Hyperbolic Secant Function Research Articles

Collective Pitch Control with Active Tower Damping of a Wind Turbine by Using a Nonlinear PID Approach

The present work describes the results obtained from a nonlinear PID control (NPID) approach for the pitch control with top tower active damping of a large wind turbine. The control algorithm is implemented by using a well-known approach based on hyperbolic secant functions. A nonlinear PID controller and a nonlinear P controller are used for the collective pitch control and the active tower damping, respectively. As simulation platform, the 5-MW reference turbine developed by NREL and the software FAST are used. Results show that the NPID approach can provide significant improvements in the control performance in comparison to the classic control approach, in particular when the wind speed goes far from its rated value.

A semi-empirical model of the aquatic plants seasonal dynamics and its application for management of perennial macrophytes

This study proposes a simple yet practical semi-empirical approach for theory-guided fitting of biomass growth curves and prediction of biomass growth rate in aquatic plants. The model is based on a hyperbolic secant function, which describes the biomass growth and provides a straightforward solution for the biomass growth rate. The approach also provides an innovative theory-based method to predict the control of succession of plants since the maximum in the absolute growth rate of the aboveground biomass corresponds to the period when perennial macrophytes allocate biomass into their overwintering parts. Field experiments indicate that the biomass allocation pattern can be strongly affected by mechanical harvesting. For instance, the cutting scenario, which yielded the smallest mass and number of Sagittaria sagittifolia tubers per plant, had been performed at the thermal time values corresponding to the peak of aboveground biomass growth rate, the timing of which can be easily predicted using the model proposed in this study.

Enhancement of elastic wave energy harvesting using adaptive piezo-lens

This paper exploits an adaptive piezo-lens to improve the harvested power (energy) from traveling waves. The piezo-lens comprises a host plate and piezoelectric patches bonded on the plate surfaces. The piezoelectric patches are shunted with negative capacitance (NC) circuits. The spatial variation of the refractive index inside the piezo-lens domain is designed to fulfill a hyperbolic secant function by tuning the NC values. This design allows the piezo-lens to continuously bend the incident waves toward a designed focal point, resulting in an energy concentration zone with a high level of energy density. This energy concentration effect may be exploited to improve the harvested power from waves. In addition, the piezo-lens is tunable - the waves can be focused at different locations by designing the NC values. This tunability may make the harvesting systems incorporating a piezo-lens be adaptable to environment changes. The above expected practical interests of the piezo-lens for wave energy harvesting are discussed and verified in the paper. Fully coupled numerical models are developed to predict the dynamical responses of the piezoelectric systems.

Islanding detection technique based on inverse hyperbolic secant function

This study presents an inverse hyperbolic secant function based islanding detection technique for distribution network containing various types of distributed generations. The proposed scheme is based on rate of change of inverse hyperbolic secant function of negative sequence voltage signal. Different non-islanding events along with various islanding conditions have been simulated by modelling IEEE 34-bus system using PSCAD/EMTDC software package. The proposed scheme is able to detect islanding condition even with zero active and reactive power mismatches. Moreover, it eliminates non-detection zone completely and at the same time also remains stable during diversified non-islanding events including change in network topology. The simulation results indicate that the proposed scheme not only detects islanding condition rapidly, but also provides better stability in case of critical non-islanding events during which most of the existing schemes issue nuisance tripping. Furthermore, comparative evaluation of the proposed scheme with the recently proposed techniques in the literature clearly indicates superiority of the proposed scheme. At the end, issues related to practical implementation of the proposed scheme have also been discussed.

MULTI-FREQUENCY RADIO PROFILES OF PSR B1133+16: RADIATION LOCATION AND PARTICLE ENERGY

ABSTRACT The pulse profile of PSR B1133+16 is usually regarded as a conal double structure. However, its multi-frequency profiles cannot simply be fitted with two Gaussian functions, and a third component is always needed to fit the bridge region (between two peaks). This would introduce additional, redundant parameters. In this paper, through a comparison of five fitting functions (Gaussian, von Mises, hyperbolic secant, square hyperbolic secant, and Lorentz), it is found that the square hyperbolic secant function can best reproduce the profile, yielding an improved fit. Moreover, a symmetric 2D radiation beam function, instead of a simple 1D Gaussian function, is used to fit the profile. Each profile with either well-resolved or not-so-well-resolved peaks could be fitted adequately using this beam function, and the bridge emission between the two peaks does not need to be a new component. Adopting inclination and impact angles based on polarization measurements, the opening angle ( θ &mgr; 0 ?> ) of the radiation beam in a certain frequency band is derived from beam-function fitting. The corresponding radiation altitudes are then calculated. Based on multi-frequency profiles, we also computed the Lorentz factors of the particles and their dispersion at those locations in both the curvature-radiation and inverse-Compton-scattering models. We found that the Lorentz factors of the particles decrease rapidly as the radiation altitude increases. Besides, the radiation prefers to be generated in an annular region rather than the core region, and this needs further validation.

A half-discrete Hardy-Hilbert-type inequality related to hyperbolic secant function

By applying weight functions and technique of real analysis, a half-discrete Hardy-Hilbert-type inequality related to the kernel of hyperbolic secant function and a best possible constant factor are given. The equivalent forms, the operator expressions with the norm, the reverses, and some particular cases are also considered.

Statistical law for tropical cyclone motion in the Northwest Pacific Ocean

The statistical law that governs the drift velocity of tropical cyclones in the Northwest Pacific Ocean is investigated. The investigation is based on data published by China Meteorological Administration for historical tracks of 2146 cyclone events that occurred during 1949–2012. Empirical formulae are obtained to relate the magnitude, the direction, the meridional and zonal components of the averaged cyclone drift velocity with latitude. As the latitude effect is excluded, it is found that the cyclone drift velocity is governed by simple statistical laws, i.e. the magnitude and direction of the deviated drift velocity approximately satisfy a gamma distribution and a symmetric bimodal distribution, respectively, while the meridional and zonal components of the deviated drift velocity satisfy the same type of symmetric probability distribution represented by the hyperbolic secant function but with different deviations. The results obtained are potentially applicable to the enhancement of current tropical cyclone track forecasting techniques. They are also useful in risk management over the coastal areas where tropical cyclones may cause serious damages.

Soliton-type and other travelling wave solutions for an improved class of nonlinear sixth-order Boussinesq equations

An improved class of nonlinear bidirectional Boussinesq equations of sixth order using a wave surface elevation formulation is derived. Exact travelling wave solutions for the proposed class of nonlinear evolution equations are deduced. A new exact travelling wave solution is found which is the uniform limit of a geometric series. The ratio of this series is proportional to a classical soliton-type solution of the form of the square of a hyperbolic secant function. This happens for some values of the wave propagation velocity. However, there are other values of this velocity which display this new type of soliton, but the classical soliton structure vanishes in some regions of the domain. Exact solutions of the form of the square of the classical soliton are also deduced. In some cases, we find that the ratio between the amplitude of this wave and the amplitude of the classical soliton is equal to 35/36. It is shown that different families of travelling wave solutions are associated with different values of the parameters introduced in the improved equations.

Interacting nonlinear wave envelopes and rogue wave formation in deep water

A rogue wave formation mechanism is proposed within the framework of a coupled nonlinear Schrödinger (CNLS) system corresponding to the interaction of two waves propagating in oblique directions in deep water. A rogue condition is introduced that links the angle of interaction with the group velocities of these waves: different angles of interaction can result in a major enhancement of rogue events in both numbers and amplitude. For a range of interacting directions, it is found that the CNLS system exhibits significantly more extreme wave amplitude events than its scalar counterpart. Furthermore, the rogue events of the coupled system are found to be well approximated by hyperbolic secant functions; they are vectorial soliton-type solutions of the CNLS system, typically not considered to be integrable. Overall, our results indicate that crossing states provide an important mechanism for the generation of rogue water wave events.

Depth profiles of radioactive cesium in soil using a scraper plate over a wide area surrounding the Fukushima Dai-ichi Nuclear Power Plant, Japan

During the Fukushima Dai-ichi Nuclear Power Plant (NPP) accident, radioactive cesium was released in the environment and deposited on the soils. Depth profiles of radioactive cesium in contaminated soils provide useful information not only for radiation protection and decontamination operations but also for geoscience and radioecology studies. Soil samples were collected using a scraper plate three times between December 2011 and December 2012 at 84 or 85 locations within a 100-km radius of the Fukushima Dai-ichi NPP. In most of the obtained radioactive cesium depth profiles, it was possible to fit the concentration to a function of mass depth as either an exponential or hyperbolic secant function. By using those functions, following three parameters were estimated: (i) relaxation mass depth β (g cm−2), (ii) effective relaxation mass depth βeff (g cm−2), which is defined for a hyperbolic secant function as the relaxation mass depth of an equivalent exponential function giving the same air kerma rate at 1 m above the ground as the inventory, and (iii) 1/10 depth L1/10 (cm), at which the soil contains 90% of the inventory. The average β value (wet weight) including ones by hyperbolic secant function in December 2012, was 1.29 times higher than that in December 2011. In fact, it was observed that depth profiles at some study sites deviated from the typical exponential distributions over time. These results indicate the gradual downward migration of radioactive cesium in the soils. The L1/10 values in December 2012 were summarized and presented on a map surrounding the Fukushima Dai-ichi NPP, and the average value of L1/10 was 3.01 cm (n = 82) at this time. It was found that radioactive cesium remained within 5 cm of the ground surface at most study sites (71 sites). The sech function can also be used to estimate the downward migration rate V (kg m−2 y−1). The V values in December 2012 (n = 25) were in good agreement with those found by a realistic approach using a diffusion and migration model. Almost all values ranged between 1.7 and 9.6 kg m−2 y−1 in this study.

Light bullets in nonlinear waveguide arrays under the influence of dispersion and the Raman effect

We study the formation and the dynamics of spatially broad light bullets generated in silica waveguide arrays. We show that these bullets are metastable even in the presence of high-order dispersion, coupling dispersion, and the Raman effect and can be approximated by the hyperbolic secant function with a high degree of accuracy. We also investigate the formation of narrow light bullets which are spatially localized in only a few adjacent waveguides and short in time. We reveal that the latter are extremely robust even in the presence of the Raman effect.

The beam propagation based on one-dimensional separation modulated photonic lattices

We numerically study the propagations of Gaussian beams in four types of separation modulated photonic lattices. The results shows that the potential wells between double positive hyperbolic secant and rectangular potential barriers and between the potential barriers in the forms of double negative hyperbolic secant and rectangular functions can both support localized linear modes. Moreover, the coupling effects between two linear modes in the potential barriers can be used to realize all-optical switch. Furthermore, the nonlinear localization can also be observed in high power. Our results supply new ideas for all optical switch, light controlling and manipulation in photonic lattices.

Light controlling in transverse separation modulated photonic lattices

We numerically study the propagation of Gaussian beams in four different types of transverse separation modulated photonic lattices. We find the modulation obeying hyperbolic secant or rectangular functions can act as optical potentials, and the light waves can be localized or recur in the regions between such two positive potentials, respectively. While the beams decay in the regions between such two negative potentials since these structures could not support localized modes. Our results provide new ways for light controlling and manipulation in photonic lattices.

Numerical Modeling of Mud Invasion Profile Based on the Well Logging with the Shallow Depth of Investigation

The drilling mud invasion is a dynamic process varying with time and reveal the permeability or the relatively permeability of the formation saturated with fluid. The mud invasion profile denotes the water saturation change with the radial distance from the borehole wall. The logging with the shallow depth of investigation (DOI), such as density logging, neutron logging and shallow electrical logging, often detects the information of the flushed zone. We chose DOI logging to model the slope invasion profile with the hyperbolic secant function, and the calculated logging reading comparing from the real logging reading to optimize the invasion profile. The results show that the mud invasion profile and the invasion depth are strongly affected by porosity, where the invasion depth reduces with porosity adding.

Human action recognition using a fast learning fully complex-valued classifier

In this paper, we use optical flow based complex-valued features extracted from video sequences to recognize human actions. The optical flow features between two image planes can be appropriately represented in the Complex plane. Therefore, we argue that motion information that is used to model the human actions should be represented as complex-valued features and propose a fast learning fully complex-valued neural classifier to solve the action recognition task. The classifier, termed as, “fast learning fully complex-valued neural (FLFCN) classifier” is a single hidden layer fully complex-valued neural network. The neurons in the hidden layer employ the fully complex-valued activation function of the type of a hyperbolic secant function. The parameters of the hidden layer are chosen randomly and the output weights are estimated as the minimum norm least square solution to a set of linear equations. The results indicate the superior performance of FLFCN classifier in recognizing the actions compared to real-valued support vector machines and other existing results in the literature. Complex valued representation of 2D motion and orthogonal decision boundaries boost the classification performance of FLFCN classifier.

Approximate homotopy symmetry and infinite series solutions to the perturbed mKdV equation

We devise a homotopy model for the perturbed mKdV equation with weak fourth order dispersion and weak dissipation, and demonstrate that the related approximate equations lead to similarity reduction equations and similarity reduction solutions of different orders which are formally coincident, respectively. Series reduction solutions for the perturbed mKdV equation are thus derived. Painlevé II type equations or hyperbolic secant function solutions are obtained for zero-order similarity reduction equations. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations.

Equivalent Pi-network model of lossy and dispersive coupled transmission lines

An equivalent circuit representation is presented for a set of coupled transmission lines. An approximation of the hyperbolic secant function allows a simple derivation of a staged model that accounts for the complex frequency dependent parameters. The model converts the T-ladder network into a Π-network with controlled sources. The equivalent circuit based approach presented here is not only intriguing but also enhances the computed accuracy and efficiency. Numerical simulations verify the accuracy of this approach for both time and frequency domain responses.

Three-dimensional Bessel light bullets in self-focusing Kerr media

We study three-dimensional (3D) Bessel light-bullet solutions of the nonlinear Schr\"odinger equation with a photonic lattice potential in the form of squared Bessel functions in polar coordinates, both analytically and numerically. Analytical solutions are obtained by the Hartree approximation, and numerical simulations are performed, to compare with the analytical solutions and to confirm the stability of localized solutions. The 3D spatial wave packets are built by the products of hyperbolic secant and Bessel functions. These solitons appear in different forms, such as 3D vortex solitons and multipole solitons.

Soliton solution for an inhomogeneous highly dispersive media with a dual-power nonlinearity law

Optical soliton propagation in an inhomogeneous highly dispersive media with a dual-power nonlinearity law is studied. The variable-coefficient nonlinear Schrödinger (NLS) equation describing the propagation in such media includes space-dependent coefficients of fourth-order dispersion, cubic–quintic nonlinearity, and attenuation. By means of a solitary wave ansatz, exact bright soliton solution is found. Importantly, we have found that it is always possible to express the propagation of bright soliton pulses, for any integer value of the exponent in the power law in the form of a power of hyperbolic secant function. All physical parameters in the soliton solution are obtained as a function of the space-dependent model coefficients. Note that, it is always useful and desirable to construct exact analytical solutions for the understanding of most nonlinear physical phenomena.

Approximate direct reduction method: infinite series reductions to the perturbed mKdV equation

The approximate direct reduction method is applied to the perturbed mKdV equation with weak fourth order dispersion and weak dissipation. The similarity reduction solutions of different orders conform to formal coherence, accounting for infinite series reduction solutions to the original equation and general formulas of similarity reduction equations. Painlevé II type equations, hyperbolic secant and Jacobi elliptic function solutions are obtained for zero-order similarity reduction equations. Higher order similarity reduction equations are linear variable coefficient ordinary differential equations.