The Earthquake Engineering Online Archive

The Causal Hysteretic Element

Makris, Nicos

UCB/SEMM-1996/11, Dept. of Civil and Environmental Engineering, University of California, Berkeley, 1996-12, 43 pages (500/C23/96/11)

In this report, the basic transfer functions and time response functions of linear phenomenological models are first revisited. The relation between the analyticity of a transfer function and the causality of the corresponding time response function is extended for the case of generalized transfer functions. Using the properties of the Hilbert transform and the associated Kramers-Kroning relations it is shown that transfer functions which have a singularity at omega = 0 in their imaginary part should be corrected by adding a delta function in their real part. This operation ensures that the resulting time response function is causal and is consistent with the theory of generalized functions. Accordingly, the transfer functions of classical viscoelastic models presented in standard vibration handbooks are revised. The addition of a delta function proposed by Crandall (1991) in the impedance of the non-causal ideal hysteretic damper is discussed. Subsequently, 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. The behavior of the proposed model is analyzed both in frequency and time domain. Finally, the response of a mass supported by the causal hysteretic element is discussed with reference to the solutions presented by Caughey.

Available online: http://nisee.berkeley.edu/documents/SEMM/SEMM-96-11.pdf (4 MB)