An Efficient Method for Estimating the Optimal Dampers' Viscosity for Linear Vibrating Systems Using Lyapunov Equation

Abstract

This paper deals with an efficient algorithm for dampers' viscosity optimization in mechanical systems. Our algorithm optimizes the trace of the solution of the corresponding Lyapunov equation using an iterative method which calculates a low rank Cholesky factor for the solution of the corresponding Lyapunov equation. We have shown that the new algorithm calculates the trace in O(m) flops per iteration, where m is a dimension of matrices in the Lyapunov equation (our coefficient matrices are treated as dense).

Model Reduction for Second‐Order Dynamical Systems Revisited

“…As a numerical example, we use the triple chain oscillator from [14]. For clarity of presentation, only the results for the approach based on [6] are shown and we restrict ourselves to the frequency-limited method for the approximation between 10 −3 and 10 −2 Hz.…”

An improved numerical method for balanced truncation for symmetric second-order systems

“…It is also possible to combine one or more of the above damping approaches with external or small-rank damping [35,36] …”

“…The scalable triple chain oscillator [35] describes three coupled chains of masses interlinked with springs and dampers. The mass and stiffness matrices M and K are symmetric and of dimension n = 150001, whereas the damping matrix D is modeled as proportional plus small rank damping…”

Optimal damping of the infinite-dimensional vibrational systems: commutative case

“…This result has been generalized in [5] to the non-modal damping, obtaining the same set of optimal matrices. The results for the optimal damping without restrictions on the structure of damping matrices usually are not very useful in practical applications, but there is some recent progress in treating such problems (see [2,18,19]). …”