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 q = q 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.
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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)
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)
Subramanian G., Raman A.: Isospectral systems for tapered beams. J. Sound Vib. 198(3), 257–266 (1996)
Ghanbari, K.: On the isospectral beams. In: Electronic Journal of Differential Equations Conference, vol. 12 (2005)
Gladwell G.M.L., Morassi A.: A family of isospectral Euler–Bernoulli beams. Inverse Prob. 26, 035006 (2010)
Gottlieb, H.: Density distribution for isospectral circular membranes. SIAM J. Appl. Math. 48, 948–951 (1988)
Gottlieb H.: Isospectral circular membranes. Inverse Prob. 20, 155–161 (2004)
McCallion H.: Vibration of Linear Mechanical Systems. Longman, London (1973)
Bokaian A.: Natural frequencies of beams under compressive axial loads. J. Sound Vib. 126(1), 49–65 (1988)
Bokaian A.: Natural frequencies of beams under tensile axial loads. J. Sound Vib. 142(3), 481–498 (1990)
Fletcher H.: Normal vibration frequencies of a stiff piano string. J. Acoust. Soc. Am. 36(1), 203–209 (1964)
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)
Panigrahi S., Chakraverty S., Mishra B.: Vibration based damage detection in a uniform strength beam using genetic algorithm. Meccanica 44(6), 697–710 (2009)
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)
Kambampati S., Ganguli R., Mani V.: Determination of isospectral nonuniform rotating beams. J. Appl. Mech. 79(6), 061,016 (2012)
Kambampati S., Ganguli R., Mani V.: Non-rotating beams isospectral to a given rotating uniform beam. Int. J. Mech. Sci. 66, 12–21 (2013)
Kambampati S., Ganguli R., Mani V.: Rotating beams isospectral to axially loaded nonrotating uniform beams. AIAA J. 51(5), 1189–1202 (2013)
Meirovitch, L.: Elements of Vibration Analysis, vol. 2. McGraw-Hill, New York (1986)
Karnovsky I., Lebed O.: Formulas for Structural Dynamics: Tables, Graphs and Solutions. McGraw-Hill, New York (2001)
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)
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Kambampati, S., Ganguli, R. Non-uniform beams and stiff strings isospectral to axially loaded uniform beams and piano strings. Acta Mech 226, 1227–1239 (2015). https://doi.org/10.1007/s00707-014-1238-6
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DOI: https://doi.org/10.1007/s00707-014-1238-6