Oded Gottlieb

Continuum model of dispersion caused by an inherent material characteristic length

Modifications of the Helmholtz free energy and the stress associated with general constitutive equations of a simple continuum are proposed to model dispersive effects of an inherent material characteristic length. These modifications do not alter the usual restrictions on the unmodified constitutive equations imposed by the first and second laws of thermodynamics. The special case of a thermoelastic compressible Newtonian viscous fluid is considered with attention focused on uniaxial strain. Within this context, the linearized problems of wave propagation in an infinite media and free vibrations of a finite column are considered for the simple case of elastic response. It is shown that the proposed model predicts the dispersive effects observed in wave propagation through a chain of springs and masses as the wavelength decreases. Also, the nonlinear problems of steady wave propagation of a soliton in the absence of viscosity and of a shock wave in the presence of viscosity are discussed. In particular it...

Nonlinear dynamics of a noncontacting atomic force microscope cantilever actuated by a piezoelectric layer

The nonlinear equations of motion for a silicon cantilever beam, covered by a piezoelectric lead–zirconate–titanate layer, subjected to a Lennard-Jones type boundary condition, are derived for voltage excitation. The Lagrangian of the system is obtained from the electric enthalpy density, including the virtual work of the Lennard-Jones potential, assuming the beam undergoes only small displacements. By application of Hamilton’s principle, the nonlinear equations of motion are consistently derived and truncated to third order for perturbation analysis. The evolution equations are obtained by the multiple scales method and periodic solutions to the equations of motion are determined and discussed with respect to different tip to sample distances. An analytically obtained frequency response function enables determination of the frequency shift of individually piezoactuated microbeams, which are proposed as fundamental elements of parallel atomic force microscopy, undergoing forced vibration in a dissipative ...

Performance of an AuPd micromechanical resonator as a temperature sensor

In this work we study the sensitivity of the primary resonance of an electrically excited microresonator for the possible usage of a temperature sensor. We find a relatively high normalized responsivity factor Rf=|TfdfdT|=0.37 with a quality factor of ∼105. To understand this outcome we perform a theoretical analysis based on experimental observation. We find that the dominant contribution to the responsivity comes from the temperature dependence of the tension in the beam. Subsequently, Rf is found to be inversely proportional to the initial tension. Corresponding to a particular temperature, the tension can be increased by applying a bias voltage.

Nonlinear dynamics of temporally excited falling liquid films

The two-dimensional spatiotemporal dynamics of falling thin liquid films on a solid vertical wall periodically oscillating in its own plane is studied within the framework of long-wave theory. A pertinent nonlinear evolution equation referred to as the temporally modulated Benney equation (TMBE) is derived and its solutions are investigated numerically. The bifurcation diagram of the Benney equation (BE) describing the film dynamics in the unforced regime is computed depicting the domains of linearly stable, linearly unstable bounded, and unbounded behaviors. The solutions obtained for film dynamics via the BE are compared to those documented for direct numerical simulations of the Navier–Stokes equations (NSE). The comparison demonstrates that the BE constitutes an accurate asymptotic reduction of the NSE in the domain preceding the transition to the regime of its unbounded solutions. It is found that periodic planar boundary excitation does not alter the fundamental unforced bifurcation structure and th...

Time resolved measurements of vortex-induced vibrations of a tethered sphere in uniform flow

The motion of a heavy tethered sphere and its wake were measured in a closed loop water channel using a time resolved, high-speed particle image velocimetry technique in a horizontal plane. Measurements were performed for nondimensional reduced velocities ranging from 2.8 to 31.1 that include three bifurcation regions. In order to analyze the vortex shedding characteristics, the directional swirling strength parameter was computed in addition to the vorticity as the former enables vortex identification. In the first bifurcation region, the sphere remained stationary and the wake was characterized by a train of hairpin vortices exhibiting symmetry in the vertical plane similar to visualization results obtained for stationary spheres. The second bifurcation region was characterized by large amplitude periodic oscillations transverse to the flow. Phase-averaged results for the swirling strength showed that although the shedding mechanism was identical for several reduced velocities, the phase at which vortic...

GLOBAL BIFURCATIONS OF NONLINEAR THERMOELASTIC MICROBEAMS SUBJECT TO ELECTRODYNAMIC ACTUATION

In this paper we formulate a nonlinear boundary-value problem describing the thermoelastic dynamics of a microbeam that is subject to a localized electrodynamic actuation and is operating in an ultra-high vacuum environment. A modal Galerkin projection reveals a planar homoclinic structure describing escape from a potential well that is perturbed by both thermoelastic damping and modulated periodic actuation. This structure is investigated via Melnikov analysis to shed light on possible existence of global bifurcations and chaotic transients.

Newtonian glass fiber drawing: Chaotic variation of the cross-sectional radius

A model of Newtonian glass fiber drawing at fixed temperature in the unsteady range (the draw ratio E>20.22) is considered. In this range under steady boundary conditions, as is well known, the draw resonance instability sets in, resulting in self-sustained oscillations. These oscillations lead to a periodic variation of the cross-sectional radius of the fiber. In the present work we consider the case where the spinline radius varies periodically. Such a variation may result from flow oscillations in the fiber forming channels in the direct-melt process, or from the variation of the preform cross-sectional radius in drawing of optical fibers. When this variation takes place in the range E>20.22, the self-sustained periodic oscillations of the draw resonance are replaced by quasiperiodic and periodic (mode-locked) subharmonic or (under the appropriate conditions) chaotic oscillations (strange attractors). The routes to chaos found in the present work include (1) smooth period doubling bifurcation of (any) mode-locked periodic solution, (2) abrupt explosions of quasiperiodic solutions. The predicted chaotic variation of the spinline radius at the winding mandrel may result in a similar variation of the cross-sectional radius of the solidified fibers.

Local and Global Bifurcation Analyses of a Spatial Cable Elastica

This paper focuses on a boundary value problem governing the equilibrium ofa slender cable subject to thrust, torsion, and gravity. In the absence of field (gravity) loading, this boundary value problem is integrable and admits periodic solutions describing planar and spatial equilibrium forms. A bifurcation analysis of the integrable problem reveals the conditions controlling local stability of periodic solutions and the existence of two limiting (bounding) homoclinic solutions. The addition of field (gravity) loading renders the boundary value problem nonintegrable. This effect is first investigated through perturbation of the limiting homoclinic solutions for weak gravity loading. Approximate existence conditions for aperiodic and spatially complex forms are determined using Melnikovs method. The effect of field loading is then re-evaluated through numerical solution of the original problem. Spatially complex solutions are determined that might mimic the loops and tangles sometimes found in underwater cables.

Internal resonance based sensing in non-contact atomic force microscopy

In this letter, the nonlinear dynamics of a non-uniform micro-cantilever for atomic force microscopy is investigated numerically for a non-contact mode of operation. A step-like heterogeneity in the cantilever longitudinal direction yields conditions for both 3:1 and 2:1 internal resonances that govern quasiperiodic energy transfer between the first and second structural bending modes. Thus, quasiperiodic micro-cantilever response can enable multiple function sensing, and possible increased accuracy of time-varying forces via single frequency base excitation.

STABILITY AND BIFURCATIONS OF PARAMETRICALLY EXCITED THIN LIQUID FILMS

We investigate the stability and bifurcations of parametrically excited thin liquid films. A recently derived nonlinear evolution equation for the two-dimensional spatio-temporal dynamics of falling liquid films on an oscillating vertical wall is expanded to low order Fourier modes. A fourth-order modal dynamical system is validated to yield the primary bifurcation structure of the fundamental falling film dynamics described by the Benney equation, and accurately predicts the quasi-periodic structure of the temporally modulated Benney equation (TMBE). The stability of fundamental steady and periodic solutions is analytically and numerically investigated so as to reveal the threshold for nonstationary and chaotic solutions corresponding to aperiodic modulated traveling waves. The reduced modal dynamical system enables construction of a comprehensive bifurcation structure, which is verified by numerical simulation of the evolution equation.