﻿Sharpness 
  of 
  Resonance 
  under 
  Sustained 
  Forcing. 
  125 
  

  

  Illustrative 
  Experiments. 
  

  

  "We 
  may 
  now 
  notice 
  several 
  experiments 
  in 
  relation 
  to 
  

   the 
  subject 
  in 
  hand. 
  Some 
  of 
  these 
  qualitatively 
  illustrate 
  

   the 
  dependence 
  of 
  range 
  of 
  resonance 
  on 
  pitch 
  as 
  well 
  as 
  

   damping. 
  Others 
  yield 
  a 
  quantitative 
  result 
  in 
  corroboration 
  

   of 
  the 
  theory. 
  While 
  yet 
  others 
  are 
  only 
  fragments 
  of 
  a 
  

   larger 
  scheme 
  which 
  is 
  just 
  outlined 
  and 
  left 
  for 
  future 
  

   examination. 
  

  

  The 
  cases 
  to 
  be 
  dealt 
  with 
  fall 
  under 
  three 
  natural 
  

   divisions 
  : 
  — 
  mechanical, 
  acoustical, 
  and 
  electrical. 
  They 
  will 
  

   be 
  noticed 
  in 
  this 
  order. 
  

  

  The 
  Double 
  Pendulum. 
  — 
  As 
  apparently, 
  the 
  simplest 
  

   possible 
  illustration 
  of 
  forced 
  vibrations, 
  one 
  is 
  apt 
  to 
  think 
  

   first 
  of 
  the 
  double 
  pendulum, 
  the 
  individual 
  pendulums 
  being 
  

   of 
  equal 
  periods 
  and 
  one 
  suspended 
  from 
  the 
  other. 
  Thus, 
  

   a 
  slight 
  motion 
  of 
  the 
  upper 
  bob 
  supplies 
  the 
  periodic 
  

   impressed 
  force 
  of 
  suitable 
  frequency 
  and 
  may 
  be 
  expected 
  

   to 
  elicit 
  and 
  maintain 
  oscillations 
  of 
  large 
  amplitude 
  of 
  the 
  

   lower 
  bob. 
  

  

  But 
  the 
  idea 
  of 
  simplicity 
  in 
  connexion 
  with 
  this 
  double 
  

   pendulum 
  is 
  a 
  delusion 
  which 
  a 
  little 
  reflexion 
  or 
  experience 
  

   may 
  serve 
  to 
  dispel. 
  

  

  In 
  the 
  first 
  place, 
  the 
  motions 
  may 
  readily 
  occur 
  in 
  various 
  

   vertical 
  planes. 
  However, 
  with 
  a 
  little 
  care 
  in 
  their 
  

   initiation 
  they 
  may 
  be 
  confined 
  to 
  a 
  single 
  plane, 
  which 
  is 
  

   preferable. 
  

  

  "We 
  may 
  next 
  notice 
  that, 
  if 
  the 
  bobs 
  are 
  of 
  nearly 
  equal 
  

   masses, 
  the 
  starting 
  of 
  the 
  lower 
  one 
  involves, 
  by 
  its 
  

   reaction, 
  the 
  stoppage 
  of 
  the 
  upper 
  one. 
  This 
  undesirable 
  

   effect 
  of 
  reaction 
  is 
  easily 
  reduced 
  to 
  a 
  negligible 
  quantity 
  

   by 
  making 
  the 
  mass 
  of 
  the 
  upper 
  bob 
  many 
  times 
  that 
  of 
  

   the 
  lower 
  one. 
  

  

  Still, 
  if 
  both 
  bobs 
  are 
  of 
  the 
  same 
  material 
  and 
  construction, 
  

   their 
  damping 
  coefficients 
  will 
  probably 
  be 
  not 
  greatly 
  

   different. 
  Hence, 
  the 
  simple 
  equations 
  for 
  sustained 
  forcing- 
  

   do 
  not 
  apply. 
  

  

  Further, 
  if 
  the 
  lower 
  bob 
  is 
  of 
  solid 
  metal, 
  or 
  even 
  dense 
  

   wood 
  (lead, 
  brass, 
  iron, 
  or 
  boxwood), 
  its 
  damping 
  coemViont 
  

   will 
  be 
  small 
  and 
  the 
  resonance 
  in 
  consequence 
  probably 
  

   inconveniently 
  sharp. 
  So 
  much 
  so 
  that 
  it 
  is 
  extremely 
  

   tedious 
  to 
  tune 
  the 
  pendulums 
  well 
  enough 
  to 
  avoid 
  beats. 
  

   These 
  beats 
  always 
  occur 
  in 
  the 
  early 
  stages 
  of 
  the 
  action 
  

   of 
  a 
  slightly 
  mistuned 
  periodic 
  force 
  on 
  a 
  system 
  initially 
  at 
  

   rest. 
  And 
  th^se 
  beats 
  must 
  last 
  till 
  the 
  vibrations 
  natural 
  

   to 
  the 
  responsive 
  system 
  have 
  been 
  practically 
  damped 
  out. 
  

  

  