Stiffness, Flexibility, Impedance, Mobility, and Hidden Delta Function

In this paper, the basic transfer functions and time-response functions of linear phenomenological models are 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. By using the properties of the Hilbert transform and the associated Kramers-Kroning relations it is shown that transfer functions that have a singularity at ω= 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. Finally, the addition of a delta function in the impedance of the noncausal ideal hysteretic dashpot that has been proposed in the literature is discussed.