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PROBLEM 15.244
A square plate of side 2r is welded to a vertical shaft that
rotates with a constant angular velocity
1
.ω
At the same
time, rod AB of length r rotates about the center of the
plate with a constant angular velocity
2
ω
with respect to
the plate. For the position of the plate shown, determine
the acceleration of end B of the rod if (a)
0,
θ
=
(b)
90 ,
θ
= °
(c)
180 .
θ
= °
PROBLEM 15.244 (Continued)
(b)
90
θ
= °
/
/
sin30 cos30
BA
BO
r
rr r
=
= + °− °
ri
ri j k
PROBLEM 15.245
Two disks, each of 130-mm radius, are welded to the 500-mm rod
CD. The rod-and-disks unit rotates at the constant rate
ω
2
3 rad/s=
with respect to arm AB. Knowing that at the instant shown
1
4 rad/s,
ω
=
determine the velocity and acceleration of (a) Point
E, (b) Point F.
PROBLEM 15.245 (Continued)
(b) Point F.
/
/
(0.25 m) (0.13 m)
(0.13 m)
FB
FD
= +
=
r ik
rk
PROBLEM 15.246
In Problem 15.245, determine the velocity and acceleration of
(a) Point G, (b) Point H.
PROBLEM 15.245 Two disks, each of 130-mm radius, are welded
to the 500-mm rod CD. The rod-and-disks unit rotates at the
constant rate
ω
2
3 rad/s=
with respect to arm AB. Knowing that at
the instant shown
1
4 rad/s,
ω
=
determine the velocity and
acceleration of (a) Point E, (b) Point F.
PROBLEM 15.246 (Continued)
Acceleration of Point G.
/G G GF c
′
=++aaa a
4 1.17 3.12
G
=−+ai j i
22
(7.12 m/s ) (1.170 m/s )
G
= −
a ij
PROBLEM 15.247
The position of the stylus tip A is controlled by the
robot shown. In the position shown the stylus moves at
a constant speed
180 mm/su=
relative to the solenoid
BC. At the same time, arm CD rotates at the constant
rate
2
1.6 rad/s
ω
=
with respect to component DEG.
Knowing that the entire robot rotates about the X axis at
the constant rate
1
1.2 rad/s,
ω
=
determine (a) the
velocity of A, (b) the acceleration of A.
PROBLEM 15.247 (Continued)
2
/
22 2
/
(480 mm/s ) [ (720 mm/s) (400 mm/s) (960 mm/s) ]
480 1.2 1.6 0
960 720 400
480 640 480 864 1536
(640 mm/s ) (960 mm/s ) (2400 mm/s )
A D CD
AD
ω
′
′
=− + ×− + +
=−+
−
=−+ − − −
=−−
a j jki
i jk
j
ji jk k
a ij k
/
22 2
(640 mm/s ) (1392 mm/s ) (2400 mm/s )
A D AD
A
′′
′
= +
=−−
aaa
a ijk
Motion of Point A relative to the frame.
/
/
(180 mm/s)
0
AF
AF
= = −
=
vu i
a
1/
(1.2 rad/s) [ (500 mm) (300 mm) (600 mm) ]
(360 mm/s) (720 mm/s)
A AG
′= × = ×− + +
= −
vωr i i j k
kj
1 1/ 1
()
A AG A
ω
′′
=×× =×
aωr ωv
PROBLEM 15.247 (Continued)
Motion of Point A relative to the frame.
/ 2/
(1.6 rad/s) [ (250 mm) (600 mm) ] (180 mm/s)
(400 mm/s) (960 mm/s) (180 mm/s)
AF AD
=×+
= ×− + −
= +−
vωr u
j ik i
kii
/
:
AF
a
(Since A moves on CD, which rotates at rate
2
,ω
we have a Coriolis term here).
/ 2 2/ 2
22
2 22
22
( )2
{(1.6 rad/s) [ (250 mm) (600 mm) ]} 2
(1.6 rad/s) [(400 mm/s) (960 mm/s) ] 2(1.6 rad/s) ( 180 mm)
(640 mm/s ) (1536 mm/s ) (576 mm/s )
(640 mm/s ) (960 mm/s )
AF AD
=×× + ×
= × ×− + + ×
= × + + ×−
=−+
= −
aω ωr ωu
ω j i k ωu
j k i ji
i kk
ik
A=−+v ijk
Coriolis acceleration:
1/
2
c AF
= ×aωv
2(1.2 rad/s) [(400 mm) (780 mm/s) ]
c=×+
a ik i
/A A AF c
432 864 640 960 960
A
=−− + − −a jk ik j
222
PROBLEM 15.248
A wheel moves in the xy plane in such a way that the location of its
center is given by the equations =123 and == 2, where xO
and yO are measured in feet and t is measured in seconds. The angular
displacement of a radial line measured from a vertical reference line is
= 84, where is measured in radians. Determine the velocity of the
point P located on the horizontal diameter of the wheel at t = 1 s.
