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

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.

Stability properties of micro-milling operations are characterized by the Stability Lobe Diagram (SLD). The material removal rates during micro-milling operations depend on the optimal values chosen for the depth of cut and also spindle speed. Theoretically, the stability boundary is calculated having the structural dynamics and the cutting parameters. However, some discrepancies are usually observed between the empirical results and the expected results that the theory supports. The driver of such a gap is that the dynamics is affected during machining operation by parameters such as the spindle speed, cutting loads, thermal changes, feed rate, etc whereas the theory is based on the structural dynamics parameters in the idle state of the machine (zero speed). Consequently, the selection of chatter-free values for cutting depth and spindle speed based on SLD in the idle state of the machine is not reliable. In addition, measuring structural dynamics parameters under cutting conditions is difficult. In this study, a novel approach is introduced to determine in-process structural dynamics parameters based on a multivariate Newton-Raphson method. Having the empirical SLD characterized by experimental data, our method tries to find the structural parameters under which the theory can support the given empirical SLD. Note that the theoretical SLD is usually characterized as a function of the cutting and structural dynamics parameters. Here our method follows the inverse flow and utilizes the empirical SLD to return the underlying parameters. The parameters returned by our method are those supported by the physics-based theories. Therefore, our approach is a hybrid method where the physics-based model is combined with the experimental results. For any given empirical SLD, with the cutting parameters fixed, the in-process structural dynamics parameters are determined using the proposed inverse approach. We use a multivariate Newton-Raphson method approach where through the iterations, an initial guess selected for the set of the parameters is adjusted step-by-step until the final set of the parameters can justify the empirical SLD based upon physics-based models.

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.

Serial industrial robots are widely in demand in the field of large-scale component processing and manufacturing. However, the structure characteristic of serial robots makes it easy to generate chatter during milling and the overhang length of milling cutter has an important influence on the stability of robotic milling. Milling cutter overhang length refers to the length from the tip of the cutter to the protrusion of the spindle, to study the influence of milling cutter overhang length on the stability behavior of robotic milling, the modal parameters of milling cutter under different overhang length were obtained by hammer experiment, and its dynamic characteristics were analyzed. The stability behavior for robotic milling under different cutter overhang lengths was analyzed by using the stability lobe diagrams, and the stability lobe diagrams were verified by robotic milling experiments. The results show that the rigidity and natural frequency of milling cutter decrease with the increase of overhang length. There is a big gap between the stability lobe diagrams and the actual machining state when the cutter overhang length is short, and the low frequency vibration is easy to occur in the robotic milling process when the spindle speed is low, which is most probably because that the absorbed vibration energy by the cutter has been transferred to the robot body before the milling cutter vibrates. When the cutter overhang is increased and the feed direction is along the y-axis of the robot global coordinate system, the stability lobe diagrams by considering the multi-mode coupling effect is more consistent with the actual milling state.

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.

The cutting stability of machine tools during operation is dependent on many parameters including the spindle speed and axial cutting depth. Cutting force coefficients may change due to the effect of heat and influence the stability lobe diagram during dry cutting aluminum alloy. Based on the identification of cutting force coefficients, the dependence of cutting force coefficients on the spindle speed and axial cutting depth are investigated. The spindle speed had a slight effect on cutting force coefficients, while the axial cutting depth greatly affected the cutting force coefficients. The cutting force coefficients were expressed as functions of axial cutting depth, and based on varying cutting force coefficients with axial cutting depths, the stability lobe diagram is modified. Then, stability lobe diagrams of varying cutting force coefficients are compared to that of constant cutting force coefficients, and the stability lobe diagram with varying cutting force coefficients was verified through a series of experiments.

The machining productivity and part quality are typically limited by the regenerative chatter induced by the dynamic interactions of spindle-holder-tool combination system. The conventional chatter stability model predicts the permissible stable axial depth of cut versus spindle speed by plotting the stability lobe diagram which represents two independent regions as absolutely stable zone and instable zone divided by the critical lobe curve. In fact, it is more reasonable to be a transition stage between the stable and instable zone. This paper introduced the grade of stability (GOS) to improve the conventional chatter stability model and study the transition zone in the stability lobe diagram. The variation of transition zone width with the stability sensibilities for different order lobe curve in milling system was analyzed. Sigmoid function was used as the membership function to develop the fuzzy stability lobe model. Then, the fuzzy stability lobe diagram with an adjustable slope coefficient was implemented to improve the mould steel milling process. The improved fuzzy stability model enhances the reliability of stability lobe diagram and guarantees the chatter-free milling process.

Sound mapping for identification of stability lobe diagrams in milling processes

Prediction of chatter stability for milling process using precise integration method

Increased stability in machining processes is highly desired by all machining industries when vibrations and specially chatter occur. This phenomenon is defined as a self-excited vibration that occurs due to the regeneration of waviness of the workpiece surface. In machining industry, the trend is to rely on the trial and error method or mere experience when deciding the machining spindle speeds, depths of cut and tool stick-outs, all of which are parameters directly related to chatter occurrence. Currently, the shortest possible tool stick-out is chosen by default, but literature has proven that longer stick-outs may bring some advantages when it comes to material removal rates. Aiming to prove this theory, this paper will discuss the influence of the tool stick-out on machining chatter occurrence. To that end, the effect of the tool stick-out on the modal parameters of the system, on the Stability Lobe Diagram (SLD) and on productivity will be analysed. Therefore, a number of Tap-Tests to different tool/tool-holder/stick-out combinations have been performed, in order to gather the data (FRFs and SLDs) where the analysis is based on. Last but not least, some machining tests have been conducted aiming to compare the theoretical chatter occurrence conditions, provided by the SLD, with the experimental ones. For that, two Al5083 workpieces have been slot milled under different cutting conditions, facilitating the unexpected results wherein the conclusions have been based upon.

Chatter vibration is an undesired and indispensable phenomenon in turning operation, which cannot be completely avoided. However, it can be suppressed by early identification and with the proper choice of input turning parameters. The key issue of chatter detection is to process the acquired signals and extract the features pertaining to it. In the present work, a methodology has been proposed for exploring tool chatter features in the incipient stage during turning on lathe. Chatter signals generated during the turning of Al 6061-T6 have been acquired using a microphone. A stability lobe diagram has been plotted to access the stability regime. Further, in order to study the effect of feed rate on stability, the recorded signals have been processed using a local mean decomposition signal processing technique, followed by the selection of dominating product functions using the Fourier transform. The decomposed signals have been used to evaluate the new output parameter, that is, chatter index. Further, the Nakagami probability distribution has been used to ascertain stability region (threshold). From the experimental validation, it has been inferred that cutting combinations obtained from the Nakagami probability distribution are significant and capable of limiting chatter vibrations. The present methodology will serve as guidelines to the researchers and machinist for the identification of tool chatter in the incipient stage, explore its severity, and finally suppress it with the proper selection of input turning parameters.

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.

In milling process, chatter is one of the most unfavorable factors, which will reduce surface quality, limit tool life, accelerate tool wear, and decrease machining efficiency. To solve this problem, a great deal of research has been done in milling dynamic modeling and chatter suppression. In this paper, a new milling force calculation method considering helix angle and bending is presented, in which the instantaneous cutting area is calculated in an improved way. The milling dynamic equations are established based on the proposed model, and the stability limit is obtained with semi discretization method (SDM). Results show that tool bending and helix play important roles in stability lobe diagram (SLD). Subsequently, the stability prediction is verified in the milling experiment. Stability analysis can just provide the guidance for selection of milling parameters. In order to get higher efficiency and larger stable region, the time-domain least mean square (LMS) adaptive algorithm is constructed and implemented for chatter suppression in this article. For the sake of applying the method to experiments, the smart toolholder equipped with piezoelectric stack actuators is designed and mounted to a three-axis milling machine. The experimental results show that this method can suppress chatter effectively.

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

Despite recent advances and improvements in modeling and prediction of the dynamics of the machining process, an efficient machining process is limited due to chatter and instability of machining system. In fact, the machining system contains various kinds of joints, which cause difficulties in dynamics modeling, simulation and prediction. Moreover, the flexible support system results in large deformation and violent vibration of the workpiece when machining, and the thin-walled workpiece easily gives rise to the chatter of the machining system. Therefore, the dynamics of the flexible support system was considered to calculate stability lobe diagram in the modeling of milling process. The whole machining system was regarded as a closed loop composed by the machine tool structures, support, workpiece and machining process. In this paper, the receptance coupling (RC) method was introduced to predict the dynamics of the closed machining system. A milling process was taken for example to predict the chatter limitations using the dynamics of closed model. The mathematical model of the machining system (machine tool structures, spindle, holder and tool), together with the details of joint contacts, was given based on the RC method. The RC model was used to obtain the dynamics of the system, while receptance of the tool point was coupled. Based on the coupling model of the machining system, the depth limitations under different speeds were estimated for the technology parameter optimization in milling process. The response was considered to be the sum of the cutting point and the support system. The flexibility of the support system was considered to be the feedback of the cutting stiffness. By this means, the traditional model was modified to calculate the stability lobe diagram based on the dynamics of the spindle and support system. Furthermore, the milling experiment was carried out to verify the prediction results, and the dominant natural frequencies of receptance at tool point were obtained by modal testing to define the stability lobe diagram. It was found that the chatter results matched well with the stability lobes. It was concluded that the support system with poor stiffness might cause violent chatter especially when the workpiece was thin-walled. The cutting depth limitations of the flexible support system were lower than that of the rigid one. Moreover, this closed model of the machining system is appropriate for the chatter prediction of the flexible support system or thin-walled workpiece, so it is helpful for a better parameter optimization.