Determination of unknown shear force in transverse dynamic force microscopy from measured final data

• Published in: Journal of inverse and ill-posed problems Vol. 32; no. 2; pp. 243 - 260
• Main Authors: Baysal, Onur, Hasanov, Alemdar, Kumarasamy, Sakthivel
• Format: Journal Article
• Language: English
• Published: Berlin De Gruyter 01.04.2024; Walter de Gruyter GmbH
• Subjects: 49J50; 65M32; 74G75; Boundary conditions; damped Euler–Bernoulli beam; Euler-Bernoulli equation; Fréchet gradient; Initial conditions; inverse boundary value problem; Inverse problems; Linear operators; Microscopy; Operators (mathematics); Shear force identification; Shear forces; solvability of the inverse problem; transverse dynamic force microscopy cantilever
• ISSN: 0928-0219; 1569-3945
• DOI: 10.1515/jiip-2023-0021

In this paper, we present a new methodology, based on the inverse problem approach, for the determination of an unknown shear force acting on the inaccessible tip of the microcantilever, which is a key component of (TDFM). The mathematical modelling of this phenomenon leads to the inverse problem of determining the shear force acting on the inaccessible boundary in a system governed by the variable coefficient Euler–Bernoulli equation subject to the homogeneous initial conditions and the boundary conditions from the final time measured output (displacement) . We introduce the input-output map , , and prove that it is a compact and Lipschitz continuous linear operator. Then we introduce the Tikhonov functional and prove the existence of a quasi-solution of the inverse problem. We derive a gradient formula for the Fréchet gradient of the Tikhonov functional through the corresponding adjoint problem solution and prove that it is a Lipschitz continuous functional. The results of the numerical experiments clearly illustrate the effectiveness and feasibility of the proposed approach.