Cutting Force Model in Micro-Dimple Pattern Process Using Two-Frequency Elliptical Vibration Texturing (TFEVT) Method

Abstract

We herein propose a cutting force model in a micro-dimple pattern process using the two-frequency elliptical vibration texturing (TFEVT) method. The TFEVT method decreases the cutting force compared to the conventional texturing (CT) method owing to the intermittent cutting behavior. The cutting force model in the TFEVT method is formulated, in which the shear angle is assumed as a transient and the transient area of cut of the micro-dimple is determined. The transient area of cut of the micro-dimple can be determined by obtaining the starting and ending time of cutting, while the shear angle can be determined by Cerniway’s hypothesis. Finally, the cutting force model was compared with the experimental cutting force value. The comparison results show that the cutting force simulation is in agreement with the experimental cutting force value. The experimental results also show that the cutting force in the TFEVT method is lower than that in the CT method.

1 Introduction

Micro-dimple fabrication demands have been increasing recently for functional surface purposes such as vibration, wear, friction [29], anti-cell adhesion [30], and noise control [1]. Friction reduction is one of the major benefits when micro-dimple exists between two lubricated contact surfaces [2]. Many sophisticated methods (e.g., abrasive jet machining (AJM) [3], electrochemical machining [4], turning [5] and laser [6, 31]) can be used to fabricate micro-dimples in advance, although the two-frequency elliptical vibration texturing (TFEVT) method is one of the methods that is characterized as fast and accurate. The TFEVT method is a method that combines the conventional texturing (CT) and the elliptical vibration cutting (EVC) method that has been proposed in a previous study [7].

The EVC method reduces cutting force significantly if compared to the conventional cutting (CC) method [8]. The reason is due to the intermittent behavior of the tool tip in the EVC method. The tool tip vibrates elliptically at ultrasonic speeds. Generally, the vibration frequency is approximately 20 kHz. The tool tip begins to cut at the cusp point and subsequently separates from the secondary deformation zone (chip-rake face contact zone). When the cutting velocity vector of the tool tip approaches in the vertical direction, subsequently, the direction of the friction force on the chip-rake face contact zone is reversed [8]. This reversal friction force was thought to have reduced the cutting force. This phenomenon is observable in the low-vibration frequency. i.e., < 10 Hz. The reversal friction force also assists the deformed chip to be pulled out, when the tool tip moves faster than the speed of the deformed chip.

Cerniway adopted Arcona’s cutting force model for the EVC study by considering the balanced cutting forces on the cutting edge, in which the friction force during cutting was considered [9]. The shear angle is considered to vary during EVC; however, the shear angle can be predicted by the value of the variation in the rake angles. Ammouri et al. [10] also adopted Arcona’s model in their EVC study during the microgrooving of Al 2024. Ma et al. [11] proposed an analytical cutting thrust force model for EVC. In Ma’s model [11], the thrust cutting force is expanded to a Fourier series owing to the periodic cutting. Moreover, the cutting force coefficient was assumed as a constant. Shamoto et al. [12] proposed the cutting force for three-dimensional EVC with a thin shear plane model, in which time instants at various critical tool locations were determined. Zhang et al. [13] proposed the cutting force model for orthogonal EVC, in which three phases of cutting (e.g., CC-like kinetic phase, static-friction phase, and reverse kinetic-friction phase) were considered during one cycle of cutting. Bai et al. [14] proposed the non-equidistant shear zone model for the orthogonal EVC process. Zhang et al. [15] proposed the cutting force model in EVC, in which the cutting force stiffness was assumed as a constant, and the cross-sectional area in a one-cycle cutting was calculated. Jieqiong et al. [16] proposed the cutting force model for oblique elliptical vibration cutting. In their cutting force model, during a one-cycle cutting, only two phase assumptions were considered (e.g., oblique cutting-like phase and reverse kinetic-friction phase).

The TFEVT method can be categorized as a two-dimensional cutting force mechanism because the tool tip vibrates simultaneously in x–y plane to create the micro-dimple shape. The tool tip does not force the workpiece along z-direction, thus the cutting force in z-direction can be neglected. Kurniawan et al. [17] adopted a two-dimensional cutting force mechanism during elliptical vibration texturing to establish the micro-dimple pattern. It is different from a milling operation to create micro-dimple [18], the tool tip in milling operation moves in three-dimensional way, thus a three-dimensional cutting force mechanism should be considered.

Based on the previous works, no information has been presented associated with the cutting force model in the micro-dimple process using the TFEVT method. The objective of this paper is to propose a cutting force model using the TFEVT method and validate the model. Owing to the process complexity, a few assumptions have been made to simplify the cutting force model as described in Sect. 2. Subsequently, the cutting force model is validated with the experimental data.

2 Cutting Force Model

In this study, a few assumptions were considered to predict the cutting force model using the TFEVT method. The assumptions are described as follows:

• 1. The tool tip is assumed to be ideally sharp; thus, the edge radius effect and the tool wear are neglected.

• 2. The motion of the tool tip ideally follows Eqs. 1 and 2.

• 3. The shear plane is assumed as a thin shear plane.

• 4. The angle of the cutting mechanic such as rake, clearance, and shear angle are measured relatively with the velocity vector of the tool tip trajectory.

• 5. The two-dimensional cutting force analysis is adopted, because the cutting force in z-direction does not exist.

In the TFEVT method, the tool tip vibrates simultaneously in a two-dimensional space in the x–y plane with two different frequencies: at an ultrasonic frequency (> 20 kHz) and at low-vibration frequencies (< 100 Hz). Figure 1 presents the illustration of the TFEVT method in which the amplitude of the sinusoidal wave must be sufficiently larger than the major amplitude of the elliptical locus (B > b). Based on the time parametric, the function of the tool tip trajectories x(t) and y(t) in the Cartesian space are defined as follows [7]:

$${\text{x}}\left( t \right) = \frac{1}{2}a \cdot \cos \left( {2\pi f_{h} \cdot t + \phi } \right) + \frac{1}{2}a + V_{f} \cdot t$$

(1)

$${\text{y}}\left( t \right) = \frac{1}{2}b \cdot cos\left( {2\pi f_{h} \cdot t} \right) + \frac{1}{2}b + \frac{1}{2}B \cdot cos\left( {2\pi f_{l} \cdot t} \right) + \frac{1}{2}B$$

(2)

where a and b are the minor and major amplitude of the elliptical locus in µm, respectively; B is the amplitude of the sinusoidal wave in µm.

Fig. 1

Micro-dimple fabrication using TFEVT method [7]

f_h is the ultrasonic vibration frequency (> 20 kHz); f_l is the low-vibration frequency; ϕ is the phase difference of the elliptical locus; V_f is the nominal cutting speed in µm/s; and t is time in seconds.

The tool tip relative velocity V_x(t) and V_y(t) can be obtained by calculating the derivation of the relative position x(t) and y(t) with respect to time, respectively [7].

$$V_{x} \left( t \right) = - a\pi f_{h} \cdot sin\left( {2\pi f_{h} \cdot t} \right) + V_{f}$$

(3)

$$V_{y} \left( t \right) = - b\pi f_{h} \cdot sin\left( {2\pi f_{h} \cdot t} \right) - B\pi f_{l} \cdot sin\left( {2\pi f_{l} \cdot t} \right)$$

(4)

The velocity angle that represents the angle of the tangential line of the tool tip trajectory with respect to the x-axis is defined as follows [7]:

$$\theta_{t} \left( t \right) = \tan^{ - 1} \left( {\frac{{V_{y} \left( t \right)}}{{V_{x} \left( t \right)}}} \right)$$

(5)

The magnitude of the transient cutting velocity of the tool tip in the Cartesian coordinate is given as follows:

$$V_{t} \left( t \right) = \sqrt {V_{x} \left( t \right)^{2} + V_{y} \left( t \right)^{2} }$$

(6)

2.1 Transient Shear Stress

The transient shear stress in the thin shear plane must be calculated to predict the transient cutting forces during the intermittent cutting process. Many researchers have used the Johnson–Cook (JC) flow stress model to predict the shear stress in the thin shear plane during the metal cutting process. The transient JC shear stress model is defined as follows [19]:

$${\text{JC}}\left( {\text{t}} \right) = \frac{1}{\sqrt 3 } \left[ {A + B\left( {\frac{\varepsilon }{\sqrt 3 }} \right)^{n} } \right]\left[ {1 + C \cdot ln\left( {\frac{{\dot{\varepsilon }}}{{\dot{\varepsilon }_{o} }}} \right)} \right]\left[ {1 - \left( {\frac{{T\left( t \right) - T_{r} }}{{T_{m} - T_{r} }}} \right)^{m} } \right]$$

(7)

where A, B, C, n, m are the material constants; T(t) is the transient temperature in the primary cutting zone, which is described in Sect. 2.1.1; T_r is the room temperature; T_m is the melting temperature of the workpiece; $\varepsilon$ is the equivalent plastic strain; $\dot{\varepsilon }$ is the equivalent plastic strain rate; $\dot{\varepsilon }_{o}$ is the absolute plastic strain rate. The values of the material constants for AISI 1045 are presented in Table 1.

Table 1 JC material constants for AISI 1045 [21]

2.1.1 Temperature in the Shear Plane

The temperature in the shear plane has to be known, such that the shear stress value in Eq. 7 can be obtained. The temperature rises in the thin shear plane depend on the given cutting energy. Because the shear plane is assumed as thin, Oxley’s model [20] is adopted to predict the temperature rise in the shear plane. Thus, the average temperature is shown in Eq. 8 [20].

$${\text{T}}\left( t \right) = T_{o} + \frac{{\left( {1 - \beta } \right)JC\left( t \right)\cos \alpha \left( t \right)}}{{\rho \cdot C_{p} \cdot \sin \phi_{s} \left( t \right) \cdot \cos \left( {\phi_{s} \left( t \right) - \alpha \left( t \right)} \right)}}$$

(8)

where T_o is the room temperature T_r before cutting, C_p is the specific heat of the workpiece, ρ is the mass density of the workpiece, and JC is the transient shear stress of Eq. 1. The transient shear angle ϕ_s(t) and rake angle α(t) are described in the Sect. 2.2.1.

The thermal properties of the workpiece AISI 1045 is presented in Table 2. β is defined as the generated energy fraction in the primary cutting zone during metal cutting, which is defined as [20]

$$\begin{array}{*{20}l} {\beta = 1 ;} \hfill & {\left( {R_{T} \cdot \tan \phi_{s} \left( t \right)} \right) < 0.04} \hfill \\ {\beta = 0.5 - 0.35 \cdot \log \left( {R_{T} \cdot \tan \phi_{s} \left( t \right)} \right);} \hfill & {0.04 \le \left( {R_{T} \cdot \tan \phi_{s} \left( t \right)} \right) \le 10} \hfill \\ {\beta = 0.3 - 0.15 \cdot \log \left( {R_{T} \cdot \tan \phi_{s} \left( t \right)} \right);} \hfill & {\left( {R_{T} \cdot \tan \phi_{s} \left( t \right)} \right) > 10} \hfill \\ \end{array}$$

(9)

where R_T is a non-dimensional thermal number that correlates with the cutting energy, and ϕ_s is the transient shear angle [20].

$$R_{T} = \frac{{\rho \cdot C_{p} \cdot V_{t} \left( t \right) \cdot TOC_{t} \left( t \right)}}{K}$$

(10)

where V_t is the magnitude of the transient cutting velocity of the tool tip (Eq. 6), TOC_t is the transient thickness of cut that was defined in Sect. 2.3, K is the thermal conductivity of the workpiece presented in Table 2, ρ is the mass density and C_p is the specific heat.

Table 2 Thermal properties of AISI 1045 [22]

As the shear stress and temperature in the thin shear plane are correlated to each other, the numerical solution (e.g., the Euler method) has been used to solve the transient JC shear stress in Eq. 7.

2.2 Downward and Upward Cutting

During the micro-dimple manufacturing process, the tool tip moves downward and upward for cutting. Figure 2 shows the downward and upward cutting. All cutting mechanics become transient procedures. In the TFEVT process, the cutting mechanics depend on time. As mentioned previously in point 4, all angles of the cutting mechanics are in degrees based on the velocity vector direction.

Fig. 2

Illustration of the downward and upward cutting in TFEVT method

2.2.1 Transient Rake and Shear Angle

Because the tool tip motion is always different relative to the machined surface, the rake angle becomes a transient value as defined in Eq. 11. Figure 3 illustrates the transient of the rake angle. At the beginning of the single cutting, the rake angle is always positive. Meanwhile, during climb cutting, the rake angle decreases gradually and its value is a large negative [23].

$$\begin{array}{*{20}l} {\upalpha\left( t \right) = \left| {\theta_{t} \left( t \right)} \right| - \alpha_{o} } \hfill & {{\text{if}}\;\theta_{t} \left( t \right) < 0} \hfill \\ {\upalpha\left( t \right) = - \alpha_{o} } \hfill & {{\text{if}}\;\theta_{t} \left( t \right) = 0} \hfill \\ {\upalpha\left( t \right) = - \left| {\theta_{t} \left( t \right) + \alpha_{o} } \right|} \hfill & {{\text{if}}\;\theta_{t} \left( t \right) > 0} \hfill \\ \end{array}$$

(11)

where α_o is the initial value of the negative rake angle, and θ_t(t) is the angle of the velocity vector (Eq. 5). In this study, negative α_o equals − 3°.

Fig. 3

Illustration of the transient rake angle

Cerniway’s hypothesis [9] was used in this study to predict the value of the transient shear angle. The transient shear angle was predicted according to the assumption of the bisector of the angle λ or the shear angle value is half of the λ value. λ is the angle measured from the velocity vector to the rake face of the tool (see Fig. 4). Thus, the transient shear angle is hypothesized as follows [9]:

$$\phi_{s} \left( t \right) = \frac{\pi }{4} - \left| {\frac{\alpha \left( t \right)}{2}} \right|$$

(12)

Fig. 4

Illustration of the transient shear angle

As hypothesized in Eq. 12, $\phi_{s} \left( t \right)$ depends on the value of the transient rake angle as described in Eq. 11.

2.2.2 Transient and Instant Cutting Time

Figure 2 illustrates the downward and upward cutting during the micro-dimple process using the TFEVT method. The tool tip cuts the workpiece elliptically in an intermittent manner. In Fig. 2, the hatched region is an area of cut in the single-cycle cutting. In a single-cycle cutting, the tool tip cuts the material at a starting point, A.

Subsequently, the tool tip passes from the starting point A to the ending point E and passes through point D during the single cutting cycle. After passing through point D, the transient thickness of cut (TOC_t) increases dramatically. The value of TOC_t is described in detail in the Sect. 2.3. Further, the terminology of the transient cutting time is shown in Table 3.

Table 3 Cutting time terminology (see Fig. 2)

The Newton–Raphson numerical method has been used to calculate the time at A. Because the tool tip at locations x and y are equal for time instants t′_A and t_A, therefore, Eq. 13 can be implemented to calculate times t′_A and t_A [24].

$$\begin{aligned} x\left( {t_{A}^{\prime } } \right) - x\left( {t_{A} } \right) &= 0 \\ y\left( {t_{A}^{\prime } } \right) - y\left( {t_{A} } \right) &= 0 \\ \end{aligned}$$

(13)

At the end of cutting, the cutting velocity is in the similar direction as the rake face surface. Thus, the cutting velocity angle value is (90° − α_o). The instant time t_F in the end cutting can be calculated using the Newton–Raphson numerical method and determined by Eq. 14.

$$\tan^{ - 1} \left( {\frac{{V_{y} \left( {t_{F} } \right)}}{{V_{x} \left( {t_{F} } \right)}}} \right) - \left( {\left( {\frac{\pi }{2}} \right) - \alpha_{o} } \right) = 0$$

(14)

The instant time t_E can be obtained by Eq. 15, where f_h is the ultrasonic vibration frequency.

$$t_{E} = t_{F} - \left( {\frac{1}{{f_{h} }}} \right)$$

(15)

The Newton–Raphson numerical method has also been used to calculate the exact time instant t_D, after obtaining the time instant t_E.

$$\left\{ {x\left( {t_{E} } \right) - x\left( {t_{D} } \right)} \right\} - \left\{ {\tan \alpha_{o} \left( {y\left( {t_{E} } \right) - y\left( {t_{D} } \right)} \right)} \right\} = 0$$

(16)

Similarly, Eq. 16 can be implemented to obtain the transient time t_P, where point P lies from point A to point E. Time t is given, which lies from point A to point D.

$$\left\{ {x\left( {t_{P} } \right) - x\left( t \right)} \right\} - \left\{ {\tan \alpha_{o} \left( {y\left( {t_{P} } \right) - y\left( t \right)} \right)} \right\} = 0$$

(17)

2.3 Transient Thickness of Cut

After the transient and instant times during a single-cycle cutting have been found, TOC_t can be easily determined. TOC_t is defined from the top workpiece surface to the point of T, as illustrated in Fig. 2. Thus, TOC_t is given by Eq. 18.

$$TOC_{t} \left( t \right) = \left\{ {\begin{array}{*{20}c} 0 & {} & {t< t_{A} , t > t_{E} } \\ {y\left( {t_{P} } \right) - y\left( t \right)} & {for} & {t_{A} \le t < t_{D} } \\ {H - y\left( t \right)} & {} & {t_{D} \le t \le t_{F} } \\ \end{array} } \right.$$

(18)

2.4 Transient Cutting Forces

Merchant’s model is adopted to determine the transient principle and thrust cutting force. Based on Merchant’s circle (Fig. 5), the transient shear force F_s on the shear plane is defined as follows [17]

$$F_{s} \left( t \right) = \frac{{JC\left( t \right) \cdot A_{t} \left( t \right)}}{{\sin \phi_{m} \left( t \right)}}$$

(19)

where JC is the Johnson–Cook of the transient shear stress as shown in Eq. 7, A_t is in which the area geometry of the rounded rake face surface is shown in Eq. 20, and ϕ_m is the Merchant’s transient shear angle which can be described as follows:

$$\begin{array}{*{20}l} {\phi_{m} = \phi_{s} - \theta_{t} } \hfill & {for\;\theta_{t} < 0^{ \circ } } \hfill \\ {\phi_{m} = \phi_{s} } \hfill & {for\;\theta_{t} = 0^{ \circ } } \hfill \\ {\phi_{m} = \phi_{s} + \theta_{t} } \hfill & {for\;\theta_{t} > 0^{ \circ } } \hfill \\ \end{array}$$

(20)

Fig. 5

Basic cutting forces vectors (Merchant’s Circle) [26]

The transient cutting area A_t can be described as follows [17]:

$$\begin{aligned} A_{t} \left( t \right) & = \left\{ {R_{n}^{2} \cdot cos^{ - 1} \left( {\frac{{R_{n} - TOC_{t} \left( t \right)}}{{R_{n} }}} \right)} \right\} \\ & \quad - \left\{ {\left( {R_{n} - TOC_{t} \left( t \right)} \right) \times \sqrt {2 \cdot R_{n} \cdot TOC_{t} \left( t \right) - TOC_{t} \left( t \right)^{2} } } \right\} \\ \end{aligned}$$

(21)

where R_n is the tool nose radius, and TOC_t is the transient thickness of cut as shown in Eq. 18. In this study, the tool nose radius R_n equals 400 µm of the standard triangular insert TCGW 110204 of polycrystalline diamond (PCD) which is manufactured by Taegu Tec Company.

According to the minimum energy principle which was proposed by Merchant, the relationship among the shear angle ϕ_s, the friction angle β(t), and the negative rake angle α(t) is defined as follows [25]:

$$\phi_{s} \left( t \right) = \frac{\pi }{4} - \frac{\beta \left( t \right)}{2} - \frac{\alpha \left( t \right)}{2}$$

(22)

Thus, the transient friction angle β from Eq. 22 can be defined as follows:

$$\beta \left( t \right) = \frac{\pi }{2} - 2\phi_{s} \left( t \right) - \alpha \left( t \right)$$

(23)

According to Fig. 5, the resultant force F_r can be defined in Eq. 24 as follows:

$$F_{r} \left( t \right) = \frac{{F_{s} \left( t \right)}}{{\cos \left( {\phi_{s} + \beta \left( t \right) + \alpha \left( t \right)} \right)}} = \frac{{JC\left( t \right) \cdot A_{t} \left( t \right)}}{{\sin \phi_{m} \cdot \cos \left( {\phi_{s} + \beta \left( t \right) + \alpha \left( t \right)} \right)}}$$

(24)

where F_s is the shear stress as given in Eq. 19. According to Fig. 5, the transient principle cutting force F_p and thrust cutting force F_t are defined by Eqs. 25 and 26, respectively.

$$F_{p} \left( t \right) = F_{r} \left( t \right) \cdot cos\left( {\beta \left( t \right) + \alpha \left( t \right)} \right)$$

(25)

$$F_{t} \left( t \right) = F_{r} \left( t \right) \cdot sin\left( {\beta \left( t \right) + \alpha \left( t \right)} \right)$$

(26)

By substituting the F_r term in Eq. 24 and the friction angle β term in Eq. 23 into Eqs. 25 and 26, the final equations of the transient principle cutting force F_p and thrust cutting force F_t are

$$F_{p} \left( t \right) = \frac{{JC\left( t \right) \cdot A_{t} \left( t \right) \cdot \cos \left( {\frac{\pi }{2} - 2\phi_{s} \left( t \right)} \right)}}{{\sin \phi_{m} \left( t \right) \cdot \cos \left( {\frac{\pi }{2} - \phi_{s} \left( t \right)} \right)}}$$

(27)

$$F_{t} \left( t \right) = \frac{{JC\left( t \right) \cdot A_{t} \left( t \right) \cdot \sin \left( {\frac{\pi }{2} - 2\phi_{s} \left( t \right)} \right)}}{{\sin \phi_{m} \left( t \right) \cdot \cos \left( {\frac{\pi }{2} - \phi_{s} \left( t \right)} \right)}}$$

(28)

where JC is the Johnson–Cook shear stress in Eq. 7, A_t is the transient cutting area in Eq. 20, ϕ_s is the hypothesized transient shear angle in Eq. 12, and ϕ_m is the transient shear angle in Eq. 20.

As the both cutting forces F_p and F_t are for a single-cycle cutting, the root mean square (rms) is used to obtain the average value of the cutting force during the micro-dimpling process.

$$\begin{aligned} F_{{p_{rms} }} \left( t \right) = & \mathop \sum \limits_{i = 0}^{n} \sqrt {\frac{1}{{t_{f} - t_{a} }}\mathop \smallint \limits_{{t_{a} }}^{{t_{f} }} \left[ {F_{p} \left( t \right)} \right]^{2} dt} \\ F_{{t_{rms} }} \left( t \right) = & \mathop \sum \limits_{i = 0}^{n} \sqrt {\frac{1}{{t_{f} - t_{a} }}\mathop \smallint \limits_{{t_{a} }}^{{t_{f} }} \left[ {F_{t} \left( t \right)} \right]^{2} dt} \\ \end{aligned}$$

(29)

where n is the number of cutting cycles during the micro-dimpling process.

3 Experimental Setup

A pre-machining using an end-mill tool with diameter of 10 mm (spindle speed is 12,000 rpm, depth of cut is 50 µm and feed rate is 50 mm/min) has been performed before the micro-dimpling process to obtain a planar surface and to minimize an error. Then, the micro-dimpling was performed using TFEVT device, where the neutral distance ∆d was set equal 5 µm, and the tool tip moves and vibrates with vibration frequencies as shown in Table 4.

Table 4 Experimental setup

The micro-dimpling using the TFEVT method has been performed using various the low-vibration frequencies of 10, 30, and 50 Hz, as shown in Table 4. The nominal cutting speed of 500 mm/min was used. An ultrasonic frequency of 24 kHz was applied. The maximum amplitude of 8 µm was used for the sinusoidal wave.

Figure 6 illustrates the TFEVT experimental setup. The TFEVT device used consists of the low-frequency displacement amplifier (LFDA) and the ultrasonic elliptical motion transducer (UEMT) [27], as shown in Fig. 6. A data acquisition (DAQ) NI 6251 model and an E.508 amplifier was used to provide the input signal of 1000 V to the piezo actuator (Physik Instrumente) P212.10, which is the primary actuator in the LFDA. Meanwhile, an ultrasonic generator (USG-110) was used to vibrate the UEMT using 300 V peak to peak.

Fig. 6

TFEVT experimental setup

During the micro-dimpling process, the cutting forces had been measured using the KISTLER dynamometer type 9272. During the data recording, an analog low pass filter of 100 Hz was used to reduce the noise due to the machine vibration. The frequency sampling is 100 kHz and the sample amount is 10,000 when recording the cutting forces. Later, the experimental data had were filtered digitally by the moving average technique in MATLAB.

4 Validation and Discussion

The cutting force simulations have been performed with the similar texturing parameter as presented in Table 4. Subsequently, the simulated cutting forces were compared with the experimental cutting forces. The transient time (such as t_a, t_f, t_e, t_d, etc.) was first calculated followed by the transient TOC_t (Eq. 18).

Figure 7 shows the TOC_t during the micro-dimple process with a vibration frequency of 10 Hz. A variation in the TOC_t value occurred from downward to upward cutting of the micro-dimple. In addition, in every single cutting, the TOC_t also varies from zero to a certain value, and subsequently goes to zero again. The zero value means the tool tip does not engage with the workpiece. The TOC_t achieves a maximum value when the tool tip reaches the bottom during the micro-dimpling process. The transient cutting area A_c is found after the TOC_t (Eq. 18) was calculated.

Fig. 7

Transient TOC in TFEVT at f_l = 10 Hz

Figure 8 shows the simulation of the hypothesized transient shear angle ϕ_s during the micro-dimpling process at a low-vibration frequency of 10 Hz. As shown, the transient shear angle decreases gradually from 58° to 0° for a single-cycle cutting. In the case of the zero shear angle, the F_P and F_T cutting force values become infinite owing to the sine term in the denominator of Eq. 27 and Eq. 28. Thus, when the transient shear angle is below 2°, the non-cutting process is assumed [9]; in other words, the F_P and F_T are set as zero.

Fig. 8

Transient shear angle ϕ_s in TFEVT at f_l = 10 Hz

The transient principle F_P (Eq. 27) and transient thrust F_T (Eq. 28) cutting force was calculated after A_c, JC, and ϕ_s have been found respectively. Figure 9 shows the simulation result of F_T and F_P during the micro-dimpling using a vibration frequency of 10 Hz. The enlarged view of the cutting forces is shown in Fig. 10. The cutting forces always achieve the maximum value at the bottom of the micro-dimple. In the case of the single-cycle cutting, at the beginning of cutting, the F_P is larger than the F_T (see Fig. 10). Then, F_P decreases gradually and F_T increases significantly. Finally, at the end cutting, the F_T is very larger than the F_P because the decreasing effect of the transient shear angle ϕ_s. This means that the tool tip does a small amount of cutting and does mostly a friction with the deformed chip at the time instant before the end cutting. Subsequently, the deformed chip is pulled easily owing to the effect of the friction at the moment before the end cutting.

Fig. 9

Transient F_P and F_T in TFEVT at f_l = 10 Hz (enlarge view in Fig. 10)

Fig. 10

Transient F_P and F_T in TFEVT at f_l = 10 Hz for a single of cutting

In our study, we did not find evidently a reversal effect of the thrust force where the previous studies [14] revealed which the thrust force becomes a negative value. Otherwise, based on experimental findings, the thrust force is significantly larger than the principle force during the micro-dimpling process. Therefore, the existed model [14] cannot be implemented in our study, thus a new analysis was developed. Because of the decreasing effect of the transient shear angle ϕ_s, the cutting force F_t is always greater than the cutting force F_p for a single cutting the micro-dimple using the TFEVT method.

Figures 11 and  12 show the cutting force comparison of F_P and F_T between simulations and experiment at f_l = 10 Hz and 30 Hz, respectively when the tool tip cuts a single micro-dimple. The simulation cutting forces have been root mean square corresponds with Eq. 29. The comparison result shows that the simulated cutting forces correspond sufficiently well with the experimental cutting forces.

Fig. 11

Cutting force comparison between experimental and simulation at f_l = 10 Hz

Fig. 12

Cutting force comparison between experimental and simulation at f_l = 30 Hz

During the micro-dimpling process, the cutting forces begin from zero and gradually increase as the tool tip performs downward cutting. The cutting forces achieve a maximum value when the tool tip is at the bottom of the micro-dimple. Subsequently, the cutting forces gradually decrease as the tool tip performs upward cutting. In the TFEVT process, the cutting force F_T is always greater than the cutting force F_P, owing to decrease in the transient shear angle for a single-cycle cutting. It should be noted that there is a difference between the simulation and experimental results as shown in Figs. 11 and 12. The first reason might be due to our model consideration of an ideal sharp tool tip. We have known that the tool edge radius effect is necessary to be considered in the micro cutting [28]. The second reason might be due to our model consideration of a hypothesized shear angle, in which the cutting force model could be improved by acknowledging the actual shear angle based on experiment result by measuring the chip thickness ratio [17].

Figure 13 shows the maximum cutting force comparison between the TFEVT and the CT method. The experimental data have been averaged for 5 samples. As shown in Fig. 13, the cutting forces in the TFEVT method is significantly lower than those in the CT method because of the transient cutting area (A_c). For the TFEVT experimental results, both cutting forces F_P and F_T decrease slightly by varying the low-vibration frequency. Meanwhile, the F_P and F_T simulation results show a constant trend when the low-vibration frequency was increased.

Fig. 13

Cutting forces comparison between TFEVT and CT method

5 Conclusion

Based on this study, the following conclusions can be drawn as:

• 1. The transient cutting force model has been proposed in this study and the transient cutting forces always depend on the transient shear stress in the thin shear plane, transient cutting area, transient shear angle, and transient velocity angle.

• 2. The comparison results show that the proposed transient cutting force model corresponds sufficiently well with the experimental results.

• 3. The cutting force F_T is always greater than the cutting force F_P for the cutting of a single micro-dimple, because of the decrease in the transient shear angle. In addition, the transient shear angle also depends on the transient rake angle.

• 4. The reduction in the cutting forces in the TFEVT might be due to the transient cutting area (A_c) that significantly reduces the cutting forces.