$$ q^{ss}(\tau) = (\mathbf{K}_{11} - n^2\sigma^2\mathbf{I})^{-1}\mathbf{f}_{11}e^{jn\sigma\tau} \quad (85) $$

where $\mathbf{I}$ is the $N \times N$ identity matrix. However, this requires inversion of the impedance matrix $\mathbf{K}_{11} - n^2\sigma^2\mathbf{I}$, which is computationally expensive for a large number of sectors, and it offers little insight into the basic vibration characteristics. In what follows, a transformation based on the cyclic symmetry of the system is exploited to fully decouple the single N-DOF system to a set of N single-DOF oscillators from which the steady-state response is easily obtained. The procedure is similar to the usual modal analysis from elementary vibration theory. However, a key difference is that the transformation matrix (and hence the system mode shapes) is known a priori and, because the transformation is unitary (thus preserving the system eigenvalues), the natural frequencies are obtained after the transformation is carried out. Moreover, due to orthogonality conditions between the normal modes and forcing vector, the steady-state response of the overall system reduces to finding the forced response of a single harmonically forced, single-DOF oscillator in modal space, which offers a clear advantage over the direct computation of Eq. (85).

As described in Sec. 3.2.2, engine order excitation can be regarded as traveling wave dynamic loading. It is therefore reasonable to expect steady-state solutions of the same type. We begin with a simple way to show the existence of such a response, and then systematically describe it using the modal analysis techniques described in Secs. 3.1.2 and 3.2.3.

*Existence of a Traveling Wave Response.* It is natural to search for steady-state solutions of the form

$$ q_i^{ss}(\tau) = A e^{i\phi_i} e^{jn\sigma\tau}, \quad i \in \mathcal{N} \quad (86) $$

which has the same traveling wave characteristics as the engine order excitation described in Sec. 3.2.2. Equation (86) assumes that each sector responds with the same amplitude $A$, but with a constant phase difference relative to its nearest neighbors, and together all $N$ such solutions form a traveling wave response among the sectors. By mapping this trial solution into Eq. (81) and dividing through by the common term $e^{i\phi_i}e^{jn\sigma\tau}$, it follows that

$$ -(n\sigma)^2 A + A + v^2(-Ae^{-j\phi_{n+1}} + 2A - Ae^{j\phi_{n+1}}) = f \quad (87) $$

where the identity $\phi_{i\pm 1} - \phi_i = \pm \phi_{n+1}$ is employed. Solving for the amplitude $A$ yields

$$ A = \frac{f}{1 + 2v^2(1 - \cos \varphi_{n+1}) - (n\sigma)^2} \quad (88) $$

from which it follows that

$$ \bar{\omega}_{n+1} = \sqrt{1 + 2v^2(1 - \cos \varphi_{n+1})} $$

is one of the $N$ natural frequencies of the coupled system corresponding to mode $p = n$ mod $N + 1$. Equation (88) shows that mode $n+1$ is excited, but the reason is not clear from this approach. A modal analysis that considers the fully coupled system is required to systematically describe the response characteristics of the cyclic system under engine order excitation.

*Modal Analysis.* Theorem 9 guarantees that circulant matrices, such as the stiffness matrix defined by Eq. (84), are diagonalizable via a unitary transformation involving the Fourier matrix, and in what follows this property is exploited to fully decouple the matrix EOM defined by Eq. (82). To this end, the change of coordinates

$$ \mathbf{q}(\tau) = \mathbf{E}\mathbf{u}(\tau) \quad \text{or} \quad q_i(\tau) = \mathbf{e}_i^T \mathbf{u}(\tau), \quad i \in \mathcal{N} \quad (89) $$

is introduced, where $\mathbf{E}$ is the $N \times N$ complex Fourier matrix (Definition 17), $\mathbf{e}_i$ is its $i$th column (Definition 18), and $\mathbf{u} = (u_1, u_2, \dots, u_N)^T$ is a vector of modal, or cyclic coordinates. Substituting Eq. (89) in Eq. (82) and multiplying from the left by $\mathbf{E}^H$ yields

$$ \mathbf{E}^H \mathbf{E} \ddot{\mathbf{u}} + \mathbf{E}^H \mathbf{K}_{11} \mathbf{E} \mathbf{u} = \mathbf{E}^H \mathbf{f}_{11} e^{jn\sigma\tau} \quad (90) $$

where $\mathbf{E}^H \mathbf{E} = \mathbf{I}$ because $\mathbf{E}$ is unitary (Theorem 6). In light of Theorem 9, it follows that

$$ \begin{bmatrix} \ddot{u}_1 \\ \ddot{u}_2 \\ \vdots \\ \ddot{u}_N \end{bmatrix} + \begin{bmatrix} \bar{\omega}_1^2 & & & 0 \\ & \ddot{\omega}_2^2 & & \\ & & \ddots & \\ 0 & & & \bar{\omega}_N^2 \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ \vdots \\ u_N \end{bmatrix} = \begin{bmatrix} \mathbf{e}_1^H \mathbf{f}_{11} \\ \mathbf{e}_2^H \mathbf{f}_{11} \\ \vdots \\ \mathbf{e}_N^H \mathbf{f}_{11} \end{bmatrix} e^{jn\sigma\tau} \quad (91) $$

where the $p$th scalar element of the $N \times 1$ modal forcing vector $\mathbf{E}^H \mathbf{f}_{11}$ is $\mathbf{e}_p^H \mathbf{f}_{11}$. Decomposition of $\mathbf{E}^H \mathbf{f}_{11}$ follows from Eq. (60) by replacing the identity matrix $\mathbf{I}_M$ with unity and the vector $\hat{\mathbf{f}}$ with $\mathbf{f}_{11}$. Equation (89) is a unitary (similarity) transformation and hence the system natural frequencies are preserved, which is guaranteed by Theorem 1. For each $p \in \mathcal{N}$, the dimensionless natural frequencies follow from Eq. (21) and are given by

$$ \begin{aligned} \bar{\omega}_p^2 &= 1 + 2v^2 - v^2 w_N^{(p-1)} + 0 + \dots + 0 - v^2 w_N^{(N-1)(p-1)} \\ &= 1 + 2v^2 - v^2 (w_N^{(p-1)} + w_N^{(N-1)(p-1)}) \\ &= 1 + 2v^2(1 - \cos \varphi_p) \end{aligned} \quad (92) $$

where $w_N$ is the primitive $N$th root of unity and the identity $w_N^{(p-1)} + w_N^{(N-1)(p-1)} = 2 \cos \varphi_p$ is employed. Equation (91) is a decoupled set of $N$ single-DOF harmonically forced modal oscillators of the form

$$ \ddot{u}_p + \bar{\omega}_p^2 u_p = \mathbf{e}_p^H \mathbf{f}_{11} e^{jn\sigma\tau}, \quad p \in \mathcal{N} \quad (93) $$

Thus, the single N-DOF system given by Eq. (82) is transformed to a system of N decoupled single-DOF systems defined by Eq. (93).

The steady-state, nonresonant modal response of the $p$th decoupled system follows from Eq. (93) using standard techniques [99]. Assuming harmonic motion, the solution is

$$ u_p^{ss}(\tau) = \frac{\mathbf{e}_p^H \mathbf{f}_{11}}{\bar{\omega}_p^2 - (n\sigma)^2} e^{jn\sigma\tau}, \quad p \in \mathcal{N} \quad (94) $$

from which the steady-state modal response vector $\mathbf{u}^{ss}(\tau) = (u_1^{ss}(\tau), u_2^{ss}(\tau), \dots, u_N^{ss}(\tau))^T$ is constructed. In physical coordinates, the steady-state response of sector $i$ follows from Eq. (89) and is given by

$$ \begin{aligned} q_i^{ss}(\tau) &= \mathbf{e}_i^T \mathbf{u}^{ss}(\tau) \\ &= \frac{1}{\sqrt{N}} (e^{j0 \cdot \varphi_i}, e^{j1 \cdot \varphi_i}, \dots, e^{j(N-1)\varphi_i}) \\ &\quad \times (u_1^{ss}(\tau), u_2^{ss}(\tau), \dots, u_N^{ss}(\tau))^T \\ &= \sum_{p=1}^{N} \frac{1}{\sqrt{N}} e^{j(p-1)\varphi_i} u_p^{ss}(\tau) \\ &= \sum_{p=1}^{N} \frac{1}{\sqrt{N}} e^{j\varphi_p(i-1)} \frac{\mathbf{e}_p^H \mathbf{f}_{11}}{\bar{\omega}_p^2 - (n\sigma)^2} e^{jn\sigma\tau} \\ &= \frac{1}{\sqrt{N}} \sum_{p=1}^{N} \frac{\mathbf{e}_p^H \mathbf{f}_{11}}{\bar{\omega}_p^2 - (n\sigma)^2} e^{j(i-1)\varphi_p} e^{jn\sigma\tau}, \quad i \in \mathcal{N} \end{aligned} \quad (95) $$

where the identity $(p-1)\varphi_i = \varphi_p(i-1)$ is employed. Equation (95) shows that there are $N$ possible resonances, depending on the details of the modal forcing terms $\mathbf{e}_p^H \mathbf{f}_{11}$. However, only a single mode survives under an engine order excitation of order $n$, which
is clear from the orthogonality condition described in Sec. 3.2.3. Noting that $\mathbf{f}_{11} = f \otimes \mathbf{f}_0 = \sqrt{N}f\mathbf{e}_{n+1}$, it follows from Eq. (72) that

$$ 
\begin{aligned}
\mathbf{e}_p^H \mathbf{f}_{11} &= \sqrt{N} f \mathbf{e}_p^H \mathbf{e}_{n+1} \\
&= \sqrt{N} f \delta_{p(n+1)} \\
&= \begin{cases} \sqrt{N} f, & p = n+1 \\ 0, & \text{otherwise} \end{cases}
\end{aligned} 
\quad (96) $$

which shows that the force vector $\mathbf{f}_{11}$ is mutually orthogonal to all but one of the modal vectors $\mathbf{e}_p$. That is, for $p \in \mathcal{N}$, the only excited mode is

$$ p = n \mod N + 1 \quad (97) $$

for an engine order $n \in \mathbb{Z}_+$ excitation. Thus, Eq. (95) reduces to

$$ q_i^{ss}(\tau) = \frac{f}{\bar{\omega}_{n+1}^2 - (n\sigma)^2} e^{i\phi_i} e^{jn\sigma\tau}, \quad i \in \mathcal{N} \quad (98) $$

where the identity $(i-1)\varphi_{n+1} = \phi_i$ is employed and

$$ \bar{\omega}_{n+1}^2 = 1 + 2v^2(1 - \cos \varphi_{n+1}) $$

from Eq. (92). Equation (98) is the same result as that obtained from Eq. (88). Indeed, the process described here is significantly more laborious than the direct approach, but many general features can be gleaned from the analysis. The eigenfrequency characteristics (Sec. 3.3.3), normal modes of vibration (Sec. 3.3.4), and resonance structure (Sec. 3.3.5) are systematically described based on the modal decomposition results of this section.

### 3.3.3 Eigenfrequency Characteristics.
The dimensionless natural frequencies follow from Eq. (92) and are given by

$$ \bar{\omega}_p = \sqrt{1 + 2v^2(1 - \cos \varphi_p)}, \quad p \in \mathcal{N} \quad (99) $$

which clearly exhibit the effect of cyclic coupling. For the special case of $v=0$, the sectors are dynamically isolated and each has the same natural frequency $\bar{\omega}_p = 1$. There are repeated natural frequencies for nonzero coupling ($v \neq 0$), a degeneracy that is due to the circulant structure of $\mathbf{K}$. This is captured by the cyclic term

$$ \cos \varphi_p = \cos \left( \frac{2\pi(p-1)}{N} \right) = \operatorname{Re} \left( w_N^{p-1} \right) \quad (100) $$

which is obtained by projecting the powers of the $N$th roots of unity onto the real axis (see Fig. 2). Multiplicity of the eigenfrequencies can also be visualized in Fig. 9, which shows the dimensionless natural frequencies in terms of the number of nodal diameters (n.d.) in their attendant mode shapes versus: the mode number $p$ for the special case of $N=10$ sectors; the wave type (i.e., BTW, FTW, or SW); the number of n.d.; and the sector number $i$.<fnref n="3" /> Results are shown for weak coupling (WC), strong coupling (SC), odd $N$ (Fig. 9(a)), and even $N$ (Fig. 9(b)). These cyclic features are described in terms of mode numbers

$$ p \in \mathcal{N} = \mathcal{P}_{\text{SW}}^{\text{O.E}} \cup \mathcal{P}_{\text{BTW}}^{\text{O.E}} \cup \mathcal{P}_{\text{FTW}}^{\text{O.E}} $$

<figure><img src="image_50.png" /><figcaption>**Fig. 9 Dimensionless natural frequencies $\bar{\omega}_p$ in terms of the number of n.d. versus mode number $p$ for WC and SC: (a) $N=11$ (odd) and (b) $N=10$ (even).** Also indicated below each figure is, for general $N$, the number of n.d. at each value of $p$ and also the mode numbers corresponding to SW, BTW, and FTW.</figcaption></figure>

where each subset is defined in Table 5. A description of the BTW, FTW, and SW designations of these sets is deferred to Sec 3.3.4.

The natural frequency corresponding to mode $p=1 \in \mathcal{P}_{\text{SW}}^{\text{O.E}}$ (zero harmonic of Eq. (100)) is distinct, but the remaining natural frequencies appear in repeated pairs, except for the case of even $N$, in which case the $p = (N+2)/2 \in \mathcal{P}_{\text{SW}}^{\text{E}}$ frequency ($N/2$ harmonic) is also distinct. There are $(N-1)/2$ such pairs if $N$ is odd, and these correspond to mode numbers in $\mathcal{P}_{\text{BTW}}^{\text{O}}$ and $\mathcal{P}_{\text{FTW}}^{\text{O}}$, respectively. For even $N$ there are $(N-2)/2$ repeated natural frequencies corresponding to mode numbers in $\mathcal{P}_{\text{BTW}}^{\text{E}}$ and $\mathcal{P}_{\text{FTW}}^{\text{E}}$. Finally, if $k \in \mathcal{P}_{\text{BTW}}^{\text{O.E}}$ then the mode number of the corresponding repeated eigenfrequency is $N+2-k \in \mathcal{P}_{\text{FTW}}^{\text{O.E}}$. The normal modes

<table><caption>**Table 5 Sets of mode numbers $p \in \mathcal{N}$ corresponding to BTW, FTW, and SW normal modes of free vibration for odd and even $N$**</caption><thead><tr><th>N</th><th>Type</th><th>Set</th></tr></thead><tbody><tr><td rowspan="3">Odd</td><td>BTW</td><td>P<sup>O</sup><sub>BTW</sub> = {p ∈ Z<sub>+</sub>: 2 ≤ n ≤ (N + 1)/2}</td></tr><tr><td>FTW</td><td>P<sup>O</sup><sub>FTW</sub> = {p ∈ Z<sub>+</sub>: (N + 3)/2 ≤ n ≤ N}</td></tr><tr><td>SW</td><td>P<sup>O</sup><sub>SW</sub> = {1}</td></tr><tr><td rowspan="3">Even</td><td>BTW</td><td>P<sup>E</sup><sub>BTW</sub> = {p ∈ Z<sub>+</sub>: 2 ≤ n ≤ N/2}</td></tr><tr><td>FTW</td><td>P<sup>E</sup><sub>FTW</sub> = {p ∈ Z<sub>+</sub>: (N + 4)/2 ≤ n ≤ N}</td></tr><tr><td>SW</td><td>P<sup>E</sup><sub>SW</sub> = {1, (N + 2)/2}</td></tr></tbody></table>

<footnote n="3">³A mode shape nodal diameter refers to a line of zero sector responses across which adjacent sectors respond out of phase. For example, in Fig. 11 of Sec. 3.3.4, mode 1 has 0 n.d., modes 2 and 100 have 1 n.d., modes 3 and 99 have 2 n.d., and so on.</footnote>
of vibration are described next, where it is shown that each can be categorized as a SW, BTW, or FTW.

### 3.3.4 Normal Modes of Vibration.
It was shown that Eq. (82) can be decoupled via a unitary transformation involving the Fourier matrix $E = (e_1, e_2, ..., e_N)$. As a consequence, $e_p$ is the $p$th normal mode of vibration corresponding to the natural frequency $\bar{\omega}_p$. In what follows these mode shapes are characterized by investigating the free response of the system, and it is shown that they are of the SW, BTW, or FTW variety.

The free response of the system in its $p$th mode of vibration can be described by

$$ \mathbf{q}^{(p)}(\tau) = a_p \mathbf{e}_p e^{i\bar{\omega}_p \tau} $$

where $a_p$ is a modal amplitude and the natural frequency $\bar{\omega}_p$ is defined by Eq. (99). There is generally a phase angle as well, which is omitted because its presence does not affect the arguments that follow. Noting that element $i$ of $\mathbf{e}_p$ can be written as $w_N^{(p=1)(i-1)} = e^{i\phi_p(i-1)}$, the free response of sector $i$ can be written in real form as

$$ \begin{aligned} q_i^{(p)}(\tau) &= a_p \cos(\varphi_p(i-1) + \bar{\omega}_p \tau) \\ &= a_p \cos \varphi_p \left( i - 1 + \frac{\bar{\omega}_p}{\varphi_p} \tau \right) \end{aligned} \quad (101) \\ = a_p \Phi_p(i-1 + C_p \tau), \quad i, p \in \mathcal{N} $$

where $C_p = \bar{\omega}_p/\varphi_p$ and the function $\Phi_p(\chi)$ is defined by Eq. (65). Equation (101) is a function of continuous time $\tau$ and it is discretized according to the sector number $i$. In this way, it is endowed with the same discrete temporal and continuous spatial duality that is described in Sec. 3.2.2 in the context of traveling wave engine order excitation. That is, it can be regarded as the time–response of individual (discrete) sectors, or a continuous spatial variation of displacements among the sectors that evolves with increasing time (i.e., a traveling wave). The propagating wave-form is strictly a BTW in the negative $i$-direction (descending sector number) with wavelength $2\pi/\varphi_p = N/(p-1)$ and speed $C_p$, an illustration of which is shown in Fig. 10. However, depending on the value of $p$, this gives rise to SW, BTW, or FTW mode shapes, a property that follows analogously from the features described in Sec. 3.2.2, where it is seen that Eq. (101) has the same form as Eq. (66).

For the special case of $p=1$ it is clear from Eq. (101) that each sector behaves identically with the same amplitude and the same phase because $\varphi_1 = 0$. An additional special case occurs when $p=(N+2)/2$ if $N$ is even. Then $\varphi_{(N+2)/2} = \pi$ and each sector has the same amplitude but adjacent sectors oscillate with a 180-deg

<figure><img src="image_51.png" /><figcaption>Fig. 10 A backward traveling wave $a_p\Phi_p(i-1 + C_p\tau) = a_p\cos(\varphi_p(i-1) + \bar{\omega}_p\tau)$ with amplitude $a_p$, wavelength $2\pi/\bar{\omega}_p = N/(p-1)$, and speed $C_p = \bar{\omega}_p/\varphi_p$</figcaption></figure>

phase difference. In this case, the vibration modes $p \in \mathcal{P}_{\text{SW}}^{\text{O.E.}}$ correspond to SW mode shapes whose characteristics can be visualized in Figs. 7(b) and 7(d) by replacing the amplitude $F$ with $a_p$. The remaining mode shapes correspond to repeated natural frequencies and are either BTWs or FTWs. In particular, the normal modes $p \in \mathcal{P}_{\text{BTW}}^{\text{O.E.}}$ (resp. $p \in \mathcal{P}_{\text{FTW}}^{\text{O.E.}}$) are backward (resp. forward) traveling waves and can be visualized in Fig. 7(a) (resp. Fig 7(c)). If mode $k \in \mathcal{P}_{\text{BTW}}^{\text{O.E.}}$ is a BTW corresponding to a natural frequency $\bar{\omega}_k$, then the attendant FTW mode is $N+2-k \in \mathcal{P}_{\text{FTW}}^{\text{O.E.}}$ with the same natural frequency $\bar{\omega}_{N+2-k} = \bar{\omega}_k$.

Figure 11 illustrates the normal modes of free vibration for a model with $N=100$ 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 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 FTWs. Finally, the number of nodal diameters can be clearly identified in Fig. 11. For example, modes 4 and 98 have 3 n.d.

### 3.3.5 Resonance Structure.
In general, there may be a system resonance if the excitation frequency matches a natural frequency, that is, if $n\sigma = \bar{\omega}_p$. These possible resonances are conveniently identified in a Campbell diagram, an example of which is shown in Fig. 12(a) for engine orders $n \in \mathcal{N}$ (the general case of $n \in \mathbb{Z}_+$ is considered below), $N=10$, and $\nu = 0.5$. The natural frequencies are plotted in terms of the dimensionless rotor speed and several engine order lines $n\sigma$ are superimposed. Possible resonances correspond to intersections of the order lines and eigenfrequency loci. There are $(N+2)/2$ such possibilities for each engine order if $N$ is odd and $(N+1)/2$ possible resonances if $N$ is even. In light of Eq. (96), however, there is only a single resonance associated with each $n$ under the traveling wave dynamic loading of Sec. 3.2, which corresponds to mode $p=n \mod N+1$. The set of $N$ resonances for a system excited by $N$ engine orders ($n=1, 2, ..., N$) are indicated by the black dots in Fig. 12(a) and the corresponding frequency response curves $|q_i^{ss}(\tau)|$ (for each $n$) are shown in Fig. 12(b) for a model with $f=0.01$. For example, a 3 e.o. excitation resonates mode 4 ($p=4$), which is a BTW with 3 n.d. Mode 8 ($p=4$) also has 3 n.d. and is excited by a 7 e.o. excitation. The TW and n.d. designations can be verified in Fig. 9.

The basic resonance structure shown in Fig. 12(a) for $n \in \mathcal{N}$ essentially aliases relative to the total number of sectors, in the sense that the excited modes for $n=mN+1, ..., (m+1)N$ with $m \in \mathbb{Z}_+$ are the same as those for $n \in \mathcal{N}$. This follows from the orthogonality condition given by Eq. (96) and is manifested in Eq. (97), which gives a relationship for the excited mode in terms of the engine order $n$ and total number of sectors $N$. Because $n>0$ by assumption (see Sec. 3.2) the first mode ($p=1$) is excited when $n=mN=10, 20, 30, ..., $, the second mode ($p=2$) is excited when $n=1+mN=1, 11, 21, ..., $, and so on. Table 6 summarizes these conditions for a model with $N=10$ sectors and the corresponding resonance structure for $n=N-1, ..., 20N$ is shown in Fig. 13(a). Each collection of resonance points $n=mN+1, ..., (m+1)N$ is qualitatively the same in structure. However, for $m>1$ the resonances become increasingly clustered, which is shown in Fig. 13(b) for $n=N, ..., 2N$. In terms of the sets defined in Tables 4 and 5, an engine order $n \mod N \in \mathcal{N}_{\text{SW}}^{\text{O.E.}}$ excites a SW mode $p \in \mathcal{P}_{\text{SW}}^{\text{O.E.}}$. Similarly, an engine order $n \mod N \in \mathcal{N}_{\text{FTW}}^{\text{O.E.}}$ (resp. $n \mod N \in \mathcal{N}_{\text{BTW}}^{\text{O.E.}}$) excites a FTW (resp. BTW) mode $p \in \mathcal{P}_{\text{FTW}}^{\text{O.E.}}$ (resp. $p \in \mathcal{P}_{\text{BTW}}^{\text{O.E.}}$).

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 frequencies and the order excitation lines leads to nontrivial resonance 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 every intersection between e.o. and natural frequency lines can lead to a resonance. These are especially important when the intersector coupling is small.
<figure><img src="image_52.png" /><figcaption>Fig. 11 Normal modes of free vibration for a model with $N=100$ sectors. Mode 1 consists of a SW, in which each sector oscillates with the same amplitude and phase. Mode 51 also corresponds to a SW, but neighboring oscillators oscillate exactly 180 deg out of phase. Modes 2–50 (resp. 52–100) consist of BTWs (resp. FTWs).</figcaption></figure>

### 3.4 Cyclic System With Two DOFs Per Sector.
This example generalizes the one DOF per sector model of Sec. 3.3 to a simple system with two DOFs per sector, which demonstrates the process of block diagonalizing the EOM when there are multiple DOFs per sector. The mathematics of the decoupling process described here applies equally as well to models with two or $N$ DOFs per sector. Of course, the nature of the natural frequencies and mode shapes depend on the details of each sector model which, for the cyclic system considered here, is discussed in Refs. [92–94]. There is much more to the topic of multiple DOFs per sector; the reader is referred to the works of Öttarsson [97,114] and Bladh [29,115–117] for more details and more complex examples.

#### 3.4.1 Equations of Motion.
The nondimensional bladed disk model shown in Fig. 14(a) consists of a rotationally periodic array of $N$ identical, identically coupled sector models (Fig. 14(b)). The

disk has radius $\delta$ and rotates with a fixed speed $\sigma$ about an axis through C. Each blade is modeled by a simple pendulum with unit mass and length, the dynamics of which are captured by the normalized angles $x_i$ with $i \in \mathcal{N}$. The blades are attached to the rotating disk via linear torsional springs with unit stiffness, and adjacent blades are elastically coupled by linear springs with stiffness $\nu$. It is assumed that the springs are unstretched when the blades are in a purely radial configuration, that is, when each $x_i=0$. As shown in the inset of Fig. 14(b), each blade is fitted with a pendulum like, circular-path vibration absorber with radius $\gamma$ and mass $\mu$ at an effective distance $\alpha$ along the blade length. The absorber dynamics are captured by the normalized pendulum angles $y_i$, which are physically limited to $|y_i| \le 1$ by stops that represent the rattling space limits imposed by the blade geometry. This feature is included for generality, but in all of what follows it is assumed that $|y_i| < 1$, i.e., that impacts do not occur. Linear
<figure><img src="image_53.png" /><figcaption>Fig. 12 (a) Campbell diagram and (b) corresponding frequency response curves $|q_i^{ss}(\tau)|$ for $N=10$, $v=0.5$, $f=0.01$, and each $n=1, 2, \dots, N$</figcaption></figure>

<table><caption>Table 6 Condition on the engine order $n \in \mathbb{Z}_+$ to excite mode $p \in \mathcal{N}$ for $N=10$</caption><thead><tr><th>Excited mode</th><th>Conditions on engine order n</th></tr></thead><tbody><tr><td>1</td><td>$mN = 10, 20, 30, \dots$</td></tr><tr><td>2</td><td>$1 + mN = 1, 11, 21, \dots$</td></tr><tr><td>3</td><td>$2 + mN = 2, 12, 22, \dots$</td></tr><tr><td>...</td><td>...</td></tr><tr><td>$N-1$</td><td>$N-2 + mN = 8, 18, 28, \dots$</td></tr><tr><td>$N$</td><td>$N-1 + mN = 9, 19, 29, \dots$</td></tr></tbody></table>

viscous damping is also included at the spring locations, but is not indicated in Fig. 14. Blade and interblade damping is captured by linear torsional and translational dampers with constants $\xi_b$ and $\xi_c$, respectively, and the absorber damping is captured by a torsional damper with constant $\xi_a$. Finally, the system is subjected to the traveling wave dynamic loading defined by Eq. (80), as shown in Fig. 14(b).

**Sector Model.** The EOM for each two-DOF sector are derived using Lagrange's method and linearized for small motions of the primary and absorber systems, that is, for small $x_i$ and $y_i$. Then for each $i \in \mathcal{N}$, the dynamics of the $i$th sector are governed by [92,93]

$$ \begin{aligned} & \mu\gamma^2(\ddot{x}_i + \ddot{y}_i) + \xi_a\dot{y}_i + \mu\gamma\delta\sigma^2(x_i + y_i) \\ & \qquad + \mu\gamma\alpha(\ddot{x}_i + \dot{\sigma}^2y_i) = 0 \quad (102a) \\ & \ddot{x}_i + \xi_b\dot{x}_i - \xi_a\dot{y}_i + x_i + \delta\sigma^2x_i \\ & \qquad + \mu\left[ \begin{aligned} & \alpha^2\ddot{x}_i + \gamma^2(\ddot{x}_i + \dot{y}_i) + \alpha\gamma(\ddot{y}_i + 2\dot{x}_i) \\ & \qquad + \alpha\delta\sigma^2x_i + \gamma\delta\sigma^2(x_i + y_i) \end{aligned} \right] \\ & \qquad + \xi_c(-\dot{x}_{i-1} + 2\dot{x}_i - \dot{x}_{i+1}) \\ & \qquad + \nu^2(-x_{i-1} + 2x_i - x_{i+1}) = f e^{i\phi_i} e^{in\sigma} \quad (102b) \end{aligned} $$

<figure><img src="image_54.png" /><figcaption>Fig. 13 (a) Campbell diagram for $N=10$, $v=0.5$, $f=0.01$, and $n=1, \dots, 20N$ and (b) the corresponding frequency response curves $|q_i^{ss}(\tau)|$ corresponding to $n=N, \dots, 2N$. Engine order lines are not shown for $n=N+1, N+2, \dots, 2N-1$, and so on.</figcaption></figure>

where Eq. (102a) describes the absorber dynamics and Eq. (102b) describes the blade dynamics. The indices $i$ are taken mod$N$ such that $x_{N+1} = x_1$ and $x_0 = x_N$, which are cyclic boundary conditions implying that the $N$th blade is coupled to the first. In matrix-vector form, and for each $i \in \mathcal{N}$, Eq. (102) becomes

$$ \left. \begin{aligned} \mathbf{M}\ddot{\mathbf{z}}_i + \mathbf{C}\dot{\mathbf{z}}_i + \mathbf{K}\mathbf{z}_i + \mathbf{C}_c(-\dot{\mathbf{z}}_{i-1} + 2\dot{\mathbf{z}}_i - \dot{\mathbf{z}}_{i+1}) \\ + \mathbf{K}_c(-\mathbf{z}_{i-1} + 2\mathbf{z}_i - \mathbf{z}_{i+1}) \\ = \mathbf{f}e^{i\phi_i}e^{in\sigma} \end{aligned} \right\} \quad (103) $$

where $\mathbf{z}_i = (x_i, y_i)^T$ captures the sector dynamics, $\mathbf{f} = (f, 0)^T$ is a sector forcing vector, and the elements of the sector mass, damping, and stiffness matrices are defined in Table 7. The matrices

$$ \mathbf{C}_c = \begin{bmatrix} \xi_c & 0 \\ 0 & 0 \end{bmatrix}, \quad \mathbf{K}_c = \begin{bmatrix} \nu^2 & 0 \\ 0 & 0 \end{bmatrix} \quad (104) $$

capture the interblade coupling and vanish if $\xi_c = \nu = 0$, in which case Eq. (103) describes the forced motion of $N$ isolated blade/ absorber systems.

**System Model.** By stacking each $\mathbf{z}_i$ into the configuration vector $\mathbf{q} = (\mathbf{z}_1, \mathbf{z}_2, \dots, \mathbf{z}_N)^T$, the governing matrix EOM for the overall $2N$-DOF system takes the form

$$ \hat{\mathbf{M}}\hat{\mathbf{q}} + \hat{\mathbf{C}}\hat{\mathbf{q}} + \hat{\mathbf{K}}\hat{\mathbf{q}} = \hat{\mathbf{f}}e^{in\sigma} \quad (105) $$

where $\hat{\mathbf{M}} \in \mathcal{B}\mathcal{C}\mathcal{B}\mathcal{S}_{2,N}$ is block diagonal with diagonal blocks $\mathbf{M}$ and $\hat{\mathbf{K}} \in \mathcal{B}\mathcal{C}\mathcal{B}\mathcal{S}_{2,N}$ has generating matrices $\mathbf{K} + 2\mathbf{K}_c, -\mathbf{K}_c, \mathbf{0}, \dots, \mathbf{0}, -\mathbf{K}_c$. The matrix $\hat{\mathbf{C}} \in \mathcal{B}\mathcal{C}\mathcal{B}\mathcal{S}_{2,N}$ is similarly defined by replacing $\mathbf{K}$ with $\mathbf{C}$ and $\mathbf{K}_c$ with $\mathbf{C}_c$ in $\hat{\mathbf{K}}$. In terms of the circulant
<figure><img src="image_55.png" /><figcaption>Fig. 14 (a) Model of bladed disk assembly and (b) sector model</figcaption></figure>

<table><caption>Table 7 Elements of the sector mass, damping, and stiffness matrices M, C, and K</caption><thead><tr><th>Matrix</th><th>Notation</th><th>Elements</th></tr></thead><tbody><tr><td>Mass</td><td><b>M</b></td><td>M<sub>11</sub> = 1 + μ(α + γ)<sup>2</sup><br>M<sub>12</sub> = μγ(α + γ)<br>M<sub>21</sub> = M<sub>12</sub><br>M<sub>22</sub> = μγ<sup>2</sup></td></tr><tr><td>Damping</td><td><b>C</b></td><td>C<sub>11</sub> = ξ<sub>b</sub><br>C<sub>12</sub> = −ξ<sub>a</sub><br>C<sub>21</sub> = 0<br>C<sub>22</sub> = ξ<sub>a</sub></td></tr><tr><td>Stiffness</td><td><b>K</b></td><td>K<sub>11</sub> = 1 + (1 + μ(α + γ))δσ<sup>2</sup><br>K<sub>12</sub> = μγδσ<sup>2</sup><br>K<sub>21</sub> = K<sub>12</sub><br>K<sub>22</sub> = μγ(α + δ)σ<sup>2</sup></td></tr></tbody></table>

operator, the system mass, damping, and stiffness matrices are defined by

$$ \left. \begin{aligned} \hat{\mathbf{M}} &= \operatorname{circ}(\mathbf{M}, \mathbf{0}, \mathbf{0}, \dots, \mathbf{0}, \mathbf{0}) \\ \hat{\mathbf{C}} &= \operatorname{circ}(\mathbf{C} + 2\mathbf{C}_c, -\mathbf{C}_c, \mathbf{0}, \dots, \mathbf{0}, -\mathbf{C}_c) \\ \hat{\mathbf{K}} &= \operatorname{circ}(\mathbf{K} + 2\mathbf{K}_c, -\mathbf{K}_c, \mathbf{0}, \dots, \mathbf{0}, -\mathbf{K}_c) \end{aligned} \right\} \quad (106) $$

The $2N \times 1$ system forcing vector is

$$ \begin{aligned} \hat{\mathbf{f}} &= (\mathbf{f}e^{j\phi_1}, \mathbf{f}e^{j\phi_2}, \dots, \mathbf{f}e^{j\phi_N})^\mathrm{T} \\ &= \mathbf{f}_0 \otimes \mathbf{f} \end{aligned} \quad (107) $$

where the $N \times 1$ vector $\mathbf{f}_0$ is defined by Eq. (69) and the interblade phase angle $\phi_i$ 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 standard techniques [99]. Its nonresonant solution in the steady-state follows in the usual way and is given by

$$ \mathbf{q}^{ss}(\tau) = \hat{\mathbf{Z}}^{-1} \hat{\mathbf{f}} e^{j n \sigma \tau} \quad (108) $$

where $\hat{\mathbf{Z}} = \hat{\mathbf{K}} - n^2\sigma^2\hat{\mathbf{M}} + jn\sigma\hat{\mathbf{C}}$ is the system impedance matrix of dimension $2N \times 2N$. However, Eq. (108) does not offer any insight into the system’s modal characteristics and it requires computation of $\hat{\mathbf{Z}}^{-1}$, 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 symmetry and the theory developed in Sec. 2. The analysis follows similarly 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

$$ \mathbf{q} = (\mathbf{E} \otimes \mathbf{I})\mathbf{u}, \quad \text{or} \quad \mathbf{z}_i = (\mathbf{e}_i^T \otimes \mathbf{I})\mathbf{u}, \quad i \in \mathcal{N} \quad (109) $$

where $\mathbf{E}$ is the $N \times N$ complex Fourier matrix and $\mathbf{e}_i$ is its $i$th column, $\otimes$ is the Kronecker product, $\mathbf{I}$ is the $2 \times 2$ identity matrix (the dimension of $\mathbf{I}$ corresponds to the number of DOFs per sector), and $\mathbf{u} = (\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_N)^\mathrm{T}$ is a vector of modal, or cyclic coordinates. Each $\mathbf{u}_p$ is $2 \times 1$ and describes the sector dynamics in modal space. Substituting Eq. (109) into Eq. (105), multiplying from the left by the unitary matrix $(\mathbf{E} \otimes \mathbf{I})^\mathrm{H} = (\mathbf{E}^\mathrm{H} \otimes \mathbf{I})$, and invoking Theorem 10 yields a system of $N$ block decoupled equations, each with two DOFs. They are

$$ \tilde{\mathbf{M}}_p \ddot{\mathbf{u}}_p + \tilde{\mathbf{C}}_p \dot{\mathbf{u}}_p + \tilde{\mathbf{K}}_p \mathbf{u}_p = (\mathbf{e}_p^\mathrm{H} \otimes \mathbf{I}) \hat{\mathbf{f}} e^{j n \sigma \tau}, \quad p \in \mathcal{N} \quad (110) $$

where $(\mathbf{e}_p^\mathrm{H} \otimes \mathbf{I})\hat{\mathbf{f}}$ is the $p$th $2 \times 1$ block of $(\mathbf{E}^\mathrm{H} \otimes \mathbf{I})\hat{\mathbf{f}}$. Equation (110) is analogous to the $N$ M-DOF systems given by Eq. (61) for the general formulation in Sec. 3.1, but in this case $M=2$ and engine order excitation is assumed from the onset. Figure 15 illustrates the transformation of the single $2N$-DOF system given by Eq. (105) to a system of $N$ block decoupled two-DOF forced oscillators defined by Eq. (110).

The $2 \times 2$ mass, damping, and stiffness matrices associated with the $p$th mode follow from Theorem 10 and are given by

$$ \left. \begin{aligned} \tilde{\mathbf{M}}_p &= \mathbf{M} \\ \tilde{\mathbf{C}}_p &= \mathbf{C} + 2\mathbf{C}_c(1 - \cos \varphi_p) \\ \tilde{\mathbf{K}}_p &= \mathbf{K} + 2\mathbf{K}_c(1 - \cos \varphi_p) \end{aligned} \right\} , \quad p \in \mathcal{N} \quad (111) $$

where $\varphi_p$ is defined by Eq. (16), the elements of $\mathbf{M}$, $\mathbf{C}$, and $\mathbf{K}$ are defined in Table 7 and the coupling matrices $\mathbf{C}_c$ and $\mathbf{K}_c$ are defined by Eq. (104). In light of Eqs. (70) and (71), the $p$th modal forcing vector takes the form

$$ \begin{aligned} (\mathbf{e}_p^\mathrm{H} \otimes \mathbf{I})\hat{\mathbf{f}} &= (\mathbf{e}_p^\mathrm{H} \otimes \mathbf{I})(\mathbf{f}_0 \otimes \mathbf{f}) \\ &= \mathbf{e}_p^\mathrm{H} \mathbf{f}_0 \otimes \mathbf{f} \\ &= \left\{ \begin{array}{ll} \sqrt{N} \mathbf{f}, & p = n + 1 \\ \mathbf{0}, & \text{otherwise} \end{array} \right. \end{aligned} \quad (112) $$
<figure><img src="image_56.png" /><figcaption>Fig. 15 The topology of a bladed disk assembly fitted with absorbers in (a) physical space and (b) modal space. The modal transformation $q(\tau) = (E \otimes I)u(\tau)$ reduces the cyclic array of $N$, two-DOF sector models ($\mathcal{B}$, $\mathcal{A}$), which together form a 2N-DOF coupled system, to a set of $N$, two-DOF block decoupled models ($\mathcal{B}_p$, $\mathcal{A}_p$).</figcaption></figure>

where $\mathbf{0} = (0,0)^T$ and the scalar product $e_p^H \mathbf{f}_0$ vanishes except for $p=n+1$. Because only mode $p=n+1$ is excited, $\mathbf{u}_{n+1}(\tau)$ is the only nonzero modal response in the steady-state.

Assuming harmonic motion, and in light of Eq. (112), the $p$th steady-state modal response follows easily from Eq. (110) and is given by

$$ \mathbf{u}_p^{ss}(\tau) = \begin{cases} \sqrt{N}\tilde{\mathbf{Z}}_{n+1}^{-1}\mathbf{f}e^{jn\sigma\tau}, & p = n+1 \\ \mathbf{0}, & \text{otherwise} \end{cases} \quad (113) $$

where

$$ \tilde{\mathbf{Z}}_p = \tilde{\mathbf{K}}_p - n^2 \sigma^2 \tilde{\mathbf{M}}_p + jn\sigma \tilde{\mathbf{C}}_p, \quad p \in \mathcal{N} \quad (114) $$

is the $p$th modal impedance matrix. The response of sector $i$ (in physical coordinates) follows from the transformation given by Eq. (109) with

$$ \mathbf{u}^{ss}(\tau) = (\mathbf{0}, \dots, \mathbf{0}, \mathbf{u}_{n+1}^{ss}(\tau), \mathbf{0}, \dots, \mathbf{0})^T $$

and is given by

$$ \mathbf{z}_i^{ss}(\tau) = \tilde{\mathbf{Z}}_{n+1}^{-1} \mathbf{f} e^{j\phi_i} e^{jn\sigma\tau}, \quad i \in \mathcal{N} \quad (115) $$

where $w^{n(i-1)} = e^{i\phi_i}$ 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 $\phi_i$. This approach offers a significant computational advantage over the direct solution to the full 2N-DOF system, as given by Eq. (108).

## 4 Conclusions

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 demonstrated by past work in this area and the review provided here, the theory of circulants provides a useful description of the fundamental structure of the mode shapes and spectrum of systems with cyclic symmetry, including those of large scale. The theory also provides a convenient means for computing the vibration response of these systems, even when the idealized symmetry is broken by mistuning or by nonlinear effects. As with any mathematical tool, the overhead in learning it must provide appropriate benefit,

whether in terms of fundamental understanding, insight, or ease of computation. We trust that the results presented here offer such benefits to readers interested in vibration analysis of cyclic systems.

It must be noted that no physical system has perfect symmetry, as assumed herein. This assumption must be examined in light of the system under consideration. One key to the suitability of a cyclically symmetric model is the intersector coupling. If the coupling is strong, so that the pairs of modes have well separated frequencies, then small imperfections will not alter the picture substantially, and one can consider each mode pair as robust against coupling to other modes. However, if the coupling is weak, so that the system frequencies are clustered near those of the isolated sector model, the possibility of localization is significantly increased. This topic has been investigated quite thoroughly, primarily in the context of the vibration of bladed disk assemblies with small blade mistuning; see, for example, [72–88]. Also, nonlinear effects can couple linear modes under certain resonance conditions, even at small amplitudes [118]. In cyclic systems this can occur for the pairs of modes with equal frequencies [119], and this possibility expands to groups of modes for the case of weak coupling, leading to extremely complicated behavior [82,84,88]. In such cases, the tools from group theory can be applied to categorize the possible modes and forced response in terms of their symmetries [65]. This topic, while interesting, is outside the scope of the present paper.

## Acknowledgment

The work of the second author related to this topic was funded by the National Science Foundation, currently by Grant No. CMMI-1100260.

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