Time-Response Functions of Mechanical Networks with Inerters and Causality

TL;DR

This paper derives causal time-response functions for three-parameter mechanical networks with inerters, addressing singularities in frequency responses and providing integral representations for improved computational accuracy in seismic signal analysis.

Contribution

It introduces a method to enhance frequency-response functions with Dirac delta functions to ensure causality and provides integral formulas for accurate time-domain responses of inerter-based networks.

Findings

Frequency-response functions require Dirac delta enhancements for causality.

Integral representations improve numerical accuracy for seismic signals.

Causal time-response functions are derived for inerter-involved networks.

Abstract

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 shows that all frequency-response functions that exhibit singularities along the real frequency axis need to be enhanced with the addition of a Dirac delta function or with its derivative depending on the strength of the singularity. In this way the real and imaginary parts of the enhanced frequency response functions are Hilbert pairs; therefore, yielding a causal time-response function in the time domain. The integral representation of the output signals 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.