﻿Vibration 
  of 
  " 
  Clamped-Directed 
  " 
  Bars. 
  459 
  

  

  assume 
  for 
  the 
  first 
  approximation 
  to 
  the 
  vibration 
  curve 
  

  

  y=y,{Z[:,llf-2{^liy\, 
  .... 
  (16) 
  

  

  which, 
  at 
  the 
  clamped 
  end, 
  satisfies 
  y 
  =-^|' 
  — 
  0, 
  and, 
  at 
  the 
  

   directed 
  end, 
  y 
  = 
  yi^ 
  y; 
  = 
  0. 
  

  

  Let 
  m 
  = 
  the 
  mass 
  concentrated 
  at 
  the 
  end, 
  

  

  M 
  = 
  the 
  couple 
  required 
  to 
  maintain 
  the 
  direction 
  at 
  

   that 
  end. 
  

  

  Then 
  the 
  ordinary 
  approximate 
  theory 
  leads 
  to 
  

  

  El4^ 
  =.^pw'^{\j=[z-x)dz-my^{l-x) 
  + 
  lsL 
  (17) 
  

  

  Inserting 
  the 
  value 
  of 
  yz 
  from 
  (16) 
  and 
  performing 
  the 
  

   integrations, 
  

  

  -EI^ 
  = 
  Wi(^^V- 
  j^?^^+ 
  120 
  F 
  - 
  420 
  i 
  ) 
  

  

  the 
  constants 
  of 
  integration 
  and 
  the 
  value 
  of 
  M 
  ha^dng 
  been 
  

   determined 
  by 
  the 
  end 
  conditions. 
  This 
  expression 
  is 
  the 
  

   second 
  approximation 
  to 
  the 
  vibration 
  type, 
  and 
  is 
  often 
  

   sufficiently 
  accurate. 
  

   Putting 
  X 
  = 
  1 
  we 
  have 
  

  

  EI 
  

   ^■'^ 
  = 
  1, 
  1 
  • 
  • 
  • 
  • 
  • 
  (19) 
  

  

  420^""^*+ 
  12'"^' 
  

  

  § 
  14. 
  Proceeding 
  to 
  a 
  still 
  closer 
  approximation 
  we 
  can 
  

   insert 
  in 
  (17) 
  the 
  value 
  of 
  y^ 
  given 
  by 
  (18) 
  and 
  again 
  

   integrate. 
  In 
  the 
  resulting 
  expression 
  for 
  y, 
  we 
  can 
  again 
  

   = 
  Z 
  and 
  get 
  

  

  ■pT 
  

  

  7.2 
  Z—1 
  (^()\ 
  

  

  ( 
  •'6^'d2p(ol 
  + 
  m 
  \ 
  13 
  u.ml' 
  ' 
  ' 
  ' 
  ^" 
  ^ 
  

   ydlUpcoUm)4.20f''^^ 
  "^ 
  12 
  

  

  The 
  portion 
  shown 
  in 
  brackets 
  is 
  the 
  correction 
  which 
  this 
  

   result 
  gives 
  to 
  that 
  first 
  obtained. 
  When 
  the 
  mass 
  of 
  the. 
  

   load 
  is 
  equal 
  to 
  that 
  of 
  the 
  bar, 
  (20) 
  reduces 
  to 
  

  

  ,,_ 
  8-730 
  EI 
  

  

  ~ 
  pcoi' 
  ' 
  

  

  212 
  

  

  