Distribution-Theoretic Basis for Hidden Deltas in Frequency-Domain Structural Modeling
Publication: Journal of Engineering Mechanics
Volume 150, Issue 11
Abstract
Frequency-domain modeling 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 frequency-domain transfer functions of certain structures, beyond those seemingly given by the structural model—e.g., in the mechanical impedance of a linear spring. Previous analyses have manually appended these “hidden deltas” to the relevant transfer functions in to ensure that they remain causal, but questions remain as to their exact origin and behavior in in noncausal 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.
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Data Availability Statement
No data, models, or code were generated or used during the study.
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Journal of Engineering Mechanics
Volume 150 • Issue 11 • November 2024
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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, https://creativecommons.org/licenses/by/4.0/.
History
Received: Mar 17, 2024
Accepted: Jun 26, 2024
Published online: Sep 13, 2024
Published in print: Nov 1, 2024
Discussion open until: Feb 13, 2025
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