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Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response

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Topics in Nonlinear Dynamics, Volume 3

Abstract

The dynamic response of structural systems commonly involves nonlinear effects. Often times, structural systems are made up of several components, whose individual behavior is essentially linear compared to the total assembled system. However, the assembly of linear components using highly nonlinear connection elements or contact regions causes the entire system to become nonlinear. Conventional transient nonlinear integration of the equations of motion can be extremely computationally intensive, especially when the finite element models describing the components are very large and detailed.

In this work, the Equivalent Reduced Model Technique (ERMT) is developed to address complicated nonlinear contact problems. ERMT utilizes a highly accurate model reduction scheme, the System Equivalent Reduction Expansion Process (SEREP). Extremely reduced order models that provide dynamic characteristics of linear components, which are interconnected with highly nonlinear connection elements, are formulated with SEREP for the dynamic response evaluation using direct integration techniques. The full-space solution will be compared to the response obtained using drastically reduced models to make evident the usefulness of the technique for a variety of analytical cases.

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Abbreviations

Symbols:

 

\( [{X_n}] \) :

Full set displacement vector

\( [{X_a}] \) :

Reduced set displacement vector

\( [{X_d}] \) :

Deleted set displacement vector

\( \left[ {{M_a}} \right] \) :

Reduced mass matrix

\( \left[ {{M_n}} \right] \) :

Expanded mass matrix

\( \left[ {{K_a}} \right] \) :

Reduced stiffness matrix

\( \left[ {{K_n}} \right] \) :

Expanded stiffness matrix

\( \left[ {{U_a}} \right] \) :

Reduced set shape matrix

\( \left[ {{U_n}} \right] \) :

Full set shape matrix

\( {\left[ {{U_a}} \right]^g} \) :

Generalized inverse

\( \left[ T \right] \) :

Transformation matrix

\( \left[ {{T_U}} \right] \) :

SEREP transformation matrix

\( [p] \) :

Modal displacement vector

\( \left[ M \right] \) :

Physical mass matrix

\( \left[ C \right] \) :

Physical damping matrix

\( \left[ K \right] \) :

Physical stiffness matrix

\( [F] \) :

Physical force vector

\( [\ddot{x}] \) :

Physical acceleration vector

\( [\dot{x}] \) :

Physical velocity vector

\( [x] \) :

Physical displacement vector

\( \alpha \) :

Parameter for Newmark integration

\( \beta \) :

Parameter for Newmark integration

\( \Delta t \) :

Time step

\( \left[ {{U_{{12}}}} \right] \) :

Mode contribution matrix

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Acknowledgements

Some of the work presented herein was partially funded by Air Force Research Laboratory Award No. FA8651-10-1-0009 “Development of Dynamic Response Modeling Techniques for Linear Modal Components”. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the particular funding agency. The authors are grateful for the support obtained.

Author information

Authors and Affiliations

  1. Structural Dynamics and Acoustic Systems Laboratory, University of Massachusetts Lowell, One University Avenue, Lowell, MA, 01854, USA

    Louis Thibault & Peter Avitabile

  2. Air Force Research Laboratory, Munitions Directorate, Fuzes Branch, Eglin Air Force Base, 306 W. Eglin Blvd., Bldg 432, Eglin AFB, FL, 32542-5430, USA

    Jason R. Foley & Janet Wolfson

Authors
  1. Louis Thibault
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  2. Peter Avitabile
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  3. Jason R. Foley
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  4. Janet Wolfson
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Corresponding author

Correspondence to Louis Thibault .

Editor information

Editors and Affiliations

  1. Purdue University, West Lafayette, 47908, USA

    D. Adams

  2. University of Liege, Liege, Belgium

    G. Kerschen

  3. LMS International, Interleuvenlaan 68, Leuven, 3001, Belgium

    A. Carrella

Appendix A: Component and System Mode Shapes

Appendix A: Component and System Mode Shapes

Fig. A.1
figure a_9

Mode shapes for unmodified components A and B

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Fig. A.2
figure b_9

Mode shapes for single beam with single soft (left) and single hard (right) contact

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Fig. A.3
figure c_9

Mode shapes for two beam system for multiple soft contact configurations

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Fig. A.4
figure d_9

Mode shapes for two beam system for multiple hard contact configurations

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© 2012 The Society for Experimental Mechanics, Inc. 2012

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Thibault, L., Avitabile, P., Foley, J.R., Wolfson, J. (2012). Equivalent Reduced Model Technique Development for Nonlinear System Dynamic Response. In: Adams, D., Kerschen, G., Carrella, A. (eds) Topics in Nonlinear Dynamics, Volume 3. Conference Proceedings of the Society for Experimental Mechanics Series. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-2416-1_9

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  • DOI: https://doi.org/10.1007/978-1-4614-2416-1_9

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  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4614-2415-4

  • Online ISBN: 978-1-4614-2416-1

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