Comparison of the full-discretization methods for milling stability analysis by using different high-order polynomials to interpolate both state term and delayed term

Abstract

This paper compares different full-discretization methods for milling stability analysis by using different high-order polynomials to interpolate both state term and delayed term (HFDMs) from the aspects of accuracy and efficiency. The dynamic model of milling process with consideration of regeneration effect is described by time periodic delay-differential equation (DDE) in the state-space. Different high-order interpolation polynomials are used to approximate the state term and delayed term. The state transition matrix is obtained on the basis of direct integration scheme. The rates of convergence of different HFDMs are compared with those of the benchmark methods using different process parameter points, the results indicate that it is difficult to evaluate the accuracy of different HFDMs through the convergence rate analysis of limited process parameter points. Then, mean differences and variances between the referenced and predicted critical depths of cut are employed for accuracy analysis. The 3rd-2nd HFDM is, on the whole, proved to be more accurate than the other methods. The efficiencies of different HFDMs are also verified through time-consuming study for both single degree of freedom (1-DOF) and two degree of freedom (2-DOF) milling system. The 3rd-2nd HFDM is proved to be an efficient method by comparing with the other methods. Besides, the HFDMs are available for predicting the stability lobe diagrams under both large immersion condition and low immersion condition.

Appendix

Combining Eqs. (3), (10), and (18), a discrete mapping can be obtained using 3rd-2nd HFDM as follow

$$ {\mathbf{x}}_{i+1}=\left(\mathbf{I}-{\mathbf{G}}_{37}{\mathbf{B}}_{\mathrm{i}+1}-{\mathbf{G}}_{38}{\mathbf{B}}_{\mathrm{i}}\right)\left[\begin{array}{l}\left({\mathbf{G}}_{35}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{36}{\mathbf{B}}_i+{\mathbf{F}}_0\right){\mathbf{x}}_i+\left({\mathbf{G}}_{33}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{34}{\mathbf{B}}_i\right){\mathbf{x}}_{i-1}+\\ {}\left({\mathbf{G}}_{31}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{32}{\mathbf{B}}_i\right){\mathbf{x}}_{i-2}-\left({\mathbf{H}}_{21}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{22}{\mathbf{B}}_i\right){\mathbf{x}}_{i+2-n}\\ {}-\left({\mathbf{H}}_{23}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{24}{\mathbf{B}}_i\right){\mathbf{x}}_{i+1-n}-\left({\mathbf{H}}_{25}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{26}{\mathbf{B}}_i\right){\mathbf{x}}_{i-n}\end{array}\right] $$

(38)

Combining Eqs. (3), (10), and (21), a discrete mapping can be obtained using 3rd-3rd HFDM as follow

$$ {\mathbf{x}}_{i+1}=\left(\mathbf{I}-{\mathbf{G}}_{37}{\mathbf{B}}_{\mathrm{i}+1}-{\mathbf{G}}_{38}{\mathbf{B}}_{\mathrm{i}}\right)\left[\begin{array}{l}\left({\mathbf{G}}_{35}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{36}{\mathbf{B}}_i+{\mathbf{F}}_0\right){\mathbf{x}}_i+\left({\mathbf{G}}_{33}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{34}{\mathbf{B}}_i\right){\mathbf{x}}_{i-1}+\\ {}\left({\mathbf{G}}_{31}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{32}{\mathbf{B}}_i\right){\mathbf{x}}_{i-2}-\left({\mathbf{H}}_{31}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{32}{\mathbf{B}}_i\right){\mathbf{x}}_{i+3-n}-\\ {}\left({\mathbf{H}}_{33}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{34}{\mathbf{B}}_i\right){\mathbf{x}}_{i+2-n}-\left({\mathbf{H}}_{35}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{36}{\mathbf{B}}_i\right){\mathbf{x}}_{i+1-n}-\\ {}\left({\mathbf{H}}_{37}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{38}{\mathbf{B}}_i\right){\mathbf{x}}_{i-n}\end{array}\right] $$

(39)

Combining Eqs. (3), (10), and (24), a discrete mapping can be obtained using 3rd-4th HFDM as follow

$$ {\mathbf{x}}_{i+1}=\left(\mathbf{I}-{\mathbf{G}}_{37}{\mathbf{B}}_{\mathrm{i}+1}-{\mathbf{G}}_{38}{\mathbf{B}}_{\mathrm{i}}\right)\left[\begin{array}{l}\left({\mathbf{G}}_{35}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{36}{\mathbf{B}}_i+{\mathbf{F}}_0\right){\mathbf{x}}_i+\left({\mathbf{G}}_{33}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{34}{\mathbf{B}}_i\right){\mathbf{x}}_{i-1}+\\ {}\left({\mathbf{G}}_{31}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{32}{\mathbf{B}}_i\right){\mathbf{x}}_{i-2}-\left({\mathbf{H}}_{41}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{42}{\mathbf{B}}_i\right){\mathbf{x}}_{i+4-n}-\\ {}\left({\mathbf{H}}_{43}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{44}{\mathbf{B}}_i\right){\mathbf{x}}_{i+3-n}-\left({\mathbf{H}}_{45}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{46}{\mathbf{B}}_i\right){\mathbf{x}}_{i+2-n}-\\ {}\left({\mathbf{H}}_{47}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{48}{\mathbf{B}}_i\right){\mathbf{x}}_{i+1-n}-\left({\mathbf{H}}_{49}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{410}{\mathbf{B}}_i\right){\mathbf{x}}_{i-n}\end{array}\right] $$

(40)

Combining Eqs. (3), (14), and (18), a discrete mapping can be obtained using 4th-2nd HFDM as follow

$$ {\mathbf{x}}_{i+1}=\left(\mathbf{I}-{\mathbf{G}}_{49}{\mathbf{B}}_{\mathrm{i}+1}-{\mathbf{G}}_{410}{\mathbf{B}}_{\mathrm{i}}\right)\left[\begin{array}{l}\left({\mathbf{G}}_{47}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{48}{\mathbf{B}}_i+{\mathbf{F}}_0\right){\mathbf{x}}_i+\left({\mathbf{G}}_{45}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{46}{\mathbf{B}}_i\right){\mathbf{x}}_{i-1}+\\ {}\left({\mathbf{G}}_{43}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{44}{\mathbf{B}}_i\right){\mathbf{x}}_{i-2}+\left({\mathbf{G}}_{41}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{42}{\mathbf{B}}_i\right){\mathbf{x}}_{i-3}-\\ {}\left({\mathbf{H}}_{21}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{22}{\mathbf{B}}_i\right){\mathbf{x}}_{i+2-n}-\left({\mathbf{H}}_{23}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{24}{\mathbf{B}}_i\right){\mathbf{x}}_{i+1-n}\\ {}-\left({\mathbf{H}}_{25}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{26}{\mathbf{B}}_i\right){\mathbf{x}}_{i-n}\end{array}\right] $$

(41)

Combining Eqs. (3), (14), and (21), a discrete mapping can be obtained using 4th-3rd HFDM as follow

$$ {\mathbf{x}}_{i+1}=\left(\mathbf{I}-{\mathbf{G}}_{49}{\mathbf{B}}_{\mathrm{i}+1}-{\mathbf{G}}_{410}{\mathbf{B}}_{\mathrm{i}}\right)\left[\begin{array}{l}\left({\mathbf{G}}_{47}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{48}{\mathbf{B}}_i+{\mathbf{F}}_0\right){\mathbf{x}}_i+\left({\mathbf{G}}_{45}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{46}{\mathbf{B}}_i\right){\mathbf{x}}_{i-1}+\\ {}\left({\mathbf{G}}_{43}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{44}{\mathbf{B}}_i\right){\mathbf{x}}_{i-2}+\left({\mathbf{G}}_{41}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{42}{\mathbf{B}}_i\right){\mathbf{x}}_{i-3}-\\ {}\left({\mathbf{H}}_{31}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{32}{\mathbf{B}}_i\right){\mathbf{x}}_{i+3-n}-\left({\mathbf{H}}_{33}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{34}{\mathbf{B}}_i\right){\mathbf{x}}_{i+2-n}-\\ {}\left({\mathbf{H}}_{35}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{36}{\mathbf{B}}_i\right){\mathbf{x}}_{i+1-n}-\left({\mathbf{H}}_{37}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{38}{\mathbf{B}}_i\right){\mathbf{x}}_{i-n}\end{array}\right] $$

(42)

Combining Eqs. (3), (14), and (24), a discrete mapping can be obtained using 4th-4th HFDM as follow

$$ {\mathbf{x}}_{i+1}=\left(\mathbf{I}-{\mathbf{G}}_{49}{\mathbf{B}}_{\mathrm{i}+1}-{\mathbf{G}}_{410}{\mathbf{B}}_{\mathrm{i}}\right)\left[\begin{array}{l}\left({\mathbf{G}}_{47}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{48}{\mathbf{B}}_i+{\mathbf{F}}_0\right){\mathbf{x}}_i+\left({\mathbf{G}}_{45}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{46}{\mathbf{B}}_i\right){\mathbf{x}}_{i-1}+\\ {}\left({\mathbf{G}}_{43}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{44}{\mathbf{B}}_i\right){\mathbf{x}}_{i-2}+\left({\mathbf{G}}_{41}{\mathbf{B}}_{i+1}+{\mathbf{G}}_{42}{\mathbf{B}}_i\right){\mathbf{x}}_{i-3}-\\ {}\left({\mathbf{H}}_{41}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{42}{\mathbf{B}}_i\right){\mathbf{x}}_{i+4-n}-\left({\mathbf{H}}_{43}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{44}{\mathbf{B}}_i\right){\mathbf{x}}_{i+3-n}-\\ {}\left({\mathbf{H}}_{45}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{46}{\mathbf{B}}_i\right){\mathbf{x}}_{i+2-n}-\left({\mathbf{H}}_{47}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{48}{\mathbf{B}}_i\right){\mathbf{x}}_{i+1-n}\\ {}\left({\mathbf{H}}_{49}{\mathbf{B}}_{i+1}+{\mathbf{H}}_{410}{\mathbf{B}}_i\right){\mathbf{x}}_{i-n}\end{array}\right] $$

(43)