938 Dr. C. Chree on the Whirling and 
These conditions are easily seen to be satisfied by 
y =(82—a0)(a@ /2a) + (a0 —z)(x* /2a°), (4) 
y! = (824+ 60)(x'/2b) —(b8 +. z)(a”/20°), 
when we treat z and 9 as constants. Under the same as- 
sumption these expressions satisfy 
adyidt—O, ay ids =O, 
equations answering to the absence of external forces on the 
shaft itself. 
_ The components of velocity at the C.G. C of the load are 
z perpendicular to AB in the plane of bending, and wz per-_ 
pendicular to the plane of bending. Thus, treating @ as small, 
we have for the kinetic energy of the system 
T=1M(z?+ 2?) +i]e?+41(e—ow6). . . (5) 
The shaft being supposed massless, contributes nothing 
to T. It is, however, the seat of the potential energy, V, 
which is given on the ordinary Euler-Bernoulli theory by 
a 6 
vegni{ ("emeerees (ceyjeerpec}.. © 
0 0 
Substituting the values of d?y/dx? and d?y//dx? from (4), 
and carrying out the integrations, we easily find 
V=4EL{3(@0—<)?/a?+3(b0+2)7/t.. 2. 2. (7 
Employing these values of T and V in the two Lagrangian 
equations 
dat GE ay 
ula) at ae 
d (dT dT dV 
alg) ata 
we have 
M(z—o%z) + 3EI<(a-3 +b-) + 3E10(6-?—a-*) =0, . (10) 
V(O+ w? 6) + 8EIz(6-?—a-”) + B8EIO(b-* +a) = 0... (11) 
For a vibration of frequency k/27 we have 
= —kz, 6= — k?@. 
Substituting for = and @ in (10) and (11), and eliminating 
z and @ between these two equations, we have, as in (8) of 
§ 13, 
§M(k? + w?) —3BI(a-8 + 0-%) } {1 (2 — ow”) —3EI (a! + 5-1) } 
=9(EL%(b-2—-a-)?_—. «(12 
