scholarly journals Noncontact Measurement for Radius of Curvature of Unpolished Lens

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Haifeng Liang
Keyword(s):  
Mathematical Model ◽  
Relative Error ◽  
Prediction Error ◽  
Convex Surface ◽  
Separation Distance ◽  
Radius Of Curvature ◽  
Maximum Relative Error ◽  
Spherical Lens ◽  
Parallel Lines ◽  
Lens Surface

A noncontact mathematical model to measure radius of curvature (ROC) of an unpolished spherical lens was proposed and also proved by experiments. This model gives ROC as a function of arcs radii and their separation distance, where the radii of the corresponding arcs could be acquired by taking coordinates of points on the arcs when two parallel lines of light project onto a lens surface. Our experiments demonstrated that the measured maximum relative error was 0.027% for a concave surface with a 38.19 mm ROC and 0.021% for a convex surface with a 97.75 mm ROC, which were all in agreement with those of theory prediction error. The suggested method presented a fast noncontact method for testing ROC of lens during coarse grinding and fine grinding.

Development of a Predictive Equation for Ventilation in a Wall-Solar Chimney System

2017 ◽  
Vol 139 (3) ◽  
Author(s):  
David Park ◽  
Francine Battaglia
Keyword(s):  
Mathematical Model ◽  
Relative Error ◽  
Natural Ventilation ◽  
Thermal Conditions ◽  
Ambient Air ◽  
Maximum Error ◽  
Scale Model ◽  
Small Scale ◽  
Maximum Relative Error ◽  
Solar Chimney

A solar chimney is a natural ventilation technique that has potential to save energy consumption as well as to maintain the air quality in a building. However, studies of buildings are often challenging due to their large sizes. The objective of this study was to determine the relationships between small- and full-scale solar chimney system models. Computational fluid dynamics (CFD) was employed to model different building sizes with a wall-solar chimney utilizing a validated model. The window, which controls entrainment of ambient air for ventilation, was also studied to determine the effects of window position. A set of nondimensional parameters were identified to describe the important features of the chimney configuration, window configuration, temperature changes, and solar radiation. Regression analysis was employed to develop a mathematical model to predict velocity and air changes per hour, where the model agreed well with CFD results yielding a maximum relative error of 1.2% and with experiments for a maximum error of 3.1%. Additional wall-solar chimney data were tested using the mathematical model based on random conditions (e.g., geometry, solar intensity), and the overall relative error was less than 6%. The study demonstrated that the flow and thermal conditions in larger buildings can be predicted from the small-scale model, and that the newly developed mathematical equation can be used to predict ventilation conditions for a wall-solar chimney.

scholarly journals Method of integro-differential equations for interpreting the results of vertical electrical sounding of the soil

2021 ◽  
pp. 67-70
Author(s):  
D. G. Koliushko ◽  
S. S. Rudenko ◽  
A. N. Saliba
Keyword(s):  
Mathematical Model ◽  
Relative Error ◽  
Power Plants ◽  
Current Source ◽  
Soil Surface ◽  
Observation Point ◽  
Practical Significance ◽  
Maximum Relative Error ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

The paper is devoted to the problem of determining the geoelectric structure of the soil within the procedure of testing the grounding arrangements of existing power plants and substations to the required depth in conditions of dense development. To solve the problem, it was proposed to use the Schlumbergers method , which has a greater sounding depth compared to the Wenner electrode array. The purpose of the work is to develop a mathematical model for interpreting the results of soil sounding by the Schlumberger method in the form of a four-layer geoelectric structure. Methodology. To construct a mathematical model, it is proposed to use the solution of a particular problem about the field of a point current source, which, like the observation point, is located in the first layer of a four-layer soil. Based on this expressions, a system of linear algebraic equations of the 7-th order with respect to the unknown coefficients ai and bi was compiled. On the basis of its analytical solution, an expression for the potential of the electric field was obtained for conducting VES (the point current source and the observation point are located only on the soil surface). Results. Comparison of the results of soil sounding by the Schlumberger installation and the interpretation of its results for the same points shows a sufficient degree of approximation: the maximum relative error does not exceed 9.7 % (for the second point), and the average relative error is 3.6 %. Originality. Based on the obtained expression, a test version of the program was implemented in Visual Basic for Applications to interpret the results of VES by the Schlumberger method. To check the obtained expressions, the interpretation of the VES results was carried out on the territory of a 150 kV substation of one of the mining and processing plants in the city of Kriviy Rih. Practical significance. The developed mathematical model will make it possible to increase the sounding depth, and, consequently, the accuracy of determining the standardized parameters of the grounding arrangements of power stations and substations.

Error-aware Design Procedure to Implement Energy-efficient Approximate Squaring Hardware

2020 ◽  
Vol 10 (4) ◽  
pp. 471-477
Author(s):  
Merin Loukrakpam ◽  
Ch. Lison Singh ◽  
Madhuchhanda Choudhury
Keyword(s):  
Energy Saving ◽  
Relative Error ◽  
Energy Efficient ◽  
Piecewise Linear ◽  
Arithmetic Operation ◽  
Design Procedure ◽  
Digital Signal ◽  
Maximum Relative Error ◽  
Square Function ◽  
Efficient Design

Background:: In recent years, there has been a high demand for executing digital signal processing and machine learning applications on energy-constrained devices. Squaring is a vital arithmetic operation used in such applications. Hence, improving the energy efficiency of squaring is crucial. Objective:: In this paper, a novel approximation method based on piecewise linear segmentation of the square function is proposed. Methods: Two-segment, four-segment and eight-segment accurate and energy-efficient 32-bit approximate designs for squaring were implemented using this method. The proposed 2-segment approximate squaring hardware showed 12.5% maximum relative error and delivered up to 55.6% energy saving when compared with state-of-the-art approximate multipliers used for squaring. Results: The proposed 4-segment hardware achieved a maximum relative error of 3.13% with up to 46.5% energy saving. Conclusion:: The proposed 8-segment design emerged as the most accurate squaring hardware with a maximum relative error of 0.78%. The comparison also revealed that the 8-segment design is the most efficient design in terms of error-area-delay-power product.

A Meshless Local Petrov-Garlerkin Method for Solving the Biharmonic Equation

2014 ◽  
Vol 931-932 ◽  
pp. 1488-1494
Author(s):  
Supanut Kaewumpai ◽  
Suwon Tangmanee ◽  
Anirut Luadsong
Keyword(s):  
Relative Error ◽  
Biharmonic Equation ◽  
Interpolation Method ◽  
Step Function ◽  
Special Treatment ◽  
Maximum Relative Error ◽  
Heaviside Step Function ◽  
Treatment Techniques ◽  
Point Interpolation Method ◽  
Point Interpolation

A meshless local Petrov-Galerkin method (MLPG) using Heaviside step function as a test function for solving the biharmonic equation with subjected to boundary of the second kind is presented in this paper. Nodal shape function is constructed by the radial point interpolation method (RPIM) which holds the Kroneckers delta property. Two-field variables local weak forms are used in order to decompose the biharmonic equation into a couple of Poisson equations as well as impose straightforward boundary of the second kind, and no special treatment techniques are required. Selected engineering numerical examples using conventional nodal arrangement as well as polynomial basis choices are considered to demonstrate the applicability, the easiness, and the accuracy of the proposed method. This robust method gives quite accurate numerical results, implementing by maximum relative error and root mean square relative error.

scholarly journals Evaluation of the Undrained Shear Strength of Organic Soils from a Dilatometer Test Using Artificial Neural Networks

2018 ◽  
Vol 8 (8) ◽  
pp. 1395 ◽  
Author(s):  
Zbigniew Lechowicz ◽  
Masaharu Fukue ◽  
Simon Rabarijoely ◽  
Maria Sulewska
Keyword(s):  
Neural Networks ◽  
Shear Strength ◽  
Artificial Neural Networks ◽  
Relative Error ◽  
Undrained Shear Strength ◽  
Organic Soils ◽  
Maximum Relative Error ◽  
Empirical Methods ◽  
Pressure Reading ◽  
Undrained Shear

The undrained shear strength of organic soils can be evaluated based on measurements obtained from the dilatometer test using single- and multi-factor empirical correlations presented in the literature. However, the empirical methods may sometimes show relatively high values of maximum relative error. Therefore, a method for evaluating the undrained shear strength of organic soils using artificial neural networks based on data obtained from a dilatometer test and organic soil properties is presented in this study. The presented neural network, with an architecture of 5-4-1, predicts the normalized undrained shear strength based on five independent variables: the normalized net value of a corrected first pressure reading (po − uo)/σ′v, the normalized net value of a corrected second pressure reading (p1 − uo)/σ′v, the organic content Iom, the void ratio e, and the stress history indictor (oc or nc). The neural model presented in this study provided a more reliable prediction of the undrained shear strength in comparison to the empirical methods, with a maximum relative error of ±10%.

scholarly journals Simultaneous Measurement of BRDF and Surface Curvature by using Pattern Illumination

2020 ◽  
Vol 2020 (28) ◽  
pp. 356-360
Author(s):  
Shinichi Inoue ◽  
Norimichi Tsumura
Keyword(s):  
Simultaneous Measurement ◽  
Convex Surface ◽  
Incident Light ◽  
Radius Of Curvature ◽  
Measurement Apparatus ◽  
Planar Surfaces ◽  
Angle Of Reflection ◽  
Bidirectional Reflection Distribution Function ◽  
Bidirectional Reflection ◽  
Normal Angle

In this study, we propose the simultaneous measurement method of the bidirectional reflection distribution function (BRDF) and the radius of curvature by using pattern illumination. For nonplanar objects, the angle of reflection light changes according to the surface normal angle of curved object. Therefore, it is necessary to consider the effects of curved surfaces when measuring the BRDF on non-planar surfaces. We suppose a convex surface that can be represented by a constant radius of curvature. The pattern of illumination was generated by placing the illumination mask with pattern apertures in the incident light path of the BRDF measurement apparatus in which the incident light is collimated light. We developed the measurement apparatus. We measured four types of sample with different BRDFs on three different radiuses of curvature. The results showed that the BRDF and the radius of curvature can be measured simultaneously by using the pattern illumination.

Constant Speed Motion Simulation and Error Analysis of Planar Non-Circular Curve Parts

2012 ◽  
Vol 220-223 ◽  
pp. 1559-1563
Author(s):  
Rui Wu ◽  
Li Bao ◽  
Yuan Kui Xu
Keyword(s):  
Mathematical Model ◽  
Error Analysis ◽  
Relative Error ◽  
Plane Curve ◽  
Simulation System ◽  
Constant Speed ◽  
Motion Simulation ◽  
Production Practice ◽  
The Mathematical Model ◽  
Circular Curve

The relative direction for a constant speed can be determined according to the planar non-circular curve parts. To establish the mathematical model, a constant speed motion simulation system is designed. The parameters of (vH=5mm/s, δ<3") is commonly used for the simulation system to simulate the movement of drawing the error curve. The results show that by controlling the movement of the plane curve parts in mathematical model can derive the basic constant speed, the relative error of constant speed is less than 3%, it provides a reliable bias when apply to production practice.

Magnetic Resonance Imaging to Investigate the Clinical Applicability of the Medial Femoral Trochlea Osteochondral Flap Within the Wrist

Hand ◽  
2022 ◽  
pp. 155894472110643
Author(s):  
Jimmy H. Daruwalla ◽  
Jan Skrok ◽  
Mitchell A. Pet ◽  
Aviram M. Giladi ◽  
James P. Higgins
Keyword(s):  
Magnetic Resonance Imaging ◽  
Magnetic Resonance ◽  
Sagittal Plane ◽  
Convex Surface ◽  
Radius Of Curvature ◽  
Resonance Imaging ◽  
Femoral Trochlea ◽  
Average Ratio ◽  
Mri Scans ◽  
Magnetic Resonance Imaging Mri

Background: The medial femoral trochlea (MFT) osteochondral flap is employed for reconstruction of unsalvageable scaphoid proximal pole nonunions. The convex surface of the cartilage-bearing proximal trochlea is used to replace the similarly contoured proximal scaphoid and articulate with the concave scaphoid fossa of the radius. A magnetic resonance imaging (MRI) comparison of the shape of the MFT as it relates to the native proximal scaphoid has not been previously performed. Our study aimed to quantifiably compare the shape of the MFT, proximal scaphoid, and scaphoid fossa. Methods: Using imaging processing software, we measured radius of curvature of the articular segments in MRI scans of 10 healthy subjects’ wrists and knees. Results: Compared with the scaphoid fossa, average ratio of the radius of circumference of the proximal scaphoid was 0.79 and 0.78 in the coronal and sagittal planes, respectively. Compared with the scaphoid fossa, average ratio of the radius of circumference of the MFT was 0.98 and 1.31 in the coronal and sagittal planes, respectively. The radius of curvature of the MFT was larger than the proximal scaphoid, in the coronal and sagittal planes. In the coronal plane, the MFT radius of curvature is nearly identical to the scaphoid fossa, a closer match than the scaphoid itself. In the sagittal plane, the radius of curvature of the MFT was larger than the radius of curvature of the scaphoid fossa. Conclusions: Our data suggest that the radius of curvature, in the sagittal and coronal planes, of the MFT and proximal scaphoid is disparate.

scholarly journals Chaotic prediction of vibration performance degradation trend of rolling element bearing based on Weibull distribution

2019 ◽  
Vol 103 (1) ◽  
pp. 003685041989219
Author(s):  
Li Cheng ◽  
Xintao Xia ◽  
Liang Ye
Keyword(s):  
Weibull Distribution ◽  
Relative Error ◽  
High Reliability ◽  
Prediction Method ◽  
Performance Degradation ◽  
Rolling Element Bearing ◽  
Maximum Relative Error ◽  
Rolling Element Bearings ◽  
Rolling Element ◽  
Vibration Performance

Rolling element bearings are used in all rotating machinery, and the degradation performance of rolling element bearings directly affects the performance of the machine. Therefore, high reliability prediction of the performance degradation trend of rolling element bearings has become an urgent research problem. However, the degradation characteristics of the rolling element bearings vibration time series are difficult to extract, and the mechanism of performance degradation is very complicated. The accurate physical model is difficult to establish. In view of the above reasons, based on the vibration performance data of rolling element bearings, a model of bearing performance degradation trend parameter based on wavelet denoising and Weibull distribution is established. Then, the phase space reconstruction of the series of bearing performance degradation trend parameter is carried out, and the prognosis is obtained by the improved adding weighted first-order local prediction method. The experimental results show that the bearing vibration performance degradation parameter can accurately depict the degradation trend of the bearing, and the reliability level is 91.55%; and the prediction of bearing performance degradation trend parameter is satisfactory: the mean relative error is only 0.0053% and the maximum relative error is less than 0.03%.

scholarly journals Two-Machine Job-Shop Scheduling Problem to Minimize the Makespan with Uncertain Job Durations

Algorithms ◽  
2019 ◽  
Vol 13 (1) ◽  
pp. 4 ◽  
Author(s):  
Yuri N. Sotskov ◽  
Natalja M. Matsveichuk ◽  
Vadzim D. Hatsura
Keyword(s):  
Relative Error ◽  
Job Shop ◽  
Sufficient Conditions ◽  
Upper Bounds ◽  
Maximum Relative Error ◽  
Scheduling Problems ◽  
Lower And Upper Bounds ◽  
Shop Scheduling ◽  
Job Duration ◽  
On Line

We study two-machine shop-scheduling problems provided that lower and upper bounds on durations of n jobs are given before scheduling. An exact value of the job duration remains unknown until completing the job. The objective is to minimize the makespan (schedule length). We address the issue of how to best execute a schedule if the job duration may take any real value from the given segment. Scheduling decisions may consist of two phases: an off-line phase and an on-line phase. Using information on the lower and upper bounds for each job duration available at the off-line phase, a scheduler can determine a minimal dominant set of schedules (DS) based on sufficient conditions for schedule domination. The DS optimally covers all possible realizations (scenarios) of the uncertain job durations in the sense that, for each possible scenario, there exists at least one schedule in the DS which is optimal. The DS enables a scheduler to quickly make an on-line scheduling decision whenever additional information on completing jobs is available. A scheduler can choose a schedule which is optimal for the most possible scenarios. We developed algorithms for testing a set of conditions for a schedule dominance. These algorithms are polynomial in the number of jobs. Their time complexity does not exceed O ( n 2 ) . Computational experiments have shown the effectiveness of the developed algorithms. If there were no more than 600 jobs, then all 1000 instances in each tested series were solved in one second at most. An instance with 10,000 jobs was solved in 0.4 s on average. The most instances from nine tested classes were optimally solved. If the maximum relative error of the job duration was not greater than 20 % , then more than 80 % of the tested instances were optimally solved. If the maximum relative error was equal to 50 % , then 45 % of the tested instances from the nine classes were optimally solved.

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