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Finite element analysis of viscous flow and rigid body interaction Irani, Mehernosh Boman 1982

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FINITE ELEMENT ANALYSIS OF VISCOUS FLOW AND RIGID BODY INTERACTION by MEHERNOSH BOMAN IRANI B.Tech., The Indian I n s t i t u t e Of Technology, Bombay, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES Department Of C i v i l E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA October 1982 © Mehernosh Boman I r a n i , 1982 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i 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 ref e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e 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 h i s 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 p e r m i s s i o n . Mehernosh Boman I r a n i Department of C i v i l E n g i n e e r i n g The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: 18 October 1982 i i ABSTRACT A f i n i t e element method i s developed to analyse the i n t e r a c t i o n of a two-dimensional v i s c o u s f l u i d and an e l a s t i c a l l y supported r i g i d body. The stream f u n c t i o n form of the Navier-Stokes equations i s d i s c r e t i z e d u sing 18 degree of freedom C 1 t r i a n g u l a r elements. For each f l u i d element i n c o n t a c t with the body c o n s i s t e n t s u r f a c e t r a c t i o n v e c t o r s are d e r i v e d i n terms of the nodal v a r i a b l e s t o represent the f o r c e c o u p l i n g terms i n the equations of motion of the body. Thus a system of n o n l i n e a r d i f f e r e n t i a l equations i s obtained f o r the coupled f l u i d - m a s s system. L For the f i r s t order s o l u t i o n of the n o n l i n e a r problem by the p e r t u r b a t i o n method, the equations are l i n e a r i z e d and the l i n e a r problem i s s o l v e d . In the present study r e s u l t s f o r the l i n e a r problem only are presented, which are v a l i d f o r "zero" Reynolds number or very slow flow. The problem of a square shaped mass which i s e x c i t e d by a harmonic f o r c e i n otherwise s t i l l f l u i d and supported by an e l a s t i c s p r i n g i n the d i r e c t i o n of i t s motion i s s t u d i e d . A simple dimensional a n a l y s i s shows that the response i s a f u n c t i o n of three b a s i c non-dimensional parameters, namely, (a) the e f f e c t i v e v i s c o s i t y , (b) the r a t i o of the f o r c i n g frequency to the n a t u r a l frequency of the spring-mass system i n vacua, and, (c) the mass to f l u i d d e n s i t y r a t i o . A i i i p a rametric study of the response i n terms of these b a s i c non-dimensional parameters i s c a r r i e d out. The i n f l u e n c e of the f l u i d on the response of the mass, represented as added mass and e f f e c t i v e damping, i s a l s o s t u d i e d as a f u n c t i o n of the frequency Reynolds number. The e f f e c t of the f l u i d , i n the form of added mass and damping, i s c l e a r l y e v i d e n t i n the s h i f t of the resonance peaks of the amplitude response curves of the mass. In the absence of known r e s u l t s no q u a n t i t a t i v e estimate of the accuracy of the method c o u l d be made. But the values of the added mass obtained at high f r e q u e n c i e s came to w i t h i n ten per cent of the p r e d i c t i o n s by i n v i s c i d flow theory. T h i s i n d i c a t e s good accuracy at l e a s t i n t h i s l i m i t i n g case. i v TABLE OF CONTENTS. Page ABSTRACT i i TABLE OF CONTENTS i v LIST OF TABLES v i LIST OF FIGURES v i i ACKNOWLEDGEMENTS ix Chapter I INTRODUCTION 1 .1.1 Background 1 1.2 Purpose And Scope 4 II THEORETICAL BACKGROUND 6 2.1 I n t r o d u c t i o n 6 2.2 Equations Of Motion 7 2.3 Comparison Of Two A s s o c i a t e d Flows 11 III FINITE ELEMENT REPRESENTATION 19 3.1 I n t r o d u c t i o n ..19 3.2 The V a r i a t i o n a l Formulation 19 3.3 The F i n i t e Element Formulation 22 3.4 Force Coupling V e c t o r s 27 3.4.1 F l u i d Forces on The Body 27 3.4.2 Forces On The F l u i d 36 V Page Chapter IV APPLICATIONS 41 4.1 I n t r o d u c t i o n 41 4.2 The L i n e a r System of Equations 43 4.3 Frequency Domain A n a l y s i s 45 4.4 Non-dimensional Parameters 46 4.5 F i n i t e Element Computation 48 4.6 Numerical R e s u l t s 53 4.6.1 Numerical Study 53 4.6.2 P h y s i c a l Study 56 V CONCLUSIONS 71 REFERENCES 73 APPENDIX A - M a t r i c e s A s s o c i a t e d with the D i s c r e t i z a t i o n of the F l u i d Domain 75 APPENDIX B - The Force Coupling V e c t o r s 81 APPENDIX C - The M o d i f i e d V a r i a t i o n a l P r i n c i p l e 85 APPENDIX D - The Beam Example 87 v i LIST OF TABLES Table Page 4.1 Added Mass C o e f f i c i e n t f o r V i b r a t i n g Square C y l i n d e r 55 A. 1 Transformation Matrix [T] f o r 18 DOF Element 80 B. 1 N o n - l i n e a r Pressure C o e f f i c i e n t Vector 83 B.2 N o n - l i n e a r Pressure Force Vector 84 v i i LIST OF FIGURES F i g u r e Page 2.1 Comparison of Two A s s o c i a t e d Flows 12 3.1 The 18 DOF Truncated Q u i n t i c Element 23 3.2 L o c a l U,n) Co o r d i n a t e s of the T r i a n g u l a r Element • 23 3.3 Normal and Shearing S t r e s s e s on an F l u i d Element 28 4.1 Spring-Mass System i n I n f i n i t e F l u i d Domain 35 4.2 Geometry of the Problem 35 4.3 T r i p l e Noding 35 4.4 F i n i t e Element G r i d s 35 4.5 Added Mass C o e f f i c i e n t f o r V i b r a t i n g Square C y l i n d e r 59 4.6 Damping C o e f f i c i e n t f o r V i b r a t i n g Square C y l i n d e r 60 4.7 Amplitude Response Curves (Density R a t i o = 1.0) 61 4.8 Phase Angle Curves (Density R a t i o = 1.0) 62 v i i i F i g u r e Page 4.9 Amplitude Response Curves (Density R a t i o = 10.0). 63 4.10 Phase Angle Curves (Density R a t i o = 10.0) 64 4.11 Flow P a t t e r n s i n a C y c l e of O s c i l l a t i o n 65 D.1 The Beam Problem 87 D.2 The Beam Element ...88 D.3 The Coupled Beam Problem ' 89 i x ACKNOWLEDGEMENTS The author would l i k e to thank h i s a d v i s o r Dr. M. D. Olson, f o r h i s i n t e r e s t and v a l u a b l e suggestions d u r i n g the res e a r c h and p r e p a r a t i o n of t h i s t h e s i s . He would l i k e to thank Dr. M. de S t . Q. Isaacson f o r h i s advice and he l p throughout t h i s work. The author i s a l s o indebted to Dr. J . Ari- G u r f o r h i s continuous support and c o n s t r u c t i v e comments on t h i s t h e s i s . The f i n a n c i a l support of the N a t u r a l S c i e n c e s and En g i n e e r i n g C o u n c i l of Canada i s g r a t e f u l l y acknowledged. 1 CHAPTER I INTRODUCTION 1.1 Background O f f s h o r e technology has experienced a remarkable growth i n t h i s century and a l o t of work has been done to develop a n a l y t i c t o o l s f o r the design and a n a l y s i s of o f f s h o r e s t r u c t u r e s . One of the f i r s t steps towards the design of a s t r u c t u r e or a s t r u c t u r a l member would r e q u i r e understanding the mechanics of the l o a d i n g on the s t r u c t u r e . There are two areas of background necessary f o r a thorough understanding of wave l o a d i n g on s t r u c t u r e s . The f i r s t concerns the fundamental f l u i d mechanics of steady and unsteady flows past b o d i e s . The second concerns the v a r i o u s wave t h e o r i e s which might be employed to pr o v i d e the d e t a i l s of the f l u i d motion due to the waves. Isaacson and Sarpkaya [1] have presented a comprehensive t e x t i n t h i s f i e l d . There are strong s i m i l a r i t i e s between the wave flow past a s e c t i o n of a s t r u c t u r a l member and a two-dimensional harmonic flow past a s i m i l a r s e c t i o n . The l a t t e r r e f e r e n c e flow, which i s c o n v e n i e n t l y represented a n a l y t i c a l l y and obt a i n e d under l a b o r a t o r y c o n d i t i o n s , i s used to generate a v a r i e t y of p e r t i n e n t r e s u l t s . 2 For most a p p l i c a t i o n s of p r a c t i c a l i n t e r e s t the f o r c e s a c t i n g on a s t r u c t u r a l s e c t i o n i n a two-dimensional flow are given by the so c a l l e d Morison's equation. T h i s equation i s e m p i r i c a l i n nature but most e x t e n s i v e l y used i n view of the f a c t t h a t even i n a steady, two-dimensional, separated flow past a smooth c y l i n d e r one does not have a t h e o r e t i c a l or numerical s o l u t i o n which e x p l a i n s a l l the c h a r a c t e r i s t i c s of the flow as a f u n c t i o n of Reynolds number. Attempts to understand the fundamental f l u i d mechanics of steady and unsteady flow past bodies can be t r a c e d back to the l a s t century when the theory of i d e a l f l u i d motion was formulated q u i t e comprehensively. Thompson and T a i t [2] presented a general theory of the motion of s o l i d s i n i d e a l f l u i d s . The t e x t of Lamb [ 3 ] , f i r s t p u b l i s h e d i n 1879, i s an i n v a l u a b l e r e f e r e n c e r e l a t i n g to t h i s kind of work. The fundamental r e s u l t of i n t e r e s t presented concerns the hydrodynamic f o r c e , termed an i n e r t i a f o r c e , which i s exerted on a body h e l d i n a u n i f o r m l y a c c e l e r a t i n g i d e a l f l u i d . , S o l u t i o n s f o r i n c o m p r e s s i b l e r e a l ( v i s c o u s ) f l u i d flow problems i n v o l v e the complete Navier-Stokes equations. The presence of the n o n l i n e a r terms in these equations makes the s o l u t i o n d i f f i c u l t . Assuming very slow motion, or motion with very l a r g e v i s c o s i t y , the n o n l i n e a r c o n v e c t i v e a c c e l e r a t i o n terms i n the Navier-Stokes equations have been n e g l e c t e d and s o l u t i o n s obtained f o r c r e e p i n g motion around a sphere (Stokes) ( r e f e r e n c e [ 4 ] ) . An improvement to Stokes s o l u t i o n 3 was given by Oseen, who took the c o n v e c t i v e a c c e l e r a t i o n terms i n the Navier-Stokes equations p a r t l y i n t o account. For very l a r g e Reynolds numbers the two-dimensional flow of a r e a l ( v i s c o u s ) f l u i d past a body began to be understood only i n the present century, f o l l o w i n g P r a n d t l ' s d i s c o v e r y of the boundary l a y e r i n 1904. A n a l y t i c s o l u t i o n s have been obt a i n e d f o r the boundary l a y e r equations f o r the flow around an o s c i l l a t i n g c y l i n d e r ( r e f e r e n c e [ 5 ] ) . These s o l u t i o n s give the same s u r f a c e s k i n f r i c t i o n as obtained by Stokes using the l i n e a r i z e d form of the N a v i e r - S t o k e s ; but the p r e s s u r e d i s t r i b u t i o n i s assumed to be, and remains, that of i n v i s c i d t heory. Hence when the primary i n t e r e s t i s i n the f o r c e s and t h e r e f o r e the p r e s s u r e , on the body, the boundary l a y e r theory cannot be used. Much work has been done i n recent years to develop numerical s o l u t i o n s f o r the two-dimensional flow around bodies using f i n i t e element and f i n i t e d i f f e r e n c e techniques. R e s u l t s of a c c e p t a b l e accuracy have been obtained f o r flows and f o r c e s on f i x e d bodies f o r a f i n i t e range of Reynolds numbers. Olson and Tuann [6] have obtained f i n i t e element s o l u t i o n s f o r steady flow around c i r c u l a r c y l i n d e r s i n the Reynolds number range from 1 to 100. More r e c e n t l y e x c e l l e n t r e s u l t s have been ob t a i n e d by Davis and Moore [7] f o r unsteady flow around squares i n the Reynolds number range from 100 to 2800 u s i n g F i n i t e D i f f e r e n c e techniques. A l o g i c a l e x t e n s i o n of these methods. would be to the study of the flow around a 4 r i g i d body which i s e l a s t i c a l l y supported, the movement of which would a f f e c t the flow f i e l d and the f l u i d f o r c e s themselves and r a i s e the q u e s t i o n of resonant o s c i l l a t i o n s of the body. 1.2 Purpose and Scope The purpose of t h i s t h e s i s i s to extend the f i n i t e element method developed by Olson [ 8 ] , f o r v i s c o u s flow problems, t o cover i n t e r a c t i o n between v i s c o u s flow and a moving s o l i d body. The problem c o n f i g u r a t i o n c o n s i d e r e d i s that of an e l a s t i c a l l y supported r i g i d body surrounded by a two-dimensional i n c o m p r e s s i b l e v i s c o u s flow. High p r e c i s i o n t r i a n g u l a r elements are used to d i s c r e t i z e the stream f u n c t i o n form of the Navier-Stokes e q u a t i o n s . And a system of n o n l i n e a r d i f f e r e n t i a l equations f o r the coupled system ob t a i n e d . The p e r t u r b a t i o n method i s intended to be used f o r s o l v i n g the h i g h l y n o n l i n e a r problem i n the time domain. The f i r s t s t e p towards the n o n l i n e a r s o l u t i o n by the p e r t u r b a t i o n method i s the z e r o t h order l i n e a r s o l u t i o n . The scope of t h i s t h e s i s i s r e s t r i c t e d to the s o l u t i o n of the l i n e a r problem o n l y . Thus the r e s u l t s are c o n f i n e d to h i g h l y v i s c o u s flow and smal l amplitude motion. The example t r e a t e d i s a square shaped e l a s t i c a l l y supported mass which i s e x c i t e d by a harmonic f o r c e i n a f l u i d 5 otherwise at r e s t . The e f f e c t of the f l u i d v i s c o s i t y and the mass to f l u i d d e n s i t y r a t i o on the response of the system i s s t u d i e d . The e f f e c t i v e damping and added mass i s a l s o o b t a i n e d as a f u n c t i o n of a frequency parameter and the high frequency r e s u l t s compared t o p r e d i c t i o n s by the i n v i s c i d flow t h e o r y . 6 CHAPTER II THEORETICAL BACKGROUND 2.1 I n t r o d u c t i o n In t h i s chapter the equations of motion f o r the coupled v i s c o u s f l u i d - s o l i d body system are presented. For the f l u i d domain the stream f u n c t i o n form of the Navier-Stokes equations f o r two-dimensional v i s c o u s flow i s used. As the numerical problems s o l v e d f o r i n t h i s t h e s i s are r e s t r i c t e d to the l i n e a r range, the c o n d i t i o n s under which the l i n e a r i z e d equations are v a l i d i s examined. The equations of motion of a e l a s t i c a l l y supported r i g i d body are a l s o presented. The r i g i d body has two t r a n s l a t i o n a l and one r o t a t i o n a l degree of freedom i n the two-dimensional f l u i d flow. The most common a p p l i c a t i o n s would i n v o l v e p r e d i c t i n g the f o r c e s on and hence the response of a s t r u c t u r a l s e c t i o n represented by a s p r i n g mass system i n a two-dimensional flow. The s t r u c t u r e i t s e l f might be i n i t i a l l y s t a t i o n a r y , and sub j e c t e d to a s p e c i f i e d time dependent flow, or i t can be sub j e c t e d to an e x t e r n a l f o r c e i n otherwise s t i l l f l u i d . I t i s shown here that these problems are i n t e r r e l a t e d and the s o l u t i o n of one leads to the s o l u t i o n of the other without much a d d i t i o n a l computational e f f o r t . 7 2.2 Equations of Motion The stream f u n c t i o n form of the Navier-Stokes equations fo r the two-dimensional flow of an i n c o m p r e s s i b l e , Newtonian f l u i d i s where v i s the kinematic v i s c o s i t y , and the stream f u n c t i o n • has the usual d e f i n i t i o n , i . e . the x- and y- v e l o c i t y components are u - * y , v - -i|» x (2.2) where the s u b s c r i p t s denote p a r t i a l d e r i v a t i v e s , i . e . v x • — . The presence of the n o n l i n e a r terms (V'yV 2^ - ^ x V 2 ^ y ) t makes the numerical s o l u t i o n very d i f f i c u l t . To e x p l o r e the c o n d i t i o n s under which these n o n l i n e a r terms can be n e g l e c t e d non-dimensional v a r i a b l e s are i n t r o d u c e d thus y' - y/L, x' = x/L, t' •= wt, = ^/UL (2.3) where L i s a r e p r e s e n t a t i v e l e n g t h , say a l i n e a r dimension of the body, t i s time, u i s a r e p r e s e n t a t i v e frequency, say the 8 frequency of o s c i l l a t i o n of the body, and U a r e p r e s e n t a t i v e v e l o c i t y . In terms of the non-dimensional q u a n t i t i e s equation 2.1 becomes V ' 2 V T , + R( + ' y » V , 2 f x . - * * x t V 2 i | > ' y i ) = V ' H ' (2.4) where R = Reynolds Number = UL/v and # = Frequency Parameter = o L 2 / v The n o n l i n e a r terms can be n e g l e c t e d i n the equation i f UL/i/ i s o(1) (tends to zero) and oL 2/y i s 0(1) (of the order of u n i t y ) . I f the r e p r e s e n t a t i v e v e l o c i t y i s put as uh, where h i s the amplitude of o s c i l l a t i o n of the body, the approximation i s v a l i d i f uhL/v i s o(1) and uL 2/v i s 0(1) i . e . the amplitude of o s c i l l a t i o n i s small compared to the dimension of the body. The equations of motion f o r the e l a s t i c a l l y supported r i g i d body can be w r i t t e n as [M b] {r} + [ K 8 ] {r} - {F} (2.5) where {r} i s the displacement v e c t o r with respect to an i n e r t i a l frame of r e f e r e n c e l o c a t e d at the c e n t r e of mass of the body at i t s e q u i l i b r i u m p o s i t i o n 9 {r} = (2.6) where * x » r y , are the t r a n s l a t i o n a l degrees of freedom i n the IT x- and y- d i r e c t i o n s , r e s p e c t i v e l y , and a i s the co u n t e r c l o c k w i s e r o t a t i o n a l degree of freedom. The mass matrix lM b] i s d e f i n e d as p 1* 0 0 = 0 M y 0 (2.7) 0 0 M a where M x and M y are i d e n t i c a l , being the mass of the body, and M a i s the r o t a t i o n a l mass moment of i n e r t i a of the body about i t s c e n t r e of mass. Note that the mass matrix i s d i a g o n a l because the equations of motions are w r i t t e n about the c e n t r e o f ma s s. The s t i f f n e s s matrix l K s l r e p r e s e n t s the s t i f f n e s s e s of the supports 10 0 0 0 (2.8) where K x » K y a n a " K ° a r e the s t i f f n e s s e s of the supp o r t i n g s p r i n g s i n the x-, y- and ° d i r e c t i o n s r e s p e c t i v e l y . The time dependent l o a d v e c t o r {F} i s represented as the sum of two components ( N F 1 X F 2 x {F} = (F x} + CF 2} - F i y + F 2 y ' F l a (2.9) where {F,} i s the v e c t o r of f l u i d f o r c e s on the body and {F 2} i s the v e c t o r of other e x t e r n a l f o r c e s on the body. For each component of the f l u i d f o r c e , a c o n s i s t e n t f o r c e v e c t o r i s d e r i v e d i n terms of the f i n i t e element nodal v a r i a b l e s by i n t e g r a t i n g the t a n g e n t i a l and normal s t r e s s e s on the boundary of each f l u i d element i n con t a c t with the body. D i s c r e t i z a t i o n of equation 2.1 y i e l d s a set of equations f o r the f l u i d domain. C o m p a t i b i l i t y c o n s t r a i n t s are imposed between the f l u i d and the body by s p e c i f y i n g the t a n g e n t i a l 11 and normal v e l o c i y components on the i n t e r f a c e to be the same. T h i s g i v e s a set of equations f o r the coupled motion of the system. The d e t a i l s of these steps are shown l a t e r i n Chapter 3. 2.3 Comparison of Two A s s o c i a t e d Flows The f l u i d body i n t e r a c t i o n problem can be s o l v e d f o r d i f f e r e n t s p e c i f i e d c o n d i t i o n s of flow or e x t e r n a l f o r c e s . A s p e c i f i c problem can be that of the f l u i d having a uniform t r a n s l a t i o n a l time-dependent v e l o c i t y f a r from the e l a s t i c a l l y supported body. Another r e l a t e d problem i s where the same s p r i n g mass system i s e x c i t e d d i r e c t l y by an e x t e r n a l f o r c e i n otherwise s t i l l f l u i d . I t i s not necessary to solve both these cases independently as the r e s u l t s o b tained f o r one case i s enough t o o b t a i n p e r t i n e n t r e s u l t s f o r the other. To show t h i s , the procedure as formulated by Isaacson [9] i s f o l l o w e d here. Consider the a s s o c i a t e d p a i r of two-dimensional flows as d e f i n e d i n f i g u r e 2.1. In case A the f l u i d remote from the spring-mass system has an uniform t r a n s l a t i o n a l v e l o c i t y U(t) and the response of the mass i s X , ( t ) . In case B the f l u i d remote from the spring-mass system i s s t a t i o n a r y . x c y c i s a c o - o r d i n a t e system f i x e d r e l a t i v e to the undisturbed f l u i d and x b y b i s a c o - o r d i n a t e system moving with v e l o c i t y U(t) as shown i n f i g u r e 2.1. The mass i s e x c i t e d d i r e c t l y by an s p e c i f i c e x t e r n a l f o r c e F e ( t ) such that the response of the 12 Case A F l u i d remote from aprlng system with uniform time dependent v e l o c i t y U ( t ) . C a s e B Stationary f l u i d remote from spring-mass system. Co-ordinate system has v e l o c i t y U(t) r e l a t i v e to f i x e d ( i n e r t i a l ) co-ordinate system x c y c . F i g u r e 2.1 - Comparison of Two A s s o c i a t e d Flows mass measured r e l a t i v e to x b y b i s the same as i n case A, i . e . = X , ( t ) . Thus the response r e l a t i v e to i n e r t i a l frame x c y c i s t X 2 = X, -Judt. (X 2=X, at t=0) The equations of motion f o r the f l u i d i n case A are 13 3U 3U a 3 U a at 3 x a a y a P 3x„ 3t + U a 3V 3^. a + V. ay, i 3 P a 9 A £ + v V 2 V P 3 x a (2.10) 3 x a 3 y a The boundary c o n d i t i o n s are V a - 0 U f l - Xlt V a • 0 — on body p e r i p h e r y U\ = U, V a - 0 — i n f a r f i e l d In case B the equations of motion with r e s p e c t to the f i x e d c o - o r d i n a t e system x c y c are 3U 3U„ 3U , 3P„ . S. + u 1 + v„ £. - - ± =\ + v v 2 u „ 3t C 3 x c c 3 y c p 3 x c 3V C 3V_ 3V i 3P (2.11) £ + U £. + V„ = - - + W 2 V „ 3t c 3 x c c 3 y c p 3 y c c 311 3V„ S. + SL . o 3 x c 3 y c 14 The v e l o c i t i e s of any p o i n t i n terms of c o - o r d i n a t e systems b and c are r e l a t e d by u c - u b - U V c » v b (2.12a) And the p r e s s u r e at any p o i n t i s independent of the frame of r e f e r e n c e P c " p b (2.12b) The displacements of the f l u i d p a r t i c l e s are r e l a t e d by x c - x b " / Q U d T ° (2.12c) Y c » Y b S u b s t i t u t i n g equations 2.12 i n t o 2.11 g i v e s 15 j I I au K i £ + b + u, dt 3t 1 3U, + V , 3U, 1 9 P K o - ± £ + v V 2 U , 3 y b *?\> p 3 ? b av, 3t + U, av, 3x, + V, av, 871 i H ± + W 2 V , p 3 y h 1 au, av. b + — b = o 9 y h or on rearrangement !!b + U h + v ^ = . i t i ! b . p duj + v v 2 u 3t b 3 x b b 3 y b P 3 U b dt (2.13) 3Uw 3V K 3V h -, 3 P b ~ + U k k + V h k - - A P- + W 2 V b at b 3 x b b 3 y b P 3 y b k + P. = 0 The boundary c o n d i t i o n s are 0 ^ „ £ i v b - 0 — on body p e r i p h e r y U b - U V b - 0 - i n f a r f i e l d 16 I t i s c l e a r that equations and boundary c o n d i t i o n s 2.10 and 2.13 are the same. There f o r e the s o l u t i o n s of the two systems are the same, hence U a " U b 1!* = 11° 3 y a 3 y b ! ! * - ! !k - p « 3 x a 3x b dt T h e r e f o r e the v e l o c i t i e s r e l a t i v e to the two f i x e d r e f e r e n c e frames 'a' and 'b' are i d e n t i c a l . The only d i f f e r e n c e i n the two cases, r e l a t i n g to the p r e s s u r e s i s the a d d i t i o n a l uniform pressure g r a d i e n t -pD i n case A and a c t i n g i n the d i r e c t i o n of a c c e l e r a t i o n . Hence the l o n g i t u d i n a l f o r c e i n case A i s g r e a t e r by an amount pAtJ where A i s the c r o s s - s e c t i o n a l area of the ' body. T h i s a d d i t i o n a l l o n g i t u d i n a l f o r c e i n case A a c t s i n the d i r e c t i o n of f l u i d a c c e l e r a t i o n . The equation of motion of the body in case A i s MX 1 + KX, f a (2.14) 17 where F f a i s the f l u i d f o r c e on the body. In case B the equation of motion of the body i s M X 2 + K X 2 - F e - F f b ( 2 i l 5 ) where F g i s the e x t e r n a l f o r c e on the body and F f b i s the f l u i d f o r c e on the body. Now 12 - i x - u x 2 - x l - /' u d T (2.16) o Thus equation 2.15 becomes M(k\ - U) + K ( X X - U d x ) - F e - F f b M X ! + K X X - MU + K / U d T + F e - F f b (2.17) From 2.14 and 2.17 i t f o l l o w s that 18 - M& + K . / * U d x + F e - F o f b * . - <Ffa + Ffb> - M " " K ^ U d T (2.18) I t has been shown that the d i f f e r e n c e i n the f l u i d f o r c e s in Hence i f the s o l u t i o n to case A f o r a given flow U(t) i s d e s i r e d , case B can be s o l v e d with the s p e c i f i c e x t e r n a l f o r c e F e given by equation 2.19. T h i s w i l l g i ve the response X 2 ( t ) , then the d e s i r e d response X , ( t ) f o r case A can be found d i r e c t l y from the two cases • P A & , from which i t f o l l o w s that F - ( P A - M) 0 - K / t U d T ( 2 . 1 9 ) o x x ( t ) - x 2 ( t ) + j t U ( t ) d x o ( 2 . 2 0 ) 19 CHAPTER III FINITE ELEMENT REPRESENTATION 3.1 I n t r o d u c t i o n In t h i s chapter the stream f u n c t i o n form of the Navier Stokes equations presented i n chapter two i s d i s c r e t i z e d u s ing the f i n i t e element method. The s t a r t i n g p o i n t i n the f i n i t e element d i s c r e t i z a t i o n i s the d e r i v a t i o n of the f u n c t i o n a l form of equation 2.1. I t i s convenient to r e c a s t the d i f f e r e n t i a l equation as a r e s t r i c t e d v a r i a t i o n a l p r i n c i p l e , which i s e x a c t l y e q u i v a l e n t to the G a l e r k i n method. T h i s process i s w e l l documented by Olson [8] and i s reproduced here fo r completeness. To c a l c u l a t e the f o r c e c o u p l i n g between the f l u i d and body motions, the f l u i d f o r c e s on the body are d e r i v e d i n terms of the f i n i t e element nodal v a r i a b l e s . T h i s i s done here by d e r i v i n g a c o n s i s t e n t f o r c e v e c t o r f o r each f l u i d element i n c o n t a c t with the body. 3.2 The V a r i a t i o n a l Formulation I t has been shown ( F i n l a y s o n [10]) that no v a r i a t i o n a l p r i n c i p l e e x i s t s c orresponding to the complete Navier-Stokes equations due to the presence of the n o n - s e l f - a d j o i n t terms. It i s convenient however to r e c a s t the Navier-Stokes equations in a r e s t r i c t e d v a r i a t i o n a l p r i n c i p l e as d e r i v e d by Olson [ 8 ] . 20 T h i s i s i d e n t i c a l t o the G a l e r k i n method but all o w s one to think i n the usual terms of seeking a s t a t i o n a r y p o i n t to some f u n c t i o n a l . B a s i c a l l y i t i s d e r i v e d by m u l t i p l y i n g equation 2.1 by a v i r t u a l "displacement" 6*, i n t e g r a t i n g by p a r t s , and then t a k i n g the 6 o p e r a t i o n o u t s i d e the i n t e g r a l by i n t r o d u c i n g barred q u a n t i t i e s where the bar i n d i c a t e s that that term i s h e l d constant with re s p e c t to the v i r t u a l "displacement" d u r i n g the v a r i a t i o n . The r e s t r i c t e d f u n c t i o n a l i s where v i s the kinematic v i s c o s i t y . N o t i c e that the barred q u a n t i t i e s are a s s o c i a t e d with the i n e r t i a f o r c e s . The p r i n c i p l e y i e l d s the f o l l o w i n g kinematic and a s s o c i a t e d n a t u r a l boundary c o n d i t i o n s I - / / A { | ( V 2 i | ; ) 2 + ( * y V 2 * ) * x ( ¥ x V 2 * ) * y + Vi|»tV4»}dA (3.1 ) (3.2) - 0 6* n - 0 v24, •= 0 (3.3) where s, n, are the l o c a l t a n g e n t i a l and outward normal c o o r d i n a t e s . 21 The n a t u r a l c o n d i t i o n s i n equations 3.2 and 3.3 have no r e a l p h y s i c a l meaning. I t seems more l o g i c a l to use the c o n d i t i o n of zero t r a c t i o n . T h i s i s done by adding a boundary i n t e g r a l to the equation 3.1. The f o r m u l a t i o n i s to be used only with a t r i a n g u l a r f i n i t e element so i t i s convenient to add the boundary i n t e g r a l on one edge of the t r i a n g l e . On the a d d i t i o n of t h i s boundary i n t e g r a l the f i n a l f u n c t i o n a l becomes T h i s f u n c t i o n a l i s to be made s t a t i o n a r y to y i e l d equation 2.1 as the E u l e r equation p r o v i d e d the barred terms are h e l d f i x e d d u r i n g the v a r i a t i o n . I t a l s o generates the f o l l o w i n g kinematic boundary c o n d i t i o n s 6 ^ - 0 and 6\|>n - 0 with the a s s o c i a t e d n a t u r a l boundary c o n d i t i o n s ( 3 . 5 ) 22 £ £ . = v V 2 \ l ) „ - U j _ + \ L r ih - \h \h (3.6) where p i s the p r e s s u r e and i s the shear s t r e s s . Note that the boundary i n t e g r a l i n equation 3.4 i s to be i n c l u d e d only when the edge 1-2 i s on a boundary along which some n a t u r a l boundary c o n d i t i o n s are to be s a t i s f i e d . A d d i t i o n a l f o r c e s a r i s e on boundaries where the kinematic boundary c o n d i t i o n s are p r e s c r i b e d . The r e s t r i c t e d v a r i a t i o n a l f o r m u l a t i o n presented here i s m o d i f i e d i n Appendix C to i n c l u d e a d d i t i o n a l terms' f o r those boundaries where the kinematic boundary c o n d i t i o n s are p r e s c r i b e d . 3.3 The F i n i t e Element Formulation The f u n c t i o n a l I c o n t a i n s products of second order d e r i v a t i v e s hence an element with C 1 c o n t i n u i t y i s r e q u i r e d f o r d i s c r e t i z i n g equation 3.4. Elements of t h i s c l a s s have been developed f o r s o l v i n g p l a t e problems i n s t r u c t u r a l mechanics. One of the most ac c u r a t e and v e r s a t i l e p l a t e elements i s the one developed by Cowper et a l [11]. Olson [8] then adapted i t to the s o l u t i o n of the v i s c o u s flow problem. T h i s method i s used here as w e l l . 23 -ft F i g u r e 3.1 - The 18 DOF Truncated Q u i n t i c Element F i g u r e 3.2 - L o c a l (£, n ) C o o r d i n a t e s of the T r i a n g u l a r Element 24 The f i n i t e element i s the 18 degree of freedom t r u n c a t e d q u i n t i c , t r i a n g u l a r element shown i n f i g u r e 3.1. ¥ i s approximated by an incomplete q u i n t i c polynomial with the term m i s s i n g ( £ , n) are the l o c a l element c o o r d i n a t e s as shown in f i g u r e 3.2, so that the normal slope i s c u b i c along the edge n=0. Two a d d i t i o n a l c o n s t r a i n t s are added so that the normal d e r i v a t i v e v a r i e s as a c u b i c on the other edges as w e l l . T h i s can be expressed as 2 o * - I a, ^ n D i ( 3 - 7 ) i=l where the a± are the polynomial c o e f f i c i e n t s , and the m.^, n± are i n t e g e r exponents which are given in appendix A. The polynomial c o e f f i c i e n t s are transformed to the 18 nodal degrees of freedom i n l o c a l (£,n) c o o r d i n a t e s by t > L , 0 , 0 > T - [ T ] U } ( 3 . 8 ) where {A} and are the v e c t o r s c o n t a i n i n g the polynomial Li c o e f f i c i e n t s and the l o c a l nodal v a r i a b l e s r e s p e c t v e l y and [T] i s the t r a n s f o r m a t i o n matrix a l s o given i n appendix B. Equation 3.8 i s then i n v e r t e d to g i v e 25 {A} - [ T ] " 1 U L , 0 , 0 } T - [ T 2 ] { * L ) ( 3 # 9 ) The v e c t o r c o n t a i n i n g the nodal v a r i a b l e s i s transformed from l o c a l ( C , n) c o o r d i n a t e s to the g l o b a l (x,y) system by " l 1 • W W (3.10) where [R] i s the r o t a t i o n matrix (see appendix A), so that the polynomial c o e f f i c i e n t s are r e l a t e d to the 18 nodal v a r i a b l e s in g l o b a l c o o r d i n a t e s through {A} = [ T 2 ] [R] {*} - [S] {>} (3.11) S u b s t i t u t i n g equation 3.7 i n t o equation 3.4 and c a r r y i n g out the i n t e g r a t i o n s leads to a matrix r e p r e s e n t a t i o n of I as 20 20 20 20 20 2 1-1 i - 1 ±2 1 J i = i i = i 1 = 1 4 i j k a i a j a k 2 0 2 0 7 (3.12) £ E D, . a., a . 1=1 1=1 * J 1 J 26 where kij» 1ijk» m i j a r e t n e l i n e a r s t i f f n e s s matrix, the n o n l i n e a r s t i f f n e s s matrix and the mass matrix f o r one element i n terms of the .polynomial c o e f f i c i e n t s (see appendix A). The a A r e p r e s e n t s the time d e r i v a t i v e . These m a t r i c e s are transformed to the g l o b a l nodal v a r i a b l e s by the equation 3.12, and then assembled i n the usual manner ( r e f e r e n c e [12]) to form the g l o b a l d i s c r e t i z e d f u n c t i o n a l f o r the f i n i t e element g r i d I - i K ± j + Q ± j k ^ + M ± j (3.13) i , j,k - 1,2, N where N i s the net s i z e of the problem which i n c l u d e s a l l nonzero nodal v a r i a b l e s , i . e . the homogeneous boundary c o n d i t i o n s are i n t r o d u c e d d u r i n g the assemblage process, and {ty} i s the g l o b a l v e c t o r of nodal v a r i a b l e s . The K ^ a n d mat r i c e s are symmetric and banded while the n o n l i n e a r matrix Qjjk i s symmetric only i n i t s f i r s t two s u b s c r i p t s . The extremum of the d i s c r e t e f u n c t i o n a l I i n equation 3.13 g i v e s the f i n i t e element r e p r e s e n t a t i o n of the Navier-Stokes equations (equation 2.1). Taking the v a r i a t i o n of equation 3.13 while h o l d i n g barred q u a n t i t i e s f i x e d and equating them to the non-barred ones afterwards g i v e s a set of n o n l i n e a r d i f f e r e n t i a l equations f o r the f l u i d domain 27 6 1 - M ± j * j + k i j * j + Q l j k 1 ^ = 0 (3. U ) i»j,k = 1,2, . . . . N 3.4 Force Coupling V e c t o r s 3.4.1 F l u i d Forces on the Body The f l u i d f o r c e on the body i s obtained by i n t e g r a t i n g the e x p r e s s i o n s f o r the t a n g e n t i a l and normal s t r e s s e s on the boundary of each f l u i d element i n contact with the body. The f o r c e e x p r e s s i o n s are d e r i v e d i n terms of the polynomial c o e f f i c i e n t s , f o r each element, then transformed in terms of the g l o b a l v a r i a b l e s (equation 3.12). The g e n e r a l i s e d f o r c e s are o b tained by m u l t i p l y i n g the f o r c e by the the appropiate d i r e c t i o n c o s i n e f o r the x and y d i r e c t i o n s (or by the proper l e v e r arm f o r the moment). The f o r c e v e c t o r s obtained fo r each element are assembled in the usual manner i n t o g l o b a l f o r c e v e c t o r s r e p r e s e n t i n g the f l u i d f o r c e s on the body. The s t a t e of s t r e s s of a f l u i d element undergoing "deformation" or r a t h e r a change.of s t r a i n with time i s shown in f i g u r e 3 ( r e f e r e n c e [ 1 3 ] ) . The shear s t r e s s i s given by 28 F i g u r e 3.3 - Normal and Shearing S t r e s s e s on an Elementary F l u i d Element T . xy 3 y 3 x (3.15) where u i s t h e ' a b s o l u t e v i s c o s i t y of the f l u i d ; Equation 3.15 can be r e w r i t t e n i n terms of the stream f u n c t i o n • and the l o c a l element c o - o r d i n a t e s £, r\ as T S n " *<*nn " * « > (3.16) The edge 1-2 of the element ( f i g u r e 3.2) i s always used on the f l u i d - b o d y i n t e r f a c e and hence the shear s t r e s s d i s t r i b u t i o n along that edge i s given by equation 3.16 f o r n=0. 29 I n t e g r a t i n g t h i s e x p r e s s i o n along t h i s edge and n o t i n g that the f o r c e on the body w i l l be opposite i n s i g n the shear f o r c e i s where F x i s the shear f o r c e on the body along edge 1-2 of the element, {h} i s the shear f o r c e v e c t o r f o r one element in terms of the polynomial c o e f f i c i e n t s (see appendix B). T h i s v e c t o r i s transformed to the g l o b a l nodal v a r i a b l e s by equation 3.12, (3.17) i 1 , 2 , 20 (3.18) i = 1 , 2 , 18 Each component of the g e n e r a l i s e d f o r c e i s ( F X ) T = (-cos6)F T ( F y ) T - ( - s i n 6 ) F T ( 3 . 1 9 ) ( F Q ) T - |r x F T I where ( F x ) x , ( F y ) x are the x- and y- components of the shear f o r c e , ( F a ) T i s the c o u n t e r c l o c k w i s e moment of the element 30 shear f o r c e about the c e n t r e of mass of the body, © i s the angle t h a t the edge 1-2 of the element makes with the p o s i t i v e x - a x i s ( f i g u r e 3 . 2 ) , and r i s the v e c t o r j o i n i n g the c e n t r e of mass of the body to the p o i n t of i n t e r s e c t i o n of the f o r c e v e c t o r and the edge 1-2 of the element, i t i s to be noted that the shear f o r c e v e c t o r and the edge 1-2 w i l l be c o l i n e a r . The f o r c e v e c t o r s f o r each element in c o n t a c t with the body are then assembled i n the usual manner to form the g l o b a l shear f o r c e v e c t o r s ( F X ) T « H* * ± ( F y ) T = H2 * ± (3.20) < Fa> T - H3 t ± i - 1,2, N where N i s the net s i z e of the problem , i s the g l o b a l v e c t o r of nodal v a r i a b l e s and ( f x ) T r ( F y ) T a r e t n e f o r c e s in the x- and y- d i r e c t i o n s , ( F 0 ) T i s the c o u n t e r c l o c k w i s e moment a c t i n g on the body due to the shear s t r e s s e s a c t i n g on the f l u i d - b o d y i n t e r f a c e . Note that the f o r c e v e c t o r s w i l l c o n t a i n non-zero terms f o r the nodal v a r i a b l e s a s s o c i a t e d with the elements on the f l u i d - b o d y i n t e r f a c e o n l y . The normal s t r e s s o y can be expressed as the sum of the p r e s s u r e p and a f r i c t i o n a l component a l s o known as the d e v i a t o r i c s t r e s s component o' v where 31 y y (3.21) where the f l u i d p ressure -p i s the a r i t h m e t i c mean of the normal s t r e s s e s on a f l u i d element. The negative sig n a r i s e s as by d e f i n i t i o n compressive pressure i s p o s i t i v e . The d e v i a t o r i c s t r e s s a ' y i s given by y 3y 5 n (3.22) f o l l o w i n g the same procedure as before the f o r c e along edge 1-2 of the element i s (3.23) i - 1,2 20 where F 0 i s the d e v i a t o r i c f o r c e along edge 1-2 of the element, {d} i s the d e v i a t o r i c f o r c e v e c t o r f o r one element i n terms of the polynomial c o e f f i c i e n t s (appendix B). This v e c t o r i s transformed i n terms of the g l o b a l v a r i a b l e s , as bef o r e , and the components of the g e n e r a l i s e d f o r c e s d e r i v e d 32 ( F x ) 0 - ( s i n 6 ) F a ( F y ) o " (-cos6)F a ( F o ) 0 - |7 x F j (3.24) The element f o r c e v e c t o r s are assembled in g l o b a l d e v i a t o r i c f o r c e v e c t o r s ( F x ) 0 = Dj * ± ( F y ) G - D 2 * ± < F a > 0 " D ^ * l ( 3 . 2 5 ) 1 = 1,2, N The e x p r e s s i o n f o r the pressure d i s t r i b u t i o n along the element edge 1-2 i s o b t a i n e d by i n t e g r a t i n g the momentum equation i n the x - d i r e c t i o n from the Navier-Stokes equations 3x 3x z 3 y z 3t 3x 3y ( 3 . 2 6 ) where p i s the f l u i d d e n s i t y . Equation 3 . 2 6 can be expressed in terms of the stream f u n c t i o n _t and the l o c a l element co-o r d i n a t e s £, n, as 33 * » 5 J C * n « + * n n n > " p * n " P < + n * 5 n " * 5 * n u > ( 3 > 2 7 ) I n t e g r a t i n g t h i s e x p r e s s i o n g i v e s the pressure d i s t r i b u t i o n along edge 1-2 of the element f o r n=0 P - [ J * [ w ( * n « + * n n n > " p * n " P < * n * 6 n " * 6 + n n > l | * £ 0 ( 3 . 2 8 ) + 8 ( n ) where g(n) i s a f u n c t i o n of n which i s a constant f o r n=0,equal to the pressure at { • o, From equation 3.28 the pressure at node 2 of an element can be expressed i n terms of the pressure at node 1 of that element. In general,p k+i» the p r e s s u r e at the second node of the kth element along the body i s expressed i n terms of P k, the pressure at node 1 of that element plu s c o n t r i b u t i o n s from the element under c o n s i d e r a t i o n Pk+l - P k + uei&i •+ p f ^ . j + p g i i i (3.29) i . J - 1,2, 20 where {e} i s the f o r c e c o e f f i c i e n t v e c t o r a r i s i n g from the v i s c o u s terms i n the equation 3.28, [ f ] i s the f o r c e c o e f f i c i e n t matrix a r i s i n g from the n o n l i n e a r terms and {g} i s 34 the c o e f f i c i e n t v e c t o r due to the time d e r i v a t i v e term (appendix B). I t i s apparent that the p r e s s u r e , Pk, at any node k w i l l depend on the datum value a s s i g n e d to the pressure at the f i r s t node of the f i r s t element. The net f o r c e on the body w i l l depend on the pre s s u r e d i f f e r e n c e r a t h e r than the ab s o l u t e p r e s s u r e , thus the datum value can be a r b i t a r i l y set to z e r o . I n t e g r a t i n g the pressure e x p r e s s i o n g i v e s the f o r c e f o r one element F p = ( a + b ) P l + V l ^ i + P W i j a ^ j + Pz±a± i , j - 1,2 20 where (a+b) i s the l e n g t h of the edge 1-2 of the element ( f i g u r e 3.2) and pl i s the pressure at node 1 of the element. The p r e s s u r e and the pr e s s u r e f o r c e f o r each element i s expressed i n terms of the g l o b a l v a r i a b l e s and the components of the g e n e r a l i s e d f o r c e s d e r i v e d < F x>p " ( - s i n 6 ) F p ( F y ) p - ( c o 8 6 ) F p < F a>p = I r x Fp"| (3.31 ) 35 The f o r c e v e c t o r s are assembled i n g l o b a l pressure f o r c e v e c t o r s ( F x ) p - L l ^  + V j + Z j ty± ( F y ) p = L * * i + + Z J * i ( Fa>p - ^ +i + *1*J + ZJ * i ( 3 * 3 2 ) i . j - 1,2 N The f l u i d f o r c e {F x) (equation 2.9) i s w r i t t e n as the sum of the f o r c e components F x 81 < F X > T + <Fx>o + ( Fx>p F y i ( F y ) T <Fy>o ( V P . F « . <Fa>x <Fa>o < Fa> P (3.33) where each f o r c e component i s given i n exp r e s s i o n s 3.20, 3.25 and 3.29. These f o r c e s are i n terms of the nodal v a r i a b l e s {ty}. Note that the f o r c e c o n t r i b u t i o n s from the shear and d e v i a t o r i c s t r e s s e s i s l i n e a r w h ile that from the pressure i n v o l v e s n o n l i n e a r and time dependent terms. Combining the l i n e a r , n o n l i n e a r and time dependent terms i n equation 3.33 the f l u i d f o r c e i s expressed as 36 IF 1} -1 + x J + ( F3 ) ± X *i ( Fy > i * i ' y J > ( F3 ) ± y a ( F 2 ) ± I I a J ( F 3 ) ± a *1 > (3.34) where t F 1 } , { p i } , { F l } are the t o t a l g l o b a l l i n e a r f o r c e x y a v e c t o r s , [ P 2 ] , [ F 2 ] , [ F 2 ] are the t o t a l g l o b a l n o n l i n e a r f o r c e x y ct m a t r i c e s , and I F 1 } , {F 2}, {F 3} are the t o t a l g l o b a l f o r c e x y a v e c t o r s from the time dependent terms. 3.4.2 Forces on the F l u i d The no s l i p c o n d i t i o n on the f l u i d - b o d y i n t e r f a c e s p e c i f i e s the normal and t a n g e n t i a l v e l o c i t i e s of the f l u i d and the body to be the same. The v e l o c i t y components of the body are assumed to be e x t e r n a l p r e s c r i b e d v a l u e s f o r the f l u i d domain. For any f l u i d element i n co n t a c t with the body these " p r e s c r i b e d " v e l o c i t y components give r i s e to a d d i t i o n a l f o r c e s on these boundaries. These " f o r c e " terms have to be added to the f l u i d equations (equations 3.14). The f o r c e s which a r i s e due to the " p r e s c r i b e d " normal and 37 t a n g e n t i a l v e l o c i t i e s (of the body), on the edge 1-2 of the f l u i d element, are d e r i v e d by the p r i n c i p l e of " v i r t u a l work". The p r i n c i p l e of " v i r t u a l work" used here i s a mixed f o r m u l a t i o n analogous to the R e i s s n e r ' s p r i n c i p l e i n s o l i d mechanics ( r e f e r e n c e [ 1 4 ] ) . I t s a p p l i c a t i o n i s i l l u s t r a t e d with a simple example from s o l i d mechanics, i n Appendix D. The i n t e r n a l v i r t u a l energy, cJw, due to the p r e s c r i b e d displacements i s 6w ( 3 . 3 5 ) where , are the p r e s c r i b e d t a n g e n t i a l and normal v e l o c i t i e s r e s p e c t i v e l y , and,"" f i x£n, ~ 6 o n are the corresponding v i r t u a l i n t e r n a l s t r e s s e s . These v i r t u a l s t r e s s e s are due to the v i r t u a l s t r a i n s compatible with the v i r t u a l displacements a p p p l i e d at the degrees of freedom where the displacements are not p r e s c r i b e d . Equating the i n t e r n a l v i r t u a l energy to the e x t e r n a l v i r t u a l work giv e s {6*} T(F} - 6W ( 3 . 3 6 ) where i s the v i r t u a l displacement a p p l i e d to the degrees 38 of freedom where the displacements are not p r e s c r i b e d , {F} i s the c o n s i s t e n t f o r c e v e c t o r which a r i s e s due to the p r e s c r i b e d d i s p l a c e m e n t s . The d e r i v a t i o n procedure of the f o r c e v e c t o r s i s i l l u s t r a t e d by c o n s i d e r i n g the f i r s t i n t e g r a l i n equation 3.35. Here ^n* i s expressed i n terms of the motion of the body, • rv cose + r s i n e (3.36) where r and' r are the x and y v e l o c i t y components of the x y body, and only t r a n s l a t i o n a l motion of the body i s c o n s i d e r e d . Hence the f i r s t term on the r i g h t hand si d e of equation 3.35 becomes 6w. I - ( r v c o s 6 + r v s i n 6 ) u J * 6 ( * n n -- b d5 n=o (3.37) which i n terms of the polynomial c o e f f i c i e n t i s 39 6 W T " _ ( r x c o s 6 + r y s i n 6 ) yb^ 6 S i i = 1 , 2 , . . . 2 0 where {h} turns out to be the same as i n equation 3.17. In terms of the g l o b a l nodal v a r i a b l e s equation 3.38 i s (3.38) {6i|>}T{F_} - - y ( r cos6 + r „ s i n 6 ) {H}{6i|>}T y (3.39) where t F T ) i s the c o n s i s t e n t elemental f o r c e v e c t o r due to the p r e s c r i b e d t a n g e n t i a l v e l o c i t y component and {H}, the fo r c e c o e f f i c i e n t v e c t o r i s again the same as i n equation 3.18. Extending t h i s procedure to the general case f o r a l l the terms of equation 3.35, and c o n s i d e r i n g the r o t a t i o n a l degree of freedom of the body a l s o , the r e p r e s e n t a t i o n of. the fo r c e on the f l u i d .due to the " p r e s c r i b e d " kinematic boundary c o n d i t i o n s on the f l u i d - b o d y i n t e r f a c e i s obtained. The summation of the f o r c e v e c t o r s so d e r i v e d f o r a l l the elements in c o n t a c t with the body g i v e s the t o t a l f o r c e {F }, 40 {F} - { F 1 } r 1 X 1 X + + f{F3}r ) x 1 x ' { F y } r y < [ F y ] r y > { F 3}r y> [ F a 2 J r a (3.38) where ^ r x » r y » r a ^ i s the displacement v e c t o r of the body as d e f i n e d i n equation 2.67. As expected, the f o r c e c o u p l i n g m a t r i c e s i n equation 3.38 are the same as those d e r i v e d i n equation 3.34. The f o r c e terms of equations 3.38 when added to equations 3.14 give the system of equations f o r the f l u i d domain. These combined with the equations of motion of the mass (equations 2.5) l e a d to a system of n o n l i n e a r d i f f e r e n t i a l equations whose s o l u t i o n , i n p r i n c i p l e , g i v e s the s o l u t i o n of the coupled system. 41 CHAPTER IV APPLICATIONS 4.1 I n t r o d u c t i o n Any s o l u t i o n procedure to be a p p l i e d to the coupled system of equations d e r i v e d i n the p r e v i o u s chapter would i n v o l v e s o l v i n g a n o n l i n e a r system i n the time domain. The presence of the n o n l i n e a r terms makes the s o l u t i o n d i f f i c u l t . The p e r t u r b a t i o n method or the combined procedure of the Newton-Raphson and the p e r t u r b a t i o n method may be used to s o l v e the n o n l i n e a r problem ( r e f e r e n c e [ 1 5 ] ) . The f i r s t s tep towards the n o n l i n e a r s o l u t i o n by the p e r t u r b a t i o n method i s the z e r o t h order l i n e a r s o l u t i o n . In t h i s t h e s i s a s o l u t i o n only f o r t h i s step i s presented. As has been d i s c u s s e d i n s e c t i o n 2.2, the v a l i d i t y of the s o l u t i o n i s t h e r e f o r e r e s t r i c t e d to h i g h l y v i s c o u s flow at high f r e q u e n c i e s . (Reynolds number = UL/v = 0 ( 1 ) and the frequency parameter = oL 2/v = 0 ( 1 ) ) . I t was a l s o shown that the l i n e a r i z i n g assumptions r e s t r i c t the amplitude of o s c i l l a t i o n s of the body to be small compared to i t s dimension. Hence i t i s c o n s i s t e n t to apply the boundary c o n d i t i o n s at the mean s p a t i a l p o s i t i o n of the body and to keep the f i n i t e element g r i d f i x e d . In the examples presented i n t h i s chapter, the l i n e a r s o l u t i o n f o r a s i n g l e degree of freedom ( f i g u r e 4.1) s p r i n g 42 mass system allowed to o s c i l l a t e i n the x d i r e c t i o n only i s presented. The mass i s sub j e c t e d t o a p e r i o d i c f a r flow i n that d i r e c t i o n which, as was shown i n s e c t i o n 2.3, i s simulated by s u b j e c t i n g i t to a p e r i o d i c f o r c e of app r o p i a t e magnitude i n otherwise s t i l l f l u i d . T h i s s i m p l i f i e s the a p p l i c a t i o n of the boundary c o n d i t i o n s and reduces the number of v a r i a b l e s i n the problem f o r the same g r i d s i z e . f.(t) 1 F i g u r e 4.1 - Spring-Mass system i n I n f i n i t e F l u i d Domain 43 4.2 The L i n e a r System of Equations For the l i n e a r problem with a s i n g l e degree of freedom s p r i n g mass system ( f i g u r e 4.1) the system of equations f o r the f l u i d domain (equations 3.14 and 3.38) become [M]{ip} + [K]{*} = ( F x } f x + ( F 3 } r x + h Q ( 4 > 1 ) where [M] i s the mass matrix, [K] i s the l i n e a r s t i f f n e s s matrix, {F 1} and t F 3 } are the f o r c e v e c t o r s d e f i n e d i n equations 3.38, ity) i s the v e c t o r of nodal v a r i a b l e s and {h 0} i s the f o r c e v e c t o r which i s e v a l u a t e d f o r boundary c o n d i t i o n s p r e s c r i b e d on boundaries other then the f l u i d - b o d y i n t e r f a c e . For t h i s case the gen e r a l equations of motion of the body (equations 2.5) become M b *x + K x r x • ( F 1 , ) 1 ! * } + {F 3 X} T{*} + F 2 x ( 4 > 2 ) where r x i s the x- displacement of the body, M b i s i t s mass, K x i s the s t i f f n e s s of the s p r i n g , t F 1 x > i s the l i n e a r f l u i d f o r c e v e c t o r , ( F 3 x} i s the time dependent f l u i d foce v e c t o r , i s the v e c t o r of the nodal v a r i a b l e s , and F 2 x i s the e x t e r n a l f o r c e on the body i n the x d i r e c t i o n . For convenience of n o t a t i o n equations 4.1 and 4.2 are r e w r i t t e n as 44 [M]U) + [K]{*} - { Fl}r - ( F 2 ) r - h (  M b * + K s r " { F L > T { * } - {F2}T{,J,} = f where r - r x , K 8 = K x, {F 1} - .{F 3 x} f {F 2} = { F 1 x> (4.3) a n d f o = F 2 x C o m p a t i b i l i t y c o n s t r a i n t s are introduced between the f l u i d and body motions by s p e c i f y i n g the normal and t a n g e n t i a l v e l o c i t y components of the f l u i d and the body t o be the same on the f l u i d - b o d y i n t e r f a c e . T h i s leads to a system of o r d i n a r y d i f f e r e n t i a l equations f o r the coupled system, (4.4) -{F 1}^ „ — r ) -1 r -{F 2}T o }' {ty} 0 > 4 1 r r < ) V ) v. { h o > 45 4.3 Frequency Domain A n a l y s i s S o l u t i o n of the system of equations 4.4 g i v e s the s o l u t i o n of the l i n e a r i z e d coupled system f o r any set of input f o r c e s {h 0} or f 0 . In the time domain, v a r i o u s techniques of time stepping procedures can be a p p l i e d to the system of equations 4.4. Since the o b j e c t i v e of t h i s i n v e s t i g a t i o n i s the s o l u t i o n of the problem f o r p e r i o d i c flows, where the p e r i o d i c f o r c i n g terms can be expressed as <h 0> - { K Q } e l w t , f » F e i w t o o (4.5) the s o l u t i o n of the l i n e a r problem may a l s o be represented in the p e r i o d i c form {*) - { $ } e l u , t , r - r e (4.6) and a l i n e a r system of equations i s obtained iw[M] + [K], -iuKF 1} - {F 2} - i w l F 1 } 1 - { F 2 } T , iu>M, + b TuT 0} r to (4.7) 46 I t i s to be noted that the bandwidth i n t h i s system of complex equations i s l a r g e due to the c o u p l i n g of a l l the i n t e r f a c e nodes. 4.4 Non-dimensional Parameters A simple dimensional a n a l y s i s y i e l d s the appropiate non-dimensional parameters as r e s p o n s e max s t Mi P L ' where r m a x i s the amplitude of o s c i l l a t i o n of the mass, r s t i s the d e f l e c t i o n of the s p r i n g i f the f o r c e f 0 were a steady f o r c e i n s t e a d of the amplitude of a s i n u s o i d a l f o r c e , 0 i s the phase angle, v i s the kinematic v i s c o s i t y of the f l u i d , P the d e n s i t y of the f l u i d , ^n , the n a t u r a l frequency of the s p r i n g mass system. R e s u l t s of response {* m a x, $} versus . frequency are r s t n presented f o r d i f f e r e n t v i s c o s i t i e s , —1 and d e n s i t y r a t i o s *b . D " p L 7 Some a n a l y t i c work has been done in the past to solve the l i n e a r Navier-Stokes equations to o b t a i n the f o r c e s on v i b r a t i n g bodies. S t u a r t [ 5 ] has presented r e s u l t s f o r v i b r a t i n g c i r c u l a r c y l i n d e r s . The r e s u l t s are presented i n the form of added mass and damping c o e f f i c i e n t s , which fo r a 47 given c r o s s s e c t i o n a l shape depend s o l e l y on one parameter, the nondimensional frequency = B = L 2 ( u / i / ) . Since no p u b l i s h e d r e s u l t s are known f o r square c y l i n d e r s to compare with, the added mass, and damping c o e f f i c i e n t i s p l o t t e d and compared with r e s u l t s f o r c i r c u l a r c y l i n d e r s . Comparison w i l l a l s o be made fo r the added mass c o e f f i c i e n t i n the l i m i t of high f r e q u e n c i e s with the value p r e d i c t e d f o r i n v i s c i d flow. To d e r i v e the added mass and damping c o e f f i c i e n t the equation of motion of the body i s r e w r i t t e n as (M. + M ) r + Cr + K„ r • f„ b a 8 ° ( 4 . 9 ) where the f l u i d f o r c e s are represented by two components, one in phase with the a c c e l e r a t i o n , which shows up as an added mass M a , and a damping f o r c e out of phase with the a c c e l e r a t i o n , which i s represented by an e q u i v a l e n t damping term C. Representing f 0 and r i n p e r i o d i c form as fo - *o e i U t . r - r e ^ ^ + •) s u b s t i t u t i n g i n equation 4 . 9 and equating the r e a l and imaginary p a r t s g i v e s 48 1. K. M a = ~ — ^ c ° 8 4 > + - 4 ~ M. <o2r b (4.10) t o 2 r I f M a = K , ( p L 3 ) where K , i s the added mass c o e f f i c i e n t , p L 3 i s the mass of f l u i d d i s p l a c e d by l e n g t h L of the square c y l i n d e r then, P L " 2 r U ' ( 4 . 1 1 ) And s i m i l a r l y p u t t i n g C/u = K 2 ( p L 3 ) where K 2 i s the damping c o e f f i c i e n t g i v e s 2 K2 " —rj [ ~ -fw sin*>] (4.12) w a ) 2 r 4.5 F i n i t e Element Computation The f i n i t e element computation i s bounded by the f l u i d domain L , x L 2 ( f i g u r e 4.2). The flow i s assumed to be symmetric with r e s p e c t to the x a x i s and so flow i s s o l v e d f o r i n the h a l f domain bounded by the f o l l o w i n g l i n e s 49 X F i g u r e 4.2 - Geometry of the Problem r g=symmetry axes r L=lower boundary r y = v e r t i c a l s e c t i o n s rb = f l u i d - b o d y i n t e r f a c e Q u a n t i t i e s L, and L 2 are to be assig n e d such that the f i n i t e boundaries are s u f f i c i e n t l y f a r away from the body to approximate an i n f i n i t e f l u i d domain. From the symmetry of the flow i t i s obvious t h a t ^ and 50 ¥ y y a r e odd, and they, together with the shear s t r e s s , v a n i s h at the symmetry axes r . From the . d e f i n i t i o n of ty , i t f o l l o w s t h at • - 0 , ifx « v - 0 , tyyy - u y - 0 , i p x x = - v x = 0 on r 8 < 4 ' 1 3 a > However, u - ty„ and u • ty , are unknowns and are allowed y y to vary along t h i s p a r t of the boundary. The s p e c i f i c a t i o n of the no s l i p boundary c o n d i t i o n s on T y and r L allow only + x x and ¥ y y r e s p e c t i v e l y as the v a r i a b l e s on these edges. The no s l i p .condition on the body i s enforced by equating the flow v e l o c i t y u - ^ y to that of the body ( r ) , whereas v=-^ x=0 everywhere on r b . On the v e r t i c a l edges of the body the stream f u n c t i o n v a r i e s l i n e a r l y as + • * r 8 ~ •<*> (4.i3b) where tyj-B i s the stream f u n c t i o n value on r , and s i s the running c o r d i n a t e as shown i n f i g u r e 4.2. On these edges ¥ x x i s a l s o unknown. S i m i l a r l y on the h o r i z o n t a l p a r t of r f e the unknown stream f u n c t i o n i s expressed i n terms of the other two v a r i a b l e s thus 51 (4.13c) In g e n e r a l , the aforementioned boundary c o n d i t i o n s are w r i t t e n as * - 0 , * x - 0 , t y - ?, * x x - 0 , * x y = 0 , * y y = 0 o n r s * - 0, * X - 0 . + y - 0, * x x - ? , * x y - 0 , * y y = 0 on r y 1> - 0, * x " 0t * y - 0 » * x x e 0 , iJ» Xy = 0 , * - * r 8 - s ( r ) , * x - o, * y - i, * x x = ? , * x y = 0 , * y y = 0 on T b ( V e r t i c a l ) * x y " ° » * y y = ? o n ^ ( H o r i z o n t a l ) (4.14) At the corners of the body there i s abrupt change i n geometry. T r i p l e noding ( f i g u r e 4.3) i s r e s o r t e d to at these c o r n e r s i n order to p r o p e r l y s p e c i f y the boundary c o n d i t i o n s along both the edges. 52 j I k fe y\ F i g u r e 4.3 - T r i p l e Noding Three nodes i , j , k ( f i g u r e 4.3) with the same c o - o r d i n a t e s are s p e c i f i e d with the f o l l o w i n g nodal parameters T s x2"' v " ' ' T x "» T y * x y " °» * y y = 0 a t i xx v t x x ' i * x y " °» * y y = ? a t J (4.15) * - * r 8 " C ^ > C r ) , t x - 0, * y - r , = 0, *xy " ° . * y y = ( * y y > j a t k T h i s p r o v i d e s c o n t i n u i t y a c r o s s each element edge by making the v a r i a b l e 1fxx at node i and node j the same, and matching the v a r i a b l e •yy at nodes j and k. Since the f o r c e s on the body are obtained by i n t e g r a t i n g the a p p r o p r i a t e expressions along the edge the s t r i c t s p e c i f i c a t i o n of the boundary 53 c o n d i t i o n s on the body i n t e r f a c e i s very important to achieve the c o r r e c t r e p r e s e n t a t i o n of the f l u i d f o r c e s . 4.6 Numerical R e s u l t s Numerical r e s u l t s obtained f o r the problem d e f i n e d i n the p r e v i o u s s e c t i o n s are presented here. F i r s t the checks made on the accuracy of the computer program are presented. Then f o r a c e r t a i n g r i d , the e f f e c t s of the g e o m e t r i c a l and p h y s i c a l parameters such as the dimensions of the f i n i t e domain chosen, the boundary c o n d i t i o n s , the v i s c o s i t y and the d e n s i t y r a t i o are a l s o s t u d i e d . A l l the computer c a l c u l a t i o n s were done on an Amdahl 470 V/8 running under the MTS o p e r a t i n g system. Double p r e c i s i o n a r i t h m e t i c was used throughout to reduce the e f f e c t of round o f f e r r o r s . 4.6.1 Numerical Study Numerical r e s u l t s f o r the response of the mass and the flow p a t t e r n s are obtained f o r the two g r i d s shown in f i g u r e 4.4. The computational domain was chosen with L t = 11.0, L 2 = 5.0' and L=1.0. An i n i t i a l check on the computer program and the accuracy of the r e s u l t s was done by o b t a i n i n g the added mass and damping c o e f f i c i e n t f o r the square c y l i n d e r f o r values of the F i g u r e 4.4 - F i n i t e Element G r i d s 55 frequency parameter p . In the absence of known r e s u l t s f o r the square shape no d i r e c t comparison c o u l d be made. But f o r a b l u f f body o s c i l l a t i n g at a high frequency with a small amplitude the added mass i s mainly given by i n v i s c i d c o n s i d e r a t i o n s , and i s not a f f e c t e d much by the v i s c o s i t y . Hence the value of obtained f o r l a r g e values of 8 should approach the value p r e d i c t e d f o r i n v i s c i d flow. Comparison i s a l s o made with the valu e s presented by Stua r t ( r e f e r e n c e [5]) f o r c i r c u l a r c y l i n d e r s . By comparing the added mass c o e f f i c i e n t , K 1 f f o r l a r g e g with that p r e d i c t e d f o r i n v i s c i d flow an estimate of the accuracy i s obtained. The v a l u e s of the added mass c o e f f i c i e n t obtained f o r l a r g e v a l u e s of P f o r both g r i d s are presented i n t a b l e 4 . 1 . G r i d e 1 2 (a) 2 (b) 2 (c) 100 1.544 1.704 1.641 1.645 125 1.390 1.5.22 1.460 1.481 150 1.288 1.398 1.338 1.368 Table 4.1 - Added Mass C o e f f i c i e n t ( K , ) f o r V i b r a t i n g Square C y l i n d e r 56 R e s u l t s 1 and 2(a) are f o r G r i d s 1 and 2 r e s p e c t i v e l y . 2(b) i s the case when the boundary c o n d i t i o n s at the f a r edges of G r i d 2 are r e l a x e d and 2(c) are r e s u l t s obtained by i n c r e a s i n g the dimensions of G r i d 2 (see s e c t i o n 4.6.1). For B=150 G r i d 1 g i v e s K,=1.288 as compared to 1.19 f o r i n v i s c i d flow, while G r i d 2 g i v e s K,=1.398 ( G r i d 2 ( a ) ) . T h e o r e t i c a l l y , i n the l i m i t of very high f r e q u e n c i e s , K, should approach the value p r e d i c t e d f o r i n v i s c i d flow. But the r e s u l t s showed that K, approached u n i t y . T h i s can be a t t r i b u t e d to the very crude nature of the g r i d s . No f u r t h e r attempts were made at g r i d refinement as the double p r e c i s i o n complex s o l v e r r o u t i n e used f o r s o l v i n g the l i n e a r system of equations (equation 4.7) d i d not make use of handedness and any f u r t h e r g r i d refinement would have i n c r e a s e d the number of v a r i a b l e s so as to exceed the a v a i l a b l e v i r t u a l memory. T h i s i s not of much concern as the main purpose of t h i s i n v e s t i g a t i o n i s to prove the a p p l i c a b i l i t y of the theory developed and to u l t i m a t e l y extend i t to the n o n l i n e a r problem r a t h e r than to o b t a i n h i g h l y a c c u r a t e r e s u l t s f o r the l i n e a r p a r t . 4.6.2 P h y s i c a l Study As i s usual i n computational f l u i d flow problems the i n f i n i t e f l u i d domain i s represented by a f i n i t e domain. A s u f f i c i e n t l y l a r g e domain should model the i n f i n i t e domain f a i r l y a c c u r a t e l y . Here the e f f e c t of the f i n i t e computation 57 s i z e was s t u d i e d by i n c r e a s i n g the dimension L 1 and L 2 by a f a c t o r of 1.5, which g i v e s K,=1.368 (Grid, 2 ( c ) ) as compared with 1^ = 1.398 ( G r i d 2 ( a ) ) . I t i s to be noted that the value does not change much as at hig h f r e q u e n c i e s the f l u i d d i s t u r b a n c e s are c o n f i n e d c l o s e to the o s c i l l a t i n g body. The e f f e c t of the boundary c o n d i t i o n s s p e c i f i e d at f a r edges i s s t u d i e d by a l l o w i n g the u - v e l o c i t y to develop on r , and both u and v - v e l o c i t y to develop on r L . T h i s r e l a x a t i o n of the boundary c o n d i t i o n s g i v e s K,=1.337 ( G r i d 2 ( b ) ) , which i s an improvement on 1.398 ( G r i d 2 ( a ) ) . F i g u r e s 4.5 and 4.6 show p l o t s of K, and K 2 obtained f o r the d i f f e r e n t g r i d s f o r the square c y l i n d e r . K, f o r the c i r c u l a r c y l i n d e r ( r e f e r e n c e [5]) i s a l s o shown i n f i g u r e 4.5. Note that both K, and K 2, decrease as g i n c r e a s e s and tend to i n f i n i t y as $ tends to zero. A l s o , as expected, K 1 r f o r the square i s always higher than f o r the c i r c l e . The study of K, and K 2 g i v e s a complete e x p l a n a t i o n of the e f f e c t of the f l u i d on the motion of the body. But a b e t t e r d e s c r i p t i o n of the p h y s i c a l phenomena i s obtained by response p l o t s of the s p r i n g mass system and the f l u i d flow p a t t e r n s generated by the o s c i l l a t i n g mass. F i g u r e 4.7 shows a s e r i e s of amplitude response curves ( r - m a x - versus ) generated f o r d i f f e r e n t v i s c o s i t i e s — n — r 6 t ^ 3 ^ % f o r a s p e c i f i c d e n s i t y r a t i o Mb' p L=1.0. They are t y p i c a l of a damped s i n g l e degree of freedom system. The e f f e c t of the f l u i d i n the form of added mass and damping i s c l e a r . A l l the 58 resonance peaks are s h i f t e d to the l e f t of v- =1.0. The n reasons f o r t h i s s h i f t being twofold, f i r s t due to the damping i t s e l f which lowers the damped n a t u r a l frequency, but the second and more pronounced e f f e c t i s due to the added mass, which lowers the n a t u r a l frequency of the system. F i g u r e 4.9 shows another set of amplitude response curves f o r a d e n s i t y r a t i o of 10.0. Since the added mass i s small r e l a t i v e to the body mass the resonance peaks are s h i f t e d towards the r i g h t compared to f i g u r e 4.7. P l o t s of the phase angle <t> versus — are shown i n f i g u r e s 4.8, 4.10 and bear out n the e x p l a n a t i o n given above. F i g u r e s 4.11 show an example of the v a r i a t i o n of the flow p a t t e r n s i n one h a l f c y c l e of o s c i l l a t i o n . These p a t t e r n s are d e p i c t e d by v e l o c i t y v e c t o r s and s t r e a m l i n e s , at time i n s t a n t s • — . In these f i g u r e s the v e l o c i t y and streamfunction are v non-dimensionalized by f o ^ p L 3 u ) n and f 0 / p L 2 % r e s p e c t i v e l y . The d e t a i l s of the flow p a t t e r n s are c l e a r l y shown i n these f i g u r e s as the body goes through h a l f a c y c l e of o s c i l l a t i o n . F i g u r e 4.11(a) shows the developed flow when the body has i t s maximum negative v e l o c i t y . F i g u r e 4.11(b) i s at the time i n s t a n t when the v e l o c i t y of the body i s n e a r l y zero but note that the f l u i d flow c o n t i n u e s i n the same d i r e c t i o n due to i t s i n e r t i a . F i g u r e s 4.11(c), (d) and (e) show the flow r e v e r s a l d e t a i l s f o r small time s t e p s . F i n a l l y f i g u r e 4.11 ( f ) shows the developed flow i n the opp o s i t e d i r e c t i o n when the body i s n e a r l y at i t s maximum p o s i t i v e v e l o c i t y . 59 i 1 1 1 r Circle (Ref .15]) Grid 1 i r Grid 2(a) Grid 2(b) Grid 2(c) 4 CO o ID J I ' I J L 0.0 2.0 4.0 6.0 8.0 10.0 Figure 4.5 - Added Mass Coefficient (Kj) for Vibrating Square Cylinder 60 Figure 4.6 - Damping Coefficient (K 2) for Vibrating Square Cylinder 61 62 Figure 4.8 - Phase Angle curves (Density Ratio=1.0) 64 Figure 4.10 - Phase Angle Curves (Density Ratio=10.0) 65 66 t =1.000 1 = 3 . 3 7 9 x 1 0 2 = 2 . 7 0 3 « 1 0 ~ 5 3 = 2 . 0 2 7 , 1 0 ~ 3 4=) . 3 5 ) x 1 0 ~ ~ 5 5 = 6 . 7 5 9 x 1 0 " 4 6 = 2 . 0 1 4 x 1 0 ~ ' ° STRERM FUNCTION CONTOURS V E L O C I T Y VECTORS sc= 0.75-= 1.059 .10'' (b) Figure 4.11 - (Cont'd) Figure 4.11 - (Cont'd) 1 6 8 i =1.100 1=-1 .883 « 10" ? 2=-l.507 — 1 « 10 ^ 3 = - l .130 . lo" * 4=-7.535 x 10" 4 5=-3.767 . 10-4-6=-l.122 . 10- 1 0 7 = 3.767 8=7.535 9=1 .130 x 10"" i S T R E R M F U N C T I O N C O N T O U R S — > 4 V E L O C I T Y V E C T O R S S C : 0.75"= 3.442 x 10 (d) Figure A.11 - (Cont'd) 69 H 1 VELOCITY VECTORS sc= 0.75-= 5.800 « io J (e) Figure 4.11 - (Cont'd) Figure 4.11 - (Cont'd) 71 CHAPTER V CONCLUSIONS A f i n i t e element method f o r the a n a l y s i s of the i n t e r a c t i o n of a v i s c o u s f l u i d and an e l a s t i c a l l y supported r i g i d body was presented. The n o n l i n e a r equations of the coupled problem were formulated f o r a two-dimensional incompressible v i s c o u s flow. As the f i r s t approximation f o r a s o l u t i o n by the p e r t u r b a t i o n method the equations were l i n e a r i z e d and the l i n e a r system was a p p l i e d to the problem of an e l a s t i c a l l y supported, square shaped, mass s u b j e c t e d to a harmonic f o r c e i n otherwise s t i l l f l u i d . I t was shown that t h i s problem i s e q u i v a l e n t to the same spring-mass system su b j e c t e d to a p e r i o d i c f a r flow. Response p l o t s f o r the coupled system were obt a i n e d . The e f f e c t of the f l u i d , i n the form of added mass and damping, was c l e a r l y e v i d e n t i n the s h i f t of the resonance peaks of the amplitude response curves of the mass. The added mass and damping c o e f f i c i e n t s f o r the square shape were presented as a f u n c t i o n of the frequency parameter. C o n s i d e r i n g the coarse nature of the g i r d s used, the added mass c o e f f i c i e n t s f o r high f r e q u e n c i e s agree w e l l with those obtained by the i n v i s c i d flow theory. The e f f e c t of r e s t r i c t i n g the numerical computation to a f i n i t e domain as an approximation to an i n f i n i t e domain was a l s o s t u d i e d . I t was done by r e l a x i n g the boundary c o n d i t i o n s 72 at the f a r boundaries and by i n c r e a s i n g the s i z e of the computational domain i t s e l f . These d i d not show much e f f e c t on the r e s u l t s suggesting that the dimensions chosen were s u f f i c i e n t l y l a r g e . T h i s was s p e c i a l l y t r u e at high f r e q u e n c i e s when most of the d i s t u r b a n c e s i n the f l u i d are c o n f i n e d c l o s e to the o s c i l l a t i n g body. Having proved the a p p l i c a b i l i t y of the theory developed f o r the l i n e a r case, f u r t h e r e f f o r t i s r e q u i r e d i n s o l v i n g the n o n l i n e a r problem by c o n s i d e r i n g the e f f e c t s of the higher order terms. 73 REFERENCES 1. Sarpkaya, T. and Isaacson, M., Mechanics of Wave Forces  on O f f s h o r e S t r u c t u r e s , Van Nostrand Reinhold, New York, 1981. 2. Thompson, W. (Lord K e l v i n ) and T a i t , P.G., T r e a t i s e on  N a t u r a l Philosophy, V o l . 1 , 1879, Vol.2, 1883, Cambridge U n i v e r s i t y P r e s s . 3. Lamb, H., Hydrodynamics , Dover P u b l i c a t i o n s , New York, 6th ed., 1932. 4. B a t c h e l o r , G.K., An I n t r o d u c t i o n to F l u i d Dynamics, Cambridge U n i v e r s i t y P r e s s , 1973. 5. S t u a r t , J.T., " P e r i o d i c Boundary Layers", F l u i d Motion  Memoirs-Laminar Boundary Layers, Rosenhead, L., Ed., Oxford U n i v e r s i t y Press, 1963, pp.390. 6. Tuann, S.Y. and Olson, M.D.,"Numerical S t u d i e s of Flow around a C i r c u l a r C y l i n d e r by a F i n i t e Element Method", S t r u c t u r a l Research S e r i e s , Report No.16, Dept. of C i v i l Eng., U n i v e r s i t y of B r i t i s h Columbia, December 1976. 7. Davis, R.W. and Moore, E.F., "The Numerical S o l u t i o n of Flow around Squares", Proceedings of Second I n t e r n a t i o n a l  Conference on Numerical Methods in Laminar and Turbulent  flow, Venice, I t a l y , 1981, pp. 279-290. 8. Olson, M.D., " V a r i a t i o n a l F i n i t e Element Methods f o r Two Dimensional and Axisymmetric Navier Stokes Equations", F i n i t e Elements on Flow Problems - V o l 1 , Eds. G a l l a g h e r , R.H., et a l . , John Wiley, New York, 1975. 9. Isaacson, M. de St.Q., "The Forces on C i r c u l a r C y l i n d e r s i n Waves", Ph.D. T h e s i s , U n i v e r s i t y of Cambridge, September 1974. 10. F i n l a y s o n , B.A., The Method of Weighted R e s i d u a l s and  V a r i a t i o n a l P r i n c i p a l s , Academic Press, New York, 1972. 11. Cowper, G.R., Kosko, E., Lindberg, G.M., and Olson, M.D., " S a t i c and Dynamic A p p l i c a t i o n s of a High P r e c i s i o n T r i a n g u l a r P l a t e Bending Element", American I n s t i t u t e of  A e r o n a u t i c s and A s t r o n a u t i c s J . , Vol.7, No. 10, Oct. 1969, pp. 1957-1965. 12. Z i e n k i e w i c z , O.C., The F i n i t e Element Method, McGraw H i l l , London, 1977. 13. S c h l i c h t i n g , H., Boundary Layer Theory, McGraw H i l l , New York, 4th. ed., 1960. 74 14. Fung, Y.C., Foundations of S o l i d Mechanics, P r e n t i c e - H a l l , New J e r s e y , 1967. 15. Kawahara, M., Yoshimura, N., Nakagawa, K. and Ohsaka, H., "Steady and Unsteady F i n i t e Element A n a l y s i s of Incompressible V i s c o u s F l u i d " , I n t e r n a t i o n a l J o u r n a l f o r  Numerical Methods i n E n g i n e e r i n g , Vol.10, No.2, 1976, pp. 437-456. 16. Savor, Z., "A Modal F i n i t e Element Method f o r the T r a n s i e n t Navier Stokes Equations", M.A.Sc. T h e s i s , U n i v e r s i t y of B r i t i s h Columbia, September 1977. 17. Leech, C. M., "Hamilton's P r i n c i p l e A p p l i e d to F l u i d Mechanics", The Q u a t e r l y J o u r n a l of Mechanics and A p p l i e d  Mathematics, Volume XXX, Part 1, February 1977, pp.107-130. 18. Cook, R. D., Concepts and A p p l i c a t i o n s of F i n i t e Element  A n a l y s i s , John Wiley and -Sons, New York, 2nd. ed., 1981. 75 APPENDIX A M a t r i c e s A s s o c i a t e d with the D i s c r e t i z a t i o n  of the F l u i d Domain The element matrices d e s c r i b e d i n s e c t i o n 3.3 which are used i n equation 3.12 are presented here. A d e t a i l e d d e s c r i p t i o n of the f o r m u l a t i o n of the 18 degree of freedom element can be found i n r e f e r e n c e [11]. D e r i v a t i o n s of the m a t r i c e s k ^ j and ^ i j k can be found i n r e f e r e n c e [12] and the d e r i v a t i o n of the matrix m . can be found i n r e f e r e n c e [16]. The geometry and c o o r d i n a t e systems used f o r t h i s element are shown in f i g u r e 3.1. The v a r i a b l e s at each of the three nodes are * ' V V *xx' *xy' *yy The i n t e g e r exponents of the incomplete q u i n t i c polynomial used to approximate the stream f u n c t i o n in equation 3.7 are m i =0, 1, 0, 2, 1, 0, 3, 2, 1, 0, 4, 3, 2, 1, 0, 5, 3, 2, 1, 0. 1^=0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, 3, 4, 0, 2, 3, 4, 5. The components of the l i n e a r s t i f f n e s s matrix k.^ , the 76 n o n l i n e a r s t i f f n e s s matrix q ^ v » a n o " the mass matrix m. . i n terms of the g e n e r a l i z e d c o o r d i n a t e s are k i j = (m i-l) (nij - 1 ) F(m i+n\j-4,n i+n ;.) + • F(mi+m:. ,n i+n : . -4) + [ I I ^ I K ( n ^ - 1 ) (n^-l) + m.^ (m_.-1) (n .-1) F(m i+m ; . -2,n i+nj -2 ) + [nun., ( I T K - 1 ) + m^ni ( i r ^ - 1 ) ] • (A.2) G (iru+itK-2 ,n i+nj-l) q i j k = [ ( n i n k - i n i n ] c ) i n j ( m j - l ) + (n.n^-m^) m. (m.-l) ] • F(m i+mj+m K -3,n i+n ;.+n k-l) /2 + [ ( n ^ - n u n ^ n_. ( n ^ - 1 ) + < n j 1 1 ^ - ^ 1 ^ ) n.^  ( r u - 1 ) ] F (i^+mj+n^-l , ^ + 1 ^ + 1 ^ - 3 ) / 2 - m.m. (m. +m . - 2 ) G (m. +m .+m,-3 ,n . +n . +n, ) / 2 1 3 1 3 1 3 k ' 1 3 k - n . n . (m. +m .) G (m. +m . +m, -1 ,n . +n .+n, - 2 ) / 2 1 3 1 3 1 j K ' 1 3 k ' (A.3) m i j = m i i n j F ( i n i + i n j ~ 2 ' n i + n j ) + n i n j F (m^+nij ,ni+n _.- 2) (A.4) where the f u n c t i o n s F and G are the expre s s i o n s obtained by i n t e g r a t i n g a general term £ m n n over the area of the t r i a n g l e and along the edge n = 0 r e s p e c t i v e l y , 77 m!n I n I F(m,n) = c n + 1 a m + 1 - ( - b ) m + 1 ( i n + n + 2 ) 1 (A. 5) G(m,n) = ^ a m + 1 - ( - b ) m + 1 i f n?0 S+X a - <-b)""* i f n=0 G(m,n) = o (A.6) and a, b r c, d e s c r i b e the geometry of the element shown i n f i g u r e 3.2. The terms i n v o l v i n g the f u n c t i o n G are a s s o c i a t e d with the boundary i n t e g r a l i n equaftion 3.4. The element m a t r i c e s are expressed i n terms of the g l o b a l nodal degrees of freedom by using the t r a n s f o r m a t i o n of equation 3.11 20 20 K i j = J l s=l S " S * J k r s (A.7) 20 20 r=l s =l r i S 3 r s (A.8) 7 8 = 20 20 20 l j k " J l sk J l S r i S s J S t k 9 r s t (A. 9) where [S] = [ T _ 1 ] t R ] (A.10) [R] i s the r o t a t i o n matrix which can be w r i t t e n i n terms of the submatrix [Rj] as ( A . 1 1 ) Ri 0 0 IR] « I 0 Rj 0 0 0 R, 79 where 1 0 0 0 0 .0 0 cose sine 0 0 0 0 -sine cose 0 0 0 0 0 0 cos 2 6 2sin6cos6 s i n 2 6 0 0 0 -sin6cos6 c o s 2 6 - s i n 2 e sin6cos6 0 0 0 s i n 2 6 -2sin6cos6 cos 2 e (A.12) The t r a n s f o r m a t i o n matrix [T] i s shown i n t a b l e A.l. It i s i n v e r t e d n u m e r i c a l l y f o r each element to y i e l d [ T ~ 1 ]. TABLE A . l TRANSFORMATION MATRIX [T] 1 -b 0 b 2 0 0 - b 3 0 0 0 b 4 0 0 0 0 - b 5 0 0 0 0 0 1 0 -2b 0 0 3b 2 0 0 0 - 4 b 3 0 0 0 0 5b 4 0 0 0 0 0 0 1 0 -b 0 0 b 2 0 0 0 - b 3 0 0 0 0 0 0 0 0 0 0 0 2 0 0 -6b 0 0 0 12b 2 0 0 0 0 -20b 3 0 0 6 0 0 0 0 0 1 0 0 -2b 0 0 0 3b 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 -2b 0 0 0 2b 2 0 0 0 - 2 b 3 0 0 0 1 a 0 2 a 0 0 3 a 0 0 0 4 a 0 0 0 0 5 a 0 0 0 0 0 1 0 2a 0 0 3a 2 0 0 0 4 a 3 0 0 0 0 5 a 4 0 0 0 0 0 0 1 0 a 0 0 2 a 0 0 0 3 a 0 0 0 0 0 0 0 0 0 0 0 2 0 0 6a 0 0 0 12a 2 0 0 0 0 20a 3 0 0 0 0 0 0 0 0 1 0 0 2a 0 0 0 3a 2 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 2a 0 0 0 2a 2 0 0 0 2 a 3 0 0 0 1 0 c 0 0 2 c 0 0 0 3 c 0 0 0 0 4 c 0 0 0 0 5 c 0 1 0 0 c 0 0 0 2 c 0 0 0 0 3 c 0 0 0 0 4 c 0 0 0 1 0 0 2c 0 0 0 3 c 2 0 0 0 0 4 c 3 0 0 0 0 5 c 4 0 0 0 2 0 0 0 2c 0 0 0 0 2 c 2 0 0 0 0 2 c 3 0 0 0 0 0 0 1 0 0 0 2c 0 0 0 0 3 c 2 0 0 0 0 4 c 3 0 0 0 0 0 0 2 0 0 0 6c 0 0 0 0 12c 2 0 0 0 0 20c 3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5a 4 c , 2 3 , 4 3a c -2a c , 4 , 3 2 -2ac -3a c 5 A 2 3 c -4a c ba c 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 5b 4 c , . 2 3 , . 4 3b c -2b c 2bc -3b c 5 ,1.2 3 c -4b c - 5 b c 4 0 0 o 81 APPENDIX B M a t r i c e s A s s o c i a t e d with the Force Coupling V e c t o r s The f l u i d f o r c e v e c t o r s d e r i v e d i n s e c t i o n 3.4 are presented here. The shear f o r c e v e c t o r d e f i n e d i n equation 3.4 i s <h} = < 0 , 0 , 0 . 2 ( a + b ) , 0 . - 2 ( a + b ) , 3 ( a ! - t > » ) , 0 , ( b ! - a ' ) , 0 , 0 , ^ ^ 4(a 3 +b 3 ) , -^(a'+b 1 ) ,0,0,5(a*-b*),±(b'-a' ) ,0,0.0} 3 2 The d e v i a t o r i c f o r c e v e c t o r d e f i n e d i n equation 3.23 i s {d} = <0 ,0 ,0 ,0 .2 (a+b ) , 0 ,0 ,2 (a ! -b ! ) , 0 , 0 , 0 .2 (a 3 +b 3 ) , 0 , 0 . 0 ,0 .0 , (g 2) 0,0,0} The l i n e a r and time d e r i v a t i v e c o e f f i c i e n t vectors,{e} and {g}, i n the pressure e x p r e s s i o n d e f i n e d in equation 3.29 are {e} = {0 .0 ,0 ,0 ,0 .0 ,0 ,2 (a+b ) ,0 ,6 (a+b ) ,0 .3 (a ' -b ' ) , 0 ,3 (a ' -b ! ) . 0 , 0 ,0 .2(a ] +b 5 ) .0 ,0} (B.3) 82 {g> = {0 ,0 . - (a+b) ,0 , ( -a '+b ' ) .0 ,0 . - (a 1 + b ] ) , 0 , 2 3 ( -a'-t -b ' ) .0 .0.0,0,0.0.0. ) (B.4) The n o n l i n e a r c o e f f i c i e n t matrix [ f ] d e f i n e d i n equation 3.29 i s given i n t a b l e B.1. The l i n e a r and time d e r i v a t i v e pressure f o r c e v e c t o r s d e f i n e d i n equation 3;30 are {a} = {0 ,0 ,0 .0 .0 ,0 ,0 , - (a ! +2ab+b ! ) ,0 , (3a ! +6ab+3b' ) ,0 . ( a ] - 3 a b ' - 2 b ' ) . 0 , ( a ' - 3 a b ' - 2 b ' ) , 0 , 0 , 0 . ( a ' + 2 a b ' + 3 b « ) , 0 . 0 , } , n c \ <z> = < 0 . 0 . ( s ' + a b + b ' ) . 0 . ( a ' - a b ' - b ' ) . 0 . 0 , ( a « + a b ' + b « ) 2 2 6 2 3 72 a" 4 0 .0 .0 , ( a j_+ab«+3b^ ) ,o . 0 . 0 ,0 ,0 0 0 0} (B.6) 20 4 10 The n o n l i n e a r pressure f o r c e matrix [w] d e f i n e d i n equation 3.30 i s given i n t a b l e B.2. TABLE B . l NON-LINEAR PRESSURE COEFFICIENT MATRIX 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -«-b 0 2 .2 - b 'o 0 - A b 3 0 0 0 - A , * 0 0 0 0 o 0 0 0 0 u W 0 0 0 **-»* 0 0 0 0 2 . W 0 0 0 0 0 0 0 0 0 o • 0 0 0 0 0 0 0 0 0 '0 0 0 0 0 0 0 0 -2(.3+b3) 0 0 -Ab* 0 0 0 -2(Ab5) 0 0 0 0 o o 0 0 0 .'-b2 0 0 0 ~~r 0 0 0 j(Afc5> T 0 0 0 0 l(.'-b') 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 D 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 :o 0 0 -* -b 0 -3<Ab*> 0 0 -3(.5+A 0 0 0 0 0 0 0 0 o o 0 0 0 . w J 0 0 T 0 0 0 0 0 0 0 iO(»7+b7> 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 - a 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 .*-** —r 0 0 0 • -b 0 0 0 0 0 0 0 !<.B-ba) 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 CO 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 co TABLE B .2 NON-LINEAR PRESSURE FORCE COEFFICIENT MATRIX 0 » ' 0 -•<•*> 0 -,'t.ti*3 0 0 -«*-.bV T 2 3 " 0 *2+Ub*b2 0 U3-Zab' w J • 20 4 5 <u -2ab -4b T I o O 0 0 0 o - • ' • . . ' • M b 3 0 - l . ' - 2 . b ' - b ' 0 0 A A n . ' - 1 . -2*bJ-  6 3 2 " " 0 0 0 0 j . ' ^ b ? ^ 0 0 0 0 0 0 0 To I 7 " T T ? " * T ' ° 0 0 0 5.'-3.b"-I0b 3a -3.b - * V 0 - t ' - b ' - J b * 5 5 « . - 4 . b 4 4 b 0 0 . 0 0 S.'+IO.bVjb* 2 1 3 3 28 T 4 2.°««b : '+2b° TT T 3 3. -3«b -411. 36 T 36 C D 85 APPENDIX C The M o d i f i e d V a r i a t i o n a l P r i n c i p l e The r e s t r i c t e d v a r i a t i o n a l p r i n c i p l e d e r i v e d by Olson [8] g i v e s the kinematic and n a t u r a l boundary c o n d i t i o n s as d i s c u s s e d i n s e c t i o n 3.2. A d d i t i o n a l terms have to be added to t h i s r e s t r i c t e d f u n c t i o n a l when p r e s c r i b e d kinematic boundary c o n d i t i o n s a r i s e . To accommadate such boundaries the r e s t r i c t e d v a r i a t i o n a l p r i n c i p l e i s m o d i f i e d here to a form of r e s t r i c t e d Hamilton's p r i n c i p l e . T h i s allows one to thin k i n the u sual terms of e x t r e m i s i n g the time i n t e g r a l of some f u n c t i o n a l (which normally r e p r e s e n t s some s o r t of k i n e t i c co-energy and v i r t u a l work) ( r e f e r e n c e [ 1 7 ] ) . The r e s t r i c t e d f u n c t i o n a l i s + J _ * { 2 v ( * u ) * n (C. 1 ) where ty*t are p r e s c r i b e d on S, and S 2 r e s p e c t i v e l y , S i s the complete boundary to the f l u i d domain, 86 T 5 n "= v ( * n n " * « > and the same n o t a t i o n as i n s e c t i o n 3.2 i s f o l l o w e d here. S e t t i n g ' 2 6 / I dt - 0 t l (C.2) while h o l d i n g the b a r r e d terms f i x e d d u r i n g the v a r i a t i o n y i e l d s equation 2.1 as the E u l e r equation and the f o l l o w i n g boundary c o n d i t i o n s , «^ = 0 o r P#5 - 0 o n S x n S * * n " 0 o r T Cn * 0 o n S 2 n s and ( C ' 3 ) * - * * . « * - 0 o n S l * - *!,*. «* n - 0 on s„ 87 APPENDIX D A Beam Bending Example The method presented i n t h i s t h e s i s , f o r f o r m u l a t i n g the equations of motion f o r the coupled fluid-mass system, i s i l l u s t r a t e d here by c o n s i d e r i n g the f a m i l i a r beam bending problem. Consider a c a n t i l e v e r beam with a t o r s i o n a l s p r i n g EI K s F i g u r e D.1 - The Beam Problem ( s p r i n g constant=K g) support at one end ( f i g u r e D.1). The p o t e n t i a l energy f u n c t i o n a l f o r t h i s problem i s 2 o d x z 2 8 dx J * " L (D. 1 ) where any e x t e r n a l loads are n e g l e c t e d f o r the purpose of t h i s e x e r c i s e . Using a s i n g l e c u b i c beam element ( f i g u r e D.2) and f o l l o w i n g standard f i n i t e element procedures ( r e f e r e n c e [18]) 88 r J ,0 r L H w(x) = a, + a 2 x + a 3 x 2 + a,x 3 F i g u r e D.2 - The Beam Element the equations of s t a t i c e q u i l i b r i u m f o r the system are where [K] EI L 7 {d> - {0} 12 6L -12 6L 6L 4L 2 -6L 2L 2 -12 -6L 1 2 -6L 6L 2L 2 -6L 4L 2+K S L EI ( D . 2 ) (D.3) and {d} = f * i l e 2 ( D . 4 ) Now c o n s i d e r the same problem as a coupled one, c o n s i s t i n g of (a) the beam, and (b) the t o r s i o n a l s p r i n g , 89 s e p a r a t e l y ( f i g u r e D.3). G, M are the r o t a t i o n and the moment EI. L F i g u r e D.3 - The Coupled Beam Problem at end x = L of the beam. To d e r i v e the equations of e q u i l i b r i u m f o r t h i s coupled problem, a procedure, s i m i l a r t o that presented i n Chapters 2 and 3 f o r d e r i v i n g the equations of motion f o r the coupled f l u i d - m a s s system, i s f o l l o w e d here. The equation of s t a t i c e q u i l i b r i u m of the s p r i n g i s K 9 + M = 0 s where M • E I d2w (D.5) i x=L In terms of the nodal v a r i a b l e s i t becomes K s § + EI{ 6L, 2 L 2 , -6L, 4L 2 }{d} «= 0 T7 (D.6) 90 For the beam a c o n s i s t e n t f o r c e v e c t o r i s d e r i v e d r e p r e s e n t i n g the f o r c e s due to the " p r e s c r i b e d " r o t a t i o n , S, at x=L. The i n t e r n a l v i r t u a l energy due to the p r e s c r i b e d G~ i s 6v = & 6M| X_ L ( D 7 ) where SM i s the v i r t u a l i n t e r n a l moment at x = L due to v i r t u a l displacements a p p l i e d at the degrees of freedom where the displacement i s not p r e s c r i b e d . Equating the i n t e r n a l v i r t u a l energy to the e x t e r n a l v i r t u a l work g i v e s , {6d} T {F} = ( E l S ) 6 l i l * ] T L d x T J x " L (D.8) T h i s g i v e s the c o n s i s t e n t l o a d v e c t o r {F}, {F} EI L 3" 6L 2 L 2 -6L 4L 2 e (D . 9 ) Hence the equations of e q u i l i b r i u m f o r the beam are 91 EI T7 12 6L -12 0 6L 4L 2 -6L 0 -12 -6L 12 0 0 0 0 1 {d} + EI IT 3 " 6L 2L 2 -6L -1 e (D.10) Combining equations D.6, D.10 and e n f o r c i n g displacement c o m p a t i b i l i t y by s p e c i f y i n g a, = §, g i v e s the system of equations f o r the "coupled" problem. EI 17 12 6L -12 6L 6L 4L 2 -6L 2L 2 -12 -6L 12 -6L 6L 2L 2 -6L 4L 2+K 5L EI {d} = 0 (D. 1 1 ) which are the same as equations D.2. In an approach analogous to the one followed i n Appendix C the system of equations (equation D.10) f o r the beam can a l s o be d e r i v e d by p r e s e n t i n g a f u n c t i o n a l whose v a r i a t i o n g i v e s the c o r r e c t E u l e r equation and boundary c o n d i t i o n s . Consider the f o l l o w i n g f u n c t i o n a l 92 J = 2 J o j2 " 2 r d w dx - ] 2 dx dx' x=L (D.12) S e t t i n g the v a r i a t i o n to zero, S J = EI d w Sw x dx' x=0 EI d w £w dx 5 L x=0 + EI I " w J o A Swdx ( wx - e )S EI djw dx 2 x=L = 0 (D.13) g i v e s dx' EI d_w = 0 0 s x i L A and Sw = 0 or x 8 w = 0 or wx = 9 o r l\ dx" EI d_w = 0 at x=0 2 EI d_w = 0 at x=0, L. 3 EI d2w dx 2 (D.14) at x=L For the s i n g l e c u b i c element example t h i s f o r m u l a t i o n g i v e s 93 12 6L -12 0 6L 4L 2 -6L 0 -12 -6L 12 0 0 0 0 _ {d} EI L 7 6L 2L 2 -6L 4L 2 e = o (D.15) which i s s i m i l a r to equation D.10. 

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