Lanczos method for dynamic analysis of damped structural systems

Abstract

SUMMARYIn this paper we extend the Lanczos algorithm for the dynamic analysis of structures' to systems with general matrix coefficients. The equations of dynamic equilibrium are first transformed to a system of first order differential equations. Then the unsymmetric Lanczos method is used to generate two sets of vectors. These vectors are used in a method of weighted residuals to reduce the equations of motion to a small unsymmetric tridiagonal system.The algorithm is further simplified for systems of equations with symmetric matrices. By appropriate choice of the starting vectors we obtain an implementation of the Lanczos method that is remarkably close to that in Reference 7, but generalized to the case with indefinite matrix coefficients. This simplification eliminates one of the sets of vectors generated by the unsymmetric Lanczos method and results in a symmetric tridiagonal, but indefinite, system. We identify the difficulties that may arise when this implementation is applied to problems with symmetric indefinite matrices such as vibration of structures with velocity feedback control forces which lead to symmetric damping matrices.This approach is used to evaluate the vibration response of a damped beam problem and a space mast structure with symmetric damping matrix arising from velocity feedback control forces. In both problems, accurate solutions were obtained with as few as 20 Lanczos vectors.

Damping optimization of parameter dependent mechanical systems by rational interpolation

“…The primary excitation comes from 10 disturbances applied at 10 consecutive masses starting with mass 471 until mass 480, in such a way that the magnitude of the excitation is highest at the centerpoint of the range of masses, dropping off at the edges. Thus, the matrix ∈ ℝ 1900×10 is defined so that (471 ∶ 480, 1 ∶ 10) = diag (10,20,30,40,50,50,40,30,20,10); (22) all other entries are zero.…”

“…A frequent context for damping optimization comes from system stabilization goals, wherein one strategically adds damping to a structure with light internal damping so as to mute resonances or move them away from the frequencies of ambient oscillatory loads or to suppressing damaging effects of external impacts on the structure. There is a vast literature in this field of research, see, e.g., [34,35,39,40,44]. Depending on the particular application, different damping criteria may be appropriate.…”

Damping optimization in mechanical systems with external force

“…Particularly the engineering literature is very rich. For a brief insight we give some selection of some older references: [2][3][4][5], as well as some less old: [6][7][8]. In references [9,10] authors studies topology optimization for the problem of damping optimization.…”

Optimization of damping positions in a mechanical system

“…A nice overview of some results related to the considered problem can be found in [36]. But, for a brief insight, one can study some older references, i.e., [8,18,25], as well as some more recent ones, i.e., [4,14,26]. Critical damping and other modal damping were considered in [9,22].…”