Identifying space-time dependent force on the vibrating Euler–Bernoulli beam by a boundary functional method

  • 1 School of Mathematics and Physics, University of Science and Technology Beijing, 100083, Beijing, P. R. China
  • 2 College of Mechanics and Materials, Hohai University, Jiangsu 210098, Nanjing, P. R. China
Chein-Shan LiuORCID iD: https://orcid.org/0000-0001-6366-3539
  • College of Mechanics and Materials, Hohai University, Nanjing, Jiangsu 210098, P. R. China, and Center of Excellence for Ocean Engineering, Center of Excellence for the Oceans, National Taiwan Ocean University, Keelung 202-24, Taiwan
  • orcid.org/0000-0001-6366-3539
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar
and Botong LiORCID iD: https://orcid.org/0000-0003-3188-9634

Abstract

In this paper we estimate an unknown space-time dependent force being exerted on the vibrating Euler–Bernoulli beam under different boundary supports, which is obtained with the help of measured boundary forces as additional conditions. A sequence of spatial boundary functions is derived, and all the boundary functions and the zero element constitute a linear space. A work boundary functional is coined in the linear space, of which the work is approximately preserved for each work boundary function. The linear system used to recover the unknown force with the work boundary functions as the bases is derived and the iterative algorithm is developed, which converges very fast at each time step. The accuracy and robustness of the boundary functional method (BFM) are confirmed by comparing the estimated forces under large noise with the exact forces. We also recover the unknown force on the damped vibrating Euler–Bernoulli beam equation.

  • [1]

    M. Abu-Hilal, Forced vibration of Euler–Bernoulli beams by means of dynamic Green functions, J. Sound Vib. 267 (2003), 191–207.

    • Crossref
    • Export Citation
  • [2]

    I. Bartoli, A. Marzani, F. L. di Scalea and E. Viola, Modeling wave propagation in damped waveguides of arbitrary cross-section, J. Sound Vib. 295 (2006), 685–707.

    • Crossref
    • Export Citation
  • [3]

    J.-D. Chang and B.-Z. Guo, Identification of variable spacial coefficients for a beam equation from boundary measurements, Automatica J. IFAC 43 (2007), no. 4, 732–737.

    • Crossref
    • Export Citation
  • [4]

    R. M. Christensen, Mechanics of Composite Materials, John Wiley and Sons, New York, 1979.

  • [5]

    B. Z. Guo, On the boundary control of a hybrid system with variable coefficients, J. Optim. Theory Appl. 114 (2002), no. 2, 373–395.

    • Crossref
    • Export Citation
  • [6]

    S. M. Han, H. Benarova and T. Wei, Dynamics of transversely vibrating beam using four engineering theories, J. Sound Vib. 225 (1999), 935–988.

    • Crossref
    • Export Citation
  • [7]

    A. Hasanov, Identification of an unknown source term in a vibrating cantilevered beam from final overdetermination, Inverse Problems 25 (2009), no. 11, Article ID 115015.

  • [8]

    A. Hasanov and O. Baysal, Identification of an unknown spatial load distribution in a vibrating cantilevered beam from final overdetermination, J. Inverse Ill-Posed Probl. 23 (2015), no. 1, 85–102.

  • [9]

    A. Hasanov and O. Baysal, Identification of unknown temporal and spatial load distributions in a vibrating Euler–Bernoulli beam from Dirichlet boundary measured data, Automatica J. IFAC 71 (2016), 106–117.

    • Crossref
    • Export Citation
  • [10]

    A. Hasanov and A. Kawano, Identification of unknown spatial load distributions in a vibrating Euler–Bernoulli beam from limited measured data, Inverse Problems 32 (2016), no. 5, Article ID 055004.

  • [11]

    R. M. Jones, Mechanics of Composite Materials, Hemisphere, New York, 1975.

  • [12]

    A. Kawano, Uniqueness in the identification of asynchronous sources and damage in vibrating beams, Inverse Problems 30 (2014), no. 6, Article ID 065008.

  • [13]

    M. Krstic, B.-Z. Guo, A. Balogh and A. Smyshlyaev, Control of a tip-force destabilized shear beam by observer-based boundary feedback, SIAM J. Control Optim. 47 (2008), no. 2, 553–574.

    • Crossref
    • Export Citation
  • [14]

    M. Krstic and A. Smyshlyaev, Boundary Control of PDEs. A Course on Backstepping Designs, Adv. Des. Control 16, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2008.

  • [15]

    C.-S. Liu, A Lie-group adaptive differential quadrature method to identify an unknown force in an Euler–Bernoulli beam equation, Acta Mech. 223 (2012), no. 10, 2207–2223.

    • Crossref
    • Export Citation
  • [16]

    C.-S. Liu, Identifying a rigidity function distributed in static composite beam by the boundary functional method, Compos. Struct. 176 (2017), 996–1004.

    • Crossref
    • Export Citation
  • [17]

    C.-S. Liu, Solving inverse coefficient problems of non-uniform fractionally diffusive reactive material by a boundary functional method, Int. J. Heat Mass Transfer 116 (2018), 587–598.

    • Crossref
    • Export Citation
  • [18]

    C.-S. Liu and J.-R. Chang, Recovering a source term in the time-fractional Burgers equation by an energy boundary functional equation, Appl. Math. Lett. 79 (2018), 138–145.

    • Crossref
    • Export Citation
  • [19]

    C.-S. Liu and Y. W. Chen, Solving the inverse problems of wave equation by a boundary functional method, J. Shipp. Ocean Eng. 6 (2017), 233–249.

  • [20]

    C.-S. Liu and B. Li, An upper bound theory to approximate the natural frequencies and parameters identification of composite beams, Compos. Struct. 171 (2017), 131–144.

    • Crossref
    • Export Citation
  • [21]

    C.-S. Liu and B. Li, Reconstructing a second-order Sturm–Liouville operator by an energetic boundary function iterative method, Appl. Math. Lett. 73 (2017), 49–55.

    • Crossref
    • Export Citation
  • [22]

    S. Nicaise and O. Zaïr, Determination of point sources in vibrating beams by boundary measurements: Identifiability, stability, and reconstruction results, Electron. J. Differential Equations 2004 (2004), Paper No. 20.

Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
Log in with your institution

Journal + Issues

This journal presents original articles on the theory, numerics and applications of inverse and ill-posed problems. These inverse and ill-posed problems arise in mathematical physics and mathematical analysis, geophysics, acoustics, electrodynamics, tomography, medicine, ecology, financial mathematics etc. Articles on the construction and justification of new numerical algorithms of inverse problem solutions are also published.

Search