Detecting a prey in a spider orb web. (English) Zbl 1428.35681
Summary: 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 modeled 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 prestress 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. The theoretical result is illustrated by means of a numerical implementation of the identification method.
MSC:
| 35R30 | Inverse problems for PDEs |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
| 74K15 | Membranes |
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