A Tool Life Prediction Model Based on Taylor’s Equation for High-Speed Ultrasonic Vibration Cutting Ti and Ni Alloys

Abstract

A high-speed ultrasonic vibration cutting (HUVC) method has been proposed for the precision machining of Ti and Ni alloys with high efficiency and fine surface quality in recent years. During the HUVC, the tool life can be enhanced significantly at a relatively high cutting speed. The effective cooling due to the tool-workpiece separation resulting from the ultrasonic vibration is regarded as the primary reason for these advantages. In order to figure out the influences of effective cooling and ultrasonic vibration for further understanding of the mechanism of HUVC and guidance of practical engineering, a quantitative relationship between the tool life and cutting conditions (including cutting, ultrasonic and cooling parameters) needs to be built. Therefore, in this paper, a tool life prediction model based on Taylor’s equation was established. Both the cooling contribution during the separation interval and tool impact resulting from the ultrasonic vibration were added to be considered. Then, experiments were conducted and the results showed that the separation effect with effective cooling was the main reason for the considerable benefits of HUVC. Although the impact was inevitable, high-speed, stable cutting regions of Ti and Ni alloys could still increase to 200–450 and 80–300 m/min, respectively. The prediction model could be used to optimize the cutting parameters and monitor the machining process according to the actual machining requirements.

1. Introduction

Owing to their distinct properties, such as superior creep resistance, corrosion resistance, high strength-to-weight ratio, and high-temperature performance, Ti and Ni alloys are now being increasingly applied in the domains of aerospace, aviation [1,2,3], and biomedical engineering [4,5]. However, as these are two typical difficult-to-cut materials, it is difficult to obtain both satisfactory machining surface quality and good cutting efficiency simultaneously. Rapid tool wear during the machining process is the core factor that influences the surface quality [6]. During the cutting process, interfacial friction, heat generation, material elastoplastic deformation, and chemical reactions can result in tool rapid wear and product quality failure [7,8]. At the same time, when cutting with a worn cutting edge, the strong cutting interfacial adhesion increases the cutting force and temperature, thus further worsening the surface quality [9,10]. In this regard, extending and predicting the tool life is of great significance in both the scientific and engineering domains to obtain quality products.

Numerous research efforts have focused on extending tool life. Efficient cooling and lubrication are critical, and researchers have used high-pressure coolant [11,12,13], minimum quantity lubrication (MQL) [14,15], cryogenic cooling [16,17], and hybrid cooling methods [18]. These methods attempted to deliver sufficient cooling/lubrication medium into the cutting interface during the cutting process, thus reducing the cutting force and temperature. Apart from the application of coolants, the cutting tool can be modified according to a specific design of textured tools to improve the tribological behavior and wear performance. A friction reduction of 16–39% can be achieved in this manner [19,20]. In addition, functional coatings [21,22] and tool geometry modifications [23,24] can both reduce the cutting force and improve tool life. Unlike the above methods, some researchers have attempted to change the cutting method and proposed an intermittent cutting method, i.e., ultrasonic vibration cutting. This can realize an extremely low cutting force and tool flank wear (12–25% of that in conventional cutting (CC)), thereby enhancing the tool life by 4–8 times [25]. This benefit can be attributed to the periodic tool-workpiece separation effect during the cutting process [26]. Therefore, when lubrication methods, such as MQL, are applied, tool wear is significantly improved through the suppression of the generation of microcracks and friction traces on the tool surface [27,28,29]. To further investigate the effect of the separation caused by ultrasonic vibration cutting, a high-speed ultrasonic vibration cutting (HUVC) method was proposed for high-speed precision machining of Ti [30,31] and Ni [32] alloys. Using this method, tool life can be extended by up to a factor of 6, as compared to that in CC. In summary, tool life can be improved by changing the cooling and lubrication states between the cutting interfaces. In this regard, for practical applications, predicting the final tool life is important.

For tool life predictions, Taylor laid the foundation of an empirical method by establishing Taylor’s equation, in which it was simply pointed out, through an exponent model, that tool life is strongly related to cutting speed [33,34]. Subsequently, this model has been improved using methods based on reference speed [33], by extending the influencing parameters to include the combination of cutting speed, feed per revolution, and depth of cut, instead of the cutting speed alone [35,36]. Furthermore, methods based on temperature-based equations [37,38], tool geometry [39], workpiece hardness [40,41], and chip groove and coating factors [42] have also been reported. Although these empirical models were proposed decades ago, they still remain useful for predicting the tool life in practical engineering. Recently, with the advent of data mining, computer technology, mathematical statistics, and other interdisciplinary developments, tool wear and tool life can be predicted in real time. By integrating the real time cutting force, tool wear, and cutting parameters, the tool life and wear procedure can be accurately predicted [43,44]. Both the former empirical and the latter developed approaches have their own advantages and disadvantages. The former is simple and easy to understand, while the latter is accurate and affords real-time, which is conducive to accurate control.

In this regard, in this study, a simple and direct method is adopted to describe the newly proposed HUVC method. Based on the decomposition of HUVC kinematics principles, the influences of relevant parameters, such as cooling and vibration parameters, are supplemented in the traditional Taylor’s equation. Through this newly proposed HUVC-Taylor’s equation, significant tool life advantages for HUVC can be predicted, while the reasons for tool life enhancement by HUVC can be analyzed quantitatively. Both these aspects can be used to guide and promote practical engineering approaches for the machining of Ti and Ni alloys

2. Tool Life Prediction Model

2.1. Basic Structure of HUVC-Taylor’s Equation

When ultrasonic vibration is added to the cutting tool, the cutting trajectory is changed significantly, which, in turn, alters the material removal process. Figure 1 demonstrates these differences. In Figure 1a,c, only a small vibration is added along the feed and axial directions in the turning process. However, when the workpiece surface is unfolded, considerable differences between HUVC (Figure 1b) trajectories are exaggerated expressions in order to clearly observe the tool motion state (not true proportions) and CC (Figure 1d) are noted. In CC, the material is removed layer by layer in accordance with the feed per revolution. The cutting thickness of each layer remains the same, and the cutting process is continuous and stable. In this regard, the cutting trajectory can be defined as:

$$Z_i=if$$

(1)

where Z_i represents the ith tool trajectory on the extended workpiece surface and f is the feed per revolution. In contrast, in HUVC process, material is removed intermittently. Once the cutting and vibration parameters are appropriately set to satisfy the separation condition [30,45], HUVC process is divided into cutting and non-cutting durations. The adjacent cutting trajectories have intersections and can be defined by:

$$Z_i=Asin\left(2\pi Ft+i\phi \right)+if$$

(2)

where A, F, $\phi$ denote vibration amplitude, frequency and phase shift between two adjacent tool trajectories, respectively. From Equation (2), it can be identified that both the phase shift and vibration amplitude affect the cutting trajectory and result in a change in cutting thickness. In this regard, HUVC process is a dynamic-stable process.

The processes of CC and HUVC can be illustrated in Figure 2a,b, respectively. In CC, there is only one state, i.e., the cutting duration in one vibration cycle from point A to point B. Hence, the tool life prediction model can be described according to Taylor’s extended equation [35,36]:

$$U_{cc}=C_d{v}_c^p{f}^q{a}_p^r$$

(3)

where $U_{cc}$, $v_c$, $f$ and $a_p$ are the tool life of CC, cutting speed, feed per revolution and cutting depth, respectively. $C_d$, $p$, $q$ and $r$ are the constants. Equation (3) can also be treated as the tool life prediction model of CC in dry machining which can be used as the basic structure for HUVC modeling.

In Figure 2b, the HUVC process consists of two successive durations. The first duration is the cutting duration (points A to B), where the cutting process is similar to that of CC. The second duration is the noncutting duration (points B to C) where sufficient coolant penetrates the tool-workpiece interval under high pressure, MQL, etc. [31,32]. Therefore, the tool life of HUVC should be the integration of these two durations and Equation (3) can be modified as:

$$U_{cc}=C_d{v}_c^p{f}^q{a}_p^r·v_c^p{}^{\prime }·D{c}^x+U_{n-c}$$

(4)

where $U_{n-c}$ denotes the tool life enhancement of the noncutting duration, $D{c}^x$ and $v_c^p{}^{\prime }$ denotes the effects of the duty cycle and coolant on the cutting duration for CC. Tool life is directly related to the cutting temperature, while cutting speed is the main factor that influences cutting temperature. In this regard, according to the temperature-based tool life equation [37,38], the effect of the coolant can be described by adding $v_c^p{}^{\prime }$ which can be combined into $v_c^p$ in the following equations.

Despite the effect of coolant owing to the noncutting duration, the impact effect should also be considered. In the CC process, the tool is kept engaged in the workpiece, and the cutting process is continuous. In this regard, the cutting process is stable. However, in the HUVC process, the tool cuts-in and disengages from the workpiece periodically in an ultrasonic frequency. Therefore, impacts are imposed on the tool when it engages and disengages from the workpiece at points A and B, respectively. This impact has a negative effect on tool life. In this regard, an exponential decay function and impulse of one period are combined to evaluate the negative effect. Equation (4) is further rewritten as

$$U_{HUVC}=\left(C_d{v}_c^p{f}^q{a}_p^{r}D{c}^x+U_{n-c}\right)·e^{\Gamma \left(I\right)}$$

(5)

where $U_{HUVC}$ is the tool life of HUVC, $\Gamma \left(I\right)$ is the influence of tool impact and $I$ is the ultrasonic impulse. Compared to the traditional Taylor’s equation (Equation (3)), Equation (5) includes both the positive and negative effects of the added ultrasonic vibration on tool life and it forms the basic structure of HUVC-Taylor’s equation from the kinematics analysis of HUVC.

2.2. Modeling of Cooling Effects

In Figure 3, during the noncutting duration, the tool and workpiece have a transient interval of 20 to 30 μm, during which the added coolant leads to a reduction in the cutting temperature. The heat transfer process can be simply treated as a heat convection process between two plates which represents the tool and workpiece surface, respectively. The heat transfer can be computed as:

$$Q=-HS\left(\Theta -{\Theta }_0\right)$$

(6)

where $Q$, $H$, $S$, $\Theta$ and ${\Theta }_0$ denote to heat flux, heat convection coefficient, heat transfer area, cutting temperature and coolant temperature, respectively. These parameters are quite difficult to be abstracted from the actual cutting conditions. According to [37,38], a temperature-based tool life equation can be simplified as:

$$\Theta U^n=C$$

(7)

where C and n are constants.

From Equations (6) and (7), it can be observed that the relationships between tool life and the cooling conditions are complex. In this regard, an empirical method based on cutting experiments should be utilized. In the heat transfer process, the coolant flow velocity and area are two main factors. Correspondingly, the cooling pressure and vibration amplitude are chosen to represent the coolant velocity and cooling area, respectively. In addition, as mentioned in [31], the tool life enhancement becomes weak when the cooling pressure reaches a certain value. In this regard, we can use an exponential function to fit the change principle of tool life. The tool life of the noncutting duration can be written as:

$$U_{n-c}=C_1\left(1-e^{-{\alpha }_1{p}_c^s}\right)\left(1-e^{-{\alpha }_2{A}^{\epsilon }}\right)ta{n}^y\left[\frac{\pi }{2}\left(1-Dc\right)\right]$$

(8)

where $C_1$, ${\alpha }_1$, ${\alpha }_2$, s, $\epsilon$, y are constants and p_c and A are cooling pressure and heat transfer area, respectively. In Equation (8), when the cooling pressure and vibration amplitude equal zero, i.e., the cooling condition and noncutting duration do not exist, the tool life is not enhanced. In addition, when the duty cycle equals 1, i.e., no tool-workpiece separation occurs despite the applied ultrasonic vibration, the tool life cannot be enhanced. However, when the duty cycle approaches zero, no cutting will occur during the tool vibration process. As a consequence, the tool life must be infinite. In this regard, tangent function is used as the basic function to describe its variation process.

Generally, the vibration amplitude during HUVC is a fixed value which is associated with the resonance of the transducer. When the vibration frequency is approximately 20,000 Hz, the vibration amplitude exhibits a small fluctuation between 16 and 20 μm. In this regard, the influence of heat transfer area can also be treated as a constant. Equation (8) can be rewritten by defining a new constant $C_l=C_1\left(1-e^{-{\alpha }_2{A}^{\epsilon }}\right)$, as the following form:

$$U_{n-c}=C_l\left(1-e^{-{\alpha }_1{p}_c^s}\right)ta{n}^y\left[\frac{\pi }{2}\left(1-Dc\right)\right]$$

(9)

In Equation (9), the influences of ultrasonic vibration and cooling conditions can be evaluated by duty cycle and cooling pressure, respectively.

2.3. Modeling of Impact Effects

Regarding the effect of impact, as illustrated in Figure 4a, once tool-workpiece separation occurs, an impulse force signal will repeat in accordance with the vibration frequency. As shown in Figure 4b, impulse, which is represented by the area below the transient cutting force signal, can be used to describe the impact effect and it can be integrated over one vibration cycle as:

$$I={{\displaystyle \int }}_{t_1}^{t_2}F\left(t\right)dt$$

(10)

where $F\left(t\right)$ is the transient cutting force signal, $t_1$ and $t_2$ are the cut-in and cut-out moments, respectively. The cutting process in one vibration cycle is demonstrated in Figure 4c. The transient cutting force can be calculated according to the traditional Merchant equation:

$$F\left(t\right)=Kh\left(t\right)\omega V$$

(11)

where K is the cutting force coefficient, $\omega$ is the cutting width and $h\left(t\right)$ is the transient cutting thickness which can be computed according to the relationship demonstrated in Figure 4c as the following form:

$$h\left(t\right)=\overline{h}+Asin\left(2\pi Ft\right)$$

(12)

Substituting Equation (12) into Equation (10), the impulse of one vibration cycle then can be modified into:

$$I={{\displaystyle \int }}_{t_1}^{t_2}K\left[\overline{h}+Asin\left(2\pi Ft\right)\right]\omega Vdt=K\overline{h}\omega v_c\mathsf{\Delta }t+\frac{KA\omega v_c}{2\pi F}\left[2sin\pi F\left(t_1+t_2\right)sin\pi F\mathsf{\Delta }t\right]$$

(13)

meanwhile, it is clear that:

$$\mathsf{\Delta }t=T·Dc=Dc/F$$

(14)

In this regard, the final expression of the impulse is:

$$I=K\overline{h}\omega v_c\frac{Dc}{F}+\frac{KA\omega v_c}{\pi F}sin\pi Dc$$

(15)

The first part in Equation (15) is the direct component of the impulse, and it can be neglected and only the alternative component is considered. In this regard, Equation (15) is modified as:

$$\tilde{I}={\alpha }_{3}Vsin\pi Dc$$

(16)

where ${\alpha }_3$ can be defined as a constant. Then, the effect of the impact can be deduced as:

$$e^{\Gamma \left(I\right)}=e^{-\beta {\left(v_{c}sin\pi Dc\right)}^{\gamma }}$$

(17)

where $\beta =-{\alpha }_3^{\gamma }$, and $\gamma$ denotes to a constant of impulse.

From Equation (17), it can be concluded that the cutting speed and duty cycle are core factors that influence tool life. The values of $e^{\Gamma \left(I\right)}$ in Equation (17) are both equal to 1 when the duty cycle is set as 0 or 1, which implies that the tool life is not shortened under the conditions of noncutting and continuous cutting, respectively. The maximum value is obtained when the duty cycle equals 0.5, which implies that, during a vibration cycle, half the time is used for cutting, while the other half is used for cooling. A duty cycle less than 0.5 implies that the impact value has not reached its maximum, while a value greater than 0.5 indicates that it is a cutting process rather than an impact. Regarding the cutting speed, the value of $e^{\Gamma \left(I\right)}$ in Equation (17) approaches zero when the cutting speed continues to increase. The limitation value is zero, which implies that the tool life is zero if a large cutting speed is applied. The results of this qualitative analysis are consistent with the change trends reported in previous studies [31]. Combined with Equations (5), (9) and (17), the tool life prediction model HUVC-Taylor’s equation can be written as:

$$U_{HUVC}=\left\{C_d{v}_c^p{f}^q{a}_p^{r}D{c}^x+C_l\left(1-e^{-{\alpha }_1{p}_c^s}\right)ta{n}^y\left[\frac{\pi }{2}\left(1-Dc\right)\right]\right\}·e^{-\beta {\left(v_{c}sin\pi Dc\right)}^{\gamma }}$$

(18)

In Equation (18), apart from the traditional cutting parameters, duty cycle and cooling pressure are introduced to characterize HUVC process. These parameters represent the additional effects of separation and cooling on the tool life, respectively.

3. Method and Experiments

In this section, tool life machining experiments were conducted to fit the constants in Equation (19) and validate its accuracy. Experiments were carried out in a CNC turning center (HASS SL40). As shown in Figure 5, the workpiece with a diameter of 120 mm was fixed on the chuck and rotated in accordance with the spindle. The constant cutting speed was maintained by adjusting the spindle speed according to the variation of the workpiece diameter when coding the CNC program. A triangular cemented carbide insert (TCMT110204) was fixed on top of the vibration transducer, which provided axial ultrasonic vibration through the stimulus of the power source. The nominal rake, clearance angle, tool nose radius, and coating material of the insert were zero, 7°, 0.4 mm, and TiAlN, respectively. The tool vibration frequency and amplitude under the resonant state were 20,153 Hz and 20 μm, respectively. High-pressure coolant, under a pressure ranging from 0 to 20 MPa, was jetted from a pump through a nozzle, which was focused on the interface between the workpiece and tool flank face. The output pressure was controlled by researchers through adjusting the flow rate and the values could be read on a meter. No coding was written into the CNC program. A common emulsion was used in the cutting experiments. During the cutting process, the tool feed and vibration were in the same direction, i.e., the axial direction. All the experiments would be conducted three time in case of randomness.

Tool life is evaluated both from the flank wear and surface roughness because HUVC is a precision machining process [30], during which the surface quality should also be taken into consideration despite of the flank wear. In this regard, the tool life can be rejected if one of the following criteria is reached: (1) Machining surface roughness, the arithmetical mean of roughness profile, Ra = 0.4 μm; (2) Flank wear VB = 0.3 mm; (3) tool breaking.

Using a digital microscope (KEYENCE VHX-6000), tool wear was monitored after each time after 30 mm width machining. Surface roughness, Ra was measured using a roughness tester (Mahr M300C) according to [30,31,32]. As typical difficult-to-cut materials, Ti-6Al-4V and Inconel 718 were chosen to validate the accuracy of the tool life prediction model, i.e., Equation (18).

The experiments were divided into two parts. The first part was to calibrate the constants in Equation (18) and the second part was to validate the accuracy of the model. In Equation (18), there are strong nonlinear terms of the transcendental equation, such as the cooling pressure and the coupling of the duty cycle and cutting speed. In this regard, the constants were solved step by step according to the cutting conditions.

Step A: Conventional dry machining experiments were conducted to calibrate the constants of $C_d$, p, q and r.

Step B: With the results of step A, HUVC with coolant was conducted to calibrate the constants of x, $C_l$, ${\alpha }_1$, s, y, β and γ.

Then, the parameters in Equation (18) were fully described and supplementary tests were conducted to validate the accuracy. The cutting parameters and cooling parameters of each step are set as follows:

For conventional dry machining (Step A), Equation (3) can be rewritten as:

$$\mathrm{In}{U}_{\mathrm{cc}}=d\mathrm{In}C+p\mathrm{In}{v}_c+q\mathrm{In}f+r\mathrm{In}{a}_p$$

(19)

The coefficients of the linear equations of Equation (19) can be determined by multiple linear regression. In order to cover the machining parameters of HUVC, orthogonal experiments of $L9\left(3^3\right)$ are designed for Ti-6Al-4V and Inconel 718, as shown in

Table 1. Orthogonal experiments $L9\left(3^3\right)$ for the conventional dry machining of Ti-6Al-4V.

	vc m/min	F mm/r	ap mm		vc m/min	F mm/r	ap mm		vc m/min	F mm/r	ap mm
1	200	0.005	0.05	4	300	0.005	0.1	7	400	0.005	0.15
2	200	0.01	0.1	5	300	0.01	0.15	8	400	0.01	0.05
3	200	0.015	0.15	6	300	0.015	0.05	9	400	0.015	0.1

and

Table 2. Orthogonal experimental $L9\left(3^3\right)$ setup for the conventional dry machining of Inconel 718.

	vc m/min	F mm/r	ap mm		vc m/min	F mm/r	ap mm		vc m/min	F mm/r	ap mm
1	80	0.005	0.15	4	160	0.005	0.2	7	240	0.005	0.25
2	80	0.01	0.2	5	160	0.01	0.25	8	240	0.01	0.15
3	80	0.015	0.25	6	160	0.015	0.15	9	240	0.015	0.2

, respectively. Then, the constants can be solved using SAS (Statistical Analysis System) software.

Based on the results of step A, HUVC experiments were conducted. Equation (18) can be rewritten as:

$$U_{HUVC}=\left\{U_{cc}D{c}^x+C_l\left(1-e^{-{\alpha }_1{p}_c^s}\right)ta{n}^y\left[\frac{\pi }{2}\left(1-Dc\right)\right]\right\}·e^{-\beta {\left(v_{c}sin\pi Dc\right)}^{\gamma }}$$

(20)

However, noting that the cutting speed and duty cycle are product forms, Equation (20) can no longer be calibrated by multiple linear regression. In this regard, this equation must be solved by non-linear regression. The experimental parameters of feed per revolution, and cutting depth for Ti-6Al-4V and Inconel 718 are 0.005 mm/r, 0.05 mm, 0.005 mm/r, and 0.2 mm, respectively, and the other parameters are shown in

Table 3. Experimental setup for HUVC machining of Ti-6Al-4V.

No.	1	2	3	4	5	6	7	8	9
$v_c$, m/min	200	200	200	200	200	200	300	300	300
pc, MPa	0.5	0.5	10	10	20	20	0.5	0.5	10
Dc	0.5	0.7	0.5	0.7	0.5	0.7	0.5	0.7	0.5
No.	10	11	12	13	14	15	16	17	18
$v_c$, m/min	300	300	300	400	400	400	400	400	400
pc, MPa	10	20	20	0.5	0.5	10	10	20	20
Dc	0.7	0.5	0.7	0.5	0.7	0.5	0.7	0.5	0.7

and

Table 4. Experimental setup for HUVC machining of Inconel 718.

No.	1	2	3	4	5	6	7	8	9
$v_c$, m/min	80	80	80	80	80	80	160	160	160
pc, MPa	0.5	0.5	10	10	20	20	0.5	0.5	10
Dc	0.5	0.7	0.5	0.7	0.5	0.7	0.5	0.7	0.5
No.	10	11	12	13	14	15	16	17	18
$v_c$, m/min	160	160	160	240	240	240	240	240	240
pc, MPa	10	20	20	0.5	0.5	10	10	20	20
Dc	0.7	0.5	0.7	0.5	0.7	0.5	0.7	0.5	0.7

. The duty cycle was set by slightly adjusting the spindle rotary speed [47].

After calibrating the constants, experiments were conducted to validate the prediction accuracy of this model. From Equation (18), it can be noted that the duty cycle and cutting speed are the two main factors that significantly influence tool life due to the cooling and impact effects resulting from the tool-workpiece separation. In this regard, the validations were focused on these two factors. The cutting speed ranged from 100 to 500 m/min and 30 to 300 m/min for Ti-6Al-4V and Inconel 718, respectively. The duty cycle was related to the phase shift and feed per revolution. The calculation results obtained using the method in [45] are shown in

Table 5. Duty cycle values (2A = 20 μm).

Phase Shift (Rad)	F (mm/r)
	0.001	0.005	0.008	0.010	0.012	0.015	0.020
0	1	1	1	1	1	1	1
π/45	1	1	1	1	1	1	1
π/9	0.3628	1	1	1	1	1	1
π/3	0.2659	0.5517	0.7517	1	1	1	1
5π/9	0.2811	0.4988	0.5994	0.7264	0.7865	0.9347	1
2π/3	0.361	0.4779	0.5972	0.6959	0.7436	0.8334	1
8π/9	0.2775	0.5817	0.6331	0.6659	0.7085	0.7755	1
π	0.516	0.5804	0.631	0.6666	0.7058	0.77	1
10π/9	0.2775	0.5817	0.6331	0.6659	0.7085	0.7755	1
4π/3	0.361	0.4779	0.5972	0.6959	0.7436	0.8334	1
13π/9	0.2812	0.4988	0.5994	0.7264	0.7865	0.9347	1
5π/3	0.2659	0.5517	0.7517	1	1	1	1
17π/9	0.3628	1	1	1	1	1	1
89π/45	1	1	1	1	1	1	1
2π	1	1	1	1	1	1	1

. The values are axisymmetric along the axis of the phase shift of π. In this regard, the feed per revolution and duty cycle were validated simultaneously. In addition, the tool life was tested thrice for each cutting condition, and the average of these tool life values, excluding obvious random errors, was considered as the final tool life.

4. Results and Discussions

4.1. Calibration of Model Constants

The cutting length is used to evaluate the tool life, and the results of step A for Ti-6Al-4V and Inconel 718 are shown in

Table 6. Tool life results of the experiments in Step A (km).

No.	1	2	3	4	5	6	7	8	9
Ti-6Al-4V	3.60	3.03	2.51	1.25	1.01	1.36	0.36	0.32	0.30
Inconel 718	2.03	1.79	1.59	1.32	1.17	1.54	0.82	1.01	0.88

. The corresponding constants can be calculated by multiple linear regression and the results are shown as:

$$\left\{\begin{array}{c}{C}_d=e^{16.7925}\left(Ti\right),e^{2.4688}\left(Ni\right)\\ p=-3.1487\left(Ti\right),-0.6065\left(Ni\right)\\ q=-0.1502\left(Ti\right),-0.0073\left(Ni\right)\\ r=-0.1629\left(Ti\right),-0.4752\left(Ni\right)\\ R^2=0.9509\left(Ti\right),0.9513\left(Ni\right)\end{array}\right.$$

(21)

From Equation (21), it is found that the significance of the multiple linear regression is greater than 95%. In this regard, Equation (3) can be clearly written under the experimental conditions as:

$$U_{cc}=\left\{\begin{array}{c}{e}^{16.7925}{v}_c^{-3.1487}{f}^{-0.1502}{a}_p^{-0.1629}\left(Ti\right)\\ e^{2.4688}{v}_c^{-0.6065}{f}^{-0.0073}{a}_p^{-0.4752}\left(Ni\right)\end{array}\right.$$

(22)

Based on the results of Equation (22), the tool life results of step B for Ti-6Al-4V and Inconel 718 are shown in

Table 7. Tool life results of the experiments in Step B (km).

No.	1	2	3	4	5	6	7	8	9
Ti-6Al-4V	33	25	116	98	162	129	28	20	92
Inconel 718	6.98	5.97	10.33	9.15	13.40	11.28	5.02	4.07	7.81
No.	10	11	12	13	14	15	16	17	18
Ti-6Al-4V	83	137	130	18	15	36	33	88	79
Inconel 718	6.72	10.85	10.09	4.96	4.98	5.12	5.23	5.47	5.56

. The corresponding constants can be calculated by non-linear regression and are given as:

$$\left\{\begin{array}{c}x=-1.0470\left(Ti\right),-0.6828\left(Ni\right)\\ C_l=1170880\left(Ti\right),5790\left(Ni\right)\\ {\alpha }_1=0.0000406\left(Ti\right),0.0011\left(Ni\right)\\ s=0.4999\left(Ti\right),0.1848\left(Ni\right)\\ y=0.5258\left(Ti\right),0.2970\left(Ni\right)\\ \beta =0.0000054\left(Ti\right),0.00013\left(Ni\right)\\ \gamma =2.0204\left(Ti\right),1.5769\left(Ni\right)\end{array}\right.$$

(23)

Based on the results of Equation (23), Equation (18) can finally be summarized as:

$$U_{HUVC}=\left\{\begin{array}{c}\begin{array}{c}\left\{\begin{array}{c}\frac{e^{16.7925}}{v_c^{3.1487}{f}^{0.1502}{a}_p^{0.1629}D{c}^{1.0470}}\\ +1170880\left(1-e^{-0.0000406p_c^{0.4999}}\right)\\ ·ta{n}^{0.5258}\left[\frac{\pi }{2}\left(1-Dc\right)\right]\end{array}\right\}·e^{-0.0000054{\left(v_{c}sin\pi Dc\right)}^{2.0204}}\left(Ti\right)\end{array}\\ \begin{array}{c}\left\{\begin{array}{c}\frac{e^{2.4668}}{v_c^{0.6065}{f}^{0.0073}{a}_p^{0.4752}D{c}^{0.6828}}\\ +5790\left(1-e^{-0.0011p_c^{0.1848}}\right)\\ ·ta{n}^{0.2970}\left[\frac{\pi }{2}\left(1-Dc\right)\right]\end{array}\right\}·e^{-0.00013{\left(v_{c}sin\pi Dc\right)}^{1.5769}}\left(Ni\right)\end{array}\end{array}\right.$$

(24)

As of now, the tool life prediction model based on Taylor’s equation has been obtained. From Equation (24), the differences in the constants of these two materials indicate property differences. It can be observed that the constants of the cutting speed are larger than those of the feed per revolution and cutting depth in Ti alloys (3.1487 to 0.1502 and 0.1629) compared to the results for Ni alloys (0.6065 to 0.0073 and 0.4752). In addition, the constants of the cooling pressure of Ti alloys are quite small compared to those of Ni alloys, while the constants of the duty cycle (x = 1.0470, y = 0.5228) of Ti alloys are larger than those of Ni alloys (x = 0.6828, y = 0.2970). These primary results can be attributed to the temperature sensitivity of Ti alloys. In this regard, the heat generation (cutting speed) and cooling effects (cooling pressure and duty cycle) for Ti alloys are more significant, while for the Ni alloys, the combination effects of the cutting temperature and cutting force should be considered (the constants of cutting speed and cutting depth are nearly the same, 0.6065 and 0.4752, respectively). The specific influences are discussed in the Section 4.2.

4.2. Validation of the Model Accuracy

In this section, the accuracy of Equation (24) is validated. It is divided into four subsections to discuss the effects of cutting speed/cooling pressure and the contributions of each component, i.e., the effects of the cooling conditions, impact, and cutting parameters.

4.2.1. Effects of Cutting Speed/Cooling Pressure

The predicted values and experimental results for Ti and Ni alloys are shown in Figure 6a,b, respectively. The two values fit well despite the 500 m/min cutting speed when machining Ti alloys. The reasons for this will be discussed in the next subsections. It can be observed that when machining both Ti and Ni alloys, the tool life, i.e., the cutting length, decreases as the cutting speed increases. This principle is in accordance with the traditional Taylor’s equation. In addition, the tool life is enhanced significantly when the cooling pressure increases compared to dry machining. Considering that all the data are obtained from HUVC, the data of CC are not shown in these figures because of their small values.

In Figure 6, three regions, i.e., the conventional region, useful high-speed stable region, and failure region are defined for both alloys.

(1)

Conventional region: In this region, the cutting speed is within the scope of conventional cutting method.

(2)

Useful high-speed stable region: In this region, although the tool life has decreased compared to the values in conventional region, the cutting speed is still usable when using HUVC method.

(3)

Failure region: In this region, the tool life decreases sharply and the cutting process cannot be kept in a stable process.

In Figure 6a, the upper boundary of the conventional region is approximately 150–200 m/min. In this region, even though the tool life is reduced sharply, the cutting length can still be used during dry machining and common fluid cooling condition. Considerable advantages are shown in the range of 200 to 450 m/min. The tool life of dry machining and common cooling tends to be kept as a constant, while the high-pressure coolant can enhance the tool life to a maximum value of seven to eight times. When the cutting length is set to the same value (black and red circles at the cutting speed of 100 m/min), once the cooling pressure is increased to 20 MPa, the cutting speed reaches approximately 340 m/min and 425 m/min compared to the cutting speed in common cooling and dry machining, respectively. Thus, the cutting efficiency can be accordingly increased to 3.4 and 4.25 times. When the cutting speed exceeds 450 m/min, the tool life is reduced sharply and the advantages of HUVC nearly vanish in the experiment results. Therefore, we define this region as the failure region. Similarly, when machining Ni alloys, the results are shown in Figure 6b, the boundaries of these three regions are ~100 m/min, 120–300 m/min, and 300~ m/min, respectively. When high-pressure cooling is applied, the cutting efficiency can be increased to four and nine times, respectively. In summary, the high-pressure cooling in HUVC can change the CC speed region into a high-speed stable region, which allows possibilities for highly efficient machining of Ti and Ni alloys.

Figure 7 demonstrates the tool life contribution components according to the HUVC-Taylor’s equation. It can be found that in the conventional region, the contribution of dry machining, i.e., the results of the traditional Taylor’s equation $U_{cc}$ decay sharply as the cutting speed increases. This is especially true for Ti alloys, which are more sensitive to the change in cutting speed (cutting temperature) than Ni alloys. The position of the upper boundary of the conventional region approaches extremely low stable values, i.e., 10 km and 3 km for Ti and Ni alloys, respectively. In this condition, CC can be considered as unstable and ineffective. However, when effective cooling is applied, the increase in tool life mainly depends on the efficient cooling method in HUVC process. This contribution $U_{n-c}$ effectively offsets the decrease in tool life caused by the increase in cutting temperature due to the cutting speed. In this regard, the tool life is extended. This can be treated as the main factor of tool life enhancement and the core principle of HUVC. As the cutting speed continues to increase, the impact $e^{\Gamma \left(I\right)}$ caused by the tool-workpiece separation dominates. The lower boundary of the failure region is defined at approximately 30% decay index. When the impact decay index is less than 30%, the enhancement of the cooling contribution is eliminated, thus the tool life of both CC and HUVC reach the same level.

In summary, it can be observed that the separation effect of the tool and workpiece during HUVC is both positive and negative. Therefore, it is important to improve the positive aspects and eliminate negative influences.

4.2.2. Effects of Cooling Conditions

For the positive influences, the tool life contribution components for the cooling conditions $U_{n-c}$ are shown in Figure 8 for Ti and Figure 9 for Ni alloys, respectively.

When machining Ti alloys, the cooling pressure has significant potential for improvement, as shown in Figure 8a. In the experiment, the maximum value of the cooling pressure can reach to 20 MPa. However, when considering the machining tool capability, the pressure usually does not exceed 10 MPa in actual engineering situation. In this regard, the improvement in the cooling pressure has a limitation owing to the machining tools and considerations of costs and safety in the real working situations. Apart from the cooling pressure, the degree of separation, i.e., the duty cycle, also has a strong influence on the tool life. As shown in Figure 8b, as the duty cycle approaches 1, i.e., no separation effect occurs, the tool life contribution is reduced sharply to zero. In a practical machining process, the duty cycle of the HUVC is usually no less than 0.5. Therefore, it also has the limitation for tool life enhancement.

Unlike the conditions of Ti alloys demonstrated in Figure 8, when machining Ni alloys, the influence of the cooling pressure has a ceiling. In Figure 9a, it can be clearly observed that when the cooling pressure reaches 10 MPa, i.e., the common region, the tool life enhancement approximately reaches its maximum value. The tool life enhancement is mainly attributed to the cooling medium rather than the pressure. Regarding the influence of the duty cycle, Figure 9b shows the same tendency as that of Ti alloys in Figure 8b.

Tool life enhancement due to the cooling pressure and duty cycle can be attributed to the significant cutting temperature reduction of HUVC. The cutting temperature results measured by self-developed tool-workpiece thermocouples [48,49] are shown in Figure 10.

The benefits of cutting temperature reduction are directly reflected in the tool wear process. As shown in Figure 11 and Figure 12, the flank wear land is smooth and regular in HUVC, while strong adhesion and irregular wear land shapes are shown in the results of CC for both Ti and Ni alloys. Moreover, when machining Ti alloys, micro tool breaking can be also observed, which aggravates the tool wear process and results in a very short cutting duration.

4.2.3. Effects of Impact

When the cutting speed enters the failure region, severe tool wear occurs in HUVC. As the impact effect dominates, tool breaking is the main tool wear form (Figure 13). In HUVC, the cutting edge forms craters, which leads to the rapid tool wear. Due to the unpredictability and randomness of tool breaking, the prediction and experimental results show a large error in Figure 6a.

The transient cutting force signal can be used to explain the reason for impact decay. Figure 14 shows the transient cutting force signal measured by a sensitive force sensor [46]. During the HUVC process, the cutting thickness variation that resulted from the tool vibration will lead to a varying cutting force. As shown in Figure 14a, when the cutting parameters are set to the same values, as the duty cycle increases, the separation duration decreases and the fluctuation between the peak values (approximately 25 N) and the average values (7.3 N, 7.4 N and 11.01 N for 0.5, 0.7, and 1, respectively) decreases accordingly, thus reducing the degree of impact. In Figure 14b, when the duty cycle is set to a fixed value of 0.6, as the cutting speed increases, the impact peak value increases significantly accordingly (25 N, 35 N and 45 N for 200, 300, and 400 m/min, respectively).

The impact decay index of different duty cycles is shown in Figure 15. When the duty cycle approaches zero, nearly no material is removed. Therefore, the influence of impact is quite small. When the duty cycle approaches one, nearly no tool-workpiece separation occurs, which weakens the effect of impact. The impact effect is maximum only when the duty cycle equals 0.5, which implies that half time is cutting and half time is separation. As the cutting speed increases, the impact decay index decreases. If the cutting speed continues to increase, the impact decay index rapidly reduces to no greater than 0.3, and the cutting process enters the failure region.

4.2.4. Effects of Variable Cutting Parameters

According to the tool-workpiece separation conditions, the feed per revolution is the key parameter that determines the separation condition [30]. When the feed per revolution is changed, the duty cycle changes simultaneously. The relationships between these parameters are shown in Table 5. Figure 16 shows the relationship between the feed per revolution and tool life. It can be observed that, during the machining of both Ti and Ni alloys, a relatively stable region exists between 0.005 and 0.015 mm/r. When the feed per revolution is less than 0.005 mm/r, the cutting process becomes unstable due to the vibration and ploughing effects resulting from the small feed per revolution. In this regard, when machining Ti and Ni alloys, the extremely small feed per revolution is meaningless. However, as the feed per revolution increases to 0.015 mm/r, the minimum value of the duty cycle increases simultaneously, which results in rapid tool life reduction. It is also noted that, when the cutting speed reaches 400 and 240 m/min for Ti and Ni alloys, respectively, the tool life may increase slightly as the feed per revolution increases. When the cutting speed increases, the impact effect is important. In this regard, when the duty cycle increases due to an increase in the feed per revolution, the impact decay index increases slightly, thus improving tool performance. The experiment results of Nos. 13 to 18 in Table 7 for Ni alloys validate this phenomenon. When the duty cycle is 0.7, the tool life becomes longer than that for the duty cycle of 0.5.

4.3. Application Ranges

From the above discussion, based on the HUVC-Taylor equation, the reasons for the excellent performance of Ti and Ni alloys exhibited during HUVC can be revealed quantitatively. The relationship between the cooling, vibration, and cutting parameters is strongly nonlinearly coupled and interrelated. However, the application ranges of HUVC can still be deduced from these complex relationships. First, the tool-workpiece separation conditions should be satisfied to create cooling medium intervals between the cutting interfaces. Then, effective cooling should be applied to reduce the interfacial temperature. Based on these two principles, the cutting parameters, duty cycle, and cooling pressure can be adjusted accordingly to avoid large impact when considering the practical machining conditions and thereby, realize a highly efficient machining process.

5. Conclusions

In this study, a simple and direct tool life prediction model for high-speed ultrasonic vibration cutting (HUVC) was proposed based on the traditional Taylor’s equation. Through this prediction model, the quantitative analysis of the performance of a comprehensive HUVC method for tool life was conducted. The main conclusions were drawn as follows:

(1)

The separation condition and effective cooling were the two key factors that influenced the tool life. The improvement of tool life was mainly attributed to the cooling condition during the separation process which could effectively reduce the cutting temperature.

(2)

Owing to the limitations of the cooling pressure and duty cycle, the useful high-speed stable regions for Ti and Ni alloys were 200–450 m/min and 80–300 m/min, respectively. In these two ranges, compared to conventional effective cutting region, the cutting efficiency was significantly improved.

(3)

The impact effect due to the tool-workpiece separation was a factor that needs to be suppressed. This was the core reason for the failure region in ultra-high-speed regions. In this regard, developing the impact-resistant tool could be seen as the next meaningful work for further cutting speed enhancement.

In summary, the correct parameter setting was of great significance in HUVC. The HUVC-Taylor’s equation developed in this study aimed to provide a comprehensive understanding of the most recently proposed high-speed ultrasonic precision machining methods and provided guidance for appropriate practical applications in future.