Pierpaolo Belardinelli

Modeling and Analysis of an Electrically Actuated Microbeam Based on Nonclassical Beam Theory

This work investigates the mechanical behavior of a clamped-clamped microbeam modeled within the framework of the strain-gradient elasticity theory. The governing equation of motion gives proper account of both the effect of the nonlinear midplane stretching and of an applied axial load. An electric-voltage difference, introducing into the model a further source of nonlinearity, is considered, including also a correction term for fringing field effects. The electric force acting on the microbeam is rearranged by means of the Chebyshev method, verifying the accuracy of the proposed approximation. The results show that a uniform error on the whole domain can be achieved. The static solution is obtained by a numerical differential quadrature method. The paper looks into the variation of the maximal deflection of the microbeam with respect to several parameters. Study of the pull-in limit on the high-order material parameters introduced by the nonclassical approach and a comparison with respect to the classical beam theory are also carried out. The numerical simulation indicates that the static response is larger, affected by the use of a nonclassical theory near the pull-in instability regime. The dynamical problem is, finally, analyzed, deriving the multi degree-of-freedom problem through a Galerkin-based approach. The study on the single degree-of-freedom model enables us to note the large influence of the nonlinear terms.

Nonlinear dynamics for estimating the tip radius in atomic force microscopy

The accuracy of measurements in Amplitude Modulation Atomic Force Microscopy (AFM) is directly related to the geometry of the tip. The AFM tip is characterized by its radius of curvature, which could suffer from alterations due to repetitive mechanical contact with the surface. An estimation of the tip change would allow the user to assess the quality during imaging. In this work, we introduce a method for tip radius evaluation based on the nonlinear dynamic response of the AFM cantilever. A nonlinear fitting procedure is used to match several curves with softening nonlinearity in the noncontact regime. By performing measurements in this regime, we are able to maximize the influence of the tip radius on the AFM probe response, and this can be exploited to estimate with good accuracy the AFM tip radius.

Size- and temperature-dependent bending rigidity of graphene using modal analysis

The bending rigidity of two-dimensional (2D) materials is a key parameter for understanding the mechanics of 2D NEMS devices. The apparent bending rigidity of graphene membranes at macroscopic scale differs from theoretical predictions at micro-scale. This difference is believed to originate from thermally induced dynamic ripples in the atomically thin membrane. In this paper, we perform modal analysis to estimate the effective macroscopic bending rigidity of graphene membranes from the frequency spectrum of their Brownian motion. Our method is based on fitting the resonance frequencies obtained from the Brownian motion in molecular dynamics simulations, to those obtained from a continuum mechanics model, with bending rigidity and pretension as the fit parameters. In this way, the effective bending rigidity of the membrane and its temperature and size dependence, are extracted, while including the effects of dynamic ripples and thermal fluctuations. The proposed method provides a framework for estimating the macroscopic mechanical properties at high frequencies in other two-dimensional nano-structures at finite temperatures.

Modal analysis for density and anisotropic elasticity identification of adsorbates on microcantilevers

Physical characteristics such as mass and stiffness of biological objects are emerging as new markers for severe diseases. Micromechanical resonators can be used to quantify multiple of these characteristics simultaneously. In this paper, we propose a methodology that utilizes higher flexural modes of vibration to perform simultaneous characterization of the density and elastic modulus of adsorbates. To demonstrate this concept, a polymeric block with a known dimension and anisotropy is written directly on the cantilever surface using a two-photon polymerization technique and characterised by modal analysis. Our method captures the effective bending stress exerted by non-isotropic materials which is masked in the atomic force microscopy indentation technique.Physical characteristics such as mass and stiffness of biological objects are emerging as new markers for severe diseases. Micromechanical resonators can be used to quantify multiple of these characteristics simultaneously. In this paper, we propose a methodology that utilizes higher flexural modes of vibration to perform simultaneous characterization of the density and elastic modulus of adsorbates. To demonstrate this concept, a polymeric block with a known dimension and anisotropy is written directly on the cantilever surface using a two-photon polymerization technique and characterised by modal analysis. Our method captures the effective bending stress exerted by non-isotropic materials which is masked in the atomic force microscopy indentation technique.

HPC Methods for Domains of Attraction Computation

The work is devoted to the development of efficient parallel algorithms for the computation of large-scale basins of attraction. Since the required computational resources increase exponentially with the dimension of a dynamical system, it is common to get into memory saturation or in a secular elaboration time.This paper presents a code, based on a cell mapping method, that evaluates basins of attraction for high-dimensional systems by exploiting the parallel programming. The proposed approach, by using a double-step algorithm, permits, i) to fully determine the basins in all the dimensions ii) to evaluate 2D Poincare sections of the system. The code is described in all its parts: the shell, in charge of the core management, permits to split over a multi-core environment the computing domain, it carries out an efficient use of the memory.A preliminary analysis of the performances is undertaken also by considering different dimensional grids; the optimal balance between computing cores and memory management cores is studied.Copyright

The Homotopy-Analysis Approach for the Dynamical Study of a Microbeam Modeled on the Basis of the Strain-Gradient Theory

The purpose of this work is to investigate the nonlinear dynamics of a slender microbeam, modeled within the framework of the strain-gradient elasticity, adopting the homotopy analysis method (HAM). The microbeam is fixed at both edges and a geometric nonlinearity is also present accounting for the axial stretch. To attain an accurate and reliable model, so that the error is spread smoothly over the domain, a Chebyshev approximation for the nonlinear electric actuation term is introduced. A reduced-order model for the governing equation of motion, represented by an high-order nonlinear partial differential equation, is obtained. Then, the single-degree-of-freedom model is studied to find an analytical approximated solution. The free vibrations of the beam are investigated and the effects of several parameters, such as the applied axial load, are analyzed. Particular attention is also paid to find the influence of the high-order length scale material parameters, introduced by the non-classical theory, that progressively modify the oscillating behaviour. The results on the nonlinear phenomena, show both an hardening and a softening behaviour, in competition between them, varying the beam parameters. A numerical solution, obtained by a 4th order Runge Kutta algorithm, is also proposed as a benchmark for the analytical results.Copyright

The Use of the Strain Gradient Elasticity Theory in the Electrically Actuated Microbeam Problem: An Investigation on the Static and the Dynamic Response

The main goal of this work is to investigate the static and the dynamic behaviour of an electrically-actuated microbeam modelled by means of the strain gradient elasticity theory. Considering the nonlinearities due to the mid-plane stretch and of the applied electric force, a numerical solution for the static response is obtained by the differential quadrature method including also the features introduced by the non-classical continuum theory. A parametric analysis confirms the well-known hardening effect generated by a tensile applied axial load and highlights the variation of the deflection with respect to the high-order material parameters. The reliability of the considered 1-d model is improved by taking also into account a correction for the electrical fringing field effects that slightly modifies the solution. Regarding the dynamics, a single-degree-of-freedom model is studied and the variation of the eigenfunction, used in the discretized problem, with respect to the higher-order length scale parameters, is analysed. The results, carried out from the 4th order Runge-Kutta numerical integration, show the softening/hardening behaviour of the device varying the beam model parameters.Copyright