Civil Engineering Reference
In-Depth Information
Following the direct assembly procedure, the global stiffness matrix is
−
10
1
2
AE
L
[
K
]
=
−
−
12
1
0
−
11
and the global consistent mass matrix is
210
141
012
[
M
]
=
AL
12
The global equations of motion are then
+
=
U
1
U
2
U
3
−
10
210
141
012
1
U
1
U
2
U
3
0
0
0
AL
12
2
AE
L
−
−
12
1
0
−
11
Applying the constraint condition
U
1
=
0,
we have
AL
12
41
12
U
2
U
3
2
U
2
U
3
0
0
2
AE
L
−
1
+
=
−
11
as the homogeneous equations governing free vibration. For convenience, the last equa-
tion is rewritten as
41
12
U
2
U
3
2
U
2
U
3
0
0
24
E
L
2
−
1
+
=
−
11
Assuming sinusoidal responses
U
2
=
A
2
sin(
t
+
)
U
3
=
A
3
sin(
t
+
)
differentiating twice and substituting results in
2
41
12
A
2
A
3
sin(
2
A
2
A
3
sin(
0
0
24
E
L
2
−
1
−
t
+
)
+
t
+
)
=
−
11
Again, we obtain a set of homogeneous algebraic equations that have nontrivial solutions
only if the determinant of the coefficient matrix is zero. Letting
=
24
E
/
L
2
, the
frequency equation is given by the determinant
=
2
−
4
2
− −
2
0
− −
2
−
2
2
which, when expanded and simplified, is
7
4
2
2
−
10
+
=
0
Treating the frequency equation as a quadratic in
2
, the roots are obtained as
2
1
2
2
=
0
.
1082
=
1
.
3204
Substituting for
, the natural circular frequencies are
E
E
1
.
611
L
5
.
629
L
1
=
2
=
rad /sec
