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FINE-TUNING ENERGY DISSIPATION IN
VIBRATING DYNAMICAL SYSTEMS RECORDED
IN INFRARED VIDEO FRAME SEQUENCES

E Cui, J F Peters

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E Cui, J F Peters. FINE-TUNING ENERGY DISSIPATION IN VIBRATING DYNAMICAL SYSTEMS RECORDED IN INFRARED VIDEO FRAME SEQUENCES. none, 2025. hal-04926879

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Submitted on 3 Feb 2025

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# FINE-TUNING ENERGY DISSIPATION IN VIBRATING DYNAMICAL SYSTEMS RECORDED IN INFRARED VIDEO FRAME SEQUENCES

E. CUI AND J.F. PETERS

**ABSTRACT.** This paper introduces a method of fine-tuning energy dissipation in vibrating dynamical systems recorded in infrared (IR) video frame sequences. The basic approach is to compare non-modulated vs. frequency modulated waveform areas in measuring energy during system vibratory motion $m(t)$ at time $t$ of a moving object such as the vibration waveforms of an off-road vehicle recorded in an infrared video. Modulation of $m(t)$ is achieved physically by adjusting the load carried by a spring system in response to the extent of frequency modulation achieved by adjusting the frequency $\omega$ in a Fourier transform (FT) of the form $m(t)e^{\pm j\omega t}$. Mersenne primes provide an effective means of adjusting the frequency $\omega$ of the FT. Expenditure of energy $E_m(t)$ by a system is measured in terms of the area bounded by the motion $m(t)$ waveform over a temporal interval. Energy dissipation is a measure of the difference between non-modulated and modulated system energy.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Method</td><td>2</td></tr><tr><td>2.1. Hilbert Envelope Lobes</td><td>3</td></tr><tr><td>2.2. Method: Tuning energy dissipation in a vibrating dynamical system</td><td>6</td></tr><tr><td>2.3. E<sub>Diss</sub></td><td>6</td></tr><tr><td>3. Results and Conclusion</td><td>7</td></tr><tr><td>3.1. Experiments</td><td>7</td></tr><tr><td>3.2. Discussion</td><td>13</td></tr><tr><td>3.3. Conclusion</td><td>14</td></tr><tr><td>Acknowledgments</td><td>15</td></tr><tr><td>4. Sample Off-road Spring System</td><td>16</td></tr><tr><td>References</td><td>17</td></tr></table>

## 1. INTRODUCTION

This paper carries forward earlier work on measuring and fine-tuning the energy characteristic of vibrating dynamical systems [4],[17],[5]. The main contribution of this paper is an effective means of measuring and smoothing dynamical system

<footnote n="2020">2020 Mathematics Subject Classification. 76F20 (Dynamical System); 74H45 (Vibrations in Dynamical Problems).</footnote>

<footnote>Key words and phrases. Characteristic, Dynamical System, Energy Dissipation, Mersenne Prime, Frequency Modulation, Motion Waveform, Vibration, Infrared Video.</footnote>
vibratory motion waveforms extracted from sequences of video frames produced by an infrared (IR) camera. This is accomplished via a cost-effective vibration monitoring and tuning technique in attenuating dynamical system energy dissipation. The focus of this paper is on the introduction of a method of diminishing energy dissipation via a correspondence between frequency-modulated motion waveforms and dampening (smoothing) vibrations of moving object due to excessive oscillating movements commonly found in the thrusting action of off-road vehicles, aircraft and rocket frames that can be recorded in sequences of IR video frames.

Frequency modulation of a motion waveform $m(t)$ is carried out via a Fourier transform (FT) $m(t)e^{j\omega t}$ (see, e.g., [16]). The focus is on fine-tuning the frequency $\omega$ in the Euler exponential $e^{\pm j\omega t}$ in modulating $m(t)$ at time $t$. Modulation is accomplished by adjusting the FT frequency $\omega$ using Mersenne $M$ primes [15], where $M = 2^p - 1 \in \{1, 3, 7, 31\}$ (for prime $p \in \{1, 2, 3, 5\}$). It has been found that Mersenne primes can generate pseudo-random sequences useful in minimizing noise and unwanted waveform oscillations in producing waveforms with long, predictable cycles with corresponding diminished energy dissipation [17],[9]. Thanks to this form of frequency modulation, highly erratic waveforms become less bumpy with more moderate up-and-down oscillations. In this work, a relation has been found between the variation in the selected Mersenne prime frequency and the weight carried by a moving spring system.

Experimental results from the Methods section of this paper provide evidence of the efficacy of the proposed approach to fine-tuning a vibrating system to achieve minimal energy dissipation. The collection of oscillatory parts of a vibrating dynamical system recorded in a single IR video frame, constructs a motion vector field [14], p.93 like the one in Fig. 1. For a detailed view of the basics of IR images, see [3]. For a typical vibratory motion vector field portrait, see Fig. 1. The framework for the IR video-based experiments introduced in this paper is shown in Fig. 2. Each one of the motion field vectors in an off-road vehicle spring system is recorded in IR video frames like those in Fig. 3. Motion waveforms of vibrating dynamical systems have varying cyclic oscillations [8], [7].

## 2. METHOD

<table><caption>TABLE 1. Principal Symbols Used in this Paper</caption><thead><tr><th>Symbol</th><th>Meaning</th></tr></thead><tbody><tr><td><i>t</i></td><td>Time</td></tr><tr><td><i>m</i>(<i>t</i>)</td><td>Motion characteristic at time <i>t</i></td></tr><tr><td><i>&omega;</i></td><td>Waveform Oscillation Frequency</td></tr><tr><td><i>M</i></td><td>Mersenne prime</td></tr><tr><td><i>E</i><sub>mod</sub></td><td>Energy of Frequency modulated waveform</td></tr><tr><td><i>E</i><sub>nmod</sub></td><td>Energy of Non-modulated waveform <i>m</i>(<i>t</i>)</td></tr><tr><td><i>E</i><sub>Diss</sub></td><td>Energy dissipation</td></tr></tbody></table>

This section introduces an application of the Parseval-Fourier transform of the frequency modulation of a vibrating system motion waveform $m(t)$ [10]. For a motion waveform $m(t)$ at time $t$, a measure of motion energy $E_{m(t)}$ is defined in terms of non-modulated energy $E_{nmod}$ and modulated energy $E_{mod}$ using a Parseval-form
<figure><img src="image_2.png" /><figcaption>FIGURE 1. Dynamical System Motion Vector Field</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>FIGURE 2. Flow chart for fine-tuning IR spring system energy dissipation $E_{Diss}$</figcaption></figure>

form of Fourier transform FT (see, e.g., Section 13.2 in [17]) in a framework that starts with an IR spring system in fine-tuning energy dissipation $E_{Diss}$ as shown in Fig. 2 (for the details of the IR spring system that is the foundation of tuning the energy dissipation of a spring system, see Fig. 4 in Appendix. 4).

2.1. **Hilbert Envelope Lobes.** Each oscillating motion waveform $m(t)$ contains peak points. To obtain a more manageable approach to measuring waveform energy, we introduce Hilbert envelopes curves on $m(t)$. A Hilbert envelop (denoted by $H_{env}(t)$ at time $t$) is a curve that is tangent to the peak points on an oscillatory
<figure><img src="image_4.png" /><figcaption>FIGURE 3. Loaded System and Unloaded System IR Videos</figcaption></figure>

waveform [2]. A $H_{env}(t)$ lobe (denoted by $H_{env_{lobe}}(t)$ is a tiny, bounded region attached to a single waveform peak point on an envelope $H_{env}(t)$ defined by [2]

$$ (1) \qquad H_{env}(m(t)) = \sqrt{m(t)^2 + (-m(t))^2}. $$

let $H_{env_{lobe}}(m(\omega_t))$ be the Hilbert envelope of the frequency modulated motion $m(t)$ at time $t$ using Parseval's theorem [18], [1]. This is explained briefly as follows. Recall that the frequency $\omega$ of a waveform $m(t)$ equals the number of time that the waveform cycle repeats itself in one second. Let $m(t) = m(t)e^{\pm j2\pi\omega t}|_{\omega=1}$ be non-modulated motion and let $m(t) = m(t)e^{\pm j2\pi\omega t}|_{\omega>1}$ be frequency modulated motion. Parseval's formula for the Fourier transform $m(t)e^{\pm j2\pi\omega t}$ over the temporal interval $[t_0, t_1]$ is defined by

$$ (2) \qquad \int_{t_0}^{t_1} |m(\omega_t)e^{\pm j2\pi\omega_t t}|^2 dt = \int_{t_0}^{t_1} |m(t)e^{\pm j2\pi t}|^2 dt. $$

Using an extension of Parseval's formula [10] in terms of the areas of Hilbert envelope lobes, the energy of a lobe is measured using:

$$ (3) \qquad E_{mod}(t) = \int_{t_0}^{t_k} |H_{env_{lobe}}(m(\omega_t))|^2 d\omega_t = \int_{t_0}^{t_k} H^2_{env_{lobe}}(m(t))dt. $$

<figure><img src="image_5.png" /><figcaption>FIGURE 4. Loaded system waveform and modulated unloaded system waveform</figcaption></figure>
<table><caption>TABLE 2. Case1: Physically modulated spring system energy with 100% loads</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>nmod1</sub></th></tr></thead><tbody><tr><td>55</td><td>0.18729641</td><td>0.04921083</td><td>0.03753135</td><td>0.03779644</td><td>0.3118</td></tr><tr><td>56</td><td>0.18729641</td><td>0.02779180</td><td>0.03753135</td><td>0.01963852</td><td>0.2723</td></tr><tr><td>57</td><td>0.18729641</td><td>0.03673046</td><td>0.02683423</td><td>0.03779644</td><td>0.2887</td></tr><tr><td>58</td><td>0.18729641</td><td>0.03979378</td><td>0.03753135</td><td>0.03779644</td><td>0.3024</td></tr><tr><td>59</td><td>0.17350394</td><td>0.02983870</td><td>0.03753135</td><td>0.03491520</td><td>0.2758</td></tr><tr><td>60</td><td>0.18729641</td><td>0.03508007</td><td>0.02120612</td><td>0.02975200</td><td>0.2733</td></tr></tbody></table>

<table><caption>TABLE 3. Case 2: Euler exponentially modulated spring system energy with $M = 1$. $*E_{Diss} = |E_{nmod1} - E_{mod}|$</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>mod</sub></th><th>E<sub>Diss</sub></th></tr></thead><tbody><tr><td>55</td><td>0.14974283</td><td>0.02468938</td><td>0.03385195</td><td>0.02858919</td><td>0.2369</td><td>0.0749</td></tr><tr><td>56</td><td>0.14974283</td><td>0.04463997</td><td>0.01985516</td><td>0.02858919</td><td>0.2428</td><td>0.0295</td></tr><tr><td>57</td><td>0.14974283</td><td>0.04257751</td><td>0.03385195</td><td>0.02742013</td><td>0.2536</td><td>0.0351</td></tr><tr><td>58</td><td>0.14974283</td><td>0.04463997</td><td>0.02269972</td><td>0.01378180</td><td>0.2309</td><td>0.0715</td></tr><tr><td>59</td><td>0.14481739</td><td>0.04463997</td><td>0.02964893</td><td>0.02858919</td><td>0.2477</td><td>0.0281</td></tr><tr><td>60</td><td>0.14974283</td><td>0.04463997</td><td>0.02883101</td><td>0.01725752</td><td>0.2405</td><td>0.0328</td></tr></tbody></table>

<table><caption>TABLE 4. Case3: Physically modulated spring system energy with 50% loads. $*E_{Diss} = |E_{nmod2} - E_{mod}|$</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>nmod2</sub></th><th>E<sub>Diss</sub></th></tr></thead><tbody><tr><td>55</td><td>0.16901949</td><td>0.02440822</td><td>0.02958366</td><td>0.02889313</td><td>0.2519</td><td>0.015</td></tr><tr><td>56</td><td>0.15029333</td><td>0.05405890</td><td>0.02221267</td><td>0.02370715</td><td>0.2503</td><td>0.0075</td></tr><tr><td>57</td><td>0.17416733</td><td>0.06644817</td><td>0.02077902</td><td>0.03706339</td><td>0.2985</td><td>0.0449</td></tr><tr><td>58</td><td>0.12269299</td><td>0.01896559</td><td>0.03288010</td><td>0.04990935</td><td>0.2244</td><td>0.0065</td></tr><tr><td>59</td><td>0.10986770</td><td>0.04257540</td><td>0.05813493</td><td>0.02972843</td><td>0.2403</td><td>0.0074</td></tr><tr><td>60</td><td>0.11176960</td><td>0.06644817</td><td>0.03770767</td><td>0.02925057</td><td>0.2452</td><td>0.0047</td></tr></tbody></table>

**Example 1** ($E_{nmod}$vs.$E_{mod}$). The Energy measured in a non-modulated spring system recorded in Table. 2& 4 is calculated via IR frames. In this example, the total energy is the sum of motion waveform lobe energy values. For example, lobes A, B, C, D are used in sampling non-FT-modulated motion waveform lobe energy derived from the off-road spring system shown Fig. 2.1.

In this example, the maximum peak considered for Lobe A includes all subregions near the peak with decreasing sizes close to Lobe A peak. Similarly, Lobes B, C and D include all subregions near the second peak with decreasing areas close to the lobe peaks.

The Energy measured in modulated spring system in Table. 3 is calculated via sampled frame waveform Hilbert envelope lobes. Similarly, in Table. 3, the modulated spring system energy is measured via the total energy of Lobes A, B, C and D from the Hilbert envelope on the waveform shown in Fig. 2.1. The modulated
system represents an ideally loaded physical system, resulting in an idealized energy dissipation measurement in which waveform modulation is via an FT at a particular frequency.

let $m(t)$ be a motion waveform derived from the movements of a physical system recorded an IR video and let $\mathbb{R}$ denote the set of real numbers. Non-modulated system energy is a mapping $E_{nmod}: \mathbb{R} \to \mathbb{R}$ defined by

$$ (4) \qquad E_{nmod}(m(t)) = \int_{t_0}^{t_k} |m(t)e^{j\omega t}|^2 dt. $$

By contrast, frequency modulated system energy is a mapping $E_{mod}: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ defined by

$$ (5) \qquad E_{mod}(m(t), \omega) = \int_{t_0}^{t_k} |m(t)e^{j\omega t}|^2 dt. $$

we adjust the frequency $\omega$ to obtain a smooth (less hilly) motion waveform $m(t)$. The basic approach is to compare non-modulated energy $E_{nmod}$ with modulated energy $E_{mod}$ either by (1) imposing a varying physical load on a vibrating spring system with resulting attenuated vibrations with corresponding $m(\text{see, Fig. 2.1})$ or (2) by using the Euler exponential $e^{\pm j\omega t}$ [6] on the motion $m(t)$, which gives us a Fourier transform to obtain modulated energy $E_{mod}(t)$ with frequency $\omega$ over the temporal interval $[t_0, t_k]$ (see, Fig. 2.1).

## 2.2. Method: Tuning energy dissipation in a vibrating dynamical system.

Many important results have been obtained in the search for an optimal form of energy dissipation for a dynamical system (see, [11], [13], [12], [19]). An overview of the method introduced in this paper in fine-turning the energy dissipation $E_{Diss}$ emanating from a vibrating system is given in Fig. 2. This is accomplished in the following steps: (1) measure the non-modulated energy $E_{nmod}(t)$ inherent in an IR-recorded motion waveform $m(t)$, (2) adjust the frequency $\omega$ in the FT on the motion waveform $m(t)$ to obtain an optimal Hilbert enveloped waveform $E_{mod}(t)$, (3) based a known correspondence between a selected frequency, adjust the load on a spring system to obtain an adjusted $E_{nmod}(t)$ that is close to $E_{mod}(t)$ to minimize the value of $E_{Diss}$. This paper reports the use Mersenne primes as a quick means of adjusting the frequency $\omega$ in the FT used to obtain the optimal $E_{mod}(t)$. We have found that this approach results in a satisfactory $E_{Diss}$ by an operational vibrating system.

### 2.3. $E_{Diss}$. let $\epsilon$ be a chosen threshold for optimal energy dissipation $E_{Diss}$ during the operation of a vibrating dynamical system. Energy dissipation is a mapping $E_{Diss}: \mathbb{R} \times \mathbb{R} \to \mathbb{R}$ is defined by:

$$ \begin{aligned} E_{Diss}(E_{nmod}(t), E_{mod}(t)) &= |E_{nmod}(t) - E_{mod}(t)| \\ &= \left| \int_{t_0}^{t_k} |m(t)|^2 dt - \int_{t_0}^{t_k} |m(t)e^{j\omega t}|^2 dt \right| \\ &< \epsilon. \end{aligned} $$
<table><caption>TABLE 5. Ex. 2 Case 1: Non-modulated Waveform with 100% loads</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>nmod1</sub></th></tr></thead><tbody><tr><td>55</td><td>0.18729641</td><td>0.04921083</td><td>0.03753135</td><td>0.03779644</td><td>0.3118</td></tr><tr><td>56</td><td>0.18729641</td><td>0.02779180</td><td>0.03753135</td><td>0.01963852</td><td>0.2723</td></tr><tr><td>57</td><td>0.18729641</td><td>0.03673046</td><td>0.02683423</td><td>0.03779644</td><td>0.2887</td></tr><tr><td>58</td><td>0.18729641</td><td>0.03979378</td><td>0.03753135</td><td>0.03779644</td><td>0.3024</td></tr><tr><td>59</td><td>0.17350394</td><td>0.02983870</td><td>0.03753135</td><td>0.03491520</td><td>0.2758</td></tr><tr><td>60</td><td>0.18729641</td><td>0.03508007</td><td>0.02120612</td><td>0.02975200</td><td>0.2733</td></tr></tbody></table>

This view of energy dissipation carries forward the original view of $E_{Diss}$ in [11]. The non-modulated energy $E_{nmod}(t)$ from case (1) and the modulated energy $E_{mod}(t)$ in case (2) are used to obtain an optimal energy dissipation $E_{Diss}$. Case (1) provides indications of the resulting $E_{nmod}(t)$ energy resulting from a physical load imposed on an operational vibrating system and case (2) provides waveform modulation $E_{mod}$ via a Fourier transform to achieve satisfactory motion waveform modulation (i.e., low oscillatory, smoother operational waveforms). The energy dissipation $E_{Diss}$ measures the difference between $E_{mod}(t)$ and $E_{nmod}(t)$ in arriving at $E_{Diss}$, which is less than the threshold $\epsilon$.

### 3. RESULTS AND CONCLUSION

The experimental results reported in this section highlights the contrast between Energy Dissipation of modulated waveforms and non-modulated motion waveforms. This experiment continues from the earlier findings in experiment of vibration detection envelope by modulated and non-modulated signals for IR video of spring system object. The goal is to study the vibration waveform measurement and construction by using IR camera in order to provide a cost-effective load simulation and monitor technique. The experiment data has been collected by Active IR camera with 720P solution under 60 FPS. Then use extraction algorithm processing the data to get the result data.

From all experiment data and observation, the loaded motion waveform can be modulated by unloaded spring system with designed model. The energy dissipation can be reduced by increasing Mersenne prime in the model. By calculation, the average total energy dissipation dropped 12.04% and the average energy dissipation dropped 11% with Mersenne prime increasing from 1 to 31 in this experiment set.

3.1. **Experiments.** The overall Mersenne modulated case energy dissipation comparison can be found in Fig. 5. There are 4 examples listed below indicates the experiment case for each Mersenne prime as $M = 1, 3, 7, 31$ energy dissipation.

**Example 2.** The energy dissipation comparison for Mersenne Prime $M = 1$ shown in Fig. 6. The vehicle here shown whole parts in the frames and moving towards left. The energy table of each lobe in Non-modulated waveform Case 1 and Case 3 see Table. 5 and Table. 7. The energy table of each lobe in $M = 1$ Modulated waveform see Table. 6.

It is observed that the modulated unloaded waveform shared multiple similarities in shapes with the non-modulated loaded waveform. Compare the energy table of both modulated waveform and non-modulated waveform we can see the lobe energy
<figure><img src="image_6.png" /><figcaption>FIGURE 5. Mersenne Modulated Experiments Energy Dissipation Comparison</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIGURE 6. Ex. 2 Non-modulated Waveform and Modulated Waveform $M = 1$</figcaption></figure>

of both waveforms sharing similar level differences. The Non-modulated average total energy is 0.64 in case 1, and Mersenne-modulated average total energy is 0.54 with $M = 1$.
<table><caption>TABLE 6. Ex. 2 Case 2: Mersenne Modulated Waveform $M = 1$</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>modM1</sub></th><th>E<sub>Diss</sub></th></tr></thead><tbody><tr><td>55</td><td>0.14974283</td><td>0.02468938</td><td>0.03385195</td><td>0.02858919</td><td>0.2369</td><td>0.0749</td></tr><tr><td>56</td><td>0.14974283</td><td>0.04463997</td><td>0.01985516</td><td>0.02858919</td><td>0.2428</td><td>0.0295</td></tr><tr><td>57</td><td>0.14974283</td><td>0.04257751</td><td>0.03385195</td><td>0.02742013</td><td>0.2536</td><td>0.0351</td></tr><tr><td>58</td><td>0.14974283</td><td>0.04463997</td><td>0.02269972</td><td>0.01378180</td><td>0.2309</td><td>0.0715</td></tr><tr><td>59</td><td>0.14481739</td><td>0.04463997</td><td>0.02964893</td><td>0.02858919</td><td>0.2477</td><td>0.0281</td></tr><tr><td>60</td><td>0.14974283</td><td>0.04463997</td><td>0.02883101</td><td>0.01725752</td><td>0.2405</td><td>0.0328</td></tr></tbody></table>

<table><caption>TABLE 7. Ex. 2 Case 3: Non-modulated Waveform with 50% loads</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>nmod2</sub></th><th>E<sub>Diss</sub></th></tr></thead><tbody><tr><td>55</td><td>0.16901949</td><td>0.02440822</td><td>0.02958366</td><td>0.02889313</td><td>0.2519</td><td>0.015</td></tr><tr><td>56</td><td>0.15029333</td><td>0.05405890</td><td>0.02221267</td><td>0.02370715</td><td>0.2503</td><td>0.0075</td></tr><tr><td>57</td><td>0.17416733</td><td>0.06644817</td><td>0.02077902</td><td>0.03706339</td><td>0.2985</td><td>0.0449</td></tr><tr><td>58</td><td>0.12269299</td><td>0.01896559</td><td>0.03288010</td><td>0.04990935</td><td>0.2244</td><td>0.0065</td></tr><tr><td>59</td><td>0.10986770</td><td>0.04257540</td><td>0.05813493</td><td>0.02972843</td><td>0.2403</td><td>0.0074</td></tr><tr><td>60</td><td>0.11176960</td><td>0.06644817</td><td>0.03770767</td><td>0.02925057</td><td>0.2452</td><td>0.0047</td></tr></tbody></table>

The total energy on average is 0.0777 lower compared to non-modulated waveform in case 1, which result in 14.45% total energy dissipation. If compare with non-modulated waveform energy in case 3, a much lower energy dissipation 1.43% can be observed.

<figure><img src="image_8.png" /><figcaption>FIGURE 7. Ex. 3 Non-modulated Waveform and Modulated Waveform $M = 3$</figcaption></figure>

**Example 3.** The energy dissipation comparison for Mersenne Prime $M = 3$ shown in Fig. 7. The vehicle here shown whole parts in the frames and moving towards left. The energy table of each lobe in Non-modulated waveform Case 1 and Case 3 see Table. 8 and Table. 10. The energy table of each lobe in $M = 3$ Modulated waveform see Table. 9.

It is observed that the modulated unloaded waveform shared multiple similarities in shapes with the non-modulated loaded waveform. Compare the energy table of both modulated waveform and non-modulated waveform we can see the lobe energy
<table><caption>TABLE 8. Ex. 3 Case 1: Non-modulated Waveform with 100% loads</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>nmod1</sub></th></tr></thead><tbody><tr><td>55</td><td>0.18729641</td><td>0.04921083</td><td>0.03753135</td><td>0.03779644</td><td>0.3118</td></tr><tr><td>56</td><td>0.18729641</td><td>0.02779180</td><td>0.03753135</td><td>0.01963852</td><td>0.2723</td></tr><tr><td>57</td><td>0.18729641</td><td>0.03673046</td><td>0.02683423</td><td>0.03779644</td><td>0.2887</td></tr><tr><td>58</td><td>0.18729641</td><td>0.03979378</td><td>0.03753135</td><td>0.03779644</td><td>0.3024</td></tr><tr><td>59</td><td>0.17350394</td><td>0.02983870</td><td>0.03753135</td><td>0.03491520</td><td>0.2758</td></tr><tr><td>60</td><td>0.18729641</td><td>0.03508007</td><td>0.02120612</td><td>0.02975200</td><td>0.2733</td></tr></tbody></table>

<table><caption>TABLE 9. Ex. 3 Case 2: Mersenne Modulated Waveform $M = 3$</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>modM3</sub></th><th>E<sub>Diss</sub></th></tr></thead><tbody><tr><td>55</td><td>0.14974098</td><td>0.04463934</td><td>0.01985489</td><td>0.02858880</td><td>0.2428</td><td>0.069</td></tr><tr><td>56</td><td>0.14974098</td><td>0.04257691</td><td>0.03385153</td><td>0.02741986</td><td>0.2536</td><td>0.0187</td></tr><tr><td>57</td><td>0.14974098</td><td>0.04463934</td><td>0.02269950</td><td>0.01378168</td><td>0.2309</td><td>0.0578</td></tr><tr><td>58</td><td>0.14481623</td><td>0.04463934</td><td>0.02964869</td><td>0.02858880</td><td>0.2477</td><td>0.0547</td></tr><tr><td>59</td><td>0.14974098</td><td>0.04463934</td><td>0.02883078</td><td>0.01725744</td><td>0.2405</td><td>0.0353</td></tr><tr><td>60</td><td>0.13141002</td><td>0.04463934</td><td>0.01751295</td><td>0.01492214</td><td>0.2085</td><td>0.0648</td></tr></tbody></table>

<table><caption>TABLE 10. Ex. 3 Case 3: Non-modulated Waveform with 50% loads</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>nmod2</sub></th><th>E<sub>Diss</sub></th></tr></thead><tbody><tr><td>55</td><td>0.16901949</td><td>0.02440822</td><td>0.02958366</td><td>0.02889313</td><td>0.2519</td><td>0.0091</td></tr><tr><td>56</td><td>0.15029333</td><td>0.05405890</td><td>0.02221267</td><td>0.02370715</td><td>0.2503</td><td>0.0033</td></tr><tr><td>57</td><td>0.17416733</td><td>0.06644817</td><td>0.02077902</td><td>0.03706339</td><td>0.2985</td><td>0.0676</td></tr><tr><td>58</td><td>0.12269299</td><td>0.01896559</td><td>0.03288010</td><td>0.04990935</td><td>0.2244</td><td>0.0233</td></tr><tr><td>59</td><td>0.10986770</td><td>0.04257540</td><td>0.05813493</td><td>0.02972843</td><td>0.2403</td><td>0.0002</td></tr><tr><td>60</td><td>0.11176960</td><td>0.06644817</td><td>0.03770767</td><td>0.02925057</td><td>0.2452</td><td>0.0367</td></tr></tbody></table>

<table><caption>TABLE 11. Ex. 4 Case 1: Non-modulated Waveform with 100% loads</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>nmod1</sub></th></tr></thead><tbody><tr><td>55</td><td>0.18729641</td><td>0.04921083</td><td>0.03753135</td><td>0.03779644</td><td>0.3118</td></tr><tr><td>56</td><td>0.18729641</td><td>0.02779180</td><td>0.03753135</td><td>0.01963852</td><td>0.2723</td></tr><tr><td>57</td><td>0.18729641</td><td>0.03673046</td><td>0.02683423</td><td>0.03779644</td><td>0.2887</td></tr><tr><td>58</td><td>0.18729641</td><td>0.03979378</td><td>0.03753135</td><td>0.03779644</td><td>0.3024</td></tr><tr><td>59</td><td>0.17350394</td><td>0.02983870</td><td>0.03753135</td><td>0.03491520</td><td>0.2758</td></tr><tr><td>60</td><td>0.18729641</td><td>0.03508007</td><td>0.02120612</td><td>0.02975200</td><td>0.2733</td></tr></tbody></table>

of both waveforms sharing similar level differences. The Mersenne-modulated average total energy is 0.46 with $M = 3$.

The total energy on average is 0.0778 lower compared to non-modulated waveform in case 1, which result in 14.45% total energy dissipation. If compare with non-modulated waveform energy in case 3, a much lower energy dissipation 2.34% can be observed. This result is nearly identical to Ex. 2.
<figure><img src="image_9.png" /><figcaption>FIGURE 8. Ex. 4 Non-modulated Waveform and Modulated Waveform $M = 7$</figcaption></figure>

<table><caption>TABLE 12. Ex. 4 Case 2:Mersenne Modulated Waveform $M = 7$</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>modM7</sub></th><th>E<sub>Diss</sub></th></tr></thead><tbody><tr><td>55</td><td>0.15338374</td><td>0.04572487</td><td>0.02033777</td><td>0.02928408</td><td>0.2487</td><td>0.0631</td></tr><tr><td>56</td><td>0.15338374</td><td>0.04361230</td><td>0.03467502</td><td>0.02808728</td><td>0.2598</td><td>0.0125</td></tr><tr><td>57</td><td>0.15338374</td><td>0.04572487</td><td>0.02325203</td><td>0.01411719</td><td>0.2365</td><td>0.0522</td></tr><tr><td>58</td><td>0.14834240</td><td>0.04572487</td><td>0.03037062</td><td>0.02928408</td><td>0.2537</td><td>0.0487</td></tr><tr><td>59</td><td>0.15338374</td><td>0.04572487</td><td>0.02953276</td><td>0.01767797</td><td>0.2463</td><td>0.0295</td></tr><tr><td>60</td><td>0.13460648</td><td>0.04572487</td><td>0.01793912</td><td>0.01528539</td><td>0.2136</td><td>0.0597</td></tr></tbody></table>

<table><caption>TABLE 13. Ex. 4 Case 3:Non-modulated Waveform with 50% loads</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>nmodM2</sub></th><th>E<sub>Diss</sub></th></tr></thead><tbody><tr><td>55</td><td>0.16901949</td><td>0.02440822</td><td>0.02958366</td><td>0.02889313</td><td>0.2519</td><td>0.0032</td></tr><tr><td>56</td><td>0.15029333</td><td>0.05405890</td><td>0.02221267</td><td>0.02370715</td><td>0.2503</td><td>0.0095</td></tr><tr><td>57</td><td>0.17416733</td><td>0.06644817</td><td>0.02077902</td><td>0.03706339</td><td>0.2985</td><td>0.062</td></tr><tr><td>58</td><td>0.12269299</td><td>0.01896559</td><td>0.03288010</td><td>0.04990935</td><td>0.2244</td><td>0.0293</td></tr><tr><td>59</td><td>0.10986770</td><td>0.04257540</td><td>0.05813493</td><td>0.02972843</td><td>0.2403</td><td>0.006</td></tr><tr><td>60</td><td>0.11176960</td><td>0.06644817</td><td>0.03770767</td><td>0.02925057</td><td>0.2452</td><td>0.0316</td></tr></tbody></table>

**Example 4.** The energy dissipation comparison for Mersenne Prime $M = 7$ shown in Fig. 8. The vehicle here shown whole parts in the frames and moving towards left. The energy table of each lobe in Non-modulated waveform Case 1 and Case 3 see Table. 11 and Table. 13. The energy table of each lobe in $M=7$ Modulated waveform see Table. 12.

It is observed that the modulated unloaded waveform shared multiple similarities in shapes with the non-modulated loaded waveform. Compare the energy table of both modulated waveform and non-modulated waveform we can see the lobe energy of both waveforms sharing similar level differences. The Mersenne-modulated average total energy is 0.472 with $M = 7$.
<table><caption>TABLE 14. Ex. 5 Case 1: Non-modulated Waveform with 100% loads</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>nmod1</sub></th></tr></thead><tbody><tr><td>55</td><td>0.18729641</td><td>0.04921083</td><td>0.03753135</td><td>0.03779644</td><td>0.3118</td></tr><tr><td>56</td><td>0.18729641</td><td>0.02779180</td><td>0.03753135</td><td>0.01963852</td><td>0.2723</td></tr><tr><td>57</td><td>0.18729641</td><td>0.03673046</td><td>0.02683423</td><td>0.03779644</td><td>0.2887</td></tr><tr><td>58</td><td>0.18729641</td><td>0.03979378</td><td>0.03753135</td><td>0.03779644</td><td>0.3024</td></tr><tr><td>59</td><td>0.17350394</td><td>0.02983870</td><td>0.03753135</td><td>0.03491520</td><td>0.2758</td></tr><tr><td>60</td><td>0.18729641</td><td>0.03508007</td><td>0.02120612</td><td>0.02975200</td><td>0.2733</td></tr></tbody></table>

<table><caption>TABLE 15. Ex. 5 Case 2: Mersenne Modulated Waveform $M = 31$</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>modM31</sub></th><th>E<sub>Diss</sub></th></tr></thead><tbody><tr><td>55</td><td>0.17140233</td><td>0.05086235</td><td>0.02272370</td><td>0.03271912</td><td>0.2777</td><td>0.0341</td></tr><tr><td>56</td><td>0.17140233</td><td>0.04872594</td><td>0.03874807</td><td>0.03139619</td><td>0.2903</td><td>0.018</td></tr><tr><td>57</td><td>0.17140233</td><td>0.05086235</td><td>0.02599143</td><td>0.01578171</td><td>0.2640</td><td>0.0283</td></tr><tr><td>58</td><td>0.16585081</td><td>0.05086235</td><td>0.03395518</td><td>0.03271912</td><td>0.2834</td><td>0.019</td></tr><tr><td>59</td><td>0.17140233</td><td>0.05086235</td><td>0.03301787</td><td>0.01977253</td><td>0.2751</td><td>0.0007</td></tr><tr><td>60</td><td>0.15041012</td><td>0.05086235</td><td>0.02004971</td><td>0.01708705</td><td>0.2384</td><td>0.0349</td></tr></tbody></table>

The total energy on average is 0.0665 lower compared to non-modulated waveform in case 1, which result in 12.36% total energy dissipation. If compare with non-modulated waveform energy in case 3, a much lower energy dissipation 2.36% can be observed. The result shows dissipation is lower in case 1 and higher in case 3 compare with Ex. 2 and Ex. 3. This implies that the Mersenne-modulated waveform with $M = 7$ is transformed towards non-modulated waveform in case 1.

<figure><img src="image_10.png" /><figcaption>FIGURE 9. Ex. 5 Non-modulated Waveform and Modulated Waveform $M = 31$</figcaption></figure>

**Example 5.** The energy dissipation comparison for Mersenne Prime $M = 31$ shown in Fig. 9. The energy table of each lobe in Non-modulated waveform Case 1 and Case 3 see Table. 14 and Table. 16. The energy table of each lobe in $M=31$ Modulated waveform see Table. 15.

Compare the energy table of both modulated waveform and non-modulated waveform we can see the lobe energy of both waveforms sharing similar level differences.
<table><caption>TABLE 16. Ex. 5 Case 3:Non-modulated Waveform with 50% loads</caption><thead><tr><th>Frames</th><th>Lobe A</th><th>Lobe B</th><th>Lobe C</th><th>Lobe D</th><th>Total E<sub>nmodM2</sub></th><th>E<sub>Diss</sub></th></tr></thead><tbody><tr><td>55</td><td>0.16901949</td><td>0.02440822</td><td>0.02958366</td><td>0.02889313</td><td>0.2519</td><td>0.0258</td></tr><tr><td>56</td><td>0.15029333</td><td>0.05405890</td><td>0.02221267</td><td>0.02370715</td><td>0.2503</td><td>0.04</td></tr><tr><td>57</td><td>0.17416733</td><td>0.06644817</td><td>0.02077902</td><td>0.03706339</td><td>0.2985</td><td>0.0345</td></tr><tr><td>58</td><td>0.12269299</td><td>0.01896559</td><td>0.03288010</td><td>0.04990935</td><td>0.2244</td><td>0.059</td></tr><tr><td>59</td><td>0.10986770</td><td>0.04257540</td><td>0.05813493</td><td>0.02972843</td><td>0.2403</td><td>0.0348</td></tr><tr><td>60</td><td>0.11176960</td><td>0.06644817</td><td>0.03770767</td><td>0.02925057</td><td>0.2452</td><td>0.0068</td></tr></tbody></table>

The Mersenne-modulated average total energy is 0.527 with $M = 31$.

On the other hand, modulated waveform with $M = 31$ express 0.011 lower energy levels on average compared to non-modulated waveform in case 1, which result in an average of 2.04% energy dissipation. If compare with non-modulated waveform energy in case 3, a much lower energy dissipation 3.35% can be observed. The result implies that the Mersenne-modulated waveform with $M = 31$ is transformed towards non-modulated waveform in case 1, further than $M = 7$. This Example result agrees with Ex. 4.

<figure><img src="image_11.png" /><figcaption>FIGURE 10. Waveform Total Energy Level Comparison</figcaption></figure>

3.2. **Discussion.** The following analysis focused on case 1 therefore consider case 1 as selected Non-modulated waveform in all examples to further study the Mersenne modulated waveform energy dissipation.

In Fig. 10 and Fig. 11, if we look at energy comparison in total and in each energy lobes for modulated unloaded system, we can see the steady increase towards the non-modulated loaded system as the Mersenne prime increases.

In Fig. 12, energy dissipation of modulations by frames are calculated. Also, in Fig. 13, energy dissipation in each lobe in all unloaded systems have been shown compared to the loaded system. these are clear indications that energy dissipation drops with frequency of modulation increases.
<figure><img src="image_12.png" /><figcaption>FIGURE 11. Waveform Lobe Energy Level Comparison</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>FIGURE 12. Waveform Total Energy Dissipation Comparison</figcaption></figure>

3.3. **Conclusion.** In conclusion, this experiment set shows the analysis of energy dissipation on different Mersenne prime:1,3,7,31 in modulated signal waveform. According to the experimental data, the loaded motion waveform can be modulated by unloaded spring system with designed model. The energy dissipation can be reduced by increasing Mersenne prime in the model.

In Table. 17, all the feature data from each lobe in different case have been listed. By calculation, the average energy dissipation dropped 11% and the average total energy dissipation dropped 12.41% with Mersenne prime increasing from 1 to 31. This modulating algorithm can put in used for payload simulation in transportation field for a variety research purpose.
<figure><img src="image_14.png" /><figcaption>FIGURE 13. Waveform Lobe Energy Dissipation Comparison</figcaption></figure>

<table><caption>TABLE 17. Data Analysis on Energy Dissipation</caption><thead><tr><th></th><th>M=1</th><th>M=3</th><th>M=7</th><th>M=31</th><th>Frame select</th></tr></thead><tbody><tr><td>Total E<sub>diss</sub></td><td>0.07774675</td><td>0.07775102</td><td>0.06654327</td><td>0.01095572</td><td>25-75</td></tr><tr><td>LobeA E<sub>diss</sub></td><td>0.03951122</td><td>0.03951307</td><td>0.02800448</td><td>0.00906185</td><td>25-75</td></tr><tr><td>LobeB E<sub>diss</sub></td><td>0.00308216</td><td>0.00308279</td><td>0.00034760</td><td>0.00574854</td><td>25-75</td></tr><tr><td>LobeC E<sub>diss</sub></td><td>0.00429869</td><td>0.00429911</td><td>0.00169741</td><td>0.00258450</td><td>25-75</td></tr><tr><td>LobeD E<sub>diss</sub></td><td>0.01030824</td><td>0.01030863</td><td>0.00811160</td><td>0.00450041</td><td>25-75</td></tr><tr><td>Lobes Average E<sub>diss</sub></td><td>0.01430008</td><td>0.01430090</td><td>0.00954027</td><td>0.00547383</td><td>25-75</td></tr><tr><td>%Lobes Average E<sub>diss</sub></td><td>18%</td><td>18%</td><td>12%</td><td>7%</td><td>25-75</td></tr><tr><td>Average Total E<sub>diss</sub></td><td>0.07774675</td><td>0.07775102</td><td>0.06654327</td><td>0.01095572</td><td>25-75</td></tr><tr><td>%Average Total E<sub>diss</sub></td><td>14.45%</td><td>14.45%</td><td>12.36%</td><td>2.04%</td><td>25-75</td></tr></tbody></table>

ACKNOWLEDGMENTS

The authors extend their profound thanks to Tane Vergili, Som (Somashekhar) Naimpally (1931-2014), Anna Di Concilio and Clara Guadagni for many of their insights concerning the topology of dynamical systems. Also, many thanks to Zdzislaw Pawlak, Andrzej Skowron, Mirek (Miroslaw) Pawlak, Ke Peng, Brent Clark, Bill (William) Hankley and Sheela Ramanna for sharing their insights concerning time-constrained dynamical systems.

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 (TÜBİTAK) Scientific Human Resources Development (BIDEB) under grant no: 2221-1059B211301223.
**Declaration of conflict of interest.** Enze Cui and James F. Peters declare that they have no conflict of interest.
4. SAMPLE OFF-ROAD SPRING SYSTEM

<figure><img src="image_15.png" /><figcaption>FIGURE 14. Two views of a vibrating system</figcaption></figure>

The spring system in the off-road vehicle in 4 helps reduce the impact caused by terrain that is traversed by such vehicles. The spring system absorbs and dampens the vibrations caused by uneven terrain. In effect, the spring system maintains vehicle stability, enhances comfort, and protects the vehicle's structure. Spring systems such as the one shown in 4 includes 2 principal parts, namely, springs and dampers. Springs are the core components that absorb shock. They compress and expand to accommodate bumps and dips in the terrain. On the other hand, dampers control the rate at which the springs compress and extend. The dampers prevent a vehicle from bouncing excessively after hitting a bump.

Coil springs in 4 are made of coiled steel and provide a balance between strength and flexibility. They compress under a load and expand to absorb impacts. In summary, when an off-road vehicle encounters rough terrain, the springs compress to absorb the initial impact and dampers then control the rebound of the springs to prevent excessive oscillation, thereby reducing vibration and providing a smoother ride.
<figure><img src="image_16.png" /><figcaption>FIGURE 15. IR Spring System</figcaption></figure>

An IR video like the one in Fig. 2 samples collected in this paper uses an active IR sensor, which is a principal component in the IR-capture system outlined in 15. A flowchart for the basic structure of the IR sensor in given in 16. The IR sensor has a short wavelength in capturing near-infrared (NIR) light (800-950 nm) to illuminate an area of interest, then collect the reflected light in NIR range of wavelength. IR sensor is the key element for collecting Infrared video. IR sensor is a device that emits infrared radiation in order to sense object of the surroundings. These types of radiations are invisible to human eyes, but the receiver unit of IR sensor can detect these radiations.

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DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING, UNIVERSITY OF MANITOBA, WPG,
MB, R3T 5V6, CANADA

Email address: cuie@umanitoba.ca

DEPARTMENT OF ELECTRICAL & COMPUTER ENGINEERING, UNIVERSITY OF MANITOBA, WPG,
MB, R3T 5V6, CANADA AND SCIENCES, ADIYAMAN UNIVERSITY, 02040 ADIYAMAN, TURKEY

Email address: james.peters3@umanitoba.ca