Transverse Vebrations of Rotating Shafts. 339 
The term $1,? contributes nothing to Lagrange’s equa- 
tions and is for this reason omitted in the text. 
Assigning any arbitrary value to. @, We eee ae “Esa 
of &2—one of which m: ay be i 
different types of vibration. 
§ 42. In general each of the equations (10) and (11) con- 
tains both < and @, and each root of k given by (12) depends 
on both Mand I’. There is obv iously, however, a complete 
separation of the transverse and oscillatory movements 
when the load is at the centre of the span. For putting 
b=a, we have < only in (10) and @ only in (11) ; while the 
right- hand side of (12) vanishes, and we have for the trans- 
verse vibration 

= 45 ( RMP yo, oo ee aa eli) 
for the oscillatory vibration 
fe ys ee ee 
Answering to #a=0 we have | 
2/2 MP/41, 
SSF 
if the load be a thin circular disk of radius +. 
Thus even when #=0, k, will exceed 4, unless the radius 
of the disk be equal to the span. 
As @ increases, k, increases while k, diminishes; thus 
under ordinary circumstances the frequency of the transverse 
vibration is much the less of the two. : 
In the above special case it is obvious that k, cannot 
vanish, and that it is only the transverse vibration in con- 
nexion with which instability can arise. The critical angular 
velocity, answering to &, becoming nil, is given by 
wo =A8(HI/MP). . 2... . (15) 
Even in the general case it is easily shown that one only 
of the two values of k? supplied by (12) can possibly vanish. 
For assuming & zero, we find the equation to reduce to 
o MI’ +? . 3EL{M(a-!4-") —I/(a-2 +5-°)} 
—9(E1)?(a+)2/a*b3=0, (16) 
a quadratic in w* whose roots ure of opposite sign. As a 
negative value of w* supplies an imaginary value of o, there 
is only one real value of » for which & can vanish. And 
as (12), regarded as an equation in 4?, cannot have equal roots, 
unless b=a, only one of the two values of k? can be made 
to vanish, 
