Clearance effects on bilinear normal mode frequencies

The effects of a clearance or interference on the normal mode frequencies of a n-dof system with bilinear stiffness and without damping are investigated through various modifications of the bilinear frequency relation. First, the exact penetration distances and bilinear natural frequencies of a single-degree-of-freedom system are analytically obtained in terms of the amount of clearance and the strength of non-linearity, and an equivalent linear system is derived. These results are in turn used to construct three methods which approximate the bilinear frequencies for the n-dof system. The specific example of a two-dof system is studied in which the resulting approximate frequencies are compared with those obtained from numerical simulations in order to determine the most accurate approximation technique. The results demonstrate how these bilinear normal mode frequencies vary with the magnitude of the clearance/interference and thus point toward the need of including such effects in methods which utilize the bilinear frequency relation.