Introduction. The impact of vibration on tool wear is the subject of numerous studies, as cutter wear determines the quality of parts and the efficiency of machining. The impact of vibrational oscillations on cutting force dynamics is comprehensively analyzed in the literature [1]. The diversity of tool and workpiece vibration sources and, correspondingly, the diversity of methods for monitoring, evaluating, and modeling them are noted [2].

At the initial stages of the development of cutting theory, vibrations were considered as a consequence of the loss of stability of the equilibrium of elastic deformations in a moving coordinate system, whose motion was determined by the trajectories of the machine actuators [3]. The loss of stability was associated with the effect of force regeneration [4], and a criterion for the stability of the equilibrium of elastic deformations [5] was proposed for its analysis. Regeneration problems were studied for turning [6], milling [7], drilling [8], and other operations.

The loss of stability was explained by:

It is known [13] that when the trajectories of the tool formative movements lose stability, the most typical attracting sets for vibrations are those of the limit cycle [14] and the chaotic attractor [15]. It is established that changes in the attracting sets of tool vibrations are determined by the cutting conditions. For example, when changing the feed rate of the tool, all types of Andronov-Hopf bifurcations can be observed [16]. Vibrations were considered as a consequence of disturbances related to the quality of the machine tool. These were primarily spindle runout [17] and kinematic disturbances [18]. Depending on the frequency of external disturbances, various effects were observed [19]. At frequencies close to the natural frequencies of the interacting subsystems, parametric phenomena such as self-excitation, oscillation synchronization [20], and others, occur.

Finally, vibrations were studied as vibrations purposefully introduced into the cutting zone to achieve a beneficial effect: chip fragmentation [21], increased tool life [22], and improved surface quality [23]. It has been shown that introducing ultrasonic vibrations (USV) into the cutting zone can improve the quality of work and reduce the intensity of tool wear [24]. To improve the efficiency of USV, USV generation systems are proposed in which feedback for self-excitation is realized through vibrations generated under the cutting process [25]. In this case, nonlinear effects of the interaction between the tool and the workpiece are taken into account, specifically, the decreasing characteristic of cutting forces with increasing velocity [26] or the amplitude-frequency modulation of the system vibrations [27].

Researchers have paid particular attention to the impact of vibrations on the tool wear rate under cutting. It is widely believed that vibrations increase wear and thereby reduce tool life [23]. However, there is evidence that with an increase in the amplitude of the introduced vibrations, an optimum amplitude is recorded at which the wear rate reaches a minimum [28]. This effect is most pronounced when ultrasonic excitation is applied through the tool [28] and when turning difficult-to-machine materials such as AISI 52100 [29] or P9M4K8F [30].

To analyze the impact of vibrations on wear rate, methods and mathematical tools are needed that can allow for the rapid assessment of wear rate. It is advisable to use the power of irreversible energy transformations (PIET) in the contact zone of the tool and the workpiece as a diagnostic indicator [31]. When turning with tools with carbide inserts, the main type of wear is observed along the flank [32]. PIET is the primary source of heat generation, so it can be argued that the optimal cutting conditions correspond to the optimal temperature [33]. Heat production and PIET are correlated and adequately reflect wear rate [34]. Therefore, when developing wear monitoring systems, temperature and PIET in the cutting zone are considered as key diagnostic parameters. Diagnostic information models employ methods of autoregressive spectral analysis [35], models based on cutting force analysis using analytical functions [36], machine learning [37], and vibroacoustic emission signal analysis [38]. When the PIET changes, the mechanisms of physical and chemical interaction (wear) in the contact between the tool and the workpiece change — from adhesive-fatigue [1] and abrasive [31] to diffusion-oxidative [39].

PIET is often considered within the entire cutting zone. However, the power distribution between the tool flank, the chip formation zone, and the secondary plastic deformation zone is disproportionate. When analyzing flank wear rate, it is required to consider irreversible energy transformations in the contact area between the tool flank and the workpiece. It is not always taken into account that vibrations cause periodic changes in the power of irreversible transformations, and depending on the current power, various wear mechanisms are activated — from frictional fatigue to diffusion-oxidation. Consequently, estimates of the velocity and rate of tool wear change periodically. Furthermore, the literature does not describe the transformation of high-frequency vibrations (HFV) introduced into the cutting zone into the trajectories of the tool flank and the workpiece, nor into the trajectories of the forces generated in the contact zone.

The objective of this study is to establish the relationship between the HFV and tool flank wear rate based on a developed theoretical model of cutting dynamics, as well as its numerical and experimental analysis. To achieve this goal, the following work is being conducted to refine the models, simulate, and conduct theoretical studies and experiments.

The developed mathematical tools and methods can be interpreted as the creation of a virtual numerical model of wear of cutting tools for the optimal selection of vibration parameters introduced into the cutting zone.

Materials and Methods

1. Problem statement. Mathematical modeling of DCS that takes into account forces on the tool flank. We consider the relationship between wear and PIET, as well as wear intensity patterns. To determine the PIET, we create a model of the DCS disturbed by the HFV. We develop algorithms and a program for calculating the PIET trajectories in the tool flank – workpiece contact (Fig. 1).

Assume that the trajectories of the machine actuators are given in the form of displacements L = {L1, L2, L3}T ∊ ℜ(3) and velocities dL/d = V(t), V(t) = {V1, V2, V3}T ∊ ℜ(3). Here, L1(t), L2(t) — trajectories of the transverse and longitudinal supports;

 — displacements of the workpiece relative to the tool at the point of contact of the tool tip with the workpiece in the direction of its rotation; D — diameter of the machining surface, mm; d — diameter of the machined workpiece, mm (рис. 1).

The disturbances are characterized by displacements

and velocities

.

Let us consider the case when additional vibrations introduced into the cutting zone are represented by a vector of periodic disturbances:

.

Also — for velocities

.

Perturbations occur in the deformation space

.

Strain rates are

.

Vector X is considered in moving coordinates defined by L (Fig. 1). The unit of measurement for L, ΔX(X) and X — mm, for V(t), ΔV(X) and V(X) — mm/s.

The workpiece is rigid, so the trajectory of the shaping movements

can be represented as:

(1)

We involve velocities dL(Ф)/dt = V(Ф) = V + ΔV(X) – V(X)in the modeling. We use work [40] to indicate the relationship between forces and deformations:

(2)

Here, m, h, c — positive-definite symmetric matrices of inertial, velocity, and elastic coefficients, m = diag{m, m, m}. The dimensions of the matrix elements m — kg∙s²/mm, h — kg∙s/mm, c — kgf/mm.

 — forces generated in the area of contact between the tool front face and the cutting zone.

They depend on the dynamic properties in the areas of primary and secondary plastic deformation (highlighted in red in Fig. 1).

 — additional forces caused by the approach of the tool flank to the workpiece (region А — В in Fig. 1).

Forces Ф(L(Ф)), introduced additionally to the model considered in [10] reveal the interaction between the tool flank and the workpiece. They limit the development of periodic movements, generate forces acting on the flank, and, together with the trajectories L(Ф), determine the work and power of irreversible energy transformations in the contact area of the tool flank. Forces F and Ф are represented as functions of trajectories L(Ф), which vary depending on ΔXi(t) and the elastic response of the tool Xi(t).

Let us analyze processing with constant modes: L1 = L1(0) = d/2 = const.

Here, d — diameter of the processed workpiece under cutting (Fig. 1), L2(t) = V2t, V2 = const, L3(t) = V3t, V3 = πDΩ = const.

Before we start modeling F(L(Ф)), let us make three statements [41].

Forces F grow monotonically with increasing cutting area S (Fig. 1), which can be represented as: S(t) = tP(t)SP(t). Here and hereafter tP(t), SP(t) — current values in mm of the cutting depth and the feed rate, respectively. Vector F has the form: F= {χ1, χ2, χ3}T, where χi — angular coefficients, and {(χ1)2+ (χ2)2 + (χ3)2} = 1. There is a delay between the variations of F and S. The delay is modeled by an aperiodic link with time constant T(0), which depends on the modes tP(t), SP(t), VP(t). Here, VP = {(V1)2 + (V2)2 + (V3)2}0.5 — cutting velocity. If there are no vibration disturbances and the deformation rates are small, then VP(t) ≈ V3, since V2 ≪ V3, and for longitudinal turning — V1 = 0. When determining time constant T(0), called the chip formation time constant in [3], it is taken into account that the path traveled by the tool tip relative to the workpiece remains approximately constant [3]. In this case, the transition from one stationary state to another is considered. The above means that T(0) depends mainly on the cutting velocity, and it can be approximated: (3) Here, T(0) — time constant in the region of low cutting speeds, s; k(T) — coefficient with the dimension s/mm. According to [10], approximation (3) is valid in a limited range of variations of the process mode. For example, when processing 45 steel, the limitations are determined as VP ∈ (0.2; 2.5) ⋅ 103 mm/s. The greatest approximation error is observed in the range of low cutting velocities and depends mainly on the properties of the limit state of the material being processed, its plasticity and thermophysical characteristics. If the parameters are specified, then the following is valid: (4) where T — workpiece turnover time, s; ρ — chip pressure on the front face of the tool, kg/mm 2. To illustrate the analysis method, we limit ourselves to the case where the main and auxiliary angles in the tool plan are equal (Fig. 1): ϕ ⇒ π/2, ϕ1 ⇒ 0.

Let us consider the disturbances:

and .

We exclude

from the analysis since SP ≪ tP. Moreover, the internal gain in the self-excitation channel varies significantly depending on the deformation direction. For the direction X1, the gain will be an order of magnitude smaller than for X2. Furthermore, with the tool geometry under consideration, it is precisely the vibrations in X2 – X3 plane that bring the tool flank and the workpiece closer together, which determines the intensity of wear and generates additional forces Ф(L(Ф)). Thus, if Ω = const, dX3/dt → 0 and dΔX3(t)/dt → 0, then T = (Ω)–1. Otherwise, T is required to calculate from the ratio:

(5)

Here, VP — law of cutting velocity variation, taking into account elastic deformations and disturbances. To study the PIET, it is necessary to know the force model Ф(L(Ф)). After cutting-in (Fig. 2 a), a trace trajectory is formed on the workpiece at angle φ = arctg(V3/V2) (Fig. 2 b).

Direction ϕ is indicated by the line А — В (Figs. 1, 2). When changing , as shown in Figure 2 b, due to , the direction of the tool movement towards or away from the workpiece changes. As the surfaces approach each other, the forces on the flank increase. When vector V(X) shifts to the area highlighted in yellow (Fig. 2 b), the tool moves away from the workpiece, and area S decreases. If the equilibrium position of elastic deformations is stable and unperturbed, then the trajectory of movement in the direction А — В (Fig. 2) is an attractor. It is displaced in space L by a constant amount of elastic deformation. Due to disturbances or loss of stability, periodic convergence or repulsion of the flanks from the workpiece occurs. To estimate the deviation of the trajectory from the projected attractor, it is convenient to consider dimensionless aggregated coordinates:

(6)

Here, υ(t) — dimensionless aggregated coordinate defining the current position of the direction of the tool tip movement; υ* — aggregated coordinate defining the desired direction of tool movement, specified by velocities V2 and V3.

The approach of the flank to the workpiece surface depends on the kinematic value of the clearance angle α. The experiments described in [39] show that forces Ф(L(Ф)) increase disproportionately with decreasing α according to an exponential law. Therefore:

(7)

where ρ0 — coefficient of conversion of contact length into force, kg/mm; ς — dimensionless parameter dependent on the tool clearance angle α in statics; kT — dimensionless coefficient of friction; kФ — dimensionless coefficient that determines the elastic recovery of the material.

From (7), it follows that there is a potential relationship between deformations and disturbances in the direction of trajectories А — В (Fig. 2), and they have virtually no effect on the forces generated at the contact between the tool flank and the workpiece. In this case, variations in cutting velocity in the range of actual values will be small. Let us denote the current variations in velocities in the directions X2 and X3:

,

.

Then from (6), we obtain:

(8)

Let us consider harmonic disturbances in two orthogonal directions, maintaining the ratio between the amplitudes of the additional vibrations. In this case, the in-phase condition of vibration (8) is V3/V2 = δV3/δV2. It is almost never satisfied in the dynamic cutting system. There are two reasons for this. Firstly, V2 ≪ V3, therefore, the direction of the total vibrations (i.e., vibrations introduced into the cutting zone with regard to deformation displacements) must be oriented along the direction of the cutting velocity. Secondly, it is important to consider the response from the cutting process. Then the total stiffness matrices of the tool subsystem become asymmetrical, even if we neglect the forces generated by the matrices m and h. Skew-symmetric components of the elasticity matrices generate circulatory forces that cause precessional movements of the tool relative to the workpiece. It has been experimentally established [39] that this form of vibrations always occurs. It provides phase displacements between vibrations in two orthogonal directions. Due to the asynchronous nature of the vibrations in the two orthogonal directions, periodically repeating sections are formed in which the tool flank and the workpiece are observed approaching each other. This occurs even with small variations in velocity relative to the established cutting velocity.

To determine the PIET, it is required to calculate the forces and vibrational velocities. Obviously, the PIET depends not only on the parameters introduced into the cutting zone of vibrations, but also on the dynamic properties of the entire USV. If we consider the interactions in terms of nonlinear acoustics, then the efficiency of ultrasonic vibration under cutting is determined by the acoustic impedance of the medium into which the ultrasonic vibrations are introduced [28]. In our case, this corresponds to the dynamic properties of the cutting system.

2. Mathematical modeling of effective parameters and forces. If we follow the paradigm of mesomechanics [42], then variation of the properties of the dynamic coupling formed by cutting, due to the introduction of HFV into the cutting zone, should be characterized by molecular-mechanical effects that change the properties of the system at the macro level. By frequency range at the macro level, we mean the range within the passbands of the interacting subsystems Ω(0) ∈ (0, Ωc), where Ωc — cutoff frequency of the DCS.

Note that the macrosystem does not perceive HFV in the frequency range Ω(0), but they do change its properties. To explain this transformation, let us recall the cutting process and the dynamic coupling in the system of mechanical interactions (2). HFV contribute to the transformation of the parameters of this coupling, and subsequently — to the change in the macrosystem. This is known from the description of the averaging method in the theory of nonlinear vibrations [43].

The basic parameters affecting the properties of the system are ρ and T(0) [6]. A typical example of effective parameters is the dimensionless effective friction coefficient kT, studied in [42]. The effective value of kT can change and even reverse sign under vibrations. This depends on the trajectory of the HFV introduced into the contact area.

Let us describe the conditions for determining the effective value of kT. For this purpose, the relationship between two factors is determined:

For analysis, a frequency region within the passband of the interacting subsystems of the tool and workpiece is selected. The analysis time is the period of high-frequency vibrations.

Let us consider parameter . Additional vibrations create a stress state that changes cyclically in the primary plastic deformation zone. The limit state of the material practically does not change, it remains close to its ultimate strength [44]. Integral and cyclic loads are redistributed, causing changes in the effective values of . Moreover, the dynamic bond formed by the cutting process lacks central symmetry with respect to deformations in the neighborhood of the equilibrium position. As a result, a complex stress state arises in the cutting zone, described by additional constant and cyclic force components. Assuming the ultimate limit state of the material is maintained in the cutting zone, we obtain the effective value of :

(9)

where

 — dimensionless parameter;

;

.

It is obvious that .

Let us estimate the effective value of  taking into account (3). The specified vibrations

are independent of the vibration power. The period of the function δV3(t) is generally determined by the period of the vibrations TΔ = (ΩΔ)–1. That is, due to the introduced vibrations, attracting sets of the limit cycle type are formed in the dynamic cutting system. Moreover, ΩΔ is at least an order of magnitude higher than the upper natural frequency of the vibrational circuits formed by the tool subsystem. We expand the nonlinear function T(0)(V3, δV3) in a Taylor series in the neighborhood of δV3:

(10)

With δV3 ≪ V3, series (10) converges quickly, therefore we limit ourselves to a linear approximation of the dependence of the time constant T0 on additional vibrations, that is, from (10) we obtain:

(11)

In (10), the first term is a constant value at V3 = const. Function

is periodic, with a period of TΔ, and (TΔ)–1 = ΩΔ ∈ ΩΔ. Therefore, to determine the effective value of , the following is valid:

(12)

where

.

Thus, we see what happens when vibrations that are not directly transmitted by the tool and workpiece subsystems are introduced into the cutting zone. In this case, the parameters of the dynamic coupling formed by the cutting process change.

The system equilibrium is asymptotically stable and unperturbed. The time constant T0 at V3 = const is also constant and is determined from the expression

.

Otherwise, the HFV is changed. This transformation is determined by the ratio of the vibrational velocity amplitude to the cutting velocity, which is taken into account by Δ in (12). Thus, HFV change the system properties in the low-frequency region. For example, increasing the time constant has two effects on equilibrium stability:

In all cases, as the amplitude increases, a decrease in the effective values of the parameters and  is observed. Their variation affects the stability of the controlled trajectories and the dynamic properties of the system in the frequency domain Ω(0). This, in turn, affects the attractive sets of deformation displacements of the tool relative to the workpiece.

3. Mathematical modeling of the impact of the HFV on the cutting forces at the flank. Vibrations change the interactions between the tool flank and the workpiece (7). They also cause reactions outside the bandwidth of the system represented by (2). Therefore, forces Ф must also be taken as averages over the vibration period. Let us analyze the effect of vibrations on Ф2. Forces Ф3 differ by a factor of kT. Consider two cases for an asymptotically stable system.

The first case: the vibrations are determined by the velocities in the feed direction and represent the difference between the vibrational velocities introduced into the cutting zone and the deformation velocities. Then from (7), we obtain:

As before, we expand

in a Taylor series:

where

;

.

The region of convergence of the series is

.

We average

over the period (Ω0)–1 and limit ourselves to the first four terms of the series:

(13)

This series always converges. The system is stable. Therefore, for ,

.

As increases, a growth of the effective value of  is observed, which depends on . This component is perceived by the subsystems and is within their bandwidth. This allows us to introduce the concept of latent force:

(14)

It is obvious that with , force .

The second case: the vibrational velocities are equal to

,

and

Expression

is expanded in a Taylor series:

(15)

where

.

Series (15) converges quickly since . We average the expression over the period (Ω0)–1 and limit ourselves to four terms:

(16)

At , expressions and  are transformed into (7) without taking into account forces kФF0. Effective values of  and  differ due to the direction of vibrations — X2 or X3. In the first case, vibrations change the proximity of the flank and the workpiece, while in the second, they change the projections of the vector onto the direction X2.

Here, we can also consider the latent force, which is zero in a stable system ():

(17)

The presented analysis allows us to formulate two conclusions that are important for further work.

4. Experiment Design and Simulation Parameters. The experiments were performed on a 16K20 machine tool with an adjustable spindle rotation and carriage feed drive. A603C01 vibration accelerometers with a sensitivity of 10.2 mV/(m/s²) and a frequency range of 0.4–15000 Hz were used as measurement interfaces. They were mounted on the tool in the longitudinal and tangential directions. The measuring stand collected data and transmitted it to the computer via an E20–10 analog-to-digital converter (ADC) with a sampling frequency of 100 kHz. The data obtained was processed using low-pass filtering algorithms to suppress noise in the measuring circuit. To determine the vibrational velocities and tool tip displacements, the vibration acceleration signal was integrated by software methods with trend removal. To measure forces, the STD.201–1 measuring system was installed in place of the support, which included:

The latter consisted of electronic units manufactured by National Instruments (USA): NI-9234, Ni-9237, and NI-9219. The sampling frequency was up to 50 kHz. The National Instruments system also measured the integral temperature value in the cutting zone. This indicator was associated with the power of irreversible energy transformations throughout the cutting zone.

To introduce USV into the cutting zone, an acoustic system based on a 500-watt magnetostrictive transducer was used, powered by a 1.5-kW ultrasonic generator. The device for automatically adjusting the generator frequency to the resonance of the acoustic system shifted when the boundary conditions of the tool – workpiece interface changed under the cutting process. Vibrations were measured with an accelerometer. Their intensity was estimated from the amplitude of harmonic displacements at a frequency of Ω0. Workpieces made of 10GN2MFA steel were machined using tools with brazed T15K6 plates without the cutting fluid.

The computer simulation considered disturbances

and , Ω0 = (5 – 20) kHz. The main angles of the T15K6 tool were ϕ = 90°, ϕ1 = 30° and α = 6° (Fig. 1). These values were selected to simplify the modeling of the dynamic cutting system, since at ϕ = 90, forces generated in the area of tool – workpiece contact produced practically zero projections in the direction X1. Process modes without considering deformations and disturbances were:

When varying , the ratio of workpiece rotation velocity to longitudinal feed rate was maintained so that . The tool subsystem parameters are given in Table 1. The total mass was m = 0.015 kg∙s²/mm.

The dynamic coupling parameters (Table 2) were determined experimentally by methods and programs described in detail for the parameters of high-speed [45] and positional [46] communication.

Work A and power N are scalar quantities. They depend on the direction of movement and are measured in kg∙mm and kg∙mm/s, respectively. Let us consider A and N in the direction А — В (Fig. 2). When turning, V2/V3 ⇒ 0. This means:

(18)

Here, a — power N and work A in the feed direction; b — power N and work A in the direction of velocity V3.

The cutting process dynamics modeling [47] demonstrated the validity of the sensitivity analysis of force variations to deformations in the feed direction. Furthermore, a regenerative self-excitation effect was formed in the direction X2, affecting the dynamics of the approach of the tool flank to the workpiece.

The numerical simulation of turning a shaft with a diameter of D = 84 mm was performed in the Simulink software package. This example can be used to study the vibration control of the PIET.

Research Results. Let us examine the results of vibration control of the power of irreversible energy transformations in the feed direction (18а) under the impact of high-frequency vibrations. First, we analyze the dependence of Ф2(t) on  without considering kФF0 (Fig. 3 а).

Note the nonlinear distortions of Ф2(t). They increase at , directed toward the flank and workpiece (section 1–2) and are practically zero when its sign changes (section 2–3). At small amplitudes (Fig. 3 a), the force variations are almost harmonic. The nonlinear distortions are due to nonlinear relationship (7), which does not have central symmetry at any point.

As we can see, the disproportionate increase in the force impulse varies depending on the clearance angle α. At small angles (large ς), a rapid increase is observed even at low amplitudes of the HFV.

Next, we analyze the relationship between the forced HFV and the tangential components of the contact interaction forces (Fig. 4).

The change in the amplitude of the HFV in Figure 4 a was specified by two piecewise constant functions with the duration of each step of Δtstep= 2 s:

for the interval t∈ [ 0;8], the change in the amplitude of the first function was specified from 0 to5·10–2 mm for t∈ [ 0;8] with a step of 0.5·10–2 mm; for the interval t∈ (8;22], the change in the amplitude of the second function was specified from 2·10–2 mm to 14·10–2 mm with a step of 2·10–2

The arrows in Figure 4 d, e indicate the increment of forces ΔФ3 with increasing HFV amplitude and the corresponding displacement of the equilibrium point of the system in the phase plane.

Figure 4 e shows an example of the phase trajectory X2 – dX2/dt, corresponding to the change in force in Figure 4 d. Transient processes are caused by a jump in the amplitude. The equilibrium point shifts because the vibrations affect and .

According to [10], the main wear mechanisms change with increasing V3 (increasing PIET). At low V3, abrasive and adhesive-fatigue are observed, and with increasing V3 — diffusion and oxidative wear. The transition from adhesive-fatigue to diffusion-oxidative wear corresponds to the minimum intensity. When vibrations are excited, the formation of PIET in the contact becomes more complex, however, wear rate can also be estimated from PIET.

Let us clarify the concepts of wear rate v(L) = dw/dL and velocity v(t) = dw/dt. The magnitude of wear on the flank is usually considered as the equivalent width of the wear band in mm, which is determined by the height of the equivalent rectangle of the wear scar on the flank. Equivalence is equality of areas, therefore:

(19)

where v(L) is a dimensionless quantity.

In the velocity range of 0.7–3 m/s, dependence v(L)(N) is well approximated by the expression:

(20)

Here, α(w) — dimensionless quantity; β(w) — parameter of dimension W2. When processing heat-resistant steels, α(w) = (0,9 – 1,1) ⋅ 10–7. Due to vibrations, N becomes a function of time N(t) with interrelated periodic and constant components. All physical interactions are inertial, that is, their manifestation also depends on frequency. Therefore, to estimate N(t), it makes sense to introduce values , depending on the direction of vibrations. For this, it is convenient to use the moving average operator for (20): (21) It is convenient to consider the averaging time as a multiple of the period (Ω0)–1. Let us analyze the change in the PIET depending on the vibration amplitude at Ω0 = 10 kHz. Let us consider its change depending on the HFV amplitude in the directions X2 (Fig. 5 a) and X3 (Fig. 5 b).

We see the HFV amplitudes at which the PIET reaches its minimum value, with the minimum depending on the degree of plate wear ς.

Figure 6 a shows the changes in v(L) depending on the PIET without additional vibrations.

Since vibration-free cutting is considered, then Vp = const. The forces on the flank are estimated by extrapolating the forces to zero cutting thickness. Dependence v(L)(N) (Fig. 6 a) has three distinct sections: in two, the power increases (clear zone), and in one, it decreases (shaded zone). Power N is estimated at a stage where wear does not exceed w = 0.2 mm. The black triangles represent experimental points, each obtained by determining the mathematical expectation from the experiments. Note that at least five experiments were conducted for each point. The well-known relationship 1 kg⋅m/s = 9.81 W was used to determine the power in watts.

Experimental data on the USV impact on cutter wear are visualized in Figure 7.

The change in wear along the cutting path is the path of the tool tip across the workpiece during each revolution of its circumference. For each point of the wear characteristics, their mathematical expectations and variances relative to the mathematical expectations are shown (represented by vertical segments). In Figure 7, value of w at each point corresponds to at least five experiments.

Discussion. Modeling has revealed that the disproportionate increase in the force impulse varies depending on the clearance angle α. At small angles (large ς), a rapid increase is observed even at low amplitudes of the HFV. Figure 3 a, c, d demonstrates the limitation of additional cutting forces relative to zero. This is due to the nonlinear dependence of cutting forces on the clearance angle of the tool in model (7). This exponential function accounts for the physical constraint on the tool continuous movement along the workpiece. Therefore, the additional forces are strictly positive. The curves in Figure 3 c, d demonstrate an asymmetry in the rise and fall of the force impulse, which is also explained by the exponential dependence in (7). If additional vibrations are absent, the force value is determined by the exponential function exp[ς(υ – υ*)] = 1. Periodic movements relative to this point cause changes in the form of the additional force impulses, moving from the exponential section with a high rate of increment (front of impulse Ф2 in Fig. 3 c, d) to a more gradual one (decay of impulse Ф2 in Fig. 3 c, d). Small variations in the vibrations relative to this point cause variations in Ф2, which, due to small deviations from the equilibrium point, can be considered in a linear approximation. Then the relationship between the vibrations and the forces remains linear, and the change in the additional forces is close to a harmonic form (Fig. 3 b).

As the amplitude increases, nonlinear properties of the force vibrations impact on the tool flank become apparent. Nonlinear interactions cause a displacement in the period-integrated vibrations Ф2, as shown by the dotted lines in Figure 3 b, c, d. The greater the vibration amplitude, the more pronounced the constant component of the forces becomes, shifting the system equilibrium point (for the case in Fig. 3 b — 8 kgf, Fig. 3 c — 22 kgf, Fig. 3 d — 50 kgf). Furthermore, with an increase in the vibrational amplitude, the force surges become closer in shape to delta functions, which, taking into account the impact of forces on the current value of the power of irreversible energy transformations, causes surges in heat production and increases tool wear.

Let us note two effects when increasing the amplitude of the simulated HFV.

First, the vibrations create a cyclically stressed state in the cutting zone, redistributing the constant and cyclic forces. This leads to a decrease in the effective values of , and therefore, of . The elastic deformations in the region adjacent to the chip formation zone depend on this force. Elastic recovery is observed in the flank contact region. This results in the generation of forces , whose magnitude depends on .

Secondly, variation of the HFV amplitude corresponds to changes in the values of additional forces Ф3 in the cutting direction. The force graphs are presented for various tool flank gradient ς taking into account changes in the current vibrational velocities, which essentially determines the degree of approach of the tool surface to the workpiece or, for example, the degree of its wear. As the HFV amplitude increases, the tool flank approaches the workpiece, resulting in an increase in Ф3 (Fig. 4 b, c, d). However, at ς = 1, the effect of force minimization is visible from Ф3 = 42 kgf to Ф3 = 30 kgf with increasing amplitude (Fig. 4 b). According to (18b), this indicates the existence of such amplitudes of the HFV at which power N(t) in the cutting zone is minimized. This effect at ς = 5 is weakly expressed and shifted to the left, towards the effect of small amplitudes of the HFV. In this case, a small change in force is noticeable from Ф3 = 19 kgf to Ф3 = 17 kgf (Fig. 4 c). At ς = 7, this effect disappears (Fig. 4 d), and Ф3 grows following the HFV amplitude to Ф3 = 103 kgf. The increment of forces ΔФ3 is proportional to each new value of the disturbance amplitude (Fig. 4 d).

Thus, as wear increases, the optimal value of the HFV amplitude, which is capable of minimizing the value of additional cutting forces on the tool flank, and, consequently, the power of irreversible energy transformations in the cutting zone, decreases.

The study allowed us to determine and visualize the amplitudes of the HFV at which the PIET takes a minimum value, and this minimum depends on parameter ς, that is, on the degree of wear of the plate. Thus, when the HFV system is disturbed in the direction V2 for ς = 1, the optimal amplitude will be mm. At this value, the power released in the cutting zone is minimized (Fig 5 a). At ς = 5, the optimum shifts in the direction of the arrow on the graphs. The minimum of power trajectory 2 occurs at  mm, and then degenerates at ς = 7. Further, even small additional vibrations correspond to an increase in the PIET. The effects noted above are neutralized if the HFV frequency exceeds the cutoff frequency of the dynamic subsystem of the cutting process, which is determined by parameter T(0).

The described effects are leveled by vibrations in the direction V3, due to which the tool deviates from the direction А — В (Fig. 5 b). This appears as a projection determined by the ratio V2/V3 (Fig. 2 b). Here, to the right of the dotted line (), a break in contact between the tool and the cutting zone is observed. This creates a cyclically stressed state, which is caused by the periodic interruption of cutting. Therefore, the effective forces and the PIET are reduced almost by half for the trajectory at ς = 1 from N = 300 W to N = 144 W, and at ς = 5 — from N = 320 W to N = 210 W. Furthermore, at low speeds, the tool and workpiece rear faces come closer together. This creates additional forces acting on the flank. This results in an effect similar to the impact of circulatory forces, which form circular tool tip trajectories.

The paper examines in detail examples of change of v(L) depending on the amplitude for two velocities (VP = 1 m/s, VP = 1.4 m/s) and the corresponding PIET (N1, N2) (Fig. 6 b, c). Point N0 = 600 W corresponds to velocity m/s at which v(L) is minimal without additional vibrations. Point N0 = 600 W lies in the velocity range of 0.7–3 m/s. It corresponds to the optimal cutting temperature. The efficiency of the vibration impact depends on VP, the direction of vibrations, and the entire dynamic cutting system, including the workpiece. The range in which these effects are detected is limited to frequencies of 15–20 kHz.

The contradictory effect of USV in the direction V2 for velocities VP = 1 m/s и VP = 1.4 m/s Fig. 7 a, b) is noticeable for the same cutting path. Figure 7 a shows a case of an extremely insignificant decrease in the maximum value of tool wear on the flank. With the introduction of USV µm, the reduction in this indicator is limited to 0.075 mm and decreases from w = 0.575 mm (curve 1) to w = 0.5 mm (curve 2). Doubling the USV amplitude (µm, Fig. 7 a) leads to intensification of wear, and its maximum value increases to w = 1.05 mm (curve 3).

Here, starting from a certain amplitude, an increase in the USV intensifies the cutting force surges along the flank edge Ф2(t). Consequently, according to (18), the energy released in this region increases, and tool wear intensifies. A similar effect was observed in numerical experiments for Ф2(t) (Fig. 3 b, c, d).

Thus, in the case of introducing USV in the feed direction, the vibration efficiency in reducing tool wear depends fundamentally on the cutting velocity. Value of Vp may initially be close to the optimal value of the PIET N0 (Fig. 6 a), its small variations in the cutting zone will increase heat generation and, accordingly, wear. With the introduction of USV, the minimum wear intensity shifts towards increasing Vp (Fig. 7 b), and the maximum wear value decreases from w = 0.55 mm (curve 1) to w = 0.35 mm (curve 2) when introducing USV with an amplitude of  µm and to w = 0.26 mm (curve 3) with an amplitude of  µm . This is due to a decrease in the cyclic components of the cutting forces.

The data in Figure 7 c suggest that at VP = 2 m/s the tool wear rate in the direction of cutting speed is virtually independent of USV. This is due to the limited ability to vary the USV amplitude during the experiment. Under our conditions, the achievable amplitude of vibrational displacements does not exceed 10–15 µm, and at high cutting speeds, the effect of USV on wear rate will be less due to the small relative variations in vibrational velocity to cutting velocity.

Setting low cutting speeds is appropriate for certain types of machining. For example, when dealing with heat-resistant steels, the introduction of USV significantly minimizes tool wear. Therefore, to effectively utilize vibration disturbances, it is required to consider the USV amplitude under the cutting process. To select the optimal ratio between the USV amplitude and the velocity of the tool vibrational displacements, an analysis of its motion is essential.

In Figure 7 c, the cutting velocity value is in the zone of small change in the PIET, that is, to the left of the dotted line, as shown in Figure 5 b. A decrease in the maximum wear value in Figure 7 c from w = 0.65 mm (curve 1) to w = 0.6 mm (curve 2) is achieved at  µm. Amplitude µm allows wear to be reduced to w = 0.48 mm.

Thus, numerical modeling and experiments have shown that vibrations along axis shift the optimum velocity VP, at which wear intensity is minimized, toward increasing cutting velocity (Fig. 6 b, c). Note that for case N1 (Fig. 6 b), velocity VP is lower than the optimum cutting velocity without vibrations. In this case, with increasing amplitude

at velocity VP, additional vibrations, as a rule, increase wear intensity. For case N2 and as the amplitude of additional vibrations increases, an extremum is observed at which wear intensity is minimized.

The overall pattern of wear rate changes depending on vibration parameters, obtained through numerical modeling, qualitatively matches the experimental results of studying the impact of USV on wear (Fig. 7). As the amplitude increases, the optimum depends not only on the process conditions but also on the initial tool geometry, such as its clearance angle. The optimum shifts as wear progresses. At a certain point, the extremum levels out, and then the introduction of additional vibrations will not increase wear resistance under any process parameters.

When additional vibrations are excited in the direction of the cutting velocity, the situation changes. Here, the extreme amplitude of the vibrational velocity is observed only in the low-speed range. With USV, the existence of an optimal amplitude in the direction of the cutting velocity is limited to a cutting velocity of 0.3–0.5 m/s.

The reduction in wear rate depends on all the basic parameters of the DCS. Here, first of all, the elements of the stiffness matrices and the generalized masses should be mentioned. Furthermore, it is important to consider the dynamic coupling parameters, whose effective values themselves depend on the vibrations. Additional vibrations from the ultrasonic acoustic system (for example, in the feed direction) change not only the spatial orientation but also the amplitude due to the reaction from the cutting process. The resulting phase shifts between vibrations in different directions depend on the amplitude. They are caused by the specific interactions between vibrations on forces and forces — on deformations. Therefore, for example, it is impossible to orient additional vibrations in the direction of the projected cutting velocity (direction А — В in Figure 2).

Thus, even small-amplitude HFV always cause periodic changes in the approach between the tool flank and workpiece. This explains the first trend — an increase in wear rate with increasing amplitude. The second trend is caused by the development of a cyclically stressed state in the primary and secondary deformation zones. As a result, the forces and the PIET decrease in the contact zone between the flank and workpiece. These two opposing trends determine the dependence of wear rate on vibrational amplitude. It should also be noted that the optimal amplitude in all cases in a given system changes as wear progresses, as this process transforms the geometry of the tool flank.

Conclusion. The findings of the research presented in this article differ from those of published studies on the impact of vibration on tool wear. The author shows changes in cutting tool wear as a function of high-frequency vibrations from a new perspective — through the relationship between the power of irreversible energy transformations at a specific location, namely, the contact area between the workpiece and the tool flank. Previously, wear was considered in the literature on the scale of the entire cutting process.

The following tasks were accomplished in the course of the work.

The proposed approach allows us to explain changes in the system properties in the low- and mid-frequency regions depending on the amplitude of the HFV. HFV introduced into the cutting zone can be considered as a control factor for:

The adequacy of the modeling results is limited by the zone of tool wear rate, in which the effect of random processes on the dynamics of the system increases and, accordingly, the assessment of the accuracy of the model is significantly reduced.

Conditions where the amplitude of vibrational displacements exceeds 10–15 µm require additional studies. Furthermore, it is necessary to elucidate the intrasystem physical processes of molecular-mechanical wear, including the physics of interactions as a whole. A promising direction for the developed models is their integration into tool wear diagnostic systems based on hybrid machine learning architectures. This approach will enable more accurate prediction of changes under the condition of cutting tools when modeling DSR.