...
An
...
overview
...
of
...
Ritz
...
and
...
Eigen
...
vectors
...
,
...
taken
...
from
...
the
...
...
...
...
(Modal
...
Analysis
...
>
...
Overview,
...
page
...
323),
...
is
...
given
...
as
...
follows:
...
- Eigenvector
...
- analysis
...
- determines
...
- the
...
...
- free-vibration
...
...
- shapes
...
- and
...
- frequencies
...
- of
...
- the
...
- system.
...
- These
...
- natural
...
- modes
...
- provide
...
- an
...
- excellent
...
- insight
...
- into
...
- the
...
- behavior
...
- of
...
- the
...
- structure.
...
- Ritz-vector
...
- analysis
...
- seeks
...
- to
...
- find
...
- modes
...
- that
...
- are
...
- excited
...
- by
...
- a
...
- particular
...
- loading.
...
- Ritz
...
- vectors
...
- can
...
- provide
...
- a
...
- better
...
- basis
...
- than
...
- do
...
- Eigenvectors
...
- when
...
- used
...
- for
...
...
...
- or
...
...
...
- analyses
...
- that
...
- are
...
- based
...
- on
...
- modal
...
- superposition.
...
You,
...
- the
...
- engineer,
...
- may
...
- determine
...
- the
...
- type
...
- of
...
- modes
...
- which
...
- you
...
- feel
...
- are
...
- the
...
- most
...
- appropriate.
...
Eigenvectors
Eigen modes are most suitable for determining response from horizontal ground acceleration, though a missing-mass (residual-mass)
...
mode
...
may
...
need
...
to
...
be
...
included
...
to
...
account
...
for
...
missing
...
high-frequency
...
effects.
...
...
participation
...
is
...
a
...
common
...
measure
...
for
...
determining
...
whether
...
or
...
not
...
there
...
are
...
enough
...
modes,
...
though
...
it
...
does
...
not
...
provide
...
information
...
about
...
localized
...
response.
...
Eigen
...
analysis
...
is
...
useful
...
for
...
checking
...
behavior
...
and
...
locating
...
problems
...
within
...
the
...
model.
...
Another
...
benefit
...
is
...
that
...
natural
...
frequencies
...
indicate
...
when
...
resonance
...
should
...
be
...
expected
...
under
...
different
...
loading
...
conditions.
...
Users
...
may
...
control
...
the
...
convergence
...
tolerance.
...
Orthogonality
...
is
...
strictly
...
maintained
...
to
...
within
...
the
...
accuracy
...
of
...
the
...
machine
...
(15
...
decimal
...
digits).
...
Sturm
...
sequence
...
checks
...
are
...
performed
...
and
...
reported
...
to
...
avoid
...
missing
...
Eigen
...
vectors
...
when
...
using
...
shifts.
...
Internal
...
accuracy
...
checks
...
are
...
performed
...
and
...
used
...
to
...
automatically
...
control
...
the
...
solution.
...
Ill-conditioned
...
systems
...
are
...
detected
...
and
...
reported,
...
then
...
still
...
produce
...
Eigen
...
vectors
...
which
...
may
...
be
...
used
...
to
...
trace
...
the
...
source
...
of
...
the
...
modeling
...
problem.
...
Ritz vectors
Load-dependent
...
Ritz
...
vectors
...
are
...
most
...
suitable
...
for
...
analyses
...
involving
...
vertical
...
ground
...
acceleration,
...
localized
...
machine
...
vibration,
...
and
...
the
...
...
...
method.
...
Ritz
...
vectors
...
are
...
also
...
efficient
...
and
...
widely
...
used
...
for
...
dynamic
...
analyses
...
involving
...
horizontal
...
ground
...
motion.
...
Their
...
benefit
...
here
...
is
...
that,
...
for
...
the
...
same
...
number
...
of
...
modes,
...
Ritz
...
vectors
...
provide
...
a
...
better
...
participation
...
factor,
...
which
...
enables
...
the
...
analysis
...
to
...
run
...
faster,
...
with
...
the
...
same
...
level
...
of
...
accuracy.
...
Further,
...
missing-mass
...
modes
...
are
...
automatically
...
included,
...
there
...
is
...
no
...
need
...
to
...
determine
...
whether
...
or
...
not
...
there
...
are
...
enough
...
modes,
...
and
...
when
...
determining
...
convergence
...
of
...
localized
...
response
...
with
...
respect
...
to
...
the
...
number
...
of
...
modes,
...
Ritz
...
vectors
...
converge
...
much
...
faster
...
and
...
more
...
uniformly
...
than
...
do
...
Eigen
...
vectors.
...
Ritz
...
vectors
...
are
...
not
...
subject
...
to
...
convergence
...
questions,
...
though
...
strict
...
orthogonality
...
of
...
vectors
...
is
...
maintained,
...
similar
...
to
...
Eigen
...
vectors.
...
Sources
...
of
...
documentation
...
on
...
Load-dependent
...
Ritz
...
vectors
...
include:
...
- Structural-analysis
...
- textbooks
...
- (Wilson,
...
- 2004)
...
- Finite-element
...
- textbooks
...
- (Cook
...
- et
...
- al.,
...
- 2001)
...
- Structural-dynamics
...
- textbooks
...
- (Chopra,
...
- 2000)
...
Tips
- Both Eigen and Ritz modes may be calculated simultaneously, in the same model, and in the same analysis run, such that their behavior may be compared.
- Sources of flexibility available during calculation of Eigen and Ritz modes include:
- Consideration of P-Delta and other nonlinear effects
- Modes at different stages of construction
- Frequency shifts for specialized loading
- Consideration of P-Delta and other nonlinear effects