of  Loaded  and  Unloaded  Bars.  361 
§  12.  Dynamical  Method. — Other  methods  for  the  solution 
of  some  of  the  problems  of  this  section  have  been  used  by 
Dr.  Chree,  but  they  depend  on  the  assumption  of  a  curve  of 
vibration. 
A  correct  type  having  been  assumed,  expressions  are 
obtained  for  the  Kinetic  and  Potential  Energies  of  the  system. 
Employing  these  in  Lagrange's  Equations  of  Motion  the 
frequency  is  readily  obtained. 
Taking,  for  example,  a  massless  bar  carrying  a  load  m,  as 
in  §8.     Assume 
t  1  t  V~>  9  /  9  \ 
y  =  r\ax  (I  — «  — X  -), 
measuring  x  and  x  from  opposite  ends  of  the   bar. 
The  Kinetic  Energy 
T  =  hny*-2malb>y2. 
The  Potential  Energy  of  Bending,  V, 
-*"{K3)M(»W 
=  6Ela2b2{a  +  b)v2. 
The  Lagrangian  Equation 
becomes 
1/BT\      BT  +  ^V=0 
4,maibi-  +  12EIa2b2(a  +  b)y  =  0; 
v  =  _  3EK 
rj  ma2b2' 
Section  IV.  Loaded  Massive  Bars. 
§  13.  When  the  mass  of  the  bar,  in  addition  to  that  of  the 
concentrated  load,  is  taken  into  account,  the  expressions  for 
the  elastic  curve  and  the  frequency  are  more  complicated. 
If,  however,  the  position  of  the  load  and  the  ratio  of  the 
masses  are  given,  the  solutions  are  simple. 
§  14.  Clamped-Free  Massive  Bar,  Ljoad  at  End. — In  this 
case,  if  we  assume 
/'  4/        l,r'\ 
^'T-37  +  ;w«)- 
Phil.  Mag.  S.  6.  Vol.  11.  No.  63.  March  1906.  2  B 
