Simultaneous determination of an unknown bending moment and shear force in the Euler–Bernoulli cantilever beam from measured boundary deflection and slope
Journal of Inverse and Ill-Posed Problems, 2026 (SCI-Expanded, Scopus)
- Yayın Türü: Makale / Tam Makale
- Basım Tarihi: 2026
- Doi Numarası: 10.1515/jiip-2026-0076
- Dergi Adı: Journal of Inverse and Ill-Posed Problems
- Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Aerospace Database, Compendex, INSPEC, MathSciNet, zbMATH, Academic Search Ultimate (EBSCO), Engineering Source (EBSCO), Materials Science & Engineering Collection (ProQuest), Technology Collection (ProQuest)
- Anahtar Kelimeler: Dirichlet-to-Neumann maps, Euler–Bernoulli beam model, Fréchet gradient, gradient formula in vector form, inverse problem, measured boundary deflection and slope, shear force and bending moment identification, Tikhonov functional
- Kocaeli Üniversitesi Adresli: Evet
Özet
It is known that the study of vibration characteristics at the tip of the micro-cantilever and its relationship to the sample plays a very important role in improving the resolution of atomic force microscopy (AFM). In this paper, within the Euler–Bernoulli beam structure, a mathematical model, defined as a model with two unknown inputs and two measured outputs, is considered for the simultaneous determination of the unknown bending moment and the shear force at the tip of the micro-cantilever from two feasible measured outputs at the same tip: the deflection and the slope. This model leads to the following inverse problem: find M(t) and g(t) in ρA(x)utt + μ(x)ut + (r(x)uxx)xx = 0, (x, t) ∈ ΩT := (0, ℓ) × (0, T) subject to the boundary conditions u(0, t) = ux(0, t) = 0, (r(x)uxx)x=ℓ = M(t), ((r(x)uxx)x)x=ℓ = g(t), and the homogeneous initial conditions u(x, 0) = ut(x, 0) = 0, from the measured outputs wℓ(t):= u(ℓ, t) and θℓ(t):= ux(ℓ, t). It is proved that the vector-form input-output map P:= (Φ, Ψ), with (Φq)(t):= u(0, t; q) and (Ψq)(t):= ux(0, t; q), where q(t):= (M(t), g(t)), corresponding to the inverse problem, is compact and Lipschitz continuous. This result allows us to prove the existence of a solution of the minimization problem for the Tikhonov functional J(q) = 12Φq − wℓ2L2(0,T) + 21Ψq − θℓ2L2(0,T). As a consequence, the existence of a quasi-solution to the inverse problem is established. Furthermore, a vector-form expression for the Fréchet gradient of the Tikhonov functional is derived, and the Lipschitz continuity of the Fréchet gradient is rigorously proven. This crucial property ensures the monotonic behavior of iterative gradient-based numerical methods. In addition, the obtained results are directly applicable to the analysis and design of both atomic force microscopes (AFMs) and their advanced counterparts, transverse dynamic force microscopes (TDFMs).