Exponential Function Research Articles

Adaptive Quick Sliding Mode Reaching Law and Disturbance Observer for Robust PMSM Control Systems

The permanent magnet synchronous motor (PMSM) has been of interest to eco-friendly industries on account of its advantages such as high performance, efficiency, and precision control. However, perturbations due to PMSM parameter uncertainty, load torque, and external disturbance interfere with the construction of PMSM precision control systems. Therefore, a robust control system is needed to avoid unnecessary system movement caused by perturbations. In this paper, sliding mode control (SMC) is adopted to implement a robust control system for the PMSM. In order to reduce the reaching time from the initial system state to the sliding surface and the chattering phenomenon that can cause the system to malfunction, the adaptive quick sliding mode reaching law based on an exponential function and power equation is proposed. Although the SMC is robust to disturbance and parameter uncertainty, unexpected disturbances can destabilize the system. To estimate the unmatched disturbance in a short time, the second-order fast terminal sliding mode observer (SFTSMO) is proposed. The results show that the motor control system based on the proposed method has a fast convergence speed to an objective value, position tracking performance, and robustness.

Influence of Eggshell Powder on the Properties of Cement-Based Materials.

Replacing cement with industrial by-products is an important way to achieve carbon neutrality in the cement industry. The purpose of this study is to evaluate the effect of eggshell powder on cement hydration properties, and to evaluate its feasibility as a substitute for cement. The substitution rates of eggshell powder are 0%, 7.5%, and 15%. Studying the heat of hydration and macroscopic properties can yield the following results. First: The cumulative heat of hydration based on each gram of cementitious material falls as the eggshell powder content rises. This is a result of the eggshell powder's diluting action. However, the cumulative heat of hydration per gram of cement rises due to the nucleation effect of the eggshell powder. Second: The compressive strengths of ES0, ES7.5, and ES15 samples at 28 days of age are 54.8, 43.4, and 35.5 MPa, respectively. Eggshell powder has a greater negative impact on the compressive strength. The effect of eggshell powder on the speed and intensity of ultrasonic waves has a similar trend. Third: As the eggshell powder content increases, the resistivity gradually decreases. In addition, we also characterize the microscopic properties of the slurry with added eggshell powder. X-ray Diffraction (XRD) shows that, as the age increases from 1 day to 28 days, hemicaboaluminate transforms into monocaboaluminate. As the content of the eggshell powder increases, FTIR analysis finds a slight decrease in the content of CSH. Similarly, thermogravimetric (TG) results also show a decrease in the production of calcium hydroxide. Although the additional nucleation effect of eggshell powder promotes cement hydration and generates more portlandite, it cannot offset the loss of portlandite caused by the decrease in cement. Last: A numerical hydration model is presented for cement-eggshell powder binary blends. The parameters of the hydration model are determined based on hydration heat normalized by cement mass. Moreover, the hydration heat until 28 days is calculated using the proposed model. The strength development of all specimens and all test ages can be expressed as an exponential function of hydration heat.

Unveiling the Power of Exponential Functions: Applications in Our Daily Lives

In this research paper, we have obtained some new applications of exponential and logarithmic functions. This report deals all the validity of previous works. This work is motivated by the work of [1-12].

On the stress-dependent tight-rock permeability under anisotropic stress conditions

Permeability is a key parameter for characterizing an unconventional tight-rock reservoir and predicting hydrocarbon production from the reservoir. Because of the technical difficulty in accurately measuring extremely low tight-rock permeability, the permeability measurements have been commonly made in laboratory under hydrostatic (or isotropic stress) conditions. For unconventional reservoirs, stress distributions, however, are generally anisotropic. This paper presents a theoretical and experimental study on stress-dependent tight-rock permeability under anisotropic stress conditions. The theoretical development is based on the concept of effective stress and expresses rock permeability as an exponential function of the effective stress defined in this paper. The laboratory permeability data obtained under anisotropic stress conditions support the usefulness of the theoretical model for the stress-dependent permeability and demonstrate the importance of stress anisotropy in determining rock permeability. Considering that the effective stress is the only independent variable for determining the relationship between rock permeability and stress, we further propose a workflow to convert tight-rock permeability data collected under hydrostatic stress conditions to those for anisotropic stress conditions that are more realistic for field applications. A theoretical relation for upscaling a parameter characterizing the impact of stress anisotropy on local-scale permeability is also discussed as the first step to address this important upscaling issue.

Safe Reinforcement Learning-Based Balance Control for Multi-Cylinder Hydraulic Press

Abstract Considering the load uncertainty and unmodeled dynamics in multicylinder hydraulic systems, this paper proposes a balance control algorithm based on safe reinforcement learning to release the restrictions of classical model-based control methods that depend on fixed gain. In this paper, the hydraulic press is controlled by a trained agent that directly maps the system states to control commands in an end-to-end manner. By introducing an action modifier into the algorithm, the system states are kept within security constraints from the beginning of training, making safe exploration possible. Furthermore, a normalized exponential reward function has been proposed. Compared with a quadratic reward function, the precision is greatly improved under the same training steps. The experiment shows that our algorithm can achieve high precision and fast balance for multicylinder hydraulic presses while being highly robust. To the best of our knowledge, this research is the first to attempt the application of a reinforcement learning algorithm to multi-execution units of hydraulic systems.

Study on the evolution of permeability properties of limestone under different stress paths

Abstract Stress path change has a great relationship with the effect of deformation and strength of rock. However, the underground rock body is in the engineering environment where the stress field, seepage field, and other fields are coupled to change, the law of fluid flow in the rock body is complex and variable. The change in the stress field has an important effect on the seepage characteristics of rock body; therefore, it is necessary to study the pattern of rock permeability evolution pattern by different stress paths. This study is based on the study of limestone, conducting triaxial unloading seepage mechanics experiments, the evolution of permeability properties of limestone specimens was analyzed based on the test results. The results show that in the conventional triaxial loading seepage test, the permeability of the limestone decreases before the rock stress reaches the peak intensity and increases after that. Increasing axial pressure unloading surrounding pressure compression section, permeability loss rate and effective stress are in line with the changing law of Gaussian distribution function. Under the action of constant axial pressure unloading surrounding pressure, with the increase in unloading amount, the permeability rate of change appears to increase slowly, and in the late stage of unloading section, the permeability rate of change appears to surge. Unloading section permeability change rate and cumulative unloading amount are in line with the law of change of the exponential function. Creep unloading pressure seepage test found that unloading pressure stage strain-time and permeability-time evolution characteristics are in line with the exponential rule of change. The experimental results of this study can provide an important experimental and theoretical basis for the permeability analysis of low-permeability rock body under complex stress conditions in underground engineering.

Superposition Formula of arbitrary functions to a (3+1)-demensional Boiti–Leon–Manna–Pempinelli equation

The (3+1)-dimensional Boiti–Leonon–Manna–Pempinelli equation was studied by using Hirota bilinear method. In this paper, we successfully derived the multi-solutions of this equation consisting of the superpositions of M arbitrary functions. A variety of new solutions can be obtained by choosing different values of the parameter M and test functions f in this paper, i.e., the localized solutions generated by M-positive quadratic functions, the mixed solutions consisting of the localized wave solutions and the kink wave solutions, and the mixed solutions consisting of a trigonometric function and exponential functions. Meanwhile, their corresponding physical plots were displayed to analyze their physical significance and dynamic properties. Some solutions from previous studies were found when M took a specific value and the test functions took particular functions. Thus, the received solutions are more general compared to the previous studies.

3D finite element performance-based study of RC interface stiffness

Although the importance of concrete-reinforcement interaction for reinforced concrete mechanics is well recognized, its intrinsic complexity needs in-depth meso-scales investigation. The contact zone between concrete and reinforcement was investigated in this study using RC ties that were short enough not to crack. The rib-scale simulation was carried out using a three-dimensional finite element technique using a simplified rib shape in the reinforcing bar. The contact between the concrete and steel was modeled using a friction cohesive zone model. An analysis was conducted to explore the impact of various contact zone parameters, suggesting numerical attributes for both normal and tangential stiffness while considering diverse rebar diameters. Concrete strain distribution and propagation of smeared splitting cracks in prisms were investigated. Additionally, a relationship between the interface's normal stiffness and rebar diameter was established through an exponential equation: Knn=2,53∙1016∙D−7,5[MN/m3]. The stiffness parameters proposed were utilized to forecast the steel strain profiles of RC ties at various load levels.

Soliton unveilings in optical fiber transmission: Examining soliton structures through the Sasa–Satsuma equation

The exploration of higher-order and multicomponent extensions of the nonlinear Schrödinger equation holds significant importance across diverse applications, notably within the field of optics. Among these equations, the integrable Sasa–Satsuma equation stands out for its captivating soliton solutions. The Sasa–Satsuma equation serves as a mathematical representation for describing the propagation of femtosecond light pulses through optical fibers. The breather, multiwave, combined dark–bright, singular, dark, bright and periodic singular optical soliton solutions are derived in this paper by using the 1φ(ϑ),φ′(ϑ)φ(ϑ) and multivariate generalized exponential rational integral function approach. Furthermore, the paper emphasizes the essential fiber parameters essential for the emergence of these structures and offers visual representations of chosen solutions to elucidate their physical characteristics. The novelty of this work lies in the inaugural application of these methods to the Sasa–Satsuma equation and gives a significant advancement in the understanding of optical phenomena. This study not only opens avenues for further research but also introduces a fresh perspective on the Sasa–Satsuma model in the field of nonlinear physics with its applications in optical systems.

An intercomparison of empirical schemes for partitioning precipitation phase

Study regionPrecipitation phase in China has changed with global warming, and accurate partitioning of the precipitation phase is crucial for understanding the hydrological processes and energy balance. Study focusBased on a 36-year daily meteorological dataset from the China Meteorological Administration, this study conducted a comprehensive evaluation of six empirical methods in estimating the precipitation phase nationwide and explored the applicability and distinction across different meteorological, topographic and geographic categories. These methods utilized inputs of air temperature, with two variants incorporating relative humidity, and encompassed a range of combined non-linear air temperature and hydrometeor temperature methods to more simple air temperature threshold methods. New hydrological insights for the regionMethods implementing non-linear functions without air temperature thresholds performed better than those relying solely on air temperature thresholds. The exponential functions with air temperature and relative humidity exhibited the best performance, while those relying only on air temperature thresholds exhibited the worst overall performance. Optimal methods were identified and recommended under different scenarios (i.e., multivariate meteorological and geographic parameters, different geographic regions). The rain–precipitation ratio exhibited a significant increasing trend, and the meteorological stations showing an increase were primarily concentrated in the three major snow-covered areas, the west and vicinity of the “Hu Huanyong Line”. This study can be applied to other regions and offers valuable insights for analyzing precipitation data and developing hydrological models.

Generation of initial debris cloud distributions for breakup events based on CARDC-SBM

With the dramatic increase in the number of resident space objects, the risk of collision between space objects and debris is growing. To protect spacecraft from such collisions, it is crucial to determine the generation patterns of space debris and analyze the evolution of debris clouds. In this paper, a continuous distribution formula is derived for each parameter of the breakup debris based on the China Aerodynamics Research and Development Center spacecraft breakup model (CARDC-SBM). Using CARDC-SBM, we theoretically establish a normal distribution for the separation velocity of the breakup debris in relation to the breakup mass center. Based on this foundation, the mean and standard deviation of the debris separation velocity distribution are empirically modeled using polynomial and exponential functions. Consequently, an empirical formula with a broad range of applicability is derived, which significantly reduces the computation time of debris cloud simulations. The initial velocity distribution given by this empirical formula is employed as the initial condition for the evolution of debris clouds from the perspective of a boundary value problem. Simulations using CARDC-SBM and subsequent analysis of the debris cloud evolution resulting from a breakup event reveal that the distribution of each parameter is determined solely by the relative collision velocity and the spacecraft's nominal density in the impact breakup scenario. Our results show that the empirical formula for the debris cloud distribution provides good agreement with Monte Carlo simulations.

A recurrence algorithm of time-dependent analysis for concrete bridges

This paper proposes an algorithm for time-dependent effects in the finite element analysis of concrete bridges. Based on the primary influence of concrete shrinkage and creep on bridge structures, the calculation formulas for the equivalent nodal load increments of beam elements caused by the shrinkage and creep effects were deduced according to the initial strain method. To simplify the calculation process and improve calculation speed, the formulas for shrinkage and creep in the bridge design code were fitted with an exponential function, and the recurrence equations for shrinkage and creep calculation were obtained. Thus, a finite element method formulation for concrete shrinkage and creep calculated according to the divided time steps was established. The algorithm was programmed into a self-developed program, and the correctness of the program's calculation was verified through 20 years of on-site monitoring results of a concrete girder cable-stayed bridge with a main span of 500 m. The method proposed in this paper can substantially improve the speed of the program to calculate the time-dependent effects during the construction and operation of large-span concrete bridges, while ensuring the reliability of the calculation results, and is easy to be applied in various types of finite element method programs.

Age-Related Constraints in the Visuomotor Plasticity of Postural Control as Revealed by a Whole-Body Mirror Learning Task.

Whether visuomotor plasticity of postural control is a trainable feature in older age remains an open question despite the wealth of visually guided exercise games promising to improve balance skill. We asked how aging affects adaptation and learning of a visual feedback (VF) reversal during visually guided weight shifting and whether this skill is modulated by explicit knowledge. Twenty-four older (71.43 ± 2.54years) and 24 young (24.04 ± 0.93years) participants were exposed to a 180° VF reversal while tracking a horizontally moving target by voluntarily weight shifting between two force platforms. An explicit strategy was available to half of the participants with detailed instruction to counter the VF rotation. Individual error data were fitted to an exponential function to assess adaptation. Fewer older (12/24) than younger (21/24) participants adapted to the VF reversal, displaying error curves that fitted the exponential function. Older adults who adapted to the VF reversal (responders, n = 12) reached an asymptote in performance in the same weight shifting cycle and displayed a similar mean asymptotic error compared with young participants. Young but not older responders exhibited an aftereffect when the VF reversal was removed. Instruction did not influence spatial error modulations regardless of age. The large individual variations within the older adults' group during early adaptation suggest age-specific limitations in using explicit cognitive strategies when older adults are exposed to an abrupt mirror feedback reversal that requires a change in weight shifting direction during whole-body postural tracking.

Assessing the Influence of Urban Lights on Night Sky Brightness with a Smartphone

The darkness of the sky is a critical parameter for assessing the suitability of an astronomical site. Among various sources of light pollution, urban lights pose the most significant threat to ground-based optical astronomical and planetary observations. Quantitatively assessing the impact of urban lights with varying scales and fluxes is indispensable for selecting an ideal optical observation site. In order to quantitatively assess the changes in Night Sky Brightness (NSB) relative to the distance from urban areas and to establish a foundation for safeguarding the light environment at the newly developed Lenghu astronomical site on the Tibetan Plateau, we employed both a Sky Quality Meter and a pre-calibrated smartphone. These instruments were used to measure the NSB in the vicinity of two cities, Da Qaidam and Delingha, which vary in size and radiant flux, on the Tibetan Plateau. The findings indicate that the NSB around both cities decreases significantly as the distance from the city center increases, although the rate of decrease varies between the two locations. This decline can be effectively modeled using an exponential decay function. Notably, the influence of city lights on NSB becomes negligible at distances exceeding 30 km from Da Qaidam, while for Delingha, this distance extends to 50 km due to its larger city size and higher total radiant flux. The methodologies and results presented in this paper offer valuable insights for the selection of astronomical observation sites and the development of light pollution management policies.

Conserved vectors and symmetry solutions of the Landau–Ginzburg–Higgs equation of theoretical physics

This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation (LGHe), which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves. The LGHe finds applications in various scientific fields, including fluid dynamics, plasma physics, biological systems, and electricity-electronics. The study adopts Lie symmetry analysis as the primary framework for exploration. This analysis involves the identification of Lie point symmetries that are admitted by the differential equation. By leveraging these Lie point symmetries, symmetry reductions are performed, leading to the discovery of group invariant solutions. To obtain explicit solutions, several mathematical methods are applied, including Kudryashov’s method, the extended Jacobi elliptic function expansion method, the power series method, and the simplest equation method. These methods yield solutions characterized by exponential, hyperbolic, and elliptic functions. The obtained solutions are visually represented through 3D, 2D, and density plots, which effectively illustrate the nature of the solutions. These plots depict various patterns, such as kink-shaped, singular kink-shaped, bell-shaped, and periodic solutions. Finally, the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors. These conserved vectors play a crucial role in the study of physical quantities, such as the conservation of energy and momentum, and contribute to the understanding of the underlying physics of the system.

Reference values for leg effort during incremental cycle ergometry in non-trained healthy men and women, aged 19-85.

Heightened sensation of leg effort contributes importantly to poor exercise tolerance in patient populations. We aim to provide a sex- and age-adjusted frame of reference to judge symptom's normalcy across progressively higher exercise intensities during incremental exercise. Two-hundred and seventy-five non-trained subjects (130 men) aged 19-85 prospectively underwent incremental cycle ergometry. After establishing centiles-based norms for Borg leg effort scores (0-10 category-ratio scale) versus work rate, exponential loss function identified the centile that best quantified the symptom's severity individually. Peak O2 uptake and work rate (% predicted) were used to threshold gradually higher symptom intensity categories. Leg effort-work rate increased as a function of age; women typically reported higher scores at a given age, particularly in the younger groups (p < 0.05). For instance, "heavy" (5) scores at the 95th centile were reported at ~200 W (<40 years) and ~90 W (≥70 years) in men versus ~130 W and ~70 W in women, respectively. The following categories of leg effort severity were associated with progressively lower exercise capacity: ≤50th ("mild"), >50th to <75th ("moderate"), ≥75th to <95th ("severe"), and ≥ 95th ("very severe") (p < 0.05). Although most subjects reporting peak scores <5 were in "mild" range, higher scores were not predictive of the other categories (p > 0.05). This novel frame of reference for 0-10 Borg leg effort, which considers its cumulative burden across increasingly higher exercise intensities, might prove valuable to judging symptom's normalcy, quantifying its severity, and assessing the effects of interventions in clinical populations.

Enhancing students’ performance on least taught topics in basic calculus through Moodle-based courseware package

This study utilized the developmental research design to identify the least taught topics in basic calculus and design and investigate the effectiveness of a Moodle-based courseware package. Data were collected from seven basic calculus teachers and five academic administrators through survey questionnaires, interviews, and validation sheets. The quasi-experimental phase of the study used a pre-/post-test control group design with a sample of 69 students, 34 students in the experimental group and 35 students in the control group. The experimental group received instruction using an asynchronous Moodle-based courseware package, while the control group received instruction through lecture notes. Results showed that the least taught topics include infinite limits, limits at infinity, and limits of trigonometric, exponential, and logarithmic functions. These topics were least taught due to time constraints, students’ capacity, scheduling, complexity, and relatability. Validation of the courseware package showed a very high validity. It was found that the use of Moodle-based courseware package and lecture notes effectively increased the students’ performance. However, one-way analysis of covariance results showed that the experimental group gained considerably greater than those in the control group. It was recommended that teachers continue to update their knowledge of different teaching methodologies to develop an encouraging and dynamic learning environment.

Temporal vortex tracking in isotropic turbulence

An algorithm to allow the temporal vortex tracking of the intense vorticity structures is introduced and applied to direct numerical simulations (DNS) of homogeneous isotropic turbulence (HIT). The algorithm allows the identification of the individual life events of the worms, including events of splits into several structures and of merging between different structures. Results obtained from a relatively small DNS, with Taylor based Reynolds number equal to Reλ = 54, for a simulation lasting more than 6 integral time scales, shows that the complement of the CDF of the life time of the IVS is well approximated by an exponential distribution function. The mean life time of the IVS which is approximately 3.6τη , where τη is the Kolmogorov time, and the birth rate of these structures is close to 4.5/τη (number of new structures per Kolmogorov time). A population model is built to predict the number of existing structures at any time, which gives a number of existing IVS equal to ≈ 16, which agrees well with the results from the DNS.

Infrared image target detection for substation electrical equipment based on improved faster region-based convolutional neural network algorithm.

Substation electrical equipment generates a massive number of infrared images during operation. However, the overall quality of the infrared images is low and it lacks image detail information. When using traditional target detection algorithms for detection, feature extraction poses great difficulties. Therefore, to address this problem, this paper proposes a target detection algorithm based on the improved faster region-based convolutional neural network (Faster R-CNN). It achieves the correct identification of different types of electrical equipment in infrared images. First, the algorithm improves the backbone network of Faster R-CNN for feature extraction. An InResNet structure is proposed to replace the residual block structure of the original ResNet-34 network, which enhances the richness of feature extraction. Second, the rectified linear unit activation function in the original feature extraction network is replaced by the exponential linear unit activation function, and group normalization is used instead of batch normalization as the network normalization method. Then, the dense connection structure is introduced into the ResNet-34 network, and the whole network is called residual dense connection network. Finally, the improved Faster R-CNN is compared to the original Faster R-CNN, a single-shot multibox detector, and you only look once v3 plus spatial pyramid pooling. The experimental results show that the improved algorithm has the highest mean average precision and average recall for most of the substation electrical equipment in infrared images. Moreover, from the confidence level of the detected electrical equipment and the accuracy of the prediction box, the improved Faster R-CNN has the best performance.

Robust Stability and Control of Fractional Polynomials Including Integer and Fractional Natural Exponential Functions

Robust Stability and Control of Fractional Polynomials Including Integer and Fractional Natural Exponential Functions