Nonlinear Harmonics Research Articles

Nonlinear viscous higher harmonics generation due to incident and reflecting internal wave beam collision

In this paper, we have considered the non-linear effects arising due to the collision of incident and reflected internal wave beams. It has already been shown analytically [Tabaei et al., “Nonlinear effects in reflecting and colliding internal wave beams,” J. Fluid Mech. 526, 217–243 (2005)] and numerically [Rodenborn et al., “Harmonic generation by reflecting internal waves,” Phys. Fluids 23, 026601 (2011)] that the internal wave beam collision generates the higher harmonics and mean flow in a linear stratification. In this paper, similar to previous analytical work, small amplitude wave theory is employed; however, it is formulated from energetics perspective which allows considering internal wave beams as the product of slowly varying amplitude and fast complex exponential. As a result, the mean energy propagation equation for the second harmonic wave is obtained. Finally, a similar dependence on the angle of incidence is obtained for the non-linear energy transfer to the second harmonic with previous analyses. A possible physical mechanism for this angle dependence on the second harmonic generation is also discussed here. In addition to previous studies, the viscous effects are also included in the mean energy propagation equation for the incident, the reflecting, and the second harmonic waves. Moreover, even though the mean flow obtained here is only confined to the interaction region, it is also affected by viscosity via the decay in the incident and the reflecting internal wave beams. Furthermore, a framework for the non-linear harmonic generation in non-linear stratification is also proposed here.

A nonlinear electromagnetics model of an asymmetrically-driven, low pressure capacitive discharge

It is well-known that standing waves having radially center-high voltage profiles exist in high frequency driven capacitive discharges. Capacitive sheaths can also nonlinearly excite driving frequency harmonics near the series resonance that can be spatially near-resonant, and therefore enhance the on-axis power deposition. The powered-electrode/plasma/grounded-electrode sandwich structure of an asymmetrically excited cylindrical discharge forms a three electrode system in which both z-symmetric and z-antisymmetric radially propagating wave modes can exist. We develop a nonlinear electromagnetics model for this system with radially- and time-varying sheath widths, incorporating both symmetric and antisymmetric modes, and the plasma skin effect. Waves generated in the electrostatic wave limit are also treated. The discharge is modeled as a uniform density bulk plasma with either homogeneous or Child law sheaths at the electrodes, incorporating their nonlinear voltage versus charge relations. The model includes a finite power source resistance and a self-consistent calculation of the dc bias voltages. The resulting set of nonlinear partial differential equations is solved numerically to determine the symmetric and antisymmetric mode amplitudes and the nonlinearly-excited radially-varying harmonics. Two examples driven at lower frequency (30 MHz) or at higher frequency (60 MHz) are examined. The first case displays the excited dominant ninth harmonic near the series resonance frequency with the fundamental frequency having a weak standing wave. The higher frequency case shows a more obvious standing wave effect, enhanced by the nonlinear harmonics. The symmetric and antisymmetric modes are of similar amplitudes over the driving electrode, summing to give a larger bottom sheath.

Studying the Effect of Non-Linear Loads Harmonics on Electric Generator Power Rating Selection

Non-linear loads connected to an electric power system produce Harmonic currents, harmonics are introduced into the system in the form of currents whose frequencies are the integral multiples of the fundamental power system frequency (50/60 Hz). The harmonic currents interact with the supply system impedance causing distortions in supply output voltage and current, which has a very bad effect on all other loads connected to the system and the power supply itself, such as overheating, increasing powers losses in the system, and malfunction of protection and control devices connected to the system. This paper presents a study to analyze the effect of voltage and current harmonics resulting from non-linear loads such as variable frequency drive, uninterruptable power supply, and battery chargers on operation and power rating of synchronous generator. The study introduces an optimized method for selecting the suitable generator power rating to withstand harmful harmonics effects for a safe operation of the generator, saving its lifetime, and to improve the power quality of the power system. The method depends on analyzing the effect of increasing the supply generator power rating on the THVD produced from non-linear loads harmonics connected to the system. By calculating the THVD for each case of a generator power rating, a mathematical relationship between generator power rating and TVHD can be found. So, the relationship between generator power rating and total harmonic distortion in the power system will be discussed clearly.

PARAMETERIZATION OF EVOLUTION OF BIPHASE DURING NONLINEAR TRANSFORMATION OF WAVES IN COASTAL ZONE

Based on experimental data, the problem of parametrization of spatial variation of the phase shift (biphase) between the first and second nonlinear harmonics of wave motion during wave transformation over sloping bottom in the coastal zone is discussed. It is revealed that the biphase values vary in the range [–π/2, π/2]. Biphase variations rigorously follow fluctuations in amplitudes of the first and second harmonics and the periodicity of energy exchange between them. The empirical relation applied in modern practice to calculate the biphase, which depends on the Ursell number, is incorrect for calculating the biphase for wave evolution in the coastal zone, because it does not take into account periodic energy exchange between the nonlinear harmonics. The new approximations of the biphase values for typical scenarios of wave transformations are suggested. It was demonstrated that the biphase of breaking waves defines breaking index and breaking type.

Influence of processes of nonlinear transformations of waves in the coastal zone on the height of breaking waves

Using data from laboratory, field, and numerical experiments, we investigated regularities in changes in the relative limit height of breaking waves (the breaking index) from peculiarities of nonlinear wave transformations and type of wave breaking. It is shown that the value of the breaking index depends on the relative part of the wave energy in the frequency range of the second nonlinear harmonic. If this part is more than 35%, then the breaking index can be taken as a constant equal to 0.6. These waves are spilling breaking waves, asymmetric on the horizontal axis, and are almost symmetric on the vertical axis. If this part of the energy is less than 35%, then the breaking index increases with increasing energy in the frequency range of the second harmonic. These waves are plunging breaking waves, asymmetric on the vertical axis, and are almost symmetric on the horizontal axis. It is revealed that the breaking index depends on the asymmetry of waves on the vertical axis, determined by the phase shift between the first and second nonlinear harmonic (biphase). It is shown that the relation between the amplitudes of the second and first nonlinear harmonics for an Ursell number less than 1 corresponds to Stokes’ second-order wave theory. The empirical dependences of the breaking index on the parameters of nonlinear transformation of waves are proposed.

Magnetic nanoparticle detection method employing non-linear magnetoimpedance effects

In this work, a sensitive tool to detect magnetic nanoparticles (Fe3O4) based on a non-linear Giant Magnetoimpedance (GMI) effect is presented. The GMI sensor is designed with four nearly zero magnetostrictive ribbons connected in series and was analysed as a function of a constant external magnetic field and exciting frequency. The influence of the magnetic nanoparticles deposited on the ribbon surface was characterized using the first (fundamental) and second (non-linear) harmonics of the magnetoinductive voltage. The results show a clear enhancement of the sensor response in the high magnetic field region (H = 1.5 kA/m) as a consequence of the stray field generated by the magnetic nanoparticles on the GMI ribbons' surface. The highest sensitivity ratios are obtained for the non-linear component in comparison with the fundamental response. The results open a new research strategy in magnetic nanoparticle detection.

Synthesis of discrete phase-coherent optical spectra from nonlinear ultrasound.

Nonlinear acoustic interactions in liquids are effectively stronger than nonlinear optical interactions in solids. Thus, harnessing these interactions will offer new possibilities in the design of ultra-compact nonlinear photonic devices. We theoretically demonstrate a new scheme for synthesis of optical spectra from nonlinear ultrasound harmonics using a hybrid liquid-state and nanoplasmonic device compatible with fibre-optic technology. The synthesised spectra consist of a set of equally spaced optical Brillouin light scattering modes having a well-defined phase relationship between each other. We suggest that these spectra may be employed as optical frequency combs whose spectral composition may be tuned by controlling the nonlinear acoustic interactions.

On the possibility of biphase parametrization for wave transformation in the coastal zone

Based on experimental data, we study the possibility of parametrizing the spatial variation in the phase shift (biphase) between the first and second nonlinear harmonics of wave motion during wave transformation over an inclined bottom in the coastal zone. It is revealed that the biphase values vary in the range [–π/2, π/2]. Biphase variations rigorously follow fluctuations in amplitudes of the first and second harmonics and the periodicity of energy exchange between them. Wave breaking influences the biphase value, retaining its variations in the negative domain in the range [–π/2, 0]. The formula applied in modern practice to calculate the biphase, which depends on the Ursell number, is incorrect for calculating the biphase for wave evolution in the coastal zone, because it does not take into account periodic energy exchange between the nonlinear harmonics. We propose a linear approximation of the biphase values from the size of the ratio of the current distance to the coast to the possible spacial duration of the exchange period, which is determined by the dispersion relation. We reveal the dependence of biphase variations on the wave transformation scenario and demonstrate the possibility of constructing a separate parameterization of the biphase for each scenario. Our research and the obtained biphase parameterizations can be used to simulate the sea state in the coastal zone, as well as in problems of predicting the development of coasts under the impact of storm waves.

Rightsizing Generators Through Harmonic Mitigation Realizes Energy, Emissions, and Infrastructure Reductions

Sizing prime and backup generator packages for nonlinear loads (such as adjustable speed drives (ASDs), solid-state uninterruptible power system, dc drives, and other SCR loads and even single phase loads, such as computer systems, ballasted lighting, and other power electronic loads) is not a simple process. Including the effects of the nonlinear load harmonics is essential to properly sizing generator capacity. Incorrectly calculating the harmonic current contribution can result in significantly undersizing the generator leading to brownout conditions, overloading of the generator, nuisance tripping, misoperation of the automatic voltage regulator, generator failures, and load equipment damage through elevated voltage distortion. However, oversizing the generator to accommodate the nonlinear load current harmonics will result in increased initial installation costs, much higher fuel and operating costs due to poor operating efficiencies and higher emissions. In an oil pipeline islanded pumping application, the problems experienced with the adjustable speed drive operation on a 200 hp pump led to the generator being significantly upsized. Not until harmonic mitigation was finally considered though, did the ASD and generator operate without issues. Computer simulations were used to demonstrate how the application of a passive wide spectrum harmonic filter would reduce the ASD harmonic currents to eliminate the voltage distortion and allow the operator to retrofit the installation with a more properly sized generator that provided energy/fuel savings and emissions reduction while maintaining good power quality.

A comparative analysis of inductance wires and tyres

Objective. To assess the effect of frequency, configuration and cross-sectional area of wires and tyres to value their inductance. Results. Obtained relative values of inductance for induction for current supplies with hollow and solid squares cross-section of round, square and rectangular configurations. The comparative analysis and conclusions about possible minimize inductances for different shapes of cross-section, as well as for low and high frequencies. Practical significance. The results can be used to calculate mutual inductance of magnetic associated elements with the aim of assessing their impedance characteristics. Dependence of inductances on the frequency and the corresponding non-linearity causes harmonics, other than the fundamental frequency, which can significantly affect electromagnetic compatibility of magnetic associated element circuit and the efficiency of the whole system in terms of energy consumption.

Physical modelling of wave scattering around fixed FPSO-shaped bodies

FPSO (Floating Production Storage and Off-loading) vessels used for offshore oil and gas production are operated in deep water, often at locations which experience severe wave loading. This paper reports on laboratory experiments on a series of simplified FPSO-shaped bodies, with the aim of understanding more about the wave-structure interaction, particularly the generation of scattered waves. These tests were carried out in the Ocean Basin at Plymouth University’s COAST Laboratory where the effects on the wave-structure interaction of model length, wave steepness and incident wave direction were investigated. All three models had semi-circular ends, separated by a box section for the 2 longer models. Input waves were based on focused wave groups generated using NewWave with an underlying JONSWAP spectrum. A general phase-based harmonic separation method was applied to separate the linear and higher-harmonic wave components of the free-surface elevation surrounding the bodies, and of relevance to the assessment of wave loads. Close to the bow of the model, the highest amplitude scattered waves are observed with the most compact model, and the third- and fourth-harmonics are significantly larger than the equivalent incident bound harmonic components. At the locations close to the stern, the linear harmonic is found to increase as the model length is decreased, although the nonlinear harmonics are similar for all three tested lengths, and the second- and third-harmonics are strongest with the medium length model. The nonlinear scattered waves increase with increasing wave steepness and a second pulse is evident in the higher-order scattered wave fields. As the incident wave angle between the waves and the long axis of the vessel is increased from 0° (head-on) to 20°, the third- and fourth-harmonic scattered waves reduce on the upstream side. These third- and fourth-harmonic diffracted waves should be considered in assessing wave run-up for offshore structure design, and may be relevant to the excitation of ringing-type structural responses in fixed and taut moored structures.

Higher harmonic generation of guided waves at delaminations in laminated composite beams

Detection and characterization of delamination damage are of great importance to the assurance of structural safety. This work investigates the potential of a baseline-free structural health monitoring technique based on higher harmonics resulting from the nonlinear interaction of guided wave and a delamination. The nonlinearity considered in this study arises from the clapping of the sub-laminates in the delaminated region, which generates contact acoustic nonlinearity. Both explicit finite element simulations and experimental tests are conducted on composite laminates containing a delamination of different sizes and at different through-thickness locations. The results show that the interaction between the fundamental asymmetric mode (A0) of guided wave and a delamination generates contact acoustic nonlinearity in the form of higher harmonics, which provides a good measure for identifying the existence of delaminations and determining their sizes in laminated composite beams. This new insight into the generation mechanisms of nonlinear higher order harmonics in composite laminates will enhance the detection and monitoring of damage in composite structures.

Nonlocality-driven supercontinuum white light generation in plasmonic nanostructures.

Structured plasmonic metals are widely employed for achieving nonlinear functionalities at the nanoscale due to their ability to confine and enhance electromagnetic fields and strong, inherent nonlinearity. Optical nonlinearities in centrosymmetric metals are dominated by conduction electron dynamics, which at the nanoscale can be significantly affected by the nonlocal effects. Here we show that nonlocal corrections, being usually small in the linear optical response, define nonlinear properties of plasmonic nanostructures. Using a full non-perturbative time-domain hydrodynamic description of electron plasma under femtosecond excitation, we numerically investigate harmonic generation in metallic Archimedean nanospirals, revealing the interplay between geometric and nonlocal effects. The quantum pressure term in the nonlinear hydrodynamic model results in the emergence of fractional nonlinear harmonics leading to broadband coherent white-light generation. The described effects present a novel class of nonlinear phenomena in metallic nanostructures determined by nonlocality of the electron response.

An Innovative Research in Power Quality Conditioner for Hospitality Sector

The usage of non-linear loads in commercial sectors, hospitality and industrial sectors is increasing drastically due to its merits. The quality of power gets affected owing to the increase in the usage of non-linear loads and other harmonics creating equipment like CFL bulbs. As a result of the poor power quality, the performance of equipment such as monitors used in hospitals gets affected and the life of the equipment connected to the system also decreases. The penalty is being imposed by the utilities for high harmonic content and low power factor. Hence, a Universal Economic Hybrid Power Quality Conditioner (UEHPQC), combining active filter, passive filter and Thyristor controlled reactor is proposed and modeled through MATLAB/Simulink tool in this paper. As a case study, a famous hotel in Tamil Nadu has been identified and various power quality issues have been analyzed. The performance of the proposed power quality conditioner has been studied with the non-linear loads through MATLAB/Simulink using two controllers, i.e., Artificial Neural Network with the Generalized Radial Basis Function model and Proportional Integral. The performance of the above controllers had been compared.

Fatigue crack detection using structural nonlinearity reflected on linear ultrasonic features

This paper theoretically and experimentally demonstrates that (1) a localized nonlinear mechanism such as a fatigue crack not only generates nonlinear modulations or harmonics, but also reduces the amplitudes of the linear responses at the excitation frequencies, (2) the energy in the linear responses is partially transmitted to the nonlinear components in the presence of structural nonlinearity, and (3) the amplitude reduction of the linear responses is larger than the amplitude increase of the nonlinear components. In addition, a baseline-free fatigue crack detection technique is proposed considering the additional amplitude reduction of the linear components due to nonlinear ultrasonic modulation.

Nonlinear Ion Harmonics in the Paul Trap with Added Octopole Field: Theoretical Characterization and New Insight into Nonlinear Resonance Effect.

The nonlinear harmonics within the ion motion are the fingerprint of the nonlinear fields. They are exclusively introduced by these nonlinear fields and are responsible to some specific nonlinear effects such as nonlinear resonance effect. In this article, the ion motion in the quadrupole field with a weak superimposed octopole component, described by the nonlinear Mathieu equation (NME), was studied by using the analytical harmonic balance (HB) method. Good accuracy of the HB method, which was comparable with that of the numerical fourth-order Runge-Kutta (4th RK), was achieved in the entire first stability region, except for the points at the stability boundary (i.e., β = 1) and at the nonlinear resonance condition (i.e., β = 0.5). Using the HB method, the nonlinear 3β harmonic series introduced by the octopole component and the resultant nonlinear resonance effect were characterized. At nonlinear resonance, obvious resonant peaks were observed in the nonlinear 3β series of ion motion, but were not found in the natural harmonics. In addition, both resonant excitation and absorption peaks could be observed, simultaneously. These are two unique features of the nonlinear resonance, distinguishing it from the normal resonance. Finally, an approximation equation was given to describe the corresponding working parameter, q nr , at nonlinear resonance. This equation can help avoid the sensitivity degradation due to the operation of ion traps at the nonlinear resonance condition.

Nonlinear standing wave excitation by series resonance-enhanced harmonics in low pressure capacitive discharges

It is well-known that standing waves having radially center-high rf voltage profiles exist in high frequency capacitive discharges. It is also known that in radially uniform discharges, the capacitive sheath nonlinearities excite strong nonlinear series resonance harmonics that enhance the electron power deposition. In this work, we consider the coupling of the series resonance-enhanced harmonics to the standing waves. A one-dimensional, asymmetric radial transmission line model is developed incorporating the wave and nonlinear sheath physics and a self-consistent dc potential, for both conducting and insulating electrode surfaces. The resulting coupled pde equation set is solved numerically to determine the discharge voltages and currents. A 10 mTorr argon plasma is chosen with density m−3, gap width 2 cm and conducting electrode radius 15 cm, driven by a 500 V rf source with resistance 0.5 . We examine a set of frequencies from near 30 MHz up to frequencies more than three times as high. For most frequencies, no harmonics correspond exactly with the series or spatial resonances, which is the generic situation. Nevertheless, nearby resonances lead to a significantly enhanced ratio of the electron power per unit area on axis, compared to the average. Nearly similar results are found for insulating electrodes. Strong effects are seen for varying source resistance: high (50 ) resistance damps out most of the harmonic activity, while zero source resistance leads to a non-steady discharge with bias voltage relaxation oscillations. Stronger harmonic effects are seen for an increased radius of 30 cm, as lower harmonics become spatially resonant at lower frequencies. The radial dependence of electron power with frequency showed significant variations, with the central enhancement and sharpness of the spatial resonances depending in a complicated way on the amplitudes of the nearby series resonance current harmonics and the phase relations among the voltage harmonics driving these current harmonics. Significant center/average electron power per unit area enhancement is found even at the lowest frequencies for both high and low densities: 4.5 at 30 MHz and m−3, and 2.2 at m−3.

Non Linear Analysis of MPPT for Power Quality Improvement

In this study the conventional inverter interfacing renewable energy sources with the grid, without any additional hardware cost. Here, the main idea is the maximum utilization of inverter rating which is most of the time underutilized due to intermittent nature of RES. Based on the non-linear characteristics of PV, these thesis designs a VSS controller to realize the maximum power output of PV arrays. The output power from renewable energy sources fluctuates because of weather variations. This study proposes an effective power quality control strategy of renewable energy sources connected to power system using Photovoltaic (PV) array. If the main controller used is a PR controller, any dc offset in a control loop will propagate through the system and the inverter terminal voltage will have a nonzero average value. In this strategy both load and inverter current sensing is required to compensate the load current harmonics. The non-linear load current harmonics may result in voltage harmonics and can create a serious PQ problem in the power system network.

INFLUENCE OF ROLLING VISCO-ELASTIC COUPLING ON NON-LINEAR DYNAMICS OF DOUBLE PLATES SYSTEM

Multi frequency vibrations of a system of two isotropic circular plates interconnected by a rolling visco-elastic layer that has nonlinear characteristics are considered. The system with two circular plates and interconnecting discrete continually distributed rolling visco nonlinear elastic rheological elements presents the model of hybrid nonlinear system. The analytical solutions of first asymptotic approximation describing stationary behavior, in the regions around the resonances, are the principal results of the author. On the basis of those results the influence on the system dynamics of rolling coupling element was numerically analyzed. A series of the amplitude-frequency and phase-frequency curves of the two-frequency like vibration regimes were numerically obtained and presented. These curves present the evolution of the first asymptotic approximation of solutions for different nonlinear harmonics obtained by changing external excitation frequencies through discrete as well as continuous values for different values of rolling elements masses. Such an analysis proves that the presence of rolling elements in the interconnected layer of two plates causes frequency overlap of resonant regions of nonlinear modes, which at the same time causes the enlargement of the mode mutual interactions

Two-layer Boussinesq models for coastal water waves

This paper presents three sets of two-layer Boussinesq models for highly dispersive and highly nonlinear water waves. These models are formulated in terms of depth-averaged velocities or velocities located at two arbitrary z locations within each layer and are fully nonlinear to the second order. Stokes-type expansions are used to theoretically analyze the linear and nonlinear properties of the models. The coefficients involved in the governing equations are determined from the minimization of the integral error between the linear wave celerity, shoaling gradient, second nonlinear harmonics of the equations and the related analytical solutions. The most promising model is applicable up to kh≈25.4 (where kh is the dimensionless water depth, k is the wave number, and h is the water depth) for the dispersive property, to kh≈6 for the second nonlinear property within 1% error, and to kh≤6 for the excellent shoaling property. The numerical implementation for one-dimensional governing equations on non-staggered grids is also presented by employing a fourth-order Adams–Bashforth–Moulton time integration and the high-accuracy finite difference method. Four demanding numerical experiments that require high accuracy of dispersion and nonlinearity are conducted to assess the performance of the models. The computational results are compared against the analytical solution and experimental data, good agreements are found.