Farbod Alijani

Bending Analysis of Symmetrically Laminated Cylindrical Panels Using the Extended Kantorovich Method

In this study, the bending of cross-ply symmetrically laminated cylindrical panels is investigated using the extended Kantorovich method (EKM). Following the classical Kirchhoff-Love assumptions, the governing partial differential equations (PDEs) of the problem are converted to a couple of systems of ordinary differential equations (ODEs). The resulted sets of ODEs are then solved, with exact analytical solutions, iteratively using arbitrary functions as initial guesses. Results for a clamped composite cylindrical panel under uniform, linear and nonlinear varying distributed loadings are presented. It is shown that the method provides accurate predictions for both displacement and stress components with very fast convergence and also initial guesses have no influence on the final results. Accuracy of the final results is investigated by comparison with the finite element and other results available in the literature.

Nonlinear dynamic characterization of two-dimensional materials

D. Davidovikj, F. Alijani, S. J. Cartamil-Bueno, H. S. J. van der Zant, M. Amabili, and P. G. Steeneken Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street W. Montreal, Quebec, Canada, H3A 2K6

CHAOTIC VIBRATIONS IN FUNCTIONALLY GRADED DOUBLY CURVED SHELLS WITH INTERNAL RESONANCE

Chaotic vibrations of functionally graded doubly curved shells subjected to concentrated harmonic load are investigated. It is assumed that the shell is simply supported and the edges can move freely in in-plane directions. Donnells nonlinear shallow shell theory is used and the governing partial differential equations are obtained in terms of shells transverse displacement and Airys stress function. By using Galerkins procedure, the equations of motion are reduced to a set of infinite nonlinear ordinary differential equations with cubic and quadratic nonlinearities. A bifurcation analysis is carried out and the discretized equations are integrated at (i) fixed excitation frequencies and variable excitation amplitudes and (ii) fixed excitation amplitudes and variable excitation frequencies. In particular, Gears backward differentiation formula (BDF) is used to obtain bifurcation diagrams, Poincare maps and time histories. Furthermore, maximum Lyapunov exponent and Lyapunov spectrum are obtained to classify the rich dynamics. It is revealed that the shell may exhibit complex behavior including sub-harmonic, quasi-periodic and chaotic response when subjected to large harmonic excitations.

Experimental characterization of graphene by electrostatic resonance frequency tuning

In the last decade, graphene membranes have drawn tremendous attention due to their potential application in Nano-Electro-Mechanical Systems. In this paper, we show that the frequency response curves of graphene resonators are powerful tools for their dynamic characterization and for extracting their equivalent Youngs modulus. For this purpose, vibrations of an electrostatically actuated circular graphene membrane are studied both experimentally and numerically. The experiments reveal the dependency of the linear and nonlinear resonance frequency of the nano-resonator on the driving DC and AC voltages. A numerical model is proposed based on the nonlinear membrane theory, and by fitting the numerically calculated change in resonance frequency due to the DC voltage to those of the experimental observations, the Youngs modulus is determined. It is shown that by using the obtained equivalent Youngs modulus, the numerical model can accurately describe the nonlinear dynamics of the graphene membrane in other...

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.

Static and Dynamic Pull-In of Electrically Actuated Circular Micro-Membranes

Micro-Electro-Mechanical devices have shown enormous popularity in engineering devices as sensors and actuators. In this paper, the instability, i.e. the dynamic pull-in behavior, of an electrically actuated circular micro-membrane is studied. In order to investigate the periodic solutions, detect bifurcations and follow branches of the solution, the non-linear equation of motion is derived using an energy approach, and, is solved by using a pseudo arc-length continuation and collocation technique. It has been shown that, both hardening and/or softening nonlinear responses could emerge depending on the applied DC voltage. The results indicate that the critical load parameters, namely DC and AC voltages and the excitation frequency, have a major influence on the pull-in characteristics of the micro-membrane. The results reveal different dynamic pull-in mechanisms. In addition, they accurately show the decrease of the pull-in voltage due to dynamic loading.The proposed approximate solution is very fast and robust for detecting the pull-in instability. It allows observation of both global and local softening behavior even close to dynamic pullin, where the resonance frequency is almost equal to zero.Copyright

Opto-thermally excited multimode parametric resonance in graphene membranes

In the field of nanomechanics, parametric excitations are of interest since they can greatly enhance sensing capabilities and eliminate cross-talk. Above a certain threshold of the parametric pump, the mechanical resonator can be brought into parametric resonance. Here we demonstrate parametric resonance of suspended single-layer graphene membranes by an efficient opto-thermal drive that modulates the intrinsic spring constant. With a large amplitude of the optical drive, a record number of 14 mechanical modes can be brought into parametric resonance by modulating a single parameter: the pre-tension. A detailed analysis of the parametric resonance allows us to study nonlinear dynamics and the loss tangent of graphene resonators. It is found that nonlinear damping, of the van der Pol type, is essential to describe the high amplitude parametric resonance response in atomically thin membranes.Robin J. Dolleman,1, ∗ Samer Houri,1, 2 Abhilash Chandrashekar,3 Farbod Alijani,3 Herre S. J. van der Zant,1 and Peter G. Steeneken1, 3, † Kavli Institute of Nanoscience, Delft University of Technology, Lorentzweg 1, 2628 CJ, Delft, The Netherlands NTT Basic Research Laboratories, NTT Corporation, 3-1, Morinosato Wakamiya, Atsugi, Kanagawa, 243-0198, Japan Department of Precision and Microsystems Engineering, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The Netherlands

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.

Chaos: The speed limiting phenomenon in dynamic atomic force microscopy

This paper investigates the closed-loop dynamics of the Tapping Mode Atomic Force Microscopy using a new mathematical model based on the averaging method in Cartesian coordinates. Experimental and numerical observations show that the emergence of chaos in conventional tapping mode AFM strictly limits the imaging speed. We show that, if the controller of AFM is tuned to be faster than a certain threshold, the closed-loop system exhibits a chaotic behavior. The presence of chaos in the closed-loop dynamics is confirmed via bifurcation diagrams, Poincare sections, and Lyapunov exponents. Unlike the previously detected chaos due to attractive forces in the AFM, which can be circumvented via simple changes in operation parameters, this newly identified chaos is seemingly inevitable and imposes an upper limit for the closed-loop bandwidth of the AFM.