UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

An experimental study of the aeroelastic instability of rectangular cylinders Smith, John David 1962

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1962_A7 S5 E9.pdf [ 6.35MB ]
Metadata
JSON: 831-1.0105634.json
JSON-LD: 831-1.0105634-ld.json
RDF/XML (Pretty): 831-1.0105634-rdf.xml
RDF/JSON: 831-1.0105634-rdf.json
Turtle: 831-1.0105634-turtle.txt
N-Triples: 831-1.0105634-rdf-ntriples.txt
Original Record: 831-1.0105634-source.json
Full Text
831-1.0105634-fulltext.txt
Citation
831-1.0105634.ris

Full Text

AN EXPERIMENTAL STUDY OF THE AEROELASTIC INSTABILITY OF RECTANGULAR CYLINDERS  by B.A.Sc.,  JOHN DAVID SMITH U n i v e r s i t y of B r i t i s h Columbia,  1960  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M.A.Sc. i n the Department Mechanical  of  Engineering  We accept t h i s t h e s i s as conforming r e q u i r e d standard  The U n i v e r s i t y of B r i t i s h Columbia August  1962  to  the  Th presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study.  I further agree that permission  for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be alloxved v/ithout my written permission.  Department of  Mechanical  Engineering  The University of British Columbia, Vancouver 3, Canada. Date A u g u s t 3 1 , 1962  ABSTRACT Dynamic w i n d rectangular  cylinders  (b/h)  and  based  on t h e  models  the  to  was  viscous with  to  to  developed  to  in  were  lateral  the  Reynolds  are  with  and c o u l d be applied  obtain  for  the  oscillated  shafts  this  To r e s t r a i n were  eddy  force  the  curves  reduced  onto  The  for  of  of  one  longer  the curve  appeared  be  strongly  i n f l u e n c e d by  not  plunge.  Strouhal cylinder,  Hot w i r e  frequency occurring  was  Direct cylinder  the  b/h  i n the  simultaneously  =  which  1.00 approach,  the  ratios  did  not  prove  of  of to  approach,  and  instantaneous b/h over  showed wake  with  tests.  using  the  measurements  present  dynamic  quasi-steady  cylinders  of  cylinders  for  rectangles  by  did  transducer  the  by  analysis  Rectangular  measured  quasi-steady  to  geometry.  were  those  be amenable  wake  damping  calibrated  square  The c u r v e s  predictions  damping.  to  current  apparatus.  range  top  hydrostatic  v a r i a t i o n with angle  number  presented.  the  attached,.  accurately  the  ratios  predictions  g u i d e d by  made f o r  plunging  to width  coupling electromagnetic  The v e l o c i t y - a m p l i t u d e  agreed  depth  Dynamic d i s p l a c e m e n t s  specifically  obtain  made o n  compared w i t h  An e l e c t r o m a g n e t i c  measurements  attack  were  approach.  transverse  a variable  force  were  quasi-steady  damping.  designed  with various  results  bearings.  device  tests  p l u n g i n g o s c i l l a t i o n they  and bottom, air  tunnel  of  4.00  that the  the plunging  the p l u n g i n g  frequency.  vi  A CKNOWLEDGEMENT  The Dr.  G.  was  performed,  thank  V.  Mr.  author  wishes  Parkinson, under  I.  for G.  his  Also,  Mechanical  Engineering  for  the  was  received  Grant  use  A-586.  of  he  wishes  their  from the  express  appreciation  whose s u p e r v i s i o n  advice  Currie for  gramming.  to  his to  the  and encouragement, assistance express  Department  facilities.  and  research and  to  i n computer  his  thanks  the  Computing  Financial  N a t i o n a l Research  to  to  prothe Centre  assistance  Council of  Canada,  CONTENTS Page No. I. II.  INTRODUCTION  1  THEORY  5  2.1 The Q u a s i - s t e a d y Approach 2.2  The E f f e c t of Afterbody Length and Re-attachment  III.  APPARATUS AND INSTRUMENTATION 3.1 General C o n s i d e r a t i o n s 3.2  IV.  V.  5  The Wind Tunnel  8 9 9 13  3.3 The Model Mounting System  14  3.4 Displacement Measurements  19  3.5 Magnetic Damping  22  3.6  23  L a t e r a l Force Transducer  3 . 7 Wake Measurement Equipment  24  3.8  25  The Models  EXPERIMENTAL PROCEDURES AND RESULTS  26  4.1  C a l i b r a t i o n Methods  26  4.2  Test Procedures  28  4.3  Experimental R e s u l t s  32  DISCUSSION AND RESULTS  38  5.1 D i s c u s s i o n  38  5.2  43  Conclusions  5.3 Recommendations f o r F u t u r e Research  44  APPENDIX  46  BIBLIOGRAPHY  49  ILLUSTRATIONS  53  iv .  ILLUSTRATIONS  F i g u r e Number 1  Wind Tunnel Aerodynamic O u t l i n e  2  A i r Bearing D e t a i l s  3  Assembled A i r Bearing  4  A i r Bearing Base  5  ' M o d e l Clamping D e t a i l s  6  A i r Supply  7  Upper Shaft  8  Test S e c t i o n and Equipment  9  Displacement  Transducer  10  Displacement  Transducer D e t a i l s  11  Displacement  Instrumentation  12  Electromagnetic  13  E l e c t r i c a l Supply f o r  14  F o r c e Measuring U n i t  15  Model Plan  16  Damping C a l i b r a t i o n  17  C a l i b r a t i o n Set-up  18  Dimensionless Amplitude v s . V e l o c i t y ; b/h =1.00  19  Reduced V e l o c i t y - A m p l i t u d e  20  Dimensionless Amplitude v s . Velocity; b/h =1.50  Dimensionless Flow  21.  Dimensionless Amplitude v s . Velocity; b/h = 2 . 0 0  Dimensionless Flow  and  Bearings  and A m p l i f i e r Schematic  Damper Dampers  Dimensionless Flow  Curve; b / h =  1.00  V  22  D i m e n s i o n l e s s Amplitude v s . D i m e n s i o n l e s s Flow V e l o c i t y ; b / h = 2.50  23  D i m e n s i o n l e s s Amplitude v s . D i m e n s i o n l e s s Flow V e l o c i t y ; b / h = 3.00  24  T i m e - A m p l i t u d e C u r v e s f o r b / h = 1.00 a n d b / h = 2.00  25  S t r o u h a l Number v s . R e y n o l d s Number f o r V a r i o u s b / h  26  S t r o u h a l Number v s . b / h r a t i o  27  Qp  28  Typical  29  Comparison o f V e l o c i t y - A m p l i t u d e tangular Sections  30  R e d u c i n g F a c t o r v s . b / h f o r Long R e c t a n g l e s  v Y  s  •  «  ^  o  r  t  n  Hot Wire  e  Square  Section  Signals Curves f o r Rec-  vii NOMENCLATURE y = l a t e r a l displacement  of o s c i l l a t i n g c y l i n d e r  h = l a t e r a l dimension of c y l i n d e r b = streamwise  section  dimension of c y l i n d e r  section  s = length of c y l i n d e r m = mass of o s c i l l a t i n g r = coefficient  system  of v i s c o u s damping of o s c i l l a t i n g  k = s p r i n g constant  of o s c i l l a t i n g  system  system  jL  CO = ( k / m ) V = air  2  = c i r c u l a r frequency of f r e e undamped o s c i l l a t i o n of system  velocity  V . = air velocity rel  r e l a t i v e to o s c i l l a t i n g  cylinder  °< = angle of a t t a c k of r e l a t i v e wind to c y l i n d e r = tan"' (y/V) p = air  section  density  p = pressure L = aerodynamic l i f t  on c y l i n d e r  D = aerodynamic drag on c y l i n d e r F = l a t e r a l component of aerodynamic f o r c e on c y l i n d e r L (p/2)V/ hs e l  5  ci  (p/2)V2 hs e l  C  F  y  =  F  (p/2)V.Zhs  U = V/(oJh)  = dimensionless  Y = A / h = dimensionless  flow  velocity  amplitude  B = r/(2m6>) = dimensionless  damping  n = p h * s / ( 2 m ) = dimensionless  coefficient  mass parameter  viii U  o  = critical  air  velocity  2B nA S = fh/V © = wt (')=  = Strouhal  = Qimensionp-^ss  derivative  V = kinematic N  R  -  number  Vh/V  time  with respect viscosity  = Reynolds  number  variable to  time  1  I.  INTRODUCTION  T h i s p r o j e c t was p a r t of a c o n t i n u i n g program s t u d y i n g the a e r o e l a s t i c bluff  cylinders.  It  i n s t a b i l i t y of a e r o d y n a m i c a l l y  i s w e l l known that b l u f f  v i b r a t e when e l a s t i c a l l y cases t h i s  c y l i n d e r s may  mounted i n a flow o f a i r .  In many  v i b r a t i o n has been r e l a t e d to a resonance  with  the eddies being shed by the model to form a Karman Vortex Street  i n the wake.  to o s c i l l a t e  Some c y l i n d e r s , however,  under c o n d i t i o n s  w i l l begin  i n which the frequency of  v o r t e x shedding i s f a r removed from any e l a s t i c n a t u r a l f r e quency of the c y l i n d e r .  These b l u f f  aerodynamically unstable, In t h i s  phase of  c y l i n d e r s are termed  and are the object  this  study.  the program, the models were r e s t r a i n e d  one degree of freedom i n order to study the o s c i l l a t i o n of b l u f f unwanted modes.  of  c y l i n d e r s without  transverse  i n t e r f e r e n c e from  A l l models were two d i m e n s i o n a l ,  the tunnel c o m p l e t e l y ,  and e f f o r t s  to  were made to  that flow was as two dimensional as p o s s i b l e .  spanning ensure  Measurements  i n c l u d e d v e l o c i t y - a m p l i t u d e r e c o r d s , amplitude-time r e c o r d s , and  shedding frequency measurements  for rectangular  cylinders  with c r o s s s e c t i o n s of v a r i o u s l e n g t h to width r a t i o s different  l e v e l s of a p p l i e d damping.  Some wake  with  measurements  were attempted. In an e a r l i e r phase of the program, N. P. H. Brooks (1)"^" experimented w i t h s i m i l a r r e c t a n g u l a r 1 Numbers i n b r a c k e t s r e f e r  to the b i b l i o g r a p h y .  cylinders  2 which were a l l o w e d contains  a fairly  problem of Parkinson mental  the (2)  degrees  complete  has  given  distinct  types  the  and c y l i n d e r  flow  survey to  the  of  mounted  vibration  vortex  2.  response  3.  flutter  thesis  bluff  theory  aeroelastic  on  also  the  cylinders. and  experi-  behaviour  of  in  towers,  the  of  design  of of  to  in uniform flow,  alternately  frequency  of  structural  result.  These  random  several  depending  types  on  are:  the  oscillation  is  Strouhal  number  considera-  frequency  which vortices  from each  is  as S  at  shear  cylinder  shedding  S defined  important periscopes  or  layer  its  the  follows: = fD/V  approaches  low,  cylinder  frequency  and  mounting.  sufficiently  The r e l a t i o n b e t w e e n  V, a n d v o r t e x  an  smokestacks,  damping  velocity  of  Resonance  vibration  the  excitation  twisting section  tall  o c c u r r i n g when  shed  natural  type  exhibit  resonance  plunging or an u n s t a b l e  This  will  the  cylinders  Vortex  level  of  properties.  1.  4.  being  His  cylinders. Elastically  tion  freedom.  i n s t a b i l i t y of  a  pertinent  of  h i s t o r i c a l background  aeroelastic  results  bluff  six  f, i s  If  are a the  oscillation size  given  D, by  flow the  Over for  a range the  from to  that  of  velocity  in  this  At  the  the  the  the  of  shedding  important  of  example  Reynolds  number  NR  above  = 2(10)  5  R  narrower move  than  on the  turbulent  of  range,  its  the  about  up t o  NR  this  cylinder. at  evidence  that  at  those  Reynolds  16  also  vortex  shedding  been  termed  "capture"  to  any  the  section in  the  with  as  the  structure  is  and higher  resonance  points  are  5).  forces  ob-  largely  but  detectable  and the  a  cylinders  show a  4,  most  becomes  forces  numbers,  (3,  the  separation  periodic  This  below  the wake  for  an  wake,  For c i r c u l a r 6  longer  out.  2(10) .  = 3.5(10) ,  no  frequency  cylinder  5  vibra-  vortex  v i b r a t i o n can  these high Reynolds  numbers  The  circular  range  constant),  the  Wake m e a s u r e m e n t s  a periodic  discuss  nearly  vibration dies  for  a n d no d e t e c t a b l e  wake  the  deviates  (proportional  vibration.  cross-section  being of  and the occur  is  control  condition  shedding  cylinder  w h i c h now r e v e r t s  outside  rearward,  served  the  cylinder,  part  to  resonant  vortex  phenomenon has  o s c i l l a t i o n can  appreciable  N  frequency  appears  of  of  the  S t r o u h a l number  and t h i s  stationary  type  the  limit  near  a stationary  since  cylinder  upper  frequency  for  with  range,  control  velocities  the  expected  coincides  tion  flow  cylinder,  flow  and  of  in  there  is  the  wake  Refs. on  6  to  cylinders  Random E x c i t a t i o n  At wakes a r e  extremely  relatively  high Reynolds  unorganized,  the  numbers  where  excitation  of  the a  cylin-  der by the random f o r c e s sidered.  exerted by the wake must be con-  It has been observed that o s c i l l a t i o n s  under these c o n d i t i o n s , and the problem has been by Fung (17)  and Davenport (18).  The response  considered  T h e i r approach has been  to c o n s i d e r a l i n e a r e l a s t i c system s u b j e c t e d random i n time.  do occur  to a f o r c e  of the c y l i n d e r was  then  c a l c u l a t e d from the power spectrum of the f o r c e and the mechanical impedance of the e l a s t i c a l l y T h i s method i s e s s e n t i a l l y the response  mounted c y l i n d e r .  s i m i l a r to the c a l c u l a t i o n of  of a tuned e l e c t r i c a l c i r c u i t to  noise.  Flutter F l u t t e r r e s u l t s when two modes of plunging and t w i s t i n g ,  oscillation,  are e x c i t e d s i m u l t a n e o u s l y  frequency u s u a l l y between the n a t u r a l f r e q u e n c i e s plunging or pure t w i s t i n g . as a p o s s i b l e  ice-coated  transmission lines  21,  22).  mechanism f o r the o s c i l l a t i o n (19)  and suspension  The aerodynamic f o r c e s  are u s u a l l y h i g h e r than f o r any of oscillation.  f o r pure  T h i s has been e x t e n s i v e l y i n -  vestigated  (20,  at a  involved in  of  bridges flutter  the other forms of  T h e r e f o r e , the b u i l d - u p times may be very  s h o r t and the onset sudden.  Plunging O s c i l l a t i o n  Plunging i s a t r a n s v e r s e cross-section,  without  twisting,  oscillation  of  the  normal to the stream d i r e c -  t i o n and i s e x h i b i t e d by s e v e r a l n o n - c i r c u l a r Only one degree of freedom i s e x c i t e d ,  sections.  oscillations  occurring above a minimum s t a r t i n g speed and the amplitudes i n c r e a s i n g w i t h a i r speed. possible. project,  Extremely l a r g e amplitudes a r e  P l u n g i n g has been the major i n t e r e s t and more w i l l be s a i d about i t  of  this  i n the next  section  I t s h o u l d be noted here that pure t w i s t i n g of an u n s t a b l e s e c t i o n may a l s o o c c u r .  II.  THEORY  ^  2.1,  The Q u a s i - s t e a d y Approach The q u a s i - s t e a d y approach, as the name i m p l i e s ,  assumes that the i n s t a n t a n e o u s  aerodynamic f o r c e s  acting  on the o s c i l l a t i n g c y l i n d e r may be approximated by the f o r c e s on the s t a t i o n a r y c y l i n d e r at an angle of a t t a c k equal to the apparent angle of a t t a c k of the o s c i l l a t i n g 1  c y l i n d e r at that i n s t a n t .  T h i s assumption i s  a r g u i n g that the v i b r a t i o n p e r i o d i s with the v o r t e x f o r m a t i o n p e r i o d , body change slowly w i t h r e s p e c t and the i n s t a n t a n e o u s  j u s t i f i e d by  long i n comparison  or t h a t c o n d i t i o n s on the  to c o n d i t i o n s i n the wake,  wake dyriamics need not be c o n s i d e r e d  Two methods of p r o c e e d i n g from the assumptions of  the  q u a s i - s t e a d y approach w i l l be d i s c u s s e d .  These a r e ;  a n a l y t i c a l approach, as used by Parkinson  (23),  the  involving  an approximate s o l u t i o n of the d i f f e r e n t i a l e q u a t i o n of motion, and the n u m e r i c a l approach.  6  The A n a l y t i c a l Approach P a r k i n s o n ' s method i n v o l v e s  the s u b s t i t u t i o n  p o l y n o m i a l approximations to the aerodynamic f o r c e i n t o the equation of motion.  curves  The e l a s t i c system i s  to be harmonic, and aerodynamic f o r c e s  assumed  are assumed to be  s m a l l compared to the i n e r t i a l and e l a s t i c f o r c e s . result  of these assumptions  and p r e d i c t s flow v e l o c i t y system,  the e f f e c t  As a  an a n a l y t i c a l . . s a l u t i o n o o f  n o n l i n e a r d i f f e r e n t i a l equation i s p o s s i b l e of K r y l o v and Bogoliubov  (24).  The s o l u t i o n  is  general  of the parameters of mass, damping,  and time on the o s c i l l a t i o n .  Thus, f o r any  p r e d i c t i o n s of amplitude as a f u n c t i o n of  speed as a f u n c t i o n of damping are p o s s i b l e . (23,25) has shown the p r e d i c t i o n s from t h i s  time,  and s t a r t i n g  Experiment method to be  compatible w i t h performance f o r the square c y l i n d e r . limitation is  the  by the method  s t a t i o n a r y amplitude as a f u n c t i o n of v e l o c i t y ,  serious  of  that the method i s no b e t t e r  One  than  the p o l y n o m i a l approximations i n v o l v e d , and some curves will  r e q u i r e h i g h order approximations i o  reasonable  o b t a i n any  fit.  Numerical and G r a p h i c a l Methods  Numerical and g r a p h i c a l methods have often used to handle problems whose n o n - l i n e a r i t i e s a n a l y t i c a l s o l u t i o n d i f f i c u l t or i m p o s s i b l e , p l u n g i n g o s c i l l a t i o n of b l u f f  cylinders.  been  have made an such as  U s i n g the  the basic  assumption motion  is  of  the  quasi-steady  sinusoidal  for  results  i n an  integral  lateral  force  coefficient  of  attack.  determination amplitude of  steady  over  this  (26)  is  However,  further.  The  Y,  B,  by  Parkinson  (23)  Rewriting  i n terms  of  a  function  velocity of  attack  of  angle  by  of  velocity The  that  s i n e  the  }  S i n 9  that  to  integral  for it  a given  the  curves  can be  (nY/B)  [nU/B :  attack  evaluating so  the apparent  a  the  graphical stationary  speed  the  as  a  function  method has  i n the  d  in  been  non-dimensional  U , mass n ,  results  = f n  p  seen  corresponding  numerically, flow  can be  of  This  and  damping  equation:  9  U  nY/B:  C It  of  velocity  Fy{rJ  C  obtain  case  intoduction  " i f  involving  starting  in this  amplitude  1*=  energy  oscillation.  described  to  that  similar!:'tdt^he^me'thodK5hown^  of  used  up an  a function  has  of  parameters as  as  and assuming set  period  integral  H i s method  to  state  the Cp  Scruton  of  appendix.  taken  of  possible  and an e s t i m a t i o n  damping.  the  cycle  is  balance  angle  one  it  approach  J  j  a given is  curve  possible  to  of  graphically  stationary  Cp^ as  obtain  maximum a p p a r e n t  integral of  .  Slne  a  angle or  amplitude  vs.  constructed. was  evaluated  numerically  for  the  )  8 square  section  University were to  of  stable  the  over  IBM 1620  Columbia.  a range  theoretical  limit  The c u r v e  the  British  obtained  yield  using  cycles  of  d i g i t a l computer Sufficient  apparent  curve  of  19,  approaches  asymptotically  a  of  attack  showing  j o i n e d by an u n s t a b l e  limit  line  the  integrals  angles  Fig.  at  two  cycle.  through  the  or i g i n . Equation approach are  is  reasonable  s m a l l over  sible  to  city  at  the  reduce  perimental  2 implies  and the  range  of  experimental  points  various  for  that  the  interest, data  levels  so  fall  on one  amplitude-velocity  one  set  of  conditions  amplitude-velocity  curve  other  This w i l l  2.2  conditions.  The E f f e c t  of  In h i s effect the  of  wake  Strouhal that  as  mation  the  of  number  the  at  afterbody first  the  curve  used  to  examined  on t h e  rectangular  length  forced  is  which  obtained  predict under  the any  Re-attachment  (2)  reviews  the  lying  coefficient  cylinder.  increased  downstream  velo^:"  stationary  cross-section drag  ex-  later.  Parkinson  a cylinder  length)  for  for  Length and  paper,  (afterbody  the is  length  curves  can be  be  reduced  curve  which would r e s u l t  Afterbody  recent  the  pos-  amplitude and flow  cylinder. under  effects  s h o u l d be  that  the  the  number  it  can be d e t e r m i n e d by Also,  force  quasi-steady  Reynolds  stationary  damping  if  the  It  and appears  vortex  and outwards,  in  as  for-  9 shown by the decrease i n S t r o u h a l number. afterbody  length i s  increased,  However, as  re-attachment  the  can occur a t  lower angles of a t t a c k u n t i l ^ f i n a l l y rek-attachment can occur at zero angle of a t t a c k .  In t h i s  would become narrower w i t h the v o r t i c e s of  the c y l i n d e r ,  event,  the wake  forming downstream  the S t r o u h a l number would i n c r e a s e ,  the s e c t i o n would have p o s i t i v e  damping.  Measurements o f  S t r o u h a l number f o r r e c t a n g u l a r s e c t i o n s i n d i c a t e t h i s would happen f o r depth to width r a t i o s b / h that 3 . 0 0 .  F i g . 26,  taken from Ref.  and  that  greater  2 shows the  varia-  t i o n of S t r o u h a l number w i t h b / h .  III.  APPARATUS AND INSTRUMENTATION  3 .1 General: C o n s i d e r a t i o n s  S i n c e t h i s p r o j e c t was p a r t of a c o n t i n u i n g program on the a e r o e l a s t i c  i n s t a b i l i t y of b l u f f  cylinders,  i n f o r m a t i o n was a v a i l a b l e on the problems l i k e l y to be e n countered, and the probable behaviour of the dynamic models.  In p r e v i o u s experiments  (1,  25),  been supported e n t i r e l y by four s p r i n g s , model s i x degrees of freedom. ficant  the model had allowing  the  However, only two s i g n i -  modes of o s c i l l a t i o n were observed.  The dominant  mode was found to be pure t r a n s l a t i o n i n a d i r e c t i o n normal to both the stream d i r e c t i o n and the model a x i s .  The other  mode was a t o r s i o n a l v i b r a t i o n i n the same plane about  the  10 mid-point  springs,  of  the  model.  As  the  m o d e l was  the weight  of  sion,  thus  limiting  model  and n e c e s s i t a t i n g  aerodynamic  drag  move d o w n s t r e a m tunnel  wall.  geometry  Both  was  the  and a  the  displacements,  measurements  of  of  the  Also,  the  model p l a n e  the  to  on  the  spring  the  light  were measured  source. off  a p o r t i o n of  i n s u c h a way length  of  a n d made  by  means  A card fastened  the  that  the  to  photocell  voltage  exposed  of  output  surface.  a l t e r e d : ' its"-:rSsp'6nse''to  the  accuracy  damping had been attached  o i l outside  satisfactory  due  to  the  o s c i l l a t i o n of  damping whose  of  model o s c i l l a t i o n .  level  The d i s p l a c e m e n t speed  meant  to  the  tunnel.  added  end o f This  s t a n d i n g waves b e i n g  linear  air  the  suspen-  of  dynamic  doubtful.  immersing a vane  with  mass  springs.  altered  i n the p h o t o c e l l  Additional  the  i n the  spring fastenings  effects  masked  dynamic  by  the  displacements  proportional to  oil  to  model b l o c k e d  w h i c h was  trough  stiff  model caused  these  inherent  a  sag  fairly  the  Effects  by  model caused  horizontal  adversely.  end of  surface  nearly  maximum a l l o w a b l e  relative  long photocell  the  the  on  Dynamic a  the  s u p p o r t e d by  that  the was  vane,  to  the  the  model  system  was  generated  resulting  dependent  system  in  on t h e  in not  in  the  nonamplitude  ^ of the  the  plane  of  p o s i t i o n of  oscillation the  displacement  11 measuring each  and damping apparatus  change  in air  speed,  The f i r s t apparatus the  was  that  the  plane  of  of  urements  and prevent  oscillation.  attempt  u n w a n t e d modes  air  the  restraint  side and  the the  tunnel  the  walls.  much f r i c t i o n t o  using  this  type  air  transmeasfrom  drag  disadvantage.  freedom;  the  they not  However,  type  static  on t h e  resulted  allow  fastened were  to  bearing could  give  friction,  shafts  three  interfere  bearing  shaft.  of  were  the  indicated  of  obvious  The m o d e l was  springs  could  slots  in  of  information  bearing  m o d e l by means  tunnel  were  The  with negligible  the  t u n n e l where springs  a  However,  lubricated bearings  low v i s c o u s  Supporting  bearings.  A preliminary test  a sliding journal  degree of  the  oscillation  abandoned.  proved that  a n d one  of  small ball  solution.  mounted o u t s i d e  to  means  the  one  the model  model by  a n d was  and a s u f f i c i e n t l y  of  the  might  to  freedom  had supported  arrangement  best  new  interest.  previous  advantages  of  of  mode o f  that, the  desired  design  the  on h y d r o s t a t i c  be  process.  of  g u i d e d by  oscillate  available  a slow  with  This would s i m p l i f y  this;, method had i n t r o d u c e d too to  degrees  restraining  with observations  aluminum shafts  model  i n the  adjusted  o s c i l l a t i o n and p e r m i t t i n g only  mode o f  A previous  number  reduced,  lational  interfering  be  making t e s t i n g  consideration  model s h o u l d be  single  had to  no  restrained  longer  with  the  the  slightly  s t i l l  required  inflow;  elastic in  the  12  top and bottom panels of the t u n n e l to admit the model attachments. To reduce the o s c i l l a t i n g weight  of the model  and mounting, an aluminum shaft was chosen.  This suggested  the p o s s i b i l i t y of r e p l a c i n g the o i l trough damping w i t h an eddy c u r r e n t damping d e v i c e .  Although t h i s  damping d e v i c e had been used e x t e n s i v e l y  type o f  to p r o v i d e damp-  ing f o r meter movements and a n a l y t i c a l b a l a n c e s ,  little  i n f o r m a t i o n c o u l d be o b t a i n e d on the p r o p e r t i e s of eddy c u r r e n t damping.  A magnetic eddy c u r r e n t damper of a form  s u i t a b l e f o r use w i t h the proposed apparatus was made and t e s t e d , u s i n g an aluminum s h a f t T h i s damper proved completely  as the moving element.  s u i t a b l e and gave almost  p e r f e c t l y v i s c o u s damping. I t was d e s i r a b l e that any method of measuring the dynamic displacement of the model should i n v o l v e no p h y s i c a l connection to the model.  S e v e r a l methods  were  considered: 1.  an a d a p t a t i o n of the p h o t o c e l l method.  2.  a d e v i c e u s i n g the p r i n c i p l e of differential  3.  a velocity  the  transformer.  s e n s i t i v e d e v i c e such as  the M i n n e a p o l i s - H o n e y w e l l LVsyn transducer. 4.  a movable c o r e v a r i a b l e r e l u c t a n c e transducer.  13 5. Each of  a  tubular  t h e s e methods  involved  2.  a  3.  an  lack  of  inherent  insufficiently  shaft  the  as:  linearity. large  transducer  model mounting  such  weight.  range.  these disadvantages  electromagnetic  3.2  disadvantages  a prohibitive additional  coupling  was  capacitor.  1.  To o v e r c o m e  of  variable  a  using  system  as  variable  the  the  aluminum  movable  element  developed.  The W i n d  Tunnel  The w i n d t u n n e l a t Columbia  is  tunnel.  Velocities  second less  and  than  tunnel  low s p e e d ,  150  0.5% as  across  pressure  can be  is  read  is  to  per  The  tunnel  motor  4 feet  drag  calibrated  the  inches  on a  by by  27  Betz  inches  6 inches for  velocity  by  a  d r i v i n g a commercial  The  level  test.  of  7:1  with  corner  is  less  by  0.25%.  direct a  cross-  growth.  than  fan with  which  fillets  inches  layer  dif-  ratio.  section  t o 4 3/4  of  The  pressure  test  horsepower  axiflow  per  micromanometer  boundary  of  15  the  section  water.  compensate  powered  against  contraction  measured  variation is  v a r i e d between  type  sphere  from 6 inches  spatial  return  i n d i c a t e d by  varying  The  British  turbulence  0 . 0 2 mm o f  to  turbulence  of  with a  is  inches  University  second  section  4 3/4  36  low  can be  feet  velocity  ferential This  a  the  current  Ward-Leonard  system of speed c o n t r o l .  The aerodynamic o u t l i n e of  the  tunnel i s given i n F i g . 1. 3 . 3 The Model Mounting System  The h y d r o s t a t i c  a i r b e a r i n g supports a load by  means of the p r e s s u r e f o r c e s  caused by i n t r o d u c i n g , h i g h  p r e s s u r e a i r between the l o a d c a r r y i n g s u r f a c e s . pressure a i r is  i n t r o d u c e d i n t o the j o u r n a l type b e a r i n g  through a number of e q u i d i s t a n t ence.  h o l e s around a c i r c u m f e r -  Each of these h o l e s c o n t a i n s  orifice.  a small  regulation  The a i r flows a x i a l l y from the h o l e s to  ends of the b e a r i n g . operates,  the  To understand how t h i s b e a r i n g  c o n s i d e r an a x i a l segment of the b e a r i n g formed  by c u t t i n g lengthwise on  either  s i d e of a h o l e along a  l i n e midway between i t and the adjacent h o l e . symmetry there w i l l be no flow across the b e a r i n g i s  centered,  these l i n e s when in this  is a f a i r approximation.  segment i s shown i n F i g . 2 ( a ) .  A i r from a r e s e r v o i r  p r e s s u r e p flows through the o r i f i c e pressure  Due to  and c o n s i d e r i n g the flow  segment as one-dimensional  where i t s  High  at  i n t o the b e a r i n g  i s p.. The a i f . then flows:: a x i a l l y  ends of the b e a r i n g which are at atmospheric p r e s s u r e Along the l e n g t h of the b e a r i n g , v i s c o u s f o r c e s progressively  This  reduced the p r e s s u r e from p  (  the  p.. 2.  have  to p^ .  As a l o a d i s a p p l i e d to the s h a f t , e c c e n t r i c w i t h r e s p e c t to the j o u r n a l .  to  i t becomes  The gap h between  15 the  shaft  load. ing  This  the  increased  through  bearing. is  and j o u r n a l decreases  the  the  reservoir  effect  decrease  the  viscous  pressure  occurs  drop  p^ i s  pressure.  the  load.  b e a r i n g was source  idea  sides  obtained  of  the  too  small a bearing  rigidity ing, s i z e . walled in  was  the  It  aluminum shaft  bearings  large  t o be  indicated large  as  missible  model,  to  the  given  the  w i t h two b e a r i n g s between  hand,  thin result  the  Two  tunnel, per  shaft.  bearings.  apparatus  amplitudes  increasing  in greater  bear-  machined.  outside  to  amplitude would r e s u l t  the  r i g i d i t y and  was  other  this  inch diameter  obtain  (27).  the  shaft  that  On t h e  of  Therefore,  the previous  desirable  Laub  result  obtained with  possible.  by  would  conveniently  shaft  sup;-  conditions  Results it  and i t  in selecting  either  to  differential  bearing  optimum  one a t  attached  bearing  dimensions  were used mounted t r a n s v e r s e l y  The m o d e l was  the  The  relative  shafts  the  Since  is  ample  the  orifice  analysis  a one  would give  enough  end o f  of  requirements)  that  of  decreased.  t o be p r a c t i c a l .  decided  flow-  p increases.  the  determining factor  was  out  the  a pressure  for  the  air  Information from  designing  (niiaximum l o a d w i t h m i n i m u m a i r in  of  from Laub.  indicated that  is  load side  A quantitative  A general  )  the  flow  through  (p^'-p  Therefore,  m a i n t a i n e d on o p p o s i t e  ports  flow  opposite  on  the  constant,  on t h e  side  forces  and decreases  volumetric  pressure  opposite  is  bearing  When t h e  reduced,  the  on t h e  the  (25) as per-  difficulties  in  16 a l i g n i n g the b e a r i n g s , sufficient  rigidity.  measuring d i s p l a c e m e n t s ,  and o b t a i n i n g  The apparatus was f i n a l l y designed  a double amplitude of s i x  inches.  Large d i a m e t r a l c l e a r a n c e s were d e s i r a b l e bearings  i n the  to make b e a r i n g alignment l e s s c r i t i c a l than f o r  bearings which gave optimum e f f i c i e n c y . i n d i c a t e that a r e l a t i v e l y Two rows of o r i f i c e s  help offset  T h i s appeared to  long b e a r i n g should be used.  with e i g h t o r i f i c e s  i n the f i n a l d e s i g n .  per row were used  These would admit s u f f i c i e n t  the e f f e c t  critical,  should not be  This m a t e r i a l would machine e a s i l y  and would  make the d r i l l i n g of the h o l e s f o r the o r i f i c e s  simpler.  the p l a n had been to use 0.020 inch diameter  but t h i s proved too d i f f i c u l t w i t h the  available.  to  the s t a t i o n a r y bushings were manufactured from  mild s t e e l .  Originally  air  of the l a r g e d i a m e t r a l c l e a r a n c e .  S i n c e the hardness of the b e a r i n g s u r f a c e s  holes,  for  A broken d r i l l  facilities  could ruin a bearing bushing.  The f i n a l bearings used a 0.040 inch diameter  hole.  The o r i f i c e was formed by g l u i n g 0.001  inch  shim s t o c k over the d r i l l e d h o l e s and punching the  orifice  h o l e with a sharpened s t e e l w i r e . The f i n a l design of the b e a r i n g i s shown i n Fig.  2(b).  The c i r c u m f e r e n t i a l r e l i e f around the  outside  of the bushing formed three s i d e s of the a i r passage which p r o v i d e d a i r to the  orifices.  The s t e e l bushings a r e clamped i n t o  split  17  aluminum blocks which both h e l d the bushing and formed the outer s i d e of the c i r c u m f e r e n t i a l a i r - passage. b l e d b e a r i n g and holder i s  IThe,' assem-  shown i n F i g . 3 .  The bores of the b e a r i n g h o l d e r s were machined parallel  to the bases and p e r p e n d i c u l a r to the f a c e s of  the h o l d e r s .  Then,  the outer s u r f a c e s  of the bushings  were  machined on a mandrel to ensure c o n c e n t r i c i t y w i t h the bushing's  bore and a c l o s e f i t  i n the  housings.  In the o r i g i n a l t e s t b e a r i n g an u n s u c c e s s f u l attempt had been made to a l l o w adjustments alignment.  T h e r e f o r e , f o r the f i n a l  to b e a r i n g  bearing support,  it  was d e c i d e d to " b u i l d i n " alignment,by machining the support to a s s u r e permanent alignment.  Warping of the base d u r i n g  machining was always a problem, u n t i l I f i n a l l y the base was made as i n F i g . 4.  The rough machining was done b e f o r e  w e l d i n g , and the f i n a l machining of the b e a r i n g s u p p o r t i n g s u r f a c e s was done with a s i n g l e s e t - u p on a v e r t i c a l m i l l i n g machine. The two channels on which the p a i r s of  bearings  were mounted b o l t onto the upper and lower panels of t e s t s e c t i o n w i t h 2\ X 2\ them at the ends.  i n c h angle i r o n b o l t e d between  Screws to a d j u s t the p a r a l l e l i s m of  s e t s of bearings are l o c a t e d at the i n t e r s e c t i o n of lower channel and the a n g l e s .  the  the  The lower channel may a l s o  be moved i n the stream d i r e c t i o n to ensure that is v e r t i c a l .  the  the model  18 The s p r i n g s mounted on the  are  angles.  fastened  Light,  provide  attachment  for  the  shaft.  D.etails  the  clamps  are  given  of  in Fig.  tubing w i t h 0;035 were  had been  Rand 2 - s t a g e into  carried ted at  by  available is  and  The a i r  foot  tunnel at  the  the  supply  test  slightly  it  air  perfectly  to  of  m o d e l s .-  the  to  the  found  round.  To o v e r c o m e  these  reduced  i n diameter  by  & 7X8  Ingersoll-  VH^-2,  A i r from the  pumping tank  a distribution manifold The maximum  i s 118 p s i . reduce  is  only  in Fig.  orifice that  gradually  the  circular.  the  of  size  the  operated  the a i r to  intermittently.  6.  was  in  the  bearings  0.005 inch  bearing  mounts  diameter.  did  deform  experienced  required alignment.  The  in alum-  t o w a r p : a r i d go ,put : 6 f  difficulties  trial  loca-  pressure  d i f f i c u l t y was  started  is  pressure  However,  diametral' clearance  deal  of  surface  s u p p l i e d by an  tank.  i s shown the  original  11 3 / 4  section,  and the  was  was is  compressor  system  also  model,  surfaces  the  model  and m a i n t a i n i n g the  inum s h a f t s  clamps  inch aluminum  a l l  manifold to  and a g r e a t  obtaining  tube  compressor the  0.003 inches  In p r a c t i c e  The o u t e r  storage  hose  Originally was  ends  made f r o m o n e  bearing  compressor,  t h r o t t l e d at  60 p s i  the  flexible  the  were  and the  a 250 c u b i c  hooks  aluminum  and the  and the  inch wall.  for  Air  soldered  springs  g r o u n d down u n t i l  removed  adjustable  5.  The s h a f t s  shafts  to  and e r r o r  :  the  shafts  until  a  were  workable  19 clearance 0.010  was  inch.  increase  achieved. The  the  larger  orifice  Fig.  7.  shown  3.4  and the The  in Fig.  of  section  transformer  wound on c o a x i a l them.  u s e d as  the  Either  of  generated shield  bearings  transducer,  is  the  transducer  and  magnetic  contained  equipment  is  in  is  essentially  the  inner  or  outer  coil  an  coils  forms w i t h an a n n u l a r  the  the  c o u p l i n g between  aluminum shaft secondary  length  i n the  the  upper  gap  may  be  coils  is  primary winding.  and hence  function  to  inch.  a n d some o f  cylindrical  i n s e r t i n g the  ling,  the  advisable  having primary and secondary  The m a g n e t i c by  0.010  it  approximately  Measurements  The d i s p l a c e m e n t  between  made  was  8.  Displacement  air-core  to  displacement  test  clearance  clearance  size  A photograph damper,  This  shaft  adjacent of  of  t h i s was  damping  i n the b e a r i n g s .  quency,  the  voltage  r a t i o between  effects  also  greater  are  the  tube  is  the the  is  the  The h i g h e r  of  Eddy  the  in  a  currents and  the  exciting and  coup-  The  frethe  shaft^positions/  the  as  coil.  compared to  shielding effect,  thickness  The  decreased  secondary  negligible  extreme  varied  from the primary  r e d u c e d w i t h an i n c r e a s e  The w a l l  annulus.  inserted.  power  p o r t i o n of  effect  the  voltage,  absorb  damping  into  the  End  frequency.  aluminum tube  affects  20 the performance of the d e v i c e ,  with t h i c k e r tubes g i v i n g  s m a l l e r end e f f e c t s and g r e a t e r s h i e l d i n g . inch w a l l tubing proved to be too t h i n f o r  The 0.035 satisfactory  o p e r a t i o n and an aluminum i n s e r t was machined to f i t the end of the s h a f t  passing  i n t o the  inside  transducer.  S i n c e the s h i e l d i n g e f f e c t does absorb some power from the primary the input impedance of the d e v i c e does not remain c o n s t a n t .  This e f f e c t  r e s u l t e d i n poor performance  when the transducer was powered by an o s c i l l a t o r with low an output  too  impedance.  Another e f f e c t  observed was that the  inductance  of the secondary c o u l d produce a resonant c i r c u i t when coupled to leads or a d d i t i o n a l c i r c u i t r y which had ficant  capacitance.  construct  a  For t h i s reason i t was d e s i r a b l e  the secondary of a s i n g l e w i n d i n g .  This  signito  resulted  i n a low impedance o u t p u t . Satisfactory  o p e r a t i o n was o b t a i n e d with f r e q u e n -  c i e s from 10 kc to the frequency h i g h enough to approach resonance oscillator  i n the c i r c u i t , u s i n g a Hewlett-Packard 202C for  excitation.  The transducer i s shown i n F i g . 9 and the fications  of the u n i t used i n the a c t u a l experimental work  are presented  i n F i g . 10.  The s i g n a l from the transducer i s  essentially  a h i g h frequency c a r r i e r amplitude modulated by the placement.  speci-  The s i g n a l from the transducer was  dis-  rectified  21 or d e t e c t e d by means of a f u l l  wave instrument r e c t i f i e r .  The f u l l wave r e c t i f i e r was used so that  the d e s i r e d  ing c o u l d be o b t a i n e d with a minimum of c a p a c i t a n c e  filteri n the  circuit. I t was d e s i r e d to use a M i n n e a p o l i s - H o n e y w e l l model 916 V i s i c o r d e r  to r e c o r d the dynamic displacement  s i g n a l f o r time-amplitude r e c o r d s . the high frequency galvanometers  The d i f f i c u l t y was  for this  that  r e c o r d e r had an  input impedance of about 35 ohms and r e q u i r e d a source resistance  of 3^100 ohms.  T h i s meant that  the  displacement  s i g n a l c o u l d not be f e d d i r e c t l y i n t o the r e c o r d e r without seriously  l o a d i n g the transducer and a f f e c t i n g  its  perform-  ance.  A t r a n s i s t o r i z e d DC a m p l i f i e r was c o n s t r u c t e d  effect  the matching.  This a m p l i f i e r had a u n i t y  gain and a c u r r e n t g a i n of approximately 5000.  to  voltage The c i r -  c u i t chosen was a balanced d i r e c t c o u p l e d , two stage emitter f o l l o w e r a m p l i f i e r . f i e r was f l a t stable.  The response  over the range of i n t e r e s t  of t h i s  ampli-  and was reasonably  One unsought advantage of the a m p l i f i e r was  that  the c u r r e n t output of the a m p l i f i e r was s h a r p l y cut o f f a safe v a l u e f o r the galvanometer.  This meant that  galvanometer c o i l s c o u l d not be burnt out by too  at  the  strong  a s i g n a l being a c c i d e n t l y a p p l i e d to the a m p l i f i e r i n p u t . S i n c e the b a s i c s i g n a l  i s of constant p o l a r i t y ,  and the V i s i c o r d e r must be d e f l e c t e d to use the f u l l  six  to both s i d e s of  inch c h a r t w i d t h , a v a r i a b l e b i a s  zero  22 circuit  was  i n s e r t e d between  The amplifier beam  transducer  11  Magnetic  shows  Eddy  currents  piece  outer  pole  the 502  form of  the  dual-  D.C.  displacement  amplifier.  means  The dampers  aluminum shaft  fehaft  damper  dissipate  magnetic the  shaft  used  to  is  shaft  passes.  while  support  a variable  Although care  the  of  create passes.  energy  from  those  possible,  ignored. coils  the  to  and the this  One  pole  passes  down  inner pole  current  taken  inside  source  choose as  was  energizes  from  l i t t l e  s y s t e m was  dampers c o u l d be  the  piece  the  residual  far  greater  r e s i d u a l magnetism a  an  rod connecting  damping e f f e c t  To o v e r c o m e of  the  which r e t a i n  the  anticipated  other  for  experimental  induced across  steel  direct was  i n the  field  the  to  materials  the  the  piece  than had been  whereby  the  the  i n t r o d u c e d by  w i n d i n g on t h e  magnetism as  be  of  A copper  damper.  available  i n the  through which  and connected  not  to  system.  the  surrounds  tube.  schematic  through which  program a s t a t i o n a r y gap  diagram of  current;dampers.  induced  oscillating  annular  block  d a m p i n g was  eddy  field  In  the  signal  amplifier.  Damping  electromagnetic a magnetic  the  and the  Additional  the  i n p u t and the  and the  oscilloscope.  instrumentation  the  detector  were monitored w i t h a T e k t r o n i x model  Fig.  3.5  the  could  arranged  switched  over  to  23 a variable a  greater  then  source.  magnetic  slowly  netism. and  A.C.  decreased,  spacers  to  residual  reduce  field  desired  during  the  voltage  than  D.C.  mounted  actual  This  the  currents, viscous damper shown  3.6  shown  in Fig.  found  the  that  in this  in  square  measurements t o p by  two  .Fig.  was  upper  end of  instrument  to  to  the  be  by  residual  mag-  bolts  removing  D.C.  was  any  current  carried  almost  to  out  viscous  eddy  12,  as  current  and the  entirely  For higher  configuration  on t h e  of  the  of  it  vis-  currents  was  for  effects  lower  are  damper.  electrical  the  model.  springs  The  supply  connected  transformer  to  the of  a  is  coefficients  direct  turntable.  a Daytronic  ± 0.040  the  The of  the  102-80  transducer.  up t o  force  mounted a t  displacement  displacement  displacements  force  obtaining  The m o d e l was  m o d e l by means  measured  lateral  w e r e made b y  i n f e r r e d by m e a s u r i n g  differential  the  and  Transducer  section  leaf  force  had g i v e n ,  13.  Lateral Force  the  set  amplitude.  close  The measurements for  give  paths.  same p r o c e d u r e  as  showing  i n nature is  of  s t i l l  thus  to  experiments.  and independent d a m p i n g was  flux  b u i l d i n g up t h e  The d a m p i n g was cous  raised  source  erasing  were always  and s l o w l y  is  using non-magnetic  extraneous  levels  value.  the  effectively  The damper was  Damping  the  field  The A . C .  This inches.  24 The s t a t i c by  friction  clamping  rested  the  on the  pan of the  and b r i n g was  pointer  to  zero.  and  transducer  displacing  the  weight  the  the  taken w i t h ness  of  the  zero  readings  the  were to  pivot  clearance  lower  end o f  between  Wake M e a s u r e m e n t  the  enough  given  t o be used  position.  the  balanced  to  balance  one  taken  pan  after  letting  it  adding  weight  side.  the to  repeated,  5 mg.  The  the  removed  the  condition  for  The  readings  establish  the  stiff-  and i n c o m b i n a t i o n w i t h  the  m o d e l was  adjusted  friction.  is  s u p p o r t e d by  vertically  model and the unit  the  tunnel  shown  to  a  vary  panels.  in Fig.  14.  Equipment  m o d e l 3CR4 c o n s t a n t  The v e l o c i t i e s  was  were  plunger  to b r i n g  plunger  Wake m e a s u r e m e n t s tion  hull  the  measuring  plunger  d i r e c t i o n and  were used  w h i c h c o u l d be  its  the  r e a d i n g was  i n the  spring,  measured  then added  from each  establish  The f o r c e  3.7  was  dial  taken  rest  the  was  Weights  balance  vertically  With  5 mg. w e i g h t  to  to  This procedure pan.  that  balance. to  in either  transducer  The conical  transducer  indicator  coming the  pah  A 5 mg. w e i g h t  readings  balance  opposite  other  plunger  i n s u c h a way  laboratory  balance  slowly.  indicator  transducer  thert. a d j u s t e d  come b a c k to  a  the  transducer  the  the  transducer  then a p p l i e d to spring  of  by  w e r e made w i t h a  resistance this  Flow  r a t i o hot  wire  instrument were not  quantitatively,  but  sufficed  Corpora-  for  anemometer.  accurate the  25 measurements instrument also  of  was  turbulence viewed  on the  If  vibration output  t h e wake  analyzer  current  lifier.  was  was  used  with  output  the  oscillator  of  frequency  by  means  of  be  measured  wire  the  tunnel.  truding  through  be p u l l e d  i n or  moved  longitudinally.  was  Vibration  recorded,  the  and  the  t r a n s i s t o r i z e d D.C. recorded were  on t h e  to  Visicorder.  b e made  on an  amp-  the  oscilloscope  to  the  unknown  The f r e q u e n c y  frequency  could  also  trace.  c o u l d be  moved  laterally  of  the  model m i d - p o i n t  The p r o b e  was  mounted on an arm  i n the out  The  side  and the  of  the  and  from  tunnel.  pro-  The  arm m o u n t i n g c o u l d  arm  be:  Models  The m o d e l s  The one  probe plane  a slot  could  and  amplifier  compared  matched  figures.  i n the  outside  the  were  7  longitudinally  The  were  f r o m :;the V i s i c o r d e r  The h o t  to be  this  H e w l e t t - P a c k a r d 202C o s c i l l a t o r .  was  Lissajous  the  measurements  signals the  was  of  directly, 1554A  a voltage  The a m p l i f i e d s i g n a l s  turbulence  longer  as  a m p l i f i e d by  filtered  thin  oscilloscope  turbulence  When f r e q u e n c y  at  The o u t p u t  f i l t e r e d w i t h a G e n e r a l R a d i o model  Analyzer.  3.8  frequency.  ends  to  a l l made f r o m w o o d w i t h m e t a l  allow attachment  inch square models  were  were  m o d e l was  to  the  s o l i d spruce,  made f r o m c e d a r ,  aluminum s t r i p s  glued  clamp on the  on t h e  shaft.  while  the  hollowed out,  and  faces.  These  tabs  strips  had  26  covered the l i g h t e n i n g holes and ensured sharp edges. A t y p i c a l model i s  shown i n F i g . 15.  IV.  EXPERIMENTAL PROCEDURES AND RESULTS  4.1  C a l i b r a t i o n Methods Wind Tunnel V e l o c i t y  A standard p i t o t  tube mounted i n the t e s t  was used to measure the v e l o c i t y  head t h e r e .  section  The p r e s s u r e  was measured on a Lambrecht i n c l i n e d manometer and compared w i t h the p r e s s u r e a c r o s s  the c o n t r a c t i o n s e c t i o n as measured  w i t h the Betz micromanometer.  The zero of both manometers  was checked between each r e a d i n g . sufficiently  It was found that  it  a c c u r a t e to r e g a r d the Betz d i f f e r e n t i a l  as equal to the v e l o c i t y  was  reading  pressure.  Displacement Transducer  The  l i n e a r i t y of the displacement  transducer  was checked by slowly moving the aluminum shaft its  full  through  t r a v e l w h i l e r e c o r d i n g the transducer output on  the V i s i c o r d e r , s e t t i n g a b l e to the shaft  speed.  the paper speed to a v a l u e comparThe paper d r i v e u n i t from a Brush  pen r e c o r d e r was used as a capstan to p u l l a c o r d a t t a c h e d to one end of the s h a f t . and  A second cord weighted  a t t a c h e d to the other end of the s h a f t  constant  on the end  maintained a  l o a d on the paper d r i v e u n i t as w e l l as m a i n t a i n i n g  27 tension found  i n the cords.  t o be l e s s  than  The d i s p l a c e m e n t  transducer  1% o f t h e maximum  Recorder  gear r e d u c e r  u n i t was employed  for  time base c a l i b r a t i o n s .  attached  contactors  Contactors  were c o n n e c t e d  was one p u l s e  pulses  were  was m e a s u r e d .  of s i x  When t h e two  i n s e r i e s so that a p u l s e  only  c l o s e d a t t h e same t i m e t h e  per second.  to the recorder  at frequencies  c y c l e s per second.  r e s u l t e d when b o t h c o n t a c t o r s  pulses  time r e f e r e n c e  t o two s h a f t s t u r n i n g a t 360 rpm a n d 300 rpm.  c y c l e s p e r second and f i v e  connected  to a multiple shaft  to give  T h e s e , b y t h e m s e l v e s , gave p u l s e s  result  displacement.  Time B a s e  A s y n c h r o n o u s motor a t t a c h e d  recorder  e r r o r was  This pulse  and the l i n e a r  I t was f o u n d t h a t  source  distance  was  between  the time bases o f  t h e V i s i c o r d e r were s t a b l e a n d a c c u r a t e .  Lateral  The with to  calibration  t h e model i n p l a c e  Transducer  of the f o r c e transducer  i n the tunnel.  was done  Threads were  fastened  t h e m i d - p o i n t o f t h e model, r u n h o r i z o n t a l l y o u t f r o m t h e  model .to e i t h e r s i d e o v e r tunnel and  Force  floor  fastened  t h e e n d o f beams p i v o t e d  a t an i n c l i n a t i o n t o s c a l e pans.  o f 45 d e g r e e s  on t h e  to the v e r t i c a l  Weights were then added  on  one s i d e , t h e n on t h e o t h e r ,  of  the transducer  to obtain a  indicator against  first  calibration  t h e f o r c e on t h e m o d e l .  28 The  c a l i b r a t i o n turned out  range  of  the  calibration  4.2  Test  to  differential set  up f o r  be  linear  transformer.  the  force  level  accomplished the  had to by  altering  was  removed  as  cleared  could  raise was the  on  the  shows  the  were  on,  i t was  Measurements  the  desired  the  settings,  supply  made. its  was  Since  was  This  After  impact  then  the  to  clear  the  to  switched  magnetism magnet  on a n d  any  on t h e  magnet  r e s i d u a l m a g n e t i s m when  desirable  was  setting  residual  switched  any  the  value.  supply  3.5.  run  supply  and the  in section  of  power  The power  power  level  the  magnet  often  test.  supply  the was  gauge a t After  tunnel  to  adjusting  the  After air  set  the  the  adjustments  the  be  outlined  small  during  17  a velocity-amplitude  required current.  without  current  to  first  off  had been  Fig.  working  Procedures  Preliminary  give  the  transducer.  Velocity-Amplitude  damping  over  air  speed  model  started  means  of  level  had been  t u r n e d on a n d t h e  the  test  starting was  the  pressure  tunnel at  increased  with f i f t y  behind the  set  for set  the for  test 60  psi  section.  oscillating.  a scale  immediately  damping  in  large  minimum speed steps  The a m p l i t u d e s d i v i s i o n s per  until  the the  were measured inch  spring mounting bracket  by  mounted on t h e  shaft.  29 One edge o f t h e b r a c k e t was c h o s e n position  at either  conditions  as a r e f e r e n c e and i t s  end o f i t s t r a v e l n o t e d .  t h e end p o s i t i o n s  U n d e r most  o f t h e b r a c k e t were  visible.  When d i f f i c u l t y was e n c o u n t e r e d  amplitude  a General Radio  S t r o b o t a c was  clearly  i n observing the  employed.  From t h e s p e e d where o s c i l l a t i o n s w e r e f i r s t o b s e r v a t i o n s were made a t s m a l l i n t e r v a l s air  speed  minutes The  until"'motion stopped.  were r e q u i r e d f o r t h e a m p l i t u d e  displacement  corder  recordings Readings  was  o f about  t o become  one m i n u t e .  f o r i n c r e a s i n g a i r speeds  with  Comparison o f t h e s e had beenireached.  limit  cycles  limit  o f a second  up t o t h e l i m i t f o r  then: s e v e r a l c h e c k  For  each p o i n t  u n t i l the  c y c l e was r e a c h e d . stable  limit  t o t h e maximum a m p l i t u d e  and  f o r the apparatus  p o i n t s w o u l d be t a k e n  over  t h e range.  t h e s c a l e r e a d i n g s were s u b t r a c t e d t o o b t a i n and added t o check  on t h e m i d - p o i n t  oscillation.  When t h e m i d - p o i n t  didn't  t h e r e a d i n g was The  I f there  cycle, readings  the amplitude  taking  I f the  ( 2 5 ) , r e a d i n g s would  then; f o r d e c r e a s i n g v e l o c i t i e s  end o f t h e upper  w o u l d be t a k e n  stable.  were then t a k e n .  be taken f o r i n c r e a s i n g v e l o c i t y  no i n d i c a t i o n  several  r e c o r d s showed a s u d d e n jump a s i n a  two s t a b l e  the apparatus, lower  low s p e e d s  showed when a s t a b l e o s c i l l a t i o n  amplitude-velocity  first  of decreasing  t r a n s d u c e r s i g n a l was r e c o r d e d on t h e V i s i V  at intervals  system  At very  noted  check  of the  an e r r o r i n  indicated.  d a t a f o r v e l o c i t y - a m p l i t u d e r u n s were p u n c h e d  30 out  on IBM c a r d s  and  the  IBM 1620  the  calculations  and  tabulate  computer  the  runs.  and i n s t r u m e n t a t i o n  funs  same a s  The  difference  measurement Visicorder placement a  large  the  the  of was  the  in this  time  transducer,  steady  velocity  amplitude.  r e c o r d was  equal  to  the  double  stopped  by  shutting  off  the  started  by  suddenly  turning  wire  the  model and  to  was  chosen  to  give  the  signal  due  The  was  filter  filter  to  one  it  that  an  accurate  means o f  allowed  input of  supply  rbime-  velocity-amplitude  to  the  the  to  the  dis-  oscillate  measured  to  amplitude.  on  the  c a l i b r a t i o n of  width  air  the  at  as  in  displacement  Visicorder The model  the  was  bearing  and  again.  Wake  Measurements  probe  was The  positioned exact  steadiest  extraneous  tuned was  side.  the  switched  was  output  the  perform  Numbers  The h o t  analyzer  so  and the  was  Strouhal  for  The a m p l i t u d e was  case  adjusted  the  m o d e l was  transducer  (a)  the  c a s e was  To s e t  the  amplitude  for  and a m p l i t u d e by  desired.  to  Records  The a p p a r a t u s were  used  results.  Time-Amp1itude  amplitude  was  to to  the give  compared  downstream  position varied  signal.  To  eliminate  turbulence,  the  frequency  narrow band f i l t e r the  on t h e  and  position.  maximum a m p l i t u d e oscilloscope  from  with  and the  the  31 signal  from the  Lissajous from (b)  Hewlett-Packard o s c i l l a t o r .  figure  the  had been  oscillator  of  was  recorded  :  traversed  the  lateral  laterally  maximum v e l o c i t y  fluctuation,  analyzer.  This  the  center  the  row as  Eskinazi  of  vortex  the  applied to  scope and vertical signals  the  the  deflection, c o u l d be  has  be  angle  between  to  of  the  ds  ducer  wire  signal  signal  the  probe  points  rms m e t e r  on  related  Schaefer  as  a  to  and  reference of  the  applied to  r e l a t i o n between  from the  Lissajous the  through  180  oscillothe  the  two  figure.  distance  l o n g i t u d i n a l l y to  these attempts  The  the  shift  the  phase  degrees  or  360  degrees.  were f r u s t r a t e d  by  the  unstead-  wake.  versus  Angle  of  Attack  * The t a k i n g  tunnel  the  shown by  i n d i c a t e d by  signals  *y  the  find  wire  p o s i t i o n c o u l d be  phase  traversed  the  Both  the  obtained  probe  b o t h by  hot  horizontal deflection  f i l t e r e d hot  spacing  CT?  to  using  displacement  longitudinal  iness  the  (28). Using  signal  spacing  i n an attempt  frequency  by  frequency  Wake Geometry  the  (c)  the  stable  dial.  To o b t a i n was  obtained  When a  the  force  readings  o s c i l l a t i n g forces  unsteadiness stopped was  of  the  adjusted.  of zero  the  With  the  complicated  f r o m t h e wake  mean  force  was  lateral  reading tunnel  of  vorticity  force. the  With  force  running at  the  and  the  transdesired  32 speed was  the  indicated angle  noted.  eral  As a check  force  curve  for  of  zero  occurred the  it  force  top  force the  flow  4.3  the  at  the  and the through  tions were  of  the  were  of  to  A series end o f  clearance bottom  of  the  at  it  top gap  the  of  those the  filled  at  end h o l e s .  w i t h i n the  but  Experimental  any  possible (b)  speed. to  This  Due t o  investigate  b / h = 1.00;  force.  large  the  to  The angle  discrepancy in  setting  run: w i t h wall  various of  the  decreased  the  end e f f e c t s  at  effect  on  For  as  the  some  s m a l l as  completely due  to  posstop  end  and the  tests  condi-  tests  top.  Results  b / h = 0.50; air  the  was  accuracy, the  and  an e r r o r  made  grease  at  measured  bottom.  Velocity-Amplitude  (a,C)  to  Any e f f e c t s  r u n w i t h a minimum gap  lat-  indicated  t o p was  the  l i m i t s of  the  model and the  m o d e l was with  in  the  m o d e l h a d a much g r e a t e r than  peak  zero  tests  the  increased  force  for  Where any  relate  lateral  the  z e r d was  determining  consistent.  the  When t h e at  6f  possible  transducer  gap  sible  was  between  clearance the  were  side  indicated angle  transducer.  clearances tunnel.  the  b o t h methods  force  the p o s i t i o n of  on e i t h e r  mean c o m p a r e d w i t h results  w h i c h g a v e no mean  model d i d not its  Results  exhibit  plunging  low n a t u r a l frequency  the  The s q u a r e  vortex  resonance  section  plunged  it  at  was  region, for  any  air  not  33 speed  above  sented  a minimum s t a r t i n g  in Fig.  Section  2.1. the  from the  Appendix.  In  closely  range  a n d show  seen  with it  to  start  section  discovery  was  speeds  this  in  amplitude  were  two  so (c) as  range  the  limit  vortex  the  two  curves  between a part  these  the  origin lowest  as  to  the  do n o t  For the  the  air  lower  or  if  and  amplitude the  amplitudes.  asymptotically become  lower  large.  show  points  and low  the of  the  velocities,  approached.  The a m p l i t u d e v e l o c i t y 20, begin  range  (1)  This  does not  the  (25)  Brooks  the higher  velocities  small amplitudes is  by  tests.  approach  the  tests  velocity  velocities  However,  of  Oscillation  amplitudes.  damping l e v e l  cycles.  most  tried  amplitude,  the  points  In e a r l i e r those  result-  and  themselves.  model c o u l d a t t a i n  appear  2.1  over  of::.  solution  equation  Section  later  initial  the  pre-  method  experimental  of  stable  are  the  the  curve  among  from higher  resonance  in Fig.  of  reduced  two  large  very  is  but would remain at  points  the  b/h = 1.50; shown  for  the  approached  represent  that  that  with a  for  stable  curve  m i d way  v e r i f i e d by  through  The c u r v e  approach  theoretical  displayed  experimental  line  19  around nU/B = 0 . 6 0 .  from r e s t ,  released  range  a  the  r e d u c e d by  in Fig.  good agreement  discovered  square  The  with  The d a t a  i n t e g r a t i o n of  form the  damping l e v e l s  was  if  numerical  this  19,  line  quasi-steady  agree  is  in Fig.  The s o l i d  o b t a i n e d by ing  18 a n d  speed.  to  appear  curves  show a c o n c a v i t y to  approach  the  for  b/h =  upwards same  1.50,  and  asymptote.  34 The curves f o r the two damping l e v e l s a r e of d i f f e r e n t  form,  with the curve f o r the higher damping l e v e l having two s t a b l e l i m i t c y c l e s over the lower p a r t of the range. l i m i t c y c l e was very d i f f i c u l t  The lower  to o b t a i n as b u i l d - u p and  decay times were of the order of f i v e minutes or more. experimental p o i n t s do not reduce to a s i n g l e .  The  curve i n  the  (  same way as f o r b / h = 1.00.  The dashed l i n e r e p r e s e n t s  the  p o s i t i o n of the h i g h damping curve i f the r a t i o of damping l e v e l s i s used i n an attempt  to reduce the high damping  curve onto the low damping c u r v e ,  the only agreement  being  in s t a r t i n g airspeed U . Q  (d)  b / h = 2.00;  The curves f o r a l l damping l e v e l s  c l o s e together but a r e s l i g h t l y fall  divergent.  fall  The curves  i n order of the magnitude o f damping, b u t , as f o r  curves f o r b / h = 1.50,  they cannot be reduced by u s i n g  damping r a t i o s  on one c u r v e .  to f a l l  of the curves has become more evident  The upward  the the  concavity  i n the lower  portions  of the curves and the upper p o r t i o n s appear to approach straight (e)  l i n e s which do not run through the o r i g i n .  b / h = 2.50;  f o r b / h = 2.00  The same g e n e r a l comments as on the  apply h e r e ,  f o r the same a m p l i t u d e s .  although a l l a i r s p e e d s  In t h i s  about i t s  r e g i o n the model was observed to be  21)  curves  are higher  S t a r t i n g at U = 14 ( F i g .  t h e r e i s a s h o r t r e g i o n where the amplitudes have off.  (Fig.  22)  fallen twisting  l o n g i t u d i n a l a x i s and i t was found f o r t h i s  the v o r t e x shedding frequency f o r the s t a t i o n a r y model  airspeed  35 approached  the  longitudinal  n a t u r a l frequency  axis  of  this  mode c o u l d b e  this  phenomenon w i l l  (f)  b/h = 3.00;  unsteady, U  = 10.3  that  the  this  15.7  frequency appears  of  to  rectangles, line  not  (g)  Longer  of  of  In  of  m o d e l was  the  6.00 X 1.00  oscillation point  at  42% b a c k  was  observed  b/h =  2.50,  natural  Otherwise the  the  stationary  the  curve  preceding to a  straight  with b/h = 4.00,  these  a l l  sections  showed  rectangle  of  the  leading  edge.  showed  when  section  showed a f l u t t e r  80.5 fps,  4.75, any  torsional  shedding frequency  of  from  origin.  None o f  vortex  from the  for  and approach  Rectangles  a velocity  it  generally  t w i s t i n g about  A g a i n as  n a t u r a l frequency inch  curve  region  shown by  plunging i n s t a b i l i t y ;  the  5.1.  coincided with the  the  tested.  approached  this  e x c i t e d when t h e  trends  rectangles;  w i t h the  i n the  in torsion.  through  resonances  The  appearing  upward c o n c a v i t y  passing  and 6 . 0 0 , were sign  23).  the model  that  s e c t i o n were  (Fig.  frequency  follow  this  gap  the  interesting  in Section  large  v i b r a t i o n was  shedding  is  the  t h e m o d e l was p l u n g i n g a n d  discussed  w i t h no t r a n s l a t i o n .  mode o f  vortex  be  t o r s i o n about  It  excited while  cross-section  mid-point  model.  Amplitudes for  with a to  the  of  in  it torsion.  type  t w i s t i n g about  a  36  Time-Amplitude  Time-amplitude 2.0 by  (Fig. 24). fairly  ingtothe  The c u r v e s  lower  limit  just  cycle  over  the  given  b / h = 1.0  a slow  initial  period  of  almost  constant  to  stable  are  b u i l d - u p was the  amplitude, For very  to  start  longer was  lower  rectangles,  just  of  which  showed v e r y  before  the  the  rapid  Curves square of  values  lower  the  of  Brooks' angle  curves  much l o w e r result, of  the  attack.  1.00  force  curve  peak There  in Fig. that  build-up with  to  time  the required  speed. inch  The  section inflection  stationary.  coefficient numbers shape  measurements  than  air  a  Curves  Reynolds  force  presented  The  any  2 . 0 0 by  Force  four  The g e n e r a l  rapidly  b u i l d - u p w i t h one  lateral  obtained from pressure  However,  with  the  were o b t a i n e d f o r  interest.  that  of  at  cycle,  f o l l o w e d by  high velocities,  a m p l i t u d e became  Lateral  At  limit  then a r a p i d  established.  oscillating varied widely  typical,  point  o s c i l l a t i o n was  and  correspond-  slow.  build-up, generally  amplitude.  b/h = 1 . 0  characterized  very  low damping the a m p l i t u d e would i n c r e a s e maximum o n c e  for  For curves  maximum f o r  was  the  for  are  slow build-up i n i t i a l l y .  velocities there  curves  Results  due  i n the appears  by  is  for  the  i n the  range  similar  Brooks  (1).  27 w e r e o b t a i n e d to  Brooks.  curve to  be  occurs  to  at  Compared at  a shift  a of  the  37 peak towards a higher angle of a t t a c k as NR which i s  c o n s i s t e n t with the d i f f e r e n c e  increases,  between these curves  and the one o b t a i n e d by Brooks. Shedding Frequency Measurements  The shedding f r e q u e n c i e s b / h = 1.50,  3.0,  f o r the r e c t a n g l e s  with  and 4.0 were measured and the S t r o u h a l  numbers c a l c u l a t e d , as shown i n F i g . 25, which a l s o shows Brooks' r e s u l t s  f o r the square.  There d i d not  appear to be any l a r g e Reynolds number e f f e c t number f o r these s e c t i o n s . and 2.5  The r e c t a n g l e s  on the S t r o u h a l  w i t h b / h = 2.0  showed i r r e g u l a r shedding f r e q u e n c i e s  but d i d appear  to agree w i t h those shown i n F i g . 26. Shedding frequency measurements  were made f o r  o s c i l l a t i n g model u s i n g the frequency a n a l y z e r . ments made i n the wake showed evidence of the  the  Measure-  vortex  shedding frequency of the s t a t i o n a r y c y l i n d e r a p p e a r i n g , a p p a r e n t l y u n d i m i n i s h e d , i n the wake of the model.  oscillating  T y p i c a l c h a r t r e c o r d s f o r the f i l t e r e d and u n -  f i l t e r e d hot wire s i g n a l s case are shown i n F i g . 28.  f o r the s t a t i o n a r y and dynamic In the o s c i l l a t i n g c a s e ,  amplitude of the component at  the s t a t i o n a r y  the  frequency  v a r i e s at the frequency of o s c i l l a t i o n of the c y l i n d e r . T h i s high frequency component was only d e t e c t e d b e h i n d the body i n or near the wake.  A hot wire b e s i d e the body only  38 p i c k e d up t h e  low frequency  components.  Wake  No a d e q u a t e  measurements  obtained.  The u n s t e a d i n e s s  vented  the  taking  ted  locate  to  a plane  the  normal to  determine  of  points  phase  hot  spacing  V.  the  moved of  conditions readings.  as  wire  vortex  a phase signal  vortices  i n the It  to  try  street  had been  test  c o u l d have  to  to  use  attemp-  this  to  using  the  compare w i t h  as  in  the  by  the hot  section  been  pre-  fluctuation  Also,  reference  were  wake  behind  i n the wake,  l o n g i t u d i n a l l y i n the the  geometry  maximum v e l o c i t y i n order  the  wake  the  the  wire  longitudinal  obtained.  D I S C U S S I O N AND R E S U L T S  5.1  Discussion  One o f imental  w o r k was  applicability for  the  b/h  = 1.00  damping (Fig. the  of  and o s c i l l a t i n g c y l i n d e r s . signal  was  of  t h e wake  displacement of  of  accurate  the width of  stationary  Geometry  of  the  19).  curve  significant  an  i n d i c a t i o n of  that the  rectangular  so  most  quasi-steady cylinder.  c o u l d be  reduced  that  the  a l l  results the  of  l i m i t s of  a p p r o a c h was  The e x p e r i m e n t a l  using  the  ratio  the  of  points applied  fell  nearly  on one  The r e d u c e d p o i n t s  fell  in the  vicinity  the  quasi-steady  the  determined  points  p r e d i c t e d by  exper-  approach.  curve of The  for  39 longer  rectangles  applied the of  damping to  results the  of  curves  origin.  could ing  be  the  as  quasi-steady  they  ing  level,  the  ratio  onto  section  those  but of  it  the  was  one  intersect  along  this  onto  one p o i n t .  points  onto  the  r a t i o b/h  would  (Fig.  indicate  if  that  section  by m u l t i p l y -  and v e l o c i t i e s  constant that  this  through  rectangles  by  to  longer  the  a curves  a p p l i e d damp-  constant  rectangles  in Fig.  these  29.  was  curves  not  of  these  these  lines,  The p o i n t s  except  c o u l d be  perhaps  curves  to  the to  the  curve were  reduce  for  a l l  c o u l d be  the of  these  the  b/h = 2.00,  plotted  obtained for for  of  points  reduce  the  reduced  The r a t i o s  used  R to  for  and an  drawn t h r o u g h  origin  c o u l d be  compared  The c u r v e s  curves.  The r a t i o s  are  appearance,  l i n e s were  curves  30).  i n good agreement  longer  the  i n t e r s e c t i o n w i t h the  from s e v e r a l  shape  quasi-steady  this  l i n e from the  points  obtained  the  for  the v a r i o u s  the  from  the  separated  Brooks  see  Also,  enough w i d e l y  the  Straight  intersection with  were  of  a l l had a s i m i l a r to  curve.  to  lengths  made  expected  levels.  for  o b t a i n e d by  externally  approach an asymptote  any  evident  damping  would be  approach.  one c u r v e  was  rectangles  attempt  the  ribt  for  not  the  obtained from the  did  to r e l a t e  The c u r v e s  origin  be  The c u r v e s  There were  any  onto  that  dimensionless amplitudes  for  longer  i n f l u e n c e d by  degree  c o u l d not  reduced  constant.  to  not  the  the  approach alone the  were  against each  b/h = 1.50. reduced  to be  b/h This in  40 very  close  agreement A curve  the  points  to be  for  almost  between  over  their  c o u l d be  b / h = 1.50  a  straight  b/h = 3.50  fitted are  ignored,  and 4 . 0 0 .  indicate  that  curve  for  rectangle  the  point  of  s e c t i o n would not  shown  be  case for  reach be  zero  argued  somewhere that  aerodynamic the  the  force  i n the  1.50  damping  level  w o u l d be  tangle,  but  as  dynamic assume comes  response that  as  greater,  ence on t h e the  shorter  is  to  the  a l l  behaviour  rectangular rectangles  of  the  of  of  the  two  the  It  length  of  a  the  it  are  high  amplitudes, long  of  rec-  each. for  the  the to  rectangles a greater  From the  might be  square  for  reasonable  the  can  have  for  results  would have  cylinder.  cylinders,  is  It  should  curve  not  from the  them.  R would  those  stable  as  the  points  should  theory  the p r o p e r t i e s  geometry  such as  the  same m e c h a n i s m g o v e r n s  afterbody  t h e wake  Since  it  the  fitted,  d i s s i m i l a r to  consider  zero  from  If  rectangle  these  inference  that  for  too  show  of  b/h = 4.00.  quasi-steady  s h o w i n g some  rectangles  inch  cycles.  did in fact  One p o s s i b l e long  by  limit  reasonable  1.00  not  square, and t h e r e f o r e two s t a b l e  of  by  v i c i n i t y of  seem  oscillate.  b/h = 4.00.  ignored and a curve  curves  exhibit  it  not  If  abscissa  i n f i n i t e distance  T h i s was  were  the  an  the  b / h = 1.50  through  lay  words,  for  curve would  intersection with  In other  the  the  A reduction factor  origin.  to  through these points.  l i n e passing  would  that  lengths.  beinflu-  behaviour  concluded that short  enough  the  that  41 they  are  not  eous wake  influenced to  dynamics,  influenced  to a great As  which  the  occurs  from the  The s e c t i o n rapid tude  attack  for  from  that  becomes  will  for  instability force  once  amplitude between  constant  cause  limiting  is  little  angle  length  air  the  result  by  the  apparent  angle  effects to  that  the  angle  of  initial damping (that  section,  the  velocity-  relationship  of  attack  and  attachment. There oscillating Strouhal  it  is does  frequency  evidence not  stop  (1,14).  that  when t h e  model  shedding v o r t i c e s This  is  for  in  square of  ampli-  differ  changes,  the  the  of  reached  The shape  apparent  This  However,  is  used for  the  b/h = 2.00.  A strong  i n f l u e n c e d by  length,  and  speed.  attack  attack  at  increases,  evidenced  The  of  attack  t h e maximum  strong p o s i t i v e  here.  c u r v e w o u l d be  afterbody  of  order  effect  the  section.  angle  the  as  of  attack  a given  reached.  this  would have  damping, of  would have  of  in amplitude.  f o l l o w e d by a v e r y a  limits  angles  instantanare  angle  i n s t a b i l i t i e s as  a stationary  re-attachment) applied  strong  re-attachment  acceleration  the  afterbody  reduced for  increase  suddenly  lower  the  geometry.  time-amplitude record for  shows  initial  t h e i r wake  occurs  the  by  rectangles  suggests,  As  at  extent  longer by  re-attachment  plunge amplitude is seen  extent  in plunging.  re-attachment  is  the  P a r k i n s o n (2)  flow  amplitude  while  a great  starts at  the  shown by b o t h  the  r e -  42 frequency  measurements  oscillating  models  started  frequency  for  frequency  in torsion for  ceased  shed at  to  oscillate, In  the  i n the  the  case  of  ported  that  the  be  reverts  one  stationary  and by  twist  when  the  the  model.  If  resonance,  the  significant  frequency  difference  when  of  the  the  natural  cylinder  have  for  that  shedding  it  had  began  been  investigators  shedding frequency  to  fact  the  the  the S t r o u h a l frequency  vortex  the  model reached  t w i s t i n g mode w o u l d n o t  the  model  to  wake,  to  excited.  have  re-  oscillating  vibration.  between  the  lateral  force  This  two  could  types  of  oscillation. A comparison of presented  here  shows  1  for  with that  that "there-is  the  square.  important  to  o b t a i n e d by  the  the  quasi-steady  aerodynamic  close  to  whose  t o be  investigated of  c y l i n d e r s show  that  appreciably  possible effect 9 , 0 0 0 .  affect  range  a decrease  if  that  The measurements stationary  number  is  the  investigated,  the  27)  other  measurements  of  the  oscillating  or  although  Reynolds  for  number up t o rectangles  had been  is for  cylinder  for  the  number d o e s the  not  Reynolds  b/h = 4 . 0 0 there  NR  =  9 , 0 0 0 .  would have  taken  it  predicted.  shedding frequency the  effect  coefficients  S t r o u h a l number o v e r  with Reynolds  that  (Fig.  approach,  force  a R e y n o l d s number response  Brooks  curves  a S i g n i f i c a n t ReynoIds.number  To u s e  obtain  the  for  shown  It  is  this  them b e l o w N R =  is  43 5.2  Conclusions  From the  experimental  work p r e s e n t e d  here,  it  may  concluded: 1.  The q u a s i - s t e a d y results width well  for  rectangles  ratios above  These  gives  whose  about  1.0,  velocity  predicted  over  two  part  range  the  amplitudes  to  velocities for  vortex  solution  two s t e a d y  pf  adequate  depth at  The q u a s i - s t e a d y  = 1.00  litudes  are  the  resonance. b/h  approach  state  velocity  were  for amp-  range.  v e r i f i e d by  ex-  periment . 2.  The dynamic b e h a v i o u r tangles  is  not  quasi-steady eous wake 3.  The  which b/h 4.  In  amenable  approach,  geometry  amplitudes  limited  by  for  the  the  case  'capture" frequency  of  does is  not  the  the  lower  oscillation.  analysis  recby  instantan-  long rectangles angles  of  to  attack  at of  plunge.  oscillation, and the  Strouhal  i n t h e wake  occurring frequency  are  Rectangles  shown not  occur  the  considered.  occurs.  observed  longer  and the  plunging  plunging model, with  to  apparent  4 . 0 0 were  the  must be  re-attachment  over  of  of  the  simultaneously from the  plunging  44 The a p p a r a t u s performed  satisfactorily,  mentation appears wake  5.3  to be  for  although  necessary  the  experimental  for  The r e s u l t s that  a  great  deal  Future  of  this  more must  a plunging model.  more s o p h i s t i c a t e d for  the  of  Research  experimental  be  if  work  l e a r n e d about  The measurements  c o u l d be a c c o m p l i s h e d  proposed  an rms v o l t m e t e r  indicate  t h e wake in  with  time constant  were used  in  with  the  hot  wire  and frequency  analyzing  equipment  tain  the  lateral  A phase output phase  p o s i t i o n of  measuring  instrument  which depended angle  signal. tudinal  between  of  the  essential  to  the  l a t i o n s h i p between and f l o w  The transducer  the  signal  Hydrostatic  air  be  hot  an  wire  the  longi-  wake. wake of  visualization  t h e wake length  to  will  and the width  re-  ratio  investigated. damping and the  but  displacement  c o u l d be d e v e l o p e d  bearings  more r e f i n e m e n t  and the  determination of  attack,  well,  fluctuation.  of  i n the  electromagnetic  worked very  although  must  ob-  time average value  understanding  re-attachment  to  give  the  of  conjunction  to  program of  angle  a  developed  reference  vortices  An e x t e n s i v e  c o u l d be  on t h e  This would permit spacing  maximum v e l o c i t y  of  section  long  ful,  instru-  measurement  sufficiently  be  work  geometry.  Recommendations  4.2  developed  proved  to  be  w o u l d be h e l p f u l ,  very  further. success-  particularly  45 of  the  struct into  bearing  supports.  It  a bearing with only  recessed  analyzed  pockets.  by Lemon  (29).  might be  four  This  advantageous  adjustable  type  of  air  to  orifices  bearing  has  con-  opening been  46 APPENDIX  y y  The d i f f e r e n t i a l my Fy  e q u a t i o n of motion i s ;  + r y + ky = F  i s the aerodynamic F  1  y  ,2  = *«» i p V M  f o r c e on  r e l  t h e body,  given  by;  h s ( C ^ c o s a < + Cosine*)  However: V  rel  cos°<  Therefore; F Def ine  Cp  y  = - |pV|hs(C  L  + C tanc«) s e c « D  by: Cpy  = — ( CL + Cj)tano<) sec<x  Substituting: Fy = * p V C h s 2  0  I n t e g r a t i n g to f i n d oscillation: ^my  F y  t h e work o v e r one c y c l e f o r a s t e a d y  dy = 0  j£k y dy = 0 This  leaves: j^r  y dy = !/>V *hs j> C 0  p  dy  47 Assuming  simple harmonic motion: y = Acos© 9  = wt  y = — Ausin6 Also: dy = ^ d t dt y  d6 = Substituting  dt  these r e l a t i o n s :  jr (Awsinef Integrating:and  =  _  -MpV^hsCp Asin©  rearranging:  ^ = - ± C sinQ pVfhs 2 T T y  d©  FF  Introducing  d9  the d i m e n s i o n l e s s parameters:  Amplitude;  Y = A/h  Velocity;  U = V /(wh)  Mass;  n = ph s/(2m)  Damping;  B = r/(2mw)  Q  5  Then: 2TT  BY 1 a= / C_ s i n e d© nU " 2TT / ^ F y a  However:  and  tan^<  =  X V  0  _ ^_ A c j s i n S V« Y . Q  T h e r e f o r e we  can s u b s t i t u t e  to obtain:  .air sew  1 1 /_ fnY/B . „"| . _ ,_ - = — —=. / f n 4 — A — s i n G - J - s m © d© nU/B nU/B 2TTy [nU/B J nY/B /,„ T  r T  48 For a numerical of  steady  oscillation  or g r a p h i c a l  nU/B c a n be d e t e r m i n e d  i n t e g r a t i o n f o r some maximum  by  angle  attackcx,. For  angles  symmetrical  of attack,  C  sections  infinitesimal  can be a p p r o x i m a t e d  F  y This  at  by  ^ F y tar>cx. dtan  gives: BY _ 1 nU 2 a  d  CF Y dtanCX U v  or:  1  ° where U start.  Q  n Ldtanoy  i s t h e minimum s p e e d  f o r which o s c i l l a t i o n  may  49 BIBLIOGRAPHY 1.  Brooks, N. P. H .  Experimental I n v e s t i g a t i o n of the A e r o e l a s t i c Instability of B l u f f Two-Dimensional Cylinders. M.A.Sc. Thesis, U n i v e r s i t y of B r i t i s h Columb i a , J u l y , 1960.  2.  P a r k i n s o n , G. V.  "Aspects of the A e r o e l a s t i c Behaviour of B l u f f C y l i n d e r s " E n g i n e e r i n g I n s t i t u t e * of Canada, 1962 Annual General Meeting Paper No. 58.  Delany, N. K. Sorensen, N. E .  "Low-Speed Drag of C y l i n d e r s of V a r i o u s Shapes." NACA T e c h n i c a l Note 3038, 1953.  Relf, E. F. Simmons, L . R. G.  "The Frequency of Eddies Generated by the Motion of C i r c u l a r C y l i n d e r s Through a Fluid." Aeronautical Research C o u n c i l , London, R. & M. No. 917, 1924.  5.  Roshko, A,  "Experiments on the Flow Past a C i r c u l a r C y l i n d e r at Very High Reynolds Number." J o u r n a l of F l u i d Mechanics, v o l . 10, p a r t 3, May, 1961. 345-356. pp.  6.  Baird,  "Wind-Induced V i b r a t i o n of a P i p e l i n e Suspension -Bridge and i t s C u r e . " ASME Paper No. 54-PET-12, September, 1954.  R. C.  Farquharson,  8.  Gerrard,  F . B.  J . H.  "The I n v e s t i g a t i o n of Models of the O r i g i n a l Tacoma Narrows B r i d g e Under the A c t i o n of W i n d . " U n i v e r s i t y of Washington Experiment S t a t i o n B u l l e t i n No. 116, Part I I I , June, 1952. "An Experimental I n v e s t i g a t i o n of the O s c i l l a t i n g L i f t and Drag of a C i r c u l a r C y l i n d e r Shedding Turbulent V o r t i c e s . " J o u r n a l of F l u i d Mechanics,  50 v o l . X I , p a r t 2, September, 1961, pp. 244-256. Goldman, R. L .  "Karman Vortex Forces on the Vanguard R o c k e t . " Shock and V i b r a t i o n B u l l e t i n s , P a r t I I , U . S. Naval Research L a b . , Washington, D. C . , December, 1958.  10.  Humphreys, J . S.  "On a C i r c u l a r C y l i n d e r i n a Steady Wind at T r a n s i t i o n Reynolds Numbers." J o u r n a l of F l u i d Mechanics, v o l . 9, p a r t 4, December, 1960, p p . 603-612.  11.  K r a l l , G.  "Forced or S e l f - E x c i t e d V i b r a t i o n of W i r e s . " Proceedings Seventh I n t e r n a t i o n a l Congress of A p p l i e d Mechanics, v o l . 4, 1948, pp. 221-225.  12.  McGregor, D. M.  "An Experimental I n v e s t i g a t i o n of the O s c i l l a t i n g P r e s s u r e s on a C i r c u l a r C y l i n d e r i n a F l u i d Stream." U n i v e r s i t y of Toronto I n s t i t u t e of A e r o p h y s i c s , TN14, June, 1957.  13.  Macovsky, M. S.  "Vortex-Induced V i b r a t i o n S t u d i e s . " David T a y l o r Model B a s i n , Report No. 1190, J u l y , 1958.  14.  Smirnov, L . P. P a v l i h i n a . , M . A.  " V o r t i c a l Traces f o r Flow around V i b r a t i n g C y l i n d e r s . " Presented at the S e s s i o n of the I . M . A . Communications, Moscow, October, 1957.  15.  Toebes, B. H. Eagleson, P. S.  " H y d r o e l a s t i c V i b r a t i o n s of F l a t Planes R e l a t e d to T r a i l ing Edge Geometry." J o u r n a l of B a s i c E n g i n e e r i n g , ASME, v o l . 83, S e r i e s D, no. 4, December, 1961, p p . 671-678.  16.  Walshe,  "The Aerodynamic I n v e s t i g a t i o n f o r a S t a c k f o r the  9.  D. E . J .  51 Canada-India Reactor Project." NPL R e p o r t A e r o 3 9 5 , National Physical Laboratory, Teddington, Middlesex, November, 1959. Y.  " F l u c t u a t i n g L i f t and Drag A c t i n g on a C y l i n d e r i n a Flow at S u p e r c r i t i c a l Reynolds Numbers." Journal of the Aerospace S c i e n c e s , v o l . 27, n o . 11, November, 1960, p p . 801 - 814.  17.  Fung,  C.  18.  Davenport,  A,  19.  Cheers,  A.  20.  Steinman,  21.  Bleich,  F.  "Dynamic I n s t a b i l i t y of Truss-Stiffened Suspension Bridges Under Wind A c t i o n . " T r a n s . A S C E , v o l . 114, 1949, p. 1177.  22.  Rocard,  F.  "Dynamic Instability." Crosby-Lockwood, London, 1957, Ch. 6.  23.  P a r k i n s o n , G. Brooks, N. P.  24.  Minorsky,  F.  D.  N.  G.  "The W i n d - I n d u c e d V i b r a t i o n of Guyed and S e l f - S u p p o r t i n g C y l i n d r i c a l Columns." Transactions of the Engineering I n s t i t u t e of Canada, vol. 3, no. 4, December, 1959, pp. 119-141. "A N o t e on G a l l o p i n g C o n ductors." National Research C o u n c i l of Canada R e p o r t No. MT-14, June 30, 1950.  B.  V.  "Aerodynamic Theory of Bridge Oscillations." T r a n s . ASCE, v o l . 115, 1950, pp 1180,  "On t h e A e r o e l a s t i c Ins t a b i l i t y of B l u f f Cylinders." J o u r n a l of A p p l i e d Mechani c s , ASME, v o l . 2 8 , S e r i e s E , No. 2, J u n e , 1961, p p . 252258. Introduction to Non-Linear Mechanics. Edwards, Ann 1  53  54  P  (a)  Section  through  (b)  Bushing  Detail  ^  Off $ig. c  Orifice  9  Figure  2-  Air,Bearing  (full  Details  size)  55  F i g u r e 3- A s s e m b l e d A i r B e a r i n g  3  P, ie a  ® To provide  sn^cf  fit  for  bearing holder <2> M/'//ed flat fo S€a."t l^ertua/  FIGURE  AIR  BEARING  BASE  anyle.  57  Figure  5-  Model Clamping  Details  II &1X&X  VHft-2  F i g u r e 6- A i r  Supply  Figure  7- U p p e r S h a f t  and Bearings  (showing  transducer  and  damper)  60  Figure 8 - Test  Section  and  Equipment  Figure  9 - Displacement  Transducer  62  Specifications Secondary d i a . 1 1/4 i n . Primary dia. 3/4 i n . Wire d i a . . 0 1 0 i n . (1 l a y e r ) Coil length 12 i n c h e s Metal'tube 65S-T6 Aluminum Dia. 1.00 i n . Wall .058 i n . O p e r a t i n g c h a r a c t e r i s t i c s a t 10 K C e x c i t a t i o n Max. o u t p u t (tube f u l l y w i t h d r a w n ) - - . 9 8 v / v o l t Min. output (fube f u l l y inserted) .023v/volt L i n e a r range \ - - a p p r o x . 10 i n c h e s S e n s i t i v i t y over l i n e a r range--approx. 0 . 0 9 v / i n / v o l t A c c u r a c y o v e r l i n e a r r a n g e . . - - A p p r o x . ^% F u l l S c a l e P o w e r c o n s u m p t i o n a t 10 K C , 5 v e x c i t a t i o n 0.17 watt  Figure  10-  Displacement  Transducer  Details  63  D.<.iAMpuTlER  IH|,|.|,L1  'I'r 3 v.  (a)  (b)  Figure  V/lSKOBDpR  OSCILLO-SCS>PF  OSCILLATOR  Displacement  D.C.  Instrumentation  Amplifier  Schematic  11-Displacement Instrumentation D.C. A m p l i f i e r Schematic  and  F i g u r e 12-  Electromagnetic  Damper  65  £>.C. fat. r/«i£T£A - ;4/y/W£T£4  ELECTS o - V/M/MIE O.CPoiA/£fi 5u/>/>iy  //OV A.C. 5lt0£  Figure  13-  Electrical  Supply  for  -UO/#£ fit/SOST/IT  Dampers  r~ " i i  i  s  y~—•  s  LIT n  V]/  '////////SSi  I  A,  £bt/£  SOCKET  F i g u r e 14- F o r c e M e a s u r i n g  Unit  TJ35  7~o  I  tt  gure  15-  Model  Plan  68  .030  0  1 0  1  1  0.10  0.20 I  Figure  16-  Damping  0.30 (amperes)  Calibration  1 0.40  1 0.50  To foAce.  TA/imDucdn  To P/VOT  e 17-  Calibration  Set-up  2 .80  s  2.40 A  D •  £4*  2.00 OD  p  6 A. 1.&0  -co-  4  ^7  57 O B B p R 35? B  1.20  A  = 1.07(10) = 1 . 9 $ (.10) - 3 = 3.64(10) -3 = 3.72(10)  0.8(  8o  h n f  x>.40-  = 1.00 4.: 6.01  in. cps  1  O 8  Figure  18-Dimensionless  Amplitude  vs.  U  10  12  Dimensionless  14  13  16  Flow V e l o c i t y ;  b/h =  20 1.00  Figure  19-  Reduced  Velocity-Amplitude  Curve  for  b/h =  1.00  fc°  00  0  o  a  0  0 0 0  4 A  0  A A  0< 0 0  A  0 0  n  A  0  A  A  0  o B = 1.03(10) A B = 5.94(10) -  o  n = 3.89(10)" f = 5.65 c p s  o  J  4  0  o  *  f' /  /  '' A  0  o  oo  b  0  2  4  6  8  F i g u r e 20- D i m e n s i o n l e s s A m p l i t u d e v s  10  12  14  18  16  U D i m e n s i o n l e s s Flow V e l o c i t y ;  b/h =  1.50  20  i  ©—v-  0  ft*  0 B A B V B h/h n f  = = = = =: =  1.03(10) -3 2.80(10) -3 5.94(10] 2.00 3.89(10) -4 5.65 cps  0  4 >  V  A?  „^  fr  0  Z  4  D  F i g u r e 21 Dimensionless Amplitude v s . Dimensionless Flow V e l o c i t y ;  LO  b / h = 2.00  2.80  A  = 5 .65 cps B = 1 . 0 3 ( 1 0 ) -3 -3 B = 5.94(10)  •  f = 3.89 cps B = 1.50(10)  O  2.40  2.00  n  A  O  A  8 O  = 3.89(10)  •1  A  O  J  1.60  o  ^  1.20 Y  o '  •A  PA  0.80  0  o  0.40  OA  ©O  0  2  4  Figure-22-Dimensionless  6  8  Amplitude vs.  10  12  Dimensionless  14  16  Flow V e l o c i t y ;  18 b/h =  2.50  20  2.80  2.40  b / h - 3 - O O  n  2.00  o o  = "5 . 8 9 x 1 0  B -  I . 0 3 x  6  n  -4-  io"  1.60  1.20'  0.80 0 .40  ,<s> 8  u  1 0  12  14  16  F i g u r e 23-Dimensiohless Amplitude v s . Dimensionless Flow V e l o c i t y ;  Ui  18  b / h = 3.00  20  2.50 b/Y-i  b A i = 1.00  =2oo  b /k> » loo ^ U  =  S.to  2.00  U  =  3.32.  0.50  0  20  40  60  80 t  Figure  24-  Time-Amplitude  Curves  for  b/h  100 (sec)  = 1.00  120  and b/h =  140  2.00  160  180  200  18  I  1  b/h O 1.56 • 3.00 A 4.00 V 1.00 (Br ooks)  a  •  n  a  •  f  A  V  1  0  o o o 0  10  8  9 (10) N  F i g u r e 25- S t r o u h a l Number v s .  a  D  1.50  °  O  2^5  R  Reynolds  Number fvori Various b / h Ratios  D  0.20  0.15  0.10 -1 i  1 0.05  1 N 0 Figure  1.0 26- Strouhal  2.0 Number v s .  3.0 b/h b/h  Ratio  R  « 19,000 4.0  5.0  6.  79  .800  • ll  X O A  .600  • O  R 22~ 3 ( 1 0 ) § 27 . 3 ( 1 0 ) 33 . 1 ( 1 0 ) 3 36 . 0 ( 1 0 ) | 66 . 0 ( 1 0 ) (Brooks) X  .400  .200  .•ft  »*«  I  6  •  X  t i  L  A  * a A O X  If *  4  /p  0  •*  200  .400 8  12 c:  Figure  27-  C  F  v s A  for  *<  16  (degrees)  Square  Section  20  80  (a) Narrow F i l t e r  (b) F i l t e r  Set f o r the Strouhal  Set f o r Frequency  of  Oscillation  (c) U n f i l t e r e d S i g n a l  Figure  28- T y p i c a l Hot  Wire  ( h o t w i r e i n wake)  Frequency  Signals  82  2.50  1 c  = 1.03(10)~ n = 3.89110) f = 5 . 6 5 c ps h = 1 . 0 0 i nc h e s  B  2.0O  o  3  1.50  R  \  \  1.00  ci  \ \  \  k 0.5C  V  "6 \  0 L.00  s  1.50  2.50  2.00  3.00  3.30  b/h  Figure  30- Reducing  Factor  vs.  b/h for  Long  Rectangles  4.00  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0105634/manifest

Comment

Related Items