APPLICATIONS OF THE EXTENDED FRACTIONAL EULER-LAGRANGE EQUATIONS MODEL TO FREELY OSCILLATING DYNAMICAL SYSTEMS

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1 APPLICATIONS OF THE EXTENDED FRACTIONAL EULER-LAGRANGE EQUATIONS MODEL TO FREELY OSCILLATING DYNAMICAL SYSTEMS ADEL AGILA 1,a, DUMITRU BALEANU 2,b, RAJEH EID 3,c, BULENT IRFANOGLU 4,d 1 Modeling & Design of Engineering Systems PhD Program, Graduate School of Natural & Applied Sciences, Atilim University, Incek-Ankara, Turkey a : adelas20@yahoo.com, agila.adel@student.atilim.edu.tr 2 Department of Mathematics and Computer, The Faculty of Arts and Sciences, Cankaya University, Ankara, Turkey and Institute of Space Sciences, Magurele-Bucharest, Romania b : dumitru@cankaya.edu.tr 3 Department of Mathematics, Atilim University, Incek-Ankara, Turkey c : reid@atilim.edu.tr, 4 Department of Mechatronics Engineering, Atilim University, Incek-Ankara, Turkey d : birfanoglu@atilim.edu.tr. Received April 29, 2015 The fractional calculus and the calculus of variations are utilized to model and control complex dynamical systems. Those systems are presented more accurately by means of fractional models. In this study, an extended version of the fractional Euler- Lagrange equations is introduced. In these equations the damping force term is extended to be proportional to the fractional derivative of the displacement with variable fractional order. The finite difference methods and the Coimbra fractional derivative are used to approximate the solution of the introduced fractional Euler-Lagrange equations model. The free oscillating single pendulum system is investigated. Key words: Single Pendulum, Fractional Euler-Lagrange Equations, Finite Difference Methods, Coimbra Fractional Derivative. 1. INTRODUCTION Fractional calculus is an emerging field in the area of applied mathematics and mathematical physics and it has many applications in many areas of science and engineering [1 11]. The calculus of variations is widely applied for some disciplines like engineering, and pure and applied mathematics. Moreover, the researches have recently proved that the physical systems with dissipation can be clearly modeled more accurately by using fractional representations [12]. The freely oscillating fractional order dissipating system is one of the important examples of complex dynamical systems. In this system the damping force is proportional to the fractional derivative of the displacement. The element that represents the damping force is a springpot a rheological element for viscoelasticity [13]. The whole fractional oscillating system that includes this element is given by the Kelvin-Voigt model [13 16]. This model obeys the Newton second law, and generates a spring mass springpot system. The RJP Rom. 61Nos. Journ. Phys., 3-4, Vol , Nos. 3-4, 2016 P , c 2016 Bucharest, - v.1.3a*

2 2 Applications of the extended fractional Euler-Lagrange equations model 351 system is considered as a very good representation for the real models, such as an earthquake model [17, 18]. On the other hand, the coefficient of the damping force generated by applying the fractional Euler-Lagrange equations FELEs [19] is a function of the Riemann-Liouville RL fractional integral order α, and the time that consists of intrinsic time τ and observing time t. In a particular case, when α and τ are constants, the damping force coefficient can be given as a function of time t. In the brush disk sliding friction [20], the friction damping force is inversely proportional to the distance between the brush and the disk. By considering the results of this experimental study and the FELEs modeling of the system, the RL fractional integral order α can be identified as a friction force coefficient between the brush and the disk. Another experimental study was done [21], in which a variable stiffness Magneto-Rheological MR fluid-based damper for vibration suppression was considered. In this experiment the damping force coefficient is given as a function of the magnetic field that is supplied to the MR valve. The magnetic field is produced by a controllable DC power. The viscosity of the liquid can be changed depending on the magnetic field intensity. Thus, the viscosity of the MR fluid is proportional to the amount of current applied to the damper [22]. So, if the DC power supply is given as a function of time then the damping force is a function of time as well. Since the viscosity of the MR fluid is affected by the magnetic field that is generated by the DC power supply, then the RL fractional order α, if the FELAs model is considered and compared with the experimental results, can be identified as a variable. This variable is contingent upon the viscosity of the MR fluid. In this paper, a model represented by FELEs is investigated. This model is a combination of the two above cases the damping force is proportional to the fractional derivative of the displacement, and the coefficient of the damping force is time varying. In addition, the fractional order of the damping term is given as a variable β t. A free oscillating single pendulum system is given as a case study. The responses of the system are obtained numerically. The finite difference techniques and the Coimbra fractional derivative are used to approximate the solution. A verification of the approximation is accomplished graphically. This is done by comparing the responses of the integer order representations and the fractional order representations at β = 1. The Runge-Kutta numerical method is used to verify the solution. Moreover the effects of α, and β on the system responses are investigated in this study. 2. BASIC DEFINITIONS In the following we give some definitions of fractional calculus and variations principles related to our topic. The RL fractional integral [1] which is used to derive

3 352 Adel Agila et al. 3 the FELEs is defined as following: and ad α t f t = a I α t f t = 1 Γα t td α b f t = t Ib α f t = 1 b Γα t a f τ 1 α dτ left, 1 t τ f τ 1 α dτ right, 2 τ t where Γ is the Gamma function and α R +. Fundamental concepts of the variational calculus and transversality conditions are introduced to derive the FELEs [23]. In order to derive the FELEs, the Lagrangian in a sub-interval [a,b] is defined as following [24 26]: J [ q t ] = b a L t, q t, a D β t q t, td γ b q t dt, 3 where J is a function to be minimized or maximized, q is an absolutely continuous function in the interval [a,b] and satisfies the boundary conditions q a = q a and q b = q b [27]. We take the RL fractional integral of the Lagrangian L to define the following action function [19]: S = a I α b Lt = 1 Γα b a L q τ, C a D β t q τ, C t D γ b q τ t τ α 1 dτ. For simplicity let the orders of the fractional derivatives β = 1, and γ = 1 [19], and consider Eq. 4 depends only on t D β b or ad γ t [19], [28] to obtain the FELE as following [19]: L q L t q 4 α 1 L t τ q = 0 t τ, 5 3. THE PROBLEM OF THE DOMAIN Consider a free oscillating single pendulum with length l and uniform mass distribution. The equation of motion of the system modeled by Eq. 5 can be represented by Eq. 6. θ t + α 1 t τ θ t + ω 2 n sinθ t = 0, 6 where ω n = 2g/l is the natural frequency of the system, τ is an intrinsic time, and t represents the observing time.

4 4 Applications of the extended fractional Euler-Lagrange equations model 353 Let the damping force in Eq. 6 to be proportional to the the fractional derivative of θ with time varying coefficient, so Eq. 6 becomes θ t + α 1 t τ 0 D βt t θ t + ω 2 n sinθ t = 0, 7 where β t represents the variable order of fractional derivative of the angular displacement θ. Now, our aim is to obtain the solution of the linearized system that is given by Eq THE METHODS Numerical techniques are used to approximate the derivatives solutions of Eq. 7. The finite difference method is used to approximate the first term and the second term is approximated by the Coimbra fractional derivative and the finite difference technique. The third term is linearized such that the oscillation angle is small. Applying the finite difference method to θ t we obtain θ t n+1 = θ t n + h = θ t n + h θ t n + h2 2! θ t n + h3 3! θ t n 1 = θ t n h = θ t n h θ t n + h2 2! θ t n h3 3! θ t n = θ t n + 0 = θ t n, where h is the time increment. Subtracting Eq. 8b from Eq. 8a yields... θ t n +...,... θ t n +..., θ t = θ n+1 θ n 1 + O 2h Adding Eq. 8a to Eq. 8b and subtracting twice Eq. 8c we obtain θ t = θ n+1 2θ n + θ n 1 h 2 8a 8b 8c h O h The fractional derivative 0 D βt t θ t can be approximated by using Coimbra fractional derivative as following [29]: 0D βt 1 t θ t n = Γ [ 1 β t n ] tn 0 t n ζ βt n θ ζ dζ ζ + θ 0 + θ 0 Γ [ 1 β t n ]. 11

5 354 Adel Agila et al. 5 The discretization of the integral that is given by Eq. 11 can be done as following for θ 0 + θ 0 [30]: 0D βt n 1 1 t θ t n = Γ [ 1 β t n ] tj+1 Applying the forward difference approximation to θζ ζ 0D βt n 1 1 t θ t n = Γ [ 1 β t n ] tj+1 n 1 1 = Γ [ 1 β t n ] θ t j+1 θ tj h where the error E 1n can be given as E 1n = t j n 1 1 Γ [ 1 β t n ] t j t n ζ βtn tj+1 tj+1 t n ζ βt n θ ζ dζ. 12 ζ yields θ tj+1 θ tj h + h.e 1 dζ t j t n ζ βt n dζ + E 1n. 13 t j t n ζ βt n h.e 1 dζ. 14 Solving the first term of the right hand side of Eq. 13 yields 0D βt t θ t n = { n 1 θt j+1 θt j h [ 1 βtn ]} 1 βtn t n t j+1 + tn t j Γ [ 2 β t n ]. 15 [ For j = 0,1,2,...,n 1 the following can be considered: tn a ] a t j+1 tn t j = h a [ n j a n j 1 a] where a R. Then, the Eq. 15 can be written as 0D βt t θ t n = h 1 θt n n 1 Γ [ 2 β t n ] { θ tj+1 θ tj h [ n j 1 βt n n j 1 1 βt n ]} = h θtn n 1 Γ [ 2 β t n ] { [ ][ n j 1 βtn n j 1 1 βt n θ ]} t j+1 θ tj. 16 Substituting Eq. 10 and Eq. 16 into Eq. 7 and considering the linearization

6 6 Applications of the extended fractional Euler-Lagrange equations model 355 we obtain θ t n+1 2θ t n + θ t n 1 h 2 + α 1 t n τ Γ [ 2 β t n ] n 1 { [ n j 1 βtn n j 1 1 βtn][ θ ]} t j+1 θ tj h θtn 17 + ω 2 nθ t n = 0. We open the summation for j = n 1 and we rewrite Eq. 17. Then we get θ t n+1 2θ t n + θ t n 1 h 2 + α 1 t n τ Γ [ 2 β t n ] n 2 { [ ][ n j 1 βtn n j 1 1 βt n θ ]} t j+1 θ tj + α 1 t n τ h θt n h θt n [ n Γ [ 2 β t n ] 1 βtn ] 1 βtn n 1 n n 1 1 [ θ t n θ t n 1 ] + ω 2 nθ t n = 0. Multiplying both [ sides of Eq. 18 by h 2, considering the initial conditions, and n 1 βtn ] 1 βtn taking into account n 1 n n 1 1 = 1 to obtain 18 h 2 βtn α 1 θ t n+1 2θ t n + θ t n 1 + t n τ {Γ [ 2 β t n ]} n 2 { [ ][ n j 1 βtn n j 1 1 βt n θ ]} t j+1 θ tj h 2 βtn α 1 + t n τ {Γ [ [ 2 β t n ]} θ t n θ t n 1 ] + h 2 ωnθ 2 t n = THE NUMERICAL RESULTS The second order homogenous variable coefficients differential equation is considered as a particular case of Eq. 7. This can be accomplished by taking β t = 1. Considering the linearity of sinθ t for small oscillation, the solutions of the integer representation and the fractional representation must be identical for β t = 1. This is shown in Fig. 1 in which the integer and fractional representations solutions are obtained by using Runge-Kutta numerical technique and the introduced approxima-

7 356 Adel Agila et al. 7 tion, respectively. The figure illustrates that both solutions are identical to each other for β = 1. Fig. 1 color online We show a comparison between the integer and fractional representations solutions that are obtained by using Runge-Kutta numerical technique and the introduced approximation for β = 1, respectively. Fig. 2 color online We illustrate the effect of β value on the system responses. For 0 β 1, selected values of β are tested for α > 1 and α < 1 as shown in Fig. 2.a and Fig. 2.b, respectively. It is concluded from Fig. 2 that the response of the system reaches the steady state earlier as β is closer to one for α > 1. For α < 1 the divergence of the system increases as β increases. The effect of α is investigated in Fig. 3 in which the following can be concluded. For α < 1, as α decreases the divergence of the system increases, while for α > 1, as α increases the convergence of the system increases for the same values of β.

8 8 Applications of the extended fractional Euler-Lagrange equations model 357 Fig. 3 color online We illustrate the effect of α value on the system responses. Fig. 4 color online We illustrate the effect of β value on the system stability.

9 358 Adel Agila et al. 9 For α = 1 there is no dissipation force acting on the system. It is inferred from Fig. 3 that the system is stable for α > 1. This is because of the positive damping force effect. The analysis of the system stability due to variation of the β values is demonstrated in Fig. 4. It is clear from the figure that for different values of α > 1 the system is stable, if 0 < β < CONCLUSIONS In this manuscript the introduced model represents an extended version of FE- LEs. This version is given as a combination of two types of damped system model. In the first model, the damping force is proportional to the fractional derivative of the displacement [13 16], and in the second one the coefficient of the damping force is time varying [20]. In addition, the fractional order of the damping term is given as a variable β t. The Coimbra fractional derivative and the finite difference technique are used to approximate the solution of the model. The verification of the approximation shows the accuracy of the approximation as illustrated in Fig. 1. The convergence and the stability of the model are affected by the values of the RL fractional integral order α and the variable order of the damping term β. These aspects are shown in Fig. 2 through Fig. 4. REFERENCES 1. S. Das, Functional Fractional Calculus Springer, New York, A.H. Bhrawy, A.A. Al-Zahrani, Y.A. Alhamed, and D. Baleanu, Rom. J. Phys. 59, A. Jafarian, P. Ghaderi, A.K. Golmankhaneh, and D. Baleanu, Rom. Rep. Phys. 66, H. Jafari, H. Tajadodi, D. Baleanu, A.A. Al-Zahrani, Y.A. Alhamed, and A.H. Zahid, Rom. Rep. Phys. 65, Gang Wei Wang and Tian Zhou Xu, Rom. Rep. Phys. 66, Gang Wei Wang and Tian Zhou Xu, Rom. J. Phys. 59, A.K. Golmankhaneh, Xiao-Jun Yang, and D. Baleanu, Rom. J. Phys. 60, M. Merdan, Proc. Romanian Acad. A 16, A.H. Bhrawy, E.H. Doha, D. Baleanu, S.S. Ezz-Eldien, and M.A. Abdelkawy, Proc. Romanian Acad. A 16, Kamel Al-Khaled, Rom. J. Phys. 60, A.H. Bhrawy, M.A. Zaky, and D. Baleanu, Rom. Rep. Phys. 67, O.P. Agrawal, J. Math. Anal. Appl L.S. Taylor, A.L. Lerner, D.J. Rubens, and K.J. Parker, A Kelvin-Voight Fractional Derivative Model for Viscoelastic Characterization of Liver Tissue, ASME International Mechanical Engineering Congress & Exposition, R. Magin, M.D. Ortigueira, I. Podlubny, and J.J. Trujillo, Signal Proc. 91,

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