Explicit Construction of Rods and Beams with Given Natural Frequencies

Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)

Abstract

In this paper we present a new method for constructing one-dimensional vibrating systems having prescribed values of the first N natural frequencies, under a given set of boundary conditions. In the case of axially vibrating rods, the analysis is based on the determination of the so-called quasi-isospectral rods, that is rods which have the same spectrum as a given rod, with the exception of a single eigenvalue which is free to move in a prescribed interval. The reconstruction procedure needs the specification of an initial rod whose eigenvalues must be close to the assigned eigenvalues. The rods and their normal modes can be constructed explicitly by means of closed-form expressions. The results can be extended to strings and to some special classes of beams in bending vibration.

Keywords

Structural identification Vibrating rods Quasi-isospectral systems Inverse problems Eigenvalues 

References

  1. 1.
    Capecchi D, Vestroni F (2000) Monitoring of structural systems by using frequency data. Earthq Eng Struct Dyn 28:447–461Google Scholar
  2. 2.
    Coleman CF, McLaughlin JR (1993) Solution of the inverse spectral problem for an impedance with integrable derivative – Part I. Commun Pure Appl Math XLVI:145–184Google Scholar
  3. 3.
    Coleman CF, McLaughlin JR (1993) Solution of the inverse spectral problem for an impedance with integrable derivative – Part II. Commun Pure Appl Math XLVI:185–212Google Scholar
  4. 4.
    Dahlberg BEJ, Trubowitz E (1984) The inverse Sturm-Liouville problem III. Commun Pure Appl Math XXXVII:255–267Google Scholar
  5. 5.
    Darboux G (1882) Sur la répresentation sphérique des surfaces. C R Acad Sci Paris 94:1343–1345Google Scholar
  6. 6.
    Deift PA (1978) Application of a commutation formula. Duke Math J 45:267–310Google Scholar
  7. 7.
    Dilena M, Morassi A (2009) Structural health monitoring of rods based on natural frequency and antiresonant frequency measurements. Struct Health Monit 8:149–173Google Scholar
  8. 8.
    Dilena M, Morassi A (2010) Reconstruction method for damage detection in beams based on natural frequency and antiresonant frequency measurements. J Eng Mech ASCE 136:329–344Google Scholar
  9. 9.
    Gesztesy G, Teschl G (1996) On the double commutation method. Proc Am Math Soc 124:1831–1840Google Scholar
  10. 10.
    Gladwell GML, Morassi A (2010) A family of isospectral Euler-Bernoulli beams. Inverse Probl 26:12, Paper n. 035006Google Scholar
  11. 11.
    Isaacson EL, Mc Kean HP, Trubowitz E (1984) The inverse Sturm-Liouville problem II. Commun Pure Appl Math XXXVII:1–11Google Scholar
  12. 12.
    Morassi A (2007) Damage detection and generalized Fourier coefficients. J Sound Vib 302:229–259Google Scholar
  13. 13.
    Morassi A (2013) Exact construction of beams with a finite number of given natural frequencies. J. Vib Control (to appear)Google Scholar
  14. 14.
    Pöschel J, Trubowitz E (1987) Inverse spectral theory. Academic, LondonGoogle Scholar
  15. 15.
    Vestroni F, Capecchi D (2000) Damage detection in beam structures based on frequency measurements. J Eng Mech ASCE 126:761–768Google Scholar

Copyright information

© The Society for Experimental Mechanics 2014

Authors and Affiliations

  1. 1.DICA – University of UdineUdineItaly

Personalised recommendations