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:
  • 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
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and Botong LiORCID iD:


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.

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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.