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Aeroelastic behavior of square section prisms in uniform flow Wawzonek, Mitchell A. 1979

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AEROELASTIC BEHAVIOR OF SQUARE SECTION PRISMS IN UNIFORM FLOW  by Mitchell A. Wawzonek B.A.Sc, University of Waterloo, 1977  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MECHANICAL ENGINEERING  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA June, 1979 ©  Mitchell Anthony Wawzonek, 1979  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r  an advanced d e g r e e a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and I f u r t h e r agree that permission f o r s c h o l a r l y p u r p o s e s may by h i s r e p r e s e n t a t i v e s .  for extensive  study.  copying of this thesis  be g r a n t e d by the Head o f my Department o r I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n  o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my written  permission.  Department n f  Mechanical  Engineering  The U n i v e r s i t y o f B r i t i s h Columbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Date  June 28,  1979  ii ABSTRACT  The w o r k p r e s e n t e d  in this  thesis  i n v e s t i g a t i o n and a n a l y s i s o f the supported  concerns  the  o s c i l l a t i o n behavior of e l a s t i c a l l y  square prisms i n a uniform flow o f a i r .  understanding  the  atmosphere  in  or  behavior of structures  i n flow  This i s situations  than about  showed t h a t when t h e  2.5,  vortex  quasi-steady  amplitude response. wind speed range,  shedding i n f l u e n c e d the theory  developed for  Some o f t h e  data  the  t h o s e due t o  vortex  i n the present  tests,  analysis  conditions.  The v e l o c i t y a t  particularly  sensitive  A that  depends i s  but  literature  the  U  0  showed t h a t  during forced  aeroelastic t o be v a l i d  the measured  so the  in  this  oscillation  behavior. for U greater  aerodynamic force  than  behavior  i n t u r n d e p e n d e n t o n R e y n o l d s number  and end  which g a l l o p i n g i s p r e d i c t e d to occur, U , 0  to these e f f e c t s ,  defines  less  g a l l o p i n g could not p r e d i c t  i n the  a n a l y s i s was v e r i f i e d  on which the  parameter  in  behavior s i g n i f i c a n t l y ,  measurements o f aerodynamic f o r c e  The q u a s i - s t e a d y 2.5  s u c h as  n o n - d i m e n s i o n a l w i n d s p e e d U was  c o u l d be u s e d t o more a c c u r a t e l y p r e d i c t  about  in  a n d a e r o d y n a m i c i n s t a b i l i t y known a s g a l l o p i n g .  The t e s t s  that the  a step  water.  The p r i n c i p a l modes o f e x c i t a t i o n s t u d i e d a r e shedding,  experimental  since the value o f the  i s not n e c e s s a r i l y  constant.  is  aerodynamic  Table of  Contents page  Abstract  i i  List  o f Tables  iv  List  of Figures  v  Acknowledgement  v i i  Nomenclature  viii  CHAPTER ONE  1  1.1  Introduction  1  1.2  Brief  1.3  Review o f the  Literature  1.3.1  Effects  of vortex  1.3.2  G a l l o p i n g from Rest  Review o f the  Quasi-Steady Theory  3 shedding  3 4  CHAPTER TWO 2.1  2.2  2  6  S t a t i c Measurements  and A n a l y s i s  6  2.1.1  Measurement  of C (a)  6  2.1.2  Application  to  7  2.1.3  Comparison to Measurements  y  the  Quasi-Steady Analysis from F o r c e d O s c i l l a t i o n  8  Dynamic B e h a v i o r E x p e r i m e n t s  9  2.2.1  Tests  9  2.2.2  Results  11  2.2.3  Discussion  12  CHAPTER THREE  17  -Conclusions  17  Bibliography  18  Appendix A  19  Appendix B  20  Appendix C  21  Appendix D  22  iv List  Table  I  of  Tables page 21  V  L i s t of Figures page 1(a)  Normal f o r c e v a r i a t i o n , square s e c t i o n  22  (b)  R e l a t i v e angle o f attack  22  (c)  Predicted  2(a) (b)  f o r moving s e c t i o n  g a l l o p i n g response f o r square s e c t i o n  Measured phase angle from dynamic measurements Measured f o r c e c o e f f i c i e n t ( f i g . 12 o f (4))  22 ( f i g . 18 o f (3))  from dynamic measurements  23 23  3  Measured l i f t  coefficients  24  4  Measured drag c o e f f i c i e n t s  25  5  Normal f o r c e c o e f f i c i e n t s  26  6  Normal f o r c e c o e f f i c i e n t s near a = 0  27  7  Equation  form  28  8  Predicted  g a l l o p i n g response o f square s e c t i o n  29  9  Schematic:  10(a)  (2.1) i n g r a p h i c  g a l l o p i n g and damping measurements  30  Damping c a l i b r a t i o n  31  V a r i a t i o n o f damping w i t h o s c i l l a t i o n amplitude  32  11  T y p i c a l t e s t model  33  12  Galloping  t e s t s , 3.3 cm model, U =0.44  34  13  Galloping  t e s t s , 3.3 cm model, U =1.38  35  14  Galloping  t e s t s , 3.3 cm model, U =1.67  36  15  Galloping  t e s t s , 3.3 cm model, U =1.77  37  16  Galloping  t e s t s , 3.3 cm model, U =1.87  38  17  Galloping  t e s t s , 3.3 cm model, U =2.07  39  18  Galloping  t e s t s , 3.3 cm model, U =2.27  40  19  Galloping  t e s t s , 3.3 cm model, U =2.29  41  20  Galloping  t e s t s , 3.3 cm model, U =2.56  42  (b)  o  0  0  0  0  o  Q  Q  0  vi page 21  Galloping tests, 3.3 cm model, U =2.77  43  22  Galloping tests, 3.3 cm model, U =3.28  44  23  Galloping tests, 3.3 cm model, U =4.07  45  24  Galloping tests, 3.3 cm model, U =4.24  46  25  Galloping tests, 3.3 cm model, U =4.76  47  26  Galloping tests, 3.3 cm model, U =5.67  48  27  Galloping tests, 3.3 cm model, U =8.91  49  28  Galloping tests, 5.1 cm model, U =0.175  50  29  Galloping tests, 5.1 cm model, U =1.25  51  30  Galloping tests, 5.1 cm model, U =1.75  52  31  Galloping tests, 5.1 cm model, U =1.76  53  32  Galloping tests, 5.1 cm model, U =2.54  54  33  Galloping tests, 5.1 cm model, U =3.06  55  34  Comparison between dynamic measurements and quasi-steady theory  56  35  Collapsed data from galloping tests  57  0  0  o  0  0  D  D  o  Q  0  0  0  o  V l l  Acknowledgement  I wish  t o t h a n k Dr. G.V.  Parkinson  f o r h i s guidance during the research.  Nomenclature  streamwise s e c t i o n viscous  damping  dimension  coefficient  measured f o r c e c o e f f i c i e n t , from natural  frequency  wake frequency  (Hz)  (Hz)  force i n y - d i r e c t i o n cross-stream model  section  dimension  length  o s c i l l a t i n g mass time flow  velocity  displacement oscillation  amplitude  angle o f a t t a c k air  density  air  viscosity  natural £^y a=0 c 2mo>  (kinematic)  frequency  „ . . „ . . galloping criterion fraction of c r i t i c a l  L %pV M  lift  coefficient  Ppy'^Yii  drag  coefficient  - — — %pV h£  normal f o r c e  2  2  ph £ 2m Vh f h v  mass parameter Reynolds number Strouhal  damping  coefficient  2  v  (rad/s)  number  (4)  IX U  = ^-r  d i m e n s i o n l e s s wind  speed  nA  critical  galloping  speed  = ^rrr  critical  resonance  speed  2B ^°  =  U_ r  y =  2TTS  dy  R  . , . section v e l o c i t y  y Y =  dimensionless  displacement  y Y = ^ • dY Y =  d i m e n s i o n l e s s amplitude . , . , . dimensionless section v e l o c i t y  Y = dY dT  dimensionless section  x = cot  dimensionless  2  2  time  acceleration  1 CHAPTER ONE  1.1  Introduction  Flow induced s t r u c t u r a l  o s c i l l a t i o n s have been observed  and h a v e b e e n a p r o b l e m i n t h e bridges,  and o t h e r  structures.  s t u d i e d e x t e n s i v e l y but about  bluff  sections,  In the  flow about  Reynolds numbers, vortex  street,  design of towers, The m e c h a n i s m s  due t o  the  many o f t h e  are  two d i m e n s i o n a l b l u f f  v o r t i c i t y i n the  separated  i m p o s i n g an u n s t e a d y ,  not  amplitude of o s c i l l a t i o n o s c i l l a t i o n have s u c h as  and t h e  damping o r t u r b u l e n c e  success  on  the present  flow  forces.  time q u i t e  effects  field  v e r y low  forms a Karman When t h i s vortex  forced  c o u p l i n g between  Studies  of this  of various  type  the  of  parameters  amplitude.  complex n a t u r e o f the  field  at  a simple  m o d e l l i n g o f v o r t e x resonance  due t o t h e  o f the  to  been  understood.  o s c i l l a t i o n s known a s  l i m i t e d by s t r o n g  fluid  m u s t be e m p l o y e d t o o b t a i n r e a s o n a b l e more d e t a i l s  body,  included observation of the  Pure mathematical limited  o f the  explanation is  except  have  flow  p e r i o d i c f o r c e on t h e b o d y .  T h i s b e a r s some s i m i l a r i t y  this  fully  shear layers  resonance  but  yet  bodies,  frequency  oscillation,  separated  time,  cables,  o f t h e s e phenomena  occurs near a natural can o c c u r .  long suspended  c o m p l e x i t y o f the details  f o r some  results.  flow  field,  only  and e m p i r i c i s m  Numerical models, which  behavior i n t o account,  c o m p l i c a t e d and r e q u i r e  h a s met w i t h  large  can be u s e d but capacity  take  are  at  computing  facilities. Another type resonance,  o f i n s t a b i l i t y , not  is galloping.  observations  necessarily related  The name o f t h i s  of e l e c t r i c a l conductors  that  i s b e l i e v e d to 'galloped'  to  vortex  stem from  first  d u r i n g h i g h winds  after  2 being coated by e f f e c t s resulting been  i n an a e r o d y n a m i c  that  supported,  will  depends on t h e  T h i s i s now known t o be  separation  shear layers  to  the  design.  Certain bluff  cross  that  sections,  structural  has  when  g a l l o p when t h e w i n d v e l o c i t y r e a c h e s a  system's  caused  body,  i n s t a b i l i t y s i m i l a r « to wing f l u t t e r  a problem w i t h a i r c r a f t  value  critical  damping and p a r t i c u l a r  shape o f  body. The d e t a i l s  vortex with  o f g a l l o p i n g behavior are  resonance,  invalid  is  to the  equation  that vortex  for a certain  Brief  review o f the  of attack  as  t h a t i n the force  is  attack  a ,  the  same as  the  The p r i n c i p a l  quasi-steady  approach  normal force  in figure  t h a t on a s t a t i o n a r y  1(a).  the  coefficient  The k e y a s s u m p t i o n  instantaneous  body at  Cy v s .  the  is  aerodynamic  same r e l a t i v e  angle  of  1(b)).  Applying the the  system.  approach,  Theory  f l o w about an o s c i l l a t i n g body,  (figure  obtains  example  those of  velocities.  knowledge o f the  for  than  apply a quasi-steady  formation can render  Quasi-Steady  requires  to  o f motion o f the  range o f flow  The a n a l y s i s angle  b e t t e r understood  m a i n l y due t o b e i n g a b l e  limitations,  limitation  1.2  during winter storms.  o f the p r o x i m i t y o f the  elastically  the  with sleet  quasi-steady  dimensionless  assumption  to  equation  2  y  differential  equation  -2BY  hand s i d e o f the  air.  In t h i s  case,  one  equation  (1.1)  c a n be s o l v e d b y a p p r o x i m a t e  method o f K r y l o v and B o g o l i u b o v b e i n g a f r e q u e n t l y right  of motion,  form:  Y + Y = n U C ( a r c t a n -gj  This non-linear  the  i s near  the motion i s n e a r l y  zero,  as  it  means,  u s e d a p p r o a c h when is  the  for oscillations  s i n u s o i d a l , so assuming  that  in  the  3 Y = Y s i n tot, w h e r e Y i s a s t a t i o n a r y a m p l i t u d e , a n d e q u a t i n g t h e w o r k done by damping and a e r o d y n a m i c f o r c e s  o v e r one c y c l e ,  one o b t a i n s  the  integral  equation:  /  If  C y ( a ) c a n be e x p r e s s e d as a p o l y n o m i a l ,  integrated  d i r e c t l y to g i v e another  shown i n  (2),  where U  = 2B/nA,  Q  and the  the  form  the  real roots Y .  result  2  0  6  It  is  vs. U / U . 0  i n uniform flow,  Smith's Cy(a) data  (1)  0,  is that  so t h a t  onset  the  e q u a t i o n can be  g a l l o p i n g o c c u r s from r e s t In the  (2),  1(c).  was w e l l  found  of a  from  square  odd-order polynomial Wind t u n n e l  CI) a g r e e d w e l l w i t h t h e Q  f o r U> Uo ,  s o l u t i o n s c o l l a p s e onto a s i n g l e  b a s e d on a 7 t h d e g r e e  i s shown i n f i g u r e  is  the r e s u l t i n g equation i s o f  The t h e o r e t i c a l g a l l o p i n g r e s p o n s e  velocity of galloping U  This  s t a t i o n a r y amplitudes are  a l s o shown t h a t t h e  experiments performed by Smith the  (1-2)  e q u a t i o n i n n , U , B , and Y .  and A i s p o s i t i v e .  a-bY +cY'*-dY =  curve of Y / U section  Cy ( a r c t a n (^j COST)} COST d T = TT ^p-  of  galloping  theory,  above t h e v o r t e x  fit  so l o n g  as  resonance  velocity U . r  The i n t e g r a l e q u a t i o n 1.2 numerical representation to find  1.3  stationary  Review o f the  1.3.1  o f CyC°9, u s i n g q u a d r a t u r e s  oscillations Several  0  scheme  amplitudes.  shedding  has been o b s e r v e d t h a t  galloping U  i n an i t e r a t i o n  a  Literature  Effects of vortex It  of  the  c a n a l s o be s o l v e d n u m e r i c a l l y , g i v e n  i n cases  where the  was l e s s t h a n t h e v o r t e x r e s o n a n c e  occurred u n t i l investigators  the  flow  C3,4,5)  t h e o r e t i c a l onset v e l o c i t y U , no r  speed approached U . r  have r e p o r t e d measurements  of  the  velocity  instantaneous sections models  i n wind tunnels.  i n order  to determine  displacement) to  By u s e  I t was n e c e s s a r y  suitable  of the  of strain  gauges a t t a c h e d  lift  the magnitude at  coefficients,  the it  i n f o r m a t i o n was  (relative  of oscillation.  was p o s s i b l e t o  to  results  to  t o be v a l i d ,  as  quasi-steady  theory p r e d i c t i o n .  expected,  h i g h e r n o n - d i m e n s i o n a l f l o w v e l o c i t i e s , and h e l p e d t o  the  at  behavior for U < U . 0  displacement of  energy  is  In f i g u r e  r  shown t o be n e g a t i v e  to the  fluid  supported  system,  be damped  out.  from t h e  stationary  o f the  velocities,  the  quasi-steady velocities sign i s  force force  f o r U<U .  This indicates  r  o s c i l l a t i o n would not  behavior is  component is  t h e phase a n g l e between  o s c i l l a t i n g p r i s m , so t h a t  Another d e s c r i p t i o n o f the magnitude  2(a),  theory  seen  s e e n t o be n e g a t i v e ,  t h e o r y would a l s o p r e d i c t  for forced o s c i l l a t i o n ,  but  the  explain  force  a net  and  transfer  i n an e l a s t i c a l l y  o c c u r and any m o t i o n w o u l d  in figure  i n phase w i t h the  force  By c o n v e r t i n g  compare t h e  These showed t h e  suitable  to the p e r i o d i c  and p h a s e a n g l e  frequency  to  useful  to apply F o u r i e r a n a l y s i s  forces  4 cross  on o s c i l l a t i n g s q u a r e a n d r e c t a n g u l a r  forced to o s c i l l a t e s i n u s o i d a l l y i n the wind,  obtained. data  aerodynamic f o r c e s  2 (b),  velocity is  retarding  shown.  the motion.  the  f o r c e t o be n e g a t i v e  the  v e l o c i t y where t h e  a c t u a l l y s t r o n g l y l i n k e d to U , a c h a r a c t e r i s t i c r  where  at  force  the A t low  The low changes  not p r e d i c t e d by  the  theory.  1.3.2  G a l l o p i n g from Rest As m e n t i o n e d ,  l o n g as  for a l l U>U  2B/nA i s p o s i t i v e .  > > 0  U , r  g a l l o p i n g s h o u l d commence f r o m r e s t  The c o n s t a n t  was f o u n d t o h a v e a w i d e r a n g e o f v a l u e s measurement different  c o n d i t i o n s and t e c h n i q u e s .  f l o w s and  geometries.  A, determined i n the  f r o m Cy  literature,  so  measurements,  d e p e n d i n g on  the  T a b l e { shows some o f t h e v a l u e s  for  5 In uniform considerably,  low t u r b u l e n c e  s i n c e a p p a r e n t l y the  number c a n h a v e l a r g e  effects.  values are  less  Brooks  t o be u n a f f e c t e d  (6)  integration  flows  scattered.  of the pressure  F o r c e measurements  b y Kwok  the values  end a n d s u r f a c e  In the uniform,  One w o u l d e x p e c t  the  (8)  on t h e o t h e r  i n t e r e s t i n g to note  effect  of small  velocity that  scale turbulence  smooth f l o w r e s u l t s  P r e l i m i n a r y experiments  that A=2.69, the v a l u e used i n  p o s s i b i l i t y was f u r t h e r  the  of from  center  a  o f the  wind  o f Kwok)  from a s m a l l d i a m e t e r r o d p l a c e d u p s t r e a m of almost four.  f o r g a l l o p i n g and t h e n f i n d i n g A u s i n g t h e Q  the  ( i n the r e s u l t s  b y N a k a m u r a (11) w e r e o b t a i n e d i n d i r e c t l y ,  i s , A = 2B/nU .  suggested  higher turbulence flows,  end l o c a t e d i n a b o u t  the m o d e l , i n c r e a s i n g the v a l u e o f A by a f a c t o r presented  c o n d i t i o n s and R e y n o l d s  hand were p e r f o r m e d w i t h  tunnel test  is  section vary  d i s t r i b u t i o n measured at mid-span o f the model.  free  It  square  b y e n d c o n d i t i o n s , s i n c e t h e v a l u e s came  c a n t i l e v e r model h a v i n g t h e section.  f o r the  investigated.  The  the of  results  using the observed  onset  known v a l u e s o f B a n d n ,  i n the present  ( 2 ) , was somewhat  research  l o w , so  this  6 CHAPTER TWO  2.1 S t a t i c Measurements  2 . 1 . 1 Measurement o f In  order  and  Analysis  C (a) v  to determine  C y ( a ) i t was n e c e s s a r y  d r a g on a s t a t i o n a r y m o d e l . galloping  tests  balance.  The model c o u l d be r o t a t e d  drag forces strain  (figure  The 3 . 3 cm„ s q u a r e  c e l l s and a m p l i f i e r s .  v e r y low t u r b u l e n c e , The m e a s u r e d  figures ones,  return  the peak  Comparing the  i n the  decreasing  i n the is  slope o f C^fa) at  also  show r e a s o n a b l e are not  intensity of-the  evident.  the  i n the  Note t h a t  the  corrected for blockage effects,  force and  of  the  s t u d y was  a  less  n  0.1%.  are p l o t t e d to the  Cn move t o t h e C  than  two l o w e r  left  measurements  in  as  increases.  o r i g i n w i t h R e y n o l d s number  For comparison, the  agreement.  f l o w was  t h r e e R e y n o l d s numbers  scatter  the  s e c t i o n 0 . 6 9 m b y 0 . 9 1 m.  h i g h R e y n o l d s number r e s u l t s  and t h e  and  lift  c a l i b r a t e d v o l t a g e output  C ^ c u r v e and t r o u g h i n t h e  R e y n o l d s number d e c r e a s e s ,  and t h e  The w i n d t u n n e l u s e d i n t h i s  and d r a g c o e f f i c i e n t s at  3 and 4.  lift  the base to a wind t u n n e l  type w i t h a t e s t  streamwise turbulence  lift  An i n c r e a s e  flow  the  s e c t i o n model used f o r  i n 0 . 1 ° increments,  c o u l d be d e t e r m i n e d from t h e  gauge  The  11) was c l a m p e d a t  t o measure  data  so the  Cn v a l u e s r e p o r t e d  o f the present CQ values are  in  (15)  measurements therefore  about  6% h i g h . In  f i g u r e 5,  determined  C (a)  from the  v  lift Cy  The  i s p l o t t e d for the  and d r a g c o e f f i c i e n t s by u s i n g  =  "(Ct  curve used to approximate  and i s r e p r e s e n t a t i v e  two h i g h e r R e y n o l d s n u m b e r s ,  o f the  +  C  n  tana)  seca  Cy(a) for the numerical c a l c u l a t i o n s i s f o r c e b e h a v i o r at  a R e y n o l d s number o f  shown,  about  7 12,000. One  can see  This a  with  A at  the  2.1.2  in figure  least  strong  from  squares  an expanded  fit  o f the is  flow,  (2)  the  for  is  fixed  the  stable  to  the  Quasi-Steady  C (a) c u r v e o f f i g u r e y  amplitudes.  C  -.9°<a<l.l? the  data  y  near  The v a l u e  one q u o t e d  apparent.  This  separation  l i n e s at  is  in  (2),  somewhat the  edges  Recall  ( C  S i n c e C (a) i s y  J  v  v  v  will  one c a n s e t  a, f o r  this  t h e n be d i v i d e d b y 4 ,  nu2  _  U2  =  left,  and t h e  4^ TTY  This represents a balance  situation,  and a f t e r  Y  between  average  for different  Intersection points  C  dynamic same  the  values  represent  .  y  test  wind  (1-2)  upper  limit  c a n be  The t e r m o n t h e  some r e - a r r a n g i n g  one -  reduced  right  hand  obtains  (2.1)  U  the v i s c o u s s t r u c t u r a l  aerodynamic force  a f u n c t i o n o f B / n and U.  integrated  the  (j2  / ^ ( a r c t a n (I COST)) COST dT 0  is depicted graphically i n figure  shown as  a p p l i e d to  solve  1.2:  TT/2 t o make t h e n u m e r i c a l i n t e g r a t i o n s h o r t e r .  s i d e must  are not  were p e r f o r m e d w i t h t h e  ( a r c t a n ( 7 7 COST"))COST'di = TT ^U " n  odd i n  up a n e q u a t i o n t o  be c o m p a r e d t o u n c o r r e c t e d  o f measurements  equation  o  5,  Analysis  Blockage corrections  since the numerical r e s u l t  tunnel.  This  comparison.  value of A increases.  comparable to  with  shown f o r  model.  Application  the  is  s c a l e p l o t o f the  data  Reynolds number-effect  and b o t h t y p e s  on  6,  R e y n o l d s number  results,  to  of Smith's data  i n such a separated  U s i n g the  data  fit  R e y n o l d s number d e c r e a s e s ,  highest  fairly  unexpected  for  as  linear  the  so t h a t  of  that  i s more e v i d e n t  =0,  of  The p o l y n o m i a l  7.  damping c o e f f i c i e n t  c o e f f i c i e n t on t h e  The d a m p i n g c o e f f i c i e n t t e r m  The a e r o d y n a m i c t e r m o n t h e  o f Y and U t o  stable  right.  get  the  amplitudes Y at  indicated the  right  is  was  curves.  corresponding  values  of  U and B / n .  F o r example the  is  thus the value of U . Q  curve  (2B/n)=6 i n t e r s e c t s  For a l l U greater  numerically,  and t h e r e s u l t i n g p l o t shown i n f i g u r e 8. from  (2),  indicating differences  b e h a v i o r due t o t h e d i f f e r e n t It  Cy(a)  of constant  On f i g u r e 8,  the  on t h e  p o i n t s o f the h y s t e r e s i s  Cy c u r v e .  For example, the  Shown f o r c o m p a r i s o n  i n predicted  hysteresis  shape o f Cy(a) a f f e c t s  the  predicted  l i n e s r a d i a t i n g from the o r i g i n  Y / U , and t h e y can r e p r e s e n t  r e a c h e d d u r i n g one c y c l e o f s t a t i o n a r y critical  determined  used.  i s i n t e r e s t i n g t o s e e how t h e  galloping response. lines  o f i n t e r s e c t i o n p o i n t s was  which  i.e.  Q  o c c u r . The c o m p l e t e s e t  the r e s u l t  U=1.5,  than U , Y i s p o s i t i v e ,  oscillations  is  Y=0 a t  t h e maximum a n g l e o f  oscillation,  attack  s i n c e tana = Y / U .  l o o p c a n be shown t o c o r r e s p o n d t o onset  are  of galloping,  (U/U )=l,  The  points  is of  0  course  dependent QJ/U ) 0  on ^ C y , a n d s i m i l a r l y , t h e jump t o t h e h i g h e r l i m i t c y c l e da a=0 = 2 . 1 seems t o c o r r e s p o n d t o a p o i n t o n C y ( a ) n e a r a = 9 . 5 ° w i t h  approximately the  same s l o p e a s a t a = 0 .  c y c l e n e a r U / U = 1.6 0  analysis,  (U,Y),  lower  t o t h e Cy p e a k n e a r a = 1 2 . 6 ° .  however, o n l y g i v e s i n f o r m a t i o n on Y / U ,  specify pairs of  2.1.3  corresponds  The jump down t o t h e  and i s n o t  f o r w h i c h more q u a n t i t a t i v e m e t h o d s  limit  This sort  enough are  of  to  required.  C o m p a r i s o n t o Measurements from F o r c e d O s c i l l a t i o n In the  c a s e o f f o r c e d o s c i l l a t i o n s , one c a n c o m p a r e some o f of figure  Cor a s s u m e s )  the phase a n g l e t o be p l u s o r minus 9 0 ° , d e p e n d i n g on t h e  Cy(°0,  w h i c h depends  quasi-steady  2 to the quasi-steady  on a = a r c t a n ( Y / U ) .  prediction.  the  measurements  of  near  T h a t i s , when Y / U i s  theory applied to forced o s c i l l a t i o n predicts  sign  large,  the angle to angle i s +90°.  the be  C o n v e r s e l y , when Y / U i s  change  i n a v e r a g e p h a s e a n g l e s i g n o c c u r s when t h e a e r o d y n a m i c f o r c e t e r m o f  figure  7 changes  and t h i s  the p r e d i c t e d phase  predicts  -90°.  sign,  small,  The t h e o r y  can occur over a wide range o f wind  The  speeds.  9 The o b s e r v e d p h a s e however,  so t h a t  magnitudes  differ  angles  in figure2(a)  for U<U , r  do n o t  vortex shedding effects  f r o m t h e p r e d i c t i o n as w e l l ,  t o between + 2 0 ° and + 8 0 ° a t  higher speeds.  a n a l o g y c a n be d r a w n t o t h e  effects  flow  theory suggests.  In the  airfoil,  t h e mass o f f l u i d  inertial  force  effect  This  is  close to  the  agreement phase  c a s e o f an o s c i l l a t i n g  at  w i t h the q u a s i - s t e a d y  scattered theory.  a n g l e s a r e between + 2 0 ° and + 8 0 ° ,  thin  low speed  airfoil  plate)  results  i n an In the  angle approaches  square  about  potential  (flat  low f l o w v e l o c i t i e s .  the p r e d i c t e d phase  angle i s  The o b s e r v e d  F o r l o w v e l o c i t y o s c i l l a t i o n s an  observed values f o r the  Around U=1.0, the phase  r  g o i n g from - 1 8 0 ° at  o s c i l l a t i n g w i t h the  zero,  dominate.  U<U ,  o f a d d e d mass t h a t u n s t e a d y  which dominates  as v e l o c i t y a p p r o a c h e s  change s i g n f o r  limit,  -180°.  s e c t i o n near U=0.7.  - 9 0 ° , i n approximate  At h i g h e r v e l o c i t y , compared t o  the  observed  the p r e d i c t e d value o f  + 90°.  2.2  Dynamic B e h a v i o r E x p e r i m e n t s  2.2.1  Tests The b a s i c s y s t e m d e s i g n e d and u s e d b y S m i t h  of  the  support  study.  It  i s b a s e d on a i r f i l m  the model, which i s  elastically  Figure 9 i s a schematic o f the  (1)  w  a  used i n t h i s  s  journal bearings  that  Instrumentation  have been r e f i n e d c o n s i d e r a b l y s i n c e S m i t h ' s experiments was p o s s i b l e t o make a c c u r a t e m e a s u r e m e n t s  at  a n d w i t h more p r e c i s e c o n t r o l o f t h e r e l e v a n t The v a r i a b l e v i s c o u s - t y p e r e s i s t a n c e d a m p e r s was m e a s u r e d w i t h t h e w i n d o f f ,  are used  c o n s t r a i n e d i n one degree  apparatus.  stage  of  to freedom.  and model d e s i g n  though,  lower o s c i l l a t i o n  and t h u s  it  amplitudes,  parameters.  a f f o r d e d by t h e D . C . eddy  using a thin  current  C-32 x 3 . 5 x 68 c m , )  10 s t r e a m l i n e d aluminum b a r  i n place o f the model. Using a Briiel  2305 l e v e l  logarithmic potentiometer  vs.  recorder  time records  with  c o u l d be o b t a i n e d .  ZR0005,  and a l l o w e d t o o s c i l l a t e w i t h no e x t e r n a l  of  amplitude trace i s  the  (log)  coefficient. current,  From r e c o r d s  that at  those obtained by Smith (1),  low v a l u e s  system mass, effects  are  reference to  of I,  shown i n f i g u r e  value  increase  I = 3 0 0 mA«, t h e  frequency 10(b)  (c @ 2y = 0 . 1  by about  based  on t h e  is  8%.  0.1  The e f f e c t  infinity.  due t o  the  turned  support  on),  some e f f e c t  Thus,  the measured system.  and s u p p l y p r e s s u r e t o on t h e  damping, but  i t was  load v a r i a t i o n the  found as  amplitude  the  damping i s  seen  o f v i s c o u s d r a g on total  fluid  i n place  (as  caused by wind  t h e s e were m i n i m i z e d t h r o u g h  at the  damping  at  fluid  b e i n g at  were a l s o  a  cm, b u t  bar  a i r bearings  )  such  velocity p r o f i l e of a viscous  damping w i t h the  Static  10(a)  cm a n d 1 0 .  near a s i n u s o i d a l l y o s c i l l a t i n g i n f i n i t e plane w i t h the at  damping  o f damping t o  t o b e a p p r o x i m a t e l y 3% o f t h e  theoretical  slope  damper  parameters  ratio  T h u s w i t h no c u r r e n t ,  double amplitudes  but  Typical  the  of  (figure  (14),  and a m p l i t u d e .  i s o n l y about  s t r e a m l i n e d b a r was e s t i m a t e d zero current,  The c u r v e  and Santosham  where c / c ^  cm).  75% b e t w e e n  change  the v i s c o u s  d a m p i n g d e p e n d e d p a r t l y on o t h e r  and o s c i l l a t i o n  amplitude  initial  o f amplitude decay f o r v a r i o u s v a l u e s I was o b t a i n e d .  type  e x c i t a t i o n , the  d i r e c t l y p r o p o r t i o n a l to  a calibration of c vs.  agrees w i t h  (log)  F o r a model g i v e n an  displacement  and K j a e r  rest  is nearly  all  being  found to  have  experimental  technique. Calibration  o f the  observing o s c i l l a t i o n this  to  the  l i n e a r displacement  transducer  amplitude with a stroboscope  RMS v o l t a g e  output  o f the  transducer  was a c c o m p l i s h e d b y  and s c a l e ,  system to  get  and  comparing  a calibration  constant. To i n v e s t i g a t e air,  the  effects  models were c o n s t r u c t e d  o f v o r t e x s h e d d i n g on g a l l o p i n g b e h a v i o r  so t h a t U  Q  c o u l d be much l e s s  t h a n U . (In r  in terms  of  the  balsa  d i m e n s i o n a l system parameters, and aluminum models were  Endplates slots into  necessary the  model i s  2.2.2  0  3.3 and 5.1  a l l o w model m o t i o n were  g a l l o p i n g response  oscillation  oscillation natural  frequency  frequency, speed,  the  and the  The  of the  since the  cm i n  length.  wind tunnel  wall  f o u n d t o be a d m i t t i n g o u t s i d e the  oscillation  behavior.  two m o d e l s was i n v e s t i g a t e d ,  several  the  dampers.  a m p l i t u d e was a l l o w e d t o average are  values  air  A  p r e d i c t i o n based  and  In U  28 t o  different  figure 0  is  approached. inset  amplitude  33 a r e  12,  less  the  than  for  for  the  the  r  r  As t h e  flow  and r e d u c e s  theoretical  the  amplitude  amplitude is  adjustable and  wind  recorded. together with 8.  e v e n when 27 a r e  the  Since  the  the for  the  3 . 3 cm  for  U =0.44 i s  shown.  Q  resonance  beat modulated, is  as  increased,  a n d n e a r U = 1 . 5 shows a s a w - t o o t h b e h a v i o r w i t h the  increases  the  o f damping  in figure  12 t o  wind speed  slow b u i l d u p and r a p i d d r o p - o f f t o speed  33,  amplitude behavior  o s c i l l a t i o n s are  amplitude o s c i l l o g r a p h . increases  were  a n d no o s c i l l a t i o n s o c c u r u n t i l  Then, near U , the  springs,  each v a l u e o f  as  two m o d e l s , figures  and  other.  experimental U  12 t h r o u g h  on n u m e r i c a l r e s u l t s ,  n o n - d i m e n s i o n a l p a r a m e t e r s were matched, model,  For given values  s t a b i l i z e at  in figures  measuring  The d a m p i n g was  or range o f amplitudes  presented  by  damping,  combinations of tension  s y s t e m c o u l d be c h a n g e d .  through  b e h a v i o r was s o m e t i m e s  the  and 68.1  a f u n c t i o n o f wind speed,  Using  current  results  theoretical  the  11 lightweight  11.  o f the  a m p l i t u d e as  frequency.  v a r y i n g the  Here,  cm s q u a r e ,  t o m i n i m i z e end e f f e c t s ,  shown i n f i g u r e  The  2  Results The  by  to  = 2c/(p£Acoh ).  r e g i o n near the model, a f f e c t i n g  typical  the  were f i t t e d  U  approached  lower amplitude. the  near U=2.5.  Further  shown b y the rather  increase  variation until Note a l s o  is  the  in  eventually variation  12 about  linear increase Figures  the  13 t o  existence  sometimes stable  o f two s t a b l e  limit  In f i g u r e  Q  s l i g h t l y g r e a t e r t h a n U , where r  c y c l e s o v e r a range o f wind speeds example,  This i s  near U=1.5, the  i n a d d i t i o n to the  is  model  is  galloping  type  observed near U=2.2.  separation  16 t o  25, U  o f resonance  observed hysteresis sort  speed.  15 f o r  b o t h n e a r Y=0 a n d Y = 0 . 2 .  In f i g u r e s  of  higher wind  15 show b e h a v i o r f o r U  apparent.  hysteresis  the  at  becomes p r o g r e s s i v e l y g r e a t e r  0  and g a l l o p i n g b e h a v i o r i s  o v e r l a p v a r i e d somewhat,  of variation.  Figures  g a l l o p i n g b e h a v i o r due t o  amplitude,  but  do i n d i c a t e  23 t o  evident.  with figures  27 do n o t  than U , and The amount  of  19 a n d 20 s h o w i n g  show t h e u p p e r  t h e p h y s i c a l l i m i t a t i o n on m o d e l  the  the  r  limit  cycles  oscillation  onset of g a l l o p i n g i n comparison to  the  theory. As p r e v i o u s l y m e n t i o n e d , of  the  U  obtained.  Q  near U  5.1  r  cm,  Over t h i s  as  for  existence  1.4<U<2. g a l l o p i n g as  Further  oscillation  with U <U , 0  o f two s t a b l e  r  higher  p o s i t i o n at  limit  observed for  cycle.  sort,  increase the  limit  the model would not  indicated, resulted  pronounced behavior o f t h i s  2.2.3  a l l experiments  from the n e u t r a l  c o n v e r g e n c e on t h e  33 show t h e  the p r e d i c t e d amplitude o n l y at  29 shows t h e  amplitudes  28 t o  F i g u r e 28 r e p r e s e n t s b e h a v i o r f o r t h e  range o f wind speed,  released  unstable  Again,  and a p p r o a c h e s  Figure  if  model.  figure  in U 3.3  rest.  lowest value o f  oscillation higher wind  begins speeds.  c y c l e s near U=1.5.  reach the upper  Manual p e r t u r b a t i o n  i n a m p l i t u d e g r o w t h and  amplitudes above  the  eventual  F i g u r e s 30 a n d 31 show m o r e  with a large 0  behavior  resulted  stable  - unstable  i n separation  loop  o f resonance  for and  cm. m o d e l .  Discussion In f i g u r e s  12 t o  33, the  'locking  in region'  described i n  (4)  is  shown,  the  upper  region  r e g i o n between  the  shaded  l i n e s near U£.  i n which the vortex frequency  and h e n c e becomes defined  independent  by power s p e c t r a  and may b e u s e f u l  'locks i n '  o f the  This represents  to the  the  oscillation  Strouhal frequency..  frequency,  T h i s r e g i o n was  a n a l y s i s o f f o r c e d o s c i l l a t i o n measurements i n  i n gaining i n t u i t i v e understanding  of vortex  (4),  shedding  characteristics. In f i g u r e observed  12, the  beat modulated amplitude behavior near U  i n v o r t e x e x c i t e d o s c i l l a t i o n s , and i s  between v o r t e x  s h e d d i n g and o s c i l l a t i o n  a m p l i t u d e do t h e  frequencies  'lock  n e a r U = 1 . 5 may i n d i c a t e t h a t  likely  frequencies.  in".  frequency.  vortex forces  Further  vortex shedding frequency  to the  forces  the  become d o m i n a n t a s  It  still  forced  behavior for U  f o r U<U  where U < U . 0  13,  15,  In f i g u r e that  is  r  amplitudes  near U  29,  Strouhal frequency,  shown t h e  represent the  Thus,  and h i g h e r  the  than  n e a r U=1.4 a r e for  taken  the  the  and  the  negative  rest  in  as  cases  in  c a n be p a r t i a l l y  7 discussed previously. d i r e c t l y from f i g u r e  stable  as a  13 o f  function  t h e maximum t h a t  forced o s c i l l a t i o n ,  are  values.  the  amplitudes,  greater than  amplitudes  the  and g a l l o p i n g  study,  o v e r one c y c l e ,  s e c t i o n w o u l d be l e s s  expected  on  observed  present  close)  in figure  aerodynamic work i n p u t  t h e o r y would p r e d i c t  the  s t a b i l i t y at  expected  r  data points  quasi-steady  apparent  theoretical  As a l r e a d y m e n t i o n e d ,  with observed  ' g r a p h i c a l a p p r o a c h as  The m a g n i t u d e s  predicts,  (4).  30, and 3 1 , f o r U > U . (but  34 a r e  3 and 4 .  and l o w e r i n t h e  0  higher  l o c k e d i n to  approach the  of higher than  o f Y and U .  between  r  Only at  mismatch  i n flow v e l o c i t y tends to r e t u r n  stationary  consistent  The e x i s t e n c e  r  explained using a  (4),  increase  o s c i l l a t i o n measurements o f  phase angle  figures  0  slight  impose a l i m i t  i s p o s s i b l e t o draw c e r t a i n p a r a l l e l s b e t w e e n  oscillation  due t o  The s a w - t o o t h m o d u l a t i o n  maximum a m p l i t u d e a n d t h a t v o r t e x s h e d d i n g i s s t i l l oscillation  i s commonly  r  the  for values of A than  the  quite plausible.  theory As  the  14 d a m p i n g - mass p a r a m e t e r quasi-steady  theory  analysis phase  (or  take  s t a b i l i t y increases,  seem t o h e l p the  e x p l a i n the  measured  Though t h i s  c o m p a r i s o n does not  field,  least  confidence  lends  forces  with displacement,  understood. at  the  lower stable  same r a n g e o f w i n d s p e e d s .  i n t o account  1 8 0 ° out o f phase)  it  and  for determining galloping i n s t a b i l i t y .  do n o t  observed for  does n o t  increased,  is valid  measurements o f f o r c e l o o p sometimes  is  to  the  in  Such  (4)  -  unstable  qualitative  f o u n d t o be  effects  give details  The  o f which are  about  t h e v a l i d i t y o f two  in  the  not  flow  independent  studies. It (4)  s h o u l d be m e n t i o n e d  took the  total  gauges a t t a c h e d  to  that  aerodynamic  the wind t u n n e l  to non-zero  aerodynamic f o r c e  phase a n g l e and m a g n i t u d e . streamlined concentrated forces  was done f o r  force  t o be t h e  the  signal difference  will  in still  air.  on t h e  A more a p p r o p r i a t e  mass i n p l a c e o f t h e  be s u b t r a c t e d  from the  technique  air,  strain  wind  T h i s method i n t r o d u c e s  'dummy' m o d e l i n s t i l l  authors of  of  two m o d e l s o s c i l l a t i n g i n p a r a l l e l , one i n t h e  one o u t s i d e  inertial  i n making t h e i r measurements,  error,  a  'dummy' m o d e l s o t h a t  s u b s e q u e n t m e a s u r e m e n t s on n o n s q u a r e  due  i n both  i s to use  a c t i v e model f o r c e  and  only  signal.  (This  rectangular  sections  in  cm, m o d e l was more u n s t a b l e  than the  other  (5) . ) I t was o b s e r v e d  that  model i n e x h i b i t i n g the is,  vortex  forces  conjectured lift  that  coefficients Once U  much e f f e c t theory data  is  0  the  higher than  expected  seemed more s i g n i f i c a n t f o r  amplitudes the  than  for the  on b e h a v i o r , accurate  lower blockage about  near resonance.  larger, model.  t h i s may b e due t o b l o c k a g e e f f e c t s ,  i s greater than  fairly  5.1  causing higher  l a r g e r m o d e l , and the  i n p r e d i c t i n g g a l l o p i n g response.  f o r a s e l e c t i o n o f t e s t s w i t h U > 2.3  is  is periodic  case.  2 . 5 , v o r t e x s h e d d i n g does not  even f o r the  It  That  shown i n f i g u r e  have  as  quasi-steady The c o l l a p s e d 35, w i t h  the  15 theoretical fairly  curve i n d i c a t e d .  good f o r  the  For higher values  lower l i m i t  o f U , the  agreement  0  cycle amplitudes,  but  not  as  is  good f o r  the  higher. A frequently cycle at figures  observed tendency  h i g h e r wind speeds to 12 a n d 15 n e a r U=3.  limit  Results presented  cycle amplitudes.  may b e a n e x t r e m e another  effect  occur near integer It  shedding,  t o n o t e some p r o g r e s s  is  for a rectangular  ( b / h = 2)  of behavior of figure  of figure  behavior  21 n e a r  T h i s b e h a v i o r may  loop i s not p r e d i c t e d . )  achieved i n  the  U=5.5  indicate  variations  (5),  using  g a l l o p i n g response. section with U  29 i s p r e d i c t e d ,  c o n t i n u i n g on i n t o g a l l o p i n g o s c i l l a t i o n - unstable  show s i m i l a r  in  seem  as U i s  0  T h i s agreement  One p r i n c i p a l  s l i g h t l y greater  with  increased. indicates  the  resonance (A  lower  stable  that  measurements u s i n g f o r c e d o s c i l l a t i o n  techniques  to p r e d i c t  a n d may p r o v e t o be a v a l u a b l e t o o l  analysis theory  o s c i l l a t i o n behavior,  i n c a s e s when U i s 0  i s not  If most  free  entirely  one c o m p a r e s  near to or  the  results  percent from the  lower data  (for of  t h a n U , where the  information for  quasi-steady  r  in  the  (1)«  A for both Smith's data  o f S m i t h (1)  to  the present r e s u l t s ,  i s the value o f A, which affects  speed where g a l l o p i n g b e g i n s . than  less  give sufficient  valid.  significant difference  R e y n o l d s numbers  to  r  result  sort  limit  f o r example  s i n c e the minima o f the  o s c i l l a t i o n measurements t o p r e d i c t  r  (1)  'hump'  same e f f e c t .  forced  than U , the  in  upper  multiples of U .  is of interest  that  amplitude o f the  w h e r e more m e a s u r e m e n t s w e r e made o f  The a m p l i t u d e  example o f the  of vortex  f o r the  show a w a v e - l i k e v a r i a t i o n ,  over a wider range o f wind speeds, upper  is  (1)>  In the present r e s u l t s ,  U , the G  g e n e r a l l y at  g a l l o p i n g began a t w i n d speeds about  the wind lower  thirty  same B / n ) t h a n one w o u l d h a v e p r e d i c t e d u s i n g A = 2 . 6 9 T h i s becomes  a p p a r e n t when one u s e s t h e  and the p r e s e n t d a t a ,  same v a l u e  of  i n reducing to a graph such  as  16 in  figure  Part to  35.  of this  The two s e t s o f d a t a  c o l l a p s e onto  s e p a r a t e but  similar  v a r i a t i o n o f A seems due t o R e y n o l d s number e f f e c t ,  end c o n d i t i o n s .  The R e y n o l d s number  effect  implies that  curves.  and p a r t  surface  due  roughness  a n d c o r n e r r a d i u s may a l s o be s i g n i f i c a n t i n d e t e r m i n i n g t h e v a l u e o f A . effects model  a l s o appear  t o be i m p o r t a n t ,  effectively increase It  between the trends,  of tests,  two g r o u p s .  even though U  are presented series,  D  t h e g a l l o p i n g measurements  a n d some d i f f e r e n c e s The d a t a p o i n t s  i s n e a r l y the  i n figures  12,  15,  18,  p e r f o r m e d s e v e r a l months  The p a r a m e t e r damping,  most d i f f i c u l t  and o n l y f o r the  investigated,  as  characteristics  were d i f f e r e n t  Though the  results  2 dimensional flow,  at  i n the  are  0  presented  if  that  the  r  scale is  structures  near resonance,  s i n c e the  U<U .  Verification  and/or  3 dimensional  0  24,  28,  The f i r s t  are presented  o f t e s t s was t h e Since i t  results  o f the  here are  i n the  f o r the  to the  damping v s .  case  the  damping  first  the  set,  two  the  apply the the  nearly  results  to  observed.behavior  critical  there i s  loading than quite  tall  wind speeds,  results,  can get  The  sets.  o f smooth and  large  o f t h i s w o u l d be w o r t h w h i l e , t h r o u g h t e s t s flows.  the  amplitude  In towers and  test  to higher  observed amplitudes  figures.  second group o f t e s t s .  may be s i g n i f i c a n t .  may be s u b j e c t e d  other  a p p a r a t u s was  f o r the  3 dimensional flow,  similar  results  The s e c o n d  is possible that  higher amplitudes  different  series  29, and 30.  s i m i l a r p r o x i m i t y can be a c h i e v e d i n t h e  the behavior i n f u l l  risk  than U  the  noticeable  18 a n d 19 show  a n d i t w o u l d b e somewhat r i s k y t o  s l i g h t l y greater  buildings,  20,  are  l i k e l y more c o m p a r a b l e b e t w e e n  predicting behavior i n turbulent, for U  in figures  t o c o n t r o l i n s e t t i n g up t h e  10 ( b ) .  h a s more c o n f i d e n c e  to  w e r e made i n two  i n response  same i n b o t h .  later,  second set  in figure  lower amplitude r e s u l t s  a d d i t i o n of endplates  s l o p e o f C y ( a ) @ a^O.  s h o u l d be m e n t i o n e d t h a t  separate series  author  the  s i n c e the  End  in  and  the  expected even  for  turbulent  CHAPTER THREE Conclusions  By e x a m i n i n g t h e in uniform flow,  1)  As e x p e c t e d , above the  a e r o e l a s t i c behavior o f a square  i t was d e t e r m i n e d  the q u a s i - s t e a d y  critical  section  prism  that:  theory i s v a l i d  resonance wind speed.  at  wind speeds  However, the  well  effects  of  R e y n o l d s number a n d end c o n d i t i o n s c a n h a v e s i g n i f i c a n t i n f l u e n c e on the the  2)  aerodynamic c h a r a c t e r i s t i c s  onset  o f the  section, especially  speed o f g a l l o p i n g Uo.  At wind speeds  less  than about  2.5,  the q u a s i - s t e a d y  n o t a c c u r a t e l y p r e d i c t t h e b e h a v i o r due t o t h e  o b s e r v e d at wind speeds above the  Dynamic f o r c e measurements used to the  reported  critical i n the  o f the  Flow v i s u a l i z a t i o n  flow f i e l d  behavior are  resonance  literature  still  not  value.  c a n be  effects.  but  known.  o f a n o s c i l l a t i n g p r i s m may b e h e l p f u l  gaining understanding of vortex  the  behavior  e x p l a i n the b e h a v i o r i n t h i s range o f wind speeds,  details  does  influence of  v o r t e x f o r m a t i o n mechanism, a l t h o u g h g a l l o p i n g - l i k e is  theory  in  18 Bibliography (1)  Smith, J . D . , "An Experimental Study o f the A e r o e l a s t i c I n s t a b i l i t y of Rectangular C y l i n d e r s " , M.A.Sc. Thesis, U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1962  (2)  P a r k i n s o n , G . V . , and S m i t h , J . D . , " T h e S q u a r e P r i s m as an A e r o e l a s t i c N o n - L i n e a r O s c i l l a t o r " , Q u a r t e r l y J o u r n a l o f M e c h a n i c s and A p p l i e d M a t h e m a t i c s , V o l . 1 7 , p a r t 2 , May 1964  (3)  N a k a m u r a , Y . N . , "Some R e s e a r c h on A e r o e l a s t i c I n s t a b i l i t i e s o f B l u f f S t r u c t u r a l S e c t i o n s " , Proceedings o f the 4th I n t e r n a t i o n a l Conference on W i n d E f f e c t s on B u i l d i n g s a n d S t r u c t u r e s , 1975, Heathrow  (4)  O t s u k i , Y . , W a s h i z u , K . - , T o m i z a w a , H . , a n d O h y a , A . , " A n o t e on t h e A e r o e l a s t i c I n s t a b i l i t y o f a P r i s m a t i c Bar w i t h Square S e c t i o n " , J o u r n a l o f Sound and V i b r a t i o n , 1 9 7 4 , 3 4 ( 2 ) , 233-248  (5)  W a s h i z u , K . , Ohya, A . , O t s u k i , Y . , and F u j i i , K . , " A e r o e l a s t i c I n s t a b i l i t y o f Rectangular C y l i n d e r s i n a Heaving Mode", J o u r n a l o f Sound a n d V i b r a t i o n , 1 9 7 8 , 5 9 ( 2 ) , 195-210  (6)  Brooks, P . N . H . , "Experimental I n v e s t i g a t i o n o f the A e r o e l a s t i c I n s t a b i l i t y o f B l u f f Two-Dimensional C y l i n d e r s ' , M.A.Sc. Thesis, U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1960  (7)  Cowdrey, C . F . , and Lawes, J . A . , " F o r c e Measurements on S q u a r e and D o d e c a g o n a l S e c t i o n a l C y l i n d e r s a t H i g h N R " , , N . P . L . - / A e r o / 3 5 1 , 1959  (8)  Kwok, K . C . S . , " C r o s s - W i n d R e s p o n s e o f S t r u c t u r e s Due t o D i s p l a c e m e n t Dependent E x c i t a t i o n s " , P h . D . T h e s i s , Monash U n i v e r s i t y , V i c t o r i a , A u s t r a l i a , 1977  (9)  D u g u n d j i , J . , a n d C h u n g , F . K . , Addendum t o V i b r a t i o n , 1978, 5 6 ( 2 ) , 309-311  (10),  Journal  o f Sound  and  (10)  Mukhopadyay, V . , and D u g u n d j i , J . , " W i n d E x c i t e d V i b r a t i o n o f a S e c t i o n C a n t i l e v e r Beam i n Smooth F l o w " , J o u r n a l o f S o u n d a n d V i b r a t i o n , 1976, 4 5 ( 3 ) , 329-339  Square  (11)  N a k a m u r a , Y . , and T o m o n a r i , Y . , " G a l l o p i n g o f R e c t a n g u l a r P r i s m s i n a S m o o t h a n d i n a T u r b u l e n t F l o w " , J o u r n a l o f S o u n d a n d V i b r a t i o n , 1977 5 2 ( 2 ) , 233-241  (12)  L a n e v i l l e , A . , " E f f e c t s o f T u r b u l e n c e on Wind Induced V i b r a t i o n s o f B l u f f C y l i n d e r s " , P h . D . T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1973  (13)  S u l l i v a n , P . P . , "Aeroelastic Galloping of T a l l Structures i n Simulated W i n d s " , M . A . S c . T h e s i s , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1977  (14)  S a n t o s h a m , T . V . , " F o r c e M e a s u r e m e n t s on B l u f f C y l i n d e r s a n d A e r o e l a s t i c Galloping of a Rectangular C y l i n d e r " , M.A.Sc. Thesis, U n i v e r s i t y of B r i t i s h C o l u m b i a , 1966  (15)  C o w d r e y , C . F . , " A N o t e on t h e U s e o f E n d P l a t e s t o P r e v e n t T h r e e D i m e n s i o n a l F l o w a t t h e Ends o f B l u f f C y l i n d e r s " , N . P . L . / A e r o / 1 0 2 5 , 1962  Appendix A Cy(a)  data,  used f o r numerical c a l c u l a t i o n s a  C  0  0 .5 1.0 2.0 3.0 4.0 4.5 5.0 6.0 7.0 8.0 8.5 9.0 9.5 10.0 10.5 11.0 11.5 12.0 12.25 12.5 12.6 12.7 12.9 13.25 14.0 15.0 16.0 17.0 18.0 19.0 20.0 21.0  y  0 .035 .0625 .118 .164 .2005 .214 .2225 .232 .238 .246 .259 .2793 .3075 .341 .378 .419 .466 .540 .578 .6045 .608 .607 .591 .552 .453 .31 .17 .0375 -.095 -.2285 -.361 -.493  A p p e n d i x B_ Predicted Galloping o f Square S e c t i o n  Lower l i m i t  u/u  0  cycle  Y/U  0  Response Prism  Upper U/U  limit  Q  cycle  Y/U  0  1.0  0.  1. 70  0.385  1.0125  0.01  1.70  0.39  1.05  0.02  1.713  0.40  1.10  0. 03  1.825  0.45  1.175  0.05  1.963  0.50  1.35  0.10  2.10  0.55  1.538  0.15  2.237  0.60  1.725  0.20  2.512  0 . 70  1.888  0 . 25  2 . 794  0.80  2.038  0. 30  3.075  0.90  2.088  0.325  3.35  1.00  2.088  0.350  (numerical r e s u l t s i n A p p e n d i x A)  b a s e d on d a t a  Appendix C Table I b/h 1.0  Reynolds No. 22000 27000 66000 66000 270000 3000 1700 3100  A 2.69 2.72 3.0 3.3 1.6 1.13 1.32 4.0 .68 .94 2.62 3.19 3.49 4.1 1.5 1.25 3.7  2.0 20400 32500 38000 33000  2.33 4.0 • 3.2 2.95 3.3  Turbulence <.l <.l <.l <.l  %  2. .1  13 7-9 12.5 9., rod b.l. b.l. 12 <. 1 <.l <.l <.l <.l  Type *  Source  I I I I I I I -II II  S m i t h ( 1 ) , p o l y n o m i a l c o e f f . , s m a l l end gaps S m i t h ( 1 ) , g r a p h , s m a l l e n d gaps Brooks ( 6 ) , g r a p h , p r e s s u r e measurements B r o o k s ( 6 ) , f r o m C L § Cn c u r v e s C7) Kwok ( 8 ) , c a n t i l e v e r , f r e e end ( 9 ) , c a n t i l e v e r , end gap ( 1 1 ) , o n s e t method ( 1 0 ) , c a n t i l e v e r , end g a p , r o u g h s u r f a c e ( 1 0 ) , c a n t i l e v e r , end g a p , r o u g h s u r f a c e  II II II II II II —  L a n e v i l l e (12), large scale L a n e v i l l e (12), small scale Laneville (12), small scale Kwok ( 8 ) , c a n t i l e v e r , f r e e Kwok ( 8 ) , c a n t i l e v e r , f r e e S u l l i v a n (13), c a n t i l e v e r , ( 1 1 ) , o n s e t method  t u r b . , s m a l l gaps t u r b . , s m a l l gaps t u r b . , s m a l l gaps end end f r e e end  Santosham ( 1 4 ) , p o l y n o m i a l c o e f f . , s m a l l gaps S a n t o s h a m ( 1 4 ) , f r o m C L £ Cn c u r v e s S a n t o s h a m ( 1 4 ) , f r o m C L 5 Cn curves S a n t o s h a m ( 1 4 ) , f r o m C L £ Cn c u r v e s Brooks ( 6 ) , g r a p h , p r e s s u r e measurements  22 Appendix D  0  1  2  3 u Uo  Figure  1(c) - Predicted Galloping Response for Square Section  A •  o •  0.025 0.050 0.100 0.150  -180  F i g u r e 2(a) - Measured phase a n g l e from m e a s u r e m e n t s ( f i g . 18 o f ( 3 ) )  dynamic  y •  2 mm 5 o 10 A 15 • 20  oA  R •  D  1.0  "  4A U  r  A  •  1.5  F i g u r e 2(b) - Measured f o r c e c o e f f i c i e n t m e a s u r e m e n t s ( f i g . 12 o f ( 4 ) )  h=150mm  2.0  from  dynamic  1.0 a 0.8  & °  Re A 8800 • 12400 o 28800  0.6  a  §  °  A •  0  a °  g  °  {}  A  9  0.4 o  0.2  A  n A  0  I  L  4  Figure  •  i  i  i  12  8  1  . 1.  16  20 a  3 - Measured  lift  coefficients  •  (degrees)  a Figure  5 - Normal f o r c e  coefficients  (degrees)  -  1  1  0  a F i g u r e 6 - Normal f o r c e  coefficients  near a = 0  2 (degrees)  Figure  7 - Equation  (2.1)  in graphic  form  00  Figure  8 - Predicted  G a l l o p i n g Response o f Square  Section  Air Bearing  Tension Spring Displacement Transducer Circuitry D.C. Power Supply (Log) Amplitude Chart  linn  gure 9 - Schematic:  g a l l o p i n g and  Recorder  RMS Voltmeter  damping measurements  Figure  10(a) - Damping  calibration  32  0.10  1.0  10.  Figure 10(b) - Variation of damping with o s c i l l a t i o n  2y (cm)  amplitude  V  Figure  11 - T y p i c a l  test  model  34  B=0.00156 f=5.25 Hz A=4.0  Figure  13 - G a l l o p i n g t e s t s ,  3.3  cm m o d e l ,  U =1.38 0  36  Y  1 r u  2  3  4  5 U  Figure  14 - G a l l o p i n g t e s t s ,  3.3  cm m o d e l ,  U =1.67 0  37  Y  Figure  15 - G a l l o p i n g t e s t s ,  3.3  cm m o d e l ,  U = 0  1.77  Figure  16 - G a l l o p i n g t e s t s ,  3.3  cm m o d e l ,  U =1.87 0  39  Figure  17 - G a l l o p i n g t e s t s ,  3.3  cm m o d e l ,  U =2.07 o  40  Figure  18  - Galloping tests,  3.3  cm m o d e l ,  U =2.27 D  Figure  19 - G a l l o p i n g t e s t s ,  3.3  cm m o d e l ,  U =2.29 D  42  Figure  20  - Galloping tests,  3.3  cm m o d e l ,  U =2.56 0  43  Y  1 r  2  u  3  4  5  6  U Figure  21 - G a l l o p i n g t e s t s ,  3.3  cm m o d e l ,  U =2.77 0  44  45  Figure  23 - G a l l o p i n g t e s t s ,  3.3  cm m o d e l ,  U =4.07 o  46  Figure  24 - G a l l o p i n g t e s t s ,  3.3  cm m o d e l ,  U =4.24 0  47  B=0.00540 f=5.25 Hz A=4.0  48  B=0.00644 f=5.25 Hz A=4.0  Figure  26 - G a l l o p i n g t e s t s ,  3 . 3 cm m o d e l , U = 5 . 6 7 0  49  B=0.00855 f=5.25 Hz A=3.5  50  Figure  28 - G a l l o p i n g t e s t s ,  5.1  cm m o d e l ,  U =0.175 o  51  Figure  29 - G a l l o p i n g t e s t s ,  5.1 cm m o d e l ,  U =1.25 0  52  Figure  30 -  Galloping tests,  5.1  cm m o d e l ,  U =1.75 0  53  Figure  31 - G a l l o p i n g t e s t s ,  5.1  cm m o d e l ,  U =1.76 0  54  Figure  32 - G a l l o p i n g t e s t s ,  5.1  cm m o d e l ,  U =2.54 0  B=0.00664 f=5.0 Hz  A=3.5  Figure  33 -  Galloping tests,  5.1  cm m o d e l ,  U =3.06 o  56  quasi-steady  theory  f r o m (4)  A  •10  •  Y  A • A o A •  Figure  0.013 0.033 0.067 0.100 0.133  34 - C o m p a r i s o n b e t w e e n d y n a m i c m e a s u r e m e n t s theory  and  quasi-steady  Figure  35 - C o l l a p s e d  d a t a from g a l l o p i n g  tests  

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