Inverse Problems for Euler-Bernoulli Beam and Kirchhoff Plate Equations

Beams and plates are important parts of main engineering constructions such as aircraft wings, flexible robotic manipulators, large space structures and robots, marine risers and moving strips. Within the scope of the linear theory of elasticity, classical Euler-Bernoulli beam and Kirchhoff plate theories have been a cornerstone tools in engineering since the development of the Eiffel Tower and the Ferris wheel in the late nineteenth century. Recently, small size nanobeams and nanoplates are found also in nanotechnology, including emerging systems for medical diagnostics and nanoscale measurements such as the Atomic Force Microscopy (AFM) and Transverse Dynamic Force Microscope (TDFM). The direct and inverse problems for the simplest Euler-Bernoulli ρ(x)utt + (EI(x)uxx)xx = F(x, t) and Kirchhoff utt + D Δ2u = g(t)f(x), (x, t) ∈ ΩT := Ω × (0, T), \(\Omega \subset \mathbb {R}^2\) equations have been extensively studied in the last 50 years due to a large number of engineering and technological applications.