The following topics are available concerning cyclically symmetric structure analysis:
Given a cyclically symmetric (periodic) structure such as a fan wheel, a modal analysis can be performed for the entire structure by modelling only one sector of it. A proper basic sector represents a pattern that, if repeated n times in cylindrical coordinate space, would yield the complete structure.
In a flat circular membrane, mode shapes are identified by harmonic indices. For more information, see the Cyclic Symmetry Analysis Guide.
Constraint relationships (equations) can be defined to relate the lower (θ = 0) and higher (θ = α, where α = sector angle) angle edges of the basic sector to allow calculation of natural frequencies related to a given number of harmonic indices. The basic sector is duplicated in the modal analysis to satisfy the required constraint relationships and to obtain nodal displacements. This technique was adapted from Dickens([148]).
Constraint equations relating the lower and higher angle edges of the two sectors are written:
 | (14–232) |
where:
| uA, uB = calculated displacements on lower angle side of basic and duplicated sectors (A and B, respectively) |
 = displacements on higher angle side of basic and duplicated sectors (A and
B, respectively) determined from constraint relationships |
|
| α = 2π/N = sector angle |
| N = number of sectors in 360° |
Three basic steps in the procedure are briefly:
The CYCLIC command in /PREP7 automatically detects the cyclic symmetry model information, such as edge components, the number of sectors, the sector angles, and the corresponding cyclic coordinate system.
The CYCOPT command in /SOLU generates a duplicated sector and applies cyclic symmetry constraints (Equation 14–232) between the basic and the duplicated sectors.
The /CYCEXPAND command in /POST1 expands a cyclically symmetry response by combining the basic and the duplicated sectors results (Equation 14–233) to the entire structure.
The mode shape in each sector is obtained from the eigenvector solution. The displacement components (x, y, or z) at any node in sector j for harmonic index k, in the full structure is given by:
 | (14–233) |
where:
| n = sector number, varies from 1 to N |
| uA = basic sector displacement |
| uB = duplicate sector displacement |
If the mode shapes are normalized to the mass matrix in the mode analysis (Nrmkey option in the MODOPT command), the normalized displacement components in the full structure is given by
 | (14–234) |
The complete procedure addressing static, modal, and prestressed modal analyses of cyclically symmetric structures is contained in the Cyclic Symmetry Analysis Guide.
A bladed disk (axial or centrifugal) is cyclically symmetric about its fundamental sector, as shown in Figure 14.16: Full Model with the Cyclic-Symmetric Sector Highlighted. These bladed disks are typically excited by rotationally symmetric excitations which produce forced vibrations.
The equation of motion for harmonically-varying forced response is:
 | (14–235) |
where:
| [Ks] = stiffness matrix of the bladed disk |
| [Ms] = mass matrix of the bladed disk |
| {Fs} = load vector |
| {us} = displacement vector |
| Ω = forcing frequency, input on the CYCFREQ command |
The superscript s implies the matrices refer to the entire 360° system. Damping is ignored in this equation and is discussed in Damping.
Equation 14–235 can be transferred into modal space by:
 | (14–236) |
where:
| [Φs] = the set eigenvectors of the full system |
| {as} = the vector of modal coordinates |
Substituting this into Equation 14–235 and pre-multiplying by [Φs]T yields the forced response equation in terms of modal coordinates:
 | (14–237) |
where:
| [Λs] = the diagonal matrix for the system natural frequencies squared |
| [I] = the identity matrix |
The number of eigenvectors (modes) used in Equation 14–237 is much less than the total number of degrees of freedom, so that Equation 14–237 is significantly smaller system than Equation 14–235.
A cyclically-symmetric structure has the following transformation from cyclic coordinates to physical coordinates:
 | (14–238) |
where:
 = cyclic (or harmonic) coordinates for harmonic index h |
| [F] = real-valued Fourier matrix |
 denotes the Kronecker product |
The system eigenvectors can also be expressed in terms of their harmonics:
 | (14–239) |
where:
 = the real-valued cyclic modes corresponding to harmonic h |
 = a pseudo-block diagonal matrix. A block is 1×1 for h = 0 and N/2 if N
is even, and 2×2 otherwise. N is the number of sectors
(CYCLIC). |
Substituting Equation 14–239 into Equation 14–237 gives the frequency response equation in cyclic coordinates:
 | (14–240) |
where the modal load {fs} is:
 | (14–241) |
The Fourier transformation expression:
 | (14–242) |
appears in Equation 14–239 and Equation 14–241
(transposed), where 
are the harmonic modes. 
has the structure:
 | (14–243) |
The harmonics h (0 < h < N/2) appear as 2×2 sub-blocks, each column being one of the repeated modes and the rows corresponding to the real and imaginary (basic and duplicate sector) solutions. The first mode in a pair is denoted with subscript A and the second mode in the pair is denoted with subscript B.
The real and imaginary terms of each double mode are related and obey the following:
 | (14–244) |
where gA = 1 if the mode shapes 
and 
have the same sign, otherwise gA = -1. Likewise:
 | (14–245) |
where gB = -gA. This property leads to the .MODE file needing to only contain the real (base) sector degrees of freedom for each mode shape. Likewise, the stresses and strains for the expanded modes on the .RST file only contain the results for the base elements (and not the duplicate elements).
The forcing function is assumed to be time harmonic as well as spatially harmonic in the circumferential direction. The force on any sector n can then be related to the force on the basic sector by only a phase shift:
 | (14–246) |
where:
| {F} = complex load vector on the basic sector |
 = phase of excitation (inter-blade phase angle, IBPA) |
| n = sector number |
| C = engine order (EO) excitation (CYCFREQ) |
| N = number of sectors |
Equation 14–246 defines the distribution of force over the blades using
the blade numbering convention shown in Figure 14.17: Forcing Sign and Numbering Convention. With this
convention, blade 2 leads blade 1 by the inter-blade phase angle 
; it is subjected to the force first and blade 1 is subjected to the same force
after a rotation of ϕ radians. The forcing wave travels in the direction shown in Figure 14.17: Forcing Sign and Numbering Convention, which is a backward traveling wave with respect to the
rotation ΩR (OMEGA or
CMOMEGA).
The right hand side of Equation 14–240 is Equation 14–241:
 | (14–247) |
Equation 14–247 can be written as the sum of each sector, if only blade loads are considered:
 | (14–248) |
where:
| [Fn] = the nth row of the Fourier matrix [F] |
| {F n} = force on blade n |
While the engine order can take any integer value, a given engine order C will only
excite a certain harmonic index. This aliased engine order 
is determined from the input engine order (for example, the
number of preceding stators) as outlined in the following table:
Table 14.2: Aliased Engine Order (Excited Harmonic Index)
| Engine Order C | Aliased Engine Order 
| |
|---|---|---|
| N Even | N Odd | |
| C ≤ (N / 2) | C ≤ (N - 1)/2 | C |
| N / 2 < C ≤ N | (N + 1)/2 ≤ C ≤ N | N - C |
| N < C ≤ (3N / 2) | (N + 1) ≤ C ≤ (3N - 1)/2 | C - N |
| (3N / 2) < C ≤ 2N | (3N +1)/2 ≤ C ≤ 2N | 2N - C |
| … | … | … |
This leads to the well-known zigzag diagram in Figure 14.18: Zigzag Diagram for an Even Number of Sectors. The positive slopes are forward traveling waves and the negative slopes are backward traveling.
Due to the orthogonality of the engine order phasing with the
sine and cosine terms of the Fourier matrix [F], only the harmonic indices h =
of [F
n] are non-zero.
The sign of
is determined
as follows:
If the aliased engine order is on a negative slope of Figure 14.18: Zigzag Diagram for an Even Number of Sectors, it is assigned a negative value.
If the rotation is opposite that illustrated in Figure 14.17: Forcing Sign and Numbering Convention, in other words a negative ΩR in the cyclic coordinate system, the value from the first step is negated.
If the engine order was input as a negative value, the value after the second step is negated again.
This process ensures that the force is correctly applied to the blades via Equation 14–247, while respecting the blade numbering, rotation direction, and the relationship between engine order and nodal diameter.
Returning to the harmonic equation of motion, Equation 14–240, the force is assumed to be harmonic and of the form:
 | (14–249) |
The excitation frequency Ω, and the corresponding frequency sweep range Ω ± ΔΩ (HARFRQ) is typically related to the rotor speed by:
 | (14–250) |
The maximum response will occur when this excitation frequency crosses a natural frequency of that nodal diameter, for instance when CΩR = ωh.
Damping may be included in three forms:
The damping matrix in cyclic modal coordinates is
 | (14–251) |
where
 = diagonal matrix of system frequencies |
 = diagonal matrix of blade modal damping terms ξi
|
 = diagonal matrix of blade modal frequencies |
The last term is the blade modal damping projected to
the system modes using the participation factors 
defined in Evaluation of Participation Factors.
Interblade aerodynamic coupling can be introduced assuming a first
order dependence on the blade motion by inserting an additional matrix 
(input using CYCFREQ,AERO) into the frequency response
equation of motion for the whole system. First consider the introduction of the aero matrix in
physical coordinates:
 | (14–252) |
where non-aero damping terms are dropped for conciseness. This equation can be transformed into modal coordinates using Equation 14–236.
 | (14–253) |
The aerodynamic coupling matrix in physical coordinates
can be written in cyclic cantilevered blade coordinates (see He et. al. [436]):
 | (14–254) |
where 
is the complex-valued Fourier matrix and 
is a matrix of cantilevered blade modes. The resulting matrix 
is in general complex since the coupling is not in phase with the motion, but
it is presumed to be cyclically symmetric and therefore block diagonal:
 | (14–255) |
Note that a sub-block 
corresponds to the interblade phase angle 
, where 
. It is common for CFD and other codes to compute the aerodynamic coefficients
that are compatible with this matrix. Each block 
is made up of coefficients 
, that capture the coupling between mode i and the force computed using mode j
in the direction of the surface normal. These values can be obtained using the following area
integral:
 | (14–256) |
where 
is the surface normal and 
is the area.
The aerodynamic term in Equation 14–253 can
be rewritten by assuming that the blade portion (
) of the system modes can be represented by a linear combination of cantilevered
blade modes:
 | (14–257) |
The aero term becomes:
 | (14–258) |
Rearranging Equation 14–254 and substituting into Equation 14–258, it can be seen that the aero term is:
 | (14–259) |
Once the vector of modal coordinates {as} is obtained for a given excitation frequency Ω, we can expand back to the physical solution quantities for the entire bladed disk. The displacements of sector n are determined by:
 | (14–260) |
Stresses and strains can be similarly evaluated by using the stress and strain mode shapes
for
.
The program employs the CMS-based Component Mode Mistuning (CMM) methodology (see Lim et. al. [430]) for small stiffness mistuning. This methodology is well suited not only to the simulation of bladed disks, but also to integrally bladed rotors and impellers where coupled disk-blade motion is important. From a tuned response of the nominal geometry, mistuning parameters are introduced into the component modes which are projected onto the system modes, yielding a compact set of equations for the harmonic frequency response of the system. This methodology builds on the theory discussed in Mode-Superposition Harmonic Analysis, and shares the basic core equations, damping, forcing, and coordinate transformations.
Stiffness mistuning can be characterized as a variation of the nominal stiffness of a system. Referring to Equation 14–235, the stiffness matrix can be written as:
 | (14–261) |
where the subscript o indicates the nominal values and 
indicates deviations from the nominal. If the mistuning is assumed to be in
the blade only, we can assume proportional mistuning:
 | (14–262) |
where 
is the nominal stiffness matrix for a blade and 
is the stiffness mistuning deviation for blade n. For a prestressed analysis, 
will include any prestress effects. This may alternatively be
considered as deviation in Young's modulus only in the case where
there are no prestress effects. Bdiag[.] denotes a block diagonal matrix.
Mode-superposition is used to project the equation of motion onto an appropriate basis and reducing it to a smaller set of degrees of freedom. In contrast to typical mode-superposition techniques where the equation of motion is projected onto the modes computed from its own system (in this case, modes of the mistuned system), the program projects onto tuned system modes. Following the same reasoning as in Transform to Modal Coordinates, the forced response equation in tuned-system modal coordinates is:
 | (14–263) |
where [Λs] is the diagonal matrix of the system natural frequencies squared and [I] is the identity matrix. The left superscript Γ indicates that only the blade DOFs are used from the eigenvectors.
Substituting Equation 14–262 into Equation 14–263 and recognizing that the global triple product can be replaced with a local sum per sector yields:
 | (14–264) |
where N is the number of sectors.
The blade matrix [Ko] in Equation 14–264 can be approximated by applying the fixed-interface Craig-Bampton (C-B) transformation:
 | (14–265) |
where {q} are the generalized C-B coordinates (modal participation factors), and the transformation [T] is given by:
 | (14–266) |
are the set of blade modes derived from assuming the blade-disk
interface is fixed (and any shroud interfaces as well), and 
are the static shapes:
 | (14–267) |
where the subscripts i and b indicate the partition into internal and boundary (interface) DOFs of the blade. Note that b corresponds to master DOFs and i corresponds to all remaining DOFs in the substructure as defined in Component Mode Synthesis (CMS). The boundary DOFs and the elements defining the blade matrices are both input on CYCFREQ,BLADE.
Applying this transformation to the blade stiffness yields:
 | (14–268) |
The number of blade modes is typically small leading to 
being a much smaller
matrix than [Ko].
Note that this is the typical Component Mode Synthesis (CMS) substructure generation analysis for the blade.
Projecting the Craig-Bampton coordinates {q} onto the system modes for a sector via:
 | (14–269) |
and substituting into Equation 14–264 yields for the blade stiffness term:
 | (14–270) |
The CMM approach to mistuning allows for the use of cyclic quantities within the reduced equations of motion. The Craig-Bampton coordinates or modal participation factors for the nth sector can be expressed in cyclic coordinates using the same transformation as in Equation 14–238:
 | (14–271) |
where {Fn} is the nth row of [F]. Substituting Equation 14–270 into Equation 14–264 and using Equation 14–271 and Equation 14–239 we arrive at:
 | (14–272) |
where the modal load {fs} is:
 | (14–273) |
Rewriting Equation 14–265 in cyclic coordinates:
 | (14–274) |
and segregating 
into
its parts yields:
 | (14–275) |
From the definition of [T] in Equation 14–266, it is clear that:
 | (14–276) |
where 
are the system modes at the blade
interface points.
Premultiplying Equation 14–274 by 
and using orthogonality of the modes, we
can solve for 
:
 | (14–277) |
where 
and 
are the blade natural frequencies squared and mode shapes from the
Craig-Bampton transformation.
The stiffness parameter 
(input on CYCFREQ,MIST) is handled as follows.
The stiffness mistuning parameter 
represents the deviation of Young’s modulus
of each blade from its nominal value:
 | (14–278) |
This leads to a modification of the blade’s natural frequencies by a common factor. In order to modify each natural frequency independently, a frequency-dependent mistuning parameter is introduced:
 | (14–279) |
where 
is the ith nominal frequency of blade
n and 
is the ith mistuned
frequency of blade n. Note that if 
is constant for all frequencies,
then 
, that is, when all frequencies are modified by the
same ratio that is identical to the modified Young’s modulus.
The application of the effective stiffness mistuning parameter 
is as follows. The
nominal Craig-Bampton reduced stiffness matrix has the following form,
after applying the transformation in Equation 14–266:
 | (14–280) |
Defining the average frequency mistuning for a blade as:
 | (14–281) |
where Nf is the number of blade frequencies, we can define the stiffness mistuning as:
 | (14–282) |
The constraint modes are modified by the average frequency mistuning, and the frequencies themselves are modified by the frequency-dependent input values.
The cyclically symmetric solution sequences consist of three basic steps. The first step transforms applied loads to cyclically symmetric components using finite Fourier theory and enforces cyclic symmetry constraint equations (see Equation 14–232) for each harmonic index (nodal diameter) (k = 0, 1, . . ., N/2).
Any applied load on the full 360° model is treated through a Fourier transformation process and applied on to the cyclic sector. For each value of harmonic index, k, the procedure solves the corresponding linear equation. The responses in each of the harmonic indices are calculated as separate load steps at the solution stage. The responses are expanded via the Fourier expansion (Equation 14–233). They are then combined to get the complete response of the full structure in postprocessing.
The Fourier transformation from physical components, X, to the
different harmonic index components,
, is given by the following:
Harmonic Index, k = 0 (symmetric mode):
 | (14–283) |
Harmonic Index, 0 < k < N/2 (degenerate mode)
For N even only, Harmonic Index, k = N/2 (antisymmetric mode):
 | (14–286) |
where:
| X = any physical component, such as displacements, forces, pressure loads, temperatures, and inertial loads |
 = cyclically symmetric component |
The transformation to physical components, X, from the cyclic
symmetry,
, components is
recovered by the following equation:
 | (14–287) |
The last term
exists only
for N even.