8.2. EQUATIONS OF MOTION OF RIGID BODIES IN 3D SPACE 137
(c) In the remaining parts of this chapter, OXYZ is applied to denote the FFR while Gxyz with
associated unit vectors
*
{ ;
*
| ;
*
k is used as the coordinate system whose origin is at the
mass center G of the rigid body.
8.2.2 EQUATIONS OF ROTATIONAL MOTION
Recall that Equation (6.28) for the angular momentum of a system of particles,
*
H
G
D
*
H
0
G
which indicates that the angular momentum is independent of whether the velocities of the
particles are referenced to the origin O of FFR or to the center of mass G. With reference to
Equation (6.28) and Figure 8.2, one can write
*
H
G
D
n
X
iD1
*
r
0
i
m
i
*
v
0
i
;
where m
i
is the mass of the particle P
i
: It should be mentioned that for a particle no rotation
is possible. However, for a rigid body which can be considered as a system with infinite number
of particles and in the above equation the summation sign can be replaced by the integral sign
and in the limit m
i
approaches to d m. e velocity
*
v
0
i
can be replaced by the cross product of
the angular velocity
*
! of the body and position vector
*
r
0
i
; which is the position vector of the
particle P
i
relative to the centroidal frame (this is the notation used in Chapter 6). en the last
equation becomes
*
H
G
D
Z
*
r
0
i
*
!
*
r
0
i
d m: (8.3)
By making use of the vector triple product A
.
B C
/
D
.
A C
/
B
.
A B
/
C , or by writ-
ing
*
H
G
D
.
H
G
/
x
*
{ C
.
H
G
/
y
*
| C
.
H
G
/
z
*
k ,
*
r
0
i
D x
*
{ C y
*
| C z
*
k and
*
! D !
x
*
{ C !
y
*
| C
!
z
*
k as well as reference to Equation (3.16), one can write
.
H
G
/
x
D
Z
y
*
!
*
r
0
i
z
z
*
!
*
r
0
i
y
d m
D
Z
y
!
x
y !
y
x
z
.
!
z
x !
x
z
/
d m
D !
x
Z
y
2
C z
2
d m !
y
Z
xy d m !
z
Z
zx d m
or
.
H
G
/
x
D !
x
N
I
x
!
y
N
I
xy
!
z
N
I
xz
; (8.4)
where the centroidal mass moments of inertia and centroidal mass products of inertia of the body are,
respectively,
N
I
x
D
R
y
2
C z
2
d m,
N
I
xy
D
R
xy d m, and
N
I
xz
D
R
zx d m.
Similarly, with reference to Equation (3.16), one can show that
.
H
G
/
y
D !
x
N
I
yx
C !
y
N
I
y
!
z
N
I
yz
(8.5)
138 8. DYNAMICS OF RIGID BODIES
r⃑
i
'
Y
y
x
z
Z
O
X
P
i
r
⃑
i
'
G
Δm
i
⃑
i
'
ω⃑
Figure 8.2: Angular momentum relative to centroidal frame.
and
.
H
G
/
z
D !
x
N
I
zx
!
y
N
I
zy
C !
z
N
I
z
: (8.6)
Equations (8.4)–(8.6) can be written in matrix form as
0
@
.
H
G
/
x
.
H
G
/
y
.
H
G
/
z
1
A
D
2
4
N
I
x
N
I
xy
N
I
xz
N
I
yx
N
I
y
N
I
yz
N
I
zx
N
I
zy
N
I
z
3
5
0
@
!
x
!
y
!
z
1
A
:
(8.7)
It is always possible to choose a system of axes, known as the principal axes of inertia of a
rigid body such that the products of inertia become zero. at is, the coefficient matrix on
the rhs of Equation (8.7) reduces to a diagonal matrix.
If the three principal centroidal moments of inertia are equal and the corresponding three
components of the angular momentum about the mass center G are proportional to the
corresponding three components of the angular velocity then the vectors
*
H
G
and
*
! are
collinear.
Having presented the angular momentum of a rigid body in 3D space, one is ready to
consider the equations of rotational motion for a rigid body. e Newtonian or inertia or fixed
frame of reference OXYZ, and the body attached frame or rotating frame of reference Gxyz, which
is generally chosen so that the moments and products of inertia are independent of time, are
shown in Figure 8.3.
With reference to Equation (6.27), the rotational equations of motion for a rigid body is
given by
n
X
iD1
*
M
G
i
D
P
*
H
G
: (8.8)
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