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10.6 Use four point Gauss quadrature to find the mass matrix for beams using the shape
functions of Eq. (10.8)
Answer:
Same as Eq. (10.10a), i.e., the usual consistent mass matrix for beams is
obtained.
10.7 Find a mass matrix for the two-dimensional eight node element of Fig. P10.7 with
thickness
t
and mass density
γ
.
FIGURE P10.7
Natural Frequencies
10.8 Find expressions for the first two natural frequencies of the spring-mass system of
Fig. P10.8.
FIGURE P10.8
sec
2
/in., find the two lowest frequencies of the three
10.9 For
k
=
100 lb/in. and
m
=
1lb
·
DOF system of Fig. P10.8.
Answer:
0.87 Hz, 1.81 Hz.
10.10 Find three natural frequencies and mode shapes for a uniform extension bar fixed at
the left end and free at the right end.
2
EA
Answer:
Exact result for first mode:
ω
1
=
π/
/(ρ
L
2
)
,
φ
1
=
sin
π
x
/(
2
L
)
10.11 Find three natural frequencies for the longitudinal motion of the rod of Fig. P10.11.
FIGURE P10.11
10.12 Find the first three natural frequencies of a cantilevered beam.
875
2
EI
694
2
EI
ω
=
.
/(ρ
L
4
)
ω
=
.
/(ρ
L
4
)
ω
3
=
Answer:
Exact
results:
1
,
4
,
855
2
EI
1
2
7
.
/(ρ
L
4
).
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