**3. Design and construction of an ultrasonic shoe**

Fig.4 (a and b) shows the ultrasonic shoe structure and the principle of generation of ultrasonic elliptic motion on the shoe end face, respectively. As shown in Fig.4 (a), the shoe is constructed by bonding a piezoelectric ceramic device (PZT) having four separated electrodes on to a metal elastic body (stainless steel,SUS304). Applying two alternating current voltages (frequency *f*, amplitude *Vp–p*, and phase difference *ψ*) generated by amplifying two AC signals from a wave function generator with power ampliers to the PZT induces simultaneous bending and longitudinal ultrasonic vibrations with amplitudes of several micrometers(see Fig.4(b)). The synthesis of vibration displacements, UB and UL, in the two directions creates an elliptic motion on the end face of the metal elastic body. Consequently, the rotation of the workpiece is controlled by the friction force between the workpiece and the shoe (see Fig.1 (a)), and the peripheral speed, *Vw*, of the workpiece is thus the same as the bending vibration speed, *Vs* (=*f*×*UB*), on the shoe end face.

Thus, it is essential that the two vibration modes, i.e., bending vibration (B-mode) and longitudinal vibration (L-mode), of the shoe must be induced simultaneously at the same frequency in order to generate an elliptic motion on the shoe end face. The present authors have pointed out in previous studies [12–15] that a combination of an even-ordered B-mode (i.e.,B2, B4,B6, B8, etc.) and an odd-ordered L-mode(i.e.,L1,L3,L5,L7,etc.), where the sole common node for the B-mode and L-mode is located at its central position, should be selected so that the ultrasonic vibrations of the shoe would not be restricted when held at the common node on the shoe holder. In addition, the simpler the vibration mode is, the easier the excitation of the shoe. From this viewpoint, a combination of L1 and B2 modes is desired. However, when the shoe is treated as a plate of length l with a uniform crosssection of width b and thickness t for simplicity, the precondition that the frequency of the rth L-mode must be the same as that of nth B-mode yields the following relationship between *l* and *t* [16]:

124 Tungsten Carbide – Processing and Applications

the gap is sufciently small. To this end, a ne vertical position adjustment mechanism composed of a vertical motion guide, a ball screw, and a table, on which the ne feed mechanism is held, was constructed in order to adjust the gap by manipulating the ball screw. Moreover, a pre-load is applied to the shoe at its left end face along its longitudinal direction using a coil spring in order to prevent the shoe from breaking due to resonance.

1.Workpiece 2.Ultrasonic shoe 3.Vertical motion guide 4.Pre-load spring 5.Ball screw for adjusting the vertical position of shoe 6.Ball screw for feeding the shoe 7.Gear head 8.Stepping motor 9.Table for holding fine feed mechanism

Fig.4 (a and b) shows the ultrasonic shoe structure and the principle of generation of ultrasonic elliptic motion on the shoe end face, respectively. As shown in Fig.4 (a), the shoe is constructed by bonding a piezoelectric ceramic device (PZT) having four separated electrodes on to a metal elastic body (stainless steel,SUS304). Applying two alternating current voltages (frequency *f*, amplitude *Vp–p*, and phase difference *ψ*) generated by amplifying two AC signals from a wave function generator with power ampliers to the PZT induces simultaneous bending and longitudinal ultrasonic vibrations with amplitudes of several micrometers(see Fig.4(b)). The synthesis of vibration displacements, UB and UL, in the two directions creates an elliptic motion on the end face of the metal elastic body. Consequently, the rotation of the workpiece is controlled by the friction force between the workpiece and the shoe (see Fig.1 (a)), and the peripheral speed, *Vw*, of the workpiece is thus

Thus, it is essential that the two vibration modes, i.e., bending vibration (B-mode) and longitudinal vibration (L-mode), of the shoe must be induced simultaneously at the same frequency in order to generate an elliptic motion on the shoe end face. The present authors have pointed out in previous studies [12–15] that a combination of an even-ordered B-mode (i.e.,B2, B4,B6, B8, etc.) and an odd-ordered L-mode(i.e.,L1,L3,L5,L7,etc.), where the sole common node for the B-mode and L-mode is located at its central position, should be selected so that the ultrasonic vibrations of the shoe would not be restricted when held at

10.Shoe holder 11.Linear motion way 12.Bolt for fixing the shoe 13.Blade 14.Blade holder

the same as the bending vibration speed, *Vs* (=*f*×*UB*), on the shoe end face.

**Figure 3.** Illustration of the new centerless grinding apparatus

**3. Design and construction of an ultrasonic shoe** 

$$l = \frac{\left(2n+1\right)^2 \pi t}{8\sqrt{3}r} \tag{1}$$

Eq. (1) gives the relationship *l*=5.7t for the L1B2(r=1, n=2) combination, but the relationship *l*=18.4t for the L1B4(r=1, n=4) combination. This suggests that a thin type shoe, the vibration excitation of which can be more easily compared with others, can be constructed based on the L1B4 combination. Thus, the L1B4 combination was selected as the ultrasonic shoe.

**Figure 4.** Structure and operating principle of the ultrasonic elliptic vibration shoe

Based on the discussion above, the structure proposed is shown in detail in Fig.5. A Tshaped extrusion is located at the center of the shoe via which the shoe can be xed on its holder by bolts. Four separate electrodes are distributed on the PZT based on the B4 mode. The dimensions of the shoe are then determined by FEM analysis followed by impedance measurement to be described later.

Fabrication of Microscale Tungsten Carbide Workpiece by New Centerless Grinding Method 127

With the exception of the length of the metal elastic body, *l2*, all dimensions were determined (see Table1) by taking into consideration the space available for installation of the proposed shoe on the existing centerless grinder. Dimension *l2* was rst predicted by nite element method (FEM) analysis under the condition *fL1* (frequency of L1 mode) = *fB4* (frequency of B4 mode). Fig.6 (a and b) shows, respectively, the L1 and B4 modes of a shoe (*l2* =96.95mm) obtained by FEM analysis for *fL1* = *fB4* =23.85 kHz. In order to conrm the generation of elliptic motion on the shoe end face having the FEM predicted dimension *l2* =96.95mm, a frequency response analysis (FRA) was carried out using piezoelectric device analysis software. Fig.7 shows the FRA results obtained for *V*p–p=50V, *f* =23.90 kHz, and *ψ*

As predicted by FEM and FRA above, *l2* must be 96.95mm in order for *fL1* to equal *fB4* and for an elliptic motion to be generated on the shoe end face. However, it is foreseen that the actual values of *fL1* and *fB4* would not agree with the predicted values due to dimensional errors associated with the metal elastic body and the PZT used. Thus, three shoes with different values of *l2*, namely 96.45, 96.95, and 97.45mm, were constructed based on the FEM and FRA results. One of these values was selected after the shoes' actual frequencies *fL1* and *fB4* were obtained by measuring their impedance characteristics. Fig.8 shows a photograph of a designed and constructed ultrasonic shoe. The shoe surface was coated with a waterproong layer in order to protect against the grinding uid during grinding. Further, the friction coefcient between the shoe and the workpiece should be large enough to prevent the workpiece from slipping on the shoe end face. Thus, a thin rubber(0.5mm in thickness) sheet made of the same materials that used in conventional regulating wheels

An impedance analyzer was used for investigating the impedance characteristics of the shoes. The results obtained for the shoe having an *l2* of 96.95mm are shown in Fig.9 (a and b) for the L1 and B4 modes, respectively. Clearly, the impedances for the B4 and L1 modes reach their minima at the frequencies of 24.13 and 24.01kHz, respectively, indicating that the respective resonant frequencies for the L1 and B4 modes are *fB4* =24.13kHz and *fL1* =24.01kHz.The impedances for the two modes reach their maxima at 24.20 and 24.22kHz, respectively, meaning the power consumption would be least when the AC voltages applied at these frequencies. This is referred to as the anti-resonance effect [17, 18]. The measured *fL1* and *fB4* are plotted against *l2* (Fig.10). It can be seen that *fL1* comes closest to *fB4* at *l2* =96.45mm.

=90°. Clearly, an elliptic motion occurs on the end face of the shoe.

was prepared and attached to the end face of the shoe.

**Figure 8.** Photograph of a fabricated shoe

Thus, *l2* was determined to be 96.45mm.

**Figure 5.** Fig.5 Detailed structure of the ultrasonic shoe


**Table 1.** The dimensions of shoe designed (mm)

(a) L1 mode (*fL1* =23.85 kHz) (b) B4 mode (*fB4* =23.85 kHz)

**Figure 6.** L1 and B4 modes obtained by FEM analysis

**Figure 7.** Elliptic motion predicted by FRA (frequency response analysis)

With the exception of the length of the metal elastic body, *l2*, all dimensions were determined (see Table1) by taking into consideration the space available for installation of the proposed shoe on the existing centerless grinder. Dimension *l2* was rst predicted by nite element method (FEM) analysis under the condition *fL1* (frequency of L1 mode) = *fB4* (frequency of B4 mode). Fig.6 (a and b) shows, respectively, the L1 and B4 modes of a shoe (*l2* =96.95mm) obtained by FEM analysis for *fL1* = *fB4* =23.85 kHz. In order to conrm the generation of elliptic motion on the shoe end face having the FEM predicted dimension *l2* =96.95mm, a frequency response analysis (FRA) was carried out using piezoelectric device analysis software. Fig.7 shows the FRA results obtained for *V*p–p=50V, *f* =23.90 kHz, and *ψ* =90°. Clearly, an elliptic motion occurs on the end face of the shoe.

As predicted by FEM and FRA above, *l2* must be 96.95mm in order for *fL1* to equal *fB4* and for an elliptic motion to be generated on the shoe end face. However, it is foreseen that the actual values of *fL1* and *fB4* would not agree with the predicted values due to dimensional errors associated with the metal elastic body and the PZT used. Thus, three shoes with different values of *l2*, namely 96.45, 96.95, and 97.45mm, were constructed based on the FEM and FRA results. One of these values was selected after the shoes' actual frequencies *fL1* and *fB4* were obtained by measuring their impedance characteristics. Fig.8 shows a photograph of a designed and constructed ultrasonic shoe. The shoe surface was coated with a waterproong layer in order to protect against the grinding uid during grinding. Further, the friction coefcient between the shoe and the workpiece should be large enough to prevent the workpiece from slipping on the shoe end face. Thus, a thin rubber(0.5mm in thickness) sheet made of the same materials that used in conventional regulating wheels was prepared and attached to the end face of the shoe.

**Figure 8.** Photograph of a fabricated shoe

126 Tungsten Carbide – Processing and Applications

**Figure 5.** Fig.5 Detailed structure of the ultrasonic shoe

**Table 1.** The dimensions of shoe designed (mm)

(a) L1 mode (*fL1* =23.85 kHz) (b) B4 mode (*fB4* =23.85 kHz)

**Figure 7.** Elliptic motion predicted by FRA (frequency response analysis)

**Figure 6.** L1 and B4 modes obtained by FEM analysis

*b t1 t2 l1 l2 h1 h2 a1 a2*

20 2 4 88.6 3 20 20 5

An impedance analyzer was used for investigating the impedance characteristics of the shoes. The results obtained for the shoe having an *l2* of 96.95mm are shown in Fig.9 (a and b) for the L1 and B4 modes, respectively. Clearly, the impedances for the B4 and L1 modes reach their minima at the frequencies of 24.13 and 24.01kHz, respectively, indicating that the respective resonant frequencies for the L1 and B4 modes are *fB4* =24.13kHz and *fL1* =24.01kHz.The impedances for the two modes reach their maxima at 24.20 and 24.22kHz, respectively, meaning the power consumption would be least when the AC voltages applied at these frequencies. This is referred to as the anti-resonance effect [17, 18]. The measured *fL1* and *fB4* are plotted against *l2* (Fig.10). It can be seen that *fL1* comes closest to *fB4* at *l2* =96.45mm. Thus, *l2* was determined to be 96.45mm.

Fabrication of Microscale Tungsten Carbide Workpiece by New Centerless Grinding Method 129

The elliptic motion of the shoe end face under various applied voltages (amplitudes, frequencies and phase differences) is investigated using a measuring system composed of two laser Doppler vibrometers (Ono Sokki Co., Ltd., LV-1610) equipped with the respective sensor heads, a vector conversion unit (Ono Sokki Co., Ltd.,), and a multi-purpose FFT(Fast

The shoe is bolted at its center (the common node for L1 and B4 mode) on the holder in order not to restrict the ultrasonic vibration. A preload is then applied to the shoe using a coil spring in order to prevent the PZT from breaking due to resonance. Two AC signals generated by a wave function generator (NF Corporation, WF1994) are applied to the PZT after being amplified by two power amplifiers (NF Corporation, 4010). During measurement, the two laser beams from the respective heads are focused at the same point near the shoe end face. The signals from the laser Doppler vibrometers are then input to the vector conversion unit for synthesis and are recorder with a digital oscilloscope (Iwatsu Co., Ltd., LT364L). The AC signal is changed by various voltages, phase differences and frequencies. From the digital oscilloscope, the trace of ultrasonic vibration will be obtained based on the different input parameters, and the relationship between the input parameters

The shoe is bolted at its center (the common node for L1 and B4 mode) on the holder in order not to restrict the ultrasonic vibration. A preload is then applied to the shoe using a

Two AC signals generated by a wave function generator (NF Corporation, WF1994) are applied to the PZT after being amplified by two power amplifiers (NF Corporation, 4010). During measurement, the two laser beams from the respective heads are focused at the same

Fourier Transform) analyzer (Ono Sokki Co., Ltd., CF-5220), as shown in Fig.11.

**4. Performance of the apparatus constructed** 

and the vibration will be clarified.

**Figure 11.** Method of measuring the ultrasonic elliptic vibration

coil spring in order to prevent the PZT from breaking due to resonance.

**4.1. Method of measuring the ultrasonic elliptic vibration** 

**Figure 9.** Impedance characteristics of the shoe

Length of metal elastic body *l2* [mm]

**Figure 10.** Measured frequencies of two modes
