Spectra and pseudospectra of neutral delay differential equations with application to real-time substructuring

This paper deals with the computation of pseudospectra of neutral delay differential equations (NDDEs) with fixed finite delays. This method provides information on the sensitivity of eigenvalues of the system under perturbations of a given size, allowing one to analyse uncertainties in, for example, structural dynamical systems. Furthermore, pseudospectra computations are a fast and efficient method by which to identify the spectra of NDDEs in any chosen part of the complex plane. This is of particular advantage as the spectra of NDDEs consists of infinitely many eigenvalues. We illustrate these methods by considering a scalar second-order NDDE with one or more fixed delays. Such systems are used to model real-time substructuring experiments, where the delay arises from the response time of actuator(s) forming a coupling between a numerical model and the substructure being tested. We identify changes in the stability and sensitivity to perturbation of this system both as the delay time and as the mass of the substructure are varied. In particular, the latter reveals an essential instability in which infinitely many eigenvalues move to the right-half plane. Finally, we investigate the relationship between pseudospectra and the effect of periodic forcing on the substructured system.