Closed-form Analytical Solution Research Articles

Harmonic Spectra Analysis of Digital SPWM in VSI With DC Bus Ripple and Dead-Time Effects

This paper presents a closed-form analytical solution to the harmonic spectrum of output voltage of a single-phase H-bridge inverter, and gives the precise harmonic spectrum considering the effects of dead-time and DC bus voltage ripple in a digitally controlled system. Since H-bridge is the basic block in converters, this solution and spectrum can be extended and applied to a variety of pulse width modulation (PWM)-based topologies. The interaction between modulation waveform and triangular during modulation process is fully analyzed from a mathematical perspective, so that the combined effects of digital sampling, dead-time and DC bus voltage ripple on the output voltage are summarized in time and frequency domains. Furthermore, the presented harmonic spectra do not need to judge the current polarity in calculation, which means that the exact amplitude and phase of each output harmonic can be obtained accurately, providing the important theoretical basis for system parameter design and specific harmonic compensation or elimination. The presented analysis results are verified by simulations and experiments.

Resource Allocation of NOMA Communication Systems for Federated Learning

Federated learning is a new communication and computing concept that allows naturally distributed data sets (e.g., as in data acquisition sensors) to be used to train global models, and therefore successfully addresses privacy, power and bandwidth limitations in wireless networks. In this paper, we study the communications problem of latency minimization of a multi-user wireless network used to train a decentralized machine learning model. To facilitate low latency, the wireless stations (WSs) employ non-orthogonal multiple access (NOMA) for simultaneous transmission of local model parameters to the base station, subject to the users’ maximum CPU frequency, maximum transmit power, and maximum available energy. The proposed resource allocation scheme guarantees fair resource sharing among WSs by enforcing only a single WS to spend the maximum allowable energy or transmit at maximum power, whereas the rest of the WSs transmit at lower power and spend less energy. The closed-form analytical solution for the optimal values of resource allocation parameters is used for efficient online implementation of the proposed scheme with low computational complexity.

Non-Newtonian Flow with Heat and Mass Transfer Over a Porous Stretching Sheet Under the Effects of Space Dependent and TGD Heat Sink, Non-Uniform Heat Source and Dissipation of Energy and CGD Mass Diffusion with Suction/Blowing

In the present paper, an analytical study of visco - elastic fluid flow with heat and mass transfer characteristics over a porous stretching sheet has been examined. Heat balance is maintained with space dependent and temperature gradient dependent heat sink/source, viscous dissipation, and non-uniform heat source on the non-Newtonian Walter’s liquid B' model. The mass balance is maintained with chemically reactive species of order I, variable mass diffusivity and concentration gradient dependent mass diffusion. Using similarity transformation technique on the highly non-linear differential equations, several closed form analytical solutions are obtained for non-dimensional temperature and concentration in both PST and PHF cases in the form of confluent hypergeometric (Kummar’s) functions. The effect of permeability parameter, viscoelastic parameter, suction parameter on velocity profiles and various physical fluid situations with different degrees of viscoelasticity, Prandtl number, Eckert number, heat source- sink strength, temperature field are discussed in detail and presented through graphs. Similarly in the mass transfer field the effects of Schmidt number, reaction rate parameters are discussed in detail and presented through graphs.

Micropolar Fluid Flow Through a Porous Stretching/Shrinking Sheet with Mass Transpiration: An Analytical Approach

In this paper, the flow of a micropolar fluid over a stretching or shrinking sheet is investigated under magnetohydrodynamic (MHD) conditions. Such a flow is described by highly nonlinear PDEs. Using the similarity transformation technique, the PDEs governing the flow are reduced to a system of nonlinear ODEs, which further allows a closed-form analytical solution. The effect of the microrotation on the skin friction coefficient, the dimensionless forms of the velocity, and the temperature flow fields in the neighborhood of the stretching or shrinking sheet are discussed for various combinations of the dimensionless parameters. The numerical results reveal that the micropolar flow may accelerate or deaccelerate depending upon the numerical values of the mass transpiration and the permeability of the porous sheet. An increase in the tangential and the angular flow velocities is found to occur with an increase in the microrotation. Further, it is observed that the increase in the microrotation increases the skin friction coefficient.

Inventory management for stockout-based substitutable products under centralised and competitive settings

Inventory planning in fashion markets is highly challenging, owing to uncertain demand; yet, in making inventory decisions, retailers may be able to capitalise on high substitutability between products. This research develops single-period inventory-management models describing a market with two substitutable products, under stockout-based substitution; i.e. when a customer’s preferred product is out-of-stock, s/he may choose to purchase the substitute. Two settings are considered: centralised (a single retailer who sells both products) and competitive (two retailers, each selling one product). For each setting, we derive closed-form analytical solutions for the inventory levels that maximise expected profit. The model is further enriched with sales data from an online apparel retailer offering substitutable products (a sneaker in different colours), and we analyse the sensitivity of the optimal inventory levels and profits to parameter values. Key findings include the following: (i) Under competitive conditions, both retailers always order positive inventory so as not to lose customers. However, in a single-retailer setting, there are situations in which the retailer orders inventory for only one product. (ii) The optimal inventory levels and corresponding profits are highly sensitive to consumers’ willingness to substitute between products. These findings provide concrete insights that can guide fashion brands’ inventory-management decisions.

The inverse Mellin transform via analytic continuation

We present a method to calculate the x-space expressions of massless or massive operator matrix elements in QCD and QED containing local composite operator insertions, depending on the discrete Mellin index N, directly, without computing the Mellin-space expressions in explicit form analytically. Here N belongs either to the even or odd positive integers. The method is based on the resummation of the operators into effective propagators and relies on an analytic continuation between two continuous variables. We apply it to iterated integrals as well as to the more general case of iterated non-iterative integrals, generalizing the former ones. The x-space expressions are needed to derive the small-x behaviour of the respective quantities, which usually cannot be accessed in N-space. We illustrate the method for different (iterated) alphabets, including non-iterative 2F1 and elliptic structures, as examples. These structures occur in different massless and massive three-loop calculations. Likewise the method applies even to the analytic closed form solutions of more general cases of differential equations which do not factorize into first-order factors.

Tangential contacts of three-dimensional power-law graded elastic solids: A general theory and application to partial slip

New technologies in additive manufacturing have enabled the cost-effective production of gradient materials with desired multifunctional properties. In mechanical engineering, such functionally graded materials (FGMs) are often used within mechanical joints, as they ensure higher strength as well as wear resistance and prevent fretting fatigue. However, due to the variations in their composition and structure, theoretical predictions of their contact-mechanical behavior are significantly complicated. In this work, a general theory for solving tangential contacts between 3D power-law graded elastic solids of arbitrary geometry is presented. For multiple contacts, such as those occurring between two nominally flat but rough half-spaces, the well-known Ciavarella–Jäger theorem is established accompanied by a discussion of tangential coupling. Nevertheless, the focus of the work is on axisymmetric single contacts under arbitrary unidirectional tangential loading, for which closed-form analytical solutions are derived based on the Mossakovskii–Jäger procedure. In comparison to the results of common approximate methods, the solutions include the non-axisymmetric components of tangential displacements, which are indispensable for the accurate determination of the relative slip components and thus the surface density of frictional energy dissipation in the partial slip regime. Although a simplified approach is used for the calculation of the dissipated energy density, the results in the limiting case of homogeneous material are in excellent agreement with those from a full numerical computation. As an application example, the complete solutions for the tangential contact of parabolically shaped power-law graded elastic solids in the partial slip regime are derived and the influence of the material gradient as well as Poisson’s ratio on the surface density of dissipated energy is investigated.

Optimal Amplitude–Density Synthesis of Linear Aperiodic Arrays

New deterministic procedures for the design of linear aperiodic arrays are described which permit exploiting, in a combined way, both the positions and the excitation amplitudes of the array elements obtaining a pattern which optimally fits, in terms of a weighted L2 norm, the pattern of a reference linear continuous aperture. A first numerical procedure is based on alternating optimization of positions and amplitudes by means of closed-form convex projectors. Sufficient conditions for the solution to be optimal are established. A second numerical procedure is based on a domino-like sequential determination of the unknowns which can be iterated to convergence. Additionally, analytical asymptotic expressions for the optimal inter-element spacing and amplitude are derived by means of variational techniques. The optimal asymptotic inter-element spacing is demonstrated to be proportional to the reference tapering to the power (-2/3) while the optimal asymptotic array amplitude is proportional to the reference tapering to the power (1/3). For aperiodic arrays with equiamplitude excitation it is proved that the optimal inter-element spacing is proportional to the reference tapering to the power (-1). Based on these asymptotic dependencies, closed-form analytical solutions are obtained for amplitudes and positions of the elements. Several numerical results confirm the effectiveness and accuracy of the new procedures.

Generalized Accurate Harmonic Calculation Method Based on Discretized Double Fourier Series to Solve Double-Pulse Phenomenon

Harmonic issues have always been paid attention to in electrical systems, and accurate calculation of the amplitude and phase of each harmonic has become one of the research focuses of scholars. This article develops a generalized harmonic closed-form analytical solution for a single-phase voltage source inverter (VSI) under digital sinusoidal pulse width modulation (DSPWM) and dead-time effects. This article is the first to point out the double-pulse phenomenon that can occur when using a double Fourier series to solve the harmonic spectrum due to the discontinuity of the dead-time effects and half-cycle integrals. Analysis and data comparison shows that this phenomenon can affect the accuracy of harmonic calculations. An discretized double Fourier series method is proposed to solve this phenomenon. Moreover, the problems of missing pulse caused by the modulation under a specific current phase and current zero-crossing are also solved, improving the harmonic calculation's accuracy and generality. Simulation analysis and the results of inverse fast Fourier transform (IFFT) proved the correctness of the optimization method. And the experimental results are obtained on the DSP and SIC platforms, which are in good agreement with the expression calculation results

Thermo-Electro-Mechanical Analysis of Micropolar FGP Cylindrical Shell Covered with Piezoelectric Actuator Layers

An investigation is performed in this paper to analyze the nonlinear thermo-electro-mechanical response of long sandwich cylindrical shells with functionally graded porous (FGP) core and thin piezoelectric actuator layers. The FGP core of the sandwich shell is assumed to be temperature- and microstructure-dependent. It is assumed that the sandwich shell is subjected to uniform lateral pressure loading in a thermo-electrical environment. The equilibrium equations of the shell with infinite length are established with the aid of the virtual displacement principle and von Kármán kinematic assumptions. The governing equations are obtained in terms of displacement components based on the first-order shear deformation model of shallow cylindrical shells and the modified couple stress theory. These nonlinear differential equations are analytically solved for a sandwich shell having both the simply-supported and clamped–clamped edge conditions by employing a two-step perturbation technique. Analytical closed-form solutions are determined as a relationship including the load parameter and the mid-span deflection. The comparison examples are made with the existing results in the literature for a simple functionally graded shell, where good agreement is obtained. It is shown that the nonlinear response of the sandwich shell is highly dependent upon the temperature variation, power law index, porosity coefficient, couple stress parameter, piezoelectric layers, and geometrical parameters of the shell.

Closed-form solutions to investigate the nonlinear response of foundations supporting operating machines under blast loads

Machine foundations are subjected to significant dynamic impacts. These impacts could spread to the surrounding regions, affecting workers, sensitive equipment in the same institution, or nearby areas. This study analyzes the response of machine-supporting foundations to harmonic and explosive loads under operational conditions and provides closed-form solutions for predicting responses in terms of displacement, velocity, and acceleration time-histories to two common types of blast loads: a more accurate typical profile and simplified triangular profile. The soil-machine foundation system is regarded as a single-degree-of-freedom (SDOF) system that exhibits elastic–perfectly flexible resistance behavior. For the analysis of the SDOF system, two cases are considered: one assumes that the supporting soil keeps elastic during the explosion, and the peak displacement is less than the elastic one, while the other assumes that the blast occurs in an elastic state, and the peak displacement occurs in a plastic state. By using the closed-form analytical solutions, a detailed parametric analysis is carried out to evaluate the impacts of significant soil-foundation system characteristics such as mass, stiffness, and damping ratio on the response-time history of machine foundations. The findings are compared to those reported in the literature, and relevant conclusions are derived. Obtained results demonstrated that, despite its simplicity and usage of only positive phase to simulate blast loads, the simplified model’s response behavior differs significantly from the typical one. Furthermore, the derived solutions are utilized to design the foundations supporting vibrating machines for both harmonic and blast loads in a variety of conceivable scenarios depending on the blast magnitude.

On the influence of edge damage on thermally induced buckling of stepped circular bi-laminates

The influence of edge damage on thermo-mechanical buckling of stepped circular bi-laminates is examined analytically. The nonlinear equations of equilibrium, the associated boundary conditions and the transversality conditions for the propagating interior boundaries of a region of sliding contact are derived via the Principal of Stationary Potential Energy as a propagating boundary problem in the calculus of variations. The problem is ultimately recast in a mixed formulation expressed in terms of the membrane forces and the transverse displacements of the composite structure. A loading parameter is identified and closed form analytical solutions to the resulting non-linear problem are determined. In addition, a stability criterion is established based on the second variation of the total potential energy of the structure, and qualitative behavior concerning contact is deduced analytically. Results of numerical simulations based on the analytical solutions established herein are generated and the phenomenon of “sling-shot buckling” is observed, whereby the structure first deflects in one direction upon heating (or cooling) and then, when a critical temperature is achieved, dynamically “sling-shots” from the current equilibrium configuration to an alternate equilibrium configuration possessing deflections in the opposite sense. In addition, the phenomenon of “buckle-trapping” is observed for stepped circular bi-laminates with hinged-fixed supports, as well as for clamped-fixed structures with relatively large damaged area. A robust mixture of contact behavior, including propagating sliding contact, edge contact and no contact, and the transition from one form of contact to another, is exhibited for both pre-buckling and post-buckling configurations. The influence of edge damage on critical load and characteristic behavior is assessed.

Asymptotically Accurate Analytical Solution for Timoshenko-Like Deformation of Functionally Graded Beams

Abstract A closed-form analytical solution is developed for a planar inhomogeneous beam subjected to transverse loading, using a variational asymptotic method (VAM). The VAM decouples the problem into a cross-sectional and an along-the-length analysis, leading to a set of ordinary differential equations. These equations along with associated boundary conditions have been solved to obtain the closed-form analytical solutions. Three distinct gradation models have been used to validate the present formulation against 3D finite element analysis and few prominent results from the literature. Excellent agreement has been obtained for all the test cases. Key contributions of the present work are (a) the solutions have been obtained without any ad hoc and a priori assumptions and (b) the ordered warping solutions result in Euler–Bernoulli type deformation in the zeroth-order, whereas the higher-order solutions provide novel closed-form expressions for transverse shear strain and stress. Finally, the effect of inhomogeneity on various field variables has been analyzed and discussed.

Low-Frequency Passivity-Based Analysis and Damping of Power-Synchronization Controlled Grid-Forming Inverter

This article presents a passivity-based analysis and damping method for the low-frequency dynamics of grid-forming (GFM) control. A closed-form analytical solution to the positive realness of control input admittance is developed, which reveals the inherent nonpassive property of GFM control. This feature gives rise to the risk of instability, especially in connection with a nearby nonpassive GFM inverter, which may induce undesired control interactions. It is further revealed that the passivity index is determined by a low-frequency resonance around 10 Hz in the dq -frame, and this resonance can be better dampened, therefore, with enhanced passivity index, by using a virtual reactance (VRA) than using a virtual resistance (VRS). Finally, the theoretical findings are verified by the experimental results.

On the thermodynamic consistency of the Drucker–Prager Cap model: Modification of the plastic potential and analysis of unloading

The Drucker–Prager Cap (DPC) model is widely used to simulate powder compaction, especially in the pharmaceutical field. However, it has long been recognized that the model may predict an excessive dilation during unloading, most notably for small values of cohesion and/or friction angle. In this article, it is shown that this phenomenon is related to an inherent inconsistency between the yield surface and the commonly used plastic potential. As a result, thermodynamically inconsistent results may be obtained for some parameter values. A modified plastic potential is proposed that guarantees thermodynamic consistency. Uniaxial unloading of a homogeneous tablet in a die with constant cross section and negligible friction is considered so that the elastoplastic equations admit exact closed-form analytical solutions. Hence, exact expressions for the dependence of the axial and radial pressure on tablet height during unloading are obtained. These can be used to analyse experimental unloading data.

Wave propagation in a non-local magneto-thermoelastic medium permeated by heat source

In this paper, we consider the new concept of non-local heat conduction equation to generalized magneto-thermoelastic problem of two dimensional isotropic and homogeneous half-space in presence of heat-flux at the boundary surface. By using the harmonic plane waves, the governing equations are transformed to the vector matrix differential equation which is then solved by eigenvalue method. The analytical closed form solutions for displacement component, temperature distribution and stress components have been made and comparisons are also illustrated graphically with the theory of non-local dual-phase-lag (NLDPL) and non-local Lord-Shulman (NLLS) theory for different values of physical parameters. The significant effects of non-local variables as well as phase lagging parameters on displacements, temperature distribution and stress components are studied by means of graphically and concluding remarks are drawn.

Closed-Form Analytical Approach for Calculating Noise Contours of Directive Aircraft Noise Sources

This paper extends the simplified airport noise model Rapid Aviation Noise Evaluator (RANE) [Torija et al., Journal of the Acoustical Society of America, Vol. 141, No. 2, 2017, pp. 981–995], adding capability of including fully nonisotropic noise sources. This extended tool, RANE v2, is developed as a part of multidisciplinary acoustic assessment of novel aircraft, in order to produce ground contours around airports and helipads. Version 2 extends the capability of RANE to accommodate predictions of future air vehicles implementing propulsion systems solution with inherent directional properties. The model uses three-dimensional noise emission surfaces around a series of discretized segments that represent the aircraft flightpath. The main inputs are the sources’ sound power level, the distance from the flightpath at which a level is observed, and the source three-dimensional directivity. The directivity function may take analytical or numerical form, allowing for experimental data inputs. This paper demonstrates the use of spherical harmonics as a form of directivity function with a closed-form analytical solution for calculating the noise exposure contours. Results and comparison against the Federal Aviation Administration’s Aviation Environmental Design Tool module for helicopter community noise indicate that exposure contour coordinates can be estimated for high and low noise exposure levels. The incorporation of source directivity allows for the assessment of lateral attenuation, engine installation effects, and transition operations (for vertical to horizontal flight and vice versa) via the assumption of individual source directivities and, therefore, complex noise surfaces. As a consequence of the analytical nature of the model, low computational requirements allow for fast exploration of the design space and parametric studies, with minimal input requirements. The capabilities of RANE v2 are demonstrated by predicting noise footprints for three helicopters, each of different size, performance, and directivity characteristics.

An analytical formulation of unidirectional composite curved beam with out-of-plane fiber waviness under bending

A closed-form analytical solution is developed for analyzing laminated composite curved beam with out-of-plane fiber waviness under bending. Timoshenko beam theory was adopted in derivation where the effects of shear deformation was considered for moderately thick beam. Explicit expressions for evaluating equivalent axial and bending stiffness are formulated based upon the modified lamination theory and taking the structural deformation characteristics of narrow section beam into consideration. Stress distribution at any given location calculated by the proposed method is investigated. Significant increase of radial normal stress and shear stresses are observed when out-of-plane fiber waviness is introduced. The comparisons between present analytical results and numerical data indicate that the present methodology have ability to capture radial normal, circumferential normal, and shear stress distributions of composite curved beam with out-of-plane fiber waviness and the level of stress predicted is satisfactory with numerical results.

Analytical approaches for growth models in economics

This study deals with the group-theoretical analysis of nonlinear optimal control problems called the optimal growth model with the environmental asset and capitalist decision model of endogenous growth, which are expressed in terms of the current and present value Hamiltonian functions. The first-order conditions for optimal control problems based on Pontryagin’s maximum principle are considered. In addition, Lie point symmetries and their relation with the Prelle-Singer, λ-symmetries, adjoint symmetries, Darboux polynomials, and Jacobi’s last multiplier approaches are studied by analyzing the coupled nonlinear first-order ordinary differential equations corresponding to the first-order conditions. For both current and present Hamiltonian cases, the first integrals and the corresponding analytical (closed-form) solutions and graphical representations for the optimal control problems are represented.

An Offline Closed-Form Optimal Predictive Power Management Strategy for Plug-In Hybrid Electric Vehicles

This article develops an optimal predictive power management strategy (OP-PMS) for plug-in hybrid electric vehicles (PHEVs) to manage the remaining battery level throughout a defined journey. Apart from using the battery before charging it again at the destination, this approach also supports transit through an Ultra Low Emission Zone (ULEZ), where a PHEV needs to operate in pure electric mode to avoid emissions. This type of PHEV power management strategy (PMS) design problem is a noncausal control problem, where the power demand of the remaining driving cycle and the final battery energy level targets influence the current power management action. Rather than solving the whole optimization problem online, this article presents a simplification that leads to a closed-form analytic solution with parameters that can be computed offline. This article makes three key contributions. First, a novel input-linearization method is developed, which converts the OP-PMS into a convex quadratic programming (QP) problem that captures the essential model nonlinearities using an input transformation and a quadratic cost function. Second, the influence of future power demands and final targets on current energy management policy is explicitly quantified from a dynamic optimal control perspective. Third, the noncausal convex QP is approximated with a closed-form analytic solution that is calculated offline. This leads to a real-time implementable OP-PMS with coefficients that can be precalculated and stored in the engine control unit. To demonstrate the viability of this approach, numerical examples are provided based on a generic PHEV model to verify the efficacy of the proposed OP-PMS.