Analytical Formulas Research Articles

Analytical and numerical optimization of broaching tool tooth distribution

This paper presents a detailed mathematical analysis of the effect of tooth distribution on the stability of broaching operations. Analytic and numeric techniques are employed to analyse a simplified one-degree-of-freedom mechanical model of variable pitch broaching, to assess its stability, and to find the optimal tooth distances maximizing robustness against harmful self-excited chatter vibrations. A novel modelling approach, considering a theoretical infinitely long broaching tool, draws parallels with variable pitch milling, and analytic formulas for tuning milling cutters are extended to broaching tools. For a further increase of robustness, and to take feasibility constraints into account, a goal function based on the semi-discretization and spectral collocation techniques is implemented in a direct numeric optimisation framework. A new, detailed derivation of the corresponding parameter gradients is presented to enable the use of gradient descent techniques. Since broaching is inherently a time-limited process, optimal parameters found in this manner are then validated via time domain simulations to show the desirable transient behaviours achievable by this ideal tuning of the tool geometry.

Kinetostatic Modeling of an Asymmetrical Double-Stepped Beam for Displacement Amplification

Abstract A kinetostatic model of an asymmetrical double-stepped beam under axial loading is developed. The beam is composed of two thick segments and three thin segments where a thick segment is between two thin segments, and the longitudinal axis of the thick segment is not colinear with that of the thin segment. The kinetostatic model based on the beam constraint model (BCM) is capable of predicting accurate bending and near-buckling behaviors of the beam. A virtual rigid link neighboring the noncolinear segments is introduced in the BCM to broaden the applicability of the BCM. An analytical formula to represent the critical load of a symmetrical double-stepped beam under axial loading is derived and the value calculated by the formula agrees with the limit load of the asymmetrical double-stepped beam within the elastic range. The investigated beam has potential applications in displacement amplifiers and robotic grippers.

Nonlinear energy harvesting system with multiple stability

The nonlinear energy harvesting systems of the forced vibration with an electron-mechanical coupling are widely used to capture ambient vibration energy and convert mechanical energy into electrical energy. However, the nonlinear response mechanism of the friction induced vibration (FIV) energy harvesting system with multiple stability and stick-slip motion is still unclear. In the current paper, a novel nonlinear energy harvesting model with multiple stability of single-, double- and triple-well potential is proposed based on V-shaped structure spring and the belt conveying system. The dynamic equations for the energy harvesting system with multiple stability and self-excited friction are established by using Euler-Lagrangian equations. Secondly, the static characteristics of the nonlinear restoring force, the friction force, and the potential energy surfaces are obtained to show the nonlinear stiffness, multiple equilibrium points, discontinuous behaviors and multiple well responses. Then, the equilibrium surface of bifurcation sets for the autonomous system is given to show the third-order quasi zero stiffness (QZS3), fifth-order quasi zero stiffness (QZS5), double well (DW) and triple well (TW). The co-dimension bifurcation sets of the self-excited vibration system are analyzed and the corresponding phase portraits for the coexistent of multiple limit cycles are obtained. Furthermore, the analytical formula of amplitude frequency response of the approximated system are obtained by the complex harmonic method. The response amplitudes of charge, current, voltage and power of the forced electron-mechanical coupled vibration system for QZS3, QZS5, DW and TW are analyzed by using the numerically solution. Finally, a prototype of FIV energy harvesting system is manufactured and the experimental system is setup. The experimental works of static restoring forces, damping forcse and the electrical outputs are well agreeable with the numerical results, which testified the proposed FIV energy harvesting model.

A subtraction scheme for processes involving fragmentation functions at NLO

We present a novel subtraction method to remove the soft and collinear divergences at next-to-leading order for processes involving an arbitrary number of fragmentation functions, where this method acts directly in the hadronic centre-of-mass frame. We provide the analytical formulae of the subtraction terms in the general case where all the final state partons fragment to hadrons and for the two special cases when one of the partons of the final state does not fragment, i.e. it is a photon or involved in a jet.

Optimal two-phase sampling for comparing correlated areas under the ROC curves of two screening tests in the presence of verification bias

ABSTRACT The accuracy of a screening test is often measured by the area under the receiver characteristic (ROC) curve (AUC) of a screening test. Two-phase designs have been widely used in diagnostic studies for estimating one single AUC and comparing two AUCs where the screening test results are measured for a large sample (Phase one sample) while the disease status is only verified for a subset of Phase one sample (Phase two sample) by a gold standard. In this paper, we consider the optimal two-phase sampling design for comparing the performance of two ordinal screening tests in classifying disease status. Specifically, we derive an analytical variance formula for the AUC difference estimator and use it to find the optimal sampling probabilities that minimize the variance formula for the AUC difference estimator. According to the proposed optimal two-phase design, the strata with the levels of two tests far apart from each other should be over-sampled while the strata with the levels of two tests close to each other should be under-sampled. Simulation results indicate that two-phase sampling under optimal allocation (OA) achieves a substantial amount of variance reduction, compared with two-phase sampling under proportional allocation (PA). Furthermore, in comparison with a one-phase random sampling, two-phase sampling under OA or PA has a clear advantage in reducing the variance of AUC difference estimator when the variances of the two screening test results in the disease population differ greatly from their counterparts in non-disease population.

Closer scrutiny of cosmic ray proton energy losses

The percent-level precision attained by modern cosmic ray (CR) observations motivates reaching a comparable or better control of theoretical uncertainties. Here we focus on energy-loss processes affecting low-energy CR protons (∼0.1–5 GeV), where the experimental errors are small and collisional effects play a comparatively larger role with respect to collisionless transport ones. We study three aspects of the problem: (i) We quantitatively assess the role of the nuclear elastic cross section, for the first time, providing analytical formulas for the stopping power and inelasticity; (ii) We discuss the error arising from treating both elastic and pion production inelastic interactions as continuous energy loss processes, as opposed to catastrophic ones. The former is the approximation used in virtually all modern numerical calculations; (iii) We consider subleading effects such as relativistic corrections, radiative and medium processes in ionization energy losses. Our analysis reveals that neglecting (i) leads to errors close to 1%, notably around and below 1 GeV, neglecting (ii) leads to errors reaching about 3% within the considered energy range, (iii) contributes to a minor effect, gauged at the level of 0.1%. Consequently, while (iii) can currently be neglected, (ii) warrants consideration, and we also recommend incorporating (i) into computations. We conclude with some perspectives on further steps to be taken towards a high-precision goal of theoretical CR predictions regarding the treatment of energy losses. Published by the American Physical Society 2024

Correlation of powers of Hüsler–Reiss vectors and Brown–Resnick fields, and application to insured wind losses

Hüsler–Reiss vectors and Brown–Resnick fields are popular models in multivariate and spatial extreme-value theory, respectively, and are widely used in applications. We provide analytical formulas for the correlation between powers of the components of the bivariate Hüsler–Reiss vector, extend these to the case of the Brown–Resnick field, and thoroughly study the properties of the resulting dependence measure. The use of correlation is justified by spatial risk theory, while power transforms are insightful when taking correlation as dependence measure, and are moreover very suited damage functions for weather events such as wind extremes or floods. This makes our theoretical results worthwhile for, e.g., actuarial applications. We finally perform a case study involving insured losses from extreme wind speeds in Germany, and obtain valuable conclusions for the insurance industry.

Analytical robust design optimization for hybrid design variables: An active-learning methodology based on polynomial chaos Kriging

In robust design optimization, statistical moments of performance are widely adopted in formulating robustness metrics. To address the high computational costs stemming from the many-query nature of such optimizations with respect to robustness metrics, analytical formulas of the statistical moments have been developed based on surrogate models. However, existing methods consider random variables as the sole model input, which excludes, from the application scope, problems that also involve deterministic design variables. To remedy this issue, this paper proposes a new Polynomial Chaos Kriging-based methodology for efficient and accurate analytical robust design optimization. The analytical solutions for the statistical moments of performance are developed considering that the Polynomial Chaos Kriging model is established in the augmented space of the deterministic design and random variables. This is achieved by systematically decoupling associations with deterministic input from random input, providing effective solutions even when the orthonormality of the basis function is not applicable in the augmented space. This work also presents an active-learning framework enabling seamless implementation of various numerical optimization methods. Several numerical examples and a practical application illustrate the performance and superiority of the proposed method.

Analytical Selection of Leakage Inductance for Single-Phase Dual Active Bridge Converters

The single-phase dual active bridge (DAB) DC/DC converter is a vital component in modern power electronics systems, facilitating efficient bidirectional power transfer between various energy sources and loads. One of the critical aspects in the design of such converters is the selection of an appropriate leakage inductance, as it significantly affects the circulating power within the converter. This study investigates the influence of inductance on the reactive power flow and introduces an analytical formula for calculating the necessary leakage inductance based on the specified rated power of the DAB converter. By employing the inductance values determined through the proposed formula, the developed power control-based modulation method, implemented in discrete time, successfully achieves zero reactive power caused by the first harmonic components under all load conditions. This selection of leakage inductance optimizes the performance of the converter ensuring efficient power delivery across various applications. The applicability of this method extends to various DAB applications, including electric vehicles, integration with asynchronous microgrids, and aerospace systems.

Waveform of the Reflected Impulse at the Oblique Sounding of the Sea Surface

The height of sea waves is one of the most important characteristics describing the wave climate of the ocean. At the present, the main radar for remote measurement of wave heights is an altimeter. Measurements are performed at the vertical sounding (incidence angle equal to zero). The Brown model was developed to describe the waveform of the reflected impulse at the vertical sounding. There is no theoretical model for the case of oblique sounding. In the Kirchhoff approximation, the theoretical task about waveform of the reflected impulse at oblique sounding was considered. In the result of the investigation, the analytical formula for the waveform of the reflected impulse for oblique sounding at the small incidence angles (< 12◦) for a microwave radar with a narrow antenna beam was obtained. The waveform of the reflected impulse depends on the width of antenna beam, incidence angle, impulse duration, significant wave height (SWH), altitude of the radar, mean square slopes of large-scale, in comparison with radar wavelength, sea waves. It is shown that possibility exist to retrieve SWH using waveform the reflected impulse at the oblique sounding.

Steady-state flow under surface recharge through unconfined aquifer with depth-decaying hydraulic conductivity: Analytical results

This paper presents some analytical results related to groundwater flow through an unconfined horizontal aquifer induced by areal recharge on the top and a decaying hydraulic conductivity. Under steady-state conditions, and in the context of Boussinesq equation, exact analytical solutions describing drainage into one or two penetrating channels at the aquifer boundaries have been derived in closed forms. The analytical expressions have been used to investigate the existence of water divide inside the unconfined aquifer. Based on the derived analytical solutions, a closed-form analytical formula is developed for calculating groundwater transit time within Dupuit-type flow system. The analytical transit time expression is simply expressed in terms of common mathematical functions. Analytical results are verified against a numerical model and show excellent agreements. The analytical transit time is validated against a numerical one obtained from a two-dimensional model representing a rectangular unconfined aquifer. Based on the explicit closed-form solution, a fully analytical inverse problem to estimate the decay parameter is proposed using a least-square objective function. The solution of the inverse problem simply corresponds to the solution of a nonlinear algebraic equation obtained from the definition of the objective function and the form of the analytical solution.

Mask 3D parameter optimization for improving imaging contrast of plasmonic lithography

Based on plasmonic lithography (PL) technology, and aiming at the special nano-optical effect of metal/dielectric multilayer composites and mask three-dimensional (M3D) effect, a method for optimizing mask parameters is proposed. As a common analytic formula, the optical transfer function method has been introduced to analyze the imaging process. In order to include the M3D effect, FDTD is used to quantitatively calculate the PL imaging results, and the aerial image (AI) intensity and the light intensity contrast of AI in the photoresist layer can be obtained. The simulation results suggest that the imaging resolution and light intensity contrast can be improved by optimizing the M3D parameters such as the sidewall angle, thickness, and material of the mask absorber. For the line space test pattern with critical dimension = 150 nm and pitch = 300 nm, the results indicate that the optimal sidewall angle is 40°, resulting in an increase in the light intensity contrast of 344%. The light intensity contrast with a mask thickness of 70 nm is improved by 11% when compared to a mask thickness of 60 nm. The use of Ta and opaque MoSi on glass as the mask absorber material improves the light intensity contrast to varying degrees compared to the Cr mask.

Estimating Shear Strength of Marine Soft Clay Sediment: Experimental Research and Hybrid Ensemble Artificial Intelligence Modeling

In the design of offshore engineering foundations, a critical consideration involves determining the peak shear strength of marine soft clay sediment. To enhance the accuracy of estimating this value, a database containing 729 direct shear tests on marine soft clay sediment was established. Employing a machine learning approach, the Particle Swarm Optimization algorithm (PSO) was integrated with the Adaptive Boosting Algorithm (ADA) and Back Propagation Artificial Neural Network (BPANN). This novel methodology represents the initial effort to employ such a model for predicting the peak shear strength of the soil. To validate the proposed approach, four conventional machine learning algorithms were also developed as references, including PSO-optimized BPANN, Support Vector Machine (SVM), BPANN, and ADA-BPANN. The study results show that the PSO-BPANN model, which has undergone optimization via Particle Swarm Optimization (PSO), has prediction accuracy and efficiency in determining the peak shear performance of marine soft clay sediments that surpass that offered by traditional machine learning models. Additionally, a sensitivity analysis conducted with this innovative model highlights the notable impact of factors such as normal stress, initial soil density, the number of drying–wetting cycles, and average soil particle size on the peak shear strength of this type of sediment, while the impact of initial soil moisture content and temperature is comparatively minor. Finally, an analytical formula derived from the novel algorithm allows for precise estimation of the peak shear strength of marine soft clay sediment, catering to individuals lacking a background in machine learning.

Coherent modulation of intense XUV pulses by ground-state coupling

We present and validate a method to modify the interaction between laser fields and an atomic three-level system based on the control of the ground state, which is interpreted with a direct analytical formula. Coherent modulation of intense extreme Ultraviolet (XUV) pulses is achieved by using near-infrared (NIR) lasers as control pulses interacting with the atomic systems, i.e. quantum transition paths are manipulated in light-matter interactions caused by the laser. We analyze the feasibility of realizing the phase reversal of the dipole response in the three-level system by controlling the interaction between the ground state and another excited state. The discussion of the modulation of intense XUV pulses complements recent studies relying on perturbative excitation. These observations provide new insights for achieving pulse shaping of the XUV frequency range by using the NIR pulse as a control pulse.

Impact of the LZ experiment on the DM phenomenology and naturalness in the MSSM

This paper uses approximate analytical formulas and numerical results with the bino-dominated dark matter (DM) as an example to analyze the impact of the LUX-ZEPLIN (LZ) experiment on the DM phenomenology and naturalness in the Minimal Supersymmetric Standard Model (MSSM). We conclude that the limitation of the latest LZ experiment worsens the naturalness of the MSSM, as the predictions of the Z-boson mass and DM relic density demonstrate, particularly in the regions where the correct DM relic density is obtained by the Z- or h-mediated resonant annihilations.

Analytic formulas for the D -mode Robinson instability

The passive superconducting harmonic cavity (PSHC) scheme is adopted by several existing and future synchrotron light source storage rings, as it has a relatively smaller R/Q and a relatively larger quality factor (Q), which can effectively reduce the beam-loading effect and suppress the mode-one instability. By using a minimum search algorithm to solve the mode-zero Robinson instability equation of uniformly filled rigid bunches, we have revealed that the fundamental mode of PSHC with a large loaded Q possibly triggers the D-mode Robinson instability [T. He , Mode-zero Robinson instability in the presence of passive superconducting harmonic cavities, ]. This instability is a mode-zero coupled bunch instability, with an oscillation frequency close to the PSHC detuning (D-mode). Uniquely, it is anti-damped by the radiation damping effect. In this paper, we derive analytical formulas for the frequency and growth rate of the D-mode Robinson instability by taking several appropriate approximations. These formulas provide crucial insights for analyzing and understanding the D-mode Robinson instability. Published by the American Physical Society 2024

Realization of planar and surface conformal mappings through stress-free growth of hyperelastic plates: Analytical formulas and numerical calculations

Conformal mapping is a well-known concept and has a long research history in mathematics. Accompanying the developments of computational science and 3D digital scanning technology, conformal mapping has also found wide applications in geometric modeling, computer graphics, medical imaging and other fields. In the virtual image or animation world, conformal mapping offers a convenient approach to achieve various shape changes of planar regions and 3D surfaces. In this work, we propose a promising approach, i.e., through the stress-free growth of hyperelastic plates, to realize arbitrary planar and surface conformal mappings in the physical world. The growth field in a hyperelastic plate can be represented by a symmetric tensor field. For any given planar or surface conformal mappings with explicit analytical expressions, the formulas of growth functions in the growth tensor are derived. In the case that the target surfaces of conformal mappings have no explicit analytical expressions, we further propose a numerical scheme to determine the growth data in the hyperelastic plates. The efficiency of the approach proposed in the current work is validated through several typical examples, where the 3D finite element simulations are conducted. The results demonstrate that, by incorporating the obtained growth functions or growth data, the target surfaces can be generated accurately through growth deformations of hyperelastic plates. As an application of the current work, we further propose an approach of shape-control for double-layer grid structures with plate forms. According to any given target shapes, the values of elongation or contraction of the bar elements are determined, then the desired deformations of the grid structures can be realized.

Deformation of solid earth by surface pressure: equivalence between Ben-Menahem and Singh’s formula and Sorrells’ formula

SUMMARY Atmospheric pressure changes on Earth’s surface can deform the solid Earth. Sorrells derived analytical formulae for displacement in a homogeneous, elastic half-space, generated by a moving surface pressure source with speed $c$. Ben-Menahem and Singh derived formulae when an atmospheric P wave impinges on Earth’s surface. For a P wave with an incident angle close to the grazing angle, which essentially meant a slow apparent velocity $c_a$ in comparison to P- ($\alpha ^{\prime }$) and S-wave velocities ($\beta ^{\prime }$) in the Earth ($c_a \ll \beta ^{\prime } \lt \alpha ^{\prime }$), they showed that their formulae for solid-Earth deformations become identical with Sorrells’ formulae if $c_a$ is replaced by $c$. But this agreement was only for the asymptotic cases ($c_a \ll \beta ^{\prime }$). The first point of this paper is that the agreement of the two solutions extends to non-asymptotic cases, or when $c_a /\beta ^{\prime }$ is not small. The second point is that the angle of incidence in Ben-Menahem and Singh’s problem does not have to be the grazing angle. As long as the incident angle exceeds the critical angle of refraction from the P wave in the atmosphere to the S wave in the solid Earth, the formulae for Ben-Menahem and Singh’s solution become identical to Sorrell’s formulae. The third point is that this solution has two different domains depending on the speed $c$ (or $c_a$) on the surface. When $c/\beta ^{\prime }$ is small, deformations consist of the evanescent waves. When $c$ approaches Rayleigh-wave phase velocity, the driven oscillation in the solid Earth turns into a free oscillation due to resonance and dominates the wavefield. The non-asymptotic analytical solutions may be useful for the initial modelling of seismic deformations by fast-moving sources, such as those generated by shock waves from meteoroids and volcanic eruptions because the condition $c / \beta ^{\prime } \ll 1$ may be violated for such fast-moving sources.

Schwarzschild black hole and redshift rapidity: a new approach towards measuring cosmic distances

Motivated by recent achievements of a full general relativistic method in estimating the mass-to-distance ratio of supermassive black holes hosted at the core of active galactic nuclei, we introduce the new concept redshift rapidity in order to express the Schwarzschild black hole mass and its distance from the Earth just in terms of observational quantities. The redshift rapidity is also an observable relativistic invariant that represents the evolution of the frequency shift with respect to proper time in the Schwarzschild spacetime. We extract concise and elegant analytic formulas that allow us to disentangle mass and distance to black holes in the Schwarzschild background and estimate these parameters separately. This procedure is performed in a completely general relativistic way with the aim of improving the precision in measuring cosmic distances to astrophysical compact objects. Our exact formulas are valid on the midline and close to the line of sight, having direct astrophysical applications for megamaser systems, whereas the general relations can be employed in black hole parameter estimation studies. We also computed the frequency shift and the redshift rapidity for emitter eccentric orbits and calculated their relative error with respect to their numerical exact value for different eccentricities.

Subsurface tidal gravity variation and gravimetric factor

SUMMARY Taking advantage of the simultaneous recording during 471 d between 2019 and 2021 by two superconducting gravimeters installed at the surface and 520 m under the surface at the Low Noise Underground Laboratory (LSBB) in Rustrel, France, we investigate whether a difference between the tidal gravity signals at the two locations can be detected. First, we model the periodical variations of the Earth’s gravity owing to the tidal influence from the Sun and Moon, at the Earth’s surface and at shallow depths. We provide analytical formulae for the Love numbers, gravimetric factor and gravity variation of simple spherical planetary models. We also numerically compute those parameters and function for a realistic spherical Earth model. We find that the fractional difference between the semi-diurnal tidal gravity variations at the surface and 520 m below is as small as 8.5 × 10$^{-5}$. We next evaluate the effect on the amplitude of the recorded gravity signal due to the calibration factors of the two superconducting gravimeters at LSBB. Finally, we compute the spectra of the difference between the gravity variations measured on and under the surface in the semi-diurnal band of the M$_2$ tidal wave. We find that the uncertainties associated to the calibration factors are larger than the theoretical or observational difference between the tidal gravity variations on the surface and at a 520-m depth.