Prediction of chatter stability for milling process using precise integration method

Chatter stability prediction for five-axis ball end milling with precise integration method

Chatter stability prediction for multi-robots collaborative milling system

Chatter stability prediction is an important technique for obtaining chatter-free cutting parameters to enhance product quality and achieve better cutting performance. Based on the exponential fitting multistep algorithms and Fibonacci search, this paper presents an implicit exponentially fitted method to efficiently and accurately determine the chatter stability limits. The dynamic model with consideration of the regeneration effect for milling processes can be expressed as delay-differential equations (DDEs) with time-periodic coefficients. The tooth-passing period can be subdivided into two distinct phases according to whether the cutting tool is contacting the machined parts. On the basis that the forced vibration phase can be discretized into time intervals of identical duration, a three-step implicit exponentially fitted method is developed to estimate the state term. Subsequently, the milling stability boundary can be obtained by using the Fibonacci search to substitute traditional sequential search, which can remarkably reduce the computational time. The effectiveness of the implicit exponentially fitted method is validated through making comparisons with the other two benchmark methods. Simulation results indicate that the implicit exponentially fitted method exhibits excellent accuracy and efficiency. Furthermore, the experimental verification was performed to further demonstrate the availability and validity of the implicit exponentially fitted method. On this basis, we extend the implicit exponentially fitted method to the variable pitch cutters case. Furthermore, a benchmark example is provided to evaluate the feasibility of the extended implicit exponentially fitted method. The results indicate that the implicit exponentially fitted method can achieve significantly better computational efficiency and accuracy.

With the rapid development of aerospace technology, Al7075 has been widely used for structural components. High-speed milling is one of the most effective ways to improve machining efficiency of Al7075. During the milling process, regenerative chatter which restricts the milling quality and productivity often occurs. With the aim of avoiding regenerative chatter, stability lobe diagram (SLD) is widely used to obtain chatter-free parameters. This work presents a stability prediction method by using shifted Chebyshev polynomials. The milling dynamics with consideration of the regenerative effect is described by time periodic delay-differential equations (DDEs). The transition matrix of the milling system is constructed with the help of Chebyshev–Gauss–Lobatto (CGL) points. In order to demonstrate the accuracy of the proposed method, the rate of convergence of the proposed method is compared with that of the classical benchmark methods. On the other hand, in the process of thin-walled workpiece milling, the dynamic behavior of the workpiece depends on the tool position. To study the influence of the tool position dependent dynamics on the chatter stability of the thin-walled workpiece, a three-dimensional SLD is obtained. The verification experiments are conducted to verify the reliability of the proposed method. The results show that the experimental results are consistent with the predicted results.

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 third-order 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 constructed 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 based on 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.

The precise integration method for semi-discretized equation in the dual reciprocity method to solve three-dimensional transient heat conduction problems

As a new type, highly efficient hole-making technique, helical milling is widely used to generate holes on composites and composite-metal compounds in the aircraft industry. The accurate prediction of cutting forces and chatter stability are essential for achieving a highly efficient and chatter-free helical milling process by cutting condition optimization. On the basis of a quantitative description of cutting zones corresponding to helical milling operation, a novel dynamic cutting force model is proposed for helical milling in this paper, in which the cutting mechanism and the cutting force contribution on both the peripheral and the front cutting edges are taken into consideration simultaneously. An analytical stability prediction method is also proposed to obtain a novel stability lobe diagram represented by the ratio of the axial to the tangential feed per tooth with regard to the spindle speed using a numerical approach to calculate the average directional cutting force coefficients. The proposed method is capable of predicting the stability lobes accurately for helical milling process where different types of cutting mechanism exist and the axial depth of cut is time varying. It is shown that the numerical predictions agree well with the cutting tests. Finally, the effect of cutting conditions on both the cutting forces and stability limits are thoroughly analyzed.

Chatter stability analysis in Micro-milling with aerostatic spindle considering speed effect

By studying the workpiece-edge-coupling (WEC) in five-axis ball-end milling, the contact characteristics between the workpiece and edge curve are analyzed, and the chip model is extracted and simplified. The edge curve involved in the cutting process of each edge are calculated at each time and a new instantaneous numerical chip thickness model is established. Then the milling forces and stability lobe diagram (SLD) are calculated in following cut process with lead and tilt angle. The milling forces and SLD of lead angle at 15° and tilt angle at −15° are verified by comparing with Otzurk model as references, and it is found the SLD of WEC model can reflect the unstable points more properly than that of Otzurk model. Also the vibration in x and y directions show a divergence trend, which proves the high precision for the new algorithm adapted to the stability prediction of five-axis ball-end milling process. In addition, the surface topography is acquired considering lead and tilt angle as well as forced vibration, and the result is consistent with the experiment in the existing literature. It is found that the milling forces, SLD and surface topography show the same variation trend with the increase of lead and tilt angle. Besides, the stability region significantly expands and the surface topography improves by applying positive tilt angle other than the negative. Then, under the conditions of positive lead and tilt angle, increasing lead angle and decreasing tilt angle reduces the milling force, expands the stability region of SLD and improve the surface topography. The optimized tool posture is acquired by the coincident analysis of milling force, SLD and surface topography under different lead and tilt angle.

The objective of the paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems. The gradient dependent model is adopted in the numerical model to overcome the result mesh-sensitivity problem in the dynamic strain softening or strain localization analysis. The equations for the dynamic elastic-plastic problems are derived in terms of the parametric variational principle, which is valid for associated, non-associated and strain softening plastic constitutive models in the finite element analysis. The precise integration method, which has been widely used for discretization in time domain of the linear problems, is introduced for the solution of dynamic nonlinear equations. The new algorithm proposed is based on the combination of the parametric quadratic programming method and the precise integration method and has all the advantages in both of the algorithms. Results of numerical examples demonstrate not only the validity, but also the advantages of the algorithm proposed for the numerical solution of nonlinear dynamic problems.

As advances and simulation prediction design become more prevalent, the methods for solving rotor dynamics response are challenged to meet higher standards for precision, efficiency, and convergence. To address the limitations of current approaches like Newmark and Runge–Kutta in solving nonlinear responses of rotor dynamics in engineering applications, this paper introduces an improved differential-quadrature precision method. This method fits the Duhamel term with a nonequidistant differential-quadrature technique and uses Taylor and Lagrange formulas for self-starting predictions, balancing accuracy with computational efficiency. Furthermore, the paper presents an iterative strategy that effectively addresses the convergence issues of precise integration methods under large step sizes for nonlinear response solutions. Comparative analysis with real aeroengine gas turbine rotor models demonstrates the performance differences between the improved precise method and Newmark method. The result shows that in steady-state response solutions the improved precise method achieves greater precision at twice the step size of the Newmark method. In transient response solutions, the improved precise method outperforms Newmark in terms of computational efficiency, and the iterative strategy proves both reliable and effective.

Forward seismic modeling is a useful tool for simulating the response of the low velocity zones near the ground surface and delicate structures at a deeper depth in northwestern China. The main objective of forward seismic modeling is to solve the wave equation accurately. The finite difference method (FDM) is a commonly used numerical method for solving the equation. It can obtain very high accuracy with high‐order difference scheme if the grid configuration is appropriate. However it shows numerical dispersion and its stability criteria is also very strict. In order to deal with these problems, a new precise integration method is advanced in this paper, using an integration scheme in the temporal domain rather than a traditional low‐order difference scheme. The precise integration method employs a difference approximation in the spatial domain, converting the acoustic or elastic wave equation into a set of differential equations with respect to time and then solves this equation system using an integration method. The final solution is attributed to an integration of a matrix of exponential functions. Theoretically this solution is accurate in the temporal domain, hence increasing the total accuracy. The whole spatial domain can be divided into several sub‐domains, considering computer capacity and speed and then the solutions put together. This revised sub‐domain precise integration method (SPIM) is more stable than the FDM and more accurate as is tested in the text. Besides it can greatly reduce the numerical dispersion with the same grid configuration as the FDM, though it may take a little more CPU time. A numerical test is presented to justify this method and then an application is shown to indicate that the SPIM is of potential value in the geophysical exploration field.

The purpose of this study is to acquire design method of machine tools with higher chatter stability. This study focuses on stiffness of main structures and friction damping in a slide of a machine tool, because the stiffness and the friction damping are considered to have major influence on the stability. A 2DOF vibration model of the machine tool with friction damping was developed, and it was applied to the chatter stability analysis. A testing machine was developed and the model parameters were identified. The chatter stability was predicted using the developed model and the identified parameters, while cutting tests were carried out using the developed machine tool to verify the model. The predictions and the experiments were in a good agreement, and therefore it is considered that the model is effective to improve the chatter stability in the machine tool design. For example, the results indicate that higher stiffness does not always lead to higher stability, and that the optimum friction force exists at every stiffness condition.

Chatter vibration in milling process is a major obstacle that limits the machining quality and productivity, which may be avoided by using stability lobe diagrams (SLDs). Many traditional models developed to predict chatter stability assume that dynamic parameters of the machine tool remain constant under operational conditions. However, these parameters such as natural frequencies, damping ratios, stiffness, and cutting force coefficients vary depending upon different aspects including spindle speed, tool wear, and machining position, reducing the accuracy of chatter prediction. In this study, a robust chatter prediction method based on conventional analytical milling stability models is presented by employing the Edge theorem and Zero Exclusion condition. In this method, optimal combinations of spindle speeds and machining positions are firstly researched to obtain higher critical depths of cut, based on the conventional stability model, modal fitting technique, Kriging model, and improved particle swarm optimization. At each combination, related nominal modal parameters and cutting force coefficients are identified, and their left and right worst-case deviations are also determined. Critical stable condition for each combination is detected by a graphic approach within the minimum and maximum bounds of uncertainties. Accordingly, a robust stability lobe diagram is obtained with related spindle speeds and critical cutting depths. The proposed method was verified by chatter tests on a real vertical machining center, demonstrating its reliability in chatter prediction compared to the conventional stability lobe diagram.

Quantification of uncertainty in machining operations based on probabilistic and robust approaches