8.2. EQUATIONS OF MOTION OF RIGID BODIES IN 3D SPACE 139
Y
y
x
z
Z
O
X
G
H
⃑
G
Ω
⃑
Figure 8.3: A rigid body with body attached frame of reference.
Note in Equation (8.3)
*
! is the angular velocity of the rigid body. However, in general, the
frame of reference Gxyz may be rotating w.r.t. the FFR such that the unit vectors
*
{ ,
*
| ,
*
k are
time dependent. Let
*
be the angular velocity of Gxyz. us, by applying Equation (7.6) in
which the vector
*
P is replaced by
*
H
G
, one has
P
*
H
G
OXYZ
D
d
*
H
G
dt
D
P
*
H
G
Gxyz
C
*
*
H
G
; (8.9)
where
P
*
H
G
OXYZ
is the rate of change of
*
H
G
of the rigid body w.r.t. the FFR,
P
*
H
G
Gxyz
is the rate of change of
*
H
G
of the rigid body with reference to the RFR, and
*
is the angular velocity of the RFR.
It may be appropriate to note that applying Equation (7.6) with the vector
*
P replaced by
*
! one
can obtain
P
*
!
OXYZ
D
d
*
!
dt
D
P
*
!
Gxyz
C
*
!
*
! D
P
*
!
Gxyz
:
is states the important fact that the angular acceleration is independent of the frame of reference.
In some problems this result can simplify the steps in the solution.
140 8. DYNAMICS OF RIGID BODIES
8.2.3 EQUATIONS OF CONSTRAINED MOTIONS
In the foregoing sections, the equations of motion apply to unconstrained rigid bodies. In this
section equations of motion of constrained rigid bodies are introduced.
Consider first the case in which the rigid body is constrained to a fixed point O. With
reference to Figure 8.4 and by making use of Equations (8.8) and (8.9), one can obtain
n
X
iD1
*
M
o
i
D
P
*
H
o
: (8.10)
P
*
H
o
OXYZ
D
d
*
H
o
dt
!
OXYZ
D
P
*
H
o
Gxyz
C
*
*
H
o
; (8.11)
where
P
*
H
o
OXYZ
is the rate of change of
*
H
o
of the rigid body w.r.t. the FFR,
P
*
H
o
Oxyz
is the rate of change of
*
H
o
of the rigid body with reference to the RFR, and
*
is the angular velocity of the RFR.
y
Y
X
x
Z
z
O
H
⃑
O
Ω
⃑
Figure 8.4: A rigid body constrained to a fixed point.
Now, consider the case in which the rigid body is constrained to rotate about a fixed axis, as
shown in Figure 8.5. For example, one is interested in setting up the equations of motion of a
rigid rotor supported by bearings.
With reference to Figure 8.5, one observes that the angular velocity of the body w.r.t.
the fixed frame OXYZ is denoted by
*
! D !
*
k . Substituting !
x
D 0; !
y
D 0, and !
z
D ! into
8.2. EQUATIONS OF MOTION OF RIGID BODIES IN 3D SPACE 141
y
Y
X
x
Z
z
O
ω⃑
Fixed Axis
Figure 8.5: A rigid body constrained to a fixed axis.
Equation (8.7) for the present problem, one obtains
.
H
o
/
x
D I
xz
!;
.
H
o
/
y
D I
yz
!;
.
H
o
/
z
D I
z
!:
Since Oxyz is attached to the rigid body such that
*
D
*
! , Equation (8.11) gives
P
*
H
o
OXYZ
D
P
*
H
o
Oxyz
C
*
!
*
H
o
D
I
xz
*
{ I
yz
*
| C I
z
*
k
d!
dt
C !
*
k
I
xz
*
{ I
yz
*
| C I
z
*
k
!
D
I
xz
*
{ I
yz
*
| C I
z
*
k
˛ C
I
yz
*
{ I
xz
*
|
!
2
D
I
xz
˛ C I
yz
!
2
*
{
I
xz
!
2
C I
yz
˛
*
| C I
z
˛
*
k :
It may be appropriate to note that if ! is constant such that
d!
dt
D ˛ D 0 then the term
P
*
H
o
Oxyz
D
I
xz
*
{ I
yz
*
| C I
z
*
k
d!
dt
D 0:
142 8. DYNAMICS OF RIGID BODIES
However, in general,
d!
dt
¤ 0. erefore, by making use of Equation (8.10) and writing compo-
nent by component, one has
n
X
iD1
.
M
o
/
x
i
D I
xz
˛ C I
yz
!
2
; (8.12a)
n
X
iD1
.
M
o
/
y
i
D I
xz
!
2
I
yz
˛; (8.12b)
n
X
iD1
.
M
o
/
z
i
D I
z
˛; (8.12c)
where the superscripts x; y, and z designate the components along the unit vector directions
*
{ ,
*
| , and
*
k , respectively, of the RFR.
If the externally applied forces are known in the body the angular acceleration ˛ can be
determined by Equation (8.12c). is acceleration ˛ can be integrated to obtain the angular
velocity !. e angular velocity and acceleration can be substituted into Equations (8.12a) and
(8.12c). ese equations, in addition to those in Equation (8.2), can be applied to find the
reactions at the bearings, B
1
and B
2
in Figure 8.5 assuming the body is a rigid rotor supported
by the bearings.
Example 8.1
A high-frequency radar used in a frigate can be considered as a rectangular plate of uniform
thickness of mass m and side lengths b and h. e plate is welded to a vertical rod AB, as shown
in Figure 8.6a. e angle between the vertical rod and the plate is D 30
ı
. If the vertical rod
rotates at a constant angular velocity
*
! , determine (a) the force and couple system representing
the dynamic reaction at point A and (b) the dynamic reaction at A when ! D 1 rad/s anti-
clockwise, m D 1;000 kg, b D 1:6 m, and h D 0:8 m.
Solution:
is solution requires use of Equations (8.1) and (8.11),
P
n
iD1
*
F
i
D m
*
a D m
*
a
G
; and
P
*
H
o
OXYZ
D
P
*
H
o
Gxyz
C
*
*
H
o
, in which the subscript o is replaced by A in the present
problem.
Angular momentum
*
H
A
Consider an auxiliary principal axes Axy
0
z
0
, as shown in Figure 8.6b such that
!
x
D 0; !
y
0
D ! cos 30
ı
; !
z
0
D ! sin 30
ı
;
I
y
0
D
1
12
mb
2
; I
z
0
D
1
12
m
b
2
C h
2
C m
h
2
2
D
1
12
mb
2
C
1
3
mh
2
;
8.2. EQUATIONS OF MOTION OF RIGID BODIES IN 3D SPACE 143
x
B
B B
y
y
y
y'
z
z z
F
⃑
A
M
⃑
A
z'
θ
θ
θ
ω⃑
ω⃑ ω⃑
b
h
A
A A
(a)
(b) (c)
Figure 8.6: (a) Rotating high-frequency rectangular radar in a frigate, (b) auxiliary principal
axes, and (c) force and couple system at A.
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