﻿Coupled 
  Vibrations 
  : 
  Unequal 
  Masses 
  or 
  Periods. 
  69 
  

  

  Accordingly 
  the 
  ratios 
  of 
  the 
  amplitudes 
  of 
  the 
  quick 
  and 
  

   slow 
  vibrations 
  in 
  the 
  y 
  and 
  z 
  traces 
  are 
  respectively 
  

  

  -(p— 
  l)rf 
  

  

  -e- 
  (p 
  - 
  l)st 
  and 
  - 
  . 
  . 
  . 
  (49) 
  

  

  P 
  

   Relation 
  of 
  Dampings 
  in 
  the 
  Vibrations 
  separate 
  and 
  

   coupled. 
  — 
  The 
  vibrations 
  of 
  a 
  separate 
  damped 
  pendulum 
  

   of 
  length 
  I 
  are 
  derived 
  from 
  the 
  equation 
  of 
  motion 
  

  

  gf 
  + 
  2 
  *f 
  + 
  ^=0, 
  .... 
  (50) 
  

  

  where 
  m?=g/l. 
  

  

  The 
  solution 
  of 
  this 
  involves 
  simple 
  harmonic 
  vibrations 
  

   of 
  approximate 
  period 
  

  

  t 
  = 
  27r/m, 
  

   and 
  of 
  damping 
  factor 
  

  

  Thus 
  the 
  ratio 
  of 
  successive 
  amplitudes 
  is 
  

  

  e 
  kT/2 
  = 
  e 
  kir/m 
  

   nearly. 
  

  

  But 
  the 
  logarithmic 
  decrement 
  \ 
  (per 
  half 
  wave) 
  is 
  

   defined 
  as 
  the 
  logarithm 
  to 
  base 
  e 
  of 
  this 
  ratio. 
  

   Hence 
  

  

  X= 
  — 
  or 
  k= 
  — 
  , 
  (51) 
  

  

  m 
  7r 
  . 
  

  

  which 
  gives 
  the 
  relation 
  between 
  damping 
  coefficient 
  and 
  

   logarithmic 
  decrement 
  for 
  a 
  separate 
  pendulum. 
  

  

  We 
  have 
  now 
  to 
  express 
  in 
  terms 
  of 
  \ 
  the 
  two 
  damping 
  

   coefficients 
  r 
  and 
  s 
  which 
  apply 
  to 
  the 
  superposed 
  vibrations 
  

   when 
  the 
  pendulums 
  are 
  coupled. 
  Thus, 
  combining 
  (23) 
  

   and 
  (24) 
  with 
  (51), 
  we 
  find 
  

  

  m 
  \ 
  (52) 
  

  

  1+P 
  

   and 
  

  

  (53) 
  

  

  1+p 
  IT 
  

  

  III. 
  Theory 
  for 
  Unequal 
  Periods. 
  

  

  Equations 
  of 
  Motion 
  and 
  Coupling. 
  — 
  Still 
  using 
  the 
  

   double-cord 
  pendulum, 
  as 
  shown 
  in 
  figs. 
  1, 
  2, 
  and 
  4 
  of 
  the 
  

   first 
  paper, 
  we 
  now 
  make 
  the 
  masses 
  of 
  the 
  bobs 
  equal, 
  but 
  

   the 
  lengths 
  of 
  the 
  suspensions 
  unequal. 
  (The 
  droops 
  of 
  

   the 
  two 
  bridles 
  always 
  remain 
  equal.) 
  In 
  other 
  words, 
  

   Q 
  = 
  P 
  or 
  p 
  — 
  l 
  3 
  while 
  the 
  lengths 
  of 
  the 
  suspensions 
  for 
  the 
  

  

  