Exact Analytical Model Research Articles

Quasi-homologous evolution of self-gravitating systems with vanishing complexity factor

We investigate the evolution of self-gravitating either dissipative or non-dissipative systems satisfying the condition of minimal complexity, and whose areal radius velocity is proportional to the areal radius (quasi-homologous condition). Several exact analytical models are found under the above mentioned conditions. Some of the presented models describe the evolution of spherically symmetric dissipative fluid distributions whose center is surrounded by a cavity. Some of them satisfy the Darmois conditions whereas others present shells and must satisfy the Israel condition on either one or both boundary surfaces. Prospective applications of some of these models to astrophysical scenarios are discussed.

Exact Analytical Model for Bose-Einstein Condensate at Negative Temperature

We present an exact analytical model of a cigar-shaped Bose-Einstein condensate at negative temperature. This work is motivated by the first experimental discovery of negative temperature in Bose-Einstein condensate by Braun et al. We have considered an external confinement which is a combination of expulsive trap, bi-chromatic optical lattice trap, and linear trap. The present method is capable of providing the exact form of the condensate wavefunction, phase, nonlinearity and gain/loss. One of the consistency conditions is shown to map onto the Schrödinger equation, leading to a significant control over the dynamics of the system. We have modified the model by replacing the optical lattice trap by a bi-chromatic optical lattice trap, which imparts better localization at the central frustrated site, delineated through the variation of condensate fraction. Estimation of temperature and a numerical stability analysis are also carried out. Incorporation of an additional linear trap introduces asymmetry and the corresponding temporal dynamics reveal atom distillation at negative temperature.

Definition of complexity factor for self-gravitating systems in Palatini f(R) gravity

The aim of this paper is to explore the complexity factor for those self-gravitating relativistic spheres whose evolution proceeds non- dynamically. We are adopting the definition of CF mentioned in (Herrera 2018 Phys. Rev. D 97, 044010), modifying it to the static spherically symmetric case, within the framework of a modified gravity theory (the Palatini f(R) theory). In this respect, we have considered radial dependent anisotropic matter content coupled with spherical geometry and determined the complexity factor involved in the patterns of radial evolution. We shall explore the field and a well-known Tolman-Oppenheimer-Volkoff equations. After introducing structure scalars from the orthogonal decomposition of the Riemann tensor, we shall calculate complexity factor. An exact analytical model is presented by considering firstly ansatz provided by Gokhroo and Mehra. The role of matter variables and f(R) terms are analyzed in the structure formation as well as their evolution through a complexity factor.

Field Prediction and Validation of a Slotless Segmented-Halbach Permanent Magnet Synchronous Machine for More Electric Aircraft

High-performance propulsion motors for more electric aircraft (MEA) are highly appreciated in transportation electrification. For the widely adopted permanent magnet synchronous machines (PMSMs) for MEA propulsion, the segmented Halbach magnetization has distinct merit that it contributes to sinusoidal air-gap magnetic fields for smooth operation. However, it is hard to fast predict the field distribution and machine performance due to the model complexity. Based on the subdomain method, this article first proposes an exact analytical model for slotless PMSM with the segmented Halbach PMs. Then, the model is applied in the fast computation of the magnetic field and performance prediction. Subsequently, the analytical results are compared with the finite element method (FEM), including the flux density, flux linkage, induced voltage, and steady-state torque at different loads. Also, the computation burden and prediction error are included to validate the model’s rapidity and accuracy. Finally, by prototype fabrication and unique field measurement methods, the flux density distribution is originally validated.

Analytical Modeling of Neutral Point Current in T-type Three-level PWM Converter

Since a T-type three-level PWM converter has several intrinsic advantages, it has been researched for various applications such as grid-connected converter systems. However, it necessarily requires an additional voltage control loop for balancing the upper and the lower DC-link voltage. To satisfy this requirement, an offset voltage is widely used to provide a neutral point current without affecting other variables such as total DC-link voltage and three-phase input current. However, these methods are mostly based on the averaged value between the applied offset voltage and the neutral point voltage or current, and there is no exact analytical model. Therefore, in this paper, the exact theoretical relationship between the offset voltage and the neutral point current is analyzed. This result can be expected to be useful for an effective voltage balancing control loop design as well as the better understanding of the whole system operation. The validity of the obtained analytical model is verified by simulation and experimental results.

Modeling and electromechanical performance analysis of frequency-variable piezoelectric stack transducers

An exact analytical model of frequency-variable piezoelectric stack transducers is proposed, and their dynamic characteristics are studied in this article. Based on the linear piezoelasticity theory, the dynamic analytical solution is first derived, and then its correctness is validated by comparing it with the results of a special example in the previous literature and the ones of the experimental study. The effects of the tuning resistance and the layer number of the active element on the dynamic characteristics are discussed. Numerical results show that tuning the resistance and the layer number of the active element can enable the multi-frequency characteristics of the piezoelectric stack transducers. A proper layer number of the active element can minimize the short-circuited resonance frequency and the open-circuited anti-resonance frequency. These findings provide guidelines to design and optimize the piezoelectric stack transducers, which have promising potential in developing the multi-frequency Langevin transducers for some underwater sound and ultrasonic applications, such as ultrasonic cleaning, ultrasonic chemistry, and sonar radiators.

Soliton dynamics for an ingenious trap combination in a Bose-Einstein condensate

We provide an exact analytical model of the 1D Gross-Pitaevskii equation for a composite trap, which is capable of producing a variety of potentials: a general power-law, harmonic, tunable double-well etc. We derive a family of exact solitary wave solution for condensate density, which can be controlled by tuning the trap parameters. The fact that the minima positions can be controlled and the potential can also be made asymmetric makes it a favorable candidate for quantum information processing. Coherent control of the condensate profile is investigated, where it shows interesting collapse and revival. Moreover, the obtained two-soliton solution follows periodic and bi-periodic trajectories. A numerical study of the condensate dynamics is carried out to verify our result, which is also supported by a numerical stability analysis.

An Analytical Subdomain Model for Dual-Rotor Permanent Magnet Motor With Halbach Array

This article describes the development of an exact analytical model for predicting magnetic field distribution within active parts of a dual-rotor (DR) permanent magnet (PM) synchronous machine with a Halbach array. The model is based on the accurate solution of Laplace’s and Poisson’s partial differential equations (PDEs) using a method of separation of variables. The model also allows the calculation of the DR machine flux linkage, back electromotive force (bEMF), and electromagnetic torque at on load and cogging torque at a no-load condition. The model can be used at the initial stage of designing a DR PM machine for the estimation of electromagnetic parameters and the optimization of machine geometry.

Effect of Slot Opening Width on the Air-Gap Magnetic Field of a Direct Drive Permanent Magnet Motor

The direct drive permanent magnet motor (DDPMM) is a promising candidate for applications because of its high efficiency, high power density, and low maintenance costs. This study focused on the effect of slot opening width on the air-gap magnetic field of a DDPMM. An exact analytical model based on Fourier analysis was established to calculate the air-gap magnetic field. The analytical general solution of the air-gap magnetic field shows that the slot opening greatly affects the air-gap magnetic field distribution when the amount of the permanent magnet is constant. Then, the air-gap magnetic fields of the DDPMM with closed and open slots were compared at different positions. Furthermore, several finite-element models of motors composed of different numbers of unit motor were established based on the different slot opening widths to study the effect of slot opening width on the air-gap magnetic field. The results obtained using the finite element method verify that the slot opening width greatly affects the air-gap magnetic field.

Non-touching confinement of ternary particle systems by electrostatic surface forces

We apply an exact analytical model to demonstrate that multiple particle systems can bind in nontouching, minimum energy equilibrium configurations. It is predicted that such systems may self-assemble due entirely to electrostatic interactions without the aid of external forces. We demonstrate the potential energy equilibria for ternary systems where the effective dipoles cancel and the anomalous electrostatic forces are described by interactions between higher order multipoles.

Exploring the role of relaxation time, bond length and the length of the polymer chain in the end-to-end looping kinetics of a long polymer chain in solution. An exact analytical model

A Smoluchowski-like equation is accounted for modeling the end-to-end looping kinetics. The phenomenon of looping is represented by a Dirac delta function sink of finite strength, placed at a point where the end-to-end distance between the monomers is zero. The sink is a physical quantity determining the possibility of looping at that point. We formulate two different rate constants, showing arrhenius type behaviour, which are obtained using the Green’s function method. The long term rate constant depends directly on relaxation time and inversely on the bond length and for shorter polymer constructs, the end-to-end looping is faster than the longer one.

An exact analytical model for fluid flow through finite rock matrix block with special saturation function

An exact analytical solution for one-dimensional fluid flow through rock matrix block is presented. The nonlinearity induced from flow functions makes the governing equations describing this mechanism difficult to be analytically solved. In this paper, an analytical solution to the infiltration problems considering non-linear relative permeability functions is presented for finite depth, despite its profound and fundamental importance. Elimination of the nonlinear terms in the equation, as a complex and tedious task, is done by applying several successive mathematical manipulations including: Hopf-Cole transformation to obtain a diffusive type PDE; an exponential type transformation to get a convective-diffusive type PDE with suitable boundary conditions; Laplace transformation; Integral transformation to homogenize the boundary condition in the Laplace domain; and finally Laplace inversion method to find the time domain solution. The obtained solution is used for developing a new matrix-fracture transfer function. The developed analytical equations are used for prediction of drying front in the case of evaporation from deep water table. The model results are in close agreement with the experimental data and numerical simulation of the process. Results of sensitivity analysis of different parameters on recovery rate and ultimate production based on design of experiment method are also discussed.

A novel and exact analytical model for determination of critical depth in trapezoidal open channels

Critical depth is a significant parameter in the designing and management of open channels and related hydraulic structures, understanding the flow characteristics and calculations of varied flows (gradually, spatially, etc.). The trapezoidal cross sections are the most commonly used geometric sections in the network of water transmission and distribution channels, thus discussing its geometrical and hydraulic parameters is inevitable. The used nonlinear and mathematical relationships governing the critical depth problem in the trapezoidal channels are implicit and complex, hence the methods of trial and error, graphical and numerical are used to calculate it. In the present study, new explicit equations are presented based on mathematical analysis of the critical depth problem in the trapezoidal channel. Mathematical analysis has led to completely mathematical and analytical solutions having a definite physical (hydraulic) concept. Having the explicit equations provided by the ease of calculation process with no limitation of the application range and high accuracy are the advantages of this analysis. The accuracy of the presented equations is desired and determined according to the required accuracy. The data used for verification of the results are based on the critical flow condition (Froude number equal to 1), which has been generated in a wide and practical range of the geometric and hydraulic characteristics of the channel. Also, the calculated values are compared with the real values of the considered parameter.

Some aspects of mathematical modelling of physical, chemical and biological processes in natural leachate treatment systems

The problem of natural ecosystems protection against pollution by waste water is very actualin many countries, including Russia. Yearly water consumption by industry and agriculture ofRussia is about 75 - 80 billion cubic meters. Only 3 - 4 % of total volume of the industrialand municipal waste water is purified to establish norms. However basic quantity of wastewater is discharged without treatment in natural reservoirs. The most dangerous pollutants ofwater are pesticides, nitrates, sulfates, phosphoric substances and salts of heavy metals[!].The methods of waste water treatment can be divided on two groups: a) the methods oftreatment in artificial plants (for example, aerated lagoons; biological filters and etc.); b) themethods of treatment in natural systems (for example, fields of irrigation; filtration fields;wetlands and etc.). The choice of the treatment depends on composition and concentration ofwaste substances in leachate and economical expediency.During projecting and building of any water protection constructions, it is necessary todevelop the theoretical basis. However, the exact calculations of hydrodynamic processes ofconvective - diffusive transfer in space and in time and physical-chemical transformation ofnon-conservative substances are impossible because of extreme complicity in most cases [2].Moreover, using of exact analytical models is non-effective because of considerableinexactitude of initial information. The investigations of processes of the convective -diffusive transfer and transformation of substance in natural conditions are very complex andexpensive task. Therefore, the methods of mathematical modeling are the most favorable inthis case

Subdomain model for predicting armature reaction field of dual-stator consequent-pole PM machines accounting for tooth-tips

This paper presents an exact analytical subdomain model of dual-stator consequent-pole permanent-magnet (DSCPPM) machines accounting for tooth-tips, which can accurately predict the armature reaction field distribution in DSCPPM machines. In the proposed subdomain model, the field domain is composed of four types of sub-regions, viz. magnets, outer/inner air gaps, slots and slot openings. The analytical expressions of vector potential in each sub-region are determined by boundary and interface conditions. In comparison to the analytically predicted results, the corresponding flux density field distributions computed by finite element (FE) method are analyzed, which confirms the excellent accuracy of the developed subdomain model.

Performance Analysis of Squirrel-Cage Induction Motors Under Broken Rotor Bar and Stator Inter-Turn Fault Conditions Using Analytical Modeling

Stator winding inter-turn (SWI) short-circuit fault and broken rotor bar (BRB) fault are two common faulty modes of the squirrel cage induction motors (SCIMs). These faults have various impacts on the motor performance and may be rooted in each other; therefore, they may be presented at the same time in an SCIM. This paper proposes an exact analytical model for the SCIMs, which is able to include the SWI and BRB faults presenting simultaneously or individually with variable severities. Multiple coupled circuit modeling is applied to achieve such analytical model. The magnetic saturation effect is included in the model by applying appropriate air gap function and the rotor bars skew effect is taken into account by using 2-D winding function theory. Simulation results as well as the corresponding experimental results approve the presence of the same harmonics in the stator line currents due to the proposed faults. However, more detailed study gives solution for discrimination of the faults. The attained analytical model can be applied to examine the performance of the related fault diagnosis methods and to design novel fault diagnosis techniques including the model-based ones.

Long-Distance Continuous-Variable Quantum Key Distribution With Quantum Scissors

The use of quantum scissors, as candidates for non-deterministic amplifiers, in continuous-variable quantum key distribution systems is investigated. Such devices rely on single-photon sources for their operation and as such, they do not necessarily preserve the Guassianity of the channel. Using exact analytical modeling for system components, we bound the secret key generation rate for the system that uses quantum scissors. We find that for non-zero values of excess noise such a system can reach longer distances than the system with no amplification. The prospect of using quantum scissors in continuous-variable quantum repeaters is therefore emboldened.

An Exact Analytical Model for the Relationship Between Flow Resistance and Geometric Properties of Tubes Used in Semi-occluded Vocal Tract Exercises

The purpose of this work was twofold. First, we aimed to develop an exact analytical model for tubes used in semi-occluded vocal tract exercises to gain a quantitative insight into the relationship between flow resistance and the tube's geometric parameters. The second goal was to provide an isometric resistance chart that can be used by clinicians to indicate either the diameter or the length of a tube, so as to provide the same flow resistance of a reference tube. The theory for confined flows based on the Darcy-Weisbach equation was used to derive the analytical model, in which the friction factor was obtained by an explicit approximation with error smaller than 0.4%. The isometric resistance chart was generated with the analytical model, assuming the volume flow to be constant and equal to 0.4 L/s. The results obtained from the model agreed very well with both experimental and theoretical results from the literature, particularly for tubes longer than 6 cm and with inner diameters greater than 4.1 mm. In general, the analytical model slightly underestimates the back pressure for shorter and thinner tubes. For these cases, the maximum difference between analytical and experimental results corresponds to ≈5%. Analysis of the equation terms indicated that the flow resistance is significantly more sensitive to variations in the inner diameter than to variations on the tube's length, which agrees with experimental results found in the literature. Moreover, the effect of the mouth configuration is negligible for tubes whose length are one order of magnitude greater the the inner diameter. Nevertheless, for short tubes the mouth configuration becomes a significant parameter and can be described by the ratio between the tube's inner diameter and the effective diameter of the patient's oral cavity. The analytical model presented in this work can be applied to the entire set of tube geometries found in clinical practice and constitutes a straightforward tool for designing tubes for semi-occluded vocal tract exercises with specific therapeutic purposes or patient needs.

Modeling and analysis of multilayer piezoelectric-elastic spherical transducers

An exact analytical model of multilayer piezoelectric-elastic spherical transducers is established in this article, and their dynamic characteristics are studied. Based on the linear theory of piezoelasticity, the dynamic analytical solution is derived in terms of Bessel functions, Lommel functions, Gamma functions, and Generalized Hypergeometric functions, and the electric impedance is obtained to determine the resonance frequencies. By proper selection of the layer number and geometric dimensions, the proposed model can approximately degrade into some special cases in the previous works, including single-layer piezoelectric spherical transducer, two-layer piezoelectric-elastic spherical transducer, and three-layer sandwiched piezoelectric-elastic-piezoelectric spherical transducer. Comparisons between the results of the proposed model and the ones of these special cases are also given and good agreements are found. Furthermore, parametric studies are conducted to discuss the effects of the key parameters on the dynamic characteristics of these special cases and a multilayer case, such as the first resonance and anti-resonance frequencies as well as the corresponding electromechanical coupling factor. This work contributes to an overall understanding of the dynamic characteristics of miniature piezoelectric hollow sphere transducers (BBs) and stacked piezoelectric spherical transducers. The developed model can be used to guide the design of the multilayer piezoelectric-elastic spherical transducers for promising applications in underwater acoustic detection, ultrasonic imaging, hydrophones, and nondestructive testing.

An analytical model for solving generalized interval eigenvalue problem

• Theorem and corollaries are derived for generalized interval eigenvalue problems . • An effective way to discriminate the sign invariance of eigenvector is presented. • The analytical model solves the exact bounds of eigenvalue with high efficiency. • Inapplicability of existing methods and superiority of proposed model are revealed. Previous algorithms on calculating the eigenvalue bounds of generalized interval eigenvalue problems usually require preconditions difficult to be met or their efficiency and accuracy are unguaranteed. To overcome these defects, an exact analytical model for obtaining the interval solution set of generalized eigenvalue problem is proposed by rigorous derivations. The eigenvalue set of generalized interval eigenvalue problem is firstly characterized according to the linear interval system and interval arithmetic. Then by introducing the Rayleigh quotient the formulation of mathematical programming on maximizing and minimizing the eigenvalue is derived for the generalized interval eigenvalue problem. The Kuhn-Tucker theorem is subsequently applied to deduce the theorem on computing the upper and lower bounds of interval solution set. By applying the proposed model in real engineering problems, it is demonstrated that the positive definiteness or non-negative decomposition of the matrix pair required by the existing analytical methods cannot be satisfied in some engineering situations where the proposed model is applicable.