Topography simulation of free-form surface ball-end milling through partial discretization of linearised toolpaths

Engineering Science and Technology, an International Journal 55 (2024) 101757 Available online 3 July 2024 2215-0986/© 2024 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Topography simulation of free-form surface ball-end milling through partial discretization of linearised toolpaths Felipe Marin a , * , Adriano Fagali de Souza b , Helton da Silva Gaspar c , Amaia Calleja-Ochoa a , Luis Norberto L´ opez de Lacalle a a Aeronautics Advanced Manufacturing Center (CFAA), University of the Basque Country (EHU/UPV), Bilbao, Spain b Federal University of Santa Catarina (UFSC), Florian´ opolis, Brazil c Federal University of Santa Catarina (UFSC), Joinville, Brazil ARTICLE INFO Keywords: Free-form 5-axis milling Ball-end milling API CAD CAM ABSTRACT Free-form surfaces are commonly present in several engineering components, from molds to more sophisticated components of aeronautical engines. These parts are usually finished by machining, more specifically, ball-end milling. In this process, the contact between the tool and the part constantly changes along the toolpath, resulting in several manufacturing problems that damage the surface and compromise the performance of the part. Besides the ordinary cutting parameters, these machined surfaces are deeply influenced by the tool tip center and the elastoplastic deformation of the material through the shearing and plowing pair. Knowing the exact orientation of the tool in relation to the surface determines all aspects of the milling process and is necessary for process modeling. The current CAD/CAM software platforms is not able to predict such surface topography. In this direction, the current work presents a software routine developed and implemented on an open interface CAD/ CAM software (Siemens® NX). The normal vectors of the surface to be machined, the cutter contact (CC), and cutting location (CL) data from the toolpath calculated by the CAD/CAM were used to obtain a complete discretization of the cutter-workpiece position along the toolpath. This information was used to predict the surface topography and identify the cutting-edge elements. Then, the developed model was used to evaluate the freeform surface of a blade milled on 5-axis together with confocal imaging analysis. The results show that the methodology developed can predict the topography aspects of a free-form machined surface and support the analysis of milling problems such as run-out. 1. Introduction Free-from surfaces are commonly present in several engineering components, from dies and tools to more sophisticated components of energy and aeronautic industries. During the manufacturing process of such parts, ball-end tooltips are recommended, which can be directly applied on the surface finish or indirectly for manufacturing tools and consumables like electrodes for electrical discharge machining (EDM). Such tools allow different contacts to produce a free-form and smooth surface. However, due to the geometry of the tooltip, it produces crest waves that significantly increase the roughness of the machined surface, which can compromise performance and must be evaluated carefully. In the design of turbines and aeronautical engines, the shape of the surface of blades, impellers, and blade-integrated disks (BLISKs) can be optimized to improve the aerodynamic flow. This results in complex shape designs, with instantaneous radius changes over the surface that make the manufacturing process challenging. Besides, such parts usually require high strength, wear, and thermal resistance, demanding hard-tocut materials and increasing design and machining complexities. The aspect of the surface generated by the milling process significantly impacts the surface integrity and performance, being the most important feature of complex high-performance parts [1]. In blades and aerofoiled components, form errors and other milling process signatures can considerably reduce the component’s performance; thus, tight design tolerances are applied to enhance its energetic efficiency. Surface integrity also plays an important aspect in the fatigue life of critical parts, and for this reason, it has been extensively investigated in the last decades [2]. 5-axis milling is preferable to finish such parts with free-form geometrical shapes. It allows wider tool access, manufacturing * Corresponding author. E-mail address: [email protected] (F. Marin). Contents lists available at ScienceDirect Engineering Science and Technology, an International Journal journal homepage: www.elsevier.com/locate/jestch https://doi.org/10.1016/j.jestch.2024.101757 Received 1 December 2023; Received in revised form 18 April 2024; Accepted 28 June 2024 Engineering Science and Technology, an International Journal 55 (2024) 101757 2 geometries that would be difficult or impossible to produce with traditional 3-axis due to gouging. Furthermore, it allows modification of the tool’s orientation and chip thickness to produce smoother toolpaths, reducing the oscillation of the cutting forces and other related problems [3–5]. Also, it reduces the number of setups, improving lead times and costs. To guarantee the position of the tool in 5-axis milling, advanced NC controllers are required together with CAM software capable of developing valid and optimized tool trajectories. In this process, the tolerance band must be carefully adjusted to maintain the performance with at least a minimum surface quality. As the tolerance band reduces, more data must be processed, and it can saturate the machine’s control loop, resulting in instantaneous reductions or process pauses that severely damage the surface [6]. Also, the data distribution (CL points) inside the tolerance band varies according to the software house, mainly because of different mathematical models used, resulting in different performances for each software in the same machine [7]. Thus, the evaluation of the process commonly considers surface errors and machining time [8]. The topographic aspect of the milled surface using the ball-end tooltip on 5-axis milling can be estimated using different geometrical models that consider the tool position in the space, its diameter, the feed per tooth, the stepover, and the tool run-out. When tool run-out is present, one cusp removes more material, causing different patterns on the surface [9]. The theoretical surface is obtained by computing the engagement between the part and piece, which is an important branch for surface evaluation and prediction. Moreover, geometrical models are required in force prediction, deflection, and form errors. For these reasons, it has been extensively investigated in the last decades [10]. Random surface patterns indicate more complex problems related to vibration, process cinematics, wear, deformations, or clamping issues and must be addressed with the support of other simulation tools. Denkena et al. [11] applied a stochastic method to predict surface topography with high accuracy, considering process imperfections and kinematics. However, it requires prior knowledge of the milling results of a surface machined under such processing conditions, which makes general implementation in software difficult. Several approaches can be considered to the geometrical model of the cutter. The most common are the analytic model of the helix, the tool dome, and the infinitesimal cutting-edge (ICE) element. The choice between them will depend on the depth and precision required for the analysis. Lazoglu and Liang [12] modelled the engagement and the chip thickness variation during convolution to predict the milling force of ball-end milling on a flat surface. Zhang et al. [13] discretized the tool edge of a ball-end tooltip with a constant helix into a series of cutting points to include the tool wear on the topography surface prediction. Although there is a considerable amount of research on ball-end milling on flat surfaces, its proposed application to milling free-form surfaces still faces several limitations due to the complexities and changes in the CWE. Ismail Lazoglu [14] solid-modelled the CWE of the 5-axis to predict the cutting force estimation. However, this approach is computationally heavy and not productive. Artetxe et al. [15] included tool run-out by adding a radial offset to the infinitesimal edge element, thus modifying the chip thickness. In these studies, the models considered the uncut chip thickness, and there is a limited dependence on the tool geometry. Layegh and Lazoglu [16] analytically modelled the surface obtained in a 5-axis ball end milling considering lead angles and tool run-out, showing that the tool orientation affects the surface texture. However, the average error obtained on a flat surface was about 20 %. Gou et al. [17] modeled 5-axis flank milling without discretization of the cutting edges. The model included variable helix angle combined with run-out to perform the real trace of 5-axis milling and precisely predict the surface, however, it is not settled so far for free-form milling with ball-end tooltips. Different approaches can be applied to model the cutter contact between the tool and the surface; the most computer-efficient method is the analytical one, which is very limited and used more for ideal cutting conditions. However, the mathematical model of the surface is protected by the core of the CAD software, and exporting this data is not trivial, so most of all studies regarding ball-end milling of free-form surfaces used monotonic or inclined planes [18]. Discrete and semi-discrete models of the surface are the most commonly used to discretize the cutter workpiece engagement (CWE) during ball-end milling. It is know that the local slope of the surface produces CWE oscillations affecting the cutting speed and oscillating the cutting forces, damaging the process [19]. Thus, discretizing the surface into small and known segments to calculate instantaneous solutions makes it possible to predict point-to-point more complex problems like free-form milling, up to now very limited [20]. Furthermore, some studies have already addressed the CWE calculation using postprocessed NC files and the CL data to compute cutting surfaces [21,22], but errors associated with the uneven point distribution are still a limitation, especially for 5-axis machining. Most of the studies of free-form surfaces are limited to the 3-axis, where the positioning of the tool is always in the z-direction, easing the identification of the pair of CL CC and respective vectors. More complex milling, considering 5-axis toolpath programming, requires detailed information on the tool orientation as well as the surface information. Tunc and Budak [23] developed a methodology for obtaining simulation data in 5-axis milling using CL data generated by CAM software. However, their proposed methodology requires comparing the CAM file with one using the same strategy generated using null tilt and lead angles to determine CC and its respective vector as well as an STL of the surface, which increases pre-work and limits its application to gauge-free surfaces. The recent growth of Industry 4.0 promotes machines and software integration to predict or control manufacturing processes, but integration with CAD/CAM software is still a limitation [24]. Thus, developing a methodology based on the CAM software, which considers the embedded CAD geometry, to obtain data for modelling 5-axis ball-end milling is still necessary for academic and industrial purposes. These difficulties lead to the wide use of empirical methods in the shopfloor and confocal microscopy imaging analysis, which is timeconsuming. This technique uses microscopy and laser light filters to reproduce the high-resolution surface texture, which is valid for inspection and surface model validation [25]. The accurate evaluation of the topography will reveal vibrational and more complex problems present in the manufacturing process. On the one hand, mixing all the damages makes hard the investigation of individual effects. On the other hand, it allows an accurate topography prediction considering regression with the geometrical model and all the noise obtained evaluating the surface [1]. Another method to evaluate the surface is comparing local results with the theoretical roughness. Although surface roughness has been more representative, lineal roughness is most commonly used to evaluate the surface due to the price and versatility of the 2D rugosimeter (ISO 1302, highlighting the 2D parameters Ra, Rt, Rz). However, it should be taken into account that diverse profiles can display identical roughness parameters and perform mechanically differently [26]. According to Mali and Gupta [27], the surface roughness of ball-end milling can be mathematically predicted with high accuracy regarding the scallop’s height. The scallop height measures the height of the highest point of the mark left by the ball-end tooltip and the lowest of the following cutting step. However, this is only valid when the material is sheared from the surface. Scandiffio et al. [28] evaluated the roughness of a free-form surface after a milling operation using a ball-end tooltip. The authors identified that the roughness is influenced mostly by the CWE and tool wear but did not consider the tool run-out, which is required to obtain a more precise simulation. Urbikain and Lacalle [29] evaluate the effect of the ICE element of end mills on the roughness. They observed that the lead F. Marin et al. Engineering Science and Technology, an International Journal 55 (2024) 101757 3 and helix angles had less influence on the average roughness for this kind of tool, while the tilt angle reduction improves surface roughness. Another aspect that is highlighted in ball-end milling is the effect of the tooltip. The tooltip can negatively affect the cutting process by increasing the components of the cutting force, increasing the tool wear, modifying the direction of its components, and other related problems that damage the surface. Furthermore, tooltip wear can lead to surface integrity issues, such as roughness, irregularity, and damage to the machined surface that facilitates the corrosion and fatigue process. Optimized feed rate and cutting speed cutting conditions can minimize the tooltip effect but cannot completely eliminate it. The cutting speed is null at the center and nearby regions, resulting in a cutting phenomenon known as plowing. When the minimum chip thickness is not achieved, the material is crushed instead of sheared. Thus, tool inclination is always preferable to avoid tooltip center contact [30]. However, due to geometrical constraints, sometimes it is impossible to evade the tooltip or nearby regions, resulting in a surface with severe plastic deformation, side flow, scratching, and cavities. Also, several pull-outs and high residual stresses can be found with different magnitudes according to the machined material [31,32]. These marks left on the surface are hard to predict but easy to identify in a postevaluation with optical imaging analysis. Knowing the idealized surface can improve the identification and a better comprehension of the process. Modeling the finish milling considering the tool’s effective radius is essential to include the plastic deformations and a proper surface prediction. Also, in many studies, the effect of the tool orientation is investigated, mitigating variations in the topography with the tool tilt and cutting direction. Oliveira et al. [33] quantified geometric errors on 4axis milling of thin-walled parts in up-cutting or down-cutting. The authors attributed the form deviation to the cutting force but did not evaluate the surface roughness and the tool run-out towards advancing the milling process simulation. Chen and Huang [34] studied the surface topography of ball-end milling considering the cutting-edge geometry. The authors evaluated the scallops of the surface for specific tool inclinations. Tilting the tool above 10◦propitiated a better surface roughness for most tool diameters. Liu et al. [35] recommend increasing the lead angles up to 45 degrees on ball-end milling to achieve better roughness values. The authors attribute this improvement to the increased effective cutting speed and decreased velocity gradient. Besides, the marks left on the feed direction were about 3 to 4 times more sensitive to tool inclination than the ones left by the lateral passes to the surface roughness. One way to investigate the quality of the manufacturing process is by directly analyzing the texture left on the machined surface. The milling process has intermittent and repetitive patterns that can characterize the surface and the process. During the cutting, the orientation of the cutting edge changes periodically with the rotation of the spindle and feed and, thus, by the feed per tooth, making periodic marks overtime on the feed direction. These marks are not well predicted in the simplistic simulation tool of commercial software (Fig. 1), making the process prediction and surface evaluations challenging and ruled by empirical knowledge. Besides the inverse cinematic of the 5-axis machining, the complexity of the engagement of the tool with the workpiece and limitations on exporting the surface data make the process modeling difficult for freeform surfaces. Nowadays, there are some models to predict topographies and other aspects of the milling process that include forces and their derived problems. However, there are many limitations when free-form surfaces and 5-axis milling are involved due to several data inputs required and boundary conditions. Additionally, studies with free-form surfaces usually use ball-end tooltips that result in more complex geometric problems, requiring surface data for modeling that is usually protected by the CAD/CAM software. So, it is common to find limited or idealized studies with specific geometries, contact conditions, or cutting parameters. To obtain a robust milled model to predict the texture of free-form surfaces, including material properties, the effective cutting speed of the tooltip, feed rate oscillation, tool run-out, cutting parameters, as well as cutting geometries and CWE geometrical model, is necessary. In this context, the current work proposes modeling and investigating the surface texture left after free-form milling with conventional ball-end tooltip. Considering the crest height on the feed direction. Therefore, a methodology for computing free-form surfaces is proposed based on the discretization of the toolpath obtained by a post-processed NC file, improving the CAD/CAM estimation and boosting industrial applications. Furthermore, a routine is developed to obtain data for general 5axis milling of free-form surfaces, using this data for surface prediction Fig. 1. Typical surface prediction by commercial CAM software. F. Marin et al. Engineering Science and Technology, an International Journal 55 (2024) 101757 4 and easy future evaluations of more complex problems that require geometrical data of the tool and the surface. 2. Materials and methods Considering the limitations of the current CAM software of simulating the real texture of a machined surface, predicting the cutting force and the geometric errors, among others, the present work presents a new approach to obtaining information directly from the CAD/CAM software by the CAD geometry and the toolpath (CC and CL data) which can be used for further developments. The current study employs this methodology to predict the machined surface texture for milling freeFig. 2. Flowchart of the proposed methodology for surface prediction. Fig. 3. NX Open routine schematic and flowchart. F. Marin et al. Engineering Science and Technology, an International Journal 55 (2024) 101757 5 form shapes in 5-axis machining. The methodology consists of discretizing the complex surface during a 5-axis milling operation and calculating the tool and surface instantaneous positions along the toolpath. This approach can also be extended to predict more complex problems dependent on the force, considering both CL (Cutter Location) and CC (Contact Condition) data. The proposed development was organized into six topics. First is presented a (2.1) overview of the proposed methodology, then the (2.2) routine for data collection developed in the software Siemens® NX is described, followed by the (2.3) geometrical modeling of the tooltip, (2.4) toolpath data treatment, and (2.5) surface prediction is presented with two modeling approaches. The first one considers the instantaneous position of the tooltip to predict the surface texture. In the second one, the trajectory of the infinitesimal elements of the tool are taken into account to improve the lateral roughness prediction of the milling process. In both cases, the effect of the tool run-out was considered. Finally, the materials and methods for evaluating the implemented routine and the mathematical modeling are presented (2.6). 2.1. Proposed methodology A programming routine was developed to obtain a set of data necessary for modeling the surface. The first step of the proposed methodology is obtaining any complex surface’s cutting contact (CC), cutting location (CL), and respective normal vectors by an implemented routine on the CAD/CAM software. A standard language C#, was used in the programming routine because it is compatible with the open interface (NX Open) of the software used – Siemens® NX. It is worth noting that the same methodology can be adapted to other software programs that have an open programming interface. Then, a geometrical model was developed based on the routine outputs to predict the ball-ended cutting-edge movement and the surface of 5-axis milling. In this second step, the geometries of the tool were mathematically modeled with a scope limited to ball-end tooltips with a constant helix angle. In this step, two approaches were considered. The first considers the dome after a complete rotation of the tool (instantaneous position), and the second one considers the trochoidal movement of the instantaneous cutting-edge (ICE) element (pondered movement), which allows to include, i.e., the tool run-out. After that, both data were used for surface texture evaluation, measuring topography, scallop height, and theoretical roughness. Fig. 2 presents the flowchart of the proposed methodology. 2.2. Routine for data collection For surface mapping, a routine was developed using an API (application programming interface) on the software Siemens® NX 1953 to obtain a list of the cutting location (CL), cutting contact (CC), and respective vectors. These data were obtained following a series of procedures in accordance with the developed routine. The first requirement is importing or modeling the workpiece geometry in the software, followed by the usual CAM programming. Thus, both geometry and trajectory will be available for computation and posterior milling. As a second requirement, it obtains the post-processed NC program file in the standard format (Cutter Location Source File − CLSF), providing normalized data to the routine containing the CC and CL data. In the case of the Siemens® NX software, the CC data were obtained through CAM post-processor once the output Contact Data is enabled. Then, with the path and directory of the exported CLSF file or the data extracted directly by the software routine, a series of filters and sorts are done while the output data missing is computed line-by-line – the surface normal vectors for a given contact point. An NX command was used to obtain the vector normal to the surface on a specific CC location. The command input requires a surface and a point to calculate the normal direction. Thus, a specific CC point is used for selecting both the position and the surface. The surface selection was automated by selecting the feature object closest to its specific CC point. Finally, the unitary vector related to the CC is calculated using a support line with its respective extreme points, and then all the features involved are destroyed; this procedure is detailed in Fig. 3. Highlight that most CAD/CAM software has closed architecture, so obtaining data relative to the surface in a specific position is not trivial. At the end of the calculation, all the non-cutting moves lines are filtered, and the other lines are presented in a standardized file with 12 columns. The first three columns represent the x, y, and z coordinates related to the CL points, columns 4, 5, and 6 present vectors i, j and k (related to the tool orientation), columns 7, 8, and 9 represent the x, y, and z coordinates of the respective CC points, and the last three columns represent the q, r, and s, vector normal to the surface on a given CC point, as exemplified in Fig. 3. 2.3. Geometrical modeling of the tooltip For the tool modeling, a reference system was defined using the right-hand rule and the center of the tooltip, with a clockwise rotation on the tool axis, as depicted in Fig. 4. Then, the tool is mathematically modeled. The local radius is calculated according to the tool’s height. Then, the axial and radial immersion angles are calculated to describe the edge position inside the tool’s dome. Finally, the effect of the tool run-out in ball-end tooltips can be included as a function of the tool’s height – r(z). Fig. 4 presents the ball-end tool schematic. Considering the given reference system at the end of the tooltip, Eq. (1) gives the coordinates of the dome of a ball-end tooltip. ⎧ ⎨ ⎩ xi=Rsin(ki)sin(θi) yi=Rsin(ki)cos(θi)for 0◦≤ki≤90◦ zi=R−Rcos(ki) (1) Where R is the radius of the tool, ki is the axial immersion angle adjacent to the vertex C (0, 0, R), shown in red in Fig. 4. For a given angular position θi, ki defines the position of the point P along the tool Fig. 4. Tool schematics for single radius tooltips. Source: https://www.geogebra.org/m/hkdr6rhj F. Marin et al. Engineering Science and Technology, an International Journal 55 (2024) 101757 6 helix using Eq. (1). To model the cutting edge, infinitesimal elements i must be defined as a function of the height. Considering that the angular position increases as the height increases, a helix can be defined at the tool dome. Defining an angular reference ∅ measured from the X-axis and the angular increment with height as φi, it gives the angular position θi, Eq. (2). θi=∅−φi(2) Where the angular increment with height φi is defined by the tangent at a given point with a plane perpendicular to the tool axis, Eq. (3). tan α =RΔφ Δz∴tan α i=Riφi zi (3) Considering that the ball-end tooltip has a constant helix angle, the infinitesimal edge increment with height can be expressed by Eq. (4). φi= (1−coski)tan α (4) Combining Eq. (1) to Eq. (4) is possible to define the edges of a ballend tooltip with a constant helix angle. Run-out ρ involves both manufacturing errors and clamping miss alignment of the tool-holder system. It can be described as a function of the local radius with the height, Eq. (5), thus dependent on the micro edge element and reliant on angular position and axial immersion angles. rj(z) = r(z) + ρ (z)(5) Where rj(z)is the local radius of the micro edge element related to the j flute, r(z)is the local radius of the tool, and ρ (z), the radial deviation of the respective micro edge element depends on the element’s position in the space. Considering an ideal manufacturing process for the tools, ρ (z)can be approximated to a linear function with a reference system as an offset (in the xy plane) and tilt ( τ ) from the reference tool axis, Eq. (6). ρ (z) = ρ 0(x,y) + f(z,θ)(6) Fig. 5 instantaneously exemplifies the linear movement of an infinitesimal edge element i with the effect of the tool run-out on the top view. Additionally, other factors, such as initial tool wear and thermal expansion, can be modeled similarly as a function of the local radius but are not considered in this work. Fig. 5 shows that in the milling process with run-out, it is possible that only one tooth generates the surface, so it is proposed a simple calculation to adjust the surface prediction based on the feed-per-tooth, tool run-out, and the number of teeth. The logic consists of comparing the bigger effective radius (r1=rmax), subtracting the theoretical crest wave (h c ) and comparing it to the other teeth; if the value of the effective radius is smaller, it will not participate in the surface generation. Also, the index number of those teeth must be saved for plotting. The average distance of the surface feed-per-tooth mark is given by Eq. (7). fz=fz(z zc)(7) Where zc is the number of teeth participating in the surface generation, given by Eq. (8). zc=∑ z−1 i=1 1+i:{i=1if rz>r1−hc;else i =0}(8) 2.4. Toolpath data treatment Using the tool geometry (section 2.3) and the toolpath generated by the CAD/CAM (section 2.2), with two procedures, it is possible to predict the surface with an error lower than the tolerance band used in the toolpath generation. The first is calculating the instantaneous positions, pondering the known contact positions (2.4.1). The second one is computing the angular position of the cutting-edge element according to the tool tilt and toolpath data (2.4.2). 2.4.1. Instantaneous position evaluation Usually, the spindle speed is much higher in the milling process than in the feed, making the spindle speed effect on the surface topography higher than the feed per tooth [36]. So, it is possible to consider that a complete rotation happens instantly, removing the material. However, in a discrete model, this assumption leads to an additional error associated with the difference between the segment’s lengths from the feedper-tooth (f z ). Thus, a good correlation between these parameters is important for the model’s accuracy. When the segment length is bigger than the f z , more material is removed, while when f z is bigger, the material is underestimated. Otherwise, it is necessary to calculate intermediary points to discretize the movement. Naturally, on complex surface milling, the distance between the NC points changes depending on the surface’s tolerance band and local curvature. So, it is necessary to interpolate the mapped surface data obtained by the routine according to the f z used to increase modeling accuracy. Fig. 6 presents a schematic of the method applied to compute intermediary points using the CL and its respective tool orientation. In the process of discretizing intermediate positions, their respective unitary vectors are also calculated. It is known that the unitary vector of the tool position (nt) obtained from the processed 5-axis CL file can be written in terms of ijk positions given the orthogonal cartesian components relative to x, y, and z, Eq. (9). nt=nxi+nyj+nzk(9) The data obtained using the routine gives the instantaneous tangent plane of the surface, described by the respective point CC and the unitary vector ns, Eq. (10). Fig. 5. Center deviation caused by run-out at a generic zi plane of the tool. Fig. 6. Schematics of toolpath adjustment for surface prediction considering instantaneous positions. F. Marin et al. Engineering Science and Technology, an International Journal 55 (2024) 101757 7 ns=nxq+nyr+nzs(10) To calculate the instantaneous direction (  fi), the product of the unitary vectors relatives to CC and CL is computed, Eq. (11). And by the cross-product of the tool position with the local surface vector, Eq. (12), it is obtained the instantaneous lateral pass gi. fi=ns×nt |ns×nt|(11) gi=ns×nt(12) 2.4.2. Adjustment of the angular position with the toolpath If the spindle speed is not considerably higher than the feed or a better estimation is required, the angular position of the tool edge must be considered. Additionally, cutting-edge orientation is required to include the effect of the tool’s helix and the run-out on the surface estimation. Thus, an additional procedure was developed, adjusting and synchronizing the angular position with the feed (axial movement of the tool). Equation (13) expresses the angular position in terms of the nominal feed per tooth, which allows the computation of the angular position for each CL (or intermediary points) according to the tool’s displacement. fz=ΔdT ti+1−ti =Δd ti+1−ti 2 π ω (13) Where Δd is the distance between consecutive CL points, T is one rotation period, ω the angular velocity, and ti+1 and ti are respectively the intermediate and the initial position during one rotation. Considering that the tool has a constant angular speed ( ω ), it is possible to indirectly correlate the feed distance covered with the angular position (θ). Splitting the toolpath into constant intermediary points will also result in a constant Δθ. With this consideration and knowing the total distance covered by the tool, it is also possible to obtain the angular position of any micro edge elements on the toolpath. Fig. 7 presents a 2D schematic of a Δθ weighted interpolation between the CL points. However, when new data points are generated, it is necessary to compute the related tool’s tilt. It is done by adjusting to the previously known unitary vectors and pondering with known positions. Furthermore, in both methods presented, instantaneous or pondered movement interpolation, the data created can be used for any surface. In the specific case of angular interpolation (Δθ), the discretization process is more critical and must be optimized to obtain a good correlation between the precision and the computational time. The smaller the Δθ, the better the precision, but the higher the computational time. Thus, the trochoidal toolpath of the ICE elements can be graphed using polynomial regressions (cubic spline) to improve processing time and maintain accuracy. 2.5. Methodology for surface evaluation First, the data were sorted according to the part’s world coordinate system (WCS) for easy graphical visualization and evaluation of the surface. To do so, a mesh at the xy plane was created considering the maximum and minimum positions of the NC code, leaving the high in z to be computed considering the instantaneous positions and the pondered movement of the tool. For each specific position, the minimum (or maximum) z hight is computed. The tools dome or infinitesimal edge element was translated and oriented according to (WCS) using the procedure described in Fig. 8. Hereafter, each geometrical feature was oriented according to the data obtained by the routine and optimized with the procedure presented in the previous section, obtaining the z height for each mesh position. Then, two different methods were applied to inspect the surface texture. The first was a simple evaluation of the scallop height to measure the theoretical roughness in the feed direction. The second is using the data cloud of the surface to compute the theoretical surface. 2.5.1. Scallop height considering instantaneous positions of the tool In order to obtain a local estimation of the surface, a simple model to calculate the scallop height is applied. It can be directly associated with the roughness and indirectly to measure the quality of the milling process. The tool’s geometry and cutting parameters feed-per-tooth (f z ) can be used to estimate the theoretical roughness of the surface considering the instantaneous positions (discretized toolpath), Eq. (14). The scallop height (hc) also can be used to calculate the roughness Rz trought Eq. (15) [37–39]. hc=R⎛ ⎝1− 1−(fz 2R)2 √⎞ ⎠(14) Rz=∑5 n=1Rzn n(15) To apply these equations throughout the 5-axis trajectory, the discretized model is used to calculate the deviations in relation to the instantaneous cutting plane for each position. Thus, the deviations are obtained as a function of the local inclination of the surface and a good approximation of the actual surface roughness. Knowing that the center of the tool has a low cutting speed and the plowing cutting phenomena is dominant, two regions were evaluated, the first in the central toolpath region where the center acts in the cut and another with a higher specific cutting speed, both on the feed direction. 2.5.2. Surface prediction using the mesh data cloud Translating the tool position and plotting the minimum z distance for each position of the equally distributed data cloud allows the simulated surface to be obtained. Then, the data were compared with confocal microscopy results of the curved surfaces (concave and convex regions) and with the form removed by Leica Map 6.2 software. Likewise, the toolpath of an infinitesimal cutting-edge element was plotted according Fig. 7. Schematics of toolpath discretization with a constant Δθ and tilt ponderation. Fig. 8. Geometrical rotation and translation procedure applied to tool dome and infinitesimal edge elements. F. Marin et al. Engineering Science and Technology, an International Journal 55 (2024) 101757 8 to a specific height, obtaining the trochoidal movement of the ICE element. The smaller the Δθ used for computation, the higher the precision of the trajectory of the element. 2.6. Case study experimental procedure A bladed shape part containing free-form concave and convex surfaces was designed using the software Siemens® NX and manufactured to evaluate the proposed methodology. The material used was a solid bulky cylinder with a nominal diameter of 46.5 mm of Waspalloy AMS 5706, the nominal composition presented in Table 1. The machine used to manufacture the blade was a multitask machining center Mazak i200 with smooth machining configuration (SMC) by default (balanced). The milling experiments were conducted utilizing the machine’s tailstock. The blade was finished using a ball-end tooltip with a 5 mm radius manufactured in micro grain solid carbide with 10 % Co coated with Table 1 Composition of WaspalloyAMS 5706. Ni Cr Co Mo Ti Fe Al Zr C Mn 58.64 19.34 12.27 3.82 3.04 1.35 1.33 0.05 0.04 0.03 Si Cu B Mg P N S Se Pb Ag 0.03 0.02 0.005 0.005 <0.001 a) b) Fig. 9. a) Part model and machining strategies. b) Presetting results. Fig. 10. Programmed routine for 5-axis data acquisition outputs. F. Marin et al. Engineering Science and Technology, an International Journal 55 (2024) 101757 9 multilayer TiSiN (3800 HV 0.05) with four flutes and a 30◦helix angle. The tool was mounted on a high-torque tool holder with a 40 mm cantilever, and a presetter Zoller® SmarTcheck 600 was used to measure tool length and the static run-out, as depicted in Fig. 9b. The milling trajectories were programmed on the software Siemens® NX, considering rough, semi-finishing, and finishing operations. A spiral toolpath with 0.4 mm of a e was used in the semi-finishing operation, leaving the part with 0.4 mm stock for the finishing operation. The finishing process was conducted in down-milling with constant a p (0.4 mm), a e (0.2 mm), f z (0.05 mm/tooth), and cutting speed of 80 m/min (parameters recommended by the tool maker). Then, three different regions with 10 mm lengths were evaluated using, in all of them, a 5-axis spiral strategy (with a 0.01 mm tolerance band). This strategy avoids tool entrances and exits that damage the surface and compromise its integrity. Also, the direct action of the tooltip was avoided by tilting the tool in the feed direction. Three cutting engagements were assessed, one in each region, 5, 15, and 25 degrees, respectively. Fig. 9a depicts the toolpath strategies. The procedure presented in section 2.5.2 was applied to inspect the manufactured parts. A confocal microscope Leica® DCM3D together with the software Leica® Map 6.2, were used for surface visualization, roughness measurement, and data cloud exportation. The surface data was treated using Matlab® 2021 software to compare the values with the simulation predictions using the proposed model. During the data treatment, the surface was analyzed as scanned and without form, where a 5th-order polynomial regression was used to ease the roughness and model evaluation. 3. Results and discussions The proposed methodology and implemented routine to extract data for predicting and evaluating free-form surface manufactured by milling is presented as follows: 3.1 Routine outputs, 3.2. Data treatment and model output; and 3.3 Study case results and modeling evaluation. 3.1. Routine outputs Fig. 10 presents an example of a 5-axis toolpath program and the results obtained using the developed methodology. As can be seen, the figure presents the temporary features (lines and points) used to support Fig. 11. a) Ball-end tool dome and helix edge; b) 5-axis milling on a free-form surface simulated by the cam software; c) toolpath data treatment for instantaneous surface modeling and application; d) toolpath data treatment for ice trochoidal movement and application. F. Marin et al.