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Biomechanics And Motor Control Of Human Movement

Advanced Mathematical Methods for Scientists and Engineers

Nonlinear Oscillations: Exact Solutions and their Approximations

Zeros of orthogonal polynomials

Polynomials orthogonal on the unit circle

This paper presents the class of nonuniform beams and nonuniform axisymmetrical circular plates whose boundary value problems of free transverse vibrations and free transverse axisymmetrical vibrations, respectively, have been identified to be eigenvalue singular problems of orthogonal polynomials. Recent published results regarding a fourth order differential equation and eigenvalue singular problem of classical orthogonal polynomials allowed this study, which extends the class of nonuniform beams and circular nonuniform plates having exact solutions for the problem of free transverse vibrations. The geometry of the elements belonging to the class presented in this paper consists of beams convex parabolic thickness variation and polynomial width variation with the axial coordinate, and plates of convex parabolic thickness variation with the radius. Two boundary value problems of transverse vibrations of beams are reported: 1) complete beam (sharp at either end) with free-free boundary conditions, and 2) half-beam, i.e. a half of the symmetric complete beam, with the large end hinged and sharp end free. The boundary value problem of circular complete plate (zero thickness at zero and outer radii) with free-free boundary conditions has been also reported. For all these boundary value problems the exact mode shapes were Jacobi polynomials and the exact dimensionless natural frequencies were found from the eigenvalues of the eigenvalue singular problems of orthogonal polynomials.Copyright © 2004 by ASME

Exact Solutions for Transverse Vibrations of Beams and Plates of Parabolic Thickness Variation

Forcespinning™ is a novel method that makes use of centrifugal forces to produce nanofibers rapidly and at high yields. A 2D computational Forcespinning™ viscous fluid dynamics model is developed, that improves on previous models. The fluid dynamics equations are solved using themethod of multiple scales along with the finite difference method, and including slender-jet theory assumptions. The effects that the Reynolds (Re) number has on the resulting fiber trajectory, radius, and tangential velocity are presented.Copyright © 2012 by ASME

Influence of Viscosity on Forcespinning™ Dynamics

This paper deals with electrostatically actuated Carbon Nano-Tubes (CNT) cantilevers using Reduced Order Model (ROM) method. Forces acting on the CNT cantilever are electrostatic, van der Waals, and damping. The van der Waals forces are significant for values of 50 nm or lower of the gap between the CNT and the ground plate. As both forces electrostatic and van der Waals are nonlinear, and the CNT electrostatic actuation is given by AC voltage, the CNT undergoes nonlinear parametric dynamics. The Method of Multiple Scales (MMS), and ROM are used to investigate the system under soft excitations and/or weak nonlinearities. The frequency-amplitude and frequency-phase behaviors are found in the case of parametric resonance.Copyright © 2013 by ASME

Reduced Order Model of Parametric Resonance of Electrostatically Actuated CNT Cantilever Resonators

This paper deals with the mass deposition influence on the natural frequencies of nonuniform cantilever resonator sensors of linear and parabolic thickness. Resonator sensitivity, defined as fraction of change in frequency per fraction of change in thickness deposition and relative density, was found. A constant thickness mass deposition on all four lateral surfaces of the cantilever of rectangular cross-section was assumed. Euler-Bernoulli theory was used, so only slender beams were considered. Mass deposition on the free end surface of the beams was neglected. The film thickness was considered very small compared to any beam dimension. The film had no contribution to the beam stiffness, only to the mass. Results show that for the same thickness deposition, the sensitivity in the first mode of beams of linear thickness is 2.5 to 3.5 higher when compared to uniform beams. For beams of parabolic thickness variation the relative sensitivity ranges between 1.5 and 2.1.Copyright © 2006 by ASME

Sensitivity of Non-Uniform Beams in Depositing Mass Process

This paper investigates the amplitude–voltage response of parametric resonance of coaxial vibrations of double-walled carbon nanotubes (DWCNTs) under electrostatic actuation. The system under investigation consists of a DWCNT parallel to a ground plate and under AC voltage. This voltage produces a nonlinear electrostatic force leading the DWCNT into vibrations. There is a nonlinear intertube van der Waals force between the two coaxial carbon-nanotubes. In coaxial vibration, the two concentric nanotubes move together synchronously. The AC frequency is near the fundamental coaxial natural frequency of the DWCNT. This leads to parametric resonance. The case of small damping, soft electrostatic actuation, and DWCNTs with a high length to diameter ratio (Euler–Bernoulli beam model), is considered. Modal analysis is performed to decouple the equations of free vibrations in their linear part. The solution of the nonlinear problem is then found in terms of modal coordinates. Reduced Order Models (ROMs) using from one to five modes of vibration are used for investigation. Three methods are used to solve these models, (1) the Method of Multiple Scales used to solve the ROM using one mode of vibration, (2) continuation and bifurcation analysis of the five modes of vibration (5T) Reduced Order Model (ROM) using AUTO-07P, and (3) numerical integration of 5T ROM using Matlab. All models and methods are in excellent agreement for amplitudes lower than 0.4 of the gap, while at amplitudes larger than 0.4 only 5T-ROM provide reliable results. The effects of detuning frequency and damping on the amplitude–voltage response of DWCNTs under electrostatic actuation are reported. The importance of the results in this paper are the effect of damping and detuning frequency on the subcritical and supercritical bifurcations, as they define the voltage intervals DWCNT reaches nonzero steady-state amplitudes. • Electrostatically actuated Double Wall Carbon Nanotube (DWCNT), soft damping. • Amplitude–voltage response of parametric resonance. • Soft excitations, AC frequency near natural frequency. • MMS, numerical integration of ROMs, continuation and bifurcation analysis. • Damping and frequency effects on the amplitude–voltage response. 

Dumitru I. Caruntu

Papers

Exact Solutions for Transverse Vibrations of Beams and Plates of Parabolic Thickness Variation

Influence of Viscosity on Forcespinning™ Dynamics

Reduced Order Model of Parametric Resonance of Electrostatically Actuated CNT Cantilever Resonators

Sensitivity of Non-Uniform Beams in Depositing Mass Process

Coaxial vibrations of electrostatically actuated DWCNT resonators: Amplitude-voltage response of parametric resonance