No abstract

We consider a mathematical model of a linear vibrational system described by the second‐order differential equation , where M and K are positive definite matrices, called mass, and stiffness, respectively. We consider the case where the damping matrix D is positive semidefinite. The main problem considered in the paper is the construction of an efficient algorithm for calculating an optimal damping. As optimization criterion we use the minimization of the average total energy of the system which is equivalent to the minimization of the trace of the solution of the corresponding Lyapunov equation AX + X AT = ‐I, where A is the matrix obtained from linearizing the second‐order differential equation. Finding the optimal D such that the trace of X is minimal is a very demanding problem, caused by the large number of trace calculations, which are required for bigger matrix dimensions. We propose a dimension reduction to accelerate the optimization process. We will present an approximation of the solution of the structured Lyapunov equation and a corresponding error bound for the approximation. Our algorithm for efficient approximation of the optimal damping is based on this approximation. Numerical results illustrate the effectiveness of our approach.

SUMMARY We consider linear vibrational systems described by a system of second‐order differential equations of the form Mtruex¨MathClass-bin+Dtruex˙MathClass-bin+KxMathClass-rel=0, where M and K are positive definite matrices, representing mass and stiffness, respectively. The damping matrix D is assumed to be positive semidefinite. We are interested in finding an optimal damping matrix that will damp a certain (critical) part of the eigenfrequencies. For this, we use an optimization criterion based on the minimization of the average total energy of the system. This is equivalent to the minimization of the trace of the solution of the corresponding Lyapunov equation AX + XAT = −GGT, where A is the matrix obtained from linearizing the second‐order differential equation, and G depends on the critical part of the eigenfrequencies to be damped. The main result is the efficient approximation and the corresponding error bound for the trace of the solution of the Lyapunov equation obtained through dimension reduction, which includes the influence of the right‐hand side GGT and allows us to control the accuracy of the trace approximation. This trace approximation yields a very accelerated optimization algorithm for determining the optimal damping. Copyright © 2011 John Wiley & Sons, Ltd.

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).

We consider the problem of determining an optimal semi‐active damping of vibrating systems. For this damping optimization we use a minimization criterion based on the impulse response energy of the system. The optimization approach yields a large number of Lyapunov equations which have to be solved. In this work, we propose an optimization approach that works with reduced systems which are generated using the parametric dominant pole algorithm. This optimization process is accelerated with a modal approach while the initial parameters for the parametric dominant pole algorithm are chosen in advance using residual bounds. Our approach calculates a satisfactory approximation of the impulse response energy while providing a significant acceleration of the optimization process. Numerical results illustrate the effectiveness of the proposed algorithm.