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An experimental and finite element investigation of added mass effects on ship structures Glenwright, David George 1987

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\ \ AN EXPERIMENTAL AND FINITE ELEMENT INVESTIGATION OF ADDED MASS EFFECTS ON SHIP STRUCTURES By DAVID GEORGE GLENWRIGHT .A.Sc., The Un i v e r s i t y of B r i t i s h Columbia, 19 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1987 © David George Glenwright, 1987 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date September 15, 1987  DE-6(3/81) ABSTRACT The Experimental and Finite Element Investigation of Added Mass Effects on Ship Structures comprised three phases : 1) investigation of the fluid modelling capabilities of the Finite Element Program VAST, 2) experimental investigation to determine the effect of the fluid on the lowest natural frequencies and mode shapes of a ship model, and 3) comparison of these experimental results with numerical results obtained from VAST. The fluid modelling capabilities of VAST were compared with experimental results for submerged vibrating plates, and the effect of fluid element type and mesh discretization was considered. In general, VAST was able to accurately predict the frequency changes caused by the presence of the fluid. Experimental work both in air and water was performed on a ship model. The lowest four modes of vertical, horizontal, and torsional vibration were identified, and the effect of draught on the frequencies and mode shapes was recorded. When the experimentally obtained frequencies and mode shapes for the ship model were compared with the numerical predictions of VAST, good agreement was found in both air and water tests for the vertical vibration modes. TABLE OF CONTENTS Page Abstract i Table of Contents i i L i s t of Tables v i L i s t of Figures i x Nomenclature x v i i Acknowledgements x i x 1. INTRODUCTION 1 1.1. Background 1 1.2. Purpose and Scope 5 2. FLUID IDEALIZATION 6 2.1. Continuum Formulation 6 2.2. F i n i t e Element Formulation 12 2.3. I n f i n i t e F l u i d Element Formulation 22 3. NUMERICAL EVALUATION OF FLUID MODELLING CAPABILITIES 25 3.1. V i b r a t i o n i n A i r 27 3.2. V i b r a t i o n i n F l u i d 35 3.2.1. E f f e c t of the Extent of the F l u i d Domain....36 i i Page 3.2.2. E f f e c t of the Number of Elements i n the F l u i d Domain 42 3.2.3. E f f e c t of D i f f e r e n t Elements 48 3.2.3.1. 8 and 20-Node F l u i d Elements 48 3.2.2. I n f i n i t e F l u i d Elements 59 3.3. Imposition of an A r t i f i c i a l Bandwidth 70 4. EXPERIMENTAL DETERMINATION OF THE VIBRATION RESPONSE OF A SHIP MODEL IN AIR AND WATER 76 4.1. Model Construction 76 4.2. Instrumentation 81 4.3. Experimental Procedure 85 4.3.1. C a l i b r a t i o n of Instrumentation 85 4.3.2. Experimental Set Up of A i r and Water Tests..88 4.3.3. D i f f e r e n t Load Conditions 92 4.3.4. Natural Frequency and Mode Shape Determination 94 4.3.5. Data A c q u i s i t i o n Program 97 4.4. Experimental Results 98 4.4.1. Frequencies 98 4.4.2. Mode Shapes 109 i i i Page 5. NUMERICAL RESULTS AND COMPARISION OF THE VIBRATION RESPONSE OF A SHIP MODEL IN AIR AND WATER 121 5.1. Ma t e r i a l Properties 122 5.2. F i n i t e Element Models 123 5.2.1. S t r u c t u r a l Models 124 5.2.2. F l u i d Models 128 5.3. Results and Comparison with the Experimental Results 130 5.3.1. Frequencies 131 5.3.1.1. Beam/Strip Theory and Experimental Results 131 5.3.1.2. F i n i t e Element and Experimental Results 133 5.3.2. Mode Shapes 141 6. CONCLUSIONS 165 7. BIBLIOGRAPHY 167 8. APPENDICES 8.1. Appendix A - V a r i a t i o n a l P r i n c i p l e 172 8.2. Appendix B - F l u i d Element Shape Functions 174 8.3. Appendix C - Thick/Thin S h e l l Element [K] and [M] 181 i v Page 8.4. Appendix D - Dimensional Analysis 190 8.5. Appendix E - Instrumentation ....197 8.6. Appendix F - Modal Analysis Program 198 8.7. Appendix G - Material Properties of A c r y l i c 203 v LIST OF TABLES Page Table 3.1 Plate Theory and F i n i t e Element Results for a/b = 0.5 b - 8.0 inches t = 0.105 inches....31 Table 3.2 Plate Theory and F i n i t e Element Results f o r a/b = 1.0 b = 8.0 inches t = 0.105 inches 31 Table 3.3 Plate Theory and F i n i t e Element Results f o r a/b - 2.0 b - 8.0 inches t = 0.105 inches 32 Table 3.4 Plate Theory and F i n i t e Element Results f o r a/b = 3.0 b - 8.0 inches t = 0.105 inches....32 Table 3.5 Average CPU Times (sec) f o r Various F l u i d Meshes Meshes fo Plate Aspect Ratio a/b = 1.0 b = 8.0 inches t •= 0.105 inches 43 Table 3.6 Comparison of Error Between 8-Noded and I n f i n i t e Element for F l u i d Domain D/b = 0.125....66 Table 3.7 Average CPU Times for F l u i d Formulation 68 Table 3.8 Solution Times i n Various Routines for Bandwidth Reduction 75 Table 4.1 Nominal Accelerometer S p e c i f i c a t i o n s 81 Table 4.2 Comparison of Experimental and F i n i t e Element Results f o r 30 l b Load Condition 90 Table 4.3 Load Conditions and Drafts of the Model 94 Table 4.4 Frequencies of the V e r t i c a l Bending Modes 99 v i Page Table 4.5 Frequencies of the Horizontal Bending Modes 99 Table 4.6 Frequenc ies of the Torsional Modes •. . .100 Table 4.7 Center of Mass and Shear f o r Various Load Conditions 101 Table 4.8 Experimental and Beam Theory Results i n A i r fo r V e r t i c a l Modes of V i b r a t i o n 106 Table 4.9 Ratios of Added Mass to St r u c t u r a l Mass f o r V e r t i c a l Bending Modes 107 Table 4.10 Ratios of St r u c t u r a l Mass to Added Mass f o r V e r t i c a l Bending Modes Using F i n i t e Elements 109 Table 5.1 Results of Tests f o r Young's Modulus 123 Table 5.2 Average CPU Times f o r S t r u c t u r a l Mesh 126 Table 5.3 CPU Times f or F l u i d Meshes 130 Table 5.4 Frequencies f o r the V e r t i c a l Bending Modes from Beam and S t r i p Theory 132 Table 5.5 Frequencies f o r the V e r t i c a l Bending Modes from F i n i t e Element 136 Table 5.6 Frequencies f o r the Horizontal Bending Modes from F i n i t e Element 137 Table 5.7 Frequencies f o f Torsional Modes from F i n i t e Element 138 Table 5.8 Frequencies Predicted by Both F i n i t e Element Models of the Weights f or 30 lb Load Condition...141 v i i Page Table G.l Results from Strain-Guage Test 205 v i i i LIST OF FIGURES Page Figure 3.1 Plate Structure 26 Figure 3.2 F i n i t e Element Plate Meshes 29 Figure 3.3 F i r s t Six Mode Shapes of Plate V i b r a t i o n 30 Figure 3.4a F i n i t e Element F l u i d Domain Meshes NFLD = 1 38 Figure 3.4b F i n i t e Element F l u i d Domain Meshes NFLD = 2 39 Figure 3.4c F i n i t e Element F l u i d Domain Meshes NFLD = 4 40 Figure 3.5 Error Between F i n i t e Element and Experimental Results Using the 8-Noded F l u i d Element f o r Mode 1 with Plate Aspect Ratio a/b - 1.0 41 Figure 3.6 8 and 20-Noded F l u i d Results f o r Mode 2 with Plate Aspect Ratio a/b - 1.0 44 Figure 3.7 Comparison of Two F l u i d Meshes with Plate Aspect Ratio a/b -1.0 NFLD - 2 with 8-Noded F l u i d Element NFLD - 1 with 20-Noded F l u i d Element....46 Figure 3.8 Comparison of Two F l u i d Meshes with Plate Aspect Ratio a/b -1.0 NFLD - 4 wiht 8-Noded F l u i d Element NFLD •= 2 with 20-Noded F l u i d Element....47 Figure 3.9 8 and 20-Noded F l u i d Results f o r Mode 1 with Plate Aspect Ratio a/b = 1.0 50 Figure 3.10 8 and 20-Noded F l u i d Results f o r Mode 3 with Plate Aspect Ratio a/b - 1.0 51 ix Page Figure 3.11 8 and 20-Noded F l u i d Results f o r Mode 4 with Plate Aspect Ratio a/b = 1.0 ....52 Figure 3.12 8 and 20-Noded F l u i d Results f o r Mode 5 with Plate Aspect Ratio a/b = 1.0 53 Figure 3.13 8-Noded F l u i d Results f o r Various Aspect Ratios with NFLD = 4 54 Figure 3.14 20-Noded F l u i d Results f o r Various Aspect Ratios with NFLD = 4 55 Figure 3.15 Comparison of 8 and 20-Noded F l u i d Results Using NFLD - 4 and Plate Aspect Ratio a/b =3.0 for Mode 1 58 Figure 3.16a I n f i n i t e Element F l u i d Domain Meshes NFLD - 1 INF = 1 61 Figure 3.16b I n f i n i t e Element F l u i d Domain Meshes NFLD - 2 INF - 1 62 Figure 3.16c I n f i n i t e Element F l u i d Domain Meshes NFLD - 4 INF - 1 63 Figure 3.16d I n f i n i t e Element F l u i d Domain Meshes NFLD = 4 INF - 2 64 Figure 3.16e I n f i n i t e Element F l u i d Domain Meshes NFLD = 4 INF = 3 65 Figure 3.17 Results f o r the I n f i n i t e Meshes f or Plate Aspect Ratio a/b = 1.0 67 Figure 3.18 Comparison of Three D i f f e r e n t F l u i d Meshes 69 x Page Figure 3.19 Error Produced by Imposing an A r t i f i c i a l Bandwidth Using the 8-Noded F l u i d Element with Plate Aspect Ratio a/b = 1.0 73 Figure 3.20 Erro r Produced by Imposing an A r t i f i c i a l Bandwidth Using the 8-Noded F l u i d Element with Plate Aspect Ratio a/b =1.0 f o r Individual Modes 1 - 5 74 Figure 4.1 Ship Model 78 Figure 4.2 Endcap and Wedge 80 Figure 4.3 Instrument Chain Schematic 82 Figure 4.4 C a l i b r a t i o n of Accelerometers 87 Figure 4.5 Power Spectrum f or the Various Tips of the Impact Hammer 89 Figure 4.6 Experimental Set Up of the A i r Tests 91 Figure 4.7 Experimental Set Up of the Water Tests.. 93 Figure 4.8 Non-Uniform Load Condition 95 Figure 4.9 V e r t i c a l Modes of V i b r a t i o n f o r the Ship Model 103 Figure 4.10 Horizontal Modes of V i b r a t i o n f o r the Ship Model 104 Figure 4.11 Torsional Modes of V i b r a t i o n f o r the Ship Model 105 x i Page Figure 4.12 Experimental V e r t i c a l Modes of the Ship Model i n A i r and Water f o r 30 l b Load Condition I l l Figure 4.13 Experimental Horizontal Modes of the Ship Model i n A i r and Water f or 30 lb Load Condition 112 Figure 4.14 Experimental Torsional Modes of the Ship Model i n A i r and Water f o r 30 l b Load Condition 113 Figure 4.15 Experimental V e r t i c a l Bending of the Ship Model i n A i r and Water f or 40 lb Load Condition 114 Figure 4.16 Experimental Horizontal Bending of the Ship Model i n A i r and Water f or 40 lb Load Condition 115 Figure 4.17 Experimental Torsional Modes of the Ship Model i n A i r and Water f or 40 l b Load Condition 116 Figure 4.18 Experimental V e r t i c a l Modes of the Ship Model i n A i r and Water f or 70 lb Load Condition 117 Figure 4.19 Experimental Horizontal Modes of the Ship Model i n A i r and Water f or 70 lb Load Condition 118 Figure 4.20 Experimental Torsional Modes of the Ship Model i n A i r and Water f o r 70 l b Load Condition 119 x i i Page Figure 5.1 S t r u c t u r a l F i n i t e Element Mesh of Ship Model...125 Figure 5.2 Various Mode Shapes ...127 Figure 5.3 F l u i d Meshes 129 Figure 5.4 Experimental and Beam Theory Results i n Air....134 Figure 5.5 Experimental and S t r i p Theory Results i n Water 135 Figure 5.6 F i n i t e Element Weight Modelling 140 Figure 5.7 F i n i t e Element V e r t i c a l Modes of the Ship Model i n Vacuum and Water f o r 30 l b Load Condition 143 Figure 5.8 F i n i t e Element Horizontal Modes of the Ship Model i n Vacuum and Water for 30 lb Load Condition 144 Figure 5.9 F i n i t e Element Torsional Modes of the Ship Model i n Vacuum and Water f o r 30 l b Load Condition 145 Figure 5.10 F i n i t e Element V e r t i c a l Modes of the Ship Model i n Vacuum and Water f o r 40 l b Load Condition 146 Figure 5.11 F i n i t e Element Horizontal Modes of the Ship Model i n Vacuum and Water for 40 lb Load Condition 147 x i i i Page Figure 5.12 F i n i t e Element Torsional Modes of the Ship Model i n Vacuum and Water for 40 lb Load Condition 148 Figure 5.13 F i n i t e Element and Experimental V e r t i c a l Modes of the Ship Model i n A i r f or 30 l b Load Condition 150 Figure 5.14 F i n i t e Element and Experimental Horizontal Modes of the Ship Model i n A i r f o r 30 l b Load Condition 151 Figure 5.15 F i n i t e Element and Experimental Torsional Modes of the Ship Model i n A i r for 30 l b Load Condition 152 Figure 5.16 F i n i t e Element and Experimental V e r t i c a l Modes of the Ship Model i n A i r for 40 l b Load Condition 153 Figure 5.17 F i n i t e Element and Experimental Horizontal Modes of the Ship Model i n A i r for 40 l b Load Condition 154 Figure 5.18 F i n i t e Element and Experimental Torsional Modes of the Ship Model i n A i r f or 40 lb Load Condition 155 Figure 5.19 F i n i t e Element and Experimental V e r t i c a l Modes of the Ship Model i n A i r f or 70 lb Load Condition 156 x i v Page Figure 5.20 F i n i t e Element and Experimental Horizontal Modes of the Ship Model i n A i r f or 70 l b Load Condition 157 Figure 5.21 F i n i t e Element and Experimental Torisonal Modes of the Ship Model i n A i r f or 70 lb Load Condition 158 Figure 5.22 F i n i t e Element and Experimental V e r t i c a l Modes of the Ship Model i n Water for 30 l b Load Condition 159 Figure 5.23 F i n i t e Element and Experimental Horizontal Modes of the Ship Model i n Water f o r 30 l b Load Condition 160 Figure 5.24 F i n i t e Element and Experimental T o r i s o n a l Modes of the Ship Model i n Water for 30 l b Load Condition 161 Figure 5.25 F i n i t e Element and Experimental V e r t i c a l Modes of the Ship Model i n Water for 40 l b Load Condition 162 Figure 5.26 F i n i t e Element and Experimental Horizontal Modes of the Ship Model i n Water f o r 40 l b Load Condition 163 Figure 5.27 F i n i t e Element and Experimental Torsional Modes of the Ship Model i n Water f o r 40 l b Load Condition 164 xv Page Figure B.l 8-Noded Fluid Element and 4-Noded Interface Element ...176 Figure B.2 20-Noded Fluid Element and 8-Noded Interface Element 178 Figure B.3 8-Noded Infinite Element 180 Figure C l Curved Shell Element 182 Figure D.l Ship Model 196 Figure G.l Strain-Guage Set Up and Bending Test of Ship Model 204 Figure G.2 Natural Frequency of Simply Supported Beam 208 xv i NOMENCLATURE Matrices and vectors are denoted by boldface type. Rectangular or Square Matrix Column Vector ] 1 Matrix Inverse ] T Matrix Transpose Time D i f f e r e n t i a t i o n ; u = — 8t d2u Time D i f f e r e n t i a t i o n ; u = a 2 t 2 V Laplacian Operator a Span Length of Plate b Cord Length of Plate D Extent of F l u i d Domain E Young's Modulus E Energy i n F l u i d g A c c e l e r a t i o n Due to Gravity I Moment of I n e r t i a [J] Jacobian Matrix [K] S t i f f n e s s Matrix N Shape Functions M Moment x v i i [M] Mass Matrix [M ] Added Mass Matrix A [M ] S t r u c t u r a l Mass Matrix s p Pressure i n the F l u i d t Plate Thickness u Displacement v V e l o c i t y x, y, z Cartesian Coordinates fi I Constant f o r V i b r a t i o n that depends on the end conditions of the beam £ S t r a i n <f> V e l o c i t y P o t e n t i a l p Density Frequency of mode i tf^ Eigenvector of mode i n 3.1415926536 . . . v Poisson's Ratio K Bulk Modulus of F l u i d u Amplitude of a Surface Wave y A Wave Length £, n, f Isoparametric Coordinates x v i i i ACKNOWLEDGEMENT I would l i k e to thank my advisor, Dr. Stanley G. Hutton, Department of Mechanical Engineering, U n i v e r s i t y of B r i t i s h Columbia f o r h i s support and guidance throughout the work on t h i s t h e s i s . I would also l i k e to thank Alan Steeves, Department of Mechanical Engineering, U n i v e r s i t y of B r i t i s h Columbia without whom much of the computer work contained i n t h i s t hesis would never have been completed. As well, I would l i k e to thank Gerry N. Stensgaard, Manager of B.C. Research, Ocean Engineering Department, f o r allowing me access to t h e i r towing tank i n which the water te s t s on the model were conducted. This work was funded through Contract S e r i a l Number 9 SC 97727-4-6248 with Defence Research Establishment A t l a n t i c (D.R.E.A.) Department of National Defence. xix 1 I N T R O D U C T I O N 11 BACKGROUND Naval a r c h i t e c t s have long recognized the f a c t that f l u i d a f f e c t s the v i b r a t i o n c h a r a c t e r i s t i c s of ships. The exact s o l u t i o n to the equations of motion of a f l e x i b l e structure involves s o l v i n g a system of equations i n which the f l u i d and structure are coupled. Not only does the f l u i d coupling introduce many complications, but the f a c t that the ship i s not uniform further complicates the task. The general equations of motion of the ship girder v i b r a t i n g i n a f l u i d have been recently developed by Bishop and Price [9-13]. The equations they present deal with the ship girder and attempt to account f o r the e f f e c t s of shear d e f l e c t i o n , rotary i n e r t i a , and warping on the frequencies and mode shapes of the beam. They assume that the e f f e c t of the f l u i d may be expressed as d i s t r i b u t e d forces exerted on the beam. Daidola [16] presents a s o l u t i o n to the equations of motion f o r a uniform c i r c u l a r Euler beam i n v e r t i c a l v i b r a t i o n , 1 ignoring the e f f e c t s of shear d e f l e c t i o n and rotary i n e r t i a . - He solves the three-dimensional Laplace equation for the v e l o c i t y p o t e n t i a l as a function of p o s i t i o n along the length of the beam, time, and the unknown d e f l e c t i o n shape (mode shape) of the beam. The e f f e c t i v e force i s then determined from the l i n e a r i z e d B e r n o u l l i equation (see Chapter 2) and inserted into the Euler equation of the beam. The concept of an added mass to account for the e f f e c t of the f l u i d , which could be incorporated into the simple beam equations, was f i r s t introduced by F.M. Lewis [27] i n 1929 and independently by J.L. Taylor [38] i n 1930. U n t i l that time, ship v i b r a t i o n had been dealt with by using data from s i s t e r ships and/or empirical formulas. Lewis and Taylor proposed a method which accounted f o r the i n e r t i a e f f e c t s of the f l u i d and allowed the two and three node v e r t i c a l v i b r a t i o n modes to be a n a l y t i c a l l y c a l c u l a t e d . Lanweber and Macagno [26] extended the method to include h o r i z o n t a l v i b r a t i o n modes. Taylor [38] s i m i l a r l y developed J - f a c t o r s f o r h o r i z o n t a l v i b r a t i o n i n the presence of a free surface. Subsequently, various authors have e i t h e r extended t h i s method by developing more a n a l y t i c procedures [25,26], or they have widened the range of a p p l i c a b i l i t y by introducing a mode dependent J - f a c t o r to account for the three-dimensional e f f e c t s of the f l u i d [2,23,40]. 2 There has been very l i t t l e experimental work- done i n t h i s f i e l d . Moullin, Browne, and Perkins [31] i n 1930 were the f i r s t to perform experiments on prisms made of wood. However, these prisms were not allowed to f l o a t f r e e l y i n water but were suspended from the underside of a s t e e l bar. This arrangement allowed various l e v e l s of immersion to be examined. They compared t h e i r r e s u l t s with those of t h e i r contemporary, Lewis and they found that the experimentally added masses were below those predicted by Lewis's method. In 1962, B u r r i l l , Robson, and Townsin [14] reported experimental work that had been performed with prismatic bars of various constant cross-sections over t h e i r length of 120 inches. These bars were made of an aluminium a l l o y so that the entrained water masses would be r e l a t i v e l y high i n r e l a t i o n to the weight of the bars i n a i r . Their experimental procedure was s i m i l a r to Moul l i n et a l , i n that these bars were suspended at t h e i r nodal lo c a t i o n s , and the l e v e l of water i n the tank was v a r i e d i n order to give the desired draughts. The r e s u l t s showed that the added mass f e l l between that derived by Lewis and that of Taylor. Further experimental work was performed by Townsin i n 1969 [40] , and he published an empirical formula f o r the three-dimensional c o r r e c t i o n f a c t o r known as the J - f a c t o r . In I960, Kuo published h i s PhD Thesis [24] i n which 3 he conducted experimental work on a scale model of a ship. Unlike the other experimental work, Kuo's model was constructed from a c r y l i c . These r e s u l t s were very i n t e r e s t i n g , as Kuo made what would seem to be the f i r s t attempt to scale the v i b r a t i o n c h a r a c t e r i s t i c s of a r e a l ship. His r e s u l t s f o r the 2-node v e r t i c a l bending mode agreed very well with Lewis's method, but as mode number was increased, the agreement began to d e t e r i o r a t e . He explained t h i s d e t e r i o r a t i o n as being due to the e f f e c t s of the r e a l three-dimensional flow around the h u l l and the lack of p r e c i s i o n i n the c a l c u l a t i o n of the added mass f o r a section. There has been tremendous progress recently i n the development of the F i n i t e Element Method [15,45], and i t now can be applied to the ship structure, y i e l d i n g complex three-dimensional f i n i t e element models in v o l v i n g many thousand degrees of freedom [4,5,35,37]. These models seem to adequately model the s t i f f n e s s , r i g i d i t y , and mass d i s t r i b u t i o n of the structure. But to extend f i n i t e elements to include the f l u i d - s t r u c t u r e i n t e r a c t i o n problem, the s t r u c t u r a l mass must be matched by a comparable technique i n the f l u i d to determine an added mass matrix. Various authors have presented d i f f e r e n t methods that allow the F i n i t e Element Method to handle the f l u i d - s t r u c t u r e i n t e r a c t i o n problem. Zienkiewicz and Bettess [46,47], Armand [4], 4 Orsero [35], and K i e f l i n g [21] have proposed a standard f i n i t e element d i s c r e t i z a t i o n of the f l u i d i n which the three-dimensional Laplace equation i s solved, from which an added mass matrix i s determined and added to the s t r u c t u r a l mass matrix of the elements i n contact with the f l u i d . As well, Beer and Meek [6], Bettess [7], and Zienkiewiz and Bettess [47] have proposed the use of i n f i n i t e f l u i d elements to reduce the size of the f i n i t e element f l u i d matrices. Hylarides and Vorus [19,43] have proposed a panel source d i s t r i b u t i o n to derive the added mass of the v i b r a t i n g ship h u l l . Deruntz and Geers [17] and Zienkiewicz, K e l l y , and Bettess [48] have also proposed boundary element s o l u t i o n procedures which can determine the added mass matrix that i n turn i s added to the s t r u c t u r a l mass matrix. A l l of these methods are very s i m i l a r ; they a l l produce an added mass matrix which i s added to the s t r u c t u r a l mass matrix, followed by sol v i n g f o r the v i b r a t i o n c h a r a c t e r i s t i c s of the structure. 12 PURPOSE AND SCOPE The purpose of the work d e t a i l e d i n t h i s thesis was thr e e - f o l d . F i r s t , the f l u i d modelling c a p a b i l i t i e s of the F i n i t e Element Program VAST were investigated. This was done by comparing the frequencies predicted by VAST with experimental 5 r e s u l t s found i n the l i t e r a t u r e f o r v i b r a t i n g c a n t i l e v e r plates i n a f l u i d . The e f f e c t of the extent of the f l u i d domain, the degree of d i s c r e t i z a t i o n , and the type of f l u i d element used were examined. Second, experimental investigations were conducted to determine the e f f e c t of the f l u i d on the lowest natural frequencies and mode shapes of a s h i p - l i k e model. F i n a l l y , these experimental r e s u l t s were compared with a n a l y t i c a l r e s u l t s obtained from VAST f o r the v i b r a t i o n c h a r a c t e r i s t i c s of t h i s s h i p - l i k e model. 6 2. THEORY This chapter w i l l present the equations of motion that describe the v i b r a t i o n response of a structure i n the presence of a f l u i d . These equations are based upon the assumption of an i n v i s c i d , incompressible, and i r r o t a t i o n a l f l u i d . Having developed the d i f f e r e n t i a l equations d e f i n i n g the continuum problem, the f i n i t e element equations that are used by VAST are presented here. 2.1 CONTINUUM FORMATION The f l u i d i s assumed to be incompressible and i n v i s c i d , and the flow i r r o t a t i o n a l . As a r e s u l t , the f l u i d motion can be described by a v e l o c i t y p o t e n t i a l <f> = ^(x,y,z), which s a t i s f i e s Laplace's equation V2<t> - 0 ( 2 . 1 ) B e r n o u l l i ' s equation, (Newman [34] and Armand [4]) , can be expressed as 2± + £ + V + 3 2 = C ( t ) ( 2 . 2 ) at p 2 /2 2 2 v + v + v the - • - x y z v e l o c i t y , V the p o t e n t i a l of the applied external forces per-unit volume, and C(t) an i n t e g r a t i o n function that may be a function of time. I f g r a v i t y i s the only external force, V i s a l i n e a r function of y and represents the hydrostatic pressure. I f only the hydrodynamic forces are of concern, V may be neglected. C(t) may be chosen a r b i t r a r i l y and can be set equal to zero, as w i l l be done i n t h i s a n a l y s i s . I f motions are small enough, such 2 that q i s a second order term, the l i n e a r i z e d B e r n o u l l i ' s equation becomes ^ + £ = 0 (2.3) at p Rearranging terms gives P = - P ^ (2.4) at From c o n t i n u i t y 3v 3v 3v .. „ _ x + _ y + _ z = I3p ( 2.5) ax ay az K at where v , v , and v are the v e l o c i t i e s i n the x, y, and z x y z d i r e c t i o n s , r e s p e c t i v e l y , and K i s the bulk modulus of the f l u i d . Noting that ^ = v ^ = v ^ = v (2.6) ax x ay y az 2 and combining equations 2.5 and 2.6 y i e l d s V 2^ - 1 ^£ = 0 (2.7) K at Taking the d e r i v a t i v e of equation 2.4 with respect to time, and 8 s u b s t i t u t i n g into equation 2.7 r e s u l t s i n V*<£ + t I _ o (2.8a) K A l t e r n a t i v e l y , equation 2.8a can be expressed i n terms of the pressure, such that V 2p + ^ p - 0 (2.8b) K Since the f l u i d i s incompressible, equations 2.8a,b reduce to V24> - 0 (2.9a) V 2p = 0 (2.9b) which must be s a t i s f i e d at every point inside the f l u i d domain. The F i n i t e Element Method w i l l be used to solve 2.9a, with the boundary conditions described below. The developed equation assumes an i n v i s c i d , incompressible, and i r r o t a t i o n a l f l u i d flow. These l i m i t a t i o n s a f f e c t the f i n i t e element's a b i l i t y to represent the r e a l flow i n the f l u i d around a v i b r a t i n g structure. (The development of the F i n i t e Element Theory i s presented i n Section 3.2.) At the f l u i d - s t r u c t u r e i n t e r f a c e , the f l u i d must always be i n contact with the structure; and hence, the v e l o c i t y of the f l u i d normal to the structure must be equal to the v e l o c i t y of the structure at that point, given by 2£ - v - u (2.10a) on or i n terms of the pressure using equation 2.4 9 an (2.10b) The equilibrium condition f o r the free surface (y = 0) i s P - Pgu y (2.11) where u i s the amplitude of a wave on the free surface. This y equation may be further developed by noting that from equation 2.4 and 2.6 1 3p u = v - - -y y P By which i n turn y i e l d s 3p 1 •• a 5 ~ l P <2-12*> A corresponding equation dealing with </> may be developed by examining the amplitude u of the waves on the free surface, such that u -$* y dy and 1 8<j> u = -y g at Combining the two above equations leads to I*--* (2.12b) 3y g iWt I f ^(x.y.z.t) = <£(x,y,z)e *-V* (2.13) w 3y which i s the free surface boundary condition expressed i n terms of the v e l o c i t y p o t e n t i a l <f>. For most s t r u c t u r a l v i b r a t i o n s , 2 2 w » g. More accurately, one should look at u 1/g (where 1 i s 10 the c h a r a c t e r i s t i c length of the body) , since w and g- are dimensional q u a n t i t i e s . A good approximation of equation 2.13 on the free surface i s <f> - 0 (2.14a) S i m i l a r l y , on the free surface p = 0 (2.14b) Equations 2.14a,b assume that any surface waves generated are n e g l i g i b l e . This i s a safe assumption i f : 1) the structure i s completely submerged at a large distance from the free surface; and 2) the structure i s f l o a t i n g and the wavelength A of the generated waves i s les s than the c h a r a c t e r i s t i c dimension 1 of the structure, which f o r ships would be the beam. For a f i n i t e element d i s c r e t i z a t i o n of the f l u i d , the i n f i n i t e boundary has to be truncated at some large distance from the structure. At t h i s imposed boundary, a s u i t a b l e co n d i t i o n i s n o n r e f l e c t i o n of waves, given by ^£ - 0 or <f> - 0 (2.15) an as these waves are generated by the structure and f i n a l l y absorbed by the f l u i d . The equations of motion f o r the coupled f l u i d - s t r u c t u r e i n t e r a c t i o n are 11 [M(x,y,z)]<{ v I + [K(x,y,z)]i v )• = F(t) w w (2.16) V cXx.y.z) = 0 (2.17) F(t) is given by the pressure distribution of the f l u i d on the structure as F ( t ) = p(x,y,z,t)dS p(x,y,z,t) = - p —<^(x,y,z) 3t F(t) = - p —<Kx,y,z)dS ,3t (2.18) When there i s no f l u i d , there i s no pressure on the structure from the f l u i d , and F(t) becomes zero. Equation 2.16 s i m p l i f i e s to the standard equation of motion f o r a structure. 2.2 FINITE ELEMENT FORMULATION I t i s possible to solve numerically using f i n i t e elements f o r the v e l o c i t y p o t e n t i a l i n the f l u i d and then use t h i s information to determine the force that i s exerted on the structure from the f l u i d , thus s o l v i n g the coupled f l u i d - s t r u c t u r e i n t e r a c t i o n problem. The theory w i l l be presented below [4,22,36,37,48]. The v a r i a t i o n a l p r i n c i p l e that governs t h i s problem 12 i s given by S ' 1 - * d<f> 2 + * 2 + d<t> 2 2 py. dz < J dxdydz d± an <j> dS - 0 (2.19) n where <j> i s the v e l o c i t y p o t e n t i a l , dS the in t e r f a c e surface area, n and v the v e l o c i t y normal to S . The v e l o c i t y p o t e n t i a l i s zero n n at the boundaries and on the free surface. (This f u n c t i o n a l i s shown to be the correc t one for t h i s problem i n Appendix A.) Proceeding with the f i n i t e element i d e a l i z a t i o n of the surrounding f l u i d , the f l u i d domain i s divided into three-dimensional f i n i t e elements. The v e l o c i t y p o t e n t i a l <f> e w i t h i n each element i s expressed i n terms of the nodal values as NNE * - I e i i i - 1 (2.22) where NNE i s the number of f l u i d nodes i n an element, the N are i the shape functions, and <f>^ i s the v e l o c i t y p o t e n t i a l at node i (which i s to be determined). The f i r s t term of equation 2.19 f o r one element may be expressed as 1 2 T ax "i ax d<f> < — — L 1 « 84> r a y 1 a y d<f> d<j> dz e dz • dxdydz (2.23) Using equation 2.22, an expression can be determined 13 B<f> ax d<j> a y 8<j> dz y = [J] aN i 3N 2 aN 3$ a? aN i aN 2 3N Br, ar? ar? 3N l 3N 2 3N ar ar ar NNE NNE NNE NNE - [B]{*> (2.24) e where [J] is the Jacobian Matrix and (f r? r) a r e the local curvilinear coordinates of the element. The volume dxdydz of one element can be expressed as dxdydz = |j|dc;dr?dr (2.25) After combining equations 2.23, 2.24, and 2.25, the first term of the functional becomes J I" [ F " [B]T[B]|j|dc;dr?dr J - i J - i J - i 2 14) (2.26) (2.27) 8{<t>) Or, setting [H] - f f f - [B]T[B] |j|d£d.,dr ° J - i J - i J - i 2 and simplifying, the first term of equation 2.19 becomes 6{4>)T[n] (</>) (2.28) e e e Integrating over the whole fluid domain, which has been divided into NFE fluid elements, the first term of the functional becomes NFE dxdydz - I 5{<4}T[H] [<f>) ' 1 o -3<i> 2 + 3<j> 2 + -3<t> 2 z ax By. 3z NFE = 1 i t ) [H]{tf] (2.29) 14 The second term of the variational principle remains to be calculated. The boundary conditions that exist for each fluid element that comes in contact with the structure's surface must be introduced, namely 8n The second term of equation 2.19 then becomes = v = u n n (2.10) an <j> dS •= 6 v <f> dS (2.30) At the interface, the velocity potential <j> and the normal velocity v can be expressed in terms of their nodal n values, using the relations NI <f> - I (2.31) i - l NI v - I N v (2.32) n i n 1 = 1 i where the are the shape functions of the interface elements, 4>^ is the velocity potential of the fluid, v is the normal velocity n i of the structure at node i , and NI is the number of nodes in an interface element. The shape functions used for the interface elements are the two-dimensional projections of those used for the ful l three-dimensional velocity potential field, with the local curvilinear coordinate f set equal to one. (See Appendix B.) Thus, equation 2.30 for one interface element can be expressed as 15 8{<t>) N NI KN N • • -N }dS 1 2 NI n {V } n e (2.33) Further, dS can be expressed in terms of the curvilinear n coordinates (i,r,) as dS = cos(n,x)dydz + cos(n,y)dxdz + cos(n,z)dxdy (2.34) = cos(n,x) | Jjdfdr7 + cos(n.y) | J 2| d£dr7 + cos (n, z) | J 31 dfdr? with r I = det l 1 | J J - det |J | = det dy/dS dz/dr) dy/dri dz/dr, dz/di dx/dr, dz/dr, dx/dr, dx/d£ dy/dr, dx/dr, dy/dr, Combining equations 2.33, 2.34, and 2.35 gives N S(4>) .1 -1 -1 N N •{N N • • -N }dS 1 2 NI n where [F] -.1 • i - l N I NI KN N • • -N }dS 1 2 NI n (2.35) {v } = 5{^}T[F] {v } (2.36) n e e e n e (2.37) It now remains to express the normal velocity in terms of the displacements of the structure. This can be done by 16 the transformation {v } - [T] {u} n e e e (2.38) where [T] = [T ] 1 e [T a]. NI e (2.39) In equation 2.39, [T^ ] contains the direction cosines for node i , such.that [T ] = [cos(n.x) cos(n.y) cos(n.z) ] i e i i i Substituting equation 2.38 into 2.36 yields 6{<f>)T[f] {v } - 6{<f>)T[F] [T] {u} = 5{^}T[F] {u} e e n e e e e e e e < with [F] = [F] [T] e e e (2.40) (2.41) (2.42) After integrating equation 2.41 over the interface surface, the second term of the variational principle becomes N IE ^ dS = 1 5{<f>}T[F] {u) - SU}T[F]{u} „ n e e e I dn e = l (2.43) where NIE is the total number of interface elements, {<f>^) the velocity potential of the interface nodes, {u} the velocity of the interface nodes in terms of the global coordinates, and [F] the assembled interface matrix. Combining equations 2.29 and 2.43 gives 5{^}T[H]{^} - 5{^}T[F]{u} =0 (2.44) and partitioning the matrices [H] and [F] with respect to the 17 i n t e r f a c e terms y i e l d s T r H H H R V I IR I I SitJ [F]{u] (2.45) A f t e r s i m p l i f i c a t i o n , equation 2.45 becomes H H H R V I IR I I 0 [F](u) (2.46) The above equation describes the coupled f l u i d - s t r u c t u r e i n t e r a c t i o n problem i n terms of the nodal v e l o c i t y p o t e n t i a l of the f l u i d [<f> ) , {<f> } and the nodal v e l o c i t i e s of the R X structure {u}. [H] has units of length (L), while [F] has those of 2 3 3 -1 area (L ) ; so, equation 2.46 has units of length/time (L T ). Equation 2.46 can be further reduced to (tfj) - [H*] _ 1[F]{i) with [H ] being the reduced f l u i d matrix given by [H*] - [H ] - [H ][H ] - 1[H ] 1 I I J IR RR L R I J (2.47) (2.48) The next step i s to solve f o r the pressure on the f l u i d - s t r u c t u r e i n t e r f a c e . Since the v e l o c i t y p o t e n t i a l } i s known at t h i s i n t e r f a c e , the pressure may be determined from (2.4) i i 8<t> {p} = - p -E at (p) _ . p 21 _ . p [H*] _ 1[F]{u} at (2.49) where {p} i s the vector containing the int e r f a c e pressures, p the f l u i d density, and {u} the vector containing the accelerations of the i n t e r f a c e nodes of the structure. 18 The equation of motion for the structure is [M ]{u} + [C]{u} + [K]{u} = {F } + {R } (2.50) s s f where [Mg] is the structural mass matrix of the dry structure, [C] the dry damping matrix of the structure, [K] the stiffness matrix of the dry ship, {F } the vector of external excitation, and {R } s f the vector of hydrodynamic nodal forces acting on the fluid-structure interface. The hydrodynamic nodal forces (Rf) may be determined from virtual work arguments. The virtual displacement of an element in the structure is given as S{u } - [N] 5{U } (2.51) where the subscript n refers to the normal of the variable, and N are the shape functions as before. The pressure distribution within each element is given as {p*} - [N]T{p} (2.52) e e e The work done by the pressure on the interface due to this virtual displacement is SW {p } S{u } dS (2.53) Replacing {p } and 6{u } by their expressions given above yields n e SW = - S{U } (N) {N} dS (P) (2.54) The work done by the nodal forces {R } moving through a virtual fn e displacement S{U } can be expressed as 19 5W - 6{U }T{R } <2.55) e n e f n e Equating equations 2.54 and 2.55 and noting that [F] = {N} {N}T dS (2.37) an expression relating pressure and the nodal forces may be obtained, namely But {R } - [T] {R } (2.57) f e e f n e so equation 2.56 becomes {R } - - [F]T{p] (2.58) re e e Integrating over the interface area and substituting the expression for {p} = - p [H*]_1[F]{uJ (2.49) gives {Rf} - - p [F]T[H*]_1[F]{u} (2.59) Finally, substituting this expression for {Rf} into equation 2.50 yields [M ]{u> + [C]{u} + [K]{u} = {F } - p[F]T[H*]"1[F] {u} (2.60) S 6 ([MJ + [MJ){u} + [C]{u} + [K]{u] - {F } (2.61) S A s where the added mass matrix [MJ is given by [M J = p[F]T[H*]"1[F] (2.62) A The interface matrix [F] has 3*NIN columns and NIN rows, while the reduced fluid matrix [H ] is square, with NIN 20 columns and NIN rows. The multiplication expressed in equation 2.62 produces an added mass matrix [M ] that is square, with 3*NIN A columns and 3*NIN rows. Since the matrix [H ] is symmetric and fu l l , the added mass matrix is also symmetric and f u l l . These characteristics have profound effects on the numerical solution of equation 2.61. The structural mass matrix [Mg] is well behaved in that i t is symmetric and banded, with the bandwidth being small in comparison to the total size of the matrix. However, the added mass matrix is not banded. Therefore, when the added mass matrix is added to the structural mass matrix, the banding is lost and the solution time greatly increased. There are two major assumptions in the above derivation of the coupled fluid-structure problem that greatly simplify the equations : 1) the fluid is incompressible, and 2) there are no waves generated on the free surface. If the fluid is compressible, then equation 2.8 cannot be simplified as before, and i t remains V2(j> + - '<j> = 0 (2.8) K Futhermore, i f surface waves are admitted, then equation 2.13 also cannot be simplified, and i t remains * = £ ^  (2.13) u> ay These two assumptions lead to the following complications of the equations of motion for the fluid. The 21 0 • \ [F]{u}/ simplified equation [H]{^ } -becomes [H]{^ } + [E]{*} (2.46) 0 [F]{u} (2.63) where [H] and [F] are as before and [E] g {N} (N}Tdr + — e e 2 C {N} {N} dO e e (2.64) where T is the free surface and fi is the whole fluid domain. The s f i r s t term of equation 2.64 accounts for surface wave considerations. The second term allows the fluid to be compressible. The inclusion of the terms in equation 2.64 introduces the dependence of the fluid solution on the frequency of vibration of the structure. 2.3 INFINITE FLUID ELEMENT FORMULATION The motivation for using infinite fluid elements is that the fluid domain in the fluid-structure dynamic interaction problem is often very large in comparison to the structure's domain. When conventional fluid elements (discussed above) are used, the fluid mesh is extended to some large distance from the body, and the boundary condition of <f> » 0 is imposed. The 22 disadvantages of this extension are two-fold. First, there-is a large increase in the size of the problem which increases CPU time. Second, i f the boundary is not far enough away, the results will be in error. A large number of nodes will be needed in modelling the region where the change in <f> is small. The infinite element can deal with this situation very well. This element extends to infinity in one direction, permitting this situation to be modelled with one infinite element. The infinite elements are very similar to the fluid elements described above, but they have two main differences. In one of the local curvilinear directions £, the shape functions have a singularity at £ — +1. Also, the velocity potential tf> is only interpolated along the finite boundary. In the infinite direction, the velocity potential <f> is assumed to decay from its value at the finite boundary to zero at the infinite boundary. Within the fluid element, the velocity potential <f> may be e expressed in terms of the nodal values, such that NNE 4 - 1 M>. (2.65) e i i i = 1 where 4 a r e the nodal values of the velocity potential and are the modified shape functions given as - Hf(q) (2.66) Here, are the shape functions, which interpolate along the finite boundary, and f(q) is the decay function, which may be of the form 23 f(q) = e Q" q f(q) - (2.67) Both of these functions tend toward zero as q -» ». The choice of which f ( q ) to use, decay length Q, and power n ( i f applicable) determines how quickly the function decays. From t h i s point on, there i s no d i f f e r e n c e between t h i s formulation and the one presented above ( i n Section 2.2). 24 3. NUMERICAL EVALUATION OF FLUID ELEMENT MODELLING CAPABILITIES This chapter w i l l describe the numerical work that was performed i n evaluating the f i n i t e element f l u i d modelling c a p a b i l i t i e s of the F i n i t e Element Program VAST. Three d i f f e r e n t f l u i d elements were investigated : the 8-noded f l u i d element, the 20-noded f l u i d element, and the 8-noded i n f i n i t e f l u i d element. (Detai l s of these elements are given i n Appendix B.) The evaluation of the f l u i d elements was conducted by comparing the numerical r e s u l t s obtained f o r d i f f e r e n t f l u i d domains, meshes, and elements with experimental r e s u l t s found i n the l i t e r a t u r e . In p a r t i c u l a r , t h i s work concentrated on modelling the experiments of a c a n t i l e v e r plate v i b r a t i n g i n a f l u i d performed by Lindholm, Kana, Chu, and Abramson [29] i n 1965. Their experiments were performed i n a 6' x 12' x 8' deep water tank, with the rectangular plate clamped to a r i g i d I-beam support structure. In these t e s t s , the cord length b of a l l plates was 8 inches, and various aspect r a t i o s and thicknesses were examined. (Figure 3.1 shows d e t a i l s of the structure used.) 25 -t Cord Length b - 8" Aspect Ratio a/b =0.5, 1.0, 2.0, Thickness t = 0.105" Figure 3.1 - Plate Structure 26 Plates with a thickness of t = 0.105 inches and aspect r a t i o ' s of a/b =0.5, 1.0, 2.0, and 3.0 were considered i n the numerical study d e t a i l e d here. The c a n t i l e v e r p l a t e was modelled using the Thick/Thin S h e l l element from the F i n i t e Element Program VAST. This element i s an isoparametric element with quadratic shape functions (completely defined i n Appendix C). I t has 8 nodes i n t o t a l , 4 corner nodes, and 4 mid-side nodes, with 5 l o c a l degrees of freedom per node, 3 t r a n s l a t i o n s , and 2 r o t a t i o n s . In these experiments, the plates were made of 1018 c o l d - r o l l e d s t e e l , and the f l u i d used was water. The material properties of s t e e l f o r the numerical work were taken as : Young's modulus E = 3.0xl0 ? p s i , Poisson's r a t i o v = 0.3, and mass density -A 2 4 / j g = 7.324x10 lb.sec / i n . The mass density of water was taken as p = 9.38xl0~ 5 l b . s e c 2 / i n * . 3.1 VIBRATION IN AIR Before attempting to model the plate's v i b r a t i o n i n f l u i d , v i b r a t i o n c h a r a c t e r i s t i c s i n a i r were examined i n order to 27 reduce the error involved i n the f l u i d analysis introduced by the d i s c r e t i z a t i o n of the plate. To compare the numerical and experimental r e s u l t s f o r the pl a t e v i b r a t i n g i n a i r , three d i f f e r e n t f i n i t e element meshes were considered f o r each of the four aspect r a t i o s (as shown i n Figure 3.2). The f i r s t mesh used four Thick/Thin S h e l l elements to model the pl a t e . The second mesh used sixteen elements, while the t h i r d mesh used thirty-two. (The r e s u l t s of these tests are summarized i n Table 3.1 - 3.4, and the modes of v i b r a t i o n are shown i n Figure 3.3). The fourth mode was a bending mode i n the cord d i r e c t i o n (as shown i n Figure 3.3). As the aspect r a t i o of the pla t e was increased, t h i s mode disappeared, and the 2-node c a n t i l e v e r mode comes into play, which has been denoted as mode 6. Thus, f o r an aspect r a t i o of a/b -=0.5 or 1.0, there were no r e s u l t s f o r mode 6, while f o r an aspect r a t i o of a/b = 2.0 or 3.0, there were no r e s u l t s f o r mode 4. 28 c) 32 Element Mesh Figure 3.2 - Finite Element Plate Meshes 29 a) Mode 1 - 1 Cantilever b) Mode 2 - 1 Torsion c) Mode 3 - 2n Cantilever d) Mode 4 - Bending b direction d) Mode 5 - 2 Cant./I Tor. e) Mode 6 - 3 Cantilever Figure 3.3 - First Six Mode Shapes of Plate Vibration 30 Mode # 1 2 3 4 5 6 Plate Theory 223 342 652 1397 1580 -CPU sec Experimental 214 339 649 1339 1518 -4-Element Mesh 224 349 727 1490 1664 - 56 16-Element Mesh 224 342 651 1401 1588 - 279 32-Element Mesh 224 341 648 1394 1573 - 1665 Table 3.1 - Plate Theory and F i n i t e Element Results a/b =0.5 b = 8.0 inches t = 0.105 inches Mode # 1 2 3 4 5 6 Plate Theory 55.6 136 341 437 496 -CPU sec Experimental 52.9 129 326 423 476 -4-Element Mesh 55.8 137 363 473 531 - 60 16-Element Mesh 55.7 136 342 440 497 - 306 32-Element Mesh 55.6 136 341 435 495 - 1954 Table 3.2 - Plate Theory and F i n i t e Element Results a/b =1.0 b = 8.0 inches t = 0.105 inches 31 Mode # 1 2 3 4 5 6 Plate Theory 13.8 59.3 85.9 - 194 234 CPU sec Experimental 12.9 58.2 80.8 - 189 228 4-Element Mesh 13.9 59.4 91.8 - 204 370 72 16-Element Mesh 13.8 59.2 86.4 - 194 250 449 32-Element Mesh 13.8 59.1 85.9 - 193 242 1369 Table 3.3 - Plate Theory and F i n i t e Element Results a/b =2.0 b = 8.0 inches t = 0.105 inches Mode # 1 2 3 4 5 6 Plate Theory 6.1 38.0 38.1 - 120 104 CPU sec Experimental 6.2 40.3 38.7 - 126 109 4-Element Mesh 6.2 40.7 37.7 - 171 124 97 16-Element Mesh 6.1 37.5 38.1 - 118 108 690 32-Element Mesh 6.1 37.5 38.1 - 118 108 1678 Table 3.4 - Plate Theory and F i n i t e Element Results a/b =3.0 b = 8.0 inches t = 0.105 inches The four element mesh had 105 degrees of freedom, the sixteen element mesh 325 degrees of freedom, and the thirty-two element mesh 665 degrees of freedom. The frequencies predicted by the sixteen element mesh were better than those of the four element mesh; and, i n turn, the r e s u l t s of the thirty-two 32 element mesh were better than those of the sixteen element - mesh (as displayed i n Tables 3.1 - 3.4). The frequencies predicted f o r the i n d i v i d u a l modes of v i b r a t i o n were also a f f e c t e d by the degree of i d e a l i z a t i o n i n the f i n i t e element model. (Figure 3.3 shows the mode shapes f o r the f i r s t s i x modes of v i b r a t i o n that the plate experienced.) As the mode number increased so d i d the complexity of the mode shape, and more degrees of freedom were required to model i t . Therefore, i n any given f i n i t e element mesh, the error i n the frequency predicted f o r a mode increased as the complexity of the modes increased. Thus, i n the numerical r e s u l t s as (displayed i n Tables 3.1 - 3.4), the four element mesh was able to p r e d i c t the frequencies of the f i r s t and second modes very well, but i t was not as accurate with higher modes. The sixteen element mesh was able to model the f i r s t f i v e or s i x modes, and the thirty-two element mesh was able to model further s t i l l but showed l i t t l e improvement i n the frequencies predicted f o r the f i r s t f i v e modes. The frequencies predicted f o r the i n d i v i d u a l modes of v i b r a t i o n were also a f f e c t e d by the aspect r a t i o of the p l a t e . (This e f f e c t can be seen i n Tables 3.1 - 3.4.) At the lower aspect r a t i o s , a/b = 0.5 and 1.0, the sixteen element mesh predicted the t h e o r e t i c a l r e s u l t s exactly, but at aspect r a t i o s of 33 2.0 or 3.0, some erro r crept i n . As the aspect r a t i o increased, the length of the element i n the span d i r e c t i o n a was af f e c t e d . As a r e s u l t , the element's a b i l i t y to model the mode shape of the higher modes was hampered, as the shape functions f o r t h i s element only modelled quadratic displacements. (See Appendix C.) The f i n i t e element and the t h e o r e t i c a l r e s u l t s were very close, but the numerical r e s u l t s were higher i n most cases than the experimental r e s u l t s by about f i v e percent. This d i f f e r e n c e was a t t r i b u t e d to two factors : 1) the f i n i t e element r e s u l t s d i d not account f o r the added mass of the a i r around the pla t e , which lowered the frequencies of v i b r a t i o n ; and 2) i f the c a n t i l e v e r support was not absolutely r i g i d , t h i s f l e x i b i l i t y also could have l e d to a decrease i n the natural frequencies f or the experimentally measured values. The r e s u l t s f o r the sixteen element mesh were not very d i f f e r e n t from those of the thirty-two element mesh. However, there was a large difference i n the CPU time, with the thirty-two element mesh taking four to s i x times longer than the sixteen element mesh, depending on which aspect r a t i o was considered. For t h i s reason, the sixteen element mesh was used for the f i n i t e element runs of the c a n t i l e v e r plate v i b r a t i n g i n the f l u i d . 34 3.2 VIBRATION IN FLUID Four plates with d i f f e r e n t aspect r a t i o s were modelled numerically to determine t h e i r v i b r a t i o n c h a r a c t e r i s t i c s i n a f l u i d . The plate structure was modelled using the sixteen element Thick/Thin S h e l l mesh, which had been found to be s a t i s f a c t o r y i n determining the lowest f i v e modes of v i b r a t i o n f or the a i r experiments. The f l u i d was modelled using the d i f f e r e n t f l u i d elements a v a i l a b l e i n VAST. These numerical r e s u l t s were compared with the experimental r e s u l t s of Lindholm et a l [29]. The numerical work conducted i n t h i s phase of the i n v e s t i g a t i o n was d i r e c t e d at determining the e f f e c t s of various f l u i d modelling parameters. These were : the extent of the f l u i d domain that needed to be d i s c r e t i z e d to give s a t i s f a c t o r y r e s u l t s , the number of f l u i d elements used i n the domain, and the type of f l u i d elements used to model the domain. Other factors that a f f e c t e d the comparison between the experimental and the numerical r e s u l t s were : the c o m p r e s s i b i l i t y of the f l u i d , the extent of the surface waves generated, and the amount of vortex shedding o f f the edges of the 35 p l a t e . These e f f e c t s could not be modelled with the present development of the f l u i d equations i n VAST. 3.2.1. E f f e c t of the Extent of the F l u i d Domain In the experimental set up, the plates were immersed i n a 6' x 12' x 8' deep water tank p a r a l l e l to the free surface. The depth of immersion for the experimental deep water r e s u l t s was not given. However, the experimental r e s u l t s showed that i f the plate was submerged a distance greater than one h a l f the span length, the r e s u l t i n g frequencies were independent of the submerged depth. The extent of the f l u i d domain was very important to the numerical r e s u l t s obtained for the v e l o c i t y p o t e n t i a l 4>. I f the f i n i t e element f l u i d domain d i d not extend f a r enough from the structure, the boundary condition of <f> = 0 was an a r t i f i c a l c onstraint on the system. This af f e c t e d the determination of the v e l o c i t y p o t e n t i a l <f> i n the i n t e r i o r of the f l u i d domain, causing the frequency to be higher than i t was normally. The range of f l u i d domains analysed were D/b - 0.125, 0.25, 0.5, 0.75, 1.0, 1.5, and 2.0. (Figure 3.4 shows three views of the three meshes used : a plan, an elevation, 36 and a r i g h t - s i d e view.) D r e f e r s to the dimension of the f l u i d domain extending out from the plate i n a l l d i r e c t i o n s , while b was the cord length of the f i x e d side of the c a n t i l e v e r p l a t e . T y p i c a l r e s u l t s f or the 8-noded f l u i d element (using three d i f f e r e n t f l u i d meshes) v i b r a t i n g i n i t s f i r s t mode of v i b r a t i o n show that as the domain of the f l u i d was enlarged, the frequencies of v i b r a t i o n converged, provided enough elements were used i n the f l u i d domain; however, the frequencies of v i b r a t i o n d i d not always converge to the experimental r e s u l t s . (See Figure 3.5). The convergence to r e s u l t s other than the experimental r e s u l t s w i l l be explained below ( i n Section 3.2.3.1). This convergence c l e a r l y displayed the adverse e f f e c t on the frequencies of having the f l u i d domain boundaries too close to the structure. In some cases, the r e s u l t s began to diverge as the domain was increased, as i s i l l u s t r a t e d by meshes that only used one element i n the f l u i d domain. This divergence was due i n part to the l i m i t e d a b i l i t y of the shape functions to model the changes i n the v e l o c i t y p o t e n t i a l <f> over one element. 37 Figure 3.4a - Finite Element Fluid Domain Meshes NFLD - 1 Mesh 38 •••• mm • • • • • • • • • • • • U l 1 • " Q J 1 • 1 • 1 1 J II 1 • • • • 4 b • 1 I I p V <*— D • < B — • ••••go • • • • • • N F L D = 2 Plate Figure 3.4b - Finite Element Fluid Domain Meshes NFLD = 2 Mesh 39 • • • • • • • • • • • • c n c n in O • • • • • • • • mm* • • • • • • • • • • • • • • • • • O • • • • • • • • • • -W4-• • D O n U D D D U Q / A • • • • • • • • O Q / r i n Q CZJ • • • • • O • Q / Q • • 1—1 c n D C 3 a • c n / a in c n ccn ^ r 1—1 c n • • D n n n n D o n c n • • • • • D a n c z j • • • D a n n n D D D n • • D D D D D D D D D D N F L D Plate • • • • • • • • • ••DDDDn • • D D D D D D Figure 3.4c - Finite Element Fluid Domain Meshes NFLD - 4 Mesh 40 0 0.5 1 1.5 2 D/b Ratio Legend A NFLD = 1 x NFLD - 2 • NFLD = 4 Figure 3.5 - Error Between F i n i t e Element and Experimental Results Using the 8-Noded F l u i d Element fo r Mode 1 with Plate Aspect Ratio a/b -1.0 41 3.2.2. E f f e c t of the Number of Elements i n the F l u i d Domain The number of f l u i d elements i n the domain also a f f e c t e d the numerical r e s u l t s . As the number of elements i n the f l u i d domain was increased, the i d e a l i z a t i o n error was reduced and the numerical r e s u l t s began to approach the exact s o l u t i o n of Laplace's equation. This phenomenon was s i m i l a r to the s t r u c t u r a l d i s c r e t i z a t i o n discussed above ( i n Section 3.1). Within each of the seven d i f f e r e n t f l u i d domains considered, three f l u i d meshes were examined. These meshes were denoted by NFLD = 1, NFLD = 2, and NFLD = 4, where NFLD was the number of f l u i d elements across the dimension D. (Figure 3.4 presents these meshes.) The f l u i d mesh NFLD = 1 had 60 elements, while NFLD = 2 had 96 elements, and NFLD = 4 had 192 elements. I t should be noted that over the pl a t e , the f l u i d mesh and the plate mesh corresponded. The elements i n the f l u i d domain were not spaced evenly but were cl o s e r together near the edges of the plat e , as t h i s was where the v e l o c i t y p o t e n t i a l <f> was changing most r a p i d l y . The r e s u l t s of the second mode of v i b r a t i o n of the pla t e (with an aspect r a t i o a/b =1.0) using the three d i f f e r e n t 42 f l u i d meshes with both 8 and 20-noded f l u i d elements 'were examined. (See Figure 3.6). Within a f l u i d mesh, the CPU times f o r each D/b r a t i o were very s i m i l a r (thus, only the averaged values are presented i n Table 3.5). Type of F l u i d Element F l u i d Mesh 8-Noded 20-Noded NFLD = 1 590 622 NFLD -= 2 614 1391 NFLD = 4 861 9952 Table 3.5 - Average CPU Time (sec) Various F l u i d Meshes f o r Plate Aspect Ratio a/b - 1.0 b = 8.0 inches t = 0.105 inches From these r e s u l t s , i t became apparent that the NFLD = 4 mesh predicted frequencies that were c l o s e s t to the experimental r e s u l t s . But the r e s u l t s of the two element f l u i d mesh NFLD = 2 were also quite good and took le s s CPU time, e s p e c i a l l y f o r the 20-noded f l u i d element meshes. At a s p e c i f i c D/b r a t i o , the frequencies predicted by the NFLD = 1 mesh were higher than the r e s u l t s of the NFLD = 2 mesh, which i n turn were higher than the NFLD = 4 mesh (as seen i n Figure 3.6). The NFLD = 1 mesh was underestimating the added mass 43 80 60 111 20-0 -20 y i y « 0 0.5 •) 8-Noded Results 80 1 1.6 D / b R a t i o 60 O 40 t_ UJ 20 H -20 0 0.5 b) 20-Noded Results 1 1.5 D / b R a t i o 2.6 5 „—i V- s A - ^ " ^ « \ * i -^=! a • • - - - - - r E 2.5 Figure 3.6-8 and 20-Noded Fluid Results for Mode 2 with Plate Aspect Ratio a/b -1.0 Legend A NFLD - 1 x NFLD • 2 D NFLD-4 44 matrix the most, while the NFLD = 4 mesh came c l o s e s t to p r e d i c t i n g the experimental r e s u l t s . I t should be noted that the r e s u l t s of the NFLD = 2 mesh f o r the 8-noded f l u i d element and the NFLD = 1 mesh of the 20-noded element took about the same CPU time (see Table 3.5), but the 20-noded r e s u l t s were s l i g h t l y better (as shown i n Figure 3.7). This f a c t i s i n t e r e s t i n g since both meshes had almost the same number of degrees of freedom; the 8-noded mesh had 105 degrees of freedom, while the 20-noded mesh had 109. At higher values of D/b the r e s u l t s of the 8-noded mesh became better than the 20-noded. mesh (see Figure 3.7), due to the high aspect r a t i o that the one 20-noded element was experiencing i n t h i s mesh. As w e l l , the r e s u l t s of the NFLD — 4 mesh for the 8-noded element and the NFLD = 2 mesh f o r the 20-noded element (as displayed i n Figure 3.8) took about the same CPU time; but, the r e s u l t s of the 8-noded mesh were worse, even though i t had more degrees of freedom. The 8-noded mesh had 539 degrees of freedom, while the 20-noded mesh had 488 degrees of freedom. The accuracy here stemmed from the f a c t that the 20-noded f l u i d elements had bett e r shape functions than those of the 8-noded elements. The shape functions f o r the f l u i d elements are discussed below (and given i n d e t a i l i n Appendix B). 45 60-1 UJ 4 0 -2 0 -- 2 0 -i d) M o d e 4 i 2 D/b Ratio Legend A HFLD-2 8-ModXl x MFLD-1 20-Mod»d Comparison of Two Fluid Meshes with Plate Aspect Ratio a/b -1.0 NFLD = 2 with 8-Node Fluid Element NFLD = 1 with 20-Noded Fluid Element 46 60 40 Ul -20 a) Mode 1 60 \ i 0 1 2 3  D/b Ratio U J 40-20 -20 1 2 D/b Ratio !r A " 60 40 U l -20 * — 0 1 2 9 2 D/b Ratio U l -20 d) Mode 4 D/b Ratio Legend A HFLD-4 6-Mo<)«d x NFLO-2 20-Nod*d D/b Ratio Figure 3.8 - Comparison of Two Fluid Meshes with Plate Aspect Ratio a/b -1.0 NFLD - 4 with 8-Noded Fluid Element NFLD - 2 with 20-Noded Fluid Element 47 3.2.3. E f f e c t of D i f f e r e n t F l u i d Elements Three types of f l u i d elements were used to construct the various f l u i d meshes : an 8-noded l i n e a r element mesh, a 20-noded quadratic isoparametric element mesh, and a mesh containing a combination of the 8-noded element and one row of the 8-noded i n f i n i t e element. The 8-noded i n f i n i t e element had the same shape functions as the normal 8-noded f l u i d element i n two of i t s d i r e c t i o n s ; however, the t h i r d d i r e c t i o n was the i n f i n i t e d i r e c t i o n , i n which the v e l o c i t y p o t e n t i a l had e i t h e r an exponential decay or an ( l / r ) n decay. (For d e t a i l s of these elements see Appendix B.) The meshes that contained the i n f i n i t e element were d i f f e r e n t from the others, as the i n f i n i t e element could be only on the f l u i d domain boundary. When the i n f i n i t e element was used, i t replaced e i t h e r the l a s t row, the l a s t two rows, or the l a s t three rows of the conventional f l u i d elements, depending on which mesh was used. 3.2.3.1. 8 and 20-Noded F l u i d Elements The r e s u l t s of the 8-noded f l u i d element and the 20-noded f l u i d element for the f i r s t f i v e modes of v i b r a t i o n with an aspect r a t i o a/b = 1.0 were examined (and are presented i n 48 Figures 3.6 and 3.9 - 3.12). In general, the 20-noded r e s u l t s p r e d i c t e d frequencies that were closer to the experimental r e s u l t s than the 8-noded r e s u l t s . This f a c t was expected since the 20-noded f l u i d element had quadratic shape functions and could represent a quadratic change i n i t s v e l o c i t y p o t e n t i a l i n any l o c a l element d i r e c t i o n , while the 8-noded element only had l i n e a r shape functions and could only represent a l i n e a r change. The r e s u l t s f o r the f i r s t mode of v i b r a t i o n with the four d i f f e r e n t p late aspect r a t i o s using the 8 and 20-noded f l u i d elements were examined. (See Figures 3.13 and 3.14.) I t was seen that as the aspect r a t i o increased, the f i n i t e element r e s u l t s began to deteriorate. This d e t e r i o r a t i o n was explained by the assumptions that were made i n the formulation of the v e l o c i t y p o t e n t i a l <f>. The flow was assumed to be i n v i s c i d , i r r o t a t i o n a l , and incompressible; however, the r e a l flow had these e f f e c t s . The d e t e r i o r a t i o n of the r e s u l t s was due to vortex shedding o f f the side of the pla t e , and i t was the viscous e f f e c t s i n the f l u i d that caused the v o r t i c e s . As the aspect r a t i o of the plate increased, the v o r t i c e s that were shed from the sides of the plate began to dominate the flow i n the area around the edges of the pl a t e . Since the f i n i t e element formulation was unable to deal with the complex flow that was produced by the v o r t i c e s , the r e s u l t s were poor. 49 a) 8 - N o d e d Resu l ts 80 1 1.5 D / b R a t i o 6 0 H O 40 i» k. U J _5 20 H -20 0 0.5 b) 20 -Noded Resu l ts 1 1.6 D / b R a t i o Figure 3 . 9 - 8 and 20-Noded F l u i d Results f o r Mode 1 with Plate Aspect Ratio a/b — 1.0 2.5 5 \ t 3 2.6 Legend A NFLD - 1 x NFLD - 8 • NFLD - 4 50 8 0 6 0 -O 4 0 UJ - 2 0 - — HL " *£7r.-r.&~T--l a- « 3 1 1.6 D/b Ratio 0 0.6 b) 20-Noded Reeultt 1 1.6 D/b Ratio Legend NFLD -1 NFLD - 2 NFLD - 4 Figure 3.10 - 8 and 20-Noded F l u i d Results f o r Mode 3 with Plate Aspect Ratio a/b -1.0 51 80 60-UJ 20-0 •20 ______ - — ' v V ^ JEk zSZt fc *•* "* % 0 0.6 i) 8-Noded Results 80 60 O 40 i _ t_ LU 20 -20 0 0.6 b) 20-Noded Results 1 1.6 D / b R a t i o 1 1.6 D / b R a t i o 2.6 2.6 Legend A NFLD - 1 x NFLD-8 • NFLD - 4 Figure 3.11 - 8 and 20-Noded Fluid Results for Mode 4 with Plate Aspect Ratio a/b =1.0 52 80 60-1LI 20H 0 -20 - ' H A > a- * D 3 - - - H 3 < 0 0.5 a) 8 -Noded Resul ts 80 60 O 40 k_ %-UJ -20 0 0.6 b) 20 -Noded Resul ts 1 1.6 D / b R a t i o 1 1.6 D / b R a t i o i 2 2.6 \ \ * — " 1 2.5 Legend A N F L D - 1 x N F L D - I • NFLD - 4 Figure 3.12 - 8 and 20-Noded F l u i d Results f o r Mode 5 with Plate Aspect Ratio a/b = 1.0 53 U l 60 40-20-•20 \ H * I — » X z — * — * D/b Ratio •) 1st Cantilever 60 ui 40 20--20 0 1 2 D/b Ratio o) 2nd Cantilever 60 \N-c 3 B---Q * 4 B B- - « U l -20 V ^ B - H — B — • 1 2 D/b Ratio D/b Ratio d) Coupled 2nd Cantilever Toralon Legend A m/b - 0.6 x m/b - to • e/b - 2.0 a a/b - S.O Figure 3.13 - 8-Noded F l u i d Results f o r Various Plate Aspect Ratios with NFLD •= 4 54 60 Ui -60-! iii A t I 3 3 D/b Ratio a) 1st Cantilever 60-1 ui -60 0 1 2 D/b Ratio c) 2nd Cantilever J Lf 1LL) Ul D/b Ratio b) 1st Torsion 60 Ul -60 B "8-B B- —8 I ' I 1 2 D/b Ratio d) Coupled 2nd Cantilever Torsion Legend e> a/b - 0.6 x a/b - 1.0 D a/b - 2.0 B a/b - 9.0 Figure 3.14 - 20-Noded Fluid Results for Various Aspect Ratios with NFLD - 4 55 To explain t h i s problem, i t was informative to examine the energy i n t h i s system. The t o t a l energy i n the f l u i d E t was constant and was produced by the v i b r a t i o n of the structure. I f the flow could be considered to be made up of two types of flow--a l o c a l flow given by the v o r t i c e s and a global flow--then, the energy i n the flow could be expressed as E - E R + E R (3.1) T V G where E^ was the t o t a l energy i n the flow, E^ was the energy i n •ft the r e a l vortex flow, and E was the energy i n the r e a l global G flow. Since the t o t a l energy i n the flow E t was constant, the presence of v o r t i c e s consumed some of the energy i n the flow, causing the energy i n the global f l u i d flow to decrease. The f l u i d elements were unable to model the e f f e c t of the v o r t i c e s ; they were able to model only the global flow i n the f l u i d . Therefore, the f i n i t e elements estimated only the energy i n the global flow given by E F E = E F E (3.2) T G since the source of energy f o r the f l u i d flow came from the s t r u c t u r a l v i b r a t i o n s i n mode i . The energy i n that mode was given by E = ( < / { * } T[M ] { * } + {* } T [ K ] { * } ) (3.3) I i i S i i I 56 which was approximately the same for the r e a l f i n i t e element model of the structure. Therefore r-FE _R E = E T T which l e d to _ F E _R „ R E = E + E G V G _ F E _ _R E > E G G Since the f i n i t e element f l u i d elements were unable to model the vortex shedding, the energy i n the global flow was overestimated. This i n turn caused the added mass matrix [M 1 to A be overestimated; thus, a lower frequency was predicted than that determined experimentally. When vortex shedding was occurring, the 8-noded f l u i d element meshes appeared to give better p r e d i c t i o n s for the frequency of v i b r a t i o n than the 20-noded f l u i d meshes, although both were below the experimental r e s u l t s . (This e f f e c t i s displayed i n Figure 3.15.) By examining the r e s u l t s of the other te s t s (given i n Figures 3.6 and 3.9 - 3.12), i t was concluded that the 8-noded f l u i d meshes always predicted frequencies that were lar g e r than those predicted by the 20-noded meshes. This f i n d i n g i n d i c a t e d that f o r any given f l u i d mesh NFLD =1, 2, or 4, the k i n e t i c energy predicted by the 8-noded f l u i d mesh was lower than that predicted by the 20-noded f l u i d mesh, or i n equation form system and the (3.4) (3.5) (3.6) 57 0 0.5 1 1.6 D/b Ratio Legend A 8-Noded x 20-Noded Figure 3.15 - Comparison of 8 and 20-Noded Fluid Result Using NFLD - 4 and Plate Aspect Ratio a/b - 3.0 for Mode 1 58 w2 {* } T[M ] } < w2 } T[M 1 {* } i i A 8 i i i 1 AJ20 i 8 20 However, the k i n e t i c energy predicted by the 8-noded f l u i d meshes was less than that predicted by the 20-noded f l u i d meshes, causing w to be higher than to , which explained the "better" r e s u l t s f o r 8 B 20 R the 8-noded f l u i d element meshes when vortex shedding was occurring. 3.2.3.2. I n f i n i t e Element The i n f i n i t e element r e s u l t s were obtained f o r plates with an aspect r a t i o of only a/b = 1.0, and they considered only the normal 8-noded f l u i d elements when other f l u i d elements were needed to model the f l u i d . The three meshes used f o r the 8 and 20-noded f l u i d elements denoted by NFLD =1, NFLD = 2, and NFLD «= 4 were examined. In the NFLD = 1 mesh, a l l the elements were replaced by the i n f i n i t e elements. This mesh was denoted as NFLD = 1 INF = 1. In the NFLD = 2 mesh, only the l a s t row of the 8-noded f l u i d elements was replaced with the i n f i n i t e elements. This mesh was denoted as NFLD = 2 INF =1. In the NFLD - 4 mesh, three d i f f e r e n t combinations of the i n f i n i t e elements were used. The f i r s t mesh replaced only the l a s t row of the 8-noded f l u i d 59 elements with the i n f i n i t e elements, and i t was denoted as NFLD = 4 INF = 1. The second mesh replaced the l a s t two rows of the 8-noded f l u i d elements with one row of i n f i n i t e elements, and i t was denoted as NFLD - 4 INF = 2. The t h i r d mesh replaced the l a s t three rows of the 8-noded f l u i d elements with one row of i n f i n i t e elements, and i t was denoted as NFLD = 4 INF = 3. (These f i v e meshes are shown i n Figure 3.16.) NFLD r e f e r r e d to the number„ of normal f l u i d elements that were used i n these meshes. INF r e f e r r e d to the number of rows i n the NFLD mesh replaced by the i n f i n i t e element. For a l l these meshes, the i n f i n i t e d i r e c t i o n was taken normal to the p l a t e . The i n f i n i t e element d i d an excel l e n t job of modelling the boundary condition of <f> = 0, even f o r small f l u i d domains, without a f f e c t i n g the determination of the v e l o c i t y p o t e n t i a l <f> i n the i n t e r i o r domain. (See Figure 3.17.) The i n f i n i t e element r e s u l t s had lower errors than the 8-noded element r e s u l t s f o r very small f l u i d domains, such as D/b = 0.125 (as i l l u s t r a t e d i n Table 3.6). 60 Figure 3.16a - Infinite Element Fluid Domain Meshes NFLD - 1 INF - 1 Mesh 61 • • • • • • • • Q [ • • • • • • • • • • • • • • Infinite Direction • • • • g a • • • • • • Infinite Direction N F L D = 2 INF = 1 Plate • • • • L f Infinite Elenent Row Figure 3.16b - I n f i n i t e Element F l u i d Domain Meshes NFLD - 2 INF - 1 Mesh 62 t • • • • • • • • • • • • i — i c n o HD • • • • • • •0 • • • • • • • • • • • • • o a a • • • • • • • • • • • • Infinite direction • • D D D D D D D D L a UD O • D O • • • i — i c o a D c n c n c n c n 3 c n o o c n c n c n c n o i n c n i — i • ••DDDDDDDDn • •D DDDDDD D • Infinite Direction N F L D = 4 INF = 1 Plate • • • [ ] • • • • • D D D D D D Infinite Elencnt Row Figure 3 . 1 6 c - Infinite Element Fluid Domain Meshes NFLD - 4 INF •= 1 Mesh 63 t • • • • • • • • • • • • I— I c n o C3 • • • • • • • D mm* • • • • • • • • • • • • C-3 C n C n Cn • • • • • • • • • • • • — D M4 b M Infinite direction Infinite Direction • • • D • • • • Q/O • i — i c n in o c n c n c n c n / b • c n C=D c n m o c n c n c n c n o i n c n i — i Infinite Element Row Figure 3.16d - Infinite Element Fluid Domain Meshes NFLD - 4 INF - 2 Mesh 64 • • • D • • • D • • • D • • • 0 ' — i cm • • • • • D • • • 0 • • D O mm • • • • • • • • • • • • O D D L T D r_f D • • • N F L D = 4 INF = 3 • o n • • • • • • • • • Plate Infinite direction Infinite Direction a • c n • • C = K D a c n • — > \ c n i n o c n a c n c n o • O Infinite Element Row Figure 3.16e - I n f i n i t e Element F l u i d Domain Meshes NFLD = 4 INF -= 3 Mesh 65 Mode # Fluid Mesh 1 2 3 4 5 NFLD=1 46.8 29.1 43.7 33.3 30.3 NFLD=2 45.5 27.2 41.8 30.8 28.1 NFLD=4 45.1 26.6 41.1 29.9 27.3 NFLD=1 INF-1 27.1 36.9 48.8 52.7 46.2 NFLD=2 INF-1 12.1 16.5 27.2 28.8 23.9 NFLD-4 INF-1 6.5 8.3 18.9 20.4 15.6 NFLD-4 INF-2 13.0 17.6 28.3 29.6 24.9 NFLD=4 INF-3 21.4 28.6 39.9 42.1 36.7 Table 3.6 - Comparison of Error Between 8-Noded and Infinite Elements for Fluid Domain D/b = 0.125 The mesh combinations of NFLD — 4 INF = 1 and NFLD - 4 INF - 2 gave the best results. These results were comparable to those obtained with the 8-noded fluid element mesh NFLD - 4. (Figure 3.18 displays the results of these three meshes for the first five modes of vibration.) It should also be noted that the CPU times of both of the infinite element meshes were lower than that of the NFLD = 4 mesh (as displayed in Table 3.7). 66 100 0 1 2 a) M o d e 1 D/b Ratio 100 so o U l •60 c) Mode 3 100 D/b Ratio e) M o d e 5 1 2 D/b Ratio 100 w 60 o U l B " 1 1 ™T 1 • 100 :5tft#dfc! i 0 1 2 S 2 D/b Ratio i . 60 O U l * 0 •60 d) Mode 4 1 2 D/b Ratio L e g e n d A NFLD-1 INF-1 x NFLD=2 INF-1 o NFLD=4 INF=1 B NFLP-4 INF-2 B NFLD-4 INF-3 • • Figure 3.17 - Results for the Infinite Fluid Meshes for Plate Aspect Ratio a/b «= 1.0 67 NFLD = 4 INF = 1 NFLD = 4 INF - 2 NFLD = 4 349 sec 284 sec 201 sec Table 3.7 - Average CPU Times f o r F l u i d Formulation Upon examining the r e s u l t s (presented i n Figure 3.18) f o r these three meshes, i t was seen that the errors f o r the 8-noded meshes were larger than the errors i n the two i n f i n i t e element meshes displayed at the higher D/b r a t i o s . In f a c t , the i n f i n i t e element r e s u l t s were lower by a constant amount fo r the l a s t two or three D/b r a t i o s , i n d i c a t i n g that the i n f i n i t e element was s t i l l modelling the boundary condition of <f> •= 0 (on the v e l o c i t y p o t e n t i a l ) better than the 8-noded element. There was another e f f e c t displayed here too; for a l l but mode 1, the r e s u l t s of the l a s t three D/b r a t i o s have increasing e r r o r . This was due to the increase i n the aspect r a t i o of the f l u i d elements as the D/b r a t i o increased. For these meshes, the length i n one d i r e c t i o n was becoming too long, and the shape functions were experiencing trouble modelling the changes i n the v e l o c i t y p o t e n t i a l <f>. 68 60 UJ 20 -20 a) Mode 1 60 :1 i 0 1 2 8  D/b Ratio 60 HJ 20 -20 40 Ul -20H 0 c) Mode 3 60 • j 60 ' I I ' 0 1 2 6 2 D/b Ratio Ul 40 20 -20 D/b Ratio d) Mode 4 ui 40-20--20 e) Mode 6 i < i 1 2 D/b Ratio i 1 i 1 2 D/b Ratio Legend A NFLD-4 l-Nod«d x NFLO-4 INF-1 • NFLO-4 INF-2 Figure 3.18 - Comparison of the Three D i f f e r e n t F l u i d Meshes NFLD - 4 with 8-Noded F l u i d Element NFLD = 4 INF - 1 I n f i n i t e Element NFLD - 4 INF - 2 I n f i n i t e Element 69 3.3 IMPOSITION OF AN ARTIFICIAL BANDWIDTH I n t h e n o r m a l f i n i t e e l e m e n t f o r m u l a t i o n , t h e s t r u c t u r a l m a s s m a t r i x [M ] a n d s t r u c t u r a l s t i f f n e s s m a t r i x [K] s w e r e s y m m e t r i c a n d b a n d e d , w h i c h g r e a t l y i m p r o v e d t h e n u m e r i c a l e f f i c i e n c y . W h i l e t h e a d d e d m a s s m a t r i x [MJ g i v e n b y [M ] - p[F] T[H*] _ 1[F] (2.62) A w a s a s y m m e t r i c m a t r i x ; u n f o r t u n a t e l y , i t w a s a l s o f u l l y p o p u l a t e d . T h e a d d e d m a s s m a t r i x c o u p l e d a l l t h e n o d e s t o g e t h e r i n c o n t a c t w i t h t h e f l u i d a n d h a d 3 * N I N c o l u m n s a n d 3 * N I N r o w s , w h e r e N I N w a s t h e n u m b e r o f n o d e s i n t h e s t r u c t u r e t h a t w e r e i n c o n t a c t w i t h t h e f l u i d . W h e n t h e a d d e d m a s s m a t r i x w a s c o m b i n e d w i t h t h e s t r u c t u r a l m a s s m a t r i x , t h e b a n d i n g w a s l o s t . T h e e x t e n t t h a t t h e b a n d w i d t h o f t h i s n e w m a s s m a t r i x [M] - [M ] + [M ] S A w a s i n c r e a s e d o v e r t h a t o f t h e s t r u c t u r a l m a s s m a t r i x d e p e n d e d o n t h e n u m b e r o f i n t e r f a c e n o d e s a n d t h e i r l o c a t i o n i n t h e s t r u c t u r e . T h e i n c r e a s e i n t h e b a n d w i d t h a d v e r s e l y a f f e c t e d t h e s o l u t i o n t i m e a n d m e m o r y r e q u i r e m e n t s o f t h e e i g e n v a l u e p r o b l e m . ( T h e f i n i t e e l e m e n t r e p r e s e n t a t i o n t h a t V A S T u s e s t o d e t e r m i n e t h e a d d e d m a s s m a t r i x [MJ i s n o t t h e o n l y m e t h o d . O t h e r s i n c l u d e t h e B o u n d a r y E l e m e n t M e t h o d a n d t h e S o u r c e 70 D i s t r i b u t i o n Method. However, both of these s u f f e r the same fate, as they produce an added mass matrix [MJ which i s very s i m i l a r to that produced by VAST's current representation of the f l u i d . The only improvement may be the CPU time taken to c a l c u l a t e the added mass matrix.) To t r y to reduce the CPU time and the memory requirements, an a r t i f i c i a l bandwidth was imposed on the added mass matrix [MJ . I t was f e l t that the mass coupling between nodes i and j would not be too strong at the f l u i d - s t r u c t u r e i n t e r f a c e , e s p e c i a l l y when the nodes were very f a r apart ( i . e . at a large bandwidth). I f t h i s was the case, the imposition of an a r t i f i c i a l bandwidth on the added mass matrix [M ] would not A a f f e c t the r e s u l t s too much. The implementation required some modifications to the source code of the subroutine MASSM i n VAST. This procedure was tested on the plate problem with an aspect r a t i o a/b =1.0. A bandwidth of 20 corresponded to the f u l l added mass matrix [MJ being used, while a bandwidth of 1 corresponded to j u s t the diagonal terms of the added mass matrix [M ] . No A attempt was made to conserve k i n e t i c energy; terms outside of the imposed bandwidth were truncated. The r e s u l t s are p l o t t e d ( i n Figures 3.19 and 3.20) as percentage err o r (between r e s u l t s with the imposed bandwidth 71 and results when the f u l l added mass matrix was used) versus percentage of bandwidth. The suspicion that the mass coupling was not that strong at large bandwidths was confirmed by the results (shown i n Figures 3.19 and 3.20). The imposition of a 70 percent bandwidth only increased the error at most by 7 percent. I t should be noted that imposing a bandwidth of 40 percent produced a l o c a l minimum, giving errors of only 10 percent for a l l but modes 2 and 3. With a bandwidth of one, the frequencies predicted were a l l above those when the f u l l added mass matrix was used. This resu l t indicated that the added mass matrix [MJ was being underestimated. However, as the bandwidth was increased, the convergence was not monotonic. In fact, i n some of the modes, as the bandwidth was increased, the frequency went below that predicted when the f u l l added mass matrix was used. This finding indicated that at some points, the added mass matrix [M ] was A overestimated. This method did not save any time i n the formation of the added mass matrix, but i t reduced solution times i n : 1) the formation of the new global mass matrix MASSM; 2) the decomposition of the global matrices DECOM; and 3) the eigenvalue problem EIGEN, since a l l of these were very dependent on the bandwidth of the matrices involved. (The percentage reduction i n the CPU times for a few of the bandwidths imposed are tabulated i n 72 80 60 40-20--20 : .« I <\ : V ft' A | / \..\ : W ^ . . \ t _ j.:.Y-.-> A k ! , _ — i 1 1 1 n " • S , 20 40 60 80 % Bandwidth 100 120 Legend A Mode 1 x Mode 2 D Mode 3 a Mode 4 n Mode 5 Figure 3.19 - Error Produced by Imposing an Artifical Bandwidth Using 8-Noded Fluid Elements with Plate Aspect Ratio a/b - 1.0 73 100 so-u l •60-A. v. Ul Ul * 0 -60 a) M o d e 1 % Bandwidth 100-1 t \ A A *.. . V -60-1 ( c) Mode 3 % Bandwidth 100-0 60 100 e) M o d e 5 % Bandwidth 100 o k. b. Ul * 0 -60 0 60 100 b) Mode 2 % Bandwidth *A  A / 100 Ul •60-********* 0 60 100 d) Mode 4 % Bandwidth Figure 3.20 Error Produced by Imposing an A r t i f i c a l Bandwidth Using 8-Noded F l u i d Elements with Plate Aspect Ratio a/b =1.0 for Individual Modes 1 - 5 74 Table 3.8.) In the tes t problem the reduction i n CPU times was not that s i g n i f i c a n t , since the bandwidth of the s t r u c t u r a l matrices was eleven, which was comparable to that of the added mass matrix. The bandwidth of the s t r u c t u r a l matrices corresponded to a bandwidth reduction of 55 percent f o r the added mass matrix. Thus, there was no decrease i n the CPU times a f t e r the bandwidth of the added mass matrix [M ] went below 55 percent. % Reduction i n Bandwidth MASSM DECOM EIGEN 5 % 33 2 21 25 % 25 2 19 50 % 17 2 12 75 % 8 1 3 100 % 0 0 0 Table 3.8 - Solution Times i n Various Routines for Bandwidth Reduction 75 4. EXPERIMENTAL DETERMINATION OF THE VIBRATION RESPONSE OF A SHIP MODEL IN AIR AND WATER This chapter w i l l describe the experimental work that was performed f o r comparison with numerical r e s u l t s . A model of a s h i p - l i k e structure was designed, b u i l t , and studied f o r i t s dynamic behaviour under various load conditions i n a i r and water. 4.1 MODEL CONSTRUCTION In order to construct a model of a ship, many va r i a b l e s had to considered. Of prime importance was the model's a b i l i t y to reproduce the dynamic behaviour of the f u l l s i z e ship. Other issues were important too, and, as i s often the case i n model s c a l i n g , not a l l of these issues could be s a t i s f i e d by the same model. In order to determine which properties were important and what material should be used, a dimensional analysis of the r e l a t i o n s that govern the v i b r a t i o n of a ship structure was conducted. (The d e t a i l s of the dimensional analysis of the model 76 are discussed i n Appendix D.) From the considerations of dimensional analysis, material constraints, and ease of construction, the dimensions of the model were determined (as shown i n Figure 4.1). The material of construction chosen was a c r y l i c . The length of the model was determined as 8 feet, since the a c r y l i c sheet was 8' x 4' and the construction of a two piece h u l l would have been very d i f f i c u l t . The c r o s s - s e c t i o n a l shape of the h u l l was chosen to be constant and semicircular over the length of the model, as t h i s made i t easy to form. Furthermore, the semicircular shape was a convenient shape f o r which the added mass could be c a l c u l a t e d using S t r i p Theory. The main h u l l was formed by heating the a c r y l i c to j u s t below i t s c r i t i c a l temperature and then bending i t over a mold of the c o r r e c t form, to produce the desired semicircular shape of the h u l l . This work was contracted out, since the U n i v e r s i t y of B r i t i s h Columbia's Department of Mechanical Engineering d i d not have the f a c i l i t i e s that would allow the sheet of a c r y l i c to be heated uniformly to the correct temperature. The bulkheads were semicircular, to match the shape of the h u l l . They were cut from a 6 mm sheet of a c r y l i c on the 77 Length (between end bulkheads) 96" (2438mm) Beam 8.391" (213mm) H u l l Thickness 0.079" (2mm) Bulkhead - Thickness 0.236" (6mm) - Diameter 8.3125" (211mm) Endcaps - Thickness 0.079" (2mm) - Radius 4.196" (107mm) Unloaded Weight 5.56 l b (2530 grams) Figure 4.1 - Ship Model 78 Department of Mechanical Engineering's XLO numerically controlled vertical milling machine. This machine was used so that a l l of the bulkheads would be exactly the same. Eleven bulkheads were used in the construction of the model (as shown in Figure 4.1). They were cemented in place by an acetate solvent. Endcaps of a semispherical design were attached to the ends of the hull to reduce any discontinuities in the fluid flow in these areas when the model was vibrating in water. If the model ended abruptly, the discontinuities would cause vortex shedding which could not be accounted for in the numerical models. The semispherical endcaps would allow for a smooth transition, reducing the effect of the discontinuities. However, the semispherical shape proved difficult to form since a sphere is an undevelopable surface. This problem was overcome by breaking the endcaps into eight sections and approximating each section as a wedge. (This procedure is illustrated in Figure 4.2.) The wedges were then bent over a circular form, and the eight wedges were cemented together using an acetate solvent, thus forming the semispherical endcaps. Finally, the endcaps were cemented to the main semicircular part of the hull, completing the construction of the model. As with the bulkheads, the wedges were cut on the numerically controlled vertical milling machine. 79 a) Assembled Endcap b) Wedge Shape Figure 4.2 - Endcap and Wedge 4.2. INSTRUMENTATION The instrumentation used to measure the frequencies and mode shapes of the ship model i s described i n t h i s section. (For a complete l i s t of the instrumentation and equipment used, see Appendix E. Figure 4.3 shows a schematic diagram of the instrument chain.) Two Bruel & Kj«r accelerometers, type 4332 and 4370, were used. These were p i e z o e l e c t r i c type accelerometers whose e l e c t r i c a l output was d i r e c t l y proportional to the ac c e l e r a t i o n they were experiencing. The nominal s p e c i f i c a t i o n s of the two accelerometers (given i n Table 4.1) needed to be confirmed before any measurements were taken. (See Section 4.3.) Type 4332 Type 4370 Weight (grams) 35 54 - 2 Voltage S e n s i t i v i t y mV/ms — 10 - 2 Charge S e n s i t i v i t y pC/ms 6.4 10 Frequency Range Hz 0.5 - 2000 0.2 - 3500 Table 4.1 - Nominal Accelerometer S p e c i f i c a t i o n s 81 Figure 4.3 - Instrument Chain Schematic 82 The e l e c t r i c a l signals from the p i e z o e l e c t r i c accelerometers were amplified by two Briiel & Kjaer type 2635 charge preamplifiers. The 2635 was only s e n s i t i v e to v a r i a t i o n s i n charge produced by the accelerometer when i t was i n motion. Thus, any length of cable might have been used without i n f l u e n c i n g the system's charge s e n s i t i v i t y . The 2635 charge a m p l i f i e r o f f e r e d comprehensive s i g n a l conditioning f o r measurement of acceleration, v e l o c i t y , and displacement. The front panel of the 2635 allowed the charge s e n s i t i v i t i e s of the accelerometers to be set between 0.1 and -2 11 pC/ms . As well, the s e n s i t i v i t y of the am p l i f i e r allowed -2 outputs of 0.1 to 1000 mV per m/s f o r measurements of v i b r a t i o n a l accelerations. Also included were high and low pass f i l t e r s which were used to prevent the influence of noise and accelerometer resonance when the measurements were made. The high pass f i l t e r s were 0.2 and 2 Hz, while the low pass f i l t e r s were 0.1, 1, 3, 10, 30, and >100 kHz. A N i c o l e t 660A Dual Channel Fast Fourier Transformer (FFT) analyser was used to analyse the accelerometer's s i g n a l s . The Ni c o l e t allowed the time domain voltage signals A(t) from two accelerometer/charge a m p l i f i e r sets to be transformed into the frequency domain by performing a f a s t Fourier 83 transformation on the incoming s i g n a l s . The instantaneous FFT was ca l c u l a t e d from the incoming time s i g n a l as S a = ?{A(t)} A S = |S |cos(cp ) + JIS |sin(<p ) A 1 A 1 A 1 A A where the subscript indicated the channel, A(t) was the time s i g n a l , } the d i r e c t Fourier transform, S the complex A spectrum, and <p the phase angle. The RMS Spectrum of channel A A was given as RMS Spectrum = / S «S* r A A where S was the complex conjugate of S . The Transfer Function A A was given as G H AB G A A where and G = S -S AA A A G = S »S* AB B A F i n a l l y , the Coherence was given by G -G 2 AB AB 7 AB G -G AA BB A PCB Piezotronics 208 A03 impact hammer was used to excite the ship model. The t i p of the impact hammer was removable, allowing d i f f e r e n t t i p s to be used. This impact hammer had three d i f f e r e n t t i p s : a rubber t i p , a p l a s t i c t i p , and a 84 s t e e l t i p . The d i f f e r e n t t i p s allowed the energy content of the impulse to be adjusted. 4.3 EXPERIMENTAL PROCEDURE The experimental procedure followed f o r the model t e s t i n g was developed over a number of t e s t s . The f i n a l procedure that was adopted w i l l be presented here. 4.3.1. C a l i b r a t i o n of Instrumentation The two accelerometers were c a l i b r a t e d using a Bruel & Kjaer type 4291 accelerometer c a l i b r a t o r . The 4291 was a small portable v i b r a t i o n e x c i t e r producing a reference a c c e l e r a t i o n of 10 ms"2 (1.02g) with i t s peak at 79.6 Hz. The nominal charge s e n s i t i v i t y ( l i s t e d i n Table 4.1) was set on the 2635, and then the voltage output of the 2635 was adjusted u n t i l a reasonable voltage s i g n a l was r e g i s t e r e d on the o s c i l l o s c o p e , i n the order of 2 v o l t s peak to peak. The charge s e n s i t i v i t y was then fine-tuned u n t i l the voltage output from the 2635 matched 85 with the readings on the os c i l l o s c o p e . (Figure 4.4 shows the os c i l l o s c o p e trace of the c a l i b r a t i o n f or the two accelerometers.) The charge s e n s i t i v i t y of the two accelerometers was found to be : -2 Bruel & Kjser Type 4332 Accelerometer 6.50 pC/ms -2 Bruel & K j s r Type 4370 Accelerometer 10.08 pC/ms These r e s u l t s were quite close to the nominal s p e c i f i c a t i o n s l i s t e d i n Table 4.1. P i e z o e l e c t r i c accelerometers are very high impedance devices and are susceptible to noise generated by the connecting cables. Any motion of the connecting cables would cause the capacitance of the cables to change. This change i n capacitance would a f f e c t the charge i n the system, causing erroneous readings. To keep t h i s e f f e c t to a minimum, the cables were clamped to prevent or l i m i t t h e i r r e l a t i v e motion. This procedure was followed for a l l the tests that were c a r r i e d out on the model. The c r i t e r i o n f o r choosing a t i p for the impact hammer was that the power l e v e l of the s i g n a l produced through the range of frequencies of i n t e r e s t should be f a i r l y l e v e l . The frequency range of i n t e r e s t f o r the model tests was 0 - 100 Hz. 86 -\ 1 r 0.998 H h -0.930 H 1 1 1 1 1 r-a) Accelerometer Type 4332 1 1 1 1-0.990 -0.972 H h H 1 1 1 1 1 h b) Accelerometer Type 4370 Figure 4.4 - Calibration of Accelerometers 87 Thus, with the Nicolet 660A FFT set to display the power spectrum of the impulse, a few tests were carried out with each tip. It was found that the plastic tip gave the best results (as shown in Figure 4.5). With the steel tip, i t was easy to get a double hit. 4.3.2. Experimental Set Up For Air and Water Tests The free-free frequencies and mode shapes of vibration for the model in air and water needed to be determined and compared. The water tests presented no problem, since the water supported the ship; but, the air tests presented a problem, since i t is difficult i f not impossible to perform free-free tests in air. For the air tests, the model needed to be supported without affecting the frequencies or the mode shapes of vibration. Clearly, this was a difficult task. A frame was constructed that would allow the model to be supported by elastic cords. (The frame and model set up for the air tests is shown in Figure 4.6.) To determine i f this set up had any effect on the vibration characteristics of the model, the tension of the elastic cords and their placement was varied, and vibration tests were performed. From these tests i t was determined that : the placement of the elastic cords had l i t t l e 88 a) Rubber Tip b) Plastic Tip c) Steel Tip 16 « A V G 6.52-09 V 2 V L G tee B000 HZ T TSU 16"-I I 1 1 1 (--I 1 1 1 A/8 HZ 16 « A V G 5.35-09 V 2 V L G 100 0000 H Z T TSU 16 A/8 HZ (6 » A V G 3.31-09 V 2 V L G 100 0000 HZ T -I 1 1 1 1 1 1— e 1A A/8 H Z -I 1 1 1-Figure A.5 - Power Spectrum for the Various Tips of the Impact Hammer 89 e f f e c t on the frequencies and mode shapes of v i b r a t i o n ; and; the tension had no e f f e c t on the v i b r a t i o n c h a r a c t e r i s t i c s , except f o r the r i g i d body modes (which were of no concern i n these t e s t s ) . However, only the v e r t i c a l modes were checked since the h o r i z o n t a l and t o r s i o n a l modes had not been i d e n t i f e d yet. Subsequently a f i n i t e element model that modelled the s t i f f n e s s of the e l a s t i c cords was examined. (These f i n i t e element r e s u l t s are displayed i n Table 4.2.) These r e s u l t s show that the e l a s t i c cords had some e f f e c t on the frequencies of v i b r a t i o n . Mode EXP OLD FE NEW FE 2NDV/2NDH 12.5 12.11 13.17 3NDH/3NDT 23.5 49.23 51.23 3NDV 33.25 32.34 34.23 4NDH/4NDT 40.75 59.94 56.96 4NDV 60.75 60.88 62.24 5NDV 92.0 95.31 93.81 Table 4.2 - Comparison of Experimental and F i n i t e Element Results f o r 30 lb Load Condition The water tests were c a r r i e d out i n the towing tank of The Ocean Engineering Center of B.C. Research. The towing tank was 220 feet i n length and 12 feet wide, with a water depth of 8 feet. At one end of the tank, there was a preparation dock. This 90 Figure 4.6 - Experimental Set Up for Air Tests 91 dock area could be i s o l a t e d from the main towing tank and was 12 feet long, 3 feet wide, and 4 feet deep, with a water depth of 37 inches. A v i b r a t i o n t e s t was f i r s t performed with the model i n the middle of the towing tank and then with the model i n the middle of the preparation dock, and the r e s u l t s were compared. I t was found that there was no difference between the r e s u l t s , so the remaining tests were performed i n the preparation dock, as i t was easier to t e s t there. (Figure 4.7 shows the experimental set up f o r the water tests.) 4.3.3. D i f f e r e n t Load Conditions The tests were conducted with s i x d i f f e r e n t load conditions. These load conditions were obtained by p l a c i n g d i s c r e t e lead weights i n 40 p o s i t i o n s along the length of the model. Each weight was 2.5" x 2" x 0.125" and had a nominal value of 0.25 l b s . Thus, p l a c i n g 40 weights along the length of the model produced a load of 10 l b s . Five of the load conditions were uniform loads, with the load being v a r i e d from 30 lbs to 70 lbs i n 10 lbs increments. These load conditions produced d i f f e r e n t d r a f t s (as shown i n Table 4.3). The s i x t h load case was a non-uniform load condition with a t o t a l weight of 70 l b s . The i n d i v i d u a l weights were d i s t r i b u t e d to model an actual mass 92 Figure 4.7 - Experimental Set Up for Water Tests 93 d i s t r i b u t i o n i n a r e a l ship; however, the model had to remain on an even keel since the free board with 70 lbs was very small. (Figure 4.8 shows t h i s d i s t r i b u t i o n . ) Load Cond. lbs 30 40 50 60 70 Drafts Inches 3.511 3.720 3.877 3.995 4.083 Table 4.3 - Load Conditions and Drafts of the Model 4.3.4. Natural Frequency and Mode Shape Determination The accelerometers were mounted at various stations on the model using bee's wax. The accelerometer type 4370 was always at s t a t i o n zero (the bow), while the accelerometer type 4332 was moved from s t a t i o n to s t a t i o n . The RMS Spectrum, Transfer Function, and the Coherence were measured at each of the twenty-one stations along the length of the model to determine the mode shapes of the model. With the two accelerometers at t h e i r respective s t a t i o n s , an impulse was applied to the model with the impact hammer and the r e s u l t s recorded. Sixteen measurements were taken and averaged to reduce the error at each of the twenty-one stati o n s numbered 0 - 2 0 along the length of the ship. A f t e r the Ni c o l e t had processed and averaged the sixteen s i g n a l s , the information was tra n s f e r r e d to floppy disks, using the program MAP 94 Sta t ion * Figure 4.8 - Non-Uniform Load Condition 95 (discussed below i n Section 4.3.5). The accelerometer type-4332 was then moved to the next s t a t i o n , and the procedure described above was repeated for a l l twenty-one s t a t i o n s . For each load condition, three separate t e s t s were conducted. Each of these tests was performed to excite a d i f f e r e n t type of mode of v i b r a t i o n . To excite the v e r t i c a l bending modes, the ship was h i t v e r t i c a l l y on the c e n t e r l i n e at s t a t i o n 8 or 12, with the two accelerometers mounted v e r t i c a l l y on the c e n t e r l i n e . To excite the h o r i z o n t a l bending modes, the ship was h i t h o r i z o n t a l l y along the starboard gunnel at s t a t i o n 8 or 12, with the two accelerometers mounted h o r i z o n t a l l y along the port gunnel. To excite the t o r s i o n a l modes, the ship was h i t v e r t i c a l l y along the starboard gunnel at s t a t i o n 20, with the two accelerometers mounted v e r t i c a l l y along the port gunnel. The natural frequencies of the ship were i d e n t i f i e d as those frequencies at which peaks occurred i n the RMS Spectrum of the N i c o l e t . (This method of determining the natural frequencies of a structure i s known as the Peak Amplitude Method.) To determine the mode shape of v i b r a t i o n , two pieces of information were needed at each of the stati o n s along the length of the ship : the magnitude of the Transfer Function 96 and the phase angle at the natural frequencies of i n t e r e s t . . The magnitude of the Transfer Function at each s t a t i o n gave the r e l a t i v e value of mode shape at that point, while the phase angle (p i n d i c a t e d i f the two accelerometers were moving i n the same o d i r e c t i o n , i n d i c a t e d by <p — 0 , or i n opposite d i r e c t i o n s , o i n d i c a t e d by <p = 180 . By p l o t t i n g the magnitude and noting the phase angle, the mode shape was obtained. 4.3.5. Data A c q u i s i t i o n Program To help with the a c q u i s i t i o n and processing of the experimental data, a computer program c a l l e d MAP (Modal Analysis Program) was written. This program allowed the information on the Ni c o l e t 660A from one s t a t i o n to be down-loaded onto floppy disks f o r storage and further processing. With t h i s information stored on floppy disks, three functions could be performed by the PC. The program could mimic the N i c o l e t by d i s p l a y i n g the RMS Spectrum, Transfer Function, or Coherence of one s t a t i o n . I t could also be used to send the information back to the N i c o l e t f o r further processing. And f i n a l l y , i t could be used to determine the mode shapes of v i b r a t i o n f o r the model i n i t s various load conditions. This program (given i n Appendix F) was written i n Turbo Pascal on an IBM Personal Computer. The PC was u t i l i z e d 97 because of its portability. 4.4. EXPERIMENTAL RESULTS Using the experimental procedure (described above in Section 4.3), the natural frequencies and mode shapes for the ship model were found. The first four modes of vertical bending, the first four modes of horizontal bending, and the first four modes of torsion for the ship model were identified both in air and water for each of the five uniform load conditions and the one non-uniform load condition. 4.4.1. Frequencies The frequencies of vibration for the various modes and load conditions were examined (and are shown in Tables 4.4 - 4.6). There were a few modes that could not be identified in certain load conditions and thus they have been left blank. 98 2-Node 3-Node 4-Node 5-Node-Load cond. A i r H 0 2 A i r H 0 2 A i r H 0 2 A i r H 0 2 30 l b 12.5 7.75 33.25 21.25 61.75 40.75 92.0 65.75 40 l b 11.25 7.25 29.5 19.75 55.25 38.0 82.0 61.0 50 l b 10.5 7.0 27.25 18.5 50.5 35.5 74.5 56.75 60 l b 10.0 6.5 25.25 17.75 46.75 33.75 67.75 53.25 70 lb 9.5 6.25 23.5 16.75 43.0 32.5 63.0 50.5 Non-Uniform 9.25 6.25 23.5 16.75 42.25 32.0 61.25 50.75 Table 4.4 - Frequencies f o r the V e r t i c a l Bending Modes 2-Node 3-Node 4-Node 5-Node Load cond. A i r H 0 2 A i r H 0 2 A i r H 0 2 A i r H 0 2 30 l b 12.0 7.75 23.5 16.0 40.75 27.75 62.75 43.75 40 l b 11.5 7.0 22.0 14.25 37.75 25.0 57.75 39.25 50 l b 10.75 6.5 20.5 13.0 35.25 22.75 52.25 35.5 60 l b 10.25 6.75 18.75 12.0 33.0 20.5 48.25 32.25 70 l b 9.5 5.75 17.75 11.0 30.25 18.75 44.0 29.75 Non-Uniform 9.25 - 17.5 11.0 29.5 18.75 - 29.5 Table 4.5 - Frequencies f o r the Horizontal Bending Modes 99 1-Node 2-Node 3-Node 4-Node. Load cond. A i r H 0 2 A i r H 0 2 A i r H 0 2 A i r H 0 2 30 lb 11.25 5.5 45.5 41.5 23.5 16.0 40.5 28.0 40 l b 12.25 5.25 43.25 37.5 22.0 14.5 37.5 25.0 50 l b 12.75 5.25 - - 20.75 13.0 34.75 22.5 60 l b 13.5 6.0 - - 19.5 12.0 32.75 20.75 70 l b 13.5 4.5 39.25 31.5 18.25 11.0 - -Non-Uniform 13.5 4.75 - 31.25 18.0 11.0 - 18.75 Table 4.6 - Frequencies f o r the Torsional Modes By examining the frequencies of v i b r a t i o n f o r the various modes, i t was determined that some of the modes were coupled together. The 2-node v e r t i c a l mode was coupled to the 2-node h o r i z o n t a l mode. (Table 4.7 displays the center of shear and the center of mass f o r the various load conditions. The values given i n Table 4.7 are distances below the top of the bulkheads.) Since the center of shear and the center of mass f o r t h i s model d i d not correspond, the h o r i z o n t a l and t o r s i o n a l modes of v i b r a t i o n were coupled, as discussed by Bishop and Price [9-13]. The 3-node h o r i z o n t a l and the 3-node t o r s i o n a l modes were coupled, while the 4-node h o r i z o n t a l and 4-node t o r s i o n a l modes were also coupled. 100 Load Cond. Center of Mass Center of Shear 30 l b 3.694 inches 5.343 inches 40 l b 3.626 inches 5.343 inches 50 l b 3.537 inches 5.343 inches 60 l b 3.437 inches 5.343 inches 70 l b 3.331 inches 5.343 inches Table 4.7 - Center of Mass and Shear f or Various Load Conditions The experimental r e s u l t s provided no information about which modes were dominate i n the coupled modes, but by examining the f i n i t e element r e s u l t s i t was possible to determine t h i s . In the 2-node v e r t i c a l - h o r i z o n t a l mode, the 2-node v e r t i c a l mode was the dominate mode. However, i n the 3-node h o r i z o n t a l - t o r s i o n a l and the 4-node h o r i z o n t a l - t o r s i o n a l modes neither mode was dominate instead both were equal i n t h e i r energy c o n t r i b u t i o n to the mode. The frequencies of the 70 lb and the non-uniform load c o n d i t i o n were very s i m i l a r . The differ e n c e between the two load conditions was the d i s t r i b u t i o n of the mass. The frequencies of the 70 l b and the non-uniform load condition were cl o s e r f o r the water t e s t s . This s i m i l a r i t y was due i n part to the d i s t r i b u t i o n of the added mass, which was uniform and on the order 101 of the s t r u c t u r a l mass of the ship; thus, the e f f e c t of the water made the t o t a l mass d i s t r i b u t i o n more uniform. The r e s u l t s of the frequency of v i b r a t i o n i n a i r and water f o r the various load conditions were examined. (See Figure 4.9 - 4.11.) I t was seen that f o r v e r t i c a l and h o r i z o n t a l motion i n any mode, the frequencies decreased as the load c o n d i t i o n increased. This was expected, since the s t i f f n e s s of the structure was not af f e c t e d by the load condition. Only the mass of the structure was affected, which lowered the frequency of v i b r a t i o n . As the mode number was increased, the slope of the l i n e s increased as we l l , and the slope of the a i r r e s u l t s were larger than the slope of the water r e s u l t s . The t o r s i o n a l modes were not as w e l l behaved. In f a c t , the 1-node t o r s i o n a l mode i n a i r increased i n frequency as the load condition was increased. By comparing the frequencies i n water with those i n a i r , and by assuming that Beam Theory held, i t was possible to c a l c u l a t e the apparent added mass i n the v e r t i c a l modes caused by the water. This was a reasonable assumption i f the complexity of the modes were low, such that shear displacement and rotary i n e r t i a e f f e c t s were not important. By comparing the experimental r e s u l t s with Beam Theory i t was seen that the frequencies compared quite well, e s p e c i a l l y f o r the lower modes (as shown i n 102 100 Load Condition Legend 2 -Node Air X 3 -Node Air • 4 -Node Air B 6 -Node Air n 2 -Node Water X 3 -Node Water • 4 iNojte Water 5 -Node Water Figure 4.9 - Vertical Modes of Vibration for the Ship Model 103 100-1 80-o-t " i 1 i ' i ' i 1 1 • 20 30 40 50 60 70 80 Load Condition Legend 2-Node Air X 3-Node Air • 4-Node Air B 5-Node Air B 2-Node Water X 3-Node Water • 4jjNojdeWater 5-Node Water Figure 4.10 - Horizontal Modes of Vibration for the Ship Model 104 100 n o-| 1 j ' i ' i ' i ' i 1  20 30 40 60 60 70 80 Load Condition Legend 1-Node Air X 2-Node Air • 3-Node Air B 4-Node Air S 1-Node Water X 2-Node Water • S^Nojde Water 4-Node Water Figure 4.11 - Torsional Modes of Vibration for the Ship Model 105 Table 4.8). The frequency of a beam was given as 1 2 , — Hz (4.1) 1 * m where /? I was a constant that depended on the end conditions of the beam and the mode of i n t e r e s t , 1 was the length of the beam, E was Young's modulus, I was the moment of i n e r t i a of the cross-section, and m was the mass per u n i t length of the beam. 2-Node 3-Node 4-Node 5-Node Load cond. EXP TH EXP TH EXP TH EXP TH 30 lb 12.5 12.98 33.25 35.78 61.75 70.15 92.0 115.9 40 l b 11.25 11.47 29.5 31.61 55.25 61.97 82.0 102.4 50 l b 10.5 10.39 27.25 28.63 50.5 56.12 74.5 92.77 60 lb 10.0 9.56 25.25 26.35 46.75 51.66 67.75 85.40 70 l b 9.5 8.91 23.5 24.55 43.0 48.12 63.0 79.55 Table 4.8 - Experimental and Beam Theory Results i n A i r For V e r t i c a l Modes of V i b r a t i o n To determine the equivalent added mass, the r a t i o of the frequency i n a i r to the frequency i n water was needed. This was determined using the r e l a t i o n s f o r a beam and noting that the frequency was in v e r s e l y proportional to the mass of the structure; thus co CO m m (4.2) 106 where m was the combined mass i n water given by m = m + m , • with w w s A mg equal to the mass of the structure and mA equal to the added mass due to the e f f e c t of the water. A f t e r s u b s t i t u t i n g the combined mass i n water, equation 4.2 became co R - -* w co W m + m s A m m 1 + m (4.3) Solving f o r the r a t i o of the added mass mw to the s t r u c t u r a l mass mg gave m R = M m - 1 w J (4.4) where mg was equal to the mass of the structure per u n i t length plus the mass of the load condition per u n i t length. The weight of the unloaded structure was 5.56 l b s . 2-Node 3-Node 4-Node 5-Node Load cond. w / A/CO m . A / m ' s w / A/CO m . A / m ' s A /CO m . A / m ' s u / A/CO m . A / m ' s 30 l b 1.61 1.60 1.57 1.45 1.52 1.30 1.40 0.96 40 lb 1.55 1.41 1.49 1.23 1.45 1.11 1.34 0.81 50 lb 1.50 1.25 1.47 1.17 1.42 1.02 1.32 0.72 60 l b 1.54 1.37 1.42 1.02 1.39 0.92 1.27 0.62 70 l b 1.52 1.31 1.40 0.97 1.32 0.75 1.25 0.56 Table 4.9 - Ratios of Added Mass to S t r u c t u r a l Mass For V e r t i c a l Bending Modes 107 Upon examining the r e s u l t s f o r the v e r t i c a l modes of v i b r a t i o n (as presented i n Table 4.9), a few general trends were determined. While i n a s p e c i f i c mode, i f the load condition of the model was increased, the r a t i o of the added mass to the s t r u c t u r a l mass was seen to decrease. Thus, the added mass e f f e c t depended on the d r a f t of the model, and i t s c o n t r i b u t i o n to the t o t a l mass of the model was not a constant proportion of the model's mass. In f a c t , at shallower draf t s the added mass e f f e c t was more s i g n i f i c a n t than at deeper d r a f t s . I t was also seen that i n a s p e c i f i c load condition, i f the mode number increased, the r a t i o of added mass to s t r u c t u r a l mass decreased. Thus, the added mass e f f e c t decreased as the complexity of the mode increased. Another method was also used to t r y to determine the added mass due to the f l u i d . This method used the f i n i t e element model of the structure i n a i r . The density of the twenty-one bulkheads was increased u n t i l there was agreement between the experimental and f i n i t e element frequencies f o r a p a r t i c u l a r mode i n the f l u i d . (These r e s u l t s are displayed i n Table 4.10.) 108 2-Node 3-Node 4-Node 5-Node. Load cond. A/W m . A/m ' s w / A/W m . A/m ' s A/W m . A/m ' s A/W m . A/m ' s 30 l b 1.58 1.50 1.52 1.33 1.50 1.33 1.48 1.17 Table 4.10 - Ratios of S t r u c t u r a l Mass to Added Mass f o r V e r t i c a l Bending Modes Using F i n i t e Elements From these f i n i t e element r e s u l t s , i t was seen that as the complexity of the modes increased, the r a t i o of the added mass to the s t r u c t u r a l mass decreased. This was the same conclusion found above using Beam Theory; however, the rate of decrease of the r a t i o of masses and frequencies was not as severe as with Beam Theory. Furthermore, (by examining the two tables) i t was seen that the f i n i t e element r e s u l t s d i d not conform to equation 4.4. Thus, the e f f e c t s of shear displacement and rotary i n e r t i a were present i n the model. 4.4.2. Mode Shapes With the Transfer Function data from the experiment, the mode shapes of the v i b r a t i o n were determined, as discussed ( i n Section 4.3.4) above. The r e s u l t i n g mode shapes i n a i r and water were compared. This was accomplished by normalizing the two modes so that they had the same energy i n a i r and water. 109 In equation form tf2 dx A * 2 dx w where and $ were the mode shapes i n a i r and water, A W r e s p e c t i v e l y , with a = 1 and the value of b determined from * 2 dx • A • dx W The experimental mode shapes were i n tabular form, a f t e r being processed by the MAP program, and, thus, needed to be numerically integrated. This was done using Simpson's r u l e 1 I - - A 3 * 2 ( x ) + 4* 2(x ) + 2* 2(x ) + 0 1 2 + 4* 2(x ) + $ 2(x ) 2N-1 2N where A was the distance between data points. (Figures 4.12 - 4.20 show the mode shapes of v i b r a t i o n of the ship i n i t s 30, 40, and 70 lb load conditions.) In general, there was l i t t l e d i f f e r e n c e between the mode shapes i n a i r and water. However, the mode shape i n water was generally smoother than the corresponding mode shape i n a i r , i n d i c a t i n g that the l o c a l bending e f f e c t s were les s i n water. Local bending began to show up i n the higher mode shapes. In the model there were eleven bulkheads. These bulkheads were located at stations 0, 2, 4, 6, 8, 10, 12, 14, 16, 110 2-Node Vertical Bending 4 -Node Vertical Bending 10 16 Station # 5 -Node Vertical Bending Legend 20 Station # Figure 4.12 - Experimental Vertical Modes of the Ship Model in Air and Water for 30 lb Load Condition Legend x M t l . O H i <• *r«t*r M.Tf N l 111 2-Node Horizontal Bending 10 Station # 15 3-Node Horizontal Bending Legend > A» 11.0 Hi < Water r.TI Hi 20 V \ 10 Station # 15 4-Node Horizontal Bending Legend x Ak WJ m * Wafer « . 0 Hz 20 V Y • 10 Station # 16 5-Node Horizontal Bending Legend x Ar 40.7* Ht * W.t.r t7.TI Hi 20 •"V • S 10 Station # Figure 4.13 - Experimental Horizontal Modes of the Ship Model i n A i r and Water f o r 30 lb Load Condition 16 Legend « A l r M . T t K l •> Mfstar «t.T6 I 20 112 1-Node Torsional 10 Station f 2-Node Torsional 15 Legend x M r W . M Hx WiMtMlb 20 k J • . — — £ i * 10 Station # 3-Node Torsional 16 Legend X A » M . * H t • W i M t 4 1 * H i 20 X 10 Station # 4-Node Torsional 16 Legend x Ak M . I M » « Wattr H.O Kz 20 / / \ \ \\ i 1 1 1 1 i Figure 4.14 Station # Experimental Torsional Modes of the Ship Model in Air and Water for 30 lb Load Condition Legend x Ak 40.1 H i 1.0 HI 113 2-Node Vertical Bending 10 Station f 3-Node Vertical Bending Legend x Air 1UI Ht •> Water T.16 Hx 20 10 Station f 4-Node Vertical Bending Legend x Afr t t . tHl •• Water 1».T6 Hi 20 10 16 Station # 5-Node Vertical Bending Legend x Air MM Hi 4 Water al.O Hx 20 10 Station # Figure 4.15 - Experimental Vertical Modes of the Ship Model in Air and Water for 40 lb Load Condition Legend « Alt M.O Hi « Water tl.O Hi 114 -1 -1 -1 2-Node Horizontal Bending ^ ^ ^ ^ ^ 10 Station # 15 3-Node Horizontal Bending Legend x J U l U H l - Water 7.0 Hx 20 ) < Station f 4-Node Horizontal Bending Legend x A » « . 0 H l •« W « t M MM Ht w 10 Station # 16 5-Node Horizontal Bending Legend x A t 1T .T1 Ml ' ar«t*r te.e H i 20 • V 1 10 Station f 16 Legend x AktT.nm 20 Figure 4.16 - Experimental Horizontal Modes of the Ship Model in Air and Water for 40 lb Load Condition 115 1-Node Torsional Station # 2 -Node Torsional Legend x Ak"IL*«Mx - Water t M Kt I 10 Station f 3-Node Torsional 18 Legend x Air *»M H» < Water *7 .l Hx 20 -1 1 1 1 1 1 1 10 Station f 4-Node Torsional 16 Legend x Ae- tl.O Hx * Water H I Hx 20 J v.. 1 ' ' ' ' I • Figure 4.17 Station # Experimental Torsional Modes of the Ship Model in Air and Water for 40 lb Load Condition Legend x Ar »7.t Hi « Water I I 0 Hx 116 2-Node Vertical Bending 10 Station # 3-Node Vertical Bending Legend x A * I t Kz water ne Hz 20 Station # 4-Node Vertical Bending Legend X Air M .I Hz * Wat* ' H .T I H i / 1 ' 1 1 ' 1 10 15 Station # 5-Node Vertical Bending Legend x A t 41.0 Hz • Water 111 Hx 20 \ \\ V 1 5 10 Station f Figure 4.18 - Experimental Vertical Modes of the Ship Model in Air and Water for 70 lb Load Condition 16 Legend x A> 01.0 HZ « Water H i Hx 20 117 2-Node Horizontal Bending 10 Station # 15 3-Node Horlzonal Bending Legend x A * te Hz •Tatar 6.TC Hi 20 , i i i i • Station f 4 -Node Horizontal Bending Legend x A* TTT6 Hi Watar n.0 Hi 1 1 1 1 I 1 1 1 1 1 • 10 Station # 15 5-Node Horizontal Bending Legend x Ak t o t I ttt - Watar ii.TI Hi 20 10 Station # Figure 4.19 - Experimental Horizontal Modes of the Ship Model in Air and Water for 70 lb Load Condition 16 Legend x A» 44.0 Hi . Watar tt.7% Hi 20 118 1-Node Torsional <• w ' — ^-10 Station i 2-Node Torsional 15 Legend x A » M . 1 K » i « l t l t 4 4 t t SO x ^ < M ' ' ' I 10 Station # 3-Node Torsional 16 Legend x At- tt.tt H i * Water t i l H i 20 J 10 Station f 16 Legend X A* W J g H l * Water TtO Ml 20 Figure 4.20 - Experimental Torsional Modes of the Ship Model in Air and Water for 70 lb Load Condition 119 18, and 20. By examining the modes that displayed l o c a l bending e f f e c t s , i t was seen that t h i s l o c a l bending was occurring between these s t a t i o n s . Furthermore, as the load condition increased, the mode shapes became smoother and there were fewer l o c a l bending e f f e c t s . 120 5. NUMERICAL RESULTS AND COMPARISON OF THE VIBRATION RESPONSE OF A SHIP MODEL IN AIR AND WATER This chapter w i l l describe the numerical work that was performed and w i l l compare these r e s u l t s with the experimental r e s u l t s of the ship model. The numerical work included both s t r u c t u r a l modelling of the ship and f l u i d modelling of the surrounding water for the v i b r a t i o n c h a r a c t e r i s t i c s of the ship model i n a i r and water. The ship f i n i t e element models were developed using a Mesh Generation Program that was written by the author. This program used the Thick/Thin S h e l l element (discussed i n Appendix C) and allowed complex f i n i t e element meshes of the ship model to be created f o r use with VAST. These meshes were permitted to have any number of elements along the ship's length and around the ship's circumference, as long as the number of nodes i n the structure d i d not exceed the maximum allowed by the VAST. The Mesh Generation Program required the load condition of the ship and then c a l c u l a t e d the displacement of the ship from t h i s information, proceeding to put only one element above the 121 water l i n e . The Mesh Generation Program also created the f l u i d mesh around the structure, using the 20-noded f l u i d elements (discussed i n Appendix B), matching the s t r u c t u r a l and f l u i d meshes on the i n t e r f a c e , and allowing the f l u i d domain to be extended r a d i a l l y out from the structure. 5.1 MATERIAL PROPERTIES To use the F i n i t e Element Program VAST, the properties of the material that was being modelled were required. These were determined through t e s t i n g of the actual material of c o n s t r u c t i o n - - a c r y l i c - - s i n c e the data supplied by the manufacturer only gave ranges f o r density p, Young's modulus E, and Poisson's r a t i o v. (The intermediate r e s u l t s of these tests are presented i n Appendix G.) The density of the material was c a l c u l a t e d by c u t t i n g a square s e c t i o n from the a c r y l i c sheet that the model was made from, measuring i t , and then weighing i t accurately. The 2 4 density of the a c r y l i c was found to be p = 0.000107 l b . sec / i n . (See Appendix G f o r d e t a i l s of the s i z e and weight of the sample.) 122 Poisson's r a t i o was determined as v = 0.368. Young's modulus was a l i t t l e more d i f f i c u l t to determine. Three d i f f e r e n t tests were performed. (They are summarized i n Table 5.1.) Test Young's Modulus P s i St r a i n Gauge 455,541 F i n i t e Element 302,000 Dynamic 622,056 Table 5.1 - Results of Tests f o r Young's Modulus The dynamic r e s u l t s f o r Young's modulus were used because these provided the best c o r r e l a t i o n with the experimental r e s u l t s . 5.2. FINITE ELEMENT MODELS The F i n i t e Element Program VAST was used to determine the frequencies and mode shapes of v i b r a t i o n numerically. F i n i t e element models of the structure and the f l u i d were made i n order to compare the predicted v i b r a t i o n 123 c h a r a c t e r i s t i c s i n a i r and water with the experimental r e s u l t s . 5.2.1. S t r u c t u r a l Models T r i a l meshes were generated using the Thick/Thin S h e l l element, and runs of VAST were executed to determine what degree of d e s c r e t i z a t i o n was required i n the s t r u c t u r a l model to obtain good r e s u l t s f o r the frequencies and mode shapes of v i b r a t i o n . I t was found that 20 elements along the length, 8 elements around the circumference, and 21 bulkheads y i e l d e d good r e s u l t s . This mesh also corresponded well with the a c r y l i c model of the ship, as there were twenty-one stations at which measurements were taken. This s t r u c t u r a l model had 3820 degrees of freedom and took an average of 3.5 hours of CPU time to solve f o r the f i r s t 20 modes of v i b r a t i o n , with 82 % of the time being spent i n the eigenvalue solver. (Figure 5.1 shows the s t r u c t u r a l mesh that was used.) 124 [SHIP MODEL WITH : .NST_=2I NBLK=19 NELM=8 NM=33 LCAD=33LBS. Figure 5.1 - Structural Finite Element Mesh of Ship Model 125 Elemental Matrix Formation 0.22hr 7% S t r u c t u r a l Matrix Assembly 0.14hr 4% Matrix Decomposition 0.27hr 8% Eigenvalue Analysis 2.63hr 81% F l u i d Matrix Formulation - -T o t a l CPU 3.27hr Table 5.2 - Average CPU Times for S t r u c t u r a l Mesh During the t r i a l runs, i t was noted that the r e s u l t s d i d not always improve with an increase i n the number of degrees of freedom. I t was very easy to create a model that had mainly l o c a l v i b r a t i o n modes and very few global modes; however, the l o c a l modes were not wanted. I t was found that introducing bulkheads every few elements removed these l o c a l modes and had l i t t l e e f f e c t on the global frequencies. I f the numerical model had very few bulkheads, the only global mode that could be i d e n t i f i e d was the 2-node v e r t i c a l mode; a l l the other modes were l o c a l or complex h o r i z o n t a l - t o r s i o n a l coupled modes. (See Figure 5.2.) 126 FREE - FREE ANALYSIS OF SHIP NST-I1 NBLKHD"1 NELMS-B L0AD°3B LBS.  NATURAL MODE SHAPE MODE NUMBER 0 I.27BE*0I CPS MAGNIFICATION FACTOR: S.8B ELEMENT TYPES: ALL 28.231 IN. FREE - FREE ANALYSIS OF SHIP NST-II NBLKHD-1 NELMS"6 L0AP-3B LBS. Figure 5.2 - Various Mode Shapes 127 5.2.2. F l u i d Meshes Two i d e a l i z a t i o n s of the f l u i d were used. (See Figure 5.3.) The f i r s t mesh used two 20-noded f l u i d elements i n the r a d i a l d i r e c t i o n ; the other used three 20-noded f l u i d elements. Both of these meshes gave very s i m i l a r r e s u l t s and used about the same CPU time (as shown i n Table 5.3), although the two element mesh had 1220 degrees of freedom and the three element mesh had 1830 degrees of freedom. However, there were modifications to some of the parameters of the computer between running the two element f l u i d mesh and the three element f l u i d mesh, and these modifications a f f e c t e d the CPU times. The reason that the f l u i d - s t r u c t u r e problem took so much more time than the structure alone was the f a c t that the added mass matrix was almost completely f u l l . When t h i s matrix was added to the s t r u c t u r a l mass matrix, which was n i c e l y banded, the banding was l o s t and the program had to deal with a matrix that had a bandwidth on the order of the number of degrees of freedom i n the f l u i d - s t r u c t u r e i n t e r f a c e . The s i z e of the s t r u c t u r a l mass matrix [MJ was 3820, but i t only had a bandwidth of 165; however, when the added mass matrix [MJ was added to the s t r u c t u r a l mass matrix [M ] the bandwidth was increased to 1389. 128 ADDED MASS MESH - NST=It NELMS°8 NELMFLDR=2 Rl"2 R2=4 ADDED HASS FINITE ELEMENT MODEL ELEMENT TYPES: ALL 1 X 5.326 IN. I ADDED MASS MESH - NELMS°8 LOAD°30 LBS. NELMFLDR=3 RI-.5 R2°1 R3°2 ADDED HASS FINITE ELEMENT MODEL ELEMENT TYPES: ALL 4V t x 3.026 IN. Figure 5.3 - Fluid Meshes 129 2 F l u i d 3 F l u i d Elemental Matrix Formation 817 - 778 -S t r u c t u r a l Matrix Assembly 1741 1% 1324 1% Matrix Decomposition 86551 34% 86411 38% Eigenvalue Analysis 67129 27% 55604 25% F l u i d Matrix Formulation 94881 38% 81590 36% To t a l CPU 69.7hr 62.7hr Table 5.3 - CPU Time f or F l u i d Meshes The f i n i t e element models of the structure and the f l u i d consumed large amounts of computer resources. Each run took between 63 and 70 hours of CPU time and used 575,000 blocks of disk space on the Department of Mechanical Engineering's VAX 11/750 computer. The harddisk on the VAX was only a 700,000 block disk. 5.3. R E S U L T S A N D C O M P A R I S O N WITH T H E EXPERIMENTAL R E S U L T S In t h i s section, the frequencies and mode shapes predicted by the various numerical methods w i l l be compared with the experimental r e s u l t s of the a i r and water t e s t s . In general, 130 there was good agreement with the v e r t i c a l modes of v i b r a t i o n and varying degrees of success with the h o r i z o n t a l and t o r s i o n a l modes. 5.3.1. Frequencies The r e s u l t s f o r the frequencies predicted by the various numerical methods were compared with the experimental r e s u l t s of v i b r a t i o n . Two d i f f e r e n t methods were used to model the e f f e c t of the f l u i d on the v i b r a t i o n c h a r a c t e r i s t i c s of the model. The f i r s t used S t r i p and Beam Theory to c a l c u l a t e the frequencies of v i b r a t i o n , while the second used the F i n i t e Element Program VAST to determine the frequencies and mode shapes. 5.3.1.1. Beam/Strip Theory and Experimental Results The r e s u l t s f o r the v e r t i c a l v i b r a t i o n c h a r a c t e r i s t i c s of the model were easy to c a l c u l a t e using Beam Theory (and were presented i n Chapter 4). The r e s u l t s from S t r i p Theory were also very simple to c a l c u l a t e , since the model had a constant c r o s s - s e c t i o n over i t s length. However, only the.results of the v e r t i c a l modes could be determined using t h i s method. (The 131 r e s u l t s are presented i n Table 5.4.) The lewis C f a c t o r f o r ' t h i s h u l l shape was 1 and was approximately the same f o r the 5 draughts considered. The J- f a c t o r s that corrected f o r the three-dimensional flow were : 0.836, 0.793, 0.771, and 0.756 f o r the 2, 3, 4, and 5-node v e r t i c a l bending modes r e s p e c t i v i l y . Thus, i t was seen by comparing these numbers with the r a t i o s of added mass to s t r u c t u r a l mass f o r the v e r i t c a l bending modes (from Table 4.9 or 4.10) that t h i s method was going to underestimate the added mass of the f l u i d . 2-Node 3-Node 4-Node 5-Node Load cond. A i r H 0 2 A i r H 0 2 A i r H 0 2 A i r H 0 2 301b EXP 12.5 7.75 33.25 21.25 61.75 40.75 92.0 65.75 B/S 12.98 9.58 35.78 26.72 70.15 52.71 115.9 87.59 401b EXP 11.25 7.25 29.5 19.75 55.25 38.0 82.0 61.0 B/S 11.47 8.46 31.61 23.61 61.97 46.57 102.4 77.39 501b EXP 10.5 7.0 27.25 18.5 50.5 35.5 74.5 56.75 B/S 10.39 7.67 28.63 21.38 56.12 42.17 92.77 70.1 601b EXP 10.0 6.5 25.25 17.75 46.75 33.75 67.75 53.25 B/S 9.56 7.06 26.35 19.68 51.66 38.82 85.4 64.54 701b EXP 9.5 6.25 23.5 16.75 43.0 32.5 63.0 50.5 B/S 8.91 6.57 24.55 18.33 48.12 36.16 79.55 60.12 Table 5.4 - Frequencies f o r the V e r t i c a l Bending Modes from Beam and S t r i p Theory 132 S t r i p Theory and experimental r e s u l t s f o r the v e r t i c a l modes of v i b r a t i o n (see Figure 5.4 and 5.5) were quite good f o r the 2-node v e r t i c a l mode; but, as the complexity of the modes increased the r e s u l t s began to deteriorate. This d e t e r i o r a t i o n occured because t h i s method d i d not account f or shear d e f l e c t i o n s and rotary i n e r t i a , although these items might have been included i n the analysis discussed by Bishop and Price [9-13]. I t should also be noted that S t r i p Theory was able to p r e d i c t the added mass f o r pure h o r i z o n t a l modes, but sa i d nothing when there was coupling with t o r s i o n a l modes. 5.3.1.2. F i n i t e Element and Experimental Results The frequencies determined i n the experiment were compared with the f i n i t e element r e s u l t s . (Tables 5.5 - 5.7 l i s t the frequencies predicted by VAST and found experimentally.) The f i n i t e element r e s u l t s were only run i n water f o r the 30 and 40 lb load conditions. As well, the 5-node h o r i z o n t a l , the 1-node t o r s i o n a l , and the 2-node t o r s i o n a l could not be i d e n t i f i e d from the f i n i t e element r e s u l t s . F i n i t e Element water runs were not performed f o r the 50, 60, and 70 lb load conditions. As well, 133 150-1 N 1 0 0 -o Load Condition Legend A 2-Node Exp x 3-Node Exp • 4-Node Exp B 6-Node Exp a 2-Node F.E. * 3jNj^e_FJE. • 4-^Node . ^ E . e 6-Node F.E. Figure 5.4 - Experimental and Beam Theory Results in Air 134 N X o c © D 100 8 0 -60 40 2 0 -20 3 0 . : ——-4 40 60 60 Load Condition —r-70 i 80 Legend A 2-Node Exp x 3-Nod« Exp • 4-Node Exp B 6-Node Exp B 2-Node F.E. * 3jN_ode_F.E. • ^ N o ^ F j ; © 6-Node F.E. Figure 5.5 - Experimental and Strip Theory Results in Water 135 some modes that were i d e n t i f i e d i n the experiment were- not i d e n t i f i e d from the f i n i t e element r e s u l t s . Thus, some places were omitted. 2-Node 3-Node 4-Node 5-Node Load cond. A i r H O 2 A i r H 0 2 A i r H 0 2 A i r H 0 2 301b EXP 12.5 7.75 33.25 21.25 61.75 40.75 92.0 65.75 FE 12.11 7.78 32.34 21.03 60.88 40.25 95.31 64.89 401b EXP 11.25 7.25 29.5 19.75 55.25 38.0 82.0 61.0 FE 10.93 7.25 29.24 19.61 55.03 37.51 86.03 60.18 501b EXP 10.5 7.0 27.25 18.5 50.5 35.5 74.5 56.75 FE 10.04 - 26.90 - 50.69 - 79.36 -601b EXP 10.0 6.5 25.25 17.75 46.75 33.75 67.75 53.25 FE 9.34 - 25.03 - 47.21 - 74.06 -701b EXP 9.5 6.25 23.5 16.75 43.0 32.5 63.0 50.5 FE 8.84 - 23.63 - 44.52 - 69.56 -Table 5.5 - Frequencies f or the V e r t i c a l Bending Modes from F i n i t e Element 136 2-Node 3-Node 4-Node 5-Node' Load cond. A i r H 0 2 A i r H 0 2 A i r H 0 2 A i r H 0 2 301b EXP 12.0 7.75 23.5 16.0 40.75 27.75 62.75 43.75 FE 12.11 7.78 49.23 34.41 59.94 42.63 - -401b EXP 11.5 7.0 22.0 14.25 37.75 25.0 57.75 39.25 FE 10.93 7.25 49.22 32.0 59.98 39.47 - -501b EXP 10.75 6.5 20.5 13.0 35.25 22.75 52.25 35.5 FE 10.04 - 49.28 - 60.29 - - -601b EXP 10.25 6.75 18.75 12.0 33.0 20.5 48.25 32.25 FE 9.34 - 49.45 - 60.85 - - -701b EXP 9.5 5.75 17.75 11.0 30.25 18.75 44.0 29.75 FE 8.84 - 50.21 - 63.87 - - -Table 5.6 - Frequencies f o r the Horizontal Bending Modes from F i n i t e Element 137 1-Node 2-Node 3-Node 4-Node-Load cond. A i r H 0 2 A i r H 0 2 A i r H 0 2 A i r H 0 2 301b EXP 11.25 5.5 45.5 41.5 23.5 16.0 40.5 28.0 FE - - - - 49.23 34.41 59.94 42.63 401b EXP 12.25 5.25 43.25 37.5 22.0 14.5 37.5 25.0 FE - - - - 49.22 32.0 59.98 39.47 501b EXP 12.75 5.25 - - 20.75 13.0 34.75 22.5 FE - - - - 49.28 - 60.29 -601b EXP 13.5 6.0 - - 19.5 12.0 32.75 20.75 FE - - - - 49.45 - 60.85 -701b EXP 13.5 4.5 39.25 31.5 18.25 11.0 - -FE - - - - 50.21 - 63.87 -Table 5.7 - Frequencies f o r the Torsional Modes from F i n i t e Element There was very good agreement between the experimental and f i n i t e element model f o r the v e r t i c a l v i b r a t i o n modes of the structures both i n a i r and water. However, there was no agreement between the experimental and f i n i t e element r e s u l t s for the frequencies of the coupled h o r i z o n t a l - t o r s i o n a l modes. One reason that the frequencies were d i f f e r e n t f o r the h o r i z o n t a l - t o r s i o n a l coupled modes was that i n the f i n i t e element model the weights were modelled as lumped masses, and they 138 were located on the c e n t e r l i n e of the structure. Thus, the center of mass f o r the f i n i t e element model and the actual structure d i d not correspond. As well, t h i s modelling d i d not account f o r the d i s t r i b u t i o n of the mass that occurred i n the r e a l model. These e f f e c t s only showed up i n the h o r i z o n t a l and t o r s i o n a l modes of v i b r a t i o n . I t was seen ( i n Tables 5.6 and 5.7) that the frequencies f o r the 3 and 4-node h o r i z o n t a l - t o r s i o n a l coupled modes i n a i r d i d not change as the load condition increased; however, they d i d change i n water. From these r e s u l t s i t was concluded that the mass modelling of the load conditions was not adequate. In order to produce better agreement between the h o r i z o n t a l and t o r s i o n a l modes, the f i n i t e element representation of the weights was improved. In t h i s new model the weights were represented using a Thick/Thin S h e l l element that had no s t i f f n e s s but was the c o r r e c t height and weight to model the 30 lb load condition. (This model i s shown i n Figure 5.6 and the r e s u l t s are tabulated i n Table 5.8.) Unfortunately, t h i s new modelling d i d not improve the frequency predictions of VAST s u b s t a n t i a l l y . 139 [SHIP MESH - NST=2I NELMS=8 NBLKHD=19 LOAD=30LB LOAD MODELED BY ELEMENT ! STRUCTURAL : FINITE ELEMENT MODEL ELEMENT TYPES: ALL z — -x 1 2.358 IN. Figure 5.6 - F i n i t e Element Weight Modelling 140 Mode EXP OLD FE NEW FE 2NDV/2NDH 12.5 12.11 11.56 3NDH/3NDT 23.5 49.23 40.27 3NDV 33.25 32.34 30.84 4NDH/4NDT 40.75 59.94 57.34 4NDV 60.75 60.88 58.01 5NDV 92.0 95.31 -Table 5.8 - Frequencies Predicted by Both F i n i t e Element Models of the Weights f o r 30 l b Load Condition This poor p r e d i c t i o n of the h o r i z o n t a l - t o r s i o n a l coupled modes seemed to be a defi c i e n c y i n the f i n i t e element's representation of the structure. The e f f e c t s that are present i n the r e a l structure are : shear d e f l e c t i o n , shear flow, rotary i n e r t i a , and warping. This f i n i t e element accounted f o r the shear d e f l e c t i o n and the rotary i n e r t i a e f f e c t s , but i t was unable to model the shear flow and warping e f f e c t s that occur i n the r e a l structure. 5.3.2. Mode Shapes The F i n i t e Element Method was able to p r e d i c t the mode shapes of v i b r a t i o n f o r the ship model. F i r s t , the mode 141 shapes i n a i r and water from the f i n i t e element r e s u l t s ' were compared f o r t h e i r s i m i l a r i t y ; then, the f i n i t e element mode shapes were compared with the experimental mode shapes f o r the a i r and water r e s u l t s . The mode shapes i n a i r and water f o r the 30 and 40 lb load conditions predicted by the F i n i t e Element Method (see Figure 5.7 - 5.12) were examined. As with the experimental mode shapes, there was l i t t l e d i f f e r e n c e between the mode shapes i n a i r and those i n water f o r the v e r t i c a l modes. However, the h o r i z o n t a l and t o r s i o n a l modes were af f e c t e d by the water, and t h e i r shape was changed considerably. This e f f e c t can be explained. Since the added mass matrix was added to the s t r u c t u r a l mass matrix, the nodes on the f l u i d - s t r u c t u r e i n t e r f a c e acquired mass, which a f f e c t e d the d i s t r i b u t i o n of mass i n the f i n i t e element model. This a d d i t i o n changed the d i s t r i b u t i o n of mass i n the structure, thus changing the mode shape. The mode shape of the h o r i z o n t a l and t o r s i o n a l modes was c l o s e r to the experimental mode shape, since the mass d i s t r i b u t i o n was c l o s e r to the a c t u a l . Local bending did not come into play i n the f i n i t e element mode shapes, since the model was developed to l i m i t t h i s e f f e c t . I f more degrees of freedom had been used i n the regions 142 2 -Node Vertical Bending 3-Node Vertical Bending —*—J 10 IS Station f 4 -Node Vertical Bending Legend x Air I I «« H i W«l»r l i o i H i 20 10 Station f 5-Node Vertical Bending Legend x Atr «C.«« H i . Wafer 40.16 HI Figure 5.7 - Finite Element Vertical Modes of the Ship Model in Vacuum and Water for 30 lb Load Condition 143 2-Node Horizontal Bending Station # 3 -Node Horizontal Bending Legend x Mr tt.11 Hi • Water 7.71 H i > Fx -1 10 Station # is 4-Node Horizontal Bending Legend x Air 4 I . » H i • Water M.41 H i 20 F i / Station # Legend x Air M . « 4 H i • Water 4 t . l t Hz Figure 5.8 - Finite Element Horizontal Modes of the Ship Model in Vacuum and Water for 30 lb Load Condition 144 3-Node Torsion 10 Station # 4 -Node Torsion 16 Legend x Air 41.11 R t • WaUr 14.41 Mt 20 y^u-—1 •ZZ i 1 1 1 1 i 10 Station f 16 Legend x Air » » . M H i - Water 4 t . « « H i 20 Figure 5.9 - Finite Element Torsional Modes of the Ship Model in Vacuum and Water for 30 lb Load Condition 145 -1 2-Node Vertical Bending 10 16 Station # 3-Node Vertical Bending Legend » Air K . ta Hi « Water TJU Ht 20 10 16 Station # 4-Node Vertical Bending Legend x Air MJ4 Hi Water tt.tl Hi 20 10 Station # 5-Node Vertical Bending r 16 Legend « Air M.M Hi - Water »7.«1 Hi 20 Figure 5.10 -Station # Finite Element Vertical Modes of the Ship Model in Vacuum and Water for 40 lb Load Condition Legend » A» aa.M Hi ' Watar M.ta Hi 146 2 -Node Horizontal Bending ,1 i—•—i T » - * t 4 10 Station t 15 3-Node Horizontal Bending Legend x Air t C . M Mi • Water 7.14 H i 20 AS Station # 4 -Node Horizontal Bending Legend x Air 4t.*l Ml « Watar (to Hi X / I I . . ^ 4 ^ > x ^ 10 Station # 16 Legend Watar »I.4T Hi 20 Figure 5.11 - Finite Element Horizontal Modes of the Ship Model in Vacuum and Water for 40 lb Load Condition 147 3 -Node Torsion }Z +. — 1 — - 1 — . • • 10 Station # 4-Node Torsion 16 Legend x Ak 41.12 Hz * W l t w U . O H i 20 ! 1 1 1 1 i i 1 1 1 1 i 1 1 ' • i 10 Station # 16 Legend x All M . M H i W«t* r %%M H> 20 Figure 5.12 - Finite Element Torsional Modes of the Ship Model in Vacuum and Water for 40 lb Load Condition 148 between the bulkheads, l o c a l bending would also have been present i n the f i n i t e element model. The mode shapes i n a i r f o r the 30, 40 and 70 lb load conditions predicted by VAST were compared with the experimental mode shapes. (See Figure 5.13 - 5.21.) In general, there was good agreement between the mode shapes f o r the v e r t i c a l modes of v i b r a t i o n . However, as with the frequencies, there was poor agreement with the h o r i z o n t a l and t o r s i o n a l mode shapes. This was due i n part to reasons explained above. The mode shapes i n water f o r the 30 and 40 l b load condtions predicted by VAST were compared with the experimental mode shapes. (See Figures 5.22 - 5.27.) The v e r t i c a l mode shapes were very s i m i l a r , but the h o r i z o n t a l and t o r s i o n a l modes shapes were not. However, the h o r i z o n t a l and t o r s i o n a l mode shapes were cl o s e r i n water than they were i n a i r . 149 2 -Node Vertical Bending 10 15 Station f 3 -Node Vertical Bending Legend x M . 11.11 Hz t i p tt.i H i Station # 4-Node Vertical Bending Legend x tl. »» .»« Hz 4 t » P »«•»« Hz Station # 5-Node Vertical Bending l i p i l , 7 i H i 10 Station # 16 Legend x fM. M .11 HI * ««« W.O Hi Figure 5.13 Finite Element and Experimental Vertical Modes of the Ship Model Air for 30 lb Load Condition 20 20 150 2 -Node Horizontal Bending 10 Station # 15 3-Node Horizontal Bending Legend * fX. tt.11 H i e»p no ra 20 4 Station # 4 -Node Horizontal Bending Legend x F X . *%a\ M» / -1 10 Station i 16 Legend x f Jt. M . I4 M l 20 Figure 5.14 - Finite Element and Experimental Horizontal Modes of the Ship Model Air for 30 lb Load Condition 151 3 -Node Torsion - i 10 Station # 4 -Node Torsion 16 Legend x F X 41.il Ml " t»P I»« Ml 20 s v ^tr- v i x^ - i i i i i i i i i i i i i I ' l l 10 Station # 16 Legend x tX. **•%* Mi * IIP «°» Mt 20 Figure 5.15 - Finite Element and Experimental Torsional Modes of the Ship Model Air for 30 lb Load Condition 152 2 -Node Vertical Bending 10 15 Station # 3-Node Vertical Bending Legend x nt. W . M H i 4 |£S JH6Jti 10 Station # 4 -Node Vertical Bending Legend X fX. 10 2* M l  4 t « P * » • ' H i 5-Node Vertical Bending Legend x P X - M i l H i •> t i p tB. I t Ml 10 Station # Figure 5.16 - Finite Element and Experimental Vertical Modes of the Ship Model Air for 40 lb Load Condition Legend x fx. M . O l Hz 4 t i p M O H i 20 20 153 2 -Node Horizontal Bending 10 Station i 3-Node Horizontal Bending Legend x fX. 1Q.W Hi * las*?*-\ Xv X 10 Station # 16 4-Node Horizontal Bending Legend x FX. 4 t J » Hi  4 t « P Hi 20 / 10 Station § 16 Legend x tX. «*.»» Ml * t*» »7.n MI 20 Figure 5.17 - Finite Element and Experimental Horizontal Modes of the Ship Model Air for 40 lb Load Condition 154 3-IMode Torsion v x 10 Station f 4-Node Torsion 15 Legend X FX. **X2 Ht 20 10 Station # 15 Legend x yx, e m M i • l i p %1X Hz 20 Figure 5.18 - Finite Element and Experimental Torsional Modes of the Ship Model Air for 40 lb Load Condition 155 2-Node Vertical Bending 4-Node Vertical Bending 10 16 Station # 5-Node Vertical Bending Legend x F X . 44 . M H J t i p 4»,C Hi Figure 5.19 6 10 Station i Finite Element and Experimental Vertical Modes of the Ship Model Air for 70 lb Load Condition Legend x F X . >*•*> Hi « t i p M O H i 20 156 2-Node Horizontal Bending 10 Station # 16 3-Node Horizontal Bending Legend x fX. i.14 Hz  4 E«P *•' Hi 20 > < Station # 4-Node Horizontal Bending Legend x FX. M . n Ml • Exp 1T.TI Hz / 1 1 1 1 1 1 1 1 1 1 i -1 Station # Legend x FX. M.ir HX  4 »»» HJa HI Figure 5.20 - Finite Element and Experimental Horizontal Modes of the Ship Model Air for 70 lb Load Condition 157 Figure 5.21 - Finite Element and Experimental Torsional Modes of the Ship Model Air for 70 lb Load Condition 158 2 -Node Vertical Bending r 10 16 Station # 3 -Node Vertical Bending Legend x F X . T.T1 H» 20 \ J 10 15 Station f 4-Node Vertical Bending Legend ^ Exp M l M l 20 \ X v 10 16 Station # 5-Node Vertical Bending Legend x F X . 4 0 . M H i  4 * » P « > • " H » 20 \ AN X - ^ j / X 10 Station # 15 Legend x F X . M . t t H x Ixp I6.T6 Hx 20 Figure 5.22 - Finite Element and Experimental Vertical Modes of the Ship Model Water for 30 lb Load Condition 159 2-Node Horizontal Bending J 1 • • 10 Station # 15 3-Node Horizontal Bending Legend x F.E- T.Ti Ml • d p T.Tt Ml SO x ^ 10 Station # 16 4 -Node Horizontal Bending Legend x fX. 14.«1 Ml • t i p W.O Hi 20 \ \ J 10 Station # 16 Legend x fX. 41.tl Hi « t i p rt.re HI 20 Figure 5.23 - Finite Element and Experimental Horizontal Modes of the Ship Model Water for 30 lb Load Condition 160 3 -Node Torsion Figure 5.24 - Finite Element and Experimental Torsional Modes of the Ship Model Water for 30 lb Load Condition 161 Figure 5.25 2 -Node Vertical Bending 10 15 Station • 3-Node Vertical Bending Legend x F X . TM H i * t i p TM H i 20 \ 10 16 Station # 4-Node Vertical Bending Legend x F X . tl.11 H i * t » P H i 20 \ 10 16 Station # 5-Node Vertical Bending Legend x F X . t T X t H i * t « P M O H i 20 10 Station # 16 Legend x F X . H i • t i p t i l H i 20 Finite Element and Experimental Vertical Modes of the Ship Model Water for 40 lb Load Condition 162 2-Node Horizontal Bending ^F Station t 3-Node Horizontal Bending Legend t i p T.O Ml Station # 4-Node Horizontal Bending Legend x M . **.OMi - t i p M »6 Ml 4 . \ Station # Legend x tJL. M.4T Ml * t i p «» C Ml Figure 5.26 - Finite Element and Experimental Horizontal Modes of the Ship Model Water for AO lb Load Condition 163 3 -Node Torsion Figure 5.27 - Finite Element and Experimental Torsional Modes of the Ship Model Water for 40 lb Load Condition 164 6. CONCLUSIONS The i n v e s t i g a t i o n of the f l u i d modelling c a p a b i l i t i e s of VAST shows that the extent of the f i n i t e element f l u i d domain i s important. I f t h i s domain i s not extended f a r enough from the v i b r a t i n g structure, the numerical r e s u l t s w i l l be higher than those that would be determined experimentally. With regard to d i s c r e t i z a t i o n of the f l u i d domain, the 20-noded f l u i d element performs better than the 8-noded f l u i d element. In meshes that contain e i t h e r the 8 or the 20-noded f l u i d element but have the same number of degrees of freedom, the 20-noded f l u i d element pr e d i c t s the experimental frequencies more accurately than the 8-noded f l u i d element. As well, meshes that contain the i n f i n i t e element i n t h e i r l a s t row run f a s t e r and have s l i g h t l y b etter r e s u l t s than meshes that do not contain the i n f i n i t e f l u i d element. However, i f the r e a l f l u i d flow has vortex shedding, the frequencies predicted by the f i n i t e element representation of the f l u i d w i l l be below the actual r e s u l t s . In the experimental t e s t i n g of the ship model, the lowest four modes of v e r t i c a l , h o r i z o n t a l , and t o r s i o n a l v i b r a t i o n were i d e n t i f i e d , and the e f f e c t of draught on the frequencies and 165 modes shapes was recorded. This experimental work w i l l prove u s e f u l when compared with numerical methods of p r e d i c t i n g the frequencies and mode shapes i n both a i r and water. The experimental r e s u l t s show that the added mass of the f l u i d has l i t t l e e f f e c t on the lowest mode shapes of v i b r a t i o n ; however, i f there i s l o c a l bending occurring i n the a i r mode shapes, the f l u i d reduces t h i s l o c a l bending. As well, these r e s u l t s show that the e f f e c t of the added mass decreases as the complexity of the mode shapes increases. When the experimentally obtained frequencies and mode shapes f o r the ship model are compared with the numerical p r e d i c t i o n s of VAST, good agreement i s found i n both a i r and water tests f o r the v e r t i c a l v i b r a t i o n modes. However, there i s l i t t l e agreement with the h o r i z o n t a l and t o r s i o n a l modes. This poor agreement i s due to the f i n i t e element's i n a b i l i t y to model the shear flow and warping e f f e c t s encountered i n the h o r i z o n t a l - t o r s i o n a l coupled modes. 166 BIBLIOGRAPHY Ahmah, S., Irons, B.M., and Zienkiewicz, O.C. "Analysis of Thick and Thin S h e l l Structures by Curved F i n i t e Elements." International Journal f o r Numerical Methods i n Engineering. 2 (1970), 419-451. Anderson, G., and Norrand, K. "A Method f o r the C a l c u l a t i o n of V e r t i c a l V i b r a t i o n with Several Nodes and Some Other Aspects of Ship V i b r a t i o n . " Transactions. 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" V i r t u a l I n e r t i a C o e f f i c i e n t s and Response Factors i n Higher Modes of Main H u l l V i b r a t i o n . " Transactions  Royal I n s t i t u t i o n of Naval A r c h i t e c t s . 119 (1977). [25] Kuo, C. " V e r t i c a l Ship V i b r a t i o n Ingestigation with an 11 f t . X ylonite Model." Doctor of Philosophy Thesis, U n i v e r s i t y of Glasgow, 1960. [26] Landweber, L., and deMacaque, M.C. "Added Mass of Two-Dimensional Forms O s c i l a t i n g i n a Free Surface." Journal of Ship Research. 1, No.3 (November 1957), 20-30. [27] Landweber, L., and Macagno, M. "Added Masses of Two-Dimensional Forms by Conformal Mapping." Journal of Ship Research. 11, No. 2 (June 1967), 109-116. [28] Lewis, F.M. "The I n e r t i a of Water Surrounding a V i b r a t i n g Ship." Transactions. Society of Naval A r c h i t e c t s and  Marine Engineers. 37 (1929), 1-20. [29] Lindholm, U.S., Kana, D.D., Chu, W.-H.,and Abramson, H.N. " E l a s t i c V i b r a t i o n C h a r a c t e r i s t i c s of Cantilever Plates i n Water." Journal of Ship Research. 9, No.l June 1965), 11-22. [30] Marcus, M.S. "A Finite-Element Method Applied fo the V i b r a t i o n of Submerged Plates." Journal of Ship Research. 22, No.2 (June 1978), 94-99. [31] Meirovitch, L. A n a l y t i c a l Methods i n V i b r a t i o n . London: Colli e r - M a c M i l l a n Ltd., 1967. [32] Moullin, E.B., Browne, A.D., and Perkins, A.J. "The Added Mass of Prisms F l o a t i n g i n Water." Proceedings. Cambridge  Phi l o s o p h i c a l Society. 1930, 258-272. [33] Murray, M.A., Agar, T.J., and Jennings, A. "Evaluation of V i r t u a l Mass Matrices f o r Ship Hulls Using F l u i d S i n g u l a r i t i e s . " Recent Advances i n S t r u c t u r a l Dynamics. 2 (1980), 403-415. [34] Newman, J.N. Marine Hydrodynamics. Cambridge, Massachusetts and London, England: MIT Press, 1977. [35] Norwood, M.E. "Analysis of S h e l l Structures by Curved F i n i t e Elements." Halifax, Nova Scotia: Martec Limited, 1975. 169 [36] Norwood, M.E. "Application of the F i n i t e Element Method f o r the C a l c u l a t i o n of the Hydrodynamic Added Mass of Marine Pr o p e l l e r Blades." Halifax, Nova Scotia: Martec Limited, 1977. [37] Orsero, P., and Armand, J.L. "A Numerical Determination of the Entrained Water i n Ship V i b r a t i o n s . " International  Journal f o r Numerical Methods i n Engineering. 13 (1978), 35-48. [38] P i e z o e l e c t r i c Accelerometers and V i b r a t i o n Preamplifiers Theory and A p p l i c a t i o n Handbook, Bruel & Kjser, 1978. [39] Restad, K. , Volcy, G.C., Gamier, H. , and Mas son, J.C. "Investigation on Free and Forced Vibrations on an LNG Tanker with Overlapping Pro p e l l e r Arrangements." Transactions. Society of Naval A r c h i t e c t s and Marine  Engineers. 81 (1973), 307-347. [40] Talyor, J.L. "Some Hydrodynamic I n e r t i a C o e f f i c i e n t s . " P h i l o s o p h i c a l Magazine. 9 (1930), 161-183. [41] Thomson, W.T. Theory of V i b r a t i o n with Ap p l i c a t i o n s . 2nd ed., Englewood C l i f f s , New Jersey: Prentice H a l l Inc., 1981. [42] Townsin, R.L. " V i r t u a l Mass Reduction Factors, ' J ' Values f o r Ship V i b r a t i o n Calculations Derivied from Tests with Beams Including E l l i p s o d s and Ship Models." Transactions. Royal I n s t i t u t i o n of Naval A r c h i t e c t s . I l l (1969), 385-397. [43] Troesh, A.W. "Wave-Induced H u l l Vibrations : An Experimental and T h e o r e t i c a l Study." Journal of Ship Reseach. 28, No.2 (June 1984), 141-150. [44] Tse, F.S., Morse, I.E., andHinkle, R.T. Mechanical Vibrations Theory and A p p l i c a t i o n s . 3rd ed. Boston: A l l y n and Bacon Inc., 1978. [45] V i b r a t i o n and Strength Analysis Program VAST : User's Manual Version #4, Martex Limited, Halifax, Nova Scotia 1986. [46] Vorus, W.S., and Hylarides, S. "Hydrodyanmic Added-Mass Matrix of V i b r a t i n g Ship Based on a D i s t r i b u t i o n of H u l l Surface Sources." Transactions. Society of Naval  A r c h i t e c t s and Marine Engineers. 89 (1981) , . [47] Zienkiewicz, O.C. The F i n i t e Element Method. 3rd ed. London: McGraw-Hill Ltd., 1983. 170 [48] Zienkiewicz, O.C., and Bettess, P. "Fluid-Structure Dynamic Int e r a c t i o n and Wave Forces, An Introduction to Numerical Treatment." International Journal f o r Numerical Methods i n Engineering.13 (1978), 1-16. [49] Zienkiewicz, O.C., and Bettess, P. " I n f i n i t e Elements i n the Study of F l u i d Structure I n t e r a c t i o n Problems." 2nd  International Symposium. Computational Methods i n Applied  Science and Engineering. V e r s a i l l e s , France 1975, 133-157. [50] Zienkiewicz, O.C., K e l l y , D.W., and Bettess, P. "The Coupling of the F i n i t e Element Method and Boundary Solution Procedures." International Journal f o r Numerical  Methods i n Engineering. 11 (1977), 355-375. 171 APPENDIX A - VARIATIONAL PRINCIPLE This appendix w i l l show that the v a r i a t i o n a l p r i n c i p l e that governs the f l u i d problem i s given by 1 o -8<f> 2 + - 2 + 8<f> 2" z dx ay. dz dxdydz - 5 ^ <t> dS = 0 (A.l) dn where </> i s the v e l o c i t y p o t e n t i a l of the f l u i d , S i s the n i n t e r f a c e area between the f l u i d and the structure, and v i s the n v e l o c i t y normal to S . The v e l o c i t y p o t e n t i a l <f> may be taken as zero at the boundaries and on the free surface (as discussed i n Section 2.1). This v a r i a t i o n a l p r i n c i p l e can be shown to be the correct one f o r the problem, i f d<f> dx d<}> d<t> dz dxdydz - d± dn 4> dS (A.2) Taking the f i r s t v a r i a t i o n of equation A. 2, the following i s obtained 51 - 4> 6<t> + 4> &4> + <t> S(j> x x y y z z dxdydz - ^ 6<t> dS = 0 (A.3) 3n where the subscript x stands f o r der i v a t i v e by x, and so on. Applying Green's theorem VuVvdV uV vdV + u(Vvn)dS to equation A.3, i t becomes 172 51 V <j> 64> dxdydz + 8<f> (V<f>-n)S<f> dS Noting that V^-n = — , then 3n 51 = V <f> S<f> dxdydz + an S<f> dS ^ S</> dS = 0 an 3n 6<f> dS =0 51 V <j> 8<f> dxdydz = 0 (A.4) and for arbitrary 8<f> V24> = 0 (A. 5) with the appropriate boundary conditions (as discussed in Section 2.1). 173 APPENDIX B - F L U I D E L E M E N T S H A P E F U N C T I O N S This appendix w i l l present the shape functions of the three f l u i d elements that were used i n the numerical work. As we l l , there were two i n t e r f a c e elements that were used to connect the f l u i d elements to the structure. These elements assured that the boundary condition of (2.10a) was met on the f l u i d - s t r u c u t r e i n t e r f a c e . B.l 8-NODED FLUID E L E M E N T This element was a three-dimensional element with 8 nodes, one at each corner. This element had l i n e a r shape functions given by N - - ( 1 + « )( 1 + i»n ) ( 1 + fC ) ( B . D i g (Figure B . l presents a general representation of t h i s element.) 174 B.2. 4-NODED INTERFACE ELEMENT This element was used to connect the 8-noded f l u i d element to the structure that the f l u i d was surrounding. The shape functions f o r t h i s element were N - - ( 1 + r,rj ) ( 1 + r r ) (B . 2 ) 1 4 (Figure B . l presents a general representation of t h i s element.) B.3. 20-NODED FLUID ELEMENT This element was a three-dimensional element with 20 nodes, one at each corner and one at each mid-side. This element had quadratic shape functions and was an isoparametric element. corner nodes: N - - ( 1 + « ) ( 1 + r/n )( « + TJT} + r r - 2 ) ( B . 3 ) i g i mid-side nodes: N. • = — ( 1 - £ 2 ) ( 1 + ijrj ) ( 1 + r r . ) £ - 0 ( B . 4 ) i , i l l 4 N - - ( 1- r,2 ) ( 1 + « ) ( 1 + r r . ) 9 . - 0 ( B . 5 ) i . i i i 4 N A - - ( l - r 2 )( i + r r t )( i + nvi ) r 4 - o (B.6 ) 4 (Figure B .2 presents a general representation of t h i s element.) 175 5 a) 8-Noded Fluid Element 1 b) 4-Noded Interface Element Figure B.l - 8-Noded Fluid Element and 4-Noded Interface Element B.4. 8-NODED INTERFACE ELEMENT This element was used to connect the 20-noded f l u i d element to the structure that the f l u i d was surrounding. The shape functions f o r t h i s element were corner nodes: N ± = - ( 1 + VVL )( + f f ± - 1 ) (B.7) 4 mid-side nodes: N - - ( 1- r?2 )( 1 + f r ) n - 0 (B.8) i 2 * i N - - ( 1- f 2 )( 1 + t,V ) f - 0 (B.9) 2 (Figure B.2 presents a general representation of t h i s element.) B.5. 8-NODED INFINITE FLUID ELEMENT This element was a three-dimensional i n f i n i t e element with 8 nodes, one at each corner. In the two n o n - i n f i n i t e coordinates, the shape functions were standard, while i n the i n f i n i t e d i r e c t i o n £ , s p e c i a l shape functions were used. The v e l o c i t y p o t e n t i a l was given as NNE 4, = I i j f (B.10) e i i 177 a) 20-Noded Fluid Element 1 b) 8-Noded Interface Element Figure B.2 - 20-Noded Fluid Element and 8-Noded Interface Element 178 where N = N.N i 1 i with N - - ( 1 + Vr, )( 1 + rr ) 8 and N a - 2 ( e + ^  ) e + i + r 2 i N 2 ( ? + ± ) + 1 + S\ * i > 0 (B.ll) (B.12) (B.13) (B.14) Note that £ N— 1, which was the normal isoparametric criterion for an element's shape functions. (Figure B.3 presents a general representation of this element.) 179 Figure B.3 - 8-Noded Infinite Fluid Element 180 APPENDIX C - THICK/THIN SHELL ELEMENT MASS AND STIFFNESS MATRIX This appendix w i l l present the isoparametric formulation of the curved s h e l l element a v a i l a b l e i n the F i n i t e Element Program VAST as presented by Norwood [35] and developed by Ahmah et a l [1] . This element i s knowng as the Thick/Thin S h e l l element IEC - 1 i n VAST. This element has 8 nodes i n t o t a l , 4 corner nodes and 4 mid-side nodes, with 5 l o c a l degrees of freedom per node, 3 t r a n s l a t i o n s , and 2 ro t a t i o n s . By drawing from the development of large c u r v i l i n e a r elements f o r three-dimensional analysis, t h i s element overcomes some of the previous approximations to geometry of the structure and the neglection of shear deformation. As well, the formation of the element takes advantage of the f a c t that even f or th i c k s h e l l s , the 'normals' to the middle surface remain p r a c t i c a l l y s t r a i g h t a f t e r deformation. Consider the three-dimensional isoparametric element (shown i n Figure C . l ) . The external faces of the element are curved, while the sections across the thickness are generated by s t r a i g h t l i n e s . I f £ and r? are two c u r v i l i n e a r coordinates i n 181 Figure C l - Curved the middle plane of the element, and f is a linear coordinate in the thickness direction varying between +1 and -1, then a relation between the local curvilinear coordinates and the xyz Cartesian coordinate system can be established as i , and zero at a l l other nodes. If the shape functions are chosen so that compatibility is achieved between elements at their interface, then the shape functions may be written as corner nodes : mid-side nodes : N - - (1 + «*<* )(1 - v2) U - ±1, rj - 0) (C.3) 2 N = — (l + !j» )(l - Z2) (n - ±1, £ - 0) (c.4) 2 It is convenient to rewrite equation C.l in a form specified by a vector connecting the upper and lower surfaces and the coordinates of the mid-surface. Thus N '. - - (1 + «,)d + r,r, ) m + rjrj -i . i i i i 4 1) (C2) m i d + I N (Cfj) £ V, 2 3 i (C.4) with (C.5) 183 Assuming the s t r a i n s normal to the mid-surface are n e g l i g i b l e , the displacement at any point i n the element can be defined by the three Cartesian components of the mid-surface node displacement i and two rotations of the modal vector V about r 3i orthogonal d i r e c t i o n s normal to i t . The displacement of the element i s given by where u, v, and w are displacements i n the Cartesian coordinate system, t i s the s h e l l thickness at node i , v and v are J i 2i l i orthogonal u n i t vectors normal to V , and and /9 are sc a l a r r o t a t i o n s about the vectors v and v , r e s p e c t i v e l y . 2i l i r J As an i n f i n i t e number of vectors V and V can be 2i l i generated normal to the vector V , a p a r t i c u l a r scheme has to be devised to ensure a unique d e f i n i t i o n , and the choice of scheme i s quite a r b i t r a r y . I f i i s a un i t vector along the x axis, then V = i x V (C.7) 1 3 Which makes the vector V perpendicular to the plane defined by the d i r e c t i o n V 3 and the global x axis. As has to be orthogonal to both and V^, i t i s given as V = V x V (C.8) 2 3 1 The required u n i t vectors v , v , and va may be formed from the 184 vectors V , V , and V3 by dividing them by their scalar length, given by i = 1. 2, 3 (C9) The strains and stresses have to be defined in order to determine the elemental stiffness matrix [k] . If the strains normal to the surface f = constant are assumed to be negligible, using the assumptions of Shell Theory, i t is necessary that the components of the strain and stress in two orthogonal directions tangent to the surface f = constant exist. To establish these directions, a normal z'can be erected to the surface and two other orthogonal axes x' and y' , but tangent to the surface. The strain components of interest are then 3u' ax' av' a y {€' } X 7 , , x y 7 , , X z 7 , , y z a u ' + av' ay' ax' aw' + au' (CIO) ax' az' aw' + av' ay' az' with the strains in the z' direction neglected, so as to be consistent with the Shell Theory assumptions. 185 The stresses corresponding to these strains- are r e l a t e d by the e l a s t i c i t y matrix [D'], such that \ = [D']{£' } ( C U ) The 5 x 5 matrix [D' ] f o r i s o t r o p i c materials i s given by [D'] -(1 - v ) 0 0 0 0 0 0 \-v V 0 0 1-1/ 0 2k sym. l - i 2k (C12) where E and v axe Young's modulus and Poisson's r a t i o , r e s p e c t i v e l y . The fa c t o r k included i n the l a s t two shear terms i s a c o r r e c t i o n to improve the shear displacement approximation and i s taken equal to 1.2. In the standard f i n i t e element way, the elemental s t i f f n e s s matrix may now be determined from [k] = [B] [D'][B]dxdydz V where [B] i s the so c a l l e d strain-displacement matrix U' } - [B]{6} (C.13) (C.14) and {6) contains the elemental displacements u , v , w , a , and e i i i i 8 f o r node i . A l l that remains i s to express equation C.13 i n 186 terms of the curvilinear coordinates i, ij, and f. This can be accompished by a number of transformations. Equation C.6 relates the global displacements u, v, and w to the curvilinear coordinates. The derivative of these displacements with respect to the global x, y, and z axes is given by du 9v 3w H 3<f 3£ au av aw ax ax ax a_u av aw a y a y a y au av 3w az az az [J] (C.15) 3u 3v 3w dr) dr] dr) 3u 3v 3w ar Bi di where [J] is the Jacobian matrix defined by a_x ay a_z di di di 5x 8y dz df) dr) dr] 3x 3y 3z sr sr 3f which can be expressed in terms of the coordinate definition of equation C.4. Thus, for every set of curvilinear coordinates, the corresponding global displacements may be calculated numerically. (C16) However, a further transformation is needed to relate the local displacement directions x' , y' , and z' to the global coordinates x, y, and z. The directions of the local axes 187 have to be determined. The vector V defines a vector that is 3 normal to the surface r = constant and can be found as the vector product of any two vectors tangent to the surface, such that 8j_ az ay az di Br, dr, 3t* ' 3x " ' 3x ' dr, I a y > *= -d£ dz a_z . an . v. (C17) dx dz dx dz dV 3£ d£ dr, dx dy dx dj_ d£ dr, dr, d£ Following the process described above, two perpendicular vectors Vz and may be formed and reduced to unit magnitude. A matrix of unit vectors may be formed. This is actually the direction cosine matrix [e] - t v v 2 , v 3 ] ( c - 1 8 ) The global derivatives of displacements u, v, and w can be transformed to the local dervatives of the local orthogonal displacements by au' 3v' a_w' dx' dx' dx' du' av' aw' dy a y a y du' av' aw' dz' dz' az' [e] au av 3w dx dx dx du a_v aw dy a y a y du av aw dz 3z az [6] (C19) The elemental stiffness matrix [k] may be written in terms of the curvilinear coordinates £, r,, and f as 188 [k] [B]T[D' ][B] I J l d ^ d r / d f (C.20) 1 - l - l where | j | i s the determinate of the Jacobian matrix. The elemental mass matrix [m] may be formed i n a s i m i l a r manner using the shape functions given above, as - i .1 - i [m] p[N] [N] J d r j d r / d f (C.21) - l - l - l 189 APPENDIX D - DIMENSIONAL ANALYSIS This appendix w i l l present the d e t a i l s of the dimensional analysis of the model that was performed. The structure of a ship i s very complex and defies rigorous closed-form a n a l y s i s . Damping, shear, and rotary i n e r t i a e f f e c t s i n the ship structure are d i f f i c u l t i f not impossible to p r e d i c t . Thus, i t was believed that the only e f f e c t i v e way to inv e s t i g a t e the dynamic behaviour of the ship h u l l was to perform experiments with a scaled model, which could be analysed for i t s v i b r a t i o n c h a r a c t e r i s t i c s both i n a i r and water. In t h i s way, the experimental r e s u l t s from the model could be compared with various numerical methods, and the accuracy and shortcomings of the various techniques could be determined. For t h i s reason, a simple model was constructed of a c r y l i c , with a constant semicircular c r o s s - s e c t i o n over i t s length of 96 inches. In order to construct a model of a ship, many va r i a b l e s had to be considered. Of prime importance was the model's a b i l i t y to reproduce the dynamic behaviour of the f u l l s i z e ship. However, other issuses were important too, and, as i s 190 often the case i n model s c a l i n g , not a l l of these issues could be s a t i s f i e d by the same model. Which of these v a r i a b l e s were most important must be determined. As well, the model was often forced to meet p h y s i c a l constraints. I t was of no use to construct a model i f i t s s i z e prevented i t from being tested or i f the d e t a i l i n the model made the t e s t i n g so complex that i t consumed large amounts of time. And f i n a l l y , the material of construction needed to be considered. I f the model was made of s t e e l , the only way the model could have s i m i l a r dynamic behaviour as the f u l l s i z e ship was i f the dimensions of the model were the same as the ship. C l e a r l y , t h i s was not reasonable. Therefore, another construction material had to be chosen. In order to determine which properties were important and what material should be used, a dimensional analysis [19,25]of the r e l a t i o n s that govern the v i b r a t i o n of a ship structure was conducted, as presented below. The following discussion applies s t r i c t l y to a simply supported beam of a uniform cross-section. The frequency of v i b r a t i o n i s given as 2 |- -| 1 / 2 El m Hz (D.l) where f} 1 i s a constant that depends on the end conditions of the beam, I i s the length of the beam, E i s Young's modulus, I i s the moment of i n e r t i a of the cross-section, and m i s the mass per un i t length of the beam. Now l e t the subscribes m and f denote the 191 model and f u l l scale ship, r e s p e c t i v e l y , and l e t the v a r i a b l e K denote the r a t i o of the f u l l scale v a r i a b l e to the model v a r i a b l e of the term of i n t e r e s t . Then co K i 1 f f 2 r co I E m f £ m I E m m m f 1/2 (D.2) or K W K 2 L K K I E K 1/2 where CO m K co CO K L 1 m K K = —- , K - — m The moment of i n e r t i a of the cro s s - s e c t i o n i s given as 1 = 1 + Ad 2 o Ad 2 - t L d 2 which i s approximately I So t L d K — — — 1 t L d 2 m m ro (D.3) (D.4) (D.5) (D.6) (D.7) which i s K = K K 3 (D.8) 1 t L i f the model i s geometrically s i m i l a r except f o r the s h e l l thickness. Now examine K . I f the mass of the structure i n water M can be expressed as m = ( 1 + C ) m w s (D.9) then 192 ms = Ap (D.10) which leads to A K - — (D.ID M A m Using equations D.8 and D.ll, equation D.3 can be expressed as K K K2 = (D.12) L Appling geometric similarity gives e - e (D.13) f m where e is the strain. Note that c = - (D.14) E and that a - ^  (D.15) I where M is the bending moment, and y is the distance from the neutral axis to the point on the section where the stress is to be determined. Assuming that the beam is simply supported, the bending moment is given by Wl2 M = — (D.16) 8 where W is the load per unit length of the beam. W is given by W = pk (D.17) By using equation D.13 and substituting in to i t equations D.15, D.16, and D.17, the following equation can be developed 193 K 5 K - — (D.18) K E or K 2 K - — (D.19) fc K E Thus, the r e l a t i o n s that hold are K K K 2 = (D.12) R K 3 L and K 2 K - — (D.19) t K E which lead to K 2 (D.20) R K L Some of these parameters are determined by p h y s i c a l l i m i t a t i o n s . The thickness of the model's h u l l i s determined by s i z e l i m i t a t i o n s , therefore t - 0.050" min and from the endeavour t =0.5" f The length of the model i s determined by the size of t e s t i n g tank, therefore I - 6' - 10' max and from the endeavour I - 300' f 194 The r a t i o of the Young's modulus i s determined by the materials a v a i l a b l e . I f the material chosen i s a c r y l i c , then K = 50 E and the maximum length of a sheet i s eight feet, and the minimum thickness of a sheet of a c r y l i c i s 0.079 inches; therefore K - 37.5 L K - 6.3 t I f the cross-section chosen i s semi-circular with D = 8.392" t - 0.079" L - 96" then the f u l l s i z e ship has Beam = 26' Length - 300' (See Figure D.l f o r a diagram of the model with dimensions.) 195 Length (between end bulkheads) 96" (2438mm) Beam 8.391" (213mm) Hull Thickness 0.079" (2mm) Bulkhead - Thickness 0.236" (6mm) - Diameter 8.3125" (211mm) Endcaps - Thickness 0.079" (2mm) - Radius 4.196" (107mm) Unloaded Weight 5.56 lb (2530 grams) Figure D.l - Ship Model 196 APPENDIX E - INSTRUMENTATION LIST This appendix gives a l i s t of the instrumentation used : 1. Bruel & Kjaer Type 4332 Piezoelectric Accelerometer 2. Bruel & Kjaer Type 4370 Piezoelectric Accelerometer 3. Bruel & Kjaer Type 2635 Charge Preamplifier 4. Bruel 6c Kjaer Type 4291 Accelerometer Calibrator 5. Nicolet 660A Dual Channel FFT Frequency Analyser 6. Tektronix 4662 Digital Plotter 7. Tektronix 5103N Digital Oscilloscope 8. Commodore PC 10-11 Personal Computer (IBM Compatible) 9. VAX 11/750 Computer 10. Digital VT 101 Terminal with Retro-Graphics 11. PCB 208 A03 Impact Hammer 197 APPENDIX F - MODAL ANALYSIS PROGRAM This appendix w i l l describe the program MAP that was wr i t t e n by the author to a s s i s t with the c o l l e c t i o n and processing of the experimental data. This program was wri t t e n i n Turbo PASCAL on an IBM Personal Computer. The MAP program i s a menu driven program with graphic c a p a b i l i t i e s and needs to be used i n conjunction with a Hercules Graphics Card. To use t h i s program, the disk with the MAP program should be put i n drive a and a blank disk or a disk with data f i l e s created by the MAP program should be put i n drive b. A f t e r logging onto the a drive, MAP should be typed to the a:> prompt. This w i l l s t a r t the program. The t i t l e page w i l l then f l a s h up on the screen. Typing any key w i l l generate the next command. The program w i l l then ask f o r the input of the f i l e p r e f i x of the f i l e to create or look at. A f t e r t h i s point the program w i l l present the main menu. This menu o f f e r s the options of : Transfer from f f t , Send to f f t , Plot mode shape, Hardcopy, Change f i l e group, and E x i t . Option one, Transfer from f f t , w i l l be high l i g h t e d . Moving around i n the menu i s achieved by pressing the up or down arrow keys. Once the option desired i s highlighted, the return key 198 should be pressed. F.1 OPTION - TRANSFER FROM F F T This option allows the t r a n s f e r of information i n the N i c o l e t onto floppy disks f o r storage and further processing. In t h i s option the baud rate of the t r a n s f e r may be set. Note that the N i c o l e t baud rate has to correspond and must be set before any information i s processed by the N i c o l e t . A baud rate of 9600 i s the f a s t e s t , and i t i s the default for the MAP program. Once t h i s option has been choosen the program w i l l ask you which f f t i s desired f o r t r a n s f e r r a l . (Only the N i c o l e t procedures have been written), the N i c o l e t f f t i s the default. Once the N i c o l e t has been chosen as the f f t , the program asks which s t a t i o n t h i s information i s for ( i . e . Station 0 - 20). Then i t asks what the baud rate i s (default 9600), and, once the return has been pressed, the t r a n s f e r i s started. The information that i s t r a n s f e r r e d i s the data from the N i c o l e t buffers 0, E, 4, 5, 6, and 7. These buffers contain : front panel s e t t i n g s , average status, averaged channel A power spectrum, averaged cross spectrum r e a l , averaged cross spectrum imaginary, and averaged channel B 199 power spectrum information, respectively. When the information has finished transferring, the program returns to the main menu. F.2 OPTION - SEND TO F F T This option is very similar to option Transfer from fft; in fact, i t is just the opposite. F.3 OPTION - PLOT MODE SHAPE This option will allow two tasks to be performed : 1) Look at data, which mimics the Nicolet functions RMS Spectrum, Transfer Function and Coherence; and 2) Plot the mode shape. If this option is chosen, the program puts up another menu that asks which task is desired. If the Look at data option is chosen, the program will ask which station is perferred and then which function is desired : RMS spectrum, Transfer function, or Coherence. Once the function is chosen, i t is displayed on the screen. A cursor may be moved back and forth, 200 using either the right and left arrow keys or the home and page up keys. The two arrow keys move the cursor one box, while the home and page up keys move the cursor 10 boxes. Exiting from this section is acheived by pressing the enter key. If the Plot mode shpae option is chosen, the program will ask how many stations there are in the datafile, and what the frequency of interest is. Once these questions have been answered, the program begins to scan the datafile for the correct information, namely the Transfer Funtion and the Phase angle. When the program has scanned throught the datafile i t will plot the mode shape on the screen. It will then ask i f this mode shape is correct. If i t is i t will write this information to a fi l e that can be used by LOTUS for further processing. If the mode shpe is not coorect, the program will ask which station is wrong and i t will then try to further smooth the data. FA OPTION - H A R D C O P Y This option has not been written, but i f i t was included in the program, i t would allow a hard copy of the mode shape and a table listing the tabular mode shape information to be 201 p l o t t e d or pr i n t e d . F.5 OPTION - CHANGE FILE GROUP This option allows a change of f i l e p r e f i x while s t i l l i n the program. F.6 OPTION - EXIT This option allows the e x i t to DOS. 202 APPENDIX G - MATERICAL PROPERTIES O F ACRYLIC G.l DENSITY The density of the acrylic used to construct the model was determined by weighing a sample with known dimensions. The sample was 12 x 12 inches, 0.079 inches (2 mm) thick, and weighed 0.00324 lb. Its density was p - 0.0369 lb/in 3 or -5 2 4 9.55x10 lb.sec /in , given as F P gV where p is the density of acrylic, F is the force or weight (lbs), g is the acceleration of gravity, and V is the sample's volume. G.2. DETERMINATION OF YOUNG'S MODULUS Three different tests were performed to determine Young's modulus. The first test involved taking a sample of the acrylic that the hull was made of and mounting two strain gauges on i t . This sample was then put into pure bending (as shown in Figure G.l). In this configuration, readings from the strain 203 ^ Guage #1 Guage #2 Figure G.l - Strain-Guage Set Up and Bending Test of Ship Model 204 gauges were measured and recorded (as shown i n Table G . l ) . S t r a i n Load grams Gauge#l Me Gauge#2 Me Moment lbs* i n Stress p s i E p s i V 0 0 0 0 0 - -157 432 -162 1.99 214 495545 0.375 270 762 -278 3.42 368 483144 0.365 385 1106 -410 4.88 525 474650 0.371 498 1458 -526 6.31 679 465736 0.361 613 1780 -656 7.77 836 469579 0.369 726 2140 -778 9.21 990 462584 0.364 841 2476 -908 10.66 1147 463141 0.367 954 2832 -1028 12.10 1300 459328 0.363 841 2554 -934 10.66 1147 448997 0.366 726 2202 -806 9.21 990 449560 0.366 613 1896 -694 7.77 836 440850 0.366 498 1528 -560 6.32 679 444400 0.367 385 1216 -448 4.88 525 431713 0.368 270 854 -320 3.42 368 431096 0.375 157 534 -202 1.99 214 400890 0.378 0 82 -70 0 0 - -Table G.l - Results from Strain-Gauge Test 205 Using these r e s u l t s , Young's modulus and Poisson's r a t i o were determined from E = - ^ -bh 2£ where E was Young's modulus, P was the load, 1 was the length between the load and the support (which was 5.750 inches), b was the width of the sample (b = 1.00 inches), h was the thickness of the sample (h = 0.23622 inches (6mm)), and e was the s t r a i n on X gauge # 1. Poisson's r a t i o was given by 6 y v = - — € x with e = s t r a i n from gauge # 2. The average Young's modulus was E =» 455,541 p s i , and the average Poisson's r a t i o was v = 0.368. The second method involved simply supporting the actual ship model at both ends, loading i t i n the middle, and then measuring the d e f l e c t i o n . Then, a f i n i t e element model was run, and Young's modulus was v a r i e d u n t i l the r e s u l t s f o r the d e f l e c t i o n matched the experimental r e s u l t s . 206 Weight lb D e f l e c t i o n inches 1 0.021 2 0.042 3 0.065 4 0.085 5 0.105 Table G.2 - Results f o r D e f l e c t i o n This method r e s u l t e d i n a Young's modulus of 302,000 p s i . The t h i r d method was a dynamic t e s t . S t r a i n gauge # 1 was connected to the N i o c l e t 660A FFT analyser, and the piece of a c r y l i c was simply supported and made to v i b r a t e i n i t s f i r s t mode by an impulse i n the middle of the beam. The frequency f o r the f i r s t mode was determined (by examining Figure G.2) as w = 29.25 Hz l From t h i s information, and using the expression _1 2 EI 2TT 1 Young's modulus was determined to be 622,056 p s i . 207 16 #AVG 2 9 . 2 5 0 0 0 HZ 1 . 5 7 - 0 3 V V L N T 0 . 2 A Figure G.2 - Natural Frequency of Simply Supported Beam 208 

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