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UBC Theses and Dissertations

Vibration studies on TRIUMF resonators Lee, Jimmy 1986

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VIBRATION  STUDIES  ON  TRIUMF  RESONATORS  by JIMMY B . A . S c ,  A  UNIVERSITY  THESIS THE  OF  SUBMITTED  MASTER  OF  BRITISH  IN  REQUIREMENTS  LEE COLUMBIA  PARTIAL FOR  THE  APPLIED  (1984)  FULFILMENT DEGREE  OF  SCIENCE  in FACULTY DEPARTMENT  We  accept to  OF  OF  thesis  required  UNIVERSITY  OF  JIMMY  as  LEE,  ENGINEERING  conforming  standard  BRITISH  SEPTEMBER  ©  STUDIES  MECHANICAL  this  the  GRADUATE  COLUMBIA  1986  1986  OF  In  presenting  requirements BRITISH freely that  this for  COLUMBIA, available  permission  scholarly  or  advanced  I  agree  for  understood  that gain  p a r t i a l degree  that  reference  for  by  in  an  extensive  purposes  Department  f i n a n c i a l  thesis  may  or  copying  or  shall  not  and  study.  granted her  OF  allowed  MECHANICAL ENGINEERING  U N I V E R S I T Y OF B R I T I S H 2075 Wesbrook Place Vancouver, Canada V6T 1W5  Date:  SEPTEMBER  1986  COLUMBIA  of by  shall I  of  of  the  UNIVERSITY make  further  this the  Head  this  without  of It  thesis my  OF it  agree  thesis  representatives.  publication be  the  Library  permission.  DEPARTMENT  at  copying  be  his  the  fulfilment  for my is for  written  ABSTRACT The meson  Cyclotron  f a c i l i t y ,  resonators the  major  which  when  beam  structure  structural  The  i n i t i a t e d  which  p a r t i c l e  D i v i s i o n  in  of  a  the this  vibration  to  determine  v i b r a t i o n . for  the  types  future, Triumf  it  The  1.  the  be  e n c i r c l i n g vibrating  out  design  funds  the  vibrating on  the  nature  originating  for  existing  hot-arm  is  of  the  from  the  A  hot-arm,  would  p a r t i c l e on  beam.  the  hot-arm  reduce  w i l l now  at  a  for  a  hot-arm  be  b e n e f i c i a l  Triumf.  proposed  construct  a  vibration  c h a r a c t e r i s t i c s  the  Kaon  kaon  In  the  Factory; producing  c y c l o t r o n .  modelled  cantilever  following  the  s t a b i l i t y  to  of  quality  thermal-related  studies  conducted  b e n e f i c i a l  the  structural  of  which  beam  RF  experiences  desirable  national  replacement  c y c l o t r o n .  encompasses  experiments  requested  transversely carried  of  the  design improve  s t a b i l i t y  the  improved  w i l l  has  factory  report  hot-arm The  by  Canada's  flow-induced  spatial  of  to  would  resonator,  scope  replacement  place  and  Triumf,  study  produced  deformation  reduces  a  of  a n a l y t i c a l l y  beam.  to  be  Investigations  a are  areas: vibration coolant  excitation  water  flowing  forces in  the  hot-arm; 2.  the  effectiveness  stiffener reducing 3.  the  and beam  of  adding  damping,  to  a  the  lumped  mass,  cantilever  a  beam  rotational model,  in  v i b r a t i o n ;  minimum-weight  v a r i a b l e - c r o s s - s e c t i o n  design  of  sandwich  i i  a  l i g h t l y  damped  cantilever  beam,  4.  subjected  to  a  unit  t i p ,  with  constraint  the  minimum-weight  harmonic on  the  subjected  5.  a  random  spectral  density  the  mean  square  the  optimal  attenuate 6.  to  the on  influence the  tip  force  the  of  force beam  at  the  free  amplitude; a  sandwich rain  over  design  the  tip  design  v a r i a b l e - c r o s s - s e c t i o n  point  with  span,  l i g h t l y  damped  cantilever  beam,  white with  noise  power  constraint  on  d e f l e c t i o n ; of  a  dynamic  hot-arm  vibration  of  shape  the  magnification  of  the  of  vibration  absorber  to  and the  coolant-flow  flow-induced  channel  excitation  in  hot-arm. A  possible  design  for  the  replacement  i i i  hot-arm  is  discussed.  a  Table  of  Contents  ABSTRACT  ii  LIST OF TABLES  vii  LIST OF FIGURES  ix  NOMENCLATURE  xiii  ACKNOWLEDGEMENT 1.  xviii  Introduction 1.1  Description  of  the  1.2  Description  of  a  1.3  Description Prototyped )  1.4  1 .5  3.  Triumf  Triumf  of the Hot-arm  Cyclotron  Existing  Hot-arm  and  Hot-arm  1.3.2  E x i s t i n g Hot-arm with Vibration Absorber Prototyped) Hot-arm  9 Attached  Dynamic 12  Hot-arm Vibration  15 on  the  Accelerating 22  Objectives  23  A Study on the Governing Pipe Conveying F l u i d  Equation  2.1  The  Motion  2.2  Possible Unstable Hot-arm Conveying  Random 3.1  5  9  Existing  Effect of Particles  1  Resonator  1.3.1  1.3.3  2.  1  Equation  Vibration  of  of  Characterizing 3.1.1  3.1.2  3.1.3  Small  Ensemble Processes  Motion  of  a 24 25  O s c i l l a t i o n of Prototyped) Coolant Water Flow  the a  of  Hot-arms  Random  29  Vibration  Averages  26  and  Process  30  Stationary 31  Power S p e c t r a l Density Stationary Process  Function  for  a  R e l a t i o n s h i p Between t h e Power Spectral Density and Autocorrelation Function for a Stationary Process  iv  31  32  3.2  3.3  3.4  3.5  4.  The Nature of the Random Excitation Forces Originating from the Coolant-Water Flow in a Prototyped ) Hot-arm  33  Transversely Vibrating a Vibrating Hot-arm  38  Response Vibrating  4.1  4.2  Study Hot-arm  for to a  Beam  Model  of  a Transversely Random R a i n F o r c e  Measurements  on  .39  the 45  on the Optimal Structural Which Minimizes V i b r a t i o n . . . . 5 0  The A d d i t i o n of D i s c r e t e Mass-Stiffness-Damping to Reduce the V i b r a t i o n D e f l e c t i o n of a Uniform Cantilever Beam  52  Optimum Design of a Hysteretically Cantilever Beam Subjected to a Unit Point Force at the Free T i p  60  4.2.1  4.2.2  4.2.3  4.3  Calculation Beam S u b j e c t e d  Random Vibration Prototyped) Hot-arm  An Analytical Geometry for a  Cantilever  Damped Harmonic  Optimality Criterion for the Minimum-weight Sandwich Cantilever Beam E x c i t e d by a U n i t Harmonic Point Force at the Free T i p Minimum-weight sandwich cantilever d e s i g n when the frequency of the harmonic point force i s much l o w e r the fundamental frequency  63  beam unit than ...66  Minimum-weight sandwich cantilever beam d e s i g n when the f r e q u e n c y of the unit harmonic point force i s the fundamental frequency  70  Minimum-Weight Design of A Hysteretically Damped S a n d w i c h C a n t i l e v e r Beam, S u b j e c t e d to A Random R a i n F o r c e with White Noise PSD, with C o n s t r a i n t on t h e T i p D e f l e c t i o n  75  4.3.1  4.3.2  Tip Deflection of H y s t e r e t i c a l l y Damped U n i f o r m Beams S u b j e c t e d t o a Random Rain F o r c e w i t h W h i t e N o i s e PSD  77  Minimum-Weight D e s i g n of a Hysteretically Damped V a r i a b l e - C r o s s - S e c t i o n Cantilever Beam, S u b j e c t e d to a Random R a i n Force with White Noise PSD, with C o n s t r a i n t on t h e MS T i p D e f l e c t i o n  79  v  4.4  4.5  5.  The E f f e c t of Adding Prototyped) Hot-arm Discussion Design  Vibration 5.1  5.2  on  Damping  to  the  Tip  of  a 83  Possible  Replacement  Strongback 84  for  Various Methods Vibration  a  Hot-arm  to  87  Attenuate  the  Hot-arm 88  An O p t i m a l Dynamic Vibration Prototyped) Hot-arm  Absorber  for  the 89  5.2.1  Lumped Parameters for a Transversely Vibrating Cantilever Beam E x c i t e d by a Random R a i n F o r c e w i t h W h i t e N o i s e PSD ....91  5.2.2  Optimal Absorber Parameters for a Main System Excited by a Random w i t h W h i t e N o i s e PSD  5.2.3  Damped Vibration Prototyped) Hot-arm 5.2.3.1  5.2.3.2  6.  a  Weights  Turbulence  Absorber  for  Generation  in  Flow  the  93 the 96  Considerations in Selecting Absorber Damping Mechanism The T e s t i n g Absorbers Hot-arm  1D0F Force  of on  the 98  Damped Vibration a Prototyped) 102  Hot-arm  Coolant  Water 110  CONCLUSIONS  113  REFERENCES  116  APPENDIX  A  121  APPENDIX  B  122  APPENDIX  C  123  APPENDIX  D  126  APPENDIX  E  1 27  vi  LIST  OF  TABLES  Table 1.1  Page A  comparison  of  the  existing  and  P r o t o t y p e d )  hot-arms. 3.1  Extreme  21  accelerations  of  the  P r o t o t y p e d )  hot-arm  t i p . 48  4.1  Vibration for  deflection  various  of  combinations  mass-stiffness-damping model 4.2  A  and  damping  comparison  cantilever  beams  4.4  Tip  subjected  noise  PSD.  rain on 5.1  Optimum main noise  5.2  MS  Test which  lumped  to  uniform  beam  excitation  a  sandwich  minimum-weight  same  t i p  uniform  amplitude.  sandwich  random  design  cantilever  force  the  beam  rain  72  cantilever  force  with  white 78  Minimum-weight sandwich  the  cantilever  55  their  the  for  beams  of  addition,  the  and  with  deflection  uniform  model.  between  counterparts 4.3  a  with t i p  a  beam,  white  v a r i a b l e - c r o s s - s e c t i o n subjected  noise  PSD,  to  a  with  random  constraint  d e f l e c t i o n .  absorber  system  of  83  parameters  excited  by  a  for  random  an  undamped  force  with  1D0F white  PSD.  96  results were  for  the  attached  damped to  the  vibration P r o t o t y p e d )  absorbers hot-arm  t i p . 103  6.1  A  comparison  of  the  hot-arm  v i i  vibration  for  the  roll-bond  panel  and  pipe-type  v i i i  panel.  112  LIST  OF  FIGURES  Figure  Page  1.1  Plan  view  1.2  A  1.3  Partial  t y p i c a l  tank  and  Exploded  1.5  Plan  1.6  Partial  view  of  RMS  spectrum  RMS  tank.  2  view  of  3 the  Triumf  Cyclotron  an  existing  roll-bond sectional cutting  of  4  the of  resonator.  hot-arm view  plane  of is  6  panel. an  8  existing  perpendicular  to  hot-arm)  the  tip  10  deflection  for  the  hot-arm.  main  vibration 1.9  a  (The  length  1DOF  of  cross  existing  Cyclotron  resonators.  view  the  A  Triumf  upper-resonator.  the  hot-arm.  1.8  the  elevation  1.4  1.7  of  system  with  absorber  spectrum  existing  13  of  attached  damped  dynamic  system.  the  hot-arm  an  tip  with  13  deflection  an  attached  for  the  vibration  absorber. 14  1.10  Plan  view  1.11  A  1.12  Partial  the A  A  cross (The  length  of  cantilevered  upstream 2.2  a  P r o t o t y p e d )  hot-arm.  2.1  of  end,  cantilevered  upstream  end,  P r o t o t y p e d )  hot-arm  panel.  16  strongback. sectional cutting the  17  view  plane  of is  a  P r o t o t y p e d )  perpendicular  to  hot-arm)  tube  conveying  vibrating tube  18  in  the  conveying  vibrating  in  ix  the  f l u i d ,  fixed  f i r s t  mode.  f l u i d ,  fixed  second  mode.  at  the 24  at  the 28  3.1  Typical a  3.2  power  spectra  c y l i n d r i c a l  Vibration  rod.  of  the  Ref.  acceleration  water-flow  noise  over  [8] of  35  the  P r o t o t y p e d )  hot-arm  t i p . 3.3  4.1  46  Probability  density  hot-arm  vibration  A  tip  uniform  excited 4.2  A  cantilever  by  uniform  a  rain  density 4.3  A  4.4  Optimal  S  0  and  point beam  with  an  added  force  with  P  an  beam  force  10"*)  free  mass  sin(w  added  damping  white  46  C°  noise  n  M°  )t.  53  rotational  excited  by  a  spectral 53  cover-plate  (T  P r o t o t y p e d )  .  cantilever  A  with  viscous  force  the  v a r i a b l e - c r o s s - s e c t i o n sandwich  static  4.5  harmonic  K°  of  acceleration. beam  cantilever  stiffenner random  function  =  point  thickness  excited and  force  by  a  for  a  =  0),  (T  for  beam. an  unit  64  undamped  harmonic  cantilever both  beam  forces  point under  at  the  t i p .  69  v a r i a b l e - c r o s s - s e c t i o n sandwich  cantilever  beam. 81  4.6  F i n i t e  element  approximation  v a r i a b l e - c r o s s - s e c t i o n 4.7  A  replacement  5.1  A  transversely  attached 5.2  A  1DOF  main  vibration  sandwich  strongback  damped  cantilever  cantilever  vibration with  the beam.  design.  vibrating  system  of  an  absorber.  86 beam  with  an  absorber. attached  81  92 damped  dynamic 94  x  5.3  Simplified vibration  5.4  drawing absorber  Dependence upon f  absorber  and 7  the  of  .  5.5  5.6a  7  5.8b  ' +'  =0.0l.  Optimal Ref.  RMS s p e c t r u m  of  P r o t o t y p e d )  hot-arm  to  the  the  the  of  P r o t o t y p e d )  hot-arm  the  attached  to  RMS s p e c t r u m  of  P r o t o t y p e d )  hot-arm  to  the  the  of  P r o t o t y p e d )  hot-arm  the  attached  to  RMS s p e c t r u m  of  P r o t o t y p e d )  hot-arm the  by  values,  of .  * " *  mass  For values  of  exceeds  than  o  p  t  r  stated T  r  =  7  a  a-opt'  [40] t i p  101  deflection  without t i p  for  the  an absorber.  deflection  with  t i p  for  an undamped  the t i p  the absorber  for  an a d j u s t e d  deflection an  for  undamped  the  t i p  damped 106 the absorber  the t i p  107  deflection  with  (v=0.1  deflection  with  f o r  an adjusted t i p .  the damped  0) for  an undamped  107 the absorber  tip.U=0.15)  RMS s p e c t r u m  of  the  P r o t o t y p e d )  hot-arm  t i p  deflection  with  xi  06  the  tip.(*i=0.05)  with  105  1  deflection  with  97  the main  response less  t i p .  tip.(<i=0.l0)  RMS s p e c t r u m  to  the hot-arm  and 7 the  damped  tip.U=0.05)  RMS s p e c t r u m  attached  to  f  contour  hot-arm  absorber 5.8a  a  P r o t o t y p e d )  attached 5.7b  parameters,  of  absorber 5.7a  attached  RMS s p e c t r u m  attached 5.6b  m  f r i c t i o n  MS d i s p l a c e m e n t  percentage. M=0.1,  the  t h e MS d i s p l a c e m e n t  within  optimal  of  107 for  an adjusted  the damped  absorber 6.1  Cross  sectional  roll-bond 6.2  Cross  attached  pipe-type  view  hot-arm  sectional  to  of  tip.(u=0.l5) the  coolant  108  channel  in  a  panel.  view  hot-arm  the  of  Ill the  panel.  coolant  channel  in  a 111  xi  i  NOMENCLATURE mass, of  a  spring 1D0F  mass, of  a  and  main  spring  damping,  respectively,  damping,  respectively,  system  and  vibration  deflection  viscous  viscous  absorber  of  a  transversely  vibrating  tube  or  beam spatial time  coordinate  p a r a l l e l  to  a  tube  or  a  beam  a  beam  coordinate  length  of  a  tube  or  a  r i g i d i t y  mass  per  unit  length  of  a  mass  per  unit  length  of  f l u i d  mean  speed  amplitude  of of  exponential frequency  the the  resultant  tube  the  imaginary  tube  or  a  in  beam  a  tube  flow  deflection of  the  argument  v i b r a t i o n ,  parameters  beam  fluctuation  v i b r a t i o n , or  beam  force f l u i d  part  define  tube  the  transversely at  tube  wall-pressure  excitation  fixed  a  f l u i d  function  of  turbulent  of  beam  flexural  real  or  of  (/)  v e l o c i t y  vibrating upstream the  part  by,  flow  2  =  the  from  conveying  a  f l u i d ,  argument  complex  argument  -1  mathematical  expectation  probability  density  x i i  tube  exit  end  complex  of  at  i  of  the  function  argument (PDF)  of  the  argument  variables  MS  mean  square  RMS  root  mean  a  variance  of  standard  deviation  2  Oy Ry(  )  cross  value  square the  value subscript of  correlation  variable  y  the  variable subscript  function  with  y  of  dependent  variable  the  y  subcript  variables  in  the  argument Sy(  )  cross  spectral  variable  y  density  with  function  dependent  of  the  subscript  variables  in  the  argument 0*  convection  c  d^  hydraulic  L  length  6(  )  Dirac  speed  of  turbulence  in  the  f l u i d  flow  turbulence  in  the  f l u i d  flow  diameter  scale  of  function  the with  dependent  parameters  in  the  argument s  stress  in  the  beam  e  strain  in  the  beam  E  e l a s t i c  modulus  c  damping  parameter  v  damping  parameter  of  the of  a  of  beam  material  viscously a  damped  h y s t e r e t i c a l l y  beam damped  beam (')  d i f f e r e n t i a t e  the  argument  once  the  argument  twice  with  respect  to  t (")  d i f f e r e n t i a t e t  f ( x , t )  forcing  function  for  xiv  a  beam  with  respect  to  h ( x , T ; a )  unit  impulse  H(x,u;a)  unit  complex  response  function  frequency  of  a  response  beam  function  of  a  beam #j(x),9j(t)  j t h  normal  mode  coordinate, vibrating hj(t)  similar for  Hj(w)  a  and  corresponding  r e s p e c t i v e l y ,  beam  to  for  generalized  a  transversely  (j=1,2,3...)  the  unit  impulse  response  function  frequency  response  1DOF s y s t e m  similar  to  function Wj  j t h  7j  damping  the  for  resonant  a  unit  complex  1DOF s y s t e m  frequency  ratio  for  for  the  a  j t h  beam normal  mode  of  the  beam IJ^(CJ)  joint  acceptance  i/>T  transpose  of  function  the  j t h  f i n i t e  element  mode  shape  vector ^ ( x ^ M ,  mode  shape  element  eigen  logarithmic vibration  (  )'  generalized  system  non-zero 6  and  mass  corresponding  of  to  the  the  f i n i t e smallest  value  decrement  of  the  damped  free  response  d i f f e r e n t i a t e  the  argument  once  the  argument  twice  with  respect  to  x (  )"  d i f f e r e n t i a t e  with  respect  to  x 0,(x)  the  f i r s t  normal  mode  #"(x)  the  f i r s t  normal  mode  xv  of of  a  beam a  beam  d i f f e r e n t i a t e d  twice  with  respect  M°  added  lumped  K°,C°  added  rotational  to  x  mass stiffener  with  lumped  viscous  damping K°,H°  added  rotational  hysteretic a  P  constant  magnitude  constant  power  0  rain I(x),m(x)  much  smaller  than  of  moment  of  spectral  inertia  per  2B  breadth  2D  constant planes  of  unit  of  the  the  separation  of  the  thickness,  sandwich p,  mass  d  constant  .,1 uni'  density  beam  point of  force  the  e l a s t i c  sandwich  random  distance in  the  the  the  mid  sandwich  and  mass  the  cover-plates  of  the  core  in  thickness  of  the  the  beam  density, in  weight  inertia  and  moment of  moment  amplitude  at  t i p  of  the  of the  uniform  inertia  of  fundamental  d e f l e c t i o n ,  xv 1  sandwich  cover-plates  sandwich  square  of  beam  between  uniform-cross-section  weight,  and  the  beam  density  3  section  beam  modulus  of  cross  respectively,  cover-plates  respectively, WorloiYorNo  harmonic  length,  sandwich  respectively,  W  L  the  v a r i a b l e - c r o s s - s e c t i o n  2  lumped  force  mass  d ( x ) , E , p  with  damping  A  S  length  stiffener  beam in  the  beams of  cross  cantilever cross  section, beams  section,  resonance  respectively,  and of  tip mean the  W . min  reference  unit-square  weight  the  of  constraint y,N  t i p  u  M  tip  a /  minimum-weight  on  amplitude  MS M  m '  a  s  s  r  a  cantilever sandwich  the  tip  amplitude  at  the  fundamental  deflection m  uniform  t  1  of 0  f  the o  r  t  e  main  beam  with  resonance  cantilever n  beam  and  beams  and  absorber  masses CJ  K  2  a  a  K  m  m  /M /M  a m  m m  f  '  m  CJ  /CJ  'a  a C a  7  C  7  m m  S^  R(/i,f ,7 , a  m  /2M  a  CJ  a  /2M u m m m m  constant  power  force  the  on  7 )dimensionless m  spectral  main MS  density  of  the  random  main  mass  mass displacement  xvi i  of  the  ACKNOWLEDGEMENT The  author  generous guidance  wishes  help  during  throughout  The  author  assistance  to the  this  is  which  he  had  have  completed  from  Mr.  Guy  Stanford  the  Cyclotron  collaboration. also  entire  Triumf.  of  grateful  who also  D i v i s i o n  for  assistance  the  was  his of  this was  his  project.  His  the  f i n a n c i a l  This  professional  to  thank  Dr.  project guidance  author's  useful  the  for  invaluable.  Triumf.  the  wishes  Hutton  for  from  without  He  The  S.G.  project  received  not  at  Dr.  course  also  could  supervisor  thank  immediate K.  Fong  discussions  Triumf's  technicians  of and is  appreciated. The  final  Engineering  of  thanks U.B.C.  go  to  and  the  the  Department  department's  xvi i i  of  Mechanical  technicians.  1.  1.1  DESCRIPTION The  OF  process  injection  of  cyclotron  vacuum  feet  in  of  four  and  twenty the  in  each  two  1.2.  Figure  sectioned  through  upper-resonator one  another  These  rows  r e s i s t o r s ,  of  (radio  alternating Dee  plane  in  radius 1  a l l  of  the  reference  resonators,  of  the  1.1,  the  view  are  giant  mega-hertz  (MeV) With are  injected  in  u n t i l kinetic  of  enveloped  1  images  of  to  accelerate  they  have  400  inside  plane. of  e l e c t r i c a l l y  ions  approximately  brackets  a  create  k i l o - v o l t  energy  a  in  tank,  c i r c u i t  (MHz)  192  the  that  beam  resonating  difference  path  of The  the  mirror  e l e c t r i c a l  The  with  view  shows  form  23.075  of  shown  of  p a r t i c l e  s p i r a l  numbers  is  accelerating  of  rows  plan  of  tank  above  1.1,  lower-resonator  capacitors  axis  the  two  d i r e c t l y  Figure  and  51.3  lower-resonators.  in  a  the  lower-resonators,  Figure  of  of  its  Inside  elevation  1  tank.  ground.  the  cylinder  with  an  [ 1] .  mega-electron-volt  the  height  with  center  shallow  1.3,  potential  a  in  the  upper-resonator  the  Gap  a  located  frequency)  the  near  begins  t y p i c a l  a  inductors  an  beam  and  to  rows  resonator  RF  inches  row.  A-A  about  at  across  A  is  rows  are  lower-resonators.  ions)  e l e c t r i c a l  two  upper-resonators  Figure  of  and  shows  acceleration  tank  v e r t i c a l  resonators  tank,  The 17.05  rows  upper-resonators  CYCLOTRON  (hydrogen  tank.  oriented  are  TRIUMF  p a r t i c l e  ions  diameter  symmetry there  H  THE  INTRODUCTION  (kV) in  gained  at  the  KeV  of  set  of  the 520  outer energy square  Figure  1.1:  Plan  view  of  the  Triumf  Cyclotron  tank.  6.5"  Figure  1.2:  A typical  upper-resonator.  CYCLOTRON  -RESONATOR l-RESONATOR  Figure tank  1.3:  and  the  Partial  elevation  resonators.  view  of  the  Triumf  Cyclotron  5  gain MHz  per  orbit,  are  energy  1250  required beam  of  turns  to  at  reach  p a r t i c l e s  is  a  rotation  520  MeV.  used  for  frequency  This  of  4.615  resulting  sub-atomic  and  high  medical  research.  1.2  DESCRIPTION A  resonator  hot-arm  and  ground-arm Figure  a  1.4.  The  length  vacuum  the  floor of  the  at  several  123  of  the to  to  hot  The  is  RF  s i g n i f i e s  R F - c a v i t i e s  electro-magnetic  allow f i e l d  is  which  two  inches  which  mode the  top  and  a  out  the  inert c a l l e d  a  is  The  MHz.  l i d  or  floor  the  points. the  The  bottom the  from  the  and  panel  strongback insufficient  Note,  voltage  is  void  between  the  RF-cavity.  the  word  potential  RF  c y c l o t r o n .  l i d  of  c i r c u l a r  ground  of  The  ground  has  at  a  approximately  inches.  formation the  is  wide.  23.075  panel 123  in  Figure  at  opposite  supported  shown  in  a  The  sub-components,  inches  RF  1.3.  is  31.5  that  inside  Figure  are  voltage  the  ground-arm,  shown  points;  the  a  are  Facing  cantilever  panel  of  either  e l e c t r i c a l l y  hot-arm  panel  which  the  which  that  in  distributed  tank.  panel  s i g n i f i e s  and  is  distributed  word  panel  several  vacuum  shown  ground  respectively  i t s e l f  p o t e n t i a l .  the  at  as  hot-arm  anchored  tank  hot-arm  r i g i d i t y  panel  is  components:  consists  and  wavelength  the  major  a  hot-arm  panels  are  is  the  of  hot-arm a  the  the  and  three  RESONATOR  levelling-arm  and  panel  ground  has  The  one-quarter  walls  TRIUMF  ground-arm  of  ground  A  consists  strongback 1.4.  OF  at  high ground  C o l l e c t i v e l y ,  an The  alternating panels  are  Figure  1.4:  Exploded  view  of  an  existing  resonator.  7  also  c a l l e d  necessary  RF-panels.  because  cannot  maintain  period  of  these of  time  This  anchored  to A  adjust B  inches as  The  levelling-arm  the  to but  shown The  coolant  The  existing  and  outlet  coolant  strongback  and  the  and  C.  A  RF  tip  water  which  in  of is  are  as  are  pipes  was  the  on  root  end  cantilevered  levelling-arm  of  the  tank  at  at  point  C  about  point  A  The  distance  the  strongback  strongback  consumption  currents  channel  Vibration  of  the  ions  three allows  and  from  is  is  point  is  132  B,  123  inches  1.2.  quantities  20  of  of  mechanism  pivoting  length  power  panel  floor  RF-cavity.  at  i n  3  of  / s  of  the  heat.  the  This  manufactured into  shown  in  connected the of the  allowed  panel  to  coolant  hot-arm  end  of  water was through  Both  looping ground  water  inlet  strongback. through  detected the  method  the  the  flows  by  panels.  a  1.5.  the  removed the  1.0  panels  roll-bond  Figure  coolant  flow  is  through by  is  plated  heat  each  root  to  cyclotron  copper  c i r c u l a t e d  incorporated  Approximately panel.  entire  panels  coolant-flow hot-arm  a  or  prolonged  metallic  root  l i d  the  of  is  supports  for  the  to  is  panel  structural  to  unsupported  water  has  and  of  High  large  B  hot-arm  clamped  hot-arm,  Figure  total  the  which  size  the  in  A,  the  the  mega-watts. generate  the  points of  of  the  properties  deposition  (analogous  either  movement  or  to  assembly  to  locations:  insulating the  of  insulated  due  strongback  beam).  the  e l e c t r i c a l l y their  supports.  the  Cantilevering  each  when  hot-arm  the  panel.  WATER  ROOT  Figure  INLET  (OUTLET)  END  1.5:  PANEL POINT  WATER  Plan  view  of  a  roll-bond  TIP  FLOW  hot-arm  ATTACHMENT  END  panel.  co  9  DESCRIPTION  1.3  OF  THE  EXISTING  HOT-ARM  AND  PROTOTYPE(1)  HOT-ARM When its  this  people  called  were  the  improved  the  this  vibration the  of  pertinent  two  and to  describes hot-arm,  the  The  layers  except a  Figure section  for  heat  only  diagrams  model  hot-arms  deformation  leakage)  vibration  with  the  the  the  hot-arm  hot-arms, w i l l  the  be  and  hot-arm  hot-arms.  optimal  to  this  shape  with  Prototyped)  The  hot-arm A l l  the  section  The of  an  design  existing  investigated.  prior  the  vibration.  resonators.  size  in  and  of  investigate  given  the  Triumf,  hot-arm  Replacement  RF  existing  existing  are  following  the  existing  attached  dynamic  hot-arm.  HOT-ARM  technique. of  is  structural  and  replacement  with  reduce:  hot-arms,  existing  roll-bond  of  constructed  absorber  association  structural  physical  EXISTING  to  minimizes  the  the  vibration  form  which  a  hot-arm.  w i l l  Prototyped)  information  two  report  his  on  needed  (source  hot-arm  1.3.1  are  flow-induced  However  and  work  thermal-related  hot-arms  a  at  started  Prototyped)  design  the  of  author  alloy  the  coolant 1.6.  The  in  the  hot-arm A  roll-bond  metal  looping  panel  sheets  pattern  channel  whose  irregular roll-bond  shape panel  is  manufactured  panel  is  by  fabricated  which  are  bonded  which  is  next  from  everywhere  inflated  cross  section  is  of  coolant  channel  the is  suspected  a  to  shown  to  in  cross  promote  Figure  1.6:  hot-arm. of  the  Partial (The  hot-arm)  cross -  cutting  sectional  plane  is  view  of  perpendicular  an to  existing the  length  11  turbulence The  related  existing  extrusion  strongback  structure.  perpendicular  to  a  five  half  of  the  strongback. p a r t i a l l y Figure  The  cut  the  than  hot-arm  which  the  inch-thick  a  a  the  reduce  the  aluminum  supported from  the  by  existing  of  the  strongback  existing is (see  free  of  tip  bending  structure  stainless  aluminum  and  span  hot-arm  overhanging  two  two  strongback  extrusion  the  the  expense  The  section  shows  of  the  half  the  panel.  webs of  aluminum  cross  strongback,  flange  at  the  p a r t i a l  I-beam-like  to  further  1.75  1.6,  gravity  is  of  is  unsupported  hot-arm  panel  weight  is  portion steel  of tubes  extrusion. and  the  The  panel  is  pounds. The  aluminum  stainless is  in  order  the  forces.  inch-wide  fact  extend  combined 280  In  excitation  of  p a r a l l e l  under  s t i f f n e s s .  length  31.5  in  deflection  Figure  the  away  1.4)  shorter  vibration  easy  steel  to  see  transversely cross shown  vibration  beam-bending  the  hot-arm  and  of  with  Figure  Acceleration hot-arm  the  the  w i l l  torsional  and  root  root  of  this  given, like  it a  non-uniform direction  as  plate-bending  significant  of the  y  r i g i d  much  with the  less  more  very  in  are  a  dimensions  beam  vibrating  The  to  behave  cantilever  The  vibration.  coincides  existing  1.4.  clamped  From  hot-arm  essentially  model B  the  vibrating  Figure of  is  levelling-arm. that  section, in  strongback  than  the  cantilever  strongback  beam  at  points  on  a  A  1.4. measurements at  the  test  have  been  f a c i l i t y  taken with  20  i n  single 3  / s  of  12  coolant  water  Vibrametric the  1030  hot-arm.  definition the  band  RMS is  mode  RMS  RMS  deflection  frequency,  1.3.2  the is  an  0.5*10 ~  EXISTING  (RMS)  1.7,  transform is  given  the  panel.  to  the  mode  of  the  A of tip  with  analyzer.  a The  Section  3.  Most  at  Hz  and  the  damping.  The  4.8  of  light  of  the  v i c i n i t y  tip  recorded  (FFT) in  indication  in  arm  spectrum  was  integration  spectrum  hot  attached  fundamental  deflection,  is  was  Figure  deflection  in  the  square  Fourier  response  one  mean  in  fast  of  response  narrow  root  shown  660A  through  accelerometer  The  d e f l e c t i o n , Nicholet  flowing  of  area  under  the  of  the  fundamental  inch.  3  HOT-ARM  WITH  ATTACHED  DYNAMIC  VIBRATION  ABSORBER A l l  of  dynamic the  the  existing  vibration  hot-arm  system  absorber  vibration  of  hot-arm  represented,  for  in  M  m  ,  K m  viscous K  and  damping the  and  C  damping C of  the  m  the  purpose mode.  its  system  system  with of  as  the hot  in  mass, arm)  M ,  representation  of  the  not  n e c e s s a r i l l y  system  which  The  of  the  absorber  are  mass  is  0.7  and  the  The  Q  viscous  dashpots  actual  pound.  1.8.  and  viscous  the  a  and  the  in  be  spring  that  mechanisms  spring  the  by  Figure  mass,  Note  can  mode,  the  absorber.  damped  attenuating  absorber  shown  (the  a  Schematically,  f i r s t  respectively, main  f i t t e d  attached  in  respectively,  diagramatic  weight  f i r s t  (2DOF)  are,  are  the  its  motion  of  are,  for  the  and  two-degree-of-freedom The  hot-arms  are  damping viscous. absorber  13  1.4  S  -  (_>  *  _i  LL. O 'o  1  "  10  FREQUENCY Figure  1.7:  existing  RMS  spectrum  of  the  15  £0  CHZ)  t i p  deflection  for  the  hot-arm.  MAIN SYSTEM  ABSORBER SYSTEM  Figure  1.8:  A  dynamic  vibration  1D0F  main  absorber  system system.  with  an  attached  damped  14  spring is  is  a  small  cantilevered  accomplished  button  on  between  a  of  an  area  t i p  under  \-  s l i d i n g  spring.  existing  hot-arm  the  dry  stainless  the  cantilever  by  to the  The  hot-arm 0.2*10-* RMS  plate.  f r i c t i o n a l  steel  plate.  surfaces damped  is  inch  as  rubbing The  the  of  of  RMS  Figure  damping a  ceramic  pressure by  force  another  attached  indicated  curve  absorber  provided  absorber  attenuates  spectrum  The  to  small  the  deflection by  the  of  tip the  smaller  1.9.  </>  (_)  *  Ld Z _l LL. « CL  ' o ^  FREQUENCY <HZ> Figure existing  1.9:  RMS  hot-arm  spectrum with  an  of  attached  the  tip  deflection  vibration  absorber.  for  the  15  1.3.3  PROTOTYPE(1) The  panel. to  a  Prototyped)  The  panel  0.032  the  vibrating with  excitation;  as  to  not  sag  under  as  three-layer  and  this  of  1.12. the  The r i g i d weight is  either  side  are  give  the  31.5  with  U-turns  The  run  design  the  along at  the  change  was  turbulent-flow  the  a  has  shape  of  roll-bond  a  riveted  the  panel  are  p a r a l l e l as  core  that  shown  r i v e t s .  in to  it  thought  aluminum  aluminum  bending  be  by  riveted  inch-wide  such  can  together  strongback the  construction.  strongback  held  e f f i c i e n t  soldered  pipes  strongback  four  of  pipes  The  I-beams  Figure the  1.11  flanges  plates.  resistance  of  These  to  the  be  more  beam. P r o t o t y p e d ) bending  in  tapered  shows The  in  of  roll-bond  These  in  the  The  is  a  generation.  prestress  beam  longer  coolant  1.10.  of  strongback  gravity.  no  reduce  section  sandwich  cover-plates  to  sandwich  I-beams  sandwich  Figure  turbulence  to  length  On  in  cross  does  the  panel  d i s c o n t i n u i t i e s  used  running  hot-arm  the  Propotyped)  of  sheet.  intent  was  core  steel  the  promote  is  copper  shown  Riveting  a  panel  stainless  the  coolant-channel  The  has  of  tip  suspected  hot-arm  inch-thick  length  made  HOT-ARM  the  upper  inch-thick  the  in  strongback the  v i c i n i t y  in basic  v i c i n i t y of  discrete  the  steps  construction  cover-plate continuous  plate  of  t i p .  constructed the The  towards of  between  is  the  root  and  to  lighter  cover-plate the  t i p .  stations  approximately  1 and 7  thickness  Figure  Prototyped)  in  1.11  strongback. 3  is  feet  in  a  0.375 length.  WATER  INLET  (OUTLET)  PANEL  ATTACHMENT  POINT T3T  TT"  _3_  CP  CP  ure  J3L  3$C  T ROOT  CD  4 TIP  END  1.10:  1  Plan  view  of  a  Prototyped)  hot-arm  panel.  LOWER  COVER-PLATE  VERTICAL 6  I  Figure  UPPER  1.11:  COVER-PLATE  A  Prototyped)  strongback.  MM  =  1  SCALE: INCH  Figure hot-arm. of  the  1.12: (The  Partial  c r o s s - s e c t i o n a l view  cutting  plane  is  of  perpendicular  a to  P r o t o t y p e d ) the  length  hot-arm)  i—»  00  19  The  lower  cover-plate  inch-thick  continuous  0.375  between  The  inch  upper  and  inch-thich  4  5,  and  inches  there  in  lap  is  stations  which  has  2  3,  and  cover-plates  plates  length  neighbouring  plate  stations  lower  0.080  between  a  30  but  is  are  no  is  a  machined of  down  3  to  inches. and  Between  upper  lower  0.625  30  station  length.  seamed  3  length  inch-thick  there  cover-plates  a  in  and  been  between  inches  0.020  1  4  are  station  cover-plate  18  cover-plate.  together  with  a  The  riveted  joint. Note  station  that  1  and  the 2  lower  accomodate  the  space  is  thicker  symmetric  cover-plate  This  u t i l i z i n g precious  than  into  the  beam  which  shows  not  protruding  panel.  s t i f f n e s s ,  section  is  inch-thick  hot-arm  cross  beam  more  is  1.85 gap.  since inches A  e f f i c i e n t  use  cut  a in  hot-arm order  about  the  core. it  of  0.625  groves  which  coolant  should  it  neutral beam  be  would  be  made axis  material  between  The  the  cannot  that  the  the  increase  between  note  of  to  strongback  into  pipes  done  small  symmetric  has  space  the  about  coolant  was the  of  P r o t o t y p e d ) the  bending  pipes.  This  built  much  not here of to  encroach that  a  bending maximize  20  the  bending The  s t i f f n e s s .  consequences  construction 1.  It  are  increases  strongback 2.  It  It  bending  It  of  cost  of  tapered  cover-plate  and  under  constructing  needed  a  under  to  counter  gravity.  e f f i c i e n c y ,  construction the  of  requirement  structural  s t i f f n e s s ,  increases  cost  riveting).  deflection  the  the  step-wise  complexity  prestressing  increases  amount 4.  the  strongback  the  the  follows.  (eg.  reduces  the 3.  as  of  static  with  loads  respect for  a  to  given  material.  fundamental  frequency  of  the  strongback. The as 1.  questions  What  is  Does  a  the  Does  a  coolant 1.1  RMS water  gives  hot-arms.  which  also  naturally  a  minimizes design  design  for  static  tip  deflection  flow  structural  minimizes  strongback  strength The  optimal  strongback  frequency 3.  arises  from  the  above  are  follows:  strongback 2.  which  rate  comparison  loads  of of  the  which tip  vibration  optimizes  vibration  structural  also  minimizes  t i p  the  3  / s  P r o t o t y p e d ) is  0.08*10  existing  ?  ? the  i n  the  fundamental  optimizes  the  of  the  which  of 20  tip  geometry  and  -  vibration hot-arm  3  inch.  ? at  Table  P r o t o t y p e d )  21  Table  1.1:  A  comparison  of  the  Existing  and  P r o t o t y p e d )  Hot-arms.  Existing  Prototype(1)  hot-arm  static  tip  weight  of  s t i f f n e s s the  fundamental damping RMS  tip  280  305  frequency  (Hz)  4.8  5.05  0.005  deflection  that  this  B  C  Figure  reports is  was  showed  similar  pressure between  were  one  vibration  that  pumping of  the in  the  from  at  a to  hot-arm  tank  water  test  the on  the pump  hot-arms beam  frame  Triumf's at  the  at  at  hot-arms  meters and  reported  mounted  c y c l o t r o n .  effects  f i r s t  the  vibration  the and  0.08*10-3  3  time,  Studies  in  0.003  measurements  Access  the  those  coolant  predominantly  hot-arm  1.4.  smoothing  cyclotron,  those  limited.  to  the  0.5*10-  vibration  report hold  cyclotron  (in)  a l l  could of  143  (lb)  which and  55  hot-arm  ratio  Note through  (lb/in)  hot-arm  Triumf  points  inside  the  the  test  frame  With  the  large  plumbing  pipes  of  hot-arms  in  not  detectable;  in  the  cyclotron  near  A,  engineering  were  mode  a l l  4.8  Hz.  the the were  22  1.4  EFFECT  OF  HOT-ARM  VIBRATION  ON  THE  ACCELERATING  PARTICLES Vibration fluctuation  of  variations of  the  of  in  the  the the  RF-cavity  voltage  resulting ions  do  path.  MeV,  the  450  millimeter. vibration p a r t i c l e A  Thus as  small  stable  for  funds  e n c i r c l i n g  the  existing  to  520  MeV  synchrotron.  The  an  function  important  cyclotron  now  at  and at  causes  the  to  Dee  Because  energy,  stationary  particle to  obtain  gives  subsequent  Gap.  kinetic  s p a t i a l l y  of  cyclotron  the  the  a  the s p i r a l  orbits  keep  of  since  hot-arm are  a  upper-resonator  is  a  is  the  1.5  hot-arm  s p a t i a l l y  c y c l o t r o n .  in  w i l l  new  hot-arm  stable  to  wall,  be  be  factory from  through  f i n a l  30  an GeV  to  serve  Factory. on  in  the  the  duration  Hot-arms the  has  produced  needed  of w i l l  Triumf  routed  w i l l  vibration  shorter  it  producing  a  is  types  future,  Factory;  and  Kaon  p a r t i c l e  much  the  the  Particle  cyclotron the  for  kaon  synchrotron  a  In  Kaon  cyclotron  anchored tank  Triumf.  construct  v i b r a t i o n .  circumferential  b e n e f i c i a l  proposed  existing  m i l l i - s e c o n d , of  be  GeV  influence  s i g n i f i c a n t  period  w i l l  existing  3  0.27  the  advantageous  the  intermediate  is  a  possible  conducted  requested  The  in  have  is  size phase  spacing  as  beam  b e n e f i c i a l  the  it  not  in  beam.  experiments be  and  variation  accelerating At  hot-arms  at  center  at  row  ends,  attached  to  the  p a r t i c l e  cyclotron  of  time  beam only  than  the  the  centre  of  the  post.  Next  to  the  the  hot-arm  hot-arm  of  of  an the  23  lower-resonator c a l l e d  a  d i r e c t l y  quadrant),  below.  located  circumferential  tank  wall,  at  The  copper  their  socket flow,  t i p s .  connectors but  quadrant the  not  hot-arms  of  study to  to  of  would  by  centre  connected  to  tulip-shaped  the  plug  of  one  of  the  matching  e l e c t r i c i t y hot-arms  of  another.  hot-arm  vibration  and  and  that  (also  neighbours  for  enough  vibrating  post  their  primarily  strong  half-row  on  the  a  a  Thus  p a r t i c l e  neighbouring  quadrant.  to  to  a  following  single  lead  the  identify  identify  identify  section  the  the  and  vibration 4.  one  a  to  study  resonator  Improved an  is  rather  design  improved  associated  of  than the  c y c l o t r o n .  a  with whole  individual The  goals  of  are:  strongback 3.  is  the  independently  resonators.  identify  hot-arm 2.  of  vibration  resonators  1.  same  effort  cyclotron  this  of  n u l l i f i e d the  are  of  OBJECTIVES The  the  be in  vibrate  effect  between  designed  connection  cannot  negative  would  1.5  the  are  Hot-arms  nature  vibration  the  which the  of  the of  optimal minimizes  optimal  a  excitation  forces  on  a  hot-arm,  structural the  tip  method  for  geometry  of  a  v i b r a t i o n , damping  the  hot-arm  and the  which  optimal  minimizes  shape the  of  t i p  a  coolant  v i b r a t i o n .  channel  cross  2.  A  S T U D Y ON  THE  GOVERNING EQUATION CONVEYING  The the  problem  problem  of  conveying  f l u i d  (1965)[2]  have  a  tube,  fixed  increased becomes  shown at  Figure  that  when  upstream  a  certain  and  small  and  the  not  p o s s i b l e .  hot-arm  of  2.1. the  end  OF  A  PIPE  s i m i l a r i t i e s  velocity and  free  and of  at  the  value,  amplitude.  o s c i l l a t i o n  In  the  o s c i l l a t i o n of  a  pipe  Paidoussis  f l u i d  perturbations  unstable  to  cantilevered  Gregory  c r i t i c a l  large  unstable  has  vibrating  random  flow-induced  reviewed be  in  the  o s c i l l a t i o n s of  vibrating  MOTION  FLUID  transversely  shown  beyond  equation  to  a  a  unstable  l a t e r a l the  of  OF  flow  in  other, the  is  system  grow  into  following, is  hot-arm  b r i e f l y is  shown  x 3y(L,t) 9t  Figure the  2.1:  upstream  A  cantilevered end,  vibrating  tube in  24  conveying the  f i r s t  f l u i d , mode.  fixed  at  25  2.1  THE  EQUATION OF  The of  a  system  of  studied  uniform,  r i g i d i t y  EI  flow  line  of  and  mass  y(x,t) beam  is  Gregory motion  of  and  mass  The  theory  x  of  3"y(x,t)  the  m U  L,  unit  with  Using  at  the the  the  +  with  a  centre  c l a s s i c a l that  f l u i d  the  f r i c t i o n ,  equation  of  small  tube:  3 y(x,t)  2  2  2m U  +  f  (m +m)  =0  f  3x3t  2  stream  deflection  established of  a  the  transverse  ox.  flexural  length  3 y(x,t)  f  3x  to  consists  conveying  per  cantilevered  2  r  m,  its  independent  2  +  length  having  3 y(x,t)  3x°  of  coincides  and  Paidoussis arrived  vibration  Paidoussis  m^  axis  tube  and  is  and  length  perpendicular  problem  EI  unit  undeflected  bending  Gregory  cantilever  per  U.  measured  dynamical  by  f l u i d  velocity  the  MOTION  tubular  incompressible  mean  SMALL  3t  2  (2.1.1) The  boundary  y  3y/3x  =  3 y/3x  conditions  = 0  =  3 y/3x 3  2  The  f i r s t  and  the  usual  s t i f f n e s s  vibration  problem.  respectively,  form:  forth  the  terms and  of  x  =  0  at  x  =  L.  equation  inertia  The  at  forces  second  centrifugal  2.1 of  and  and  (2.1.2) are,  respectively,  the  c l a s s i c a l  beam  third  terms  are,  c o r i o l i s  forces  of  the  of  the  f l u i d .  Solution A  3  = 0  2  flowing  are:  Procedure  solution  procedure  is  to  assume  a  deflection  26  y(x,t) where  =  JJe[Y(x)exp(/ ]  Re[  arguement. number. co  is  denotes  In  The  negative.  imaginary that  zero,  the  For  given  a  increased second w i l l  as  the and  become  the  in  Gregory  Paidoussis.  2.2  and  s t i l l  UNSTABLE  CONVEYING  COOLANT  is  flow  velocity its  an  assumed  was  checked  for  the  The  c o r i o l i s  term  in  for  for  c r i t i c a l  the  f l u i d  small  random  large  amplitudes.  the  coolant  length  of  pipe  gives  water  the  In  the  flows  hot-arm;  the  grow  case in  thus  complex part  s t a b i l i t y  if  Paidoussis slowly  in  more  is  one  or  flow  OF  f i r s t  mode. not  the have  modes.  in  further  The  of  from  velocity  increased  were  is its it  unstable  studied  PROTOTYPE(1)  analytical  of  two  tube  by  HOT-ARM  the  into  net  of  When  this  unstable  damping  vibration  directions of  the  coolant  occurs  P r o t o t y p e d )  effect  of  equation  p o s i t i v e  lateral  opposing  the  of  d i f f e r e n t i a l  effect  the  model  conveying  p o s s i b i l i t y  s i t u a t i o n .  perturbations  a  imaginary  unstable  cantilevered  flow  is  complex  FLOW  water  motion  the  increased  the  modes  hot-arm,  of  a  w  and  fourth  P r o t o t y p e d )  o s c i l l a t i o n .  as  become  WATER  the  Gregory  OSCILLATION  completeness,  flow,  if  w i l l  in  of  frequency  unstable  higher  POSSIBLE  For  zero.  part  neutral  (m^+m)/m,  unstable  o s c i l l a t i o n  in  velocity  ratio  if  is  become  system  real  unstable  is  flow  w i l l  mass  be  w  the  c i r c u l a r  system  the  the  system  mode  also  of  the  w i l l  The  part  shown  taking  general  system  (2.1.3)  cot)],  the  the of  hot-arm, along  the  c o r i o l i s  27  force  is  zero.  Inspection system  also  cannot  occur.  is  that  energy tube and  the from  the  shows  that  A  the  f l u i d  flow.  f l u i d  flow  f l u i d  the  at  velocity forces that  U.  when  from  tube  gained water  vibrating  is of  inlet  must  go  the  be  and  ,  and  capable  the  be  to  vibrating f l u i d  velocity  in at  U.  transferred of  these  is  p o s i t i v e . are  cannot  extract  o s c i l l a t i o n  fixed  is  not  hot-arms.  is  of  the  the inlet  external 2.2  to  U  r  ,  shows  of  when  the  can  occur,  vibrating  modes  energy  tube  second mode,  the  Since  f i r s t  the  Figure  this  to  the  outlet,  o s c i l l a t i o n  outlet  P r o t o t y p e d )  two  to  its  than  its  For  in  v e l o c i t y  from  the  o s c i l l a t i o n  cantilevered  when  f l u i d .  hot-arm  the  larger  energy  the  a  the  accumulating  only  that  in  of  resultant  occur,  the  that  shows  of  unstable  vibrate  can  unstable  f l u i d  hot-arm  unstable  existing  tube  under  inlet  r  to  interaction  from  2.1  transferred  the  f l u i d  the  w i l l  this  velocity  than  the  During  be  U  for  Assume  mode,  outlet,  the  resultant  f i r s t  For  must  smaller  its  Figure  be  can  mechanism  o s c i l l a t i o n  condition  system  a  transfer  unstable  necessary  second modes. in  energy  vibration  conveying  vibrating  Thus  of  be  energy system.  v i b r a t i o n , net  energy  a  hot-arm's  coolant  the  r i g i d  from  the  possible  the  the  for  tank,  water both  a  flow. the  28  Figure the  2.2:  upstream  A  cantilevered end,  vibrating  tube in  conveying the  second  f l u i d , mode.  fixed  at  3. In  a  RANDOM V I B R A T I O N  previous  turbulent  f l u i d  source  random  of  o s c i l l a t i o n . random  in  this  flow-induced (1984)[8],  Bakewell  given  excited  by  C y l i n d r i c a l spectral  the  mode  f l u i d  by  of  were  c y l i n d r i c a l  (1971)[12] motion the rod  of  and a  random  of  function of  the  hot-arm  on  is  this  a  recent of  theory  l i s t s  of  Accounts are  rods  and  o n many  given  by  types  by  fluctuations  of  Blevins on  Corcos  induced  were  i s  bodies  (1962)[9],  rod  29  by  i s  Chen  the  to  of  the the  wall-pressure a  the be  fluctuation.  wall-pressure,  In  and  At  postulated  function  rod.  under  turbulent  (1974)[13].  fluctuating  vibrates  the  wall-pressure a  vibrating  by  studies  Paidoussis  the  is  (1958,1963,1979)[3,4,5],  fluctuating  of  focus  unstable  ( 1 968)[11].  turbulent  the  the  the  study  the  c y l i n d r i c a l  deflection  of  as  a  established  studied  and C l i n c h  of  merely  vibration  well  wall-pressure  flows  of  main  Crandall  fluctuations  shapes  P r o t o t y p e d )  the  problems  density  correlation  to  onset  random  vibration  wall-pressure  the  the  of  excitation  considered the  (1984)[7],  Vibration  outset,  is  and Nigam  (1 9 6 8 ) [ 1 0 ]  Wambsganss  study  random  at  Accounts  Turbulent  to  tube  comparison  are  (1967)[6]  a  section,  The  vibration.  references  exposed  on  the  perturbation  excitation.  c l a s s i c a l  Lin  flow  In  development  section  OF T H E HOT-ARMS  similar random  power cross and  manner,  a  excitation  30  induced  by  The hot-arm 1.  2.  A  the  study is  given.  The  nature the  For as  4.  5.  the  For  beam  is  span random  each  t i p .  model  The  ensemble  of  1 , 2 , 3 , . . . , n . functions  of  number  with  an  of n  the  is  hot-arm  needed. is  modelled  beam. c a l c u l a t i o n  rain  measurements  of  on  force  for  over  a the  a  P r o t o t y p e d )  random  the  dependent one  necessary  e s t a b l i s h But  a  randomly  mounted  process  for  random  multivariate  usually  it  is  the  (t)  t, a  vibrating  for  variables  of  vibrating  hot-arm  y  variable  vibration  to a  signals,  independent  random  PROCESS  similar,  accelerometer  The  (PDF).  random  hot-arm  optimal  hot-arm  response  accelerometer  the  function  the  forces  RANDOM V I B R A T I O N  characterize to  originating  P r o t o t y p e d )  cantilever  a  terminologies  analyzed.  n  hot-arms  to  follows.  shown.  CHARACTERIZING A Consider  a  vibrating  the  subjected  vibration  are  in  excitation  vibrating  as  P r o t o t y p e d )  forces  determine  a  a  vibration  excitation  to  studies,  of  sections  random  flow  studies,  analytical beam  The  order  transversely  vibrating  the  water  of  vibration  smaller  random  knowledge  analytical a  of  the  in  flow.  random  into  coolant  a  hot-arm  3.1  of  studied;  design 3.  water  d e f i n i t i o n  are  from  of  organized  brief  are  coolant  an  r  =  y (t)  are  r  the  time.  hot-arm  probability  s u f f i c i e n t  is  to  it  To is  density establish  31  only  the  p ( y i f  v  f i r s t  2 ) r  where  order  and  y^  y(t^)  =  the  second  and tj  order  are  PDF, p C y ^  discrete  and  values  of  time.  ENSEMBLE  AVERAGES  AND STATIONARY  Important  averages  of  3.1.1  an y  ensemble. at  a  Consider  fixed  mean  value:  mean  square  root  mean  time  t  a  random  the ensemble  =  square  value  £ [ y (RMS):  o  If  these  averages  of  j ) ,  the  S t i l l function.  Consider  respectively, process  i s  at  and  not a  £ [ y i y  3.1.2  2  J  ] =  said  of to  important  average  two  of  sets  two f i x e d  y J  of  2  p(y)dy  (3.1.1)  y p(y)dy  (3.1.2) (3.1.3)  ] -  S t y ]  times  of  = £ [ y ( t ) y ( t + r ) ]  SPECTRAL  i s or  =  2  be  (3.1.4)  2  t,  a  (3.1.5)  ( i e .  independent  stationary. the  and  average  just t  time  i s  autocorrelation  values  y,  t  the  2  .  If  yiY2  of  function  individually,  T  of  and  =  y  2  ,  random ° r t  2  the -  t,  i e . ,  R(T).  DENSITY  simply  y a r e :  2  ensemble  the  function,  function  POWER  i s  stationary,  autocorrelation  /  = £ [ y  2  independent  process  another  averages  or  = /variance.  are  random  2  y ( t , )  across  RMS = / M S  variance: deviation:  =  are taken  values  The ensemble  (MS):  standard  process of  E[y] value  PROCESSES  (3.1.6)  FUNCTION  FOR  A  STATIONARY  PROCESS Another frequency  type  domain.  of Any  ensemble stationary  average  i s  done  member  of  the  in  the  ensemble  32  y(t),  represented  expanded y(t) y  in =  s  for  into the  £ C n= 1  time  interval  function  interval,  exp(/  n  the  periodic  time  large  y  -s/2  which  to  is  s/2  can  identical  be with  i e . ,  ncj t)  (3.1.7)  0  k.  The process  a  in  temporal y(t)  is  mean  given  square  value  of  the  stationary  by:  {y (t)J = i sjj^ yMt) dt !  where  the  member In  brackets  of  the  the  summation  in  following  form:  {y (t)}  /  =  where  3.1.3  Zo,  S  (a>)  f l u i d  easily  =  0  Aco a n d  approaches  becomes  the  power  t h i s ,  the If  t In  spatial y(x,t)  averages,  updating  the  THE  FUNCTION  wall-pressure  flow,  by  s  co  an  s  a  single  =  27r/Aco.  i n f i n i t y  integral  the  of  the  (3.1.9)  BETWEEN  considered.  s t a t i s t i c a l  when  co,  along  spectral  density  (PSD)  y(t).  to  variables.  =  o  averaging  dco,  AUTOCORRELATION  turbulent  nco  time  3.1.8  c a l l e d  RELATIONSHIP  variable  Let  process  is  of  Prior  denotes  equation  S^(co)  function  }  ensemble.  limiting  2  {  is  POWER FOR  (time) many  A  SPECTRAL STATIONARY  was  the  random on  coordinates  also  random  independent  are  and  PSD  d e f i n i t i o n s  such  exposed  the  space-time  autocorrelation previous  only  bodies  the to  a  independent  process, can  to  AND  PROCESS  processes  fluctuation  a  DENSITY  be  its  defined  include  the  33  variable  x.  Fourier  transform  c o r r e l a t i o n are, R  y (  S  y  x  i »  x  and cross  =  /  u )  =  (1/27T)  2  f  T H E NATURE  To  a l l ,  pair  2 / f )  FROM  the  additional  s p a t i a l c a l l e d  spectral  _  t  S  t  OF  ( x J  1  , x  Tec  determine  excitation the  the  density  vibration the  on  To  coolant  x  gives  space-time (CSD)  a  cross  functions  which  measure  would  Fortunately,  it  i s  p o s s i b l e  shedding,  excitation  the  The  cross  HOT-ARM  knowledge  necessary; i s  i s  inner  direct  form  of  undoubtedly  surface  d e l i c a t e to  of  after  the  c o r r e l a t i o n  complex  to  ORIGINATING  a  The  many  too  (3.1.11)  the a  of  the  of  the  pressure  c h a r a c t e r i z e .  characterize  the  random  q u a l i t a t i v e l y .  BLT  turbulent  separation.  i s  hot-arm  the  Boundary-layer-turbulence  of  design,  hot-arm  require be  FORCES  f o r c e s .  on  CJT) 6T  PROTOTYPE(1)  a  the  may  sources  A  (3.1.10)  exp(-/  hot-arm  the  it  flow-induced  , r )  IN  of  on  and  f i e l d  2  6CJ  WT)  hot-arm  the  transducers  pressure  x  f  FLOW  wall-pressure,  pipes,  l  excitation  complex  fluctuating  exp(/  ( x  response  excitation  one.  , w )  optimal  turbulent-flow  BLT  the  T H E RANDOM E X C I T A T I O N  forces  of  2  R  THE COOLANT-WATER  consequence  of  variable  r e s p e c t i v e l y :  ( x , , x  3.2  The  studied  noise  are  generating  following  wall-pressure  (BLT)  i s  by  some  flow  bends,  empirical  fluctuation  wall-pressure  i s  a  type  researchers.  The  p u l s a t i o n , c a v i t a t i o n  model  of  proposed  by  the  vortex and  CSD f o r  Corcos:  flow the  34  S(co,x,z)  =  S(co) A ( c o x / U )  B(coz/U )  C  cos(cox/U ),  C  (3.2.1)  C  where, x  =  x , - x  and z  2  distances  in  =  the  z , - z  respectively  2  d i r e c t i o n  of  flow  are  the  separation  and perpendicular  to  flow. cu i s U  the  i s  C  S(co)  frequency  the i s  speed the  of  at  the  which  PSD  wall-pressure the  BLT i s  function  of  f l u c t u a t i o n .  convected.  the  BLT  wall-pressure  f l u c t u a t i o n . A(CJX/U )  and  c  c o r r e l a t i o n  B(coz/U )  amplitudes  perpendicular In  to  this  empirical  i s  independent  CSD  separation  For  low  c y l i n d r i c a l  1  )  model  =  of  for  the  S(co,x  frequency r o d , Chen  A(wx/U  ) =  e x p ( - 0 . 1 |cox/U  B(coz/U ) C  where U  i s  =  free  Typical are  shown  in  the  surface  thickness stream  power Figure of  a  of  cross  flow  and  homogeneous  turbulence,  the  locations  and  dependent  , z  S(co).  on  the  2  2  , z  2  )  =  p a r a l l e l  (3.2.2)  water  flow  over  a  suggest:  e x p [ - 2 . 2 ( c o » c / U )]  C  | )  C  the  a  and  exp(-0.55|coz/U |  K i s the  0.4  d i r e c t i o n  a n d Wambsganss  +  /U  , x  2  0.6  c  the  the  i e . ,  =  U  in  are  flow.  distances,  S (co, x , , x , , z , , z  respectively  C  1  ) ,  of  (3.2.3)  the  turbulent  boundary  layer  and  v e l o c i t y .  spectra  measured  3.1  [8].  inch  rod  The in  a  by  Chen  spectra 2  inch  and  Wambsganss  were  measured  on  water  channel.  In  35  Figure over  a  3.1:  Typical  c y l i n d r i c a l  rod.  power  spectra  of  the  water-flow  noise  36  the  Strouhal  are  nearly  approximate S(«)  =  Q  2  ( p  w  number  range  constant  and  constant  PSD:  U  2  )  d  2  10  >  tod^/2ir\J  Blevins  >  0.1,  suggests  the  spectra  the  following  ,  3  (3.2.4)  where Q2  _  9.4*10"  is  the  s e c / f t  7  hydraulic  Approximate  3  ,  p  is  w  the  density  of  the  in  a  water  and  d^  diameter.  Length  Scale  of  the  BLT  Prototyped)  Hot-arm  The  Reynold's  number  of  the  Prototyped)  hot-arm  panel  at  evaluated  in  Appendix  A,  given  Reynold's  No.  The  =  Strouhal  evaluated given  the  f  No. is  hydraulic  =  the  of  s l i g h t l y  lower  length  scale  hot-arm  of  the  fd  h  /U  the  of  since  come  from  =  / s  of  coolant  a  water,  by:  BLT  wall-pressure  frequency  5.05(0.31)/40  fundamental  coolant  Wambganss.  would  of  fundamental  diameter  velocity  and  3  in  of  fluctuation  the  hot-arm  is  by:  Strouhal where  i n  flow  1.1*10*.  number  at  is  20  turbulent  than But the most the  fundamental  frequency  (0.31  in),  water the the BLT  (40  and i n / s ) .  experimental most at  important the  of  the  frequency.  0.04, (5.05  U  is  This  to  Hz), the  is  the  hot-arm in  the  the stream  number  frequency  wall-pressure  is  free  studied  number  the  d^  Strouhal  cases  fundamental  contribution PSD  =  by  is  Chen  integral of  the  vibration v i c i n i t y  37  The percent  convection of  distance is  U.  for  The  the  are  length  the  as  scale  is  energy  L  «  The hot-arm  0.6U/w  which  form  one  rain  S ( w , x , , x f  where  6  is  may  )  = a  be  for  Summary  on  Prototype(1) The excitation  is  case a  2  the  approximate  zero  function  is  integral  length  components  since  frequency.  The  0.8  in  the  «  L.  length is  of  not  correlation  white  noise  points  d i r e c t i o n ,  in  the  the very  length, with  no  space.  CSD  of  For this  as: ) ,  function.  (3.2.5) For  d i s t r i b u t i o n a n a l y t i c a l  Nature  correlation  situation  between  flow  written  of  0.37,  by:  than  spatial  excitation in  This  =  or  the  the  given  60  of  the  just  an  the  long  beam-like  of  excitation  hot-arm  forces  is  studies.  Flow-induced  Excitation  in  a  Hot-arm  above on  r a i n ,  further  the  to  smaller  inches.  S(CJ) 6 ( x , - x Dirac  scale  limiting  case  dimensional  adequate  A  2  123  the  dimensional  random  one  of  frequency  proportional  much  The  to  separation 1/e  c o r r e l a t i o n  distance.  higher  is  fraction  functions,  cross  equal  the  Since  0.6(40)/(2)TT(5.05)  random  correlation  a  c  for  length  scale  the  to  U /o>.  decay  is  =  is  from of  to  the  integral  length  different a  for  d i s s i p a t i o n  «  distance,  attenuate  L,  eddy  smaller  U / u  approximately  decay  equal  scale,  the  approximate  to  is  c  exponential-decay  integral same  u"  eddy  CSD  approximately  amplitudes  speed  the  was  hot-arm  is  example.  l i k e l y  to  be  The more  flow-induced complex  than  38  the  BLT  excitation.  However,  flow-induced  excitation  total  length  of  f l u i d  and  stream.  thus The  Wambganss  content took  data  that  the  lower  the for  hot-arm's  PSD  a  the  would  density  flow-induced  be  VIBRATING  is  section,  has  a  it and  random  a  Clinch,  the  a  BLT  in  and  frequency  none  with of  of  them  v i c i n i t y to  varying  model  down  wall-pressure  the  force  the  Chen  reasonable  slowly  to  viscous  wide-band  is  rain  a  the  quickly  Although  possible on  is  very  frequencies  of  assume  in  those  white  noise  the  actual  hot-arm.  CANTILEVER  studies,  vibrating  of  a  hot-arm  cantilever  coolant  material  dimensional  of  of  comparison  water  decay  PSD  scale  BEAM  MODEL  OF  A  with  a  HOT-ARM  conveying  structural  a  the  range.  VIBRATING  analytical  transversely  flow  wide-band  Thus  in  Bakewell,  frequency,  excitation  TRANSVERSELY  would  the  low  length  small  since  frequency very  be  by  that  water  low  frequencies.  For  taken  fundamental  spectral  3.3  hot-arm  indicate  of  in  should  disturbances data  a l l  fluctuation  the  the  water  the  s t r e s s - s t r a i n  beam  beam  is with  within is  relation  modelled a  its  the  cross  structure.  postulated of  uniform  to  obey  The  a  v i s c o e l a s t i c  one type,  i e . ,  S  =  E(e  where  s  +  ce),  and  modulus  of  and  dot  the  e  (3.3. are,  respectively,  e l a s t i c i t y denotes  and  c  is  stress  the  differentiation  and  viscous with  strain, damping respect  E  is  1) the  constant, to  time.  39  The  bending  moment  of  3 y(x,t) =  +  Modifying damped  ).  undamped of  motion  is  +  m,U  2  derivatives.  = 3t  because  of  c o r i o l i s  force,  terms  of  centrifugal comparison  to  usual  the  3.4  c l a s s i c a l  RESPONSE  a  excitation to  force,  at  c a n be  i e . ,  =  no  force,  obtained  and t  t  by  0  For  a  i s  and  to  are  with  i t s  normal force  third  Triumf  taken  mode  and  hot-arm,  be  small  in  neglected;  the  or  without  difference. shapes  the  Now  and  the  their  employed.  A  TRANSVERSELY  VIBRATING  BEAM  FORCE the  response  superposition  excitations. =  second,  are  y  c l a s s i c a l  the  detectable  c a n be  of  centrifugal  and they  beam  structure,  unit a  possess  hot-arm,  T O A RANDOM R A I N  linear  x  term  CALCULATION FOR  standardized  applied  not  terms  a  undamped  conditions  SUBJECTED For  c o r i o l i s  showed  (3.3.3)  independent  (3.3.3).  of  3x3t  f ( x , t ) ,  is  damping  stiffness  flow,  orthogonality  the  f  £  respectively,  frequency  water  does  equation  and the  fundamental coolant  the  2m U  2  2  function  system  2  y(x,t)  (m,+m)  This  the  3 y(x,t) +  3x  2.1,  by:  2  1  forcing  equation  2  3x"3t  a  motion,  3 y(x,t)  cEI  f l  f ( x , t ) ,  forth  given  5  t  and  of  3 y(x,t)  +  modes  (3.3.2)  equation  3  where  by:  2  equation  +  given  3x 3t  3"y(x,t) 3x  c  2  the  EI  i s  3  -EI( 3x  beam  3 y(x,t)  2  M  the  a  A  unit  standard  to of  a  general  the  response  impulse  response  form  of  excitation  40  f(x,t) The  =  unit  y(x,t)  the  r  t - t  unit  The  at =  0  x  .  =  =  Since  exp(z unit  exp(/ 6(x-a)  transform  of  h(x,co;a).  R  denoted  as  of  at  a  the  excitation  fixed  i s  frequency  co  (3.4.3) frequency  response  function  is  equation  it  space-time  (3.4.4)  cot). in  (3.4.1),  as  be  a s :  term  by  form  force  complex  of  obtained  to  cot).  transform  The  i s  i e . ,  H(x,co;a)  the  standard  harmonic  a,  state  denoted  y(x,t)  function  (3.4.2)  Another  6(x-a)  steady be  (3.4.1)  response  complex  applied  ) .  0  h ( x , r ; a ) , =  f(x,t)  6 ( t - t  impulse  =  where  to  6(x-a)  follows  cross  (3.4.3)  that  superposition  =  Z  of  the  the  Fourier  i s  Fourier  H(x,co;a)  correlation  the  is  of  the  unit  response  impulse  is  responses  follows:  y  ( x  1  r  x  2  , T )  /  a cU>i  J  Z  M x , , * ? , ; * , ) where  0,  and  2  coordinate, spatial  the  cross S (x y  1  f  and  the £  2  coordinate  correlation of  are  8  of  the  response  x ,co) 2  =  /Q  as d £ ,  2  , 0  2  ; £  integration  are  the  and  R^  obtained  R U,  d £ ,  2  h ( x  excitation  i s  correlation  ad 0  2  F  )  d £  2  the  force. by  S ,T+e,-d ) 2  2  ,  (3.4.5)  variables  integration i s  r  for  the  variable  time  for  space-time  the cross  The c o r r e s p o n d i n g  Fourier  transforming  CSD the  follows: /jj H ( x , , - c o ; £ , ) H ( x  2  ,co; £  2  )S  f  U,  ,£ ,co) 2  d £  (3.4.6) where  Sr  i s  the  CSD of  the  excitation.  2  41  The can  be  impulse  found  h(x,t;a)  =  in  Z  #j(x)  terms  g.(t;a)  j where  response of  of  i t s  the  jth  normal  normalized  that  such as  dx  k  their  unit  and  length  integrate  +  The  ccj2g  g  k  (t;a)  =  impulse  response  for  h(x,t;a)  =  H  1 k  (w)  j  *  k,  the  <6j(x)  be  generalized  =  are  the  the  x  from  of  the  =  1,2...n)  length  a n d mass  equation resulting  0  to  normal  L  (3.4.7)  and  modes  apply  [14]  to  equation  same  beam  as  (3.4.9)  the  with  zero  one-degree-of-freedom  function.  i s  given  R  Thus  the  unit  the  = mL[toJ-£J  2  i n i t i a l  as  given  by:  +  /  3.4.11  gives: (3.4.12)  the  27 w co], k  k  impulse  (3.4.11)  R  same  (1DOF)  by:  k  R  function  y i e l d : (3.4.9)  # (x)0 (a)H (w),  is  the  (3.4.10)  response  R  by  by:  the  the  per and  equation  k  k  Z k  masses  (3.4.8)  Substitute  Z 0 ( x ) 0 ( a ) h ( t ) . k transform of equation  H (u>)  response  Let  i s  0 (a)5(t)/mL.  given  i s  k  k  and gj(t)  k  unit  where  =  k  k  h (t)  H(x,w;a)  (3.4.7)  h (t)c6 (a),  where  Fourier  over  to  is  shape  (k  multiply  conditions  solution  conditions  beam.  (3.3.3),  + o,2g  k  k  the  tf> (x),  k  for  =  of  into  g  0  j  respectively,  orthogonality  mode  corresponding  m,  (3.4.1)  k  1,2...n),  coordinate.  = mL f o r =  L  =  i e . ,  follows:  m^j(x)tf> (x)  where  modes,  (j  generalized  /Q  normal  beam  3  i s  defined  uniform-cross-section  4>.{x)  1  corresponding  are  a  1DOF u n i t  complex  frequency  (3.4.13)  42  and  t h e damping  27  = ecu ..  7  i s given by:  k  (3.4.14)  2  k  Substituting S ( x ,  ,x ,o>) 2  equation = Z Z  y where, I  ratio  j k  (  w  The  )  j  ^  function  into  (3.4.6)  0 • ( x , )<t>, ( x ) H . ( - w ) H .  J  K  k  (« )S 2  Ij (<j)  i s  k  f  U, , *  y i e l d s : .. (u)  2  i  'o^O * j U i ) 0  =  (3.4.12)  (3.4.15)  K _ J K  2  ,")  known  d £ , d £  as  the  .  2  (3.4.16)  joint  acceptance  function.  Vibration  Response  The  to a  joint  k  =  used  D  X  F  O  by a  random  i s evaluated R  :5=  f o r rain  of  equation  a  to be:  K  (3.4.8)  (3.4.17)  the  integral  in  equation  to mL/m.  non-uniform  to evaluate  finite  a  forcing  f o r j * k .  i s equal  For  Force function  excited  (3.2.5)  0  virtue  3.4.17  be  equation  beam  = S(w) JQ  Ij (w)  By  of  Rain  acceptance  uniform-cross-section function  Random  beam, the  element  the  finite  reponse.  Consider  discretized  uniform-cross-section  beam  element  method  the  can  following  system  f o r  a  given b y :  [M]v + [ K ] v = 0 , where  v ,  [M] a n d  coordinate denote inner of  vector,  [K] a r e , mass  the normalized product  equation  i//j[M]\J>j  (3.4.8).  respectively,  matrix  finite  and stiffness element  i s analogous Usually  the  mode  displacement  matrix.  shapes.  Let  Then  to the generalized  the finite  element  the mass  generalized  43  mass  is  normalized  element  case,  to  the  equal  analogy  of  acceptance  function  of  The  reason  for  existence  the  individual  matrix  are  (3.4.8). with  the terms  the  dissimilar  product"  ^j[M]\//j  (3.4.17), the  system  are  set  of  the from  sectional  [M]  is  a  when  the  mass  matrix per  with  well  resonance  peaks  into  (3.4.15)  =* S ( u ) L  For  a  random  S ,  integrating  residue  Z  c6 (  rain  of  the  equation  beam  elements  the  of  in  mass a l l  "inner equation  matrix  the  of  elements  mode  damped  structure  substituting  (3.4.8)  equation  gives: (3.4.18)  2  k  u  yields  the  a  noise  range  following  PSD,  where >  S(w)  - «  >  MS  deflection:  <J  °° b y  = the  2  Z  —  k  .  (3.4.19)  CJ£  approximation  is  summation. s o l u t i o n  mode  approximation f i r s t  applying  on  <6 (x)  intuitive  the  the  lightly  the  [4]  mode  a  over  *  one  for  S (x,w)  also  of  to  discussion  white  2  a  to  length  with  cm L Often  mass  |H ( ) | .  x )  y  theorem  ( x , t ) ]  equal  force  0  £ [ y  element  integral  separated,  and  2  S ir 2  that  properties,  the  with  0  i s ,  similar  the  unit  1/m.  unity.  beam,  y  to  joint  to  analogy  many  finite  the  equal  finite  composed of  analogous  uniform-cross-section  S (x,u)  integrals  the  in  is  simple  consistent  cross  Continuing  (3.4.17)  a  is  where  to  of  for  integral  (3.4.17)  system  is  Then  the  equation  evaluated  When  unity.  that  most would  resonant  of come  is  the from  frequency,  adequate; of  the  the  order  contribution the to,.  S(CJ)  in  Thus  for  convergence  1A>£« to  the  the S(co)  It  is first  v i c i n i t y slowly  44  varying  OJ, ,  near  deflection  is  given S(tO,  £ [ y  2  ( x , t ) ]  the  one  mode  approximation  of  the  MS  by:  )7TC/>  (x)  2  »  .  (3.4.20)  m Lca>i 2  The  energy  vibratory when  the  d i s s i p a t e d  system  increases  amplitude  of  e l a s t i c  bodies,  energy  d i s s i p a t i o n  frequency, [14],  For  is  the  a  £ [ y  v  is  The given  per  a  viscously  frequency  of  is  constant.  But  damping is  the  MS  damped vibration  for  than  viscous  deflection  the  f i n i t e  L  whose  of  is  the  damping given  by:  (x) ,  2  many  mechanism,  independent  approximation  damping, 2  by  the  cycle  * m  where  with  hysteretic  hysteretic  ( x , t ) ]  cycle  vibration  better  Stw, )ir0 2  per  (3.4.21)  vco  3  hysteretic element  damping  constant.  equivalence  of  equation  3.4.21  is  by:  S f w , ) * U T [ M ] t f , )H ( x ) £ [ y  2  ( x , t ) ]  «  , ?M w 2  where  M,  u s a l l y  is  a  vibrating  w i l l  modal  normalized  Now  force  the  over be  design.  one beam the  needed  to  mode  later  3  mass  in  the  fundamental  mode  which  is  unity. vibration  subjected beam  (3.4.22)  to  a  response one  span  has  been  in  the  study  of  a  dimensional e s t a b l i s h e d . for  the  transversely random These  optimal  rain  results hot-arm  45  3.5  RANDOM  VIBRATION  MEASUREMENTS  ON  THE  t i p  was  PROTOTYPE(1)  HOT-ARM Random the  vibration  following  signal  spectra and  Krohn-Hite  analyzer.  showed  higher  that  beam  transducer  signal  at  and  5.05  The  Hz  f i l t e r i n g  acceleration The  in  damped  1DOF  average equal  system The  the  expected of  are  response  the the  are  small.  envelope  were  analysis analyzer.  tip  small,  The  deflection their  tip  large. the  tip  typical by  for  any  a  second  mode  the  660A  deflection  signal  although  a  Gaussian  of  zero  a  of  the  l i g h t l y  crossings  Because two  a  noise or  of  the  in  a  whose hot-arm  envelope random  is beat  harmonics  beating  the  system  narrow-band  varying  y(t)  white  narrow-band  similar  slowly  acceleration  response  frequency.  shows  f i r s t  Nicholet  together,  figure  for  modes  with  close  hot-arm  frequency  fundamental  frequencies  of  the  vibration  associated  amplitude  the  of a  Nicholet  from  the  excited  phenomenon  in  is  and  contributions  were  plot  p i e z o e l e c t r i c  tip  higher  3.2  of  with  the  because  contributions  frequency  to  the  measured  1030  f i l t e r  f i l t e r e d  into  needed  Figure  excitation.  of  band  domain  band  Inspection  channeled  time  plotted  Vibrametric  vibration  was  from  hot-arm  3750  modes  was  contributions  the  apparatus:  accelerometer, FFT  of  with  fashion  [15]. The a  normal  flow  PDF or  of  y,  p(y),  Gaussian  turbulence  is  a  plotted  in  Figure  3.3  approximates  d i s t r i b u t i o n .  This  is  process  many  independent  with  plausible  since random  46  cu  Ld O U  TIME ( S E C ) Figure  3.2:  hot-arm  Vibration  acceleration  of  the  P r o t o t y p e d )  t i p .  5.66  : >> CL  0.0 -0.854  -r 0  0.854  T I P A C C E L E R A T I O N , y CIN/S > Figure hot-arm  3.3: t i p  Probability vibration  density  function  a c c e l e r a t i o n .  of  the  P r o t o t y p e d )  47  sources A  and  Gaussian  in  a  such input  Gaussian  following  a  random  linear  output.  A  single-variable  of  vibratory  system  w i l l  Gaussian  PDF  }.  o~  to  and  standard  0.08*10~  3  Figure  3.3,  the  mean  deviation  crossings.  0.078  where  Note  of  of  deviation  variance  is  it  is  would  equal  be  bigger  acceleration. one  extreme  interval.  that  A  set  value  The  y  denoted  y  by  denoted  y by  the than of  by  is  mean  to  value.  at  the  time  extreme  extreme sampled  equal  RMS  standard data for  is  every  a»  of  is  plot  the  zero  a.. y  is  of  the  /  w  of  2  = zero the  and  the  of  the  y,  acceleration  deviation shown  is  zero,  value  domain  values  to  frequency value  the  MS  the  deviation  expected  the  equal  look  2  standard  the  since  the  is  .  is  is  another  evident  to,  to  2  denoted  that  standard  Taking  i n / s  deflection  i n ,  the  (3.5.1)  1  t i p  has  2aj  According  hot-arm  result  2  exp{  approximately  Gaussian.  d i s t r i b u t i o n :  - ( y - y )  =  the  approximately  a  form  • (2*)  is  into  1 p(y)  process  in  twenty  of  the  Table  3.1;  second  time  48  Table  3.1;  t i p . ( i n / s  2  Extreme  accelerations  of  the  P r o t o t y p e d )  hot-arm  )  0.20  0.12  0.15  0.13  0.74  0.13  0.11  0.20  0.13  0.15  0.092  0.074  0.15  0.13  0.12  0.092  0.11  0.092  0.12  0.15  0.  0.11  0.092  0.15  0.12  0.13  0.13  0.12  0.13  0.092  0.18  0.17  0.18  0.13  0.18  0.16  0.21  0.18  0.11  0.15  0.12  0.15  0.061  0.13  0.12  0.074  0.18  0.12  0.13  0.13  12  Mean-extreme This times good  shows  the  estimate,  vibration  Estimate The to  on  be  of  in  the  a  a  the  y =*  note  a  one  as  3a».  e  theory  2  usual  of  by  Excitation  as  i n / s  deviation  P r o t o t y p e d )  to  0.13  g  i e . , the  modelled  subjected  y =  that  standard  references to  value:  practice the  The  hot-arm  on  dimensional  three  value  find  with  is  a  further  application  [16].  a  excited  transversely  can  values  Davenport  Force  taking  mean-extreme reader  extreme  of  P r o t o t y p e d ) by  the  vibrating  random  rain  Hot-arm  coolant  flow  cantilever force  over  is  beam the  49  beam  span.  As mentioned  this  cantilever  1.4.  Vibration  beam  T h e beam  PSD  random  from  this  the  f i r s t  measured  mode  function f i n i t e a  of  B.  that  material  rain MS  with  the  force  section,  points  f i r s t  mode  at  o) , of  Stw,), the  root  alone  h y s t e r e t i c a l l y  y  the  A and B of  is  deflection  method.  of  Figure  i s  to  damped.  c a n be  hot-arm  material.  be The  estimated  t i p  attributed fact  that model  actual  =  in  the  the  the  riveted  hot-arm  has  a  i s  t i p  has a  o n e mode  are  in  the  parameters  of  model  i s  same  for  to  of  as  that  of  model  i s  for  aluminum  moduli  be  a  f i n i t e  can  be  and  the  cantilever  l b / i n  where  of  140  element  in  selected  P r o t y p e d )  s t i f f n e s s  the  shown  of  109  the  modelled  the  p s i  e l a s t i c  approximated  t i p  as  6  with  i s  beam  the  10.6*10  construction  measured  hot-arm  is  the  s t i f f n e s s  evaluated  modulus  usual  acceptance  beam.  as  the  l b / i n . model  are  by: 1  i/>,(L)  a  frequency  discrepancy  hot-arm  The given  to  of  The e l a s t i c than  joint  cantilever  modulus  fundamental  This  and  hot-arm  element  e l a s t i c  hot-arm.  mass  The P r o t o t y p e d )  f i n i t e  p s i , lower  6  modal  Prototyped)  The  i t s  actual  6.5*10  M,  the  non-uniform  such  The  in  shape,  element  Appendix  the  previous  mode. The  as  a  coincides  response  considered. of  in  (*![M]*i)  l b * i n , =  2.95  i n ,  =  31.7  For  variance  of  the  t i p  the  strength  of  the  excitation  (3.4.22)  to  be:  Siujv/v  =  255  =  8. 9* 1 0 "  3  / s  2  ,  rad/s  d e f l e c t i o n : i s  i n  o*  =  6.0*10  approximated 8  l b * i n .  -  9  by  i n  2  ,  equation  4.  AN  A N A L Y T I C A L STUDY FOR  An  A  study  structural  reduce  hot-arm  by  transversely  a  a n a l y t i c a l is  to  determine  what  stiffener,  response  a  the  or  It  was  shown  dimensional  random  rain  possible  model  hot-arm.  The  harmonic  point  the  tip  free The  addition  of  the  other,  a  force  of  the  result study  structural  of  better  like  at  model  of  minimum-weight  either  the  the  is  design  unit  hot-arm, than  harmonic  is  made  the  beam  that  noise  a  PSD  excitation  frequency  is  a  one is  a  on  a  unit  applied  find  the  to  i e . ,  beam.  a  continuous continuously  For  an  hysteretic viscous  optimal  e l a s t i c  damping damping.  h y s t e r e t i c a l l y cantilever  point  50  of  involving  the  a  sandwich  vibration  excitation,  to  material,  of  the  mass,  mass-stiffness-damping  hot-arm  cantilever  damping  for  beam.  extended a  discrete  section  fundamental  structural  the  of  model  investigation  models  white  would  replaced  comparison  random  type  then  Triumf  A  two  with  discrete  of  of on  previous  the  v a r i a b l e - c r o s s - s e c t i o n to  force  cantilever  of  the  a  for  flow-induced  v a r i a b l e - c r o s s - s e c t i o n structure  in  this  has  beam.  is  beam  of  the  which  structure  adding  damping  standard  geometry  d i s t r i b u t i o n  that  determine  hot-arm  point  responses  e x c i t a t i o n .  to  cantilever  cantilever  beam  a  hot-arm  starting  effect  uniform  analytical  The  for  GEOMETRY  VIBRATION  conducted  vibrating The  STRUCTURAL  MINIMIZES  geometry  rotational  of  OPTIMAL  is  vibration.  studies.  of  THE  H O T - A R M WHICH  analytical  desirable  ON  force  or  beam, the  is  a  Thus damped  subjected  random  rain  51  force,  with  constraint  investigated. designs  is  material: Size in and  to  The to  make  put  the  r e s t r i c t i o n  the  form  beam  reported analyses.  of  a  depth. on  rational  on  on for  the  the  hot-arm  use  where is  after  possible a  careful  of it  w i l l  replacement  construction do  onto  between  most the  the  hot-arm  consideration  is  minimum-weight  the  imposed  dimensionless ratio A  v i b r a t i o n ,  investigating  e f f i c i e n t material  t i p  of  beam  good. designs length  design the  is  above  52  4.1  THE  ADDITION  REDUCE  THE  OF  DISCRETE  MASS-STIFFNESS-DAMPING  VIBRATION  D E F L E C T I O N OF  models  the  A  UNIFORM  TO  CANTILEVER  BEAM Let models A1  A  three  ' A T , uniform  fixed A2  A  at  A  labelled B1  A  A  as  C1  as  the  Let  two  'C1' light  the  shown  A  light  into  of  the  at  end  added  to  model  shown  mass  be  labelled  unit  length  as  per  m  is  x=L.  in  and  A1  at  x=a  as  shown  in  of  4.1,  with  the  analytical and  force  modified  span  in  is  attached  to  model  A1  Figure  4.2,  where  of  beam  P  sin(oj t)  A  <<  L.  excitation  be  'B2'.  Figure  the  K°  models  point  force  beam  of  where  beam white as  models  and  co  is  n  P  noise  shown of  is  n  the  could PSD,  in  applied  fundamental  be S  0  ,  Figure  damping  be  at  unity. is  applied  4.2. labelled  as  ' C 2 ' .  viscous via  damping complex  damping  beam  parameter modulus  parameter  hysteretic  hysteretic K°.  as  rain  beam  viscous C2  free  stiffener  'B1'  harmonic  over  A  with  analytical  models  random  models  x=a+A  two  frequency B2  is  rotational  and  unit  x=L,  beam  and  M°  beam  4.1.  x=a  Let  x=0  cantilever  ' A 3 ' .  cantilever  mass  lumped  at  and  end  lumped  Figure A3  'A2'  of  via  damping  C°  damping  complex  is  added  parameter  H°  is  introduced  approach  modulus  parameter  c  is  in  and  p a r a l l e l  v  is  approach added  a  in  into lumped to  K°.  introduced  and  a  lumped  p a r a l l e l  to  53  Figure  4.1:  A  excited  by  harmonic  Figure  4.2:  rotational random  a  rain  uniform  A  point  uniform  stiffenner force  cantilever  with  K°  force  beam P  sin(a>  cantilever and  white  viscous noise  with  an  mass  M°  )t.  beam damping  spectral  added  with C°  an  added  excited  density  S  0  .  by  a  54  For M ° «  2  ( a )  small «  mL,  H A #" (a)  «  c  v  0  2  <<  2  1 and  accuracy in  i s  reference  rate  of  method  the can  fundamental simple the  mass-stiffness-damping  K°A 07 (a) 2  2  vco mL) 1)  a  needed, [17], mode  the  vibration  more  model.  but  analysis  to  ^  2  ( a )  in i s  « the  beams  method,  T h e mode  does  are  a  for beam  lend  o n e mode the  of itself  analysis  combinations  excitation  tabulated  in  more  acceleration  the  shows  If  convergence  approximation  C  ( i e . ,  described  the  not  and  2  accurate method  ( i e . ,  cco mL,  adequate.  increase  procedure.  addition,  results  2  acceleration  deflection  The  C ° A  damping  used  Appendix  mass-stiffness-damping damping  mode  c a n be  frequency  solutions.  2  o n e mode  summation  give  <j mL,  and l i g h t  2  «  «  addition  Table  model 4.1.  the to of of and  55  Table  4.1;  for  Vibration  various  deflection  Model  y(x)  B1;  -  Model  addition,  uniform of  beam  cantilever the  excitation  P  K°=0,  sin(cj t) n  C°=0,  model  H°=0  Model  CI:  viscous  damping  Model  C2:  hysteretic  Model  CI:  viscous  Model  C2: hysteretic  P0,(L)«,(x) • mLcw i  B1 :  P  sin(a> t) n  damping  Pt6, ( L ) * , ( x ) y(x)  m L vix>\  Model  B2:  S  0  S ?r0i o  £ [ y  2  damping  (x)  ( x , t ) ] m Lcco^ 2  Model  B2:  S  0  S JT<6 0  £ [ y  2  ( x , t ) ]  2  (x)  * m Lfwi 2  beam lumped  model  A1: M°=0,  Model  a  combinations  mass-stiffness-damping damping  of  damping  and  continue  Model  table  A2:  Model  . . .  M°#0,  B1:  P  K°=0,  C°=0,  sin(a> t)  H°=0  Model  n  C1 :  viscous  damping  P*, ( L ) * , (x)^{l + (M°/mL)<6 (a)} 2  y(x)  mLco) 3  Model  B1:  P  sin(a> t) n  Model  C2:  hysteretic  Model  C1:  viscous  Model  C2:  hysteretic  damping  P<4, ( L ) ^ , ( x ) y(x)  m L I>CJ  Model  B2:  S  2  0  S TTC4 0  £ [ y  2  ( x , t ) ]  2  damping  (x)  * m Lcw? 2  Model  B2:  S  0  S 7T0 O  £ [ y  2  2  (x)  ( x , t ) ] m L v c o V ( 1+ ( M ° / m L ) 0 2  2  (a)}  damping  57  continue  Model  table  . . .  A3: M°=0,  Model  B1 :  P  K°*0,  C°#0  sin(co t)  or  H°#0  Model  n  C1 :  viscous  damping  Ptf, ( U t f , ( x ) y(x) m L { c w + ( C ° A / m L ) c 4 7 ( a ) }v/{cj + ( K ° A 2  Model  B1 :  P  2  2  sin(a> t)  2  Model  n  2  / m L ) 0 V  C2: hysteretic  2  ( a ) }  damping  P0, ( L ) $ , ( x ) y(x)  mL{^cj  Model  B2:  S  2  + (H°A /mL)0V (a)} 2  2  Model  0  C1: viscous  S 7T0  2  O  £ [ y  2  2  Model  B2:  S  2  0  2  Model  0  ( a ) }{u> + ( K ° A / m L ) 0','  2  2  O  ( x , t ) ]  2  C2: hysteretic  S 7T0 £ [ y  (x)  ( x , t ) ] m L{cu> + (C A /mL)<f>'J  2  damping  2  (a)}  2  damping  (x)  m L{^co 2  2  + (H A /mL)0V o  2  (a) }/{cj  2  2  +(K°A /mL)c6^ 2  2  (a)}  where, a>,  :  the  0 (x)  :  2  M°=K°=0; of  x  mode  frequency  the  of  it  square i s  positive  the  of  the  f i r s t  beam  normal  and increases  with  for  M°=K° = 0.  mode  shape  increasing  for value  [17].  <67 (x) 2  fundamental  :  the  shape  for  square  of  M°=K°=0;  the it  curvature i s  of  positive  the  first  and decreases  normal with  58  increasing The  value  of  following  model  x  [17],  observations  A2-B1-C1  can  be  M°  increases  shows  that  and  A2-B2-C1  made: the  resonant  amplitude. models  A2-B1-C2  effect  on  model  A2-B2-C2  the  resonant  and  shows  show  MS  that  M°  has  l i t t l e  deflection.  that  M°  attenuates  the  MS  deflection. models that  A3-B1-C1,  C°  model  and  models  Added deflection  A3-B1-C1, the  in  deflection  and  damping.  a l l  H°  For  models  at  x=0  (extension  is  of  of  H°  MS  and  A3-B2-C2  and  MS  l i t t l e  show  deflection,  effect  on  elastic  of the  is  the  on  the  that  K°  and  MS  the  effects model  of  of  that  suggests  resonant  independent  and  effective  discretely  the  the  like  amplitudes.  This  deflection.  damping  M°  show  structure  resonant  that  A3-B2-C2  attenuate  However  indictates  most  x=A.  and  which  the  2  and  an  shows  0" (x) H°  has  and  and  dependent  model  attenuates  damping  C°  damping,  better  function  resonant  K°  A3-B2-C1  cases.  are  Inspection damped  the  that  resonant  dampings  the  hysteretic  shows  A3-B2-C1  amplitude.  attenuate  the  attenuate  A3-B1-C2  resonant  is  H°  A3-B1-C2,  K°  have  The  when that  modified)  M°  and  beam  on  excitation  the the  K°  hot-arm, frequency,  [13].  amplitudes  a  of  of  hysteretically  l i t t l e  effect  characteristic  rotational its the  while of  stiffener  attachment  points  continuously  minimum-weight  the with are  modified  cantilever  59  beam be  design  one  i e . ,  with  which  a  beam  is  on  the  stiffened  in  bending  which  of  tapering,  i e . ,  is  not  at  known  models  d e f l e c t i o n .  is  tapered  l i n e a r l y ,  this  Inspection damped  constraint  of  shows The  K°  effects  regions.  At  frequencies  is  damping The  regions that  of  attenuation. 0" (x) 2  the  MS  the  most  cantilever  that  M°  when  be  (extension beam one  has  This of  design  which  attached  is  of  in  the  the  MS  three  resonance, force) from  inertia  is  is the  force)  resonance  the  dominant. a l l  three  with  the  fact  independent  of  the  a l l  give  K°  and  H°  functions at  and  x=L, x=A  that  on  a  the  tf> (x) 2  and  to  K°  and with  attenuate  continuously  modified)  constraint  MS  from  the  x=0  the  into  away  force)  effective at  e t c . ,  frequency  the  near  M°,  suggests  heavier  divided  and  is  discretely with  The  combined  why  on  s t i f f n e s s  and  force  most  H°  contributions  explains  is  attenuate  from  damping  c h a r a c t e r i s t i c s  d e f l e c t i o n .  modified  should  The  on  geometry  h y s t e r e t i c a l l y  generalized  response,  damping  vibration,  away  above  (or  generalized deflection  effective  MS  and  end,  l a t e r .  of  be  fixed  The  follows. can  should  c u b i c a l l y  and  generalized  force  frequency  dictate  system  frequencies  (or  t i p .  a l l  K°  as  frequencies  hysteretic  frequency  H°  At  that of  at  H°  M°,  below  (or  inertia  force  fact  1DOF  force  dominant.  of  and  explained  damped  the  discussed  M°,  a  the  is  that  of  resonance  it  deflections  be  while  the  MS  response  dominant  near  the  can  s t i f f n e s s  towards  amplitude  p a r a b o l i c a l l y ,  point?  deflection  the  resonant  minimum-weight MS  v i c i n i t y  deflection of  the  free  60  tip  and  geometry is  4.2  stiffened of  this  in  l a t e r .  OPTIMUM  DESIGN  SUBJECTED  FREE  OF  A  TO  A  fixed  end.  minimum-weight  DAMPED  HARMONIC  and  A  POINT  Buckling  of  The beam  CANTILEVER  FORCE  and  AT  THE  Shape  found  of  d i s t r i b u t i o n  of  c o l l e c t i o n  of  in  plates,  fundamental  a  two  Distributed  type  as  and  of  volume Parameter  l i t e r a t u r e  is  follows.  shells  frequency  of  bars,  beams,  and  compliance  of  structures  response  of  two  and  Multi-purpose category  sub-categories, forced  be  design  shells  Deflection Dynamic  of  optimum  optimization  problem  columns,  Maximization plates  can  The  to  the  continuous  Optimization  according  on  comprehensive  references  t i t l e d  organized  exists  involving  Structures(1980)[18].  dynamic  dimensional and  elements  special  response  can  problems be  divided  into  two  follows. state  dynamic  a r t i c l e s  chronological  three  structures  as  steady  transient Published  UNIT  material.  l i t e r a t u r e  The  the  HYSTERETICALLY  l i t e r a t u r e  structures  structural  edition  of  TIP  Substantial e l a s t i c  v i c i n i t y  v a r i a b l e - c r o s s - s e c t i o n  discussed  BEAM  the  order  on  o s c i l l a t i o n  response forced are  as  steady  state  follows.  o s c i l l a t i o n  Icerman  in  (1969)[19]  61  formulated  a  constraint  that  the  given  deflection a  minumum-weight  of  at  the  for  bending  a  axial  plate  Plaut  deflection  to  the  of  a  the  (1973)[22] optimal  two  authors  The  of  a  and response  of  of  a  design  of  on  structures  the  (1970)[20]  used  to  the  broader  optimality of  a  periodic  beam,  the  loading,  structure.  potential  bending  both  of  prescribes  of  technique  for  a  problem  that  mutual  Rayleigh-Ritz  of  of  obtained  under  for  work  truss.  constraint  condition  design  amplitude  rod, bending a  a  virtual  applicable  structures  point  with  Mroz  formulated  stationary  optimality  the  the  i s  specified  used  on  which  design  of  on  application.  motion  design  p r i n c i p l e  derive  of  problem  bound  load  (1971)[21]  minimum-weight subjected  of  point  structures.  of  upper  formulation  conditions  the  an  amplitude  variational  class  sets  design  a  He  used  energy beam.  to  static  the  to Plaut  approximate and  dynamic  d e f l e c t i o n s . In with  a r t i c l e s  undamped  by  e l a s t i c  damping  should  hot-arm  vibration,  resonance  cantilever the  free  be  is  minimum-weight  investigated.  and Plaut  structures.  In  more  In  effective  damping  design  interest. of  subjected with  Mroz  included.  of  beam, t i p ,  Icerman,  a  the  In  a  constraint  unit on  have  dealt  r e a l i s t i c  cases  particular at  the  the  h y s t e r e t i c a l l y to  they  case  fundamental  following damped  the  sandwich  harmonic  point  the  amplitude,  t i p  of  load  at is  62 The a  governing  h y s t e r e t i c a l l y  beam 3  i s  given 3  2  E  2  equation damped  3x  Take  t r i a l  y(x,t)  uniform  y(x,t)  3 +  2  the  transverse  or  vibration  non-uniform  of  cantilever  by:  [I(x) 3x  for  2  I(x)  3 y(x,t) 2  ]  + m(x)  3x 3t  u  solution  = y(x)  -  y(x,t)  3  3t  2  of  =  f(x,t) .  the  form  2  (4.2.1)  cos(cot),  (4.2.2)  for H(-co) f ( x , t )  =  P  6(x-L)  cos(cot)  .  (4.2.3)  |H(w)| For  co < co  H,(co), with H  T  (  -  H(co)  1f  the unit  O  )  (  C  at O  a  function  complex  resonance C  i s  2  -  C  co,, O  2  )  frequency  =  response  for a  i  +  /{(co -co )  that  i s  co,  y ( x ) ,  I  =  Substituting  2  2  »>co  2  (4.2.4)  + Uco ) } 2  an unknown  I(x)  by  1D0F s y s t e m  . 2  |H,(w)|  y  c a n be a p p r o x i m a t e d  i e . ,  =  Note  which  2  at  present.  For convenience  l e t  and m = m(x).  (4.2.2),  (4.2.3)  and  (4.2.4)  into  (4.2.1)  y i e l d s :  H(-co) E ( 1 + / v) [ l y " ]  B  -  co my 2  =  P  5(x-L)  .  (4.2.5)  |H(co) | where the  4.2.1  (  )'  and  arguement  (  )",  once  respectively,  and twice  OPTIMALITY  CRITERION  CANTILEVER  BEAM  AT  THE FREE  TIP  with  denotes  respect  to  differentiating x.  FOR T H E MINIMUM-WEIGHT  EXCITED  BY A UNIT  HARMONIC  SANDWICH  POINT  FORCE  63  If then  y  y  is  gives  principle,  E  , Jn  a  kinematically a  stiffer co <  when  „ Iy" dx  0  and  by  analogy  to  y(L)  y  Rayleigh's  P  (4.2.6)  |H(u)| the  principle  of  minimum  potential  energy  Im{H(-co)}  Iy" dx  £  2  y(L)  P  /  (4.2.7)  |H(u)|  0  where  by  of  Re{H(-co)} £  2  .  fn  vE  and  Consequently  0  j i  „ my dx  Sn  2  system.  approximation  co^ ,  T  co  -  2  admissible  }  Re{  and  imaginary A  }  Jm{  part  sandwich  denote,  of  the  beam  respectively,  taking  the  real  has  the  arguement.  depicted  in  Figure  4.3  properties: m(x)  =  where core the  4B{p D+p d(x)} 1  I(x)  2  p,  and  p ,  and  the  cover-plates  respectively,  2  cover-plate.  between  the  to  bending  the  designs which  I  are  The  with  constrained  densities  maintains  the  physical  it of  does a  not  beam.  corresponding to  mass  elastic  have  the  modulus  Consider  tip  the for  separation  contribute  deflections  same  of  directly two y  amplitude,  such  and  y,  i e . ,  Re{li(-co)}  ReiH(-co)} y(L)  the the  but  E  (4.2.8)  2  is  stiffness I,  4BD d(x)  are  and  core  cover-plates  and  =  =  P  |H(w)|  y(L)  P  |H(co)|  or E/Q  Iy  n  2  dx  co j£  my dx  2  2  =  Ej|j  Iy" dx 2  and Im{H{-oo) y(L)  }  Im{U(-co)} =  P  |H( )| W  y(L)  P  |H(«)|  -  W 2 /Q  my dx 2  (4.2.9)  64  Figure  4.3:  A  v a r i a b l e - c r o s s - s e c t i o n  sandwich  beam.  or uE  /Q  Iy" dx  =  2  * E /Q  Combining /Q  ( I - I )  If  y  of  Ey"  2  -  with  [Ey"  2  ( p  2  -  2  / D  where  t J  I 2  ) y  kinematic  y(0)=0 2  i s  2  =  2  a  design  with  the  Combining  If  y  of  G  2  ) y  2  ]  and  >  (4.2.6)  y i e l d s :  0.  (4.2.11)  ,  2  I  (4.2.12)  2  I  (4.2.13)  same  £  constant,  cannot t i p  (4.2.10),  y" dx  design  / H  positive  design  (I-I)  (4.2.8)  y(L)=X  that  vEf^  2  (4.2.10)  constraints:  (4.2.11) I  ( p  d x .  s a t i s f i e s  y'(L)=0 G  2  (4.2.9),  design w  l y "  be  s a t i s f i e s  heavier  it  follows  than  any  from other  amplitude.  (4.2.8)  0.  then  and •  (4.2.7)  y i e l d s : (4.2.14)  65  vEy"  =  2  with  G l ,  (4.2.15)  kinematic  y(0)=0 where  y'(0)=0 G  i s  2  that  design  with  I  For  thus  design the  I  same  constant, cannot  t i p  near  the  1f  optimality  be  it  heavier  than  (4.2.12)  co  then  follows  than  from  any  other  amplitude.  smaller  c r i t e r i o n  co a p p r o a c h e s  (4.2.16)  positive  much  a  optimality  y(L)=X  a  (4.2.14)  and  constraints:  and  co,  alone  is  damping  c r i t e r i o n  light  damping,  adequate.  force  However  becomes  (4.2.15)  as  important  should  be  used  instead. The into  real  E[ly"]" with  governing  -  force  and  equation  imaginary  parts.  cj 4B(p D+p d)y  =  2  1  of  2  constraints:  at  motion The  (4.2.5)  real  part  c a n be i s  divided  given  0,  by: (4.2.17)  x=L  Re{H(-u>) } EI(x)y"(x)=0  E[I(x)y"(x)]'=  -P  (4.2.18)  |H(u)| and /  the  imaginary  i>E[ly ]"  with  n  force  =  part  i s  given  by:  0  (4.2.19)  constraints:  at  x=L /m{H(-u)}  ?E[I(x)y"(x)]'=  vEl(x)y"(x)=0  -P  .  (4.2.20)  |H(w)| For  y  obtained  minimum-weight (4.2.19).  design  I(x)  The problem  as  approximations For  w much  from  are  smaller  (4.2.12) must  and  satisfy  (4.2.15), both  quite  (4.2.17)  formulated  i s  needed  to  find  the  minimum-weight  than  co,  and  light  the and  complicated;  hysteretic  design. damping,  66 design  I(x)  which  approximates approaches thus  the  near  design  co,, the  FREQUENCY  LOWER  THAN  (4.2.12)  design. force  s a t i s f i e s  minimum-weight  MINIMUM-WEIGHT THE  damping  which  the  equations  minimum-weight  l(x)  approximates  4.2.2  s a t i s f i e s  (4.2.15)  SANDWICH  CANTILEVER  OF T H E UNIT  HARMONIC  the  dimensionless  y  =  y / L  D  =  D/L  d  =  d/L  I  =  4B/L  T  =  w  M  =  M/EL  s  =  s / E  P  =  P/EL  of M  the is  the  given  by:  For  CJ  damping,  the  and  amplitude  sandwich  beam  sandwich  given  by:  D  -  2  T y  2  =  a  important and  and  (4.2.19),  BEAM POINT  DESIGN FORCE  WHEN  IS  MUCH  2  p  2  stress  given  by:  L  2  / E (4.2.21)  2  in s  the =  cover-plates  - (1 +/ » > ) E D y " ,  and  4BDds. than  beam  forms  u  design  approximates  dimensionless  y "  =  the  is  smaller  their  2  of  which  (1+/*>)M  much  (4.2.17)  CJ  quantities:  x / L  i s  as  design.  =  s  However  becomes  x  where  (4.2.17),  THE FUNDAMENTAL FREQUENCY  Introducing  3  and  the the  1  and  which  light s a t i s f i e s  minimum-weight two  hysteretic  optimality  (4.2.12)  design.  equations  In are  (4.2.22)  2  MTy =  0  (4.2.23)  D»/(a +Ty ) 2  where  ai  i s  and  is  given  d  a  constant by:  2  which  c a n be  expressed  in  terms  of  X  67  M d  = —  — .  (4.2.24)  BD/(a +Ty ) 2  The at  dimensionless x=0  "at  2  boundary  y(x)=0  —  ~ —  x=1  y(x)=X  conditions  are:  y'(x)=0  ~  ~ M(x)=0  Re{H{~co)}  M'(x)=P  .  (4.2.25)  |H(co) | For  an  found  undamped a  series  damped  beam  Mroz's  only  solutions small 5o  =  M  = M  Note  +  0  +  for  of  of  TM,  +  T  co m u c h  y,  2  a  here  boundary  i s  parameter.  (4.2.27)  Thus  are  and The  different  conditions.  and  T,  small  method  from series  found  for  i e . ,  +  . . .  (4.2.26)  +  . . .  (4.2.27)  smaller  than  into  the  the  (4.2.22), over  two  co, ,  terms  x  parameter equating  i s  terms  and applying  solution  T  given  the  small. of  like  boundary  by:  „ 2  „  x D  of  6  .  (4.2.28)  3  Substitution  the  M  of  solved  (4.2.26)  perturbation  D  240  terms  be  integrating  y i e l d  a,  =  2  2  the  terms  parameter  (4.2.26)  T,  Yo = — ~ x 2  y  2  in  to  kind the  T  used  forced  the  +  conditions  —  the  T y ,  of  a,  solution  in  Substituting powers  Mroz  problem  values  Y  beam  like  boundary  (4.2.27) powers  and of  conditions  T,  (4.2.28)  into  integrating  y i e l d :  (4.2.23), over  x  and  equating applying  68  _ M  =  0  Re{H(-o)}  -P  (1-x)  |H(u)I Ba,  M,  p,  _  =  P  ( 4 x - x « - 3 ) 24  p  + 120  2  (  D  H  (  -  C  J  )  }  _  (5x"-3x -5x+3) . 5  |H(u)|  2  (4.2.29) To  specify  guess  H^ico)  Rayleigh's ji a)  =  i t e r a t i v e  c o r r e c t l y .  fundamental  2  an  For  frequency quotient,  computer  a  co,  scheme  tentative can  be  is  required  to  design  I(x),  the  evaluated  from  the  i e . ,  EBD dy"dx 2  -z 7---Z—-. L / ! B(p D+p d)y dx 2  (4.2.30)  2  (  In  )  1  the  2  case  of  an  undamped  beam  v=0,  the  term |H(w)|  is  equal  needed.  to  unity  In  the  corresponds the  to  variable  =  a, T  10" =  =  1  1010-"  which  an  case  of  beam  loaded  plate  dimensionless P  a  and  5  =  frequency  s t a t i c a l l y .  thickness  d(x)  T  =  1  1/120  p,/p  7  could  zero  scheme  to  guess  p C J  2  2  =  120  =  2  / E  =  =  the  problem  Figure  4.4  presents  for  the  following  =1/2  0 to  2D  =  2  the  in  2.54*10""/10.3*10  280  s "  2  ,  not  1/10  correspond  in  is  T=0,  data  typical  2B  30  of  a  i e . , L  CJ,  data:  3  and  i t e r a t i v e  and  for  P  6  ( s  =  1 . 5* 1 0 "  2  / i n  2  ) 2  =  in  (aluminum) lb.  and  hot-arm,  69  Figure  4.4:  cantilever (T  =10"")  (T  =0),  Optimal beam  and  both  for  cover-plate  excited a  forces  by  cantilever at  the  free  thickness a  unit  beam t i p .  for  harmonic  under  static  an  undamped  point  force  point  force  70  4.2.3  MINIMUM-WEIGHT THE  FREQUENCY  FUNDAMENTAL For and D  (4.2.19) 2  =  =  0  where  a  and  i s  2  y "  vM"  study  d  CANTILEVER  OF THE UNIT  HARMONIC  BEAM POINT  DESIGN FORCE  WHEN  IS  THE  FREQUENCY  at  in  SANDWICH  fundamental  their  resonance,  dimensionless  equation  forms  are  given  (4.2.15) by: (4.2.31)  a \  (4.2.32)  i s  2  a  constant  given  which  c a n be  expressed  in  terms  of  X  by:  M d  =  — — . BDa  The  (4.2.33)  2  dimensionless  at  x=0  y(x)=0  at  x=1  y(x)=X  Equations those  p r i n c i p l e optimality  for  Integrating  y  = — 2  ~ M  the  x  2  and  minimum  loaded  and  are:  Shield  potential for  sandwich equations  dimensionless  t-M' ( x ) = P .  (4.2.32)  minimum-weight  c r i t e r i o n  s t a t i c a l l y  applying  M(x)=0  Prager of  conditions  y'(x)=0  (4.2.31)  equations  s t a t i c a l l y .  boundary  are  (4.2.34)  similar  design  of  a  (1968)[23] energy  to  minimum-weight  in  forms  beam  to  loaded  applied  the  derive  the  design  of  a  beam. (4.2.31) boundary  and  (4.2.32)  conditions  then  y i e l d :  (4.2.35)  D  P = — (1-x) .  v  (4.2.36)  71  where The  2  2XD.  minimum-weight  same  as  factor ~ ~ d(x)  For  =  a  that 1 /V,  weight  the  the  resonance  s t a t i c a l l y  loaded  excitation case  is  except  ~ (1-x).  — v BDa  the  of  for  the  for  a  i e . ,  P =  d(x)  (4.2.37)  2  case  in  of  the  cover-plates p gPL  which  p D/p d 1  is  2  minimum-weight  small  and  beam  is  n e g l i g i b l e , that  the  of  the  a l o n e , i e . , 2  2  ".in • 7 - r where  g A  is  <*- 2  the  gravitational  typical  between  a  line  of  counter  the  same  tip  amplitude.  are  for  the  uniform  minimum-weight both  uniform  Table  and  to  t i p  where  c o / s  the  beams.  cantilever  a  4.2  is  are  part.  not  The  and  the  harmonic the  the the  both  a  comparison  beam  are  and  constrained  its to  f i r s t  three  columns  of  Table  4.2  the  sixth  column  is  for  the  forth  force  fundamental as  and  beams.  respective  same  represents cantilever  where  minimum-weight  unit  Note  part  beams,  beam,  subjected  counter  acceleration.  uniform-cross-section  minimum-weight  38)  of  f i f t h Both  P  of  frequency of  its  common  to  beams  are  the  sinfo^t)  fundamental  that  are  at  the  frequencies of  a  free of  uniform  minimum-weight  72  Table  4.2:  A comparison  cantilever the  same  beams  and  resonant  their  t i p  uni form  uniform  w  uniform-cross-section counterparts  with  amplitude  common  common  D  I  uni  the  minimum-weight  uniform  d  W  between  minimum-weight W . mm  y  Io  W  yo  0  0  1.0  N/A  1 .000  N/A  1.000  N/A  0.5  0.25  0.844  0.375  1 .185  0.386  0.4  0.20  0.768  0.400  1 .302  0.309  0.3  0.15  0.650  0.425  1 .538  0.232  0.2  0.10  0.486  0.450  2.058  0 . 155  0.1  0.05  0.271  0.475  3.693  0.077  the  cover-plate  where, N/A  :  W  0  :  p  2  :  g I  :  y  0  x=L. the  applicable  60p g 2  mass  density  gravitational :  0  not  :  of  material  acceleration  0.08333 8.3868*10 [P/^E] 5  A l l  the  A l l  sandwich  type  cantilever  shown  beams, in  beams  are  uniform  Figure  4.3  fixed  at  x=0  and  and minimum-weight, and described  by  free are  at of  equation  73  (4.2.8).  The  negligible damped  the  kind  section length  of  60  of  Table I  0  inertia  reference The  y  of  is  the  length I  and  cover-plate moment  of  uniform The depth the  The  one  beam  core  be  units.  of  beams  are  resonant  mode  to  tip  approximation  the  unit-square  reference  depth  to  hot-arms.  this  the  and  t o t a l  tip  cross  beam.  The  length  ratio  f i r s t  entry  The  unit-square  respectively, section  with  This  Triumf  uniform  the  of  no-core  beam.  weight,  moment  amplitude  l i s t e d  reference  inclusion  of  the  of  a  boundary  the  sandwich  core.  The  is  maintained  of  with  increasing  core  thickness.  inertia  respectively,  of  cantilever  radius  cross  of  of  a  beam  decreases The  and  width  2B=1.  the  total  weight,  gyration  section  and  of  t i p  through  beams  Let  w u  n  ^  uniform  cross  section,  amplitude  of  the  beams.  minimum-weight 1  2D+d=1  Table  where  area  depth  in  beam  sectional  thickness,  radius  the  sections  cross  be,  >  cross  exterior  L=60, y  2D+d(0)  same  A l l  assumed  beam.  thus  D,  be,  unit-square  of  60  cross  proportionately  d,  is  are  is  4.1.  cantilever  modifications  i n i t i a l  core  cover-plates.  beams  sandwich  0  the  Table  corresponds  modification  are  a  subsequent  are  out;  in  beam  and  0  to  sandwich  h y s t e r e t i c a l l y .  c h a r a c t e r i s t i c 4.2  the  uniform  uniform  the  is  W ,  4.2  the  without  of  of  and  found  the  of  comparison  for  Let  Let  in  l i g h t l y  amplitude of  weight  and  cantilever  width  gyration  2B=1. D  as  beams A  are  of  length  minimum-weight  its  beam  L=60, has  uniform-cross-section  74  counter  be, at  with  the  same  respectively,  the  total  x=0  part  for  Note I(x)  =  the that  tip  weight  minimum-weight equation  amplitude.  and  Let  W . min  cover-plate  cantilever  and  d(0)  thickness  beams.  (4.2.8),  4BD d(x) 2  becomes  more  inaccurate  in  comparison  to  the  more  conventional, I(x) as  =  2B[(2D+d(x)}  d(x)  increases  D=0.375 almost less  is zero  the  beam  case  the of  error  excluded  at  at  d=0.1  and  uniform Table  Equation  d=0.25  and  D=0.475  beams 4.2  (4.2.39)  is  with  with was  d the  used  beam.  the  i l l u s t r a t i o n ,  the  at  x=0  exceeds  i e . ,  unity,  error  from  beam.  reference  The  reason,  In  (3.5160) d(0)  D.  this  reference  the  to  The  For  are  (4.2.39)  3  percent.  0.25 of  {2D-d(x)} ]/l2,  relative  percent.  than  exception in  3.5  -  3  depth  of  2D+d(0)  the >  1,  minimum-weight where  2  =  d.  (4.2.40)  8 The  maximum  depth  not  have  exceed  D's  for  to the  two  of  the  the  beams  minimum-weight  depth are  of  the  choosen  beam,  uniform  2D+d(0), beam,  appropiately;  2D+d, but  does if this  75  i l l u s t r a t i o n Table damping and  at  would  4.2,  for  subjected the  be  to  and  uniform .  beam,  point  force  with  light at  hysteretic  the  free  constraint  on  the  tip  saving  are  beam  possible  is  used  when  the  instead  of  a  i e . , 2  16  W  .  W  material  the  smaller  the tip  structural  to  e f f i c i e n t  the use  neutral of  the  axis  material  of to  amplitude. with  larger  e l a s t i c  modulus  E  give  amplitude. materials  smaller  tip  MINIMUM-WEIGHT  WHITE  with  larger  damping  parameter  v  amplitude.  DESIGN  CANTILEVER  WITH  nearest  least  materials  tip  SANDWICH  0  located  shows  structural  FORCE  tip  (4.2.41)  minimize  4.3  with  «  bending  give  beams  frequency,  sandwich  (3.5160)  0  beam  harmonic  material  minimum-weight  W  a  complex.  shows:  weight  -Ei*  more  cantilever  fundamental  amplitude,  W  s l i g h t l y  OF  BEAM,  NOISE  A  HYSTERETICALLY  SUBJECTED  PSD,  WITH  TO  A  DAMPED  RANDOM  C O N S T R A I N T ON T H E  RAIN TIP  DEFLECTION There structural Nigam with In  are  just  optimization  (1972)[24] weight  Nigam's  a  and  studied fatigue  few  references  in the  random  i l l u s t r a t i o n ,  a  as  the  the  vibration  structural  damage  on  subject  enviroment.  optimization objective  viscously  of  damped  problem  functions. uniform  76  cantilever cross  beam  with  section  and  constant-thickness a  tip  stationary-random-process and  a  prescribed  expected  rate  frequency  box  represents  in  to  a  general  random  design  a  Both  depth the  of  a r t i c l e s  and  thin  a  Rao  structural  weight,  width  water  This  tank  tower  studied  optimization with  of  wall.  (1984)[25]  dealt  The  a  problem  p r o b a b i l i s t i c  problems.  locatable.  suitable In  the  h y s t e r e t i c a l l y beam  noise  S  PSD,  method.  The  d i r e c t l y  0  ,  damped  over  value  is  the of  weight  and  The  length,  out  dimensional  31.5  static  beam  the  deflection  span  study  is  the  found  using  the  feasible  constraints  be  with  no  width to  on  on  larger and  1.85  gravity  beam.  regime the  the  is  for  cantilever  white direct  evaluated a  t r i a l  The  iterative  defined  by  the  structure.  strongback than  123  inches can  of  sandwich  force  function  were  design  rain  deflection)  the  due  under  minimum-weight  random  constraints  in  problem  objective  of in  should  inches  a  t i p  dimensional  strongback  to  an  MS  carried  the  v a r i a b l e - c r o s s - s e c t i o n  subjected  (ie.  search  for  following  v a r i a b l e - c r o s s - s e c t i o n  the  model  acceleration.  environment.  cantilever  the  of  under  constraints.  the  thickness  multiobjective  References not  minimized  simplified  ground  The  is  are  a  weightless.  damage  stress  to  mean  assumed  with  box  zero  is  the  a  subjected  beam  parameters  and  is  acceleration  p r o b a b i l i s t i c  structural  subjected  ground  The  fatigue  section  problem  more  of  and  optimized the  PSD.  mass  thin-walled  be  in kept  are  that  inches  in  depth.  If  small  by  77  prestressing  the  depth  be  should The  uniform  f i r s t  strongback, u t i l i z e d  of  sandwich  the  TIP  Consider sandwich section force  in  the  on  of  s e c t i o n ,  the  which  beam  cantilever  beam.  be,  respectively,  thickness, cross of  If random the  cover-plate  section,  the  uniform the rain  beams,  the  unit  sandwich  W ,  W  MS • ,  force Table  with 4.3  mode  solution  for  kind  found  Table  frequency  white  gives  the 4.1  light is  PSD, MS  t i p  hysteretic used.  '  MS  of tip  N  0  of  deflection I,  <J,  ' '  1  and  cover-plate inertia  of  deflection  beams.  beams  noise their  and  inertia  2D,  uniform  moment  10  tip d,  and  CJ  0  of  the  evaluated  I ,  0  moment  weight,  cantilever  point  be  and  the  harmonic  w i l l  Let  cantilever  in  section Let  PSD  uniform  however,  separation,  fundamental  uniform  in  total  the  described  um' N  NOISE  time  frequency  to  BEAMS  This  weight,  to  pertains  WHITE and  were  (4.2.39).  total  fundamental  reference  cross  equation the  pertains  beams.  WITH  for  of  s t i f f n e s s .  second  reference  frequency.  with  the  FORCE  fundamental  respectively,  cross  beams  a  inches  bending  cantilever  RANDOM R A I N  of  1.85  H Y S T E R E T I C A L L Y DAMPED UNIFORM  unit-square  inertia  the  the  and  design  the  of  beams  sandwich  of  investigations  minimum-weight  accordance  be,  A  cantilever  at  moment  TO  a l l  maximize  cantilever  D E F L E C T I O N OF  SUBJECTED  to  following  v a r i a b l e - c r o s s - s e c t i o n  4.3.1  then  are  subjected  S  over  0  ,  the  d e f l e c t i o n s . damping  case  to  a  span  of  A of  one the  78  Table  4.3;  beams  subjected  w  T i p  .  to  a  random  d  uni  W  deflection  for rain  D  0  uniform force  Io  N  0  0  0.250  1 .000  1 .000  1 .000  0.9  0.45  0.275  0.999  1 .054  1 .056  0.8  0.40  0.300  0.992  1.114  1 . 1 32  0.7  0.35  0.325  0.973  1 .179  1 .245  0.6  0.30  0.350  0.936  1 .249  1 .426  0.5  0.25  0.375  0.875  1 .323  1 .728  0.4  0.20  0.400  0.784  1 .400  2.278  0.3  0.15  0.425  0.657  1 .480  3.428  0.2  0.10  0.450  0.488  1 .562  6.559  0.1  0.05  0.475  0.271  1 .646  22.41  0.08333 :  o  :  2.8l94*l0- [E/p ]2" a  2  white  noise  PSD  or  constant  3 1 N  PSD  0.50  :  0  noise  N  1  S  white  1.0  0  w,  with  cantilever  I  where, I  sandwich  :  2.9747*l0 [SoirAE2pf ] 9  PSD  79  Inspection  of  Table  with  light  hysteretic  with  white  noise  the  fundamental the MS  tip  the  bending  mode,  the  no-core  shows to  tip  located  the  minimize  d e f l e c t i o n ,  MS  N,  a.  structural  materials  give  and  mass  i e . ,  MS  with  tip  density is  RANDOM  RAIN  CONSTRAINT  p  give  2  DESIGN  case  h y s t e r e t i c a l l y  THE  of  force the  However,  tip  for  MS  the  damped  point on  FORCE  ON  to  vibration  has  the  the  in  smallest  neutral  use  of  axis  the  of  beam  d e f l e c t i o n ,  inversely  proportional  smaller  OF  e l a s t i c MS  t i p  to  parameter  v  modulus  E  WITH TIP  A  HYSTERETICALLY  WHITE  rain  DAMPED  SUBJECTED  NOISE  PSD,  TO  A  WITH  DEFLECTION  minimum-weight  the  d e f l e c t i o n ,  aluminum.  CANTILEVER BEAM,  beam,  design  a  with  to  a  light a  unit  frequency,  with  parametric  force  of  subjected  fundamental  amplitude,  random  damping  larger  over  cantilever at  larger  with  VARIABLE-CROSS-SECTION  found.  for  force  d e f l e c t i o n .  preferred  MINIMUM-WEIGHT  constraint  random  pf.  materials  steel  t i p is  and  smaller  structural  harmonic  beams  J  v,  the  a  beam  e f f i c i e n t  the  parameters  In  to  span,  nearest  least  the  4.3.2  beam  cantilever  3  b.  subjected  cantilever  shows:  material  material MS  over  uniform  d e f l e c t i o n .  beam  the  for  damping  PSD  unit-square  4.3,  solution  white  noise  was PSD,  80  S  0  ,  a  parametric  approximate general and  numerical  shape  the  of  the  values  of  i t e r a t i v e l y . beam  In  design  deflection near  the  solution  C(x),  with  the  free  was  free  4.1  i s  be  was  optimal  Thus  parameters  beam  with  f i r s t  b  were  that  free  an  instead.  t i p  The f i r s t  optimized  a  cantilever  respect  for was  author;  was g u e s s e d  shown  the  a,  this  found  near  cover-plate  to  parameters  it  heavier  end.  variable-thickness  known  minimum-weight  which  fixed  not  solution  section  should  is  to  the  and  MS  stiffened  approximation,  given  and c  a  as  parabolic shown  the shape  Figure  4.5  where, C(x) A l l  =  - a ( x - b )  the  For  are a  Segments the  to  the  of  a  automate  f i l e  for  beams  to  mass for  computer  task  mode  of  of  the  VAST  program  used,  solution  of  the  kind  given  equation  cubic  beam  deflection  finite f i e l d s ,  cross  parameters. (residing  computer) shapes  were  and  new  elements  of  were  used  (3.4.22) equal to  used joint  program, input  data  with  the  compatible  was w r i t t e n .  and c  computer  VAST  a  b  A  three  creating  segments  Twenty  a,  A preprocessing  iteration,  in  beam  exists  c a l l e d  Department  beam.  the  minimum.  these  program  matrix, the  there i s  search  and  square.  beam,  deflection  repetitive  each  L=60  unit  the  element  function  of  a  Engineering  the  the  of  was u s e d  f i n i t e  were  within  MS t i p  Mechanical  evaluate  (4.3.1)  weight  scheme  acceptance to  c .  confined  given  which  iterative  in  +  cantilever  sections  for  2  A was  one  mode  used.  lengths, approximate  with the  81  parabolic  p r o f i l e  had  a  of  the  non-uniform  set  at  two  to  the  Gauss  inertia  uniform  cross  sectional  beam  over  integration  the  two  Gauss  small  C(x=L). with  those  with  the  was  was  mass  of  evaluated  was  the  average  element  The  average  section  assigned  moment  of  evaluated  at  was  assigned  do  design  to on  cover-plates  a  random the  which  stiffened  and  that  equal-weight  improvements  not  C(L)  the of  rain  to  heavier  the  the  with  v i c i n i t y  the of  are  their  very  small.  noise  v i c i n i t y  the  that  cantilever  should  fixed  of  over  complicated  white  are  Comparison  emphasize  d e f l e c t i o n , in  symbols  improvements  sandwich  force  are  other  counterparts  a  sandwich  respectively,  sections.  results  tip  are,  of  justify  MS  in  minimum-weight  previous  shows  minimum-weight  and  the  d e f i n i t i o n s in  4.3  uniform  C(0)  The  However  constraint  cross  three  where  d i s t r i b u t i o n s .  subjected  length  convenience,  and  shows  beams  4.4  respective  points  4.4  and  consistent  For  beam  properties.  element's and  Each  4.6.  element.  Table cantilever  Figure  points  non-uniform  f i n i t e  t i p  an  of  the  The  in  element.  same  Tables  shown  f i n i t e  the  C(x=0)  as  mass the beam,  PSD,  with  one  with  the  free  be of end.  82  SECTION A-A  Figure  A  4.5:  v a r i a b l e - c r o s s - s e c t i o n  sandwich  cantilever  beam.  Figure  4.6:  Finite  v a r i a b l e - c r o s s - s e c t i o n  element  approximation  sandwich.cantilever  beam.  of  the  83  4.4:  Table  sandwich with  Minimum-weight cantilever  white  noise  design  of  beam,  subjected  PSD,  with  a  v a r i a b l e - c r o s s - s e c t i o n  to  a  random  constraint  on  rain  force  the  MS  t i p  deflection  w W  b  a  N  'D  N  W  0  0.5  2.16*10"  0.3  2 . 51 * 1 0 -  0.2  1.84*10-  4.4  THE  5  5  5  EFFECT  PROTOTYPE(1) So been  far  presented.  result  which  placing  A  lumped  one mass  noise  placed  1 .657  66.0  0.390  0.281  0.389  1 .762  3.253  70.0  0.440  0.345  0.433  1 .234  6.002  OF  ADDING  of at  WEIGHTS  TO  THE  TIP  OF  A  HOT-ARM  mode  with  were  2.037  analytical would  be  the  be  at  beam  Blocks  0 . 105  could  cantilever white  0 . 136  comfirm  weights  hot-arm.  0.315  It  to  0  28.8  several  experiments  a  C(L)  C(0)  c  analytical  tip  analysis  attenuates is  computer  reassuring  checked  the  which  and  of has the  if  one  was  the  vibrating  shown  that  the  vibration  of  subjected  to  a  have  could  results.  easily a  results  The effect  do one of  Prototyped) addition a  random  of  uniform  rain  force  PSD. steel the  each t i p  weighing of  a  approximately  Prototyped)  1.27  hot-arm  kg with  84  coolant added from MS  water  the  drawn  5.35  in  the  trend.  There  except  that  currents, could  •4.5  made  point  force  force  with  at  the as  a  the  guide  on  numbers  of no  has  steel  were  decreased  showed for  experiment  are  could  be  no  clear  the  above  (eg.,  less  mode  blocks  conclusion  explanations  and  blocks  fundamental  which  of  a  less  human  air  t r a f f i c )  y i e l d  to  or the  Thus  a a  either  core  Table  core, V  axis  0  50 ,  4.2  of  in 0.5  d i f f e r i n g  optimal  two  the  bending  has  4.3  volume not  show of  the  lead  to  comparison V  0  may  be  to used  strongback. designs  cantilever to  rain  of  and  % the  deflection volume  random  of  does  with  harmonic  presence  the  minimum-weight  subjected  unit  that  replacement  sandwich  beams  or  0.5 t i p  a  DESIGN  frequency  neutral  uniform  STRONGBACK  cantilever  vibration.  in  v a r i a b l e - c r o s s - s e c t i o n  excitation  show  the  beam  on  damping  uniform  PSD  tip  designing  hysteretic  REPLACEMENT  subjected  to  the  beam.  Studies  The  supply  on  noise  increase  in  Hz.  fundamental  nearest  reference  3.25  data  POSSIBLE  no-core  appreciable  hot-arm  quieter  damping  inclusion  reference  34  difference.  white  effect  the  a  of  the  definite  studies  hysteretic  total  Unfortunately  water  ON A  light  that  no  a  Parametric  minimal  D.  a  of  various  possibly  material  final  experimental  are  DISCUSSION  beam  at  smoother  have  to  Appendix  from  When  frequency  Hz  deflections  recorded  inside.  fundamental  i n i t i a l  tip  flowing  beams  models  cover-plate  of  with the  of light beam  geometries.  85  The  minimum-weight  subjected at  the  are  to  unit  fundamental  tapered  near  a  the  tip  subjected  to  the  beam  span  the  free  tip  random is  one  the  an  excitation more  on  with  r e a l i s t i c  Triumf  has  manufactured  aluminum is  its  11  cover-plates  are  s t i f f n e s s  is  overly  subjected  to  a  be  used  at  in  the  form  Figure the  4.7.  of The  effective  damping vibration section.  a  capacity  fixed  a force  feet  near  the  are  and  a  at of  fixed  the the are  that  in  weight  end.  noise  are  The beam  PSD  over  heavier  end.  excitation  as  But in  near  in  light  Section  3,  should  be  material  of  content  the  of  at  the  the  force,  mass  and  can  dual  hot-arm  8-foot  tip  entire discussed  be  for  feet.  not a  bending strongback  added  purpose? it  vibratory further  A  aluminum  cover-plates  absorber  and  commonly 8  The  thinner  mass  Most  length  tip  vibration serves  chief  property.  Since  lumped  dynamic  white that  length.  feet.  rain  and  model.  in 3  t i p  cantilever  frequency  important  absorbers  with  beam  cover-plates  sandwich  aluminum  absorber  mass  the  plates  short  tip  near  non-magnetic  random  the  lighter  excitation  for  strongback  are  wide-band  construction  free  they  cover-plate  specified  the  that  flow-induced  a  at  with  rain  stiffened  discussion  force  cantilever  one  of  with  sandwich  is  stiffened  design  and  of  such  a  point  frequency  and  a  of  harmonic  linearly  minimum-weight  the  design  as it  to  can  the  tip  shown  in  increases  increases  the  system.  The  in  the  next  86 The the  crude  section has  above  attached  design  gradually  to  its  strongback  both  harmonic  replacement  characteristics  tapers  proposed from  proposed  the point  for  a  Furthermore,  tip  of  a  random the  the  dynamic  the  rain  proposed  wing  tip.  But  vibration  encompasses  minimum-weight at  airplane  towards  design  force  an  strongback  designs,  the  the  design whose  a  absorber.  design  for  force  white  is  The  c h a r a c t e r i s t i c s  frequency  design  cross  strongback  fundamental with  has  f a i r l y  a  unit  and  noise  the PSD.  simple  construct.  CDRE CDVER-PLATES ABSORBER  RDDT  TIP NDT TD SCALE  Figure  4.7:  A  replacement  strongback  design.  to  5. Triumf indicated various  VIBRATION  hot-arms  by  their  methods  vibrating  to  hot-arm  reducing  hot-arm  that  amplitudes  the  vibration  designing most its  of  a  vibration.  f i r s t  an  amplitudes  of  a  an  to  maximum.  The  original  hot-arms  built  for  heavier  The by  quickly  are  light A  were are to  showed  a  effective  absorbers. out  tested.  random  In  large  to  for force  addition, transient  a  makes  to  from  mode  and  for the  effect  the total  of  a  were  of  were  the given  1D0F  noise  hot-arm  vibration.  is  absorbers  white  damped  Since  deflection  the  of  the  comes  absorbers  case  such  since  which  absorbers  with  for  than  f i r s t  mode  and  the  a  d i f f i c u l t .  heavier  reduction  87  small,  comparison  The  that  designed  larger  the  vibration  hot-arm  a  device  vibration  f i r s t  of  in  consideration  hot-arm  in  following,  found  be  times  are  tuned  as  capacity  can  mechanism  series  optimal  many  the  damped  P r o t o t y p e d )  that  damp  where  hot-arm.  subjected  damped  is  tip  weights  testing  absorber  the  p r a c t i c a l  hot-arm  the  In  was  important  to  existing  the  parameters system  the  It  absorber  damping  structures  damping  a  amplified  contribution  a  An  is  mode,  of  is  This  attached  weight  are  HOT-ARM  v i b r a t i o n .  investigated.  secondary  the  A  damped  the  absorber  amplitudes.  FOR  l i g h t l y  increase are  vibration  hot-arm  are  narrow-band  dynamic  its  DAMPING  absorber  main  PSD,  S^.  vibration should  88  5.1  VARIOUS The  devices  METHODS TO  two  the  inside  the  are  simple The  and  two  [27]  vitreous it  is  be  to  The  avoid  coating, in  a  in  a  of  more  are  out  a  to  onto and  survey  period  v i s c o e l a s t i c  acquisition  testing  is  devices  vibration  clean The  in  of  a  do  devices  (1980)[26].  materials  a p p l i c a t i o n . coating  the  heat  l i f e  of  vacuum  particles such  molecular  is  the  to  release  cyclotron  coating  within  However,  not  undesirable  the  and  material,  would  radioactive  s p e c i f i c to  damping  ASME  service  unlink  under  of  by  which into  are:  and  v i s c o e l a s t i c  to  and  d i f f i c u l t  power  devices.  inexpensive  time  predict.  devices  damping  kept  material,  prime  passive  devices  molecules  intensity  The  for  arcing. of  on  on  hot-arm,  e l e c t r i c a l  magnetic  passive  papers  the  be  of  of  an  must  the  hand  absorption  find  other  with  since  r e l i a b l e .  damping  materials  ruled  modes  other  distributed  unspecified  f i e l d  attenuating  other  the  c o l l e c t i o n  made  vacuum  the  and  On  vibration  number  enamel  gas  use.  vibration  d i f f i c u l t  retained tank.  vibration  interfere  s u b - c l a s s i f i c a t i o n s  had  painted  w i l l  possibly  concentrated  reviewed  devices  cyclotron to  distributed  large  active  devices  impractical  Jones  on  VIBRATION  and  beginning  are  are  HOT-ARM  devices.  electromagnetic  A  c l a s s i f i c a t i o n s  devices  passive  f i e l d  THE  are:  active  From  major  ATTENUATE  a  chains  enviroment  of  d i f f i c u l t  to  materials  for  available  time  89  frame.  S l i p  considered helpful to  by  Triumf  feature  promote A  if  Various  mass,  forms  of  But  such  can  devices  designing  a  to  hot-arm  where  hot-arm  vibration  hot-arm  has  be  an  place  the  of  Prototyped)  Hz)  is  damped  mode  it  w i l l  the  other  5.2  AN  operate  such  l i g h t l y ,  in  the  is  needed. has  a  dynamic  vibration A  its  the  are  the  dynamic amplitudes  hot-arm  tip  since  the  small,  free  tip  maximum  f i r s t  mode.  ( f i r s t  absorber  at  tuned  l i t t l e  which  d i f f i c u l t .  is  respectively  with  a  damping.  arm  at  is  secondary  mechanism is  been  smooth  device  that  resonances  an  made  damping  than  deflection  e f f i c i e n t l y  of  for  Hz  to  the the  Because  and 5  The  a  second and  15  the  first  interaction  with  modes.  OPTIMAL  PROTOTYPE(1) For  absorber  are  are  (1979)[29].  hot  damping  separated  This  consideration  a  predominantly  resonances and  of  vibration  well  more  larger  important  an  designs.  called  designed  secondary  has  contact  and  Hunt  amplitudes  joint  absorption  spring  times  is  vibration  in  s t i l l ,  by  lap  hot-arm  vibration  many  This  location  new  surfaces  reviewed  magnified  i n i t i a l  riveted  its  [28],  absorber  amplitudes.  a  secondary  are  vibration  best  in  the  s l i d i n g  absorbers  makes  in  concentrated  secondary  are  damping  many  the  design  to  a  of  DYNAMIC  ABSORBER  FOR  THE  HOT-ARM  years  there  dynamic  vibratory  VIBRATION  has  been  vibration  system,  w i l l  considerable  absorbers reduce  that,  interest when  s i g n i f i c a n t l y  in added the  90  vibration  response.  Ormondroyd absorber, damper, mass  and  Den  which  Hartog  consists  attached  is  The c l a s s i c a l  to  subjected  an to  made  design  parameters  when  the  Snowdon  to  beams.  a l l  is  In  subjected  and  Hoppe  Warburton main It  is  to  excited  reference  important  The  [36]  a  random  which  references  fraction  of  i s given  the  i s  an  that  spring  the  above,  used  above  numerous  with  most  as  the  damped rods  main  and  system Jacquot  (1974)[35], parameters  white  studies  a  and  for  noise  extensively  represent  al  damping.  excitation.  absorber  force  et  of  such  the  absorber  some  and Campbell the  which  Randall  application  harmonic  viscous  optimum  systems,  of  and of  contains  mentioned  optimized by  for  e l a s t i c  Wirsching  by  excitation.  the  deterministic  investigated  1DOF s y s t e m  system  studies  (1982)[36]  mass,  charts  showed  (1973)[34],  system  following.  a  a  umdamped  continuous the  of  main  (1968)[33]  absorbers  (1928)[30,31]  harmonic  (1978)[32]  problem  a  PSD.  in  the  small  but  done  on  the  91  dynamic  vibration  The  basic  continuous for  strategy  e l a s t i c  vibration  in  continuous with  absorbers.  system  the  f i r s t  system  is  equivalent  f i r s t  of  designing  such mode  as  the  is  as  replaced  lumped  an  absorber  for  Prototyped)  a  hot-arm  follows:  with  parameters  a  1D0F  for  main  system  vibration  in  the  mode.  optimal found  absorber  parameters  for  the  1D0F  main  system  are  constructing  an  a n a l y t i c a l l y .  a n a l y t i c a l  results  are  implemented  in  absorber. a.  consider the  b.  5.2.1  the  test  the  LUMPED  WHITE  constructed  PARAMETERS  be  of  undamped  in  constructing  of  the  attached the  beam from  beam  vibration  expressed  absorber.  FOR  EXCITED  A BY  TRANSVERSELY A  RANDOM  RAIN  VIBRATING FORCE  WITH  PSD shows  approximated  energy  be  5.1  an  parameters  BEAM  NOISE  Figure and  limitations  absorber.  CANTILEVER  beam  practical  a  transversely  damped  absorber  for  vibration  the  kinetic  vibrating deflection  vibrating  in in  at in  the  energy  the the  x=L.  f i r s t f i r s t  cantilever The  f i r s t and  mode  the  mode. mode  lumped  at  can strain  The x  =  free L  can  as:  y(L,t)  =  (L)g,  (t).  (Note,  the  previous  (5.2.1 ) definitions  of  symbols  are  valid  unless  92  otherwise The  stated.)  effective  which  when  absorber,  mass,  vibrating  M  = M,/^?(L).  Similarly lumped of  the  t  n  point  kinetic  f i r s t  beam  e  of  It  a  lumped  attachment  energy  mode.  i s  is  as given  the  absorber,  2  effective  given  by:  mass, of  the  cantilever  by:  (5.2.2)  which  beam  r  s t i f f n e s s ,  when  gives  vibrating  placed the in  at  f £ »  for  the  point  K e  same the  strain  f i r s t  mode.  the  beam  of energy It  i s  a  as given  the by:  (5.2.3)  2  viscous  i s  attachment  = M,^ /!// ( L ) .  e f f  The  same  the  f ° the  effective  spring,  cantilever  K  in  the  at  the  beam  e f £  e  placed  gives  f f  M  damping  ratio,  7 j£f e  of  the  beam  i s  93  7  * 6/2*  eff  where  6  i s  vibration can  be  lumped with  the  in  logarithmic  the  f i r s t  substituted spectral  white  given  (5.2.4)  mode.  for  c u ,  density,  noise  PSD,  For  in S  for  decrement  e  of  response  equation ^ ,  for  3.4.19.  The  random  in  beam  the  rain  f i r s t  force  mode,  i s  by:  5.2.2  OPTIMAL EXCITED Having  cantilever  absorber lumped  ABSORBER  beam  'm'  PARAMETERS  the  system denotes  parameters with  The of  FOR A  WITH  as  system  shown  main  lumped  Figure  system  the  thus  subscript  ' m ' .  i s  the  SYSTEM  PSD  parameters, modelled  5.2,  and subscript  1DOF m a i n beam;  NOISE  c a n now b e  in  system  1D0F MAIN  WHITE  necessary  and absorber  system.  replaced  (5.2.5)  BY A RANDOM F O R C E  isolated  discrete  subscript  be  e  effective  = S o U T l M ] * , )/*?(L) •  2DOF  for ^y ff  c a l c u l a t i o n ,  the  vibration  the  given  where ' a '  the  subscript  the as  a  the  denotes effective  ' e f f  can  94  Introducing  the  non-dimensional mass  ratio  "I  - V  co = m 2  forced  K  frequency  tuning absorber  =  n  parameters:  M  /M a m  a  M  /M m' m ratio  ratio  f  damping  7  =  r co  =  system  damping ^  J  3  white The  non-dimensional  given R U , f  noise  ;  7 m  C  m  /2M co o  =  =  =  a  a  C / 2 M co m m m  PSD =  MS d i s p l a c e m e n t  "A>  /co a m  a  main  =  S„ m  of  the  main  mass,  R,  is  by:  ,7 ,7 ) a  m  -  (1/2TT)  /  Z  a  H(r)H(-r)  dr  (5.3.1)  MAIN SYSTEM  ABSORBER SYSTEM  Figure  5.2:  A  dynamic  vibration  1D0F  main  absorber.  system  with  an  attached  damped  95  where of  H(r)  the  is  main  dimensional  the mass  MS  dimensionless  complex  displacement,  given  displacement  of  the  frequency in  main  response  Appendix mass,  N,  E. is  The given  by:  N  =  2TTS <U R / K . For  for  the  3R/37  =  a  These be  the Q  m  given  t  two  with  of the  u  nonlinear  ,  7  following  the  optimality  p a r t i a l  conditions  derivatives:  mass,  Table  5.1  main  ratio which shows  system  S . M  increasing  The a.  (5.3.2)  equations,  simultaneously  PSD,  and  3R/3f = 0 .  frequency  undamped noise  are  = 0  main  p .  values  absorber  solved  ratio,  R  (5.3.2)  2  mm  and are the  given  to  in  obtain  Appendix the  dimensionless  optimal  excited response  by R  Q  absorber a  p  t  random  have  optimal  MS  respectively,  E,  7  to  damping  displacement a  -  D  p  r  t  ^opt  parameters force  decreases  with  of a  for  n  d  an  white  monotonically  96  Table main  5.1;  Optimal  system  absorber  excited  M  f  by  a  opt  parameters  random  7  for  force  a'opt  with  R  0.9926  0.0498  9.988  0.10  0.9315  0.1525  3.126  0.20  0.8740  0.2087  2.189  0.30  0.8249  0.2479  1 .772  0.40  0.7825  0.2782  1 .524  0.50  0.7454  0.3028  1 .354  0.60  0.7126  0.3234  1 .229  0.70  0.6835  0.3410  1 . 132  0.80  0.6573  0.3564  1 .054  0.90  0.6338  0.3699  0.990  1 .00  0.6124  0.3819  0.935  DAMPED V I B R A T I O N From  only  this were  previous  parameter  damped small  the  which  vibration weight  section, built  in  section can  absorber  weighs  on  a  the  absorbers  tested  is  apparent  chosen  to  a  white  noise  FOR THE P R O T O T Y P E p ) it  be  comparison  damped and  ABSORBER  undamped  of  1D0F PSD.  opt  0.01  5.2.3  an  that  freely.  mere weight  n  The pound,  of  a  Prototyped)  hot  sizes  is  the  existing  0.7  various  HOT-ARM  of  a arm. n  hot-arm  very In that are  97  reported. absorber  Figure  constructed  cantilever The  spring  absorber  a  steel  a  the  basic  tested.  It  design  of  contains  a  s l i d i n g - f r i c t i o n  and  spring  mechanism  plate.  s l i d i n g - f r i c t i o n  shows and  and  weight  s l i d i n g - f r i c t i o n on  5.3  was  The  damping  were a  damping  made  ceramic  of  mechanism  is  also  of  damped  weight,  a  mechanism.  steel  button  effectiveness  a  and  the  which  s l i d  employing  discussed  in  a the  following.  Figure  5.3:  vibration  Simplified  absorber  drawing  attached  to  the  of  the  hot-arm  f r i c t i o n t i p .  damped  98  5.2.3.1  Considerations  in  Selecting  the  Absorber  Damping  Mechanism In  the  parameters with  a  analytical the  dashpot.  My  +  a  discussion  =  +  the  Ky  plus  is  damping  damping cycle  no  c  s l i d i n g  or  the  the  is  to  equal and  the  excited  My  +  =  no  the  1D0F by P  damping  system  a  The  and and  with  f r i c t i o n  sign is  of  the other  f r i c t i o n  with  is  amount  equivalent  energy  When is  but  displacement  P  damping  is  per  the  solution viscous  work  viscous smaller  than  When  cycle  with  per  damping  of  coordinate  system force:  if  of  dissipated.  dissipation  harmonic  velocity  equivalent  of  f(y)  approximately  method  an  force,  the  nonlinear  usual  discrete  sincot.  the  4F/7TCJY.  energy  to  of  The  same  The  a  mechanism.  advantages  motion  damping  motion.  F  constructed  force:  the  damping  sincot.  gives  dashpot  Ky  sign on  f r i c t i o n  Consider  cy  harmonic  minus  occurs  than  modelled  sincut,  Y  proportional  +  P  =  which of  bigger is  y  the  parameter  a  small,  sinosoidal, replaces  by  F r i c t i o n  is  the  absorber  was  actual  damping  discrete  dependent  coordinate.  on  the  s l i d i n g - f r i c t i o n  1D0F  =  optimal  damping.  a  excited  ±F,  the  absorber  f(y)  where  However  is  of  the  mechansism  s l i d i n g - f r i c t i o n  Consider damping  damping  a  disadvantages of  for  uses  following  kinds  absorber  viscous  absorber  study  P  F is  motion  [37], a  viscous  99  With  a  cycle  true of  motion  velocity system since  the  more  the  viscous  force  large.  slight  the  viscous  damping  type  in  the  absorber  [26]  of  radiation  w i l l  well  out-gassing.  The  absorber  mass  sloshing  is  good  the  viscous about  the  container  simple  damping  has  some  velocity  the  damping  but  to  in  velocity in  the  cannot  is  below with  a  the  cyclotron  has  not  materials,  a  viscous heat  and  vacuum  and  been  found.  for  coating  constrained-layer-damping  contaminate  d i s s i p a t i o n is  to  spring,  alternative  vacuum-sealing Another  as  the  quickly  resistant  for  absorber  increase  damping  preferred  v i s c o e l a s t i c or  co-ordinate  coordinate.  is  is  system  coordinate  adjust  the  vibratory  damped  to  viscous  velocity  which  as  a  when  per  of  increase  displacement  damping  spring  the  f r i c t i o n  construction  Energy-dissipating  damped  proportional  force  square  corresponding  non-contaminating  simple  the  corresponding  the  dissipation  displacement  a  is  the  of  to  a  the  example,  mechanism  radiation,  of  the  viscous  increase  A  yet  For  than  which  With  energy  viscously  without  dissipated,  optimal  A  stable  large  the  proportional  magnitude  co-ordinate.  grow  is  grow  energy  dashpot,  co-ordinate.  is  cannot  viscous  can  deteriorate  the  cyclotron  damping  produced  in  a  f l u i d - f i l l e d  if is  the  in  vacuum by  by the  container  complexity  of  justifed. mechanism  relationship produced  to by  whose  the  the  square  impacting  energy of of  the the  100  absorber impact tip  mass  damping  barely  absorber spring  main  as  In  the  It  excited  by  energy-dissipating and  multi-layer  a  at  simple  construction  The  depends  adjustment vibration  and  of  the  levels  f r i c t i o n  vibration the 7_  damped  care  unit.  The to  i s  by  absorber, for  R  Warburton the small  a  riveted  There  mechanism  the  the  from  be  to  5.4 R  R  deviations  was  which and  is yet  Q  the  shows p  t  damping  for  and  adjusted  operating  (1982)[40].  response  are  construction  must  sensitive  in  of  spring  found.  in  Figure  a  the  absorber  s l i d i n g - f r i c t i o n  unit  for  include  mechanism  been  taken  similar  increase  not  damper  investigation  damping  the  amplitudes.  reported  appreciably  has  damping  percent  A  a  [39].  spring. The  no  becomes  force  the  the  has  absorber  mechanisms  damping.  of  on  it  was  of  Lanchester  harmonic  s l i d i n g - f r i c t i o n  r e l i a b i l i t y  mechanism  since  the  weight  damped  of  hot-arm  spring  absorber  j o i n t - s l i p p i n g  point.  to  of  viscously  a  cantilever  superior in  that  damping  on  this  the  the  with  reduce  only),  root-slipping  variations  to  absorber  when  steady  absorber  discontinued  the  a  An absorber  balance  known  energy-dissipating  endless  to  motion  i s  possible  the  limit  as  [38]. a b i l i t y  when  horizontal  damper.  Other  Its  enough  efficient  mass  mass  tested.  strong  ( i . e .  not  main  increases  mass.  Lanchester  the  was  vibration  made  is  on  condition  magnitudes the a  at  contours  range  For  of  a  of  f  dashpot  does  not  increase  from  the  optimal  101  Figure mass .  7  5.4:  upon  Dependence absorber  within  a  of  the  parameters,  contour  the  MS f  response  displacement and  7  .  For  exceeds  of  the  values the  of  main f  and  optimal  MS  a, displacement values,  f-f  by  less 7  A  - 7  A  .  than  O  P  T  ,  stated  *=0- ' 1  percentage.  7 -0.01. M  '+'  Ref.[40]  optimal  102  absorber  parameters.  allowable  errors  5.2.3.2  The  for  M  K  7  m  lumped  m  Damped  Vibration  =  M,/WU  =  1/(2.9457)  =  2  of  the  f i r s t  model,  damping  the  Absorbers  on  a  are  0.115  P r o t o t y p e d )  mode,  given  l b * s  2  evaluated  hot-arm from  i t s  parameters  are  solving  two  by:  / i n  = M,co /V/ (L) 2  =  1(32) /(2.9457)  *  £/2TT  <*  0.003.  2  this  small  not  much  different  nonlinear  f  of  the  For  to  f r i c t i o n  smaller.  parameters in  element  2  m  for  Hot-arm  vibration  f i n i t e  be  Testing  Prototype(1) The  may  But  1.0)  opt  a  n  ,  7  no  116  l b / i n  the  optimal  from  the  equations  shows  =  2  of  absorber  case  Appendix  difference  7 E  for  m  =  for  three  0; a  range  of  decimal  the u  (0.01  places  in  give  an  Vopf  d  Table  5.2  and  the  experiments  account  of  each  u,  the  the  pressure  s t a i n l e s s  absorber  adjusted  was  made  as  was  force  steel  was  the  small  of  the as  with  f i r s t  between  plate  u n t i l  accompanying  the  four tuned  the  values to  f  Q  p  ceramic  f r i c t i o n  RMS d e f l e c t i o n  possible.  figures  damping of  the  of  t  n.  and  button  For then and  mechanism hot-arm  t i p  103  Table were  5.2:  Test  attached  to  M  Fig.  l b * s  No.  results the  the  damped  Prototyped)  f  M  a 2  for  absorbers  hot-arm  t  opt  7  which  tip.  a  N  4.  R  opt  / i n  in  5.5  0.00  5. 6a  0.57*10-  2  0.05  0.965  free  5.6b  0.57*10-  2  0.05  0.965  0 . 109  5.7a  1.14*10"  2  0.10  0.932  free  5.7b  1.14*10"  2  0.10  0.932  0.152  5.8a  1 .71*10"  2  0.15  0.903  free  5.8b  1.71*10"  2  0.15  0.903  0 . 183  N/A  N/A  N/A  2  15*10"  9  N/A  14*10"  9  4.49  4 .7*10"  9  N/A  7 .0*10"  9  3 .1*10-  9  3 .9*10"  9  5.25  3.14 N/A  noi se  2.26  where, 7  :  3  the  absorber are R  opt  mass N  :  not :  damping the  t  *  i  e  from the  with  an  free  :  from  the  analytical  one but  attached  MS peak two  are  the  the  desired  analytical  optimal  study;  they  values.  dimensionless  alone  the  values  ratio  measured  measured  5Hz), system  numerical  tip in  MS  displacement  of  the  main  study. deflection the  peaks  for  spectrum  near  u  m  for  vibration at  CJ^  the  for main  near the  main  system  absorber.  f r i c t i o n  damping  mechanism  is  disengaged  but  104 it  takes  no  account  of  the  structural  damping  in  the  from  the  absorber. noise  :  the  vibration N/A  :  small  measuring  not  5.5  deflection  hot-arm  when  panel.  It  Figure  spectrum  of  accounting different two  and f,  <  in  the  f  for  In  2  f  the  hot-arm the  proportion  peaks  7  t i p  enviroment.  the  in  hot-arm and  to  in  the  absorber  to  the  mass.  co  of  the  the  RMS  the  and  Let peaks,  mass  moved  hot-arm  and  Both  motion  was  of They  two  f r i c t i o n  motion  (not  5Hz).  d i r e c t i o n s .  When  of  t i p .  (=  m  the  r e l a t i v e  the  absorbers  hot-arm  mode  2  CJ^  undamped  dissipation  relative  at  absorber  opposite  energy  this  f  through  when  below  and  hot-arm  show  the  one  the  5.8a  frequencies  contributed  of  peak  damping)  to  and  centre  moved  rate  and  deflection  above  of  flowing  prominent  attached  mode  and  was  structural  undamped  attenuation  2  test  f,  between  damping  was  increased  in  the  two  peaks  attenuated. With  f  5.7a  direction  modes  added,  were  f,  mass  and  2  originating  spectrum  coolant  5.6a,  one the  same  absorber  and  RMS  a  were  be  2  the  the  jz's  the  shows  the  peaks  f .  equipment  shows  hot-arm.  show  noise  applicable.  Figure tip  background  as  mode  of  shown was  could  the in  peaks Figures  s t r i c t l y be  absorbers, and  5.6a,  relative  increased  amplification 5.7a  motion,  further  M  increased  and  5.8a.  with  before  of  gave the  f  2  Since  the  increased  u,  causing  over  105  0  5  10  15  20  FREQUENCY <HZ> Figure  5.5:  Prototyped)  RMS  spectrum  hot-arm  without  of an  the  tip  absorber.  deflection  for  the  1C6  0  5  10  15  FREQUENCY Figure  5.6a:  P r o t o t y p e d ) the  RMS hot-arm  spectrum with  an  of  the  undamped  20  <HZ>  t i p  d e f l e c t i o n  absorber  for  the  attached  to  tip.(M=0.05)  FREQUENCY Figure  5.6b:  RMS  P r o t o t y p e d ) attached  to  spectrum  hot-arm the  of  with  tip.(M=0.05)  the an  t i p  <HZ> d e f l e c t i o n  adjusted  damped  for  the  absorber  107  FREQUENCY Figure  Prototyped)  the  RMS  5.7a:  hot-arm  spectrum with  an  of  the  undamped  <HZ>  t i p  deflection  absorber  for  attached  the to  tip.(u=0.10)  FREQUENCY  Figure  5.7b:  RMS s p e c t r u m  Prototyped) attached  to  hot-arm the  of  with  tip.(M=0.l0)  the an  t i p  <HZ)  deflection  adjusted  damped  for  the  absorber  108  FREQUENCY CHZ) Figure  5.8a:  RMS  P r o t o t y p e d ) the  spectrum  hot-arm  of  with  the an  t i p  d e f l e c t i o n  undamped  absorber  for  attached  the to  tip.(M=0.15)  FREQUENCY <HZ> Figure  5.8b:  P r o t o t y p e d ) attached  to  RMS  spectrum  hot-arm the  t i p .  with  (/z=0.1 5)  of an  the  tip  adjusted  d e f l e c t i o n damped  for  the  absorber  109  damping in  the  in  the  f r i c t i o n  d i s s i p a t i o n effect the  of  of  mostly  the  entire  5.7b  5.2.  and  For  background  noise  measuring  equipment  vibration  was  Over  addition  the  form  the  steady  should transient  of  the  of a  a  state  shorten  vibrations.  _  2  of  energy  The  overall  increasing  MS  ,  s l i d i n g  as  shown  tip  5.8b  the  enviroment;  by  deflections  Figure from  was  7  shows  vibration  the  hot-arm  s i g n i f i c a n t l y . vibration  was  attenuated  amount  of  secondary  absorber.  In  addition  random the  the  * 10  test  hot-arm  damped  and  rate  peaks  originating  and  small  the  stop  system.  both  1.71  attenuated  a l l ,  the  of  =  would  optimally  5.8b  n  reduce  vibratory  and  n  damping  and  attenuation  5.6b,  Table  Over  mechanism  increasing  further  Figures in  absorber.  time  vibration, needed  a to  damping  by in  to  reducing  damped  absorber  attenuate  large  6.  TURBULENCE The  it  can  G E N E R A T I O N IN  concensus be of  irregular  shape  the  turbulence  the  in  a  in  roll-bond  P r o t o t y p e d )  did  hot-arm used.  from  a  roll-bond  with  coolant  A  flow  the not  y i e l d  channel  from  trace  the  same  it  was  soldered  form  the  one-third-length  to  time,  the  hot-arm  and  deflection mode  at  two  panels the  of  looping  coolant  the  several  were  hot-arm  flow  rates  roll-bond of  the  on  pattern to  a  as  pipe the  panel.  the  water  let  through.  were  110  N,  for  vibration  recorded  in  the hot-arm  was  cut  size  but  pipe was  as bent  roll-bond  onto  t i p ,  of  panel  i n s t a l l e d was  section  of  copper-plated  pipe-type  brief  answers.  flatten  in  these  generation  same  The  The  A  cross  types  constructed.  flow  avoid  levels  s l i g h t l y  figure  plate  was  a  the  and  6.2  would  of  one.  conclusive  panel  and  subject  nebulous  different  Another  made  The  vibration  two  wall  turbulence  any  path,  coolant-channel  coolant-channel  in  panel  to  that  flow  c h a r a c t e r i s t i c s .  one-third-length  panel.  a  FLOW  is  channel the  panel  the  when  of  is  contribution of  the  panel.  generating  the  in the  6.1)  hot-arm  compared  were  to  a  panel  panels  shown  water  shape  study  Figure  on  WATER  excitation  bends  roll-bond  of  irregular  The  in  coolant  determine  the  roughness  a  turbulence  to  by  in  designer  potential study  flow,  HOT-ARM COOLANT  turbulent-flow  by  (shown  section  prudent  the  aggravated  branching  cross  on  THE  aluminum One  at  a  Prototyped)  in  Table  The the 6.1.  MS f i r s t  •0.3'  Figure  6.1:  roll-bond  Cross  hot-arm  sectional  view  of  the  coolant  channel  in  a  coolant  channel  in  a  panel.  CD •0.3'  Figure  6.2:  pipe-type  Cross hot-arm  sectional panel.  view  of  the  1 12  Table  6.1:  roll-bond  type  of  A  comparison  panel  and  of  pipe-type  panel  water  hot-arm  vibration  for  the  panel.  flow  rate  N  l i t r e s / m i n  i n  pipe-type  16  26*10"  9  pipe-type  35  43*10"  9  roll-bond  18  10*10  roll-bond  36  35*10"  From  Table  panel  is  better.  It  d i f f i c u l t  is  differentiate  6.1  two  it  The to  make  very  test  basis  d i f f i c u l t  design  is  which  not  levels  specialized it  was  to  vibration  accurate  similar  f a c i l i t y  minimizes  of  possible  and  especially  hot-arm  the  determine  are  vibration  vibratory  determine  to  very  -  9  9  which s i m i l a r .  measurement  systems. without  optimal  vibration.  2  Without  to a  analytical  coolant  flow  CONCLUSIONS  The a  wide-band  simple  random  rain  analytical  excitation  in  discussion  force  model  a  Triumf  on  the  of  with the  white  noise  actual  hot-arm  as  PSD  is  flow-induced  shown  by  the  boundary-layer-turbulence  wall-pressure. The  analyses  that  beam  bending  a  uniform  material  has  v i b r a t i o n . their  with  nearest  minimal Thus  to  the  contribution  sandwich  u t i l i z a t i o n  cantilever  in  beams  of  the  beam  a  uniform  beam  model  neutral  axis  minimizing  are  most  material  show  the  tip  e f f i c i e n t  to  of  minimize  in tip  v i b r a t i o n . The  analyses  that  when  section  to  no-core  with  the  only  fit  inside  unit  vibration  for  constraint  frequency  white  PSD,  The  minimum-weight  with  light  type  excitation and  t i p  amplitude,  tapers  with  linearly  to  minimum-weight light  gives  the  harmonic the  of  a  in  with  zero  design  hysteretic  the of  a  damping,  113  model  the  beam  the  smallest force  point  tip  at  the  force  with mode.  cantilever  subjected  with  damped  fundamental  sandwich  show cross  l i g h t l y  rain  to  force  a  whose  beam  standard  at  constraint  cover-plates  at  beam  point  random  harmonic  frequency,  one  for the  damping,  unit  fundamental  is  square,  vibration  design  of  is  unit and  hysteretic  t i p  The  for  unit beam  the  fundamental noise  a  square both  cantilever  the  free  on  the  thickness  t i p . sandwich  cantilever  subjected  to  a  beam random  1 14  rain  force  mean  with  square  fundamental heavier 6.  7.  The  with for  is  cover-plates  t i p  one and  with  stiffened  smaller  cantilever  beam  vibration  models,  the  unit  harmonic  frequency  and  The  materials  beam  The  random  cantilever rain  beam  force  cantilever  models,  the  The a  and  proposed  previous  both  point  The  strongback  t i p  but the  a  is  be  t i p .  The  to  damped  increases  the  e f f e c t i v e  increases  the  damping  The  predominantly  vibration absorber  of  a  tuned  with  strongback  mass  f i r s t  an  for  can  at  noise  PSD. p  case  give  2  of  the  both  excitation  the  fundamental  noise  for  PSD.  the  white  is  to  dual  entire  PSD.  near  the  be  attached  purpose; t i p  main  and  it it  structure.  narrow-band  attenuated  unit  noise  weight  hot-arm  1DOF  give  of  in  the  v  c h a r a c t e r i s t i c s  the  the  effective  fundamental  in  serves  be  give  described  absorber  mode,  E  parameter  design  of  end.  the  design  the  at  are  excitation  white  lighter  capacity  hot-arm to  force  absorber  which fixed  the  for  force  vibration  the  PSD.  point  made  in  density  damping  design  the  both  white  for  noise  designs, and  at  mass  encompasses  force  dynamic  with  vibration  rain  replacement section  force  larger  harmonic  random  force  on  modulus  for  point  white  the  e l a s t i c  larger  beam  vibration  near  vibration  with  minimum-weight  harmonic  to  with  with  smaller  unit  rain  beam  materials  constraint  d e f l e c t i o n ,  larger  frequency  10.  the  PSD,  with  random  9.  mode,  noise  materials  smaller  8.  t i p  near  beam  white  by  random a  damped  system.  The  115  11.  optimal  absorber  excited  by  heavier  absorber  a  parameters  random mass  the  t i p  The  contributions  coolant irregular a  those  for  force  with  white  w i l l  give  bigger  main  noise  system  PSD.  The  attenuation  of  vibration.  flow  nebulous  to  turbulence  path,  shape  roll-bond  too  are  of  panel to  the and  study.  branches  generation in  coolant  coolant-channel roughness  on  the  cross  by  bends  flow  path,  section  channel  wall  in  in are  REFERENCES  1.  Brackhaus, megawatts thesis,  of  R.W.  o s c i l l a t i o n  of  Royal  generation for  the  Department  Canada,  Gregory,  Proc.  "The  RF power  Physics  Columbia, 2.  K . H . ,  and  Triumf  of  the  control  of  Cyclotron",  University  1.5  Ph.  of  D.  B r i t i s h  1975. and  Paidoussis,  tubular  M . P . ,  cantilevers  Society,  London,  "Unstable  conveying  1966,  v o l .  f l u i d " ,  293,  pp.  513-542. 3.  Crandall, Cambridge,  4.  Crandall, Mechanical  S . 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Journal,  and  Quarterly  pp.535-539. design  Solids  in  1972,  of  Structures,  optimization  AIAA  for  problems",  R.T.,  "Structural  enviroment",  v o l .  random 10,  No.  551-553.  S . S . ,  design  v o l .  to  315-318.  design  Shield,  1968,  4,  ZAMM,  Quarterly  solutions  v o l .  structures",  v o l .  pp.  structural  multi-purpose  Nigam,  subject  design  loading",  1971,  Mathematics,  W.  vibration  25.  periodic  Plaut,  Prager,  loads",  structural  v o l .  of 23.  "Optimal  Mathematics,  dynamic  structures  303-309.  deflection  22.  of  harmonically-varying  pp.  Plaut,  design  "Multiobjective  with  process",  optimization  uncertain  AIAA  Journal,  parameters 1984,  in and  v o l .  22,  structural stochastic  No.  11,  pp.  1670-1678. 26.  Torvik, Control,  27.  P . J . ,  editor,  1980,  ASME,  Applied  Mechanics  Jones,  D . I . G . ,  applications", Control, pp.  P . J .  27-51.  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C ,  Systems,  Wiley,  Jacquot,  R.G.  absorber", pp. 35.  Wirsching,  v o l .  Vibration New  Hoppe,  for 103,  and  York,  Engrg.  pp.  P.H.  under  absorber", 2,  D.M.  absorbers 198V,  and  J .  III,  and  linear pp.  Shock  Taylor, damped  D . L . ,  systems",  908-913.  in  Damped  Mechanical  1968. D . L . ,  Mech.  "Optimal  D i v . ,  random  ASCE,  1973,  vibration v o l .  99,  612-616.  response  36.  Halsted  Campbell,  random  Earthquake  G.W.,  excitation Engrg.  "Minimal using  Struct.  structural  the  Dyn.  ,  vibration 1974,  v o l .  parameters  for  303-312.  Warburton, various  and  G . B . ,  "Optimum  combinations  parameters",  Earthquake  of  absorber response  Engrg.  Struct.  and Dyn.,  excitation 1982,  v o l .  120  10, 37.  pp.  Den  381-401.  Hartog,  "Systems  c h a r a c t e r i s t i c s " , edn., 38.  impact J .  J . P . ,  McGraw-Hill,  Thomas,  M.D.  and  damper  Mech.  New  a  variable  Mechanical York,  Sadek,  with  Eng.  with  1956,  M.M.,  pp.  "The  1974,  non-linear  Vibrations,  4  th.  of  the  335-379.  effectiveness  spring-supported  Science,  or  a u x i l i a r y  V o l .  16,  mass",  No.  2,  pp.  109-116. 39.  Hunt,  editor,  "Acceleration  vibration  absorbers",  Dynamic  Vibration  Mechanical  Engineering  Publication  L t d . ,  pp. 40.  J . B . ,  dampers  or  impact  Absorbers,  London,  1979,  87-97.  Warburton, Dynamic  G . B . ,  Vibration  Engineering  "A  design  Isolation  Publications  procedure and  L t d . ,  for  absorber",  Absorption,  Mechanical  Worthing,  England,  P h i l l i p s ,  R . S . ,  1982,  pp.59-69. 41.  James,  H.M.,  Theory  of  Nichols,  N.B.  Servomechanisms,  and  M.I.T.  Press,  1964.  editors,  APPENDIX  Reynold's  number  of  the  water  A  flow  in  the  Prototype(1)  hot-arm. q  =  flow  rate  of  flow  area  of  of  coolant  water  in  a  P r o t o t y p e d )  hot-arm  panel A  =  f c  n  =  number  U  =  mean  d  t  p M  w  a  tubes  speed  of  diameter  =  density  of  =  absolute  20 =  =  i n  2  w  =  1.94  =  2.36*10- .  = =  The  flow  the  in  a  tube  s l i g h t l y  flattened  tube  of  of  the  water  water  flow  2  7 0.5  U R  7.5*10"  =  v  for  panel  / s  f c  P u  3  water  v i s c o s i t y  =  i n  a  the  water  q  d  panel  average  R e y n o l d ' s No.  n  in  =  =  t  in  the  R  A  tube  s l u g / f t 5  q/7A U  in  d  t  p  w  =  t  /  value  M  w of  40  3  l b * s / f t  2  i n / s ( U / 1 2 ) ( d / l 2 ) p  =  t  R  indicates  w  that  /  M  y  ,  =  this  121  1.1*10" flow  is  a  turbulent  one.  APPENDIX  The  cross  model  sectional  of  P r o t o t y p e d )  The  P r o t o t y p e d )  vibrating =  0  of  properties  inch the  f i n i t e and  f i n i t e  t i p  hot-arm  x  =  element  the  x  is  finite  treated  cantilever 122.75  beam  Sect. No.  for  element  beam  hot-arm  element at  B  beam  inches.  are  given  as with  transversely its  Sectional  in  the  A  2  a  root  at  x  properties  following  I  table.  P  in  in  1  122.75  104.75  3.660  0.843  0.590*10-*  2  104.75  75.00  9.756  4.256  0.387*10-3  3  75.00  45.00  23.460  8.883  0.315*10-  4  45.00  1 1 .93  28.960  12.722  0.305*10-3  5  1 1 .93  0.00  36.029  13.570  0.297*10-3  i n  in"  2  l b * s  2  / i n  f  t  3  where, x,,  x  2  :  x  A  :  cross  I  :  moment  p  :  mass  A  lump  122.75  coordinates sectional of  per  two  ends  of  a  beam  element  area  inertia unit  for  of  cross  section  volume  weight  of  5  pounds  inches  to  account  is for  also the  122  attached hot-arm  to  the  t i p  accessories.  at  x  =  APPENDIX  The  effect  of  adding  which  is  subjected  beam free  lump to  mass a  to  a  harmonic  uniform point  cantilever  force  at  the  t i p Consider  beam  a  subjected  concentrated in  a  C  terms  y(x,t)  of  =  to  a  moment the  J  vibrating  harmonic  M(a,t)  normal  Lg.(t)0.(x) j  where  transversely  uniform  point  applied  force  at  x=a.  cantilever  F(a,t)  Assume  and  a  a  solution  modes  f  3  the  generalized  coordinate  gj  must  satisfy  the  equation g j (t) and  +  oj?gj(t)  mL  is  If  a  shown  in  given  by: =  If  is  [F(a,t)*j(a)  generalized  lump  mass  Figure  F(a,t)  - M ° y ( a , t ) small  adequate.  The  one  frequency  is  2  *  of  4.1,  a  w  M°  the  =  mass  M°  is  the  =  +  M(a,t)*J(a)]/mL  of  the  attached  force  - M ° I g . <6.  given  to  exerted  the  by  M°  on  at  the  x=a,  as  beam  is  a  one  approximation  mode  analysis  is  of  the  fundamental  is  subjected  by:  co /{l + (M°/mL)0 (a)}. 2  If  2  the  beam  harmonic  point  analysis  gives  y(x,t)  =* P  and  force the  sin(<jt)<6  beam  1  attached of  P  mass  sin(cot)  at  x  deflection  (L)<)>,  (x)H,  (co) ,  where Hi (co) 1  beam  (a)  modification mode  beam.  =  mL[co -{ 1 + (M°/mL)c4 2  2  ( a ) }co 123  2  +  /cco co]. 2  =  L,  one  to  a  mode  124  If  the  deflection is  given  for  point  l i g h t l y  force  damped  is  beam  Psin(co t),  the  n  with  viscous  resonant  parameter  c  by: ( L ) 0 , (x)v/{l + ( M / m L ) 0 ( a ) }  P 0 , y(x)  harmonic  o  2  —  mLccj, The in  beam  deflections  Table  The  of  adding  viscous  subjected If  to  a  the  moment =  lumped to  harmonic  lumped  cantilever  a  damping  a  the  M(a,t)  various  other  situations  a  uniform  point  rotational  beam  exerted  at by  x=a the  rotational  cantilever  force  at  the  stiffener and  of  x=a+A  stiffener  K ° { y ' ( a + A , t ) - y ' ( a , t ) }  =  stiffener  as is  beam  free K°  If  the  stiffener  is  adequate.  The CJ  2  One  =  P  mode  small  shown given  approximation must  - ( K ° / m L ) {cV, ( a + A j - c V , ( a )  «  - ( K ° A  2  / m L ) 0 7  a  lumped and  sin(cot)  the at  ( a ) g  frequency  2  2  2  2  of  x=L,  «  which  is  attached  in  to  figure  4.2,  by:  i a  gives  one the  mode  analysis  equation  that  s a t i s f y } g, 2  . the  stiffened  is a  damping  C°  is  subjected  to  one  analysis  mode  deflection y(x,t)  a  beam  is  given  by:  (a).  viscous beam  1  with  K°{Lg.0!(a+A)-Zg.*!(a)}.  modification  *  £0 + ( K ° A / m L ) c 4 7  p a r a l l e l  a  coordinate  fundamental  If  of  is  generalized  gi+u?g,  given  t i p  is  i  the  are  4.1.  effect  lumped  for  P s i n t u t ) * , ( L ) 0 , ( x ) H , ( u ) ,  a  added  harmonic  to point  gives  the  K°  in  force beam  125  where Hi'(u)  «  mL[{co +  + If  (K°A /mL)0l (a)}-£j } 2  2  the  2  / {cu + (C°A /mL)0l 2  2  harmonic  beam  deflection  y(x)  =  is  point  given  2  2  force  2  The  beam  Table  2  deflections  4.1.  for  is  Psin(to t),  the  n  resonant  by:  P0, ( L ) 0 , m L { c e o + ( C ° A / m L ) 01  ( a ) }co].  2  (x)  ( a ) } v / { ^ + ( K ° A / m L ) 01 2  various  other  2  2  (a)}  situations  are  l i s t e d  APPENDIX  Experimental weights  at  The  study  the  let  through  arm  panel  were of  N,  at  a  at  block No.  of  placing has  34  on flow  Hz  of  the  steel at  fundamental  t i p  and  blocks  of  of  of  twos.  the  of  f  At  a  was  water  was  ground  inlet  weighing  frequency, in  t i p  deflection  between  each  increments  the  an e x i s t i n g  10 p s i  vibration  at  litres/min  blocks  the  for  60  of  blocks  MS  hot-arm  drop  placing  hot-arm.  steel  of  t i p ,  recorded n  of  the  deflection  f  effect  Prototyped)  pressure  A total  t i p  a  Prototyped)  two b l o c k s  were  that  the  A combined  a  placed  square  of  hot-arm  experimented.  outlet.  t i p  effect  Prototyped)  on  D  and  1.27  kg  increments  and the  mean  fundamental  mode,  • N in  block No.  2  f  n  Hz  N in  2  18  3.900  11.9*10-  20  3.800  12.8*10-  22  3.700  17.3*10"  24  3.600  10. 1 * 1o-  9  26  3.500  10.9*10-  11.2*10-  9  28  3.450  9.3*10"  4.250  5.5*10-  9  30  3.350  9.1 * 1 0 ~  14  4 . 125  6.3*10"  9  32  3.300  8.9*10-  16  4.000  5.7*10-  9  34  3.250  7.2*10"  0  5.350  10.8*10-  2  5 . 100  9.2*10-  4  4.900  8.9*10-  9  6  4.700  10.8*10-  9  8  4.550  10.6*10-  10  4.375  12  9  9  126  9  9  9  9  9  9  9  9  9  APPENDIX  The  optimum  excited  by  absorber a  random  Consider  a  damped  absorber  random  force  parameters found  with  which  with  which  main  the  white  1D0F  noise  PSD  system  and  main  noise  minimizes  a  white  10DF  non-dimensional  main  H(r)  force  for  this  mass  is  main  system  an  attached  excited  by  PSD. The  optimal  main  displacement  mass  a  absorber are  below. The  for  parameters  damped of  E  =  mass  complex  displacement  y K / P •* m m  =  is  frequency  given  response  function  by:  (A+/B)/(C+/D)  where A  =  f  - r  B  =  2  C  =  ( f  D  =  27 rfd-r  2  7  a  2  2  f r  - r  2  ) ( 1 - r 2  2  ) - M f  r  2  2  7  2  2  Ul  MS d i s p l a c e m e n t  of  the  main  mass  is  by:  R U , f , 7 The  7 f r a m r ( f - r )  2  non-dimensional  given  - 4  -Mr )+27  a  The  2  a  , 7  m  )  =  (1/2TT)  integral  was  evaluated  again  recorded.  The  on  UBC  reference  by [4]  be  program  MTS  by  was  that  "Reduce"  network.  calculus  of  residue  and  to  y i e l d :  [41]  127  [36].  no e r r o r s  evaluated  called  computer  dr.  Warburton  certain  integrand  algebra  integrated  H(r)H(-r)  reported  to  symbolic the  /  with  the  which The  integral  is  It  was  have  been  a i d  of  available  integrand tables  a  found  was in  128  R U  ' ' a' m f  7  7  )  n ,y ),  KM,f,7 ,7 )/L(M,f  =  a  a  m  m  where L(M,f ,7 ,7 ) = 4[Mf7| 7 7 {l"2f + +  a  2  m  a  m  f  f  t  (l+/i)  2  + 4f 7 fa 0 M ) + 4 f 7 a7mT J 1+f ( 1 + M ) ]+4f 7 2  +  2  2  2  =  = 7 [ l - f ( 2 +M ) + f ( l + M ) ] + M f 7  KM,f,7 ,7 ) a  2  2  3  +  a  m  +  4f7 7 [l f d M)]+4f 7 7 2  +  2  +  2  m  m  } + Mf m 7 m] 3  2  4f 7|(l M) 2  +  2  a  The o p t i m a l c o n d i t i o n s w h i c h  m i n i m i z e s t h e MS  displacement  o f t h e m a i n mass a r e g i v e n b y : 3R/3f = 0 =  2 +  7 (l+M) f +[37 (l+M) +M7 2  47 7 (l M)]f 2  2  +  3  2  2  m  7 =  4f (l+y)7«+8f 7 2  a  +[ 4  2 7  These p a r t i a l T h e s e two with a  m  (1+M)7 a 3  3  m  f (l+<.)-1+f ( 2 + A t ) " f " ( l + ^ ) a  2  d e r i v a t i v e s were  non-linear  standard  computer  2  37|-2Mf 7 7 " M f 3  m  a  a l s o checked with  equations  were s o l v e d  solution  routine  fl  7  2  "Reduce".  simultaneously for f ^ opt  and  a"opt* For t h e s p e c i a l case of 7  m  = 0 , simple s o l u t i o n s a r e as  follows: f  2 +  = 0  7 a  =  7  2  a  2 7 a 7 ui f M + 4 7 fa (1+M) ] f + 7M47!(1+M)-(2 + ^) ] f - 2 a7 mf - 7a a a a  3R/3  5  m  7 a  opt  7 R  =  2 o  p  t  (1 ^/2) /(1 M) +  2  +  = M(1+3M/4)/[4(1+M)(1+/I/2)]  . = (1/M)^[(1+3M/4)/(1+M)]2  

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