Technical Briefs

Analytical Solution for Whirling Speeds and Mode Shapes of a Distributed-Mass Shaft With Arbitrary Rigid Disks

[+] Author and Article Information
Jong-Shyong Wu

e-mail: jswu@mail.ncku.edu.tw

Huei-Jou Shaw

Department of Systems and Naval
Mechatronic Engineering,
National Cheng-Kung University,
Tainan 701, Taiwan

1Corresponding author.

Manuscript received November 16, 2011; final manuscript received May 12, 2013; accepted manuscript posted May 29, 2013; published online September 19, 2013. Assoc. Editor: Alexander F. Vakakis.

J. Appl. Mech 81(3), 034503 (Sep 19, 2013) (10 pages) Paper No: JAM-11-1434; doi: 10.1115/1.4024670 History: Received November 16, 2011; Revised May 12, 2013; Accepted May 29, 2013

The purpose of this paper is to present an approach for replacing the effects of each rigid disk mounted on the spin shaft by a lumped mass together with a frequency-dependent equivalent mass moment of inertia so that the whirling motion of a rotating shaft-disk system is similar to the transverse free vibration of a stationary beam and the technique for the free vibration analysis of a stationary beam with multiple concentrated elements can be used to determine the forward and backward whirling speeds, along with mode shapes of a distributed-mass shaft carrying arbitrary rigid disks. Numerical results reveal that the characteristics of whirling motions are significantly dependent on the slopes of the associated natural mode shapes at the positions where the rigid disks are located. Furthermore, the results obtained from the presented analytical method and those obtained from existing literature or the finite element method (FEM) are in good agreement.

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Grahic Jump Location
Fig. 1

A multistep bearing-support shaft composed of n uniform shaft segments (denoted by (1), (2), …, (i − 1), (i), (i + 1), …, (n)) separated by n − 1 nodes (denoted by 1, 2, …, i − 1, i, i + 1, …, n − 1) and carrying a rigid disk md,i (with equivalent mass moments of inertia Jeq,i) at each node i, for i = 1 to n − 1

Grahic Jump Location
Fig. 2

The coordinate systems for a rotating shaft with spin speed Ω about the a-axis and whirling speed ω˜ about the x-axis with xyz, x˜y˜z˜, and abc denoting the space-fixed, shaft-fixed and cross-sectional coordinate systems, respectively

Grahic Jump Location
Fig. 3

Free-body diagrams for the “shaft segment” dx on (a) the xy-plane, and (b) the xz-plane

Grahic Jump Location
Fig. 4

Free-body diagrams for the “rigid disk” i located at node i (with axial coordinate x = xi) joining shaft segments (i) at left side and (i + 1) at the right side on (a) the xy-plane, and (b) the xz-plane. The superscripts L and R refer to the left and right sides of disk i, respectively.

Grahic Jump Location
Fig. 5

Influence of the inertia ratio μJ on the lowest four nondimensional whirling speed coefficients βrL (r = 1 − 4), for the P-P shaft carrying a single disk at x = x1 = 0.25 L (cf., Fig. 6(a)) with the speed ratio λ = Ω/ω˜r = 1.0

Grahic Jump Location
Fig. 6

A uniform P-P shaft carrying (a) one single disk (with thickness h = 0.012 m) at its center (x1 = L/2), and (b) three identical rigid disks (with h1 = h1 = h1 = h/3 = 0.004 m) at x1 = L/4, x2 = L/2 and x3 = 3 L/4, respectively

Grahic Jump Location
Fig. 7

The lowest five natural mode shapes for the P-P shaft carrying a single central rigid disk (see Fig. 6(a)) (with the speed ratio λ = 0) obtained from the presented method (denoted by solid lines: —) and the FEM (denoted by dashed lines: - - -)

Grahic Jump Location
Fig. 8

The lowest five (whirling) mode shapes of the P-P shaft carrying a single rigid disk (see Fig. 6(a)) obtained from the presented method (denoted by the solid lines: —) and the FEM (denoted by the dashed lines: - - -) with the speed ratio λ = 1.0 for (a) forward whirling, and (b) backward whirling

Grahic Jump Location
Fig. 9

Influence of the spin speeds Ω on the lowest four whirling speeds (ω˜1-ω˜4) for the P-P shaft carrying 1 central rigid disk (see Fig. 6(a)) obtained from the presented method

Grahic Jump Location
Fig. 10

The lowest five natural mode shapes for the P-P shaft carrying three identical rigid disks (see Fig. 6(b)) (with the speed ratio λ = 0). The legends are the same as those of Fig. 7.



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