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

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 $t$.