﻿Sharpness 
  of 
  Resonance 
  under 
  Sustained 
  Forcing, 
  121 
  

  

  Hence, 
  if 
  A, 
  be 
  written 
  for 
  the 
  logarithmic 
  decrement 
  of 
  

   the 
  vibrations 
  per 
  complete 
  period, 
  we 
  have 
  

  

  x= 
  rf 
  = 
  , 
  ^ 
  ^ 
  , 
  m 
  m 
  (25) 
  

  

  Thus, 
  when 
  G 
  2 
  is 
  small 
  enough 
  to 
  make 
  the 
  quantity 
  

   under 
  the 
  root 
  sensibly 
  equal 
  to 
  unity, 
  we 
  may 
  use, 
  instead 
  

   of 
  G, 
  the 
  approximate 
  value 
  G' 
  defined 
  by 
  (25). 
  

  

  Or, 
  in 
  other 
  words, 
  when 
  the 
  range 
  of 
  resonance 
  is 
  small, 
  

   it 
  is 
  almost 
  proportional 
  to 
  the 
  logarithmic 
  decrement 
  of 
  the 
  

   natural 
  vibrations 
  of 
  the 
  responding 
  system. 
  The 
  distinction 
  

   is 
  shown 
  by 
  the 
  following 
  numerical 
  values 
  : 
  — 
  

  

  True 
  Range 
  G 
  =0'1, 
  0% 
  0*3, 
  0*4, 
  0*5 
  "j 
  

  

  Approximate 
  do. 
  G' 
  =0-1001, 
  0-201, 
  0-303, 
  0*408, 
  0*517 
  l(2G) 
  

   Percentage 
  Difference 
  = 
  0*1, 
  0*5, 
  1, 
  2, 
  34 
  J 
  

  

  The 
  possible 
  variations 
  of 
  the 
  essential 
  factors 
  in 
  the 
  range 
  

   of 
  resonance 
  may 
  be 
  now 
  conveniently 
  summed 
  up 
  as 
  

   follows 
  : 
  — 
  

  

  For 
  p 
  constant, 
  G 
  oc 
  h 
  (27) 
  

  

  For 
  h 
  constant, 
  G 
  oc 
  1/p 
  (28) 
  

  

  For 
  k 
  oc 
  p, 
  G 
  remains 
  constant 
  . 
  . 
  (29) 
  

  

  The 
  first 
  of 
  these 
  illustrates 
  what 
  appears 
  to 
  be 
  the 
  

   common 
  idea, 
  that 
  the 
  character 
  of 
  the 
  resonance 
  depends 
  

   on 
  the 
  damping 
  present 
  in 
  the 
  responding 
  system. 
  The 
  

   second 
  shows 
  under 
  what 
  condition 
  the 
  sharpness 
  of 
  resonance 
  

   varies 
  directly 
  as 
  the 
  frequency. 
  While 
  the 
  last 
  is 
  the 
  

   special 
  case 
  in 
  which 
  the 
  character 
  of 
  the 
  resonance 
  remains 
  

   unchanged 
  in 
  spite 
  of 
  changes 
  occurring 
  proportionally 
  in 
  

   both 
  damping 
  and 
  frequency 
  coefficients. 
  

  

  Some 
  vibrational 
  systems 
  may 
  fall 
  naturally 
  (or 
  even 
  

   inevitably) 
  into 
  one 
  or 
  other 
  of 
  the 
  above 
  classes. 
  Others 
  

   may 
  lend 
  themselves 
  to 
  modifications 
  at 
  the 
  experimenter's 
  

   disposal 
  which 
  enable 
  him 
  to 
  relegate 
  them 
  to 
  one 
  class 
  or 
  

   another 
  at 
  will. 
  Examples 
  in 
  illustration 
  of 
  these 
  cases 
  

   occur 
  later 
  in 
  the 
  paper. 
  

  

  If, 
  in 
  any 
  vibrational 
  system, 
  we 
  are 
  unable 
  to 
  modify 
  

   /,■ 
  at 
  will, 
  but 
  know 
  how 
  it 
  varies, 
  if 
  at 
  all, 
  with 
  p 
  ; 
  we 
  can 
  

   then 
  predict 
  the 
  corresponding 
  changes 
  in 
  G 
  and 
  H 
  and 
  may 
  

   be 
  able 
  to 
  put 
  them 
  to 
  the 
  test 
  experimentally. 
  Conversely, 
  

   in 
  a 
  vibrational 
  system 
  for 
  which 
  we 
  cannot 
  easily 
  obtain 
  

   any 
  direct 
  experimental 
  knowledge 
  as 
  to 
  any 
  variation 
  of 
  /\ 
  

  

  