The following is a brief introduction to multistage cyclic symmetry equations.
For simplicity, a 2-stage example will be used to demonstrate the system equations, but this can be generalized to more stages. Similarly, while this discussion is limited to the stiffness matrix, it can be extended to other system matrices, for example, the mass or damping matrices. The starting point of the multistage analysis is two or more independent stage meshes resulting in system matrices and load vectors that are partitioned into two or more uncoupled submatrices.
The following equations describe displacement, stiffness, and force of the entire multistage system before coupling at the interstage boundaries:
 | (1–1) |
 | (1–2) |
 | (1–3) |
where
 = displacements of stage 
|
 = stiffness of stage 
|
 = force of stage 
|
Cyclic Transformation
For each stage, the equations are transformed into cyclic symmetry space by
applying the real-valued Fourier matrix 
, which decouples the system into harmonic index blocks:
 | (1–4) |
 | (1–5) |
 | (1–6) |
where
 , which represents the Kronecker product of the real-valued
Fourier matrix and an identity matrix with the size of the number of DOFs of
a single sector stage 
|
 = real-valued Fourier matrix for sector stage
|
 = displacements of sector stage  in cyclic symmetry coordinates |
 = stiffness of the uncoupled multistage system transformed
into cyclic symmetry coordinates |
 = force vector of the uncoupled multistage system
projected into cyclic symmetry coordinates |
At this point, it is possible to consider one or more harmonic index blocks of each stage as needed.
Cyclic Base/Duplicate Architecture and Cyclic Constraint Equations
In practice, the prior cyclic transformation is achieved by creating a base and duplicate sector for each sector stage and then applying cyclic constraint equations (see Coupling and Constraint Equations (CEs) in the Cyclic Symmetry Analysis Guide). These constraint equations are applied to each sector stage separately.
The physical stage degrees of freedom, 
and 
, can be split into internal and interstage boundary degrees of
freedom. The coupling between the stages occurs on the interstage boundary
(IB) degrees of freedom, 
and 
. To ensure compatibility in the physical domain, the boundary
degrees of freedom between the stages are enforced to be equal:
 | (1–7) |
If the interstage boundary degrees of freedom in the physical domain are not aligned, a mapping matrix is applied to relate interstage boundary degrees of freedom.
These boundary degrees of freedom are approximated using the real-valued Fourier coefficients of the base associated with the number of sectors of the respective cyclic stages
 | (1–8) |
where 
and 
represent the harmonic index of interest for each stage to be
coupled. In practice, at least one multistage cyclic set of equations with
is required. The constraint equation now only contains cyclic
quantities and can be applied to the multistage cyclic system:
 | (1–9) |
The final coupled multistage equations are formed by applying this constraint.