Nonlinear dynamics of milling processes

In the space of system parameters, the closedform stability chart is determined for the delayed Mathieu equation defined as (t)(cost)x(t) bx(t2). This stability chart makes the connection between t...

Chatter stability prediction is of great practical importance for stable machining because regenerative chatter in the milling process will result in poor surface quality and low machining efficiency. Full-discretization method and its variants have been demonstrated to be effective for the prediction of milling stability. However, the main shortcoming of such methods is that they can predict milling stability but involve inverse matrix calculation, which would lead to increases in computational complexity and reductions in numerical stability. In addition, there may not necessarily exist good inverse matrix for these methods. This study proposes a precise integration-based third-order full-discretization method that can be both accurate and efficient in milling stability prediction without the need of any inverse matrix calculation. The performance evaluation performed by the simulation demonstrates that the proposed method outperforms the conventional methods with respect to stability prediction accuracy and speed. Extensive simulation is also carried out to investigate the effects of interpolation order for the simplified state term on the performance of the proposed method. Three demonstrative examples are employed to demonstrate how the proposed method can function effectively in the prediction of milling chatter stability. Although a chatter stability prediction tool for the milling process is the particular application presented here, the proposed method can be applied to other machining processes, such as turning, boring, and drilling.

This article analyses the dynamics of a resonantly excited single–degree–of–freedom linear system coupled to an array of nonlinear autoparametric vibration absorbers (pendulums). Each pendulum is also coupled to the neighbouring pendulums by linear elastic springs. The case of a 1:1:…:2 internal resonance between pendulums and the primary oscillator is studied for stationary (harmonic) and non–stationary (slow frequency sweep) excitations. The method of averaging is used to obtain amplitude equations that determine the first–order approximation to the nonlinear response of the system. The amplitude equations are analysed for their equilibrium as well as non–stationary solutions as a function of the parameters associated with the absorber pendulums. For stationary excitation, most steady–state solutions correspond to modes in which only one pendulum and the primary system execute coupled motions. Conditions for the existence of manifolds of equilibria are revealed when the averaged equations are expressed in modal coordinates. In the non–stationary case with linear frequency sweep through the primary resonance region, delays through pitchforks, smooth but rapid transitions through jumps, and transitions from one stable coupled–mode branch to another are studied using numerical simulations of the amplitude equations. The array of autoparametric pendulums is shown to effectively attenuate the large–amplitude resonant response of structures over a wide band of excitation frequencies.

Undesirable chatter is one of the key problems that restrict the improvement of robot milling quality and efficiency. The prediction of chatter stability, which is used to guide the selection of process parameters, is an effective method to avoid chatter in robot milling. Due to the weak stiffness of the robot, deformation caused by milling forces becomes an unavoidable problem, which will change the tool–workpiece contact area and affect the stability prediction. However, it is often simplified and neglected. In this paper, a multipoint contact dynamic model of robot milling is established, which considers the influence of force-induced deformation on the regenerative effect and process damping. The tool–workpiece contact area is discretized into a finite number of nodes along the axial direction so that the force and deformation at each node can be calculated separately. The different contact forms of the tool–workpiece under different process parameters are discussed in different cases, and the interaction process between cutting force and force-induced deformation is analyzed in detail. An iterative strategy is used to calculate the deformation of each node and the result of the tool–workpiece contact boundary. Finally, chatter stability of robot milling is predicted by a fully discrete method. Robot milling experiments were carried out to verify the predicted results. The results show that force-induced deformation is an important factor improving the stability prediction accuracy of robot milling, and a more accurate prediction result can be obtained by simultaneously considering force-induced deformation and process damping.

A Limit Theorem for the Solutions of Differential Equations with Random Right-Hand Sides

Generalized model for dynamics and stability of milling of titanium alloys by integrating process damping, multiple modes and multiple delays

The chatter stability in milling severely affects productivity and quality of machining. Tool wear causes both the cutting coefficient and the process damping coefficient, but also other parameters to change with cutting time. This variation greatly reduces the accuracy of chatter prediction using conventional methods. To solve this problem, we consider the cutting coefficients of the milling system to be both random and time-varying variables and we use the gamma process to predict cutting coefficients for different cutting times. In this paper, a time-varying reliability analysis is introduced to predict chatter stability and chatter reliability in milling. The relationship between stability and reliability is investigated for given depths and spindle speeds in the milling process. We also study the time-varying chatter stability and time-varying chatter reliability methods theoretically and with experiments. The results of this study show that the proposed method can be used to predict chatter with high accuracy for different cutting times.

The prediction of regenerative chatter stability has long been recognized as an important issue of concern in the field of machining community because it limits metal removal rate below the machine’s capacity and hence reduces the productivity of the machine. Various full-discretization methods have been designed for predicting regenerative chatter stability. The main problem of such methods is that they can predict the regenerative chatter stability but do not efficiently determine stability lobe diagrams (SLDs). Using third-order Newton interpolation and third-order Hermite interpolation techniques, this study proposes a straightforward and effective third-order full-discretization method (called NI-HI-3rdFDM) to predict the regenerative chatter stability in milling operations. Experimental results using simulation show that the proposed NI-HI-3rdFDM can not only efficiently predict the regenerative chatter stability but also accurately identify the SLD. The comparison results also indicate that the proposed NI-HI-3rdFDM is very much more accurate than that of other existing methods for predicting the regenerative chatter stability in milling operations. A demonstrative experimental verification is provided to illustrate the usage of the proposed NI-HI-3rdFDM to regenerative chatter stability prediction. The feature of accurate computing makes the proposed NI-HI-3rdFDM more adaptable to a dynamic milling scenario, in which a computationally efficient and accurate chatter stability method is required.

Stabilization of Tethered Tug–Debris System with Residual Liquid Fuel

During the bull-nose end milling operations of thin-walled structures, chatter usually occurs and adversely affects cutter performance, finished surface quality, and production efficiency. To accurately predict chatter stability, a suitable dynamic model with effective system parameters is required. In this article, a three-degree-of-freedom (3-DOF) dynamic model is developed to analyze the milling stability of the thin-walled cylinders, in which the dynamics of the bull-nose end mill along the x-axis and y-axis directions and the dynamic of the workpiece along the z-axis direction are taken into account. Then, the cutter-workpiece engagement (CWE) is extracted by employing a slice-intersection-based approach. And the layered cutting force coefficients are identified by considering the influences of varying cutter diameters on the cutting speed. Thereafter, the semi-discretization method (SDM) is adopted to compute the stability lobe diagram (SLD). In the end, a group of milling tests are carried out on a thin-walled cylinder to validate the accuracy and reliability of the proposed model, and the results show that the model predictions agree well with the experimental data.

Adaptive Robust Constraint-Following Control for Satellite Formation Flying with System Uncertainty

A new analytical method of chatter stability prediction in milling is presented. A general formulation for the dynamic milling system is developed by modeling the cutter and workpiece as multi-degree-of-freedom structures. The dynamic interaction between the milling cutter and workpiece is modeled considering the varying dynamics in the axial direction. The dynamic milling forces are governed by a system of periodic differential equations with delay whose stability analysis leads to an analytical relation for chatter stability limit in milling. The model can be used to determine the chatter free axial and radial depth of cuts without resorting to time domain simulations.

A numerical differentiation method is presented to predict the high speed milling stability of a two degrees of freedom (DOF) system based on the finite difference method and extrapolation method. The milling dynamics taking the regenerative effect into account are represented as linear periodic delayed differential equations (DDE) in the state space form. Then, each component of the first derivative of the state function versus time at the discretized sampling grids is approximated as a weighted linear sums of the state function values at its neighboring grid points, where the weight coefficients are calculated based on the extrapolation method. As such, the DDE on the forced vibration duration is approximately discretized as a series of algebraic equations. Thereafter, the Floquet transition matrix can be constructed on one tooth passing period by combining the analytical solution of the free vibration and the algebraic equations of the forced vibration. Finally, the milling stability is determined according to Floquet theory. The stability diagrams and convergence of critical eigenvalues in comparison with the benchmark algorithms (the semi-discretization method and numerical integration method) via experimentally verified examples are utilized to demonstrate the effectiveness and efficiency of the proposed method.

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.

Prediction of chatter stability for milling process using precise integration method