Nonlinear Error Analysis Research Articles

Simulation of Motion Nonlinear Error Compensation of CNC Machine Tools With Multi-Axis Linkage

In order to solve the problem of nonlinear error for a dual rotary table five-axis CNC machine tool due to the linkage of rotary and translational axes, the simulation of motion nonlinear error compensation for a multi-axis linkage CNC machine tool is proposed. The adjacent points in the tool position file are selected as the tool position points for building the model, and then the nonlinear error model resolved by the harmonic function is established according to the error distribution in the classical post-processing. The nonlinear error between the two tool position points is quickly predicted by the analytical expression of this model, and the real-time error compensation of the intermediate interpolation points is realized. Finally, MALTLAB simulation analysis is performed on the tool position file of an impeller part machining to verify the effectiveness of the proposed algorithm. The experimental results show that it can be seen from the distribution curve of the nonlinear error that it is about 10% after compensation as before compensation, thus verifying the effectiveness of the nonlinear error compensation mechanism. The correctness of the nonlinear error analysis and compensation method and the effectiveness of post-processing are verified.

Nonlinear error analysis of fast optical delay lines.

Optical delay lines have wide applications in terahertz time-domain spectroscopy and optical coherence tomography. In this study, a fast-rotating optical delay line (FRODL) with 24 turntable reflection surfaces was designed. By analyzing the working principle of the FRODL, a mathematical model was established for the nonlinear parameter error of the FRODL delay time. By constructing the polarization Michelson interference system and testing the FRODL structure, the error of actual assembly parameters of the FRODL was approximately 0.015 mm, the actual delay time of the FRODL was greater than 43.5 ps, and the linearity was 99.785%.

Removal of Brilliant Green Dye from Aqueous Solutions Using Multi-walled Carbon Nanotubes (MWCNTs): Linear and Nonlinear Isotherm Models and Error Analysis

Current research explains the comparison of linear and nonlinear regression methods for finding the optimal isotherm study using experimental data for the adsorption of BG on multi-walled carbon nanotubes MWCNTs. BG dye maximum adsorption onto MWCNTs occurred at pH 2 and 35°C, with the apparent equilibrium reached after 15 min. In this study, five error functions were used: ERRS, Hybrid, Chi-square (χ2), ARE, and EABS. The values of error functions suggest that the Langmuir Linear type 3 is a suitable isotherm to describe the adsorption of BG on MWCNTs. The results showed that the Langmuir isotherm is a fit good isotherm to describe the adsorption process. The coefficient of non-determination (K2) showed Hybrid, and ERRS were the preferable error functions used to predict the fit of linear and nonlinear isotherm models. Compared with other studies, MWCNTs can be used as a low-cost adsorbent with low contact time for the removal of BG dye from an aqueous solution.

Error Analysis of Heterodyne Interferometry Based on One Single-Mode Polarization-Maintaining Fiber.

Using polarization-maintaining fiber (PMF) in dual-frequency heterodyne interferometry has the advantages of reducing the laser's own drift, obtaining high-quality light spots, and improving thermal stability. Using only one single-mode PMF to achieve the transmission of dual-frequency orthogonal, linearly polarized beam requires angular alignment only once to realize the transmission of dual-frequency orthogonal, linearly polarized light, avoiding coupling inconsistency errors, so that it has the advantages of high efficiency and low cost. However, there are still many nonlinear influencing factors in this method, such as the ellipticity and non-orthogonality of the dual-frequency laser, the angular misalignment error of the PMF, and the influence of temperature on the output beam of the PMF. This paper uses the Jones matrix to innovatively construct an error analysis model for the heterodyne interferometry using one single-mode PMF, to realize the quantitative analysis of various nonlinear error influencing factors, and clarify that the main error source is the angular misalignment error of the PMF. For the first time, the simulation provides a goal for the optimization of the alignment scheme of the PMF and the improvement of the accuracy to the sub-nanometer level. In actual measurement, the angular misalignment error of the PMF needs to be smaller than 2.87° to achieve sub-nanometer interference accuracy, and smaller than 0.25° to make the influence smaller than ten picometers. It provides theoretical guidance and an effective means for improving the design of heterodyne interferometry instruments based on PMF and further reducing measurement errors.

Analysis and Suppression of Nonlinear Error of Pendulous Integrating Gyroscopic Accelerometer at Instrument Level.

The error coefficients of the pendulous integrating gyroscopic accelerometer (PIGA) mainly include the bias, scale factor, and nonlinear error. Previous works have fully studied and suppressed the bias and scale factor of PIGAs. At present, the nonlinear error is the most critical factor restricting the measurement accuracy of PIGAs. To address this barrier, a study on the analysis and suppression of the nonlinear error of PIGAs at the instrument level was carried out. Firstly, the error model of a PIGA is established by kinematics and dynamics analyses. Then, nonlinear error is analyzed based on the established model. Finally, a suppression method for the nonlinear error is proposed based on the analysis results. The nonlinear error analysis found that (1) the nonlinear error includes a quadratic term error caused by unequal inertia and the inertia product, cross-coupling error is caused by lateral accelerations, and error is caused by unequal stiffness; (2) unequal inertia and the inertia product were the most critical factors resulting in nonlinear error. Based on the results in the nonlinear error analysis, the suppression method for error focuses on unequal inertia and the inertia product. The proposed method of analysis and suppression was validated experimentally as the quadratic term coefficient was reduced by an order of magnitude from 1.9 × 10-6/g0 to 1.91 × 10-7/g0.

Mathematical Simulation of Heat Transfer in Thermally Magnetised Oldroyd-B Fluid in Sakiadis Rheology with a Heat Reservoir

Sakiadis rheology of a generalised polymeric material, as well as a heat source or sink and a magnetic field, are all part of this study. Thermal radiations have been introduced into the convective heating process. The translation of a physical situation into a set of nonlinear equations was achieved through mathematical modelling. To convert the resulting partial differential equation into a set of nonlinear ordinary differential equations, appropriate transformations have been used. The velocity and temperature profiles are generated both analytically by HAM and numerically by the Runge–Kutta method (RK-4). In order to analyse the behaviour of the physical quantities involved, numerical and graphical depictions have been offered. To show that the acquired findings are correct, a nonlinear system error analysis has been offered. The heat flux study has been shown using bar charts. For the essential factors involved, the local Nusselt number and local Skin friction are calculated in tabular form. The fluid particles’ molecular mobility was slowed due to the magnetic field and porosity, and the heat transfer rates were demonstrated to be lowered when magnetic and porosity effects are present. This magnetic field and porosity effects regulating property has applications in MHD ion propulsion and power production, the electromagnetic casting of metals, etc. Furthermore, internal heat absorption and generation have diametrically opposed impacts on fluid temperature. The novelty of the present study is that no one has investigated the Sakiadis flow of thermal convection magnetised Oldroyd-B fluid in terms of a heat reservoir across a porous sheet. In limited circumstances, a satisfactory match is revealed when the collected values are compared to the existing work published corroborating the current attempt. The findings of this study are expected to be applicable to a wide range of technical and industrial processes, including steel extrusion, wire protective layers, fiber rolling, fabrication, polythene stuff such as broadsheet, fiber, and stainless steel sheets, and even the process of depositing a thin layer where the sheet is squeezed.

A new characteristic numerical approach with evolutionary residual error analysis to nonlinear boundary value problems occurring in heat and mass transfer via combinatoric Mittag-Leffler polynomial

This study focuses on new numerical approach to the solutions of nonlinear boundary value problems occurring in heat and mass transfer, constructing a matrix-combinatorial method collocated by the Chebyshev-Lobatto points and based on the Mittag-Leffler polynomial. For the first time, a matrix-collocation method is coupled with a combinatoric polynomial. In view of this combination, the method converts the linear and nonlinear terms to the matrix forms and then gathers them to a fundamental matrix equation. In addition to the novelty, an inventive nonlinear residual error analysis of general type is firstly theorized and adapted for improving the solutions to the problems in question and also, it allows to regard the nonlinear terms as an operator in calculations. The obtained solutions are thereby corrected. Numerical and graphical illustrations are provided to scrutinize the accuracy, productivity and comparability of the method. Upon evaluations of all these tasks, one can admit that the method is comprehensible, consistent and easily programmable.

Adsorption of Zn2+ from Synthetic Wastewater Using Dried Watermelon Rind (D-WMR): An Overview of Nonlinear and Linear Regression and Error Analysis

Sustainable wastewater treatment is one of the biggest issues of the 21st century. Metals such as Zn2+ have been released into the environment due to rapid industrial development. In this study, dried watermelon rind (D-WMR) is used as a low-cost adsorption material to assess natural adsorbents’ ability to remove Zn2+ from synthetic wastewater. D-WMR was characterized using scanning electron microscope (SEM) and X-ray fluorescence (XRF). According to the results of the analysis, the D-WMR has two colours, white and black, and a significant concentration of mesoporous silica (83.70%). Moreover, after three hours of contact time in a synthetic solution with 400 mg/L Zn2+ concentration at pH 8 and 30 to 40 °C, the highest adsorption capacity of Zn2+ onto 1.5 g D-WMR adsorbent dose with 150 μm particle size was 25 mg/g. The experimental equilibrium data of Zn2+ onto D-WMR was utilized to compare nonlinear and linear isotherm and kinetics models for parameter determination. The best models for fitting equilibrium data were nonlinear Langmuir and pseudo-second models with lower error functions. Consequently, the potential use of D-WMR as a natural adsorbent for Zn2+ removal was highlighted, and error analysis indicated that nonlinear models best explain the adsorption data.

Rapid, -insensitive, dual-band quasi-adiabatic saturation transfer with optimal control for complete quantification of myocardial ATP flux.

PurposePhosphorus saturation‐transfer experiments can quantify metabolic fluxes noninvasively. Typically, the forward flux through the creatine kinase reaction is investigated by observing the decrease in phosphocreatine (PCr) after saturation of γ‐ATP. The quantification of total ATP utilization is currently underexplored, as it requires simultaneous saturation of inorganic phosphate (Pi) and PCr. This is challenging, as currently available saturation pulses reduce the already‐low γ‐ATP signal present.MethodsUsing a hybrid optimal‐control and Shinnar‐Le Roux method, a quasi‐adiabatic RF pulse was designed for the dual saturation of PCr and Pi to enable determination of total ATP utilization. The pulses were evaluated in Bloch equation simulations, compared with a conventional hard‐cosine DANTE saturation sequence, before being applied to perfused rat hearts at 11.7 T.ResultsThe quasi‐adiabatic pulse was insensitive to a >2.5‐fold variation in B1, producing equivalent saturation with a 53% reduction in delivered pulse power and a 33‐fold reduction in spillover at the minimum effective B1. This enabled the complete quantification of the synthesis and degradation fluxes for ATP in 30‐45 minutes in the perfused rat heart. While the net synthesis flux (4.24 ± 0.8 mM/s, SEM) was not significantly different from degradation flux (6.88 ± 2 mM/s, P = .06) and both measures are consistent with prior work, nonlinear error analysis highlights uncertainties in the Pi‐to‐ATP measurement that may explain a trend suggesting a possible imbalance.ConclusionsThis work demonstrates a novel quasi‐adiabatic dual‐saturation RF pulse with significantly improved performance that can be used to measure ATP turnover in the heart in vivo.

Nonlinear error analysis and calibration model for cyclic ADCs in large array CMOS image sensors

A mathematical model is established to study the impact of nonlinear errors on the performance of cyclic analog-to-digital converters (ADCs) and imaging quality of CMOS image sensors (CIS). And a radix-based foreground digital calibration method for cyclic ADCs in large array CIS is proposed and modeled. The nonlinear errors caused by capacitor mismatch and finite operational amplifier (OPAMP) gain are mainly studied. Simulation results show that the proposed calibration method based on the statistical characteristics of large array CIS can effectively reduce the influence of nonlinear errors, greatly improve ADC linearity under large capacitor mismatch and finite OPAMP gain. In a CIS with 1500 columns of parallel ADCs, when the capacitor mismatch and OPAMP gain are 1% and 40 dB respectively, the effective number of bits (ENOB) of the calibrated ADC can be increased from about 6bit to 13bit. These results provide a new guideline for the calibration of cyclic ADCs in large array CMOS image sensors.

Linear and Nonlinear Isotherm Models and Error Analysis for the Sorption of Kresoxim-Methyl in Agricultural Soils of India.

Kresoxim methyl sorption in soils of five agro-climatic zones of Indiavaried from 41.6% to 84.7%. Highest sorption was recorded in organic carbon rich Almora soil. Isotherm parameters for linear and non-linear Freundlich and Temkin models were almost same, whereas Langmuir parameter Q0, for linear (1.60 to 9.434μgg-1) and non-linear (8.48 to 17.129μgg-1) models were quite different. For isotherms optimization different error functions such as sum of squares error (SSE), root mean square error (RMSE), Chi square error, hybrid fractional error (HYBRID) and average relative error (ARE) were calculated. Lowest error function values were obtained for Freundlich isotherm in all the soils except inceptisol (Kolkata) for which Langmuir isotherm gave the best fit. Statistical analysis using SAS 9.3 software and Tukey's HSD test revealed the significant effect (p < 0.001) of soil type on sorption. Sorption correlated positively with the organic carbon and clay contents of the soil.

Analysis and Control Algorithm of Non-linear Error for Nine-axis Synchronize Milling Blades with Twin-cutters

Analysis and Control Algorithm of Non-linear Error for Nine-axis Synchronize Milling Blades with Twin-cutters

Removal of Eriochrom Black T from aqueous solution using Al2O3 surface: Linear and non-linear isotherm models, error analysis and thermodynamic studies

Abstract A comparison of linear and non-linear regression method to find the best fitting isotherm studied using the experimental data for adsorption of Eriochrom Black T on the Al2O3 surface. So three error function used: Standard Deviation, Sum of the squares of the error, and Root mean square error. The error values suggest that the non-linear method is an appropriate method to describe the adsorption of the Eriochrom Black T on the Al2O3 surface. The results showed that Langmuir isotherm was the best isotherm to describe the adsorption process. The thermodynamic parameters appeared that the process was spontaneous and endothermic under the studied conditions.

Effective and fast adsorptive removal of fluoride on CaAl2O4:Ba nanoparticles: isotherm, kinetics and reusability studies

Present systematic investigation, effective and rapid adsorptive removal of fluoride has been studied experimentally using optimized Barium doped calcium aluminate. Synthesized BCA nanoparticle characterized with analytical techniques like PXRD, FTIR, BET, pHPZC, SEM and EDS. During the adsorption process, to optimize dosage, various influencing parameters are carried out like pH (2–12), adsorbent dosage (0–50) mgl−1, contact time (0–50) min and adsorbate strength (0–30) mgl−1. Nonlinear modeling and statistical error analysis were computed and curve fitted using experimentally obtained equilibrium and kinetic data. Entire nonlinear parameters and statistical error functions solved using excel solver-2019. Among the experimentally computed data, Freundlich isotherm and pseudo-second-order kinetic models best fitted with R2 > 0.98. Maximum Langmuir adsorption capacity is 125 mgg−1 and free mean energy of fluoride adsorption EDR is 7.9 kJmol−1: Linear model suggests physisorption mechanism. From the statistical error analysis, PSO kinetic and D-R equilibrium model strongly recommended. Spent adsorbent reused for six cycles which are regenerated using 0.1 M NaOH. Adsorption of fluoride at 5 mgl−1 concentration is efficient in the presence of competing anions like SO42−, NO3−, Cl− and HCO3−. The experimental analysis concludes that adsorbent is an effective and well-organized adsorbent for the removal of fluoride from groundwater.

Nonlinear phase error analysis of equivalent thickness in a white-light spectral interferometer

Abstract A white light spectral interferometry based on a Linnik type system was established to accurately measure the thin film thickness through transparent medium. In practical work, the equivalent thickness of a beam splitter and the mismatch of the objective lens introduce nonlinear phase errors. Adding a transparent medium also increases the equivalent thickness. The simulation results show that the equivalent thickness has a significant effect on thin film thickness measurements. Therefore, it is necessary to perform wavelength correction to provide a constant equivalent thickness for beam splitters. In the experiments, some pieces of cover glasses as the transparent medium were added to the measured beam and then a standard thin film thickness of 1052.2±0.9 nm was tested through the transparent medium. The results demonstrate that our system has a nanometer-level accuracy for thin film thickness measurement through transparent medium with optical path compensation.

Periodic Nonlinear Error Analysis and Compensation of a Single-Excited Petal-Shaped Capacitive Encoder to Achieve High-Accuracy Measurement.

The measurement results of a single-excitation petal-shaped capacitive encoder show strong periodic characteristics for nonlinear errors. This paper presents the analysis of periodic nonlinear errors in a single-excitation petal-shaped encoder in terms of three main aspects—sensitive structure processing error, circuit demodulation error, and installation error. Analytical and simulation results confirm that the first-, second-, and fourth-periodic electrical errors are caused by the misalignment of circuit parameters, non-uniform segmentation of the processing error, and cross interference of the electric field, respectively. Further experimental investigation reveals that the mechanical periodic error is caused by installation misalignment. Based on these analytical, simulation, and experimental results, the design of the capacitive encoder was optimized and a method based on harmonic components was applied to compensate the periodic nonlinear error of the encoder. Measurement results shows that the prototype which has 180 petal-shaped numbers can achieve a reduction of periodic nonlinear errors to less than 0.02° and its accuracy can be improved to 0.0006° after compensation over the full measurement range.

Nonlinear error analysis and experimental measurement of Birefringence-Zeeman dual-frequency laser interferometer

Abstract After a long-term research, the birefringence-Zeeman dual-frequency (BZDF) laser has entered the practical stage, and the BZDF laser interferometer has more powerful displacement and size measurement functions, especially the near-zero nonlinearity. This paper theoretically analyzes the mechanism of the near-zero nonlinear error, including theoretical simulation and experimental measurements. The measurement results show that the nonlinear error of the BZDF laser interferometer is only 0.3 nm, which is one order of magnitude smaller than that of traditional Zeeman dual-frequency (ZDF) laser interferometer. It has broad application prospects in the field of ultra-precision metrology.

Comparing Anisotropic Adaptive Strategies on the Second AIAA Sonic Boom Workshop Geometry

The recent release of the second AIAA sonic boom geometry offers the opportunity to review the classical anisotropic adaptive strategies for complex geometries. The design of Mach-aligned tailored grids is also a great challenge to see how adaptivity can compete with user-defined tailored grids. Two classical adaptive strategies, multiscale and goal-oriented, are compared with the results obtained on tailored grids. For the flow solver, several low-dissipation numerical schemes of order 4 and 6 are discussed with respect to regular second-order scheme both on inviscid and Reynolds-averaged Navier–Stokes flow models. Finally, a nonlinear error analysis is performed to assess the convergence of the sequence of adaptive meshes with respect to tailored grids. All results and discussions are based on the C25D baseline geometry provided for the Second AIAA Sonic Boom Prediction Workshop.

Study of Isothermal, Kinetic, and Thermodynamic Parameters for Adsorption of Cadmium: An Overview of Linear and Nonlinear Approach and Error Analysis.

Reports about presence and toxicity of Cd2+ in different chemical industrial effluents prompted the researchers to explore some economical, rapid, sensitive, and accurate methods for its determination and removal from aqueous systems. In continuation of series of investigations, adsorption of Cd2+ onto the stem of Saccharum arundinaceum is proposed in the present work. Optimization of parameters affecting sorption potential of Cd2+ including pH, contact time, temperature, sorbent dose, and concentration of sorbate was carried out to determine best suited conditions for maximum removal of sorbate. To understand the nature of sorption process, linear and nonlinear forms of five sorption isotherms including Freundlich and Langmuir models were employed. Feasibility and viability of sorption process were evaluated by calculating kinetics and thermodynamics of the process, while error analysis suggested best fitted sorption model on sorption data. Thermodynamic studies demonstrated exothermic nature of reaction, while kinetic studies suggested pseudo-second order of reaction.

Analysis of Nonlinear Error Caused by Motions of Rotation Axes for Five-Axis Machine Tools with Orthogonal Configuration

Since a nonlinear relationship exists between the position coordinates of the rotation axes and components of the tool orientation, the tool will deviate from the required plane, resulting in nonlinear errors and deterioration of machining accuracy. Few attempts have been made to obtain a general formula and common rules for nonlinear error because of the existence of various kinematic structures of machine tools with orthogonal configuration. This paper analyzes the relationship between the deviation of cutter location points and motion of tool orientations. Five-axis CNC machine tools are divided into two groups according to the configuration of the two rotational joints and the home position. Motions of the tool are regarded as a combination of translation and rotation. A model for error calculation is then built. The maximum deviation of the tool with respect to the reference plane generated by the initial and the final orientation is used to quantify the magnitude of the errors. General formulas are derived and common change rules are analyzed. Finally, machining experiment is conducted to validate the theoretical analysis. The research has important implications on the selection of a particular kinematic configuration that may achieve higher accuracy for a specific machining task.