Equality-Based Weighted Residual Formulation for the Forced Vibration of a Beam with an Attached Dry Friction Damper

Abstract An equality-based weighted residual formulation is proposed to analyze the periodic responses of continuous systems subject to discrete frictional occurrences. A cantilever beam with an attached friction damper is considered to evaluate the ability of friction to mitigate response amplitudes. The system is a common simple model of a bladed-disk sector with an attached platform wedge or ring damper. Friction, governed by Coulomb's classical law, is expressed as a nonsmooth equality condition that augments the equations of motion into a mixed displacement-contact force formulation, and periodic solutions are obtained via a Ritz–Galerkin procedure using Fourier basis functions. The formulation employs an exact equality representation of Coulomb's law for interfaces with mass and operates entirely in the frequency domain, thereby eliminating the need for time-domain calculations of contact forces and avoiding common assumptions like regularization, penalization, or massless interfaces. Coulomb's law and intricate stick-slip behavior at low-frequency sub-resonances are captured accurately, as validated by time-integration results. The effectiveness of the friction damper at mitigating the beam's vibration response and the effect of damper mass are assessed. Results confirm that the damper is optimal for an intermediate range of excitation amplitudes, at which response behavior is strongly nonlinear. The effect of damper mass on the response becomes significant when it is more than 20% of the beam mass. Overall, the approach proves to be compact, robust, computationally efficient, and free of convergence issues up to large numbers of harmonics.