Time-response functions of mechanical networks with inerters and causality

This paper derives the causal time-response functions of three-parameter mechanical networks that have been reported in the literature and involve the inerter—a two-node element in which the force-output is proportional to the relative acceleration of its end-nodes. This two-terminal device is the mechanical analogue of the capacitor in a force–current/velocity–voltage analogy. The paper employs a methodology that was recently presented for simpler, two-parameter mechanical networks (Makris in J Eng Mech 143:04017123-1–04017123-13, 2017) with which the real and imaginary parts of the enhanced frequency response functions are Hilbert pairs within the context of generalized functions; therefore, yielding causal time-response functions in the time domain. The integral representation of the output signals presented in this paper offers an attractive computational alternative given that the constitutive equations of the three-parameter networks examined herein involve the third derivative of the nodal displacement which may challenge the numerical accuracy of a state-space formulation when the input signal is only available in digital form as in the case of recorded seismic accelerograms.