Skip to main content
SpringerLink
Log in

The recovery of external force in nonlinear system by using a weak-form integral method

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

Abstract

For the recovery of unknown external force in the inverse vibration problems, we first transform the linear ordinary differential equation of motion into a linear parabolic-type partial differential equation and then use the Green second identity to derive a boundary integral equation in terms of the adjoint Trefftz test functions. It results in a weak-form integral method to recover the external force for nonlinear structures, of which only the data of displacements and two velocities at initial and final times are specified. The advantages gained in this transformation to a weak-form integral formulation can be seen when we test some nonlinear inverse vibration problems within a long time span and under a large noise.

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

Similar content being viewed by others

References

  1. Chen, Y., Xu, J.: Applications of the integral equation method to delay differential equations. Nonlinear Dyn. 73, 2241–2260 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Chen, Y.W.: Application of the characteristic time expansion method for estimating nonlinear restoring forces. J. Appl. Math. 2013, ID 841690, p 13 (2013)

  3. Crawley, E.F., Aubert, A.C.: Identification of nonlinear structural elements by force-state mapping. AIAA J. 24, 155–162 (1986)

    Article  Google Scholar 

  4. Crawley, E.F., O’Donnell, J.J.: Identification of nonlinear system parameters in joints using the force-state mapping technique. AIAA Paper 86(1013), 659–667 (1986)

    Google Scholar 

  5. Doebling, S.W., Farrar, C.R., Prime, M.B.: A summary review of vibration-based damage identification methods. Shock Vib. Dig. 30, 91–105 (1998)

    Article  Google Scholar 

  6. Dong, L., Alotaibi, A., Mohiuddine, S.A., Atluri, S.N.: Computational methods in engineering: a variety of primal and mixed methods, with global and local interpolations, for well-posed or ill-posed BCs. CMES Comput. Model. Eng. Sci. 99, 1–85 (2014)

    MathSciNet  Google Scholar 

  7. Duym, S., Schoukens, J., Guillaume, P.: A local restoring force surface method. Int. J. Anal. Exp. Modal Anal. 11, 116–132 (1996)

    Google Scholar 

  8. Farooq, U., Feeny, B.F.: Smooth orthogonal decomposition for modal analysis of randomly excited system. J. Sound Vib. 316, 137–146 (2008)

    Article  Google Scholar 

  9. Feldman, M.: Consider high harmonics for identification of non-linear systems by Hilbert transform. Mech. Syst. Signal. Process. 21, 943–958 (2007)

    Article  Google Scholar 

  10. Haykin, S., Sayed, A.H.H., Zeidler, J.R., Yee, P., Wei, P.C.: Adaptive tracking of linear time-variant systems by extended RLS algorithms. IEEE Trans. Signal Process. 45, 1118–1128 (1997)

    Article  Google Scholar 

  11. Huang, C.H.: A generalized inverse force vibration problem for simultaneously estimating the time-dependent external forces. Appl. Math. Model. 29, 1022–1039 (2005)

    Article  MATH  Google Scholar 

  12. 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)

    Article  Google Scholar 

  13. Law, G., Yong, D.: Identification of multistory shear buildings under unknown earthquake excitation using partial output measurements: numerical and experimental studies. Math. Probl. Eng. 2014, Article ID 987694, p. 10 (2014)

  14. Liu, C.S.: Identifying time-dependent damping and stiffness functions by a simple and yet accurate method. J. Sound Vib. 318, 148–165 (2008)

    Article  Google Scholar 

  15. Liu C.S.: Solving an inverse Sturm–Liouville problem by a Lie-group method. Bound. Value Probl. 2008, Article ID 749865, p. 18 (2008)

  16. Liu, C.S.: An iterative GL(\(n\), R) method for solving nonlinear inverse vibration problems. Nonlinear Dyn. 74, 685–699 (2014)

    Article  Google Scholar 

  17. Liu, C.S.: An LGDAE method to solve nonlinear Cauchy problem without initial temperature. CMES Comput. Model. Eng. Sci. 99, 371–391 (2014)

    MathSciNet  Google Scholar 

  18. Liu, C.S., Atluri, S.N.: A fictitious time integration method for the numerical solution of the Fredholm integral equation and for numerical differentiation of noisy data, and its relation to the filter theory. CMES Comput. Model. Eng. Sci. 41, 243–261 (2009)

    MathSciNet  Google Scholar 

  19. Liu, C.S., Chang, C.W.: A real-time Lie-group differential algebraic equations method to solve he inverse nonlinear vibration problems. Inverse Probl. Sci. Eng. (2016). doi:10.1080/17415977.2015.1130043

  20. Ma, C.K., Tuan, P.C., Lin, D.C.: A study of inverse method for the estimation of impulse loads. Int. J. Syst. Sci. 29, 663–672 (1998)

    Article  Google Scholar 

  21. Ma, C.K., Tuan, P.C., Chang, J.M., Lin, D.C.: Adaptive weighting inverse method for the estimation of input loads. Int. J. Syst. Sci. 34, 181–194 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Maccari, A.: Vibration amplitude control for a van der pol-Duffing oscillator with time delay. J. Sound Vib. 317, 20–29 (2008)

    Article  Google Scholar 

  23. Masri, S.F., Chassiakos, A.G., Caughey, T.K.: Identification of nonlinear dynamic systems using neural networks. ASME J. Appl. Mech. 60, 123–133 (1993)

    Article  Google Scholar 

  24. Namdeo, V., Manohar, C.S.: Force state maps using reproducing kernel particle method and Kriging based functional representations. CMES Comput. Model. Eng. Sci. 32, 123–159 (2008)

    MathSciNet  MATH  Google Scholar 

  25. Ni, Y.Q., Ye, X.W., Ko, J.M.: Monitoring-based fatigue reliability assessment of steel bridges: analytical model and application. J. Struct. Eng. 136, 1563–1573 (2010)

    Article  Google Scholar 

  26. Shiguemori, E.H., Chiwiacowsky, L.D., Campos Velho, H.F., Silva, J.D.S.: An inverse vibration problem solved by an artificial neural network. TEMA Tendencies Comput. Appl. Math. 6, 163–175 (2005)

    MathSciNet  MATH  Google Scholar 

  27. Wei, L., Griffin, J.: The prediction of seat transmissibility from measures of seat impedance. J. Sound Vib. 214, 121–137 (1998)

    Article  Google Scholar 

  28. Wu, A.L., Loh, C.H., Yang, J.N.: Input force identification: application to soil–pile interaction. J. Struct. Control Health Monit. 16, 223–240 (2009)

    Article  Google Scholar 

  29. Xu, J., Chen, Y., Chung, K.W.: An improved time-delay saturation controller for suppression of nonlinear beam vibration. Nonlinear Dyn. 82, 1691–1707 (2015)

    Article  MathSciNet  Google Scholar 

  30. Yang, J.N., Lin, S.: On-line identification of non-linear hysteretic structures using an adaptive tracking technique. Int. J. Non Linear Mech. 39, 1481–1491 (2004)

    Article  MATH  Google Scholar 

Download references

Acknowledgments

The project NSC-102-2221-E-002-125-MY3 and the Chair Professor of Hohai University for the first author are highly appreciated.

Author information

Authors and Affiliations

  1. Center for Numerical Simulation Software in Engineering and Sciences, College of Mechanics and Materials, Hohai University, Nanjing, 210098, Jiangsu, China

    Chein-Shan Liu

  2. Department of Civil Engineering, National Taiwan University, Taipei, 10617, Taiwan

    Chein-Shan Liu

  3. Department of Systems Engineering and Naval Architecture, National Taiwan Ocean University, Keelung, 20224, Taiwan

    Jiang-Ren Chang

  4. Department of Marine Engineering, National Taiwan Ocean University, Keelung, 20224, Taiwan

    Yung-Wei Chen

Authors
  1. Chein-Shan Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiang-Ren Chang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yung-Wei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yung-Wei Chen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, CS., Chang, JR. & Chen, YW. The recovery of external force in nonlinear system by using a weak-form integral method. Nonlinear Dyn 86, 987–998 (2016). https://doi.org/10.1007/s11071-016-2939-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-016-2939-2

Keywords

Search

Navigation