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

No Thumbnail Available
Hasanov, Alemdar
Romanov, Vladimir
Journal Title
Journal ISSN
Volume Title
IOP Publishing Ltd.
Research Projects
Organizational Units
Journal Issue

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.

Damped Euler-bernoulli and Wave Equations , Singular Values
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.