Three Classes of Fractional Oscillators

Abstract

This article addresses three classes of fractional oscillators named Class I, II and III. It is known that the solutions to fractional oscillators of Class I type are represented by the Mittag-Leffler functions. However, closed form solutions to fractional oscillators in Classes II and III are unknown. In this article, we present a theory of equivalent systems with respect to three classes of fractional oscillators. In methodology, we first transform fractional oscillators with constant coefficients to be linear 2-order oscillators with variable coefficients (variable mass and damping). Then, we derive the closed form solutions to three classes of fractional oscillators using elementary functions. The present theory of equivalent oscillators consists of the main highlights as follows. (1) Proposing three equivalent 2-order oscillation equations corresponding to three classes of fractional oscillators; (2) Presenting the closed form expressions of equivalent mass, equivalent damping, equivalent natural frequencies, equivalent damping ratio for each class of fractional oscillators; (3) Putting forward the closed form formulas of responses (free, impulse, unit step, frequency, sinusoidal) to each class of fractional oscillators; (4) Revealing the power laws of equivalent mass and equivalent damping for each class of fractional oscillators in terms of oscillation frequency; (5) Giving analytic expressions of the logarithmic decrements of three classes of fractional oscillators; (6) Representing the closed form representations of some of the generalized Mittag-Leffler functions with elementary functions. The present results suggest a novel theory of fractional oscillators. This may facilitate the application of the theory of fractional oscillators to practice.

1. Introduction

Any systems that consist of three elements, namely, inertia, restoration, and damping, may oscillate. Therefore, oscillations are common phenomena encountered in various fields, ranging from physics to mechanical engineering, see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17].

Fractional oscillators and their processes attract the interests of researchers, see, e.g., [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53]. There are problems worth studying with respect to fractional oscillators. On the one hand, the analytical expressions in the closed forms of responses to certain fractional oscillators, e.g., those described by (42) and (43) in Section 2, remain unknown. In addition, closed form representations of some physical quantities in fractional oscillators, such as mass, damping, natural frequencies, in the intrinsic sense, are lacking. On the other hand, technology and analysis methods, based on 2-order linear oscillations, almost dominate the preference of engineers although nonlinear oscillations have been paid attention to. Therefore, from a view of engineering, it is meaningful to establish a theory to deal with fractional oscillators with equivalent linear oscillation systems of order 2. This article contributes my results in this aspect.

This research studies three classes of fractional oscillators.

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).

By fractional oscillating in this research, we mean that either the inertia term (31) or the damping (42) or both (43) are described by fractional derivative. Thus, this article studies all described above from Class I to III except those fractional nonlinear ones, such as fractional van der Pol oscillators (Leung et al. [47,55], Xie and Lin [52], Kavyanpoor and Shokrollahi [56], Xiao et al. [57]), fractional Duffing ones (Xu et al. [45], Liu et al. [54], Chen et al. [58], Wen et al. [59], Liao [60]). Besides, the meaning of fractional oscillation in this research neither implies those with fractional displacement such as Abu-Gurra et al. [61] discussed nor those in the sense of subharmonic oscillations as stated by Den Hartog ([3], Sections 8–10, Chapter 4), Ikeda [62], Fudan Univ. ([63], pp. 96–97), Andronov et al. ([64], Section 5.1).

Fractional differential equations represented by (31), (42), and (43) are designated as fractional oscillators in Class I, II, and III, respectively, in what follows. Note that closed form analytic expressions for the responses (free, impulse, step, frequency, and sinusoidal) to fractional oscillators in Class II and III are rarely reported. For oscillators in Class I, analytic expressions for the responses (free, impulse, step) are only represented by a type of special functions called the Mittag-Leffler functions but lack in representing the intrinsic properties, such as damping. This article aims at presenting a unified approach to deal with three classes of fractional oscillators.

The present highlights are as follows.

• 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.

Note that this article studies fractional oscillators by the way of dealing with fractional inertia force and or fractional friction equivalently using inertia force and or fractional friction of integer order. In doing so, methodologically speaking, the key point is about three equivalent oscillation models, which transform fractional inertia force and or fractional friction equivalently into inertia force and or fractional friction of integer order, which we establish with Theorems 1–7. Though they may yet imply a novel way to study fractional derivatives from the point of view of mathematics, my focus in this research is on treating fractional oscillators from a view of physical or engineering oscillations (vibrations).

The rest of the article is organized as follows. Section 2 is about preliminaries. The problem statement and research thoughts are described in Section 3. We establish three equivalent 2-order oscillation equations respectively corresponding to three classes of fractional oscillators in Section 4. The analytical representations of equivalent masses, equivalent dampings, equivalent damping ratios, equivalent natural frequencies for three classes of fractional oscillators are proposed in Section 5. We present the analytic expressions of the free responses to three classes of fractional oscillators in Section 6, the impulse responses to three classes of fractional oscillators in Section 7, the step responses in Section 8, the frequency responses in Section 9, and the sinusoidal ones in Section 10. Discussions are in Section 11, which is followed by conclusions.

2. Preliminaries

This Section consists of two parts. One is to describe the basic of linear oscillations and fractional ones related to the next sections. The other the solutions to fractional oscillators in Class I based on the generalized Mittag-Leffler functions.

2.1. Brief of Linear Oscillations of Order 2

2.1.1. Simple Oscillation Model

The simplest model of an oscillator of order 2 is with single degree of freedom (SDOF). It consists of a constant mass m and a massless damper with a linear viscous damping constant c. The stiffness of spring is denoted by spring constant k. That SDOF mass-spring system is described by

$$\{\begin{array}{l}m\frac{d^{2}q(t)}{d{t}^2}+c\frac{dq(t)}{dt}+kq(t)=e(t)\\ q(0)=q_0,q^{\prime }(0)=v_0,\end{array}$$

(1)

where e(t) is the forcing function. The solution q(t) may be the displacement in mechanical engineering [1,2,3,4,5,6,7] or current in electronics engineering [8].

In physics and engineering, for facilitating the analysis, one usually rewrites (1) by

$$\{\begin{array}{l}\frac{d^{2}q(t)}{d{t}^2}+\frac{c}{m}\frac{dq(t)}{dt}+\frac{k}{m}q(t)=\frac{e(t)}{m}\\ q(0)=q_0,q^{\prime }(0)=v_0,\end{array}$$

(2)

and further rewrites it by

$$\{\begin{array}{l}\frac{d^{2}q(t)}{d{t}^2}+2\varsigma {\omega }_n\frac{dq(t)}{dt}+{\omega }_n^{2}q(t)=\frac{e(t)}{m}\\ q(0)=q_0,q^{\prime }(0)=v_0,\end{array}$$

(3)

where ${\omega }_n$ is called the natural angular frequency (natural frequency for short) with damping free given by

$${\omega }_n=\sqrt{\frac{k}{m}},$$

(4)

and the parameter ς is the damping ratio expressed by

$$\varsigma =\frac{c}{2\sqrt{mk}}.$$

(5)

The characteristic equation of (3) is in the form

$$p^2+2\varsigma {\omega }_{n}p+{\omega }_n^2=0,$$

(6)

which is usually called the frequency equation in engineering [1,2,3,4,5,6,7]. The solution to the above is given by

$$p_{1,2}=-\varsigma {\omega }_n\pm i{\omega }_n\sqrt{1-{\varsigma }^2},$$

(7)

where $i=\sqrt{-1}.$ Taking into account damping, one uses the term damped natural frequency denoted by ${\omega }_d.$ It is given by

$${\omega }_d={\omega }_n\sqrt{1-{\varsigma }^2}.$$

(8)

Note 2.1: All parameters above, namely, m, c, k, ζ, ${\omega }_n,$ and ${\omega }_d,$ are constants.

2.1.2. Responses

The free response, meaning that the response with e(t) = 0, is driven by initial conditions only. It is given by

$$q(t)=e^{-\varsigma {\omega }_{n}t}\left(q_0\cos {\omega }_{d}t+\frac{v_0+\varsigma {\omega }_n{q}_0}{{\omega }_d}\sin {\omega }_{d}t\right),\hspace{1em}t\ge 0.$$

(9)

If e(t) = δ(t), where δ(t) is the Dirac-delta function, the response with zero initial conditions is called the impulse response. In the theory of linear systems (Gabel and Roberts [65], Zheng et al. [66]), the symbol h(t) is used for the impulse response. Thus, consider the equation

$$\frac{d^{2}h(t)}{d{t}^2}+2\varsigma {\omega }_n\frac{dh(t)}{dt}+{\omega }_n^{2}h(t)=\frac{\delta (t)}{m}.$$

(10)

One has

$$h(t)=\frac{e^{-\varsigma {\omega }_{n}t}}{m{\omega }_d}\sin {\omega }_{d}t,\hspace{1em}t\ge 0.$$

(11)

Let u(t) be the Heaviside unit step (unit step for short) function. Then, the response to (3) with zero initial conditions is called the unit step response. As usual, it is denoted by g(t) in practice. Thus, consider

$$\frac{d^{2}g(t)}{d{t}^2}+2\varsigma {\omega }_n\frac{dg(t)}{dt}+{\omega }_n^{2}g(t)=\frac{u(t)}{m}.$$

(12)

One has

$$g(t)={\displaystyle \underset{0}{\overset{t}{\int }}h(\tau )d\tau }=\frac{1}{k}\left[1-\frac{e^{-\varsigma {\omega }_{n}t}}{\sqrt{1-{\varsigma }_{}^2}}\cos \left({\omega }_{d}t-\varphi \right)\right],$$

(13)

where

$$\varphi ={\tan }^{-1}\frac{\varsigma }{\sqrt{1-{\varsigma }_{}^2}}.$$

(14)

Denote by H(ω) the Fourier transform of h(t). Then, H(ω) is usually called the frequency response to the oscillator described by (3). It is in the form

$$H(\omega )=\frac{1}{m\left({\omega }_n^2-{\omega }^2+i2\varsigma {\omega }_n\omega \right)}=\frac{1}{m{\omega }_n^2\left(1-\frac{{\omega }^2}{{\omega }_n^2}+i2\varsigma \frac{\omega }{{\omega }_n}\right)}.$$

(15)

With the parameter γ defined by

$$\gamma =\frac{\omega }{{\omega }_n},$$

(16)

which is called frequency ratio, H(ω) may be rewritten by

$$H(\omega )=\frac{1}{m{\omega }_n^2\left(1-{\gamma }^2+i2\varsigma \gamma \right)}.$$

(17)

The amplitude of H(ω) is called the amplitude frequency response. It is in the form

$$\left|H(\omega )\right|=\frac{1}{m{\omega }_n^2\sqrt{{\left(1-{\gamma }^2\right)}^2+{\left(2\varsigma \gamma \right)}^2}}.$$

(18)

Its phase is termed the phase frequency response given by

$$\phi (\omega )={\tan }^{-1}\frac{2\varsigma \gamma }{1-{\gamma }_{}^2}.$$

(19)

When the oscillator is excited by a sinusoidal function, the solution to (3) is termed the sinusoidal or simple harmonic response. Suppose the sinusoidal excitation function is Acosωt, where A is a constant. Then, the solution to

$$\{\begin{array}{l}\frac{d^{2}q(t)}{d{t}^2}+2\varsigma {\omega }_n\frac{dq(t)}{dt}+{\omega }_n^{2}q(t)=\frac{A\cos \omega t}{m}\\ q(0)=q_0,q^{\prime }(0)=v_0,\end{array}$$

(20)

is the sinusoidal response in the form

$$q(t)=\frac{\frac{A}{m{\omega }_d}}{{\left({\omega }_n^2-{\omega }^2\right)}^2+{\left(2\varsigma {\omega }_n\omega \right)}^2}\left\{\begin{array}{l}\left({\omega }_n^2-{\omega }^2\right)\cos \omega t+2\varsigma {\omega }_n\omega \sin \omega t\\ +e^{-\varsigma {\omega }_{n}t}\left[\left({\omega }_n^2-{\omega }^2\right)\cos {\omega }_{d}t-\frac{\varsigma }{\sqrt{1-{\varsigma }_{}^2}}\left({\omega }_n^2+{\omega }^2\right)\sin {\omega }_{d}t\right]\end{array}\right\}.$$

(21)

The responses mentioned above are essential to linear oscillators. We shall give our results for three classes of fractional oscillators with respect to those responses in this research.

2.1.3. Spectra of Three Excitations

The spectrum of δ(t) below means that δ(t) contains the equal frequency components for ω ∈ (0, ∞).

$${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\delta (t)e^{-i\omega t}dt}=1.$$

(22)

The spectrum of u(t) is in the form

$${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}u(t)e^{-i\omega t}dt}=\pi \delta (\omega )+\frac{1}{i\omega }.$$

(23)

The Fourier transform of $\cos {\omega }_{1}t$ is given by

$${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\cos {\omega }_{1}t{e}^{-i\omega t}dt}=\pi \left[\delta (\omega +{\omega }_1)+\delta (\omega -{\omega }_1)\right].$$

(24)

Three functions or signals above, namely, δ(t), u(t), and sinusoidal functions, are essential to the excitation forms in oscillations. However, their spectra do not exist in the domain of ordinary functions but they exist in the domain of generalized functions. Due to the importance of generalized functions in oscillations, for example, δ(t) and u(t), either theory or technology of oscillations nowadays is in the domain of generalized functions. In the domain of generalized functions, any function is differentiable of any times. The Fourier transform of any function exists (Gelfand and Vilenkin [67], Griffel [68]).

2.1.4. Generalization of Linear Oscillators

Let us be beyond the scope of the conventionally physical quantities, such as displacement, velocity, acceleration in mechanics, or current, voltage in electronics. Then, we consider the response of the quantity $q^{(n)}(t),$ where n is a positive integer. Precisely, we consider the following oscillation equation

$$\{\begin{array}{l}\frac{d^2}{d{t}^2}\left[\frac{d^{n}q(t)}{d{t}^n}\right]+2\varsigma {\omega }_n\frac{d}{dt}\left[\frac{d^{n}q(t)}{d{t}^n}\right]+{\omega }_n^2\frac{d^{n}q(t)}{d{t}^n}=\frac{e(t)}{m}\\ q^{(n)}(0)=q_0,q^{(n+1)}(0)=v_0.\end{array}$$

(25)

The above may be taken as a generalization of the conventional oscillator described by (3). Another expression of the above may be given by

$$\{\begin{array}{l}\frac{d^n}{d{t}^n}\left[\frac{d^{2}q(t)}{d{t}^2}\right]+2\varsigma {\omega }_n\frac{d^n}{d{t}^n}\left[\frac{dq(t)}{dt}\right]+{\omega }_n^2\frac{d^{n}q(t)}{d{t}^n}=\frac{e(t)}{m}\\ q^{(n)}(0)=q_0,q^{(n+1)}(0)=v_0.\end{array}$$

(26)

Alternatively, we have a linear oscillation system described by

$$\{\begin{array}{l}\frac{d^{n+2}q(t)}{d{t}^{n+2}}+2\varsigma {\omega }_n\frac{d^{n+1}q(t)}{d{t}^{n+1}}+{\omega }_n^2\frac{d^{n}q(t)}{d{t}^n}=\frac{e(t)}{m}\\ q^{(n)}(0)=q_0,q^{(n+1)}(0)=v_0.\end{array}$$

(27)

Physically, the above item with $q^{(n+2)}(t)$ corresponds to inertia, the one with $q^{(n)}(t)$ to restoration, and the one with $q^{(n+1)}(t)$ damping.

Note that (27) remains a linear oscillator after all. Nevertheless, when generalizing n to be fractions, for instance, considering −1 < ${\epsilon }_1$ ≤ 0 and −1 < ${\epsilon }_2$ ≤ 0, we may generalize (27) to be

$$\{\begin{array}{l}m\frac{d^{{\epsilon }_1+2}q(t)}{d{t}^{{\epsilon }_1+2}}+c\frac{d^{{\epsilon }_2+1}q(t)}{d{t}^{{\epsilon }_2+1}}+kq(t)=e(t)\\ q(0)=q_0,q^{\prime }(0)=v_0.\end{array}$$

(28)

Then, we go into the scope of fractional oscillations.

2.2. Three Classes of Fractional Oscillators

Denote by $\frac{d^{\nu }}{d{t}^{\nu }}={}_{-\infty }D_t^{\nu }$ the Weyl fractional derivative of order ν > 0. Then (Uchaikin [38], Miller and Ross [69], Klafter et al. [70]),

$${}_{-\infty }D_t^{\nu }f(t)=\frac{1}{\mathsf{\Gamma }(-\nu )}{\displaystyle \underset{-\infty }{\overset{t}{\int }}\frac{f(u)du}{{(t-u)}^{1+\nu }}},$$

(29)

where Γ(ν) is the Gamma function. The Weyl fractional derivative is used in this research because it is suitable for the Fourier transform in the domain of fractional calculus (Lavoie et al. ([71], p. 247)).

The Fourier transform of $\frac{d^{\nu }f(t)}{d{t}^{\nu }},$ following Uchaikin ([72], Section 4.5.3), is given by

$${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\frac{d^{\nu }f(t)}{d{t}^{\nu }}{e}^{-i\omega t}dt}={\left(i\omega \right)}^{\nu }F(\omega ),$$

(30)

where F(ω) is the Fourier transform of f(t).

This article relates to three classes of fractional oscillators as follow. We denote the following oscillation equation as a fractional oscillator in Class I.

$$\{\begin{array}{c}m\frac{d^{\alpha }{y}_1(t)}{d{t}^{\alpha }}+k{y}_1(t)=e(t)\\ y_1(0)=y_1{}_0,{y^{\prime }}_1(0)={y^{\prime }}_1{}_0\end{array},\hspace{1em}1<\alpha \le 2.$$

(31)

The free response to (31) is in the form (Mainardi [25], Achar et al. [33], Uchaikin ([38], Chapter 7))

$$y_1(t)=y_{10}{E}_{\alpha ,1}\left[-{\left({\omega }_{n}t\right)}^{\alpha }\right]+{y^{\prime }}_{10}t{E}_{\alpha ,2}\left[-{\left({\omega }_{n}t\right)}^{\alpha }\right],\hspace{1em}1<\alpha \le 2,t\ge 0,$$

(32)

where $E_{a,b}(z)$ is the generalized Mittag-Leffler function given by

$$E_{a,b}(z)={\displaystyle \sum _{k=0}^{\infty }\frac{z^k}{\mathsf{\Gamma }(ak+b)}},\hspace{1em}a,b\in C,\mathrm{Re}(a)>0,\mathrm{Re}(b)>0.$$

(33)

The Mittag-Leffler function denoted by E_a(t) is in the form

$$E_a(z)={\displaystyle \sum _{k=0}^{\infty }\frac{z^k}{\mathsf{\Gamma }(ak+1)}},\hspace{1em}a\in C,\mathrm{Re}(a)>0,$$

(34)

referring Mathai and Haubold [73], or Gorenflo et al. [74], or Erdelyi et al. [75] for the Mittag-Leffler functions.

Denote by $h_y{}_1(t)$ the impulse response to a fractional oscillator in Class I. Then (Uchaikin ([38], Chapter 7)),

$$h_y{}_1(t)=t^{\alpha -1}{E}_{\alpha ,\alpha }\left[-{\left({\omega }_{n}t\right)}^{\alpha }\right],\hspace{1em}1<\alpha \le 2,t\ge 0.$$

(35)

Let $g_{y1}(t)$ be the step response to a fractional oscillator of Class I type. Then,

$$g_{y1}(t)=t^{\alpha }{E}_{\alpha ,\alpha +1}\left[-{\left({\omega }_{n}t\right)}^{\alpha }\right],\hspace{1em}1<\alpha \le 2,t\ge 0.$$

(36)

For a fractional oscillator in Class I, its sinusoidal response driven by sinωt is expressed by

$$y_1(t)=A_1\sin (\omega t-{\theta }_1)+A_2{e}^{-\beta t}\cos \left[{\omega }_{n}t\sin \frac{\pi }{\alpha }-{\theta }_2\right]+{\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-st}{K}_{\alpha }(s)ds},$$

(37)

where

$$\begin{gathered} A_1=\frac{1}{\sqrt{{\omega }_n^{2\alpha }+{\omega }^{2\alpha }+2{\omega }_n^{\alpha }{\omega }^{\alpha }\cos \frac{\alpha \pi }{2}}}, \\ {A}_2=\frac{2\omega }{\alpha {\omega }_n^{\alpha -1}\sqrt{{\omega }_n^4+{\omega }^4+2{\omega }_n^2{\omega }^2\cos \frac{2\pi }{\alpha }}}, \end{gathered}$$

(38)

$$\beta =-{\omega }_n\cos \frac{\pi }{\alpha },$$

(39)

$$\begin{gathered} {\theta }_1={\tan }^{-1}\frac{{\omega }^{\alpha }\sin \frac{\alpha \pi }{2}}{{\omega }_n^{\alpha }+{\omega }^{\alpha }\cos \frac{\alpha \pi }{2}}, \\ {\theta }_2={\tan }^{-1}\left[\frac{{\omega }_n^2\sin \frac{(1+\alpha )\pi }{\alpha }-{\omega }^2\sin \frac{(1-\alpha )\pi }{\alpha }}{{\omega }_n^2\cos \frac{(1+\alpha )\pi }{\alpha }+{\omega }^2\cos \frac{(1-\alpha )\pi }{\alpha }}\right], \end{gathered}$$

(40)

$$K_{\alpha }(s)=\frac{\omega \sin (\pi \alpha )}{\pi \left(s^2+{\omega }^2\right)\left(s^{2\alpha }+2s^{\alpha }{\omega }_n^2\cos (\pi \alpha )+{\omega }_n^{2\alpha }\right)}.$$

(41)

An oscillator that follows the oscillation equation below is called a fractional oscillator in Class II.

$$m\frac{d^2{y}_2(t)}{d{t}^2}+c\frac{d^{\beta }{y}_2(t)}{d{t}^{\beta }}+k{y}_2(t)=0,\hspace{1em}0<\beta \le 1.$$

(42)

The equation below is called an oscillation equation of a fractional oscillator in Class III.

$$m\frac{d^{\alpha }{y}_3(t)}{d{t}^{\alpha }}+c\frac{d^{\beta }{y}_3(t)}{d{t}^{\beta }}+k{y}_3(t)=0,\hspace{1em}1<\alpha \le 2,0<\beta \le 1.$$

(43)

2.3. Equivalence of Functions in the Sense of Fourier Transform

Denote by $F_1(\omega )$ and $F_2(\omega )$ the Fourier transforms of $f_1(t)$ and $f_2(t),$ respectively. Then, if

$$F_1(\omega )=F_2(\omega ),$$

(44)

one says that

$$f_1(t)=f_2(t),$$

(45)

in the sense of Fourier transform (Gelfand and Vilenkin [67], Papoulis [76]), implying

$${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left[f_1(t)-f_2(t)\right]e^{-i\omega t}dt}=0.$$

(46)

The above implies that a null function as a difference between $f_1(t)$ and $f_2(t)$ is allowed for (45). An example relating to oscillation theory is the unit step function.

Denote by $u_1(t)$ in the form

$$u_1(t)=\{\begin{array}{c}1,t\ge 0\\ 0,t<0\end{array}.$$

(47)

Let $u_2(t)$ be

$$u_2(t)=\{\begin{array}{c}1,t>0\\ 0,t\le 0\end{array}.$$

(48)

Clearly, either $u_1(t)$ or $u_2(t)$ is a unit step function. The difference between two is a null function given by

$$u_1(t)-u_2(t)=\{\begin{array}{cc}1,\hfill & \hfill t=1\\ 0,\hfill & \hfill \mathrm{elsewhere}\end{array}.$$

(49)

Thus, $u_1(t)=u_2(t).$ In fact, the Fourier transform of either $u_1(t)$ or $u_2(t)$ equals to the right side on (23).

Similarly, if $f_1(t)=f_2(t),$ we say that (44) holds in the sense of

$${\displaystyle \underset{-\infty }{\overset{\infty }{\int }}\left[F_1(\omega )-F_2(\omega )\right]e^{i\omega t}d\omega }=0.$$

(50)

3. Problem Statement and Research Thoughts

We have mentioned three classes of fractional oscillators in Section 2. This section contains two parts. One is the problem statement and the other research thoughts.

3.1. Problem Statement

We first take fractional oscillators in Class I as a case to state the problems this research concerns with.

The analytical expressions with respect to the responses of free, impulse, step, to the oscillators of Class I are mathematically obtained (Mainardi [25], Achar et al. [33], Uchaikin ([38], Chapter 7)), also see Section 2.2 in this article. All noticed that a fractional oscillator of Class I is damping free in form but it is damped in nature due to fractional if 1 < α < 2. However, there are problems unsolved in this regard.

Problem 1.

How to analytically represent the damping of Class I oscillators?

In this article, we call the damping of fractional oscillators in Class I equivalent damping denoted by $c_{eq1}.$

It is known that damping relates to mass. Therefore, if we find $c_{eq1}$ in a fractional oscillator in Class I, its intrinsic mass must be different from the primary one m unless α = 2. We call it equivalent mass and denote it by $m_{eq1}.$

Problem 2.

How to analytically represent $m_{eq1}?$

Because a fractional oscillator in Class I is damped in nature for α ≠ 2, there must exist a damped natural frequency. We call it equivalent damped natural frequency, denoted by ${\omega }_{eqd,1}.$ Then, comes the problem below.

Problem 3.

What is the representation of ${\omega }_{eqd,1}?$

As there exists $m_{eq1}$ that differs from m if α ≠ 2, the equivalent damping free natural frequency, we denote it by ${\omega }_{eqn,1},$ is different from the primary damping free natural frequency ${\omega }_n=\sqrt{\frac{k}{m}}.$ Consequently, the following problem appears.

Problem 4.

What is the expression of ${\omega }_{eqn,1}?$

If we find the solutions to the above four, a consequent problem is as follows.

Problem 5.

How to represent response (free, or impulse, or step, or sinusoidal) with $m_{eq1},$ $c_{eq1},$ ${\omega }_{eqn,1},$ and ${\omega }_{eqd,1}$ to a fractional oscillator in Class I?

If we solve the above problems, the solution to the following problem is ready.

Problem 6.

What is the physical mechanism of a fractional oscillator in Class I?

Note that the intrinsic damping for a Class II fractional oscillator must differ from its primary damping c owing to the fractional friction $c\frac{d^{\beta }{y}_2(t)}{d{t}^{\beta }}$ for β ≠ 1. We call it the equivalent damping denoted by $c_{eq2}.$ Because $c_{eq2}$≠ c if β ≠ 1, the equivalent mass of a fractional oscillator in Class II, denoted by $m_{eq2},$ is not equal to the primary m for β ≠ 1. Thus, the six stated above are also unsolved problems for fractional oscillators in Class II. They are, consequently, the problems unsolved for Class III fractional oscillators.

Note that there are other problems regarding with three classes of fractional oscillators. For example, the explicit expression of the sinusoidal response (37) in closed form needs investigation because of the difficulty in finding the solution to ${\displaystyle \underset{0}{\overset{\infty }{\int }}{e}^{-st}{K}_{\alpha }(s)ds}.$ We shall deal with them in separate sections. The solutions to the problems described above constitute main highlights of this research.

We note that the damping nature of a fractional oscillator in Class I was also observed by other researchers, not explicitly stated though, as can be seen from, e.g., Zurigat ([26], Figure 1), Blaszczyk et al. ([28], Figure 2), Al-rabtah et al. ([29], Figure 2), Ryabov and Puzenko ([36], Equation (5)), Uchaikin ([38], Chapter 7), Duan et al. ([39], Equation (4.3), Figure 2), Gomez-Aguilar et al. ([53], Equation (15), Figures 2 and 3), Chung and Jung ([77], Figure 1). One thing remarkable is by Tofighi, who explored the intrinsic damping of an oscillator in Class I, see ([35], pp. 32–33). That was an advance regarding with the damping implied in (31) but it may be unsatisfactory if one desires its closed form of analytic expression.

3.2. Research Thoughts

Let us qualitatively consider possible performances of equivalent mass and damping. In engineering, people may purposely connect an auxiliary mass $m_a$ to the primary mass m so that the equivalent mass of the total system is related to the oscillation frequency ω (Harris ([4], p. 6.4)). In the field of ship hull vibrations, added mass has to be taken into account in the equivalent mass (i.e., total mass) of a ship hull (Korotkin [78]) so that the equivalent mass is ω-varying. In fact, the three dimensional fluid coefficient with respect to the added mass to a ship hull relates to the oscillation frequency, see, e.g., Jin and Xia ([79], pp. 135–136), Nakagawa et al. [80].

In addition, damping may be also ω-varying. A well-known case of ω-varying damping is the Coulomb damping (Timoshenko ([2], Chapter 1), Harris ([4], Equation (30.4))). Frequency varying damping is a technique used in damping treatments, see, e.g., Harris ([4], Equation (37.8)). Besides, commonly used damping assumptions in ship hull vibrations, such as the Copoknh’s, the Voigt’s, the Rayleigh’s, are all ω-varying (Jin and Xia ([79], pp. 157–158)). Therefore, with the concept of ω-varying mass and damping, I purposely generalize the simple oscillation model expressed by (1) in the form

$$\{\begin{array}{l}{m}_{eq}(\omega )\frac{d^{2}q(t)}{d{t}^2}+c_{eq}(\omega )\frac{dq(t)}{dt}+kq(t)=e(t)\\ q(0)=q_0,q^{\prime }(0)=v_0.\end{array}$$

(51)

The above second-order equation may not be equivalent to a fractional oscillator unless $m_{eq}$ and or $c_{eq}$ are appropriately expressed and properly related to the fractional order α for Class I oscillators, or β for Class II oscillators, or (α, β) for oscillators in Class III. For those reasons, we further generalize (51) by

$$\{\begin{array}{l}{m}_{eq1}(\omega ,\alpha )\frac{d^2{x}_1(t)}{d{t}^2}+c_{eq1}(\omega ,\alpha )\frac{d{x}_1(t)}{dt}+k{x}_1(t)=e(t)\\ x_1(0)=x_{10},{\dot{x}}_1(0)=v_{10},\end{array}$$

(52)

for Class I oscillators. As for Class II oscillators, (51) should be generalized by

$$\{\begin{array}{l}{m}_{eq2}(\omega ,\beta )\frac{d^2{x}_2(t)}{d{t}^2}+c_{eq2}(\omega ,\beta )\frac{d{x}_2(t)}{dt}+k{x}_2(t)=e(t)\\ x_2(0)=x_{10},{\dot{x}}_2(0)=v_{20}.\end{array}$$

(53)

Similarly, for Class III oscillators, we generalize (51) to be the form

$$\{\begin{array}{l}{m}_{eq3}(\omega ,\alpha ,\beta )\frac{d^2{x}_3(t)}{d{t}^2}+c_{eq3}(\omega ,\alpha ,\beta )\frac{d{x}_2(t)}{dt}+k{x}_3(t)=e(t)\\ x_3(0)=x_{30},{\dot{x}}_3(0)=v_{30}.\end{array}$$

(54)

Three generalized oscillation Equations (52)–(54), can be unified in the form

$$\{\begin{array}{l}{m}_{eqj}\frac{d^2{x}_j(t)}{d{t}^2}+c_{eqj}\frac{d{x}_j(t)}{dt}+k{x}_j(t)=e(t)\\ x_j(0)=x_{j0},{\dot{x}}_j(0)=v_{30}\end{array},\hspace{1em}j=1,2,3.$$

(55)

By introducing the symbols ${\omega }_{eqn,j}=\sqrt{\frac{k}{m_{eqj}}}$ and ${\varsigma }_{eqj}=\frac{c_{eqj}}{2\sqrt{m_{eqj}k}}$ for j = 1, 2, 3, we rewrite the above by

$$\{\begin{array}{l}\frac{d^2{x}_j(t)}{d{t}^2}+2{\varsigma }_{eqj}{\omega }_{eqn,j}\frac{d{x}_j(t)}{dt}+{\omega }_{eqn,j}^2{x}_j(t)=\frac{e(t)}{m_{eqj}}\\ x_j(0)=x_{j0},{\dot{x}}_j(0)=v_{30}\end{array}, j=1,2,3.$$

(56)

Let $Y_j(\omega )$ be the Fourier transform of $y_j(t),$ where $y_j(t)$(j = 1, 2, 3) respectively corresponds to the one in (31), (42), and (43). Denote by $X_j(\omega )$ the Fourier transform of $x_j(t).$ Then, if we find proper $m_{eqj}$ and $c_{eqj}$ such that

$$Y_j(\omega )=X_j(\omega ), j = 1, 2, 3,$$

(57)

the second-order equation (52), or (53), or (54) is equal to the fractional oscillation Equation (31), or (42), or (43), respectively.

Obviously, once we discover the equivalent equations of the fractional oscillation Equations (52), or (53), and (54), all problems stated previously can be readily solved.

4. Equivalent Systems of Three Classes of Fractional Oscillators

In this section, we first present an equivalent system and then its equivalent mass and damping in Section 4.1, Section 4.2 and Section 4.3, respectively for each class 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 gives the equivalent oscillator with the integer order for the fractional oscillators in Class I.

Theorem 1 (Equivalent oscillator I).

Denote a fractional oscillator in Class I by

$$m\frac{d^{\alpha }{y}_1(t)}{d{t}^{\alpha }}+k{y}_1(t)=0,\hspace{1em}1<\alpha \le 2.$$

(58)

Then, its equivalent oscillator with the equation of order 2 is in the form

$$-m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}\frac{d^2{x}_1(t)}{d{t}^2}+m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}\frac{d{x}_1(t)}{dt}+k{x}_1(t)=0,\hspace{1em}1<\alpha \le 2.$$

(59)

Proof. 

Consider the frequency response of (58) with the excitation of the Dirac-delta function δ(t). In doing so, we study

$$m\frac{d^{\alpha }{h}_{y1}(t)}{d{t}^{\alpha }}+k{h}_{y1}(t)=\delta (t),\hspace{1em}1<\alpha \le 2.$$

(60)

Doing the Fourier transform on the both sides of (60) produces

$$\left[m{\left(i\omega \right)}^{\alpha }+k\right]H_{y1}(\omega )=1,\hspace{1em}1<\alpha \le 2,$$

(61)

where $H_{y1}(\omega )$ is the Fourier transform of $h_{y1}(t).$ Using the principal value of i, we have

$$i^{\alpha }=\cos \frac{\alpha \pi }{2}+i\sin \frac{\alpha \pi }{2}.$$

(62)

Thus, (61) implies

$$\begin{array}{l}\left[m{\left(i\omega \right)}^{\alpha }+k\right]H_{y1}(\omega )=\left\{m\left(\cos \frac{\alpha \pi }{2}+i\sin \frac{\alpha \pi }{2}\right){\omega }^{\alpha }+k\right\}{H}_{y1}(\omega )\\ =\left(m{\omega }^{\alpha }\cos \frac{\alpha \pi }{2}+im{\omega }^{\alpha }\sin \frac{\alpha \pi }{2}+k\right)H_{y1}(\omega )=1.\end{array}$$

(63)

Therefore, we have the frequency response of (60) in the form

$$H_{y1}(\omega )=\frac{1}{m{\omega }^{\alpha }\cos \frac{\alpha \pi }{2}+im{\omega }^{\alpha }\sin \frac{\alpha \pi }{2}+k}.$$

(64)

On the other hand, for 1 < α ≤ 2, we consider (59) by

$$-m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}\frac{d^2{h}_{x1}(t)}{d{t}^2}+m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}\frac{d{h}_{x1}(t)}{dt}+k{h}_{x1}(t)=\delta (t).$$

(65)

Performing the Fourier transform on the both sides of (65) yields

$$\begin{array}{l}\left[-m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}\left(-{\omega }^2\right)+m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}\left(i\omega \right)+k\right]H_{x1}(\omega )\\ =\left(m{\omega }^{\alpha }\cos \frac{\alpha \pi }{2}+im{\omega }^{\alpha }\sin \frac{\alpha \pi }{2}+k\right)H_{x1}(\omega )=1,\end{array}$$

(66)

where $H_{x1}(\omega )$ is the Fourier transform of $h_{x1}(t).$ Therefore, we have

$$H_{x1}(\omega )=\frac{1}{m{\omega }^{\alpha }\cos \frac{\alpha \pi }{2}+im{\omega }^{\alpha }\sin \frac{\alpha \pi }{2}+k}.$$

(67)

By comparing (64) with (67), we see that

$$H_{y1}(\omega )=H_{x1}(\omega ).$$

(68)

Thus, (59) is the equivalent equation of (58). The proof completes. □

4.1.2. Equivalent Mass of Fractional Oscillators in Class I

From the first item on the left side of (59), we obtain the equivalent mass for the fractional oscillators of Class I type.

Theorem 2 (Equivalent mass I).

The equivalent mass of the fractional generators in Class I, denoted by $m_{eq1},$ is expressed by

$$m_{eq1}=m_{eq1}(\omega ,\alpha )=-\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}\right)m,\hspace{1em}1<\alpha \le 2.$$

(69)

Proof. 

According to the Newton’s second law, the inertia force in the system of the fractional oscillator (58) corresponds to the first item on the left side of its equivalent system (59). That is, $-m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}\frac{d^2{x}_1(t)}{d{t}^2}.$ Thus, the coefficient of $\frac{d^2{x}_1(t)}{d{t}^2}$ is an equivalent mass expressed by (69). Hence, the proof finishes. □

From Theorem 2, we reveal a power law phenomenon with respect to $m_{eq1}$ in terms of ω.

Remark 1.

The equivalent mass I, $m_{eq1},$ follows the power law in terms of oscillation frequency ω in the form

$$m_{eq1}(\omega ,\alpha )~{\omega }^{\alpha -2}m,\hspace{1em}1<\alpha \le 2.$$

(70)

The equivalent mass $m_{eq1}$ relates to the oscillation frequency ω, the fractional order α, and the primary mass m. Denote by

$$R_{m1}(\omega ,\alpha )=-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2},\hspace{1em}1<\alpha \le 2.$$

(71)

Then, we have

$$m_{eq1}=m_{eq1}(\omega ,\alpha )=R_{m1}(\omega ,\alpha )m,\hspace{1em}1<\alpha \le 2.$$

(72)

Note 4.1: Since

$$R_{m1}(\omega ,2)=1,$$

(73)

$m_{eq1}(\omega ,\alpha )$ reduces to the primary mass m when α = 2. That is,

$$m_{eq1}(\omega ,2)=m.$$

(74)

In the case of α = 2, therefore, both (58) and (59) reach the conventional harmonic oscillation with damping free in the form

$$m\frac{d^2{x}_1(t)}{d{t}^2}+k{x}_1(t)=0.$$

Note 4.2: If α → 1, we have

$$\underset{\alpha \to 1}{\lim }{m}_{eq1}(\omega ,\alpha )=0 \mathrm{for} \omega \ne 0.$$

(75)

The above implies that $m_{eq1}$ vanishes if α → 1. Consequently, any oscillation disappears in that case.

Note 4.3: When 1 < α ≤ 2, we attain

$$0<R_{m1}(\omega ,\alpha )\le 1 \mathrm{for} \omega > 1.$$

(76)

Thus, we reveal an interesting phenomenon expressed by

$$m_{eq1}(\omega ,\alpha )\le m \mathrm{for} 1 < \alpha \le 2, \omega > 1.$$

(77)

The coefficient $R_{m1}(\omega ,\alpha )$ is plotted in Figure 1.

Remark 2.

For α ∈ (0, 2), we have

$$\underset{\omega \to \infty }{\lim }{m}_{eq1}(\omega ,\alpha )=0.$$

(78)

The interesting and novel behavior, described above, implies that a fractional oscillator in Class I does not oscillate for ω → ∞ because it is equivalently massless in that case.

Remark 3.

For α ∈ (0, 2), we have

$$\underset{\omega \to 0}{\lim }{m}_{eq1}(\omega ,\alpha )=\infty .$$

(79)

The interesting behavior, revealed above, says that a fractional oscillator of Class I type does not oscillate at ω = 0 because its mass is equivalently infinity in addition to the explanation of static status conventionally described by ω = 0.

4.1.3. Equivalent Damping of Fractional Oscillators of Class I

We now propose the equivalent damping.

Theorem 3 (Equivalent damping I).

The equivalent damping of a fractional oscillator in Class I, denoted by $c_{eq1},$ is expressed by

$$c_{eq1}=c_{eq1}(\omega ,\alpha )=\left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}\right)m,\hspace{1em}1<\alpha \le 2.$$

(80)

Proof. 

The second term on the left side of (59) is the friction with the linear viscous damping coefficient denoted by (80). The proof completes. □

Denote

$$R_{c1}(\omega ,\alpha )={\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2},\hspace{1em}1<\alpha \le 2.$$

(81)

Then, we have

$$c_{eq1}(\omega ,\alpha )=R_{c1}(\omega ,\alpha )m.$$

(82)

The coefficient $R_{c1}(\omega ,\alpha )$ is indicated in Figure 2.

Remark 4.

The equivalent damping I relies on ω, m, and α. It obeys the power law in terms of ω in the form

$$c_{eq1}(\omega ,\alpha )~{\omega }^{\alpha -1}m,\hspace{1em}1<\alpha \le 2.$$

(83)

Note 4.4: Because

$${c_{eq1}(\omega ,\alpha )|}_{\alpha =2}=0,$$

(84)

we see again that a fractional oscillator of Class I type reduces to the conventional harmonic one when α = 2.

Remark 5.

An interesting behavior of $c_{eq1},$ we found, is expressed by

$$\underset{\omega \to \infty }{\lim }{c}_{eq1}(\omega ,\alpha )=\infty ,\hspace{1em}1<\alpha <2.$$

(85)

The above says that the equivalent oscillator (59), as well as the fractional oscillator (58), never oscillates at ω → ∞ for 1 < α < 2 because its damping is infinitely large in that case. Due to

$$\underset{\omega \to 0}{\lim }{c}_{eq1}(\omega ,\alpha )=0,\hspace{1em}1<\alpha <2,$$

(86)

we reveal a new damping behavior of a fractional oscillator in Class I in that it is equivalently dampingless for 1 < α < 2 at ω = 0.

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 below describes the equivalent oscillator for the fractional oscillators of Class II type.

Theorem 4 (Equivalent oscillator II).

Denote a fractional oscillator in Class II by

$$m\frac{d^2{y}_2(t)}{d{t}^2}+c\frac{d^{\beta }{y}_2(t)}{d{t}^{\beta }}+k{y}_2(t)=0,\hspace{1em}0<\beta \le 1.$$

(87)

Then, its equivalent 2-order oscillation equation is given by

$$\left(m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\frac{d^2{x}_2(t)}{d{t}^2}+\left(c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)\frac{d{x}_2(t)}{dt}+k{x}_2(t)=0,\hspace{1em}0<\beta \le 1.$$

(88)

Proof. 

Consider the following equation:

$$m\frac{d^2{h}_y{}_2(t)}{d{t}^2}+c\frac{d^{\beta }{h}_y{}_2(t)}{d{t}^{\beta }}+k{h}_y{}_2(t)=\delta (t),\hspace{1em}0<\beta \le 1.$$

(89)

Denote by $H_{y2}(\omega )$ the Fourier transform of $h_y{}_2(t).$ Then, it is its frequency transfer function. Taking the Fourier transform on the both sides of (89) yields

$$\left[-m{\omega }^2+c{\left(i\omega \right)}^{\beta }+k\right]H_{y2}(\omega )=1,\hspace{1em}0<\beta \le 1.$$

(90)

With the principal value of $i^{\beta },$ (90) becomes

$$\begin{array}{l}\left[-m{\omega }^2+c{\left(i\omega \right)}^{\beta }+k\right]H_{y2}(\omega )=\left\{-m{\omega }^2+c\left(\cos \frac{\beta \pi }{2}+i\sin \frac{\beta \pi }{2}\right){\omega }^{\beta }+k\right\}{H}_{y2}(\omega )\\ =\left(-m{\omega }^2+c{\omega }^{\beta }\cos \frac{\beta \pi }{2}+k+ic{\omega }^{\beta }\sin \frac{\beta \pi }{2}\right)H_{y2}(\omega )=1.\end{array}$$

(91)

The above means

$$H_{y2}(\omega )=\frac{1}{-m{\omega }^2+c{\omega }^{\beta }\cos \frac{\beta \pi }{2}+k+ic{\omega }^{\beta }\sin \frac{\beta \pi }{2}}.$$

(92)

On the other hand, we consider the equivalent oscillation equation II with the Dirac-δ excitation by

$$\left(m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\frac{d^2{h}_{x2}(t)}{d{t}^2}+\left(c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)\frac{d{h}_{x2}(t)}{dt}+k\frac{d^2{h}_{x2}(t)}{d{t}^2}=\delta (t),\hspace{1em}0<\beta \le 1.$$

(93)

Performing the Fourier transform on the both sides of the above produces

$$\begin{array}{l}\left[-m{\omega }^2+c{\omega }^{\beta }\cos \frac{\beta \pi }{2}+ic{\omega }^{\beta }\sin \frac{\beta \pi }{2}\left(i\omega \right)+k\right]H_{x2}(\omega )\\ =\left(-m{\omega }^2+c{\omega }^{\beta }\cos \frac{\beta \pi }{2}+k+ic{\omega }^{\beta }\sin \frac{\beta \pi }{2}\right)H_{x2}(\omega )=1,\end{array}$$

(94)

where $H_{x2}(\omega )$ the Fourier transform of $h_x{}_2(t).$ Thus, from the above, we have

$$H_{x2}(\omega )=\frac{1}{-m{\omega }^2+c{\omega }^{\beta }\cos \frac{\beta \pi }{2}+k+ic{\omega }^{\beta }\sin \frac{\beta \pi }{2}}.$$

(95)

Equations (92) and (95) imply

$$H_{y2}(\omega )=H_{x2}(\omega ).$$

(96)

Hence, (88) is the equivalent oscillation equation of the fractional oscillators of Class II. This completes the proof. ☐

4.2.2. Equivalent Mass of Fractional Oscillators of Class II

The equivalent mass of the fractional oscillators of Class II type is presented in Theorem 5.

Theorem 5 (Equivalent mass II).

Let $m_{eq2}$ be the equivalent mass of the fractional oscillators of Class II type. Then,

$$m_{eq2}=m_{eq2}(\omega ,\beta )=m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2},\hspace{1em}0<\beta \le 1.$$

(97)

Proof. 

Consider the Newton’s second law. Then, we see that the inertia force in the equivalent oscillator II is $\left(m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\frac{d^2{x}_2}{d{t}^2}.$ Therefore, (97) holds. The proof completes. □

From Theorem 5, we reveal a power law phenomenon with respect to the equivalent mass II.

Remark 6.

The equivalent mass $m_{eq2}$ obeys the power law in terms of ω in the form

$$m_{eq2}~-c{\omega }^{\beta -2},\hspace{1em}0<\beta \le 1.$$

(98)

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.

For 0 < β ≤ 1, we have

$$\underset{\omega \to \infty }{\lim }{m}_{eq2}(\omega ,\beta )=m.$$

(99)

Figure 3 shows its plots for m = c = 1 with the part of $m_{eq2}(\omega ,\beta )$> 0.

Remark 8.

For 0 < β < 1, we have

$$\underset{\omega \to 0}{\lim }{m}_{eq2}(\omega ,\beta )=-\infty .$$

(100)

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

Figure 4 exhibits the negative part of $m_{eq2}(\omega ,\beta ).$

Remark 9.

We restrict our research for $m_{eq2}(\omega ,\beta )$ > 0.

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

$${m_{eq2}(\omega ,\beta )|}_{\beta =1}=m.$$

(101)

In fact, a fractional oscillator in Class II reduces to the conventional oscillator below if β = 1

$$m\frac{d^2{x}_2}{d{t}^2}+c\frac{d{x}_2}{ct}+k{x}_2=0.$$

4.2.3. Equivalent Damping of Fractional Oscillators in Class II

Let $c_{eq2}$ be the equivalent damping of a fractional oscillator in Class II. Then, we put forward the expression of $c_{eq2}$ with Theorem 6.

Theorem 6 (Equivalent damping II).

The equivalent damping of the fractional oscillators in Class II is in the form

$$c_{eq2}=c_{eq2}(\omega ,\beta )=c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2},\hspace{1em}0<\beta \le 1.$$

(102)

Proof. 

The second term on the left side of (88) is the friction force with the linear viscous damping coefficient denoted by (102). The proof completes. □

Denote by

$$R_{c2}(\omega ,\beta )={\omega }^{\beta -1}\sin \frac{\beta \pi }{2},\hspace{1em}0<\beta \le 1.$$

(103)

Then, we have

$$c_{eq2}(\omega ,\beta )=R_{c2}(\omega ,\beta )c.$$

(104)

Figure 5 indicates $R_{c2}(\omega ,\beta ).$

Remark 10.

The equivalent damping $c_{eq2}$ is associated with the oscillation frequency ω, the primary damping c, and the fractional order β. It follows the power law in terms of ω in the form

$$c_{eq2}(\omega ,\beta )~{\omega }^{\beta -1}c,\hspace{1em}0<\beta \le 1.$$

(105)

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

$${c_{eq2}(\omega ,\beta )|}_{\beta =1}=c.$$

(106)

Remark 11.

The equivalent damping $c_{eq2}$ has, for β ∈ (0, 1), the property given by

$$\underset{\omega \to \infty }{\lim }{c}_{eq2}(\omega ,\beta )=0.$$

(107)

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 $\underset{\omega \to \infty }{\lim }{c}_{eq1}(\omega ,\beta )=\infty$ 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.

The equivalent damping $c_{eq2}$ has, for β∈ (0, 1), the asymptotic property for ω → 0 in the form

$$\underset{\omega \to 0}{\lim }{c}_{eq2}(\omega ,\beta )=\infty .$$

(108)

The above property implies that a fractional oscillator in Class II does not oscillate at ω → 0 because not only it is in static status but also its equivalent damping is infinitely large.

4.3. Equivalent Oscillation System for Fractional Oscillators of Class III

4.3.1. Equivalent Oscillation Equation of Fractional Oscillators in Class III

We present Theorem 7 below to explain the equivalent oscillation equation for the fractional oscillators of Class III.

Theorem 7 (Equivalent oscillator III).

Denote a fractional oscillation equation in Class III by

$$m\frac{d^{\alpha }{y}_3(t)}{d{t}^{\alpha }}+c\frac{d^{\beta }{y}_3(t)}{d{t}^{\beta }}+k{y}_3(t)=0,\hspace{1em}1<\alpha \le 2,0<\beta \le 1.$$

(109)

Then, its equivalent oscillator of order 2 for 1 < α ≤ 2 and 0 < β ≤ 1 is in the form

$$\begin{array}{l}-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\frac{d^2{x}_3(t)}{d{t}^2}\\ +\left(m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)\frac{d{x}_3(t)}{dt}+k{x}_3(t)=0.\end{array}$$

(110)

Proof. 

Let us consider the equation

$$m\frac{d^{\alpha }{h}_y{}_3(t)}{d{t}^{\alpha }}+c\frac{d^{\beta }{h}_y{}_3(t)}{d{t}^{\beta }}+k{h}_y{}_3(t)=\delta (t),\hspace{1em}1<\alpha \le 2,0<\beta \le 1.$$

(111)

Let $H_y{}_3(\omega )$ be the Fourier transform of $h_y{}_3(t).$ Doing the Fourier transform on the both sides of the above results in

$$\left[m{\left(i\omega \right)}^{\alpha }+c{\left(i\omega \right)}^{\beta }+k\right]H_y{}_3(\omega )=1,\hspace{1em}1<\alpha \le 2,0<\beta \le 1.$$

(112)

Taking into account the principal values of $i^{\alpha }$ and $i^{\beta },$ (112) becomes

$$\begin{array}{l}\left[m{\left(i\omega \right)}^{\alpha }+c{\left(i\omega \right)}^{\beta }+k\right]H_y{}_3(\omega )\\ =\left[m\left(\cos \frac{\alpha \pi }{2}+i\sin \frac{\alpha \pi }{2}\right){\omega }^{\alpha }+c\left(\cos \frac{\beta \pi }{2}+i\sin \frac{\beta \pi }{2}\right){\omega }^{\beta }+k\right]H_y{}_3(\omega )\\ =\left[m{\omega }^{\alpha }\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta }\cos \frac{\beta \pi }{2}+k+i\left(m{\omega }^{\alpha }\sin \frac{\alpha \pi }{2}+c{\omega }^{\beta }\sin \frac{\beta \pi }{2}\right)\right]H_y{}_3(\omega )=1.\end{array}$$

(113)

Consequently, we have

$$H_y{}_3(\omega )=\frac{1}{m{\omega }^{\alpha }\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta }\cos \frac{\beta \pi }{2}+k+i\left(m{\omega }^{\alpha }\sin \frac{\alpha \pi }{2}+c{\omega }^{\beta }\sin \frac{\beta \pi }{2}\right)}.$$

(114)

On the other hand, considering the equivalent oscillator III driven by the Dirac-δ function, we have

$$\begin{array}{l}-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)\frac{d^2{h}_x{}_3(t)}{d{t}^2}+\left(m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)\frac{d{h}_x{}_3(t)}{dt}+k{h}_x{}_3(t)\\ =\delta (t).\end{array}$$

(115)

When doing the Fourier transform on the both sides of the above, we obtain

$$\begin{array}{l}\left(m{\omega }^{\alpha }\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta }\cos \frac{\beta \pi }{2}\right)H_x{}_3(\omega )+i\left(m{\omega }^{\alpha }\sin \frac{\alpha \pi }{2}+c{\omega }^{\beta }\sin \frac{\beta \pi }{2}\right)H_x{}_3(\omega )+k{H}_x{}_3(\omega )\\ =1,\end{array}$$

(116)

where $H_x{}_3(\omega )$ is the Fourier transform of $h_x{}_3(t).$ Therefore, from the above, we get

$$H_x{}_3(\omega )=\frac{1}{m{\omega }^{\alpha }\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta }\cos \frac{\beta \pi }{2}+k+i\left(m{\omega }^{\alpha }\sin \frac{\alpha \pi }{2}+c{\omega }^{\beta }\sin \frac{\beta \pi }{2}\right)}.$$

(117)

Two expressions, (114) and (117), imply that

$$H_x{}_3(\omega )=H_y{}_3(\omega ).$$

(118)

Thus, Theorem 7 holds. □

4.3.2. Equivalent Mass of Fractional Oscillators in Class III

From Section 4.3.1, we propose the equivalent mass of the fractional oscillators in Class III type by Theorem 8.

Theorem 8 (Equivalent mass III).

Let $m_{eq3}$ be the equivalent mass of the fractional oscillators in Class III. Then, for 1 < α ≤ 2 and 0 < β ≤ 1,

$$m_{eq3}=m_{eq3}(\omega ,\alpha ,\beta )=-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right).$$

(119)

Proof. 

When considering the Newton’s second law in the equivalent oscillator III (110), we immediately see that Theorem 8 holds. □

Remark 13.

The equivalent mass $m_{eq3}$ obeys the power law in terms of ω.

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.,

$${m_{eq3}(\omega ,\alpha ,\beta )|}_{\alpha =2,\beta =1}=m.$$

(120)

As a matter of fact, a fractional oscillator of Class III reduces to the ordinary oscillator when α = 2 and β = 1.

Remark 14.

In the case of ω → ∞, we obtain

$$\underset{\omega \to \infty }{\lim }{m}_{eq3}(\omega ,\alpha ,\beta )=0,\hspace{1em}1<\alpha <2,0<\beta <1.$$

(121)

Therefore, we suggest that a fractional oscillator in Class III does not oscillate for ω → ∞ because its equivalent mass disappears in that case. Figure 6 shows its positive part for α = 1.5, β = 0.9, m = c = 1.

Remark 15.

In the case of ω → 0, we obtain

$$\underset{\omega \to 0}{\lim }{m}_{eq3}(\omega ,\alpha ,\beta )=-\infty ,\hspace{1em}1<\alpha <2,0<\beta <1.$$

(122)

In fact, if ω is small enough, $m_{eq3}(\omega ,\alpha ,\beta )$ will be negative, see Figure 7.

Remark 16.

This research restricts $m_{eq3}(\omega ,\alpha ,\beta )$ ∈ (0, ∞).

4.3.3. Equivalent Damping of Fractional Oscillators in Class III

Let $c_{eq3}$ be the equivalent damping of a fractional oscillator of Class III type. Then, we propose its expression with Theorem 9.

Theorem 9 (Equivalent damping III).

The equivalent damping of the fractional oscillators in Class III is given by, for 1 < α ≤ 2 and 0 < β ≤ 1,

$$c_{eq3}=c_{eq3}(\omega ,\alpha ,\beta )=m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}.$$

(123)

Proof. 

The second term on the left side of the equivalent oscillator III is the friction force with the linear viscous damping coefficient denoted by (123). Thus, the proof completes. □

Remark 17.

The equivalent damping $c_{eq3}$ relates to ω, m, c, and a pair of fractional orders (α, β). It obeys the power law in terms of ω. It contains two terms. The first term is hyperbolically increasing in ${\omega }^{\alpha -1}$ as α > 1 and the second hyperbolically decayed with ${\omega }^{\beta -1}$ since β < 1.

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

$${c_{eq3}(\omega ,\alpha ,\beta )|}_{\alpha =2,\beta =1}=c.$$

(124)

Remark 18.

One asymptotic property of $c_{eq3}$ for ω → ∞, due to $\underset{\omega \to \infty }{\lim }{\omega }^{\alpha -1}=\infty$ for 1 < α ≤ 2, is given by

$$\underset{\omega \to \infty }{\lim }{c}_{eq3}(\omega ,\alpha ,\beta )=\infty .$$

(125)

The above says that a fractional oscillator of Class III does not vibrate for ω → ∞.

Remark 19.

Another asymptotic property of $c_{eq3}$ in terms of ω for ω → 0, owing to $\underset{\omega \to 0}{\lim }{\omega }^{\beta -1}=\infty$ for 0 < β < 1, is expressed by

$$\underset{\omega \to 0}{\lim }{c}_{eq3}(\omega ,\alpha ,\beta )=\infty .$$

(126)

A system does not vibrate obviously in the case of ω → 0 but Remark 19 suggests a new view about that. Precisely, its equivalent damping is infinitely large at ω → 0. Figure 8 and Figure 9 illustrate $c_{eq3}(\omega ,\alpha ,\beta )$ for m = c = 1.

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

$${c_{eq3}(\omega ,\alpha ,\beta )|}_{\alpha =2,c=0}=0.$$

(127)

4.4. Summary

We have proposed three equivalent oscillation equations with order 2 to equivalently characterize three classes of fractional oscillators, opening a novel way of studying fractional oscillators. The analytic expressions of equivalent mass $m_{eqj}$ and damping $c_{eqj}$ (j = 1, 2, 3) for each equivalent oscillator have been presented. One general thing regarding $m_{eqj}$ and damping $c_{eqj}$ is that they follow power laws. Another thing in common is that they are dependent on oscillation frequency ω and fractional order.

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

We have presented three equivalent oscillation equations corresponding to three classes of fractional oscillators in the last section. Functionally, they are abstracted in a unified form

$$m_{eqj}\frac{d^2{x}_j(t)}{d{t}^2}+c_{eqj}\frac{d{x}_j(t)}{dt}+k{x}_j(t)=f(t),\hspace{1em}j=1,2,3.$$

(128)

In each equivalent oscillator, either $m_{eqj}$ or $c_{eqj}$ is not a constant in general. Instead, either is a function of the oscillation frequency ω and the fractional order α for $m_{eq1}$ and $c_{eq1},$ β for $m_{eq2}$ and $c_{eq2},$ (α, β) for $m_{eq3}$ and $c_{eq3}.$ Consequently, natural frequencies and damping ratios of fractional oscillators should rely on ω and fractional order. We shall propose their analytic expressions in this section.

5.1. Equivalent Natural Frequency I

Definition 1.

Denote by ${\omega }_{eqn,j}$ a natural frequency of a fractional oscillator in the jth class (j = 1, 2, 3). It takes the form

$${\omega }_{eqn,j}=\sqrt{\frac{k}{m_{eqj}}},\hspace{1em}j=1,2,3,$$

(129)

where $m_{eqj}$ is the equivalent mass of the fractional oscillator in the jth class.

With the above definition, we write (128) by

$$\frac{d^2{x}_j(t)}{d{t}^2}+\frac{c_{eqj}}{m_{eqj}}\frac{d{x}_j(t)}{dt}+\frac{k}{m_{eqj}}{x}_j(t)=\frac{d^2{x}_j(t)}{d{t}^2}+\frac{c_{eqj}}{m_{eqj}}\frac{d{x}_j(t)}{dt}+{\omega }_{eqn,j}^2{x}_j(t)=\frac{f(t)}{m_{eqj}},\hspace{1em}j=1,2,3.$$

(130)

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

$${\omega }_n=\sqrt{\frac{k}{m}},$$

(131)

as a special case.

Corollary 1 (Equivalent natural frequency I1).

The equivalent natural frequency I1, which we denote it by ${\omega }_{eqn,1},$ of a fractional oscillator in Class I is given by

$${\omega }_{eqn,1}=\frac{{\omega }_n}{\sqrt{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}},\hspace{1em}1<\alpha \le 2.$$

(132)

Proof. 

According to (129), we have, for 1 < α ≤ 2,

$${\omega }_{eqn,1}=\sqrt{\frac{k}{m_{eq1}}}=\sqrt{\frac{k}{-m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}}=\sqrt{\frac{1}{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}}\sqrt{\frac{k}{m}}=\frac{{\omega }_n}{\sqrt{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}}.$$

(133)

The proof finishes. □

Figure 10 shows the plots of ${\omega }_{eqn,1}$.

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,

$${{\omega }_{eqn,1}|}_{\alpha =2}={\sqrt{\frac{k}{m_{eq1}}}|}_{\alpha =2}={\frac{{\omega }_n}{\sqrt{-{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}}|}_{\alpha =2}={\omega }_n.$$

(134)

Corollary 2 (Equivalent natural frequency I2).

The natural frequency I2, ${\omega }_{eqn,2},$ of a fractional oscillator in Class II is given by

$${\omega }_{eqn,2}=\frac{{\omega }_n}{\sqrt{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}.$$

(135)

Proof. 

Following (129), we have

$${\omega }_{eqn,2}=\sqrt{\frac{k}{m_{eq2}}}=\sqrt{\frac{k}{m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}=\sqrt{\frac{k}{m\left(1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}}=\frac{{\omega }_n}{\sqrt{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}.$$

Hence, the proof completes. □

Figure 11 indicates the curves of ${\omega }_{eqn,2}.$

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,

$${{\omega }_{eqn,2}|}_{\beta =1}={\frac{{\omega }_n}{\sqrt{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}|}_{\beta =1}={\omega }_n.$$

(136)

Corollary 3 (Equivalent natural frequency I3).

The natural frequency I3, denoted by ${\omega }_{eqn,3},$ of a fractional oscillator in Class III is given by

$${\omega }_{eqn,3}=\frac{{\omega }_n}{\sqrt{-\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}}.$$

(137)

Proof. 

With (129), we write

$${\omega }_{eqn,3}=\sqrt{\frac{k}{m_{eq3}}}=\sqrt{\frac{k}{-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}}=\frac{{\omega }_n}{\sqrt{-\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}}.$$

(138)

The above completes the proof. □

Figure 12 gives the illustrations of ${\omega }_{eqn,3}.$

Note 5.6: Figure 12 exhibits that ${\omega }_{eqn,3}$ is an increasing function in terms of ω.

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

$${{\omega }_{eqn,3}|}_{\alpha =2,\beta =0}={\sqrt{\frac{k}{-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}}|}_{\alpha =2,\beta =1}=\sqrt{\frac{k}{m}}={\omega }_n.$$

(139)

5.2. Equivalent Damping Ratio

Definition 2.

Let ${\varsigma }_{eqj}$ be the equivalent damping ratio of the equivalent system of a fractional oscillator in Class j. It is defined by

$${\varsigma }_{eqj}=\frac{c_{eqj}}{2\sqrt{m_{eqj}k}},\hspace{1em}j=1,2,3.$$

(140)

Corollary 4 (Equivalent damping ratio I).

The equivalent damping ratio of a fractional oscillator in Class I is expressed by

$${\varsigma }_{eq1}={\varsigma }_{eq1}(\omega ,\alpha )=\frac{{\omega }^{\frac{\alpha }{2}}\sin \frac{\alpha \pi }{2}}{2{\omega }_n\sqrt{-\cos \frac{\alpha \pi }{2}}},\hspace{1em}1<\alpha \le 2.$$

(141)

Proof. 

Replacing $m_{eq1}$ and $c_{eq1}$ in the expression below with the equivalent mass I and the equivalent damping I described in Section 4

$${\varsigma }_{eq1}=\frac{c_{eq1}}{2\sqrt{m_{eq1}k}}$$

(142)

yields

$${\varsigma }_{eq1}=\frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}}{2\sqrt{\left(-m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}\right)k}}=\frac{{\omega }^{\frac{\alpha }{2}}\sin \frac{\alpha \pi }{2}}{2\sqrt{-\cos \frac{\alpha \pi }{2}}}\sqrt{\frac{m}{k}}=\frac{{\omega }^{\frac{\alpha }{2}}\sin \frac{\alpha \pi }{2}}{2{\omega }_n\sqrt{-\cos \frac{\alpha \pi }{2}}},\hspace{1em}1<\alpha \le 2.$$

(143)

The proof finishes. □

Remark 20.

The damping ratio ${\varsigma }_{eq1}$ follows the power law in terms of ω.

Remark 21.

The damping ratio of fractional oscillators in Class I relates to the oscillation frequency ω and the fractional order α. It is increasing with respect to ω.

$${\varsigma }_{eq1}(0,\alpha ) =0\mathrm{and}{\varsigma }_{eq1}(\infty ,\alpha )=\infty .$$

(144)

Figure 13 shows the curves of ${\varsigma }_{eq1}(\omega ,\alpha ).$

Note 5.8: Figure 13 indicates that the smaller the α the greater the ${\varsigma }_{eq1}.$

Corollary 5 (Equivalent damping ratio II).

The damping ratio of a fractional oscillator in Class II is given by

$${\varsigma }_{eq2}={\varsigma }_{eq2}(\omega ,\beta )=\frac{\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}}{\sqrt{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}},\hspace{1em}0<\beta \le 1,$$

(145)

where $\varsigma =\frac{c}{2\sqrt{mk}}.$

Proof. 

When replacing the $m_{eq2}$ and $c_{eq2}$ in the following expression by the equivalent mass II and the equivalent damping II proposed in Section 4, we attain

$$\begin{array}{l}{\varsigma }_{eq2}=\frac{c_{eq2}}{2\sqrt{m_{eq2}k}}=\frac{c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}}{2\sqrt{\left(m-c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)k}}=\frac{c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}}{2\sqrt{\left(1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)mk}}\\ =\frac{c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}}{2\sqrt{mk}\sqrt{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}=\frac{\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}}{\sqrt{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}},\hspace{1em}0<\beta \le 1.\end{array}$$

This finishes the proof. □

Remark 22.

The damping ratio ${\varsigma }_{eq2}$ obeys the power law in terms of ω.

Remark 23.

The damping ratio ${\varsigma }_{eq2}$ is associated with ω and the fractional order β. It is decreasing in terms of ω.

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

$${{\varsigma }_{eq2}(\omega ,\beta )|}_{\beta =1}={\frac{\varsigma {\omega }^{\beta -1}\sin \frac{\beta \pi }{2}}{\sqrt{1-\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}}}|}_{\beta =1}=\varsigma .$$

(146)

Figure 14 indicates the plots of ${\varsigma }_{eq2}(\omega ,\beta )$ in the case of m = 1, c = 1, and k = 1.

Corollary 6 (Equivalent damping ratio III).

Let ${\varsigma }_{eq3}$ be the damping ratio of a fractional oscillator in Class III. Then, for 1 < α ≤ 2, 0 < β ≤ 1,

$${\varsigma }_{eq3}={\varsigma }_{eq3}(\omega ,\alpha ,\beta )=\frac{{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}}{2{\omega }_n\sqrt{-\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)}}.$$

(147)

Proof. 

If replacing the $m_{eq3}$ and $c_{eq3}$ below with the equivalent mass III and the equivalent damping III presented in Section 4, we obtain

$$\begin{array}{l}{\varsigma }_{eq3}=\frac{c_{eq3}}{2\sqrt{m_{eq3}k}}=\frac{m{\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+c{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}}{2\sqrt{-\left(m{\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+c{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)k}}\\ =\frac{m\left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+\frac{c}{m}{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}{2\sqrt{-\left({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}+\frac{c}{m}{\omega }^{\beta -2}\cos \frac{\beta \pi }{2}\right)mk}}=\frac{m\left({\omega }^{\alpha -1}\sin \frac{\alpha \pi }{2}+2\varsigma {\omega }_n{\omega }^{\beta -1}\sin \frac{\beta \pi }{2}\right)}{2\sqrt{mk}\sqrt{-({\omega }^{\alpha -2}\cos \frac{\alpha \pi }{2}}}\end{array}$$