Civil Engineering Reference
In-Depth Information
In this case mode shapes have been normalised with the displacement component, and therefore
the rotation component in the mode shape vector has the unit m
-1
while the modal mass has the
unit kg.
In a general structural eigen-value problem
(
)
KMΦ 0
(4.4)
−
ω
⋅
=
the modes are usually defined M-orthonormal, i.e. such that
T
Φ M Φ I
(4.5)
⋅
⋅
=
where
I
is the (diagonal) identity matrix. It should be acknowledged that prior to any
scaling of the modes their components have units meters or radians, and that after any
scaling has taken place relative units must be maintained (as shown in the example
above). It should also be noted that from a finite element solution the eigen-vectors will
emerge in accordance with the chosen degrees of freedom in the system. Below, these
original mode shape vectors have been rearranged into separate
compo-
nents, each associated with the
r
,
r
and
r
θ
displacement components as illustrated in
Fig. 4.3. The reason for this choice is that in Chapters 6 and 7 it will facilitate an effec-
tive solution strategy focusing directly on the important
φ
,
φ
and
φ
y
z
q
,
q
and
q
load components
θ
and corresponding displacement degrees of freedom.
In the mathematical development of a frequency domain response calculation theory
that follows below, the cross sectional displacement and load components are as men-
tioned above formally taken as continuous function. The motivation behind this choice is
mainly convenience, but it is also for practical reasons as spatial load integration will
most often require mode shape vectors in a considerably finer element mesh than what is
considered sufficient for the eigen-value solution from which they have been obtained.
After the theory has been developed the return to discrete vectors will be shown wher-
ever this is necessary for a convenient numerical solution. The basic assumption behind
a modal approach is that the structural displacements
(
)
may be represented by the
r
x t
sum of the products between natural eigen-modes
T
()
φ
x
=
⎣
⎡
φφφ
⎤
(4.6)
⎦
i
y
z
θ
i
()
and unknown exclusively time dependent functions
η
i
t
, i.e.
