5_5_Excercise 5

Problem 18.157
A
600 mm
.
φ
30o
.
ψ
A
.
φ
B
30o
A 2-kg disk of 150-mm diameter is
attached to the end of a rod AB of
negligible mass which is supported
by a ball-and-socket joint at A. If the
disk is observed to precess about the
vertical in the sense indicated at a
constant rate of 36 rpm, determine
.
th rate
the
t off spin
i ψ off the
th disk
di k about
b t AB.
AB
Problem 18.157
Solving Problems on Your Own
A 2-kg disk of 150-mm diameter is attached
600 mm to the end of a rod AB of negligible mass
which is supported by a ball-and-socket
jjoint at A. If the disk is observed to p
precess
B
about the vertical in the sense indicated at
.
a constant rate of 36 rpm, determine the
ψ
.
rate of spin ψ of the disk about AB.
1. Determine the angular velocity ω of the body and the angular
velocity Ω of the rotating frame: ω is the angular velocity of the
body with respect to a fixed frame of reference. The vector ω may
be resolved into components along the rotating axes. The angular
velocity is often obtained by adding two components of angular
velocities ω1 and ω2. Ω is the angular velocity of the rotating
frame. If the rotating frame is rigidly attached to the body, Ω = ω .
A
30o
.
φ
Problem 18.157
Solving Problems on Your Own
A 2-kg disk of 150-mm diameter is attached
600 mm to the end of a rod AB of negligible mass
which is supported by a ball-and-socket
jjoint at A. If the disk is observed to p
precess
B
about the vertical in the sense indicated at
.
a constant rate of 36 rpm, determine the
ψ
.
rate of spin ψ of the disk about AB.
2. Determine the mass moments and products of inertia of the
y For a three dimensional bodyy these are the q
quantities Ix,
body:
Iy, Iz, Ixy, Ixz, and Iyz, where xyz is the rotating frame. If the rotating
frame is centered at G (mass center) and is in the direction of the
principal axes of inertia (Gx’y’z’), then the products of inertia are
zero and Ix, Iy, and Iz, are the principal centroidal moments of
inertia.
A
.
φ
30o
Problem 18.157
Solving Problems on Your Own
A 2-kg disk of 150-mm diameter is attached
600 mm to the end of a rod AB of negligible mass
which is supported by a ball-and-socket
jjoint at A. If the disk is observed to p
precess
B
about the vertical in the sense indicated at
.
a constant rate of 36 rpm, determine the
ψ
.
rate of spin ψ of the disk about AB.
3. Determine the angular momentum of the body: The angular
momentum HG of a rigid body about point A can be expressed in
terms of the components of its angular velocity ω and its moments
and products of inertia.
( HA )x = + Ix ωx - Ixy ωy - Ixz ωz
( HA )y = - Iyx ωx + Iy ωy - Iyz ωz
( HA )z = - Izx ωx - Izy ωy + Iz ωz
A
30o
.
φ
Problem 18.157
Solving Problems on Your Own
A 2-kg disk of 150-mm diameter is attached
600 mm to the end of a rod AB of negligible mass
which is supported by a ball-and-socket
jjoint at A. If the disk is observed to p
precess
B
about the vertical in the sense indicated at
.
a constant rate of 36 rpm, determine the
ψ
.
rate of spin ψ of the disk about AB.
4. Compute the rate of change of angular momentum : The rate
of change of HA with respect to a fixed frame is given by
.
.
HA = ( HA )Oxyz + Ω x HA
.
where ( HA )Oxyz is the rate of change of HA with respect to the
rotating frame, and Ω is the angular velocity of the rotating frame.
If the rotating frame is rigidly attached to the body, Ω is equal to ω,
the angular velocity of the body.
A
.
φ
30o
Problem 18.157
Solving Problems on Your Own
A 2-kg disk of 150-mm diameter is attached
600 mm to the end of a rod AB of negligible mass
which is supported by a ball-and-socket
jjoint at A. If the disk is observed to p
precess
B
about the vertical in the sense indicated at
.
a constant rate of 36 rpm, determine the
ψ
.
rate of spin ψ of the disk about AB.
5. Draw the free-body-diagram: The diagram shows the system
of the external forces exerted on the body.
6. Write equation of motion: For a body rotating about point A :
.
Σ MA = H A
Z
z
φK
.
ψk
A
Problem 18.157 Solution
Determine the angular velocity ω of the body.
.
.
y
x
θ
.
φ = - 36 rpm = - 3.770 rad/s
.
. .
ω = - φ sinθ i + ( ψ + φ cosθ ) k
L
Determine the angular velocity Ω of the
rotating frame.
B
.
.
Ω = -φ sinθ i + φ cosθ k
θ = 30o
Determine the mass moments of inertia.
L = 600 mm
Ix = 41 m r2 + m L2
Ix = 41 2 ( 0.075 )2 + 2 ( 0.6 )2 = 0.7228 kg. m2
Iz = 21 m r2 = 21 2 ( 0.075 )2 = 0.005625 kg. m2
z
A
.
θ
.
Determine the angular momentum of the body.
.
L
.
HA = Ix ωx i + Iz ωz k
.
. .
HA = - Ix φ sinθ i + Iz ( ψ + φ cosθ ) k
B
Compute the rate of change of angular
.
.
momentum.
Ω = -φ sinθ i + φ cosθ k
HA = ( HA )oxyz + Ω x HA
.
.
( HA )oxyz = 0, since ψ = constant
j
. k
. i
0
φ cosθ
HA = Ω x HA = -φ sinθ
.
. .
- Ix φ sinθ 0 Iz (ψ + φ cosθ )
.
.
. .
HA = φsinθ [ Iz (ψ + φ cosθ ) - Ix φ cosθ ] j
.
.
ω = - φ sinθ i + ( ψ + φ cosθ ) k
Angular momentum about A :
y
x
.
.
φK
.
ψk
Problem 18.157 Solution
Z
Z
z
φK
.
ψk
Problem 18.157 Solution
.
A
.
Draw the free-body-diagram.
y
x
θ
L
B
Z
z
Az
A
x
A
x
θ
.
B
Problem 18.157 Solution
Write equation of motion.
Ay
y
Recall:
.
.
.
.
.
HA = φsinθ [ Iz (ψ + φ cosθ ) - Ix φ cosθ ] j
L
Ax
θ
L
W = mg
Z
Az
Ay
y
Ax
The y axis and Ay
are in a direction
perpendicular (out)
to the plane of the
figure.
z
.
B
Sum of moments about A :
.
Σ MA = HA :
W = mg
.
.
.
.
((- L k ) x ( -mg
mg K ) = φsinθ [ Iz (ψ + φ cosθ ) - Ix φ cosθ ] j
.
.
.
.
- mg L sinθ j = φsinθ [ Iz (ψ + φ cosθ ) - Ix φ cosθ ] j
I -I .
mgL
.
.
ψ = x I z φ cosθ Iz φ
z
x
A
.
φK
.
ψk
Problem 18.157 Solution
Z
z
.
y
θ = 30o
Ix = 0.7228 kg. m2
L = 600 mm
Iz = 0.005625 kg. m2
L
.
φ = - 3.77 rad/s
m = 2 kg
k
θ
B
I -I .
mgL
.
.
ψ = x I z φ cosθ Iz φ
z
.
0.7228 - 0.005625
ψ =
( 3.77
(3 77 ) cos 30o 0.005625
.
ψ = 138.9 rad/s
(2)(9.81)(0.6)
0.005625 (- 3.77)

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