Axial Vibration Confinement in Nonhomogenous Rods

A design methodology for the vibration confinement of axial vibrations in nonhomogenous rods is proposed. This is achieved by a proper selection of a set of spatially dependent functions characterizing the rod material and geometric properties. Conditions for selecting such properties are established by constructing positive Lyapunov functions whose derivative with respect to the space variable is negative. It is shown that varying the shape of the rod alone is sufficient to confine the vibratory motion. In such a case, the vibration confinement requires that the eigenfunctions be exponentially decaying functions of space, where the notion of spatial domain stability is introduced as a concept dual to that of the time domain stability. It is also shown that vibration confinement can be produced if the rod density and/or stiffness are varied with respect to the space variable while the cross-section area is kept constant. Several case studies, supporting the developed conditions imposed on the spatially dependent functions for vibration confinement in vibrating rods, are discussed. Because variation in the geometric and material properties might decrease the critical buckling loads, we also discuss the buckling problem

Axial vibration confinement in nonhomogeneous rods

Abstract. A design methodology for the vibration confinement of axial vibrations in nonhomogenous rods is proposed. This is achieved by a proper selection of a set of spatially dependent functions characterizing the rod material and geometric properties. Conditions for selecting such properties are established by constructing positive Lyapunov functions whose derivative with respect to the space variable is negative. It is shown that varying the shape of the rod alone is sufficient to confine the vibratory motion. In such a case, the vibration confinement requires that the eigenfunctions be exponentially decaying functions of space, where the notion of spatial domain stability is introduced as a concept dual to that of the time domain stability. It is also shown that vibration confinement can be produced if the rod density and/or stiffness are varied with respect to the space variable while the cross-section area is kept constant. Several case studies, supporting the developed conditions imposed on the spatially dependent functions for vibration confinement in vibrating rods, are discussed. Because variation in the geometric and material properties might decrease the critical buckling loads, we also discuss the buckling problem

Confinement of Vibrations in Variable-Geometry Nonlinear Flexible Beam

In this paper, we propose a novel strategy for controlling a flexible nonlinear beam with the confinement of vibrations. We focus principally on design issues related to the passive control of the beam by proper selection of its geometrical and physical parameters. Due to large deflections within the regions where the vibrations are to be confined, we admit a nonlinear model that describes with precision the beam dynamics. In order to design a set of physical and geometrical parameters of the beam, we first formulate an inverse eigenvalue problem. To this end, we linearize the beam model and determine the linearly assumed modes that guarantee vibration confinement in selected spatial zones and satisfy the boundary conditions of the beam to be controlled. The approximation of the physical and geometrical parameters is based on the orthogonality of the assumed linear mode shapes. To validate the strategy, we input the resulting parameters into the nonlinear integral-partial differential equation that describes the beam dynamics. The nonlinear frequency response curves of the beam are approximated using the differential quadrature method and the finite difference method. We confirm that using the linear model, the strategy of vibration confinement remains valid for the nonlinear beam

Nonlinear Dynamical analysis of an AFM tapping mode microcantilever beam

We focus in this paper on the modeling and dynamical analysis of a tapping mode atomic force microscopy (AFM) microcantilever beam. This latter is subjected to a harmonic excitation of its base displacement and to Van der Waals and DMT contact forces at its free end. For AFM design purposes, we derive a mathematical model for accurate description of the AFM microbeam dynamics. We solve the resulting equations of motions and associated boundary conditions using the Galerkin method. We find that using one-mode approximation in tapping mode operating in the neighborhood of the contact region one-mode approximation may lead to erroneous results