A matrix framework is developed for single and multispan micro-cantilevers Timoshenko beam models of use in atomic force microscopy (AFM). They are considered subject to general forcing loads and boundary conditions for modeling tipsample interaction. Surface effects are considered in the frequency analysis of supported and cantilever microbeams. Extensive use is made of a distributed matrix fundamental response that allows to determine forced responses through convolution and to absorb non-homogeneous boundary conditions. Transients are identified from intial values of permanent responses. Eigenanalysis for determining frequencies and matrix mode shapes is done with the use of a fundamental matrix response that characterizes solutions of a damped second-order matrix differential equation. It is observed that surface effects are influential for the natural frequency at the nanoscale. Simulations are performed for a bi-segmented free-free beam and with a micro-cantilever beam actuated by a piezoelectric layer laminated in one side.

Vibration dynamics of elastic beams that are used in nanotechnology, such as atomic force microscope modeling and carbon nanotubes, are considered in terms of a fundamental response within a matrix framework. The modeling equations with piezoelectric and surface scale effects are written as a matrix differential equation subject to tip-sample general boundary conditions and to compatibility conditions for the case of multispan beams. We considered a quadratic and a cubic eigenvalue problem related to the inclusion of smart materials and surface effects. Simulations were performed for a two stepped beam with a piezoelectric patch subject to pulse forcing terms. Results with Timoshenko models that include surface effects are presented for micro- and nanoscale. It was observed that the effects are significant just in nanoscale. We also simulate the frequency effects of a double-span beam in which one segment includes rotatory inertia and shear deformation and the other one neglects both phenomena. The proposed analytical methodology can be useful in the design of micro- and nanoresonator structures that involve deformable flexural models for detecting and imaging of physical and biochemical quantities.

Microbeam models with surface and piezoelectric effects are considered for atomic force microscopy (AFM). These models include rotatory inertia and shear deformation as proposed by Timoshenko and they are subject to forcing loads. Eigenanalysis of the free dynamic matrix model is performed with the use of a fundamental matrix response to determine the modal frequency equation and matrix mode shapes. The fundamental response governs the behaviour of a non-classical damped second-order matrix differential equation. It was observed that surface effects are influential for the natural frequency at the nanoscale. When the beam length increases from nanometers to microns, the surface effects disappear and the results converge into natural frequencies of classical Timoshenko model. Simulations with the piezoelectric model were performed to observe the effects of forcing pulses located at different positions of the microbeam.Keywords-Dynamic of nanomaterials, Atomic force microscopy, surface and piezoelectric effects, Timoshenko beam model.

Resumo Nesse trabalho considera-se o comportamento dinâmico de vigas, que são utilizadas em nanotecnologia, tais como, modelagem em microscopia de força atômica, nanotubos de carbono e micro/nano dispositivos eletromecânicos. O modelo clássico de Timoshenko sofre alterações, baseadas na teoria do gradiente de deformação, a fim de capturar os efeitos da mudança de escala. Soluções espaciais e frequências naturais são obtidas utilizando o método modificado da decomposição de Adomian. Resultados comparativos mostram que os efeitos não-locais tem mais influência sobre as frequências naturais quando a espessuraé comparável a medida característica de comprimento interno. Palavras-chave: vibração de nanovigas, modelos não-locais, método da decomposição de Adomian.