Two types of inverse source problems of identifying asynchronously distributed spatial loads governed by the Euler–Bernoulli beam equation ρ ( x ) w tt + μ ( x ) w t + ( EI ( x ) w xx ) xx − T r u xx = ∑ m = 1 M g m ( t ) f m ( x ) , ( x , t ) ∈ Ω T ≔ ( 0 , l ) × ( 0 , T ) , with hinged–clamped ends ( w ( 0 , t ) = w xx ( 0 , t ) = 0 , w ( l , t ) = w x ( l , t ) = 0 , t ∈ ( 0 , T ) ), are studied. Here g m ( t ) are linearly independent functions, describing an asynchronous temporal loading, and f m ( x ) are the spatial load distributions. In the first identification problem the values ν k ( t ) , k = 1 , K ¯ , of the deflection w ( x , t ) , are assumed to be known, as measured output data , in a neighbourhood of the finite set of points P ≔ { x k ∈ ( 0 , l ) , k = 1 , K ¯ } ⊂ ( 0 , l ) , corresponding to the internal points of a continuous beam, for all t ∈ ] 0 , T [ . In the second identification problem the values θ k ( t ) , k = 1 , K ¯ , of the slope w x ( x , t ) , are assumed to be known, as measured output data in a neighbourhood of the same set of points P for all t ∈ ] 0 , T [ . These inverse source problems will be defined subsequently as the problems ISP1 and ISP2. The general purpose of this study is to develop mathematical concepts and tools that are capable of providing effective numerical algorithms for the numerical solution of the considered class of inverse problems. Note that both measured output data ν k ( t ) and θ k ( t ) contain random noise. In the first part of the study we prove that each measured output data ν k ( t ) and θ k ( t ) , k = 1 , K ¯ can uniquely determine the unknown functions f m ...

We consider distributions $u\in\mathscr{S}'(\mathbb{R})$ of the form $u(t)=\sum_{n\in\mathbb{N}}a_n\mathrm{e}^{\imath\,\lambda_{n}t}$, where $(a_n)_{n\in\mathbb{N}}\subset\mathbb{C}$ and $\Lambda=(\lambda_n)_{n\in\mathbb{N}}\subset\mathbb{R}$ have the following properties: $(a_n)_{n\in\mathbb{N}}\in s'$, that is, there is a $q\in\mathbb{N}$ such that $(n^{-q}\,a_n)_{n\in\mathbb{N}}\in\ell^1$; for the real sequence $\Lambda$, there are $n_0\in\mathbb{N}$, $C>0$, and $\alpha>0$ such that $n\geq n_0\Rightarrow\left|\,\lambda_n\,\right|\geq Cn^\alpha$. Let $I_\epsilon\subset\mathbb{R}$ be an interval of length $\epsilon$. We prove that for given $\Lambda$, (1) if $\Lambda=\mathrm{O}(n^\alpha)$ with $\alpha0$ such that $u|_{I_\epsilon}=0\Rightarrow u\equiv0$; (2) if $\Lambda=\mathrm{O}(n)$ is uniformly discrete, then $\exists\epsilon>0$ such that $u|_{I_\epsilon}=0\Rightarrow u\equiv0$; (3) if $\alpha>1$ and $\Lambda$ is uniformly discrete, then for all $\epsilon>0$, $u|_{I_\epsil...

We consider the inverse problem of localizing a prey hitting a spider Orb Web from dynamic measurements taken near the center of the web, where the spider is supposed to stay. The actual discrete O...

The theme of this work is related to the field of vibration and source detection, which is important in naval, aerospace and civil engineering industries. The detection of unexpected vibration sources, in general, signals malfunctioning, or even an undesired presence in the case of defense systems. The focus will be on thin plates, which are among the basic building blocks of large complex structures. Here, we consider loads acting on a rectangular plate R of the product form g(t)Q(x), where the function of time g has a continuous first derivative and the spatial load distribution Q is a square-integrable function over R. We prove that the observation of the displacement of a line segment with arbitrary length parallel to one of the sides of the plate is enough for the determination of Q, provided that the interval of time is long enough. We also prove that the normal derivative along a side of the rectangle measured for an arbitrarily small interval of time is sufficient to determine the spatial load distribution Q. The method used to obtain the results is based on the series decomposition of the dynamic response and an analysis of the almost periodic distribution that arises from it.