Modeling of cutting forces in trochoidal milling with respect to wear-dependent topographic changes

Nickel-based super-alloys are widely used in the energy and aerospace industry. Difficult-to-cut materials, like Inconel 718, are characterized by a high strength and corrosion resistance, even at high temperatures, which make them especially popular in these sectors [1, 2]. However, the machining of such materials is difficult as the material tends to cause high process forces and high tool wear [3]. One possibilty to improve the productivity and the tool life is the utilization of alternative path strategies, e.g., trochoidal milling [4, 5], which exhibit more favorable wear progressions than linear path milling strategies [6]. Regarding the complex engagement situation in trochoidal milling and optimization of the process, a simulation-based analysis is required, especially when the consideration of tool wear is of special interest. Therefore, models for turning and drilling were presented in several studies that take tool wear into account by using finite element analysis [7, 8].

While simulation systems have been used to predict force evolution for milling processes or evaluate the dynamic process behavior [9, 10], the developed models are restricted to specific cases for which they have been calibrated. An essential process characteristic is the resulting chip thickness during machining, which can be used to predict process forces. Especially for the analysis of process stability and tool wear, a precise prognosis of the occurring chip thicknesses is needed [11‐13]. The determination of so called undeformed chip thicknesses during cutting operations has been subject to various abstractions to ensure feasibility in regard to simulations of whole processes [14]. Empirical models can be used, based on assumptions derived from orthogonal cutting operations and their prevalent chip thickness [15]. These models can be applied to a discretized tool model to result in a global process force calculated for each time step. The prediction of chip thicknesses during milling operations based on the Constructive-Solid-Geometry technique [10] can also be used, when neglecting effects of tool wear or variations of the cutting edge. Other approaches considered tool wear by modifying force model parameter values for each tool wear state  [16, 17] or extending the force model with an additional wear-dependent parameter [18]. The calculation of process forces for trochoidal milling strategies have been investigated using numerical models, which derived parameter values specifically for trochoidal path strategies [19, 20], since the transferability of force parameter values from linear strategies to trochoidal milling could not be achieved [21]. However, tool wear was not considered in these studies. Due to the complex engagement situation of trochoidal paths and the wear-dependent change of the shapes of cutting tools, the prediction of process forces is limited with previous methods, when considering tool wear-specifically to trochoidal milling strategies. In this investigation, a new modeling approach is presented, which utilizes different states of tool wear derived from trochoidal milling strategies to increase the precision of the calculated process forces in an existing geometric physically-based simulation system. Instead of using wear-specific process force parameter values, the topography of the tool is taken into account, allowing for a calibration based on linear path strategy experiments. A time-discrete envelope model is applied to predict process forces for linear path and trochoidal path strategies in Inconel 718 using a novel tool model. The new model is based on the consideration of individual cutting segments in combination with a discrete workpiece representation.

In this section, the experimental setup and the measuring equipment used is described. In addition, the method for digitizing cutting insert is explained in detail.

The motion of the trochoidal path can be described as a superposition of a circular motion with a linear movement of the tool. In literature, different approaches for the calculation of trochoidal path were compared, e.g., a circular and trochoidal model by Rauch et al. [23] or the epicycloidal model by Salehi et al. [24]. In this paper, the calculation of X- and Y-coordinates of the tool center point along the path is given bywith a tool diameter \({{\hbox{D}}} = \,{16}\,{{\hbox{mm}}}\), the step over \({\textit{S}_{\hbox{o}}} = {1}\,{{\hbox{mm}}}\) and the path angle \({\theta }\). B can be calculated with \({{\hbox{B}}} = {(\textit{S}_{w}- {\hbox{D}})/{\hbox{2}}}\), where \({\textit{S}_{w}}\) is the slot width of \({50}\,{{\hbox{mm}}}\). The movement of the cutting edges is realized by the rotational axis of the tool spindle. The use of trochoidal path strategies results in a complex engagement situation which leads to variable undeformed chip thicknesses. While linear machining (Fig. 4a) creates constant maximum chip thicknesses, the use of trochoidal paths (Fig. 4c) generates varying maximum chip thicknesses along the path. In Fig. 4b a comparison of both strategies shows a reduction of the resulting maximum chip thicknesses through the process.

In order to predict process forces during cutting operations, a geometric physically-based simulation system [9] was extended with a new tool model. The simulation system utilizes a time-step discretization of NC programs to calculate the TCP for cutting tools. For a given set of process parameter values, the NC program is subdivided into discrete time steps, which correspond to engagement situations along the given tool path. Instead of directly using the TCP to position a tool model, an interpolation along the trajectory of the cutting edge is applied [11, 25]. This is necessary due to the superposition of TCP and tool rotation. For each of these time steps, the swept envelope volume regarding the flank face shape of the tool was calculated. The swept volume was generated by connecting corresponding points of the flank face along the movement path of the cutting edge for two consecutive time steps (Fig. 5). The tool model is based on a structured grid which allows for an accessible assignment of isolated cutting segments along the tool [26]. The structured grid can be implemented as a triangular mesh, which enables numerically stable intersection tests with discrete workpiece representations [27]. The workpiece is modelled by using three perpendicular dexelboards, each with a defined spatial precision. The respective dexelboards are modeled using line segments arranged over a two dimensional grid, representing the volume of the workpiece [28]. Using these discrete models, the uncut chip thickness was approximated by calculating the discretized removed material intervals of the workpiece for every simulation step. These removed intervals were sub-sampled to discrete interval points, which were then projected to the corresponding cutting segments of the cutting edge to determine the uncut chip thicknesses per cutting segment (Fig. 5b).

For the validation of the presented simulation approach, two force measurements of single arcs of a trochoidal milling path were selected from the milling experiments (Sect. 2). The first measurement is a single trochoidal path corresponding to a steady state stage of the cutting insert to validate the new cutting edge state after a material removal of approximately \({\textit{V}_{w}} = {1000}\,{{\hbox{mm}}^{3}}\). The second measurement was selected after machining of 135 paths with a material removal of approximately \({\textit{V}_{w}} = {6500}\,{{\hbox{mm}}^{3}}\), showing noticeable changes in force amplitudes (Fig. 9b). A comparable runout error can be observed, which corresponds to the previous considered worn state. The process forces showed an increase of force amplitudes of up to 30N. For these measurements, the first half of the arc (Fig. 9a \(\textcircled {1}\)) was simulated in five exemplary regions along the path using the novel empirical force model taking topographic information into account (Fig. 10).

For the calculation of process forces, the modeled structured grids for a new and worn cutting inserts were used (Sect. 4.1). The runout was set to \(RO_{radial}\) = 0.013 mm and \(RO_{axial}\) = 0.005 mm, according to measurements conducted before milling experiments. The time discretization was set to 600 steps per tool rotation, resulting in a time-step of \(\varDelta _t\) = 0.124 ms. The discretization of the dexel grids was set to 0.005 mm in each direction, while the cut intervals were sub-sampled to 0.001 mm each to increase the chip sampling accuracy. The force coefficient values were set to \(k_c\) = 1926.08,  \(k_n\) = 502.64 and m = 0.28 as concluded in the initialization experiments (Sect. 4.2).

The calculated forces for the cutting insert in a new state show a good correspondence regarding the overall force progression for a trochoidal path (Fig. 10a) in X- and Y-direction. Process forces in Z-direction, however, were underestimated, which corresponds to the observed behaviour during initialization tests. Looking at two exemplary tooth engagements (Fig. 11a), a slight underestimation of the resulting contact length for the second tooth can be identified. This can be caused by small variations between cutting edges and minor differences in runout throughout the process. Overall, an equally good correspondence can be seen between measured and simulated process forces. Especially the differences of approx. 45 N in Z-direction are primarily seen at the beginning of the engagement. After initial contact, the force level is showing a good agreement. For the worn state of a cutting insert, a similar pattern in the overall force progression can be observed (Fig. 10b). While the force amplitudes show a good agreement for booth teeth (Fig. 11b), a slight deviation in X- and Y-direction, especially for the first tooth is noticeable after initial contact. These differences are caused by dynamic effects, as the force measurements are subjected to the transfer behaviour of the dynamometer. These variations can be observed between two tooth engagements (see Fig. 9b \(\textcircled {2}\)). The second tooth exhibits a shorter contact length, presumed to be a deviation in runout error. The process forces in Z-direction are equally underestimated compared to the differences during the initialization experiment (Sect. 4.2). In summary, it can be concluded that an increase in the process forces due to a topographical change caused by tool wear could be reproduced successfully.

In this work, an extension of a time-domain simulation system was presented which enables the usage of digitized cutting edges for the prediction of process forces. With this extension, shape- and wear-specific information of cutting tools can be considered by structured grids. This was shown for cutting inserts, which were worn during milling of Inconel 718. The predicted process forces were determined with the extended process force model. Since runout can have a significant impact on the distribution of process forces when using multi-insert tools, the measured runout was applied for both tool wear states. For both wear states, a good accordance in linear path milling experiments could be achieved with one set of parameter values. Even when using non-linear strategies like in trochoidal milling experiments, which show high variance regarding chip thickness and engagement situations, the force prediction were still in good agreement with measured forces for both wear states. This methodology allowed wear-specific changes in process forces to be modeled without having to adjust force coefficients throughout the process. Slight deviations due to the occurrence of dynamic effects were neglected, as they did not play a decisive role in this investigation.

In future work, further investigations of model initialization will be conducted to ensure an increased approximation quality of process forces for cutting operations regarding different tool wear states, especially regarding the process forces occuring in Z-direction. An adaptive compensation of the runout error will be investigated to account for differences in contact length when using modular tool holders for cutting inserts.