26 Model Reduction and Lumped Models for Jointed Structures 275 Fig. 26.1 Sub-structuring: the interfaces are not the interfaces of the joints. ƒa includes the joint. More over the domain is not limited to the non-linear DoFs, but extended to the region around the joint. ƒb is assumed to behave linearly We define the Principal Joint Strain Basis as a small size basis Vin which we can decompose the local displacements ua. ua DVp (26.6) Of course ub can be decomposed on a modal basis. This leads to: ua ub D V 0 0 ˆbb p q (26.7) We assume that if two macro-models of the same joint are associated with deformations vectors which are LOCALLY orthogonal, then these two models are UNCOUPLED. Unfortunately, the modal vectors are GLOBALLY orthogonal but their restrictions to some DoF are not. Let us decompose the modal basis: ua ub D ˆaa ˆab ˆba ˆbb qa qb (26.8) Our purpose is to compute a SVD-basis from the rectangular matrixˆaa such as: ˆaa DVSTW (26.9) Thematrix Vis constituted with orthogonal vectors that span the subspace of modal deformations of the joint. The number of “important” vectors in this basis is less or equal to the number of modes in the modal basis. In this basis, as the vectors are orthogonal, we assume that non-linear forces are uncoupled, so that: f DTVF.Vp;r/ D 2 66 64 f1.p1;r1/ f2.p2;r2/ : : : fm.pm;rm/ 3 77 75 (26.10) 26.2.3 How to Build the Macro-models Associated with the PJSB The PJSB can be used as a load case in a numerical study assuming quasi-static conditions to compute the reduced order non-linear force, see Festjens et al. [18] or Caignot et al. [9]. The PJSB can also be used to design an experimental setup that allow to obtain the behavior of the joint under a load that corresponds to one of the vectors of the PJSB.
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