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Hydrodynamic coefficients for axisymmetric bodies at finite depth Chan, Johnson L. K. 1984

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HYDRODYNAMIC COEFFICIENTS FOR AXISYMMETRIC BODIES AT FINITE DEPTH by JOHNSON L.K. CHAN Bachelor of Applied Science, U n i v e r s i t y of Toronto, 1982 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE l n THE FACULTY OF GRADUATE STUDIES Department of Mechanical Engineering We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1984 © Johnson L.K. Chan, 1984 In presenting t h i s t h e s i s in p a r t i a l f u l f i l l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree that permission f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by his or her r e p r e s e n t a t i v e s . I t i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Johnson L.K. Chan Department of Mechanical Engineering The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, B r i t i s h Columbia Canada V6T 1W5 Date: January 31, 1984 ABSTRACT A numerical procedure tor the c a l c u l a t i o n of v e l o c i t y p o t e n t i a l , f o r axisymmetric shapes i s st u d i e d . The procedure incorporates l i n e a r i z e d f r e e surface and r a d i a t i o n c o n d i t i o n s . Forced heave, surge, p i t c h induced v e l o c i t y p o t e n t i a l s and wave d i f f r a c t i o n p o t e n t i a l s are considered as s p e c i f i c a p p l i c a t i o n s . D e t a i l s of the a n a l y t i c a l steps necessary to reduce the three dimensional i n t e g r a l equations to two dimensional e q u i v a l e n t equations are given in the appendix. The hydrodynamic c o e f f i c i e n t s and e x i s t i n g forces f o r four well studied examples are presented and compared to published r e s u l t s by other numerical methods. The method studied here o f f e r s c o n s i d e r a b l e savings i n computer time, as the input requirements and the computational e f f o r t s are reduced by intermediate a n a l y t i c a l steps. iii TABLE OF CONTENTS Page ABSTRACT i i CHAPTER I -- INTRODUCTION 1.1 ' CHAPTER II — MATHEMATICAL FORMULATION 2.1 4 1. D e f i n i t i o n of the Coordinate System and Major V a r i a b l e s 2.1 * 2. General Formulation 2.2 ^ 3. Method of S o l u t i o n 2.4 7 CHAPTER III -- HYDRODYNAMIC COEFFICIENTS 3.1 /4 1. General Formulation of Hydrodynamic C o e f f i c i e n t s 3.1 /4 2. Computation of Heave Added Mass and Damping C o e f f i c i e n t 3.5 »8 3. Computation of Surge Added Mass and Damping C o e f f i c i e n t 3.6 l9 4. Computation of P i t c h Added Mass and Damping . C o e f f i c i e n t 3.8 Z l CHAPTER IV — EXCITING FORCE AND MOMENT 4.1 *S 1. General Formulation of E x c i t i n g Force and Moment due to Incoming Waves 4.1 25" 2. Heave E x c i t i n g Force 4.5 29 3. Surge E x c i t i n g Force 4.7 52 4. P i t c h E x c i t i n g Moment 4.9 33 5. An A l t e r n a t i v e Method f o r Computing E x c i t i n g Forces 4.10 34 CHAPTER V — RESULTS 5.1 36 1. S i n g l e C y l i n d e r at F i n i t e Water Depth 5.1 36 2. Composite C y l i n d e r 5.3 58 3. Hemisphere 5.5 4o iv Table of Contents (continued) Page 4. Axisymmetric F l o a t i n g Rig 5.6 4' 4 2 5. Discussions and Conclusion 5.7 LIST OF FIGURES F . l 44-NOMENCLATURE N.l 7 2 APPENDIX A A.l 74 APPENDIX B A.2 7S APPENDIX C A.4 7 g APPENDIX D A.6 8o APPENDIX E A.13 87 APPENDIX F A.22 9£ APPENDIX G A.24 98 APPENDIX H A.26 /oo REFERENCES R.l /o3 - 1.1 -! I. INTRODUCTION The motions of a platform at sea i s one of the most important t o p i c s t h a t Naval A r c h i t e c t s are concerned about. In the l i n e a r frequency domain a n a l y s i s of a f l o a t i n g o b j e c t , the computation of the added masses, the damping c o e f f i c i e n t s , and the e x c i t i n g forces i s necessary. Two well developed t h e o r i e s can be applied to solve the problem. One of these t h e o r i e s i s the c l a s s i c a l S i n g u l a r i t y Method. Using conformal mapping and a d i s t r i b u t i o n of sources and sinks of various strength i n s i d e or on the f l o a t i n g o b j e c t , an a n a l y t i c f u n c t i o n d e s c r i b i n g the f l u i d flow can be obtained. The a l t e r n a t e method i s to solve the Integral Equation of the system coupled with the Green's Function. The v e l o c i t y p o t e n t i a l value on the boundary surface and in the f l u i d f i e l d can then be found. In 1949, U r s e l l s u c c e s s f u l l y solved the forced o s c i l l a t i o n problem of a semi-submerged c i r c u l a r c y l i n d e r f o r the deep water case. The a n a l y s i s was done using the S i n g u l a r i t y Method and the waves were assumed to be long. However, only the heave motion was considered. His work was followed by T a i s a i [1959] and Porter [1960] who extended h i s method with the assumption of John [1950] that the body's surface must be perpendicular to and at the f r e e s u r f a c e . In 1969, Kim obtained the added masses and damping c o e f f i c i e n t s by using the Lewis Transform f o r f i n i t e water depth. Three modes of motion were considered. The p o t e n t i a l of Integral Equation Method was not obvious u n t i l the development of the computers. In 1953, U r s e l l developed a theory f o r an o s c i l l a t i n g c y l i n d e r at high frequency with heave motion and shallow draught approximation. His theory was f i r s t a p p l i e d s u c c e s s f u l l y by - 1.2 - 2 MacCamy in 1961. However, the Integral Equation Method had obtained i t s success three years p r i o r to that, when an acoustic problem was solved by Smith and Hess. In 1967, Frank's removed the geometric body r e s t r i c t i o n s from the problem which John [1950] had o r i g i n a l l y imposed. The disadvantage of the c l a s s i c a l approach i s that a long s e r i e s of terms i s required f o r convergence and the a n a l y s i s i s r e s t r i c t e d to two dimensions o n l y . On the other hand, the Integral Equation Method also has i t s r e s t r i c t i o n s . The bottom of the control volume must be f l a t and the Green's Functions are hard to compute. Moreover, the body geometry r e s t r i c t i o n was not removed u n t i l Frank's work in 1967. In 1955, Havelock had solved the problems of a semi-submerged sphere by the Integral Equation Method. In 1966, S. Wang extended h i s work from deep water to f i n i t e water depth. The f i r s t complete three-dimensional a n a l y s i s was done by Kim [1965], who used an e l l i p s o i d a l - p o l a r coordinate system. S o l u t i o n s f o r several spheroids and e l l i p s o i d s with surge, heave, and p i t c h motions were obtained. In 1972, Bai obtained successful r e s u l t s by using the F i n i t e Element Method with an approximate r a d i a t i o n boundary c o n d i t i o n . One year l a t e r , Yeung solved s i m i l a r problems in two and three dimensions with no r e s t r i c -t i o n on the body's geometry. A general form of the Integral Equation f o r f o r c e d - o s c i l l a t i o n problems was then set up. In 1979, K r i t i s a p p l i e d Yeung's equation to obtain a s a t i s f a c t o r y r e s u l t f o r heave motion of some axisymmetric o b j e c t s . - 1.3 - s In 1981, S.H. C a l i s a l and Sabuncu developed a matching technique f o r composite c i r c u l a r c y l i n d e r problems. In the f o l l o w i n g chapters, the author presents more s p e c i a l i z e d Inte-gral Equations f o r s o l v i n g axisymmetric body f l u i d flows in f i n i t e water depth. The i n t e g r a l equations given in the fo l l o w i n g chapters are a modi-f i c a t i o n of Yeung's equations. These enable one to solve a three-dimen-sion a l problem in two dimensions. The only r e s t r i c t i o n i s that the body must be axisymmetric about the v e r t i c a l a x i s . Equations f o r determining hydrodynamic c o e f f i c i e n t s as well as e x c i t i n g forces are developed. - 2.1 - 4 I I . MATHEMATICAL FORMULATION I I . l D e f i n i t i o n o f the Co-ordinate System and Major V a r i a b l e s The f i r s t step in s o l v i n g the problem i s to set up a co - o r d i n a t e system, so that the geometry of the whole system can be c l e a r l y d e s c r i b e d . The o r i g i n of the co-ordinate axes i s l o c a t e d i n s i d e the f l o a t i n g body. The x and z axes are at the undisturbed free surface while the y axis i s p o i n t i n g upward. Angle 6 i s measured from the x axis as shown in F i g . A. h i s defined as l o c a l water depth. A c y l i n d r i c a l outer boundary i s chosen to define the c o n t r o l volume which can be described by a r a d i a l d i s t a n c e R D. Any a r b i t r a r y point can be descri b e d by (x,y,z) or by (R,y ,6) c o o r d i n a t e s . They are r e l a t e d by Let P be an "observation point" l o c a t e d e i t h e r i n s i d e or outside the con t r o l volume. The p o t e n t i a l value at P i s to be found. A l s o , l e t point Q be a "control p o i n t " l o c a t e d on the c o n t r o l s u r f a c e . Q i s a running parameter when taking area i n t e g r a t i o n f o r computing p o t e n t i a l value at P . Let the observ a t i o n point P be descr i b e d by (R^, y^, 0) and the c o n t r o l point Q be (RcosQ, y, -Rsin6). The dis t a n c e r between P and Q i s given by x = RcosO z = Rsine . (1) (2) the gradient of r , Vr i s (Rcos6- Rp, y-y , -Rsine) Vr = (3) - 2.2 -5 where Vr is given in x-y-z co-ordinates. F i n a l l y , le t n be the unit normal vector on the control surface pointing out of the control volume. 11.2 General Formulation If an i n v i s c i d and incompressible f l u i d flow ex i s t s , a complex potential function <& can be found. In order to predict the response of an arb i t rary f loat ing body in waves, the computation of the added masses, damping coe f f i c i en t s , and exci t ing forces is necessary. (See Wehausen, 1971.) These hydrodynamic coef f ic ients which are related to the hydrodynamic pressure force can be calculated i f the motion induced ve loc i ty po ten t i a l , $ is known. $ in general is a function of posit ion as well as time and must sat i s fy the Laplace Equation because of continuity of incompressible f l u i d flow, i . e . V 2 * = 0. (4) This is an e l l i p t i c par t ia l d i f f e r e n t i a l equation to be solved with the appropriate boundary condit ions . For this type of problem with a f i n i t e depth and free surface, there are four di f ferent kinds of boundary condit ions . The f i r s t one is the r i g i d impermeable boundary condit ion. That i s , the component of f l u i d ve loc i ty in the d irect ion normal to the boundary surface is zero. * , = V" • n S o = 0 , (5) where V is the ve loc i ty of f l u i d at the boundary, and SQ is the impermeable boundary surface. - 2.3 - 6 The second boundary c o n d i t i o n i s of the f l u i d moving in contact with the body. In t h i s case, the component of f l u i d v e l o c i t y in the d i r e c t i o n normal to the body's surface must be equal to the component of the body's v e l o c i t y along the d i r e c t i o n n. This i s given by ••n = V „ , (6) where V n i s the v e l o c i t y of the boundary in the n d i r e c t i o n and i s the surface area of the body. On the fr e e surface, two boundary c o n d i t i o n s have to be s a t i s f i e d at the same time. The kinematic boundary c o n d i t i o n which says that the f l u i d p a r t i c l e s at the surface w i l l remain at the surface or Dt <*-y) S f = n , t + U T 1,x " v + W V ( 7' a ) = 0 on y = n as given in Mewman (1977). The second c o n d i t i o n on the fr e e surface i s the dynamic boundary c o n d i t i o n which i s a B e r n o u l l i Equation and which gives the pressure at the surfa c e . This c o n d i t i o n can be wr i t t e n as $ . + £- + i ( u 2 + v 2 + w 2) + gy = 0, on y = n (7.b) , t p c. where n i s the wave e l e v a t i o n , p i s the pressure, p i s the f l u i d d e n sity, and y = n on the free surface. The l a s t boundary c o n d i t i o n i s the r a d i a t i o n c o n d i t i o n . This boundary c o n d i t i o n concerns about the propagation of energy in wave form away from the body. The l i n e a r i z e d form of t h i s boundary c o n d i t i o n i s given by Wehausen (1950) as: l i m R > o o (*,R + 1k*) = 0 , (8) where k, the wave number, f o r p e r i o d i c waves, can be found by s o l v i n g the surface wave equation k tanh(kh) = |^ . (9) u> i s the angular frequency of the waves. 11.3 Method of S o l u t i o n In order to solve (4) with the appropriate boundary c o n d i t i o n s , a s u i t a b l e s o l u t i o n method must be chosen. Numerous methods have been developed f o r the s o l u t i o n of p o t e n t i a l problems, and most of them can be used s a t i s f a c t o r i l y . A n a l y t i c a l s o l u t i o n methods are p r e f e r a b l e in general f o r simple geometries but they become highly complicated f o r problems which w i l l be considered here. The F i n i t e Element Method i s the most popular to use but the r e s u l t s are expensive to o b t a i n . This i s due to two main reasons. The f i r s t reason i s that the control volume must be d i v i d e d into a l a r g e number of f i n i t e elements; the input preparation takes much computer time. The second reason i s that a l i n e a r equation with a large matrix has to be solved during computation. As the cost of computing time v a r i e s with the t h i r d power of the order of the matrix, the computing charge w i l l be rather high. The Surface S i n g u l a r i t y Method and the Boundary Element Method each o f f e r an a l t e r n a t i v e s o l u t i o n , though they are not so commonly used. However, the Boundary Element Method fo r p o t e n t i a l problems has c e r t a i n advantages over the F i n i t e Element Method in terms of the computer input time and computational time. Moreover, the reduction of number of the elements can decrease the complexity of the problem. - 2.5 - 6 Before going any further into the computing technique, some assumptions have to be made to s implify the problem defined above. It is obviously non-l inear because of the free surface boundary condit ions . One major assumption can be made about the complex potential function. It is assumed that $ can be represented by a series of terms of dif ferent orders of magnitude. That i s , • = * ( 0 > + ^ + e 2 * ^ + e%<n> , (10) where e is a small number and the superscript inside the bracket indicates the order of that term. For e, one can take i t equal to A/A, the non-dimensional wave amplitude which is the wave amplitude divided by the wave-length. It can be proved that each of these terms should sat i s fy (4) and the following statement can be shown to be true 3 x m 8 x m That i s , the order of the derivat ive of a term is the same as the order of that term. Since the veloci ty component, U. , is given by $, , i t is also true to assume that U- = U - ( 0 ) + s U . ( i ) + £ 2 U . ( 2 ) + •1 -1 - 1 - "1 - * <12> Moreover, the free surface and pressure functions are assumed to exist respectively in the same form. (0) A (1) A 2 (2) A n = n + en + e n + • • • • , (13) p = p ^ + e p ^ + e 2 p ^ + • • • • . (14) - 2.6 - 9 One of the major r e s u l t s of l i n e a r i z a t i o n i s that the boundary c o n d i t i o n s are s a t i s f i e d at the mean p o s i t i o n of the body's surface, and at the free surface (see Appendix A). The mean p o s i t i o n of the free surface i s taken as the undisturbed f r e e s u r f a c e . One can show that « ^ i n (10) has to be equal to a constant to s a t i s f y the boundary c o n d i t i o n s and i s taken to be zero since the reference atmospheric pressure i s also taken as zero. Therefore, the zeroth order terms of p, IK n can be equalled to zero. Thus, $ = e 2J2"> + £ $ + • • • • p = e e p + • • • • U. = e eU.j + •••• n = e 2 (2) . e n +  (15) I t must be noted that p i s the change in pressure due to o s c i l l a t i o n of the body and does not include the h y d r o s t a t i c pressure. By b r i n g i n g (15) i n t o (4), (5), (6), (7.a), (7.b) and keeping only the terms of order e, i . e . the f i r s t order terms, the l i n e a r i z e d problem can be wr i t t e n as 'n = 0, = o , (16.a) (16.b) 'n = V (1) .(1) Al) S f (1) ' t + gy 'y = o , (1) - n fl) at y = 0, at y = 0, (16.c) (16.d) (16.e) but in (16.e), i s equal to zero at y = n. t h e r e f o r e , + g n ( 1 ) = 0 at y = 0. (17) s And since (16.d) and (17) are boundary c o n d i t i o n s f o r the free surface, they can be combined as a s i n g l e equation as $ U ) ' t t | + 9 $ U ) ' y = 0 a t y = 0 * { 1 8 ) |S f Now, with the understanding that only the f i r s t order terms are to be solved, a l l the s u p e r s c r i p t s w i l l be dropped. For the study of p e r i o d i c motions, the body i s assumed to be o s c i l l a t i n g at a c e r t i n frequency, u>, and the induced waves are generated at the same frequency. Therefore, the form of the expected $ can be imposed as: $ = <|>(x,y,z)e"iu)t , (19) where <)> i s the amplitude of $ and i s a f u n c t i o n of p o s i t i o n only. S i m i l a r l y , n and can be expressed as re" 1 ^  and e""1^. Thus, the f i n a l s i m p l i f i e d form of the equations i s as f o l l o w s : From (16.a), the p a r t i a l d i f f e r e n t i a l equation can be r e w r i t t e n as: v2^ = 0 , (20) the impermeable boundary condition. (16.b) becomes * , J = 0 , (21) the boundary c o n d i t i o n on the body's surface (16.c) becomes ••nL = V n • ( 2 2 )  b and f o r the f r e e surface (18) -o>2<t> + g<hy|sf = 0 • ( 2 3 ) - 2.8 -In general, (20), (21), (22) and (23) are f o r any a r b i t r a r y f l o a t i n g body i n p e r i o d i c waves. The r a d i a t i o n boundary c o n d i t i o n i s dependent on the type of fi^w. For d i f f e r e n t problems, there i s a numerically d i f f e r e n t form f o r the r a d i a t i o n c o n d i t i o n . In some a p p l i c a t i o n s , some of the f l o a t i n g objects are axisymmetric i n shape. Included are composite c y l i n d e r s and objects that are c y l i n d r i c a l , s p h e r i c a l , and coni c a l . For any axisymmetrical body, the r a d i a t i o n c o n d i t i o n i s given by Bai(1972) as 4>,R = (- + 1k) 4> , (24) S „ R where RD i s defined as in F i g . A, S„ i s the outer boundary surfac e , and k i s defined by (9). The equations (20), (21), (22), (23) and (24) form a complete set of equations f o r the f l u i d flow system under the assumption of simple harmonic o s c i l l a t i o n of an axisymmetric body. The s o l u t i o n of (20) can be obtained by using an i n t e g r a l equation suggested by Brebbia (1978). The i n t e g r a l equation i s given as CQ*(P) + / s 4>(Q)^ ( P' Q ) dS = J s G(P,Q)dS , (25) where S t i s the t o t a l surface area bounding the control volume. P and Q are points defined in se c t i o n II. 1 . Q i s equal to 4u i f P i s i n s i d e the control volume and equal to zero when P i s outside the control volume. When P i s at the boundary surfac e , Q i s equal to 2 it. C i s a constant equal to 1 in three-dimensional problems and equal to 1/2 f o r two-dimensional ones. G i s the well-known Green's Function which i s equal to 1/r in three-dimensional formulations and equal to l n ( l / r ) i n two-dimensional ones, r i s as defined in se c t i o n II.1 and i s a f u n c t i o n of P and Q. Therefore, G i s a fun c t i o n of P and Q only. For area i n t e g r a t i o n i n (25), a numerical computation i s req u i r e d . In gener a l , the control surface i s d i v i d e d i n t o elements of f i n i t e area. C i r c u l a r r i n g elements are chosen f o r t h i s study as the v a r i a t i o n of po t e n t i a l value on each of these elements i s a known fu n c t i o n of e. The mid point of each element i s defined as the node point of that element.(Fig. B) For three-dimensional problems with P loc a t e d on the boundary, equation (25) can be rewritten as 2n*(P) + Js • ( Q ^ W e d * = Js M i l I Rdedi . (26) This equation can be rewritten as a matrix equation [ ^ [ • j ] = [B,] (27) as shown in Appendix B. By i n c l u d i n g the boundary c o n d i t i o n s (21), (22), (23) and (24) i n t o (26) , the f o l l o w i n g equation i s obtained. (See Appendix C) 2*<fr(P) + / s | ? r { ^ ) < D ( Q ) R d e d x + J s !^)<|,(Q)RdFdx • K + / S f & (7> " F ^  » ( Q ) R d * d R = J $ b 7 V n RdedJl . (28) Although (28) i s d i f f e r e n t i n form from (26), i t can als o be wr i t t e n as a matrix equation s i m i l a r to (27). A f t e r the matrix equation i s set up, the 4> value at the node of each element can be solved by the well-known methods - 2.10 »3 of l i n e a r algebra. In general, $ w i l l be a complex number since some of the matrix e n t r i e s are complex in value because of the r a d i a t i o n boundary c o n d i t i o n s . Equation (28) seems to have surface i n t e g r a l s , but f o r axisymmetric cases, i t can be s i m p l i f i e d i n t o a l i n e i n t e g r a l equation. The d e t a i l s of the s i m p l i f i c a t i o n are explained i n the next chapter. - 3.1 - 14 I I I . HYDRODYNAMIC COEFFICIENTS I I I . l General Formulation of Hydrodynamic C o e f f i c i e n t s The motion of a body can be described with the supe r p o s i t i o n of 6 independent motions. Three of these are t r a n s l a t i o n a l motions i n the x, y, and z d i r e c t i o n s and the three others are r o t a t i o n a l motions about the axes. The three t r a n s l a t i o n a l motions represented by x x , x 2 , and x 3 are known as surge, heave and sway. The r o t a t i o n a l motions x^, x 5 , and x 6 are named r o l l , yaw, and p i t c h . With the above n o t a t i o n , the most general dynamic equation of the body i s 6 j = l K-j + a i j ) x i + b i j x i + c i j x i (29) where m.^ i s e i t h e r the mass or the mass moment of i n e r t i a of the body, a., and b.. are known as the hydrodynamic c o e f f i c i e n t s . F' i s the e x c i t i n g f o r c e or moment in the i d i r e c t i o n . [See Wehausen 1971]. m.. i s the actual mass of the body when i i s equal to j and l e s s than 4. I t i s equal to zero when e i t h e r i or j i s l e s s than 4 and i i s not equal to j . When neit h e r i nor j i s l e s s than 4, m.. i s the mass moment of i n e r t i a . For s i m p l i c i t y , m.. i s u s u a l l y expressed in matrix fo * J rm. m.. = U 0 0 0 0 0 0 M 0 0 0 0 0 0 M 0 0 0 0 0 0 0 0 0 0 0 0 Ixx Ixy Ixz Iyx Iyy Iyz Izx Izy Izz (30) - 3.2 - 15 a., i s known as the "added mass" which when m u l t i p l i e d by the corresponding a c c e l e r a t i o n gives the hydrodynamic pressure force component in phase with the a c c e l e r a t i o n , b ^ i s the "damping c o e f f i c i e n t " which when m u l t i p l i e d by the corresponding v e l o c i t y gives the hydrodynamic pressure force i n phase with the v e l o c i t y , c . i s the " r e s t o r i n g force c o e f f i c i e n t " and when m u l t i p l i e d by the corresponding body displacement (from the e q u i l i b r i u m p o s i t i o n ) gives the h y d r o s t a t i c force component opposite to the displacement. A general r e l a t i o n s h i p between the hydrodynamic forces and the added masses and damping c o e f f i c i e n t s i s given by Newman [1977] as where U. and II. are the v e l o c i t y and a c c e l e r a t i o n of the body in the j J J > t I I I I I d i r e c t i o n . ( E ^ E 2 , E 3) are the hydrodynamic pressure forces and (E^, E 5, i i E 6 ) are the hydrodynamic pressure moments. E^ i s induced by the motion of the body and the h y d r o s t a t i c pressure force i s not included because of l i n e a r i z a t i o n . For p e r i o d i c motion, l e t u. be the amplitude of the body's v e l o c i t y and 3 E.. be the force amplitude. That i s , U. = u e ~ i a ) t , (32) J J U. . = -iuAJ. , (33) and E^ = E . e ~ i a ) t . (34) Therefore, (31) can be rewritten as - 3.3 -6 b.. E. s (a.. + i - L i ) u. = • Another formula f o r computing added masses and damping c o e f f i c i e n t s i s given by Wehausen [1971] i j + f T T - IK ' • j " t d S • , 3 6 > where p i s the f l u i d density, <)>'. i s the p o t e n t i a l f u n c t i o n induced by u n i t 3 v e l o c i t y of the body in the j d i r e c t i o n and n i i s the i component of the u n i t normal vector. From (16.e), only the hydrodynamic pressure, p, can be a f u n c t i o n of time and i t can be equated to the negative of the time d e r i v a t i v e of the p o t e n t i a l f u n c t i o n , i . e . P = " P * , t = iu><|>e~1a)t . (37) Hence, the hydrodynamic pressure force or moment can be computed as: E. = L p n. dS i = 1 to 6 , (38) 1 5 b 1 where ( n 1 , n 2 , n 3) = ( n x > n y » n z ) (39.a) and ( n 4 , n & , n g) = (r x n) . (39.b) r Q i s the p o s i t i o n vector measured from the o r i g i n . If (37) i s s u b s t i t u t e d into (38), the reader can prove (36) e a s i l y . Moreover, i f i s used to represent ^, (36) can be rewritten as 3 - 3.4 - 17 a . j + i ^ - = n.dS . (40.a) The non-dimensional form of the above equation can be obtained by d i v i d i n g i t by the f l u i d density and the body's d i s p l a c e d volume, v. w + 4p4 - 4 v ^ - d s • (4o-b> Equation (36) and (40.b) are the general'.forms f o r computing added masses and damping c o e f f i c i e n t s . The area i n t e g r a t i o n i s u s u a l l y done by a numerical method. However, the value of <j>. on each element must be known J before the area i n t e g r a t i o n i s done. Thus, (28) must be solved f o r the c a l c u l a t i o n of hydrodynamic c o e f f i c i e n t s . Before proceeding on to the next s e c t i o n , some assumptions about $ must be made. I t was mentioned by Wehausen [1971] t h a t $ can be expressed as the sum of three d i f f e r e n t terms, i . e . , $ = * + $ D + $ F (41) $j i s the p o t e n t i a l value of incoming waves, ^ i s the p o t e n t i a l value r e s u l t e d from r e f l e c t i o n of $>j on the body's surface , and ^ i s the po t e n t i a l value induced by the motion of the body. $j and $ are important i n e x c i t i n g f o rce and moment computation. In l i n e a r i z e d c a l c u l a t i o n s , the body i s assumed to be f i x e d in space and experiences a force due to $j and For the computation of the added masses and damping c o e f f i c i e n t s , ^ i s induced by the motion of the body. The d i s t r i b u t i o n of <s>p on the submerged surface of the body w i l l determine the value of a., and b^.. - 3.5 - /8 The p o t e n t i a l f u n c t i o n , $, which appears in t h i s chapter i s a c t u a l l y the $p; the i n t e r a c t i o n between $ , $ D and ^ i s assumed to be n e g l i g i b l e . III.2 Computation of Heave Added Mass and Damping C o e f f i c i e n t For an axisymmetric f l o a t i n g body o s c i l l a t i n g i n the y d i r e c t i o n , a ^ and b ^ can be computed as: a „ b y=0 2 it <u where y t i s the keel of the body. Since the body i s axisymmetrical about the y a x i s , <}>2 can be assumed independent of 0; that i s , ^ i s a constant value on a c i r c u l a r r i n g . Then, (28) can be s i m p l i f i e d as y=0 -4 R 2n + (y-y_)n ) 2n 2 * U P ) + / [ 5 E L J L E + —B- ( - i - E - F ) ] c y=y + (a-b)/a+F /(a+F) (a-b) 6 R D 4(R 2n + (y-y ) n j 2n„ + J R [- _ P _ J L E + (— r _ E - F ) ] M A 0 (a-b) / a+F /a+F (a-b) L y=-h /a+b (a-b)/a+b RR ,„#.. ..^  » 2 r + j r . 5 l [ y i y p J _ E . i £ L i _ F ] M R RC /a+TTla-b) g/a+F - 3.6 - /9 ry=° 4R = / ™ _ F u n dA , (42) y=yt /a+b" c y where F and E are Complete E l l i p t i c Functions of the f i r s t and second kind, a and b are constant values. D e t a i l s of the s i m p l i f i c a t i o n are shown i n Appendix D. (41) can e a s i l y be rewritten as a 22 boo i y=o <t>o y " t As <t>2 on each element can be found by s o l v i n g (42), and can be evaluated from (43). A computer program, named HEAVE.FTN, was w r i t t e n to solve f o r <^  and a22 ^22 then compute and . This computer program sets up the matrix a 0 0 equation (42), and then solves f o r <|>0 on each c i r c u l a r r i n g element. -=^- and c pV b 2 2 are then computed by (43). I t must be noted that (42) and (43) are now l i n e i n t e g r a l equations instead of area i n t e g r a t i o n as in (28). III.3 Computation of Surge added Mass and Damping C o e f f i c i e n t I f the body i s o s c i l l a t i n g i n the x.jdirection, the normal v e l o c i t y component on the moving boundary i s in the form $ I = u, n ncose . (44.a) ,n c 1 R That i s , the motion of the body has a maximum e f f e c t when e i s equal to 0 and % and has a minimum e f f e c t when 9 i s equal to ± This suggests ^ w i l l be in the f o l l o w i n g form - 3.7 - 20 <t>1 = <J>^  cose . (44.b) By taking (44.b) into (28), (28) becomes 2TUD1'(P) + J | n ( l / r ) cos9 RdedA S o + / - | r ( l / r ) (Dj cose RdedA S b . 2 + J [ -^f (1/r) - |- l/r]<^cose RdedR S f + ' f | n " ( 1 / r ) " ^ + J o c o s e Rdedy R = / t^cose n R 1/r RdedA . (45) S b The f i r s t term i n (45) i s 2n<^  instead of 2it<|>Jcose because point P i s on the x-y reference plane. (See Section I I . l ) . Equation (45) can be s i m p l i f i e d i n t o a l i n e i n t e g r a l equation by i n t e g r a t i n g with respect to 9. The d e t a i l s are in Appendix E and the r e s u l t i s 2n*J ( P ) + [-4= ("RR2 + n (y-y )R)(F - * E) y=y t b/a+b K y p u DJ 2 + j n nf/a+¥ E - ^ — F + a E ) ] ^ d A D K /a+b /a+b(a-b) 1 RR 0 b/a+F K y P 1 3 0 1 + £ n D (/a+b" E - -2-A- F + — E ) ] M A D K /a+b /a+b (a-b) 1 - 3.8 - 21 RR - 2 2 c ( y - y J a R c 9 p 9 /a+b /i+bla-b) ^ 0 2 r r 2(2R -a) r c a r >, , 4ikR , ,—r- r a c^ n ' ., J L — — — (F - r r ^ y - E) + — g - (/a+b E - F) fcdy y=-h b/a+F u D' D /a+b 1 y=0 4u, n DR / 1 , R [ - A — F - E/a+F ] d i . (46) y=y t /a+F A f t e r on each element i s known by s o l v i n g (46), the p o t e n t i a l value at any point on the boundary elements i s given by <t>'^cose. By b r i n g i n g (44.b) into (40.b), the f o l l o w i n g equation can be obtained. a n b n 1 y=0 2u <t>-!cose -4r + -4r~ = i / / -r, n D c o s e Rdedi pV pVu v y = y^ o u l R 1 ,y=0 *i = ^ / ™ R TT R d* • (47) v y=yt u i A computer program named SWAY.FTN was w r i t t e n to solve equation (46) a l l b l l and then evaluate -^=- and •—^ . The r e s u l t s computed by SWAY.FTN are shown in Chapter V. III.4 Computation of P i t c h Added Mass and Damping C o e f f i c i e n t In computing agg and bgg, the body i s assumed to p i t c h about a f i x e d p o i n t and induces the p i t c h p o t e n t i a l value $g. The choice of t h i s centre of r o t a t i o n i s a r b i t r a r y . ( O r i g i n of coordinate axis i s chosen here). In t h i s case the z-axis i s taken as the axis of r o t a t i o n . I f the angular v e l o c i t y of the body i s assumed to be y e _ U ) t , the boundary c o n d i t i o n (22) on the body becomes 6 ' n (y x r Q ) • n S b = [(0,0,y) x (Rcose,y,-Rsine) ] • ( n R c o s e , n y , - n R s i n e ) = (-rij^y + n y R ) Y cose , (48) where r o i s the p o s i t i o n vector from the o r i g i n . Equation (48) suggests the term 'cose' w i l l be present in <t>g, t h e r e f o r e , ^ i s assumed to have the form 4>6 = <t>g cose (49) where <|>g w i l l be at i t s maximum value at e equal to zero. Since both the p i t c h p o t e n t i a l f u n c t i o n and the surge p o t e n t i a l f u n c t i o n have the 'cose' term, i t i s reasonable to say that the p i t c h motion of the body has induced a surge moment on the body and the coupled hydrodynamic c o e f f i c i e n t s , a g ^ and b ^ , should e x i s t . By taking (48) and (49) i n t o (28), the i n t e g r a l equation f o r p i t c h i n g motion i s 2Ti<Dg (P) + / f n - d / r H g - cose RdedA + / - | r ( l / r ) <DgCOS e RdedA S o S b a 2 + / [ Ipj-d/r) - |-(1/D ] <D6cose RdedR S f + / [|„ d / r ) - ( i + i k ) l / r J o c o s e Rdedy S r R J (-n^y + n R ) Y c o s e ( l / r ) R d e d i . (50) r iv y b b Equation (50) i s s i m i l a r to (45) except that the term on the right-hand side i s d i f f e r e n t . The evaluation of the l e f t - h a n d side i s e x a c t l y s i m i l a r - 3.10 -to (E.2), (E.3), (E.4) and (E.5) mentioned i n Appendix E. The i n t e g r a t i o n of the right-hand side of (50) i s shown i n Appendix F. (50) can be re w r i t t e n in the fo l l o w i n g form. 2 7 t < f , 6 ( P ) + L ° ( n R R 2 + n y { y " V R ) ( F " i ? F E ) y=y t b/a+b 2 + I n J / i + F E - ^ - F + — E)]o>' d l 0 K /a+b /a+b(a-b) 0 2 + ?- n D ( / a + F E — F + E)]<t>'di D K / a + b / a + b ( a - b ) ° RR A 2 2 p ( y - y n ) a R„ 9 P 9 /5+b" /aTF(a-b) ^ + j° [ 2 ( 2 R 2 - a ) ( r a r ) + 4ikR ( / i T F £ _ _j_ p ) ] , y=y h b/a+ F ^ / i + F J 6 = / -4(-n Dy + n R ) J ^ [/a+F E — - — F ] d i . (51) y=y t K y /a+ F A f t e r <))g i s obtained by s o l v i n g (51), the added masses and damping c o e f f i c i e n t s can be computed by using (40). For the eva l u a t i o n of a „ and 00 bgg, Uj i s equal to y, i s equal to ^ c o s e , and n.. i s equal to (r x n)•(0,0,1). Therefore, (40) becomes - 3.11 -a66 b66 1 y = 0 2 % *6 W+ 1 = v / J — cos e (Rn - n y) cos e Rd 9d H M y=y t o y * + 1 / -4 (Rn - r y ) RTI dx . (52) y=yt Y The non-dimensional form of the above equation i s obtained by d i v i d i n g 2 2 both sides by R^ • R M i s a length parameter of the body. F i n a l l y , the non-dimensional form of a „ and b „ becomes 66 66 _ 2 | + A ' / _ | ( R N . N „) D J L . (53) PVRM PVwRM VRM y=y For a computation of ag.| and bg.|, the u n i t normal vector component, n. i s equal to n Rcose, and (40) becomes a t l b,, , y=0 it i>l W+i-M, - 1 I * I % ^ n R c o s e R d e d * H y=yt o Y l r° *e cose . p H o - J n R Rn dA . y=yH t The non-dimensional form i s given as a.-, b ^ 1 y=0 <D6' fii fii l J r u ita - § 5 - + 1 -?T5- = ITS" / — n D Rit d l . (54) pVR M uVa,RM VR M ^  . R Since (51) i s s i m i l a r to (46), the computer program SWAY.FTN was converted to solve (51) and give <|>g. a g ^ , a g g , b g^ and b g g can be computed by (53) and (54). A computer program PITCH.FTN was w r i t t e n to do the j o b . Results and output of a l l the hydrodynamic c o e f f i c i e n t s mentioned in Sections III.2, III.3, and III.4 w i l l be discussed in the l a s t chapter. - 4.1 -25" IV. EXCITING FORCE AND MOMENT IV.1 General Formulation o f E x c i t i n g Force and Moment Due to Incoming Waves For an axisymmetric f l o a t i n g body, the external f o r c e s and moments a c t i n g on i t are mainly due to surface waves. The surface waves can be considered as u n i d i r e c t i o n a l . They i n t e r a c t with the f l o a t i n g body, and are r e f l e c t e d and r e f r a c t e d from the body's s u r f a c e . These i n c i d e n t waves and r e f l e c t e d waves i n d i c a t e a t r a n s f e r of wave momentum between a s o l i d o b j e c t and the f l u i d flow. I t i s the t r a n s f e r of momentum th a t sets the body i n t o motion; and the f o r c e s experienced by the body at t h i s i n s t a n t are known as the e x c i t i n g f o r c e s or moments. And they are represented by F.J on the r i g h t -hand side of (29). The p o t e n t i a l f u n c t i o n of incoming waves, $ j , i s assumed to be a two-dimensional p e r i o d i c f u n c t i o n . I f the waves are approaching from the p o s i t i v e x - d i r e c t i o n with an angular frequency w, $j i s u s u a l l y given as in Newman [1977] as, * . Ag cosh k(y+h) J(kx+cot) ®T ~ ~ 1 L,U 6 > (00) I to cosh kh ' where A i s the wave amplitude and g i s the g r a v i t a t i o n a l a c c e l e r a t i o n . $ D i s the r e f l e c t e d wave p o t e n t i a l f u n c t i o n due to the presence of the body. T h i s r e f l e c t e d wave p o t e n t i a l f u n c t i o n i s a l s o assumed to be p e r i o d i c because <t>j i s a p e r i o d i c f u n c t i o n . In most cases, the amplitude of $ D i s not known and i s obtained as par t of the s o l u t i o n f o r the given geometry o f the f l o a t i n g body. When the amplitude of e x c i t i n g f o r c e s i s computed by a l i n e a r theory, the body i s assumed f i x e d i n space. The o s c i l l a t i n g frequency of the e x c i t i n g f o r c e w i l l be the same as the waves' frequency though there may be a phase d i f f e r e n c e between them. -4.2 - 26 When s o l v i n g $ D from (26), i t must be noted that on the surface of the body the f l u i d v e l o c i t y i s equal to zero. This boundary c o n d i t i o n implies that on the body's surface the normal d e r i v a t i v e of the t o t a l p o t e n t i a l i s zer o , t h a t i s , '^b and i t can be rewri t t e n as D,n = _ a> c I.n b b (56) S b For an axisymmetric body, 4^  must s a t i s f y the i n t e g r a l equation (26) as well as the boundary c o n d i t i o n s (21), (23), (24) and (56). These f i v e equations w i l l be combined to give a s i n g l e i n t e g r a l equation s i m i l a r to (28). By assuming ^ as the time independent part of $ D, the i n t e g r a l equation (26) becomes * l o<t>n(Q) -, 2* <KD(P) + J 4>D -§jj (-p) RdedA = j—iL—1 RdedA . (57) S t S t The impermeable boundary c o n d i t i o n becomes *D,n L " 0 • ( 5 8 ) >bo The free surface boundary c o n d i t i o n (23) i s now writte n as - w 2 < t > D + 9< t > D j V j = 0 . (59) ' S f The r a d i a t i o n c o n d i t i o n i s • D . R s " (- 2TR + ik) V ( 6 0 ) r - 4.3 - 27 For the boundary c o n d i t i o n on the body, the right-hand side of (56) has to be evaluated. Since $j i s given i n (55) and i f ^  i s assumed to be the amplitude of $ j , <J>J i s given i n the form o f < Ag cosh k(y+h) ikx *I " " 1 ui COsh kh e • ( 6 1 ) ikx A s e r i e s representation of the term e i s given by Abramowitch and Stegun [1964] as . Ag cosh k(y+h) ikRcose $ I " 1 a) cosh kh e m=U where J i s the Bessel Function of order m and B i s equal to one when m i s m "m ^ equal to zero, otherwise, 6m i s given by 2 i m . Therefore, (56) can be rewr i t t e n as pD,n where and = 1 ^ c ^ s T T h Jn ^ c o s m 9 t J m ( k R ) nR c o s h k (* + h ) m=0 + J n ( k R ) n si n hk(y+h) ] , (63) 1' ^m-l ^ m+1 m = 2 { 6 4 ) J _ = (-1) J m . (65) - m m D e t a i l s f o r the d e r i v a t i o n of (63) are shown i n Appendix G. With (58), (59), (60) and (63), the reader can e a s i l y rewrite (57) in t o the f o l l o w i n g form 2 ^ D ( P ) + / ^ ( 1 ) <t>p(Q) RdedA + / ^ ( 1 ) <t>D(Q) RdedA S o S b + / [|n <^ " ( 1 / r ) + i k ) l % R d 6 d y b r K + / $ - i«b «QJ RDEDR = / 1 ? O T i v o s m e t j ; ( k R ) v o s h k(y+h> m=0 + J m ( k R ) n y s i n h k(y+h) ]-p R dedA . (66) From (66), one can observe that i n order to solve f o r 4 ,^ a large number of terms must be used on the right-hand side of the equation. The higher the value of m, the more p r e c i s e the value of 4^  w i l l be. Moreover, since 4>j appears as a functi o n of cos me, (66) can no longer be s i m p l i f i e d unless an assumption i s made on 4^. The most straightforw 3'"* way of doing t h i s i s to assume that 4>D i s also e x p r e s s i b l e as a s e r i e s of terms i n v o l v i n g cosme. That i s , I 4>n cos me . (67) m=0 m Then, (66) can be written as 2u4>D (P) + / | r 7 ( 1 / r ) , t b (Q)cosme RdedA + / ' - ^ l / r ) (Q)cosme RdedA m S„ m S. m 0 b + / fel/r) - d / r H ik)]4h (Q)cosme Rdedy S a n ^ KR Um r 2 + / [|n" ( 1 / r ) " f~ ( 1 / r ) ] " b (Q)cosmeRdedR 9 m - 4.5 -2 9 = / 1 £ c F i O h P mcosme[j' m(kR)n Rcosh k(y +h) + J m ( k R ) n y s i n h k(y+h)jl/rRdedA . (68) Equation (68) has to be solved f o r d i f f e r e n t m values, and 4^  can be computed by (67). For the c a l c u l a t i o n of e x c i t i n g forces and the p i t c h i n g moment, evaluation of the pressure d i s t r i b u t i o n on the body's surface i s necessary. From (37), the expression of pressure i s w r i t t e n as P = I - P U T + <t>D cosme) iwe l a ) m=0 m m • t * - « « "Hit" - **o rfc^ee11* • (69) m=0 m The e x c i t i n g forces are given by s u b s t i t u t i o n of (69) into (38) as - U l P["AS V- lkR> m=0 - i4>D oj ] cos me n.R dedjt. } e i ( i ) t , (70) m where ( n ^ n 2 , n 3 ) = (n R c o s e , n y , -n R s i n e ) (71a) and ( n 4 , n g , n g) = (Rcose, y,-Rsine) x ( n R c o s e , n y , - n R s i n e ) . (71b) IV.2 Heave E x c i t i n g Force For the heave e x c i t i n g force computation, (70) can be used by l e t t i n g n.. be equal to n y . From Appendix H, F ^ i s given as 30 4.6 -F 2 = { f °-2n PRn [Ag " g ^ f f 0 ,J 0<kR) + 1 » D »] d i j e 1 ^ . 72) y=yt ^ o That i s , F 2 depends only on the terms with m equal to zero. Therefore, only the value of <t>D w i l l be necessary f o r computation of F' . Then o Equation (68) can be w r i t t e n as 2 % % ( P ) + 'In" ( 1 / r ) < t b ( Q ) R d e d A + / "IT (7) <trj (Q)RdedA 0 S Q 0 S b 0 + / " ^ + i k ) ( ? n v ( Q ) R d e d 5^ K 0 = | A 1 "^w cosfi~~kh~ P 0 [ j ; ( k R ) n R c o s h k ( y + h ) b b + J Q ( k R ) n y s i n h k(y+h) ](l/r)Rdedjt . (73) Equation (73) i s s i m i l a r to (28) except that a d i f f e r e n t f u n c t i o n independent of 9 i s on the right-hand s i d e . Therefore, by f o l l o w i n g the procedure described in Appendix D, the reader can e a s i l y obtain y=0 -4(R 2n + (y-y_)n 2 7 % (P) + / [ = E u o y=yt (a-b)/a+F \ -4(R 2n + ( y - y J n R ) + J [ P—K- E 0 (a-b)/a+b~ + 3i (T^T E- F»1 V Q , < U • r ^p . m - R p-^ -v 2 E H d ( Q l d y y=-h /i+F (a-b)/a+F u o RR 4R(y-y n) - 2 D + / [ E - — F ] «u dR R c /a+F~(a-b) g/a+F u o y-y t + J ( k R ) n s i n h k(y+h)] F d* . (74) 0 y /a+F Now, <{>n can be obtained by s o l v i n g (74) and F' can be computed by o using (72). A computer program, F2.FTN has been w r i t t e n to evaluate the heave e x c i t i n g f o r c e experienced by an axisymmetric body due to incoming waves. IV.3 Surge E x c i t i n g Force To compute the surge e x c i t i n g f o r c e , equation (70) should be used by l e t t i n g n. equal to n D c o s e . T h e r e f o r e , (70) i s r e w r i t t e n as 1 K - i<t»D to] cosmecose n Rdedi} e 1 u ) t (75) m By i n t e g r a t i n g with r e s p e c t to e, (75) can be expressed as F J = { f ° - upn R R[Ag ^ o s ^ k h ^ ¥ l ( k p ) + HD ^ d * } e 1 U j t ( 7 6 ) y=yt l - 4.8 - 32 D e t a i l s of the proof are given i n Appendix H. From Appendix I, i t i s known that F' i s dependent only on ^  a n d <t>T • For <|> and <{>.. with m not equal to one, there w i l l be no c o n t r i b u t i o n to m m the surge e x c i t i n g f o r c e . Therefore, by keeping only the terms of m = 1 (68) becomes 2lt*D ( P ) + J f n ( 1 / r ) < , b (Q)cos9Rded* + /|n" ( 1 / r H (Q)coseRd9d*-1 S o 1 Sb 1 + / [fn" (1/r) ~ (" + 1 k ) ( l / r ) ] ^  (Q)cosGRdedy S^ , R 1 - 2 + / [|jr - | - ( 1 / r ) ] ^ (Q)coseRdedR S r 1 = / 1 ^ H s T 1 ^ P l t j ; ( k R ) V o s h k ( y + h )  b b + J 1 ( k R ) n y s i n h k ( y + h ) ] ( l / r ) RcosededA . (77) (77) i s s i m i l a r to (45) except that a d i f f e r e n t f u n c t i o n independent of e i s on the right-hand side of (77). Therefore, (77) can be in t e g r a t e d with respect to e and can be s i m p l i f i e d i n exa c t l y the same way as explained i n Appendix E to give 2*VP1 * Xt feiS ( n« R 2 + V ' - V W " T^ T E> 2 - 4.9 2 + | n (/i+¥ E - 4 = F + - = E ) ] * j d x D K /a+b /a+b(a-b) u l RR A 2 . 2 . _ ( y - y n ) a c 0 2 , t r2(2R - a) (c aE > . 4ikR , r- c a r n A , + y i - h [-T^r Ta^ rST^  5 W - — F ) ] ^ d y 33 j ^ 1 c^osTTTrT hWk*)nRC05h k ( y + h ) + J.(kR)n slnh k(y+h)] £*- [ — — F - /a+E" E] djj. . (78) 1 y D /a+b" Now, <t>_ can be obtained by so l v i n g (78) and F' can be computed by u l 1 using (76). IV.4 P i t c h E x c i t i n g Moment In the computation of the p i t c h e x c i t i n g moment, n. i n (70) i s equal to n c which i s given by (71 .b) as b "S = ( n y R " C 0 S Q » ( 7 9^ and (70) becomes • n S b J P [ . » , f i ! | h ^ h i ^ 0 ( k R ) b m=0 - i 4>D oo] (n R - npy) cosmeRdedx } e 1 a ) t . (80) m * 4.10 As in Appendix I, i t can be proved that the only c o n t r i b u t i o n to Fg i s from the terms with m equal to one. Therefore, Fg i s w r i t t e n in the f o l l o w i n g form: F6 " < yCt - *PR <V " V H * C°cSsh(rhh) ¥l™ + i <un co ] d* } e i t j t . (81) 1 i A f t e r <)>D i s solved from (78), F g can be computed by (81). As both F-| and Fg are dependent only on 4^  , they are computed at the same time in computer program Fl.FTN. IV.5 An A l t e r n a t i v e Method f o r Computing E x c i t i n g Forces Wehausen [1971] gives a r e l a t i o n s h i p between the damping c o e f f i c i e n t and the e x c i t i n g force due to incoming waves f o r three-dimensional f l ow. k l F J / A 12 b.. = 1 ' 1— (82) D i i 2 Pg vg ' ( * a where V i s the group v e l o c i t y of waves, b.. i s the damping c o e f f i c i e n t , |F. | i s the amplitude of e x c i t i n g f o rce or moment and A i s the wave amplitude. Vg can be found in Newman [1977] as % • - . k where V p i s the phase v e l o c i t y , k i s the wave number and h i s the l o c a l water depth. - 4.11 -Equation (82) i s a good way of checking the amplitude of the e x c i t i n g f o r c e computed using the formulations of t h i s chapter. Chapter V w i l l e x p l a i n the r e s u l t s of the computation by (72), (76), (80), and (82). - 5.1 -36 V. RESULTS In t h i s chapter, the hydrodynamic c o e f f i c i e n t s and e x c i t i n g forces of some axisymmetric bodies are computed using the equations presented in the previous chapters. The r e s u l t s are then compared to those obtained by some other s o l u t i o n methods. The d i f f e r e n c e s between a l t e r n a t e methods are also discussed. The r e s u l t s presented by the author in t h i s t h e s i s are under the assumption that the p o t e n t i a l value i s a constant on each r i n g element. Therefore, when the hydrodynamic c o e f f i c i e n t s and e x c i t i n g f o r c e s are computed by taking the i n t e g r a l of the p o t e n t i a l values on the body's surface, the p o t e n t i a l value at the node of each r i n g element i s used. Although, a more pre c i s e r e s u l t could be achieved by c o n s i d e r i n g the p o t e n t i a l value varying across an element, the constant p o t e n t i a l value assumption i s the simplest assumption that one can make. The examples studied and the comparisons done f o r each p a r t i c u l a r geometry are explained below. V.l S i n g l e C y l i n d e r a t F i n i t e Water Depth The f i r s t example to study i s a s i n g l e c y l i n d e r at f i n i t e water depth. The radius of the c y l i n d e r i s designated by R, the draught by T, and the water depth by D. ( F i g . C) The hydrodynamic c o e f f i c i e n t s f o r heave, surge and p i t c h cases are computed and given in non-dimensional form (41). Results are p l o t t e d against the non-dimensional frequency, 2 to R/g, and are shown in F i g s . 1, 2 and 3. In F i g . 1, the heave hydrodynamic c o e f f i c i e n t s f o r T/R = 0.5 and D/R = 1.5 are c a l c u l a t e d and p l o t t e d against the published r e s u l t s of K r i t i s [1979]. The r e s u l t s by K r i t i s and the author are both obtained - 5.2 - 37 through the Integral Equation Method discussed in previous chapters. Good agreement between the r e s u l t s i s observed. In F i g . 2, the surge hydrodynamic c o e f f i c i e n t s f o r T/R = 0.5 and D/R =2.0 are p l o t t e d and compared to the values obtained by the Matching Technique (M.T.) of Sabuncu and C a l i s a l [1981]. The r e s u l t s obtained usingthe Boundary Element Method (B.E.M.) are lower than those of the M.T. There i s a d i f f e r e n c e of about 6% and 10% at the peak values of the added mass and damping c o e f f i c i e n t r e s p e c t i v e l y . The p i t c h added mass and damping c o e f f i c i e n t f o r a simple c y l i n d e r are p l o t t e d in F i g . 3 and compared to the values given by M.T. The added mass by B.E.M. has a lower value over the low and high frequency range. The damping c o e f f i c i e n t by B.E.M. i s lower than that of M.T. at low frequency but i s higher at the high frequency end. The maximum d i f f e r e n c e between the two curves i s at the peak value and i s l e s s than 5% f o r both the added mass and the damping c o e f f i c i e n t . The heave e x c i t i n g force experienced by the s i n g l e c y l i n d e r due to incoming waves i s shown in F i g . 9. The heave e x c i t i n g force per u n i t wave amplitude, f ^ , i s non-dimensionalized by the weight of the f l u i d d i s p l a c e d by the c y l i n d e r . Another way to p r e d i c t f^ i s by using the r e l a t i o n s h i p in (82) and the pr e v i o u s l y computed b^ values shown in F i g . 1. F i g . 9, shows that the B.E.M.1s r e s u l t i s lower than the r e s u l t obtained by using equation (82). T h e i r maximum d i f f e r e n c e i s l e s s than 5%. The surge e x c i t i n g f o r c e , F n , i s non-dimensionalized and p l o t t e d i n F i g . 10. Result p r e d i c t e d by equation (82)(with b ^ on F i g . 2) are als o shown on the same f i g u r e . Both r e s u l t s e x h i b i t the same peak value but 2 there i s a d i f f e r e n c e of about 10% at u R/g = 3.0. The p i t c h e x c i t i n g moment i s non-dimensionalized by the weight of d i s p l a c e d f l u i d and the radius of the c y l i n d e r . The moment i s c a l c u l a t e d about the z - a x i s . Both the r e s u l t s of B.E.M. and of equation (82) (with b g 6 on F i g . 3) are p l o t t e d in F i g . 11. Peak values are comparable but o there i s a maximum d i f f e r e n c e of about 15% at to R/g = 3.0. From F i g s . 9, 10, and 11, one can see that the e x c i t i n g forces p r e d i c t e d by B.E.M. have a b e t t e r agreement with those from (82) in the low frequency range. At the high-frequency end, the r e s u l t s of the two methods, though they e x h i b i t some d i f f e r e n c e , show the same trend and are comparable in magnitude. V.2 Composite C y l i n d e r As a second example, the hydrodynamic c o e f f i c i e n t s of a composite c y l i n d e r are computed. The dimensions of the composite c y l i n d e r are shown in F i g . D. The s o l u t i o n method the author w i l l compare with i s the Matching Technique of Sabuncu and C a l i s a l [1983]. The non-dimensional hydrodynamic c o e f f i c i e n t s are p l o t t e d against the 2 2 parameter to /g. to /g i s used instead of the non-dimensional frequency because in t h i s case i t i s hard to f i n d a s i g n i f i c a n t length parameter as the c r i t i c a l dimension of the composite c y l i n d e r . However, to non-dimensional i z e the p i t c h hydrodynamic c o e f f i c i e n t s (53), the l a r g e r radius of the composite c y l i n d e r i s used. F i g . 4 shows the heave added mass and damping c o e f f i c i e n t . The r e s u l t by B.E.M. e x h i b i t s an o s c i l l a t i n g behavior both in the added mass and damping c o e f f i c i e n t . This o s c i l l a t i o n i s b e l i e v e d to be caused by numerical problems in the computer program and w i l l be discussed l a t e r . I r r e s p e c t i v e of the o s c i l l a t i n g behavior, the r e s u l t s by B.E.M. e x h i b i t a s l i g h t l y higher value in the added mass computation. The damping c o e f f i c i e n t by B.E.M. agrees quite well with that computed by M.T. The maximum d i f f e r e n c e between the added mass value computed by the two 2 methods i s l e s s than 10% and occurs at to /g = 1.5. The surge hydrodynamic c o e f f i c i e n t s are shown in F i g . 5. The maximum d i f f e r e n c e in the added masses computed by the B.E.M. and M.T. i s at the peak value and i s l e s s than 5%. The damping c o e f f i c i e n t s obtained by the two methods e x h i b i t a d i s p a r i t y of about 10% at the peak value. The non-dimensional added mass approaches a constant value of about 3.75 at high frequency and the damping c o e f f i c i e n t goes to zero. F i g . 6 i s a p l o t of the p i t c h hydrodynamic c o e f f i c i e n t s ( a gg and bgg). The values by B.E.M. are lower in magnitude in an average sense. The maximum d i f f e r e n c e of damping c o e f f i c i e n t s i s observed at the peak value and i s about 5%. However, the added mass has a large d i f f e r e n c e of 2 about 35% at to /g = 1.5. The non-dimensional added mass value has a l i m i t equal to 0.4 at the high frequency while the damping c o e f f i c i e n t goes to zero. R e f e r r i n g to the f i g u r e s mentioned above, one can see that the o s c i l l a t i n g behavior of r e s u l t s by B.E.M. i s always about some mean value. I t i s bel i e v e d that t h i s numerical o s c i l l a t i o n i s caused by the constant p o t e n t i a l value assumption discussed in the e a r l y s e c t i o n s , and i s r e l a t e d to the s i z e of the r i n g elements chosen. In F i g . 12, the heave hydrodynamic c o e f f i c i e n t s of a s i n g l e c y l i n d e r are shown f o r two d i f f e r e n t element s i z e s . The r e s u l t with a l a r g e r - 5.5 -40 element s i z e e x h i b i t s o s c i l l a t i o n s , while the r e s u l t s with a smaller element s i z e does not. A p o s s i b l e explanation f o r t h i s i s that as the element s i z e becomes l a r g e r and l a r g e r , the constant p o t e n t i a l value assumption can no longer represent the actual p o t e n t i a l value on the surface of the body. This i s e s p e c i a l l y true f o r an object with complicated geometry. Therefore, in order to suppress t h i s o s c i l l a t i n g behavior, a f i n e r element s i z e must be used. V.3 Hemisphere In the t h i r d example, the heave and surge hydrodynamic c o e f f i c i e n t s of a hemisphere are computed. As in our f i r s t example, general notations such as R and D are used to represent the radius of the hemisphere and the water depth r e s p e c t i v e l y . In t h i s example, the published data of Garrison [1974] by Surface S i n g u l a r i t y Method (S.S.M.) are used f o r comparison. The heave hydrodynamic c o e f f i c i e n t s of a hemisphere with D/R equal to 2 2.0 are p l o t t e d against the non-dimensional frequency (co R/g) in F i g . 7. I t can be seen that the added mass computed by B.E.M., though o s c i l l a t i n g about some value, has in general a magnitude c l o s e to the r e s u l t by S.S.M.'s value in general. For the surge case ( F i g . 8 ) , the hydrodynamic c o e f f i c i e n t s obtained by the two methods e x h i b i t e d a d i f f e r e n c e of about 10%. The r e s u l t s by B.E.M. are lower than the r e s u l t s by S.S.M. V.4 Axisymmetric F l o a t i n g Rig The l a s t example studied i s an axisymmetric f l o a t i n g platform with a c i r c u l a r d r i l l i n g well at i t s centre ( F i g . F ) . The hydrodynamic c o e f f i c i e n t s as well as the e x c i t i n g forces are computed and compared to the unpublished r e s u l t s of Bai who used a F i n i t e Element Method (F.E.M.) in 1981. The r e s u l t s are presented in real value instead of non-dimensional 3 value. The hydrodynamic c o e f f i c i e n t s are given in un i t s of volume ( f t ) and the e x c i t i n g forces i n pounds per u n i t wave amplitude. The heave hydrodynamic c o e f f i c i e n t s are p l o t t e d against OJ in F i g . 13a. The r e s u l t s by B.E.M. e x h i b i t an o s c i l l a t i n g behavior while the r e s u l t s by F.E.M. seem to e x h i b i t an " i r r e g u l a r frequency" at co = 1.0. The B.E.M. r e s u l t s are lower than those by F.E.M. i n the low frequency range but higher in the high frequency range. Since both r e s u l t s show some numerical e r r o r , a general remark about the heave hydrodynamic c o e f f i c i e n t s i s hard to make. The surge hydrodynamic c o e f f i c i e n t s are shown in F i g . 13b. The r e s u l t s by B.E.M. are lower in value throughout the frequency range. The d i f f e r e n c e between the two methods i s about 10% f o r the added mass and l e s s than 8% f o r the damping c o e f f i c i e n t . The p i t c h hydrodynamic c o e f f i c i e n t s shown in F i g . 13c are in much bette r agreement. The r e s u l t s by B.E.M. are lower than those by F.E.M. However, the maximum d i f f e r e n c e i s l e s s than 3%. The heave e x c i t i n g force presented in F i g . 14a shows some agreement between the two methods. The r e s u l t by F.E.M. seems to show an " i r r e g u l a r frequency" at w c l o s e to one. The r e s u l t by B.E.M. i s lower - 5.7 - 42 than that of F.E.M. by 4% in the low frequency range but the two approach each other at high frequency. The p r e d i c t i o n of the phase angle between the heave e x c i t i n g force and the incoming waves using two methods i s shown in F i g . 14b. One can see that the r e s u l t s are extremely c l o s e to each other. The surge e x c i t i n g force and phase angle between e x c i t i n g force and waves are shown in F i g . 15a and F i g . 15b. F i g . 15a shows that the surge e x c i t i n g force by B.E.M. i s lower by 10% at the peak value but the phase angles p r e d i c t e d by the two methods are almost i d e n t i c a l . The p i t c h e x c i t i n g moment and i t s phase angle are c a l c u l a t e d and shown in F i g . 16a and F i g . 16b. In F i g . 16a, the p i t c h i n g moment by B.E.M. i s observed to be 4% lower than the values c a l c u l a t e d by F.E.M. The phase angles p r e d i c t e d by the two methods show an extremely good agreement. V.5 Discussions and Conclusion Following the r e s u l t s of the examples presented in the previous s e c t i o n s , a few remarks can be made. 1. The magnitude of the hydrodynamic c o e f f i c i e n t s , a., and b.. i s 1 J 1 J frequency dependent. 2. The r e s u l t s computed by B.E.M. are us u a l l y lower in magnitude value than the r e s u l t s by other methods. 3. With a constant p o t e n t i a l assumption, the r e s u l t s computed by B.E.M. e x h i b i t an o s c i l l a t i n g behavior which i s be l i e v e d to be as s o c i a t e d with the s i z e of surface elements. T h i s o s c i l l a t i n g behavior i s l e s s s i g n i f i c a n t as the element s i z e i s decreased. ( F i g . 12) - 5.8 - 43 4. The o s c i l l a t i n g behavior of B.E.M.'s r e s u l t s can be suppressed by decreasing the s i z e of the surface elements. ( F i g . 12) 5. I r r e s p e c t i v e of the o s c i l l a t i n g behavior of the r e s u l t s by B.E.M., the method gives good p r e d i c t i o n s f o r hydrodynamic c o e f f i c i e n t s as well as e x c i t i n g f o r c e s . 6. The pr e d i c t e d phase angles between the e x c i t i n g forces and waves are extremely close to those p r e d i c t e d by F.E.M. In general, i t i s hard to state that any s o l u t i o n method i s superior to the others, f o r every s o l u t i o n method may have i t s own advantages and disadvantages. However, the B.E.M. i s a ge n e r a l l y accepted method f o r so l v i n g forced o s c i l l a t i o n problems and p r e d i c t i n g e x c i t i n g f o r c e s . In t h i s t h e s i s , though the d i s c u s s i o n i s r e s t r i c t e d to axisymmetric bodies, the general i n t e g r a l equation, (28), can be ap p l i e d to f l o a t i n g objects of a r b i t r a r y shape. The cost of the computation i s u s u a l l y lower than f o r other s o l u t i o n methods. A problem of B.E.M. presented in t h i s t h e s i s i s the i r r e g u l a r o s c i l l a t i n g behavior of hydrodynamic c o e f f i c i e n t s e x h i b i t e d in the r e s u l t s under the constant p o t e n t i a l value assumption. In f a c t , as i n d i c a t e d before, t h i s o s c i l l a t i n g behavior can be decreased by decreasing the s i z e of the surface elements bounding the control volume. However, f o r some f l o a t i n g objects with complicated geometries are complicated such as the f l o a t i n g r i g studied ( F i g . F ) , i t i s extremely c o s t l y to decrease the element s i z e even by 50%. Therefore, i n order to avoid the o s c i l l a t i n g behavior of the r e s u l t s , as well as to achieve a more p r e c i s e r e s u l t , the po t e n t i a l value should be assumed varying l i n e a r l y across an element i n s t e a d of taking i t as a constant. In other words, a higher l e v e l s o l u t i o n technique ( l i n e a r elements) seems to be p r e f e r a b l e . - F . l - 44 LIST OF FIGURES Figure Page A. Coordinate System F.2 4-5 B. Axisymmetry of Geometry F.3 4 6 C. A S i n g l e C y l i n d e r F.4 47 D. A Composite C y l i n d e r F.5 48 E. A Hemisphere F.6 49 F. A D r i l l i n g Platform F.7 So 1. A 2 2 and B 2 2 of S i n g l e C y l i n d e r F.8 s i 2. A-JI and B ^ of Sin g l e C y l i n d e r F.9 52 3. A g 6 and B g 6 of Single C y l i n d e r F.10 S3 4. A 2 2 and B 2 2 of Composite C y l i n d e r F . l l 54 5. A ^ and B ^ of Composite C y l i n d e r F.l2 55 6. Ag 6 and Bg g of Composite C y l i n d e r F.13 56 7. A 2 2 and B 2 2 of Hemisphere F.l4 57 8. A ^ and B^ of Hemisphere F.l5 SB 9. f 2 of Single C y l i n d e r F.16 SB 10. f| of S i n g l e C y l i n d e r F.17 Bo 11. fg of S i n g l e C y l i n d e r F.18 61 12. A 2 2 and B 2 2 of Single C y l i n d e r F.l9 62. 13a. A 2 2 and B 2 2 of D r i l l i n g Platform F.20 €3 13b. A-JI and B ^ of D r i l l i n g Platform F.21 6* 13c. Agg and B gg of D r i l l i n g Platform F.22 65" 14a. F 2 of D r i l l i n g Platform F.23 66 14b. Phase D i f f e r e n c e between F 2 and Wave F.24 67 15a. F 1 of D r i l l i n g Platform F.25 6g 15b. Phase D i f f e r e n c e between F-j and Wave F.26 69 16a. F 6 of D r i l l i n g Platform F.27 7o 16b. Phase D i f f e r e n c e between F g and Wave F.28 11 - F.2 -- F.3 -I - F.4 - 47 FIG D A . COMPOSITE CYLINDER 00 - F . 6 -49 FIG F A DRILLING PLATFORM o 1 . A 2 2 & of 7 S i n g l e C y l i n d e r ^ 1. 60^_ © 0 1.40J. 0 0 1.201 TJ ^ 1.00L "TJ C 0- 80}. 0 0 0 ^ 0. 40|. a. E U 0. 201 0. 00L 0. 00 _L_J I L I J I L J I I L 0. 50 1. 00 1. 50 w * w * R / g L E G E N D o A22 b y B . E . M-*• A22 b y K n i t i e + B22 b y B . E . M . x B22 b y K n i t i e J I L J I I L J I I L 2. 00 2. 50 3. 00 F i g 2 . A l l . S B l l o f S i n g l e C y l i n d e r - C T / R = 0 , 5 , D / R = 2 , 0 ) 0. 80 L E G E N D © A l l b y B. E . M . * A l l b y M . T . + B l l b y B. E . M . x B l l b y M . T . 0. 00 0. 00 1.00 2. 00 4. 00 5. 00 6. 00 w*w*R/g i I F i g 3 , A 6 6 8, B66 o f S i n g l e C y l i n d e r - C T / R = l . 0 , D / R = 2 , 0 ) 0. 20^_ • 0 0 0. 17 0 j? 0. 15| 0 TJ 0 0. 12L_ "0 C 0 . 101 0 ^ 0.08 (D 0 0 P 0.051 •r>( £ 0 ~0 0.03! 0. 00! 0. 00 L E G E N D A66 b y B . E . M . * A66 b y M . T . + B66 b y B . E . M. x B66 b y M . T . 1.00 2. 00 3. 00 w*w*R/cj 4. 00 -©-5. 00 -€5 6700 F i g 4 , A 2 2 8c B 2 2 o f C o m p o s i t e C y l i n d e r - 1 0. 80r ® 0 0- 70i 0 CO 0 0.60J TJ 0 TJ TJ 0 0. 50 ~C 0. 40, 0 0. 00 1. 00 2. 00 w*w/g 3. 00 Cl. / f t . ) L E G E N D © A22 b y B. E . M . * A22 b y M . T . + B22 b y B . E . M . x B22 b y M . T . F i g 5 . A l l & B l l o f C o m p o s i t e C y l i n d e n 1 1.20, 0. 00L 0. 00 L E G E N D © A l l b y B . E . M . * A l l b y M . T . + B l l b y B . E . M . x B l l b y M . T . 1.00 2. 00 w * w / C J r * H - * i x1 * i i i >k i i i i ik 3. 00 4. 00 5. 00 6. 00 < 1 . . / f t . ) F i g 6 . A 6 6 & B 6 6 o f C o m p o s i t e C y l i n d e r - 1 . L E G E N D w*w/g CI. / f t . ) F 7. A22 & o f h e m i s p h e r e ( D /R = 2 . 0 ) 0. 80 0 0. 70^_ 0 r-(0 0 0. 60. 0 0.50 C 0. 40 0.301— 0 0 (P 0. 20L CL o U 13 0. 1EL-0. 001 I I. 0. 00 J — I J 0. 50 L „ i i i J L 1.00 1.50 J I L E G E N D A 2 2 b y B . E . M . * A 2 2 b y G A R R I S O N + B 2 2 b y B . E . M . x B 2 2 b y G A R R I S O t U H 2. 00 2. 50 . J L 3. 00 w * w * R / g F i 3 . _ a . A l l 8, _B1 1_ o_f_ h e m i s p h e r e C _D/R=2. 0 ) 0. 80^ _ L E G E N D A l l A l l B l l B l l b y B . E . M . b y G A R R I S O N b y B . E . M . b y G A R R I S O N 0. 00 0. 00 0. 50 —I- t—1—I I X. 1. 50 w * w * R / g j L—i~_ i 2. 00 I L 2. 50 A I I L 3. 00 I 1 f 2 o f S i n q l e C y l i n d e r - C T / R = 0 . 5„ D / R = 1 . 5 ) 2. 00 1. 75! 1. 50 -P 1. 25 <+-\ 1. 00! < OJ 0. 75] 0. 50 0. 25 L E G E N D © f 2 b y B. E . M. * f 2 f r o m B 2 2 0. 001 J i I I I 0. 00 0. 50 J I L L _ 1 J I 1- 1. J L__ 1 — 1 1. 00 1. 50 2. 00 — J I L„..i I I 2. 50 3. 00 ! i F i q 10. f l o f S i n q l e C y l i n d e r - C T / R = 0 . 5 . D / R = 2 . 0 ) 1. 20, 1. 05j 0. 90 •P 0.75| E* < 0. 601 0. 45L_ 0. 30L_ 0. 15 0. 001 I 1 I I J L L L 0.00 1.00 _ 1 _ L 2. 00 - I I L 1. L E G E N D co f 1 b y B . E . M. * f l f r o m B l l J I l J J i L 3. 00 W * W * R /C | 4 . 00 5. 00 6. 00 F i g 1 1 . f 6 o f S i n g l e C / y l i n d » 0. 30r 0. 26L 0. 221 •P 0. 19U *+• \ C 0. lSf < (0 0 . u } _ 0. 08L. 0. 04; 0. 00L 0. 00 - i J I I .1 L L 1. 00 J J I I L 1 X 2. 00 3. 00 w * w * " R / g C T / R = l . 0 , D / R = 2 . 0 ) LEGEND © f 6 by B. E . M. * f 6 f r o m B 6 6 ! 1 J I L 1 I X 4. 00 5. 00 6. 00 i F i g 12 , A 2 2 & B22 o f S i n g l e C y l i n d e r - C T / R = 0 . 5 . D / R = l . 5 ) 1. 60 <D 0 1. 40 0 0 1.20 "0 <D ^ 1. 00 0 £ 0.80 0 0.60 <D 0 0 ? 0. 40 •H CL E 0 "D 0.20 0. 00 0. 00 co L E G E N D A 2 2 * A 2 2 + B 2 2 x B 2 2 d L = 0 . 2 d L = 0 . 4 d L = 0 . 2 d L = 0 . 4 •—e-J I L 0. 50 I I I L J L J I L J I L 1.00 1.50 w*"w*-R/g 2. 00 2. 50 3. 00 0. 24 r 0. 2 l L_ 0. 18L 0. 15L_ 0. 12f__ (\l CM OQ 0. 0 9 C <\1 ^ 0 . 0 6 L 0. 03L_ + III 0. 00L F i a 13 . a A 2 2 & B 2 2 o f D r i l l i n q P l a t f o r m 0. 00 L E G E N D © A 2 2 b y B . E . M . ^ A 2 2 b y F . E . M . + B 2 2 b y B . E . M . x B 2 2 b y F . E . M . J I L 1 0.25 J I I L J I L 0. 50 0. 75 w CI. / e e c ) J — I 1 — L 1 1.00 J I I L 1 1. 25 J I I L 1.50 i O 0 . 2 4 _ 0. 2lL_ ro * 0 . 18 » -p (D + LLJ 0 . 15 ^ 0 . 12 i — i «—i CD 0 . 09 cx3 «—• < 0 . 06 0 . 03j F i q 13 . b A l l & B l l o f D r i l l i n g P l a t f o r m L E G E N D © A l l b y B . E . M . ^ A l l b y F . E . M . + B l l b y B . E . M . x B l l b y F . E . M . 0 . 00L_J__i__l I I I I — I — I — I — I — l — ' — L I I 1 1 1 1 I I I 1 1 1 1 1 L 0. 00 0. 2 5 0. 50 0 . 75 w ( 1 - / s e c ) 1. 00 1. 25 1. 50 F i g 1 3 . c A 6 6 & B 6 6 o f D r i l l i n g P l a t f o r m 0. 56 r 0. 49. in * 0 . 4 4 • +> tr-IS 0. 35U r - l + LL) 0. 28 _ (0 (D m 0-21 (0 (0 0. 14. 0. 07. 0. 00. J L 0. 00 L E G E N D © A 6 6 b y B . E . M . ^ A 6 6 b y F . E . M . + B 6 6 b y B . E . M . x B 6 6 b y F . E . M . 1 J L J I I L J I L J I L J I I L 0. 25 0. 50 0. 75 w Cl. /sec) 1. 00 1. 25 1. 50 1 F i g 14 , g F2 o f D r i l l i n g P l a t f o r m 0. 24r L E G E N D F2 b y B. E . M. F2 b y F . E . M. 0. 21 0. 18L_ 0. 15 + LU . 0. 12L ^ 0. 09i \ C\J LL 0. 06L 0. 03L 0. 00L J I L J I L J I L J I L J I 1 L J I I L 0. 00 0. 25 0. 50 w 0. 75 C1. /sec) 1. 00 1. 25 1. 50 1 4 . b P h a s e D i f f © n o n e © b e t w e e n F 2 a n d Wave L E G E N D © b y B . E . M . ^ b y F . E . M . 0. 180, 0. 135 Q CO + Ul 0. 045 0. 000[_ •H D-0. 045| <D (0 0 Q_-0- 0901 -0. 135L_ -0. 1801 J I L 0. 000 J I L 0. 250 J I I L 1 J I L 0. 500 w 0. 750 CI. /sec) 1. 000 1 I L J I I L 1. 250 1. 500 F i g 15 . a F l o f D r i 1 1 i n g P l a t f o r m L E G E N D © F l b y B . E . M . * F l b y F . E . M . w C1. / s o c ) OS 0 0 15. b P h a s e D i f f e r e n c e b e t w e e n F l a n d Wave 0. 180_ 0. 135 0*0. 090 Q CO LU 0. 045 0. 000| ^0. 045U <D (0 0 J : Qr-0. 090! -0. 135 -0. 1801 0. 000 L E G E N D © b y B . E . M. b y F . E . M. J I I L J L J I I L J I L J I 1 L J I I L 0. 250 0. 500 w 0. 750 (1. /sec) 1. 000 1. 250 1. 500 F i g 16 , q F 6 o f D r i l l i n g P l a t f o r m , - I L E G E N D w C I . / e e c ) F i q 16 . b P h a s e D i f f e r e n c e b e t w e e n F6 a n d Wave 0. 180_ 0. 135 ^0. 090| a co LU 0. 0451 0. 000 t <+• •ri C5-0. 045|_ (D 0 Ct-0. 0901 •0. 135 -0. 1801 I 1—L 0. 000 L E G E N D co b y B . E . M. b y F . E . M. i i 00 J I L J I L L J I I L . 1 J I L J L 0. 250 0. 500 w 0. 750 CI. /sec) 1. 000 1. 250 1. 500 LIST OF SYMBOLS wave amp!i tude constant water depth complete E l l i p t i c Function of f i r s t and second kind hydrodynamic pressure force e x c i t i n g force time independent part of F'. J Green's Function Bessel Function observation point co n t r o l point length parameter f o r non-dimensioning radius of control volume r a d i a l measurement from z axis surface area area of r a d i a t i o n boundary area of free surface area of f l o a t i n g body area of r i g i d wall t o t a l surface area draught of f l o a t i n g body v e l o c i t y of f l u i d p a r t i c l e added mass in i d i r e c t i o n induced by motion in j d i r e c t i o n damping c o e f f i c i e n t r e s t o r i n g force c o e f f i c i e n t g r a v i t a t i o n a l constant wave-number integer actual mass of f l o a t i n g body normal un i t vector on the control surface pressure or hydrodynamic pressure d i s t a n c e between P and Q p o s i t i o n measured from o r i g i n 73 - N.2 -t time u.j ve loc i ty component of f loa t ing body Vg group veloci ty v phase ve loc i ty p R ,y radial and ver t i ca l measurement of point p n D ,n radial and y component of unit normal vector K y a,b constant u,v,w ve loc i ty component of f l u i d par t i c le s X,Y,Z coordinate axes $ tota l ve loc i ty potential <|>, <t>', ^ time independent part of $ 4j incoming wave potential $D refracted and ref lected potential $P potential value induced by motion of f loat ing body e , n constant m a) angul ar frequency p f l u i d density e small parameter for l i n e a r i z a t i o n e angular measurement n wave elevation v f l u i d volume displaced by f loa t ing object Y angular ve loc i ty of f loat ing object. X wave-length APPENDIX A L i n e a r i z a t i o n of the Kinematic Boundary Cond i t i o n From (16.d), which i s the free surface boundary c o n d i t i o n , rj - 4 = 0 . ... .(16.d) »t »y P h y s i c a l l y , i t means t h i s equation i s to be s a t i s f i e d at the p o s i t i o n of y = TI. However, by (15) T) = e n ( 1 ) + e 2 n ( 2 ) + . . . . . . . .(15) (1) 2 (2) $ can be writt e n as $(x , e T ] •+ E n + z , t ) . I f $ i s expanded about y = 0 as: (1) 2 2 $ = <s>(x,o,z,t) + st) $ (x,o,z,t) + e n $ u v / ( x , o , z , t ) + •< »y >yy Then, to the lowest power i n e, $ = $ (x,o,z,t) + e n ( 1 ) $ v v ( x , o , z , t ) + ... Therefore, $ i s evaluated at the mean p o s i t i o n , y = 0, »y APPENDIX B Matrix Representation of the Integral Equation From (26), 2ir<|.(P) + /s *(Q) - | r (7) M e d * = J s |* (Q)-Jr RdedA • ( 2 6 ) Now, suppose the control surface i s d i v i d e d into n c i r c u l a r r i n g elements, n equations can be written f o r each point P located on a d i f f e r e n t element. For each of these equations, there are n terms on the r i g h t hand side of the equation as well as n terms on the l e f t hand s i d e . When P and Q are on d i f f e r e n t elements, the term on the r i g h t hand side i s ^ ¥ Q ) I n (F> R d e d * = A i j * j ' When P and Q are on the same element, 2wfj + /* V,(Q) ^  (1) Rde d* = A j j*. . On the l e f t hand s i d e , n 2 TT 9<|>. , ^Q Hn~ 7 R d 9 d * = B i • Therefore, the equations can be represented by (27) [A..] [•.] = [B.] i = 1, ...n, 1 J J 1 j = 1, ..-n. (27) In some cases, not a l l <|>. are known on each element, then the term J »n j ,n i s expressed as a fun c t i o n of 4>. and moved to the r i g h t hand side (as shown in Appendix C). I f <j>. cannot be expressed as a fun c t i o n of A., J ,n j some reasonable constant are assumed to represent , and exchange J p o s i t i o n with <{>. on the l e f t hand s i d e . I t must be noted that, the j , n - A.2 -APPENDIX B Matrix Representation of the Integral Equation 7 6 From (26), 2*<t>(P) + JS *(Q) IJJ- (1) RdecU = JS |4 (Q)l Rdedi . (26) t t Now, suppose the control surface i s d i v i d e d i n t o n c i r c u l a r r i n g elements, n equations can be written f o r each point P located on a d i f f e r e n t element. For each of these equations, there are n terms on the r i g h t hand side of the equation as well as n terms on the l e f t hand s i d e . When P and Q are on d i f f e r e n t elements, the term on the r i g h t hand side i s C • j W In ( 7 } R d e d * " A i j * j ' When P and Q are on the same element, 2«*S + C^ ( Q ) In (?) R d 9 d * = A j j * j ' On the l e f t hand s i d e , n 2n a*. , s / - r - i 1 Rdedi = B. . j=l 0 5 n r 1 Therefore, the equations can be represented by (27) CA..] [*.] = [B.] i = l,...n, 1 J J 1 j = l , . - n . (27) In some cases, not a l l (t>. are known on each element, then the term *. i s expressed as a fun c t i o n of A. and moved to the r i g h t hand side (as J shown i n Appendix C ) . I f <)>. cannot be expressed as a functi o n of A., J , n j some reasonable constant are assumed to represent A.., and exchange p o s i t i o n with *. on the l e f t hand s i d e . I t must be noted that, the J»n - A.3 - 77 matrix equation must have a column vector, which e n t r i e s are known, on the r i g h t hand s i d e . The only unknown i s the column vector on the l e f t hand s i d e , of which i t s e n t r i e s may be A. or <(>. . - A.4 -APPENDIX C Combining the Boundary Conditions with the Integ r a l Equation 78 The i n t e g r a l equation i s 2Tc<t»(P) + 4>(Q) IJJ- (1) RdedA = J, ^ 1 On the f i x e d boundary (at the bottom) S , S T an ( ( » r R d 9 ( U * (26) (21) On the moving boundary, S^, = V. (22 ) On the free surface, S f ! OJ 9 (23) and on the r a d i a t i o n control surface, S^, '.R + Ik) (24) Now, with S o + S b + S r + S f equation (26) can be decomposed i n t o b ,1 9 ,1 2rc<t> + J s 4> (7) RdedA + J s <|) (^) RdedA 0 b + J c • h <i> RdedR + L 4) £ r (±) Rdedy a , 1 . 'S f T an l r an r 's Sf 7 R d e d * + /• ^ 1 RdedA S. an r W S f ^ RdedR • ^ | i I Rdedy . - A.5 - 79 By s u b s t i t u t i n g Equations (21), (22), (23), and (24) i n t o the r i g h t hand si d e of the above equation and moving the unknown expressions to the l e f t hand s i d e , the f o l l o w i n g equation i s obtained. 2*» + 's0 Ir (?> *R d 9 d* + kbh <?> •Rded* + V i ' ? ' " (4+ 1k)^] *Rd9dy = Jo i (v • n) R dedA . (28) b b r Bo - A.6 -APPENDIX D S i m p l i f i c a t i o n of Equation (28) i n t o (42) Equation (28) i s given i n Appendix C as: 2*+2 * >s0 h <?> *2 R D E D * * kb h <?> ^Rd9d^ 2 + /Sf[lr7 (F} " F f" ] «feRdedR = 'sb F {* ' " ) R d e c U ' ( 2 8 ) Let R be the radius of the body at the free surface, y. be the point on the keel of the body, and the normal u n i t vector, n, be in the f o l l o w i n g form: n = ( n R cos e, n , - n R s i n e ) , (D.l) where n D and n be the R and y component of n. Then, with r and vr as K y s i 1 given in (2) and (3), -^ r(^ r) i s equal to — ^ V r * " which can be represented by the f o l l o w i n g expression: a 1 (Rcose - R p ) n R c o s e + ( y - y p ) n y + Rsine n R s i n e ^ ~ " { [(R cose - R p ) 2 + ( y - y p ) Z + (-R s i n e ) 2 ] 3 / 2 1 Rn R + (y-y )n - R n R c o s e = - { , o o o o / o } • (D.2) ( R 2 + R 2 + ( y - y D ) 2 - 2RR c o s e ) 3 / 2 r r r By l e t t i n g a = R 2 + R 2 + (y - y ) 2 (D.3) P P and - A . 7 - Bl b = 2 RRp , ( D . 4 ) (D.2) can be rewri t t e n as: a i R nR + ( y _ y n ) n v " R n n R C O s e IJJ (1) = - { — P Y } • ( D . 5 ) 9 n r ( a - b cos Q ) 6 / d ( i ) By s u b s t i t u t i n g ( D . 5 ) into the second term of (28), /s0 In (7} h R d6 d* RR 2TT R 2 n D + ( y - y n ) R n w - R R n B c o s e = / / " { — P y 3/2 P J *2 d 9 d* 0 0 (a - b cos %V,C L R r ? 217 rift = J - (R n R + (y-y )Rn \ J ^ ™ d£ 0 K P Y ^ 0 ( a - b c o s e r ' * D R 2u , ( a - b cose - a ) n R + / / - I j T P — A, de dA 0 0 * ( a - b cose) 1 R R = / - 2(R 2n R + (y-y )Rn ) <fr2 f — — - J J ^ de dz 0 K P Y ^ 0 (a - b c o s e r ' * R D 1 a 0 K L 0 (a - b c o s e ) 1 ^ (a - b c o s e r ' * From page 154 and 156 of Gradshteyn and Ryzhik [ 1 9 6 5 ] , one has I d x = 2 F (J.6) ( 0 . 6 ) 0 / a - b cosx / a + b and / D X = 2 E 6 ) , ( D . 7 ) 0 ' (a - b c o s x ) 3 ( a - b ) / a + b 82 - A.8 -where 6 = / 2b • a+b Therefore, the integration can be further s impl i f ied into RR 9 2E <|u RR 2n p F $ / - 2 ( R S R + ( y - y j R n ) d£ + / - ^ dA 0 (a-b) / a + b 0 / a + b RR 2anD A„ E + / R 2 dA , 0 / a + b (a - b) that i s , a i RR - 4 ( R 2 n D + ( y - y j R n ) E / s ^ ( l ) , 2 R d e d A = J [ o ° " ' ^ o ( a - b ) r m 2nD c (D.8) ( i i ) The integration of the th i rd term of (28) is s imi lar to that of the second term, therefore, from (D.8) , 3 1, y = 0 - 4 ( R 2 n R + ( y - y j R n l E ' s k I n ^ *2 R d 9 d* " / t " y=y ( a - b ) / a + b 2nr / a + b i a D ; * ( i i i ) For the fourth term, - A.9 - 83 /s t f e ^ - F ^ + 1 ' k ) ] ^ R d e d y r K 0 2it R 2 n R + R ( y - y p ) n y - RR pn Rcos 9 -h 0 (R^ + R^ + ( y - y D ) - 2RR cos e y ' c r r r -R/2RR + ikR (R + R 2 + ( y - y p ) 2 - 2RR pcos 9 ) 1 / Z 2 but on S . n = 0, n D = 1 and R = R D. Then, the above expression v y K K becomes 2 0 2TC R - RR p cos 9 / / [ - —* * * ~TT7 -h 0 (IT + IT + ( y - y n r - 2RRcos ^ p p p ' j - i k R + — 5 Q 5 "T79 ] <t>o d9 dy . (R + R 2 ( y - y p ) 2 - 2RR pcos e ) 1 / z 2 Using (D.3) and (D.4), the above expression becomes Zn ^ 1 R 2 + R 2 + ( y - y p ) 2 - 2RR pcos 9 - R 2 - ( y - y p ) 2 + R 2 -h 0 ? ( a - b cos 9 ) 3 / 2 i - i k R •jjZ ] *2  d e dy ( a - b cos 9) -h 0 * ( a - b cos 9 ) ' ^ ( a - b cps e r ' * i - 1 k R + ] * d e dy (a - b cos 9 ) ' ^ L - A.10 -o i t R 2 - R 2 - (y-y J 2 2 i k R = / / [ E _ __ ] * de dy -h 0 ( a - b cos er u ( a - b cos e ) 1 ^ c 0 (R 2 - R 2 - ( y - y J 2 ) E 4 i k R F • / t " 2 P , P ' ^ ] *2 dy . -h (a - b) / a + b / a + b c Therefore, r K 0 2 ( R 2 - R 2 - ( y - y J 2 ) E 4 1 k R F = / [ E B- _ _ _ _ _ ] . d y > ( D > 1 0 ) -h (a - b) / a + b / a + b * ( i v ) I n t e g r a t i n g the f i f t h term, RR 2* Rn R + ( y - y p ) n y - R p n R cose J- J |_ - - o 5 5 Q/o K c o ( R * + R „ + (y-yJ - 2RR cos e ) J / i 2 1 - — 5 * 5 T-75- ] <tv>R de dR. 9 (R + Rp + (y-y r - 2RR p cos e ) l / 2 1 On S^, n R i s equal to zero and n^ i s equal to one. A l s o , by using (D.3) and (D.4), the above expression can be r e w r i t t e n as: RR ii (y-y.) R 2 R / 2/ [ E " R ] , de dR 0 0 (a - b cos %r ,c g(a - b cos e) l c c - A.11 -R (y-yJ R E 2 R F / 4 ( . P M R F ) 4, dR . 0 / a + b ( a - b ) g / a + b That i s , / S f t Sir & " F H R d e d R RR (y-yJ R E 2 R F = Jo 4 ( - P M R F ) dR . ( D . l l ) K c / a + b ( a - b ) g / a + b (v) The f i n a l term to be considered i s the only term on the r i g h t hand si d e . L -J- (v • n) R de dA 5 b r y=0 2TC R U „ n = 1 1 _ de dA y=y t o (R + Rp + (y-y p) - 2RR pcos e ) 1 ^ y=0 TC . = J 2 R u 2n / ^ m dA y=yt * y 0 (a - b cos e) x'c y=0 4 R u ?n F = / L ? dA . (D.12) y=yt / a + b Therefore, (28) can be rewri t t e n as: 2n4> + (D.8) + (D.9) + (D.10) + ( D . l l ) = (D.12) , and becomes - A.12 -86 2 n* 2 + y=0 4(R n + (y-y )Rn ) E J [ H 3  y=y + ( a + b ) / a + b 2n / a + b R C T a ^ b T " M ] *2 dx ;R R 4 ( R 2 n R + ( y - y p ) R n y ) E 0 ( a - b ) / a + b 2n R ( - n r r r r - F ) ] * 9 d i y r ° r 4 k R F 2 ( r 2 - Rp - t y - y 2 E . a . J L- , - v v — J *? dy y=-h / a + b (a - b) / a + b + ;R R r. 4 R ( ^ P ) E 4a)2RF lc / a + b ( a - b ) g / a + b) ] *2 dR 4 R F 7 y=y t / a + b u 2 n y d i . (42) 87 - A.13 -APPENDIX E S i m p l i f i c a t i o n of Equation (45) i n t o (46) Equation (45) i s given in Chapter 111.3 as: 2 7 l < t >l + ^S 0 fn & *} c o s 9 R d e d i + kh In & < h ' c o s 0 R d 0 ( U + / S f [ In & " F ] *i c o s 9 R d e d R + 's t In " C Ik) 1 3 *J cos 9 R d e d y r R J s Ujcos 9 nR 1 R d9 dA . (45) ( i ) The second term in (45) w i l l be considered f i r s t . RR 2TI ' ' * IrT & *i c o s e R d9 d i 0 o RR 2TC (n DR + n ( y - y ) - n D R „ cose)*; R cos 9 * = / / - R 2 y 2 P - P ] 3 / 2 - d9 dA 0 0 (R + R„ + (y-Yn' - 2 R R n C 0 S G ) By using (D . 3 ) and ( D . 4 ) , the expression i s r e w r i t t e n as: R 2 R 2TC (n DR + n , ( y - y J R - nDRR cos e ) * i cos 9 / / - - i y _ _ £ P 3 / 2 — J d e d i 0 0 (a - b cos 9)°'* RR 2u ( n D R 2 + n ( y - y n)R) cos 9 • ' I I ' — - y P M 0 0 (a - b cos 9)' 88 - A.14 -2 bn R cos 6 + ^ ] de d i 2(a-b cos 6) 6/d 1 RR TC ( n D R 2 + n (y-y )R) ( a - b cose - a) = J 2 J [ - —5 * 2 0 0 b (a - b cos e r ' * (a - b cos e - a) cos e -> , . . - n„ — T ^ — J A. de d i R 2(a - b cos e ) 3 / 2 1 . j \ [ ^ - ') (  0 0 D (a - b cos e ) 1 7 ^ a ) nR_ p i (a - b cos e - a) ( a - b cos e ) 3 / Z " 2 l F " ( a - b cos e ) 1 / Z , a (a - b cos e - a) -> -, .... + j A, de d i D (a - b cos e ) 3 / ^ 1 . ,\ [ - v y-vR ( — J — M 0 0 D (a - b cos e) W £ - ^ ) + R ( ( A . B C O S Q)V2 ( a - b cos er u 60 ) - X (-  1 (a - b cos e ) 1 / 2 " (a - b cos e ) 1 / 2 - - 0 7 0 ) ] <th de dA . ( a - b cos er /d 1 From p. 156 of Gradshteyn and Ryzhik [1965], one has / / a - b cos x dx = a / a + b E {j, 6 ) , ( E . l ) where - A.15 - 89 6 = / _2b_ • a+b With the help of (D.6), (D.7) and ( E . l ) , the above integral expression can be rewritten as: ;R R 2 { (n RR 2 + n y ( y - y p ) R) ^ 2 F _ 2 a E 0 b / a + b / a + b (a - b) + ^ ( 2 / T T T E - 2 a F ) / a + b n„a f y ;  / a + b / a + b (a - b) R r __1L_ 2 a E i i i J o RR / I 4 ( " / + n ( y - y j R ) (F - a E 0 b / a + b R y p ( a - b ) 2 + | n D ( / a + b E - 2 a F + a E )} d A. (E.2) D K / a + b / a + b (a - b) 1 ( i i ) The th i rd term in (45) is exactly the same as the second term except the l i m i t is d i f f e rent , therefore, J s (7) cos e R de d£ b I 0 [ — 4 = ( » / - n ( y - j , )R) (F - ) y=y t b / a + b ^ K v ' - A.16 -2 9 L ) ] 0>; d i . ( E / a + b ( a - b ) 1 ( i i i ) The fourth term i s s i m p l i f i e d as the f o l l o w i n g way: 5 A ^ u ) 2 >Sf I ?» ^  " ?F 1 *i c° s 9R d e d R R R TC ( n R R + n y ( y - y p ) - n R R P cose) R cose = / 2 / [ - - — 7 j 5 : • l / o o o (R + + (y-y J - 2RR cos s y ' c r r r 2 QJ R cos e -j ^ <j0 dR 9 ( R 2 + R 2 ( y - y n ) 2 - 2RR cos e ) 1 / 2 ^ but f o r Sf, n R = 0, n = 1. Thus, the expression becomes R R TC (y-^p^ R c o s 9 J 2 / [ - ——.5 x o T77 o o ( R + R P + ( y - y p ) - 2RR p cos e ) l / < : 2 a) R cos e • K i u b o -j i de dR 9 (R 2 + R 2 + ( y - y n ) 2 - 2RR cos e ) 1 / 2 ^ p -p- p By (D . 3 ) , (D.4) the above expression i s given as: RR TC ( y - y p ) R ( a - b cos e - a) J 2 / [ 0 7 5 0 0 b (a - b cos e ) ' 2 R(a - b cos e - a) + £ L ] d e d i 9 b(a - b cos e) ]/d 1 RR TC (y-y )R 1 = / 2 / [ rf-— ( fpr - o / o 0 0 D (a - b cos e) (a - b cos e r ^ a - A.17 -2 + ((a - b cos 9 ) 1 / 2 ^ )] <jH de dA . 9 b (a - b cos e ) 1 / 2 ^ With the help of (D.6), (D.7) and ( E . l ) , the expression can be re w r i t t e n as: RR ( y - y j R or ? a F / 2 [ P — ( 2 F 2 3 E 0 b / a + b / a + b (a - b) 2 + |_R ( 2 / T + T E - ) ] ^  dA 0 D 9 P 9 / a + b (y-y ) a E P ] *; dA . (E.4) / a + b ( a - b ) 1 ( i v ) For the f i f t h term of (45), 's C In ^ ' [ 2 T " + 1 k > 7 ] *i cos 9 R ^  dy x* R 0 2TC ( n R R + n y ( y - y p ) - n R R P cos e)R cos e -h o (R 2 + R p + (y-y p ) 2 - 2RR p cos e ) 3 / 2 2 R R ( R * + R „ + (y-yJ - 2RR cos e ) 1 ^ 1 With (D.3) and (D.4), n = 0 and n = 1 on S . the above expression y K r can be writte n as: - A.18 - 92 o TC ( R - R R cos e) cos e / 2 / [ - - — * -h 0 ( a - b cos e) 172 I "I -I 1 R COS e "1 | A rt A ' 2R7 1 k ) ~ — 1 / 2 J <h d e d y ( a - b cos e) 0 *r ( r 2 + R p + ( y - y p ) 2 - 2 R R P c o s e + R 2 - Rp - ( y - y p ) 2 ) c o s e / 2/ [ -h 0 2(a - b cos 0) 17? r -1 , . . •> R ( a - b cos e - a) n , . „ . ^ K R D (a - b cos e ) 1 7 ^ 1 0 it 2 r o (% r ( a - b cos 9 + 2R - a) cos e J 2 J L - ; 7372 -h 0 2(a - b cos B y " + ( ijT + i k ) £ ( ( a " b c o s e ) 1/2 ( a - b cos e ) y/2 ) ] ^ de dy r 9 f 1 1 r (a - b cos e - a) . (2R - a ) f a - b cos 9 - a) -h 0 2b(a - b cos e ) l / d 2b(a - b cos e ) J / + ( ^ - + 1k) £ ((a - b cos e) 1/2 ( a - b cos e ) T 7 2 ) ] * i d e d^ On S_, R = R R, and the above expression can be decomposed i n t o - A.19 J°2 j% [ + L (a - b cos e ) 1 / 2 5 -h 0 ^D 2b(a - b cos e) ' + (2R 2 - a) _ (2R 2 - a) a 2b(a - b cos e ) 1 / 2 2b(a - b cos e ) 3 / 2 + (- Y + 1kR) ((a - b cos e ) 1 / 2 95 ( a - b cos 9) 172 ) ] 4>J d9 dy - / ° 2 /" [ J ^ ^ - l ( 3 _ _ ^ ) -h 0 ^D ( a - b cos QVU (a - b cos % ) i U + i-M ((a - b cos e ) 1 / 2 a ^ )] *; de dy D ( a - b cos e ) s u 1 Using (D.6), (D.7) and ( E . l ) , the above expression becomes j° [ (2R 2 - a) 2F 2aE -h u / a + b /a + b (a - b) + ^ (2 / a + b E - 2 a F )] * i de dy D / a + b 1 j° r_ 2(2R 2 - a) ( p _ _ _ J E _ ] + _4ikR_ ( / - F T T . -h b / a + b / a - b aF / a + b ) ] <t>} dy . (E.5) A.20 (v) The only term on the r i g h t hand side of (45) i s y=0 2rt R cos 0 / / u,n R — * j-{ de dA . y=y t 0 (R + Rp + ( y - y p r - 2RR p cos e ) " * By using (D.3) and (D.4), the above expression becomes y=0 TX - u,n DR ( a - b cos e - a) / 2 / [ — L B ] de d A y=y t 0 b(a - b cos e) ' y=0 TI fTf ipR = / 2 / - — L [ / a - b cos e 3 ] de dA y=y t 0 0 (a - b cos e ) 1 ^ With (D.6) and ( E . l ) , the above equation can be rewr i t t e n as: y=0 2u.n_R , _ / . _ I L . [ 2 / T H b E - 2 a F ] dA y=y + / a + b y=0 4u.n R / ^ L [ / a + b E - a h 1 dA . (E.6) y=y t / a + b Therefore, equation (45) i s eq u i v a l e n t to the f o l l o w i n g equation: 2 ™ * i < P > + 7 t (" r R 2 + " y ( y - y P m » F Ta?ET E ' + -I n D (/ a + b E - 2 9 F + 3 E )] Ai dA D K / a + b / a + b ) ( a - b ) 1 9 4 - A.21 - 9S + / [ 4 ( n D R 2 + n ( y - y n ) R ) ( F a 0 b / a + b 'IT ' ' V J V n a - b E ) + | n R (/ a + b E ^ — F + a / a + b / a + b ( a - b ) + / F R t f - y a + b E + ( y- y F a 7 r ) - = = 9 / T T T (y - y n ) aE e ] d R / a + b (a - b) + /°[212BL^|1(F..^E) y=-h b / a + b u D ) + ( y-g- r-b E . — J — F ) ] <t>l dy D / a + b 1 y=0 4u.n RR J b y =y+ [ a F - E / a + b ] d i . / a + b (46) - A.22 -APPENDIX F S i m p l i f i c a t i o n of the Right Hand Side of (50) The s i m p l i f i c a t i o n of the l e f t hand side of (50) i s e x a c t l y s i m i l a r to (E.2), (E.3), (E.4) and (E.5) in Appendix E. The r i g h t hand si d e of (50) i s given as: J s (- V + nyR) Y cos e R 1 de di y=0 2TC m cos e y=y t 0 y (R + Rp + (y-y p) - 2RR pcos e)1^ y=0 / 2(-n Ry + n R) YR y=y t TX r cos e .„ . x j — 5 9 5 j-pr de di . o (R + R^ + (y - y r - 2RR COS e)1^ r r r By (D.3) and (D.4), the above expression becomes y = 0 . 7 1 r n c ft / 2(- n Ry + n R) YR / iy» de d i ' R -*- " V K J Y K J ; 7T72y=y t K y 0 (a - b cos e) 1 y7° o r ^ „i yR r 7 1 (a - b cos e - a) , J " 2(- npy + n R) \ j - de di y=y. ^ y . 0 (a - b cos e) 1 y=0 « R TT / - 2(- n R y + n i ) \ j [/ a - b cos e y=y t y 0 5 m ] de di ( a - b cos e)1^ A.23 - ^ F i n a l l y , w i th ( E . l ) and (D.6), the i n t e g r a l has the form y=0 / - 4(- n y + n R ) l£ [ /~a~+~¥ E - a F ] d i . ( F . l ) y=y* R y b / T + T - A.24 -APPENDIX G The S e r i e s Representation of 4^  n From (62), 4>j i s given as: i M cosh k (y + h) ( k R ) C Q s mQ { 6 2 ) co cosh kh r. rm m ' m=Q and with >I,n = V * i ' " = ( W' ay» R 5 0 H ' ( nR» ny» n 9 ) ' (--D However, f o r axisymmetric body, n i s always equal to zero. Therefore, • i . n = •i.R n R + • i . y n y ' ( G' 2 ) By taking the d e r i v a t i v e of 4>j with respect to R and y, one can get = _ i M cos;1 k(y+h) * ,( , I,R 00 cosh kh m_Q pm m and _ . i _ k sinh k(y+h) " J ( k R ) c o s m 0 . I,y OJ cosh kh __Q Km m - A.25 - 99 Therefore, and • = _ i M k z B cos me [J'(kR)n Dcosh k(y+h) I,n (j cosh kh n rm L m R m=0 + J m ( k R ) n y sinh k(y + h) ] , (G.3) D,n to cosh kh m_Q L m R + J m ( k R ) n y s i n h k(y + h) ] . (63) - A.26 -APPENDIX H S i m p l i f i c a t i o n of E x c i t i n g Force and Moment /oo ( i ) From (71) one has y=y t 0 M=0 i <t>D_ w ] cos me n yR de dA } e 1 w t . (71) However, only the term 'cos me' in (71) i s a function of e. A l l other v a r i a b l e s i n s i d e the i n t e g r a l are functions of R and y only. Therefore, by taking i n t e g r a t i o n with respect to e, one can get F2 = t P [" Ag C ° S h C 0 ^ y k ; h ) P0 J 0 ( k R ) - i *QQW]2nn R dA y=y+ J + y / ° s P [- Ag c o s h k l \ t h ) p J (kR) v_ v m = i c o s n k n m mv •'t Hn-W ] s i n me Dm" J m 2 U n R dA } e 1 u ) t . (H.l) 0 y I t i s obvious from (H.l) that f o r m greater than zero, the i n t e g r a l i s equal to zero. Therefore, (71) can be s i m p l i f i e d as: F' = {V - 2* PRn y[Ag ^ ^ l ^ h ) P 0 J 0 ( k R ) + ^DcH { 7 2 ) y=y t - A.27 ( i i ) From (75), the surge e x c i t i n g f o r c e i s given as: j y = f ° r 2 % " r fln cosh k(y + h) a , 0, F 1 = * J J n s P L- Ag c o s h kh pm Jm { k R ) y=y t 0 m=0 2u . t - i * D m to ] n RR / cos me cos e de d i } e w y=y t m=0 2n . t i * D m u ] n RR / cos me cos e de d i } e w However, 2n / cos me cos e de = n m = 1 0 = 0 m * 1 Therefore, Fj - ( T - * R p [ Ag *> M l ( k R ) y=y H t + i * D 1 to ] d i } e 1 u t ( i i i ) From (80), the e x c i t i n g moment Fg i s A.28 102 i<t>D_w ](n R - n ^ j c o s me cos e Rde dA } e 1 w t . (80) f 2 Jo - R(v - v> ^  c o v i \ : h ) iw.<«) y y^. 2TC i<l>D_ w ] / cos me cos e de dA } e By using (H.2), the above equation becomes y y t + i< t . D 1 o) ] dA } e 1 w t . (81) - R.l -REFERENCES BAI, K.J., A V a r i a t i o n a l Method i n P o t e n t i a l Flows with a Free Surface, U n i v e r s i t y of C a l i f o r n i a , College of Engineering, Rep. NA 72-2 (Sept. 1972), vi + 137 pp. BREBBIA, C.A., The Boundary Element Method f o r Engineering, Pentech Press, 1978, pp. 46-72. FRANK, W. " O s c i l l a t i o n of C y l i n d e r s i n or Below the Free Surface of Deep F l u i d s , " Naval Ship Research and Development Center, Rep. 2375 (1967), vi + 40 pp. FRANK, W., "The Heave Damping C o e f f i c i e n t s of Bulbous C y l i n d e r s , P a r t i a l l y Immersed i n Deep Water," Journal of Ship Research, V o l . 11, (1967), pp. 151-153. GARRISON, C.J., Hydrodynamic Loading o f Large Offshore S t r u c t u r e : Three  Dimensional Source D i s t r i b u t i o n Methods, John Wiley & Sons, 1974, pp. 130-131. GRADSHTEYN, R., Table of I n t e g r a l s , S e r i e s , and Products, Academic Press, 1983, pp. 153-156. JOHN, F., "On the Motion of F l o a t i n g Bodies. I I . Simple Harmonic Motions," Comm. Pure Appl. Math., V o l . 3, (1950), pp. 45-101. KEULEGAN, G.H. and CARPENTER, L.H., "Forces on C y l i n d e r s and Plates i n an O s c i l l a t i n g F l u i d , " Journal of Research of the National Bureau o f  Standard, 1958. KIM, W.D., " O s c i l l a t i o n s of a R i g i d Body in a Free Surface," Journal o f F l u i d Mechanics, V o l . 21 , (1965), pp. 427-451. KRITIS, I r . B., "Heaving Motions of Axisymmetric Bodies," Journal of Ship  Research, 1979, pp. 26-27. MacCAMY, R.C., "On the Heaving Motion of Cy l i n d e r s of Shallow D r a f t , " Journal of Ship Research, V o l . 5, No. 3, (1961), pp. 34-43. NEWMAN, J.N., Marine Hydrodynamics, The MIT Press, 1980. References (continued) PORTER, W.R., "Pressure D i s t r i b u t i o n s , Added Mass, and Damping C o e f f i c i e n t s f o r C y l i n d e r s O s c i l l a t i n g in a Free Surface," U n i v e r s i t y of Cali f o r m ' a , Berkeley, I n s t i t u t e of Engineering Research Rep. 8216 ( J u l y 1960), x + 181 pp. SABUNCU, T. and CALISAL, S.M., "Hydrodynamic C o e f f i c i e n t s f o r V e r t i c a l C i r c u l a r C y l i n d e r s at F i n i t e Depth," Ocean Engineering, V o l . 8, 1981, pp. 25-63. TASAI, F., "Damping Force and Added Mass of Ships Heaving and P i t c h i n g , " Rep. Research I n s t i t u t e of Applied Mechanics, Kyushu U n i v e r s i t y , V o l . 7 (1959), pp. 131-152. URSELL, F., "On the Heaving Motion of a C i r c u l a r C y l i n d e r in the Free Surface of a F l u i d , " Quart. J . Mech. Appl. Math., V o l . 2 (1949), pp. 218-231. URSELL, F., "On the R o l l i n g Motion of C y l i n d e r s i n the Surface of a F l u i d , " Quart. J . Mech. Appl. Math., V o l . 2, (1949), pp. 335-353. WEHAUSAN, J.V., "The Motion of F l o a t i n g Bodies," Annual Review of F l u i d  Mechanics, V o l . 3, 1971, pp. 237-267. YEUNG, R.W., A S i n g u l a r i t y -- D i s t r i b u t i o n Method f o r Free Surface Flow  Problems with an O s c i l l a t i n g Body, U n i v e r s i t y of C a l i f o r n i a , Berkeley, 1973, pp. 9-18. 

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