1. Introduction

In this document, we summarize the methods of cyclic dynamic substructuring. We give a definition of cyclic repetitiveness (or cyclic symmetry) and we present the main effects of this property on the dynamic behavior of the structure (nodal circles and diameters, double modes). Then, we explain, in some detail, the two methods of cyclic dynamic substructuring, implemented in Code_Aster. Improvements have been made to the classical methods by taking into account the presence of axis nodes.

These methods assume that the mesh of the base sector is such that its traces on the right and left interfaces are coincident (compatible meshes).

General notes:

\({\omega }_{m}\)	:	Maximum pulsation of a system (\({\mathrm{rad.s}}^{-1}\))
\(M\)	:	Mass matrix from finite element modeling
\(K\)	:	Stiffness matrix from finite element modeling
\(q\)	:	Vector of degrees of freedom from finite element modeling
\({f}_{\mathit{ext}}\)	:	Vector of forces external to the system
\({f}_{L}\)	:	Vector of bond forces applied to a substructure
\(\Phi\)	:	Matrix containing the vectors of a projection base organized in columns
\(\eta\)	:	Vector of generalized degrees of freedom
\(B\)	:	Extraction matrix for interface degrees of freedom
\(L\)	:	Link matrix
\(T\)	:	Kinetic energy
\(U\)	:	Deformation energy
\(\mathit{Id}\)	:	Identity matrix
\(\lambda\)	:	Diagonal matrix of generalized rigidities
\({R}_{e}(\omega )\)	:	Residual dynamic flexibility matrix
\({R}_{e}(0)\)	:	Residual static flexibility matrix

Notations specific to cyclic substructuring:

\(N\)	=	number of sectors
\(\alpha\)	=	angle formed by the base sector
\(\beta\)	=	inter-sector phase shift
\(\mathrm{Oz}\)	=	cyclic symmetry axis
\(\theta\)	=	rotation of angle \(\alpha\) and axis \(\mathrm{Oz}\)
\(\text{Re}(Z)\)	=	real part of the \(Z\) complex
\(\text{Im}(Z)\)	=	imaginary part of the \(Z\) complex
\(\theta\)	=	matrix for moving from right nodes to left nodes
\({\theta }_{a}\)	=	sector change matrix for the nodes of the axis

Note:

The index	d	is	is	relative	to right-wing degrees of liberty
“	g	“	“	at left degrees of liberty
“	a	“	“	at the degrees of freedom of the axis
“	1	“	“	with identified specific modes
“	2	“	“	with unknown proper modes