oj  Loaded  and  Unloaded  Bars.  873 
Or,  for  an  approximate  solution,  starting  with 
dx~ 
we  may  put  n         3  ij\ 
e=--2T 
EIi/=^1(.t-02{|(,'  +  20  +  2'}, 
Whence 
and  t/i  _  3EI 
'WW 
§  28.   When  the  mass  of  the  bar  is  taken  into  account,  and 
Rayleigh's  method  is  used,  the  kinetic  energy  is 
T  -  s{«*,+1'  bSJ+ J«^+ J>G£s)  * }> 
where  the  part  in  square  brackets  is  to  be  taken  for  the 
position  of  the  load,  and  those  under  the  integral  signs 
are  between  the  limits.  The  potential  energy  of  bending 
is,  as  before, 
-5M3> 
In    general     the   evaluation    of    these,    though   perfectly 
(Til 
simple,  is  tedious.     The  values  of  y  and  -^  are  taken  from 
earlier  parts  of  this  paper,  and  the  energy  expressions  then 
used  in  the  Lagrangian  Equation  just  as  in  the  example 
already  given. 
§  29.  It  is,  however,  simpler  merely  to  add  the  terms  due 
to  Hotatory  Inertia  to  the  previous  solutions.  A  general 
investigation  of  these  terms  would  occupy  too  much  space, 
but  in  a  numerical  example  the  work  is  not  difficult. 
Referring  to  §  14  for  the  case  of  a  loaded  massive  bar,  the 
term  to  be  added  to  the  value  of  —  EI  -~  for  the  rotatory 
inertia  of  m  is 
_t  d*('jk\  ■ 
dAdxlJ 
and  for  an  approximate  solution  we  can  write 
MA         _-±  //i 
