where u is a vector of dependent variables,
r is the position, and the vector
kF represents the spatial periodicity of the excitation.
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