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Acta Mechanica

, Volume 226, Issue 4, pp 1227–1239 | Cite as

Non-uniform beams and stiff strings isospectral to axially loaded uniform beams and piano strings

  • Sandilya Kambampati
  • Ranjan GanguliEmail author
Article

Abstract

In this paper, we derive analytical expressions for mass and stiffness functions of transversely vibrating clamped–clamped non-uniform beams under no axial loads, which are isospectral to a given uniform axially loaded beam. Examples of such axially loaded beams are beam columns (compressive axial load) and piano strings (tensile axial load). The Barcilon–Gottlieb transformation is invoked to transform the non-uniform beam equation into the axially loaded uniform beam equation. The coupled ODEs involved in this transformation are solved for two specific cases (pq z k 0 and qq 0), and analytical solutions for mass and stiffness are obtained. Examples of beams having a rectangular cross section are shown as a practical application of the analysis. Some non-uniform beams are found whose frequencies are known exactly since uniform axially loaded beams with clamped ends have closed-form solutions. In addition, we show that the tension required in a stiff piano string with hinged ends can be adjusted by changing the mass and stiffness functions of a stiff string, retaining its natural frequencies.

Keywords

Axial Load Transverse Vibration Compressive Axial Load Uniform Beam Loaded Beam 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Gottlieb H.: Isospectral Euler–Bernoulli beams with continuous density and rigidity functions. Proc. R. Soc. Lond. A Math. Phys. Sci. 413(1844), 235–250 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Barcilon V.: Inverse problem for the vibrating beam in the free-clamped configuration. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Sci. 304(1483), 211–251 (1982)CrossRefzbMATHGoogle Scholar
  3. 3.
    Subramanian G., Raman A.: Isospectral systems for tapered beams. J. Sound Vib. 198(3), 257–266 (1996)CrossRefGoogle Scholar
  4. 4.
    Ghanbari, K.: On the isospectral beams. In: Electronic Journal of Differential Equations Conference, vol. 12 (2005)Google Scholar
  5. 5.
    Gladwell G.M.L., Morassi A.: A family of isospectral Euler–Bernoulli beams. Inverse Prob. 26, 035006 (2010)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Gottlieb, H.: Density distribution for isospectral circular membranes. SIAM J. Appl. Math. 48, 948–951 (1988)Google Scholar
  7. 7.
    Gottlieb H.: Isospectral circular membranes. Inverse Prob. 20, 155–161 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    McCallion H.: Vibration of Linear Mechanical Systems. Longman, London (1973)Google Scholar
  9. 9.
    Bokaian A.: Natural frequencies of beams under compressive axial loads. J. Sound Vib. 126(1), 49–65 (1988)CrossRefGoogle Scholar
  10. 10.
    Bokaian A.: Natural frequencies of beams under tensile axial loads. J. Sound Vib. 142(3), 481–498 (1990)CrossRefGoogle Scholar
  11. 11.
    Fletcher H.: Normal vibration frequencies of a stiff piano string. J. Acoust. Soc. Am. 36(1), 203–209 (1964)CrossRefGoogle Scholar
  12. 12.
    Zhang Y., Liu G., Han X.: Transverse vibrations of double-walled carbon nanotubes under compressive axial load. Phys. Lett. A 340(1), 258–266 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Panigrahi S., Chakraverty S., Mishra B.: Vibration based damage detection in a uniform strength beam using genetic algorithm. Meccanica 44(6), 697–710 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Farrar C.R., Doebling S.W., Nix D.A.: Vibration-based structural damage identification. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 359(1778), 131–149 (2001)CrossRefzbMATHGoogle Scholar
  15. 15.
    Kambampati S., Ganguli R., Mani V.: Determination of isospectral nonuniform rotating beams. J. Appl. Mech. 79(6), 061,016 (2012)CrossRefGoogle Scholar
  16. 16.
    Kambampati S., Ganguli R., Mani V.: Non-rotating beams isospectral to a given rotating uniform beam. Int. J. Mech. Sci. 66, 12–21 (2013)CrossRefGoogle Scholar
  17. 17.
    Kambampati S., Ganguli R., Mani V.: Rotating beams isospectral to axially loaded nonrotating uniform beams. AIAA J. 51(5), 1189–1202 (2013)CrossRefGoogle Scholar
  18. 18.
    Meirovitch, L.: Elements of Vibration Analysis, vol. 2. McGraw-Hill, New York (1986)Google Scholar
  19. 19.
    Karnovsky I., Lebed O.: Formulas for Structural Dynamics: Tables, Graphs and Solutions. McGraw-Hill, New York (2001)Google Scholar
  20. 20.
    Chen R.: Letter to the editor: Evaluation of natural vibration frequency of a compression bar with varying cross-section by using the shooting method. J. Sound vib. 201(4), 520–527 (1997)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Wien 2014

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringThe Pennsylvania State UniversityUniversity ParkUSA
  2. 2.Department of Aerospace EngineeringIndian Institute of ScienceBangaloreIndia

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