## Frequency response computation for complex structures with damping and acoustic fluid

##### Abstract

Modal frequency response analysis is a very economical approach for large
and complex structural systems since there is an enormous reduction in dimension
from the original finite element frequency response problem to the number of modes
participating in the response. When damping does not exist, the modal frequency
response problem is inexpensive to solve because it becomes uncoupled. However,
when damping exists, the modal damping matrices can become fully populated,
making the modal frequency response problem expensive to solve at many frequencies.
The conventional approach to solve the modal frequency response problem
with damping is to use either direct methods with O(n
3
) operations at each frequency,
or iterative methods with O(n
2
) operations per iteration and numerous
iterations at each frequency, where n is the number of modes used to represent the
response. Another approach is to use a state space formulation and an eigensolution
to uncouple the damped modal frequency response problem, but this doubles the
dimension of the problem. All of the existing traditional methods are very expensive
for systems with many modes.
In this dissertation, a new algorithm to solve the modal frequency response
problem for large and complex structural systems with structural and viscous damping
is presented. The newly developed algorithm, fast frequency response analysis
(FFRA), solves the damped modal frequency response problem with O(n
2
) operations
at each frequency.
The FFRA algorithm considers both structural damping and viscous damping
for structural systems. When only structural damping exists, the modal frequency
response problem is uncoupled by applying the eigensolution of the complex
symmetric modal stiffness matrix. A complex symmetric matrix eigensolver
(CSYMM) has been developed to solve the complex symmetric matrix eigenvalue
problem efficiently. If a viscous damping matrix is also present, the algorithm handles
viscous damping by noting that the rank of the viscous damping matrix is
typically very low for the problems of interest in the automobile industry because of
the small number of viscous damping elements such as shock absorbers and engine
mounts. This algorithm has also been applied to the coupled response of systems
consisting of a light acoustic fluid and structure, and systems with enforced motion.
Also, the algorithm is implemented in parallel on shared memory multiprocessor
machines for performance improvement.
The FFRA algorithm is evaluated for several industry finite element models
which have millions of degrees of freedom. The FFRA algorithm produces outstanding
performance compared to the methods available in the commercial finite
element software MSC.Nastran or NX.Nastran in terms of analysis time, since the
new algorithm is many times faster while obtaining almost the same accuracy as
MSC.Nastran. Therefore, the new FFRA algorithm makes inexpensive high frequency
analysis possible and extends the capability of solving modal frequency response analysis to higher frequencies.

##### Department

##### Description

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