﻿Vibration 
  of 
  ^^Clamped-Directed'^ 
  Bars. 
  461 
  

  

  Equation 
  (22) 
  shows 
  that 
  P 
  = 
  and 
  vibration 
  ceases 
  when 
  

   7il 
  = 
  TT, 
  that 
  is, 
  when 
  

  

  T 
  = 
  El7r7^\ 
  

  

  After 
  expansion, 
  (22) 
  becomes 
  

   12EI/, 
  4 
  

  

  ?j(i-i.«-.....> 
  

  

  ?«/ 
  tan 
  i-?2Z 
  — 
  4 
  ?i/' 
  

  

  agreeing 
  with 
  (21) 
  when 
  T, 
  and 
  therefore 
  n, 
  is 
  zero. 
  

  

  When 
  nl 
  = 
  2z7r 
  (i 
  being 
  an 
  integer) 
  the 
  right-hand 
  side 
  of 
  

   (22) 
  is 
  indeterminate 
  but 
  has 
  the 
  limit 
  —T/ml, 
  vibration 
  

   being 
  impossible 
  if 
  T 
  is 
  positive. 
  

  

  § 
  17, 
  The 
  result 
  may 
  also 
  be 
  written 
  

  

  T 
  jnl 
  

  

  from 
  which 
  (nl 
  being 
  positive) 
  the 
  expression 
  for 
  the 
  

   frequency 
  is 
  real 
  when 
  tan 
  ^ 
  nl 
  > 
  ^ 
  nl, 
  that 
  is, 
  for 
  values 
  of 
  

   nl 
  between 
  and 
  tt, 
  and 
  for 
  decreasing 
  intervals 
  in 
  the 
  

   neighbourhood 
  of 
  Stt, 
  ott, 
  &c. 
  ; 
  vibration 
  always 
  ceasing 
  

   when 
  7iZ 
  = 
  (2i 
  — 
  l)7r. 
  

  

  The 
  first 
  of 
  these 
  intervals 
  is 
  from 
  nl 
  = 
  S'986S 
  ... 
  to 
  

   711=377, 
  and 
  this 
  corresponds 
  to 
  the 
  first 
  harmonic. 
  

  

  With 
  the 
  massless 
  bar 
  harmonics 
  are 
  impossible 
  with 
  low 
  

   values 
  of 
  T. 
  As 
  T 
  is 
  increased 
  the 
  frequency 
  falls, 
  becoming- 
  

   zero 
  when 
  nl='jT,'y 
  being 
  then 
  zero. 
  Between 
  tt 
  and 
  8*9868 
  

   vibration 
  is 
  impossible, 
  the 
  deflexion 
  increasing 
  with 
  the 
  

   time. 
  If 
  this 
  region 
  be 
  safely 
  passed 
  that 
  between 
  8'9868 
  

   and 
  Stt 
  is 
  reached, 
  during 
  which 
  vibration 
  may 
  again 
  occur, 
  

   T 
  being 
  now 
  sufficient 
  to 
  enable 
  the 
  bar 
  to 
  assume 
  the 
  curve 
  

   of 
  the 
  first 
  harmonic 
  type. 
  

  

  Fia-. 
  6. 
  

  

  I 
  

  

  § 
  18. 
  If 
  the 
  mass 
  be 
  situated 
  at 
  any 
  point 
  in 
  the 
  length, 
  

   let 
  it 
  divide 
  the 
  bar 
  into 
  segments 
  a 
  and 
  b, 
  as 
  indicated 
  in 
  

   fig. 
  6. 
  Then 
  for 
  x<a 
  the 
  differential 
  equation 
  becomes 
  

  

  Elg 
  = 
  - 
  mjj„{a 
  -ic) 
  + 
  T(y, 
  -y) 
  -M, 
  

  

  of 
  which 
  the 
  solution 
  is 
  

  

  // 
  = 
  Ai 
  sm 
  nx 
  + 
  jj^ 
  cos 
  nx-] 
  — 
  rr^C^*'— 
  ^O+yi"' 
  'Y' 
  

  

  