An improved three-point method based on a difference algorithm

Huang, Ran; Pan, Wei; Lu, Changhou; Zhang, Yixin; Chen, Shujiang

doi:10.1016/j.precisioneng.2020.01.008

Cited by 6 publications

(5 citation statements)

References 17 publications

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“…The barycentric Lagrange interpolation of a four variable function can be written as By combining (31) and ( 14), and substituting the interpolation nodes (x p , y q , z v , t r ) , we can obtain Equation (32) can be written as matrix form and further abbreviated as where By combining (31) and (15), and substituting the interpolation nodes (x p , y q , z v , t r ) , we can obtain…”

Section: (3+1) Dimensional Nonlinear Wave Equation Directly Linear It...mentioning

confidence: 99%

“…Among them, inverse scattering method [6], straight line method based on cubic recursion relation [7], hyperbolic tangent function expansion method [8], Laplace transform method [9] are studied to obtain the approximate solution of Sine/Klein-Gordon equation. Homotopy analysis method [10,11], optimal homotopy analysis method [12], finite difference method [13][14][15], direct meshless local Petrov-Galerkin method [16,17] are used to study some nonlinear wave problems in engineering problems. In addition, there are numerical methods such as modified simple equation (MSE) method [18] and spectral method based on Legendere wavelet basis [19].…”

Section: Introductionmentioning

confidence: 99%

“…The error of the threepoint central difference method[15] at t = 3Δt and the order of convergence of E a in Example 4…”

mentioning

confidence: 99%

See 2 more Smart Citations

Solving Nonlinear Wave Equations Based on Barycentric Lagrange Interpolation

Yuan,

Wang,

2024

J Nonlinear Math Phys

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In this paper, we deeply study the high-precision barycentric Lagrange interpolation collocation method to solve nonlinear wave equations. Firstly, we introduce the barycentric Lagrange interpolation and provide the differential matrix. Secondly, we construct a direct linearization iteration scheme to solve nonlinear wave equations. Once again, we use the barycentric Lagrange interpolation to approximate the (2+1) dimensional nonlinear wave equations and (3+1) dimensional nonlinear wave equations, and describe the matrix format for direct linearization iteration of the nonlinear wave equations. Finally, the comparative experiments show that the barycentric Lagrange interpolation collocation method for solving nonlinear wave equations have higher calculation accuracy and convergence rate.

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Section: (3+1) Dimensional Nonlinear Wave Equation Directly Linear It...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Solving Nonlinear Wave Equations Based on Barycentric Lagrange Interpolation

Yuan,

Wang,

2024

J Nonlinear Math Phys

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“…Gao et al [ 22 , 23 ] proposed several new multi-probe methods, which can effectively separate the roundness of the measured workpiece from the spindle radial error, thereby avoiding the problem of harmonic suppression. In addition, some scholars have also systematically studied the three-point method [ 11 , 24 , 25 ], but most of them have only carried out theoretical research and verified it in the laboratory environment. To sum up, many scholars have conducted in-depth research on the three-point method, but there are still few practical applications of the three-point method in ultra-precision machining measurement.…”

Section: Introductionmentioning

confidence: 99%

In Situ Measurement of Spindle Radial Error for Ultra-Precision Machining Based on Three-Point Method

Sun

Dai

et al. 2023

Micromachines

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The radial error is an important parameter to evaluate the performance of ultra-precision spindles. The three-point method has not yet been well applied in nanometer-scale measurement due to its disadvantages of harmonic suppression and the complicated error separation process. In order to verify that the three-point method can realize the nanometer-scale measurement of the radial error in the machining environment, an in situ measurement and evaluation system is established. Experiments are performed using the system, and a comparative experiment is conducted to verify the accuracy of the system. The average value and standard deviation of the measurement results are 23.096 nm and 0.556 nm, respectively. The in situ measurement result was in good agreement with the Donaldson reversal method using a commercially available spindle analyzer.

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“…Several methods for efficiently separating the artifact’s form error signals from the runout signals can be used: reversal, 16 multistep, 17,18 and multi probe methods. 11–13,15,16,19–21 Compared with other methods, the multi probe method allows for real-time or on-machine measurements without additional precision devices.…”

Section: Introductionmentioning

confidence: 99%

Measurement of five-degrees-of-freedom spindle error motion using multi probe error separation

Baek

Hwang

Cho

2021

Proceedings of the Institution of Mechanical Engineers, Part B:

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Herein, a precision measurement method is proposed to evaluate the radial, axial, and tilt error motions of rotating devices such as precision spindles. To improve the accuracy of the estimated multiple-degree-of-freedom error motion components, form error signals in the runout signals are separated and precompensated for before the calculation of the five-degrees-of-freedom error motion components. Fourier model-based multi probe error separation (MPES) techniques, which can prevent the occurrence of harmonic distortions, are applied to separate the form error signals. A three-probe method is applied to layers on the side surface of the cylindrical artifact, and a modified two-probe method is developed and applied to the upper surface. The radial, tilt, and axial error motions are calculated using the runout signals that do not contain the separated form error signals. The measurement system uses eight capacitive probes to detect the runout signals of the cylindrical artifact mounted at the center of the rotating device. To compare the proposed method with the five-probe-based conventional measuring method, an evaluation test simulation is conducted repeatedly five times. Results indicate that the proposed MPES method calculated the uncertainty using the deviation between the computed results from the existing and novel methods; additionally, the input signal in terms of the radial, tilt (layers 1 and 2), and axial error motions are [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text], respectively. It is confirmed that the undesirable effects of the form error signals are successfully removed and that the accuracies of the measured spindle error motion components improve.

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An improved three-point method based on a difference algorithm

Cited by 6 publications

References 17 publications

Solving Nonlinear Wave Equations Based on Barycentric Lagrange Interpolation

Solving Nonlinear Wave Equations Based on Barycentric Lagrange Interpolation

In Situ Measurement of Spindle Radial Error for Ultra-Precision Machining Based on Three-Point Method

Measurement of five-degrees-of-freedom spindle error motion using multi probe error separation

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