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


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.


Axial Load Transverse Vibration Compressive Axial Load Uniform Beam Loaded Beam 
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© 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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