﻿Vibration 
  of 
  '^Clamped- 
  Directed" 
  Bars. 
  45^ 
  

  

  § 
  6. 
  The 
  values 
  of 
  fil 
  given 
  above 
  for 
  the 
  lower 
  harmonics 
  

   may 
  be 
  readily 
  calculated 
  by 
  the 
  method 
  explained 
  below 
  

   for 
  ^2^. 
  

  

  We 
  know 
  that 
  yu,o^= 
  {'2 
  — 
  \)7r 
  approximately. 
  Hence 
  

  

  tanhyci2^= 
  -999967, 
  (5 
  a) 
  

  

  to 
  the 
  limits 
  of 
  accuracy 
  of 
  the 
  tables 
  at 
  my 
  disposal. 
  By 
  

   equation 
  (3) 
  therefore 
  

  

  tan 
  yLt2^= 
  --•999967 
  approximately. 
  

   .-. 
  ;ci2^=:27r--7853815 
  = 
  5-4978038, 
  

  

  which 
  is 
  more 
  than 
  sufficiently 
  accurate*. 
  Even 
  when 
  this 
  

   method 
  is 
  used 
  for 
  the 
  fundamental, 
  the 
  error 
  is 
  less 
  than 
  

   O'Ol 
  per 
  cent. 
  

  

  § 
  7. 
  To 
  examine 
  the 
  curve 
  assumed 
  by 
  the 
  centre 
  line 
  of 
  

   the 
  bar, 
  we 
  obtain 
  a 
  further 
  relation 
  by 
  putting 
  ?/=yi 
  when 
  

   ,r=l 
  ; 
  whence 
  it 
  can 
  easily 
  be 
  shown 
  that 
  

  

  ?/ 
  = 
  i^^[(sinh 
  /jlI+ 
  sin 
  fil) 
  (sin 
  /jlx— 
  sinh 
  /j.v) 
  ~ 
  (cosh 
  /jlI 
  

  

  — 
  cos 
  fil) 
  (cos 
  fjix— 
  cosh 
  /jlx)'] 
  -t- 
  (1— 
  cos 
  yLtZ 
  cosh 
  /xZ). 
  (6) 
  

  

  The 
  values 
  of 
  /ulI 
  to 
  be 
  used 
  in 
  this 
  equation 
  are 
  those 
  given 
  

   in 
  § 
  5. 
  

  

  § 
  8. 
  When 
  dealing 
  with 
  harmonics 
  the 
  positions 
  of 
  the 
  

   nodes 
  are 
  to 
  be 
  found 
  from 
  (6) 
  by 
  putting 
  3/ 
  = 
  0. 
  We 
  thus 
  

   get, 
  at 
  each 
  node, 
  

  

  sin 
  /-6.1' 
  — 
  sinh 
  //,.y 
  cosh^tZ 
  — 
  cos 
  /u-Z 
  ._. 
  

  

  cos 
  fXcC 
  — 
  cosh 
  ficC~ 
  sinh 
  fil 
  + 
  sin 
  fjbl 
  

  

  The 
  fundamental 
  is 
  free 
  from 
  nodes. 
  For 
  the 
  first 
  harmonic 
  

   the 
  value 
  of 
  the 
  right-hand 
  side 
  of 
  (7) 
  is 
  1-000033. 
  For 
  

   higher 
  tones 
  it 
  may 
  be 
  taken 
  as 
  unity. 
  

  

  Solving 
  (7) 
  by 
  trial, 
  we 
  find 
  in 
  the 
  case 
  of 
  the 
  first 
  

   harmonic 
  that 
  the 
  distance 
  of 
  the 
  node 
  from 
  the 
  clamped 
  

  

  * 
  This 
  metliod 
  depends 
  on 
  the 
  fact 
  that 
  a 
  slight 
  difference 
  between 
  

   two 
  numbers 
  of 
  the 
  magnitudes 
  involved 
  makes 
  no 
  appreciable 
  difference 
  

   in 
  their 
  hyperbolic 
  tangents. 
  If 
  h 
  is 
  small 
  

  

  tanh 
  {x-i-h)~tQ.Tihx=Ji{l— 
  tanh^ 
  a) 
  

  

  approximately. 
  Iq 
  the 
  example 
  in 
  the 
  text 
  the 
  result 
  shows 
  that 
  

  

  tanh/Lt,/=tanh(2-i)7r 
  

  

  to 
  at 
  least 
  eight 
  decimal 
  places. 
  There 
  can 
  be 
  no 
  hesitation, 
  therefore, 
  

   in 
  accepting 
  equation 
  (o 
  a). 
  The 
  method 
  appears 
  more 
  natural 
  than 
  the 
  

   normal 
  one, 
  which 
  would 
  be 
  to 
  put 
  /Z2^=(2— 
  ^)'r+r 
  in 
  tan 
  fi.J 
  and 
  

   tanh 
  nJ 
  (where 
  x 
  is 
  small), 
  and 
  to 
  proceed 
  by 
  approximations. 
  

  

  