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EML 6267 Assignment #1 Spring 2009
Please remember to put your name on your assignment.
Due:
1/21/09
1.
(25 pts.) For a given single degreeoffreedom (SDOF) lumped parameter system
(undamped), the free vibration,
x
(
t
), is shown below.
a)
(20 pts.) If the motion was initiated with a velocity of 2
π
mm/s and zero initial displacement,
express the motion in the forms show below (use units of mm):
i)
(5 pts.)
x
(
t
) =
A
cos(
ω
n
t
) +
B
sin(
n
t
)
Ans.
()
B
x
A
x
t
B
t
A
x
t
B
t
A
x
n
n
n
n
n
n
n
π
=
=
=
=
+
−
=
+
=
2
0
0
0
cos
sin
sin
cos
&
&
From the plot, the period of the signal is 0.1 s, so the frequency in Hz is 1/0.1 = 10 Hz.
Therefore,
n
= 2
π
*10 rad/s. You can now solve for
B
= 2
π
/
n
= 0.1.
t
x
10
2
sin
1
.
0
⋅
=
mm
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
0.1
0.08
0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
0.1
Time (s)
x (mm)
HW 1, problem 1
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View Full Document This matches the plot (zero phase shift sine wave with amplitude of 0.1 mm).
ii)
(5 pts.)
x
(
t
) =
C
cos(
ω
n
t
+
Φ
c
)
Ans.
()
() ( )
c
n
c
c
n
n
c
n
C
x
C
x
t
C
x
t
C
x
φ
π
sin
2
0
cos
0
0
sin
cos
−
=
=
=
=
+
−
=
+
=
&
&
Initial conditions satisfied for
C = B
and
c
= 270 deg = 90 deg.
−
⋅
=
+
⋅
=
2
10
2
cos
1
.
0
2
3
10
2
cos
1
.
0
t
t
x
mm
Note the 90 deg phase shift between sin and cos.
iii)
(5 pts.)
x
(
t
) =
C
sin(
n
t
+
s
)
Ans.
Same as part i).
t
x
10
2
sin
1
.
0
⋅
=
mm
iv)
(5 pts.)
x
(
t
) =
De
Ee
it
nn
ωω
+
−
.
Ans.
E
D
i
x
E
D
x
Ee
i
De
i
x
Ee
De
x
n
t
i
n
t
i
n
t
i
t
i
n
n
n
n
−
=
=
+
=
=
−
=
+
=
−
−
2
0
0
0
&
&
Multiply
x
(0) expression by
i
n
and sum with initial velocity condition. Solve for
D =
/i
n
,
then
E = D
. Substitute
n
= 2
π
*10 to obtain:
t
i
t
i
e
i
e
i
x
10
2
10
2
20
20
⋅
−
⋅
+
−
=
mm
b)
(5 pts.) Given your expression for
x
(
t
) in part ai), write an expression for the velocity,
dx
(
t
)/
dt
, (mm/s). Plot your result over the same time interval (0 to 0.5 s) shown above. Verify
that the peak velocity is
n
B
. Please include the computer code used to generate the plot.
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This note was uploaded on 04/23/2009 for the course EML 6267 taught by Professor Schmitz during the Spring '08 term at University of Florida.
 Spring '08
 Schmitz

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