Energy Dissipation in Hilbert Envelopes on Motion Waveforms Detected in Vibrating Dynamical Systems: An Axiomatic Approach
Abstract
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 $m(t)$.
Keywords
Dissipation, Energy, Frequency, Hilbert envelope, Mersenne prime, Motion waveform, Vibrating Dynamical System, Video Frames
Supporting Institution
This research has been supported by the Natural Sciences & Engineering Research Council of Canada (NSERC) discovery grant 185986 and Instituto Nazionale di Alta Matematica (INdAM) Francesco Severi, Gruppo Nazionale per le Strutture Algebriche, Geometriche e Loro Applicazioni grant 9 920160 000362, n.prot U 2016/000036 and Scientific and Technological Research Council of Turkey (TUBİTAK) Scientific Human Resources Development (BIDEB) under grant no: 2221-1059B211301223.
Ethical Statement
All authors approve this manuscript. This paper is an original researh article and has not been submitted or published elsewhere. It is declared that during the preparation process of this study, scientific and
ethical principles were followed and all the studies benefited from are stated in the bibliography.
Thanks
The authors extend their profound thanks to the reviewers, who make very helpful suggestions. We also wish to thank Tane Vergili for sharing her insights concerning the underlying topology and proximity space theory in this paper. In addition, we extend our thanks to Andrzej Skowron, Mirosław Pawlak, Divagar Vakeesan, Enze Cui, Younes Shokoohi, William Hankley, Brent Clark and Sheela Ramanna for sharing their insights concerning time-constrained dynamical systems. In some ways, this paper is a partial answer to the question ’How [temporally] Near?’ put forward in 2002 [25].
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