Analysis of the Effect of Machining Parameters on the Cutting Tool Deflection in Curved Surface Machining

Abstract

The aim of this study is to investigate the impact of machining parameters on the deflection of a cutting tool (i.e., end mill) in the milling of a surface with a curvilinear profile. Test samples were made of aluminium alloy EN AW-7075 T651. Experiments were conducted using the Gocator 2530 laser line profile sensor for real-time measurement of dynamic tool displacement with an inspection speed up to 10 kHz at resolution ranging from 0.028 to 0.054 mm. Response surface methodology was used. Five main technological factors were analysed: cutting speed, feed per tooth (cutting parameters), amplitude, term (curvilinear profile parameters), and the number of flutes (end mill parameter). Obtained data were filtered and visualised as 3D plots. The results showed that cutting speed and amplitude had the greatest impact on tool deflection, while feed per tooth also played a significant role in process stability. In particular, the use of tools with a higher number of flutes led to a considerable reduction in tool deflection, confirming their positive effect on the stability of the machining process. These findings may serve as a basis for the optimisation of machining parameters by taking into account the dynamic deformation of cutting tools.

1. Introduction

Milling is one of the most popular machining methods, valued for its ability to precisely shape even the most complex details. This process is widely used in various industries—from automotive and aerospace to the production of injection moulds and medical devices [1,2]. Its versatility allows for the production of both large structural elements and precision micro components [3]. Combined with CNC technology, milling enables mass production with high repeatability and quality [4,5,6]. That is why, it is an indispensable element of modern manufacturing processes.

Milling is the most widely used machining method for manufacturing thin-walled parts [7]. Thin-walled structures are typically produced by subtractive machining processes in which up to 95% of the initial material volume is removed, leading to a significant decrease in structural mass while maintaining the desired geometry and functionality of manufactured parts [8,9]. However, given their low structural rigidity, these parts are prone to deformation during the milling process, which can greatly reduce the accuracy of the final product [10,11].

Milling process is characterized by cyclic contact between the tool blade and the material, as well as variable cross-section of the cutting layer. A significant challenge arises when the milling process is conducted with the use of end mills, because the cutting forces acting on the tool can cause it to deflect or deviate from its nominal axis, directly affecting the accuracy of the component [12]. In practice, this implies the risk of dimensional inaccuracies or surface irregularities [13]. Therefore, it is crucial to properly select the machining parameters and tool design to suit the material type and product requirements. Bolar et al. [14] studied thin-wall milling and optimized the process parameters to minimize wall deflection caused by tool bending. Sivaprakash et al. [15] used 3D finite element simulations to analyze deformation and force distribution during milling of AISI 1045 steel.

Both experimental studies have been conducted and prediction models have been developed relating to the problem of curved surface machining. Li et al. [16] found that the quality of the machined surface depended on the length of an interpolation line segment. Sharp corners were formed at the intersection of straight sections, leading to increased stress concentration. To investigate the relationship between surface quality and interpolation line length in surface machining, they prepared a mathematical model of the surface topography in 3-axis ball-end milling of curved surfaces, based on acceleration and deceleration control. They found that when the length of the interpolation lines and the radius of curvature were constant, the stress concentration factor would decrease with increasing the central angle. When the radius of curvature and the central angle were maintained constant, the stress concentration factor increased with increasing the interpolation line length. A study [17] modelled process geometry (including feed direction, equivalent feed per tooth, and cutter entry/exit angle) in curved surface machining. The entire milling process was discretized into a series of small, straight surface milling cuts at feed per tooth intervals along the tool path, for which mathematical models of the process geometry were developed. Based on the conducted tests, it was found that the simulated cutting forces were consistent with the measurements in both magnitude and shape, with a difference of less than 12%, which confirmed the validity of the proposed mathematical models for calculating process geometry in curved surface machining. A study by Desai et al. [18] focused on modelling the process geometry in machining curved surfaces, specifically considering the effect of cutter runout. A mathematical model was introduced to calculate key process geometry parameters such as cutter–workpiece engagement and instantaneous uncut chip thickness, while accounting for cutter runout. Unlike simpler models, this approach incorporated the interaction between the current cutting tooth trajectory and those of previous teeth, resulting in a more accurate and realistic representation. Simulation results highlighted the importance of considering these trajectory interactions for precise prediction of machining parameters. The results clearly demonstrated that accurately determining process geometry under these conditions required a structured and comprehensive modelling approach, as proposed by the work.

Luo et al. [19] based their research on real-time deflection monitoring in the milling of a thin-walled element using thin-film PVDF (polyvinylidene fluoride) sensors with a cantilevered beam. The sensor was attached to the non-machined surface of the workpiece to copy the deflection excited by the dynamic cutting force. The correlation between input deflection and output voltage in the monitoring system was established through calibration testing. An analysis of the measured workpiece deflection indicated significant vibration as the cutting tool engaged in the material, and continuous vibration during machining was effectively captured by tracking workpiece deflection. Upon tool exit, the workpiece initially underwent forced vibration, followed by free vibration that gradually diminished until it was fully damped. The machining tests validated the effectiveness of the deflection monitoring approach, demonstrating the viability of using PVDF sensors for real-time vibration assessment in thin-walled component machining. Ma et al. [20] also applied a sensor module integrating a thin-film PVDF piezoelectric strain sensor with an in situ data logging system, which was developed to monitor feed and transverse forces in peripheral end milling. Mounted on the tool shank, the module captured dynamic tool strains and recorded data locally. Its close proximity to the signal source ensured minimal distortion and high signal fidelity. Using the Euler–Bernoulli beam theory and piezoelectric constitutive equations, they found that the measured signals were correlated to cutting forces. Experimental results showed high agreement with the measurements from a platform-type piezoelectric dynamometer. Huo et al. [21] analysed tool deflection in micro-milling and its influence on cutting forces and surface formation. A mathematical model was employed to calculate the distribution of cutting forces on the tool, incorporating the effects of tool elasticity and runout to determine the resulting deflection. Enhanced models for cutting force prediction and sidewall surface generation were developed, explicitly accounting for tool deflection. Experimental validation through micro-end-milling tests demonstrated a strong correlation between the simulated results and the experimental data, confirming the accuracy of the proposed models. Dépincé and Hascoet [22] based their research on prediction accounting for surface errors using a contact point method, considering tool deflection caused by the cutting force. To enable fast yet accurate surface evaluation, a cantilever beam model with an equivalent tool diameter was employed in their study. The milled surface geometry was determined using a contact point approach, showing its reliable performance for both constant and variable radial depths of cut. The accurate estimation of the surface profile allowed for offline prediction of the final machined geometry. Rao et al. [23] analysed the compensation of surface errors caused by tool deflection due to cutting forces in the peripheral milling of complex shapes with continuously varying curvature. Their work proposed a CNC-based tool path compensation strategy tailored for curved geometries, utilizing a mechanistic model for cutting force prediction and a cantilever beam model to estimate tool deflection. Experimental results across various geometries confirmed that the proposed approach significantly enhanced dimensional accuracy in the milling of curved surfaces.

Given that milling is characterized by intermittent tool operation and variable cross-section of the cut layer, the use of end mills poses a significant problem related to the deflection (from the nominal axis) of the tool due to the impact of the cutting force generated during machining [24]. Denkena et al. [25] introduced a sensory spindle slide system for real-time tool deflection control during high-speed milling. Lyu et al. [26] modelled 3D surface topographies in slot milling by incorporating time-dependent tool deflection into their simulation framework. Safari and Razfar [27] combined milling simulation with a genetic algorithm to predict cutting forces and minimize tool deflection.

Despite extensive research on tool deflection and machining dynamics, there remains a clear gap in experimental studies that simultaneously consider both technological parameters and tool path geometry in the milling of curvilinear surfaces. Most existing works have concentrated either on modelling cutting forces or on analysing process parameters in isolation, while the combined effect of surface profile geometry (amplitude and term) and tool characteristics (number of flutes) on tool deflection has not been comprehensively investigated. The research motivation is to experimentally evaluate these interactions under controlled conditions. The results of this study are expected to provide practical guidelines for minimizing tool deflection and improving dimensional accuracy in precision milling of curvilinear surfaces.

The objective of this study is to investigate end mill deflection in the milling of a curvilinear profile surface. Specifically, a relationship between tool deflection δ [mm] and time t [s] is examined, as well as its maximum value is determined for the following variable parameters: cutting speed v_c [m/min], feed per tooth f_z [mm/tooth], term T [mm], amplitude A [mm], and the number of tool blades (z = 2 and z = 3).

2. Materials and Methods

Experiments were performed to measure real-time tool deflection with the use of a high-resolution laser sensor. In addition, test samples of specified curvilinear geometry were designed, which made it possible to determine the effect of tool path geometry on obtained results.

Experiments were conducted on the samples made of aluminium alloy EN AW-7075 T651, a grade which is widely used in the aerospace industry and the design of structures requiring high strength-to-weight ratio. This material has very good machinability, high mechanical strength and moderate fatigue corrosion resistance. The chemical composition of this alloy is listed in

Table 1. Chemical composition of aluminium alloy EN AW-7075 T651 (wt%) (own elaboration, based on [28]).

Element	Zn	Mg	Cu	Cr	Fe	Si	Mn	Ti	Other (Total)
Content [%]	5.6–6.1	2.1–2.5	1.2–1.6	0.18–0.28	≤0.5	≤0.4	≤0.3	≤0.2	≤0.15

.

For purposes of the experiments, a set of test samples was prepared. The exemplary geometry of these samples is shown in Figure 1. The choice of such geometry resulted from its practical application, e.g., as turbine blades. Figure 1a shows a 3D model of a curvilinear profile sample after milling, while Figure 1b shows a dimensioned drawing of a curvilinear part for T = 30 mm and A = 3 mm.

Test samples were fabricated for different amplitude/term combinations. Five different amplitudes were tested: A = 1, 2, 3, 4, and 5 mm, and five different variants of the term: T = 20, 25, 30, 35, and 40 mm. As a result, a total of 25 unique geometric configurations were obtained reflecting different combinations of tool path parameters (e.g., A = 3 mm and T = 30 mm). These parameters made it possible to make a systematic assessment of the impact of variable curvilinear profile surface on the dynamic characteristic of the milling process, that is—tool deflection. The tested variants made it possible to reproduce actual machining conditions for a curvilinear surface as well as to determine the impact of sudden changes in a milling path direction on the generated cutting forces and resulting tool deflection.

Machining tests were conducted on a vertical machining centre, CNC Avia VMC 800 HS, which is dedicated to 3-axis precision machining. The machine has a high rigidity of the kinematic system, which allows for tool deflection measurements with minimum system errors resulting from the machine design. Basic parameters of this machining centre are:

• work tool surface: 1000 × 540 mm,
• displacement in the axes X/Y/Z: 800/540/620 mm,
• maximum spindle rotation: 24,000 rev/min,
• maximum feed rate: 40 m/min.

The milling process was conducted using two SECO cutting tools (Figure 2) designed for ISO N materials, including aluminium alloys. Specifically, the following end mills were employed:

• a 2-flute end mill JS412120D2SZ2.0 (Figure 2a),
• a 3-flute end mill JS413120D2SZ3.0 (Figure 2b).

They were selected to investigate the impact of the number of flutes on the process dynamics and tool defection in the milling of curvilinear surfaces. These tools have optimized geometry to be suitable for high-performance machining of lightweight alloys and their coated blades ensure increased wear resistance and improved chip removal.

Table 2. Technical parameters of cutting tools.

Technical Parameters	JS412120D2SZ2.0	JS413120D2SZ3.0
Cutting diameter [mm]	12
Shank diameter [mm]	12
Length of cut [mm]	24
Overall length [mm]	88
Number of flutes [-]	2	3

presents technical parameters of cutting tools.

The experiments were conducted under dry and down milling conditions, including constant and variable technological parameters of finishing operation. Prior to the actual milling process, test samples were rough milled in order to achieve the preliminary profile of the desired curvilinear surface. The milling process diagram is presented in Figure 3.

The following parameters were maintained constant during finishing:

• the axial depth of cut: a_p = 15 mm,
• the radial depth of cut: a_e = 1 mm.

The following variables were investigated in terms of their impact on tool deflection:

• cutting speed v_c: five values were tested ranging 300–900 m/min, changed every 150 m/min (i.e., v_c = 300, 450, 600, 750, 900 m/min),
• feed per tooth f_z: five values were tested ranging 0.025–0.125 mm/tooth, changed every 0.025 mm/tooth (i.e., f_z = 0.025; 0.050; 0.075; 0.100; 0.125 mm/tooth).

In addition to the variable parameters of machining, two cutting tools with different numbers of flutes (2 and 3 flutes) and variable tool path geometry (amplitude and term) were used in the experiments, which made it possible to assess the impact of the tested combinations of technological parameters and tool trajectories on tool deflection in milling.

Tool deflection was measured with a laser linear profile sensor, Gocator 2530A-2-B-01-S, from LMI Technologies (Burnaby, Canada), operated with a network controller, Gocator Master 810. The Gocator 2530 profiler (Figure 4) allows for measurements with inspection speeds up to 10 kHz at resolution ranging 0.028–0.054 mm.

Figure 5 shows schematically the phenomenon of tool deflection in milling. Figure 5a shows a real situation when the cutting tool is impacted by the cutting force causing its deflection from the nominal axis. This type of deformation causes the tool trajectory to be deflected from the planned path, which may result in dimensional errors and reduced surface quality. Figure 5b shows the shape of a workpiece after milling with tool deflection. One can see a difference between the nominal and the actual profile of the surface, which is due to the tool deflection during milling.

Obtained data were used to design 3D plots illustrating interactions between the machining parameters, the number of flutes, and the tool deflection. An analysis of these relationships provided a basis for the assessment of the effect of technological parameters and machining path geometry on the dynamics of the machining process. The experimental results also served as a starting point for further optimisation of the milling strategy in terms of minimizing tool deflection and improving dimensional accuracy of machined surfaces. For the purpose of creating 3D plots and performing a statistical analysis, the experiments were conducted in compliance with the design of experiments (DoE) approach. To that end, we used the specialist software package Design-Expert^® (23.1.8) allowing experiment design and analysis by advanced statistical modelling tools. Relationships between the input data and the output tool deflection were described by response surface methodology (RSM), which helps describe nonlinear relationships between parameters of the process and its results. This approach allows not only for data interpolation but also for setting the directions of technological parameters optimization. The experiments were conducted in compliance with the central composite design (CCD), which made it possible to assess both linear and quadratic effects together with interactions between the input variables. The experimental design covered all tested range variants of cutting speed, feed per tooth, term, amplitude, and the number of flutes.

3. Results

The study investigated the impact of parameters such as cutting speed, feed per tooth, and the number of flutes on the maximum tool deflection. Recorded signals made it possible to identify characteristic deflection areas and to assess the variations in deflection for different configurations of machining parameters. The characteristics of tool deflection were found to correspond to the periodic change in the workpiece profile described by the parameters: term and amplitude.

Below is given an example of signal analysis. Figure 6 shows an example of tool deflection δ [mm] as a function of t [s], measured in the milling process for a sample with a curvilinear geometry described by:

• amplitude A = 3 mm,
• term T = 30 mm.

The milling process was conducted with the following:

• cutting speed v_c = 600 m/min,
• feed per tooth f_z = 0.075 mm/tooth,
• number of flutes z = 2.

Tool deflection was measured toward the Y-axis of the machine. The signals clearly show areas reflecting subsequent stages of the machining process, where the deflection values significantly increase at the relative extreme of the curvilinear profile

A key parameter of the analysis was the maximum tool deflection as a difference between the greatest and the smallest measured tool deflection in a given trial (1):

δ_max = δ_{max_measurement} − δ_{min_measurement}

(1)

Based on the plot (Figure 6), the following conclusions can be drawn:

• deflection occurs in cycles, which suggests the occurrence of repeated maximum deflection values in specified time intervals consistent with the term T,
• the presence of these clearly local maximum deflection values may indicate temporary increases in the cutting force,
• deflection measurements did not consider the impact of the tool’s axial run-out, which may affect the interpretation of results,
• the tool deflection exhibits repeatability in specified time intervals, which may result from regular variations in the cutting force when the tool passes through the workpiece,
• the observed peaks may results from temporary variations in the cutting forces caused by, e.g., variable chip thickness, workpiece material inhomogeneity, free vibrations of the machine tool system,
• the results can help optimize the cutting parameters in order to minimize tool deformation and improve process stability.

In effect, this parameter makes it possible to clearly assess the extent of tool deflection in the machining process. Obtained signal waveforms were preliminarily filtered in order to eliminate random noise and single brief peaks which could considerably distort the actual tool deflection value and thus lead to erroneous conclusions. To eliminate outliers, a filter was applied that removed 0.5% of the maximum signal values (the 99.5th percentile).

Filtered data are shown as 3D plots illustrating the effect of the tested variables on the tool deflection δ. The plots are shown in Figure 7, Figure 8, Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13. For comparison purposes, the plots show the tool deflection results for two variants of tool geometry:

• a tool with two flutes (z = 2),
• a tool with three flutes (z = 3).

This comparison made it possible to determine the impact of the number of flutes on the stability of the machining process and the range of generated dynamic deformation. It must be stressed that the comparison was made for the intermediate values of the tested machining parameters, which means that the combinations including the extreme values (minimum and maximum) of cutting speed and feed per tooth were omitted in the analysis. This was done to prevent boundary effects and to ensure the comparison’s reliability.

The experimental results, in accordance with research plan, are presented in

Table A1. Experimental results (in accordance with the research plan).

Std	Run	Cutting Speed [m/min]	Feed Per Tooth [mm/tooth]	Amplitude [mm]	Term [mm]	Number of Flutes [-]	Tool Deflection [mm]
70	1	900	0.075	3	30	2	0.1091
106	2	750	0.05	2	25	3	0.0644
126	3	450	0.1	4	25	3	0.0906
42	4	450	0.1	2	35	2	0.069
157	5	450	0.1	4	35	3	0.0525
98	6	600	0.075	3	30	2	0.0679
155	7	750	0.05	4	35	3	0.0853
138	8	750	0.05	2	35	3	0.0717
30	9	750	0.1	4	25	2	0.1099
29	10	750	0.1	4	25	2	0.1061
33	11	450	0.05	2	35	2	0.052
88	12	600	0.075	5	30	2	0.0568
110	13	450	0.1	2	25	3	0.0724
27	14	450	0.1	4	25	2	0.054
181	15	600	0.075	1	30	3	0.0791
178	16	600	0.125	3	30	3	0.0435
74	17	600	0.025	3	30	2	0.06
174	18	600	0.025	3	30	3	0.0545
128	19	450	0.1	4	25	3	0.0837
102	20	450	0.05	2	25	3	0.0474
125	21	450	0.1	4	25	3	0.0912
61	22	750	0.1	4	35	2	0.1005
147	23	750	0.1	2	35	3	0.0528
50	24	450	0.05	4	35	2	0.0503
184	25	600	0.075	1	30	3	0.0788
122	26	750	0.05	4	25	3	0.063
123	27	750	0.05	4	25	3	0.0645
196	28	600	0.075	3	40	3	0.066
163	29	750	0.1	4	35	3	0.1008
56	30	750	0.05	4	35	2	0.0647
95	31	600	0.075	3	40	2	0.078
18	32	450	0.05	4	25	2	0.0645
58	33	450	0.1	4	35	2	0.06
26	34	450	0.1	4	25	2	0.0615
96	35	600	0.075	3	40	2	0.0773
156	36	750	0.05	4	35	3	0.0585
45	37	750	0.1	2	35	2	0.0889
39	38	750	0.05	2	35	2	0.0533
120	39	450	0.05	4	25	3	0.0525
171	40	900	0.075	3	30	3	0.052
141	41	450	0.1	2	35	3	0.048
66	42	300	0.075	3	30	2	0.054
40	43	750	0.05	2	35	2	0.054
65	44	300	0.075	3	30	2	0.0525
80	45	600	0.125	3	30	2	0.1206
170	46	900	0.075	3	30	3	0.0337
3	47	450	0.05	2	25	2	0.048
154	48	750	0.05	4	35	3	0.063
21	49	750	0.05	4	25	2	0.0608
192	50	600	0.075	3	20	3	0.069
38	51	750	0.05	2	35	2	0.0525
168	52	300	0.075	3	30	3	0.066
162	53	750	0.1	4	35	3	0.0855
161	54	750	0.1	4	35	3	0.0801
177	55	600	0.125	3	30	3	0.0465
137	56	750	0.05	2	35	3	0.0767
117	57	450	0.05	4	25	3	0.0535
59	58	450	0.1	4	35	2	0.0615
132	59	750	0.1	4	25	3	0.0915
109	60	450	0.1	2	25	3	0.0765
63	61	750	0.1	4	35	2	0.1005
107	62	750	0.05	2	25	3	0.063
75	63	600	0.025	3	30	2	0.0615
140	64	750	0.05	2	35	3	0.0705
16	65	750	0.1	2	25	2	0.069
71	66	900	0.075	3	30	2	0.1088
5	67	750	0.05	2	25	2	0.0552
142	68	450	0.1	2	35	3	0.052
35	69	450	0.05	2	35	2	0.054
7	70	750	0.05	2	25	2	0.0525
34	71	450	0.05	2	35	2	0.0495
131	72	750	0.1	4	25	3	0.0825
185	73	600	0.075	5	30	3	0.1041
152	74	450	0.05	4	35	3	0.057
115	75	750	0.1	2	25	3	0.0503
183	76	600	0.075	1	30	3	0.06
104	77	450	0.05	2	25	3	0.051
100	78	600	0.075	3	30	2	0.06
153	79	750	0.05	4	35	3	0.0627
182	80	600	0.075	1	30	3	0.06
121	81	750	0.05	4	25	3	0.0632
199	82	600	0.075	3	30	3	0.1017
133	83	450	0.05	2	35	3	0.048
179	84	600	0.125	3	30	3	0.0465
41	85	450	0.1	2	35	2	0.0735
114	86	750	0.1	2	25	3	0.0514
23	87	750	0.05	4	25	2	0.069
22	88	750	0.05	4	25	2	0.0615
94	89	600	0.075	3	40	2	0.0765
134	90	450	0.05	2	35	3	0.048
180	91	600	0.125	3	30	3	0.0509
145	92	750	0.1	2	35	3	0.0555
172	93	900	0.075	3	30	3	0.0441
24	94	750	0.05	4	25	2	0.074
186	95	600	0.075	5	30	3	0.0879
46	96	750	0.1	2	35	2	0.102
187	97	600	0.075	5	30	3	0.099
64	98	750	0.1	4	35	2	0.099
6	99	750	0.05	2	25	2	0.0507
17	100	450	0.05	4	25	2	0.051
72	101	900	0.075	3	30	2	0.1065
112	102	450	0.1	2	25	3	0.072
25	103	450	0.1	4	25	2	0.063
150	104	450	0.05	4	35	3	0.0495
166	105	300	0.075	3	30	3	0.0555
101	106	450	0.05	2	25	3	0.0478
76	107	600	0.025	3	30	2	0.063
73	108	600	0.025	3	30	2	0.0645
68	109	300	0.075	3	30	2	0.0495
175	110	600	0.025	3	30	3	0.046
164	111	750	0.1	4	35	3	0.0764
51	112	450	0.05	4	35	2	0.0525
85	113	600	0.075	5	30	2	0.0546
188	114	600	0.075	5	30	3	0.1155
77	115	600	0.125	3	30	2	0.1024
57	116	450	0.1	4	35	2	0.054
32	117	750	0.1	4	25	2	0.1028
37	118	750	0.05	2	35	2	0.054
14	119	750	0.1	2	25	2	0.063
97	120	600	0.075	3	30	2	0.066
176	121	600	0.025	3	30	3	0.0575
116	122	750	0.1	2	25	3	0.0551
28	123	450	0.1	4	25	2	0.0555
62	124	750	0.1	4	35	2	0.1095
200	125	600	0.075	3	30	3	0.0765
148	126	750	0.1	2	35	3	0.063
111	127	450	0.1	2	25	3	0.0765
19	128	450	0.05	4	25	2	0.0555
36	129	450	0.05	2	35	2	0.0514
12	130	450	0.1	2	25	2	0.0766
124	131	750	0.05	4	25	3	0.066
60	132	450	0.1	4	35	2	0.0615
48	133	750	0.1	2	35	2	0.0945
136	134	450	0.05	2	35	3	0.0501
49	135	450	0.05	4	35	2	0.0525
9	136	450	0.1	2	25	2	0.066
86	137	600	0.075	5	30	2	0.0555
79	138	600	0.125	3	30	2	0.087
81	139	600	0.075	1	30	2	0.0735
13	140	750	0.1	2	25	2	0.0735
2	141	450	0.05	2	25	2	0.0525
108	142	750	0.05	2	25	3	0.0555
173	143	600	0.025	3	30	3	0.046
149	144	450	0.05	4	35	3	0.052
159	145	450	0.1	4	35	3	0.0489
105	146	750	0.05	2	25	3	0.054
191	147	600	0.075	3	20	3	0.0666
146	148	750	0.1	2	35	3	0.051
1	149	450	0.05	2	25	2	0.051
78	150	600	0.125	3	30	2	0.1016
91	151	600	0.075	3	20	2	0.078
89	152	600	0.075	3	20	2	0.069
193	153	600	0.075	3	40	3	0.0615
69	154	900	0.075	3	30	2	0.0866
20	155	450	0.05	4	25	2	0.057
87	156	600	0.075	5	30	2	0.0593
82	157	600	0.075	1	30	2	0.075
103	158	450	0.05	2	25	3	0.06
92	159	600	0.075	3	20	2	0.0799
83	160	600	0.075	1	30	2	0.0679
43	161	450	0.1	2	35	2	0.0723
167	162	300	0.075	3	30	3	0.102
44	163	450	0.1	2	35	2	0.0705
90	164	600	0.075	3	20	2	0.0645
127	165	450	0.1	4	25	3	0.0922
151	166	450	0.05	4	35	3	0.0491
53	167	750	0.05	4	35	2	0.0585
31	168	750	0.1	4	25	2	0.0919
194	169	600	0.075	3	40	3	0.063
47	170	750	0.1	2	35	2	0.0984
198	171	600	0.075	3	30	3	0.069
139	172	750	0.05	2	35	3	0.075
118	173	450	0.05	4	25	3	0.052
15	174	750	0.1	2	25	2	0.0735
165	175	300	0.075	3	30	3	0.048
113	176	750	0.1	2	25	3	0.0435
8	177	750	0.05	2	25	2	0.048
119	178	450	0.05	4	25	3	0.0536
67	179	300	0.075	3	30	2	0.0534
144	180	450	0.1	2	35	3	0.0505
160	181	450	0.1	4	35	3	0.051
189	182	600	0.075	3	20	3	0.0783
55	183	750	0.05	4	35	2	0.06
129	184	750	0.1	4	25	3	0.0895
195	185	600	0.075	3	40	3	0.0603
190	186	600	0.075	3	20	3	0.0777
11	187	450	0.1	2	25	2	0.0705
99	188	600	0.075	3	30	2	0.066
93	189	600	0.075	3	40	2	0.0735
4	190	450	0.05	2	25	2	0.0525
158	191	450	0.1	4	35	3	0.0437
169	192	900	0.075	3	30	3	0.043
84	193	600	0.075	1	30	2	0.066
130	194	750	0.1	4	25	3	0.0942
52	195	450	0.05	4	35	2	0.0525
135	196	450	0.05	2	35	3	0.053
54	197	750	0.05	4	35	2	0.057
197	198	600	0.075	3	30	3	0.075
143	199	450	0.1	2	35	3	0.0415
10	200	450	0.1	2	25	2	0.0654

.

3.1. Effect of Cutting Speed and Feed per Tooth on Tool Deflection

Figure 7 shows the effect of the cutting speed v_c and the feed per tooth f_z on the tool deflection δ. The results were obtained for the constant intermediate amplitude A = 3 mm and the term T = 30 mm, in compliance with the experiment design.

Figure 7. Cutting speed (v_c) and feed per tooth (f_z) versus the tool deflection δ in a milling process conducted with A = 3 m and term T = 30 mm, for: (a) two-flute end mill (z = 2), (b) three-flute end mill (z = 3).

Figure 7. Cutting speed (v_c) and feed per tooth (f_z) versus the tool deflection δ in a milling process conducted with A = 3 m and term T = 30 mm, for: (a) two-flute end mill (z = 2), (b) three-flute end mill (z = 3).

For the two-flute end mill (Figure 7a), one can observe a clear increase in the tool deflection with increasing the cutting speed and feed per tooth. The highest deflection values exceeding δ = 0.12 mm were obtained at the maximum values of the tested technological parameters. The minimum deflections occurred at low cutting speeds and feeds, with their values never dropping below 0.04 mm over the entire tested range. For the three-flute end mill (Figure 7b), the deflection values are significantly lower compared to those obtained for the two-flute tool. The deflection mainly increases with the cutting speed, while the impact of the feed per tooth on the deflection value is minimal for this configuration. The maximum deflection value did not exceed δ = 0.09 mm, which indicates higher stability of the milling process conducted with the three-flute tool. The higher number of flutes has a positive effect on reducing tool deflection during machining. The three-flute end mill ensures more stable cutting conditions, especially at higher cutting speeds. The cutting speed v_c is the dominant factor affecting deflection, while the impact of the feed per tooth f_z is more pronounced for the two-flute tool.

3.2. Effect of Cutting Speed and Amplitude on Tool Deflection

An analysis of the data in Figure 8 reveals that the cutting speed v_c and the curvature amplitude A affect the tool deflection δ. The analysis was performed by maintaining the other parameters constant, i.e., the feed per tooth f_z = 0.075 mm/tooth and the term T = 30 mm.

Figure 8. Cutting speed (v_c) amplitude (A) versus the tool deflection δ in a milling process conducted with f_z = 0.075 mm/tooth and T = 30 mm, for: (a) two-flute end mill (z = 2), (b) three-flute end mill (z = 3).

Figure 8. Cutting speed (v_c) amplitude (A) versus the tool deflection δ in a milling process conducted with f_z = 0.075 mm/tooth and T = 30 mm, for: (a) two-flute end mill (z = 2), (b) three-flute end mill (z = 3).

For the two-flute end mill (Figure 8a), the tool deflection clearly increases with increasing the cutting speed and amplitude. The maximum deflection of approximately 0.12 mm was obtained at the highest tested amplitude A = 5 mm and the highest cutting speed v_c = 900 m/min. In contrast, the minimum deflection value of below 0.04 mm was obtained at the lowest amplitude A = 1 mm and the cutting speed v_c = 300 m/min. For the three-flute end mill (Figure 8b), the obtained deflection values are noticeably lower over the entire tested range when compared to those for the two-flute tool. The main trend indicates that the tool deflection increases with the cutting speed v_c, while the effect of the amplitude A is less pronounced and does not cause such rapid changes in deflection values. The maximum deflection of the three-flute tool also occurs at high amplitudes and high cutting speeds, but these deflection values are lower than those observed for the two-flute tool, which confirms a more stable operation of the tool with a larger number of flutes.

3.3. Effect of Cutting Speed and Term on Tool Deflection

Response surface plots shown in Figure 9 were used to determine the effect of the variable cutting speed v_c and curvature term T on the tool deflection δ in milling.

The results are shown for a milling process conducted with the constant amplitude A = 3 mm and the constant feed per tooth f_z = 0.075 mm/tooth. For the two-flute end mill (Figure 9a), an increase in the cutting speed causes a gradual increase in the tool deflection, which is particularly visible at higher term values. The use of a lower term results in reduced deflection of the tool, with the lowest deflection observed at the lowest term and the lowest cutting speed. The highest deflection occurs when the maximum cutting speed and the maximum term are applied at the same time. For the three-flute tool (Figure 9b) a similar trend can be observed, however the deflection values are lower over the entire tested range. The cutting speed has the main impact on tool deflection, while the impact of the variable term is less pronounced. The use of a higher term augments the impact of cutting speed increase on tool deflection, nevertheless this impact is smaller than that observed for the tool with two flutes.

Figure 9. Effect of cutting speed (v_c) and curvature term (T) on the tool deflection δ in a milling process conducted with f_z = 0.075 mm/tooth, and A = 30 mm, for: (a) two-flute end mill (z = 2), (b) three-flute end mill (z = 3).

Figure 9. Effect of cutting speed (v_c) and curvature term (T) on the tool deflection δ in a milling process conducted with f_z = 0.075 mm/tooth, and A = 30 mm, for: (a) two-flute end mill (z = 2), (b) three-flute end mill (z = 3).

3.4. Effect of Feed per Tooth and Profile Amplitude on Tool

3D plots in Figure 10 illustrate the impact of the variable feed per tooth f_z and the variable amplitude A on the tool deflection δ during machining. The results are shown for a milling process conducted with the constant cutting speed v_c = 600 m/min and the constant curvature term T = 30 mm.

For the two-flute end mill (Figure 10a) the highest tool deflection was observed when the feed per tooth and amplitude values were increased simultaneously. The lowest deflection was observed for the minimum values of both, which indicates that the milling process is more stable under limited dynamic load. The impact of amplitude is particularly important due to the fact that an increase in amplitude led to a considerable increase in deflection, this impact increasing with feed. For the three-flute end mill (Figure 10b) the deflection values are clearly lower over the entire tested range of parameters than those observed for the two-flute tool. Increased feed and amplitude have a smaller impact on tool deflection than was the case with the two-flute tool. One can only observe a slight increase in tool deflection at high amplitude values, particularly with higher feeds. The results confirm that the use of a higher number of flutes reduces the impact of the dynamic variables on the stability of the milling process, which helps minimize tool deflection.

Figure 10. Effect of feed per tooth (f_z) and curvature amplitude (A) on tool deflection δ in a milling process conducted with v_c = 600 m/min and T = 30 mm, for: (a) two-flute end mill (z = 2), (b) three-flute end mill (z = 3).

Figure 10. Effect of feed per tooth (f_z) and curvature amplitude (A) on tool deflection δ in a milling process conducted with v_c = 600 m/min and T = 30 mm, for: (a) two-flute end mill (z = 2), (b) three-flute end mill (z = 3).

3.5. Effect of Feed per Tooth and Term on Tool Deflection

Figure 11 shows the effect of the feed per tooth f_z and the curvature term T on the tool deflection δ during machining. The results are shown for a milling process conducted with the constant cutting speed v_c = 600 m/min and the constant amplitude A = 3 mm.

Figure 11. Effect of feed per tooth (f_z) and term (T) on tool deflection δ in a milling process conducted with v_c = 600 m/min and A = 3 mm, for: (a) two-flute end mill (z = 2), (b) three-flute end mill (z = 3).

Figure 11. Effect of feed per tooth (f_z) and term (T) on tool deflection δ in a milling process conducted with v_c = 600 m/min and A = 3 mm, for: (a) two-flute end mill (z = 2), (b) three-flute end mill (z = 3).

For the two-flute end mill (Figure 11a), the tool deflection clearly tends to increase with increasing feed per tooth and term. The highest deflection values oscillating about 0.12 mm were obtained for the combinations of maximum feeds and terms. This is due to increasing dynamic loads on the tool-workpiece system. The lowest deflection of 0.04 mm was obtained at low values of both analysed parameters, which confirms that the milling process is more stable under such conditions. For the three-flute end mill (Figure 11b), the impact of parameter variations is noticeably smaller, and the deflection values over the tested range remain lower than for the two-flute tool. The use of higher feed per tooth and term also results in higher tool deflection, albeit these changes are smaller. The plot characteristics confirm the greater stability of the machining process conducted with the three-flute tool.

3.6. Effect of Amplitude and Term on Tool Deflection

Figure 12 shows relationships between the tool deflection δ as a function of the curvature amplitude A and the curvature term T for the fixed cutting speed v_c = 600 m/min and feed per tooth f_z = 0.075 mm/tooth. For the two-flute end mill (Figure 12a), one can observe a clear tendency for the deflection to increase as the amplitude and term are increased.

Figure 12. Effect of amplitude (A) and term (T) on tool deflection δ in a machining process conducted with v_c = 600 m/min and f_z = 0.075 mm/tooth, for: (a) two-flute end mill (z = 2), (b) three-flute end mill (z = 3).

Figure 12. Effect of amplitude (A) and term (T) on tool deflection δ in a machining process conducted with v_c = 600 m/min and f_z = 0.075 mm/tooth, for: (a) two-flute end mill (z = 2), (b) three-flute end mill (z = 3).

The highest deflection values were obtained for the combinations with the maximum values of the tested parameters. This may indicate an increase in the dynamic tool deflection when the tool path geometry is significantly changed. The smallest deflection values were observed for a low amplitude of A = 1 mm and a short term of T = 20 mm, which confirms the beneficial effect of these parameters on the stability of the machining process. For the three-flute end mill (Figure 12b) the tool deflection values are considerably lower over the entire tested range. This confirms the more favourable machining conditions and greater rigidity of the three-flute tool. The use of higher amplitude and term also results in higher deflection, but these values remain noticeably lower than those obtained for the two-flute end mill. Such distribution of the results indicates lower dynamic loads and more stable conditions in the machining process conducted with the three-flute end mill, which may result in better surface quality of the workpiece and reduced tool wear.

Table 3. Statistical analysis (ANOVA) results of tool deflection.

Source	Sum of Squares	Df	Mean Square	F-Value	p-Value
Model	0.0479	29	0.0017	17.93	<0.0001
A-Cutting speed	0.0057	1	0.0057	61.77	<0.0001
B-Feed per tooth	0.0091	1	0.0091	98.82	<0.0001
C-Amplitude	0.0023	1	0.0023	25.46	<0.0001
D-Term	0.0002	1	0.0002	1.76	0.1861
E-Number of flutes	0.0005	1	0.0005	5.27	0.0230
AB	0.0004	1	0.0004	3.98	0.0477
AC	0.0020	1	0.0020	21.67	<0.0001
AD	0.0016	1	0.0016	17.05	<0.0001
AE	0.0035	1	0.0035	37.58	<0.0001
BC	0.0006	1	0.0006	7.00	0.0089
BD	0.0002	1	0.0002	2.46	0.1189
BE	0.0035	1	0.0035	37.91	<0.0001
CD	0.0007	1	0.0007	7.61	0.0064
CE	0.0016	1	0.0016	16.88	<0.0001
DE	0.0008	1	0.0008	8.55	0.0039
A2	0.0005	1	0.0005	5.11	0.0250
B2	0.0006	1	0.0006	6.04	0.0150
C2	7.015 × 10−6	1	7.015 × 10−6	0.0761	0.7829
D2	0.0001	1	0.0001	1.23	0.2686
ABE	0.0021	1	0.0021	23.09	<0.0001
ACE	0.0001	1	0.0001	1.13	0.2886
ADE	0.0005	1	0.0005	5.80	0.0171
BCE	0.0011	1	0.0011	12.15	0.0006
BDE	0.0017	1	0.0017	18.49	<0.0001
CDE	7.605 × 10−6	1	7.605 × 10−6	0.0825	0.7742
A2E	0.0016	1	0.0016	17.06	<0.0001
B2E	0.0029	1	0.0029	31.03	<0.0001
C2E	0.0002	1	0.0002	1.73	0.1904
D2E	0.0005	1	0.0005	5.04	0.0260
Residual	0.0157	170	0.0001
Lack of Fit	0.0085	20	0.0004	9.01	<0.0001
Pure Error	0.0071	150	0.0000
Cor Total	0.0636	199

lists the results of statistical analysis (ANOVA) for tool deflection. The ANOVA results confirm the significant impact of the selected machining parameters and interactions between them on tool deflection in milling.

The most significant main factors with the greatest impact on the deflection value are the cutting speed (A), feed per tooth (B), and curvature amplitude (C), for which the value of p < 0.0001, which clearly indicates their statistical significance. The number of flutes (E) also proved to be a significant factor (p = 0.0230), although its impact was slightly smaller. The results of the analysis showed the significance of a series of interactions between the parameters under study. A particularly strong effect was observed for the interactions AE, BE, CE, ABE, and BDE, which indicates a high sensitivity of the measurement system to the number of flutes in combination with other machining parameters. The value p < 0.0001 for these interactions confirms their significant impact on the deflection of the tool. In turn, the interactions such as AB, AD, CD, and DE also reached a level of significance, albeit with less impact. The analysis also showed a significant impact of quadratic effects, especially A² and B², which confirms the nonlinear nature of changes in tool deflection in relation to these parameters. It is worth stressing the high value of the F-statistic for the main model (F = 17.93), which confirms the good quality of the model’s fit to the experimental data and allows it to be considered statistically significant (p < 0.0001). At the same time, the Lack of Fit test also proved to be significant (p < 0.0001), which may suggest the need for further analysis of the nonlinear relationships or for the consideration of additional factors affecting the process. The statistical analysis results clearly confirm the significant impact of cutting speed, feed per tooth, amplitude, and the number of flutes on tool deflection. Furthermore, the demonstrated presence of significant interactions between the parameters indicates their interdependence and the complexity of phenomena occurring during machining. These results are consistent with the observations obtained from the analysis of the 3D plots, which also indicated a strong influence of the number of flutes and the complex interactions between the machining conditions on the stability of the milling process.

Table 4. Regression equations for tool deflection amplitude.

Number of Flutes	Equation
2	y = 0.220783 − 0.000307 × A − 1.70242 × B − 0.008519 × C − 0.003539 × D − 0.001538 × AB + 0.000032 × AC + 1.94583 × 10−6 × AD − 0.028500 × BC + 0.018550 × BD − 0.000517 × CD + 8.38889 × 10−8 × A2 + 5.03000 × B2 − 0.001606 × C2 + 0.000046 × D2
3	y = −0.161680 + 0.000117 × A + 3.00246 × B − 0.015168 × C + 0.007351 × D − 0.000636 × AB + 0.000020 × AC + 7.39583 × 10−6 × AD + 0.208125 × BC − 0.039825 × BD − 0.000419 × CD − 2.86852 × 10−7 × A2 − 12.97167 × B2 + 0.001049 × C2 − 0.000136 × D2

shows regression equations for tool deflection amplitude.

Figure 13 shows the main effects plots for the tested number of flutes. The first plot (Figure 13a) shows a very slight effect of changing the number of flutes from two to three on the tool deflection value. The average deflection value remains almost unchanged, suggesting that under the given cutting conditions, the number of flutes does not play a significant role in tool deflection.

The deflection values for both variants are practically the same, and the differences are within the acceptable margin of error. In the second plot (Figure 13b), the impact of the number of flutes on tool deflection becomes more visible. An increase in the number of flutes from 2 to 3 causes a slight increase in the deflection value. When milling is conducted with the intermediate values of the tested parameters, the number of flutes begins to affect the dynamics of the milling process. The increase in the deflection value with increasing the number of flutes can result from more complex interactions of the cutting force, particularly at higher dynamic loads. In the final plot (Figure 13c) one can observe an opposite relationship, i.e., as the number of flutes is increased, the tool deflection is reduced. At the maximum values of the tested parameters, the use of the three-flute end mill causes the tool deflection to decrease. This may results from a more uniform distribution of the cutting forces due to the use of more flutes, which, consequently, reduces the deflection value. The observed differences between the plots confirm the strong impact of machining conditions on the significance of the number of flutes in tool deflection control. With moderate parameters, the number of flutes plays no significant role. At higher parameters, however, the number of flutes becomes more significant depending on a given configuration of parameters—in one case, it increases tool deflection, but in another case, it reduces it. It can also be seen that the scatter of results was greater when using the three-flute tool. This indicates greater changes in the dynamics of the cutting process. However, most results were concentrated at similar deflection value. This proves that the phenomena occurring during machining processes are very complex and that the interactions between process parameters must be taken into account for process optimization.

Figure 13. Main effects plots showing the impact of the number of flutes: (a) v_c = 450 m/min, f_z = 0.05 mm/tooth, A = 2 mm, T = 25 mm; (b) v_c = 600 m/min, f_z = 0.075 mm/tooth, A = 3 mm, T = 30 mm; (c) v_c = 750 m/min, f_z = 0.1 mm/tooth, A = 4 mm, T = 35 mm.

Figure 13. Main effects plots showing the impact of the number of flutes: (a) v_c = 450 m/min, f_z = 0.05 mm/tooth, A = 2 mm, T = 25 mm; (b) v_c = 600 m/min, f_z = 0.075 mm/tooth, A = 3 mm, T = 30 mm; (c) v_c = 750 m/min, f_z = 0.1 mm/tooth, A = 4 mm, T = 35 mm.

4. Discussion

The results clearly demonstrate that tool deflection in curvilinear surface milling significantly depend on both technological parameters (e.g., cutting speed and feed per tooth) and tool features (number of flutes). The results agree with the findings of previous studies [1,2,3], which showed that the cutting force and dynamic phenomena had a significant impact on the dimensional accuracy and process stability in the milling of parts with complex geometries.

Specifically, the cutting speed (v_c) was identified as the main factor affecting tool deflection, which agrees with the findings of previous studies [4,5]. An increase in the cutting speed leads to higher dynamic loads, which results in greater tool deformation, particularly in the case of less rigid end mills. In a similar fashion, the feed per tooth (f_z) has a strong impact on the deflection of the two-flute end mill, whereas this impact is clearly smaller on the tool with three flutes, which proves that the tool with a greater number of flutes is more resistant to dynamic load.

A novel contribution of this study is that it takes into account the impact of tool path geometry—i.e., the amplitude (A) and term (T) of a surface profile—on tool deflection. Previous studies have rarely considered this aspect, focusing mainly on the impact of machining parameters. The results have demonstrated that an increase in amplitude causes a significant increase in tool deflection, particularly when combined with increased feed. The impact of the term T is moderate, although it increases when high values of other parameters are used. This behaviour pattern confirms the hypothesis saying that sudden changes in tool path direction lead to increased dynamic loads.

An analysis of interactions between the technological and geometric parameters (e.g., AE, BE, CE, ABE, BDE) has revealed the presence of strong nonlinear interdependent relationships between the variables. These complex interdependencies have been confirmed by the results of both response surface analysis and ANOVA statistical analysis. It should be highlighted that previous studies seldom considered the aspects of tool path geometry and the number of flutes at the same time, an undertaking which is a significant contribution of this work.

The nonlinear interactions observed between machining parameters can be interpreted through the principles of classical cutting mechanics. The end mill can be approximated as a cantilever beam subjected to distributed cutting forces, where deflection depends on both tool rigidity and dynamic loading [25]. Cutting speed (v_c) directly influences the dynamic cutting force amplitude, while feed per tooth (f_z) determines the instantaneous uncut chip thickness, which in turn modifies the distribution of cutting forces [9]. An increase in profile amplitude (A) amplifies chip thickness variation due to the changing tool engagement angle, thereby intensifying the cyclic bending load on the tool [1]. The combination of these effects explains why nonlinear interactions such as AE (cutting speed × number of flutes) or BE (feed × number of flutes) were statistically significant in the ANOVA. These observations are consistent with established cutting force models and predictive approaches for tool deflection in milling [27].

From a practical point of view, the results demonstrate that the optimization of a milling process for a curvilinear surface must take into account both standard parameters (v_c, f_z) and tool path geometry. It has been shown that tools with three flutes ensured higher stability of the machining process under dynamic conditions, which is of crucial importance in high precision applications or at high cutting speeds.

Future directions of research in this area include:

• extending the scope of analyses to include 5-axis machining and 3D trajectories,
• considering the effects of overhang length and tool holder rigidity,
• developing methods for compensating tool deflection in real time,
• the use of artificial intelligence methods (i.e., neural networks and machine learning) to optimize tool path in terms of minimized deflection (it is an interesting research area).

Summing up, this study has provided in-depth insight into the effects of machining parameters and tool path geometry on tool deflection, and the proposed approach to modelling this problem can serve as an auxiliary tool for optimizing milling processes for parts with complex shapes.

5. Conclusions

The results of the experiments, deflection signals, ANOVA, and response surface models (RSMs) led to the following conclusions:

• Tool deflection in curvilinear surface milling is cyclic, which results from periodic variations in tool path profile (parameter T). The signal characteristics clearly show the presence of local maximum deflections occurring at regular time intervals.
• The maximum tool deflection (δmax) increases primarily with the cutting speed (v_c) and the amplitude (A). These two factors have the greatest impact on the dynamic deflection range of the tool.
• Feed per tooth (f_z) has a significant effect on tool deflection, particularly with respect to the two-flute end mill. For a tool with three flutes, this impact is weaker, which indicates higher dynamic rigidity and stability of the machining process conducted using tools with a greater number of flutes.
• The number of flutes (z) plays an important role in the behaviour pattern of deflection:

• the use of a three-flute tool results in lower deflection and high stability of the milling process,
• the effect of a greater number of flutes depended on the configuration of parameters– in some cases, the greater number of flutes reduced tool deflection, but in other cases, this caused a slight increase in deflection.

5.

The obtained tool deflection signals indicate the presence of local peaks that can result from:

• temporary increases in the cutting force,
• variable chip thickness and material inhomogeneity,
• free vibrations of the machine tool system and micro vibrations of the tool.

6.

The high signal oscillation frequency may indicate the presence of micro vibrations that negatively affect surface quality and tool wear.

7.

Tool deflection measurements were made in the Y-axis direction. The deflection increases visible in the signal characteristics correlate with the position of the tool in the workpiece, an important observation which can be used when designing adaptive methods of deflection compensation.

8.

The effect of axial run-out was not considered in the measurements, a fact which must be taken into consideration when interpreting the results. Since axial run-out may affect temporary tool deflection, it is worth investigating its compensation in future studies.

9.

The use of design of experiment methods (DoE) and response surface methodology (RSM) made it possible to create reliable regression models and identify linear effects affecting the interactions between parameters.

10.

The results provide a basis for the optimization of milling strategies for curvilinear surfaces by offering guidance on the selection of conditions that minimize tool deflection and increase process stability.