LIANG Zu-feng, TANG Xiao-yan. Analytical Solution of a Fractionally Damped Beam by Using Adomian Decomposition Method[J]. Applied Mathematics and Mechanics, 2007, 28(2): 200-208.
Citation: LIANG Zu-feng, TANG Xiao-yan. Analytical Solution of a Fractionally Damped Beam by Using Adomian Decomposition Method[J]. Applied Mathematics and Mechanics, 2007, 28(2): 200-208.

Analytical Solution of a Fractionally Damped Beam by Using Adomian Decomposition Method

  • Received Date: 2006-03-13
  • Rev Recd Date: 2006-10-24
  • Publish Date: 2007-02-15
  • The analytical solution of a viscoelastic continuous beam whose damping characteristics are described in terms of a fractional derivative of arbitrary order was derived by means of the Adomian decomposition method.The solution contains arbitrary initial conditions and zero input.For specific analysis,the initial conditions were assumed homogeneous,and the input force was treated as a special process with a particular beam.Two simple cases,step and impulse function responses,were considered respectively.Subsequently,some figures were plotted to show the displacement of the beam under different sets of parameters including different orders of the fractional derivatives.
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  • [1]
    Deng R,Davies P,Bajaj A K.A case study on the use of fractional derivatives: the low-frequency viscoelastic uni-directional behavior of polyurethane foam[J].Nonlinear Dynamics,2004,38(1/4):247-265. doi: 10.1007/s11071-004-3759-3
    [2]
    Rossikhin Y A,Shitikova M V.Analysis of the viscoelastic rod dynamics via models involving fractional derivatives or operators of two different orders[J].The Shock and Vibration Digest,2004,36(1):3-26. doi: 10.1177/0583102404039131
    [3]
    Agrawal O P.Analytical solution for stochastic response of a fractionally damped beam[J].ASME J Vibr Acoust,2004,126(4):561-566. doi: 10.1115/1.1805003
    [4]
    Oldham K B,Spanier J.The Fractional Calculus[M].New York:Academic Press,1974.
    [5]
    Podlubny I.Fractional Differential Equations[M].San Diego:Academic Press,1999.
    [6]
    Suarez L E,Shokooh A.Response of systems with damping materials modeled using fractional calculus[J].ASME J Appl Mech Rev,1995,48(11):118-127. doi: 10.1115/1.3005059
    [7]
    Samko G,Kilbas A A,Marichev O I.Fractional Integrals and Derivatives: Theory and Applications[M].Yverdon:Gordon & Breach,1993.
    [8]
    Kemple S,Beyer H.Global and causal solutions of fractional differential equations[A].In:Transform Methods and Special Functions: Varna96, Proceedings of 2nd International Workshop[C].Singapore:Science Culture Technology Publishing,1997,210-216.
    [9]
    Kilbas A A, Pierantozzi T,Trujillo J J,et al.On the solution of fractional evolution equations[J].J Phys A: Math Gen,2004,37(9):3271-3283. doi: 10.1088/0305-4470/37/9/015
    [10]
    Luchko Y, Srivastava H M.The exact solution of certain differential equations of fractional order by using operational calculus [J].Comput Math Appl,1995,29(8):73-85.
    [11]
    Adomian G.A new approach to nonlinear partial differential equations[J].J Math Anal Appl,1984,102(2):420-434. doi: 10.1016/0022-247X(84)90182-3
    [12]
    Adomian G.Solving Frontier Problems of Physics: The Decomposition Method[M].Boston:Kluwer Academic Publishers,1994.
    [13]
    Wazwaz A M.Exact solutions for variable coefficients fourth-order parabolic partial differential equations in higher-dimensional spaces[J].Appl Math Comput,2002,130(2/3):415-424. doi: 10.1016/S0096-3003(01)00109-6
    [14]
    Momani S, AlKhaled K. Numerical solutions for systems of fractional differential equations by the decomposition method[J].Appl Math Comput,2005,162(3):1351-1365. doi: 10.1016/j.amc.2004.03.014
    [15]
    Vadasz P, Olek S. Convergence and accuracy of Adomian's decomposition method for the solution of Lorenz equations[J].Internat J Heat Mass Transfer,2000,43(10):1715-1734. doi: 10.1016/S0017-9310(99)00260-4
    [16]
    Chen W H, Lu Z Y.An algorithm for Adomian decompostion method[J].Appl Math Comput,2004,159(1):221-235. doi: 10.1016/j.amc.2003.10.037
    [17]
    Chen Q S, Suki B, An K N.Dynamic mechanical properties of agarose gels modeled by a fractional derivative model[J].ASME J Biomech Eng,2004,126(5):666-671. doi: 10.1115/1.1797991
    [18]
    Saha Ray S, Poddar B P,Bera R K.Analytical solution of a dynamic system containing fractional derivative of order one-half by Adomian decomposition method[J].ASME J Appl Mech,2005,72(2):290-295. doi: 10.1115/1.1839184
    [19]
    Saha Ray S, Bera R K. Analytical solution of the Bagley Torvik equation by Adomian decomposition method[J].Appl Math Comput,2005,168(1):398-410. doi: 10.1016/j.amc.2004.09.006
    [20]
    Daftardar-Gejji V, Jafari H.Adomian decomposition: a tool for solving a system of fractional differential equations[J].J Math Anal Appl,2005,301(2):508-518. doi: 10.1016/j.jmaa.2004.07.039
    [21]
    Shawagfeh N T. The decomposition method for fractional differential equations[J].J Frac Calc,1999,16:27-33.
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