Structural Engineering and Mechanics

  • 제25권6호
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  • Pages.717-732
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  • 2007
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  • 1225-4568(pISSN)
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  • 1598-6217(eISSN)
테크노프레스 (Techno-Press)
권호 표지
DOI QR코드

DOI QR Code

An approximate formula to calculate the fundamental period of a fixed-free mass-spring system with varying mass and stiffness

  • Kim, Juwhan (School of Civil & Environmental Engineering, College of Engineering, Yonsei University) ;
  • CoIIins, Kevin R. (5 Greystone Court Ledyard) ;
  • Lim, Yun Mook (School of Civil & Environmental Engineering, College of Engineering, Yonsei University)
  • 투고 : 2006.06.23
  • 심사 : 2007.01.24
  • 발행 : 2007.04.20
⟨ 이전 논문 다음 논문 ⟩

초록

A formula to approximate the fundamental period of a fixed-free mass-spring system with varying mass and varying stiffness is formulated. The formula is derived mainly by taking the dominant parts from the general form of the characteristic polynomial, and adjusting the initial approximation by a coefficient derived from the exact solution of a uniform case. The formula is tested for a large number of randomly generated structures, and the results show that the approximated fundamental periods are within the error range of 4% with 90% of confidence. Also, the error is shown to be normally distributed with zero mean, and the width of the distribution (as measured by the standard deviation) tends to decrease as the total number of discretized elements in the system increases. Other possible extensions of the formula are discussed, including an extension to a continuous cantilever structure with distributed mass and stiffness. The suggested formula provides an efficient way to estimate the fundamental period of building structures and other systems that can be modeled as mass-spring systems.

키워드

참고문헌

  1. Abrate, S. (1986), 'Vibration of non-unifonn rods and beams', J. Sound Vib., 185, 703-716 https://doi.org/10.1006/jsvi.1995.0410
  2. Artin, M. (1991), Algebra. Prentice-Hall, New Jersey, First edition
  3. Bapat, C.N. (1995), 'Vibration of rods with uniformly tapered sections', J. Sound Vib., 185, 185-189 https://doi.org/10.1006/jsvi.1995.0371
  4. Braun, M. (2003), 'On some properties of the multiple pendulum', Archive of Appl. Mech., 72, 899-910
  5. Cha, P. (2005), 'A general approach to formulating the frequency equation for a beam carrying miscellaneous attachments', J. Sound Vib., 286, 921-939 https://doi.org/10.1016/j.jsv.2004.10.012
  6. Chen, D.-W. (2006), 'An exact solution for free torsional vibration of a uniform circular shaft carrying multiple concentrated elements', J. Sound Vib., 291, 627-643 https://doi.org/10.1016/j.jsv.2005.06.034
  7. Chen, S., Guo, K. and Chen, Y. (2004), 'A method for estimating upper and lower bounds of eigenvalues of closed-loop systems with uncertain parameters', J. Sound Vib., 276, 527-539 https://doi.org/10.1016/j.jsv.2003.08.011
  8. Chopra, A.K. (1995), Dynamics of Structures. Prentice Hall
  9. Elliot, J.F. (1953), 'The characteristic roots of certain real symmetric matrices', Master's thesis, University of Tennessee
  10. Gokda, H. and Kopmaz, O. (2005), 'Eigenfrequencies of a combined system including two continua connected by discrete elements', J. Sound Vib., 284, 1203-1216 https://doi.org/10.1016/j.jsv.2004.08.022
  11. Goldberg, J.L. (1992), Matrix Theory with Applications. McGraw-Hill
  12. Gregory, R.T. (1978), A Collection of Matrices for Testing Computational Algorithms. R.E. Krieger Pub. Co
  13. Gupta, A. and Krawinkler, H. (2000), 'Behavior of ductile special moment resisting frames at various seismic hazard levels', J. Struct. Eng., 126(1), 98-107 https://doi.org/10.1061/(ASCE)0733-9445(2000)126:1(98)
  14. Gtirgoze, M. (1996), 'On the eigenfrequencies of cantilevered beams carrying a tip mass and spring-mass inspan', Int. J. Mech. Sci., 28(12), 1295-1306
  15. Gtirgoze, M. (2005), 'On the eigenfrequencies of a cantilever beam carrying a tip spring-mass system with mass of the helical spring considered', J. Sound Vib., 282, 1221-1230 https://doi.org/10.1016/j.jsv.2004.04.020
  16. Gtirgoze, M. (2006), 'On some relationships between the eigenfrequencies of torsional vibrational systems containing lumped elements', J. Sound Vib., 290, 1322-1332 https://doi.org/10.1016/j.jsv.2005.05.011
  17. Gtirgoze, M. and Zeren, S. (2006), 'On the eigencharacteristics of an axially vibrating viscoelastic rod carrying a tip mass and its representation by a single-degree-of-freedom system', J. Sound Vib., 294, 388-396 https://doi.org/10.1016/j.jsv.2005.11.006
  18. Iwan, W.D. (1995), 'Drift demand spectra for selected northridge sites', SAC Final Report, number 95-07 in Earthquake Engineering Research Laboratory Report, California Institute of Technology, Pasadena, California
  19. Jacobsen, L.S. and Ayre, R.S. (1958), Engineering Vibrations. McGraw-Hill
  20. Kim, J. (2003), Performance-Based Building Design Using Wave Propagation Concepts. Department of Civil and Environmental Engineering, The University of Michigan at Ann Arbor, Ann Arbor, Michigan
  21. Kim, J, Collins, K.R. and Lim, Y.M. (2006), 'Application of internally damped shearbeam model to analysis of buildings under earthquakes: Robust procedure for quick evaluation of seismic performance', J. Struct. Eng., 132(7), 1139-1149 https://doi.org/10.1061/(ASCE)0733-9445(2006)132:7(1139)
  22. Kumar, B.M. and Sujith, R.I. (1997), 'Exact solutions for the longitudinal vibration of nonuniform rods', J. Sound Vib., 207, 721-729 https://doi.org/10.1006/jsvi.1997.1146
  23. Lee, K.-Y and Renshaw, A.A. (2002), 'A numerical comparison of alternative galerkin methods for eigenvalue estimation', J. Sound Vib., 253(2), 359-372 https://doi.org/10.1006/jsvi.2001.4046
  24. Li, Q.S., Wu, J.R. and Xu, J. (2002), 'Longitudinal vibration of multi-step non-uniform structures with lumped masses and spring support', Appl. Acoust., 63, 333-350. technical note https://doi.org/10.1016/S0003-682X(01)00034-2
  25. Li, W. (2003), 'A degree selection method of matrix condensations for eigenvalue problems', J. Sound Vib., 259(2), 409-425 https://doi.org/10.1006/jsvi.2002.5336
  26. Low, K.H. (2000), 'A modified dunkerley formula for eigenfrequencies of beams carrying concentrated masses', Int. J. Mech. Sci., 42,1287-1305 https://doi.org/10.1016/S0020-7403(99)00049-1
  27. Meirovitch, L. (1986), Elements of Vibration Analysis. McGraw-Hill, second edition
  28. Press, W.H., Teukolsky, S.A, Vettering, W.T. and Flannery, B.P. (1992), Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, second edition
  29. Rajendran, S. (2002), 'Computing the lowest eigenvalue with rayleigh quotient iteration', J. Sound Vib., 254(3), 599-612 https://doi.org/10.1006/jsvi.2001.4102
  30. Strobach, D. and Braun, M. (2003), 'On some property of the natural frequencies of elastic chains', Proc. in Appl. Math. Mech., 3, 128-129
  31. Structural Engineers Association of California (1997), Uniform Building Code. International Conference of Building Officials, Whittier, California
  32. Thomson, W.T. (1981), Theory of Vibration with Applications. Prentice-Hall, second edition

피인용 문헌

  1. A simplified method for estimating fundamental periods of pylons in overhead electricity transmission systems vol.19, pp.2, 2007, https://doi.org/10.12989/eas.2020.19.2.119