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Review Article

Circulant Matrices and Their Application to Vibration Analysis

[+] Author and Article Information
Brian J. Olson

Applied Physics Laboratory,
Air and Missile Defense Department,
The Johns Hopkins University,
Laurel, MD 20723-6099
e-mail: brian.olson@jhuapl.edu

Steven W. Shaw

University Distinguished Professor
Department of Mechanical Engineering,
Michigan State University,
East Lansing, MI 48824-1226
e-mail: shawsw@egr.msu.edu

Chengzhi Shi

Department of Mechanical Engineering,
University of California, Berkeley,
Berkeley, CA 94720
e-mail: chengzhi.shi@berkeley.edu

Christophe Pierre

Professor and Vice President
for Academic Affairs
Department of Mechanical Science
and Engineering,
University of Illinois at Urbana-Champaign,
Urbana, IL 61801
e-mail: chpierre@uillinois.edu

Robert G. Parker

L.S. Randolph Professor and Department Head
Department of Mechanical Engineering,
Virginia Polytechnic Institute
and State University,
Blacksburg, VA 24061
e-mail: r.parker@vt.edu

Equation (52a) also holds if N is odd, but the Nyquist component is repeated with multiplicity of two.

A mode shape nodal diameter refers to a line of zero sector responses across which adjacent sectors respond out of phase. For example, in Fig. 11 of Sec. 3.3.4, mode 1 has 0 n.d., modes 2 and 100 have 1 n.d., modes 3 and 99 have 2 n.d., and so on.

1Corresponding author.

Manuscript received January 12, 2014; final manuscript received May 7, 2014; published online June 19, 2014. Editor: Harry Dankowicz.

Appl. Mech. Rev 66(4), 040803 (Jun 19, 2014) (41 pages) Paper No: AMR-14-1006; doi: 10.1115/1.4027722 History: Received January 12, 2014; Revised May 07, 2014

This paper provides a tutorial and summary of the theory of circulant matrices and their application to the modeling and analysis of the free and forced vibration of mechanical structures with cyclic symmetry. Our presentation of the basic theory is distilled from the classic book of Davis (1979, Circulant Matrices, 2nd ed., Wiley, New York) with results, proofs, and examples geared specifically to vibration applications. Our aim is to collect the most relevant results of the existing theory in a single paper, couch the mathematics in a form that is accessible to the vibrations analyst, and provide examples to highlight key concepts. A nonexhaustive survey of the relevant literature is also included, which can be used for further examples and to point the reader to important extensions, applications, and generalizations of the theory.

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Figures

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Fig. 1

(a) Finite element model of a bladed disk assembly [1] and (b) general cyclic system with N identical sectors and nearest-neighbor coupling

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Fig. 2

Example plots of the distinct Nth roots of unity

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Fig. 3

Arrays showing the (i, k) elements of the (a) identity, (b) flip, and (c) cyclic forward shift matrices of dimension N = 3 for i, k = 1, 2, 3

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Fig. 5

The axial gas pressure p(θ): ideal and (notional) actual conditions

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Fig. 6

Example illustration of the discrete temporal and continuous spatial variations of the traveling wave excitation defined by Eq. (63) in real form: (a) the discrete dynamic loads with amplitude F and period T = 2π/nΩ applied to each sector; and (b) the continuous BTW excitation with wavelength N/n and speed C = NΩ/2π relative to the rotating hub

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Fig. 4

Topology diagram of a general cyclic system

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Fig. 7

Engine orders n mod N corresponding to BTW, FTW and SW applied dynamic loading for (i) odd N and (ii) even N (see also Table 4); example plots of applied dynamic loading (represented by the dots) for a model with N = 10 sectors and with (a) n = 1 (BTW), (b) n = 5 (SW), (c) n = 9 (FTW), and (d) n = 10 (SW). The BTW engine order excitation is represented by the solid lines.

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Fig. 8

Linear cyclic vibratory system with N sectors and one DOF per sector

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Fig. 9

Dimensionless natural frequencies ω¯p in terms of the number of n.d. versus mode number p for WC and SC: (a) N = 11 (odd) and (b) N = 10 (even). Also indicated below each figure is, for general N, the number of n.d. at each value of p and also the mode numbers corresponding to SW, BTW, and FTW.

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Fig. 10

A backward traveling wave apΦp(i-1 + Cpτ) = apcos(ϕp(i - 1) + ω¯pτ) with amplitude ap, wavelength 2π/ϕp = N/(p - 1), and speed Cp = ω¯p/ϕp

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Fig. 11

Normal modes of free vibration for a model with N = 100 sectors. Mode 1 consists of a SW, in which each sector oscillates with the same amplitude and phase. Mode 51 also corresponds to a SW, but neighboring oscillators oscillate exactly 180 deg out of phase. Modes 2–50 (resp. 52–100) consist of BTWs (resp. FTWs).

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Fig. 12

(a) Campbell diagram and (b) corresponding frequency response curves |qiss(τ)| for N = 10, ν = 0.5, f = 0.01, and each n = 1,2,…,N

Grahic Jump Location
Fig. 13

(a) Campbell diagram for N = 10, ν = 0.5, f = 0.01, and n = 1,…,20N and (b) the corresponding frequency response curves |qiss(τ)| corresponding to n = N,…,2N. Engine order lines are not shown for n = N + 1,N + 2,…,2N - 1, and so on.

Grahic Jump Location
Fig. 14

(a) Model of bladed disk assembly and (b) sector model

Grahic Jump Location
Fig. 15

The topology of a bladed disk assembly fitted with absorbers in (a) physical space and (b) modal space. The modal transformation q(τ) = (E⊗I)u(τ) reduces the cyclic array of N, two-DOF sector models (B,A), which together form a 2N-DOF coupled system, to a set of N, two-DOF block decoupled models (Bp,Ap).

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