Dynamic Response Analysis of Fractionally Damped Beams Subjected to External Loads using Homotopy Analysis Method

Document Type: Research Paper


National Institute of Technology Rourkela, Department of Mathematics, Odisha, Rourkela, 769008, India


This paper examines the solution of a damped beam equation whose damping characteristics are well-defined by the fractional derivative (FD). Homotopy Analysis Method (HAM) is applied for calculating the dynamic response (DR). Unit step and unit impulse functions are deliberated for this analysis. Acquired results are illustrated to show the movement of the beam under various sets of parameters with different orders of the FDs. Here FD is defined in the Caputo sense. Obtained results have been compared with the solutions achieved by Adomian decomposition method (ADM) to show the efficiency and effectiveness of the presented method.


Main Subjects

[1] Machado, J.T., Kiryakova, V., Mainardi, F., Recent history of fractional calculus, Communication in Nonlinear Science and Numerical Simulation, 16 (2011) 1140–1153.

[2]Rossikhin, Y.A., Shitikova, M.V., Application of fractional calculus for dynamic problems of solid mechanics: Novel trends and recent results, Applied Mechanics Reviews, 63 (2010) 1–51.

[3] 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, Nonlinear Dynamics, 38 (2004) 247–265.

[4] Rossikhin, Y.A., Shitikova, M.V., Analysis of the viscoelastic rod dynamics via models involving fractional derivatives or operators of two different orders, The Shock and Vibration Digest, 36(1) (2004) 3–26.

[5] Agrawal, O.P., Analytical solution for stochastic response of a fractionally damped beam, ASME J Vibr Acoust, 126(4) (2004) 561–566.

[6] Kiryakova, V.S., Generalized Fractional Calculus and Applications, Longman Scientific and Technical, Longman House, Burnt Mill, Harlow, England, 1993.

[7] Golmankhaneh, A.K., Investigations in Dynamics: With Focus on Fractional Dynamics, Academic Publishing, Lap Lambert, 2012.

[8] Baleanu, D., Machado, J.A.T., Luo, A.C.J., Fractional Dynamics, and Control, Springer, 2012.

[9] Baleanu, D., Diethelm, K., Scalas, E., Trujillo, J.J., Fractional Calculus: Models and Numerical Methods, World Scientific Publishing Company, 2012.

[10] Miller., Ross., An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley and Sons, New York, 1993.

[11] Oldham, K.B.,  Spanier, J., The Fractional Calculus, Academic Press, New York, 1974.

[12] Podlubny, I., Fractional Differential Equations, Academic Press, New York, 1999.

[13] Samko, S.G., Kilbas, A.A., Marichev, O.I., Fractional Integrals and Derivatives-Theory and Applications, Gordon and Breach Science Publishers, Langhorne,1993.

[14] He, J.H., Homotopy perturbation technique, Comput Methods Appl Mech Eng, 178 (1999) 257–62.

[15] Zhang, B.G., Li, S.Y., Liu, Z.R., Homotopy perturbation method for modified Camassa-Holm and Degasperis-Procesi equations, Phys. Letter A, 372 (2008) 1867-1872.

[16] Sushila., Jagdev, S., Yadvendra, S.S., A new reliable approach for two dimensional and axisymmetric unsteady flows between parallel plates, Z Naturforsch A, 68a (2013) 629–634.

[17] Singh, J., Kumar, D., Kilicman, A., Numerical solutions of nonlinear fractional partial differential equations arising in spatial diffusion of biological populations, Abstr Appl Anal, 2014 (2014) 12 pages.

[18] Birajdar, G.A., Numerical solution of time fractional Navier–Stokes equation by discrete Adomian decomposition method, Nonlinear Eng, 3 (2014) 21–26.

[19] Momani, S., Odibat, Z., Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method, Appl Math Comput, 177 (2006) 488–94.

[20] Wazwaz, A.M., Solitary wave solutions for modified forms of Degasperis-Procesi and Camassa-Holm equations, Phys. Lett. A, 352(6) (2006) 500–504.

[21] Wazwaz, A.M., The Camassa-Holm-KP equations with compact and noncompact travelling wave solutions, Appl. Math. Comput, 170 (2005) 347–360.

[22] Bagley, R.L., Torvik, P.J., A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983) 201–210.

[23] Bagley, R.L., Torvik, P.J., Fractional calculus––a different approach to the analysis of viscoelastically damped structures, AIAA J., 21, 1983, pp. 741–748.

[24] Koeller, R.C., Application of fractional calculus to the theory of viscoelasticity, ASME J. Appl. Mech., 51 (1984) 299–307.

[25] Gaul, L., Klein, P., Kemple, S., Impulse response function of an oscillator with fractional derivative in damping description, Mech. Res. Commun., 16 (1989) 4447–4472.

[26] Gaul, L., Klein, P., Kemple, S., Damping description involving fractional operators, Mech. Syst. Signal Process. 5(2) (1991) 8–88.

[27] Gorenflo, R., Fractional calculus: some numerical methods, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag, New York, 1997.

[28] Shokooh, A., Suarez, L.E., A comparison of numerical methods applied to a fractional model of damping, J. Vib. Control, 5 (1999) 331–354.

[29] Enelund, M., Olsson, P., Time-domain modeling of damping using anelastic displacement fields and fractional calculus, Int. J. Solids Struct., 36 (1999) 939–970.

[30] Enelund, M., Lesieutre, G.A., Damping described by fading memory-analysis and application to fractional derivative models, Int. J. Solids Struct. 36 (1999) 939–970.

[31] Yuan, L., Agrawal, O.P., A numerical scheme for dynamic systems containing fractional derivatives, J. Vib. Acoust., 124 (2002) 321–324.

[32] Liang, Zu-feng., tang, Xiao-yan., Analytical solution of fractionally damped beam by Adomian decomposition method, Applied Mathematics and Mechanics, 28(2) (2007) 219–228.

[33] Chakraverty, S., Behera, D., Dynamic responses of fractionally damped mechanical system using homotopy perturbation method, Alexandria Engineering Journal, 52 (2013) 557-562.

[34] Behera, D., Chakraverty, S., Numerical solution of fractionally damped beam by homotopy perturbation method, Cent. Eur. J. Phys. 11(6) (2013) 792-798.

[35] Mareishi, S., Kalhori, H., Rafiee, M., Hosseini, S.M., Nonlinear forced vibration response of smart two-phase nano-composite beams to external harmonic excitations, Curved and Layer. Struct. 2 (2015) 150–161.

[36] He, X.Q., Rafiee, M., Mareishi, S., Liew, K.M., Large amplitude vibration of fractionally damped viscoelastic CNTs/fiber/polymer multiscale composite beams, Composite Structures.131 (2015) 1111–1123.

[37] Lewandowski, R., Wielentejczyk, P., Nonlinear vibration of viscoelastic beams described using fractional order derivatives, Journal of Sound and Vibration, 399 (2017) 228–243.

[38] Freundlich, J., Transient vibrations of a fractional Kelvin-Voigt viscoelastic cantilever beam with a tip mass and subjected to a base excitation, Journal of Sound and Vibration, 438 (2019) 99-115.

[39] Liao, S.J., The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University, 1992.

[40] Liao, S.J., Beyond perturbation: an introduction to the homotopy analysis method, Boca Raton: CRC Press, Chapman & Hall, 2003.

[41] Liao, S., Homotopy analysis method: a new analytical technique for nonlinear problems, Commun Nonlinear Sci Numer Simulat, 2(2) (1997) 95–100.

[42] Freundlich, J., dynamic response of a simply supported viscoelastic beam of a fractional derivative type to a moving force load,Journal of theoretical and applied mechanics,54(4) (2016) 1433-1445.