Variation Of Parameters Research Articles

Sample Size Reestimation in Stochastic Curtailment Tests With Time-to-Events Outcome in the Case of Nonproportional Hazards Utilizing Two Weibull Distributions With Unknown Shape Parameters.

Stochastic curtailment tests for Phase II two-arm trials with time-to-event end points are traditionally performed using the log-rank test. Recent advances in designing time-to-event trials have utilized the Weibull distribution with a known shape parameter estimated from historical studies. As sample size calculations depend on the value of this shape parameter, these methods either cannot be used or likely underperform/overperform when the natural variation around the point estimate is ignored. We demonstrate that when the magnitude of the Weibull shape parameters changes, unblinded interim information on the shape of the survival curves can be useful to enrich the final analysis for reestimation of the sample size. For such scenarios, we propose two Bayesian solutions to estimate the natural variations of the Weibull shape parameter. We implement these approaches under the framework of the newly proposed relative time method that allows nonproportional hazards and nonproportional time. We also demonstrate the sample size reestimation for the relative time method using three different approaches (internal pilot study approach, conditional power, and predictive power approach) at the interim stage of the trial. We demonstrate our methods using a hypothetical example and provide insights regarding the practical constraints for the proposed methods.

The Influence of Variable Plasma Welding Parameters on Weld Geometry, Dilution Factor, and Microhardness

Weld surfacing is the process of applying a layer of metal to the surface of metal objects by simultaneously melting the substrate. As a result of this process, the metal content of the padding weld can be as high as several tens of percents. It is a method used to regenerate machine parts and improve the properties of the surface layer, increasing its resistance to abrasion, corrosion, erosion, and cavitation. It also supports the repair and creation of permanent protective coatings in the engineering, automotive, energy, and aerospace industries. This makes it possible to repair damaged parts instead of completely replacing them, saving time and production costs. Plasma surfacing technology is used for components that require high hardness and corrosion resistance under various environmental conditions. Plasma wire surfacing is not sufficiently presented and described in the current literature, which creates problems in determining the appropriate process parameters. The influence of variable plasma surfacing parameters on steel C45 significantly affects surfacing weld geometry, the dilution factor, and microhardness. Higher currents can increase the dilution factor, integrating more base metal into the weld pool, which may alter the chemical composition and mechanical properties of the weld. Variations in surfacing speed and heat input also affect the microhardness of the surfaced joint, with higher heat inputs potentially leading to softer welds due to slower cooling rates. Optimizing these parameters is essential to achieving desired surfacing weld characteristics and ensuring the structural integrity of C45 steel joints. This paper presents the influence of varying plasma surfacing parameters on the surfacing geometry, the dilution factor, and microhardness. The tests were carried out on a Panasonic TM-1400 GIII automated surfacing machine with CastoMag 45554S solid wire as the filler material. Flat bars of C45 steel were prepared, and then the variable parameters of the surfacing process were developed. Tests were carried out to determine the dilution factor, followed by microhardness measurements. The results showed a significant dependence of the effect of the parameters on the surfacing geometry and the dilution factor.

Chemically reactive flow beyond constant thermophysical characteristics

Hydromagnetic chemically reactive flow over stretching wall is discussed. Formulation is based upon consideration of Reiner-Rivlin constitutive relation. Variable thermophysical characteristics are under consideration. Radiation, heat generation and dissipation impacts exist in thermal relation. Concentration expression subject to reaction with first order is considered. Modeling with entropy optimization is made. Appropriate transformations lead to relevant differential systems. Numerical study by ND-solve is carried out. Quantities under interest are graphically analyzed. An opposite behavior detected for magnetic parameter on entropy rate and velocity. Velocity field versus magnetic field parameter decreases whereas reverse impact witnessed for variable viscosity parameter. An enhancement in thermal distribution and entropy detected for radiation. Thermal distribution enhancement is observed for generation of heat and thermal conductivity variables. Chemical reaction for concentration has decaying impact. Results of Schmidt number for concentration are similar to that of chemical reaction. Brinkman number leads to entropy rate improvement. Entropy rate upsurges against variable viscosity and mass diffusivity parameters.

Performance optimization of triple tube heat exchanger using aluminum metal foam: A comprehensive numerical investigation

Metal foam is a porous medium with high porosity, convoluted flow pathways, strength-to-weight ratio, and thermal conductivity employed in heat exchanges and other engineering applications because of its properties. Therefore, a novel circular tube filled with aluminum metallic foam is proposed to improve hydraulic thermal performance and reduce pumping power losses and metal foam volume. The current work considers the Brinkman-Forchheimer Extended Darcy model for forced convection flow and the local thermal equilibrium (LTE) model for heat transfer under a constant thermal boundary condition. The governing equations are solved using the finite volume method (FVM). The variable parameters are the pore per inch (PPI) of 10–50, the porosity εrange of 0.79–0.95, and the Reynolds number range of 2500–7000. The results show that the performance index (PI) has its maximum value of 6.8 at 10PPI and ε =0.95 in the heat exchanger; this enhancement at 10PPI is 13% and 34.48% more as compared to the PI value at ε =0.85 and ε =0.79 and at a fixed inner annulus Re of 7000. The average Nusselt number (Nu) increased by 81.63%, 104.3%, and 120% compared to the foamless tube at 10, 20, and 30PPI andε =0.95, respectively, in an intermediate tube. Heat transfer coefficients (HTC) considerably increase compared to foamless tubes nearly 3.2 times, 3.8 times, and 4.2 times at 10, 20, and 30PPI, respectively, at 2500 Re and 144%, 175% and 200% more at 7000 Re. The maximum reduction ratio in normalized friction factor ratio (fP/fNoP) at 10PPI is approximately 50% compared to (fP/fNoP) at 30PPI and 25% at 20PPI; it occurs at 7000 Re. The normalized area goodness factor ratio has the maximum value at 10PPI and ε =0.95, which is 5.8, and for 20 and 30PPI, it is 3.5 and 2.3, respectively. This shows that aluminum metal foam in triple tube heat exchanger's intermediate tube at 10PPI and porosity of 0.95 can be the optimized results for performance enhancement.

Warm inflation with bulk viscous pressure for different solutions of an anisotropic universe

We study a warm inflationary model for different expansions assuming an anisotropic universe described by Bianchi I metric. The universe is filled with a scalar field or inflaton, radiation, and bulk viscous pressure. We carry out the inflationary analysis for different solutions of such universe in two different cases of the coefficient of dissipation Γ and the coefficient of bulk viscosity ξ as constant and variable parameters, respectively. We compare the obtained results with the recent observations, in order to find the observational constraints on the parameter space of the models. Moreover, we attempt to present a better judgment among the considered models by calculation of the non-linear parameter f NL describing the non-Gaussianity property of the models.

Forecasting the output of high-tech industry in China: A novel nonlinear grey time-delay multivariable model with variable lag parameters

Under the rapidly developing economy in China, accurate forecasting holds vital significance for policymaking and operational planning within the high-tech industry. However, the influencing factors affecting the output, accompanied by the time-delay effect, could be nonlinear, and uncertain. Thereby, this paper proposes a new nonlinear grey multivariable model with time-varying lag parameters. To be specific, the newly designed time delay function and power exponent are introduced, which can significantly enhance the adaptability and flexibility of the proposed method. The Grey Wolf Optimization algorithm is utilized to calculate the dynamic time lag parameters and power exponent to improve the prediction reliability. Furthermore, this new approach is applied to predict the high-tech industry’s output in China, Shanghai Municipality, and the Eastern Region, with due consideration given to the time-delay effect between input factors and outputs. To assess its efficacy, some leading models are selected for comparison to the proposed model. Furthermore, the utilization of Monte-Carlo simulation, the Probability Density Analysis, and the simulations are used to demonstrate the robustness and stability of this new method. The findings show that the proposed model is a feasible and applicable approach for prediction, exhibiting outstanding accuracy.

Propagation properties of the partially coherent radially polarized twisted beam through optical system in turbulence atmosphere

In this paper, the transmission properties of the partially coherent radially polarized twisted (PCRPT) beam propagating in the turbulence atmosphere are investigated. The analytical formulas for the components of the cross-spectral density matrix for the PCRPT beam passing through the optical system in the turbulent atmosphere are developed using the Collins integral and aperture function. Research results indicate that modifying the variable parameters and dimensions of the optical system can control the near-field and far-field distributions of the beam, while providing a more flexible choice of receivers for the PCRPT beam in the receiving plane. By utilizing the Cassegrain reflector system and adjusting the optical system parameters, it is possible to achieve collimated transmission of the PCRPT beam and significantly enhance the beam transmission efficiency in turbulent atmospheric conditions. The derivation process and the research results presented in this paper can be expanded to analyze the application of optical systems to control high-dimensional beam field variations. The envisioned utilization of the results obtained from this research investigation pertains to the fields of beam shaping and optical communications.

Synthesis of hardware and software for adaptive discrete high-order control systems based on a model of the desired transient process with a guaranteed degree of stability

In this work, analytical expressions for the connection between optimization criteria andsettings parameters of the adaptive PI-PID controller with variable parameters of the object andadaptive filter have been removed. A method for synthesizing a model of discrete controllersbased on given dynamic characteristics has been developed. A method has been developed forsynthesizing a model of a discrete PID controller based on given dynamic characteristics basedon the criterion of the transient process of a guaranteed level of stability. 3 figures, 0 tables, 32formulas, 19 bibliographic sources..Analytical expressions have been removed to link the optimization criteria andadjustment parameters of the adaptive PI-PID controller with the variable parameters of theobject and adaptive filter.A method for synthesizing a model of discrete controllers from specified dynamiccharacteristics has been developed. The average warehouse of output pulses at the output of anonlinear channel, which is formed at the output of such a regulator and is seen as an inertialinduced uninterrupted part, changes according to the law, which is designated as a reversetransmission the function of the lanyard of the new molded filter model at the collar of the non-linear element.A method has been developed for synthesizing a model of a discrete PID controller fromspecified dynamic characteristics based on the criterion of the required transient process of aguaranteed stability level.The modeling showed that while the parameters of the object and the adaptive filter willchange during the ASC operation, then, obviously, the optimal parameters for adjusting theadaptive PID controller will immediately change.

Effects of viscous dissipation, temperature dependent thermal conductivity, and local thermal non‐equilibrium on the heat transfer in a porous channel to Casson fluid

AbstractThe current paper deals with viscous dissipation effects in a permeable (or porous) channel filled with non‐Newtonian Casson fluid by considering the local thermal non‐equilibrium (LTNE) model. The dependency of the effective thermal conductivities of the solid and fluid phases on the respective temperatures has been studied along with the spatially varying Biot number. The Brinkman number Casson fluid parameter , thermal conductivity variation parameter , porosity Darcy number , and the ratio of fluid and solid phase thermal conductivities are the main governing parameters. The Darcy–Brinkman model is employed to govern the fluid flow in permeable media and the velocity profile has been obtained analytically. Moreover, the energy equations for both phases along with suitable boundary conditions are derived and solved with the fourth order boundary value solver. The findings of the current study depict that the Nusselt number increases with the increment in Casson fluid parameter and decreases with the increment in Brinkman number and thermal conductivity variation parameter. Overall, the heat transmission between the solid and fluid phases increases with the decrement in Brinkman number and thermal conductivity variation parameter. On the other hand, the heat transmission between both the phases magnifies by increasing the value of Casson fluid parameter.

Tuning equations for sliding mode controllers: An optimal multi-objective approach for non-minimum phase systems

This article proposes an approach to create tuning equations based on optimal multi-objective design specifically for Sliding Mode controllers applied to non-minimum phase processes that can be approximated to a First-Order-Plus-Deadtime class of systems. This is achieved by taking into account various performance criteria, including tracking, controller action, and transient response. The proposed tuning equations aim to achieve a well-balanced trade-off among competing objectives. The paper presents a systematic framework for formulating the multi-objective optimization problem and derives analytical expressions for the tuning parameters based on the desired performance specifications. The effectiveness of the proposed method is demonstrated through simulations in three dynamic processes: a high-order linear system with dominant delay, a high-order plus inverse response system, and a nonlinear process with variable parameters and dominant deadtime. A comparative analysis with an existing SMC tuning technique shows superior performance results. This paper provides valuable information on the optimization procedure for obtaining analytical tuning equations for controllers, offering a promising avenue for improving the performance of control systems in practical applications.

Rapid Inverse Parameter Inference Using Physics-Informed Neural Networks

As Li-ion batteries become more essential in today's economy, tools need to be developed to accurately and rapidly diagnose a battery's internal state-of-health. Using a Li-ion battery's (high-rate) voltage response, it is proposed to determine a battery's internal state through Bayesian calibration.However, Bayesian calibration is notoriously slow and requires thousands of model runs. To accelerate parameter inference using Bayesian calibration, a surrogate model is developed to replace the underlying physics-based Li-ion model. Developing a surrogate model for rapid Bayesian calibration analysis is discussed for both the single particle model (SPM) and the pseudo two-dimensional (P2D) model.Surrogate models are constructed using physics-informed neural networks (PINNs) that encode the influence of internal properties on observed voltage responses. In practice, a neural network can be trained by: 1) using simulation results of the physics-based model (i.e., a data-loss approach); 2) using the residuals of the governing equations themselves (i.e., a physics-loss approach); or 3) using a combination of simulation results and governing equation residuals. In the present work, PINNs are developed using a variety of training losses and neural network architectures. In this analysis, it is shown that a PINN surrogate model can be reliably trained with only physics-informed loss. However, using a coupled data-informed and physics-loss approach produced the most accurate PINNs. Figure~\\ref{fig:spm_2d} illustrates the absolute relative errors of trained PINN networks using several different training losses and neural network architectures.After determining a consistent training strategy for both the SPM and P2D PINN surrogate models, the PINNs are extended to determine additional internal state-of-health parameters. As more and more parameters were introduced, the PINN training suffered from ``the curse of dimensionality", which was mitigated by using a hierarchical training approach (where a PINN trained with fewer variable model parameters was used to train a PINN with more variable model parameters).Next, the high-dimensionality PINN surrogates are then integrated into Bayesian calibration schemes to identify internal Li-ion battery properties from experimentally measured voltages. Interpreting the high-dimensional parameter posteriors is discussed with respect to model error, parameter prior choices, and experimental errors. Figure 1

Reconstructing the Pseudo-2-Dimensional (p2D) Model in Comsol Multiphysics Using Mathematical Interfaces

The classic ‘pseudo-2-dimensional’ (p2D) model of Newman and co-workers (also known as the DFN model), with its different efficient implementations, is the state-of-the-art physics-based electrochemical model for lithium-ion batteries. COMSOL Multiphysics (COMSOL) is a popular modeling software for multiphysics simulations, and its Battery Design Module for Li-ion batteries is widely used in the battery modeling community. This work presents a step-wise model development for reconstructing the detailed p2D model equations. This includes variable model parameters and weak-form implementation for the solid-phase diffusion, eliminating the need for non-local couplings and approximations for modeling the pseudo radial-dimension in the solid-phase.To model and understand the effect of capacity degradation on lithium-ion cells, the battery models in COMSOL need to incorporate additional degradation mechanisms. In this work, we have followed the approach of equation-based modeling in COMSOL, which enables us to formulate the p2D model using the mathematical PDE interfaces. This approach provides us with a lot of flexibility in terms of battery degradation model formulation and helps develop a clearer understanding of the p2D model equation in light of the assumptions involved.The steps in the model development involved writing the conservation equations in the generic mathematics interfaces coupled with the additional equations for the degradation mechanisms. The efforts were centered on developing cycling simulations based on Events for practically relevant operating conditions. This custom model was validated against the results from COMSOL’s Battery Interface, in-house codes, and baseline cycle data from the experiments and further developed as a physicochemical degradation model for automotive, large-format Lithium-ion batteries.This equation-based COMSOL model with cycling works without the requirement for Battery Design Module, LiveLink and Application Builder and can be adopted for a variety of electrochemical modeling applications by the electrochemical modeling community.

Numerical study of third‐grade nanofluid flow with motile microorganisms under the mixed convection regime over a stretching cylinder

AbstractThe investigation of mixed convective flow involving microorganisms in nanofluid has garnered considerable interest in recent times owing to its extensive applicability in the biomedical domain. However, there has been a lack of study investigating a comprehensive parameter analysis for the flow of a third‐grade nanofluid around a cylinder in the presence of microorganisms. This study focuses on numerically investigating the flow of nanofluid around a stretched cylinder in the presence of motile microorganisms. The study also considers convective boundary conditions. Moreover, this study explores the nanofluid qualities related to Brownian motion and thermophoresis diffusion characteristics. The variable parameters include the Prandtl number (Pr), Peclet number (Pe), buoyancy ratio parameter (Nr), mixed convection parameter (), bioconvection Lewis number (Lb), Biot number (Bi), and Marangoni number (Ma). The bvp4c issue solver tool in MATLAB is used to numerically solve the nonlinear governing differential equations. The numerical model has been validated using prior papers. Graphical representations are created to depict several important measurements, such as velocity streamlines, velocity profiles, temperature distributions, nanoparticle concentrations, densities of gyrotactic motile microorganisms, local Nusselt numbers, skin friction, and Sherwood numbers. The link between the Nusselt number and the parameters Nt and Rd suggests that as Nt falls and Rd increases, the Nusselt number increases. The skin friction value is directly proportional to the values of Nr and Nc. There is a positive correlation between the rise in the local mass transfer rate and the values of Rd and Nt. The population of mobile microorganisms grows as the values of Lb and Pe decrease.

Analytical Study of R C C Deck Slab Bridge with Variable Parameters

Bridges ware always proved to be the key elements for the infrastructural development of country .While designing the bridges ,three major aspects were always to be considered. These three factors were strength , stability and cost effectiveness .It was observed that cost economy was based on width to span ratio of bridge. Another important aspect of economics of bridge was the thickness of deck slab. The length/span of bridge was kept constant and magnitude of width was selected as variable parameter. Four different width to span ratio were taken as 0.75,1, 1.5 and 2.0 respectively. Keeping the same width to span ratios , design calculations were carried out for 0.6 m and 0.7 m deck slab thickness Design was carried out for Mix M-30 and steel as Fe -415. Total eight cases were studied to observe an economy of deck slab bridge Four cases were considered for different parameters of design of deck slab . For all the four width to span ratios ,mentioned above for 0.6 m deck slab thickness and four cases for 0.7 m deck slab thickness were attempted. It was concluded that for width to span ratio=1.0 , was suitable for minimum cost . The results thus obtained were shown in tabular form .The results thus shown were useful for optimizing the design of deck slab bridges . This analysis is extremely useful to civil engineers, practicing engineers, contractors, research scholars and budding engineers.

Influence Analysis of Material Parameter Uncertainties on the Stability Safety Factor of Concrete Gravity Dams: A Probabilistic Method

Anti-sliding stability safety is a critical issue that must be given sufficient and widespread attention during the entire lifecycle of gravity dams. The calculated anti-sliding stability safety factor (ASS-SF) is usually compared with the allowable value required by the standards in the traditional method, which ignores the influence of material parameter uncertainties and leads to unreasonable safety evaluation results. Therefore, the nonlinear functional relationship between the stability safety factor (SF) and the random variable parameters is constructed based on the response surface equations, and the distribution types of SF sequences calculated by the Monte Carlo sampling are determined, then a probabilistic stability evaluation method for concrete gravity dams is proposed. Engineering application shows that the calculated SF obeys the normal distribution; the minimum guaranteed rate of different sliding paths in a gravity dam is 86.66%, and the guaranteed rate for the overload safety factor (OSF) is 36.00%. The results imply that a guaranteed rate for the allowable value of the ASS-SF should be provided when making the stability safety evaluation of the dams, especially the OSF. The outcome of this research will advance the understanding of stability evaluation of concrete dams, and reduce the potential risk of sliding instability of concrete dams.

Efficiency Measurements of Energy Harvesting from Electromagnetic Environment for Selected General Purpose Telecommunication Systems

The results of measurements of the efficiency of energy harvesting from commonly available general-purpose telecommunications systems, divided into typical bands available under European conditions, have been presented in this paper. Specially designed harvesters were used, dedicated to powering autonomous semi-passive Radio Frequency Identification (RFID) tags. For the assumed resistive loads, the achievable output voltage values of the harvesters were measured across a wide spectrum of electromagnetic field strengths, simulating real conditions. The performance and dynamics of the energy storage process with fixed parameters were studied at an intermediate stage, before the energy conditioning process. The harvesters were treated as typical energy sources with unknown but variable parameters, so their dynamic parameters and instantaneous energy supply were also analyzed. These activities will enable the final development of a power supply system with parameters acceptable for the planned applications and whose efficiency will be maximized under the given conditions. For this purpose, the energy harvesting systems were designed, a suitable laboratory stand was built, and the elaborated circuits were measured to determine the expected parameters of energy harvesting.

Computational analysis of nanoparticles and waste discharge concentration past a rotating sphere with Lorentz forces

Abstract As industries rely more and more on magnetohydrodynamic (MHD) systems for different uses in power, production, and management of the environment, it becomes essential to optimize these operations. The study seeks to improve the effectiveness and productivity of cooling structures, chemical reaction reactors, and contaminant control methods by investigating these intricate interconnections. Because of this, the work scrutinizes the endothermic/exothermic (EN/EX) chemical processes, convective boundary conditions, and pollutant concentration impacts on MHD nanofluid circulation around a rotating sphere. The governing equations based on the above assumptions are reduced into a system of ordinary differential equations and solved numerically with Runge–Kutta Fehlberg’s fourth- and fifth- order schemes. The obtained numerical outcomes from the numerical scheme are presented with the aid of graphs, and the results show that the rate of mass transfer decreases with an increase in the external pollutant local source and solid volume percentage. For changes in the values of the activation energy parameter and solid fraction, the rate of thermal dispersion drops for the EN case and upsurges for the EX case. The concentration profile shows increment with the addition of the external pollutant source variation parameter and local pollutant external source parameter. The outcomes of the present work can be helpful in cooling equipment, developing advanced methods for controlling pollution, environmental management, MHD generators, and various industrial contexts.

Multi-objective Optimization of Process Parameters for Surface Quality and Geometric Tolerances of AlSi10Mg Samples Produced by Additive Manufacturing Method Using Taguchi-Based Gray Relational Analysis

AbstractIn this study, it has been focused on examining the effects of production parameters on quality parameters such as surface roughness and geometric tolerances in the production of AlSi10Mg samples by the additive manufacturing method. The experimental design has been prepared according to the Taguchi L27 orthogonal array. As a result, in the production of samples, increasing laser power (P) contributed positively to surface roughness and diameter change, and increasing scanning distance (SD) negatively contributed to circularity change and concentricity. Further, it has been determined that increasing the scanning speed (SS) negatively affects the concentricity change of the produced samples. The optimum production parameters for surface roughness and diameter variation has been determined as A1B1C3. The optimum production parameters for circularity variation and concentricity have been determined as A3B3C1 and A3B1C1, respectively. According to the ANOVA analysis results, the most effective parameters for surface roughness, diameter change, circularity change and concentricity have been 53.22% P, 62.45% SD, 37.23% SS and 40.41% SD, respectively. Furthermore, as a result of the gray relationship analysis (GRA) performed for the output parameters, the optimum production parameter has been determined as A2B1C3.

A yarn-to-fabric multiscale modeling strategy for auxetic behavior analysis of negative Poisson’s yarn-based fabric

Auxetic behavior is critical to determining the deformation and mechanical performance of materials. Auxetic mechanisms of the woven fabric have previously not been well understood, especially for those containing negative Poisson’s ratio yarn (NPRY). Most theoretical models were developed from the macro geometric structure which lacks analysis of yarn levels. This paper aims to use numeral analysis combined with finite element modeling to investigate the auxetic behavior of the NPRY-based fabric. Firstly, the theoretical model that reveals the relationship between the auxetic behaviors of yarn and NPRY-based fabric was established based on component yarn interweaving rules, deformation, and stress. Following, a strain-dominated deformation simulation was further performed to clarify the micro stress and auxetic mechanism of fabric by calculating the stress–strain response and NPR value. Moreover, the experiments of NPRY-based fabric were also developed to validate the theoretical and simulation results, Results indicate that the model can well predict the auxetic behavior of fabric with an error under 7%. The micro–macro analysis of the auxetic behavior from yarn to fabric provides a fundamental understanding of the auxetic mechanism of NPRY-based fabric. Additionally, the yarn-to-fabric model can used to study the auxetic behavior of NPR fabric with variable parameters thus supplying theoretical guidance for engineering.

Mapping structured Laguerre–Gaussian beam states onto the orbital Poincaré sphere in the form of controllable spatial trajectories

The structured Laguerre–Gaussian (LG) beam is a two-parameter superposition of 2n+ℓ+1 Hermite–Gaussian modes (where n and ℓ are a radial number and a topological charge of the initial LG beam) whose orbital angular momentum oscillations are controlled by phases and amplitude parameters. But we succeeded in reducing its representation to a simple sum of a standard LG mode and a hybrid Hermite–Laguerre–Gaussian (HLG) beam that is a key point in understanding a hidden geometry of the structured LG (sLG) beams and implementations of its unique prosperities. In assents, the hybrid HLG beam is mapped onto the orbital Poincaré sphere in the form of a plane trajectory along a main meridian of the sphere. However, the most intriguing thing is as follows. First, once we slightly perturb the HLG beam with a single LG mode, the flat trajectory turns into a complex multi-petalled tracery with multiple self-intersections due to cyclic variation of the phase parameter of the sLG beam. Moreover, the shape of the tracery as well as the birth and destruction of the self-intersection points can be controlled with the amplitude parameter. However, it is worth noting that when changing the beam parameters cyclically, the area outlined by the trajectory on the sphere is directly related to the geometric phase acquired by the sLG beam that can be treated as an additional degree of freedom for transmitting big data. In the article, we study the sLG beam properties and its mapping onto the orbital Poincarè sphere in the framework of a symplectic 4×4 matrix formalism while the orbital Stokes parameters are experimentally measured, and we have found good agreement between theory and experiment.