Suppression of hysteresis in a forced van der Pol–Duffing oscillator

This paper examines the suppression of hysteresis in a forced nonlinear self-sustained oscillator near the fundamental resonance. The suppression is studied by applying a rapid forcing on the oscillator. Analytical treatment based on perturbation analysis is performed to capture the entrainment zone, the quasiperiodic modulation domain and the hysteresis area as well. The analysis leads to a strategy for the suppression of hysteresis occurring between 1:1 frequency-locked motion and quasiperiodic response. This hysteresis suppression causes the disappearance of nonlinear effects leading to a smooth transition between the quasiperiodic and the frequency-locked responses. Specifically, it appears that a rapid forcing introduces additional apparent nonlinear stiffness which can effectively suppress hysteresis in a certain range of the rapid excitation frequency. This work was motivated by the important issue of controlling and eliminating hysteresis often undesirable in mechanical systems, in general, and in application to microscale devices, especially.     

Hysteresis suppression for primary and subharmonic 3:1 resonances using fast excitation

Nonlinear Dynamics - We analyze the effect of a fast harmonic excitation on hysteresis and on entrainment area in a forced van der Pol–Duffing oscillator near the primary and the 3:1...     

2:1 and 1:1 frequency-locking in fast excited van der Pol–Mathieu–Duffing oscillator

The frequency-locking area of 2:1 and 1:1 resonances in a fast harmonically excited van der Pol–Mathieu–Duffing oscillator is studied. An averaging technique over the fast excitation is used to derive an equation governing the slow dynamic of the oscillator. A perturbation technique is then performed on the slow dynamic near the 2:1 and 1:1 resonances, respectively, to obtain reduced autonomous slow flow equations governing the modulation of amplitude and phase of the corresponding slow dynamics. These equations are used to determine the steady state responses, bifurcations and frequency-response curves. Analysis of quasi-periodic vibrations is carried out by performing multiple scales expansion for each of the dependent variables of the slow flows. Results show that in the vicinity of both considered resonances, fast harmonic excitation can change the nonlinear characteristic spring behavior from softening to hardening and causes the entrainment regions to shift. It was also shown that entrained vibrations with moderate amplitude can be obtained in a small region near the 1:1 resonance. Numerical simulations are performed to confirm the analytical results.     

Predicting Homoclinic Bifurcations in Planar Autonomous Systems

An analytical method to predict the homoclinic bifurcation in a planar autonomous self-excited weakly nonlinear oscillator is presented. The method is mainly based on the collision between the periodic orbit undergoing the homoclinic bifurcation and the saddle fixed point. To illustrate the analytical predictive criteria, two typical examples are investigated. The results obtained in this work are then compared to Melnikov's technique and to a previous criterion based on the vanishing of the frequency. Numerical simulations are also provided.     

Heteroclinic connections in the 1:4 resonance problem using nonlinear transformation method

In this paper, the method of nonlinear time transformation is applied to obtain analytical approximation of heteroclinic connections in the problem of stability loss of self-oscillations near 1:4 resonance. As example, we consider the case of parametric and self-excited oscillator near the 1:4 subharmonic resonance. The method uses the unperturbed heteroclinic connection in the slow flow to determine conditions under which the perturbed heteroclinic connection persists. The results show that for small values of damping, the nonlinear time transformation method can predict well both the square and clover heteroclinic connection near the 1:4 resonance. The analytical finding is confirmed by comparisons to the results obtained by numerical simulations.     

Diacetylenes with Ionic‐Liquid‐Like Substituents: Associating a Polymerizing Cation with a Polymerizing Anion in a Single Precursor for the Synthesis of N‐Doped Carbon Materials

Imidazolium‐ and benzimidazolium‐substituted diacetylenes with bromide or nitrogen‐rich dicyanamide and tricyanomethanide anions were synthesized and used as precursors for the preparation of N‐doped carbon materials. On pyrolysis under argon at 800 °C both halide precursors afforded graphite‐like structures with nitrogen contents of about 8.5 %. When the dicyanamide and tricyanomethanide precursors were thermolyzed at the same temperature, graphite‐like structures were obtained that exhibit nitrogen contents in the range 17–20 wt %; thereby, the benefit of associating a polymerizing cation with a polymerizing anion in a single precursor was demonstrated. On pyrolysis at 1100 °C the nitrogen contents of the latter pyrolysates remain high (ca. 6 wt %). Adsorption measurements with krypton at 77 K indicated that the materials are nonporous. The highest electrical conductivity was observed for a pyrolysate with one of the lowest nitrogen contents, which also has the highest degree of graphitization. Thus, the quest for N‐rich carbons with high electrical conductivities should include both maximization of the nitrogen content and optimization of the degree of graphitization. Crystallographic investigation of the precursors and spectroscopic characterization of the pyrolysates prepared by heating at 220 °C indicate that construction of the final carbon framework does not involve the intermediate formation of a polydiacetylene.     

Stabilised finite-element methods for solving the level set equation with mass conservation

Finite-element methods are studied for solving moving interface flow problems using the level set approach and a stabilised variational formulation proposed in Touré and Soulaïmani (2012; Touré and Soulaïmani To appear in 2016 Touré, Mamadou Kabirou, and Azzeddine Soulaïmani. To appear in 2016. “Stabilized Finite Element Methods for Solving the Level Set Equation without Renitialization.” Computers &; Mathematics with Applications. doi:10.1016/j.camwa.2016.02.028[Crossref] , [Google Scholar]), coupled with a level set correction method. The level set correction is intended to enhance the mass conservation satisfaction property. The stabilised variational formulation (Touré and Soulaïmani 2012; Touré and Soulaïmani, To appear in 2016 Touré, Mamadou Kabirou, and Azzeddine Soulaïmani. To appear in 2016. “Stabilized Finite Element Methods for Solving the Level Set Equation without Renitialization.” Computers &; Mathematics with Applications. doi:10.1016/j.camwa.2016.02.028[Crossref] , [Google Scholar]) constrains the level set function to remain close to the signed distance function, while the mass conservation is a correction step which enforces the mass balance. The eXtended finite-element method (XFEM) is used to take into account the discontinuities of the properties within an element. XFEM is applied to solve the Navier–Stokes equations for two-phase flows. The numerical methods are numerically evaluated on several test cases such as time-reversed vortex flow, a rigid-body rotation of Zalesak's disc, sloshing flow in a tank, a dam-break over a bed, and a rising bubble subjected to buoyancy. The numerical results show the importance of satisfying global mass conservation to accurately capture the interface position.     

Numerical investigations of the XFEM for solving two-phase incompressible flows

This paper discusses the application of the extended finite element method (XFEM) to solve two-phase incompressible flows. The Navier–Stokes equations are discretised using the Taylor–Hood finite element. To capture the different discontinuities across the interface, kink or jump enrichments are used for the velocity and/or pressure fields. However, these enrichments may lead to an inappropriate combination of interpolations. Different polynomial enrichment orders and different enrichment functions are investigated; only the stable combination will be used afterward.

In cases with a surface tension force, the accuracy mainly relies on the precise computation of the normal and curvature. A novel method for computing normal vectors to the interface is proposed. This method employs successive mesh refinements inside the cut elements. Comparisons with analytical and numerical solutions demonstrate that the method is effective. Moreover, the mesh refinement improves the sub-integration in the XFEM and allows for a precise re-initialisation procedure.     

Effect of fast harmonic excitation on frequency-locking in a van der Pol–Mathieu–Duffing oscillator

We study the effect of high-frequency harmonic excitation on the entrainment area of the main resonance in a van der Pol–Mathieu–Duffing oscillator. An averaging technique is used to derive a self- and parametrically driven equation governing the slow dynamic of the oscillator. The multiple scales method is then performed on the slow dynamic near the main resonance to obtain a reduced autonomous slow flow equations governing the modulation of amplitude and phase of the slow dynamic. These equations are used to determine the steady state response, bifurcation and frequency–response curves. A second multiple scales expansion is used for each of the dependent variables of the slow flow to obtain slow slow flow modulation equations. Analysis of non-trivial equilibrium of this slow slow flow provides approximation of the slow flow limit cycle corresponding to quasi-periodic motion of the slow dynamic of the original system. Results show that fast harmonic excitation can change the nonlinear characteristic spring behavior and affect significantly the entrainment region. Numerical simulations are used to confirm the analytical results.