Abstract
Thin beams and panels subjected to large loadings will behave nonlinearly due to membrane stretch effects as they approach deflections on the order of their thickness; this behavior can be efficiently and accurately modeled using nonlinear reduced order models based on the structure’s linear normal modes. However, the complexity of such reduced order models grows cubically with the number of linear modes in the basis set, making complicated geometries prohibitively expensive to compute. Component mode synthesis techniques may be used to reduce this cost by assembling a set of smaller nonlinear subcomponent models, each of which can be more quickly computed than a nonlinear model of the entire structure. Since geometric nonlinearity is heavily dependent on each structure’s boundary conditions, however, subcomponents of an assembly which are constrained only at their interfaces – such as panels mounted to an underlying frame – prove difficult to treat using existing nonlinear modeling techniques. This work uses Craig-Bampton dynamic substructuring combined with characteristic constraint modes for interface reduction to examine the challenges associated with panel and frame assemblies, with a simple example motivating a discussion of current solutions and future challenges.
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Notes
- 1.
If the \(\boldsymbol{\Phi }^{\,j}\) is orthogonal through the mass matrix M j, then \(\mathbf{q}^{\,j} = (\boldsymbol{\Phi }^{\,j})^{T}\mathbf{M}^{\,j}\mathbf{Y}\). This is not the case when a Craig-Bampton transformation is used.
- 2.
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Acknowledgements
This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1256259. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the authors(s) and do not necessarily reflect the views of the National Science Foundation.
Support was also provided by the Graduate School and the Office of the Vice Chancellor for Research and Graduate Education at the University of Wisconsin – Madison with funding from the Wisconsin Alumni Research Foundation.
The authors also acknowledge Joseph Hollkamp from the Air Force Research Laboratory’s Structural Sciences Center, for the insights that he shared during the conduct of this and other work.
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Appendix
Appendix
Various details related to the linear models of the substructure components are given below. Retained fixed interface modes of the panel are given in Table 27.5.
Retained fixed interface modes of the hat stiffeners for soft and stiff assemblies are shown in Table 27.6.
A modal assurance criterion (MAC) check, along with corresponding frequency errors, between the substructured model and its full-order FEA counterpart is given in Fig. 27.8.
(Top): Cross-MAC between full-order FEA truth modes and substructure modes. (Bottom): Frequency errors between truth and substructure modes, based on matching MAC value; frequency values along the horizontal axis are full-order values while those adjacent to error bars are the substructured values
Nonlinear normal modes generated using various NLROMs of each assembly, along with the resultant periodicity errors from each model, are shown in Figs. 27.9 and 27.10 for the stiff and soft assemblies, respectively.
(Top) NNMs computed using one, three, and five-mode ROMs of a fully assembled FEA model with “stiff” supports. (Bottom) Periodicity errors associated with each computed backbone curve
(Top) NNMs computed using one, three, and five-mode ROMs of a fully assembled FEA model with “soft” supports. (Bottom) Periodicity errors associated with each computed backbone curve
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Schoneman, J.D., Allen, M.S., Kuether, R.J. (2017). Nonlinear Modal Substructuring of Panel and Stiffener Assemblies via Characteristic Constraint Modes. In: Allen, M., Mayes, R., Rixen, D. (eds) Dynamics of Coupled Structures, Volume 4. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, Cham. https://doi.org/10.1007/978-3-319-54930-9_27
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