Civil Engineering Reference
In-Depth Information
resulting equations of motion are the same as those given by both the
Lagrangian
approach and
variational principles
[5].
Examination of Equation 10.95 in light of known facts about the stiffness and
mass matrices reveals that the differential equations are coupled, at least through
the stiffness matrix, which is known to be symmetric but not diagonal. The phe-
nomena embodied here is referred to as
elastic coupling,
as the coupling terms
arise from the elastic stiffness matrix. In consistent mass matrices, the equations
are also coupled by the nondiagonal nature of the mass matrix; therefore, the term
inertia coupling
is applied when the mass matrix is not diagonal. Obtaining solu-
tions for coupled differential equations is not generally a straightforward prode-
cure. We show, however, that the modal characteristics embodied in the equations
of motion can be used to advantage in examining system response to harmonic
(sinusoidal) forcing functions. The so-called
harmonic response
is a capability of
essentially any finite element software package, and the general techniques are
discussed in the following section, after a brief discussion of natural modes.
In the absence of externally applied nodal forces, Equation 10.95 is a system
of
P
homogeneous, linear second-order differential equations in the independent
variable time. Hence, we have an eigenvalue problem in which the eigenvalues
are the natural circular frequencies of oscillation of the structural system, and
the eigenvectors are the amplitude vectors (mode shapes) corresponding to the
natural frequencies. The frequency equation is represented by the determinant
|−
2
[
M
]
0
(10.96)
If formally expanded, this determinant yields a polynomial of order
P
in the vari-
able
+
[
K
]
|=
2
. Solution of the frequency polynomial results in computation of
P
natural
circular frequencies and
P
modal amplitude vectors. The free-vibration response
of such a system is then described by the sum (superposition) of the natural
vibration modes as
P
A
(
j
)
i
i
(
t
)
=
sin(
j
t
+
j
)
i
=
1,
P
(10.97)
j
=
1
Note that the superposition indicated by Equation 10.97 is valid only for linear
differential equations.
In Equation 10.97, the
A
(
j
)
i
and
j
are to be determined to satisfy given initial
conditions. In accord with previous examples for simpler systems, we know that
the amplitude vectors for a given modal frequency can be determined within a
single unknown constant, so we can write the modal amplitude vectors as
1
(
i
)
2
A
(
i
)
=
A
(
i
1
(
i
)
3
i
=
1,
P
(10.98)
.
(
i
)
P
