References
- Abu-Mallouh, R., I. Abu-Alshaikh, H. S. Zibdeh, and K. Ramadan. 2012. Response of fractionally damped beams with general boundary conditions subjected to moving loads. Shock and Vibration 19 (3):333–47. doi:https://doi.org/10.1155/2012/321421.
- Adolfsson, K., M. Enelund, and P. Olsson. 2005. On the fractional order model of viscoelasticity. Mechanics of Time-Dependent Materials 9 (1):15–34. doi:https://doi.org/10.1007/s11043-005-3442-1.
- Agrawal, O. P. 2004. Analytical solution for stochastic response of a fractionally damped beam. Journal of Vibration and Acoustics 126 (4):561–6. doi:https://doi.org/10.1115/1.1805003.
- Bagley, R. 2007. On the equivalence of the Riemann–Liouville and the Caputo fractional order derivatives in modeling of linear viscoelastic materials. Fractional Calculus and Applied Analysis 10 (2):123–6.
- Bagley, R. L., and P. J. Torvik. 1983. A theoretical basis for the application of fractional calculus to viscoelasticity. Journal of Rheology 27 (3):201–10. doi:https://doi.org/10.1122/1.549724.
- Bagley, R. L., and P. J. Torvik. 1985. Fractional calculus in the transient analysis of viscoelastically damped structures. AIAA Journal 23 (6):918–25. doi:https://doi.org/10.2514/3.9007.
- Banerjee, S., S. Shaw, and B. Mukhopadhyay. 2019. Memory response on thermoelastic deformation in a solid half-space with a cylindrical hole. Mechanics Based Design of Structures and Machines 1–23. doi:https://doi.org/10.1080/15397734.2019.1686989.
- Caputo, M., and F. Mainardi. 1971. A new dissipation model based on memory mechanism. Pure and Applied Geophysics (Pageoph) 91 (1):134–47. doi:https://doi.org/10.1007/BF00879562.
- Datta, N., and A. W. Troesch. 2012. Dynamic response of Kirchhoff’s plates to transient hydrodynamic impact loads. Marine Systems & Ocean Technology 7 (2):79–94. doi:https://doi.org/10.1007/BF03449302.
- Ding, H., L.-Q. Chen, and S.-P. Yang. 2012. Convergence of Galerkin truncation for dynamic response of finite beams on nonlinear foundations under a moving load. Journal of Sound and Vibration 331 (10):2426–42. doi:https://doi.org/10.1016/j.jsv.2011.12.036.
- Eldred, L. B., W. P. Baker, and A. N. Palazotto. 1995. Kelvin–Voigt versus fractional derivative model as constitutive relations for viscoelastic materials. AIAA Journal 33 (3):547–50. doi:https://doi.org/10.2514/3.12471.
- Enelund, M., and P. Olsson. 1999. Damping described by fading memory—analysis and application to fractional derivative models. International Journal of Solids and Structures 36 (7):939–70. doi:https://doi.org/10.1016/S0020-7683(97)00339-9.
- Foda, M. A., and Z. Abduljabbar. 1998. A dynamic green function formulation for the response of a beam structure to a moving mass. Journal of Sound and Vibration 210 (3):295–306. doi:https://doi.org/10.1006/jsvi.1997.1334.
- Freundlich, J. 2013. Vibrations of a simply supported beam with a fractional viscoelastic material model–supports movement excitation. Shock and Vibration 20 (6):1103–12. doi:https://doi.org/10.1155/2013/126735.
- Freundlich, J. 2019. 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 438:99–115. doi:https://doi.org/10.1016/j.jsv.2018.09.006.
- Freundlich, J. K. 2016. Dynamic response of a simply supported viscoelastic beam of a fractional derivative type to a moving force load. Journal of Theoretical and Applied Mechanics 1433. doi:https://doi.org/10.15632/jtam-pl.54.4.1433.
- Frýba, L. 1972. Vibration of Solids and Structures under Moving Loads. Groningen, The Netherlands: Noordhoff International Publishing.
- Ghayesh, M. H., and M. Amabili. 2013. Parametric stability and bifurcations of axially moving viscoelastic beams with time-dependent axial speed. Mechanics Based Design of Structures and Machines 41 (3):359–81. doi:https://doi.org/10.1080/15397734.2013.771093.
- Gürgöze, M. 1987. Parametric vibrations of a viscoelastic beam (Maxwell model) under steady axial load and transverse displacement excitation at one end. Journal of Sound and Vibration 115 (2):329–38. doi:https://doi.org/10.1016/0022-460X(87)90476-7.
- Hedrih, K. S. 2006. The transversal creeping vibrations of a fractional derivative order constitutive relation of nonhomogeneous beam. Mathematical Problems in Engineering 2006:1–18. doi:https://doi.org/10.1155/MPE/2006/46236.
- Hilal, M. A., and H. S. Zibdeh. 2000. Vibration analysis of beams with general boundary conditions traversed by a moving force. Journal of Sound and Vibration 229 (2):377–88. doi:https://doi.org/10.1006/jsvi.1999.2491.
- Hosseinkhani, A., D. Younesian, R. Shakeri, and S. Farhangdoust. 2019. Vibro-acoustic response analysis of fractional railpads in frequency domain. Mechanics Based Design of Structures and Machines 1–18. doi:https://doi.org/10.1080/15397734.2019.1688169.
- Lewandowski, R., and B. Chorążyczewski. 2010. Identification of the parameters of the Kelvin–Voigt and the Maxwell Fractional models, used to modeling of viscoelastic dampers. Computers & Structures 88 (1–2):1–17. doi:https://doi.org/10.1016/j.compstruc.2009.09.001.
- Liang, Z-f, and X-y Tang. 2007. Analytical solution of fractionally damped beam by Adomian decomposition method. Applied Mathematics and Mechanics 28 (2):219–28. doi:https://doi.org/10.1007/s10483-007-0210-z.
- Lorenzo, S. D., F. P. Pinnola, and A. Pirrotta. 2012. On the dynamics of fractional visco-elastic beams. Paper presented at the ASME 2012 International Mechanical Engineering Congress and Exposition, 1273–81. American Society of Mechanical Engineers. doi:https://doi.org/10.1115/IMECE2012-86566.
- Mahmoodi, S. N., S. E. Khadem, and M. Kokabi. 2007. Non-linear free vibrations of Kelvin–Voigt visco-elastic beams. International Journal of Mechanical Sciences 49 (6):722–32. doi:https://doi.org/10.1016/j.ijmecsci.2006.10.005.
- Mainardi, F. 2010. Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models. Singapore: World Scientific.
- Mainardi, F., and G. Spada. 2011. Creep, relaxation and viscosity properties for basic fractional models in rheology. The European Physical Journal Special Topics 193 (1):133–60. doi:https://doi.org/10.1140/epjst/e2011-01387-1.
- Mallik, A. K., S. Chandra, and A. B. Singh. 2006. Steady-state response of an elastically supported infinite beam to a moving load. Journal of Sound and Vibration 291 (3-5):1148–69. doi:https://doi.org/10.1016/j.jsv.2005.07.031.
- Marynowski, K., and T. Kapitaniak. 2007. Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension. International Journal of Non-Linear Mechanics 42 (1):118–31. doi:https://doi.org/10.1016/j.ijnonlinmec.2006.09.006.
- Mondal, S., A. Sur, and M. Kanoria. 2019. Transient heating within skin tissue due to time-dependent thermal therapy in the context of memory dependent heat transport law. Mechanics Based Design of Structures and Machines 1–15. doi:https://doi.org/10.1080/15397734.2019.1686992.
- Paola, M. D., R. Heuer, and A. Pirrotta. 2013. Fractional visco-elastic Euler–Bernoulli beam. International Journal of Solids and Structures 50 (22–23):3505–10. doi:https://doi.org/10.1016/j.ijsolstr.2013.06.010.
- Paola, M. D., A. Pirrotta, and A. Valenza. 2011. Visco-elastic behavior through fractional calculus: an easier method for best fitting experimental results. Mechanics of Materials 43 (12):799–806. doi:https://doi.org/10.1016/j.mechmat.2011.08.016.
- Podlubny, I. 1999. Fractional Differential Equations, Vol. 198 of Mathematics in Science and Engineering. San Diego, CA: Academic Press .
- Podlubny, I. 2000. Matrix approach to discrete fractional calculus. Fractional Calculus and Applied Analysis 3 (4):359–86.
- Podlubny, I., A. Chechkin, T. Skovranek, Y. Chen, and B. M. Vinagre Jara. 2009. Matrix approach to discrete fractional calculus II: partial fractional differential equations. Journal of Computational Physics 228 (8):3137–53. doi:https://doi.org/10.1016/j.jcp.2009.01.014.
- Rao, S. S. 2017. Mechanical Vibrations in SI Units. Pearson Higher ed. London.
- Rossikhin, Y. A., and M. V. Shitikova. 2001. A new method for solving dynamic problems of fractional derivative viscoelasticity. International Journal of Engineering Science 39 (2):149–76. doi:https://doi.org/10.1016/S0020-7225(00)00025-2.
- Rossikhin, Y. A., and M. V. Shitikova. 2010. Application of Fractional Calculus for Dynamic Problems of Solid Mechanics: Novel Trends and Recent Results. Applied Mechanics Reviews 63 (1):10801. doi:https://doi.org/10.1115/1.4000563.
- Thomson, W. 2018. Theory of Vibration with Applications. Boca Raton, FL: CRC Press.
- Wang, X., X. Liang, and C. Jin. 2017. Accurate dynamic analysis of functionally graded beams under a moving point load. Mechanics Based Design of Structures and Machines 45 (1):76–91. doi:https://doi.org/10.1080/15397734.2016.1145060.
- Zibdeh, H. S., and R. Rachwitz. 1996. Moving loads on beams with general boundary conditions. Journal of Sound and Vibration 195 (1):85–102. doi:https://doi.org/10.1006/jsvi.1996.0405.