The periodic response of a fractional oscillator with a spring-pot and an inerter-pot

The fractional oscillation system with two Weyl-type fractional derivative terms −∞ D β t x (0 < β < 1) and −∞ D α t x (1 < α < 2), which portray a “spring-pot” and an “inerter-pot” and contribute to viscoelasticity and viscous inertia, respectively, was considered. At first, it was proved that the fractional system with constant coefficients under harmonic excitation is equivalent to a second-order differential system with frequency-dependent coefficients by applying the Fourier transform. The effect of the fractional orders β (0 < β < 1) and α (1 < α < 2) on inertia, stiffness and damping was investigated. Then, the harmonic response of the fractional oscillation system and the corresponding amplitude– frequency and phase–frequency characteristics were deduced. Finally, the steady-state response to a general periodic incentive was obtained by utilizing the Fourier series and the principle of superposition, and the numerical examples were exhibited to verify the method. The results show that the Weyl fractional operator is extremely applicable for researching the steady-state problem, and the fractional derivative is capable of describing viscoelasticity and portraying a “spring-pot”, and also describing viscous inertia and serving as an “inerter-pot”.


INTRODUCTION
Fractional calculus that offers an appropriate mathematical instrument of describing the memory-dependent phenomena and the intermediate process has penetrated increasingly into various areas of science and engineering covering rheology and viscoelasticity, system identification, signal processing, relaxation and oscillation, anomalous diffusion, dielectric polarization, chaos and so on [1][2][3][4][5][6][7][8]. A number of remarkable discussions, devoted specifically to the subject, can be consulted in [3,[9][10][11]. The criteria for the existence and uniqueness of the solutions and different solving methods for fractional-order equations were presented in [10,11].
In theory of viscoelasticity, fractional calculus has been extensively used for modeling of real viscoelastic materials for structural and vibration control engineering due to the heredity and nonlocality of the fractional operator [3,8]. The fractional stress-strain relationship of some viscoelastic materials like polymeric damping materials was introduced in very readable publications [3,[12][13][14][15][16]. Scott-Blair [14,15] pioneered the work in characterizing a viscoelastic body with a fractional equation σ (t ) = b · 0 D v t ε(t ) (0 < v < 1) to simulate the mechanical property between a Hookean solid (v = 0) and a Newtonian fluid (v = 1). Such fractional dashpot is also referred as a "springpot" or the Scott-Blair model [3,16].
Considerable attention has also been paid to the theoretical analysis, dynamical behaviors and numerical simulation of linear and nonlinear fractional oscillators. Fractional oscillators and related mechanical models were reported by Mainardi [3], Gorenflo and Mainardi [17], Rossikhin and Shitikova [18,19], Narahari Achar et al. [20] and others [21][22][23][24][25][26][27][28][29][30][31][32]. In [1,18], dynamical responses of linear and nonlinear fractional oscillation systems wherein the viscoelastic damping forces have been depicted by fractional constitutive relationship have been reviewed. In [19], the solution to viscoelastic oscillators constructed by some fractional rheological models was obtained via the Laplace transformation technique. The Green's function solution for the fractional-order equation of motion with the aid of the Laplace transform and Mittag-Leffler function was obtained in [20]. In [21], the causal impulse response of the fractional oscillator was analyzed by means of the Fourier transform. In [22], the prehistorical effect in the initial functions of fractional oscillators was taken into account.
It has been shown in [23,24] that the orders of fractional derivatives are associated with the quality factor of an oscillating system. Li [24] investigated equivalent two-order oscillation representations of three kinds of fractional systems. Li et al. [25], Duan et al. [26] and Huang and Duan [27] centered around the elaboration of dynamical analysis and resonance characteristic for fractional oscillation. The harmonic response and its frequency dependence of fractional oscillation equation were discussed in [28,29], and the distributed order derivatives were involved in [30]. Lim and Eab [31] studied two classes of fractional oscillator processes and the corresponding fractional Brownian motion on the basis of Riemann-Liouville and Weyl fractional operators. Shen et al. [32] extended the averaging method to research analytically a Duffing oscillator with two fractional orders.
Moreover, the stability analysis of fractional oscillators was done by Li et al. [33], Wang and Hu [34] and Li and Ma [35]. The phenomena of stochastic P-bifurcation and resonance for a fractional system whose order value is limited in the interval (0, 2] were investigated by Yang et al. [36]. The chaotic behavior of the fractional Duffing-type oscillator [37] and chaos synchronization of the fractional Chua system [38] were analyzed numerically. In [39], the two numerical schemes for a fractional oscillator equation were developed and convergence and stability of the schemes were illustrated. In [40], fractal behavior of a vibration equation involving local fractional derivative was shown.
Most literatures involved the fractional derivatives with the order satisfying 0 < v < 1, and this case describes viscoelasticity of materials and portrays a "spring-pot". In [24,29,36], the case of the order satisfying 1 < v < 2 was considered. The terminology "inerter-pot" was used in [29] considering that the viscous inertia of materials can be characterized by the fractional element of order 1 < v < 2. We note that inerter [41][42][43][44] is a mechanical two-terminal element with the property that the externally impressed force is proportional to the relative acceleration across its terminals, and is employed in structure vibration, suppression and the design of vehicle suspension systems. In [44], the performance advantages and the influence on natural frequencies of inerter in mechanical systems were well demonstrated.
For convenient reference, let us retrospect the related definitions of fractional derivatives and integrals. Suppose f(t) is piecewise continuous on (a, +∞) and integrable on any subinterval (a, t). Then, the αth-order Riemann-Liouville fractional integral of f(t) is defined as [9][10][11] a J α t f (t ) := (1) where ( · ) is the Euler's gamma function. The αth-order fractional derivative of f(t) in the Riemann-Liouville sense, Let f (n) (t) be piecewise continuous on (a, +∞) and integrable on any subinterval (a, t). The αth-order fractional derivative of When a = −∞, (2) and (3) are identical in the above two definitions of fractional derivatives. In this case, they are uniformly referred to as the Weyl fractional derivative of order α, and denoted as −∞ D α t f (t ). From the definitions of the fractional integral and derivatives, we notice that the fractional calculus is related to the whole change history of the function. So, the fractional calculus is more suitable for describing the hereditary and memory phenomena in mechanical engineering. Such examples of applications cover the constitutive equations for viscoelastic bodies, and the related forced oscillations, the dissipation in vibrations, impact and wave propagation in viscoelastic materials [1,3,18,45].
We mention that Podlubny [46] gave geometric and physical interpretations of fractional integration and fractional differentiation. By introducing a transformation of time history and an adding dimension, the fractional integral represents an area. By using two kinds of time, the equably flowing homogeneous individual time and the cosmic inhomogeneous time, like the idea in the theory of relativity, the fractional derivative describes the relationship between the real distance and the individual speed for a moving object.
The Fourier transform will be used in this work. We adopt the Fourier transform and its inverse as follows: where i is the imaginary unit. The following Fourier transform formulas [8,10,47] will be used: where δ( · ) is the Dirac's delta function. The Dirac's delta function satisfies the sampling property for any continuous function f and real number ω. For easy reference, the classic integer-order oscillation subjected to a harmonic driving is given as where m, k, c, A and ω are positive constants. Its steady-state periodic response is where θ, the range of which is restricted to 0 < θ < π as usual, means the phase difference between the excitation and the response and is written as The case of the lower limit a of the integration taken as 0 appears in many listed literatures. For this kind of problem, it is necessary to consider suitable initial-valued conditions for resolution and generally there exist no periodic solutions [48,49]. We remark that Kang et al. [49] gave a masterly proof for the nonexistence of non-constant exact periodic solutions in a type of fractional-order dynamical systems.
Actually, the steady-state response of an oscillatory system after a long enough time should be of more interest. When a = −∞, it physically is the same as taking the initialized time to be at −∞, which eliminates the transient effects and enables the system to reach equilibrium state [22,31].
In this work, we investigate the steady-state periodic response of a fractional oscillator system with two Weyl-type fractional , which represent a "spring-pot" and an "inerter-pot", respectively. In the next section, we consider the fractional oscillation system subject to the complex harmonic incentive. The equivalent integer-order system is verified and the expressions of equivalent mass, equivalent stiffness and equivalent damping are presented. Further, we analyze the contribution of two fractional derivative terms. In Section 3, the response to the complex harmonic incentive is considered, and then the amplitude-frequency relation and phase-frequency relation are discussed. In Section 4, we obtain the periodic solution of the system under general periodic excitation by using the Fourier series and superposition principle, and display two numerical examples.

THE EQUIVALENT INTEGER-ORDER SYSTE M OF THE FR ACTIONAL OSCILL ATOR
We consider a fractional oscillation system with two Weyl fractional derivative terms under the harmonic excitation Ae iωt , where m, c, k, b 1 , b 2 , A and ω are positive real constants and 1 < α < 2 and 0 < β < 1; the terms b 1 −∞ D α t x (1 < α < 2) and b 2 −∞ D β t x (0 < β < 1) denote forces pertinent to the whole deformational history caused by viscous inertia and viscoelasticity of materials.
First, we give the equivalence between Eq. (12) and a secondorder equation with ω-varying coefficients by using the method of the Fourier transform. Theorem 1. The second-order oscillator equation equivalent to (12) is Proof. By performing the Fourier transform on both sides of Eq. (12), we have where X(λ) represents the Fourier transform of x(t). Using the equality i γ = e i(π/2)γ = cos π 2 γ + i · sin π 2 γ , γ = α or β, and the sampling property of the Dirac's delta function (8), Eq. (14) is written as On the other hand, applying the Fourier transform on the both sides of (13) and utilizing sampling property (8), we solve for the Fourier transform X 1 (λ) of x 1 (t). We find that X 1 (λ) has the same expression with X(λ), i.e.
Consequently, (13) is the equivalent equation of (12). The proof is completed.
By comparing (13) with classical integer-order case, the equivalent mass m eq , the equivalent stiffness k eq and the equivalent damping c eq of fractional oscillator (12) are proposed and denoted, respectively, as That is, the equivalent integer-order equation of the fractional oscillation system (12) is Obviously, the solution of the system (12) is consistent with the steady-state response of the integer-order system (20). Next, we consider the contributions of fractional derivative terms. The three expressions are derived from Eqs (17)- (19) as follows: We call m c (ω,α), k c (ω,β) and c c (ω,α, β) as mass contribution, stiffness contribution and damping contribution of fractional derivative terms, respectively, which are all positive. By comparison with the integer-order case, the contributions of −∞ D α t x (1 < α < 2) to the mass and damping result in the viscous inertia changes of the original system. −∞ D β t x (0 < β < 1) contributes  to the stiffness and damping causing the viscoelastic changes of the original system. It is noticed that either of the two fractional derivative terms contributes to the damping. The mass contribution m c (ω, α) is monotonically decreasing for the driving frequency ω. It is vanishing for ω → +∞ and approaches to infinity for ω → 0 + . Setting b 1 = 1, the curves of m c (ω,α) versus ω for different α and m c (ω, α) versus α for different ω are plotted in Figs 1 and 2, respectively.

PERIODIC RESPONSE TO THE HAR MONIC E XCITATION
By calculating inverse Fourier transform of Eq. (15), or using the result of the equivalent integer-order equation, we obtain the harmonic response as follows.
Theorem 2. The response of the fractional oscillation system (12) with the complex harmonic incentive is Obviously, (31) signifies a periodic response with the same circular frequency as the input Ae iωt . To simplify the above formula, we set R(ω) = (P(ω)) 2 + (Q (ω)) 2 1/2 .
Then, the response (31) of the fractional system has the expression and the phase difference between the excitation and the response is rewritten as From (31), we derive the amplitude magnification factor Taking m = k = b 1 = b 2 = 1 and c = 0.2, the amplitudefrequency characteristics of η(ω) versus ω are displayed in Fig. 8a for β = 0.1 and different values of α, and in Fig. 8b for α = 1.9 and different values of β. From Fig. 8, the increase of α or the decrease of β causes increase of resonance peak values. The reason is that as β approaches 0 or α approaches 2, the damping of the fractional system becomes smaller.
The phase-frequency curves of θ (ω) versus ω are shown in Fig. 9 for the same parameter values as in Fig. 8. In Fig. 9a, the larger the value of α, or in Fig. 9b, the lower the value of β, the more rapidly the shift from in-phase to anti-phase with the driving frequency ω takes place.

RESPONSE TO GENER AL PERIODIC E XCITATION AND NUMERICAL E X A MPLES
In this section, we consider the fractional oscillation system under a general periodic incentive f(t), (38) First, from the complex harmonic response (31), we deduce the responses of system to real harmonic incentives cos(ω t) and sin(ω t): For a general periodic excitation f(t), we decompose it into a superposition of harmonic incentives by employing the Fourier series and then use the responses of the cosinusoidal incentives cos(ω t) and the sinusoidal incentives sin(ω t) in this process. Assuming f(t) to be a real periodic incentive with the period 2l, the circular frequency ω is π/l. The trigonometric Fourier series of f(t) is The response of the system (38) to a constant incentive A 0 is A 0 /k. The result for harmonic incentives cos(nωt) and sin(nωt ) can be obtained from (39) and (40), respectively, as cos(nωt ) : x sin(nωt ) : where R(nω) = (P(nω)) 2 + (Q (nω)) 2 1/2 . where θ((2u − 1)ω) and R((2u − 1)ω) are determined by Eqs (46) and (47).

CONCLUSIONS
In this paper, the steady-state periodic response to periodic incentive in a fractional oscillation system with two Weyl fractional derivative operators −∞ D β t (0 < β < 1) and −∞ D α t (1 < α < 2) was considered. Here, the lower terminal a = −∞ signifies that the steady-state response of the system is directly triggered regardless of initial conditions, and two fractional derivatives model a "spring-pot" and an "inerter-pot" mechanical element, respectively.
In Section 2, we validated that a second-order differential equation with ω-varying coefficients is equivalent to the designated fractional system with constant coefficients by applying the Fourier transform, and obtained the expressions of the equivalent mass, equivalent stiffness and equivalent damping. Based on the equivalent integer-order oscillation equation, the effect of the two fractional derivative terms on the inertia, stiffness and damping was discussed. We confirmed that the change of viscoelasticity is caused by the contribution of −∞ D β t (0 < β < 1) to stiffness and damping, while the viscous inertia variation is caused by the contribution of −∞ D α t (1 < α < 2) to mass and damping. The relevance of the three contribution functions of mass, stiffness and damping with respect to the driving frequency ω and the orders α and β was also revealed.
In Section 3, we investigated the response to the harmonic incentive Ae iωt . The amplitude-frequency relation and phasefrequency relation were given explicitly. The amplitude magnification factor and the phase difference with respect to the driving frequency ω and the orders α and β were also derived. These characteristics were shown in figures for different values of ω, α and β.
In Section 4, we obtained the steady-state response to the general periodic excitation by decomposing the driving function into the Fourier series and employing the principle of superposition, and the results on harmonic incentives were obtained. Then, finally, two numerical examples were considered and the steady-state periodic responses were obtained accurately, which validate the effectiveness of the proposed method.