Simple Equation Research Articles

Adaptive Controller Based on Estimated Parameters for Quadcopter Trajectory Tracking

This paper presents a trajectory control system design for a quadcopter, an unmanned aerial vehicle (UAV), which is based on estimated parameters that are assumed to exhibit random walk behavior. Initially, the rotational dynamic model of the UAV is formulated using the Newton Euler method in terms of angular velocity about the x, y, and z axes. This model is then simplified into three separated-first-order linear differential equations, with coefficients derived from the combined effects of inertia, aerodynamic drag, and gyroscopic effects, referred to as lumped parameters. A Proportional-Integral (PI) controller with feed-forward design is then developed to control this simplified model. To adapt the controller to the lumped parameters that exhibit random walk behavior, each simplified equation is restructured into a processing and measurement model. The states of these models are estimated by using the Unscented Kalman Filter (UKF). These estimated values are then utilized to adjust the PI gains and compensate the signal of the designed angular velocity controller, transforming it into an adaptive controller. The entire UAV controller comprises two main parts, an inner loop for adaptive angular rate control and an outer loop serving as an attitude-thrust controller. The proposed controller is simulated using Simulink, with circular and square trajectories. The simulation results demonstrate that the quadcopter successfully follows the desired circular and square paths. The steady-state error for the x and y axes in the square trajectory is less than 0.05 meters within 5 seconds, and for the z axis, it is less than 0.02 meters within 2.5 seconds. The controller gains do not require adjustment when changing trajectories. Moreover, the estimated parameters remain nearly constant at steady state.

THE INFLUENCE OF PEER ENVIRONMENT ON STUDENT LEARNING ACHIEVEMENT MAJORING

This research aims to determine the influence of the peer environment on the learning achievement of students majoring in accounting at SMK Negeri 2 Sidrap. The variables in this research are the peer environment as the independent variable (X) and learning achievement as the dependent variable (Y). The population of this study were all students majoring in Accounting at Negen 2 Sidrap Vocational School, totaling 122 students, while the sample used proportionate stratified random sampling technique with a sample size of 55 students. The data collection technique used was a questionnaire and documentation. The data analysis technique used was descriptive analysis of the percentage of instrument tests and test the hypothesis using SPSS 25 for windows. Based on the results of the data analysis that has been carried out, a simple linear regression equation model Y-63.760+0.351 From the results of the analysis of the coefficient of determination (R), the value R2-45% is obtained, which means that the peer environment contributes to student learning achievement by 45% and the remaining 55% is influenced by natural factors. Meanwhile, from the results of the t-test, a significant value of 0.000 0.05 is obtained, which means Peer environment has a significant effect on student achievement, thus the 'accepted hypothesis.Keywords: Peer Environment; Learning Achievement

Significance of heat generation and thermophoretic particle deposition in Marangoni convective driven boundary layer flow of cross nanofluid with activation energy

The thermo-solutal Marangoni boundary layer flow has been studied because it has a number of real-world uses, such as drying silicon wafers, applying thin coats of paint or glue, employing adhesive in heat exchangers, and growing crystals in space. The physical phenomenon of Cross (FeSO4−(CMC−H2O(<0.4%)) nanofluid flow in a porous medium with thermophoretic particle deposition and Marangoni convective flow is explored. Non-Newtonian Cross nanofluid flow is explored in relation to the effect of activation energy. The process of thermophoretic particle deposition, which is important in both electrical engineering and aero-solution, is one of the most fundamental methods for transferring small particles across a heat gradient. Cross nanofluid containing FeSO4 as nanoparticle, and based fluid is (CMC- water). Using the appropriate similarity variables, the set of governing PDEs is transmuted into a set of ODEs. The RKF-45th method is used to numerically solve these simplified equations. The graphic examination of the concentration, velocity and thermal fields is developed for the embedded non-dimensional characteristics. By raising the Marangoni ratio parameter, the surface tension gradient gets stronger, causing the induced flows to get stronger through the liquid more effectively. The concentration and thermal profiles fall with an enhancement in Marangoni ratio parameter.

Trajectory Tracking Control of Mobile Manipulator Based on Improved Sliding Mode Control Algorithm

Research on trajectory tracking control for climbing welding robots holds significant importance in the field of automated welding. However, existing trajectory tracking methods suffer from issues such as jitter and slow speed. In this paper, an improved sliding mode control strategy is proposed based on the self-designed wall-climbing welding mobile manipulator. Firstly, a new adaptive sliding mode control strategy is proposed for the mobile platform based on the kinematic model. By introducing a new approach law, the controller is designed when the distance between the center of mass is unknown. Secondly, regarding the manipulator, we analyze simplified dynamic equations, extract uncertain components, and utilize a CNN for compensation. This compensation strategy is integrated into the sliding mode control law, achieving precise control over the manipulator and effectively resolving issues like slow tracking speeds, large errors, and chattering. The stability of the robot control system is proved by the Lyapunov function. Through simulation analysis and experimental validation, the proposed control method is confirmed to be feasible and superior.

Efficient Modeling of Distributed Energy Resources’ Impact on Electric Grid Technical Losses: A Dynamic Regression Approach

Technical losses in electrical grids are inherent inefficiencies induced by the transmission and distribution of electricity, resulting in energy losses that can reach up to 40% of the generated energy. These losses pose significant challenges to grid operators regarding energy sustainability, reliability, and economic viability. Distributed Energy Resources (DERs) offer promising solutions to lower technical losses by decentralizing energy generation and consumption, reducing the need for long-distance transmission and optimizing grid operation. Hence, estimating the impact of DERs on grid technical losses becomes paramount for grid operators and planners. In response, this article proposes the application of regression modeling and nonlinear curve fitting algorithms to provide a more nuanced understanding and better characterize the intricate interplay between DER deployment and technical losses. Through a comprehensive case study based on more than 1080 computer simulations, we demonstrate the effectiveness of our proposed dynamic polynomial varying coefficient regression model in estimating the impact of DERs on technical losses within electrical grids. The proposed model offers a simple and effective methodology that allows grid operators to gain insights into the nonlinear dynamics of DER integration and make quicker and more informed decisions regarding grid management strategies, infrastructure investments, and policy interventions. Also, this research contributes to advancing the field of grid optimization by offering a simple equation that enhances our ability and haste to assess and mitigate technical losses in the context of an evolving energy landscape characterized by increasing DER adoption.

Solving a Predictive Model of Infectious Disease in Epidemiology and Epizootiology by Assigning Scores to Individuals

Every mathematical solution to a problem corresponds to a model that more or less corresponds to reality. One of the many possibilities is the solution to epidemiological problems (processes) using the susceptible-infected-removed (SIR) model. Like any model, this model also has its advantages and benefits that allow us to close such a complicated reality as the origin and course of the epidemic into simple differential equations (DE) and through the model’s parameters modify it and compare it with reality. This is possible only in the case of a comprehensive understanding of reality in all its breadth and depth. This study focuses on another solution to the epidemiological or epizootiological processes based on the newly designed simulation, which assigns individuals a certain score in time intervals. In this case, the score from the simulation has a normal probability distribution with certain parameters N (µ, σ) in given time intervals. This score level also divides the given individuals into certain groups based on the given (selected) critical level. This determines whether individuals are with manifestations of the disease or without signs of the disease. In the study, the epidemic curve (EC) simulation results could be used to solve the current pandemic caused by COVID-19. It can be stated that the newly proposed model could be more suitable because it permanently confirms the agreement of experimental and expected frequencies.

Relaxation oscillations in the Darie wind power plant model

The article discusses the mathematical model of the Daria small wind power plant. This installation is a type of vertical axis wind turbine named after its inventor, Georges Jean Marie Darrieux. The design consists of a vertically oriented shaft with curved blades or airfoils attached to it, forming a shape similar to an egg whisk. In today’s world, against the backdrop of climate change and steadily increasing energy demand, wind energy acts as a critical pillar of the transition to renewable energy sources. This technology helps reduce carbon emissions and mitigate humanity’s impact on the environment. In this context, wind energy is emerging not only as a means of supplying electricity, but also as a powerful catalyst for building a more sustainable and energy-efficient future. The equation of stationary modes is studied at the value of the external resistance of the dynamic model specified by the simplest equation. Conditions have been found under which relaxation oscillations are observed in the system.

Imaging n-Dimensional Spaces Within m-Dimensional Spaces: An Extension of Hinton’s Method

We derive a Euclidean manifold that is virtually cyclic using a simple equation based on a Euclidean geometry and related to Hinton&amp;rsquo;s method. The derived equation is simple to understand, but able to project n-dimensional spaces into m-dimensional spaces. In addition, the method produces exact images of rectangular cuboids as elements of a vector space, implying that information is a vector and not a scalar.

In-plan distribution of peak floor acceleration demands in torsionally irregular buildings

The precise estimation of peak floor acceleration demands is essential to ensure the seismic safety of building contents and attachments. The present study aims to investigate the distribution of the peak floor acceleration demands across the floor plan in torsionally irregular buildings. To achieve this objective, 28 mid-rise reinforced concrete buildings with torsional irregularities are analyzed under 5600 bidirectional earthquake excitations, considering various strength ratios in buildings. A total of 2,59,200 in-structure amplification factors are estimated at multiple locations within the building plan and also at different floors. It is observed that current seismic design codes tend to underestimate the in-structure amplification factor for torsionally irregular buildings, particularly at locations distant from the center of rigidity (towards flexible frames), by up to 137 % for elastic buildings, and up to 54 % for moderately inelastic buildings. Even in the case of frames closer to the stiff frames, the seismic design codes underestimated the in-structure amplification factor by up to 68 % for elastic buildings. It is observed that a strong relationship exists between the torsional amplification of the peak floor acceleration and displacement amplification due to building torsion at the floor of interest. The torsional amplification of peak floor acceleration is found to be approximately equal to torsional displacement amplification for the flexible frames at the floor of interest. Simplified equations are developed for their integration with existing building codes to precisely estimate the in-plan distribution of peak floor acceleration demands in torsionally irregular buildings.

Practical approaches for evaluating radiative heat from high pressure hydrogen flame

As the construction of hydrogen stations and other hydrogen-related facilities progresses, it becomes increasingly crucial for safety designs to account for scenarios where leaked hydrogen might ignite. Since a hydrogen flame is usually invisible, detection must rely on radiative heat or temperature. While it is possible to predict accurate radiative heat flux from a high pressure hydrogen jet flame through detailed numerical calculations of hydrogen combustion with its radiation, the objective of this study is to provide a simpler and more accurate prediction method. Previous experimental data of radiative heat flux, Qr from high pressure hydrogen jet flames were compiled, and a simple experimental equation Qr=1463.3(Lo/Lf)−2.32 was derived, where Lf and Lo represent the flame length and the distance perpendicular to the flame axis from the midpoint of the flame length to the observer, respectively. Radiative heat fluxes were computed through detailed numerical simulations using three types of water vapor radiation models: Goody's narrow random band model, the exponential broad band model by Edward and Elsasser's narrow regular band model. The temperature and water vapor partial pressure distributions obtained from numerical simulations were used as input conditions. The results were found to be in good agreement with the experimental equation. Moreover, a simple diagram representing the radiative heat flux from a hydrogen flame was derived using the method of cylinder-like approximation. The results of this study show that if the hydrogen pressure and leakage aperture diameter are given, the radiative heat flux at a certain distance can be predicted with high accuracy. Additionally, the average value of emissivity for the hydrogen diffusion flame was found to be on the order of 0.03.

Data-based modelling of arrays of wave energy systems: Experimental tests, models, and validation

One of the key steps towards economic feasibility of wave energy conversion technology concerns scaling up to farms of multiple devices, in the attempt to reduce installation costs by sharing infrastructure, and a consequent drop in levelised cost of energy. Moreover, whenever wave energy systems are deployed in proximity (in so-called arrays), the exploitation of the hydrodynamic interactions between single devices is fully enabled, potentially increasing the final energy outcome. To achieve this in real (operational) time, the employed energy-maximising control strategies require control-oriented array models, able to efficiently describe the dynamics of these interconnected systems in a representative fashion. This can be, nonetheless, a difficult task when considering first principles alone, under small motion assumptions, for modelling purposes. Recognising the uncertainty associated to array numerical models obtained from the linearisation of simplified system equations around their equilibria, this paper presents models of several array configurations identified following a frequency domain approach on the basis of experimental data. Tailored tests on laboratory-scale devices have been designed and conducted in the Aalborg University (Denmark) wave tank facility, with the purpose of performing representative system identification of the wave energy systems arrays. The obtained models are validated on different representative sea states configurations, in controlled and uncontrolled motion operational conditions. The validation results are fully discussed and analysed in terms of standard error measures and time lag, while the obtained models are made freely accessible via a linked repository (named OCEAN), in the attempt to openly provide validated models for different array configurations.

A Note on a Simple Equation for Solving Nonlinear Undamped Oscillations

A Note on a Simple Equation for Solving Nonlinear Undamped Oscillations

Investigation on the influence of previous in-plane loading on the out-of-plane behavior of the innovative damped masonry infill wall

The damped masonry infill wall (DMIW) is an innovative masonry infill wall (MIW), wherein multiple horizontal subpanels, also called masonry units, are subdivided by inserting the horizontal damping layer joints (DLJs). While existing research on DMIW primarily focuses on its pure in-plane (IP) and pure out-of-plane (OOP) behavior, it is crucial to account for the interaction effect between IP and OOP behaviors to accurately assess OOP performance, Failure to consider this interaction may lead to an overestimation of OOP capacity, a concern previously highlighted in traditional MIW studies. Addressing this gap, this study conducts a sequential experiment involving IP cyclic quasi-static loading followed by OOP monotonic quasi-static loading. The aim is to examine the influence of previous IP cyclic loading on OOP behavior, considering factors such as OOP damage evolution, performance parameters, deformation patterns, and resistance mechanisms. The results show that, in DMIW, the effect of previous IP cyclic loading on the reduction of OOP performance is significantly lower compared to traditional MIW. Furthermore, the fundamental OOP resistance mechanism of DMIW remains unchanged under IP-OOP sequential loading. However, the OOP deformation pattern is altered due to the previous horizontal cracks developed in the masonry units under IP cyclic loading. Finally, a simple bi-linear equation is derived to quantify the effect of the previous IP cyclic loading on the primary OOP performance parameters of the DMIW.

Predicting out-of-plane bending strength of cross laminated timber: Finite element simulation and experimental validation of homogeneous and inhomogeneous CLT

The strength of cross laminated timber (CLT) depends on the stiffness and strength of the lamellas and on the strength of the finger joints. A model for how stiffness and strength vary along and between lamellas is used in combination with a finite element model of CLT and Monte Carlo simulations to calculate out-of-plane bending strength of homogeneous and inhomogeneous CLT. Calculated and experimentally obtained results of characteristic bending strengths, coefficient of variation of bending strength and the proportion of finger joint failures, agree very well for both types of CLT. The characteristic out-of-plane bending strength and the mean bending stiffness were 23% and 16% higher, respectively, for inhomogeneous CLT with outer layer lamellas graded in the strength class C35, compared to homogeneous CLT with all lamellas graded in the class C24. Simulation results give basis for simple equations by which bending strength of CLT can be determined as function of the layup, the strength class of outer layer lamellas and characteristic strength of the finger joints. Furthermore, system effects are investigated. For inhomogeneous CLT, with outer layer lamellas of high strength class, the system effects turn out to be quite different from those of ordinary, homogeneous CLT.

Estimating elastic properties of sediments by pseudo-wells generation utilizing simulated annealing optimization method

The hydrate concentration model considerably affects elastic properties, including bulk and shear modulus. Defining seismic properties of sediments, such as compressional and shear wave velocity and density, provides valuable information to identify rock facies and fluid types. This information commonly results from pre-stack seismic inversion, while post-stack seismic information provides acoustic impedance as a layer-based property. Traditionally, seismic inversion requires well logs to produce an initial guess of inversion routines and provide a low-frequency part of the amplitude spectrum. Eventually, seismic inversion methods could not be performed in the areas without well-log data, such as deep sea areas. In such cases, pseudo-well logs derived from pre-stack seismic data are a solution. Pseudo-well generation is a title used to estimate the elastic parameters of sediments in areas, such as deep marine environments, where drilled wells are absent or sparse. Metaheuristic optimization algorithms are suitable tools for minimizing the cost function as they best match real and synthetic seismic data. In this study, the SEAM earth model has been used as a reference to investigate the quality of pseudo-well generation utilizing a simulated annealing (SA) algorithm as an optimization method of property model change, which minimizes the cost function of seismic inversion. As a result, considering an initial model type of the SEAM model, simultaneous seismic inversion introduced the best compressional and shear wave velocities and density logs, which provide the best real and synthetic seismic data match when synthetic data is calculated from the simplified Zoeppritz equation.

Ballast Settlement Accumulation in Zones with Unsupported Sleepers

The high influence of impact and vibration on the behavior of crushed stone and ballast materials has been known for a long time. The zones with unsupported sleepers, which are always present in transition zones, crossings, welds, etc., are typically characterized by impact interaction, ballast full unloading, and additional preloading. However, no studies on ballast layer settlements consider impact vibration loading. Moreover, the influence of the cyclic loading on the ballast settlement intensity is considered ambiguously, with both decelerating and accelerating trends. The comprehensive literature review presents the influence of factors on settlement intensity. The present study aims to estimate the long-term processes of sleeper settlement accumulation depending on the loading factors: impact, cyclic loading, and preloading. The typical for a void zone ballast loading pattern was determined for various void sizes and the position along the track by using a model of vehicle-track interaction that was validated by experimental measurements. The loading patterns were parametrized with four parameters: maxima of the cyclic loading, impact loading, sleeper acceleration, and minimal preloading. A specially prepared DEM simulation model was used to estimate the ballast settlement intensity after initial settlement stabilization for more than 100 loading patterns of the void zone cases. The settlement simulation results clearly show that even a low-impact loading pattern causes many times increased settlement intensity than ordinary cyclic loading. Moreover, the initial preloading in the neighbor-to-void zones can cause even a decrease in the settlement intensity compared to the full ordinary or partial unloading. A statistical analysis using a machine learning approach and an analytic one was used to create the model for the intensity prediction regarding the loading patterns. The analytic approach demonstrates somewhat lower prediction quality, but it allows to receive plausible and simple analytic equations of the settlement intensity. The results show that the maximal cyclic loading has a nonlinear influence on the settlement intensity that corresponds to the 3–4 power function, and the impact loading is expressed by the linear to parabolic function. The ballast’s minimal preloading contributes to the reduction of the settlement intensity, especially for high cyclic loadings that are typical for neighbor-to-void zones. The results of the present study could be used for the complementing of the present phenomenological equations with the new factors and further application in the algorithms of the settlements accumulation prediction.

New momentum integral equation applicable to boundary layer flows under arbitrary pressure gradients

By incorporating the traditionally overlooked advective term in the wall-normal momentum equation, a new momentum integral equation is developed for two-dimensional incompressible turbulent boundary layers under arbitrary pressure gradients. The classical Kármán's integral arises as a special instance of the new momentum integral equation when the pressure gradient is weak. The new momentum integral equation's validity is substantiated by direct numerical simulation data. Unlike the classical Kármán's integral, which is limited to predicting wall shear stress within mild pressure gradients, the new momentum integral equation accurately computes wall shear stress across a broad range of pressure gradients, even in the presence of strong adverse pressure gradients that lead to flow separation. Moreover, a new pressure parameter $\beta _\kappa$ is introduced through examining terms in the new momentum integral equation. This parameter naturally quantifies the pressure gradient's influence on turbulent boundary layers and offers guidance for applying the classical Kármán's integral. Additionally, to facilitate experimental determination of wall shear stress under strong pressure gradients, an approximate integral equation is proposed that relies solely on easily measurable variables. Validation against direct numerical simulation data demonstrates that this simplified equation provides reasonably accurate estimates of wall shear stress in turbulent boundary layers experiencing strong pressure gradients.

Experimental analysis of the flexural behaviour of precast concrete composite beams with discontinuous connections

Full-depth precast deck panels hold the potential to enhance constructability and productivity, and reduce the costs associated with highway bridges. This paper investigates the influence of discontinuous shear connections between girders and panels on the flexural behaviour of composite beams. Experimental findings from precast composite beams and direct shear tests are presented, including L-shape push-off tests and double shear tests to determine resistance at rough concrete-to-concrete plain interfaces and shear pocket connections. Composite beams with a full-depth precast slab connected through a shear pocket with shear keys into the upper face of a precast beam exhibit equivalent flexural resistance to monolithic beams when shear pocket spacing remains below the effective depth of the composite beam. However, composite beams with shear pockets featuring plain interfaces show a reduction in flexural resistance of up to 30% compared to monolithic beams. Moreover, composite beams with shear keys demonstrate interface shear stiffness up to four times greater than those with plain interfaces, indicating full interaction. The paper proposes simplified equations to estimate flexural resistance for composite beams with full-depth precast slabs joined by shear pockets, showing a coefficient of variation of only 12% compared to experimental results, offering a reliable method for predicting flexural resistance in composite beams with discontinuous connections.

On some novel solitonic structures for the Zhiber–Shabat model in modern physics

Abstract In this article, the modified Kudryashov and extended simple equation methods are employed to obtain analytical solutions for the Zhiber–Shabat problem. The outcomes of this study clearly indicate that the provided methodologies are appropriate techniques for generating some new exact solutions for nonlinear evolution equations. Furthermore, the nature of the solutions would be presented in three dimensions for various parameters applying the most advanced scientific instruments. The physical behavior of the solutions are graphically displayed, and it is established that the acquired solutions are newly constructed in the form of bright, dark, optical, singular, and bell-shaped periodic soliton wave structures. The properties of the nonlinear model have been illustrated using 3D, 2D, and contour plots by selecting an appropriate set of parameters, which is demonstrated to visualize the physical structures more productively. Finally, it is concluded that similar strategies can also be implemented to study many contemporary models. To the best of our knowledge, the current work presents a novel case study that has not been previously studied in order to generate several new solutions to the governing model appearing in diverse disciplines. The results show that the strategies that have been employed are more effective and capable than the traditional methods found in previous research.

THE INFLUENCE OF TEACHER CREATIVITY IN MANAGING LEARNING ON MOTIVATION STUDENT LEARNING

This research is quantitative research with an ex post facto type of research, the research location is located in Cinnong Village, Sibulue District, namely SMA Negeri 12 Bone and the data sources are economics subject teachers and students in class XI IPS 1 and XI IPS 2. Data collection process, researchers used observation, documentation and questionnaires with follow-up of the data using descriptive analysis techniques with validity and reliability testing stages, as well as hypothesis testing with simple regression equations and t tests (partial). The results of research using questionnaires related to teacher creativity and student learning motivation, the validity test carried out on variable table 0.05. In variable Y, the sig value of each item is the highest value 0.042 &lt; 0.05 from number 12 and the calculated r value &gt; r table with the lowest value 0.271, r table 0.05, namely 0.260. The reliability test for In hypothesis testing, the constant number 154,448 means that if there is no variable .05 with the conclusion that H0 is rejected. In the t test the calculated t value is -4.166 with df = n-2 (10 – 2 = 8) with a value of 1.860 which has a significant effect.