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Accurate and efficient analysis of rotorcraft
blades has long been a challenge. Using classic
finite-element methods, it is possible to model blades
of geometry, but a high price is paid for this capability
in the form of large models with correspondingly
large computational costs. It is possible in many
applications to reduce computational costs by
employing a technique called modal reduction,
which seeks to collapse the analytical model to a
handful of coordinates. Unfortunately, the modal
reduction process is often ineffective for a rotor blade
composed of finite elements that parameterize all
unknown quantities as displacements. The reason is
that the blade's axial force, which is the most critical
factor in determining the blade's bending stiffness, is
composed of contributions that nearly cancel, one
from the axial displacement, and one from the
bending displacements. Thus, the error in the axial
force can be expected to be significantly larger than
the error in either of the contributions; this is an
example of the classic "small difference of large
quantities" conundrum that often confronts numerical
analysts.
One approach to modally reducing rotor blades
is to express the blade's axial displacement in terms
of the axial force. This eliminates the error-prone step
of summing contributions that nearly cancel to the
axial force, but does so at the expense of imposing
restrictions on the blade geometry that can be
modeled. An alternative approach, and one that has
been the subject of this research, is to represent the
axial force and the axial displacement in the blade
model. Such an analytical procedure, which represents
displacements as well as forces, is referred to as
a mixed method. In addition to the anticipated
improvement modal reduction accuracy, numerical
analysts have long known that mixed methods bring
about a synergy that results in better accuracy, for a
given number of model degrees of freedom, than
could be obtained with methods that employ only
displacements or only forces. Apart from these
accuracy considerations, mixed finite elements have
several important analytical advantages: for example,
they can be used in software developed for the more
commonly used displacement elements, and they can
be coupled to displacement elements.
FY99 saw the completion of a software prototype
for studying the modal reduction properties of mixed
finite elements. This task applied a mixed treatment,
in the axial direction only, to the principal rotor-blade
modeling component, the Nonlinear Beam
Element, of the Second Comprehensive Helicopter
Analysis System (2GCHAS). A preliminary evaluation
of the element's modal reduction accuracy was
conducted by studying the periodic solution of an
articulated blade. Figure 1 compares the blade's flap
response when calculated in modal and in finite-element
coordinates. Note that the finite-element
model has 92 degrees of freedom, whereas the two
sets of modal coordinates employ only one each or
two each of the bending, axial, and torsion
eigenmodes. It may be seen that the modal results are
nearly identical to the finite-element results when
only two each of the eigenmodes are employed as
model coordinates.
Point of Contact: G. C. Ruzicka
(650) 604-3919
gruzicka@mail.arc.nasa.gov
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Fig. 1. Comparison of blade-tip response using finite-
element and modal coordinates.
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