A non-iterative method for nonlinear modal analysis of frictional systems using contact-state interpolation

This paper presents a non-iterative approximative nonlinear mode method based on linear modes for systems frictional stick-slip interfaces under the single resonant mode assumption. The proposed method effectively captures the influence of contact states on nonlinear dynamic behavior and addresses the need for efficient predictive analysis in frictionally damped systems. The core idea is to characterize contact states via stick-slip transitions at frictionally damped systems. Nonlinear modes are constructed by interpolating between linear modes associated with piecewise contact states, ranging from fully stuck to fully slipped. Under the weakly nonlinear assumption, interpolation functions for nonlinear modal frequencies are first established for one single frictional interface, and then generalized to multiple contact interfaces. The interpolation leverages linear modes across varying contact states to construct a piecewise function describing the dependency of nonlinear modal frequencies on relative displacement amplitudes on the interface. This non-iterative formulation avoids computationally expensive iterative procedures determining nonlinear frequencies. Moreover, corresponding nonlinear mode shapes are derived using the interpolated frequencies, and damping ratios are computed via an energy-based approach. The accuracy and efficiency of the proposed framework are demonstrated through three academic and one engineering numerical case studies.