Determining Frequency-Energy Dependency of Nonlinear Normal Modes and Internal Resonances Using a Numerical Independent Approach

  • Arash Ghariblou Shahid Rajaee Teacher Training University, Tehran, Iran
  • Mussa Mahmoudi Shahid Rajaee Teacher Training University, Tehran, Iran

Abstract

Application of linear normal modes to the nonlinear area provides an in-depth investigation of structures. In this paper, a straightforward approach is proposed to investigate nonlinear normal modes (NNMs) thoroughly and focus on all possible solutions and bifurcations, independent of all initial assumptions and prior solutions. In this context, after discretization of the response domain over an appropriate resolution, a periodicity algorithm is suggested to capture the solutions that meet the NNMs criteria. Afterward, the frequency and energy of the system during accepted responses and degrees of freedom (DOFs)’ relations are derived. Finally, after verifying the proposed approach and acquiring new internal resonances, the frequency-energy plots and NNMs of a nonlinear elastic system with more substantial nonlinearities and a two-story steel structure with nonlinear material are studied. It is worth noting that the periodicity algorithm and capturing all possible solutions and bifurcations are among the apparent novelties of the current paper.

Keywords

Nonlinear Normal Modes, Frequency-Energy dependency, Independent Approach, Internal Resonances, Bifurcations, Nonlinear dynamic analysis,

References

1. A.C. Altunisik, E. Kalkan, F.Y. Okur, K. Ozgan, O. Karahasan, A. Bostanci, Nondestructive modal parameter identification of historical timber bridges using ambient vibration tests after restoration, Measurement (London), 146: 411–424, 2019, doi: 10.1016/j.measurement.2019.06.051.
2. B. Bhowmik, T. Tripura, B. Hazra, V. Pakrashi, Real time structural modal identification using recursive canonical correlation analysis and application towards online structural damage detection, Journal of Sound and Vibration, 468: 22 pages, 2020, doi: 10.1016/j.jsv.2019.115101.
3. E.T. Chipato, A.D. Shaw, M.I. Friswell, Nonlinear rotordynamics of a MDOF rotor–stator contact system subjected to frictional and gravitational effects, Mechanical Systems and Signal Processing, 159: 107776, 2021, doi: 10.1016/j.ymssp.2021.107776.
4. T. Chujo, O. Mori, J. Kawaguchi, Normal mode analysis of rubble-pile asteroids using a discrete element method, Icarus, 321: 458–472, 2019, doi: 10.1016/j.icarus.2018.12.011.
5. T. Detroux, L. Renson, L. Masset, G. Kerschen, The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems, Computer Methods in Applied Mechanics and Engineering, 296: 18–38, 2015, doi: 10.1016/j.cma.2015.07.017.
6. T. Dossogne et al., Nonlinear ground vibration identification of an F-16 aircraft – Part II: Understanding nonlinear behaviour in aerospace structures using sine-sweep testing, [in:] IFASD 2015 – International Forum on Aeroelasticity and Structural Dynamics, 2015.
7. E. Ellobody, Interaction of buckling modes in railway plate girder steel bridges, Thin-Walled Structures, 115: 58–75, 2017, doi: 10.1016/j.tws.2016.12.007.
8. E. Ferhatoglu, E. Cigeroglu, H.N. Özgüven, A novel modal superposition method with response dependent nonlinear modes for periodic vibration analysis of large MDOF nonlinear systems, Mechanical Systems and Signal Processing, 135: 106388, 2020, doi: 10.1016/j.ymssp.2019.106388.
9. M. Jahn, S. Tatzko, L. Panning-von Scheidt, J. Wallaschek, Comparison of different harmonic balance based methodologies for computation of nonlinear modes of nonconservative mechanical systems, Mechanical Systems and Signal Processing, 127: 159–171, 2019, doi: 10.1016/j.ymssp.2019.03.005.
10. L. Jiang, C. Liu, L. Peng, J. Yan, P. Xiang, Dynamic analysis of multi-layer beam structure of rail track system under a moving load based on mode decomposition, Journal of Vibration Engineering & Technologies, 9: 1463–1481, 2021, doi: 10.1007/s42417-021-00308-8.
11. G. Kerschen, M. Peeters, J.C. Golinval, A.F. Vakakis, Nonlinear normal modes, Part I: A useful framework for the structural dynamicist, Mechanical Systems and Signal Processing, 23(1): 170–194, 2009, doi: 10.1016/j.ymssp.2008.04.002.
12. S.E. Kim, K.W. Kang, D.H. Lee, Full-scale testing of space steel frame subjected to proportional loads, Engineering Structures, 25(1): 69–79, 2003, doi: 10.1016/S0141-0296(02)00119-0.
13. M.E. King, A.F. Vakakis, An energy-based formulation for computing nonlinear normal modes in undamped continuous systems, Journal of Vibration and Acoustics, Transactions of the ASME, 116(3): 332–340, 1994, doi: 10.1115/1.2930433.
14. M. Krack, Nonlinear modal analysis of nonconservative systems: Extension of the periodic motion concept, Computers and Structures, 154: 59–71, 2015, doi: 10.1016/j.compstruc.2015.03.008.
15. R.J. Kuether, MS. Allen, A numerical approach to directly compute nonlinear normal modes of geometrically nonlinear finite element models, Mechanical Systems and Signal Processing, 46(1): 1–15, 2014, doi: 10.1016/j.ymssp.2013.12.010.
16. W. Lacarbonara, G. Rega, A.H. Nayfeh, Resonant non-linear normal modes. Part I: Analytical treatment for structural one-dimensional systems, International Journal of Non-Linear Mechanics, 38(6): 851–872, 2003, doi: 10.1016/S0020-7462(02)00033-1.
17. S. Lotfan, Nonlinear modal interactions in a beam-mass system tuned to 3:1 and combination internal resonances based on correspondence between MTS and NSI methods, Mechanical Systems and Signal Processing, 164: 108221, 2022, doi: 10.1016/j.ymssp.2021.108221.
18. A.M. Lyapunov, The general problem of the stability of motion, International Journal of Control, 55(3): 531–534, 1992, doi: 10.1080/00207179208934253.
19. L. Meyrand, E. Sarrouy, B. Cochelin, G. Ricciardi, Nonlinear normal mode continuation through a Proper Generalized Decomposition approach with modal enrichment, Journal of Sound and Vibration, 443: 444–459, 2019, doi: 10.1016/j.jsv.2018.11.030.
20. K.J. Moore, M. Kurt, M. Eriten, D.M. McFarland, L.A. Bergman, A.F. Vakakis, Timeseries-based nonlinear system identification of strongly nonlinear attachments, Journal of Sound and Vibration, 438: 13–32, 2019, doi: 10.1016/j.jsv.2018.09.033.
21. A. Nayfeh, R. Ibrahim, Nonlinear interactions: Analytical, computational, and experimental methods, Applied Mechanics Reviews, 54(4): B60–B61, 2001, doi: 10.1115/1.1383674.
22. J.P. Nöel, L. Renson, G. Kerschen, B. Peeters, S. Manzato, J. Debille, Nonlinear dynamic analysis of an F-16 aircraft using GVT data, [in:] IFASD 2013 – International Forum on Aeroelasticity and Structural Dynamics, 2013.
23. K. Park, M.S. Allen, Quasi-static modal analysis for reduced order modeling of geometrically nonlinear structures, Journal of Sound and Vibration, 502: 116076, 2021, doi: 10.1016/j.jsv.2021.116076.
24. M. Peeters, G. Kerschen, J.C. Golinval, Modal testing of nonlinear vibrating structures based on nonlinear normal modes: Experimental demonstration, Mechanical Systems and Signal Processing, 25(4): 1227–1247, 2011, doi: 10.1016/j.ymssp.2010.11.006.
25. M. Peeters, R. Viguié, G. Sérandour, G. Kerschen, J.C. Golinval, Nonlinear normal modes, Part II: Toward a practical computation using numerical continuation techniques, Mechanical Systems and Signal Processing, 23(1): 195–216, 2009, doi: 10.1016/j.ymssp.2008.04.003.
26. L. Renson, G. Deliége, G. Kerschen, An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems, Meccanica, 49(8): 1901–1916, 2014, doi: 10.1007/s11012-014-9875-3.
27. L. Renson, G. Kerschen, B. Cochelin, Numerical computation of nonlinear normal modes in mechanical engineering, Journal of Sound and Vibration, 364: 177–206, 2016, doi: 10.1016/j.jsv.2015.09.033.
28. R.M. Rosenberg, Normal modes of nonlinear dual-mode systems, ASME Journal of Applied Mechanics, 27(2): 263–268, 1960, doi: 10.1115/1.3643948.
29. R.M. Rosenberg, On nonlinear vibrations of systems with many degrees of freedom, Advances in Applied Mechanics, 9(C): 155–242, 1966, doi: 10.1016/S0065-2156(08)70008-5.
30. S. Shaw, C. Pierre, Non-linear normal modes and invariant manifolds, Journal of Sound and Vibration, 150(1): 170–173, 1991, doi: 10.1016/0022-460X(91)90412-D.
31. S.W. Shaw, C. Pierre, Normal modes of vibration for non-linear continuous systems, Journal of Sound and Vibration, 169(3): 319–347, 1994, doi: 10.1006/jsvi.1994.1021.
32. Y. Shen, N. Béreux, A. Frangi, C. Touzé, Reduced order models for geometrically nonlinear structures: Assessment of implicit condensation in comparison with invariant manifold approach, European Journal of Mechanics – A/Solids, 86: 104165, 2021, doi: 10.1016/j.euromechsol.2020.104165.
33. S. Sikdar, W. Ostachowicz, P. Kudela, M. Radzienski, Barely visible impact damage identification in a 3D core sandwich structure, Computer Assisted Methods in Engineering and Science, 24(4): 259–268, 2017, doi: 10.24423/cames.187.
34. C.S.M. Sombroek, P. Tiso, L. Renson, G. Kerschen, Numerical computation of nonlinear normal modes in a modal derivative subspace, Computers and Structures, 195: 34–46, 2018, doi: 10.1016/j.compstruc.2017.08.016.
35. M. Song, L. Renson, J.P. Noël, B. Moaveni, G. Kerschen, Bayesian model updating of nonlinear systems using nonlinear normal modes, Structural Control Health Monitoring, 25(12): e2258, 2018, doi: 10.1002/stc.2258.
36. M. Strozzi, V.V. Smirnov, L.I. Manevitch, F. Pellicano, Nonlinear normal modes, resonances and energy exchange in single-walled carbon nanotubes, International Journal of Non-Linear Mechanics, 120: 103398, 2020, doi: 10.1016/j.ijnonlinmec.2019.103398.
37. MATLAB (R2015a), The MathWorks Inc., 2015.
38. R. Walentynski, Description of large deformations of continuum and shells and their visualisation with Mathematica, Computer Assisted Methods in Engineering and Science, 26(3–4): 191–209, 2019, doi: 10.24423/cames.272.
39. F. Wang, Bifurcations of nonlinear normal modes via the configuration domain and the time domain shooting methods, Communications in Nonlinear Science and Numerical Simulation, 20(2): 614–628, 2015, doi: 10.1016/j.cnsns.2014.06.008.
40. B. Xu et al., Nonlinear modal interaction analysis and vibration characteristics of a Francis hydro-turbine generator unit, Renewable Energy, 168: 854–864, 2021, doi: 10.1016/j.renene.2020.12.083.
41. L. Xu, Y. Hui, W. Zhu, X. Hua, Three-to-one internal resonance analysis for a suspension bridge with spatial cable through a continuum model, European Journal of Mechanics – A/Solids, 90: 104354, 2021, doi: 10.1016/j.euromechsol.2021.104354.
42. Q. Zhao, T. Han, D. Jiang, K. Yin, Application of variational mode decomposition to feature isolation and diagnosis in a wind turbine, Journal of Vibration Engineering and Technologies, 7(6): 639–646, 2019, doi: 10.1007/s42417-019-00156-7.
Published
Sep 12, 2022
How to Cite
GHARIBLOU, Arash; MAHMOUDI, Mussa. Determining Frequency-Energy Dependency of Nonlinear Normal Modes and Internal Resonances Using a Numerical Independent Approach. Computer Assisted Methods in Engineering and Science, [S.l.], v. 29, n. 3, p. 261–292, sep. 2022. ISSN 2299-3649. Available at: <https://cames.ippt.pan.pl/index.php/cames/article/view/523>. Date accessed: 08 dec. 2022. doi: http://dx.doi.org/10.24423/cames.523.
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Articles