Free Response of a Rotational Nonlinear Energy Sink Coupled to a Linear Oscillator: Fractality, Riddling, and Initial-Condition Sensitivity at Intermediate Initial Displacements

Abstract

Abstract Free response of a rotational nonlinear energy sink (NES) inertially coupled to a linear oscillator is investigated for dimensionless initial rectilinear displacements ranging from just above the smallest amplitude at which nonrotating, harmonically rectilinear motion is unstable absent direct rectilinear damping, up to the next-largest amplitude at which such motion is orbitally stable. With motionless initial conditions (MICs), i.e., initial velocity of the primary mass and initial angular velocity of the NES mass both zero, predicted behavior for two previously investigated combinations of the dimensionless parameters (characterizing rotational damping, and coupling of rectilinear and rotational motions) differs strongly from that found at smaller initial displacements (Ding and Pearlstein, 2021, “Free Response of a Rotational Nonlinear Energy Sink: Complete Dissipation of Initial Energy for Small Initial Rectilinear Displacements,” J. Appl. Mech., 88(1), p. 011005). For both combinations, a wide range of MICs leads to solutions displaying transient chaos and depending sensitively on initial conditions, giving rise to fractality and riddling in the relationship between initial conditions and asymptotic solutions. Absent direct rectilinear damping of the linear oscillator, for one combination of parameters there exists a wide range of MICs with trajectories leading to time-harmonic, orbitally stable “special” solutions with a single amplitude, but no MICs are found for which all initial energy is dissipated. For the other combination, no such special solutions are found, but there exist MICs for which all initial energy is dissipated. With direct rectilinear damping, sensitivity extends to a measure of settling time, which can be extremely sensitive to initial conditions. A statistical approach to this sensitivity is discussed, along with implications for design and implementation.