COUPLED-OSCILLATOR MODELS FOR VORTEX-INDUCED OSCILLATION OF A CIRCULAR CYLINDER BY KELVIN NORMAN WOOD B.A.Sc, University of B r i t i s h Columbia, 1973 A.THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of Mechanical Engineering We accept t h i s thesis as conforming required to the standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1 9 7 6 (cT) Kelvin Norman Wood, 1976 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mechanical Engineering The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date August 17, 1976 (i) ABSTRACT The vortex-induced o s c i l l a t i o n of a c i r c u l a r cylinder i s modelled by a non-linear system with two degrees of freedom. The periodic l i f t acting on the cylinder due to the vortex-street wake i s represented by a s e l f - e x c i t e d o s c i l l a t o r , which i s coupled to the cylinder motion. Approximate solutions and s t a b i l i t y c r i t e r i a are presented which are v a l i d over r e s t r i c t e d i n t e r v a l s . Changes to the form of the coupled-oscillator model and i t s approximate solution are examined i n order to improve the comparison between predicted model and experimental r e s u l t s . The changes are motivated by the study of experimental evidence, and by comparison with the known properties of s i m i l a r systems of non-linear equations. ~- S i g n i f i c a n t improvement i n the coupled-oscillator model performance i s obtained through the i n c l u s i o n of an e f f e c t i v e s t r u c t u r a l damping term which i s dependent on wind speed and cylinder displacement. TABLE OF CONTENTS Page 1. INTRODUCTION 1 2. PRELIMINARY 3 3. MODEL FORMULATION . 3.1 7 Higher Order Non-linearity 7 3.2 Combination-Oscillation 8 3.3 Variable Damping (i) (ii) 4. . Solution 11 Harmonic Solution Combination-Oscillation .13 Solution 17 DISCUSSION 22 REFERENCES APPENDIX A 23 ; Hartlen and Currie's O r i g i n a l System of ELifferential Equations 24 APPENDIX B Extension to Seventh Order Non-linearity i n C 1 ........ 30 J-i APPENDIX C Combination-Oscillation Solution to Hartlen and Currie's O r i g i n a l System 36 APPENDIX D Variable S t r u c t u r a l Damping 39 (iii) LIST OF FIGURES Page Figure I Figure II Figure I I I Figure IV Figure V Figure VI Figure VII ~" Figure VIII Experimental Results f o r Vortex-Induced O s c i l l a t i o n of a C i r c u l a r Cylinder (Feng) 42 Schematic Diagram of Experimental Configuration 43 C h a r a c t e r i s t i c Domains of Vortex-Induced Oscillation 44 Theoretical Predictions for Hartlen and Currie's O r i g i n a l Model 45 Theoretical P r e d i c t i o n for a Higher Order Non-linearity i n C' 46 Theoretical Predictions for CombinationO s c i l l a t i o n Solution Applied to Hartlen and Currie's O r i g i n a l Model 47 Theoretical Predictions of the E f f e c t of Variable S t r u c t u r a l Damping-Harmonic Solution 48 T h e o r e t i c a l Predictions of the E f f e c t of Variable Structural Damping-CombinationO s c i l l a t i o n Solution 49 (iv) LIST OF TABLES . ... Page Table I Table I I E f f e c t i v e Structural Damping During Vortex-Induced O s c i l l a t i o n 11 Damping Parameter Determination 12 (v) LIST OF SYMBOLS Non-dimensional transverse cylinder displacement amplitude. Component of A at ft) (free component) F Component of A at U)^ (harmonic component) Instantaneous l i f t c o e f f i c i e n t Amplitude of l i f t c o e f f i c i e n t Amplitude of l i f t c o e f f i c i e n t f o r stationary Amplitude of the component of C J_i Amplitude of the component of C cylinder at w (harmonic component) c at oo V (free component) F h <o v g 2 IT V Sizrouhal number Free stream v e l o c i t y Instantaneous transverse cylinder displacement X Non-Dimensional c transverse cylinder displacement = — — 2 Mass parameter = ^ ^ Coupling parameter Damping parameter Cylinder diameter Cylinder mass per unit length Detuned frequency of cylinder o s c i l l a t i o n (wind-on) Natural frequency of spring-cylinder system (still-air) Vortex formation frequency f o r the e l a s t i c a l l y mounted cylinder (vi.) Vortex formation frequency for stationary cylinder Vortex formation frequency approximately at to^ ( e l a s t i c a l l y mounted cylinder) s - to • c CO n to V s CO n to to n C r i t i c a l damping r a t i o (wind-on) C r i t i c a l damping r a t i o (wind-off) C o e f f i c i e n t s of non-linear damping terms Phase angle by which C leads X J_i Detuning parameter f o r cylinder o s c i l l a t i o n frequency F l u i d density Non-dimensional time = to t n ACKNOWLEDGEMENT The author would l i k e to thank Dr. G.V. Parkinson f o r h i s advice and guidance i n the course of t h i s research. F i n a n c i a l support was received from the National Research Council of Canada, Grant A586. 1. 1. INTRODUCTION Dating from the early 1960's, there has been an active program i n t h i s department to study the e f f e c t s on f i x e d or e l a s t i c a l l y supported b l u f f bodies of the wakes produced by them. In the Reynolds Number range 4 which i s of i n t e r e s t [0(10 ) ] , the wake i s characterized by p e r i o d i c a l l y shed v o r t i c e s , the frequency of which i s governed by the Strouhal r e l a t i o n ship. This work i s concerned with the i n t e r a c t i o n of an e l a s t i c a l l y mounted c i r c u l a r cylinder with i t s wake, f o r the case i n which the Strouhal frequency i s close to the resonance frequency, of the cylinder-mounting system. Detailed experimental studies have been carried out by Ferguson and Feng (2) to document the vortex-induced o s c i l l a t i o n of j u s t such a system. - As d i r e c t s o l u t i o n of the governing dynamic equations f o r the cylinder and i t s wake i s not f e a s i b l e at present, a v a r i e t y of s i m p l i f i e d mathematical models have been suggested to describe the i n t e r a c t i o n [a summary of the more promising suggestions i s given by Parkinson (3)]. A proposal by Hartlen and Currie (4) seems to have p a r t i c u l a r merit. consider the l i f t acting on the cylinder They (due to i t s periodic wake) to be governed by a second order non-linear d i f f e r e n t i a l equation (of the type studied by van der Pol) which i s coupled to the cylinder motion. r e s t r i c t e d i n t e r v a l , the r e s u l t s predicted Over a by t h e i r model bear good r e - semblance to c e r t a i n of the experimentally observed features. They f a i l to produce some important c h a r a c t e r i s t i c s however. Using the coupled-oscillator concept i t i s the intention of t h i s work to suggest changes i n the form of non-linear terms and examine the e f f e c t s on the solution. The stimulus f o r t h i s comes from the need (1) to obtain better c o r r e l a t i o n between model predictions and experimental results. 3. 2. PRELIMINARY Figure I provides a summary of Feng's r e s u l t s f o r the vortexinduced o s c i l l a t i o n of a c i r c u l a r cylinder (for given input c o n d i t i o n s ) . As Feng determined only three values of l i f t c o e f f i c i e n t transient in C amplitude, behaviour was used i n e s t a b l i s h i n g the l o c a t i o n of the jumps [Parkinson (5)]. The r e s u l t s demonstrate that over a d i s c r e t e range of flow speeds (the l o c k - i n range), cylinder displacement and f l u c t u a t i n g l i f t are p e r i o d i c i n time, with the same frequency, which i s close to that of the natural frequency of the spring-cylinder system. The amount by which the phase of the e x c i t i n g force leads the c y l i n d e r displacement i s measured as w e l l . Important features to note are the hysteresis loops which exist f o r both amplitude (of displacement and l i f t ) and phase. Also s i g n i f i c a n t i s the response f o r u)„ > 1.4 (outside of l o c k - i n ) , where c y l inder o s c i l l a t i o n s p e r s i s t at frequency close to while the frequency of the predominant e x c i t a t i o n i s considerably higher ( w ) . r Figure I I describes the configuration and the important of the spring-cylinder system. elements With the e f f e c t of the vortex-street wake on the cylinder included as a f o r c i n g function, the d i f f e r e n t i a l equation for transverse displacement X i s : c mX + 2g0) mX + mu) X = C (§V h) c n c n c L I. - • 2 2 T To nondimensionalize the equation, introduce T = to t n 4. h CO •- S V = (Strouhal Relationship) and obtain . r X" + 23X' + X = aco C (2.1) 2 0 L For modelling purposes, the problem now reduces to determining pression f o r C . Ju an ex- ' Hartlen and Currie o r i g i n a l l y suggested that the l i f t coe f f i c i e n t be governed by the following d i f f e r e n t i a l C" - atOoC; + ^ C' + w i-i Jj C 0 Li 3 o 2 0 equation C = bX' Jj (2.2) This form was chosen because of i t s s i m p l i c i t y , and because away from resonance of the spring-cylinder system (bX' •> 0 ) , s e l f - e x c i t e d osd i l a t i o n of amplitude and frequency approximately equal toy— respectively i s predicted f o r C is consistent with experimental (provided a, y are small). L observation i f w — and co 0 3 y This behaviour — ' i s set equal to the amY p l i t u d e of the l i f t c o e f f i c i e n t f o r a stationary c y l i n d e r (C ). L 0 The coupling term fliX ') was included to provide a dependence of on c y l i n d e r motion. C Li I t s presence leads to the p r e d i c t i o n of i n t e r e s t i n g behaviour f o r C0 c l o s e to co . n o Drawing a comparison between t h i s system and the well-studied forced o s c i l l a t i o n of the van der P o l equation (6)], one would expect a range of co for which C 0 [Stoker and X have the same os- Li d i l a t i o n frequency ( l o c k - i n ) , bounded by a range of co f o r which C has 0 components close to co and to (combination-oscillation). n 0 Figure I I I dem- onstrates that the postulated regions of c h a r a c t e r i s t i c response are consistent with experimental evidence - region A being associated with the 5. t y p i c a l forced response of an e l a s t i c system, region B with the transi t i o n a l range i n which frequency components close to co and 0) are present, n 0 and region C with the l o c k - i n range. I t i s not possible to make further assumptions concerning the d e t a i l e d nature of the response, as the f o r c i n g function i s i t s e l f dependent on C through Equation (2.1). Hartlen and Currie obtained an approximate s o l u t i o n to the system of coupled d i f f e r e n t i a l equations [Equations (2.1) and v a l i d w i t h i n the l o c k - i n region, by assuming X and (2.2)] to be given as follows (method of van der Pol) X = C T s i n fix = C„ s i n (fix + <{>„) (2.3) The~actual a n a l y s i s and a summary of r e s u l t s i s included i n Appendix A. Figure IV summarizes model p r e d i c t i o n s f o r the indicated input values. The r e s u l t s demonstrate the model's a b i l i t y to generate c e r t a i n of the features of vortex-induced oscillation. The s t a b i l i t y of the approximate s o l u t i o n i s not given d i r e c t l y by the method of van der P o l . formation i s the K-B method and developed i n Appendix A. An a l t e r n a t e method^ which does provide such i n - [Minorsky (7)]. This analysis i s introduced The r e s u l t s obtained allow one to confirm that the solutions summarized by Figure IV are stable, and that the two approximate methods of solution y i e l d i d e n t i c a l r e s u l t s provided that fi, ft = 1. 2 The r e s u l t s duce a double-amplitude obtained are encouraging. The model f a i l s to pro- response, however, and since the approximate s o l - ution i s v a l i d only within the l o c k - i n region, the system behaviour f o r 0) > 1.4 cannot be produced. o The following work i s concerned with an i n v e s t i g a t i o n of the form of model and solution used, with a view to improving the comparison between predicted and experimental r e s u l t s . 7. 3. 3.1 MODEL FORMULATION HIGHER_ORDER_NON L^ = f " It was decided to investigate the e f f e c t of increasing the order of n o n - l i n e a r i t y i n the governing equation f o r C . T Following a suggestion by Landl (8), odd power terms to seventh order i n C' were included. The equation f o r C Li then takes the form J-i C" ~ O0) C' + X_ ( c ' ) ( C M +'7^f ( C M + 0 ) L L 0J L C0 3 L l0~> L 3 5 7 o o o o 2 o C = bX* L (3.1) where a, y, n, <5 > 0 The j u s t i f i c a t i o n f o r including f i f t h and seventh powers of C' Ju comes from examining the homogeneous form of Equation (3.1) For a, y, (bX - y 0). 1 r\, & small, then C T = C_, s i n O J T 0 and C„ may have one or three p o s i t i v e r e a l roots. F In the l a t t e r case the middle root would be unstable, and the t r i v i a l s o l u t i o n C^, = 0 i s unstable i n either case. Considering the inhomogeneous form, i t was hoped that the increase i n n o n - l i n e a r i t y would r e s u l t i n the e x i s t ence of two stable C amplitudes for a given co within the l o c k - i n region; 0 Li a hysteresis e f f e c t possibly r e s u l t i n g from the manner of the dependence on 0) . o Approximate solutions (by the methods of van der Pol and K-B) to the system of Equations (2.1) and (3.1) are included i n Appendix B. Values for the non-linear c o e f f i c i e n t s a, y, n, 6 are determined by r e q u i r i n g that three p o s i t i v e r e a l roots C e x i s t within l o c k - i n (two of 8. which are known from experiment), and that one r e a l root C e x i s t away from ° l o c k - i n (bX' 0). r In order to match predicted with experimental values of l i f t c o e f f i c i e n t amplitude within l o c k - i n , the non-linear c o e f f i c i e n t s necessary were found to be of 0 (10). The e f f e c t of the magnitude of a, y» T), 6 on the approximate solution of Equation (3.1) has not been examined. Figure V shows numerical r e s u l t s f o r the indicated input values. The s t a b i l i t y analysis confirms that the middle amplitudes of C„ and H are unstable, and that the other amplitudes are stable. The r e s u l t s demonstrate the system's a b i l i t y to model the behaviour of C T reasonably w e l l within l o c k - i n (as i t was designed t o ) . The frequency and phase v a r i a t i o n s remain a problem, however, as to a f i r s t order approximation they are independent of C and thus do not r e f l e c t jumps i n amplitude which the system produces. The behaviour of the pre- dicted cylinder amplitude i s c l e a r l y a problem as w e l l . The predicted r e s u l t s i n d i c a t e that an extension to seventh order n o n - l i n e a r i t y i n C' r e s u l t s i n only marginal improvement of the system behaviour, while introducing further complications i n doing so. 3.2 COMBINATION-OSCILLAT Currie and Oey (9) proposed that the double amplitude response could be accounted f o r by the existence of d i f f e r e n t solutions to the system of Equations (2.1) and (2.2) for harmonic, or combination-type forms of s o l u t i o n ; that i s , whether X and C are assumed to be of form given by Equation (2.3), or as shown below (combination-type) 9. C L = C H + s i n Qr X = s i n («x + cp ) + C R s i n aye p (3.2) s i n (oyx + ^ They draw comparisons between the coupled-oscillator system and the forced o s c i l l a t i o n of the van der Pol equation. Actual r e s u l t s of a detailed analysis have yet to be published. Experimental evidence supports a combination-oscillation form of s o l u t i o n over a range of U) adjacent to the l o c k - i n region (Figure I'll, 0 region B). There i s no evidence for a s o l u t i o n of t h i s form within the l o c k - i n region, however. A study was carried out to see whether or not a s o l u t i o n of t h i s form could r e a l i s t i c a l l y account for one of the amplitudes within lock— in,-or the system behaviour outside of i t . cluded i n Appendix C. The actual analysis i s i n - A s t a b i l i t y analysis was not c a r r i e d out, as the approximations which are required i n order to combine the K-B method with a combination-oscillation form of s o l u t i o n are not at a l l obvious. Figure VI i l l u s t r a t e s the important numerical r e s u l t s f o r the indicated input values. The phase and frequency v a r i a t i o n s for and $ are i d e n t i c a l to those f o r the harmonic case and thus have not been shown. Away from the neighbourhood of to = 1, the forced cylinder response at 0 oy i s n e g l i g i b l e , thus A^, and cf> have not been shown as well. F The r e - s u l t s demonstrate the p o s s i b i l i t y of the existence of a combination-type o s c i l l a t i o n within l o c k - i n . Unfortunately, the analysis predicts a s o l u t i o n v a l i d only within l o c k - i n , and a complicated C t h i s range - C behaviour over i s predicted to have components of approximately equal Li magnitude at frequencies of Q and o j . 10. It would appear that the governing equations as formulated are not capable of accommodating a combination-type solution. 11. If one assumes the cylinder motion to be governed by Equation (2.1), and that within l o c k - i n X and C may be approximated then by substituting f o r X and C by; Equation ( 2 . 3 ) , i n Equation (2.1) and applying the p r i n c i p l e of harmonic balance, the following r e s u l t may be obtained: Since a i l the quantities on the right-hand-side of the equation are known or are measurable, the apparent s t r u c t u r a l damping during vortex-induced cylinder o s c i l l a t i o n may be calculated. These calculated values are then to be compared with the value measured i n s t i l l - a i r (which i s the value given by Feng). Table I summarizes the experimental r e s u l t s and the calculated r a t i ° 23 (23^)' w h e r e C 3o) 2 i s the wind-off s t r u c t u r a l damping. The e f f e c t i v e s t r u c t u r a l damping appears to depend on c y l i n d e r o s c i l l a t i o n amplitude as w e l l as wind speed. 23 Wo .98 1.06 1.12 1.21 H .03 .11 .21 .48 .3 TABLE I 23o • .45 .8 1.5 1.91 .5 4° .57«-> 1.7 1.1 2 «-KL4 9° .3 •«-* 2 . 2 1.5 10 +-+16 11° 37 -e-^-59 37 102 2.8 1.7 a 23 a 1.9 4 -w 5.6 4 2.7 E f f e c t i v e Structural Damping During Vortex-Induced O s c i l l a t xon. c = = .002: .002 = .97 12. I t i s clear that any model which f a i l s to take t h i s e f f e c t into account w i l l have l i t t l e chance of success i n p r e d i c t i n g experimental behaviour. It i s proposed that the e f f e c t i v e s t r u c t u r a l damping 'be. approximated by a r e l a t i o n s h i p of form: 23 = 2g 0 (1 + fu)02 A J J ) 2 The 0J o and provide a dependence of system damping on the wind force acting on the c y l i n d e r , and c y l i n d e r displacement respectively. One would expect the constant f to depend on the experimental configuration. An appropri- ate value can be calculated from the experimental r e s u l t s as follows: f 23 too 23o .98 .03 1.1 3.5 1.06 .11 1.5 4.0 1.12 .21 1.9 3.4 1.21 .48 .3 4 2.7 4.3 3.9 TABLE I I Damping Parameter Determination A value of f = 4 would seem to be indicated. The modified equation governing cylinder response then i s X" + 23 0 (1 + fw 2 0 A J J ) X ' + X = ato 2 D C (3.3) L In order to assess the e f f e c t of the proposed v a r i a b l e damping term, the system of Equations ( 2 . 2 ) and ( 3 . 3 ) has been solved approximately, assuming harmonic and combination-type forms of s o l u t i o n for X and C ^ . A stability analysis has been carried out f o r the harmonic s o l u t i o n and i s included 13. i n Appendix D. (i) Harmonic Solution Within the l o c k - i n range, assume X and C to be given by Li Equation (2.3). I f one substitutes f o r X and C into Equations (2.2) and Li (3.3) and neglects terms i n A^, C^, <f>^ and higher harmonics, the following system of equations can be obtained by applying the p r i n c i p l e of harmonic balance: aw aco 2 C 0 2 0 R C R cos (p = (1 - f t ) 2 H s i n <J> = A^, ftB (1 + fco H 0 o : / ' " " . . ^ " ' , ' 1 2 0 A^) - ^ ' . ' - 1 ....(3.4) (0J - ft ) . , , (1 - ft ) H - - —~ s i n <p - cos A — * p„ = — aw ft H H • 2 HH a co C to H 2 v 2 2 b A so 0 0 0 where B c 5 23 0 To proceed, i t i s necessary to make an assumption concerning the frequency behaviour ft(co ) (which i s close to 1 throughout 0 duce where |X| = 0(1) and make the assumption that the l o c k - i n region). Intro- 14. 1 - ft = X B„ - X 2 ^ — 2 S XB 0 fi, fi = 1 both of which are reasonable, since B =0(10 c ). From Equation (3^4) then, one obtains ao) XB. 0 ao) 2 0 ^ o C C sin ^ S y ^ l + f 0) R ° S *HH + r S ± *H n ( 1 0)H 0 -A- s i n * - cos * ~ ^ ( 1 > u) where A = 0) 2 o A^) '= (3.5) 0 S H 0 - 1 o From Equations (3.5.1 and 2) tan a 2 ,B n. = 1 + f "° -\ A X r 2 H + (1 + f 2 W o A^) 2 ... (3.6) aw, From Equations (3.5.3 and 4) X 2 = (1 + f o ) 2 0 A ) H ., _ ab where n = —— Bo no) ^ - - a + fa).' y . (3.7) Substituting f o r A i n Equation (3.6.2) aw n » u 0 Substituting f o r tan (f> i n Equation ( 3 . 5 . 3 ) 1 + f0) o — - cxw ^ < + r } (1 - 2 w ) = 0 0 then substituting f o r A and p _A_) cta) (from Equations(3.7 and 8)) -ml A 2 ( 0 _ ( i + fo) A 2 0 (!•:+ fa>„ AJJ) [ 1 - where C, = f 1 V —-) ~2 H ) ) ^ A ^ (1 + — i F A ) O 2 - ) n A which can be expanded to y i e l d 0 - S \ X 7 + 8 2 ^ + + where = C g ; L 2 1 (f 0) )' 2 o 8, = 3C (f0) ) 2 8 3 g 4 5 = 3C 1 = C^ 2 o f 2 W ° 2 - 2 C (fw ) 2 1 1 o o g 7' A H one + 8 8 ^ AJJ) 16. H g = - 7 ' 2c = f u„ i 2 A (1 + ( — ) aw 2 ) ' 0 The seventh order polynomial i n A^ can be solved approximately as a f u n c t i o n of 0) and the input parameters (n,b, C o L , f ) . Once the roots A^ have been 2 can be determined from Equation(3. 8), and A 2 from i B Equation (3.7). The sign of A^^ (and thus = 1 - A -y) can be determined 1 determined, values C 1 2 by s u b s t i t u t i n g f o r C„ and tan <p'„ i n Equation (3.5.3). 1 1 Figure VII shows the r e s u l t s of such an analysis f o r the indicated input values. The r e s u l t s demonstrate the system's a b i l i t y to.generate multiple amplitudes i n A^, C , (p and ft with varying to . The p o s s i b i l i t y R 0 of producing a hysteresis e f f e c t e x i s t s as the upper branch of A ^ o O v a l i d f o r ft < 1 only, and the two lower branches f o r ft > 1 only. is The p r i n c i p l e r e s u l t of the s t a b i l i t y a n a l y s i s (Appendix D) i s that the middle branch of Ay.(u)0) i s unstable, while the upper and lower branches are stable. The arrows on Figure VII incorporate t h i s information i n describing p o s s i b l e behaviour for increasing or decreasing w • 0 Although there are s t i l l remaining d i f f i c u l t i e s with the amplitudes of X and C , and with trends i n the phase angle f o r ft > 1, the i n c l u s i o n of the v a r i a b l e damping term has resulted i n a s i g n i f i c a n t improvement i n model performance within the l o c k - i n range. (ii) Combination-Oscillation Solution If one assumes X and C^ to be given by Equation substituting into Equations (3.2), then (2.2) and (3.3) and neglecting terms such as A^, (pp, higher harmonics and combination tones and f i n a l l y applying the p r i n c i p l e of harmonic balance, one obtains the following system of equations: 2 ato 2 C 0 aco 2 C 0 (1 -a> ) F F 2 s i n <j> = Ap o^B. (1 + fw j ^ ) p ato cos (f> = p F C 2 0 0 cos (J> = A H h (1 - ft ) 2 H ato„ ... (3.10) (C0 :) o (Wo aoj 0 to W p . " F F T S ± ^ F N 2 - ST) ~~o^ft - G O *F S 2 , (Wo c (u 2 2 - ST) 0 ato ft o - Sr> s . + S l .ft . , n P F ( P H (if) + 2 H F au) C„ 3 A + 2P„) H' 2 0 = 0 P: ^ F (ft")) J A H = 0 P > 2 + 1 s i n <J> - cos <J> H ( o 2 - C—) (P* 1 « o . o L 0J„ cos cJ) + s i n (p 1 ( H ato C 0 0 Next introduce 2 a 2 H 18. 2 "H = ^2 o OX0 ft 0 then from Equations ( 3 . 1 0 . 5 and 6) CT ( 1 + tan b a ? 0J < j >) = F' ~ ~ Ba 2 tan cp o F OJ 0 bAp bat0 s i n cp^ which uses — — — = ~ F (1 + f . 5 u C - ) 3 11 V o from Equation ( 3 . 1 0 . 2 ) . : To pro- ceed, i t i s necessary to make assumptions concerning the frequency behaviour ft (too) (which i s close to 1 ) and co„(co ) (which i s close to U) ). 0 = 0J , then from Equations ( 3 . 1 0 . 1 and 2 ) Assume that w . t a n 0 o * 3 (1 + ^ o 2 V 1 - too 2 2 thus, tan < J > < < 1 f o r to away from the immediate neighbourhood 0 of i0 = 1 . o From Equation ( 3 . 1 1 ) then 2 where A = 0) o Substituting - 1 f o r O and tan d) i n Equation ( 3 . 1 0 . 5 ) y i e l d s P F ( ? ) 2 + 2 P H ^ ( 1 - £ O I ^ > < ^ ) . . - . . ( 3 . 1 2 . 1 ) 2 From Equation ( 3 . 1 0 . 7 ) P H then solving f o r p„ + Vft^ ) 2 = ( 1 + Otoo'tan * > ^ H from Equations ( 3 . 1 2 . 1 and 2 ) 2 •••• ( - - ) 3 1 2 2 From Equations (3.10.7 and 8) (a„ H p r - r - — — — - — - .) tan 6 , + aft (1 + fca O H 2 XB introduce ft = 1 - —^- = o a H 7 Y 0 and assume that 1 - ft = X B 2 - 0 X B 2 = XB 2 F ft,ft = 1 2 then examine 2 2 - ft ~, A ao) ft aw H 0 -D for a „ 2 1 + f 0) o 2 A x — - 2 and tan (J) i n Equation (3.14), one obtains H X 0 V r 3 ^ ( i + f -o v = E t a n Substituting Q Z 0 ) / E H = ( > U ) ° 2 - ? ]) (1 + f t o A ) 2 G 2 A (1 + f t o / A ) H From Equations (3.10.1 and 2) 2 (aw ) C 2 0 2 R = A ^ Bo 2 2 ( (1 - ft ) +• (1 + f too Ag) ) 2 2 2 2 „ 2 ,2 . „ . . 2 A " B " ( X ' + (1 + fto " A J J ) ; ) 2 # H 0 0 2 then substituting f o r X : h 2 C „ " = " 2 nto A 0 or 2 (1 + f a ) , AJJ) 20. P H = A H 2 I 2 . nco C A 2 ( 1+ f ° w 2 (3.16) V 0 Equating (3.13.1) and (3.16), and s u b s t i t u t i n g f o r tan <j> and X one obtains H . nto A ( l + fw aio 1) n 0 / 0 3b , 2 ,., , _ 2 . v nA (C co ) ^ 0 i-"o + 2 nB 0J - 1) cxA Q o which can be expanded to y i e l d 0 = g A l 7 H +g A 2 H + . . . 6 +gg where & 1 = 9 C (fto ) 2 2 1 0 g 2 = 27 C^ g 3 = 27 C g 4 5 9 C g 5 5 12 H E *7 * 6 c (f 2 2 2 o + 6 C ± C i C ) (ft0 ) 2 x 1 2 W o foj 2 C (fco ) 2 2 0 0 2 < 2 C 2 + > Y (3.17) 21. L 0 2 n B to °2 ~ a A r - 0 The seventh order polynomial i n q ~ 1 can be solved i n a manner similar to 2 Equation (3.9). Once the roots have been determined, values f o r 2 can be determined from Equation (3.16), and A. from Equation (3.15).. The A. sign of X. (and thus ft^ = 1 — ) can be determined by s u b s t i t u t i n g f o r 1 i Bo 2 2 C and tan cp i n Equation (3.13.1). C i s then given by Equation (3.13.2). H. H. h. 1 1 x and Aj, and (p^, from Equations (3.10.1 and 2). Figure VIII shows the r e s u l t s of such an analysis for the i n dicated input values. Since the forced c y l i n d e r response at to i s n e g l i g i b l e F away from the neighbourhood of to - 1, A^, and cp^ have not been shown. D The r e s u l t s demonstrate the system's a b i l i t y to generate a combination-type s o l u t i o n v a l i d only at the extremes of the resonance region, and r e a l i s t i c behaviour of C f o r to < 1.15 or to > 1.38. These features are both characterQ e i s t i c of vortex-induced cylinder o s c i l l a t i o n . There i s no s o l u t i o n f o r 1.15 < to < 1.28 as C i s imaginary r . 1 0 over t h i s range. There i s no s o l u t i o n f o r to < 1.05 as the r e s u l t s are i n 0 v a l i d i n the neighbourhood of to = 1. Q The i n c l u s i o n of the v a r i a b l e damping term i n the d i f f e r e n t i a l equation governing cylinder displacement appears to allow f o r the r e a l i s t i c accommodation of a combination-oscillation form of s o l u t i o n . This has the e f f e c t of extending the range of a p p l i c a b i l i t y of the coupled-oscillator model outside of the l o c k - i n region. 22. 4. DISCUSSION Several changes i n form of the governing equations of Hartlen and Currie's o r i g i n a l coupled-oscillator model f o r vortex-induced osc i l l a t i o n have been suggested and examined. Various forms of s o l u t i o n to the modified equations and the question of t h e i r s t a b i l i t y have been investigated as well. Predicted r e s u l t s have been compared with exper- imental information, i n order to obtain a measure of t h e i r usefulness. The r e s u l t s of t h i s work show the a p p l i c a t i o n of a combinationo s c i l l a t i o n form of s o l u t i o n to Hartlen and Currie's o r i g i n a l model, and the extension to a seventh order n o n - l i n e a r i t y i n C ' to be unproductive. A p o s i t i v e contribution has been made, however, with the i n c l u s i o n of an e f f e c t i v e s t r u c t u r a l damping term dependent on wind speed and c y l i n d e r displacement. The modified governing equations then produce a hysteresis mechanism within the l o c k - i n region (harmonic s o l u t i o n ) , and r e a l i s t i c system behaviour outside of l o c k - i n (combination-oscillation form of solution). The i n c l u s i o n of a v a r i a b l e s t r u c t u r a l damping term (which i s consistent with experimental evidence) has the e f f e c t of improving trends i n the coupled-oscillator model performance, and extending i t s range of applicability. I t i s proposed that the r e s u l t s are encouraging enough to warrant further i n v e s t i g a t i o n of t h i s form of n o n - l i n e a r i t y . 23. REFERENCES 1. Ferguson, N., "The Measurement of Wake and Surface E f f e c t s on the S u b - c r i t i c a l Flow Past a C i r c u l a r Cylinder at Rest and,in VortexExcited O s c i l l a t i o n " , M.A.Sc. Thesis, U.B.C, 1965. 2. Feng, C.C, "The Measurement of Vortex Induced E f f e c t s i n Flow Past Stationary and O s c i l l a t i n g C i r c u l a r and D-section Cylinders", M.A.Sc. Thesis, U.B.C, 1968. 3. Parkinson, G.V., "Mathematical Models of Flow-Induced Vibrations", Symposium on Flow-Induced S t r u c t u r a l Vibrations, Karlsruhe, August 1972. 4. Hartlen, R.T., Baines, W.D., and Currie, I.G., "Vortex-Excited of a C i r c u l a r Cylinder", UTME - TP 6809, November 1968. 5. Parkinson, G.V., "Wind-Induced I n s t a b i l i t y of Structures", P h i l Trans. Roy. Soc. Lond. A, 269, 1971, 395 - 409. 6. Stoker, J . J . , "Non-linear Vibrations i n Mechanical and E l e c t r i c a l Systems", Interscience Publishers, Inc., New York, 1950. 7. Minorsky, N., "Non-linear O s c i l l a t i o n s " , van Nostrand, 1962. 8. Landl, R., " T h e o r e t i c a l Model for Vortex-Excited O s c i l l a t i o n s " , •International Symposium V i b r a t i o n Problems i n Industry, Keswick, England. A p r i l 1973. 9. Currie, I.G., Leutheusser, H.J. and Oey, H.L., "On the Double-Amplitude Response of C i r c u l a r Cylinders Excited by Vortex Shedding", Proc. CANCAM '75, Fredericton, New Brunswick, May, 1975. Oscillation 24. APPENDIX A Hartlen and Currie's o r i g i n a l system of d i f f e r e n t i a l equation-soluti on \ by the methods of van der P o l and K-B Governing System X " + 23 X' + X = au) 0 2 0 C A.l 3 C " J- (i) - aa) C + 0 1 (C ') + u)„ CL = bX' u) L L 2 T Li 0 Solution after van der P o l Assume X = A^ s i n fir ...» A. 2 C L = C H + S±n then,,.substituting f o r X and C V i n Equation A . l and neglecting terms such as A^, <p^, and higher harmonics, one obtains the following system of equations a f t e r applying the p r i n c i p l e of harmonic balance: au) 2 C 0 2 R 2 a<jj 0 cos <j> = (1 - ft ) H C s i n <p = B fi H 0 ... A.3 2 2 O0) owe fl o 2 c o s * + S l n H ^H s i n AH - cos <f> (1 " " H " fl V VJ ( 1 - t > (7—) K P p ) H' TI H } =aco = c ° CL H where B 0 — 23 0 25. C C L V 0 _ 3 Y C 2 HV-) P From Equations A.3.1 and 2 , B ft tan (p Q 1-ft H 2 2 ao) 2 •1 + C o t 2 <f> n 2 From Equations A.3.3.. and 4 ... A. 4 2 ft 1 - n 2 1 + G ot 2 <j> H 2 C = ( too C H L ) 2 ( ^ 0 1 + (to p 2 - ft ) ) 2 '•" ctto ft tan * 0 n , _ ab where n = — (ii) Solution by the K-B method (to a s c e r t a i n the s t a b i l i t y of the approximate solutions to Equation A.1). Rewriting Equation A . l X ' ' + X = atoo 2 C - | * L b (CJ ' L - Oto 0 c ' + XL to 0 c; L 3 + Wo C 2 T L ) ... A.5 assuming X = (x) s i n (x + 8 H (x)) .. A.6 X' = A J J (x) cos (x + e R (x)) 26. which implies that (x) sin(T + e R (x)) + ^ 6' or 6 (T) (x) cos(x + 6 r sin(x + 6 (x) = - ^ C O ^ (x) H (x)) = 0 (x)) H A. 7 cos(x+ 9 (x)) R then multiplying Equation A.5 by X', one can determine V ( T ) - ( a L - C ¥ ( C L *' " ^ C L t + L C > 3 ^ + W C ° S ^ + 8 H ( T ) ) and from Equation A.7 e' / 2 „ (ato C (x) = Since a, B 0 c B - ^ c L (C- - aco C 0 L + J - C' 3 L + 0) o 2 c )) L sin(x + 6 are 0 (10~ ) 2TT •» ^ 2 (a w ~ 2TT C L 0 i T B - Tf- ( • • • )) D cos ip dip A. 9 2TT (aw 2lT 0 where If one assumes that C 2 C L g -'-jp ( ... )) s i n ip dip ip = x + 8. H = C (x) s i n ( X + cp (x)) TT L H H and that C', cp*, are 0 (10~ ), then from Equation A. 9 n. n a C H - (w 2 o 1 2 . + - (1 - " ) ) s i n C n ato + — n (1 - H .) cos C 2 H (x)> 27, r -a C, * 2 3' H (w + ^ (1 - w )) 2 cos C + 2 D a ( i - s i n c too ... A.10 where £ = 9 R - <J> H 2 2 C„ C H ._ H 4. a ~ C 2 3 y = H ~ L Stationary K, = K, o solutions to Equations A.10 exist f o r = 0 i n which case , C„ = C and 6 ' = - K which implies 8 = -K x. H H H H . s S order approximation £ must not be a function of X, thus <jj• H H H S s H - K„ X - r where r f H ^s s s = r s CO. (w 0 2 +•£(!- w ) ) n s i n ? + ^JSifl: s n 2 0 = 8 - Z, H s s s In which case one obtains P - To f i r s t . H ! _ ( H _|) os C Wo 2. C = 0 s ... A.11 0 (Wo 2 1 0 P + f- ( 1 - w ) ) cos ? + ^ 2 0 —V"- 2K H ( 1 - ~ § ) sin K = ^5 s 0 Two further equations are obtained by r e q u i r i n g that the stationary solutions s a t i s f y Equation A . l . Substituting f o r X and C : T X = AJJ. s i n (x + e R ) = A J J s i n (1 - K ) x E ^ R C L = C H s i n (x + cp h ) E C.I xi s i n ( (1 - K )x - C ) = where 11 ft s = 1 - K H provides s i n fix ... A.12 C s i n (ftx - X, ) H s 28. at0 C o sin £ H = - B„ ft A^ g S £ ... A. 13 aco 2 0 C cos fl Cg 2 (1 - ft ) = s s Solving Equations A.11 and A.13 one obtains 2 1 - n 1 + cot C 2 c 2 H = r 2 H au _ BO s A.-14 s (1 + cot £ ) s tan - (Co,. ) 2 aoj (1 + a> 0 (n - 1)) c S where tan £ i-ft which can be shown to be i d e n t i c a l to the r e s u l t s obtained by the method 2 ~ of van der P o l (Equation A. 4) provided that ft, ft = 1. To determine the s t a b i l i t y of a p a r t i c u l a r solution, one need H examine the sign of - j — H d A only i n the neighbourhood of the root A^ . s the expression f o r A^ (Equation A.10.1) one can determine d A i d = dA thus H ^ (— dAjj C C H a r a P H — R n u 0 ^ cos C) ^ From 2 9 . -aa H d C, H P s n as. cos r -T-7- s d h A. 1 5 From Equation A . 1 4 . 2 H ____ d _= , B^Jl C , ( 2 } (. l. + o t L C O. 2 A aco H c The s t a b i l i t y c r i t e r i o n i s d A' d C H Examining Equation A . 1 5 , since 7 - 7 - , < 0 Stable > 0 Unstable a, a , p , n, co , are a l l p o s i t i v e 0 quantities, then the question of s t a b i l i t y i s decided by - cos? or - cos ( Thus ( - C ) < T / 2 •> (•-5) w i l l be stable > n / 2 - > A J J w i l l be unstable 30. APPENDIX B Extension to 7th order non-linearity i n C' - s o l u t i o n by the methods of van der P o l and K-B. Governing System X'* + Bo X' + X = a w C L* - ™° L C ^ + L ( C ( - ^3 ) 3 0J C V 5 + A C 2 0 5 ( L C 0J o ) 7 + C L = ' b X o B.l (i) Solution a f t e r van der P o l Assume X = Ag s i n ftx • • • B. 2 C = C L Substituting f o r X and C s i n (fix + cb ) R R into Equation B . l , and applying the p r i n c i p l e of harmonic balance one obtains: aco 2 0 C 2 (1 - ft ) cos cp = R H 2 , aco ; (co a n ^ iy 4a ~ 2 C0 r 2 H C 0 2 0 s i n o> = A R R R B ft ... B.3 0 - ft ) cos cp + to ft 2 a 5 n > ft,4 8 a w7 4 H 6 ~ 64 ( ) aw7 n 6 o (COQ 9 2 2 ft ) . , P^- s i n <p„ ao) ft H 0 ! _ 31 ( A ) 2 2 r 6 C i ^ H c 3 s l 6 n - n *H " 0 31. I H rJL) 8 a ^w ' 2 _ 15 6 _ft 4 c 64 0 Equations B . 3 . 3 6 6 a ^ W/ U H cos A = H aa) C 0 H and 4 are obtained by assuming that C* = C ft cos (ftr + L ( C L ) (C[) ( From Equation B.3 C L ) " 3 5 ri I = | = 7 A) H If fi3 C C ft 5 flT+ cos (ftx + 5 H H ( V cos ? H V COS (fiT+ FI ? (J) ) one can determine: B ft tan 0 1 - ft' r 2 ( 2 H B ft (1 + cot Wo 1 \ B.4 G <|>) H ft' — n 1 + cot A H 6 8 ]1 Wp_. 2 7 4 r ' ; L H + 48 35 j_ 6 .u^ 6 ft * 6 ; | ( " Q 2 - " > 3 6w fttan A 0 = 0 H In order to provide f o r a double amplitude response within the l o c k - i n region, three r e a l roots of the cubic polynomial i n C must exist. H For a p a r t i c u l a r value of ft (and thus c o ) within the region, values of 0 32. and C are a v a i l a b l e from experimental data and provide two max min equations f o r the determination of the non-linear c o e f f i c i e n t s . The choice of a t h i r d root (C Y Ct •g, ~f so that ) i s made i n order to e s t a b l i s h "unstable a single r e a l root of predetermined amplitude (C ) i s lo T predicted outside of the l o c k - i n region. of an appropriate value of 6 as w e l l . This requires the s e l e c t i o n Once the various parameters have 2 been s p e c i f i e d , Equation B.4.4 can be solved f o r C methods. (ii) (w ) by standard c H, 1 Solution by the K-B method Rewrite Equations B . l X' ' + x = a W„ C - T L <V> , „ 3 V b + v ~ i W„ ' V + 7 + k C C L , ) 3 .... B.5 "' °i 2 W 0 assume X = X' = then proceeding (T) s i n (T + 0 H (x)) (T) cos R (T)) (X + 9 i n a manner i d e n t i c a l to that introduced i n Appendix A ( i i ) one obtains T ~ I 0.H a C H 2 -a C H - (Wo 2 ( w 2 0 + ~ (1 - Wo )) n 2 + n (1 - w ) ) 2 0 s i n - C - r - ^ G (C„) cos £ n n cosX + — n G (C„) s i n X n B.6 33. where r ( G K c ) B = . 3 i 4a 1 J c t, = r x. - (p C C I n J L _ 3 5 6 S i _ 8 a 4 64 a i.6 2 4 + 2 0) W o 6 0) 0 o Stationary solutions to Equation B.6 exist f o r A ' =0 i n which case R A = R s C = C H H s 0 * = - K H H To a f i r s t order approximation, £ A = H 9 s s -^s 0 which implies = - K H T = - K x H s H must not be a function of - ? s W h e r e C s ^ s X, thus ( T ) In which case one obtains (0) 9 o Z 1 9 + j- (1 - to/)) s i n r XI OUJ + ^ o XJL G (C Jtl ) cos r o ' = 0 s ' <. u 2 + ± ( i - a,. )) cos 2K 2 ? s + ^ c (c H \ .in c . s <r± H ••• " B 7 I t i s required as w e l l that the stationary s o l u t i o n s a t i s f y Equation B . l for X = A C L = C H H s i n (x + 01T ) E A s i n (1 - K„)x E A H 11 n n s i n (X + <p ) E C s i n ( (1 - K ^ x - r, ) E C s s H s R g s i n fix s i n (fix - R ? s g ) 34. which provides ato C Q a sinC u = - B ft A s s 0 H s B.8 a0J C 2 o cos £ u ri S S = A (1 -ft ) 2 u n S From Equations B.7 and 8 one obtains: 2 1 1 - n 1 + cot r, B.9 9 H C 64 35 M 2 H 2 ^Bft ' S H 7 ° 6 . 2 , + cot £ ) (1 L H 35 + 6 u ° C H . 6 W = 0 ° -ftB where tan £ 1 - ft' which can be shown to be i d e n t i c a l to the r e s u l t s obtained by the method 2 4 of van der P o l (Equation B . 4 ) provided that ft, ft , ft , 6 ~ ft =1. To determine the s t a b i l i t y of a root A^ , examine the sign of d V From Equation B . 6 . 1 one has that A^' = F ( C ) , therefore R A H = A H d d A V dF H " d C H d d A C H H which can be evaluated from Equations 35. B.6.1 and 9.2. The c r i t e r i o n of s t a b i l i t y being < 0 stable > 0 unstable 36. APPENDIX C Combination-oscillation s o l u t i o n applied to Hartlen and Currie's o r i g i n a l system (solution by the method of van der P o l ) . Assume X = A^ s i nfix+ Ap s i n to^x ... C . l C L = ° H S V + ± n(fiT then s u b s t i t u t i n g f o r X and C + C F S L N ( W F V' T + into Equation A . l and neglecting terms such as A ', (f> ' , higher harmonics and combination tones, the following r e s u l t s H r are obtained a f t e r applying the p r i n c i p l e of harmonic balance: aw C 2 0 2 0 2 c F c s iin n * ao> ao) cos cf> = Ap (1 - Up ) F C cos R A H = Ap, B F = A R 0 2 (1 - fi ) C.2 ato, 0 C R s i n 4> = ^ B H 0 2 / 2 2. (w - Wp ) 2 0 aw . w„ cos <p_ + s i n <J) 1 ( F P 0 2 (Wo 2 - cy Wp^) S ^ T T CnO s i n Ap - cos cp i CO,/ ( P F (TT> + 2 PH> 2 P 2 + H ) aa) C Q T 2 (LOO 2 - ft ) ato ft ft i C O , S + S > l n 0 < ( l0 2 o ^ - ft ) aa) ft o 0 S l , ft , ~ n c o W F 2 = 0 *H 2 . 2 i 2 U F 2 ao) C 0 s 2 where 2 p = H _4 a 3 y C L J ° 2 F 4 3 H. 2 a ~C. Y V Note that Equations C.2.5-7 assume that 'V* ~i 3 + T a 3. C 0) o F <r <V V cos + C H O + 2 C 2 <<T> °F ? ft C„ cos (ftx:+ cb ) w F W F V> + 2 C H 2 From Equations C.2.5 and 6 one obtains tOo 1 n 1 • OL + cot CO 2 CO 1 + • • * C• 3 ao) Wp tan cP 0 From Equations C.2.7 and 8 COo ft 2 1 -• n 1 2 + cot cb. H F 38. 2 2 1 +. (Mo - « ) au) ft tan <J> H J to„ P + H 2 p F<fiT> = <^> 0 From Equations C.2.1-4 tO tan <p. F - ( B U> F D B 2 1 - w„ 2 .a to„ . 2 p 2 F (1 + cot cf> ) fiB tan <p = 0 1 - JT 2 B N 0 n 2 ( i + cot <{>) H If one assumes that fi ? 1 and to = 0) then Equations C.3 can o solved for p„ „, A. „, cp „ as functions of w T H, r ii, r 0> a,y and r\. fi(to ) 0 and H, r to (to ) are given by the appropriate roots of Equations G. 3.3, and• 1. F 0 39. APPENDIX D Variable s t r u c t u r a l damping - s o l u t i o n by the K-B method. r''" Governing system X" + B 0 (1 + f03 y 2 o X' + X = aw Cj 2 0 ... D . l C " T Y - au> C ' + - - (C ') 2 + a> 1 0 T T C 0 = b X' T Assume X = Ag (x) sin(x + 9 (x)) R X' C L AJJ (x) cos (x + 9 (x)) = C R R s i n (x + <|> (x)) H Then proceeding i n a manner i d e n t i c a l to that introduced i n Appendix A ( i i ) one obtains 2 1 2 s i n £ (u) o - - (1 + f0) ^ V - cos C H i 2 (0J - 1)) o _ 2 2 _ OU) „ (1 + f 0) A^) — " H P ) : D.2 o cos C ( aj + sin £ ( 1 o 2 0 (1 + fw 0 1 - - (1 + fw 2 Ay) . * «o — ( 1 0 0 A^) (u> 2 0 "0)V 2 o - 1)) 4 0 . Stationary solutions to Equation D.2 exist f o r AJJ' = 0 i n which case A ^ = C H = C and 8 ' = - K which implies 8 H H H £ £ s H = - K T. H To a f i r s t order s must not be a function of T , thus A H ^ £ (x). g H approximation K T - r where H s ^s Two further equations are obtained by r e q u i r i n g that the stationary s o l u t i o n s a t i s f y Equation D . l . l f o r X = A C L = C H S ± n S ± n ( 1 H ( ( 1 s " H K ) T ~ H K " s C = ) T = } C s i n fir *H H S l n ( f i T s " s C } The expressions which are derived f o r A , C , fi and H H S £ s s are i d e n t i c a l to those obtained by the method of van der P o l (Equations(3.6-9)) where - - K \B C H To determine the s t a b i l i t y of a root A ^ , examine the sign of From Equation D.2.1 one has that A ^ ' A H = = F (A^, C , 0, R therefore \ V d d A H = 9 F _ 9 A H 3 F d C ' H C H d A 3F H d£_ H d A . D.3 H 41. The p a r t i a l d e r i v a t i v e s can be obtained from Equation D.2.1, and the exact d i f f e r e n t i a l s from Equations (3.5.2) and (3.8). s t a b i l i t y being d V a. < 0 stable > 0 unstable The c r i t e r i o n f o i 42. FIGURE I: Experimental Results For Vortex-Induced Oscillation of a Circular Cylinder (Feng) 43. FIGURE II: Schematic Diagram of Experimental Configuration 45. FIGURE IV: Theoretical Predictions from Currie's Original Model Hartlen and 47. FIGURE VI: Theoretical Predictions for CombinationOscillation Solution Applied to Hartlen and Currie's Original Model 48. 120 -80 1-40 1.2H Q H •8H -.8 C /3 = . 0 0 1 / a = .0022 C. = .5 b =1.24 a = .55 -.4 { = 4 fr-2 ~V5 1.8 o FIGURE VIS = Theoretical Variable Solution Predictions ot the Structural Effect of Damping — Harmonic FIGURE VIII: Theoretical Predictions of the Effect of Variable Structural Damping—CombinationOscillation
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Coupled-oscillator models for vortex-induced oscillation of a circular cylinder Wood, Kelvin Norman 1976
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Title | Coupled-oscillator models for vortex-induced oscillation of a circular cylinder |
Creator |
Wood, Kelvin Norman |
Date Issued | 1976 |
Description | The vortex-induced oscillation of a circular cylinder is modelled by a non-linear system with two degrees of freedom. The periodic lift acting on the cylinder due to the vortex-street wake is represented by a self-excited oscillator, which is coupled to the cylinder motion. Approximate solutions and stability criteria are presented which are valid over restricted intervals. Changes to the form of the coupled-oscillator model and its approximate solution are examined in order to improve the comparison between predicted model and experimental results. The changes are motivated by the study of experimental evidence, and by comparison with the known properties of similar systems of non-linear equations. Significant improvement in the coupled-oscillator model performance is obtained through the inclusion of an effective structural damping term which is dependent on wind speed and cylinder displacement. |
Subject |
Coupled mode theory |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2010-02-09 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080800 |
URI | http://hdl.handle.net/2429/19909 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of Mechanical Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
Aggregated Source Repository | DSpace |
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