# Entropy Generation in a Mass-Spring-Damper System Using a Conformable Model

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary

#### 2.1. Conformable Fractional Operator

**Definition**

**1.**

**Theorem**

**1.**

- 1.
- ${T}_{\gamma}\{a\}(t)=0$;
- 2.
- ${T}_{\gamma}\{{t}^{b}\}(t)=b\xb7{t}^{b-\gamma}$;
- 3.
- ${T}_{\gamma}\{a\xb7f\pm b\xb7g\}(t)=a\xb7{T}_{\gamma}\{f\}(t)\pm b\xb7{T}_{\gamma}\{g\}(t)$;
- 4.
- ${T}_{\gamma}\{f\circ g\}(t)={\displaystyle \frac{df}{dg}}\xb7{T}_{\gamma}\{g\}(t);$
- 5.
- ${T}_{\gamma}\{f\xb7g\}(t)={T}_{\gamma}\{f\}(t)\xb7g(t)+f(t)\xb7{T}_{\gamma}\{g\}(t);$
- 6.
- ${T}_{\gamma}\left\{f/g\right\}(t)={\displaystyle \frac{g(t)\xb7{T}_{\gamma}\{f\}(t)-f(t)\xb7{T}_{\gamma}\{g\}(t)}{{g}^{2}(t)}};$
- 7.
- ${T}_{\alpha}\{f(t)\}={t}^{n+1-\alpha}{\displaystyle \frac{{d}^{n+1}f}{d{t}^{n+1}}},\phantom{\rule{0.277778em}{0ex}}\alpha \in ]n,n+1],\phantom{\rule{0.277778em}{0ex}}n\in {\mathbb{Z}}_{+}$; and f is $(n+1)$-differentiable at $t>0$.

#### 2.2. Entropy Generation of Mechanical Systems

## 3. Mathematical Models

#### 3.1. Ordinary Model

#### 3.2. Fractional Model

#### 3.3. Conformable Model

#### 3.4. Entropy Production Rate Model

^{2}) from Equation (9) gives

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Displacement $x(t)$ of the MSD system using the conformable model, considering the four typical damping cases and varying the fractional order $\gamma $.

**Figure 3.**Phase planes $x(t)$ vs. $v(t)$ of the MSD system using the conformable model, considering the four typical damping cases and varying the fractional order $\gamma $.

**Figure 4.**Settling time ${t}_{s}$ (s) of the MSD system using the conformable model, determined by varying the fractional order $\gamma $ and the damping ratio $\zeta $.

**Figure 5.**Entropy generation rate ${\dot{S}}_{gen}(t)$ of the MSD system using the conformable model, considering the four typical damping cases and varying the fractional order $\gamma $.

**Figure 6.**Total entropy generation $\Delta {S}_{gen}$ (J/K) of the MSD system using the conformable model, varying the damping ratio $\zeta \in [0,2]$ and the fractional order $\gamma \in ]0,1]$.

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**MDPI and ACS Style**

Cruz-Duarte, J.M.; Rosales-García, J.J.; Correa-Cely, C.R.
Entropy Generation in a Mass-Spring-Damper System Using a Conformable Model. *Symmetry* **2020**, *12*, 395.
https://doi.org/10.3390/sym12030395

**AMA Style**

Cruz-Duarte JM, Rosales-García JJ, Correa-Cely CR.
Entropy Generation in a Mass-Spring-Damper System Using a Conformable Model. *Symmetry*. 2020; 12(3):395.
https://doi.org/10.3390/sym12030395

**Chicago/Turabian Style**

Cruz-Duarte, Jorge M., J. Juan Rosales-García, and C. Rodrigo Correa-Cely.
2020. "Entropy Generation in a Mass-Spring-Damper System Using a Conformable Model" *Symmetry* 12, no. 3: 395.
https://doi.org/10.3390/sym12030395