Analysis of Hysteretic Damping Using Analytic Signals

The purpose of this technical note is to present a time-domain formulation for linear hysteretic damping. The integro-differential equations that govern the dynamics of structures with linear hysteretic damping are transformed into ordinary differential equations in analytic signals—that is, complex-valued signals in which the real and imaginary parts are a Hilbert transform pair. The poles of this type of system show radial symmetry in the complex plane, determining that for each stable pole in the left-hand half of the complex plane, there is an “unstable” pole in the right-hand half. The impulse response functions of these unstable poles are bounded but noncausal. To illustrate the formulation, the analytic impulse response of a single-degree-of-freedom oscillator with linear hysteretic damping is obtained. The response of the structure to any loading signal can be obtained using time convolution of this impulse response function and the corresponding analytic excitation signal.