Harmonic Type Research Articles

On the analytical solution of the Schrödinger equation in ellipsoidal coordinates with constant potential

This paper solves the Schrödinger equation in confocal ellipsoidal coordinates under a constant potential. An auxiliary approach is introduced to separate the equation, which allows for the analytical derivation of energy levels using a series solution method. The results show that the four types of ellipsoidal harmonics lead to the same energy expressions. The energy levels are analyzed in terms of their dependence on the geometrical properties of the coordinates and their degeneracy patterns. This study highlights the role of ellipsoidal geometry in defining energy spectra and demonstrates its usefulness for describing quantum systems with non-spherical symmetry. The findings are relevant to diatomic molecules, quantum chemistry, and nanoscale physics, particularly nanorods. The method provides a foundation for future work on more complex potentials, which could extend the range of problems addressed using this approach.

Research on Electromagnetic Performance and Rotor Eddy Current Loss Suppression of Ultra‐Low Temperature and High‐Speed Permanent Magnet Motor Materials for LNG Pumps

AbstractThere is no unified material selection principle for ultra‐low temperature and high‐speed permanent magnet motors used in liquefied natural gas (LNG) pumps, and high rotor eddy current losses can easily lead to vaporization of the transported LNG. This article first studies the electromagnetic performance of motor structural component materials at room temperature and ultra‐low temperature environments, and proposes the material selection principles for this type of motor. Based on the experimental measurement results of material electromagnetic performance, the loss characteristics of motors under ultra‐low temperature and room temperature environments were compared. Based on the mechanism of rotor eddy current loss and considering the influence of environmental temperature on the material properties of motors, three optimization methods for rotor eddy current loss are proposed. The impact of different suppression measures on rotor eddy current loss is revealed. The suppression of magnetic conduction harmonics can be achieved by opening stator auxiliary slots and optimizing their size and position. High conductivity metal shielding layers are used and their materials and sizes are optimized, And the use of magnetic slot wedges can simultaneously suppress three types of harmonics. Finally, the suppression principles of different optimization methods on rotor eddy current losses were analyzed, and the optimization effects and advantages and disadvantages of different methods were discussed, providing theoretical reference for the optimization design of high‐speed permanent magnet motors operating in ultra‐low temperature environments. © 2024 Institute of Electrical Engineers of Japan and Wiley Periodicals LLC.

Optimal control strategies for toxoplasmosis disease transmission dynamics via harmonic mean-type incident rate

Toxoplasma infection in humans is considered due to direct contact with infected cats. Toxoplasma infection (an endemic disease) has the potential to affect various organs and systems (brain, eyes, heart, lungs, liver, and lymph nodes). Bilinear incidence rate and constant population (birth rate is equal to death rate) are used in the literature to explain the dynamics of Toxoplasmosis disease transmission in humans and cats. The goal of this study is to consider the mathematical model of Toxoplasma disease with harmonic mean type incident rate and also consider that the population of humans and cats is not equal (birth rate and the death rate are not equal). In examining Toxoplasma transmission dynamics in humans and cats, harmonic mean incidence rates are better than bilinear incidence rates. The disease dynamics are first schematically illustrated, and then the law of mass action is applied to obtain nonlinear ordinary differential equations (ODEs). Analysis of the boundedness, positivity, and equilibrium points of the system has been analyzed. The reproduction number is calculated using the next-generation matrix technique. The stability of disease-free and endemic equilibrium are analyzed. Sensitivity analysis is also done for reproduction number. Numerical simulation shows that the infection is spread in the population when the contact rate βh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta_{h}$$\\end{document} and βc\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta_{c}$$\\end{document} increases while the infection is reduced when the recovery rate δh\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta_{h}$$\\end{document} increases. This study investigates the impact of various optimal control strategies, such as vaccinations for the control of disease and the awareness of disease awareness, on the management of disease.

Elliptic Rigid Inclusion in Soft Materials of Harmonic Type

Abstract We investigate finite plane deformations of an elliptic rigid inclusion embedded in a soft matrix that is made of a particular class of harmonic-type hyperelastic materials. The inclusion is assumed to be perfectly bonded to the matrix, which is subjected to a constant remote in-plane loading. Utilizing the Cauchy integral techniques associated with conformal mappings, we derive closed-form solutions for the full-field deformation, Piola stress, and Cauchy stress in the entire matrix. Numerical examples are presented to illustrate the current solutions in comparison with those established from linear elasticity theory. We find that in terms of the Cauchy stress around the inclusion, the maximum normal stress component always appears at the endpoints of the major axis of the inclusion, irrespective of the magnitude of the remote loading, while the maximum hoop stress component occurs not exactly at the above-mentioned endpoints when the remote loading exceeds a certain value. In particular, we identify an exact explicit formula for determining the relative rotation of the inclusion during deformation induced by a remote uniaxial loading of arbitrarily given magnitude and direction.

Cyclic variations of the structure and energetics of solar magnetic fields

ABSTRACT The solar cycle is a complex phenomenon, a comprehensive understanding of which requires the study of various tracers. Here, we consider the solar cycle as manifested in the harmonics of the solar large-scale surface magnetic field, including zonal, sectorial, and tesseral harmonics, divided into odd and even relative to the solar equator. In addition to considering the amplitudes of the harmonics, we analyse their contribution to the magnetic energy. It turns out that the relative contribution of different types of harmonics to the magnetic energy is virtually independent of the cycle height. We identify different phases of the activity cycle using harmonics of different symmetries. A possible way to incorporate the obtained result into the solar dynamo theory is proposed.

Dynamics analysis of dengue fever model with harmonic mean type under fractal-fractional derivative

<abstract><p>Dengue is a viral illness transmitted by Aedes mosquitoes and is a significant global threat. In this study, we developed a model of the dengue epidemic that incorporates larvicide and adulticide, as well as the harmonic mean incidence rate under fractal-fractional derivatives. We examined various theoretical aspects of the model, including nonnegativity, boundedness, existence, uniqueness, and stability. We computed the basic reproduction number $ \Re _{0} $ using the next-generation matrix. The model has two disease-free equilibriums, a trivial equilibrium, and a biologically realistic, along with one endemic equilibrium point. These findings enhanced our understanding of dengue transmission, providing valuable insights for awareness campaigns, control strategies, intervention approaches, decision support, guiding public health planning, and resource allocation to manage dengue effectively.</p></abstract>

Study of optical properties of DM–DM planar interface by incorporating non-linearity and NID space

Study of the optical characteristics of materials, objects and geometries has wider role and applications in science and engineering. Reflection and transmission coefficients are used to study the optical properties of planar geometries. Primary aim of the presented work is to study the reflection and transmission of second harmonic generation in non-integer dimensional environment. Nonlinear medium in our considered geometry is of second harmonic generation type hence we only studied the second order reflection and transmission characteristics. Solution of the nonlinear wave equation is obtained by using slowly varying amplitude approximation and method of separation of variables in NID space is used to study how the presence of NID medium alter the electromagnetic waves. As a special case of the original geometry we also studied the changes in the behavior of the coefficients when an epsilon negative and a mu negative NID mediums are used as a reflected half space. In all the cases impact of the non-linearity and the noninteger dimensions on the unknown coefficients are studied separately. It is studied that higher amplitude of the reflection coefficient can be achieved by using a dielectric magnetic NID medium compared to epsilon negative NID medium. On the other hand it is also noted that for epsilon near zero medium, amplitude of the reflection coefficient reaches zero for specific values of the nonlinear permeability and incident angle whereas, for mu negative NID medium amplitude of both the coefficients increases (decreases) by increasing the values of ϵic(ϵrc). It is also noted that dimensions of the NID mediums can be used as a controlling factor for the amplitude of the coefficients by keeping the same shape. For the verification of the analytical results nonlinear power law is used.

An incommensurate fractional order model for complex dynamics of viral infection with immunity

This paper deals with an incommensurate fractional order mathematical model for dynamic analysis of viral infection with immunity. The primary focus of the work is to explore stability analysis of this version of incommensurate fractional order model with harmonic mean type incidence function and fractional derivative in Caputo sense. First, well-posed ness of the model has been established by analyzing existence and uniqueness of the solution. In the next section, stability analysis of the equilibrium points has been caried out based on the basic reproduction number. Sensitivity analysis of the threshold parameter have been performed in the following sections. Finally, rigorous numerical simulations have been performed to support the theoretical findings as well as to observe the effect of various fractional orders and incidence function.

Adaptive error feedback regulator problem for a 1-D wave equation with velocity recirculation

In this paper, we consider the error feedback regulator problem for a 1-D wave equation with velocity recirculation. There are harmonic disturbances in all channels of the system and the reference signal is also a harmonic type. We first introduce an invertible transformation such that the disturbances are only on the left boundary, which is non-collocated with the control. Then we propose an adaptive error-based observer by utilizing the measurable non-collocated tracking error and its derivative. Next, an observer-based error feedback controller is constructed to regulate the tracking error to zero asymptotically and make all internal signals bounded. Finally, the results are illustrated with simulations.

A Deep Learning Based Enhancing the Power by Reducing the Harmonics in Grid Connected Inverters

The increasing use of renewable energy systems has led to a rise in the number of grid-connected inverters, which can have a detrimental effect on the superiority and constancy of grid electricity due to the injected current harmonics. In this study, the proportional integral (PI) and proportional resonant (PR) controllers have been investigated for their effectiveness in reducing harmonics in grid-connected inverters. The study also investigates the impact of harmonics compensators (HC) on the control strategies. The results of the study suggest that the implementation of PI and PR controllers in the synchronous frame can effectively reduce the injected current harmonics in grid-connected inverters. The use of harmonics compensators can further enhance the performance of the controllers by reducing the distortion and improving the stability of the grid. The efficiency of the regulator strategies be contingent on the type and level of harmonics in the grid, as well as the design and tuning of the controllers and compensators. The statement that the “PR+HC controller has a superior quality output current” is more specific and suggests that this control method may be more effective than the others in reducing harmonics and enlightening the value of the productivity current. The comparison of the IEEE 1547 standard by three viable inverters from diverse constructors is also noteworthy, as it can provide insights into the compatibility and performance of different types of inverters with the standard. The use of deep learning with the RCNN network for analyzing harmonics and providing information about power is an interesting application of machine learning in power systems research. This approach may have the probable to development the accuracy and competence of harmonics analysis as well as power monitoring in grid-connected inverters. Overall, the study highlights the importance of effective control strategies for managing harmonics in grid-connected inverters, particularly in the context of the increasing usage of renewable energy systems. The findings of the study can inform the development of more efficient and reliable grid-connected inverters, which are essential for the incorporation of renewable energy systems into the power grid.

Harvesting vibration energy by branch structures and amplified inertial force acting on piezoelectric sheets

To improve the efficiency of ambient vibration energy harvesting, a novel vibration energy harvester with branch structures is proposed, which primarily generates electric output through dynamic force rather than deflection. When excited, the branch can amplify the inertial force and acts on piezoelectric sheets perpendicularly. This capability allows the harvester to generate a substantial output without undergoing excessive deflection. Theoretical models are established, and corresponding dynamic equations are derived using the Lagrange equation. The simulation results show that there exists super-harmonic resonance in the response. The validation experiments were carried out, in which the excitation types were chosen as the sweeping-frequency, harmonic and random types sequentially. The experiment results for sweeping-frequency and harmonic excitations demonstrate the occurrence of super-harmonic vibration and resonance in the response. These phenomena lead to a significant increase in output voltages. The experiments involving stochastic excitations demonstrate the harvester's capability to generate large outputs under weak random excitations, with a piezoelectric material with dimensions of 10 × 10 × 0.2 mm3. The root mean square of output power could reach 46.24 μW under the excitation of power spectrum density of 0.06 g2/Hz, about 80 % higher than the classical inverted-beam harvester.

Vibration response of viscoelastic nanobeams including cutouts under moving load

This manuscript presents a mathematical model and numerical solution to predict vibration responses of nonlocal strain gradient (NLS) perforated viscoelastic nanobeam under dynamic moving loads. The microstructure and size length scales are considered in the frame of NLS theory. The viscoelastic damping of the structure is considered by linear Kelvin Voigt viscoelastic model. Timoshenko and Euler Bernoulli beam theories are developed to consider both thin and thick structure response. The moving load is portrayed by point load and harmonic type. The dynamic equations of motion incorporating viscoelasticity and size dependency effects is derived by using Hamilton principle. A differential quadrature method (DQM) with Chebyshev–Gauss–Lobatto formula is used to discretize the space domain and solve the model numerically. The obtained results are validated and compared with available literature. Numerical studies are conducted to show the influence of the material viscosity parameter, perforation parameters, shear deformation effect, nonlocal strain gradient, moving load velocity, and beams slenderness ratio on the dynamic behavior of viscoelastic perforated nanobeams under moving load. The material viscosity parameter can effectively control the magnitude and stability of the magnification factor profiles. Higher filling ratios result in larger values of the dynamic magnification factor, whereas larger numbers of hole rows lead to smaller values. The proposed procedure is supportive in the analysis and design of perforated viscoelastic NEMS structures under moving load.

Exploring the effectiveness of control measures and long-term behavior in Hepatitis B: An analysis of an endemic model with horizontal and vertical transmission

This study presents an optimal control and stability analysis of a mathematical model for the spread of Hepatitis B with a harmonic mean type incidence rate, providing insights into the effectiveness of control measures and the long-term behavior of the disease dynamics. The hepatitis B virus causes hepatitis B infection. It is one of the most serious viral infections out there, as well as a global health issue. Various stages, such as chronic and acute carrier stages, serve an essential role in the transition of hepatitis B infection. Chronic disease is characterized by the presence of individuals who do not show any symptoms but are nevertheless able to spread the illness. In this study, we focused on the infectiousness of hepatitis B at different stages of illness and created an endemic model by means of a nonlinear occurrence rate. To accomplish so, we start by dividing the infectious group into two further sub-classes: i.e., acute infected and chronic carriers, both of which can transmit horizontally and vertically. The suggested model’s basic characteristics are provided. The method of the next-generation matrix is utilized to compute the basic reproduction number. The biological significance of the threshold state is thoroughly researched and addressed. We also discover the criteria for investigating all of the model’s potential equilibria in terms of the fundamental reproduction number. Finally, to supplement our analytical work, we do the numerical estimation.

The evaluation of the influence of Gaussian noise, harmonic type noise and combined noise to short range wireless devices

Modern short-range wireless devices are widely used for data transmission and / or control actions in various radio engineering and information-measuring networks. Such a device in the process of use is in the conditions of a potential violation of the permissible type, which limits the possibilities of their application. The aim of the work is to evaluate the Gaussian noise, harmonic type noise and combined noise on short-range wireless devices using pseudo-random frequency conversion and frequency ripple. An estimate of the influence of various types of noise on the named wireless devices is obtained, as well as the dependence of the average error probability on the signal-to-noise and signal-to-interference ratios. It is shown that devices in which the frequency span between channels is continuously changing have the highest noise immunity, and this is also typical for devices with the highest signal-to-noise ratio. The increase in the noise immunity of devices increases with the size of the signal alphabet. It is shown that the noise immunity of the device when exposed to combined noise, which is a superposition of noise and a harmonic signal, differs little when exposed to one of the named noises. It is noted that the use of binary block codes increases the noise immunity of the device. In the course of the work, practically significant results were obtained, which make it possible to evaluate the noise immunity of these devices for various values of the signal-to-noise and signal-to- interference ratios, as well as the size of the signal alphabet. The results obtained can be used in the construction and analysis of wireless devices operating in the unlicensed bands below 1 GHz.

An Analytic Solution for 2D Heat Conduction Problems with Space–Time-Dependent Dirichlet Boundary Conditions and Heat Sources

This study proposes a closed-form solution for the two-dimensional (2D) transient heat conduction in a rectangular cross-section of an infinite bar with space–time-dependent Dirichlet boundary conditions and heat sources. The main purpose of this study is to eliminate the limitations of the previous study and add heat sources to the heat conduction system. The restriction of the previous study is that the values of the boundary conditions and initial conditions at the four corners of the rectangular region should be zero. First, the boundary value problem of 2D heat conduction system is transformed into a dimensionless form. Second, the dimensionless temperature function is transformed so that the temperatures at the four endpoints of the boundary of the rectangular region become zero. Dividing the system into two one-dimensional (1D) subsystems and solving them by combining the proposed shifting function method with the eigenfunction expansion theorem, the complete solution in series form is obtained through the superposition of the subsystem solutions. Three examples are studied to illustrate the efficiency and reliability of the method. For convenience, the space–time-dependent functions used in the examples are considered separable in the space–time domain. The linear, parabolic, and sine functions are chosen as the space-dependent functions, and the sine, cosine, and exponential functions are chosen as the time-dependent functions. The solutions in the literature are used to verify the correctness of the solutions derived using the proposed method, and the results are completely consistent. The parameter influence of the time-dependent function of the boundary conditions and heat sources on the temperature variation is also investigated. The time-dependent function includes exponential type and harmonic type. For the exponential time-dependent function, a smaller decay constant of the time-dependent function leads to a greater temperature drop. For the harmonic time-dependent function, a higher frequency of the time-dependent function leads to a more frequent fluctuation of the temperature change.

Identification of Control-Related Signal Path for Semi-Active Vehicle Suspension with Magnetorheological Dampers

This paper presents a method for the identification of control-related signal paths dedicated to a semi-active suspension with MR (magnetorheological) dampers, which are installed in place of standard shock absorbers. The main challenge comes from the fact that the semi-active suspension needs to be simultaneously subjected to road-induced excitation and electric currents supplied to the suspension MR dampers, while a response signal needs to be decomposed into road-related and control-related components. During experiments, the front wheels of an all-terrain vehicle were subjected to sinusoidal vibration excitation at a frequency equal to 12 Hz using a dedicated diagnostic station and specialised mechanical exciters. The harmonic type of road-related excitation allowed for its straightforward filtering from identification signals. Additionally, front suspension MR dampers were controlled using a wideband random signal with a 25 Hz bandwidth, different realisations, and several configurations, which differed in the average values and deviations of control currents. The simultaneous control of the right and left suspension MR dampers made it necessary to decompose the vehicle vibration response, i.e., the front vehicle body acceleration signal, into components related to the forces generated by different MR dampers. Measurement signals used for identification were taken from numerous sensors available in the vehicle, e.g., accelerometers, suspension force and deflection sensors, and sensors of electric currents, which control the instantaneous damping parameters of MR dampers. The final identification was carried out for control-related models evaluated in the frequency domain and revealed several resonances of the vehicle response and their dependence on the configurations of control currents. In addition, the parameters of the vehicle model with MR dampers and the diagnostic station were estimated based on the identification results. The analysis of the simulation results of the implemented vehicle model carried out in the frequency domain showed the influence of the vehicle load on the absolute values and phase shifts of control-related signal paths. The potential future application of the identified models lies in the synthesis and implementation of adaptive suspension control algorithms such as FxLMS (filtered-x least mean square). Adaptive vehicle suspensions are especially preferred for their ability to quickly adapt to varying road conditions and vehicle parameters.

An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case of space–time dependence. First, the physical system is decomposed into two one-dimensional subsystems, each of which can be solved by combining the proposed shifting function method with the eigenfunction expansion theorem. Therefore, through the superposition of the solutions of the two subsystems, the complete solution in the form of series can be obtained. Two numerical examples are used to investigate the analytic solution of the 2D heat conduction problems with space–time-dependent boundary conditions. The considered space–time-dependent functions are separable in the space–time domain for convenience. The space-dependent function is specified as a sine function and/or a parabolic function, and the time-dependent function is specified as an exponential function and/or a cosine function. In order to verify the correctness of the proposed method, the case of the space-dependent sinusoidal function and time-dependent exponential function is studied, and the consistency between the derived solution and the literature solution is verified. The parameter influence of the time-dependent function of the boundary conditions on the temperature variation is also investigated, and the time-dependent function includes harmonic type and exponential type.

Karakteristik Pasang Surut Air Laut di Peairan Belawan Menggunakan Metode Admiralty

The purpose of this study is to understand the tidal characteristics using the Admiralty method, namely to determine the tidal harmonic constants and types of tides based on the forrmzahl value in Belawan waters. This research was conducted from March 1 to 29, 2022. The information was obtained from the Meteorology, Climatology and Geophysics Agency (BMKG) at the Belawan Maritime Station in Medan. Information analysis using the Admiralty method. The results showed that the value of the harmonic constant was 28.745. The results of these calculations belong to the category of single daily tidal type. While the evaluation of the height of the water level, the average water level at the time of the full moon (MHWL) is 77.2 cm, the average (MLWL) is 44.2 cm. From the results of this study, it can be determined that Belawan waters have a single daily tidal type.

Parametric design of the human bone with the aim of promoting dynamic resistance

In sports practices, the intensity and type of forces exerted on the human body are totally different from everyday living. Harsh movements and impact loadings are very common in sports and sometimes cause severe injuries to both muscles and bones. In a specific case, in volleyball games, the lower arm of the player is continuously exposed to heavy impacts by the ball. The main part of the body enduring these impact loads are the two ulna and radius bones in the lower arm. Therefore, in this type of ball impacts it is important to investigate the effect of ball movement and condition on the stress-induced in bones. In the present study, using numerical finite element and generalized differential quadrature (GDQ) methods, stability conditions in the ulna and radius bones in volleyball players under several static and harmonic loading types are investigated. Furthermore, change in the ball mass in the standard range is also considered in analyses. The bone structure is modeled as a cylindrical shell with classical elasticity theory and for inhomogeneous material properties. The results obtained using the numerical method are presented with a focus on the state of stability of the both ulna and radius bones.

A Numerical Investigation of Sloshing in a 3D Prismatic Tank with Various Baffle Types, Filling Rates, Input Amplitudes and Liquid Materials

Partially filled liquid-carrying tanks have been used in many engineering applications, such as ships, vehicle fuel tanks, rockets, and drink or petroleum tankers. Liquid sloshing is an exciting phenomenon that researchers are investigating because of its complex behavior specifications. In this study, the sloshing responses of a prismatic tank with the approximate volume of an automobile fuel tank under different laterally harmonic excitation amplitudes, baffle structures, filling rates, and different types of liquid were investigated numerically. The computational fluid dynamics method (CFD) was used to solve fluid dynamics equations, and the volume of fluid method was applied to simulate two-phase flow in the tank. A validation study was performed by a literature study. Later, the effect of large and small excitation amplitudes, filling rates and fluid types on sloshing behavior were investigated and comparatively analyzed in the tank system with various baffle types.