Publication:
Unique Recovery of Unknown Spatial Load in Damped Euler-Bernoulli Beam Equation From Final Time Measured Output

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Date
2021
Authors
Hasanov, Alemdar
Romanov, Vladimir
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Publisher
IOP Publishing Ltd.
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Abstract

In this paper we discuss the unique determination of unknown spatial load F(x) in the damped Euler-Bernoulli beam equation rho(x)u(tt)+mu u(t)+(r(x)u(xx))(xx)=F(x)G(t) 0, the damping coefficient mu > 0 and the temporal load G(t) > 0. As an alternative method we propose the adjoint problem approach (APA) and derive an explicit gradient formula for the Frechet derivative of the Tikhonov functional J(F)=parallel to u(.,T; F) - u(T)parallel to(2)(L2(0,l)). Comparative analysis of numerical algorithms based on SVE and APA methods are provided for the harmonic loading G(t) = cos(omega t), omega > 0, as a most common dynamic loading case. The results presented in this paper not only clearly demonstrate the key role of the damping term mu u (t) in the inverse problems arising in vibration and wave phenomena, but also allows us, firstly, to find admissible values of the final time T > 0, at which a final time measured output can be extracted, and secondly, to reconstruct the unknown spatial load F(x) in the damped Euler-Bernoulli beam equation from this measured output.

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Damped Euler-bernoulli and Wave Equations , Singular Values
Citation
Hasanov, A., Romanov, V., & Baysal, O. (2021). Unique recovery of unknown spatial load in damped Euler–Bernoulli beam equation from final time measured output. Inverse Problems, 37(7), 075005.