Abstract
Recently a time-domain substructuring method was proposed based on the assembly of series of impulse responses of components: the Impulse Base Substructuring (IBS). Although theoretically the IBS is the time-domain equivalent to the Frequency Based Substrcutring method (FBS), it has several advantages when computing shock responses for instance. However a major drawback of the IBS is the rapid increase of computational costs when the simulated time increases. In this contribution we propose a truncation and windowing procedure in order to limit the cost involved by the discretized convolution product inherent to the IBS method. We describe how to truncate the Impulse Response Functions of floating and non-floating substructures, and describe a cosine windowing function to improve the accuracy and stability of the dynamic response obtained from the superposition of truncated impulse responses. A simple bar example is used to illustrate the numerical performance of the truncated IBS.
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Notes
- 1.
In the time-discrete form λ are chosen to represent interface impulses.
- 2.
In case of substructures, one must also add the interface forces \( {\mathbf{{\mathit{B}}}^{{(s)}^{T}}}\lambda \) in these equations. Nevertheless, to simplify the text, we will omit the interface forces in this section.
- 3.
In our work the oscillatory amplitude is detected by evaluating the peak values of the IRFs.
- 4.
This assumption means that the damping in the (sub)structure is internal and does not produce any damping force when no deformations are present.
- 5.
Let us note that since the impulse response for a linear dynamic system can be seen as the superposition of its impulse modal responses, an equivalent way to compute the vibrational part of the IRF is to project the IRF M-orthogonal to the rigid body modes, namely
$$ \mathbf{{\mathit{H}}}^{vib} = (\mathbf{{\mathit{I}}} -\mathbf{{\mathit{R}}} \mathbf{{\mathit{M}}}_{tot}^{-1} \mathbf{{\mathit{R}}}^{T})\mathbf{{\mathit{H}}} $$and the rigid part is
$${{{\mathit{\mathbf{H}}}}}^{rig} = {{\mathit{\mathbf{R}}}}{{\mathit{\mathbf{M}}}}_{tot}^{-1}{{\mathit{\mathbf{R}}}}^{T}{\mathit{\mathbf{H}}}$$
References
Klerk DD, Rixen DJ, Voormeeren SN (2008) General framework for dynamic substructuring: history, review and classification of techniques. AIAA J 46(5):1169–1181
Rixen DJ (2010) Substructuring technique based on measured and computed impulse response functions of components. In: Sas P, Bergen B (eds) ISMA, KU Leuven
Rixen DJ (2010) Substructuring using impulse response functions for impact analysis. In: IMAC-XXVIII: international modal analysis conference, Jacksonville, Society for experimental mechanics, Bethel
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Rixen, D., Haghighat, N. (2012). Truncating the Impulse Responses of Substructures to Speed Up the Impulse-Based Substructuring. In: Mayes, R., et al. Topics in Experimental Dynamics Substructuring and Wind Turbine Dynamics, Volume 2. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-2422-2_14
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DOI: https://doi.org/10.1007/978-1-4614-2422-2_14
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