Delay differential equation models for real-time dynamic substructuring

Real-time dynamic substructuring is a testing technique that models an entire system through the combination of an experimental test piece, representing part of the system, with a numerical model of the rest of the system. Delays can has a significant effect on the technique, as signals are passed between the two parts of the system in real-time. The focus of this paper is the influence of the delay on the dynamics of the substructured system. This is addressed using a linear example which may be described by a delay differential equation (DDE) model. This type of analysis allows critical delay values for system stability to be computed, which in turn can be used to help design the substructuring test system. Two methods are presented for the example considered. The first makes use of an analytical approach and the second of a numerical software tool, DDE-BIFTOOL. Normally, in substructuring tests, the actuators response time exceeds the critical delay time and the substructured system is unstable. It is demonstrated that the system can be stabilized using an adaptive delay compensation technique based on forward polynomial prediction. Finally we outline how these techniques may be applied to an industrial example of modelling a nonlinear spring.