Simultaneous determination of an unknown bending moment and shear force in the Euler–Bernoulli cantilever beam from measured boundary deflection and slope

Abstract 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 ) {M(t)} and g ⁢ ( t ) {g(t)} in ρ A ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = 0 {\rho_{A}(x)u_{tt}+\mu(x)u_{t}+(r(x)u_{xx})_{xx}=0} , ( x , t ) ∈ Ω T := ( 0 , ℓ ) × ( 0 , T ) {(x,t)\in\Omega_{T}:=(0,\ell)\times(0,T)} subject to the boundary conditions u ⁢ ( 0 , t ) = u x ⁢ ( 0