J Austral Math Soc Ser B 32 pp207--222, 1990.

Reduction of second order linear dynamical systems, with large dissipation, by state variable transformations

R. B. Leipnik

Abstract

Linear dynamical systems of the Rayleigh form Mq^·· + Cq^· + Kq = f are transformed by linear state variable transformations w = Aq^· + Bq, where A and B are chosen to simplify analysis and reduce computing time. In particular, A is essentially a square root of M, and B is a Lyapunov quotient of C by A. Neither K nor C is required to be symmetric, nor is C small. The resulting state-space systems are analysed by factorisation of the evolution matrices into reducible factors. Eigenvectors and eigenvalues are determined by these factors. Conditions for further simplification are derived in terms of Kronecker determinants. These results are compared with classical reductions of Rayleigh, Duncan, and Caughey, which are reviewed at the outset.

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Author

R. B. Leipnik

Department of Mathematics, University of California, Santa Barbara, U.S.A.