Milling stability prediction based on the hybrid interpolation scheme of the Newton and Lagrange polynomials

Abstract

The stability lobe diagram (SLD) is commonly used to determine the suitable cutting parameters of the machining system in order to achieve a chatter-free machining process. An improved full discretization method (FDM) is proposed to predict the SLD based on the hybrid interpolation scheme of the Newton and Lagrange polynomials. In order to solve the SLD, a thirdorder Newton polynomial is employed to interpolate the state term of the physical space equation of the system. Meanwhile, to investigate the influence of the interpolation order on predicting the SLD, the delayed term is estimated using the Lagrange polynomials of orders one to four successively. Subsequently, after constructing the transition matrix, a series of calculation for the stability prediction are carried out by applying Floquet theory. The calculated results from these proposed methods demonstrate that the FDM with a second-order Lagrange polynomial is optimal, and further has better computational performance compared with some existing discretization methods. Lastly, the influences of the dynamic parameters on chatter stability are analyzed according to the proposed FDM. When the stiffness and damping ratio increase, the limit cutting depth will be enhanced. The increasing natural frequency not only causes an obvious shift of the lobes to the right, but also raises the limit cutting depth to some extent. These theoretical analyses can guide the prediction and improvement of the chatter stability of a machining system.

Prediction method of cutting stability of roadheader based on the Newton–Lagrange mixed discrete method

“…Niu et al 25 adopted the generalized Runge-Kutta method and time-domain simulation technology uncover the dynamic mechanism of distinct chatter behaviors in general milling scenarios, and obtained the distribution rule of chatter patterns in stability lobe diagrams for milling processes with general flute-spacing tools considering runout. Xia et al 26 improved the full discrete method by Newton and Lagrangian interpolation methods, and discussed the optimal order of interpolation by comparing and verifying the methods. The scholars also studied the prediction of surface roughness in cutting chatter machining through machine learning algorithm.…”

End-supporter path scheduling for robot-assisted asymmetrical support machining of thin-walled parts with non-equal thickness and closed section

“…Guo et al [4] combined theoretical analyses with milling tests, and obtained SLD based on milling dynamics model by semi-discretization method based on improved Runge-Kutta method. Xia et al [5] proposed an improved fully discretization method (FDM) to predict SLD based on a hybrid interpolation scheme of Newton and Lagrange polynomials, and analyzed the influence of dynamic parameters on chatter stability by using the proposed FDM. Karandikar et al [6] established the SLD of the milling process with unknown tool dynamics or material cutting force coefficients through a novel Bayesian learning approach, and obtained the optimal parameters.…”

An improved numerical integration method to predict the milling stability based on the Lagrange interpolation scheme

“…Compared with the zeroth-order SDM, the first-order FDM converged faster. Subsequently, the improved FDMs with higher computational accuracy were put forward by using the higher-order interpolation polynomials [19][20][21][22][23][24][25][26][27]. As a result, the second-order Lagrange polynomial [19,20], third-order Newton polynomial [21][22][23][24], and third-order Hermite polynomial [25,26] were separately employed to estimate the system state item.…”

“…Subsequently, the improved FDMs with higher computational accuracy were put forward by using the higher-order interpolation polynomials [19][20][21][22][23][24][25][26][27]. As a result, the second-order Lagrange polynomial [19,20], third-order Newton polynomial [21][22][23][24], and third-order Hermite polynomial [25,26] were separately employed to estimate the system state item. Correspondingly, the delayed item was interpolated by the secondorder or third-order Lagrange polynomial [20,24,25], the second-order Hermite polynomial [23], and the third-order Newton polynomial [22,26], respectively.…”

“…As a result, the second-order Lagrange polynomial [19,20], third-order Newton polynomial [21][22][23][24], and third-order Hermite polynomial [25,26] were separately employed to estimate the system state item. Correspondingly, the delayed item was interpolated by the secondorder or third-order Lagrange polynomial [20,24,25], the second-order Hermite polynomial [23], and the third-order Newton polynomial [22,26], respectively. Furthermore, the fourth-order and fifth-order methods were proposed to further increase the precision of the stability prediction [27,28].…”