A reduction procedure for joint models that was developed in earlier work is extended to allow for relative motion between surfaces, and the effect of this procedure on timestep issues is considered. A general one-dimensional structure containing a frictional interface is considered. Coulomb friction is approximated with nonlinear springs of large but finite stiffness. The system of equations describing this structure is reduced in a procedure similar to Guyan reduction by assuming that the system deforms only in the shapes that it takes when the interface is massless. The result of this procedure is that the dynamics associated with the interface region are removed from the analysis. Following the development of the reduction procedure, the reduced formulation is specialized to the case of a simple lap joint. A numerical example problem is considered in which both the full and reduced equations of motion are integrated over time. It is seen that, for the example problem considered, the reduction procedure results in tremendous computational savings with little loss of accuracy. Based on the results of the simple example problem, it appears that the proposed reduction procedure has potential to be an accurate and effective method of alleviating the timestep difficulties associated with direct finite element analysis of joints in structural dynamics applications.

Active vibration isolation from an arbitrarily, structurally complex receiver is considered with respect to the impacts of structure flexibility on the open- and closed-loop system characteristics. Specifically, the generally weak influence of flexibility on the open-loop transfer function in the case of total force feedback, in contrast to acceleration feedback, is investigated. The open-loop system characteristics are analyzed based on open-loop transfer function expressions obtained using modal expansion and on modal model order reduction techniques. To closely demonstrate and illustrate the impacts of flexibility on the closed-loop system performance and stability, a problem of automotive engine vibration isolation from a flexible subframe is presented where the neglected dynamics are represented as an output multiplicative model perturbation. A physical explanation as to why the contribution of flexibility to the open-loop transfer function could be neglected in the case of total force feedback in contrast to acceleration feedback is given. Factors for an individual eigenmode to not significantly contribute to the total force output are presented where the deviation of the mode direction relative to the actuator force direction is pointed out as a key one in addition to modal mass and damping coefficient. In this context, the inherent differences between model order reduction by modal and by balanced truncation are being stressed. For the specific automotive vibration isolation application considered, the degradation of robust performance and stability is shown to be insignificant when obtaining a low-order controller by using total force feedback and neglecting flexibility in the design phase.