﻿114 
  Prof. 
  E. 
  H. 
  Barton 
  on 
  Range 
  and 
  

  

  forced 
  and 
  natural 
  vibrations 
  of 
  the 
  system. 
  We 
  may 
  thus 
  

  

  (5) 
  

  

  1 
  

  

  write 
  it 
  

  

  where 
  

  

  3/i 
  = 
  

   in 
  which 
  

  

  */ 
  = 
  */l 
  + 
  */2, 
  • 
  

  

  fe-** 
  sin 
  (nt 
  — 
  $) 
  

  

  </{h(h-k)+p 
  2 
  -n*\ 
  2 
  + 
  (k 
  

   (Jc-2h)n 
  

  

  2A) 
  

  

  2^,2' 
  

  

  tan 
  S= 
  

  

  and 
  

  

  in 
  which 
  

  

  ?/ 
  2 
  = 
  A^~ 
  w/2 
  sin 
  (qt 
  

  

  q 
  2 
  =p 
  2 
  - 
  

  

  4 
  

  

  «)>) 
  

  

  (6) 
  

  

  (7) 
  

  

  A 
  and 
  a 
  are 
  arbitrary 
  constants 
  which 
  must 
  be 
  chosen 
  in 
  

   accordance 
  with 
  the 
  initial 
  conditions. 
  

  

  We 
  are 
  chiefly 
  concerned 
  with 
  equations 
  (6), 
  which 
  express 
  

   the 
  forced 
  vibrations 
  or 
  the 
  response 
  of 
  the 
  system 
  to 
  the 
  

   impressed 
  forces. 
  

  

  Response 
  under 
  Damped 
  and 
  Sustained 
  Forcing. 
  — 
  As 
  a 
  

   gauge 
  of 
  the 
  fulness 
  of 
  resonance 
  or 
  response 
  we 
  may 
  take, 
  

   as 
  done 
  by 
  Lord 
  Rayleigh 
  (* 
  Theory 
  of 
  Sound/ 
  vol. 
  i. 
  p. 
  47, 
  

   1894), 
  the 
  kinetic 
  energy 
  possessed 
  by 
  the 
  responding 
  system 
  

   at 
  the 
  instant 
  of 
  its 
  passage 
  through 
  the 
  position 
  of 
  equilibrium 
  

   expressed 
  by 
  ?/i 
  = 
  0. 
  To 
  compute 
  this 
  we 
  need 
  the 
  first 
  

   differential 
  of 
  #, 
  and 
  must 
  then 
  take 
  its 
  value 
  when 
  yi 
  = 
  0. 
  

   Thus 
  

  

  fivA 
  n 
  f 
  e 
  ~ 
  M 
  /ox 
  

  

  Then, 
  for 
  the 
  kinetic 
  energy, 
  T' 
  say, 
  we 
  take 
  half 
  the 
  

   square 
  of 
  this 
  quantity, 
  since 
  it 
  denotes 
  the 
  velocity 
  of 
  unit 
  

   mass 
  "or 
  the 
  electric 
  current 
  through 
  unit 
  inductance. 
  Hence 
  

   we 
  find 
  

  

  1 
  p 
  p 
  -2ht 
  

  

  ■ 
  ■ 
  (9) 
  

  

  T'= 
  

  

  h(h—k) 
  

  

  y- 
  

  

  n 
  z 
  \ 
  2 
  

  

  But 
  the 
  numerator 
  on 
  the 
  right 
  is 
  the 
  mean 
  square 
  of 
  the 
  

   impressed 
  force. 
  Hence, 
  dividing 
  out 
  by 
  this 
  we 
  obtain 
  

   the 
  kinetic 
  energy 
  per 
  unit 
  forcing. 
  This 
  may 
  be 
  termed 
  the 
  

   response 
  of 
  the 
  system 
  under 
  damped 
  forcing 
  and 
  denoted 
  

   by 
  N'. 
  It 
  is 
  obviously 
  given 
  by 
  

  

  ?> 
  2 
  

   W= 
  . 
  . 
  no) 
  

  

  