Vibrations of the Mass Variable Systems
31
Using the notation
v S1 = v, v S2 = u
(28)
and the angular momentum of the separating body
L S2 = 2 I S2 = 2 I
(29)
where 2 is the angular velocity of the separating body and I is the tensor of
moments of inertia, principles of linear and angular momentums (7) and (23) are
M
v
t
−
M
t
(v − u) = F r
(30)
L
t
+
I
t
2 − ρ S2 ×
M
t
(v − u) = M
Fr
S + M
(31)
If the time tends to zero, relations transform into
M
dv
dt
= F r +
dM
dt
(u − v)
(32)
dL S
dt
= M
Fr
S + M + ρ S2 ×
dM
dt
(u − v) +
dI
dt
2
(33)
For L S = I, where I = I S is the moment of inertia and is the angular
velocity of rotation, the first time derivative is
dL S
dt
= I
d
dt
+ × I +
dI
dt
(34)
Substituting (27) into (26), we obtain
M
dv
dt
= F r + φ
(35)
I
d
dt
+ × I = M
Fr
S + M + M
φ
S + + S
(36)
where
φ =
dM
dt
(u − v)
(37)
is the well-known reactive force introduced by Meshchersky [13],
M
φ
S = ρ S2 ×
dM
dt
(u − v)
(38)