Identifying space-time dependent force on the vibrating Euler–Bernoulli beam by a boundary functional method
-
Chein-Shan Liu
and
Botong Li
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.
Funding statement: The Thousand Talents Plan of China under the Grant Number A1211010 and the Fundamental Research Funds for the Central Universities under the Grant Number 2017B05714 for the financial support to the first author are highly appreciated. The work of Li is supported by the Fundamental Research Funds for the Central Universities.
References
[1] M. Abu-Hilal, Forced vibration of Euler–Bernoulli beams by means of dynamic Green functions, J. Sound Vib. 267 (2003), 191–207. 10.1016/S0022-460X(03)00178-0Search in Google Scholar
[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. 10.1117/12.640032Search in Google Scholar
[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. 10.1016/j.automatica.2006.11.002Search in Google Scholar
[4] R. M. Christensen, Mechanics of Composite Materials, John Wiley and Sons, New York, 1979. Search in Google Scholar
[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. 10.1023/A:1016039819069Search in Google Scholar
[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. 10.1006/jsvi.1999.2257Search in Google Scholar
[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. 10.1088/0266-5611/25/11/115015Search in Google Scholar
[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. 10.1515/jiip-2014-0010Search in Google Scholar
[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. 10.1016/j.automatica.2016.04.034Search in Google Scholar
[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. 10.1088/0266-5611/32/5/055004Search in Google Scholar
[11] R. M. Jones, Mechanics of Composite Materials, Hemisphere, New York, 1975. 10.1115/1.3423688Search in Google Scholar
[12] A. Kawano, Uniqueness in the identification of asynchronous sources and damage in vibrating beams, Inverse Problems 30 (2014), no. 6, Article ID 065008. 10.1088/0266-5611/30/6/065008Search in Google Scholar
[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. 10.1137/060676969Search in Google Scholar
[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. 10.1137/1.9780898718607Search in Google Scholar
[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. 10.1007/s00707-012-0707-zSearch in Google Scholar
[16] C.-S. Liu, Identifying a rigidity function distributed in static composite beam by the boundary functional method, Compos. Struct. 176 (2017), 996–1004. 10.1016/j.compstruct.2017.06.003Search in Google Scholar
[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. 10.1016/j.ijheatmasstransfer.2017.08.124Search in Google Scholar
[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. 10.1016/j.aml.2017.12.010Search in Google Scholar
[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. Search in Google Scholar
[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. 10.1016/j.compstruct.2017.03.014Search in Google Scholar
[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. 10.1016/j.aml.2017.04.023Search in Google Scholar
[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. Search in Google Scholar
© 2020 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Identification of the interface between acoustic and elastic waves from internal measurements
- A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem
- Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues
- The enclosure method for inverse obstacle scattering over a finite time interval: VI. Using shell-type initial data
- Identifying space-time dependent force on the vibrating Euler–Bernoulli beam by a boundary functional method
- A stability result for the determination of order in time-fractional diffusion equations
- Recovery of non-smooth radiative coefficient from nonlocal observation by diffusion system
- An inverse problem of triple-thickness parameters determination for thermal protective clothing with Stephan–Boltzmann interface conditions
- Direct and inverse source problems for degenerate parabolic equations
- Partial inverse problems for quadratic differential pencils on a graph with a loop
Articles in the same Issue
- Frontmatter
- Identification of the interface between acoustic and elastic waves from internal measurements
- A numerical method for an inverse source problem for parabolic equations and its application to a coefficient inverse problem
- Determination of the impulsive Sturm–Liouville operator from a set of eigenvalues
- The enclosure method for inverse obstacle scattering over a finite time interval: VI. Using shell-type initial data
- Identifying space-time dependent force on the vibrating Euler–Bernoulli beam by a boundary functional method
- A stability result for the determination of order in time-fractional diffusion equations
- Recovery of non-smooth radiative coefficient from nonlocal observation by diffusion system
- An inverse problem of triple-thickness parameters determination for thermal protective clothing with Stephan–Boltzmann interface conditions
- Direct and inverse source problems for degenerate parabolic equations
- Partial inverse problems for quadratic differential pencils on a graph with a loop
