Efficient Higher-Order Shear Deformation Theories for Instability Analysis of Plates Carrying a Mass Moving on an Elliptical Path

Document Type: Research Paper


1 Young Researchers and Elite Club, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Isfahan,Iran

2 Department of Mechanical Engineering, Khomeinishahr Branch, Islamic Azad University, Khomeinishahr, Isfahan, Iran



The dynamic performance of structures under traveling loads should be exactly analyzed to have a safe and reasonable structural design. Different higher-order shear deformation theories are proposed in this paper to analyze the dynamic stability of thick elastic plates carrying a moving mass. The displacement fields of different theories are chosen based upon variations along the thickness as cubic, sinusoidal, hyperbolic and exponential. The well-known Hamilton’s principle is utilized to derive equations of motion and then they are solved using the Galerkin method. The energy-rate method is used as a numerical method to calculate the boundary curves separating the stable and unstable regions in the moving mass parameters plane. Effects of the relative plate thickness, trajectories radii and the Winkler foundation stiffness on the system stability are examined. The results obtained in this research are compared, in a special case, with those of the Kirchhoff’s plate model for the validation.                    


Main Subjects

[1] Karimi A.H., Ziaei-Rad S., 2015, Nonlinear coupled longitudinal–transverse vibration analysis of a beam subjected to a moving mass traveling with variable speed, Archive of Applied Mechanics 85: 1941-1960.
[2] Lv J., Kang H., 2018, Nonlinear dynamic analysis of cable-stayed arches under primary resonance of cables, Archive of Applied Mechanics 88: 1-14.
[3] Babagi P.N., Neya B.N., Dehestani M., 2017, Three dimensional solution of thick rectangular simply supported plates under a moving load, Meccanica 52: 3675-3692.
[4] Ozgan K., Daloglu A.T., Karakaş Aİ., 2013, A parametric study for thick plates resting on elastic foundation with variable soil depth, Archive of Applied Mechanics 83: 549-558.
[5] Rofooei F.R., Enshaeian A., Nikkhoo A., 2017, Dynamic response of geometrically nonlinear, elastic rectangular plates under a moving mass loading by inclusion of all inertial components, Journal of Sound and Vibration 394: 497-514.
[6] Enshaeian A., Rofooei F.R., 2014, Geometrically nonlinear rectangular simply supported plates subjected to a moving mass, Acta Mechanica 225: 595-608.
[7] Wang Y.Q., Zu J.W., 2017, Nonlinear steady-state responses of longitudinally traveling functionally graded material plates in contact with liquid, Composite Structures 164: 130-144.
[8] Wang Y.Q., Zu J.W., 2017, Nonlinear dynamic thermoelastic response of rectangular FGM plates with longitudinal velocity, Composites Part B: Engineering 117: 74-88.
[9] Niaz M., Nikkhoo A., 2015, Inspection of a rectangular plate dynamics under a moving mass with varying velocity utilizing BCOPs, Latin American Journal of Solids and Structures 12: 317-332.
[10] Frýba L., 1999, Vibration of Solids and Structures under Moving Loads, Thomas Telford House, London.
[11] Nikkhoo A., Rofooei F.R., 2012, Parametric study of the dynamic response of thin rectangular plates traversed by a moving mass, Acta Mechanica 223: 15-27.
[12] Esen I., 2013, A new finite element for transverse vibration of rectangular thin plates under a moving mass, Finite Elements in Analysis and Design 66: 26-35.
[13] Nikkhoo A., Hassanabadi M.E., Azam S.E., Amiri J.V., 2014, Vibration of a thin rectangular plate subjected to series of moving inertial loads, Mechanics Research Communications 55: 105-113.
[14] Hassanabadi M.E., Attari N.K.A., Nikkhoo A., 2016, Resonance of a rectangular plate influenced by sequential moving masses, Coupled Systems Mechanics 5: 87-100.
[15] Gbadeyan J.A., Dada M.S., 2006, Dynamic response of a Mindlin elastic rectangular plate under a distributed moving mass, International journal of mechanical sciences 48: 323-340.
[16] Amiri J.V., Nikkhoo A., Davoodi M.R., Hassanabadi M.E., 2013, Vibration analysis of a Mindlin elastic plate under a moving mass excitation by eigenfunction expansion method, Thin-Walled Structures 62: 53-64.
[17] Ansari M., Esmailzadeh E., Younesian D., 2010, Internal-external resonance of beams on non-linear viscoelastic foundation traversed by moving load, Nonlinear Dynamics 61: 163-182.
[18] Rao G.V., 2000, Linear dynamics of an elastic beam under moving loads, Journal of Vibration and Acoustics 122: 281-289.
[19] Nelson H.D., Conover R.A., 1971, Dynamic stability of a beam carrying moving masses, Journal of Applied Mechanics 38: 1003-1006.
[20] Pirmoradian M., Keshmiri M., Karimpour H., 2014, Instability and resonance analysis of a beam subjected to moving mass loading via incremental harmonic balance method, Journal of Vibroengineering 16: 2779-2789.
[21] Pirmoradian M., Keshmiri M., Karimpour H., 2015, On the parametric excitation of a Timoshenko beam due to intermittent passage of moving masses: instability and resonance analysis, Acta Mechanica 226: 1241-1253.
[22] Yang T-Z., Fang B., 2012, Stability in parametric resonance of an axially moving beam constituted by fractional order material, Archive of Applied Mechanics 82: 1763-1770.
[23] Torkan E., Pirmoradian M., Hashemian M., 2018, On the parametric and external resonances of rectangular plates on an elastic foundation traversed by sequential masses, Archive of Applied Mechanics 88: 1411-1428.
[24] Torkan E., Pirmoradian M., Hashemian M., 2017, Occurrence of parametric resonance in vibrations of rectangular plates resting on elastic foundation under passage of continuous series of moving masses, Modares Mechanical Engineering 17: 225-236.
[25] Pirmoradian M., Torkan E., Karimpour H., 2018, Parametric resonance analysis of rectangular plates subjected to moving inertial loads via IHB method, International Journal of Mechanical Sciences 142: 191-215.
[26] Karimpour H., Pirmoradian M., Keshmiri M., 2016, Instance of hidden instability traps in intermittent transition of moving masses along a flexible beam, Acta Mechanica 227: 1213-1224.
[27] Reddy J.N., 1984, A simple higher-order theory for laminated composite plates, Journal of Applied Mechanics 51: 745-752.
[28] Shi G., 2007, A new simple third-order shear deformation theory of plates, International Journal of Solids and Structures 44: 4399-4417.
[29] Touratier M., 1991, An efficient standard plate theory, International Journal of Engineering Science 29: 901-916.
[30] Karama M., Afaq K.S., Mistou S., 2003, Mechanical behaviour of laminated composite beam by the new multi-layered laminated composite structures model with transverse shear stress continuity, International Journal of Solids and Structures 40: 1525-1546.
[31] El Meiche N., Tounsi A., Ziane N., Mechab I., 2011, A new hyperbolic shear deformation theory for buckling and vibration of functionally graded sandwich plate, International Journal of Mechanical Sciences 53: 237-247.
[32] Torkan E., Pirmoradian M., Hashemian M., 2019, Instability inspection of parametric vibrating rectangular mindlin plates lying on Winkler foundations under periodic loading of moving masses, Acta Mechanica Sinica 35: 242-263.
[33] Jazar G.N., 2004, Stability chart of parametric vibrating systems using energy-rate method, International Journal of Non-Linear Mechanics 39: 1319-1331.
[34] Torkan E., Pirmoradian M., Hashemian M., 2019, Dynamic instability analysis of moderately thick rectangular plates influenced by an orbiting mass based on the first-order shear deformation theory, Modares Mechanical Engineering 19: 2203-2213.
[35] Pirmoradian M., Karimpour H., 2017, Parametric resonance and jump analysis of a beam subjected to periodic mass transition, Nonlinear Dynamics 89: 2141-2154.
[36] Pirmoradian M., Karimpour H., 2017, Nonlinear effects on parametric resonance of a beam subjected to periodic mass transition, Modares Mechanical Engineering 17: 284-292.