3. A short reminder of the methodology

Only the fundamental principles of the technique are mentioned here. For more details, the reader is invited to consult the reference document [R4.06.02].

Projecting the substructure on a reduced base necessarily leads to an approximation of the results obtained. The role of the modeler is to choose a fairly rich base in order to be able to understand the dynamic behavior of the substructure. The quality of the results depends on the choice of this projection base. The most used techniques for choosing this base are the Craig Bampton technique and the Mac Neal technique. In addition to these two techniques, there is the use of interface modes, which is a variant of these two techniques where the degrees of freedom of connection are condensed.

• The Craig-Bampton technique consists in choosing normal modes and constrained modes as the basis for projection. Normal modes are modes specific to the substructure where all the nodes of the interfaces with the adjacent substructures are blocked. A constrained mode is defined by the static deformation obtained by imposing a unit displacement on one degree of freedom of connection, the other degrees of freedom of connection being blocked. The projection base contains normal modes and all the all the constrained modes associated with all the degrees of bond freedom.

• Mac Neal’s technique consists in choosing normal modes and attachment modes as the projection base. In this case, the normal modes are modes specific to the free interface substructure. A mode of attachment is defined by the static deformation obtained by imposing a unitary force on one degree of freedom of connection, the other degrees of freedom of connection being free. The projection base contains normal modes and all attachment modes associated with all bond degrees of freedom.

• Use of interface modes: It can be seen that the dimension of the projection base can become very large if the number of degrees of freedom of connection is large. The concept of interface mode is then defined, which consists in condensing the behavior of the structure on the degrees of freedom of the interface and solving the associated eigenvalue problem. This makes it possible to express the behavior of the reduced structure at the interface. We limit ourselves to keeping as base vectors the first modes calculated in addition to the normal modes. In fact, if we remain within the framework of low-frequency dynamics, all the constrained (or attachment) modes are not necessary to represent the dynamics of the problem. However, it is recommended to check the relevance of the selected base afterwards by verifying, for example, the continuity of movements at the interfaces.

The various steps of the process can be listed below:

• For each substructure:

  • Calculation of the projection base: normal modes and link modes or interface modes

  • Calculation of generalized substructure matrices: projection of substructure mass and stiffness matrices on the reduced base

• Assembly of substructures by the formulation of the continuity of movement at the interfaces and the nullity of the work of the connecting forces. A generalized model of the assembled structure is thus obtained.

• Calculation of the response in generalized coordinates

• Restitution of results on a physical basis

• control of the results obtained

• post-treatment (stress calculation, energy calculation,…)