Semi‐discretization method for delayed systems

The paper presents an efficient numerical method for the stability analysis of linear delayed systems. The method is based on a special kind of discretization technique with respect to the past effect only. The resulting approximate system is delayed and also time periodic, but still, it can be transformed analytically into a high‐dimensional linear discrete system. The method is applied to determine the stability charts of the Mathieu equation with continuous time delay. Copyright © 2002 John Wiley & Sons, Ltd.     

A Generalised Conjugate Residual method for the solution of non‐symmetric systems of equations with multiple right‐hand sides

This paper presents an iterative algorithm for solving non‐symmetric systems of equations with multiple right‐hand sides. The algorithm is an extension of the Generalised Conjugate Residual method (GCR) and combines the advantages of a direct solver with those of an iterative solver: it does not have to restart from scratch for every right‐hand side, it tends to require less memory than a direct solver, and it can be implemented efficiently on a parallel computer. We will show that the extended GCR algorithm can be competitive with a direct solver when running on a single processor. We will also show that the algorithm performs well on a Cray T3E parallel computer. Copyright © 1999 John Wiley & Sons, Ltd.     

Updated semi‐discretization method for periodic delay‐differential equations with discrete delay

An updated version of the semi‐discretization method is presented for periodic systems with a single discrete time delay. The delayed term is approximated as a weighted sum of two neighbouring discrete delayed state values and the transition matrix over a single period is determined. Stability charts are constructed for the damped and delayed Mathieu equation for different time‐period/time‐delay ratios. The convergence of the method is investigated by examples. Stability charts are constructed for 1 and 2 degree of freedom milling models. The codes of the algorithm are also attached in the appendix. Copyright © 2004 John Wiley & Sons, Ltd.