion. The results from the harmoni analysis show that no single model or ombination of models an repliate the fored responses of the full plate. Symmetri modeling is not aurate in many ases and should only be used very arefully. In ertain ases the omplexity and size of the model an be redued but the omputation time annot. Referenes 3.1. Chandrupatla, T. R., Belegundu, A. D., Introdution to Finite Elements in Engineering, 2002, Prentie-Hall In., Upper Saddle River, New Jersey 3.2. Cook, R. D., Malkus, D. S., Plesha, M. E., Witt, R. J., Conepts and Appliations of Finite Element Analysis, 2004, John Wiley & Sons 3.3. Timoshenko, S, Vibration Problems in Engineering, 1937, D. Van Nostrand Company In., New York 3.4. Denhartog, J. P., Mehanial Vibrations, 1985, Dover Books 3.5. Soedel, W., Vibrations of Shells and Plates, 2004, Marel Dekker
83 APPENDIX A1 FULL DEVELOPMENT OF KACZKOWSKI S SOLUTION The equation of the transverse vibration of a triangular segment of a hexagonal plate and the proposed solution are given by Kazkowski as: D 4 w(x, y) + N 2 w(x, y) (μω 2 K)w(x, y) = 0 w(x, y) = m=1,3,5, C m osh mπψ m a A m osh mπφ m a y + D m sinh mπψ m a y + B m sinh mπφ m y + a y sin mπ a x (A1.1) (A1.2) The four oeffiients require four distint boundary onditions in order to reah an expliit solution. The solution is formulated suh that symmetri boundary onditions are true on the line x=a/2 in Fig. 2.2. The equations for the boundary onditions are developed in Appendix A2. FIGURE 2.2. The regular hexagon and the triangular setion used in the development of the analytial equation.
84 A1.1 Free Boundary Conditions When the plate is freely held on the edge y=0 from Fig 2.2 the boundary onditions for that line are as follows: V y = Q y + M xy = 0 x 3 w y 3 + (2 ν) 3 w x 2 y y=0 = 0 (A1.3a) Applying Kazkowski s solution to this ondition gives: mπφ m a 3 A m sinh mπφ m mπψ m a a 3 D m osh mπψ m y + mπφ m a y sin mπ x a a 3 B m osh mπφ m a y + mπψ m a 3 C m sinh mπψ m y + (2 ν) mπ a 2 mπφ m A a m sinh mπφ m y + mπφ m B a a m osh mπφ m y + a mπψ m C a m sinh mπψ m y + mπψ m D a a m osh mπψ m y sin mπ x = 0 a a a (φ m 3 (2 ν)φ m )B m + (ψ m 3 (2 ν)ψ m )D m (A1.3b) The seond ondition omes from the onsideration that the plate setion is not symmetri about the line y=0. This means that the even funtions an be disarded while keeping the odd funtions. From this onsideration the following is true. A m = C m = 0 (A1.4) The two equations, (A1.3b) and (A1.4), along with the two from the symmetri boundary onstitute four onditions that give a system of linear equations whih an be solved. A1.2 Fixed Boundary Conditions When the plate has a fixed boundary on the line y = 0 in Fig. 2.2 the boundary onditions are as follows: (w) y=0 = 0 (A1.5a)
85 Applying Kazkowski s solution to the first ondition gives: w = 0 (A1.6a) y y=0 A m + C m = 0 (A1.5b) Applying Kazkowski s solutionto the seond ondition gives: mπφ m a A m sinh mπφ m mπψ m a 0 a C m sinh mπψ m y + mπφ m B a m osh mπφ m a y + mπψ m a a y + D m osh mπψ m a y sin mπ a x = φ m B m + ψ m D m = 0 (A1.6b) These two equations along with the two from the symmetri boundary onstitute four onditions that give a system of linear equations whih an be solved as: A1.3 Symmetri Boundary Conditions Used by Kazkowski The line y = 2βx from Fig. 2.2 an support two different boundary onditions when it is onsidered in the ontext of the entire hexagonal plate. This boundary an either be symmetri or anti-symmetri. Here the symmetri ondition will be onsidered. In order to apply the boundary equations, a new oordinate system must be defined. x = s a 2βa 2βa n, y = s + n a, where = 2 a 2 2 + (βa) 2 ξ = σ 2βζ, η = 2βσ + ζ, σ = s, ζ = n With these substitutions, the proposed solution beomes w(σ, ζ) = m=1,3,5, A m osh mπφ m (2βσ + ζ) + B m sinh mπφ m (2βσ + ζ) + C m osh mπψ m (2βσ + ζ) + D m sinh mπψ m (2βσ + ζ) sin mπ(σ 2βζ) (A1.7)
86 And the Laplaian of the proposed solution beomes: 2 w(σ, ζ) = m=1,3,5, ΦA m osh mπφ m (2βσ + ζ) + ΦB m sinh mπφ m (2βσ + ζ) + ΨC m osh mπψ m (2βσ + ζ) + ΨD m sinh mπψ m (2βσ + ζ) sin mπ(σ 2βζ) (A1.8) Using trigonometri and hyperboli identities, the solution beomes: w = m=1,3,5, {[A m (osh(mπφ m 2βσ) osh(mπφ m ζ) + sinh(mπφ m 2βσ) sinh(mπφ m ζ)) + B m (sinh(mπφ m 2βσ) osh(mπφ m ζ) + osh(mπφ m 2βσ) sinh(mπφ m ζ)) + C m (osh(mπψ m 2βσ) osh(mπψ m ζ) + sinh(mπψ m 2βσ) sinh(mπψ m ζ)) + D m (sinh(mπψ m 2βσ) osh(mπψ m ζ) + osh(mπψ m 2βσ) sinh(mπψ m ζ))][sin(mπσ) os( mπ2βζ) + os(mπσ) sin( mπ2βζ)]} (A1.9) And the Laplaian of the solution beomes: 2 w = π a 2 m=1,3,5, {[ΦA m (osh(mπφ m 2βσ) osh(mπφ m ζ) + sinh(mπφ m 2βσ) sinh(mπφ m ζ)) + ΦB m (sinh(mπφ m 2βσ) osh(mπφ m ζ) + osh(mπφ m 2βσ) sinh(mπφ m ζ)) + ΨC m (osh(mπψ m 2βσ) osh(mπψ m ζ) + sinh(mπψ m 2βσ) sinh(mπψ m ζ)) + ΨD m (sinh(mπψ m 2βσ) osh(mπψ m ζ) + osh(mπψ m 2βσ) sinh(mπψ m ζ))][sin(mπσ) os( mπ2βζ) + os(mπσ) sin( mπ2βζ)]} (A1.10) The boundary onditions are given as w = 0 ζ ζ=0 2 w = 0 ζ ζ=0 (A1.10a) (A1.11a) Applying the modified proposed solution to the first ondition yields: π m=1,3,5, m{[φ m A m (osh(mπφ m 2βσ) sinh(mπφ m ζ) + sinh(mπφ m 2βσ) osh(mπφ m ζ)) + φ m B m (sinh(mπφ m 2βσ) sinh(mπφ m ζ) +
87 osh(mπφ m 2βσ) osh(mπφ m ζ)) + ψ m C m (osh(mπψ m 2βσ) sinh(mπψ m ζ) + sinh(mπψ m 2βσ) osh(mπψ m ζ)) + ψ m D m (sinh(mπψ m 2βσ) sinh(mπψ m ζ) + osh(mπψ m 2βσ) osh(mπψ m ζ))][sin(mπσ) os( mπ2βζ) + os(mπσ) sin( mπ2βζ)] 2β[A m (osh(mπφ m 2βσ) osh(mπφ m ζ) + sinh(mπφ m 2βσ) sinh(mπφ m ζ)) + B m (sinh(mπφ m 2βσ) osh(mπφ m ζ) + osh(mπφ m 2βσ) sinh(mπφ m ζ)) + C m (osh(mπψ m 2βσ) osh(mπψ m ζ) + sinh(mπψ m 2βσ) sinh(mπψ m ζ)) + D m (sinh(mπψ m 2βσ) osh(mπψ m ζ) + osh(mπψ m 2βσ) sinh(mπψ m ζ))][sin(mπσ) os( mπ2βζ) + os(mπσ) sin( mπ2βζ)][ sin(mπσ) sin( mπ2βζ) + os(mπσ) os( mπ2βζ)]} = 0 m=1,3,5, m[a m (φ m sinh(mπφ m 2βσ) sin(mπσ) 2β osh(mπφ m 2βσ) os(mπσ)) + B m (φ m osh(mπφ m 2βσ) sin(mπσ) 2β sinh(mπφ m 2βσ) os(mπσ)) + C m (ψ m sinh(mπψ m 2βσ) sin(mπσ) 2β osh(mπψ m 2βσ) os(mπσ)) + D m (ψ m osh(mπψ m 2βσ) sin(mπσ) 2β sinh(mπψ m 2βσ) os(mπσ))] = 0 (A1.10b) Applying the Laplaian of the modified solution to the seond ondition gives: π π a 2 m=1,3,5, m{[φ m ΦA m (osh(mπφ m 2βσ) sinh(mπφ m ζ) + sinh(mπφ m 2βσ) osh(mπφ m ζ)) + φ m ΦB m (sinh(mπφ m 2βσ) sinh(mπφ m ζ) + osh(mπφ m 2βσ) osh(mπφ m ζ)) + ψ m ΨC m (osh(mπψ m 2βσ) sinh(mπψ m ζ) + sinh(mπψ m 2βσ) osh(mπψ m ζ)) + ψ m ΨD m (sinh(mπψ m 2βσ) sinh(mπψ m ζ) + osh(mπψ m 2βσ) osh(mπψ m ζ))][sin(mπσ) os( mπ2βζ) + os(mπσ) sin( mπ2βζ)] 2β[ΦA m (osh(mπφ m 2βσ) osh(mπφ m ζ) + sinh(mπφ m 2βσ) sinh(mπφ m ζ)) + ΦB m (sinh(mπφ m 2βσ) osh(mπφ m ζ) + osh(mπφ m 2βσ) sinh(mπφ m ζ)) + ΨC m (osh(mπψ m 2βσ) osh(mπψ m ζ) + sinh(mπψ m 2βσ) sinh(mπψ m ζ)) + ΨD m (sinh(mπψ m 2βσ) osh(mπψ m ζ) +
88 osh(mπψ m 2βσ) sinh(mπψ m ζ))][sin(mπσ) os( mπ2βζ) + os(mπσ) sin( mπ2βζ)][ sin(mπσ) sin( mπ2βζ) + os(mπσ) os( mπ2βζ)]} = 0 m=1,3,5, m[φa m (φ m sinh(mπφ m 2βσ) sin(mπσ) 2β osh(mπφ m 2βσ) os(mπσ)) + ΦB m (φ m osh(mπφ m 2βσ) sin(mπσ) 2β sinh(mπφ m 2βσ) os(mπσ)) + ΨC m (ψ m sinh(mπψ m 2βσ) sin(mπσ) 2β osh(mπψ m 2βσ) os(mπσ)) + ΨD m (ψ m osh(mπψ m 2βσ) sin(mπσ) 2β sinh(mπψ m 2βσ) os(mπσ))] = 0 (A1.11b) A1.4 Symmetri Boundary Conditions from Appendix A2 Using slightly different oordinate transformation and different boundary onditions, i.e. those developed in Appendix A2, slightly different equations for the oeffiients are obtained. If the oordinate transformation used is: x = a Then the proposed solution beomes 2βa 2βa t n, y = t + a n, where = 2 a 2 2 + (βa) 2 w(n, t) = m=1,3,5, A m osh mπφ m 2βt) + C m osh mπψ m 2βt) sin mπ (t 2βn) (n + 2βt) + B m sinh mπφ m (n + (n + 2βt) + D m sinh mπψ m (n + (A1.12) Applying the identities mentioned above transforms the solution into the following: w = m=1,3,5, sinh mπφ m B m sinh mπφ m osh mπφ m C m osh mπψ m A m osh mπφ m 2βt sinh mπφ m 2βt osh mπφ m 2βt sinh mπφ m 2βt osh mπφ m n + n + n + 2βt osh mπψ m n + n + (A1.13)
89 sinh mπψ m D m sinh mπψ m osh mπψ m os mπ 2βt sinh mπψ m n + 2βt osh mπψ m 2βt sinh mπψ m 2βn t sin mπ The symmetri boundary onditions are: n + n sin mπ mπ t os 2βn + w n y=2βx = 0 V n = Q n + M nt t = 0 3 w n 3 + (2 ν) 3 w n t 2 y=2βx = 0 (A1.14a) (A1.15a) Applying Kazkowski s solution to the first symmetry ondition on n = 0 gives: m=1,3,5, m=1,3,5, mπφ m sinh mπφ m osh mπφ m sinh mπψ m osh mπψ m os mπ sinh mπφ m osh mπφ m sinh mπψ m osh mπψ m os mπ A m osh mπφ m 2βt osh mπφ m 2βt osh mπφ m 2βt osh mπψ m 2βt osh mπψ m 2βt sinh mπφ m n + n + mπφ m n + mπψ m n + mπψ m n sin mπ B m sinh mπφ m C m osh mπψ m D m sinh mπψ m t os mπ t sin mπ 2βn 2β mπ A m osh mπφ m 2βt sinh mπφ m n + B m sinh mπφ m 2βt sinh mπφ m 2βt sinh mπψ m 2βt sinh mπψ m mπ t os 2βn = 0 A m φ m sinh mπφ m 2β osh mπφ m B m φ m sinh mπφ m C m ψ m sinh mπφ m 2β osh mπφ m D m ψ m sinh mπφ m 0 n + C m osh mπψ m n + D m sinh mπψ m n sin mπ 2βt sin mπ t 2βt os mπ t + 2βt sin mπ 2βt sin mπ 2βt os mπ t + 2βt sin mπ t sin mπ 2βt sinh mπφ m n + 2βt sinh mπψ m n + 2βt sinh mπψ m n + 2βn + 2βt osh mπφ m n + 2βt osh mπφ m 2βt osh mπψ m 2βt osh mπψ m 2βn + n + n + n + t 2βosh mπφ m 2βt os mπ t t + t 2βosh mπφ m 2βt os mπ t = Applying the solution to the seond symmetry ondition on n = 0 gives: (A1.14b)
90 m=1,3,5, mπφ m sinh mπφ m osh mπφ m mπψ m mπψ m osh mπψ m os mπ sinh mπφ m osh mπφ m sinh mπψ m osh mπψ m os mπ sinh mπφ m osh mπφ m mπψ m mπψ m osh mπψ m os mπ sinh mπφ m osh mπφ m mπψ m sinh mπψ m osh mπψ m os mπ 3 A m osh mπφ m 2βt osh mπφ m 2βt osh mπφ m 3 C m osh mπψ m 3 D m sinh mπψ m 2βt osh mπψ m 2βt sinh mπφ m n + n + mπφ m n + 2βt sinh mπψ m 2βt sinh mπψ m n + n sin mπ t sin mπ 2βn (2β)3 mπ 2βt sinh mπφ m 2βt sinh mπφ m 2βt sinh mπψ m 2βt sinh mπψ m t os mπ 2βt sinh mπφ m 2βt sinh mπφ m 2 C m osh mπψ m 2 D m sinh mπψ m 3 B m sinh mπφ m n + sinh mπψ m mπ t os 2βn + 3 A m osh mπφ m n + B m sinh mπφ m n + C m osh mπψ m n + D m sinh mπψ m n sin mπ 2βn 6β mπφ m 2βt sinh mπψ m t os mπ 2βt osh mπφ m 2βt osh mπφ m C m osh mπψ m 2βt osh mπψ m 2βt osh mπψ m n + mπφ m n + 2βt osh mπψ m 2βt osh mπψ m n + n sin mπ 2βn 12β2 mπφ m mπ t sin 2βn = 0 n + mπφ m n + 2βt sin3 h mπψ m n + mπψ m n sin mπ t sin mπ 2 A m osh mπφ m 2βt sinh mπφ m n + 2βt osh mπψ m n + 2βt osh mπφ m n + n + 2βt osh mπφ m 2 B m sinh mπφ m 2βt osh mπψ m 2βt osh mπψ m 2βn n + sinh mπψ m mπ t sin 2βn + A m osh mπφ m B m sinh mπφ m n + D m sinh mπψ m t os mπ n + n + 2βt osh mπφ m n + 2βt osh mπφ m n + 2βt sinh mπψ m n + 2βt sinh mπφ m n + n + 2βt sinh mπφ m 2βt sinh mπψ m n + 2βn + m=1,3,5, A m (φ 3 m 3φ m (2β) 2 ) sinh mπφ m 2βt sin mπ (2β) 3 + 3φ m 2 (2β) osh mπφ m 3φ m (2β) 2 ) osh mπφ m (2β) 3 + 3φ m 2 (2β) sinh mπφ m 3ψ m (2β) 2 ) sinh mπψ m (2β) 3 + 3ψ m 2 (2β) osh mπψ m 2βt os mπ 2βt sin mπ t + 2βt os mπ 2βt sin mπ t + 2βt os mπ t + t + B m (φ 3 m t + C m (ψ m 3 t + D m (ψ m 3 (A1.15b)
91 3ψ m (2β) 2 ) osh mπψ m 2βt sin mπ (2β) 3 + 3ψ m 2 (2β) sinh mπψ m t + 2βt os mπ t = 0
92 APPENDIX A2 Development of Basi Plate Equations Some basi moment-defletion equations: M x = D 2 w + ν 2 w x 2 y 2 M y = D 2 w + ν 2 w y 2 x 2 M xy = M yx = D(1 ν) 2 w x y (A2.1) D = Eh 3 12(1 ν 2 ) A2.1 Equilibrium Summing all of the fores on the element in the x diretion gives ΣF x = 0 = Q x dy Q x + Q x x dx dy Q ydx + Q y + Q y dy dx qdxdy y Q x x dxdy Q y dxdy qdxdy = 0 y Q x x + Q y y + q = 0 (A2.2) Similarly summing the fores in the y diretion gives Q x x + Q y y q = 0 (A2.3) Summing all of the moments of the element about the x-axis gives ΣM x = 0 = M y dx M y + M y y dy dx + M yxdx M yx + M yx x + Q x + Q x dx dydy + qdxdy x M y y dxdy M yx x dxdy + Q y = 0 dx dy + Q y + Q y y dy dxdy Q xdydy Q y = M y y + M xy x (A2.4)
93 Similarly summing the moments about the y-axis gives Q x = M x x + M xy y (A2.5) An equilibrium equation an be expressed solely in terms of the moment derivatives by substituting expressions (---.4) and (---.5) into (---.2) and eliminating Q x and Q y. x M xy y + M x x + y M xy x + M y y + q = 0 2 M x x 2 + 2 M y y 2 2 2 M xy x y = q (A2.6) A2.2 Differential Equation of Transverse Defletion The differential equation for the transverse defletion of the plate an be obtained by substituting the moment-urvature relationships (A2.1) into the equilibrium equation (A2.6). Here the plate bending stiffness is assumed to be onstant. 2 x 2 D w 2 x 2 + ν 2 w 2 + y2 y 2 D w 2 y 2 + ν 2 w x 2 2 2 x y D(1 ν) 2 w D 4 w x 4 Dν 4 w x 2 y 2 D 4 w y 4 Dν 4 w 4 w x 2 2D(1 ν) y2 x 2 y 2 = q 4 w x 4 + 4 w y 4 + ν 4 w 4 w 4 w x 2 y 2 + ν x 2 + (2 2ν) y2 x 2 y 2 = q D = q x y 4 w x 4 + 2 4 w + 4 w x 2 y 2 y 4 = 4 w = q D (A2.7) This equation for the transverse defletion does not inlude any mid-plane stress and is essentially for a freely bound plate.
94 A2.3 Mid-Plane Stress Mid-plane stress develops diretly from the appliation of fores at the boundary of the plate. Fig. A2.1 shows the fores per unit length of the mid-plane. FIGURE A2.1. Fores per unit length of the mid-plane Equilibrium of the mid-plane fores on element δxδy gives the following equations. F x = 0 = N x dy + N x + N x x dx dy N xydx + N xy + N xy dy dx y N x x dxdy + N xy y dxdy = 0 N x x + N xy y = 0 (A2.8) F y = 0 = N y dx + N y + N y y dy dx N xydy + N xy + N xy dx dy x N y y dxdy + N xy x dxdy = 0 N y y + N xy x = 0 (A2.9) If a fore funtion, Φ, is introdued the onditions (A2.8) and (A2.9) are satisfied and the quantities N x, N y, and N xy an be alulated.
95 N x x + N xy y = 0 = 2 Φ 1 x y + 2 Φ 1 x y N x = Φ 1 y, N xy = Φ 1 x N y y + N xy x = 0 = 2 Φ 2 x y + 2 Φ 2 x y N y = Φ 2 x, N xy = Φ 2 y N xy N xy = 0 = Φ 1 x + Φ 2 y Φ 1 = Φ y, Φ 2 = Φ x N x = 2 Φ y 2 N y = 2 Φ x 2 N xy = 2 Φ x y (A2.10) If the strains in the mid-plane are due solely to the mid-plane fores then the following are true aording to the Theory of Elastiity. u = N x νn y x Eh v = N y νn x y Eh u + v = 2N xy (1+ν) y x Et (A2.11) The ompatibility ondition requires the following to be true. Whih an be expressed as: 2 y 2 u 2 + x x 2 v y 2 x y u y + v x = 0 2 y 2 N x νn y Eh + 2 x 2 N y νn x 2 Eh x y 2N xy(1 + ν) = 0 Et 2 y 2 1 Eh 2 Φ 2 ν y2 y 2 1 Eh or when h is onstant over the plate: 2 Φ 2 + x2 x 2 1 Eh 2 Φ 2 ν x2 x 2 1 Eh 2 Φ y 2(1 + ν) 4 Φ + 2 Eh x 2 y 2 = 0 4 Φ + 4 Φ x 4 4 Φ + 2 = y 4 x 2 y 2 4 Φ = 0 (A2.12)
96 A2.4 Z Component Mid-Plane Fores FIGURE A2.2. A plate setion under defletion. The mid-plane fore exhibits a z-omponent. It an be seen from Fig A2.2 that the mid-plane fores have a z-omponent, whih an be written by inspetion as w @x = 0 N x x dy N w xy y dy w @x = dx N x x dy + x N w x x dxdy + N w xy @y = 0 N y w y dx N xy w @y = dy N y y dx + y N w y y dxdy + N xy y dy + x N xy w x dx w x dx + y N xy w dxdy y w dxdy x The resultant fore in the z diretion (inluding a fore applied to the fae of the plate) an be written as F z = q w = N x x dy N w xy y dy + N w x x dy + x N x N y w y dx N xy w x dx + N y w y dx + y N y w x dxdy + N w xy y dy + x N xy w y dxdy + N xy x N w x x + N w xy y dxdy + y N w y y + N w xy dxdy = q x w x dx + y N xy w dxdy y w dxdy x N 2 w + 2N 2 w x x 2 xy + N 2 w x y y = y 2 q 2 Φ 2 w 2 2 Φ 2 w + 2 Φ 2 w = x 2 x 2 x y x y y 2 y 2 q (A2.13) Combining (A2.7) and (A2.13) gives an expression for the defletion of a plate that inludes mid-plane fores.