Fast Computation of Optimal Damping Parameters for Linear Vibrational Systems

Nevena Jakovčević Stor, Ivan Slapničar and Zoran Tomljanović
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Nevena Jakovčević Stor: Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Rudjera Boškovića 32, 21000 Split, Croatia
Ivan Slapničar: Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture, University of Split, Rudjera Boškovića 32, 21000 Split, Croatia
Zoran Tomljanović: Department of Mathematics, J. J. Strossmayer University of Osijek, Trg Ljudevita Gaja 6, 31000 Osijek, Croatia

Mathematics, 2022, vol. 10, issue 5, 1-17

Abstract: We propose a fast algorithm for computing optimal viscosities of dampers of a linear vibrational system. We are using a standard approach where the vibrational system is first modeled using the second-order structure. This structure yields a quadratic eigenvalue problem which is then linearized. Optimal viscosities are those for which the trace of the solution of the Lyapunov equation with the linearized matrix is minimal. Here, the free term of the Lyapunov equation is a low-rank matrix that depends on the eigenfrequencies that need to be damped. The optimization process in the standard approach requires O ( n 3 ) floating-point operations. In our approach, we transform the linearized matrix into an eigenvalue problem of a diagonal-plus-low-rank matrix whose eigenvectors have a Cauchy-like structure. Our algorithm is based on a new fast eigensolver for complex symmetric diagonal-plus-rank-one matrices and fast multiplication of linked Cauchy-like matrices, yielding computation of optimal viscosities for each choice of external dampers in O ( k n 2 ) operations, k being the number of dampers. The accuracy of our algorithm is compatible with the accuracy of the standard approach.

Keywords: linear vibrational system; quadratic eigenvalue problem; diagonal-plus-low-rank matrix; Cauchy-like matrix (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
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