Cyclic Symmetry and Floquet Periodic Conditions
These boundary conditions are based on the Floquet theory which can be applied to the problem of small-amplitude vibrations of spatially periodic structures. It is also applicable when performing a linear buckling analysis.
If the problem is to determine the frequency response to a small-amplitude time-periodic excitation that also possesses spatial periodicity, the theory states that the solution can be sought in the form of a product of two functions. One follows the periodicity of the structure, while the other one follows the periodicity of the excitation. The problem can be solved on a unit cell of periodicity by applying the corresponding periodicity conditions to each of the two components in the product.
This section describes the theory for solids in 3D, but is equally applicable to shells. In the case of shells, the periodicity condition is applied to edges, and the rotational degrees of freedom (displacements of the normal vector) are treated in the same way as the translational displacements.
The problem can be modeled using the full solution without applying the above described multiplicative decomposition. For such a solution, the Floquet periodicity conditions at the corresponding boundaries of the periodicity cell are expressed as
where u is a vector of dependent variables, r is the position, and the vector kF represents the spatial periodicity of the excitation.
The cyclic symmetry boundary condition presents a special but important case of Floquet periodicity, for which the unit periodicity cell is a sector of a structure that consists of a number of identical sectors. The frequency response problem can then be solved in one sector of periodicity by applying the periodicity condition. The situation is often referred to as dynamic cyclic symmetry.
For an eigenfrequency study, all the eigenmodes of the full problem can be found by performing the analysis on one sector of symmetry only and imposing the cyclic symmetry of the eigenmodes with an angle of periodicity φ = mθ, where the cyclic symmetry mode number m can vary from 0 to N/2, with N being the total number of sectors so that θ = 2π/N.
The Floquet periodicity conditions at the sides of the sector of symmetry can be expressed as
where the u represents the displacement vectors with the components given in the default Cartesian coordinates. Multiplication by the rotation matrix given by
makes the corresponding displacement components in the cylindrical coordinate system differ by the factor exp(−iφ) only. For scalar dependent variables, a similar condition applies, for which the rotation matrix is replaced by a unit matrix.
The angle φ represents either the periodicity of the eigenmode for an eigenfrequency analysis or the periodicity of the excitation signal in case of a frequency-response analysis. In the latter case, the excitation is typically given as a load vector
when modeled using the Cartesian coordinates. The parameter m is often referred to as the azimuthal wave number.
When using cyclic symmetry, you may get computed modes having a mode number other than the one you asked for. Such modes are also correct solutions. This phenomenon is an effect of the fact that the trigonometric boundary conditions are only unique up to a factor 2π. The general rule is that if you search for modes with mode number m, you can get any mode with mode number |m + kN|. Here, N is the number of sectors for a full revolution, and k is any integer, positive or negative.
If your model is fully rotationally symmetric, then it is a good idea to use a very small sector in the analysis. Not only will you have a low number of degrees of freedom, but since N is then large, you will usually get the modes you asked for. Modes with high mode numbers have complicated patterns that will cause high natural frequencies (or, in the case of buckling, high critical load factors).
Ref. 158 contains more information about cyclic symmetry conditions.