From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: (a) Finite element model of a bladed disk assembly [1] and (b) general cyclic system with N identical sectors and nearest-neighbor coupling

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: Example plots of the distinct Nth roots of unity

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: 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

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: Topology diagram of a general cyclic system

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: The axial gas pressure p(θ): ideal and (notional) actual conditions

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: 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

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: 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.

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: Linear cyclic vibratory system with N sectors and one DOF per sector

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: 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.

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: 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

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: 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).

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: (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

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: (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.

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: (a) Model of bladed disk assembly and (b) sector model

From: Circulant Matrices and Their Application to Vibration Analysis Date of download: 10/29/2017 Copyright © ASME. All rights reserved. From: Circulant Matrices and Their Application to Vibration Analysis Appl. Mech. Rev. 2014;66(4):040803-040803-41. doi:10.1115/1.4027722 Figure Legend: 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).