Basic Response Functions of Simple Inertoelastic and Inertoviscous Models

Abstract

Motivated by the growing interest in suppressing vibrations with supplemental rotational inertia, this paper examines and constructs the basic frequency-response functions and subsequently derives the corresponding causal time-response functions of elementary mechanical networks that involve the inerter, a two-node element in which the force-output is proportional to the relative acceleration of its end-nodes. This is achieved by extending the relationship between the causality of a time-response function and the analyticity of its corresponding frequency-response function to the case of generalized functions. This 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. It is shown that because of the inerter, some basic time-response functions exhibit causal oscillatory response, in contrast to the decaying exponentials that originate from dashpots. Most importantly, the inerter emerges as an attractive response-modification element because in some cases it absorbs the singular response of the solitary spring or dashpot.