12 O. M. Zobel et al. Theory of Impulse-Based Substructuring Here, only a short summary of the dually assembled Impulse-Based Substructuring (IBS) theory proposed by RIXEN in 2010 in [2] is given. Further details can be found there or in the authors’ previous work, ‘Enabling Experimental Impulse-Based Substructuring through Time Domain Deconvolution and Downsampling’ [4]. The time-continuous IBS method is given by: Definition Impulse-Based Substructuring Method[2] y(s)(t)=Z t 0 H(s)(t −τ) f(s)(τ)+B (s) f T λf(τ) dτ Equations of Motion NSX s=1 B(s) y y (s)(t)=0 Compatibility (1) where: y(t) System response H(t) Matrix of impulse response functions (s) Substructure index f(t) Externally applied forces B Signed Boolean constraint matrix ∗f Force mapping NS Number of substructures λ(t) Global vector of Lagrange multipliers ∗y Response mapping These equations must be discretized in time, and then a time-marching scheme over the discrete time steps k has to be evaluated. The aforementioned scheme first calculates a prediction of the system responses using a convolution product between the impulse response functions (IRFs) and the applied forces while neglecting the compatibility condition for the current time stepk. Then, based on the existing interface gap, Lagrange multipliers λare calculated that, for a linear system, exactly close the gap when applied to the system in the corrector step [2, 4]. This is summarized in fig. 1. Predictor Step Without λ[k−1] y[k] = ˜y(s)[k] Calculate λ[k−1] to Close Gaps g[k] ! =0 Corrector Step for λ[k−1] y[k]+=y (s) λ [k] Calculation of Gaps at Time k g[k] = NSX s=1 B(s) ˜y(s)[k] Increment Discrete Time Stepk Fig. 1 Flowchart of IBS scheme calculations within each time step The IRFs are determined from impact measurements using the time domain convolution procedure proposed by [6]. This procedure allows averaging of multiple impacts and essentially is the time domain counterpart of the H1 FRF estimator. Depending on the excitation bandwidth compared to the full measurement bandwidth, the system might not be properly excited at all frequencies, see fig. 2. Contrary to FBS, where the higher frequencies can even be omitted after the substructuring procedure, for IBS, the improperly excited frequency content must be removed before the IRF calculation. The authors proposed in [4] to do this with downsampling after applying an appropriate low-pass filter, limiting the considered bandwidth. Virtual Point Transformation for IBS For three-dimensional structures, coupling two triaxial sensors on the interface, one on each substructure, is not sufficient as this does not account for the rotational dofs and only couples the three measured translations, while in reality, the interface connection is a line or surface contact, rather than a point contact. Measuring additional rotational dofs is very difficult because the required torque excitation and rotational sensors are not standard measurement equipment [7]. Instead, it was proposed by DE KLERK ET AL. in [7] to determine the rotational dofs by combining multiple triaxial sensors. Nevertheless, this approach was found to be non-ideal, as the interface problem could be overdetermined, and measurement errors can lead to unwanted stiffening and spurious peaks in the coupled FRFs.
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