New Approach to High-Speed Multi-Coordinate Milling Based on Kinematic Cutting Parameters and Acoustic Signals

Abstract

In this work, a new approach to high-speed multi-coordinate milling was developed. The new approach is based on a new model of trochoidal machining; this is, in turn, based on the theoretical thickness of a chip and its ratio to the cutting edge’s radius, allowing us to establish the vibroacoustic indicators of cutting efficiency. The new model can be used for the real-time assessment of prevailing cutting mechanisms and chip formation. A set of new indicators and parameters for trochoidal high-speed milling (HSM), which can be used to calculate tool paths during technological preparation of slotting, was determined and verified. The size effect in the multi-coordinate HSM of slots on cast iron was identified based on the dependency of vibroacoustic signals on the cutting tooth’s geometry, HSM’a operational machining modes, theoretical chip thicknesses, the sizes of the cut chips, and the quality/roughness of the surface being machined. Based on the analysis of vibroacoustic signals, a set of the most important indicators for monitoring HSM and determining cutting and crack-formation mechanisms during chip deformation was derived. Based on the new model, recommendations for monitoring HSM and for assigning the tool path relative to the workpiece during production preparation were developed and validated.

1. Introduction

The high-speed cutting of materials is one of the most common types of high-performance machining [1]. HSM differs from traditional machining in that HSM cutting speeds are 8–15 times higher [2]. Recent developments in HSM pave the way to machining alloys of high hardness, as well as cast iron, with high accuracy and low roughness [3]. In high-speed machining, productivity, defined as the material removal rate (MRR), increases due to high feeds; an increase in cutting speeds and cutting depth, a_p, occurs, while the cutting width, a_e, decreases, which reduces the area and cross-section of chips and thus reduces the cutting tool’s load [4,5]. In this case, the chip’s thickness is low. During high-speed machining, the radius of the cutting edge is often comparable to the chip’s thickness [6]. This intensifies material deformation by the cutting edge when cutting into a workpiece and is known as the size effect [7], which is observed during the machining of both brittle [8,9] and plastic [10,11,12] materials.

One of the most common machining operations is slotting. Box-type products with slots are usually made of cast iron [13]. There are two different schemes of contact between the cutting tool and the workpiece: (a) The cutting width is equal to the tool’s diameter. With these cutting parameters, thermomechanical loads on the cutting edges increase [14], which is caused by an increase in the average thickness and angle of contact of the end mill with the workpiece. (b) Cutting with a cutting width smaller than the tool’s diameter results in a small contact angle between the tool and the workpiece [15], which improves chip and heat removal from the cutting zone. The second scheme is preferable for HSM. The recommended contact angle between the tool and the workpiece for HSM is 10–20 degrees. Thus, one of the most promising ways of developing HSM technologies is to improve machining strategies by distributing the allowance across the width in the radial direction between cutting cycles. This can be achieved by using special tool paths.

The use of high-precision machine mechanics and the development of special machining strategies with the distribution of the cut-off allow for a significant increase in HSM productivity [16]. One of the most common machining strategies involves moving the cutting tool along a trochoidal trajectory [17]. In some cases, it is also possible to decrease the roughness and, thus, improve the quality of the surface being machined by distributing the cut-off allowances during different trochoidal machining cycles [18]. There exist many publications dedicated to studying the dependence of the geometric parameters of the cut-off layer on the cutting modes and on the kinematics of the tool’s movement [19,20,21,22]. They show clearly that the size of the machined layer in the cutting zone, the shape and size of the tool section in contact with the layer of the material being machined, and the mechanisms of the surface’s formation are extremely important. Therefore, the cutting mechanism of the cutting wedge that is in contact with the workpiece directly affects the product’s quality. To evaluate the cutting mechanisms, it is necessary to know the minimum uncut chip thickness in the cutting zone. This allows the direct assessment of such an important indicator as a fraction of the material removed with an increased degree of deformation. Based on this indicator, operational modes in which the width of the machined layer of material is higher than the minimum uncut chip’s thickness are assigned. Otherwise, increased forces occur during cutting, and significant residual irregularities on the surface being machined are formed [23,24,25]. A comprehensive study of the aforementioned processes during the cutting of non-plastic materials was performed [26].

The surface of a slot machined with a smaller-than-permissible cutting depth is subject to edge cracking and chipping when it intersects the end face of the product. The kinematics of the tool’s movement during trochoidal machining causes a decrease in the chip’s width due to the decreased contact angle between the cutter and the workpiece [27], which, in turn, significantly increases the size effect. In this case, when the contact angle between the end mill and the workpiece decreases, the chip’s thickness decreases. Thus, in some cases, the chip’s thickness may be insufficient for the effective cutting of the cutting wedge into the workpiece material to form a chip element [28]. In this case, the adverse effect manifests itself via the elastic deformation of the material’s surface, the increased cutting force acting on the workpiece’s surface, and increased surface heating. Chip cutting efficiency can also be assessed using indirect indicators such as the chip’s morphology, size, and shape [29,30], the quality of the surface being machined, and MRR [31].

To reduce the risk of premature loss of tool functionality and to increase the tool’s life in difficult cutting conditions, various wear-resistant coatings are used [32]. The best efficiency in difficult cutting conditions was demonstrated by coatings with a multilayer structure, which prevents the cracks propagation and cutting part’s edge chipping. In this case, instead of a direct transverse crack through a monolayer coating towards the substrate surface, a more complex nature of crack propagation is shown [33,34]. Coating and tool wear increase the radius of the cutting edge. In [35], it was shown that the use of a cut layer width that is greater than the cutting edge radius provides a more stable cutting process. It was also noticed that more intense wear of an uncoated tool with a smaller radius causes the same adverse effect as machining with a large cutting edge radius due to the formation of a zone of elastic deformation of a workpiece material by a cutting wedge due to the absence of a clearance angle in the wear area along a flank surface of the cutting tool [36].

Monitoring of high-precision slotting is extremely important due to the high cost associated with the replacement of defective products. Obtaining products with inappropriate surface quality entails the need to increase production time and, thus, leads to profit loss. There exist various approaches to monitoring and data collection, including deterministic methods such as support vector machine [37], artificial intelligence methods [38], and stochastic methods such as hidden Markov models [39], Kalman, and particle filters [40]. Chips with a thickness less than the critical value of the non-deformable chips’ thicknesses are not removed [41]. Therefore, one of the most important tasks in high-speed trochoidal machining with a chip width comparable to the cutting edge radius is to determine whether material is being cut off. It is also known that it is quite difficult to identify the cutting mechanism using force analysis methods due to changes in the cutting force not being an unambiguous indicator of the formation of a plunger and effective zones, which severely complicates the cutting mechanisms identification [42]. Monitoring of the HSM of products with small cut-off allowances using the analysis of vibroacoustic signals is more sensitive compared to the analysis of cutting forces [43]. HSM operation monitoring may be integrated into the pre-production process as a part of the CAM software.

Special tool trajectories for HSM are developed with accounting for the state of the cutting tool depending on the tool properties and its geometric parameters. However, this area is still largely by-way and underexplored. Technological pre-production for trochoidal milling always includes chip thickness modeling [44]. There exist methods with simplified schemes formation an overlength by an arc of a circle [18]. Numerical methods for calculating the chip thickness, in which calculations of the chip thickness, including the instantaneous thickness of the plunger zone and the instantaneous shear thickness, are also known. In [45], the minimum undeformed chip thickness (MUCT) was calculated for the first time by establishing and solving the equations for the cutting edge trajectory. In [46], a model for predicting milling forces for cutting modes with a predominance of the shear zone during material removal and for cutting parameters with a predominance of plunger cutting was developed. In [47], a model of the chip layer that takes into account the tool edge radius, size effects, tool deflection, tool runout, the actual trochoidal tool path, and the cutter tooth tip trajectory’s interaction was proposed. Moreover, it was reported that elastic recovery can significantly affect the calculations of MUCT and, thus, the accurate prediction of the cutting force. However, researchers have yet to obtain a solution in analytical form for the problem of allowance chip formation across a wide range of cutting parameters when modeling the rotational motion of two adjacent cutter teeth.

The obvious advantage of the analytical mathematical model of the chip thickness is the possibility to assign HSM cutting parameters based on monitoring outcomes, which opens up a new perspective of easy and efficient HSM technology tuning based on efficient plunging criteria, obtained through analyses of vibroacoustic signals.

The main goal of this work is to establish and quantify the size effect during high- speed multi-coordinate slotting of cast iron based on the analysis of the vibroacoustic signal; this depends on factors such as the cutter tooth geometry, high-speed operational modes, theoretical chip thickness, measured chip shape, and the surface quality of the workpiece. Dependencies identified in this study form the theoretical foundation for the development a new approach to improving HSM technologies via the real-time adjustment of operational modes using an analytical model for calculating the rational sizes of chip layers based on the monitoring and processing of vibroacoustic signals, which enable the direct assessment and control of the nature and material removal uniformity of the real-time regime.

2. Materials and Methods

2.1. Methods of Determining Cutting Mechanisms

The methods for determining the cutting mechanisms can be divided into two groups: (1) prediction of the cutting process depending on the static parameters of the technological system; (2) dynamic indicators of the cutting process in the form of actual monitoring results. In this section, a new model of allowance formation is developed. The general structure of the chip thickness model was shown in [48]. To develop a new theoretical instrument for assessing the impact of the cut layer geometrical parameters, we first considered and then assessed the known methods for determining the minimum thickness of the uncut chip, or the cutting depth (is it from this that the chip is separated). An increase in the parameters affecting the cutting forces that occur with small cut layers strictly depends on the ratio between the cutting depth, h, and the minimum chip thickness of the unprocessed chip thickness, h_CZII. At h > h_CZII, the cutting forces are mainly affected by chip separation, occurring between the tool front surface and the workpiece and, to a smaller extent, by smoothing and friction between the flank surface of the tool (chip contact at h < h_CZI) and the workpiece (Figure 1). In this case, the effect of the cutting force components with different cutting mechanisms cannot be determined using cutting force analysis, because it is impossible to separate and to identify impacts of each component; thus, only monitoring the cutting force and the indicators cannot solve this problem. The main disadvantage of the plunger cutting mechanism is the significant stresses that are normal to the surface being machined, which cause microcracks on the cast iron surface.

Now, we consider a cutting mechanism with an actual cutting depth that is comparable to or less than the cutting edge radius (h < h_CZII). In this case, the kinematic rake angle of the tool is negative, which causes the simultaneous plowing and shear of the workpiece material. The material in the shear zone can be partially removed into chips and partially pressed into the surface. Chip formation may not occur when the cutting depth is less than h_CZI, defined as the minimum undeformed chip thickness (MUCT). As a result of plowing (the indentation of the material into the workpiece’s surface), the material is also squeezed out beyond the cutting edge; then, the deformed material begins to accumulate in front of the cutting edge tip, which can lead to chip formation. This indicates that the reverse process of transition from the plunger zone to the shear zone is also possible [49]. The periodic occurrence of cyclically repeating plunger cutting mechanisms and unstable chip formation at the elastoplastic stage leads to the changing integrity of the machined surface with increased residual stresses and the deteriorating micro-texture of the surface layer [50]. The minimum undeformed chip thickness during small-overcut cutting depends on the type of workpiece material, defined by the rotation angle of the end mill (θ_m) and the cutting edge corner radius, re. The term “stagnation point” refers to a region in which the material deforms or flows with minimal deformation or kinematic energy; this is because, at the stagnation point, the power spent on material removal is minimal. Therefore, a common strategy of identification includes determining the total energy as a function of stresses. This function determines the point of the material being cut with a minimum energy. Based on the aforementioned principle, the rotation angle, θ_m, is assumed to be the same as the friction angle (β) between the workpiece and the rake face [50]. The analytical notation for determining the minimum undeformed chip thickness, h, is described by the following equation:

$${\mathrm{h}}_{\mathrm{m}}={\mathrm{r}}_{\mathrm{e}}(1-\mathrm{c}\mathrm{o}\mathrm{s} ({\mathrm{\theta }}_{\mathrm{m}}\left)\right),$$

(1)

Using the Johnson–Cook fracture model [51], the machining of AL2024-T3 was simulated to determine the minimum uncut chip thickness by analyzing the material flow stress due to deformation induced in the workpiece by the pressing cutting edge. However, the model is applicable only to materials with ductile fracture. The authors of [52] focused on the simulation of orthogonal cutting of Ti-6Al-4V to estimate the minimum uncut chip thickness by varying the thickness of the cut material. Changing the chip-formation mechanisms affects the degree of elastic recovery of the material being processed; the minimum thickness of uncut chips can vary in the range from 0.125 to 0.5r_e. Many factors used in these models have to be measured. This complicates the calculations of the minimum uncut chip thickness. In view of these circumstances, the best way to determine the actual minimum uncut chip thickness is to measure it.

Changes in cutting mechanisms during the machining of brittle materials are shown in [53]. By analyzing the surface roughness, they found h_min = 0.293r_e for hardened SAE 1045 steel. Similar work was performed in [54] on the micro-milling of OFHC pure copper, where h_min was estimated to b_e ≈ 0.38r_e. By analyzing the cutting force behavior, the minimum uncut chip thickness for brass [55] and AISI 1045 steel [56] was found to be 0.11–0.4r_e and 0.3r_e, respectively. Both studies showed that, at cut thicknesses of less than the minimum uncut chip thickness, the axial component of the cutting force accounts for most of the total cutting force. This is the result of a predominant plunger cutting mechanism without observable chip formation. Based on this, we limit the cutting depth in the plunger zone by the ratio to h_CZI < 0.25r_e from the cutting edge rounding radius, r_e, by the maximum critical value (Figure 1). This allows us to establish a plunger cutting indicator with a sufficient degree of confidence. A fraction of the chips formed in the transitional shear zone are efficiently separated from the workpiece, while their remains are pressed into the workpiece surface. The transitional shear zone is limited by the cutting depth, h_CZI − h_CZII = 0.25–0.6r_e. The chip thickness being greater than the cutting edge’s radius characterizes the efficient chip separation from the workpiece without the size effect.

2.2. Methods for Determining the Chip Thickness in Trochoidal Milling

Predicting the mechanisms of the cutting process, depending on the static parameters of the HSM technological system, primarily requires a focus on the distribution of the thickness of the cut chips during the entire machining cycle. The specificity of the HSM implies a uniform distribution of the allowance removed with a high feed and cutting speed. Improving the technological preparation of production with a rational chip thickness can be implemented by calculating the nature of the distribution of the cut layers between the teeth of the mill. The most common tool path when implementing HSM is trochoid. The main path to a relationship between the stage of technological preparation of production and the cutting mechanism is the determination of the chip geometry and its thickness during trochoidal machining. The geometry of uncut chips changes significantly during slot milling with a trochoidal tool path due to the impacts of circular and linear motions. In addition, the implementation of the trochoidal tool path is associated with complex calculations of the chip thickness.

Existing methods for calculating the chip thickness during trochoidal milling approximate the actual trajectory or the sequential cutter positions by circular arcs, which leads to significant errors [18]. The work presented in [57] most adequately reflects the tooth trajectory in trochoidal milling. In this study, a numerical algorithm was proposed and developed based on the search for self-intersection and cross-intersection points of the trochoidal tool trajectory, and the correct strategy for connecting the intersection points and constructing the chip area was determined. In addition, the numerical algorithm was generalized to model the uncut chip thickness of multi-tooth and/or variable-pitch cutting tools. However, iteratively searching for the allowance thickness is associated with high computational costs and a need to compute for each ratio for different materials. Therefore, this model is not suitable for searching for a universal indicator linking the production preparation system and dynamic indicators obtained during real-time HSM monitoring.

2.3. Methods of Monitoring High-Speed Machining

Monitoring cutting forces disallows the unambiguous identification of chip-formation mechanisms. Based on the macrotexture of the surface being machined, the size of uncut chips can be indirectly estimated [31]. The analysis of acoustic emissions during the cutting of various materials allows the identification of general patterns of changes in the spectra of the vibroacoustic signals that are associated with changes in the energetic impacts on the surface of materials being cut [58]. In this work, it was established that a change in the quality of machining in the form of increased wear leads to an alternate predominance of the ratio of the amplitudes of acoustic signals in the low-frequency and high-frequency ranges. The frequency range of high sensitivity of accelerometers to activity voltage is up to 20 kHz, but in practice, relevant data were obtained for a frequency range of up to 40 kHz. When monitoring the process of mechanical processing, the change in AE signals to changes in the state of the tool during cutting was studied. These include the wear of the cutting edges, chipping, and breakage. If the relationship between the nature of the change in the parameters of AE signals, the change in the power density, and, accordingly, the proportion of evaporated material during processing with concentrated energy flows is clear [59], then the relationship of the acoustic parameters with the wear of the blade tool needs to be substantiated. In this case, there are no thermal pulses, the power density of which can be associated with a change in the parameters of AE signals. From the theory of spectral analysis of pulse signals, it is known that the frequency of the upper limit of the decomposition of a pulse signal into a spectrum containing the main part of the pulse energy is inversely proportional to the pulse duration [60]. Of great importance in assessing the trochoidal processing is the radial depth of cut on tool wear was studied considering cutting speed, and a corresponding tool wear model related with radial cutting depth was established [61].

The mechanisms of pulse time change during cutting may have different causes. Increase in the load in the contact of the tool with the workpiece surface is accompanied by an increase in friction forces causing dissipation of vibrational energy, which is especially noticeable in the high-frequency range. An increase in friction forces between the surfaces of the tool and the workpiece causes the heating of the surfaces. This may lead to a decrease in the mechanical properties of the workpiece, which reduces the share of brittle fracture in favor of the viscous mechanism during chip formation. Brittle cracks are characterized by high crack velocity and short pulses that create amplitude spectra, extending into the high-frequency range of the vibroacoustic signal. Viscous cracks have a significantly lower development rate and form longer pulses, creating smaller amplitudes at high frequencies. With an increase in the contact density of the cutter tooth with the workpiece, the surface irregularities pass into a state close to all-round compression, which prevents the formation of brittle cracks, and the pulses become longer. In this case, the interaction energy is distributed over a longer period, which reduces the specific power. The second mechanism of change in the vibroacoustic characteristics of the signal during cutting is associated with a change in temperature, which increases with friction. An increase in temperature affects the mechanical properties of materials, such as hardness and elastic modulus. In metals, an accelerated drop in mechanical properties begins in the temperature range of 0.5 of the melting point [62]. Changes in the parameters of AE signals with a change in the temperature of the tool and the workpiece under the influence of friction forces can be traced when turning steel workpieces at different speeds. Thus, it can be concluded that an increase in friction forces and increased heat generation accompanying the wear of cutting tools determine a drop in the amplitudes of acoustic signals at high frequencies.

In [58], it was revealed that the main reason for the impact on the change in the characteristics of the acoustic signal is the change in the power density of the energy impact on the workpiece material. The power density, defined as the impact power per unit surface, changes with a change in the power of the energy source, with an increase in the surface area of the impact, and with an extension of the duration of the energy pulses accompanying the processing. A decrease in the power density entails a drop in the amplitude of the high-frequency component of the acoustic signals compared to the low-frequency component. For practical use of the parameters of the acoustic signal, it is possible to control the ratio of the root mean square amplitudes of the acoustic signals in the low-frequency and high-frequency ranges. An increase in this ratio indicates that the pulses of the energy impact on the material have become smaller in amplitude or have become more stretched over time. That is, a change in the technological process has led to a drop in the power density of the impact on the material. During cutting, this is explained by the increase in tool wear, an increase in the area of the cutting edge impact and, accordingly, the transition to a viscous mechanism of crack formation during the displacement of chip elements. The ratio of the amplitudes of acoustic signals in the low- and high-frequency ranges is described by the coefficient K_f, which is sufficiently informative for measuring tool wear and should be based on preliminary experiments. There are no strict boundaries for choosing the ranges due to the change in the initial conditions of the natural oscillations of the machine tool technological system, but it significantly depends on the cutting speed. When changing the cutting speed within the selected frequency ranges; they should include areas of natural frequencies with a wide range, to monitor the condition of the tool at different cutting speeds, the most noticeable change in amplitudes with increasing wear is necessary. The change in cutting depends less on the processing modes if they do not change too widely; the depth and feed of the tool have a significantly smaller effect on K_f. The choice of frequency ranges for forming K_f depends on the natural frequencies of the elastic system of the machine and on the location of the accelerometer and should be based on preliminary experiments. The assignment of the boundaries of the analyzed signals includes the allocation of areas of natural frequencies with the most noticeable change in amplitudes with increasing wear [63]. As a result of the above-described works, it was proven that the ratio of the amplitudes of acoustic signals can be the contact area of the tool with the workpiece per unit of surface area by the power of the impact of the cutting wedge on the material; therefore, this indicator can presumably be used as a criterion for assessing the cutting mechanism during high-speed multi-coordinate processing with a trochoidal trajectory of the tool.

3. Mathematical Modeling of Chip Formation During High-Speed Milling

3.1. Analytical Model for Calculating Chip Thickness

The cutter path is a trochoid, which increases the material removal rate by increasing the depth, a_e, the feed rate per tooth, f_z, and the cutting speed, V (Figure 2). It is known that the axial cutting depth during trochoidal milling can be increased by up to ten times compared to traditional milling.

This section describes a new analytical model of chip formation in trochoidal cutting in detail. The trochoidal tool path can be characterized by three parameters: microphone speed, which is the speed of the router in rpm; nutation speed, which is the center tool rotational motion; and feed rate units in mm/min, which is the speed of the center of the end mill. In developing the new model, a relationship is established between the nutation speed and the angular position in the rotation of the mill, which is represented as the nutation angle through the angle describing the angular between the OX axis and the position passing through the contact point of the cutter with the machined surface. The combination of these three components of motion gives a trochoidal trajectory of the end mill. The mathematical model for calculating the sizes and shapes of chips in trochoidal milling height observes the relationships between the geometric parameters of the mills and the parameters of the sharp edges, cutting modes, and the design parameters of the machined slot.

The trajectories of movement of two teeth forming a turn of chips are described by the following equations:

$${\mathrm{F}}_{\mathrm{x}\mathrm{Z}1}\left(\mathrm{\theta }\right)={\displaystyle \frac{\mathrm{D}·\cos \left({\mathrm{\beta }}_1+\mathrm{N}·\mathrm{\theta }\right)}{2}}-{\displaystyle \frac{{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }\right)}{2}}+{\displaystyle \frac{{\mathrm{a}}_{\mathrm{e}}·\mathrm{\theta }}{2\mathrm{\pi }}},$$

(2)

$${\mathrm{F}}_{{\mathrm{y}}_{\mathrm{Z}1}}\left(\mathrm{\theta }\right)={\displaystyle \frac{{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }\right)}{2}}-{\displaystyle \frac{\mathrm{D}·\sin \left({\mathrm{\beta }}_1+\mathrm{N}·\mathrm{\theta }\right)}{2}},$$

(3)

$${\mathrm{F}}_{\mathrm{x}\mathrm{Z}2}\left(\mathrm{\theta }\right)={\displaystyle \frac{\mathrm{D}·\cos \left({\mathrm{\beta }}_2+\mathrm{N}·\mathrm{\theta }\right)}{2}}-{\displaystyle \frac{{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }\right)}{2}}+{\displaystyle \frac{{\mathrm{a}}_{\mathrm{e}}·\mathrm{\theta }}{2\mathrm{\pi }}},$$

(4)

$${\mathrm{F}}_{{\mathrm{y}}_{\mathrm{Z}2}}\left(\mathrm{\theta }\right)={\displaystyle \frac{{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }\right)}{2}}-{\displaystyle \frac{\mathrm{D}·\sin \left({\mathrm{\beta }}_2+\mathrm{N}·\mathrm{\theta }\right)}{2}},$$

(5)

where N is the number of cutter strokes for half the length of the cutter’s trajectory in contact with the workpiece, determined by the following expression:

$$\mathrm{N}={\displaystyle \frac{{\mathrm{\pi }}^2·{\mathrm{D}}_{\mathrm{t}}^2+{{\mathrm{a}}_{\mathrm{e}}}^2}{16\mathrm{\pi }·{\mathrm{f}}_{\mathrm{z}}·\mathrm{Z}}},$$

(6)

where β₁, β₂—initial angular positions of the cutter teeth, forming the angular pitch of the teeth; D—end mill diameter; D_t—slot width.

The main parameters of the cut chips during trochoidal machining are shown in Figure 3.

The distance between the points of the trajectories of the cutting edges of two adjacent teeth of mill is determined by the following equations:

$${\mathrm{F}}_{\mathrm{t}}={\mathrm{A}}_{\mathrm{t}}\left(\mathrm{\theta }\right)+{\mathrm{C}}_{\mathrm{t}}\left(\mathrm{\theta }\right),$$

(7)

$${\mathrm{A}}_{\mathrm{t}}\left(\mathrm{\theta }\right)={\displaystyle \frac{\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }\right)-\mathrm{\pi }·\mathrm{D}·\mathrm{N}·\mathrm{c}\mathrm{o}\mathrm{s}({\mathrm{\beta }}_2+\mathrm{N}·\mathrm{\theta })}{{\mathrm{a}}_{\mathrm{e}}+\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }\right)-\mathrm{\pi }·\mathrm{D}·\mathrm{N}·\mathrm{s}\mathrm{i}\mathrm{n}({\mathrm{\beta }}_2+\mathrm{N}·\mathrm{\theta })}}$$

(8)

$${\mathrm{C}}_{\mathrm{t}}\left(\mathrm{\theta }\right)={\displaystyle \frac{{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }\right)}{2}}-{\displaystyle \frac{\sin \left({\mathrm{\beta }}_2+\mathrm{N}·\mathrm{\theta }\right)}{2}}-{\displaystyle \frac{\left(\frac{{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }\right)}{2}-\frac{\mathrm{D}·\mathrm{N}·\cos \left({\mathrm{\beta }}_2+\mathrm{N}·\mathrm{\theta }\right)}{2}\right)+\left({\mathrm{a}}_{\mathrm{e}}+\mathrm{\pi }·\mathrm{D}·\cos \left({\mathrm{\beta }}_2+\mathrm{N}·\mathrm{\theta }\right)-\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }\right)\right)}{{\mathrm{a}}_{\mathrm{e}}+\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }\right)-\mathrm{\pi }·\mathrm{D}·\mathrm{N}·\sin \left({\mathrm{\beta }}_2+\mathrm{N}·\mathrm{\theta }\right)}},$$

(9)

The distance between the points of the paths of the vertices of two adjacent teeth of the cutter with an offset of the original position of the cutting tooth angular tooth pitch is expressed as

$${\mathrm{h}}_{\mathrm{\delta }}\left(\mathrm{\theta }\right)=\sqrt{{\left({\displaystyle \frac{\mathrm{D}·\cos \left({\mathrm{\beta }}_1+\mathrm{N}·\left(\mathrm{\Theta }+\mathrm{\theta }\right)\right)}{2}}-{\displaystyle \frac{{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\Theta }+\mathrm{\theta }\right)}{2}}+{\displaystyle \frac{{\mathrm{a}}_{\mathrm{e}}·\left(\mathrm{\Theta }+\mathrm{\theta }\right)}{2\pi }}-\left({\displaystyle \frac{\mathrm{D}·\cos \left({\mathrm{\beta }}_2+\mathrm{N}·\mathrm{\theta }\right)}{2}}-{\displaystyle \frac{{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }\right)}{2}}+{\displaystyle \frac{{\mathrm{a}}_{\mathrm{e}}·\mathrm{\theta }}{2\mathrm{\pi }}}\right)\right)}^2+{\left({\displaystyle \frac{{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\Theta }+\mathrm{\theta }\right)}{2}}-{\displaystyle \frac{\mathrm{D}·\sin \left({\mathrm{\beta }}_1+\mathrm{N}·\left(\mathrm{\Theta }+\mathrm{\theta }\right)\right)}{2}}\right)}^2},$$

(10)

$$\mathrm{\Theta }={\displaystyle \frac{2·\mathrm{\pi }}{\mathrm{N}·\mathrm{Z}}},$$

(11)

Correction of the initial points of the second tooth on the trajectory of movement determines the average value of the displacement relative to the view of the first tooth:

$${\mathrm{h}}_{\mathrm{t}}={\displaystyle \frac{\left|{\mathrm{A}}_{\mathrm{t}}\left(\mathrm{\theta }\right)·\left(\frac{\mathrm{D}·\cos \left({\mathrm{\beta }}_1+\mathrm{N}·\mathrm{\theta }\right)}{2}-\frac{{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }\right)}{2}+\frac{{\mathrm{a}}_{\mathrm{e}}·\mathrm{\theta }}{2\mathrm{\pi }}\right)-\left(\frac{{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }\right)}{2}-\frac{\mathrm{D}·\sin \left({\mathrm{\beta }}_1+\mathrm{N}·\mathrm{\theta }\right)}{2}\right)+{\mathrm{C}}_{\mathrm{t}}\left(\mathrm{\theta }\right)\right|}{\sqrt{{\left({\mathrm{A}}_{\mathrm{t}}\left(\mathrm{\theta }\right)\right)}^2+1^2}}},$$

(12)

Then, the theoretical chip thickness ${\mathrm{h}}_{\mathrm{\delta }}\left(\mathrm{\theta }\right)$ in analytical form is determined by the following equation:

$${\mathrm{h}}_{\Sigma }\left(\mathrm{\theta }\right)={\mathrm{h}}_{\mathrm{t}}\left(\mathrm{\theta }\right)-\left({\mathrm{h}}_{\mathrm{\delta }}\left({\mathrm{\beta }}_2\right)-{\mathrm{h}}_{\mathrm{\delta }}\left({\mathrm{\beta }}_1\right)\right)·\sin \left({\displaystyle \frac{\mathrm{\pi }}{2}}+\left|{\displaystyle \frac{\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }\right)}{{\mathrm{a}}_{\mathrm{e}}+\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }\right)}}\right|\right).$$

(13)

The envelope of the continuous position of a cutter with a diameter of the mill, D, when moving along a trochoid with a transverse slot, D_t, is determined by a system of parametric results:

$${\mathrm{F}}_{\mathrm{x}\mathrm{\beta }}\left(\mathrm{\theta }\right)={\displaystyle \frac{-{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }\right)}{2}}+{\displaystyle \frac{{\mathrm{a}}_{\mathrm{e}}·\mathrm{\theta }}{2\mathrm{\pi }}}-{\displaystyle \frac{\mathrm{D}}{2}}·\sin \left(\mathrm{atan}\left({\displaystyle \frac{\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }\right)}{{\mathrm{a}}_{\mathrm{e}}+\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }\right)}}\right)\right),$$

(14)

$${\mathrm{F}}_{\mathrm{y}\mathrm{\beta }}\left(\mathrm{\theta }\right)={\displaystyle \frac{{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }\right)}{2}}+{\displaystyle \frac{\mathrm{D}}{2}}·\cos \left(\mathrm{atan}\left({\displaystyle \frac{\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }\right)}{{\mathrm{a}}_{\mathrm{e}}+\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }\right)}}\right)\right).$$

(15)

The tangent to the envelope at each point is determined by the following expression:

$${\mathrm{F}}_{\mathrm{e}}\left(\mathrm{\theta }\right)={\mathrm{A}}_{\mathrm{e}}\left({\displaystyle \frac{{\mathrm{d}\mathrm{F}}_{\mathrm{x}\mathrm{\beta }}\left(\mathrm{\theta }\right)}{\mathrm{d}\mathrm{t}}};{\displaystyle \frac{{\mathrm{d}\mathrm{F}}_{\mathrm{y}\mathrm{\beta }}\left(\mathrm{\theta }\right)}{\mathrm{d}\mathrm{t}}}\right)+{\mathrm{C}}_{\mathrm{e}}\left({\displaystyle \frac{{\mathrm{d}\mathrm{F}}_{\mathrm{x}\mathrm{\beta }}\left(\mathrm{\theta }\right)}{\mathrm{d}\mathrm{t}}};{\displaystyle \frac{{\mathrm{d}\mathrm{F}}_{\mathrm{y}\mathrm{\beta }}\left(\mathrm{\theta }\right)}{\mathrm{d}\mathrm{t}}};{\mathrm{F}}_{\mathrm{x}\mathrm{\beta }}\left(\mathrm{\theta }\right);{\mathrm{F}}_{\mathrm{y}\mathrm{\beta }}\left(\mathrm{\theta }\right)\right).$$

(16)

The chip thickness in the contact zone of the cutter with the workpiece when moving along the trochoid is determined by the following equation:

$${\mathrm{h}}_{\mathrm{e}}\left(\mathrm{\theta }\right)={\displaystyle \frac{\left|{\mathrm{A}}_{\mathrm{e}}\left(\frac{{\mathrm{d}\mathrm{F}}_{\mathrm{x}\mathrm{\beta }}\left(\mathrm{\theta }\right)}{\mathrm{d}\mathrm{t}};\frac{{\mathrm{d}\mathrm{F}}_{\mathrm{y}\mathrm{\beta }}\left(\mathrm{\theta }\right)}{\mathrm{d}\mathrm{t}}\right)·\left(\frac{-{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }+2\mathrm{\pi }\right)}{2}+\frac{{\mathrm{a}}_{\mathrm{e}}·\left(\mathrm{\theta }+2\mathrm{\pi }\right)}{2\mathrm{\pi }}-\frac{\mathrm{D}}{2}·\sin \left(\mathrm{atan}\left(\frac{\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }+2\mathrm{\pi }\right)}{{\mathrm{a}}_{\mathrm{e}}+\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }+2\mathrm{\pi }\right)}\right)\right)\right)-\left(\frac{{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }+2\mathrm{\pi }\right)}{2}+\frac{\mathrm{D}}{2}·\cos \left(\mathrm{atan}\left(\frac{\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\cos \left(\mathrm{\theta }+2\mathrm{\pi }\right)}{{\mathrm{a}}_{\mathrm{e}}+\mathrm{\pi }·{\mathrm{D}}_{\mathrm{t}}·\sin \left(\mathrm{\theta }+2\mathrm{\pi }\right)}\right)\right)\right)+{\mathrm{C}}_{\mathrm{e}}\left(\frac{{\mathrm{d}\mathrm{F}}_{\mathrm{x}\mathrm{\beta }}\left(\mathrm{\theta }\right)}{\mathrm{d}\mathrm{t}};\frac{{\mathrm{d}\mathrm{F}}_{\mathrm{y}\mathrm{\beta }}\left(\mathrm{\theta }\right)}{\mathrm{d}\mathrm{t}};{\mathrm{F}}_{\mathrm{x}\mathrm{\beta }}\left(\mathrm{\theta }\right);{\mathrm{F}}_{\mathrm{y}\mathrm{\beta }}\left(\mathrm{\theta }\right)\right)\right|}{\sqrt{{\left({\mathrm{A}}_{\mathrm{e}}\left(\frac{{\mathrm{d}\mathrm{F}}_{\mathrm{x}\mathrm{\beta }}\left(\mathrm{\theta }\right)}{\mathrm{d}\mathrm{t}};\frac{{\mathrm{d}\mathrm{F}}_{\mathrm{y}\mathrm{\beta }}\left(\mathrm{\theta }\right)}{\mathrm{d}\mathrm{t}}\right)\right)}^2+{\left(1\right)}^2}}},$$

(17)

Equation (17), for the first time, enables us to establish the dependence of the theoretical chip thickness on the technical parameters of trochoidal high-speed milling in an analytical form without simplifications, obtained on the basis of new dependencies (2–16).

Dependences of the thickness of the cut chips during trochoidal high-speed milling on the angular position of the cutter under different cutting conditions are shown in Figure 4: (a) f_z 0.03 mm, ae 1.2 mm; (b) f_z 8 mm, a_e 10 mm.

Based on the dependences of the width of the machined layer on the angular position of the tooth during cutting per cycle of the trochoidal feed motion during slot machining shown in Figure 4, it can be concluded that, when the trajectory approaches the slot walls, the chip width decreases significantly (up to 3–4 times). This complicates the formation of chips with a thickness exceeding the typical size of the cutting edge rounding radius for uniform chip separation without indentation.

The proposed model, for the first time, enables us to solve the problem of finding the thickness chip during trochoidal cutting in an analytical form, which will enable the adaptation of the criteria of cutting efficiency without numerical iterative recalculations and using this model to interpret the signal in real time.

3.2. Determining the Kinematic Parameters of the Cutter–Workpiece Contact During Trochoidal High-Speed Milling

Let us consider the parameters describing the conditions of the cutter–workpiece contact. To assess the correlation of the cutting mechanisms relative to a wider range of process parameters, including the kinematic geometry of the tool. The angle, $\mathrm{\psi }\left(\mathrm{\theta }\right)$, of contact between the cutter and the workpiece depends on the diameter of the cutter and the width of its contact zone with the workpiece (the dependence graph is shown in Figure 5a); this is determined by the following equation:

$$\mathrm{\psi }\left(\mathrm{\theta }\right)=\mathrm{acos}\left({\displaystyle \frac{\frac{\mathrm{D}}{2}-{\mathrm{h}}_{\mathrm{e}}\left(\mathrm{\theta }\right)}{\frac{\mathrm{D}}{2}}}\right)$$

(18)

The number of teeth of the cutter with the workpiece, ${\mathrm{Z}}_{\mathrm{c}}$, that depends on the diameter of the cutter and the thickness of its contact zone with the workpiece (graph is shown in Figure 5b), is determined by the following equation:

$${\mathrm{Z}}_{\mathrm{c}}={\displaystyle \frac{\mathrm{\psi }\left(\mathrm{\theta }\right)·\mathrm{Z}}{2\pi }},$$

(19)

According to the dependencies of the cutter contact angle with the workpiece and the number of teeth in the cutter in contact during cutting, it follows that, during trochoidal milling, this contact area is small and does not exceed 50 degrees. During high-speed milling during the entire trochoidal milling cycle, no more than one tooth is in contact with the workpiece. These indicators show the importance of identifying the cutting mechanisms during the entire milling cycle. Therefore, each tooth when cutting into the workpiece determines the mechanics of cutting and chip formation, so the relevance of this study is quite high. The kinematic front angle, ${\mathrm{\gamma }}_{\mathrm{k}}\left(\mathrm{\theta }\right)$, can be determined by the following equation:

$${\mathrm{\gamma }}_{\mathrm{k}}\left(\mathrm{\theta }\right)=-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}\left({\displaystyle \frac{{\mathrm{r}}_{\mathrm{e}}-{\mathrm{h}}_{\mathrm{e}}\left(\mathrm{\theta }\right)}{{\mathrm{r}}_{\mathrm{e}}}}\right),$$

(20)

During the machining process, the static geometry of the cutting tool undergoes changes. The angle by which the rake and clearance angles change is determined by the difference between the tangent to the cutting surface and the tangent to the trajectory of the milling cutter tooth, this angle $\mathrm{\kappa }\left(\mathrm{\theta }\right)$ is described by the following equations:

$$\mathrm{K}\mathrm{c}\left(\mathrm{\theta }\right)=\mathrm{a}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{\left(\mathrm{D}\mathrm{t}·\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathrm{\theta }\right)-\mathrm{D}·\mathrm{s}\mathrm{i}\mathrm{n}\left({\mathrm{\beta }}_2+\mathrm{N}\left(\mathrm{a}\mathrm{e}\right)·\mathrm{\theta }\right)-\mathrm{D}\mathrm{t}·\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathrm{\theta }\right)\right)}{\mathrm{D}·\mathrm{c}\mathrm{o}\mathrm{s}\left({\mathrm{\beta }}_2+\mathrm{N}\left(\mathrm{a}\mathrm{e}\right)·\mathrm{\theta }\right)}}\right)-90,$$

(21)

$$\mathrm{K}\mathrm{t}\left(\mathrm{\theta }\right)=\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{\mathrm{\pi }·\mathrm{D}\mathrm{t}·\mathrm{c}\mathrm{o}\mathrm{s}\left(\mathrm{\theta }\right)-\mathrm{\pi }·\mathrm{D}·\mathrm{N}\left(\mathrm{a}\mathrm{e}\right)·\mathrm{c}\mathrm{o}\mathrm{s}\left({\mathrm{\beta }}_2+\mathrm{N}\left(\mathrm{a}\mathrm{e}\right)·\mathrm{\theta }\right)}{\mathrm{a}\mathrm{e}+\mathrm{\pi }·\mathrm{D}\mathrm{t}·\mathrm{s}\mathrm{i}\mathrm{n}\left(\mathrm{\theta }\right)-\mathrm{\pi }·\mathrm{D}·\mathrm{N}\left(\mathrm{a}\mathrm{e}\right)·\mathrm{s}\mathrm{i}\mathrm{n}\left({\mathrm{\beta }}_2+\mathrm{N}\left(\mathrm{a}\mathrm{e}\right)·\mathrm{\theta }\right)}}\right),$$

(22)

$$\mathrm{\kappa }\left(\mathrm{\theta }\right)=(\mathrm{K}\mathrm{c}-\mathrm{K}\mathrm{t}){\displaystyle \frac{\mathrm{\psi }\left(\mathrm{\theta }\right)}{\mathrm{\psi }\left(\mathrm{\beta }/2\right)}}.$$

(23)

The dependence of the kinematic front angle on the angular position of the cutter, shown through the point of contact of the cutter with the machined surface during movement along the trochoid during high-speed multi-coordinate machining with a chip thickness equal to the radius of the cutting edge, is shown in Figure 6. The kinematic front angle of the tool takes negative values in the area of the size of the cutting edge.

Based on the dependencies on the graph in Figure 6a, it follows that the kinematic front angle in the studied range of values with a chip thickness not exceeding the radius of the cutting edge leads to the simultaneous occurrence of plunger cutting and shear of the workpiece material. The angles of the cutting wedge during high-grade milling with trochoidal cutting change in accordance with the trend of the dependencies shown in Figure 6b,c. From these dependencies it follows that the kinematic parameters change significantly from the periphery to the center in the region with an increase in the thickness of the cut chip, while kinematic sharpening is more characteristic of up-cut milling (highlighted in red) than of down-cut milling (highlighted in blue).

3.3. Identification of Cutting Mechanisms, Uncut Chip Thickness and Geometry of the Cutter Blade in the Cutting Zone

To identify the cutting mechanisms and chip formation, we introduce the relationship between the uncut chip thickness he and the cutting edge rounding radius, r_e, we call this relationship or parameter, ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$:

$${\mathrm{K}}_{\mathrm{r}\mathrm{e}}\left(\mathrm{\theta }\right)={\displaystyle \frac{{\mathrm{h}}_{\mathbf{e}}\left(\mathrm{\theta }\right)}{{\mathrm{r}}_{\mathrm{e}}}}·100\%,$$

(24)

As was substantiated in the description of the method, h_CZI < 0.25r_e; that is, at ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ < 25%, chip formation may not occur due to the predominance of the plunger mechanism of chip formation. The choice of the upper limit of the uncut chip thickness with a plunger cutting mechanism, taken from open sources, will allow us to establish a certain value of the safety margin for a more reliable criterion for identifying the cutting mechanism. In addition, the value corresponds to an angle from −42° to −58°, i.e., the range of the transition of a stagnant zone of the chip with the minimum material removal rate, as was revealed in [46]. These results also correlate with the results of changing the kinematic geometry, obtained from the analytical model in this paper (Equation (20)), which is demonstrated in the graph of Figure 6. With an increase in the ratio of the chip thickness to the cutting edge radius of more than 40%, effective plunger cutting will be observed, accompanied by chip formation, which occurs due to the division of the material movement into two equal flows relative to the cutting edge in the range of the coefficient 25% < ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ < 60%. The chip thickness in the transition zone of shear during cutting combines the mechanism of plunger cutting and cutting with the formation of a material shear area; that is, this can occur in the range of the chip thickness in which part of it is effectively separated from the workpiece, and the other part is pressed into the machined surface. The value of the chip thickness, exceeding the radius of the cutting edge ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ > 60%, characterizes the process with effective separation of the chip from the workpiece without the characteristic size effect. Figure 7 shows the changes in the coefficient ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ depending on the angular position of the contact point of the cutter with the workpiece, θ, during one cycle of trochoidal high-speed milling.

These theoretical results of the ratio of the chip thickness to the radius of cutting edge for one cycle of trochoidal motion are compared with the characteristic of the ratio of the frequencies of vibroacoustic emission of the low-frequency range of the signal spectrum to the high-frequency range. For this purpose, an experimental verification of theoretical and practical results is carried out in the work.

4. Experimental Validation

4.1. Experimental Setup

The machining test was performed on DMG DMC Ecoline 635v (Ulyanovsk Machine Tool Plant, DMG-RUS, Ulyanovsk, Russia) three-coordinate milling machine with maximum spindle speed of 12,000 rpm and maximum spindle power of 13 kW (Figure 8). The CNC program was created in Autodesk Fusion 360 CAM module. For HSM processes, trochoidal toolpaths were generated. During the experiment, the straight-through slots were machined, each with a width of 16 mm and a depth of 21 mm. When machining, a workpiece was clamped in generic machine vise. The milling cutters were clamped in a HydroForce DVHCTHT20070M G5 NG01 hydraulic chuck (Kennametal, Latrobe, PA, USA). To improve chip removal and reduce cutting tooth load, compressed air at pressure of 0.8 MPa was applied as a coolant.

To improve chip removal and reduce the load on the cutting teeth, compressed air supplied under a pressure of 0.8 MPa was used in the cooling medium.

The chemical composition of the high-strength cast iron to be machined in the experimental part of the work is shown in

Table 1. Chemical composition of workpiece.

Element	Content, %
Carbon, C	3.2–3.6
Si	2.6–2.9
Manganese, Mn	0.4–0.7
Nickel, Ni	<0.6
Sulfur, S	<0.01
Phosphorous, P	<0.1
Chromium, Cr	<0.15
Copper, Cu	<0.6
Iron, Fe	~91

.

The physical and mechanical properties of the workpiece are presented in

Table 2. Mechanical properties of high-strength cast iron used in experiment.

Hardness, Brinell, MPa	Tensile Strength, MPa	Yield Strength, MPa	Elongation at Break, %	Workpiece Dimensions, mm
270	800	480	2	270 × 145 × 40

.

4.2. Milling Cutter and Coating

The material of the end mills is solid carbide type HM CK10-30-UF (DIN 50 049-2.2): monocarbide (WC—93.00%; Co—7.00%), with a density of 14.60 g/cm³, a hardness of Rockwell 93 HRA, a transverse rupture strength of 3600 N/mm², an impact strength of 0.2 Kgf-m/cm², and a porosity of A01.3. The experiments were carried out using the typical designs of cutters made of the solid carbide end-milling cutters with a corner radius produced by Cerin S. p. A. (Verona, Italy), 64 THR.1201222105 (Figure 9).

The milling cutter and its design parameter values are represented in Figure 9 and Figure 10 and in

Table 3. Geometrical parameters of the end mill.

Dc, h10 mm	Number of Flutes, Z	L2, mm	L1, mm	D2, h6 mm	D3, mm	L3, mm	r, mm	Helix Angle ω, °
12	4	22	105	12	11.8	22	1.0	30

.

A total of four milling cutters with the same product code were used. Each of them was numbered as follows: No. 1 and No. 2 are uncoated mills; No. 3 is a mill with multilayer titanium–aluminum nitride (TiAlN/TiN) coating; No. 4 is a mill with CrN-AlTiCrN/SiN–AlTiN (nACRo) coating [8,9,10]. Milling cutter No. 1 was used for the traditional milling strategy, and cutters No. 2, No. 3, and No. 4 were used for the high-speed milling (HSM) strategy.

Uncoated cutter No. 1 was tested using the modes and trajectory of motion corresponding to traditional mechanical processing. The tests of cutters No. 2, No. 3, and No. 4 were carried out with a trajectory of motion in the form of a trochoid (Figure 7).

4.3. Coating of the Cutting Part of the Cutter

To increase wear resistance, functional coatings were selected on the surface of the cutting part of the cutters.

The multilayer TiAlN coating consists of alternating layers of nitride phase with different Al and Ti contents grown on the adhesion layer of TiN. Upon these layers, a gradient (TiAl)N layer with concentration of AlN constantly increasing towards the coating surface was applied. The nACRo3 coating consists of a sequence of CrN, AlTiN, and nanocomposite AlTiCrN/SiN. The properties of the coatings used in the experiment are represented in

Table 4. Coatings’ properties.

Coating	Nano-Hardness, GPa, min	Thickness, μm	Coefficient of Friction	Working Temperature, °C, Max
TiAlN/TiN	36	1…4	0.5	700
CrN-AlTiCrN/SiN–AlTiN (nACRo)	45	1…7	0.45	1100

.

These coatings have very high resistance to abrasive wear and are recommended for use in particularly difficult conditions with high cutting speeds.

These coatings have been proven to be good choices in the industry today; let us consider the effectiveness of their applications in conditions of high-speed trochoidal processing of cast iron with high productivity.

4.4. Cutting Edge Rounding Radii Measurement

The control procedure was performed using the MikroCAD 3D surface metrology system. MikroCAD 3D scanners use structured light fringe projection profilometry to achieve sub-micron-level 3D edge scans. This technology allows MikroCAD to capture full point clouds of the cutting edge geometry after multiple fringe patterns are projected onto a single plane. For each tooth, the average values of the cutting edge rounding were measured for 4 sections in the working area of the end mill. For this purpose, the end mill tooth profiles were scanned in these sections. The images of the measurements of the radii of the cutting edges of all the cutter teeth for 4 end mills in the measurement plane (Figure 11a–e), normally to the cutting edges, are pre-set, as shown in Figure 11. The results of the measurement of the radii of the sharp edges, including the average value for all the teeth, are shown in

Table 5. Radius of rounding of cutting edges in two sections for 4 experimental samples of cutters, μm.

No. of Cutter	Tooth No. 1	Tooth No. 2	Tooth No. 3	Tooth No. 4	re s
1—Non-coated	7.35	7.4	7.35	7.4	7.375
2—Non-coated	7.75	7.65	8.35	7.35	7.78
3—TiN–Al/TiN	9.25	9.55	8.55	8.2	8.88
4—CrTiN–AlTiN–AlTiCrN/SiN	8.55	9.6	8.4	8.85	8.85

. All measurements were taken at an equal distance of 1.0, …, 2.0 mm from the tooth top, which is ensured by the high resolution of the measuring device.

4.5. Cutting Modes

The following machining paths were used in the 4 experiments:

Uncoated cutter #1 was used with the traditional technology of machining the slot over the entire cutting width.

Uncoated cutter #2 and cutter #3, with a TiAlN/TiN coating, and cutter #4, with a CrN–AlTiCrN/SiN–AlTiN (nACRo), coating were used in trochoidal machining.

HSM can be defined as machining with cutting speeds that are significantly higher than those used in conventional machining. Although the term high-speed machining does not imply strict restrictions on the range of cutting speeds, an approximate range of speeds in conventional and high-speed machining is shown in Figure 12 [2].

All assigned cutting modes for conventional and trochoidal machining for the experiment are given in

Table 6. Cutting modes.

Cutting Parameters	Traditional Machining	Trochoidal Machining
Cutting speed, V, m/min	120	300
Spindle speed, n, rpm	3185	7961
Width of cut, ae, mm	12	1.2
Depth of cut, ap, mm	7	21
Feed per tooth, fz, mm	0.011 (T1, 1 mill)–0.03 (T2, T3, 1 mill)	0.011 (T1, 2 mill); 0.12 (T1, 3 4 mills), 0.03 (T2, T3, 2,3,4 mill)
Feed per minute, f, мм/мин	382	955
Number of depth cuts	3	1
Number of side cuts	2	1
Material removal rate, Q, cm3/min	22	24.08

.

4.6. Monitoring of Vibroacoustic Emission

The scheme of monitoring and measuring the main indicators (Figure 13a) of the high-speed milling process represents a complex system for diagnosing (Figure 13b) the high-speed milling process in real time with subsequent analysis of the quality of the machined surface. The system includes several interconnected components that work together to collect, process, and analyze data.

To record the acoustic emission parameters, accelerometers of the AP2037-100 and KD-35 models were used (GlobalTest, Sarov, Russia). These sensors can be installed at various points of the equipment or part and are designed to measure the vibrations that occur during processing. The recorded signals are then transmitted to the Acoustic Emission Amplifiers (Acoustic Emission Amplifiers) of the MODEL VShV003 and ADC E440 models. The vibroacoustic emission recording device consists of several main components. The system of the process (Figure 13a) equipment (1) serves as a basis for fastening the remaining elements. A magnet (2) or a threaded connection is used to install the accelerometer. The accelerometer (3) records the vibroacoustic signals. Then, the signal goes to the preamplifiers (4) and then to the VShV003 model amplifiers (5). The amplified signals are transmitted to the E440 analog-to-digital converter (6). After digitalization, the signals are recorded on the device (7). The operating principle of the system is based on measuring the vibroacoustic activity of the process equipment. The signals are digitalized and then subjected to time and spectral processing. When constructing a high-speed milling monitoring system, the level is equal to half the sensor supply voltage (+7.5 V) and the maximum amplitude of use immediately before the wire electrode breaks (2): detailed amplitude spectra; 1/3 octave bands with numbers show a useful signal from the sensor of 29.7 mV. Since the central frequency is 29.7 mV, the input range of the analog-to-digital converter (ADC) is ±10 V, and the bit depth is 14 bits, it is necessary to match the signal from the CS with the ADC input by offsetting. Control of the electric current parameters ensured a constant level (+7.5 V) to zero and amplification to +10 V, as monitored using a software gain amplifier (PGA). The CS sends a signal proportional to the measured current with a constant used for offsetting. The CDR value changes discretely from 0 to 10 kOhm with a level width that is equal to half the sensor supply voltage (+7.5 V) and a maximum amplitude of 8 bits via the I2C interface. PGAs are instrumentation amplifiers with a gain of 29.7 mV (at a measured current of 100 A) from the sensor; these are used because they are controlled using a parallel code and provide gain in a range from 1 to 8000. The input range of the analog-to-digital converter (ADC) is ±10 V.

The microcontroller (MC) calculates the required offset and gain “in real time” based on the current signal from the ADC and generates a control signal for the CDR and PGA. This automatic control enables the full use of the ADC input range and minimizes digital noise. Digitized ADC signals are transmitted via the MC to the computer for analysis. The AE signal recorded by the accelerometer mounted on the vice holding the workpiece was used as the output signal. In practice, data were obtained for a frequency range of up to 40 kHz of activity; a more complete range was used, including the resonant frequency and higher. In cases in which plunger cutting prevails in comparison with effective chip separation, then friction forces increase, the power density of energy impact changes, the area of impact surface decreases, and the duration of energy pulses decreases. These factors lead to a drop in the amplitude of the high-frequency component of acoustic signals in comparison with the low-frequency component. Spectral analysis allows us to evaluate the condition of the tool or cutting mechanisms in the range of signal amplitudes most susceptible to change. The adjustment of the frequency ranges for the above-described milling machine configuration was carried out on a trial pass of slot cutting. The active region of the low-frequency component of the acoustic signal was revealed in a range of 2–5 kHz; the active region of the high-frequency component of the acoustic signal was revealed in the range of 8–13 kHz for this machine configuration, cutting conditions, measuring system, and technological parameters of processing. In the active regions, narrow-band frequency filtering and the smoothing of the records were performed by constructing root mean square values of the signal gain with different averaging periods. MICRO-CAD GFM MICROCAD PREMIUM (GFMesstechnik GmbH, Berlin, Germany) was used to measure the radius of the rounding of the cutting edges. Additionally, the system used the Hommel Tester T8000 profilometer (Hommelwerke GmbH, Jena, Germany), which measures the surface profile of the processed part, allowing the operator to evaluate the quality of processing after the process is completed. All components of the system were interconnected via digital interfaces and communication lines that provided data transfer from sensors and measuring devices to a single device. Thus, the system provided comprehensive monitoring of the processing, which is necessary in evaluating the mechanisms of cutting and chip formation. This section describes the main elements of the technological system and the monitoring system; in the next section, we will consider the results obtained from measuring the characteristics and the results from the technological process.

5. Results and Discussion

5.1. Roughness of the Machined Surface of the Slot Walls in High-Speed Milling

The roughness of the slot surfaces machined in the course of the experiment was measured with a Hommel Tester T8000 measuring station (Hommelwerke GmbH, Jena, Germany). The measurements are shown in Figure 14. Figure 14 shows the results for four milling cutter samples. The results are given for sample #1–#4 in sequence from left to right, and from top to bottom for the three processing stages (T1 = left; T2 = center; T3 = right).

From the measurements of the residual unevenness of the bottom and walls of the machined slot shown in Figure 14, it can be concluded that sample No. 1 showed the worst indicators of the quality of the machined surface. There are many points of deep unevenness on the machined surface and the entire surface as a whole has a level line with differences. The roughness of the surface machined by cutter No. 2 with a trochoid trajectory is lower than that of the sample machined using cutter No. 1. It should be noted that, over time, with the formation of wear, the quality of the machined surface deteriorated; this indicator has the highest value on the surface machined by cutters No. 1 and No. 2. The surfaces of the slots machined by cutter samples No. 3 and No. 4 have the best indicators of the roughness of the slot walls, and the bottom of the slot has the best quality when machined along the trochoid by cutter sample No. 4.

5.2. Wear Results of Experimental Milling Cutter Samples

The wear value on the flank surface was measured using a ZEISS SteREO Discovery V12/V20 digital optical microscope (Oberkochen, Germany). The measurement results are shown in Figure 15. It was found that milling cutter No. 1 with characteristic chips along an extended section on the cutting edge and flank surface has the highest wear value. Milling cutter No. 2, with a wear-resistant coating, used in the traditional slot cutting scheme, has less wear than milling cutter No. 1. The lowest wear was shown by milling cutter samples No. 3 and 4 with wear-resistant coatings in the trochoidal processing scheme.

Having analyzed the nature and magnitude of wear of the flank surfaces of the cutters with the same volume of cut material, it was found that the wear value during traditional processing is higher compared to that obtained through the trochoidal processing for the samples of cutters without a coating. In addition, an increase in the contact area of the cutter with the workpiece (performed in order to achieve comparable productivity with trochoidal processing) caused significant chipping of the cutting wedge produced using cutter No. 1 (Figure 16a). The abrasive wear produced using cutter No. 2 (Figure 16b) is less than the wear produced using cutter No. 1. The parametric wear produced using cutter No. 2 has an abrasive nature, expressed in the form of the localization of a chamfer along the flank surface. Such wear allows further operation of the cutter without significant loss of performance (Figure 14: dependence graphs on the right above a, b, and c); meanwhile, cutter No. 1 loses its functional purpose and does not enable the processing of the surfaces of parts with the required quality (Figure 14: dependence graphs on the left above a, b, and c). At the same time, restoring the cutting capabilities of this class of cutters requires significant effort and financial costs, due to the need to cut off the worn edge and re-sharpen the end of the tool. The wear of cutters No. 3 and No. 4 after slot processing has comparable values. The wear of cutter sample No. 4 (Figure 16c) with a TiN–Al/TiN coating is lower compared to that of sample No. 3, which indicates a more suitable coating for these processing conditions. The wear of cutter No. 4 (Figure 16d) is presented in the form of micro-chips, which may indicate greater hardness and brittleness of the cutting wedge of the cutter with a CrTiN–AlTiN–AlTiCrN/SiN coating.

5.3. Shape and Size of Chip Elements

Having analyzed the nature and size of the formed chip elements with different processing parameters and different experimental samples of cutting tools, it was found that segmented chips with different thicknesses were formed throughout the cutting process. As a result of cutting with cutter No. 1, chips were formed with maximum thickness (Figure 17a). Chips produced by cutter No. 2 have a smaller thickness with a high longitudinal duration (Figure 17b). When using cutter No. 3 and cutter No. 4, the number of small chip elements increased (Figure 17c,d), which indicates the good sharpness of the cutting wedge throughout the milling cycle. When cutting with cutter sample No. 4, the chips had low thickness but were long.

5.4. Results of Evaluation of Vibroacoustic Characteristics During Cast Iron Machining by Traditional and Trochoidal Milling with Coated and Uncoated Cutters

Vibration signals were measured continuously to study the mechanism of the cutter cut-in mechanism during trochoidal machining with different cutting parameters. The cutting depth during the operation of the first sample of the cutter with a linear feed path was significantly greater than the edge rounding radius; therefore, the effective detachment of cutting prevails throughout the entire cutting period, which corresponds to the h_CZIII region. In this regard, the vibroacoustic characteristics of this cutting cycle were of no interest and were not considered. The nature of chip removal in the trochoidal milling cycle clearly shows the uniformity of the distribution of the volume of cut chips and the efficiency of the distribution of cutting energy during material removal (Figure 7). The chip width varied during each trochoidal machining cycle and passed through all three ranges with different cutting mechanisms; in connection with this, the experimental study compared the parameter ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ and the parameter k_f, which describe the ratios of the amplitudes of acoustic signals in the low-frequency and high-frequency ranges of the acoustic signal components.

The active regions of the low-frequency range of 2–5 kHz (Figure 18, green signal) and the high-frequency range of 8–13 kHz (Figure 18, blue signal) of the acoustic signal components are shown in Figure 18. Also, at the initial moment of processing, when the cutter cuts into the workpiece, the feed was reduced to attain a smooth running-in process during the first three cycles. The acoustic sensor was located near the right side of the workpiece; as a result, the amplitude of the vibroacoustic emission is higher at the first stage of slot processing and decreases by the third. The appearance of the total signals in everything when cutting three slots along the entire length with three samples of cutters during trochoidal high-speed milling is shown in Figure 19.

Three stages of slot machining were selected for the analysis of cutting mechanisms. The slot machining was divided into three time series: T1—at the beginning of the slot machining process; T2—in the middle of the slot machining process; T3—at the end of the slot machining process. For each stage of machining, one trochoid cycle was selected for the analysis of the low-frequency and high-frequency components of the acoustic signal. The start/end time of the cutting cycle and the start/end of the transition zone of cutting in the trochoid cycle are described by the following:

$$\mathrm{Tp}·{\mathrm{K}}_{\mathrm{r}\mathrm{e}}/100+{\mathrm{T}}_0$$

(25)

where $\mathrm{Tp}·{\mathrm{K}}_{\mathrm{r}\mathrm{e}}$/100 is the plunger cutting cycle time and T0 is the start time of the cutter tooth cutting into the workpiece in the considered trochoidal milling cycle. Based on the results obtained using the new analytical model (expressions 2–17), theoretical values of the trochoidal milling cycle time ranges with different cutting mechanisms were established. In each trochoid cycle, the theoretical duration of different cutting mechanisms was estimated. Comparison of the cutting mechanisms, the chip formation ploughing zone (CZI), the shearing zone (CZII), and the cutting zone (CZIII) by comparing the theoretical value of the k_re parameter, defined as the ratio of the thickness of the chip, h_CZ, to the cutting edge radius, r, for each of the cutter samples and the kf parameter, defined as the ratio of the root mean square amplitudes in the frequency ranges of 2–5 kHz and 8–13 kHz.

In this part of the section, important indicators of monitoring high-speed multi-axis milling are noted. All three samples of the cutter at the first stage of slot machining (Figure 20a,d,g) had a similar cutting mechanism over the entire range of values of the ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ parameter of the trochoidal cutting cycle. The parameter ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ was determined by the average value of the measurement radius for four teeth of each of the cutter patterns (Table 5, r_e s). The value of the thickness of the cut layer during the transition of the plunger zone to the shearing zone (from CZI to CZII ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ < 40%) and the reverse transition (from CZII to CZI ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ > 40%) on all samples of cutters corresponded to an equal value of the amplitudes of the acoustic signals in the low-frequency (marked with number 1, red line, Figure 20) and high-frequency ranges (marked with number 2, green line, Figure 20), corresponding to the parameter k_f, close to 1.

In the shear zone of CZII during cutting in the range of parameter values 25% < ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ < 60%, the mechanism of plunger cutting and cutting with the formation of a material shear region can be combined. The low-frequency root mean square component of the signal (2) exceeds the high-frequency (1) component with peak values of the parameter K_f for the entire cutting cycle (Figure 20). This can be explained by the fact that the thickness of the chip increases, and the chip separation mechanism has not yet settled down, while the friction of the edge and the pressing of the material by the edge into the machined surface are active (Figure 14, sample 3 and 4), which can be a pattern of the plunger cutting mechanism and cutting with the formation of a material shear region in the CZII zone.

The chip thickness area corresponding to the parameter ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ > 60% is characterized by approximately equal values of root mean square amplitudes in the frequency ranges of 2–5 kHz and 8–13 kHz. The cutting conditions are efficient and have high specific power. The main part of the power is spent on chip removal, and not on the friction of the flank and cutting edge. At the second stage slot machining, with an increase in feed, cutting is accompanied by chipping of the edges, characterized by a high propagation rate of brittle cracks with short pulses over the entire duration of the section with up-cut milling (including up-cut milling areas CZI, CZII, and up to 50% of the CZIII area), which create amplitude spectra extending into the high-frequency range. With down-cut milling, a decrease in power density is observed, which entails a drop in the amplitude of the high-frequency component of the signal compared to the low-frequency component. This can be explained by the fact that the contact of the flank surface of the cutter tooth with the machined surface is more rigid than in up-milling. At the same time, the degree of parametric wear on the back surface is low; therefore, in this regard, an increase in the amplitudes in the low-frequency range of the signal at the second stage of processing is not expected. The amplitude of the signals increased from the first to the third stage of processing due to the fact that the tool was approaching the accelerometer. At the first and second stages of processing in the plunger cutting zone of CZI, their alternation is observed on the graphs of the dependencies of the root mean square amplitudes in the frequency ranges of 2–5 kHz (1) and 8–13 kHz (2). The nature of the change in the root mean square amplitudes is characteristic of the second stage of processing, which is especially well expressed when the fourth sample, with less resulting wear, is working. Therefore, such a feature can be attributed to a characteristic indicator of plunger cutting. At the first stage of cutting, the wear of the cutter tooth does not yet have a significant effect on the change in the k_f parameter; the cutting edges have the original geometry of the rounding radius.

Important mechanisms and indicators were noted in this section in the trochoidal high-speed milling at the first and second stages of machining, which can be used as fundamental parameters for monitoring or integration into the structure of computer-aided manufacturing for rational calculation of special processing trajectories.

The recommended values of the k_f parameter depend on the change in the trochoid pitch, the angle between the line passing through the point of contact of the cutter with the workpiece and the center of the cutter, and the tangent to the machined surface of the slot wall (Figure 21). Figure 21 shows three ranges of cutting mechanisms based on the ratio of the chip thickness to the edge radius (CZI, CZII, and CZIII) [4,23] and the characteristic values of the K_f parameter in these ranges.

Analysis of the signal at the first stage of cutting allows us to conclude that, in the region of the thickness of the cut chip CZI (highlighted in red in Figure 21), the increase in the kf parameter is predominant, proportional to the change in the ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ (θ) parameter. In the zone of cutting mechanics, CZII, the recommended range of the K_f parameter is from 0.85 to 1.15, with sharp jumps in its change over a short time interval. The cutting zone with a predominantly effective chip separation mechanism, CZIII, has a k_f parameter value close to 1 with a 10% spread.

At the third stage of processing, an increase in the root mean square amplitudes in the frequency ranges of 2–5 kHz is observed (1, Figure 20c,g,j). The increase in the ratio of the root mean square amplitudes of acoustic signals in the low- and high-frequency ranges k_f has increased twofold in almost the entire range.

The reason for the effect on the change in the characteristics of the acoustic signal is the wear of the cutter tooth with an increase in the surface area of the wear chamfer and an increase in the duration of energy pulses. Due to the change in the shape of the tooth after wear, it is difficult to unambiguously establish the mechanism of chip formation in the plunger zone and shear zone. However, it can be assumed that the wear chamfer can act as an edge and form a plunger cutting mechanism due to the kinematic change in the geometric parameters. During counter-milling, an increase (decrease) in the kinematic clearance angle is observed, while the velocity vector of the resulting movement of the cutter tooth will not be tangent to the cutting surface; thus, the plunger thickness of the chip of the worn tooth will increase. As a result, the low-frequency range of the root mean square amplitudes of the vibroacoustic signal during counter-milling increases significantly.

As can be seen from the diagram in Figure 22, the up-cut milling area in the first half of the trochoidal cycle is characterized by an increase in the clearance angle. The worn section of the cutter tooth results in the formation of a plunger cutting zone. Although the clearance angle also increases, which reduces the friction of the flank surface against the machined material, the wear chamfer area, due to the change in the direction of the cutting vector, can function as the rake surface when the cutter tooth cuts into the workpiece material. In the second part of the trochoidal cutting cycle, a down-cut milling zone is formed in the contact zone of the cutter with the workpiece.

The kinematic clearance angle decreases (Figure 6c), although friction can increase, the wear chamfer, with a high degree of probability, should not press the removed workpiece material into the machined surface, since it is located tangentially to the tooth movement trajectory. Based on these considerations, we will make an adjustment to the value of the parameter ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ (θ):

$${\mathrm{K}}_{\mathrm{r}\mathrm{e}}\left(\mathrm{\theta }\right)={\displaystyle \frac{{\mathrm{h}}_{\mathrm{e}}\left(\mathrm{\theta }\right)}{{\mathrm{r}}_{\mathrm{e}}+{\mathrm{V}\mathrm{B}}_{\mathrm{B}}\mathrm{t}\mathrm{a}\mathrm{n}\left(\mathrm{\kappa }\right)}}·100\%,$$

(26)

Considering the fact that chip thickness with a plunger cutting mechanism increases only due to the positive component of the change in the kinematic angle, κ, only this will be taken into account in Equation (25). The dependence of the change in the thickness of the uncut chip and the parameter ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ (θ) on the angle of the angle between the line passing between the point of contact of the cutter with the workpiece and its center and the tangent to the machined surface of the slot wall is shown in Figure 23a and Figure 23b, respectively.

According to the nature of the change in the kf parameter and the root mean square amplitudes of the acoustic signals in the low-frequency and high-frequency ranges of the signal at the third stage of processing TIII (Figure 20c,g,j), it follows that the k_f coefficient in the first half of the trochoidal cycle exceeds 1.5 and reaches 2. The reason for this may be a complex combination of the plunger cutting mode and the friction of the flank surface against the workpiece [44,58]. The rigid contact of the tool with the workpiece during cutting is characterized by the high amplitude of the acoustic signal in the low-frequency range of the signal. In the second half of the trochoidal cycle, the contact area of the cutter with the workpiece decreases and the nature of the change in the k_f parameter corresponds to the nature of its change for the first stage of processing in zone CZIII. The reason for this may be the predominance of down-cutting milling in the contact zone of the tool with the workpiece [59,61]. Based on the identified features of cutting with a tool with wear, we will form the recommended ranges of the k_f parameter, taking into account the reduction in chip thickness due to the interaction of the plunger’s wear chamfer with the workpiece material. A graph of this dependence is shown in Figure 24.

Figure 24 shows three ranges of cutting mechanisms based on the ratio of the chip thickness to the edge radius (CZI, CZII, and CZIII) at the stage of increased parametric wear. From the analysis of the signal at the first stage of cutting, it can be concluded that, in the area of the thickness of the cut chip, CZI (highlighted in red in Figure 21), the increase in the kf parameter prevails in proportion to the change in the ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ (θ) parameter and that the value at the boundary of the plunger zone CZI is less than 1; this is due to a decrease in the actual thickness of the cut chip due to the increase in tool wear [30,61]. The recommended values for monitoring the change in the k_f coefficient are from 0.7 to 1 in the CZII area and can take a value of up to 2–2.5. The CZIII cutting area is characterized by a k_f parameter value from 1.5 to 2.3 until the middle of the trochoidal cutting cycle and a range of the k_f parameter from 1 to 1.7 in the second half of the cycle. In this section, the important mechanisms and indicators of the trochoidal high-speed multi-coordinate milling process are presented. We formulate our recommendations for assigning a trochoid step when writing control programs for a CNC machine. Based on the results obtained from distributing the cutting mechanisms, we set the value of the change in the ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ parameter depending on the trochoid step for the same cycle time. A graph of this dependence is shown in Figure 25. The graphs show the dependences of the parameter on 1/30, 1/20, 1/10, ¼, 1/3, and ½ stages of the cycle. From the results of changing ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ it follows that with a trochoid step ap of 0.1 mm, the entire cycle is located in the plunger cutting area CZI and cutting with a shear area CZII, which are unfavorable conditions. Starting with a trochoid step value of 0.6 mm, more than half of the trochoid cycle is located in the area with an effective thickness of the cut chip, more than h_CZIII. Therefore, the recommended minimum value of the range for the given process conditions is the trochoid pitch, ae, from 0.7 mm.

A new model for determining the mechanism of cutting wedge cut-in in trochoidal machining based on the theoretical chip thickness and its ratio to the cutting edge radius allows us to establish criteria for vibroacoustic monitoring to assess effective cutting with chip formation. New analytical models for calculating theoretical chip thickness and evaluating cutting mechanisms make it possible to expand the scope of their application for high-speed milling in real time. Based on this, it is possible to identify the component of the vibroacoustic characteristics that increases the vibration value and the cause of their change for various cutting mechanisms. As a result, this section notes indicators and parameters that are important in trochoidal high-speed milling; these can be used as fundamental parameters for monitoring the process, offering important criteria for calculating tool paths during the technological preparation for the production of slots on machine parts.

6. Conclusions

The present study establishes and identifies the size effect in the high-speed multi-coordinate machining of slots on cast iron workpieces by the characteristics of the vibroacoustic signal; this depends on the high-speed machining mode used, the theoretical chip thickness obtained, the experimental size of the cut chips obtained, the quality of the machined surface, and the geometry of the end mill tooth. These can be expressed in a stable correlation between the ratio of the root mean square amplitudes in the frequency ranges of 2–5 kHz and 8–13 kHz and the ratio of the cutting edge radius to the theoretical chip thickness, K_re. For example, loading the cutting edge radius, r_e, relative to the theoretical chip thickness, K_re, which is equal to 40%, the ratio coefficient of the root mean square amplitudes, k_f, can be varied in a range from 0.96 to 1.02.

The identified new analytic dependences (2–24) allow us to improve high-speed machining technologies through the assignment of modes and the determination of rational tool movement trajectories. This can be achieved by using an analytical model for calculating the rational sizes for chips based on the established indicators during vibroacoustic signal monitoring; mo9reover, it is possible to establish the nature of chip formation in real time at various stages of the trochoidal cycle of high-speed machining.

As a result of studying high-speed machining, features of changes in the spectral composition of the acoustic signals that are characteristic of various material removal mechanisms were established; these were shown to depend on the ratio of the chip thickness to the cutting edge radius during the trochoidal high-speed machining process. The qualitative nature of the influence of cutter tooth wear on the change in the k_f coefficient was also established, from 1 at the beginning of processing to 0.5 at the last stage of machining. Based on the analysis of vibroacoustic characteristics data, the most important indicators for monitoring high-speed milling (with a recognition of cutting mechanisms and crack-formation mechanisms) during chip deformation were presented, and these were analytically described through ${\mathrm{K}}_{\mathrm{r}\mathrm{e}}$ and using dependence (25).

Based on the use of the new model, the formed recommendations for the monitoring indicators of the high-speed processing process were applied; here, practical recommendations were made for the assignment of the tool path relative to the workpiece. This model is fully consistent with the current trends in modern production and the formation of a virtual production space that adequately reflects the state of the technological system in real time.

Parameters for the trajectory with a lower probability of the formation of plunger cutting mechanisms during high-speed machining were proposed. For feed modes of 0.03 mm/Z and 955 mm/min with a cutting speed of 300 m/min, the recommended trochoid pitch is in the range of 0.9 mm and higher; this is because a 20% reduction will occur in the duration of high-probability plunger cutting (i.e., 20% of the duration of the first half of the trochoidal cutting cycle).