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Estimating stability boundaries of distributed-delay oscillators via two-stage numerical integration

A procedure based on the concept of full-discretization and numerical integration is established in this work for the stability analysis of periodic distributed-delay oscillators governed by delay integro-differential equations (DIDEs). DIDEs can be found as models of mechanical systems suffering from distributed-delay feedback, such as regenerative machine tool vibrations modeled by distributed force and wheel shimmy. Unstable vibrations in such systems are systematically avoided/controlled if the boundaries between the stable and unstable subspaces are established. The presented method involves the two-stage application of numerical integration to the governing DIDE. While the first-stage application discretizes and converts the distributed delay to fine series of short discrete delays, the second-stage application results in discrete solutions paving the way for a new method of constructing a finite monodromy operator. The error and convergence of the method are studied. It is found that the presented method is of the same convergence as that of the well-accepted first-order semi-discretization method, but more computationally efficient in terms of time savings. A number of case study DIDEs that have already been studied in the literature using methods of semi-discretization and spectral finite elements are studied with the presented method. It is seen that the presented method is valid as it produces stability results that compare well with those of the earlier works.