Lateral  Vibration  of  Loaded  and  Unloaded  Bars.       355 
A  reference  should  be  made  here  to  the  important  papers 
by  Professor  Dunkerley  *  and  by  Dr.  Chree  f  on  the  "  Whirl- 
ing" of  Shafts.  The  relationship  of  the  whirling  to  the 
vibrational  problem  was  very  clearly  brought  out  by  Chree, 
and  under  certain  circumstances  the  two  problems  are 
identical. 
§  2.  The  notation  used  is  similar  to  that  of  the  paper 
previously  cited;  x  and  z  are  taken  parallel,  and  y  perpendi- 
cular, to  the  undisturbed  position  of  the  axis,  bending  occurring 
in  the  xy  plane. 
E  =  Young's  Modulus  for  the  material  (assumed  homo- 
geneous and  isotropic), 
p  =  density  of  material  of  bar, 
ft>  =  sectional  area, 
I  =  ft)/c2  =  Geometrical    Moment    of    Inertia    of    cross- 
section  about  the  neutral  axis, 
I  =  length  or  span  of  bar, 
/E 
U  =  \/  —  =  Velocity   of    transmission   of   longitudinal 
vibrations  in  the  bar, 
yi  ya  &c.  =  displacements  of  given  points  in  the  length 
of  the  bar, 
M0  Mi    =  bending  couples  required  to  fix  the  ends. 
We  have  also 
h  —  R. 
yi     y' 
and  for  the  frequency 
2tt  V         yi 
The  value  of  N  is,  however,  not  always  recorded,  as  it  is 
sufficient  to  find  the  expression  for  — . 
Vi 
Section  II.    Unloaded  Bars. 
§  3.  Uniform  Bar  Clamped  at  Both  Ends. — Taking  the 
origin  at  one  end,  the  terminal  conditions  are 
*  Phil.  Trans.  Roy.  Soc.  A  1894,  p.  279. 
t  Phil.  Map;.  May  1904,  p.  504. 
