﻿722 
  Mr. 
  G. 
  Green 
  on 
  Flexuml 
  

  

  (2) 
  That 
  the 
  principal 
  cause 
  o£ 
  the 
  difference 
  is 
  the 
  fact 
  

   that 
  the 
  arc 
  behaves 
  as 
  i£ 
  it 
  has 
  sel£-inductance, 
  which 
  is 
  

   constant 
  during 
  any 
  particular 
  train 
  o£ 
  oscillations 
  

   (though 
  different 
  under 
  different 
  circumstances), 
  and 
  

   which 
  must 
  be 
  taken 
  into 
  account 
  in 
  calculating 
  the 
  

   frequencies. 
  

  

  (3) 
  That 
  any 
  one 
  of 
  the 
  harmonics 
  of 
  the 
  arc-note 
  up 
  to 
  

   the 
  sixth 
  can 
  be 
  strongly 
  reinforced 
  by 
  suitably 
  adjusting 
  

   the 
  ratio 
  of 
  the 
  two 
  frequencies 
  of 
  the 
  system. 
  

  

  (4) 
  That 
  if 
  the 
  two 
  notes 
  of 
  the 
  arc 
  are 
  equally 
  stable, 
  and 
  

   if 
  the 
  interval 
  between 
  them 
  is 
  a 
  perfect 
  fifth, 
  a 
  third 
  

   note 
  may 
  be 
  heard, 
  which 
  corresponds 
  to 
  an 
  electrical 
  

   oscillation 
  of 
  frequency 
  equal 
  to 
  the 
  difference 
  of 
  the 
  

   frequencies 
  of 
  the 
  two 
  primes^ 
  

  

  Bangor, 
  July 
  1909. 
  ; 
  

  

  " 
  \ 
  ^= 
  

  

  LXXIY. 
  Flexural 
  Vibrations 
  of 
  Thin 
  Rods. 
  By 
  George 
  

  

  Green, 
  M.A.^ 
  B.Sc, 
  Assistant 
  to 
  the 
  Professor 
  of 
  Natural 
  

  

  Philosophj 
  in 
  the 
  University 
  of 
  Glasgow^. 
  

  

  § 
  1. 
  rilHE 
  main 
  object 
  of 
  this 
  paper 
  is 
  to 
  point 
  out 
  a 
  

   JL 
  method 
  of 
  applying 
  hydrodynamic 
  solutions 
  

   already 
  obtained 
  to 
  the 
  solution 
  of 
  problems 
  relating 
  to 
  

   flexural 
  vibrations 
  of 
  thin 
  elastic 
  rods. 
  

  

  The 
  results 
  found 
  apply 
  to 
  rods 
  not 
  subjected 
  to 
  permanent 
  

   tension, 
  and 
  vibrating 
  so 
  that 
  one 
  principal 
  axis 
  of 
  each 
  

   transverse 
  section 
  lies 
  in 
  the 
  plane 
  of 
  vibration. 
  The 
  central 
  

   line 
  of 
  the 
  rod 
  is 
  assumed 
  to 
  remain 
  unaltered 
  in 
  length 
  ; 
  

   and 
  particles 
  lying 
  in 
  a 
  plane 
  transverse 
  section, 
  when 
  un- 
  

   disturbed, 
  remain 
  always 
  in 
  a 
  plane 
  normal 
  to 
  the 
  central 
  

   line. 
  For 
  convenience 
  in 
  what 
  follows 
  we 
  may 
  here 
  derive 
  

   the 
  equation 
  of 
  motion 
  of 
  a 
  rod 
  subject 
  to 
  these 
  conditions. 
  

   Taking 
  o) 
  as 
  the 
  area 
  of 
  each 
  section, 
  k^(o 
  its 
  moment 
  of 
  

   inertia, 
  q 
  as 
  Young's 
  modulus, 
  dx 
  as 
  the 
  length 
  of 
  the 
  central 
  

   line 
  of 
  a 
  small 
  portion, 
  R 
  its 
  radius 
  of 
  curvature, 
  and 
  N 
  as 
  

   the 
  total 
  tangential 
  force 
  acting 
  on 
  the 
  cross-section 
  at 
  x, 
  

   we 
  obtain 
  the 
  equation 
  of 
  angular 
  motion 
  of 
  the 
  element 
  

  

  where 
  y 
  is 
  the 
  vertical 
  displacement 
  oF 
  the 
  element 
  and 
  p 
  is 
  

   its 
  density. 
  The 
  equation 
  o£ 
  vertical 
  motion 
  is 
  

  

  3T='""§# 
  ^^) 
  

  

  * 
  Reprinted 
  from 
  Proc. 
  Roy. 
  Soc. 
  Edin. 
  vol. 
  xxix. 
  1909, 
  from 
  a 
  separate 
  

   copy 
  communicated 
  by 
  the 
  author. 
  

  

  