Problems
15.19
15.20
The shaft of the distributed mass system is simply
supported by the bearings at A and B and has
a constant speed of 550 rev/min. The reaction
forces acting on the bearings at A and B are FA =
−17.5î + 30.3ĵ N and FB = −25.0î − 43.3ĵ N,
respectively. Using the graphic approach determine
the magnitudes and angular orientations of the
reaction forces at bearings A and B.
879
the system acting on the ground at bearings A and
B are (F21 )A = 250î + 75ĵ N and (F21 )B = 80î −
125ĵ N, respectively. Determine the magnitudes
and orientations of the correcting masses that must
be removed in the correction planes 1 and 2 to
ensure moment (dynamic) balance of the system.
15.21
The distributed mass system, denoted as body 2,
is simply supported bearings at A and B and has
a constant speed of 240 rev/min. The forces from
The shaft, simply supported by bearings at A and
B, has a constant speed of 300 rev/min. Using
the graphic approach determine the magnitudes
and orientations of the reaction forces at bearings
A and B.
y
a
m2
b
c
y
d
m2
40°
R2
m1
R1
20° x
70°
ω
z
m1
B
A
1
R3
1
1
m3
m3
Figure P15.19 a = 1.2 in, b = c = 1.0 in, d = 0.8 in, R1 = 0.80 in, R2 = 0.80 in,
R3 = 0.60 in, m1 = 15.4 lb, m2 = 26.5 lb, and m3 = 17.6 lb.
x
a
b
c
ω
z
2
B
1
1
A
1
2
Figure P15.20 a = 250 mm, b = 200 mm, c = 75 mm, and RC1 = RC2 = 30 mm.
y
y
a
m2
m1
80°
R1
R2
R3
60°
m3
1
30°
x
b
ω
z
m2
A
1
c
m3
Figure P15.21 a = 40 mm, b = c = 15 mm, d = 25 mm, R1 = 10 mm,
R2 = 35 mm, R3 = 15 mm, m1 = 17 kg, m2 = 4 kg, and m3 = 8 kg.
d
m1
B
1
880
15.22
BALANCING
The angular speed of the continuous mass system,
denoted as 2, in the simply supported bearings at
A and B is a constant 360 rev/min. The forces
from the system acting on the ground at bearings
A and B are specified as (F21 )A = 73î − 79ĵ lb
and (F21 )B = 63î+118ĵ lb, respectively. Determine
the magnitudes and orientations of the masses that
must be removed in the correcting planes 1 and 2
to ensure dynamic balance of the system.
15.23
The constant angular velocity of the two-cylinder
engine crankshaft is ω = 200 rad/s counterclockwise. Determine the x and y components of the
primary shaking force acting on the crankshaft
bearing in terms of the crank angle θ. Then,
determine the magnitudes and orientations of the
correcting masses that must be added at the radial
distance RC = 40 mm from the crankshaft axis.
Determine the answers when the reference line
(attached to crank 1) is specified at the crank angle
θ = 30◦ , as shown on the figure to the right.
y
a
b
c
2
ω
z
B
A
1
2
1
1
Figure P15.22 a = 30 in, b = 27 in, c = 7 in, and RC1 = RC2 = 2 in.
y
y
1
m1
–ω
45°
L
Ref. Line
Ref. Line
θ
30°
R
1
x
120°
ω
x
60°
R
L
m2
1
Figure P15.23 R1 = R2 = R = 80 mm, L1 = L2 = L = 160 mm, and m1 = m2 = m = 15 kg.
Problems
15.24
15.25
The two-cylinder engine crankshaft is rotating
counterclockwise with a constant angular velocity ω = 45 rad/s. Determine the magnitude and
direction of the primary shaking force in terms of
crank angle θ . If correcting masses are required to
balance the primary shaking force, then determine:
(a) the magnitudes and orientations of the inertial
forces created by these correcting masses, and (b)
the magnitudes and orientations of the correcting
masses if RC1 = RC2 = 6 in. Determine the answers
for parts (a) and (b) when the reference line
attached to crank 1 is at crank angle θ = 210◦ .
881
primary shaking force, then determine (a) the
magnitudes and orientations of the inertial forces
created by these correcting masses, (b) the magnitudes and orientations of the correcting masses
if RC1 = RC2 = 200 mm, and (c) determine
the answers when the reference line (attached to
crank 1) is specified at crank angle θ = 0◦ .
15.26
The crankshaft of the two-cylinder engine (Fig.
P15.25, next page) is rotating counterclockwise
with a constant angular velocity ω = θ̇ = 45 rad/s.
Determine the magnitude and orientation of the
primary shaking force in terms of crank angle θ.
If correcting masses are required to balance the
The crankshaft of the three-cylinder engine (Fig.
P15.26, next page) is rotating with a constant
angular velocity ω = 50k̂ rad/s. Determine the x
and y components of the primary shaking force,
the magnitude(s) of the correcting force (or forces)
created by the correcting mass (or masses), and the
orientation(s) of the correcting force (or forces).
Determine the answers when the reference line
(attached to crank 1) is specified at crank angle
θ = 0◦ .
y
y
1
θ = 210°
x
R2
θ
ω
x
Ref. Line
150°
–ω
R1
45°
60°
L2
L1
Ref. Line
m2
m1
1
1
Figure P15.24 R1 = R2 = R = 3 in, L1 = L2 = L = 24 in, and m1 = m2 = m = 13.25 lb.
882
BALANCING
m1
1
y
m2
Ref.
Line
1
L
y
L
45°
45°
R
–ω
θ
x
1
ω
R
Ref. Line
ω
x
Figure P15.25 R1 = R2 = R = 100 mm, L1 = L2 = L = 550 mm, and m1 = m2 = m = 5 kg.
y
m1 = 2m
1
Ref. Line
L
y
–ω
R
θ
x
ω
45°
45°
1
Ref. Line
x
R
1
L
L
m2 = m
1
m3 = m
Figure P15.26 R1 = R2 = R3 = R = 6 in, L1 = L2 = L3 = L = 30 in, m1 = 2m = 22 lb, and m2 = m3 = m = 11 lb.
Problems
883
y
1
y
Ref. Line
m1 = 3m
L
–ω
1
R
60°
θ
x
ω
Ref. Line
x
1
m2 = m
150°
L
R
Figure P15.27 R1 = R2 = R = 120 mm, L1 = L2 = L = 450 mm, m1 = 3m = 60 kg, and m2 = m = 20 kg.
15.27
15.28
The crankshaft of the two-cylinder engine (Fig.
P15.27, page 883) is rotating counterclockwise
with a constant angular speed ω = 330 rev/min.
Determine the x and y components of the primary
shaking force on the crankshaft bearing, and the
magnitudes and orientations of the inertial forces
created by the correcting masses that balance the
primary shaking force. Determine the magnitude
and orientation of the primary shaking force when
the reference line (attached to crank 1) is specified
at crank angle θ = 0◦ .
The two cranks of the two-cylinder engine (Fig.
P15.28, next page) are oriented at 240◦ to each
other, and the crankshaft is rotating counterclockwise with a constant angular speed ω =
690 rev/min. Both pistons are in the same xy plane.
Determine the x and y components of the primary
shaking force acting on the ground bearing, and
the magnitudes and orientations of the correcting
masses that balance the primary shaking force.
The radial distances of the correcting masses are
RC1 = RC2 = 16 in. Determine the answers when
the reference line (attached to crank 1) is specified
at the crank angle θ = 60◦ .
15.29
The constant angular velocity of the crankshaft of
the two-cylinder engine (Fig. P15.29, next page) is
ω = 250k̂ rad/s. Determine the x and y components
of the primary shaking force in terms of the
crank angle θ, and determine the magnitudes and
orientations of the correction masses. The correction masses are to be added at a radial distance
RC = 50 mm from the crankshaft axis. Determine
the answers when the reference line (attached to
crank 1) is specified at the crank angle θ = 30◦ .
884
BALANCING
y
45°
m1
Ref. Line
L1
1
y
–ω
R1
m2
θ
240°
1
x
ω
R2
L2
Ref.
Line
60°
x
1
Figure P15.28 R1 = 12 in, R2 = 6 in, L1 = L2 = L = 26 in, m1 = 22 lb, and m2 = 110 lb.
y
y
75°
Ref. Line
–ω
R
θ
Ref. Line
R
ω
x
30°
L
1
L
30°
m2
45°
1
1
m1
Figure P15.29 R1 = R2 = R = 150 mm, L1 = L2 = L = 300 mm, and m1 = m2 = m = 20 kg.
x