Natural frequencies and internal resonance of beams with arbitrarily distributed axial loads

Document Type : Special Issue Paper


Department of Civil and Building Engineering and Architecture, Polytechnic University of Marche, 60131 Ancona, Italy


An exact analytical solution for transversal free vibrations of a beam subjected to an arbitrary distributed axial load and a tip tension is obtained by means of a power series representation, whose coefficients are determined recursively in an easy way. The dependence on the natural frequencies on the load is then investigated, and the buckling load (corresponding to vanishing frequency) is also discussed. Next, the 1:3 internal resonance between the first and the second mode is deeply studied, and an interesting (and unexpected) property is found for linearly distributed axial loads


Main Subjects

[1] Patel, M.H., Seyed, F.B., Review of flexible riser modelling and analysis techniques, Eng. Structures, 17(4), 1995, 293-304.
[2] Franzini, G.R., Santos, C.C.P., Mazzilli, C.E.N., Pesce, C.P., Parametric excitation of an immersed, vertical and slender beam using reduced-order models: influence of hydrodynamic coefficients, Marine Systems and Ocean Technology, 11(1-2), 2016, 10-18.
[3] Franzini, G.R., Mazzilli, C.E.N., Non-linear reduced-order model for parametric excitation analysis of an immersed vertical slender rod, Int. J. NonLinear Mech., 80(1), 2016, 29-39.
[4] Vernizzi, G.J., Franzini, G.R., Lenci, S., Reduced-order models for the analysis of a vertical rod under parametric excitation, Int. J. Mech. Sciences, 163, 2019, 105122.
[5] Latalski, J., Warminski, J., Nonlinear vibrations of a rotating thin-walled composite piezo-beam with circumferentially uniform stiffness (CUS), Nonlinear Dynamics, 98(4), 2019, 2509-2529.
[6] Sterken, P., A guide for tree-stability analysis. No publisher, 2005. ISBN: 9090193774.
[7] Abdel-Gawwad, A.K., El-Kady, H.M., Shalaby, A.M., Three Dimensional Dynamic Analysis Of Ancient Egyptian Obelisks. To Investigate Their Behavior Under Aqaba Earthquake, J. Emerging Trends in Engineering and Applied Sciences, 2(2), 2011, 266-272.
[8] Virgin, L., Vibration of axially loaded structures, Cambridge University Press, 2007. ISBN-13 978-0-521-88042-8.
[9] Shaker, F.J., Effect of axial load on mode shapes and frequencies of beams, NASA Lewis Research Centre Report NASA-TN-8109, 1975.
[10] Timoshenko, S., Vibration problems in engineering, Nostrand, 1955.
[11] Rao, S.S., Vibrations of continuous systems, John Wiley & Sons, Hoboken, New Jersey, 2007. ISBN-13: 978-0-471-77171-5
[12] Bokaian, A., Natural frequencies of beams under compressive axial load, J. Sound Vibr., 126(1), 1988, 49-65.
[13] Bayon, A., Gascon, F., Medina, R., Nieves, F.J., Salazar, F.J., On the flexural vibration of cylinders under axial loads: Numerical and experimental study, J. Sound Vibr., 331(10), 2012, 2315-2333.
[14] Carpinteri, A., Malvano, R., Manuello, A., Piana, G., Fundamental frequency evolution in slender beams subjected to imposed axial displacements, J. Sound Vibr., 333, 2014, 2390-2403.
[15] Love, A.E.H., A treatise on the mathematical theory of elasticity, Dover Publication, 1944. ISBN 0-486-60174-9
[16] Villaggio, P., Mathematical models for elastic structures, Cambridge University Press, Pisa, 1997. ISBN: 0-521-57324-6
[17] Greenhill, A.G., Determination of the greatest height consistent with stability that a vertical pole or mast can be made, and of the greatest height to which a tree of given proportion can grow, Cambridge Phil. Soc. Proc., 4, 1881, 65-73.
[18] Laird, W.M., Fauconneau, G., Upper and lower bounds for the eigenvalues of vibrating beams with linearly varying axial load. NASA-Cr-653, 1966.
[19] Pilkington, D.F, Carr, J.B., Vibration of beams subjected to end and axially distributed loading, J. Mech. Eng. Sci., 12, 1970, 70-72.
[20] Kim, Y.C., Natural frequencies and critical buckling loads of marine risers, J. Offshore Mech. Arct. Eng., 110(1), 1988, 2-8.
[21] Sparks, C.P., Transverse modal vibrations of vertical tensioned risers. A simplified analytical approach, Oil & Gas Science and Technology, 57(1), 2002, 71-86.
[22] Senjanovic, I., Ljusutina, A.M., Parunov, J., Natural vibration analysis of tensioned risers by segmentation method, Oil & Gas Science and Technology, 61(5), 2006, 647-659.
[23] Mazzilli, C.E.N., Lenci, S., Demeio, L., Non-linear free vibrations of tensioned vertical risers, in Ecker H., Steindl A., Jakubek S. (Eds) ENOC Proceeding of the 8th European Nonlinear Dynamics Conference, Vienna, Austria, 6-11 July 2014, 2014.
[24] Alfosail, F.K., Nayfeh, A.H., Younis, M.I., A state space approach for the eigenvalue problem of marine risers, Meccanica, 53, 2018, 747-757.
[25] Huang, T., Dareing, D.W., Buckling and lateral vibration of drill pipe, ASME J. Eng. for Industry, 90(4), 1968, 613-619.
[26] Huang, T., Dareing, D.W., Bucking and frequencies of long vertical pipes, ASCE J. Eng. Mechanics Division, 95(1), 1969, 167-181.
[27] Paidoussis, M., Dynamics of flexible slender cylinders in axial flow. Part 1. Theory, J. Fluid Mech., 26, 1966, 717–736.
[28] Bajaj, A.K., Davies, P., Chang, S.I., On internal resonances in mechanical systems, Chapter 3 of “Nonlinear Dynamics and Stochastic Mechanics”, W. Kliemannin (Ed.), CRC Press, 1995 (e-version 2018). e-ISBN: 9781351075053.
[29] Manevich, A.I., Manevitch, L.I., Mechanics of nonlinear systems with internal resonances, Imperial College Press, 2003. ISBN: 978-1-86094-682-0.
[30] Clementi, F., Lenci, S., Rega, G., 1:1 internal resonance in a two d.o.f. complete system: a comprehensive analysis and its possible exploitation for design, Meccanica, 55, 2020, 1309-1332.
[31] Arioli, G., Gazzola, F., A new mathematical explanation of what triggered the catastrophic torsional mode of the Tacoma Narrows Bridge, Applied Mathematical Modelling, 39, 2015, 901–912.
[32] Nayfeh, A.H., Mook, D., Nonlinear oscillations, Wiley, 1979.