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Engineering Mechanical Engineering
Mechanical Vibrations
Q) Consider Spring-Mass-Damper system with K= 4000 N/m, W= 100 N and C= 40 N.s/m. Find the Steady-
state and the total response of the system under the harmonic force F= 200 sin 10t, for initial conditions x= 10
cm at t=0 and x' = 0 at t=0.
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Transcribed Image Text
Q) Consider Spring-Mass-Damper system with K= 4000 N/m, W= 100 N and C= 40 N.s/m. Find the Steady-state and the
total response of the system under the harmonic force F= 200 sin 10t, for initial conditions x= 10 cm at t=0 and x' = 0 at
t=0.
Givendata
SpringconstantK = 4000N/m
WeightW = 100N
DampingconstantC = 40N
⎯
s/m
Todeterminethesteadystateandthetotalresponseofthesystemunder
harmonicf orceF = 200 sin 10t
fortheinitialconditionsx = 10cmandatt = 0and = = 0att = 0?
x
.
dx
dt
Calculationofmassusingweight
W = mg = 100N
m =
W
g
g = Accelerationduetogravity = 10m/se
c
2
m =
100
10
massm = 10kg
Equationofmotionforthesystemandgivenforcesystem
m + C + Kx = F
(
t
)
x
d
2
d
t
2
dx
dt
10 + 40 + 4000x = 200sin10t
x
d
2
d
t
2
dx
dt
+ 4 + 400x = 20sin10t
x
d
2
d
t
2
dx
dt
Calculation
of
general
solution
for
the
obtained
differential
equation
+ 4 + 400x = 20sin10t
x
d
2
d
t
2
dx
dt
General
solution
consists
of
homogeneous
solution
and
perticular
integral
x
(
t
)
= +
x
c
x
t
Homogeneous
solution
+ 4 + 400x = 0
x
d
2
d
t
2
dx
dt
+ 4m + 400 = 0
m
2
m =
⎯
4
±
16
⎯
4
(
1
)(
400
)
√
2
m =
⎯
4
±
16
⎯
1600
√
2
m =
⎯
2
±
6i
11
⎯ ⎯⎯⎯
√
= A + B
x
c
e
(
⎯
2+6i
)
t
11
√
e
(
⎯
2
⎯
6i
)
t
11
√
Calculationofperticularintegral
=
[
D = , =
]
x
p
20 sin 10t
+4D+400
D
2
d
dt
D
2
d
2
d
t
2
=
x
p
20sin10t
⎯
100+4D+400
=
x
p
20sin10t
4D+300
=
x
p
5sin10t
D+75
Applycompdendoanddividendo
=
×
x
p
5sin10t
D+75
(D
⎯
75
)
(D
⎯
75
)
=
x
p
5(D
⎯
75
)
sin10t
⎯
D
2
75
2
=
x
p
5(D
⎯
75
)
sin10t
⎯
10
2
75
2
=
x
p
(D
⎯
75
)
sin10t
⎯
1145
=
x
p
10 cos 10t
⎯
75 sin 10t
⎯
1145
=
x
p
15sin10t
⎯
2cos10t
229
(
t
)
= sin10t
⎯
cos10t
x
p
15
229
2
229
General
solution
x
(
t
)
=
(
t
)
+
(
t
)
x
c
x
p
(
t
)
= A + B
x
c
e
(
⎯
2+6i
)
t
11
√
e
(
⎯
2
⎯
6i
)
t
11
√
(
t
)
= sin10t
⎯
cos10t
x
p
15
229
2
229
x
(
t
)
=
(
t
)
+
(
t
)
x
c
x
p
x
(
t
)
=
[
A + B
]
+
[
sin10t
⎯
cos10t
]
e
(
⎯
2+6i
)
t
11
√
e
(
⎯
2
⎯
6i
)
t
11
√
15
229
2
229
Applythegivenboundaryconditions
Att = 0; x = 10cm = 0. 1m
A + B
⎯
= 0. 1
2
229
A + B =
⎯⎯⎯
(
1
)
249
2290
Att = 0, = 0
dx
dt
(
6i
⎯
2
)
A
⎯
(
6i + 2
)
B +
11
⎯ ⎯⎯⎯
√
11
⎯ ⎯⎯⎯
√
150
229
6i (A
⎯
B
)
⎯
2
[ ]
=
⎯
11
⎯ ⎯⎯⎯
√
249
2290
150
229
A
⎯
B =
⎯ ⎯ ⎯ ⎯
(
2
)
167i
2290
11
√
Addtheequations
(
1
)
and
(
2
)
2A = +
249
2290
167i
2290
11
√
A = +
249
4580
167i
4580
11
√
B =
⎯ ⎯
249
2290
249
4580
167i
4580
11
√
B =
⎯
249
4580
167i
4580
11
√
x
(
t
)
=
[
A + B
]
+
[
sin10t
⎯
cos10t
]
e
(
⎯
2+6i
)
t
11
√
e
(
⎯
2
⎯
6i
)
t
11
√
15
229
2
229
x
(
t
)
=
[(
+
)
+
(
⎯
) ]
249
4580
167i
4580
11
√
e
(
⎯
2+6i
)
t
11
√
249
4580
167i
4580
11
√
e
(
⎯
2
⎯
6i
)
t
11
√
+
[
sin10t
⎯
cos10t
]
15
229
2
229
Steady
state
and
total
response
of
the
given
system
x
(
t
)
=
[(
+
)
+
(
⎯
) ]
249
4580
167i
4580
11
√
e
(
⎯
2+6i
)
t
11
√
249
4580
167i
4580
11
√
e
(
⎯
2
⎯
6i
)
t
11
√
+
[
sin10t
⎯
cos10t
]
15
229
2
229
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Q) Consider Spring-Mass-Damper system with K= 4000 N/m, W= 100 N and C= 40 N.s/m. Find the Steady-state and the total respons…/
Q) Consider Spring-Mass-Damper system with K= 4000 N/m, W= 100 N and C= 40 N.s/m. …
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