ME163 Mechanical Vibrations (Winter 2009)
HW1 Due 1.15.08 in class
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1. This problem is a review of some necessary background from mathematics. Consider the following
differential equation
¨
x
(
t
) + 2 ˙
x
(
t
) + 2
x
(
t
) = ˙
y
(
t
) + 2
y
(
t
)
Such equation are often solved by taking the Laplace transform of the
input
y
(
t
) and
output
x
(
t
)
which is defiend as follow
X
(
s
) =
integraldisplay
∞
0
e

st
x
(
t
)
dt
Y
(
s
) =
integraldisplay
∞
0
e

st
y
(
t
)
dt
where
s
is a complex number. Take the Laplace transform of both sides of the equation with
x
(0) =
˙
x
(0) = 0
, y
(0) = 0 to obtain
X
(
s
)(
s
2
+ 2
s
+ 2) =
Y
(
s
)(
s
+ 2)
Divide
X
(
s
) by
Y
(
s
) to obtain a quantity called the
Transfer Function
H
(
s
).
H
(
s
) =
X
(
s
)
Y
(
s
)
=
s
+ 2
s
2
+ 2
s
+ 2
Later we will see that it determines how the
output
x
(
t
) depends on the
input
y
(
t
).We shall see more
about
H
(
s
) later in the course, for now we would like to play with complex number
•
We call the roots of the polynomial in the numerator of the transfer functin as
zeors
and that
of the denominator as
poles
. Find the zeros and poles of the transfer function.
Zero is at
s
=

2
Poles is at
s
=

1
±
ı
•
Represent the complex poles obtained above in the
Euler
form (Hint:
re
ıθ
) and compute the
sum, the product and the ratio of the two.
In Polar coordinates
Z
1
=
√
2
e
ı
3
π
4
Z
2
=
√
2
e
ı

3
π
4
Z
1
+
Z
2
=
√
2
e
ı
3
π
4
+
√
2
e
ı

3
π
4
=

2
Z
1
·
Z
2
=
√
2
e
ı
3
π
4
·
√
2
e
ı

3
π
4
= 2
Z
1
Z
2
=
e
ı
π
2
=

ı
ME163 Mechanical Vibrations
1
HW1
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•
Express the sum, product and ratio in Cartesian coordinates.
In Cartesian coordinates
Z
1
=

1 +
ı
1
Z
2
=

1

ı
1
Z
1
+
Z
2
=

2
Z
1
·
Z
2
=
2
Z
1
Z
2
=

ı
the ratio can be the different sign depending on which you choose as the numerator
2. The purpose of this problem is to remind you on how to work with the
freebody diagrams
and derive
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 Winter '08
 Mezic,I
 Equations, Angular Momentum, Moment Of Inertia

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