# Non-rotating beams isospectral to a given rotating uniform beam

Kambampati, Sandilya and Ganguli, Ranjan and Mani, V (2013) Non-rotating beams isospectral to a given rotating uniform beam. In: INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES, 66 . pp. 12-21.

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## Abstract

In this paper, we seek to find non-rotating beams with continuous mass and flexural stiffness distributions, that are isospectral to a given uniform rotating beam. The Barcilon-Gottlieb transformation is used to convert the fourth order governing equation of a non-rotating beam, to a canonical fourth order eigenvalue problem. If the coefficients in this canonical equation match with the coefficients of the uniform rotating beam equation, then the non-rotating beam is isospectral to the given rotating beam. The conditions on matching the coefficients leads to a pair of coupled differential equations. We solve these coupled differential equations for a particular case, and thereby obtain a class of non-rotating beams that are isospectral to a uniform rotating beam. However, to obtain isospectral beams, the transformation must leave the boundary conditions invariant. We show that the clamped end boundary condition is always invariant, and for the free end boundary condition to be invariant, we impose certain conditions on the beam characteristics. We also verify numerically that the frequencies of the non-rotating beam obtained using the finite element method (FEM) are the exact frequencies of the uniform rotating beam. Finally, the example of beams having a rectangular cross-section is presented to show the application of our analysis. Since experimental determination of rotating beam frequencies is a difficult task, experiments can be easily conducted on these rectangular non-rotating beams, to calculate the frequencies of the rotating beam. (c) 2012 Elsevier Ltd. All rights reserved.

Item Type: Journal Article INTERNATIONAL JOURNAL OF MECHANICAL SCIENCES PERGAMON-ELSEVIER SCIENCE LTD Copyright for this article belongs to the PERGAMON-ELSEVIER SCIENCE LTD, ENGLAND. Free vibration; Rotating beam; Isospectral; Modal analysis; Finite element; Eigenvalue Division of Mechanical Sciences > Aerospace Engineering(Formerly Aeronautical Engineering) 09 May 2013 11:19 09 May 2013 11:19 http://eprints.iisc.ac.in/id/eprint/46505