Modeling and experimental study of cutting forces of a variable pitch ball-end cutter in five-axis milling

Abstract

This study develops a cutting force prediction model for variable-pitch ball-nose cutters used in multi-axis milling. It starts by defining the five-axis machining coordinate system and creating a geometric model of the cutter. Then, using micro-element cutting force theory, the study establishes a cutting force model specific to ball-nose milling.To simplify the complexity from dynamic tool orientations, the research proposes a practical method. It projects the engagement region onto a plane that is perpendicular to the tool axis. This approach accurately defines the boundary curve. After calculating the engagement region with a fixed tool axis, a rotation transformation matrix is applied. This helps determine the engagement domain in the tool coordinate system for any tool orientation.The study also finds the instantaneous entry and exit angles for each element. It does this by identifying intersection points between the infinitesimal projection circle and the engagement domain boundary, along with the lines connecting their centers.Lastly, the research introduces a method for identifying milling force coefficients based on the average milling force per tooth cycle. Through systematic milling experiments, these coefficients are calibrated. The results indicate that the predicted milling forces closely match the measured ones, with a maximum root mean square error of 6.74%. This validates the model’s effectiveness for variable-pitch ball-nose cutters.

Introduction

In high-end aerospace equipment manufacturing, components such as structural parts, engine blisks, and casings often have complex curved surfaces and thin walls. These features typically include large torsion angles and small curvatures. The final geometric accuracy, surface quality, and performance of these parts - including fatigue life - heavily depend on the quality of the final machining operation, which is multi-axis CNC finishing^1–4.Ball-end mills are essential for achieving high precision and excellent surface quality in these tasks. Their ability to adapt to complex shapes comes from the various ways the spherical cutter head interacts with the workpiece. However, ball-end milling is more complex than flat-end milling due to the changing cutting speeds, which range from zero at the tool tip to maximum at the edges. As the tool moves, the entry and exit angles of the cutting teeth, as well as the effective cutting engagement length, also change.Relying solely on experience or simple models is not enough to accurately predict or control cutting forces. Therefore, it is crucial to develop precise cutting force models that reflect the complex mechanics of ball-end mills, in order to improve machining quality and efficiency in these components.

Within the historical context of milling force modeling, researchers have primarily pursued two main avenues: geometric analysis and kinematic refinement. Early work by Yucesan et al.⁵ laid a theoretical foundation by systematically analyzing the influence of key geometric parameters, such as undeformed chip thickness and rake angle in three-dimensional cutting, on chip flow direction and stress distribution on the rake face, proposing a more universal 3D mechanistic modeling framework. A milestone contribution came from Altintas et al.⁶, who developed a comprehensive time-domain simulation model by coupling instantaneous regenerative cutting load, experimentally calibrated specific cutting force coefficients, the structural transfer function of the machine-tool system, and precise tool geometry, based on a detailed analysis of the true trochoidal kinematics of the ball-end milling process. Lamikiz et al.⁷ adopted a semi-empirical approach, inversely identifying the shearing and ploughing specific coefficients from slot milling experiments on 7075-T6 aluminum alloy, thereby constructing a predictive mechanistic model.As demand for prediction accuracy grew, the critical role of the undeformed chip thickness (hC) as the bridge between tool kinematics and force generation became apparent. Its precise calculation is paramount to model success, leading to two predominant technical approaches: the vector projection method and the geometric simulation method. The vector projection method essentially approximates hC by projecting the feed motion vector onto the normal direction of the infinitesimal cutting edge element at a given instant. Bolsunovsky et al.⁸ pioneered the discretization of the resultant force into infinitesimal force vectors and systematically investigated the significant influence of tool orientation on macroscopic cutting force characteristics in 3- and 5-axis milling, highlighting the modulation of hC by the time-varying cutter-workpiece engagement (CWE) geometry. Wang et al.⁹ further incorporated dynamic factors, such as tool vibration and runout, into the hC calculation, establishing a more comprehensive cutting force prediction model. Urbikain et al.¹⁰ employed a fine mesh discretization of the cutting edge, comprehensively considering tool geometry, lead and tilt angles, and runout to develop a force model suitable for complex 5-axis ball-end milling. Research by Zhang et al.¹¹ demonstrated that proper application of such methods can yield satisfactory agreement between simulation and experimental data even in complex 5-axis surface machining.

In contrast, the geometric simulation method derives hC based on the underlying logic of solid Boolean operations, either by calculating the normal distance between successive tooth trajectory surfaces or by employing a dexel-based representation of the tool-workpiece interaction process. Liang et al.¹² proposed an integrated method combining a 3D trochoidal tooth trajectory model with precise engagement boundary determination to define the instantaneous chip thickness for 5-axis ball-end mills. Dambly et al.¹³ utilized an efficient hybrid dexel-based analytical approach for rapid and accurate calculation of hC under complex conditions. The instantaneous CWE region, dictated by the combined effect of tool orientation and machining parameters, directly determines the machined surface quality and tool load. Consequently, significant research efforts have been dedicated to its precise prediction. Lazoglu et al.¹⁴ developed a cutting force model for complex free-form surface machining by directly extracting the CWE boundary along the tool path. Li et al.¹⁵ introduced the concept of the “Effective Feasible Contact Arc,” vividly illustrating the significant impact of runout. Qin et al.¹⁶ adapted the classical Z-map method for rapid numerical calculation of instantaneous hC, with simulations consistently identifying tool orientation as a key controllable factor affecting process stability. Li et al.¹⁷ similarly utilized the Z-map method to identify engaged cutting edge elements within the instantaneous contact zone, combined with geometric hC calculation, to establish a force model for longitudinal-torsional ultrasonic vibration-assisted 5-axis milling considering runout. To balance accuracy and computational efficiency, Zhu et al.¹⁸ explored the application of an integral method for rapid prediction of 5-axis milling forces, achieving significant gains in computational speed.An alternative strategy for quality enhancement focuses on optimizing the tool geometry itself to actively suppress process vibrations¹⁹. Research in this area originated with Slavicek’s concept of irregular pitch cutters²⁰. Altintas²¹ subsequently provided quantitative validation of their significant vibration suppression capability by comparing the frequency-domain stability lobe diagrams (SLDs) of variable-pitch tools against regular-pitch tools via extensive time-domain simulations and experiments. Budak^22,23 advanced the field substantially by systematically proposing optimization methodologies for the pitch angles of variable-pitch mills and successfully applying them in practice. Domestic scholars have also contributed deeply; spectral analysis of milling forces by Xu and Liu et al.^24,25 revealed the modulation effect of pitch angle distribution on specific force frequency components, providing data support for the damping effect. Guo²⁶ experimentally confirmed that optimized variable-pitch-variable-helix mills achieve broader absolute stability regions compared to conventional tools. Cui²⁷ explained the underlying mechanism from the perspective of chip thickness modulation, suggesting that variable-pitch tools enhance stability thresholds by reducing the periodic modulation of chip thickness inherent in the regenerative effect. Li²⁸ and Liu²⁹, through stability theory and milling experiments, respectively validated the effectiveness of their designed variable-pitch end mills and taper ball-end mills in reducing vibration amplitude and cutting forces.Optimization theories have continuously incorporated more advanced mathematical tools. Olgac and Sipahi³⁰ innovatively applied the cluster treatment of characteristic roots method to analyze the influence of pitch angle variation on the system’s characteristic root distribution, constructing a corresponding optimization model. Song et al.³¹ pursued a more comprehensive objective, proposing a multi-objective optimization method for the structural parameters of variable-pitch mills aimed at achieving high dynamic performance. Comak and Budak³² contributed two efficient optimization strategies: one involving direct traversal and comparison of SLDs for different pitch angle combinations, and another, less computationally intensive iterative method, albeit based on the assumption of a single alternating pitch variation pattern between adjacent teeth. Stepan et al.³³ proposed a numerical iterative optimization method based on precise multi-delay dynamic models. Iglesias et al.³⁴ successfully identified optimal pitch combinations using a direct exhaustive search method, with experimental validation for production applications. The work of Mei et al.³⁵ and Nie et al.³⁶ focused on analytical approaches, proposing more general principles for determining optimal pitch angles by analyzing the inherent relationship between the phase difference of adjacent teeth, the system time delay, and the stability limit.

In summary, while the vibration damping efficacy of variable-pitch cutters has been fully validated for flat-end mills, their underlying mechanisms—particularly the dynamic behavior in ball-end milling scenarios characterized by significant multiple time-delay coupling and nonlinear effects—still lack systematic and in-depth investigation. The existing theoretical foundations and optimization models are largely built upon the relatively simple geometry and kinematics of end mills, failing to fully encompass key features specific to ball-end mills, such as time-varying geometric engagement and complex spatial force loads. Therefore, integrating the variable-pitch vibration suppression strategy with the complex ball-end mill to conduct deeper research into its cutting force theory holds significant theoretical value and aligns with the urgent industrial need for high-quality machining of complex curved components in the aerospace sector.

Modeling of cutting forces for five-axis milling with variable pitch ball-end milling cutter

Definition of the coordinate system for the five-axis milling process system

In five-axis milling operations, the precise characterization of tool position and orientation, as well as the optimization of computational procedures, necessitates the establishment of four Cartesian coordinate systems: the machine coordinate system MCS, the workpiece coordinate system WCS, the tool coordinate system TCS, and the feed coordinate system FCS, as illustrated in Fig. 1. The MCS is defined with reference to the independent motion axes of the machine tool. The WCS is anchored at a point determined by the specific geometric features of the workpiece and typically functions as the programming reference coordinate system, which is configured by the programmer. The TCS is centered at the tool tip or cutting edge and serves to describe the spatial orientation of the tool, encompassing both its position and directional vectors. The FCS shares its origin with the TCS and is employed to characterize the local motion attributes of the tool at the cutting point, including direction and velocity.

Fig. 1.

Coordinate system definition of five-axis milling process system.

Within the FCS, the feed axis F denotes the principal feed direction of the tool and is tangential to the machining trajectory. The normal axis (N) is defined as the direction of the machined surface normal; the lateral axis (C) is defined as the transverse feed direction, perpendicular to the feed direction F and, together with the F and N axes, forms a right-handed orthogonal coordinate system. The orientation of the tool is quantified by specific angular parameters: the lead angle α, defined as the rotation of the tool axis vector about the lateral axis C of the FCS, and the lateral tilt angle β, defined as the rotation of the tool axis vector about the feed axis F. The transformation from the FCS to the TCS is achieved through a defined rotational operation, the mathematical relationship of which is expressed in Eq. 1.

1

Geometric definition of variable pitch ball-end milling cutter

Considering the geometric characteristics of the ball-end milling cutter, the principal structural dimensions encompass the tool diameter D, the arc radius R, and the cutting edge length H. In the case of variable pitch ball-end cutters, the tooth angle between consecutive cutting edges, denoted as φ_{p, j} for the angle between cutting edge j-1 and cutting edge j, varies such that the angles φ_p,1, φ_p,2, φ_p,3, and φ_p,4 between adjacent cutting edges are distinct, where k(z) is the axial immersion angle, θ(z) is the angle between the cutting element and the initial unknown angle. As the cutting edge rotates around the tool axis during cutting, the value of θ corresponding to the cutting element will continuously change, as illustrated in Fig. 2.

Fig. 2.

Geometric model of variable pitch ball-end milling cutter.

Furthermore, the coordinate system O-X_TY_TZ_T is defined such that its origin coincides precisely with the tool tip. Utilizing these geometric relationships, the position of point P on the lth milling microelement of the jth milling edge can be represented by Eq. (2):

2

In this context, z_i denotes the axial height of point P relative to the origin, while the local radius R(z_i) represents the distance from point P to the coordinate axis Z_T (i.e., the tool axis). This relationship is mathematically defined by Eq. (3):

3

The radial immersion angle ψ_ji(θ,z) is expressed as:

4

Where φ_i is the spiral lag angle,φ_i(z_i) = z_i∗tanβ₀/R andβ₀ is the nominal spiral angle of the tool.

Micro-element milling force model of variable pitch ball-end milling cutter

In the context of variable-pitch ball-end milling, precise determination of the milling force is achieved by discretizing the cutting edge along the tool axis into a sequence of milling disks, following the mechanical model established by Lee and Altintas^21–39. The milling force is computed for each individual microelement, and the aggregate force exerted on the variable-pitch ball-end cutter is subsequently obtained by summing the contributions of all these microelements, as illustrated in Fig. 3.

Fig. 3.

Discrete micromechanical model of milling force.

Within the mechanical mechanics framework, the milling force acting on the milling element is resolved into shear and plowing components. The tangential, axial, and radial milling forces exerted on the element are formulated as functions of the element width, milling edge length, milling force coefficient, and the undeformed milling thickness (illustrated in Fig. 4), as presented in Eq. (5)⁴⁰:

5

Fig. 4.

Undeformed milling thickness and width.

In the equation, dF_t, dF_r and dF_a represent the tangential, radial, and axial elemental milling forces, respectively;K_tc, K_rc and K_ac represent the tangential, radial, and axial shear force coefficients, respectively; K_te, K_re, and K_ae represent the plowing force coefficients in the tangential, radial, and axial directions, respectively; h represents the unmodified milling thickness; db represents the discrete element width; and ds represents the discrete element length.

The discrete element length ds can be expressed by formula (6):

6

g(ψ_ji(θ,z)) is a window function used to determine whether the cutting edge element participates in milling:

7

In Eq. (7), φ_st(z) and φ_ex(z)represent the cutting angle and exit angle corresponding to the milling element at the tool axial height z, respectively. The solution process is described in detail in Sect. 2.4. h(ψ,θ,κ) is the instantaneous undeformed milling thickness, and db is the discrete element width. If the tool tooth motion trajectory is approximated as an arc, based on the geometric relationships in Fig. 4, h(ψ,θ,κ) and db can be expressed by formula (8):

8

Among them, κ_i is the axial immersion angle, expressed as , f_z is the feed per tooth, which can be expressed as f_z=F/(N_TΩ), where F corresponds to the feed rate. In the case of variable pitch tools, the feed per tooth varies across different milling edges; therefore, Eq. (8) requires modification, as illustrated in Eq. (9):

9

Among them, f_t(j) represents the feed rate of the jth milling edge, which can be calculated based on the corresponding tooth angle φ_{p, j} and feed rate per tooth f_z:

Using the transformation matrix T and formula (2), the micro-element cutting forces in the X_T, Y_T, and Z_T directions in the tool coordinate system can be obtained:

10

Among them, the transformation matrix T can be expressed as:

11

By integrating the micro-element forces in the X_T, Y_T, and Z_T directions along the tool axis according to Eq. (11), we can obtain the milling force acting on the k-th milling edge:

12

Within this context, the parameters z_min and z_max denote the lower and upper axial boundaries of the tool engaged in the milling process. A comprehensive explanation of the solution methodology is provided in Sect. 4. Based on Eq. 13, the total milling force exerted by the ball end mill with N_t tooth pitches is expressed as follows:

13

Identification of cutting force coefficients for variable pitch ball-end milling cutter

Thin-walled components employed in the aerospace industry are primarily fabricated from materials that are challenging to machine, including titanium and high-temperature alloys. During precision machining with ball-end mills, the typical milling depth is generally less than 0.5 mm, with the majority of operations involving single-tooth engagement. Consequently, conventional slot or half-slot cutting experiments are unsuitable for this context. Moreover, in precision milling using ball-end cutters, the tip of the cutter lacks milling capability and cannot participate in the milling process. Therefore, this study utilizes experiments conducted at minimal cutting depths to accurately calibrate the milling force coefficients associated with ball-end cutters.

The ball-end cutter milling force coefficient identification model developed in this section is specifically designed for single-tooth milling conditions, defined as N ≤ 1, where N denotes the number of milling edges engaged in the milling process at any given time. Consequently, it is imperative to first verify whether the milling condition corresponds to single-tooth milling; if not, the model cannot be applied. As outlined in Sect. 2.3, the milling force is computed through a discrete element approach. For each element, the maximum milling angle θ is determined based on entry angle of the axial contact angle θ_st and exit angle of the axial contact angle θ_ex, which together define the angular span between the entry and exit points of a single tooth. The precise formulation is presented in Eq. (14):

14

The operational mode of the tool, specifically whether it is engaged in single-tooth milling, can be determined by analyzing the relationship between the maximum milling angle, θ, and the individual tooth angles of the variable-pitch tool. If the maximum milling angle is smaller than any of the tooth angles, denoted as θ < ϕ_gj, the tool operates in single-tooth milling mode. Conversely, if this condition is not met, the tool functions in multi-tooth milling mode. When focusing on the milling zone, the cutting force associated with a single tooth of the ball-end cutter can be mathematically represented by the following formula:

15

Due to the distinctive spherical geometry of the ball-end mill’s head, employing a single constant to characterize the milling force coefficient across varying tool positions results in considerable inaccuracies. Consequently, this study models the shear force coefficient as a cubic polynomial function of the axial position angle, while considering the plowing force coefficient as a constant. This approach facilitates the development of a calibration model for the milling force coefficient corresponding to an individual tooth element.The shear force coefficient is expressed as a function of the axial position angle κ and coordinate transformation methods are employed to enhance the model’s adaptability to different tool orientations. This approach provides methodological support for subsequent optimisation studies. Binding the shear force coefficient to the tool’s intrinsic geometric parameter, the axial position angle κ, enables the coefficient to reflect the spatial orientation and cutting state of different points on the ball-nose cutting edge. This approach offers a degree of universality and reduces the experimental workload^{41]– [42}.

Specifically, the shear force coefficient is formulated as a cubic polynomial function of the axial position angle^43–45, whereas the plowing force coefficient is treated as a constant. The detailed mathematical representation of the model is provided in Eq. 16:

16

Substituting formula (16) into formula (15) gives us:

17

Where T is the aforementioned transformation matrix and M₁, M₂, M₃, M₄, and M₅ are as shown in the formula.

18

令:

19

Substituting Eqs. (18) and (19) into Eq. (17) gives us the following result:

20

Among them:

21

The existing literature^32–37demonstrates that slot or semi-slot cutting experiments commonly employ the average milling force measured over the entire machining cycle to determine the milling force coefficient. This approach is justified by the fact that, during a complete cycle, at least two cutting edges engage simultaneously in the milling process, and there are no intervals without cutting activity. In contrast, during finishing operations with ball-end tools, typically only one tooth is engaged in cutting at any given time. Consequently, utilizing the average milling force over the entire cycle to calibrate the milling force coefficient does not accurately represent the actual cutting forces experienced. To overcome this limitation, the present study proposes a calibration method based on the average milling force of a single tooth, thereby enhancing the precision of the coefficient estimation. The calibration model for the average milling force coefficient of a single-tooth micro-element is formulated as follows:

22

Among these, andrepresent the average forces in the x, y, and z directions, respectively. Equation (22) has three equations and 15 unknowns. Since the number of unknowns exceeds the number of equations, it is an indeterminate system. Therefore, a single set of milling force data cannot solve the equations. Theoretically, at least five experiments are required to obtain 15 independent equations for solution. To minimize systematic errors, this study conducted six experiments to calibrate the milling force coefficients.

Solving the contact area between workpiece and cutting tool

Due to the inherently complex curved surface geometry of ball-nose cutters and the variable tool orientation during five-axis machining, the contact area between the cutter and workpiece constitutes a three-dimensional surface that evolves dramatically both temporally and spatially. Without precisely defining this boundary, it becomes impossible to determine the “cutting-in” and “cutting-out” states of each discrete cutting edge element. This leads to the introduction of numerous non-existent “idle cutting” elements or the omission of elements actually participating in cutting during cutting force calculations, resulting in severely distorted prediction results. Therefore, the precise determination of the contact area is crucial: First, it provides the entry angle (θ_st) and exit angle (θ_ex) for calculating the undeformed chip thickness on each cutting edge microelement, which is the primary source of cutting force excitation. Second, it defines the upper and lower limits for integrating instantaneous cutting forces. For variable-pitch ball-end mills, the non-uniform distribution of inter-tooth angles causes instantaneous changes in the rake conditions of each cutting edge, further exacerbating the time-varying complexity of the contact area.

The contact area solution method described here is primarily suitable for the finishing stage of part machining. It also offers higher accuracy for relatively flat curved surfaces or planar regions on part surfaces. For the calibration of free-form surfaces, refer to references^43,44.

Solving the fixed-axis cutting contact area

As shown in Fig. 5(a), surface S is the cutting contact area between the tool and the workpiece under a fixed tool axis, composed of boundary curves 1, 2, and 3. The plane projection of the cutting contact area is shown in Fig. 5(b).

Fig. 5.

Schematic diagram and plane view of the cutting area of a ball-end mill. (a) Schematic diagram of the contact area, (b) Planar projection diagram of the contact area.

As shown in the Fig. 5, establish the analytical equations for boundary curves 1, 2, and 3:

Boundary curve 1 is the intersection line between the tool hemisphere and the upper surface of the workpiece, which can be expressed as:

23

Boundary curve 2 is the intersection line between the contour circular cross section of the ball-end cutter perpendicular to the feed direction and the transition surface of the workpiece, which can be expressed as:

24

The boundary curve 3 is the intersection line between the circular cross section perpendicular to the feed direction of the tool hemispherical surface and the contact area, which can be expressed as:

25

In Eqs. (23), (24), and (30), R₁ is the maximum effective milling radius at a given cutting depth, e is the distance between boundary curve 1 and boundary curve 2 of the projection plane XZ of the cutting contact area and the tool axis, a_p is the milling depth, and a_e is the milling pitch.

Solving the free-axis cutting contact area

Following the determination of the cutting contact area corresponding to the fixed tool axis, a sequence of two rotational transformations and one translational transformation is applied, as illustrated in Fig. 6. In these rotational transformations, the rotation angles are defined with reference to the positive directions of the XYZ coordinate axes, measured toward the origin; counterclockwise rotations are assigned positive values, whereas clockwise rotations are considered negative. This methodology facilitates the conversion of the cutting contact area S_g, initially obtained under the fixed tool axis, into the cutting contact area S_gzc corresponding to an arbitrary tool axis orientation.

Fig. 6.

Coordinate frame rotation step.

The detailed transformation procedure is as follows: As depicted in Step 2 of Fig. 6, the coordinate system O-XYZ is rotated clockwise about the Y-axis by an angle α, resulting in the intermediate coordinate system O-X_PY_PZ_P. Subsequently, according to Step 3, a rotation about the X_P-axis by an angle β is performed, yielding the coordinate system O-X_mY_mZ_m. Finally, as shown in Step 4, the coordinate system O-X_mY_mZ_m undergoes a translation along the negative direction of the Z_m-axis by a distance R, producing the tool coordinate system O-X_TY_TZ_T. This transformation sequence is mathematically represented by Eq. 26.

26

The rotation transformation matrices T_t and T₁ are specifically expressed as follows:

27

Simulation and experimental study of milling cutting force

Experimental design and conditions

To determine the milling force coefficients and to validate the accuracy and reliability of the multi-axis milling variable pitch ball-end cutter cutting force model, an experimental study was conducted. Milling tests were performed on a rectangular workpiece measuring 100 mm × 70 mm × 30 mm to acquire milling force data. The workpiece material selected was titanium alloy TC4, chosen for its superior corrosion resistance, high-temperature tolerance, and favorable toughness characteristics. The experimental setup is illustrated in Fig. 7. The milling operations were carried out on a 700U machine tool, with cutting forces measured by a Kistler 9255B fixed three-axis force transducer. Data acquisition was facilitated through a Kistler Type 5010 charge amplifier and signal acquisition unit, with measurements recorded using DEWESoft testing software at a sampling frequency of 20,000 Hz. The testing principle and detailed experimental configuration are depicted in Fig. 7(a-b).

Fig. 7.

Milling force coefficient calibration experiment site.

In the experiment, to effectively extract the cutting force signal, the fundamental frequency of the cutting force was first calculated based on the spindle speed and the number of cutting edges. The spectral characteristics of the signal were then analyzed. Based on this analysis, the upper and lower cutoff frequencies of the bandpass filter were set to effectively preserve the fundamental frequency and harmonic components of the dynamic cutting force while suppressing high-frequency noise and low-frequency drift interference.

Solution of cutting force coefficient

The experiment employed a four-tooth carbide ball-end milling cutter characterized by an 8 mm tool diameter, a helix angle of 40°, and sequential tooth angles of 85°, 95°, 85°, and 95°. Prior to experimentation, the tool underwent inspection using the Zoller high-precision tool measurement and analysis system. The measured tool radius deviated from the theoretical value by approximately 0.01 mm, while the measured tooth angles differed by about 0.3°, both deviations falling within acceptable experimental tolerances. Experimental conditions included a spindle speed of 3000 rpm, a front tilt angle of 0°, a side tilt angle of 20°, milling depths of 0.3 mm and 0.5 mm, and feed rates per tooth (f_z) of 0.03, 0.04, and 0.05 mm/tooth. Six experimental groups were established by systematically combining these parameter variations.This study fixed the helix angle at 40°, which was established as the optimal angle through prior systematic parameter optimization experiments. Under this premise, the core objective of the research is to focus on investigating the independent influence of the variable pitch angle as the central innovative variable.

In each series of experiments, the component forces F_x, F_y, and F_z along the x, y, and z axes were measured independently. Subsequently, the average milling force about per individual tooth of the tool was determined. The detailed data are presented in Table 1.

Table 1.

Experimental test milling force data.

Experiment number	Milling depth ap (mm)	Feed per tooth fz (mm/tooth)
1	0.3	0.03	-5.13	6.76	4.72
2	0.3	0.04	-6.84	9.01	6.29
3	0.3	0.05	-8.55	11.26	7.86
4	0.5	0.03	-12.03	12.68	9.77
5	0.5	0.04	-16.04	16.91	13.03
6	0.5	0.05	-20.05	21.14	16.29

By substituting the six sets of milling force data presented in the preceding table into Eq. (22), the resulting calculations produce the expressions for the shear force coefficient and the plowing force coefficient, which are summarized in Table 2.

Table 2.

Expression of milling force coefficient.

Shear force coefficient	Ploughing force coefficient
Ktc=-1106.94κ3-1940.81κ2-2616.18κ + 3018.44	Kte = 86.65
Krc = 8244.34κ3-2454.69κ2 + 2000.59κ + 148.66	Kre=-60.97
Kac = 7596.21κ3-1335.60κ2-181.65κ + 588.99	Kae=-58.02

Tool-workpiece contact area solution

Drawing upon the calculations of the cutting contact area for the ball-end cutter presented in Sect. 3, the cutting contact areas for both fixed-axis and free-axis conditions were determined independently, utilizing the cutting parameters detailed in Table 3. The corresponding plane views are illustrated in Fig. 8. As the lead and lateral tilt angles change, it can be seen that the engagement area also changes accordingly.

Table 3.

Contact area calculation processing parameters table.

Category	Diameter(Φ/mm)	Depth(ap/mm)	Step(ae/mm)	lead angle (α/°)	Lateral tilt angle(β/°)
Fixed-axis	8	0.6	0.8	0	0
Free-axis	8	0.6	0.8	20	0
	8	0.6	0.8	0	20
	8	0.6	0.8	20	20

Fig. 8.

Comparison of cutting contact areas under different tool postures. (a)Cutting area under fixed-axis, (b) Cutting area under α = 20°,β = 0, (c) Cutting area under α = 0,β = 20°, (d) Cutting area under α = 20°,β = 20°.

Figure 9 shows the projection of the cutting contact area in the direction perpendicular to the tool axis under one posture orientation. The projection of the cutting contact area can be used to determine the tool’s entry and exit angles, as well as the upper and lower limits z_min and z_max of the axial integral corresponding to the micro-element force. Project the l milling micro-elements along the tool’s axial direction onto a plane perpendicular to the tool axis. The radius of the projected circle corresponds to the milling radius of the milling micro-elements at different z-coordinates. As shown in Fig. 9, each axial integral micro-element’s projected circle intersects the contact area at two points (i.e., the entry point and exit point). The angles formed by the lines connecting the entry point and exit point to the center of the projected circle (i.e., the tool tip) are different (i.e., each micro-element has different entry and exit angles). The projection circles corresponding to the nearest and farthest points from the center in the shaded region are the milling micro-elements at the upper and lower limits of the axial integration, respectively. Their coordinates in the tool axis direction are z₁ and z₂, respectively, and k₁ and k₂ are the axial immersion angles corresponding to the nearest and farthest points, respectively.

Fig. 9.

Schematic diagram of cutting-in and cutting-out angles and upper and lower limits of axial integration.

By solving the equations, the cutting angles (both entry and exit) corresponding to the cutting edge micro-elements under different tool orientations, along with their axial positions, can be obtained, as shown in Fig. 10. The tool diameter, cutting depth, and feed rate are consistent with those in Table 3. It is clearly evident that, under the same machining parameters but different tool orientations, the cutting angles (entry and exit) and their corresponding axial positions exhibit significant changes in response to variations in tool posture orientation.

Fig. 10.

Cutting-in and cutting-out angles corresponding to cutting edge microelements under different tool postures. (a) α = 0°,β = 0°, (b) α = 20°,β = 0°, (c) α = 0°,β = 20°, (d) α = 20°,β = 20°.

Cutting force simulation and result analysis

To verify the accuracy of the milling force coefficients, the milling parameters used to measure the milling force in Table 1 were substituted into the milling force coefficient expression in Table 2 to calculate the milling force coefficients. Then, using the milling force coefficient and the lead angle and clearance angle obtained in Sect. 2.3, combined with the ball-end milling force model established in Sect. 2.2, milling force prediction was performed. The experimentally obtained milling force was compared with the predicted milling force, as shown in Fig. 11. The red solid line represents the experimentally obtained milling force, while the blue dashed line represents the simulated milling force. Figure 11(a) shows the results for a milling depth of 0.5 mm and a feed rate of 0.03 mm/tooth, while Fig. 11(b) shows the results for a milling depth of 0.5 mm and a feed rate of 0.05 mm/tooth.

Fig. 11.

Comparison of milling force experiment and simulation.

As can be seen from Fig. 11, the milling force obtained from simulation is in good agreement with the milling force obtained from experiment. To calculate the degree of deviation between the milling force obtained from simulation and the milling force obtained from experiment, the root mean square error is introduced for quantitative analysis. The calculation formula is given by Eq. (28):

28

In the equation, F_{x_max}, F_{y_max}, and F_{z_max} are the average values of the maximum cutting forces per tooth obtained from the experiment, as expressed in (34). ΔFs is the sum of the squares of the interpolations between the simulated maximum cutting forces in each direction and the corresponding experimental maximum cutting forces in the same direction, as expressed in the following equation:

29

In the equation, j denotes the number of teeth on the tool, and F_{xi_max}, F_{yi_max} and F_{zi_max} represent the measured maximum milling forces for the ith tooth in the X, Y, and Z directions, respectively. (Note that the maximum milling force here refers to the magnitude of the force, which is a scalar.)

30

In the equation, F_{x_sm}, F_{y_sm} and F_{z_sm} represent the maximum simulated milling forces in the X, Y, and Z directions, respectively.

According to formulas (33), (34), and (35), the processing parameters can be calculated as follows: when the milling depth is 0.5 mm and the feed rate per tooth is 0.03 mm/tooth, the root mean square error is 3.11% and 6.74%, respectively. This indicates that the milling forces obtained from simulation are in good agreement with those obtained from experiments.

Conclusion

Variable-pitch ball-nose end mills optimise cutting dynamics by actively regulating the pitch of the tool geometry, which effectively suppresses regenerative chatter. They offer significant advantages for high-speed, high-efficiency and high-precision machining, particularly for difficult-to-cut materials and thin-walled, complex structural components. The key to optimising their vibration-damping performance lies in establishing an accurate cutting force model and optimising tooth spacing. Based on the classical infinitesimal milling force mechanical model, this paper presents an accurate model for variable-pitch ball end mills that can be applied to multi-axis finishing operations. The main achievements are as follows:

1. A geometric cutting dynamics model for variable pitch ball end mills under five-axis finishing conditions is established based on the infinitesimal cutting concept.

2. A cutting force coefficient calibration method based on the average single-tooth milling force is proposed which takes into account practical conditions in precision machining, such as a small radial depth of cut, single-tooth cutting and tip avoidance due to the lead and tilt angle. Experimentally calibrated coefficients applied to cutting force simulation yield results that are highly consistent with measured values, with a maximum root mean square error of 6.74%.

3. An efficient and accurate method for calculating the cutting contact area is presented. First, the contact area is computed under a fixed tool axis. Then, coordinate transformation is applied to solve for the contact area under any tool axis vector. This significantly enhances the accuracy of cutting force prediction.

Author contributions

Tian weijun: Project Manager and First Contact Person，Methodology, investigation, formal analysis and draft writing. Review and editing. Zhou Jinhua and Ren Junxue: Project administration，formal analysis, review, and editing. Wang Yizhuo: Project administration, Supervision and review. Bai zerui: Supervision, project administration, and review.

Funding

This research has been supported by the National Natural Science Foundation of China under Grant nos. 52375465,, and the Natural Science Basic Research Program of Shaanxi (Program No.2024JC-YBMS-288;2025JC-YBMS-474).

Data availability

All data and materials used or analyzed during the current study are included in this manuscript.

Declarations

Competing interests

The authors declare no competing interests.

Ethical approval

The manuscript is approved by all authors for publication. This manuscript is original and the entire paper or any part of its content has never been published in other journals. The data and results are true and clear. We guarantee the transparency and objectivity of research, and strictly abide by the ethical and professional standards of the industry. All the authors listed agree to participate in this manuscript.Consent for publication.

All co-authors agree to publish the version of this work in Scientific Reports.