**2. Analytical prediction of chatter stability in ball end milling with tool inclination [7]**

A new analytical model to predict chatter vibration in ball end milling with consideration of tool inclination is introduced in this section.

#### **2.1. Introduction**

The result shows that picture images can be printed successfully on hardened steel as nano‐

Micro-Nano Mechatronics — New Trends in Material, Measurement, Control, Manufacturing and Their Applications in

(a)gray scale image (b)microphotograph (c)measured profile

Nano sculpturing experiments involving dimple patterns were also carried out with the developed sculpturing system. Sinusoidal commands to control the vibration amplitude were input to the elliptical vibration control system during machining, and the phase of the sinusoidal commands was changed by 180 degrees in every cutting feed, so that precisely

Figure 6 shows microphotographs. The hexagonal dimple patterns, whose borders are sharp, can be observed on the left. On the other hand, isolated circular dimple patterns were also sculptured successfully on the right, as the maximum depth of cut was considerably smaller than the amplitude variation.The results show that a variety ofdimple patterns can be obtained

Novel ultra‐precision sculpturing technology for difficult‐to‐cut materials at micro/nano scale was proposed by utilizing elliptical vibration cutting technology. In the proposed method, the depth of cut is controlled without the conventional FTS technology by actively manipulating the vibration amplitude in the depth of cut direction. In order to verify the proposed method, the vibration amplitude control system and a high performance micro/nano sculpturing system were developed and applied to sculpturing experiments on hardened steel. Conse‐ quently, micro textured grooves, an image of a picture and various dimple patterns were manufactured on the hardened steel workpiece successfully as nano‐scale sculptures. These were done by merely combining a simple feed motion at a constant speed with high‐speed

**Figure 5.** Gray scale image for amplitude command, machined surface and measured surface profile

ultra‐precisely on the steel materials by using the developed sculpturing system.

scale sculptures.

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54

aligned patterns were sculptured.

**1.6. Conclusion**

depth of cut control.

The ball end milling is an important precision machining process, which is used in production of dies, molds, impellers, screw propellers and parts with free‐form surfaces. As the slender ball end mills or the thin workpiece structures are flexible, the self‐excited chatter vibration often occurs and causes severe problems such as short tool life and deterioration of surface quality. Many researchers investigated prediction of the self‐excited chatter vibration in the milling process. Altintas and Budak [8] developed an analytical model to solve the chatter stability forthe cylindrical end mills. Altintas, Shamoto, Lee and Budak extended the analytical model for the cylindrical end mills to predict the stability limits in ball end milling [9], but the tool inclination has not been considered due to geometric complexity of the ball end milling process.

On the other hand, 5‐axis machining technology has been widely spread recently, and it enables arbitrary tool inclination for better machining efficiency, accuracy and stability.

Therefore, an analytical model of the ball end milling process with the self‐excited chatter vibration is developed and verified with consideration of the tool inclination.

### **2.2. Outline of analytical model to predict chatter stability in ball end milling with tool inclination**

The ball end milling process with the tool inclination is illustrated in Figure 7. The Cartesian coordinates *xyz* are fixed to the ball end mill and aligned with the workpiece coordinates *uvw* before the inclination. The origins of both the coordinate systems are placed at the present ball

center. *u*, *v* and *w* are the cutting feed direction, the pick feed direction and the normal direction to the finished surface respectively. The tool is inclined around *x* by *ix* and then around *y* by *iy*, as shown in the figure. The sky‐blue region shows engagement of the ball end mill with the workpiece. If vibration occurs in the process, it changes the uncut chip thickness and generates the dynamic cutting force in this engagement region. This dynamic cutting force excites the mechanical structure and generates vibration again.

period. The overlap factor *μ* can be approximated by 1, since the feed rate is usually small

The multi‐directional vibrations can be coupled when their modes in different directions have close resonant frequencies. The present study, therefore, deals with the *xyz* vibrations and their

The transfer functions are supposed to be measured, while the directional milling force coefficients, which are called process gains in the present paper, need to be calculated for the prediction of the chatter stability. The process gains are time‐dependent in the milling process, but it is known that they can be approximated by their averages, i.e. DC components [8]. Therefore, the average process gains are derived numerically by assuming small displacement

> Ball end milling process gain

> > *Pxx Pxy Pxz Pyx Pyy Pyz Pzx Pzy Pzz*

Transfer function

*Gxy* (*s*) *Gxz*(*s*) *Gyx*(*s*) *Gyy* (*s*) *Gyz*(*s*) *Gzx* (*s*) *Gzy* (*s*) *Gzz* (*s*)

*Gxx*(*s*)

 Dynamic cutting force *f f x y*

Precision Micro Machining Methods and Mechanical Devices 57

*fz*

enough compared with the ball radius *r*.

in each direction [7].

mode coupling effect as shown in the diagram.

+

Delay

*<sup>e</sup>Ts*

**Figure 8.** Block diagram of ball end milling process with regenerative and mode-coupling chatter vibrations.

The machining experiments were conducted on the inclined workpiece surface with a slender ball end mill and a vertical machining center (Millac 3VA‐DS, Okuma Corp.), as shown in Figure 9. The vibration was measured at the tool holder with the displacement sensor. The transfer function matrix was measured by the impulse response method, and the specific cutting forces, which are required to compute the process gains, are identified using a

Figure 10 shows the predicted gain margins *gm*, the predicted stability limits, i.e. contour lines of *gm*=1, and experimental results of chatter vibrations. The experimental vibrations were classified by using the chatterfrequency component ofthe measured displacement*s*0, as shown

Previous dynamic displacement

**2.3. Experimental setup and measurement**

dynamometer and an oblique cutting model [10].

**2.4. Analytical and experimental results**

Present dynamic displacement

Regenerative displacement *x y z*

**Figure 7.** Ball end milling process with tool inclination.

This closed loop system can be expressed by the block diagram shown in Figure 8. If the vibration grows up through this closed loop, the system is unstable and generates the self‐ excited chatter vibration. The diagram contains two self‐excitation mechanisms, i.e. regener‐ ation and mode coupling. Note that the original or static milling forces and the forced vibrations, which change in synchronization with the tooth passage, are neglected in the present study, because they do not basically affect the self‐excited chatter vibration.

In the ball end milling process, most of the surface removed by the present tooth has been cut by the previous tooth, as shown in the *x‐y* cross section in Figure 7, and thus the present uncut chip thickness fluctuates by not only the present vibration but also the previous vibration. This regenerative effect is represented generally by the delay term *μe‐Ts*, where *T* is the tooth passing period. The overlap factor *μ* can be approximated by 1, since the feed rate is usually small enough compared with the ball radius *r*.

The multi‐directional vibrations can be coupled when their modes in different directions have close resonant frequencies. The present study, therefore, deals with the *xyz* vibrations and their mode coupling effect as shown in the diagram.

The transfer functions are supposed to be measured, while the directional milling force coefficients, which are called process gains in the present paper, need to be calculated for the prediction of the chatter stability. The process gains are time‐dependent in the milling process, but it is known that they can be approximated by their averages, i.e. DC components [8]. Therefore, the average process gains are derived numerically by assuming small displacement in each direction [7].

**Figure 8.** Block diagram of ball end milling process with regenerative and mode-coupling chatter vibrations.

#### **2.3. Experimental setup and measurement**

center. *u*, *v* and *w* are the cutting feed direction, the pick feed direction and the normal direction to the finished surface respectively. The tool is inclined around *x* by *ix* and then around *y* by *iy*, as shown in the figure. The sky‐blue region shows engagement of the ball end mill with the workpiece. If vibration occurs in the process, it changes the uncut chip thickness and generates the dynamic cutting force in this engagement region. This dynamic cutting force excites the

Micro-Nano Mechatronics — New Trends in Material, Measurement, Control, Manufacturing and Their Applications in

This closed loop system can be expressed by the block diagram shown in Figure 8. If the vibration grows up through this closed loop, the system is unstable and generates the self‐ excited chatter vibration. The diagram contains two self‐excitation mechanisms, i.e. regener‐ ation and mode coupling. Note that the original or static milling forces and the forced vibrations, which change in synchronization with the tooth passage, are neglected in the

In the ball end milling process, most of the surface removed by the present tooth has been cut by the previous tooth, as shown in the *x‐y* cross section in Figure 7, and thus the present uncut chip thickness fluctuates by not only the present vibration but also the previous vibration. This regenerative effect is represented generally by the delay term *μe‐Ts*, where *T* is the tooth passing

present study, because they do not basically affect the self‐excited chatter vibration.

mechanical structure and generates vibration again.

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56

**Figure 7.** Ball end milling process with tool inclination.

The machining experiments were conducted on the inclined workpiece surface with a slender ball end mill and a vertical machining center (Millac 3VA‐DS, Okuma Corp.), as shown in Figure 9. The vibration was measured at the tool holder with the displacement sensor. The transfer function matrix was measured by the impulse response method, and the specific cutting forces, which are required to compute the process gains, are identified using a dynamometer and an oblique cutting model [10].

#### **2.4. Analytical and experimental results**

Figure 10 shows the predicted gain margins *gm*, the predicted stability limits, i.e. contour lines of *gm*=1, and experimental results of chatter vibrations. The experimental vibrations were classified by using the chatterfrequency component ofthe measured displacement*s*0, as shown

> stabilityatvariedspindlespeedandtoolinclinationinthepresentresearch.Thepredictedchatter stability agreed well with the experimental results, and it is expected that the present analysis will contribute to improve machining efficiency, surface quality and tool life in the free‐form

Precision Micro Machining Methods and Mechanical Devices 59

A new non‐contact fluid bearing utilizing traveling waves is proposed in this research. Conventional hydrostatic bearings utilize externally compressed fluid, which requires plumbing and compressors. In contrast, on proposed bearing system the moving part is supported with a thin fluid film compressed by the waves traveling radially on the bearing surface. The proposed bearing realizes non‐contact smooth motion without such a large apparatus, and furthermore it has a capability to electrically control the bearing force or

Bearings are one of the most fundamental and necessary elements for machines to generate motion between to surfaces smoothly with low friction. There have been constant demands

As for non‐contact bearings, L. ‐D. Girard invented a hydrostatic bearing in 1865 [15]. An active magnetic bearing was invented around the same time, and its actual system was developed as early as 1950 [16]. The feasibility of a squeeze bearing was first demonstrated and reported in 1964 [17]. These non‐contact bearings have been improved and utilized in practice according to their different characteristics. However, it seems that no fundamental principles of non‐

A new principle of non‐contact fluid bearings, which utilizes traveling waves, is proposed in this research, and a prototype device is developed on the basis of the proposed principle.

According to this new principle the non‐contact fluid bearing is realized by generating the

The working principle of the first prototype is illustrated in Figure 11a. Three sets of piezo actuators are placed radially around the bearing surface, and sinusoidal voltage is applied to the actuators with a phase shift to generate traveling waves on the flexible surface (Figure 11b). As the waves are generated radially, the fluid is transported from outside to the center, which in return generates pressure and a floating force to support an object. When the amplitude and frequency of the voltage are increased, more fluid is pumped underthe bearing

The following key properties show that this new bearing design is promising for high precision

machining with the flexible structures like slender ball end mills and thin workpieces.

**3. A new fluid bearing utilizing traveling waves [11]**

for low friction or non‐contact bearings for many years [12‐14].

contact bearings have been proposed after those inventions.

**3.2. Working principle of the new fluid bearing system**

surface allowing higher loads to be supported.

applications:

traveling waves radially on the bearing surface as shown in Figure 11.

clearance.

**3.1. Introduction**

**Figure 9.** Ball end milling experiment with tool inclination.

under the figure. For example, severe chatter was detected at *d*>2 mm and *ix*=10 deg, where the measured displacement has a large chatter frequency component of *s*0>1.3 μm. On the other hand, only the spindle speed harmonics were observed at *d*=2.5 mm and *ix*=45 deg. The analytical and experimental results are all in a good agreement as shown in the figure. There were no chatter vibrations at *gm*>1.2, while severe chatter vibrations were detected at *gm*<0.6.

**Figure 10.** Predicted gain margins and chatter stability limits, and experimental results at varied tool inclination angle ix. Workpiece: aluminum alloy (JIS:A5052); Cutter: HSS ball end mill (EBD80820, OSG Corp.), Cutter: number of flutes nf = 2, radius r = 10 mm; Cutting conditions: iy = 0 deg, pick feed p = 1 mm, n = 6240 min-1, feed rate of 0.01 mm/tooth; Cutting fluid: soluble. Chatter vibrations were classified as follows; ○: no chatter (s0 ≤ 0.12 μm), ∆: slight chatter (0.12 μm < s0 ≤ 0.24 μm), ×: chatter (0.24 μm < s0 ≤ 1.2 μm), \*: severe chatter (1.2 μm < s0).

#### **2.5. Conclusion**

The analytical model of the ball end milling process with the self‐excited chatter vibration was developed with consideration of the tool inclination, and it was applied to predict the chatter stabilityatvariedspindlespeedandtoolinclinationinthepresentresearch.Thepredictedchatter stability agreed well with the experimental results, and it is expected that the present analysis will contribute to improve machining efficiency, surface quality and tool life in the free‐form machining with the flexible structures like slender ball end mills and thin workpieces.
