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Hydroelastic oscillations of square cylinders Bouclin, Denis N. 1979

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HYDROELASTIG OSCILLATIONS OF SQUARE CYLINDERS by DENIS N. BOUCLIN B.A.Sc,, U n i v e r s i t y of Toronto, 1975 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF • THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the department of Mechanical Engineering We accept t h i s t h e s i s as conforming t o the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA May, 1977 (c) Denis N. Bouclin, 1979 I n 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 o f t h e r e q u i r e m e n t s f o r a n a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l m a k e i t f r e e l y a v a i l a b l e f o r r e f e r e n c e a n d s t u d y . I f u r t h e r a g r e e 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 c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may b e g r a n t e d b y t h e H e a d o f my D e p a r t m e n t o r b y h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 D a t e / T ? ^ & } ? ? D E N I S B O U C L I N ABSTRACT A program of research has been undertaken t o examine the i n t e r -a c t i o n between v o r t e x shedding and the g a l l o p i n g type o s c i l l a t i o n which square c y l i n d e r s are subject t o when immersed i n a water stream. I t i s p o s s i b l e t h a t the f l u c t u a t i n g f o r c e from the vortex shedding c o u l d quench the g a l l o p i n g o s c i l l a t i o n i f i t a c t s as a f o r c e d o s c i l l a t i o n (independent of c y l i n d e r motion). An experiment was designed where a square c y l i n d e r w i t h one degree of freedom c o u l d o s c i l l a t e t r a n s v e r s e l y t o the water f l o w . The amplitude and frequency of the c y l i n d e r osc-i l l a t i o n were measured. By u s i n g a hot f i l m anemometer s p e c t r a of the f l u c t u a t i n g v e l o c i t y i n the wake were taken t o determine what f r e -quencies v o r t e x shedding occurred a t . The r e s u l t s show t h a t f o r v e l o c i t i e s g r e a t e r than the resonant v e l o c i t y the g a l l o p i n g o s c i l l a t i o n i s dominant and the c y l i n d e r motion c o n t r o l s the frequencies of the wake. For v e l o c i t i e s l e s s than the resonant v e l o c i t y no g a l l o p i n g occurs and the v o r t e x shedding seems t o c o n t r o l any c y l i n d e r motion which occurs. To e x p l a i n t h i s type of response a mathematical model has been c o n s t r u c t e d . The model i s a s e t of two coupled s e l f e x c i t e d o s c i l l a t o r s } one w i t h the c h a r a c t e r i s t i c s of the g a l l o p i n g o s c i l l a t i o n and the other w i t h the c h a r a c t e r i s t i c s of the f l u c t u a t i n g l i f t f o r c e from the v o r t e x shedding. Using the model some aspects of the obser-ved i n t e r a c t i o n are e x p l a i n e d . - i i -CONTENTS Page No. I . INTRODUCTION 1 I I . EXPERIMENTAL PROGRAM 6 2.1 Apparatus and Instrumentation 6 2.2 Experimental Procedure 10 2.3 Experimental R e s u l t s 14 i ) 3/4 i n . square c y l i n d e r 14 i i ) 1 i n . square c y l i n d e r 16 i i i ) 1/2 i n . square c y l i n d e r 18 i v ) Rectangular 2»1 C y l i n d e r 19 I I I . THE QUASI - STEADY MATHEMATICAL MODEL 22 IV. FLUID OSCILLATOR MATHEMATICAL MODEL 27 4.1 Model Synt h e s i s 27 4.2 Model Compared t o Experimental R e s u l t s 32 4.3 E f f e c t of Changing Damping and Mass Parameter 36 V. CONCLUSIONS 38 VI. RECOMMENDATIONS FOR FURTHER WORK 40 BIBLIOGRAPHY 41 APPENDIX A: SANTOSHAM'S EXPERIMENTAL RESULTS 43 APPENDIX B i ANALYTICAL SOLUTION TO THE FORCED VAN 47 DER POL EQUATION - i i i -ILLUSTRATIONS FIGURE NUMBER PAGE No. 1. Physical Model of Flow Past a Square Cylinder 54 2. Water Channel Layout 55 3. Model Mounting Apparatus 56 ka.. Block Diagram of Displacement Apparatus 58 kt. Voltage Displacement Calibration 58 5a. Block Diagram of Wake Frequency Apparatus 59 5b. Sample Spectrum of Wake Frequencies Behind 59 Oscillating Cylinder 6. Different Types of Cylinder Oscillation Observed 60 7. Amplitude of Cylinder Oscillation| d«.75 i n . 61 8. Frequency of Cylinder Oscillation; d«=.75 in,, 62 9. Wake Frequency; da a.75 i n . 63 10. Amplitude of Cylinder Oscillation? d=l i n . 64 11. Frequency of Cylinder Oscillation; d«l i n . 65 12. Wake Frequency; d=l i n , 66 13. Amplitude of Cylinder Oscillation; d=.5 i n . 68 lk. Frequency of Cylinder Oscillation; d«*.5 i n . 69 15. Wake Frequency; d"»5 i n . 70 16. Amplitude of Oscillation for 2 i l Rectangular 71 Cylinder 17. Frequency of Oscillation for 2»1 Rectangular 72 Cylinder 18. Independence of Damping from Oscillation Amplitude 73 f or U > 2 i - i v -PAGE No. 19. Width of Resonance Region 74 20. Phase Angle f o r Forced O s c i l l a t i o n 74 21. L i f t Force C o e f f i c i e n t C L and V e l o c i t y Component 75 C L I f o r Forced O s c i l l a t i o n 22. E f f e c t of Damping f o r Small U l F l u i d O s c i l l a t o r 76 Theory 23. E f f e c t of Mass Parameter n f o r Small U« F l u i d 77 O s c i l l a t o r Theory A l Amplitude of O s c i l l a t i o n f o r Square C y l i n d e r on 78 End of C a n t i l e v e r Beam A2 Frequency of O s c i l l a t i o n f o r Square C y l i n d e r on 79 End of C a n t i l e v e r Beam A3 Amplitude of O s c i l l a t i o n f o r Rectangular 2»1 80 C y l i n d e r on End of C a n t i l e v e r Beam A k Frequency of O s c i l l a t i o n f o r Rectangular 2 i l 81 C y l i n d e r on End of C a n t i l e v e r Beam LIST OF SYMBOLS a l as fundamental component of C ^ i n resonance r e g i o n ( f o r c e d c y l i n d e r motion) a 2 sc fundamental component of i n nonresonance r e g i o n ( f o r c e d c y l i n d e r motion) r f o u r i e r c o e f f i c i e n t s obtained from Fast F o u r i e r Transform G F y ss q u a s i - steady f o r c e / l P V^dZ 2 1 C L f F l u c t u a t i n g L i f t Force due t o v o r t e x shedding/ 1 pV^dZ 2 amplitude of f l u c t u a t i n g l i f t f o r c e c o e f f i c i e n t f o r s t a t i o n a r y c y l i n d e r G L o ss C L S3 f o r c e component a t frequency c y l i n d e r i s f o r c e d at/l^^^Z^ C L I SS C^sinjrf ° component of C^ i n phase w i t h c y l i n d e r v e l o c i t y CLM S3 C^cos^ « component of C^ i n phase w i t h c y l i n d e r a c c e l e r a t i o n d S S c y l i n d e r width G = e m p i r i c a l constant H SE e m p i r i c a l constant k S S s t i f f n e s s of s p r i n g s K s S S w s «= 2TTSU «* dimensionless s t r o u h a l frequency w n dimensionless frequency observed i n wake K S S m - mass/length of e l a s t i c system M ss mass of e l a s t i c system M a ss added mass of c y l i n d e r i n s t i l l water M e - mass of moving p a r t s of apparatus n N l p d ^ z mass parameter 2' M number of p o i n t s c a l c u l a t e d n u m e r i c a l l y R e - Vd « Reynolds number - v i -S « S t r o u h a l number X * A z N - p e r i o d of Fast F o u r i e r Transform U = V dimensionless v e l o c i t y w d n U - nU U:: • 1 m resonant v e l o c i t y U • t h e o r e t i c a l onset v e l o c i t y of g a l l o p i n g o s c i l l a t i o n V » f l u i d v e l o c i t y w n 9. K -» n a t u r a l frequency of e l a s t i c system i n s t i l l water M w a = n a t u r a l frequency of e l a s t i c system i n a i r w = s t r o u h a l frequency of v o r t e x shedding s y » displacement of c y l i n d e r y = £ o dimensionless displacement of c y l i n d e r d Y - nY Y* = amplitude of f o r c e d c y l i n d e r motion Y - dY z *» c y l i n d e r l e n g t h < m a r c t a r i (Y/U) « angle of a t t a c k Z = w t *« dimensionless time n /3 « non dimensional damping r a t i o y° » f l u i d d e n s i t y J m kinematic v i s c o s i t y 0 * phase of r e l a t i v e t o c y l i n d e r displacement = phase of C T_ r e l a t i v e t o c y l i n d e r displacement i n resonance i Lt r e g i o n Q = phase of r e l a t i v e t o c y l i n d e r displacement i n nonresonance r e g i o n AZ = time increment of numerical s o l u t i o n - v i i -ACKNOWLEDGEMENT The author would l i k e t o express h i s a p p r e c i a t i o n t o Dr. G.V, Parkinson, under whose auspices t h i s research was performed. A l s o , he would l i k e t o thank the t e c h n i c a l s t a f f i n the Mechanical Eng-i n e e r i n g Department f o r t h e i r advice and he l p i n making the exper-imental apparatus and i n s t r u m e n t a t i o n . A s p e c i a l thanks i s deserving t o the personnel of the H y d r a u l i c s Laboratory of the C i v i l E ngineering Department f o r making a v a i l a b l e the water flume f o r experimental t r i a l s . T h i s research has been f i n a n c e d by the N a t i o n a l Research C o u n c i l of Canada, grant A - 586. I . INTRODUCTION I t i s w e l l known th a t e l a s t i c a l l y mounted two dimensional b l u f f bodies are sub j e c t t o motion or o s c i l l a t i o n i f placed i n a f l u i d stream. I n such a f l u i d e l a s t i c system energy i s e x t r a c t e d from the f l u i d stream by the . s - f c r u c t u r e and o s c i l l a t i o n s o f c o n s i d e r -a b l e magnitude can b u i l d up. The aim of the engineer i n the a n a l y s i s of a dynamical system whether i t be f l u i d e l a s t i c or not i s t h r e e f o l d (Ref, l ) i F i r s t , the p h y s i c a l and geometrical parameters of the system need t o be recognized i n order t o e s t a b l i s h a p h y s i c a l s t r u c t u r a l model. Second, the aim i s t o l e a r n what f o r c e s are operating l n order t o s e l e c t the a p p r o p r i a t e a c t i o n scheme from which the equations of motion f o r the model can be w r i t t e n . F i n a l l y the a n a l y s i s should aim at a mathematical model, the s o l u t i o n of which w i l l demonstrate the e x i s t e n c e and uniqueness of the p h y s i c a l motion. I n the p a r t i c u l a r case of the two dimensional square c y l i n d e r i n a f r e e stream, many previous experiments ( b i b l i o g r a p h y i n r e f , 2,3) have e s t a b l i s h e d a p h y s i c a l model as shown i n f i g u r e 1, The f l o w about the c y l i n d e r i s c h a r a c t e r i z e d by two f i x e d s e p a r a t i o n p o i n t s from which shear l a y e r s develop. The shear l a y e r s separate the higher v e l o c i t y i n v i s c i d main flow from the low v e l o c i t y f l u i d i n the neighbourhood of the c y l i n d e r . These shear l a y e r s are i n h e r e n t l y unstable and r o l l up behind the c y l i n d e r t o form an a l t e r n a t i n g v o r t e x s t r e e t . I n the p h y s i c a l model the motion of the c y l i n d e r occurs i n the d i r e c t i o n y, normal t o the oncoming f l u i d v e l o c i t y V, The motion i s dependent upon a number of parameterst y - f ( d , m, wn, p , V,^ , w s) (1.1) - 2 -d «= c y l i n d e r width m « M/Z = c y l i n d e r mass/length Wn= c y l i n d e r n a t u r a l frequency (radians/sec) ft m non dimensional damping r a t i o p «= f l u i d d e n s i t y W « s t r o u h a l frequency of v o r t e x shedding s Performing a dimensional a n a l y s i s on the above v a r i a b l e s r e s u l t s i n the s m a l l e r set of parameters Y - //d - f (U - V/W d, n- 5- /> d 2Z , /3 , K « )[s ) (1 . 2 ) n M s W n I f the c y l i n d e r i s not two dimensional or the f l u i d stream i s t u r b u l e n t and subject t o v i s c o s i t y i n f l u e n c e other parameters are needed. These would be the aspect r a t i o Z/d, turbulence l e v e l u'/V and Reynolds number R «• Vd/v> The f o r c e s a c t i n g on the c y l i n d e r w i l l be the i n e r t i a f o r c e due t o i t s mass, the s p r i n g r e s t o r i n g f o r c e , i n t e r n a l damping f o r c e and a f l u i d f o r c e due the pressure d i s t r i b u t i o n on the c y l i n d e r . Because the flow i s separated t h i s pressure d i s t r i b u t i o n cannot be deduced from the f i e l d equations of f l u i d mechanics. The pressure d i s t r i b u t i o n g i v i n g r i s e t o the f l u i d f o r c e w i l l depend upon the p o s i t i o n of the shear l a y e r s and the nature of the o s c i l l a t o r y wake. The p o s i t i o n of the shear l a y e r s depends upon the v e l o c i t y of the f l u i d r e l a t i v e t o the c y l i n d e r so a f o r c e r e l a t e d t o the phase of the c y l i n d e r motion can be expected. The near wake of the c y l i n d e r i s dominated by the r o l l i n g up of the shear l a y e r s t o form v o r t i c e s i n a formation r e g i o n . The shedding of the v o r t i c e s causes - 3 -pressure f l u c t u a t i o n s and thus f o r c e s on the c y l i n d e r a t the v o r t e x -shedding frequency. The a c t i o n of these d i f f e r e n t types of f o r c e s g i v e s r i s e t o d i f f e r e n t types of f l u i d e l a s t i c o s c i l l a t i o n s . Vortex e x c i t e d o s c i l l a t i o n s occur a t a v e l o c i t y U where the r e s wake frequency i s equal t o the frequency of the e l a s t i c system. The a c t i o n of the f o r c e due t o vo r t e x shedding i s not l i k e the resonance of a s t r u c t u r e t o a f o r c e d o s c i l l a t i o n but r a t h e r represents a coupled f l u t t e r of the f l u i d - s t r u c t u r e system. No mathematical model i s yet s u c c e s s f u l i n e x p l a i n i n g the r e l a t i o n s h i p s between the c y l i n d e r motion, f l u i d f o r c e s , and t h e i r a s s o c i a t e d phase r e l a t i o n s h i p . G a l l o p i n g o s c i l l a t i o n s have been observed t o occur a t v e l o c i t i e s s u b s t a n t i a l l y g r e a t e r than U r e s « The i n s t a b i l i t y i s a r e s u l t of the a c t i o n of t h a t f o r c e dependent on the p o s i t i o n of the shear l a y e r s , A s u c c e s s f u l mathematical model f o r the g a l l o p i n g o s c i l l a t i o n s of square c y l i n d e r s i n a i r has been obtained ( r e f , 4), The f l u i d f o r c e c o e f f i c i e n t C p v (<) « Force*2/f> V^dZ i s measured f o r v a r i o u s angles of a t t a c k «=arctan(y/V) while the c y l i n d e r i s a t r e s t and assumed t o be the same p e r t u r b a t i o n a l f o r c e a c t i n g on the c y l i n d e r when i n motion. Summing the f o r c e s a c t i n g on the c y l i n d e r and i n t r o d u c i n g the dimensionless parameters o f equation 1,2 r e s u l t s i n the non l i n e a r d i f f e r e n t i a l equation, Y + */?Y + Y - n U 2 G p y ( Y / u ; ( U 3 ) For n « 1 t h i s weakly non l i n e a r equation can be solved t o show the e x i s t i ence and uniqueness of the p h y s i c a l motion. The g a l l o p i n g o s c i l l a t i o n s - 4 -w i l l e x i s t f o r b l u f f bodies f o r which ~t,Q„ I >0. The t h e o r e t i c a l v e l o c i t y U q a t which t h i s o s c i l l a t i o n w i l l s t a r t i s p r o p o r t i o n a l t o the r a t i o /^/n. The mathematical model of g a l l o p i n g o s c i l l a t i o n s i s s u c c e s s f u l i f U > 2 or 3 U „ For U s m a l l e r the i n t e r a c t i o n of the g a l l o p i n g O 273 S • O w i t h the v o r t e x shedding becomes important. I t has been observed e x p e r i m e n t a l l y i n a i r fl o w s ( r e f , 5» 6) t h a t i n these cases the nature of the I n t e r a c t i o n i s such t h a t the o s c i l l a t i o n s t a r t s a t U . r e s When a c y l i n d e r i s g a l l o p i n g a t v e l o c i t i e s removed from U , r© s there appears t o be no i n t e r a c t i o n between the g a l l o p i n g and vo r t e x shedding. Experimental r e s u l t s ( r e f , 7) show t h a t the frequency of the vortex shedding continues t o f o l l o w the s t r o u h a l r e l a t i o n s h i p . The a c t i o n of the v o r t e x shedding i n t h i s r e g i o n c o u l d be thought of as an added ha r m o n i c a l l y f o r c e d p e r t u r b a t i o n n*U * C L ^ . * c o s ( K s ^ ) , T h i s added term has no e f f e c t on the s o l u t i o n t o equation 1.3 i f the mass parameter n i s s m a l l , n ~ l O i n a i r . For water flows n ^ . O l and s o l v i n g equation 1.3 ( r e f , 7) shows t h a t the added p e r t u r b a t i o n due t o the v o r t e x shedding i s important i f i t i s taken as a f o r c e d o s c i l l a t i o n . I f i s l a r g e enough the g a l l o p i n g o s c i l l a t i o n w i l l be quenched a l t o g e t h e r . I f the o s c i l l a t i o n i s not quenched i t s amplitude w i l l be reduced and s i g n i f i c a n t resonance e f f e c t s should occur a t U = 3 * ^ r e s O a n i n t e g e r ) . F u r t h e r -more f o r the same damping l e v e l s as experiments i n a i r U < U g i v i n g o r e s r i s e t o the previous observed i n t e r a c t i o n a t U . The o b j e c t i v e of t h i s research then, i s t o i n v e s t i g a t e exper-i m e n t a l l y ! - 5 -i ) the p o s s i b i l i t y of quenching or r e d u c t i o n i n the g a l l o p i n g o s c i l l a t i o n due t o the vo r t e x shedding a c t i o n f o r U > U , r e s i i ) the i n t e r a c t i o n between the g a l l o p i n g and the v o r t e x shedding i n the r e g i o n o f U . r e s and t h e o r e t i c a l l y H i ) a mathematical model t o demonstrate the i n t e r a c t i o n between the g a l l o p i n g and vo r t e x e x c i t a t i o n types of o s c i l l a t i o n * mt ^ mt I I . EXPERIMENTAL PROGRAM 2,1 Apparatus and Instrumentation The experiments were performed u s i n g a 22 i n , wide open channel water flume, A maximum water v e l o c i t y of 2 f t , / s e c was a v a i l a b l e . I n keeping w i t h o b j e c t i v e ( l ) above i t was d e s i r a b l e t o have as l a r g e a dimensionless v e l o c i t y U = V/Wnd as p o s s i b l e , w h i l e not making the mass parameter n, or Reynolds number too s m a l l . T h i s c o u l d be done by having an e l a s t i c system w i t h a low n a t u r a l frequency. The l a y o u t of the water channel i s shown i n f i g u r e 2, Water from a h o l d c i s t e r n i s pumped t o a tank a t the head of the channel. Here a b a f f l e and s e t of double screens are employed t o reduce the e f f e c t s of pipe discharge. The water flows from the tank t o the channel proper where a 6 i n , t h i c k 3/8 i n , aluminum honeycomb was i n s t a l l e d t o reduce the turbulence i n the f l o w . The t e s t s e c t i o n was 27 f t . downstream from the honeycomb where the boundary l a y e r on the s i d e s and bottom was s t i l l o n l y a few Inches t h i c k . The f l o w r a t e through the channel was con-t r o l l e d by a gate valve a t the head of the channel. The depth of water i n the channel was kept a constant 22 i n , by an a d j u s t a b l e t a i l g a t e a t the channel o u t l e t . The primary i n v e s t i g a t i o n was on three square aluminum c y l i n d e r s . Data r e l a t i n g t o them i s presented belowt C y l i n d e r d blockage r a t i o aspect r a t i o Reynolds number 1 1 i n . 17 3800 - 15200 2 3 3 A i n . 1/2 i n . 2.25 % 23 2800 - 11400 1900 - 7600 - 7 -C y l i n d e r 1 was hollow w i t h a w a l l t h i c k n e s s of 3/32 i n . whereas c y l i n d e r s 2 and 3 were s o l i d aluminum. I n a d d i t i o n t o the square c y l i n d e r s a i i n . by 1/2 i n , s o l i d aluminum r e c t a n g u l a r s e c t i o n was t e s t e d . Damping c a l i b r a t i o n s were performed on a streamlined aluminum model of c r o s s s e c t i o n 1.4 i n . by .15 i n . Only the bottom 1? i n . of each model was immersed i n the water channel. The top of the c y l i n d e r , above water, was clamped t o the model mounting apparatus. The model mounting apparatus t h a t was designed i s shown i n f i g u r e 3, The top of the model i s clamped t o two t h i n aluminum tubes. Each tube i s supported by two a i r bearings designed by Smith ( r e f , 8 ) , The a i r bearings c o n s t r a i n the motion t o a transverse d i r e c t i o n w h i l e p r o v i d i n g n e a r l y f r i c t i o n l e s s support of the l o a d . The bearings are mounted t o a s t e e l p l a t e frame which i n t u r n i s mounted t o the water channel. The bottom set of bearings are b o l t e d t o the s t e e l p l a t e through o v e r s i z e d h o l e s . By s l i g h t l y a l t e r i n g t h e i r p o s i t i o n the bottom tube can be a l i g n e d r e l a t i v e t o the top tube. The model i s p o s i t i o n e d i n the d i r e c t i o n normal t o the flow by the clamps h o l d i n g the model t o the tubes. Springs are attached t o e i t h e r s i d e of the model by means of a t h i r d clamp. The low design n a t u r a l frequency r e q u i r e d weak and thus long extension s p r i n g s . The other ends of these s p r i n g s are h e l d v i a eyeb o l t s t o handy angles which extended over the s i d e s of the channel. Model displacement measurements were made w i t h the a i r core transformer used i n previous g a l l o p i n g experiments ( r e f , 8) . The a i r core transformer i s mounted t o the handy angle and i t s aluminum s h a f t - 8 -attached t o the model by the clamp h o l d i n g the s p r i n g s . A 10 k c , s i g n a l from a Wavetek i s the input t o the primary windings of the transformer. The displacement of the s h a f t i n the annulus of the a i r core transformer v a r i e s the magnetic c o u p l i n g between the c o i l s . Thus, the output s i g n a l from the secondary windings of the transformer i s a 10 k c , c a r r i e r , amplitude modulated by the model displacements, A f u l l wave r e c t i f i e r demodulates the high frequency s i g n a l , A low pass f i l t e r w i t h 20 db, g a i n was used t o get r i d of noise i n the c i r c u i t and a m p l i f y the s i g n a l . Output from the f i l t e r was measured by a d i g i t a l r.m.s. voltmeter which when c a l i b r a t e d y i e l d e d the r.m.s. amplitude of the model o s c i l l a t i o n . The s i g n a l from the f i l t e r was a l s o f e d t o a c h a r t r e c o r d e r t o o b t a i n the frequency of the o s c i l l a t i o n . A block diagram o u t l i n i n g the displacement measurements i s shown i n f i g u r e 4a. A c a l i b r a t i o n of the displacement transducer was performed by a t t a c h i n g a p i n t o the model and a 6 i n . r u l e r t o the frame between the a i r bearings. With the a i r bearings o f f the model was d i s p l a c e d at 1/4 i n . i n t e r v a l s and the corresponding v o l t a g e s measured by a D.C, voltmeter. The transducer was l i n e a r and the voltage - displacement r e l a t i o n s h i p i s shown i n f i g u r e 4b, The v e l o c i t y of water i n the channel was measured u s i n g an A, Ott Kempter c u r r e n t meter i n s t a l l e d 24 f t , behind the t e s t s e c t i o n a r e a . The c u r r e n t meter c o n s i s t s o f a p r o p e l l e r probe w i t h a counter t o measure i t s r e v o l u t i o n s per minute. Standard c a l i b r a t i o n curves f o r d i f f e r e n t p r o p e l l e r s g ive v e l o c i t y ( f t / s e c ) l i n e a r l y r e l a t e d t o r.p.m. The l a s t stage of the experimental program was t o measure the frequency of v o r t e x shedding from the o s c i l l a t i n g c y l i n d e r , A DISA - 9 -constant temperature anemometer u n i t and a Thermo Systems I n c . wedge type hot f i l m probe.were used t o measure the v e l o c i t y f l u c t u a t i o n s i n the wake. The s i g n a l from an anemometer u n i t i s p r o p o r t i o n a l t o the square r o o t of v e l o c i t y and normally a l i n e a r i z e r i s used t o o b t a i n a v e l o c i t y - voltage r e l a t i o n s h i p . Since only the frequencies of the v e l o c i t y f l u c t u a t i o n s were needed the s i g n a l d i d not have t o be l i n e a r i z e d . The c h a r t r e c o r d e r c o u l d not be used t o o b t a i n the frequency of the v e l o c i t y f l u c t u a t i o n s as i t had been used t o f i n d the frequency of c y l i n d e r o s c i l l a t i o n s f o r two reasons! Since the hot f i l m probe was f i x e d t o the l a b frame of reference and not t o the frame of reference of the o s c i l l a t i n g c y l i n d e r v e l o c i t y f l u c t u a t i o n s a t two frequencies co u l d be expected^ The frequency of c y l i n d e r o s c i l l a t i o n and the frequency of vortex shedding. A l s o , when the anemometer s i g n a l was recorded f o r a s t a t i o n a r y c y l i n d e r the s i g n a l was not s i n u s o i d a l , i n d i c a t i n g t h a t the f l u c t u a t i n g v e l o c i t y a s s o c i a t e d w i t h each vortex shed was not constant. A l t e r n a t i v e l y t h i s c o u l d be caused by the f a c t t h a t a wedge type hot f i l m i s not as s e n s i t i v e as a c o n i c a l or c y l i n d e r type probe. These problems n e c e s s i t a t e d some s o r t of s p e c t r a l a n a l y s i s of the f l u c t u a t i n g v e l o c i t y s i g n a l . No spectrum a n a l y z e r was a v a i l a b l e t h a t went as low as 1 Hz. so one was constructed from a band pass f i l t e r and r.m.s. voltmeter* The B r u e l & K j a e r v a r i a b l e band pass f i l t e r used had a band width 3 % of the frequency. The v a r i a b l e s c a l e was l o g a r i t h e m t i c and went from . 2 - 2 Hz, 2 - 20 Hz so frequencies towards the bottom end of the s c a l e were more a c c u r a t e l y measured. The output from the f i l t e r was f e d t o 10 -the d i g i t a l r.m.s. voltmeter. For each frequency s e l e c t e d a number of voltmeter readings were taken a t equal time l e n g t h and averaged. By changing the frequency a crude spectrum was obtained from which the dominant frequencies of the s i g n a l c o u l d be found. A b l o c k diagram of the wake frequency measurement system i s shown i n f i g u r e 5&* A r e p r e s e n t a t i v e spectrum i s shown i n f i g u r e 5^. 2.2 Experimental Procedure The equipment de s c r i b e d above provided the means of measuring the amplitude and frequency of c y l i n d e r o s c i l l a t i o n s , the wake frequencies and channel v e l o c i t i e s . Two other parameters of equation 1,1, the c y l i n d e r mass M and the non dimensional damping r a t i o p needed t o be known. Before d e s c r i b i n g the procedure f o r f i n d i n g these f i x e d parameters t h e i r d e f i n i t i o n r e l a t i n g t o the present flow s i t u -a t i o n r e q u i r e s e l a b o r a t i o n . I n the Theory of I d e a l F l u i d s , Milne - Thomson (ref,9) show th a t a l l bodies moving i n the presence of a f l u i d are a f f e c t e d by the f l u i d i n t h a t the body i s given an added mass. T h i s added mass depends upon the shape of the body and the nature of the motion. For example, a c i r c u l a r c y l i n d e r moving i n an i d e a l f l u i d has an added mass equal t o the mass 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 . I n r e a l f l u i d s and flow s i t u a t i o n s the added mass cannot be c a l c u l a t e d t h e o r e t i c a l l y . I t i s advantageous ( r e f , 24), t o t h i n k of the f o r c e due t o the f l u i d a c t i n g on the c y l i n d e r undergoing p e r i o d i c motion as being d i v i d e d i n t o two components, one component i n phase with c y l i n d e r a c c e l e r a t i o n and the - 11 other component I n phase w i t h c y l i n d e r v e l o c i t y . The former has the e f f e c t of i n c r e a s i n g the c y l i n d e r i n e r t i a and the l a t t e r the e f f e c t of i n c r e a s i n g the damping. I n the flo w s i t u a t i o n s s t u d i e d here the the nature o f motion and thus f o r c e s a c t i n g on the c y l i n d e r i s expected t o change w i t h v e l o c i t y so t h a t a t best the added mass can be given only as a reference added mass M . T h i s reference value w i l l be taken Si as the added mass of the c y l i n d e r i n s t i l l water (V=0), The mass M i n equation 1,1 w i l l t h e r e f o r e be taken t o be the mass of the moving p a r t s of the apparatus M Q and the added mass M & which the c y l i n d e r experiences i n s t i l l water. The non dimensional damping r a t i o f i n equation 1.1 i s a measure of the energy d i s s i p a t e d by vi s c o u s e f f e c t s . For t h i s experiment these are the vi s c o u s f r i c t i o n i n the a i r bearing and the v i s c o u s f o r c e s a c t i n g on the c y l i n d e r when immersed i n water. Again a t b e s t , these can only be approximated by those experienced by a streamlined c y l i n d e r i n s t i l l water. The procedure f o r measuring the added mass was t o f i r s t f i n d the mass of the moving p a r t s on a balance-; \ and the s t i f f n e s s of the sp r i n g s by measuring t h e i r e xtension t o known weights. The n a t u r a l frequency of the assembled system was measured i n a i r ( a) and water ( n) from whence 2 M - ( V - 1 ) M (2.1) a W 2 6 n or M - W ~ 2 * s t i f f n e s s of sp r i n g s - M Q (2.2) 12 -A more accurate agreement between the t h e o r e t i c a l and experimental W was obtained i f 1/3 the mass of the sp r i n g s i s not i n c l u d e d i n M a e Therefore the values of 1/3 the mass of the sp r i n g s are not i n c l u d e d i n M g given below and f o r the cases examined are l e s s than % of M g. Correspondence and accuracy was obtained t o two d i g i t s . I t was observed t h a t t h i s added mass measured was s l i g h t l y l e s s than the mass of the water d i s p l a c e d by the c y l i n d e r (p d Z ) . C y l i n d e r d M e ( s l u g s ) M a ( s l u g s ) d^Z ( s l u g s ) n= d 2Z/2M 1 1 i n . .0793 .0231 .0191 .093 2 3/4 i n« .09106 .0113 .0107 .0516 3 1/2 i n . .06688 .00592 .00477 .0328 The streamlined model was used t o measure the non dimensional damping r a t i o . A d i f f e r e n t clamp was cons t r u c t e d f o r the str e a m l i n e d model t o enable weights t o be added t o the model. Enough weights were added t o give the same n a t u r a l frequency i n s t i l l water as was measured f o r the corresponding c y l i n d e r . A t r a c e of the decaying o s c i l l a t i o n was made on the char t recorder from which & - 1 In ( Y / Y- ) • ( i s number of c y c l e s (2 . 3 ) Having found values f o r the mass and damping parameters the next step of the experimental program was t o vary the v e l o c i t y and r e c o r d the amplitude of the o s c i l l a t i o n . Experiments were p e r f -ormed on each c y l i n d e r twice i n order t o keep the Reynolds number as la r g e as p o s s i b l e while examining the two regions of o s c i l l a t i o n . F i r s t a set of sp r i n g s ( s t i f f n e s s = 28 .06 l b / f t ) was used t o f i n d the - 13 -amplitude of o s c i l l a t i o n i n the r e g i o n of resonance. The h i g h e r s t i f f n e s s r e s u l t e d i n a l a r g e r n a t u r a l frequency and thus an upper l i m i t of approximately U m 2. For each v e l o c i t y step the average of a number of amplitude readings from the d i g i t a l voltmeter was taken. A t r a c e of the o s c i l l a t i o n was recorded on the c h a r t r e c o r d e r from which the frequency c o u l d be found. Next the weak set of s p r i n g s ( s t i f f n e s s - 1.5 l b / f t ) was attached t o the model t o examine the r e g i o n away from resonance. I n these cases a maximum dimensionless v e l o c i t y of U = 5 - 11 c o u l d be obtained depending upon the c y l i n d e r i n v o l v e d . The same procedure was used t o f i n d the amplitude and frequency as f o r the s t i f f s p r i n g s . Wake frequency measurements were performed on each c y l i n d e r f o r each set of sp r i n g s a t a l a t e r date. The hot wedge probe was p o s i t i o n -ed i n the wake t o one si d e o f the c e n t e r l i n e o f the c y l i n d e r . With the a i r t o the bearings turned o f f so th a t the c y l i n d e r would not o s c i l l -a te a spectrum was taken t o determine the s t r o u h a l frequency f o r t h a t v e l o c i t y . The a i r would then be turned on so t h a t the c y l i n d e r c o u l d o s c i l l a t e and another spectrum taken. As the amplitude of the o s c i l l a t i o n i n c r e a s e d the probe was moved over so t h a t i t was always t o one si d e of the c y l i n d e r c e n t e r l i n e . T h i s was done so as t o ensure t h a t o nly the v e l o c i t y f l u c t u a t i o n s experienced on the si d e of the c y l i n d e r were obtained. I f the probe was placed a t the c e n t e r l i n e of the c y l i n d e r , frequencies of twice the vor t e x shedding c o u l d be expected. A problem arose f o r l a r g e c y l i n d e r o s c i l l a t i o n amplitudes because the probe would be outside the shear - 14 -l a y e r s and exposed t o the f r e e stream f o r a major p o r t i o n of the c y l i n d e r o s c i l l a t i o n p e r i o d . The r.m.s. s i g n a l from the anemometer would get weaker the l a r g e r the amplitude of the o s c i l l a t i o n and e v e n t u a l l y became too weak t o d i s t i n g u i s h the f r e q u e n c i e s . T h i s l i m i t e d the upper value of U f o r which wake frequencies c o u l d be obtained, 2,3 Experimental R e s u l t s i ) 3 A i n . square c y l i n d e r ! The experimental r e s u l t s f o r the amplitude and frequency of c y l i n d e r o s c i l l a t i o n are presented i n f i g u r e s 7 and 8, The wake frequencies observed are shown i n f i g u r e 9, Figure 6 shows c h a r t recorder t r a c e s of the d i f f e r e n t types of o s c i l l a t i o n observed. I n each of these f i g u r e s the r e s u l t s from both values of s p r i n g s t i f f -nesses are presented by the same symbol (+). Two regions of c y l i n d e r o s c i l l a t i o n occur. The f i r s t r e g i o n i s centered around 1/3 U . Here the c y l i n d e r i s s t a b l e a t r e s t and ' r e s has t o be given an i n i t i a l amplitude t o begin o s c i l l a t i o n . The f r e -quency of the o s c i l l a t i o n i s the n a t u r a l frequency of the c y l i n d e r i n water. Throughout t h i s r e g i o n of o s c i l l a t i o n the wake frequencies t h a t were measured were at 1/2 Wr and Wn . The amplitude of the o s c i l l a t i o n i n c r e a s e d w i t h U and a b r u p t l y disappeared a t U = .421, When the o s c i l l a t i o n stopped the frequency of the wake returned t o the s t r o u h a l v a l u e . When the v e l o c i t y was in c r e a s e d f u r t h e r a second r e g i o n of c y l i n d e r v i b r a t i o n s t a r t e d from r e s t and a t the s t r o u h a l frequency of - 1 5 -the corresponding water v e l o c i t y . As the v e l o c i t y i n c r e a s e d the l o c k i n continued between the frequency of vo r t e x shedding and the frequency of c y l i n d e r o s c i l l a t i o n . T h i s l o c k i n frequency was de-tuned from the s t r o u h a l r e l a t i o n s h i p . The amplitude of the o s c i l l -a t i o n i n t h i s r e g i o n was q u i t e modulated not as a r e s u l t of there being two frequencies i n the system, but more than l i k e l y because of the i n a b i l i t y of the f l u c t u a t i n g l i f t from the vortex shedding t o maintain a constant v a l u e . J u s t before U another frequency somewhat l a r g e r than the J T G S s t r o u h a l frequency began t o appear i n the wake. J u s t a f t e r U a t r e s U « 1.47 the frequency of c y l i n d e r o s c i l l a t i o n jumped t o a lower valu e . As the v e l o c i t y was in c r e a s e d more t h i s second frequency i n the wake in c r e a s e d u n t i l i t became twice the c y l i n d e r frequency. I n t h i s r e g i o n of v e l o c i t i e s the amplitude of the c y l i n d e r o s c i l l a t i o n continued t o i n c r e a s e w i t h v e l o c i t y . The modulation of the c y l i n d e r decreased i n d i c a t i n g t h a t the e x t r a c t i o n of energy from the flo w was due more t o the g a l l o p i n g mechanism than a resonance w i t h the vo r t e x shedding. At higher v e l o c i t i e s past t h i s resonance r e g i o n the amplitude of the o s c i l l a t i o n i n c r e a s e d i n the manner of a g a l l o p i n g o s c i l l a t i o n . The c y l i n d e r o n ly needed one or two c y c l e s t o b u i l d up and the shape of the waveform became l e s s harmonic. T h i s i s t y p i c a l of a system not weakly non l i n e a r . The frequency of the c y l i n d e r l e v e l e d o f f a t ,85 WR and a t the l a r g e s t values of v e l o c i t i e s measured began t o decrease a g a i n . - 16 -Around U » 2.8 the frequency In the wake changed from 2 to 3 times the frequency of cylinder osci l l a t i o n . Around U • 3.5 another frequency at 4 times the frequency of cylinder oscillation appeared. Whenever a change occurred i n the frequency of the wake i t did not do so abruptly but gradually. This made i t d i f f i c u l t to determine exactly what frequency was dominant. For U> 4.2 the signal from the anemometer was too weak to determine accurately the frequencies which occurred i n the wake. The results from this cylinder indicate that the galloping type oscillation i s dominant for U > U and the amplitude of the oscillation i s not reduced or quenched. Furthermore the galloping oscillation controls the frequency which occurs i n the wake so the vortex shedding does not act as a forced oscillation. The results that were obtained for the wake frequency when the cylinder was held stationary are not plotted in figure 9. The frequency vs, velocity relationship was linear as the strouhal relationship predicts. A linear regression of the data points gave S, the strouhal number to be .13 • i i ) 1 i n . square cylinderi Experimental results for the 1 in, square cylinder (ft = ,093 ) are quite similar to those for the 3 A l n » square cylinder. These results of the cylinder amplitude, frequency and wake frequency are shown i n figures 10, 11, 12 respectively. The f i r s t region of in s t a b i l i t y centered about 1/3 U i n fact starts at U = .328 or 1/4 U , At the beginning of this os c i l l a t i o n - 17 -r e g i o n frequencies a t 1/4 W r and 1/3 W R are observed i n the wake. Towards the end of the o s c i l l a t i o n r e g i o n , where the higher amplitudes of c y l i n d e r response occurred, the frequencies l n the wake were 1/2 W N and W„ . The o s c i l l a t i o n i n c r e a s e d i n amplitude t o Y «* .036 a t U » .505 where i t ended a b r u p t l y . The frequency of the c y l i n d e r o s c i l l a t i o n was s l i g h t l y l e s s than W„at the beginning o f the o s c i l l a t i o n r e g i o n and i n c r e a s i n g t o s l i g h t l y more than W„ a t the end of the r e g i o n . The second r e g i o n of o s c i l l a t i o n s t a r t e d around U = ,55 and had the same l o c k i n c h a r a c t e r i s t i c s as f o r the 3/4 i n . square c y l i n d e r . A t U r e g there was a t r a n s i t i o n i n the wake frequency from the frequency which the c y l i n d e r o s c i l l a t e d a t t o a frequency somewhat l a r g e r than the s t r o u h a l v a l u e . A l s o a t U r e s the c y l i n d e r frequency took a jump downward i n d i c a t i n g t h a t t h i s was the end of the l o c k i n r e g i o n . I n the g a l l o p i n g regime the c y l i n d e r frequency i n c r e a s e d t o ,8W„ where i t l e v e l e d o f f . J u s t past U the wake frequency i n c r e a s -ed u n t i l i t was twice the c y l i n d e r frequency. At U •= 2,3 another frequency a t 3 times the c y l i n d e r frequency appeared. Around U « 3 another frequency a t 4 times the c y l i n d e r frequency appeared g i v i n g a spectrum w i t h three frequencies o c c u r r i n g a t once ( f i g u r e 5b)» For v e l o c i t i e s g r e a t e r than U • 4 . 6 the s i g n a l became weak. At these l a r g e v e l o c i t i e s the frequency of the c y l i n d e r o s c i l l a t i o n appeared i n the spec t r a a g a i n . T h i s i s due t o the l a r g e displacements of the c y l i n d e r and not t o vortex shedding. The r e s u l t i n g wake frequencies when the c y l i n d e r was h e l d a t r e s t give a s t r o u h a l number, S 0 ,12 , - 18 -i i i ) 1/2 i n . square c y l i n d e r ! Experimental r e s u l t s f o r c y l i n d e r amplitude and frequency and wake frequency of the 1/2 i n . square c y l i n d e r (n » .0328) are shown i n f i g u r e s 13» 14, and 15 r e s p e c t i v e l y . An o s c i l l a t i o n e x i s t e d f o r .303^0* ^ , 4 , The amplitude of t h i s o s c i l l a t i o n was low, Y< . 0 2 , and a t the frequency of the c y l i n d e r i n s t i l l water. The high n a t u r a l frequency of the 1/2 i n . square c y l i n d e r compared t o the other c y l i n d e r s f o r the same s p r i n g s r e s u l t e d i n a low channel v e l o c i t y , .25 f t / s e c < V < ,33 f t / s e c i n t h i s f i r s t r e g i o n of o s c i l l a t i o n . Because of t h i s the s i g n a l from the anemometer was too weak t o measure the frequencies i n the wake. The s m a l l e r width of the c y l i n d e r r e s u l t s i n a Reynolds number around 2000 f o r t h i s range so v i s c o u s e f f e c t s would a l s o be important. The second r e g i o n of o s c i l l a t i o n s t a r t e d around U = .95 . The frequency a t which the c y l i n d e r s t a r t e d o s c i l l a t i n g was again the s t r o u h a l frequency of the corresponding v e l o c i t y . The l o c k i n of the c y l i n d e r frequency and the wake frequency a t a frequency detuned from the s t r o u h a l r e l a t i o n s h i p continued up t o U » 2.2 • T h i s was a notable departure from the other square c y l i n d e r s where the l o c k i n ended around U . When the l o c k i n ended the frequency of the re s c y l i n d e r jumped t o a lower v a l u e . Past the l o c k i n r e g i o n the amplitude of the c y l i n d e r continued t o increase i n the manner of a g a l l o p i n g o s c i l l a t i o n . The frequency of the c y l i n d e r i n c r e a s e d t o ,92 Wn and a t the highest values of U s t a r t e d t o decrease a g a i n . When the l o c k i n ended the frequency of the wake jumped t o 3 times the c y l i n d e r frequency w i t h no frequency at 2 times the c y l i n d e r frequency being observed. At U « 4.5 a frequency a t 4 times the c y l i n d e r frequency began t o appear. Around U • 5«3 a frequency at 5 times the c y l i n d e r frequency a l s o appeared i n the wake g i v i n g a spectrum w i t h three frequencies dominant. For U> 6.5 the amplitude of the o s c i l l a t i o n was l a r g e and consequently the f l u c t u a t i n g s i g n a l from the anemometer became too weak t o determine the f r e q u e n c i e s , A l i n e a r r e g r e s s i o n of the wake fr e q u e n c i e s measured when the c y l i n d e r was h e l d at r e s t y i e l d e d a s t r o u h a l number S «= .127 , Previous measurements of the s t r o u h a l number f o r square c y l i n d e r s ( r e f , 2) g i v e S - ,135 • The s t r o u h a l numbers measured i n t h i s experiment S » ,12, ,127, «13 are below t h a t . T h i s c o u l d be a t t r i -buted t o the lower Reynolds number which Shaw ( r e f , 23) shows has the e f f e c t of decreasing S, Other experiments ( r e f , 6) have used S « .125 • i v ) Rectangular 2:1 C y l i n d e r Amplitude - v e l o c i t y and frequency - v e l o c i t y r e s u l t s f o r the 2 t l r e c t a n g u l a r c y l i n d e r are shown i n f i g u r e s 16 and 17* Here, as w i t h the square c y l i n d e r s two regions of o s c i l l a t i o n occur. The f i r s t r e g i o n of o s c i l l a t i o n occurs f o r ,606<U<1,28. A maximum amplitude o f Y * ,25 occurs around 1/2 U , T h i s maximum x*@s amplitude i s c o n s i d e r a b l y higher than f o r the same r e g i o n of the square c y l i n d e r s . The o s c i l l a t i o n does not end a b r u p t l y but decreases i n amplitude u n t i l i t becomes a random v i b r a t i o n . The frequency at which the c y l i n d e r o s c i l l a t e s i n t h i s r e g i o n i s - 20 -dependent on the v e l o c i t y . At the beginning of the o s c i l l a t i o n r e g i o n i t i s s l i g h t l y l e s s than W r and i n c r e a s e s t o 1.3 W„ a t the end of the o s c i l l a t i o n r e g i o n . T h i s means t h a t a t the beginning of the o s c i l l a t i o n the phase angle by which the f l u i d f o r c e leads the c y l i n d e r displacement i s l e s s than 90°. The component of t h i s f o r c e i n phase w i t h c y l i n d e r a c c e l e r a t i o n a c t s t o Increase the i n e r t i a and thus decrease the c y l i n d e r frequency. When the phase angle i s g r e a t e r than 90° the opposite i s t r u e . T h i s same change of phase angle was observed by Nakamura ( r e f , 6) when the c y l i n d e r was f o r c e d e x t e r n a l l y . The second r e g i o n of o s c i l l a t i o n s t a r t s before U a t U » 1,71 r e s and i n c r e a s e s i n the manner of a g a l l o p i n g type o s c i l l a t i o n . The frequency of the o s c i l l a t i o n i n t h i s second r e g i o n s t a r t s a t around 1/2 W„ and i n c r e a s e s t o .75 W^ a t U « 8 , There i s not p e r f e c t agreement of frequencies i n the r e g i o n of overlapping v e l o c i t i e s f o r the two d i f f e r e n t s p r i n g s . For U>8 there i s a decrease i n the frequency a t which the c y l i n d e r o s c i l l a t e s . Complete wake frequency measurements c o u l d not be made f o r the r e c t a n g u l a r c y l i n d e r . When the c y l i n d e r was h e l d a t r e s t the s p e c t r a measured were q u i t e broad band w i t h wide peaks; i n d i c a t i n g more than one main s t r o u h a l frequency. I n the f i r s t r e g i o n of o s c i l l a t i o n the wake frequency occurred a t the frequency of c y l i n d e r o s c i l l a t i o n . Throughout the second r e g i o n of o s c i l l a t i o n the s p e c t r a were of low I n t e n s i t y and too broad band t o make any c o n c l u s i o n s about what wake frequencies occurred. S i m i l a r r e s u l t s t o these have been obtained by Novak ( r e f . 10). - 2i -He used a r e c t a n g u l a r 2 t l c y l i n d e r (n«= .00386) which c o u l d p i v o t about i t s base. Two regions of o s c i l l a t i o n were found i n t h a t ex-periment. The f i r s t around 1/2 U had a peak amplitude dependent on r@s damping l e v e l . The second r e g i o n s t a r t e d a t U » 1.72 and was independent 3 of damping. For a Reynolds number of 7*5*10 the wake of the s t a t i o n a r y c y l i n d e r had broad band e x c i t a t i o n w i t h low peaks corresponding t o S o .078, .018 . I n the f i r s t r e g i o n o f o s c i l l a t i o n a wake frequency equal t o t h a t of the c y l i n d e r frequency was a l s o found. - 22 -I I I . THE QUASI - STEADY MATHEMATICAL MODEL The q u a s i - steady mathematical model w i l l be the f i r s t model used t o t r y and e x p l a i n the p h y s i c a l motion. C o r r e l a t i o n between the s o l u t i o n of t h i s model and the experimental r e s u l t s would v e r i f y t h a t the c o r r e c t operating f o r c e s have been chosen. The p h y s i c a l motion observed f o r the case of the o s c i l l a t i n g square c y l i n d e r when U> U shows an increase i n amplitude w i t h v e i -o c i t y l i k e a g a l l o p i n g o s c i l l a t i o n . The frequency of t h i s o s c i l l a t i o n changes w i t h v e l o c i t y and the waveform i s t h a t of an o s c i l l a t o r which i s not weakly non l i n e a r . The frequencies of vo r t e x shedding observed are c o n t r o l l e d by the c y l i n d e r o s c i l l a t i o n and do not a c t as a f o r c e d o s c i l l a t i o n . I n the quas i - steady model the e l a s t i c system i s modeled by a damped simple harmonic o s c i l l a t o r w i t h a mass equal t o t h a t measured when the c y l i n d e r i s o s c i l l a t e d i n s t i l l water. A c t i n g on t h i s simple harmonic o s c i l l a t o r w i l l be the p e r t u r b a t i o n a l q u a s i - steady f o r c e 2 nU *G Fy (<) measured, when the c y l i n d e r i s a t r e s t , f o r d i f f e r e n t o angles of a t t a c k < « a r c t a n (Y/U), The r e s u l t i n g non l i n e a r equation of motion i s Y + i / 3 Y + Y - nU 2 C ( Y ) (3 .1 . ) *y u The p e r t u r b a t i o n a l force*:' used i s t h a t f o r non t u r b u l e n t flow about a square c y l i n d e r from reference 4. C„ (X - Y ) - 2.69 x - 169 x 3 + 6270 x 5 - 59900 x 7 (3.2) Fy TJ An averaging method or asymptotic theory of the f i r s t approx-2 i m a t i o n cannot be used t o so l v e equation 3,1 because nU f o r the cases examined i s not always l e s s than one. Rather than take a h i g h e r approximation i t i s e a s i e r t o solve the equation n u m e r i c a l l y . T h i s has been done u s i n g a f o u r t h order Runge Kutta a l g o r i t h m ( r e f . 11). Since the r.m.s. value o f the o s c i l l a t i o n was measured e x p e r i m e n t a l l y , the r.m.s, of an i n t e g e r number of c y c l e s of the s o l u t i o n was c a l c u -l a t e d u s i n g a Simpson's Rule i n t e g r a t i o n , i s l e s s than the peak value f o r the nonharmonic wave form obtained. The r e s u l t s c a l c u l a t e d f o r the amplitude and frequency of the square c y l i n d e r o s c i l l a t i o n s are presented along w i t h the experimental r e s u l t s i n f i g u r e s 7, 8, 10, 11, 13, and 14. The amplitudes c a l c u -l a t e d by the quasi-steady model agree w e l l w i t h experimental r e s u l t s f o r U>2. The c a l c u l a t e d frequency of o s c i l l a t i o n does not agree w i t h t h a t measured ex p e r i m e n t a l l y except f o r l a r g e U. T h i s i s not s u r p r i s i n g when one considersfethe type of non l i n e a r equation t h a t the quasi - steady model i s . The amplitude o f the o s c i l l a t i o n i s e s s e n t i a l l y s et by the component of the p e r t u r b a t i o n a l f o r c e ( f l u i d f o r c e ) i n phase w i t h the c y l i n d e r v e l o c i t y . The frequency of the o s c i l l a t i o n w i l l depend on t h a t component of the p e r t u r b a t i o n a l f o r c e 2 i n phase w i t h c y l i n d e r a c c e l e r a t i o n . Around U = 2 when nU < 1 the quasi - steady model p r e d i c t s the f l u i d f o r c e t o be 90° out of phase w i t h c y l i n d e r displacement. Thus there i s no component of f l u i d f o r c e i n phase w i t h c y l i n d e r a c c e l e r a t i o n t o reduce the frequency. I n the a c t u a l flow s i t u a t i o n there i s , and the phase between the f l u i d - 24 -f o r c e and c y l i n d e r displacement i s not 90°. Although the q u a s i -steady model p r e d i c t s the c o r r e c t magnitude of the f l u i d f o r c e a c t i n g i n phase w i t h the c y l i n d e r v e l o c i t y i t i s not the absolute magnitude of the a c t u a l p e r t u r b a t i o n a l f o r c e a c t i n g on the c y l i n d e r . When U 2 i s l a r g e r , nU > 1 and the equation becomes more non l i n e a r and w i l l p r e d i c t the change i n phase with amplitude of o s c i l l a t i o n , although not e n t i r e l y a c c u r a t e l y f o r a l l the c y l i n d e r s . At these higher values of v e l o c i t y surface e f f e c t s began t o be n o t i c e a b l e . On the f r o n t of the c y l i n d e r the water would r i s e t o an i n c h above the f r e e s u r f a c e . On the back of the c y l i n d e r a s e p a r a t i o n c a v i t y would form a l s o t o about one i n c h below the f r e e s u r f a c e . T h i s would e f f e c t i v e l y decrease the l e n g t h of the c y l i n d e r over which the quasi - steady f o r c e s would a c t , thus g i v i n g a lower value of n. A decrease i n the value of n f o r a c e r t a i n U of the quasi - steady model would tend t o decrease the amplitude and i n c r e a s e the frequency of o s c i l l a t i o n . T h i s i s what appears t o be happening e x p e r i m e n t a l l y f o r these l a r g e values of U, A p p lying the q u a s i - steady model t o the r e c t a n g u l a r c y l i n d e r can be u s e f u l i n determining the l e v e l of turbulence i n the oncoming flow. The l e v e l of turbulence could not be measured by the hot f i l m anemometer because of the e l e c t r o n i c d r i f t t o which wedge type f i l m s are subject t o i n flows contaminated by p a r t i c l e s . The G„ (<=<) curve i s a l t e r e d by the i n t e n s i t y of turbulence and f o r a l a r g e enough i n t e n s i t y the r e c t a n g u l a r 2«1 c y l i n d e r w i l l not g a l l o p . The C„ (<) curve f o r a r e c t a n g u l a r c y l i n d e r i n smooth flow can - 25 -be approximated by the polynomial ( r e f . 12) ,33 x + 1100 y? - 74 1 . 6 6 x l 0 6x 7 - 1 .607x10 7x 9 * 5 . 7 3 x l 0 7x 1 1 (X - Y/U) = 2. 3 00 x J  240 x 5 ^ ^ The c Fy(°0 curve f o r a turbulence i n t e n s i t y of 7#, the lowest a v a i l a b l e i n the l i t e r a t u r e , can be approximated by the polynomial ( r e f . 13) C p y (X - Y/U) - 5.52 x - 161 x 3 - 11299 x 5 + 3 . 7 x l 0 5 x 7 - 1.062xl0 6 x 9 - 4 . 5 9 l x l 0 7 x 1 1 The numerical s o l u t i o n f o r the amplitude and frequency of o s c i l l a t i o n w i t h the non l i n e a r i t i e s of equations 3*3 and 3«^ are shown i n f i g u r e s 16, and 17 along w i t h the experimental r e s u l t s o f the r e c t a n g u l a r c y l i n d e r . The numerical c a l c u l a t i o n suggests t h a t there i s some turbulence i n the oncoming flow which reduces the amplitude from the t h e o r e t i c a l case f o r smooth f l o w . The turbulence l e v e l , however, i s not as l a r g e as 7$. An important f a c t t h a t can be deduced from the quasi - steady model i s t h a t damping can be neglected f o r U>2« I n equation 3*1 the maximum value of G„ i s approximately .5 . Y i s the same order o f magnitude as U. Th i s leaves nU » 2y3 f o r water f l o w s , thus the damping term can be neglected. By a change of v a r i a b l e s 0 - nU , Y - nY (3.5) and n e g l e c t i n g the damping term the d i f f e r e n t i a l equation 3*1 becomes ? + Y = U 2 C ^ ( Y/ 0 ) (3.6) The s o l u t i o n of t h i s one equation g i v e s values of amplitude v s , v e l o c i t y f o r any n. - 2 6 -The r e s u l t s f o r the d i f f e r e n t square c y l i n d e r s can be compared by u s i n g the t r a n s f o r m a t i o n 3*5 • T h i s i s shown i n f i g u r e 18 and compared t o the s o l u t i o n of equation 3*6 . Along w i t h the r e s u l t s f o r the three square c y l i n d e r s obtained here are the r e s u l t s from a p r e v i o u s l y unpublished experiment by Santosham (see Appendix A ) , The r e s u l t s from Santosham's experiment are high compared t o the theory because h i s r e s u l t s g ive peak amplitudes compared t o f o r the r e s t . I n summary, the quasi - steady model i s u s e f u l i n showing the amplitude of the o s c i l l a t i o n f o r U>2, the change of the frequency o f the o s c i l l a t i o n f o r U l a r g e , and the n e g l i g i b l e e f f e c t of damping when nU >> 2 /3 . - 27 -IV FLUID OSCILLATOR MATHEMATICAL MODEL 4.1 Model Synthesis As shown above f o r U> 2 U the g a l l o p i n g o s c i l l a t i o n i s dominant and c o n t r o l s the frequency of the wake. For U< 2 U the 3TGS opposite seems t o be t r u e . When the c y l i n d e r begins t o o s c i l l a t e i n the second r e g i o n i t does so a t the s t r o u h a l frequency. As the l o c k i n continues the f l u i d - s t r u c t u r e i n t e r a c t i o n i s such t h a t the l o c k i n frequency i s n e i t h e r the s t r o u h a l nor the n a t u r a l frequency of the c y l i n d e r . The experimental r e s u l t s may be e x p l a i n a b l e by t h i n k i n g o f the separated o s c i l l a t o r y wake as a f l u i d o s c i l l a t o r . Such an approach has been taken w i t h p a r t i a l success t o e x p l a i n vortex e x c i t e d o s c i l l -a t i o n s of c i r c u l a r c y l i n d e r s ( r e f , 15» 16), A c h a r a c t e r i s t i c f e a t u r e of t h i s approach i s the attempt t o model the f o r c e s a c t i n g on the c y l i n d e r e m p i r i c a l l y r a t h e r than from a flow f i e l d model, A flow c h a r t of the reasoning used t o o b t a i n the model i s shown below. f r e e o s c i l l a t i o n ^ a c t u a l f l u i d s t a t i o n a r y c y l i n d e r , f o r c e d c y l i n d e r m o t i o n s y s t e m c y l i n d e r r e s p o n s e m e a s u r e d f o r c e s o n c y l i n d e r ; m a g n i t u d e s f r e q u e n c i e s , p h a s e s . m o d e l o f f o r c e s a c t i n g o n c y l i n d e r I d e a l l y the best model would be one g i v i n g the f l u i d f o r c e s as measured on the c y l i n d e r d u r i n g f r e e o s c i l l a t i o n . These f o r c e s are d i f f i c u l t t o measure e x p e r i m e n t a l l y so a model i s con s t r u c t e d u s i n g i n f o r m a t i o n about the f l u i d f o r c e s a c t i n g upon a s t a t i o n a r y c y l i n d e r or c y l i n d e r under f o r c e d o s c i l l a t o r y motion. The r e s u l t i n g model can be compared t o the a c t u a l c y l i n d e r response when f r e e l y o s c i l l a t i n g . The equation of motion f o r the s t r u c t u r e w i l l be represented by equation 3,1 w i t h not only the quasi - steady p e r t u r b a t i o n a l f o r c e but an added f l u c t u a t i n g l i f t c o e f f i c i e n t C L f = F l u c t u a t i n g L i f t Force/§p ^ d Z due t o the vortex shedding or f l u i d o s c i l l a t o r , V° + Y + Y - nU 2 [Cpy (Y ) + G L^ (4.1) A d i f f e r e n t i a l equation can be const r u c t e d from the knowledge of the f l u c t u a t i n g l i f t f o r c e when the c y l i n d e r i s s t a t i o n a r y or f o r c e d s l n u s o i d a l l y . The f l u c t u a t i n g l i f t f o r c e on a s t a t i o n a r y c y l i n d e r i s e s s e n t i a l l y s i n u s o i d a l and a t a frequency p r o p o r t i o n a l t o v e l o c i t y according t o the s t r o u h a l r e l a t i o n s h i p K • 2 7 7 SU, An o s c i l l a t o r s having such p r o p e r t i e s would be C, « + K C T - • 0 } i n i t i a l c o n d i t i o n s i C T „ » C, L f s L f L f Lo (4 2) Kl * 0 The s t a t e of such an o s c i l l a t o r as given by equation 4,2 i s dependent on the i n i t i a l s t a t e whereas the f l u i d o s c i l l a t o r i s s e l f e x c i t e d and s e l f l i m i t i n g t o some amplitude C L q . T h i s can be achieved i n the model by adding a non l i n e a r damping term C L f " G 2 s, 2 G L o - fi£ s Kf + K s 2 °Lf * G <*-3) - 29 -If|G|<1 the oscillator 4 ,3 w i l l oscillate at frequency K and amplitude s C L q regardless of what i n i t i a l conditions i t i s started at. In other words the linear oscillator (equation 4,2) can exist at an i n f i n i t y of energy levels whereas the oscillator i n equation 4 , 3 with that particular nonlinear function i s constrained to oscillate at one energy l e v e l . The response of C L f when the cylinder i s not stationary w i l l depend upon the state of the cylinder. This effect can be included by a r b i t r a r i l y exciting the f l u i d oscillator by the cylinder velocity, > 2 2 L f + K s C L f °Lf -G r 2 cT f 2 U. — i lit \ Lo (-j-) s G T- + K 2 C T_ - HY (4.4) This form of the f l u i d oscillator suggested by Currie (15) w i l l be used here. It contains two empirical constants H and G which have to be determined. H and G can be related to properties of the fluctuating l i f t force when the cylinder i s sinusoidally oscillated externally. C^ can be determined experimentally. The value of the fluctuating l i f t coefficient when the cylinder i s stationary depends upon the aspect ratio, intensity of turbulence, end conditions and Reynolds number. Vickery (ref, 18) obtained experimental values of C L q * 1.8? for smooth flows and G^Q - .96 for turbulent flows. As there i s a spanwise correlation of the pressure fluctuations those results apply to a thin s t r i p approaching zero width. The same author (ref. 19) measured the fluctuating l i f t on a square cylinder i n a turbulent flow (10$ intensity) for different aspect ratios. For an aspect ratio of 20, where the end conditions begin to have l i t t l e effect he obtains a value of C^ » ,65 . Vickery's 30 -experiments were conducted i n a Reynolds number range 10^  - 10"\ Other experimenters ( r e f , 20) show th a t decreasing the Reynolds number decreases the value of G^Q . For an experiment of aspect r a t i o 3 ( b u i l t i n ends) i n smooth flow a t R of 5000 they o b t a i n a value of © G^ o = 1,2? • The experimental r e s u l t s obtained i n t h i s research are 3 4 i n the Reynolds number range 2*10 t o 2*10 . I n the present e x p e r i -ment the aspect r a t i o s are 17* 23» 3^  and some turbulence was present. I n view of t h i s a value of «• ,7 w i l l be used i n the model when i t i s compared t o the experimental r e s u l t s f o r t h i s experimental setup. Experiments have been done ( r e f , 5» 6) where the c y l i n d e r was f o r c e d e x t e r n a l l y ( Y • Y*cos T ) and the p e r t u r b a t i o n a l f o r c e s a c t i n g on the c y l i n d e r measured. At U the p e r t u r b a t i o n a l f l u i d r e s f o r c e was found t o be a maximum and i n a s m a l l r e g i o n about U r e s occurs only a t the frequency the c y l i n d e r was o s c i l l a t e d a t . Outside t h i s narrow r e g i o n the f l u c t u a t i n g l i f t f o r c e occurred a t two main fr e q u e n c i e s ! t h a t a t which the c y l i n d e r was f o r c e d a t , and the s t r o u h a l frequency. The s t r i k i n g s i m i l a r i t y between t h i s type of response and t h a t of a s i n u s o i d a l l y f o r c e d s e l f e x c i t e d o s c i l l a t o r of the type equation 4,3 suggests the form of c o u p l i n g i n equation 4,4 , I n t h a t experiment a f o u r i e r a n a l y s i s of the f l u c t u a t i n g f o r c e was done t o o b t a i n the component a t the frequency t h a t the c y l i n d e r was f o r c e d a t . T h i s component can be d e s c r i b e d by a magnitude C L and phase ^ ( r e l a t i v e to c y l i n d e r displacement). A l t e r n a t i v e l y i t can be represented by two q u a n t i t i e s } the component of C^ i n phase w i t h c y l i n d e r v e l o c i t y and C T M the component i n phase w i t h c y l i n d e r a c c e l e r a t i o n . - 31 -The analog of e x p e r i m e n t a l l y f o r c i n g the c y l i n d e r e x t e r n a l l y would be t o s u b s t i t u t e the time dependent Y «» Y*cos? r i n the model equations 4.1, 4.4. . The t h e o r e t i c a l f o r c e measured on the c y l i n d e r would be Force on C y l i n d e r - C F y ^ - Y * s l n r ^ + C L f (4.5) £fV*dZ U The experimental and $ are analogous t o the harmonic components s i n ? and cos *, of equation 4.5 f C L c o s ( t + $ ) - harmonic components of C F y (-Y»slnt ) + C L fj " (4.6) To compare the t h e o r e t i c a l C^ and $ w i t h the corresponding experimental r e s u l t s i t i s necessary t o o b t a i n the s o l u t i o n C ^ of equation 4,4 . Equation 4,4 i s the f o r c e d Van der P o l equation, the s o l u t i o n of which can be e a s i l y obtained p r o v i d i n g G and HY* are l e s s than one. I n the resonance r e g i o n ( K « 1 ) the s o l u t i o n has the form G L f * a l C G S ^'ir*G' ) + ni6her harmonics (4.7) Outside the resonance r e g i o n the s o l u t i o n has the form -HY*sln K - 1 G L f " a2 c o s ^ K s t t l 9 x ) -HY slne+ higher harmonics (4,8) s The t h e o r e t i c a l a^ and 0^ are obtained i n Appendix B by a standard s o l u t i o n ( r e f , 1.7). T h i s s o l u t i o n shows th a t the r a t i o H/G w i l l change the peak amplitude a^ and thus the magnitude of C^at exact resonance (K " 1 ) . G w i l l change the width of the resonance s r e g i o n . The t h e o r e t i c a l C^ and 0 can be obtained by equating the s i n t and cos ^ terms of equation 4,6 , u s i n g the appropriate C ^ - 32 -i n the resonance and the non resonance domains. The value of a.^ and ©2 need not be s o l v e d f o r as the i n t e r e s t i s not i n these f r e q u e n c i e s . * Outside the resonance r e g i o n only the term -HY s i n - ^ of GT „ w i l l K — 1 c o n t r i b u t e t o C^ and 0 • s A value of G «=» .05 was chosen t o maintain agreement w i t h the observed width of the resonance r e g i o n f o r i n c r e a s i n g Y ( r e f . 5 )• T h i s i s shown i n f i g u r e 19, A value of H/G •= 35 was chosen t o give best agreement f o r a t K g = 1. The t h e o r e t i c a l C L i s compared t o the experimental i n f i g u r e 21, A l s o shown, i n f i g u r e 21, i s C T T i l l • C^ sin0 compared w i t h the experimental. The p o s i t i v e part of the curve represents energy flow t o the c y l i n d e r . The phase angle 0 i s shown i n f i g u r e 20. With the exception of the phase angle 0 these ; f i g u r e s show th a t the model synt h e s i z e d has the same f e a t u r e s as observed e x p e r i m e n t a l l y when the c y l i n d e r i s f o r c e d s l n u s o i d a l l y . I t should be noted t h a t any experimental r e s u l t s from f o r c i n g the c y l i n d e r e x t e r n a l l y w i l l be independent of n and p . S e t t i n g G and H from these r e s u l t s prevents any dependence of t h e i r values on n a n d ^ • 4.2 Model Compared t o Experimental R e s u l t s To compare the f l u i d o s c i l l a t o r model w i t h the experimental r e s u l t s d e s c r i b e d i n s e c t i o n I I r e q u i r e s the simultaneous s o l u t i o n o f 2 equations 4.1 and 4.4 , For nU < 1 an a n a l y t i c a l s o l u t i o n i s p o s s i b l e , however i t i s d i f f i c u l t and r e q u i r e s knowledge about the type of 2 s o l u t i o n expected. For nU > 1 no s o l u t i o n i s p o s s i b l e beyond e x i s t -ence and uniqueness p r o o f s . - 33 -Because of t h i s a numerical s o l u t i o n has been sought u s i n g a system subroutine f o r d i f f e r e n t i a l equations on the IBM 3?0, Once the s o l u t i o n curve f o r Y was obtained an i n t e g e r number of c y c l e s was squared then i n t e g r a t e d by Simpson's r u l e t o o b t a i n the r.m.s, value of the o s c i l l a t i o n . Y = Id J~Y2 d t (4.9) rms vj o v 7 / I t was a l s o r e q u i r e d t o o b t a i n the frequency a t which occurred. Since a number of frequencies occurred i n t h i s c o u l d be done by u s i n g the Fast F o u r i e r Transform t o f i n d the F o u r i e r s e r i e s of a p e r i o d T » AtH of the s o l u t i o n C L f . CTJ?) - O o . + £ Q - e o s ^ w j t + L s i n t (4.10) A £ i s the i n t e r v a l of the N p o i n t s c a l c u l a t e d . The j ' s f o r which the q u a n t i t y ^2 ^ i7 a. + b. was maximum would correspond t o a frequency 0 3 K - 2£j[ of G U N L 1 * The accuracy of any frequency obtained would t h e r e f o r e be + 2TT/A? N. I n computing the s o l u t i o n t o the system 4.1, 4.4 i t was necessary t o • keep N constant. For s m a l l U, s m a l l K values were expected thus * t c o u l d be made l a r g e r and the frequencies obtained more accurate than a t l a r g e U, U < .5 * * - ,4 K - K - .065 .5< U< 2. &c- - .3 K - K- .0873 2<U< 4 . 5 - .15 K - K - .175 5<U< 7 . 5 A Z - .1 K - K - .262 U>8 ^ - .075 K - K - .349 - 3 4 -The coupled equations of the f l u i d o s c i l l a t o r model were solved w i t h the corresponding values of n, y3 and S obtained i n the experimental program. The r e s u l t s c a l c u l a t e d f o r the c y l i n d e r amplitude and frequency and the frequencies of C L f are shown i n the same f i g u r e s as the experimental r e s u l t s . F i r s t c o n s i d e r the model a p p l i e d t o the 3 / 4 " square c y l i n d e r . For t h i s model as f o r the experiment two d i f f e r e n t regions of o s c i l l -a t i o n occur, A low amplitude resonance peak i s observed a t 1 / 3 U s» The frequency of the c y l i n d e r o s c i l l a t i o n here i s the n a t u r a l frequency of the c y l i n d e r i n s t i l l water. Two frequencies occur i n C ^ i the frequency of the c y l i n d e r and the s t r o u h a l frequency. The c y l i n d e r s t i l l o s c i l l a t e s a t dimensionless v e l o c i t i e s U between t h i s low ampl-i t u d e resonance peak and the onset of the second r e g i o n of o s c i l l a t i o n , although i t i s an i n t e r m i t t e n t o s c i l l a t i o n . When the second r e g i o n of o s c i l l a t i o n begins the c y l i n d e r frequency jumps down t o the s t r o u h a l value g i v i n g the l o c k i n c o n d i t i o n . When the v e l o c i t y i s increased the amplitude r i s e s q u i t e s h a r p l y and the l o c k i n frequency becomes detuned from the s t r o u h a l r e l a t i o n s h i p . T h i s l o c k i n frequency i s l a r g e r than the experimental value i n d i c a t i n g t h a t the model p r e d i c t s the value of t o be too s m a l l . The amplitude of the c y l i n d e r here, i s not modulated as the peak value of CL^ does not f l u c t u a t e from c y c l e t o c y c l e . The amplitude has a higher value than measured ex p e r i m e n t a l l y . T h i s i s because the model p r e d i c t s a value of G^j too l a r g e . The l a r g e value of C T_ and sm a l l value of C, M are because o f the q u a n t i t a t i v e l y - 35 -i n c o r r e c t phase angle $ i n t h i s r e g i o n between the s t a r t of o s c i l l -a t i o n and U •» 2, For U > 2 the amplitude and frequency of the c y l i n d e r o s c i l l -a t i o n i s only a l i t t l e d i f f e r e n t from the q u a s i - steady model. Around U » 2 a s t r o u h a l frequency appears i n G^ ,^ along w i t h the frequency of the c y l i n d e r f o r a l l U> 2 , For 2 ,5<U < 5 a frequency a t 3 u occurs i n C ^ , however no frequencies a t 2w or 4w appear as the experimental r e s u l t s show. Except f o r the frequencies of G L f the r e s u l t s of t h i s model f o r the 3/4 i n . square c y l i n d e r agree q u a l i t -a t i v e l y w i t h what i s observed e x p e r i m e n t a l l y . The r e s u l t of the f l u i d o s c i l l a t o r model a p p l i e d t o the 1 i n . square c y l i n d e r i s shown i n f i g u r e s 10,11,12. I t i s e s s e n t i a l l y the same as f o r the 3/4 i n , c y l i n d e r w i t h a few notable exceptions. The s t a r t of the second r e g i o n of o s c i l l a t i o n occurs a t a lower U as i s the case e x p e r i m e n t a l l y . At U •> 1 there i s a jump downwards i n the l o c k i n frequency from the s t r o u h a l value t o some value detuned from i t , A much more n o t i c e a b l e resonance e f f e c t i s shown a t U « 3*5 • A jump occurs here i n the frequency of the c y l i n d e r which does not occur e x p e r i m e n t a l l y . The r e s u l t s f o r the 1/2 i n , square c y l i n d e r are shown l n f i g u r e s 13,14,15, Noticeable d i f f e r e n c e from the previous cases are t h a t the second r e g i o n of o s c i l l a t i o n begins a t a higher U and t h a t around U • 4 there i s only a narrow r e g i o n where a frequency of 3 occurs i n C ^ , The major discrepancy between the f l u i d o s c i l l a t o r model and - 36 -the experimental r e s u l t s i s i n p r e d i c t i n g the frequencies of C L^. The frequencies a t 1 / 2 W^ , 2 W , and 4^ do not appear. An u n e x p l a i n -a b l e frequency atu/ occurs i n f o r l a r g e U as i t does i n the experiment. I n the experimental case t h i s cannot be a t t r i b u t e d t o v o r t e x shedding but i n s t e a d t o the hot f i l m passing through the shear l a y e r . The f l u i d o s c i l l a t o r model was not a p p l i e d t o the r e c t a n g u l a r c y l i n d e r because of the two s t r o u h a l frequencies which occur i n the wake. A l s o the e x p e r i m e n t a l l y observed response of C^ t o e x t e r n a l l y f o r c e d c y l i n d e r motion i s d i f f e r e n t than t h a t f o r the square c y l i n d e r ( r e f . 6) . 4,3 E f f e c t of Changing Damping and Mass Parameter For U< 1 the damping of the c y l i n d e r w i l l become an important parameter i n the o s c i l l a t i o n . A r t i f i c i a l damping was not used i n the experiment so i t s e f f e c t i n t h i s r e g i o n c o u l d not be determined. The t h e o r e t i c a l e f f e c t of damping was looked a t u s i n g the f l u i d o s c i l l a t o r theory. T h i s i s shown i n f i g u r e 2 2 . I n c r e a s i n g the damping decreases the amplitude of o s c i l l a t i o n a t 1 /3 U u n t i l f o r ft - . 0 2 the o s c i l l -a t i o n does not occur at a l l . The damping does not a f f e c t the s t a r t of the second r e g i o n of o s c i l l a t i o n t h i s being dependent on the mass parameter n .only. The experimental r e s u l t s and the f l u i d o s c i l l a t o r model both i n d i c a t e t h a t the onset of the second r e g i o n of o s c i l l a t i o n depends on n. T h i s was examined f u r t h e r by u s i n g the f l u i d o s c i l l a t o r model. Using the experimental values of S and {3 f o r the 3/4 i n . square c y l -i n d e r the s o l u t i o n was obtained f o r d i f f e r e n t values of n between - 37 -.0516 and .00732 . The Amplitude - V e l o c i t y curves are shown i n f i g u r e 23• Decreasing n in c r e a s e s the U a t which the second r e g i o n of o s c i l l a t i o n begins. For n sma l l enough the onset o f the o s c i l l -a t i o n i s U . T h i s i s a l s o observed i n references 5 and 6 f o r r e s values of/2/n of 1.59 and .353 r e s p e c t i v e l y . They do not give the s p e c i f i c values of n r e l a t i n g t o t h e i r observations though. For n < < l as i n a i r f l o w s equations 4.1 takes more than one or two c y c l e s t o b u i l d up and a numerical s o l u t i o n i s not s u i t a b l e . I n t h i s case the equations 4.1 and 4.4 are e s s e n t i a l l y uncoupled and C T „ has no e f f e c t on Y f o r U > U . A s o l u t i o n has been sought f o r Lt r e s ° n = ,00043, U q «= 3»39 corresponding t o the experiment i n a i r flow ( r e f , 7) where the wake frequency was measured e x p e r i m e n t a l l y . The f l u i d o s c i l l a t o r only s l i g h t l y a f f e c t s the values o f Y obtained. The response of i s a l s o a t the s t r o u h a l frequency as observed e x p e r i m e n t a l l y i n t h a t case. - 38 -CONCLUSIONS Transverse g a l l o p i n g type o s c i l l a t i o n s occur f o r e l a s t i c a l l y mounted square c y l i n d e r s i n water f o r U> 2. The o s c i l l a t i o n . 2 i s not weakly non l i n e a r and f o r nU l a r g e , w i l l become a r e l a x -a t i o n o s c i l l a t i o n . The c y l i n d e r motion c o n t r o l s the frequency of the wake r a t h e r than v i c e - v e r s a , t h e r e f o r e the v o r t e x shedding does not a c t l i k e a f o r c e d o s c i l l a t i o n . The s t r u c t u r a l damping of the e l a s t i c system, i f s m a l l , has no e f f e c t on the o s c i l l -a t i o n i n t h i s r e g i o n . For U>2 the amplitude of the o s c i l l a t i o n can be p r e d i c t e d from the q u a s i - steady mathematical model. The frequency of o s c i l l -a t i o n i s not p r e d i c t e d c o r r e c t l y thus showing the q u a s i - steady p e r t u r b a t i o n a l f o r c e does not give the c o r r e c t magnitude of the component i n phase w i t h c y l i n d e r a c c e l e r a t i o n . The onset of the c y l i n d e r i n s t a b i l i t y which develops i n t o the g a l l o p i n g o s c i l l a t i o n occurs before U . I t i s a d i r e c t r e s u l t r e s of the i n t e r a c t i o n w i t h vortex shedding and depends upon the value of the mass parameter n. A second i n s t a b i l i t y occurs near 1/3 U . I f the s t r u c t u r a l damping i s low enough an I n i t i a l d e f l e c t i o n o f the c y l i n d e r can impose frequencies i n the wake. The s y n c h r o n i z a t i o n of these frequencies w i t h the c y l i n d e r motion s u s t a i n s the o s c i l l a t i o n . The f l u i d o s c i l l a t o r mathematical model can d e s c r i b e the i n t e r a c t i o n which occurs between the g a l l o p i n g o s c i l l a t i o n and v o r t e x e x c i t e d o s c i l l a t i o n i n the resonance r e g i o n . The model i s mathematically - 39 -a system of coupled s e l f e x c i t e d o s c i l l a t o r s . O s c i l l a t o r A i s the g a l l o p i n g o s c i l l a t i o n of the c y l i n d e r which e x i s t s f o r n < < l and U>U . O s c i l l a t i o n B i s the f l u c t u a t i n g l i f t f o r c e from 3 T 6 S vortex shedding which e x i s t s when the c y l i n d e r i s a t r e s t . The r e s u l t a n t i n t e r a c t i o n (A-»B) occurs when the mass parameter n = 0 ( . l ) . The o s c i l l a t i o n (A-ffl) e x i s t s and q u a l i t a t i v e l y agrees w i t h the c y l i n d e r amplitude and frequency measured e x p e r i m e n t a l l y . The model as cons t r u c t e d i s not capable of p r e d i c t i n g a l l the frequencies t h a t occur i n or the q u a n t i t a t i v e values of the amplitude and frequency of the c y l i n d e r o s c i l l a t i o n . - 40 -RECOMMENDATIONS FOR FURTHER WORK Exp e r i m e n t a l l y , the frequency of v o r t e x shedding d u r i n g l a r g e amplitudes of o s c i l l a t i o n should be i n v e s t i g a t e d . T h i s would i n v o l v e d e s i g n i n g an experiment u s i n g a pressure tap on the face of the c y l i n d e r or a hot f i l m probe attached t o the c y l i n d e r . An extension of the f l u i d o s c i l l a t o r model c o u l d be looked a t ; agreement w i t h the ex p e r i m e n t a l l y measured f a c t t h a t C^ 0 v a r i e s w i t h angle of a t t a c k <= a r c t a n (Y/U) when the c y l i n d e r i s at r e s t , A polynomial f u n c t i o n w i t h even terms i n i t w i l l probably cause C L f t o have a response a t frequencies 2W and 4 ^ as observed e x p e r i m e n t a l l y . The i n t e r e s t i n g r e s u l t s which Santosham obtained f o r the r e c t -angular c a n t i l e v e r e d system should be pursued f u r t h e r as these types of s t r u c t u r e would occur more oft e n i n p r a c t i c e . The i n t e r a c t i o n which occurs between the f i r s t two n a t u r a l frequencies of the beam and the two main frequencies i n the wake seem t o make the g a l l o p i n g type o s c i l l a t i o n disappear. s A c o r r e c t form of the f u n c t i o n f(Y/U) can be chosen t o g i v e BIBLIOGRAPHY 1. Skowronski, J.M. " M u l t i p l e Nonlinear Lumped Systems", P o l i s h S c i e n t i f i c P u b l i s h e r s , I969, PP 18. 2. Parkinson, G.V. "Wind - induced i n s t a b i l i t y of s t r u c t u r e s " , P h i l . Trans. Roy. Soc. Lond. A.269, PP395 - 409. 3. P a r k i n s o n, G.V. "Flow - Induced S t r u c t u r a l V i b r a t i o n s " , IUTAM -IAHR Symposium Karlsruhe (Germany), August 14 - 16, 1972. 4. Parkinson, G.V. Smith, J.D, "The Square Prism as an A e r o e l a s t i c Nonlinear O s c i l l a t o r " , Quart. J o u r n . Mech, and A p p l i e d Math., V o l . 17, P a r t 2, May 1964. 5. O t s u k i , Y., et a l , "A note on the A e r o e l a s t i c i n s t a b i l i t y of a p r i s m a t i c bar w i t h square s e c t i o n " , J o u r n a l of Sound and V i b r a t i o n (1974) 34 ( 2 ) , 233 - 248. 6. Nakamura, Y., M i z o t a , T. "Unsteady L i f t s and Wakes of O s c i l l -a t i n g Rectangular Prisms", J o u r n a l of the Engineering Mechanics Division,ASCE, V o l . 101, EM6 Dec. 1975. 7. Parkinson, G.V., Santosham, T.V., " C y l i n d e r s of Rectangular S e c t i o n as A e r o e l a s t i c Nonlinear O s c i l l a t o r s " , presented a t the V i b r a t i o n s Conference of the ASME, h e l d a t Boston, Mass. March 19 - 31, I967. 8. Smith, J.D. "An experimental study of the A e r o e l a s t i c I n s t a b i l i t y of Rectangular C y l i n d e r s " , M.A.Sc. T h e s i s , U. of B r i t i s h Columbia, I962. 9. Milne - Thomson, L.M, " T h e o r e t i c a l Hydrodynamics", 5th ed., M c M i l l a n I968, pp. 246. 10. Novak, M,, " G a l l o p i n g and Vortex Induced O s c i l l a t i o n s of S t r u c t u r e s " , Proc. 3rd I n t . Conf. Wind E f f e c t s on B u i l d i n g s and S t r u c t u r e s , Tokyo, 1971, IV, pp 804 - 808. 11. Conte, S.D., Boor, C , "Elementary numerical A n a l y s i s " , 2nd ed. McGraw H i l l 1972, pp 338. 12. Santosham, T.V. "Force Measurements on B l u f f C y l i n d e r s and A e r o e l a s t i c G a l l o p i n g of a Rectangular C y l i n d e r " , M.A.Sc. T h e s i s , U of B r i t i s h Columbia, 1973. - 42 13. L a n e v i l l e , A., " E f f e c t s of Turbulence on Wind Induced V i b r a t i o n s of B l u f f C y l i n d e r s " , PhD. T h e s i s , U. of B r i t i s h Columbia, 1973. 14. Thomson, W.T., "Theory of V i b r a t i o n s w i t h A p p l i c a t i o n s " , P r e n t i c e -H a l l 1972, pp 305. 15. C u r r i e , H a r t l e n , R.T., " L i f t O s c i l l a t o r Model of Vortex Induced V i b r a t i o n " , J o u r n a l of the Eng. Mechanics D i v i s i o n , ASCE V o l . 96, No. EM5, Oct. 1970. 16. Skop, R.A., G r i f f i n , O.M., "A Model f o r the Vortex E x c i t e d Resonant Response of B l u f f C y l i n d e r s " , J o u r n a l of Sound and V i b r a t i o n (1973) 27 (2) 225 - 233. 17. Dinca, F., Teodosiu, C , "Nonlinear and Random V i b r a t i o n s " Academic P r e s s , 1973* PP I69. 18. V i c k e r y , B.J., " F l u c t u a t i n g l i f t and drag on a l o n g c y l i n d e r of square c r o s s - s e c t i o n i n a smooth and i n a t u r b u l e n t stream", J/. F l u i d Mech. (1966), V o l . 25, p a r t 3» PP 481 - 494. 19. Vickery,. B.J., "Load F l u c t u a t i o n s i n Turbulent Flow", J o u r n a l of Eng. Mech. D i v i s i o n ASCE, V o l . 9^ # No. EMI, Feb. 1968. 20. C h a p l i n , J.R.,Shaw, T.L., "Flow - Induced Dynamic Pressures on Square S e c t i o n C y l i n d e r s " , Proc. IAHR Congress V o l . I I , P a r i s , 1971. 21. Minorsky, N,, "Nonlinear O s c i l l a t i o n s " , Van Nostrand, 1962. 22. Bogoliubov, N», M i t r o p o l s k y , Y,, "Asymptotic Methods i n the Theory of Non L i n e a r O s c i l l a t i o n s " , Hinhustan P u b l i s h i n g Corp., I96I. 2 3 . Shaw, T.L., "Wake Dynamics of Two Dimensional S t r u c t u r e s i n Confined Flows", Proc. IAHR Congress, V o l . I I , P a r i s , 197L 24. Amey, H.B., Pomonik, G., "Added Mass and Damping of Submerged Bodies O s c i l l a t i n g Near the Surface", presented at the Fourth Annual Offshore Technology Con-ference, Houston, Texas, May 1972. - 43 -APPENDIX A J SANTOSHAM'S EXPERIMENTAL RESULTS A s e r i e s of experiments were done by Santosham on square and r e c t a n g u l a r c y l i n d e r s u s i n g the DREP water t u n n e l i n Esquimalt, The a n a l y s i s of those r e s u l t s r e l e v e n t t o t h e i r use here i s presented i n t h i s appendix. The experimental setup c o n s i s t e d o f a 1/2 i n . by 13.75 i n . square aluminum c y l i n d e r attached t o the end of a 47 i n , c a n t i l e v e r beam. The c a n t i l e v e r beam was used t o o b t a i n a low n a t u r a l frequency i n the e l a s t i c system. Only the aluminum c y l i n d e r was exposed t o the flow i n the water t u n n e l . Displacement measurements were made by s t r a i n gauges attached t o the base of the c a n t i l e v e r beam. The r e s u l t s f o r the amplitude and frequency of c y l i n d e r o s c i l l a t i o n f o r a number of d i f f e r e n t experiments are shown i n f i g u r e s A l and A 2 , The amplitudestand f requencies of o s c i l l a t i o n obtained are q u a l i t a t i v e l y s i m i l a r t o those d e s c r i b e d i n t h i s t h e s i s . The q u a s i steady theory can be a p p l i e d t o these experimental r e s u l t s i f the experimental setup i s modeled as a beam w i t h a non l i n e a r shear f o r c e and moment a p p l i e d t o i t s end. y 4. 1/2,0 V*dzC (y/V) d z V - 44 -The assumed mode method ( r e f , 14) i s used t o f i n d the d i f f e r e n t i a l equation governing the motion of the beam. The beam v i b r a t e d i n i t s f i r s t mode t h e r e f o r e a s o l u t i o n i s sought of the form y ( x , t ) = 0(x)q(t) where the f i r s t mode, 0(x), f o r a c a n t i l e v e r beam i s ; 0(x) « c o s h ^ - COB f x - ( c o s / L + c o s h / L) (sinh<fx - s i n / x ) s i n / L •+ s i n h ^ L fL - 1.8751 ; / " « M. 4 L*k (Al) The procedure i s t o f i n d expressions f o r the k i n e t i c and p o t e n t i a l energy of the beam and the g e n e r a l i z e d damping f o r c e s a c t i n g on the end of the beam, (damping f o r c e s i n the beam w i l l be neglected as small) The equation of motion can be found by sub-s t i t u t i n g these expressions i n Lagrange's equation. The k i n e t i c energy of the beam w i l l be} L a. K . E . - 1 5" / ( x >"0 M ( x ) dx - UM o (A2) 2 » L 2 ^ The p o t e n t i a l energy w i l l be P.E. - 1 f k L 3 d£ y ( x , t ) dx 2 -> 3 . 2 o dx 2 A 1 M W 9 2 n Z (A3) The g e n e r a l i z e d f o r c e due t o the shear f o r c e a c t i n g on the end of the beam can be found from the v i r t u a l work p r i n c i p l e } S W, - 1 i= V*dZ C 0(L) 1 . £y(L,t) - 1 p / d Z j2f(L)J_ v -I 21 r y L V J Q - £ ^ = 1 o V ^ d Z G 1 £? 0 1 Fy WO X V 0(L) ( A 4 ) - 45 -Similarly for the moment M SW - !i (L,t) d % (L.t) - M (L,t) d ff(L)$Z (A5 ) Q2 = -_2 o £ dx 1 2-V 2 dZ 'Py dx JZf(L) i_ 1 djZf(L) V i dx The differential equation of motion becomes 1 p V^dZ 2 r M JZf (L) (0Cu) ] ) + H 0(L) GPv(jZf(L) £) V 2 dx Fy or i n nondimensional quantities Y *» 0(L)q/d, t « WRt Y + Y - nU20 (L) f 0(L) t Z d 0(L) 1 C - /YN L 2 dx • J * y MJ ; (A6) (A7) The only difference between this and the quasi steady model of section 3.1 i s the addition of the term 0(L) which 0(L) + Z d 0(L) 2 dx effectively increases n. This i s true only because the nonlinear generalized damping force acts at the end of the beam. I f i t acts a l l along the beam (as the case would be i f the whole beam was exposed to the flow) the generalized force would be a more complicated expression. For the f i r s t mode the quantities 0(L) = 2.0 and d 0 (L) = dx 1,4682 are known (ref. 14). The f i r s t natural frequency of the cylinder i s W„ =2.03 (IT* T n e stiffness of the cantilever beam was measured J M to be ,515 l b / i n . The experimental natural frequency i n s t i l l water was known to be 2.7 cps thus M and n could be found. Equation A7 can be solved numerically and the results for the peak amplitudes of oscillation and the frequency are shown i n figures Al and A2. As with the square cylinder results discussed previously the amplitude corresponds to that observed but the frequency does not. The results Santosham obtained for the rectangular cylinder on - 4-6 -a d i f f e r e n t beam are shown i n f i g u r e s A3 and A4. TWO resonant type are observed; No g a l l o p i n g type of o s c i l l a t i o n i s observed. The o s c i l l a t i o n s t a r t s around U • 1 and occurs a t the f i r s t n a t u r a l frequency of the beam. As U i s increased the frequency of the o s c i l l a t i o n i n c r e a s e s . At U = 3»5 the beam s t a r t s o s c i l l a t i n g a t the beam's second n a t u r a l frequency and again i n c r e a s e s w i t h v e l -o c i t y . The increase of frequency of o s c i l l a t i o n w i t h U i s l i n e a r a t a s t r o u h a l number corresponding t o S = ,165 • I f t h i s i s a l s o the wake frequency then i t would appear t h a t a l o c k i n i s o c c u r r i n g between the v i b r a t i n g beam and the o s c i l l a t o r y wake, T h i s l o c k i n frequency would correspond t o the other main frequency which occurs i n the spectrum of the s t a t i o n a r y r e c t a n g u l a r c y l i n d e r ' s wake a t S • ,18 , The response then, of an e l a s t i c system w i t h two or more frequencies and a f l u i d system w i t h two frequencies can be q u i t e d i f f e r e n t than t h a t of a s i n g l e degree of freedom e l a s t i c system and an o s c i l l a t o r y wake at one frequency. - 47 -APPENDIX B: ANALYTICAL SOLUTION TO THE FORGED VAN DER POL EQUATION The a n a l y t i c a l s o l u t i o n of equation 4 .4 given here i s based on the asymptotic method of K r y l o v and Bogoliubov as given i n reference 17. V a r i a t i o n s on t h i s method and other methods of s o l u t i o n are described i n references 21 and 22. The equation f o r the f l u c t u a t i n g l i f t f o r c e c o e f f i c i e n t when the c y l i n d e r i s e x t e r n a l l y f o r c e d ( Y= Y*cos t ) i s J °Lf - G [ =Lo - I (jjj£) G T „ + K 2 C T . =-HY*sin-c: ( B l ) For ease of n o t a t i o n the f l u c t u a t i n g l i f t f o r c e c o e f f i c i e n t « C*Lf w i l l be represented by x, x denotes d i f f e r e n t i a t i o n w i t h respect to dimensionless time t . I f G <= e g and HY* =» e-hy are small and the same order of magnitude then x + e f (x) + K 2 x « - f h y s i n t s f (x) "3 2 1. * 2 n  x - 4 / X \ • ° 3 (K } . s -1 (B2) In the r e g i o n of resonance (K l ) a frequency l o c k i n can be s expected. A s s o c i a t e d w i t h the l o c k i n i s the detuning K g - 1 = £ A . Equation B2 then becomes x + x = - e h y s i n ^ + A X + f ( x ) (B3) An asymptotic form of s o l u t i o n i s sought where x = a^ cos ct L +• € u ^ ( a ^ , ct , t ) 06 - r + 0 ( B 4 ) I n the case of resonance, the amplitude of o s c i l l a t i o n and the phase d i f f e r e n c e between the n a t u r a l o s c i l l a t i o n and the e x t e r n a l a c t i o n e x e r t s i n f l u e n c e on the change i n the amplitude and frequency of the -48 -o s c i l l a t i o n . I t i s d e s i r a b l e then t o seek an expansion where a x = £A 1 (a,, O, ) S, = £Bj_ ( a ( ) 9, ) (B5) By s u b s t i t u t i n g B4 then B5 i n t o equation B3 and c o l l e c t i n g terms of the same order of magnitude, £ , g i v e s the p a r t i a l d i f f e r -e n t i a l equation 3L a. a. ^ ui + 3L<L±L + ^ u, + u = _f C_ a s in a l l a*. 1 a o t a r a t 1 ui ±i ^ ai 1 ; -hy s i n t + 2A> s i n <=t + (2 a,B, - a. A ) cos <=c 1 1 1 1 ( B 6 ) By n e c e s s i t a t i n g t h a t "a^" be the t o t a l amplitude of the fundamental harmonic of the s o l u t i o n , the f u n c t i o n u^ ( a ^ , t ) should have no fundamental harmonics of dL, No terms s i n c t , cos d-should appear on the r i g h t hand s i d e of equation B6 as they w i l l be p a r t i c u l a r s o l u t i o n s t o u^. Therefore j ~ [ - f j (-ax s i n c e ) - hy s i n ( * - s,) + e\ A 1 s i n ct O + U a ^ - a i A ) c o s ^ j j j ^ j d o t = 0 (B7) Performing the i n t e g r a t i o n s g i v e s the f u n c t i o n s A 1 and B 1 and th e r e f o r e aw 1 l k1^eki •= JL ^  [ f j _ ( - a ^ l n rf. ) + hy s i n (<* - e) J s i n c t d °L = 1 [ G C 2 q a 1 - G_a 3 + H Y * cos ©,] (B8a) 2 2 0, «= G B 1 j [ b y s i n + a 1 ^  cos ct J cos <=t d cc - 1 2a, ( K 2 - / ) a. - HY* s i n e . (B8b) s * "1 Since the R,H,S, of equation B6 i s equal t o the sum of a f u n c t i o n depending only on a^ and <=c and another f u n c t i o n depending only on t , a s o l u t i o n t o u^ can be sought i n the form. - 49 -( a 1 # <t, t ) - u x ( a t , cc ) + ( r ) (B9) The requirement B7 ensures there are only higher harmonics of cc t h e r e f o r e U j ( f ) «= 0 "u^ ( a ^ , c t ) « J_ £ s i n C ^ ) C (-a^ s i n °c) s i n ( <,<*.) d o t . . - 3_. — ' 1 1 i II . i man • - 1 (BIO) To compare the experimental C^ t o the t h e o r e t i c a l the only i n t e r e s t i s i n the harmonics of the frequency which the c y l i n d e r i s f o r c e d a t so u^ does not have t o he known i n the resonance r e g i o n . What i s needed i s the values of a^ and 9/ • These w i l l correspond t o steady s t a t e p e r i o d i c s o l u t i o n s where a^» © =0 i n equations B8« Combining equations B8 t o get r i d of the s i n s, , cos s, terms g i v e s the polynomial i n "a^" f o r the steady s t a t e values of amplitudes of the fundamental harmonic. C T a« + Lo 1 L s s ^ 2 4 ( K , - i ) + G Z C L ; & 12 - ( H Y * ) 2 - 0 (Bll) All o-,*s s a t i s f y i n g B l l may not correspond t o s t a b l e s o l u t i o n s t o B8. The s t a b i l i t y o f the p e r i o d i c v i b r a t i o n can be examined by stu d y i n g the motion of the o s c i l l a t o r c l o s e t o the p e r i o d i c v i b r a t i o n , a / (r) - a , * S i . (?) i6*(r) - 8. + S £>, ( z ) By s u b s t i t u t i n g equations B12 i n t o B 8 a system of l i n e a r d i f f e r e n t i a l equations f o r the v a r i a t i o n 6 a^, $&, are obtained. d ( i a j d ( 6<9,) • dc-a l ^ + h da^ ( K 3 - 1 ) £ a t -2a: c5a^  - HY*sin©, 2 So, HY*cos 2aj (B13) - 50 -where - i ( - c c: Lo + G K 2 2 ) dh da. Ga K2 1 .2 The c h a r a c t e r i s t i c equation of B13 i s a. A + \ HY*cos s> 2a, + a 1 dh + h da. 1 + ( a 1 dh + h ) HY* cos ©, da 2a, •+ HY*sln e, (K~ - 1) - s 4a, 0 (B14) "1 "1 I f the r o o t s of the c h a r a c t e r i s t i c equation are not equal the general s o l u t i o n t o B13 i s 6 aj_ - c x e ' c 2 e a ; ^ © - D t e + D2 a I n order f o r the p e r i o d i c v i b r a t i o n t o be s t a b l e i t i s s u f f i c i e n t t h a t both X, and A^  have negative r e a l p a r t s . Thus the sum of the r o o t s must be negative and t h e i r products p o s i t i v e . T h i s g i v e s the two requirements f o r s t a b i l i t y i ) HY*. cos e, + & 1 2a dh da. + h > i i ) ( a t dh + h) HY» cos 0 + HY»sln£>r]C - 1 da. 2a, 2a. > 0 or ir) 2^G 'Lo > 0 (B15) i i ) 1 ( G a / 2 K - G. C L 2 ) • ( ] G ^ 2 K 2 2 G L o 2 ) + ( K 2 - l ) 2 > 0 - 51 -For each amplitude, a^, s a t i s f y i n g B H , the corresponding s t a b i l i t y can be determined from B15, Once the values of a^ and 6, f o r the f l u c t u a t i n g l i f t c o e f f i c i e n t have been determined the t h e o r e t i c a l l i f t measured on the c y l i n d e r a t the frequency t h a t the c y l i n d e r i s for c e d a t can be found from equation 4 . 6 G cos ( t + P ) - - [ A Y* - 3 B ( Y * ) 3 + £ G ( Y * ) 5 - 35 D (Y*)' 1 L U 4 U 8 U 6 4 * U s i n t + a x cos C^ + e,) (B16) The c o e f f i c i e n t s A,B,C,D, are the qua s i steady c o e f f i c i e n t s from equation 3*2 . The l i f t f o r c e on the c y l i n d e r a t the frequency o f the c y l i n d e r motion i s due p a r t l y t o the quasi steady c o n t r i b u t i o n and p a r t l y from the f l u i d o s c i l l a t o r . Equating s i n f and cos Z components gives} C T T - CTsin0 - [ A Y* - 2 B ( Y * ) 3 + £ C ( Y * ) 5 - 35. D (Y*) h l h L u 4 U 8: U 6 4 U GLM = a l 0 0 8 e> ^ B 1 7 ^ G + a ^ s i n Q / L ,2 ~ ? r G L I + GLM ' t a n ^ = G L I / G L M T h i s i s what can be compared t o the corresponding experimental r e s u l t i n the resonance r e g i o n . The resonance c o n d i t i o n w i l l occur p r o v i d i n g both c o n d i t i o n s B15 h o l d . Thus the width of the resonance r e g i o n can be determined f o r d i f f e r e n t amplitudes of f o r c e d c y l i n d e r o s c i l l a t i o n . r Outside the resonance r e g i o n ( f o r a detuning l a r g e enough) no l o c k i n occurs. The f o r c e d f l u i d o s c i l l a t o r ' s main frequency w i l l be the s t r o u h a l frequency. Only the amplitude of the o s c i l l a t i o n w i l l e x e r t an i n f l u e n c e on the change of amplitude and frequency o f the o s c i l l a t i o n . The asymptotic form of s o l u t i o n sought w i l l be - 52 -x *» a„cos d- + e u 1 (a a 2 ° 6 V ( B l od = K + € B, ( o t ) s 1 ' S u b s t i t u t i n g B18 i n t o B2 and c o l l e c t i n g terms of order of magnitude g i v e s the p a r t i a l d i f f e r e n t i a l equation. u. cannot c o n t a i n the fundamental harmonics K and t h i s c o n d i t i o n can l s be used t o f i n d and B 2 by e l i m i n a t i n g terms c o n t a i n i n g sin<* and cos ct as i n the resonance case. I n the non resonance case however, the only i n t e r e s t i s i n the terms of u^ c o n t a i n i n g s i n t or cos t components. Having e l i m i n a t e d s i n and cos <*- terms from the R.H.S, of B19 a s o l u t i o n t o u 1 can be sought i n the form K' hy s i n X + 2K g s i n et + 2K a^B cos oC (B19) u x ( a 2 , ct , z) - u x ( a 2 , *•) where u l ^ a 2 ' * ) " TT 2 K - 1 s hy s l n ^ - 5 3 -The t h e o r e t i c a l l i f t measured on the c y l i n d e r a t the frequency the c y l i n d e r i s f o r c e d a t i s then C cos ( K + 0) - - [ A Y*- £ B ( Y * ) 3 + £ C ( Y * ) 5 - ,35 D ( Y * ) 7 U U 8 U - HY*sln X K 2 - 1 s Equating s i n -tr and c o s t c o e f f i c i e n t s g i v e s 'LI LM G. s i n 0 B ( Y * ) 3 + 5. C ( Y * ) 5 -'LI 0 » + 90° U sin'C (B21) 31 D (Y*)' % U + HY* 2 K -1 s Equations B22 f o r the non resonance r e g i o n and equation B17 f o r the resonance r e g i o n are what have been compared t o the exper-ime n t a l r e s u l t s i n f i g u r e s 20,21, v o r t e x s t r e e t F I G U R E 1: P H Y S I C A L MODEL OF FLOW P A S T A SQUARE C Y L I N D E R - 5 5 -g a t e v a l v e b a f f l e d o u b l e s c r e e n s b a f f l e h o n e y c o m b 27 f t . m o d e l t e s t a r e a 2 4 f t , v e l o c i t y m e a s u r e m e n t a p p a r a t u s | 5 f t . a d j u s t a b l e t a i l g a t e F I G U R E 2 : WATER C H A N N E L L A Y O U T FIGURE 3: MODEL MOUNTING APPARATUS - 5 7 -F I G U R E 3 C o n t i n u e d 1 . s t e e l p l a t e f r a m e 2 . a l u m i n u m t u b e s 3 . a i r b e a r i n g s 4 . c l a m p s h o l d i n g m o d e l 5 . m o d e l 6 . 6 0 - 7 0 p s i a i r s u p p l y 7 . a i r c o r e t r a n s f o r m e r 8 . t r a n s f o r m e r c o r e 9 . c l a m p h o l d i n g s p r i n g s a n d t r a n s f o r m e r c o r e 1 0 . h a n d y a n g l e s 1 1 . s p r i n g s - 5 8 --» y 10 kHz o s c i l l a t o r V2ZZZZZZA V//////Z77 V//////77Z. A i r core Transformer r e c t i f i e r & bias f i l t e r r .m. s . oltmeter c h a r t Recorder FIGURE 4a: BLOCK DIAGRAM OF DISPLACEMENT MEASUREMENT APPARATUS --2.5 -1.5 ' -0.5 0.5 1 .5 2.5 IN D I 5 P FIGURE 4b: VOLTAGE - DISPLACEMENT CALIBRATION -59-hot f i l m probe A anemometer -7 u n i t v a r i a b l e band pass f i l t e r r.m.s. voltmeter FIGURE 5a: BLOCK DIAGRAM OF WAKE FREQUENCY MEASUREMENT APPARATUS . C 3 ' 4-1 1 o . -a . .. J. a • cn +4--K + + + i I i -I- l + 1 4- j :-h I ; 'a I.J.. 1 .5 2.0 _ ] — 2.5 K _ T__ 3.0 3.5 FIGURE 5b: SAMPLE SPECTRA OF WAKE FREQUENCIES BEHIND OSCILLATING CYLINDER; d = 1 i n . , U = 2.91, n = .093, S = .12 .... \\imM H i - f i H f } f 4 r .TfTT m U-[..i-i. t-Xh JTjX] Ix [ i Mill I HI S H! f-I i-t ! s ! r " l i i !:>?i-i i. i i I l l Xl»j tiL?r 111 H-X-1-M • H X X H X rn ....... r..!.L...U...;.t... i X i H t 4111 If J I L -r . i. L.[..[.-! !.. u i . .L i-'-i-i i j . i •..•IX ; u . L , M 5 i l l ' I. 1J...1.; L.LI_!..LX.LL.i.L.vJ_ r i X r h n 1X1 i i i i X X Mr M l XI }X L i . n r r r . .i.i.'(Xf> u s LIICIJ i < u.\.u 14M.;.. J J . . .L- . .L-UL_U.! ,i„Ll.L.Li.l I i l-t-t-i-j-U-pl- f +• l--X4-Ur X-1 i i i 1 \ I Hi. ILL l-i.l. : i U FIGURE 10: AMPLITUDE OF CYLINDER OSCILLATION; d=1.0 In., n=.093, jS =.0044, S=.12 , - r?i 1.T.M I ill !-!•• H M a: • r [ . T W m ri I-! ' I m i r 4-1-i 14' It FT t i l -i : i r t i t fli " T T T r m T T r r r -1 trv\ i H i m r.Xn i i U JTT tp Tf m ^jftx^n i j i f . ' !OU-l1]41tr-W -t-1 t t-i-;11.ij:H-In -.LU •ii! M i t I! I ! .ij i-T*lf^f!ni-[i!i! t - ' T i I t ' ! 1:1 ! i i l t r : h 0.D 0.2E I t ' i : f t r ir -H i r • i ' i m • "X 1 1 ri'TTt 'f TJTTT ' r fTT i III f-t.J-i. . 4-• I M-i 1 i r H i 0.5 u '[•ha -[."""! n-i j i n . t i t i i j ' i (f..i i it.i'i r jT i " ' " t 4 ' L T ' iV iTn i '•• f i u j . r u . | H l n i . U . t i ^ 1 111 [li J l [r -|Hi- -U; |H; MHtMiiM!Ht;-!X> t i h t+t-f-f i r r - r i -. L L i i t I i . I i 1.1.1 I ( i i f ! : L i . l J M i l . i . M ) M I K i i i i I I n i l Ti^! X , 1 V ! , , . , . , . . . , , . +L-|4-+-H-0.0 0.5 ] .0 tilETOi L-TLL i i i i i . M M U . i 1 V : 0 M r ] ' ( i t t T r i"H-r.n~ t"t"rtriXTtT'rLTiTr.lrri'~"!""'~~ i'4+iulimniti-. tni Hi! 1.5 I 1 LEAF 67 OMITTED IN PAGE NUMBERING. i « u -74-F I C U R E 1 9 : WIDTH OF K K S O H A N C K K E C I O N 7 $ ' j .0 Wma 2 ITU E x p e r i m e n t : T h e o r y : c o m p l e t e s y n c h r o n i z a t i o n i n c o m p l e t e s y n c h r o n i z a t i c — ; G = .05, K =1.75 , C F I G U R E 2 0 : P H A S E A N G L E FOR F O R C E D O S C I L L A T I O N 5 1 0 1 5 — A _ ^ 2 f l l (-U _ J 1 0 1 5 75 F I G U R E 2 3 : E F F E C T O F M A S S P A R A M E T E R n F O R S M A L L U ; F L U I D O S C I L L A T O R THEORY 2.0 r e s D i m e n s i o n l e s s V e l o c i t y U 79 FIGURE A 3 : AMPLITUDE OF OSCILLATION FOR RECTANGULAR 2 :1 CYLINDER ON END OF CANTILEVER BEAM; d - . 5 i n . , w - 2 . 0 COS. n i • " ! ' + ! + + + + * ; + + i + t + i 1 — 5 . 0 6 . 2 5 U + + + + + 0 . 0 1 . 2 5 I— 2.5 3 . 7 5 7.5 8 . 7 5 1 0 . 0 

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