...
Modal
...
analysis
...
,
...
or
...
the
...
mode-superposition
...
method,
...
is
...
a
...
...
...
procedure
...
which
...
evaluates
...
and
...
superimposes
...
free-vibration
...
mode
...
shapes
...
to
...
characterize
...
displacement
...
patterns.
...
Mode
...
shapes
...
describe
...
the
...
configurations
...
into
...
which
...
a
...
structure
...
will
...
naturally
...
displace.
...
Typically,
...
lateral
...
displacement
...
patterns
...
are
...
of
...
primary
...
concern.
...
Mode
...
shapes
...
of
...
low-order
...
mathematical
...
expression
...
tend
...
to
...
provide
...
the
...
greatest
...
contribution
...
to
...
structural
...
response.
...
As
...
orders
...
increase,
...
mode
...
shapes
...
contribute
...
less,
...
and
...
are
...
predicted
...
less
...
reliably.
...
It
...
is
...
reasonable
...
to
...
truncate
...
analysis
...
when
...
the
...
number
...
of
...
mode
...
shapes
...
is
...
sufficient.
...
A
...
structure
...
with N degrees of freedom will have N corresponding mode shapes. Each mode shape is an independent and normalized displacement pattern which may be amplified and superimposed to create a resultant displacement pattern, as shown in Figure 1:
Figure 1 - Resultant displacement and modal components
Numerical evaluation proceeds by reducing the equations of motion (N simultaneous differential equations coupled by full mass and stiffness matrices) to a much smaller set of uncoupled second order differential equations (N independent normal-coordinate equations). The orthogonality of mode-shape relations enables this reduction.
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