32
L. Cveticanin and D. Cveticanin
is the moment of the reactive force according to the mass center S and
S =
dI
dt
( 2 − )
(39)
is the reactive torque. The reactive force and the reactive torque are the results of
continual mass and moment of inertia variation in time. Reactive force is the product
of mass variation and of relative velocity of mass variable body. Reactive torque
is obtained by multiplying of time variation of the moment of inertia with relative
angular velocity of mass variable body. Reactive force and reactive torque do not
exist in dynamics of body with constant mass. Equation (28) defines the translational
motion, while (29) the rotation around mass center S.
For practical reasons, it is convenient to rewrite vector differential (28) and (29)
into scalar ones. Introducing the fixed coordinate system Oxyz, components v x , v y ,
v z of the velocity v, components u x , u y , u z of the velocity u and components F x , F y ,
F z of the resultant F r , we rewrite (28) into
M
dv x
dt
= F x + φ x , M
dv y
dt
= F y + φ y , M
dv y
dt
= F z + φ z ,
(40)
where the projections of the reactive force are
φ x =
dM
dt
(u x − v x ), φ y =
dM
dt
u y − v y
, φ z =
dM
dt
(u z − v z )
(41)
For the reference system S ξηζ , fixed to the body with the origin S in center of body
mass, inertial tensor I has nine components, but only six of them are independent: I ξ ,
I η , I ζ are the moments of inertia, and I ξη , I ηζ , I ζ ξ are centrifugal moments of inertia.
If axes are principal and centrifugal moments of inertia that are zero, inertial tensor
I has only three principal moments of inertia I ξ , I η , I ζ . Angular velocity has three
components Ω ξ , Ω η and Ω ζ , in this frame. If Ω 2ξ , Ω 2η , Ω 2ζ are components of the
angular velocity 2 , M
ξ , M
η and M
ζ are body-axis components of M
φ
S , M ξ , M η
and M ζ are body-axis components of M
Fr
S and M ξ , and M η and M ζ are projections
of the torque vector M, relation (29) gives equations
I ξ
dΩ ξ
dt
+
I ζ − I η
Ω η Ω ζ = M ξ + M ξ + M
φ
ξ + + ξ
I η
dΩ η
dt
+
I ξ − I ζ
Ω ξ Ω ζ = M η + M η + M
φ
η + + η
I ζ
dΩ ζ
dt
+
I η − I ξ
Ω ξ Ω η = M ζ + M ζ + M
φ
ζ + + ζ
(42)
where the components of the reactive torque S are
L. Cveticanin and D. Cveticanin
is the moment of the reactive force according to the mass center S and
S =
dI
dt
( 2 − )
(39)
is the reactive torque. The reactive force and the reactive torque are the results of
continual mass and moment of inertia variation in time. Reactive force is the product
of mass variation and of relative velocity of mass variable body. Reactive torque
is obtained by multiplying of time variation of the moment of inertia with relative
angular velocity of mass variable body. Reactive force and reactive torque do not
exist in dynamics of body with constant mass. Equation (28) defines the translational
motion, while (29) the rotation around mass center S.
For practical reasons, it is convenient to rewrite vector differential (28) and (29)
into scalar ones. Introducing the fixed coordinate system Oxyz, components v x , v y ,
v z of the velocity v, components u x , u y , u z of the velocity u and components F x , F y ,
F z of the resultant F r , we rewrite (28) into
M
dv x
dt
= F x + φ x , M
dv y
dt
= F y + φ y , M
dv y
dt
= F z + φ z ,
(40)
where the projections of the reactive force are
φ x =
dM
dt
(u x − v x ), φ y =
dM
dt
u y − v y
, φ z =
dM
dt
(u z − v z )
(41)
For the reference system S ξηζ , fixed to the body with the origin S in center of body
mass, inertial tensor I has nine components, but only six of them are independent: I ξ ,
I η , I ζ are the moments of inertia, and I ξη , I ηζ , I ζ ξ are centrifugal moments of inertia.
If axes are principal and centrifugal moments of inertia that are zero, inertial tensor
I has only three principal moments of inertia I ξ , I η , I ζ . Angular velocity has three
components Ω ξ , Ω η and Ω ζ , in this frame. If Ω 2ξ , Ω 2η , Ω 2ζ are components of the
angular velocity 2 , M
ξ , M
η and M
ζ are body-axis components of M
φ
S , M ξ , M η
and M ζ are body-axis components of M
Fr
S and M ξ , and M η and M ζ are projections
of the torque vector M, relation (29) gives equations
I ξ
dΩ ξ
dt
+
I ζ − I η
Ω η Ω ζ = M ξ + M ξ + M
φ
ξ + + ξ
I η
dΩ η
dt
+
I ξ − I ζ
Ω ξ Ω ζ = M η + M η + M
φ
η + + η
I ζ
dΩ ζ
dt
+
I η − I ξ
Ω ξ Ω η = M ζ + M ζ + M
φ
ζ + + ζ
(42)
where the components of the reactive torque S are