Characteristics of Vibrating Systems having Time-Constrained Energy

This paper introduces an axiomatic basis for measuring the energy characteristic of vibrating dynamical systems. The basic approach is to compare non-modulated vs. modulated waveforms in measuring energy during the vibratory motion $m(t)$ at time $t$ of moving object such as off-road vehicle oscillating movements recorded in an infrared (IR) video. Modulation of $m(t)$ is achieved either physically by adjusting the load on a spring system or geometrically by adjusting the frequency $\omega$ of the Euler exponential in $m(t)e^{\pm j \omega t}dt$. 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 $t$.