Predicting S4 Beam Joint Nonlinearity Using Quasi-Static Modal Analysis

  • Mitchell Wall
  • Matthew S. Allen
  • Iman ZareEmail author
Conference paper
Part of the Conference Proceedings of the Society for Experimental Mechanics Series book series (CPSEMS)


Recently, a new algorithm was presented (Festjens et al., Int J Mech Sci 75 (2013) 170–177) that allows one to predict the effective natural frequency and damping ratio as a function of amplitude for a structure with bolted joints. This paper applies a variant on that algorithm to a finite element model of the “S4 Beam” (two C-shaped beams bolted together on their ends) and compares the results with the measurements described in (Singh et al., IMAC 2019). The algorithm, which is here referred to as Quasi-Static Modal Analysis (QSMA), is applied to a detailed finite element model of the beam in the commercial software package, Abaqus®. Coulomb friction is assumed to govern the contact interface. Amplitude dependent damping and natural frequency curves are calculated for the structure and compared to experimental measurement. Several studies are included which explore the solver tolerances and preload values needed to reach agreement with experimental measurements. Additionally, the shape of the actual contact interfaces is measured using a profilometer and fed into the model to quantify the effect of slight curvature in the contact.


Quasi-static modal analysis Nonlinearity FEA Damping Natural frequency 



This material is based in part upon work supported by the National Science Foundation under Grant Number CMMI-1561810. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.


  1. 1.
    Berger, E.: Friction modeling for dynamic system simulation. Appl. Mech. Rev. 55(6), 535 (2002)CrossRefGoogle Scholar
  2. 2.
    Hartwigsen, C.J., Song, Y., McFarland, D.M., Bergman, L.A., Vakakis, A.F.: Experimental study of non-linear effects in a typical shear lap joint configuration. J. Sound Vib. 277(1–2), 327–351 (2004)CrossRefGoogle Scholar
  3. 3.
    Groper, M.: Microslip and macroslip in bolted joints. Exp. Mech. 25(2), 171–174 (1985)CrossRefGoogle Scholar
  4. 4.
    Groper, M., Hemmye, J.: The dissipation of energy in high strength friction grip bolted joints. In: Presented at the SESA Spring Conference, Cleveland, OH (1983)Google Scholar
  5. 5.
    Hartwigsen, K.: Handbook of Lubrication, ll—Theory and Design, Part l—Frictio. CRC Press Inc., Boca Raton, FL (1983)Google Scholar
  6. 6.
    Festjens, H., Chevallier, G., Dion, J.: A numerical tool for the design of assembled structures under dynamic loads. Int. J. Mech. Sci. 75, 170–177 (2013)CrossRefGoogle Scholar
  7. 7.
    Quinn, D.D.: Modal analysis of jointed structures. J. Sound Vib. 331(1), 81–93 (2012)CrossRefGoogle Scholar
  8. 8.
    Segalman, D.J.: A Modal Approach to Modeling Spatially Distributed Vibration Energy Dissipation. Sandia National Laboratories, Albuquerque, NM.” SAND2010-4763, 993326, Aug (2010)CrossRefGoogle Scholar
  9. 9.
    Allen, M.S., Lacayo, R.M., Brake, M.R.: Quasi-Static Modal Analysis Based on Implicit Condensation for Structures with Nonlinear Joints, p. 15. Sandia National Laboratories, Albuquerque, NM (2016)Google Scholar
  10. 10.
    Segalman, D.J.: A four-parameter Iwan model for lap-type joints. J. Appl. Mech. 72(5), 752 (2005)CrossRefGoogle Scholar
  11. 11.
    Deaner, B.J., Allen, M.S., Starr, M.J., Segalman, D.J.: Investigation of modal Iwan models for structures with bolted joints. In: Topics in Experimental Dynamic Substructuring, vol. 2, pp. 9–25 (2014)CrossRefGoogle Scholar
  12. 12.
    Lacayo, R.M., Allen, M.S.: Updating structural models containing nonlinear Iwan joints using quasi-static modal analysis. Mech. Syst. Signal Process. 118, 133–157 (2019)CrossRefGoogle Scholar
  13. 13.
    Jewell, E.A., Allen, M.S., Lacayo, R.: Predicting damping of a cantilever beam with a bolted joint using quasi-static modal analysis. V008T12A019 (2017)Google Scholar
  14. 14.
    Zare, I., Allen, M.S., Jewell, E.: An Enhanced Static Reduction Algorithm for Predictive Modeling of Bolted Joints. Springer International Publishing, Cham (2019)CrossRefGoogle Scholar
  15. 15.
    Zare, I., Allen, M.S.: A Block-Gauss Seidel algorithm with static reduction to predict damping in bolted joints. In: Presented at the in International Seminar on Modal Analysis (ISMA), p. 12. Leuven, Belgium (2018)Google Scholar
  16. 16.
    Ahn, Y.J., Barber, J.R.: Response of frictional receding contact problems to cyclic loading. Int. J. Mech. Sci. 50(10–11), 1519–1525 (2008)CrossRefGoogle Scholar
  17. 17.
    Flicek, R.C., Hills, D.A., Barber, J.R., Dini, D.: Determination of the shakedown limit for large, discrete frictional systems. Eur. J. Mech. – A/Solids. 49, 242–250 (2015)CrossRefGoogle Scholar
  18. 18.
    Masing, G., Mauksch, W.: Eigenspannungen und Verfestigung des plastisch gedehnten und gestauchten Messings. In: Wissenschaftliche Veröffentlichungen aus dem Siemens-Konzern: IV. Band. Zweites Heft, pp. 244–256. Springer, Berlin, Heidelberg (1925)CrossRefGoogle Scholar
  19. 19.
    Singh, A., et al.: Experimental characterization of a new benchmark structure for prediction of damping nonlinearity. Nonlinear Dynam. 1, 57–78 (2019)Google Scholar
  20. 20.
    Jewell, E., Allen, M.S., Zare, I., Wall, M.: Application of quasi-static modal analysis to a finite element model with experimental correlation. J. Sound Vib. Submitted for Publication (2019)Google Scholar
  21. 21.
    Lyndon, B.: Criteria for Preloaded Bolts, p. 38. NASA, Houston, TX (1998)Google Scholar
  22. 22.
    Motosh, N.: Development of design charts for bolts preloaded up to the plastic range. J. Eng. Ind. 98(3), 849 (1976)CrossRefGoogle Scholar
  23. 23.
    Fronk, M., et al.: Inverse methods for characterization of contact areas in mechanical systems. Nonlinear Dynam. 1, 45–56 (2019)Google Scholar
  24. 24.
    Ames, N.M., et al.: Handbook on Dynamics of Jointed Structures. Sandia National Laboratories, Albuquerque, NM.,” SAND2009-4164, 1028891 (2009)CrossRefGoogle Scholar

Copyright information

© Society for Experimental Mechanics, Inc. 2020

Authors and Affiliations

  1. 1.Department of Engineering PhysicsUniversity of WisconsinMadisonUSA

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