Research On The Complex Eigenvalue Of The Large Finite Element Model

With the development of modern space technology, larger and larger spacecrafts are built. The structure of a large-scale spacecraft is very complex and it tends to vibrate drastically under external and internal disturbances. Such vibration loads might lead to malfunction of equipment, or even failure of the whole system. To increase the damping level of some equipment or the whole structure is an efficient method. But these damping forms, such as viscoelastic damping, fluid damping, will introduce complex eigenvalue problems, into system analysis. Complex modal analysis, which obtains the natural frequencies and loss factors by solving eigenvalue problems, is the main technique to analyze a damped structure. In this paper, complex modal analysis algorithms of damped structure are extensively researched and programmed. The work of this paper may constitute a core numerical solver of structural finite element analysis.Several eigenvalue algorithms, such as subspace iteration, power algorithm, inverse-power algorithm, Lanczos algorithm and QR iteration, are first surveyed and compared. The application range, merit and limitation of each algorithm are also expounded.The Lanczos-QR iteration algorithm, which combines integrates the efficiency of the algorithm and the stability of QR algorithm, is studied. The using an improved Lanczos algorithm, a large-scale system matrix is reduced to a low-dimension tri-diagonal matrix that includes the low-order eigenvalues. Then a two-step QR algorithm is applied to solve for all eigenvalues and eigenvectors of the low-dimension tri-diagonal matrix.Finally, the Lanczos-QR algorithm is programmed in FORTRAN. Two examples, a standard cantilever beam with constraint damping layer and a complex satellite, are presented to validate this algorithm. Through comparison with output data from other algorithm and from experiments, it is shown that the Lanczos-QR algorithm is accurate and efficient.