Civil Engineering Reference
In-Depth Information
a system of springs and masses having
N
degrees of freedom, as depicted in
Figure 10.5, and apply the assembly procedure for a finite element analysis, the
finite element equations are of the form
k
1
{
U
[
M
]
}+
[
K
]
{
U
}={
0
}
(10.47)
m
1
where
[
M
]
is the system mass matrix and
[
K
]
is the system stiffness matrix. To
determine the natural frequencies and mode shapes of the system's vibration
modes, we assume, as in the 1 and 2 degrees-of-freedom cases, that
k
2
U
i
(
t
)
=
A
i
sin(
t
+
)
(10.48)
m
2
Substitution of the assumed solution into the system equations leads to the fre-
quency equation
k
3
2
[
M
]
|
[
K
]
−
|=
0
(10.49)
2
. The solution of Equation 10.49
results in
N
natural frequencies
j
, which, for structural systems, can be shown to
be real but not necessarily distinct; that is, repeated roots can occur. As discussed
many times, the finite element equations cannot be solved unless boundary condi-
tions are applied so that the equations become inhomogeneous. A similar phe-
nomenon exists when determining the system natural frequencies and mode
shapes. If the system is not constrained, rigid body motion is possible and one or
more of the computed natural frequencies has a value of zero. A three-dimensional
system has six zero-valued natural frequencies, corresponding to rigid body trans-
lation in the three coordinate axes and rigid body rotations about the three coor-
dinate axes. Therefore, if improperly constrained, a structural system exhibits
repeated zero roots of the frequency equation.
Assuming that constraints are properly applied, the frequencies resulting
from the solution of Equation 10.49 are substituted, one at a time, into Equa-
tion 10.47 and the amplitude ratios (eigenvectors) computed for each natural
mode of vibration. The general solution for each degree of freedom is then
expressed as
which is a polynomial of order
N
in the variable
k
N
m
N
Figure 10. 5
A
spring-mass system
exhibiting arbitrarily
many degrees of
freedom.
N
A
(
j
)
i
U
i
(
t
)
=
sin(
j
t
+
j
)
i
=
1,
N
(10.50)
j
=
1
illustrating that the displacement of each mass is the sum of contributions from
each of the
N
natural modes. Displacement solutions expressed by Equa-
tion 10.50 are said to be obtained by
modal superposition.
We add the indepen-
dent solutions of the linear differential equations of motion.
EXAMPLE 10.3
Determine the natural frequencies and modal amplitude vectors for the 3 degrees-of-
freedom system depicted in Figure 10.6a.
