26 Model Reduction and Lumped Models for Jointed Structures 277 Fig. 26.4 1D Finite Element Model of a jointed structure, see [14] Bernoulli formulation. Thejth element has two transverse displacement DoFs denotedu j andujC1 and two rotational DoFs denoted j and jC1. Non-linearities in real joints have a local extent and thus contains both non-linear as well as linear elements. The fifth element of the whole structure is fitted with a nonlinear hysteretic model. This model, referred to as an Adjusted Iwan Beam Element (AIBE), see [10], rely on the reduction of the 2D linear elastic beam element to rigid bars and linear springs. The adjusted Iwan beam element is obtained by replacing these two springs with one-dimensional hysteretic models. The model is first reduced using a modal basis. The obtained ROM is called ROM1. The model is secondly reduced using the PJSB. The obtained ROM is called ROM2. To evaluate the accuracy of both model reduction methods, the following load case is studied on the 1D-model. A sinusoidal force of frequency close to the one of the second mode is applied until the steady state regime is reached; Then a wavelet excitation is applied to excite the first mode and the unstationary response is observed. The response of the structure is multi-modal. This load case highlights the coupling effects that may occur in jointed structures. On the one hand, the global amplitude of ROM1 decreases faster than the Full Order Model. This ROM1 stands for the situation where no coupling effect is considered and is not able to take into account the coupling effect due to non-linear damping predicted by the full order model. On the contrary, ROM2 is able to take into account this effect. ROM1 may strongly underestimate the actual amplitude of vibration. In a conception process, this situation can lead to an underdimensioning of the structure. The presented load case may correspond to excessively severe conditions, but in general, the actual dissipation generated by joint depends on the real amplitude. As a the projection on a global Ritz basis is not suitable contrary to the use of a local basis (Fig. 26.5). 26.4 Conclusions This paper provides a pragmatic and accurate way for reducing the models of jointed structures. The method is founded on the intuition that the modes of a “local” basis are adequate to generate the principal movements that the joints may carry out under low frequency dynamic excitations. The proposed formulation couples together the modes of the whole structure. It also keeps a geometrical meaning and thus the reduced order model is able to take into account the local damping as well as the softening effect induced by the joints. Finally the joints macro-models are directly updated from experimental data. As a result, the obtained models are very light but nevertheless accurate.
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