# Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel

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## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts

## 3. Mass-Spring-Damper System

- Mass-spring system, $\beta =0$$${}_{0}^{C}{D}_{t}^{2\gamma}x\left(t\right)+\frac{k}{m}{\sigma}^{2(1-\gamma )}x\left(t\right)=\frac{F\left(t\right)}{m}{\sigma}^{2(1-\gamma )},\phantom{\rule{2.em}{0ex}}0<\gamma \le 1,$$$${}_{0}^{CF}{\mathcal{D}}_{t}^{2\gamma}x\left(t\right)+\frac{k}{m}{\sigma}^{2(1-\gamma )}x\left(t\right)=\frac{F\left(t\right)}{m}{\sigma}^{2(1-\gamma )},\phantom{\rule{2.em}{0ex}}0<\gamma \le 1.$$
- Damper-spring system, $m=0$:$${}_{0}^{C}{D}_{t}^{\gamma}x\left(t\right)+\frac{k}{\beta}{\sigma}^{1-\gamma}x\left(t\right)=\frac{F\left(t\right)}{\beta}{\sigma}^{1-\gamma},\phantom{\rule{2.em}{0ex}}0<\gamma \le 1,$$$${}_{0}^{CF}{\mathcal{D}}_{t}^{\gamma}x\left(t\right)+\frac{k}{\beta}{\sigma}^{1-\gamma}x\left(t\right)=\frac{F\left(t\right)}{\beta}{\sigma}^{1-\gamma},\phantom{\rule{2.em}{0ex}}0<\gamma \le 1.$$

#### 3.1. Mass-Spring System

#### 3.2. Damper-Spring System

#### 3.3. Mass-Spring-Damper System

## 4. Conclusion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Gómez-Aguilar, J.F.; Yépez-Martínez, H.; Calderón-Ramón, C.; Cruz-Orduña, I.; Escobar-Jiménez, R.F.; Olivares-Peregrino, V.H.
Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel. *Entropy* **2015**, *17*, 6289-6303.
https://doi.org/10.3390/e17096289

**AMA Style**

Gómez-Aguilar JF, Yépez-Martínez H, Calderón-Ramón C, Cruz-Orduña I, Escobar-Jiménez RF, Olivares-Peregrino VH.
Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel. *Entropy*. 2015; 17(9):6289-6303.
https://doi.org/10.3390/e17096289

**Chicago/Turabian Style**

Gómez-Aguilar, José Francisco, Huitzilin Yépez-Martínez, Celia Calderón-Ramón, Ines Cruz-Orduña, Ricardo Fabricio Escobar-Jiménez, and Victor Hugo Olivares-Peregrino.
2015. "Modeling of a Mass-Spring-Damper System by Fractional Derivatives with and without a Singular Kernel" *Entropy* 17, no. 9: 6289-6303.
https://doi.org/10.3390/e17096289