﻿618 
  Mr. 
  A. 
  Stephenson 
  on 
  Periodic 
  

  

  The 
  graphs 
  o£ 
  the 
  required 
  integral 
  and 
  its 
  square 
  are 
  

   OPQRS 
  . 
  . 
  . 
  and 
  OP'Q'PS' 
  . 
  . 
  . 
  respectively; 
  CP 
  = 
  OA 
  . 
  OC, 
  

   and 
  CP' 
  = 
  CP 
  2 
  . 
  The 
  limiting 
  case 
  of 
  impulsive 
  disturbance 
  

   is 
  reached 
  by 
  keeping 
  OA 
  . 
  OC 
  constant, 
  =^yu,, 
  say, 
  and 
  

   continually 
  diminishing 
  OC 
  towards 
  zero. 
  The 
  mean 
  value 
  

   of 
  the 
  squared 
  integral 
  is 
  then 
  CjV 
  = 
  ^/jl 
  2 
  ; 
  and 
  therefore 
  

  

  Thus 
  a 
  spring 
  disturbance 
  made 
  up 
  of 
  equal 
  and 
  opposite 
  

   impulses 
  of 
  magnitude 
  /n, 
  following 
  at 
  equal 
  intervals, 
  

   increases 
  the 
  effective 
  spring 
  by 
  /4 
  2 
  /4, 
  provided 
  the 
  frequency 
  

   of 
  the 
  disturbance 
  is 
  large 
  compared 
  with 
  that 
  of 
  the 
  

   system. 
  

  

  If 
  X 
  is 
  zero 
  we 
  have 
  c 
  = 
  fju/2. 
  This 
  case 
  is 
  realised 
  

   mechanically 
  by 
  pivoting 
  a 
  body 
  so 
  that 
  it 
  is 
  free 
  to 
  rotate 
  

   horizontally 
  and 
  driving 
  the 
  pivot 
  to 
  and 
  fro 
  in 
  a 
  small 
  

   horizontal 
  path 
  with 
  constant 
  speed, 
  V 
  say. 
  The 
  amplitude 
  

   of 
  the 
  imposed 
  motion 
  being 
  small, 
  the 
  action 
  at 
  the 
  pivot 
  is 
  

   confined 
  to 
  the 
  impulsive 
  forces 
  at 
  the 
  ends 
  of 
  the 
  path. 
  

   Let 
  h 
  be 
  the 
  distance 
  of 
  the 
  mass 
  centre 
  from 
  the 
  pivot, 
  and 
  

   h 
  the 
  radius 
  of 
  gyration 
  about 
  the 
  vertical 
  axis 
  through 
  the 
  

   mass 
  centre. 
  If 
  the 
  angular 
  displacement 
  from 
  the 
  position 
  

   of 
  relative 
  equilibrium 
  is 
  6 
  when 
  the 
  pivot 
  is 
  at 
  the 
  remote 
  

   end 
  of 
  the 
  path, 
  and 
  if 
  co 
  and 
  co 
  x 
  are 
  the 
  angular 
  velocities 
  

   before 
  and 
  after 
  the 
  reversing 
  impulse, 
  then 
  since 
  the 
  

   angular 
  momentum 
  about 
  the 
  pivot 
  is 
  unchanged 
  by 
  the 
  

   action, 
  

  

  (P 
  + 
  /*>! 
  -Yh0 
  = 
  [V 
  + 
  h 
  2 
  )co 
  + 
  Vh0, 
  

  

  2Yh 
  

  

  i. 
  e. 
  co 
  x 
  — 
  co 
  + 
  i*,™ 
  = 
  0. 
  

  

  Similariyy-if 
  X 
  is 
  the 
  angular 
  displacement 
  on 
  reaching 
  

   the 
  other 
  end, 
  and 
  co 
  2 
  the 
  angular 
  velocity 
  after 
  the 
  impulse 
  

   there, 
  

  

  Thus 
  the 
  spring 
  is 
  made 
  up 
  of 
  equal 
  and 
  opposite 
  impulses 
  

   of 
  magnitude 
  2\h/(k 
  2 
  -^ 
  A 
  2 
  ) 
  following 
  at 
  equal 
  intervals, 
  and 
  

   therefore 
  by 
  the 
  preceding 
  the 
  mean 
  motion 
  is 
  a 
  simple 
  

   oscillation 
  

  

  '""Kf^'+O- 
  

  

  If 
  a 
  is 
  the 
  length 
  of 
  the 
  pivot 
  path 
  c/n 
  = 
  fi/2n 
  = 
  ali/ir^k 
  2 
  + 
  A 
  2 
  ), 
  

   which 
  is 
  small, 
  and 
  the 
  conditions 
  necessary 
  for 
  the 
  approxi- 
  

   mation 
  are 
  therefore 
  satisfied. 
  

  

  