# Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Generalized Fractional Derivative Operators

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

**Definition**

**4.**

**Definition**

**5.**

**Definition**

**6.**

## 3. Fractional Mass-Spring-Damper Systems

#### 3.1. Caputo Generalized Fractional Derivative

#### 3.1.1. Absence of Mass

**Theorem**

**1.**

**Proof.**

#### 3.1.2. Absence of the Spring Coefficient

**Theorem**

**2.**

**Proof.**

#### 3.1.3. In the Presence of Mass and Spring Coefficients

#### 3.2. Left Generalized Fractional Derivative

#### 3.2.1. Absence of Mass

**Theorem**

**3.**

**Proof.**

#### 3.2.2. Absence of the Spring Coefficient

**Theorem**

**4.**

**Proof.**

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Numerical simulation for Equation (14) considering different values of $\alpha $, arbitrarily chosen.

**Figure 2.**Numerical simulation for Equation (17) considering different values of $\alpha $, arbitrarily chosen.

**Figure 3.**Numerical simulation for Equation (19) considering different values of $\alpha $, arbitrarily chosen.

**Figure 4.**Numerical simulation for Equation (26) considering different values of $\alpha $, arbitrarily chosen.

**Figure 5.**Numerical simulation for Equation (29) considering different values of $\alpha $, arbitrarily chosen.

**Figure 6.**Numerical simulation for Equation (31) considering different values of $\alpha $, arbitrarily chosen.

**Figure 7.**Numerical simulation for Equation (35) considering different values of $\alpha $, arbitrarily chosen.

**Figure 8.**Numerical simulation for Equation (42) considering different values of $\alpha $, arbitrarily chosen.

**Figure 9.**Numerical simulation for Equation (43) considering different values of $\alpha $, arbitrarily chosen.

**Figure 10.**Numerical simulation for Equation (45) considering different values of $\alpha $, arbitrarily chosen.

**Figure 11.**Numerical simulation for Equation (51) considering different values of $\alpha $, arbitrarily chosen.

**Figure 12.**Numerical simulation for Equation (53) considering different values of $\alpha $, arbitrarily chosen.

**Figure 13.**Numerical simulation for Equation (55) considering different values of $\alpha $, arbitrarily chosen.

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**MDPI and ACS Style**

Sene, N.; Gómez Aguilar, J.F.
Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives. *Fractal Fract.* **2019**, *3*, 39.
https://doi.org/10.3390/fractalfract3030039

**AMA Style**

Sene N, Gómez Aguilar JF.
Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives. *Fractal and Fractional*. 2019; 3(3):39.
https://doi.org/10.3390/fractalfract3030039

**Chicago/Turabian Style**

Sene, Ndolane, and José Francisco Gómez Aguilar.
2019. "Fractional Mass-Spring-Damper System Described by Generalized Fractional Order Derivatives" *Fractal and Fractional* 3, no. 3: 39.
https://doi.org/10.3390/fractalfract3030039