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Dynamics of neutrally buoyant inflatable viscoelastic cantilevers in the ocean environment Poon, David Tat-Sang 1976

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DYNAMICS OF NEUTRALLY BUOYANT INFLATABLE VISCOELASTIC CANTILEVERS IN THE OCEAN ENVIRONMENT by DAVID TAT-SANG POON B.A.Sc., Univers i ty of B r i t i s h Columbia, 1974 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of Mechanical Engineering •We accept th i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA October, 1976 . * 0 David Tat^Sang Poon, 1976 In presenting this thesis in part ia l fulfilment of the requirements for an advanced degree at the University of Br i t i sh Columbia, I agree that the Library shal l make i t 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 representatives. It is understood that publication, in part or in whole, or the copying of this thesis for f inancial gain shal l not be allowed without my written permission. •j DAVID TAT-SANG POON Department of Mechan ica l Eng inee r ing The U n i v e r s i t y of B r i t i s h Columbia , Vancouver, B . C . , Canada, V6T 1W5. Date i i ABSTRACT St a t i c s and dynamics of the n e u t r a l l y buoyant i n f l a t e d v i s c o -e l a s t i c c a n t i l e v e r s c o n s t i t u t i n g a submarine detection system i s i n -vestigated. Two geometries of the thin-walled beams are considered: uniform c i r c u l a r c y l i n d r i c a l and c i r c u l a r tapered. The s t a t i c f l e x u r a l behaviour of the beam i s studied using the three parameter v i s c o e l a s t i c s o l i d model which y i e l d s material properties f o r the mylar-polyethylene-mylar p l a s t i c f i l m used. Results of a d e t a i l e d experimental program are also presented to substantiate v a l i d i t y of the a n a l y t i c a l model. This i s followed by free v i b r a t i o n analyses of the i n f l a t e d c a n t i l e v e r s i n the ocean environment accounting f o r the added i n e r t i a and nonlinear hydrodynamic drag. For the uniform c y l i n d r i c a l beam, t h i n - s h e l l theories are employed to account f o r the i n f l a t i o n e f f e c t s on the free v i b r a t i o n c h a r a c t e r i s t i c s . A s i g n i f i c a n t feature of the analysis i s the reduction of the s h e l l equations (the membrane, Fliigge's, and Herrmann-Armenakas') into a s i n g l e equation which i s s i m i l a r i n form to that f o r a v i b r a t i n g beam with rotary i n e r t i a e f f e c t s . The natural frequencies obtained are compared with the experimental r e s u l t s and those predicted by the Rayleigh-Ritz method i n conjunction with the Washizu and membrane s h e l l theories. The analyses show, and experi-mental program confirms, that Fliigge's s h e l l equation i n the reduced form i s capable of p r e d i c t i n g free v i b r a t i o n behaviour quite accurately. However, the reduction technique should be applied with care, since i i i i n several cases i t leads to misleading r e s u l t s (e.g. i n the case of Herrmann-Armenakas theory). For the tapered case the elementary beam theory i s used to predict t h e i r n a t u r a l frequencies. Next, the dynamical response of the uniform and tapered c a n t i l e v e r s to root e x c i t a t i o n , at the fundamental wave frequency and i t s second harmonic, i s studied. The governing nonlinear equations are analyzed by taking two terms of the assumed Fourier se r i e s s o l u t i o n . Results suggest that f o r the case of the simple harmonic e x c i t a t i o n , the nonlinear hydrodynamic drag introduces no superharmonic components into the response. For low f o r c i n g frequencies t y p i c a l of the ocean environment, an increase i n taper r a t i o tends to reduce the t i p amplitudes. However, for frequencies above the fundamental, the response c h a r a c t e r i s t i c s are completely reversed. The analysis provides valuable information concerning the system parameters leading to c r i t i c a l response and hence should prove u s e f u l i n the design of i n f l a t a b l e members employed i n the submarine detection system. i v TABLE OF CONTENTS Chapter Page 1. INTRODUCTION . . . . . 1 1.1 Preliminary Remarks 1 1.2 Literature Review . 4 1.3 Purpose and Scope of the Study 15 2. STATICS OF NEUTRALLY BUOYANT INFLATED VISCOELASTIC CANTILEVERS 17 2.1 Theoretical Analysis 17 2.1.1 Uniform Cyl indrica l Beam 17 2.1.2 Tapered Beam 23 2.2 Experimental Program 25 2.2.1 Test Equipment and Procedures 25 2.2.2 Test Models 27 2.3 Results and Discussion 27 2.3.1 Uniform Cyl indrical Beam 30 2.3.2 Tapered Beam 32 2.4 Concluding Remarks 36 3. FREE VIBRATION OF NEUTRALLY BUOYANT INFLATED CANTILEVERS 41 3.1 Uniform Cyl indrical Beam Analysis 42 3.1.1 Reduced Shell Equation Approach 42 (a) Formulation 42 (b) Solution for Zero Drag 54 (c) Perturbation Solution 57 3.1.2 Rayleigh-Ritz Method 65 V C h a p t e r Page 3.2 T a p e r e d Beam A n a l y s i s 72 3.3 R e s u l t s and D i s c u s s i o n 77 3 .3 .1 U n i f o r m C y l i n d r i c a l C a n t i l e v e r 78 3 .3 .2 Tape red C a n t i l e v e r 81 3.4 C o n c l u d i n g Remarks 84 4. FORCED VIBRATION OF NEUTRALLY BUOYANT INFLATED VISCOELASTIC CANTILEVERS 87 4.1 U n i f o r m C y l i n d r i c a l Beam 88 4.2 T a p e r e d Beam 96 4 . 3 R e s u l t s and D i s c u s s i o n 101 4 .4 C o n c l u d i n g Remarks . 107 5. CLOSING COMMENTS 109 5.1 Summary o f C o n c l u s i o n s 109 5.2 Recommendat ion f o r F u t u r e Work 110 BIBLIOGRAPHY . 112 APPENDIX I - DERIVATION OF WATER INERTIA TERM IN SHELL EQUATIONS 120 APPENDIX I I - ORTHOGONALITY CONDITION FOR EQUATION (3 .22b ) . . 123 APPENDIX I I I - RAYLEIGH-RITZ MATRIX ELEMENTS FOR WASHIZU'S SHELL THEORY 125 APPENDIX IV - FREQUENCY EQUATION FOR TAPERED BEAMS USING 1 - , 2 - , AND 3-TERM, APPROXIMATIONS 131 APPENDIX V - REDUCED MEMBRANE AND HERRMANN-ARMENAKAS EQUATIONS 136 APPENDIX V I - POTENTIAL ENERGY EXPRESSION FOR THE MEMBRANE THEORY 141 v i L I ST OF ILLUSTRATIONS FIGURES Page 1-1 S c h e m a t i c d i a g r a m o f a s ubmar i ne d e t e c t i o n s y s t e m u s i n g an a r r a y o f n e u t r a l l y b u o y a n t i n f l a t a b l e c a n t i l e v e r s . . . 3 1 - 2 S c h e m a t i c d i a g r a m o f t h e p r o p o s e d p l a n o f s t u d y . . . . 16 2 - 1 Geometry o f f l e x u r e o f an i n f l a t e d c i r c u l a r c y l i n d r i c a l beam 18 2-2 Th ree p a r a m e t e r v i s c o e l a s t i c s o l i d mode l 22 2 -3 Geometry o f f l e x u r e o f an i n f l a t e d t a p e r e d beam . . . . 24 2-4 E x p e r i m e n t a l s e t - u p 26 2-5 Sandwiched m a t e r i a l and d e t a i l s o f t h e end cap 28 2-6 A t y p i c a l c r e e p - r e l a x a t i o n c u r v e f o r t h e s a n d w i c h e d i n f l a t e d beam 29 2-7 Compa r i s on o f a n a l y t i c a l and e x p e r i m e n t a l r e s u l t s f o r t h e s t a t i c d e f l e c t i o n o f u n i f o r m c y l i n d r i c a l beams 31 2 -8 E f f e c t o f i n t e r n a l p r e s s u r e on t h e s t a t i c d e f l e c t i o n s o f u n i f o r m c y l i n d r i c a l beams 33 2-9 T i p d e f l e c t i o n h i s t o r i e s f o r two L/d r a t i o s 34 2-10 Compa r i s on o f a n a l y t i c a l and e x p e r i m e n t a l r e s u l t s f o r t h e s t a t i c d e f l e c t i o n s o f t a p e r e d beams 35. 2-11 E f f e c t o f i n t e r n a l p r e s s u r e on t h e s t a t i c d e f l e c t i o n s o f t a p e r e d beams 37 2-12 I n s t a n t a n e o u s t i p d e f l e c t i o n as a f u n c t i o n o f L/d 38 r 2 - 13 T i p d e f l e c t i o n a t t = 30 m i n . as a f u n c t i o n o f L/d 39 r 3 - 1 Geometry and c o - o r d i n a t e s y s t e m f o r a c i r c u l a r c y l i n d r i c a l s h e l l 44 v i i F i g u r e s Page 3-2 V e r t i c a l and h o r i z o n t a l d i s p l a c e m e n t components . . . . 50 3-3 Geometry o f m o t i o n o f a t a p e r e d c a n t i l e v e r 73 3 - 4 V a r i a t i o n o f e i g e n v a l u e s w i t h t a p e r r a t i o 82 4 - 1 S c h e m a t i c r e p r e s e n t a t i o n o f t h e ene r g y c o n t a i n e d i n t h e s u r f a c e waves o f t h e oceans ~ ( f r o m R e f e r e n c e 86) 89 4 -2 Response o f v i s c o e l a s t i c c a n t i l e v e r s t o s i m p l e h a r m o n i c r o o t e x c i t a t i o n 103 4 - 3 Response o f v i s c o e l a s t i c c a n t i l e v e r s t o e x c i t a t i o n w i t h a s m a l l s e c o n d - h a r m o n i c component : (a) £ = 0 . 4 , 0 .8 104 (b) £ = 1 105 4 -4 E f f e c t o f t e m p e r a t u r e on t h e s t i f f n e s s o f p o l y e t h y l e n e 106 I - l Geometry o f l a t e r a l d i s p l a c e m e n t o f a s h e l l s e c t i o n . . 120 TABLES 3.1 Compar i s on be tween a n a l y t i c a l l y and e x p e r i m e n t a l l y o b t a i n e d f r e q u e n c i e s f o r u n i f o r m beams (Hz) . . . . . 79 3.2 C o m p a r i s o n be tween a n a l y t i c a l l y and e x p e r i m e n t a l l y o b t a i n e d f r e q u e n c i e s f o r t a p e r e d beams (Hz) 83 v i i i ACKNOWLEDGEMENT The author wishes to thank h i s colleagues f o r t h e i r valuable suggestions and c r i t i c i s m s . He i s p a r t i c u l a r l y g r a t e f u l to Dr. Arun K. Misra f o r the s e l f l e s s sharing of h i s knowledge and experience. Above a l l , the author i s deeply indebted to Dr. V.J. Modi, h i s supervisor, f o r the guidance given throughout the preparation of th i s t h e s i s . His help and encouragement have made the accomplishment possible. F i n a l l y , to h i s wife A l i c e , the author owes h i s deep gratitude for her u n f a i l i n g patience and understanding during the many cheerless moments of t h i s undertaking. The i n v e s t i g a t i o n reported herein was supported by the Defence Research Board of Canada, Grant No. 9550-38, and the National Research Council of Canada, Grant No. A-2181. ix LIST OF SYMBOLS coefficients in the reduced shel l Equation (3.7) cross-sectional area of water inside an inflated beam cross-sectional area of water inside an inflated beam at root nondimensionalizing constants, Equation (3.16) 2 shel l bending st iffness, Eh/(1-V ) coefficients in solution for zero drag, Equations (3.22) and (3.24) drag and added inert ia coefficients, r e s p e c t i v e l y „ d *. coefficients in the eigenfunction expansion of s , Equation (3.33b) cosine-componeht of forced response, Equation (4.6a) cosine-component of excitation, Equation (4.6c) determinant of operator [L], Equation (3.2) minors of determinant D Young's modulus parameters in l inear viscoelastic so l id model complex modulus tip load hydrodynamic drag force acting on an element of abeam, Equation (3.15) total hydrodynamic force acting on an element of a beam, Equation (3.58b) parameter, pa (1-v )/E shear modulus amplitudes of forced response, Equation (4.10a) moment of i n e r t i a of the beam cross-section i n t e g r a l s , Equations ( 4 . 8 ) moment of i n e r t i a of the beam cross-section at root creep compliance i n shear normalizing m u l t i p l i e r , Equation (3.25) length of c a n t i l e v e r beam matrix elements of d i f f e r e n t i a l operator [L], Equation (3.1) a x i a l prestress due to i n t e r n a l pressure, pa/2 circu m f e r e n t i a l prestress due to i n t e r n a l pressure, dimensionless pressure, pa/(Eh) amplitudes of forced response, Equation (4.10b) sine-component of forced response, Equation (4.6b) sine-component of e x c i t a t i o n , Equation (4.6d) k i n e t i c energy of s h e l l p o t e n t i a l energy of s h e l l radius of beam trac e r s , i = 1, 2, 3 diameter of beam diameter of beam at root diameter of beam at t i p x i a x i a l and ci r c u m f e r e n t i a l s t r a i n s of a s h e l l s e c t i o n , r e s p e c t i v e l y , Equations (3.50) thickness of beam taper r a t i o , (d -d )/d r t r i n t e r n a l pressure dimensionless a x i a l co-ordinate, x/a, Chapter 3 time time v a r i a b l e , Equation (3.29) transverse v e l o c i t y of a beam f l e x u r a l displacement of beam v i s c o e l a s t i c f l e x u r a l displacement of beam orthogonal displacement components i n a x i a l , circum-f e r e n t i a l , and r a d i a l d i r e c t i o n s , r e s p e c t i v e l y , Figure 3-1 a x i a l co-ordinate f o r c y l i n d r i c a l s h e l l , Figure 3-1 distance from the middle surface of a s h e l l section p o t e n t i a l function eigenfunctions of a c a n t i l e v e r eigenfunctions of a clamped-pinned beam dimensionless frequency, Equation (3.57b) damping parameter of beam, Equation (3.17b) mode shape of zero drag s o l u t i o n , Equation (3.22b) constant, Equation (3.40b) dimensionless v i s c o e l a s t i c damping c o e f f i c i e n t x i i Y(w) v i s c o e l a s t i c loss factor, Equation (4.Id) 6 r e a l part of complex modulus, Equation (4.1c) £ , e A a x i a l and circumferential middle surface s t r a i n s , x o respectively T) dimensionless transverse displacement, w/d 6 circumferential co-ordinate, Figure 3-1 K , K- a x i a l and circumferential curvatures, respectively, x 0 Equations (3.52) dimensionless natural frequency of tapered beam, ((1+C )p A lAd2/(EI ) ) 1 / 2  v m w w l r • r A, A', A" solutions to frequency equations (3.23) p r i n c i p a l stretches, k = .1, 2, 3 eigenvalues of a cantilever V Poisson's r a t i o £ dimensionless distance from root (clamped end) of a cantilever p, P w mass densities of tube w a l l and water, respectively O^, 0"Q , a^g vibratory stresses, Equations (3.47) i i i 0"x, OQ , O^ g i n i t i a l stresses due to i n t e r n a l pressure, Equation (3.49) a.. stress tensor T dimensionless time, Equation (3.16) eigenvalues of a clamped-pinned beam ' in t e g r a l s , p. 100 0) frequency Dots indicate d i f f e r e n t i a t i o n with respect to T. Primes, unless other-wise s p e c i f i e d , denote d i f f e r e n t i a t i o n with respect to 1 1. INTRODUCTION 1.1 Preliminary Remarks There has been considerable attention given l a t e l y to the behaviour of t h i n p r e s s u r e - s t a b i l i z e d s h e l l structures, commonly c a l l e d i n f l a t a b l e s h e l l s . Besides t h e i r a b i l i t i e s to r e s i s t loads e f f i c i e n t l y by normal t e n s i l e stresses, i n f l a t a b l e s h e l l s have the advantages of being lightweight, compact, and c o l l a p s i b l e , implying ease i n transportation and erection f o r service. The state-of-the-art i n i n f l a t a b l e s h e l l research i s summarized by Leonard 1, who, i n hi s conclusions, recommended more e f f o r t s to study: (i ) i n t e r a c t i o n of the s h e l l s to t h e i r embedding media; ( i i ) material r e l a t e d problems, e.g., determining the best materials from both the strength and the environmental s t a b i l i t y aspects. I n f l a t a b l e s h e l l s have already exhibited t h e i r p o t e n t i a l i n the design of s t r u c t u r a l components f o r aerospace and oceanographic systems. Brauer has discussed i n considerable d e t a i l s , which i n -clude design and performance data, a v a r i e t y of i n f l a t a b l e structures having aerospace a p p l i c a t i o n . They include the propellent tank of the Atlas i n t e r c o n t i n e n t a l b a l l i s t i c m i s s i l e , the gigantic U.S. balloon s a t e l l i t e s Echo I and I I , and the experimental paragliders as a decelerating device for atmospheric re-entry. On the other hand, n e u t r a l l y buoyant i n f l a t e d structures have been proposed for 2 u n d e r w a t e r a p p l i c a t i o n s l i k e s u b m a r i n e d e t e c t i o n , o c e a n o g r a p h i c s u r v e y , l i f t i n g s u r f a c e s o f h y d r o f o i l s , e t c . Of p a r t i c u l a r i n t e r e s t i s t h e u se o f sonobuoys i n s u b m a r i n e d e t e c t i o n s y s t e m s . Sonobuoys a r e p a s s i v e l i s t e n i n g d e v i c e s c o n v e n -t i o n a l l y hou sed i n a c y l i n d e r about 0.9m (3 f t . ) l o n g and 12.7 t o 15 .3 cm (5 t o 6 i n . ) i n d i a m e t e r . The c o n t a i n e r s a r e d e p o s i t e d f r o m an a i r c r a f t i n t h e a r e a o f i n t e r e s t . Upon h i t t i n g t h e w a t e r , a hyd rophone a t t a c h e d by a c a b l e t o t h e f l o a t i n g c o n t a i n e r i s r e l e a s e d . A l l a c o u s t i c s i g n a l s r e c e i v e d a t t h e hyd rophone a r e r e l a y e d b a c k t o t h e a i r c r a f t v i a a t r a n s m i t t e r . S i n c e t h e t a r g e t e m i t s s i g n a l s on an unknown t i m e - b a s e , a t l e a s t t h r e e o r f o u r hyd rophones a r e needed t o l o c a t e i t i n two o r t h r e e d i m e n s i o n s , r e s p e c t i v e l y . The sonobuoy has a p r e s e t l i f e t i m e a f t e r w h i c h i t t u r n s i t s e l f o f f and s i n k s . E x t e n s i v e r e s e a r c h ha s e s t a b l i s h e d t h a t t h e e f f i c i e n c y o f t h i s o p e r a t i o n can be i m p r o v e d by u s i n g an a r r a y o f i n f l a t a b l e t u b e s , e a ch c a r r y i n g a hyd rophone a t one end and j o i n e d t o a pump-equ ipped c e n t r a l head a t t h e o t h e r ( F i g u r e 1 - 1 ) . The t a r g e t c an t h e n be l o c a t e d t h r o u g h p r o c e s s i n g o f s i g n a l s r e c e i v e d by t h e a r r a y , p r o v i d e d t h e p o s i t i o n and o r i e n t a t i o n o f t h e a r r a y a r e known. The o p t i m a l d e s i g n o f s u c h a s u b m a r i n e d e t e c t i o n s y s t e m r e q u i r e s a know ledge o f i t s r e s p o n s e t o e n v i r o n m e n t a l l o a d s s u ch as ocean c u r r e n t s , .waves and o t h e r l o c a l d i s t u r b a n c e s . The g e n e r a l T o a i r c r a f t T r a n s m i t t e r Figure 1-1 Schematic diagram of a submarine detection system using an array of n e u t r a l l y buoyant i n f l a t a b l e c a n t i l e v e r s motion of the system is quite complex because of the large number of degrees of freedom involved 3 : the r o l l i n g and spatial osc i l lat ions of the buoy superimposed on i t s dr i f t ing , three dimensional motion of the f lexible cable, and the coupled inplane and out of plane f lexural-rotational motions of the array i t s e l f . 1.2 Literature Review • The interest in inflatable structures is of re lat ive ly recent or ig in . Leonard, Brooks and McComb1* were probably the f i r s t to calculate the collapse and buckling loads for inflated cy l indr ica l cantilever beams. Buckling in the form of wrinkles occurs when compressive bending stress balances the tensile stress due to internal pressure. As the load is increased the wrinkles progress around the cross section, and collapse occurs when they propagate a l l the way to the other extreme. Assuming the collapsing root to fold l ike 3 a plast ic hinge, the c r i t i c a l t ip load was found to be F=TTpa / L , where p i s internal pressure, and a and L are the radius and length of the beam, respectively. The test data presented showed good agreement with this equation. Stein and Hedgepeth5 derived a theory to predict the structural behaviour of wrinkled membranes. It was shown that membrane structures retain much of their stiffness a t loads substantially above that at which wrinkling f i r s t occurs. Comer and Levy 6 studied the t ip deflections and maximum stresses of inflated cy l indr ica l cantilever beams for loads between incipient buckling and f i n a l collapse. Topping 7 analyzed the buckling of 5 i n f l a t e d plates and columns by deriving a r e l a t i o n between shear s t i f f n e s s and i n f l a t i o n pressure accounting for beam edge e f f e c t s . He concluded that the i n f l a t i o n pressure can be treated as an e f f e c t i v e shear modulus. A l l these inv e s t i g a t o r s observed that the f l e x u r a l s t i f f n e s s i s e s s e n t i a l l y independent of i n t e r n a l pressure. This i s true i f the deformation i s small and the y i e l d strength of the material i s not exceeded. A theory for the case of small deformation superposed on known f i n i t e deformations of a t h i n , homogeneous, e l a s t i c membrane was formulated by Corneliussen and S h i e l d 8 . Small deformations of a c i r c u l a r c y l i n d r i c a l tube subjected to f i n i t e homogeneous ex-tension and i n f l a t i o n were considered as an example. A c i r c u l a r membrane s h e l l prestressed by i n t e r n a l pressure and by a x i a l tension has been studied by Huang 9. The behaviour of the membrane subjected to a r a d i a l l i n e load was considered i n d e t a i l . Using the theory of incremental deformations, Douglas 1 0 investigated the e f f e c t of f i n i t e i n f l a t i o n on the subsequent response of a c i r c u l a r c y l i n d r i c a l c a n t i -lever to bending loads. The analysis accounted for changing geometry and material properties during i n f l a t i o n . It was observed that f or small i n f l a t i o n the pressure and i n f l a t i o n s t r e t c h are nearly l i n e a r l y r e l a t e d . The e f f e c t of i n t e r n a l pressure on the influence c o e f f i c i e n t s of c y l i n d r i c a l s h e l l s subjected to axisymmetric edge loads was studied by Narasimhan 1 1. Using the stress-deformation equations formulated by Nachbar 1 2, curves_of influence c o e f f i c i e n t s f o r various edge loads were obtained. Koga 1 3 also presented a system of l i n e a r c o n s t i t u t i v e 6 equations f o r small s t r a i n deformations superimposed upon a known state of f i n i t e deformation. The equations were applied to an i n -f l a t e d c i r c u l a r c y l i n d r i c a l membrane subjected to pure bending. These in v e s t i g a t i o n s are l i m i t e d to materials with time-independent properties. A knowledge of the hydrodynamic forces acting on a v i b r a t i n g c y l i n d e r i s e s s e n t i a l to the study of i t s underwater dynamics. A number of model and prototype investigations devoted to force data i s summarized by W i e g e l 1 4 . The conventional Morison's type equation, derived independently by Morison et a l . 1 5 and Keulegan and Carpenter 1 6, assumes that the t o t a l hydrodynamic force can be obtained by l i n e a r l y com-bining the drag and added i n e r t i a components, and i s v i r t u a l l y u n i v e r s a l l y used i n t h i s class of i n v e s t i g a t i o n s . Keulegan and Carpenter investigated the drag and i n e r t i a c o e f f i c i e n t s of cylinders i n simple s i n u s o i d a l currents and correlated them with the period parameter U T/D, where U i s the maximum i n t e n s i t y of the s i n u s o i d a l m ' m J current, T i s the period of the wave, and D i s the diameter of the cyl i n d e r . They observed that the drag and mass c o e f f i c i e n t s show opposite behaviours over the range of the period parameter considered, but the sum of the two forces deviates r e l a t i v e l y l i t t l e from the 1 1 7 average value. Stelson and Mavis studied the v i r t u a l mass of long c i r c u l a r cylinders o s c i l l a t i n g i n water and found that f o r cylinders with large length to diameter r a t i o s the added mass approaches u n i t y , as predicted by p o t e n t i a l flow analysis. J e n 1 8 observed, experimentally, 7 that the forces exerted by uniform periodic waves i n r e l a t i v e l y deep water give an average added mass c o e f f i c i e n t of 1.04. L a i r d et a l . 1 9 and Toebes et a l . assumed a constant mass c o e f f i c i e n t and included a l l i t s deviations from unity i n the v a r i a t i o n of the drag c o e f f i c i e n t . The forces on cylinders having constant acceleration and deceleration have been measured by L a i r d et a l . Although the drag c o e f f i c i e n t was found to change, the v a r i a t i o n s were not s u b s t a n t i a l . Toebes et a l . measured the hydrodynamic forces on a transversely o s c i l l a t i n g c y l i n d e r with i t s axis perpendicular to the mean flow d i r e c t i o n . The drag c o e f f i c i e n t was observed to deviate s u b s t a n t i a l l y from the t h e o r e t i c a l value i f the v i b r a t i o n a l frequency was close to the Strouhal frequency. However, the deviations were small for frequencies f a r from the Strouhal 2 1 frequency. Using a s i m i l a r concept, Protos et a l . also considered a fixed apparent mass and studied the v a r i a t i o n of the remaining force with the frequency r a t i o ( r a t i o of the natural frequency of the c y l i n d e r to the Strouhal frequency). In another study, L a i r d et a l . 2 2 demonstrated that wave forces on a c i r c u l a r c y l i n d e r could be influenced s i g n i f i c a n t l y by eddy-shedding from neighbouring cy l i n d e r s . In contrast to the extensive l i t e r a t u r e on apparent mass e f f e c t s f o r a r i g i d c y l i n d e r i n a f l u i d , the corresponding studies for a f l e x i b l e c y l i n d e r are r e l a t i v e l y scarce. Landweber 2 3' l h and Warnock 2 5 investigated dynamics of an e l a s t i c c ylinder i n an incom-p r e s s i b l e , i n v i s c i d f l u i d to determine the apparent mass e f f e c t s . However, the p o t e n t i a l flow assumed discounted any hydrodynamic damping forces. The f l e x u r a l v i b r a t i o n of an i n f l a t e d c y l i n d r i c a l 8 c a n t i l e v e r i n a i r has been studied by Douglas 2 6, and Corneliussen and S h i e l d 8 . M i s r a 3 , on the other hand, investigated an i n f l a t e d v i s c o e l a s t i c c a n t i l e v e r v i b r a t i n g i n water. In the study a d e t a i l e d analysis of the c y l i n d r i c a l c a n t i l e v e r i n the presence of hydrodynamic drag and a t e n s i l e follower force was given. However, the elementary beam theory employed does not account f o r the c i r c u m f e r e n t i a l stress induced by the i n t e r n a l pressure. Modifications of the Timoshenko beam equations f o r thin-walled tubes to account f o r normal pressure and Poisson r a t i o e f f e c t s were made by Simmonds 2 7. The study was con-fined to open-ended tubes as there was no a x i a l i n i t i a l stress terms present. To f u l l y account f o r the stresses a r i s i n g due to i n t e r n a l pressure, one has to resort to t h i n s h e l l theory. However, only a small portion of the vast amount of a v a i l a b l e l i t e r a t u r e on s h e l l v i -brations i s concerned with the beam-bending mode of i n t e r e s t here. The reason may be that, as pointed out by F o r s b e r g 2 8 , f o r r e l a t i v e l y long s h e l l s without i n t e r n a l pressure, the beam-bending mode analysis can be considerably s i m p l i f i e d without much loss of accuracy by considering the s h e l l as a thin-walled beam and applying the beam theory. Kornecki has shown that, for the beam-bending mode of 3 0 s h e l l s without i n t e r n a l pressure, the Goldenveizer s h e l l equations reduce to an equation very s i m i l a r to the one for the transverse vi b r a t i o n s of a beam with rotary i n e r t i a included. Incorporating i n i t i a l s tress e f f e c t s due to i n t e r n a l pressure 9 requires a generalization of the equations of motion f o r t h i n s h e l l s . Various s h e l l equations accounting f o r i n i t i a l stresses have been c o l l e c t e d and l i s t e d i n a monograph by L e i s s a 3 1 . Because of the r e l a t i v e mathematical s i m p l i c i t y , the vast majority of studies made to date have dealt with s h e l l s having t h e i r boundaries supported by "Shear-pjLaphragms" (SD). Straight-forward methods for handling other edge conditions, in c l u d i n g an exact procedure, are av a i l a b l e but have been sparingly applied because of the great deal of e f f o r t required. R e i s s n e r 3 2 and V l a s o v 3 3 have independently shown that i n the case of the SD-SD s h e l l s , use of the Donnell-Mushtari theory and neglecting tangential i n e r t i a , give r i s e to simple formulas, which d i r e c t l y r e l a t e frequencies f o r the unloaded and uniformly prestressed s h e l l s . Using the Flugge theory, Greenspon 3 1* also arr i v e d at a r e l a t i o n between the natural frequencies of the pressurized and unloaded s h e l l s , and concluded that an i n t e r n a l pressure w i l l always give an increase i n nat u r a l frequency. This i s i n variance with the r e s u l t s given by Baron and B l e i c h 3 5 f o r the beam-bending mode. However, f o r other modes t h e i r r e s u l t s approach the values predicted by Greehspon. DiGiovanni and Dugundji 3 6, using the Washizu s h e l l e q u a t i o n s 3 1 , analyzed pressurized SD-SD s h e l l s by the exact method. For the beam-bending mode the frequency was found to be v i r t u a l l y independent of i n t e r n a l pressure, e s p e c i a l l y f o r short s h e l l s . Even for long s h e l l s the increase i n frequency was very moderate and of minor importance. Fung et a l . investigated the e f f e c t of pressure on the frequency of vib r a t i o n s of SD-SD c y l i n d r i c a l s h e l l s . The 10 frequency equation was established on the basis of the Timoshenko-V o s s 3 1 theory. However, for c e r t a i n ranges of s h e l l parameters and for modes having small number of circumferential waves t h e i r r e s u l t s , as claimed by Herrmann and Shaw 3 8, cannot be r e l i e d on to y i e l d more accurate r e s u l t s than those obtained by R e i s s n e r 3 9 using a shallow s h e l l theory. In these i n v e s t i g a t i o n s , where the s h e l l s are supported at both ends by shear-diaphragms (SD-SD), the equations of motion and the end conditions are exactly s a t i s f i e d by simple displacement functions. For other boundary conditions the problem i s considerably more complicated and r e l a t i v e l y few r e s u l t s are a v a i l a b l e . The exact procedure to determine frequency parameters for clamped-clamped s h e l l s was described by Seggelke 1* 0. Unfortunately, the r e s u l t s presented do not specify the s h e l l theory used, thus making usefulness of the data questionable. Experimental r e s u l t s f o r clamped-clamped c i r c u l a r c y l i n d r i c a l s h e l l s subjected to i n t e r n a l pressure were given by Mixson and Heer**1, Miserentino and Vosteen 1* 2, and N i k u l i n 1 * 3 . The Rayleigh-i • Ritz method or i t s equivalent have been used by several in v e s t i g a t o r s to study the motion of s h e l l s with various boundary conditions'*'*" 5 2. Sewall and Naumann"*1* used the Rayleigh-Ritz technique with beam functions and the Goldenveizer-Novozhilov s h e l l theory to obtain frequencies for clamped-free s h e l l s and compared them with experimental r e s u l t s . They employed seven terms i n the assumed mode shapes to obtain conver-gence of the R i t z procedure. Results were also obtained by Resnick and Dugundji 1* 5 using an energy method equivalent to Rayleigh-Ritz, i • 11 beam functions, and the Sanders s h e l l theory. A good agreement between theory and experiment was found only f o r modes with more than f i v e c i r cumferential waves. Extensive numerical r e s u l t s f o r clamped-free s h e l l s were obtained by Sharma and Johns'* 6 - 4 8 using the R i t z method and the Fliigge s h e l l equations. Displacement functions were assumed to be a combination of the clamped-free and clamped-pinned beam functions. I t may be pointed out that the above approximate i n v e s t i -gations were confined to the case of zero i n i t i a l s t r e s s . The energy method was also used by Arya et a l . 1 * 9 to study the dynamic c h a r a c t e r i s t i c s of f l u i d - f i l l e d s h e l l containers. Open c y l i n d r i c a l s h e l l s clamped at the base and free at the top were studied under the empty condition as w e l l as when f i l l e d to varying water depths. The mode shapes and frequencies of free v i b r a t i o n of c a n t i l e v e r c y l i n d r i c a l s h e l l s p a r t i a l l y or completely f i l l e d with f l u i d were studied by Baron and S k a l a k 5 0 . Numerically obtained v i r t u a l mass c o e f f i c i e n t s were presented as functions of the height of the f l u i d i n a tank. The free v i b r a t i o n s of orthotropic and i s o t r o p i c f l u i d - f i l l e d c y l i n d r i c a l s h e l l s using the Rayleigh-Ritz method were also studied by J a i n 5 1 and S t i l l m a n 5 2 . A number of workers have used s h e l l theories to study the added mass e f f e c t s of c y l i n d r i c a l s h e l l s i n a f l u i d media. Workers l i k e Greenspon 5 3, Warburton 5 1*, B l e i c h and B a r o n 5 5 , Herrmann and R u s s e l 5 6 , and B e r g e r 5 7 are amongst the i n t e r e s t e d i n v e s t i g a t o r s . From space consideration, i t i s a t t r a c t i v e to look at the 1 2 array configuration, mentioned i n the context of submarine detection, consisting of tapered i n f l a t a b l e c a n t i l e v e r s . Hence a b r i e f review of the l i t e r a t u r e on tapered beams, which have interested researchers for a long time, would be appropriate. Conway et a l . 5 8 calculated the frequencies f o r truncated-cone c a n t i l e v e r bars for a number of d i f f e r e n t boundary conditions. The f i r s t three natural frequencies and the corresponding modes of v i b r a t i o n of c a n t i l e v e r beams for numerous d i f f e r e n t tapers were presented by Housner and K e i g h t l e y 5 9 . Gaines and V o l t e r r a 6 0 derived approximate formulas for upper and lower values of the three lowest n a t u r a l frequencies f o r transverse v i b r a t i o n s of c a n t i l e v e r bars of v a r i a b l e cross sections. Other investigations on the free v i b r a t i o n s of taper c a n t i l e v e r beams include those by Cranch and A d l e r 6 1 , S i d d a l l and I s a k s o n 6 2 , and P i n n e y 6 3 . A l l these workers made use of elementary beam theory and e f f e c t s of i n i t i a l stress were not considered. If the i n i t i a l stress e f f e c t s are to be included, s h e l l theory has to be used. In t h i s case the tapered c a n t i l e v e r beam becomes a c o n i c a l s h e l l , the analysis of which i s f a r from simple. Weingarten 6 investigated the case of the simply-supported c o n i c a l s h e l l frustum subjected to i n t e r n a l and external pressures. The Galerkin method was applied to.reduce the Donnell c o n i c a l s h e l l equations to the form of a frequency determinant. The agreement between t h e o r e t i c a l and experi-mental frequencies was poor f o r modes with a small number of circum-f e r e n t i a l waves, mainly due to the nature of the Donnell theory used. 13 Goldberg et a l . 6 5 ' 6 6 developed a general numerical i n t e g r a t i o n computer program and demonstrated i t s a p p l i c a b i l i t y to the problem of the clamped-clamped c o n i c a l s h e l l subjected to pressure. Unfortunately whether the pressure was i n t e r n a l or external was not indicated. The i n f l a t a b l e members of the array under study are generally made of p l a s t i c films which are v i s c o e l a s t i c and ex h i b i t time-dependent behaviour. This i n t e r n a l damping e f f e c t has received l i t t l e a ttention i n the past while studying the dynamic c h a r a c t e r i s t i c s of i n f l a t e d beams. The forced l a t e r a l v i b r a t i o n of a uniform c a n t i l e v e r Timoshenko beam with i n t e r n a l damping was studied by L e e 6 7 . On the other hand, Lei s s a and Hwee 6 8 investigated the problem of forced v i b r a t i o n s of simply-supported Timoshenko beams with viscous damping. Numerical r e s u l t s showed that the amplitude responses f o r the Timoshenko beam to be considerably larger than the corresponding simple beam predictions. Baker et a l . 6 9 introduced various i n t e r n a l and external damping forces into the equations f o r free v i b r a t i o n of elementary beams. Solutions to the equations were obtained by energy methods i n conjunction with a d i g i t a l computer. The free v i b r a t i o n of a c a n t i l e v e r was also studied by Paidoussis and des Troi s Maisons 7 0, who represented the v i s c o e l a s t i c e f f e c t s by the two parameter Kelvin-Voigt model. F i n a l l y , the steady-state v i b r a t i o n of beams with nonlinear material damping was analyzed 7 1 by Fu using a perturbation technique. Numerical r e s u l t s obtained were compared to the ones given by P i s a r e n k o 7 2 . Under operating conditions the submarine detection system w i l l 14 be continually disturbed by the marine environment. The buoy at the water surface i s excited by wind, surface waves, and currents. The cable and the hydrophone array are perturbed by currents, sub-surface turbulence, and i n t e r n a l waves. Each subsystem responds to these disturbances with d r i f t motion together with t r a n s l a t i o n a l and r o t a -t i o n a l o s c i l l a t i o n s . The cable subsystem also s u f f e r s from mechanical deformation. The general response of the whole assembly i s governed by i n t e r a c t i o n s between the coupled subsystems. The equilibrium configurations and dynamical behaviour w i l l be s i m i l a r to those of a towed v e h i c l e system although the d r i f t i n g v e l o c i t y i s usually much lower. The towed v e h i c l e problem has drawn considerable i n t e r e s t for the l a s t two decades. Applications of t h i s system range from mooring of buoys to towing of g l i d e r a i r c r a f t s . A v a r i e t y of techniques have been employed i n studying these systems —• methods of c h a r a c t e r i s t i c s , l i n e a r i z a t i o n procedures, equivalent lumped mass approach, f i n i t e element method, etc. A survey of these a n a l y t i c a l methods for the dynamic simulation of cable-buoy systems i s given by Choo and C a s a r e l l a 7 3 . An i n t e r e s t i n g d e s c r i p t i o n on the advances i n Canadian towed system research i s presented by Eames and Drummond7lf. The f a i r i n g of the cable to reduce drag and i t s associated problems are discussed i n depth. An extensive l i t e r a t u r e review on cable dynamics and towed systems i s given by M i s r a 3 . The configuration of the towed body i n the submarine detection system under consideration i s more complicated than previous studies 15 due to coupling of the array of f l e x i b l e v i s c o e l a s t i c tubes. The f l e x i b i l i t y of the array legs i s an important parameter i n the analysis and can a f f e c t the s t a b i l i t y of the system considerably. 1.3 Purpose and Scope of the Study In the dynamical study of any system, a knowledge of the s t i f f n e s s and response c h a r a c t e r i s t i c s of the constituent members nece s s a r i l y forms a pr e r e q u i s i t e . With t h i s i n mind and because of the complex nature of the problem, a small subsystem i s selected. An e f f o r t i s made to investigate i n d e t a i l the s t a t i c and dynamical be-haviour of the n e u t r a l l y buoyant, i n f l a t e d , v i s c o e l a s t i c beams forming the hydrophone array. Two geometries of the thin-walled v i s c o e l a s t i c beams are considered: uniform c i r c u l a r c y l i n d r i c a l and c i r c u l a r tapered. The s t a t i c solutions of the e l a s t i c , i n f l a t e d c a n t i l e v e r s are f i r s t extended to the v i s c o e l a s t i c case f o r moderately large i n f l a t i o n . This i s followed by free v i b r a t i o n analyses of the i n f l a t e d c a n t i l e v e r members i n the ocean environment. For the uniform c y l i n d r i c a l beam, the s h e l l theory i s used to account f o r the i n t e r n a l pressure e f f e c t s on the s t a t i c s and free v i b r a t i o n c h a r a c t e r i s t i c s . For the tapered case elementary beam theory i s used to pr e d i c t t h e i r n a t u r a l frequencies. Next, the dynamical response of the v i s c o e l a s t i c , i n f l a t e d c a n t i l e v e r s (uniform c y l i n d r i c a l and tapered) to root e x c i t a t i o n i s studied. The governing nonlinear equations are analyzed by taking two terms of the assumed Fourier s e r i e s s o l u t i o n . Experimental r e s u l t s are obtained to substantiate the s t a t i c and free v i b r a t i o n analyses. Figure 1-2 i l l u s t r a t e s schematically the plan of study. VISCOELASTIC NEUTRALLY BUOYANT INFLATABLE TUBES Uniform C y l i n d r i c a l S t a t i c s Experiment Tapered Dynamics 1 Theory E l a s t i c Solution Free V i b r a t i o n Analysis Forced V i s c o e l a s t i c Response V i s c o e l a s t i c Solution Theory Experiment Figure 1-2 Schematic diagram of the proposed plan of study 17 2. STATICS OF NEUTRALLY BUOYANT INFLATED VISCOELASTIC CANTILEVERS The present chapter investigates s tat ic deflections of neutrally buoyant inflated viscoelastic c ircular beams. It is assumed that internal pressure effects are relat ively small such that the change in material properties is ins ign i f i cant 7 . Consequently only geometrical variations due to inf lat ion are treated. The viscoelastic deflections are obtained using the three parameter sol id model in conjunction with the correspondence principle . The analysis is substantiated through an experimental program employing a large number of uniform and tapered beam models made of mylar-polyethylene sand-wiched films. The objectives of this chapter are: (i) to predict stat ic deflections of c ircular cy l indr ica l uniform and tapered beams made from plast ic films; ( i i ) to obtain information concerning material constants for the specified mylar-polyethylene sandwiched films; ( i i i ) to investigate the internal pressure effects on the stat ic behaviour of the inflated cantilevers. 2.1 Theoretical Analysis 2.1.1 Uniform Cyl indrical Beam Consider a thiri-walled inflated c ircular cy l indr ica l cantilever 18 (Figure 2-1) of i n i t i a l length , diameter d^, wa l l thickness h^, and i n t e r n a l pressure p. The dimensions at any instant during i n f l a t i o n are r e l a t e d to t h e i r i n i t i a l values by the p r i n c i p a l stretches as follows, * * * L = AJLQ > d = ^2^0 A N C * ^ = 3^0 ' (2.1) As the bulk modulus of the materials under study i s r e l a t i v e l y large, i n c o m p r e s s i b i l i t y can be sa f e l y assumed, and hence, A * A * A * = 1 . (2.2) The p r i n c i p a l stresses f o r the c y l i n d r i c a l beam can be shown to b e 1 0 z , w ^ Internal pressure = p l Figure 2-1 Geometry of flexure of an i n f l a t e d c i r c u l a r c y l i n d r i c a l beam 19 2 -2 * . . * a n = W i + * - i x i = p d / 4 h ' ( 2 , 3 a ) 2 -2 CT22 = V"*1 X2 + < f ) - l X 2 = p d / 2 h ' ( 2 , 3 b ) * 2 *~ 2 a 3 3 - * 0 + * l X 3 4 * - l X 3 = 0 ( P ) ' ( 2 - 3 c ) where <f>^  are sca l a r functions of the diagonal s t r e t c h matrix. Note that i s small compared to and 0^ (the r a t i o being of the order of h/d). Setting 0".^ to zero, M i s r a 3 has shown that, f o r moderate stretches up to 40% increase i n diameter, A* = 1 (2.4a) X2 - (l-pd 0/2Gh 0) ± / H , (2.4b) A* = 1/X* , (2.4c) where G i s the shear modulus of the undeformed material. As _ * * pdg/2GhQ « 1, the expressions for and A^  can be s i m p l i f i e d to get A2 = l+pd 0/8Gh 0 , (2.5a) A* = l-pd 0/8Gh 0 . (2.5b) In actual p r a c t i c e , a change i n length w i l l be small compared to 20 changes i n diameter and thickness. Hence Equations (2.4) represent a good approximation to changes i n dimensions due to i n f l a t i o n . To account f o r the time dependent properties of the material, Equations (2.5) can be modified using a concept s i m i l a r to the corres-pondence p r i n c i p l e 7 5 , A*(t) = l + p d 0 J s ( t ) / 8 h Q , (2.6a) A*(t) = l - p d 0 J s ( t ) / 8 h 0 , (2.6b) where p i s the step pressure applied at t=0 and J g ( t ) i s the shear creep compliance of the material. The dimensions a f t e r a long time are thus given by L f = L Q , (2.7a) d f = d 0[l+pd 0J s(«>)/8h 0] , (2.7b) h f = h 0[l-pd 0J s(°°)/8h 0] . (2.7c) The c a n t i l e v e r beam i s now allowed to undergo bending de-formations. I t i s assumed that the i n t e r n a l pressure i s s u f f i c i e n t l y large to make the resultant stress t e n s i l e everywhere so that no wrinkles appear on the beam. The resultant a x i a l stress on an element with coordinates (x,y,z) i s obtained by superposing the stresses due 21 to bending and i n f l a t i o n pressure, i . e . , 0 U = F(L-x)z/I+pd f/4h f , (2.8) where F i s the load and I the c r o s s - s e c t i o n a l moment of i n e r t i a about a transverse axis, I = Trd^hf/8 From elementary beam theory the curvature i s given by .2 — | = -F(L-x)/EI . (2.9) dx Integrating twice and applying the boundary conditions at x = 0, leads to the s t a t i c d e f l e c t i o n expression of an e l a s t i c c a n t i l e v e r , w(x) = -(FL 3/6EI)[(x/L) 2(3-x/L)] = W(x)/E . (2.10) For step loads the Laplace transforms of the v i s c o e l a s t i c and e l a s t i c d e f l e c t i o n s are r e l a t e d v i a the correspondence p r i n c i p l e : w v e > ( x , s ) = w(x,s)E/sE(s) , (2.11) where w^  g (x,s), w(x,s) and E(s) are the Laplace transforms of the v i s c o e l a s t i c s o l u t i o n , e l a s t i c s o l u t i o n , and the relaxation modulus 22 of the material, r e s p e c t i v e l y . Noting that w(x,s) = W(x)/sE , from Equation (2.11) one obtains w (x,s) = W(x)/s 2E(s) . (2.12) For r e l a t i v e l y low st r e s s l e v e l s the beam material under study i s found to exhibit a small long-time creep and behaves l i k e a l i n e a r v i s c o e l a s t i c s o l i d . Hence the three parameter v i s c o e l a s t i c s o l i d model (Figure 2-2) can be used to represent i t s behaviour f a i r l y w e l l . For a three parameter s o l i d , -AAAAAAAA-Figure 2-2 Three parameter v i s c o e l a s t i c s o l i d model 23 s 2 E ( s ) = sE 1(E 2+V 2s)/(E 1+E 2+V 2s) , (2.13) where E^, E 2 and V,, are the three parameters defining the material behaviour. Substituting Equation (2.13) into Equation (2.12) and i n v e r t i n g into time domain, gives w (x,t) = W(x)J(t) , (2.14) V • " • where J ( t ) = l / E 1 + ( l / E 2 ) [ l - e x p ( - E 2 t / v 2 ) ] and W(x) i s obtained from Equation (2.10). 2.1.2 Tapered Beam For a tapered c a n t i l e v e r (Figure 2-3), the diameter and length are l i n e a r functions of x, and,thus the curvature r e l a t i o n i s = -F(L-x)/EI dx where I = I(xX = I r [ l - k ( x / L ) ] 3 k = taper r a t i o = (d -d }/d r r t r I I f 24 z,w I n t e r n a l p r e s s u r e p Figure 2-3 Geometry of flexure of an i n f l a t e d tapered beam I = moment of i n e r t i a at root, i . e . , at x = 0. Integrating twice and using the boundary conditions at x = 0, leads to w(x) = W(x)/E (2.15) where W(x) = - F L 3 { [ ( k - l ) / ( l - k x / L ) - ( k + l ) ] / 2 - l n ( l - k x / L ) } / k 3 I Analogous to the uniform c y l i n d r i c a l beam, the v i s c o e l a s t i c s o l u t i o n 25 may be w r i t t e n as w ( x , t ) = W ( x ) J ( t ) , ( 2 .16 ) V • c • where J ( t ) and W(x) a r e g i v e n i n E q u a t i o n s ( 2 .14 ) and ( 2 . 1 5 ) , r e s p e c t i v e l y . 2.2 E x p e r i m e n t a l P rog ram 2 .2 .1 T e s t Equ ipment and P r o c e d u r e s To a s s e s s v a l i d i t y o f t h e a n a l y s i s and t o g e n e r a t e r e l e v a n t d e s i g n i n f o r m a t i o n , an e x p e r i m e n t a l programme was u n d e r t a k e n . M o d e l t e s t were p e r f o r m e d i n a 1 .83x0.91x1.22m ( 6 ' x 3 ' x 4 ' ) r e c t a n g u l a r w a t e r t a n k ( F i g u r e 2 -4 ) made o f w a t e r p r o o f p l ywood w i t h f r o n t and s i d e p l e x i g l a s p a n e l s t o f a c i l i t a t e o b s e r v a t i o n . A compres sed a i r b o t t l e p r e s s u r i z e d an i n t e r m e d i a t e w a t e r t a n k f o r i n f l a t i n g a mode l a f t e r t h e t e s t t a n k had been f i l l e d w i t h w a t e r . A p r e s s u r e gauge i n t h e i n t e r -c o n n e c t i n g p i p i n g i n d i c a t e d t h e i n f l a t i o n p r e s s u r e . A t r o l l e y s y s t e m e n a b l e d s t a t i c l o a d i n g a t any d e s i r e d s t a t i o n a l o n g t h e t u b e . As t h e s t a t i c d e f l e c t i o n s a r e t i m e v a r y i n g and t h e m e a s u r e -ments a t d i f f e r e n t s t a t i o n s a l o n g t h e t ube have t o be t a k e n s i m u l -t a n e o u s l y , p h o t o g r a p h i c t e c h n i q u e was employed t o r e c o r d t h e t i m e h i s t o r y o f a beam u n d e r g o i n g c r e e p i n g d e f o r m a t i o n . 35mm p i c t u r e s were t a k e n , i n i t i a l l y 30 seconds a p a r t w i t h t h e i n t e r v a l g r a d u a l l y i n c r e a s i n g t o 5 m i n u t e s as t h e c r e e p r a t e d i m i n i s h e d . A t h i n w i r e 27 strung above the beam served as a reference during these measurements. The d e f l e c t i o n data were obtained from the p r o j e c t i o n of the pictures on a screen. 2.2.2 Test Models A large number of uniform and tapered c y l i n d r i c a l tubes were made from t h i n sandwiched films of mylar and polyethylene. For the tapered beams only the case of 0.5 taper r a t i o , i . e . , d r = 2d f c, i s considered. Two sheets of the commercially a v a i l a b l e p l a s t i c f i l m (Nap-Lam clear laminating f i l m by General Binding Corporation) were pressed together by a heat tacking i r o n . The heat melted the poly-ethylene layers and fused them together (Figure 2-5a). The sandwiched sheet was then wrapped around an appropriate c y l i n d r i c a l blank to form the desired cross-section, and the edges sealed with a piece of mylar heat-sealing tape (Schjel-Bond GT-300 Thermoplastic Adhesive by Schjeldahl). One end of the tube was closed using a t h i n p l e x i g l a s cap epoxy-glued to the end (Figure 2-5b). Each tube was divided i n t o 10.16 cm (4 iiO sections at which de f l e c t i o n s were measured. 2.3 Results and Discussion Although a vast amount of experimental information was generated, only a few of the t y p i c a l r e s u l t s h e l p f u l i n i d e n t i f y i n g trends are presented here. Figure 2-6 shows a t y p i c a l creep-relaxation curve for the sandwiched material under study. An instantaneous d e f l e c t i o n followed by creep i s apparent. The creep rate decreases and becomes S 28 M y l a r j Typical Plastic Film Polyethylene Sandwiched Material (a) Sandwiched material made from two layers of p l a s t i c films f Tube / 0.3175 cm (1/8") / Plexiglass * / (b) End d e t a i l s Figure 2-5 Sandwiched material and d e t a i l s of the end cap vv, in. L=1.02 m (40") d=5.08 cm (2.0") h= 0.008 cm (0.003") F=1.11 N (0.25 lb.) 6.0 4.0 cm 2.0 80 0 ho VO Figure 2-6 A t y p i c a l creep-relaxation curve for the sandwiched i n f l a t e d beam 30 almost n e g l i g i b l e a f t e r about 40 minutes. Removal of the load causes an instantaneous drop i n the d e f l e c t i o n , of the magnitude equal to the i n i t i a l one. The beam asymptotically returns to the o r i g i n a l p o s i t i o n following e s s e n t i a l l y the same behaviour as that observed during the loading cycle. 2.3.1 Uniform C y l i n d r i c a l Beam Figure 2-7 compares some of the tes t r e s u l t s with a n a l y t i c a l p r e d i c t i o n s . It i s i n t e r e s t i n g to note that the behaviour can be described very w e l l by the three parameter s o l i d model. Average values of the three material constants, E^, IL^, and have been obtained to give the a n a l y t i c a l curves. In the tests was found to vary s l i g h t l y , but E^, the instantaneous modulus of e l a s t i c i t y , was very 9 2 5 nearly constant (==1.65x10 N/m or 2.4x10 p s i ) . It should be emphasized that f o r higher stress l e v e l s the long time s t r a i n has a nonlinear r e l a t i o n s h i p with the s t r e s s . K a l i n n i k o v 7 6 observed the creep r e l a t i o n f o r polyethyleneterephthalate (mylar) to be of the form . . ~ni n e = e n+ao t , c cO where a, m, n are material constants. On the other hand, Findley and K h o s l a 7 7 have found the creep of polyethylene to follow the equation e = e'^sinh(a/a )+m'sinh(a/a ) t n , c cU e m 31 L= 1.02m (40") 6.0 2.0 •• • • F=1.11 N (0.25 lb.) d = 5.08cm (2.0") 1.5 - h= 0.008cm (0.003") 4.0 w in. cm 1.0 L= 0.61m (24") . A . —i F=2.22 N (0.5 lb.) 2.0 0.5 5.0 a h=0.008 cm (0.003D A • • • - • ft- •—s • 1.5 • m. — if • 0 V V cm w in. 1.0 F= 8.90 N (2.01b.) L=0.61m (24") d=7.62cm (3.0") • * —i 2.5 — ; r ~ « • • h=0.015cm (0.006") 0.5 0 0 10 20 30 Time (min.) Figure 2-7 Comparison of a n a l y t i c a l and experimental r e s u l t s f o r the s t a t i c d e f l e c t i o n of uniform c y l i n d r i c a l beams 32 where e'„, m', n, O and 0 are constants. In the study of dynamics cO e m J J of these structures, however, only the short time creep i s of s i g n i -ficance, since the period of most of the n e u t r a l l y buoyant beams i s very small (around one second). The e f f e c t of i n t e r n a l pressure on the s t a t i c d e f l e c t i o n s i s shown i n Figure 2-8. For the range of pressures considered, the i n f l a t i o n causes l i t t l e change i n the s t a t i c behaviour of the c a n t i -* * lever beam. This means that the deviations from unity of and i n Equations (2.5) are small. In the actual c a l c u l a t i o n i t i s found that, since the shear modulus of the material i s r e l a t i v e l y large, the i n t e r n a l pressure induces geometrical changes of le s s than 1%. On the other hand, M i s r a 3 has found that, for polyethylene, the e f f e c t may be s i g n i f i c a n t and changes i n dimensions can reach as high as 15%. In the design of these i n f l a t e d c a n t i l e v e r s the L/d^ r a t i o i s an important parameter. Figure 2-9 shows the d e f l e c t i o n h i s t o r i e s for two L/dg r a t i o s . Good agreement between theory and experiments confirms the cubic power v a r i a t i o n of d e f l e c t i o n with L/d^ r a t i o predicted by the t h e o r e t i c a l a n a lysis. 2.3.2 Tapered Beam Figure 2-10 shows some of the t y p i c a l d e f l e c t i o n h i s t o r i e s and t h e i r corresponding a n a l y t i c a l p r e d i c t i o n s . Again the v a l i d i t y the of the t h e o r e t i c a l analysis i s confirmed by i t s good agreement with experimental data. 1.0 w, in. 0.5 0 L = 0.61m (24") d =7.62 cm (3.0") , h = 0.008 cm (0.003") F=4.45N (1.0 lb.) 10 4- A o A 3.0 psi - 2.07 . 5.0 psi - 3.45 [ x i 0 4 N / m 2 o 6.0 psi -4.14 t .min, 20 3.0 2.0 cm 1.0 30 u> u> Figure 2-8 E f f e c t of i n t e r n a l pressure on the s t a t i c d e f l e c t i o n s of uniform c y l i n d r i c a l beams L/d = 12 4 — * 1- 6.0 h = 0.008 cm (0.003") F=4.45 N (1.0 lb.) -RO L / d = 8 T ! 2.0 0 10 20 t, min. 30 Figure 2-9 Tip d e f l e c t i o n h i s t o r i e s f o r two L/d r a t i o s 35 1.0 w, in. 0.75 0.25 1.0 w, in. 0.75 h = 0.008cm (0.003") l3.0 L = 0.61 m (24") d r = 7.62 cm (3.0") F=4.45 N (1.0 lb.) h = 0.015cm (0.006") L = 1.02m (40") F = 2.23N (0.51b.) iii . . . . . . — » I e • dr=10.16 cm (4.0") h = 0.008 cm (0.003") L = 0.61m (24") F= 4.45N (1.01b.) cm 2.0 1.0 3.0 cm 2.0 « ~Q- 0 1.0 0.25 0 10 20 30 t, min. Figure 2-10 Comparison of analytical and experimental results for the static deflections of tapered beams 3 6 The d e f l e c t i o n s of three tapered beams subjec ted to d i f f e r e n t i n t e r n a l pressures are p l o t t e d i n F i g u r e 2-11 . The exper imenta l data a l l f a l l very c l o s e to the z e r o - i n f l a t i o n t h e o r e t i c a l c u r v e , and no s i g n i f i c a n t e f f e c t s of i n t e r n a l p ressure i s observed. The t i p d e f l e c t i o n s at t = 0 ( ins tantaneous) and t = 30 minutes are p l o t t e d as func t ions o f L / d ^ i n F i g u r e s 2-12 and 2-13 , r e s p e c t i v e l y , f o r the s t r u c t u r a l model hav ing a w a l l t h i cknes s of 0.008cm (0.003") and t i p load of 4.45N (1 .0 l b ) . The l i n e s represent the a n a l y t i c a l r e s u l t s as g iven by Equat ion (2.16) w h i l e the i s o l a t e d p o i n t s i n d i c a t e the t e s t da t a . P o t e n t i a l of the a n a l y t i c a l approach becomes apparent as i t i s ab le to p r e d i c t w i t h accuracy even l a r g e d e f l e c t i o n s . Th i s suggests that the curva tu re can be represented by A 1 d w — r wi thou t much e r r o r even though the d e f l e c t i o n s are l a r g e , dx 2.4 Conc lud ing Remarks From the preced ing s t a t i c a n a l y s i s s e v e r a l important con-c l u s i o n s can be summarized as f o l l o w s : ( i ) The a n a l y s i s suggests tha t the three parameter s o l i d model can be used to y i e l d s u f f i c i e n t l y accura te r e s u l t s u s e f u l i n the des ign of n e u t r a l l y buoyant i n f l a t e d s t r u c t u r e s made from the v i s c o e l a s t i c sandwiched f i l m s s p e c i f i e d . Even l a r g e d e f l e c t i o n s can be p r e d i c t e d w i t h accuracy by the s imple a n a l y s i s g i v e n , ( i i ) P rov ided the i n t e r n a l p ressure i s moderate and no w r i n k l e s w. in. d r = 7.62cm (3.0") J6.0 L = 0.81 m (32") h =0.008 cm (0.003") F=4.45 N (1.0 lb.) pcoo - A • o * 3.0 psi - 2.07 5.0 psi - 3 . 4 5 f x 1 0 4 N / m ; o 6.0 psi -4.14 d r =10.16 cm (4.0") o o T d r = 12.70cm(5.0") 4.0 cm 2.0 •3 oi 0 10 20 t, min, 0 30 OJ Figure 2-11 E f f e c t of i n t e r n a l pressure on the s t a t i c d e f l e c t i o n s of tapered beams 38 Figure 2-13 Tip d e f l e c t i o n at t = of L/d r 30 min. as a function 40 occur, i n f l a t i o n has n e g l i g i b l e e f f e c t s on the s t a t i c bending s t i f f n e s s of the i n f l a t a b l e beams considered. At high i n t e r n a l pressures, however, changes i n both the geometric and the material properties may be s i g n i f i c a n t and the d e f l e c t i o n r e l a t i o n s h i p w i l l become nonlinear, ( i i i ) The instantaneous modulus of e l a s t i c i t y , E^, for the v i s c o -e l a s t i c material considered i s found to be nearly constant 9 2 5 at 1.65x10 N/m (2.4x10 p s i ) . Average values for and \>2 are experimentally found to be 1.24x10''"^  N/m2 (1.8x10^ psi) 9 2 5 and 3.72x10 N-sec/m (5.4x10 p s i - s e c ) , r e s p e c t i v e l y . This information on the material properties w i l l be useful i n the forthcoming study of the dynamic behaviour of the v i s c o e l a s t i c c a n t i l e v e r s . 41 3. FREE VIBRATION OF NEUTRALLY BUOYANT INFLATED CANTILEVERS S t a t i c behaviour of the n e u t r a l l y buoyant i n f l a t e d v i s c o -e l a s t i c c a n t i l e v e r s having been studied, the next l o g i c a l step would be to analyze dynamic response of the beams v i b r a t i n g i n the ocean environment. The object of t h i s chapter i s to in v e s t i g a t e free 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 uniform c i r c u l a r c y l i n d r i c a l and tapered beams i n water. F i r s t , the f l e x u r a l free v i b r a t i o n of the i n f l a t e d uniform c y l i n d r i c a l c a n t i l e v e r i s considered. S h e l l equations are used to incorporate the i n i t i a l stresses due to i n t e r n a l pressure. For r e l a t i v e l y l o n g t s h e l l s v i b r a t i n g i n the beam-bending mode, the governing s h e l l equations are found to be reducible to a s i n g l e equation very s i m i l a r to the one for the transverse v i b r a t i o n s of a beam with rotary i n e r t i a included. Three d i f f e r e n t approaches are studied: the membrane, Fliigge's, and Herrmann and Armenakas'. The natural frequencies obtained are compared with the experimental r e s u l t s and those predicted by the Rayleigh-Ritz method i n conjunction with the Washizu and membrane s h e l l theories. The presence of hydrodynamic forces introduces n o n l i n e a r i t i e s into the governing equation of motion. E f f e c t of t h i s nonlinear drag force on the free response, as given by the reduced Fliigge equation, i s studied using the perturbation technique. 42 T h i s i s f o l l o w e d by a f r e e v i b r a t i o n a n a l y s i s o f t h e i n f l a t e d t a p e r e d beam. The e l e m e n t a r y beam t h e o r y i s u s e d and t h e n a t u r a l f r e q u e n c i e s a r e f ound u s i n g t h e m o d e - a p p r o x i m a t i o n p r o c e d u r e . V a l i d i t y o f t h e a p p r o x i m a t e method i s e xam ined by c o m p a r i n g t h e r e s u l t s w i t h t h e e x p e r i m e n t a l d a t a . As t h e m a t e r i a l unde r s t u d y has r e l a t i v e l y s m a l l damp ing , t h e t i m e dependence o f t h e n a t u r a l f r e q u e n c i e s i s o f s e c o n d a r y i m p o r t a n c e i n t r a n s i e n t r e s p o n s e s t u d i e s p r e s e n t e d h e r e . Hence , no a t t e m p t i s made t o i n c o r p o r a t e t h e v i s c o e l a s t i c p r o p e r t i e s i n t h e i n v e s t i g a t i o n , and t h e m a t e r i a l i s t r e a t e d as e l a s t i c w i t h Y o u n g ' s modulus E^. 3.1 U n i f o r m C y l i n d r i c a l Beam A n a l y s i s 3 .1 .1 Reduced S h e l l E q u a t i o n A p p r o a c h (a) F o r m u l a t i o n The c o n v e n t i o n a l beam e q u a t i o n , b e c a u s e o f i t s i n a b i l i t y t o a c c o u n t f o r s t r e s s e s i n d u c e d by t h e i n t e r n a l p r e s s u r e , c a n n o t be a p p l i e d t o i n f l a t e d t h i n - w a l l e d beams. I n o r d e r t o i n v e s t i g a t e i n i t i a l s t r e s s e f f e c t s on t h e dynamic b e h a v i o u r o f t h e i n f l a t e d c a n t i l e v e r s one ha s t o r e s o r t t o t h i n s h e l l t h e o r y . S h e l l s have a l l t h e c h a r a c t e r i s t i c s o f p l a t e s b u t w i t h one d i f f e r e n c e — c u r v a t u r e . I n o t h e r w o r d s , p l a t e r e p r e s e n t s a l i m i t i n g ca se o f t h e s h e l l w i t h no c u r v a t u r e . B e s i d e s t h e added c o m p l e x i t y 43 of curvature, shells are more d i f f i c u l t to analyze than plates since their bending cannot, in general, be separated from stretching. Thus, a c lass ica l bending theory of shells is governed by a system of eighth order par t ia l d i f ferent ia l equations, while the corresponding plate equations are of the fourth order. A further challenge enters the problem through the boundary conditions as well . Four specified conditions, as compared to two in plate theory, are required here. To complicate matters further, whereas a l l academicians agree on the form of the c lass i ca l , fourth order plate equations, such agreement does not exist in shel l theory. Numerous different theories have been derived and are in use. The equations of motion for a thin c ircular cy l indr ica l shel l may be written conveniently in matrix form as L l l L12 L13 L21 L22 L23 L31 L32 L33 {0} (3.1) where x, y, and z are the orthogonal components of displacement in the x (axial) , 6 (circumferential), and radial directions, respectively (Figure 3-1). For the most commonly used Flugge's shel l theory, the d i f f er -ential operators in Equation (3.1), after neglecting the small terms 2 2 involving h /(12a ) , a r e 3 1 Figure 3-1 Geometry and co-ordinate system c i r c u l a r c y l i n d r i c a l s h e l l 6 s 6 t) d t 12 2 dsdQ L 13 = ( V - V C ) = 1 + v 9 J21 2 3s38 22 9s a 2 C 2 G ~ 2 9t L 23 = ( 1 + V c ) | e L 31 = CV-NQ/C)^ 45 L 3 2 = (l+Ng/cOfg L „ - 1 + c.G3 9 8 x 3 3 3 3 V C 362 C 3 s 2 where s = x/a , G = pa 2(l-V 2)/E , C = Eh/(1-V2) , and c^, c 2 , are tracers (=0 or 1) introduced to i d e n t i f y the effect of the different i n e r t i a l forces, c^ also incorporates the added mass effect of the surrounding water. The problem of determining the functions x, y, and z s a t i s f y -ing Equation (3.1) can be reduced to solution of a single equation involving one potential function T(s, 9, t ) . Let D be a determinantal operator D = det L , (3.2) i t can be shown 3 0 that each solution of the equation D(H = 0 (3.3) corresponds to an i n t e g r a l of the system of homogeneous equations of motion [Equation (3.1)] given by 46 y = T=vD23(r> > (3.4b) ? - T=v D 33 ( r ) ' ( 3 ' 4 c ) where denote the corresponding minors of the determinant in Equation (3.2). It should be noted that in the above derivation the differentiation symbols are treated as algebraic quantities. This is applicable only i f their coefficients are constants. Assuming solutions of the form x(s, 6, t) = ]Tx (s,t)cos(n6) , (3.5a) n=0 n y(s, 6, t) = £ y (s,t)sin(n0) , (3.5b) n=0 z(s, 6, t) = Jjz. (s,t)cos(n6) , (3.5c) n=0 oo T(s, 0, t) = £r n ( s , t )cos(ne) , (3.5d) n=0 and substituting into Equations (3.4) yields: 2 2NflN Nfl V 8 ' ^ " i ? v ( 1 + V c ) b>-i-2KQ/c)r + ( - J U - +...-2 2VN 2c 9G 9 2 r ' 47 2N„ 2nc,G 3 2T y n ( s , t ) - - n ^ l + ^ / c ) { l + w ^ ) T n - ^ ( l ^ / c ) — | at „ (3+v)N„ N , . , 1 J a . v 2N z n ( s , t ) = n 4 ( l + N e / C ) ( l - r T I - I ^ ) r n - ^ ( ^ [ l + ^ / C H l - f f l g / C ] 2 N A 1 N 2 N 2Nfl 2c G 3 2r 2c c G 23 4r 2N 2c G 3 2 r " " K G ^ + n ^ c l + T Z v [ l + N ^ / C ] ) — a - , (3.6c) i n which the primes denote d i f f e r e n t i a t i o n with respect to s. The p o t e n t i a l function must s a t i s f y Equation (3.3) which appears i n e x p l i c i t form as 3 2r 3 4r 3 6r 3 2r 3 4r A„T + A,—=2 + A „ — + A.—~ + A,—=2 + A n 0 n W 2 3 s 4 3 3 s 6 4 3 t 2 5 3 s 2 3 t 2 3 6r 3 4r 3 6r 3 6r + A ^ + A ^ + A ^ + A v n = 0 • < 3 - 7 ) The following assumptions are now made: (i ) Only the lowest derivatives of T with respect to time are important. ( i i ) Two adjacent a x i a l nodes are very f a r apart. This implies that the most s i g n i f i c a n t terms i n Equations (3.6) and (3.7) 48 are those with the lowest de r i v a t i v e of T i n s, i . e . , 29 m+lT 9 5 > > 9 ^ T ~ m ~ m+1 ds ds Hence the higher d e r i v a t i v e s of T with respect to s may be neglected as a f i r s t approximation. For the beam-bending mode of i n t e r e s t here, n = 1, and the i n i t i a l stresses induced by the i n t e r n a l pressure are N Q = pa , N x = pa/2 Defining a dimensionless pressure P = pa/(Eh), and keeping i n mind the aforementioned assumptions, Equation (3.7) can be reduced to 4 2 4 3 r. a r 3 r A 2 ~ r + \ - r + V 2 - 2 " 0 • ( 3 - 8 ) ds 3t 3s at where A = l i ^ V p 3 , (13+ 7 V)(1 - V 2 ) 2 2 ( 1 4 - 2 v - 1 2 v 2 ) ( l - y 2 ) „ , ( l - v ) ( l - v 2 ) 2 A 4 = G ( c 2 + c 3 ) [ ( l - v 2 ) 2 P 2 + ( 3 1 ^ ) ( 1 - V 2 ) P + i ^ ] 49 2 A5 = G ( ( C l + c 3 ) ( l - V 2 ) 2 P 2 - [7(l+v)c 2+2(3-V)c 1+3(3-v)c 3 ]( i^-)P ^Y1^ - (i-v 2)c 2 - (i-v)c 3) Retaining the first-order terms only in this approximation, Equations (3.6) become 8r x x = ((l+v)P[V-3-2(l-V2)P]-l)^-i , (3.9a) y 1 = -tl+(l-v 2)P][l+2(l+v)P]r i , (3.9b) Zj_ = -Y1 (3.9c) and, by Equations (3.5), x = x^osS , (3.10a) y = y ; Lsin9 , (3.10b) z = z^cosO = -y^cosO . (3.10c) Multiplying Equations (3.10b) and (3.10c) by sih6 and cosO, respectively, and subtracting gives y^(s,t) = ysinB - zcos0 . (3.11a) 50 On the other hand, multiplying them by cos6 and sin9, respectively, and adding gives 0 = zsinO + ycos9 . (3.11b) The right-hand side of Equation (3.11a) denotes the ver t ica l while that of (3.11b) the horizontal displacements of the points of the middle surface of the shel l (Figure 3-2). Since they do not depend upon 9, the shel l behaves l ike a beam. The cross sections remain r ig id in their planes and are displaced vert ica l ly by y i ( s , t ) > w(s,t) . (3.12) Figure 3-2 Vert ica l and horizontal displacement components 51 rewritten as 9 w 5,a N2 9 w , 2.a..49 w ,„ • . ^  + \V^h?  + \V^ •  (3-13) By Equations (3.9a) and (3.12) the l o n g i t u d i n a l displacement i s v _ l-(l+V)P[v-3-2(l-V 2)P] 8w A x - r jj—cosU [1+(1-V )P][1+2(1+V)P] d S which corresponds to the ro t a t i o n of the cross section about i t s h o r i z o n t a l diameter. Equations (3.9) suggest that the r a d i a l and ci r c u m f e r e n t i a l displacements z and y are of the same order, thus has to be set to unity. If the rotary i n e r t i a i s to be taken i n t o account as w e l l , c^ must also be unity. To incorporate the added i n e r t i a of the water both i n s i d e and outside the tube, c^ i s adjusted to 1+(1+C m)P wa/(ph) (Appendix I ) . The value of the added mass c o e f f i c i e n t C , as predicted by m the p o t e n t i a l t h e o r y 7 8 , i s 1.0. This was confirmed experimentally by 1 7 Stelson and Mavis for cylinders with large length to diameter r a t i o s . Hence, i n the analysis here the added mass c o e f f i c i e n t i s taken to be unity. Note that f o r a s h e l l v i b r a t i n g i n a i r with no i n t e r n a l pressure, c^ = 1 and P = 0, and Equation (3.13), a f t e r some rearranging, reduces to 52 4; + - (5:2V) i \ - ° • ^ 9t 2 P a ^ f z 9 0 t Equation (3.14) i s i d e n t i c a l to the one derived by Korneclci 2 9. It i s inte r e s t i n g to compare th i s with the elementary beam equation for a thin-walled tube including rotary i n e r t i a , 2 4 4 9 w _j_ _E 9 w _ 1 9 w _ 9 t 2 2pa 29£ 4 2 9 ? 2 9 t 2 I t i s apparent that the reduced s h e l l equation takes into account the Poisson r a t i o effect which the simple beam theory ignored. To account for the viscous effect of the surrounding water a hydrodynamic resistance force i s added. The hydrodynamic resistance i s generally taken as a drag force proportional to the square of the v e l o c i t y , i . e . , F = dp v |v | , (3.15) d 2 d w r e l 1 r e l ' where C , i s the drag c o e f f i c i e n t , and v ., the v e l o c i t y of the s h e l l d ° r e l element r e l a t i v e to the f l u i d . The value of i n the f l a t portion of the s u b c r i t i c a l region i s approximately 7 9 1.18, and t h i s value i s used i n the present analysis. Defining n = w/d , B_ = Or) '3 L ^ 2 pa Z[2+(l+C m)(^w) (p}' P 53 Equation (3.13) can be nondimensionalized as where A + B _ | ! n _ + B ^ ^ j a a , , 0 f ( 3 . i 7 a ) 9 £ 4 Z 9 r o T I 9 T 3 T 9 T a - ( | ) c d ( r ) ( f ) / [ 1 + ( 1 + c m ) ( p w / p ) ( a / h ) ] ' 2C, d (3.17b) TT(1+C ) m since (1+C )(p /p)(a/h) » 1. m w It may be noted that the damping parameter a i s i d e n t i c a l to the one formulated by M i s r a 3 and i s independent of the geometrical dimensions of the cy l i n d e r i f (a/h) >> 1. The boundary conditions are given by n ( 0 f T ) - M ° * l 2 = i J l % l ) - £LJ1%I) = o . (3.17c) 35- . a r 3 £ This nonlinear p a r t i a l d i f f e r e n t i a l equation does not seem to have any known closed form s o l u t i o n . Hence one i s forced to approach the problem through numerical analysis or approximate procedures. 54 (b) Solution f o r Zero Drag For determination of the n a t u r a l frequency i t can be assumed, at l e a s t up to the f i r s t order approximation, that the period of o s c i l l a t i o n i s e s s e n t i a l l y unaffected by the presence of hydrodynamic drag 3. Dropping the drag term, Equation (3.17a) becomes —4 + B 2 J T L 2 + B l ^ = ° • (3-18) Assuming s o l u t i o n of the form n(5,T) = 8(€)f(x) , (3.19) and s u b s t i t u t i n g into Equation (3.18) gives or 4 2 2 ? -J? + B 2 T 2 ~ 2 + B 1 B 7 1 " 0 dfc, dt, dx dx 6" " f ,4 B 3''+B 3 = ~ f = = constant. (3.20) This leads to two ordinary d i f f e r e n t i a l equations: ~ + A 4 f = 0 , (3.21a) dx & - - B / 8 - 0 , (3.21b) dC d5 55 with solutions f = f cos(A2T) + f sin(A2T) , (3.22a) B = djCosMA'O + C2sinh(A*C) + GyosU"?) + C4sin(A"5) . (3.22b) Applying the boundary conditions (3.17c) gives the following equations relating A, A', and A": B 2A 4 + 2^(l+coshA'cosA") -v^B^sinhA'sinA" = 0 , (3.23a) A» 2 - A"2 = B 2A 4 , (3.23b) A'A" = v^A 2 . (3.23c) By solving Equations (3.23) the natural frequencies and the associated mode shapes can be found. The general solution is then given by oo na,x) = £ s r ( O f r0t) , (3.24) r=l where B^(£) is the normalized mode shape given by 6 (£) = K [coshA'£-CcsinhA'5-cosA"£+C,sinA"S] r r r 5 • r"• r^ 6 r 56 with X , 2coshX'+X , , 2cosX" c = _r_ r r r 5 A^sinhX'+A'V'sinX" r r r r r X , 2coshX'+X , , 2cosX" c = r r r r 6 A ' A - s i n h A ' + A ' ^ s i n A " r r r r r Here is chosen to normalize 3 R ( 0 > i . e . 1 ,2. folios = 1 0 Analysis showed to be (cl-ch sinhA* K . = ( l + — 2^~^~ + 2\'R[(1+C5)coshA^-2C5sinhA^] r sinX* + - 2 P S [ a - c 6 ) c o 8 r - 2 c 6 8 i n x ; ] + _ i _ [ ( c 5 A ; - c 6 A » ) * T A +A r r coshA'cosA"- (X»+C c C,A , ) coshX , s inX"+(C r C^X , , -A' ) * r r r 5 6 r r r 5 6 r r _1 sinhX'cosXM+(C cX ,'+C,X ,)sinhX'sinX"+C cX'-C,A"]) 2 . (3.25) r r 5 r 6 r r r 5 r 6 r 1 The natural frequencies are thus given by 0) 2 = / B T X 2 , ( 3 . 2 6 ) r 3 r ' ' where X^ _ is the solution of Equations (3.23), and is given by 57 E q u a t i o n ( 3 . 1 6 ) . The o r t h o g o n a l i t y c o n d i t i o n f o r t h e n o r m a l i z e d mode shape 3(£) i s f o und t o be ( A p p e n d i x I I ) 1 B„ 1 B . , f B B ^ d C - ^ f g ' g ' d g - I [ X V ( D 3 ( l ) - x X ( D 3 ' ( l ) 3 , .J m n B. V m n „ 4N m m n n m n 0 1 0 B (A -A ) 1 m n f o r m ^ n . ( 3 . 27 ) ( c ) P e r t u r b a t i o n S o l u t i o n To i n v e s t i g a t e e f f e c t o f t h e h y d r o d y n a m i c d r a g on t h e f r e e r e s p o n s e , E q u a t i o n ( 3 .17a ) must be s o l v e d . The n o n l i n e a r n a t u r e o f t h e e q u a t i o n does n o t p e r m i t an e x a c t s o l u t i o n and one i s f o r c e d t o r e s o r t t o an a p p r o x i m a t e a n a l y s i s . I n t h i s s e c t i o n t h e p e r t u r b a t i o n t e c h n i q u e i s u sed t o s t u d y t h e r e s p o n s e o f t h e i n f l a t e d c a n t i l e v e r s u b j e c t e d t o h y d r o d y n a m i c d r a g . The g o v e r n i n g E q u a t i o n ( 3 .17a ) may be r e w r i t t e n as where t h e a p p r o p r i a t e s i g n f o r t h e d r a g t e r m i s cho sen so as t o oppose t h e m o t i o n . I t i s s u f f i c i e n t t o s o l v e t he above e q u a t i o n e i t h e r f o r t h e p o s i t i v e o r n e g a t i v e s i g n o v e r h a l f a c y c l e , s o l u t i o n f o r t h e o t h e r h a l f o b t a i n e d s i m p l y by r e v e r s i n g t h e s i g n o f a w i t h new i n i t i a l c o n d i t i o n s . 58 The s o l u t i o n for the negative sign i s sought i n the form n(?,x) = n 0 ( £ , T ) + o m 1 ( 5 , T ) + a 2 n 2(£,T) + . . . . (3.28) A new time va r i a b l e t defined by t = T[1 + ab1 + a 2 b 2 + . . .] (3.29) i s introduced where b^ and b 2 are slowly varying function of £, to account f o r the period of o s c i l l a t i o n which may vary along the length. Substituting from Equations (3.28) and (3.29), the governing equation takes the form 3 4 2 2 2 3 4 — j ( T i 0 4 o n 1 4 a n 2 + . . . ) +-{HXxb 14<x b 2 + . . . } { B 2 — 2 T 2 ( n o + a T 1 l 3 £ 3 £ 3 t 2 g 2 2 3 2 +a n 2 + . • . ) + B 1 — - ^ ( n 0 + a T i ; L + a n 2 + . • . ) - a [ ^ ( n 0 + a n 1 + a n 2 3 t +. . .)] 2} = 0 Equating the c o e f f i c i e n t s of the d i f f e r e n t powers of a to zero one obtains 0 d \ 3 \ * \ CT : 7 + B - , " + B- 5 = 0 (3.30a) 4 4 2 4 2 1 *\ , 3 D , 3 n , a V 3 n n a : — ? + B 2 - 2 ^ 2 + B l " 4 " " 2 b l ( ^ C B 2 ~ ~ 2 ~ ^ + B l — V dE, dZ dt 1 3 t Z X Z 3 0 t 3 t 59 3n 2 + ( ) , (3.30b) 3 t 2 3S *\ *\ 2 3 \ 3 \ a : 1? + B W + B i — > = - W H - ^ - g i . a \ 9 4 n 1 3n 0 2 3n 3r, -2b (?) [B 2—s-^- + B V i ] + 2b .<e)C—)• + 2 - 2 — i , z 9 r 9 t z 1 9 t Z 1 9 t 9 t 3 t (3.30c) etc. The objective is to solve this system of equations such that a l l n^'s conform to the boundary conditions (3.17c) with the prescribed i n i t i a l conditions satisf ied by n 0 alone, and zero i n i t i a l conditions for other n^'s. Equation (3.30a) is identical in form to Equation (3.18) whose solution has been shown to be (3.24). However, the rather complex orthogonality condition [Equation (3.27)] for the exact mode function does not permit decoupling of the equations later on in the perturbation analysis. Hence an approximate solution to Equation (3.30a) is sought in the form 00 n0 = Z $ r ( 5 ) f r ( t ) . • (3.31) r=l where $ (£ ) are the eigenfunctions of a cantilever beam, 60 « r(£) = cosh ( y r C ) - cos (y^£) - o*^  [sinh ( y ^ £ ) — sin ( y ^ _ £ ) ] Here ° r ~ (coshy^ + c o s y r ) / ( s i n h y r + siny^,) , and y r are the roots of the frequency equation 1 + coshycosy = 0 Substitution of Equation (3.31) into Equation (3.30a) gives 4 2 2 9 oo d * d * d f d f Noting that d 4$ 7 " ' M r ' (3.33a) dC d 2$ and that 5 can be expressed i n terms of $ (£) by the seri e s d r r d 2$ - 4 c r i * i < « > < 3' 3 3 b> where C ^ i s given b y 8 0 61 r i 1 2 f 4(y ra r-y 1a 1)/[(-l) i + r-(y i/a r) 2] , i 4 r I V r ( 2 _ y r a r ) » 1 Equation (3.32) may be rewritten as d 2 f r=J ( y 4 $ f + [B0 V c .*.+B.« ] )^ v r r r 2 f-< r i 1 1 r ,~2J i=l dt = 0 (3.32a) Multiplying by $^ , integrating with respect to K over the length, and noting that where 6^ is the Kronecker delta, Equation (3.32a) becomes d 2 f . V i + tB2 SLV 8! 1!? = ° j = 1, 2, (3.34) The s o l u t i o n of t h i s equation i s f . ( t ) = F c . c o s [ y 2 t / ( B 1 + B 2 g .^) 2 ] + F s . s i n [ y 2 t / ( B 1 + B 2 g^ .) (3.35) Let the i n i t i a l conditions be 62 Tl(5,0) = A n ( ? ) and ^ ( 5 , 0 ) = 0 , (3.36) U 3t the zeroth order so lu t ion is.;-then oo •2-n 0 ( C , t ) = 2^A 0 j $ j (5)cosy^t , (3.37a) j = l where A 0 j = I A 0 ( ? ) $ j ( C ) d 5 ' (3.37b) 0 and *2 = ^ v-^Ev2 • (3-37c) r=l With t h i s , Equation (3.30b) becomes 3 4 n 3 2n-, 1 + B % + B 5 = q ( £ , t ) , (3.38) 3 T 3? 3t 3t where ^ -4 d 2 $ i ( ? ) - 2 . q(?,t) = 2b (g) 2^, A y^[B $ (£)+B ] c o s y Z t • p i d£ J + -2 Z E v # j ( 5 ) v ^ M 8 ^ r t j = l k=l -2 -2 ~ cos(y +u, ) t ] J k 63 A solution to Equation (3.38) can be taken in the form n 1 ( ? , t ) = 2 V ? ) f m ( I ) ' ( 3 - 3 9 ) m=l Substituting Equation (3.39) into Equation (3.38), multiplying by $ n ( D » integrating with respect to £ over the length and using the orthogonality condition leads to 4 d 2 f (t) V n r t ) + [ h + h I l C ^ ] - i p T - = » n - 1 , 2, ...... _ _ n dt m _ 1 (3.40) where 1 Q n (t) = J q ( C , t ) * n ( C ) d ? 0 Equation (3.40) can be rewritten as ~ d + W n f n " V * ' • " " I . 2. •••• dt where Qn(t) - Q nCt)/tB 1 +B 2 V c j - [1/(B 1 +B 2 Tc^)] * m=l m=l d $. j=l J V d5 J 0 oo oo j=l k=l J J J J J (3.40a) 64 and 3.. « / * . ( ? ) * . ( O * <5)<£ . (3.40b) jkn J j k n 0 As the term involving b^(£) in Equation (3.40a) gives r ise to secular quantities, i t must vanish for a l l j , b x ( £ ) = 0 (3.41) i . e . the natural frequencies are independent of the nonlinear hydro-dynamic drag up to the f i r s t order. This confirms the approximation made in the previous section that the period of osc i l la t ion is un-affected by the presence of the hydrodynamic drag. The solution to Equation (3.40) can now be written as oo f (t) = cosy 2* + D , s i n y 2 t + |[l/(B n+B V c )] * n In n In n z 1 ^^pY j=l k=l cos(y 2 +y 2 ) t / [y 4 - (y 2 +y 2 ) 2 ] ) . (3.42) Applying the zero i n i t i a l conditions gives qo oo co c, = ~[1/(B.+B. T * c ) ] Y n y ^ 3 . v y 2 y 2 A . . A n , * In 2 1 2JLf0 i m i ' V / / J jkn j k Oj Ok m = 1 ~ k=l 65 D = 0 (3.43b) Substitution of Equation (3.43) into (3.42) yields j -1 k - l 1 ? l B 1 + B 2 gCmn ([ cosy 21- cos (u 2-\) t ] / [ y 4 - (vi 2-\) 2 ] - [cosy 2 t -cos(y 2 +y 2 ) t ] / [y 4 - (y 2 +y 2 ) 2 ] ) . (3.44) Thus the solution, up to the f i r s t order of approximation, is n(£,t) = ri0(£,t) + an1(?,t) . (3.45) 3.1.2 Rayleigh-Ritz Method Consider a uniform cy l indr ica l shel l in equilibrium acted upon i i i by stat ic i n i t i a l stress O ^ , O Q , and 0"^Q. During vibration the internal stresses in the shel l consist of the i n i t i a l stresses and the additional vibratory stresses 0"^ , Og, and O * X Q - Assuming there is no interaction between the prestress displacements and the vibratory stresses, the internal strain energy of the she l l , taking the prestressed equilibrium 3 1 state as the reference leve l , can be written as 66 u = i / ( a x e x + a e e e + axeYxe ) d ( V o l- ) Vol. + / ( a x e x + a e e e + *ie Yxe ) d ( V o l-> <3-46> Vol. = U l + U2 The vibratory stresses O , Oa, and a Q are related to the x 8 X0 vibratory strains by Hooke's Law a - — S-(e + Ve f l) , (3.47a) 1-V 0 0 e = 3 ( e Q + V 6 x ) ' ( 3 ' 4 7 b ) •Y„a • (3.47c) x9 2(1+V)'x9 Substituting Equations (3.47) together with the strain-displacement relations of a given shel l theory into (3.46) and integrating over the thickness yields the strain energy. Because the i n i t i a l stresses may be large i t is necessary to use the second-order, nonlinear s tra in-displacement equations in the of Equation (3.46) while using only the l inear relations in U^. This maintains the proper homogeneity 3 1 in the orders of magnitude of the terms in the integrands is made up of two parts, one due to stretching (membrane) and the other due to the addition of bending stiffness, i . e . , 67 u i " u u + U, . (3.48) 1 membrane bending ' The membrane component i s given by 8 1 w « . • ^  /fife2 + W^1 + 3 while u b e n ( j i n g contains small terms proportional to (h/a) that are n e g l i g i b l e for modes with small number of circ u m f e r e n t i a l waves 8 2. It should be noted that a l l the e x i s t i n g s h e l l theories lead to the i d e n t i c a l expression f o r U , the differences occur i n the membrane ^bending" For pressurized tubes the i n i t i a l stresses are given by < • -2- S - T . . • < & - 0 • « • « > The s t r a i n e of an element at a distance z from the middle surface consists of the stretching of the middle surface and that due to r o t a t i o n of the element. Accordingly, e x = £ x + Z K x ' (3.50a) e Q = e Q•+ zKg , (3.50b) where e , e Q denote the middle surface s t r a i n s and K , Kn the changes x y x 6 68 in curvature. Note that e's and K ' S are not functions of z. Equations (3.50) may also be derived from strain-displacement relations of the three-dimensional theory of e las t i c i ty . In order to satisfy the Kirchhoff hypothesis, which states that the normals to the undeformed middle surface remain straight and normal to the deformed middle surface and suffer no extension, the displacements are restricted to l inear relationships. Using the l inear relations and -neglecting the (z/a) terms in comparison with unity the s tra in-displacement relations simplify to Equations (3.50). The second order strain-displacement relations according to Washizu's shel l theory 8 3 are 9x , 1, ,9x..2 , ,9y s2 , /9zv2, c : i . e x " 5 5 + 2 1 ( S i > + (3x> + % > 1 • ( 3 - 5 1 a ) 2a a 90 Since the i n i t i a l stresses are assumed to be due to membrane action, i . e . , uniform through the thickness, i t i s sufficient to retain only linear terms in the expressions relating curvature changes to displace-... ments8"*. There is general agreement among the shel l theories for expressions of the middle surface curvatures and K Q , usually taken as 3 2z K = - 2_ , (3.52a) X 3x 69 - _ 1 3 2 { g ( 3 . 52b ) K 0 " a 2 3 9 " a 2 3 9 2 S u b s t i t u t i o n o f E q u a t i o n s ( 3 . 49 ) t o ( 3 . 52 ) i n t o u"2 and i n t e -g r a t i o n t h r o u g h t h e t h i c k n e s s g i v e s ,dXx2 . x , 3 y s 2 , x . d z . 2 x d z / Y ( u 9 * 4. X . 9X . 2 , x . 3 y . 2 " x , & z . 2 4 d x 2 2a 2a 2a N 6 h 9 y N 6 h 3 2 Z N 6 h 9 y N 6 h V , 3 x . 2 4a 36 4a 3 9 z 4a 36 4a 8a 36 - - ^ d ? + z ] 2 ~ ^ [ y " l | ] 2 ) a d x d 9 . ( 3 . 53 ) 8a 8a The k i n e t i c ene r gy o f t h e i n f l a t e d s h e l l a c c o u n t i n g f o r t h e added i n e r t i a i s T = | P h ff K ^ V > ( | f ) 2 + (|f ) 2 ] a d 9 d x + 5 ( 1 + C m ) p w J J j f f ? s i n 6 + f f c o s 6 ] 2 r d 9 d x d r 1 v r C r ^ 2 1 ,3Vx2 . . 3 z N 2 . ( 1 + C m ) p w a / 3 y , a h a - 2 co s 9 ) 2 ] ad9dx . ( 3 . 54 ) 3 t The assumed d i s p l a c e m e n t s i n t h e beam b e n d i n g mode a r e 70 x = [ a ^ C C ) + a2^(?)]cosecosfc)t , (3.55a) y = [a 3 * r (?) + a 4f r(?)]sin6coswt , (3.55b) z = [a c * (?) + a,V (?)]cos9coscot , (3.55c) 5 r o r where $ r ( £ ) are the characteristic beam functions for cantilevers and Y ( ? ) = c o s h U ? - c o s u £ - a ( s i n h U £ - s i n U £ ) r r r r r r are the characteristic functions for a clamped-pinned beam. Here a = c o t U L r r and are the roots of the frequency equation tanUL-tanhuL = 0 The modal forms used in Equations (3.55) are quite rea l i s t i c as in practice behaviour of an inflated shel l suggests boundary conditions o n between the two sets mentioned above According to the Rayleigh-Ritz Method I— (U-T) = 0 (3.56) 9a. Substituting the assumed modes (3.55) i n t o the energy expressions, i n t e g r a t i n g over the period, and applying the Rayleigh-Ritz procedure one obtains a s i x t h degree frequency equation, which can be rearranged to form an eigenvalue problem of the type [M](a) = fi2[N](a) , (3.5Za) 2 where Q i s the dimensionless frequency given by Q2 m Pa2(1-V V m ( 3 5 7 h ) Hi The order of the matrices w i l l be 3n i n general, where n i s the number of mode shapes i n the assumed so l u t i o n . Premultiplying Equation (3.57a) by [N] gives [N] 1[M](a) = fl2(a) or [Q](a) = ft2(a) , (3.57c) where [Q] = [N] _ 1[M] 72 The system of Equations (3.57c) can now be solved by an i t e r a t i o n procedure (e.g. UBC DREIGN) to obtain the frequencies and mode shapes. The elements of the matrices [M] and [N] are given i n Appendix I II for Washizu's s h e l l theory. 3.2 Tapered Beam Analysis For a tapered beam one has to resort to the c o n i c a l s h e l l theory to account f o r i n i t i a l stresses induced by i n f l a t i o n pressure. The c o n i c a l s h e l l theory i s a simple generalization of the uniform c y l i n d r i c a l s h e l l theory. Uniform c y l i n d r i c a l s h e l l s are a s p e c i a l case of c o n i c a l s h e l l s with zero vertex angle. Due to varying radius along the lengthy the i n f l a t i o n prestresses are functions of x, the a x i a l co-ordinate. Hence the p o t e n t i a l reduction technique employed i n the previous section f o r the uniform c y l i n d r i c a l s h e l l s cannot be applied i n the present case. The exact s o l u t i o n of the co n i c a l s h e l l equations themselves i s f a r from simple. In general, approximate and numerical techniques have been used i n the i n f l a t e d c o n i c a l s h e l l studies to date. As pressure e f f e c t s on the natural frequencies of the c o n i c a l s h e l l s are expected to be small f o r the beam-bending mode of i n t e r e s t here, a f i r s t s i m p l i f i c a t i o n would be to neglect the i n t e r n a l pressure e f f e c t s and study the tapered c a n t i l e v e r using the elementary beam theory. Equilibrium of the forces acting on a section of a tapered beam (Figure 3-3) o s c i l l a t i n g i n water leads to 73 z , w Figure 3-3 Geometry of motion of a tapered c a n t i l e v e r (3.58a) where F^ i s the t o t a l hydrodynamic force on the element. It may be noted that the second term representing the i n e r t i a force of the section i s p r i m a r i l y due to the water in s i d e the c a n t i l e v e r since the mass of the w a l l material i s very small. The resistance F„ i s taken to be of the Morison t y p e 1 5 , i . e . , made up of a drag force proportional to the square of the v e l o c i t y and an added i n e r t i a force caused by the acceleration of the surrounding water, , dv F u - ^C,d(x)p v .|v .I + C A (x)p — H 2 d w r e l ' r e l 1 m w w < r e l dt (3.58b) 74 where C, and C are the drag and added mass coefficients, respectively, d m and v - the velocity of the cantilever relative to the fluid. This rel J assumes that the drag and the inertia effects are free of appreciable mutual interference. In the study the values of C, and C are assumed J d m to be constant and equal to 1.18 and 1.0, respectively, as in the uniform cylindrical beam analysis. From quations (3.58a) and (3.58b) one obtains dx ox dt (3.58c) Defining n = w/d r , £ = x/L , k = (d r-d t)/d r , and noting that I(x) = I ( l - k £ ) 3 , A (x) = A ( 1 - k C ) 2 r w wr d(x) = d r(l-k?) , (3.59) Equation (3.58c) can be rewritten as ^ ^ [ ( l - k £ ) 3 ^ ] + (l+C ) P A (l-k£) 2^ L 8£2 3£2 m W w r 3t 2 75 For the purpose of finding natural frequencies, Equation (3.60) may be simplified considerably i f i t i s assumed that the nonlinear drag term has only a second order effect and may be neglected as a f i r s t approximation. Discarding the nonlinear term, Equation (3.60) can be rewritten as , o o ( 1 + c )P A L 4 „ ~3^ o i2„ m w w r.1 a - k e ) 2 3 - ^ - 6ku-ke)?-3 + 6k 2 5 -2 + — j L - d - k g ) ! ^ dt H 9? r 3 t z - 0 . . (3.61) In absence of any known closed form solution an approximate procedure has to be used. The solution i s assumed to be of the form 1 oo n(?,t) = £ ) * ± ( 5 ) f i ( t : ) ' ( 3 , 6 2 ) i=l where, as before, $. are the characteristic beam functions for a cantilever. Substituting (3.62) into Equation (3.61), multiplying by $ (5) and integrating over the length gives oo i A 1 2 2 ^ /(a-k5> — f V i " 6 k ( 1 " k } I V i + 6 k 5 * r f ± d r d r r 0 (l+C )p A L 4 ,2. m w w d f. + EI —(1-^5)^* |)d? - 0 . (3.63a) r dt 76 For natural v i b r a t i o n s , f ' vary harmonically with the same c i r c u l a r frequency U). Thus d 2 f * V Noting that d V d £ 4 1 1 Equation (3.63a) becomes l - J . n 3 2 6 k ( l - k £ ) 1* + 6k^ 1* + ( l - k £ ) [ u ? ( l - k £ ) 0 d?J r d r r 1 (l+C )p A L4oo2 m w^ w x ___E ]* ±* )f d£ = 0 , r = 1, 2, i . r (3.63b) The above set containing an inf in i te number of frequency equations gives r ise to the eigenvalue problem of the form [S].(f) = X 2 [V](f) , (3.64a) or [Z](f) = X 2 ( f ) , (3.64b) where [S], [V], and [Z] are square matrices of order i , with 77 [Z] = [V] X [ S ] 2 and X i s the dimensionless frequency given by (1+C )p A LV „ m w w 1 A i " - E I A d e t a i l e d c a l c u l a t i o n procedure and the elements of the matrices [S] and [V] are given i n Appendix IV for i = 1, 2, and 3, respectively. 3.3 Results and Discussion To assess v a l i d i t y of the a n a l y t i c a l procedures, i t was thought appropriate to experimentally determine natural frequency over the range of system parameters of i n t e r e s t i n p r a c t i c e . The tests were c a r r i e d out i n the hydraulic tank, (Figure 2-4), described i n Chapter 2. A seri e s of cantilevers of varying radius, length, taper, f i l m thickness and i n t e r n a l pressure were tested. Natural frequencies were monitored through two waterproof s t r a i n gauges attached to the top and bottom side of an i n f l a t e d beam near i t s root (clamped end). Free v i b r a t i o n s were triggered through i n i t i a l displacement and release, and the c y c l i c s t r a i n recorded on the o s c i l l o s c o p e indicated the natural frequency. A t y p i c a l trace on the o s c i l l o s c o p e screen i s shown i n Figure 2-4. In general, the tests f o r a given s e t t i n g were repeated at least f i v e times and the average was used. The 78 tests can be repeated with deviations less than 2%. 3.3.1 Uniform C y l i n d r i c a l Cantilever Table 3.1 compares the experimentally measured frequencies of f i v e i n f l a t e d c a n tilevers with various t h e o r e t i c a l p r e d i c t i o n s . Experimental r e s u l t s show a s l i g h t increase i n frequency with pressure. Although the increase i s quite small and almost n e g l i g i b l e f o r a l l p r a c t i c a l purposes, i t i s s i g n i f i c a n t to recognize that t h i s trend i s co r r e c t l y predicted by the solutions to Fliigge's reduced equation (3.13). The difference between the exact frequency [Equation (3.26)] and that obtained by the mode-approximation [Equation (3.37c)] i s n e g l i g i b l e for a l l cases considered. Both solutions are capable of p r e d i c t i n g the natural frequencies with excellent accuracy. Besides Fliigge's theory, the reduction procedure was also applied to the membrane and Herrmann-Armenakas theories (Appendix V). The r e s u l t s obtained from these reduced equations are included i n the table f o r comparison. Both the methods tend to overstress the pressure e f f e c t s , with the Herrmann-Armenakas theory erroneously p r e d i c t i n g a decrease i n frequency with i n t e r n a l pressures. In a few cases the frequencies drop to zero and turn imaginary, rendering the v a l i d i t y of the reduced Herrmann-Armenakas equation questionable. The reduced membrane equation predicts a much larger increase i n frequency with i n t e r n a l pressure than that observed. On the other hand, the much simpler elementary beam theory, despite i t s i n a b i l i t y to predict the pressure e f f e c t s on natural f r e -quencies, gives r e s u l t s of reasonable accuracy. 79 Pressure Experi- Reduced Equations Rayleigh--Ritz TUBE N/m mental Beam Mem- Herr.- Flugge Washizu Membrane SIZES (psi)• Data Theory brane Armen. I II-. 1-term 2-term 2-term 0 0.69 0.69 0.69 0.69 0.69 1.05 1.02 1.02 L = 1.02m (40") d =5.08cm (2.0") h=0.008cm (.003") 2.07xl0 4 (3.0) 3.45x10 (5.0) 4.14x10 (6.0) 0.70 0.70 0.70 1.05 1.23 1.31 0.20 Im. Im. 0.69 0.69 0.69 0.69 0.69 0.69 2.40 2.97 3.22 2.22 2.65 2.89 1.47 1.69 1.78 0 1.04 1.04 1.04 1.05 1.04 1.58 1.54 1.54 L = 0.91m (36") d =7.62cm (3.0") h=0.008cm (.003") 2.07xl0 4 (3.0) . 3.45x10 (5.0) 4.14x10 (6.0) 1.06 1.07 1.08 1.36 1.54 1.62 0.74 0.45 0.16 1.05 1.05 1.06 1.04 1.05 1.05 2.86 3.45 3.71 2.71 3.19 3.39 1.94 2.16 2.26 L = 0.61m 0 A 1.91 1.91 1.91 1.92 1.91 2.90 2.82 ; 2.82 (24") d =5.08cm 2.07x10 (3.0) 2.01 2.32 1.56 1.93 1.91 4.61 4.44 3.34 (2.0") h=0.008cm 3.45x10 (5.0) 2.01 2.56 1.28 1.93 1.92 5.45 5.14 3.64 (.003") 4.14xl0 4 (6.0) 2.02 2.67 1.12 1.93 1.92 5.82 5.44 3.77 L = 0.61m 0 2.34 2.34 2.34 2.38 2.34 3.52 3.41 3.41 (24") d =7.62cm 2.07x10 (3.0) 2.38 2.70 2.07 2.39 2.35 4.99 4.82 3.85 (3.0") h=0.008cm 3.45x10 (5.0) 2.39 2.91 1.87 2.39 2.36 5.76 5.48 4.10 (.003") 4.14x10 (6.0) 2.39 3.01 1.77 2.39 2.36 6.11 5.76 4.22 L = 0.61m 0 3.31 3.31 3.31 3.36 3.31 4.97 4.81 4.81 (24") d =7.62cm 2.07x10 (3.0) 3.36 3.57 3.12 3.37 3.32 6.10 5.93 5.13 (3.0") h=0.015cm 3.45x10 (5.0) 3.37 3.73 3.00 3.37 3.32 6.75 6.53 5.33 (.006") 4.14x10 (6.0) 3.37 3.81 2.93 3.37 3.32 7.05 6.78 5.43 I - Exact s o l u t i o n [Equation J(3.26)] II - Mode-approximation s o l u t i o n [Equation (3.37c)] Table 3.1 Comparison between A n a l y t i c a l l y and Experimentally Obtained Frequencies f o r Uniform Beams (Hz) Also shown i n Table 3.1 are the t h e o r e t i c a l predictions based on the Rayleigh-Ritz method applied to the Washizu and the membrane s h e l l theories. The d e r i v a t i o n of the energy expressions for the membrane theory i s presented i n Appendix VI. For the Washizu theory r e s u l t s f or one- and two-term approximations are shown for comparison. Their agreement with experimental r e s u l t s i s poor, and the large diff e r e n c e between the one- and two-term approximations indicates a slow convergence of the natural frequencies. The fa c t that the convergence of r e s u l t s can be very slow has been observed by other investigators such as Sewall and Naumann1*1*, and Resnick and Dugundji 1* 5. Sewall and Naumann compared a n a l y t i c a l frequencies with experimental r e s u l t s f or clamped-free s h e l l s without prestress. They used seven terms i n the assumed mode shapes to obtain convergence of the R i t z procedure. Resnick and Dugundji, using an energy approach, found that the t h e o r e t i c a l and experimental data agreed only for modes with more than f i v e c i r c u m f e r e n t i a l waves. Thus a large number of terms w i l l be required to converge to the r i g h t value, but the amount of algebra involved i s great since the order of the governing matrices increases r a p i d l y with the number of terms used i n the approximation (order of matrices equals three times the number of terms used). 81 3.3.2 Tapered Cantilever V a r i a t i o n of the fundamental and second mode eigenvalues (v^X?) with the taper r a t i o k i s shown i n Figure 3-4 for the cases of one-, two-, and three-term approximations. For the fundamental mode the three-term approximation gives eigenvalues that l i e i n between the ones predicted by the one- and two-term approximations. If more terms are taken i n the evaluation, r e s u l t s w i l l probably converge to intermediate values bounded by the two- and three-term approximations. For the second mode, the eigenvalue remains r e l a t i v e l y unchanged up to a taper r a t i o of about 0.5. For larger tapers the two-term approximation f a i l s to give accurate p r e d i c t i o n s . On the other hand, the one-term approximation, employing the second-mode ca n t i l e v e r beam function, provides estimates deviating le s s than 2.5% from t h e i r corresponding three-term approximation values. Nevertheless, the deviations among the three approximations shown are n e g l i g i b l e f o r small tapers ( k £ 0 . 5 ) . Table 3.2 shows the experimental r e s u l t s and the associated t h e o r e t i c a l p r edictions. The agreement i s very good and the simple beam theory used i s capable of accurate predictions. It i s obvious from the experimental r e s u l t s that the increase i n frequency due to i n t e r n a l pressure may be considered n e g l i g i b l e from p r a c t i c a l design considerations. 5. Or 4.0 J/2 1- t e r m 2- t e r m 3- t e r m 3.0 1.0 0 0.2 0.4 0.6 T a p e r r a t i o k 0.8 Figure 3-4 Variation of eigenvalues with taper ratio 83 PRESSURE 1 N /m 2 ( p s i ) TUI 0.61m ( 24 " ) JE LENGTH 0.81m ( 32 " ) 1.02m ( 40 " ) Beam E x p t . Theo r y D a t a Beam E x p t . Theo r y D a t a Beam E x p t . Theo r y Da t a d =7.62cm r ( 3 . 0 " ) d =3.81cm t ( 1 . 5 " ) h=0.008cm ( . 003 " ) 0 2 . 0 7 x l 0 4 ( 3 . 0 ) 3 . 4 5 x l 0 4 ( 5 . 0 ) 4.14x10 4 ( 6 . 0 ) 3.27 3.22 3.25 3.25 1.84 1.84 1.84 1.85 1.18 1.21 1.22 1.22 d =10.16cm r ( 4 . 0 " ) d =5.08cm C ( 2 . 0 " ) h=0.008cm ( . 003 " ) 0 2 . 0 7 x l 0 4 ( 3 . 0 ) 3 . 4 5 x l 0 4 ( 5 . 0 ) 4 . 1 4 x l 0 4 ( 6 .0 ) 3.74 3.74 3.76 3.77 2.13 2.13 2.14 2.15 1.36 1.30 1.32 1.33 d =12.70cm r ( 5 . 0 " ) d =6.35cm C ( 2 . 5 " ) h=0.008cm ( . 003 " ) 0, 2 . 0 7 x l 0 4 ( 3 . 0 ) 3 . 4 5 x l 0 4 ( 5 . 0 ) 4 . 1 4 x l 0 4 ( 6 . 0 ) 4.22 4 .23 4 .24 4 .24 2.38 2.39 2.40 2.40 1.52 1.53 1.53 1.54 d =7.62cm r ( 3 . 0 " ) d =3.81cm ( 1 . 5 " ) h=0.015cm ( 0 . 0 0 6 " ) 0 2 . 0 7 x l 0 4 ( 3 . 0 ) 3 . 4 5 x l 0 4 ( 5 . 0 ) 4 . 1 4 x 1 0 4 ( 6 . 0 ) 4 .63 4 .64 4.65 4.66 2.60 2.49 2.52 2 .53 1.67 1.64 1.65 1.65 T a b l e 3.2 Compar i s on Between A n a l y t i c a l l y and E x p e r i m e n t a l l y O b t a i n e d F r e q u e n c i e s (Hz) f o r T ape red Beams (Hz) 84 3.4 Concluding Remarks The s i g n i f i c a n t conclusions based on the free v i b r a t i o n analysis can be summarized as follows: ( i ) For the uniform c y l i n d r i c a l i n f l a t e d c a n t i l e v e r s v i b r a t i n g i n the beam-bending mode, the governing three-dimensional s h e l l equations do not permit simple s o l u t i o n s . Although an exact procedure i s a v a i l a b l e , i t has been sparingly applied because of the great amount of work required. Numerical and approximate techniques have mostly been used i n the s h e l l v i b r a t i o n studies to date. For the present study, i t i s found that the s h e l l equations can be reduced to a s i n g l e equation s i m i l a r i n form to the one f o r the transverse v i b r a t i o n s of a beam with rotary i n e r t i a included. Flugge's s h e l l equation i n reduced form i s capable of p r e d i c t i n g the v i b r a t i o n a l behaviour of uniform c y l i n d r i c a l beams subjected to i n t e r n a l pressure. Accurate predictions are possible even with the approximate s o l u t i o n of the equation discussed here. However, the reduction technique should be applied with care, since various s h e l l theories give r e s u l t s which may be s i g n i f i c a n t l y d i f f e r e n t . The reduced equations f o r the membrane and Herrmann-Armenakas theories f a i l to give reason-able r e s u l t s . I t should also be noted that f o r c e r t a i n s h e l l theories the equations are nonsymmetric (e.g., the Timoshenko-Voss equations used by Fung et a l . 3 7 ) and the 85 potential function T w i l l not be simply related to the la tera l displacement w. In these cases the reduction process w i l l not give meaningful results . ( i i ) The elementary beam theory gives predictions of reasonable accuracy for both the uniform cy l indr ica l and the tapered beams although the theory does not incorporate internal pressure effects. Fortunately the effect of pressure, at 4 2 least in the range investigated here (< 4.14x10 N/m or 6.0 psi) appears to be insignif icant. Even with the internal 4 2 pressure of 4.14x10 N/m or 6.0psi (P = 0.02), the increase in frequency would amount to less than 2%. ( i i i ) The Rayleigh-Ritz method does not give accurate results in the present investigation. The convergence of the results is slow and the relat ively great amount of work required to achieve acceptable accuracy cannot be j u s t i f i e d . (iv) The hydrodynamic drag damping causes only an amplitude decay and does not affect the resonant frequencies of the canti-levers up to the f i r s t order approximation. (v) For the tapered beam the fundamental frequency increases with the amount of taper. On the other hand, the second natural frequency stays relat ively constant up to a taper ratio of about 0.5. For beams with taper ratios less than 86 0.5, there i s no apparent advantage i n employing more than one term i n the assumed mode so l u t i o n (3.62). 87 4. FORCED VIBRATION OF NEUTRALLY BUOYANT VISCOELASTIC INFLATED CANTILEVERS The previous chapter investigated free response of the i n -f l a t e d c a n t i l e v e r s . It was noted that with increasing pressure, the increase i n resonant frequencies i s so small that the pressure e f f e c t s may be neglected. The object of t h i s chapter i s to study the steady state response of the v i s c o e l a s t i c c a n t i l e v e r to wave e x c i t a t i o n . A preliminary study on the coupled motion of the submarine detection system was made by M i s r a 3 , who concluded that displacements of the c a n t i l e v e r t i p s , where the hydrophones are located, may be reduced by using an e l a s t i c cable with small s t i f f n e s s and a heavy ce n t r a l head. With a s o f t cable the transmission to the array of the buoy movements due to the surface waves w i l l be r e l a t i v e l y small. Of course, i n general, such a submerged platform i s subjected to a v a r i e t y of disturbances i n c l u d i n g those due to surface and i n t e r n a l waves, and ocean currents. The configuration of the submarine detection system of i n t e r e s t here suggests that wave e x c i t a t i o n f e l t by the buoy and transmitted by the cable are ultimately experienced by the ce n t r a l head. Hence i n the present i n v e s t i g a t i o n , the disturbance i s taken to be a generalized known displacement objective being the r e s u l t i n g response of the i n f l a t e d c a n t i l e v e r . 88 To incorporate viscoelastic nature of the beam material, equivalent dissipative terms are included in the governing equation. This can be achieved quite readily by replacing the modulus of e las t ic i ty by the complex Young's modulus 3 ' 8 5 . For a three parameter so l id the complex Young's modulus can be represented a s 7 5 E*(w) = E 1 (E 2 +lV 2 u) / (E 1 +E 2 +iV 2 o>-) . (4.1a) Equation (4.1a) may be rewritten as a sum of real and imaginary parts, i . e . , E*(u>) = LE 1(6+iojY) - E ^ S + Y ! ^ ) , (4.1b) where 6 = 1 - E 1 ( E 1 + E 2 ) / [ ( E 1 + E 2 ) 2 + V 2 C 0 2 ] , (4.1c) and y is the loss factor given by Y = E ^ / K E ^ E j ) 2 ^ 2 ] . (4. Id) 4.1 Uniform Cyl indrical Beam For forced vibration, the root (clamped end) of the cantilever is assumed to be displaced periodical ly by 89 TI = > (n c o s m U T + r) s inmiOT ) r c , m s , m m = l ( 4 . 2 ) where T i s r e l a t e d t o t h e r e a l t i m e t by (3 .16 ) w i t h t h e Y o u n g ' s modulus E r e p l a c e d by t h e i n s t a n t a n e o u s modu lus E^. I t s h o u l d be n o t e d t h a t E q u a t i o n ( 4 .2 ) i s a F o u r i e r s e r i e s and i s c a p a b l e o f r e -p r e s e n t i n g any p e r i o d i c e x c i t a t i o n . I n t h e a n a l y s i s , h oweve r , o n l y t he f i r s t two te rms a r e i n c l u d e d as s p e c t r a l a n a l y s i s o f a t y p i c a l ocean wave shows a s t e e p r e d u c t i o n i n ene r gy c o n t e n t a t h i g h e r h a r m o n i c s ( F i g u r e 4 - 1 ) . R e c o g n i z i n g t h a t t h e maximum ene r gy c o n t e n t ! L o n g p e r i o d ! G r a v i t y W a v e s i C a p i l l a r y W a v e s ! s W a v e s U ^ T i d e s 1 / ~ x ; 1! •• In, ^ ^T^I~~*r^7^rrm.iLw»i . -CD i_ <D C LU 1 0 , - 3 ,-1 1 0 ' 1 0 ' F r e q u e n c y ( H z ) 2 4 h r 5 m i n 0.1 s e c P e r i o d F i g u r e 4 -1 S c h e m a t i c r e p r e s e n t a t i o n o f t h e ene r gy c o n t a i n e d i n t h e s u r f a c e waves o f t h e o c e a n s - - ( R e f e r e n c e 86) 90 of the surface waves is at around 0.1 Hz while the natural frequency of the inflated beam is in the range of 0.7 to 4 Hz (table 3.1, p. 79) the effect of higher harmonics is expected to be negligible. The absolute lateral displacement of the beam is n(5,T) = n r + n(5,T) , (4.3a) where 1"|(£,T) is the displacement of the cantilever relative to the root. In general the relative displacement can be represented by ,xx> ri(5,T) = ^ ^ [ n c n(Ocosnon: + n s n(£)sinn<DT] . (4.3b) n=l Substituting Equations (4.1) and (4.3) into the nondimensionalized governing equation of motion (3.17a) leads to oo oo 4^ 4^ > / {[6+Y^-][ -^cosnoox + ^-sinnon] - n w B * i ? f n A d X dC dK A2 A2 d n d ri _ „ . r : c,n , 's,n . , 2„ , 2, * [ s-cosncoT H s-sxnncoT] - w B, [m (ri cosmurr 1 c,m d£ d£ * 2 2 * +n sinmorr)+n (n cosnwx+n sinnoox)] + w a[m(ri cosmtox c,m c,n s,n s,m * i * -ri sinma)x)+n(n cosnaix-n sinnwx)] m(n cosmaix •c,m s,n c,n ' s,m * i -n sinma)x)+n(n cosnwx-ri sinnoox) } = 0 , (4.4) c,m s,n c,n 1 91 where Y =>^ 3Y Taking two terms (n=2) i n the assumed so l u t i o n (4.3b) the nonlinear drag term i n Equation (4.4) can be rewritten as +w2a[A2siii2(coT-cJ>1)+2ABsin^ ] , where A2 B2 h M u l t i p l y i n g Equation (4.4) by cosunr, cos2u)T, sinwx, and sin2u)T, r e s p e c t i v e l y , and int e g r a t i n g with respect to U)T over the period gives four independent equations. It should be noted that the hydrodynamic drag force represented by the l a s t term i n Equation (4.4) i s i n phase with the v e l o c i t y and changes d i r e c t i o n twice during a cycle. Hence ^ c , l + T 1 c , l > + ' V l * ^ ] 4 [ ( V 2 + \ , 2 > 2 + <2 +\,2> 2] n arctan /VL + V A V c , l + n c , J * r arctan / y 2 + y 2 \ \ nc,2 + n c , 2 / 92 the correct sign should be chosen accordingly. However, the p o s i t i v e and negative i n t e r v a l s of t h i s term i s dependent on the amplitudes A and B. Here, as a f i r s t approximation, i t i s assumed that A » B such 2 that the d i r e c t i o n of the force i s governed by the term i n v o l v i n g A . This i s j u s t i f i e d as the second harmonic response w i l l be i n general much smaller than the fundamental one. A better estimate may be obtained by examining the first-approximation solutions of A and B and adjust the sign accordingly to give more accurate r e s u l t s . Assuming the sign of the drag term to be governed by 2 2 2 w aA s i n (u)T-<}>^ ) and carrying out the i n t e g r a t i o n r e s u l t s i n the following four equations: 4 4 2 d ^c 1 d n s 1 2 d n c 1 2 * dC dc L dr ' • 2 i 2£L{|A2sin<|>1+B2 [2sin())1-|sin(3(|)1-2(()2)~sin(5(f»1-2(()2) ]} = 0 , (4.5a) 4 4 2 d n c 2' d n s 2 2 2 d n c 2 2 * 5 — - S f + 2 y o > — * f - 4 ( o V — £ ^ - 4</B (r, 2 + n c 2 ) + dr dr dr ' • |AB0«iJ 2[2sin(}) 24isin(2(() 1+()) 2)~sin(4(() 1-(}) 2)] = 0 , (4.5b) ,4_ ' 4 ,2v d n . d n . , d n 6 — ¥ - Y * > — ¥ - 0) 2B O —-2±i - a>V(n* ,+n J -d £ 4 ' d ? 4 2 dK1 ^ ' • . i - ' - . i ' ooi32r8.2 ^{|A 2cos(() 1+B 2[2cos(j) 1^cos(3()) 1-2* 2)-icos(5(J) 1-2(() 2)]} = 0 , (4.5c) 93 4 4 2 dn D N 0 ? D T 1 9 9 * d £ 4 d? 4 2 d £ 2 1 s ' 2 s ' 2 | A B O O > 2 [COS<(»2+J |COS (4<P1-<()2) ] = 0 . (4.5d) A * The quantities r\ , T) , r) , and X] can be represented as c ,n ' s,n c ,n ' s,n r ^ C j n s ,n Substituting these into Equations (4.5), multiplying by <I>^  and $ 2 > respectively, and integrating with respect to £ over the length, one obtains eight simultaneous algebraic equations: [ M 46 - t o 2 ( B 2 c 1 1 + B 1 ) ] c 1 1 - o ) 2 B 2 c 2 1 c 1 2 + y 4 ywS 1 1 2 * 0) B l C u + SffiLr = o TT 1 (4.7a) [y j6 -4u 2 (B 2 C 1 1 +B 1 ) ]C 2 1 - ^ W 2 B 2 C 2 1 C 2 2 + 2y 4 Y wS 2 1 - ^2\C*21 + 2 ^ I 0 = 0 TT 2 (4.7b) 94 y J y M c u - [ y 4 6 - a ) 2 ( B 2 c 1 1 + B 1 ) ] s 1 1 + O ) 2 B 2 C 2 1 S 1 2 2 * oxo + W B 1 S 1 1 + — J 3 = 0 ( 4 . 7 c ) 2 y > c 2 1 - [ y 4 6 - 4 a ) 2 ( B 2 c 1 1 + B 1 ) ] s 2 1 + 4 a ) 2 B 2 c 2 1 s 2 2 2 * au + ^\S21 + 4 — 1 4 - 0 , (4.7d) c o 2 B 2 C 1 2 C 1 1 - t y 4 6 - a ) 2 ( B 2 C 2 2 + B 1 ) ] C 1 2 - y j y w s ^ 2 * ato + U B 1 C 1 2 " = ° ( 4 . 7 e ) 4 a ) 2 B 2 C 1 2 C 2 1 - [ y 4 6 - 4 a ) 2 ( B 2 C 2 2 + B 1 ) ] C 2 2 - 2vfas22 2 * ao) + 40) B l C 2 2 - 2 — l g - 0 (4 .7 f ) y ^ C 1 2 + a ) 2 B 2 C 1 2 S n - [ y 4 6 - c o 2 ( B 2 C 2 2 + B 1 ) ] s 1 2 2 * OUti + W B 1 S 1 2 + I T h - 0 (4.7g) 2 y 2 V c 2 2 + 4 c o 2 B 2 C 1 2 S 2 1 - [ y 4 6 - 4 a ) 2 ( B 2 C 2 2 + B l ) ] S 2 2 + 4 0 ) 2 B l S * 2 + 4 ^ - I 8 - 0 (4.7h) 95 where B 2 [ 2 s i n * - |sin(3$ ± - 2 $ 2 ) isin(5$ 1-2$ 2)]}d5 , (4.8a) 1 = ^ A B * ^ I 2 - j A B 1 [ 2 s i n * 2 + |sin(2$ 1+$ 2) - ^ s i n ^ - ^ ) ]'d£ , (4.8b) 0 .1 =J { | A 2 C O S $ 1 I 3  I { ^ J A ' c o s ^ + B 2 [ 2 c o s $ 1 + |cos(3$ 1-2$ 2) - • i c o s C S * ^ ^ ) ] } * ^ , (4.8c) 1 I 4 = ^y*AB[cos* 2 + ^ 0 0 8 ( 4 * ^ 2 ) ] * ^ , (4.8d) 1 I c = | { | A 2 s i n * . + B 2 [ 2 s i n * 1 - |sin(3*..-2* 0) 0 - ^ 8 ^ ( 5 * ^ 2 * 2 ) ] } * ^ , (4.8e) I g = J [2sin$ 2 + | s i n ( 2 * ^ 2 ) - ^ s i n ^ * . ^ ) ]AB* 2d£ , (4.8f) 0 I ? = y ^ | A 2 c o s 2 * 1 + B 2 [ 2 c o s * 1 + |cos(3* 1-2* 2) 0 - 5 0 0 3 ( 5 * ^ 2 * 2 ) ]}* 2d£ , (4.8g) X8 = J t c o s $ 2 + ? i c o s ( 4 $ r $ 2 ) ] A B < | , 2 d ? • (4.8h) 96 The set of Equations (4.7) has to be solved simultaneously to obtain the response amplitudes. Here i t was accomplished numerically. 4.2 Tapered Beam The governing equation based on the elementary beam theory was given previously in Section 3.2, ^ ^ [ ( i - k 5 ) 3 3 - ^ ] + (l+C )p A ( i - k S ) 2 ^ L K2 m W W r d t 2 Accounting for the viscoelastic dissipation the equation may be nondimensionalized as [S+Y|=][(l-kS)2^5 - 6 k ( l - k ? ) ^ + 6 k 2 ^ ] + ( 1 - k ? ) 3 ^ dC 3 5 3 5 3 T + a ^ l ^ l = 0 With a periodic root excitation given by (4.2) and an assumed solution similar to that for the uniform cy l indr ica l beam eight simultaneous equations are obtained after following the anlyt ical procedure similar to that for the uniform case: 97 {yjs - w2 + [(w 2-2u 46)x 3 - 66X;L]k + [y 4 x 4 + 6 ( x 2+C n) ]6k 2}C 1 1 + {[w2X7 - 6(2y2X7+6X5)]k + [y 2X 8 + 6(X 6+C 2 1 ) ] 6 k 2}c i 2 l x . l i k 2 + {yjyw - [ 2 y j x 3 + 6x1]ywk + [ y x x 4 + 6 ( c 1 1 + x 2)]ywk } s n - yco{[2y2x7 + 6x5]k - [ y 2 x g + 6(x 6+C 2 1) ]>S12 - w 2 { c * x - [ x ^ + X7C*2]k - ^ I1> = 0 , (4.9a) { y 46 - 4OJ2 + [2x3(2032-yj6) - 6 6 x x]k + [y*X 4 + 6 ( x 2+C n) ] 6 k 2 } c 2 1 + { [ 2 x 7(2co 2 - y 46) - 6 6 x 5 l k + [ y 2X 8 + 6(x 6+C 2 1) ]6k2>C22 +' {2y4 - 4 [ y j x 3 - 3x±]k + 2 [ y * x 4 + 6 ( x 2+C n) ]k2}yws21 {4 [ y 2 x 7 + 3x 5l - 2 [ y 4 x 8 + 6(x 6+C 2 1)]}yws 2 2 W ^ S l " [X 34 +-X7C*2]k + 2^-2I2} = 0 , (4.9b) { y * - 2 [ y 4 x 3 + 3 x 1 ] k + [ y * x 4 + 6 (X 2+C u) 1 >YwCu {2 [ y 2 x 7 + 3x5] - [ y 2 x 8 + 6(X 6 + C 2 1 ) ] } Y ^ C 1 2 - { y * 6 - co2 - [ ( 2 y j 6-w 2 ) x 3 + 6 6 X ; L]k + [ y j x 4 + 6(X 2+C U) ] 6 k 2 } s u + {t(2y45-co2)xy + 66x5]k - [u^ Xg + 6(x6+C21) ]6k 2}s i 2 + ^ { S ^ - + S*2xy]k + ^I3> =0 , . (4.9c) {2y4 - 4[yjx 3 + 3xx]k + 2[yjx 4 + 6(x2+C1;L) ]k2}YwC21 r - U[y 2x ? + 3x5]k - 2[y 4x g + 6(X 6+C 2 1)]k 2}ywC 2 2 - {y^ 6 - 4w2 - 2[(y46-2o)2)x3 + 66X;L]k + [y*x4 + 6(x2+C1;L) ]6k 2}S 2 1 + {2[(y46-2a)2)x7 + 36x5lk - [y 2X g + 6(x6+C21) ]6k2}s"22 + 4to2{sJ1 - [S 2 1x 3 + sj 2x 7]k + ^I4> = 0 , (4.9d) { [ ( (0 2-2yj6)x 7 - 66x9]k + [y?_x8 + 6(X10+C12) ]<5k2}cu + {y46 - a)2 + [(w2-2y46)x13 - 66x n]k + [y 2X l 4 + 6(x 1 2+C 2 2) ]6k2}c - {2[y4x7 + 3x9]k - [yjx 8 + 6 ( X 1 0 + C 1 2 ) 1 } y W ^ l l + ^ 2 " 2 ^ 2 X13 + 3 x l l ] k + [ y2 X14 + 6 ( X12 + C22 ) ] } Y^12 - 0)2{C*2 - [C* l X ? + cJ 2X 1 3]k - ^ I5> = 0 , (4.9e) {2[(2w2-yj6)x7 - 36Xg]k + [yJ X g + 6<X10+C12)]<5k2}c21 99 + {y*6 - 4co2 - 2[(u^-2o) 2)x 1 3 + 36xulk + [ y 2 X 1 4 + 6(x 1 2 +C 2 2)]6k 2 }C 2 2 - {4[yjx 7 + 3xg]k - 2[y*xg + 6 ( X 1 0 + C 1 2 ) ] * k 2}YC0S 2 1 + 2 y * - 4[y 2x 1 3 + 3x u]k + 2[y 2 X 1 4 + 6 ( X 1 2 + C 2 2 ) ] * k 2 } Y cuS 2 2 - 4 o , 2 { C 2 2 - [ C 2 1x ? + cj 2x 1 3]k - |-i 6} = 0 , ( 4 . 9 f ) {2[yjx 7 + 3x9]k - [yjx8 + 6(x 1 0 + C 1 2 ) ]k 2 } y a )C n - {y^  -2[y 2 X l 3 + 3xu]k + [y 2x 1 4 + 6(x 1 2 + C 2 2)]k 2 } Y w C 1 2 + {[(co2-2yj6)x7 - 66x9]k + [yjxg + 6(x 1 0 + C 1 2 ) ]6k 2 } S n + {y*6 - co2 - [(2yj6-a)2)x13 + 6 6 X l l]k + [y 2 X 1 4 + 6(x 1 2 + C 2 2 ) ] * 6k 2 }S 1 2 - c o 2 { S * 2 - [ S * l X 7 + S * 2 X 1 3 l k + ^ I ? > = 0 , ( 4 .9g ) {4[yjx 7 + 3x9]k - 2[yJXg + 6 (x 1 0 +C 1 2 ) ]k 2 }yaJC 2 1 - { 2 y 2 - ^ 2 X 1 3 + 3 x l l ] k + 2 [ y 2 X 1 4 + 6(x 1 2 +C 2 2)]k 2 }YO)C 2 2 + .'{2[(2a3 2-y^)x 7 - 3<$x9]k + [yjx8 + 6 ( X 1 0 + C 1 2 ) ]<$k2}s"21 100 + {u^S - 4to 2 + 2[(2w2-y2<S)X13 - 3 6 x n ] k + [ y * x l 4 + 6(X 1 2+C 2 2 ) ] 6 k 2}s 2 2 - 40) 2{S 2 2 - [ S ^ x , + S ^ x ^ l k + } = 0 , ( 4 . 9 h ) TT o where 1 1 X ] _ - / - ^ j / ' d S , X2 = / ^ i $ l " d ? o •"• •"• 0 1 „ 1 U Q 1 1 2 X5 = o / * 1 * 2 " d 5 , X6 = / ^ " d 5 0 X7 = , Xg - /V*^ 1 1 x9 = / $ i " V ^ > x 1 0 - ; c « i " * 2 d 5 i - 1 X i r o / V 2 " d € ' X12= / C * 2 ^ " d C xi3= . x M - A 2 » 2 de As i n the case of the uniform c y l i n d r i c a l beams, Equations ( 4 . 9 ) were solved numerically using a simultaneous-equation computer subroutine 101 (e.g. UBC NDINVT). 4.3 Results and Discussion In order to evaluate the response, the r e l a t i v e magnitudes i n the chosen harmonics and the modes are rewritten i n two d i f f e r e n t forms as fi = H1sin(a)T+^1) + H2sin(2o)T+^2) , (4.10a) and ri = tR11sin(a)T+A1) + R 1 2sin(2o)T+A 2) ] $ 1 + [R21sin(o)T+A3) + R22sin(2a)T+A4) ] $ 2 . (4.10b) Here H^ and H 2 correspond to amplitude responses i n f i r s t and second harmonics of the fo r c i n g frequency, r e s p e c t i v e l y , while R_ refer s to amplitude response i n the mode due to j * " * 1 harmonic. Figures 4-2 and 4-3 show the response amplitudes at three d i f f e r e n t points along the beams of various degrees of taper subjected to e x c i t a t i o n s at the root. I t should be noted that <o i s dimensionless and proportional to the frequency r a t i o Oi^/ti (co^ = f o r c i n g frequency, = natural frequency), co and co^ are related by 1 0 2 to = w f/v^ where i s defined i n ( 3 . 1 6 ) . For a p a r t i c u l a r beam / B ^ can be calculated and to may be found i f the f o r c i n g frequency i s known. Figure 4 - 2 shows the response to a one-term harmonic e x c i t a t i o n . There i s no second-harmonic response ( i . e . H 2 = 0 ) -*-n t h i s case. With an increase i n taper r a t i o , the peak tends to occur at a larger to but has a smaller magnitude. The f i r s t and second mode contributions, R^^ and R ^ l ' a r e a ^ s o shown as functions of to. These are constant along the length of the beam. For an increase i n taper r a t i o , the R ^ response peaks at a larger to with a smaller amplitude. On the other hand, the R ^ amplitude i s l a r g e r f or a higher taper r a t i o . The i n t e r n a l pressure e f f e c t s f o r the range of i n t e r e s t here are less than 1% and can c e r t a i n l y be considered n e g l i g i b l e . Figure 4 - 3 shows the responses for an e x c i t a t i o n including a second harmonic of a representative amplitude. Here the second-harmonic response i s apparent. The response H^ i s i d e n t i c a l to that i n Figure 4 - 2 f o r simple harmonic e x c i t a t i o n . The fundamental response components, R ^ and a r e a ^ s o v e r Y much si m i l a r except f o r minor magnitude dif f e r e n c e s . The superharmonic response components, R^2 and R-22> a r e d u e t o t' l e P r e s e n c e °f t n e second harmonic e x c i t i n g term. I t i s apparent that the response amplitude decreases with an increase i n taper r a t i o . It should be noted that the amplitude components R - Q > R 2 1 ' A N D R 2 2 a r e ^ n d e P e n d e n t °^ t* i e locations along the beam and are thus i d e n t i c a l f o r a l l values of £. Figure 4-2 Response of v i s c o e l a s t i c c a n t i l e v e r s to simple harmonic root e x c i t a t i o n 104 Figure 4-3 Response of v i s c o e l a s t i c c a n t i l e v e r s to e x c i t a t i o n with a small second-harmonic component: (a) £ = 0.4, 0.8 105 03 Figure 4-3 Response of v i s c o e l a s t i c c a n t i l e v e r s to e x c i t a t i o n with a small second-harmonic component: (b) K = 1 106 Before closing a comment concerning the effect of low temp-erature on response of the system would be appropriate. This is quite relevant because of the interest in i t s performance in the arct ic region. Although material properties of polymers vary with temperature, the glass transition temperature is expected to be very low (below -50°C 8 7 \ for polyethylene ) , hence the material is l ike ly to behave as visco-elast ic under almost a l l pract ical conditions. However, at low temp-eratures, the stiffness of the material increases while the creep rate decreases. Hence for certain applications, e .g . , submarine detection in the Arct i c , the material properties w i l l be different from the aforementioned ones (Section 2.4, p. 36), which are measured at a temperature of around 10°C. A typical stiffness-temperature plot for polyethylene is presented in Figure 4-4. It is apparent that with a | ^ A.SX.M.-D747| - 4 0 -20 0 20 40 60 Temperature fC) Figure 4-4 Effect of temperature on the stiffness of polyethylene 107 decrease i n temperature from 10°C to 0*0 the r i s e i n s t i f f n e s s can be by around 20%. This would be r e f l e c t e d i n the corresponding reduction i n the frequency r a t i o (u^/u )'-and hence a reduction i n the response amplitude. Thus the system performance i s expected to improve i n the low temperature environment. However, t h i s does not n e c e s s a r i l y imply better performance i n the a r c t i c region as now the character of the f o r c i n g function i s expected to be s u b s t a n t i a l l y d i f f e r e n t due to f l o a t i n g - c o l l i d i n g i c e masses. L i t e r a t u r e survey reveals considerable e f f o r t i n progress to assess t h i s . Unfortunately, no pr e c i s e information i s yet a v a i l a b l e . 4.4 Concluding Remarks The above analysis of the v i s c o e l a s t i c c a n t i l e v e r s subjected to p e r i o d i c root e x c i t a t i o n leads to the following conclusions: ( i ) Within the range of i n t e r n a l pressures studied the i n f l a t i o n has n e g l i g i b l e e f f e c t s on the forced response of the v i s c o -e l a s t i c c a n t i l e v e r s i n water, ( i i ) The analysis enables the p r e d i c t i o n of the response of the v i s c o e l a s t i c c a n t i l e v e r s to p e r i o d i c e x c i t a t i o n s that are expressible i n terms of the fundamental f o r c i n g frequency and i t s second harmonic, ( i i i ) For the case of the simple harmonic e x c i t a t i o n , the nonlinear hydrodynamic drag introduces no superharmonic components into the response. 108 (iv) At low e x c i t i n g frequencies t y p i c a l i n the ocean an increase i n taper r a t i o w i l l reduce the displacements of the leg t i p s . However, for frequencies above the fundamental resonance, a high taper r a t i o w i l l increase the t i p displacement amplitudes, (v) Dynamical response of the uniform c y l i n d r i c a l and tapered v i s c o e l a s t i c beams to root e x c i t a t i o n accounting for the hydrodynamic drag should prove useful i n the design of an underwater submarine detection system. 109 5. CLOSING COMMENTS 5.1 Summary of Conclusions As stated at the outset, the objective of t h i s i n v e s t i g a t i o n has been to gain an understanding of the s t a t i c s and dynamics of the n e u t r a l l y buoyant i n f l a t e d v i s c o e l a s t i c s t r u c t u r a l members c o n s t i t u t i n g a submarine detection system. The emphasis has been on the deter-mination of trends rather than presenting massive data. The important conclusions based on the study can be summarized as follows: ( i ) The s p e c i f i e d materials of the i n f l a t a b l e members are v i s c o -e l a s t i c and can be described with good accuracy by the three parameter s o l i d model, ( i i ) The elementary beam theory, with the three parameter s o l i d model, i s capable of p r e d i c t i n g accurately the s t a t i c be-haviour of both the uniform c y l i n d r i c a l and the tapered c a n t i l e v e r s . ( i i i ) The free response of the uniform c y l i n d r i c a l beams may be studied using the reduced s h e l l equation derived i n th i s d i s s e r t a t i o n . The s h e l l theory includes the i n i t i a l stress e f f e c t s due to i n f l a t i o n , and the reducing technique described combines the three s h e l l equations i n t o one that gives r e s u l t s of adequate accuracy f o r the beam-bending mode of i n t e r e s t here. The elementary beam theory i s adequate i n 110 p r e d i c t i n g t h e d y n a m i c a l b e h a v i o u r o f t h e i n f l a t e d t a p e r e d beams. E x p e r i m e n t a l d a t a o b t a i n e d c o n f i r m s t h e v a l i d i t y o f t he a n a l y s i s . The r e s u l t s s h o u l d be u s e f u l i n t h e s ub sequen t s t u d y o f t h e s ubmar i ne d e t e c t i o n s y s t e m , ( i v ) F o r t h e r ange o f i n f l a t i o n o f i n t e r e s t h e r e , t h e i n t e r n a l p r e s s u r e ha s a n e g l i g i b l e e f f e c t on t h e d y n a m i c a l b e h a v i o u r o f t h e n e u t r a l l y buoyan t i n f l a t e d c a n t i l e v e r s . The p r e s s u r e , howeve r , s h o u l d be a d e q u a t e t o p r e v e n t w r i n k l i n g o f t h e c a n t i l e v e r s a t a l l t i m e s , (v ) The f o r c e d r e s p o n s e o f t h e c a n t i l e v e r t o r o o t e x c i t a t i o n s h o u l d p r o v e u s e f u l i n t h e d e s i g n o f t h e s y s t e m . The a n a l y s i s e n a b l e s an e s t i m a t e o f t h e beam r e s p o n s e t o d i s p l a c e m e n t e x c i t a t i o n s e x p r e s s i b l e by t he f i r s t two te rms o f t h e F o u r i e r s e r i e s . 5.2 Recommendat i o n f o r F u t u r e Work The re a r e numerous p o s s i b i l i t i e s f o r e x t e n s i o n o f t h e p r e s e n t i n v e s t i g a t i o n . Some o f t h e i m p o r t a n t a r e a s o f i n t e r e s t a r e i n d i c a t e d b e l o w : ( i ) An i n t e n s i v e e x p e r i m e n t a l p rog ram t o d e t e r m i n e t h e a p p a r e n t mass c o e f f i c i e n t , one o f t h e u n c e r t a i n p a r a m e t e r s i n t h e s t u d y , s h o u l d p r o v e v e r y u s e f u l , ( i i ) P r o t o t y p e t e s t s i n t h e ocean w o u l d u n d o u b t e d l y p r o v i d e v a l u a b l e i n s i g h t . The p o s s i b i l i t y o f e x c e s s i v e f l e x u r a l I l l displacements f o r legs with very large L/d r a t i o s may necessitate a more refined large-amplitude theory as the slope e f f e c t s may become s i g n i f i c a n t , ( i i i ) The current study could f i r s t be extended to the coupled motion of three s i m i l a r f l e x i b l e i n f l a t e d c a n t i l e v e r s placed around a c e n t r a l head to form an array. 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L e e , H .C . , " F o r c e d L a t e r a l V i b r a t i o n o f a U n i f o r m C a n t i l e v e r Beam W i t h I n t e r n a l and E x t e r n a l D a m p i n g , " J o u r n a l o f A p p l i e d  M e c h a n i c s , V o l . 27 , September 1960, pp . 551 -556 . 68. L e i s s a , A.W., and Hwee, M.O. , " F o r c e d V i b r a t i o n s o f T imoshenko Beams w i t h V i s c o u s D a m p i n g , " Deve lopment s i n M e c h a n i c s , J o h n W i l e y & Sons , I n c . , N .Y . , V o l . 3 , P a r t 2, 1965 , p p . 7 1 - 8 1 . 69. B a k e r , W.E . , Woolam, W.E . , and Young , D., " A i r and I n t e r n a l Damping o f T h i n C a n t i l e v e r Beams , " I n t e r n a t i o n a l J o u r n a l o f  M e c h a n i c a l S c i e n c e s , V o l . 9 , 1967, pp . 743 -766 . 70. P a i d o u s s i s , M . P . , and des T r o i s M a i s o n s , P . E . , " F r e e V i b r a t i o n o f a Heavy , Damped C a n t i l e v e r i n a P l a n e I n c l i n e d t o t h e V e r t i c a l , " M . E .R . L . R e p o r t No. 6 9 - 6 , Depar tment o f M e c h a n i c a l E n g i n e e r i n g , M c G i l l U n i v e r s i t y , M o n t r e a l , Quebec, Augu s t 1969. 71 . F u , C . C . , "Beam V i b r a t i o n s w i t h N o n l i n e a r D a m p i n g , " Deve l opment s  i n M e c h a n i c s , J ohn W i l e y & Sons , I n c . , N.Y. , V o l . 3 , P a r t 2 , 1965 , pp . 131 -139 . 118 72. P i s a r e n k o , G . S . , O s c i l l a t i o n s o f E l a s t i c Systems w i t h S c a t t e r i n g  o f Ene rgy i n t h e M a t e r i a l , A c a d . S c i . U k r . SSR., 1955 ( R u s s i a n ) . 73. Choo, Y . , and C a s a r e l l a , M . J . , " A . S u r v e y o f A n a l y t i c a l .Methods f o r Dynamic S i m u l a t i o n o f C a b l e - B o d y S y s t e m s , " J o u r n a l o f H y d r o -n a u t i c s , V o l . 7, No. 4 , O c t o b e r 1973 , pp . 137 -144 . 74. Eames, M .C . , and Drummond, T . G . , " H y d r o n a u t i c s i n C a n a d a , " C a n a d i a n A e r o n a u t i c s and Space J o u r n a l , V o l . 17 , No. 9 , November 1971 , pp . 381 -389 . 75. F l i i g g e , W., V i s c o e l a s t i c i t y , B l a i s d e l l P u b l i s h i n g Company, Wa l tham, M a s s a c h u s e t t s , 1967, pp . 3 2 - 50 . 76. K a l i n n i k o v , A . E . , " C r e e p and A f t e r e f f e c t o f PET F i l m s Under C o n d i t i o n s o f U n i a x i a l S t r e s s , " M e k h a n i k a P o l i m e r o v , V o l . 1, No. 2, 1965, pp . 5 9 - 6 3 . 77. F i n d l e y , W.N . , and K h o s l a , G . , " A n E q u a t i o n f o r T e n s i o n C reep o f Th ree U n f i l l e d T h e r m o p l a s t i c s , " S o c i e t y o f P l a s t i c E n g i n e e r s  J o u r n a l , V o l . 12 , No. 1 2 , December 1956, pp . 2 0 - 2 5 . 78. Lamb, H . , H y d r o d y n a m i c s , Cambr idge U n i v e r s i t y P r e s s , L o n d o n , 6 t h E d i t i o n , 1932, p. 644. 79. H o e r n e r , S . F . , F l u i d Dynamic D r a g , P u b l i s h e d by t h e A u t h o r , M i d l a n d P a r k , N . J . , 1965 , p p . 3 - 8 . 80. G r e g o r y , R.W., and P a i d o u s s i s , M . P . , " U n s t a b l e O s c i l l a t i o n o f T u b u l a r C a n t i l e v e r s C o n v e y i n g F l u i d , " P r o c e e d i n g s o f t h e R o y a l  S o c i e t y , L o n d o n , V o l . 2 9 3 ( A ) , 1966, pp . 5 12 - 542 . 8 1 . B l e i c h , H.H.", and D i M a g g i o , F. > " A S t r a i n - E n e r g y E x p r e s s i o n f o r T h i n C y l i n d r i c a l S h e l l s , " J o u r n a l o f A p p l i e d M e c h a n i c s , V o l . 20 , No. 3 , 1 953 , p p . 448 -449 . 82. Sharma, C . B . , " C a l c u l a t i o n o f N a t u r a l F r e q u e n c i e s o f F i x e d - F r e e C i r c u l a r C y l i n d r i c a l S h e l l s , " J o u r n a l o f Sound and V i b r a t i o n , V o l . 3 5 , No. 1, 1974, p p . 5 5 - 7 6 . 83 . W a s h i z u , K., V a r i a t i o n a l Methods i n E l a s t i c i t y and P l a s t i c i t y , Pergamon P r e s s , New Y o r k , 1968. 84. Sampath, S . G . , " V i b r a t i o n s o f C i r c u l a r C y l i n d r i c a l S h e l l s Under S p a c e - V a r y i n g I n i t i a l S t r e s s e s and Body F o r c e s , " Ph .D . D i s s e r t a t i o n , Oh i o S t a t e U n i v e r s i t y , O h i o , 1970. 85. Snowdon, J.C.. , V i b r a t i o n and Shock i n Damped M e c h a n i c a l S y s t ems , J o h n W i l e y and Sons I n c . , New Y o r k , 1968. 119 86. K i n s m a n , B., Wind Waves - T h e i r G e n e r a t i o n and P r o p a g a t i o n on  t h e Ocean, P r e n t i c e - H a l l , I n c . , Eng lewood C l i f f s , New J e r s e y , 1965 , p. 23 . 87. R a f f , R .A .V . , and A l l i s o n , J . B . , P o l y e t h y l e n e , I n t e r s c i e n c e P u b l i s h e r s , I n c . , New Y o r k , 1956. 120 APPENDIX I DERIVATION OF WATER INERTIA TERM IN SHELL EQUATIONS Figure 1-1 Geometry of l a t e r a l displacement of a s h e l l section Let q be the normal pressure acting on s h e l l due to i n e r t i a of water ins i d e , q i s defined p o s i t i v e outwards. From geometry, 2 2 9 w 9 y . Q m — r = m — ' s i n B W 9 t 2 W 9 t 2 9 2z m —?rcos8 w 9 t 2 (1.1) A force balance i n the v e r t i c a l d i r e c t i o n gives 2ir %2 2ir a ~2 / qacosGdG = / m 2-S = p / .,../ [ ^ s i n S 0 W 9 t 2 W 0 0 9 t 2 a 2  9 z 3t J cos6]adrd0 (1.2) 121 Assuming q = qf.ir'(x)cos0cosCi)t and y = Yip(x)sin9coso)t , z = ZifKx)cos0cosU)t Equation (1.2) becomes 2TT p w0) 2a 2i|;(x) 2TT a 2 / q nip(x)cos 0coswtad0 = - „• / /(Ysin 0-Zcos 0)cosO)td0 . 0 0 0 (1.3) For the beam-bending mode, Y =-Z, and Equation (1.3) becomes q 0 = P waa) 2Z Thus ? 2 2 28 -qa = - p a w Zi|;(x) cos0cosu)t = p a — r . (1.4) w w 3 t 2 2 2 In the s h e l l equations the force term i s -qa*ji{;""' ) t and i s equivalent to Eh E h a t 2 ' 122 or o \t2 To account f o r the i n e r t i a of the water surrounding the » beam, an added i n e r t i a c o e f f i c i e n t , C , i s introduced. The term m to be added to the s h e l l equations i s then ( 1 + c ) G - ^ r 9 - f m P hg f c2 123 APPENDIX II ORTHOGONALITY CONDITION FOR EQUATION (3.22b) Equation (3.21b) may be rewritten, f o r modes m and n, as B'"' = X 4(B B ' V B . B J , (II. la) m m 2 m l m B" " = X 4(B.B"+B 1B ) ( i i . l b ) n n 2 n 1 n Mul t i p l y i n g Equations ( I I . l a ) and (II.lb) by BR and BM, r e s p e c t i v e l y , subtracting and in t e g r a t i n g over the length, one obtains 1 l i t ! t i l l X A I I • / (B B -B B )d? = / [A ( B i B +B.B B ) 0J m n m n ' Q ; m 2 m n 1 m n' - X4(B_B V'+B-S e'')]dC . ( I I . 2) n z m n I n n Integrating by parts, Equation (II.2) becomes 0 = B.[XVB -X4B B']J + (X4-X4)B, / B B d? 2 m m n n m n 0 m n 1 ^ m n - B_(X4-X4) /Va'd? . ( i i . 3 ) L m n Q m n Applying the c a n t i l e v e r boundary condition, Equation (II.3) can be rearranged to give 124 1 B, 1 , , B,[AV(l)g (1)-A ( l )e ' ( l ) ] / 3 B dC - J J B B d5 + 2 m m n A ; m 2 0 m n , B 1 0 m n B.CA4-^4) 1 m n = 0 for m ^ n (II.4) This is the condition of orthogonality of the eigenfunctions B ( £ ) . APPENDIX III RAYLEIGH-RITZ MATRIX ELEMENTS FOR WASHIZU'S SHELL THEORY The eigenvalue problem given by Equation (3.57a) i s [M](a) = ft2[N](a) The elements of [M] and [N] are: •IIL. = (1-V) / V 2 d 5 + 2 ( f ) 2 / $ " 2 d £ + (1 - V 2)P * 1 1 0 L 0 [ ( 2 — + - ^ ) / V 2 d £ + ( T-) 2 A " 2 d ? ] ; 2 a 6a 2 0 L 0 1 FL 9 1 ? m 1 0 = (1-V) / * f1" d £ + 2 ( a ) Z / $ , nr"'d£ + (1 - V Z ) P * 1 2 0 L 0 I ( 2 ~ + - ^ ) / WdZ + (|) Z / * " Y " d ? ] ; 6a 0 0 1 1 m.„ = (T-)[2V / $"$d£ - (1-V) / $' Zd£] ; 1 J L 0 0 m14 ~ VL 1 1 ( a)[2v / r'VdZ - (i-v) / $ , vi"dS] o o l m-,. = 2 v ( a ) / $"$d£ IS L Q 1 m i 6 = 2 v ( ! 5 ' $ , , V F d ? 21 1 , 1 , (1-V) / S ' r d C + 20) j * , n f " d 5 + ( 1 - V Z ) P * 0 L 0 h h2 1 9 1 6a 0 0 m, 22 (1-V) / r2 d c j +. 2 ( a ) 2 / V"ZdZ + ( 1 - V 2 ) P * 2 1 1 2 a 6a2 0 1 1 m 2 3 = ( a ) [ 2 v / W d ? - ( i - v ) / s ' r d c ; ] = ( a ) [2V / W » ' d £ - ( 1 - V ) / r 24 V L 25 2V(S ) / $>T 'd£ L 0 m 2 6 = 2V(5) / W " d 5 m = ( f ) [ 2 v / $ $ " d ? - ( 1 - V ) / $ ' d ? ] 31 X L m = (a ) [ 2 v / * f ' d ? - ( 1 - V ) / fc'rdS] 32 V L 0 0 126 m 33 1 2 1 2 + ( i - v ) ( a ) 2 / r2dc; + ( i - v 2 ) p [ 4 - ^ + - ^ + ( a ) 2 / $ ' 2 dt;] L 0 3 3a2 L 0 1 a 9 1 9 t n 3 4 = 2 / WdZ + ( 1 - V ) ( a ) / * ' rdc- + ( 1 - V )P * 0 0 h h2 1 „ 9 ! [ (4-f+-^) J cWdS + (V / $ » r d £ ] a 3aZ 0 L 0 2 + (1-V 2)P[4 - - + -£R] a . 2 3a ' 1 „ , 2 1 2 / sWd£ T (1-V Z)P (4 - - + -==) / m e 0 a 3a Z 0 ( a )[2v / #•*YdC - (l - v ) / t ' f 'dS] 1 1 9 (a)[2V f W d S - (1-V) j 1" cl£] L 0 0 1 9 1 9 2 / M ' d C + ( i - v ) ( a r / s - r d e ; + ( i - v z ) p * 0 L 0 1 , 1 Ja 0 0 2, + ( i - v ) ( f ) 2 / r 2 d ? + ( i - v 2 ) p [ 4 - - + H 3a 9 1 9 +• (fr / r 2 d£] L o ^ 9 V, h 2 1 2 / WdS + ( i - v z ) p ( 4 - - + - i L j / $y d£ 0 a 3a 0 2:4- ( l - v 2 ) P ( 4 - - + -A5) a 3a 2 2v( a ) / **'*ds L 0 2 v ( f ) / $ ¥ " d C L 0 128 9 h hZ 2 + ( l V ) (4 --2 + -2-)P 3 3a2 1 2 1 2•/ *¥d£ + ( l V)P(4 - - + / 0 a 3aZ 0 2 2 + (1-V2)P[4 - ^  + A + a 3a2 ( a ) 2 / V 2 d ? ] L 0 1 9 h 2 / $ Y d ? + (1-VZ)P[(4 - -0 a h2 1 + A ) / wit + 3a 0 9 1 z / a ' r d ? ] 0 1 2v(a) / ^"i-dS ; 0 1 2v(a) / "Fd£ ; L 0 1 9 h h 2 1 2 / Wd? + ( l V)P(4 - - + J L ) / CTdC 0 a 3aZ 0 9 1, >, 2 2 + ( l V ) ( 4 - § + J L ) P 3 3a2 1 9 h 2 / *Yd£ + (1-VZ)P[(4 - -0 h2 1 + J L ) / wat + 3a 0 9 i z / $'4"dc;] 0 9 h h 2 2 + (1-VZ)P[4 - - + J L + 3 3aZ a}2 j \ , ,*-d5] L 0 > 1 9 2 / $ , Z d £ ; 0 1 2 / * " i " d 5 ; 0 n14 = n15 = n16 " n = 2 / *«r<is 0 22 x 9 2 / r dg o n23 n24 n25 n26 n31 3(1+C)pa n 3 3 _ 2 + y - — 3(1+C )p a 1 "34 - '2 + - T V E 1 , ' W « n (1+C )p a _ m w 35 ~ ' 4 P h n (1+C )p a 1 m W / WdZ 36 4 p h Q n41 = n42 = 0 3(1+C )p a 1 "43 • v + —TVEV " * 3(1+C )p a 0 . m w n, . = 2 + : -44 4 p h (1+C )p a 1 45 4 p h Q J ( 1 + C ) p a _ m w n46 4 p~h 130 n 5 1 = n 5 2 = 0 n (1+C )p a _ m w 53 " 4 p h n (1+C ) p a 1 m W / Wd? 54 4 p h Q 3(1+C )p a «, , m w n c c = 2 + — . .-55 4 p h 3(1+C )p a 1 n 5 6 = E 2 + " T T h l / n 6 1 = ti62 " ° (1+C )p a 1 " 6 3 " -rVV T (1+C )p a n 6 4 4 p h 3(1+C )p a 1 131 APPENDIX IV FREQUENCY EQUATION FOR TAPERED BEAMS USING 1-, 2-, AND 3-TERM APPROXIMATIONS 1=1: For one-term approximation Equation (3.59b) becomes 1 d 3$ d 2$ / {-6k(l-k£) h + 6kV i + a-kOiuUl-kO -) del 1 dZ X2]^}dZ = 0 (IV. 1) The s o l u t i o n can be written as i ° t 9 1 2 0 X = u {- ,% [1 + k(12-2 - 21, ) + k^Cl. =2 + m *m l - k l . 2 lm v 2m 3 m m 9C - f > l M , (IV. 2) ym where 1 ? I- - / d£ lm Q ; m 1 9 9 i , = / r* d? 2m Q J ^ m ^ and C ^ i s given by Equation (3.33c). 132 1=2: Two equations are obtained from Equation (3.59b) f o r the two-term approximation: 1 d 3$ dV / {(-6k ( l - k c : ) $ 1 i + 6k Z$. i + ( 1 - ^ ) ^ ( 1 - ^ ) - A z ] $ z ) f o 1 dr 1 dK d\ 7 dv + (-6k( l -kc;)$ 1 f + 6kV Z + ( l - k O t y ^ l - k C ) 1 dr 1 d5 z z - A 2 ] $ 1 $ 2 ) f 2 > d c ; = 0 . , (IV. 3a) 1 d 3$ dV 2 / {(-6k ( l -kc ; ) 1$ + 6k Z i $ 0 + ( l - k c ; ) [ y ^ ( l - k c : ) - X ]«,*,) o dr z dr 3 2 d $ d $ + (-6k(i-k?)*_—4 + 6 k V — § + ( i - k ? ) [ y 4 ( i - k c ; ) dr z dc z - A 2]<l 2)f 2 } d c : - 0 . (IV. 3b) Equations (IV.3) may be put i n the form of (3.64a), i . e . , where [S](f) = A 2 [ V ] ( f ) S l l = y l " [ 6 X 1 + 2 y i x 3 ] k + ( 6 x 2 + 6 C 1 1 + y i X 4 ) k 2 133 S12 = " [ 6 X 5 + 2 y 2 X 7 ] k + [ 6 X 6 + 6 C 2 1 + y 2 X 8 ] k 2 521 = " [ 6 X 9 + 2 y i X 7 ] k + [ 6 xio + 6 C12 + y J X 8 ] k 2 522 " A ~ [ 6 X 1 1 + 2 y 2 X 1 3 ] k + [ 6 X 1 2 + 6 C 2 2 + y 2 X 1 4 ] k 2 ' v n = 1 - X 3k v 1 2 ; - v 2 1 = - X ? k , V22 " 1 " X 1 3 k ' 1=3: For the three-term approximation three equations are obtained from Equation (3.59b) which can be written i n the matrix form (3.64a). The elements of the two matrices are: s l l = y l [ _ 6 x l + 2 y l x 3 ] k + [ 6 x 2 + 6 C 1 1 + y l x 4 ] k 2 S12 = ~ [ 6 X 5 + 2 ^ 2 X 7 ] k + [ 6 X 6 + 6 C 2 1 + ¥ 8 ] k 2 5 1 , 1 1 S l „ = -[6 / *.*:"d? + 2uZ j c j ^ - d S l k + [6 / 54 j ; n d ? 0 0 0 + 6 C 3 1 + j V ^ d S l k 2 ; - [ 6 x 9 + 2u4x7]k + [6x 1 Q + 6C 1 2 + yjx 8 ]k 2 4 4 4 2 P2 " [ 6 X 11 + 2 p 2 X 1 3 ] k + [ 6 X12 + 6G22 + V 1 2 X l4 ] k [6 / $ 2 $ 3 " d c ; + 2y4 / s*9$,d£]k + [6 / es^V 'dS 2 3 + 6 C 3 2 + V l j \ \ * 3 W 2 1 , 1 1 [6 / dC + 2y? / * d?]k + [6 / ?*'"*,d? 0 J 0 0 1 I n + 6C 1 3 + yj J ?z*1*3d5]k" -[6 / <D*"$3dc; + 2y4 / C$ 9$,dc;]k + [6 / ?$;"*,d5 2 "3 + 6C 2 3 + y 4 / 1C 2* 2* 3dC]k 2 ? A 1 A 1 o 1 uZ - [6 / $ * ' " d £ + 2y^  / €$,d£]k + [6 / C$ $ ' " d 0 0 o + 6 C 3 3 + y 4 / 5 2* 3d?]k 2 1 " X 5 k -X y k -k / C$ $ d? ? 0 '21 v 22 i - x 1 3 k 23 -k / $ d5 o J v 31 -k / P $ 0 32 -k / £*9$,d£ 0 J v 3 3 = i - k / e$*ds 136 APPENDIX V REDUCED MEMBRANE AND HERRMANN-ARMENAKAS EQUATIONS (i ) Reduced Membrane Equation The d i f f e r e n t i a l operators i n Equation (3.1) f o r the membrane 2 2 3 1 theory, a f t e r neglecting terms inv o l v i n g h /(12a ), are 2 2 2 1 1 " 3 s 2 2 se2 " C l V 12 2 9s96 13 9s _ .. 1+v a 2 21 2 9s98 2 2 2 J22 2 . 2 + „fl2 C 2 V ~ 2 9s 96 9t L = i -23 36 3 L31 " VTs L - D-32 36 > Nfl .2 N f l„2 N a2 T - i 6 a. „ ~ 3 6 3 x 3 L „ „ = 1 - - r c 33 * '. C 3" 2 C M 2 C a 2 3t 96 3s 137 Carrying out the reduction procedure the following equation i s obtained for the beam-bending mode: . d 2w , . d 4w . . 32w . . 3 ^w f, i v A l 7 2 + A27-4 + \ ~ 2 + A 5 7 2 T 2 = ° ' ^ d s d s d t d s d t where . _ (3+V)(l-V 2). A l 4 P . _ (5-4v-v2)(l-v2)„ (1-v) (1-v 2) A2 "8 * 2 A 4 = G ( i ^ ) ( c 2 + c 3 ) 2 A = -G{ ( 1" V )[2c 1 + ( l - V ) c 2 ] P + • ^ Y ^ c 1 + ( 1 - V 2 ) c 2 + (1 - V)c 3> ( i i ) Reduced Herrmann-Armenakas Equation For the l i n e a r i z e d Herrmann-Armenakas theory, the d i f f e r e n t i a l 2 2 3 1 operators, a f t e r neglecting the small terms inv o l v i n g h /(12a ), are Ci • d . (-1-V . V\ d _c3 i+v a 2 J12 2 3s30 138 L 1-W 92 21 2 9s99 L 2 2 = ( - + G - X ) 3 V 2 + ( l + ^ ^ 2 - C 2 G - 2 - r N 9 , 3 L 2 3 = ( 1 + 2-^9-0 L31 = V b Ne a L32 = ^ + 2 C~^ 96 1 + V r 9 i !e9 i V _ L33 = C C3 & 3 t2 " C 3 02 " C 3 g2 Applying the reduction technique outlined i n section 3.1.1 and s e t t i n g n=l, one obtains an equation s i m i l a r to (V.l) with the c o e f f i c i e n t s given by A 1 = -(2(1-V 2) 3P 3 + (3-2v)(l-V 2) 2P 2 + (1-V)(1-V 2)P) A, = (1-V 2) 3P 3 + ( 27-7v)(l-V 2) 2 p2 + ( 9 - 7 V - 2 V 2 ) ( l - V 2 ) p 2 3 1-V-V +V A 4 = G[2(c 2 + c 3 ) ( l - V 2 ) 2 P 2 + (2-v)(c 2 + c 3 ) ( l - V 2 ) P + Ar) ( C 2 + C3) 139 3 2 2 2 (11-v) A 5 = -G{[2c± 4 | ( c 2 + c 3 ) ] ( l - V Z ) V + [(2-V)c± + " ^ ^ X 2 ^ + a > 3 v ) C 3 ] ( 1 _ v 2 ) P + ii^>Cl + ( 1-V 2)c 2 + (l-v)c 3} ( i i i ) Solution L e t t i n g £ = x/L , n = w/d Equation (V.l) may be rewritten as .2 A 0 4.4 A, 2„2^ A. 2 .4 —2 + r<!>-M + r<!> H + r<!> H r i = 0 • ( v - 2 ) 3 t 2 A4 L 3 ? 4 A4 L 3C A 4 L 3c; 23t To f i n d the fundamental frequency an approximate s o l u t i o n i s sought i n the form n = A* (C)coswt . (V.3) Substituting (V.3) into Equation (V.2), multiplying by ^ ( 5 ) and i n t e -grating with respect to ? over the length one obtains ii ii ^ 140 Equation (V.4) may be solved to give the frequency 0) = A4 + V L A I 141 t h e o r y a r e APPENDIX V I POTENTIAL ENERGY EXPRESSION FOR THE MEMBRANE THEORY The s t r a i n - d i s p l a c e m e n t r e l a t i o n s a c c o r d i n g t o t h e membrane 3 1 e = p + ±[U)2 , ( V I . l a ) x 3 x 2Kd-xf e = + 5 + - A - f f - l ) 2 . (vi.lb) ee a ^ e ; a 0 2 w La The m i d d l e s u r f a c e c u r v a t u r e s K and K Q a r e i d e n t i c a l t o t he ones x o i n ( 3 . 5 2 ) . S u b s t i t u t i n g t h e above r e l a t i o n s i n t o u"2 and i n t e g r a t i n g t h r o u g h t h e t h i c k n e s s g i v e s r r t„ 3x N x r 3 z ^ 2 , N 9 8 y N 0 r r i z ^ 2 , ~ fi u 2 • / (: { N x 9 x + rfe) + r s i - 27W }adxd9 As suming mode shapes g i v e n by (3 .55 ) and a p p l y i n g t h e R a y l e i g h -R i t z t e c h n i q u e ( 3 . 56 ) t h e f o l l o w i n g e l e m e n t s a r e o b t a i n e d f o r t h e m a t r i x [M] i n E q u a t i o n ( 3 . 5 7 a ) : -m.. = (1-V) A , 2 d c : + 2(f)2 / V ' 2 d £ 1 1 0 0 m 1 9 = (1-V) / * ' 1 " d « i + 2 ( - ) z / ^ ' r ' d e ; l z 0 0 1 1 9 ( a)[2v / $ * ' d c ; - ( l - v ) / <D , zdS] (f)[2v / <S>"Vdc: - (1-v) / Q'V'dtZ] L 0 0 2v( a) / $ * ' $ d c ; L 0 2v( a) / $ ' * ¥ d c ; ( i - v ) /-t'y'd? + 2 ( a ) 2 / *"Y"de ( i - v ) / V 2 d £ ; + 2 ( a ) 2 /V' 2 de; o o l I ( a)[2v / * ¥ " d 5 - (1-v) / *-"F»-d?] L 0 0 1 1 9 ( a)[2v J W ' d S - (1-v) / r <£] 2v( a) / W"d? 2v( a) / W"d? L 0 1 1 -( a)[2v J $ $ " d c ; - (1-v) / * ' <i€] L 0 0 1 1 ( a)[2V / J f ' d S - (1-V) / fl'Y'dd 0 0 2 + ( i - v ) ( T - ) 2 / $ , 2 d f ; L 0 2 / Wd? + (1 - V ) ( a ) 2 / $,vi"d? 0 L 0 2 / WdE, 0 1 1 ( a)[2v / *"!Fdci - (1-V) J *"TdC] L 0 0 1 1 9 ( a)[2v / W , f d £ - (1-v) / ¥'<!«;] r 1 a 2 r 1 2 / OTdS + (1-V)(f) / S ' VdJ ; 0 0 2 + (1 - V ) ( f ) 2 / Y ' 2 d £ 2 / <S>fdc; 0 2v( a) / $$"d£ L 0 2v( a) / W'd? 2 [ m&z 0 144 m 5 5 = 2 + (1 - V 2)P[(|) 2 / V 2 d H m56 m62 m63 1 9 a 9 1 2 / *Yd£ + ( l - V Z ) ( a ) Z P / $ '1"d? • 0 L 0 m61= 2 V ( f ) Q A"^ 1 2v(a) / V ' Y d C - ; L 0 1 2 / $Yd? ; o m64 = 2 > 1 1 m, = 2 / W d ? + (iV)(arp / fc'Y'd? " o L 0 m , , = 2 + ( l - V 2 ) ( a ) 2 P /V2d£ 66 L Q T h e m a t r i x [N ] i s i d e n t i c a l t o t h e o n e g i v e n i n A p p e n d i x I I I f o r W a s h i z u ' s s h e l l t h e o r y . 

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