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ADVANCED DYNAMICS OF STRUCTURES / Midterm Exam / December 16, 2009
HBoduro
ğ
lu/ZCelep
Problem # 1
Consider the system of two degreesoffreedom shown:
a.
Evaluate the flexibility
d
matrix, the mass matrix
m
and the rigidity matrix
k = d
1
and the load vector
p
.
b.
Determine the circular frequencies and the periods of the free vibration
i
ω
and
i
T
in terms of
EI
,
m
and
h
. Obtain the
corresponding two mode shapes
φ
i
and give their graphical representation (
i =1, 2
),
c.
Check the orthogonality of the modes with respect to the mass matrix and the stiffness matrix
φ
1
T
m
φ
2
,
and
φ
1
T
k
φ
2.
d.
Evaluate the generalized masses and stiffness
M
i
=
φ
i
T
m
φ
i
,
and
K
i
=
φ
i
T
k
φ
i
,
and assess the relationship
i
2
= K
i
/ M
i
. (
i =1, 2
),
h
v
(t)
2
v
(t)
1
m
3m
2h
p
(t)
2
p
(t)
1
EI
EI
Pa
3EI
EI
a
P
3
Pa
2EI
2
Ma
2EI
EI
a
M
2
Ma
EI
v(x, t)
x
V(x, t)
M(x, t)
m, EI
k
l
M
k
v
t
M
Problem # 1
Problem # 2
h
d
12
2h
1
EI
d
11
h
d
22
2h
1
EI
d
21
Problem # 2:
Consider the distributed parameter system shown where
m
is the mass per unit length and
EI
is the bending rigidity of the cross
section. The beam has two lumped masses of
M
at the two ends. Write down the boundary conditions for the free vibration of the
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This note was uploaded on 11/23/2010 for the course DEP 503E 10852 taught by Professor Prof.zekaicelep during the Fall '10 term at Istanbul Technical University.
 Fall '10
 Prof.ZekaiCELEP

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