Nonlinear Dynamics

, Volume 88, Issue 4, pp 2403–2416 | Cite as

A generalized bilinear amplitude and frequency approximation for piecewise-linear nonlinear systems with gaps or prestress

Original Paper


Analysis of piecewise-linear nonlinear dynamical systems is critical for a variety of civil, mechanical, and aerospace structures that contain gaps or prestress that are caused by cracks, delamination, joints or interfaces among components. Recently, a technique referred to as bilinear amplitude approximation (BAA) was developed to estimate the response of bilinear systems that have no gap or prestress. The method is based on an idea that the dynamics of a bilinear system can be treated as a combination of linear responses in two time intervals both of which the system behaves as a distinct linear system: (1) the open state and (2) the closed or sliding state. Both geometric and momentum constraints are then applied as compatibility conditions between the states to couple the linear vibrational response for each time interval. In order to estimate the response for more general cases where there are either gaps or prestress in the system, a generalized BAA method is proposed in this paper. The new method requires inclusion of contact stiffness and damping to model contact behavior in the sliding state, and new equilibrium positions for each state to establish proper coordinates. The new method also finds the bilinear frequency of the system, which cannot be computed using the bilinear frequency approximation method previously developed since that method is only accurate for the zero gap and no prestress case. The generalized BAA method is demonstrated on a single degree of freedom system, a three degree of freedom system, and a cracked cantilever beam model for various gap sizes and prestress levels.


Bilinear system Bilinear amplitude approximation Bilinear frequency approximation Piecewise-linear nonlinearity 


  1. 1.
    Allemang, R.: Investigation of some multiple input/output frequency response experimental modal analysis techniques. Ph.D. Thesis. Mechanical Engineering Department, University of Cincinnati (1980)Google Scholar
  2. 2.
    Banea, M.D., da Silva, L.F.M., Campilho, R.D.S.G.: Effect of temperature on the shear strength of aluminium single lap bonded joints for high temperature applications. J. Adhes. Sci. Technol. 28(14–15), 1367–81 (2014)CrossRefGoogle Scholar
  3. 3.
    Bennighof, J.K., Lehoucq, R.B.: An automated multilevel substructuring method for eigenspace computation in linear elastodynamics. SIAM J. Sci. Comput. 25(6), 2084–2106 (2004). doi: 10.1137/S1064827502400650 MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bilotta, A., Faella, C., Martinelli, E., Nigro, E.: Indirect identification method of bilinear interface laws for FRP bonded on a concrete substrate. J. Compos. Constr. 16(2), 171–184 (2012)CrossRefGoogle Scholar
  5. 5.
    Brownjohn, J.M.W., De Stefano, A., Xu, Y.L., Wenzel, H., Aktan, A.E.: Vibration-based monitoring of civil infrastructure: challenges and successes. J. Civ. Struct. Health Monit. 1(3), 79–95 (2011). doi: 10.1007/s13349-011-0009-5 CrossRefGoogle Scholar
  6. 6.
    Burlayenko, V.N., Sadowski, T.: Influence of skin/core debonding on free vibration behavior of foam and honeycomb cored sandwich plates. Int. J. Non-Linear Mech. (2009). doi: 10.1016/j.ijnonlinmec.2009.07.002 Google Scholar
  7. 7.
    Chati, M., Rand, R., Mukherjee, S.: Modal analysis of a cracked beam. J. Sound Vib. 207(2), 249–270 (1997)CrossRefMATHGoogle Scholar
  8. 8.
    Della, C.N., Shu, D.: Vibration of delaminated composite laminates: a review. Appl. Mech. Rev. 60(1), 1–20 (2007). doi: 10.1115/1.2375141 CrossRefGoogle Scholar
  9. 9.
    Dimarogonas, A.D.: Vibration of cracked structures: a state of the art review. Eng. Fract. Mech. 55(5), 831–857 (1996)CrossRefGoogle Scholar
  10. 10.
    Doebling, S.W., Farrar, C.R., Prime, M.B.: A summary review of vibration-based damage identification methods. Shock Vib. Dig. 30(2), 91–105 (1998)CrossRefGoogle Scholar
  11. 11.
    Doebling, S.W., Farrar, C.R., Prime, M.B., Shevitz, D.W.: Damage identification and health monitoring of structural and mechanical systems from changes in their vibration characteristics: a literature review. Los Alamos National Laboratory Report LA-13070-MS, Los Alamos, NM (1996)Google Scholar
  12. 12.
    D’Souza, K., Epureanu, B.I.: Multiple augmentations of nonlinear systems and generalized minimum rank perturbations for damage detection. J. Sound Vib. 316(1–5), 101–121 (2008). doi: 10.1016/j.jsv.2008.02.018 CrossRefGoogle Scholar
  13. 13.
    D’Souza, K., Tien, M.H.: Bilinear amplitude and frequency approximation for nonlinear systems with gaps or prestress. In: Proceedings of the ASME 2016 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, Charlotte, NC (2016)Google Scholar
  14. 14.
    Ewins, D.J.: Modal Testing: Theory and Practice. Research Studies Press, Taunton (1984)Google Scholar
  15. 15.
    Farrar, C.R., Doebling, S.W., Nix, D.A.: Vibration-based structural damage identification. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 359(1778), 131–149 (2001)CrossRefMATHGoogle Scholar
  16. 16.
    Friswell, M.I.: Damage identification using inverse methods. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 365, 393–410 (2007)CrossRefGoogle Scholar
  17. 17.
    Friswell, M.I., Penny, J.E.T., Garvey, S.D.: Using linear model reduction to investigate the dynamics of structures with local non-linearities. Mech. Syst. Signal Process. 9(3), 317–328 (1995)CrossRefGoogle Scholar
  18. 18.
    Guyan, R.J.: Reduction of stiffness and mass matrices. AIAA J. 3(2), 380 (1965). doi: 10.2514/3.2874 CrossRefGoogle Scholar
  19. 19.
    Ha, N.V., Golinval, J.C.: Damage localization in linear-form structures based on sensitivity investigation for principal component analysis. J. Sound Vib. 329(21), 4550–4566 (2010). doi: 10.1016/j.jsv.2010.04.032 CrossRefGoogle Scholar
  20. 20.
    Hein, H., Feklistova, L.: Computationally efficient delamination detection in composite beams using haar wavelets. Mech. Syst. Signal Process. 25(6), 2257–2270 (2011). doi: 10.1016/j.ymssp.2011.02.003 CrossRefGoogle Scholar
  21. 21.
    Hestenes, M.R.: Multiplier and gradient methods. J. Optim. Theory Appl. 4, 303–320 (1969)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Irons, B.: Structural eigenvalue problems: elimination of unwanted variables. AIAA J. 3(5), 961–962 (1965). doi: 10.2514/3.3027 CrossRefGoogle Scholar
  23. 23.
    Jaumouill, V., Sinou, J.J., Petitjean, B.: An adaptive harmonic balance method for predicting the nonlinear dynamic responses of mechanical systems—application to bolted structures. J. Sound Vib. 329(19), 4048–4067 (2010). doi: 10.1016/j.jsv.2010.04.008 CrossRefGoogle Scholar
  24. 24.
    Jiang, L.J., Wang, K.W.: An experiment-based frequency sensitivity enhancing control approach for structural damage detection. Smart Mater. Struct. 18(6), 1–12 (2009). doi: 10.1088/0964-1726/18/6/065005 Google Scholar
  25. 25.
    Jung, C., D’Souza, K., Epureanu, B.I.: Nonlinear amplitude approximation for bilinear systems. J. Sound Vib. 333(13), 2909–19 (2014)CrossRefGoogle Scholar
  26. 26.
    Kim, T.C., Rook, T.E., Singh, R.: Super- and sub-harmonic response calculations for a torsional system with clearance nonlinearity using the harmonic balance method. J. Sound Vib. 281(3–5), 965–993 (2005)CrossRefGoogle Scholar
  27. 27.
    Ma, O., Wang, J.: Model order reduction for impact-contact dynamics simulations of flexible manipulators. Robotica 25(04), 397–407 (2007)CrossRefGoogle Scholar
  28. 28.
    Newmark, N.M.: A method of computation for structural dynamics. J. Eng. Mech. 85(EM3), 67–94 (1959)Google Scholar
  29. 29.
    O’Callahan, J., Avitabile, P., Riemer, R.: System equivalent reduction expansion process (SEREP). In: Proceedings of the 7th International Modal Analysis Conference, pp. 29–37. Las Vegas, NV (1989)Google Scholar
  30. 30.
    Poudou, O.: Modeling and analysis of the dynamics of dry-friction-damped structural systems. Ph.D. thesis, The University of Michigan (2007)Google Scholar
  31. 31.
    Sairajan, K.K., Aglietti, G.S.: Robustness of system equivalent reduction expansion process on spacecraft structure model validationi. AIAA J. 50(11), 2376–2388 (2012)CrossRefGoogle Scholar
  32. 32.
    Saito, A., Castanier, M.P., Pierre, C.: Estimation and veering analysis of nonlinear resonant frequencies of cracked plates. J. Sound Vib. 326(3–5), 725–739 (2009). doi: 10.1016/j.jsv.2009.05.009 CrossRefGoogle Scholar
  33. 33.
    Saito, A., Castanier, M.P., Pierre, C., Poudou, O.: Efficient nonlinear vibration analysis of the forced response of rotating cracked blades. J. Comput. Nonlinear Dyn. Trans. ASME 4(1), 011005 (2009)CrossRefGoogle Scholar
  34. 34.
    Saito, A., Epureanu, B.I.: Bilinear modal representations for reduced-order modeling of localized piecewise-linear oscillators. J. Sound Vib. 330(14), 3442–57 (2011)CrossRefGoogle Scholar
  35. 35.
    Shaw, S.W., Holmes, P.J.: A periodically forced piecewise linear oscillator. J. Sound Vib. 90(1), 129–155 (1983)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Shiiryayev, O.V., Slater, J.C.: Detection of fatigue cracks using random decrement signatures. Struct. Health Monit. 9(4), 347–360 (2010)CrossRefGoogle Scholar
  37. 37.
    Theodosiou, C., Natsiavas, S.: Dynamics of finite element structural models with multiple unilateral constraints. Int. J. Non-Linear Mech. 44(4), 371–382 (2009). doi: 10.1016/j.ijnonlinmec.2009.01.006 CrossRefMATHGoogle Scholar
  38. 38.
    Żak, A., Krawczuk, M., Ostachowicz, W.: Vibration of a laminated composite plate with closing delamination. J. Intell. Mater. Syst. Struct. 12(8), 545–551 (2001). doi: 10.1177/10453890122145320 CrossRefGoogle Scholar
  39. 39.
    Zucca, S., Epureanu, B.I.: Bi-linear reduced-order models of structures with friction intermittent contacts. Nonlinear Dyn. 77(3), 1055–1067 (2014). doi: 10.1007/s11071-014-1363-8
  40. 40.
    Zucca, S., Firrone, C.M., Gola, M.M.: Numerical assessment of friction damping at turbine blade root joints by simultaneous calculation of the static and dynamic contact loads. Nonlinear Dyn. 67(3), 1943–1955 (2012). doi: 10.1007/s11071-011-0119-y MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Department of Mechanical and Aerospace EngineeringThe Ohio State UniversityColumbusUSA

Personalised recommendations