Cutting Force Model of Ultrasonic Elliptical Vibration-Assisted Helical Milling of SiCp/Al Composites

Abstract

SiC particle-reinforced Al metal matrix (SiCp/Al) composites are more and more widely used in the aerospace field due to their excellent properties, and the realization of high-quality drilling of SiCp/Al composites has an important impact on improving the performance of parts. In this paper, ultrasonic elliptical vibration-assisted helical milling (UEVHM) is applied to the machining of SiCp/Al composites. Firstly, the kinematic analysis of UEVHM is carried out, and then the cutting force model is established, which takes into account the interaction between particles and the cutting edge, and calculates the crushing force, pressing force, and debonding force of the particles. Finally, the UEVHM tests are conducted to verify the accuracy of the model and to analyze the influence of process parameters on the cutting force. It was found that the radial and axial forces decreased by 34% and 39%, respectively, when the spindle speed was increased from 2000 r/min to 10,000 r/min; the radial and axial forces increased by 200% and 172%, respectively, when the pitch increased from 0.1 mm to 0.4 mm; and the radial and axial forces increased by 29% and 69%, respectively, when the rotational speed increased from 30 r/min to 70 r/min. The maximum error between the cutting force model and the experimental values is 19.06%, which has a good accuracy. The research content of this paper can provide some guidance for the high-quality hole-making of SiCp/Al composites.

1. Introduction

With the development of science and technology, people’s demand for high-performance materials is getting higher and higher, and composite materials have also emerged, among which SiCp/Al composites, which have the advantages of wear resistance, a low coefficient of expansion, and high specific strength [1,2], are widely used in the aerospace field [3,4]. However, due to the high hardness of SiC, the machining process is prone to serious tool wear and machining defects. Especially in hole machining, stress concentration and fatigue cracks are easily generated at the holes, and the surface quality of the machined holes has a great influence on the life of the parts; so, the quality of hole machining will directly affect the service performance of the parts. How to suppress the hole damage by reducing the cutting force is a difficult problem to solve at this stage, and modeling and analyzing the cutting force are particularly important.

Ultrasonic-assisted machining is considered to be an effective method to improve material machining quality and reduce tool wear. Therefore, several scholars have investigated the ultrasound-assisted machining of SiCp/Al composites. The ultrasonically assisted scratching experiments found that the scratch force and surface damage can be reduced with the assistance of ultrasonic vibration [5,6,7]. Ultrasonic vibration-assisted machining of SiCp/Al composites can realize intermittent cutting, which reduces the cutting force, and it can also change the removal of SiC particles and reduce the machining damage [8,9].

Helical milling (HM) has been increasingly applied to the machining of composites and difficult-to-machine materials due to the advantages of a small axial force and good heat dissipation [10,11,12,13,14]. However, there are few studies on the application of ultrasonic elliptical vibration and helical milling to the processing of SiCp/Al composites. Some researchers have applied ultrasonic vibration-assisted helical milling to the hole-making of other difficult materials. Liu et al. [15] applied ultrasonic-assisted helical milling to the hole machining of carbon fiber-reinforced polymer (CFRP) composites, and found that ultrasonic vibration-assisted milling can reduce the cutting force and reduce the hole exit burr. Chen et al. [16] compared the quality of holes machined by helical milling and ultrasonic vibration helical milling (UVHM) for a titanium alloy and found that the hole diameter error and surface roughness of UVHM is less than HM, and the residual stress of UVHM is increased by 63.5% compared with HM.

The cutting force is an important factor affecting the machining quality, and the prediction of the cutting force can efficiently guide machining production, which is important for realizing high-quality machining; so, many scholars have studied the cutting force model. Liu et al. [17] considered the deformation resistance of the machining material and the friction between the tool and the material and established a SiCp/Al ultrasonic vibration-assisted scratch force model. The accuracy of the model was verified using experiments, and the maximum error between the experimental value and the predicted value was 33%. Wang et al. [18] calculated the cutting force during SiCp/Al turning machining from three deformation zones; considered shear, friction, and plow deformation forces; and put forward a prediction model for the cutting force based on the geometric and mechanical properties of the tool and the workpiece. Through the experimental verification, the average percentage prediction errors of the Fz, Fx, and Fy components were 1.93%, 6.20%, and 10.48%, respectively. Chang et al. [19] proposed an energy-based mechanistic modeling method for the drilling cutting force of SiCp/Al composites, which considered different ways of removing silicon carbide particles, and verified the accuracy of the model through experiments on materials with different particle diameters, where the average error of the drilling force model was 6.55%. Lu et al. [20] established a prediction model for the axial force and torque when grinding holes in SiCp/Al composites by taking into account the plastic deformation force of the matrix material, the friction between the abrasive particles and the workpiece material, and the particle removal force, and proposed a new method for calculating the thickness and cross-sectional area of the undeformed chips. A new method for calculating the thickness and cross-sectional area of undeformed chips was proposed. It was found that the prediction deviations of axial force and torque were 7.8% and 5.2%, respectively, through experimental validation. However, the above study did not take into account the removal mode of particles and the friction between particles and the front cutter surface.

At present, there are fewer studies on the ultrasonic elliptical vibration helical milling (UEVHM) of SiCp/Al composites at home and abroad, and there are fewer studies on its mechanical modeling. Therefore, in this paper, UEVHM is applied to the hole-making machining of SiCp/Al composites, the cutting-edge trajectory is modeled, and the effect of elliptical ultrasound on the interaction between the tool and the workpiece is analyzed. The cutting force model of UEVHM is established by considering the friction, particle crushing, pressing, debonding force, and material rebound force to explore its mechanical law, and the reliability of the model is verified by experiments.

2. Kinetic Analysis

The schematic diagram of UEVHM machining is shown in Figure 1. There are five kinds of movements: tool rotation around the spindle, tool rotation around the center of the hole, axial feed, and x and y axis ultrasonic vibrations. The interaction between the tool and the workpiece is complicated. Therefore, the modeling and analysis of the UEVHM trajectory can help to understand the cutting process and provide theoretical support for cutting force modeling.

The applied two-dimensional ultrasonic equations of motion are given in Equation (1),

$$\left\{\begin{array}{c}{x}_1(t)=A_x\sin (2\pi ft+{\theta }_0)\\ y_1(t)=A_y\sin (2\pi ft+{{\theta }_0}^{\prime })\end{array}\right.$$

(1)

where f is the ultrasonic vibration frequency, $A_x$ and $A_y$, respectively, are the x, and y direction ultrasonic vibration amplitudes, and ${\theta }_0$ and ${{\theta }_0}^{\prime }$ are the initial phases of ultrasonic vibration. The equation for the motion of the tool around the hole axis in the HM is shown in Equation (2), and the equation for the motion of a point on the cutting edge around the tool axis is shown in Equation (3).

$$\left\{\begin{array}{c}{x}_2(t)=e\cos ({\omega }_{g}t+{\theta }_1)=e\cos ({\displaystyle \frac{2\pi n_g}{60}}t+{\theta }_1)\\ y_2(t)=e\sin ({\omega }_{g}t+{\theta }_1)=e\sin ({\displaystyle \frac{2\pi n_g}{60}}t+{\theta }_1)\\ z(t)=v_{fa}\cdot t={\displaystyle \frac{S{n}_g}{60}}\cdot t\end{array}\right.$$

(2)

$$\left\{\begin{array}{c}{x}_3(t)=R_T\cos ({\omega }_{z}t+{\theta }_2)=R_T\cos ({\displaystyle \frac{2\pi n_z}{60}}t+{\theta }_2)\\ y_3(t)=R_T\sin ({\omega }_{z}t+{\theta }_2)=R_T\sin ({\displaystyle \frac{2\pi n_z}{60}}t+{\theta }_2)\end{array}\right.$$

(3)

where $n_z$ and ${\omega }_z$ are the spindle rotation speed and angular velocity, $n_g$ and ${\omega }_g$ are the tool revolution speed and angular velocity, ${\theta }_1$ and ${\theta }_2$ are the initial angle of tool rotation, S is the pitch of the helical trajectory, $R_T$ is the tool radius, and e is the eccentricity.

Therefore, the trajectory equation of a point on the side edge of UEVHM can be expressed as follows:

$$\left\{\begin{array}{c}x(t)=e\cos ({\displaystyle \frac{2\pi n_g}{60}}t+{\theta }_1)+A_x\sin (2\pi ft+{\theta }_0)+R_T\cos ({\displaystyle \frac{2\pi n_z}{60}}t+{\theta }_2)\\ y(t)=e\sin ({\displaystyle \frac{2\pi n_g}{60}}t+{\theta }_1)+A_y\sin (2\pi ft+{{\theta }_0}^{\prime })+R_T\sin ({\displaystyle \frac{2\pi n_z}{60}}t+{\theta }_2)\\ z(t)=v_{fa}\cdot t={\displaystyle \frac{S{n}_g}{60}}\cdot t\end{array}\right.$$

(4)

The cutting edge trajectory of UEVHM and helical milling (HM) are shown in the Figure 2. Figure 2a shows an overall photograph of the cutting edge trajectory, and Figure 2b,c shows the local magnification. As in Figure 2c, the red curve is the cutting edge trajectory of UEVHM and the green curve is the cutting edge trajectory of HM. It is known that the cutting edge can realize intermittent cutting under the action of ultrasonic elliptical vibration.

The tool helical path can be decomposed into the axial feed speed $v_{fa}$ and the tangential feed speed $v_{ft}$. To carry out the modeling and analysis of undeformed chip and cutting force, it is necessary to calculate the axial feed per tooth $f_{za}$ and the tangential feed per tooth $f_{zt}$. These are given by Equations (5) and (6), respectively:

$$f_{za}={\displaystyle \frac{a_p{n}_g}{n_{z}N}}={\displaystyle \frac{v_{fa}}{n_{z}N}}$$

(5)

$$f_{zt}={\displaystyle \frac{f_{za}}{\tan \alpha }}={\displaystyle \frac{2\pi e{n}_g}{n_{z}N}}={\displaystyle \frac{v_{ft}}{n_{z}N}}$$

(6)

where N is the number of tool teeth and $\alpha$ is the helix angle of the cutting edge trajectory.

The speed generated by ultrasonic vibration will change the cutting speed and feed speed. As shown in Figure 3, the tangential feed speed $v_{ft-u}$ and cutting speed $v_{c-u}$ of UEVHM are calculated as follows:

$$v_{ft-u}=v_{ft}+v_{x-u}\sin \delta +v_{y-u}\cos \delta$$

(7)

$$v_{c-u}=v_c+v_{x-u}\sin \delta +v_{y-u}\cos \delta$$

(8)

where $v_{x-u}$ and $v_{y-u}$ are the vibration velocities in the x and y directions, respectively, which are calculated by Equation (9). $\delta$ is the angle of the tool center turning along the helical track, which is calculated by Equation (10).

$$\left\{\begin{array}{l}{v}_{x-u}=2\pi f{A}_x\cos (2\pi ft+{\theta }_0)\hfill \\ v_{y-u}=-2\pi f{A}_y\sin (2\pi ft+{{\theta }_0}^{\prime })\hfill \end{array}\right.$$

(9)

$$\delta ={\omega }_{g}t$$

(10)

Due to the changes in the cutting speed and feed speed, the concepts of equivalent spindle speed $n_{ze}$ and equivalent revolution speed $n_{ge}$ are introduced, which are calculated by Equations (11) and (12).

$$n_{ze}={\displaystyle \frac{v_{c-u}}{2\pi R_T}}$$

(11)

$$n_{ge}={\displaystyle \frac{v_{ft-u}}{2\pi e}}$$

(12)

Therefore, the feed per tooth $f_{zt-u}$ of the side blade in UEVHM is calculated as follows:

$$f_{zt-u}={\displaystyle \frac{v_{ft-u}}{n_{ze}N}}$$

(13)

Therefore, the undeformed chip thickness $h_{cp-s}$ of the side edge under ultrasonic elliptical vibration is calculated as follows:

$$h_{cp-s}=f_{zt-u}\sin \phi$$

(14)

The undeformed chip thickness $h_{cp-b}$ of the bottom edge is calculated as follows:

$$h_{cp-b}=f_{za-u}={\displaystyle \frac{a_p\cdot n_{ge}}{n_{ze}\cdot N}}$$

(15)

3. Cutting Force Model

3.1. Force on the Tool Rake Face

The friction between the rake face and the chip is an important source of the cutting force. The tool–chip contact state of SiCp/Al composites is shown in Figure 4. The existence of SiC particles causes three kinds of friction in the contact area, which are the sliding friction $F_{fm}$ between the rake face and the aluminum matrix, the sliding friction $F_{2-body}$ between the rake face and the particles, and the three-body rolling friction $F_{3-body}$ between the rake face-chip-particle. Therefore, the total friction is calculated as follows:

$$F_f=F_{fm}+F_{2-body}+F_{3-body}$$

(16)

3.1.1. Sliding Friction between the Rake Face and Matrix

The contact area between the rake face and the aluminum matrix will produce sliding friction. The sliding friction between the tool and the matrix can be expressed as follows:

$$F_{fm}={\tau }_c(L_{tc}{l}_{ce}-A_{SiC})$$

(17)

where $l_{ce}$ is the length of the cutting edge involved in cutting, $L_{tc}$ is the knife–chip contact length calculated from Equation (18), ${\tau }_c$ is the average shear stress on the knife–chip contact surface calculated from Equation (19), and $A_{SiC}$ is the area of SiC particles in the knife–chip contact area calculated from Equation (20).

$$L_{tc}=h_{cp}\cdot {\varsigma }^{1.5}$$

(18)

$${\tau }_c=0.28{\sigma }_R$$

(19)

$$A_{SiC}=\pi {r_g}^2{N}_p\varphi$$

(20)

In the equations, $h_{cp}$ is the undeformed chip thickness, $\varsigma$ is the chip compression ratio, ${\sigma }_R$ is the ultimate tensile strength of the aluminum matrix, and $\varphi$ is the percentage of particles involved in the wear of the tool–chip section. According to the particle size, $\varphi$ is taken as 50% [21]. $r_g$ is the radius of the groove formed by the particles embedded in the rake face, which is calculated by Equation (21). As shown in Figure 5, ${\delta }_0$ is the depth of the groove, calculated by Equation (22).

$$r_g=\sqrt{R^2-{(R-{\delta }_0)}^2}$$

(21)

$${\delta }_0={({\displaystyle \frac{9\pi }{4}})}^2{({\displaystyle \frac{{\sigma }_{y-tool}}{E^{*}}})}^{2}R$$

(22)

In the equations, R is the radius of the particle, ${\sigma }_{y-tool}$ is the yield strength of the tool material, and $E^{*}$ is the elastic modulus of the composite material, which is calculated by Equation (23):

$${\displaystyle \frac{1}{E^{*}}}={\displaystyle \frac{{(1-{\upsilon }_{tool})}^2}{E_{tool}}}+{\displaystyle \frac{{(1-{\upsilon }_p)}^2}{E_p}}$$

(23)

where $E_{tool}$ and $E_p$ are the elastic modulus of the tool and particle, respectively, and ${\upsilon }_{tool}$ and ${\upsilon }_p$ are the Poisson ratio of the tool and particle, respectively.

3.1.2. Sliding Friction between the Rake Face and Particle

To calculate the two-body sliding friction force in the tool–chip contact area, the number of particles in the contact area needs to be calculated first. The area of the contact area of the knife chip is calculated as follows:

$$A_{tc}=L_{tc}\cdot l_{ce}$$

(24)

Therefore, the number of SiC particles in the contact area is calculated as follows:

$$N_p={\displaystyle \frac{V_p\cdot A_{tc}}{\pi R^2}}$$

(25)

The sliding friction between the tool and the particles can be expressed as follows:

$$F_{f-2body}=3{\sigma }_{y-tool}\cdot N_p\cdot A_i\cdot \varphi$$

(26)

In the equations, $A_i$ is the area of the embedded rake face, as shown in Figure 5. $A_i$ can be calculated from Equation (27) based on the particle radius R and the contact angle ${\theta }_p$:

$$A_i={\displaystyle \frac{R}{2}}\left[{\displaystyle \frac{\pi }{180}}(2{\theta }_p)-\sin (2{\theta }_p)\right]$$

(27)

$${\theta }_p=\arcsin {\displaystyle \frac{r_g}{R}}$$

(28)

3.1.3. Three-Body Rolling Friction of the Rake Face, Chip, and Particle

The rake face–chip–particle three-body rolling friction can be expressed as follows:

$$F_{f-3body}=\mu F_N$$

(29)

In the equation, $\mu$ is the friction factor, calculated by Equation (30), $F_N$ is the total normal force generated by multiple particles, and the normal force in the tool–chip contact area is composed of the normal force on many particles, calculated by Equation (31):

$$\mu ={\displaystyle \frac{k_{tool}}{\pi \cdot H_t}}{({\displaystyle \frac{2R}{r_g}})}^2\cdot \left\{1-{\left[1-{({\displaystyle \frac{r_g}{R}})}^2\right]}^{{\displaystyle \frac{1}{2}}}\right\}$$

(30)

$$F_N=F_{N1}\cdot N_p$$

(31)

where $k_{tool}$ is the shear stress of the tool material, calculated by Equation (32), and $F_{N1}$ is the normal force of a single SiC particle, calculated by Equation (33).

$$k_{tool}={\displaystyle \frac{{\sigma }_{y-tool}}{2}}={\displaystyle \frac{H_t}{6}}$$

(32)

$$F_{N1}=2.9\pi R{\sigma }_{y-tool}{\delta }_0$$

(33)

In summary, the total friction force is calculated as follows:

$$F_f={\tau }_c(L_{tc}{l}_{ce}-A_{SiC})+3{\sigma }_{y-tool}\cdot N_p\cdot A_i\cdot \varphi +\mu \cdot F_N$$

(34)

3.2. Force on the Cutting Edge

When processing SiCp/Al composites, SiC particles can be pressed into an aluminum matrix, broken, and debonded. When the cutting edge is located within the upper critical height and the lower critical height ${\delta }_0$, the particles are pressed in and pulled out, respectively. When the cutting edge is located within the middle $2(R-{\delta }_0)$ height, the particles break, as shown in Figure 6. The probability of the three removal methods is as follows [17]:

$$P_e={\displaystyle \frac{{\delta }_0}{2R}}$$

(35)

$$P_f={\displaystyle \frac{R-{\delta }_0}{R}}$$

(36)

$$P_d={\displaystyle \frac{{\delta }_0}{2R}}=P_e$$

(37)

where $P_e$ is the probability of particle pressing in, $P_f$ is the probability of particle breaking, and $P_d$ is the probability of particle pulling out.

The number of particles in contact with the cutting edge can be calculated from the knife–chip contact length and particle radius, as shown in Equation (38):

$$N_{pc}={\displaystyle \frac{l_c}{2R}}$$

(38)

Therefore, the particle crushing force can be expressed as follows:

$$F_{pf}={\sigma }_{SiC}\cdot S_{pf}\cdot N_{pc}\cdot P_f$$

(39)

where $S_{pf}$ is the average broken area of a single SiC particle, calculated from Equation (40). The volume of silicon carbide fracture is equivalent to a cylinder of the same height, and the cross-sectional area of the cylinder is taken as the average area of SiC fracture.

$$S_{pf}={\displaystyle \frac{\pi \left[2R^3-{{\delta }_0}^2(3R-{\delta }_0)\right]}{3(R-{\delta }_0)}}$$

(40)

The particle pressing force can be expressed as follows:

$$F_{pe}={\sigma }_R\cdot S_{pe}\cdot N_{pc}\cdot P_e$$

(41)

where $S_{pe}$ is the average pressed area of a single particle calculated from Equation (42).

$$\begin{array}{ll}{S}_{pe}& =4\pi R^2-{\displaystyle \frac{1}{2}}{\displaystyle {\int }_{\theta }^{\frac{\pi }{2}}2\pi rRd\theta }=4\pi R^2-{\displaystyle {\int }_{\theta }^{\frac{\pi }{2}}\pi R^2\cos \theta d\theta }\\ & =4\pi R^2-\pi R^2(1-\sin \theta )=4\pi R^2-\pi R{\delta }_0\end{array}$$

(42)

The debonding force of a single particle is calculated as follows [22]:

$$F_{pd1}={\displaystyle {\int }_1^{2\pi R}(\frac{1-{{\upsilon }_{SiC}}^2}{E_{SiC}}})\pi {{\sigma }_{SiC}}^{2}awda$$

(43)

In the equation, w is the crack width, which we approximately take as the particle radius, and a is the crack length. We assume that the crack initiation is 1 μm, and the crack propagation length at complete debonding is $2\pi R$ [23]. The total debonding force of all particles in contact with the cutting edge is calculated as follows:

$$F_{pd}=N_{pc}\cdot P_d\cdot {\displaystyle {\int }_1^{2\pi r_g}(\frac{1-{{\upsilon }_{SiC}}^2}{E_{SiC}}})\pi {{\sigma }_{SiC}}^{2}awda$$

(44)

3.3. Force on the Tool Flank Face

The force of the flank face is mainly the normal force generated by the spring back of the aluminum substrate on the machined surface. As shown in Figure 7, due to the good ductility of the aluminum matrix, a rebound will occur, ignoring the very small rebound generated by the particles. Therefore, the rebound area $A_r$ is calculated as follows:

$$A_r=l_{ce}\cdot (R_c\cdot \sin \gamma +{\displaystyle \frac{s}{\tan {\alpha }_0}})\cdot V_p$$

(45)

In the equation, ${\alpha }_0$ is the tool back angle, and s is the spring back height of the aluminum matrix. It is calculated from Equation (48).

Therefore, the rebound force can be expressed as follows:

$$F_{re}={\sigma }_{fs}\cdot A_r$$

(46)

In the equation, ${\sigma }_{fs}$ is the average stress at the flank face, which is calculated by Equation (47):

$${\sigma }_{fs}=k_1\cdot H_v\sqrt{{\displaystyle \frac{H_v}{E^{*}}}}$$

(47)

$$s=k_2\cdot R_c\cdot {\displaystyle \frac{H_v}{E^{*}}}$$

(48)

k₁ and k₂ are 4.1 and 43 [23], respectively, and $H_v$ is the Vickers hardness of the composite. Combined with the above equation, the force generated by the spring back of the machined surface can be expressed as follows:

$$F_{re}=k_1\cdot H_v\sqrt{{\displaystyle \frac{H_v}{E^{*}}}}\cdot l_c\cdot (R_c\cdot \sin \gamma +{\displaystyle \frac{k_2\cdot R_c\cdot H_v}{E^{*}\cdot \tan {\alpha }_0}})\cdot V_p$$

(49)

3.4. Radial and Axial Forces

Since the two cutting edges at the bottom of the tool produce two radial forces of equal size and opposite direction, the radial force comes from the side edge and is substituted into the cutting edge length $l_{ce-s}$ of the side edge participating in the cutting, as shown in Equation (50):

$$l_{ce-s}={\displaystyle \frac{S}{{180}^{\circ }}}\phi$$

(50)

In the equation, $\phi$ is the rotation angle of a single cutting edge from entering the cutting to leaving the cutting around the spindle. The radial resultant force can be expressed as follows:

$$F_R=\sqrt{{(F_f+F_{re}+F_{pe})}^2+{(F_{pd}+F_{pf})}^2}$$

(51)

The axial force mainly comes from the bottom edge. Substitute the length $l_{ce-b}$ of the bottom edge to participate in the cutting in Equation (52) as follows:

$$l_{ce-b}=R_t$$

(52)

Therefore, the axial force can be expressed as follows:

$$F_A=2(F_f+F_{re}+F_{pe})$$

(53)

4. Experimental Procedures

The UEVHM test setup is shown in Figure 8. The workpiece material is a 65% volume fraction SiCp/Al composite material, and the specimen size is 100 mm × 80 mm × 6 mm. Workpiece purchased from Henan Han Yin Optoelectronics Technology Co., Ltd., Luoyang City, Henan Province. The aluminum matrix is 2024 aluminum alloy and the average particle size of SiC is 20 μm. The test was carried out on the TC500R vertical drilling and tapping center produced by Shenyang Machine Tool Works. The amplitude was measured using a laser vibrometer, as shown in Figure 8a, and the power of the ultrasonic generator was adjusted to ensure that the amplitude reached the desired value. Due to the high hardness of SiC, the SiCp/Al composites were machined using a polycrystalline diamond (PCD) tool with a diameter of 6 mm and a double-flute straight flute end mill, as shown in Figure 8c. The ultrasonic generator used was purchased from Hangzhou Chenrong Ultrasonic Equipment Co., Ltd., Hangzhou City, Zhejiang Province, model JZT2, with a power of 500 W, a frequency range of 15–60 kHz, and an adjustable step of 0.1 kHz.

The processing parameters are shown in

Table 1. Machining parameters.

Test Number	Spindle Speed (r/min)	Pitch (mm)	Revolution Speed (r/min)	X Amplitude (μm)	Y Amplitude (μm)
1	2000	0.2	50	4	4
2	5000	0.2	50	4	4
3	7000	0.2	50	4	4
4	10,000	0.2	50	4	4
5	7000	0.1	50	4	4
6	7000	0.3	50	4	4
7	7000	0.4	50	4	4
8	7000	0.2	30	4	4
9	7000	0.2	40	4	4
10	7000	0.2	60	4	4

, and the cutting force is measured using a Kistler dynamometer.

5. Results and Discussion

The cutting force in the stable cutting area during the machining process is selected, and the average is calculated as the actual value of the cutting force. The curve of the cutting force changing with the spindle speed is shown in Figure 9. The cutting force decreases with the increase in the spindle speed. When the spindle speed increases from 2000 r/min to 10,000 r/min, the radial force and axial force decrease by 34% and 39%, respectively. From the basic theory of cutting, the thickness of undeformed chip decreases with the increase in the spindle speed. The main sources of cutting force are material deformation resistance and friction between the tool and the workpiece. The decrease in undeformed chip thickness reduces the contact length between tool and workpiece, thus reducing the friction force, and so the cutting force decreases. When the spindle speed increases from 2000 r/min to 4000 r/min, the cutting force changes most obviously. This is because the cutting speed has a great influence on the ultrasonic vibration. When the spindle speed is 7000 r/min and 10,000 r/min, the cutting speed is very high. The separation time between the cutting edge and the workpiece is reduced within the same cutting distance, which is not conducive to reducing the cutting force; therefore, the reduction of the cutting force is smaller.

As shown in Figure 10, as the pitch increases from 0.1 mm to 0.4 mm, the cutting force increases gradually, the radial and axial forces increase by 200% and 172%, respectively, and the effect of pitch change on the cutting force is very significant. When the pitch increases from 0.1 mm to 0.4 mm, the axial feed rate increases, which will increase the contact length between the side edge and the workpiece and the undeformed chip thickness of the bottom edge. The contact thickness between the tool and the workpiece increases exponentially. The cutting deformation resistance of the processed material to be overcome by the tool increases. Meanwhile, the friction between the workpiece and the tool increases, which leads to an increase in the cutting force. The cutting force in the machining process comes from the interaction between the tool and the workpiece. As the pitch increases, the tool will have more contact with the SiC particles, which will also increase the cutting force.

As shown in Figure 11, when the revolution speed increases from 30 r/min to 70 r/min, the radial force and axial force increase by 29% and 69%, respectively. When the revolution speed increases, the tangential feed speed and the axial feed speed will increase. Because the speed limit of ultrasonic vibration is much larger than the tangential feed speed, the increase in the tangential feed speed can still realize the separation and re-contact of the tool and the material. However, the bottom edge still contacts with the workpiece at all times, and the radial force mainly comes from the side edge. Therefore, compared with the axial force, the change in the revolution speed has little effect on the radial force. The increase in the revolution speed will increase the undeformed chip thickness of the side edge and the bottom edge, and the cutting distance of the tool rotation will increase, which will increase the material deformation resistance. Additionally, the cutting edge will contact more SiC particles, and so the cutting force increases. However, compared with the influence of the pitch change on the cutting force, the change in the revolution speed has less influence on the cutting force.

Table 2. Comparison of the experimental values and predicted values of the cutting force.

No.	Spindle Speed (r/min)	Pitch (mm)	Revolut-ion Speed (r/min)	FR (N)		FZ (N)		Error (%)
				Predicted Value	Experime-ntal Value	Predicted Value	Experime-ntal Value	FR	FZ
1	2000	0.2	50	38.14	40.22	55.91	61.21	−5.17	−8.66
2	5000	0.2	50	31.95	30.36	44.15	46.22	5.23	−4.48
3	7000	0.2	50	28.98	26.41	41.84	39.21	9.73	6.71
4	10,000	0.2	50	25.25	21.71	34.04	30.23	16.3	12.6
5	7000	0.1	50	19.64	22.36	22.08	25.74	−12.16	−14.2
6	7000	0.3	50	49.64	55.65	50.11	47.21	−10.80	6.14
7	7000	0.4	50	60.12	63.82	59.75	57.03	−5.80	4.77
8	7000	0.2	30	23.74	19.94	32.84	29.71	19.06	10.54
9	7000	0.2	40	26.32	22.98	39.06	34.16	14.53	14.34
10	7000	0.2	60	31.13	36.20	56.17	48.19	−14.00	16.56

lists the error between the predicted value and the experimental value of the cutting force model in UEVHM under different process parameters, and the maximum error is 19.06%. Since the broken particles in the processing of SiCp/Al composites will cause more uncertainty in the processing process, it is considered that the error is within a reasonable range. The experimental results show that the predicted values of the model are in good agreement with the experimental values, and the model can predict the force in the UEVHM process of SiCp/Al composites well.

6. Conclusions

In this paper, the cutting force model in the UEVHM process is established, and the validity of the model is verified by experiments. The influence of process parameters on cutting force is analyzed. The main conclusions are as follows:

(1)

The cutting edge motion trajectory model of UEVHM was established, and the comparison of the motion trajectories of HM and UEVHM revealed that UEVHM could realize intermittent cutting. In addition, a theoretical model of the maximum undeformed cutting thickness for spiral milling considering the effect of ultrasonic elliptical vibration was established.

(2)

An axial and radial helical milling force model that considers the three removal methods of particles and the effect of SiC particles on the friction of the front face, as well as the rebound force of the back face, was developed. The accuracy of the axial and radial force model was verified by tests, and the results showed that the model had a good accuracy with a maximum error of 19.06%.

(3)

The ultrasonic elliptical vibration helical milling force increases with an increase in the spindle speed and decreases with increases in the pitch and revolution speed. The change in pitch has the most obvious influence on the cutting force. A larger cutting speed has no obvious effect on reducing the cutting force. Therefore, the revolution speed can be appropriately increased in the UEVHM process, which can improve the processing efficiency while maintaining a small cutting force.