# The Maximum Flywheel Load: A Novel Index to Monitor Loading Intensity of Flywheel Devices

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Experimental Approach to the Problem

#### 2.2. Subjects

^{2}p), the minimum required sample size was 8 subjects.

#### 2.3. Procedures

^{2}for FW-load).

#### 2.3.1. Incremental Load Test

^{2}, followed by increments of 0.025 kg·m

^{2}until a maximum value of 0.125 kg·m

^{2}. For further analyses, the three repetitions with the highest mean concentric speed were selected in both tests. If any subject showed considerable variability between the mean concentric speed (i.e., denoting a bad execution technique), that load was not considered for further analyses. Only subjects that completed all of the five loads were considered in the final analyses (n = 21).

#### 2.3.2. Data Acquisition

^{2}) and angular (rad/s

^{2}) accelerations, as the time derivative (incremental ratio) of filtered speed. Each repetition was detected during the FW-load condition using the change from negative (eccentric phase) to positive (concentric phase) acceleration values. For FW-load, we also calculated torque as the product of the flywheel moment of inertia times angular acceleration. Finally, we calculated power as the product of force (for ISO-load) or torque (for FW-load) and speed. We only used the concentric phase values for the analyses (propulsive phase (34) for ISO-load and whole concentric phase for FW-load).Before any intervention, subjects performed a standardized warm-up consisting of 5 min cycling on a mechanical ergometer at a self- selected submaximal pace, followed by 5 min of lower limb mobilization, five countermovement jumps, and six submaximal repetitions on the squat device, using the lowest external load from the incremental load test (20 kg for ISO-load and 0.025 kg·m

^{2}for FW-load).

#### 2.3.3. Mechanical Variables

#### 2.3.4. Flywheel Workload Indexes

#### 2.3.5. Statistical Analyses

## 3. Results

#### 3.1. Relationships between Mechanical Variables and Moment of Inertia

#### 3.2. Flywheel Training Intensity Index

^{2}. Figure 2 shows the association between the MAA- and MT-%MFL. When MT was calculated using the logarithmic fit model between the MAA-moment of inertia association, the highest MT was observed at 36.8% ± 0.7 of MFL, in accordance with the theoretical model. Finally, the total PTL average resulted in an average of 0.08 ± 0.02 kg·m

^{2}. To illustrate the concept of this methodology, Figure 3 shows a comparison between subjects with a notable difference in both MFL and PTL.

#### 3.3. Relationships and Differences between Loading Conditions

## 4. Discussion

^{2}), T = torque (N·m), and I = moment of inertia (kg·m

^{2}).

^{2}. Individuals with a higher MFL might not be able to work at a high relative load because it could exceed the machine’s load capability. This is not a problem per se, since the logarithmic model loses validity close to 100%MFL (see Appendix A for more information), and since the optimal load (PTL) is invariably achieved at ≈37% MFL. Therefore, it is still unknown if higher relative loads (e.g., >70–80% MFL) have practical interest and applications. Finally, the resolution in inertia adjustment is different for each FRTD, since the numbers and types of flywheel are usually limited. The device used in this study allowed for small increments down to 0.005 kg·m

^{2}. However, when the range of adjustment is limited, it might be necessary to adopt an available inertia different from the theoretically desired value. Future studies may analyze the reproducibility and validity of these indexes at different strength levels, in different exercises, and on different FRTD machines (e.g., with different shafts and loading ranges).

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Rationale of the Study and Physics of Flywheel Resistance Training Devices

^{2}, is the gravity acceleration on Earth. In Physics, the mass M is also referred to as the linear moment of inertia since it creates a force (M·a) that opposes movement whenever there is a change in its status, i.e., acceleration or deceleration.

^{2}).

#### Appendix A.2. Selection of the Analytical Model: Logarithmic

#### Appendix A.3. Analytical Calculation of Maximum Flywheel Load (MFL)

^{2}(and, in any case, no worse than other models) but always allows analytical computation of the curve intercept on the horizontal axis, i.e., a theoretical maximum inertia value corresponding to null acceleration (no movement): Maximum Flywheel Load, MFL. Its value is given by solving the logarithmic formula e) for y = 0 (1):

#### Appendix A.4. Validity of the Logarithmic Model and MFL

- imaginary numbers $i\equiv \sqrt{-1}$;
- exponentials of complex numbers representing sinusoidal waves;
- n-dimension spaces with n > 3;
- Fourier transforms used to represent signals in the frequency domain;
- Dirac’s delta (δ) (signal with infinite amplitude and infinitesimal duration but unitary energy), used to model signal sampling;
- quantum mechanics and quantum tunnel effects;
- etc.

#### Appendix A.5. Peak Torque Load (PTL)

**Figure A1.**Acceleration (

**upper**) and torque (

**lower**) using the logarithmic model. MFL = maximum flywheel load. PTL = peak torque load.

^{1}: $a+b\cdot I\frac{d}{dI}\cdot \left(\mathrm{ln}(I)\right)+b\cdot \mathrm{ln}(I)\cdot \frac{d}{dI}\left(I\right)=0$

^{2}: $a+b\cdot I\cdot \frac{1}{I}+b\cdot \mathrm{ln}(I)=0$

#### Appendix A.6. Calculation Notes

^{1}In this step, the derivative of the product of two functions: f(x) = x and g(x) = ln(x) has been computed using the rule:$$\frac{d}{dx}\left(f\left(x\right)\cdot g\left(x\right)\right)=f\left(x\right)\frac{d}{dx}g\left(x\right)+g\left(x\right)\frac{d}{dx}f\left(x\right)$$^{2}In this step, the derivative of logarithm has been computed as:$$\frac{d}{dx}\mathrm{ln}(x)=\frac{1}{x}$$

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**Figure 1.**Association between each mechanical variable and the moment of inertia used, with different fitting models. r

^{2}shows the mean ± standard deviation coefficient of determination for each model. The lines represent the regression line for each model, while the gray points represent individual values for each subject.

**Figure 2.**Association between mean angular acceleration and mean torque versus relative flywheel load. Lines represent mean ± standard deviation (dotted lines) of all the subjects’ individual regression lines (logarithmic fit for angular acceleration and second-degree polynomial for mean torque). The vertical black line represents the relative inertia at which peak torque load (PTL) is attained as a function of maximum flywheel load (MFL).

**Figure 3.**Illustration of the maximum flywheel load index concept. The upper graph shows the acceleration regression line (continuous line) and the torque regression line (dashed line) of a subject with a lower maximum flywheel load (MFL, black color) against a subject with a higher MFL (gray color). MFL is calculated as the intercept of the acceleration curve on the horizontal axis. The lower graph compares both subjects’ profiles using relative intensities instead.

**Figure 4.**Association between maximum repetition (ISO-load) and the maximum flywheel load (FW-load). Dotted lines represent 95% of the confidence intervals.

**Figure 5.**(

**A**) Estimated data normalized from maximum speed; (

**B**) acceleration; (

**C**) vertical force, and (

**D**) power for five relative intensities (training intensity continuum) for the flywheel (FW-load) and weight load types (ISO-load). Data points represent mean values, and error bars the standard deviation. Between loading condition differences are shown as: * = p < 0.05, ** = p < 0.001.

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**MDPI and ACS Style**

Muñoz-López, A.; Floría, P.; Sañudo, B.; Pecci, J.; Carmona Pérez, J.; Pozzo, M.
The Maximum Flywheel Load: A Novel Index to Monitor Loading Intensity of Flywheel Devices. *Sensors* **2021**, *21*, 8124.
https://doi.org/10.3390/s21238124

**AMA Style**

Muñoz-López A, Floría P, Sañudo B, Pecci J, Carmona Pérez J, Pozzo M.
The Maximum Flywheel Load: A Novel Index to Monitor Loading Intensity of Flywheel Devices. *Sensors*. 2021; 21(23):8124.
https://doi.org/10.3390/s21238124

**Chicago/Turabian Style**

Muñoz-López, Alejandro, Pablo Floría, Borja Sañudo, Javier Pecci, Jorge Carmona Pérez, and Marco Pozzo.
2021. "The Maximum Flywheel Load: A Novel Index to Monitor Loading Intensity of Flywheel Devices" *Sensors* 21, no. 23: 8124.
https://doi.org/10.3390/s21238124