In this paper we analyze the problem of identification of the distribution of asynchronous vibration sources and rigidity perturbations in Euler-Bernoulli beams. Here it is proved that the displacement of a beam, even in an engineered continuous beam resting upon several supports, measured for an arbitrary small interval of time over an arbitrary small open set is enough for the unique determination of vibration sources. Using this result, we prove that perturbations in rigidity can be uniquely determined in a linearized problem.

Abstract. We consider distributions u ∈ S (R) of the form u(t) = n∈N ane ı λnt , where (an) n∈N ⊂ C and Λ = (λn) n∈N ⊂ R have the following properties: (an) n∈N ∈ s , that is, there is a q ∈ N such that (n −q an) n∈N ∈ 1 ; for the real sequence Λ, there are n 0 ∈ N, C > 0, and α > 0 such that n ≥ n 0 ⇒ | λn | ≥ Cn α . Let I ⊂ R be an interval of length . We prove that for given Λ, (1)is uniformly discrete, then ∃ > 0 such that u| I = 0 ⇒ u ≡ 0; (3) if α > 1 and Λ is uniformly discrete, then for all > 0, u| I = 0 ⇒ u ≡ 0. Since distributions of the above mentioned form are very common in engineering, as in the case of the modeling of ocean waves, signal processing, and vibrations of beams, plates, and shells, those uniqueness and nonuniqueness results have important consequences for identification problems in the applied sciences. We show an identification method and close this article with a simple example to show that the recovery of geometrical imperfections in a cylindrical shell is possible from a measurement of its dynamics.

Two types of inverse source problems of identifying asynchronously distributed spatial loads governed by the Euler–Bernoulli beam equation , with hinged–clamped ends ( ), are studied. Here are linearly independent functions, describing an asynchronous temporal loading, and are the spatial load distributions. In the first identification problem the values , of the deflection , are assumed to be known, as measured output data, in a neighbourhood of the finite set of points , corresponding to the internal points of a continuous beam, for all . In the second identification problem the values , of the slope , are assumed to be known, as measured output data in a neighbourhood of the same set of points P for all . 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 and contain random noise. In the first part of the study we prove that each measured output data and can uniquely determine the unknown functions . In the second part of the study we will introduce the input–output operators , and  , , corresponding to the problems ISP1 and ISP2, and then reformulate these problems as the operator equations: and , where and . Since both measured output data contain random noise, we use the most prominent regularisation method, Tikhonov regularisation, introducing the regularised cost functionals and . Using a priori estimates for the weak solution of the direct problem and the Tikhonov regularisation method combined with the adjoint problem approach, we prove that the Fréchet gradients and of both cost functionals can explicitly be derived via the corresponding weak solutions of adjoint problems and the known temporal loads . Moreover, we show that these gradients are Lipschitz continuous, which allows the use of gradient type iteration convergent algorithms. Two applications of the proposed theory are presented. It is shown that solvability results for inverse source problems related to the synchronous loading case, with a single interior measured data, are special cases of the obtained results for asynchronously distributed spatial load cases.

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.

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 orb-web, formed by a finite number of radial and circumferential threads, is modelled as a continuous membrane. The membrane has a specific fibrous structure, which is inherited from the original discrete web, and it is subject to tensile pre-stress in the referential configuration. The transverse load describing the prey's impact is assumed of the form g(t)f (x), where g(t) is a known function of time and f (x) is the unknown term depending on the position variable x. For axially-symmetric orb-webs supported at the boundary and undergoing infinitesimal transverse deformations, we prove a uniqueness result for f (x) in terms of measurements of the transverse dynamic displacement taken on an arbitrarily small and thin ring centered at the origin of the web, for a sufficiently large interval of time.

No abstract

In this paper, we consider the uniqueness issue for the inverse problem of load identification in a nanoplate by dynamic measurements. Working in the framework of the strain gradient linear elasticity theory, we first deduce a Kirchhoff-Love nanoplate model, and we analyze the well-posedness of the equilibrium problem, clarifying the correct Neumann conditions on curved portions of the boundary. Our uniqueness result states that, given a transverse dynamic load ∑ M m=1 g m (t)𝑓 m (x), where M ≥ 1 and {g m (t)} M m=1 are known time-dependent functions, if the transverse displacement of the nanoplate is known in an open subset of its domain for any interval of time, then the spatial components {𝑓 m (x)} M m=1 can be determined uniquely from the data. The proof is based on the spherical means method. The uniqueness result suggests a reconstruction technique to approximate the loads, as confirmed by a series of numerical simulations performed on a rectangular clamped nanoplate.

No abstract