Structural Engineering and Mechanics

  • 제22권6호
  • /
  • Pages.701-717
  • /
  • 2006
  • /
  • 1225-4568(pISSN)
  • /
  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

On the natural frequencies and mode shapes of a multiple-step beam carrying a number of intermediate lumped masses and rotary inertias

  • Lin, Hsien-Yuan (Dept. of Mechanical and Electro Mechanical Engineering, National Sun Yat-Sen University) ;
  • Tsai, Ying-Chien (Dept. of Mechanical and Electro-Mechanical Engineering, National Sun Yat-Sen University, Department of Mechanical Engineering, Cheng Shiu University)
  • 투고 : 2005.08.24
  • 심사 : 2005.12.09
  • 발행 : 2006.04.20
⟨ 이전 논문 다음 논문 ⟩

초록

In the existing reports regarding free transverse vibrations of the Euler-Bernoulli beams, most of them studied a uniform beam carrying various concentrated elements (such as point masses, rotary inertias, linear springs, rotational springs, spring-mass systems, ${\ldots}$, etc.) or a stepped beam with one to three step changes in cross-sections but without any attachments. The purpose of this paper is to utilize the numerical assembly method (NAM) to determine the exact natural frequencies and mode shapes of the multiple-step Euler-Bernoulli beams carrying a number of lumped masses and rotary inertias. First, the coefficient matrices for an intermediate lumped mass (and rotary inertia), left-end support and right-end support of a multiple-step beam are derived. Next, the overall coefficient matrix for the whole vibrating system is obtained using the numerical assembly technique of the conventional finite element method (FEM). Finally, the exact natural frequencies and the associated mode shapes of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and substituting the corresponding values of integration constants into the associated eigenfunctions, respectively. The effects of distribution of lumped masses and rotary inertias on the dynamic characteristics of the multiple-step beam are also studied.

키워드

참고문헌

  1. Balasubramanian, T.S. and Subramanian, G. (1985), 'On the performance of a four-degree-of-freedom per node element for stepped beam analysis and higher frequency estimation', J. Sound Vib., 99(4), 563-567 https://doi.org/10.1016/0022-460X(85)90541-3
  2. Balasubramanian, T.S., Subramanian, G. and Ramani, T.S. (1990), 'Significance of very high order derivatives as nodal degrees of freedom in stepped beam vibration analysis', J. Sound Vib., 137(2), 353-356 https://doi.org/10.1016/0022-460X(90)90803-8
  3. 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 system', J. Sound Vib., 255(2), 299-322 https://doi.org/10.1006/jsvi.2001.4156
  4. Chen, D.W. (2003), 'The exact solutions for the natural frequencies and mode shapes of non-uniform beams carrying multriple various concentrated elements', Struct. Eng. Mech., 16(2), 153-176 https://doi.org/10.12989/sem.2003.16.2.153
  5. De Rosa, M.A. (1994), 'Free vibrations of stepped beams with elastic ends', J. Sound Vib., 173(4), 557-563 https://doi.org/10.1006/jsvi.1994.1246
  6. De Rosa, M.A., Belles, P.M. and Maurizi, M.J. (1995), 'Free vibrations of stepped beams with intermediate elastic supports', J. Sound Vib., 181(5), 905-910 https://doi.org/10.1006/jsvi.1995.0177
  7. Epperson, J.F. (2003), An Introduction to Numerical Methods and Analysis, John Wiley & Son, Inc.
  8. Hamdan, M.N. and Abdel Latif, L. (1994), 'On the numerical convergence of discretization methods for the free vibrations of beams with attached inertia elements', J. Sound Vib., 169(4), 527-545 https://doi.org/10.1006/jsvi.1994.1032
  9. Jang, S.K. and Bert, C.W. (1989a), 'Free vibrations of stepped beams: Exact and numerical solutions', J. Sound Vib., 130(2), 342-346 https://doi.org/10.1016/0022-460X(89)90561-0
  10. Jang, S.K. and Bert, C.W. (1989b), 'Free vibrations of stepped beams: Higher mode frequencies and effects of steps on frequency', J. Sound Vib., 132(1), 164-168 https://doi.org/10.1016/0022-460X(89)90882-1
  11. Ju, F., Lee, H.P. and Lee, K.H. (1994), 'On the free vibration of stepped beams', Int. J. Solids Struct., 31, 3125-3137 https://doi.org/10.1016/0020-7683(94)90045-0
  12. Laura, P.A.A., Rossi, R.E., Pombo, J.L. and Pasqua, D. (1994), 'Dynamic stiffening of straight beams of rectangular cross-section: A comparison of finite element predictions and experimental results', J. Sound Vib., 150(1), 174-178 https://doi.org/10.1016/0022-460X(91)90413-E
  13. Lee, J. and Bergman, L.A. (1994), 'Vibration of stepped beams and rectangular plates by an elemental dynamic flexibility method', J. Sound Vib., 171(5), 617-640 https://doi.org/10.1006/jsvi.1994.1145
  14. Lin, S.Y. and Tsai, Y.C. (2005), 'On the natural frequencies and mode shapes of a uniform multi-span beam carrying multiple point masses', Struct. Eng. Mech., 21(3), 351-367 https://doi.org/10.12989/sem.2005.21.3.351
  15. Maurizi, M.J. and Belles, P.M. (1994), 'Natural frequencies of one-span beams with stepwise variable crosssection', J. Sound Vib., 168(1), 184-188 https://doi.org/10.1006/jsvi.1993.1399
  16. Naguleswaran, S. (2002a), 'Natural frequencies, sensitivity and mode shape details of an Euler-Bernoulli beam with one-step change in cross-section and with ends on classical supports', J. Sound Vib., 252(4), 751-767 https://doi.org/10.1006/jsvi.2001.3743
  17. Naguleswaran, S. (2002b), 'Vibration of an Euler-Bernoulli beam on elastic end supports and with up to three step changes in cross-section', Int. J. Mech. Sci., 44, 2541-2555 https://doi.org/10.1016/S0020-7403(02)00190-X
  18. Subramanian, G. and Balasubramanian, T.S. (1987), 'Beneficial effects of steps on the free vibration characteristics of beams', J. Sound Vib., 118(3), 555-560 https://doi.org/10.1016/0022-460X(87)90373-7
  19. 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. Methods Eng., 50, 1039-1058 https://doi.org/10.1002/1097-0207(20010220)50:5<1039::AID-NME60>3.0.CO;2-D
  20. Wu, J.S. and Chou, H.M. (1998), 'Free vibration analysis of a cantilever beams carrying any number of elastically mounted point masses with the analytical-and-numerical-combined method', J. Sound Vib., 213(2), 317-332 https://doi.org/10.1006/jsvi.1997.1501
  21. 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

피인용 문헌

  1. On the natural frequencies and mode shapes of a multi-span and multi-step beam carrying a number of concentrated elements vol.29, pp.5, 2008, https://doi.org/10.12989/sem.2008.29.5.531
  2. Dynamic analysis of a multi-span uniform beam carrying a number of various concentrated elements vol.309, pp.1-2, 2008, https://doi.org/10.1016/j.jsv.2007.07.015
  3. The influence of bending and shear stiffness and rotational inertia in vibrations of cables: An analytical approach vol.33, pp.3, 2011, https://doi.org/10.1016/j.engstruct.2010.11.026
  4. Differential transform method and numerical assembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and rotary inertias vol.53, pp.3, 2015, https://doi.org/10.12989/sem.2015.53.3.537
  5. On the natural frequencies and mode shapes of a multispan Timoshenko beam carrying a number of various concentrated elements vol.319, pp.1-2, 2009, https://doi.org/10.1016/j.jsv.2008.05.022
  6. Free vibration of axial-loaded multi-step Timoshenko beam carrying arbitrary concentrated elements using continuous-mass transfer matrix method vol.38, 2013, https://doi.org/10.1016/j.euromechsol.2012.08.003
  7. Free vibration analysis of rotating tapered blades using Fourier-p superelement vol.27, pp.2, 2007, https://doi.org/10.12989/sem.2007.27.2.243
  8. A modified modal perturbation method for vibration characteristics of non-prismatic Timoshenko beams vol.40, pp.5, 2006, https://doi.org/10.12989/sem.2011.40.5.689