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  616 
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  L. 
  On 
  Periodic 
  Won 
  generating 
  Force 
  of 
  High 
  Frequency. 
  

   By 
  Andrew 
  Stephenson 
  *. 
  

  

  1. 
  npHE 
  influence 
  of 
  periodic 
  nongenerating 
  force 
  of 
  

   _L 
  high 
  frequency 
  has 
  been 
  investigated 
  in 
  the 
  two 
  

   cases 
  when 
  the 
  force 
  is 
  simple 
  oscillatory 
  f 
  and 
  impulsive 
  % 
  

   respectively. 
  We 
  now 
  take 
  up 
  the 
  general 
  problem 
  of 
  

   determining 
  the 
  effective 
  spring 
  of 
  a 
  system 
  about 
  a 
  position 
  

   of 
  equilibrium 
  when 
  acted 
  on 
  by 
  any 
  rapid 
  periodic 
  dis- 
  

   turbance 
  of 
  the 
  above 
  type. 
  

  

  In 
  the 
  first 
  place 
  we 
  deal 
  with 
  the 
  case 
  in 
  which 
  the 
  

   disturbance-time 
  graph 
  has 
  an 
  axis 
  of 
  symmetry: 
  the 
  spring 
  

   is 
  then 
  made 
  up 
  of 
  a 
  series 
  of 
  simple 
  elements 
  which 
  are 
  in 
  

   zero 
  phase 
  simultaneously. 
  The 
  equation 
  of 
  motion 
  under 
  

   such 
  disturbance 
  is 
  

  

  lc 
  + 
  (\ 
  + 
  2n 
  2 
  2 
  ot 
  r 
  cos 
  rnt) 
  se=0 
  9 
  

   1 
  

  

  where 
  n 
  is 
  the 
  frequency 
  of 
  the 
  force 
  per 
  2ir 
  units 
  of 
  time. 
  

  

  The 
  complete 
  solution 
  is 
  given 
  by 
  

  

  00 
  

  

  ,r= 
  X 
  A 
  r 
  sin 
  \(c—m)t-\-e}, 
  

  

  — 
  CO 
  

  

  where 
  e 
  is 
  arbitrary, 
  and 
  

  

  A,{X~( 
  c 
  -rn) 
  2 
  } 
  +n* 
  %« 
  8 
  (A 
  r 
  - 
  8 
  + 
  A 
  r+8 
  ) 
  = 
  0. 
  . 
  (r) 
  

  

  s=l 
  

  

  We 
  limit 
  our 
  enquiry 
  to 
  the 
  case 
  when 
  the 
  a's 
  are 
  small, 
  

   and 
  the 
  applied 
  frequency 
  so 
  large 
  that 
  X 
  is 
  negligible 
  

   compared 
  with 
  n 
  2 
  . 
  Under 
  these 
  conditions 
  it 
  will 
  be 
  observed 
  

   that 
  a 
  sound 
  approximation 
  is 
  given 
  by 
  

  

  A 
  — 
  A 
  — 
  ar 
  A 
  

   when 
  products 
  of 
  the 
  a'& 
  are 
  neglected. 
  Thus 
  A 
  is 
  large 
  

  

  CO 
  

  

  compared 
  with 
  2(A 
  r 
  + 
  A_ 
  r 
  ), 
  and 
  the 
  mean 
  motion 
  is 
  there- 
  

  

  i 
  

   fore 
  approximately 
  simple 
  oscillatory 
  

  

  «r 
  = 
  A 
  sin 
  (ct 
  + 
  e). 
  

  

  On 
  substituting 
  for 
  the 
  A's 
  in 
  (0) 
  we 
  find 
  

  

  c 
  2 
  = 
  \ 
  + 
  2n 
  2 
  X(j\\ 
  

  

  an 
  equation 
  determining 
  the 
  effective 
  spring 
  of 
  the 
  motion. 
  

   For 
  any 
  given 
  disturbance 
  the 
  coefficients 
  a 
  can 
  be 
  found 
  

  

  * 
  Communicated 
  by 
  the 
  Author, 
  

   f 
  "On 
  Induced 
  Stability/' 
  Phil. 
  Mag. 
  February 
  1908. 
  

   t 
  "On 
  a 
  New 
  Type 
  of 
  Dynamical 
  Stability/' 
  Manchester 
  Memoirs, 
  

   vol. 
  lii. 
  1908, 
  no. 
  8. 
  

  

  