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Substructuring in Engineering Dynamics

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 594))

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Abstract

This section highlights some specialized substructuring methods, such as methods for estimating the fixed-interface modes of a substructure from measurements of the free–free structure with a fixture attached at the interface, and also highlights some industrial examples. Transfer path analysis is reviewed, elaborating some of the similarities in the theoretical foundations. Most information about TPA has been obtained from van der Seijs et al. (2016), please refer to the original paper for a more elaborate discussion. — Chapter Authors: Maarten van der Seijs, Randall Mayes and Matt Allen.

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Notes

  1. 1.

    The signed Boolean matrix \(\mathbf {B}\) establishes the relations for all interface DoFs of A and B that are vectorially associated, e.g. \(u^\text {A}_{2x}\) and \(u^\text {B}_{2x}\). Guidelines on the construction and properties of the signed Boolean matrix are found in de Klerk et al. (2008).

  2. 2.

    In this framework, the resulting responses \(\mathbf {u}_3\) are formulated as a matrix-vector product, namely the sum of the partial responses. Techniques to evaluate the individual transfer paths contributions are discussed in van der Seijs et al. (2016).

  3. 3.

    Note that the terms in the dynamic stiffness matrix \(\mathbf {Z}^\text {mt}\) correspond to differential displacements of the associated interface DoFs \(\mathbf {u}^\text {A}_2 - \mathbf {u}^\text {B}_2\) and not the coordinates of both A and B. Some implications for the terms in \(\mathbf {Z}^\text {mt}\) are discussed in Voormeeren (2012), Barten et al. (2014).

  4. 4.

    If the admittances of the subsystems are available separately, one may also apply the free velocities directly, as shown in van Schothorst et al. (2012), Rixen et al. (2015).

  5. 5.

    A intuitive presentation of the range between the two limit cases is given in Moorhouse (2001).

  6. 6.

    This is standard practice for FEM assembly.

  7. 7.

    Similar ideas are used in the field of experimental substructure decoupling: the identification of the force that decouples the residual substructure can be improved by defining an “extended interface”, adding some additional DoFs on the structure of interest distant from the interface (Sjövall and Abrahamsson 2008; D’Ambrogio and Fregolent 2010; Voormeeren and Rixen 2012).

  8. 8.

    This was erroneously stated in the original work (Janssens and Verheij 2000).

  9. 9.

    An example with two connection points was chosen here merely to provide better insight into some important cross-correlation properties. There is no fundamental consequence for the generality of the methods derived.

  10. 10.

    This decorrelation has been approached in several ways: for instance, based on so-called Global and Direct Transmissibility Functions (GTDT) in the work of Magrans (1981), Guasch and Magrans (2004); Guasch (2009); Guasch et al. (2013) or along conventional FRFs as seen in Varoto and McConnell (1998), Ribeiro et al. (1999, 2000), Maia et al. (2001).

  11. 11.

    In Roozen and Leclère (2013), it is suggested that external excitation on the source structure, e.g. by use of a non-instrumented hammer, leads to better conditioning of the transmissibility matrix compared to a sequence of operational excitations. This resembles the approach depicted by Fig. 5.7b.

  12. 12.

    The \(H_1\) estimator is a well-known principle in experimental modal analysis to determine FRFs from a Multiple Input–Output (MIMO) test, see for instance, Bendat and Piersol (1980), Ewins (2000). Alternative ways to obtain the transmissibility matrix have recently been explored, such as the \(H_2\) or \(H_s\) estimator to balance the error contributions between the inputs and outputs (Leclère et al. 2012).

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Authors and Affiliations

  1. Department of Engineering Physics, University of Wisconsin–Madison, Madison, WI, USA

    Matthew S. Allen

  2. Lehrstuhl für Angewandte Mechanik, Technische Universität München, Munich, Bayern, Germany

    Daniel Rixen

  3. Vibes Technology, Delft, The Netherlands

    Maarten van der Seijs

  4. D-MAVT, ETH Zürich, Zürich, Switzerland

    Paolo Tiso

  5. Chalmers University of Technology, Göteborg, Sweden

    Thomas Abrahamsson

  6. Sandia National Laboratories, Albuquerque, NM, USA

    Randall L. Mayes

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  1. Matthew S. Allen
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  2. Daniel Rixen
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  3. Maarten van der Seijs
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  4. Paolo Tiso
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  5. Thomas Abrahamsson
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  6. Randall L. Mayes
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Correspondence to Matthew S. Allen .

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Allen, M.S., Rixen, D., van der Seijs, M., Tiso, P., Abrahamsson, T., Mayes, R.L. (2020). Industrial Applications & Related Concepts. In: Substructuring in Engineering Dynamics. CISM International Centre for Mechanical Sciences, vol 594. Springer, Cham. https://doi.org/10.1007/978-3-030-25532-9_5

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