Circulant Matrices and Their Application to Vibration Analysis Brian Olson, Steven Shaw, Chengzhi Shi, Christophe Pierre, Robert Parker
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Brian Olson, Steven Shaw, Chengzhi Shi, Christophe Pierre, Robert Parker. Circulant Matrices and Their Application to Vibration Analysis. Applied Mechanics Reviews, American Society of Mechanical Engineers, 2014, 66 (4), 10.1115/1.4027722. hal-01378829
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Distributed under a Creative Commons Attribution| 4.0 International License Circulant Matrices and Their Application to Vibration Analysis
Brian J. Olson Steven W. Shaw Chengzhi Shi Applied Physics Laboratory, University Distinguished Professor Department of Mechanical Engineering, Air and Missile Defense Department, Department of Mechanical Engineering, University of California, Berkeley, The Johns Hopkins University, Michigan State University, Berkeley, CA 94720 Laurel, MD 20723-6099 East Lansing, MI 48824-1226 e-mail: [email protected] e-mail: [email protected] e-mail: [email protected]
Christophe Pierre Robert G. Parker Professor and Vice President for Academic Affairs L.S. Randolph Professor and Department Head Department of Mechanical Engineering, Department of Mechanical Science and Engineering, Virginia Polytechnic Institute and State University, University of Illinois at Urbana-Champaign, Blacksburg, VA 24061 Urbana, IL 61801 e-mail: [email protected] e-mail: [email protected]
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, applica-tions, and generalizations of the theory.
systems possess special properties, such as symmetries, which aid in the analysis by enabling significant reduction of the fi- nite element models (Fig. 1(a)). This is particularly true for systems with cyclic symmetry. 1 Introduction The main goal of this tutorial is to consider the vibrations of structural systems with cyclic symmetry, also known as rotation- “The theory of matrices exhibits much that is visually attractive. Thus, diagonal matrices, symmetric matrices, (0, 1) matrices, and ally periodic systems. A useful geometric view of these systems is the like are attractive independently of their applications. In the that of a circular disk (a pie) split into N equal sectors (i.e., equally same category are the circulants.” sized pieces of the pie), each of which contains an identical Philip J. Davis mechanical structure with identical coupling to forward-nearest- neighbors and to ground (Fig. 1(b)). The modeling and analysis of structural vibration is a The theory of circulants also applies to more general forms mature field, especially when vibration amplitudes are small of coupling with non-nearest-neighbors, for example through and linear models apply. For such models, the powerful tools a base substructure, as long as the rotational symmetry is of modal analysis and superposition allow one to decompose preserved. These structural arrangements arise naturally in the system and its response into a set of uncoupled single certain types of rotating machines. Turbomachinery examples degree-of-freedom (DOF) systems, each of which captures the include bladed disks, such as fans, compressors, turbines, and motion of the overall system in a given normal mode. For impeller stages of aircraft, helicopter engines, power plants, geometrically simple continua, the natural frequencies, vibra- as well as propellers, pumps, and the like. Rotational perio- tion modes, and response to known excitations can be analyti- dicity also arises in some stationary structures such as satel- cally determined. When discrete models are developed, matrix lite antennae. In some systems, such as planetary gears and methods are readily applied for both natural frequency and rotors with pendulum vibration absorbers, the overall system response analyses. For general system models with no special is not necessarily cyclic, but subcomponents of it may be. properties, which occur in the majority of applications, large- When perfect cyclic symmetry of a model is assumed, special scale computational models must be developed, typically modal properties exist that significantly facilitate its vibration using finite element methods. However, certain classes of analysis.
1 diagonal and also shares this cyclic property. The size of the ele- ments of K is equal to the number of DOFs per sector, and is denoted by M. Thus, a system with N sectors and M DOFs per sec- tor has a total of NM DOFs. The most important utility of the theory of circulants in analyzing rotationally periodic systems is that they enable a NM-DOF system to be decomposed to a set of NM-DOF uncoupled systems using the appropriate coordinate transformation. Admittedly, the same can be accomplished using brute-force methods to uncouple the entire system using modal analysis, but such an approach overlooks fundamental properties that are crucial to understanding the free and forced response of these systems and requires significantly more computational power. This is the central motivation for understanding and utiliz- ing circulants to analyze cyclic systems. The vibration modes of rotationally periodic systems consist of multiple pairs of repeated natural frequencies (eigenvalues) that lead to pairs of degenerate normal modes (eigenvectors). The number and nature of such pairs depend on whether N is even or odd. Each mode pair is characterized as a pair of standing waves (SWs) with different spatial phases, or a pair of traveling waves, labeled as a forward traveling wave (FTW) and backward travel- ing wave (BTW) when following the terminology used in applica- tions to rotating machinery. The choice of formulation is based on convenience for a given application, which depends on the nature of the system excitation. For example, the excitation frequency is proportional to the engine speed for many cyclic rotating systems, which leads to the so-called engine order (e.o.) excitation, and often the spatial nature of the excitation (in the rotating frame of reference) is in the form of a traveling wave. When such excita- tion is applied to systems with cyclic symmetry, the response also has special properties that can be easily uncovered by making use of the system traveling wave vibration modes. The strength of the intersector coupling is an important parame- ter in rotationally periodic systems. When the intersector coupling is strong, the frequencies of the mode pairs are well separated. In contrast, weak intersector coupling yields closely spaced frequen- cies, high modal density, and large sensitivity to cyclic-symme- try-breaking imperfections. A wave representation of the response [2] shows that the strength of the coupling determines frequency Fig. 1 (a) Finite element model of a bladed disk assembly [1] passband widths, wherein unattenuated propagation of waves and (b) general cyclic system with N identical sectors and nearest-neighbor coupling takes place. Weak intersector coupling leads to narrow passbands, and the passbands widen as the coupling strength increases. Another important parameter for cyclically symmetric structures The nature of rotationally periodic systems imposes a cyclic is the total number of sectors. The modal density is larger for large structure on their mass and stiffness matrices, which are block cir- N, which corresponds to more natural frequencies within each fre- culant for systems with many DOFs per sector (Fig. 1(a)) and cir- quency passband. In all cases, the modes are spatially distributed, culant for the special case of a single DOF per sector (Fig. 1(b)). or extended, for models of cyclic systems. That is, the pattern of By denoting the stiffness of the internal elements of each sector displacements in a modal response is uniformly spread around the by K0 and the coupling stiffness between sectors as –K1, the stiff- circumference of the structure. ness matrices of rotationally periodic structures with nearest- Systems with cyclic symmetry have been studied in the context neighbor coupling have the general form of vibration analysis for over 40 yr. Early work considered proper- 2 3 ties of the vibration modes [3,4] and the steady-state response to K0 K1 0 … 0 K1 harmonic excitation [5–7] of tuned and mistuned turbomachinery 6 7 6 K1 K0 K1 … 007 rotors. Many of these contributions were motivated by vibration 6 7 6 0 K1 K0 … 007 studies of general rotationally periodic systems [8–20], bladed 6 7 K ¼ 6 ...... 7 disks [1,3,21–30], planetary gear systems [31–43], rings [44,45], 6 ...... 7 4 5 circular plates [46–48], disk spindle systems [49–51], centrifugal 000… K0 K1 pendulum vibration absorbers [52–56], space antennae [57], and K1 00… K1 K0 microelectromechanical system frequency filters [58]. Implicit in these investigations is the assumption of perfect symmetry which, where K0 and K1 are themselves matrices for the complex model of course, is an idealization. Perfect symmetry gives rise to well- shown in Fig. 1(a) and scalars for the simplest prototypical model structured vibration modes [9,31,33–37,39,53,56], which are char- shown in Fig. 1(b). A key property of K is that the elements of acterized by certain phase indices that define specific phase rela- each row are obtained from the previous row by cyclically per- tionships between cyclic components in each vibration mode [18]. muting its entries. That is, for j ¼ 2; 3; …; N, row j is obtained This vibration mode structure is critical in the investigation of from row j – 1 by shifting the elements of row j – 1 to the right by dynamic response of cyclic systems using modal analysis one position and wrapping the right-end element of row j – 1 into [54]. These special properties of rotationally periodic structures the first position. This is precisely the form of a circulant matrix, save tremendous calculation effort in the analysis of the system which is formally defined in Sec. 2.2. The mass matrix of a rota- dynamics [59–61]. The properties of cyclic symmetry are not only tionally periodic structure with nearest-neighbor coupling is block used in the study of mechanical vibrations, they are also important
2 to the analysis of elastic stress [62,63] and coupled cell the seminal work by Davis [96] and is presented using mathemat- networks [64]. ics and notation that should be familiar to the vibrations engineer. The special properties of systems with cyclic symmetry extend Selected topics from linear algebra are reviewed in Sec. 2.1 to nonlinear systems, where they are expressed quite naturally in to introduce relevant notion and support the theoretical terms of symmetry groups: the cyclic group, in particular [65–68]. development of circulant matrices in Secs. 2.2–2.8. This material The group theoretic formulation can also be applied to the linear is included for completeness; the apprised reader can skip directly vibration problems considered in this paper [69,70], but the to Secs. 2.2 and 2.3, where circulant and block circulant matrices approach presented here is more approachable to readers with a (also referred to as circulants and block circulants) are defined. standard engineering background in linear algebra. Representations of circulants are discussed in Sec. 2.4. Diagonal- The extension to systems with small imperfections that perturb ization of circulants and block circulants is discussed at length in the cyclic symmetry has led to important results related to mode Sec. 2.5, which begins with a treatment of the Nth roots of unity localization, which arises in systems with high modal density in Sec. 2.5.1 and the Fourier matrix in Sec. 2.5.2. It is subse- caused by weak intersector coupling or a large number of sectors. quently shown how to diagonalize the cyclic forward shift matrix In particular parameter regimes, the mode shapes are highly sensi- in Sec. 2.5.3 a circulant in Sec. 2.5.4, and a block circulant in tive to small, symmetry-breaking imperfections among the nomi- Sec. 2.5.5. Some generalizations of the theory are discussed in nally identical sectors, and the spatial nature of the vibration Sec. 2.6, including the diagonalization of block circulants with modes can become highly localized. For these cases, the vibration circulant blocks. Relevant mathematics of the DFT and IDFT are energy is focused in a small number of sectors, and sometimes summarized in Sec. 2.7. Finally, the circulant eigenvalue problem even a single sector. This behavior, which stems from the seminal (cEVP) is discussed in Sec. 2.8, including the eigenvalues and work of Anderson on lattices [71], was originally recognized to be eigenvectors of circulants and block circulants, their symmetry relevant to structural vibrations by Hodges and Woodhouse characteristics, and connection to the DFT process. [72,73] and Pierre and Dowell [74], and has been extensively studied from both fundamental [75] and applied [76–78] points of view. The phenomenon of mode localization is also observed in 2.1 Mathematical Preliminaries. Definitions and relevant the forced response and has practical implications for the fatigue properties of special operators and matrices are discussed in life of bladed disks in turbomachinery [79,80]. It is interesting to Secs. 2.1.1 and 2.1.2, respectively, including the direct (Kro- note that localization also arises in nonlinear systems with perfect necker) product, and Hermitian, unitary, cyclic forward shift, and symmetry, where the dependence of the system natural frequen- flip matrices. This is followed in Sec. 2.1.3 with a treatment of cies on the amplitudes of vibration naturally leads to the possibil- matrix diagonalizability. ity of mistuning of frequencies between sectors if their amplitudes 2.1.1 Special Operators. Let C denote the set of complex are different [81–88]. numbers and Zþ be the set of positive integers. Another topic central to vibration analysis that relies on the DEFINITION 1 (Direct Sum). For each i ¼ 1; 2; …; N and pi pi theory of circulants is the discrete Fourier transform (DFT) pi 2 Zþ, let Ai 2 C . Then the direct sum of Ai is denoted by [89,90]. The DFT was known to Gauss [91], and is the most com- mon tool used to process vibration signals from experimental N i¼1 Ai ¼ A1 A2 … AN measurements and numerical simulations. The DFT and inverse DFT (IDFT) provide a computationally convenient means of and results in the block diagonal square matrix determining the frequency content of a given signal. Because the 2 3 mathematics of circulants is at the heart of the computation of the A1 0 … 0 DFT, we include a brief introduction to the relationship between 6 7 6 0A2 … 0 7 the DFT and IDFT, and its connection to the theory of circulants. A ¼ 6 . . . . 7 The goal of this paper is to provide a detailed theory of circu- 4 . . .. . 5 lant matrices as it applies to the analysis of free and forced struc- 00… AN tural vibrations. Much of the material was developed as part of the Ph.D. research of the lead author [21,22,92–95]. References to of order p1 þ p2 þ þpN, where each zero matrix 0 has the other relevant work are included throughout this paper, but we do appropriate dimension. ᭝ not claim to provide an exhaustive survey of the relevant It is convenient to define the operator diagð Þ that takes as its literature. The remainder of the paper is organized as follows. argument the ordered set of matrices A1; A2; …; AN and results in Section 2 gives a quite exhaustive and self-contained treatment of the block diagonal matrix given in Definition 1, that is, the theory of circulants, which is distilled from the seminal work by Davis [96]. We adopt a presentation style similar to that of A ¼ diagðA1; A2; …; ANÞ¼ diag ðAiÞ Ottarsson [97], one that should be familiar to an analyst in the i¼1;…;N vibrations engineering community. This section is meant to act simultaneously as a detailed reference and tutorial, including For the case when each Ai ¼ ai is a scalar (1 1), the direct sum proofs of the main results and simple illustrative examples. of ai is denoted by the diagonal matrix Section 3 provides three examples that make use of the theory, including ordinary circulants and the more general block circulant diagða1; a2; …; aNÞ¼ diag ðaiÞ matrices. Particular attention is given to cyclic systems under trav- i¼1;…;N eling wave engine order excitation because this type of system n forcing appears naturally in many relevant applications of rotating DEFINITION 2 (Direct Product). Let a; b 2 C . Then the direct prod- machinery. The apprised reader, or the reader who wishes to learn uct (or Kronecker product) of a and bT is the square matrix by example, can skip directly to Sec. 3, depending on their back- 2 3 ground, and revisit Sec. 2 as warranted. The paper closes with a a1b1 a1b2 a1bn 6 7 brief summary in Sec. 4. 6 a2b1 a2b2 a2bn 7 a bT ¼ 6 . . . . 7 4 . . . . . 5 2 The Theory of Circulants anb1 anb2 anbn This section details the theory and mathematics of circulant matrices that are relevant to vibration analysis of mechanical where ð ÞT denotes transposition. If A 2 Cm n and B 2 Cp q are structures with cyclic symmetry. The basic theory is distilled from matrices, then the direct product of A and B is the matrix
3 2 3 a11B a12B a1nB (6) The transpose or conjugate transpose of a direct product 6 7 6 a21B a22B a2nB 7 yields the direct product of two transposes or conjugate A B ¼ 6 . . . . 7 transposes. If A and B are square matrices, then 4 . . .. . 5 a B a B a B T m1 m2 mn ðA BÞ ¼ AT BT (6a) of dimension mp nq. ᭝ ðA BÞH ¼ AH BH (6b) Example 1. Consider the matrices where ð ÞH ¼ ð Þ T is the conjugate transpose and ð Þ denotes 12 complex conjugation. A ¼ ½ 123 and B ¼ 34 (7) If A and B are square matrices with dimensions n and m, respectively, then
Then the direct product of A and B is given by m n detðA BÞ¼ðdet AÞ ðdet BÞ (7a) 12 A B ¼ ½ 123 trðA BÞ¼trðAÞtrðBÞ (7b) 34 12 12 12 where detð Þ and trð Þ denote the matrix determinant and ¼ 1 ; 2 ; 3 trace. 34 34 34 12 24 36 2.1.2 Special Matrices. The definitions and relevant proper- ¼ ; ; ties of selected special matrices are summarized. Hermitian and 34 68 912 unitary matrices are defined first (see Table 1), followed by a brief 12243 6 treatment of two important permutation matrices: the cyclic for- ¼ 3468912 ward shift matrix and the flip matrix. The details of circulant mat- rices and the Fourier matrix, which are employed extensively throughout this work, are deferred to Secs. 2.2, 2.3, and 2.5.2. Because A is 1 3 and B is 2 2, the direct product A B has N N dimension 1 2 3 2, or 2 6. DEFINITION 3 (Hermitian Matrix). A matrix H 2 C is Hermi- H ᭝ Some important properties of the direct product are as follows: tian if H ¼ H . The elements of a Hermitian matrix H satisfy hik ¼ h ki for all (1) The direct product is a bilinear operator. If A and B are i; k ¼ 1; 2; …; N. Thus, the diagonal elements hii of a Hermitian square matrices and a is a scalar, then matrix must be real, while the off-diagonal elements may be com- plex. If H ¼ HT then H is said to be symmetric. N N aðA BÞ¼ðaAÞ B ¼ A ðaBÞ (1) DEFINITION 4 (Unitary Matrix). A matrix U 2 C is unitary if UHU ¼ I, where I is the N N identity matrix. ᭝ (2) The direct product distributes over addition. If A, B, and C Real unitary matrices are orthogonal matrices. If a matrix U is are square matrices with the same dimension, then unitary, then so too is UH. To see this, consider ðUHÞHðUHÞ H ðA þ BÞ C ¼ A C þ B C (2a) ¼ UU ¼ I, from which it follows that A ðB þ CÞ¼A B þ A C (2b) UHU ¼ UUH ¼ I (8)
(3) The direct product is associative. If A, B, and C are square H 1 matrices, then Finally, if U is unitary and nonsingular, then U ¼ U . A general permutation matrix is formed from the identity A ðB CÞ¼ðA BÞ C (3) matrix by reordering its columns or rows. Here, we introduce two such matrices: the cyclic forward shift matrix and the flip matrix. (4) The product of two direct products yields another direct DEFINITION 5 (Cyclic Forward Shift Matrix). The N N cyclic product. If A, B, C, and D are square matrices such that AC forward shift matrix is given by and BD exist, then 2 3 010 00 A B C D AC BD (4) 6 7 ð Þð Þ¼ð Þ ð Þ 6 001 007 6 ...... 7 6 ...... 7 (5) The inverse of a direct product yields the direct product of rN ¼ 6 7 6 000 107 two matrix inverses. If A and B are invertible matrices, 4 5 then 000 01 100 00N N ðA BÞ 1 ¼ A 1 B 1 (5) which is populated with ones along the superdiagonal and in the 1 where ð Þ denotes the matrix inverse. (N, 1) position, and zeros otherwise. ᭝ The cyclic forward shift matrix plays a key role in the represen- tation and diagonalization of circulant matrices, which are dis- cussed in Secs. 2.4 and 2.5. Table 1 Selected special matrices Example 2. Let a ¼ (a, b, c) be a three-vector. Then the operation Type Condition 2 3 010 Symmetric A AT 6 7 ¼ ar abc4 0015 Hermitian A ¼ AH 3 ¼ ½ Orthogonal ATA ¼ I (or) AT ¼ A 1 100 Unitary AHA ¼ I (or) AH ¼ A 1 ¼ðc; a; bÞ
4 Table 2 Selected types of linear transformations Proof. Let Q be an arbitrary nonsingular matrix. Then
Type Condition Transformation pðBÞ¼pðQ 1AQÞ XN Equivalence P and Q are nonsingular B ¼ PAQ 1 k Congruence Q is nonsingular B ¼ QTAQ ¼ ckðQ AQÞ Similarity Q is nonsingular B ¼ Q 1AQ k¼0 T 1 1 1 1 Orthogonal Q is nonsingular and orthogonal B ¼ Q AQ ¼ Q AQ ¼ c0I þ c1Q AQ þ c2Q AQQ AQ þ Unitary Q is nonsingular and unitary B QHAQ Q 1AQ ¼ ¼ 1 1 þ cNQ AQ Q AQ ¼ c I þ c Q 1AQ þ c Q 1A2Q þ þc Q 1ANQ 0 1 2 N 1 2 N ¼ Q c0I þ c1A þ c2A þ þcNA Q cyclically shifts the entries of a by one entry to the right. That is, ¼ Q 1pðAÞQ the ith entry of a is shifted to entry i þ 1, except for entry N ¼ 3, which is placed in position 1 of a. which completes the proof. DEFINITION 6 (Flip Matrix). The N N flip matrix is given by k If p(t) ¼ t with k > 0 in Theorem 2, then we have the following 2 3 result. 1 k 1 k 100 00 COROLLARY 2. If B ¼ Q AQ, then B ¼ Q A Q for any ٗ 7 6 6 000 017 k 2 Zþ. 6 7 6 000 107 DEFINITION 8 (Diagonalizable Matrix). A square matrix A is jN ¼ 6 ...... 7 6 ...... 7 diagonalizable if there exists a nonsigular matrix Q and a diago- 4 5 nal matrix D such that Q 1AQ ¼ D. ᭝ 001 00 Thus, a matrix is diagonalizable if it is similar to a diagonal ma- 010 00N N trix. If A is diagonalizable by Q, we say that Q diagonalizes A and that Q is the diagonalizing matrix. which is populated with ones in the (1, 1) position and along the THEOREM 3. An N N matrix A is diagonalizable if it has N lin- ٗ .subantidiagonal, and zeros otherwise. ᭝ early independent eigenvectors COROLLARY 1. Let jN be the flip matrix. Then Proof. Suppose A has N linearly independent eigenvectors and denote them by q1; q2; …; qN. Let ki be the eigenvalue of A corre- ) sponding to for each 1; …; . Then if is the matrix that 2 qi i ¼ N Q j ¼ IN N has as its ith column the vector qi, it follows that H T 1 jN ¼ jN ¼ jN ¼ jN AQ ¼ ðÞAq1; Aq2; …; AqN
¼ ðÞq1k1; q2k2; …; qNkN ٗ .where IN is the N N identity matrix ¼ ðÞq1; q2; …; qN diag ðkiÞ 2.1.3 Matrix Diagonalizability. Matrix diagonalization is the i¼1;…;N process of taking a square matrix and transforming it into a diago- QD nal matrix that shares the same fundamental properties of the underlying matrix, such as its characteristic polynomial, trace, and 1 determinant. This section defines matrix diagonalizability in terms Because Q is nonsingular by hypothesis, D ¼ Q AQ. of similarity, provides necessary conditions for a matrix to be diagonalizable, and summarizes relevant properties of diagonaliz- 2.2 Circulant Matrices able matrices. Diagonalization of circulant matrices is deferred to Sec. 2.5. DEFINITION 9 (Circulant Matrix). AN N circulant matrix (or DEFINITION 7 (Similarity Transformation). Let Q be an arbitrary circulant, or ordinary circulant) is generated from the N-vector 1 nonsingular matrix. Then B ¼ Q AQ is a similarity transforma- fc1; c2; …; cNg by cyclically permuting its entries, and is of the tion and B is said to be similar to A. ᭝ form 1 2 3 If B is similar to A, then A ¼ Q 1 BQ 1 is similar to B. c1 c2 cN It therefore suffices to say that A and B are similar. A summary of 6 7 6 cN c1 cN 1 7 selected additional linear transformations is provided in Table 2. C ¼ 6 . . . . 7 D If B is orthogonally (resp. unitarily) similar to A, then we say that 4 . . . . . 5 A and B are orthogonally (resp. unitarily) similar matrices. c2 c3 c1 THEOREM 1. If A and B are similar matrices, then they have the ٗ .same characteristic equation and hence the same eigenvalues Theorem 1 guarantees that the eigenvalues of a matrix are DEFINITION 10 (Generating Elements). Let the N N circulant preserved under a similarity transformation. A proof can be found matrix C be given by Definition 9. Then the elements of the in any standard textbook on linear algebra [98,99]. Because N-vector QT ¼ Q 1 for orthogonal Q and QH ¼ Q 1 for unitary Q, the eigenvalues are also preserved under orthogonal and unitary fc1; c2; …; cNg transformations. THEOREM 2. Let the matrices A and B be similar. Then if are said to be the generating elements of C. ᭝ Thus, a circulant matrix is defined completely by the generating XN elements in its first row, which are cyclically shifted to the right k pðtÞ¼ ckt by one position per row to form the subsequent rows. The set of k¼0 all such matrices of order N is denoted by CN. A matrix contained in CN is said to be a circulant of type N. denotes a finite polynomial in t with arbitrary coefficients It is convenient to define the circulant operator circð Þ that ck ðk ¼ 1; 2; …; NÞ, the matrix polynomials p(A) and p(B) are takes as its argument the generating elements c1; c2; …; cN and ,similar. ٗ results in the array given in Definition 9, that is
5 2 3 2 3 C ¼ circðc1; c2; …; cNÞ (9) c11 c12 c1N c11 c12 c1N 6 7 6 7 6 c21 c22 c2N 7 6 c1N c11 c1ðN 1Þ 7 6 7 6 7 An N N circulant is also characterized in terms of its (i, k) entry C ¼ 4 . . . . 5 ¼ 4 . . . . 5 ...... by ðCÞik ¼ ck iþ1ðmodNÞ for i; k ¼ 1; 2; …; N. Example 3. The circulant array formed by the generating ele- cN1 cN2 cNN c12 c13 c11 ments a, b, c, d can be written as which is a N N circulant matrix with generating elements 2 3 abcd c11; c12; …; c1N. 6 7 Any matrix that commutes with the cyclic forward shift matrix 6 dabc7 circða; b; c; dÞ¼4 5 2 C4 is, therefore, a circulant. Theorem 4 also says that circulant cdab matrices are invariant under similarity transformations involving bcda the cyclic forward shift matrix. That is, C is similar to itself for a similarity transformation using rN. which is a circulant matrix of type 4. Example 5. Consider the 3 3 matrix If a matrix is both circulant and symmetric, its generating ele- 2 3 ments are abc ( A ¼ 4 cab5 c1; …; cN ; cNþ2; cN ; …; c3; c2; N even bca 2 2 2 (10) c1; …; cN 1; cNþ1; cNþ1; cN 1; …; c3; c2; N odd 2 2 2 2 Then which are necessarily repeated. Only (N þ 2)/2 generating ele- 2 32 3 2 3 ments are distinct if N is even and (N þ 1)/2 are distinct if N is abc 010 cab odd. The set of all N N symmetric circulants is denoted by 6 76 7 6 7 4 cab54 0015 ¼ 4 bca5 SCN. A matrix contained in SCN is said to be a symmetric circu- lant of type N. bca 100 abc 2 32 3 Example 4. The 5 5 matrix 010 abc 6 76 7 2 3 ¼ 4 00154 cab5 abccb 6 7 100 bca 6 babcc7 6 7 circða; b; c; c; bÞ¼6 cbabc7 2 SC5 4 5 ccbab which implies that Ar3 ¼ r3A. Thus, A ¼ circða; b; cÞ2C3 is a bccba circulant matrix of type N ¼ 3. Next we introduce block circulant matrices, which are natural generalizations of ordinary circulants. is both symmetric and circulant. Because N ¼ 5 is odd, it has (N þ 1)/2 ¼ 3 distinct elements. The matrix defined in Example 3 is not a symmetric circulant 2.3 Block Circulant Matrices. A block circulant matrix because its generating elements are distinct. Next, we give a nec- is obtained from a circulant matrix by replacing each entry ck in essary and sufficient condition for a square matrix to be circulant. Definition 9 by the M M matrix Ci for i ¼ 1; 2; …; N. DEFINITION 11 (Block Circulant Matrix). Let Ci be a M M THEOREM 4. Let rN be the cyclic forward shift matrix. Then a N N matrix C is circulant if and only if Cr ¼ r C. ٗ matrix for each i ¼ 1; 2; …; N. Then a NM NM block circulant N N matrix (or block circulant) is generated from the ordered set Proof. Let C be an N N matrix with arbitrary elements cik for i; k ¼ 1; 2; …; N. Then fC1; C2; …; CNg, and is of the form 2 3 2 3 C1 C2 CN c1N c11 c12 c1ðN 1Þ 6 7 6 7 6 CN C1 CN 1 7 6 c2N c21 c22 c2ðN 1Þ 7 6 7 C ¼ 6 . . . . 7 D CrN ¼ . . . . . 4 . . .. . 5 4 ...... 5 C2 C3 C1 cNN cN1 cN2 cNðN 1Þ
DEFINITION 12 (Generating Matrices). Let the NM NM block and circulant C be given by Definition 11. Then the elements of the 2 3 ordered set c21 c22 c23 c2N 6 7 6 c31 c32 c33 c3N 7 fC1; C2; ; CNg rNC ¼ 6 . . . . . 7 4 ...... 5 are said to be the generating matrices of C. ᭝ c11 c12 c13 c1N A block circulant is therefore defined completely by its generat- ing matrices. The matrix array given by Definition 11 is said to be These matrices are equal if and only if the equalities a block circulant of type (M, N). The set of all such matrices is denoted by BCM;N. A matrix C 2 BCM;N is not necessarily a cir- c1N ¼ c21; c11 ¼ c22; c1ðN 1Þ ¼ c2N culant, as the following example demonstrates. Example 6. Let c2N ¼ c31; c21 ¼ c32; c2ðN 1Þ ¼ c3N . . . . . . . . . 2 1 10 A ¼ and B ¼ cNN ¼ c11; cN1 ¼ c12; cNðN 1Þ ¼ c1N 12 0 1 are satisfied. Then C can be written as Then
6 2 3 AB0B A ¼ ar0 þ br1 þ cr2 6 7 3 3 3 6 BAB07 2 C ¼ 6 7 ¼ aI3 þ br3 þ cr3 4 0BAB5 2 3 2 3 2 3 100 010 001 B0BA 2 3 6 7 6 7 6 7 ¼ a4 0105 þ b4 0015 þ c4 1005 2 1 10 0 0 10 6 7 6 12 0 1000 1 7 001 100 010 6 7 2 3 6 7 abc 6 10 2 1 10 0 07 6 7 6 7 6 0 1 12 0 10 07 ¼ 4 cab5 6 7 ¼ 6 7 6 00 10 2 1 107 bca 6 7 6 000 1 12 0 1 7 6 7 4 5 where I and r are the 3 3 identity and cyclic forward shift 1000 10 2 1 3 3 matrices. 0 10 0 0 1 12 Corollary 3 is exploited in Sec. 2.5.4 to diagonalize a general circulant matrix, and can be generalized to represent a general block circulant matrix in terms of the cyclic forward shift matrix is a block circulant of type (2, 4), but it is not a circulant. and its powers. Next we give a necessary and sufficient condition for a matrix COROLLARY 4. Let C 2 BC be a block circulant matrix of to be block circulant. M;N type (M, N) with generating matrices C1; C2; …; CN. Then C can THEOREM 5. Let rN be the cyclic forward shift matrix of dimen- be represented by the matrix sum sion N and IM be the identity matrix of dimension M. Then a NM NM matrix C is a block circulant of type (M, N) if and only 0 1 N 1 C ¼ rN C1 þ rN C2 þ þrN CN ٗ .if CðrN IMÞ¼ðrN IMÞC The proof of Theorem 5 follows similarly to that of Theorem 4 ٗ .where rN is the cyclic forward shift matrix by replacing the scalar elements cik with M M matrices Cik for i; k ¼ 1; 2; …; N. The reader can verify that the matrix C in Corollaries 3 and 4 motivate the following result, which cap- Example 6 satisfies the condition in Theorem 5, but not that in tures the representation of circulant and block circulant matrices Theorem 4. in terms of the cyclic forward shift matrix, and facilitates their A block circulant, block symmetric matrix of type (M, N) has diagonalization in Secs. 2.5.4 and 2.5.5. generating matrices of the same form as Eq. (10), and is obtained DEFINITION 13. Let t and s be arbitrary square matrices. Then the function by replacing each entry ck by the M M matrix Ck for k 1; 2; …; N. The set of all such matrices is denoted by ¼ XN BCBSM;N. The matrix C in Example 6 is recognized to be a qðt; sÞ¼ tk 1 s block symmetric, block circulant matrix of type (2, 4), that is, it is k¼1 contained in BCBS2;4, which is a subset of BC2;4. is a finite sum of direct products. ᭝ COROLLARY 5. Let rN be the N N cyclic forward shift matrix. 2.4 Representations of Circulants. It is clear from Defini- Then a circulant matrix with generating elements c1; c2; …; cN tion 5 that the N N cyclic forward shift matrix is a circulant with and a block circulant matrix with generating elements N generating elements 0; 1; 0; …; 0; 0. The integer powers of r N C1; C2; …; CN can be represented by the matrix sums can be written as XN 9 qðr ; c Þ¼ rk 1c ¼ circðc ; c ; …; c Þ 0 > N k N k 1 2 N rN ¼ circð1; 0; 0; 0; 0; …; 0; 0Þ¼IN > k¼1 > 1 > XN rN ¼ rN ¼ circð0; 1; 0; 0; 0; …; 0; 0Þ > > q r ; C rk 1 C circ C ; C ; …; C 2 => ð N kÞ¼ N k ¼ ð 1 2 NÞ rN ¼ circð0; 0; 1; 0; 0; …; 0; 0Þ k¼1 (11) . > ٗ < . > where the function qð Þ is given by Definition 13. N 1 > r ¼ circð0; 0; 0; 0; 0; …; 0; 1Þ > What is meant by the notation qðrN; ckÞ, for example, is to sub- N > N 0 ; stitute t with rN and s with ck in Definition 13 and then perform rN ¼ circð1; 0; 0; 0; 0; …; 0; 0Þ¼rN ¼ IN the summation observing any indices k introduced by the substitution. where each successive power cyclically permutes the generating elements. This enables the representation of a general circulant in 2.5 Diagonalization of Circulants. Any circulant or block terms of a finite matrix polynomial involving the cyclic forward circulant matrix can be represented in terms of the cyclic forward shift matrix and its powers. shift matrix according to Corollary 5. The diagonalization of a general circulant begins, therefore, by finding a matrix that diago- COROLLARY 3. Let C 2 CN be a circulant matrix of type N with nalizes rN. Together with some basic results from linear algebra generating elements c1; c2; …; cN. Then C can be represented by the matrix sum (these are summarized in Sec. 2.1), this leads naturally to the diagonalization of an arbitrary circulant. Regarding a suitable diagonalizing matrix, there are a number of candidates 0 1 2 N 1 C ¼ c1rN þ c2rN þ c3rN þ þcNrN [10–12,16,17,100], but all feature powers of the Nth roots of unity or their real/imaginary parts. In this work, we employ an array composed of the distinct Nth roots of unity (Sec. 2.5.1) and their where rN is the N N cyclic forward shift matrix. ٗ integer powers in the form of the complex Fourier matrix Example 7. The matrix A ¼ circða; b; cÞ from Example 5 can (Sec. 2.5.2). A unitary transformation involving the Fourier matrix be represented by the matrix sum is used to diagonalize the cyclic forward shift matrix (Sec. 2.5.3),
7 general circulant matrices (Sec. 2.5.4), and general block circulant DEFINITION 16. Let wN be the primitive Nth root of unity. Then matrices (Sec. 2.5.5). 2 3 10 6 7 2.5.1 Nth Roots of Unity. A root of unity is any complex num- 6 w 7 6 N 7 ber that results in 1 when raised to some integer N 2 Zþ [101]. 6 2 7 jho 6 w 7 More generally, the Nth roots of a complex number zo ¼ roe are XN ¼ 6 N 7 given by a nonzero number z ¼ rejh such that 6 . 7 4 . . 5
N N jNh jho N 1 z ¼ zo or r e ¼ roe (12) 0 wN N N 2 N 1 ffiffiffiffiffiffiffi ¼ diagð1; wN; wN; …; wN Þ p N where j ¼ 1. Equation (12) holds if and only if r ¼ ro and Nh ¼ ho þ 2pk with k 2 Z. Therefore, is the N N diagonal matrix formed by placing the distinct Nth 9 2 N 1 ᭝ pffiffiffiffi roots of unity 1; wN; wN; …; wN along its diagonal. r N r = ¼ o The matrix XN appears naturally in the diagonalization of cir- h þ 2pk ; k 2 Z (13) culants, which is discussed in Secs. 2.5.3–2.5.5. h ¼ o ; N 2.5.2 The Fourier Matrix. This section introduces the com- and the Nth roots are plex Fourier matrix and its relevant properties, including the sym- metric structure of the N-vectors that compose its columns. A key ffiffiffiffi p ho þ 2pk result is that the Fourier matrix is unitary, which is systematically z ¼ N r exp j ; k 2 Z (14) o N developed and proved. ffiffiffiffi DEFINITION 17 (Fourier Matrix). The N N Fourier matrix is pN Equation (14) shows that the roots all lie on a circle of radius ro defined as 2 3 centered at the origin in the complex plane, and that they are 11 1 1 6 7 equally distributed every 2p=N radians. Thus, all of the distinct 6 1 w w2 wN 1 7 roots correspond to k ¼ 0; 1; 2; …; N 1. 6 N N N 7 1 6 2 4 2ðN 1Þ 7 DEFINITION 14 (Distinct Nth Roots of Unity). The distinct Nth ffiffiffiffi 6 1 w w w 7 EN ¼ p 6 N N N 7 roots of unity follow from Eq. (14) by setting ro ¼ 1 and ho ¼ 0 N 6 . . . . . 7 and are denoted by 4 ...... 5 N 1 2ðN 1Þ ðN 1ÞðN 1Þ ðkÞ 2pk 1 w w w j N N N N N N wN ¼ e Z ᭝ for integers k ¼ 0; 1; 2; …; N 1. ᭝ where wN is the primitive Nth root of unity and N 2 þ. DEFINITION 15 (Primitive Nth Root of Unity). The primitive Nth root of unity is denoted by
j 2p wN ¼ e N which corresponds to k ¼ 1 in Definition 14. ᭝ k COROLLARY 6. The integer powers w of the primitive Nth root N ðkÞ of unity are equivalent to the distinct Nth roots of unity wN for ٗ .k ¼ 1; 2; …; N k j 2p k j 2pk ðkÞ Proof. Consider wN ¼ðe N Þ ¼ e N ¼ wN , which follows from Definition 14. Thus, the powers
1 2 N 1 1; wN; wN; …; wN are equivalent to the distinct Nth roots of unity
ð1Þ ð2Þ ðN 1Þ 1; wN ; wN ; …; wN
0 ð0Þ where wN ¼ wN ¼ 1. Example plots of the distinct Nth roots of unity are shown in ðkÞ Fig. 2, where wN are arranged on the unit circle in the complex plane (centered at the origin) for N ¼ 1; 2; …; 9. Note that ð0Þ ðN=2Þ wN ¼ 1 is real, as is wN ¼ 1ifN is even. The remaining roots appear in complex conjugate pairs. Thus, the distinct Nth roots of unity are symmetric about the real axis in the complex plane. Example 8. Let N ¼ 4. Then the distinct Nth roots of unity are given by the set
2p 2p 2p 2p 0 1 2 3 j 4 0 j 4 1 j 4 2 j 4 3 fw4; w4; w4; w4g¼fe ; e ; e ; e g 0 j p j p j 3p ¼fe ; e 2; e ; e 2 g ¼f1; j; 1; jg
These four distinct roots of unity can be visualized in Fig. 2 for the case of N ¼ 4. Fig. 2 Example plots of the distinct Nth roots of unity
8 Example 9. For the special case of N ¼ 4, the Fourier matrix is The columns ei of the Fourier matrix EN ¼ðe1; …; eNÞ exhibit given by a symmetric structure with respect to the index i ¼ (N þ 2)/2 for 2 3 even N. In Example 9, for instance, the vectors e and 11 1 1 1 6 7 eðNþ2Þ=2 ¼ e3 are real and distinct, and the vectors e2 and e4 1 6 1 j 1 j 7 appear in complex conjugate pairs. This same structure generally E4 ¼ pffiffiffi 6 7 4 4 1 11 1 5 holds for any EN with even N. To see this, consider 1 j 1 j T 1 ðN6qÞ 1 ðN6qÞ N 1 ffiffiffiffi 2 2 eNþ26q ¼ p 1; wN ; …; wN 2 N Clearly the Fourier matrix is symmetric, but generally it is not 1 1 N 1 T Hermitian. It can be written element wise as ¼ pffiffiffiffi 1; w6q ; …; w6q (17) N N N 1ffiffiffiffi ði 1Þðk 1Þ ðENÞik ¼ p wN N where the integers 6q correspond to vector pairs relative to the 1 index i ¼ðN þ 2Þ=2 and the identity ¼ pffiffiffiffi ejði 1Þuk N N6q N w2 ¼ w2 w6q ¼ w6q 1 N N N N ¼ pffiffiffiffi ejðk 1Þui ; i; k ¼ 1; 2; …; N (15) N is employed. The case of q ¼ 0 corresponds to i ¼ðN þ 2Þ=2 and where yields the real vector
2p 1ffiffiffiffi T ui ¼ ði 1Þ (16) eNþ2 ¼ p ð1; 1; 1; …; 1; 1Þ (18) N 2 N is the angle subtended from the positive real axis in the complex which has the same value for each element with alternating signs plane to the ith power of w , which is also the ði þ 1Þth of the N N from element to element. For q 0 the terms w6q are complex roots of unity according to Definition 14 and the numbering 6¼ N conjugates according to the proof of Corollary 7 such that scheme in Fig. 2. H e and e are complex conjugate pairs. There are COROLLARY 7. The matrix E is obtained from E by changing ðÞþðNþ2Þ=2 q ðÞ ðNþ2Þ=2 q N N (N 2)/2 such pairs corresponding to q 1; 2; …; N 2 =2 in the signs of the powers of each element. ٗ ¼f ð Þ g Eq. (17). Finally, the case of i 1 always yields the real vector Proof. The Fourier matrix is unaffected by transposition ¼ H because it is symmetric. Thus, the (i, k) element of EN is 1ffiffiffiffi T e1 ¼ p ð1; 1; 1; …; 1; 1Þ (19) H 1 ði 1Þðk 1Þ 1 ði 1Þðk 1Þ N E ¼ ðENÞ ¼ pffiffiffiffi w ¼ pffiffiffiffi w N ik ik N N N N for even and odd N. A similar formulation for odd N shows that e1 where the identity is real and distinct and the remaining (N – 1)/2 vectors appear in 2p 2p 2p k complex conjugate pairs. This is shown by example in Sec. 3.3 in k j k j N k j N k wN ¼ e N ¼ e ¼ e ¼ wN ; k 2 Z the context of vibration modes for a cyclic structure with a single DOF per sector. H is employed. It follows that EN can be obtained from EN by A key feature of the Fourier matrix is that it is unitary. This is changing the sign of the powers of the Nth roots of unity. essentially a statement of orthogonality of each column of EN and It is shown in Sec. 2.8 that all circulant matrices contained in is captured by the following lemmas. CN share the same linearly independent eigenvectors, the ele- LEMMA 1 (Finite Geometric Series Identity). Let N 2 Zþ and ments of which compose the N columns (or rows) of EN. q 2 C. Then DEFINITION 18. Let wN be the primitive Nth root of unity. Then sþXN 1 for i ¼ 1; 2; …; N the columns of the Fourier matrix EN are qsð1 qNÞ qr ¼ defined by 1 q r¼s T 1 ði 1Þ 2ði 1Þ ðN 1Þði 1Þ ei ¼ pffiffiffiffi 1; w ; w ; …; w ٗ .N N N N for any s 2 Z and q 6¼ 1 1 T Proof. Consider the finite geometric series ¼ pffiffiffiffi 1; ejui ; ej2ui ; …; ejðN 1Þui N sþXN 1 r s sþ1 sþ2 sþN 1 ᭝ q ¼ q þ q þ q þ q where ui is given by Eq. (16). r¼s Example 10. The third column of the Fourier matrix E4 is given by s 2 N 1 T ¼ q ð1 þ q þ q þ þq Þ 1 ð3 1Þ 2ð3 1Þ 3ð3 1Þ e3 ¼ pffiffiffi 1; w ; w ; w 4 4 4 4 Multiplying from the left by q yields 1 2 4 6 T ¼ 1; w4; w4; w4 2 sþXN 1 T r s 2 3 N 1 j 2p 2 j 2p 4 j 2p 6 q q ¼ q ðq þ q þ q þ þq Þ ¼ 1; e 4 ; e 4 ; e 4 2 r¼s 1 T ¼ 1; ejp; ej2p; ej3p 2 Subtraction of the second equation from the first results in 1 ¼ ð1; 1; 1; 1ÞT 2 sþXN 1 ð1 qÞ qr ¼ qsð1 qNÞ for the special case of N ¼ 4. r¼s
9 from which the proof is established by division of the term (1 – q) Lemma 2 allows for representations of the N N identity, flip, because q 6¼ 1 by restriction. and cyclic forward shift matrices in terms of certain conditions on Lemma 1 is used to establish the following result, which is their indices relative to N. required to show that the Fourier matrix is unitary. The orthogon- COROLLARY 8. For i; k ¼ 1; …; N and any integer m, the (i, k) ality condition is fundamental to the diagonalization of circulants elements of the N N identity, flip, and cyclic forward shift matri- in Secs. 2.5.3–2.5.5, and the relationship between the DFT and ces can be represented by the summations IDFT in Sec. 2.7. ( LEMMA 2. Let wN be the primitive Nth root of unity with 1 XN 1 1; i k ¼ mN N 2 Zþ. Then rði kÞ dik ¼ðINÞik ¼ wN ¼ N 0; otherwise sþXN 1 N; i k ¼ mN r¼0 wrði kÞ ¼ ( N 0; otherwise 1 XN 1 1; i þ k 2 ¼ mN r¼s ðj Þ ¼ wrðiþk 2Þ ¼ N ik N N r¼0 0; otherwise ) ٗ .for i; k 2 Z and any s; m 2 Z ði kÞ j 2pði kÞ N N XN 1 1; i k þ 1 ¼ mN Proof. Let q ¼ wN ¼ e and note that q ¼ 1. If 1 rði kþ1Þ j2pm i k ¼ mN, then q ¼ e ¼ 1 for any integer m, and it follows that ðrNÞik ¼ wN ¼ N r¼0 0; otherwise sþXN 1 sþXN 1 rði kÞ r wN ¼ q r¼s r¼s ٗ .where dik is the Kronecker delta ¼ð1Þs þð1Þsþ1 þ þð1ÞsþN 1 The reader can verify Corollary 8 for the special case of N ¼ 3 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} by inspection of the arrays in Fig. 3. N terms We are now ready to state the key result required to diagonalize ¼ N a general circulant matrix. Corollaries 7 and 8 are used to show that the Fourier matrix is unitary. For the case of i k 6¼ mN it follows from Lemma 1 that THEOREM 6 (Unitary Fourier Matrix). The Fourier matrix EN is ٗ .unitary XN 1 XN 1 H rði kÞ r Proof. For 1 i; k N, the (i, k) entry of EN EN is given by wN ¼ q r¼0 r¼0 XN 1 qN EHE ¼ ðEHÞ ðE Þ ¼ (Lem. 1) N N ik N ir N rk 1 q r¼1 1 1 ¼ 1 q XN 1ffiffiffiffi ði 1Þðr 1Þ ¼ p wN ¼ 0 r¼1 N which completes the proof. 1ffiffiffiffi ðr 1Þðk 1Þ Example 11. Consider the orthogonality condition given by p wN (Eq. 15 and Cor. 7) Lemma 2 for s ¼ 0. Then if i k ¼ N ¼ 5, N X5 1 X4 5r XN r 5 j 2p 1 w e 5 ðr 1Þði kÞ 5 ¼ ¼ wN r¼0 r¼0 N r¼1 X4 j 2p 5r ¼ e 5 XN 1 1 rði kÞ r¼0 ¼ w N N ¼ e0 þ ej2p þ ej4p þ ej6p þ ej8p r¼0 ¼ 1 þ 1 þ 1 þ 1 þ 1 ¼ ðÞIN (Cor. 8) ¼ 5 ik which is numerically equal to N, as expected. If instead we set H from which it follows that E EN ¼ IN. i k ¼ 5 but N ¼ 4, then N Example 12. Consider the matrix E4 from Example 9. Because X4 1 X3 the Fourier matrix is unitary, it follows that 2p 5r r 5 j 4 w4 ¼ e r¼0 r¼0 X3 j 2p 5r ¼ e 4 r¼0 X3 j 5pr ¼ e 2 r¼0 5p 0 5p 1 5p 2 5p 3 ¼ e 2 þ e 2 þ e 2 þ e 2 0 5p 5p 15p ¼ e þ e 2 þ e þ e 2 ¼ 1 þ j 1 j ¼ 0 Fig. 3 Arrays showing the (i, k) elements of the (a) identity, (b) flip, and (c) cyclic forward shift matrices of dimension N 5 3for sums to zero. i, k 5 1, 2, 3
10 2 3 2 3 H 1111 1111 expanding the product ei ek according to Eq. (15) and invoking 6 7 6 7 6 7 6 7 Lemma 2, as is done in the proof of Theorem 6. The same result 6 1 j 1 j 7 6 1 j 1 j 7 also follows by expanding H 1ffiffiffi 6 7 1ffiffiffi 6 7 E E4 ¼ p 6 7 p 6 7 4 4 6 7 4 6 7 2 3 4 1 11 1 5 4 1 11 1 5 eH 6 1 7 6 7 1 j 1 j 1 j 1 j 6 . 7 6 . 7 2 3 6 . 7 4000 6 7 6 7 H 6 H 7 6 7 EN EN ¼ 6 ei 7½ e1 ek eN 6 04007 6 7 1 6 7 6 7 ¼ 6 7 6 . 7 4 6 7 6 . 7 4 00405 4 5 H 0004 eN 2 3 H H H ¼ I e e1 e ek … e eN (20) 4 6 1 1 1 7 6 7 6 . . . . 7 where I is the 4 4 identity matrix. 6 . . . . 7 4 6 . . . . 7 The following corollaries follow from Theorem 6. 6 7 6 H H H 7 COROLLARY 9. If EN is the unitary Fourier matrix, then EN is ¼ 6 e e1 e ek e eN 7 6 i i i 7 also unitary. ٗ 6 7 Proof. Note that H T because T is symmet- 6 . . . . 7 EN ¼ EN ¼ EN EN ¼ EN 6 . . . . . 7 ric. It follows that 4 5 eHe … eHe … eHe H T N 1 N k N N EN EN ¼ ENEN ¼ IN ¼ ENE N
H which is a statement of unitary EN. The matrix equality holds only ¼ ENEN H if each ei ek ¼ dik for i; k ¼ 1; 2; …N. T H Example 14. Consider the vector e ¼ 1 ð1; 1; 1; 1Þ from ¼ E EN (Eq. 8) 3 2 N Example 10. Then the product ¼ IN (Thm. 6) 2 3 1 6 7 which implies that EN is unitary. 6 7 6 1 7 Example 13. Consider the unitary matrix E4 from Example 12. H 1 1 6 7 Corollary 9 guarantees that E is also unitary. Thus, e3 e3 ¼ ½ 1 11 1 6 7 4 2 2 6 1 7 2 3 2 3 4 5 1111 1111 6 7 6 7 1 6 7 6 7 6 1 j 1 j 7 6 1 j 1 j 7 1 H 1ffiffiffi 6 7 1ffiffiffi 6 7 E E4 ¼ p 6 7 p 6 7 ¼ ð1 þ 1 þ 1 þ 1Þ 4 4 6 7 4 6 7 4 4 1 11 1 5 4 1 11 1 5 ¼ 1 1 j 1 j 1 j 1 j 2 3 4000 corresponds to d33 ¼ 1 in Corollary 10. However, the product 6 7 6 7 2 3 1 6 04007 1 6 7 6 7 ¼ 6 7 6 7 4 6 00407 6 j 7 4 5 H 1 1 6 7 e e2 ¼ ½ 1 11 1 6 7 3 2 2 6 7 0004 4 1 5
¼ I4 j 1 as expected. ¼ ð1 j 1 þ jÞ 4 COROLLARY 10. Let ei denote the ith column of the Fourier matrix EN and dik be the Kronecker delta. Then ¼ 0 H ei ek ¼ dik vanishes because e2 and e3 are mutually orthogonal. for i; k ¼ 1; 2; …; N. ٗ COROLLARY 11. If EN is the N N Fourier matrix and IM is the H Proof. For i; k ¼ 1; 2; …; N, the (i, k) entry of EN EN can be identity matrix of dimension M, then the NM NM matrix ٗ .written as EN IM is unitary Proof. Consider the matrix product H H ei ek ik ¼ EN EN ik
¼ ðÞIN ik (Thm. 6) H ðEN IMÞ ðEN IMÞ
H H H where IN is the N N identity matrix. Thus, ei ek ¼ dik. ¼ðEN IMÞðEN IMÞ (Eq. 6b) Corollary 10 shows that the columns (and rows) of the Fourier H H matrix are mutually orthogonal. The result can also be proved by ¼ðEN ENÞ ðIMIMÞ (Eq. 4)
11 ¼ IN IM A number of properties follow directly from Theorem 7. ¼ I (Thm. 6) COROLLARY 13. Let EN and jN be the N N Fourier and flip NM matrices. Then where IN and INM are identity matrices of dimension N and NM, (a) ENjN ¼ jNEN; pffiffiffiffiffi 2 respectively. (b) jN ¼ IN or jN ¼ pIffiffiffiffiffiN; and 4 4 COROLLARY 12. If ei denotes the ith column of the Fourier matrix (c) EN ¼ IN or EN ¼ IN EN, IM is the identity matrix of dimension M, and dik is the Kro- ٗ .where IN is the N N identity matrix necker delta, then the NM M matrices ei IM are such that Property (a) of Corollary 13 says that the flip and Fourier matri- ces commute or, since EN is unitary, that jN is invariant under a ðe I ÞHðe I Þ¼d I i M k M ik M unitary transformation with respect to EN. Thus, jN is not diago- nalizable by EN. Properties (b) and (c) give alternative definitions for i; k ¼ 1; 2; …; N. ٗ of the flip and Fourier matrices. Moreover, because the power of a Proof. Consider the matrix product diagonal matrix is obtained by raising each diagonal element to the power in question, if follows that the eigenvalues of jN are H H H ðei IMÞ ðek IMÞ¼ðei IMÞðek IMÞ (Eq. 6b) 61 and those of EN are 61 and 6j, each with the appropriate H H multiplicities. ¼ðei ekÞ ðIMIMÞ (Eq. 4)
¼ dik IM (Cor. 10) 2.5.3 Diagonalization of the Cyclic Forward Shift Matrix. In ¼ dikIM light of Corollary 5, diagonalization of a general circulant or block circulant matrix begins by diagonalizing the cyclic forward which completes the proof. shift matrix. Corollary 12 can also be obtained directly from Corollary 11 by THEOREM 8. Let EN be the N N Fourier matrix and rN be the writing 2 3 2 3 N N cyclic forward shift matrix. Then H H e1 e1 IM 6 7 6 7 EHr E ¼ X 6 7 6 7 N N N N 6 eH 7 6 eH I 7 6 2 7 6 2 M 7 ٗ .EH I ¼ 6 7 I ¼ 6 7 is a diagonal matrix, where X is given by Definition 16 N M 6 . 7 M 6 . 7 N 6 . 7 6 . 7 Proof. For i; k ¼ 1; 2; …; N, the (i, k) entry of E X EH is given 4 . 5 4 . 5 N N N by H H eN eN IM XN XN H H and ENXNEN ik ¼ ðENÞipðXNÞprðEN Þrk r¼1 p¼1 EN IM ¼ðe1; e2; …; eNÞ IM XN XN 1 ¼ pffiffiffiffiwði 1Þðp 1Þd wðr 1Þ ¼ðe1 IM; e2 IM; …; eN IMÞ N pr N r¼1 p¼1 N expanding these matrices similarly to Eq. (20), and setting the 1 pffiffiffiffi w ðr 1Þðk 1Þ (Eq. 15) result equal to INM ¼ diagðIM; IM; …; IMÞ. N N Next we derive a relationship between the Fourier and flip matrices. 1 XN ¼ wði 1Þðr 1Þwðr 1Þw ðr 1Þðk 1Þ THEOREM 7. Let EN and jN denote the N N Fourier and flip N N N matrices, respectively. Then r¼1 1 XN H 2 ٗ ¼ wðr 1Þði kþ1Þ 2 EN ¼ jN ¼ EN : N N r¼1
1 XN 1 Proof. We first shown that E2 ¼ E E ¼ j . For any inte- ¼ wrði kþ1Þ N N N N N N ger m and i; k ¼ 1; 2; …; N, the (i, k) entry of ENEN is given r¼0 by ¼ ðÞrN ik (Cor. 8) XN ðÞENEN ¼ ðENÞ ðENÞ H ik ir rk from which it follows that ENXNEN ¼ rN. The desired result fol- r¼1 H lows by multiplying from the left by EN , multiplying from the XN right by E , and invoking Theorem 6. 1ffiffiffiffi ði 1Þðr 1Þ 1ffiffiffiffi ðr 1Þðk 1Þ N ¼ p wN p wN (Eq. 15) Theorem 8 implies that r is unitarily similar to a diagonal ma- N N N r¼1 trix whose diagonal elements are the distinct Nth root of unity 1 XN 1 (i.e., Definition 16). Because the eigenvalues of a matrix are pre- ¼ wrðiþk 2Þ N N served under such a transformation (this is guaranteed by Theo- r¼0 rem 1), it follows that ¼ ðÞjN ik (Cor. 8) a r Þ¼aðX Þ¼f1; w ; w2 ; …; wN 1g ð N N N N N 2 H 2 from which it follows that EN ¼ jN. The result jN ¼ EN follows from complex conjugation and transposition of where að Þ denotes the matrix spectrum. The eigenvectors of the H circulant matrix rN are the linearly independent columns of jN ¼ ENE N, and by invoking the properties jN ¼ jN and H 2 EN ¼ðe1; e2; …; eNÞ, which are given by Definition 18. In fact, E2 ¼ EH . N N all circulant matrices contained in CN share the same eigenvectors
12 ei, which is shown in Sec. 2.8. In light of Corollary 2, we have the Proof. Consider the representation following results. COROLLARY 14. Let EN and rN be the N N Fourier and cyclic C ¼ qðrN; ckÞ (Cor. 5) forward shift matrices. Then for any n 2 Zþ,
H n n EN rNEN ¼ XN H ¼ qðENXNEN ; ckÞ (Thm. 8)
ٗ .where XN given by Definition 16 COROLLARY 15. Let ei be the ith column of the Fourier matrix H ¼ ENqðXN; ckÞEN (Thms. 2 and 6) EN and rN be cyclic forward shift matrix. Then for any n 2 Zþ and i; k ¼ 1; 2; … N,
H n i 1 n where the last step follows directly from the proof of Theorem 2 ei rNek ¼ðwN Þ dik and the polynomial term ¼ wnði 1Þd N ik ! XN ðk 1Þði 1Þ ٗ where dik is the Kronecker delta. qðXN; ckÞ¼ diag ckwN Corollary 15 shows that the columns of the Fourier matrix are i¼1;…;N k¼1 orthogonal with respect to the cyclic forward shift matrix and its powers. It is shown in Sec. 2.5.4 that they are in fact orthogonal i 1 ¼ diag ðqwN ; ckÞÞ with respect to any circulant matrix. i¼1;…;N 2.5.4 Diagonalization of a Circulant. Corollary 16. Let the matrix XN be populated with the distinct follows from Corollary 16 and Definition 13. The desired result is Nth roots of unity according to Definition 16 and s be an arbitrary obtained by multiplying from the left by EH, multiplying from the square matrix. Then N right by EN, and invoking Theorem 6. Theorem 9 shows that the Fourier matrix E diagonalizes any i 1 N qðXN; sÞ¼ diag ðqðwN ; sÞÞ N N circulant matrix. As discussed in Sec. 2.8, the columns i¼1;…;N e1; e2; …; eN of EN are the eigenvectors of C. The scalars ki in Theorem 9 are the eigenvalues of C. Unlike the eigenvectors, they ٗ where qð Þ is given by Definition 13. depend on the elements (i.e., generating elements) of C. In fact, Proof. Consider the representation Eq. (21) has the same form as the DFT of a discrete time series, which is clear by comparing it to Definition 20 of Sec. 2.7. In this qðX ; sÞ¼qðdiagð1; w ; w2 ; …; wN 1Þ; sÞ N N N N case, ck ðk ¼ 1; 2; …; NÞ and kiði ¼ 1; 2; …; NÞ are analogous to a XN discrete signal and its DFT, and are related by ¼ diagðw0 ðk 1Þ; w1 ðk 1Þ; …; wðN 1Þðk 1ÞÞ s N N N 2 3 2 3 k¼1 k1 c1 XN 6 7 6 7 6 k 7 ffiffiffiffi 6 c 7 ¼ diagðsw0 ðk 1Þ; sw1 ðk 1Þ; …; swðN 1Þðk 1ÞÞ 6 2 7 p 6 2 7 N N N 6 7 ¼ NEN6 7 (22) k¼1 6 . 7 6 . 7 ! 4 . 5 4 . 5 XN ði 1Þðk 1Þ kN cN ¼ diag swN i¼1;…;N k¼1
i 1 which has the same form as the matrix–vector representation of ¼ diag ðqðwN ; sÞÞ i¼1;…;N the DFT defined by Eq. (27) of Sec. 2.7. Thus, the eigenvalues ki can be calculated directly using Eq. (21) or Eq. (22), which are which is diagonal when s is a scalar and block diagonal when s is equivalent. a matrix. If the circulant matrix in Theorem 9 is also symmetric, the eigenvalues are real-valued and certain ones are repeated, as THEOREM 9 (Diagonalization of a Circulant). Let C 2 C have N shown in Corollary 17. generating elements c1; c2; …; cN. Then if EN is the N N Fourier matrix, COROLLARY 17. If C 2 SCN is a symmetric circulant, Eq. (21) reduces to 2 3 8 k1 0 6 7 > XN=2 6 7 > 2pðk 1Þði 1Þ 6 k2 7 > c1 þ 2 ck cos H > N E CEN ¼ 6 7 > k¼2 N 6 .. 7 < 4 . 5 i 1 ki ¼ þð 1Þ cNþ2; N even > 2 0 kN > > ðNXþ1Þ=2 > 2pðk 1Þði 1Þ :> c1 þ 2 ck cos ; N odd is a diagonal matrix. For i ¼ 1; 2; …; N, the diagonal elements N are k¼2 ٗ .(XN where the generating elements are given by Eq. (10 i 1 ðk 1Þði 1Þ ki ¼ qðwN ; ckÞ¼ ckwN (21) Proof. If N is even, the generating elements of C are given by k¼1 c1; c2; …; cN ; cNþ2; cN ; …; c3; c2 2 2 2 where wN is the primitive Nth root of unity and the function qð Þ is given by Definition 13. ٗ and Eq. (21) reduces to
13 XN ðk 1Þði 1Þ is a diagonal matrix. The diagonal elements can be computed ki ¼ ckwN directly using Eq. (21) for N ¼ 4. For example, k¼1 0 ði 1Þ 1 ði 1Þ 2 ði 1Þ X4 ðk 1Þð3 1Þ ¼ c1wN þ c2wN þ c3wN þ k3 ¼ ckw4 N Nþ2 k¼1 ðÞ2 1 ði 1Þ ðÞ2 1 ði 1Þ þ cN w þ cNþ2w 2 N 2 N ð1 1Þð3 1Þ ð2 1Þð3 1Þ ¼ 4w4 þð 1Þw4 Nþ4 ðÞ2 1 ði 1Þ þ cN wN þ ð3 1Þð3 1Þ ð4 1Þð3 1Þ 2 þ 0 w4 þð 1Þw4 ðN 1 1Þði 1Þ ðN 1Þði 1Þ j 2p 2 j 2p 6 þ c3wN þ c2wN 4 1 e 4 0 e 4 ¼ ð Þ þ ði 1Þ ðN 1Þði 1Þ jp j 3p ¼ c1 þ c2 wN þ wN ¼ 4 e þ 0 e 2ði 1Þ ðN 2Þði 1Þ ¼ 4 ð 1Þþ0 ð 1Þ¼6 þ c3 wN þ wN þ N 2 N N 2 ði 1Þ N which is recognized to be the third diagonal element of the matrix 2 ði 1Þ ðÞ2 2ði 1Þ þ cN w þ w þ cNþ2w H 2 N N 2 N product E4 CE4. Because C is a symmetric circulant matrix, Corol- lary 17 can also be used. Observing that 4 is even, it follows that N ¼ XN=2 2pðk 1Þði 1Þ i 1 ¼ c1 þ 2 ck cos þ cNþ2ð 1Þ X4=2 N 2 2pðk 1Þð3 1Þ ð3 1Þ k¼2 k3 ¼ c1 þ 2 ck cos þð 1Þ c4þ2 4 2 k¼2 where the identity 2pð2 1Þð3 1Þ ¼ 4 þ 2 ð 1Þ cos þð 1Þ2 0 kði 1Þ ðN kÞði 1Þ 2pk 4 wN þ wN ¼ 2 cos ði 1Þ N ¼ 4 2 cos p þ 0 is employed. If N is odd, the generating elements of C are given by ¼ 4 2ð 1Þþ0 ¼ 6
c1; c2; …; cN 1; cNþ1; cNþ1; cN 1; …; c3; c2 2 2 2 2 as before. The eigenvalues and eigenvectors of C are discussed in and Eq. (21) reduces similarly to the case for even N, which is left Example 24 of Sec. 2.8. as an exercise for the reader. Example 16. Let C ¼ circð4; 1; 0; 1Þ2C4 be a nonsymmetric The cosinusoidal nature of ki in Corollary 17 implies that k1 is circulant matrix. Then the product distinct for a symmetric circulant matrix C 2 SC , but the 2 32 3 N 11 1 1 4 10 1 remaining elements ki ¼ kNþ2 i appear in repeated pairs. How- 6 76 7 6 76 7 ever, kðÞNþ2=2 is also distinct if N is even. 1 6 1 j 1 j 76 14 107 Example 15. Let C circ 4; 1; 0; 1 be a symmetric H ffiffiffi 6 76 7 ¼ ð Þ2SC4 E4 CE4 ¼ p 6 76 7 circulant matrix. Then the product 4 4 1 11 1 54 014 1 5 2 32 3 1 j 1 j 10 1 4 1111 4 10 1 2 3 6 76 7 6 76 7 11 1 1 6 1 j 1 j 76 14 107 6 7 H 1ffiffiffi 6 76 7 6 7 E4 CE4 ¼ p 6 76 7 1 6 1 j 1 j 7 4 6 76 7 ffiffiffi 6 7 4 1 11 1 54 0 14 1 5 p 6 7 4 4 1 11 1 5 1 j 1 j 10 14 1 j 1 j 2 3 2 32 3 1111 6 7 1111 44 2j 44þ 2j 6 7 6 76 7 6 1 j 1 j 7 6 76 7 1 6 7 1 6 1 j 1 j 76 42þ 4j 42 4j 7 pffiffiffi 6 7 ¼ 6 76 7 4 6 7 4 6 76 7 4 1 11 1 5 4 1 11 1 54 4 4 þ 2j 4 4 2j 5 1 j 1 j 1 j 1 j 4 2 4j 4 2 þ 4j 2 32 3 2 3 11 1 1 24 6 4 4000 6 76 7 6 7 6 76 7 6 04 2j 007 6 1 j 1 j 76 24j 6 4j 7 6 7 1 6 76 7 ¼ 6 7 ¼ 6 76 7 6 00407 4 6 76 7 4 5 4 1 11 1 54 2 46 4 5 0004þ 2j 1 j 1 j 2 4j 64j 2 3 is a diagonal matrix. The diagonal elements are the eigenvalues of C, 2000 6 7 which is stated explicitly for general C in the following corollary. They 6 7 6 04007 maybecomputeddirectlyusingEq.(21), as it is done in Example 15, 6 7 ¼ 6 7 but Corollary 17 cannot be used because C is not symmetric. 6 7 4 00605 The eigenvalue magnitudes of the nonsymmetric matrix C in Example 16 are observed to exhibit the same multiplicity (and 0004 symmetry) as the eigenvalues in Example 15 for a symmetric cir- culant. It is shown in Sec. 2.8.3 that the eigenvalues of any circu- ¼ diagð2; 4; 6; 4Þ lant matrix C 2 CN with real-valued generating elements exhibit
14 a certain symmetry about the so-called “Nyquist” component, XN i 1 ðk 1Þði 1Þ which is analogous to the DFT of a real-valued sequence. This is Ki ¼ qðwN ; CkÞ¼ CkwN (23) because Eq. (22) represents the DFT of the generating elements k¼1 c ; c ; …; c . 1 2 N where w is the primitive Nth root of unity and the function qð Þ is COROLLARY 18. Let e be the ith column of the Fourier matrix N ٗ .i given by Definition 13 E and C be a N N circulant matrix. Then for i; k ¼ 1; 2; …; N, N Proof. Consider the representation H ei Cek ¼ kidik XN k 1 C ¼ rN Ck (Cor. 5) where dik is the Kronecker delta and ki is defined by Eq. (21) for k¼1 N C 2 CN or Corollary 17 if C 2 SCN. ٗ X k 1 H Thus, the columns of the Fourier matrix are mutually orthogo- ¼ ðENXN EN Þ Ck (Cor. 14) nal (Corollary 10) and orthogonal with respect to any circulant k¼1 matrix (Corollary 18), not just the cyclic forward shift matrix and XN k 1 H its integer powers (Corollary 15). ¼ ðEN IMÞ XN Ck ðEN IMÞ (Eq. 4) Example 17. Consider the circulant C ¼ circð4; 1; 0; 1Þ k¼1 from Example 15. Then H ¼ðEN IMÞqðXN; CkÞðE IMÞ (Def. 13) 2 3 2 3 N 4 10 1 1 6 7 6 7 where ! 1 6 14 107 1 6 1 7 H ffiffiffi 6 7 ffiffiffi 6 7 XN e3 Ce3 ¼ p ½ 1 11 1 4 5 p 4 5 ðk 1Þði 1Þ 4 0 14 1 4 1 qðXN; CkÞ¼ diag CkwN i¼1;…;N 10 14 1 k¼1 2 3 i 1 6 ¼ diag ðqðwN ; CkÞÞ 6 7 i¼1;…;N 1 6 6 7 ¼ ½ 1 11 1 6 7 4 4 6 5 follows from Corollary 16 and Definition 13. The desired result follows by multiplying from the left by 6 H H 1 ðEN IMÞ¼ðEN IMÞ ¼ 24 ¼ 6 4 multiplying from the right by ðEN IMÞ, and invoking Corollary 11. which is recognized to be the third diagonal element k3 of the ma- Thus, the unitary matrix E I reduces any NM NM block H N M trix E4 CE4 in Example 15. However, the scalar circulant matrix with M M blocks to a block diagonal matrix 2 3 2 3 4 10 1 1 with M M diagonal blocks. 6 7 6 7 Example 18. Consider C ¼ circðA; B; 0; BÞ2BC from 6 7 6 7 2;4 1 6 14 107 1 6 1 7 Example 6. It can be block diagonalized via the transformation eHCe ¼ pffiffiffi ½ 1 11 1 6 7 pffiffiffi 6 7 H 3 1 6 7 6 7 ðE4 I2ÞCðE4 I2Þ. That is, 4 4 0 14 1 5 4 4 1 5 2 3 10 14 1 10101010 2 3 6 7 6 7 2 6 01010101 7 6 7 6 7 6 7 6 7 1 6 2 7 6 7 ¼ ½ 1 11 1 6 7 6 10 j 0 10 j 0 7 4 6 7 6 7 4 2 5 6 7 1 6 01 0 j 0 10 j 7 ffiffiffi 6 7 2 p 6 7C 4 6 10 10 1 0 107 1 6 7 6 7 ¼ 0 ¼ 0 6 7 4 6 01 0 10 1 0 1 7 6 7 6 7 vanishes, as expected, because i 6¼ k in Corollary 18 such that 4 10 j 0 10 j 0 5 dik ¼ 0. 01 0 j 0 10 j 2.5.5 Block Diagonalization of a Block Circulant. Theorem 9 2 3 is generalized to handle block circulants using the Fourier and 10101010 6 7 identity matrices together with the Kronecker product. The choice 6 7 H 6 01010101 7 of diagonalizing matrix EN IM is discussed in Sec. 2.6, where 6 7 6 7 generalizations of Theorem 10 are considered. 6 7 6 10 j 0 10 j 0 7 THEOREM 10 (Block Diagonalization of a Block Circulant). Let 6 7 C and denote its M M generating matrices by 6 7 2 BCM;N 1 6 01 0 j 0 10 j 7 C ; C ; …; C . Then if E is the N N Fourier matrix and I is ffiffiffi 6 7 1 2 N N M p 6 7 the identity matrix of dimension M, 4 6 10 10 1 0 107 6 7 2 3 6 7 K 0 6 7 1 6 01 0 10 1 0 1 7 6 7 6 7 6 K2 7 6 7 H 6 7 4 10 j 0 10 j 0 5 ðEN IMÞCðEN IMÞ¼6 . 7 4 . . 5 01 0 j 0 10 j 0 KN "#"#"#"#! 0 1 2 1 4 1 2 1 is a NM NM block diagonal matrix. For i ¼ 1; 2; …; N, the ¼ diag ; ; ; M M diagonal blocks are 10 12 14 12
15 which is a block diagonal matrix with 2 2 diagonal blocks. The of Theorem 10, and aids in proving orthogonality relationships for eigenvalues and eigenvectors of C are discussed in Example 25 of the cyclic eigenvalue problems described in Sec. 2.8. Sec. 2.8. In light of Corollary 14, it is clear that the choice of A ¼ EN H COROLLARY 19. Let C 2 BCM;N have M M generating matri- and ð Þ ¼ ð Þ accomplishes block diagonalization of a matrix ces C1; C2; …; CN and Ki be defined by Eq. (23). Then if each Ci C 2 BC . Then if B ¼ I , the appropriate diagonalizing matrix M;N M ٗ is symmetric, it follows that Ki is symmetric for i ¼ 1; 2; …; N. to block decouple C without operating on its generating matrices Proof. If each Ck is symmetric for k ¼ 1; 2; …; N, then so too # ðk 1Þði 1Þ ðk 1Þði 1Þ is EN IM (see Theorem 10). However, if B and ð Þ are kept are the matrices CkwN for each i because wN is a general, we have the following result. scalar. Moreover, the sum and difference of two symmetric matri- THEOREM 11. Let C 2 BC have M M generating matrices ces is again symmetric. If follows that M;N C1; C2; …; CN and EN be the N N Fourier matrix. Then for an arbitrary matrix B 2 CM M and operator ð Þ#, XN 2 3 K wðk 1Þði 1Þ i ¼ Ck N W1 0 k¼1 6 7 6 W2 7 H # 6 7 ðEN B ÞCðEN BÞ¼ . 4 .. 5 is symmetric for i ¼ 1; 2; …; N. 0 WN COROLLARY 20. Let C 2 BCM;N be a NM NM block circulant matrix. Then for i; k ¼ 1; 2; …; N, is a block diagonal matrix, where H ðei IMÞCðek IMÞ¼Kidik XN i 1 # # ðk 1Þði 1Þ Wi ¼ qðwN ; B CkBÞ¼ B CkBwN (24) k¼1 ٗ .(where Ki is defined by Eq. (23 Example 19. Consider C ¼ circðA; B; 0; BÞ2 from ٗ BC2;4 Examples 6 and 18. Then is the ith M M diagonal block for i ¼ 1; 2; …; N. COROLLARY 21. Let C 2 BCM;N be a NM NM block circulant matrix, B 2 CM M be an arbitrary square matrix, and ð Þ# denote 10 H 1 an arbitrary operation that takes a square matrix as its argument ðe I2ÞCðe3 I2Þ¼ pffiffiffi ½ 1 11 1 C 3 4 01 and returns another square matrix with the same dimension. Then 0 2 3 1 1 for i; k ¼ 1; 2; …; N, B 6 7 C B 1 6 1 7 10C H # Bpffiffiffi 6 7 C ðei B ÞCðek BÞ¼Widik @ 4 4 1 5 01A ٗ .(where W is defined by Eq. (24 1 i 4 1 Theorem 11 is useful if there exists an equivalence transforma- ¼ tion B#C B that simplifies each of the generating matrices. For 14 k example, if each Ck is a circulant of type M, then the additional # H choice of B ¼ EM and ð Þ ¼ ð Þ fully diagonalizes a block cir- is the third 2 2 block of the block diagonal matrix obtained in culant matrix C 2 BCN;M with circulant blocks. Example 18. COROLLARY 22. Let C 2 BCM;N have generating matrices C1; C2; …; CN 2 CM and denote the generating elements of each ð1Þ ð2Þ ðMÞ Ci by c ; c ; …; c . Then 2.6 Generalizations. Let ð Þ denote an arbitrary operation i i i that takes a square matrix as its argument and returns another 2 3 ð1Þ square matrix with the same dimension. For example, the opera- k 0 6 i 7 tion could denote a matrix inverse such that ð Þ ¼ ð Þ 1. Let ð Þ# 6 7 6 kð2Þ 7 be another arbitrary matrix operation with the same restrictions H H 6 i 7 ðEN EMÞCðEN EMÞ¼ diag 6 7 (i.e., returns another square matrix with the same dimension). i 1;…;N6 . 7 ¼ 4 . . 5 Then if C 2 BCM;N has generating matrices C1; C2; …; CN, it fol- lows that ðMÞ 0 ki
A B# C A B ð Þ ð Þ is a NM NM diagonal matrix, where ! XN ¼ðA B#Þ rk 1 C ðA BÞ (Cor. 5) XN XM N k ðpÞ ðlÞ ðl 1Þðp 1Þ ðk 1Þði 1Þ k¼1 ki ¼ ck wM wN (25) k¼1 l¼1 XN k 1 # ¼ ððA rN Þ ðB CkÞÞðA BÞ (Eq. 4) k¼1 is the pth diagonal element of the ith M M block for ٗ XN i ¼ 1; 2; …; N and p ¼ 1; 2; …; M. k 1 # ðpÞ ¼ ðA rN AÞ ðB CkBÞ (Eq. 4) Corollary 22 shows that the NM-dimensional eigenvectors qi k¼1 of C are the columns of EN EM. This is in contrast to rotation- ally periodic structures, where q is partitioned into NM-vectors for any matrices A 2 CN N and B 2 CM M. The importance of corresponding to each sector and decomposed into a set of N this result is that C can be decomposed into a summation of direct reduced-order eigenvalue problems described in Sec. 2.8. products of two separate equivalence transformations, one that Example 20. Consider C ¼ circðA; B; 0; BÞ2BC2;4 from operates on the cyclic forward shift matrix and the other on the Examples 6, 18, and 19. Because each of its generating matrices is generating matrices of C. This decomposition justifies the diago- a circulant, that is, ðA; B; 0Þ2C2, the block circulant C is diagon- nalizing matrix used in Sec. 2.5, motivates some generalizations alized via the transformation
16 0 2 3 1 1111 computation is exactly analogous to the determination of the B 6 7 "#C B 6 7 C eigenvalues of a circulant matrix given its generating elements. B 1 61 j 1 j 7 11C We begin by defining the DFT sinusoids, which provide a conven- EH EH C E E Bpffiffiffi6 7 CC ð 4 2 Þ ð 4 2Þ¼B 6 7 C ient means of representing the DFT of a discretized signal, and @ 441 11 15 1 j A then develop the relationships of interest for the DFT and the 1 j 1 j IDFT. We present only the basic results as they relate to the 0 2 3 1 theory and mathematics of circulants. The reader can find a vast 1111 B 6 7 "#C literature on related topics [89–91,102–110]. k B 6 7 C DEFINITION 19 (DFT Sinusoids). Let w denote the distinct Nth B 1 61 j 1 j 7 11C 2p N B ffiffiffi6 7 C j N Bp 6 7 C roots of unity, where wN ¼ e is the primitive root. Then the @ 441 11 15 1 j A DFT sinusoids are
1 j 1 j k r kr j 2pkr 2 3 S ðrÞ¼ðw Þ ¼ w ¼ e N k N N 10000000 6 7 6 7 for k; r 0; 1; …; N 1. ᭝ 6 0 1000000 7 ¼ ٗ .COROLLARY 23. The DFT sinusoids are orthogonal 7 6 6 7 6 0 0100000 7 Proof. Consider the DFT sinusoids 6 7 6 7 ) 6 0 0030000 7 j 2pir ¼ 6 7 SiðrÞ¼e N 6 7 ; i; k; r ¼ 0; 1; …; N 1 6 0 0003000 7 j 2pkr 6 7 SkðrÞ¼e N 6 7 6 0 0000500 7 6 7 6 7 The inner product of S (r) and S (r) is given by 4 0 0000010 5 i k 0 0000003 XN 1 hSiðrÞ; SkðrÞi ¼ SiðrÞSkðrÞ r¼0 from which is follows that aðCÞ¼f 1; 1; 1; 3; 3; 5; 1; 3g. Observ- ing that XN 1 ir kr 9 ¼ wNwN (Def. 19) ð1Þ ð1Þ ð1Þ ð1Þ = r¼0 c1 ¼ 2; c2 ¼ 1; c3 ¼ 0; c4 ¼ 1 XN 1 ð2Þ ð2Þ ð2Þ ð2Þ ; ir kr c1 ¼ 1; c2 ¼ 0; c3 ¼ 0; c4 ¼ 0 ¼ wNwN (Cor. 7) r¼0 are the generating elements of C, the NM ¼ 4 2 ¼ 8 diagonal XN 1 rði kÞ elements can be calculated directly using Eq. (25). For example, ¼ wN the second diagonal element (p ¼ 2) of the third 4 4 diagonal r¼0 block (i 3) is given by ( ¼ N; i ¼ k (Lem. 2) ð2Þ ð1Þ ð1 1Þð2 1Þ ð1 1Þð3 1Þ ð2Þ ð2 1Þð2 1Þ ð1 1Þð3 1Þ ¼ k3 ¼ c1 w2 w4 þ c1 w2 w4 0; otherwise ð1Þ ð1 1Þð2 1Þ ð2 1Þð3 1Þ ð2Þ ð2 1Þð2 1Þ ð2 1Þð3 1Þ þ c2 w2 w4 þ c2 w2 w4 which shows that the DFT sinusoids are orthogonal. The Fourier matrix in Definition 17 can be written element cð1Þwð1 1Þð2 1Þwð3 1Þð3 1Þ cð2Þwð2 1Þð2 1Þwð3 1Þð3 1Þ þ 3 2 4 þ 3 2 4 wise as ð1Þ ð1 1Þð2 1Þ ð4 1Þð3 1Þ ð2Þ ð2 1Þð2 1Þ ð4 1Þð3 1Þ þ c4 w2 w4 þ c4 w2 w4 1 ðk 1Þði 1Þ ðENÞ ¼ pffiffiffiffi w ¼ð2Þð1Þð1Þþð 1Þð 1Þð1Þ ik N N
þð 1Þð1Þð 1Þþð0Þð 1Þð 1Þ 1 j 2pðk 1Þði 1Þ ¼ pffiffiffiffi e N N þð0Þð1Þð1Þþð0Þð 1Þð1Þ 1ffiffiffiffi 1ffiffiffiffi þð 1Þð1Þð 1Þþð0Þð 1Þð 1Þ ¼ p Si 1ðk 1Þ¼p Sk 1ði 1Þ N N ¼ 5 for each i; k ¼ 1; 2; …; N. It follows that The reader can compare this matrix decomposition to the results 2 3 p in Example 25 of Sec. 2.8.1. Finally, the eigenvectors qð Þ are the S0ð0Þ S0ð1Þ … S0ðN 1Þ i 6 7 columns of E4 E2 and are stated explicitly in Example 25. 6 7 6 S1ð0Þ S1ð1Þ … S1ðN 1Þ 7 6 7 1 6 7 ffiffiffiffi 6 S2ð0Þ S2ð1Þ … S2ðN 1Þ 7 EN ¼ p 6 7 2.7 Relationship to the Discrete Fourier Transform. N 6 7 Before turning to the circulant eigenvalue problem, we consider a 6 . . . . 7 4 . . . . . 5 somewhat tangential but relevant subject on the DFT and its inverse, which are central to the analysis of experimental data in a SN 1ð0Þ SN 1ð1Þ … SN 1ðN 1Þ wide range of fields, including mechanical vibrations. In this 1 detour we show that computation of the DFT is, in fact, a multipli- ¼ pffiffiffiffi WN (26) cation of an N-vector of discrete signal samples by the Fourier N matrix and a constant cf. The IDFT is similarly defined using the Hermitian of the Fourier matrix and a constant ci, where is a representation of the Fourier matrix in terms of the DFT sinu- cf ci ¼ 1=N. More importantly, it is shown that the DFT soids, where WN is the DFT sinusoid matrix. The DFT is formally
17 pffiffiffi defined next and subsequently reformulated in terms of a matrix where W4 ¼ 4E4 follows from Example 9 or by elementwise multiplication involving WN. direct computation according to Definition 19. Alternatively, each DEFINITION 20 (DFT). Let x(i) denote a finite sequence with indi- element X(k)ofX4 can be computed using the summation given ces i ¼ 1; 2; …; N. Then for k ¼ 1; 2; …; N, the DFT of x(i) is the in Definition 20. If k ¼ 3, for example, sequence X4 XN j 2pði 1Þð3 1Þ 2p Xð3Þ¼ xðiÞe 4 j N ði 1Þðk 1Þ XðkÞ¼ xðiÞe i¼1 i¼1 XN X4 ði 1Þðk 1Þ ¼ xðiÞejpði 1Þ ¼ xðiÞwN i¼1 i¼1 XN ¼ xð1Þejp 0 þ xð2Þejp 1 þ xð3Þejp 2 þ xð4Þejp 3 ¼ xðiÞSi 1ðk 1Þ i¼1 ¼ð0Þð1Þþð1Þð 1Þþð2Þð1Þþð3Þð 1Þ ik where wN is the primitive Nth root of unity and SiðkÞ¼wN is a DFT sinusoid. ᭝ ¼ 0 1 þ 2 3 The DFT preserves the units of x(i). That is, if x(i) has engineer- ing units EU, then the units of X(k) are also EU. This is clear from ¼ 2 Definition 20, where the exponential function is dimensionless. Expanding each X(k) in Definition 20 yields which is the third element of X4, as expected. Results for k ¼ 1, 2, 9 4 follow similarly. Xð1Þ¼xð1ÞS ð0Þþ þxðNÞS ðN 1Þ > 0 0 > If XN is known, the N-vector xN is recovered from Eq. (27) by > multiplying from the left by EH and invoking Theorem 6. Then Xð2Þ¼xð1ÞS1ð0Þþ þxðNÞS1ðN 1Þ > N > > . => H H . EN XN ¼ EN WNxN (Eq. 27) pffiffiffiffi > H XðkÞ¼xð1ÞSi 1ð0Þþ þxðNÞSi 1ðN 1Þ > ¼ EN NEN xN (Eq. 26) > . > pffiffiffiffi . > NEHE x . > ¼ N N N ; pffiffiffiffi XðNÞ¼xð1ÞSN 1ð0Þþ þxðNÞSN 1ðN 1Þ ¼ NINxN (Thm. 6) pffiffiffiffi If the discrete samples x(i) and corresponding sequence of DFTs ¼ NxN X(k) are assembled into the configuration vectors ) and it follows that T xN ¼ðxð1Þ; xð2Þ; …; xðNÞÞ 1 H 1 H T xN ¼ pffiffiffiffi E XN ¼ W XN (28) XN ¼ðXð1Þ; Xð2Þ; …; XðNÞÞ N N N N then the DFT of xN can be represented in the matrix–vector form is a matrix–vector representation of the IDFT of XN. That is, the IDFT computation is simply a matrix multiplication of XN with X ¼ W x (27) N N N the Hermitian of WN and constant ci ¼ 1=N such that cf ci ¼ 1=N. pffiffiffiffi In light of Corollary 7, Eq. (28) can be written as where WN ¼ NEN follows from Eq. (26). Thus, the DFT com- 2 3 2 32 3 putation is simply a multiplication of the configuration vector xN xð1Þ S ð0Þ S ð1Þ … S ðN 1Þ Xð1Þ 6 0 0 0 7 with the DFT sinusoid matrix WN and constant cf ¼ 1. Equation 6 7 6 76 7 6 xð2Þ 7 6 76 Xð2Þ 7 (27) has exactly the same form as Eq. (22), where the generating 6 7 1 6 S1ð0Þ S1ð1Þ … S1ðN 1Þ 76 7 elements of a circulant matrix are analogous to the sequence of 6 7 ¼ 6 76 7 6 . 7 6 . . . . 76 . 7 signals x(i) and the resulting eigenvalues are analogous to the 6 . 7 N 6 . . . . 76 . 7 4 . 5 4 . . . . 54 . 5 sequence of DFTs X(k). Example 21. Consider the configuration vector of samples xðNÞ SN 1ð0Þ SN 1ð1Þ … SN 1ðN 1Þ XðNÞ T x4 ¼ð0; 1; 2; 3Þ . The DFT of x4 follows from Eq. (27) with N ¼ 4 and is given by Expanding each row yields 9 X4 ¼ W4x4 1 > 2 32 3 xð1Þ¼ ðXð1ÞS0ð0Þþ þXðNÞS0ðN 1ÞÞ > N > S0ð0Þ S0ð1Þ S0ð2Þ S0ð3Þ xð1Þ > > 6 76 7 1 > 6 S1ð0Þ S1ð1Þ S1ð2Þ S1ð3Þ 76 xð2Þ 7 xð2Þ¼ ðXð1ÞS1ð0Þþ þXðNÞS1ðN 1ÞÞ > 6 76 7 N > ¼ 6 76 7 > 4 S2ð0Þ S2ð1Þ S2ð2Þ S2ð3Þ 54 xð3Þ 5 . => . S3ð0Þ S3ð1Þ S3ð2Þ S3ð3Þ xð4Þ > 2 32 3 1 > xðiÞ¼ ðXð1ÞSk 1ð0Þþ þXðNÞSk 1ðN 1ÞÞ > 11 1 1 0 N > 6 76 7 > 6 1 j 1 j 76 1 7 . > 6 76 7 . > ¼ 6 76 7 . > 4 1 11 1 54 2 5 > 1 ;> xðNÞ¼ ðXð1ÞS ð0Þþ þXðNÞS ðN 1ÞÞ 1 j 1 j 3 N N 1 N 1 T ¼ð6; 2 þ 2j; 2; 2 2jÞ which provides an alternative representation of the IDFT.
18 DEFINITION 21 (IDFT). Let the sequence X(k) be defined accord- positive. Then each entry Sk(r)inWN is replaced with SkðrÞ to ing to Definition 20. Then for each i ¼ 1; 2; …; N, the IDFT of produce WN (see Corollary 7), and the DFT is instead given by X(k) is the sequence
XN ¼ WNxN (29) 1 XN xðiÞ¼ XðkÞS ði 1Þ N k 1 k¼1 The corresponding IDFT takes the form XN 1 ði 1Þðk 1Þ 1 ¼ XðkÞw H N N xN ¼ WN XN (30) k¼1 N
XN 1 j 2pði 1Þðk 1Þ which follows in the same way as Eq. (28) by replacing EN with ¼ XðkÞe N N EN and invoking Corollary 9. Similarly, the multiplicative con- k¼1 stants that define the DFT/IDFT pair are arbitrary as long as the
ik product of the constants is equal to 1/N. This is important for where wN is the primitive Nth root of unity and SiðkÞ¼wN is a applications involving a transformation of data to the frequency ᭝ DFT sinusoid. domain for analysis and then back to the time domain for results. The IDFT preserves the units of X(k). If X(k) has engineering However, the multiplicative constant is irrelevant if the analysis units EU, then the units of x(i) are also EU. This is clear from Def- goal is only to identify periodicities in a data set (e.g., frequencies inition 21, where the exponential function is dimensionless, as is that correspond to amplification of a structural response). the number N. Thus, Definitions 20 and 21 form a representation of the DFT/ Example 22. Reconsider Example 21, where it is shown that the IDFT pair, where Eqs. (27) and (28) are the corresponding T DFT of x4 ¼ð0; 1; 2; 3Þ is the four-vector X4 ¼ð6; 2 þ 2j; matrix–vector forms. The DFT and IDFT pair can be written in T 2; 2 2jÞ . The IDFT of X4 follows from Eq. (28) with N ¼ 4 the general form and is given by XN 6j 2pði 1Þðk 1Þ 1 XðkÞ¼cf xðiÞe N (31a) x ¼ WHX 4 N 4 4 i¼1 2 32 3 XN 2p S ð0Þ S ð1Þ S ð2Þ S ð3Þ Xð1Þ j N ði 1Þðk 1Þ 6 0 0 0 0 76 7 xðiÞ¼ci XðkÞe (31b) 6 76 7 k 1 6 76 Xð2Þ 7 ¼ 1 6 S1ð0Þ S1ð1Þ S1ð2Þ S1ð3Þ 76 7 ¼ 6 76 7 N 6 76 7 where ci and cf are such that cf ci ¼ 1=N but are otherwiseffiffiffiffi ffiffiffiffi arbi- 4 S2ð0Þ S2ð1Þ S2ð2Þ S2ð3Þ 54 Xð3Þ 5 p p trary. Common choices are fcf ; cig¼f1= N; 1= Ng or S3ð0Þ S3ð1Þ S3ð2Þ S3ð3Þ Xð4Þ fcf ; cig¼f1; 1=Ng, where the multiplicative constants in the 2 32 3 matrix–vector formulation (i.e., Eqs. (27) and (28) or Eqs. (29) 1111 6 and (30)) are adjusted accordingly. 6 76 7 6 76 7 If the generally complex sequences x(i) and X(k) are dissected 6 1 j 1 j 76 2 þ 2j 7 1 6 76 7 into their real and imaginary parts, the DFT/IDFT pair is repre- ¼ 6 76 7 4 6 76 7 sented by 4 1 11 1 54 2 5
1 j 1 j 2 2j xðiÞ¼xRðiÞþjxIðiÞ (32a)
1 T XðkÞ¼X ðkÞþjX ðkÞ (32b) ¼ ð0; 4; 8; 12Þ R I 4 where the subscripts R and I denote the real and imaginary parts ¼ð0; 1; 2; 3ÞT and each sequence pair ðxRðiÞ; xIðiÞÞ and ðXRðkÞ; XIðkÞÞ is real for i; k ¼ 1; 2; …; N. An alternative representation of the DFT and which is the same four-vector from Example 21. Alternatively, IDFT is obtained by substituting Eq. (32) in Definition 20, writing each element x(i)ofx4 can be computed using the summation the complex exponential in terms of sines and cosines, and col- given in Definition 21. If i ¼ 3, for example, lecting the real and imaginary terms. The result is X4 XN 1 j 2pð3 1Þðk 1Þ 2pði 1Þðk 1Þ xð3Þ¼ XðkÞe 4 X ðkÞ¼ x ðiÞ cos 4 R R k¼1 i¼1 N 1 jp 0 jp 1 jp 2 jp 3 2pði 1Þðk 1Þ ¼ Xð1Þe þ Xð2Þe þ Xð3Þe þ Xð4Þe x ðiÞ sin (33a) 4 I N 1 XN ¼ ðÞð6Þð1Þþð 2 2jÞð 1Þþð 2Þð1Þþð 2 þ 2jÞð 1Þ 2pði 1Þðk 1Þ 4 X ðkÞ¼ x ðiÞ sin I R N 2 i¼1 ¼ 2pði 1Þðk 1Þ þ xIðiÞ cos (33b) which is the third element of x4, as expected. Results for k ¼ 1, 2, N 4 follow similarly. There are other suitable definitions of the DFT/IDFT pair. For which together form the DFT of x(i) according to Eq. (32b).A example, the signs of the exponents are irrelevant as long as they similar relationship for the IDFT is obtained by substituting are opposite in the DFT and IDFT. To see this, suppose that the Eq. (32) into Definition 21 and collecting the real and imaginary sign of the exponential in Definition 20 is negative instead of terms. Then
19 XN 1 2pði 1Þðk 1Þ of the scalar eigenvalues k and NM 1 eigenvectors q of C such x ðiÞ¼þ X ðkÞ cos R N R N that i¼1 2pði 1Þðk 1Þ 0NM ¼ ðÞC kINM q (35) þ XIðkÞ sin (34a) N where 0 is a NM 1 vector of zeros and I is the identity ma- NM NM XN trix of dimension NM. The problem can be simplified consider- 1 2pði 1Þðk 1Þ xIðiÞ¼ XRðkÞ sin ably by exploiting the block circulant nature of C. To this end, N N T i¼1 partition q ¼ ðÞq1; q2; …; qN into M 1 vectors qi ði ¼ 1; …; NÞ 2pði 1Þðk 1Þ and introduce the change of coordinates XIðkÞ cos (34b) N q ¼ðEN IMÞu (36)
where IM is the M M identity matrix (same dimension as the gener- together form the IDFT of X(k) according to Eq. (32a). The DFT/ T IDFT pair defined by Eqs. (32)–(34) are equivalent to Definitions ating matrices of C)andu ¼ ðÞu1; u2; …; uN is composed of NM- 20 and 21. vectors ui. Substituting Eq. (36) into Eq. (35) and multiplying from H H Example 23. Reconsider Example 21, where it is shown that the the left by the unitary matrix ðEN IMÞ ¼ðEN IMÞ yields T DFT of x4 ¼ð0; 1; 2; 3Þ is the four-vector X4 ¼ð6; 2 2j; T H 2; 2 þ 2jÞ . Because x is real, Eq. (33) can be used to com- 0NM ¼ðEN IMÞðÞðC kINM EN IMÞu 4 pute each XðkÞ¼XRðkÞþjXIðkÞ for k ¼ 1, 2, 3, 4. If k ¼ 2, for H ¼ðEN IMÞCðEN IMÞ example, the real and imaginary DFT components are given by H kðE IMÞINMðEN IMÞ u X4 N 2pði 1Þð2 1Þ 02 3 2 312 3 X ð2Þ¼ x ðiÞcos R R 4 K1 0 IM 0 u1 i¼1 B6 7 6 7C6 7 B6 7 6 7C6 7 B6 K2 7 6 IM 7C6 u2 7 2pð1 1Þð1Þ 2pð2 1Þð1Þ B6 7 6 7C6 7 ¼ 0 cos þ 1 cos ¼ B6 7 k6 7C6 7 4 4 B6 . . 7 6 . . 7C6 . 7 @4 . 5 4 . 5A4 . 5 2pð3 1Þð1Þ 2pð4 1Þð1Þ þ 2 cos þ 3 cos 0 KN 0IM uN 4 4 0p 1p 2p 3p which follows from Theorem 10, and from ¼ 0 cos þ 1 cos þ 2 cos þ 3 cos 2 2 2 2 H H ðEN IMÞINMðEN IMÞ¼ðEN IMÞ ðEN IMÞ ¼ð0Þð1Þþð1Þð0Þþð2Þð 1Þþð3Þð0Þ ¼ INM ¼ 2 ¼ diag ðIM; IM; …; IMÞ X4 |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} 2pði 1Þð2 1Þ X ð2Þ¼ x ðiÞ sin N terms I R 4 i¼1 0p 1p 2p 3p in light of Corollary 11. Thus, the single eigenvalue problem ¼ 0 sin þ 1 sin þ 2 sin þ 3 sin defined by Eq. (35) is decomposed into the N reduced-order stand- 2 2 2 2 ard EVPs ¼ð0Þð0Þþð1Þð1Þþð2Þð0Þþð3Þð 1Þ 0M ¼ ðÞKi kIM ui; i ¼ 1; 2; …; N (37) ¼ 2 where 0M is a M 1 vector of zeros and each Ki is defined by Eq. such that Xð2Þ¼ 2 2j, as expected. Results for k ¼ 1, 3, 4 fol- (23) in terms of the generating matrices of C. Because the transforma- low similarly. tiondefinedbyEq.(36) is unitary, it preserves the eigenvalues of C. Thus, the NM eigenvalues of C are the eigenvalues of the NM M 2.8 The Circulant Eigenvalue Problem. The eigenvalues matrices Ki, which follow from the characteristic polynomials and eigenvectors of circulants and block circulants with circulant blocks have thus far been inferred from Theorem 9 (circulant matrix) detðKi kIMÞ¼0; i ¼ 1; 2; …; N (38) and Corollary 22 (block circulant matrix with circulant blocks), ðpÞ where these matrices are fully diagonalized. Determination of the If k ¼ k denotes the pth eigenvalue of the ith matrix Ki for i ðpÞ eigenvalues and eigenvectors of a block circulant matrix with gener- p ¼ 1; 2; …; M, then the attendant M 1 eigenvector ui is ally noncirculant and nonsymmetric blocks is systematically obtained from Eq. (37). The corresponding NM 1 eigenvector of described here, which reinforces the results already obtained for the C follows from Eq. (36) and is given by special cases of C 2 CN and C 2 BCM;N with circulant generating T matrices. The standard cEVP is discussed in Sec. 2.8.1,wherean ðpÞ ðpÞ qi ¼ðEN IMÞ 0M; …; ui ; …; 0M eigensolution is obtained for a general matrix C 2 BCM;N.Section 2.8.2 generalizes these results to handle (M,K) systems with system T ðpÞ matrices contained in BCM;N, which arise naturally in vibration stud- ¼ ðÞðÞe1; …; ei; …; eN IM 0M; …; ui ; …; 0M ies of rotationally periodic structures. The structure of the eigenval- ues and eigenvectors is discussed in Sec. 2.8.3, which builds upon ¼ ðÞe1 IM; …; ei IM; …eN IM the relationship to the DFT described in Sec. 2.7.Section2.8.4 intro- (39) T duces fundamental orthogonality conditions for the special case of 0 ; …; uðpÞ; …; 0 block circulants with symmetric generating matrices. M i M ðpÞ 2.8.1 Standard Circulant Eigenvalue Problem. Consider the ¼ðei IMÞui block circulant matrix C 2 BCM;N with arbitrary generating mat- ðpÞ ¼ ei u rices C1; C2; …; CN. The standard cEVP involves determination i
20 ðpÞ for i ¼ 1; 2; …; N and p ¼ 1; 2; …; M. If the scalars ai are such The same basic results also hold for the nonsymmetric circulant that each C ¼ circð4; 1; 0; 1Þ2C4 in Example 16, where it is shown that EHCE diag 4; 4 2j; 4; 4 2j . In this case, the eigenvalues ðpÞ ðpÞ ðpÞ 4 4 ¼ ð þ Þ u~i ¼ ai ui (40) are given by is orthonormal with respect to Ki, then the corresponding ortho- aðCÞ¼f4; 4 2j; 4; 4 þ 2jg normal eigenvectors of C are given by and the corresponding eigenvectors are the same as those in ðpÞ ðpÞ q~i ¼ ei u~i Example 24 because all circulants contained in CN share the same linearly independent eigenvectors. ðpÞ ðpÞ ¼ ei ai ui Example 25. Consider C ¼ circðA; B; 0; BÞ2BC2;4 from (41) Examples 6, 18, 19, and 20, where it is shown that aðpÞ ðpÞ ¼ i ei ui H ðE4 I2ÞCðE4 I2Þ ðpÞ ðpÞ "#"#"#"#! ¼ ai qi 0 1 2 1 4 1 2 1 ¼ diag ; ; ; for i ¼ 1; 2; …; N and p ¼ 1; 2; …; M. 10 12 14 12 If C is an ordinary (not block) circulant with M ¼ 1, Eq. (41) reduces to diagðK1; K2; K3; K4Þ
ð1Þ ð1Þ q~i ¼ qi ¼ ei 1 ¼ ei; ðM ¼ 1Þ (42) The eigenvalues of C are obtained from the reduced-order matri- ces Ki ði ¼ 1; 2; 3; 4Þ. For example, the eigenvalues of K3 follow which confirms that all circulants contained in CN share the same from the characteristic polynomial N 1 eigenvectors e ; e ; …; e , and each e is orthonormal with "#"# 1 2 N i respect to C. The eigenvalues of a matrix C 2 CN with generating 4 1 10 det K kI k elements c1; c2; …; cN are given by Eq. (21), or equivalently by ð 3 2Þ¼ 14 01 Eq. (22).IfC 2 SCN, the eigenvalues follow from Corollary 17. Example 24. Consider C ¼ circð4; 1; 0; 1Þ2C from 2 4 ¼ k 8k þ 15 Examples 15 and 17, where it is shown that H ¼ðk 3Þðk 5Þ¼0 E4 CE4 ¼ diagð2; 4; 6; 4Þ Thus, the eigenvalues of C are 2, 4, 6, and 4. Because N ¼ 4is and are given by aðK3Þ¼f3; 5g. Similarly, aðK1Þ¼f 1; 1g and aðK2Þ¼aðK4Þ¼f1; 3g. Because N ¼ 4 is even, aðK1Þ and aðK3Þ even, k1 ¼ 2 and kðNþ2Þ=2 ¼ k3 ¼ 6 are distinct and k2 ¼ k4 ¼ 4 are repeated. The reason for this eigenvalue symmetry is dis- are distinct sets of eigenvalues and aðK2Þ¼aðK4Þ are repeated cussed in Sec. 2.8.3. The eigenvalues can be verified using sets. Section 2.8.3 discusses the symmetry characteristics and multiplicities of these groups of eigenvalues. The eigenvectors of p C follow from the eigenvectors of each Ki according to Eq. (39). ki ¼ 4 2 cos ði 1Þ; i ¼ 1; 2; 3; 4 ð1Þ T For example, for the eigenvector u ¼ð1; 1Þ of K3 with eigen- 2 ð1Þ 3 value k3 ¼ 3, the corresponding eigenvector of C is which follows from Corollary 17 for the case of even N because C ð1Þ ð1Þ is also contained in . Because C is an ordinary circulant, its q3 ¼ e3 u3 SC4 "# orthonormal eigenvectors are simply the columns of the Fourier 1 matrix E , and are given by Definition 18 according to Eq. (42). 1ffiffiffiffi 2 4 6 T 4 ¼ p ð1; w4; w4; w4Þ For example, N 1 "# 1 1 ffiffiffi 1 2 3 T 1 T e2 ¼ p 1; w4; w4; w4 ¼ pffiffiffi ð1; 1; 1; 1Þ 4 4 1 T 1 j 2p 1 j 2p 2 j 2p 3 "#"#"#"#! ¼ 1; e 4 ; e 4 ; e 4 T 1 1 1 1 2 1 T ¼ ; ; ; 1 j p j p j 3p ¼ 1; e 2 ; e ; e 2 2 1 1 1 1 2 1 1 T ¼ ð1; j; 1; jÞT ¼ ð1; 1; 1; 1; 1; 1; 1; 1Þ 2 2 which can be visualized in Fig. 2 for the case of N ¼ 4. The com- The entire set of NM ¼ 8 reduced-order eigenvectors and eigen- values is given by plete set of eigenvectors is given by 9 2 3 2 3 2 3 2 3 uð1Þ 1; 1 T; kð1Þ 1 > 1 1 1 1 1 ¼ð Þ 1 ¼ > > 6 7 6 7 6 7 6 7 ð2Þ T ð2Þ > 6 1 7 6 j 7 6 1 7 6 j 7 u1 ¼ð 1; 1Þ ; k1 ¼ 1 > 1 6 7 1 6 7 1 6 7 1 6 7 > e1 ¼ 6 7; e2 ¼ 6 7; e3 ¼ 6 7; e4 ¼ 6 7 ð1Þ T ð1Þ > 2 4 1 5 2 4 1 5 2 4 1 5 2 4 1 5 u ¼ð1; 1Þ ; k ¼ 1 > 2 2 > ð2Þ T ð2Þ => 1 j 1 j u2 ¼ð 1; 1Þ ; k2 ¼ 3 ð1Þ T ð1Þ > u3 ¼ð1; 1Þ ; k3 ¼ 3 > The eigenvectors e1 and e3 are real and correspond to the distinct > ð2Þ ð2Þ > eigenvalues k1 ¼ 2 and k3 ¼ 6. The eigenvectors e2 and e4 are T > u3 ¼ð 1; 1Þ ; k3 ¼ 5 > complex conjugates according to Eq. (17) and correspond to the > ð1Þ T ð1Þ > repeated eigenvalue k2 ¼ k4 ¼ 4. Example 29 of Sec. 2.8.4 dis- u4 ¼ð1; 1Þ ; k4 ¼ 1 > > cusses orthogonality of the orthonormal eigenvectorsðÞ 1=2 ei with ð2Þ T ð2Þ ; respect to the matrix C. u4 ¼ð 1; 1Þ ; k4 ¼ 3
21 where each pair satisfies Eq. (37). The corresponding eigenvectors composed of N sectors with M DOFs per sector, where M and K of C are are NM NM block circulant mass and stiffness matrices. Several 9 examples are discussed in Sec. 3, where the eigenvectors q are the ð1Þ 1 T > system mode shapes and the eigenvalues k correspond to the sys- q ¼ ð1; 1; 1; 1; 1; 1; 1; 1Þ > 1 2 > tem natural frequencies. The generalized cEVP defined by Eq. > 1 > (44) is handled in the same way as the standard cEVP of Sec. qð2Þ ¼ ð 1; 1; 1; 1; 1; 1; 1; 1ÞT > T 1 > 2.8.1. To this end, partition q ¼ ðÞq1; q2; …; qN into M 1 vec- 2 > > tors qi ði ¼ 1; 2; …; NÞ, transform to a new set of coordinates ð1Þ 1 T > T q ¼ ð 1; 1; j; j; 1; 1; j; jÞ > u ¼ ðÞu1; u2; …; uN by substituting Eq. (36) into Eq. (44), and 2 2 > H > left-multiply the result by the unitary matrix EN IM. It follows 1 > that qð2Þ ¼ ð 1; 1; j; j; 1; 1; j; jÞT => 2 2 H 1 > 0NM ¼ðEN IMÞðÞðK kM EN IMÞu ð1Þ 1; 1; 1; 1; 1; 1; 1; 1 T > q3 ¼ ð Þ > H H 2 > ¼ðEN IMÞKðEN IMÞ kðEN IMÞMðEN IMÞ u > 02 3 2 31 ð2Þ 1 T > e e q ¼ ð 1; 1; 1; 1; 1; 1; 1; 1Þ > K1 0 M1 0 3 2 > B6 e 7 6 e 7C > B6 K2 7 6 M2 7C 1 > B6 7 6 7C ð1Þ ; ; ; ; ; ; ; T > ¼ B6 . 7 k6 . 7C q4 ¼ ð 1 1 j j 1 1 j jÞ > @4 . 5 4 . 5A 2 > . . > e e ð2Þ 1 T ;> 0 KN 0 MN q4 ¼ ð 1; 1; j; j; 1; 1; j; jÞ 2 3 2 u1 6 7 6 u 7 The eigenvectors are made orthonormalffiffiffi according to Eqs. (40) 6 2 7 ðpÞ p (45) and (41) by setting each a ¼ 1= 2 for i ¼ 1; 2; …; N and 6 . 7 i ðpÞ 4 . 5 p ¼ 1; 2; …; M. Orthogonality of the eigenvectors q~i with respect to C is discussed in Example 30 of Sec. 2.8.4. uN The determinant of any matrix is the product of its eigenvalues, where the decomposed M M matrices which yields the following result. 9 COROLLARY 24. Let C 2 BCM;N be a block circulant matrix. XN > Then the determinant of C is given by e ðk 1Þði 1Þ > Mi ¼ MkwN => k¼1 YNM ; i ¼ 1; 2; …; N (46) XN > det C ¼ ki > e ðk 1Þði 1Þ > i¼1 Ki ¼ KkwN ; k¼1 ٗ .where ki is the ith eigenvalue of C The determinant of C 2 CN follows from Corollary 24 by set- follow from Theorem 10. Equation (45) represents a set of N ting M ¼ 1. In light of Eq. (21), reduced-order generalized EVPs YN XN e e ðk 1Þði 1Þ 0M ¼ Ki kMi ui; i ¼ 1; 2; …; N (47) det C ¼ ckwN ðC 2 CNÞ (43) i¼1 k¼1 which are analogous to the reduced-order standard EVPs defined for an ordinary circulant matrix with generating elements by Eq. (37). The eigenvalues are determined from the characteris- c1; c2; …; cN. For the special case of C 2 SCN, the ith eigenvalue tic polynomials ki in Corollary 24 can be replaced by the result given in Corollary e e 17. Similarly, ki can be replaced with Eq. (25) if C 2 BCM;N has detðKi kMiÞ¼0; i ¼ 1; 2; …; N (48) circulant generating matrices. Example 26. Consider C ¼ circðA; B; 0; BÞ2BC2;4 from ðpÞ ðpÞ and are denoted by k ¼ ki for p ¼ 1; 2; …; M. Each ki also Example 25, where it is shown that the eigenvalues are given by satisfies Eq. (44) because the transformation to new coordi- the set aðCÞ¼f 1; 1; 1; 3; 3; 5; 1; 3g. The determinant of C fol- nates via Eq. (36) is unitary, and hence preserves the system lows from Corollary (24) and is given by ðpÞ eigenvalues. The reduced-order eigenvectors ui ¼ ui are obtained from Eq. (47) for each kðpÞ. The eigenvectors of the det C ¼ 1 1 1 3 3 5 1 3 ¼ 135 iðpÞ ðpÞ full system (M,K) are given by qi ¼ ei ui , which is the same as Eq. (39). The relationships defined by Eqs. (40) and which is simply the product of the eigenvalues of C. (41) also hold for the generalized cEVP to make the eigen- vectors orthonormal. 2.8.2 Generalized Circulant Eigenvalue Problem. The gener- If M and K are ordinary circulants (i.e., M ¼ 1), then the system alized cEVP involves determination of the scalar eigenvalues k ð1Þ ð1Þ eigenvectors reduce to q~i ¼ qi ¼ ei 1 ¼ ei, which is the and NM 1 eigenvectors q of a cyclic system (M,K) such that same as Eq. (42) and shows that e1; e2; …; eN are the orthonormal eigenvectors of all generalized eigensystems defined by 0NM ¼ ðÞK kM q (44) M; K 2 CN.IfM1; M2; …; MN and K1; K2; …; KN are the generat- ing elements of M and K, respectively, the corresponding eigen- where values are ) XN M ¼ circðM1; M2; …; MNÞ ðk 1Þði 1Þ KkwN K ¼ circðK1; K2; …; KNÞ k ¼ k¼1 ; i ¼ 1; 2; …; N (49) i XN ðk 1Þði 1Þ are block circulant matrices contained in BCM;N. This type of MkwN problem arises in the study of cyclic vibratory mechanical systems k¼1
22 e XN 1 which follows from Eq. (45) by replacing each Mk with Mk and 2ppr e Y ðrÞ¼ yðpÞ cos (51a) Kk with Kk. For the special case of M ¼ IN with generating ele- R N ments 1; 0; …; 0 (i.e., for the standard cEVP), Eq. (49) reduces to p¼0 the form shown in Theorem 9, as expected. XN 1 2ppr It is clear from this formulation that the same basic steps are Y ðrÞ¼ yðpÞ sin (51b) I N followed to solve the standard and generalized cEVPs. For the p¼0 ðpÞ ðpÞ standard cEVP, we say that ki and qi are the eigenvalues and eigenvectors of the matrix C. For the generalized cEVP, the follow from the formulation of Eq. (33) in Sec. 2.7 for a real- eigenvalues and eigenvectors correspond to the system (M,K), not valued signal where, recall, the eigenvalue expression defined by the individual matrices M and K. Eq. (21) has exactly the same form as the DFT in Definition 20. Example 27. Consider the generalized cEVP defined by Eq. That is, Eq. (51) also represents the real and imaginary parts of (44). Let the generating elements of M 2 C2 be M1 ¼ 2 and the DFT of a real-valued signal, where the sequence of generating M2 ¼ 1 and the generating elements of K 2 C2 be K1 ¼ 5 and elements y(p) is analogous to a real signal and the eigenvalues K2 ¼ 1 such that Y(r) are analogous to its DFT. It is shown that the symmetry of the DFT about the so-called Nyquist component also exists for the 2 1 5 1 M ¼ and K ¼ eigenvalues of a circulant matrix with real generating elements. 12 15 As is customary in signal processing, we restrict the formulation to even N. The case of odd N also yields symmetric eigenvalues, Then the eigenvalues of the system (M, K) follow from Eq. (49) but with multiplicity of the Nyquist component. This is handled with N ¼ 2 and are given by by example in Sec. 3 (for instance, see Fig. 9). For even N, the zeroth eigenvalue Y(0) and “Nyquist” eigen- X2 value YNðÞ=2 are always real because ðp 1Þði 1Þ Kpw N XN 1 XN 1 p¼1 k ¼ Y Y ð0Þ¼ yðpÞ cosð0Þ¼ yðpÞ i X2 0 R ðp 1Þði 1Þ p¼0 p¼0 MpwN (52a) XN 1 p¼1 YIð0Þ¼ yðpÞ sinð0Þ¼0 0 i 1 p¼0 5w2 þð 1Þw2 ¼ XN 1 XN 1 2w0 þð 1Þwi 1 N 2 2 Y Y ¼ yðpÞ cosðppÞ¼ yðpÞð 1Þp N=2 R 2 jpði 1Þ p¼0 p¼0 5 e (52b) ¼ XN 1 2 ejpði 1Þ N YI ¼ yðpÞ sinðppÞ¼0 2 p¼0 for i ¼ 1, 2. It follows that k1 ¼ 4 and k2 ¼ 2. The corresponding orthonormal eigenvectors are given by are such that the imaginary parts vanish.2 The remaining eigenval- "# 1 ues appear in complex conjugate pairs, as the following corolla- ð1Þ 1ffiffiffi ries show. q~1 ¼ e1 ¼ p 2 1 COROLLARY 25. Let y(p) be the real-valued generating elements "# "# of a circulant matrix for p ¼ 0; 1; 2…; N 1 and Y(r) denote its 1 1 eigenvalues according to Eq. (21). Then ð1Þ 1ffiffiffi 1ffiffiffi q~2 ¼ e2 ¼ p ¼ p 2 w2 2 1 YðN rÞ¼YðrÞ¼Yð rÞ ٗ which follow from Definition 18 according to Eq. (42). for r ¼ 0; 1; 2…; N 1. Proof. The eigenvalues of y(p) are given by 2.8.3 Eigenvalue and Eigenvector Structure. For real-valued YðrÞ¼YRðrÞþjYIðrÞ, where generating elements, such as those that arise in models of physical systems with cyclic symmetry, the eigenvalues of a circulant ma- XN 1 2ppr trix C 2 are endowed with certain symmetry characteristics, YRðrÞ¼ yðpÞ cos CN N and the same is true for the eigenvalues of systems M; K 2 . p¼0 CN XN 1 However, we do not require the circulants to be symmetric, as it is 2ppð rÞ (Eq. 51a) assumed in Corollary 17. We begin by systematically describing ¼ yðpÞ cos N the eigenvalue structure of an ordinary circulant with real generat- p¼0 ing elements. The results are generalized by inspection to handle ¼ YRð rÞ block circulants and (M,K) systems composed of circulant or block circulant matrices. and the property cosð hÞ¼cosðhÞ is employed. Similarly, The eigenvalues ki of a circulant matrix C 2 CN with generat- ing elements c1; c2; …; cN are given by Eq. (21), or the equivalent XN 1 matrix–vector form in Eq. (22). It is convenient to re-index each 2ppr YIðrÞ¼ yðpÞ sin ck and ki such that p¼0 N XN 1 ) 2ppð rÞ (Eq. 51b) yðpÞj ¼ ckj ¼ yðpÞ sin p¼0;1;2;…;N 1 k¼1;2;3;…;N N (50) p¼0 YðrÞjr¼0;1;2;…;N 1 ¼ kiji¼1;2;3;…;N ¼ YIð rÞ which facilitates the results for real generating elements and clari- fies their interpretation. If the eigenvalues are dissected according 2Equation (52a) also holds if N is odd, but the Nyquist component is repeated to YðrÞ¼YRðrÞþjYIðrÞ, then the real and imaginary components with multiplicity of two.
23 where the property sinð hÞ¼ sinðhÞ is employed. It follows that (frequencies) are symmetric about the Nyquist component YN=2, as are the arguments (phase angles) but with the opposite sign for YðrÞ¼YRðrÞ jYIðrÞ indices r > N=2. The eigenvalue symmetries can be observed in the examples of ¼ YRð rÞþjYIð rÞ Sec. 2.5.4. The eigenvalues of the matrix C ¼ circð4; 1; 0; 1Þ ¼ Yð rÞ in Example 15 are aðCÞ¼f2; 4; 6; 4g, where Y(0) ¼ 2 and YN=2 ¼ Yð2Þ¼6 are real and distinct and Y(1) ¼ Y(3) ¼ 4 are real which completes the right-hand side of the proof. To prove the and repeated. Similarly, for the matrix C ¼ circð4; 1; 0; 1Þ in left-hand side, consider Example 16, the eigenvalues are aðCÞ¼f4; 4 2j; 4; 4 þ 2jg, where Yð1Þ¼4 2j and Yð3Þ¼4 þ 2j are complex conjugates XN 1 2ppðN rÞ and Yð0Þ¼YN=2 ¼ Yð2Þ¼4 are real-valued. As expected, these YRðN rÞ¼ yðpÞ cos same symmetries are also observed in Example 21 for the DFT of p¼0 N a real-valued signal. XN 1 2ppr A similar formulation shows that the eigenvalues of real-valued ¼ yðpÞ cos 2pp N (M,K) systems exhibit the same symmetry characteristics because p¼0 the numerator and denominator of Eq. (49) have exactly the same XN 1 2ppr form as Eq. (21). If the scalars Mk and Kk in Eq. (49) are re- ¼ yðpÞ cosð2ppÞ cos N indexed and restricted to be real-valued, then Corollaries 25 and p¼0 26 are generalized accordingly. 2ppr For signal processing applications, the symmetry characteristics sinð2ppÞ sin N summarized in Table 3 have practical significance because the subset of ðN þ 2Þ=2 frequency-domain components 0 r N=2 XN 1 2ppr 2ppr ¼ yðpÞ 1 cos 0 sin is endowed with the same basic “information” contained in all N N N time-domain signal samples for 0 p N 1. That is, when p¼0 transformed to the frequency domain by the DFT process, any XN 1 2ppð rÞ real-valued signal has a zero-frequency or dc component (r ¼ 0), ¼ yðpÞ cos distinct magnitude and phase information for 0 r N=2, and p 0 N ¼ repeated magnitude and phase information for r > N=2. For circu- ¼ YRð rÞ (Eq. 51a) lant matrices that describe physical systems with cyclic symmetry, Y0 and YN=2 correspond to standing wave vibration modes and the and a similar expansion shows that YIðN rÞ¼ YIðrÞ. It follows remaining eigenvalues are associated with traveling wave modes. that This is discussed in Sec. 3. Equation (42) shows that e1; e2; …; eN are the eigenvectors of YðN rÞ¼YRðN rÞ jYIðN rÞ any circulant matrix C,andthesameistruefor(M,K)systemscom- posed of circulant matrices. Each eigenvector e is associated with an ¼ YRðrÞ jYIðrÞ i eigenvalue Y(r)(i.e.,ki) according to the indices defined by Eq. (50). ¼ YðrÞ For the special case of real generating elements, the eigenvalues summarized in Table 3 have exactly the same symmetry characteris- which completes the proof. ticsasthevectorse1; e2; …; eN, which are discussed in Sec. 2.5.2. Corollary 25 establishes the following result. For example, if N is even, the real eigenvalues YN/2 (i.e., k(N+2)/2)and COROLLARY 26. Let y(p) be the real-valued generating elements Y(0) (i.e., k1) correspond to the real eigenvectors eðÞNþ2=2 and e1 of a circulant matrix for p ¼ 0; 1; 2…; N 1 and Y(r) denote its defined by Eqs. (18) and (19), respectively. The remaining eigenval- eigenvalues according to Eq. (21). Then ues appear in complex conjugate pairs and are associated with the complex conjugate eigenvectors according to Eq. (17). jYðN rÞj ¼ jYðrÞj If the scalar sequences y(p) and Y(r) for a circulant matrix are ffYðN rÞ¼ ffYðrÞ replaced by a sequence of matrices y(p) and Y(r), it is clear that the formulation given above also holds for Eq. (23), which defines the eigenvalues for block circulant matrices. In this case, the ٗ .for integers r ¼ 0; 1; 2…; N 1 groups of eigenvalues associated with each Ki are endowed with The magnitudes jYðrÞj and arguments ffYðrÞ of the eigenvalues the symmetry properties given in Table 3. This is confirmed by Y(r) are listed in Table 3 for the special case of even N ¼ 8, where Example 25 where, using the indexing scheme of that section, YðrÞ¼YðN rÞ follows from complex conjugation of the left- aðK Þ¼f 1; 1g and aðK Þ¼f3; 5g are distinct sets of eigenval- hand equality given by Corollary 25. The zeroth eigenvalue 1 3 ues and aðK2Þ¼aðK4Þ¼f1; 3g are repeated. Y(0) ¼ Y0 is real and generally distinct, as is YNðÞ¼=2 YN=2 for even N, but the remaining eigenvalues generally appear in com- 2.8.4 Eigenvector Orthogonality. Here, we consider eigen- plex conjugate pairs. It follows that the eigenvalue magnitudes vector orthogonality relationships for block circulant matrices C and systems (M,K) contained in BCM;N for the special case of Table 3 Magnitudes and arguments of Y(r) for even N 5 8 symmetric generating matrices. However, we do not restrict either C or (M,K) to be symmetric or block symmetric. We require only Index Eigenvalue Magnitude Argument that the generating matrices C1; C2; …; CN of C are symmetric for rY(r) YðN rÞjYðrÞj ffYðrÞ the standard cEVP, which guarantees that each Ki is symmetric according to Corollary 19. Similarly, we require that the generat- 0 Yð0Þ¼Y0 Yð8ÞjYð0Þj ¼ Y0 ffYð0Þ ing matrices M1; M2; …; MN and K1; K2; …; KN of (M,K) are 1 Y(1) Yð7ÞjYð1Þj ffYð1Þ e e symmetric for the generalized cEVP, which implies that ðKi; MiÞ 2 Y(2) Yð6ÞjYð2Þj ffYð2Þ are symmetric. Symmetric generating matrices commonly arise in 3 Y(3) Y 5 Y 3 Y 3 ð Þjð Þj ff ð Þ models of rotating flexible structures, including the ones consid- 4 ¼ ðÞN=2 Yð4Þ¼Yð4Þ¼YN=2 Yð4ÞjYð4Þj ¼ YN=2 ffYð4Þ 5 Yð5Þ¼Yð3Þ Yð3ÞjYð3Þj ffYð3Þ ered in Sec. 3, where the sector models and intersector coupling 6 Yð6Þ¼Yð2Þ Yð2ÞjYð2Þj ffYð2Þ are described by symmetric matrices. Then the usual orthogonal- 7 Yð7Þ¼Yð1Þ Yð1ÞjYð1Þj ffYð1Þ ity relationships hold for the reduced-order eigenvectors defined ðpÞ in Secs. 2.8.1 and 2.8.2. It is first shown that each u~i is
24 ð1Þ ð2Þ orthogonal with respect to Ki for the standard cEVP. Proofs can has eigenvalues k ¼ 3 and k ¼ 5 as its diagonal elements. be found in any standard textbook on linear algebra [98,99]. To- 3 3 ~ð1ÞTK ~ð1Þ ~ð2ÞTK ~ð2Þ gether with generic orthogonality conditions on the basic circulant The reader can verify that u3 3u3 ¼ 3 and u3 3u3 ¼ 5 according to Corollary 27. structure of a matrix C 2 BCM;N, this gives rise to an orthogonal- ðpÞ ðpÞ ðpÞ ðpÞ Orthogonality of an eigenvector q~i ¼ ei u~i with respect to ity condition on the eigenvector q~i ¼ ei u~i with respect to C. ðpÞ C 2 BCM;N is essentially decomposed into orthogonality of ei Orthogonality of q~i with respect to the system M; K 2 BCM;N is handled similarly. with respect to the circulant structure of C and orthogonality of ~ðpÞ The results of this section, and the requirement of symmetric gener- each ui with respect to the symmetric matrices Ki. These indi- ating matrices, are meant to highlight how orthogonality of an eigen- vidual orthogonality conditions are captured by Corollaries 18 and ðpÞ ðpÞ 27, which lead to the following fundamental result. vector q~i ¼ ei u~i is essentially dissected into the orthogonality COROLLARY 28. Let C 2 BCM;N be a block circulant matrix with of ei with respect to the circulant structure of C 2 BCM;N and ortho- ðpÞ symmetric generating matrices, ei be the ith column of the N N gonality of u~i with respect to the generating matrices (e.g., Ki), ðpÞ Fourier matrix EN, and u~ be the pth reduced-order orthonormal where the latter requires symmetric C1; C2; …; CN. It should be i noted that none of the results in Sec. 2, aside from this section, require eigenvector corresponding to Ki defined by Eq. (23). Then symmetric generating matrices. In particular, Theorems 9 and 10, ðpÞ T ðqÞ ðpÞ ðeH ðu~ Þ ÞCðe u~ Þ¼k d d upon which block reduction of the cEVPs in Secs. 2.8.1 and 2.8.2 are i i k i i ik pq based, are valid for arbitrary M M generating matrices. ðpÞ for i; k ¼ 1; 2; …; N and p; q ¼ 1; 2; …; M, where k is the eigen- COROLLARY 27. Suppose each K defined by Eq. (23) is symmetric. i i ðpÞ ٗ . ðpÞ ðpÞ ~ d Let u~ be the pth orthonormal eigenvector of K and k be the value associated with ui and ik is the Kronecker delta i i i e ð1Þ ð2Þ ðMÞ e ð1Þ ð2Þ ðMÞ Proof. Let Ui ¼ðu~i ; u~i ; …; u~i Þ be the orthonormal modal corresponding eigenvalue. Then if Ui ¼ðu~i ; u~i ; …; u~i Þ is the matrix associated with Ki and C1; C2; …; CN be the symmetric M M orthonormal modal matrix associated with Ki, it follows that 2 3 generating matrices of C. Corollary 19 guarantees that Ki is sym- kð1Þ 0 metric because the generating matrices are symmetric. By setting 6 i 7 e # T 6 ð2Þ 7 B ¼ U and ð Þ ¼ ð Þ in Corollary 21, it follows that eT e 6 ki 7 i Ui KiUi ¼ . ; i ¼ 1; 2; …; N 4 . 5 H eT e . ðe U ÞCðek UiÞ¼Widik ðMÞ i i ( 0 ki Wi; i ¼ k is a diagonal matrix with eigenvalues kðpÞ along its diagonal for ¼ i 0; otherwise p ¼ 1; 2; …; M and
ðpÞT ðqÞ ðpÞ where u~i Kiu~i ¼ ki dpq XN # ðn 1Þði 1Þ (where dpq is the Kronecker delta. ٗ Wi ¼ B CnBwN (Eq. 24 Example 28. Consider the eigensolutions n¼1 9 XN eT e ðn 1Þði 1Þ ð1Þ 1ffiffiffi T ð1Þ > ¼ Ui CnUiwN (by substitution) u~3 ¼ p ð1; 1Þ ; k3 ¼ 3 = 2 n¼1 ! XN ð2Þ 1 T ð2Þ > u~ ¼ pffiffiffi ð 1; 1Þ ; k ¼ 5 ; ¼ UeT C wðn 1Þði 1Þ Ue 3 2 3 i n N i n¼1 corresponding to the symmetric matrix ¼ UeT K Ue (Eq. 23) 2 i i i 3 4 1 kð1Þ 0 K ¼ 6 i 7 3 14 6 ð2Þ 7 6 ki 7 ¼ 4 . 5 (Cor. 27) . . from Example 25, where the reduced-order eigenvectors are in ðMÞ orthonormal form. The corresponding reduced-order modal matrix 0 ki is denoted by e ð1Þ ð2Þ for i ¼ 1; 2; …; N. Expanding the M M matrix product U3 ¼ðu~3 ; u~3 Þ H eT e ðei Ui ÞCðek UiÞ yields "# 0 2 31 1 1 1ffiffiffi ðu~ð1ÞÞT ¼ p B 6 i 7C 2 11 B 6 7C B 6 . 7C B 6 . 7C B 6 7C It follows from Corollary 27 that the diagonal matrix B H 6 ðpÞ T 7C ð1Þ ðqÞ ðMÞ Be 6 ðu~ Þ 7CCek ðu~ ; …; u~ ; …; u~ Þ B i 6 i 7C i i i "#"#"# B 6 7C T B 6 . 7C 1 1 4 1 1 1 @ 4 . 5A eT e 1ffiffiffi 1ffiffiffi U3 K3U3 ¼ p p ðMÞ T 2 11 14 2 11 ðu~i Þ "#"# 2 3 eH ðu~ð1ÞÞT 1 113 5 6 i i 7 ¼ 6 7 2 6 . 7 11 35 6 . 7 "# 6 7 6 H ðpÞ T 7 ð1Þ ðqÞ ðMÞ 60 ¼ 6 e ðu~ Þ 7Cðek u~ ; …; ek u~ ; …; ek u~ Þ 1 6 i i 7 i i i ¼ 6 7 2 6 . 7 010 4 . 5 "# 30 eH ðu~ðMÞÞT ¼ i i 05 which produces an M M array with scalar elements
25 H ðpÞ T ðqÞ ð2ÞH ð2Þ 1 ðei ðu~i Þ ÞCðek u~i Þ q Cq ¼ pffiffiffi ½ 111 1 111 1 3 3 2 2 2 3 in the (p, q) position for p; q ¼ 1; 2; …; M. However, in light of 2 1 10 0 0 10 6 7 the diagonal structure of each Wi, only the diagonal elements sur- 6 7 vive in this expansion. That is, 6 12 0 1000 1 7 6 7 6 7 ( 6 10 2 1 10 0 07 kðpÞd ; p ¼ q 6 7 H ðpÞ T ðqÞ i ik 6 0 1 12 0 10 07 ðei ðu~i Þ ÞCðek u~i Þ¼ 6 7 6 7 0; otherwise 6 7 6 00 10 2 1 107 ðpÞ 6 7 ¼ ki dikdpq 6 000 1 12 0 1 7 6 7 6 7 4 1000 10 2 1 5 which completes the proof. If i ¼ k in Corollary 28, the orthogonality condition can be 0 10 0 0 1 12 ðpÞ ðpÞ 2 3 stated in terms of the eigenvectors q~i ¼ ei u~i . That is, 1 6 7 ðpÞ H ðqÞ ðpÞ 6 7 ðq~i Þ Cq~i ¼ ki dpq; i ¼ 1; 2; …; N (53) 6 1 7 6 7 6 7 ðpÞ 6 1 7 For an ordinary circulant, each u~i ¼ 1 and Corollary 28 reduces to 6 7 6 7 1 6 1 7 H pffiffiffi 6 7 ei Cek ¼ kidik; i ¼ 1; 2; …; N (54) 2 2 6 7 6 1 7 6 7 6 1 7 which is the same result given by Corollary 18. 6 7 6 7 Example 29. Consider C ¼ circð4; 1; 0; 1Þ2C4 from 4 1 5 Examples 15, 17, and 24, where it is shown that 1 T 1 e2 ¼ 2 ð1; j; 1; jÞ is an eigenvector of C corresponding to the eigenvalue k ¼ 4. Thus, the product 1 2 ¼ ½ 111 1 111 1 2 3 2 3 8 4 10 1 1 6 7 6 7 ð 5; 5; 5; 5; 5; 5; 5; 5ÞT 6 7 6 7 6 14 107 6 j 7 H 1 6 7 1 6 7 ¼ 5 e Ce2 ¼ ½ 1 j 1 j 6 7 6 7 2 2 6 7 2 6 7 4 0 14 1 5 4 1 5 ð2Þ which is numerically equal to the eigenvalue k3 . However, the 10 14 j matrix product
ð1Þ ð2Þ 1 T H ¼ ½ ð1 j 1 j 4; 4j; 4; 4jÞ ðe3 u3 Þ Cðe3 u3 Þ 4 ¼ðeH ðuð1ÞÞHÞCðe uð2ÞÞ 1 3 3 3 3 ¼ ð4 þ 4 þ 4 þ 4Þ H ð1Þ T ð2Þ 4 ¼ðe3 ðu3 Þ ÞCðe3 u3 Þ ¼ 4 1 ¼ pffiffiffi ½ 11 1 111 1 1 C 2 2 is numerically equal to k2 according to Eq. (54). However, the 1 product pffiffiffi ð 1; 1; 1; 1; 1; 1; 1; 1ÞT 2 2 1 1 eHCe ¼ ½ ð1 j 1 j 4; 4j; 4; 4jÞT ¼ ½ 11 1 111 1 1 4 2 4 8 T 1 ð 5; 5; 5; 5; 5; 5; 5; 5Þ ¼ ð4 4 þ 4 4Þ 4 ¼ 0 ¼ 0 vanishes because dpq ¼ 0 ðp 6¼ qÞ in Corollary 28. Similarly, vanishes because i 6¼ k. ð2Þ H ð2Þ ðe2 u Þ Cðe3 u Þ Example 30. Consider C ¼ circðA; B; 0; BÞ2BC2;4 from 3 3 Examples 6, 18, 19, 20, and 25, where it is shown that 1 ¼ pffiffiffi ½ 11 jj1 1 j j C 1 2 2 ~ð2Þ ffiffiffi ð2Þ q3 ¼ p q3 1 2 pffiffiffi ð 1; 1; 1; 1; 1; 1; 1; 1ÞT 2 2 1 T ¼ pffiffiffi ð 1; 1; 1; 1; 1; 1; 1; 1Þ 1 2 2 ¼ ½ 11 jj1 1 j j 8 is an orthonormal eigenvector of C corresponding to the eigen- ð 5; 5; 5; 5; 5; 5; 5; 5ÞT ð2Þ value k3 ¼ 5. Because the generating matrices A, B, 0, B are symmetric (see Example 6), Corollary 28 guarantees that each ¼ 0 ðpÞ ðpÞ q~i ¼ ei u~i is mutually orthogonal with respect to C. For example, it follows from Eq. (53) that because dik ¼ 0 ði 6¼ kÞ.
26 If instead we set A ¼ 4 and B ¼ 1 such that C 2 C4 is an ordi- DOFs per sector, which easily extends from two to M DOFs per nary circulant, we recover the orthogonality condition in Example sector using the theory presented in Sec. 2. 15 for i ¼ 3. The orthogonality conditions used in Example 30 for the special 3.1 General Cyclic System case of an ordinary circulant does not require that the circulant structure is symmetric (i.e., C need not be contained in SCN). 3.1.1 Equations of Motion. Consider the general cyclic vibra- The reader can verify that Eq. (54) also holds for nonsymmetric tory system shown schematically in Fig. 4, which consists of N matrices C 2 CN by inspection of Example 16. sectors with coupling (elastic and damping) to adjacent neighbors. The fundamental orthogonality relationship given by Corollary The topology diagram only indicates nearest-neighbor coupling, 28 also holds for M; K 2 BCM;N systems with symmetric generat- but more general coupling is admissible as long as the cyclic sym- ing matrices, as the following corollary shows. metry is preserved. If there are M DOFs per sector, each M 1 Corollary 29. Let M; K 2 BCM;N be block circulant with vector qi describes the dynamics of the ith sector for symmetric generating matrices, ei be the ith column of the N N i 2f1; 2; …; Ng N. Then the linear EOM takes the form ðpÞ Fourier matrix EN, and u~i be the pth M 1 reduced-order e e b orthonormal eigenvector corresponding to the ith system ðMi; KiÞ Mq€ þ Cq_ þ Kq ¼ f (55) defined by Eq. (46). Then 9 T where q ¼ðq1; q2; …; qNÞ is a NM 1 configuration vector, the H ðpÞ T ðpÞ = ðei ðu~i Þ ÞMðei ui Þ¼dikdpq system matrices are block circulant with M M blocks, and over- H ðpÞ T ðpÞ ðpÞ ; dots denote differentiation with respect to time. If the M 1 vec- ðe ðu~ Þ ÞKðei u Þ¼k dikdpq i i i i tor fi denotes the component of forcing on the ith sector, then b T f ¼ðf1; f2; …; fNÞ is a NM 1 general forcing vector. The ðpÞ for i; k ¼ 1; 2; …; N and p; q ¼ 1; 2; …; M, where ki is the eigen- NM NM system mass, damping, and stiffness matrices are of the ٗ ðpÞ value associated with u~i and dik is the Kronecker delta. form In practice, of course, the M 1 reduced-order eigenvectors 9 ðpÞ u~i are not known a priori. Instead, Theorem 10 is used in Sec. 3 M ¼ circðM1; M2; …; MNÞ2BCM;N => to decouple the NM-DOF system equations into a set of N C ¼ circðC ; C ; …; C Þ2BC ; (56) reduced-order M-DOF standard vibratory problems, from which 1 2 N M N ;> the system eigenvalues (natural frequencies) and eigenvectors K ¼ circðK1; K2; …; KNÞ2BCM;N (normal modes) are extracted.
where the generating matrices Mi, Ci, and Ki depend on the M M sector mass, damping, and stiffness matrices and the inter- 3 Example Applications sector coupling (stiffness and damping). Equation (55) is a general In this section we apply the results developed in Sec. 2 to vibra- model for any linear, lumped-parameter, conservative, nongyro- tion models of systems with cyclic symmetry. For each model scipic, cyclic vibratory system with N sectors and M DOFs per considered, we begin by formulating the equations of motion sector. For example, a linearized lumped-parameter model of a (EOM) and then use the theory of circulants to diagonalize or bladed disk assembly under engine order excitation is captured by block diagonalize the governing equations. This is achieved by a Eq. (55), where N is the number of blades, M is the number of coordinate transformation that exploits the special relationship DOFs per blade, f has the special properties described in Sec. 3.2, between circulant matrices and the Fourier matrix. The process and the system matrices depend on the structural details of each also shows how external forces are projected on the resulting blade (i.e., sector) and its connectivity to adjacent blades and block diagonal EOM. The special case of traveling wave engine rotating hub. order excitation is also presented in some detail because it appears 3.1.2 Modal Transformation. Of course, one can apply stand- in many relevant applications of rotating machinery. ard techniques [99] to investigate the free and forced response of Three examples are presented. The first example (Sec. 3.1) con- the model given by Eq. (55). However, this requires solving an siders the structure of the EOM for a general cyclic system with N NM NM eigenvalue problem to determine the modal properties, sectors, M DOFs per sector, and arbitrary excitation. It is shown or inversion of an NM NM impedance matrix to determine the how to block diagonalize the system equations via a modal trans- response to harmonic excitation. Such an approach may be pro- formation involving the Fourier matrix, which results in NM-DOF hibitive or computationally expensive for practical models with reduced-order vibratory systems. If engine order excitation is assumed (Sec. 3.2), it is shown that the steady-state forced response of the NM-DOF system can be obtained from a single M- DOF harmonically forced, reduced-order system in modal space. The mathematical and physical details of engine order excitation are discussed, including its temporal and spatial duality. The sec- ond example (Sec. 3.3) considers a cyclic system with one DOF per sector under engine order excitation. This system is fully dia- gonalized by the Fourier matrix. The example is presented in detail, showing the nature of the natural modes and frequencies, and the resonant response to excitation of various engine orders. The third example (Sec. 3.4) has two DOFs per sector and demon- strates the block diagonalization process for a perfectly cyclic sys- tem with specified sector models, as opposed to the general sector models in the first example. In each sector, one DOF is due to flexure, and thus has a constant frequency, while the other DOF is a centrifugally driven pendulum whose frequency is proportional to the rotor speed. The coupling between these DOFs leads to some interesting behavior in both the free and forced vibration of the system, which is discussed in Refs. [21,22] and [92–95]. More importantly, this example shows the process of handling multiple Fig. 4 Topology diagram of a general cyclic system
27 many sectors and many DOFs per sector, nor does it highlight or and make no assumptions on the nature of the applied forcing. exploit the underlying features of the cyclic system. It is precisely Thus, the solution to the NM-DOF matrix EOM given by Eq. (58) these special features that are brought to light by the properties of reduces to solving NM-DOF uncoupled systems the circulants or block circulants that describe the system. Solving for the system response is significantly facilitated by a e e e H b Mpu€p þ Cpu_ p þ Kpup ¼ðep IMÞf; p 2N (61) modal transformation that exploits the cyclic symmetry among the N sectors. Specifically, it is straightforward to block decouple for the modal solutions u . These N reduced-order EOMs of order the EOM into a set of N systems, each with M DOFs. To this end, p M offer a substantial computational savings compared to the full we introduce the change of coordinates NM-DOF system defined by Eq. (55). The reduced equations can q ¼ðEN IMÞu (57) be solved directly or with standard modal analysis by solving a set of N generalized eigenvalue problems, each of order M, from T where u ¼ðu1; u2; …; uNÞ is a NM 1 vector of modal coordi- which one can form the global eigenvectors of the system. Specifi- nates. Each ui is M 1 and describes the sector dynamics in cally, solving the N eigenvalue problems associated with Eq. (61) modal space, where the EOM are block decoupled (as described yields a set of normalized M M modal matrices Up, which define below). In this formulation the physical coordinates q are a change to reduced modal coordinates via up ¼ Upsp. The sp are expressed in terms of the modal coordinates ui and the Fourier modes that define the behavior of the internal sector dynamics and elements of length N, which accounts for the overall cyclic nature account for the manner in which these are coupled to each other of the EOM. Substituting Eq. (57) into Eq. (55) and multiplying in a given Fourier mode of the overall system. The EOM for the si H H from the left by ðEN IMÞ ¼ðEN IMÞ yields form a set of M uncoupled equations, and these are related to the H H physical coordinates of the original system by q ¼ðEN IMÞu ðE IMÞMðEN IMÞu€ þðE IMÞCðEN IMÞu_ T N N where u ¼ðU1s1; U2s2; …; UNsNÞ . This process is general, and it þðEH I ÞKðE I Þu allows one to decouple the full EOM in two steps, one which N M N M accounts for the global cyclic nature of the system, and the other H b which accounts for the details of the sector model. ¼ðEN IMÞf Equation (61) can be simplified even further if the system is or subjected to the so-called engine order excitation. The mathemat- 2 3 2 3 ics and physics of this type of excitation are developed in the next e 2 3 e 2 3 M 0 u€1 C 0 u_ 1 section, which closes with a treatment of the general cyclic system 6 1 7 6 1 7 6 e 76 7 6 e 76 7 governed by Eq. (61) under engine order excitation. 6 M 76 u€2 7 6 C 76 u_ 2 7 6 2 76 7 6 2 76 7 6 76 7 þ 6 76 7 6 . 76 . 7 6 . 76 . 7 4 . . 54 . 5 4 .. 54 . 5 3.2 Engine Order Excitation. Traveling wave engine order excitation arises in rotating machinery and is a primary source of e e 0 MN u€N 0 CN u_ N forced vibration response in bladed disk assemblies [11,111]. A 2 3 2 3 e 2 3 H b mathematical model of this common form of excitation is devel- K1 0 u1 ðe1 IMÞf oped in Sec. 3.2.1 and its traveling wave characteristics are 6 76 7 6 7 6 e 7 6 H b7 described in Sec. 3.2.2. While this material is known, the discus- 6 K 76 u2 7 6 ðe I Þf 7 6 2 76 7 6 2 M 7 þ 6 76 7 ¼ 6 7 (58) sion that follows is unique because it provides physical insights 6 . 76 . 7 6 . 7 and a systematic explanation of the temporal and spatial duality of 4 .. 54 . 5 4 . 5 . engine order excitation. The general cyclic system of Sec. 3.1 is e 0 K uN H b reconsidered in Sec. 3.2.3 under engine order excitation, where it N ðeN IMÞf is shown that the steady-state forced response of the NM-DOF The block diagonal structure on the left-hand side of Eq. (58) fol- system reduces to that of a single sector in modal space. lows from Theorem 10, where the M M modal mass, damping, and stiffness matrices associated with the pth mode follow from 3.2.1 Mathematical Model. Ideally, the steady axial gas pres- Eq. (23) and are given by sure in a jet engine might vary with radius but is otherwise uni- 9 form in the circumferential direction, thus resulting in an identical XN > force field on each blade in a particular fan, compressor, or turbine e ðk 1Þðp 1Þ > Mp ¼ MkwN > within the engine. In practice, however, flow entering an engine > k¼1 > inlet invariably meets static obstructions, such as struts, stator XN = e ðk 1Þðp 1Þ vanes, etc., in addition to rotating bladed disk assemblies in its Cp ¼ CkwN >; p 2N (59) path to the exhaust. Even in steady operation, therefore, the flow > k¼1 > slightly upstream of these bladed assemblies is nonuniform in XN > ðk 1Þðp 1Þ > pressure, temperature, and so on. This results in a static pressure Ke ¼ K w ;> p k N (effective force) field on the blades that vary circumferentially, a 1 k¼ notional example of which is shown in Fig. 5. The forcing terms on the right-hand side of Eq. (58) follow from Consider, for example, an engine in steady operation with n the decomposition evenly spaced struts slightly upstream (or downstream) of a bladed assembly. As explained in Ref. [3], these obstructions produce a b T b circumferential variation upon the mean axial gas pressure that ðEH I Þf ¼ðeH; eH; …; eHÞ I f N M 1 2 N M is essentially proportional to cos nh, where h is an angular posi- H H H Tb tion. Thus, a blade rotating through this static pressure field expe- ¼ e IM; e IM; …; e IM f 2 1 2 3 N riences a force proportional to cos nXt, where X is the constant H b angular speed of the bladed disk assembly and t is time. An ðe IMÞf 6 1 7 (60) adjacent blade experiences the same force, but at a constant frac- 6 b7 6 ðeH I Þf 7 6 2 M 7 tion of time later. This type of excitation is defined as engine order ¼ 6 7 (e.o.) excitation and n is said to be the order of the excitation. 6 . 7 4 . 5 To be more precise, the axial gas pressure of a steady flow H b through a jet engine may be described by the function ðeN IMÞf pðhÞ¼pðh þ 2pÞ, where h is an angular coordinate measured
28 waveform, or traveling wave. If a single observer was placed on the rotating hub and recorded the strength of this traveling wave as a function of i (taken here to be continuous), it would resemble the curve shown in Fig. 6(b). In this context, the instantaneous loading applied to individual blades is obtained by essentially “sampling” the continuous traveling wave at each sector i 2N and, as time evolves, these sampled points define N time profiles of the force amplitudes, which is equivalent to the discrete tempo- ral interpretation described above. However, the latter interpreta- tion illuminates some important traveling wave characteristics of the engine order excitation that are otherwise difficult to explain, and in what follows these are systematically described. To explain the traveling wave mathematically, it is convenient to define the function 2pðk 1Þ U ðvÞ¼cos v ¼ cosðu vÞ (65) Fig. 5 The axial gas pressure pðhÞ: ideal and (notional) actual k N k conditions
which is a harmonic waveform with wavelength 2p=uk. Then for relative to a fixed origin on the machine. That is, the pressure field i 2N and noting that /i ¼ unþ1ði 1Þ, Eq. (63) can be written is rotationally periodic and can therefore be expanded in a Fourier in real form as series with terms of the form po cos nh. If the angular position of F ðtÞ¼F cosðu ði 1ÞþnXtÞ the ith blade relative to the same origin is defined by i nþ1 nX 2p ¼ F cos unþ1 i 1 þ t (66) h ðtÞ¼Xt þ ði 1Þ; i 2N unþ1 i N ¼ FUnþ1ði 1 þ CtÞ where N is the total number of blades and N¼f1; 2; …; Ng is the set of blade, or sector numbers, it follows that the total effective which is a harmonic function with a wavelength of force exerted on blade i due to the nth harmonic of the pressure 2p=unþ1 ¼ N=n (unþ1 is the wave number) and angular frequency field pðhÞ is captured by nX. Equation (66) shows that engine order excitation is a TW in the negative i-direction (descending blade number) with speed n F cos nXt þ 2p ði 1Þ ; i 2N (62) C ¼ nX=unþ1 ¼ NX=2p, measured in sectors per second.An N example plot of this continuous BTW is shown in Fig. 6(b) and, as described above, the applied loads can be obtained from this Upon complexifying, figure by continuously “sampling” the waveform at the discrete sector numbers as time evolves. Then the engine order excitation j/ jnXt FiðtÞ¼Fe i e ; i 2N (63) applied to the individual blades consists of a wave composed of these N discrete points, examples of which are shown in is a model for the nth predominant component of the excitation. Figs. 7(a)–7(d). Interestingly, this gives rise to discrete SW or Eq. (63) has period T ¼ 2p=nX, strength F, and is said to have even FTW applied dynamic loads (depending on the value of n angular speed X. The so-called interblade phase angle is defined relative to N), even though Eq. (66) is strictly a BTW relative to by the rotating hub. These additional possibilities arise due to alias- ing of the “sampled points” just as it occurs in elementary signal ðnÞ n processing theory [112,113]. Before characterizing the traveling / ¼ / ¼ 2p ði 1Þ¼nu ; i 2N (64) i i N i and standing waveforms, it is shown that one need only consider engine orders n 2N. where n 2 Zþ and ui is the angle subtended from blade 1 to blade The traveling wave nature of the discrete applied loads (i.e., i and is defined by Eq. (16). Equation (63) is defined as nth engine SW, BTW, or FTW) depends only on the value of n relative to N. order, or traveling wave excitation. The traveling wave character- To see this, let istics of this type of excitation are considered next. n ¼ n mod N 2N; n 2 Zþ (67) 3.2.2 Traveling Wave Characteristics. The function defined by Eq. (63) is continuous in time and discretized in space via the and assume n ¼ n þ mN for some integer m. Then index i. This gives rise to two interpretations of engine order exci- tation relative to the rotating hub, one discrete and the other con- Un þmNþ1ðvÞ¼Un þ1ðvÞ tinuous. These can be visualized in Fig. 6, which shows a dissection of the excitation amplitudes along time and sector axes. That is, if n ¼ n corresponds to a SW, BTW, or FTW, then so In the first and usual sense, Eq. (63) is a discrete temporal varia- does n þ mN for any m 2 Zþ. In this sense, the traveling wave na- tion of the dynamic loading applied to individual blades. That is, ture of the applied dynamic loads is seen to alias relative to N. under an engine order n excitation, each sector is harmonically These features are characterized for engine orders forced with strength F and frequency nX, but with a fixed phase difference relative to its nearest neighbors. Physically, one can n 2N ¼NO;E [NO;E [NO;E think of this as placing N different observers at the discrete sectors BTW FTW SW and having the ith observer record the excitation strength applied where it is understood that the results apply to any n > N simply to sector i as a function of time. Their recorded time traces would O;E O;E resemble those shown in Fig. 6(a). In the second and more general by taking n modulo N, as appropriate. The subsets N BTW, N FTW, O;E sense, Eq. (63) can be viewed as a continuous spatial variation of and N SW are defined in Table 4 and discussed below. the excitation strength relative to the rotating hub (along the sector For the special case of n ¼ N the rotating blades become ðNÞ axis) that evolves with increasing time, i.e., it is a propagating entrained with the excitation because /i ¼ 2pnði 1Þ with
29 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 5 2p=nX applied to each sector; and (b) the continuous BTW excitation with wavelength N/n and speed C 5 NX=2p relative to the rotating hub i; n 2 Zþ, and hence each is forced with the same strength and 3.2.3 Forced Response of a General Cyclic System Under phase. As illustrated in Fig. 7(d), this is a SW excitation where Engine Order Excitation. The general cyclic system governed by each blade is harmonically forced according to FiðtÞ¼F cos nXt. Eq. (55) is reconsidered here under engine order excitation. Using Entrainment also occurs when n ¼ N/2 if N is even, in which case the notation of Sec. 3.1, a model for the nth engine order excita- ðN=2Þ /i ¼ pnði 1Þ, where (i 1) is odd (resp. even) for even tion is (resp. odd) sector numbers i 2N. Accordingly, all blades with jnu jnXt odd sector numbers are driven by FiðtÞ¼F cos nXt, as are the fi ¼ fe i e ; i 2N (68) blades with even sector numbers, but with a 180-deg phase shift. As shown in Fig. 7(b), this amounts to the same standing wave ex- where f is a constant M M vector of sector force amplitudes, t is citation as the n ¼ N case, except for a phase reversal in the excita- time, ui is the angle subtended from sector 1 to sector i and is tion among adjacent blades. The engine orders corresponding to defined by Eq. (16), n is the order of the excitation, nX is the SW excitations for odd and even N are denoted by the sets angular frequency of the excitation, and X is the angular speed of N O;E N, and all other values of n 2N correspond to traveling SW O;E O;E the system relative to the excitation. Under this type of excitation, waves. Engine orders n 2NBTW (resp. n 2NFTW) correspond to the system forcing vector becomes BTW (resp. FTW) excitation, an example of which is shown in O;E O;E 2 3 2 3 jnu jnXt Fig. 7(a) (resp. Fig. 7(c)), where N BTW and N FTW are also f1 fe 1 e defined in Table 4. These sets can be visualized in Figs. 7(i) and 6 7 6 7 6 7 6 7 7(ii) for odd and even N, respectively. 6 7 6 jnu jnXt 7 6 f2 7 6 fe 2 e 7 The manner in which cyclic systems respond to this type of b 6 7 6 7 jnXt f ¼ 6 7 ¼ 6 7 ¼ f0 fe excitation is considered in the examples that follow. In Sec. 3.2.3, 6 . 7 6 . 7 6 . 7 6 . 7 we prove the most important result related to the forced response 4 5 4 5 of cyclic systems, namely, that in the case of perfect symmetry jnu jnXt each engine order excites only a single mode. This is clear mathe- fN fe N e matically and will be explored with more physical insight in the examples presented in Secs. 3.3 and 3.4. where
30 H which is a direct product of the scalar product ep f0 with the nth order harmonic excitation fejnXt imparted to each sector. This dis- section of the modal forcing term highlights an orthogonality con- dition that is shared by all cyclic systems under engine order excitation. In particular, H 1 j 0 u j 1 u jðN 1Þu e f0 ¼ pffiffiffiffi e p ; e p ; …; e p p N T ejnu1 ; ejnu2 ; …; ejnuN
1 XN ¼ pffiffiffiffi e jðk 1Þup ejnuk N k¼1
XN 1 jðk 1Þ2pðp 1Þ jn2pðk 1Þ ¼ pffiffiffiffi e N e N N k¼1 (71) XN 1ffiffiffiffi ðk 1Þðp 1Þ nðk 1Þ ¼ p wN wN N k¼1
XN 1ffiffiffiffi ðk 1Þðnþ1 pÞ ¼ p wN N k¼1 ( pffiffiffiffi N; p ¼ n þ 1 ¼ 0; otherwise
which follows from Lemma 2 and shows that the force vector f0 is mutually orthogonal to all but one of the modal vectors ep. This Fig. 7 Engine orders n mod N corresponding to BTW, FTW and result is obtained more directly from Corollary 10 by observing SW applied dynamic loading for (i)oddN and (ii)evenN (see also that Table 4); example plots of applied dynamic loading (represented by the dots) for a model with N 5 10 sectors and with (a) n 5 1 jnu jnu jnu T f0 ¼ e 1 ; e 2 ; …; e N (BTW), (b) n 5 5(SW),(c) n 5 9(FTW),and(d) n 5 10 (SW). The BTW engine order excitation is represented by the solid lines. T ¼ ej 0 unþ1 ; ej 1 unþ1 ; …; ejðN 1Þunþ1 Table 4 Sets of engine orders nðmod NÞ ‰ N corresponding to pffiffiffiffi BTW, FTW, SW dynamics loads applied to the blades for odd ¼ Nenþ1 and even N. These can be visualized in Figs. 7(i) and 7(ii).
where enþ1 is the ðn þ 1Þth column of the Fourier matrix. It fol- N Type Set lows that N 1 pffiffiffiffi Odd BTW N O ¼ n 2 Z : 1 n H H BTW þ ep f0 ¼ ep Nenþ1 2 pffiffiffiffi FTW O N þ 1 H N ¼ n 2 Z : n N 1 ¼ Nep enþ1 (72) FTW þ 2 pffiffiffiffi ¼ Ndpðnþ1Þ SW N O ¼ fgN SW Even BTW N 2 which is the same orthogonality condition given by Eq. (71). N E ¼ n 2 Z : 1 n BTW þ 2 Thus, for p 2N, the only excited mode is FTW E N þ 2 p n mod N 1 (73) N ¼ n 2 Zþ : n N 1 ¼ þ FTW 2 N SW N E ¼ ; N for an engine order n 2 Zþ excitation. Then Eq. (70) becomes SW 2 ( pffiffiffiffi jnXt H b Nfe ; p ¼ n þ 1 ðep IMÞf ¼ (74) jnu jnu jnu T 0; otherwise f0 ¼ðe 1 ; e 2 ; …; e N Þ T (69) ¼ðej/1 ; ej/2 ; …; ej/N Þ and the forced response of the NM-DOF matrix EOM given by Eq. (58) reduces to solving a single, M-DOF system is a vector of constant intersector phase angles /i ¼ nui. Thus, pffiffiffiffi e e e jnXt the pth modal forcing term on the right-hand side of Eq. (61) Mnþ1u€nþ1 þ Cnþ1u_ nþ1 þ Knþ1unþ1 ¼ Nfe (75) reduces to in modal space. Assuming harmonic motion, the steady-state H b H jnXt ðep IMÞf ¼ðep IMÞðf0 fÞe modal response is given by ffiffiffiffi H jnXt p ¼ðe f0Þ ðIMfÞe (70) ss e jnXt p unþ1ðtÞ¼ NZnþ1fe (76) eHf fejnXt; p ¼ð p 0Þ 2N where
31 e e e 2 e Znþ1 ¼ Knþ1 þ jnXCnþ1 ðnXÞ Mnþ1 is the ðn þ 1Þth modal impedance matrix. All other steady-state modal responses are zero because only mode p ¼ n þ 1isexcited. In light of the decomposition shown in Eq. (39), the forced response in physical coordinates follows from Eq. (57) and is given by ss ss q ðtÞ¼enþ1 unþ1ðtÞ (77) where enþ1 is the ðn þ 1Þth column of the Fourier matrix and uss ðtÞ is given by Eq. (76). Expanding Eq. (77) into its sector nþ1 Fig. 8 Linear cyclic vibratory system with N sectors and one components yields DOF per sector 2 ss 3 2 3 q1 ðtÞ ej 0 unþ1 6 ss 7 6 7 where f is the strength of the excitation, the interphase blade angle 6 q2 ðtÞ 7 6 ej 1 unþ1 7 6 7 6 7 /i is defined by Eq. (64), n 2 Zþ is the excitation order, r is the 6 . 7 6 . 7 angular speed, and s is time (all dimensionless). 6 . 7 6 . 7 pffiffiffiffi 6 7 1ffiffiffiffi 6 7 e jnXt The linear dynamics of the ith sector are obtained using New- 6 ss 7 ¼ p 6 7 NZnþ1fe (78) 6 q ðtÞ 7 N 6 ej ði 1Þ unþ1 7 6 i 7 6 7 ton’s laws and are governed by 6 7 6 . 7 4 . 5 4 . 5 . . 2 j/i jnrs q€i þ qi þ t ð qi 1 þ 2qi qiþ1Þ¼fe e ; i 2N (81) ss ej ðN 1Þ unþ1 qN ðtÞ where overdots denote differentiation with respect to dimension- Thus, the steady-state forced response of the ith sector in physical less time s. In Eq. (81), the qi61 terms arise from the left-nearest- coordinates is given by neighbor (i 1) and right-nearest-neighbor (i þ 1) elastic cou- pling. By stacking the N coordinates qi into the configuration pffiffiffiffi T ss 1 j i 1 u e jnXt vector q ¼ðq ; q ; …; q Þ , the governing EOM for the overall q ðtÞ¼pffiffiffiffi e ð Þ nþ1 NZ fe 1 2 N i nþ1 N-DOF system takes the form N (79) jnrs e jn/i jnXt q€ þ K11q ¼ f11e ; i 2N (82) ¼ Znþ1fe e ; i 2N j/ j/ j/ T where f11 ¼ðfe 1 ; fe 2 ; …; fe N Þ is the system forcing vector, where /i ¼ 2pðn=NÞði 1Þ. The response of each sector is identi- which accounts for the constant phase difference in the dynamic cal but simply shifted in time by a constant phase relative to its loading from one sector to the next. The N N matrix nearest neighbors. 2 3 1 þ 2t2 t2 0…0 t2 6 7 3.3 Cyclic System With One DOF Per Sector. This exam- 6 t2 1 þ 2t2 t2 …0 07 ple considers the simplest prototypical model for vibrations of a 6 7 6 2 2 7 bladed disk with only one DOF per sector, nearest-neighbor cou- 6 0 t 1 þ 2t …0 07 6 7 pling, and perfect symmetry. Despite the simplicity of the model, K11 ¼ 6 ...... 7 6 ...... 7 we show that the resonant response can be quite complicated 6 7 when the system is subjected to engine order excitation with mul- 4 000…1þ 2t2 t2 5 tiple orders. We begin by formulating the EOM, and then consider t2 00… t2 1 þ 2t2 a direct (traditional) approach to deriving the response to traveling (83) wave excitation. The forced response is then derived using the modal analysis based on the Fourier matrix, which decouples the EOM. The nature of the natural frequencies and modes is consid- captures the nondimensional stiffness of each sector relative to the ered next, which sets the stage for examining the resonance hub (additive unity along its diagonal) and the intersector cou- pling (t2 along the super-diagonal and subdiagonal). The elements behavior of the system when subjected to engine order excitation. 2 This provides a quite general view of the forced response of these t appearing in the (1, N) and (N, 1) positions of K11 are due to models. the cyclic boundary conditions q0 ¼ qN and qNþ1 ¼ q1. In the ab- sence of these cyclic coupling terms, the system represents a finite 3.3.1 Equations of Motion. The undamped cyclic system to chain of N oscillators. Thus, in addition to being symmetric, Eq. be considered is shown in Fig. 8 in dimensionless form. It consists (83) is also a circulant and can be written as of a cyclic chain of N identical and identically coupled single- 2 2 2 DOF oscillators with unit mass, the dynamics of which are cap- K11 ¼ circð1 þ 2t ; t ; 0; …; 0; t Þ2SCN (84) tured by the dimensionless transverse displacements q for i 2N. i 2 2 2 The oscillators are uniformly attached around the circumference where 1 þ 2t ; t ; 0; …; 0; t are the N generating elements. of a stationary rigid hub via linear elastic elements with unit stiff- In the absence of coupling (that is, if t ¼ 0) K11 is diagonal and ness and unit effective length. Adjacent masses are elastically Eq. (82) represents a decoupled set of N harmonically forced, coupled via linear springs, each with nondimensional stiffness . single-DOF oscillators. It is assumed that the elastic elements are unstressed when the The forced response of Eq. (82) is considered next with empha- sis on a modal analysis whereby the fully coupled system (that is, oscillators are in a purely radial configuration, that is, when qi ¼ 0 for each i 2N. An individual oscillator, together with the for- one in which t 6¼ 0) is reduced to a set of N single-DOF oscilla- ward-nearest-neighbor elastic coupling, forms one fundamental tors, only one of which is harmonically excited. The approach sector and there are N such sectors in the overall system. The os- taken here, and a generalization in which each sector has multiple DOFs, is applied to the linear system in Sec. 3.4 to block decouple cillator chain has cyclic boundary conditions such that q0 ¼ qN the system matrices as it is done in Sec. 3.1. and qNþ1 ¼ q1. Finally, the system is subjected to engine order ex- citation (Sec. 3.2) according to 3.3.2 Forced Response. The steady-state forced response of
j/ jnrs Eq. (82) can be obtained using standard techniques [99] and, for fiðtÞ¼fe i e ; i 2N (80) nonresonant forcing, is given by
32 ss 2 2 1 jnrs H H H jnrs q ðsÞ¼ðK11 n r IÞ f11e (85) E Eu€ þ E K11Eu ¼ E f11e (90) where I is the N N identity matrix. However, this requires inver- where EHE ¼ I because E is unitary (Theorem 6). In light of 2 2 sion of the impedance matrix K11 n r I, which is computation- Theorem 9, it follows that ally expensive for a large number of sectors, and it offers little 2 3 2 32 3 2 3 insight into the basic vibration characteristics. In what follows, a 2 H u€1 x 1 0 u1 e1 f11 transformation based on the cyclic symmetry of the system is 6 7 6 76 7 6 7 6 7 6 2 76 7 6 H 7 exploited to fully decouple the single N-DOF system to a set of N u€2 x 2 u2 e2 f11 6 7 6 76 7 6 7 jnrs single-DOF oscillators from which the steady-state response is 6 7 þ 6 76 7 ¼ 6 7e (91) 6 . 7 6 . . 76 . 7 6 . 7 easily obtained. The procedure is similar to the usual modal analy- 4 . 5 4 . 54 . 5 4 . 5 sis from elementary vibration theory. However, a key difference 2 H u€N 0 x uN e f is that the transformation matrix (and hence the system mode N N 11 shapes) is known a priori and, because the transformation is uni- tary (thus preserving the system eigenvalues), the natural frequen- where the pth scalar element of the N 1 modal forcing vector EHf is eHf . Decomposition of EHf follows from Eq. (60) by cies are obtained after the transformation is carried out. 11 p 11 11 b Moreover, due to orthogonality conditions between the normal replacing the identity matrix IM with unity and the vector f with modes and forcing vector, the steady-state response of the overall f11. Equation (89) is a unitary (similarity) transformation and system reduces to finding the forced response of a single harmoni- hence the system natural frequencies are preserved, which is guar- cally forced, single-DOF oscillator in modal space, which offers a anteed by Theorem 1. For each p 2N, the dimensionless natural clear advantage over the direct computation of Eq. (85). frequencies follow from Eq. (21) and are given by As described in Sec. 3.2.2, engine order excitation can be 2 2 2 ðp 1Þ 2 ðN 1Þðp 1Þ regarded as traveling wave dynamic loading. It is therefore rea- x p ¼ 1 þ 2t t wN þ 0 þ þ0 t wN sonable to expect steady-state solutions of the same type. We 2 2 ðp 1Þ ðN 1Þðp 1Þ begin with a simple way to show the existence of such a response, ¼ 1 þ 2t t wN þ wN (92) and then systematically describe it using the modal analysis tech- niques described in Secs. 3.1.2 and 3.2.3. 2 ¼ 1 þ 2t ð1 cos upÞ Existence of a Traveling Wave Response. It is natural to search for steady-state solutions of the form where wN is the primitive Nth root of unity and the identity ðp 1Þ ðN 1Þðp 1Þ wN þ wN ¼ 2 cos up is employed. Equation (91) is a ss j/i jnrs qi ðsÞ¼Ae e ; i 2N (86) decoupled set of N single-DOF harmonically forced modal oscil- lators of the form which has the same traveling wave characteristics as the engine order excitation described in Sec. 3.2.2. Equation (86) assumes 2 rs u€ þ x u ¼ eHf ejn ; p 2N (93) that each sector responds with the same amplitude A, but with a p p p p 11 constant phase difference relative to its nearest neighbors, and to- gether all N such solutions form a traveling wave response among Thus, the single N-DOF system given by Eq. (82) is transformed to the sectors. By mapping this trial solution into Eq. (81) and divid- a system of N decoupled single-DOF systems defined by Eq. (93). ing through by the common term ej/i ejnrs, it follows that The steady-state, nonresonant modal response of the pth decoupled system follows from Eq. (93) using standard techniques 2 ðnrÞ A þ A þ t2 Ae junþ1 þ 2A Aejunþ1 ¼ f (87) [99]. Assuming harmonic motion, the solution is H ss ep f11 jnrs where the identity /i61 /i ¼ 6unþ1 is employed. Solving for u ðsÞ¼ e ; p 2N (94) p 2 2 the amplitude A yields x p ðnrÞ f A ¼ (88) from which the steady-state modal response vector 2 2 1 þ 2t ð1 cos unþ1Þ ðnrÞ ss ss ss ss T u ðsÞ¼ u1 ðsÞ; u2 ðsÞ; …; uN ðsÞ is constructed. In physical from which it follows that coordinates, the steady-state response of sector i follows from Eq. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (89) and is given by 2 x nþ1 ¼ 1 þ 2t ð1 cos unþ1Þ qssðsÞ¼eT ussðsÞ i i is one of the N natural frequencies of the coupled system corre- 1 ffiffiffiffi j 0 ui j 1 ui jðN 1Þui sponding to mode p n mod N 1. Equation (88) shows that ¼ p e ; e ; …; e ¼ þ N mode n þ 1 is excited, but the reason is not clear from this ss ss ss T approach. A modal analysis that considers the fully coupled sys- u1 ðsÞ; u2 ðsÞ; …; uN ðsÞ tem is required to systematically describe the response character- N X 1 istics of the cyclic system under engine order excitation. ffiffiffiffi jðp 1Þui ss ¼ p e up ðsÞ (95) p¼1 N Modal Analysis. Theorem 9 guarantees that circulant matrices, XN H such as the stiffness matrix defined by Eq. (84), are diagonalizable 1 ep f11 ¼ pffiffiffiffiejupði 1Þ ejnrs via a unitary transformation involving the Fourier matrix, and in N x 2 ðnrÞ2 what follows this property is exploited to fully decouple the matrix p¼1 p XN H EOM defined by Eq. (82). To this end, the change of coordinates 1 ep f11 ¼ pffiffiffiffi ejði 1Þup ejnrs; i 2N 2 2 T N p¼1 x p ðnrÞ qðsÞ¼EuðsÞ or qiðsÞ¼ei uðsÞ; i 2N (89) is introduced, where E is the N N complex Fourier matrix (Defini- where the identity ðp 1Þui ¼ upði 1Þ is employed. Equation T tion 17), ei is its ith column (Definition 18), and u ¼ðu1; u2; …; uNÞ (95) shows that there are N possible resonances, depending on the H isavectorofmodal,orcyclic coordinates. Substituting Eq. (89) in Eq. details of the modal forcing terms ep f11. However, only a single (82) and multiplying from the left by EH yields mode survives under an engine order excitation of order n, which
33 is clear from the orthogonalitypffiffiffiffi condition described in Sec. 3.2.3. Noting that f11 ¼ f f0 ¼ Nf enþ1, it follows from Eq. (72) that pffiffiffiffi H H ep f11 ¼ Nf ep enþ1 pffiffiffiffi ¼ Nf dpðnþ1Þ (96) ( pffiffiffiffi Nf ; p ¼ n þ 1 ¼ 0; otherwise which shows that the force vector f11 is mutually orthogonal to all but one of the modal vectors ep. That is, for p 2N, the only excited mode is
p ¼ n mod N þ 1 (97) for an engine order n 2 Zþ excitation. Thus, Eq. (95) reduces to f qssðsÞ¼ ej/i ejnrs; i 2N (98) i 2 2 x nþ1 ðnrÞ where the identity ði 1Þunþ1 ¼ /i is employed and
2 2 x nþ1 ¼ 1 þ 2t ð1 cos unþ1Þ from Eq. (92). Equation (98) is the same result as that obtained Fig. 9 Dimensionless natural frequencies x p in terms of the from Eq. (88). Indeed, the process described here is significantly number of n.d. versus mode number p for WC and SC: (a) 5 5 more laborious than the direct approach, but many general fea- 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 tures can be gleaned from the analysis. The eigenfrequency char- and also the mode numbers corresponding to SW, BTW, and acteristics (Sec. 3.3.3), normal modes of vibration (Sec. 3.3.4), FTW. and resonance structure (Sec. 3.3.5) are systematically described based on the modal decomposition results of this section. where each subset is defined in Table 5. A description of the 3.3.3 Eigenfrequency Characteristics. The dimensionless nat- BTW, FTW, and SW designations of these sets is deferred to ural frequencies follow from Eq. (92) and are given by Sec 3.3.4. O;E qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The natural frequency corresponding to mode p ¼ 1 2PSW 2 (zero harmonic of Eq. (100)) is distinct, but the remaining natural x p ¼ 1 þ 2t ð1 cos u Þ; p 2N (99) p frequencies appear in repeated pairs, except for the case of even E N, in which case the p ¼ðN þ 2Þ=2 2PSW frequency (N/2 har- which clearly exhibit the effect of cyclic coupling. For the special monic) is also distinct. There are (N 1)/2 such pairs if N is odd, case of t ¼ 0, the sectors are dynamically isolated and each has and these correspond to mode numbers in PO and PO , the same natural frequency x ¼ 1. There are repeated natural fre- BTW FTW p respectively. For even N there are (N 2)/2 repeated natural fre- quencies for nonzero coupling ðt 6¼ 0Þ, a degeneracy that is due to E E the circulant structure of K. This is captured by the cyclic term quencies corresponding to mode numbers in PBTW and PFTW. Finally, if k 2PO;E then the mode number of the corresponding BTW 2pðp 1Þ repeated eigenfrequency is N þ 2 k 2PO;E . The normal modes cos u ¼ cos ¼ Re wp 1 (100) FTW p N N which is obtained by projecting the powers of the Nth roots of Table 5 Sets of mode numbers p ‰ N corresponding to unity onto the real axis (see Fig. 2). Multiplicity of the eigenfre- BTW, FTW, and SW normal modes of free vibration for odd and quencies can also be visualized in Fig. 9, which shows the dimen- even N sionless natural frequencies in terms of the number of nodal diameters (n.d.) in their attendant mode shapes versus: the mode N Type Set number p for the special case of N ¼ 10 sectors; the wave type (i.e., BTW, FTW, or SW); the number of n.d.; and the sector num- Odd BTW N þ 1 PO ¼ p 2 Z : 2 n ber i.3 Results are shown for weak coupling (WC), strong cou- BTW þ 2 pling (SC), odd N (Fig. 9(a)), and even N (Fig. 9(b)). These cyclic FTW O N þ 3 features are described in terms of mode numbers P ¼ p 2 Zþ : n N FTW 2 O;E O;E O;E SW O 1 p 2N ¼P [P [P PSW ¼ fg SW BTW FTW Even BTW N PE ¼ p 2 Z : 2 n BTW þ 2 FTW N þ 4 PE ¼ p 2 Z : n N FTW þ 2 3 A mode shape nodal diameter refers to a line of zero sector responses across SW N 2 which adjacent sectors respond out of phase. For example, in Fig. 11 of Sec. 3.3.4, E þ PSW ¼ 1; mode 1 has 0 n.d., modes 2 and 100 have 1 n.d., modes 3 and 99 have 2 n.d., and so 2 on.
34 O;E of vibration are described next, where it is shown that each can be phase difference. In this case, the vibration modes p 2PSW corre- categorized as a SW, BTW, or FTW. spond to SW mode shapes whose characteristics can be visualized in Figs. 7(b) and 7(d) by replacing the amplitude F with ap. The 3.3.4 Normal Modes of Vibration. It was shown that Eq. (82) remaining mode shapes correspond to repeated natural frequen- cies and are either BTWs or FTWs. In particular, the normal can be decoupled via a unitary transformation involving the Fou- O;E O;E rier matrix E ¼ðe ; e ; …; e Þ. As a consequence, e is the pth modes p 2PBTW (resp. p 2PFTW) are backward (resp. forward) 1 2 N p traveling waves and can be visualized in Fig. 7(a) (resp. Fig 7(c)). normal mode of vibration corresponding to the natural frequency O;E x . In what follows these mode shapes are characterized by inves- If mode k 2PBTW is a BTW corresponding to a natural frequency p O;E tigating the free response of the system, and it is shown that they xk, then the attendant FTW mode is N þ 2 k 2PFTW with the are of the SW, BTW, or FTW variety. same natural frequency x Nþ2 k ¼ x k. The free response of the system in its pth mode of vibration can Figure 11 illustrates the normal modes of free vibration for a be described by model with N ¼ 100 sectors. In this figure, the extent of the radial lines represents sector displacements. Those appearing outside the hub are to be interpreted as being positively displaced relative to their qðpÞðsÞ¼a e ejx ps p p zero positions, and the opposite is true for lines inside the hub. Modes 1 and 51 are SWs, modes 2–50 are BTWs, and modes 52–100 are where ap is a modal amplitude and the natural frequency x p is defined FTWs. Finally, the number of nodal diameters can be clearly identi- by Eq. (99). There is generally a phase angle as well, which is omitted fiedinFig.11. For example, modes 4 and 98 have 3 n.d. because its presence does not affect the arguments that follow. Noting ðp 1Þði 1Þ jupði 1Þ that element i of ep can be written as wN ¼ e ,thefree 3.3.5 Resonance Structure. In general, there may be a system response of sector i can be written in real form as resonance if the excitation frequency matches a natural frequency, that is, if nr ¼ x p. These possible resonances are conveniently qðpÞðsÞ¼a cos ðu ði 1Þþx sÞ i p p p !!identified in a Campbell diagram, an example of which is shown in Fig. 12(a) for engine orders n 2N (the general case of n 2 Zþ x p ¼ ap cos up i 1 þ s (101) is considered below), N ¼ 10, and ¼ 0:5. The natural frequen- up cies are plotted in terms of the dimensionless rotor speed and sev- ¼ a U ði 1 þ C sÞ; i; p 2N eral engine order lines nr are superimposed. Possible resonances p p p correspond to intersections of the order lines and eigenfrequency loci. There are ðN þ 2Þ=2 such possibilities for each engine order where Cp ¼ x p=up and the function UpðvÞ is defined by Eq. (65). if N is odd and ðN þ 1Þ=2 possible resonances if N is even. In light Equation (101) is a function of continuous time s and it is discre- of Eq. (96), however, there is only a single resonance associated tized according to the sector number i. In this way, it is endowed with each n under the traveling wave dynamic loading of Sec. 3.2, with the same discrete temporal and continuous spatial duality which corresponds to mode p ¼ n mod N þ 1. The set of N that is described in Sec. 3.2.2 in the context of traveling wave resonances for a system excited by N engine orders engine order excitation. That is, it can be regarded as the time–res- ðn ¼ 1; 2; …; NÞ are indicated by the black dots in Fig. 12(a) and ponse of individual (discrete) sectors, or a continuous spatial vari- ss the corresponding frequency response curves jqi ðsÞj (for each n) ation of displacements among the sectors that evolves with are shown in Fig. 12(b) for a model with f ¼ 0.01. For example, a increasing time (i.e., a traveling wave). The propagating wave- 3 e.o. excitation resonates mode 4 (p ¼ 4), which is a BTW with form is strictly a BTW in the negative i-direction (descending sec- 3 n.d. Mode 8 (p ¼ 4) also has 3 n.d. and is excited by a 7 e.o. exci- tor number) with wavelength 2p=up ¼ N=ðp 1Þ and speed Cp, tation. The TW and n.d. designations can be verified in Fig. 9. an illustration of which is shown in Fig. 10. However, depending The basic resonance structure shown in Fig. 12(a) for n 2N on the value of p, this gives rise to SW, BTW, or FTW mode essentially aliases relative to the total number of sectors, in the shapes, a property that follows analogously from the features sense that the excited modes for n ¼ mN þ 1; …; ðm þ 1ÞN with described in Sec. 3.2.2, where it is seen that Eq. (101) has the m 2 Zþ are the same as those for n 2N. This follows from the same form as Eq. (66). orthogonality condition given by Eq. (96) and is manifested in For the special case of p ¼ 1 it is clear from Eq. (101) that each Eq. (97), which gives a relationship for the excited mode in terms sector behaves identically with the same amplitude and the same of the engine order n and total number of sectors N. Because phase because u1 ¼ 0. An additional special case occurs when n > 0 by assumption (see Sec. 3.2) the first mode (p ¼ 1) is excited p ¼ (N þ 2)/2 if N is even. Then uðNþ2Þ=2 ¼ p and each sector has when n ¼ mN ¼ 10; 20; 30; …, the second mode (p ¼ 2) is excited the same amplitude but adjacent sectors oscillate with a 180-deg when n ¼ 1 þ mN ¼ 1; 11; 21; …, and so on. Table 6 summarizes these conditions for a model with N ¼ 10 sectors and the corresponding resonance structure for n ¼ N 1; …; 20N is shown in Fig. 13(a). Each collection of resonance points n ¼ mN þ 1; …; ðm þ 1ÞN is qualitatively the same in structure. However, for m > 1 the resonances become increasingly clustered, which is shown in Fig. 13(b) for n ¼ N; …; 2N. In terms of the O;E sets defined in Tables 4 and 5, an engine order n mod N 2NSW excites a SW mode p 2PO;E. Similarly, an engine order O;E SW O;E n mod N 2NFTW (resp. n mod N 2NBTW) excites a FTW (resp. O;E O;E BTW) mode p 2PFTW (resp. p 2PBTW). While each engine order excites only a single mode, realistic excitation it composed of multiple harmonics (that is, orders), so that many modes can be excited. The nature of the natural fre- quencies and the order excitation lines leads to nontrivial reso- nance behavior even in the case of perfect symmetry. Of course, as noted elsewhere in this paper, imperfections that disturb the symmetry lead to even more complicated responses, in which ev-
Fig. 10 A backward traveling wave ap Up ði 1 1 Cp sÞ ery intersection between e.o. and natural frequency lines can lead 5 apcosðupði 2 1Þ 1 x p sÞ with amplitude ap, wavelength 2p=up to a resonance. These are especially important when the intersec- 5 N=ðp 2 1Þ, and speed Cp 5 x p=up tor coupling is small.
35 Fig. 11 Normal modes of free vibration for a model with N 5 100 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).
3.4 Cyclic System With Two DOFs Per Sector. This exam- disk has radius d and rotates with a fixed speed r about an axis ple generalizes the one DOF per sector model of Sec. 3.3 to a sim- through C. Each blade is modeled by a simple pendulum with unit ple system with two DOFs per sector, which demonstrates the mass and length, the dynamics of which are captured by the nor- process of block diagonalizing the EOM when there are multiple malized angles xi with i 2N. The blades are attached to the rotat- DOFs per sector. The mathematics of the decoupling process ing disk via linear torsional springs with unit stiffness, and described here applies equally as well to models with two or N adjacent blades are elastically coupled by linear springs with stiff- DOFs per sector. Of course, the nature of the natural frequencies ness . It is assumed that the springs are unstretched when the and mode shapes depend on the details of each sector model which, blades are in a purely radial configuration, that is, when each for the cyclic system considered here, is discussed in Refs. [92–94]. xi ¼ 0. As shown in the inset of Fig. 14(b), each blade is fitted There is much more to the topic of multiple DOFs per sector; the with a pendulum like, circular-path vibration absorber with radius reader is referred to the works of Ottarsson [97,114]andBladh c and mass l at an effective distance a along the blade length. The [29,115–117] for more details and more complex examples. absorber dynamics are captured by the normalized pendulum angles yi, which are physically limited to jyij 1 by stops that 3.4.1 Equations of Motion. The nondimensional bladed disk represent the rattling space limits imposed by the blade geometry. model shown in Fig. 14(a) consists of a rotationally periodic array This feature is included for generality, but in all of what follows it of N identical, identically coupled sector models (Fig. 14(b)). The is assumed that jyij < 1, i.e., that impacts do not occur. Linear
36 Fig. 12 (a) Campbell diagram and (b) corresponding frequency ss response curves jqi ðsÞj for N 5 10, m 5 0:5, f 5 0.01, and each Fig. 13 (a) Campbell diagram for N 5 10, m 5 0:5, f 5 0.01, and n 5 1; 2; ...; N n 5 1; ...; 20N and (b) the corresponding frequency response ss curves jqi ðsÞj corresponding to n 5 N; ...; 2N. Engine order Table 6 Condition on the engine order n ‰ Zþ to excite mode lines are not shown for n 5 N 1 1; N 1 2; ...; 2N 2 1, and so on. p ‰ N for N 5 10
Excited mode Conditions on engine order n where Eq. (102a) describes the absorber dynamics and Eq. (102b) describes the blade dynamics. The indices i are taken modN 1 mN ¼ 10; 20; 30; … such that xNþ1 ¼ x1 and x0 ¼ xN, which are cyclic boundary con- 21þ mN ¼ 1; 11; 21; … ditions implying that the Nth blade is coupled to the first. In 32þ mN ¼ 2; 12; 22; … matrix–vector form, and for each i 2N, Eq. (102) becomes . . . . 9 N 1 N 2 þ mN ¼ 8; 18; 28; … M€zi þ Cz_i þ KziþCcð z_i 1 þ 2z_i z_iþ1Þ => NN 1 þ mN ¼ 9; 19; 29; … þ K ð z þ 2z z Þ (103) c i 1 i iþ1 ;> j/ jnrs viscous damping is also included at the spring locations, but is not ¼ fe i e indicated in Fig. 14. Blade and interblade damping is captured by T T linear torsional and translational dampers with constants nb and nc, where zi ¼ ðÞxi; yi captures the sector dynamics, f ¼ ðÞf ; 0 is a respectively, and the absorber damping is captured by a torsional sector forcing vector, and the elements of the sector mass, damp- damper with constant na. Finally, the system is subjected to the ing, and stiffness matrices are defined in Table 7. The matrices traveling wave dynamic loading defined by Eq. (80),asshownin "# "# 2 Fig. 14(b). nc 0 0 Cc ¼ ; Kc ¼ (104) Sector Model. The EOM for each two-DOF sector are derived 00 00 using Lagrange’s method and linearized for small motions of the primary and absorber systems, that is, for small xi and yi.Thenfor capture the interblade coupling and vanish if nc ¼ ¼ 0, in which each i 2N, the dynamics of the ith sector are governed by [92,93] case Eq. (103) describes the forced motion of N isolated blade/ 2 2 absorber systems. lc ðx€i þ y€iÞþnay_i þ lcdr ðxi þ yiÞ 2 System Model. By stacking each zi into the configuration vector þ lcaðx€i þ r yiÞ¼0 (102a) T q ¼ ðÞz1; z2; …; zN , the governing matrix EOM for the overall 2 2N-DOF system takes the form x€i þ nbx_i nay_i þ xi þ dr xi "# 2 2 b b b b jnrs a x€i þc ðx€i þ y€iÞþacðy€i þ 2x€iÞ Mq€ þ Cq_ þ Kq ¼ fe (105) þ l 2 2 þadr xi þ cdr ðxi þ yiÞ b where M 2 BCBS2;N is block diagonal with diagonal blocks M b þ n ð x_ þ 2x_ x_ Þ and K 2 BCBS2;N has generating matrices K þ 2Kc; Kc; c i 1 i iþ1 b 0; …; 0; Kc. The matrix C 2 BCBS2;N is similarly defined by 2 j/ jnrs þ ð xi 1 þ 2xi xi 1Þ¼fe i e (102b) b þ replacing K with C and Kc with Cc in K. In terms of the circulant
37 The 2N 1 system forcing vector is b j/ j/ j/ T f ¼ fe 1 ; fe 2 ; …; fe N (107) ¼ f0 f
where the N 1 vector f0 is defined by Eq. (69) and the interblade phase angle /i is given by Eq. (64). 3.4.2 Forced Response. The forced response of the overall system defined by Eq. (105) can be handled directly using stand- ard techniques [99]. Its nonresonant solution in the steady-state follows in the usual way and is given by
ss 1b jnrs q ðsÞ¼Zb fe (108) b where Zb ¼ Kb n2r2Mb þ jnrC is the system impedance matrix of dimension 2N 2N. However, Eq. (108) does not offer any insight into the system’s modal characteristics and it requires computation of Zb 1, which can be prohibitive for practical bladed disk models with many sectors and many DOFs per sector. We thus turn to a decoupling strategy that exploits the system symme- try and the theory developed in Sec. 2. The analysis follows simi- larly to that presented in Sec. 3.1, except in this case the single coupled 2N-DOF system is transformed into a set of N block decoupled two-DOF systems. To this end, we introduce the change of coordinates
T q ¼ðE IÞu; or zi ¼ðei IÞu; i 2N (109)
where E is the N N complex Fourier matrix and ei is its ith col- umn, is the Kronecker product, I is the 2 2 identity matrix (the dimension of I corresponds to the number of DOFs per sec- T tor), and u ¼ ðÞu1; u2; …; uN is a vector of modal, or cyclic coor- dinates. Each up is 2 1 and describes the sector dynamics in modal space. Substituting Eq. (109) into Eq. (105), multiplying from the left by the unitary matrix ðE IÞH ¼ðEH IÞ, and invoking Theorem 10 yields a system of N block decoupled equa- Fig. 14 (a) Model of bladed disk assembly and (b) sector tions, each with two DOFs. They are model e e e H b jnrs Mpu€p þ Cpu_ p þ Kpup ¼ðep IÞfe ; p 2N (110)
Table 7 Elements of the sector mass, damping, and stiffness b b where ðeH IÞf is the pth 2 1 block of ðEH IÞf. Equation matrices M, C, and K p (110) is analogous to the NM-DOF systems given by Eq. (61) for Matrix Notation Elements the general formulation in Sec. 3.1, but in this case M ¼ 2 and engine order excitation is assumed from the onset. Figure 15 illus- 2 Mass M M11 ¼ 1 þ lða þ cÞ trates the transformation of the single 2N-DOF system given by M12 ¼ lcða þ cÞ Eq. (105) to a system of N block decoupled two-DOF forced oscil- M21 ¼ M12 lators defined by Eq. (110). 2 M22 ¼ lc The 2 2 mass, damping, and stiffness matrices associated
Damping C C11 ¼ nb with the pth mode follow from Theorem 10 and are given by C12 ¼ na 9 e C21 ¼ 0 Mp ¼ M > n = C22 ¼ a e 2 Cp ¼ C þ 2Ccð1 cos upÞ ; p 2N (111) Stiffness K K11 ¼ 1 þ ðÞ1 þ lða þ cÞ dr > 2 ; K12 ¼ lcdr e Kp ¼ K þ 2Kcð1 cos upÞ K21 ¼ K12 2 K22 ¼ lcðÞ a þ d r where up is defined by Eq. (16), the elements of M, C, and K are defined in Table 7 and the coupling matrices Cc and Kc are defined by Eq. (104). In light of Eqs. (70) and (71), the pth modal forcing vector takes the form operator, the system mass, damping, and stiffness matrices are defined by H b H ðe IÞf ¼ðe IÞðf0 fÞ 9 p p (112) b H M ¼ circðM; 0; 0; …; 0; 0Þ > ¼ ep f0 f = ( ffiffiffiffi b p C ¼ circðC þ 2Cc; Cc; 0; …; 0; CcÞ (106) Nf; p ¼ n þ 1 > ¼ b ; K ¼ circðK þ 2Kc; Kc; 0; …; 0; KcÞ 0; otherwise
38 Fig. 15 The topology of a bladed disk assembly fitted with absorbers in (a) physical space and (b) modal space. The modal transformation qðsÞ 5 ðE IÞuðsÞ reduces the cyclic array of N, two-DOF sector models ðB; AÞ, which together form a 2N-DOF coupled system, to a set of N, two-DOF block decoupled models ðBp; Ap Þ.
T H where 0 ¼ (0,0) and the scalar product ep f0 vanishes except for whether in terms of fundamental understanding, insight, or ease of p ¼ n þ 1. Because only mode p ¼ n þ 1 is excited, unþ1ðsÞ is the computation. We trust that the results presented here offer such only nonzero modal response in the steady-state. benefits to readers interested in vibration analysis of cyclic Assuming harmonic motion, and in light of Eq. (112), the pth systems. steady-state modal response follows easily from Eq. (110) and is It must be noted that no physical system has perfect symmetry, given by as assumed herein. This assumption must be examined in light of ( pffiffiffiffi the system under consideration. One key to the suitability of a e 1 jnrs cyclically symmetric model is the intersector coupling. If the cou- ss NZnþ1fe ; p ¼ n þ 1 up ðsÞ¼ (113) pling is strong, so that the pairs of modes have well separated fre- 0; otherwise quencies, then small imperfections will not alter the picture substantially, and one can consider each mode pair as robust where against coupling to other modes. However, if the coupling is weak, so that the system frequencies are clustered near those of e e 2 2 e e Zp ¼ Kp n r Mp þ jnrCp; p 2N (114) the isolated sector model, the possibility of localization is signifi- cantly increased. This topic has been investigated quite thor- is the pth modal impedance matrix. The response of sector i (in oughly, primarily in the context of the vibration of bladed disk physical coordinates) follows from the transformation given by assemblies with small blade mistuning; see, for example, [72–88]. Eq. (109) with Also, nonlinear effects can couple linear modes under certain res- onance conditions, even at small amplitudes [118]. In cyclic sys- ussðsÞ¼ð0; …; 0; uss ðsÞ; 0; …; 0ÞT tems this can occur for the pairs of modes with equal frequencies nþ1 [119], and this possibility expands to groups of modes for the case of weak coupling, leading to extremely complicated behavior and is given by [82,84,88]. In such cases, the tools from group theory can be applied to categorize the possible modes and forced response in ss e 1 j/i jnrs zi ðsÞ¼Znþ1fe e ; i 2N (115) terms of their symmetries [65]. This topic, while interesting, is outside the scope of the present paper. where wnði 1Þ ¼ ej/i is employed. From Eq. (115) it is clear that each blade/absorber combination behaves identically except for a constant phase shift from one sector to another, which is captured by the interblade phase angle /i. This approach offers a signifi- cant computational advantage over the direct solution to the full Acknowledgment 2N-DOF system, as given by Eq. (108). The work of the second author related to this topic was funded by the National Science Foundation, currently by Grant No. 4 Conclusions CMMI-1100260. The goal of this paper is to provide the mathematical tools for handling circulant matrices as they apply to the free and forced vibration analysis of structures with cyclic symmetry. As demon- References strated by past work in this area and the review provided here, the [1] Bladh, J. R., Pierre, C., Castanier, M. P., and Kruse, M. J., 2002, “Dynamic theory of circulants provides a useful description of the fundamen- Response Predictions for a Mistuned Industrial Turbomachinery Rotor Using Reduced-Order Modeling,” ASME J. Eng. Gas Turbines Power, 124(2), tal structure of the mode shapes and spectrum of systems with pp. 311–324. cyclic symmetry, including those of large scale. The theory also [2] Brillouin, L., 1953, Wave Propagation in Periodic Structures, Dover, New provides a convenient means for computing the vibration response York. of these systems, even when the idealized symmetry is broken by [3] Ewins, D. J., 1973, “Vibration Characteristics of Bladed Disc Assemblies,” J. Mech. Eng. Sci., 15(3), pp. 165–186. mistuning or by nonlinear effects. As with any mathematical tool, [4] Ewins, D. J., 1976, “Vibration Modes of Mistuned Bladed Disks,” ASME J. the overhead in learning it must provide appropriate benefit, Eng. Power, 98(3), pp. 349–355.
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