On Time-Domain Solutions to the Free Vibration of SDOF Systems With Linear Hysteretic Damping

This article focuses on the free vibration of single-degree-of-freedom (SDOF) systems with linear hysteretic damping and investigates the consistency and convergency of the solution to the free vibration in time domain. The criteria associated with the unit-impulse response function at t = 0 are suggested to check the consistency of possible solutions. Exhaustive and systematic derivations are presented to clarify the confusing results in the literature. Subsequently, the integration in the Hilbert term proves to be nonconvergent when an exponential term (e(st)) is assumed to be the general form of displacement solutions. Hence, it can be concluded that the exact solution to the free vibration of SDOF systems with linear hysteretic damping is still an open question. At last, a numerical verification method is proposed to assess the accuracy of possible solutions.