Civil Engineering Reference
In-Depth Information
in more detail in conjunction with specific examples to follow, a general rule of
thumb for frequency analysis is as follows: If the finite element analyst is inter-
ested in the first
P
modes of vibration of a structure, at least
2
P
modes should
be calculated. Note that this implies the capability of calculating a subset of
frequencies rather than all frequencies of a model. Indeed, this is possible and
extremely important, since a practical finite element model may have thousands
of degrees of freedom, hence thousands of natural frequencies. The computa-
tional burden of calculating all the frequencies is overwhelming and unnecessary,
as is discussed further in the following section.
10.6 MASS MATRIX FOR A GENERAL ELEMENT:
EQUATIONS OF MOTION
The previous examples dealt with relatively simple systems composed of linear
springs and the bar and beam elements. In these cases, direct application of
Newton's second law and Galerkin's finite element method led directly to the for-
mulation of the matrix equations of motion; hence, the element mass matrices. For
more general structural elements, an energy-based approach is preferred, as for
static analyses. The approach to be taken here is based on
Lagrangian mechanics
and uses an energy method based loosely on
Lagrange's equations of motion
[4].
Prior to examining a general case, we consider the simple harmonic oscilla-
tor of Figure 10.1. At an arbitrary position
x
with the system assumed to be in
motion, kinetic energy of the mass is
1
2
m
x
2
T
=
˙
(10.80)
and the total potential energy is
1
2
k
(
x
)
2
U
e
=
st
+
−
mg
(
st
+
x
)
(10.81)
therefore, the total mechanical energy is
1
2
m
1
2
k
(
x
2
x
)
2
E
m
=
T
+
U
e
=
˙
+
st
+
−
mg
(
st
+
x
)
(10.82)
As the simple harmonic oscillator model contains no mechanism for energy
removal, the principle of conservation of mechanical energy applies; hence,
d
E
m
d
t
=
0
=
m
x
˙
x
¨
+
k
(
st
+
x
)
x
˙
−
mg
x
˙
(10.83)
or
mg
(10.84)
and the result is exactly the same as obtained via Newton's second law in Equa-
tion 10.2.
m
x
¨
+
k
(
st
+
x
)
=
