﻿Deflexion 
  and 
  Vibration 
  of 
  ^'' 
  Clamped-Directed 
  " 
  Bars. 
  453 
  

  

  the 
  technologist, 
  but 
  rather 
  in 
  the 
  fact 
  that 
  the 
  problems 
  that 
  

   have 
  been 
  solved 
  are 
  not 
  necessarily 
  those 
  of 
  which 
  the 
  

   solutions 
  are 
  most 
  required 
  in 
  practice. 
  

  

  § 
  2. 
  As 
  an 
  important 
  example, 
  in 
  which 
  the 
  terminal 
  

   conditions 
  here 
  considered 
  occur, 
  one 
  might 
  mention 
  the 
  

   case 
  of 
  the 
  cylinder 
  of 
  a 
  steam-engine 
  supported 
  on 
  two 
  or 
  

   more 
  mild 
  steel 
  standards. 
  It 
  will 
  be 
  recognized 
  that 
  the 
  

   upper 
  ends 
  of 
  the 
  standards 
  must 
  be 
  treated 
  as 
  " 
  directed." 
  

  

  It 
  is 
  worthy 
  of 
  notice 
  that 
  these 
  terminal 
  conditions 
  are 
  

   mentioned 
  by 
  Lord 
  Eayleigh 
  in 
  his 
  ' 
  Theory 
  of 
  Sound 
  ' 
  * 
  

   but 
  that 
  the 
  directed 
  end 
  is 
  at 
  once 
  dismissed 
  from 
  con- 
  

   sideration 
  with 
  the 
  remark 
  that 
  ^' 
  there 
  are 
  no 
  experimental 
  

   means 
  by 
  which 
  the 
  contemplated 
  constraint 
  could 
  be 
  

   realized.^' 
  

  

  § 
  3. 
  Notation 
  and 
  End 
  Conditions. 
  — 
  

  

  Let 
  ?/, 
  y.- 
  = 
  deflexions 
  at 
  points 
  x 
  and 
  z 
  in 
  the 
  length 
  of 
  

   the 
  bar 
  ; 
  

   0), 
  I 
  = 
  area 
  and 
  moment 
  of 
  inertia 
  of 
  cross-section, 
  

   the 
  cross-dimensions 
  being 
  supposed 
  small 
  

   compared 
  with 
  the 
  length 
  ; 
  

   Z 
  = 
  length 
  of 
  bar 
  ; 
  

   p 
  = 
  density 
  of 
  the 
  material 
  ; 
  

  

  E 
  = 
  Young^s 
  Modulus 
  for 
  the 
  material 
  (when 
  there 
  

   is 
  an 
  axial 
  pull 
  P 
  in 
  the 
  bar, 
  E 
  stands 
  for 
  

   P/a) 
  + 
  Young's 
  Modulus) 
  ; 
  

   f 
  = 
  time, 
  measured 
  from 
  any 
  instant 
  at 
  which 
  y 
  

   is 
  everywhere 
  zero 
  ; 
  

   N 
  = 
  number 
  of 
  complete 
  vibrations 
  per 
  second. 
  

  

  The 
  symbol 
  y 
  is 
  written 
  for 
  d^y'dt^ 
  ; 
  and 
  if 
  yi 
  is 
  the 
  in- 
  

   stantaneous 
  deflexion 
  at 
  some 
  particular 
  point 
  in 
  the 
  length 
  

   o£ 
  the 
  bar, 
  we 
  have, 
  for 
  simple 
  harmonic 
  vibrations, 
  

  

  -ijh=-iii!yi=^{;^, 
  . 
  . 
  . 
  . 
  (1) 
  

  

  where 
  

  

  N 
  = 
  A-/27r. 
  

  

  The 
  curve 
  assumed 
  by 
  the 
  elastic 
  central 
  line 
  at 
  any 
  

   instant 
  is 
  given 
  in 
  terms 
  of 
  y^. 
  11 
  a 
  is 
  the 
  amplitude 
  at 
  the 
  

   point 
  in 
  the 
  length 
  at 
  which 
  _?/i 
  is 
  the 
  instantaneous 
  deflexion, 
  

   the 
  value 
  of 
  yi 
  is 
  given 
  by 
  

  

  yi 
  = 
  a^'mkt 
  (2; 
  

  

  The 
  origin 
  will 
  always 
  be 
  taken 
  at 
  the 
  clamped 
  end. 
  At 
  

   :c 
  = 
  we 
  have, 
  thereloTe,y 
  = 
  dy/dx 
  = 
  ; 
  whilst 
  at 
  the 
  directed 
  

  

  * 
  See 
  Rayleio^h's 
  ' 
  Sound/ 
  vol. 
  i. 
  p. 
  2o9, 
  1894 
  edition. 
  

  

  