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UBC Theses and Dissertations

Aeroelastic instability of a structural angle section Slater, Jonathan Ernest 1969

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AEROELASTIC  INSTABILITY  STRUCTURAL  ANGLE  OF  A  SECTION  by  JONATHAN B.A.Sc,  A THESIS THE  University  SUBMITTED  ERNEST  of British  Columbia,  1964  IN PARTIAL FULFILMENT  REQUIREMENTS  FOR T H E D E G R E E  DOCTOR OF  in  SLATER  OF  PHILOSOPHY  the Department of  Mechanical  We  accept  required  THE  this  thesis  Engineering  as conforming  to the  standard  UNIVERSITY  OF  March,  BRITISH 1969  COLUMBIA  OF  In p r e s e n t i n g an the  inpartial  advanced degree a t the U n i v e r s i t y Library  I further for  this thesis  f u l f i l m e n t of the requirements of British  s h a l l make i t f r e e l y a v a i l a b l e  agree that  permission  for  Columbia,  I agree  for  that  r e f e r e n c e and S t u d y .  for extensive copying of t h i s  thesis  s c h o l a r l y p u r p o s e s may be g r a n t e d b y t h e Head o f my D e p a r t m e n t o r  by  his  of  this thesis  written  representatives.  It i sunderstood  for f i n a n c i a l gain  M.EcUAui':*'-  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  s~h/c:/^^B.lZW<s Columbia  copying or p u b l i c a t i o n  s h a l l n o t be a l l o w e d w i t h o u t my  permission.  Department o f  that  ABSTRACT  Angle tures, when  have  section  been  exposed  has been  natural  frequency  aims  It  angle  reported. make  plunging-torsional From  various  f o r the safe  The  vortex  large  on model  aerodynamic  loadings,  experiment stability very  low  with  low  to aeroelastic  nature.  The  forces  of the  thesis  and t h e  structures.  and dynamics  of the  t o r s i o n a l and  combined  angle  resonant  variations  the dependence Incorporating  the quasi-steady  analysis  models, wind  speeds f o r  o f the unsteady of the resonant the  stationary  i s able  instability  and r e s u l t i n g amplitude  The absence  of torsional galloping  only  by t h e t h e o r y  a t high  wind  which  speeds  i t i s  to preand b u i l d -  during  shows t h e  the  i n -  o r f o r systems  with  damping. The  sections torsional such  orientation.  i s substantiated to occur  design  instances  together  susceptible  on s t a t i o n a r y  the c r i t i c a l  of attack.  response.  oscillations  conditions.  instability  time  geometry  plunging,  coefficients indicate  up  struc-  a n d i n a few  on t h e aerodynamics  stationary,  to predict  the galloping  amplitude  or galloping  aerodynamic  dict  engineering  of the aerodynamic  t h e measurements  angles  large  winds,  members  resonant  the nature  during  i n open  The b l u f f  these  information  section  possible  atmospheric  instabilities  presents  used  to experience  of a vortex  at studying  resulting  known  to normal  failure  vibrations  members,  dynamical  study  are susceptible,in vibrations.  system  parameters  demonstrates general,  The n a t u r e as damping,  that  structural  t o combined  plunging  of the i n s t a b i l i t y natural  frequency,  angle and  depends angle  on of  attack,  section  distinct  families  represent  and  those to  etc.  the  single  hinge  of the  are  theoretically,  the p l u n g i n g  two  t o be  similar  the  to  T h i s makes i t p o s s i b l e  s t u d y i n g , both  and  torsion.  m o t i o n as w e l l as  found  freedom.  o b t a i n p e r t i n e n t i n f o r m a t i o n by  and  plunging or  coupled  instability  degree of  to the e x i s t e n c e of  p o i n t s , i t i s p o s s i b l e to  predominantly  frequency  range o f the  i n the  However, due  of v i r t u a l  t h e m o t i o n as  Furthermore, type  size,  torsional  experimentally  degrees  of  freedom,  separately. During  plunging resonance,  a vortex capture trolled On  the  trol  by  the  phenomenon where t h e  c y l i n d e r motion over  o t h e r hand, the  c o n d i t i o n over  shedding  torsional  frequency  results,  the  with  less  amplitude  the unsteady  modulation  During  velocity  and  resonance  longitudinal  however, t h e wake w i d t h plunging motion. no  and  in either  experiences  on  is  and  p r e s s u r e s on  con-  vortex  f o l l o w s the stationary  the  larger  angle  surface  Consequently,  increase with  this  insta-  degree o f freedom, the  t h a t the  torsional  the v o r t e x shedding  and  i n magnitude  phase v a r i a t i o n .  substantial  con-  range.  shows a v o r t e x  s p a c i n g remain e s s e n t i a l l y  I t appears  effect  wind speed  Compared t o t h e  substantially  experiences  frequency  r a n g e where t h e  aerodynamic c o e f f i c i e n t s  bility.  virtually  are  shedding  of o s c i l l a t i o n  fluctuating  d u r i n g p l u n g i n g resonance  section  a finite  a large velocity  governs the  angle  vibration  s t a t i o n a r y model S t r o u h a l c u r v e . torsional  the  vortex  unaltered,  increase with resonance  o r wake  has  characteristics.  TABLE OF CONTENTS Chapter 1  2  Page Introduction  1  1.1  Preliminary  1.2  Literature  1.3  Purpose  Aerodynamics  Remarks  . . .  1  Survey  and Scope  4 o f the I n v e s t i g a t i o n  of a Stationary Angle Section  . . .  8  . . . .  12  2.1  Preliminary  Remarks  12  2.2  M o d e l s , A p p a r a t u s , I n s t r u m e n t a t i o n and Calibration  12  2.2.1  A n g l e Models  12  2.2.2  Fluctuating  Pressure Transducer  and C a l i b r a t i o n 2.2.3 2.3  Wake P r o b e  and T r a v e r s i n g G e a r  . . . .  16  Test Procedures  19  2.3.1  B a l a n c e Measurements  19  2.3.2  V o r t e x Shedding Frequency  19  2.3.3  Mean S t a t i c Surface Fluctuating  19  2.3.4  Model 2.3.5 2.4  15  P r e s s u r e on M o d e l Static  Surface  2.4.2  22  Wake S u r v e y  23  Experimental Results 2.4.1  P r e s s u r e on  and D i s c u s s i o n  . . . . .  S t e a d y L i f t , D r a g and P i t c h i n g Moment D i s t r i b u t i o n s V o r t e x S h e d d i n g F r e q u e n c y and S t r o u h a l Number  2.4.3  Mean S t a t i c  Pressure Distributions  2.4.4  Fluctuating Static Pressure Distributions  25  25 28 . .  34 38  V  Chapter  Page 2.4.5  2.4.6 2.5 3  Fluctuating Lift, Coefficients  a n d Moment 47  Wake G e o m e t r y  Concluding  Dynamics  Drag  50  Remarks  o f an A n g l e  66  Section  70  3.1  Preliminary  3.2  Experimental  Arrangement  72  3.2.1  Model  Mounting  72  3.2.2  Instrumentation Procedures  3.3  3.4  Response Plunging  Remarks  70  System and T e s t  75  o f an A n g l e S e c t i o n w i t h Combined and T o r s i o n a l Degrees o f freedom . .  Theoretical  Development  3.4.1  Vortex  3.4.2  Galloping  88  Resonance  3.4.2.1  3.4.2.2  89  Instability Singularities in the Small Limit  Cycles  90 and  Stability 91  and B u i l d - u p  Time 3.5  Results 3.5.1  and D i s c u s s i o n Vortex  Resonance  3.5.1.1  3.5.1.2  3.5.1.3  3.5.2  Galloping 3.5.2.1  80  Model A m p l i t u d e - V e l o c i t y Measurements  94 95 95 95  S u r f a c e P r e s s u r e a n d Wake Characteristics with Oscila t i n g Angle Model  105  Resonant Theory Predictions  117  Motion Theoretical Predictions f o r P l u n g i n g Degree o f Freedom  118  118  vi Chapter  Page 3.5.2.2  P l u n g i n g A m p l i t u d e and B u i l d - u p Time Measurements  3.5.2.3  .  Theoretical Predictions f o r T o r s i o n a l Degree o f Freedom  3.6 4  127  C o n c l u d i n g Remarks  Recommendations  122  132  f o r F u t u r e Work  13 8  Bibliography  140  Appendices I II III IV  Geometric  Properties  of Angle  S e c t i o n Members  . .  150  Wind T u n n e l W a l l C o r r e c t i o n s  15 3  E l e c t r o n i c Instruments  166  Theory f o r P l u n g i n g o r T o r s i o n a l Degree o f Freedom  16 8  L I S T OF  TABLES  Table 2-1 1-1  Page Wake Geometry P a r a m e t e r s Bodies Geometric  Features  f o r Various  of Angle  64 Sections  152  L I S T OP  FIGURES  Figure 2-1  2-2  Page P r e s s u r e t a p a n g l e m o d e l and n u m b e r i n g p r e s s u r e h o l e s and c o n t o u r s i d e s C a l i b r a t i o n curves arrangement  of 14  for Barocel transducer 17  2-3  D i m e n s i o n s and  2-4  Instrumentation l a y o u t f o r vortex shedding f r e q u e n c y and f l u c t u a t i n g p r e s s u r e measurements . . .  20  Typical fluctuating pressure signals (a) wake p r o b e (b) m o d e l s u r f a c e  21  2-5 2-6 2-7  2-8 2-9 2-10 2-11  2-12  S c h e m a t i c o f i n s t r u m e n t a t i o n f o r wake measurements .  2-14  probe  . .  18  from survey 24  D i s t r i b u t i o n o f s t e a d y l i f t , d r a g and p i t c h i n g moment c o e f f i c i e n t s f o r b a l a n c e m o d e l B  27  Comparison of aerodynamic c o e f f i c i e n t s t r u c t u r a l angle s e c t i o n s  29  for various  V a r i a t i o n of vortex shedding frequency with v e l o c i t y f o r p r e s s u r e tap a n g l e model V a r i a t i o n o f S t r o u h a l number and wind speed w i t h angle of a t t a c k  vortex  wind 30  resonant 30  S t r o u h a l number d i s t r i b u t i o n s f o r (a) d i f f e r e n t s i z e angle models (b) v a r i o u s s t r u c t u r a l a n g l e sections  32  Midspan d i s t r i b u t i o n s coefficient i -45° £ a 1 35°  35  ii 2-13  c a l i b r a t i o n data of disc  40°  o f mean s t a t i c  pressure  <_ a <_ 135°  36  C o m p a r i s o n o f p r e s s u r e i n t e g r a t e d and b a l a n c e measured s t e a d y aerodynamic c o e f f i c i e n t s  37  Typical fluctuating pressure signals from v a r i o u s model t a p s i n d i c a t i n g (a) random a m p l i tude m o d u l a t i o n (b) p h a s e v a r i a t i o n  39  ix Figure 2-15  Page Midspan d i s t r i b u t i o n s o f f l u c t u a t i n g s t a t i c p r e s s u r e c o e f f i c i e n t a n d a m p l i t u d e modulation ratio i - 4 5 ° 1 a <_ 0° ii iii  2-16  ii iii 2-17 2-18  2-19  2-20 2-21  2-23 2-24 2-25 2-26  2-27  41  90° <_ a <_ 135°  42  o f midspan  fluctuating  -45° < a < 0°  44  15° <_ a <_ 60° . . . . . . . . . . . . . 75° <_ a <_ 1 3 5 °  .  . . . . . . . . . . . . .  Spanwise v a r i a t i o n of f l u c t u a t i n g c o e f f i c i e n t and p h a s e . Comparison o f f l u c t u a t i n g coefficients  46 48  and s t e a d y  V a r i a t i o n o f peak f l u c t u a t i n g downstream c o o r d i n a t e  45  pressure aerodynamic • •  L a t e r a l v a r i a t i o n of f l u c t u a t i n g p r e s s u r e a m p l i t u d e i n t h e wake o f (a) 3 i n . a n g l e (b) 1 i n . a n g l e m o d e l pressure  49  model 52  with 53  L a t e r a l p o s i t i o n o f v o r t e x rows b e h i n d 3 i n . a n g l e models i - 4 5 ° ± a <_ 3 0 ° . ii  2-22  15° <_ a <_ 75° . . . .  Phase v a r i a t i o n pressure i  40  1 i n . and 54  45° <_ a <_ 1 3 5 ° .  55  Variation of l a t e r a l vortex spacing f o r 1 i n . and 3 i n . a n g l e m o d e l s  57  L o n g i t u d i n a l v a r i a t i o n o f phase angle of s t a t i o n a r y 3 i n . a n g l e model  59  i n wake  Streamwise v a r i a t i o n o f (a) L o n g i t u d i n a l vortex spacing (b) v o r t e x v e l o c i t y . .  60  L o n g i t u d i n a l d i s t r i b u t i o n s o f wake g e o m e t r y r a t i o f o r 1 i n . and 3 i n . a n g l e models  61  D i s t r i b u t i o n s o f t h e 'near i n f i n i t y values of t h e wake s u r v e y p a r a m e t e r s f o r 1 i n . and 3 i n . a n g l e models  63  C o m p a r i s o n o f S t r o u h a l number,wake g e o m e t r y and drag c o e f f i c i e n t f o r angel section  65  1  X  Figure 3-1  3-2  3-3  Page D e t a i l s o f model s u p p o r t system w i t h p l u n g i n g and t o r s i o n a l d e g r e e s o f f r e e d o m (a) p l u n g ing arrangement (b) t o r s i o n a l a s s e m b l y  73  Instrumentation layout for plunging s i o n a l d i s p l a c e m e n t measurements  76  and  tor-  Displacement measurements f o r angle model a t a = -45° w i t h combined p l u n g i n g and torsional degrees of freedom i p l u n g i n g g a l l o p i n g i n i t i a t e d below t o r s i o n a l resonance Q  i i  3-4  Response plunging  3-5  3-8  o f angle model a t degree of freedom  Response curve f o r angle with t o r s i o n a l degree of r o t a t i o n a l axis at shear  3-6  3-7  plunging galloping initiated t o r s i o n a l resonance  •  a = only 0  -45°  above 81 with 84  model a t a = -45° f r e e d o m o n l y and centre Q  3-10  95  Typical torsional displacement signals for angle model e x p e r i e n c i n g v o r t e x e x c i t e d motion at v a r i o u s wind speeds near resonance  97  S e p a r a t i o n o f v o r t e x r e s o n a n c e and galloping t y p e o f o s c i l l a t i o n by c h o i c e o f damping o r mass p a r a m e t e r f o r a = -45°  98  Plunging resonant curves at v a r i o u s o r i e n t a t i o n s  99  Torsional resonant at a = -45° w i t h centre Q  3-11  85  T y p i c a l displacement signals f o r angle model e x p e r i e n c i n g v o r t e x e x c i t e d p l u n g i n g or t o r s i o n a l motion  0  3-9  80  for 3  i n . angle' model  curves f o r 3 i n . angle model axis of r o t a t i o n at shear 100  T o r s i o n a l r e s o n a n t curves f o r 3 i n . angle model at a = -45° and 135° w i t h a x i s o f r o t a t i o n a t centre of gravity  101  S t a b i l i t y diagram f o r 3 i n . angle model iencing vortex excited plunging motion a = -45°  103  Q  3-12  experat  G  3-13  S t a b i l i t y diagram f o r 3 i n . angle model experi e n c i n g v o r t e x e x c i t e d t o r s i o n a l motion at a = -45° w i t h a x i s o f r o t a t i o n a t s h e a r centre 0  104  xi Figure 3-14  Page V a r i a t i o n o f c y l i n d e r and v o r t e x s h e d d i n g f r e q u e n c i e s , p h a s e , and d i s p l a c e m e n t amplitude near v o r t e x resonance ( a = -45°) . . . . . . . .  107  V a r i a t i o n o f c y l i n d e r and v o r t e x s h e d d i n g c h a r a c t e r i s t i c s w i t h w i n d s p e e d f o r a = -45' i t o r s i o n a l a m p l i t u d e and f r e q u e n c y results . . . . . .  108  0  3-15  0  ii  f r e q u e n c y , p h a s e and mean amplitude near resonance  torsional . . . . . . . .  109  3-16  M i d s p a n and s p a n w i s e d i s t r i b u t i o n s o f f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t , amplitude modulat i o n r a t i o a n d p h a s e d u r i n g s t a t i c and d y n a m i c c o n d i t i o n s o f t h e model . . . . . . . . . . . . . I l l  3-17  Comparison o f f l u c t u a t i n g aerodynamic c o e f f i c i e n t s f o r s t a t i o n a r y and v o r t e x e x c i t e d c o n d i t i o n s o f t h e a n g l e m o d e l a t a = -45° . . . . .  112  V a r i a t i o n o f a m p l i t u d e and p h a s e o f f l u c t u a t i n g p r e s s u r e i n wake o f a n g l e m o d e l e x p e r i e n c i n g v o r t e x e x c i t e d m o t i o n a t o =» -45° (a) p l u n g i n g (b) t o r s i o n . . . . . . . . . . . .  114  0  3-18  0  3-19  L o n g i t u d i n a l v a r i a t i o n o f wake s u r v e y p a r a . m e t e r s f o r p l u n g i n g and t o r s i o n a l c o n d i t i o n s Of m o d e l a t a = -45° . . . . . . . . . . . . .  115  C o m p a r i s o n o f 'near i n f i n i t y ' v a l u e s o f t h e wake s u r v e y p a r a m e t e r s f o r s t a t i o n a r y and v o r t e x e x c i t e d c o n d i t i o n s o f t h e a n g l e model a t a = -45°  116  Polynomial curve c o e f f i c i e n t data  120  Q  3-20  Q  3-21 3-22  3-23  f i t of typical  lateral  force  V a r i a t i o n o f g a l l o p i n g plunging amplitude with w i n d v e l o c i t y and model a t t i t u d e as p r e d i c t e d by t h e q u a s i - s t e a d y t h e o r y . . . . . . . . . . .  121  G a l l o p i n g a m p l i t u d e - v e l o c i t y r e s u l t s f o r angle m o d e l a t a = - 4 5 ° and t h e i r c o m p a r i s o n w i t h theory . .  123  Galloping model a t •theory  124  0  3-24  a m p l i t u d e - v e l o c i t y r e s u l t s f o r angle a = 90° and t h e i r c o m p a r i s o n with Q  xii Figure 3-25  Page C o m p a r i s o n o f e x p e r i m e n t a l and t h e o r e t i c a l a m p l i t u d e b u i l d - u p time f o r a n g l e model at a = 90°  X26  Moment c o e f f i c i e n t d i s t r i b u t i o n m o d e l a t a = 45°  128  0  3-26  f o r stationary  Q  3-27  C o n t o u r p l o t o f t o r s i o n a l moment c o e f f i c i e n t as a f u n c t i o n o f e and G f o r a = -45°  129  Q  3-2 8  P o l y n o m i a l c u r v e f i t o f t o r s i o n a l moment c o e f f i c i e n t d a t a f o r a n g l e model a t a = - 4 5 ° Q  3-29  . .  130  V a r i a t i o n o f g a l l o p i n g a m p l i t u d e and r e d u c e d frequency w i t h wind v e l o c i t y f o r angle s e c t i o n at a = - 4 5 ° as p r e d i c t e d by t h e q u a s i steady theory  131  C r o s s - s e c t i o n s o f a n g l e member (a) ' i d e a l i z e d ' angle s e c t i o n (b) t y p i c a l s t r u c t u r a l a n g l e . . .  151  Percentage c o r r e c t i o n a p p l i c a b l e t o 3 i n . angle s e c t i o n t e s t e d i n departmental wind tunnel  156  c  1-1  II-l  II-2  V a r i a t i o n o f S t r o u h a l number w i t h b l o c k a g e f o r v a r i o u s o r i e n t a t i o n s o f t h e a n g l e members  . .  164  ACKNOWLEDGEMENT  The appreciation given tion  wishes t o express  h i s s i n c e r e thanks and  t o D r . V . J . Modi f o r t h e g u i d a n c e and a s s i s t a n c e  t h r o u g h o u t t h e r e s e a r c h programme a n d d u r i n g of the thesis.  invaluable. Dr.  author  His help  The c o n s t a n t  G.V. P a r k i n s o n  during  special  a n d e n c o u r a g e m e n t have b e e n  interest  a n d e n c o u r a g e m e n t shown by  the experimental  comments on t h e t h e o r e t i c a l  aspects  work, a n d h e l p f u l  of this  study  Engineering  f o r t h e use o f t h e i r  nical  members f o r t h e i r  staff  facilities,  assemblying  of electronic  Reduction  and t o t h e t e c h the construc-  systems, and t h e  instrumentation.  o f some o f t h e e x p e r i m e n t a l  and t y p i n g o f t h e o r i g i n a l  f o r m e d b y my w i f e .  Mechanical  valuable help during  o f t h e wind t u n n e l models and s u p p o r t  output,  require  appreciation. T h a n k s a r e a l s o due t o t h e D e p a r t m e n t o f  tion  the prepara-  data  a n d computer  t h e s i s manuscript  were  per-  Her hours o f a s s i s t a n c e a r e g r e a t f u l l y  appreciated. Financial  a s s i s t a n c e was r e c e i v e d f r o m t h e N a t i o n a l  R e s e a r c h C o u n c i l o f Canada i n t h e form o f d i r e c t scholarship  and o p e r a t i n g g r a n t  (No. A - 2 1 8 1 ) .  postgraduate  LIST OF SYMBOLS Steady l i f t , drag and p i t c h i n g coefficients, respectively  moment  L a t e r a l force c o e f f i c i e n t P i t c h i n g moment c o e f f i c i e n t S e c t i o n a l steady l i f t , drag and p i t c h i n g moment coefficients, respectively Average peak, f l u c t u a t i n g s e c t i o n a l l i f t , drag and p i t c h i n g moment c o e f f i c i e n t s , r e s p e c t i v e l y 1  2  Mean s t a t i c p r e s s u r e c o e f f i c i e n t , ( p - p ) / j pV 0  Average peak f l u c t u a t i n g p r e s s u r e c o e f f i c i e n t ,  P 7  j  PV  Drag,  2  j V hlC 2  P  D  1 L a t e r a l force  f o r p l u n g i n g system,  2 pV h l C y  Mass moment o f i n e r t i a about e l a s t i c a x i s Mass moment o f i n e r t i a about i n e r t i a l a x i s Reduced frequency parameter i n g theory Lift,  j  V hlC  for torsional  2  P  L  P i t c h i n g moment,  1 2 2 j pV h l C ^  P i t c h i n g moment f o r t o r s i o n a l 1 2 2  l"  V  h  l  C  gallop-  M  system,  e  Degree o f p o l y n o m i a l curve f i t Reynolds number, Vh/v Dimensionless r e p r e s e n t a t i v e r /h  radius  parameter,  r  S t r o u h a l numbers, f^e/V and f^h/V, r e s p e c t i v e l y Dimensionless f l u i d v e l o c i t y , V / u ^ h  XV Dimensionless  critical  Dimensionless  resonant  velocity,  2g/na^  wind v e l o c i t y ,  1/(2TTS^)  Fluid  velocity  f a r upstream o f angle  model  Fluid  velocity  relative to o s c i l l a t i n g cylinder  R e s o n a n t w i n d v e l o c i t y , w hu •* n res Streamwise v o r t e x v e l o c i t y Dimensionless model, y/h  l a t e r a l displacement of o s c i l l a t i n g  Dimensionless  amplitude  of l a t e r a l displacement,  y/h Dimensionless displacement Longitudinal  initial  amplitude  s p a c i n g between  of l a t e r a l  vortices  (i=0 ,1,2 ,3 , • • • • • ,N) c o e f f i c i e n t s o f p o l y n o m i a l curve f i t Lateral  s p a c i n g between  vortices  cos X Projected width  o f a n g l e model  Frequency of c y l i n d e r  oscillations  Natural  frequency  i n plunging,  Natural  frequency  i n torsion,  co /2TT n  y  Frequency of v o r t e x  GO e  shedding  Maximum w i d t h o f a n g l e m o d e l P l u n g i n g and t o r s i o n a l s t i f f n e s s e s respectively Length  of angle  /2TT  o f the system,  model  Mass o f o s c i l l a t i n g s y s t e m Dimensionless  2 mass p a r a m e t e r , ph 1/(2m)  D i m e n s i o n l e s s mass moment o f i n e r t i a p a r a m e t e r , ph l/(2l) 4  Mean s t a t i c p r e s s u r e  S t a t i c p r e s s u r e f a r upstream Fluctuating  static  o f angle  model  pressure  A v e r a g e peak o f f l u c t u a t i n g  static  pressure  axis  to surface  R a d i a l d i s t a n c e from e l a s t i c element  P l u n g i n g a n d t o r s i o n a l v i s c o u s damping cients, respectively  coeffi-  sin A Real  time  Velocity  of vortex relative  Surface element  velocity  t o surrounding  for torsional  Downstream c o o r d i n a t e f r o m a n g l e axis  fluid  oscillation  section  inertial  Tranverse c o o r d i n a t e from angle s e c t i o n i n e r t i a l axis o r instantaneous l a t e r a l displacement of o s c i l l a t i n g model, normal t o flow d i r e c t i o n Amplitude  of lateral  displacement  Coordinate along e l a s t i c  axis  from model midspan  Torsional displacement of o s c i l l a t i n g Amplitude Initial  of torsional  amplitude  m o d e l , =8  displacement  of torsional  displacement  Phase l a g between model d i s p l a c e m e n t and f l u c t u a t i n g f o r c e o r moment Vortex  frequency  System f r e q u e n c y  ratio, ^^/^ ratio,  n  w ^/<») n  n  Instantaneous angle o f attack o f o s c i l l a t i n g s y s t e m o r a t t i t u d e o f s t a t i o n a r y model Mean a n g l e o f a t t a c k o f o s c i l l a t i n g D i m e n s i o n l e s s damping p a r a m e t e r system, r /2mu) y D i m e n s i o n l e s s damping p a r a m e t e r system, r/2Iu  system  for plunging  n  y  for torsional  X V I I  A n g l e b e t w e e n r e l a t i v e and a p p r o a c h i n g v e l o c i t i e s f o r o s c i l l a t i n g system  fluid  Increment i n angle of a t t a c k o f o s c i l l a t i n g s i o n a l system, a - a  tor-  Q  Coordinate origin  along contourline  o f C^  , passing  through  6  A n g l e b e t w e e n s u r f a c e e l e m e n t v e l o c i t y and a p p r o a c h ing f l u i d velocity f o r o s c i l l a t i n g torsional system Value  of n  Torsional Angle  r  a=0o  at  displacement of o s c i l l a t i n g  between  C and 8/U  Kinematic v i s c o s i t y  system  axis  of the  fluid  Coordinate perpendicular to C axis Density  of the  fluid  D i s t a n c e between i n e r t i a l  and e l a s t i c  axes~  Dimensionless  time  for oscillating  Dimensionless  time  f o r vortex shedding,  Reduced d i m e n s i o n l e s s time system, S T Phase  system,  u t n  u t v  for oscillating  angle  A v e r a g e p h a s e a n g l e between signals Circular  fluctuating  frequency of c y l i n d e r  Natural circular  frequency  pressure  oscillations  of o s c i l l a t i n g  system 1/2  Natural  circular  frequency  i n plunging,  (k^/m)  Natural  circular  frequency  i n torsion,  (k./I)  Circular  frequency  of vortex  shedding  1 / / 2  xviii .Subscripts F  Value o f the parameter conditions  e  Parameter  m  P r e s s u r e on m o d e l  max  maximum  r  Average v a l u e o f t h e parameter o s c i l l a t i o n s o f angle s e c t i o n  s  Value o f the parameter  f o r stationary  w  Value o f the parameter  i n wake  y  Value o f the parameter  i n plunging  6  Value o f the parameter  i n torsion  00  Constant v a l u e o f the parameter d i s t a n c e downstream  based  under  free  stream  on p r o j e c t e d w i d t h e surface  for torsional  model  at a large  Superscripts (•)  Derivative with respect  t o d i m e n s i o n l e s s time T  (o)  Derivative with respect  to real  *  time t  Reduced v a l u e o f t h e d i s p l a c e m e n t o r v e l o c i t y , parameter/U 0  1 1.1  Preliminary The  Remarks  oscillations  exposed  to a f l u i d  study.  To e n g i n e e r s ,  transmission interest. shedding  INTRODUCTION  suspension  In general,  corresponding  during  static  bodies,  when  t h e a e r o e l a s t i c v i b r a t i o n s o f smoke s t a c k s , bridges,  the nature  buildings, etc., are of  o f the wind l o a d i n g ,  and wake g e o m e t r y f o r m t h r e e  m e t e r s i n an a e r o e l a s t i c i n s t a b i l i t y the  bluff  stream, have been a s u b j e c t o f c o n s i d e r a b l e  lines,  frequency  o f aerodynamically  information  study.  vortex  important  para-  The d e t e r m i n a t i o n  associated with  a structural  of  angle,  and d y n a m i c c o n d i t i o n s , f o r m s t h e s u b j e c t o f t h i s  thesis. Structural o f open c i v i l mission  angles  engineering  of  computers  relatively together ularly long  structural  Furthermore, recent  through experimental aid  s t r u c t u r e s , such as h i g h  t o w e r s , a n t e n n a m a s t s , and b r i d g e s .  s e c o n d a r y members, t h e s e ible.  a r e f r e q u e n t l y used i n the c o n s t r u c t i o n  more f l e x i b l e  with  s e c t i o n s may be l o n g  and i n e n g i n e e r i n g  encouraged  the  individual  low n a t u r a l f r e q u e n c y  to experience  angle  make t h e s e induced  as and f l e x -  of  with the  lighter  Bluff members  and  geometry partic-  vibrations.  Some  members i n t r a n s m i s s i o n  t o w e r s have been known  l a r g e amplitude o s c i l l a t i o n s  when e x p o s e d t o n o r m a l  a t m o s p h e r i c w i n d s , and i n a few i n s t a n c e s reported.  use  design  components.  susceptible to aerodynamically  slender  Incorporated  trans-  advances i n the m e t a l l u r g i c a l s c i e n c e  research have  voltage  failure  has been  I t i s , t h e r e f o r e , d e s i r a b l e to understand  the nature  of  2 the unsteady  forces  design of these  and  e.g.,  the  vortex resonance, classical  turbulence.  oscillations galloping one  the n a t u r e o f the aerodynamic  s t r u c t u r a l member may  galloping,  beam.  geometric-aerodynamic  or s t a l l  of b l u f f  flutter,  cylinders  or torsion  Furthermore,  occur.  excitation,  a  vibration,  instability  o r random m o t i o n  called  excited  are o f the v o r t e x resonant  by  about  since  the  or  occurring, predominately, i n  o f freedom; f l e x u r e t r a n s v e r s e t o the longitudinal elastic  the e l a s t i c  members a r e n o t c o i n c i d e n t , may  safe  However, i n g e n e r a l , t h e a e r o d y n a m i c a l l y i n d u c e d  degrees  direction,  f o r the  e x h i b i t v a r i o u s forms o f  type w i t h the v i b r a t i o n s  o f two  instabilities  structures.  D e p e n d i n g on flexible  the r e s u l t i n g  and  inertial  axis of  axes o f  coupled t o r s i o n a l - f l e x u r a l  N e v e r t h e l e s s , as d e t e r m i n e d  flow  by K o s k o ^ and  the  angle  vibrations observed  by  2 Wardlaw,  the coupled t o r s i o n a l - f l e x u r a l  considered produces  as r o t a t i o n a l m o t i o n  two  distinctly  l o w e r modes, one other  family  different appears  natural  families  predominantly  be  hinge p o i n t s which  of v i b r a t i o n .  turbulence of atmospheric  s t r u c t u r a l members a r e e x p o s e d i n the  atmospheric be  virtual  can  flexural  At  and  the  the  torsional. The  that  about  oscillations  a large  bulence  i s n o t , i n g e n e r a l , comparable  to The  w i n d s have f l u c t u a t i n g  may  fraction  velocity  o f t h e mean w i n d s p e e d . of the v e l o c i t y  of t y p i c a l cannot  be  angle  expected  s e c t i o n beams, t o cause  components w h i c h However, s i n c e fluctuations  a t a f r e q u e n c y , w h i c h i s much l o w e r 3  quencies  the  steady a i r stream o f c o n v e n t i o n a l wind t u n n e l s .  peak o f t h e power s p e c t r u m occurs  winds t o which  than  generally  the n a t u r a l f r e -  the atmospheric  serious  the  resonant  tur-  vibration  3 of  i n d i v i d u a l s t r u c t u r a l members.  by  highly  likely  turbulent  to  Nevertheless,  reduces  the  phenomenon and the  influence  of  significant  thereby  distinct  i s not  yet  hence the  s h o u l d be  sists of  the  the  i s referred  alternating force e v e n when t h e  confined  fre-  that  turbu-  shedding  resonant  vibrations.  mentioned t h a t  of vortex e x c i t e d The  the  may  be  and  galloping  former i s e s s e n t i a l l y a  vortex formation f r e q u e n c y ,  forcing function,  coincides  system under c o n s i d e r a t i o n . t o as  a forced  motion i s stopped.  This the  m o t i o n and  Although  any  the  on  available  circular  g e o m e t r i c s i m p l i c i t y as w e l l  as  the  literature  the type  of  sustainper-  b l u f f member  when s u i t a b l y mounted, w o u l d  oscillation,  t o such s t u d i e s  the  and  with  vibration since  e x i s t s independent of  arbitrary cross-section,  vortex excited  natural  f u l l y u n d e r s t o o d and  emphasized.  f r e q u e n c y of  oscillation  the  circumstances.  character  f r e q u e n c y of  vortex  the  i t s h o u l d be  r e s o n a n c e phenomenon where the  ing  i s more  reports  the  i n suppressing  instability,  turbulence  oscillations  available literature  aids  u n d e r some  The  have peak(s) n e a r the  spanwise c o r r e l a t i o n of  galloping  natural  o t h e r hand, b u f f e t t i n g  wakes f r o m o t h e r b l u f f s t r u c t u r e s  energy s p e c t r u m can  quencies.  For  the  c a u s e e x c i t a t i o n o f members l y i n g downstream s i n c e  turbulent  lence  On  exhibit  is largely  c y l i n d e r s because of p r a c t i c a l importance  the of  the  section. The represents fluid by  the  second an  forces  form of  important  type of  which create  f a c t that  the  instability,  r e f e r r e d t o as  galloping,  self-excited vibration.  a condition  cross-section  of  of the  instability body i s  are  The generated  aerodynamically  4 unstable  to small disturbances.  t i o n s w h i c h grow i n a m p l i t u d e the  fluid  stream balances  o f damping. excited" created  the f l u i d  controlled  stops,  the unsteady  vortex  resonance.  vibration state  amplitude  instability be  by  forces  tends  itself,  "self-  the motion  and  i f the  t h e combined e f f e c t  judicious  choice of e i t h e r  frequency  i s required  motion  o f freedom  are that and  the  t o i n c r e a s e w i t h i n c r e a s i n g wind  at a given f l u i d  are  This i s i n contrast  as i s t h e c a s e w i t h a s t r u c t u r a l  Literature  t o as  main f e a t u r e s o f g a l l o p i n g degree  oscilla-  t h r o u g h v a r i o u s forms  that sustain  the motion  in  e x t r a c t e d from  are r e f e r r e d  stream v e l o c i t y  the steady-  velocity.  and  angle of attack and  galloping.  forms o f e x c i t a t i o n ,  damping, a n g l e s e c t i o n  t o s e p a r a t e t h e two  to  a n g l e beam, t h e  of both v o r t e x resonance  To p e r m i t t h e s t u d y o f t h e i n d i v i d u a l  1.2  result  the energy  forces disappear. The  forces  that dissipated  can o c c u r i n a s i n g l e  Often,  may  until  Galloping oscillations  because and  These  size  or  the natural  phenomena.  Survey 4  Strouhal  was  the f i r s t  to correlate  the v o r t e x shedding w i t h the diameter and  velocity  of the f l u i d  stream.  the p e r i o d i c i t y  of the c i r c u l a r  T h i s was  of  cylinder  f o l l o w e d by  numerous  5 e x p e r i m e n t s on wake g e o m e t r y by B e n a r d , the c l a s s i c a l study 6 7 s t a b i l i t y by Von Karman, and wake a n a l y s i s by H e i s e n b e r g . Ever  since,  sulted  interest  i n the v o r t e x shedding  i n many t h e o r e t i c a l  and  phenomenon has  experimental investigations  reby  g  Roshko, K o v a s n a y , R o s e n h e a d , E s k i n a z i  and  presented  literature.  an e x c e l l e n t  review  of t h i s  others.  Marris  has  of  The  mechanism o f g a l l o p i n g e x c i t a t i o n  of b l u f f  cylinders  g  was  probably  first  d e s c r i b e d by  Den  Hartog  determination of h i s plunging s t a b i l i t y h o w e v e r , i t was linear  theory  "sustained" classical activity  Lord  and  Rayleigh^  proposed  oscillations.  this  criterion.  indicated  the  a nonlinear equation Van  nonlinear equation on  who  0  together with  the  Originally, inadequacy  of  t o e x p l a i n the  der P o l ' s development of h i s  i n 1920  led to a f l u r r y  s u b j e c t by A p p l e t o n ,  G r e a v e s and  of  research  others.  E x c e l l e n t r e v i e w s o f t h e s e d e v e l o p m e n t s a r e s u m m a r i z e d by v a n d e r 11 12 Pol and L e C o r b e i l l e r , w i t h many r e c e n t r e f e r e n c e s f o u n d i n 13 2 Minorsky. Wardlaw e x t e n d e d Den H a r t o g " s a n a l y s i s by d e v e l o p i n g a generalized stability motion.  Sisto,^ 20  Dicker, solve  f o r coupled  Parkinson,  torsional-flexural  et a l , ^  Ii>"^  21 Novak,  e t c . , a p p l i e d the q u a s i - s t e a d y problems  transmission Richardson, 26 Davenport  oscillations  of e x i s t i n g  approach  f o r bodies  shapes w i t h p l u n g i n g and/or t o r s i o n a l  galloping  tall  Scruton,^  the n o n l i n e a r v i b r a t i o n  metric The  criterion  to  o f v a r i o u s geo-  degrees  of  freedom.  s t r u c t u r e s , mainly  of  c o n d u c t o r l i n e s , were o b s e r v e d and s t u d i e d by S c r u t o n , 23 24 25 et a l , Cheers, D r y d e n and H i l l and o t h e r s . suggested  the p o s s i b i l i t y  t h i n b u i l d i n g s w h i c h may  be  of galloping  instability  c o n s t r u c t e d i n the near  of  future.  27 A p a p e r by bluff  Parkinson  cylinders  and  Scruton, gated  in detail  structures  such  d i s c u s s e s the provides  aeroelastic  a good s u r v e y  as w e l l as D a v e n p o r t and t h e w i n d l o a d i n g and as  s t a c k s , towers,  behaviour  of the  literature.  a s s o c i a t e s have  dynamics o f c e r t a i n  m a s t s and  of  invesitcomplete  buildings.  On  the  their  to  the  28 o t h e r hand, D a l e ,  et a l ,  have c o n c e n t r a t e d  study  6  dynamic behaviour o f hydrophone c a b l e s .  Intensive  investigations  i n t o the aerodynamic i n s t a b i l i t y o f suspension b r i d g e s s p e c i a l reference  with  t o the o r i g i n a l Tacoma Narrows Bridge have 29  been conducted by Farquharson, e t a l ,  30 Kelley,  etc.  In a d d i t i o n t o t h i s , the unsteady f o r c e s and wake geometry a s s o c i a t e d w i t h two-dimensional b l u f f c y l i n d e r s have been 31 32 intensively studied. McGregor and G e r r a r d have conducted e x p e r i m e n t a l i n v e s t i g a t i o n s o f the f l u c t u a t i n g p r e s s u r e s on 33 stationary  circular cylinders.  More r e c e n t l y , Ferguson  wake survey as w e l l as f l u c t u a t i n g s u r f a c e on  the same s e c t i o n .  made  p r e s s u r e measurements  The c o r r e s p o n d i n g r e s u l t s f o r s t a t i o n a r y  square, r e c t a n g u l a r , and e l l i p t i c a l c y l i n d e r s were presented by 34 35 36 Modi and Heine, and Wiland. S i m i l a r l y , Grove, e t a l , 37 38 39 Bishop and Hassan, Humphreys, and Fung measured the f l u c t u a t i n g f o r c e s on s t a t i o n a r y c i r c u l a r c y l i n d e r s o v e r d i f f e r e n t 3 ranges o f Reynolds number. In a d d i t i o n , Wardlaw and Davenport, 40 and  Vickery  conducted some f l u c t u a t i n g f o r c e measurements on  d i f f e r e n t shapes i n laminar as w e l l as i n t u r b u l e n t  flow.  On the o t h e r hand, i n v e s t i g a t i o n o f wake and s u r f a c e c o n d i t i o n s on o s c i l l a t i n g b l u f f c y l i n d e r s i s l e s s complete. The study o f f l u c t u a t i n g l i f t  and drag  f o r c e s by  Bishop and  41 Hassan  showed t h a t the v o r t e x frequency i s c o n t r o l l e d over  a range o f  c i r c u l a r c y l i n d e r frequencies;  while  Ferguson  42 and  Parkinson  metry r e l a t e d  measured f l u c t u a t i n g to a c i r c u l a r  duced o s c i l l a t i o n s .  Molyneux  cylinder 43  pressures  and wake geo-  experiencing  vortex i n -  developed techniques f o r measur-  i n g the aerodynamic f o r c e s on o s c i l l a t i n g a i r f o i l s .  The t h r e e -  d i m e n s i o n a l s t r u c t u r e o f the wake and c o r r e l a t i o n along a c i r c ular c y l i n d e r during  s t a t i c or dynamic c o n d i t i o n s  have been  investi-  g a t e d by  Gerrard,  44  Toebes,  45  Prendergast,  46  Feng  47  and  others.  40 Vickery  measured the  a stationary  square  c o r r e l a t i o n of  lift  a l o n g the  surface  cylinder.  Dynamic a m p l i t u d e m e a s u r e m e n t s o f  two-dimensional b l u f f  c y l i n d e r s o f v a r i o u s c r o s s s e c t i o n s have been i n v e s t i g a t e d 48 49 50 Brooks. Both Smith and S a n t o s h a m concentrated t h e i r to  aeroelastic  galloping  of  rectangular  cylinders  51 degree of  Eagleson, et vibrations  On t h e 54 al,  of  The  dealt with  flat  are  plates  But  literature  i s devoted  the  oscillations  as  of  airfoils  and  to  the  to  emphasized  the  trailing  that  the  circular section.  investigations  plunging  investi-  hydroelastically self-excited  related  the  studies  prismatic bars, 53 Eagleson, and  f o r bodies of various  i t s h o u l d be  fact that  present  o f wake-body i n t e r a c t i o n on  reported  geometry.  Otsuki  o t h e r h a n d , T o e b e s and  influence  instability  by  and  t o r s i o n a l o s c i l l a t i o n s of  respectively.  in a  by  52  f r e e d o m ; w h i l e Chuan  g a t i o n s on  of  geometry.  aeroelastic  cross-sectional bulk of  the  This  indicated  into aerodynamically  s t r u c t u r a l a n g l e s were n o t  edge  initiated  is  excited until  1962.  55  Thornton  c o n d u c t e d e x p e r i m e n t s on  single  double angle s e c t i o n  able  and  to  suppress the  More r e c e n t l y ,  m o t i o n by 2  Wardlaw  has  the  v i b r a t i o n of  members i n s t e a d y the  addition  reported  the  of  several  f l o w and  flat  plate  o s c i l l a t i o n s of  of  structural  consideration,  in  still  sectional  a i r of  lift, the  equal-legged  drag  and  coupled and  p i t c h i n g moment.  disFrom  a  flexural-torsional vibrations  unsymmetrical  beams were e x a m i n e d t h e o r e t i c a l l y and  spoilers. a  3x3x3/16 i n . a l u m i n u m s t r u c t u r a l a n g l e beam t o g e t h e r w i t h tributions  was  angle  section  e x p e r i m e n t a l l y by  Kosko.  1  8  1.3  P u r p o s e and The  bluff and  Scope o f t h e  problem  c y l i n d e r s has  theoretically,  Investigation  of a e r o e l a s t i c been a c t i v e l y in this  instability  of  s t u d i e d , both experimentally  d e p a r t m e n t s i n c e 1958.  A review  56 the p r o g r e s s  i s r e p o r t e d i n two  atic  investigations  eral  study  programme. angle  on b u i l d i n g s  d e s c r i b e d here  section,  the wind speed  and  the e f f e c t s  and  a source  structures  system-  and  structures.  gen-  The  on  the aerodynamics o f  angle of a t t a c k ranges the  In p a r t i c u l a r ,  of impor-  T h i s i s i n t e n d e d t o p r o v i d e , even-  simple  section.  the i n v e s t i g a t i o n s examined the  dynamics o f e q u a l - l e g g e d , angle  aero-  s e c t i o n models i n  a c o n v e n t i o n a l low  t u r b u l e n c e , r e t u r n - t y p e wind t u n n e l w i t h  test  i n . x 27  section  m o d e l s had uniform The  Being  and  3x3  i n . x 8 2/3  i n . and  s u r f a c e s were smooth w i t h i n themselves,  stationary  ft.  The  1/2  i n . , respectively.  sharp  contour  s u p p o r t i n g systems w i t h  mean a e r o d y n a m i c c o n d i t i o n s b e i n g e s s e n t i a l l y  A comparison m o d e l s and Appendix  I.  o f the geometric  commercially  with  edges.  t h e m o d e l s were mounted n o r m a l t o  or spring  f e a t u r e s o f the  available  angle  a  experimental  i n . c r o s s - s e c t i o n a l dimensions  l e g t h i c k n e s s e s o f 1/6  rigid  f l o w on  o f 36  l x l i n . and  exterior  the  o f i n f o r m a t i o n f o r t h e s a f e d e s i g n o f open  composed o f t h i s  d y n a m i c s and  These  o f t h e model d y n a m i c s on  aerodynamic parameters.  tually,  '  forms a p a r t o f t h i s c o n t i n u i n g  I t presents information  instability, tant  survey papers.  of  57  have c o n t r i b u t e d p e r t i n e n t d a t a t o t h e  o f wind e f f e c t s  investigation  two-dimensional  the  structural  two-dimensional.  'idealized'  angle  sections i s presented i n  The tigating and  r e s e a r c h programme was  the  angle  section  divided  (i)  the  steady  (ii)  lift,  mean and  (iv)  For the  drag  and p i t c h i n g moment  fluctuating  lift,  static  drag  and  pressure  on:  coefficients;  o f S t r o u h a l number w i t h R e y n o l d s fluctuating  torsion,  stationary  thesis presents experimental results  variation  (iii)  four stages i n v e s -  during stationary, plunging,  combined p l u n g i n g - t o r s i o n c o n d i t i o n s .  model t e s t s ,  into  number;  distributions;  p i t c h i n g moment  coefficients;  (v) wake g e o m e t r y as  f u n c t i o n s of angle of attack.  I n most t e s t s : t h e 4  number was  c o n f i n e d t o the range  static  force  ducted  on  and  vortex shedding  commercially  The  results  available  in  one  chapter of the t h e s i s  of  t h e m o d e l d y n a m i c s and  lateral plunging  investigations. ities for  of  .  a l u m i n u m and  steel  three stages the  of the  section  and  (i) (ii) (iii)  on  such  vortex shedding fluctuating  frequency  static  wake g e o m e t r y  parameters and  pressure;  features  The  existing  study  i n the  f o r the  galloping  resonance,  as: phase;  and  torsional  instabil-  damping  modes o f v i b r a t i o n .  experiencing vortex excited  the model motion  combined  important  f o r the  con-  members.  s p r i n g mounting system  the vortex induced  three o s c i l l a t o r y  angle  are  a s s o c i a t e d aerodynamics. suitable  comparison,  measurements were  i n s t r u m e n t a t i o n were d e s i g n e d  Both  For  were e x a m i n e d a t v a r i o u s a n g l e s o f a t t a c k and  each  angle  frequency  o f f r e e d o m , b u t a new  a u x i l i a r y measuring  t o 10  to correlate  s u p p o r t i n g e q u i p m e n t was degree  5  2x10  of the remaining  Reynolds  For the  levels the  effects  were d e t e r m i n e d . were p r e d i c t e d ating  force  obtained was  model  from resonant theory.by  data.  static  applied  Peak v o r t e x r e s o n a n t d i s p l a c e m e n t  In  aerodynamic  to provide  unless  otherwise  number o f  not  in-Appendix  to  and on  and  fluctu-  quasi-steady of  the  analysis galloping  the  wind tunnel  uncorrected  the  results  on  the  presented,  f o r these e f f e c t s .  to e s t a b l i s h trends of  t h i s d a t a t o g e t h e r w i t h a summary confinement c o r r e c t i o n s  comment c o n c e r n i n g  the  applicability  a p r a c t i c a l situation i s pertinent r e l a t e to a r i g i d ,  a n g l e s e c t i o n beam, w i t h t h e externally.  p o t e n t i a l e n e r g y i s due the  walls  are  A the  of  presented  II.  damping a p p l i e d  with  the  experimentally  the  established,  are  wall  sented i n t h i s thesis long  of  e x p e r i m e n t s were c o n d u c t e d  theories  A  well  stated,  interference,  existing  a  the  theoretical predictions  influences  measured d a t a are  tion  loadings,  incorporating  dynamics. Since the  wall  addition, using  amplitudes  t o the  internal friction  of  of  here.  The  investigaresults  pre-  two-dimensional element  s t i f f n e s s lumped as  However, f o r an s t r a i n s and  the  this  material.  of  springs  a c t u a l beam,  damping i s  :  the  associated  Nevertheless,  21 Novak  has  reported  between v i b r a t i o n s o f procedure  an  rigid  sections  f o r model s i m u l a t i o n  p r e s e n t e d by  Whitbread.  r e q u i r e m e n t s h a v e t o be exists,  i n v e s t i g a t i o n showing  the  static  For  continuous systems.  of p h y s i c a l  structures  has  s e c t i o n a l models, c e r t a i n  satisfied.  aerodynamic  and  compatability  When g e o m e t r i c  r e s u l t s are  e v e n a t o t h e r R e y n o l d s numbers w i t h i n  directly  A  been similarity  similitude applicable,  a c e r t a i n range, i f flow  11 separation studies,  i s f i x e d by edge c o n d i t i o n .  the requirements  involve  parameters,  such  as s t i f f n e s s ,  represented  i n nondimensional  In t h e s e e x p e r i m e n t s , chosen of  1 2 s e c t i o n beams. '  inertia  a n d damping.  form as  n and.0,  the spring  stiffnesses  u> /u)  =  Q  values  limit  solids  I ) make t h e f o r m e r f o rtorsional  6 0  '  e  t  a  f o r convenience,  ^ have shown t h a t t h e i n t e r n a l  models s o t h a t  t h e decay  known t h a t t h e i n t e r n a l i t s particular  while  friction  energy  i s logar-  62 '  indicate  that the  as w e l l ,  with  I n any c a s e , i t i s w e l l  i s a function of the material  c h e m i c a l and p h y s i c a l p r o p e r t i e s .  Typical  f o r aluminum w o u l d be o f t h e o r d e r o f 0.001 < 8 < 0.01, f o rsteel  0.0G08 < £ < 0.006.  the v a l u e o f i n t e r n a l duct  inves-  f r i c t i o n of  of the v i b r a t i o n  However, o t h e r i n v e s t i g a t i o n s  damping i n c r e a s i n g w i t h d i s p l a c e m e n t .  values  suit-  Several inves-  by v i s c o u s r e s i s t a n c e o r o t h e r  l o g a r i t h m i c d e c r e m e n t i s d e p e n d e n t on a m p l i t u d e  and  of the  b u t does n o t  of the r e s u l t s obtained.  61 ithmic.  which i s t y p i c a l  3 times h i g h e r than those o f the  c a n be a p p r o x i m a t e d  dissipative  have b e e n  (Appendix  T h i s was s e l e c t e d  the a p p l i c a b i l i t y  tigators^'  respectively.  F o r t h e 3 i n . models i n t h e p l u n g i n g c a s e , t h e n^  are approximately  prototype.  3  They a r e  ^ characteristics  f o rplunging studies, while the l a t t e r  tigations.  of structural  The i n e r t i a  1 i n . and 3 i n . a n g l e models able  correspondence  t o provide a frequency r a t i o  angle  However, f o r d y n a m i c a l  With  this  damping, i t was t h o u g h t  t h e t e s t s w i t h damping l e v e l s  to occur i n the f u l l - s c a l e  uncertaintyi n a d v i s a b l e t o con-  i n and below t h e range  structure.  likely  2  2.1  AERODYNAMICS OF  A STATIONARY ANGLE SECTION  P r e l i m i n a r y Remarks I n an  aeroelastic  investigations oscillatory c a n be  instability  provide v i t a l  c o n d i t i o n s such  predicted.  t o e x t e n s i v e wind i n g mean w i n d  study,  stationary  i n f o r m a t i o n from which as w i n d s p e e d  A s e t of angle tunnel testing  and  model  model  critical orientation  s e c t i o n m o d e l s were s u b j e c t e d to obtain information concern-  l o a d i n g , S t r o u h a l number, f l u c t u a t i n g  static  p r e s s u r e and wake g e o m e t r y . This apparatus, also  form  mental 2.2  i n s t r u m e n t a t i o n , and  The A  test  list  results  study  i n the  a r e d i s c u s s e d and  o f the e l e c t r o n i c  which  following  conclusions pre-  instruments used  i n the  experi-  programme i s g i v e n i n A p p e n d i x I I I .  Models, Apparatus,  2.2.1  Angle  I n s t r u m e n t a t i o n and  Calibration  Models  D e p e n d i n g on angle  experimental procedures  the b a s i s f o r the dynamical  chapter. sented.  c h a p t e r d e s c r i b e s model c o n s t r u c t i o n , n e c e s s a r y  the type of e x p e r i m e n t a l  s e c t i o n m o d e l s a r e c a t e g o r i z e d as  test  proposed,  the  follows:  (i) p r e s s u r e tap model; (ii) (iii) The  dynamic model; balance  designed  model.  a n g l e m o d e l s had  smooth f a c e s t o f a c i l i t a t e of  surface conditions.  usually  sharp  contour  c o n s t r u c t i o n and  edges and  provide uniformity  However, c o m m e r c i a l l y  have r o u n d e d e d g e s and  rough  relatively  available  surfaces.  To  angles  determine  the  13 influence on  o f the d i s p a r i t i e s  certain  aerodynamic c h a r a c t e r i s t i c s ,  were made f r o m s t e e l To the  hollow  examine t h e mean and f l u c t u a t i n g  the pressure angle  model,  fastening tab plates  bonded t o a c r y l i c  0.025 i n . d i a m e t e r  (Figure 2-1).  taps,  of the angle.  wise d i r e c t i o n , section,  to e x t e r n a l l y diameter  pressures  and dynamic  on  con-  The 3x3x1/2 i n . 0.020  A 39 h o l e p r e s s u r e  ring  l o c a t e d a t t h e midspan o f t h e model, distributions  Two t a p s were p r o v i d e d  around  i n the span-  a t d i s t a n c e s o f 4 1/2 i n . and 9 i n . f r o m t h e mid-  a t t h e same c o n t o u r  surface pressure  members.  b u l k h e a d s and 1/4 i n . t h i c k  a means o f e x a m i n i n g t h e p r e s s u r e  contour  models  26 3/4 i n . l o n g , was c o n s t r u c t e d f r o m  end  the  static  t a p model was d e s i g n e d .  aluminum s h e e t  provided  angle  section during s t a t i c  in.  of  three balance  and aluminum s t r u c t u r a l  s u r f a c e o f an a n g l e  ditions  between t h e m o d e l s and p r o t o t y p e s  position  as t a p number 5.  The  s i g n a l s were t r a n s m i t t e d f r o m t h e p r e s s u r e  located transducers  "Intramedic"  through  polyethylene  taps  5 f t . l o n g , 0.066 i n .  tubing.  The 1 i n . and 3 i n .  d y n a m i c m o d e l s , w h i c h d i d n o t have p r e s s u r e  t a p s , were o f i d e n t i c a l  geometry. Six different designed  and b u i l t  strain-gauge leg width  s e c t i o n models  balance.  model D and E h a d h a l f  M o d e l s A and B d i f f e r e d  pressure steel  the e f f e c t s  the l e g  o n l y by t h e 1/4 i n . t h i c k  model.  T h i s was i n t e n d e d t o  o f t h e same end p l a t e s mounted on t h e  t a p and dynamic a n g l e  structural  s i x component,  M o d e l s A, B and C h a d t h e same n o m i n a l  p l a t e s mounted on t h e l a t t e r  determine  (Appendix I) were  f o r m o u n t i n g on t h e A e r o l a b  and t h i c k n e s s , w h i l e  thickness. end  angle  angles  with  models.  M o d e l s C and D were  rounded c o r n e r s  and r o u g h  sur-  Figure  2-1  P r e s s u r e s i d e s  t a p  angle  model  and  n u m b e r i n g  o f  p r e s s u r e  h o l e s  and  c o n t o u r  15 faces.  A  tively by  typical  smooth  model  E.  surface Model  cross-sectional models,  commercial  the  F  effective  was  2.2.2  Fluctuating  nominally  new  pressure  ducing  System  power  instrument ing  pressure  ally  of  a  diaphragm ditioner on  the  output  voltage  ±0.1%  for  diaphragm response step  of  has  to  be  pressure  frequency  of  diaphragm  was  The  two  B  with  wind  with  a  a  pressure  input.  sensor  From  cavity to  of  head  ±0.1%  2500  cps  i s less the  connection  approximately  on  signal  mm.  of  on  most The  and  stability  and  2 ms  8 of  transient to  a  resonator  side of  cps.  the  prestressed  the  Helmholtz  210  the  head.  The  con-  mercury  of  than  one  steel  any  changes;  experiment  and be  of  on  basic-  The  sensor  scale  fluctuat-  consists  mercury  pressure  sensitive  intended  0-10  Trans-  conditioner  stainless  from of  temperature  n a t u r a l frequency  a  developed  Pressure  chambers.  full  has  signal  sensor  with  ranges  linearity  ambient  Calibration  f o r the  0 - 0 . 0 0 1 mm.  d.c.  found  the  precision,  pressure  pressure  volts  the  to  sensor,  high  divider  sensitivity to  and  suitable  i s 0-5  the  model  a l l balance  Barocel Modular  is a  to  a  exposed  a pressure  coupled  ±15°F  of  called  when  ranges  was  rela-  represented  to  For  with  Waltham, M a s s a c h u s e t t s ,  voltage  sensitive  scale  in.  Transducer  of  s e p a r a t i n g the 8  r a d i u s was  2x2x1/3  Barocel  proved  capacitive  sensitivity  of  measurements.  least  sensitive  Inc.  The  provides  section  27 i n .  consisting  which  corner  length which  transducer  supply.  angle  geometrically similar  Pressure  Datametrics  and  was  small  dimensions  stream  a  and  aluminum  the  The  Barocel i s accurately calibrated  However, f o r f l u c t u a t i n g pressures  f o r steady  t r a n s m i t t e d through r e l a t i v e l y  long, small diameter tubes c o n s i d e r a b l e a t t e n u a t i o n Therefore,  pressures.  occurred.  the output e l e c t r i c a l s i g n a l r e q u i r e s c a l i b r a t i o n  a g a i n s t known i n p u t f l u c t u a t i n g pressure T h i s was achieved  a t the model s u r f a c e .  u s i n g the c a l i b r a t i o n system developed by  Wiland. The  e f f e c t o f amplitude and frequency  o f the source  sure on the output s i g n a l i s shown i n F i g u r e 2-2(a). curves  indicate  the l i n e a r i t y o f the system.  c a l i b r a t i o n curves  These  For convenience the  i n Figure 2-2(a) were r e p l o t t e d  as a r a t i o o f output to i n p u t .  pres-  This eliminates  i n Figure  2-2(b)  frequency  interpolation. 2.2.3  Wake Probe and T r a v e r s i n g Gear The wake geometry survey was performed u s i n g a d i s c probe  c o n s t r u c t e d by Ferguson  33  and d e s c r i b e d i n d e t a i l by Bryer e t a l .  I t was mounted on a 1 i n . hypodermic needle which i n turn was connected to a 14 i n . long, 1/4 i n . diameter s t i n g . pressure  c a l i b r a t i o n data were obtained  from wind tunnel t e s t s  (Figure 2-3).  Static  f o r t h i s p a r t i c u l a r probe  The measurements  indicate  the probe t o be r e l a t i v e l y i n s e n s i t i v e to a p i t c h o f ± 5 ° and yaw of ±20°. section  To enable the wake probe t o be p o s i t i o n e d i n the t e s t of the tunnel with c o n t r o l  longitudinal  of movement i n a l a t e r a l and  d i r e c t i o n , the wake t r a v e r s i n g  gear designed by  33 Ferguson  was used.  approximately  The accuracy  0.02 i n .  i n positioning  the probe was  63  17  Input,  volt  3  a. 3  - o  Frequency , cps  Figure  2-2  C a l i b r a t i o n curves arrangement  f o r Barocel  transducer  Figure  2-3  Dimensions  and c a l i b r a t i o n  data.of disc  probe  2.3  Test  2.3.1  Procedures  Balance After  Measurements setting  the  angle  o f a t t a c k and  wind speed t o a p r e s e l e c t e d v a l u e , the moment on then  a balance  m o d e l were r e c o r d e d .  i n c r e a s e d t o two  obtained. models  2.3.2  Vortex  over  the  Shedding  was  entire  the  instrumentation  p r o b e was  l o c a t e d a t an  where t h e  fluctuating  was  typical  from the probe. m o d e l s and vortex  2.3.3  the balance  shedding  Reynolds  Using  five of  data  balance  attack.  was  of the  Illustrated  this  2-4.  i n the  The  clear.  a band  pass  i n Figure 2-5(a),  f l u c t u a t i n g pressure 1 i n . and  (B,C,D,E, and  v a r i a t i o n with  wake  mid-plane  relatively  signal,  system, the  models  accomplished  are  signals  3 i n . angle  F) were t e s t e d f o r  angle  of attack  and  number.  The  Pressure  mean p r e s s u r e  on M o d e l  ethyl  alcohol  the  column caused was  facilitate  reduced  by  Surface  d i s t r i b u t i o n was  L a m b r e c h t manometer w i t h  To  range of angle  s i g n a l was  filtered  frequency  Mean S t a t i c  pressure  f o r the  l a y o u t shown i n F i g u r e  pressure  and  was  corresponding  appropriate position  incorporated.  unfiltered  pitching  wind speed  Strouhal frequency  However, t o i m p r o v e t h e q u a l i t y filter  and  Frequency  Measurement o f the using  repeated  drag  The  f u r t h e r s e t t i n g s and  This procedure  (A t o E)  lift,  i n c r e a s i n g the  alcohol.  fluctuating  using a r e s t r i c t i o n  r e d u c t i o n of the  data,  obtained  The  using  oscillation  component o f t h e i n the p r e s s u r e  the p r e s s u r e  on  the  a of  the  static line. model  20  Probe  Polyethylene tubes K l = 5'  ;  d 0.066" » |=  Barocel Damping bottle  Signal conditioner  Visicorder  gure  2-4  I n s t r u m e n t a t i o n l a y o u t q u e n c y and f l u c t u a t i n g  f o r v o r t e x s h e d d i n g f r e p r e s s u r e measurements  Figure 2 - 5  T y p i c a l f l u c t u a t i n g pressure (b) model s u r f a c e  s i g n a l s from  (a) wake probe  22 was of  measured r e l a t i v e the wind t u n n e l .  pressure  to the t o t a l  changes d u r i n g the  Fluctuating  the  measurements. ations, that  the  the  For  and  produced  angle  Pressure  of the  a n g l e m o d e l was  and  always p o s i t i v e .  Static  Investigation of  tests  f o r a range o f wind speed  2.3.4  settling  T h i s e l i m i n a t e d the e f f e c t  d i f f e r e n t i a l w h i c h was ed  head i n t h e  The  of  s t u d y i n g the  a pressure  r e s u l t s were o b t a i n -  of attack.  on M o d e l  Surface  into  amplitude  amplitude  and  taken  from  the phase  t h e model t a p s  was  and  used  As  fluctuating frequency tude  d i s c o v e r e d by  pressure  and  t h e model s u r f a c e . the  c y c l e s was  determined  signals  average  peak was  peak v a l u e o v e r from  the  a second  d u c e d by  charts.  the  '  the the  random  ampli-  unfiltered,  signals  approximately  the r a t i o  a  35  fluctuating  chart records.  between t h e  d i f f e r e n t model t a p s was  Barocel transducer  Visicorder  typical  on  from pressure  150  to  In a d d i t i o n ,  o f maximum  250 to  amplitude  obtained.  phase d i f f e r e n c e  from  seemingly  modulated p r e s s u r e  e v a l u a t e the s i g n a l modulation,  The  had  For p r e s e n t a t i o n of the  amplitude,  to average  and  F i g u r e 2-5(b) i l l u s t r a t e s  random a m p l i t u d e  except  t h e model s u r f a c e were a t  of the v o r t e x shedding,  modulation.  filtered  other i n v e s t i g a t o r s ,  s i g n a l s on  fluctu-  recorded 33  Visicorder.  surface  of the p r e s s u r e  i n s t r u m e n t a t i o n shown i n F i g u r e 2-4  s i g n a l was  atmospheric  f l u c t u a t i n g p r e s s u r e on  separated  section  To  fluctuating  o b t a i n e d by i n c o r p o r a t i n g  s y s t e m and  comparing the  e l i m i n a t e the e f f e c t  i n s t r u m e n t a t i o n , the  pressure  signals  o f phase s h i f t  f l u c t u a t i n g pressure  from  on  introone  23 m o d e l t a p was a r b i t r a r i l y phase  selected  as a r e f e r e n c e .  a n g l e was a v e r a g e d o v e r 10 t o 20 c y c l e s .  The r e l a t i v e  The a m p l i t u d e  and p h a s e  m e a s u r e m e n t s f o r t h e 41 p r e s s u r e t a p s on t h e model were  performed  a t t h r e e wind  The wind  speed  band pass  filter  and f i l t e r signal  Visicorder  calibration  2.3.5  and v a r i o u s  angles o f attack.  a n t e n u a t i o n was d e t e r m i n e d  cut-off  sinusoidal  using  speeds  f o r each  f r e q u e n c y s e t t i n g s by u s i n g a  from a low f r e q u e n c y f u n c t i o n g e n e r a t o r . a n d impedance a t t e n u a t i o n were  The  determined  t h e same p r o c e d u r e . Wake  Survey  Wake s u r v e y m e a s u r e m e n t s were a c c o m p l i s h e d b y e x a m i n i n g the  fluctuating pressure f i e l d  from t h e model u s i n g Figure  2-6.  independent to  shed  t h e i n s t r u m e n t a t i o n s e t u p shown i n  S i n c e t h e wake r e s u l t s were f o u n d t o be s u b s t a n t i a l l y o f Reynolds  o n l y one wind  speed  Traversing and  associated with the vortices  recording  number, t h e measurements were for various  the disc  t h e average  probe  peaks  confined  angles o f attack. laterally  a t various  x-stations  of the fluctuating pressure  s i g n a l s p r o v i d e d a s e t o f c u r v e s e a c h h a v i n g two maxima n e a r t h e vortex centrelines.  T h e y - d i s t a n c e b e t w e e n t h e s e maxima a t e a c h  x - s t a t i o n was t a k e n t o be a measure o f t h e l a t e r a l s p a c i n g b e t w e e n t h e two rows o f v o r t i c e s convenient of tap  probe  to plot  the results  t o model t a p average  selected  f o r t h e probe  represented  a position  fluctuating  pressure value.  shed  from t h e model.  o f the l a t e r a l  traverse  fluctuating pressures.  I t was as a r a t i o The model  r a t i o was somewhat a r b i t r a r y , b u t  on t h e model c o n t o u r h a v i n g a n e a r  maximum  A t r u e rms v o l t m e t e r was u s e d f o r  Damping bottle  Damping bottle  Signal  Signal conditioner  conditioner  Power supply  Filter  Filter  RMS voltmeter  R-C damping unit  VTVM  Oscilloscope  Visicorder  F i g u r e 2-6  Schematic of i n s t r u m e n t a t i o n f o r wake survey measurements  25 averaging of  of the f l u c t u a t i n g pressure  t h e peak v a l u e s w i t h  decay o f t h e v o r t i c e s Longitudinal one  i n t h e downstream d i r e c t i o n . s p a c i n g , a , between c o n s e c u t i v e  row o f t h e v o r t e x  s t r e e t was o b t a i n e d  a 360° p h a s e d i f f e r e n c e  signals  associated  Traversing the  averaged  was d e t e r m i n e d  pressure  o f a v o r t e x row,  f r o m t h e p r o b e a n d a r e f e r e n c e model on a V i s i c o r d e r .  10 t o 20 c y c l e s , was p l o t t e d  t h e downstream c o o r d i n a t e  was i n t e r p r e t e d .  longi-  i n t h e same row.  the centreline  simultaneously  over  from  corresponds  between t h e f l u c t u a t i n g  consecutive vortices  f l u c t u a t i n g pressures  data, of  with  vortices i n  indirectly  The s p a c i n g d i s t a n c e  t h e wake p r o b e a l o n g  t a p were r e c o r d e d  The v a r i a t i o n  x c o o r d i n a t e g a v e an i n d i c a t i o n o f t h e  t u d i n a l phase measurements. to  signals.  The p h a s e as a  angle  function  from which t h e v o r t e x s p a c i n g , a,  In a d d i t i o n ,  the vortex  streamwise v e l o c i t y  from the l o n g i t u d i n a l spacing u s i n g the r e l a t i o n  (2.1)  2.4  Experimental  2.4.1  Steady L i f t , The  by  static  various angle 4  4x10 The  and D i s c u s s i o n  D r a g a n d P i t c h i n g Moment D i s t r i b u t i o n s lift,  sections  drag  a n d p i t c h i n g moment  over  experienced  t h e R e y n o l d s number r a n g e o f  4 to  11x10  are presented  i n the following  illustrations.  p i t c h i n g moment h a s b e e n m e a s u r e d r e l a t i v e t o an a x i s  coinciding with that  Results  over  the centroid  o f an a n g l e  section.  t h e wind speed range i n v e s t i g a t e d ,  I t was  the force  observed  and moment  26 c o e f f i c i e n t s were i n d e p e n d e n t Figure  2-7, p r e s e n t s t h e v a r i a t i o n  c o e f f i c i e n t s with B  angle  of attack,  which i s g e o m e t r i c a l l y s i m i l a r  angle models.  I t i s apparent  t h e maximum n e g a t i v e l i f t and  = -45°  variation reduced  force  values near  section  c o e f f i c i e n t with  dant  o f model o r i e n t a t i o n .  test  results  the  1/4 i n . t h i c k  section  experiences a = 10°  a = 4 0 ° . On t h e  i s maximum a t t h e s y m m e t r i c  coefficient  angle  i s open u p s t r e a m .  on t h e p r o j e c t e d f r o n t a l  d o e s n o t become c o m p l e t e l y  o f model A i n d i c a t e d  The  a n g l e o f a t t a c k c a n be  A comparison  end p l a t e s  model  t o t h e p r e s s u r e t a p and d y n a m i c  a n d moment a t a p p r o x i m a t e l y  by b a s i n g t h e c o e f f i c i e n t  However, t h e d r a g  o f t h e aerodynamic  that the angle  where t h e m o d e l a n g l e o f the drag  number.  o , f o r the balance  c o r r e s p o n d i n g maximum p o s i t i v e  o t h e r hand, t h e d r a g a  o f t h e Reynolds  of this  indepen-  data with the  no s i g n i f i c a n t  on t h e o v e r a l l  area.  influence of  aerodynamic  characteristics. The butions  abrupt  at certain  are a t t r i b u t e d contour.  on and  finally  shifted  orientation  a = 1 0 ° , 40° and 104°  such  as  and moment  t o major changes i n t h e f l o w f i e l d  forming  the separated  a s e p a r a t i o n bubble  disappeared producing  In the v i c i n i t y from  over study  distri-  the angle using a  With the i n c r e a s e o f angle o f attack i n the  5° t o 25° and 97° t o 1 2 5 ° ,  one s i d e  side.  drag  T h i s was s u b s t a n t i a t e d by a q u a l i t a t i v e  smoke t u n n e l . ranges  changes i n t h e l i f t ,  of  the junction  flow reattached  which decreased  attached flow over  a = 40°, the forward  i n length  the e n t i r e  stagnation point  o f s i d e s 3 and 4 t o a p p r o x i m a t e l y t a p  F i g u r e 2-7  D i s t r i b u t i o n of steady l i f t , f o r balance model B  drag and p i t c h i n g  moment c o e f f i c i e n t s to  28 number 21 c a u s i n g a change i n t h e p o s i t i o n shear  layers  downstream.  Figure radius able  o f the separated  2-8  illustrates  some o f t h e e f f e c t s  and s u r f a c e r o u g h n e s s a s s o c i a t e d w i t h  a n g l e members.  The r e s u l t s  show t h a t  commercially  a n g l e members C, D and E h a v e r e a s o n a b l e  similarity  even  features.  I n F i g u r e 2-8(c) s l i g h t  moment d i s t r i b u t i o n s due  to differences  comparison thickness  are  are apparent  minor d i f f e r e n c e s  i n geometric  d e v i a t i o n s i n the p i t c h i n g over  suggests  t h e range  that  o r corner radius decreases This variation  pressure d i s t r i b u t i o n s The  aerodynamic  o f - 4 5 ° t o 15°  i n l e g t h i c k n e s s and c o r n e r c o n d i t i o n s .  of the r e s u l t s  i n g moment.  avail-  a n g l e model B and  structural  though t h e r e  o f edge  present l i f t  a reduction i n l e g  t h e magnitude o f the p i t c h -  i s apparent  given l a t e r  A  from  t h e mean  static  i n F i g u r e 2-12.  and d r a g d i s t r i b u t i o n s  compare w e l l  with  2 Wardlaw's m e a s u r e m e n t s angles quoted the  a n d t h e few d r a g r e s u l t s f o r s t r u c t u r a l 64 by Hoerner. F o r t h e a n g l e models a t a = - 4 5 ° ,  corrected drag  coefficient  value of approximately  2.0 i s 64  similar Using  to that  o f a normal f l a t  plate  t h e measured base p r e s s u r e v a l u e  section  at  o r 90° wedge m o d e l . f o r the pressure tap angle  a = 135°, Roshko's ^ notched 6  a 90° wedge, p r e d i c t s  a value of  with  drag  the experimental  C  D  hodograph s o l u t i o n f o r  = 1.8 w h i c h compares  coefficient  o f 1.7 f o r b a l a n c e  well model  B. 2.4.2  Vortex The  speed  Shedding  Frequency  a n d S t r o u h a l Number  dependence o f t h e v o r t e x shedding  frequency  on w i n d  i s shown i n F i g u r e 2-9 f o r v a r i o u s a n g l e s o f a t t a c k o f  the p r e s s u r e  t a p angle model.  The l i n e a r i t y  of the p l o t s i n -  29  Figure  2-8  Comparison structural  of aerodynamic c o e f f i c i e n t s angle s e c t i o n s  for various  Figure  Figure  2-9  2-10  V a r i a t i o n of vortex shedding frequency with wind v e l o c i t y f o r p r e s s u r e t a p a n g l e model  V a r i a t i o n o f S t r o u h a l number and v o r t e x r e s o n a n t wind speed w i t h angle o f a t t a c k f o r 3 i n . angle model  dicates  that the Strouhal  number i s i n d e p e n d e n t o f t h e R e y n o l d s 4  number o v e r t h e r a n g e o f Using tion with (Figure  this  angle  10  t o 15x10  and s i m i l a r  of attack  2-10).  4  Basing  data,  investigated.  the Strouhal  f o r the pressure  the dimensionless  number  t a p m o d e l was  frequency  p r o j e c t e d model w i d t h e, r a t h e r than t h e c o n s t a n t  h,  reduces  it  completely  in  the Strouhal  of Strouhal  a =15°  the  reattachment of the flow  the  balance  in  general,  Strouhal  measurements.  characteristic  Therefore,  length,  ation  o f t h e mean b a s e p r e s s u r e , data  number may have 66  Roshko's  velocity,  result  angles  wake  of attack  i n the  less  as w e l l as merit.  section w i l l  have  equal  1, 2 and 3 i n . a n g l e  to the  number d i s t r i b u t i o n s f o r  m o d e l s t e s t e d i n t h e same w i n d  2-11(a)) i n d i c a t e s the presence o f t u n n e l w a l l  ference  effects.  blockage.  values  o f 0.164.  (Figure  with  Examin-  g e o m e t r y and s h e d d i n g  approximately  A comparison o f the S t r o u h a l the  which,  concept of a  has c o n s i d e r a b l e  i n d i c a t e d t h a t the angle  o f S* f o r v a r i o u s  increase  A  layer separation  accepted  i n the d i s c u s s i o n o f  S , b a s e d on wake w i d t h  shear  frequency  i s attributed to  i f t h e wake w i d t h i s u s e d as t h e  the Strouhal  S t r o u h a l Number,  increase  r e d u c e s t h e wake w i d t h  d e p e n d e n c y on m o d e l o r i e n t a t i o n . Universal  105°  i s a c c o m p a n i e d by a s i m u l t a n e o u s  frequency.  dimension  The s u d d e n  and  as e x p l a i n e d  This  The t r e n d  i s f o r Strouhal  For comparison,  on  number b u t d o e s n o t make  independent of o r i e n t a t i o n . number n e a r  obtained  parameter  the  the v a r i a t i o n  varia-  inter-  number t o i n c r e a s e  the estimated  stream c o n d i t i o n i s a l s o i n c l u d e d  tunnel  curve  (Appendix I I ) .  f o r free  32  F i g u r e 2-11  S t r o u h a l number d i s t r i b u t i o n s f o r (a) d i f f e r ent s i z e angle models (b) v a r i o u s s t r u c t u r a l angle s e c t i o n s  The models  (B,C,D, and  similarity angle the  corresponding are  between the  w o u l d be  size,  similar.  For  i s presented  f o r the  shown.in F i g u r e 2-11(b).  s h a r p - e d g e d m o d e l B and  s e c t i o n s i s apparent.  same n o m i n a l  curve  E)  S t r o u h a l number d a t a  3x3 angle  Since a l l the i n . , the W a l l  the  Aerodynamic commercial  a n g l e members a r e confinement  sections of t h i s width,  i n Appendix  I I , Figure  Comparison of the p r e s e n t  balance  of  effects a  correction  II-l.  S t r o u h a l number d a t a w i t h  the  2 few  results  p u b l i s h e d by Wardlaw  more, t h e  angle  cylinders  of d i f f e r e n t  several  section  results  shows good a g r e e m e n t . f o l l o w the  geometric  form.  r e p r e s e n t a t i v e s e c t i o n s are  t r e n d e s t a b l i s h e d by  The  listed  Further-  S t r o u h a l numbers f o r below: S _fF  (i)  angle  (ii)  flat  (iii)  90°  (iv)  plate  circular  in  results  a t a - -45°  0.135  normal to flow  0.145  wedge and  (v) f l a t The  section  angle  section  cylinder a t a = 40°  suggest  a tendency  a decrease  0.18 0.20  plate  magnitude w i t h  a t a = 135°  0.23 f o r t h e S t r o u h a l number t o i n c r e a s e  in bluffness.  Further  comparison  64 with  other published data  liability  of the  From t h e regarding  the  aeroelastic represents  c o n f i r m a t i o n as t o t h e  re-  measurements. S t r o u h a l number r e s u l t s ,  resonant  instability the  provided  critical  wind speed may wind  occur, speed  (V can  important  information  ), a t which v o r t e x be  obtained.  induced  v  a t which the v o r t e x  r  e  s  shedding  frequency  coincides with  Expressed  i n nondimensional  obtained  from  the  the n a t u r a l frequency form, the  resonant  S t r o u h a l number u s i n g t h e  r  e  2US  s  of the  system.  wind speed  can  be  expression  (2.2)  U  n Shown i n F i g u r e 2-10  i s the v a r i a t i o n  3  3 i n . angle  section.  susceptible  to v o r t e x resonance  a =  I t i s apparent  t h a t an  a t the  with  res  angle  lowest  a  f o r the  member i s  velocity  near  30° .  2.4.3  Mean S t a t i c The  m o d e l was  Pressure  mean s t a t i c  found  t o be  range  independent  10  to  12x10  of attack.  stagnation d a t a on  and  The  .  The  the  s u r f a c e of the  angle  o f t h e R e y n o l d s number  indicate  separation points.  of the  pressure  distribution  i s shown i n F i g u r e 2-12,  results  the base p r e s s u r e  evaluation  on  over  4  the model midspan s e c t i o n angles  Distributions  pressure  4 the  of U  the  for various  location  In a d d i t i o n ,  the  of  shear  layer  velocity  and  the  curves  c o e f f i c i e n t which i s u s e f u l  separated  around  provide  i n the wind  tunnel  wall correction. F o r t h e same o r i e n t a t i o n s , i n v e s t i g a t i o n o f spanwise C d i s t r i b u t i o n showed i t t o be r e l a t i v e l y c o n s t a n t , P substantiating  two-dimensionality  of the  A comparison o f the p r e s s u r e measured  aerodynamic  observation.  As  coefficients  shown, b o t h  good a g r e e m e n t o v e r  the  lift  entire  flow.  i n t e g r a t e d and  ( F i g u r e 2-13) and  drag  balance  confirms  coefficients  range of angle  p i t c h i n g moment measurements show s i m i l a r  the thus  above  are i n  of attack.  trend except  the  The  f o r the  35  Figure  2-12-il  Midspan d i s t r i b u t i o n s s u r e c o e f f i c i e n t (40°  o f mean s t a t i c < a < 135°)  pres-  r  T  D (  Balance model B  Pressure tap angle model  Figure  90°  45°  -45  2-13  C o m p a r i s o n o f p r e s s u r e i n t e g r a t e d and b a l a n c e dynamic c o e f f i c i e n t s  1-0.4 o  measured  steady  aero-  38 d i s c r e p a n c y i n magnitude over  2.4.4  Fluctuating  The  from  from  two  two  (i) neighbouring the (ii)  from  Over the range o f  from  from  26x10  the  pressure  average are  amplitude comparable values and At  75°  63x10  fluctuating  o f the  pressure  and  can  but  ratios  was  the  experiment  s t r e n g t h o f t h e v o r t e x wake  observed by  system.  the  the  average  distributions modulation that  the  a magnitude The  maximum a =  average  C-,  -45°  between 1.5  and  2.5.  coefficient  the modulation  This variation  w i t h model o r i e n t a t i o n flow v i s u a l i z a t i o n  reach  level.  the  that  angle.  i n the v i c i n i t y o f  135°,  on  I t i s apparent  corresponding modulation  2.0.  observed  c o e f f i c i e n t and  t o t h e mean s t a t i c p r e s s u r e  approximately  model.  significant effect  pressure  fluctuating  o f a t t a c k o f 45°  model;  o f a t t a c k , the midspan  d i m i n i s h e s t o a l e v e l b e l o w 0.2 at  i t was  r  c o e f f i c i e n t o r phase  a r e o f t h e o r d e r o f 0.8  angles  as f o l l o w s :  same s i d e o f  same s i d e o f t h e  i l l u s t r a t e d i n F i g u r e 2-15.  with  signals  3  to  For various angles  ratio  the  pressures  t y p i c a l phase  from o p p o s i t e s i d e s of the  R e y n o l d s number d i d n o t have any  the  The  of  various pressure taps  3  of  modulation  2-14.  model;  spanwise taps  fluctuating  pressure  are i l l u s t r a t e d i n F i g u r e  midspan taps  midspan taps  (iii)  fluctuating  spanwise t a p s /  (b) were o b t a i n e d  25°.  Distributions  (a) show t h e a m p l i t u d e  m i d s p a n and  shown i n  a = ±  traces of t y p i c a l  v a r i o u s model taps  photographs i n  range  S t a t i c Pressure  Oscilloscope signals  the  ratio  of f l u c t u a t i n g  remains  pressure  q u a l i t a t i v e l y as w e l l d u r i n g n o t i n g the  change i n t h e  Tap no.  111 , 111. Iill! 10  41  •» • i  r  LyilUULllLiailBiil  (a)  F i g u r e 2-14  T y p i c a l f l u c t u a t i n g pressure s i g n a l s from v a r i o u s model taps i n d i c a ting (a) random amplitude modulation (b) phase v a r i a t i o n  40  Figure  2-15-i  Midspan d i s t r i b u t i o n s of f l u c t u a t i n g static p r e s s u r e c o e f f i c i e n t a n d a m p l i t u d e modul a t i o n r a t i o (-45° < a < 0°)  41  P'  'max  F  Figure 2 - 1 5 - i i  3 4 5 Model contour sides Midspan d i s t r i b u t i o n s of f l u c t u a t i n g s t a t i c p r e s s u r e c o e f f i c i e n t and amplitude modulat i o n r a t i o (15° < a < 75°)  42  Figure  2-X5-iii  Midspan d i s t r i b u t i o n s of f l u c t u a t i n g s t a t i c p r e s s u r e c o e f f i c i e n t and a m p l i t u d e m o d u l a t i o n r a t i o (90° < a < 135°)  43 For of  the  near  t h e p o r t i o n o f t h e body i n t h e wake, t h e  fluctuating  the  forward  pressure  i s comparatively  stagnation point, S .  A  T  larger  similar  magnitude than  that  reduction i n  Li  C-,  occurs  S .  I t seems r e a s o n a b l e  T  the  two  i n the v i c i n i t y  s t a g n a t i o n areas  fluctuations,  from  the  of phase.  o f the  that this i s due  two  r e a r " s t a g n a t i o n " r e g i o n marked tendency  to the  cancellation  out  However, t h i s  rear  o f t h e body b e c a u s e o f i r r e g u l a r i t i e s observations indicated  the p r e s s u r e frequency  fluctuations  fluctuations  seemingly  shear  vorticity  generated  cylinder  layers  From t h e  exhibit  and  20%  of the  the  turbulence.  are at  the  Quanof  fundamental  as  are of d i f f e r e n t  from  on  180°  as  2-16).  These phase v a r i a t i o n s  are  d o w n s t r e a m and  behind the  tend  lose  to  t h e body i n c r e a s e s . i t is  observed  o p p o s i t e s i d e s of the  model  phase d i f f e r e n c e .  the  t a p s on  l a r g e as  50°  attributed  a r o u n d t h e m o d e l due  formation  the  forming  On  the  of the next  t o the  vortex  t o 100° to the shedding  core.  other  same s i d e  frequently reported i n  have p h a s e d i f f e r e n c e s  field  of  immediately  i n F i g u r e 2-14,  neighbouring  i n p h a s e as  pressure  Much o f t h e ,  s t r e n g t h s and  the d i s t a n c e behind  familiar  signals  of the  Even the v o r t i c e s ,  b u t may  flow  modulation  associated vortices.  fluctuations  are not n e c e s s a r i l y  the  pressure  h a r m o n i c component  t o the g e n e r a l i n s t a b i l i t y  sample p h a s e d a t a  the pressure  hand, p r e s s u r e  of  than  complete  i n t h e wake.  a t t h e body i s d i s s i p a t e d  through  individuality  contour  less  random a m p l i t u d e  Karman v o r t e x s t r e e t ,  that  i s less  the second  t o be  i s attributed  separated  their  of the  value.  The  the  effect  near  (  s i d e s o f t h e wake s y s t e m , w h i c h  180°  titative  o f v a n i s h i n g C-  literature, (Figure  adjustment of a  vortex  Recent measure-  44  Figure  2-16-i  Phase v a r i a t i o n o f m i d s p a n p r e s s u r e (-45° < a < 0°)  fluctuating  Figure  2-16-ii  Phase v a r i a t i o n o f m i d s p a n p r e s s u r e (-15° <_ a < 60°)  fluctuating  ments  by W i l a n d  phase  phenomenon. The  shown  spanwise  i n Figure  efficient  along  inconsistency attack,  Vickery,  40  t h e model  i n phase  flow  exists  over  circular  the order  the  correlation  the  order  in the  trated sure  i n Figure  2.4.5  grated ing  been  moment  points  are plotted  measured  lations,  phase  however,  suffers  motion.  for,a  substantial  Drag  by o t h e r  a n d Moment  moment  f o r each  i s about  take  into  flow  by  Vickery,  and i s o f  square  member  conditions,  reduction.  model  As  illus-  of the pres-  a r e i n phase  o f the model.  Similar 31  33 35 ' '  d i s t r i b u t i o n was  inte-  investigators. Coefficients  l i f t ,  angle  drag  section  the e.g. a x i s .  a t the midspan  account  increased  bodies  coefficient to indicate  difference  For  i s typically  with  modulations  the fluctuating  by t h e s t a t i o n a r y  by  correlation of  stationary  angle  I t seems  As r e p o r t e d  turbulent  the length  trend.  length  f o r sharp-edged  the amplitude  to obtain  The p i t c h i n g  and  o f t h e model.  diameters  fluctuating pressure  experienced  pressure co-  spanwise  length  under  reported  Lift,  pressure,  reported  the correlation  cylinder  and a l o n g  midspan  numerically  2-18).  the  have  that  on t h e s t a t i o n a r y  Fluctuating The  47  cylinder  However,  2-14(a),  the contour  observations  o f t h e work  a finite  improves  correlation  similar  the scatter  any d e f i n i t e  t o s i x diameters  fluctuations  around  during  stream.  spanwise  and Feng  cylinders  length  of five  a smooth  only  uniform  However,  on t h e b a s i s 46  a  p a r t i c u l a r l y a t some a n g l e s o f  o f two t o t h r e e  two-dimensionality  reasonably  length.  data,  indicated  of the fluctuating  to establish  Prendergast  stationary of  2-17, s u g g e s t  t o assume  cylinders  variations  are too large  reasonable  the  on e l l i p t i c  taps.  t h e 180° phase  Two  and p i t c h (Figure data  the effect of Both  calcu-  between t h e  50  <x = - 4 5 °  •*$  ; •  1  -30"  i  50  -15° 1  ' ' 100  o 30  15° I  o  50  « »  0 50 45°'  50  60° (r  •  1  •  i  r  1-50 75° (  1  50  90° 1  I  (  -  >  ( 1  i  •50 i  105°  •  V 135°  ,  •  T  120°.  <  1  40  41  5  40  41  Spanwise tap numbers Figure  2-17  Spanwise v a r i a t i o n o f f l u c t u a t i n g c o e f f i c i e n t and phase  pressure  F i g u r e  2-18  C o m p a r i s o n o f f l u c t u a t i n g d y n a m i c c o e f f i c i e n t s  and  s t e a d y  a e r o -  fluctuating  pressures  from the  r e s e a r c h programme was ations,  this  effect  comparison,  the  from F i g u r e  2-13  (i) (ii)  of  the  aimed a t  studying  i s not  included  i n the  distributions are  indicates  sides  not  of  the  model.  the  spanwise  vari-  above r e s u l t s .  For  static  the  Since  sectional  coefficients  included.  Examination of ients  two  the  fluctuating  forces  and  moment c o e f f i c -  that:  Cj, s  i s of  the  same o r d e r as  C^,  i s approximately  1/10  C^; of  C^;  s (iii) The  C-, m  i s about m  s  fluctuating  1/2  This  In  also  determination of i s less  coefficients.  the  Study of  less  at  the  than  the  variation  the  final  fluctuating  pressure  the  evaluation pressure  and  change  accurate  sides of  attack  45°  phase i s to  Thus,  a l o n g the  a =  fluctuating  ±10%.  each angle of  of  model the  amplitude  reveals  that  and  the  o c c u r o v e r r e g i o n s w i t h n e a r minimum  thus c o n t r i b u t i n g  little  to  the  final  C-,  summation.  Wake G e o m e t r y Using  the  g e o m e t r y d a t a was pressure except  show l a r g e  e f f e c t of  phase angle  phase d i f f e r e n c e s  values, 2.4.6  by  the  s i g n i f i c a n t i n the  phase d i s t r i b u t i o n s large  s u g g e s t e d by  general,  above c o e f f i c i e n t s  contour  .  r e a c h i n g minimum v a l u e s n e a r  t r e n d was  distributions. the  C  aerodynamic c o e f f i c i e n t s  w i t h model o r i e n t a t i o n 135°.  of  tap  for  techniques described obtained  models.  one  for  A l l tests  additional  the  in section  2.3.5, wake  1 i n . d y n a m i c and  were p e r f o r m e d a t  examination of  the  N  lateral  3 in. R  =  59,500  vortex  51 spacing  f o r the p r e s s u r e  a = 135°.  Within  significant  pressure  angles  2-19  butions  are  shows t h e  amplitude  of attack.  As  similar  =26,800  R  lateral  on  The  variations  both  the  fluctuating  wake c e n t r e l i n e  x-axis.  However, f o r t h e m o d e l s a t o t h e r a n g l e s  the  with  fluctutypical  pressure  distri-  s i d e s o f t h e wake f o r s y m m e t r i c a l l y  models.  wake i s u n s y m m e t r i c a l  o f the  a t v a r i o u s x - s t a t i o n s f o r two  expected,  no  observed.  oriented  on  with  r a n g e o f R e y n o l d s number i n v e s t i g a t e d ,  c h a n g e i n wake g e o m e t r y was  Figure ating  the  t a p a n g l e model a t N  i s then  coincident with of attack  the peaks of the p r e s s u r e  the  curves  s i d e f o r w h i c h t h e p o i n t o f s e p a r a t i o n i s most  the  higher  rearward.  67 The  experimental  vorticity  i s shed  results from  by  t h e u p p e r and  asymmetric model a t the  same r a t e .  t h e h i g h e r p r e s s u r e p e a k s on by  the  fact  F a g e and  one  Johansen lower  showed t h a t  surfaces of  T h e r e f o r e , the presence  s i d e o f t h e wake may  t h a t the  corresponding  vortex travels  s h o r t e r d i s t a n c e and  hence s u f f e r s  less  From l a t e r a l those in The  indicate  between p r e s s u r e  and  results,  and  an  o b t a i n e d as  approximate i n v e r s e  explained  dispersion.  similar  t h e d e c a y o f t h e peak p r e s s u r e  t h e d o w n s t r e a m d i r e c t i o n was curves  be  of  relatively  dissipation  pressure d i s t r i b u t i o n  i n F i g u r e 2-19,  an  to  amplitude  shown i n F i g u r e  2-20.  proportionality  downstream d i s t a n c e w h i c h a g r e e s  w i t h the 68 a n a l y t i c a l p r e d i c t i o n g i v e n by S c h a e f e r and E s k i n a z i s vortex s t r e e t model. E x p e r i m e n t a l measurements w i t h c i r c u l a r and e l l i p t i c 33 35 c y l i n d e r s by F e r g u s o n and W i l a n d , respectively, indicate 1  similar  pressure Taking  the p o s i t i o n s  decay  curves.  the peaks of the of  the  two  lateral  pressure d i s t r i b u t i o n s  v o r t e x rows, F i g u r e 2-21  shows t h e  as  stream-  52  1  Figure  2-20  V a r i a t i o n o f peak f l u c t u a t i n g downstream c o o r d i n a t e  pressure  with  Figure  2-21-ii  L a t e r a l p o s i t i o n s o f v o r t e x rows b e h i n d a n g l e models (45° <_ a < 135°)  1 i n . and 3 i n .  wise v a r i a t i o n angle  o f t h e wake d a t a  models a t v a r i o u s a n g l e s  f o r both  t h e 1 i n . and 3 i n .  of attack.  However, as p o i n t e d  69 o u t by H o o k e r pressure centres  t h e maximum v e l o c i t y  fluctuations,  do n o t o c c u r  a s some e x p e r i m e n t e r s  fluctuations  and, t h e r e f o r e ,  along the path  of the vortex  have a s s e r t e d b u t r a t h e r d e v e l o p i n  t h e n e i g h b o u r h o o d o f t h e edge o f t h e c o r e centreline. lie  inward  radius  Using  viscous vortex, Schaefer  an e x p r e s s i o n f o r c o r r e c t i n g  For  t h e angle models, t h i s  with  x  from  distance x/h  o f the pressure boundaries  of the vortex cores.  isolated at  Hence, t h e a c t u a l p o s i t i o n s  of  this  virtually x/h = 1 0 .  correction  from  of the vortex  the street centres  by an amount e q u a l  the mathematical  t o the  model o f an  and E s k i n a z i were a b l e t o a r r i v e  the experimental  measurements.  g i v e s a c o r r e c t i o n which i n c r e a s e s  z e r o a t t h e model t o l e s s I t i s apparent,  than  4% a t a  however, t h a t a t l a r g e  i s n o t a d e q u a t e as i t d o e s n o t a c c o u n t f o r  t u r b u l e n c e o r wake i n s t a b i l i t y of  farthest  w h i c h may i n f l u e n c e t h e e x p a n s i o n  the vortex cores or the p o s i t i o n  of their  centres.  This 70 7  observation  i s i n agreement w i t h  w h i c h show t h a t t h e v o r t i c e s in parallel  the experimental  measurements  do n o t f l o w downstream  rows b u t a l w a y s move away f r o m  '  indefinitely  the c e n t r e l i n e  with  increasing x e v e n when an i n t e r m e d i a t e s e q u e n c e o f v o r t i c e s have some u n i f o r m i t y o f c o n f i g u r a t i o n . From F i g u r e 2-21, t h e d i s t r i b u t i o n separation  o f t h e v o r t e x rows i s p l o t t e d  1 i n . and 3 i n . m o d e l s . the downstream d i r e c t i o n , dicated. on model  i n F i g u r e 2-22 f o r t h e  The i n c r e a s e o f t h e l a t e r a l t e n d i n g t o some u n i f o r m  I t i s shown t h a t t h e p a r a m e t e r orientation.  of the transverse  b/h  spacing i n  value, i s i n -  i s also  dependent  Figure  2-22  V a r i a t i o n of l a t e r a l vortex and 3 i n . a n g l e models  spacing  for  1 in.  58  From t h e  results,  ference  effects  ment o f  t h e wake f o r t h e  influence  on  do  on  appears  t h a t wind t u n n e l  indicated  by  l a r g e r model.  A  phase a n g l e  t a p model are  phase v a r i a t i o n that  as  wake g e o m e t r y i s p r e s e n t e d  Results pressure  exist,  i t i s apparent  the  i n Appendix I I .  2-23.  i n Figure  t o become l i n e a r  i n t h e wake f o r The  vortices  have r e a c h e d  a uniform  streaming  thereby,  constant  longitudinal  spacing.  The  tunnel  signals Using  test  s e c t i o n and  restricted  the  fact  the  pressure  in  one  the  as  shown i n F i g u r e  of  the  a/h  signals  2-24(a).  The  trend  value  velocity as  the  directly  at large  curves  by  Combining  various  x/h.  in  the  to  attain  the  spacing  x,  of the  vicinity  of  s i n c e the  2-22  and  x/h  =  2-24(a), b/a,  2.5  decrease  value.  parameter  dimensionless a  similar  v a r i a b l e s are  models  i n Figure  According  of  quite ^  t o Karman,  the  at  2-25.  to reach  ^  some l i m i t i n g  t h a t the  f o r both  b/a  then  approaches  distribution  trend i s for and  spacing  [equation(2.1)].  a t t a c k , i s summarized general  two  the  vortices  t h a t the  The  13.  =  obtained  2-24(b) e x h i b i t  S t r o u h a l number  Figures  s p a c i n g was  of attack.  of  between  consecutive  t h e body and  and,  pressure  x/h  exists  I t i s apparent  wake g e o m e t r y p a r a m e t e r ,  angles  increasing  360°  indicate  behind  shown i n F i g u r e  longitudinal  related  classical  measurements t o  curves  increases rapidly  length  fluctuating  longitudinal  i s a l s o a f u n c t i o n of angle  vortex  the  associated with  corresponding  vortices  a constant  longitudinal  of  t h a t a phase d i f f e r e n c e o f  fluctuating row,  clarity  indicating  velocity  limited  the  longitudinal  f a r downstream  the  the  confine-  d i s c u s s i o n of w a l l  distribution  presented  relative  inter-  With  a.maximum rapidly  6  b/a  should  F i g u r e  2-23  L o n g i t u d i n a l v a r i a t i o n 3 i n . a n g l e model  o f  p h a s e  a n g l e  i n wake  o f  s t a t i o n a r y  F i g u r e  2-25  L o n g i t u d i n a l d i s t r i b u t i o n s o f wake g e o m e t r y r a t i o f o r 1 i n . and 3 i n . a n g l e models  be a p p r o x i m a t e l y  equal  i s h w i t h downstream  t o 0.36  behind  t h e o b s t a c l e and t h e n  d i s t a n c e to the c l a s s i c a l  However, f o r some o r i e n t a t i o n s  dimin-  v a l u e o f 0.2 81.  o f t h e a n g l e model  an i n c r e a s e i n  5 b/a  with  x  indicating  observed.  a similar  The distances  was  Benard  as w e l l h a s p r e s e n t e d  r e v e r s a l of the t r e n d .  c o n s t a n t v a l u e s o f t h e wake g e o m e t r y v a r i a b l e s  downstream  are p l o t t e d  o f the model, r e f e r r e d  i n F i g u r e 2-26.  t o as " n e a r  The g r a p h s c l e a r l y  each o f the l a t e r a l  and l o n g i t u d i n a l  reduce  Note t h a t  the v a r i a t i o n  It  i s of i n t e r e s t  in  the v i c i n i t y  t h e use o f  in literature.  characteristics  behind  and  have been l i s t e d  at  a = -45°  a circular  bodies angle  curves f o r  non-dimensional-  angle of attack. of  (b/a)  s e c t i o n have n o t b e e n  o b s t a c l e s o f s i m p l e r shape a r e 2-1.  cylinder.  The  similarity  results with  a = 135°  However, s u c h  c a n n o t be g e n e r a l i z e d due  is  v a l u e o f 0.281.  However, numerous p u b l i c a t i o n s  i n Table  either  significantly  the d i s t r i b u t i o n  and t h e d a t a a t  on wake  available  f o r the angle  the f l a t  compares  similarity  model  plate  with  that  between  the  t o t h e complex g e o m e t r y o f t h e  section. As a f i n a l  stationary  angle  wake g e o m e t r y tation  does n o t  f o r an a n g l e  have r e a s o n a b l e  characteristics, for  that  o f Karman's s t a b i l i t y  Wake g e o m e t r y d a t a reported  e  as t h e  o f t h e wake p a r a m e t e r w i t h  to observe  the de-  s p a c i n g s a r e shown w i t h  t h e maximum o r p r o j e c t e d h e i g h t o f t h e model length.  Two  at large  infinity,"  indicate  p e n d e n c e o f wake g e o m e t r y on m o d e l o r i e n t a t i o n .  izing  data  summary o f t h e e x p e r i m e n t a l  section,  and d r a g  are presented  results  f o r the  a c o m p a r i s o n o f t h e S t r o u h a l number,  coefficient  v a r i a t i o n s w i t h model  i n F i g u r e 2-27.  It i s interesting  orient o note  F i g u r e 2-26  D i s t r i b u t i o n s of the 'near i n f i n i t y ' values of the wake survey parameters f o r 1 i n . and 3 i n . angle models  TABLE  2-1  Wake Geometry P a r a m e t e r s f o r V a r i o u s  body  investigator  circular cylinder  Benard  (a/e)„  (E)  4.3  Fage and J o h a n s e n  (E) (E)  Schaefer  and E s k i n a z i  (E)  Ferguson  (E)  (T)  Fage and J o h a n s e n (E) angle  present data  section  experiment  1.2  0.28  (u/v)„  (E) a = - 4 5  c  (T)  (  v » v)  0.14  0.23 = 1.6 1.3-1.8  = 0.32  0.80 0.225  0.24-0.28 0.82  5.50  1-7  0.31  0.20  5.45  1.54  0.283  0.229  5.25  a = 135°  (E)  m  0.32  (E)  Heisenberg  =4.8  (b/a)  0.15-0.49 average=0.32  4.27  Rosenhead and Schwabe  Karman  m  4.68-6.43  Karman (E)  normal flat plate  (b/e)  Bodies  0.77  5.58  1.77  0.32  0.77  4.64  1.36  0.29  0.79  theory <7\  66 the s i m i l a r i t y  o f the d i s t r i b u t i o n s .  near  d i m i n i s h t o almost  a =25°,  a = 60° near  and  9 0 ° , and  relationship  the remaining  drag  2.5  a = 100°  range.  This  to  attain  suggests body  I n t r o d u c t i o n of the U n i v e r s a l S t r o u h a l  several  i n v e s t i g a t o r s may  o b s e r v a t i o n s o f t h e wake-body  Concluding  r e m a r k s c a n be  be  attributed  of to  interaction.  Remarks  B a s e d on  angle  between  r e p r e s e n t a t i o n o f S t r o u h a l number as a f u n c t i o n  c o e f f i c i e n t by  similar  at  t o peak  between t h e wake g e o m e t r y and  aerodynamic c h a r a c t e r i c s . Number, and  tend  constant levels  increase rapidly  uniform values over  significant  A l l curves  the experimental  results  made c o n c e r n i n g t h e  the  following  aerodynamics of  general  stationary  sections: ( i ) The  distributions  of the  steady  drag  and  pitching  moment c o e f f i c i e n t s  o b t a i n e d from b a l a n c e  measurements  for  a n g l e m o d e l s and  commercial  the sharp-edged  the  structural  angles  variations  i n t h e a e r o d y n a m i c f o r c e s and  model o r i e n t a t i o n aeroelastic  a r e i n good a g r e e m e n t .  suggest  Significant moment w i t h  the p o s s i b i l i t y  instability.  The  compare w e l l w i t h  results,  essentially  thus  indicating  The  sharp-edged  comparable  and  of  galloping  pressure integrated,  aerodynamic c o e f f i c i e n t s  c o n d i t i o n o f t h e m o d e l and (ii)  lift,  the  steady  balance  two-dimensional  mean f l o w ,  structural  a n g l e models  S t r o u h a l number d i s t r i b u t i o n s , .  exhibit The  Strouhal  67 number v a r i e s is  significantly  essentially  independent  4 range  o f Reynolds  number o v e r t h e  4  10  t o 15x10  .  p r o j e c t e d model w i d t h  B a s i n g t h e S t r o u h a l number on reduces  of a t t a c k o n l y s l i g h t l y . wind speed,  i t s d e p e n d e n c y on  The n o n d i m e n s i o n a l  being inversely  number, i s a f u n c t i o n  related  angle  resonant  to the Strouhal  o f model a t t i t u d e h a v i n g a minimum  a = 30°.  near (iii)  w i t h model o r i e n t a t i o n b u t  The p r e s s u r e f l u c t u a t i o n s  on t h e s u r f a c e o f t h e model  change w i t h a n g l e o f a t t a c k b u t a r e i n d e p e n d e n t  of  3  Reynolds  number o v e r  vestigated. around  The v a r i a t i o n  the contour  minimum v a l u e n e a r level  close  modulation large  the range  26x10  3  t o 63x10  of the pressure  of the s e c t i o n the forward  as 2.5  indicating  coefficient  i s s i g n i f i c a n t with  The  c a n be as  s u b s t a n t i a l peak p r e s s u r e s . the  p h a s e d i f f e r e n c e between t h e two a p p a r e n t  o f t h e model c o n t o u r . differences  as l a r g e  as 50° t o 100° may  t a p s on e a c h  Fortunately,  the e f f e c t  the i n t e g r a t e d than  familiar sides  I n a d d i t i o n , however, p h a s e  neighbouring  less  random  pressure signal  The f l u c t u a t i n g p r e s s u r e d a t a c o n f i r m s 180°  unsteady  exist  side of the angle  of these force  between  section.  l a r g e phase a n g l e s  on  and moment c o e f f i c i e n t s i s  10%.  ( i v ) The s p a n w i s e d i s t r i b u t i o n coefficient  a  s t a g n a t i o n p o i n t and peak  to the p o i n t s of s e p a r a t i o n . of the f l u c t u a t i n g  in-  of the f l u c t u a t i n g  i s reasonably uniform over  pressure  the length of  68 the model examined.  However, v a r i a t i o n  p r e s e n t which would a f f e c t fluctuating  forces along  dimensionality w i l l induced (v) The  the model.  and  longitudinal  downstream function  constant  (x/h>6).  of the  spacings,  3  to  The  As  values  expected,  attitude  Of  the  60x10  3  and  dependence of the  section i s similar  drag  coefficient.  substantiates  wake s u r v e y a = -45°  results  and  distribution stability Any  design  tions of  each parameter i s a  angle  section.  the  forcing  0.281  angle  values  the  S t r o u h a l number similarity  cylinder, spacing  i s of  to those  between  ratio  at  around  The  Karman's  interest,  to minimize vortex  From t h e  The  f o r a normal  respectively.  induced  vibra-  c o n s i d e r the  f u n c t i o n , n a t u r a l frequency  of the  of  of  s e c t i o n mounted  s e c t i o n beams s h o u l d  damping.  investigation  range  aerodynamic c h a r a c t e r i s t i c s .  are comparable  criterion  of angle  structural  This variational  of the v o r t e x of  However,  the o r i e n t a t i o n  t o t h a t of the  a circular  value  body  at a large distance  'near i n f i n i t y '  f o r the  135°  p l a t e and  with  the  the e x i s t e n c e of a r e l a t i o n s h i p  t h e wake g e o m e t r y and  flat  two-  investigated.  t h e wake g e o m e t r y p a r a m e t e r s on  (vi)  together  i s i n d e p e n d e n t o f R e y n o l d s number i n t h e  angle  the  This lack of  increase r a p i d l y behind  gradually attain  26x10  of  a i d i n reducing possible vortex  the v o r t e x v e l o c i t y  it  correlation  vibrations,  lateral  and  the  i n phase i s  and  s t a t i o n a r y angle  S t r o u h a l number and  nature  inherent model  aerodynamic  characteristics, for  the f i r s t  the ranges be  the  d  ,, s  attack near  in  and  guide  C-,) m s  are near  wind speed  50° <_ a <_ 100° s h o u l d  maximum.  curve  forces  On t h e o t h e r  indicates  30° s h o u l d n o t be u s e d  of vortex vibrations.  lines  Any o r i e n t a t i o n s i n  s i n c e t h e magnitude o f t h e unsteady  resonant  onset  two c o n s i d e r a t i o n s .  - 4 5 ° <_ a <_ 10°  avoided  (Cr,. 1 s  i t i s p o s s i b l e t o suggest  hand,  that angles o f  i n order t o avoid the  Hence, t h e u s e o f a n g l e beams  open e n g i n e e r i n g s t r u c t u r e s s h o u l d i n v o l v e  sufficient  damping o r h i g h n a t u r a l f r e q u e n c y  t o prevent p o s s i b l e  structural  resonance.  failure  due t o v o r t e x  Moreover, the angle experience indicated  oscillations  section  as a b l u f f  of a galloping nature.  by t h e d i s t r i b u t i o n s  o f the steady  large variations  over  the  c o m p l e t e r a n g e o f model o r i e n t a t i o n .  ing  chapter  rence  aerodynamic data of the galloping  This i s drag  p i t c h i n g moment w h i c h e x h i b i t  the quasi-steady  may  lift,  and  steady  cylinder  theory  In the f o l l o w -  i n c o r p o r a t i n g the  i s employed t o p r e d i c t instability.  the occur-  3. 3.1  Preliminary As  DYNAMICS OF AN  Remarks  indicated  susceptible  t o two  r e s o n a n c e and dominately,  i n the i n t r o d u c t i o n , b l u f f  distinct  forms  i n one  degree  o f freedom.  position  t o s t u d y the dynamics  an a n g l e s e c t i o n ,  the wind  ranges  In a d d i t i o n , analytical  i t was  occurring, pre-  e x t e n t of the and  associated  desirable  s p e e d and and  distinct  ing  and  families  p r e d o m i n a n t l y p l u n g i n g and 1  studying,  res-  system.  Furthermore,  torsional  modes  of coupled  the o t h e r  torsion,  2 and Wardlaw.  information  both experimentally  T h e r e f o r e , i t was  about c r i t i c a l  the c h a r a c t e r i s t i c  torsional  of the  appropriate  degree o f freedom.  t h e c o n c e p t o f two  by  attack  were f o u n d t o be  confirmed  and  freedom.  the frequency of the coupled motion,  phenomenon b e t w e e n t h e p l u n g i n g and  speeds  angle of  of the  the phase  wind  of  and wake p a r a m e t e r s .  the dynamics  t o those i n the s i n g l e  to obtain pertinent  of  the e f f e c t  ranges of i n s t a b i l i t i e s  as r e p o r t e d by K o s k o  axes.  aerodynamics  to introduce certain  models t o p r e d i c t  v i b r a t i o n s , one  coupling  inertial  d e g r e e Is)  torsional  on t h e i m p o r t a n t a e r o d y n a m i c  t h e t y p e s and  similar  and  i n t e r e s t were t h e w i n d  Experimentally, and  The  of e l a s t i c  c o n d u c i v e t o model v i b r a t i o n ,  onant motion  are  t u n n e l m o d e l s were mounted on a s u p p o r t  system p r o v i d i n g p l u n g i n g and/or Of p a r t i c u l a r  cylinders  of i n s t a b i l i t y , vortex  g a l l o p i n g , with the v i b r a t i o n s  d e p e n d s on t h e r e l a t i v e Therefore,  ANGLE SECTION  features  and  degrees o f freedom,  possible  orientations,  of the  theoretically, separately.  instabilities the plung-  In g e n e r a l , the both was  v o r t e x resonance observed  and  damping  o r m o d e l s i z e was phenomena, t h u s instability.  and  divided  angle  section  in torsion  h i g h wind speed  judicious  by  a theoretical  steady  approach.  Of  study  over  damping  o f each form  the  two  of  investigated, i s not  likely  unless very  this  damping o r  This observation i s on  g r e a t e r importance w h i c h was  with  a n g l e m o d e l s a t v a r i o u s damping  that form  relatively substan-  the n o n l i n e a r , q u a s i -  i s the v i o l e n t ,  type  of  of wind  suggesting  to exhibit  low  a n a l y s i s based  of i n s t a b i l i t y  thus  inves-  and  degree  the wide ranges  resonant the  choice of  p o r t i o n s ; vortex resonance,  i s encountered.  tiated  situation  instances to i s o l a t e  observed  of  freedom a t s e v e r a l  However, f o r t h e t o r s i o n a l  model o r i e n t a t i o n  instability  A  This  c o n s i d e r a t i o n , the experimental  two  g a l l o p i n g was  a structural of  into  a combined e f f e c t  excitation.  permitting separate this  be  degree of  levels.  oscillations.  f r e e d o m , no  may  galloping  r e q u i r e d i n such  B a s e d on  t i g a t i o n was  speed  and  f o r the p l u n g i n g  orientations  galloping  instability  observed levels  vortex  and  investigated  and  model  orientations. Since quasi-steady  galloping  theory  f o r the  plunging  17 case  i s w e l l e s t a b l i s h e d , more a t t e n t i o n  t h e d e v e l o p m e n t o f an The  analogous  theory  p r o b l e m i s somewhat c o m p l i c a t e d  linear  forcing  due  i s directed  f o r the  torsional  to the  f u n c t i o n (moment) d e p e n d s on  towards  fact  the  mode.  t h a t the  instantaneous  14 angular p o s i t i o n  as w e l l as t h e v e l o c i t y .  Sisto  analyzing  the problem of s t a l l - f l u t t e r ,  cation  a s s u m i n g a mean moment d i s t r i b u t i o n  by  non-  19 and I i ,  eliminated this  compli-  as a f u n c t i o n o f  the to  instantaneous satisfy  empirical ever,  angle  approach p a r a l l e l s  since the flow  is  theory  w h i c h i n t u r n was  the c l a s s i c a l during s t a l l  flutter  chosen  point.  This  theory.  and c l a s s i c a l  How-  flutter  t h e r e i s some d o u b t as t o t h e s i g n i f i c a n c e point i n s t a l l  f o r the g a l l o p i n g  flutter  instability  analysis.  of bluff  A  cylinders  presented. This  resonant the  fields  the three-quarter chord  modified  a  t h e downwash a t t h e t h r e e - q u a r t e r c h o r d  are q u i t e d i f f e r e n t , of  of attack  chapter  outlines  and q u a s i - s t e a d y  specific  galloping  the a d d i t i o n a l  is  presented,  summarizes  the useful  under dynamical  angle  results.  A concluding  information concerning  procedure  angle  section sections  System  investigations  into  s e c t i o n were c o n d u c t e d  plunging  description  Arrangement  Model Mounting  t h e dynamic i n s t a b i l i t y  by m o u n t i n g t h e a n g l e  d e s c r i b e d , on a s u i t a b l y  schematically  A  condition.  Experimental  viously  application to  f o l l o w e d by a c o m p a r i s o n a n d d i s c u s s i o n o f t h e  and e x p e r i m e n t a l  The  theories with  i n s t r u m e n t a t i o n and e x p e r i m e n t a l  analytical  3.2.1  aspects of the vortex  c o n f i g u r a t i o n under i n v e s t i g a t i o n .  of  3.2  the important  i n F i g u r e 3-1.  and t o r s i o n a l  degrees  designed  support  o f an  models,  pre-  s y s t e m shown  The m o u n t i n g a r r a n g e m e n t p r o v i d i n g o f f r e e d o m was  essentially  49  composed o f a l a t e r a l pivots.  a i r b e a r i n g system  The s y s t e m o f j o u r n a l  encircling  the tunnel t e s t  and c r o s s - s p r i n g  a i r - b e a r i n g s l o c a t e d on a frame  section  gave t h e model a p l u n g i n g  degree o f freedom normal t o the wind d i r e c t i o n .  The c r o s s e d  Torsional pivot unit ^101 Lateral displacement  Air bearing frame  transducer  o ° o  Counter weight  Torsional damper  \  Air  bearing  frame  — Air  bearing  and  ] Wind  tunnel  • Model  Lateral  shaft,  Scale.  •i Lateral  spring  JTT ^]—*(TO BAM-I) Angular  Lateral  Torsional damper  (b) 3-1  D e t a i l s o f model s u p p o r t of freedom (a) p l u n g i n g  displacement  transducer  damper  Fiaure  shaft  s y s t e m w i t h p l u n g i n g and t o r s i o n a l d e g r e e s arrangement (b) t o r s i o n a l a s s e m b l y  74 beam p i v o t  units, providing  r o t a t i o n a l s t i f f n e s s f o r the  s i o n a l mode, were f i x e d t o t h e suitable springs  brackets  and  held  lateral  transversely  t o make,the t w o . v i b r a t i o n a l  model was L-shaped  fastened fingers.  to  the  by  bridge  respectively.  electromagnetic of  the  the  the  s y s t e m and  The  by  two  the  strain-gauge  damping was  intro-  freedom system  through  To  permit  investigation  i n d i v i d u a l degrees of  to eliminate  by  coil  pivot units  Variable  e d d y - c u r r e n t dampers..  a i r bearing  lateral  a i r - c o r e t r a n s f o r m e r and  model dynamics i n the  lateral  the  i n t o each degree of  c l a m p s were p r o v i d e d  the  angular displacements of  m o d e l were ' d e t e c t e d  duced i n d e p e n d e n t l y  by  shafts  systems independent.  f r e e ends o f  L a t e r a l and  transducers,  a i r bearing  tor-  freedom,  unwanted mode.  auxiliary  Details  instrumentation 49  d e s i g n e d by  S m i t h have b e e n r e p o r t e d  However, s i n c e particularly of  the  is  cross-spring  f o r the  pertinent  crossed parts  the  pivot  present research  constructional  Basically,  the  pivots  thin-metal  strips  i n the  50  literature.  u n i t s were  '  designed  programme a b r i e f summary  details  c o n s i s t of  rigidly  is  provided.  two  fixed.:, a t  pairs  of  uniform,  t h e i r ends t o t h e  b e t w e e n w h i c h r e l a t i v e a n g u l a r movement i s r e q u i r e d .  s i g n i f i c a n t that  damping, and  t h i s device  for small  movement  of  introduces (<15°), the  a minimum o f axis  of  two It  inherent  rotation  remains e s s e n t i a l l y s t a t i o n a r y at the i n t e r s e c t i o n of 72 f l e e t e d crossed s t r i p s . Theoretical investigations  the in  undethe 73  design  of c r o s s - s p r i n g p i v o t s 74 75 76 Haringx, Wittrick ' and e x p e r i m e n t a l i n v e s t i g a t i o n and  have been p r e s e n t e d by E a s t m a n , 77 others. Young c o n d u c t e d an a r r i v e d at s e v e r a l  empirical  75  r e l a t i o n s h i p s f o r the design c h a r a c t e r i s t i c s .  Based on  the  76 t h e o r e t i c a l development by W i t t r i c k  and the  experimental  i n f o r m a t i o n g i v e n by Young, a workable assembly was for  s u p p o r t i n g the angle models w i t h a t o r s i o n a l degree of f r e e -  dom.  The  two  beams, connected  o r t h o g o n a l l y a t 87.3% the moving end. of  a r r i v e d at  to s u i t a b l e  end b r a c k e t s , crossed  ( i . e . , 1/2(1+/5/3)) of the d i s t a n c e from  T h i s c r o s s over p o s i t i o n was  s e l e c t e d because  i t s s u p e r i o r performance c h a r a c t e r i s t i c s as p o i n t e d out by  Wittrick.  The  t h i n metal beams, made from blue tempered s p r i n g  s t e e l , have a f r e e working l e n g t h of 2.80 c r o s s - s e c t i o n a l dimensions,  in.  particular  0.030x0.500 i n . , were s e l e c t e d from  c o n s i d e r a t i o n of r i g i d i t y and s t i f f n e s s . (Figure 3-1(b)) was  The  A counter weight  added to produce a p i v o t assembly f o r which  the o v e r a l l centre of g r a v i t y of the o s c i l l a t i n g components of the p i v o t c o i n c i d e d with the i n e r t i a l a x i s of the model. 3.2.2  Instrumentation The  and T e s t  Procedures  l a t e r a l and angular displacements  of the angle models  were recorded u s i n g the i n s t r u m e n t a t i o n l a y o u t shown i n F i g u r e 3-2. to  The  top a i r b e a r i n g s h a f t was  p r o j e c t i n t o the l a t e r a l displacement  of s u f f i c i e n t length 49  transducer.  With a  10 kc s i n u s o i d a l s i g n a l , the i n t e r f e r e n c e of the aluminum s h a f t with the magnetic c o u p l i n g between the c o - a x i a l c o i l s gave r i s e to an amplitude to  the model displacement.  c i r c u i t was The  cylindrical  modulated output p r o p o r t i o n a l  A f u l l wave r e c t i f i e r and RC  used f o r demodulating the high-frequency  t o r s i o n a l displacement  filter  carrier.  of the o s c i l l a t i n g model was  measured  u s i n g a strain-gauge type of transducer i n c o r p o r a t e d i n the  76  Shaft  Lateral displacement transducer.  Function generator  F i g u r e 3-2  Instrumentation l a y o u t f o r p l u n g i n g and t o r s i o n a l displacement measurements  77 lower  cross-spring pivot  b o n d e d t o t h e beams and bridge  connected  an e l e c t r i c a l  angular displacement. recorded e i t h e r amplitude  on  The  t o form  Controlled individual  a 4-arm W h e a t s t o n e  signal  corresponding to  the  signals  were  or V i s i c o r d e r .  signals  ( u s i n g t h e rms  When  o c c u r r e d , maximum  voltmeter  circuit)  recorded. damping, i n a d d i t i o n  oscillating  eddy c u r r e n t s .  portional  gauges,  an E l l i s B r i d g e - A m p l i f i e r -  of the response  means o f e l e c t r o m a g n e t i c through  strain  model d i s p l a c e m e n t  minimum as w e l l as mean were  foil  a storage o s c i l l o s c o p e  modulation  displacements  the  Metallic  c i r c u i t in conjunction with  Meter, produced  and  unit.  support  to that inherent with  systems,  was  i n t r o d u c e d by  dampers, w h i c h d i s s i p a t e d The  amount o f m a g n e t i c  t o the i n p u t d.c.  current.  For the  energy  damping was lateral  pro-  system,  49 t h e dampers d e s i g n e d  by  Smith  mounting s h a f t s b e i n g used horseshoe  s h a p e d dampers w i t h  the magnetic assembly. each  field  A variable  ism induced  i n the  technique  dampers were  the d i s s i p a t i v e thin  copper  the  aluminum  medium.  strips  Analogous,  intersecting  i n t h e gaps were c o n s t r u c t e d f o r t h e a.c.  damper a r r a n g e m e n t t o  standard  as  were e m p l o y e d w i t h  iron of  c u r r e n t s o u r c e was erase  cores of  over  incorporated in  the u n d e s i r a b l e r e s i d u a l the electromagnets.  logarithmic decrement,  calibrated  torsional  a range of  d.c.  magnetthe  Using  the electromagnetic current using  suit-  50 able  s t r e a m l i n e d models  viscous  nature of  the  linearity  of  values  t h e damping  of  the  in place  of  the angle  m a g n e t i c d a m p i n g was  The  i n d i c a t e d by t h e  l o g a r i t h m i c decrement c u r v e s  coefficients  section.  from which  were d e t e r m i n e d .  To  the  introduce  78 f u r t h e r damping d u r i n g t h e t o r s i o n  experiments,  the  e l e c t r o m a g n e t s were r e p l a c e d by p e r m a n e n t magnets Corp.,  T y p e 6.30540) w i t h a p p r o x i m a t e  Gauss each.  The  aluminum s t r i p s  field  (Cinaudagraph  s t r e n g t h o f 5225  m a g n i t u d e o f t h e damping was of various thicknesses into  horseshoe  v a r i e d by  t h e gap  placing  between  the  poles. F r o m t h e S t r o u h a l number d a t a g i v e n i n F i g u r e 2-11 using  equation  resonance The  (2*1), the approximate  of the  1 i n . and  model o r i e n t a t i o n ,  were t h e p a r a m e t e r s s i v e p l u n g i n g and around  levels.  No  observed  d u r i n g any The  an  3-2  was  used  coincided.  f o r measuring  tigations  damping was  associated  2  (section  with  were o b t a i n e d 2.3).  A  l a y o u t s shown i n F i g u r e s 2-4  and  the f r e q u e n c i e s of v o r t e x shedding  s i g n a l s was  charts.  as t h e a m p l i t u d e  s u r f a c e p r e s s u r e s f o r the speeds  the  v o r t e x shedding frequencies  on V i s i c o r d e r  o f t h e wake g e o m e t r y , as w e l l  a t wind  and  o b t a i n e d over the wind  and  s i g n a l s were a v e r a g e d  phase o f the f l u c t u a t i n g  and  tests.  characteristics  f o r which the c y l i n d e r  m o d e l were c o n d u c t e d  and  C o r r e s p o n d i n g p h a s e d a t a between  displacement  The  conducted  or h i g h e r - harmonic resonance  of the instrument  and  ranges  f o r v a r i o u s model a t t i t u d e s  d e s c r i b e d i n Chapter  pressure  size  Exten-  measurements were  e x p e r i e n c i n g v o r t e x e x c i t e d motion  oscillation.  speed  f r e q u e n c y and m o d e l  resonant conditions with increasing  aerodynamic  cylinder  calculated.  the resonant v e l o c i t i e s .  of the amplitude  wake and  the procedure  combination  natural  amplitude  d e t e c t a b l e sub-  angle s e c t i o n  using  system  torsional  d e c r e a s i n g wind speeds  f o r vortex  3 i n . a n g l e m o d e l s were  affecting  the fundamental  wind speeds  and  Inves-  and  oscillating  c o r r e s p o n d i n g t o peak  reson-  ance.  Due  t o the dynamical  vortex velocity  was  The  final  for  the  calculated  _ / T" / f  JV  condition  v  (3.1)  the corresponding  instability,  investigation  d y n a m i c s a t v a r i o u s a n g l e s o f a t t a c k and over  occurred, and  the wind speed  further  the e x i s t i n g  characteristics initial attain  the of  intensive  a larger,  95%  signal the  3.3  to s t a r t  stable  on  final  stable  ft/sec.  oscillating The  of the  time  was  data  for galloping  initial  o b t a i n e d by  ( F i g u r e 3-3)  3 i n . d y n a m i c a n g l e model mounted a t  the  axis  spring  or  oscillator  T h i s gave an  the  shear  rest  displacement displaying  indication  instability.  model a m p l i t u d e  of r o t a t i o n  from  taken  R e s p o n s e o f an A n g l e S e c t i o n w i t h Combined and T o r s i o n a l D e g r e e s o f F r e e d o m The  increasing  a f t e r which i t would  a predetermined  amplitude  con-  oscillations  for  the  model  t h e m o d e l r e q u i r e d an  a storage o s c i l l o s c o p e .  severity  If  conducted  D e p e n d i n g on  amplitude.  of the  damping l e v e l s was  the model s t a r t i n g  a t some o r i e n t a t i o n s ,  t o b u i l d - u p from  o f the  0-70  s t u d y was  displacement.  displacement  oscillations to  range  d e c r e a s i n g wind speeds w i t h  from  valmes  model.  For the g a l l o p i n g  ducted  the  formula  ,a\  r e s u l t s were compared w i t h  stationary  section,  u s i n g the m o d i f i e d  \ r  n  of the angle  of the t o r s i o n a l  c e n t r e of the angle  section.  s t i f f n e s s e s were s e l e c t e d  a  D  system  The  Plunging  was  obtained for  = -45°  and  coinciding  p l u n g i n g and  with with  torsional  to provide a frequency  ratio  80  0.151  1  1  1  1  n = 0.01952 0 P = 0.00513 0  Sl = 2.92  0.10Motion  with  0  0.05-  0  1  2  3 u  Figure  3-3-i  4  5  y  D i s p l a c e m e n t measurements f o r a n g l e model at a = - 4 5 ° w i t h combined p l u n g i n g and t o r s i o n a l degrees o f freedom (plunging g a l l o p i n g i n i t i a t e d below t o r s i o n a l resonance) 0  81  i-ll-t  0 0.15  /  M  A  I  X  1  I  1  1 T i j  n = 0.01952  i i  P = 0.00513  i  e  0.10  e  0  Torsional resonance  '  s  1  !  '  '"e  j  I 1 |A  A maximum ? • mean T minimum  1  i  A  0.05  Motion with  j  i  2.92  j  1  A  '  A  1  f  ,  _  A A  i  •  S •  A  X  T  * ^  V  y^  AAj^-T^ « ^  V  A  A  A  Jkv*-^, 1  Figure 3 - 3 - i i  u. Displacement measurements f o r angle model a t a = -45° with combined p l u n g i n g and t o r s i o n a l degrees o f freedom (plunging g a l l o p i n g i n i t i a t e d about t o r s i o n a l resonance) G  82 representative  of t y p i c a l  structural  a n g l e beams,  (OJ_ / O J ^ 6 y n  Two  clamping  levels  as t o i n i t i a t e b e l o w and a study ance.  galloping  The  (dotted  the t o r s i o n a l  elec-  natural  and  torsional  frequency  exhibited  galloping  The  The  result  i s similar 2  Both  presence  ranges  c a n be  predominant degree  of  ampli-  i s indicated  o f the  angle the  follows:  plunging; plunging;  torsion.  observed  instability  over  to the v i b r a t i o n a l i n that  i n the  the wind speed  of the d i f f e r e n c e  To  better  frequency  characteristics  understand  investi-  resonant  i n the n a t u r a l  c a s e many d i s c r e t e  torsional  range  torsional  o b t a i n e d because of the i n f i n i t e  frequencies.  plunging  o f f r e e d o m as  wind speed principal  induced  c a t e g o r i z e d by  except  be  system  values.  uous a n g l e beams may  the  the response  s e p a r a t i o n o f t h e p l u n g i n g and  conditions,a values,  that  violent,  o f the s u p p o r t i n g  components f o r t h e  of natural galloping  mode o f v i b r a t i o n was  the  a t p a r t i c u l a r wind speeds,  t o note  induced  vortex excited  occurrence  limit  are p l o t t e d .  d i s c r e t e wind speed and  to r e s t r i c t  minimum d i s p l a c e m e n t  i s of i n t e r e s t  (i) vortex e x c i t e d  gated.  to the  displacement  t h e maximum, mean and  over  sufficient  i n Figure 3-3-ii).  oscillations  type of i n s t a b i l i t y  No  not  & = 0.14  modulation,  (iii)  This permitted  maximum damping o b t a i n a b l e w i t h  l i n e at  (ii)  speeds  reson-  dampers was  effects  so  the t o r s i o n a l  and  section  a t wind  condition.  2.92).  on  v o r t e x resonance  It  were c o n s i d e r e d  oscillations  resonant  o f the g a l l o p i n g motion  torsional  by  lateral  above t h e t o r s i o n a l  tromagnetic  tude  f o r the p l u n g i n g system  =  n  of  contin-  ranges  of  number o f  the nature  of  the  83 instability  of the r i g i d  angle  i n d i v i d u a l modes o f v i b r a t i o n f o r m e d by ly, in  thus  clamping  the  obtaining  F i g u r e s 3-4  section,  torsional  3-5.  The  amplitudes  were m e a s u r e d .  p l u n g i n g and  and  the  and  tests  were  per-  l a t e r a l systems c o n s e c u t i v e -  torsional  l a t e r a l and  were i d e n t i c a l t o t h e v a l u e s u s e d  The  i n the  amplitude torsional  f o r t h e two  d a t a as damping  degree  of  shown levels  freedom  investigation. The inertial is  results  indicate  axes f o r the  a t the n a t u r a l  that,  section  do n o t  f r e q u e n c i e s f„ n v  of  freedom.  T h i s i s due  comparatively ling  to the  s m a l l f o r an  J  f_, "e  and  the coupled  of the s i n g l e  fact that  angle  the e l a s t i c  coincide,  and  t h e r a t i o ma  section,  thus  2  motion  degrees — /I  is  Q  r e d u c i n g the  coup-  effect. Comparison of the response  system that  exhibits  p l u n g i n g motion  f o r the s i n g l e  torsional  (Figure those  icant  3-3-i), curve  motion.  The  which i s almost  by  the o t h e r hand,  do  The  induced 6 curves condition.  characteristic near  U  =1  the level similar  a s h a r p peak and  signif-  lower  damping  not resemble  of the p l u n g i n g ,  r e s o n a n t peak i s s u b s t a n t i a l l y  the modulation  of  the  are  However, a t t h e results  coupled  i d e n t i c a l to  resonant displacements  to the presence  and  the  F o r t h e h i g h e r damping  characterized  the t o r s i o n a l  On  that  the r e l a t i v e p o s i t i o n  instability.  modulation.  due  data reveals  freedom.  d e p e n d s on  i n F i g u r e 3-5,  resonant  resonant  of  3 - 3 - i i ) , the t o r s i o n a l  amplitude  (Figure  degree  amplitude  plunging galloping  to  although  reduced  the  normal  galloping i n magnitude  i s v i r t u a l l y non-existent. are a r e s u l t o f the l a t e r a l  y For h i g h e r wind speeds,  the  lateral  gallop-  Figure  3-4  Response o f dom only  angle  model  at  a  0  =  -45°  with  plunging  degree  of  free-  Figure  3-5  Response c u r v e f o r a n g l e model a t a = -45° w i t h t o r s i o n a l o f f r e e d o m o n l y and r o t a t i o n a l a x i s a t s h e a r c e n t r e Q  degree CD LT1  86 ing  i n s t a b i l i t y produces  a t o r s i o n a l motion  which i n c r e a s e s i n magnitude p r o p o r t i o n a l displacement.  However, f o r t h e l o w e r  i s c o n t r o l l e d by t h e n a t u r a l  the range  o f v o r t e x resonance  of  oscillation  the a i r - b e a r i n g  resonance.  induced  values  p l u n g i n g motion  frequency  until  U n f o r t u n a t e l y , the  allowable  displacement  d u r i n g t h e range  Therefore, the t o r s i o n a l  sent only the a n t i c i p a t e d Similar  i s exceeded.  system  y  to the plunging  torsional  r o s e t o t h e maximum  support  n  damping c a s e , t h e i n d u c e d  amplitude  plunging  of frequency f  curves  forU  for unrestricted of frequency  f  of torsional > 2.5 r e p r e -  y  galloping  mode.  i s observed a t  n  6  t h e h i g h e r damping  level  during the t o r s i o n a l  resonance.  A comment c o n c e r n i n g t h e p h a s e o f t h e c o u p l e d m o t i o n i s appropriate here. displacements ing  resonant  I t was o b s e r v e d  that  the l a t e r a l  o f t h e same f r e q u e n c y were i n p h a s e f o r t h e p l u n g and g a l l o p i n g  oscillations,  while  o n a n c e t h e modes were 180° o u t o f p h a s e . of  virtual  former  centres of rotation,  case,  the r e s u l t s  centre of rotation x/h  =30.  axis.  oscillations  ing  motion,  just  case  while  the v i r t u a l  (Figure 3-3-ii)  forward  o f the i n e r t i a l  on t h e p r e d o m i n a n t mode, t h e c o u p l e d  c a n be c o n s i d e r e d as r o t a t i o n a l  centres.  For the  f a r downstream a t a p p r o x i m a t e l y  point located  depending  that  res-  the presence  as h i n g e p o i n t s .  i n Figure 3-3-iindicate  i s located  t o a hinge  Thereby,  different  denoted  at. t o r s i o n a l  T h i s suggests  On t h e o t h e r hand, t h e l a t t e r  corresponds  and t o r s i o n a l  The f i r s t  corresponds  motion  about  to essentially  the other represents t o r s i o n .  two plung-  Thus, the p r e s e n t  measurements c o n f i r m t h e e x i s t e n c e o f t h e two f a m i l i e s o f v i r t u a l 1 2 h i n g e p o i n t s p r e d i c t e d by Kosko and o b s e r v e d by Wardlaw f o r an  87 angle and  s e c t i o n beam.  experimental  Garland  results  78  has p r e s e n t e d  similar  for a cantilever,  theoretical  extended-channel  beam. It  i s of interest  t o note,  that the experimental  r e p o r t e d by Wardlaw on t h e d y n a m i c s o f an a n g l e patible,  i n general, with  investigations. the  the trends obtained  The r e s o n a n t  vibrations  beam a r e com-  from  the present  c a n be p r e d i c t e d  S t r o u h a l d a t a , w i t h o n l y a few e x c e p t i o n s .  o f attack are predominantly  galloping. tions  over  galloping  Nevertheless, at certain d i s c r e t e wind v e l o c i t y  tially of  and t o r s i o n  and p h a s e f u r t h e r  Thereby, the type corresponding  critical  o f an a n g l e  o f freedom.  wind speed  relative  Theoretical  by s t u d y i n g  o f freedom.  t o understand  In  the nature  instabilities.  of the governing  o f m o t i o n and t h e i r  plunging  solutions  a summary o f t h e b a s i c r e l a t i o n s  linear  and t h e  Development  Derivation  with  and g a l l o p i n g  essen-  observation.  amplitude  degrees  occur  combined  The measurements  c a n be d e t e r m i n e d  approach i s adopted  the vortex resonant  equations  than  to exhibit  section with  substantiate this  system dynamics i n t h e i n d i v i d u a l  3.4  appeared  rather  the o s c i l l a -  that the o s c i l l a t i o n s  of i n s t a b i l i t y ,  what f o l l o w s , t h i s of  indicates  i n one o f t h e two d e g r e e s  frequency  the  i n nature  orientations,  ranges  a t many  characteristics.  I n summary, t h e r e s p o n s e plunging  resonant  from  In a d d i t i o n ,  Wardlaw's measurements r e v e a l t h a t t h e o s c i l l a t i o n s angles  results  and t o r s i o n a l  a r e g i v e n i n A p p e n d i x IV  provided here.  Assuming  s p r i n g and v i s c o u s damping and e x p r e s s i n g t h e a p p l i e d  88 aerodynamic f o r c e  F  o r moment M. y  equations  of motion  m  V  +  ie Evaluation  r  y  y  form,  the  =  i ^ V ' h l C ^ C y . y - t )  0.2)  k.e » i ^ t c ^ e ^ i )  +  forms the  during particular subject of this  0.3)  types of  aero-  analysis.  Resonance t h e p l u n g i n g and  forcing  torsional  equations  i s presented.  function, equation  (3.2) , i n  are  similar  Assuming  a  nondimensional  becomes  + giving  y  only the p l u n g i n g a n a l y s i s  sinusoidal form,  y  of these equations  Since in  k  +  • ce  Vortex  form,  c a n be w r i t t e n as  dynamic e x c i t a t i o n s 3.4.1  in coefficient  o  Y  +  Y  n.U Cp  =  a  the s t e a d y - s t a t e s o l u t i o n  v =  Evaluation  of  2  ^c?/(i-n;r+ (A^/i/  (3.5)  damping c o e f f i c i e n t amplitude  as  U  r  (3.4)  sint  g i v e s the  as t h e d i s t r i b u t i o n  ( f t = 1.0), v  familiar  equation  (3.5)  <3  resonance  curves  parameter.  reduces  -5)  with  For the  peak  to  79 Parkinson et a l to obtain a s o l u t i o n  considered a s l i g h t l y  of the  form  different  approach  89  Yma*  =  ^  i V  C  r  S  ' *AF  < '  n  3  which i n c o r p o r a t e s the e x p e r i m e n t a l l y observed v o r t e x phenomenon through  the v a r i a b l e s * AF  3.4.2  7 )  shedding  and w /u> n c y y  Galloping Instability 17 Using the q u a s i - s t e a d y approach,  which assumes no aero-  dynamic h y s t e r e s i s e f f e c t s i n the f o r c e and moment c h a r a c t e r i s t i c s and the v o r t e x shedding  frequency to,be  f a r removed  from  the c y l i n d e r v a l u e s , the governing d i f f e r e n t i a l equations o f motion  (3.2) and (3.3) can be put i n the nondimensional  Y  + Y  =  ® ® +  where the v a r i a b l e s i n e r t i a parameters,  y  form  f,(Y)  /^V®'®'  s  and u  y  (3.8)  Q  '  (3 9)  are m o d i f i e d mass and moment of  respectively.  The f u n c t i o n f ( Y ) , i n c o r p o r y  a t i n g the aerodynamic and v i s c o u s f o r c e s , i s r e p r e s e n t e d as a p o l y n o m i a l i n Y u s i n g the steady l i f t larly,  f  (0, 0)  and drag r e s u l t s .  i s a p o l y n o m i a l r e l a t e d t o the steady  moment w i t h an assumption  t h a t the c o n t o u r l i n e s  of C  Simipitching (0, 0)  M  8  are l i n e a r and p a r a l l e l . << 1  making equations  solution  techniques.  F o r a system i n a i r , both y  y  and p  Q  (3.8) and (3.9) amenable t o q u a s i - l i n e a r  90 3.4.2.1  Singularities  and  Analytically,  the  S t a b i l i t y i n the  Small  c o n d i t i o n of s t a b i l i t y  of the motion i n  80 the  s m a l l c a n be  determined  in  the phase p l a n e .  be  reduced  For  by  investigating  the p l u n g i n g  the  singularities  system, equation  (3.8)  can  to  v = z (3.10)  z  = -Y  +  /.Jiz)  giving  az  -Y + M/UZ)  —  =  <JY It  i s apparent  at  the o r i g i n  equation  S  (3.11)  Z  t h a t the o n l y s i n g u l a r i t y o f the phase p l a n e .  (3.9)  S  f o r the system i s l o c a t e d  Similarly,  the  torsional  becomes  X  -  X = -0  +  (3.12)  yu f (©,x) e  e  giving  x  de The  singularities  points  g i v e n by  ®n U*l-(a,C* e  Lf) 9  Again  the o r i g i n  of  the  (3.13) a r e  l o c a t e d on  the 0 a x i s at  r o o t s of the p o l y n o m i a l  + QC e-ac e 1  3  , +  r e p r e s e n t s one  ....  +  equation  (.,)-a c ®- ) = 0 N  N  the  (3.14)  H  '  o f the s i n g u l a r i t i e s .  ing  r o o t s o f the e q u a t i o n , b e i n g r e l a t i v e l y l a r g e ,  the  condition of s t a b i l i t y  are  severe.  unless  the  torsional  The  do n o t  remainaffect  disturbances  91 The  nature  of the  tories  in their vicinity  of the  system u s i n g the  M,»  =  (  for  the p l u n g i n g  (XJ for  the  Since the  ^ (  -  '  singularity  d e p e n d i n g on  (3.9),  1,  the  values  o n s e t of the  or U /s,  -  1  r e p r e s e n t the  instability.  by  the  e  t o be  ) ]  -(»+yu U c;  2  (3.16)  2  e  either  indicating  a centre or a For  the  focus  particular  singular point i s a centre. initial  wind v e l o c i t i e s  b a s e d on  the  s i g n of the  These  f o r the  A l l other values of U give r i s e  to  real part, i s  criterion  u < u0 plunging:  (3.15)  r o o t s a r e complex c o n j u g a t e s  the  Q  a f o c u s , whose s t a b i l i t y , determined  2  (3.11).  a t the o r i g i n  0  a linear analysis  roots  the magnitude of the r e a l p a r t .  c a s e when U = U  trajec-  and  ) i / [ £ ( U s - U  0  the phase  s t u d i e d through  ±/C^(U-U„)]  relation  <<  g  be  characteristic  system  torsional p  can  o )  U  = M U s - U 2  and  critical  U  s i n g u l a r p o i n t s and  stable  (3.17)  u > ue unstable  U < torsion;  U /5 0  ,.  U  > LL/*  stable  (3.18)  unstable T h e s e c o n d i t i o n s a r e b a s e d on U /s 0  are p o s i t i v e .  negative  the o r i g i n ,  the and  the  s l o p e of the on  that U  However, i t i s p o s s i b l e f o r them t o  as g o v e r n e d by  designates  the assumption  the  s i g n of the  coefficient  and  0  be  a^ , w h i c h  a e r o d y n a m i c f o r c e o r moment c u r v e  s i g n o f t h e t o r s i o n a l p a r a m e t e r s.  The  at  system i s c a l l e d because  a "soft"  for sufficiently  critical  velocity),  s e n c e o f an  arbitrarily  oscillation required to  the  by  many  may  occur  concept  of negative  about the  obtained  by  and  the a  i n pre-  other  "hard"  hand,  oscillator  However, s u s t a i n e d  disturbance exceeding This criterion  positive  stability  some  i s analogous  a e r o d y n a m i c damping  critical  given  section i n plunging,  Hartog's  9  a n a l y s i s i n the  angles  examining the  moment d i s t r i b u t i o n s angle  On  the  used  investigators.  information be  from r e s t  commence f r o m r e s t .  i s provided.  Thus, from the  can  oscillate  initial  is positive  ( g r e a t e r than  s y s t e m i s d e n o t e d as  i f an  minimum v a l u e  of U  small disturbance.  the  vibration will  i f a ^ o r a^*s  large value  the model w i l l  when i t i s n e g a t i v e , s i n c e no  oscillator  small, pertinent  o f a t t a c k and  wind  s t a t i o n a r y aerodynamic  i n Chapter stability  2, can  Figure be  2-8.  speeds force  For  the  studied using  Den  and  criterion > 0  stable (3.19)  < 0 which  i n c o r p o r a t e s the  apparent slope,  t h a t , i f the  the  angle  steady lift  unstable  lift  curve  s e c t i o n may  be  has  drag  self-excited.  a simple  study  system parameters i s r e q u i r e d .  -45°  <_ aof_  Therefore,  45°, the  s is positive s i g n of  a^•s  coefficients.  sufficiently  motion, such of the  relation  and  i s not  but  large  For  available,  the  hence  In the  becomes n e g a t i v e  depends on-a^.  As  C  It is  negative torsional direct  range for  ot  0  >  45°.  i s proportional  M  6 to  -C ,  -45°  M  the  <_ a <_ 0  system w i l l 45°  i f the  b e h a v e as  slope of C  M  a soft plot  oscillator  i n the  range  i s negative. Likewise,  for  a> Q  4 5 ° , an a n g l e s e c t i o n  t h e moment c u r v e 2-8(c),  i s a soft  i s positive.  a structural  studied  plunging  <_ 10° and 40° <_ a  0  g a l l o p i n g motion  i n Figure  from r e s t  system,  the s o l u t i o n  7  -YSJ9)  =  <_ 6 0 ° .  Q  o f Parameters.  (3.20)  6y(Y) i s a p o l y n o m i a l r e l a t e d does n o t reduce  c a s e b u t becomes a f u n c t i o n  to f (Y). y  For the t o r s i o n a l  t o z e r o as i n t h e p l u n g i n g  o f 0 and U g i v i n g  t h e s o l u t i o n as  Q * -©4(©)  The setting  0  amplitude  parameter.  o f the s u s t a i n e d motion  , of the algebraic  i s e v a l u a t e d by  the r e a l p o s i t i v e  r o o t s , Y_. o r  equations  i^(Y)  =0  S (e)  =  <- > 3 22  0  nature o f the response  criterion  frequency  Y = 0 = 0, and o b t a i n i n g  B  The  t o f ( 0 , 0 ) . The t e r m  O  (1-K.) r e p r e s e n t s a r e d u c e d 6  0j  (3.21)  6.(0) a n d K. a r e p o l y n o m i a l s r e l a t e d D  Forthe  c a n be w r i t t e n as  0  mode, h o w e v e r , 0  where  in a  and i t s s t a b i l i t y i n t h e l a r g e c a n  u s i n g t h e method o f V a r i a t i o n  (p where  T h u s , as i l l u s t r a t e d  L i m i t C y c l e s a n d B u i l d - u p Time The  be  i f the slope of  a n g l e member may o s c i l l a t e  t o r s i o n a l mode f o r - 4 5 ° <_ a  3.4.2.2  oscillator  (3.23) i s determined  by L y a p u n o v ' s  Stability  94  m M  T 5 _  plunging:  or,  -  i n the light  t > O o  stable  4©  o  unstable  of  (3.20)  and  time  and  (3.21)  (3.25)  (3.21) stable  f > o Y = Y;  i:  40  of amplitude  (3.26)  < o  J  The  unstable  3®  plunging:  torsion:  (3.24)  1  'Y = Yj  torsion:  stable  <0  build-up  by e v a l u a t i n g t h e  unstable  o  stable  o  unstable  (3.27)  i s readily  obtained  from  (3.20)  integrals  A  plunging:  (3.28)  A  (3.29)  torsion:  where are  Y , 0  some  e  id represent  0  small  initial  prescribed fractions  displacements  a n d Y.., Q..  ( s a y 9 5 % ) o f t h e Y_. , 0..  values,  respectively.  3.5  Results  3.5.1  and D i s c u s s i o n  Vortex  3.5.1.1  Model Shown  displacement peak  Resonance  vortex  Amplitude i n Figure  signals  - Velocity 3-6  are typical  obtained  resonance.  The  Measurements  f o r wind  signals  lateral speeds  indicate  and  torsional  corresponding the sinusoidal  to wave-  (b)  (a;  K/ res U  = 0  '  9 8 3  )  •.Hililli  IM  8  (C)  llllfltllllAI  •••MB! m  •  -  1,  wm  (d)  (4,/ res= U  Figure 3-6  (f)  (e) °-  9  8  0  )  T y p i c a l displacement s i g n a l s f o r angle model e x p e r i e n c i n g v o r t e x e x c i t e d plunging or t o r s i o n a l motion  form and c o n s t a n t f r e q u e n c y o f t h e d i s p l a c e m e n t s . steady-state slight  amplitude  variations  The p l u n g i n g  i s u n i f o r m a t peak r e s o n a n c e  occurring  o n l y a t wind speeds  above and b e l o w  resonance.  On t h e o t h e r h a n d , t h e t o r s i o n a l m o t i o n  substantial  random, a m p l i t u d e  may a t t a i n trated  a definite  modulation  (Figure  wind v e l o c i t y  again  shows c o n s i d e r a b l e random m o d u l a t i o n .  (Figure  f o r the t o r s i o n a l  3-7(c)), the t o r s i o n a l  a s t h e mean a m p l i t u d e s  Vortex resonance  amplitude  s u p p o r t c o n d i t i o n s a r e summarized horizontal  the  limits  dotted lines  to record  t h e average results  f o r t h e 1 i n . and 3 damping l e v e l s and  a t Y = 0.70 a n d 0 = 0.14  indicate  F i g u r e 3-8 i l l u s t r a t e s t h e a t the symmetrical  attack o f -45°.  The p l o t s  the presence  vortex resonance  and g a l l o p i n g  indicate  excitation,  damping on t h e ~ t w o t y p e s o f i n s t a b i l i t y .  ively  shifts  the v e l o c i t y amplitude.  i n model  the axis  entire  size  (i.e.,  galloping  and t h e e f f e c t o f An i n c r e a s e i n damp-  mass  curve  F o r t h e l o w e r damping l e v e l s ,  large  exceeding  the  lateral  system.  the l i m i t i n g  angle of  o f combined  parameter) to  the  effect-  right  together with a reduction i n vortex  very  since the  i n F i g u r e s 3-8 t h r o u g h 3-11.  o f t h e mounting systems.  or decrease  the  level.  plunging amplitude-velocity results  ing  illus-  displacement  of o s c i l l a t i o n ,  a n g l e models a t v a r i o u s o r i e n t a t i o n s ,  The  as  which  Thus, i n p r e s e n t i n g  mode, i t i s i m p o r t a n t  peak d i s p l a c e m e n t may be many t i m e s  in.  3-6(f)),  When f u r t h e r above t h e r e s o -  nant  maximum a s w e l l  exhibits  f r e q u e n c y a t o t h e r wind speeds  i n F i g u r e 3-7(a) a n d ( b ) .  results  w i t h some  resonant  the amplitude  displacement  on  became  attainable  Of c o u r s e , t h e a n g l e s e c t i o n  exhibits  with  97  I  min r « ii III ii mill iiin'iiiii iii in mini nm uiMI Him II II in m minim llllilIIIIM  PII nil  vv  urn  'r t.  H ^ ^ H l ^ ^ ^ n H H H r e wwWf9 in/Ww* inrwlNI WR^PI WHHHIWIi^^H  1  ii Ii •? _  ilJUUIIK (a)  (  U  9  /  •m I f f l l t f f pitiip HlijuL  T  t  m  TI  •  '|  I  I  4  ,1 i l l  MUli J J i J l d l J III (b)  U  r «  =  0  -  8  4  ° )  •r  nun  IH PT H'T if" 111 n I'li.ilit i uniJidM.ifiu^y i i L » 1  3-7  1  lill  (c) (  Figure  •i  U  e /  U  r e s = ' « )  T y p i c a l t o r s i o n a l displacement s i g n a l s f o r angle model e x p e r i e n c i n g v o r t e x e x c i t e d m o t i o n a t v a r i o u s wind speeds near resonance  98  —i  0.81  r  -45°  n  y  = 0.00648  P = 0.00618 0.4  ->—id  Figure  3-8  S e p a r a t i o n o f v o r t e x r e s o n a n c e and g a l l o p i n g t y p e o f o s c i l l a t i o n by c h o i c e o f damping o r mass parameter f o r a = -45° Q  99  F i g u r e 3-9  P l u n g i n g resonant curves f o r 3 i n . angle model at v a r i o u s o r i e n t a t i o n s  100 0.15  (a)  CX= -45 o  n = 0.01952 Q  p = 0.002055 Q  A maximum  o mean v minimum  0.5 Figure  3-10  1.0  , U /U  1.5  6 res T o r s i o n a l r e s o n a n t c u r v e s f o r 3 i n . a n g l e model a t a == - 4 5 ° w i t h a x i s o f r o t a t i o n a t s h e a r centre Q 0  101  Figure  3-11  T o r s i o n a l r e s o n a n t curves f o r 3 i n . angle model at a = -45° and 135° w i t h a x i s o f r o t a t i o n at centre of gravity G  102 v o r t e x resonance but be  the r e s u l t s  at other orientations  show t h e peak d i s p l a c e m e n t s  s m a l l e v e n a t low  forcing  function,  manner s i m i l a r  damping.  t o the unsteady  Besides  (Figure  results  of the resonant  ence o f a s i g n i f i c a n t l y  characteristic,  condition  velocity  at  a  and  the t o r s i o n a l  which of  is typical  rotation  gravity  range  c u r v e s may  ics  be  f o r the t o r s i o n a l  direct  small  3-11(a)) The  has  associated with  oscillations  spread over  resonance  The  influence  v o r t e x resonance  i t is  a  levels.  rela-  case,  to  be  apparent  at  a  =  c  2-18  135°  t h e dynamis a  moment  occurrence of  (Figure  the  3-11 (b)) i s  of the unsteady d u r i n g the  forcing stationary  (c)).  o f t h e mass and  c a n be  on  i n amplitude  The  axis  to the c e n t r e of  fluctuating  a transfer.  f u n c t i o n w i t h a n g l e o f a t t a c k as o b s e r v e d (Figure  3-10)  decrease  a similar variation  model measurements  the  w i t h the p r e s -  considerable influence  substantial  accompanying such  amplitude  indicate  T r a n s f e r o f the  (Figure  consequence o f a r e d u c t i o n i n the  coefficient  3-11  d i s p l a c e m e n t d e c r e a s e s w i t h i n c r e a s e d damping,  of a resonant condition.  o f the system.  torsional  as compared t o t h e p l u n g i n g  from the shear c e n t e r  (Figure  for  the  = -45°  0  c r i t i c a l . . From t h e d a t a g i v e n i n F i g u r e 3-10, that  in a  s h a r p peak e v e n a t m o d e r a t e damping  tively  a possibility  to the  characteristics  i n F i g u r e s 3-10  the displacement  suggesting  related  may  2-18(a)).  Furthermore, large  3-9)  vary with attitude  aerodynamic  the modulation  amplitude-velocity  (Figure  i n these cases  Being d i r e c t l y  t h e peak a m p l i t u d e s  the s t a t i o n a r y model  severity  as w e l l  summarized  damping p a r a m e t e r s  i n the  form of a  on  the  stability  22  diagram.  F i g u r e s 3-12  and  3-13  illustrate  such p l o t s  f o r the  103  Figure  3-12  S t a b i l i t y d i a g r a m f o r 3 i n . a n g l e model e x p e r i e n c i n g v o r t e x e x c i t e d p l u n g i n g motion at a = -45° c  104  Fiqure  3-13  S t a b i l i t y diagram f o r 3 i n . angle model experi e n c i n g vortex e x c i t e d t o r s i o n a l motion at a = -45° w i t h a x i s o f r o t a t i o n a t shear c e n t r e  105 3 in.angle section degrees of  at  the upper  and  a s m a l l e r angle vibrations.  The  graphs  The  the v e l o c i t y  as w e l l  Reduction  of the i n s t a b i l i t y  proportional  effectively  domain w i t h  i s apparent.  t o t h e model  reduce  the  to 8 =  3-13  0.0045 r e p r e s e n t i n g t h e v a l u e f o r w h i c h  bounds were d e t e r m i n e d .  Furthermore,  the upper  Q  non-existent.  I t s h o u l d be  noted,  amount o f damping i s r e q u i r e d  the g i v e n s e t of data, a damping was  3.value  g r e a t e r than  n e c e s s a r y w i t h an i n e r t i a  the i n v e s t i g a t i o n critical  suggests  and may  lead  that  o f the  be a  sub-  condition. of  the  parameter  of  0.01  aluminum a n g l e s .  the t o r s i o n a l  to f a i l u r e  8%  and  Hence,  however, t h a t  to reach t h i s  which i s r e p r e s e n t a t i v e of 3 i n . s t r u c t u r a l  3.5.1.2  size,  d o t t e d p o r t i o n s of the curves i n F i g u r e  Q  critical  As  resonant  s h o u l d meet a t t h e r e s o n a n t peak v e l o c i t y . 2 values of 2 3 /n U > 20, t o r s i o n a l r e s o n a n c e w i l l y o res  stantial  as  permis-  limits  virtually  Thus,  mode i s l i k e l y  to  structure.  S u r f a c e P r e s s u r e and Wake C h a r a c t e r i s t i c s A s s o c i a t e d w i t h O s c i l l a t i n g Angle Model To u n d e r s t a n d  the  variation  bounds f o r some l i m i t i n g ,  is directly  peak d i s p l a c e m e n t  be  show t h e  c o r r e s p o n d i n g wind speed,  section w i l l  torsional  2 t o 2g./n„U = 20 were o b t a i n e d by p r o j e c t i n g t h e mean 9' 6 r e s ^  extended  For  f o r t h e p l u n g i n g and  damping o r d e c r e a s i n g mass p a r a m e t e r  t h e mass p a r a m e t e r  for  and  lower v e l o c i t y  displacement.  increasing  lower  = -45°  Q  o f freedom, r e s p e c t i v e l y .  t h e peak a m p l i t u d e  sible  a  fundamental  the v o r t i c e s ,  the i n f l u e n c e  parameters,  resulting  such  unsteady  o f the resonant motion  as t h e s h e d d i n g p r o p e r t i e s s u r f a c e p r e s s u r e s and  g e o m e t r y , measurements were c o n d u c t e d  w i t h the p r e s s u r e  wake tap  on of  106 angle this  modellat form  of  a  = -45°  Q  plunging  and  plunging  case,  3-14  around  the  cylinder the  the v o r t e x from  and  torsional  c a p t u r e where t h e the  the wind speed  3-15  present  degrees  graph  of the  oscillations value.  frequency  these  over  shedding a finite  The  cylinder  frequencies  closely  over  velocity  capture  region obtained  frequency g e n e r a l l y  o f t h e undamped formation  a large velocity  f o l l o w i n g the  system. dominates  range w i t h  Strouhal curve.  the  This  tor-  s i o n a l phenomenon, i n c o n t r a s t t o v o r t e x c a p t u r e , i s d e n o t e d vortex control. division  o f the  distinctly  However, f o r cylinder  different  the v o r t e x shedding displacements  u  /  u  > r  e  s  frequency  1-2,  into  there develops  two  corresponds  having  B r a n c h 1, f o l l o w i n g  to small  amplitude  o c c u r r i n g d u r i n g motion b u i l d - u p or decay.  the  large amplitudes,  the  torsional  the  frequency  n a t u r a l frequency,  f_  as  a  branches each  periods of o s c i l l a t i o n . frequency,  the  is controlled  range of wind  t o r s i o n a l mode, however, t h e v o r t e x frequency  For  vortex  Strouhal value  cylinder  remains c l o s e to the n a t u r a l frequency  the  vortices  below t h i s  to f o l l o w the  s t a t i o n a r y model t e s t s .  the  to  f o r the  t h e phenomenon o f  Above and  tends  results  o f freedom, r e s p e c t i v e l y .  illustrates  frequency  resonant  For  range conducive  instability.  Figures  by  over  of o s c i l l a t i o n s  For  approaches  , thus momentarily  destroy-  "e ing  t h e phenomenon o f v o r t e x c o n t r o l .  shedding the  the  i n p u t of energy,  control This  initiates  that vortex  t o r s i o n a l v i b r a t i o n s which b u i l d  but  leading f i n a l l y  I t appears  larger  amplitudes  cause  loss  t o the d i m i n i s h i n g o f the  e x p l a i n s the h i g h l y modulated d i s p l a c e m e n t  of  up  with  vortex  oscillations.  signals  obtained  107  Figure  3-14  Variation quencies, resonance  o f c y l i n d e r and v o r t e x s h e d d i n g frep h a s e , and d i s p l a c e m e n t a m p l i t u d e near (a = -45°) D  108  Figure  3-15-i  V a r i a t i o n o f c y l i n d e r and v o r t e x s h e d d i n g c h a r a c t e r i s t i c s w i t h wind speed f o r a = -45° ( t o r s i o n a l a m p l i t u d e , and f r e q u e n c y r e s u l t s ) Q  109  Figure  3-15-ii  V a r i a t i o n o f c y l i n d e r and v o r t e x s h e d d i n g c h a r a c t e r i s t i c s w i t h wind speed f o r a = -45° ( f r e q u e n c y , p h a s e and mean t o r s i o n a l a m p l i tude near resonance) 0  110 for  w i n d s p e e d s w e l l above U  Figure  3-7 ( c ) .  motion  and  at u"  The  the  r e g  ,  distribution  forcing  the b e g i n n i n g is typical  by  damped, o s c i l l a t o r  the  f o r the  variations;  and  ratio  angle  amplitude,  coefficient  on  and  the  similar final  at  a  modulation  For  the  cylinder  the  math-  plunging  moment a t t h e  ratio  the  stationary  model  For  the  and  degrees over  the  phase  to almost  the  o t h e r hand, the  fluctuating  model  of  freedom,  the  entire  modulation f o r the  reduction.  still  3-16  o f t h e mean  present  The  span-  and  have  curves.  i n f l u e n c e of vortex  aerodynamic c o e f f i c i e n t s  torsional  centre of g r a v i t y  plunging values  are  to  Figure  stationary  larger  show o n l y s l i g h t  summary s h o w i n g t h e  i n F i g u r e 3-17.  = -45°.  G  torsional  amplitude  surface  corresponding  are c o n s i d e r a b l y l e s s ; while  o f p h a s e and t o the  fluctuating  the p l u n g i n g case,  the magnitude o f the unsteady  given  to p r e d i c t  i s comparatively  phase v a r i a t i o n  wise v a r i a t i o n s  section  t h e p l u n g i n g and  of the model.  A  torsional  spanwise d i s t r i b u t i o n s  t o r s i o n a l mode, t h e r e s u l t s  treads  fails  i n c l u d e , f o r comparison,  In b o t h ,  and  experiencing  i s consistent with  s t u d i e d a t the wind v e l o c i t y  pressure  surface  theory  t h e m i d s p a n and  the p r e s s u r e  c o n t r o l o f the  o f m o d e l d y n a m i c s on  fluctuating  results.  when above  approaching  phenomenon.  peak a m p l i t u d e summarizes  180°  of a l i n e a r ,  effect  p r e s s u r e s was  in-phase  and  The  the  nearly  of resonance  e m a t i c a l model, but  The  sample t r a c e i n  o f t h e p h a s e l a g between  the v o r t e x e x c i t a t i o n  vortex capture  the  f u n c t i o n v a r y i n g from  f o r c e d harmonic v i b r a t i o n . frequency  as shown by  r e s  case,  decrease  is  the unsteady  from  the  shear  lift  large  s t a t i o n a r y model r e s u l t s .  moment a b o u t t h e  resonance  On  the  c e n t r e shows  Ill  Model  Figure  3-16  contour  sides  Spanwise tap numbers  M i d s p a n and spanwise d i s t r i b u t i o n s of fluctuating pressure coefficient, amplitude modulation r a t i o and phase d u r i n g s t a t i c and dynamic c o n d i t i o n s o f the model  112  oi = - 4 5  ith <(> Wll  ACT  • O O  Q  C  without (j)  1.2  0.8  C7  C  n7 0.4  about e.g. position Stationary model  y  F i g u r e  3-17  Torsional motion at  Plunging motion at U /U =0.983 res  U  e  / res = U  0  -  9  8  0  C o m p a r i s o n o f f l u c t u a t i n g a e r o d y n a m i c c o e f f i c i e n t s f o r s t a t i o n a r y and v o r t e x e x c i t e d cond i t i o n s o f t h e a n g l e m o d e l a t a = -45° G  113 a  substantial  axis  position  amplitudes  a l l cases.  for  coefficients  torsional  distributions 3-18)  i n Chapter  ity,  a n d wake  also  show  inal  i n Figure spacing  the other  the  vortex  with  spacing. the as  static  velocity  spacing  the stationary  model  the plots  and dynamic  remain that  the e f f e c t i v e paremeter  or torsional  (Figure  of vortex  veloc-  (Figure values  3-19)  may  infinity'  be  values  conditions i s  model,  the longitud-  essentially the l a t e r a l  unchanged. spacing of  increase during  plunging  increase i n the spacing  i s used  feature  becomes  i n conjunction  nondimensionalizing blockage  reduces model  pressure  measurements  coordinate  an i n t e r e s t i n g  when  obtained  the pres-  directions  the l i m i t i n g  displacement  width  on t h e  plunging  The f l u c t u a t i n g  substantial  Nevertheless,  peak  using  o f t h e 'near  i s a simultaneous  By t a k i n g  lateral  summary  i t i s apparent  frontal  performed  For the o s c i l l a t i n g  i f the plunging  t h e model  although  experiences  There  during  and l o n g i t u d i n a l  during  3-20.  hand,  as w e l l .  apparent  A  and v o r t e x  rows  resonance. ratio  section  geometry  downstream  trends  phase  i s zero  2-18).  a result,  with  coefficient  pressure  to the stationary As  i n magnitude.  the angle  shown  On  geometry  comparable  different for  2.  = -45°, t h e  0  t h e 10% d i f f e r e n c e  = -45°.  Q  i n the lateral i n form  drag  with  i n t h e peak  a n d 3-11 (a) . A t a  with  c o n d i t i o n s was  oriented at a  are similar  presented  (Figure  moment  difference  o f the midspan  o f t h e wake  resonant  t a p model  3-10 (a)  compares  model  Investigation  sure  of pitching  o f the f l u c t u a t i n g  The e f f e c t  the stationary  and  variation  i n Figures  component  aerodynamic  This  substantiates the observed  reported  fundamental for  increase.  t o equal  t o almost result,  t h e row  (2y + h ) ,  t h e same  thus  value  suggesting  114  F i g u r e  3-18  V a r i a t i o n of a m p l i t u d e and p h a s e of f l u c t u a t i n g pressure i n wake of angle model e x p e r i e n c i n g v o r t e x e x c i t e d m o t i o n at a = -45° (a) p l u n g i n g (b) t o r s i o n Q  Figure  3-19  L o n g i t u d i n a l v a r i a t i o n o f wake s u r v e y p a r a m e t e r s f o r p l u n g i n g and t o r s i o n a l c o n d i t i o n s o f model at a = -45° D  116  r  'igure  3 = 20  Comparison of 'near i n f i n i t y ' v a l u e s of the wake s u r v e y parameters f o r s t a t i o n a r y and v o r t e x e x c i t e d c o n d i t i o n s o f the angle model a t a = -45° 0  that  the  i n c r e a s e i n wake w i d t h  amplitude  of the p l u n g i n g  is effectively  section  has  indicate  virtually  t h a t the  no  effect  and wake c h a r a c t e r i s t i c s .  noted  t h a t these  and  and wake g e o m e t r y may  the  on  and  results  n o t be  frequency  t o r s i o n a l motion o f the the v o r t e x shedding  vortex capture  the plunging resonant limited  vortex shedding  motion.  f o r the  entirely  angle  frequency  T h i s i s i n c o n t r a s t to the  e n l a r g e m e n t o f t h e wake w i d t h which occurs w i t h  by  oscillations.  I n summary, t h e wake s u r v e y investigations  caused  pronounced  phenomenon I t should  fluctuating  be  pressure  r e p r e s e n t a t i v e o f what  may  33 o c c u r a t o t h e r w i n d s p e e d s and damping c o n d i t i o n s . Ferguson, 47 74 Feng and P a r k i n s o n , e t a l , d u r i n g t e s t s on c i r c u l a r and Dsection  cylinders,  parameters near on  observed  the resonant  decrease peak.  The  the magnitude o f the o s c i l l a t i o n s  level.  S i m i l a r behaviour  where, l i k e  of the pressure  and  parameters a l s o  c o n t r o l l e d by  i s anticipated  f o r the  wake depended  the  damping  angle  section  t h e D - s e c t i o n , s e p a r a t i o n i s c o n t r o l l e d by  the  edge  geometry. 3.5.1.3  Resonant Theory P r e d i c t i o n s The  by  applicability  comparing the  o f t h e a n a l y t i c a l m o d e l s c a n be  theoretically  p r e d i c t e d and  measured v a l u e s .  However, i t may  experimental  i s i n c o r p o r a t e d i n the  the on  final the  0  validity  accuracy  consider ct  data  = -45°  of the  be  analytical  of the e x p e r i m e n t a l  the p l u n g i n g motion of the ( F i g u r e s 3-14  and  3-17).  experimentally  p o i n t e d out  solutions  angle Here  that since  theoretical  results.  rests For  calculations, partially  example,  s e c t i o n mounted 6  y  tested  = 0.00414. n  y  at = 0.00505 ,  118 c  f  /  y  Ct, x =  f  n  =  y  =  1-0.0, U / r e s U  1.057.  0.63  max  analytical tex  etical  =0.39  effects  predictions  analytical  cussion with  steady  =  C-,  =  Therefore,  t h e peak  tively.  Good  substantiates  6  with  the  n  the  not included  of discrepancy,  vor-  theor-  amplitude  that  max  the  however, the  the observed  an i n c r e a s e  d  Y  e t a l , since  Note,  contribute  a  experimental  with  3  factors  ° '  7  (3.7),  agreement  o f spanwise  the  suggest-  i n the  wind  since  tunnel  the  dis-  i n the unsteady  c o r r e l a t i o n of the to reduction  force un-  i n the ex-  0.980, from  with  consider g  Q  =  0.0235,  U = res  =  n  0.986,  equations are G  the data  T  =  Q  the experimental  =  66° and  to  (3.6) a n d  and 0.039, value  of  respec0.041  Motion Predictions  f o r Plunging  Degree  a p p l i c a t i o n of the t h e o r e t i c a l analysis as an a n g l e  oscillations,  in  0.01013,  AF  analogous 0.045  given  analyses.  Theoretical Freedom  such  Here  displacements  agreement  Galloping  The system  a n d 3-17. Q  (3.7),  3.5.2.1  also  =  A  I t i s anticipated  Lack  1.01, U / U = ' %' r e s  0.213.  3.5.2  than  t h e t o r s i o n a l mode,  f  / f n  ' $ F  6  amplitude.  3-15-ii  1  8  by P a r k i n s o n ,  II indicates  will  9  (3.6) a n d  a closer  i s one s o u r c e  aerodynamics  Figures c  are larger  confinement.  For  n  indicates  considerations.  perimental  °'  =  are incorporated.  i n Appendix  flow  res  Comparison  of additional  interference  U  equations  respectively.  the influence  wall  from  s o l u t i o n developed  capture  ing  Therefore,  a n d 0.55, of Y  value  0.983,  =  y  requires  section,  for prediction  aerodynamic  force  of  to a  physical  of the galloping  distributions at  various  orientations  as  tude-velocity  the  curves  starting velocity polynomial. velocity the  input data.  U  Q  can  be  and  the  Determination  provides  obtained  tributions 3-21)  calculation  o f t h e model b u i l d - u p  time the  of  the  6^(Y)  a p p r o p r i a t e r o o t s of the  u s e f u l information concerning  the  a c c u r a t e l y measured steady  (Figure 2-7) the  rized  f o r c e curves  f  were e x p r e s s e d  velocity  curves  i n Figure  as p o l y n o m i a l s  were o b t a i n e d . 3-22  i n the  versus  severity  Y as  amplitude  mass and  damping p a r a m e t e r s .  Figure  For 2-10  regions  The  angle  resonance together with oscillations. f i r m e d by  equation  of  where t h e 3 .  shown by  results  amplitude-  data  0  i s summa-  two  graph  f o r a set of  smaller areas  resonant  be  are  a t U*  build-up  time  equal with  velocity  in  curve  that there  transformed U*,  by  are  i s taken  the  con-  governing  system parameters initiates  a f u n c t i o n of the t o be  vertex  3-8.  In a d d i t i o n , v a r i a t i o n  i s only  from  a universal solution  a m p l i t u d e - v e l o c i t y curve  velocity  be-  introducing stretched  T* to provide  to u n i t y .  typical  and g a l l o p i n g  i n Figure  investigators,"^'"^  independent of the  Y * when Y . * Q  given  of  analysis predicts  o f combined v o r t e x  results  previous  reduced  abscissa  displacement  with  I t i s apparent  areas  v a r i a b l e s Y*>  The  a  dis-  (Figure  D  T h i s phenomenon o f combined e x c i t a t i o n s was  o f motion can  or reduced  a  drag  s e c t i o n i s s u s c e p t i b l e to pure  the e x p e r i m e n t a l  As  analytical  quasi-steady  the v o r t e x  i s also included.  i n which the  at various  a f u n c t i o n o f U and  comparison,  and  from which the  The  large regions of i n s t a b i l i t y  tween.  lift  form of a t h r e e - d i m e n s i o n a l  plunging  and  from the  ampli-  galloping oscillations. Using  two  Knowledge o f t h e model  95%  n  y  from of  the  the  initial  o r some o t h e r  con-  F i g u r e 3-21  Polynomial curve f i t of t y p i c a l  l a t e r a l force c o e f f i c i e n t  data  i—• to  o  QJ  tr,  122 stant  value of  velocity tained  Comparison o f the t h e o r e t i c a l  and b u i l d - u p t i m e  results  3.5.2.2  j * -  Y  curves with  i s presented  Plunging Amplitude Figures  3-23  t h e e x p e r i m e n t a l l y ob-  i n the f o l l o w i n g s e c t i o n . and B u i l d - u p Time  and 3-24  summarize  f o r the 1 i n . a n g l e model o r i e n t e d a t  with  v a r i o u s damping results  3-8)  obtained with  are i n c l u d e d .  sionless  levels.  velocity  the i n i t i a l a  = -45°  D  Note t h a t a s m a l l e r model extends and a m p l i t u d e  ranges,  at  indicate  tion,  = -45° and 90°  0  t h e 3 i n . model a t  instabilities.  orientations  oscillator  a  F o r comparison,  v o r t e x and g a l l o p i n g both  Measurements  the a m p l i t u d e - v e l o c i t y  data  ing  amplitude-  undergoing  besides  a discontinuous  section  (Figure  the dimen-  s e p a r a t i n g the  The e x p e r i m e n t a l  the angle  gallop-  results  t o be a  jump phenomenon.  soft In a d d i -  an i n c r e a s e i n 3 o r a r e d u c t i o n o f n delays the y y  initi-  1  ation  of galloping  larger velocity  and h a s a t e n d e n c y  and d i s p l a c e m e n t  s t a r t i n g wind v e l o c i t i e s , form to  of reduced  approximately  theory. culty of  A slight  the exact  starting  the a p p r o p r i a t e i n the  minor  fluctuations.  i s due t o t h e  velocity,  diffi-  and t h e vortex  at small displacement  and m e a s u r e d c u r v e s may  slight  i n a c c u r a c i e s i n the steady  origin  and t h e c o r r e s p o n d i n g  shed-  o f the  be a t t r i b u t e d  force distribution  polynomial  presence  and i t s  The d i f f e r e n c e between t h e p o s i t i o n s  knee o f t h e t h e o r e t i c a l  data  the quasi-steady  as t u n n e l v i b r a t i o n ,  and u n c e r t a i n t y o f damping  analysis.  confirms  i n c o n s i s t e n c y a t low Y*  i n f l u e n c e s such  toward  C o l l a p s i n g of the experimental  ding,  the  Using  the curves  are also p l o t t e d  t h e same d i s t r i b u t i o n  i n determining  extraneous  values.  the r e s u l t s  parameters.  to shift  near  the  r e p r e s e n t a t i o n used i n  to  123  Figure  3-23  Galloping amplitude-velocity model a t a = -45° and t h e i r theory Q  r e s u l t s f o r angle comparison with  124  F i g u r e  3-24  G a l l o p i n g a m p l i t u d e - v e l o c i t y r e s u l t s f o r angle m o d e l a t " a = 90° and t h e i r c o m p a r i s o n w i t h t h e o r y Q  125 F u r t h e r measurements were attack which g e n e r a l l y confirmed galloping One  instability  over  the presence  interest  the theory p r e d i c t s  experiment  a t other angles of o r absence o f the  as p r e d i c t e d by t h e t h e o r y  region of p a r t i c u l a r  where  conducted  i s the range  s u b s t a n t i a t e d the e x i s t e n c e of the g a l l o p i n g  approximately  t h e same r a n g e  were  t o g e t h e r t o f o r m one c o n t i n u o u s  connected  slightly  The b u i l d - u p t i m e = 90° a r e p r e s e n t e d  conducted  and t h e i n i t i a l  to give approximately Figure with  levels  increasing  £ y  y  amplitude (Figure  similar  characteristics.  3-25(b)),  the data  etical  prediction.  i n t h e model  displacements,  shift  and e x p e r i m e n t a l  value  i n the corresponding amplitude  time w i t h  values are not i d e n t i c a l  i n general, indicate increasing velocity.  a r e more s e v e r e  Y* .  curve  f o r t h e model  damping  i n position  d i s c o n t i n u o u s jumps d a t a may  selected  variable  results  3-24(b)).  be a t t r i b u t e d results Y  the theor-  of the theor-  (Figure  for a l l B  plot  levels  o f the experimental  differences  results,  , were  and compare w e l l w i t h  o f b u i l d - u p time  t h e Y*  Q  amplitude  time-velocity  f o r the d i f f e r e n t  The s c a t t e r  that  Y  were  of the t i m e - v e l o c i t y  In the reduced  a t s m a l l U* i s due t o t h e d i f f e r e n c e etical  The measurements  to that observed  The l o w e r  zones  f o r t h e 1 i n . a n g l e model a t  used  c o l l a p s e v e r y n e a r l y t o one c u r v e  that  region.  t h e same v a l u e o f t h e r e d u c e d  3-25(a) shows an upward  motion  i n m a g n i t u d e and t h e two  i n F i g u r e 3-25.  a t t h e damping  investigations,  larger  results  The  of angle of attack except  were  0  -15° <_ a <_ 15°  two s m a l l z o n e s o f i n s t a b i l i t y .  the amplitudes  a  ( F i g u r e 3-22) .  t o the s l i g h t and t h e f a c t  levels.  The  a monotonically decreasing build-up Therefore, galloping  a t h i g h e r wind speeds s i n c e t h e time  oscillations for build-  126 60 (a)  n = 0.000733 y  Y =0.01107 0  40  o  i  o  20  0  2  4  6  y 60  1  1  1  1  1  ./Theory  40 h  Y*= 0.00500 o  °  o  0.000844  2.20  0.00503  *  0.001313  3.52  0.00500  0.001452  3.60  0.00459  0.00177  4.94  0.00536  ?  A  (b)  %  Py  •  -  • \  20  —  X  \ .  o  -  — 1.0  1  i  1.4  i  •  1  i  1.8  1  2.2  ft  Q. 1  l  •  2.6  u Figure  3-25  C o m p a r i s o n o f e x p e r i m e n t a l and t h e o r e t i c a l a m p l i t u d e b u i l d - u p t i m e f o r a n g l e model a t a = 90° D  3.0  up  i s shorter  3.5.2.3  analysis  discussed  -45° ±  orientations damping  in  the torsional  limited  of  tunnel  a  Q  given Figure which  <_ 6 0 ° .  D  absence  support  Evaluation  moment  simplification  with  Extraneous at high  distribution  linear  introduced  instability  50  speeds  ft/sec.  c a n be  of  curves  from  C..  portion  the stationary  using  3-26  for  equations  ( 0 , 0) a s s h o w n i n are lines  and p a r a l l e l , by t a k i n g  explained  the relevant  coefficient  from  vibration of  2-7) i s r e p l o t t e d i n F i g u r e  a graph  f o r various  a variety  wind  approximately  o f t h e moment  IV g i v e s  nearly  models  F o r example,  The r e p r e s e n t a t i v e  appear  However,  be i n d u c e d . e i t h e r  system  below  theory.  i n v e s t i g a t i o n (Figure  3-27.  oscillations i n  of the galloping oscillations  aerodynamic  i n Appendix  could  amplitude.  t o a range  i s larger.  the s t a b i l i t y  self-excited a  o f freedom  initial  the quasi-steady  = -45°.  predicts  3.4.2.1,  c o n f i g u r a t i o n s , no g a l l o p i n g  and model  the tests  the steady  model  i n section  <_ 1 0 ° a n d 4 0 °  degree  large  The using  0  and mounting  or with wind  a  amplitude  f o r T o r s i o n a l Degree o f  o f t h e 1 i n . and 3 i n . angle  of  the  before  of the singularity  ranges  rest  displacement  Theoretical Predictions Freedom As  the  and t h e r e s u l t i n g  thus  of constant  C  substantiating the  as a f u n c t i o n  of 5 .  The  JXLQ  resulting  distribution  approximated  by a 1 7 t h degree  of  £ a r e shown  a  =-45°.  c  Using polynomials C  curves  o f C..  since  polynomial.  t h e moment  Only  i s symmetric  f o r 6„ a n d K  3-28 a n d  the p o s i t i v e  about  values  the origin f o r  U ) expression e . the displacement and reduced  the coefficients  8  (£) i s p l o t t e d i n F i g u r e  of the C  i n the frequency  0  c a n be d e t e r m i n e d .  Figure  3-29 s u m m a r i z e s  the  results  128  129  F i g u r e 3-27  Contour p l o t of t o r s i o n a l moment c o e f f i c i e n t as a f u n c t i o n of 6 and 0 f o r a = 45° 0  130  F i g u r e 3-28  Polynomial curve f i t of t o r s i o n a l moment c o e f f i c i e n t data f o r angle model a t a = -45° 0  Figure  3-29  V a r i a t i o n of g a l l o p i n g amplitude v e l o c i t y f o r angle section at a quasi-steady theory  0  and r e d u c e d f r e q u e n c y w i t h w i n d = - 4 5 ° as p r e d i c t e d by t h e  i—  1  132 for  the  the  system parameters.  at  3 i n . angle  this  model a t a  orientation slightly  case  1,  galloping  7.76  with  lation  oscillator  h i g h e r than  U /s 0  = 590.  The  model a t a results  From F i g u r e 3-29, nature The  of the g a l l o p i n g  damping o r r e d u c e d frequency directly  direct  galloping as  3.6  for K .  = -45°  0  examination  i n the p l u n g i n g  Concluding  following  initial will  can  for 6 , Q  the  be  obtained.  with  increased  the  as  g  by  low  interest.  Likewise,  suggested  =  0  reduced  the  the  and  is  indicated curves  torsional  or u n i v e r s a l  form  Remarks the e x p e r i m e n t a l  instability  s e c t i o n s are  galloping  types  be  and  of angle of  theoretical  made c o n c e r n i n g  and  the  the nature  of  plunging  freedom:  of a e r o e l a s t i c  combined p l u n g i n g  results  sections with  s u s c e p t i b l e to vortex  torsional  by  or  case.  and/or t o r s i o n a l degree(s) (i) Angle  U /s  f o r systems of v e r y  c o l l a p s e to a reduced  g e n e r a l r e m a r k s can  aeroelastic  of  that galloping  parameter n  of the e q u a t i o n  system does not  g i v e s an  in torsion  inertia  section  calcu-  information concerning  However, as  Q  B a s e d on  the  Similar  i n c r e a s e s w i t h wind v e l o c i t y  Q  p r o p o r t i o n a l to the  the e x p r e s s i o n by  as  shown.  of  For  velocity  parameter.  (1-K )  angle  a frequency  to higher v e l o c i t i e s  inertia  parameter  with  indicate  instability  g a l l o p i n g motion s h i f t s  t h a t the  range of p r a c t i c a l  further  values  a starting  o n l y a t v e r y h i g h wind speeds or  damping, w h i c h a r e o u t s i d e t h e  typical  the n a t u r a l frequency.  theory predicts  1 i n . angle  with  indicate  a m p l i t u d e - v e l o c i t y curve  f o r the  velocity occur  an  plots  i s a soft  oscillation the  The  = -45°  Q  induced  instabilities.  and  The  motion i n d i c a t e s  the  133 existence  o f two d i s t i n c t  substantiating lateral is  the v i r t u a l  resonance  predominantly  distinct  Therefore,  hinge  and g a l l o p i n g , plunging.  torsional  principally  centres of rotation,  about  Furthermore, essentially  the coupled  motion  occurs, the v i b r a t i o n i s  p o i n t near  the o s c i l l a t i o n s  type o f i n s t a b i l i t y  During  On t h e o t h e r h a n d , when  resonance a hinge  concept.  thus  the e l a s t i c  axis.  may be c a t e g o r i z e d by t h e  and p r e d o m i n a n t mode o f v i b r a t i o n .  the frequency the n a t u r a l  o f the coupled motion i s  frequencies of the i n d i v i d u a l  modes. In g e n e r a l , t h e p l u n g i n g - t o r s i o n a l amplitude r e sults  a r e comparable w i t h  measurements. hinge  elastic  vibration  concept,  the investigation  important  degrees  the r e s u l t i n g  associated modulation (ii)  In p l u n g i n g degree ceptible exist  t o both  ranges  Nevertheless,  the range  will  be s u b s t a n t i a l l y  o f freedom, angle  v o r t e x resonance  reduced,  s e c t i o n s are sus-  and g a l l o p i n g .  o f angle o f a t t a c k o f pure  vortex  of the g a l l o p i n g  There  reso-  excitations.  E x p e r i m e n t a l measurements a t v a r i o u s the presence  of t o r s i o n a l  a n g u l a r d i s p l a c e m e n t and  n a n c e o r c o m b i n e d v o r t e x and g a l l o p i n g  confirmed  This  t h e d y n a m i c s and  section.  when p l u n g i n g , g a l l o p i n g p r e c e d e s  approximated  o f freedom.  i n f o r m a t i o n about  aerodynamics of the s t r u c t u r a l  resonance,  o f freedom  of the aero-  o f a n g l e members c a n be  studying the i n d i v i d u a l  provides  degree  Therefore, with c o n s i d e r a t i o n o f the  virtual  by  the single  orientations  instability  134 as p r e d i c t e d by t h e q u a s i - s t e a d y tiating  the a p p l i c a b i l i t y  indicates with  of the a n a l y s i s .  or hard  i n between.  oscillator  substan-  The t h e o r y  instability  Exhibiting  characteristic,  either  the angle  section experiences  an i n c r e a s e i n g a l l o p i n g  w i t h wind v e l o c i t y ,  and u n d e r g o e s a d i s c o n t i n u o u s  at several orientations. up  reduces  the  with  severity  The t i m e  o f the g a l l o p i n g  amplitude  f o r amplitude  i n c r e a s e d wind speed, thus  jump build  increasing  instability.  Nevertheless,  a h i g h e r damping o r r e d u c e d  mass p a r a m e t e r s h i f t s t h e  amplitude  and b u i l d - u p t i m e  curves  velocity,  thus  d e l a y i n g the onset  i n g o f the experimental the  reduced  Angle  t h e damping l e v e l ,  resonant increased  the concept  of a theory.  vortex excitation  at a l l  depends lift  d i a g r a m shows r e d u c t i o n i n with  mass p a r a m e t e r ,  degree o f freedom, the v o r t e x induced  of aeroelastic  the quasi-steady  ing w i l l  Collaps-  curve i n  o f the resonance  damping o r d e c r e a s e d  However, t h e s e c t i o n  by  to a single  r a n g e and peak d i s p l a c e m e n t  n a n c e may be s e v e r e  type  of galloping.  mass p a r a m e t e r and u n s t e a d y  The s t a b i l i t y  velocity  In t o r s i o n a l  wind  as p r e d i c t e d by t h e q u a s i - s t e a d y  but the s e v e r i t y  coefficient.  toward h i g h e r  confirms  sections experience  orientations on  results  v a r i a b l e plane  universal plot  (iii)  thus  two l a r g e r e g i o n s o f g a l l o p i n g  two s m a l l e r a r e a s  a soft  theory,  occur  e v e n a t m o d e r a t e damping appears  t o be f r e e  instability.  from  levels. galloping  This i s substantiated  a n a l y s i s which p r e d i c t s  only a t r e l a t i v e l y  reso-  h i g h wind  that  gallop-  velocities  135 or  f o r systems w i t h v e r y  galloping of  the  theory  torsional  suggests  no  the  steady with  i n the  the p l u n g i n g  torsional  promising  based  modulation  plete  range of wind v e l o c i t y  conducive  The  maximum as w e l l as  recorded  failure lent, ly, ( i v ) The  the  attain  of the  may  s t r u c t u r a l members.  plunging  exhibits  resonance  c a p t u r e phenomenon where t h e  On  the  a large velocity tion  obtained  a t wind  speeds  should be many  cause  from  the  the  familiar  the vio-  vortex  frequency  a finite  torsional  of o s c i l l a t i o n . f o l l o w i n g the  is  wind  resonant  conspeed  vibration  T h i s extends Strouhal  shows  shedding over  distribu-  s t a t i o n a r y model t e s t s .  f a r above t h e  times  i s , relative-  phenomenon where t h e v o r t e x  range  be  T h e r e f o r e , the  shedding  t h e o t h e r hand, t h e  frequency  below  galloping,  c y l i n d e r motion over  a vortex control governs the  com-  frequencies, ^  type of i n s t a b i l i t y than  by  the  ex-  resonant  definite  ultimately  o f more i m p o r t a n c e  range.  to  t h e mean a m p l i t u d e s  average,  v o r t e x resonance  trolled  over  s i n c e t h e peak v a l u e s , w h i c h may  g r e a t e r than  i t s success  a t c e r t a i n w i n d s p e e d s above and  does t h e m o d u l a t i o n  res  on  i n general,  l a r g e random a m p l i t u d e  U  theory, since  analysis.  hibit  Only  frequency  e x p e r i m e n t a l l y , the q u a s i -  vortex excited o s c i l l a t i o n s ,  vibration.  the  i t i s not p o s s i b l e to  of the  observed  approach appears  The  Furthermore,  a variation  Although,  applicability  g a l l o p i n g was  damping.  o s c i l l a t i o n which i s i n c o n t r a s t to  the p l u n g i n g case. verify  low  v a l u e , the  However, cylin-  136 der  frequency  appears  that  i s a f u n c t i o n o f model a m p l i t u d e . the vortex shedding  tions but loses  control  up w i t h a r e s u l t i n g the n a t u r a l  initiates  the o s c i l l a -  as t h e model a m p l i t u d e  drop  in oscillation  torsional value.  builds  frequency t o  Nevertheless, the d i s p l a c e -  ment e v e n t u a l l y d i m i n i s h e s and t h e c y l i n d e r returns  It  frequency  to the Strouhal value.  During  p l u n g i n g o r t o r s i o n a l motion,  the d i s t r i -  b u t i o n o f t h e p h a s e l a g b e t w e e n t h e d i s p l a c e m e n t and forcing over  function varies,  approximately,  the region of vortex  tially tion,  and t o r s i o n a l  i n magnitude w i t h  and reduced  midspan t a p s .  Consequently  coefficient  center. local  from  Similar  amplitude  between  the unsteady  modula-  neighbouring lift,  d r a g and  increase during plunging  the g r e a t e r t o r s i o n a l of axis  less  phase v a r i a t i o n  L a r g e r p i t c h i n g moment a b o u t  fer  r e s u l t s , the  p r e s s u r e s on t h e p l u n g i n g model a r e s u b s t a n -  larger  pitching  0°-180°  resonance,  (v) Compared t o t h e s t a t i o n a r y fluctuating  from  the e l a s t i c  resonance  observed  the centre of gravity  resonance.  axis explains with  the trans-  t o the shear  t o t h e s t a t i o n a r y model r e s u l t s , t h e  phase d i f f e r e n c e  around  t h e model c o n t o u r  the dynamic c o n d i t i o n s has c o m p a r a t i v e l y (< 10%) on t h e s e c t i o n a l ,  unsteady  little  during effect  aerodynamic  coefficient. ( v i ) The wake s u r v e y stream  results  show s i m i l a r  coordinate f o r both  e x c i t e d models.  trend with  the stationary  During resonance  down-  and v o r t e x  i n either  degree o f  137 freedom, the v o r t e x v e l o c i t y remain e s s e n t i a l l y experiences  effective sionalized  spacing  model v a l u e .  motion has v i r t u a l l y  vortex  shedding  o r wake  s e c t i o n beams i n open e n g i n e e r i n g  of  of a vortex  structural  design  should  frequency.  failure  nondimen-  to the s t a t i o n -  no e f f e c t  on t h e  characteristics,  From t h e d y n a m i c i n v e s t i g a t i o n s ,  t o prevent  motion.  I t appears t h a t the t o r -  resonant  Therefore,  width  i s b a s e d on t h e  identical  sional  oscillations  plunging  (2y + h ) , t h e r e s u l t i n g  parameter i s almost  ary o r t o r s i o n a l  (vii)  increase with  i f the l a t e r a l  blockage  spacing  u n a l t e r e d , however, t h e wake  substantial  Nevertheless,  and l o n g i t u d i n a l  i t i s shown t h a t  angle  s t r u c t u r e s may  exhibit  resonance or g a l l o p i n g nature.  vibration  and p o s s i b l e  occurrence  o f t h e i n d i v i d u a l members,  involve sufficient  damping o r h i g h n a t u r a l  T h i s i s i n agreement w i t h  f r o m t h e s t a t i o n a r y model  study.  the  the p r e d i c t i o n  4. Possible  RECOMMENDATIONS theoretical  FOR FUTURE WORK  and e x p e r i m e n t a l  p r e s e n t work may be summarized  t o the  as f o l l o w s :  (i) A n a l y s i s o f the coupled  system d u r i n g g a l l o p i n g u s i n g ,  probably,  t h e method o f V a r i a t i o n  certainly  desirable.  force  extensions  o f Parameters i s  The e x p r e s s i o n s  f o r the aerodynamic  a n d moment r e q u i r e d i n t h e a n a l y s i s w i l l  g e n e r a l , q u i t e complex even w i t h  be, i n  the quasi-steady  approach. (ii)  Experimental firm  measurements o f t o r s i o n a l  the a p p l i c a b i l i t y  of the quasi-steady  w o u l d be a v a l u a b l e s t u d y . u s i n g an a n g l e  i n g moment d a t a  Determination unsteady  T h i s may be c a r r i e d o u t  F o r such  a study  accurate  as a pitch-  i s required.  o f spanwise v a r i a t i o n s  o f t h e s t e a d y and  a e r o d y n a m i c s a l o n g n o r m a l a n d yawed  s e c t i o n s w o u l d be u s e f u l . ating  analysis  s e c t i o n o r some s i m p l e r m o d e l s u c h  rectangular cylinder.  (iii)  g a l l o p i n g t o con-  the o v e r a l l  This i s important  three-dimensional  nature  angle i n evalu-  of the  excitation.  (iv) A flow v i s u a l i z a t i o n standing  study  should prove u s e f u l i n under-  the character of the flow,  reattachment,  such  as s e p a r a t i o n ,  wake g e o m e t r y , e t c .  (v) I n g e n e r a l , i n f o r m a t i o n c o n c e r n i n g  wind  tunnel wall  139 interference  effects  on  dynamic c h a r a c t e r i s t i c s bodies  i s lacking.  static  and  the  steady  and  unsteady  aero-  and wake g e o m e t r y f o r b l u f f  T h e r e f o r e , a systematic study  during  d y n a m i c c o n d i t i o n s o f t h e model w o u l d be  of  value. (vi)  Since angle  s e c t i o n s , when u s e d  structures,  are exposed t o atmospheric  is  o f paramount importance  the a e r o e l a s t i c conducted  turbulence, i t  to determine  vibrations.  The  i t s effect  investigation  i n the wind t u n n e l under c o n t r o l l e d  be  tests  field  aerodynamic c h a r a c t e r i s t i c s  on  steady  angle  and  a natural  Theoretical predictions ratified  exposed  by  unsteady  s e c t i o n s of  l e g s o r o t h e r s t r u c t u r a l members u n d e r  c o n d i t i o n s w o u l d be  be  terrain  should  t u r b u l e n t winds.  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Report MA-245, 1962; a l s o Movie F i l m , " A e r o e l a s t i c I n s t a b i l i t y o f Aluminum Angle S e c t i o n , " N a t i o n a l Research C o u n c i l o f Canada, A e r o n a u t i c a l Establishment. 56. P a r k i n s o n , G.V., " A e r o e l a s t i c G a l l o p i n g i n One Degree of Freedom," P r o c . F i r s t I n t . Conf. on Wind E f f e c t s on Bldgs. and S t r u c t u r e s , NPL, Teddington, Vol.11, 1965, pp.581-609. 57. P a r k i n s o n , G.V. and Modi, V . J . , "Recent Research on Wind E f f e c t s on B l u f f Two-Dimensional Bodies," I n t . Research Seminar: Wind E f f e c t s on B l d g s . and S t r u c t u r e s , NRC, Ottawa, September 196 7. 58. Whitbread, R.E., "Model S i m u l a t i o n o f Wind E f f e c t s on Structures," P r o . F i r s t I n t . Conf. on Wind E f f e c t s on Bldgs. and S t r u c t u r e s , NPL, Teddington, V o l . 1 , 1965, pp.283-306. [ " 59. Soroka, W.W., "Note on the R e l a t i o n between Viscous and S t r u c t u r a l Damping C o e f f i c i e n t s , " J . A e r o n a u t i c a l Sciences, Vol.16, 1949, pp.409-410. 60. K i m b a l l , A.L. and L o v e l l , D.E., " I n t e r n a l F r i c t i o n i n S o l i d s , " P h y s i c s Review, Vol.30, S e r i e s 2, December 1927, p.948. 61. I o k i b e , K. and S a k a i , S., "The E f f e c t o f Temperature on the Modulus o f R i g i d i t y and on the V i s c o s i t y of S o l i d Metals," P h i l . Mag., Vol.42, S e r i e s 6, 1921, pp.397-418.  146 62. O c k l e s t o n , A . J . , M i l d S t e e l Bar," pp.705-712.  "The Damping o f t h e L a t e r a l V i b r a t i o n P h i l . Mag., V o l . 2 6 , S e r i e s 7, 1938,  of a  63. B r y e r , D.W., W a l s h e , D.E. a n d G a r n e r , H.C., "Pressure P r o b e s S e l e c t e d f o r T h r e e - D i m e n s i o n a l F l o w Measurement," A e r o n a u t i c a l R e s e a r c h C o u n c i l , R. a n d M. No.3037, 1958.  64. H o e r n e r , S.F., F l u i d - D y n a m i c D r a g , pp.6-18 a n d C h a p t e r 4, p . 6 .  1965, C h a p t e r 3,  65. R o s h k o , A., "A New H o d o g r a p h F o r F r e e - S t r e a m l i n e T h e o r y , " NACA, T e c h . N o t e 3168, J u l y 1954. 66. R o s h k o , A. "On t h e Wake a n d D r a g o f B l u f f B o d i e s , " J . A e r o n a u t i c a l S c i e n c e s , V o l . 2 2 , No.2, F e b r u a r y 1955, pp.124-133. 67. F a g e , A. a n d J o h a n s e n , F.C., "The S t r u c t u r e s o f V o r t e x Sheets," P h i l . Mag., S e r i e s 7, V o l . 5 , No.28, 1928, pp.317-440.  68. S c h a e f e r , J.W. a n d E s k i n a z i , S., "An A n a l y s i s o f t h e V o r t e x S t r e e t Generated i n a Viscous F l u i d , " J . F l u i d Mech., V o l . 6 , 1959, pp.241-260. 69. H o o k e r , S.G., "On t h e A c t i o n o f V i s c o s i t y i n I n c r e a s i n g the Spacing R a t i o o f a V o r t e x S t r e e t , " P r o c . Roy. S o c . o f L o n d o n , S e r i e s A, V o l . 1 5 4 , 1936, pp.67-89. 70. F a g e , A. a n d J o h a n s e n , F.C., "On t h e F l o w o f A i r B e h i n d an I n c l i n e d F l a t P l a t e o f I n f i n i t e Span," P r o c . Roy. S o c . o f L o n d o n , S e r i e s A, V o l . 1 1 6 , 1927, pp.170-197. 71. R o s e n h e a d , L . a n d Schwabe, M., "An E x p e r i m e n t a l I n v e s t i g a t i o n o f t h e Flow B e h i n d C i r c u l a r C y l i n d e r s i n Channels of D i f f e r e n t Breadths," P r o c . Roy. S o c . o f L o n d o n , S e r i e s A, V o l . 1 2 9 , 1930, p p . l l 5 - I T 5 " : 72.  " A p p l i c a t i o n o f Spring S t r i p s t o Instrument Design," N o t e s on A p p l . S c . , No.15, 4 t h e d . , 1965, pp.2-5.  NPL,  147  .73. Eastman, F.S., "The D e s i g n o f F l e x u r e P i v o t s , " J . Aeron a u t i c a l Sciences, V o l . 5 , No.11, November 193 7, pp.16-21. 74. H a r i n g x , J.A., "The C r o s s - S p r i n g P i v o t as a C o n s t r u c t i o n a l Element," A p p l . S c i . Res., V o l . A l , 1947-49, pp.313-332. 75. W i t t r i c k , W.H. "The Theory o f S y m m e t r i c a l C r o s s e d F l e x u r e P i v o t s , " A u s t r a i l i a n J . S c i . Res., A, V o l . 1 , No.2, 1948, pp.121-134. 76. W i t t r i c k , W.H., "The P r o p e r t i e s o f C r o s s e d F l e x u r e P i v o t s , and t h e I n f l u e n c e o f t h e P o i n t a t which t h e S t r i p s C r o s s , " The A e r o n a u t i c a l Q u a r t e r l y , V o l . 1 1 , F e b r u a r y 1951, pp.272-292. 77. Young, W.E., "An I n v e s t i g a t i o n o f t h e C r o s s - S p r i n g J . A p p l . Mech., V o l . 1 1 , June 1944, pp.A113-A120.  Pivot,"  78. G a r l a n d , C.F., "The Normal Modes o f V i b r a t i o n s o f Beams Having N o n c o l l i n e a r E l a s t i c and Mass Axes," J . A p p l . Mech., ASME T r a n s . , V o l . 6 2 , 1940, pp.A97-Al05. 79. P a r k i n s o n , G.V., Feng, C.C. and Ferguson, N., "Mechanisms of V o r t e x - E x c i t e d O s c i l l a t i o n o f B l u f f C y l i n d e r s , " Symposium on Wind E f f e c t s on B l d g s . and S t r u c t u r e s , NPL/Roy. Aeron a u t i c a l S o c i e t y , London, March 1968. 80. Cunningham, W.J., I n t r o d u c t i o n t o N o n l i n e a r A n a l y s i s , M c G r a w - H i l l , New York, 195 8. 81. P a n k h u r s t , R.C. and H o l d e r , D.W., Wind Tunnel Technique, Pitman and Sons L i m i t e d , London, 1952, Chapter 8. 82. G l a u e r t , M., "The I n t e r f e r e n c e o f Wind Channel W a l l s on t h e Aerodynamic C h a r a c t e r i s t i c s o f an A e r o f o i l , " B r i t i s h A.R.C., R. and M. 867, March 1923; and, B r i t i s h A.R.C., R. and M. 889, 1923-24. 83. Fage, A., "On t h e Two-Dimensional Flow P a s t a Body o f S y m m e t r i c a l C r o s s - S e c t i o n Mounted i n a Channel o f F i n i t e B r e a d t h , " B r i t i s h A.R.C., R. and M. 1223, 1928-29.  1 4-J  '84. Durand, W.F., Aerodynamic Theory, Durand R e p r i n t i n g Committee, C a l i f o r n i a , V o l . I l l , D i v . I , P a r t 1, Chapter I I I , " I n f l u e n c e on the Dimensions of the A i r Stream,'' 194 ->, pp.280-319. 85. A l l e n , H.J. and V i n c e n t i , W.G., "Wall I n t e r f e r e n c e i n a Two-Dimensional-Flow Wind Tunnel, w i t h C o n s i d e r a t i o n o f the E f f e c t of Compressibility," NACA 13th Annual Report, No. 782, 1944, pp. 155-184. 86. Lock, M., "The I n t e r f e r e n c e o f a Wind Tunnel on a Symmetr i c a l Body," B r i t i s h A.R.C., R. and M. 1275, 1929. 87. G l a u e r t , H., "The C h a r a c t e r i s t i c s o f a Karman Vortex S t r e e t i n a Channel o f F i n i t e Breadth," B r i t i s h A.R.C., R. and M. 1151, 1928-29. 88. M a s k e l l , E.C., "Theory o f the Blockage E f f e c t s on B l u f f Bodies and S t a l l e d Wings i n a C l o s e d Wind Tunnel," British A.R.C., R. and M. 3400, November 1963. . 89. Bearman, P.W., "The Flow Around a C i r c u l a r C y l i n d e r i n t h e C r i t i c a l Reynolds Number Regime," NPL, Aerodynamics D i v . , Report No. 1257, January 1968. 90. Jones, W.P., "Wind Tunnel I n t e r f e r e n c e E f f e c t s on t h e Values o f E x p e r i m e n t a l l y Determined D e r i v a t i v e C o e f f i c i e n t s for O s c i l l a t i n g Aerofoils," B r i t i s h A.R.C., R. and M. 1912, 1943. 91. Jones, W.P., "Wind Tunnel I n t e r f e r e n c e E f f e c t s on M e a s u r e ments o f Aerodynamic C o e f f i c i e n t s f o r O s c i l l a t i n g A e r o f o i l s , " B r i t i s h A.R.C., R. and M. 2786, 1958. 92. Acum, W.E.A., "Wall C o r r e c t i o n s f o r Wings O s c i l l a t i n g i n Wind Tunnels o f C l o s e d Rectangular S e c t i o n , " "Part I - T h e o r y and T a b l e s , " B r i t i s h A.R.C., Report No. 19593, O c t o b e r 1957; and, "Part I I - A p p l i c a t i o n t o a D e l t a Planform i n a 9 x 7 tunnel," B r i t i s h A.R.C., Report No. 19756, J a n u a r y 1958. 93. Runyan, H.L., Woolston, D.S. and Rainey, A.G., "Theoretical and E x p e r i m e n t a l i n v e s t i g a t i o n o f the E f f e c t of T u n n e l W a l l s on the Forces A c t i n g on the O s c i l l a t i n g A i r f o i l i n TwoDimensional Compressible Flow," NACA, Report No. 1262,1956.  149  '94.  M o l y n e u x , W.G., Measurements," M a r c h 1964.  "Wind T u n n e l I n t e r f e r e n c e i n Dynamic R.A.E., T e c h . Memo. No. S t r u c t u r e s 604,  95.  R o b e r t s , B.W., "The N a t u r e o f t h e F r e e - S t r e a m C o n d i t i o n s A p p r o a c h i n g a n O s c i l l a t i n g Body i n a Low Speed Wind T u n n e l , " The A e r o n a u t i c a l Q u a r t e r l y , V o l . 1 7 , P a r t 4, November 1966.  96.  R o s s , D., " V o r t e x - S h e d d i n g Sounds o f P r o p e l l e r s , " B e r a n e k a n d Newman, R e p o r t No.1115, M a r c h 1964.  97.  Cheng, S., "An E x p e r i m e n t a l I n v e s t i g a t i o n o f t h e A u t o rotation of a Flat Plate," Univ. o f B r i t i s h Columbia, M.A.Sc. T h e s i s , November 1966.  Bolt,  APPENDIX I Geometrical  The  Properties  cross-sectional features  somewhat  'idealized' since  smooth.  T h i s was  the  other  well  as  the  are w e l l  features  commercially  but  included  various  do  commercially  1-1  wind t u n n e l  t a b l e were n o t  On  compares  the  tested during  The  extending  sections  experimentally  geometric  and  the  across  four angle  tested  s t r u c t u r a l angles  as  models i n c l u d i n g  f o r f u r t h e r comparison of the available  and,  have r o u n d e d edges  Table  of the wind t u n n e l . the  of  sharp-edged models.  angles  1-1).  surfaces  the  features  wind  models.  tap,  as w e l l  with  the the  ever,  of the  the  a v a i l a b l e s t r u c t u r a l angles  Survey of  for  (Figure  bottom of  of commercially tunnel  the  construction  Nevertheless,  a p p r o x i m a t e d by  height  a t the  are  the  A l l m o d e l s were s i m i l a r i n l e n g t h  effective  listed  facilitate  hand, s t r u c t u r a l s t e e l  experiment. the  d e s i g n e d models were  edges were s h a r p and  h o l l o w models.  rough s u r f a c e s  geometric  to  of the  Members  aluminum a n g l e s e c t i o n s have s i m i l a r geometry  therefore, the  their  necessary  thin-walled,  available  of Angle Section  as  the  tabulated  1 i n . and  structural  importance  3 i n . dynamic models compare  corner  on  difference i n inner since  the  almost stagnant. substantiated  this  i n d i c a t e s t h a t the  aluminum a n g l e s  f i l l e t e d inner  this  data  the  area  visualization  observation.  similar size  favourably except  c o m m e r c i a l members.  radius w i l l  f l u i d near t h i s  Flow  of  pressure  be on  using  of minor the  aerodynamic  section  a smoke  How-  is  tunnel  F i g u r e 1-1  C r o s s - s e c t i o n s of angle member (a) ' i d e a l i z e d ' angle s e c t i o n (b) t y p i c a l s t r u c t u r a l angle  Geometric Model d in.  TABLE 1-1 Features o f Angle  Sections  Nominal dimensions length t . o . i in. in. in. in.  e.g. a in.  I shear X centre 4 a - i n . in.  R  R  system m lbm/ft  system  I Material  1  lbm-in?/ft  pressure tap  3  1/2  26  3/4  0  0  0.93  0.96  2.3  0.91  8.2  aluminum,acrylic hollow section  large dynamic  3  1/2  26  3/4  0  0  0.92  0.96  2.1  0.71  7.9  acrylic section  small dynamic  1  26  3/4  0  0  0.31  0.32  0.027  0.70  7.0  aluminum section  solid  1/6  balance A  1/2  27  0  0  0.90  0.93  1.95  2.92  7.2  aluminum section  solid  3  balance B  Identical  t o model A e x c e p t  f o r 1/4 i n . t h i c k  hollow  end p l a t e s  balance C  1/2  27  1/8  5/16  0.93  0.98  2.2  9.4  24.0  structural angle  steel  3  balance D  1/4  27  1/16  5/16  0.84  1.03  1.2  4.9  13.4  structural angle  steel  3  balance E  3  1/4  27  1/64  7/32  0.84  1.01  1.2  1.71  4.5  s t r u c t u r a l aluminum a n g l e  balance F  2  1/3  27  0  0  0.62  0.64  0.44  1.42  1.6  aluminum section  -  1/2  -  1/64  1/4  0.93  0.96  2.2  3.30  8.3  structural inum  alum-  3  -  3/16  -  1/16  3/16  0.31  0.32  0.030  1.16  0.32  structural angle  steel  1  -  1/8  -  1/16  3/16  0.30  0.33  0.022  0.80  0.24  structural angle  steel  1  -  1  3/16  -  1/6 4  5/32  0.31  0.32  0.030  0.41  0.11  s t r u c t u r a l aluminum a n g l e  solid  (— 1  to  APPENDIX Wind  There wind  tunnel  steady  is a  In  investigations  with  the  These and  of  considerable  the  general,  express model  the  the  to  analysis confined  be  Corrections  body  of  information  wall  on  wall  the  first  found  as  a  of  cylinders,  confinement  or  but  the  8 1  8  7  e  t  a  models  in  series  in  dimension  second  ascending ratio  order  i n most  terms. situations  confidence. there  i s some  theoretical  experimental  analysis is  81 relatively provide bluff More flow, a  less  techniques  circular  i s the  Maskell's tions  to  it  the  near drops  the to  Pankhurst  and  under  but  their  and body  applicability  Strouhal flow  critical  model,  number  around  a  and  Vickery  analysis  i s rather  to  limited.  force  For  where  the  blockage  a  and  circular  drag  correc-  applied cylinder  coefficient  making the s o l i d blockage as 89 i m p o r t a n t as wake b l o c k a g e , Bearman has s u g g e s t e d the a p p l i c a t i o n 85 of t h e method o f A l l e n and V i n c e n t i developed f o r low drag, 87 83 streamlined  airfoils.  theoretically,  the  0.2  l i f t  cylinder.  number,  completely  obtained  fluctuating  square  Reynolds  approximately  c o n d i t i o n of  Durand  separated 5 8 s i m p l i f i e d e x p r e s s i o n q u o t e d by W h i t b r e a d which i s 88 c a s e o f a g e n e r a l a n a l y s i s by M a s k e l l . Using 40  mathematical  this  84  Holder,  for extrapolating streamline  cylinders  appropriate,  particular  to  complete.  on  l  semi-empirical  satisfactory  some m e a s u r e  stationary bluff  information  effects  and  characteristic  to  been  applied with  For  theoretical  tunnel  c o r r e c t i o n s have  can  Wall  interference for stationary streamlined  flow.  powers  Tunnel  II  thus  Glauert  two-dimensional  and  Fage  flow  have  around  considered,  cylinders  in  154  channels  of f i n i t e  R o s e n h e a d and information  breadth.  The  S c h w a b e , ^ and  concerning  the  F a g e and  Test results  during  of t h i s  course  on  stationary  regards  by  useful on  a n g l e models  flow  taken  infor-  t o the departmental  wind  system. The  is  Johansen^ provide  r e s e a r c h programme, gave some  m a t i o n on w a l l i n t e r f e r e n c e w i t h tunnel  measurements  i n f l u e n c e of w a l l confinement  characteristics. the  experimental  analysis  o f w a l l i n t e r f e r e n c e on o s c i l l a t i n g  c o n s i d e r a b l y more c o m p l e x , and  predicting  the  Nevertheless,  to date, e f f e c t i v e  c o r r e c t i o n s have n o t b e e n c o m p l e t e l y for oscillating  airfoils  models  methods f o r evolved.  i n wind t u n n e l s ,  there 90-93  are v a r i o u s t h e o r e t i c a l et a l 94 Molyneux On  has  and  experimental  presented  a good r e v i e w  the o t h e r hand, f o r o s c i l l a t i n g  wall  confinement  i s almost  approaches  bluff  completely  reported  of t h i s  bodies,  literature,  i n f o r m a t i o n on  lacking.  81 As from wind divided  indicated  Pankhurst  and  tunnel w a l l s during steady  Holder,  the i n t e r f e r e n c e  f l o w c o n d i t i o n may  be  into:  (i) (ii) (iii) (iv)  solid  blockage;  wake  blockage;  lift  or c i r c u l a t i o n  boundary  layer  (v) s t r e a m w i s e For  an  not  exist.  effect;  interference;  static  and  pressure gradient  o b j e c t s y m m e t r i c a l l y p l a c e d irt t h e  Columbia, is  by  For  compensate  flow f i e l d ,  the wind t u n n e l a t the U n i v e r s i t y  the w a l l boundary  relatively  influence.  small partly  f o r boundary  layer due  layer  of  t h i c k n e s s i n the  to the growth.  filleted  ( i i i ) does British  test  corners  Furthermore,  the  section which pressure  155 integrated 2-13)  and b a l a n c e m e a s u r e d a e r o d y n a m i c c o e f f i c i e n t s  showed good c o r r e l a t i o n .  negligible.  S i m i l a r l y , f o r the tunnel  minor s i g n i f i c a n c e  ( l e s s than 1%).  blockages represent characteristics. stationary following  Therefore,  angle  ( i v ) a p p e a r s t o be  (v) was f o u n d t o be o f  Therefore,  t h e major i n t e r f e r e n c e s  Available  (Figure  corrections  solid  and wake  on model  appropriate  aerodynamic f o r the  s e c t i o n measurements a r e summarized  i n the  pages.  (1)  Mean F r e e  Stream V e l o c i t y 81  According correction  i s a sum o f t h e s o l i d y  where V  °s  =  +  velocity;  °w '  c  o  r  r  e  ^  c  o  n  value;  f a c t o r d e p e n d i n g on model and  geometry.  Another estimate o f the v e l o c i t y c o r r e c t i o n tained  from M a s k e l l ' s  Illustrated  i n Figure  the  I I - l , i s a c o m p a r i s o n between t h e p e r c e n -  is  given  by t h e above two methods when  t o t h e 3x3 i n . a n g l e m o d e l s .  two c o r r e c t i o n s more a p p l i c a b l e  for  separated  and  Vincenti  Pankhurst  c a n be ob-  s i m p l i f i e d r e l a t i o n (equation ( 2 ) ) .  tage v e l o c i t y c o r r e c t i o n s applied  by (1)  stream o r c o r r e c t e d  tunnel  speed  and wake b l o c k a g e s g i v e n  V = measured a p p r o a c h i n g v e l o c i t y °  t h e wind  ( I + cr )  =  F  = free  p  t o P a n k h u r s t and H o l d e r ,  c a n be e x p l a i n e d to streamlined  flow c o n d i t i o n . as s t a t e d  The d i s c r e p a n c y by t h e f a c t t h a t  bodies while  between the former  the l a t t e r i s  The v e l o c i t y c o r r e c t i o n o f A l l e n  by Bearman i s i d e n t i c a l t o  that  of  and H o l d e r .  (2)  Fluctuating  F r e e Stream  Conditions  95 Roberts  investigated  the flow c o n d i t i o n s  ahead o f an  ~T  Velocity  (Pankhurst  Velocity  ( Maskell )  Strouhal  \  1  1  1  and  Holder)  number , S,  \  Lateral vortex spacing  -45  Figure  135  I I - l  Percentage c o r r e c t i o n applicable t o 3 i n . angle t e s t e d i n departmental wind tunnel  section  157 oscillating  model by  equation.  Two-dimensional c i r c u l a r  harmonically  i n the  .flow v a r i a b l e s of and  u s i n g the unsteady  static  For  f u n c t i o n s of the amplitudes  enon t o be  T h u s , one  would expect  thin  airfoil  of  angle  velocity ever,  and  diameter,  the v e l o c i t y  l a r g e as the  s e c t i o n s , both  o f a t t a c k by  field  20  and  40  ;  percent,  t i m e - d e p e n d e n t phenomlow  frequency,  D u r a n d , as w e l l as  c o n s i d e r i n g an e q u i v a l e n t  s e c t i o n no  Allen  for wall interference corrections  o f a s y s t e m o f images on  for a bluff  amplitude  Attack  Vincenti provide expressions the  upstream  exist.  (3) E f f e c t i v e A n g l e For  Bernoulli's  oscillating  showed t h e  s i g n i f i c a n t where l a r g e a m p l i t u d e ,  model o s c i l l a t i o n s  to  o f one  o f the  frequency  p r e s s u r e v a r i a t i o n s were as  respectively.  and  cylinders  streamwise d i r e c t i o n  t o be  the motion.  form  induced  the base p r o f i l e .  appropriate analysis  How-  i s apparently  available. (4) Mean S t a t i c  Pressure  and  R e s u l t a n t Steady  Forces  88 Using the  steady  corrected  the  static  free  pressure  =  p  C  p  drag  c  i-c V  and  on b l u f f  has  shown t h a t  s t r u c t u r e s can  be  as f o l l o w s :  ' - c  where C_  s t r e a m l i n e model, M a s k e l l  ,  , C  D  c  =  D  = free °F  I  -FT  drag  stream  ,  c  (2)  ~  0  c  s c  or c o r r e c t e d s t a t i c  coefficients,  = measured s t a t i c  pressure  and  respectively;  pressure  and  drag  coefficients;  158 C  = free  stream  base p r e s s u r e  coefficient,  2  -(k - 1 ) ;  % S = m o d e l a r e a on w h i c h C C = tunnel test  i s based;  Q  section area. 40  This  i s identical  t o the blockage  c o r r e c t i o n s g i v e n by V i c k e r y .  The e x p r e s s i o n i s a l s o a p p l i c a b l e t o t h e c o r r e c t i o n f o r c e a n d moment c o e f f i c i e n t s .  of other  87 Glauert for  d e r i v e d an e x p r e s s i o n  a body f o r m i n g  f i n i t e width.  a wake o f a l t e r n a t e v o r t i c e s  coefficient  i n a channel o f  T h e a n a l y s i s , b a s e d on L o c k ' s image method a n d  Karman's v o r t e x to  f o r the drag  street  t h e o r y , was d e v e l o p e d  f o ra plate  normal  t h e f l o w b u t , a s s t a t e d by D u r a n d , i s a p p l i c a b l e f o r o t h e r  bodies for  o f abrupt o r sharp-edged  form.  The t h e o r e t i c a l  t h e d r a g - c o e f f i c i e n t i n c o n f i n e d f l o w i s g i v e n by  expression the  equation 0  V've  v  where u = W - V  V  y  fluid  superimposed  = vortex  of  expression  a plane  unlimited  C where  i s similar  n  =  velocity  (equation  streamwise  b = measured l a t e r a l This  (3)  y  W = surrounding is  We/ H on w h i c h t h e v o r t e x  system  (15));  velocity; vortex  row s p a c i n g .  t o t h a t g i v e n by Karman f o r t h e c a s e  flow,  (5.656  -  2.24  ttr)^f  (4)  159 and  a  F  l° 9i  =  n  a. p r i o r i , flow.  of  the  vortices.  =  C  b/e  - *  b„/e  C  n  the drag  =  C  0  +  n  D  provided  expression  32  However, t h e  the  F  form  f o r the  (5)  1  F  to the u n l i m i t e d flow.  In  practice,  i s generally negligible  reduces  and,  to  / ^ V f e  VV  r  this  as  F  correspond  by  an e x p r e s s i o n  V i-2u /v Hv;e/ H  second term i n the p a r e n t h e s i s  therefore,  generated  to  p a s s i n g downstream i n t h e  F  where u_/V_ and F i t  solution  ifc^'-4Ur/V,  1  t o know.  f o r t h e body i n a c o n s t r a i n e d  rate of v o r t i c i t y  T h i s study  + I 32  n  D  i t i s necessary  D  i n constrained flow  '  0  the  and  amount o f v o r t i c i t y  coefficient  C  for C  approximate t h e o r e t i c a l  c o n s i d e r i n g the  discrete  drag  o f u/V  G l a u e r t gave an  and  vortices  s p a c i n g between c o n s e c u t i v e  the e x p r e s s i o n  the v a l u e s  p r o b l e m by body  linal  strength T .  of However, t o use  t u c  (6)  e / H  theory  is still  incomplete,  s i n c e i t does 7  not p r e d i c t  u /V F  complete the F  Heisenberg's  plate  and  theory  L__ If  p  b /e. F  for a flat  0.2295 results  normal to the  An  flow  a n a l y s i s by plate,  and are  gave t h e  4f]V =  adopted, drag  becomes  Heisenberg,  to  values  0.3535  correction for a  (7)  flat  160  k- - • - a w h i c h c a n be predictions  referred  t o as G l a u e r t - H e i s e n b e r g ' s  are i n agreement w i t h e x p e r i m e n t a l  formula.  results  Its  given  by  70 Fage and  Johansen.  o f t h e same f o r m -C /Cp D  I t s h o u l d be n o t e d  as M a s k e l l ' s s i m p l i f i e d  i s r e p l a c e d by  b  Applicability C  D  4/C  the angle  and  approximate  ( b / e ) ^ = 1.77,  C  section  C  D  and  corrections  that  (6) f o r  shown by  the  values of  -45°,  (u/V}^  t h e d r a g e x p r e s s i o n becomes  0  H (8) f o r t h e  Bearman, as w e l l as P a n k h u r s t velocity  equation  c a n be  experimental  w h i c h compares w e l l w i t h e q u a t i o n  their  (2) e x c e p t  1 i n . a n g l e model mounted a t a =  For the  and  expression i s  D<  f o l l o w i n g example.  = 0.201  equation  of Glauert's simplified  t o t h e measurements on  c o n s i d e r i n g the  that this  can  and  a l s o be  flat  Holder  applied  plate.  suggested  to s t a t i c  that  pressures  f o r c e s by  ~ JV C  s  £d  =  F  S = F  (5) F l u c t u a t i n g  _Sv =  !  k  i  F o r c e s C a u s e d by V o r t e x  -2<r  ( 1 0 )  Shedding  40 Vickery fluctuating  lift  has based  presented on  the  a blockage  a n a l y s i s by  c o r r e c t i o n f o r the 96 Ross, which p r e d i c t s  t h a t the r a t i o C][,/C ment.  i s almost independent o f the w a l l c o n f i n e -  D  T h e r e f o r e , t o the f i r s t  approximation,  f o r c e s can be c o r r e c t e d u s i n g the steady  the f l u c t u a t i n g  force expressions.  (6) S t r o u h a l Number and Vortex V e l o c i t y Using the same model as employed by M a s k e l l and i n t r o d u c 6 6  ing  Roshko's  U n i v e r s a l S t r o u h a l Number  (S*), Vickery  developed  an e x p r e s s i o n f o r the c o r r e c t i o n o f the S t r o u h a l number, Q J L _F = S.w £ F  d -5— w  (11)  w  where  d  i s the e f f e c t o f w a l l c o n s t r a i n t on the wake width;  F  k = (1-C  and  p  1/2  )  On the o t h e r hand, the S t r o u h a l number, which i s a f u n c t i o n o f the f l u i d v e l o c i t y and v o r t e x shedding by t h e i r v a r i a t i o n s .  frequency,  may be a f f e c t e d  Bearman assumed t h a t the c o r r e c t i o n f o r  the S t r o u h a l number was a f u n c t i o n of the v e l o c i t y c o r r e c t i o n o n l y , and n e g l e c t e d any e f f e c t o f w a l l i n t e r f e r e n c e on the shedding  frequency. To i n t r o d u c e the c o r r e c t i o n f o r the v o r t e x  frequency,  i t i s a p p r o p r i a t e t o c o n s i d e r the e x p r e s s i o n {  and, and  shedding  thereby,  =  d2)  v / a  i n v e s t i g a t e the v a r i a t i o n s o f the v o r t e x  velocity  l o n g i t u d i n a l spacing with w a l l i n t e r f e r e n c e e f f e c t s .  The  v o r t e x v e l o c i t y i s a f u n c t i o n o f the a b s o l u t e v e l o c i t y of the surrounding  f l u i d and the r e l a t i v e v e l o c i t y o f the v o r t i c e s i n  the wake, and i s g i v e n by V  v  = W-u.  T h e o r e t i c a l l y , f o r an  162 unconfined  f l o w , W w o u l d e q u a l Vp,  Karman's vortex v e l o c i t y  and  relation.  u w o u l d be  For the  given  flow i n a  by  channel,  87 Glauert the  has  d e r i v e d u s i n g the  image method an e x p r e s s i o n f o r  r e d u c t i o n of the v o r t e x v e l o c i t y  by  the presence  of  the  walls,  —  =  I •- 8 e  c o s h HP  a  F  where  ^  2 TT M  _ "  a  However, s i n c e e * i s g e n e r a l l y v e r y walls  appears  then  be  t o be  expressed lv  Using  ( 1 3 )  negligible.  i n the -  Jr  V  5  s m a l l , the  The  i n f l u e n c e of  frequency  correction  a  ^ - U r / Q  flow c o n t i n u i t y  i  ,  as s u g g e s t e d  by  Glauert,  W on w h i c h t h e v o r t e x s y s t e m i s s u p e r i m p o s e d  than  approaching  backward  flow induced  Hence, v e l o c i t y  Using  stream by  velocity  W s V  by  the  the v o r t e x s t r e e t  W i s determined  Karman's s t a b i l i t y  by  the  image  4  )  the  amount e q u a l and  1  i s greater to  the  system.  equation  results,  + J/Tu i  substituted  can  form  velocity the  the  d6)  p  which can  be  into  The  final  e x p r e s s i o n f o r the  the  form  the free  frequency stream  correction  equation.  S t r o u h a l number i s o f  163  (17)  Although  t h e S t r o u h a l number f o r m s one o f t h e i m p o r t a n t  parameters  i n aeroelastic  literature  for i t scorrection  a  series  and  sections.  show t h e v a r i a t i o n  gives  studies,  the a v a i l a b l e  do n o t c o r r e l a t e w e l l .  Therefore,  o f e x p e r i m e n t a l measurements were c o n d u c t e d  3 i n . angle  various  instability  The r e s u l t s ,  plotted  with  1, 2,  i n Figure II-2,  o f t h e S t r o u h a l number w i t h b l o c k a g e f o r  angles of attack.  The e x t r a p o l a t i o n  of these  results  an e s t i m a t e o f t h e e q u i v a l e n t S t r o u h a l number i n an un-  limited  flow.  The p e r c e n t a g e  plotted  i n Figure I I - l ,  curves,  i s generally larger  correction  though s i m i l a r  f o r t h e 3 i n . model as  i n form  to the v e l o c i t y  o v e r most o f t h e a n g l e o f a t t a c k  range. (7) Wake G e o m e t r y 71 The measurements by R o s e n h e a d and Schwabe both  the l o n g i t u d i n a l  i n c r e a s i n g blockage etry  a t such  r a t i o b/a e s s e n t i a l l y  model t o t u n n e l w i d t h likely on the  and l a t e r a l  vortex spacings decrease  a rate  as t o m a i n t a i n  constant.  ratios  indicate  T h i s was  o f up t o 1/3  characteristics  with  t h e wake geom-  determined f o r  and, t h e r e f o r e , i s a l s o 87  t o be r e p r e s e n t a t i v e o f t h e a n g l e m o d e l wakes.  t h e o t h e r hand, has d e v e l o p e d  that  Glauert,  an e x p r e s s i o n f o r d e t e r m i n i n g  of the vortex s t r e e t  in a  channel,  Figure  II-2  V a r i a t i o n o f S t r o u h a l number w i t h v a r i o u s o r i e n t a t i o n s of the angle  blockage models  for  165 which suggests flow  t h a t the dimensions  are g r e a t e r than  those  o f t h e wake i n a c o n s t r a i n e d  i n an u n l i m i t e d f l o w .  T h i s i s an  opposite  t r e n d t o t h e e x p e r i m e n t a l measurements by R o s e n h e a d and  Schwabe,  and c o n t r a d i c t s  general physical  intuition.  88 Maskell  ^w  a indicating However, the  _  F  ,  Co ~ C o  +  _S  (19)  a  F  the influence  o f w a l l c o n s t r a i n t on t h e wake  width.  on t h e e x p e r i m e n t a l d a t a f o r t h e 3 i n . a n g l e m o d e l ,  term  i n t h e above e q u a t i o n i s o f t h e o r d e r 0.02 a n d ,  therefore, Maskell's analysis approximately  r  (k -0(k -i) c  "  w  based  second  has d e r i v e d t h e r e l a t i o n s h i p ,  independent  To e s t a b l i s h  suggests  of the w a l l  the c o r r e c t  t h a t t h e wake w i d t h i s constraints.  t r e n d f o r w a l l i n t e r f e r e n c e on  wake -geometry, a s e r i e s o f wake measurements were c o n d u c t e d the that  1 i n . and 3 i n . a n g l e s e c t i o n s . the l a t e r a l  the  agrees  with the trend of the experimental  o f R o s e n h e a d and Schwabe.  the percentage  indicates  v o r t e x s p a c i n g i s c o n f i n e d by t h e p r e s e n c e o f  the w a l l s and, t h e r e b y , results  This s e t of data,  with  correction  3 i n . model i s p l o t t e d  An a p p r o x i m a t e  f o r the l a t e r a l  estimate of  spacing i n the w a k e of  i n F i g u r e I I - l f o r comparison.  APPENDIX I I I Electronic  Following electronic  is a list  instruments  used  Instruments  o f t h e r e c o r d i n g and i n the c a l i b r a t i o n  auxiliary-  and  experimental  tests:  Pressure  Transducer:  D a t a m e t r i c , B a r o c e l P r e s s u r e S e n s o r , Type 511-10; S i g n a l C o n d i t i o n e r , Type 1015; Power S u p p l y , T y p e 700.  Filter:  K r o h n - H i t e , band pass c p s - 2 K c , model 330B.  Oscilloscope:  T e k t r o n i x , Type 564, d u a l beam s t o r a g e oscilloscope.  Chart  H o n e y w e l l , 906C V i s i c o r d e r ; S u b - m i n i a t u r e G a l v a n o m e t e r s , T y p e M100-120 and M200120; S t a n d a r d , s p e c . 2, V i s i c o r d e r R e c o r d ing Paper.  Recorder:  variable  filter  0.02  Voltmeters:  H e w l e t t P a c k a r d , HP-3400A t r u e rms v o l t m e t e r ; and HP-412 vacuum t u b e v o l t m e t e r .  Function  H e w l e t t P a c k a r d , low f r e q u e n c y f u n c t i o n g e n e r a t o r , model 202A; H e a t h k i t , a u d i o f r e q u e n c y g e n e r a t o r , model 1G-72.  Vibration  Generators:  Generator:  A m p l i f i e r and Supplies: R-C  damping  Power  circuit:  V47.  Low f r e q u e n c y , t r a n s i s t o r i z e d power a m p l i f i e r w i t h two 12 v o l t d . c . power s u p p l i e s , b u i l t i n t h e d e p a r t m e n t . [97] R e s i s t o r - c a p a c i t o r system, v a r i a b l e time c o n s t a n t 0 t o 60 s e c o n d s , b u i l t i n t h e d e p a r t m e n t . [35] A e r o l a b , 6 component, p y r a m i d a l s t r a i n gauge b a l a n c e ; two c o r r e s p o n d i n g 3 c h a n n e l readout e l e c t r o n i c cabinets with b u i l t - i n d.c. v o l t m e t e r s .  Balance:  Time-Mark  Goodmans, T y p e  Generator:  T e k t r o n i x , T y p e 184, 2 n a n o s e c o n d s t o 5 seconds c r y s t a l o s c i l l a t o r .  167 L a t e r a l Displacement Transducer:  Air-core transformer, b u i l t partment. [49]  L a t e r a l Transducer Demodulator:  F u l l wave r e c t i f i e r and RC i n the department. [49]  Dampers and Supplies:  E l e c t r o m a g n e t i c dampers; v a r i a b l e d . c . and a . c . power s u p p l i e s , b u i l t i n t h e department. [49]  Power  Power  Supply:  i n the filter,  Electro, variable, filtered, s u p p l y , M o d e l D-612T.  d.c.  Variac:  G e n e r a l R a d i o Company, Type W5M, a b l e c i u t o t r a n s f ormer.  Strobotachometer:  General Radio 1531.  Angular Displacement Transducer: Angular Transducer Power S u p p l y and Output System:  built  power adjust-  Company, S t r o b o t a c , Type  4-arm s t r a i n - g a u g e b r i d g e , b u i l t department. Ellis,  de-  i n the  B r i d g e - A m p l i f i e r - M e t e r , M o d e l BAM-1.  APPENDIX Theory  1  General  f o r Plunging  Equations  of  IV  or Torsional  Degree' o f Freedom  Motion  / / / / / /  v  The ing  degree  subjected dynamic degree  system  to linear  loading F o f freedom  express  spring  .  may  be  and  t.  Using  time  m  (Figure IV-1),  restrained  perpendicular to the flow  A  and v i s c o u s  corresponding  i s shown  the aerodynamic  general,  motion  o f mass  o f freedom  IV-2  Figure  IV-1  Figure  i n Figure  terms  a function  o f model  direction, i s f o r c e s and  IV-2.  I t i s convenient form  a  aero-  with  displacement  torsional  degrees  to  which, i n and  velocity,  formulation, the equation  and t o r s i o n a l  plung-  system  in coefficient  the Lagrangian  f o r the plunging  damping  to a  o f freedom  of  become  (1)  (2)  169 These e q u a t i o n s are a n a l y z e d w i t h v o r t e x r e s o n a n t or  galloping  excitations  of  to determine  the a e r o e l a s t i c  instability  angle  sections. 2  Vortex  Resonance  The since  analysis  the t o r s i o n a l  c o n s i d e r s p l u n g i n g degree solution  a t i o n due  to vortex shedding,  imated  a sinusoidal  to .  by  Equation  (1) t h e n  w h i c h c a n be w r i t t e n  fi'Y giving  V  =  i^l/Cji *in f  'max  2  frequency  form  ( 4 )  w  as  U  J2y rP  y  J'  For a p h y s i c a l cylinder  and  approx-  •2  (5)  the resonant amplitude  - 1  C^,  is  becomes  i n the nondimensional  Y c „„ C ,  near  the f o r c e c o e f f i c i e n t  the s t e a d y - s t a t e amplitude  Therefore,  only  For t r a n s v e r s e e x c i t -  f u n c t i o n of amplitude  • Y  *2^Sl<{  i s similar.  of freedom  U  2  r c , s  system  ( 6 )  c o n s i s t i n g o f an e l a s t i c a l l y ' m o u n t e d  i n an a i r f l o w , i t has  the resonance  simple mathematical  becomes  been o b s e r v e d  that  the  oscillations  peak do n o t comply w i t h t h e p r e d i c t i o n model, but r a t h e r e x h i b i t  of  a phenomenon  the  called  79 vortex capture. Parkinson,et  al  solved equation  i m a t e l y , u n d e r t h e c o n d i t i o n s o f v o r t e x c a p t u r e by  (1),  approx-  assuming  170  S  and  The  final  expression c  3  (7)  J  Galloping  3.1  f o r the resonant  of oscillation  forces  o r moment.  aerodynamic amount  tinue  its  until  steady  same  acting  value  apparent  aerodynamic  ever,  certain  problem  since:  may  exhibit  with  then  effects  model  a  0  , the  attitude  force  oscillations  and t h e  exceeds will  that  con-  balance  i s established.  The  so t h a t  the instantaneous  forc-  model  on t h e s t a t i o n a r y  of attack.  galloping  of attack  by t h e f l u i d  damping,  a  o f the aerodynamic  angle  on t h e o s c i l l a t i n g  This exist  c a n be r e p l a c e d  model  assumption  oriented implies  i n the force  shedding  frequency  or  by  at the no  that  moment  i s f a r removed  frequency.  the nature  moment  input  and t h e v o r t e x  the cylinder  twisting  an i n c r e a s e  i s adopted  hysteresis  From  t h e mean  a n e t energy  acting  angle  characteristics from  shows  approach  function  Section  of the nature  I f , from  by t h e v i s c o u s  t o grow  Unstable  sections  because  loading  quasi-steady ing  bluff  o f r e s u l t i n g energy  dissipated  becomes  Remarks  Geometrically type  now  Py  7  of Aerodynamically  Preliminary  displacement  turn  o f the problem,  o u t t o be h i g h l y  the lateral  nonlinear  simplifications are possible  force  functions.  or How-  i n the a e r o e l a s t i c  171 (i)  the r a t i o  o f a i r d e n s i t y t o model d e n s i t y i s s m a l l  consequently, with (ii)  the aerodynamic  the e l a s t i c  the frequency  and i n e r t i a l  of o s c i l l a t i o n  force i s small f o r c e s of the  and  compared system;  i s c l o s e t o the n a t u r a l  frequency; (iii) Under  the e q u i l i b r i u m motion i s n e a r l y  these  can be  a s s u m p t i o n s t h e p r o b l e m becomes q u a s i - l i n e a r  s o l v e d by v a r i o u s a n a l y t i c a l For  has been  Sisto  and  simplify cylinders  I i ,  and P a r k i n s o n , e t a l .  ^  t h e downwash c o n d i t i o n a t t h e t h r e e - q u a r t e r p o i n t t o  the a n a l y s i s .  velocity. and  For the t o r s i o n a l  a modified theory  more c o m p l i c a t e d  function  However, t h e t o r s i o n a l  A derivation  angular p o s i t i o n  of  bluff  quasi-steady  analysis  s i n c e the n o n l i n e a r aerodynamic  of the instantaneous  torsional  galloping  f o l l o w i n g the plunging  is slight-  moment i s a  as w e l l as t h e  of the b a s i c expressions  f o r the  plung-  degrees of freedom i s g i v e n i n the f o l l o w i n g  sections. 3.2  theory  c o n s i d e r i n g the problem of s t a l l - f l u t t e r , i n -  approach i s presented.  ing  the q u a s i - s t e a d y  19  corporated  ly  which  techniques.  the p l u n g i n g degree of freedom,  e s t a b l i s h e d by S c r u t o n " ^  14  sinusoidal.  P l u n g i n g Degree  o f Freedom  Figure  IV-3  1 7 2  Following  the  analysis  established  by  P a r k i n s o n and  1 7  associates, is  related  to  the  instantaneous  the  steady  lift  aerodynamic  and d r a g  force  (Figure  IV-3)  by  F = -(LcosY + DsinV) y  (9)  where  and  the  V = V  r e l  instantaneous cosy,  the  angle  final  of  attack  expression  a = a  + y.  0  for  the  force  Noting  that  coefficient  becomes  C = - ( C + C tanT) Stc T F  (ID  D  1  The  change  component  i n model  7  the  governing  +  It the  is  related  form of  +  has  equation  the  symmetrical tion.  the  transverse  velocity  ( 1 ) becomes  Sv j?v hlc (Ci) (i3; =  been  customary  2  to  a polynomial using C  T  F  obtain  an e x p r e s s i o n  and C  experimental  It  analyze  to  by  Therefore,  111  attitude  particular angle  Santosham  50  of  case where  attack.  considered  in  Cp  data,  y  and  u  the  model  is  In t h i s  case,  C  the  for  problem of  mounted a t  a  i s an odd f u n c y aeroelastic insta-  173 bility  of  rectangular  polynomials  of  for  analyzing  the  t h e o r y has  at  11th the  mial  containing  actual least  degree square  degree  a l l terms  used  to  attack has  and  been  criterion.  functioned  the of  the  to or  the  of  However,  section,  model  degree  i s 25  Thereby,  angle  the  extended  Chebyshev  solutions. an  consider  the p o l y n o m i a l used  error  odd  oscillations  generalized  angles of  of  and  for obtaining  galloping  been  unsymmetrical  sections  oriented  the  25.  polyno-  However,  less  based  expression  the  on  a  for  C  p  y can  be  written  i n the  polynomial  form  (14)  |sl <  where Note  that  apparent  a  25 .  Q  =  0 because  zero p o s i t i o n  y  represents  governed  by  the  the  displacement from  steady  l i f t  force.  the C  is  F  y o  generally are  plotted  directly  related  Equation  ¥ or  •  Y  combining  as  a  function  by  (12).  (13)  can  = -20 Y y  the  two  be  +  terms  of  tan  y,  since  J  (y/V)  and  tan  nondimensionalized to  lyUX/Y) on  the  right  ,15)  hand  side  to  give  (16)  (17)  Based (16)  on  the  aforementioned  i s a quasi-linear  assumption,  differential  y  v  < <  equation of  1; the  therefore, autonomous  y  type. right ing,  Actually, hand  the  quasi-linear  side  of  equation  implicity,  an  upper  Using approximate  the  limit  Variation  solution  of  (16) to  of  form to  requires  remain  the  equation  complete  s m a l l , thus  velocity  Parameter  the  U.  method,  ( 1 6 ) , when  specify-  y  the  =  0,  first  order  i s of  the  form  Y therefore  = Y s i n ( x • <p)  .  Y = Y For  y  <  ing  (16)  <  1,  to  the  a  COS  first  system  of  ( T  ( 1 8 )  0)  +  order  solution  first  order  i s determined  differential  by  reduc-  equations  Y = z considering  Y = Y  sin ( T  Z = Y  COS ( T  + = and  assuming  reduces  Y  and  Since  be  0)  (20)  0  to  be  functions of  time  T,  equation  (19)  to  ^  be  +  x + 4>  dx  will  +  slowly  considered,  y  '  dY  Y and  varying  (21)  f [ Ycos^J s i n t y  d<6  are  p r o p o r t i o n a l to  functions of  approximately,  time.  constant  Hence,  during  one  —  y  ,Y  Y  and  cycle.  and  p  jz$ c a n Equa-  tion  (21) c a n t h e n be r e p l a c e d  by t h e i r a v e r a g e  values  ay Eii  =  f [ Y c o s ^ ] cos*/' (22)  By e x a m i n i n g $ and r e c a l l i n g t h e p o l y n o m i a l function  f^(Y),  sine  series.  and,  therefore,  the integrand  Upon i n t e g r a t i o n  dx =  by a  term i n t h i s s e r i e s  Fourier vanishes  as  -Y<5 (Y)  -  dj>  each  (22) c a n be w r i t t e n  -  il  c a n be r e p l a c e d  form o f the  *  (23)  0  d t  where  J (Y)  =  -  C {  J ^ l  [  Y  cos  +1  cos  +  (24!  0  On s u b s t i t u t i n g  ^  )  .  .  k  =  where  (17) i n t o  ^  ^  ( 2 4 ) , 5^ becomes  Y  • I*,*  r M  for  M o<dd  M-l  for  W  b^Y'"'}  ( 2 5 )  even  and  b,  The  , i  displacement  o b t a i n e d by d e t e r m i n i n g  amplitudes  of the l i m i t  the r e a l p o s i t i v e  roots  cycles  are  Y_. o f t h e  (Y)  176 polynomial,  from which  the s t a b i l i t y  of the sustained  motions  c a n be a n a l y z e d . 3.3  T o r s i o n a l Degree  o f Freedom  Figure  For excitation steady  the a n a l y s i s  IV-4  of the t o r s i o n a l  i s t h e i n s t a n t a n e o u s moment M.,  a p p r o a c h c a n be r e l a t e d  degree o f freedom, which  the  u s i n g the quasi-  t o t h e e x p e r i m e n t a l v a l u e , M,  by  .2  c  = -  c  (°0  (26)  where  Q It  i s apparent that V  r e  ^  «S  i s different  (  from V i n both  2  7  )  magnitude  o  and d i r e c t i o n each  because  s u r f a c e element  relative centre  velocity  of the angular v e l o c i t y on  the contour e x p e r i e n c e s a  g o v e r n e d by  of r o t a t i o n .  9.  i t s position  I t i s assumed t h a t  Furthermore, different  with respect  an e f f e c t i v e  to the  relative  177  velocity  can  be  written  as  = e n e Ir tT  r where tion  i s the of v  gravity  .  at  effective  For a  Q  ( 2 8 ;  r  the  = 0°,  nondimensionalized  radius  n  o f r o t a t i o n and  r  a n g l e model s u s p e n d e d a b o u t the the  representative  i s the  direc-  centre  parameter values  of  suitably  are  -  R  0.3015 ( 2 9 )  For  different  angle  n  r  can  angles be  of  attack,  obtained  using  the the  representative direction relation  (30)  The an  effective  This , by  a p p r o a c h when a p p l i e d  downwash v e l o c i t y a t t h e  agrees with  the  value  plate a i r f o i l  three-quarter  adopted i n the  stall  gives  chord p o i n t .  flutter  analysis  _. . 14 , _. 19 Sisto and I i . For  the  d i a g r a m i s as ity  to a f l a t  can  be  oscillating  c y l i n d e r , the  shown i n F i g u r e  written  IV-4.  overall  Hence, t h e  v e l o c i t y vector  relative  veloc-  as  (31)  (If and  the  -  , -  representative 1  ill  X  angle  V -  •  cos y  r  becomes  er cosT] f r  r  tyf (32)  178 In  nondimensional (2)  motion  form,  the governing  differential  equation  of  becomes  ©  • e  .  • n9u'c  -2pe  (»,*)  (33)  9  with  auxiliary  expressions  transformed  to  (34)  Upon  equations almost result, lished to  ( 3 4 ) , i t was  linear a new  observed  cartesian  a t an  lines  angle  C  M  q  and  6  by  and ©  X to the 0  system,  c a n be  of 0  of constant  5 and  the c o n t o u r l i n e C..  approximation, to 0  that  coordinate  along  i s related  as a f u n c t i o n  e  and p a r a l l e l  5 axis  with  a first  E, w h i c h  evaluation of  C  axis. c, , was  =0.  expressed  from  ^ were As  a  estab-  Therefore,  as a  the coordinate  M  function  of  transform-  ation  (35) where  s  and  c =  Similar.to is  =  sin  A  cos  A  the plunging  expressed  case,  as a p o l y n o m i a l  the aerodynamic in £  moment  coefficient  179  C  = a,|  M  + a% z  + a ?  2  3  3  . . .+ a  +  N  ?  (36)  M  where Substituting on  (35) and  the r i g h t  reduces  hand  (36) i n t o  side,  (33) and c o m b i n i n g  t h e two  the nondimensional equation  terms  of motion  t o the form  © + © = /^M©,©) J^Q  where  Q  - ^ g ^ i  = U*{( J o)@  f {® Q)  and  Vs  f  (37)  U  2  ~ C&  + ^ ( § 5 -  ©c)' (38)  +  For  a system i n a i r , y  <<  1.  6  Using plunging  t h e method o f V a r i a t i o n  analysis,  i®  =  the s o l u t i o n of  -  §  i  o f Parameters  the  as i n  (37) c a n be w r i t t e n  as  ( © )  d *  6  (39)  ^ dt  =  -  (8)  k e  where  2TT© ;  K ( © ) = -A 0  .air  2TT© J  0  <> 4 0  f [© s i n f , 0 c o s ^ ] sin 4" 4w> 9  180 Note, the v a r i a t i o n of phase w i t h the p l u n g i n g the  case.  integration,  time  Substituting  6.  and  K  (38)  w h i c h was into  (40)  not present i n and  carrying  out  become  + ... +  (41)  -+(o) K (§)-£l/[c +  where  The  |3{(§fct  •  e  • • • + - 1 1 - 1  L  -  3,  S  coefficients  --.-.•.-71  ,7,  b.  j  ;  . and  =  i - I  • -  t.  j  =  -  j+ |  (  4  2  )  »,3, 5", • • ,1  . ,, a r e g i v e n  by  1,1+1  if]  b-  c ' t j ®  M +  c.  . d.  . (43)  Here t h e  c.  equation  (38) , and  the  . are the b i n o m i a l c o e f f i c i e n t s o f the  integration  determined  from  d^  ^  + 1  and  of e q u a t i o n the  s^  + 1  (40).  following  are  constants  terms i n  obtained  T h e s e c o e f f i c i e n t s can  expressions:  from be  181  -(k-ij  9ij+2 "(i+O-O"-')  e.  =  9iJ  e. •  J 2 +  (44)  l,J + Z  I « I, 3, 5,-•  where  The (39)  j  amplitudes  j = \, 3,5, • • •, I  of the l i m i t  using the c o n d i t i o n ,  dQ/dr = 0.  ;  cycles  k = 1,2 ,3, • •  t- I .  c a n be o b t a i n e d  from  The s t a b i l i t y  tained  m o t i o n s c a n be a n a l y z e d by e x a m i n i n g t h e  96-/9 0  evaluated a t these  limit  of the sus-  sign of  cycles.  o  On d e t e r m i n i n g uated  using equation  the l i m i t  O  a m p l i t u d e , ft c a n be  eval-  (42) as  <p = - K X  where 0  cycle  + <f>  0  i s the constant of i n t e g r a t i o n .  (45) Thus, the s t e a d y - s t a t e  182 amplitude  where  of oscillation  ®  (46)  =  ^  = T + 0 = ( | - f < ) - C  (47) i t a p p e a r s  from  the natural  meter  lp  ©  From  second  i s g i v e n by  sin  that  +  the frequency  frequency  0  (47)  o  of o s c i l l a t i o n  by t h e amount K . Q  i s reduced  Therefore, the  e q u a t i o n o f (39) i n t e r m s o f t h e r e d u c e d  frequency  para-  becomes  I + #  =  I -  K  (48)  

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