The R-x, R2-x Planes; Some New Ideas for the Study of Nonlinear Systems

Use of some new planes such as the R-x, R2-x (where R represents in the n-dimensional phase space, the radius vector from the origin to any point on the trajectory described by the system) is suggested for analysis of nonlinear systems of any kind. The stability conditions in these planes are given. For easy understanding of the method, the transformation from the phase plane to the R-x, R2-x planes is brought out for second-order systems. In general, while these planes serve as useful as the phase plane, they have proved to be simpler in determining quickly the general behavior of certain classes of second-order nonlinear systems. A chart and a simple formula are suggested to evaluate time easily from the R-x and R2-x trajectories, respectively. A means of solving higher-order nonlinear systems is also illustrated. Finally, a comparative study of the trajectories near singular points on the phase plane and on the new planes is made.

The x¿n-x Plane for Analysis of Certain Second-Order Nonlinear Systems

Analysis of certain second-order nonlinear systems, not easily amenable to the phase-plane methods, and described by either of the following differential equations xÿn-2ÿ+ f(x)xÿ2n+g(x)xÿn+h(x)=0 ÿ+f(x)xÿn+h(x)=0 n≫0 can be effected easily by drawing the entire portrait of trajectories on a new plane; that is, on one of the xÿnÿx planes. Simple equations are given to evaluate time from a trajectory on any of these n planes. Poincaré's fundamental phase plane xÿÿx is conceived of as the simplest case of the general xÿnÿx plane.

Techniques for Analysis of Certain Nonlinear Systems

This paper suggests the use of simple transformations like ÿ=kx, kx2 for second-order nonlinear differential equations to effect rapid plotting of the phase-plane trajectories. The method is particularly helpful in determining quickly the trajectory slopes along simple curves in any desired region of the phase plane. New planes such as the tÿ-x, tÿ2-x are considered for the study of some groups of nonlinear time-varying systems. Suggestions for solving certain higher-order nonlinear systems are also made.

Mechanisms of summer intraseasonal sea surface temperature oscillations in the Bay of Bengal

Intraseasonal variations (ISV) of sea surface temperature (SST) in the Bay of Bengal (BoB) is highest in its northwestern part. An Indian Ocean model forced by QuikSCAT winds and climatological river discharge (QR run) reproduces ISV of SST, albeit with weaker magnitude. Air-sea fluxes, in the presence of a shallow mixed layer, efficiently effect intraseasonal SST fluctuations. Warming during intraseasonal events is smaller (<1°C) for June - July period and larger (1.5° to 2°C) during September, the latter due to a thinner mixed layer. To examine the effect of salinity on ISV, the model was run by artificially increasing the salinity (NORR run) and by decreasing it (MAHA10 run). In NORR, both rainfall and river discharge were switched off and in MAHA10 the discharge by river Mahanadi was increased tenfold. The spatial pattern of ISV as well as its periodicity was similar in QR, NORR and MAHA10. The ISV was stronger in NORR and weaker in MAHA10, compared to QR. In NORR, both intraseasonal warming and cooling were higher than in QR, the former due to reduced air-sea heat loss as the mean SST was lower, and the latter due to enhanced subsurface processes resulting from weaker stratification. In MAHA10, both warming and cooling were lower than in QR, the former due to higher air-sea heat loss owing to higher mean SST, and the latter due to weak subsurface processes resulting from stronger stratification. These model experiments suggest that salinity effects are crucial in determining amplitudes of intraseasonal SST variations in the BoB.

Novel nonlinear opticalphenomenain nematic liquid crystals

In normal materials, the nonlinear optical effects arise from nonlinearities in the polarisabilities of the constituent atoms or molecules. On the other hand the nonlinear optical effects in liquid crystals arise from totally different processes. Also they occur at relatively low laser intensities. In a laser field a liquid crystal exhibits many novel and interesting nonlinear optical effects. In addition we also find laser field induced effects that are peculiar to liquid crystals, like structural transformations, orientational transitions, modulated structures and phase transitions, to name a few. Here we dwell upon a few of these interesting and important nonlinear optical phenomena that exist in nematic liquid crystals.

Smooth DMS-FEM: A new approach to solving nearly incompressible nonlinear elasto-static problems

The DMS-FEM, which enables functional approximations with C(1) or still higher inter-element continuity within an FEM-based meshing of the domain, has recently been proposed by Sunilkumar and Roy [39,40]. Through numerical explorations on linear elasto-static problems, the method was found to have conspicuously superior convergence characteristics as well as higher numerical stability against locking. These observations motivate the present study, which aims at extending and exploring the DMS-FEM to (geometrically) nonlinear elasto-static problems of interest in solid mechanics and assessing its numerical performance vis-a-vis the FEM. In particular, the DMS-FEM is shown to vastly outperform the FEM (presently implemented through the commercial software ANSYS (R)) as the former requires fewer linearization and load steps to achieve convergence. In addition, in the context of nearly incompressible nonlinear systems prone to volumetric locking and with no special numerical artefacts (e.g. stabilized or mixed weak forms) employed to arrest locking, the DMS-FEM is shown to approach the incompressibility limit much more closely and with significantly fewer iterations than the FEM. The numerical findings are suggestive of the important role that higher order (uniform) continuity of the approximated field variables play in overcoming volumetric locking and the great promise that the method holds for a range of other numerically ill-conditioned problems of interest in computational structural mechanics. (C) 2011 Elsevier Ltd. All rights reserved.

Synthesis and nonlinear optical properties of Lead Telluride nanorods

Lead Telluride (PbTe) nanorods have been uniformly grown on silicon substrates, using the thermal evaporation technique under high vacuum conditions. The structural and morphological studies are done using X-ray diffraction and scanning electron microscopy. Optical nonlinearity studies using the open aperture z-scan employing 5 ns and 100 fs laser pulses reveal a three-photon type absorption. For nanosecond excitation the nonlinear absorption coefficients (gamma) are in the order of 10(-22) m(3) W-2 and for femtosecond excitation it is in the order of 10(-29) m(3) W-2. The role of free carriers and excitons in causing the nonlinearity in both excitation time domains is discussed. Results indicate that PbTe nanorods are good optical limiters with potential device applications. (C) 2011 Elsevier B.V. All rights reserved.

Fuzzy logic based TCSC supplementary controller for damping of inter area oscillations in power system

This paper describes an application of a FACTS supplementary controller for damping of inter area oscillations in power systems. A fuzzy logic controller is designed to regulate a thyristor controlled series capacitor (TCSC) in a multimachine environment to produce additional damping in the system. Simultaneous application of the excitation controller and proposed controller is also investigated. Simulation studies have been done with different types of disturbances and the results are shown to be consistent with the expected performance of the supplementary controller.

Beat frequency oscillations in the output voltage of high voltage DC power supplies under pulsed loading

High voltage power supplies for radar applications are investigated, which are subjected to pulsed load (125 kHz and 10% duty cycle) with stringent specifications (<0.01% regulation, efficiency>85%, droop<0.5 V/micro-sec.). As good regulation and stable operation requires the converter to be switched at much higher frequency than the pulse load frequency, transformer poses serious problems of insulation failure and higher losses. Few converter topologies are proposed to tackle these problems. A study is made regarding the beat frequency oscillations that may exist with pulsed loading. It is illustrated with respect to the proposed converter topologies. Methods are proposed to eliminate or minimize these oscillations.

Dynamical behavior in the nonlinear rheology of surfactant solutions

Several surfactant molecules self-assemble in solution to form long, flexible wormlike micelles which get entangled with each other, leading to viscoelastic gel phases. We discuss our recent work on the rheology of such a gel formed in the dilute aqueous solutions of a surfactant CTAT. In the linear rheology regime, the storage modulus G′(ω) and loss modulus G″(ω) have been measured over a wide frequency range. In the nonlinear regime, the shear stress σ shows a plateau as a function of the shear rate math above a certain cutoff shear rate mathc. Under controlled shear rate conditions in the plateau regime, the shear stress and the first normal stress difference show oscillatory time-dependence. The analysis of the measured time series of shear stress and normal stress has been done using several methods incorporating state space reconstruction by embedding of time delay vectors. The analysis shows the existence of a finite correlation dimension and a positive Lyapunov exponent, unambiguously implying that the dynamics of the observed mechanical instability can be described by that of a dynamical system with a strange attractor of dimension varying from 2.4 to 2.9.

Nonlinear cyclic load analysis for lateral response of batter piles in soft clay with a rigorous degradation model

Nonlinear analysis of batter piles in soft clay is performed using the finite element technique. As the batter piles are not only governed by lateral load but also axial load, the effect of P- Delta moment and geometric stiffness matrix is included in the analysis. For implementing the nonlinear soil behavior, reduction in soil strength (degradation), and formation of gap with number of load cycles, a numerical model is developed where a hyperbolic relation is adopted for the soil in static condition and hyperbolic relation considering degradation and gap for cyclic load condition. The numerical model is validated with published experimental results for cyclic lateral loading and the hysteresis loops are developed to predict the load-deflection behavior and soil resistance behavior during consecutive cycles of loading. This paper highlights the importance of a rigorous degradation model for subsequent cycles of loading on the pile-soil system by a hysteretic representation.

Structural, vibrational and thermal studies of a new nonlinear optical material: L-Asparagine-L-tartaric acid

Crystals of a new nonlinear optical (NLO) material, viz., L-asparagine-L-tartaric acid (LALT)(1) were grown by slow evaporation of an aqueous solution containing equimolar concentrations of L-asparagine and t-tartaric acid. The structure of the title compound which crystallizes in the non-centrosymmetric monoclinic space group P2(1) consists of a molecule of L-asparagine and a molecule of free L-tartaric acid both of which are interlinked by three varieties of H-bonding interactions namely O-H center dot center dot center dot O, N-H center dot center dot center dot O and C-H center dot center dot center dot O. The UV-Vis-NIR spectrum of 1 reveals its transparent nature while the vibrational spectra confirm the presence of the functional groups in 1. The thermal stability and second harmonic generation (SHG) conversion efficiency of 1 were investigated. (C) 2012 Elsevier B.V. All rights reserved.

Nonlinear viscoelasticity of freestanding and polymer-anchored vertically aligned carbon nanotube foams

Vertical arrays of carbon nanotubes (VACNTs) show unique mechanical behavior in compression, with a highly nonlinear response similar to that of open cell foams and the ability to recover large deformations. Here, we study the viscoelastic response of both freestanding VACNT arrays and sandwich structures composed of a VACNT array partially embedded between two layers of poly(dimethylsiloxane) (PDMS) and bucky paper. The VACNTs tested are similar to 2 mm thick foams grown via an injection chemical vapor deposition method. Both freestanding and sandwich structures exhibit a time-dependent behavior under compression. A power-law function of time is used to describe the main features observed in creep and stress-relaxation tests. The power-law exponents show nonlinear viscoelastic behavior in which the rate of creep is dependent upon the stress level and the rate of stress relaxation is dependent upon the strain level. The results show a marginal effect of the thin PDMS/bucky paper layers on the viscoelastic responses. At high strain levels (epsilon - 0.8), the peak stress for the anchored CNTs reaches similar to 45 MPa, whereas it is only similar to 15MPa for freestanding CNTs, suggesting a large effect of PDMS on the structural response of the sandwich structures. (C) 2012 American Institute of Physics. http://dx.doi.org/10.1063/1.3699184]

Wrinkled and slack membranes: nonlinear 3D elasticity solutions via smooth DMS-FEM and experiment

The smooth DMS-FEM, recently proposed by the authors, is extended and applied to the geometrically nonlinear and ill-posed problem of a deformed and wrinkled/slack membrane. A key feature of this work is that three-dimensional nonlinear elasticity equations corresponding to linear momentum balance, without any dimensional reduction and the associated approximations, directly serve as the membrane governing equations. Domain discretization is performed with triangular prism elements and the higher order (C1 or more) interelement continuity of the shape functions ensures that the errors arising from possible jumps in the first derivatives of the conventional C0 shape functions do not propagate because the ill-conditioned tangent stiffness matrices are iteratively inverted. The present scheme employs no regularization and exhibits little sensitivity to h-refinement. Although the numerically computed deformed membrane profiles do show some sensitivity to initial imperfections (nonplanarity) in the membrane profile needed to initiate transverse deformations, the overall patterns of the wrinkles and the deformed shapes appear to be less so. Finally, the deformed profiles, computed through the DMS FEM-based weak formulation, are compared with those obtained through an experiment on an ultrathin Kapton membrane, wherein wrinkles form because of the applied boundary displacement conditions. Comparisons with a reported experiment on a rectangular membrane are also provided. These exercises lend credence to the feasibility of the DMS FEM-based numerical route to computing post-wrinkled membrane shapes. Copyright (c) 2012 John Wiley & Sons, Ltd.

Synchronization of globally coupled two-state stochastic oscillators with a state-dependent refractory period

We present a model of identical coupled two-state stochastic units, each of which in isolation is governed by a fixed refractory period. The nonlinear coupling between units directly affects the refractory period, which now depends on the global state of the system and can therefore itself become time dependent. At weak coupling the array settles into a quiescent stationary state. Increasing coupling strength leads to a saddle node bifurcation, beyond which the quiescent state coexists with a stable limit cycle of nonlinear coherent oscillations. We explicitly determine the critical coupling constant for this transition.