PROBLEM 18.100
An experimental Fresnel–lens solar–energy concentrator can rotate
about the horizontal axis AB, which passes through its mass center
G. It is supported at A and B by a steel framework, which can rotate
about the vertical y axis. The concentrator has a mass of 30 Mg, a
radius of gyration of 12 m about its axis of symmetry CD, and a
radius of gyration of 10 m about any transverse axis through G.
Knowing that the angular velocities
1
ω
and
2
ω
have constant
magnitudes equal to 0.20 rad/s and 0.25 rad/s, respectively, determine
for the position
60
θ
= °
(a) the forces exerted on the concentrator at
A and B, (b) the couple
2
Mk
applied to the concentrator at that
instant.
PROBLEM 18.100 (Continued)
Angular momentum about Point G.
22 2
21
[(10) (12) (10) ]
(100 cos 144 sin 100 )
G x x yy zz
G xyz
z
I II
m
m
ω ωω
ωωω
ωθ ωθ ω
′′ ′
′′
′
= ++
′′
= ++
′′
= ++
H i jk
H i jk
i jk
PROBLEM 18.100 (Continued)
Equate like components:
63
3
2
2
PROBLEM 18.101
A 6–lb homogeneous disk of radius 3 in. spins as shown at the
constant rate
1
60
ω
=
rad/s. The disk is supported by the fork-
ended rod AB, which is welded to the vertical shaft CBD. The
system is at rest when a couple
0
(0.25 ft lb)= ⋅Mj
is applied to
the shaft for 2 s and then removed. Determine the dynamic
reactions at C and D after the couple has been removed.
2
zz
C D mc
ω
+=−
PROBLEM 18.101 (Continued)
/
2
D D A AD
m
Σ==+×
M H Hr a
PROBLEM 18.102
A 6–lb homogeneous disk of radius 3 in. spins as shown at the
constant rate
1
60
ω
=
rad/s. The disk is supported by the fork-
ended rod AB, which is welded to the vertical shaft CBD. The
system is at rest when a couple
0
M
is applied as shown to the
shaft for 3 s and then removed. Knowing that the maximum
angular velocity reached by the shaft is 18 rad/s, determine
(a) the couple
0
,M
(b) the dynamic reactions at C and D after
the couple has been removed.
2
zz
C D mc
ω
+=−
PROBLEM 18.102 (Continued)
/
2
22
22
0 22 2
2( )
D D A AD
A
xz A
m
M b C C mbc mc mbc
ωωω
Σ==+×
+× + = − + +
M H Hr a
j j i kH i j k
2 22 2 2
0 12 2 2 1 2
1 11
22
2 42
zx
bC M bC m r bc m r c m r bc
ωω ω ω ω ω
+− = − + + + +
ij k i j k
22
02
1
:4
M mrc
ω
= +
j
(1)
22 22
12 12
11
:22 2 2
xx
mm
C r bc D r bc
bb
ωω ωω
=− + =−− +
k
(2)
22
12 2 12 2
11
:22 22
zz
mm
C r bc D r bc
bb
ωω ω ωω ω
= −=− +
i
(3)
Data:
11
6 lb, 3 in. 0.25 ft, 4 in. 0.33333 ft,
5 in. 0.41667 ft, 60 rad/s, 0
Wr b
c
ωω
= = = = =
= = = =
(a) While the couple is applied
2
2
2
18 6 rad/s
3t
ω
ω
= = =
22
1
PROBLEM 18.103
A 2.5 kg homogeneous disk of radius 80 mm rotates with an
angular velocity
1
ω
with respect to arm ABC, which is welded
to a shaft DCE rotating as shown at the constant rate
2
12
ω
=
rad/s. Friction in the bearing at A causes
1
ω
to
decrease at the rate of
2
15 rad/s .
Determine the dynamic
reactions at D and E at a time when
1
ω
has decreased to
50 rad/s.
x yxy A
DDEE m
+ ++=
i j i ja
PROBLEM 18.103 (Continued)
Resolve into components.
2
22
2
22
/
2 2 22
0 22 2 2 2
()
()
()
2( ) ( ) ( ) ( )
xx
yy
E E A AE A A A
xy A
D E mc b
D E mb c
m bcl m
l D D M m bl cl m cl bl m b c
ωω
ωω
ωω ωω ω
+= −
+= +
Σ = = + × = + −+ ×
×++ =+ − + + + +
M H H r a H i jk a
k i j kH i j k
22
1
22
2.5 1 (0.08) (12)(50) 0 (0.06)(0.15)(12) 5.20 N
(2)(0.15) 2
y
E
= − ++ =−
(21.2 N) (5.20 N)=−−E ij
PROBLEM 18.104
A 2.5-kg homogeneous disk of radius 80 mm rotates at the
constant rate
1
50
ω
=
rad/s with respect to arm ABC, which is
welded to a shaft DCE. Knowing that at the instant shown shaft
DCE has an angular velocity
2
ω
=
(12 rad/s)k and an angular
acceleration
2
2(8 rad/s ) ,
α
=k
determine (a) the couple which
must be applied to shaft DCE to produce that acceleration,
(b) the corresponding dynamic reactions at D and E.
x yxy A
DDEE m
+ ++=
i j i ja
PROBLEM 18.104 (Continued)
Resolve into components.
2
22
2
22
/
2 2 22
0 22 2 2 2
()
()
()
2( ) ( ) ( ) ( )
xx
yy
E E A AE A A A
xy A
D E mc b
D E mb c
m bcl m
l D D M m bl cl m cl bl m b c
ωω
ωω
ωω ωω ω
+= −
+= +
Σ = = + × = + −+ ×
×++ =+ − + + + +
M H H r a H i jk a
k i j kH i j k
22
1
PROBLEM 18.105
For the disk of Problem 18.99, determine (a) the couple M1j
which should be applied to arm ABC to give it an angular
acceleration
2
1
(7.5 rad/s )
α
= − j
when
1
5 rad/s,
ω
=
knowing that
the disk rotates at the constant rate
215 rad/s,
ω
=
(b) the force–
couple system representing the dynamic reaction at A at that
instant. Assume that ABC has a negligible mass.
PROBLEM 18.99 A thin disk of mass
4 kgm=
rotates with an
angular velocity
2
ω
with respect to arm ABC, which itself rotates
with an angular velocity
1
ω
about the y axis. Knowing that
ω
1
5 rad/s=
and
2
15 rad/s
ω
=
and that both are constant,
determine the force–couple system representing the dynamic
reaction at the support at A.
SOLUTION
Angular velocity of the disk.
12
(5 rad/s) (15 rad/s)
ωω
=+= +ωjk j k
Moments of inertia about principal axes passing through the mass center.
2
22
22
1
4
1(4)(0.150 m) 0.0225 kg m
4
10.045 kg m
2
xy
z
I I mr
I mr
= =
= = ⋅
Angular momentum about mass center C.
C xx yy zz
II I
ωωω
′′ ′′ ′′
=++
H i jk
PROBLEM 18.105 (Continued)
Acceleration of Point C.
1/ 1
22
( 7.5 ) (0.45 0.225 ) 5 ( 2.25 )
(3.3750 m/s ) (11.25 m/s )
C CA C
αω
= × + × =− × + + ×−
= −
a jr jv j i j j k
ki
(0.450 0.225 ) ( 45 13.5 ) 0.16875 3.375
6.0750 10.125 3.0375 0.16875 3.375
6.4125 6.2438 10.125
A
= + ×− + − +
=−+ +− +
=−+
M i j ik j i
j k i ji
i jk
(a) Required couple.
1
(6.24 N m)=−⋅Mj j
PROBLEM 18.106*
A slender homogeneous rod AB of mass m and length L is made to
rotate at the constant rate
ω
2 about the horizontal z axis, while frame
CD is made to rotate at the constant rate
ω
1 about the y axis. Express as
a function of the angle
θ
(a) the couple M1 required to maintain the
rotation of the frame, (b) the couple M2 required to maintain the
rotation of the rod, (c) the dynamic reactions at the supports C and D.
1
PROBLEM 18.106* (Continued)
Rate of change of .
G
H
We note that
1
ω
and
2
ω
are constant, while
θ
varies with t, with
2
.
θω
=
Eq. (18.22) yields
22
11
22
11 2
22
12 12
22 2
1 12
22
12 12
11
( ) (2cos 2 ) ( 2 cos sin )
24 12
11
sin 2
24 12
11
cos 2 sin 2
12 12
11
sin 2
24 12
11
(1 cos 2 ) sin
12 12
G G Gxyz G
G
mL mL
mL mL
mL mL
mL mL
mL mL
ω θθ ω θ θθ
ω ωθ ω
ωω θ ωω θ
ω θ ωω
ωω θ ωω
= +× =− + −
+ ×− +
=−−
++
= −−
HH ΩH i j
j ik
ij
ki
Hi
22
1
2
1sin 2
24 mL
θ
ωθ
+
j
k
(1)
PROBLEM 18.106* (Continued)
22
1sin 2
12
6mL
12
6mL
PROBLEM 18.107
A uniform thin disk of 6-in. diameter is attached to the end of a rod AB
of negligible mass which is supported by a ball-and–socket joint at
point A. Knowing that the disk is observed to precess about the vertical
axis AC at the constant rate of 36 rpm in the sense indicated and that
its axis of symmetry AB forms an angle
60
β
= °
with AC, determine
the rate at which the disk spins about rod AB.
PROBLEM 18.107
(Continued)
( )
2
22
10.25 0.03125 ft
22
Ir
m
1
= = =
( ) ( )
22
22 2
11
0.25 2 4.015625 ft
44
Irc
m
′= += + =
60 , 36 rpm 3.7699 rad/s
βφ
=°= =
Substituting into (1),
( )( )
2
0.03125 3.7699 4.015625 0.03125 3.7699 cos 60 10
ψ
−°
PROBLEM 18.108
A uniform thin disk of 6-in. diameter is attached to the end of a rod AB
of negligible mass which is supported by a ball-and–socket joint at
point A. Knowing that the disk is spinning about its axis of symmetry
AB at the rate of 2100 rpm in the sense indicated and that AB forms an
angle
45
β
= °
with the vertical axis AC, determine the two possible
rates of steady precession of the disk about the axis AC.
PROBLEM 18.108
(Continued)
( )
2
22
10.25 0.03125 ft
22
Ir
m
1
= = =
( ) ( )
22
22 2
11
0.25 2 4.015625 ft
44
Irc
m
′= += + =
45 , 2100 rpm 219.91 rad/s
βψ
=°= =
Substituting into (1),
( )( )
( )( ) ( )
( )( )
2
0.03125 219.91 4.015625 0.03125 cos 45 10
32.2 2 32.2 2
φφ
−°
+ −=
( )
2
0.106711 0.043748 1 0
φφ
+ −=
6.1537 rad/s or 3.7145 rad/s
φφ
=−=
58.8 rpm, 35.5 rpm
φ
= −