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Research Papers

Determination of Isospectral Nonuniform Rotating Beams

[+] Author and Article Information
Sandilya Kambampati

Department of Aerospace Engineering,  Indian Institute of Science, Bangalore 560012, Indiasandilya@aero.iisc.ernet.in

Ranjan Ganguli1

Department of Aerospace Engineering,  Indian Institute of Science, Bangalore 560012, Indiaganguli@aero.iisc.ernet.in

V. Mani

Department of Aerospace Engineering,  Indian Institute of Science, Bangalore 560012, Indiamani@aero.iisc.ernet.in

1

Corresponding author.

J. Appl. Mech 79(6), 061016 (Sep 21, 2012) (14 pages) doi:10.1115/1.4006460 History: Received July 27, 2011; Revised March 20, 2012; Posted March 21, 2012; Published September 21, 2012; Online September 21, 2012

In this paper we look for nonuniform rotating beams that are isospectral to a given uniform nonrotating beam. A rotating nonuniform beam is isospectral to the given uniform nonrotating beam if both the beams have the same spectral properties, i.e., both the beams have the same set of natural frequencies under a given boundary condition. The Barcilon-Gottlieb type transformation is proposed that converts the governing equation of a rotating beam to that of a uniform nonrotating beam. We show that there exist rotating beams isospectral to a given uniform nonrotating beam under some special conditions. The boundary conditions we consider are clamped-free and hinged-free with an elastic hinge spring. An upper bound on the rotation speed for which isospectral beams exist is proposed. The mass and stiffness distributions for these nonuniform rotating beams which are isospectral to the given uniform nonrotating beam are obtained. We use these mass and stiffness distributions in a finite element analysis to show that the obtained beams are isospectral to the given uniform nonrotating beam. A numerical example of a beam having a rectangular cross section is presented to show the application of our analysis.

Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Schematic of a rotating beam

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Figure 2

Variations of m(x), f(x), t(x), and p(x) with x of isospectral cantilever beams

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Figure 3

Plots of x(z) and x′(z) for different rotation speeds

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Figure 4

Plot of minimum of xz versus the rotation speed

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Figure 5

Variations of m(x), f(x), t(x), and p(x) of isospectral cantilever beams for z0  = 0.6, 0.7, 0.8

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Figure 6

Variations of m(x), f(x), t(x), and p(x) of isospectral hinged-free beams with a root spring

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Figure 7

Plot of q02 versus λ

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Figure 8

Variations of m(x), f(x), t(x), and p(x) of isospectral hinged-free beams with a root spring for z0  = 0.65, 0.7, 0.75

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Figure 9

Convergence plots of ηk /ηexact versus number of finite elements for the cantilever beam

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Figure 10

Convergence plots ηk /ηexact versus number of finite elements for the hinged-free beam with a root spring

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Figure 11

First eight mode shapes of the rotating cantilever beam (exact and FEM)

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Figure 12

First eight mode shapes of the rotating spring hinge-free beam (exact and FEM)

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Figure 13

Breadth and height distributions for isospectral cantilever beams of equal length

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Figure 14

Breadth and height distributions for rotating cantilever beams isospectral to a uniform nonrotating beam of smaller length

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Figure 15

Breadth and height distributions of the isospectral rotating hinged-free beams with a root spring

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Figure 16

Breadth and height distributions of the rotating hinged-free beams with a root spring isospectral to a uniform nonrotating beam of smaller length

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