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

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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.