1. Introduction

Faced with the complexity of mechanical structures, often consisting of an assembly of several components, classical numerical or experimental methods of vibration mechanics prove to be expensive, sometimes even unusable. In perfect coherence with the modular organization of major projects, substructuring methods appear to be the most effective way to carry out the vibratory study of the whole based on the dynamic behavior of the components [bib4].

In this report, we present, first of all, the theoretical bases of modal synthesis methods. They combine substructuring and modal recombination techniques. Each substructure is represented by a projection base composed of dynamic eigenmodes and static deformations at the interfaces.

Then, we present the two classical substructuring calculation techniques, implemented in Code_Aster [bib5]: the Craig-Bampton methods, Mac Neal methods and interface modes. They are essentially distinguished by the use of different bases for the substructures.

The specific case of modal analysis of structures with cyclic symmetries by dynamic substructuring is addressed in the document [R4.06.03].

General notes:

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

\(\stackrel{ˉ}{\mathrm{M}}\mathrm{=}{\varphi }^{T}\mathrm{M}\varphi\)	:	is the generalized mass matrix,
\(\stackrel{ˉ}{\mathrm{C}}\mathrm{=}{\varphi }^{T}\mathrm{C}\varphi\)	:	is the generalized depreciation matrix,
\(\stackrel{ˉ}{\mathrm{K}}\mathrm{=}{\varphi }^{T}\mathrm{K}\varphi\)	:	is the generalized stiffness matrix,
\({\{{\stackrel{ˉ}{f}}_{\text{ext}}\}}_{o}={\mathrm{\Phi }}^{T}{\{{f}_{\text{ext}}\}}_{o}\)	:	is the vector of the generalized external forces applied
\({\{{\stackrel{ˉ}{f}}_{L}\}}_{o}={\mathrm{\Phi }}^{T}{\{{f}_{L}\}}_{o}\)	:	is the vector of the generalized bond forces applied,
\(\eta ,\dot{\eta },\ddot{\eta }\)	:	is the vector of generalized movements, speeds, and accelerations.

Note:

The exponent :math:`k`*characterizes the quantities relating to the substructure:math:`{S}^{k}`*and the generalized quantities are surmounted by a bar: for example*:math:`{stackrel{ˉ}{M}}^{k}`*is the generalized mass matrix of the substructure*:math:`{S}^{k}`* . *