﻿Vibration 
  of'Clamped-Dire^ied'" 
  Bars. 
  457 
  

  

  general 
  solution 
  is 
  then 
  of 
  the 
  form 
  

  

  j/ 
  = 
  A 
  cosh 
  u.i' 
  -f 
  B 
  sinh 
  cc.v 
  4- 
  C 
  cos 
  j3ai 
  + 
  D 
  sin 
  /3.i', 
  

   in 
  which, 
  whatever 
  the 
  end 
  conditions, 
  

  

  

  (10)^ 
  

  

  The 
  terminal 
  conditions 
  give 
  four 
  equations 
  for 
  which 
  it 
  is 
  

   necessarv 
  that 
  

  

  atanha^ 
  = 
  -/3tan/5/ 
  (11) 
  

  

  Elimination 
  of 
  a 
  and 
  j3 
  from 
  the 
  equations 
  (10) 
  and 
  (11) 
  

  

  would 
  give 
  an 
  expression 
  for 
  — 
  — 
  and 
  enable 
  the 
  frequency 
  

   to 
  be 
  calculated. 
  -^^ 
  

  

  § 
  10. 
  The 
  form 
  of 
  the 
  centre 
  line 
  of 
  the 
  bar 
  at 
  any 
  instant 
  

   is 
  then 
  given 
  by 
  

  

  y 
  — 
  y^ 
  (tanh 
  ul 
  sinh 
  ^Cc/; 
  — 
  cosh 
  olx— 
  -^ 
  tanh 
  cd 
  sin 
  ^x 
  -J- 
  cos 
  ^x) 
  

  

  4^ 
  i_\ 
  

  

  \cos/3Z 
  cosh 
  a// 
  

   whilst 
  the 
  couple 
  M 
  at 
  the 
  directed 
  end 
  can 
  be 
  obtained 
  from 
  

  

  -fP 
  = 
  — 
  pQ)--( 
  -7r:iC0Sh5jZ+ 
  -^COS/SZ 
  | 
  — 
  (cOsh. 
  Otl 
  — 
  COS 
  SI). 
  

  

  § 
  11. 
  Failing 
  an 
  exact 
  solution 
  of 
  equations 
  (10) 
  and 
  (11), 
  

   a 
  useful 
  approximate 
  one 
  can 
  be 
  obtained 
  by 
  limiting 
  the 
  

   longitudinal 
  force 
  to 
  such 
  values 
  as 
  may 
  occur 
  in 
  practice. 
  

  

  Writing 
  </> 
  = 
  5c/, 
  Q 
  — 
  /3/, 
  equation 
  (11) 
  is 
  

  

  </) 
  tanh 
  <\> 
  — 
  — 
  Q 
  tan 
  Q. 
  

  

  'd6__ 
  ^ 
  + 
  tan^-f6'tair^ 
  

   cW 
  ~~ 
  <t>-\- 
  tanh 
  ^ 
  — 
  <p 
  tanh- 
  cf) 
  ' 
  

  

  Also 
  (10) 
  gives 
  

  

  P/2 
  

  

  ^' 
  = 
  ^'-m 
  ^^-^ 
  

  

  and 
  when 
  P 
  is 
  zero 
  (equation 
  (1)) 
  

  

  cl>= 
  6 
  = 
  2-36502 
  

   for 
  vibrations 
  of 
  the 
  fundamental 
  type. 
  

  

  * 
  Cf. 
  Eayleigh's 
  'Theory 
  of 
  Sound/ 
  vol. 
  i. 
  1894, 
  p. 
  299. 
  

   Phil. 
  Mag. 
  S. 
  6. 
  Vol. 
  18. 
  No. 
  lOG. 
  Oct. 
  1909. 
  2 
  I 
  

  

  