Energy Dissipation in Hilbert Envelopes on Motion Waveforms Detected in Vibrating Dynamical Systems: An Axiomatic Approach
James Francis Peters, Tharaka U. Liyanage · Communications in Advanced Mathematical Sciences · 2024
This paper introduces an axiomatic approach in the theory of energy dissipation in Hilbert envelopes on motion waveforms emanating from various vibrating dynamical 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 a moving object such as a walker, runner, biker or the action of any spring system recorded in a video. Modulation of $m(t)$ is achieved by using Mersenne primes to adjust the frequency $\omega$ in the Fourier transform $m(t)e^{\pm j2\pi \omega t}$ on motion waveform $m(t)$, where the frequency $\omega$ is a Mersenne prime. Expenditure of energy $E_{m(t)}$ by a system is measured in terms of the area bounded by the motion $m(t)$ waveform at time $t$. Energy dissipation is measured in terms of the difference between modulated and non-modulated $m(t)$.