Distribution-theoretic basis for hidden deltas in frequency-domain structural modelling

Abstract

Frequency-domain modelling is a core tool for the analysis of linear time-invariant structures. In a process that has been unclear, additional Dirac delta distributions can arise in the frequencydomain transfer functions of certain structures, beyond those seemingly given by the structural model—for instance, in the mechanical impedance of a linear spring. Previous analyses have manually append these "hidden deltas" to the relevant transfer functions in order to ensure that they remain causal, but questions remain as to their exact origin, and behaviour in in non-causal models. Here, we demonstrate that these hidden deltas arise from the theory of distributions, and the solution of the distributional division equation. We demonstrate a rigorous and reliable method for deriving these hidden deltas in which the role of causality constraints are made clear. Furthermore, we demonstrate that the appropriate frequency-domain conditions for causality in such systems are generalized—not, classical—Hilbert transform relations, and that the process of appending delta distributions is related to the analysis of causality via these generalized relations.