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[     354     ] 
XXVIII.  On  the  Lateral  Vibration  of  Loaded  and  Unloaded 
Bars.  By  John  Morrow,  2L.Sc.  (  Vict.),  M.Eng.  (Liver- 
pool), Lecturer  in  Engineering,  University  College, 
Bristol  *. 
Contexts. 
Introduction. 
Unloaded  Bars. 
Loaded  Bars  of  Negligible  Mass. 
Loaded  Massive  Bars. 
Practical  Formulas  for  Loaded  Bars. 
Correction  for  Rotatory  Inertia. 
Section  I.   Introduction  and  Notation. 
§  1.  A  METHOD  o£  calculating  the  frequency  o£  the 
J-JL  lateral  vibration  o£  bars  has  been  described  in  a 
recent  paper,  "  On  the  Lateral  Vibration  of  Bars  o£  Uniform 
and  Varying  Sectional  Area "  (see  Philosophical  Magazine, 
July  1905). 
It  is  an  important  feature  of  this  method  that  it  gives,  in 
a  simple  form,  the  equation  of  the  elastic  central  line  of  the 
displaced  bar,  and  thus  provides  data  from  which  the  stresses 
and  strains  in  all  parts  may  be  readily  calculated.  The 
method  lends  itself  to  many  cases  of  loaded  bars  which  have 
not  hitherto  been  solved,  and  the  present  paper,  after  dealing 
with  some  cases  of  unloaded  bars  under  different  end  condi- 
tions, and  bars  of  negligible  mass  carrying  concentrated 
loads,  gives  more  particularly  the  solutions  for  some  im- 
portant problems  of  loaded  bars  of  appreciable  mass. 
It  Avill  be  seen  that  a  full  consideration  of  the  simpler 
cases  treated  first  very  materially  lessens  the  labour  involved 
in  solving  the  more  complex  problems  in  which  both  the  load 
and  the  mass  of  the  bar  itself  are  taken  into  account. 
These  solutions  are,  in  general,  obtained  by  a  process  of 
continuous  approximation.  Each  approximation  depends  on 
the  principle  that,  at  any  point  in  the  length  of  the  bar,  the 
curvature  is  equal  to  the  couple  due  to  the  reversed  effective 
forces  divided  by  the  flexural  rigidity. 
To  estimate  the  value  of  the  couple  a  vibration-curve  must 
be  assumed.  The  above  principle  then  gives  an  expression 
for  the  curvature  at  all  points. 
The  process  of  continuous  approximation  to  the  exact 
solution  is  based  on  the  fact  that  the  expression  for  the 
deflexion,  as  obtained  from  that  for  the  curvature,  is  a  much 
closer  approximation  to  the  truth  than  is  that  originally 
assumed  for  the  purpose  of  calculating  the  effective  forces. 
*  Communicated  by  the  Physical  Society  :  read  November  24,  1905. 
