# Three Classes of Fractional Oscillators

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

**Class I:**The first class contains oscillators with fractional inertia force $m\frac{{d}^{\alpha}x(t)}{d{t}^{\alpha}}(1<\alpha \le 2)$ only. Its oscillation equation is in the form of (31), see, e.g., Duan ([24], Equation (3)), Mainardi ([25], Equation (27)), Zurigat ([26], Equation (16)), Blaszczyk and Ciesielski ([27], Equation (1)), Blaszczyk et al. ([28], Equation (10)), Al-rabtah et al. ([29], Equation (3.1)), Drozdov ([30], Equation (9)), Stanislavsky [31], Achar et al. ([32], Equation (1), [33], Equation (9), [34], Equation (2)), Tofighi ([35], Equation (2)), Ryabov and Puzenko ([36], Equation (1)), Ahmad and Elwakil ([37], Equation (1)), Uchaikin ([38], Chapter 7), Duan et al. ([39], Equation (4.2)).

**Class II:**The second consists of oscillators only with fractional damping term $c\frac{{d}^{\beta}x(t)}{d{t}^{\beta}}(0<\beta \le 1),$ see, e.g., Lin et al. ([40], Equation (2)), Duan ([41], Equation (31)), Alkhaldi et al. ([42], Equation (1a)), Dai et al. ([43], Equation (1)], Ren et al. ([44], Equation (1)), Xu et al. ([45], Equation (1)), He et al. ([46], Equation (4)), Leung et al. ([47], Equation (2)), Chen et al. ([48], Equation (1)), Deü and Matignon ([49], Equation (1)), Drăgănescu et al. ([50], Equation (4)), Rossikhin and Shitikova ([51], Equation (3)), Xie and Lin ([52], Equation (1)), Chung and Jung [53]. That takes the form of (42) in the next Section.

**Class III:**The third includes the oscillators with both fractional inertia force $m\frac{{d}^{\alpha}x(t)}{d{t}^{\alpha}}(1<\alpha \le 2)$ and fractional friction $c\frac{{d}^{\beta}x(t)}{d{t}^{\beta}}(0<\beta \le 1),$ see, e.g., Liu et al. ([54], Equation (1)), Gomez-Aguilar ([53], Equation (10)), Leung et al. ([50], Equation (3)). This class of oscillators is expressed by (43).

- Establishing three equivalent 2-order differential equations respectively corresponding to three classes of fractional oscillators.
- Presenting the analytical representations, in the closed form, of equivalent masses, equivalent dampings, equivalent damping ratios, equivalent natural frequencies, and equivalent frequency ratios, for each class of fractional oscillators.
- Proposing the analytic expressions, in the closed form by using elementary functions, of the free, impulse, step, frequency, and sinusoidal responses to three classes of fractional oscillators.
- Revealing the power laws of equivalent mass and equivalent damping for each class of fractional oscillators.
- Representing some of the generalized Mittag-Leffler functions by using elementary functions.

## 2. Preliminaries

#### 2.1. Brief of Linear Oscillations of Order 2

#### 2.1.1. Simple Oscillation Model

**Note 2.1:**All parameters above, namely, m, c, k, ζ, ${\omega}_{n},$ and ${\omega}_{d},$ are constants.

#### 2.1.2. Responses

#### 2.1.3. Spectra of Three Excitations

#### 2.1.4. Generalization of Linear Oscillators

#### 2.2. Three Classes of Fractional Oscillators

_{a}(t) is in the form

#### 2.3. Equivalence of Functions in the Sense of Fourier Transform

## 3. Problem Statement and Research Thoughts

#### 3.1. Problem Statement

**Problem**

**1.**

**Problem**

**2.**

**Problem**

**3.**

**Problem**

**4.**

**Problem**

**5.**

**Problem**

**6.**

#### 3.2. Research Thoughts

## 4. Equivalent Systems of Three Classes of Fractional Oscillators

#### 4.1. Equivalent System for Fractional Oscillators in Class I

#### 4.1.1. Equivalent Oscillation Equation of Fractional Oscillators in Class I

**Theorem**

**1**(Equivalent oscillator I)

**.**

**Proof.**

#### 4.1.2. Equivalent Mass of Fractional Oscillators in Class I

**Theorem 2**(Equivalent mass I)

**.**

**Proof.**

**Remark**

**1.**

**Note 4.1**: Since

**Note 4.2:**If α → 1, we have

**Note 4.3:**When 1 < α ≤ 2, we attain

**Remark**

**2.**

**Remark**

**3.**

#### 4.1.3. Equivalent Damping of Fractional Oscillators of Class I

**Theorem 3**(Equivalent damping I)

**.**

**Proof.**

**Remark**

**4.**

**Note 4.4:**Because

**Remark**

**5.**

#### 4.2. Equivalent Oscillation System for Fractional Oscillators of Class II Type

#### 4.2.1. Equivalent Oscillation Equation of Fractional Oscillators in Class II

**Theorem 4**(Equivalent oscillator II)

**.**

**Proof.**

#### 4.2.2. Equivalent Mass of Fractional Oscillators of Class II

**Theorem 5**(Equivalent mass II)

**.**

**Proof.**

**Remark**

**6.**

**Note 4.5:**Equation (97) exhibits that ${m}_{eq2}$ is related to the oscillation frequency ω, the fractional order β, the primary mass m, and the primary damping c.

**Remark**

**7.**

**Remark**

**8.**

**Note 4.6:**The equivalent mass II is negative if ω is small enough.

**Remark**

**9.**

**Note 4.7:**The equivalent mass II reduces to the primary mass m for β = 1 as indicated below.

#### 4.2.3. Equivalent Damping of Fractional Oscillators in Class II

**Theorem 6**(Equivalent damping II)

**.**

**Proof.**

**Remark**

**10.**

**Note 4.8:**The following says that ${c}_{eq2}$ reduces to the primary damping c if β = 1.

**Remark**

**11.**

**Note 4.9:**The equivalent oscillation equation of Class II fractional oscillators reduces to $m\frac{{d}^{2}{x}_{2}(t)}{d{t}^{2}}+k{x}_{2}(t)=0$ in the two cases. One is ω → ∞, see Remark 7 and Remark 12. The other is c = 0.

**Note 4.10:**Remark 5 for $\underset{\omega \to \infty}{\mathrm{lim}}{c}_{eq1}(\omega ,\beta )=\infty $ and Remark 11 just above suggest a substantial difference between two types of fractional oscillators from the point of view of the damping at ω → ∞.

**Remark**

**12.**

#### 4.3. Equivalent Oscillation System for Fractional Oscillators of Class III

#### 4.3.1. Equivalent Oscillation Equation of Fractional Oscillators in Class III

**Theorem 7**(Equivalent oscillator III)

**.**

**Proof.**

#### 4.3.2. Equivalent Mass of Fractional Oscillators in Class III

**Theorem 8**(Equivalent mass III)

**.**

**Proof.**

**Remark**

**13.**

**Note 4.11:**The equivalent mass ${m}_{eq3}$ is related to ω, m, and c, as well as a pair of fractional orders (α, β).

**Note 4.12:**If α = 2 and β = 1, ${m}_{eq3}$ reduces to the primary m, i.e.,

**Remark**

**14.**

**Remark**

**15.**

**Remark**

**16.**

#### 4.3.3. Equivalent Damping of Fractional Oscillators in Class III

**Theorem 9**(Equivalent damping III)

**.**

**Proof.**

**Remark**

**17.**

**Note 4.13:**From (123), we see that ${c}_{eq3}$ reduces to the primary damping c for α = 2 and β = 1. That is,

**Remark**

**18.**

**Remark**

**19.**

**Note 4.14:**The equivalent damping ${c}_{eq3}$= 0 if both α = 2 and c = 0:

#### 4.4. Summary

## 5. Equivalent Natural Frequencies and Damping Ratio of Three Classes of Fractional Oscillators

#### 5.1. Equivalent Natural Frequency I

**Definition**

**1.**

**Note 5.1:**${\omega}_{eqn,j}$ may take the conventional natural frequency, denoted by

**Corollary 1**(Equivalent natural frequency I1)

**.**

**Proof.**

**Note 5.2:**From Figure 10, we see that ${\omega}_{eqn,1}$ is an increasing function with ω. Besides, the greater the value of α the smaller the ω

_{eqn}

_{,1}.

**Note 5.3:**${\omega}_{eqn,1}$ becomes ${\omega}_{n}$ if α = 2. In fact,

**Corollary 2**(Equivalent natural frequency I2)

**.**

**Proof.**

**Note 5.4:**Figure 11 shows that ${\omega}_{eqn,2}$ is a decreasing function with ω. The greater the value of β the smaller the ${\omega}_{eqn,2}.$

**Note 5.5:**${\omega}_{eqn,2}$ takes ${\omega}_{n}$ as a special case for β = 1. As a matter of fact,

**Corollary 3**(Equivalent natural frequency I3)

**.**

**Proof.**

**Note 5.7:**${\omega}_{eqn,3}$ takes ${\omega}_{n}$ as a special case for α = 2 and β = 1. Indeed,

#### 5.2. Equivalent Damping Ratio

**Definition**

**2.**

**Corollary 4**(Equivalent damping ratio I)

**.**

**Proof.**

**Remark**

**20.**

**Remark**

**21.**

**Corollary 5**(Equivalent damping ratio II)

**.**

**Proof.**

**Remark**

**22.**

**Remark**

**23.**

**Note 5.9:**${\varsigma}_{eq2}$ takes ζ as a special case for β = 1. In fact,

**Corollary 6**(Equivalent damping ratio III)

**.**

**Proof.**