An improved semi-analytical approach for modeling of process damping in orthogonal cutting considering cutting edge radius

Abstract

Process damping generated between the tool flank face and the wavy finish workpiece surface has a non-negligible effect on cutting dynamics and chatter stability, especially at low cutting speeds, resulting in higher stability limits. In modeling of process damping, the calculation of extruded volume is one of the most critical challenges, especially in machining with honed tools due to the complex and time-variable contact condition between the arc cutting edge and the finite amplitude wave surface. In this study, a semi-analytical method with high computational efficiency is proposed to calculate the extruded volume in cutting with honed tools. Based on this method, we construct the stability lobes under the condition of finite vibration amplitude accurately and efficiently, which overcomes the limitation of analytical methods based on the assumption of small amplitude vibrations and the low computational efficiency of numerical method. The predicted cutting stability is verified against both the experimental results and the time-domain simulation results.

Introduction

Chatter is an unstable phenomenon in machining process when improper cutting conditions are selected, which leads to detrimental effects such as tool breakage and severe vibration marks. So, chatter avoidance should better be taken into account at the process planning stage, as seen in the literature.^1–6 Process damping which is always ignored in the literature on chatter stability analysis affects the chatter stability significantly, especially in machining of some hard-to-machine aerospace materials such as titanium and nickel super alloys where low cutting speeds must be satisfied.

The primary source of process damping is due to the indentation of tool cutting edge and clearance face into wavy finish workpiece surface. To understand the process damping, pioneers have already done some excellent works trying to explain how process damping generates and affects the machining stability.^7–14 Moreover, process damping models for specific issues, such as 2-degree-of-freedom (DOF) regenerative chatter, ¹⁵ different tool geometries, ¹⁶ influence of feed rate on process damping, ¹⁷ micro-milling operations, ¹⁸ mode coupling, ¹⁹ and thin-wall milling, ²⁰ were also discussed. To identify the coefficients related to process damping, several methods were established, including fast tool servo device, ²¹ energy dissipation principle, ²² inverse stability analysis method,^23,24 operational modal analysis method,^25,26 turning process excitation method, ²⁷ and the generalized method with static milling forces. ²⁸

Due to the nonlinear property of process damping, the cutting process can be classified into three vibration states: ¹² fully stable state, finite amplitude state, and fully unstable state. The phenomenon of finite amplitude state caused by process damping is an intermediate state between fully stable and fully unstable states in which the vibration amplitude grows and stabilizes at a certain amplitude before the tool separates from the cutting process, which has been demonstrated by numerical simulations15,29 and experiments. ³⁰ This phenomenon derives amplitude-dependent stability region ¹² which is a group of the stability boundaries under different vibration amplitudes instead of the traditional stability boundary represented by a single boundary between stable and unstable regions. Hence, stability analysis under the condition of finite vibration amplitude is necessary.

The vibration amplitude is closely related to the surface roughness of the machined workpiece. By establishing the stability lobes under the condition of finite vibration amplitude, that is, the finite amplitude stability, chatter can be eliminated at early stages when the vibration amplitude is still small and the produced surface quality can be ensured. Under the finite amplitude state, the calculation of extruded volume is one of the most critical challenges, especially in machining with honed tools due to the complex and time-variable contact condition between the arc cutting edge and the finite amplitude wave surface. Analyticalmethods^7,8,25,31 based on the assumption of small amplitude vibrations have been demonstrated to be conservative under the finite amplitude vibration state. ¹² Numerical methods^22,23,32 can reach high precision and perform with low computational efficiency that hinders its application. How to calculate the extruded volume efficiently is still an essential problem in predicting the stability of honed tools under the finite amplitude vibration state. To this end, an efficient semi-analytical method is proposed so that the stability of cutting with honed tools considering the process damping is accurately and efficiently obtained in this work.

The equivalent process damping coefficient, a key parameter, is derived for honed tools based on energy dissipation principle which is applied to establish the relationship between the extruded volume and the equivalent process damping coefficient. Since the extruded volume can be formulated analytically, the computations of extruded volume, equivalent process damping coefficient, and stability lobes are much more efficient. This superiority is verified by comparing with the pure numerical method, experimental results in Budak and Tunc, ²² and the time-domain simulation method. Moreover, the proposed method is applicable in stability analysis for both the cases of small amplitude and finite amplitude vibrations. This article is organized as follows: Calculation of the extruded volume and establishment of the stability prediction algorithm for honed tools with cutting edge radius are presented in section “Calculation of the extruded volume and prediction of stability.” Section “Simulations and experiments” offers the simulation and experimental results to validate the semi-analytical method, and section “Conclusion” draws the conclusions.

Calculation of the extruded volume and prediction of stability

In this section, the semi-analytical method for calculation of the extruded volume is first constructed. Then, the equivalent process damping coefficient as well as the algorithm is presented for prediction of the stability with honed tools.

The semi-analytical method for calculation of the extruded volume

Existing analytical methods^7,8,25,31 to calculate the extruded volume are conservative under the finite amplitude state since they are based on the assumption of small amplitude vibrations. To overcome the limitation of analytical methods applied in finite amplitude state and the low computational efficiency of numerical methods,^22,23,32 a semi-analytical method, which abandons the assumption of small amplitude vibrations, is proposed to calculate the extruded volume efficiently and accurately for honed tools.

As shown in Figure 1, the honed tool with cutting edge radius $r$ and clearance angle $\gamma$ moves from left to right horizontally at a cutting speed $v$ . The tool is assumed to be rigid in the direction of $x$ , and the workpiece is assumed to be rigid relative to the compliant tool, so as to simplify the analysis and focus the emphasis on the modeling of process damping itself. The governing equation of this cutting system can be established as follows

$$M\stackrel{··}{y}\left(t\right)+C\stackrel{·}{y}\left(t\right)+\mathrm{Ky}\left(t\right)=F_r$$

(1)

where $M$ , $C$ , and $K$ are the modal parameters of the system. The cutting force $F_r$ exerted on the tool in the feed direction is composed by the shearing force $F_{\mathrm{rs}}$ and the ploughing force $F_{\mathrm{rp}}$

$$\begin{array}{c}{F}_r=F_{\mathrm{rs}}+F_{\mathrm{rp}}\hfill \\ F_{\mathrm{rs}}=b{K}_{f}h\hfill \\ h=\max (0,s_t+y(t-\tau )-y\left(t\right))\hfill \end{array}$$

(2)

Figure 1.

Schematic diagram of orthogonal cutting process with honed tool.

In equation (2), $b$ , $K_f$ , $h$ , and $s_t$ represent the width of cut, cutting force coefficient in the feed direction, chip thickness, and feed per revolution, respectively. $y(t)$ and $y(t-\tau )$ are, respectively, the inner and outer amplitudes of vibration between tool and workpiece, where $\tau$ is the time delay between them. The total instantaneous uncut chip thickness $h$ is composed of two components: $s_t$ and regeneration term $(y(t-\tau )-y(t))$ . $h$ equals zero when the tool separates from the workpiece due to the amplitude of vibration reaching the critical value. In the evaluation of ploughing force, the ploughing force in the feed direction is widely evaluated by the volume $V$ of the work material extruded by the tool, ⁷ as follows

$$F_{\mathrm{rp}}=K_{\mathrm{sp}}V,V=b*S\left(x\right)$$

(3)

where $S(x)$ is the area of the extruded volume, that is, the indentation area, and $K_{\mathrm{sp}}$ is the specific indentation coefficient which depends on the mechanical properties of the material. This indentation model was shown to be the primary source of process damping and a nonlinear model where the extruded volume depends on the vibration amplitude; the bigger the vibration amplitude, the higher the effect of process damping. It can be seen that the ploughing force is determined according to equation (3) for considering the effect of process damping, where an explicit form of the extruded volume $V$ needs to be developed to calculate the ploughing force.

The extruded volume is modeled based on the existence of a separation point on the arc cutting edge. As illustrated in Figure 1, $A$ is the separation point and the trajectory of the point $A$ is the boundary $L$ which forms the wavy work surface after machining. The work material is partitioned at the separation point $A$ into two parts where the material above the boundary $L$ flows up to be the chip in the rake face and leads to the generation of shearing cutting forces, while the material below the boundary $L$ is away from the rake face and extruded downward causing the occurrence of ploughing forces in the flank face eventually. $B$ is the lowest point on the honed tool edge, and $O$ is the center of the arc cutting edge so that the length of the segment $\mathrm{OB}$ is the hone radius $r$ . $C$ is the intersection between the tool clearance face $\mathrm{BC}$ and the boundary $L$ . The length of the segment $\mathrm{BD}$ and the angle between segments $\mathrm{OA}$ and $\mathrm{OB}$ are regarded as the penetration depth $\eta$ and the separation angle $\beta$ , respectively. As shown, the shaded area surrounded by the surface $\mathrm{ABC}$ and the boundary $L$ represents the indentation area $S(x)$ which can be divided by the segment $\mathrm{BD}$ into two subareas $S(\mathrm{ABD})$ and $S(\mathrm{BCD})$ . Finally, the extruded volume can be calculated as

$$V=b*S\left(\mathrm{ABCD}\right)=b*[S\left(\mathrm{ABD}\right)+S\left(\mathrm{BCD}\right)]$$

(4)

The vibration of tool is assumed to be a sinusoidal wave with amplitude $Y$ and length $L_1$ , thus the displacement $y$ and the vibration velocity $\stackrel{·}{y}$ of the tool tip $(x,y)$ can be described as

$$\begin{array}{c}y=Y\sin \left(\lambda x\right);\lambda =\frac{2\pi }{L_1}\hfill \\ \stackrel{·}{y}=\frac{\mathrm{dy}}{\mathrm{dt}}=\frac{\mathrm{dy}}{\mathrm{dx}}*\frac{\mathrm{dx}}{\mathrm{dt}}=Y\lambda \cos \left(\lambda x\right)*v\hfill \end{array}$$

(5)

where $F$ is an arbitrary point on the boundary $L$ , which is on the left of point $B$ with a horizontal distance $\xi$ . The displacement of point $F$ can be represented accurately as follows

$$y_F(x,\xi )=Y\sin \left[\lambda (x-r\sin \beta -\xi )\right]$$

(6)

Therefore, the height $\mathrm{FG}$ of the area $S(\mathrm{BCD})$ , that is, $h(\xi )$ in Figure 1, can be defined as

$$\begin{array}{c}h\left(\xi \right)=y_F(x,\xi )-y_D+\eta -\xi \tan \gamma \hfill \\ =Y\sin \left[\lambda (x-r\sin \beta -\xi )\right]-Y\sin \left[\lambda (x-r\sin \beta )\right]\hfill \\ \multicolumn{1}{c}{+\eta -\xi \tan \gamma ,0\le \xi \le {\xi }_c}\end{array}$$

(7)

where ${\xi }_c$ determines the location of the intersection $C$ . Due to the boundary condition of $h({\xi }_c)=0$ at point $C$ , ${\xi }_c$ is the solution of the equation as follows

$$\begin{array}{c}\multicolumn{1}{c}{h\left({\xi }_c\right)=Y\sin \left[\lambda (x-r\sin \beta -{\xi }_c)\right]-Y\sin \left[\lambda (x-r\sin \beta )\right]}\\ \multicolumn{1}{c}{+\eta -{\xi }_c\tan \gamma =0}\end{array}$$

(8)

where ${\xi }_c$ can be calculated analytically under the assumption of $\lambda {\xi }_c\approx 0$ . ⁷ However, that assumption is not always valid especially for the condition of low cutting speed and high vibration frequency while the effect of process damping is more significant at this condition. Thus, ${\xi }_c$ cannot be obtained analytically by equation (8) without any approximations, hence we solve it by the method of bisection numerically. Once ${\xi }_c$ is determined, based on equation (7), the indentation area $S(\mathrm{BCD})$ can be obtained as

$$\begin{array}{c}\multicolumn{1}{c}{S\left(\mathrm{BCD}\right)=\underset{0}{\overset{{\xi }_c}{\int }}h(\xi )d\xi =Y\sin \left[\lambda (x-r\sin \beta )\right]}\\ \multicolumn{1}{c}{*(\frac{\sin \left(\lambda {\xi }_c\right)}{\lambda }-{\xi }_c)}\\ \multicolumn{1}{c}{-\frac{Y\cos \left[\lambda (x-r\sin \beta )\right]}{\lambda }*[1-\cos \left(\lambda {\xi }_c\right)]}\\ \multicolumn{1}{c}{+\eta {\xi }_c-\left({\xi }_c^2\tan \gamma \right)/2}\end{array}$$

(9)

As shown in equation (9), in order to calculate the indentation area, the vibration amplitude $Y$ needs to be preset.

Under the finite amplitude vibration state, the indentation area, $S(\mathrm{ABD})$ , which is not a constant area, cannot be ignored for calculation of the extruded volume. Because points $A$ and $D$ are very close, the segment of the wave $\mathrm{AD}$ can be assumed to be linear. Therefore, the indentation area $S(\mathrm{ABD})$ can be calculated as

$$\begin{array}{c}\multicolumn{1}{c}{S\left(\mathrm{ABD}\right)=S\left(\mathrm{OAB}\right)-S\left(\mathrm{OAD}\right)=\frac{1}{2}\beta r^2}\\ \multicolumn{1}{c}{-\frac{1}{2}(r-\eta )*r\sin \beta }\end{array}$$

(10)

Based on equations (4), (9), and (10), the extruded volume can be formulated analytically as

$$V=b*\left\{\begin{array}{c}Y\sin \left[\lambda (x-r\sin \beta )\right]*(\frac{\sin \left(\lambda {\xi }_c\right)}{\lambda }-{\xi }_c)-\frac{Y\cos \left[\lambda (x-r\sin \beta )\right]}{\lambda }*[1-\cos \left(\lambda {\xi }_c\right)]\hfill \\ \multicolumn{1}{c}{+\eta {\xi }_c-\left({\xi }_c^2\tan \gamma \right)/2+\frac{1}{2}\beta r^2-\frac{1}{2}(r-\eta )*r\sin \beta }\end{array}\right\}$$

(11)

where the penetration depth $\eta$ needs to be determined.

Figure 1 shows the tool penetration geometry; the penetration depth $\eta$ is composed of the length of segment $\mathrm{DH}$ and $\mathrm{BH}$ . Therefore, $\eta$ can be obtained as a function of separation angle $\beta$ , hone radius $r$ , and vibration amplitude $Y$ as follows

$$\begin{array}{c}\multicolumn{1}{c}{\eta =\mathrm{BH}+\mathrm{DH}=r-r\cos \beta +Y\sin \left[\lambda (x-r\sin \beta )\right]}\\ \multicolumn{1}{c}{-Y\sin \left(\lambda x\right)}\end{array}$$

(12)

where the separation angle $\beta$ is modeled by the cutting speed and the hone radius according to Budak and Tunc ²² as follows

$$\beta ={\beta }_0+{\beta }_{1}v+{\beta }_{2}r$$

(13)

The extruded volume $V$ can be formulated by equation (11) analytically, while ${\xi }_c$ , which is the upper limit of integral in equation (9), needs to be calculated numerically. Thus, the extruded volume $V$ calculated by equation (11) can be regarded as a semi-analytical method.

The semi-analytical method is established without the assumption of small amplitude vibrations. It can calculate the extruded volume accurately under the finite amplitude state, where the indentation area $S(\mathrm{BCD})$ is accurately formulated, the upper limit of integral ${\xi }_c$ is calculated numerically, and the indentation area $S(\mathrm{ABD})$ is included. Moreover, the extruded volume can be formulated analytically, which ensures the high computational efficiency. In the meantime, the calculated extruded volume is time-variable (equation (5)), which well reflects the dynamic contact between the arc cutting edge and the finite amplitude wave surface.

Stability prediction algorithm

First, equivalent process damping coefficient, a key parameter for modeling of process damping, is presented for honed tools based on the extruded volume calculation. Then, the algorithm is established to predict the stability for honed tools based on the equivalent process damping coefficient.

The equivalent viscous damping approach ³³ is widely utilized to quantitatively estimate the nonlinear effect of process damping, where the equivalent process damping coefficient $C_p$ is derived based on the energy dissipation principle which assumes that the total energy dissipated by ploughing force $F_{\mathrm{rp}}$ is equated to the total energy dissipated by the equivalent viscous damper with coefficient $C_p$ for one cycle ²²

$$C_p(Y,\gamma ,r,L_1,\beta )=\frac{-K_{\mathrm{sp}}b}{v}\frac{1}{Y\pi }\underset{0}{\overset{L_1}{\int }}S\left(\mathrm{ABCD}\right)\cos \left(\lambda x\right)\mathrm{dx}$$

(14)

Since the transcendental equations (8) and (11) cannot be analytically solved, $C_p$ is calculated numerically with the discrete step $\Delta x=L_1/50$ in this article.

Based on the equivalent process damping coefficient $C_p$ in equation (14), a stability prediction algorithm is established to compute the stability lobes considering process damping for honed tools.

The equation of motion in the feed direction, that is, equation (1), can be transformed under the effect of the equivalent process damping coefficient $C_p$ as follows

$$\begin{array}{c}M\stackrel{··}{y}\left(t\right)+C_t\stackrel{·}{y}\left(t\right)+\mathrm{Ky}\left(t\right)=K_{f}b[s_t+y(t-\tau )-y\left(t\right)]\hfill \\ C_t=C+C_p\hfill \end{array}$$

(15)

where $C_p$ , determined by equation (14), contributes to the total damping $C_t$ with the structural damping $C$ to act on the system. The critical stability solution of equation (15) is expressed as follows ³⁴

$$\begin{array}{c}b=\frac{-1}{2K_f\mathrm{Re}\left(\varphi \right)},\varphi =\frac{1}{K-M{\omega }_c^2+C_{t}i{\omega }_c}\hfill \\ {\omega }_{c}T=2k\pi +\epsilon ,k=0,1,2,\dots \epsilon =3\pi +2\phi \hfill \\ \phi ={\tan }^{-1}\frac{\mathrm{Im}\left(\varphi \right)}{\mathrm{Re}\left(\varphi \right)}\hfill \\ n=\frac{60}{T}\hfill \end{array}$$

(16)

where $\varphi$ is the frequency response function (FRF) of the vibratory system, $k$ is the lobe number, ${\omega }_c$ is chatter frequency which is in the vicinity of the system natural frequency, $\phi$ is the phase angle between the vibratory displacement and the exciting force, $\epsilon$ is the phase angle between the inner and the outer waves, $n$ (rev/min) is the spindle speed, and $T$ is the spindle rotation period.

Since the cutting width $b$ and the equivalent process damping coefficient $C_p$ are affected by each other as shown in equation (14), an iterative solution based on equation (16) and the equivalent process damping coefficient $C_p$ will be applied to obtain the stability lobes under the effect of process damping. Eventually, the stability prediction algorithm for honed tools can be established as follows.

Algorithm (stability prediction procedures)

Input: Chatter frequency ${\omega }_c$ , lobe number $k$ , initial cutting width $b^0$ , and equivalent process damping coefficient $C_p^0$ . Output: Convergent cutting width $b^{*}$ and spindle speed $n^{*}$ . Step: (1) Choose ${\omega }_c$ and $k$ . (2) Set $m=0$ , $b^m=0$ , $C_p^m=0$ . (3) Calculate $b^{m+1}$ , $n^{m+1}$ by equation (16) using $C_p^m$ . (4) Update ${\beta }^{m+1}$ by equation (13) and ${L_1}^{m+1}=v/f_c$ . (5) Update $C_p^{m+1}$ by equation (14) using ${\beta }^{m+1}$ , ${L_1}^{m+1}$ , and $b^{m+1}$ . (6) If $|\min (b^m)/\min (b^{m+1})-1|>\mathrm{err}$ , set $m=m+1$ and go to step 3. Else exit and report $(b^{*},n^{*})$  =  $(b^{m+1},n^{m+1})$ .

The threshold err is sufficient to be set as 1% for the convergence of cutting width. The increment of $\Delta {\omega }_c$ depending on the lobe number $k$ is performed to get one single lobe by repeating the procedure above. The whole procedure is repeated again with different $k$ to obtain different lobes.

Simulations and experiments

This section presents the simulations and experiments to validate the proposed semi-analytical method. First, the indentation area $S(x)$ calculated by thesemi-analytical method is compared with that of the numerical integration method in terms of accuracy and efficiency. The applicability of the semi-analytical method dealing with different cutting parameters and tool geometries is revealed. Second, the ploughing force is measured experimentally to verify the ploughing force predicted by the semi-analytical method. Finally, the stability lobes computed by the proposed semi-analytical method are presented and the advantages are discussed. The simulation results in sections “Indentation area calculation” and “Stability lobes construction” are programed by MATLAB R2014a on a PC with a 3.6-GHz processor.

Indentation area calculation

The key for computing the stability lobes accurately and efficiently focuses on the calculation of indentation area $S(x)$ . The indentation area $S(x)$ calculated by the semi-analytical method should be evaluated in terms of accuracy and efficiency by comparing with that from traditional numerical integration method.^22,23

Indentation area calculated by the numerical integration method with high precision can be regarded as the exact value. For convenience, $S(\mathrm{ABD})$ and $S(\mathrm{BCD})$ in the numerical integration method are both divided by the same discrete number $N$ which is the number of the divided area unit. With the high-enough discrete number $N=100$ , the indentation area $S(x)$ calculated by the numerical integration method is illustrated in Figure 2(a). The calculation conditions, which are set by choosing one of the parameters in the computation of the finite amplitude stability in section “Finite amplitude stability,” are applied as follows ²²

$$\begin{array}{c}{L}_1=1\mathrm{mm},v=75\mathrm{m}/\min \hfill \\ Y=3\mathrm{\mu }\mathrm{m},r=60\mathrm{\mu }\mathrm{m},\gamma =3°,{\beta }_0=27.5,\hfill \\ \multicolumn{1}{c}{{\beta }_1=-0.164,{\beta }_2=0}\end{array}$$

Figure 2.

(a) The indentation area calculated by the numerical integration method with $N=100$ and the semi-analytical method; (b) The absolute errors of the indentation area calculated by the numerical integration method with $N=3,5,10$ and the semi-analytical method compared with the indentation area calculated by the numerical integration method with $N=100$ .

To evaluate the accuracy of the semi-analytical method, ${\xi }_c$ is also calculated by the method of bisection under the same precision as the numerical integration method. The indentation area $S(x)$ calculated by the semi-analytical method and the absolute error of the indentation area calculated by the semi-analytical method compared with the exact value of indentation area calculated by the numerical integration method with discrete number $N=100$ are illustrated in Figure 2(a) and (b), respectively. The low absolute error demonstrates the accuracy to calculate the indentation area by the semi-analytical method. The error of the semi-analytical method to calculate the indentation area is mainly from the linear assumption of $\mathrm{AD}$ in calculation of the area $S(\mathrm{ABD})$ .

To evaluate the efficiency of the semi-analytical method to calculate the indentation area, different discrete numbers $N$ are applied in the numerical integration method to search which discrete number has the same range of errors as the semi-analytical method to calculate the indentation area. The absolute errors of the indentation area calculated by the numerical integration method with discrete numbers $N=3,5,10$ compared with the exact value of indentation area calculated by the numerical integration method with discrete number $N=100$ are drawn in Figure 2(b). It can be found in Figure 2(b) that the absolute error of the indentation area calculated by the numerical integration method with discrete number $N=10$ is slightly larger than that of the semi-analytical method. Thus, the appropriate discrete number in the numerical integration method to compare the computational efficiency with the semi-analytical method is $N=10$ . To reduce the error of running time, the program computing the indentation area is run repeatedly for 1000 times. It takes 2.01 s for the semi-analytical method while 7.73 s for the numerical integration method with $N=10$ . The direct running time comparison demonstrates that at the same range of computational errors, the efficiency of the indentation area calculated by the semi-analytical method is much higher than the numerical integration method.

To demonstrate the applicability of the semi-analytical method dealing with different cutting parameters, tool geometries, and work materials, the average indentation area over one wavelength and the average specific process damping coefficient calculated by the semi-analytical method under different cutting parameters and tool geometries are illustrated in Figures 3 and 4, respectively. The average specific process damping coefficient in Figure 4, which determines the stability limit directly, ²² is obtained under the condition of the blue curve’s parameters in Figure 3 for AISI 1050 where $K_{\mathrm{sp}}=70,000$ $\mathrm{N}/\mathrm{m}{\mathrm{m}}^3$ and ${\beta }_0=24,{\beta }_1=-0.15,{\beta }_2=0.05$ . ²² Larger cutting edge radius, lower cutting speed, smaller clearance angle, and higher vibration amplitude and frequency will increase indentation area, which will lead to more significant process damping effect and higher stability limit. The separation angle $\beta$ and the cutting edge radius $r$ are the geometric parameters for honed tools, like the parameter of wear land width for tools with wear land. No matter, in the proposed semi-analytical method or the numerical integration method, the separation angle $\beta$ , cutting edge radius $r$ , vibration amplitude $Y$ , and the indentation coefficient $K_{\mathrm{sp}}$ need to be determined. These parameters can be identified by experiments in advance.

Figure 3.

The average indentation area calculated by the semi-analytical method with different cutting parameters and tool geometries.

Figure 4.

The average specific process damping coefficient calculated by the semi-analytical method with different cutting speeds and (a) cutting edge radii, (b) clearance angles, (c) vibration ampltitudes, and (d) vibration frequencies.

Verification of ploughing force predicted by the semi-analytical method

In order to verify the accuracy of the ploughing force predicted by the semi-analytical method, cutting tests are conducted. The measurement of cutting edge radius is critical for the calculation of ploughing force. Thus, the tool edges, which are symmetrically rounded, are measured by Alicona INFINITEFocus G5. The measurement report for one of tools in the test as an example is shown in Figure 5, where the cutting edge radius is measured as $r\approx 60\mathrm{\mu }\mathrm{m}$ .

Figure 5.

Alicona INFINITEFocus G5 to measure the tool edge radius and experimental setup of cutting tests.

The experimental setup on a computer numerical controlled (CNC) lathe is also illustrated in Figure 5. Carbide tools and AL7075 workpiece are adopted where $K_{\mathrm{sp}}=10,000\mathrm{N}/\mathrm{m}{\mathrm{m}}^3$ ²² can be set. The inserts have 0° rake and 11° clearance angle. The cylindrical workpiece with 100 mm diameter and 300 mm length is fixed in the chuck. To ensure only the straight part of tool edges engage in cutting, the end of workpiece is machined to discs with different thicknesses where the feed is in radial direction. The tool is mounted on a Kistler 9257B table dynamometer which is used to measure the cutting force. The extrapolation method on zero uncut thickness, a common approach for identification of ploughing force, is applied to determine the ploughing force in the cutting tests. To investigate the effect of the width of cut on the ploughing force, tests 1–4 at coolant condition with tools of one single cutting edge radius are conducted at different widths of cut as shown in Table 1. To further investigate the effect of cutting edge radius on the ploughing force, tests 5–8 without coolant with tools of different cutting edge radii are conducted at a constant width of cut as shown in Table 1. The cutting parameters in the tests should satisfy chatter-free condition.

Table 1.

Cutting conditions of the tests.

Tests	Honeradius, $r$ ( $\mathrm{\mu }\mathrm{m}$ )	Widthof cut, $b$ (mm)	Feed per revolution, $s_t$ (mm/rev)	Cuttingspeed, $v$ (m/min)
1	60	2	0.05, 0.1, 0.15, 0.2	60
2	60	2.5	0.05, 0.1, 0.15, 0.2	60
3	60	2.8	0.05, 0.1, 0.15, 0.2	60
4	60	3	0.05, 0.1, 0.15, 0.2	60
5	20	2	0.05, 0.1, 0.12, 0.15	60
6	30	2	0.05, 0.1, 0.12, 0.15	60
7	40	2	0.05, 0.1, 0.12, 0.15	60
8	50	2	0.05, 0.1, 0.12, 0.15	60

In the cutting tests, commercial finite element analysis ³⁵ is adopted to obtain the separation angle. The effects of width of cut and feed rate on the separation angle are neglected. Figure 6 shows the vertical velocity components of the workpiece around the honed tool edge simulated at the condition of tests 1–4 by the Deform v11.0 software. The boundary line $E$ represents the line with zero material vertical velocity based on the velocity spectrum. The position $A$ with zero vertical velocity, which is the intersection of line $E$ and the cutting edge boundary, indicates the material separation point. Therefore, the separation angle is identified as 55° in Figure 6 by measuring the angle between lines $\mathrm{OA}$ and $\mathrm{OB}$ . With this method, the separation angles for tests 5–8 are obtained, as shown in Figure 8(b).

Figure 6.

Finite element result of separation angle for tests 1–4.

The vibration amplitude in the cutting tests is actually small. Therefore, the predicted ploughing force can be calculated by the semi-analytical method according to equations (3) and (11) under the condition of vibration amplitude $Y=0$ . The thrust forces at different cutting widths measured in tests 1–4 are illustrated in Figure 7(a). The measured ploughing force and the predicted ploughing force at different widths of cut are compared in Figure 7(b). The predicted ploughing forces are less than the measured ploughing forces at different widths of cut. The reasons may lie in two points. On one hand, the extrapolation method on zero uncut thickness may amount to considerable overestimation of ploughing forces.^36,37 On the other hand, the predicted ploughing force is calculated under totally vibration-free conditions which ignores the component of ploughing force generated by the effect of vibration. In view of the above-mentioned reasons, we consider the predicted ploughing force is reasonable based on the comparison with the measured ploughing force. It also reflects good linear relationship between the measured ploughing forces and the width of cut, which can be explicitly observed in the formula of the predicted ploughing force (equation (3)). The thrust forces at different cutting edge radii measured in tests 5–8 are illustrated in Figure 8(a). The measured ploughing force and the predicted ploughing force at different cutting edge radii are compared in Figure 8(b), where the ploughing forces increase with increasing cutting edge radius. The predicted ploughing forces are also less than the measured ploughing forces at different cutting edge radii. The reason may also lie in the above two points.

Figure 7.

At different widths of cut: (a) the measured thrust force versus uncut chip thickness and (b) the measured and predicted ploughing forces.

Figure 8.

At different cutting edge radii: (a) the measured thrust force versus uncut chip thickness and (b) the measured and predicted ploughing forces and the simulated separation angle for tests 5–8.

Stability lobes construction

Stability lobes considering the effect of process damping are computed based on the established stability prediction algorithm in section “Stability prediction algorithm.” For consistency, the computation of stability in the algorithm where the indentation area is calculated by the semi-analytical method is also designated as the semi-analytical method, while the indentation area calculated by the numerical integration method is designated as the numerical method.

Finite amplitude stability

To compare with the experimental result of finite amplitude stability with vibration amplitude $Y=3\mathrm{\mu }\mathrm{m}$ in Budak and Tunc, ²² the same simulation conditions to predict the cutting stability are applied in Table 2.

Table 2.

The simulation parameters.

Modal parameters	$K=15.9\mathrm{MN}/\mathrm{m},M=0.28\mathrm{kg},\zeta =1.49\mathrm{\%}$
Tool geometry	$r=60\mathrm{\mu }\mathrm{m},\gamma =3^{\mathrm{o}}$
Indentation coefficient	$K_{\mathrm{sp}}=10,000\mathrm{N}/\mathrm{m}{\mathrm{m}}^3$
Cutting force coefficient	$K_f=158\mathrm{MN}/{\mathrm{m}}^2$
Separation angle	${\beta }_0=27.5,{\beta }_1=-0.164,{\beta }_2=0$
Vibration amplitude	$Y=3\mathrm{\mu }\mathrm{m}$

With the semi-analytical method, the finite amplitude stability lobes with vibration amplitude $Y=3\mathrm{\mu }\mathrm{m}$ are plotted in Figure 9. For comparison, the experimental results in Budak and Tunc ²² and the finite amplitude stability lobes with $Y=3\mathrm{\mu }\mathrm{m}$ computed by the numerical method are also illustrated. Figure 9 shows that the predicted stability computed by the semi-analytical method is consistent with that of the numerical method, which demonstrates the accuracy of the predicted stability computed by the semi-analytical method. However, there are small errors between the predicted stability computed by the semi-analytical method and the experimental results in Budak and Tunc. ²² Moreover, the computational efficiency of the semi-analytical method is much higher than the numerical method. It takes 2.51 s to get one single lobe using the semi-analytical method which is much shorter than the time 7.84 s using the numerical method.

Figure 9.

Finite amplitude stability lobes with $Y=3\mathrm{\mu }\mathrm{m}$ computed using the semi-analytical method and the numerical method (experimental results from Budak and Tunc ²² ).

The three-dimensional (3D) stability lobes depending on the vibration amplitude predicted by the proposed semi-analytical method for honed tools are illustrated in Figure 10(a). The amplitude-dependent stability lobes with finite vibration amplitude form the finite amplitude stability region in the stability diagrams. The cutting process will be in the finite amplitude state if the cutting parameters locate in this region. For comparison, the two-dimensional (2D) stability lobes with vibration amplitude $Y=6\mathrm{\mu }\mathrm{m},3\mathrm{\mu }\mathrm{m},\mathrm{and}0.1\mathrm{\mu }\mathrm{m}$ are also plotted in Figure 10(b). The same conditions in Table 2 are applied except the vibration amplitude.

Figure 10.

With the semi-analytical method: (a) 3D finite amplitude stability lobes and (b) 2D finite amplitude stability lobes established at specific vibration amplitudes.

Verification by time-domain simulations

To verify specific vibration amplitudes, such as $0.1\mathrm{\mu }\mathrm{m},3\mathrm{\mu }\mathrm{m},\mathrm{and}6\mathrm{\mu }\mathrm{m}$ , at which the stability lobes are established, the points ( $P_i,Q_i,R_i;i=1,2$ ) in Figure 10(b) are chosen close to the stability lobes established at these specific vibration amplitudes for time-domain simulations. At each time step, the ploughing force is calculated according to equation (3) where the indentation area is calculated by the numerical integration method and contributed to the dynamic cutting force. The dynamic equation is solved by the fourth-order Runge–Kutta method, and the simulation results are illustrated in Figure 11. The corresponding vibration amplitudes of $P_i,Q_i,R_i;i=1,2$ at steady state are listed in Table 3. The cutting width $b_0$ at the $A_s=0.1\mathrm{\mu }\mathrm{m}$ lobes and the cutting width $b_3$ at the $A_s=3\mathrm{\mu }\mathrm{m}$ lobes as well as the cutting width $b_6$ at the $A_s=6\mathrm{\mu }\mathrm{m}$ lobes are used to select the points ( $P_i,Q_i,R_i;i=1,2$ ), which are also listed in Table 3. The corresponding vibration amplitudes at these cutting widths are $0.1,3,\mathrm{and}6\mathrm{\mu }\mathrm{m}$ , respectively, where the corresponding amplitude-dependent stability lobes are established. From Table 3, it can be found that the vibration amplitudes of points $P_1,P_2$ below the $A_s=0.1\mathrm{\mu }\mathrm{m}$ lobes are less than $0.1\mathrm{\mu }\mathrm{m}$ , the vibration amplitude of point $Q_1$ or $R_1$ close to the $A_s=3\mathrm{\mu }\mathrm{m}$ lobes is slightly less or larger than $3\mathrm{\mu }\mathrm{m}$ , and the vibration amplitude of point $Q_2$ or $R_2$ close to the $A_s=6\mathrm{\mu }\mathrm{m}$ lobes is slightly less or larger than $6\mathrm{\mu }\mathrm{m}$ . It demonstrates the accuracy in computing the finite amplitude stability lobes with different vibration amplitudes by the semi-analytical method.

Figure 11.

Time-domain simulation result at points $P_i,Q_i,R_i;i=1,2$ .

Table 3.

Cutting widths at points $P_i,Q_i,R_i;i=1,2$ and their corresponding amplitudes of vibration obtained from time-domain simulations.

Cutting speed (m/min)	80					85
	$P_1$	$b_0$	$Q_1$	$b_3$	$R_1$	$P_2$	$b_0$	$Q_2$	$b_6$	$R_2$
Cutting width (mm)	5.700	5.773	6.200	6.282	6.400	4.700	4.935	6.200	6.330	6.400
Vibration amplitude ( $\mathrm{\mu }\mathrm{m}$ )	0.095	0.100	2.909	3.000	3.290	0.028	0.100	5.906	6.000	6.171

Application of the semi-analytical method in construction of lower bound lobes

Lower bound lobes are developed for fully stable state based on the assumption of small amplitude vibrations. ¹² Generally, the stability determined by the lower bound lobes is conservative for case of the finite amplitude state due to the underestimation of the extruded volume by the assumption of small amplitude vibrations. The stability lobes computed by the analytical methods^7,8,25,31 based on the assumption of small amplitude vibrations can be regarded as the lower bound lobes. The vibration amplitude $Y$ can be set as $Y=0.1\mathrm{\mu }\mathrm{m}$ in the semi-analytical method to establish the lower bound of the stability lobes, which is small enough to satisfy the assumption of small amplitude vibrations. A vibration amplitude smaller than $Y=0.1\mathrm{\mu }\mathrm{m}$ can also be set in the semi-analytical method, but the computed stability lobes will not change. Figure 12 shows the lower bound lobes computed by the semi-analytical method with vibration amplitude $Y=0.1\mathrm{\mu }\mathrm{m}$ . It is computed under the same conditions in Table 2 except the vibration amplitude. For comparison, the critical stability limits computed by the analytical method in Ahmadi and Altintas ²⁵ are also illustrated in Figure 12. It shows that the lower bound lobes using the semi-analytical method coincide with the stability limits computed by the analytical method in Ahmadi and Altintas. ²⁵ It demonstrates the ability in computing the lower bound of the stability lobes by the semi-analytical method.

Figure 12.

Lower bound lobes computed using the semi-analytical method and the analytical method in Ahmadi and Altintas. ²⁵

Discussion on the efficiency and applicability of the semi-analytical method

The proposed semi-analytical method will promote the computational efficiency of extruded volume since the extruded volume can be formulated analytically by geometric modeling of the complex and time-variable contact condition between the arc cutting edge and the finite amplitude wave surface. At the same time, the geometric modeling of contact condition is elaborated to ensure the calculation accuracy of the semi-analytical method. Therefore, in the same stability prediction algorithm, the semi-analytical method will have advantage over the numerical integration method in terms of the computational efficiency. The direct computation time comparisons are given as follows.

The efficiency of indentation area calculated by the semi-analytical method is greatly promoted by 74% (from 7.73 to 2.01 s) compared with the numerical integration method. Consequently, under the condition of finite vibration amplitude $Y=3\mathrm{\mu }\mathrm{m}$ , the efficiency of one single lobe computed by the semi-analytical method is greatly promoted by 68% (from 7.84 to 2.51 s) compared with the numerical method.

Based on the demand to describe the contact condition between the arc cutting edge and the finite amplitude wave surface, the extruded volume calculated by the semi-analytical method is dependent on the vibration amplitude. As a result, the proposed semi-analytical method is dependent on the vibration amplitude. Also, the dependent vibration amplitude can satisfy the assumption of small amplitude vibrations. Therefore, with the semi-analytical method, the lower bound of the stability and the finite amplitude stability region considering the effect of process damping for honed tools can be defined. With these defined stability regions, chatter can also be eliminated at early stages when the vibration amplitude is still small. In fact, the boundaries computed with the assumption of small amplitude vibrations are commonly adopted in the literature to determine stability limits. In addition, the semi-analytical method for the issue of process damping is applicable to deal with different cutting parameters, tool geometries, and work materials, which will save a lot of resources without further testing.

Conclusion

Under the finite amplitude vibration state for honed tool with cutting edge radius, analytical methods based on the assumption of small amplitude vibrations are conservative and numerical method performs with low computational efficiency in calculating the extruded volume, which both hinder the prediction of stability considering process damping. This article develops an efficient semi-analytical method to calculate the extruded volume so that the stability of cutting with honed tools considering process damping can be accurately and efficiently obtained. The main conclusions can be summarized as follows:

1. The complex and time-variable contact condition between the arc cutting edge and the finite amplitude wave surface makes the efficient calculation of extruded volume and stability difficult. With the analytical formulation of contact geometry and the numerical calculation of contact boundary condition, the semi-analytical calculation of the extruded volume for honed tool is achieved. The semi-analytical method abandons the assumption of small amplitude vibrations although the numerical calculation of contact boundary condition is included, which overcomes the limitation of analytical methods based on the assumption of small amplitude vibrations and the low computational efficiency of numerical method.

2. With the proposed semi-analytical method depending on the vibration amplitude, the extruded volume and the finite amplitude lobes can be calculated efficiently and accurately. The direct computation time comparisons demonstrate that the proposed semi-analytical method possess much higher efficiency compared with the numerical method. At the same time, the elaborate modeling of contact geometry and the accurate determination of boundary condition ensure the calculation accuracy of the semi-analytical method.

3. Since the dependent vibration amplitude established for stability lobes in the semi-analytical method satisfies the assumption of small amplitude vibrations, the lower bound of stability lobes can also be defined by the proposed semi-analytical method. Both the results of the lower bound of stability lobes and the finite amplitude lobes show great promise for understanding and quantifying the effect of process damping.

The efficient semi-analytical method makes it possible to offer a quick guidance for predicting of the stability under the effect of process damping in cutting with honed tools. Future work will focus on evaluating the relationship between the vibration amplitude and the surface roughness so that the produced surface quality can be controlled with the finite amplitude stability.