Causal Hysteretic Element

In this paper the “causal hysteretic element” is constructed and analyzed. The dynamic stiffness of the proposed hysteretic model has the same imaginary part as the “ideal” hysteretic damper, but has the appropriate real part that makes the model causal. The proposed model is constructed by requiring that the real and imaginary parts of its transfer functions satisfy the Kramers-Kroning relations. This condition ensures that the corresponding time-response functions of the proposed model are zero at negative times. The causal hysteretic element is physically realizable at finite frequencies, but is not definable at ω= 0. The behavior of the proposed model is analyzed both in frequency and time domain. It is shown that the causal hysteretic element is the limiting case of a linear viscoelastic model with nearly frequency-independent dissipation. Finally, the response of a mass supported by the causal hysteretic element is discussed.