Abstract
Experimental dynamic substructures in both modal and frequency response domains using the transmission simulator method have been developed for several systems since 2007. The standard methodology couples the stiffness, mass and damping matrices of the experimental substructure to a finite element (FE) model of the remainder of the system through multi-point constraints. This can be somewhat awkward in the FE code. It is desirable to have an experimental substructure in the Craig-Bampton (CB) form to ease the implementation process, since many codes such as Nastran, ABAQUS, ANSYS and Sierra Structural Dynamics have CB as a substructure option. Many analysts are familiar with the CB form. A square transformation matrix is derived that produces a modified CB form that still requires multi-point constraints to couple to the rest of the FE model. Finally the multi-point constraints are imported to the modified CB matrices to produce substructure matrices that fit in the standard CB form. The physical boundary degrees-of-freedom (dof) of the experimental substructure matrices can be directly attached to physical dof in the remainder of the FE model. This paper derives the new experimental substructure that fits in the CB form, and presents results from an analytical and an industrial example utilizing the new CB form.
Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy National Nuclear Security Administration under Contract DE-AC04-94AL85000.
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Abbreviations
- CB:
-
Craig-Bampton method of substructuring
- CMIF:
-
Complex mode indicator function
- FE:
-
Finite element model
- FRF:
-
Frequency response function
- MCFS:
-
Method of constraint for fixture and subsystem
- MPC:
-
Multi-point constraint
- TS:
-
Transmission simulator – the fixture attached to the experimental substructure of interest
- dof:
-
Degree of freedom
- sdof:
-
Single degree of freedom
- p :
-
Modal dof of the experimental substructure with fixed boundary
- q :
-
Modal dof of free modes extracted from experimental substructure with TS attached
- s :
-
Free modal dof of the transmission simulator
- x :
-
Physical displacement dof
- ω :
-
Frequency in radians per second
- ζ :
-
Modal damping ratio
- K :
-
Stiffness matrix
- L fix :
-
Reduction matrix applying fixed boundary constraint to experimental equations of motion
- M :
-
Mass matrix
- T :
-
Transformation matrix to convert free modal model to modified CB model
- Φ:
-
Mode shape matrix extracted for experimental substructure with TS attached
- Ψ:
-
Free mode shape matrix of the TS
- Γ:
-
Eigenvectors resulting from fixed boundary constraint of experimental equations of motion
- b :
-
Subscript for the fixture or boundary
- fix :
-
Subscript for the fixed boundary modes of the experimental substructure with TS as the boundary
- free :
-
Subscript for the free modes obtained in the modal test of the experimental substructure with TS
- +:
-
Superscript indicating the Moore-Penrose pseudo-inverse of a matrix
References
Allen MS, Mayes RL, Bergman EJ (2010) Experimental modal substructuring to couple and uncouple substructures with flexible fixtures and multi-point connections. J Sound Vib 329:4891–4906
Allen MS, Kammer DC, Mayes RL (2014) Experimental based substructuring using a Craig-Bampton transmission simulator model. In: Proceedings of the 32nd international modal analysis conference, Orlando, February 2014, paper number 171
Mayes RL (2012) Refinements on estimating fixed base modes on a slip table. In: Proceedings of the 30th international modal analysis conference, Jacksonville, February 2012, paper number 162
De Klerk D, Rixen DJ, Voormeeren SN (2008) General framework for dynamic substructuring: history, review, and classification of techniques. AIAA J 46(5):1169–1181
Technical conversation with Todd Simmermacher, Sandia National Laboratories Albuquerque, Simmermacher suggested moving the multi-point constraint inside the experimental substructure, so that it could connect directly to physical dof as the normal Craig-Bampton substructure does. I pointed out that this would introduce more singularities in the CB stiffness matrix since there were more physical connection dof than generalized connection dof. Simmermacher reminded me that the FE substructure should provide adequate stiffness and mass for the physical dof so that the full system matrices would not be unduly singular. March 2014
Hensley DP, Mayes RL (2006) Extending SMAC to multiple references. In: Proceedings of the 24th international modal analysis conference, pp 220–230, St. Louis, MO, February 2006
Mayes RL, Allen MS, Kammer DC (2013) Correcting indefinite mass matrices due to substructure uncoupling. J Sound Vib 332:5856–5866. doi:10.1016/j.jsv.2013.05.025
Acknowledgments
Notice: This manuscript has been authored by Sandia Corporation under Contract No. DE-AC04-94AL85000 with the U.S. Department of Energy. The United States Government retains and the publisher, by accepting the article for publication, acknowledges that the United States Government retains a non-exclusive, paid-up, irrevocable, world-wide license to publish or reproduce the published form of this manuscript, or allow others to do so, for United States Government purposes.
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Mayes, R.L. (2015). A Craig-Bampton Experimental Dynamic Substructure Using the Transmission Simulator Method. 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/10.1007/978-3-319-15209-7_13
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DOI: https://doi.org/10.1007/978-3-319-15209-7_13
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