Approximate Analytical Equations Research Articles

Derivation of Analytical Equations for the Fundamental Period of Framed Structures Using Machine Learning and SHAP Values

The fundamental period is one of the most important parameters for the design of new structures as well as for estimating the capacity of existing ones. Thus, to estimate it, various design codes and researchers have adopted several approximate analytical equations based on a number of key structural parameters. To this end, the present study introduces a novel methodology for deriving the analytical equations for the fundamental period of reinforced concrete structures. The methodology is based on machine learning explainability techniques, specifically the so-called SHapley Additive exPlanations values. These values are commonly employed as an explainability tool. However, in the proposed novel approach they are employed as a basis to fit analytical curves, which allows the resulting equations to be constructed sequentially and in an informed manner while controlling the balance between accuracy and complexity. An extended dataset consisting of 4026 data points is employed, on which a Gradient Boosting Machine model is fitted. The model achieves excellent accuracy, with a coefficient of determination R2≈0.99, while the equations derived from the proposed formulation achieve an R2≈0.95 and Mean Absolute Error ≈0.12. This demonstrates the potential applicability of the proposed methodology in a wide array of similar engineering challenges.

Approximate analytical algorithm for pull-out resistance-displacement relationship of combined anchor system

Building upon an evaluation of the distinct advantages and limitations of anchor rods and anchorage plates, this paper introduces an integrated anchor-tensioning system that combines these elements in series. This innovative approach significantly enhances the pullout bearing capacity of the overall anchorage system. Based on the hyperbolic model of the distribution of lateral friction resistance of the anchor solid and the Winkler′s foundation model of load transfer of the anchorage plate, the theoretical analysis method for the analysis of the pullout bearing capacity and deformation of the combined anchorage system was established, and an approximate analytical equation for the relationship between pullout force and displacement was obtained. The results from indoor model tests, numerical simulations, and theoretical analyses were compared, verifying the reliability of the theoretical methods employed. The findings of this study significantly advance the discourse on the innovation of structural support configurations, thereby enhancing the operational efficiency of anchor-pull support systems.

Non-relativistic energy equations for diatomic molecules constrained in a deformed hyperbolic potential function.

In this paper, the approximate analytical energy equations for the deformed hyperbolic potential have been obtained for arbitrary parameters of the potential. The potential function was transformed to a molecular potential by subjecting it to the Varshni conditions which allows for the determination of the energy levels of diatomic molecules. The molecular vibrational energy spectra for [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] diatomic molecules were obtained and found to match with the results obtained with another analytical approach, potential functions, and experimental data. The noticeable slight differences in the approximate energy spectra obtained in this work and existing literature may be ascribed to the analytical method, computational approach, and the accuracy of the molecular potential functions. The obtained energy equations were used to determine the energy of a particle for arbitrary parameters of the potential function. The obtained energy is bounded and increases with the increase in the quantum numbers. The results conformed to the ones obtained via the path integral approach and numerical solutions obtained via the MATHEMATICA program. The energy spectra equations were obtained via the Nikiforov-Uvarov approach and semi-classical WKB approximation. The Pekeris approximation has been applied to resolve the difficulty in solving the complete energy spectrum of the non-relativistic wave equation for the potential function. The numerical data of the energy spectra was obtained using the MATHEMATICA program.

Fractional Trigonometric Function Approximations in a Regular System

Introduction: Linear fractional differential equations of the commensurable order, specifically those related to fractional trigonometry, pose challenges in terms of analytical solutions. background: This study presents an approach to the solution of fractional differential equations with commensurate order, specifically those associated with fractional trigonometry. Methods: The purpose of this article is to present a novel approximate analytical technique to solve linear fractional differential equations of the commensurable order, which are related to fractional trigonometry. This method is used to obtain approximate solutions by forming linear combinations of appropriate fractional basic functions, for which Laplace transforms are irrational functions. This approximation technique enables us to represent these functions using timeinvariant linear system models. Also, the implementation of analog circuits of these basic fractional order systems can be obtained using approximation of rational functions. Results: The research demonstrates the efficacy and precision of this method through illustrative examples, showcasing its effectiveness in solving linear fractional systems. Conclusion: The newly introduced approximate analytical method presents a promising approach to solving linear fractional differential equations of the commensurable order, providing accurate solutions and possibly offering applications in various fields.

Characteristics Analysis of Current-Controlled VSCs for Periodic Forced Oscillations Excited by Grid Frequency Disturbance

Uncontrolled forced oscillations can jeopardize the power system security. Most modeling and analysis methods for forced oscillations in previous publications were linearization methods such that some new nonlinear oscillation phenomenon can not be reproduced powerfully. Based on the considerations above, this paper hopes to provide clearer understandings and recognitions to forced oscillations by theoretically reveal the characteristics. First, a dynamic model of the current-controlled VSCs system based periodic forced oscillations is established to describe the relationships between system parameters and intrinsic quantities. Then, the multi-scale method is introduced to derive the mathematical expressions of approximate analytical solutions and relationship equation. Next, these characteristics, containing jumping phenomenon, frequency shift, doubling frequency response, and frequency coupling, are found and described by analyzing the mathematical expressions. Final, all the results are tested and verified on simulations and experiments.

Energy spectrum and zero-temperature magnetic functions of a position-dependent mass system in a Pöschl-Teller-type potential constrained by a vector magnetic potential field

In this work, the position-dependent mass Schrödinger equation is solved with the Pöschl-Teller-like potential in the presence of magnetic and Aharonov–Bohm (AB) flux fields. The BenDaniel-Duke ambiguity parameter ordering is used to formulate the Hamiltonian operator for the system. An approximate analytical equation of the bound-state energy spectrum is obtained using the parametric Nikiforov-Uvarov solution technique along with a Pekeris-like approximation scheme. With the aid of the obtained equation for the energy levels, analytical formulas of magnetization and magnetic susceptibility at zero-temperature are derived and subsequently used to predict the physical properties of diatomic substances including the ground state H2, HCl, CO and LiH molecules. The expression for the bound-state-energy spectrum is used to generate numerical data for the molecules. The computed energy eigenvalues agree with the literature on diatomic molecules. The study revealed that in the absence of the external fields, the energy eigenvalues and magnetic susceptibility of the system are degenerate. However, with only a low intensity AB field, the degeneracy is completely eliminated from the energy states of the molecules.

Activity coefficients at infinite dilution via a perturbation method of NRHB model

Activity coefficients of solutes at infinite dilution play a central role in molecular thermodynamics of phase equilibria, solvation, solubility and related properties. Numerous equation-of-state models highly appropriate for concentrated systems have been developed in the open literature. Quite often, however, their equations for the chemical potential or the activity coefficient are not analytical and recursive numerical methods are needed for their use. This is the case for the versatile and widely used Non-Randomness with Hydrogen-Bonding equation of state model and, in the present work, a straightforward perturbation method is used for the derivation of analytical expressions for the chemical potential or the activity coefficient of solute at infinite dilution. The derivations are validated and compared with the full numerical calculations as well as with relevant experimental data. It is shown that calculations with the approximate analytical equations are essentially identical with the full numerical ones. These derivations are of a general character and may be used in a variety of other analogous thermodynamic models.

Forced Vibration of Axially Accelerating three Parameter Beam Constituted by Fraction Alderivative Model

The forced vibration characteristics of axially variable speed viscoelastic beams are studied. The material is described by the three parameter model of Poynting-Thompson beam. According to Newton's second law, the fractional derivative is introduced to deduce the governing equation of the beam. The approximate analytical solution and amplitude frequency equation are obtained by multi-scale method. The amplitude frequency equation is divided into real part and imaginary part by separating variables. According to the Routh-Hurwitz criterion, the Jacobi matrix and characteristic equation are established to determine the stable region and unstable region obtained under different parameters. Numerical examples show that the instability region shows a downward trend with the increase of the order of fractional derivative.

The use of photoionization cross section for evaluating contribution of continuum to the blackbody radiation induced shift and broadening of Rydberg-state energy levels of group IIb ions

General properties of n-dependence at the threshold and regularities of above-threshold ionization cross sections for states with high principal quantum numbers n and arbitrary angular momenta l are determined and used for evaluating the blackbody-radiation(BBR)-induced ionization broadening of nS, nP, nD and nF Rydberg-state energy levels of the group IIb ions Zn+, Cd+, Hg+ at temperatures from T=100 K to T=3000 K. The contributions to the energy level shifts from transitions into continuum are also determined with the use of the ionization cross sections. The single-electron quantum defect method (QDM) was used, which appears rather precise and suitable for calculating the amplitudes of radiation transitions from Rydberg states, and specifically for summation of matrix-element-dependent terms over infinite number of discrete states. The dependencies on temperature and binding energy are determined and approximate analytic equations are proposed for simplified evaluations of the BBR-induced ionization broadening and shifts. Numerical parameters of approximate equations are tabulated for S, P, D and F Rydberg states of the group IIb ions.

Self-consistent surface-temperature boundary condition for liquefying-fuel-based hybrid rockets internal-ballistics simulation

A novel methodology for the calculation of the surface temperature of liquefying fuels typically burned in hybrid rockets is proposed. This procedure stems from the formulation of a fuel in-depth pyrolysis model coupled with the resolution of the thermo-fluid-dynamic field in the rocket combustion chamber, which allows for the characterization of the unstable liquid layer formed on top of the fuel surface. The aim is the simulation of the internal ballistics of hybrid rocket engines fed by paraffin-based fuels without the need for parametrically assigning the surface temperature to match the experimental data as, indeed, required in the authors’ previous work. With the presented technique, surface temperature and fuel vaporization rate are calculated locally along the wall, and, with the integration of a liquid fuel entrainment model, which requires the tuning of just one parameter (i.e. the so-called entrainment factor), the fuel regression rate is determined. The overall numerical approach, upon the assumption that the liquid fuel is in the supercritical pressure regime, is based on the solution of the Reynolds-averaged Navier–Stokes equations for single-phase multicomponent turbulent reacting flow. A series of numerical simulations are carried out to unveil the effect of the oxygen mass flux, which allowed deriving an approximate analytical equation for the regression rate prediction. A set of hot fires of a laboratory-scale hybrid rocket are reproduced through single numerical simulations carried out on the fuel port average geometry in the burn to validate the computational model, showing deviations between the measured and predicted average regression rate less than 4.5%. In order to fairly match also the fuel consumption axial profile, transient numerical simulations over the entire engine firing are conducted with which the post-burn port shape is captured with maximum error of 8%.

Theoretical analysis of micro/nano electrochemical machining with ultra-short voltage pulses

The effect of pulse-on and pulse-off times on the efficiency of micro/nano ECM with ultra-short voltage pulses is theoretically analyzed. A new approximate analytical equation for the time of charging electrical double layer (EDL) is obtained. The equation takes into account all main parameters of the process. It enables one to determine the pulse-on time that provides a high localization of Faradaic process in the zones with small interelectrode gap and a high productivity of ECM process. The results of numerical calculations agree well with the data calculated by the proposed approximate analytical equation. The numerical calculations of the pulse-off time are performed. The effect of reverse voltage pulse imposed in the pauses on the EDL relaxation time is analyzed. The amplitude of reverse voltage pulse is limited by the condition of the absence of the tool-electrode wear. The proposed algorithm for calculating the pulse-off time and amplitude of reverse voltage pulse provides the required high coefficient of metal dissolution localization in the zone of small interelectrode gaps at a given pulse-on time.

Effect of aggregation on the simple ion transfer across oil|water interfaces

The theory of cyclic voltammetry of simple ion transfer across a liquid|liquid (L|L) interface including the effect of aggregation is developed. We focus on two different situations: ion transfer followed by an aggregation process in the organic phase, and a disaggregation process in the aqueous phase followed by ion transfer. We use parameters typical of the areas of study of molecular aggregation and polymer formation, namely average aggregate size and polydispersity index. These parameters allow us to analyse, in a simple and rigorous way, the interconversion between monomer and aggregate species, both at the aqueous or organic side of the L|L interface and in the interfacial region during the potential scan. Our results show that aggregation occurring in the organic phase facilitates ion transfer across the L|L interface. In contrast, when the aggregates are present in the aqueous phase, they must be disaggregated so that the monomers can be transferred, hence the transfer energy increases. Both processes significantly alter the shape of voltammograms. The peak current and mid-peak potential are analysed as a function of initial concentrations, aggregation constants and maximum number of monomers forming each aggregate. These results are compared with approximate analytical equations, which can be used to guide further experiments and extract information from experimental responses.

Analytical evaluation of third and fourth virial coefficients for hard disk fluids in narrow channels and equation of state

The third and fourth virial coefficients of hard disks in narrow channels have been evaluated as a series in powers of the channel width deviations from the one dimensional limit. We have obtained analytically the coefficients of the power series to very high order. These virial coefficients are used in truncated virial series to provide approximate analytical equation of state (EOS). The EOS is shown to accurately describe Monte Carlo simulations over a wide range of pressures and densities near the one dimensional limit. Explicit expressions for the free energy, chemical potential, compressibility and EOS as a function of the channel widths are given. These results represent progress toward understanding the long standing problem of the thermodynamic crossover from a one dimensional to quasi one dimensional fluid. They are also useful in a number of applications. We mention the examples of thermodynamic perturbation theory and stochastic dynamic mobility in anomalous single-file diffusion.

Approximate Analytical Equations for the Stirrer Angular Correlation in a Reverberation Chamber

In a reverberation chamber (RC), the angular correlation coefficient of a stirrer is an important parameter. It has been used to evaluate the performance of a stirrer or to estimate the number of independent samples in a measurement. In the previous work, the angular correlation coefficient (ACC) was evaluated numerically and no analytical equation was proposed. In this study, we propose an approximate analytical equation to fit the measured angular correlation that shows good agreements with measurement results. General properties of ACC are explored with physical insights; the equivalency of the mean value of the angular correlation and the $K$ -factor is revealed. This study provides further understandings on the control of the stirrer angular correlation and the $K$ -factor in an RC.

Study of substrate temperature dependence upon nozzle traverse speed at cold spraying

The paper presents the results of investigation of dependence of substrate temperature upon nozzle traverse velocity under cold spray conditions. A model is proposed for prediction of substrate temperature. Experimental and simulated data were obtained in the typical range of change in the nozzle traverse speed (1-400 mm/s) and air stagnation temperatures (200-600°C), encountered in the practice of cold spraying. The simulated data are in good agreement with experimental. An approximate analytical equation is proposed for estimation of substrate temperature during cold spraying in a wide range of gas stagnation temperatures and nozzle traverse velocities.

Sagittal and tangential foci produced by tilted plane wavefronts refracted through simple lenses.

We study the formation of caustic and wavefront surfaces produced by a tilted plane wavefront propagating through spherical positive lenses. The shape of the caustic surface is a function of the indices of refraction, the geometrical parameters of the lens involved in the process of refraction, and the obliquity angle with respect to the optical axis, as we expect. We provide exact and approximate analytic equations for tangential and sagittal focal surfaces and also for Petzval field curvature considering arbitrary lenses.

Crystallization kinetics of the Fe40Ni40P14B6 metallic glass in an extended range of heating rates

The effect of heating rate (in the range of 0.083–3.333 K s−1) on the parameters of thermal stability and crystallization kinetics of the Fe40Ni40P14B6 metallic glass has been investigated by differential scanning calorimetry and X-ray diffraction. An analytical model of glass crystallization describing the homogeneous nucleation rate, the velocity of interface-limited growth, the number of crystal volume density in crystallized samples as well as the crystallization kinetics at constant rate heating is presented. A modified procedure of estimation of thermodynamic and kinetic parameters governing the rate of crystal nucleation and growth based on the use of the measured heating rate-dependent variations of the volume fraction crystallized and the average grain size is proposed. The values of the effective diffusivity have been estimated by the Kissinger-like isoconversional method accounting the contribution of both the free energy difference and the specific interfacial nucleus-melt energy changes estimated from the structural data. It has been shown for the first time that the shapes of the non-isothermal experimental kinetic crystallization curves measured at the heating rates ≤ 1.333 K s−1 are well described by the approximate analytical equation and the values of the Avrami exponent lowering from 5.5 to 2.3 with the heating rates increasing have been estimated. The evaluated ranges of the nucleation rate (1.4 × 1016–2.5 × 1017 m−3 s−1) and growth velocity of crystals (2.05 × 10−9–3.0 × 10−7 m s−1) in Fe40Ni40P14B6 glass at temperatures from 653 to 714 K are in reasonable agreement with the experimental data. Possible variations of the established eutectic crystallization mechanism revealed by changes of the Avrami exponent with the heating rate increasing are discussed.

Oxygen concentration dependence of silicon oxide dynamical properties

To understand oxidation in three-dimensional silicon, dynamic characteristics of a SiOx system with various stoichiometries were investigated. The calculated results show that the self-diffusion coefficient increases as oxygen density decreases, and the increase is large when the temperature is low. It also shows that the self-diffusion coefficient saturates, when the number of removed oxygen atoms is sufficiently large. Then, approximate analytical equations are derived from the calculated results, and the previously reported expression is confirmed in the extremely low-SiO-density range.

Studying about applied force and the output performance of sliding-mode triboelectric nanogenerators

Triboelectric nanogenerator (TENG) as a powerful mechanical energy harvesting technology has major applications as micro/nano-power source, for self-powered sensors and even large-scale blue energy, which would impact the world likely development for the future. In this work, a theoretical model for the sliding-mode TENG with considering the external force applied onto the TENG was presented. Through approximate analytical equations derivation, the output characteristics of TENG with arbitrary load resistance were calculated, including the output power and energy. Based on the relationships between the output characteristics and load resistance or sliding velocity, the force applied on the sliding component of TENG and the friction force were investigated for two cases, i.e., without load resistor and with load resistor. Also, the influences of the resistance and sliding velocity on the forces were investigated. Furthermore, the corresponding experiments were carried out to measure the force applied on the TENG as well as the output performance of the fabricated sliding-mode TENG. The experimental results are in good agreement with the theoretical predictions, which can effectively reveal the principle of applied force on the TENG and facilitate the understanding of the relationship between the power, energy and applied force.

Approximate analytical calculations of photon geodesics in the Schwarzschild metric

We develop a method for deriving approximate analytical formulae to integrate photon geodesics in a Schwarzschild spacetime. Based on this, we derive the approximate equations for light bending and propagation delay that have been introduced empirically. We then derive for the first time an approximate analytical equation for the solid angle. We discuss the accuracy and range of applicability of the new equations and present a few simple applications of them to known astrophysical problems.