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10.13 Suppose a cantilevered extension bar of mass/length
ρ
carries a mass
m
at the free
end. Find four natural frequencies and mode shapes.
860
EA
Answer:
Exact result for first mode with
ρ
L
/
m
=
1;
ω
1
=
0
.
/(ρ
L
2
)
,
φ
1
=
860
L
.
10.14 Find the natural frequencies of a clamped free beam using a determinant search and
the dynamic stiffness matrix of Table 10.1. Suppose the cross-section is circular of
diameter
d
. Let
L
sin 0
.
7800 kg/m
3
.
=
1m,
d
=
0
.
1m,
E
=
207 GPa,
ν
=
0
.
3 and
γ
=
i
Answer:
ω
=
λ
(
EI
/γ
AL
4
)
with
λ
=
1
.
8699
,
λ
=
4
.
6065
,
λ
=
7
.
5313
.
i
1
2
3
10.15 Find the fundamental natural frequency of the system shown in Fig. P10.15. Let
L
sec
2
/in.,
E
10
7
psi,
I
2 in.
4
, and the weight density of the
=
20 in.,
m
=
10 lb
·
=
=
beam material equals 1.5 lb/in.
(a) Model the beam with distributed mass.
(b) Neglect the weight of the beam.
(c) Lump all the weight of the beam at midspan and treat the system as a two-DOF
system.
(d) Use Guyan reduction to reduce the system to a single-DOF model.
FIGURE P10.15
Answer:
(b) 27.39 rad/sec
(c) 27.46 rad/sec
(d) 27.37 rad/sec
10.16 Find the first ten natural frequencies and mode shapes for the bending of the stepped
beam of Fig. P10.16. Use consistent mass matrices. Also, condense out the rotational
DOF. Find the first five natural frequencies and mode shapes.
Answer:
Mode
Mode
No.
Consistent Mass
No.
Condensed Case
1
6
.
251946
×
10
1
62.520
10
2
2
1
.
058243
×
2
105.840
3
2
.
222744
×
10
2
3
222.369
10
2
4
3
.
616329
×
4
361.800
10
2
5
5
.
487418
×
5
548.834
6
6
.
676266
×
10
3
10
3
7
7
.
858440
×
8
8
.
973045
×
10
3
10
4
9
1
.
012506
×
10
1
.
097755
×
10
4
10.17 Find the natural frequencies of the frame of Fig. P10.1. Use the mass and stiffness
matrices obtained in Problem 10.1.
Answer:
ω
=
663
,
ω
=
1807
,
ω
=
9819
,
ω
=
25 385
,
ω
=
28 777
,
ω
=
59 439
.
1
2
3
4
5
6
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