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Based on the theory of complex variable function, the propagation properties of anti-plane shear wave in inhomogeneous right-angle space are analysed. The inhomogeneity of the medium is reflected in the fact that the shear modulus is a function related to spatial coordinates. A displacement auxiliary function and a pair of mapping functions are introduced to assist in deriving the governing equation. Meanwhile, the transformation between coordinates can be more convenient in the complex coordinate system. The wave field expressions of reflected and scattering waves are constructed by using the image method. The unknown coefficients in the scattering waves are solved according to the boundary condition of the circular hole, and the analytical expressions of the wave field in the right-angle space are given. In dynamic analysis, rigidity can be expressed by shear modulus. In the analysis of numerical examples, the dependence of displacement amplitude and dynamic stress concentration factor (DSCF) on the inhomogeneous parameter, reference wave number, and the position of circular hole are discussed. [ABSTRACT FROM AUTHOR]

Purpose. Determination in the analytical form of the maximum limiting influence of the non-identity of the resistive elements of the high-voltage arm on the amplitude-frequency characteristic and phase-frequency characteristic of the voltage divider with parallel-series connection of R-, C-elements of the high-voltage arm. Methodology. Based on the previously developed theory of broadband voltage dividers with parallel-series connection of R‑, C-elements, analytical expressions for amplitude-frequency and phase-frequency characteristics of the voltage divider are obtained and investigated taking into account the limit case of non-identical resistive elements of high-voltage arm. Results. The nature of the dependencies of the frequency characteristics of the broadband voltage divider on the value of the tolerance of the resistive elements of the high-voltage arm, the division factor of the voltage divider in a wide range of frequency changes are determined. Simplified approximating expressions for the maximum values of frequency characteristics of the voltage divider are proposed and their error is determined. Originality. For the first time in the analytical form the limiting influence of non-identity of resistive elements of a high-voltage arm of a voltage divider on its frequency characteristics is considered. A mathematical model of this influence is constructed and the limit values of frequency characteristics of the voltage divider are determined. Practical value. It is recommended to introduce into the normative documentation of broadband voltage dividers the corrected value of the division factor, which allows to significantly reduce the deviation of the actual value of the division factor of the voltage divider from the normalized value in a wide range of frequency changes.

Purpose. Determination in the analytical form of the maximum limiting influence of the non-identity of the resistive elements of the high-voltage arm on the amplitude-frequency characteristic and phase-frequency characteristic of the voltage divider with parallel-series connection of R-, C-elements of the high-voltage arm. Methodology. Based on the previously developed theory of broadband voltage dividers with parallel-series connection of R-, C-elements, analytical expressions for amplitude-frequency and phase-frequency characteristics of the voltage divider are obtained and investigated taking into account the limit case of nonidentical resistive elements of high-voltage arm. Results. The nature of the dependencies of the frequency characteristics of the broadband voltage divider on the value of the tolerance of the resistive elements of the high-voltage arm, the division factor of the voltage divider in a wide range of frequency changes are determined. Simplified approximating expressions for the maximum values of frequency characteristics of the voltage divider are proposed and their error is determined. Originality. For the first time in the analytical form the limiting influence of non-identity of resistive elements of a high-voltage arm of a voltage divider on its frequency characteristics is considered. A mathematical model of this influence is constructed and the limit values of frequency characteristics of the voltage divider are determined. Practical value. It is recommended to introduce into the normative documentation of broadband voltage dividers the corrected value of the division factor, which allows to significantly reduce the deviation of the actual value of the division factor of the voltage divider from the normalized value in a wide range of frequency changes. [ABSTRACT FROM AUTHOR]

The resistance–capacitance (RC) model is one of the most applicable circuits for modeling the charging and discharging processes of supercapacitors (SCs). Although this circuit is usually used in the electric and thermal investigation of the performance of SCs, it does not include leakage currents. This paper presents exact analytical formulas of leakage-current-based supercapacitor models that can be used in industrial applications, i.e., constant-power-based applications. In the proposed model, current and voltage are represented as a solution of nonlinear equations that are solved using the standard Newton method. The proposed expressions' accuracy is compared with the results obtained using traditional numerical integration methods with leakage current formulation and other methods, found in the literature, with no leakage current formulation. The results confirm that including leakage current represents a more accurate and realistic manner of modeling SCs. The results show that the derived expressions are precise, allowing the generation of results that closely match those obtained using traditional numerical-based methods. The derived expressions can be used to investigate SCs further and achieve more accurate and efficient regulation and control of SCs in different applications. [ABSTRACT FROM AUTHOR]

Compartment fire is an important research area in fire science and fire safety. Owing to its simplicity, the zone model remains attractive in studying compartment fire. Although computer software packages for the two‐layer zone are in popular use, a basic understanding of the physical phenomena involved and the mathematical formulation thereon would benefit users of the zone model in terms of critical judgement of software outputs, which could be unreasonable, and insight of the general trends of important variables. In general, the differential equations based on a two‐layer zone model have no analytical solutions and numerical methods are required. However, stiffness is present in using numerical methods to solve these differential equations, and reliable results are obtained only when very small time steps are used in computation. Nevertheless, in some special cases, analytical solution for the clear height in the zone model exists, as demonstrated in the present study. Predicted values of smoke layer height in the present approach are shown to be in good agreement with experimental results reported in the literature. The present work would provide insight into the tenability condition of a compartment on fire and is an important issue in fire safety management. [ABSTRACT FROM AUTHOR]

The numerical study on the ultimate bearing capacity of the tubular T-joints are presented at elevated temperatures when the brace is compressed. A numerical method is given for analyzing the load bearing capacities of the tubular T-joints at elevated temperatures, which is realized by using the finite element software ABAQUS. The shell element is adopted to enable a realistic modeling of tubular T-joints subjected to fire. During the analysis, the influence of geometric parameters on the joints strength in fire is studied, including β (diameter ratio) and γ (chord radius/thickness ratio). Based on the analysis of the geometric parameters, a predictive analytical expressions of the ultimate bearing capacity of the tubular T-joints in fire are proposed on the basis of the observed trends of the ultimate bearing capacity reduction with the temperatures rising. [ABSTRACT FROM AUTHOR]

This study presents the analytical expressions of drainable and fillable porosity for layered soils under shallow groundwater environments. The expressions of drainable porosity λd and fillable porosity λf for two‐layered soils are first derived with water table depth change ∆d→0 under dynamic soil moisture conditions. The expressions of λd and λf consider the dynamic soil moisture conditions through the assumption of vertical fluxes in the unsaturated zone, varying as functions of water table elevation h. The analytical expressions of λd and λf have the same formula form but different values under evapotranspiration and recharge conditions, which is also verified using the numerical calculation. The case comparisons indicate that the vertical distribution of soil layers and the fluxes have significant effects on the performance of λd and λf. Our proposed expressions could effectively characterize the effects of soil layering and vertical fluxes on λd and λf. They could improve the accuracy of the previous expressions for single‐layer soils, and complement the lack of description for layered soils under dynamic soil moisture conditions in previous expressions. Furthermore, under the hydrostatic assumption (static equilibrium of soil water pressure distribution) in the unsaturated zone, the expressions of drainable porosity λd* and fillable porosity λf* for two‐layered soils with large water table fluctuations are derived based on λd and λf (asterisks in λd* and λf* is to distinguish them from λd and λf derived based on ∆d→0). Verification with drainage experimental data indicates that the proposed expressions can be extended to well capture the λd* and λf* changes for two‐layered soils. Application scenarios suggest that our analytical expressions of drainable and fillable porosity are useful to improve the application performance of related subsurface modeling and groundwater estimation for layered soils under shallow water table environments. Key Points: Analytical expressions of drainable and fillable porosity for layered soils were proposed under vertical fluxes conditionsExpressions of drainable/fillable porosity for layered soils with large water table fluctuations were derived under hydrostatic conditionsExpressions were useful to improve the application performance of subsurface modeling and groundwater estimation for layered soils [ABSTRACT FROM AUTHOR]

The magnetotelluric (MT) sounding of a layered Earth model involving a transitional layer has been widely studied, and MT responses of the model with dipping anisotropic conductivity have also been treated. However, a model incorporating both a transitional layer and dipping anisotropy has seldom been considered. The analytical solution of such a geoelectrical model including three layers was derived in this study. The middle layer was a transitional layer with conductivity exponentially varying with depth, which was covered by a homogeneous layer and underlaid by a dipping anisotropic half-space. The electromagnetic (EM) fields in the transitional layer were explicitly solved with modified Bessel functions. The surface impedance was calculated recursively. The dependence of the apparent resistivity and impedance phase as well as the EM fields on different model parameters were investigated in detail. We believe that our analytical solution provides a useful complement to the theory of the one-dimensional (1D) inversion and interpretation based on the layered model with fixed conductivity. [ABSTRACT FROM AUTHOR]

Purpose. Determination in the analytical form of the maximum limiting influence of the non-identity of the capacitive elements of the high-voltage arm on the amplitude-frequency characteristic and phase-frequency characteristic of the voltage divider with parallel-series connection of R-, C-elements of the high-voltage arm. Methodology. Based on the previously developed theory of broadband voltage dividers with parallel-series connection of R-, C-elements, analytical expressions for amplitude-frequency and phase-frequency characteristics of the voltage divider are obtained and investigated taking into account the limit case of non-identical capacitive elements of high-voltage arm. Results. The nature of the dependencies of the frequency characteristics of the broadband voltage divider on the value of the tolerance of the capacitive elements of the high-voltage arm, the division factor of the voltage divider in a wide range of frequency changes is determined. Simplified approximating expressions for the maximum values of frequency characteristics of the voltage divider are proposed and their error is determined. Originality. For the first time in the analytical form the limiting influence of non-identity of capacitive elements of a high-voltage arm of a voltage divider on its frequency characteristics is considered. A mathematical model of this influence is constructed and the limit values of frequency characteristics of the voltage divider are determined. Practical value. It is recommended to introduce into the normative documentation of broadband voltage dividers the corrected value of the division factor, which allows to significantly reduce the deviation of the actual value of the division factor of the voltage divider from the normalized value in a wide range of frequency changes.

For Arc High-Impedance Faults (AHIFs) in Active Distribution Networks (ADNs) with neutral grounding via Petersen coil, both nonlinear characteristics caused by arc and the influence of zero-input response have been not considered in previous research. Quantitative analysis methods based on the threshold setting comparison is mostly adopted in current field applications, which cannot balance the detection reliability and sensitivity simultaneously. In response to this problem, a higher precision nonlinear AHIF equivalent model is established based on the logarithmic arc model. State equations describing the occurrences of AHIFs are solved by using piecewise linear fitting technique, and analytical expressions of zero-sequence voltage and currents are present in paper. Furthermore, the regular volt-ampere characteristic dynamic trajectory between the zero-sequence voltage and current in characteristic frequency band is analyzed, and a tangent-inverse method is proposed to extract the direction of dynamic trajectory, then an effective fault detection algorithm is designed. The validity of the proposed nonlinear equivalent model and detection algorithm are verified by various cases’ simulations and field test data. [ABSTRACT FROM AUTHOR]

Bearing stiffness, as one of the most important service characteristics for ball bearing, plays a crucial role in the bearing design and rotor dynamic analysis. To rapidly and accurately calculate the stiffness matrix of ball bearing under the arbitrary load conditions, a 5-DOF analytical model for bearing stiffness matrix analysis has been established by the ball–raceway contact analysis, implicit/explicit differential method, and matrix operations. The model has been validated comparing with the previous methods and experimental results. Based on this, the model has been used to investigate the influences of the load and operation conditions, the structural parameters variation on stiffness of ball bearing. The results show that property increasing axial preload can inhibit the attenuation of speed-varying stiffness, and the contact states between balls and raceways also have significant influence on the change in the stiffness of ball bearings. Besides, a larger curvature coefficient of inner raceway and a small curvature coefficient of outer raceway can effectively improve the stiffness of ball bearing at high speed. Therefore, the proposed method can be a useful tool in bearing optimize design and performance analysis of ball and rotor system under various load conditions. [ABSTRACT FROM AUTHOR]

Objectives: The reduced latitude can simplify many complex problems in ellipsoid geodesy. In addition to the reduced latitude, geodetic, geocentric, rectifying, authalic, and conformal latitudes are commonly used in map projection transformation. The differences between commonly used latitudes and reduced latitude are generally expressed in numerical form, and there are no corresponding analytical expressions. Analytical expressions of differences between commonly used and reduced latitudes can visually display the differences between latitudes. Methods: Starting from the definition of commonly used and reduced latitudes, with the aid of computer algebra analysis methods, the function relations between commonly used latitudes and reduced latitude are derived, then the differences between commonly used and reduced latitudes can be obtained by finding the extreme points. Analytical expressions of differences between them are derived and their coefficients are expressed in a power series of the eccentricity e, and the third flattening n. Results: Theoretical and numerical analysis results show that the analytical expressions in terms of n are more concise than those in terms of e, the absolute value of the differences between commonly used latitudes and reduced latitude first increase and then decrease, and the extreme points are all near Ï€/4, the reduced latitude is closer to commonly used latitudes in numerical values. Conclusions: Expressions of differences between commonly used latitudes and reduced latitude are more intuitively and applicable to any referenced ellipsoid, to some extent, they could enrich the analytical theory of the various commonly used latitudes. Objectives: The reduced latitude can simplify many complex problems in ellipsoid geodesy. In addition to the reduced latitude, geodetic, geocentric, rectifying, authalic, and conformal latitudes are commonly used in map projection transformation. The differences between commonly used latitudes and reduced latitude are generally expressed in numerical form, and there are no corresponding analytical expressions. Analytical expressions of differences between commonly used and reduced latitudes can visually display the differences between latitudes. Methods: Starting from the definition of commonly used and reduced latitudes, with the aid of computer algebra analysis methods, the function relations between commonly used latitudes and reduced latitude are derived, then the differences between commonly used and reduced latitudes can be obtained by finding the extreme points. Analytical expressions of differences between them are derived and their coefficients are expressed in a power series of the eccentricity e, and the third flattening n. Results: Theoretical and numerical analysis results show that the analytical expressions in terms of n are more concise than those in terms of e, the absolute value of the differences between commonly used latitudes and reduced latitude first increase and then decrease, and the extreme points are all near π/4, the reduced latitude is closer to commonly used latitudes in numerical values. Conclusions: Expressions of differences between commonly used latitudes and reduced latitude are more intuitively and applicable to any referenced ellipsoid, to some extent, they could enrich the analytical theory of the various commonly used latitudes. [ABSTRACT FROM AUTHOR]

The resistance–capacitance (RC) model is one of the most applicable circuits for modeling the charging and discharging processes of supercapacitors (SCs). Although this circuit is usually used in the electric and thermal investigation of the performance of SCs, it does not include leakage currents. This paper presents exact analytical formulas of leakage-current-based supercapacitor models that can be used in industrial applications, i.e., constant-power-based applications. In the proposed model, current and voltage are represented as a solution of nonlinear equations that are solved using the standard Newton method. The proposed expressions’ accuracy is compared with the results obtained using traditional numerical integration methods with leakage current formulation and other methods, found in the literature, with no leakage current formulation. The results confirm that including leakage current represents a more accurate and realistic manner of modeling SCs. The results show that the derived expressions are precise, allowing the generation of results that closely match those obtained using traditional numerical-based methods. The derived expressions can be used to investigate SCs further and achieve more accurate and efficient regulation and control of SCs in different applications.

This paper presents new analytical expressions for the maximum power point voltage, current, and power that have an explicit dependence on the series resistance. An explicit expression that relates the series resistance to well-known solar cell parameters was also derived. The range of the validity of the model, as well as the mathematical assumptions taken to derive it are explained and discussed. To test the accuracy of the derived model, a numerical single-diode model with solar cell parameters whose values can be found in the latest installment of the solar cell efficiency tables was used. The accuracy of the derived model was found to increase with increasing bandgap and to decrease with increasing series resistance. An experimental validation of the analytical model is provided and its practical limitations addressed. The new expressions predicted the maximum power obtainable by the studied cells with estimated errors below 0.1 % compared to the numerical model, for typical values of the series resistance. [ABSTRACT FROM AUTHOR]

The proliferation of photovoltaic (PV) can cause several operational problems in distribution systems. In this paper, comprehensive analytical expressions (CAEs) are proposed for maximizing the technical benefits of multiple PV units to distribution systems considering the uncertainty of PV generation and load profiles. Specifically, the proposed CAEs quantify and optimize the following five vital indices with multiple PV units: 1) active energy losses, 2) reactive losses, 3) voltage deviations, 4) line congestion margin, and 5) voltage stability index. The smart functions of the PV inverter (i.e., reactive power support and active power curtailment) are also incorporated in the CAEs, complying with the revised IEEE 1547:2018 standard. Further, various PV tracking options are considered, including fixed, one-axis, and two-axis trackers. Unlike existing approaches, the CAEs can simultaneously solve the optimal allocation problem of multiple PV units in a direct manner without needing optimization algorithms, iterative processes, or simplifying procedures. The calculated results reveal the high performance of the CAEs in terms of accuracy, flexibility and computational speed while providing further PV planning options. Moreover, CAEs are effectively utilized for two other applications with promising computational performance, i.e., rapid assessment of PV impacts with annual datasets and optimal centralized/decentralized inverter control. [ABSTRACT FROM AUTHOR]

Purpose. Determination in the analytical form of the maximum limiting influence of the non-identity of the capacitive elements of the high-voltage arm on the amplitude-frequency characteristic and phase-frequency characteristic of the voltage divider with parallelseries connection of R-, C-elements of the high-voltage arm. Methodology. Based on the previously developed theory of broadband voltage dividers with parallel-series connection of R-, C-elements, analytical expressions for amplitude-frequency and phasefrequency characteristics of the voltage divider are obtained and investigated taking into account the limit case of non-identical capacitive elements of high-voltage arm. Results. The nature of the dependencies of the frequency characteristics of the broadband voltage divider on the value of the tolerance of the capacitive elements of the high-voltage arm, the division factor of the voltage divider in a wide range of frequency changes is determined. Simplified approximating expressions for the maximum values of frequency characteristics of the voltage divider are proposed and their error is determined. Originality. For the first time in the analytical form the limiting influence of non-identity of capacitive elements of a high-voltage arm of a voltage divider on its frequency characteristics is considered. A mathematical model of this influence is constructed and the limit values of frequency characteristics of the voltage divider are determined. Practical value. It is recommended to introduce into the normative documentation of broadband voltage dividers the corrected value of the division factor, which allows to significantly reduce the deviation of the actual value of the division factor of the voltage divider from the normalized value in a wide range of frequency changes. [ABSTRACT FROM AUTHOR]

The modular multilevel converter (MMC) based multiport energy router (MER) provides multiple connecting points such as HVAC terminal, HVDC terminal and dc terminals of submodules (SMs). The uneven power distribution of SM to distributed energy resources (DERs) is allowed and required for MMC-MER but may cause imbalance on capacitor voltages, further leading to SM overvoltage. The existing circulating current injection methods have the problems of significantly increasing the current stress or voltage stress of SM power devices, which will increase the power rating and cost of power devices. This paper develops a novel reactive fundamental frequency circulating current injection method that can suppress the high voltage stress resulted by static capacitor voltage difference and high current stress caused by second-order circulating current injection. The key of the proposed method is the determination of reactive fundamental frequency circulating current reference combing with power distribution of SM to DER, which is on the basis of derived analytical expressions between power limits and reactive fundamental frequency circulating current. Finally, simulation and experimental results validate the feasibility of proposed method. [ABSTRACT FROM AUTHOR]

High frequency excitation in the electro-hydraulic mode, especially kilohertz regime, has already been realized owing to the merits of specially designed two-dimensional rotary valve (2D rotary valve). However, some sacrifice in the excitation amplitude is inevitable as pursuing the higher excitation frequency. A scheme for an electro-hydraulic excitation system working at resonant is proposed, which could accomplish high excitation frequency and large vibration amplitude simultaneously. More specifically, the resonant frequency of this system can be regulated as loads of the excitation system, which would greatly facilitate fatigue testing of different elastomeric materials. This paper investigates more details of the excitation characteristics in whole frequency domain, including non-resonance and variable resonance. The excitation waveforms, variable resonant frequency and resonant peak are derived and validated by experimental data. This work provides clear analytical expressions of the mathematical model, offers important guidance for the design of the system, and paves the way for the potential engineering application of the variable resonant excitation technology.

Bearing stiffness, as one of the most important service characteristics for ball bearing, plays a crucial role in the bearing design and rotor dynamic analysis. To rapidly and accurately calculate the stiffness matrix of ball bearing under the arbitrary load conditions, a 5-DOF analytical model for bearing stiffness matrix analysis has been established by the ball–raceway contact analysis, implicit/explicit differential method, and matrix operations. The model has been validated comparing with the previous methods and experimental results. Based on this, the model has been used to investigate the influences of the load and operation conditions, the structural parameters variation on stiffness of ball bearing. The results show that property increasing axial preload can inhibit the attenuation of speed-varying stiffness, and the contact states between balls and raceways also have significant influence on the change in the stiffness of ball bearings. Besides, a larger curvature coefficient of inner raceway and a small curvature coefficient of outer raceway can effectively improve the stiffness of ball bearing at high speed. Therefore, the proposed method can be a useful tool in bearing optimize design and performance analysis of ball and rotor system under various load conditions.

Analytical expressions for the on-axis normalized transmittance and total transmittance z-scan of thick nonlocal nonlinear media illuminated by a Gaussian beam are obtained in this paper. The corresponding thin media transmittance expressions and the distributed lens method are used together with the assumption that the nonlocal media exhibits weak nonlinear absorption and refraction in order to obtain the formulas. The obtained expressions enable the consideration of different degrees of nonlocality and the reproduction of the results reported for local media. [ABSTRACT FROM AUTHOR]

New expressions in terms of stress components of the Karafillis and Boyce (1993) yield criterion are derived. First, it is demonstrated that the isotropic version of this criterion is a homogeneous polynomial in J 2 and J 3 2 , where J 2 and J 3 are the invariants of the stress deviator. Moreover, it is shown that irrespective of the value of the interpolation parameter, it has the same mathematical expression as Hershey–Hosford isotropic yield criterion. Making use of these results, new expressions of the Karafillis and Boyce (1993) yield function for orthotropic BCC and FCC materials are derived. This enables to demonstrate that Barlat et al. (1991) and Karafillis and Boyce (1993) orthotropic criteria have the same mathematical form. Most importantly, the benefits of using these new expressions of the Karafillis and Boyce (1993) orthotropic yield criterion for the evaluation of the anisotropy coefficients are demonstrated. It is shown that these coefficients can be evaluated by solving simple algebraic systems of equations involving only experimental flow stresses, or alternatively involving only Lankford coefficients for three orientations. This makes the parameterization of an advanced non-quadratic criterion reliable even when the available mechanical test data are limited. [ABSTRACT FROM AUTHOR]

Electromigration (EM) is considered to be one of the most important reliability issues for current and future ICs in 10-nm technology and below. In this article, we propose a fast analytic solution to compute the stress evolution in the confined multisegment interconnect wires. The new method, called the accelerated separation of variables (ASOV) method, aims to find the analytic solutions of the partial differential equations of stress in confined interconnect metals based on the SOV method. It offers several improvements over the existing plain SOV-based method. First, we show that the accuracy of the solution depends on the structure of the interconnects. As a result, the number of required eigenvalues is structure and problem dependent, instead of fixed numbers used by the existing SOV method. Second, for the straight line multisegment and star-structured multiterminal interconnects, analytical expressions are formulated to calculate the eigenvalues directly instead of using numerical methods as in the existing SOV method. Third, we propose a linear Gaussian elimination (GE) algorithm by exploiting the banded structure with the serrated-edge form of the transcendental matrix, which can significantly speed up GE process, and is the key computing step in the SOV-based solution framework. Fourth, instead of using the simple bisection search, we propose to use an enhanced determinant-based secant iterative method to find the eigenvalues of the transcendental matrix. Numerical results show that a good agreement is achieved between analytical and numerical results on two special cases, and the resulting algorithm can lead to 3–5X speedup over the existing plain SOV-based solution on a number of multisegment interconnects benchmarks. [ABSTRACT FROM AUTHOR]

This paper presents new analytical expressions for the maximum power point voltage, current, and power that have an explicit dependence on the series resistance. An explicit expression that relates the series resistance to well-known solar cell parameters was also derived. The range of the validity of the model, as well as the mathematical assumptions taken to derive it are explained and discussed. To test the accuracy of the derived model, a numerical single-diode model with solar cell parameters whose values can be found in the latest installment of the solar cell efficiency tables was used. The accuracy of the derived model was found to increase with increasing bandgap and to decrease with increasing series resistance. An experimental validation of the analytical model is provided and its practical limitations addressed. The new expressions predicted the maximum power obtainable by the studied cells with estimated errors below 0.1% compared to the numerical model, for typical values of the series resistance.

This Letter presents the closed‐form expressions of the unloaded Q‐factor in metallic cavities with equilateral triangular cross section, computed from its analytical resonant mode solutions. The obtained analytical formulas for the Q‐factor extend the very restricted range of problems with closed‐form solutions, complementing the classic cases of the rectangular and circular cavities found in the literature. The presented derivation provides extremely rapid reference solutions for diverse applications such as filter design by test cases for microwave characterisation systems or numerical full‐wave solvers. A universal Q‐chart, valid for any aspect ratio, is given and the achieved analytical results for the first resonant modes are tested with numerical commercial software, showing excellent agreement. [ABSTRACT FROM AUTHOR]

Voltage dips/sags are one of the major concerns for electricity consumers as well as utility service providers. Therefore, the characterisation of voltage dips/sags is required. This study presents a set of mathematical expressions for characterising different types of voltage dips/sags and their associated phase‐angle jumps, which are typically found due to faults and/or disturbances in electricity networks. The expressions are derived analytically from the model of the power network containing generators, transmission and/or distribution lines, transformers etc. Four types of voltage dips, namely, A, B, E, and G, which are associated with four major types of faults including balanced three‐phase faults, single line‐to‐ground, double line‐to‐ground, and line‐to‐line faults, are considered to derive the analytical expressions. Dynamic simulation results, using a test distribution system, approve the validity as well as the accuracy of the developed expressions. The influence of fault‐types and fault‐locations is investigated from the mathematical expressions; further, validation is conducted through a simulation study. The analytical expressions, presented in this study, are a valuable tool in the planning stage since the expressions can be employed to characterise during‐fault voltage dips at different buses in electricity network without conducting a large number of repeated dynamic simulations. [ABSTRACT FROM AUTHOR]

• A propagating tangential crack whose interior contains medium in multiferroic composites is considered. • The analytical solutions of the thermal field and magnetic-electric-elastic coupling fields are obtained in terms of elementary functions. • The influences of thermal effect, crack moving velocity and tangential force on fracture behavior are revealed. • The thermal fracture behaviors depend on the magneto-electro-elastic properties of crack interior, external loadings and inherent material properties. • The shear loading escalating leads to intensifying the thermal stress intensity factor. The purpose of the present work is to study the thermal fracture behaviors of multiferroic composites weakened by a propagating tangential crack containing a medium. The thermally, magnetically and (or) electrically impermeable/permeable crack models in the static state are the special cases of the present dynamic, partially permeable one. The heat conduction equation and magneto-electro-elastic governing equations with boundary conditions are converted analytically into singular integral equations which are solved exactly to yield the full plane solutions. Three cases of the eigenvalue properties are considered depending on material properties. Based on those solutions, various loadings generated by the crack interior are determined. Numerical experiments are conducted, which show that the thermal fracture behaviors depend on the magnetic, electric and elastic properties of the crack interior as well as the external loadings and inherent material properties characterized by the crack moving velocity. [ABSTRACT FROM AUTHOR]

In this paper, we present a probabilistic fracture mechanics methodology to analyze elastic and elastic-plastic fracture of semi-elliptical longitudinal cracks in pipes under internal pressure. Numerical results are acquired using three-dimensional finite element simulations. Analytical expressions are proposed with unknown coefficients obtained by nonlinear fitting to the numerical results. For the elastic case, results of the shape function using the newly proposed expression are found to be in a good agreement with those found in the literature. In the elastic-plastic case, estimates of the J-integral are presented for various ratios including crack depth to pipe thickness (a/t), reference stress to material yield stress (σref/σy) and mean pipe radius to its thickness (Rm/t). It is found that the range of applicability of the proposed expressions is extended even beyond those found in the literature. Finally, failure probability is accessed by a statistical analysis for uncertainties in loads and material properties, and structural reliability. The probability density function is estimated by the Monte Carlo Method. It is shown from the present results that the crack size is an important factor influencing the distribution function of (J/Je), failure and reliability rates. [ABSTRACT FROM AUTHOR]

This article conducts an exact analysis of a thermal dielectric crack moving in piezoelectric materials. Self-generating thermal and electric loadings by the crack interior are exerted on the crack surfaces as well as various external loadings including a shearing force. Fundamental solutions of the thermal and electro-elastic coupling fields are given by determining a temperature function and a harmonic function with eigenvalues properties due to material properties considered. Analytical expressions are obtained benefiting evaluation of key parameters. Numerical analysis is done and some interesting observations are found. There is a critical crack velocity within and beyond which the electric loading exerts different influences on the thermal flux of crack interior and the thermal stress intensity factor. [ABSTRACT FROM AUTHOR]