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Coupled-oscillator models for vortex-induced oscillation of a circular cylinder Wood, Kelvin Norman 1976

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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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