Structural Engineering and Mechanics

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Free vibration analysis of a Timoshenko beam carrying multiple spring-mass systems with the effects of shear deformation and rotary inertia

  • Wang, Jee-Ray (Institute of Mechatronoptic Systems, ChienKuo Technology University) ;
  • Liu, Tsung-Lung (Department of Naval Architecture and Marine Engineering, Chung Cheng Institute of Technology, National Defense University) ;
  • Chen, Der-Wei (Department of Naval Architecture and Marine Engineering, Chung Cheng Institute of Technology, National Defense University)
  • Received : 2005.01.27
  • Accepted : 2006.11.14
  • Published : 2007.05.10
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Abstract

Because of complexity, the literature regarding the free vibration analysis of a Timoshenko beam carrying "multiple" spring-mass systems is rare, particular that regarding the "exact" solutions. As to the "exact" solutions by further considering the joint terms of shear deformation and rotary inertia in the differential equation of motion of a Timoshenko beam carrying multiple concentrated attachments, the information concerned is not found yet. This is the reason why this paper aims at studying the natural frequencies and mode shapes of a uniform Timoshenko beam carrying multiple intermediate spring-mass systems using an exact as well as a numerical assembly method. Since the shear deformation and rotary inertia terms are dependent on the slenderness ratio of the beam, the shear coefficient of the cross-section, the total number of attachments and the support conditions of the beam, the individual and/or combined effects of these factors on the result are investigated in details. Numerical results reveal that the effect of the shear deformation and rotary inertia joint terms on the lowest five natural frequencies of the combined vibrating system is somehow complicated.

Keywords

References

  1. Abramovich, H. and Elishakoff, I. (1990), 'Influence of shear deformation and rotary inertia on vibration frequencies via Love's equations', J. Sound Vib., 137(3), 516-522 https://doi.org/10.1016/0022-460X(90)90816-I
  2. Abramovich, H. and Hamburger, O. (1991), 'Vibration of a cantilever Timoshenko beam with a tip mass', J. Sound Vib., 148(1), 162-170 https://doi.org/10.1016/0022-460X(91)90828-8
  3. Abramovich, H. and Hamburger, O. (1992), 'Vibration of a uniform cantilever Timoshenko beam withtranslational and rotational springs and with a tip mass', J. Sound Vib., 154(1), 67-80 https://doi.org/10.1016/0022-460X(92)90404-L
  4. Cha, P.D. (2001), 'Natural frequencies of a linear elastica carrying a number of sprung masses', J. Sound Vib., 247, 185-194 https://doi.org/10.1006/jsvi.2001.3623
  5. Chen, D.W. and Wu, J.S. (2002), 'The exact solutions for the natural frequencies and mode shapes of nonuniform beams with multiple spring-mass systems', J. Sound Vib., 255, 299-232 https://doi.org/10.1006/jsvi.2001.4156
  6. Faires, J.D. and Burden, R.L. (1993), Numerical Method, PWD Publishing Company, Boston, U.S.A
  7. Gurgoze, M. (1998), 'On the alternative formulation of the frequency equation of a Bernoulli-Euler beam to which several spring-mass systems are attached in-span', J. Sound Vib., 217, 585-595 https://doi.org/10.1006/jsvi.1998.1796
  8. Laura, P.A.A., Susemihl, E.A., Pombo, J.L., Luisoni, L.E. and Gelos, R. (1977), 'On the dynamic behaviour of structural elements carrying elastically mounted concentrated masses', Appl. Acoust., 10, 121-145 https://doi.org/10.1016/0003-682X(77)90021-4
  9. Meirovitch, L. (1967), Analytical Methods in Vibrations, Macmillan Company, London, U.K
  10. Qiao, H., Li, Q.S. and Li, G.O. (2002), 'Vibratory characteristic of non-uniform Euler-Bernoulli beams carrying an arbitrary number of spring-mass systems', Int. J. Mech. Sci., 44, 725-743 https://doi.org/10.1016/S0020-7403(02)00007-3
  11. Rossi, R.E., Laura, P.A.A., Avalos, D.R. and Larrondo, H. (1993), 'Free vibrations of Timoshenko beam carrying elastically mounted concentrated masses', J. Sound Vib., 165(2), 209-223 https://doi.org/10.1006/jsvi.1993.1254
  12. Thomson, W.T. (1981), Theory of Vibration with Application, Englewood Cliffs, New Jersey, Prentice-Hill
  13. Wu, J.S. and Chen, D.W. (2001), 'Free vibration analysis of a Timoshenko beam carrying multiple spring-mass systems by using the numerical assembly technique', Int. J. Numer. Eng., 50, 1039-1058 https://doi.org/10.1002/1097-0207(20010220)50:5<1039::AID-NME60>3.0.CO;2-D
  14. Wu, J.S. and Chou, H.M. (1998), 'Free vibration analysis of a cantilever beam carrying any number of elastically mounted point masses with analytical-and-numerical-combined method', J. Sound Vib., 213(2), 317-332 https://doi.org/10.1006/jsvi.1997.1501
  15. Wu, J.S. and Chou, H.M. (1999), 'A new approach for determining the natural frequencies and mode shapes of a uniform beam carrying any number of sprung masses', J. Sound Vib., 220(3), 451-468 https://doi.org/10.1006/jsvi.1998.1958

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