XiaoJian Zhang

Stability Analysis of Milling Via the Differential Quadrature Method

This paper presents a time-domain semi-analytical method for stability analysis of milling in the framework of the differential quadrature method. The governing equation of milling processes taking into account the regenerative effect is formulated as a linear periodic delayed differential equation (DDE) in state space form. The tooth passing period is first separated as the free vibration duration and the forced vibration duration. As for the free vibration duration, the analytical solution is available. As for the forced vibration duration, this time interval is discretized by sampling grid points. Then, the differential quadrature method is employed to approximate the time derivative of the state function at a sampling grid point within the forced vibration duration by a weighted linear sum of the function values over the whole sampling grid points. The Lagrange polynomial based algorithm (LPBA) and trigonometric functions based algorithm (TFBA) are employed to obtain the weight coefficients. Thereafter, the DDE on the forced vibration duration is discretized as a series of algebraic equations. By combining the analytical solution of the free vibration duration and the algebraic equations of the forced vibration duration, Floquet transition matrix can be constructed to determine the milling stability according to Floquet theory. Simulation results and experimentally validated examples are utilized to demonstrate the effectiveness and accuracy of the proposed approach.

Stability analysis in milling of thin-walled workpieces with emphasis on the structural effect

Abstract Regenerative chatter easily occurs in milling and has become the common limitation to achieve good surface quality and high productivity. For the purpose of chatter avoidance, the structural effect of the thin-walled part should be considered for the milling chatter stability analysis for the optimization of axial cutting depth and spindle speed pairs. The main objective of this paper is to examine the link between the structural modes (i.e. modal shapes) and the chatter stability limits in the case of finish milling thin-walled workpieces. In this paper, the dynamic stability of the milling process of thin-walled workpieces is investigated through a two-degree-of-freedom mechanical model. The mathematical relationship between the critical axial depth and the thin-walled part modal shapes is deduced and an optimal calculation process of milling stability lobes is presented. Peripheral milling of aluminium alloy (2A70 Al) plates is carried out on a computer numerically controlled (CNC) five-axis super high-speed machining centre to validate the method. The experimental results agree with the prediction by the presented method. Additionally, the experimental results show that the cutting stability is also influenced by the modal frequencies of the thin-walled part, which have a great influence on the milling stability analysis when the tool passing frequency (i.e. the inverse of the tooth passing period) harmonics are close to the modal frequencies of the part. The presented method is effective in the prediction of milling chatter limits in the thin-walled case for the optimization of machining parameters.

Improved full-discretization method for milling chatter stability prediction with multiple delays

This paper presents an analytical stability prediction method in milling chatter analysis with multiple delays. The improved full-discretization method is used to determine the critical boundary of chatter stability. The proposed method can take many practical factors into consideration, such as helix angle, non-constant pitch, and cutter runout, etc. The influence of the multiple delays is explored in detail and the simulation results show that the cutter runout has the great influence on the elevation of the stability boundary. The finding proves the validity of the proposed method by experiment verification.

A Legendre Polynomials Based Method for Stability Analysis of Milling Processes

This paper presents a time-domain approach for a semi-analytical prediction of stability in milling using the Legendre polynomials. The governing equation of motion of milling processes is expressed as a delay-differential equation (DDE) with time periodic coefficients. After the DDE being re-expressed in state-space form, the state vector is approximated by a series of Legendre polynomials. With the help of the Legendre–Gauss–Lobatto (LGL) quadrature, a discrete dynamic map is formulated to approximate the original DDE, and utilized to predict the milling stability based on Floquet theory. With numerical examples illustrating the efficiency and accuracy of the proposed approach, an experimental example validates the method.

A new solution for stability prediction in flexible part milling

The machining instability (chatter phenomenon) easily occurs when flexible parts are machined. This paper predicts the milling stability domain of the flexible part with multiple structure mode interaction induced by the cutting force. First, the dynamic milling process of a flexible part is modeled as a multiple modal degree-of-freedom mechanical model. Then, the full-discretization method is employed to calculate the stability boundary and the cases with different factors are compared, the simulation shows that the cutting position has the dominating effect on the stability analysis of the flexible part milling. The chatter experiment verifies the validity of the proposed method. The proposed method can be used to improve the machining efficiency of flexible parts in aerospace and power industries.

Response sensitivity analysis of the dynamic milling process based on the numerical integration method

As one of the bases of gradient-based optimization algorithms, sensitivity analysis is usually required to calculate the derivatives of the system response with respect to the machining parameters. The most widely used approaches for sensitivity analysis are based on time-consuming numerical methods, such as finite difference methods. This paper presents a semi-analytical method for calculation of the sensitivity of the stability boundary in milling. After transforming the delay-differential equation with time-periodic coefficients governing the dynamic milling process into the integral form, the Floquet transition matrix is constructed by using the numerical integration method. Then, the analytical expressions of derivatives of the Floquet transition matrix with respect to the machining parameters are obtained. Thereafter, the classical analytical expression of the sensitivity of matrix eigenvalues is employed to calculate the sensitivity of the stability lobe diagram. The two-degree-of-freedom milling example illustrates the accuracy and efficiency of the proposed method. Compared with the existing methods, the unique merit of the proposed method is that it can be used for analytically computing the sensitivity of the stability boundary in milling, without employing any finite difference methods. Therefore, the high accuracy and high efficiency are both achieved. The proposed method can serve as an effective tool for machining parameter optimization and uncertainty analysis in high-speed milling.

Spectral method for prediction of chatter stability in low radial immersion milling

The aim of this paper is to develop an integral equation based spectral method for prediction of chatter stability in low radial immersion milling. First, the delay-differential equation with time-periodic coefficients governing the dynamic milling process is transformed into the integral equation. Then, the duration of one tooth period is divided into the free vibration and the forced vibration processes. While the former one has an analytical solution, the discretization technique is explored to approximate the solution of the latter one. After the forced vibration duration being equally discretized, the Gauss-Legendre formula is used to discretize the definite integral, in the meantime the Lagrange interpolation is adopted for approximating the state item and the time-delay item by using the corresponding discretized state points and time-delay state points. The approximate Floquet transition matrix is thereafter constructed to predict the milling stability based on the Floquet theory. The benchmark examples are utilized to verify the proposed method. Compared with previous time domain methods, the proposed method enables higher rate of convergence. The results also demonstrate that the proposed method is high-effective.

Dynamic cutter runout measurement with laser sensor

The cutter runout is very common in machine milling and has a great effect on the surface accuracy. In this paper, a measurement of radial cutter runout in revolving milling tool is proposed by using the laser sensor. A laser beam is projected onto the milling tool edge and subsequently reflected. The diffuse reflection is captured by the sensor and the displacement between the cutter and the laser sensor is obtained. Based on the dynamic displacement, the cutter runout is calculated. The experimental results show that the radial cutter runout is dynamically varying in the constant rotation speed and the runout fluctuation largens with the increasing speed.

Force prediction in plunge milling of inconel 718

In manufacturing, plunge milling is one of the most effective methods and widely used for material removal in rough and semi-rough machining while machining hard material. The cutting force in milling is very difficult and complex to predict, especially in plunge milling since there are many parameters acting as the design variables in the cutting force formulation. In this paper, an empirical formula is used for reference and a new cutting force model is developed in plunge milling of Inconel 718. The coefficients in the new formula are calibrated by the plunge milling cutting test, then an experiment is designed to confirm the new model. By using the proposed model, its easy to predict the cutting force in plunge milling with considerable accuracy. Furthermore, the new model provides advice for the selections of machining parameters, machine tools and cutters to ensure the safety and high-quality of manufacturing process.

Improving the Computational Efficiency of Stability Prediction in Milling Employing the Two-Dimensional Bisection Method

This paper presents an effective method to improve the computational efficiency of stability prediction in milling based on the two-dimensional bisection method. Contrasted with the traditional semi-analytical time-domain methods, the proposed method for stability prediction only checks the eigenvalues of less nodes on the parameter plane with the two-dimensional bisection method, so that, the computational efficiency of stability can be improved. The novel method for milling stability calculation is comprised of the bisection method in two dimensions and the numerical integration method [NIM], its validity is testified by the comparison of the stability diagram and computation time in contrast to the NIM. The computation demonstrates that the calculated stability diagram by using the presented method agrees well with the result of NIM, while the computation time of the stability diagram can reduce to 1/4 to 3/4 compared with the original methods.Copyright