Vibration Analysis of Multi-Step Bernoulli-Euler and Timoshenko Beams Carrying Concentrated Masses

Subject Areas : Engineering

K Torabi ¹ , H Afshari ² , H Najafi ³

1 - Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan
2 - Department of Solid Mechanics, Faculty of Mechanical Engineering, University of Kashan
3 - Department of Solid Mechanics, Faculty of Mechanical Engineering, Politecnico di Milano, Milan, Italy

Received: 2013-05-20 Accepted : 2013-08-18 Published : 2013-12-30

Keywords: Concentrated mass, Bernoulli-Euler beam, Timoshenko beam, Multi-Step beam, Rotary inertia,

Abstract :

In this paper, vibration analysis of multiple-stepped Bernoulli-Euler and Timoshenko beams carrying point masses is presented analytically for various boundary conditions. Each attached element is considered to have both translational and rotational inertias. The method of solution is “transfer matrix method” which is based on the changes in the vibration modes at the vicinity of any discontinuity in geometrical and natural parameters; these changes are shown by transfer matrices depended on the geometry of each step or value of the translational and rotational inertias of each attached element. First, natural frequencies and corresponding normal mode shapes are obtained by implementation of the compatibility conditions and external boundary conditions; Then, the precision of the proposed method is checked by comparison of the results with other exact solutions; Finally, the effect of the translational and rotational inertias and position of the attached elements on the natural frequencies of multi-stepped beams are investigated for various boundary conditions.

References:

[1] Chen Y., 1963, On the vibration of beams or rods carrying a concentrated mass, Journal of Applied Mechanics 30:310-311.
[2] Laura P., Maurizi M.J., Pombo J.L., 1975, A note on the dynamics analysis of an elastically restrained-free beam with a mass at the free end, Journal of Sound and Vibration 41:397-405.
[3] Laura P., Verniere de Irassar P.L., Ficcadenti G.M, 1983, A note on transverse vibration of continuous beams subjected to an axial force and carrying concentrated masses, Journal of Sound and Vibration 86:279-284.
[4] Gurgoze M., 1984, A note on the vibrations of restrained beams and rods with point masses, Journal of Sound and Vibration 96:461-468.
[5] Gurgoze M., 1985, On the vibration of restrained beams and rods with heavy masses, Journal of Sound and Vibration 100:588-589.
[6] Liu W.H., Wu J.R., Huang C.C., 1988, Free vibrations of beams with elastically restrained edges and intermediate concentrated masses, Journal of Sound and Vibration 122:193-207.
[7] De Rosa M.A., Franciosi C., Maurizi M.J., 1955, On the dynamics behaviour of slender beams with elastic ends carrying a concentrated mass, Computers and Structures 58:1145-1159.
[8] Rossit C.A., Laura P., 2001, Transverse vibrations of a cantilever beam with a spring mass system attached on the free end, Ocean Engineering 28:933-939.
[9] Rao G.V., Saheb K.M., Janardhan G.R., 2006, Fundamental frequency for large amplitude vibrations of uniform Timoshenko beams with central point concentrated mass using coupled displacement field method, Journal of Sound and Vibration 298:221-232.
[10] Rossit M.C.A., Laura P., 2001, Transverse normal modes of vibration of a cantilever Timoshenko beam with a mass elastically mounted at the free end, Journal of the Acoustical Society of America 110:2837-2840.
[11] Laura P., Filipich C.P., Cortinez V.H., 1987, Vibrations of beams and plates carrying concentrated masses, Journal of Sound and Vibration 117:459-465.
[12] Rossi R.E., Laura P, 1990, Vibrations of a Timoshenko beam clamped at one end and carrying a finite mass at the other, Applied Acoustics 30:293-301.
[13] Chang C.H., 2000, Free vibration of a simply supported beam carrying a rigid mass at the middle, Journal of Sound and Vibration 237:4733-4744.
[14] Maiz S., Bambill D., Rossit C., Laura P., 2007, Transverse vibration of Bernoulli–Euler beams carrying point masses and taking into account their rotary inertia, Journal of Sound and Vibration 303:895-908.
[15] Lin H.Y., 2009, On the natural frequencies and mode shapes of a multi-span Timoshenko beam carrying a number of various concentrated elements, Journal of Sound and Vibration 319:593-605.
[16] Demirdag O., Murat Y.S., 2009, Free vibration analysis of elastically supported Timoshenko columns with attached masses using fuzzy neural network, Journal of Scientific and Industrial Researches 68:285-291.
[17] Demirdag O., Yesilce Y., 2011, Solution of free vibration equation of elastically supported Timoshenko columns with a tip mass by differential transform method, Advances in Engineering Software 42:860-867.
[18] Gutierrez R.H., Laura P., Rossi R.E., 1991, Vibrations of a Timoshenko beam of non-uniform cross-section elastically restrained at one end and carrying a finite mass at the other, Ocean Engineering 18:129-145.
[19] Naguleswaran S., 2002, Vibration of an Euler–Bernoulli beam on elastic end supports and with up to three step changes in cross-section, International Journal of Mechanical Science 44:2541-2555.
[20] Kukla Y., Zamojska I., 2007, Frequency analysis of axially loaded stepped beams by Green’s function method, Journal of Sound and Vibration 300:1034-1041.
[21] Lu Z.R., Huang M., Liu J.K., Chen W.H., Liao W.Y., 2009, Vibration analysis of multiple-stepped beams with the composite element model, Journal of Sound and Vibration 322:1070-1080.
[22] Mao Q., 2011, Free vibration analysis of multiple-stepped beams by using Adomian decomposition method, Mathematical and Computer Modelling 54:756-764.
[23] Timoshenko S., Weaver W., Young D.H., 1974, Vibration Problems in Engineering, Wiley, New York.
[24] Hutchinson J.R., 2001, Shear coefficients for Timoshenko beam theory, ASME Journal of Applied Mechanics 68:87-92.