scholarly journals An Investigation on using Lagrange, Newton and Least Square Methods for Generating Nonlinear Interpolation Function for the Measuring Instruments

2021 ◽  
Vol 14 ◽  
pp. 1-8
Author(s):  
Gouda Mohamed Mahmoud ◽  
Shaker Gelany
Keyword(s):  
Polynomial Function ◽  
Least Square Method ◽  
Polynomial Equation ◽  
Interpolation Polynomial ◽  
Least Square ◽  
Measuring Instruments ◽  
Interpolation Function ◽  
Realistic Approach ◽  
Nonlinear Interpolation

This research is considered the milestone for metrologists to choose the appropriate method for determination of the nonlinear interpolation function for the measuring instruments. Three methods of generating the interpolation polynomial equations were investigated; Newton, Lagrange, and Least Square method. The response of the measuring instruments under investigation was calculated and compared with the experimental results. Least Square method was found that it is the most accurate and most realistic approach to determine the interpolation polynomial function for the measuring instruments. It is recommended to use Least Square method rather than other methods to interpolating the polynomial equation. This recommendation is very important for metrologist as well as for measuring instruments applicant. This article is millstone to determine the response of the measuring instrument at non calibrated points in the calibrated range.

X-ray Diffractometric Determination of Lattice Misfit Between γ and γ' Phases in Ni-Base Superalloys

1988 ◽  
Vol 32 ◽  
pp. 365-375 ◽  
Author(s):  
Katsumi Ohno ◽  
Hirosi Harada ◽  
Toshihiro Yamagata ◽  
Michio Yamazaki ◽  
Kazumasa Ohsumi
Keyword(s):  
Lattice Misfit ◽  
Least Square Method ◽  
Least Square ◽  
X Rays ◽  
Deconvolution Method ◽  
X Ray ◽  
Window Functions ◽  
Profile Fitting ◽  
Overlapping Peaks

AbstractThe lattice misfits between γ and γ' phases in Ni-base superalloys (single crystal) were accurately determined for filings of specimens by using both a conventional X-ray tube focusing diffractometer(CXRFD) and a synchrotron-radiation parallel beam X-ray diffractometer (SRPXRD). All reflection peaks measured with the CXRFD were in a cluster of overlapping peaks because of the very small differences in the lattice parameters of both phases and the instrumental broadening due to X-ray optics including the spectral distribution of Xray source such as CuKα doublet. The deconvolution method was applied to remove the instrumental broadening from the peaks measured with the CXRFD. The window functions for the deconvolution method were calculated from CuKα doublet reflection of Si standard by a nonlinear least-square method.The instrumental broadening of SRPXRD was much smaller than that of CXRFD since the monochromatic X-rays produced single peak profiles and constant profile shape over a wide 2θ range. A profile fitting with a pseudo-Voigt function was used to determine 2θ angles to 0.0005 deg. for the synchrotron powder data. The peak angle and shape reflected from γ' phases in γ-matrix and those fron electrochemically extracted γ'-phase were significantly different.

Asymmetric Prandtl-Ishlinskii Hysteresis Model for Giant Magnetostrictive Actuator

2016 ◽  
Vol 20 (2) ◽  
pp. 223-230 ◽  
Author(s):  
Zhuoyun Nie ◽  
◽  
Chanjun Fu ◽  
Ruijuan Liu ◽  
Dongsheng Guo ◽  
...  
Keyword(s):  
Linear Function ◽  
Polynomial Function ◽  
Least Square Method ◽  
Least Square ◽  
Hysteresis Model ◽  
Hysteresis Behavior ◽  
Magnetostrictive Actuator ◽  
Simulation Results

An asymmetric Prandtl–Ishlinskii (API) hysteresis model for a giant magnetostrictive actuator (GMA) is proposed in this paper. The classical Prandtl–Ishlinskii (PI) model is analyzed and divided into two parts: linear function and operator summation. To enhance model asymmetry, a polynomial function is used in the API model as the center curve of the hysteresis instead of the linear function. The remaining curve of the hysteresis is modeled by a new operator that provides some basic asymmetric hysteresis. In this manner, the proposed API model requires relatively less operators and fewer parameters to describe the asymmetric hysteresis behavior of the GMA. All parameters of the API model are identified by a standard least square method. Simulation results show that the API model is very successful in formulating an asymmetric hysteresis of the GMA. In addition, it provides better identification results compared with the classical PI model.

The Application of Cubic Spline Interpolation Function in Determining Rock Rheological Long-Term Strength

2014 ◽  
Vol 580-583 ◽  
pp. 205-208
Author(s):  
Mo Li Zhao ◽  
Qiang Yong Zhang
Keyword(s):  
Spline Interpolation ◽  
Least Square Method ◽  
Cubic Spline ◽  
Least Square ◽  
Maximum Deviation ◽  
Interpolation Function ◽  
Term Strength ◽  
Cubic Spline Interpolation ◽  
Dagangshan Hydropower Station

The rheological long-term strength is determined according to the triaxial rheological test data of diabase at the dam area of Dagangshan Hydropower Station. Firstly, based on the stress-strain isochronous curve method and connected the test points with cubic spline interpolation function, the maximum deviation point in the long-term interval is determined as the turning point and established the long-term strength by nonlinear least square method. The results show that this method is consistent with the other methods. Finally, the advantage and disadvantage of this method is analyzed. This method can overcome the randomness of artificial selecting the turning points. Therefore, maximum deviation point method is relatively a reasonable and effective method to determine the rheological long-term strength of rock.

scholarly journals Numerical evaluation of the modulus of longitudinal elasticity in structural round timber elements of the Eucalyptus genus

2011 ◽  
Vol 31 (5) ◽  
pp. 1007-1014 ◽  
Author(s):  
André L. Christoforo ◽  
Túlio H. Panzera ◽  
Fabiano B. Batista ◽  
Paulo H. R. Borges ◽  
Francisco A. R. Lahr
Keyword(s):  
Inverse Analysis ◽  
Numerical Evaluation ◽  
Least Square Method ◽  
Least Square ◽  
Analysis Method ◽  
The Finite Element Method ◽  
Eucalyptus Citriodora ◽  
Inverse Analysis Method ◽  
Round Timber

Currently, the standards that deal with the determination of the properties of rigidity and strength for structural round timber elements do not take in consideration in their calculations and mathematical models the influence of the existing irregularities in the geometry of these elements. This study has as objective to determine the effective value of the modulus of longitudinal elasticity for structural round timber pieces of the Eucalyptus citriodora genus by a technique of optimization allied to the Inverse Analysis Method, to the Finite Element Method and the Least Square Method.

scholarly journals Determination of the Sediment Carrying Capacity Based on Perturbed Theory

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Zhi-hui Ni ◽  
Qiang Zeng ◽  
Wu Li-chun
Keyword(s):  
Carrying Capacity ◽  
Influencing Factor ◽  
Least Square Method ◽  
Least Square ◽  
New Method ◽  
Linear Regression Method ◽  
Average Flow Velocity ◽  
New Formula ◽  
Sediment Carrying Capacity

According to the previous studies of sediment carrying capacity, a new method of sediment carrying capacity on perturbed theory was proposed. By taking into account the average water depth, average flow velocity, settling velocity, and other influencing factors and introducing the median grain size as one main influencing factor in deriving the new formula, we established a new sediment carrying capacity formula. The coefficients were determined by the principle of dimensional analysis, multiple linear regression method, and the least square method. After that, the new formula was verified through measuring data of natural rivers and flume tests and comparing the verified results calculated by Cao Formula, Zhang Formula, Li Formula, Engelung-Hansen Formula, Ackers-White Formula, and Yang Formula. According to the compared results, it can be seen that the new method is of high accuracy. It could be a useful reference for the determination of sediment carrying capacity.

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