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Research Papers

# Rotating Beams and Nonrotating Beams With Shared Eigenpair

[+] Author and Article Information
Ananth Kumar

Department of Aerospace Engineering, Indian Institute of Technology Madras, Chennai 600036, Indiaae05b004@iitm.ac.in

Ranjan Ganguli1

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

1

Corresponding author.

J. Appl. Mech 76(5), 051006 (Jun 18, 2009) (14 pages) doi:10.1115/1.3112741 History: Received July 11, 2008; Revised February 05, 2009; Published June 18, 2009

## Abstract

In this paper, we look for rotating beams whose eigenpair (frequency and mode-shape) is the same as that of uniform nonrotating beams for a particular mode. It is found that, for any given mode, there exist flexural stiffness functions (FSFs) for which the $jth$ mode eigenpair of a rotating beam, with uniform mass distribution, is identical to that of a corresponding nonrotating uniform beam with the same length and mass distribution. By putting the derived FSF in the finite element analysis of a rotating cantilever beam, the frequencies and mode-shapes of a nonrotating cantilever beam are obtained. For the first mode, a physically feasible equivalent rotating beam exists, but for higher modes, the flexural stiffness has internal singularities. Strategies for addressing the singularities in the FSF for finite element analysis are provided. The proposed functions can be used as test-functions for rotating beam codes and for targeted destiffening of rotating beams.

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## Figures

Figure 1

Schematic of a rotating beam

Figure 2

EI1(x) variation for first four modes for a beam with properties given in Table 1

Figure 3

Beam element used for p-version FEM

Figure 4

Gauss-quadrature points of order 30 for the domain (0,0.6) and the singularity points of the flexural stiffness functions

Figure 5

Convergence of the p-version finite element code for first four modes using derived flexural stiffness functions

Figure 6

Residue of mode-shapes obtained from p-version FEM

Figure 7

Convergence characteristics and residue of nodal displacements for first mode with h-version finite element code

Figure 8

Fundamental mesh for the second mode h-version FEM, and the first two refinements

Figure 9

Convergence characteristics and residue of nodal displacements for second mode with h-version finite element code

Figure 10

Fundamental mesh for the third mode h-version FEM, and the first two refinements

Figure 11

Convergence characteristics and residue of nodal displacements for third mode with h-version finite element code

Figure 12

Fundamental mesh for the fourth mode h-version FEM, and the first two refinements

Figure 13

Convergence characteristics and residue of nodal displacements for fourth mode with h-version finite element code

Figure 14

Variation in height and breadth of a rotating beam equivalent to a uniform nonrotating beam for the first mode

Figure 15

Top and side views of a rotating beam equivalent to a uniform rotating beam for the first mode

Figure 16

Realistic beams: approximations to the exact function for the second mode

Figure 17

Variation in height and breadth of a rotating beam equivalent to a uniform nonrotating beam for the second mode

Figure 18

Realistic beams: approximations to the exact function for the third mode

Figure 19

Variation in height and breadth of a rotating beam equivalent to a uniform nonrotating beam for the third mode

Figure 20

Realistic beams: approximations to the exact function for the fourth mode

Figure 21

Variation in height and breadth of a rotating beam equivalent to a uniform nonrotating beam for the fourth mode

Figure 22

Destiffening effects of flexural stiffness variations for various modes compared with uniform rotating beam (EI1(r))

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