1. Introduction
Titanium alloy has a number of excellent properties, such as corrosion resistance, high strength and good heat resistance, and is widely used in medical, aerospace and other fields [
1]. It is considered a typical difficult-to-cut material due to its chemical, physical and mechanical properties; for instance, the cutting temperature and cutting force is high, the friction force is large, and tool wear is serious [
2,
3]. Currently, a number of difficulties and challenges still exist in the traditional processing technologies (turning, grinding and milling, etc.) for such difficult-to-cut materials, particularly while maintaining high quality, high efficiency and low cost [
4,
5].
Ultrasonic-assisted machining technology is a non-traditional machining technology. It applies vibrations of an ultrasonic frequency to the tool or workpiece along a certain direction. Compared with traditional cutting technologies, such as turning, grinding and milling, ultrasonic machining technology has obvious technological advantages, including a small cutting force, low cutting temperature, high surface quality and high machining precision [
6,
7,
8].
Ultrasonic vibration-assisted milling technology started relatively recently, with its literature mainly focusing on the early 21st century. At present, research on ultrasonic vibration-assisted milling is in the developing stages. In 2006, Chern and Chang [
9] studied the machinability of ultrasonic vibration-assisted micro-milling of aluminum alloy, and found that vibration milling meaningfully improved the surface quality. This was the first time that ultrasonic vibration was introduced into milling. Shen et al. [
10] carried out a detailed study on ultrasonic vibration-assisted milling of aluminum alloy. They found that ultrasonic vibration-assisted milling transformed the continuous cutting process into discontinuous differential cutting. In addition, the cutting process produced a cutting force similar to that of the pulse type, effectively reducing the average cutting force. Zarchi et al. [
11] studied ultrasonic vibration-assisted milling of AISI 420 stainless steel and found that for a small feed rate, it reduced the cutting force in up milling, while during a larger feed rate, it reduced the cutting force in down milling. Zhao et al. [
12] investigated the cutting force differences between rotary ultrasonic machining and conventional diamond side grinding, drilling and face grinding on K9 glass, and found that rotary ultrasonic machining reduced the cutting force to a certain extent.
However, in one-dimensional ultrasonic-assisted machining, with the necessary high-frequency vibration of the tool, the tool flank generates extrusion friction with the machined surface. This is seriously harmful to the life of the tool and the machined surface quality [
13,
14], and limits further application of ultrasonic vibration cutting technology. On the contrary, in longitudinal-torsional ultrasonic vibration-assisted milling (LTUM), due to the cutting edge moving in three-dimensional space, extrusion friction between the tool flank and the machined surface is effectively avoided. In addition, the shear angle is increased and the average cutting force and cutting temperature in processing are (further) reduced [
15,
16], which is important in anti-fatigue manufacturing of high performance components.
An accurate prediction of cutting force is conducive to the evaluation of machine tool power, main bearing pressure, parts deformation, control of processing quality, and so on. At present, the most commonly used methods for predicting milling force are as follows: Micro-element model, finite element model, multiple regression analysis model and artificial neural network prediction model.
Liu et al. [
17] conducted verified experiments and presented a mechanistic model for cutting force in rotary ultrasonic machining of brittle materials, and found that the model predicted cutting force with high precision. Zhang et al. [
18] developed a mechanistic model to predict the cutting force in rotary ultrasonic drilling of brittle materials, showing that the model could be successfully applied to evaluate cutting force. Xiao et al. [
19] developed a theoretical model of cutting force both in the axial and feed directions in rotary ultrasonic machining. Yuan et al. [
20], Cong et al. [
21], and Wang et al. [
22] each presented a predictive model for cutting force, wherein their calculated results were subsequently verified by experimentation.
In the literature, there have been numerous valuable researches into cutting force; however, in terms of the outstanding advantages of LTUM (in particular, the tool as a vibration carrier), few of these investigations have been in relation to milling of difficult-to-cut materials, such as titanium alloys. Therefore, in the present work, in order to meet the compressive stress and anti-fatigue manufacturing requirements of titanium alloy Ti-6Al-4V, an LTUM machining method (with the tool as a vibration carrier) was developed. Models of the cutting edge trajectory and undeformed chip thickness (UCT) were also constructed. In addition, a cutting force model of LTUM was established and then verified through a series of experiments. Based on this established model, the influence of several parameters on cutting force was analyzed.
2. Materials and Methods
2.1. Trajectory Model of Cutting Edge in LTUM
The principles of LTUM are shown in
Figure 1. The cutting edge trajectory employed in LTUM makes it very different to traditional milling (TM), as the trajectory used has a direct impact on the cutting force, as well as other processing procedures and results.
For TM, the cutting edge trajectory equation (Equation (1)) is as follows:
where v
f is feed rate, R
t is tool radius, and n is spindle speed.
From this, an equation (Equation (2)) concerning the cutting edge trajectory for LTUM may be established:
where
f is ultrasonic frequency, A
l is longitudinal amplitude in LTUM, and ω
n-t is the actual rotation angle of the tool, obtained from Equation (3).
where, ω
l-t is torsional angle, obtained from Equation (4).
where, A
t is torsional amplitude and φ
l-t is the phase difference for the longitudinal-torsional vibration.
Based on Equations (1) and (2), the trajectories of the cutting edges in LTUM and TM are illustrated in
Figure 2.
In
Figure 2, A1, A2,…, An points represent the starting separation positions between the tool and workpiece, while B1, B2,…, Bn points represent the starting contact points, called the feature points. It can be seen that the cutting edge moves along the blue line in LTUM, and that the tool–workpiece undergoes periodic changes of separation-contact-separation with ultrasonic vibration. This caused a large difference from the cutting edge trajectory of TM (red line).
According to the solution model of feature points, the separation and contact points of LTUM were able to be calculated, as shown in
Figure 3. In the figure, A1, A2, and A3 are the separation points between the tool and workpiece, and B1, B2 and B3 are the contact points. In the process, the cutting edge is separated from the workpiece from point A1 to point B1, and contacting the workpiece from point B1 to point A2.
Thus, we concluded that for LTUM, the tool and workpiece exhibit periodic separation, while at the same time perform a cutting edge motion in 3D space. This significantly avoids the extrusion friction between the tool flank and machined surface, thereby reducing the cutting force and improving the quality of the machined surface.
2.2. Undeformed Chip Thickness Model of LTUM
Undeformed chip thickness (UCT) greatly influences both the cutting force and machining results; however, compared with TM, UCT is more complicated in LTUM. As such, it is helpful to establish a UCT model of LTUM to reveal the mechanism of the cutting force.
As shown in
Figure 1, a tool coordinate system was established, wherein the feed direction was represented by the
x axis, the width of cut direction was represented by the
y axis, the spindle axis was represented by the
z axis, and the origin was set on the tool rotation center. From this, a UCT model could be established, as shown in
Figure 4.
At any time, t, the position of the j-th teeth is given by the solid line in
Figure 4. The angular displacement of the tool teeth and the Y
t axis is expressed in Equation (5):
From this, UCT hj(t) at time t is equal to the distance between the position Q of the j-th teeth and the position P of the (j−1)-th teeth at the time (t−ṷj(t)), where ṷj(t) is the time delay between the j-th teeth and (j−1)-th teeth.
At time
t, the Q point coordinate value may be expressed in Equation (6):
In Equation (6), is the coordinates of a moving coordinate system origin in the fixed coordinate system XOY, where is the coordinates of the tool center in the moving coordinate system.
At time (t−ṷ
j(t)), the point P coordinate may be calculated from Equation (7):
P point coordinate may also be calculated using Equation (8):
Obviously, Equations (7) and (8) are equal, therefore,
From Equations (7)–(10):
where
As a result, UCT may be calculated using Equation (13):
and
If the tool moves uniformly with feed rate, the UCT h
j(t) is composed of a static cutting thickness h
js and a dynamic cutting thickness h
jd. Thus:
In actual fact, for LTUM, the cutting edge trajectory is similar to sub-cycloid motion. The cutting edge generates sub-cycloid motion with a longer period of spindle rotation and a shorter period of ultrasonic vibration.
Thus, at time t, UCT h
jlt(t) of LTUM is given by Equation (16):
2.3. Cutting Force Model of LTUM
On the basis of Merchant’s [
23] cutting force model, we transformed LTUM into an oblique cutting model, as shown in
Figure 5. The geometric relationship of this cutting force model is shown in
Figure 6.
According to Ref. [
24], the relationship between the chip flow direction angle and the inclination angle (helix angle) is:
where β
sf is the friction angle and the direction of cutting speed and force is determined by ϕ
sn, ϕ
si, θ
sn, θ
si.
Then, the normal force F
n may be obtained using:
where Fx and Fy represent the cutting forces in the feed direction and the width of cut direction, respectively.
A non-linear cutting force model [
25] is adopted. For any teeth, j, the instantaneous cutting forces are related to the UCT, as in Equation (21):
Then, Fx and Fy may be solved by applying Equation (22):
where
is the window function. To judge whether or not the teeth are cutting in, Equation (23) is applied:
where Φ
st, Φ
ex is the cutting-in and out angle.
From Equations (21)–(23), cutting force at any time is solved using Equation (24):
where Kt, Kr are cutting force coefficients, calculated by Equation (25):