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.

Fractional oscillators and their processes attract the interests of researchers, see, e.g., . 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.
• 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.

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.

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 dt + kq(t) = e(t) q(0) = q 0 , q (0) = v 0 , (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   and further rewrites it by    d 2 q(t) where ω n is called the natural angular frequency (natural frequency for short) with damping free given by and the parameter ς is the damping ratio expressed by The characteristic equation of (3) is in the form p 2 + 2ςω n p + ω 2 n = 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 where i = √ −1. Taking into account damping, one uses the term damped natural frequency denoted by ω d . It is given by Note 2.1: All parameters above, namely, m, c, k, ζ, ω n , and ω d , are constants.

Responses
The free response, meaning that the response with e(t) = 0, is driven by initial conditions only. It is given by 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 One has h(t) = e −ςω n t mω d sin ω d t, t ≥ 0.
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 d 2 g(t) dt 2 + 2ςω n dg(t) dt + ω 2 n g(t) = u(t) m .
One has where 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 With the parameter γ defined by which is called frequency ratio, H(ω) may be rewritten by . (17) The amplitude of H(ω) is called the amplitude frequency response. It is in the form |H(ω)| = 1 Its phase is termed the phase frequency response given by 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 is the sinusoidal response in the form 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.

Spectra of Three Excitations
The spectrum of δ(t) below means that δ(t) contains the equal frequency components for ω ∈ (0, ∞).
The spectrum of u(t) is in the form The Fourier transform of cos ω 1 t is given by 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]).

. 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    d 2 dt 2 d n q(t) dt n + 2ςω n d dt d n q(t) dt n + ω 2 n d n q(t) dt n = e(t) m q (n) (0) = q 0 , q (n+1) (0) = v 0 .
The above may be taken as a generalization of the conventional oscillator described by (3). Another expression of the above may be given by Alternatively, we have a linear oscillation system described by 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.

Equivalence of Functions in the Sense of Fourier Transform
Denote by F 1 (ω) and F 2 (ω) the Fourier transforms of f 1 (t) and f 2 (t), respectively. Then, if F 1 (ω) = F 2 (ω), (44) one says that f 1 (t) = f 2 (t), (45) in the sense of Fourier transform (Gelfand and Vilenkin [67], Papoulis [76]), implying 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 Let u 2 (t) be Clearly, either u 1 (t) or u 2 (t) is a unit step function. The difference between two is a null function given by 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

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.

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 . e −st K α (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 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.

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 m eq (ω) d 2 q(t) dt 2 + c eq (ω) dq(t) dt + kq(t) = e(t) q(0) = q 0 , q (0) = v 0 . (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 for Class I oscillators. As for Class II oscillators, (51) should be generalized by Similarly, for Class III oscillators, we generalize (51) to be the form Three generalized oscillation Equations (52)- (54), can be unified in the form By introducing the symbols ω eqn,j = k m eqj and ς eqj = c eqj 2 √ m eqj k for j = 1, 2, 3, we rewrite the above by Let Y j (ω) 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 (ω) the Fourier transform of x j (t). Then, if we find proper m eqj and c eqj such that 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.

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 Sections 4.1-4.3, respectively for each class of fractional oscillators.
Then, its equivalent oscillator with the equation of order 2 is in the form Proof. Consider the frequency response of (58) with the excitation of the Dirac-delta function δ(t).
In doing so, we study Doing the Fourier transform on the both sides of (60) produces where H y1 (ω) is the Fourier transform of h y1 (t). Using the principal value of i, we have Thus, (61) implies Therefore, we have the frequency response of (60) in the form On the other hand, for 1 < α ≤ 2, we consider (59) by Performing the Fourier transform on the both sides of (65) yields −mω α−2 cos απ 2 −ω 2 + mω α−1 sin απ where H x1 (ω) is the Fourier transform of h x1 (t). Therefore, we have By comparing (64) with (67), we see that Thus, (59) is the equivalent equation of (58). The proof completes.

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 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, Thus, the coefficient of dt 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 The equivalent mass m eq1 relates to the oscillation frequency ω, the fractional order α, and the primary mass m. Denote by Then, we have m eq1 (ω, α) reduces to the primary mass m when α = 2. That is, In the case of α = 2, therefore, both (58) and (59) reach the conventional harmonic oscillation with damping free in the form 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 Thus, we reveal an interesting phenomenon expressed by The coefficient R m1 (ω, α) is plotted in Figure 1.
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.
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.
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. □

Remark 2.
For α ∈ (0, 2), we have lim 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.
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.

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 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.
Then, we have The coefficient R c1 (ω, α) is indicated in Figure 2.
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 1 , eq c we found, is expressed by 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 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.

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

Note 4.4:
Because 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 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 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.

Equivalent Oscillation
Then, its equivalent 2-order oscillation equation is given by Proof. Consider the following equation: Denote by H y2 (ω) the Fourier transform of h y2 (t). Then, it is its frequency transfer function. Taking the Fourier transform on the both sides of (89) yields With the principal value of i β , (90) becomes The above means On the other hand, we consider the equivalent oscillation equation II with the Dirac-δ excitation by Performing the Fourier transform on the both sides of the above produces −mω 2 + cω β cos βπ 2 + icω β sin βπ where H x2 (ω) the Fourier transform of h x2 (t). Thus, from the above, we have Equations (92) and (95) imply Hence, (88) is the equivalent oscillation equation of the fractional oscillators of Class II. This completes the proof.

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, Proof. Consider the Newton's second law. Then, we see that the inertia force in the equivalent oscillator 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 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. Figure 3 shows its plots for m = c = 1 with the part of m eq2 (ω, β)> 0.
Note 4.6: The equivalent mass II is negative if ω is small enough. Figure 4 exhibits the negative part of m eq2 (ω, β). Symmetry 2017, 9,

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

Note 4.7:
The equivalent mass II reduces to the primary mass m for β = 1 as indicated below.
In fact, a fractional oscillator in Class II reduces to the conventional oscillator below if β = 1

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 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.
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. □

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

Note 4.8:
The following says that c eq2 reduces to the primary damping c if β = 1.
Remark 11. The equivalent damping c eq2 has, for β ∈ (0, 1), the property given by dt 2 + kx 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 lim ω→∞ c eq1 (ω, β) = ∞ 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 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.

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 Then, its equivalent oscillator of order 2 for 1 < α ≤ 2 and 0 < β ≤ 1 is in the form Proof. Let us consider the equation Let H y3 (ω) be the Fourier transform of h y3 (t). Doing the Fourier transform on the both sides of the above results in Taking into account the principal values of i α and i β , (112) becomes Consequently, we have On the other hand, considering the equivalent oscillator III driven by the Dirac-δ function, we have When doing the Fourier transform on the both sides of the above, we obtain where H x3 (ω) is the Fourier transform of h x3 (t). Therefore, from the above, we get H x3 (ω) = 1 mω α cos απ 2 + cω β cos βπ 2 + k + i mω α sin απ 2 + cω β sin βπ 2 . (117) 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., As a matter of fact, a fractional oscillator of Class III reduces to the ordinary oscillator when α = 2 and β = 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. 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.

Equivalent Damping of Fractional Oscillators in Class III
Let 3 eq c 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,

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, 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 ω α−1 as α > 1 and the second hyperbolically decayed with ω β−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, Remark 18. One asymptotic property of c eq3 for ω → ∞, due to lim ω→∞ ω α−1 = ∞ for 1 < α ≤ 2, is given by 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 lim 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. Figures 8 and 9 illustrate c eq3 (ω, α, β) for m = c = 1.

Remark 18. One asymptotic property of
The above says that a fractional oscillator of Class III does not vibrate for ω → ∞.

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

Summary
We have proposed three equivalent oscillation equations with order 2 to equivalently

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.

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 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.

Equivalent Natural Frequency I
where m eqj is the equivalent mass of the fractional oscillator in the jth class.
With the above definition, we write (128) by Note 5.1: ω eqn,j may take the conventional natural frequency, denoted by as a special case.
Corollary 1 (Equivalent natural frequency I1). The equivalent natural frequency I1, which we denote it by ω eqn,1 , of a fractional oscillator in Class I is given by Proof. According to (129), we have, for 1 < α ≤ 2, The proof finishes. Figure 10 shows the plots of ω eqn,1 .
Hence, the proof completes. □ Figure 11 indicates the curves of ,2 .
Corollary 2 (Equivalent natural frequency I2). The natural frequency I2, ω eqn,2 , of a fractional oscillator in Class II is given by Proof. Following (129), we have Hence, the proof completes. Figure 11 indicates the curves of ω eqn,2 . Note 5.4: Figure 11 shows that The above completes the proof. □ Figure 12 gives the illustrations of ,3 . Note 5.4: Figure 11 shows that ω eqn,2 is a decreasing function with ω. The greater the value of β the smaller the ω eqn,2 .
Note 5.5: ω eqn,2 takes ω n as a special case for β = 1. As a matter of fact, Corollary 3 (Equivalent natural frequency I3). The natural frequency I3, denoted by ω eqn,3 , of a fractional oscillator in Class III is given by Proof. With (129), we write The above completes the proof.   Note 5.6: Figure 12 exhibits that ω eqn,3 is an increasing function in terms of ω. Note 5.7: ω eqn,3 takes ω n as a special case for α = 2 and β = 1. Indeed,

Equivalent Damping Ratio
Definition 2. Let ς eqj be the equivalent damping ratio of the equivalent system of a fractional oscillator in Class j. It is defined by Corollary 4 (Equivalent damping ratio I). The equivalent damping ratio of a fractional oscillator in Class I is expressed by 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 The proof finishes.

Remark 20.
The damping ratio ς 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.
The proof finishes. □ Remark 20. The damping ratio 1 eq ς 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. Note 5.8: Figure 13 indicates that the smaller the α the greater the 1 . eq ς Corollary 5 (Equivalent damping ratio II). The damping ratio of a fractional oscillator in Class II is given by Note 5.8: Figure 13 indicates that the smaller the α the greater the ς eq1 .
Corollary 5 (Equivalent damping ratio II). The damping ratio of a fractional oscillator in Class II is given by This finishes the proof. □ Corollary 6 (Equivalent damping ratio III). Let ς eq3 be the damping ratio of a fractional oscillator in Class III. Then, for 1 < α ≤ 2, 0 < β ≤ 1, 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 Thus, we finish the proof.

Remark 24.
The damping ratio ς eq3 follows the power law in terms of ω.
Proof. If replacing the 3 eq m and 3 eq c below with the equivalent mass III and the equivalent damping III presented in Section 4, we obtain Thus, we finish the proof. □ Remark 24. The damping ratio 3 eq ς follows the power law in terms of ω.

Note 5.10:
3 eq ς regards ζ as a special case for α = 2 and β = 1. As a matter of fact,

Equivalent Natural Frequency II
Now, with two parameters , eqn j ω and eqj ς presented above, we rewrite the equivalent oscillator (130) by

Equivalent Natural Frequency II
Now, with two parameters ω eqn,j and ς eqj presented above, we rewrite the equivalent oscillator (130) by The characteristic equation of (149) is given by The characteristic roots are in the form Functionally, we utilize the symbol ω eqd,j for Thus, the characteristic roots are Note that, in practice, 0 ≤ ς eqj < 1 because 1 ≤ ς eqj means no oscillation at all. We write those above for the sake of applying the theory of linear oscillations to fractional ones. Now, we discuss ω eqd,j .

Corollary 7 (Equivalent natural frequency II1).
Let ω eqd,1 be the functional damped natural frequency of a fractional oscillator in Class I. It may be termed the equivalent natural frequency II1. Then, Proof. Note that Using the above ς ed1 , we have This finishes the proof.
The parameter ω eqd,1 functionally takes the form of damped natural frequency as in the conventional linear oscillation theory. In this research, we do not distinguish the natural frequencies with damped or damping free. At most, we just say that it is a functional damped one. It relates to the oscillation frequency ω and the fractional order α.
The parameter ,1 eqd ω functionally takes the form of damped natural frequency as in the conventional linear oscillation theory. In this research, we do not distinguish the natural frequencies with damped or damping free. At most, we just say that it is a functional damped one. It relates to the oscillation frequency ω and the fractional order α.

Corollary 8 (Equivalent natural frequency II2).
Let ω eqd,2 be the functional damped natural frequency of a fractional oscillator in Class II. Term it with the equivalent natural frequency II2. Then, for 0 < β ≤ 1, (157)

There Exists Infinity of Natural Frequencies of a Fractional Oscillator
The previous discussions imply that there exists infinity of natural frequencies, for either We functionally derived the two characteristic roots of the frequency equation (151), namely, ,1,2 , j s actually stand for infinity of roots owing to ω Taking a fractional oscillator in Class I into account, its frequency equation is given by Figure 19.

There Exists Infinity of Natural Frequencies of a Fractional Oscillator
The previous discussions imply that there exists infinity of natural frequencies, for either ω eqn,j or ω eqd,j , because each is dependent on ω ∈ (0, ∞). We functionally derived the two characteristic roots of the frequency equation (151), namely, s j,1,2 , actually stand for infinity of roots owing to ω ∈ (0, ∞).
Taking a fractional oscillator in Class I into account, its frequency equation is given by Then, it is easy to see that there exists infinitely many characteristic roots in the above, also see Li et al. [18]. A contribution in this work in representing characteristic roots of three classes of fractional oscillators is that they are expressed analytically. Moreover, functionally, they take the form as that in the theory of conventional linear oscillations, making it possible to represent solutions to three classes of fractional oscillators by using elementary functions, which are easier for use in both engineering applications and theoretic analysis of fractional oscillators.

Free Responses to Three Classes of Fractional Oscillators
We put forward the free responses in this section to three classes of fractional oscillators based on their equivalent oscillators presented in Section 4. Since the equivalent oscillators are expressed by using second-order differential equations in form, in methodology, therefore, it is easy for us to find the responses we concern with. Note that the equivalence explained in Section 4 says that where the subscript j stands for the Class I to III. Consequently, Therefore, our research implies three advances.
• First, proposing the free responses to three classes of fractional oscillators using the way of solving conventional oscillators. • Then, since the responses to conventional oscillators are represented by elementary functions while those to fractional ones are expressed by special functions, such as the Mittag-Leffler function and its generalizations, we shall present novel representation to a certain special functions by elementary ones. • Finally, analytic expressions of the logarithmic decrements, which are useful in practice, of three classes of fractional oscillators are proposed.

General Form of Free Responses
Consider the free response to the functional equivalent oscillator in Class j in the form Following the representation style in engineering, we rewrite it by The above may be rewritten in the form where the equivalent amplitude A eqj is given by and the equivalent phase θ eqj is Note that, for ω eqn,j , ς eqj , A eqj , and θ eqj , each is not constant for fractional oscillators. Instead, each is generally a function of oscillation frequency ω and fractional order.

Free Response to Fractional Oscillators in Class I
We state the free response to a fractional oscillator in Class I by Theorem 10.
Theorem 10 (Free response I). Let x 1 (t) be the free response to a fractional oscillator in Class I. Then, for 1 < α ≤ 2, x 1 (t) is given by Proof. For t ≥ 0, consider In the above, replacing ω eqn,1 by the one in (132), ς eq1 with that in (141), and ω eqd,1 by the one in (154) yields This completes the proof. ς ω α also see Figure 10. When α = 2, 1 ( ) x t reduces to the free response to the ordinary harmonic oscillation with damping free in the form (also see Figure 20d) The free response to a fractional oscillator in Class I is presented in (172). It uses elementary functions instead of special functions.  Figure 20, both oscillation frequency ω and the fractional order α have affects on the damping ς eq1 (ω, α), also see Figure 10. When α = 2, x 1 (t) reduces to the free response to the ordinary harmonic oscillation with damping free in the form (also see Figure 20d)

Note 6.1: As indicated in
The free response to a fractional oscillator in Class I is presented in (172). It uses elementary functions instead of special functions.
Since there exists infinity of natural frequencies for a fractional oscillator, as we explained in Section 5, x 1 (t) is actually a function of both t and ω as can be seen from (172). In Figure 20, plots are only specifically with fixed ω. Its plots with varying ω are viewed by Figure 21.  Figure 20, both oscillation frequency ω and the fractional order α have affects on the damping 1 ( , ), eq ς ω α also see Figure 10. When α = 2, 1 ( )

Note 6.1: As indicated in
x t reduces to the free response to the ordinary harmonic oscillation with damping free in the form (also see Figure 20d) The free response to a fractional oscillator in Class I is presented in (172). It uses elementary functions instead of special functions.
Since there exists infinity of natural frequencies for a fractional oscillator, as we explained in Section 5, 1 ( ) x t is actually a function of both t and ω as can be seen from (172). In Figure 20, plots are only specifically with fixed ω. Its plots with varying ω are viewed by Figure 21. When emphasizing the point of time-frequency behavior, we view it in t-ω plane as Figure 22 shows. When emphasizing the point of time-frequency behavior, we view it in t-ω plane as Figure 22 shows. 1, Let ti and ti + 1 be two time points where ( ) Let t i and t i + 1 be two time points where x j (t i ) reaches its successive peak values of x j (t i ) and x j (t i+1 ), respectively. Let ∆ eqj be the logarithmic decrement of x j (t i ). Then, from (178), we immediately obtain Corollary 10 (Decrement I). Let x 1 (t) be the free response of a fractional oscillator in Class I. Then, its logarithmic decrement is given in the form Proof. According to (174), we have The proof finishes.

Free Response to Fractional Oscillators in Class II
We state the free response to a fractional oscillator in Class II by Theorem 11. Theorem 11 (Free response II). Denote by x 2 (t) the free response to a fractional oscillator of Class II type. Then, it is, for t ≥ 0 and 1 < β ≤ 2, in the form Proof. Note that, for t ≥ 0, x 2 (t) = e −ς eq2 ω eqn,2 t x 20 cos ω eqd,2 t + v 20 + ς eq2 ω eqn,2 x 20 ω eqd,2 sin ω eqd,2 t .
Symmetry 2017, 9, x FOR PEER REVIEW 46 of 106    Similar to 1 ( ), x t 2 ( ) x t is also with the argument ω. Its plots with variable ω are demonstrated in Figure 25. Figure 26 shows its plots in t-ω plane.  Similar to x 1 (t), x 2 (t) is also with the argument ω. Its plots with variable ω are demonstrated in Figure 25. Figure 26 shows its plots in t-ω plane. Similar to 1 ( ), x t 2 ( ) x t is also with the argument ω. Its plots with variable ω are demonstrated in Figure 25. Figure 26 shows its plots in t-ω plane.

Corollary 11 (Decrement II).
Denote by x 2 (t) the free response to a fractional oscillator in Class II. Then, for 0 < β ≤ 1, its logarithmic decrement ∆ eq2 , is in the form Proof. According to (174), we have Replacing the above ς eq2 with that in (145) produces This finishes the proof.

Free Response to Fractional Oscillators in Class III
We now present the free response to a fractional oscillator in Class III by Theorem 12.
Theorem 12 (Free response III). Let 3 ( ) x t be the free response to a fractional oscillator in Class III. Then, for t ≥ 0, 1< α ≤ 2, 0 < β ≤ 1, it is given by

Free Response to Fractional Oscillators in Class III
We now present the free response to a fractional oscillator in Class III by Theorem 12.
Theorem 12 (Free response III). Let x 3 (t) be the free response to a fractional oscillator in Class III. Then, for t ≥ 0, 1< α ≤ 2, 0 < β ≤ 1, it is given by Proof. Note that, for t ≥ 0, In (182), when substituting ς eq3 , ω eqd,3 , and ω eqn,3 with those explained in Section 5, we have (181). The proof finishes. Note 6.6: If (α, β) = (2, 1), x 3 (t) returns to be the free response to an ordinary oscillator with the viscous damping. As a matter of fact, where ω d = ω n 1 − ς 2 . Figure 28 indicates Note that the plots regarding with 3 ( ) x t in Figure 28 are with fixed ω. However, actual 3 ( ) x t is frequency varying. Figure 29 shows its frequency varying pictures in time domain and Figure 30 in t-ω plane.  Note that the plots regarding with x 3 (t) in Figure 28 are with fixed ω. However, actual x 3 (t) is frequency varying. Figure 29 shows its frequency varying pictures in time domain and Figure 30 in t-ω plane. Note that the plots regarding with 3 ( ) x t in Figure 28 are with fixed ω. However, actual 3 ( ) x t is frequency varying. Figure 29 shows its frequency varying pictures in time domain and Figure 30 in t-ω plane.

Corollary 12 (Decrement III). Denote by 3 ( )
x t the free response to a fractional oscillator in Class III.

Corollary 12 (Decrement III).
Denote by x 3 (t) the free response to a fractional oscillator in Class III. Then, for 1 < α ≤ 2 and 0 < β ≤ 1, its logarithmic decrement, denoted by ∆ eq3 , is given by Proof. Note that Replacing the above ς eq3 with that in (147) yields (184). This completes the proof.

Application to Representing Generalized Mittag-Leffler Function (1)
The previous research (Mainardi [25], Achar et al. [33], Uchaikin ([38], Chapter 7)) presented the free response to fractional oscillators of Class I type by using a kind of special function, called the generalized Mittag-Leffler function, see (32). The novelty of our result presented in Theorem 10 is in that Equation (172) or (173) is consistent with the representation style in engineering by using elementary functions. Thus, we obtain novel representations of the generalized Mittag-Leffler functions as follows.

Application to Representing Generalized Mittag-Leffler Function (1)
The previous research (Mainardi [25], Achar et al. [33], Uchaikin ([38], Chapter 7)) presented the free response to fractional oscillators of Class I type by using a kind of special function, called the generalized Mittag-Leffler function, see (32). The novelty of our result presented in Theorem 10 is in that Equation (172) or (173) is consistent with the representation style in engineering by using elementary functions. Thus, we obtain novel representations of the generalized Mittag-Leffler functions as follows.

Impulse Responses to Three Classes of Fractional Oscillators
In this section, we shall present the impulse responses to three classes of fractional oscillators using elementary functions.
In Section 4, we have proved that where H yj (ω) is the frequency response function solved directly from a jth fractional oscillator while H xj (ω) is the one derived from its equivalent oscillator. Doing the inverse Fourier transform on the both sides above, therefore, we have where h yj (t) is the impulse response obtained directly from the jth fractional oscillator but h xj (t) is the one solved from its equivalent one. In that way, therefore, we may establish the theoretic foundation for representing the impulse responses to three classes of fractional oscillators by using elementary functions. The main highlight presented in this section is to propose the impulse responses to three classes of fractional oscillators in the closed analytic form expressed by elementary functions. As a by product, we shall represent a certain generalized Mittag-Leffler functions using elementary functions.

General Form of Impulse Responses
Given a following functional form of equivalent oscillators for finding their impulse responses, we denote by h j (t) the impulse response to the equivalent oscillator in Class j in the form Rewrite the above in the form According to the results in the previous sections, we have Therefore, functionally, we have h j (t) = e −ς eqj ω eqn,j t m eqj ω eqd,j sin ω eqd,j t, t ≥ 0.
Equation (196) is a general form of the impulse response to fractional oscillators for Class j (j = 1, 2, 3). Its specific form for each Class is discussed as follows.

Impulse Response to Fractional Oscillators in Class I
Theorem 13 (Impulse response I). Let h 1 (t) be the impulse response to a fractional oscillator in Class I. Then, for t ≥ 0 and 1 < α ≤ 2, we have When replacing m eq1 by that in Section 4, ς eq1 and ω eqd,1 as well as ω eqn,1 with those in Section 5, respectively, we obtain This finishes the proof. Figure 33 shows the plots of h 1 (t), where the oscillation frequency ω is fixed. Note that ω is an argument of h 1 (t). Therefore, its pictures in time domain are indicated in Figure 34. Figure 35 indicates its figures in t-ω plane. This finishes the proof. □ Figure 33 shows the plots of h1(t), where the oscillation frequency ω is fixed. Note that ω is an argument of h1(t). Therefore, its pictures in time domain are indicated in Figure 34.

Impulse Response to Fractional Oscillators in Class II
Theorem 14 (Impulse response II). Denote by h 2 (t) the impulse response to a fractional oscillator in Class II. For t ≥ 0 and 1 < β ≤ 2, therefore, it is given by Proof. From (196), we have sin ω eqd,2 t, t ≥ 0.
By replacing m eq2 with that in Section 4, ς eq2 , ω eqn,2 , and ω eqd,2 by those in Section 5, we obtain This is (200). Hence, the proof completes. Figure 36 illustrates h 2 (t) with fixed ω. Its plots with variable ω are shown in Figure 37. Its pictures in t-ω plane are indicated in Figure 38.

Impulse Response to Fractional Oscillators in Class III
We present the impulse response to fractional oscillators in Class III with Theorem 15.

Note 7.2:
The impulse response h 2 (t) reduces to the conventional one if β = 1. As a matter of fact, (202)

Impulse Response to Fractional Oscillators in Class III
We present the impulse response to fractional oscillators in Class III with Theorem 15.
In the above expression, substitute m eq3 with the one in Section 4, ς eq3 , ω eqd,3 , ω eqn,3 by those in Section 5, respectively, we have, for t ≥ 0, The right side on the above is (203). Thus, the proof completes.
The plots of h 3 (t) with fixed ω are shown in Figure 39, with variable ω in Figure 40, and in t-ω plane by Figure 41.
The right side on the above is (203). Thus, the proof completes. □ The plots of 3 ( ) h t with fixed ω are shown in Figure 39, with variable ω in Figure 40, and in t-ω plane by Figure 41.    (c) (d)

Note 7.3:
The impulse response h 3 (t) degenerates to the conventional one when α = 2 and β = 1. Indeed, 7.5. Application to Represetenting Generalized Mittag-Leffler Function (2) The impulse response to fractional oscillators in Class I by using the generalized Mittag-Leffler function is in the form (Uchaikin ([38], Chapter 7)) In this section, we propose the representation of (206) by elementary functions.

Corollary 16.
The generalized Mittag-Leffler function in the form (206) can be expressed by the elementary functions in Theorem 13, for 1 < α ≤ 2 and t ≥ 0, in the form The proof is straightforward from Theorem 13 and (206).

Step Responses to Three Classes of Fractional Oscillators
In this section, we shall put forward the unit step responses to three classes of fractional oscillators in the analytic closed forms with elementary functions. Besides, we shall suggest a novel expression of a certain generalized Mittag-Leffler function by using elementary functions.

General Form of Step Responses
Denote by g j (t) (j = 1, 2, 3) the step response to a fractional oscillator in the jth Class. Then, it is also the step response to the jth equivalent oscillator. Precisely, g j (t) is the solution to the jth equivalent oscillator expressed by The solution to the above equation is given by where

Step Response to a Fractional Oscillator in Class I
Theorem 16 (Step response I). Let g 1 (t) be the unit step response to a fractional oscillator in Class I. For t ≥ 0 and 1 < α ≤ 2, it is given by where (212) Proof. Note that Substituting ς eq1 with the one in (141) into the above produces Replacing ω eqn,1 and ω eqd,1 with those in Section 5 in the above yields (211) and (212). The proof finishes. Figure 42 shows the unit step response g 1 (t) with fixed oscillation frequency ω. Note that g 1 (t) takes ω as an argument. Thus, we use Figure 43 to indicate g 1 (t) with variable ω in time domain. Its plots in t-ω plane are shown in Figure 44.

Note 8.1:
If α = 2, g 1 (t) reduces to the conventional step response with damping free. In fact, and

Step Response to a Fractional Oscillator in Class II
Theorem 17 (Step response II). Denote by g 2 (t) the unit step response to a fractional oscillator in Class II. It is in the form, for t ≥ 0 and 0 < β ≤ 1, where Proof. ; Substituting ς eq2 with that in Section 5 into the following expression On the other side, replacing ω eqn,2 by the one in (135) in the above results in Finally, substituting ω eqd,2 by that in (157) in the above produces (217) and (218). Hence, we finish the proof.
We use Figure 45 to indicate g 2 (t) with fixed ω. When considering variable ω, we show g 2 (t) in Figure 46 in time domain and Figure 47 in t-ω plane.
Step response g 2 (t) to a fractional oscillator in Class II with variable ω for m = c = k = 1. where

Note 8.2:
When β = 1, g 2 (t) turns to be the ordinary step response. As a matter of fact, where

Step Response to a Fractional Oscillator in Class III
Theorem 18 (Step response III). Let g 3 (t) be the unit step response to a fractional oscillator in Class III. It is in the form, for t ≥ 0, 1 < α ≤ 2, and 0 < β ≤ 1, where Proof. Replacing ς eq3 by that in (147) on the left side of the following produces the right side in the form Further, replacing ω eqn,3 with the one in (137) in the above yields Finally, considering ω eqd,3 expressed by (160), we have (223) and (224). Hence, the proof finishes. where

Application to Represetenting Mittag-Leffler Function (3)
The step response to fractional oscillators in Class I by using the generalized Mittag-Leffler function is in the form (Uchaikin ([38], Chapter 7)) In the following corollary, we propose the representation of (229) by elementary functions.

Corollary 17.
The generalized Mittag-Leffler function expressed by (229) can be represented by using the elementary functions described in Theorem 16. Precisely, for t ≥ 0 and 1 < α ≤ 2, we have where φ 1 is given by (212).

Frequency Responses to Three Classes of Fractional Oscillators
We put forward frequency responses to three classes of fractional oscillators in this section. They are expressed by elementary functions based on the theory of three equivalent oscillators addressed in Section 4.

General Form of Frequency Responses to Three Classes of Fractional Oscillators
Denote by H j (ω) the Fourier transform of the impulse response h j (t) to a fractional oscillator in Class j (j = 1, 2, 3), where h j (t) is given by (196). Then, it is the frequency response function to a fractional oscillator in Class j (j = 1, 2, 3).
In fact, doing the Fourier transform on the both sides of (195) produces Thus, we have Note that Therefore, by letting γ eqj be the equivalent frequency ratio of a fractional oscillator in Class j, H j (ω) may be expressed by The amplitude of H j (ω) is Its phase frequency response is given by Proof. In the equation below, when replacing γ eq1 by and 2ς eq1 γ eq1 by we have (237). This completes the proof.
From Theorem 19, we have the amplitude of H 1 (ω) given by and the phase in the form (242)

Note 9.1 (Equivalent frequency ratio I):
The equivalent frequency ratio γ eq1 is a function of oscillation frequency ω and the fractional order α. It may be denoted by γ eq1 (ω, α). Figure 51 shows the plot of γ eq1 . Figure 52 indicates the illustrations of H 1 (ω).
( ) and the phase in the form

Sinusoidal Responses of Three Classes of Fractional Oscillators
When the excitation force takes the sinusoidal one in the form of cosωt or sinωt, the response is termed sinusoidal response, which plays a role in the field of oscillations.

Stating Problem
Note that the sinusoidal response to fractional oscillators attracts research interests but it is yet a problem that has not been solved satisfactorily. In fact, the existence of the sinusoidal response to fractional oscillators remains a problem. In mathematics, it is regarded as a problem of periodic solution to fractional oscillators. Kaslik and Sivasundaram stated that the exact periodic solution does not exist ( [81], p. 1495, Remark 5). The view of Kaslik and Sivasundaram's in [81] is also implied in other works of researchers. Taking fractional oscillators in Class I as an example, Mainardi noticed that the solution to fractional oscillators for 1 < α < 2, when driven by sinusoidal function, does not exhibit permanent oscillations but asymptotically algebraic decayed ( [25], p. 1469), also see Achar et al. ( [33], lines above Equation (14)), Duan et al. ( [39], p. 49).
As a matter of fact, when considering a fractional oscillator of Class I type for 1 < α < 2 without the case of α = 2 in the form Note 9.6: If (α, β) = (2, 1), H 3 (ω) reduces to the ordinary one given by

Sinusoidal Responses of Three Classes of Fractional Oscillators
When the excitation force takes the sinusoidal one in the form of cosωt or sinωt, the response is termed sinusoidal response, which plays a role in the field of oscillations.

Stating Problem
Note that the sinusoidal response to fractional oscillators attracts research interests but it is yet a problem that has not been solved satisfactorily. In fact, the existence of the sinusoidal response to fractional oscillators remains a problem. In mathematics, it is regarded as a problem of periodic solution to fractional oscillators. Kaslik and Sivasundaram stated that the exact periodic solution does not exist ( [81], p. 1495, Remark 5). The view of Kaslik and Sivasundaram's in [81] is also implied in other works of researchers. Taking fractional oscillators in Class I as an example, Mainardi noticed that the solution to fractional oscillators for 1 < α < 2, when driven by sinusoidal function, does not exhibit permanent oscillations but asymptotically algebraic decayed ( [25], p. 1469), also see Achar et al. ( [33], lines above Equation (14)), Duan et al. ( [39], p. 49).
As a matter of fact, when considering a fractional oscillator of Class I type for 1 < α < 2 without the case of α = 2 in the form it is obvious that y 1 (t) must contain steady-state component that is not equal to 0 for t → ∞ no matter what value of α ∈ (1, 2) is. Otherwise, the conservation law of energy would be violated. The problem is what the complete solution of y 1 (t) should be. The actual solution y 1 (t) should, in reality, consist of two parts. One is the steady-state part, denoted by y 1s (t), where the subscript s stands for steady-state, which is not equal to 0 for t → ∞ and for any value of α ∈ (1, 2). The other is the transient part, denoted by y 1tr (t), where the subscript tr means transient. Thus, the complete solution should, qualitatively, be in the form We contribute the complete solutions to three classes of fractional oscillators regarding their sinusoidal responses in this section. Our results will show that there exist steady-state components for fractional oscillators in either class with any value of α ∈ (1, 2) for those in Class I, or β ∈ (0, 1) in Class II, or any combination of α ∈ (1, 2) with β ∈ (0, 1) for those in Class III.

Stating Research Thought
Consider the sinusoidal responses to three classes fractional oscillators based on the equivalent oscillation equation in the form The complete response x j (t) consists of the zero state response, denoted by x jzs (t), and zero input response denoted by x jzi (t), according to the theory of differential equations. Therefore, where x jzi (t) is solved from On the other hand, x jzs (t) is the solution to Note that x jzi (t) is actually the free response to the fractional oscillators in Class j. It has been solved in Section 6. Thus, the focus of this section is on (263).
Denote by x 1zs,s (t) and x 1zs,tr (t) the steady component and the instantaneous one, respectively. Then, we have and x 1zs,

Remark 29.
We found that the sinusoidal response to fractional oscillators in Class I for any value of α ∈ (1, 2) does have steady-state component x 1zs,s (t) expressed by (267), also see Figure 59.  The illustration of x 1zs,tr (t) is indicated in Figure 60.  cos ω n t = A mω n (cos ω t + cos ω n t). (269)
) Note that power laws plays a role in understanding the nature in general, see, e.g., Gabaix et al. [82], Stanley [83]. As a matter of fact, the fractional order α relates to the fractal dimension, see Lim et al. [20][21][22]. Thus, my study of the power laws previously stated is quite beginning in the aspect of fractional oscillations. Further research is needed in future. In addition to that, our future work will consider the applications of the present equivalent theory of the fractional oscillators to fractional noise in communication systems (Levy and Pinchas [84], Pinchas [85]), partial differential equations, such as transient phenomena of complex systems or fractional diffusion equations (Toma [86], Bakhoum and Toma [87], Cattani [88], Mardani et al. [89]).

Conclusions
We have established a theory of equivalent oscillators with respect to three classes of fractional oscillation systems. Its principle is to represent a fractional oscillator with constant coefficients (mass and damping) by a 2-order oscillator equivalently with variable mass and damping. The analytic expressions of equivalent masses, equivalent dampings, equivalent damping ratios, equivalent natural frequencies, and equivalent frequency ratios have been presented. We have revealed that the equivalent masses and dampings of three classes of fractional oscillators follow power laws in terms of oscillation frequency. By using elementary functions, we have put forward the closed form representations of responses (free, impulse, step, frequency, sinusoidal) to three classes of fractional oscillators. Additionally, analytic expressions of the logarithmic decrements of three classes of fractional oscillators have been proposed. As by products, we have stated the representations of four types of the generalized Mittag-Leffler functions in the closed form with elementary functions.