Skip to main content

Advertisement

SpringerLink
Log in

Nonlinear model updating of frictional structures through frequency–energy analysis

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Recently, some researchers have suggested using frequency–energy representations of nonlinear normal modes (NNMs) for nonlinear model updating of conservative systems. The current work investigates this idea for frictional structures. The studied structure undergoes friction under variable normal contact force. Based on the invariance principle, the NNM-branch extension of the experimental structure’s first linear mode was obtained in a new way: The frequency–energy plot (FEP) was approximated by measuring the instantaneous frequencies and total energy of the system’s nonlinear decaying response. The wavelet-bounded empirical mode decomposition (WBEMD) method was used to obtain the first isolated frequency component of the experimental signals. On the other hand, theoretical FEPs were computed for every candidate nonlinear model’s equivalent conservative system. To do this, the extended periodic motion concept (EPMC) was used while introducing the multi-harmonic balance (MHB) equations into a continuation algorithm. The proposed model updating method considers the overlapping of such FEPs as a measure of similarity between dynamic behaviors of design model and the real structure. Finally, after performing sensitivity analysis for the named criterion, the updated nonlinear model was verified against the experimental data. This survey concludes that the new technique is capable of successfully updating the nonlinear finite element model of the structures that contain frictional contact undergoing variable normal force. Therefore, it can be used for bladed-disk systems, in which frictional contacts serve as energy dissipators. The proposed method’s applicability is restricted to a certain band of system’s mechanical energy starting from linear regimes of motion. By increasing this energy level, the occurrence of first strong modal interactions would prevent the decomposition of response to isolated frequency components.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig.17

Similar content being viewed by others

Data availability

The experimental data that support the findings of this study are available from Mr. Amir Golestaneh, but restrictions apply to the availability of these data, which were used under license for the current study, and so are not publicly available. Data are, however, available from the authors upon reasonable request and with permission from Mr. Amir Golestaneh.

References

  1. Kerschen, G., Worden, K., Vakakis, A.F., Golinval, J.-C.: Past, present and future of nonlinear system identification in structural dynamics. Mech. Syst. Signal Process. 20, 505–592 (2006). https://doi.org/10.1016/j.ymssp.2005.04.008

    Article  Google Scholar 

  2. Noël, J.P., Kerschen, G.: Nonlinear system identification in structural dynamics: 10 more years of progress. Mech. Syst. Signal Process. 83, 2–35 (2017). https://doi.org/10.1016/j.ymssp.2016.07.020

    Article  Google Scholar 

  3. Gaul, L., Lenz, J.: Nonlinear dynamics of structures assembled by bolted joints. Acta Mech. 125, 169–181 (1997). https://doi.org/10.1007/BF01177306

    Article  MATH  Google Scholar 

  4. Valanis, K.C.: Fundamental consequences of a new intrinsic time measure. Plasticity as a limit of the endochronic theory. Arch. Mech. 32, 171–191 (1980)

    MathSciNet  MATH  Google Scholar 

  5. Abad, J., Medel, F.J., Franco, J.M.: Determination of Valanis model parameters in a bolted lap joint: experimental and numerical analyses of frictional dissipation. Int. J. Mech. Sci. 89, 289–298 (2014). https://doi.org/10.1016/j.ijmecsci.2014.09.014

    Article  Google Scholar 

  6. Renson, L.: Nonlinear modal analysis of conservative and nonconservative aerospace structures. Universite de Liege. Doctoral Thesis (2014)

  7. Rosenberg, R.M.: On nonlinear vibrations of systems with many degrees of freedom. Adv. Appl. Mech. 9, 155–242 (1966). https://doi.org/10.1016/S0065-2156(08)70008-5

    Article  Google Scholar 

  8. Peeters, M., Viguié, R., Sérandour, G., Kerschen, G., Golinval, J.C.: Nonlinear normal modes, part II: toward a practical computation using numerical continuation techniques. Mech. Syst. Signal Process. 23, 195–216 (2009). https://doi.org/10.1016/j.ymssp.2008.04.003

    Article  Google Scholar 

  9. Shaw, S.W., Pierre, C.: Non-linear normal modes and invariant manifolds. J. Sound Vib. 150, 170–173 (1991). https://doi.org/10.1016/0022-460X(91)90412-D

    Article  Google Scholar 

  10. Kuether, R.J., Renson, L., Detroux, T., Grappasonni, C., Kerschen, G., Allen, M.S.: Nonlinear normal modes, modal interactions and isolated resonance curves. J. Sound Vib. 351, 299–310 (2015). https://doi.org/10.1016/j.jsv.2015.04.035

    Article  Google Scholar 

  11. Ehrhardt, D.A., Kuether, R.J., Allen, M.S.: Nonlinear normal modes in finite element model validation of geometrically nonlinear flat and curved beams. In: 56th AIAA/ASCE/AHS/ASC Struct. Struct. Dyn. Mater. Conf., pp. 1–12 (2015). https://doi.org/10.2514/6.2015-0689

  12. Kerschen, G., Peeters, M., Golinval, J.C., Stéphan, C.: Nonlinear modal analysis of a full-scale aircraft. J. Aircr. 50, 1409–1419 (2013). https://doi.org/10.2514/1.C031918

    Article  Google Scholar 

  13. Craig, R.R., Bampton, M.C.C.: Coupling of substructures for dynamic analyses. AIAA J. 6, 1313–1319 (1968). https://doi.org/10.2514/3.4741

    Article  MATH  Google Scholar 

  14. Renson, L., Noël, J.P., Kerschen, G.: Complex dynamics of a nonlinear aerospace structure: numerical continuation and normal modes. Nonlinear Dyn. 79, 1293–1309 (2014). https://doi.org/10.1007/s11071-014-1743-0

    Article  Google Scholar 

  15. Kurt, M., Eriten, M., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Methodology for model updating of mechanical components with local nonlinearities. J. Sound Vib. 357, 331–348 (2015). https://doi.org/10.1016/j.jsv.2015.07.012

    Article  Google Scholar 

  16. Kurt, M., Moore, K.J., Eriten, M., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Nonlinear model updating applied to the IMAC XXXII Round Robin benchmark system. Mech. Syst. Signal Process. 88, 111–122 (2017). https://doi.org/10.1016/j.ymssp.2016.10.016

    Article  Google Scholar 

  17. Van Damme, C.I., Alleny, M.S., Hollkampz, J.J.: Nonlinear structural model updating based upon nonlinear normal modes. In: AIAA/ASCE/AHS/ASC Struct. Struct. Dyn. Mater. Conf., Kissimmee, Florida, pp. 1–17 (2018). https://doi.org/10.2514/6.2018-0185

  18. Moore, K.J., Kurt, M., Eriten, M., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Time-series-based nonlinear system identification of strongly nonlinear attachments. J. Sound Vib. 438, 13–32 (2019). https://doi.org/10.1016/j.jsv.2018.09.033

    Article  Google Scholar 

  19. Delli Carri, A., Ewins, D.J.: Nonlinear identification of a numerical benchmark structure: from measurements to FE model updating. In: Proc. ISMA 2014—Int. Conf. Noise Vib. Eng. USD 2014—Int. Conf. Uncertain. Struct. Dyn., pp. 2437–2452. KU Leuven (2014)

  20. Peter, S., Grundler, A., Reuss, P., Gaul, L., Leine, R.I.: Towards finite element model updating based on nonlinear normal modes. In: IMAC, A Conf. Expo. Struct. Dyn., Orlando, pp. 209–217 (2015). https://doi.org/10.1007/978-3-319-15221-9_20

  21. Peeters, M., Kerschen, G., Golinval, J.C.: Modal testing of nonlinear vibrating structures based on nonlinear normal modes: experimental demonstration. Mech. Syst. Signal Process. 25, 1227–1247 (2011). https://doi.org/10.1016/j.ymssp.2010.11.006

    Article  Google Scholar 

  22. Peeters, M., Kerschen, G., Golinval, J.C.: Dynamic testing of nonlinear vibrating structures using nonlinear normal modes. J. Sound Vib. 330, 486–509 (2011). https://doi.org/10.1016/j.jsv.2010.08.028

    Article  Google Scholar 

  23. Ahi, M.: Nonlinear model updating of frictional structures in presence of variable-normal-load, through frequency-energy analysis. Iran University of Science and Technology (IUST). MSc. Thesis in Farsi (2019)

  24. Londoño, J.M., Cooper, J.E., Neild, S.A.: Identification of systems containing nonlinear stiffnesses using backbone curves. Mech. Syst. Signal Process. 84, 116–135 (2017). https://doi.org/10.1016/j.ymssp.2016.02.008

    Article  Google Scholar 

  25. Peter, S., Leine, R.I.: Excitation power quantities in phase resonance testing of nonlinear systems with phase-locked-loop excitation. Mech. Syst. Signal Process. 96, 139–158 (2017). https://doi.org/10.1016/j.ymssp.2017.04.011

    Article  Google Scholar 

  26. Peter, S., Schreyer, F., Leine, R.I.: A method for numerical and experimental nonlinear modal analysis of nonsmooth systems. Mech. Syst. Signal Process. 120, 793–807 (2019). https://doi.org/10.1016/j.ymssp.2018.11.009

    Article  Google Scholar 

  27. Moore, K.J., Kurt, M., Eriten, M., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Wavelet-bounded empirical mode decomposition for measured time series analysis. Mech. Syst. Signal Process. 99, 14–29 (2018). https://doi.org/10.1016/j.ymssp.2017.06.005

    Article  Google Scholar 

  28. Flandrin, P., Rilling, G.: http://perso.ens-lyon.fr/patrick.flandrin/emd.html (2007)

  29. Deering, R., Kaiser, J.F.: The use of a masking signal to improve empirical mode decomposition. In: Int. Conf. Acoust. Speech, Signal Process., pp. 485–488 (2005)

  30. Wang, X.: Construction of frequency-energy plots for nonlinear dynamical systems from time-series data. University of Illinois. MSc. Thesis (2010)

  31. Petrov, E.P., Ewins, D.J.: Analytical formulation of friction interface elements for analysis of nonlinear multi-harmonic vibrations of bladed disks. J. Turbomach. 125, 364–371 (2003). https://doi.org/10.1115/1.1539868

    Article  Google Scholar 

  32. Neglia, S.G., Culla, A., Fregolent, A.: Non-linear dynamics of jointed systems under dry friction forces. In: IMAC, A Conf. Expo. Struct. Dyn., pp. 11–21 (2015). https://doi.org/10.1007/978-3-319-15221-9_2

  33. Do, N.B.: Modeling of frictional contact conditions in structures. Georgia Institute of Technology. http://hdl.handle.net/1853/7123. MSc. Thesis (2005)

  34. Yang, B.D., Chu, M.L., Menq, C.H.: Stick-slip-separation analysis and non-linear stiffness and damping characterization of friction contacts having variable normal load. J. Sound Vib. 210, 461–481 (1998). https://doi.org/10.1006/jsvi.1997.1305

    Article  Google Scholar 

  35. Wright, J.P., Pei, J.S.: Solving dynamical systems involving piecewise restoring force using state event location. J. Eng. Mech. 138, 997–1020 (2012). https://doi.org/10.1061/(ASCE)EM.1943-7889.0000404

    Article  Google Scholar 

  36. Krack, M.: Nonlinear modal analysis of nonconservative systems: extension of the periodic motion concept. Comput. Struct. 154, 59–71 (2015). https://doi.org/10.1016/j.compstruc.2015.03.008

    Article  Google Scholar 

  37. Krack, M., Gross, J.: Harmonic Balance for Nonlinear Vibration Problems. Springer (2019). https://doi.org/10.1007/978-3-030-14023-6

    Book  MATH  Google Scholar 

  38. Hadji, A., Mureithi, N.: Validation of friction model parameters identified using the IHB method using finite element method. Shock Vib. (2019). https://doi.org/10.1155/2019/3493052

    Article  Google Scholar 

  39. Bazrafshan, M., Jalali, H., Ahmadian, H.: Modeling the interaction between contact mechanisms in normal and tangential directions. Int. J. Non-linear Mech. 58, 111–119 (2014)

    Article  Google Scholar 

  40. Pourahmadian, F., Ahmadian, H., Jalali, H.: Modeling and identification of frictional forces at a contact interface experiencing micro-vibro-impacts. J. Sound Vib. 331, 2874–2886 (2012). https://doi.org/10.1016/j.jsv.2012.01.032

    Article  Google Scholar 

  41. Golestaneh, A.: Nonlinear structural model updating using instantaneous frequencies and amplitudes of the dynamic response. Iran University of Science and Technology (IUST). MSc. Thesis in Farsi (2019)

Download references

Acknowledgements

The authors of this paper would like to thank the IUST alumni, Mr. Amir Golestaneh, for allowing them to use his experimental test data and his MSc. Thesis for this study. The authors are also thankful to Prof. Malte Krack and Dr. Johann Groß for sharing their “NLvib” codes on https://www.ila.uni-stuttgart.de/nlvib for all to use and modify.

Funding

The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

    Mahdi Ahi

  2. Center of Excellence in Experimental Solid Mechanics and Dynamics, School of Mechanical Engineering, Iran University of Science and Technology, Narmak, Tehran, 16848, Iran

    Hamid Ahmadian

Authors
  1. Mahdi Ahi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Hamid Ahmadian
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

MA was involved in methodology, software, validation, formal analysis, data curation, writing—original draft preparation, visualization, project administration. HA contributed to supervision, conceptualization, investigation, resources, writing— reviewing and editing, funding acquisition.

Corresponding author

Correspondence to Mahdi Ahi.

Ethics declarations

Competing interests

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supplementary Information

Below is the link to the electronic supplementary material.

Supplementary file1 (DOCX 1604 kb)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ahi, M., Ahmadian, H. Nonlinear model updating of frictional structures through frequency–energy analysis. Nonlinear Dyn 110, 95–116 (2022). https://doi.org/10.1007/s11071-022-07660-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-07660-8

Keywords

Search

Navigation