HW8 Solutions Mechanical Vibrations
Transcript of HW8 Solutions Mechanical Vibrations
Introduction to Dynamics (N. Zabaras)
HW8 SolutionsMechanical Vibrations
Prof. Nicholas Zabaras
Warwick Centre for Predictive Modelling
University of Warwick
Coventry CV4 7AL
United Kingdom
Email: [email protected]
URL: http://www.zabaras.com/
May 3, 2016
1
Introduction to Dynamics (N. Zabaras)
Problem 1
k
A cylinder of weight W is
suspended as shown.
Determine the period and natural
frequency of vibrations of the
cylinder.
SOLUTION:
• From the kinematics of the system,
relate the linear displacement and
acceleration to the rotation of the
cylinder.
• Based on a free-body-diagram
equation for the equivalence of the
external and effective forces, write the
equation of motion.
• Substitute the kinematic relations to
arrive at an equation involving only the
angular displacement and
acceleration.
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Introduction to Dynamics (N. Zabaras)
Problem 1
SOLUTION:
• From the kinematics of the system, relate the linear
displacement and acceleration to the rotation of the cylinder.
rx rx 22
rra
ra
• Based on a free-body-diagram equation for the equivalence of
the external and effective forces, write the equation of motion.
: effAA MM IramrTWr 22
rkWkTT 2but21
02
• Substitute the kinematic relations to arrive at an equation
involving only the angular displacement and acceleration.
21 12 2
2 2
80
3
Wr W kr r m r r mr
k
m
m
kn
3
8
k
m
nn
8
32
2
m
kf nn
3
8
2
1
2
3
Introduction to Dynamics (N. Zabaras)
Problem 2
s13.1
lb20
n
W
s93.1n
The disk and gear undergo torsional
vibration with the periods shown.
Assume that the moment exerted by the
wire is proportional to the twist angle.
Determine a) the wire torsional spring
constant, b) the centroidal moment of
inertia of the gear, and c) the maximum
angular velocity of the gear if rotated
through 90o and released.
SOLUTION:
• Using the free-body-diagram equation
for the equivalence of the external
and effective moments, write the
equation of motion for the disk/gear
and wire.
• With the natural frequency and
moment of inertia for the disk known,
calculate the torsional spring
constant.
• With natural frequency and spring
constant known, calculate the
moment of inertia for the gear.
• Apply the relations for simple
harmonic motion to calculate the
maximum gear velocity.
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Introduction to Dynamics (N. Zabaras)
Problem 2
s13.1
lb20
n
W
s93.1n
SOLUTION:
• Using the free-body-diagram equation for the
equivalence of the external and effective moments,
write the equation of motion for the disk/gear and wire.
: effOO MM
0
I
K
IK
K
I
I
K
nnn
2
2
• With the natural frequency and moment of inertia for
the disk known, calculate the torsional spring
constant.2
22
21 sftlb 138.0
12
8
2.32
20
2
1
mrI
K
138.0213.1 radftlb27.4 K
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Introduction to Dynamics (N. Zabaras)
Problem 2
s13.1
lb20
n
W
s93.1n
radftlb27.4 K
K
I
I
K
nnn
2
2
• With natural frequency and spring constant known,
calculate the moment of inertia for the gear.
27.4293.1
I 2sftlb 403.0 I
• Apply the relations for simple harmonic motion to
calculate the maximum gear velocity.
nmmnnmnm tt sinsin
rad 571.190 m
s 93.1
2rad 571.1
2
nmm
srad11.5m
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Introduction to Dynamics (N. Zabaras)
Problem 3
7
The 10-kg rectangular plate shown.is
suspended at its center from a rod
having a torsional stiffness k=1.5 N · m
/ rad. Determine the natural period of
vibration of the plate when it is given a
small angular displacement in the
plane of the plate.
The torsional restoring moment created by the rod is
M = k . This moment acts in the direction opposite to
the angular displacement . The angular acceleration
acts in the direction of positive .
Equation of Motion.
0 0
0
O n
kM I k I
I
2 2 2 2 2 2
0
1 110 0.2 0.3 0.1083 .
12 12I m a b kg m kg m
02 0.10832 2 1.69
1.5n
Is
k
Introduction to Dynamics (N. Zabaras)
Problem 4
8
The bent rod shown has a negligible
mass and supports a 5-kg collar at its
end. If the rod is in the equilibrium
position shown. determine the natural
period of vibration for the system.
Since the spring is subjected to an initial
compression of xst for equilibrium, then
when the displacement x > xst the spring
exert a force of Fs=k (x -xst) on the rod.
5ay must act upward, in accordance with
positive
Introduction to Dynamics (N. Zabaras)
Problem 4
9
Equation of Motion.
:
(0.1 ) (0.1 ) 49.05 (0.2 ) (5 ) (0.2 )
B k B
st y
M
kx m kx m N m kg a m
M
-k xst(0.1m), represents the moment created by the
spring force which is necessary to hold the collar in
equilibrium, i.e., at x = 0. This moment is equal and
opposite to 49.05 N(0.2 m) created by the weight
of the collar. Thus
(0.1 ) (5 ) (0.2 )ykx m kg a m
Introduction to Dynamics (N. Zabaras)
Problem 4
10
Kinematics. Since is small,
x = (0.1 m) and y = (0.2 m) .
Therefore:
2
(0.1 ) (5 ) (0.2 ) 5(0.2 )0.2
20 0 20 4.47 /
2 / 2 / 4.47 1.40
y
n n
n
kx m kg a m
rad s
s
Introduction to Dynamics (N. Zabaras)
Problem 5
11
A 10-lb block is suspended from a cord that passes
over a 15-lb disk. The spring has a stiffness k = 200
lb/ft. Determine the natural period of vibration for
the system.
:O k OM M
.. ..
0.75 0.75bs a s
200 / (0.75 ) 10sF lb ft ft lb
2
368 0
368 19.18 /
2 / 2 /19.18 0.328
n n
n
rad s
s
..2
2 2
1 15 1010 (0.75 ) (0.75 ) (0.75 ) (0.75 )
2 32.2 / 32.2 /s b
lb lblb ft F ft ft a ft
ft s ft s
Introduction to Dynamics (N. Zabaras)
Problem 6
12
The thin hoop show is supported by the peg at O .
Determine the natural period of oscillation for small
amplitudes of swing. The hoop has a mass m.
2 2 2 2 2 2
0
2 2
1 1( )
2 2
( cos )
cos
nT I mr mr mr
V mg r
T V mr mgr
Take time derivative:
2 2 sin 0
2 sin 0
sin 0 ( )2
20 2 / 2
2 2n n
mr mgr
mr r g
gsmall
r
g g r
r r g
Introduction to Dynamics (N. Zabaras)
Problem 7
13
A 10-kg block is suspended from a cord that passes
over a 5-kg disk. The spring has a stiffness k = 200
N/m. Determine the natural period of vibration for
the system.
2
2 2 2 2
0
2 2
2 2
1 1 1 1 1(10 ) (0.15 ) ( 5 ) (0.15 ) 0.1406
2 2 2 2 2
1 1( ) (200 / )( 0.15 ) 98.1 (0.15 )
2 2
0.1406 100( 0.15 ) 14.715
b b d
st st
st
T m v I kg m kg m
V k s s Ws N m s N m
T V s
Take time derivative:
0.28125 200( 0.15 )0.15 14.715 0
98.1/ 200 0.4905
16 0 16 4 / 2 / 4 1.57
st
st
n
s
s m
rad s s
0.15m
0.15m
0.15
98.1N
0.15s
Datum
sts s
200 /k N m
200 /k N m