Approximate and exact stability analysis of 1-DOF and 2-DOF single-actuator real-time hybrid substructuring

This paper presents approximate and exact stability analysis to determine the critical time delays of one and two degree of freedom (DOF) single-actuator real-time hybrid substructuring (RTHS). The approximate stability analysis is based on applying the Routh-Hurwitz stability criterion using a first-order Taylor expansion of the actuator dynamics represented by a constant gain and pure time delay. The exact stability analysis involves solving the critical frequencies first by canceling the exponential delay term of the closed-loop characteristic equation through multiplication by its complex conjugate. The exact critical time delays are then found using the phase of the closed-loop characteristic equation evaluated at the critical frequencies. These stability analysis techniques are applied to several 1-DOF and 2-DOF mass-spring configurations of single-actuator RTHS. Results show that the critical time delays are highly dependent on the mass, stiffness, and damping partitioning of the physical and numerical substructures as well as the actuator gain. This information provides useful insight into the stability behavior of a particular substructure partitioning configuration to evaluate the feasibility of conducting RTHS tests of these systems.