Civil Engineering Reference
In-Depth Information
weight is an option that must be selected by the user of the software. Whether to
include gravitational effects is a judgment made by the analyst based on the
specifics of a given structural geometry and loading.
The system of second-order, linear, ordinary, homogeneous differential
equations given by Equation 10.34 represents the free-vibration response of the
2 degrees-of-freedom system of Figure 10.4. As a freely oscillating system, we
seek solutions in the form of harmonic motion as
U
2
(
t
)
=
A
2
sin(
t
+
)
(10.35)
U
3
(
t
)
=
A
3
sin(
t
+
)
where
A
2
and
A
3
are the vibration amplitudes of nodes 2 and 3 (the masses at-
tached to nodes 2 and 3);
is an unknown, assumed harmonic circular frequency
of motion; and
is the phase angle of such motion. Taking the second derivatives
with respect to time of the assumed solutions and substituting into Equation 10.34
results in
2
m
A
2
A
3
sin(
5
k
A
2
A
3
sin(
0
0
0
−
2
k
−
t
+
)
+
t
+
)
=
0
m
−
2
k
2
k
(10.36)
or
5
k
A
2
A
3
sin(
0
0
2
−
m
−
2
k
t
+
)
=
(10.37)
2
−
2
k
2
k
−
m
Equation 10.37 is a system of two, homogeneous algebraic equations, which
must be solved for the vibration amplitudes
A
2
and
A
3
. From linear algebra, a
system of homogeneous algebraic equations has nontrivial solutions if and only
if the determinant of the coefficient matrix is zero. Therefore, for nontrivial
solutions,
=
2
5
k
−
m
−
2
k
0
(10.38)
2
−
2
k
2
k
−
m
which gives
2
)(2
k
2
)
4
k
2
0
(10.39)
Equation 10.39 is known as the
characteristic equation
or
frequency equation
of
the physical system. As
k
and
m
are known positive constants, Equation 10.39 is
treated as a quadratic equation in the unknown
(5
k
−
m
−
m
−
=
2
and solved by the quadratic
formula to obtain
two
roots
k
m
2
1
=
(10.40)
6
k
m
2
2
=
