Four different sets of JIM will be numerical investigated, namely: x POD based on linear discretization of the joint area x POD with weighted inner product based on linear discretization of the joint area x POD based on nonlinear discretization of the joint x POD with weighted inner product based on nonlinear discretization of the joint 5.1. POD based on linear discretization of the joint area As explained in detail in [10] the joint area is subdivided into r subareas. Each subarea is loaded with a unit pressure distribution on both sides of the joint. This leads to r linear static response computations and the deformations due to this loads are collected in the (n x r) matrix Lin X . The POM are computed via the eigenvalue problem ª º O ¬ ¼ T Lin Lin Lin Lin Lin i i i X X v v for i = 1, … , r. (19) Using Lin V as the (r x k) matrix of considered POM, the according JIM can be given as the (n x k) matrix Lin Lin Lin ĭ X V . (20) 5.2. POD with weighted inner product based on linear discretization of the joint area In difference to the latter approach the stiffness matrix K is considered for the POM computation in the form of ª º O ¬ ¼ T Lin Lin Lin,K Lin,K Lin,K i i i X KX v v for i = 1, … , r. (21) Using Lin,K V as the (r x k) matrix of considered POM, the according JIM can be given as the (n x k) matrix Lin,K Lin Lin,K ĭ X V . (22) 5.3. POD based on nonlinear discretization of the joint area In contrast to the approaches before, the nonlinear contact is considered during the r static response computations and the resulting deformations are collected in the (n x r) matrix NoLin X . The POM are computed via the eigenvalue problem ª º O ¬ ¼ T NoLin NoLin NoLin NoLin NoLin i i i X X v v for i = 1, … , r. (23) Using NoLin V as the (r x k) matrix of considered POM, the according JIM can be given as the (n x k) matrix NoLin NoLin NoLin ĭ X V . (24) 5.4. POD with weighted inner product based on linear discretization of the joint area The POM are computed along ª º O ¬ ¼ T NoLin NoLin NoLin,K NoLin,K NoLin,K i i i X KX v v for i = 1, … , r. (25) Using NoLin,K V as the (r x k) matrix of considered POM, the according JIM can be given as the (n x k) matrix NoLin,K NoLin NoLin,K ĭ X V . (26) In the following, these four methods are applied at different structures and compared with respect to each other and with the former approach which based on a generalized eigenvalue problem. 6. Examples 6.1. Generic beam example 6.1.1. FE model A generic cantilever beam has been modelled according to Figure 2. The entire structure consists of two beamlike substructures, which are denoted with orange and blue colour in Figure 2. The two sub-structures are connected by two beams (Ø 8mm) which could represent two rivets or weld spots. Around the latter connectors 23
RkJQdWJsaXNoZXIy MTMzNzEzMQ==