Evaluation of Non-Planar Tool Interaction in Milling of Shaped Surfaces Using a Copy Milling Cutter

Abstract

When milling complex parts or curved surfaces, we encounter several problems that must be addressed in the production process. Various factors affect the quality and accuracy of production. The main objective of this paper was to analyse the size of the effective tool diameter when machining a shaped surface concerning the selected position, namely at the perpendicular position of the tool. At the same time, the distribution of the engagement area on the tool surface was evaluated by extracting the area content and volume data at the point of contact. This study highlights the importance of the choice of finishing strategy in a CAM system. The results showed that the tool engagement size corresponded to the extracted data describing the area and volume for each tool position with regards to the curvature of the surface. The negative deviations obtained by the scanning method were due to machining near the tool centre, which was affected by the changing effective tool diameter.

1. Introduction

The manufacturing process of machining is a specific process mainly found in automotive and aircraft parts production, as well as plastics processing in the area of shaped cavities production. Input factors entering the machining process affect the resulting quality of the machined surface, as well as shape and dimensional accuracy, and include cutting conditions, type of choice of machine tool, type of fixture, size of the machined sample, type of tool, selected milling strategy, clamping of the part, unloading of the tool, cooling medium and, last but not least, the post-processor and the control system of the machine [1].

In the production of shaped parts, a larger portion of the process is focused on optimizing production from the point of view of productivity with regard to the quality and accuracy of production. Production optimization can be addressed by correctly selecting milling strategy, machines, etc. However, due to the production of these shaped surfaces, it is necessary to understand the behaviour of the tool in the cutting process. The milling of shaped surfaces is characterized by non-planar interactions between the tool and workpiece [2,3]. In the case of machining free form surfaces, which are complex surfaces, it is important to select a suitable milling strategy to achieve not only efficient production in terms of time, but also surface quality in the form of dimensional and shape accuracy. Therefore, great importance is placed on appropriate finishing strategies, as well as their use [4]. Machining shaped surfaces is very time-consuming. In the industrial sector, these shapes are machined on 3-axis machines, which leads to some inaccuracies. This requires special positioning and repositioning of the workpiece for machined surfaces. Machining is only possible with a certain degree of accuracy on these machines. For general machining of surfaces, only one tool is often used, when it is necessary to consider higher machining time and possible errors. Undesirable effects arising during the cutting process can be eliminated using a tool with a suitable diameter. In 3-axis milling, the tool is controlled in three directions, the position of the tool relative to the general position of the surface is determined by three coordinates. In conventional milling, the diameter of the ball mill is limited by the curvature of the convex shapes. Exceeding the used diameter of the tool is most often manifested by undercutting the machined surface. Therefore, it is necessary to ensure the correct ratio between the radius of the used tool and that of the curvature of the machined surface [5].

CAx systems are widely used to simulate various production processes such as machining, forming, joining, welding, or additive manufacturing processes and other production methods [6,7,8,9]. The selection of suitable strategies was implemented in the SolidCAM CAM system. With the appropriate selection of milling strategies, we can machine different types of surfaces, which is greatly helped by the simulation modes in the CAM system. Hence, the CAM system plays an important role in the field of machining parts with complex shapes, where it is not possible to use classic workshop programming containing only certain cycles intended for 2.5D milling.

It is important to define the tool used to machine a complex shape (general surface), considering the maximum possible usable diameter. It is important to prevent local undercutting of the workpiece. Therefore, it is necessary to determine whether the shape of the general area is concave and saddle-like and what is the degree of curvature. This control is usually performed in CAM systems where it is possible to practically control undercutting or the collision of the tool with the workpiece [10]. The use of copying tools in the field of finishing operations has its advantages due to adaptability to changes in surface curvature. When using the finishing strategies applied to the 3-axis machining process, the cutting tool must always be tangent to the machined surface. Using a copying cutter is a position that is given by the contact point of the cutter, the centre of the tool, the position of the cutter, the radius of the tool, and the vector of the tool orientation [11]. When machining shaped parts, the area of engagement of the cutter when the tool is in contact with the machined surface is also important, which changes from the point of view of the curvature of the surface during the cutting process. It is difficult to predict and analyse the area of capture in a specific position of the tool to the machined surface. The use of copying tools in the field of finishing operations has its advantages due to adaptability to changes in surface curvature. Nevertheless, the tool grip changes during the manufacturing process, resulting in process instability [12,13].

Many studies have been devoted to applying multi-axis machining in the production of shaped surfaces due to less tool wear owing to its tilting, and to avoid the negative effect known as material ploughing, which appears when the tool is in a perpendicular position relative to the machined surface. An accompanying phenomenon of material ploughing is the action of the vertical position of the tool, while there is no shearing mechanism, as the action of the centre of the tool does not create enough chips for removal, thus there is a zero-cutting speed at the cutting point [14]. In his research, Varga [15] assessed the effect of the vertical position of the tool on the quality and topography of the machined flat surface, evaluated the geometric characteristics of the surface, and compared four strategies. The author also evaluated the roughness of the surface, where the best results were obtained with the circular pocket and Constant step pocket auto boundary strategies. He expanded his research with knowledge regarding the influence of finishing milling strategies on the production quality of the shaped surface applied in 3-axis milling of aluminium samples, while it was a combination of convex and concave curves.

Varga also assessed the roughness and choice of a milling strategy, where the Constant Z strategy achieved the best results in terms of roughness [16]. The author proposed a method of comprehensive assessment of the quality of the machined surfaces (surface topography, surface roughness, shape deviation, etc.), useful in deciding on the production method and the use of appropriate strategies in 3-axis milling. For the experiment, he selected an aluminium alloy that had a parabolic shape and evaluated the elements of the surfaces with respect to the chosen height of the sample using a digital microscope, where the regular arrangement of the tool paths along the contour lines was achieved only with the Constant Z strategy [17]. In their works, Kaymakci [18] and Diciuc [19] evaluated the methods dealing with the interaction between the tool and the machined surface. Since CAM systems are important in the production of shaped surfaces, Bagci [20] analysed the accuracy of their use due to the difference in the generation of tool paths compared to real production. The difference in the results was up to 30%.

Wojciechowski [21] investigated the vibrations during the production of shaped surfaces and their estimation, and emphasized the influence of tool overhang. In his results, Tuysuz [22] reported effective cutting speeds in areas where the interaction of the tool with respect to the machined surface reached a tangential level. From the point of view of dimensional and shape accuracy assessment, various surface assessment methods are used, resulting in various analyses and reports providing information on production accuracy, shape deviations, etc. [23]. Mizugaki et al. [24] investigated the design of an algorithm designed to predict the quality of the machined surface where the authors used a copy cutter for the finishing process. Capla [25] investigated the contact point in the interaction of the cutting tool and the machined surface in multi-axis machining, where he changed the angle of inclination of the tool in the cutting process.

The comparison of milling finishing strategies in the production of shaped surfaces and the analysis of different methods for the evaluation of surface topography with respect to the effective tool diameter for three different heights on the workpiece was investigated by Varga [26]. The author showed that the changes in the observed parameters were due to the variation of the tool effective diameter as well as the influence of the strategy used. The best results were obtained with the Constant Z strategy, in which uniform toolpaths were achieved over the whole height of the specimen. The cutting edge was displaced in height during machining when the setting angle was changed stepwise. This resulted in the approximation of a semi-spherical shape by a series of low tapered surfaces. Bou-jelbene [27] studied the influence of the effective tool diameter in the production of dies, in which the material investigated was Super Plast ^® 300 (SP300) grade steel and where a Ti(CN) coated copy tool was used. The cutting mechanism based on the changing effective diameter of the cutter when milling free-form surfaces was also investigated, in which the authors focused mainly on the change of cutting forces when the tool contacts the workpiece [28,29].

The optimisation of the selection of finishing strategies was studied by Toh [30], focusing on high-speed machining for HSM. Ramos et al. [31] investigated the machining of a part that equally contained concave and convex surfaces, compared different strategies such as linear strategy, beams, and 3D offset, and evaluated their effect on texture and roughness. The worst results for machined surface texture and roughness were obtained with the linear strategy and the best with the 3D offset strategy. The effect of toolpath on milling a low curvature convex surface using hardened material was investigated by Shaghayegh et al. [32]. The authors also investigated the influence of milling strategies, namely linear, beam, 3D offset, and helix in the workpiece-tool relationship, evaluating cutting forces, surface texture, and machining time. The results showed that the beam strategy achieved the best surface quality, while the spiral strategy was the worst machined surface quality. Matras and Kowalczyk [33] analysed the effect of four milling strategies (circle, square, beams, and Z level) on the production of a freeform surface where the material was an aluminium alloy. The lowest roughness parameter was obtained only for the circular toolpath due to the defined requirement of achieving roughness. Sadeghi et al. [34] investigated the effect of different strategies (linear, 3D offset, spiral, and beam) and machining parameters on the microhardness of a convexly curved surface of 1.4903 stainless steel, and the design of experiments was carried out using the Taguchi’s method. The results showed that different toolpaths have different effects on the microhardness of the milled surfaces, where the spiral strategy reached the maximum hardness and the beam strategy reached the minimum hardness. The tool-workpiece contact relationship in freeform machining was investigated by Arruda [35], who compared descending, ascending milling, and tool-centred machining.

In addition to virtual modelling, a reverse engineering method is also used to obtain a 3D model from a physical (real) part. One of the possible methods is digitisation of the physical part using a non-contact optical scanner. The output format is the so-called cloud of points. Currently, various aspects focused on the milling of shaped surfaces and their evaluation are being investigated. Previous studies have assessed the selection of a suitable copying tool based on a cloud of points, without considering the specific shape of the machined surface [36,37,38].

An important output of the evaluation is data that contribute to the optimisation of the machining process. Data evaluation using the scanning method allows us to verify the correctness of the selection of the milling strategy as well as the influence of the effective diameter of the tool on the quality or accuracy of the machined surface [39]. When milling free-form surfaces, it is possible to use 3-axis and 5-axis machining, although the choice of machine is conditioned by the shape of the future part and the required accuracy. In his research, Sadílek [40] compared 3-axis and 5-axis machining in the production of shaped surfaces and achieved the highest accuracy precisely using 5-axis machining. The author also compared shape deviations between these two methods of machining, where better surface quality and smaller deviations were demonstrated in 5-axis machining. The deviation of the produced shape was evaluated, as well as the effect of the effective diameter with respect to the contact point of the tool and the workpiece in the cutting process. Grešová [41] evaluated the shape deviations of a free-form surface using a 3D laser scanner. Better results in terms of the quality of the machined surface in 5-axis machining were achieved with linear and spiral strategies and in 3-axis machining with an offset strategy. Tran [42] focused on the evaluation of the machined surface based on the cloud of points obtained by different scanning devices where the evaluation parameters were the roughness and shape deviation of the samples. The results pointed to the importance of the data obtained from the point cloud. Masood et al. [43] and Bey [11] assessed the design of a mathematical model to generate tool paths for the production of a shape sample intended for 3-axis machining and evaluate the individual roughness parameters from the obtained point cloud data using scanning.

Li and Gu [44] compared machined surfaces using the scanning method, where they used a formula based on an extended Gaussian curve and normal orientation as an evaluation module. Deng [45] focused on the selection of characteristic points that allowed for defining the limitation zone for the assessment of surface quality, where, thanks to this factor, he defined the minimum zone for evaluating the machined surface. Yuwen [46] evaluated the quality of shaped surfaces obtained by milling using the SQP (sequential quadratic programming) method, which he applied to process optimisation. This method is used to evaluate the quality of the surface at a large set of sampling points. The 5-axis surface inspection technology known as sweep scan was applied by Zhang [47] who distributed the free form of the machined surface into parts of elementary shapes; based on this, the necessary algorithms were designed that defined the optimal scanning path. Therefore, the control of machined surface is improved. Chen [48] pointed to the analysis of the cutting effect, where he compared different positions of the tool while applying the method of geometric modelling and numerical simulation.

When milling shaped surfaces, the area of engagement of the tool with the machined surface changes along the path of the cutter. It is therefore difficult to analyse its distribution in terms of size, area, or volume. The tool feed when milling shaped surfaces can be determined based on the start and exit angles as a function of the axial height along the tool axis. In the case of a possible evaluation, we may encounter several evaluation procedures, such as Boolean operations, spatial operations, and the use of discrete vectors. Applying the subtraction method between the model of the tool and the machined surface enabled through Boolean operations is considered the most accurate [49,50]. The application of scanning methods was also investigated by Kender [51], who analysed several reverse engineering techniques used for mechanical systems and described the benefit of their use for various components in the field of automotive design [52].

The study focused on the selection of the finishing strategy and the consideration of its selection through the display of residual materials in the CAM system. Since in the machining process the position and orientation of the tool is important, the authors proposed a methodology for evaluating the effective diameter of the tool when using a copying cutter in the production of a shaped surface. An analysis of the changing contact zone in the tool-workpiece relationship was conducted, as changing the position of the cutter during the milling of shaped surfaces changes the distribution of the area of engagement on the surface of the tool. Among other things, this analysis provides data from the section area in the form of area content and volume at the point of the contact surface. Kender also described the benefit of using various components in the field of automotive design. To date, data on this transformation are lacking; therefore, further research is warranted to better understand the changing area of the tool grip when milling shaped surfaces. Likewise, based on a broader review of the literature, there is a lack of research on the analysis of the effective diameter of the tool when machining the shaped surface in the point of contact of the perpendicular position of the tool–workpiece. Understanding the contact zone in the tool–workpiece relationship when milling shaped surfaces can contribute to a better analysis of the cutting forces in the machining process, which have a significant impact on the machining accuracy, as well as the chip thickness that corresponds to the cutting edge of the tool used.

2. Materials and Methods

To adequately illustrate the possibility of producing complex and free-form surfaces, the profile of a human face (Figure 1) created in the Rhinoceros 2020 graphics program was selected as a CAD model. The model was extracted from the Rhino environment into STL format and saved in STEP format using the STL2STEP converter. This format is supported by the CAD system SolidWorks 2020, which forms the basis for the chosen CAM system SolidCAM 2022, which allowed us to define the necessary milling strategies for machining the desired shape and dimensions.

An aluminium alloy was chosen as a semi-finished product AlCu4Mg (Slovalco, a. s., Žiar nad Hronom, Slovakia), tensile strength = 420 MPa; yield strength = 240 MPa; hardness = 120 HB. A 3-axis milling machine was used to produce a test sample with dimensions of 100 × 80 × 40 mm EMCO Mill 155 ((EMCO MAIER Ges.m.b.H., Hallein, Austria) with a control system Heidenhain TNC 426. The maximum speed of the machine spindle is 5000 rpm. Production took place in six operations, where three types of tools were used as shown in

Table 1. Cutting parameters used in the experiment.

Tool Diameter [mm]	Cutting Speed [m.min −1]	Feed per Tooth [mm]	Tooth Number	Tool Code
End Mill D 18	237	0.25	4200	AMS2018S
Ball Nose End Mill D6	94	0.015	4900	S510602.060
Ball Nose End Mill D4	63	0.008	4900	S510602.040

. All tools were from the Korloy company, while the cutting conditions corresponded to the manufacturer’s recommendations.

Semi-finish and finish operation were tools clamped in a BT-40 clamping system by a mechanical collet chuck. Overhang of the tool was 40 mm. A mineral emulsion-based coolant was used for cooling during production. For the clamping of the specimen, a vise type KSX 125 (Schunk GmbH & Co. KG, Brackenheim, Germany) was used, with the specimen clamped in the middle of the vise jaws.

The choice of finishing strategies was conditioned by previous studies, as well as the most frequent use in the production of the mentioned shapes. The milling strategies linear, constant step and constant Z in the simulation mode that SolidCAM 2023 includes were selected as finishing operations and compared with each other. This is the SolidVerify mode, in which, based on the consideration of the residual material after machining with the chosen strategy, the most suitable one for production was selected.

The implementation of the experiment included the following sequence:

• Roughing: face cylindrical cutter D18 mm with two interchangeable cutters plates marked APXT11T3PDR-MA, axial depth of cut a_p = 1 mm, radial depth of cut a_e = 0.6 mm, tool path tolerance T = 0.1 mm, surface allowance P = 0.5 mm
• Semi-finishing—ball nose end mill D6 mm, cutting material solid carbide, number of teeth 2, coating AlTiN, tool geometry λ = 30°, γ = 12°, machining strategy—linear, axial depth of cut a_p = 0.5 mm, radial depth of cut a_e = 0.5 mm, surface allowance P = 0.2 mm
• Finishing—ball nose end mill D4 mm, cutting material solid carbide, number of teeth 2, coating AlTiN, tool geometry λ = 30°, γ = 12°, machining strategy—constant Z, radial depth of cut a_e = 0.2 mm, tool path tolerance T = 0.01 mm

The following sequences were chosen to evaluate the results of the experiment:

• Comparison of machined surfaces between CAM system and real production
• Evaluation of the effective diameter of the tool Deff with respect to the contact of the tool and the workpiece
• Evaluation of tool surface area distribution using areal content and volume data extraction at the contact patch location
• Assessment of surface deviations by the 3D scanning method—scanner FARO Laser ScanArm V3 (FARO Technologies Italy S.r.l., Rezzato, Italy)

2.1. Methodology for Evaluating the Effective Diameter of the Tool with Regard to the Contact between the Tool and the Workpiece

The evaluation of the effective diameter of the Deff cutter was carried out directly in the CAD system, where it was possible to perform a measurement based on the contact surface of the tool with respect to the workpiece in a specific position. The selection of individual positions on the shape surface is shown in Figure 2a and an example of displaying the contact of the tool with the workpiece for position no. 8 shows Figure 2b.

2.2. Methodology for Assessing the Distribution of the Engagement Area on the Tool Surface

Using the functions that are part of the Solidworks 2022 CAD system (Boolean operations), it was possible to analyse the contact surface forming a tangent in the tool-workpiece relationship. By obtaining a CL point for each tool position, it was possible to extract the contact area between the tool and the workpiece using the Body Boolean function, which gave the resulting 3D contact area [36]. This contact surface was subsequently projected onto a plane perpendicular to the tool axis. Through the application of Boolean functions, the boundaries and curves of the contact area were obtained. By creating perpendicular cuts to the axis of the tool, individual cuts in the form of disks, circles were projected onto a plane perpendicular to the cutter axis, as shown in Figure 3.

To compare the amount of tool engagement in the cutting process, which is affected by the contact area of the tool with respect to the machined surface, the positions were chosen as shown in Figure 2.

2.3. Methodology for Evaluating Surface Deviations Using the Scanning Method

The choice of this method was based on its effective scanning of the surface and obtaining results that would not be possible to achieve with the touch method. For evaluation purposes, a 3D scanning methodology was used. The FARO Laser ScanArm V3 scanner was chosen for scanning purposes, which combines CMM touch scanner technology with a non-contact laser scanning method. The device works in seven axes. The laser scanning method was used for the digitisation of the machined part. The measuring system was connected to a PC via a USB port, where it operates with Polyworks software, but compatible with Geomagic or Rapidform software. At the ideal distance of the scanning head from the body, the system can capture up to 640 points in one laser beam line, up to 30 images per second. The scanning took place in the work module IM Inspect project of the Polyworks software, where the software automatically recalculated the overlap of the scan with the 3D model. The accuracy of the scanner measurement is given by the manufacturer and its calibration, while the value is ±35 µm.

The machined surface of the face was oriented upwards. The facial profile was scanned by a laser scanning head, where the resulting 3D scan contained a total of 130.844 points as shown in Figure 4.

This was followed by a comparison of the 3D scan of the face profile with its original 3D model. After importing the point cloud of the face scan and defining the X, Y, and Z coordinate system (Figure 5a), the 3D model of your face was also imported (Figure 5b).

In the next procedure, the 3D model was overlaid with a scan (Figure 6), where it was necessary to identify all dimensional deviations of the manufactured component compared to the original 3D model, from which the CNC program was initially prepared and generated. This operation took place in the work module IM Inspect project of the Polyworks software with automatic recalculation of the overlap of the scan with the 3D model. Such a comparison of dimensions revealed in the results is the final accuracy of machining and thus helped to identify possible shortcomings or deviations from the designed dimensions. In the analysis process of evaluating surface deviations, a positive and negative deviation of ±0.5 mm of the produced dimension from the original 3D model was set, where the reference surface was the 3D model.

3. Results

3.1. Comparison of Machined Surfaces between CAM System and Real Production

A visual comparison of the machined surface in the Solid Verify simulation mode and the real state after the roughing operation (Figure 7) and the pre-finishing operation is shown in Figure 8, where individual tool paths, the simulation mode, and the real state of production are plotted.

The comparison shows that the obtained shape of the machined surface in the simulation corresponds with the obtained shape on the CNC milling machine. After roughing, it was possible to visually see an irregular surface, for the removal of which a pre-finishing operation was used.

The final state after the pre-finishing operation was the machined surface that corresponded to the simulation mode, while an allowance of 0.2 mm was left for the last production operation, the so-called finishing. The choice of the final strategy corresponded to three strategies, namely linear, constant step over, and constant Z, which were compared with each other. The most suitable one was chosen for the production process and that was the constant Z strategy. A comparison of the display of the tool paths in the simulation mode and the residual material in the CAM system during the finishing operation for the linear strategy (Figure 9), the constant step over strategy (Figure 10) and the constant Z strategy shows Figure 11.

From the given display of residual machining after choosing one of the finishing strategies, the constant Z strategy proved to be effective, leaving a small amount of material for finishing. Subsequently, it was used for the production process, where the resulting part after milling is shown in Figure 11.

3.2. Evaluation of the Effective Diameter of the Tool Deff with Respect to the Contact of the Tool and the Workpiece

Figure 12 shows the resulting interactions between the tool and the machined surface for specific positions.

To obtain the necessary data considering the changing effective diameter of the tool with respect to the machined surface and its curvature, every single interaction in the tool-workpiece relationship according to Figure 12 was gradually analysed. At the beginning of the analysis, visual maps were made showing the border distances within this interaction, the contact area as shown in

Table 2. Visual maps showing tool-workpiece interaction boundary/contact area distances.

Position	1	2	3	4	5	6	7	8	9

in a specific plane.

Obtained maximum and minimum values of the effective diameter of the tool at the point of contact with the workpiece relative to the position of the tool are shown in

Table 3. Measured maximum and minimum values for the parameter Deff.

	Position 1	Position 2	Position 3	Position 4	Position 5	Position 6	Position 7	Position 8	Position 9
Deff max [mm]	1.252	1.605	1.442	1.081	1.843	1.474	1.235	0.992	1.189
Deff min [mm]	0.398	1.049	0.802	0.410	1.425	0.796	0.471	0.174	0.393

.

A graphic representation of the maximum and minimum values of the Deff parameter with respect to the position of the tool and the machined surface is shown in Figure 13.

From the graphic comparison of the Deff parameter depending on the position of the tool and the contact machined surface, a decreasing tendency was demonstrated in all positions of the tool. The most significant difference with respect to Deff max and Deff min for a specific position of the instrument was manifested in the position of instrument No. 8. Within this interaction of the tool with the workpiece, the Deff min parameter was almost 6 times smaller compared to the Deff max value. The smallest difference ratio was recorded in the position of tool No. 2.

3.3. Evaluating Tool Surface Area Distribution Using Data Extraction

The tool engagement when milling shaped surfaces can be determined based on the start and exit angles as a function of the axial height along the tool axis. Since the capture area is formed by a combination of start and exit angles at the point of contact with the machined surface, the next procedure was to assign these angles to each projected circle in a plane perpendicular to the tool axis. The extracted values of surface content and volume of the cutting part of the tool at the point of contact with the machined surface are described in

Table 4. Extracted values of surface content and volume of the cutting part of the tool at the point of contact with the machined surface.

	Position 1	Position 2	Position 3	Position 4	Position 5	Position 6	Position 7	Position 8	Position 9
Surface [mm2]	8.467	7.197	5.751	5.541	5.513	5.096	5.080	2.894	0.921
Volume [mm3]	0.806	0.513	0.397	0.373	0.329	0.269	0.256	0.092	0.009

. A graphical comparison of these data depending on the position of the tool and the contact machined surface is described in Figure 14.

The graphic comparison of surface content and volume depending on the position of the tool and the contact machined surface shows that in both cases a decreasing tendency of these parameters was achieved. It is possible that the highest value of surface content was achieved at the position of tool no. 1 and the smallest position of the tool with respect to the machined surface was at position no. 9. The same was the case with the volume comparison. Figure 15 shows the overall comparison of all extracted data (Deff max, Deff min, area content, and volume) at the point of contact of the tool with respect to the curvature of the surface.

When milling shaped surfaces, not only the value of the effective diameter of the tool changes due to the curvature of the shaped surface, but also the start and exit angle changes in the area of the cutting part of the tool. Individual angles are described in

Table 5. Values of the start and exit angle define the grip of the tool at the point of contact with the machined surface.

	Position 1	Position 2	Position 3	Position 4	Position 5	Position 6	Position 7	Position 8	Position 9
Start angle	71.29	27.95	71.25	69.05	50.81	35.12	41.74	20.77	55.81
Exit angle	178.26	102.69	157.58	165.44	110.66	116.94	125.54	91.95	90.99

.

Figure 16 shows the individual positions for which the start and exit angles were analysed at the point of tool engagement with respect to the curvature of the surface. Their closer evaluation and comparison in the form of a graph, when the tool and workpiece are in contact with respect to a specific position of the tool, are shown in Figure 17.

The size and shape of the tool engagement in the cutting process affect the length and time of the engagement in relation to the cutting edge of the tool during one revolution of the tool. As shown in Figure 17, the start and exit angles defining the tool engagement determine which part of the cutting tool participates in the cutting process. The smallest engagement of the tool due to the curvature of the shaped surface corresponded to position no. 9 and the largest to position no. 1.

The obtained image sizes also corresponded to the extracted data describing the size of the surface and volume for individual positions of the tool with respect to the curvature of the surface (Figure 18). For position no. 9, an area with a value of 0.921 mm² was measured, which represented the smallest value among all positions, and for position no. 1, an area with a value of 8.467 mm² was measured in comparison with the other positions. As for the volume, for position no. 9, a capture volume of 0.009 mm³ was achieved, which represented the smallest value among all positions. For position no. 1, compared to the other positions, the obtained volume was 0.806 mm³. The obtained information about the start angle at the location of the tool engagement defines its input position.

3.4. Evaluation of Surface Deviations by the 3D Scanning Method

The position of the scanned points corresponded to the applied deviation range as shown in Figure 19.

Nine points of contact between the tool and the machined surface were identified and selected, where the 3D dimensional deviations of the selected 3D scan points were plotted in the X, Y, and Z coordinate system. These control points are shown in Figure 20. Based on the set deviation and the colour coding of the measured points, it was possible to identify the points that characterize the uncut or undercut points to the final dimension. The measured surface deviations by the scanning method are shown in

Table 6. Measured surface deviations for specific tool positions with respect to surface curvature.

Position 1	Position 2	Position 3	Position 4	Position 5	Position 6	Position 7	Position 8	Position 9
−0.036	−0.146	−0.002	−0.139	0.114	0.028	−0.075	−0.102	−0.127

.

Analysis of the deviations of the surface showed that the places defined the undercut in the machining process, which are characterized by negative deviations compared to the initial 3D model. The maximum negative deviation was reached at position no. 2 (value −0.146 mm) and the smallest negative at position no. 3 (value −0.002 mm). The highest positive deviation was measured at position no. 5 (value 0.114 mm) and the smallest positive at position no. 6. (value 0.028 mm). The negative deviations obtained by the scanning method were achieved due to machining near the centre of the tool, which was affected by the changing effective diameter of the tool for a given position due to the curvature of the surface. As a result, this has been shown to adversely affect production accuracy. The highest negative deviations for position no. 4 (−0.139 mm), position no. 8 (−0.102 mm), position no. 9 (−0.127 mm) corresponded to the maximum and minimum values of the Deff parameter compared to positive deviations for tool position no. 5 (0.114 mm) and for position no. 6 (0.028 mm). A graphical comparison between the obtained surface deviations given in Table 6 and the values of the effective diameter parameter Deff max and Deff min (Table 3), for specific positions of the tool with respect to the curvature of the surface is shown in Figure 21.

It can be concluded that the negative deviations obtained by the scanning method were due to machining near the tool centre, which was affected by the changing effective tool diameter for that position with respect to the curvature of the surface. As a result, an adverse effect on production accuracy was demonstrated. The highest negative deviations for position 4 (−0.139 mm), position 8 (−0.102 mm), and position 9 (−0.127 mm) corresponded to the maximum and minimum values of the Deff parameter compared to the positive deviations for tool position 5 (0.114 mm) and for position 6 (0.028 mm).

4. Discussion

The specimen used for the experiment represented a shape complex surface precisely due to showing the non-planar interaction between the tool and the machined surface. The results of the experiment showed the possibilities of evaluating this interaction and outputs that can be obtained by extracting different data from the point of contact between the tool and the machined surface. This was an assessment of the effective tool diameter, area, and volume parameter at the cutter engagement point.

The experiment demonstrated the importance of applying the CAM system due to the variability in the use of possible milling strategies. Constant Z milling strategy was the most suitable in terms of surface quality and residual material displayed in the CAM system with respect to the shape area.

The toolpaths generated in the SolidCAM corresponded to the toolpaths in the manufacturing process. It is therefore possible to conclude that the SolidCAM is suitable for the production of parts with shaped surfaces.

Different finishing milling strategies with a defined geometry of the cutting edges of the tools impact the selection of the optimal milling strategy and allow for the prediction of machining errors as well as the amount of machining in the form of residual material that remains after the given strategy.

The experiment proposed a methodology for comprehensive assessment of the area of the tool’s engagement with the machined surface using Boolean operations, where it is difficult to analyse its distribution in terms of size, area, or volume.

The selection of the milling strategy intended for finishing operations considered the size of the chip removal, which was defined by the milling cutter at the point of contact between the tool and the workpiece, which manifested itself in the form of changing maximum and minimum of the effective diameter of the tool, as well as considering the parameters of the chip in the form of the volume of material removed and surfaces.

In the case of the evaluation of shape deviations, it can be concluded that using the constant Z strategy was conditioned by the display of the smallest amount of the area of residual material after milling compared to the Linear and Constant step over strategies, when the deviation in the form of undercutting by more than −0.15 mm was not exceeded.

The negative deviations obtained by the scanning method were achieved due to machining near the centre of the tool, which was affected by the changing effective diameter of the tool.

Larger distortion could be seen in the places where the tool interacted with the machined surface and in places near the boundary corners or corner areas. Based on the experiments, the number of places listed was 5–8.

The aim of the contribution was to clarify the changing effective diameter of the tool with respect to the curvature of the machined surface and that in the specific point of contact of the tool with the surface. This experiment also points to the obtained information about the area and volume of the cutting surface at the point of contact of the tool with the workpiece, and identifies the cut points located at the output of the area cutter.

5. Conclusions

The aim of this study was to analyse the choice of finishing strategy in machining a shape complex surface and evaluate the size and effect of effective tool diameter on the accuracy of the shape complex surface. For the purpose of the experiment, aluminium alloy was chosen, and the machining process was carried out on a 3-axis milling machine. For the analysis of the evaluation of the effective tool diameter, 9 positions at different heights of the machined surface were selected. The individual analyses were extended to evaluate the cutter engagement at the tool contact point with respect to the curvature of the machined surface. Boolen operations were used for this analysis.

The results confirmed the suitability of using Boolean operations for determining the size of the change in the cutter engagement during “tool–workpiece” contact considering the curvature of the shaped surface. The size of the cutter engagement of the tool corresponded to the extracted data describing the size of the area and volume for individual positions of the tool. Evaluation of the size of the cutter engagement for a specific contact zone “tool–workpiece” with respect to the curvature of the surface proved the importance in the changing of tool effective diameter, as well as the size of the area and volume. The position of the cutter when machining shaped surfaces impacts the size of the area and volume of the future chip in the cutting process due to the changing start and exit angle at the location of the cut. The negative deviations obtained by the scanning method were achieved due to machining near the centre of the tool, which was affected by the changing effective of the tool diameter. The applied method contributes to the prediction of the chip size at the location of the milling cutter when milling a shaped surface.

One of the limits that can be stated within the experiment was the large sample used. As it is convenient to determine that in practice we will encounter larger dimensions, for example in the case of injection moulds, this experiment is only a part of further research, which is necessary to better understanding of the meaning of the contact zone between the tool and the workpiece. It is necessary to change the tool when milling shaped surfaces, which affects the achievement of sufficient surface quality and minimal adjustment in the case of production. This contact zone should be designed with the curvature of the surface in mind to allow optimal contact and minimize deviation from the desired shape. Consequently, the choice of milling strategy is important.

The following outlines are offered for further research:

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Analysis of the impact and efficiency of the machining process due to the change in the milling method (downward and upward milling).

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The effect of tilting the tool on the wear of the cutting edge during the machining of shaped surfaces with the support of cutting force measurement.

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Decomposition of shaped surfaces focused on deeper knowledge of the impact of milling strategies on surface topography.

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Comparison of the effectiveness of the length of the tool extension in processing shaped surfaces, where the stiffness, cutting forces, and deflection of the tool in contact with the workpiece would be evaluated.

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Comparison of high-speed milling strategies known as HSM with conventional milling strategies.

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A certain shortcoming is the lack of comparison of 3- and 5-axis machining. For this reason, we intend to conduct further research where 5-axis milling is applied, which will provide more detailed data not only on the influence of the tilting of the tool but also on the contact zone at the cutting point between the tools and the machined surface. It will also be possible to assess the size of the maximum and minimum average diameter of the tool, i.e., according to the measurement parameters in the form of the volume of the removed material and surfaces.

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There is also the possibility to examine the methodology used to assess the quality of the curved/shaped surface in the case of milling other materials, e.g., alloys that are difficult to machine, such as nickel, titanium, and stainless steel alloys.

The study was performed to verify our assumptions and obtain data for further research.