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In this paper, a dynamic model of reconstruction of the shear force $g(t)$ inthe Atomic Force Microscopy (AFM) cantilever tip-sample interaction isproposed. The interaction of the cone-shaped cantilever tip with the surface ofthe specimen (sample) is modeled by the damped Euler-Bernoulli beam equation$\rho_A(x)u_{tt}$ $+\mu(x)u_{t}+(r(x)u_{xx}+\kappa(x)u_{xxt})_{xx}=0$,$(x,t)\in (0,\ell)\times (0,T)$, subject to the following initial, $u(x,0)=0$,$u_t(x,0)=0$ and boundary, $u(0,t)=0$, $u_{x}(0,t)=0$, $\left(r(x)u_{xx}(x,t)+\kappa(x)u_{xxt} \right )_{x=\ell}=M(t)$, $\left(-(r(x)u_{xx}+\kappa(x)u_{xxt})_x\right )_{x=\ell}=g(t)$ conditions, where$M(t):=2h\cos \theta\,g(t)/\pi$ is the momentum generated by the transverseshear force $g(t)$. For the reconstruction of $g(t)$ the measured displacement$\nu(t):=u(\ell,t)$ is used as an additional data. The least square functional$J(F)=\frac{1}{2}\Vert u(\ell,\cdot)-\nu \Vert_{L^2(0,T)}^2$ is introduced andan explicit gradient formula for the Fr\'echet derivative through the solutionof the adjoint problem is derived. This allows to construct a gradient basednumerical algorithm for the reconstructions of the shear force from noise freeas well as from random noisy measured output $\nu (t)$. Computationalexperiments show that the proposed algorithm is very fast and robust. Thisallows to develop a numerical "gadget" for computational experiments of genericAFMs.

Reconstruction of shear force in Atomic Force Microscopy from measured displacement of the cone-shaped cantilever tip