8.2. EQUATIONS OF MOTION OF RIGID BODIES IN 3D SPACE 145
(b) Substituting for the given data, the dynamic force and couple system at A are obtained as
*
F
A
D m!
2
h
2
sin 30
ı
*
k D 200
*
k N;
*
M
A
D m!
2
1
3
h
2
sin 30
ı
cos 30
ı
*
{ D 92:3760
*
{ Nm.
8.2.4 EULERIAN ANGLES AND MOTION OF A GYROSCOPE
A gyroscope is generally defined as a rotating body having one axis of symmetry and the rotation
about the axis of symmetry is relatively large compared with the rotation about any other axis.
e term gyroscope was introduced by Foucault to denote a device that can be applied to prove
the movement of the earth [1]. A modern example is a rotor mounted in a Cardan’s suspension,
as shown in Figure 8.7a. A gyroscope can assume any orientation. However, its mass center must
be stationary in space. e two chosen frames of reference are the inertial frame of reference
OXYZ and the body fixed or body attached frame Oxyz which is frequently called the RFR.
To completely characterize the position of the gyroscope at any given moment, the so-
called Eulerian or Euler angles are used. e commonly adopted notation of Eulerian angles (in
radian) in sequence is ; , and, where
is the rotation of the outer gimbal about O
1
O
2
(from the reference position in Figure 8.7a
to the position in Figure 8.7b),
is the rotation of the inner gimbal about A
1
A
2
, and
is the rotation of the rotor about B
1
B
2
.
It may be appropriate to mention that the sequence of these three angles must be main-
tained [1, 2] because finite rotations are not vectors (of course, for infinitesimal small rotations
they can be expressed as vectors).
Recall, the unit vectors
*
{ ;
*
| ; and
*
k , respectively, along the x; y, and z axes of the RFR are
attached to the rotor. Note that the y-axis along A
1
A
2
and the z-axis along B
1
B
2
are principal
axes of inertia for the gyroscope. One can write the angular velocity
*
! of the gyroscope w.r.t.
the FFR as
*
! D
P
*
K C
P
*
| C
P
*
k (8.15)
in which
*
K is the unit vector along the Z-axis of the FFR while
P
D
d
dt
;
P
D
d
dt
, and
P
D
d
dt
are, respectively, the precession, nutation, and spin. As the vector components for
*
! in Equa-
tion (8.15) are not orthogonal
*
K is expressed in components parallel to the x and z axes. at
is,
*
K D sin
*
{ C cos
*
k :
146 8. DYNAMICS OF RIGID BODIES
Z
Y
X
A
1
O
1
O
2
A
2
B
1
B
2
H
2
H
1
O
Outer Gimbal
Inner Gimbal
O
Z
Y
A
2
B
2
O
2
O
1
H
2
H
1
A
1
B
1
θ
ϕ ψ ψ
O
z
Z
Y
y
x
A
2
B
2
O
2
O
1
H
2
H
1
A
1
B
1
θ
ϕ
ϕ̇K
⃑
ψ
̇k
⃑
θ̇ j
⃑
(a)
(b) (c)
Figure 8.7: Gyroscope with (a) FFR, (b) Eulerian angles, and (c) angular velocities.
8.2. EQUATIONS OF MOTION OF RIGID BODIES IN 3D SPACE 147
Substituting this equation into (8.15) and rearranging, one obtains
*
! D
P
sin
*
{ C
P
*
| C
P
C
P
cos
*
k (8.16)
which can be identified with the angular velocity in Equation (8.3),
*
! D !
x
*
{ C !
y
*
| C !
z
*
k
so that
!
x
D
P
sin ; !
y
D
P
; !
z
D
P
C
P
cos : (8.17)
us, if one denotes the moment of inertia of the rotor about its spin axis by I
z
; moment of
inertia about a transverse axis through mass center O by I
x
, and disregarding the mass of the
gimbals as well as making use of Equation (8.7), the angular momentum of the gyroscope about
O becomes
0
@
.
H
o
/
x
.
H
o
/
y
.
H
o
/
z
1
A
D
2
4
I
x
0 0
0 I
x
0
0 0 I
z
3
5
0
@
!
x
!
y
!
z
1
A
;
or in vector form
*
H
o
D
I
x
P
sin
*
{ C
I
x
P
*
| C I
z
P
C
P
cos
*
k : (8.18)
Before the equations of motion for the above gyroscope are derived, one needs to obtain the
angular velocity
*
of the RFR which, in the present case, is attached to the inner gimbal and
therefore,
*
D
P
*
K C
P
*
|
which can be identified as
*
! with its spin component equal to zero. us, by Equation (8.16)
one has
*
D
P
sin
*
{ C
P
*
| C
P
cos
*
k : (8.19)
Substituting Equations (8.18) and (8.19) into (8.11), and operating on the resulting equation as
well as rearranging, one can obtain the equations for the motion of the gyroscope as
n
X
iD1
.
M
o
/
x
i
D I
x
R
sin C 2
P
P
cos
C I
z
P
P
C
P
cos
; (8.20a)
n
X
iD1
.
M
o
/
y
i
D I
x
R
P
2
sin cos
C I
z
P
sin
P
C
P
cos
; (8.20b)
n
X
iD1
.
M
o
/
z
i
D I
z
d
P
C
P
cos
dt
: (8.20c)
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