Dynamic behaviour of functionally graded Timoshenko beams on a four parameter linear elastic foundation due to a high speed travelling mass with variable velocities

Year 2021, Volume: 2 Issue: 1, 48 - 75, 23.06.2021

İsmail Esen Mehmet Akif Koç Mustafa Eroğlu

Abstract

This study presents a four-parameter linear basis model to analyse and control the dynamic response of an FGM Timoshenko beam exposed to the accelerating / decelerating mass using the finite element method. The dynamic effects of the foundation's mass and damping are taken into account and the foundation is assumed to consist of four parts: mass, spring, viscous damper and shear layer. Considering the actual physical neutral axis, the combined motion equations of the FGM beam-mass-base-base system are obtained by combining terms of first order shear deformation (FSDT) and mass and base interactions. In view of the resulting high-speed motion and acceleration conditions of the moving mass, some new findings are presented for both the moving load and the moving mass assumptions to highlight the differences that may be useful in the analysis of new high-speed transport applications today. Due to their effects, the frequency change of the FGM Timoshenko beam-base system is emphasized to show the main cause of the changes.

Keywords

FGM beams, mass inertia, high-speed moving mass, four-parameter linear foundation, finite element

References

• [1]M.A. Koç, İ. Esen, Modelling and analysis of vehicle-structure-road coupled interaction considering structural flexibility , vehicle parameters and road roughness †, J. Mech. Sci. Technol. 31 (2017) 1–18. doi:10.1007/s12206-017-0913-y.
• [2]M.A. Koc, I. Esen, H. Dal, Solution of the Moving Oscillator Problem using the Newmark ’ s β Method, Int. J. Innov. Res. Sci. Eng. Technol. 5 (2016) 16–23.
• [3]M.A. Koç, Finite Element and Numerical Vibration analysis of a Timoshenko and Euler-Bernoulli beams traversed by a moving high-speed train, J. Brazilian Soc. Mech. Sci. Eng. 7 (2021). doi:10.1007/s40430-021-02835-7.
• [4]C. Mızrak, İ. Esen, The optimisation of rail vehicle bogie parameters with the fuzzy logic method in order to improve passenger comfort during passage over bridges, Int. J. Heavy Veh. Syst. 24 (2017) 113. doi:10.1504/IJHVS.2017.083057.
• [5]İ. Esen, M.A. Koç, Dynamics of 35 mm anti-aircraft cannon barrel durig firing, in: Int. Symp. Comput. Sci. Eng., Aydın, 2013: pp. 252–257.
• [6]İ. Esen, M.A. Koç, Optimization of a passive vibration absorber for a barrel using the genetic algorithm, Expert Syst. Appl. 42 (2015) 894–905. doi:10.1016/j.eswa.2014.08.038.
• [7]M.A. Koç, İ. Esen, Y. Çay, Tip deflection determination of a barrel for the effect of an accelerating projectile before firing using finite element and artificial neural network combined algorithm, Lat. Am. J. Solids Struct. 13 (2016) 1968–1995. doi:http://dx.doi.org/10.1590/1679-78252718.
• [8]M.A. Koç, I. Esen, Y. Çay, Dynamic analysis of gun barrel vibrations due to effect of an unbalanced projectile considering 2-D transverse displacements of barrel tip using a 3-D element technique, Lat. Am. J. Solids Struct. 15 (2018). doi:10.1590/1679-78254972.
• [9]S.J. Singh, S.P. Harsha, Nonlinear dynamic analysis of sandwich S-FGM plate resting on pasternak foundation under thermal environment, Eur. J. Mech. -A/Solids. 76 (2019) 155–179. doi:https://doi.org/10.1016/j.euromechsol.2019.04.005.
• [10]Q. Li, Q. Wang, D. Wu, X. Chen, Y. Yu, W. Gao, Geometrically nonlinear dynamic analysis of organic solar cell resting on Winkler-Pasternak elastic foundation under thermal environment, Compos. Part B Eng. 163 (2019) 121–129. doi:https://doi.org/10.1016/j.compositesb.2018.11.022.
• [11]M.T.A. Robinson, S. Adali, Buckling of nonuniform and axially functionally graded nonlocal Timoshenko nanobeams on Winkler-Pasternak foundation, Compos. Struct. 206 (2018) 95–103. doi:https://doi.org/10.1016/j.compstruct.2018.07.046.
• [12]C.K. Rao, L.B. Rao, Torsional post-buckling of thin-walled open section clamped beam supported on Winkler-Pasternak foundation, Thin-Walled Struct. 116 (2017) 320–325. doi:https://doi.org/10.1016/j.tws.2017.03.017.
• [13]D. Shahsavari, M. Shahsavari, L. Li, B. Karami, A novel quasi-3D hyperbolic theory for free vibration of FG plates with porosities resting on Winkler/Pasternak/Kerr foundation, Aerosp. Sci. Technol. 72 (2018) 134–149. doi:https://doi.org/10.1016/j.ast.2017.11.004.
• [14]Q. Li, D. Wu, X. Chen, L. Liu, Y. Yu, W. Gao, Nonlinear vibration and dynamic buckling analyses of sandwich functionally graded porous plate with graphene platelet reinforcement resting on Winkler–Pasternak elastic foundation, Int. J. Mech. Sci. 148 (2018) 596–610. doi:https://doi.org/10.1016/j.ijmecsci.2018.09.020.
• [15]F. Najafi, M.H. Shojaeefard, H.S. Googarchin, Nonlinear dynamic response of FGM beams with Winkler–Pasternak foundation subject tononcentral low velocity impact in thermal field, Compos. Struct. 167 (2017) 132–143. doi:https://doi.org/10.1016/j.compstruct.2017.01.063.
• [16]K. Gao, W. Gao, D. Wu, C. Song, Nonlinear dynamic stability of the orthotropic functionally graded cylindrical shell surrounded by Winkler-Pasternak elastic foundation subjected to a linearly increasing load, J. Sound Vib. 415 (2018) 147–168. doi:https://doi.org/10.1016/j.jsv.2017.11.038.
• [17]A. Deniz, Z. Zerin, Z. Karaca, Winkler-Pasternak foundation effect on the frequency parameter of FGM truncated conical shells in the framework of shear deformation theory, Compos. Part B Eng. 104 (2016) 57–70. doi:https://doi.org/10.1016/j.compositesb.2016.08.006.
• [18]A.M. Zenkour, A.F. Radwan, Compressive study of functionally graded plates resting on Winkler–Pasternak foundations under various boundary conditions using hyperbolic shear deformation theory, Arch. Civ. Mech. Eng. 18 (2018) 645–658. doi:https://doi.org/10.1016/j.acme.2017.10.003.
• [19]B. Mechab, I. Mechab, S. Benaissa, M. Ameri, B. Serier, Probabilistic analysis of effect of the porosities in functionally graded material nanoplate resting on Winkler–Pasternak elastic foundations, Appl. Math. Model. 40 (2016) 738–749. doi:https://doi.org/10.1016/j.apm.2015.09.093.
• [20]J.K. Lee, S. Jeong, J. Lee, Natural frequencies for flexural and torsional vibrations of beams on Pasternak foundation, Soils Found. 54 (2014) 1202–1211. doi:https://doi.org/10.1016/j.sandf.2014.11.013.
• [21]B. Akgöz, Ö. Civalek, Vibrational characteristics of embedded microbeams lying on a two-parameter elastic foundation in thermal environment, Compos. Part B Eng. 150 (2018) 68–77. doi:https://doi.org/10.1016/j.compositesb.2018.05.049.
• [22]G. Chen, Z. Meng, D. Yang, Exact nonstationary responses ofrectangular thin plate on Pasternak foundation excited by stochastic moving loads, J. Sound Vib. 412 (2018) 166–183. doi:https://doi.org/10.1016/j.jsv.2017.09.022.
• [23]N.D. Duc, J. Lee, T. Nguyen-Thoi, P.T. Thang, Static response and free vibration of functionally graded carbon nanotube-reinforced composite rectangular plates resting on Winkler–Pasternak elastic foundations, Aerosp. Sci. Technol. 68 (2017) 391–402. doi:https://doi.org/10.1016/j.ast.2017.05.032.
• [24]R.T. Corrêa, A.P. da Costa, F.M.F. Simões, Finite element modeling of a rail resting on a Winkler-Coulomb foundation and subjected to a moving concentrated load, Int. J. Mech. Sci. 140 (2018) 432–445. doi:https://doi.org/10.1016/j.ijmecsci.2018.03.022.
• [25]M.M. Keleshteri, H. Asadi, M.M. Aghdam, Nonlinear bending analysis of FG-CNTRC annular plates with variable thickness on elastic foundation, Thin-Walled Struct. 135 (2019) 453–462. doi:https://doi.org/10.1016/j.tws.2018.11.020.
• [26]Y. Fan, Y. Xiang, H.-S. Shen, Nonlinear forced vibration of FG-GRC laminated plates resting on visco-Pasternak foundations, Compos. Struct. 209 (2019) 443–452. doi:https://doi.org/10.1016/j.compstruct.2018.10.084.
• [27]F. Najafi, M.H. Shojaeefard, H.S. Googarchin, Nonlinear low-velocity impact response of functionally graded plate with nonlinear three-parameter elastic foundation in thermal field, Compos. Part B Eng. 107 (2016) 123–140. doi:https://doi.org/10.1016/j.compositesb.2016.09.070.
• [28]R. Benferhat, T.H. Daouadji, M.S. Mansour, Free vibration analysis of FG plates resting on an elastic foundation and based on the neutral surface concept using higher-order shear deformation theory, Comptes Rendus Mécanique. 344 (2016) 631–641. doi:https://doi.org/10.1016/j.crme.2016.03.002.
• [29]E. Bahmyari, M.R. Khedmati, Vibration analysis of nonhomogeneous moderately thick plates with point supports resting on Pasternak elastic foundation using element free Galerkin method, Eng. Anal. Bound. Elem. 37 (2013) 1212–1238. doi:https://doi.org/10.1016/j.enganabound.2013.05.003.
• [30]R. Pavlovic, I. Pavlovic, Dynamic stability of Timoshenko beams on Pasternak viscoelastic foundation, Theor. Appl. Mech. i Primenj. Meh. 45 (2018) 67–81. doi:10.2298/tam171103005p.
• [31]H. Ding, K.L. Shi, L.Q. Chen, S.P. Yang, Dynamic response of an infinite Timoshenko beam on a nonlinear viscoelastic foundation to a moving load, Nonlinear Dyn. 73 (2013) 285–298. doi:10.1007/s11071-013-0784-0.
• [32]M.K. Ahmed, Natural frequencies and mode shapes of variable thickness elastic cylindrical shells resting on a Pasternak foundation, J. Vib. Control. 22 (2016) 37–50. doi:10.1177/1077546314528229.
• [33]A.H. Sofiyev, On the buckling of composite conical shells resting on the Winkler–Pasternak elastic foundations under combined axial compression and external pressure, Compos. Struct. 113 (2014) 208–215. doi:10.1016/j.compstruct.2014.03.023.
• [34]W.H. Lee, S.C. Han, W.T. Park, A refined higher order shear and normal deformation theory for E-, P-, and S-FGM plates on Pasternak elastic foundation, Compos. Struct. 122 (2015) 330–342. doi:10.1016/j.compstruct.2014.11.047.
• [35]H. Matsunaga, Vibration and buckling of deep beam-columns on two-parameter elastic foundations, J. Sound Vib. 228 (1999) 359–376. doi:10.1006/jsvi.1999.2415.
• [36]N.R. Naidu, G. V. Rao, Vibrations of initially stressed uniform beams on a two-parameter elastic foundation, Comput. Struct. (1995). doi:10.1016/0045-7949(95)00090-4.
• [37]H.P. Lee, Dynamic Response of a Timoshenko Beam on a Winkler Foundation Subjected to a Moving Mass, Appl. Acoust. 55 (1998) 203–215. doi:10.1016/S0003-682X(97)00097-2.
• [38]S.C. Dutta, R. Roy, A critical review on idealization and modeling for interaction among soil-foundation-structure system, Comput. Struct. 80 (2002) 1579–1594. doi:10.1016/S0045-7949(02)00115-3.
• [39]A.D. Senalp, A. Arikoglu, I. Ozkol, V.Z. Dogan, Dynamic response of a finite length euler-bernoulli beam on linear and nonlinear viscoelastic foundations to a concentrated moving force, J. Mech. Sci. Technol. 24 (2010) 1957–1961. doi:10.1007/s12206-010-0704-x.
• [40]M.H. Kargarnovin, D. Younesian, D.J. Thompson, C.J.C. Jones, Response of beams on nonlinear viscoelastic foundations to harmonic moving loads, Comput. Struct. 83 (2005) 1865–1877. doi:10.1016/j.compstruc.2005.03.003.
• [41]W. Luo, Y. Xia, S. Weng, Vibration of Timoshenko beam on hysteretically damped elastic foundation subjected to moving load, Sci. China Physics, Mech. Astron. 58 (2015) 84601. doi:10.1007/s11433-015-5664-9.
• [42]M. Seguini, D. Nedjar, Nonlinear Analysis of Deep Beam Resting on Linear and Nonlinear Random Soil, Arab. J. Sci. Eng. 42 (2017) 3875–3893. doi:10.1007/s13369-017-2449-7.
• [43]İ. Esen, M.A. Koç, Y. Çay, Finite element formulation and analysis of a functionally graded timoshenko beam subjected to an acceleratingmass including inertial effects of the mass, Lat. Am. J. Solids Struct. 15 (2018) 1–18. doi:10.1590/1679-78255102.
• [44]Y.H. Chen, Y.H. Huang, C.T. Shih, Response of an infinite tomoshenko beam on a viscoelastic foundation to a harmonic moving load, J. Sound Vib. 241 (2001) 809–824. doi:10.1006/jsvi.2000.3333.
• [45]S.C. Mohanty, R.R. Dash, T. Rout, Static and dynamic analysis of a functionally graded Timoshenko beam on Winkler’s elastic foundation, J. Eng. Res. Stud. 1 (2010) 149–165.
• [46]S.C. Mohanty, R.R. Dash, T. Rout, Parametric instability of a functionally graded Timoshenko beam on Winkler’s elastic foundation, Nucl. Eng. Des. 241 (2011) 2698–2715. doi:10.1016/j.nucengdes.2011.05.040.
• [47]S.-H. Ju, H.-T. Lin, A finite element model of vehicle–bridgeinteraction considering braking and acceleration, J. Sound Vib. 303 (2007) 46–57. doi:https://doi.org/10.1016/j.jsv.2006.11.034.
• [48]P.O. Azimi H, Galal K, A numerical element for vehicle-bridge interaction analysis of vehicles experiencing sudden deceleration., Eng Struct. 49 (2013) 792–805.
• [49]D.K. NGUYEN, B.S. GAN, T.H. LE, Dynamic Response of Non-Uniform Functionally Graded Beams Subjected to a Variable Speed Moving Load, J. Comput. Sci. Technol. (2013). doi:10.1299/jcst.7.12.
• [50]M. Şimşek, T. Kocatürk, Free and forced vibration of a functionally graded beam subjected to a concentrated moving harmonic load, Compos. Struct. 90 (2009) 465–473. doi:10.1016/j.compstruct.2009.04.024.
• [51]S.M.R. Khalili, A.A. Jafari, S.A. Eftekhari, A mixed Ritz-DQ method for forced vibration of functionally graded beams carrying moving loads, Compos. Struct. 92 (2010) 2497–2511. doi:https://doi.org/10.1016/j.compstruct.2010.02.012.
• [52]S.R. Mohebpour, P. Malekzadeh, A.A. Ahmadzadeh, Dynamic analysis of laminated composite plates subjected to a moving oscillator by FEM, Compos. Struct. (2011). doi:10.1016/j.compstruct.2011.01.003.
• [53]J.-J. Wu, Transverse and longitudinal vibrations of a frame structure due to a moving trolley and the hoisted object using moving finite element, Int. J. Mech. Sci. 50 (2008) 613–625. doi:https://doi.org/10.1016/j.ijmecsci.2008.02.001.
• [54]I. Esen, A new finite element for transverse vibration of rectangular thin plates under a moving mass, Finite Elem. Anal. Des. 66 (2013) 26–35. doi:10.1016/j.finel.2012.11.005.
• [55]G.T. Michaltsos, A.N. Kounadis, The Effects of Centripetal and Coriolis Forces on the Dynamic Response of Light Bridges Under Moving Loads, J. Vib. Control. 7 (2001) 315–326. doi:10.1177/107754630100700301.
• [56]İ. Esen, A new FEM procedure for transverse and longitudinal vibration analysis of thin rectangular plates subjected to a variable velocity moving load along an arbitrary trajectory, Lat. Am. J. Solids Struct. 12 (2015) 808–830.
• [57]I. Esen, A modified FEM for transverse and lateral vibration analysis of thin beams under a mass moving with a variable acceleration, Lat. Am. J. Solids Struct. 14 (2017). doi:10.1590/1679-78253180
• [58]H.P. Lee, Transverse vibration of a Timoshenko beam acted on by an accelerating mass,Appl. Acoust. 47 (1996) 319–330. doi:10.1016/0003-682X(95)00067-J.
• [59]J.-H. Kim, G.H. Paulino, Finite element evaluation of mixed mode stress intensity factors in functionally graded materials, Int. J. Numer. Methods Eng. 53 (2002) 1903–1935. doi:10.1002/nme.364.
• [60]P.J.W. Markworth AJ, Ramesh KS, Modelling studies applied to functionally graded materials, J. Mater. Sci. 30 (1995) 2183–2193.
• [61]N.M. Wakashima K, Hirano T, Space applications of advanced structural materials, ESA, 1990.
• [62]M.F. Eltaher MA, Alshorbagy AE, Determination of neutral axis position and its effect on natural frequencies of functionally graded macro/nanobeams, Compos Struct. 99 (2013) 193–201.
• [63]D.-G. Zhang, Y.-H. Zhou, A theoretical analysis of FGM thin plates based on physical neutral surface, Comput. Mater. Sci. 44 (2008) 716–720. doi:10.1016/J.COMMATSCI.2008.05.016.
• [64]J. Kosmatka, An improve two-node finite element for stability and natural frequencies of axial-loaded Timoshenko beams., Comput. Struct. 57 (1995)141–149.
• [65]A. Chakraborty, S. Gopalakrishnan, J.N. Reddy, A new beam finite element for the analysis of functionally graded materials, Int. J. Mech. Sci. (2003). doi:10.1016/S0020-7403(03)00058-4.
• [66]S.A. Sina, H.M. Navazi, H. Haddadpour, An analytical method for free vibration analysis of functionally graded beams, Mater. Des. 30 (2009) 741–747. doi:10.1016/J.MATDES.2008.05.015.
• [67]C. Zhang, A. Savaidis, G. Savaidis, H. Zhu, Transient dynamic analysis of a cracked functionally graded material by a BIEM, Comput. Mater. Sci. 26 (2003) 167–174. doi:https://doi.org/10.1016/S0927-0256(02)00395-6.
• [68]L. Fryba, Vibration solids and structures under moving loads, Thomas Telford House, 1999.