Order reduction of bilinear systems containing a clearance

A technique for order reduction of nonsmooth bilinear systems of arbitrary dimension containing a clearance is presented. First, a linear-based order reduction transformation is applied. When only one master coordinate is retained, the frequency of the reduced order model may be used to approximate the bilinear normal mode (BNM) frequency of the full model. By utilizing other approximation methods previously developed via the well-known bilinear frequency relation (BFR), this result is in turn used to construct improved reduced order models whose frequencies are even better approximations to the BNM frequencies for the full model. The technique is applied to two and four degree-of-freedom bilinear systems with nonvanishing clearance or interference. The resulting approximate frequencies are compared with those obtained from numerical simulations. The advantages of the present technique include a reduced order model which uses a subset of the original physical coordinates, contains the form of the nonsmooth nonlinearity of the full model, and whose time series closely tracks that of the corresponding BNM in the full model.