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EXPERIMENTAL INVESTIGATION OF NONLINEAR COUPLED VIBRATIONS OF COLUMNS  by  Dale P. Johnson  B . A . S c , University of B r i t i s h Columbia, 1968  A'Thesis Submi tted-i-n Partial FulfiJJment Requirements for.the Degree <3f  of the —  Master of Applied Science In the Department of Mechanical Engineering  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA May, 1970  In  presenting  this  thesis  an a d v a n c e d d e g r e e the L i b r a r y I  further  for  agree  scholarly  by h i s of  shall  at  the U n i v e r s i t y  make i t  that  written  thesis  freely  permission  for  It  fulfilment  of  of  Columbia,  British  available  by  gain  shall  requirements  reference copying  that  not  copying  I agree and  of  this  or  Department  of  Mechanical  The U n i v e r s i t y o f B r i t i s h V a n c o u v e r 8, Canada  May  25,  1970  Engineering Columbia  that  study. thesis  my  permission.  P.  for  or  publication  be a l l o w e d w i t h o u t  Dale  Date  the  t h e Head o f my D e p a r t m e n t  is understood  financial  for  for extensive  p u r p o s e s may be g r a n t e d  representatives.  this  in p a r t i a l  Johnson  TABLE  OF  CONTENTS  ABSTRACT ACKNOWLEDGEMENT... LIST  OF  FIGURES...  LIST  OF  TABLES  LIST  OF  APPENDICES  NOMENCLATURE  CHAPTER  I  .  INTRODUCTION P r e l i m i n a r y Remarks L i t e r a t u r e Review Limitations of Investigation,.  CHAPTER  II  THEORY D i f f e r e n t i a l Equations of M o t i o n f o r a Column.... Theoretical Predictions.. Torsional Coupling  CHAPTER  III  APPARATUS  AND  INSTRUMENTATION  General Outline E l e c t r o n i c System . L o a d i n g Frame and Column Description D e t a i l s of Measuring System...  CHAPTER  IV  TEST  PROCEDURE Calibration Testing.... Photography  Page  CHAPTER  V  RESULTS  AND  DISCUSSION.  3  I n t e r p r e t a t i o n o f Frequency Spectra Identification of Strain Peaks Results of Flexural Strain Record Results o f Axial Strain Record  CHAPTER  VI  SUMMARY  AND  CONCLUSIONS  Summary Conclusions Suggestions search  8  38 39 42 48  59 59 61  f o r Further  Re64  BIBLIOGRAPHY  67  APPENDICES  68  ABSTRACT  Coupling vibration loading The  of the flexural,  modes  was  o f a column  analytically  initial  crookedness  inertia  give  tions.  Further,  result  rise  subjected  and  o f t h e column  t h e Weber  and  axial  investigated. longitudinal  flexural-longitudmal  effect  between  torsional  to periodic  and e x p e r i m e n t a l l y  to coupled  i n coupling  longitudinal,  vibra-  and l o n g i t u d i n a l  longitudinal  and  inertia  torsional  oscillations.  To  assess  the validity  apparatus  was  vibration  control  The  s e t up t o a x i a l l y  experimental  theoretical exhibiting flexural ratio  results  results  a frequency  longitudinal  that  coupled  1:3 w a s o b s e r v e d , oscillation  when  was  coupled  a column  agreement  longitudinal  o f 1:2 w e r e  also  observed,  "Further,  were  experimental using  present. with  shaker.  with the vibrations  observed.  Coupled  though  frequency  a  the experimental  vibrations  vibration  a  other  than  those  In p a r t i c u l a r , a frequency  and a c o r r e s p o n d i n g  coupled  a  ratio of flexural  present.  torsional  the applied  sional  ratio  coupled  expected  i n good  Coupled  not established.  suggest  excite  an  a n d an e l e c t r o m a g n e t i c  were  predictions.  theoretically  A  generator  o s c i l l a t i o n s were  was  of the theory,  frequency.  mode  frequency A second  was  was  experimentally  twice  coupled  observed  the fundamental torsional  mode  torappeared  - 11 when  the  excitation  torsional  The was to  phase  three  times  relationship  between  the  The  coupled  vibrations  significant  \  was  the  fundamental  frequency.  observed. be  frequency  resonant at  certain  coupled  frequencies.  vibrations were  found  ACKNOWLEDGEMENT  My visors, to  particular gratitude Dr. C R .  participate i n this  out  the duration  I cular,  also  Mr. P h i l  spent Sing  work  Ramsay,  and f o r t h e i r  f a c u l t y ad-  f o ri n v i t i n g assistance  t o thank Hurren  the technical staff,  and Mr. J o h n  Hoar  me  through-  Diane  Johnson  Leim  i n assisting  This  project  with  i n typing  provided  by t h e D e f e n s e  invaluable  The care  the thesis,  t h e photography,  was made p o s s i b l e  and i n p a r t i -  f o rtheir  i n s e t t i n g up t h e a p p a r a t u s .  by M i s s  66-9510  a n d D r . H.  t o my  of the project.  wish  co-operation  No.  Hazell  i s extended  through Research  i s  and time  a n d b y Mr.  appreciated.  Research Board  Grant  o f Canada.  -  LIST  i v -  OF  FIGURES  Figure  Fig..  I I - l  Page  Reference  Axis  For Displacement  Measurements  9  Fig.  II-2  Elastic  Fig.  III-l  Signal  Fig.  III-2  Photograph o f Experimental Apparatus  Fig.  III-3  B a r Under Flow  Schematic  Axial  Loading  Chart  View  19  o f Lower  20  Column  Mount  21  Fig.  III-4  Column  Fig.  III-5  Sectional  Mounting  and Loading  Drawing  Alignment  of Loading  System..  Mechanism  24  III-6  Schematic  Fig.  III-7  Fig.  III-8  Photograph o f Specimen Loading Frame S c h e m a t i c S t r a i n Gauge Arrangement F o r F l e x u r a l S t r a i n Measurement...  Fig.  III-9  Fig.  Fig.  Fig.  Fig.  V - l  V-2  V-3  V-4  V-5  View  22  and  Fig.  Fig.  15  o f Column  Specimen...  25  26 28  S c h e m a t i c S t r a i n Gauge Arrangement For L o n g i t u d i n a l S t r a i n Measurement  29  F o r c e d Resonant F r e q u e n c i e s P l o t t e d V e r s u s Mode N u m b e r F o r V a r i o u s End C o n d i t i o n s  40  F l e x u r a l S t r a i n Record M i d p o i n t o f Column  42  Possible Flexural Axially Excited  Obtained At  Mode S h a p e s F o r Column  43  Frequencies of Experimentally Observed F l e x u r a l Resonant Strain Peaks  45  Waveforms f o r O b s e r v e d F l e x u r a l Resonances  46  Coupled  -  v -  Figure  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  Fig.  V-6  V-7  V-8  V-9  V-10  V - l l  V-12  V-13  A - l  Page  A x i a l S t r a i n Record Obtained at M i d p o i n t o f Column.  49  A Portion o f a Column Twisting  51  Undergoing  Frequencies of Experimentally Observed Resonant A x i a l Strain Peaks  51  Waveforms a t Fundamental n a l Resonance  52  Waveforms Mode  a t Second  Waveforms Mode  at Third  Longitudi-  Coupled  Axial 54  Coupled  Axial 55  Phase R e l a t i o n s h i p Between Vibration Components a t Second C o u p l e d A x i a l Mode  56  Waveforms Modes  57  f o r Coupled  E x t e n s i o n and R o t a t i o n Plane Fiber  Torsional  of  Central 69  -  LIST  v i -  OF  TABLES Page  Table  V - l  Permissable A c c e l e r a t i o n Levels Over V a r i o u s F r e q u e n c y Ranges F o r Column S p e c i m e n  35  -  v i i -  Page APPENDIX A  APPENDIX B  APPENDIX  C  S t r a i n E x p r e s s i o n f o r Column Some I n i t i a l C r o o k e d n e s s  With 68  Details of Electronic Equipment Used f o r V i b r a t i o n C o n t r o l System..  71  Linear D i f f e r e n t i a l Equations of M o t i o n f o r a Column  73  -  viii  -  NOMENCLATURE  Symbol P P  = total 0  applied  = constant  axial  applied  load  axial  R.  = amplitude  of variable  V"  = frequency  of applied  6  X  = strain  load applied  axial  of c e n t r a l plane  axial  load  load  fiber  o f column  SIQ  = bending  6  = total  u  - l o n g i t u d i n a l displacement  w  = initial  transverse  displacement  w  = dynamic t r a n s v e r s e  displacement  0  = rotation of cross-section-  x,y,z  = coordinate  V  = strain  T  = kinetic  W  = work done by e x t e r n a l  v  - volume  A  = area  b  = width  h  = heighth  L  = length  s  = fiber  1  = minimum moment o f i n e r t i a o f column section  I  = constant section  strain  strain  o f a column  fiber  distances  energy energy loading  o f column  of cross-section  o f column  o f column o f column o f column length  d e p e n d i n g on d i m e n s t i o n s  cross-  of cross-  -  I  o f i n e r t i-3 a b o u t  moment  2  i x -  Io  polar  moment  E  elastic  G  shear  <t>  density  t  time  OC  phase  *tion  angle longitudinal  transverse  freqi  frequency ancies  integer  i w  o f cross-s<.  modulus  resonant  i  nertia  axis  modulus  fundamental p  of  y  resonant  t  torsional  acceleration  g H,  functions  H (x) , 1  H (X)  of  f r e q u e ;cy  gravity  of variable  x c ly  , H (x)  2  3  X(x) , F ( x ) , G (y.) 1  G  t  w, w, x  x  2  (x)  u  x  partial respect  d e r i v a t i v e o f a d splacement to the variable x  Abbrevi at ions  BAM  Bridge  amplifier  CRO  Cathode  Hz  Cycles  RMS  Root-mean-square  a n d metex  ray oscilloscope per  second value  of a  function  with  CHAPT  ntrocSuctiort  -  1  -  CHAPTER I  INTRODUCTION  Preliminary  In  Remarks  t h e modern  spacecraft, ment,  a n d many  the struggle  intensified.  dynamic  excitation  response  becomes  o f a column  important does  from  often  mation  at frequencies  investigation column  can  An form  lar  frequency  of  D  made these  vibrations  which  equip-  i s being provide  as columns.  to periodic  viewpoint.  10 K H z .  order  speed  vibration  sufficiently  of order  Simple  at frequencies  loading  linear  accurate  The focus  nonlinear  Thus  axial  infor-  of  coupled  this  effects  o f s e v e r a l KHz  sustaining a periodic  + P, c o s V t ,  V,  may  where  t i s time  be e x c i t e d i n t o Parametric  i n the axial  on s m a l l  direction  These  free  researchers  have  been  and  column.  interested  circu-  torsional  studies on t h e  dynamic made  periodic  by a number focused  vibrations i n the f i r s t  straight  load o f  and P has a  i n v e s t i g a t i o n s have  lateral  f o r an i n i t i a l l y  under  have  axial  transverse  stability  v i b r a t i o n s o f a column  investigators.  marily  a design  column  of vibration.  loading  frequency  aircraft,  significant.  P = P  transverse  o f high  undesirable  subjected  i s on second  elastic  the  varieties  not provide  vibrations,  be q u i t e  modes  High  of turbines,  t o s t r u c t u r e s such  theory  in  other  against  rapidly  the  day p r o d u c t i o n  As a  themselves  prispatial  consequence i n relatively  - 2low  angular  frequencies  Towards encountered theory  the upper  coupling  phenomena  However,  a linear  similar  i n order  Schneider  [9]  h a s shown  discrete  with  i s shown  that  with  excite  of other  does  point  causes  namely, that  become high  reasonable  to  of oscillation. frequencies  longitudinal  i n a bar. coupling  In t h i s  effect  work,  works i n t h e  i s excited  i s one h a l f  o r one  frequency,  of  oscilla-  i f a column  longitudinal flexural  t h e wave-  relatively  at particular  nonlinear  the bar into  vibrations  t h e two modes  that  a frequency  resonant  out that  I t i s therefore  frequencies  also;  nonlinearities.  at these  coupling  this  direction  fundamental to  parametric  between  tions  tudinally  inertia,  frequencies.  excitation  reverse  due t o l o n g i t u d i n a l  and l o n g i t u d i n a l  transverse  it  linear  o f magnitude  interaction  range  a simple  analysis  of flexural  frequency  investigation  and a h o s t  lengths  mechanical  excitation.  end o f the a c o u s t i c  i n the present  i s inaccurate  expect  of  t h i r d the  i t i s  o s c i l l a t i o n s with  longi-  possible  significant  amplitudes.  Torsion a  steel  column  direction the  can r e s u l t  Tso  due  of the cross  i n a shortening  of the square  parametric  movements  apparent.  i n rotation  of the order  possible  tudinal  resulting  coupling  that  i n the  between  torsional effect  of  axial  of the rotation.  to the shortening  [ 8 ] h a s shown  sections  Thus  and  longi-  becomes  i f a b a r i s under  forced  -  longitudinal twice  excitation  a natural  excite  the  The  into  equations  solution.  The  that  the  order  An of dic tor  flexural axial the  forced  a  and  column  study  with  those  particular  initial  in and  Chapter Y can  feature  allows  Several  methods  and  obtain the  to  Literature  made  to  can  on  be  the  of  a  determine  resonances.  and  arranged  f o r more were  to  a closed-  however,  so  conveniently  the  under  perio-  to  moni-  frequency resonant  vibrations  from  are  that  realistic  i s that  admission  longitudinal  so  response  attempted  experiments  independently  frequency  column  Observed  apparatus,  attempted  dynamic  theoretical considerations.  arising  experimental  controlled  without  manipulated,  vibrations  crookedness  I I I , was be  was  i n these  due  The  are  p r e d i c t e d by  c o u p l i n g phenomena  considered.  presented  nonlinear coupled  linear to  are  and  importance  to  viewpoint.  coupled and  i s close  oscillations.  S e v e r a l methods were  responses  vibrations  which  i t i s possible  c o u p l i n g terms  torsional  f o r these  compared  torsional  physical  loading.  frequency  frequency,  equations  experimental  spectrum  Of  from  a  of motion  form  second  -  with  torsional  column  interpreted  3  the of  inertia  which  parameters each  spectrum  strains were  i s described P  other.  theoretical  t o monitor  of  non-  , P, This  modelling.  coupled  f o r these  0  vibrations,  oscillations.  Review  Previous  research  connected  with  the  parametric  response  - Aof bars and columns has focused primarily on parametric vibrations associated with small free l a t e r a l vibrations in the f i r s t s p a t i a l mode.  The investigations have been  r e s t r i c t e d to r e l a t i v e l y low angular frequencies of  excita-  t i o n , not more than three or four times the fundamental l a t e r a l resonant Beliaev  frequency.  [2] was the f i r s t to analyze the parametric  response of a column under time-dependent citation,  longitudinal ex-  and he reduced the equation of motion to the stan-  dard Mathieu-Hill equation. of longitudinal i n e r t i a .  He did not consider the  Somerset  influence  [ 4 ] and Bolatin [6]  experimentally v e r i f i e d that the s t a b i l i t y of the column described by Beliaev could be analyzed by investigating  the  s t a b i l i t y of the solutions of the Mathieu-Hill equation of motion for the column.  An unstable region was found  characterized by l a t e r a l column o s c i l l a t i o n s h a l f the excitation  frequency.  at exactly  Experimentally observed  oneef-  fects not anticipated by this equation were attributed to longitudinal i n e r t i a , l i n e a r and nonlinear damping, nonlinear elasticity,  rotary i n e r t i a , and i n t e r n a l f r i c t i o n .  Somerset and Evan-Iwanowski perimentally investigated  [ 7 ] theoretically  and ex-  the parametric i n s t a b i l i t y of  straight columns sustaining a periodic axial load of low c i r c u l a r frequency.  They also b r i e f l y considered the i n -  fluence of damping and of nonlinearities which are amplitude dependent,  such as nonlinear e l a s t i c i t y  and longitudinal  inertia. regions  They of  presence  of  Numerous  metric  render  amplitude  of  lateral  r e f e r e n c e s on  found  in  a  the  survey  stability  of  cross-section that  with  torsional  into  a  an  under  over  certain  that  the  the  column  vibration  topic  of  para-  article  by  Evan-  oscillations.  the  stability  axial  i s under  which  close  forced  i s close  the  to  of  the  to  the  has  of He  longitudi-  to  twice  excite the  a  the  frequency  fundamental  longitudinal  bar  para-  loading.  F u r t h e r , when  effect of  of  cantilever  i t is possible  l o a d i n g becomes  torsional  dynamic  frequency  frequency,  problem  elastic  i f a bar  frequency,  torsional  applied  the  be  and  to  torsional  longitudinal on  can  analytically  the  other  space,  tend  companion  natural  of  the  may  studied a  excitation  bar  factors as  i s unstable  parametric  has  rectangular showed  column  [5 ] .  [8]  Tso  the  PI,V")  unstable  vibrations  Iwanowski  that  nonlinear  increases.  nal  (PQ,  the  increasingly  metric  found  inertia  to  be  taken  thfe r e s p o n s e  of  a  into  consideration.  The clamped  present at  equations  both of  longitudinal above the  the  been a  bar  ends  motion  considers with  inertia  from  treated  of  the  are  transversely  lower  author,  by  of  axially  for i n i t i a l at  strains  investigated  central  i n the  up  to  past.  coupled  i n bars plane He  The  and  frequency.  vibrations  the  excited.  frequencies  i n the  [9]  column  crookedness  longitudinal  coupled  Schneider  and  end  considered  resonant  admission only  the  accounting  fundamental  knowledge  sulting  work  and To re-  have  excited  longitudinal  - 6oscillations. tions the  occur  He  The  that  a t 1 a n d 1/2  fundamental  Limitations  found  of  investigation  form  solution  o f coupled i s said  oscillations. limiting column  about  along  stresses.  symmetrical  about  in  plane.  stant  along  respect linearly The  vibrations  not serious  Internal  i s used,  isotropic  of developing  i s assumed  o f l o a d i n g so that  t o be bending  occurs  The m a t e r i a l p r o p e r t i e s a r e assumed  o f shear  were  elastic  and capable  fre-  resonant  of flexure  to a nearly straight  t h e l e n g t h o f t h e column  applications, are  theory  the  Analytically,  o f coupled  The c r o s s - s e c t i o n  proportional  transverse  so only  are indicated.  the amplitudes  the plane  limitations.  derived are not exact.  i s not available,  i t s length  to the plane  effects  at  vibration.  important  equations  vibrations  the analysis  t h e same  resonant  The B e r n o u l l i - E u l e r  uniform  bending  oscilla-  the transverse excitation  has s e v e r a l  A  nothing  longitudinal  Investigation  the theoretical  quencies  times  longitudinal  Firstly, closed  resonant  of loading.  Bending  to distance deformation  from  limitations  stiff  F o r most columns  plane.  inertia  on  engineering a r e used,  modes  and t r a n s v e r s e  with  i s assumed  the central  at t h e lower  and e x t e r n a l damping  strain  and o f r o t a r y  are ignored.  relatively  and s y m m e t r i c a l  con-  of  these  oscillation,  loading are not  considered.  Secondly, tions.  the experimental  I t is. not possible  study  to realize  has numerous in practice  restricthe  boundary Clamped  conditions column  experimental mate,  to  amount and  of  signal as  acceleration power' to  an  gauges  were  coupled  was  magnetism exact  exciter  the  To  were  to  used  of  noise  the  an  Finally,  1 0 KHz  by  purely found  electronic  to  were the  level  tests  noise  very  small.  measured  level  as  high  sinusoidal  'approximate  the  approxi-  considerable  signal  the  an  means  levels  constant  achieve  spectra.  frequency  of  A  minimize  strain  maintain  bands  i s not  sensitive  c o n t r i b u t i o n of  frequency  control.  the  to  in  i s always  vibrations.  required since  only  development.  approximated  excitation  excitation upper  only  column  the  is difficult.  possible,  best  that  caution  the  at  analytical  alignment  nonlinear  induced  Assessing  are  the  Further,  Strain  monitor  ends  in  set-up.  meaning  axial.  used  constant  were  automatic  limited  vibration  CHAPTI  Theory  - 8CHAPTER I I  THEORY  Differential  Linear for  plane  assume no s t r a i n s  motion,  this  means  column  crookedness.  placement  The dynamic  account  Lagrangian deflections  deflection  curve.  description expression A).  plane  where,  u  x  referring u  =  w =  +  assumption  exhibit  i n the  a strain  some  differential  expression  or static  of strain  Mettler with  w  t o an  +  [ 3 ] developed  some  the strain  *x x  relative  i s used;  which  dis-  that i s , initial  [ 1 ] d e r i v e d an a n a l a g o u s  Love  o f t h e column =  f o r t h e un-  This  crookedness  are measured  obtained  £x  from  definition  f o r a column  fiber  arise  i n the  Considering  axis  a l l columns  used  or bar.  of strain.  He  since  any i n i t i a l  o f a column  commonly  exist  segment.  The n o n l i n e a r i t i e s  derived herein  into  the neutral  line  i n practice  o f motion  loading i s zero.  that  i s a straight  limitations  dix  C)  the axial  equations  Column  (Appendix when  initial  for a  equations  plane  loaded  takes  o f Motion  differential  columns  neutral  has  Equations  initial  Eulerian  the  strain  crookedness  expression  at the  central  as (V2) w 2 x  (  to Fig. I I - l  longitudinal  displacement  initial  transverse  displacement  w - dynamic  transverse  displacement  2  (Appen-  - i )  - 9and  t h e s u b s c r i p t s denote  pect  t o that  partial  differentiation  with  res-  variable.  -INITIAL DEFLECTION CURVE •DYNAMIC DEFLECTION CURVE  z,w Fig.  I I - l .  Reference  From  simple  bending  at  a distance 6  This  b  expression  sections mation  w  derived,  x  coupled a s was  (slenderness  plane  axial  +6  fiber  axis i s  assuming that  during  bending;  The t o t a l  that  strain  plane  cross-  i s , shear  defor-  i s therefore (2-3)  b  done  equations  by M e t t l e r  greater  than  o f motion  [3]. A slender 50) i s . c o n s i d e r e d  can be n e g l e c t e d ,  loading  of a  (2-2)  differential  ratio  inertia  the neutral  xx  i s neglected.  The  rotary  z  Measurements  the additional strain  i s obtained  remain  6=6  Only  "  f o r Displacement  theory,  z from =  Axis  i s considered.  and damping  i s  c a n now  column so  that  Ignored.  be  - 10 From usual beam bending theory, assuming that stresses occur only in the x - d i r e c t i o n , the e l a s t i c V = | £  e  d  s t r a i n energy i s (2-4)  v  where, E = modulus of  elasticity  v = volume {2-1)  Substituting equations  and (2-2)  into equation  (2-3),  substituting the resultant expression into equation (2-4) integrating p a r t i a l l y over the cross-section  and  of the column,  the s t r a i n energy becomes V =  f J x *x x + 'J xx 0 L(u  +  0  EI  +(v2)w Vdx  w  Lw  x  (2-5)  2 d x  2. J  where, A = cross-sectional  area of column  I = minimum moment of i n e r t i a of column cross-section The approximate k i n e t i c energy T =  ~ J  L q  (u  2 t  is  + w )  dx  2  t  (2-6)  where, 0 = density of column material t = t ime Hamilton's p r i n c i p l e for a conservative system states that (V - T) dt = 0 '1 Substituting equations (2-5) and (2-6) ing  into (2-7)  and employ-  calculus of variations provides the coupled d i f f e r e n t i a l  equations of plane motion for the free vibrations of a column. EA(u Elw  x  + (l/2)w  x x x x  + ww)  " EA[(u  + d>Aw = 0 tt  2 x  x  x  +  x  ( ) 1/2  +  x  Wx  2  Au +  w  = 0  t t  ^  x  ) (  ^  (2-8) +  _^ .  }  (2-9).  If the  t h e n o n l i n e a r terms  familiar  columns  equation  results.  provides  some  which  f o r the longitudinal  equation  linearizing for lateral  treatment  are dropped,  vibration equation  vibration  of the equations  p r e d i c t i o n s t o b e made  oscillations at  (2-8)  of  (2-9) o f beams.  Predictions  qualitative  allows  i n equation  Similarily,  t h e common  Theoretical  A  11 -  o f t h e column.  coupled  terms  concerning  In p a r t i c u l a r ,  are l i k e l y  of motion the  resonant  the frequencies  t o be most  significant  are  revealed.  Expanding, (2-9)  xxxx  EA[u form  the  x x  level  x  E  +  A  [ u  u  x  xx x W  i s also  expect  the axial  vary  x  x  u  x xx]  equation  w  + A4w t t  =  ]  (2-10)  excitation  of the acceleration displacement  +  w  sinusoidal  column  to  w  "  of the axial  constant tion  and r e a r r a n g i n g  gives E I w  The  differentiating  i n the tests  acceleration. with  respect  A double  t o time  time-variation at the excited  sinusoidal.  indicates  a t any p o i n t  along  ting  into  differentiating  equation  to  t h e column  i n time,  (2-11)  V i s the applied frequency  Appropriately  that  end of the  u = U ( x ) c o s ( r t + OC ) where  integra-  I t i s therefore reasonable  displacement  sinusoidally  i s that of  (2-10)  and  equation  yields  i s some p h a s e (2-11)  and  angle.  substitu-  -  E  I  w  xxxx  12 -  - EA[u  x x  w  + u w  x  x  x x  ]  + A^wtt =  F(x) cos(Yt+CQ where  F ( x ) i s some  by  the s t a t i c  of  equation  Then  (2-12)  o f frequency  are  therefore  in  V .  while  modes,  these  they  oscillations, for  a column  since  time  being,  response can  the  equation  (2-8)  c  length  f>  resonant flexural  linear  theory  into  equation  that  terms  the  transverse  f o r the  transverse response  (2-8) d e s c r i b i n g  mathematical  the  description of  D i f f e r e n t i a t i n g and  rearranging  yields + i - utt = w  u  the  A sinusoidal  t o g e t an improved  " xx where  o f the coupled  indicates  longitudinal behavior.  equation  t o avoid  excited.  i s sinusoidal.  motion  t o occur  resonant  from  since  t othe  However,  flexural  can be p r e d i c t e d  (2-12)  back  excited,  normal  t o as ' l i n e a r '  the influence  be s u b s t i t u t e d  axial  they  transversly  Neglecting  speaking,  are understood  coupled  i n -  Strictly  t othe e x c i t a t i o n .  be r e f e r r e d  function.  oscillations  i n directions  with  side  flexural  are parametrically  oscillations  resonances  will  Ther i g h t  a sinusoidal transverse  resonant  occur  forced  x determined  as a f o r c i n g  t obe e x c i t e d .  oscillations  directions parallel  confusing  has  'Linear'  oscillations  excitation,  o f t h e column.  can be regarded  expected  flexural  o f the coordinate  effectively  put  parametric  function  displacement  t h e column  these  (2-12)  1  3  t  i  l  e  x  W  x  x  + w w x  x x  + w w x  x x  (2-13)  v e l o c i t y o f l o n g i t u d i n a l waves  o f t h e column.  Substituting  a sinusoidal  along trans-  verse  response  equation  into  (2-13),  of x only,  13 -  the coupled  t e r m s on t h e r i g h t  and remembering t h a t w  and w  x  are f u n c t i o n s  x x  gives = H(x)cos2tft  (2-14)  = H (x)cos^t 1  (2-15)  ^xWxx = H ( x ) c o s t f t  (2-16)  w  xw  x x  w xw X  x  2  cos2 Yi  Since  =  w w x  +  c  o  s  2  = H (x)  x x  t h e sum  +  + H  (x)(H-cos2*t)  3  first  ~2 tt  (2-17)  different  into  t  (2-18)  with  longitudinal  longitudinal  vibration frequency.  vibration  of the transverse v i b r a t i o n  frequency  vibration a  frequency  vibrations,  f r e q u e n c i e s are expected.  as t h e e x c i t a t i o n  a t an e x c i t a t i o n  excitation  costft  X  that the l o n g i t u d i n a l  resonant  resonant  resonant  frequency  2  These are t h e n o n l i n e a r c o u p l e d  same f r e q u e n c y  occurs  (2-16) and  o f two s i n u s o i d a l o s c i l l a t i o n s  coupled  coupled  (2-17)  ={H!(x) + H ( ) }  u  indicates  r a t i o o f 1:2. two  (2-15),  ~ xx  This equation  and  (2-14) becomes  t  (2-13) , one o b t a i n s u  is  y ) , equation [1 + cos2tft]  3  Substituting expressions equation  side of  frequency  f o r the f i r s t  The  i s at the The  second  i s at twice the  o f t h e column.  o f j&V  It  where oo, i s t h e  coupled  longitudinal  vibration.  To c o n t i n u e axial  response  verse  response.  this  somewhat i t e r a t i v e  procedure,  the  c a n be u s e d t o o b t a i n a more p r e c i s e t r a n s The f i g h t  side of equation  (2-18) c a n be  -  regarded with be  as two  equation  • from jit),  +  resonant  questionable, effect  each  those  iteration  can i n  on t h e r i g h t  two..flexural  side  of  form (2-19)  2  transverse  turn  coupled  G (x)cos2JTt frequencies  analysis  a t a> a n d x  due  twisted  close  with.  terms  does  The  phenomena  additional  decreasingly not indicate  discussed.  i s a  terms  further  second provided  significant.  The  the significance of  The e x p e r i m e n t a l  investi-  value.  between  longitudinal  to the "shortening  s t e e l b a r o r column  shortening  sections. forced  the coupling  the i t e r a t i o n s  Coupling  Coupling exists  f o r continuing  become  i s of obvious  Torsional  axial  since  to start  coupled  gation  For  t  of the  functions  expressions  terms  a consequence  justification  theoretical  to  These  forcing  arise.  order by  two  .  predicted  x  sinusoidal  the coupled  As  G ( )cos 1  and  into  c a n be  which  The is  Y  (2-10) .  vibrations  -  longitudinal  frequencies  substituted  14  Tso  i n addition [8] s h o w s  effect". subjected  a natural  parametrically  excite  simplicity, torsional  dissociated  from  each  An  of the  a n a l y t i c a l l y that with  torsional  a  frequency  torsional  i n the present  undergoes cross-  which i s  i t i s possible  and f l e x u r a l c o u p l i n g  other  un-  i f a b a r i s under  frequency,  the bar into  motion  initially  t o torques  to rotation  longitudinal, vibrations to twice  and t o r s i o n a l  work.  oscillations. are  treated  Following  Tso's nal  development  displacement  roidal  axis  can  and of  be  referring  a point written  u(x,y,t)  =  at  to F i g . II-2, the a  distance y  from  longitudi the  cent-  as  U(x,t)  -  P(y.t)  Fig. Warping in  as  i n St.  equation  lation,  The  Elastic Venant  (2-20).  however. €  where  II-2.  =  The  torsion  i s not  Loading  taken  axial  strain  i s given  rj - \ y O 2 x  0  rotation  of  strain  GAh"  e  displacement cross  energy  2 x  }dx  into  i n the  account formu-  by (2-21)  2  x  longitudinal  elastic  Under A x i a l  It i s considered later  U = =  Bar  of  cross  section  section  becomes  (2-22)  -  16  where A = c r o s s - s e c t i o n a l I  -  area  = moment o f i n e r t i a  2  about 2 - a x i s  1-^ = c o n s t a n t d e p e n d i n g 1 The  2  where I  a c c o u n t s f o r S t . Venant  X  torsion.  Q  IJQ  =  [ A U  t  +  - « t ]dx I  e  (2-23)  2  = p o l a r moment o f i n e r t i a  work done by t h e a p p l i e d  loads  of cross-section. a t t h e end i s  W = f^p^tjud^tjhdy where p ( y , t ) that  represents  (2-24)  t h e end l o a d  on t h e column.  t h e mathematical form o f t h e end l o a d p(y,t)  where ^ ( y ) (2-20)  1  S  a  n  even  l A  (2-25)  f u c t i o n o f y.  Substituting  i n t o e q u a t i o n (2-24) L  2  2  x  1 f b/2 R-,= - j h ^ ( y ) dy 1  A j  -b/  R =r f  equations  yields  - jR I P(t)j © 2  P(t)U(l)  Assume  i s separable,  = P(t)*My)  and ( 2 - 2 5 ) W = R  where  The  energy i s T  The  7  t e r m 3- GAh 6  kinetic  on d i m e n s i o n s o f c r o s s - s e c t i o n  d x  { 2  _  2 6 )  0  2  2  hyV(y)dy  Using Hamilton's P r i n c i p l e , S f ( T - V + W)dt = 0  (2-27)  t2  with equations (2-22), variational motion  (2-23)  and ( 2 - 2 6 ) ,  and e m p l o y i n g a  procedure, the coupled d i f f e r e n t i a l , equations o f  f o r t o r s i o n a l and l o n g i t u d i n a l v i b r a t i o n s <DAUTT  - EAU  <H>©tt  - t^GAh  - |^Ii©xx©x  2  -  X X  +  2  *  EI 0 e  0  2  x  x x  = 0  R i p )]e 2  2  ( t  result;  x x  (2-28) +  Ei  2 ( U x  e  x ) x  ( " > 2  2 9  1 7  -  -  These e q u a t i o n s o f motion d e s c r i b e t h e l o n g i t u d i n a l and t o r s i o n a l response o f a u n i f o r m Column o f t h i n  rectangular  c r o s s s e c t i o n b by h and o f f i x e d l e n g t h L , l o a d e d symmetr i c a l l y about t h e OX-axis and u n i f o r m l y o v e r t h e t h i c k n e s s o f the  strip.  tropic.  The column m a t e r i a l i s assumed l i n e a r and i s o -  I f t h e n o n l i n e a r terms are n e g l e c t e d i n e q u a t i o n s  (2-28) and (2-29) t h e e q u a t i o n s become u n c o u p l e d . (2-2 8) nal  Equation  t a k e s t h e form o f t h e f a m i l i a r e q u a t i o n f o r l o n g i t u d i -  v i b r a t i o n s o f a r o d , w h i l e e q u a t i o n (2-29) Ol^tt  " [GAh /3 + 2 l 2  R  P ( t ) ] 0 2  xx  =  becomes  0  ( - > 2  30  T h i s e q u a t i o n i s d i f f e r e n t from t h e u s u a l form f o r t o r s i o n a l v i b r a t i o n o n l y i n t h e s t i f f n e s s term, which accounts f o r t h e a x i a l end l o a d .  I t i s seen t h a t a compressive end l o a d such  as i s used i n t h i s experiment w i l l decrease t h e t o r s i o n a l s t i f f n e s s o f t h e column. the  U s i n g a p p r o p r i a t e end c o n d i t i o n s ,  approximate r e s o n a n t t o r s i o n a l f r e q u e n c i e s can be c a l c u -  lated  (Appendix C ) . Tso p o i n t s o u t t h e e x i s t e n c e o f two c r i t i c a l  frequency  ranges where t o r s i o n a l c o u p l i n g i s most l i k e l y t o o c c u r .  The  f i r s t range appears when t h e e x t e r n a l l y a p p l i e d f r e q u e n c y i s close t o twice the n a t u r a l frequency of a p a r t i c u l a r mode.  torsional  The second c r i t i c a l range appears when t h e a p p l i e d  frequency i s c l o s e to the l o n g i t u d i n a l frequency, p a r t i c u l a r l y i f t h e dimensions o f t h e column are such t h a t the f u n damental l o n g i t u d i n a l f r e q u e n c y f r e q u e n c y f o r a t o r s i o n a l mode.  i s close t o the n a t u r a l  CHAPTI  Apparatu  CHAPTER I I I  APPARATUS  General  A tation of  AND  Outline  signal  flow  diagram  o f the apparatus  i s shown i n F i g . I I I - l .  the actual  apparatus  The e x p e r i m e n t a l  parts.  The f i r s t  includes  part  t h e feedback  generator, rometer.  to deliver  ponding  t o the type  transmitting  specimen.  signal. circuit  circuit,  Feedback  unit  and making  this  with  exciter signal  the shaker since  considered  instruments  necessary  of  specimen.  accele-  i s procorres-  The s h a k e r table  responds  to the  the vibration the excitation  control i t  The  between  i s  to the output feedback  accelerometer  a n d the. e x c i t e r  of the apparatus  Strain  a n d an  control  which  signal  an a c c e l e r o m e t e r  unit.  t o gather  i n two  system,  of a  t h e programmed e x c i t a t i o n  part  i n this  t o the shaker  wanted.  using  level,  f o r any d i f f e r e n c e  second  photograph  control  corrections  control  compares  i s a  amplifiers  'checking'  i s accomplished  the excitation  the t e s t  of  suitable  to the exciter  compensate  through  i s essential  i s capable  This  vibration  of excitation  a force  instrumen-  and c o n s i s t s  an e l e c t r o n i c  monitors  The  can be  i s the vibration  The a u t o m a t i c  providing  apparatus  electromagnetic shaker,  grammed  exciter  F i g .III-2  and  and i n s t r u m e n t a t i o n used  study.  by  INSTRUMENTATION  control  so t h a t  unit  i tcan  t h e two.  consists  of monitoring  i n f o r m a t i o n on t h e b e h a v i o r  gauges  attached  t o the specimen  FOUR BEAM A  OSCILLOSCOPE  LEVEL  LU ACCELEROMETER  Feedback  FOTONIC SENSOR  Jb-o-  RECORDER  PREAMPLIFIER  GENERATOR (Feedback and Synchronization)  POWER AMPLIER  Fig. IH - 1. Signal Flow Diagram  For  Synchronization 1>  - 20 -  Fig.  III-2.  1.  Frequency  2.  Time  3.  Amplifier Analyzer  4.  and  Spectrum  of Experimental  9.  Wavetek  Apparatus  Signal  10.  Power  11.  Accelerometer  12.  Specimen  13.  Column  14.  Spring  15.  Fotonic  16.  Shaker  Generator  Amplifier Preamplifier  Frame  CRO Level  6.  Vibration Control  8.  Counter  Switch  5.  7.  Photograph  Fotonic BAM  Specimen  Recorder Exciter  Sensor  Unit  17.  Roots  Sensor  Blower  Holder  indicate  flexural  erometer  used  leration  level  displayed spectral  on  the  The by  automatic  Fig.  lower  and  strain  circuit  levels.  also  column mount.  level  the  recorder,  signals  vibration  generator)  the  desired signal  shaker.  s p e c i m e n mount actual  feedback  at the  a CRO  amplified the  longitudinal  were  and  The  monitored  accel-  the  acce-  These  signals  were  where  appropriate,  performed.  System  control  m i n e d by  i n the  analyses of  Electronic  The  and  An  III-3.  at  control  (hereafter  p r o v i d e d a pre-programmed  vibration was  used  level to  that  Schematic  a power  t o produce  accelerometer  ( F i g . II1-3)  vibration  exciter  rigidly  monitored  the  signal  of  mechanical m o u n t e d on  vibration the  acceleration  Lower Column  deter-  amplifier.  point.  View  called  Mount  lower of  the  - 22 Following rometer  amplification  preamplifier,  a n d impedance m a t c h i n g  thevibration  t h e programmed s e t - p o i n t control  generator then  correcting and of  the instruments  contained Loading The  Fig.  i nthecontrol  completed  any d i f f e r e n c e  t h e preprogrammed  level  was compared  generator.  t h e feedback  i n the vibration  signal  specification  control  loop i s  column m o u n t i n g  Description and l o a d i n g  system  i s shown i n  III-4. FRAME UPPER MOUNT COLUMN SPECIMEN SPRING STRAIN GAUGES  - ACCELEROMETER -LOWER MOUNT SHAKER  BED  BLOWER  N Fig.  The  i n A p p e n d i x B.  Frame and Column  III-4.  CONCRETE BASE  Column M o u n t i n g a n d L o a d i n g  with  c i r c u i t by  between t h e a c c e l e r o m e t e r  s i g n a l . A more d e t a i l e d  used  of the accele-  System  The  column  was  arranged  dently. two  the  the  ments  upper  shaker  quite  the  uin)  above  were  always  complete lower column  of  the  lower  the  column the  much  or  Y could  be  was  applied  to the  on  P =  lower  tests  system,  and  constrained arrangement  the  +  The  very  specimen  and  by  attached  always  of  suffi-  applied  cyclic  small  displace-  (of the  force  P  The  order  was  Q  were  conducted  at  the  operating  frequencies  natural  consisting  t o move  indepen-  P! c o s Y t was  static  Hz,  the  shaker  shown  0  were  Since the  than  P  was  It  i n compression.  mount.  of  hundred  and  frame.  varied  side  column  variation  greater  shaft  either  mounts,  springs  five  i n the  and  load  the  and  spring-mass  was  P„  vertically  to maintain the  four  mount,  t  mounted  through  so  , P  0  -  loaded  load  and  negligible.  cies  P  springs  component  of  200  this,  that  magnitude  varying  was  static  parallel  cient  of  so  The  between  by  specimen  23  table.  of  The  vertically.  i n F i g . III-5  frequency the  was  of  column,  lower To  frequen-  end  of  the springs, the  accomplish used.  - 24 -  Fig. Sectional  Drawing  111-5  of Loading  and A l i g n m e n t  Mechanism  3 A  — inch  diameter  Atlas  table  t o the lower  lower  mount  bushing the  directly  The attached  with  i s shown  upper  mount  t o t h e frame  mount.  using  the shaft.  so t h a t  i n line  mount  specimen  prevented  to guide  specimen  column  was  Superior shaft  The  t h e moving the lower in Fig. was  connected  the  shaker  Transverse motion  a Thompson shaker shaker  adjustable  was head  specimen  of the  situated was  ball below  positioned  support.  The  lower  Ill-3.  similar  except  that  and i t d i d n o t have  i t was  rigidly  the accelerometer  - 25 attachment.  The mounts were i n t e n d e d t o p r o v i d e  conditions.  As shown i n F i g . I I I - 3 , s p a c e r s were f o r c e d i n  on each s i d e between t h e column and mount.  clamped end  The top i n s i d e  edges o f t h e wedges were b e v e l l e d , and t h e specimen for  length  f l e x u r a l o s c i l l a t i o n s was t h e d i s t a n c e between t h e e x t r e -  m i t i e s o f t h e b e v e l s on t h e upper and l o w e r mounts.  To s e c u r e  3  the specimen i n p l a c e , a t i g h t f i t t i n g  i n c h diameter m i l d  s t e e l p i n was i n s e r t e d t h r o u g h t h e mount, wedges and specimen. The e x p e r i m e n t s were performed on a s t e e l specimen o f r e c t a n g u l a r c r o s s s e c t i o n f a b r i c a t e d from hot r o l l e d carbon steel, flat  Fig.  stock  (Fig. III-6) .  III-6.  Schematic View o f Column Specimen  The column specimen has a l e n g t h of 11.625" between and a 0.375" by 0.125" r e c t a n g u l a r c r o s s s e c t i o n .  fillets, The  dimen-  s i o n s o f t h e specimen were chosen t o produce a measurable s t r a i n l e v e l w i t h t h e f o r c e a v a i l a b l e , and t o p r o v i d e  the  f i r s t c o u p l e d l o n g i t u d i n a l r e s o n a n t v i b r a t i o n i n the 8 Khz region.  The f i r s t E u l e r b u c k l i n g l o a d o f t h e specimen f o r  clamped end c o n d i t i o n s was 534 pounds.  The s u r f a c e s were  l i g h t l y ground t o s i z e on a s u r f a c e g r i n d e r , which gave t h e column a s l i g h t i n i t i a l c u r v a t u r e , t a k e n care o f i n t h e t h e o r e t i c a l model.  The specimen had a w i d e r s e c t i o n on each  -  end t o accomodate  26  -  a h o l e f o r the p i n and t o a l l o w a  closer  approximation to clamped ends. The column specimen was mounted i n the upper p o r t i o n o f the  heavy frame, as shown i n F i g . I I I - 4 .  to  a s t e e l bed.  the  shaker was  On t o p o f the bed was supported.  which c i r c u l a t e d  The frame was  bolted  a block o f wood on which  Below the bed was  a Roots blower  a i r through the shaker t o prevent o v e r h e a t i n g .  The blower and i t s d r i v i n g motor were i s o l a t e d  from the t e s t  bed and frame to a v o i d any unnecessary t r a n s m i s s i o n of e x t r a neous v i b r a t i o n the  to the specimen.  t e s t bed was mounted was  minimize e x t e r n a l v i b r a t i o n  isolated sources.  graph of the specimen, l o a d i n g  Fig.  III-7.  The concrete base on which from the b u i l d i n g  F i g . I I I - 7 i s a photo-  frame, shaker and t e s t  Photograph o f Specimen  to  Loading  Frame  bed.  - 27 D e t a i l s of Measuring  System  A p a r t from t h e a c c e l e r o m e t e r , t h e use o f two t y p e s o f transducers was attempted t o m o n i t o r the b e h a v i o r o f t h e specimen.  The f i r s t of t h e s e was  a non-contacting displace-  ment t r a n s d u c e r , c a l l e d a F o t o n i c S e n s o r , manufactured by M e c h a n i c a l Technology, L i m i t e d .  The F o t o n i c Sensor i s a  s o l i d s t a t e e l e c t r o n i c i n s t r u m e n t w i t h a probe c o n s i s t i n g o f a packed bundle o f s p e c i a l l y c o n s t r u c t e d g l a s s  fibres  a r r a n g e d i n a random t r a n s m i t - a n d - r e c e i v e c o n f i g u r a t i o n .  The  t r a n s m i t t i n g f i b r e s carry l i g h t to the t a r g e t ; the r e f l e c t e d l i g h t i s returned through the r e c e i v i n g f i b r e s t o i l l u m i n a t e a l i g h t s e n s i t i v e diode.  Though t h e i n s t r u m e n t o f f e r s many  d e s i r a b l e f e a t u r e s , i t was not found v e r y s u c c e s s f u l i n t h i s work.  The i n s t r u m e n t probe must be mounted v e r y r i g i d l y so  t h a t i t undergoes no movements.  With t h e apparatus used, i t  was not found p o s s i b l e t o p r o v i d e a mount s u f f i c i e n t l y  iso-  l a t e d from a l l s o u r c e s o f v i b r a t i o n . The second type o f t r a n s d u c e r , the s t r a i n gauge, was more successful.  Four BLH E l e c t r o n i c s SR-4  t y p e FAP-12-12 f o i l  gauges were a t t a c h e d t o t h e m i d d l e o f t h e column specimen; two s i d e by s i d e a l l i g n e d a x i a l l y on each o f t h e w i d e r f a c e s o f t h e column specimen. the arrangement  To measure bending s t r a i n  levels,  shown s c h e m a t i c a l l y i n F i g . I I I - 8 was  where t h e specimen was  used,  l a t e r a l l y d i s p l a c e d t o produce d e c r e a s e d  compression i n two gauges on one s i d e and i n c r e a s e d compression i n t h e two gauges on t h e o t h e r s i d e .  A f o u r arm b r i d g e o f the  - 28  -  Fig. III-8 Schematic S t r a i n Gauge Arrangement f o r F l e x u r a l S t r a i n Measurement. type shown was  used f o r s e v e r a l r e a s o n s .  s e n s i t i v e bridge  c o n f i g u r a t i o n , and was  t u r e compensating.  T h i s b r i d g e was  I t was  the most  completely  i n s e n s i t i v e t o any  form l o n g i t u d i n a l s t r a i n s i n t h e specimen.  Since  s t r a i n l e v e l s were e n c o u n t e r e d , a f u r t h e r i n c r e a s e s e n s i t i v i t y was voltage  obtained  temperauni-  small in  by i n t r o d u c i n g as much a d d i t i o n a l  as p o s s i b l e i n s e r i e s w i t h the i n t e r n a l e x c i t a t i o n  of the bridge  amplifier.  The  A m p l i f i e r and Meter used was  E l l i s A s s o c i a t e s BAM  c a p a b l e o f measuring dynamic  s i g n a l s o v e r the range e n c o u n t e r e d .  I t s frequency response  i s such t h a t t h e a t t e n u a t i o n a t 10 Khz L o n g i t u d i n a l s t r a i n was  1 Bridge  i s approximately  measured u s i n g the b r i d g e  f i g u r a t i o n shown i n F i g . I I I - 9 .  3%. con-  L o n g i t u d i n a l s t r a i n cannot  be measured u s i n g f o u r a c t i v e arms, so two on each s i d e o f the specimen) and two  a c t i v e arms  (one  dummy gauges p r o v i d i n g  -  Schematic  Strain  sensitivity  in  a  arm  to  torsional  torsion this  equal  of magnitude  strain  sensitivity.  strain  oscilloscope. the at  level  The  The  of  Frequency  2107,  which  zer.  The  quency  gauge  served  level  Analyzer  and  middle  than  was  strain  taken  also  spectral  sensitive  gauges.  sensitivity  longitudinal  was  also  used  cross-section, since  i n both  strain  were  i s several or  from  torsional  t h e BAM  a m p l i f i e d and  analyses  For  were  to  an  recorded  also  on  recorded  interest.  analyzer as  used  a voltage  recorder was  the  signal  signal  recorder,  frequencies  lower  compensation  c o n f i g u r a t i o n was  strain  bending  orders  temperature  This  rotations of  arrangement,  The  and  bridge.  produces  -  Fig. III-9. Gauge A r r a n g e m e n t for Longitudinal S t r a i n Measurement  increased four  29  used  a Bruel  was  a Bruel  a m p l i f i e r and  i n conjunction  and  Kjaer  Type  and  Kjaer  spectrum with  2 305,  the  Type analyFre-  offering  a  - 30 -  r a n g e o f p a p e r and w r i t i n g  speeds.  means o f an i n k pen on 100 mm. brated paper. RMS  RMS,  f u n c t i o n was  R e c o r d i n g s were made by  logarithmic  frequency  DC o r peak v a l u e s c o u l d be  plotted;.the  chosen t o minimize t h e i n f l u e n c e  extraneous s i g n a l s .  F o r most t e s t s ,  of  the recording  Meter,  A G e n e r a l R a d i o Company D i g i t a l  Type 1151-A,  excitation forms and yzer  measured t h e average  t o a t l h*z. a c c u r a c y .  of interest  were done u s i n g  Time and  and l e v e l  Frequency  frequency of the  Spectral  analysis  o f wave-  frequency c a l i b r a t e d  paper anal-  recorder.  displayed  on a T e k t r o n i x  s c o p e , where t h e y c o u l d be p h o t o g r a p h e d photographs  was  frequency  a m e c h a n i c a l f r e q u e n c y l i n k a g e between t h e f r e q u e n c y  The waveforms were  X  sudden  paper  c a l i b r a t e d b y means o f an e v e n t m a r k e r and a d i g i t a l counter.  cali-  were t a k e n w i t h a P e n t a x  directly.  camera u s i n g  film.  )  565  oscillo-  These Kodak  Plus-  CHAPTER  Test Piro<  4  -  31  -  CHAPTER I V  TEST  PROCEDURE  Calibration  The  performance  instruments generator checks  was  checked.  at the upper  generator, between  b y means  Since  scale  the counter  frequency  of the control  and t h e f r e q u e n c y  was  a  accura-  on t h e c o n t r o l  used.  scale  which  well  be r e a d  scale  counter  circuit  against  f r e q u e n c i e s cannot  end o f t h e frequency  a digital  scale  of a built-in  on t h e f r e q u e n c y  frequency.  of the experimental  The frequency  calibrated  two p o i n t s  defined tely  was  and c a l i b r a t i o n  Agreement  was g o o d  ata l l  frequencies.  A [9]  factory  found  output, The  that  with  favourable response  BAMs  noise  frequency  spectral  was made indicated  were  over  the frequency analysis  was  the accelerometer.  used.  within the  involved.  analyzer  tests  the accelerometer  range  The  t h e most  frequency  manufacturer's of the  experiments.  a n a l y z e r was  Frequency  and t h e l e v e l  was e x c e l l e n t  agreed  generator.  and t h e one w i t h  o f the frequency  t h e range  Schneider  by t h e a c c e l e r o m e t e r  the frequency  response  used.  s e t on t h e c o n t r o l  characteristics  over  to cool  evaluated,  was w e l l  was  d i dnot affect  the acceleration  specifications  between  level  o f t h e BAM  flat  accelerometer  cooling  so no p r o v i s i o n  Several  tely  forced  acceleration  exactly  The  calibrated  comple-  synchronization recorder f o r  provided  they  were  - 32 accurately of  the frequency  reference  The  signal speed  records with  was  The  it  the  was  capable  response  of  during  an a c c u r a t e  speeds  providing  o f measured the  extent t o minimize  signals  sensitivity  zener-diode.  writing  the  influence  on  the s t r a i n  record.  before  and d u r i n g  the  o f one o f t h e o s c i l l o s c o p e  BAM.  often  necessary. A l l cables  pick  twisting  up  the course  The  experiment.  plug-in  units  of the experiments,  which  developed from  Fresh and  the  leads used  were  were  gauge  noisy  ground  signals  l o o p s was  were  so  a floating  minimize  always  signals  used i n  by  ground  was  Electro-  minimized  t h e gauges  promptly  eliminated  to  shielded.  electromagnetic fields.  i n s t r u m e n t s up w i t h  through  dry c e l l s  t h e l e a d s t o g e t h e r and k e e p i n g  out of strong  arising  o c c u r r e d , and e f f o r t s  i n the strain  leads  the  using  internal  found  calibrated  signals  were  magnetic by  an  The  replaced.  Noisy noise  t o some  unacceptable  was  high  and e x t r a n e o u s  performance  became  calibrated  by  was  beforehand.  In the present experiments,  reduced  oscilloscope  was  produced  recorder  levels.  noise  analyzer  voltage  level  accurate  of  synchronized manually  Any  and  strain  replaced. hooking  gauges Noise  most  and g r o u n d i n g  of them  oscilloscope.  Testing  The place and  specimen  with  i n the upper  the bevelled  and  strain lower  spacers  g a u g e s a t t a c h e d was (Figs.  III-3  i n on e a c h  side.  mounts  driven  put i n and The  III-5)  - 33 specimen  was  replaced. by  the  Any  output  correctly and  the  was  a  were  installed  and  of  strain  the  as  leads were  on  in  the  leakage  specimen  to  batteries of  gauge  bridge  least to  turn  advance  of  to  adjusted  were  in  coil  rigidity  to  the  specimen.  resistance  and  for  the  and  the  gauge  balanced.  before  the  this  point  to  testing.  instrumentation anticipated  on  the A  time.  from  wires  and The  excited. the  a  was few  of the BAM  By  strain  position  switch  the  gauges  a l l instruments  automatically  testing  The  ground.  BAM,  standby  time  excita-  possibility  lead  circuit  within  At  turned  the  in  e f f e c t of  of  resistances  BAM  extended  the  oscilloscope  and  the  leads  the  checked  combined  external  minimized  of  was  lead.  the  The  shields  compressive  preamplifier.  the  connected  III.  which  to  its  including  Chapter  specimen  accelerometer  according and  the  constant  accelerometer  accelerometer  leads  sharp  of  to  was  the  the  The  When  level  side a  detected  acceleration  connected  hours  the  column.  be  be  clean  The  connected  two  the  their  variable  properly  given  provide  on  to  and  ground.  were  means  a  could  became  to  loose  for  were  achieve  had  amplifier.  tightened  flexural  checked  to  configuration  a  power  either  gauge  described  gauges  the  gauge  shaker  on  was  the  the  power waveform  connected  appropriate,  tion,  any  and  strain  springs  pounds  capacitance  an  The  preamplifier  The  the  a  of  from  required  minimum.  64  when  misalignment  aligned,  power  of  only  waveform  attached  force  The  removed  for  were at  connected hours  in  - 34 Preliminary acceleration for  tests  level  levels next  necessary  c o u l d be used  the particular  control  were  over  system  studied.  could not provide  constant  over  the entire  choice would  frequency  be a c o n s t a n t  which  displacement  range level  acceleration  level  used  would  record over  most  of the frequency  best the  procedure  would  experimental  trol.  A  The  was  that  t h e power  some  point  these  the into  range  are used input  power  same t i m e t o o many  segments  into  range  bei n -  The  power  of this  that the  would  noise.  a constant  next  plotj but type  of  con-  'approximated  constant  power  i s the division  a number  level  over  t o t h e specimen segment.  i s an o p t i m i z a t i o n delivered  power  Table  and c o r r e s p o n d i n g  stant,  maximum  In this power  reaches  plot  was  such  a maximum a t  chosen  and s i z e  of  so t h a t t h e  i s maximum w h i l e a t  of the frequency  IV-1 i n d i c a t e s  a first  of  accelera-  segments  T h e number  acceleration  way  each  Different  frequency  t o t h e specimen  segments.  studied.  o f parts over  process  avoiding the division  column  spectra  i s constant.  different  i n the frequency  segments  average  an  constant  the acceleration levels  t o be so s m a l l  i s not capable  called  approximated  tion  been  acceleration  used.  the frequency  which  have  The  unsuitable since the  inherent electronic  system  compromise,  spectra',  of  from  exciter  or velocity  sinusoidal  however,  have  range  investigated.  This,  distinguishable  also  what  frequency  The v i b r a t i o n  excitation.  strain  was  to ascertain  the frequency  levels  used  approximation  achieved while  range  making  f o r the to a  con-  use o f  -  the experimental control.  system c a p a b i l i t y f o r constant  The a m p l i t u d e  as p o s s i b l e  35 -  so t h a t  o f v i b r a t i o n was  as h i g h  ACCELERATION LEVEL  750 Hz  7.0  750  850  4.0  850  1200  20.0  1200  2500  15.0  2500  5100  40.0  5100  8000  20.0  8000  9000  55.0  9000  10200  20 .0  to  kept  t h e s t r a i n s were more r e a d i l y m e a s u r e d .  FREQUENCY RANGE  250  thus  acceleration  T a b l e IV-1 P e r m i s s i b l e A c c e l e r a t i o n L e v e l s Over V a r i o u s F r e q u e n c y Ranges f o r Column S p e c i m e n The  t e s t i n g was now  ready  to begin.  The  following  s t e p s were a d h e r e d t o : (1)  the generator deliver  (2)  the desired  the frequency generator  c o n t r o l was  programmed t o  acceleration  scanning  speed  and c o r r e s p o n d i n g  level  r e c o r d e r were c h o s e n  speed  was u s u a l l y  state  conditions).  used  level.  on t h e c o n t r o l  paper speed (the lowest  t o approximate  on t h e scanning  steady-  (3)  36 -  the compressor stability  (4)  (5)  i n the  the frequency was  speed  with  unit  chosen  to provide  feedback.circuit  indicated  synchronized  the control  was  was  by t h e l e v e l  the control put i n i t s  recorder  unit  excitation  mode (6)  the proper analyzer  (7)  i n the frequency  was s e t  the attenuation recorder  (8)  amplification  and w r i t i n g  speed  of the level  were s e t  t h e Roots  blower  was  activated  to cool the  shaker (9)  (10)  These frequency from  of interest  signals  the scanning  steps  were  extended  possibility  At  analysis  were  over  was made  performed over  was  each  to another  a few H e r t z  of losing  activated.  segment  of the  of the actual due t o l a r g e  so t h a t their  information  display  change transient  the frequency  end points  at the  to  avoid  acceleration  frequency.  particular  made.  record level  to appropriately  mechanism  repeated  No  The t e s t s  changeover  that  the  one a c c e l e r a t i o n  segments  were  adjusted  spectra.  voltages.  the  t h e CRO w a a  frequencies  The p r o c e d u r e  was much  the frequency  of interest,  followed  t h e same scanning  as t h a t  spectral  i n obtaining outlined  mechanism  a  above,  was n o t  analyses  spectral except  activated,  the  frequency  mechanical level  frequency  recorder  quency test  a n a l y z e r was  s y n c h r o n i z a t i o n was  and frequency  of interest  was  put i n the analysis  was  analyzer.  mode,  and a  p r o v i d e d between t h e The e x c i t a t i o n  s e t on t h e c o n t r o l  generator  fre-  and t h e  carried out.  Photography  Photographs were  obtained  particular  over  interest  Photographs oscilloscope Spotmatic of  of strain, most  o f the frequency  a r e shown  i n Chapter  o f waveforms were  i n the normal  Camera w i t h  second,  acceleration  with  taken  triggering  and power range.  waveforms  Those o f  IV.  directly  mode  using  from the a  Pentax  an f / s t o p o f 2.8, an a p e r a t u r e  a 55 mm.  lens  and a no. 3 c l o s e - u p  speed lens.  Oscosslooi ©f R e s u l t ;  -  38  -  CHAPTER  RESULTS  Interpretation  The a  chart  Fig. the  DISCUSSION  of Frequency  strain  versus  produced  V-3).  AND  To  by  V  Spectra  frequency  the level  p l o t s were  recorder  correlate the record  following parameters  must  (a)  t h e gauge  (b)  the strain  be  obtained  ( s e e F i g . V-2  with  and  the measured  strain,  known:  f a c t o r and b r i d g e gauge  on  bridge  arrangement  excitation  voltage  used (c)  the magnification  o f t h e BAM  and  voltage  amplifier (d)  the attenuation  The  bridge  The  magnification  pendent.  e x c i t a t i o n voltage o f t h e BAM  I t i s apparent  voltage  level  reason,  and because  to  study  to a true  was  carried  out at resonant  cludes this peaks  that  not carried  of these  out.  was  was  small  minimized (less  intent  that  and  dependent.  frequency  de-  o f the measured For  this  of the experiments i s  v i b r a t i o n s , such  a  con-  c a l c u l a t i o n s were  of interest Care must  records,  since  electromagnetic and t h e i n f l u e n c e 1 db).  time  i s tedious.  Approximate  vibrations.  a c o n t r i b u t i o n from  effect  record  frequencies  of the strain  time  the conversion  strain  recorder  is slightly  i s both  of predicted  version  interpretation  used  the primary  the existence  significance  of the level  be  t o study  the  exercised i n  the signal i n induction. on  the  However,  resonant  - 39 The records  frequencies  a t which peaks o c c u r r e d  were s t u d i e d ,  and t h e s t r a i n  q u e n c i e s were a n a l y z e d . sented  and d i s c u s s e d  Identification The several to  between  By  to identify  'linear  linear  verse  records  obtained  oscillations  Resonant  1  Some o f t h e s e  t h e peaks i n o r d e r  flexural  i n order  analytically.  o b j e c t i v e was n o t r e a l i z e d . where between c l a m p e d frequencies  frequencies  of various  transverse  of strain frequencies  mental points  resonant  resonant  the experimental  setup this  F i g . V - l i n d i c a t e s the  modes theory  f o r pinned accounting  and c l a m p e d f o r t h e con-  A l s o p l o t t e d on t h e g r a p h a r e t h e  peaks b e l i e v e d t o c o r r e s p o n d o f t h e a c t u a l specimen.  are obtained  trans-  A knowledge o f t h e e n d  clamped end c o n d i t i o n s ,  and p i n n e d .  compressive end l o a d .  first.  The end c o n d i t i o n s were some-  ends as c a l c u l a t e d f r o m l i n e a r stant  peaks.  t o determine the  Although  was d i r e c t e d t o w a r d s o b t a i n i n g  resonant  to distinguish  are considered  c a n be c a l c u l a t e d .  i s necessary  correspond  I t i s therefore  p e a k s and n o n l i n e a r  oscillations  peaks a t  peaks  (Appendix C ) , t h e approximate  frequencies  frequencies  have r e p r o d u c e a b l e  o f t h e specimen.  resonant  theory  conditions  are pre-  chapter.  discrete frequencies.  desirable  results  fre-  o f S t r a i n Peaks  strain  resonant  waveforms a t t h e s e  The e x p e r i m e n t a l  i n this  on t h e s t r a i n  from t h e f l e x u r a l  quency r e c o r d .  The e x p e r i m e n t a l  curve  the  frequencies  o f clamped end o s c i l l a t i o n s  end  vibrations.  to  The e x p e r i -  strain  versus  i s seen t o f a l l  The n a t u r a l f r e q u e n c i e s  resonant  and t h o s e  of transverse  fre-  between of  pinned  vibra-  - 40 -  Fig. V - l . Forced  Resonant  Frequencies  for  Various  Plotted  Versus  End C o n d i t i o n s  Mode  Number  tion  as t h e y  test  o f the boundary  are plotted  Resonant following.  end  loading,  a t 8789  strain  record  Hz. the  This  mental  Using  linear  theory,  Hz  (Appendix  occurs  would  C).  required  A  a t 8 700 H z . level  indicate  desire  as a q u a l i t a t i v e  and a c c o u n t i n g  spike  f o r the pre-  at  condition  to oscillate increase  the acceleration  required  increased  resonant  would  i s  t h e power  of acceleration  o f t h e column  in  on t h e l o n g i t u d i n a l  Further,  a probable  to l i m i t  are considered  l o n g i t u d i n a l resonance  l o n g i t u d i n a l frequency  loading  V - l serve  conditions.  a constant  natural  in Fig.  the fundamental  dicted  sustain  -  longitudinal oscillations  the  to  41  at the the  8700  since  funda-  external  to the  preset  level.  Finally, using  linear  stiffness  t o r s i o n a l resonances theory,  considering  and t h e compressive  frequencies  are c a l c u l a t e d  are considered.  the decrease  end  load,  (Appendix  C).  t o r s i o n a l resonance  occurs  second  t o r s i o n a l harmonics  are calculated  on  No  the l o n g i t u d i n a l s t r a i n  3380 H z , nance appear at  respectively.  so i t appears  i s not excited.  10000  torsional  Hz.  These  vibrations  record  that  will  be  with  strain  computed  Hz;  the f i r s t  a t 6 760 Hz  strain  peak  fundaand  and  i s observed  i n the neighbourhood torsional  of reso-  similar characteristics record  discussed  i n a later  resonant  The  the fundamental  Peaks  on t h e e x p e r i m e n t a l  a t 3380  significant  in torsional  the predicted  mental  10140 Hz  Again,  a t 6920  Hz  i n connection  section  of this  and with  chapter.  again  Results  of Flexural  Fig. for  identified section that  Strain  V-2 s h o w s  the midpoint  42  strain  o f t h e column.  versus  Most  flexural  strain  peaks  i s expected  • -t <  < i i  than  even  those  a t t h e column  i ( ( ( i if f  made  resonances. with  frequency  of the strain  according t o the deductions  transverse resonances  This  Record  aflexural  concerning  smaller  -  peaks a r e  i n the previous  I t i s  noteworthy  mode n u m b e r s  having  record  o d d mode  appear  as  numbers.  midpoint.  f -c —f—r — r —r—r—r — r— r— r— r — r  ed Flexural Mode -Fifth Flexural Mode E,i .,„ Second Coupled Flexural Mode= 1  —r —r—r—r—r—r —r—r—r—r—r—r—r—i' —r —r —r —t —r —i  Tenth-ElexyraLModp^^ m i n l a r l Flexural F l A v u m l Mqde M o d e ^ First Coupled EXCITATION ERECIUENCY^  Flexural  As  shown  specimen column the  spike  F i g . V-2 Obtained  i s at o r near  midpoint  exhibits  strain  number,  larger  Record  9 10  kHz  at Midpoint  o f Column  s c h e m a t i c a l l y i n F i g . V-3, t h e midpoint  flexural  mode a  Strain  will  result.  f o r an e v e n  relatively  midpoint  little  i s near  a n d an a t t e n d a n t  of the  mode n u m b e r .  F o r an o d d  an a n t i n o d e  larger  The  c u r v a t u r e , and  i s c o r r e s p o n d i n g l y low.  t h e column  curvature,  a node  1—r —r —r—r-  —i  Ninths Fl exural Mode  ein  amplitude  having strain  -  43  -  Mode  S h a p e at M i n i m u m  Mode  Shape at M a x i m u m  Extension  Extension  Midpoint Flexural Displacement  ^ M i d p o i n t Flexural ,' Displacement  P  Q  ODD  Possible  not  F i g . V-3. Shapes F o r A x i a l l y  Mode  yet accounted f o r . frequency  frequency  digital the  EVEN  NUMBER  significant flexural  citation tion  MODE  Flexural  Two  P  Rcosrt  +  counter.  vibration  Except  of  at  low  'fluttering' oscilloscope;  of  The  about  strain  first 2900  approximately  spikes  4 350  the Hz  occurs  second as  excitation  levels,  feedback  the v i b r a t i o n  r e s p o n s e waveforms circuit  i s not  Column  a t an  a t an  indicated  a t e n d e n c y t o become  The  NUMBER  appear which  system has  the  MODE  In the neighbourhood of these  rapidly.  PcosTt  Excited  of these Hz;  +  D  two  are ex-  excitaby  the  frequencies,  unstable. system flutter  capable  of  begins on  the  stabili-  zing  the  oscillation.  citation the  i s lowered,  response  the  stability,  on  system  described  earlier.  s t i l l  same  the  phenomena  strain  waveform  frequency  approximately on  the  tion  a  strain  record.  The  when  fundamental excitation  in  the  are  observed  bars  at  spikes on  the  and  studied.  of  system  jump  to  waveforms response  this  the  is  jump If  flexural  excitation.  passes  through  vibration  at  spike  appears  those  excita-  to  Hz,  fle-xural while  and  of  when  the  at  resonances  as  cross-hatched  strain  non-  peaks  are  coincides.with the 8700  the the  to  strain  associated with  frequency  half  flexural  flexural  frequency  i s one  corresponding  the  theory,  Nonlinear  excitation  4 350  of  of  indicates  frequencies  longitudinal  frequency  so  longitudinal  correspond  linear  peaks.  frequency  be  oscillation.  frequency  F i g . V-4  strain  excitation  resonant  unstable  nonlinear strain  which  ordinary  indicate  present  nal  solid  of  ex-  record.  shown  at  a sudden  fluttering  that  excitation  small  in  frequency  one-half  can  condition  of the  type  8700  frequencies  linear  be  a  the  the  becomes s t a b l e  frequency,  fundamental  resonant  bars  frequency  o c c u r r i n g , the  Hz  from  by  level.of  4350. Hz  results  excitation  the  and  level  indicated  would  system  Hz  f o r the  frequencies  by  2900  a snap-through  histogram  predicted  the  hand,  The  when  flexural  The  as  the  were  Finally, the  as  i s not  snap-through  at  other  instability  acceleration  however,  acceleration the  -  I f the  waveforms  Increasing  44  Hz,  when  fundamental excitation  the longitudi-  frequency  - 45 is  one t h i r d  2900  of the fundamental  longitudinal  frequency  Hz.  EXCITATION FREQUENCY Frequencies  The  resonant  F i g . V-4. o f Experimentally Observed Resonant S t r a i n Peaks  transverse  quencies  are referred  coupled  f l e x u r a l modes  histogram at  at  which  i n F i g . V-4 coupled  condition  respectively. indicate  f l e x u r a l modes  Flexural  occurring  t o as t h e f i r s t ,  those  kHz  at these  second, Arrows  and t h i r d  below t h e  excitation  are expected  fre-  frequencies  from  theoretical  considerations.  Shown  i n F i g . V-5 a r e p h o t o g r a p h s  corresponding The  upper  of several  to o s c i l l a t i o n s o f the coupled  waveform  i n each  picture  waveforms  flexural  i s the excitation  modes. voltage  - 4 6 -  • First  Coupled  Flexural  Mode  t  (Excitation Order  Frequency=8700  of Signals(top Excitation Flexural  to  Hz  bottom)  Voltage  Strain  Acceleration  Second  Coupled  Excitation Order  Flexural  Mode  Frequency=4350  of Signals(top Excitation Flexural  Hz  t o bottom)  Voltage  Strain  Acceleration  Third  Coupled  Excitation Order  Flexural  Frequency=2900  of Signals(top Excitation Flexural  Strain  F i g . V-5 Corresponding t o Coupled  Flexural  Modes  Hz  t o bottom)  Voltage  Acceleration  Waveforms  Mode  supplied to the shaker, the middle waveform is the flexural s t r a i n level at the midpoint of the column, and the lower waveform i s the acceleration monitored at the lower end mount of the column.  The time base i s the same for a l l the wave-  forms in each picture.  The s t r a i n at the column midpoint  corresponding to the second and t h i r d flexural modes are comparable in magnitude to the strains associated with l i n e a r flexural resonances,  while the strain level at the  first  coupled flexural mode i s somewhat less. The nonlinearity at an excitation  frequency of approxi-  mately 8700 Hz may be the result of longitudinal i n e r t i a forces,  which can influence the dynamic behavior of a column  when the frequency of the^external  force is near the l o n g i -  tudinal natural frequency of the column; that i s , when the longitudinal vibrations have a resonance  character.  In other-  words, t h i s nonlinearity may represent the parametric influenc of resonant longitudinal vibrations which give rise to a f l e x u r a l vibration as indicated in the theory by the term (u w ) . x  x  x  The second and t h i r d coupled flexural modes may be  parametrically excited in a s i m i l a r manner. Recalling the theoretical predictions,  the  transverse  response was seen to- be,";appT5S?mately sinusiodal on the substitution  (equation 2-12) .  This apparently describes  first the.  transverse motion quite well except when the fundamental longitudinal frequency i s excited.  On the t h i r d  substitution  the transverse motion included a sinusoidal term with twice  the  excitation  describes the  the  frequency response  fundamental  third from  coupled the  ( e q u a t i o n 2-19)  when  the  longitudinal  flexural  term  appearance  of  This  excitation  frequency.  resonances  sinusoidal  .  having  twice  frequency  The  exhibit  second  no  the  better equals and  contribution  excitation  fre-  quency .  The indicates do  occur.  flexural by  that  the  vibrations  arrows  lower  not is  compared  the  provided  the  on  Axial  V-6  However,  a  f o r the  than  second  and  at  the second  coupled  frequencies  i n the  flexural  strain  below  1000  amplitude  of  these  the  amplitude,  of  the  of  the  excitation.  frequency  indicated  Hz.  are  irregularities waveform  Further, No  strain  noted  These  primary  response.  flexural  were  the  tests  explanation is  peak  which  is  not  Hz.  Strain  Record  axial  strain  addition  small  to  record the  influence  bridge  is directly  vibration  the  900  In  flexural  V-4.  remaining  i s an  column.  contains  voltage  high  f o r at  of  Fig.  also  frequency  f o r one  accounted  to  first  anticipated  nonlinearities  focused  Results  the  frequencies, particularly  small  were  of  in Fig.  coupled  vibrations  only  were  discussed, since  having  third  coupled  Theoretically,  Small at  higher  the  due  axial to  configuration  proportional  to  f o r the strain,  the used,  the  midpoint the  flexural the  axial  of  record strain.  measured  strain  and  the  -  Axial  Strain  of the flexural  strain  a n d maximum  the to  order  strain.  axial  record. mately about  The  4350 H z ,  these third  Fig. three  spikes  coupled  V-6.  frequencies  when  axial  strain  o f coupled  that  i s  and a r e o f  the contribution  negligible.  on t h e a x i a l  strain  frequency  the excitation  occurs  when  flexural  the  of approxi-  frequency i s excitation  The r e s o n a n t  conditions  are referred  t o as t h e f i r s t ,  modes  The e x c i t a t i o n  resonant  strain  appear  and t h e t h i r d  i s 2900 H z . strain  are comparable  i s a t an e x c i t a t i o n  8700 Hz, t h e s e c o n d  frequency  and  first  spikes  o f Column  t h e maximum  i t i s apparent  t h e r e c o r d by t h e f l e x u r a l  prominent  Since  strain  o f 200 u i n / i n ,  Three  to  -  F i g . V-6. Obtained at Midpoint  Record  square  49  respectively,  frequencies  spikes  resonances  second  as i n d i c a t e d  associated with  coincide with  flexural  corresponding  the  in  these  excitation  discussed  earlier.  -  In  general,  quickly peaks  than  resonant  V-6  these of  a t two  these  away  Hz,  response  the central  if  i t i s originally  is  under  it  c a n be  strain AB  1  that  i n those  Fiber  through  AB  angle  metrically  ©.  6920  Hz  a n d 10000  The  histogram  shown  are observed  indicate  torsional  Hz.  The  shape  sweeps  excitation  twisting  of  through  of  puts  torsional  the  fibres  Remembering t h a t  t h e column  will  the central when  to Fig.  reduce  fibre,  t h e column  torsional  oscillations  strain  will  level  at excitation  V-7  the  such  as  i s twisted  be  are  para-  reduced.  frequencies of  Hz.  frequencies  bars  first  i n tension,  t o AB'  the axial  at which  bars  The  o f t h e column  from  when  frequencies  hatched  have  record i n  system  o f t h e column  i n F i g . V-6  more  strain  l o a d i n g , and r e f e r r i n g  elongate  excited,  i s t h e case  fibre  away  Thus,  The  strain  i s a t 10000  as s t e e l ,  twisting  fibres  must  Such  axial  up  oscillations  parametric  unloaded.  compressive seen  peaks.  as t h e v i b r a t i o n  plane  build  o f magnitude.  the second  For m a t e r i a l s such  from  peaks  on t h e a x i a l  frequencies indicates  modes.  strain  and a x i a l  order  appear  strain  frequencies of excitation.  i s a t 6920  the s t r a i n  axial  flexural  peaks  -  flexural  i n t h e same  Inverted Fig.  resonant  f o r resonant  amplitudes  50  strain  correspond  on  spikes  arrows  strain  axial  frequencies  The  indicates  those  corresponding  the axial  t o coupled  excitation  resonances.  i n F i g . V-8  to  record.  modes  while  associated with  below  the histogram  excitation resonant The the  crosssolid  coupled indicate  - 51  A  Portion  of A  -  Fig. V-7. Column U n d e r g o i n g  Twisting  A I  8?  8  EXCITATION FREQUENCY  Frequen c i e s  Fig. V-8. of Experimentally Observed  Axial  kHz Strain  Peaks  those  excitation  modes.were record. tions  of  Two  axial  a t which  anticipated  discrepancies  between  results  i s significant  the theoretical  considerations.  torsional  mode,  when t h e e x c i t a t i o n the fundamental  However, frequency  i t appears  on t h e a x i a l  though  frequency  The  third  the second  t o be e x c i t e d ,  frequency  to the third  was  expec-  i n the neighbourhood  t o be p a r a m e t r i c a l l y  corresponding  predic-  i t i s not present  Further,  was  excited  strain  the theoretical  i f i t were  longitudinal  parametrically  are apparent.  mode  coupled ted  theoretically  and e x p e r i m e n t a l  coupled in  frequencies  a t 8700 H z . excited  torsional  First  mode.  Coupled  (Fundamental Resonance) Excitation  at the  Axial  Longitudinal  Frequency=8700  Order o f Signals bottom) Excitation Axial  Mode  (topt o  Voltage  Strain  Acceleration  Waveforms  Fig. an  V-9 i s a p i c t u r e  excitation  excitation  Fig. V-9. a t Fundamental Longitudinal  frequency  frequencies  Resonance  o f t h e waveforms  o f 8700  Hz.  Similar  i n the neighbourhood  corresponding  to  waveforms f o r o f 4 350 Hz a n d  -  2900  Hz  a r e shown  A spectral  53  i n F i g . V-10  analysis,  f r e q u e n c i e s was  revealed:  a t an e x c i t a t i o n  tion  frequency  quency 2900 The  vibration  8700  Hz  Hz  o f 4350  Hz  exists;  and  terms  and  longitudinal  having twice mode  The  response  the e x c i t a t i o n  at e x c i t a t i o n  and  frequency.  i s therefore expected  a  was resonant  a t an of  fre-  frequency 8700  t o be  excita-  of  Hz  exists.  significant  frequencies  of  development, suggested  t h e sum  frequency  Hz,  vibration  (2-8) a p p e a r  was  these  exists;  of frequency  theoretical  the excitation  o f 3700  8700  at  following  a t an e x c i t a t i o n  i n equation  Hz the  frequency  vibration  predicted theoretically Hz.  The  a longitudinal  as  4350  signal  performed.  of frequency  a longitudinal  coupled  and V - l l r e s p e c t i v e l y .  of the strain  excitation  longitudinal  -  a sinusoidal  The  t o be  of a sinusoidal  excited  when  that term  term  fundamental  8700  having  longitudinal  the  excitation  '.'i •  r  freq.uency  i s one h a l f  o f "the  frequency.  excitation Hz  soidal  tal when  frequencies  term  vibration phase  of the strain  having term  system  moves  relationship  longitudinal t h e two  points  A  phase  the excitation through  frequency  As  in Fig.V - l l  change  the excitation  this  i s illustrated  sinusoidal  a n d A'.  <9  twice  having  waveforms  frequencies i n the neighbourhood  reveals a continuous  sinusoidal  longitudinal  *  Observation  4350  fundamental  are  the frequency  frequency  frequency  frequency  excited  ' i n phase' range  the  of sinu-  and t h e  as t h e  range.  i n F i g . V-12.  i s first  terms  between  for  The  in Fig.  This fundamenV-12(a)  as i n d i c a t e d  i s scanned,  by  t h e two  -  54  -  ! Excitation I , Order  of  Frequency  Signals  Excitation Axial  =  4320  (top to  Hz  bottom)  Voltage  Strain  A c c e l e r a t ioi.  Excitation Order  of  Frequency  Signals  Excitation Axial  =  4345  (top to  Hz  bottom)  Voltage  Strain  Accele ration  Excitation Order  of  Frequency  Signals  Excitation Axial  =  4360  (top t o Voltage  Strain  Acceleration  Waveforms  Fig. V-10. Corresponding to Second  Coupled  Axial  Mode  Hz  bottom)  - 55-  Excitation Order  Frequency  of Signals Excitation Axial  =  2890  Hz  ( t o p t o bottom) Voltage  Strain  Acceleration  Excitation Order »  Frequency  of Signals Excitation Axial  =  2905  Hz  ( t o p t o bottom) Voltage  Strain  Acceleration  Excitation Order  Frequency  of Signals Excitation Axial  Fig. V - l l . Corresponding to Third  Coupled  Voltage  Strain  Axial  2910  Hz  ( t o p t o bottom)  Acceleration  Waveforms  =  Mode  Increasing  (N  Excitation Frequency Y  (c)  lb)  F i g . V-12. I l l u s t r a t i o n Showing Changing Phase R e l a t i o n s Between T h e Two V i b r a t i o n s C o m p r i s i n g t h e S e c o n d C o u p l e d A x i a l Mode  waveforms point  A  move  i n F i g . V-12(b).  two waveforms of  'out o f phase'  Finally,  shown  by  . -S7  between  of a complete  i n F i g . V-12(c),  o f the fundamental l o n g i t u d i n a l  discontinues.  t h e movement  t h e phase  has changed by one h a l f  the excitation,  citation  as shown  and h e r e resonant  of  the cycle  t h e excondition  First  Coupled  Excitation Order  Torsional  Frequency  of Signals  = 6920  Hz  ( t o p t o bottom)  Excitation Axial  Mode  Voltage  Strain  Acceleration  Second  Coupled  Excitation Order  Torsional  Frequency  of Signals Excitation Axial  Mode  = 10000  Hz  ( t o p t o bottom) Voltage  Strain  Acceleration  Waveforms  The coupled and ted,  C o r r e s p o n d i n g t o F i r s t and Second T o r s i o n a l Modes  waveforms torsional  10000  Hz  motion  modes a t e x c i t a t i o n  respectively  the strain  coupled  corresponding to the f i r s t  waveform  torsional  mode  t h e second  and  frequencies  second o f 6920  a r e shown i n F i g . V - 1 3 .  remains  sinusoidal  i s excited,  h a s t h e same f r e q u e n c y  p o n s e when  Coupled  coupled  since  when  As  the  load.  torsional  i s  mode  expec-  first  the resultant  as t h e a p p l i e d  Hz  axial  The  excited,  res-  - 58 however, able  i s somewhat  here,  strain will  due t o t h e p a r a m e t r i c  strain  excitation by the  since the frequency  be d i f f e r e n t  axial  nonlinear*  than  frequency  of the variation  excitation  the frequency  due t o t h e l o n g i t u d i n a l  of the third  observing  A nonlinearity  that  a  scale.  torsional  "linear  1  i n the axial  of a torsional  of the variation excitation.  mode  flexural  i s reason-  might  be  resonance  mode, of the  Parametric accounted f o r i s nearby  on  AjeuuuuoiQ  - 59 CHAPTER V I  SUMMARY, &  CONCLUSIONS  Summary  An  experimental  behavior  o f a column  Observed  resonant  vibrations  investigation  was  made  o f t h e dynamic  subjected to periodic  forced  a r e compared  vibrations with  those  axial  loading.  and n o n l i n e a r p r e d i c t e d by  coupled  theoretical  considerations.  To  accomplish  differential to  this  equations  study,  o f motion  describe the relationship  oscillations particular plane  o f t h e column  relating from  and between  importance  to axial  inertia.  interpreting  longitudinal  were  vibrations Finally,  having  the second  expected  t o be  f o r t h e column  and  strains  motion. initial  These  flexural  vibrations. i n the  Of  central  terms,  In a d d i t i o n , ratio  and p o s s i b l y  a  arise  and  longi-  the equations  several  predicted. with  strains  crookedness  manipulating  were  a frequency  axial  partial  i n formulating the equations  vibrations  anticipated.  derived  initial  a r e assumed  vibrations  nonlinear  and t o r s i o n a l  some o f t h e c o u p l e d  coupled  were  i s that  By s u i t a b l y  resonant  1:2  axial  and f l e x u r a l  coupled  were  between  considerations regarding  tudinal  coupled  nonlinear  Firstly,  frequency  two  ratio  of  two c o u p l e d  flexural  o f 1:2 w e r e  expected.  the third  parametrically excited.  and  torsional  modes  - 60 The with  experimental  the theoretical  was  found  the  excitation  mode the  when  when  flexural  the excitation  longitudinal mental  longitudinal  indicated  that  with  frequency.  parametric  coincided  resonant  These  response  influence  resonant  was  with  axial  fundamental  o f 1:2.  the  vibrations fundamental  of the  funda-  A waveform  t o that  of  response  resonant  at these  equal  when  one h a l f  of the  and one h a l f  frequency.  a frequency  coupled  ratio  exhibited  response  fundamental  and t h e  a frequency  frequency  the resonant  sinusoidal  tion  frequency  resonant  frequency  the frequency  also  agreement  resonance  to the  frequency,  hence  response  axial  The second  the excitation  longitudinal  good  The l o n g i t u d i n a l  coupled  frequency.  resonance;  provided  corresponded  a vibration'having  longitudinal  was  frequency  fundamental  contained  the f i r s t  resonant  appeared  The  predictions.  to exhibit  longitudinal  investigation  two  analysis  frequencies  of the excita-  o s c i l l a t i o n s represent the  of longitudinal  o s c i l l a t i o n s on  flexural  oscillations.  Parametrically ved  when  the longitudinal  fundamental frequency third  torsional  was  equal  torsional  Further and  flexural  quency  excited  excitation  frequency,  resonances frequency  and again  t o the frequency  when  were  was  twice the  the  corresponding  obser-  applied  to the  mode.  coupled response  equaled  torsional  o s c i l l a t i o n s were when  one t h i r d  observed  the longitudinal  of the fundamental  on t h e a x i a l  excitation  fre-  longitudinal  resonant axial  resonance  ration nal  frequency.  resonance.  the  The  that  analysis  of  coupled  having  of  the  the•response  frequency  third  vibration  -  waveform  indicated  exhibiting  sinusoidal  A  61  the  the  coupled  contained  fundamental  flexural  same  third  vib-  longitudi-  resonance  frequency  a  as  was  the  a  applied  loading.  Both the  the  axial  quickly  coupled  dynamic  than  Coupled  those  'linear'  with  coupled  fundamental rations modes  of  strain  resonant  record. of  different phase  frequency  hibiting  the  the  i s less  with was  passed.  primary  when  two  vibrations  were  discontinued tudinal  higher  each  other  ' i n phase  at  1  f r e q u e n c i e s the  with  the  component  frequency  90°  'behind'  having the  strain  two  vib-  coupled the  the  axial  ex-  appeared  excitation  the •  coupled  waveform  became  to  associated at  as  frequency  amplitude  more  level  The the  on  comparable  strain  the  out  level  incipient  having  resonant  at  strain the  peaks  dynamic  frequency.  longitudinal  parametric  and  than  The  waveform  died  amplitudes  The  respect to  The  frequency,  and  frequencies comprising  quency. the  up  strain  flexural  modes.  resonant  fundamental  the  on  modes h a v e  flexural modes  resonant  record built  peaks  longitudinal  resonant  of  'linear'  flexural  axial  changed  'ahead'  and  90°  fre-  maximum  resonant  resonant  condition  the  fundamental  primary  vibration.  longi-  Conclusions  The  following  theoretical  and  c o n c l u s i o n s were  experimental  drawn  analysis  on  from the  the  foregoing  flexural,  longitudinal, to  periodic  1.  and t o r s i o n a l  axial  coupled  anticipated  longitudinal  and f l e x u r a l  longitudinal  vibration  applied  loading  resonant  frequency  as t h e e x t e r n a l  which  derations.  frequency quency . ing  the frequency  ponse  frequency though of  ratio  i t may  a resonant  frequency tains tion  n o t be  ratio  a third  o f 1:3  flexural load  This  o f two  having  a  t h e same  resonance  also  fre-  exhibit-  exists.  indicates  that  vibrations  the  having  seems  the axial three  excitation  longitudinal  The e x p e r i m e n t a l  that  consi-  fundamental  the  c o n t a i n i n g two v i b r a t i o n s suggests  experi-  the  when  interpretation  oscillation exhibiting  frequency.  frequency  that  i s excited  development  complete.  condition  of the  longitudinal  i s observed  suggests  coupled  i s t h e sum  o f 1:2.  fundamental  f o r by t h e t h e o r e t i c a l  of the applied  o f t h e column  response  o f the fundamental  The a n a l y t i c a l  between  the frequency  excitation  vibration  vibration  i s one t h i r d  The  as  loading.  analysis  A corresponding  3.  same  i s not accounted  resonant  to exist  o f the fundamental  coupled  A waveform  longitudinal  when  on t h e f l e x u r a l  resonant  appear  oscillations.  At t h i s  appears  mentally  subjected  due t o t h e i n t e r a c t i o n  i s excited  resonance  A  vibrations  i s one h a l f  frequency.  2.  o f a column  loading:  Resonant  theoretically  response  a  correct,  observation with  response  times  res-  the  a con-  excita-  4.  Theoretically,  the f l e x u r a l response  sum  o f two v i b r a t i o n s  with  the  predicted  resonances  results  coupled  reveal  no c o n t r i b u t i o n  the  vibration  is,  the f l e x u r a l response  frequency  having  over  5.  twice  to  the fundamental  torsional ted  that  mode higher  with  the excitation  range  'linear'  6. quite of  f l e x u r a l resonance  amplitudes. slightly peaks  less.  build  resonances. resonance other  axial  phase  as t h e r e s o n a n t  fundamental,  was  also  more  vibrations  frequency  that  a  axial  flexural  are  resonant  are i n general  resonant  than  comprising  appear  amplitudes  amplitudes  sharply  by a t o t a l  noted  nearby.  'linear'  resonant  be  This  to the fact  f l e x u r a l resonant as  expec-  corresponding t o  t o r s i o n a l mode.  be a t t r i b u t e d  The two v i b r a t i o n s  change  frequency  and ' l i n e a r '  up a n d d e c l i n e  the  was  i s close  t o r s i o n a l mode was  resonant  o f magnitude  Coupled  could  of a l l coupled  Coupled  Coupled  frequency,  A coupled  The amplitudes  frequency  similarily  e x c i t a t i o n might  t h e same o r d e r  the applied  resonances  f o rthe third  significant.  torsional vibrations  I t i s t o be  a t an e x c i t a t i o n  That  investigated.  parametrically.  experimentally  parametric  from  i s sinusoidal  excited.  frequency  the experimental  to t h e f l e x u r a l response  torsional  torsional  While  frequency.  When  i s excited  o f 1:2.  load  parametrically  the  do o c c u r ,  The e x i s t e n c e o f c o u p l e d experimentally.  ratio  the external  the frequency  verified twice  a frequency  i s also the  strain  do f l e x u r a l  a coupled  axial  o f 180° r e l a t i v e t o each  i s passed.  T h e maximum  - 64 - • amplitude are  resonant  condition  appears  vibrations  r  First would done  f o r Future  Research  considerations  probably to date.  i n suggesting  aim a t overcoming To t h i s  equations 'derived  predictions  could  coupled  oscillations;  excitation tional could  then  could  variables be  be  b e made  similar  t o t h e one. u s e d .  and/or  damping  Though solution,  meters  through  i t would  might  damping  damping  n o t be  oscillations  apparatus  be  extended  and ' l i n e a r ' of viscoelastic  to the  and i s o l a t i o n  surfaces properties  i n evaluating  the  materials.  obtained  describing  results.  addi-  theoretical  then  layers  as f r u i t f u l  solution  of  and  temperature  the application  be o f i n t e r e s t  equations  experimental  on c o u p l e d  and  of  response  an e x p e r i m e n t a l  parameters  would  a numerical  excited  damping  The study  o f various  differential  plement ally  probably  layers  the amplitudes  and t h e i n f l u e n c e  of c o n t r o l l i n g coupled  Optimizing  effectiveness  the  using  elastic-viscoelastic  the bar.  for  be n e c e s s a r y  f o rt h e  Analytical  V e r i f i c a t i o n f o r these  would  oscillations,  concerning  as i n t e r n a l  results  research  solution  i s desirable.  investigated;  such  means  form  t h e r e l a t i o n s h i p between  considered.  evaluate  further  t h e l i m i t a t i o n s o f t h e work  end, a c l o s e d  theoretical  of  t h e two  ' i n phase .  Suggestions  to  when  as an a n a l y t i c a l on a c o m p u t e r f o r  t h e column  The s i g n i f i c a n c e  and t h e i n f l u e n c e  v i b r a t i o n s :fcould be  could of  parametric-  of various  investigated.  com-  para-  The  - 65 major not  shortcoming  say  anything  to  the  type  is  much more  Other ratus  the In  would  could tor  wise  be  of  Boundary practical  the  at  of  axial  and  higher  of  control  Attention  should  be  column  behavior.  BAM  is essential  invesigated  favourable  signals are a  using  are  small.  sensitive  would  at  amplifier  Fotonic  the  Sensor  the  the  Fotonic  to Con-  pieceThe  corresponding  unknown.  means  of  superior to are  the  the  be have  measured  noise  and  to  must  Sensor.  i s required,  genera-  work.  amplifier  electrical  column  useful.  various  since  of  control  amplitudes  The  clamped  control  is  An  appro-  than  frequencies  the  appa-  simultaneously.  of  for this  on  characteristics,  of  be  be  response  b e t t e r than  vibrations  gauges.  use  i n the  points  behavior  used  Alternatively,  e l i m i n a t e d through  very  be  focused  however,  multichannel  displacement  i f response  strain  noise  the  vibration  KHz  would  longitudinal  the  unit  A  oscillations  monitoring  very  monitor  A  rise  solution.  inherent  interest.  acceleration  coupled  harmonic  10  give  solution,  conditions other  flexural  than  might  analytical  are  i t does  conditions should  providing constant  level  a  several different  frequencies  constant  an  study  i s desirable to  displacement  1  the  Such  than  traced simultaneously.  importance to  observed.  on  is that  mechanisms which  obtained  closely.  specimen  higher  stant  the  solution  d e s i r e d boundary  system  capable  much  The  way,  be  about  response  also  column this  numerical  limitations  more  recording  a  readily  used.  ximated ends  of  of  problems However,  great  care  - 66 must the  be e x e r c i s e d Sensor  developed  probe.  in. p r o v i d i n g A t some  to accurately  particular  frequencies  measuring  t h e column  urements  To  and o p t i c a l  further  oscillations, axis  o f t h e column  sional  movement  conducted of  oscillations subjected  <7  of interest.  gauges  be  since  improved other  Finally,  on t h e b e h a v i o r  holder  be  sound  at  of  level  meas-  torsional  a t 45°to  the major  the s e n s i t i v i t y Tests  cross-section the effect  excitation will  be  considered.  thereby.  o f more  could  methods  excited  positioned  help,  having  Other  should  system f o r  o f t h e column  including  parametrically  interest.  t o dynamic  the length  techniques  would  on columns  practical  scan  might  mounting  expense,' a p r o b e  vibration,  study  strain  a rigid  of  complicated deserve  t o tor-  could  be.  geometries coupled structures  attention.  -  67  -  BIBLIOGRAPHY  1.  L o v e , A . E . H . , "A T r e a t i s e o n t h e M a t h e m a t i c a l E l a s t i c i t y " , 3rd Ed., Cambridge, U n i v e r s i t y  Theory Press,  of BT20  2.  B e l i a e v , N.M., " S t a b i l i t y of Prismatic Rods.Subject to Variable Longitudinal Forces", Collection of Engineering C o n s t r u c t i o n and S t r u c t u r a l M e c h a n i c s (Inzhinernye s o o r z h e i i a i s t r o i t e l ' n a i a mekhanika), L e n i n g r a d , Put, 1924  3.  M e t t l e r , E . , Dynamic B u c k l i n g , "Handbook o f E n g i n e e r i n g M e c h a n i c s , 1 s t Ed.., F l u e g g e , W., editor, McGraw-Hill B o o k Company, I n c . 1962  4.  Somerset, J.H., "Parametric Instability of E l a s t i c Columns", S y r a c u s e U n i v e r s i t y R e s e a r c h I n s t i t u t e TR SURI no. 1 0 5 3 - 7, J a n u a r y , 1963  5.  Evan-Iwanoski, R.M., " P a r a m e t r i c (Dynamic) S t a b i l i t y o f E l a s t i c Systems", Proceedings of the F i r s t Southeastern C o n f e r e n c e on T h e o r e t i c a l a n d A p p l i e d Mechanics", P l e n u m P r e s s , 1 9 6 3 , p p . I l l - 130  6.  B o l o t i n , T.V., "Dynamic S t a b i l i t y o f E l a s t i c ( t r a n s l a t e d from R u s s i a n ) , H o l d e n - D a y , San Calif., 1964  7.  Somerset, J.H., and E v a n - I w a n o s k i , R.M., "Experiments on P a r a m e t r i c I n s t a b i l i t y o f C o l u m n s " , P r o c e e d i n g s o f the S e c o n d S o u t h e a s t e r n C o n f e r e n c e on T h e o r e t i c a l and A p p l i e d M e c h a n i c s , A t l a n t a , G a . , M a r c h , 196 4, p p . 503 525  Systems", Fransisco,  S t a b i l i t y of a Bar Applied Mechanics,  -  8.  T s o , W.K., Parametric Torsional Axial Excitation", Journal of M a r c h 196 8, p p . 13 - 19  Under  9.  S c h n e i d e r , B.C., "Experimental Investigation of Nonlinear C o u p l e d V i b r a t i o n s o f B a r s and P l a t e s " , M.A.Sc. T h e s i s , The U n i v e r s i t y o f B r i t i s h C o l u m b i a , A p r i l , 1969  Append.c<  - 68 APPENDIX  DEVELOPMENT  OF S T R A I N  ACCOUNTING  The [3]  results  into the  following  column  derivation  exhibit,  small.  The Lagrangian  dynamic  deflections  lection  curve.  cription tions  of strain.  lies  dynamic  provided definition  that  curve,  while  coordinate  system  i n which  Fig. plane  represent show  A - l shows  o f a column  relative  Love's  plane  between  relative  deduction  the i n i t i a l  takes  displacement are  i s used;  that i s ,  Eulerian  t h e two  defdes-  deriva-  presented  t o an assumes  initial a  deflection  fixed o f any  i s zero.  undergoing  plane  displacement  displacements  which  t o an i n i t i a l  schematically a fiber  the s t a t i c  dynamic  of strain  i n the derivation  are measured  by M e t t l e r  displacements  [ 1 ] d e r i v e s an a n a l o g o u s  i n the fact  i n the neutral  formulated  or static  these  The d i f f e r e n c e  deflections  COLUMN  f o r a column  crookedness  deflection  point  originally  are measured  Love  FOR A  CROOKEDNESS  expression  any i n i t i a l  may  EXPRESSION  FOR I N I T I A L  i n a strain  account  A  motion. while  relative  of the central The b a r r e d  the ordinary  letters  letters  to the i n i t i a l  deflection  i n the central  plane  curve.  An  element  o f t h e column  before- deformation ds  fiber  has a l e n g t h (A-l)  - 69  Extension  The  deformed ds  The  total  length  =V  strain p  =  ds  each  theorem  as  by c  +  U  x  the  u  and  x  )  2  A-l. of C e n t r a l  element <*  +  Plane  P'iber  is  2  (A-2)  5  central  fiber  is  - ds ds  equation =  the  (A-3)  radical  binomial given  of  (1 of  x  Expanding  Fig. Rotation  and  -  + w  1 +  in equations  retaining (A-3) w x x  +  4^ 2  x  (A-l)  second  and  order  (A-2)  terms,  by the  becomes 2  1 —w  22  xv  (A-4)  the strain  -  Assuming  small i n i t i a l  -  70  crookedness or s t a t i c  transverse  displacements, "  and  the  2 x  «  (A-5)  1  s t r a i n expression  becomes  1 x  x  Equation f o r the static  strains  i n the  strains  central  due  plane  fiber.  plane i n two  to bending  f o r i n o b t a i n i n g the  central  (.A-bj  i s t h e most g e n e r a l  o r dynamic d e f l e c t i o n s  Additional ted  (A-6)  o  2 x  xx  strain  strain  f i b e r of  expression a column  orthogonal  or t w i s t i n g  directions.  must be  a t some d i s t a n c e  for  from  accouthe  -  71  -  APPENDIX  DETAILS  OF  ELECTRONIC  INSTRUMENTATION  VIBRATION  The  following  electronic vibration  The  level  to  peak  100  o f 0.1  i s also inches  1025  up  t o peak;  a maximum  constant  to the earth's  ward  or reverse  over  has  frequency  any  synchronized speeds. dial  A  level  fixed  quencies frequency  speeds  recorder,  a fixed  actually dial  up  to  change  or  speed  angular  132  con-  value  of  control where  KHz.  I t can  The  force.  with  other  implies  the  velocity;  logarithmically.  range,  frequency  the  scanning frequency  the scanned with  for-  generator  fixed  that  scan  scan  frequency  control  is  1 g  to automatically  logarithmically  i s calibrated  peak  gravitational  10  i f used  a maximum  velocity  value,  the whole  o f the range.  scanning  peak  i s equipped  spectrum  scanning  moves w i t h  g's  continuously over  segments  six fixed  1000  peak  Constant  acceleration  due  entire  Automatic  with  constant  the acceleration  generator  applied  of providing  t o 2 KHz  and  control  of the  the  or acceleration.  t o 2KHz w i t h and  Kjaer  capable  KHz  the  to  peak  to control  and  10  The  or  possible  up  work  i s available  inches  a description  i s a Bruel  velocity,  per second;  attainable is  control  FOR  specimen.  C o n t r o l Type  displacement,  amplitude  in this  generator  Exciter  displacement  trol  used  USED  SYSTEM  provide  t o t h e column  control  Vibration  CONTROL  paragraphs  apparatus  B  time,  fre-  since  the  The the  vibration  input to the  control that  generator.  bias  in  The  signal  the  level  circuit  table  i s r e g u l a t e d back  tion  at  regulation  of the  m o u n t e d on  dual  channel  10  mv  the  lower  sensed  slow  of the  constant i t deter-  at the  as  correct  shaker compressor as  possible  desired  providing  vibra-  adequate  a Bruel  and  K j a e r Type  The  Preamplifier  Type  2622 has  attenuator, which,  when  correctly  output by  was  column mount.  Accelerometer  p r o v i d e s an g  a  shaker.  sensitivity  per  since  such  and  the  time  choice of  speed  frequencies, while s t i l l  stud  justed,  compressor  A  is  great,  until  important change  the  circuits  integration  to normal.  a c c e l e r o m e t e r used  built-in  compressor  a sudden  in  circuits  becomes t o o  i n order to avoid d i s t o r t i o n  low  The  A  The  i s very  with which  i s available.  chosen  i n the  accelerometer i s  circuits  c o r r e s p o n d i n g l y drops  the  is  vibration  appears  mines  speeds  speed  the  or compressor  i s obtained.  compressor  from  operation of these  table  generator output  vibration  signal  regulation  i f the shaker  larger the  feedback  the  signal  on  the  accelerometer.  Bruel  43 35  and  Kjaer a ad-  voltage channel  of  - 73 APPENDIX C LINEAR EQUATIONS Equation  o f Motion  The bending an  linear  for Flexural Vibrations  partial  vibrations  axial  FOR COLUMN VIBRATIONS  differential  o f a uniform,  elastic  column s u s t a i n i n g  0  Proceeding  xxxx  +  P  oxx W  +  A  to a solution  <  t t .=  W  0  of equation  assumed t o v i b r a t e h a r m o n i c a l l y  the  describing the  load P i s E I  in  equation  a normal c o n f i g u r a t i o n .  {  (C-l),  C  ~  l  )  t h e column i s  a t a n a t u r a l f r e q u e n c y and  Thus a s o l u t i o n  i s assumed  having  form w = X(x)(Acospt  + B s i n pt)  (C-2)  where X(x) i s a f u n c t i o n o f x and A and B a r e c o n s t a n t s . Substituting ordinary  equation  (C-2) i n t o e q u a t i o n  differential d X 4  EI d  equation  2 Po—TT d  7 +  x  2  =-Ad>p  s o l u t i o n s of equation  end  conditions furnish  The  s i m p l e s t case  (C-3) s a t i s f y i n g  the p r e s c r i b e d  t h e a p p r o p r i a t e normal f u n c t i o n s .  results  i  -  S  l  n  i f t h e ends o f t h e b a r a r e s i m p l y  "T—  where i i s an i n t e g e r .  resultant  (C-3)  T h e s e c o n d i t i o n s a r e s a t i s f i e d by x  equation  X  dx^  x  The  supported.  (C-l) provides a n  (C-4) The f r e q u e n c y  equation  (C-4) i s s u b s t i t u t e d i n t o e q u a t i o n resonant  flexural  results i f  (C-3),  and t h e  f r e q u e n c i e s a r e g i v e n by  -  74  -  (Cwhere  c  2  EI  =  .  The c o m p r e s s i v e  decrease  the resonant  resonant  transverse  work V-1  f o r pinned , using  transverse  by t h e s p r i n g s  Equation  of Motion  The  linear  vibration  2 where tes  c  end load  used  of P  in Fig. = 64  0  pounds  set-up.  of motion  f o r the  axial  column i s  ux x  (  C  u  c  i n this  Vibrations  equation  elastic  calculated  "  6  )  =  as b e f o r e .  Assuming modes  that  t h e column  of v i b r a t i o n ,  oscilla-  a solution i s  o f t h e form  where  = X(x)(c cos  procedure t h e same  x = 0,  level tions,  pt + c s i n  3  X(x) i s a  For at  2  The  to  EI  u  is  =  i n one o f i t s n a t u r a l  taken  The  f o r Longitudinal  differential  i s seen  0  are plotted  i n the experimental  of a straight, utt  ends  compressive  provided  P  f o r t h e column  and clamped  a constant  load  frequencies.  frequencies  ends  axial  5)  4  function  followed  pt)  (C-7)  o f x and c^ and c^ a r e  i n obtaining  the frequency  as t h a t  outlined  above  t h e column  studied,  the deflection  and at x = L t h e a p p l i e d  sinusoidal  acceleration.  the frequency S c o s 1ft =  equation  - p 2 c -ss ii n z  C t 5  k  &-  c  constants.  for flexural  load  Using  vibrations.  remained  provided  these  equation  zero  constant  boundary  condi-  becomes: cos pt  (C-8)  -  where  S = amplitude  mode o c c u r s , resonant  of applied  the e x c i t a t i o n  frequency  u  =  p.  frequency  ^  sin  x = L, t h e l o n g i t u d i n a l  by  static  of  25 p o u n d s ,  uin.  considerations.  tion  (C-9) a l o n g  the  experimental  nal  resonant  The  with  calculated  by more  forces  and a p p l i e d  Equation  The  0  a  load  i s about  f o ru into  level  200  equa-  S provided i n longitudi-  results.  resonant  frequency  a f e w Hz  f o r a wide  of  8790 Hz d o e s n o t  range  of end  loading  levels.  f o rT o r s i o n a l  Vibrations  equation  elastic  describing  column  the torsional  sustaining  an a x i a l  i s  2  ~ ae Z  tt  = ^ -  xx  ^Ah^  =0  Assuming  that  A  the b a r performs  o f frequency t ^ — , w  1  conditions,  (C-10)  +  3  torsion  Hz  9)  a n d an e n d  tested  value  '-  approximated  I V - 1 ) an a p p r o x i m a t e  o f 8790  linear differential  ®  where  acceleration  o f a uniform,  Law  1  o f t h e column  acceleration  o f Motion  oscillations P  than  c a n be  Hooke s  normal  becomes  (c  deflection  (Table  frequency  then  a  with the  )  the acceleration  tests  change  load  this  c o s p t  When  t coincides  response  Using  the extension  Substituting  acceleration.  The a x i a l  e^(  "p2 L s  When  75 -  a natural  , . and u s i n g  mode o f v i b r a t i o n i n  . ,' .. , . the following c  , , boundary  U  =  U the  0; =  t t  frequency  0  76  = 0 at  x = 0  Xcosft, Q=  equation  -  0 at x = L  results  nrra w  In  l  " ~  calculating  a, t h e f o l l o w i n g  G =  11.5  x 10 p s i  A  =  4.66  x IO"  h  = 0.125 i n . 1  I.  -  5.5  P  = 64 l b s . 0  equation  for  the f i r s t  and  10140 H z .  values  i  2  n  .  2  ( f o r uniform end ,,.-4 . 4 x 10 . i  = 6 . 1 1 x 1 0 (C-ll), three  a r e used:  6  1*2 =  I From  (C-ll)  -4  i n .  4  the resonant  modes  loading)  n  torsional  are calculated  frequencies  a s : 3380 H z ,  6760  Hz