Skip to main content
SpringerLink
Log in

Nonlinear constitutive force model selection, update and uncertainty quantification for periodically sequential impact applications

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

Abstract

Contact–impact events frequently occur between solid contact interfaces in complex dynamic mechanical systems and engineering applications. Impact force models are used to determine the responses of contacting solids. The selection of the impact force model strongly affects the predictions of the dynamic behavior of the contacting bodies, mainly in sequential repeated impacts. In this work, the strong nonlinear dynamic behavior of a cantilever steel beam, repeatedly impacting a steel rigid stop, is investigated applying experimental and numerical methodologies. The applicability of a nonlinear constitutive force model selection and model-updating framework at engineering problems where contact–impact phenomena happen in a sequential periodic manner is presented. At first, the dynamic behavior of the examined problem due to variation in impact parameters, excitation frequency and applied force model is examined. The general viscoelastic impact force law based on the Hertz contact theory, augmented with a damping term is integrated in the FE model. Experimental trials of the impacting cantilever beam under base excitation were carried out, and a model selection process between nine different expressions of the viscous term at a constant impact parameter is performed. Inadequacy of the updated selected model dictated a new model selection approach including the constant impact parameter. A new model selection and updating process is finally presented in order to tune and study in detail the effect of impact parameters and clearance gap between the contacting bodies in a specific sinusoidal excitation frequency of the impacting cantilever beam, validating the experimental results.

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

Similar content being viewed by others

References

  1. Yu, T.X., Qiu, L.X.: Introduction to Impact Dynamics, 1st ed. Wiley, Singapore (2018)

  2. Rahnejat, H.: Multi-body dynamics: historical evolution and application. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 214(1), 149–173 (2000)

    Google Scholar 

  3. Flores, P., Ambrósio, J., Claro, J.C.P., Lankarani, H.M.: Influence of the contact–impact force model on the dynamic response of multi-body systems. Proc. Inst. Mech. Eng. K J. Multi-body Dyn. 220(1), 21–34 (2006)

    Google Scholar 

  4. Khulief, Y.A., Shabana, A.A.: Dynamic analysis of constrained system of rigid and flexible bodies with intermittent motion. J. Mech. Transm. Autom. Des. 108(1), 38–45 (1986)

    Google Scholar 

  5. Seifried, R., Schiehlen, W., Eberhard, P.: Numerical and experimental evaluation of the coefficient of restitution for repeated impacts. Int. J. Impact Eng. 32(1), 508–524 (2005)

    Google Scholar 

  6. Alves, J., Peixinho, N., da Silva, M.T., Flores, P., Lankarani, H.M.: A comparative study of the viscoelastic constitutive models for frictionless contact interfaces in solids. Mech. Mach. Theory 85, 172–188 (2015)

    Google Scholar 

  7. Masri, S.F.: Analytical and experimental studies of a dynamic system with a gap. J. Mech. Des. 100(3), 480–486 (1978)

    Google Scholar 

  8. Davtalab, S., Woo, K.C., Pagwiwoko, C.P., Lenci, S.: Nonlinear vibrations of a multi-span continuous beam subject to periodic impact excitation. Meccanica 50(5), 1227–1237 (2015)

    Google Scholar 

  9. Lenci, S., Rega, G.: Regular nonlinear dynamics and bifurcations of an impacting system under general periodic excitation. Nonlinear Dyn. 34(3–4 SPEC. ISS.), 249–268 (2003)

    MathSciNet  MATH  Google Scholar 

  10. Lenci, S., Rega, G.: Numerical control of impact dynamics of inverted pendulum through optimal feedback strategies. J. Sound Vib. 236(3), 505–527 (2000)

    Google Scholar 

  11. Brach, R.M.: Mechanical Impact Dynamics: Rigid Body Collisions. Raymond M. Brach (2007)

  12. Ghaednia, H., Marghitu, D.B., Jackson, R.L.: Predicting the permanent deformation after the impact of a rod with a flat surface. J. Tribol. 137(1), 011403-011403-8 (2015)

    Google Scholar 

  13. Garland, P.P., Rogers, R.J.: Recent advances in contact mechanics: papers collected at the 5th contact mechanics international symposium (CMIS2009), April 28–30, 2009, Chania, Greece. In: Stavroulakis, G.E. (Ed.) Comparisons of Contact Forces during Oblique Impact: Experimental vs. Continuum and Finite Element Results, pp. 239–255. Springer, Berlin (2013)

  14. Schiehlen, W., Seifried, R.: Three approaches for elastodynamic contact in multibody systems. Multibody Syst. Dyn. 12(1), 1–16 (2004)

    MATH  Google Scholar 

  15. Machado, M., Moreira, P., Flores, P., Lankarani, H.M.: Compliant contact force models in multibody dynamics: evolution of the Hertz contact theory. Mech. Mach. Theory 53, 99–121 (2012)

    Google Scholar 

  16. Pham, H.-T., Wang, D.-A.: A constant-force bistable mechanism for force regulation and overload protection. Mech. Mach. Theory 46(7), 899–909 (2011)

    MATH  Google Scholar 

  17. Gonzalez-Perez, I., Iserte, J.L., Fuentes, A.: Implementation of Hertz theory and validation of a finite element model for stress analysis of gear drives with localized bearing contact. Mech. Mach. Theory 46(6), 765–783 (2011)

    MATH  Google Scholar 

  18. Dopico, D., Luaces, A., Gonzalez, M., Cuadrado, J.: Dealing with multiple contacts in a human-in-the-loop application. Multibody Syst. Dyn. 25(2), 167–183 (2011)

    MathSciNet  Google Scholar 

  19. Ceanga, V., Hurmuzlu, Y.: A new look at an old problem: Newton’s cradle. J. Appl. Mech. 68(4), 575–583 (2000)

    MATH  Google Scholar 

  20. Stoianovici, D., Hurmuzlu, Y.: A critical study of the applicability of rigid-body collision theory. J. Appl. Mech. 63(2), 307–316 (1996)

    Google Scholar 

  21. Theodosiou, C., Natsiavas, S.: Dynamics of finite element structural models with multiple unilateral constraints. Int. J. Non-Linear Mech. 44(4), 371–382 (2009)

    MATH  Google Scholar 

  22. Metallidis, P., Natsiavas, S.: Vibration of a continuous system with clearance and motion constraints. Int. J. Non-Linear Mech. 35(4), 675–690 (2000)

    MATH  Google Scholar 

  23. Sanders, J.W., Dankowicz, H., Lacarbonara, W.: Design and analysis of a microelectromechanical device capable of testing theoretical models of impact at the microscale. In: Proceedings of the ASME Design Engineering Technical Conference (2012)

  24. Hertz, H.: Ueber die Berührung fester elastischer Körper. In: Journal für die reine und angewandte Mathematik (Crelle’s Journal), p. 156 (1882)

  25. Goldsmith, W.: Impact: The Theory and Physical Behaviour of Colliding Solids, 1st ed. E. Arnold, London (1960)

  26. Flores, P., Machado, M., Silva, M.T., Martins, J.M.: On the continuous contact force models for soft materials in multibody dynamics. Multibody Syst. Dyn. 25(3), 357–375 (2011)

    MATH  Google Scholar 

  27. Gharib, M., Hurmuzlu, Y.: A new contact force model for low coefficient of restitution impact. J. Appl. Mech. 79(6), 064506-064506-5 (2012)

    Google Scholar 

  28. Lankarani, H.M., Nikravesh, P.E.: A contact force model with hysteresis damping for impact analysis of multibody systems. J. Mech. Des. 112(3), 369–376 (1990)

    Google Scholar 

  29. Herbert, R.G., McWhannell, D.C.: Shape and frequency composition of pulses from an impact pair. J. Eng. Ind. 99(3), 513–518 (1977)

    Google Scholar 

  30. Marhefka, D.W., Orin, D.E.: A compliant contact model with nonlinear damping for simulation of robotic systems. IEEE Trans. Syst. Man Cybern. A Syst. Hum. 29(6), 566–572 (1999)

    Google Scholar 

  31. Zhiying, Q., Qishao, L.: Analysis of impact process based on restitution coefficient. J. Dyn. Control 4, 294–298 (2006)

    Google Scholar 

  32. Hunt, K.H., Crossley, F.R.E.: Coefficient of restitution interpreted as damping in vibroimpact. J. Appl. Mech. 42(2), 440–445 (1975)

    Google Scholar 

  33. Dyagel, R.V., Lapshin, V.V.: On a nonlinear viscoelastic model of Hunt–Crossley impact. Mech. Solids 46(5), 798–806 (2011)

    Google Scholar 

  34. Gonthier, Y., McPhee, J., Lange, C., Piedbœuf, J.-C.: A regularized contact model with asymmetric damping and dwell-time dependent friction. Multibody Syst. Dyn. 11(3), 209–233 (2004)

    MATH  Google Scholar 

  35. Lee, T.W., Wang, A.C.: On the dynamics of intermittent-motion mechanisms. Part 1: dynamic model and response. J. Mech. Transm. Autom. Des. 105(3), 534–540 (1983)

    Google Scholar 

  36. Lee, L.S.S.: Mode responses of dynamically loaded structures. J. Appl. Mech. 39(4), 904–910 (1972)

    Google Scholar 

  37. Lee, L.S.S., Martin, J.B.: Approximate solutions of impulsively loaded structures of a rate sensitive material. Zeitschrift für angewandte Mathematik und Physik ZAMP 21(6), 1011–1032 (1970)

    MATH  Google Scholar 

  38. Ahmad, M., Ismail, K.A., Mat, F.: Impact models and coefficient of restitution: a review. ARPN J. Eng. Appl. Sci. 11, 6549–6555 (2016)

    Google Scholar 

  39. Giagopoulos, D., Dertimanis, V., Chatzi, E., Spiridonakos, M.: Online state and parameter estimation of a nonlinear gear transmission system. In: 34th IMAC, a Conference and Exposition on Structural Dynamics, 2016. Springer, New York (2016)

  40. Morin, D., Kaarstad, B.L., Skajaa, B., Hopperstad, O.S., Langseth, M.: Testing and modelling of stiffened aluminium panels subjected to quasi-static and low-velocity impact loading. Int. J. Impact Eng. 110, 97–111 (2017)

    Google Scholar 

  41. Gruben, G., Sølvernes, S., Berstad, T., Morin, D., Hopperstad, O.S., Langseth, M.: Low-velocity impact behaviour and failure of stiffened steel plates. Mar. Struct. 54, 73–91 (2017)

    Google Scholar 

  42. Gruben, G., Langseth, M., Fagerholt, E., Hopperstad, O.S.: Low-velocity impact on high-strength steel sheets: an experimental and numerical study. Int. J. Impact Eng. 88, 153–171 (2016)

    Google Scholar 

  43. Natsiavas, S., Giagopoulos, D.: Non-linear dynamics of gear meshing and vibro-impact phenomenon. In: Tribology and Dynamics of Engine and Powertrain: Fundamentals, Applications and Future Trends, pp. 773–792. Elsevier, Amsterdam (2010)

  44. Natsiavas, S.: On the dynamics of oscillators with bi-linear damping and stiffness. Int. J. Non-Linear Mech. 25(5), 535–554 (1990)

    MATH  Google Scholar 

  45. Lenci, S., Demeio, L., Petrini, M.: Response Scenario and Nonsmooth Features in the Nonlinear Dynamics of an Impacting Inverted Pendulum (2006)

  46. Perlikowski, P., Woo, K.C., Lenci, S., Kapitaniak, T.: Special issue: dynamics of systems with impacts. J. Comput. Nonlinear Dyn. 12(6), 060301 (2017)

  47. MSC.Software, NASTRAN’s Reference Guide. 2019: USA

  48. Newmark, N.M.: A Method Of Computation for Structural Dynamics. American Society of Civil Engineers (1959)

  49. Ewins, D.J.: Modal Testing: Theory and Practice. Research Studies Press, Somerset (1984)

    Google Scholar 

  50. Mohanty, P., Rixen, D.J.: Identifying mode shapes and modal frequencies by operational modal analysis in the presence of harmonic excitation. Exp. Mech. 45(3), 213–220 (2005)

    Google Scholar 

  51. Richardson, M.H., Formenti, D.L.: Global curve fitting of frequency response measurements using the rational fraction polynomial method. In: Third IMAC Conference, Orlando, FL (1985)

  52. Spottswood, S.M., Allemang, R.J.: On the investigation of some parameter identification and experimental modal filtering issues for nonlinear reduced order models. Exp. Mech. 47(4), 511–521 (2007)

    Google Scholar 

  53. Mthembu, L., Marwala, T., Friswell, M.I., Adhikari, S.: Model selection in finite element model updating using the Bayesian evidence statistic. Mech. Syst. Signal Process. 25(7), 2399–2412 (2011)

    Google Scholar 

  54. Beck, J.L., Yuen, K.-V.: Model selection using response measurements: Bayesian probabilistic approach. J. Eng. Mech. 130(2), 192–203 (2004)

    Google Scholar 

  55. Ching, J., Chen, Y.-C.: Transitional Markov chain Monte Carlo method for bayesian model updating, model class selection, and model averaging. J. Eng. Mech. 133(7), 816–832 (2007)

    Google Scholar 

  56. Giagopoulos, D., Natsiavas, S.: Hybrid (numerical-experimental) modeling of complex structures with linear and nonlinear components. Nonlinear Dyn. 47(1), 193–217 (2007)

    MATH  Google Scholar 

  57. Giagopoulos, D., Natsiavas, S.: Dynamic response and identification of critical points in the superstructure of a vehicle using a combination of numerical and experimental methods. Exp. Mech. 55(3), 529–542 (2015)

    Google Scholar 

  58. Hansen, N.: The CMA Evolution Strategy: A Tutorial. Research Centre Saclay—Île-de-France Université Paris-Saclay, LRI (2011)

  59. Hansen, N.: The CMA evolution strategy a comparing review. Towards New Evol. Comput. 192(1), 75–102 (2006)

    Google Scholar 

  60. Hansen, N., Müller, S.D., Koumoutsakos, P.: Reducing the time complexity of the derandomized evolution strategy with covariance matrix adaptation (CMA-ES). Evol. Comput. 11(1), 1–18 (2003)

    Google Scholar 

  61. Giagopoulos, D., Arailopoulos, A.: Computational framework for model updating of large scale linear and nonlinear finite element models using state of the art evolution strategy. Comput. Struct. 192, 210–232 (2017)

    Google Scholar 

  62. Hadjidoukas, P.E., Angelikopoulos, P., Papadimitriou, C., Koumoutsakos, P.: \(\varPi \)4U: a high performance computing framework for Bayesian uncertainty quantification of complex models. J. Comput. Phys. 284, 1–21 (2015)

    MathSciNet  MATH  Google Scholar 

  63. Hadjidoukas, P.E., Lappas, E., Dimakopoulos, V.V.: A runtime library for platform-independent task parallelism. In: 20th Euromicro International Conference on Parallel, Distributed and Network-based Processing. Garching, Germany (2012)

  64. Beck, J.L.: Bayesian system identification based on probability logic. Struct. Control Health Monit. 17(7), 825–847 (2010)

    Google Scholar 

  65. Beck, J.L., Katafygiotis, L.S.: Updating models and their uncertainties. I: Bayesian statistical framework. J. Eng. Mech. 124(4), 455–461 (1998)

    Google Scholar 

  66. Yuen, K.-V., Kuok, S.-C.: Bayesian methods for updating dynamic models. Appl. Mech. Rev. 64(1), 010802 (2011)

    Google Scholar 

  67. Yuen, K.V.: Bayesian Methods for Structural Dynamics and Civil Engineering. Wiley, Singapore (2010)

    Google Scholar 

  68. Katafygiotis, L.S., Beck, J.L.: Updating models and their uncertainties. II: Model identifiability. J. Eng. Mech. 124(4), 463–467 (1998)

    Google Scholar 

  69. Papadimitriou, C.: Bayesian uncertainty quantification and propagation in structural dynamics simulations. In: Proceedings of the 9th International Conference on Structural Dynamics, EURODYN: 2014, pp. 111–123. Department of Mechanical Engineering, University of Thessaly, Porto, Portugal (2014)

  70. Giagopoulos, D., Arailopoulos, A., Dertimanis, V., Papadimitriou, C., Chatzi, E., Grompanopoulos, K.: Structural health monitoring and fatigue damage estimation using vibration measurements and finite element model updating. Struct. Health Monit. 18(4), 1475921718790188 (2018)

    Google Scholar 

  71. Arailopoulos, A., Giagopoulos, D., Zacharakis, I., Pipili, E.: Integrated reverse engineering strategy for large-scale mechanical systems: application to a steam turbine rotor. Front. Built Environ. 4(55), 1–6 (2018)

    Google Scholar 

  72. BETA CAE Systems, S.A., ANSA & META-Post. BETA CAE Systems, S.A.: Thessaloniki, Greece (2018)

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Western Macedonia, Bakola & Sialvera, 50100, Kozani, Greece

    Alexandros Arailopoulos & Dimitrios Giagopoulos

Authors
  1. Alexandros Arailopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dimitrios Giagopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dimitrios Giagopoulos.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Arailopoulos, A., Giagopoulos, D. Nonlinear constitutive force model selection, update and uncertainty quantification for periodically sequential impact applications. Nonlinear Dyn 99, 2623–2646 (2020). https://doi.org/10.1007/s11071-019-05444-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-05444-1

Keywords

Search

Navigation