356  Mr.  J.  Morrow  on  the  Lateral  Vibration 
and  hence  the  equation  to  be  assumed  as  the  first  approxima- 
tion to  the  type  of  displacement  is 
16?/,/   ,      %v>       x±\ 
The  ordinary  Euler-Bernoulli  theory  leads  to 
dry  Cx  Cl 
—  EI^  =  M0  +  /og>1    0?— £)yzdz— \pa>x\    ydx. 
Substituting  for  y,  integrating,  and  determining  M0  by  the 
all 
condition  that  -~ -  =0   when  x  =  l,  we  find 
dx 
+  •059,524^; 
...    h=  -494-7  A 
Vi  P<»1 
Similarly,  using  these  values  of  y  and  ^,  we  arrive  at  a 
second  approximation,  in  which 
-y  =  '079^(^1  ( -35572  IV -  -55113  Va*  +  -49603  ZV 
r12\ 
501V) 
-  -33069  /V  -f  -05511  a10-  -03006  ^  +  -0( 
and  ••  -T.T 
_  21  =  500-4  EI 
agreeing  well  with  the  value  of  500*6  obtained  for  the  con- 
stant by  the  exact  solution. 
§4.    Clamped-supported  Bar. — In  this   case  let  y{  be  the 
displacement  at  such  a  distance  from  the  fixed  end  (taken  as 
origin)  that  ~~  —  0  there.     The  initial  type  is  then 
/  3  y"       o  oc        cc\ 
y  =  7-6931^^-2^  +  ^ 
and 
w,^?  £  yi  JO  l  Ul   JQ> 
)dx. 
