Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-02T23:04:16.507Z Has data issue: false hasContentIssue false
  • English
  • Français

Model reduction strategies for nonlinear beams subjected to large rotary actuations

Published online by Cambridge University Press:  03 February 2016

B. Stanford ,
P. Beran  and
M. Kurdi
B. Stanford
Affiliation:
[email protected], Air Vehicles Directorate, Air Force Research Laboratory, Wright Patterson AFB, USA
P. Beran
Affiliation:
[email protected], Air Vehicles Directorate, Air Force Research Laboratory, Wright Patterson AFB, USA
M. Kurdi
Affiliation:
[email protected], Air Vehicles Directorate, Air Force Research Laboratory, Wright Patterson AFB, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

The solution to nonlinear structural dynamics problems with time marching schemes can be very expensive, particularly if the desired time-periodic response takes many cycles to form. Two cost reduction methods, which need not be considered separately, are formulated in this work. The first projects the nonlinear system of equations onto a reduced basis defined by a set of modes computed with proper orthogonal decomposition. The second utilises a monolithic time spectral element method, whereby the system of ordinary differential equations is converted into a single algebraic system of equations. The spectral element method can be formulated such that only the time-periodic response is computed. These techniques are implemented for a planar elastic beam, actuated at its base to emulate a flapping motion. Nonlinear elastic terms are computed with a corotational finite element method, while inertial terms are computed with a standard multibody dynamics formulation. For a variety of actuation frequencies and kinematic motions, results are given in terms of POD modes, reduced order model accuracy, and computational cost, for both the time marching and the monolithic time schemes.

Type
Research Article
Information
The Aeronautical Journal , Volume 113 , Issue 1150: Flight structures fundamental research in the United States of America: A Special Issue – PART II , December 2009 , pp. 751 - 762
Copyright
Copyright © Royal Aeronautical Society 2009 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Wasfy, T. and Noor, A., Computational strategies for flexible multibody systems, Applied Mechanics Reviews, 2003, 56, (6), pp 553–613.Google Scholar
2. Wang, D. and Meng, M., Frequency analysis of a flexible robot manipulator, IEEE Conference on Electrical and Computer Engineering, Edmonton, Alberta, Canada, 9-12 May, 1999.Google Scholar
3. Murtagh, P., Basu, B. and Broderick, B., Along-wind response of a wind turbine tower with blade coupling subjected to rotationally sampled wind loading, Engineering Structures, 27, (8), pp 1209–1219, 2005.Google Scholar
4. Barut, A., Das, M. and Madenci, E., Nonlinear deformations of flapping wings on a micro air vehicle, AIAA Structures, Structural Dynamics, and Materials Conference, Newport, RI, USA, 1-4 May, 2006.Google Scholar
5. Madenci, E. and Barut, A., dynamic response of thin composite shells experiencing nonlinear elastic deformations coupled with large and rapid overall motions, Int J for Numerical Methods in Engineering, 1998, 39, (16), pp 2695–2723.Google Scholar
6. Cook, R., Malkus, D., Plesha, M. and Witt, R., Concepts and Applications of Finite Element Analysis, Wiley, New York, USA, 2002.Google Scholar
7. Venkataraman, S. and Haftka, R., Structural optimization complexity: What has Moore’s Law done for us? Structural and Multidisciplinary Optimization, 2004, 28, (6), pp 375–387.Google Scholar
8. Trier, S., Marthinsen, A. and Sivertsen, O., Design sensitivities by the adjoint variable method in nonlinear structural dynamics, SIMS Simulation Conference, Trondheim, Norway, June 11-13, 1996.Google Scholar
9. Lucia, D., Beran, P. and Silva, W., Reduced-order modelling: new approaches for computational physics, Progress in Aerospace Sciences, 2004, 40, (1), pp 51–117.Google Scholar
10. Kerschen, G., Golinval, J., Vakakis, A. and Bergman, L., The method of proper orthogonal decomposition for dynamical characterization and order reduction of mechanical systems: an overview, Nonlinear Dynamics, 2005, 41, (1), pp 147–169.Google Scholar
11. Weickum, G., Eldred, M. and Maute, K., Multi-point extended reduced order modelling for design optimization and uncertainty analysis, AIAA Structures, Structural Dynamics, and Materials Conference, Newport, RI, USA, May 1-4, 2006.Google Scholar
12. Krysl, P., Lall, S. and Marsden, J., Dimensional model reduction in nonlinear finite element dynamics of solids and structures, Int J Numerical Methods in Engineering, 2001, 51, (4), pp 479–504.Google Scholar
13. Amsallem, D. and Farhat, C., Interpolation method for adapting reduced-order models and application to aeroelasticity, AIAA J, 2008, 46, (7), pp 1803–1813.Google Scholar
14. Sen, S., Veroy, K., Juynh, D., Deparis, S., Nguyen, N. and Patera, A., Natural norm a posteriori error estimators for reduced basis approximations, J Computational Physics, 2006, 217, (1), pp 37–62.Google Scholar
15. Spiess, H. and Wriggers, P., Reduction Methods for FE Analysis in Nonlinear Structural Dynamics, Proceedings in Applied Mathematics and Mechanics, 5, (1), pp 135–136, 2005.Google Scholar
16. Meyer, M. and Matthies, H., Efficient model reduction in non-linear dynamics using the Karhunen-Loève expansion and dual-weighted-residual methods, Computational Mechanics, 2003, 31, (1), pp 179–191.Google Scholar
17. Slaats, P., De Jongh, J. and Sauren, A., Model reduction tools for nonlinear structural dynamics, Computers and Structures, 1995, 54, (6), pp 1155–1171.Google Scholar
18. Ma, X., Vakakis, A. and Bergman, L., Karhunen-Loève modes of a truss: Transient response reconstruction and experimental verification, AIAA J, 2001, 39, (4), pp 687–696.Google Scholar
19. Steindl, A. and Troger, H., Methods for dimensional reduction and their application in nonlinear dynamics, Int J Solids and Structures, 2001, 38, (10), pp 2131–2147.Google Scholar
20. Lenci, S. and Rega, G., Dimension reduction of homoclinic orbits of buckled beams via the nonlinear normal modes technique, Int J Nonlinear Mechanics, 2007, 42, (3), pp 515–528.Google Scholar
21. Kurdi, M. and Beran, P., Spectral element method in time for rapidly actuated systems, J Computational Physics, 227, (3), pp 1809–1835, 2008.Google Scholar
22. Lau, S. and Zhang, W., Nonlinear vibrations of Piecewise-Linear systems by incremental harmonic balance method, J Applied Mechanics, 1992, 59, (1), pp 153–160.Google Scholar
23. Chen, S., Cheung, Y. and Xing, H., Nonlinear vibration of plane structures by finite element and incremental harmonic balance method, Nonlinear Dynamics, 2001, 26, (1), pp 87–104.Google Scholar
24. Zhou, J. and Zhang, L., Incremental harmonic balance method for predicting amplitudes of a multi-D.O.F. Nonlinear wheel shimmy system with combined coulomb and quadratic damping, J Sound and Vibration, 2005, 279, (1), pp 403–416.Google Scholar
25. Beran, P., Parker, G., Snyder, R. and Blair, M., Design Analysis Strategies for Flapping Wing Micro Air Vehicles, International Forum on Aeroelasticity and Structural Dynamics, Stockholm, Sweden, 18-20 June, 2007.Google Scholar
26. Gopinath, A., Beran, P. and Jameson, A., Comparative Analysis of Computational Methods for Limit-Cycle Oscillations, AIAA Structures, Structural Dynamics, and Materials Conference, Newport, RI, 1-4 May, 2006.Google Scholar
27. Bar-Yoseph, P., Fisher, D. and Gottlieb, O., Spectral element methods for nonlinear spatio-temporal dynamics of an Euler-bernoulli beam, Computational Mechanics, 1996, 19, (2), pp 136–151.Google Scholar
28. Mahapatra, D. and Gopalakrishnan, S., A spectral finite element model for analysis of axial-Flexural-Shear coupled wave propagation in laminated composite beams, Composite Structures, 2003, 59, (1), pp 67–88.Google Scholar
29. Krawczuk, M., Palacz, M. and Ostachowicz, W., The dynamic analysis of a cracked Timoshenko beam by the spectral element method, J Sound and Vibration, 2003, 264, (5), pp 1139–1153.Google Scholar
30. Rankin, C. and Brogan, F., An element independent corotational procedure for the treatment of large rotations, J Pressure Vessel Technology, 1986, 108, (2), pp 165–174.Google Scholar
31. Nour-Omid, B. and Rankin, C., Finite rotation analysis and consistent linearization using projection, Computer Methods in Applied Mech and Eng, 1991, 93, (5), pp 353–384.Google Scholar
32. Galvanetto, U. and Crisfield, M., An energy-conserving corotational procedure for the dynamics of planar beam structures, Int J Numerical Methods in Eng, 1998, 39, (13), pp 2265–2282.Google Scholar
33. Elkaranshawy, H. and Dokainish, M., Corotational finite element analysis of planar flexible multibody systems, Computers and Structures, 1995, 54, (5), pp 881–890.Google Scholar
34. Belytschko, T. and Schoeberle, D., On the unconditional stability of an implicit algorithm for nonlinear structural dynamics, J Applied Mechanics, 1975, 42, (4), pp 865–869.Google Scholar
35. Pozrikidis, C., Introduction to Finite and Spectral Element Methods Using Matlab, CRC Press, Boca Raton, 2005.Google Scholar
36. Coelho, R., Breitkopf, P. and Knopf-Lenoir, C., Model reduction for multidisciplinary analysis – Application to a 2D wing, Structural and Multidisciplinary Analysis, 2008, 37, 1, pp 29–48.Google Scholar
37. Yoo, H. and Chung, J., Dynamics of rectangular plates undergoing prescribed overall motion, J Sound and Vibration, 2001, 239, (1), pp 123–137.Google Scholar
38. Feeny, B. and Kappagantu, R., On the physical interpretation of proper Orthogonal Modes in Vibration, J Sound and Vibration, 1998, 211, (4), pp 607–616.Google Scholar
39. Bendat, J. and Piersol, A., Random Data: Analysis and Measurement Procedures, Wiley, New York, USA, 2000.Google Scholar
40. Lau, S., Cheung, Y. and Wu, S., Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear systems, J Applied Mechanics, 1983, 50, (4), pp 871–876.Google Scholar