Energy Dissipation in Hilbert Envelopes on Motion Waveforms Detected in Vibrating Systems: An Axiomatic Approach

TL;DR

This paper presents an axiomatic method to analyze energy dissipation in Hilbert envelopes of waveforms from vibrating systems, comparing modulated and non-modulated signals to measure energy loss.

Contribution

It introduces a novel axiomatic framework for quantifying energy dissipation in waveforms using Hilbert envelopes and modulation via Mersenne primes.

Findings

Energy expenditure is measured by the area under the waveform.

Modulation of waveforms affects energy dissipation measurements.

The approach provides a new perspective on vibratory energy loss analysis.

Abstract

This paper introduces an axiomatic approach in the theory of energy dissipation in Hilbert envelopes on waveforms emanating from various vibrating systems. A Hilbert envelope is a curve tangent to peak points on a motion waveform. The basic approach is to compare non-modulated vs. modulated waveforms in measuring energy loss during the vibratory motion $m(t)$ at time $t$ of moving object such as a walker, runner or biker recorded in a video. Modulation of $m(t)$ is achieved by using Mersenne primes to adjust the frequency $\omega$ of the Euler exponential in $m(t)e^{\pm j2\pi \omega t}{\rm d}t$. Expediture of energy $E_{m(t)}$ by a system is measured in terms of the area bounded by the motion $m(t)$ waveform at time $t$.